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Chapter 3 Unusual anharmonic local mode systems
CHAPTER 3
Unusual Anharmonic Local Mode Systems
A.J. SIEVERS
J.B. PAGE
Laboratory of Atomic and Solid State Physics and the Materials Science Center Cornell University Ithaca, NY 14853-2501 USA
Department of Physics and Astronomy Arizona State University Tempe, Arizona 85287-1504 USA
Dynamical Properties of Solids, edited by G.K. Horton and A.A. Maradudin
9 Elsevier Science B.V., 1995
137
Contents 1. Introduction
141
1.1. Impurity modes in crystals 141 1.2. Localized modes in perfect anharmonic lattices ,
143
Experimental and theoretical studies of a thermally anomalous nearly unstable impurity system: KI:Ag + 147 2.1. Initial experiments 147 2.2. Basic shell model description of the T = 0 K nearly unstable lattice dynamics 182 2.3. Pocket gap mode experiments 193 2.4. The (3, 3', 3") and the quadrupolar deformability models 199 2.5. Discussion and conclusions 203
,
Intrinsic localized modes in perfect anharmonic lattices 3.1. One-dimensional monatomic lattices 207 3.2. One-dimensional diatomic lattices 239
4. Speculation
243
4.1. ILMs and anomalous defect properties 4.2. ILMs and other properties 247 5. Conclusion
251
6. Acknowledgements Note added in proof References
251 251
252
139
244
206
We dedicate this chapter to LUDWIG GENZEL and the late HEINZ BILZ for their pioneering experimental and theoretical studies of numerous key aspects of lattice dynamics in crystals
1.
Introduction
1.1. Impurity modes in crystals The early far infrared measurements with grating instruments on the low temperature properties of defects in alkali halides produced a variety of sharp features associated with local modes (Sch~ifer 1960), gap modes (Sievers et al. 1965) and resonant modes (Sievers 1964; Weber 1964). The sharp resonant modes were somewhat of a surprise, even though the possibility of strongly coupled impurity-induced lattice resonances had been proposed in earlier theoretical work (Brout and Visscher 1962; Visscher 1963; Dawber and Elliott 1963). The observation of these low-lying sharp spectral features was quickly followed by more accurate measurements using the fourier transform IR technique (Sievers 1969) and somewhat later by Raman scattering studies. (Klein 1990 has recently reviewed this topic.) A variety of defect systems has been studied in detail, and much of that work has been reviewed from both the experimental (Barker and Sievers 1975; Bridges 1975) and theoretical (Stoneham 1975; Bilz et al. 1984) points of view. The findings are that most localized vibrational modes in defect-lattice systems behave in the expected fashion; that is, the modes identified with impurity motion can be interpreted from the experimental studies as harmonic or slightly anharmonic oscillators. However, for some systems the modes have anomalous properties that place them outside of the bounds of that interpretation. The vibrational properties of the light Li + impurity in KC1 or KBr are two defect-lattice systems that have received a great deal of attention over the years. KCI:Li + has been studied mainly because the impurity ion is off-center in the (111) directions so that low frequency tunneling states play a prominent role (Narayanamurti and Pohl 1970; Kirby et al. 1970; Devaty and Sievers 1979; Wong and Bridges 1992). On the other 141
142
A.J. Sievers and J.B. Page
Ch. 3
hand, an equal effort has gone into examining the resonant mode properties of KBr:Li + because now the impurity ion is barely stable at a normal lattice site (Sievers and Takeno 1965; Page 1974; Kahan et al. 1976). The near instability demonstrated by the single ion spectrum is fullfilled for Li + pairs in KBr which are found to be off-center and to tunnel coherently from site to site (Greene and Sievers 1982, 1985). Although these point defect systems have received ample attention in the literature and are thought to be understood there are other anomalous systems which have not played a central role since the paucity of available data did not allow their unusual features to be recognized as crucial. There are two "simple" heavy defect ion-lattice systems in particular, which display unusual vibrational properties: KI:Ag + and RbCI:Ag +. At low temperatures both of these systems show defectinduced lattice modes in the far IR. Early measurements showed that the Ag + impurity in KI was on-center at a K + lattice site, while the Ag + impurity in RbC1 was off-center in a (110) direction and exhibited tunneling between equivalent sites (Barker and Sievers 1975). The dynamics of this heavy impurity had been assumed to be similar to those found for the Li + impurity in KBr and KC1, respectively. In the last decade, paraelectric resonance studies of the tunneling modes in RbCI:Ag + as a function of hydrostatic pressure indicate unusual intensity dependences for some transitions. These have been interpreted as a signature for the existence of an on-center configuration nearby in energy (Bridges et al. 1983; Bridges and Chow 1985; Bridges and Jost 1988). For KI:Ag +, the early observation that the IR active resonant mode and the later observation that the entire T = 0 K impurity-induced spectrum disappear as the temperature is increased to 25 K has been shown to be associated with the Ag + ion moving from the on-center to an off-center position with increasing temperature (Sievers and Greene 1984). These results for KI:Ag + may indicate that at T = 0 K there is both an on-center configuration and an off-center one with nearly the same energy. The very unusual behavior of these two lattice-defect systems has called into question the underlying fundamentals of standard defect phonon theory. Where is the flexibility in the Lifshitz theory (Lifshitz 1956) for the existence of multiple elastic configurations? To address this fundamental lattice dynamics question, much theoretical and experimental effort has gone into reexamining in greater detail the T = 0 K properties of the simplest of the two systems, namely on-center KI:Ag +, which may have at least two low-lying configurations. The approach that has been used is first to apply the harmonic shell model in an attempt to explain the observed spectroscopic properties of this lattice defect system, excluding the temperature dependence. This has proved to be quite successful, and with the addition of some anharmonicity as described
w1
Unusual anharmonic local mode systems
143
here, it gives a quantitative description of the local dynamics of the KI:Ag + system at T = 0 K. The resulting comparison between theory and experiment in unprecedented detail has led to the prediction and verification of an unexpected new class of defect modes, called "pocket" gap modes (Sandusky et al. 1991). These defect modes have the unusual property that the maximum vibrational amplitude is not at the impurity but is localized at lattice sites well removed from the impurity. The experimental investigation of the isotope effect (Sandusky et al. 1991; Sandusky et al. 1993a), stress shifts (Rosenberg et al. 1992) and electric field Stark effect (Sandusky et al. 1994) provide sufficient experimental information to demonstrate that a dynamically-induced electronic deformability of the Ag + impurity plays an essential role in the dynamics. Upon analysis of much of the data, one finds that a convincing explanation of the T = 0 K experimental far IR and Raman defect-induced spectra can be made within the Lifshitz perturbed phonon framework, in terms of a slightly anharmonic shell model in which the coupled defect/host system is nearly unstable. In contrast, while a variety of phenomenological anharmonic models have been introduced to account for the observed temperature effects, it is still not possible to present a consistent dynamical picture of the temperaturedependent behavior of this system. One purpose of this review is illustrate how the temperature dependence models fail. The end result of these tightly woven arguments will be that the observed anomalous temperature dependence of the spectra appears to fall truly outside the bounds of current defect mode theory. A primary function of this review is to gather together all of the experimental and theoretical data on this particular on-center system, so that the reader can inspect, understand and appreciate what lies beyond the framework of current impurity dynamics theory. 1.2. Localized modes in perfect anharmonic lattices While it is not surprising that the loss of periodicity in defect crystals leads to localized vibrational phenomena, the standard description of the dynamics of defect-free periodic lattices in terms of plane wave phonons is so deeply ingrained that it was surprising to many researchers when it was argued theoretically in 1988 (Sievers and Takeno 1988; Takeno and Sievers 1988) that the presence of strong quartic anharmonicity in perfect lattices can also lead to localized vibrational modes, henceforth called "intrinsic localized modes" (ILMs). These papers studied the classical vibrational dynamics of a simple one-dimensional monatomic chain of particles interacting via nearestneighbor harmonic and quartic anharmonic springs, and they used a "rotating wave approximation" (RWA), in which just one frequency component was
144
A.J. Sievers and J.B. Page
Ch. 3
kept in the time dependence. For the case of sufficiently strong positive quartic anharmonicity, it was found that the lattice could sustain stationary localized vibrations having the approximate mode pattem A(..., 0 , - 1 / 2 , 1 , - 1 / 2 , 0 . . . . ), where A is the amplitude of the "central" atom. This pattem is odd under reflection in the central site and is "optic mode"-like, in that adjacent particles move 7r out of phase. As with all nonlinear vibrations, the ILM frequencies are amplitude dependent. The ILMs can be centered on any lattice site, giving rise to a configurational entropy analogous to that for vacancies, and some thermodynamic ramifications were explored in the preceding two references and by Sievers and Takeno (1989), along with speculations about the possible presence of ILMs in strongly anharmonic solids such as solid He and ferroelectrics. A provocative, but still unproven speculation was that these modes might be involved in the anomalous thermally-driven low temperature on-center --+ off-center transition of the nearly unstable KI:Ag + system, which is the other main subject of this chapter. It is straightforward to check the theoretical predictions using molecular dynamics simulations to solve the equations of motion numerically. This of course avoids the RWA and other approximations used by Sievers and Takeno (1988), and fig. 1 shows the results for a 21-particle monatomic chain with periodic boundary conditions and harmonic plus quartic anharmonic nearest-neighbor interactions. Each panel shows the time-evolution of the particle displacements for the same initial displacement pattem, namely that of the predicted ILM pattem given above, centered on the middle particle. The initial velocities were zero. The two panels give the subsequent displacements for the case of (a) purely harmonic interactions and (b) harmonic plus strong quartic anharmonic interactions. As expected, in (a) the initial displacements of the purely harmonic system spread into the lattice, since they are a linear combination of independently evolving homogeneous plane-wave normal modes. In sharp contrast, for the anharmonic system (b), the initial displacement pattem is seen to persist, and the theoretical predictions are strikingly verified. In the few years since the appearance of the 1988 Sievers and Takeno papers, there has been a rapidly increasing number of theoretical studies published related to ILMs. Interestingly, during the preparation of the present chapter, we discovered an earlier, brief paper (Dolgov 1986), which obtains the Sievers-Takeno solution and the analogous even-parity ILM solution, derived independently by Page (1990). The Dolgov paper has gone unnoticed by subsequent workers.
w
Unusual anharmonic local mode systems
145
(a) harmonic 0
o it} o
EL
9
.... J T
T - - _ -
....
-
5 0
r
tO
EL
----------x
10 t 0
~
5
-
~ 10
15
(b) anharmonic 10 rr
._o O
EL to
"s
5 0
-5
EL
-10 0
~i
1'0
15
time (units of 2~'(Om)
Fig. 1. Molecular dynamics simulations for a monatomic chain with (a) purely harmonic and (b) harmonic plus quartic anharmonic nearest-neighbor interactions. For both panels, the initial configuration is the odd-parity ILM displacement pattern A( . . . . 0, - 1/2, 1, - 1/2, 0 . . . . ), with the amplitude on the central particle being 0.1 of the equilibrium nearest-neighbor distance a. The initial velocities are zero. The masses are 39.995 amu, the lattice constant is 1 ,~ and k2 = 10 eV/,~2. The dashed lines in (a) are guides which delineate the spreading of the initial localized displacement pattern into the harmonic lattice. For (b) the value of the anharmonicity parameter A4 = k4A2/k2 is 1.63, well within the range when this ILM should be a valid solution. An implementation of the fifth-order Gear predictor-corrector method was used for the MD runs (Allen and Tildesley 1987), with the time step taken to be 1/180 of the period of the maximum harmonic frequency ~Om. In both panels, the particle displacements are magnified, for clarity.
T h e I L M p a p e r s p u b l i s h e d since 1988 r o u g h l y divide into t w o b r o a d and o v e r l a p p i n g c a t e g o r i e s . T h e first o f these i n v o l v e s the relation o f the n e w excitations to the g e n e r a l b e h a v i o r o f discrete n o n l i n e a r systems, with e m p h a s i s on c o n n e c t i o n s to soliton-like b e h a v i o r in lattices. T h e s e c o n d c a t e g o r y is m o r e tightly f o c u s e d on g e n e r a l i z a t i o n s and e x t e n s i o n s o f the 1988 papers, in o r d e r to d i s c o v e r the p r o p e r t i e s o f h i g h l y - l o c a l i z e d l a r g e - a m p l i t u d e I L M s and also to assess their likely i m p o r t a n c e for real solids. T h e general diffi-
146
A.J. Sievers and J.B. Page
Ch. 3
culty of dealing with complex nonlinear dynamical systems is reflected in the diversity of theoretical approaches used, which range from purely analytic studies to completely numerical molecular dynamics simulations. In our opinion the former should, whenever possible, be accompanied by the latter, since the unfamiliarity of the phenomena can easily lead to unwarranted approximations being made in analytical work. Computer studies of the dynamics of nonlinear lattices extend back to the pioneering work on one-dimensional chains by Fermi, Pasta and Ulam (1955), and they have been a vital adjunct to much of the subsequent work on solitons in anharmonic lattices. Discussions of both analytic and numerical aspects of soliton behavior in one-dimensional lattices with intersite cubic and quartic anharmonic interactions are found in Flytzanis et al. (1985) for the monatomic case and in Pnevmatikos et al. (1986) for the diatomic case. Of particular interest for ILMs is their relationship to lattice envelope solitons. Stationary lattice solitons were studied theoretically more than 20 years ago by Kosevich and Kovalev (1974) for anharmonic chains with cubic and quartic onsite and intersite anharmonicity. The emphasis was on onsite anharmonicity, and solutions were obtained for low-amplitude stationary envelope solitons whose spatial extent is broad compared with the lattice constant. Later MD simulations by Yoshimura and Watanabe (1991) for linear chains with quadratic plus quartic interactions showed that the stationary envelope soliton solutions account well for ILMs whose spatial widths are sufficiently broad, e.g., 10 or more lattice constants. However, as was subsequently emphasized by Kosevich (1993a), the stationary envelope soliton solutions do not describe the large-amplitude highly localized ILMs, such as that of fig. 1. Some recent papers dealing with general aspects of localized dynamics in discrete nonlinear systems and ILM-related soliton studies are: Flach and Willis (1993), Flach et al. (1993), Kivshar and Campbell (1993), Chubykalo and Kivshar (1993), Claude et al. (1993), Kosevich (1993b, c), and Cai et al. (1994). Some representative papers on ILM solutions in lattices with nearest-neighbor intersite anharmonicity and not already cited are: Takeno et al. (1988), Burlakov et al. (1990a, b, d), Kiselev (1990), Takeno (1990), Bickham and Sievers (1991), Takeno and Hori (1991), Takeno (1992), Fischer (1993), Aoki et al. (1993), Chubykalo et al. (1993), and Kiselev et al. (1993). Some representative studies discussing traveling ILMs are given in Takeno and Hori (1990), Bickham et al. (1992), Hori and Takeno (1992), Sandusky et al. (1993b), Bickham et al. (1993) and Kiselev et al. (1994b). A detailed study of the stability of ILMs was given by Sandusky et al. (1992), and the relationship between ILMs and the stability of homogeneous lattice phonon modes has been investigated by Burlakov et al. (1990c), Burlakov and Kiselev (1991), Dauxois and Peyrard (1993), Kivshar (1993),
w
Unusual anharmonic local mode systems
147
Sandusky et al. (1993b), and Sandusky and Page (1994). Energy transport and/or the effects of defects or disorder are discussed by Bourbonnais and Maynard (1990a, b), Takeno and Homma (1991), Kivshar (1991), Kiselev et al. (1994a, b) and Zavt et al. (1993). An important recent development has been the generation of stable ILMs in the presence of cubic anharmonicity (Bickham et al. 1993), and the subsequent inclusion of realistic potentials such as Lennard-Jones, Morse, or Born-Mayer plus Coulomb (Kiselev et al. 1993; Sandusky and Page 1994; Kiselev et al. 1994a, b). A primary effect of including the odd-order potential energy terms is that the ILMs are accompanied by localized DC lattice distortions. It is not our aim here to give a critical assessment of the rapidly growing literature in this area. Rather, we wish to describe some of the important phenomena and results as simply as possible, referring to the literature for details. We will emphasize the work of our own groups, and we will restrict our attention to one-dimensional monatomic or diatomic chains of particles interacting via quadratic, cubic and quartic springs, or via realistic potentials. Throughout, we will stress the phenomena themselves rather than the theoretical methods, which are well-described in the original papers.
0
Experimental and theoretical studies of a thermally anomalous nearly unstable impurity system: KI:Ag +
2.1. Initial experiments
2.1.1. Intrinsic temperature dependent absorption in KI A number of investigators have shown that the far infrared transmission of pure alkali halide crystals at frequencies below the reststrahlen region is controlled by two-phonon difference band energy-conserving absorption processes (Stolen and Dransfeld 1965; Eldridge and Kembry 1973; Eldridge and Staal 1977; Hardy and Karo 1982). The probability of absorption of a phonon is proportional to the phonon occupation number n~ and that of emission of a phonon is proportional to (1 + n~). For a two-phonon difference process in the absorption spectrum, the absorption of a photon is accompanied by the emission of one phonon (1 + n l) and the absorption of another phonon n2, giving a temperature-dependent factor (1 + n l)n2. The net absorption is obtained by correcting for spontaneous photon emission by subtracting the reverse process, namely (1 + n2)nl, so that the resulting temperature dependence is proportional to In2- nl I. Since thermal phonons are required to generate such an absorption process, pure crystals are extremely transparent at low temperatures and low frequencies, far from the
A.J. Sievers and J.B. Page
148
Ch. 3
reststrahlen band. However, since we are interested in examining the temperature dependence of defect induced absorption, these temperature-dependent properties of the pure crystal need to be quantified as well. As long as the sample does not have parallel sides, the absorption coefficient c~(w) is determined from the transmission expression It --
(1 =
-
R) 2
exp[-c~(w)/]
=
t(w)
Io
,
1 --
R 2
( 2 . 1 a)
exp[-2o~(w)/]
where the transmission coefficient t(w) is defined as the ratio of the transmitted to the incident intensity, R is the reflectivity coefficient and l is the sample length. When R is small and/or ~(w)l is large, the second term in the denominator can be ignored and eq. (2.1a) reduces to
t(w) ~ (1
-
R) 2
exp[-a(w)/].
(2.1b)
Figure 2 shows the temperature-dependent absorption coefficient a(w) for KI over a wide range by using three different ordinate scales versus frequency (Love et al. 1989). For the samples used here, the frequency and temperature-dependent absorption coefficient a(w, T) is obtained by dividing the spectra taken at the higher temperature by one at 4.2 K. From eq. (2.1b) this gives 1
c~(w, T) - c~(w,4.2 K) = -7 In[It(w, T)/It(w, 4.2 K)].
(2.2)
l
In fig. 2(a), (b) the absorption coefficient at 4.2 K is small enough on both of these scales that it can be ignored, hence these data give the temperature dependence of the difference band absorption coefficient directly. The observed spectral structure agrees with that predicted by early workers, including the weak difference band contribution at 11.3 cm -1 which stems from vertical transitions between the s163 symmetry branches in the frequency region where the dispersion curve slopes are nearly parallel. The sharp edge at 31 cm -1 in fig. 2(c) is due to relatively strong ZI(LA)-Z4(TO) two-phonon difference band transitions across the acousticoptic phonon gap.
2.1.2. Resonant mode IR absorption Because of the freezing out of the intrinsic difference band absorption processes for frequencies below the reststrahlen region in alkali halides at
w
Unusual anharmonic local mode systems i
i
i
i
I
I
. (0)
I
I
/82K
40-
-
.
/64K
20
~/47'
K.-
,
"T
(b)
E
2
149
55K
5K
8
C
._~
-
llJ O O
/
4-
J]
e--
O .IQ
<
S
O
(c) 1.6-
82K
0.8-
0
20
40 Frequency
60
80
( c m -a)
Fig. 2. Temperature dependence of the absorption coefficient of pure KI at different temperatures. The sample temperature is varied from 4.2 to 82 K. The spectral resolution is 0.44 cm-1 for frequencies between 3 and 26 cm -1 , and is 0.35 cm -1 at higher frequencies. The spectral features are produced by two-phonon energy-conserving difference band processes. (After Love et al. 1989).
low temperatures, transmission measurements on impurity-doped crystals at p u m p e d liquid helium temperatures give directly the impurity induced absorption coefficient c~i(w). The Ag + ion was the first impurity to be studied at far infrared frequencies in these hosts below the reststrahen region (Sievers 1964; Weber 1964). A simple method in which the lattice-constant
A.J. Sievers and J.B. Page
150
Ch. 3
dependence of an impurity resonant mode can be seen is to plot the log of the observed mode frequency for a fixed impurity ion versus the log of the host lattice constant. This is presented for Ag + in fig. 2. The straight line through all but one of these frequencies indicates that this defect resonant mode obeys a Mollwo-Ivey-like rule (Hayes and Stoneham 1985) up to at least the large lattice constant region (Sievers 1968). This straight line implies that the dependence of the frequency upon the lattice constant can be written as
0 lnw 01na
3A = w '
(2.3)
where a is the lattice constant, w is the resonant frequency and A is the hydrostatic coupling coefficient. For five of the alkali halide hosts in fig. 3, the quantity 3A/w = 3; however, KI:Ag + is seen to give a much lower frequency than that projected from this straight line. The A coefficient of KI:Ag + can be estimated from uniaxial stress measurements, with the result that the quantity 3A/w in eq. 2.3, is equal to 68. This corresponds to a very different slope, as shown by the line at the KI:Ag + point in fig. 3. It is perhaps fortuitous that the resonant mode for KBr falls on the 3A/w = 3
! 601
~
I
I- oc,
I
,o F
I >'351
"-"
~,,~No I
"~Br
~ 3o
25
20-
16 z'~
. I
I
,
3.2s LATTICE CONSTANT(A) 3.00
,
t KI 3.50
Fig. 3. Center frequency of the Ag+ resonant mode for different alkali halide crystals as a function of the host lattice constant. For five different alkali halide lattices the resonant mode frequency can be fit to a straight line. The line centered at the KI:Ag+ point is from uniaxial stress measurements. (After Sievers 1968).
w
Unusual anharmonic local mode systems
151
line, since RbC1, which has nearly the same lattice constant, is an off-center system (Kirby et al. 1970). [The multiple absorption bands associated with excited tunneling transitions in RbCI:Ag + (Barker and Sievers 1975) are not included in fig. 3]. The resonant mode absorption bands associated with the Ag + ion, excluding KI, are found to be relatively temperature independent, at least for thermal energies up to the resonant mode energy. One possible reason that the KI:Ag + resonant mode frequency does not follow the 3A/w = 3 line in fig. 3 is that KI:Ag + might not be an on-center defect system. This possibility has been eliminated by studying the uniaxial stress dependence of this absorption line. A typical splitting for stress applied in the [100] direction for two different polarizations of the radiation is shown in fig. 4. The area under each of the three curves appears to be roughly the same. By making such measurements for uniaxial stress along the [100], [110] and [111] crystal directions, it is possible to distinguish between a frozen off-center system, a tunneling system or an on-center system (Nolt and Sievers 1968). From the number of components observed and the absence of any stress-induced dichroism, it can be concluded that the Ag + defect has either Oh or Td symmetry. In addition, these measurements provide experimental values for the three even-symmetry stress coupling coefficients between the resonant mode and the lattice: A(hydrostatic), B(tetragonal), and C(trigonal). 2
v
I
i
!
I
q
I
.--,.
.
E
~' [o~o]
I
'
I
K I : Ag
4-
(b)
[I 00]
Stress
Zero Stress
(J
---I I'--
r
._e .u_ ' r. - I
(o)
/
0
(.)
~' Doo]
\ \ 0 I
I 16
,
n
IB Frequency
20
J22
(cm"l)
Fig. 4. KI:Ag + absorption coefficient for a [100] stress of 3.3 kg/mm 2. The sample temperature is 1.5 K. The instrumental resolution is 0.3 cm -1. The polarizations are identified in the figure. The integrated line strength is 1.03, 1.23 and 1.1 4-0.1 cm -2 for curves (a), (b) and (c), respectively. (After Nolt and Sievers 1968).
A.J. Sievers and J.B. Page
152
Ch. 3
Electric field measurements on the IR active resonant mode confirmed that the defect had Oh symmetry but in addition produced surprises (Kirby 1971). The electric field induced absorption coefficient for KI:Ag + is shown in fig. 5. The negative Aa region identifies components associated with the zero-field spectrum that have changed. The complex difference spectrum in the resonant mode region arises from the Tlu mode at 17.3 cm -1 mixing with an even mode of Eg symmetry at 16.35 cm -1. These measurements also show another E-field induced line at 25 cm -1, which has been interpreted as an A lg symmetry mode. In addition, absorption features at larger frequencies, namely 30 and 44 cm -1, decrease in strength with applied Efield. The presence of the Eg symmetry mode was confirmed by Raman scattering studies, but no trace of the Alg mode was found. An example of the temperature-independent behavior found for the Ag + doped hosts that obey the Mollwo-Ivey-like rule is seen in fig. 6, which presents the spectrum of NaI:Ag + at 1.2 and 24.2 K (Greene 1984). At the higher temperature the resonant mode centered at 36.7 cm -1 appears on a two-phonon difference-band background spectrum. The resonant mode strength itself is expected to be temperature independent if the mode behaves as a harmonic oscillator. From the Kramers-Kronig dispersion relations, a general f sum rule for the absorption coefficient can be found (Smith 1985):
cx3
f0
71"~ dw c~i(w) = 2c = constant,
(2.4a)
where the exact expression for the plasma frequency squared or oscillator strength depends on the specific model employed. As long as the magnitude of the effective charge which produces the absorption coefficient is temperature independent and the integral is taken over the complete IR absorption spectrum, then the above sum rule holds. In particular, it applies to anharmonic as well as to harmonic oscillators. When anharmonicity is included the transition of interest may shift, broaden and change in strength, but the anharmonicity will also produce other lines in the spectrum (e.g. sidebands, overtones) which will ensure the constant sum rule (Bilz et al. 1984). A specific form of eq. (2.4a), appropriate to impurity-induced absorption over a frequency interval encompassing a line for the case of a host with dielectric constant e0 is (Sievers 1964)
dw Oq(W)
--
2c
=
CMQx/C~
3
'
(2.4b)
A.J. Sievers and J.B. Page
154 32 F
KBr:Li//'~
30~
/''o jw " ~ ' ~ l :cu "//~'~~
-'" 281 o ' E 26
~
P
~' 241-
," 18e~
I I
A
/o
_.I
i6~- ,,e~,~ 14~"~ 0
Ch. 3
~
~h--e-.-14
I , 2 Pressure,
I 3
,
I I 4 in kilobars
CsI : TI I 5
i
I 6
I
I 7
I
Fig. 7. Hydrostatic pressure-induced shift in the resonant mode frequency of four different resonant mode systems. The experimental uncertainties are +0.1 cm-] in the resonant mode frequency and -t-10% in the pressure. The sample temperature is 4.2 K. (After Patterson 1973). where e* is the effective charge and MQ is an effective mass associated with the mode. The fact that the resonant mode shown in fig. 6 is nearly temperature independent indicates that this defect system is not anharmonically coupled to other degrees of freedom, for if it were the area under the resonant mode transition would decrease with increasing temperature and the area difference would transfer into its sidebands such that the total area remained constant (Klein 1968; Barker and Sievers 1975). Hydrostatic pressure measurements up to 7 kbars on the resonant mode feature provide additional evidence that the simple absorption spectrum is not associated with a tunneling defect, since none of the complex spectral behavior previously found for the tunneling system KCI:Li + was observed here (Kahan et al. 1976). Figure 7 shows the hydrostatic frequency shifts observed for different on-center defect systems (Patterson 1973). For all of these systems only one IR active resonant mode is found at each pressure. One curious feature is that the effective resonant mode force constant for KI:Ag + varies nonlinearly with strain (for Aa//a up to 1.6%), while all other low lying resonant modes show a linear dependence over the same range. Figure 8 shows the resonant mode absorption spectrum c~(w) for a KI:Ag + sample at atmospheric pressure and at 1.9 kbars, respectively, as measured with respect to an empty pressure cell (Patterson 1973). Because of the different transmission charactistics of the empty cell, the ordinate gives the absorption coefficient in arbitrary units; however, since the absorption spectra at the two pressures were measured sequentially, the absorption strengths can be compared. Within experimental accuracy the area under the line
w
Unusual anharmonic local mode systems I
........
'
I
'
! 0.3 . - .
I
o.~
'
KI:Ag+ 175 kV/cm 4.2OK -'11"-
1
0.2
I
153
-
-
j
J
tJ
o
/'-"~_/~
'"
-o.i -0.2
!
IO
i
20
I
t
I
=
30 40 FREQUENCY ( cm-I)
~
-
50
Fig. 5. Electric-field-induced absorption coefficient for KI:Ag +. A negative Aa indicates a decrease in absorption with applied field, while a positive Aa indicates a field-induced increase. The complex line shape in the 17 cm -1 region indicates that two resonant modes of opposite parity are being mixed by the odd-parity electric field. Here Ei~llEdcll[100], and the instrumental resolution is 0.55 cm -1. (After Kirby 1971).
3.0
!
NaI +
A
3E v
0.2 %
I
AgI
..... 1 . 2 K
........ 24.2 K
O
..,./-
,.e...
,-" 2 . 0
.~. r
-
NI,-
O (..) C:
.9 .a,...
1.0
O. i,_ 0 U') ,JO
<:[ 0 ,, 20
I
30 Frequency
1
40
50
(cm -I)
Fig. 6. Temperature dependent Ag+-induced absorption in the resonant mode frequency region of NaI. The strength of the resonant mode centered at 36.7 cm -1 is relatively temperature independent, while the two phonon difference band absorption clearly increases between the two temperatures shown. The instrumental resolution is 0.52 cm -1. (After Greene 1984).
w
Unusual anharmonic local mode systems
r 0 "On O. t,. 0 r JO
9
155
~
9 00 4~ OIbO0000 O0 O0
O0
Oo~
Ollo
| OOO O
13
I, 15
1 7 3 c m -I
216cm -t
I 17
I 21
I 19
Frequency,
I 25
co
I 25
I 27
I 29
(cm -I)
Fig. 8. Strength of the KI:Ag + resonant mode at two different hydrostatic pressures. The pressures are 0 and 1.9 kbars, respectively. (After Patterson 1973).
appears to scale with frequency over this pressure range. This pressuredependent area is a surprising result since according to eq. (2.4a, b) for a single absorption line the area should remain fixed, independent of frequency (ignoring the small decrease in e0 with increased pressure). The uniaxial stress measurements, admittedly at lower pressures, did not appear to give such an effect. Although the eigenvector of the resonant mode would be expected to change with hydrostatic pressure, the magnitude of the effective charges in eq. (2.4b) are expected to remain fixed over this pressure range. Clearly, the possibility of this pressure-dependent effect should be examined with more experimental precision. 2.1.3. Anomalous temperature dependence of the resonant and gap modes
Figure 9 clearly indicates that the temperature dependence of the KI:Ag + resonant mode strength is qualitatively different from that observed for the other Ag + resonant mode systems. The strength of the mode decreases rapidly with increasing temperature until by 21.6 K it seems to have disappeared into a slightly larger background spectrum (Sievers and Greene 1984). This temperature dependence of the line strength is independent of the Ag + concentration. Notice that between 1.2 and 9.3 K where about half of the strength has been lost, the center frequency of the mode remains nearly unchanged.
A.J. Sievers and J.B. Page
156
I
1.0
I
I
Ch. 3 I
_ KI" AgI
E O.8 1.2 K
"5 0.6
--it-
U
~,
0.4
9.3 K
o
<~ 0.2
f2,.6 K ,,~..~.,.....~...~.~....,r I0
15 Frequency
20 (cm "l)
25
Fig. 9. Temperature dependence of the absorption spectrum of KI + 0.2 mole% AgI in the frequency region of the T]u resonant mode. The three temperatures are identified in the figure. At high temperatures the broad nonresonant absorption coefficient is about 3% of the low temperature resonant mode peak value. (After Sievers and Greene 1984). For emphasis it is helpful to contrast this unusual temperature dependence of the resonant mode strength with the temperature-independent properties of a related defect system. Each of the five temperature-independent Ag + induced resonant mode frequencies in fig. 3, when normalized to the appropriate host lattice Debye frequency, is much larger than observed for KI:Ag + (Wr/WD = 17.3/91 = 0.19). Perhaps below a certain value of this ratio a temperature-dependent strength occurs? The NaCI:Cu + resonant mode (Weber and Nette 1966) centered at 23.5 cm -1 provides a good test of this idea since now the ratio Wr/WD is 23.5/223 = 0.11. The temperature dependence of this mode's absorption is shown in fig. 10, where the temperature ranges from 4.2 K for curve (a) up to 77 K for curve (d). The resonant mode peak can easily be seen at each temperature. The center frequency and the linewidth are temperature dependent, as expected for a slightly anharmonic oscillator, but for at least the two lowest temperatures the strength appears to increase slightly with increasing temperature 8(21.5 K) /S(4.2 K) = 1.15. Because of the large width at still higher temperatures it is difficult to separate unambiguously the resonant mode and the two-phonon difference band absorption. At 42.6 K (curve c) the intrinsic two-phonon difference band absorption at the resonant mode center frequency is only 0.25 cm -1 (Love et al. 1989), compared to the resonant mode peak value of
w
Unusual anharmonic local mode systems 8
'i .........
157
I
._
89
caL , , "
:
:"lt
j
'G
~ u 0
9
.o
(b) 0
**
(a) .
!o
.
.
.
.
.
.
.
.
.
.
.
, i , ,
20
5o Frequency (crn-a)
|
40
Fig. 10. Temperature dependence of the absorption coefficient of NaCI+0.5 mole% CuC1 in the frequency region of the resonant mode. The four temperatures are (a) 4.2 K; (b) 21.5 K; (c) 42.6 K and (d) 77 K. At high temperatures the resonant mode resides on top of the intrinsic two phonon difference band absorption. The resolution is 0.43 cm-I . (After Sievers, unpublished). about 4 cm-1 but it is nevertheless clearly observable on the high frequency of the line in fig. 10. The unusual temperature-induced absorption strength change of the KI:Ag + resonant mode can be examined in a number of different ways to separate out the different contributions to the spectrum. One of these is to examine the strength change over AT jumps. Such changes can be seen with precision in fig. 11, where the absorption coefficient difference between neighboring temperatures as a function of frequency is presented (Greene 1984). The largest strength changes are seen to occur in the 6 K to 13 K region. Compared to the rapid resonant mode strength change with temperature, the line broadening and center frequency shift in this representation are again seen to be quite small. The broadening and shift also appear comparable to those measured in fig. 10 for the resonant mode of NaCI:Cu + at the two lowest temperatures. This strength change was first observed in 1965, with less precision (Takeno and Sievers 1965). In that work it was proposed that the resonant mode, through anharmonicity, is linearly coupled to other low-lying phonon
A.J. Sievers and J.B. Page
158
:(a)
Ch. 3
; all.2)-a(2.91
I I I
/,,%.~A
__1_
A -~'-
-]
~
; ~
:
@
9
:~
-
(c)
l
IE rJ
I
I
.
I
I
a(8.2)-a(9.3)
%;-
-
I
I
,
',
, x .
eJ
.(d) .; .[ .,L
"1"
tl
II
a (10.9)'-a 3.7)ll (I .,,..,~
V V .,_,-
-
A i!
IIII1~ ~ l l
Ii
. . . . . . . . . . . . . .
V
] i I
r
t I
~
(2(18.1)-Cl(21.6)
I
I
I
!
I
I
,
1
~5
20
I 0 - ~
I /~
L~-'"-,,~J_
-'
I
.
, I
g
, I
~o
.....
Frequency
I ....
--
I :
. v-
l
(cm -I)
Fig. 11. Absorption coefficient differences between adjacent temperatures, vs. frequency for KI:Ag +. The instrumental resolution is 0.47 cm -]. (After Greene 1984).
modes. Such a coupling could remove strength from the "zero phononlike" resonant mode transition without contributing to the temperature dependence of the center frequency and linewidth. The temperature-dependent strength data fit a Debye-Waller-like factor with a T 2 in the exponent. If the sidebands extended over a large enough frequency interval they would be negligible in the experimental measurement. The temperature dependence produced by such a Debye-Waller-like factor has the form
I(T) = exp
-
SO~ 2
1+ 4
~
JO
e :~ -- 1
'
(2.5)
where S'0 = 9W2c/4Mv2hw 3 contains an acoustic spectrum cutoff frequency We which is less than the Debye frequency coD. Here 0c = hwc/kB, v is
w
Unusual anharmonic local mode systems
159
the Debye sound velocity, and M is the average ion mass appropriate to the Debye modes. For Oh symmetry the effective strain coupling parameter A is determined by the three coupling coefficients A(hydrostatic), B(tetragonal), C(trigonal) for the even-parity symmetry types. Note that it is only at very low temperatures that the T 2 behavior contributes to the exponent. When the measured strain coupling coefficients from the uniaxial stress measurements are used to determine A, the value is an order of magnitude too small to agree with the temperature dependent strength data (Alexander et al. 1970). Besides linear coupling between a resonant mode and a Debye spectrum, Alexander et al. (1970) also attempted to fit the temperature dependence of the absorption strength with a linear coupling to a single even-parity resonant mode of frequency DE. In this case the intensity of the zero phonon line is given by I(T) = e-'YEIo(CE),
(2.6)
where 7E = SE(2~E + 1)
(2.7)
CE = SEcsch(hf2E/2kBT).
(2.8)
and
Because Sz is now of order unity instead of N -1, where N is the number of atoms as for extended lattice modes, one must include the modified Bessel function Io(Co) in the modulation analysis. One of the conclusions of the work of Alexander et al. (1970) was that eq. (2.6) provides a much better fit to the observed temperature dependence than does eq. (2.5), given the known strain coupling coefficients. This work, together with the experimental work of Kirby (1971) on the presence of both odd and even-parity resonant modes in the KI:Ag + spectrum, seemed to indicate that the vibrational properties of point defects were indeed well understood. But it was then discovered by Greene (1984) in the early 1980's that the reported good agreement between the linearly coupled odd and even mode analysis of the temperature-dependent spectrum was in fact due to an erroneous numerical calculation. No set of model parameters in eqs (2.6), (2.7) and (2.8) can produce agreement with the observed temperature dependence of the KI:Ag + resonant mode. In retrospect, it is now easy to see why the linear coupling model cannot explain the observed temperature dependence. For linear coupling to either extended band modes or to localized modes, the temperature
160
A.J. Sievers and J.B. Page
Ch. 3
dependence in the exponent comes from a (2n + 1) occupation factor. At temperatures large compared to the characteristic modulation frequency of the phonons or the low lying even mode this term must vary as kBT which, even in the exponent, is too slow to account for the rapid disappearance of the line strength by 25 K. When the E-field dependence of the resonant mode was investigated by Kirby (1971), other silver-induced features in the absorption spectrum were found, most notably a gap mode centered at 86.2 cm -l but also other much weaker but identifiable peaks located at 29.8, 44.4, 51.0, 55.9 and 63.6 cm -1. The interconnection between these different features remained unexplored until temperature-dependence measurements on the complete impurity-induced spectrum were made (Greene 1984; Sievers and Greene 1984). Some key features can be identified in the Ag+-induced absorption spectra shown in fig. 12. The solid curve in fig. 12(a) shows the impurity-induced absorption coefficient c~(w) versus frequency at 1.2 K. The assignment of the different features is the same as that given earlier by Kirby (1971). One surprising result is that the entire impurity-induced spectrum is very temperature dependent, as shown by the dotted absorption curve for 11 K in fig. 12(a). With increasing temperature the strength of the resonant mode, combination bands, and gap mode all decrease, while a broad absorption band associated with intrinsic two-phonon difference band processes increases at high temperatures. Precise measurements of the temperature dependence of the absorption coefficient in the low-frequency region around the resonant mode show that the decrease in the resonant mode strength is accompanied by a corresponding increase in a broad absorption which appears to be nearly frequency independent from the smallest wave number measured (3 cm -1) up to at least 25 cm -1. The magnitude of this non-resonant absorption is proportional to the low temperature strength of the IR active resonant mode over a factor 40 in Ag + concentration (Sievers and Greene 1984). There are other spectral features which appear at elevated temperatures, as can be seen in fig. 12(b), where the difference in absorption coefficient Ac~(w) between a given temperature and the 1.2 K reference temperature is plotted versus frequency. The solid curve is for T = 3.4 K, and the dashed one is for 10 K. A positive Aa(w) in this figure indicates that the sample absorbs more at high temperatures than at 1.2 K at that frequency. In addition to the nonresonant absorption (n) at low frequencies there are two additional new impurity-induced features which appear at elevated temperatures, namely a density of states peak (d) at 69 cm -1, and a gap mode (g) at 78.6 cm -1. Note that the KI phonon gap extends from 70 to 96 cm -1. The evolution of the excited state transitions with temperature can be seen more clearly in fig. 13, which gives absorption difference traces at a number of temperatures
A.J. Sievers and J.B. Page
162
0.6
----r
~
!
!
Ch. 3
r'""'T'-
q
/ 0.4 IE r
<1
0.2
-
s TK
30
J / !
50 70 Frequency (era-I)
Fig. 13. Temperature dependence study of the excited-state transitions for KI:Ag+. The ordinate shows the absorption coefficient difference between the temperature given in the figure and 1.2 K. The two features at 69 and 78.6 cm-1 are seen to grow with increasing temperature, as does the intrinsic two phonon difference band absorption. (After Greene 1984). (Greene 1984). All three of the high-temperature absorption processes (n, d and g) have similar temperature dependences: they are not observable at 1.2 K and grow in magnitude with increasing temperature. The common low-temperature behavior indicates that all of these absorptions may stem from the same elevated energy state. An important experimental finding is that the temperature dependence of the two strong modes, labeled r and g in fig. 12(a) are essentially the same. The strength I(T) of the IR active gap or resonant mode at a particular temperature T is given by eq. (2.4a, b), but with the frequency integration only across the line itself. The temperature dependence of the measured strength I(T) of each line is plotted in fig. 14(a). The growth of the excited state gap mode at 78.6 cm-1 is shown in fig. 14(b). The strength of this "hot" band is normalized to the low temperature strength of the gap mode at 86.2 cm -1. At first sight it might appear that these temperature dependences could follow from simple Boltzmann population effects for anharmonic oscillators, but this is not the case. One of the simplest explanations would be to ascribe the temperature dependence to the anharmonicity of the resonant mode itself, treated as an independent oscillator. If the 0 --+ 1 transition does not coincide with the 1 --+ 2 and higher transitions then only the 0 --+ 1 transition contributes to the unshifted resonant mode transition, and its intensity would decrease as the ground state is thermally depopulated (Alexander et al. 1970). Assuming that the anharmonicity is not too large
w
Unusual anharmonic local mode systems
161
Fig. 12. Temperature dependence of the Ag+-induced absorption spectrum in KI. The vertical dot-dashed lines divide the figure into three parts: The concentration in the center region (2 • 1018 Ag+/cm 3) is twice that of the two end regions. Assignment of the different features: r = resonant mode, c = combination band, d = density of states peak, g = gap mode, and n = nonresonant absorption. (a) The absorption coefficient vs. frequency for two temperatures: solid curve, 1.2 K; dashed curve, 10 K. (b) absorption coefficient difference between a given temperature and 1.2 K vs. frequency for two temperatures: solid curve, 3.4 K; dashed curve, 10 K. Note that the ordinate in the center region of (a) is expanded 5 times. (After Sievers and Greene 1984).
w
Unusual anharmonic local mode systems
163
Fig. 14. Normalized absorption strength vs. temperature for the Tlu resonant and gap modes in KI:Ag +. (a) Gap and resonant mode data for a number of Ag + concentrations. Solid and dashed curves are the predictions of the three dimensional double anharmonic oscillator model. The dotted curve, which is 1 minus the nonresonant growth curve data, is taken at a frequency of 4 cm -1. (b) Excited-state gap mode data. The solid curve shows the predicted temperature dependence of the excited-state strength for the same anharmonic model as used in (a). (After Sievers and Greene 1984). so that the h a r m o n i c oscillator partition functions can still be used, then for a t h r e e - d i m e n s i o n a l oscillator in w h i c h only o n e e x c i t e d state is i m p o r t a n t for e a c h polarization, o n e has
I(T)/I(O) - (1 - e - h l 2 r / k T ) 4
_ Zr4
(2.9)
w h e r e Zr is the partition function of a o n e - d i m e n s i o n a l oscillator. We n o w s u p p o s e that there is a t h r e e - d i m e n s i o n a l d o u b l e a n h a r m o n i c oscillator with e x c i t a t i o n n u m b e r s labeled (g,r). T h e t e m p e r a t u r e - d e p e n d e n t intensity o f two transitions are o f interest:
A.J. Sievers and J.B. Page
164
Ch. 3
(a) resonant mode (0, 0) ~ (0, 1) with intensity -,~ (Zg)-3(Zr) -4, (b) gap mode (0, 0) ~ (1,0) with intensity ~ (Zg)-n(Zr) -3. The dashed curve in fig. 14(a) gives the predicted temperature dependence for the gap mode and the solid curve gives the predicted dependence for the resonant mode. The solid line in fig. 14(b) shows the corresponding prediction for the excited state gap mode (0, 1) ~ (1, 1) transition. In each case the temperature dependence is not fast enough to fit the experimental data. Another missing feature required by this model at elevated temperatures is the (0, 1) ~ (0, 2) transition. Figure 15 shows that such a feature does not appear in the spectrum; instead the high temperature component consists of nonresonant absorption over the entire frequency region (Greene 1984). Here the difference between the absorption coefficient at the temperatures shown and that at 20.6 K is plotted versus temperature. Since the nonresonant absorption is not present at 1.2 K and the resonant mode absorption is essentially gone by 20.6 K, the lowest spectrum in fig. 15, cff l.2 K ) - c~(20.6 K), shows both the resonant mode and the nonresonant absorption components fully developed. 1
I
T
13.7 K
\
o
U
'
0
0 5
I0
15 20 Frequency (cm-I)
25
Fig. 15. Temperature-induced change in the absorption coefficient for KI:Ag+. Ac~ = c~(T) - c~(20.6 K) and T for each trace are given. Both the resonant mode and nonresonant absorption are evident in the lowest trace. The spectral resolution is 0.43 cm-1. (After Sievers et al. 1984).
Unusual anharmonic local mode systems
w
9
0.6
9
9
0..0~0
165 1.0
O0
A
o, 0.4
A
w, to to
/
A
e-
0.5 ~ , I--v
/"
_
i-
i:I
/"
0.2
,L" 0
-.,~-.~'~
0
I
I
I0
I
20
I
i
30
Temperoture (K) Fig. 16. Normalized nonresonant far IR absorption coefficient for KI:Ag + at 4 cm -1 vs. temperature (solid curve). The normalization factor is the high temperature value. Also shown (dots) is the measured temperature dependence of T A 1 defined later by eq. (2.15) in the text. This quantity is proportional to the population in the off-center configuration. The initial increase of the data with temperature follows an energy gap law with a gap of 24 K. (After Hearon and Sievers 1984).
In order to understand better the role of the nonresonant absorption, its contribution was examined at a frequency far removed from the resonant mode itself. A broad band millimeter wave spectrum with intensity maximum centered near 4 cm-] and a full width at half maximum of 4 cm-1 was used to measure the nonresonant absorption growth curves as a function of temperature. The experimental results normalized to the high temperature data are represented by the solid line shown in fig. 16 (the ordinate for this trace is on the right). By subtraction of the normalized growth curve from unity, the data can be compared directly with the temperature dependence of the resonant and gap mode strengths. These data are represented by the dotted curve in fig. 14 (a). It appears that the strength lost by the gap and resonant modes with increasing temperature is transferred to the nonresonant absorption (Sievers et al. 1984).
2.1.4. Two elastic configuration model When the unusual strong temperature dependent properties of KI:Ag + are contrasted with the temperature-independent properties of NaI:Ag +, it seems clear that the original Lifshitz approach (1956) would have qualitative difficulties in serving as a basis to explain these temperature dependent features,
A.J. Sievers and J.B. Page
166
I cz
Cl
I
[ i|
Ch. 3
ii
I
I
I
I I I d g
I
I I
I I I
i
i
I 0
I
Fig. 17. Schematic representation of the two configuration model. C1 identifies the on-center elastic configuration, with only the gap mode transition shown. C2 shows the higher energy (~) off-center elastic configuration with the excited state density of states and the gap mode transitions. One possible source for the relaxation time r is shown.
since the same impurity is surrounded by the same nearest neighbors in two alkali halides having the same symmetry. To describe in a phenomenological manner the temperature-dependent data, it was proposed that under certain circumstances a second elastic configuration could occur in addition to the ground state configuration of the defect-lattice system (Sievers and Greene 1984). Figure 17 illustrates such a possible two-configuration arrangement. Here C] identifies the ground state on-center configuration and C2 the offcenter one at a higher energy ~. A few of the assignments of the observed transitions are also shown. The rapid temperature dependence observed for KI:Ag + requires that the second configuration contain a low-lying multiplet with a large degeneracy in comparison to the on-center configuration, perhaps related to the displacive tunneling of the Ag + defect among many equivalent minima centered about a normal lattice site. This possibility is not so unusual, since a twelve-component spectrum with a tunneling splitting of ,,~ 0.1 cm -1 has been identified for the normal ground state configuration of Ag + in RbC1 (Kirby et al. 1970; Kapphan and Luty 1972; Holland and Luty 1979). As long as the thermal and electromagnetic energies are much larger than the tunnel splitting, the frequency dependence of the absorption coefficient for such an arrangement could have the simple Debye form considered below.
w
Unusual anharmonic local mode systems
167
The Clausius-Mossotti equation for a medium with intrinsic dielectric constant eo that contains Ni polarizable defects per cm 3 is (Hearon and Sievers 1984) A
~'- 1 ~"+ 2
=
eo + , 4 - 1 eo + ,~ + 2
=
eo- 1
47rNi~i
)
eo + 2
,
(2.10)
3
A
where ,4 is the defect-induced contribution to the complex dielectric constant and ~i is the dipolar polarizability of the defect. In the limit of a small offcenter concentration Ni, the impurity-induced dielectric constant becomes (Sievers et al. 1984)
Z~ = 47rNi~i ( e~ + 2 ) 2.
(2.11)
3
In the Debye approximation, the dielectric response at frequency w is
3 = A 1 -I--iA2 -- A0(1 - iw'r)- 1,
(2.12)
where
A0 = 4"n'Ni ~
3
'
(2.13)
p is the dipole moment, k is Boltzmann's constant, T is the temperature and ~- is the relaxation time. According to eq. (2.12) for the far IR limit (wT >> 1), the nonresonant absorption coefficient is
O~n
CV~
~
,
(2.14)
which is independent of frequency and is proportional to the number density of off-center ions. If the Debye model describes the nonresonant absorption data, then in addition to the far IR signature in the temperature-dependent absorption coefficient there should also be a radio frequency signature in the real part of the impurity-induced dielectric function.
A.J. Sievers and J.B. Page
168
Ch. 3
2.1.5. Radio frequency dielectric constant measurements Measurements of the real and imaginary parts of the dielectric function at 10 kHz as a function of temperature are shown in fig. 18 (Hearon and Sievers 1984). An important observed feature of these measurements is that only the real part of the dielectric function is temperature dependent; no temperature dependence is found for the imaginary part. Although e l ( T ) - ~1(1.4 K) shows a maximum at about 10 K, no corresponding loss peak is observed in e2(T). For ground state tunneling systems such as RbCI:Ag +, a peak in el (T) as a function of temperature is always accompanied by a concomitant peak in e2(T) as r(T) is swept through the wr = 1 condition with increasing temperature (Holland and Luty 1979); moreover, the radio frequency spectra shown in fig. 18 are independent of excitation frequency w. Both of these results indicate that wr << 1 over the entire temperature region of these KI:Ag + measurements. Equations (2.12) and (2.13) show that in this limit
A I = A oo(
Ni(T) T
and
A2=0,
0.04
(2.15)
I
z~
0.02 I
x~ x
F-x
X X
0.00 0
20
40
Temperature (K) Fig. 18. Temperature dependence of the impurity-induced dielectric constant for KI:Ag + as obtained from radio frequency measurements, el ( T ) - e1(1.4 K) is plotted vs. temperature for the single-crystal samples of 0.5 and 0.2 mole% AgI in KI in the melt (triangles and crosses respectively.) Also shown is the temperature dependence of e o ( T ) - e0(1.4 K) for pure KI (the z's). e 2 ( T ) is temperature independent and has the same value for all three samples (,-~ 1.5 x 10-3). The measuring frequency is 10 kHz. (After Hearon and Sievers 1984).
w
Unusual anharmonic local mode systems
169
so that the temperature dependence measurement of the real part of the dielectric function provides a direct determination of the population in the off-center configuration. The temperature dependence of the population in this second configuration is found as follows. The data in fig. 18 for the 0.5 mole% AgI are subtracted from those for pure KI, the result is multiplied by T and plotted vs. temperature, resulting in fig. 16. At high temperatures the data points in this figure indicate that the off-center population approaches a near constant value. Information about the reorientational relaxation time for Ag + in this offcenter state can be obtained by combining the experimental results for the dipole moment from the radio-frequency dielectric constant measurement with the measured temperature dependence of the nonresonant absorption coefficient c~n in the far IR spectral region. Inspection of fig. 16 shows that the far IR nonresonant absorption data follow Ni(T), while according to eq. (2.14) the data should vary as Ni(T)/[TT(T)]. The consequence is that TT(T) -= constant. These radio-frequency measurements appear to support the picture that at least two elastic configurations exist for this lattice defect system. At low temperature the Ag + ion is on-center with its associated on-center spectral features. When the temperature is increased, a second elastic configuration becomes populated. This configuration has an energy with respect to the on-center ground state of at least 24 K. The large increase in the defect contribution to the dielectric constant indicates that the Ag + ion is off-center in this second, higher-energy elastic state. Because the off-center population appears to approach a constant value at elevated temperatures, the available data seem to indicate that only these two elastic configurations are close by in energy. The large difference in the temperature-dependent properties of Ag + in the hosts NaI and KI may indicate that low-lying configurations can only occur when the impurity ion is smaller than the host ion it replaces (Na + < Ag + < K + < Rb+). This idea is supported by three different hydrostatic pressure measurements which have been used to transform off-center systems such as RbCI:Ag + (Holland and Luty 1979) and KCI:Li + (Devaty and Sievers 1980) to on-center ones. The almost discontinuous change observed in the impurity ion position at low temperatures and at finite temperatures would be a natural consequence if two different elastic configurations could coexist for a defect lattice system. The apparent transition would occur whenever the pressure-tuned ground states for the two configurations have approximately the same energy, so that direct configurational tunneling becomes possible. The detailed paraelectric resonance studies over a narrow hydrostatic pressure range for Ag + in the RbC1 host provide an experimental probe of the dynamics in this nearly degenerate two-configuration region. It has been
170
A.J. Sievers and J.B. Page
Ch. 3
shown that a modest hydrostatic pressure tunes the levels of what is interpreted as the on-center configuration into the same energy range as the off-center one, so that a microwave transition can be observed between these configurations (Bridges et al. 1983; Bridges and Chow 1985; Bridges and Jost 1988). It is noteworthy that the integrated strength of this transition decreases much more rapidly with increasing temperature than Boltzmann population effects predict (Bridges and Jost 1988). 2.1.6. Microwave absorption measurements
Absorption measurements on KI:Ag + as a function of temperature from 5 K to 300 K have been made at three different frequencies in the microwave region, in order to explore the off-center Debye model in more detail. According to eq. (2.14) the model predicts a very simple result for the absorption coefficient in the limit w~- >> 1. These absorption measurements were made with an oversized nonresonant cavity technique (Hearon 1986) which was developed at the Max Planck Institute in Stuttgart (Kremer 1984; Poglitsch 1984). Backward-wave oscillators are used to span the frequency range from 40 GHz to 160 GHz. In order to approach an isotropic field in the cavity, mode stirrers are used to vary the effective shape of the cavity as a function of time, so that a large number of modes interact with the sample. The sample is surrounded by an unsilvered fused quartz double-walled cold finger which hangs in the cavity, and it is cooled to low temperatures by a continuous flow of cryogenic helium gas. The temperature dependence of the absorption coefficient for samples of KI +0.2 mole% AgI and KI +0.03 mole% AgI are shown in fig. 19. The impurity-induced absorption coefficient is given by the difference between the two temperature dependent curves at 160 GHz and 80 GHz. (The low concentration sample was destroyed before the 40 GHz data set could be completed.) To obtain these results, the detector signal for the cavity with the sample is divided by that without the sample at the same temperature to determine the absorption coefficient for a fixed frequency. The estimated uncertainty in the data is represented by the cross-hatched region around each set of data. The qualitative temperature dependence of the impurityinduced nonresonant absorption (the difference between the two curves in a given panel) for the limit a~- >> 1 can be seen most clearly for the highest frequency data shown in fig. 19 (a). These data, which appear to be temperature independent over much of the temperature interval studied, indicate that Nip2/T~ - is essentially a constant for temperatures large enough to favor the off-center configuration. One possible conclusion is that T~- = constant, just as was found earlier by comparing the far IR and the radio frequency data.
w
Unusual anharmonic local mode systems
0.4-
171
o) f=160
0.2-
0
IE 0.2
-
t
b) f: 80 GHz
I
c) f=40GHz
v
E ~0.1
g := O
a= ,,~
0
0. IO
,/ / ,~/ ,,,'/',, / /j...~
0.0S
0 I
0
I00 200 Temperoture (K)
300
Fig. 19. Temperature dependence of the impurity-induced microwave absorption coefficient of KI:Ag + at three different frequencies. (a) 160 GHz (5.33 cm-]), (b) 80 GHz (2.67 cm - ] ) and (c) 40 GHz (1.33 cm-1). The upper curve in each panel is for KI + 0.2 mole% AgI and the lower ones in frames (a) and (b) are for KI + 0.03 mole% AgI. The estimated errors for the oversized microwave cavity technique are identified by the cross hatched regions in each frame. (After Hearon 1986).
A.J. Sievers and J.B. Page
172
Ch. 3
The lower frequency data in fig. 19 (b) and (c) show additional temperaturedependent structure at low temperatures, which probably indicates that both resonant and nonresonant absorption are being observed for the off-center configuration. It now appears clear that given sufficient nonresonant absorption data versus temperature for different frequencies, both the applicability of the Debye model and the temperature dependent behavior of the different factors in Nip2/TT could be isolated and explored, identifying important aspects of the mysterious off-center configuration.
2.1.7. Rotational motion in the off-center configuration Perhaps the simplest way to produce two configurations, one of which has a much larger effective degeneracy than the other, is to assume that the off-center configuration of the Ag + ion contains a large number of free rotor states. In this model we assume that the crystal field splitting of such free rotor states is negligible. In addition, we assume that the rotational moment of inertia I of the Ag + ion about the normal lattice site is large enough so that the spacing between the rotational levels is small compared to kT at all temperatures for which the upper configuration is populated. In this case the rotational levels are dense and the rotational motion can be treated by classical statistical mechanics. The rotational partition function becomes Zr ~
2IkT h2
•
T Tr
.
(2.16)
If we assume that the vibrational degrees of freedom are roughly the same in both configurations, then the on-center population for the two-configuration model displayed in fig. 17 is simply
Pon(T) =
l+
T -~-~ Trr e
.
(2.17)
The surprising result is that with both the energy gap ~ and the moment of inertia Tr treated as free parameters, it is still not possible to fit the data shown in fig. 14. When Tr is made sufficiently small (generating a high density of states) so that at 25 K a small value for Pon(T) can be produced, then the abrupt knee at about 5 K is missed. Apparently, even this limiting case of free rotor states without the quenching produced by crystalline electric field effects does not produce a dense enough set of levels to match the measured temperature dependence. A better fit to the data of fig. 14 is obtained simply by replacing (T/Tr) in eq. (2.17) by a constant degeneracy factor g t> 100.
w
173
Unusual anharmonic local mode systems
2.1.8. Ag + in KI alloys
To test rotor-like models, disorder has been introduced into the lattice to see if the temperature-dependent properties of the resonant mode are changed. Early work on Li + ions in KBr alloys (Clayman et al. 1967) has shown that a resonant mode can be shifted both to smaller and larger frequencies, since an alloy of two alkali halides has an average lattice constant intermediate to those of the two constituents (Havighurst et al. 1925). The dependence of the mode frequency and linewidth on lattice constant was found to be linear in Clayman's work. Since the average lattice constant (a) is calculated from the measured alloy concentration by the Vegard relation, namely (a) = al + ( a 2 - al)z where al and a2 are the lattice constants of the two components and z is the molar concentration of component 2 (Vegard 1921), the centroid frequency of the resonant mode is expected to be a linear function of the molar alloy concentration. Since A V / V -- 3Aa/(a), the frequency shift can be expressed as -Aw = A ( A V / V ) cm -1. Because RbI has a larger lattice constant than KI, alloying some RbI with KI:Ag + would be expected to decrease the resonant mode frequency, making the defect system more unstable. The far IR temperature dependent spectrum of a KI + 1 mole% RbI + 0.2 mole% Ag + sample is presented in fig. 20
1.0
m
A
"T i!
- o.s
.J // i
l 0
Z>v'"-7"~ C)"
I0
.
~
lyi'~" i
20
30
40
Frequency (cm "l) Fig. 20. Temperature dependence of the Ag+-induced resonant mode absorption coefficient for a KI alloy. The sample is KI + 1 mole% RbI + 0.2 mole% AgI. The solid line is for T = 1.2 K, the dot-dashed line is for T = 10 K and the dashed line is for T = 20 K. The center frequency of the resonant mode is shifted slightly to the red and one half of the low temperature resonant mode area is missing when compared to an unalloyed sample grown at the same time. (After Hearon 1986).
174
A.J. Sievers and J.B. Page
Ch. 3
(Hearon 1986). At 1.2 K (solid line) the resonant mode centroid is shifted to a smaller frequency and has lost half of its strength when compared to a KI + 0.2 mole% Ag + sample, grown at the same time. The dot-dashed trace shows the same spectrum for I0 K. The temperature-induced area loss is very similar to that found for the unalloyed crystal: compare this result with the 9.3 K trace shown in fig. 9. At the highest temperature of 20 K (dashed curve), the resonant mode peak has nearly vanished. These temperature dependent results appear very similar to those presented earlier for the unalloyed samples. As the concentration of RbI is increased from 1 mole% to 5%, the frequency shift is found to obey the relation -Aw = 300(AV/V) cm -1. For each alloy, the strength of the mode appears to have the same temperaturedependent properties as described above. Unfortunately, the alloy-induced shift of the resonant mode to smaller frequencies is accompanied by a rapid decrease in the low temperature line strength. As a result, for 3 mole% RbI the strength is about 1/20 of that for the unalloyed crystal, and by 5 mole% RbI, the strength ratio is a barely observable 1/50. Is this rapid change in the low temperature strength with increasing alloy lattice constant a consequence of the off-center configuration coming into resonance with the on-center one? Probably not. For the different alloy concentrations, the observed strength most likely stems from the Ag + centers that have unperturbed neighbors. If the Ag + centers are in a random distribution and it is the next-neighbor Rb + ions that destroy the mode, then the absorption strength should vary as ,-~ (1 - : c ) 12 where z is the alloy mole concentration. The rapid strength change observed here with increasing z could be interpreted as evidence for some degree of nonuniformity in the Ag + :Rb + distribution. When the crystal is grown, the Ag + ions may prefer to remain in a substitutional lattice site near the Rb + ions in the crystal. A remarkably similar strength change with alloy composition has been found previously for the KBr:Li + resonant mode. In this case expanding the alloy lattice by 5 mole% KI also causes the mode strength to disappear (Clayman and Sievers 1968). Although the strength of the Ag + resonant mode is strongly influenced by lattice disorder, its temperature dependent properties appear to remain unchanged, within the experimental uncertainties generated by the large strength change.
2.1.9. Temperature dependence of the Ag + electronic transitions in KI Another way that the temperature dependence of the position of the Ag + ion can be monitored is by making use of the known optical properties of this ion in alkali halide crystals. This behavior can be understood in
w2
Unusual anharmonic local mode systems
175
terms of the 4d 1~ --+ 4d95s parity-forbidden electronic transitions of the Ag + defect (Fussg~inger 1969; Kojima et al. 1968; Holland and Luty 1979) being made allowed by dynamic or static symmetry breaking. When the ion is a substitutional defect in a cubic crystal these electronic transitions are made somewhat allowed by vibronic coupling to odd-parity vibrational modes, or if the defect is off center, by the static odd-parity lattice distortion. In each case the larger the odd-parity contribution, the larger the line strength. A schematic representation of the temperature dependence of the oscillator strength of the transition for the different possibilities is given in fig. 21. Four cases as a function of temperature are shown: (A) gives the optical signature for an on-center defect coupled to a single Tlu symmetry harmonic vibration; (B) is for an on-center defect coupled to a single Tlu symmetry anharmonic vibration; (C) is for an off-center impurity with a shallow barrier and (D) is for a frozen off-center impurity with a large potential barrier. Little optical work has been done on KI:Ag + because this particular system has the unusual property of "aging" with time at room temperature much faster than for the other alkali halide hosts (Fussg~inger 1969). A systematic way to reverse the aging process and hence to reactivate the samples was found during the radio frequency studies of KI:Ag + by Hearon (1986). I
to-'
D
I
,
f
~
cons,
'
[x,] Id 2
td 3
,
0
I
|
200 Temperature (K) I00
Fig. 21. Schematic temperature-dependence signatures for the electronic oscillator strength of the Ag + ion coupled to different strength odd-parity distortions. (A) on-center ion coupled to a harmonic Ttu vibrational mode. (B) on-center ion coupled to an anharmonic Tlu vibrational mode. (C) off-center ion with a shallow potential barrier. (D) off-center ion with a large potential barrier. (After Holland and Luty 1979).
A.J. Sievers and J.B. Page
176
Ch. 3
o) ? Jan.
0.02 oo
OO
O Oo
O
O O O
O.OI
O O O
""
21 Jan. A&
& "~' a a t=
aa~ Aaat=a
A
,r
o ~ ~
.........................
b)
I
h-- 0 . 0 4
After ,txtxt•t="at=tbt= at=a ~ a
tx
0.02
t=
o 0
ooo
Before
oooooo,oooooooo 8 16 Temperature (K)
~ 24
Fig. 22. Aging and reactivation of isolated Ag + defect centers in KI. (a) The temperature dependent radio frequency dielectric constant versus temperature for an as-grown sample (circles) of KI + 0.2 mole% AgI and the aging observed in the same sample after it has been refrigerated for 14 days (triangles). The number of isolated defect centers has decreased by a factor two. (b) Radio frequency dielectric constant data (circles) for a KI + 0.5 mole% AgI sample maintained at room temperature for several months. The data for the same sample after heat treating at 200 C (triangles) shows that the isolated defect density is fully restored. (After Hearon 1986).
Such effects had been observed previously in the far IR, and samples were refrigerated between runs to slow the process. New crystals were always grown once previous samples had aged. Hearon (1986) found that the reactivation procedure for a Ag + doped KI crystal which has been stored at room temperature is first to heat it to 200 ~ in air and then cool it back down to room temperature in a few minutes. Examples of this aging and reactivation effect are shown in fig. 22. Here the temperature-dependent behavior of the dielectric constant is monitored to determine when the isolated Ag + defect state is present. Figure 22(a) shows the aging effect as measured via the radio frequency dielectric constant for a freshly grown sample
w
Unusual anharmonic local mode systems
177
of KI + 0.2 mole%AgI (circles) and for the same sample after it had been refrigerated for 14 days (triangles). The concentration of isolated centers has decreased by a factor 2. Figure 22(b) illustrates how the sample can be reactivated. A KI + 0.5 mole% AgI sample maintained at room temperature for several months is nearly devoid of isolated centers, as indicated by the circles. After cycling the sample to 200 ~ the temperature-dependent dielectric constant associated with isolated centers is fully restored (Hearon 1986). This aging effect is also quite pronounced in the UV absorption spectrum of the doped crystal. Figure 23 shows the spectrum of an aged KI:Ag + sample before (dashed) and after (dot-dashed) the heat treatment. Before heat treating, the UV spectrum has a strong edge near 420 nm that is in the same frequency interval as the band gap of pure AgI (Kleppmann 1976). After heat treatment, the AgI-like spectrum in the UV disappears, and the crystal becomes transparent. An Ag+-like absorption spectrum is now found at much higher frequencies. Apparently, at room temperature the Ag + ion is fairly mobile in the alkali halide host and tends to organize into silver 1.0
,
w
,..........
after before 0.8
.,,,..,,.v,
,,,-.--
9 - - . ~9 . , . ' . . ~ " ~- 9 - . - ~ , . ~ . , ~ . .
/.r ,,
I I ! I ! I I
! !
.s 0.6
! !
t
,..., 0 . 4 [.-,
/
i /
t
'--
/
i i
/ / /
!
0.2
,-. /
J
/
i
/
I /
t
i 0.0
I
/ /
......... 250
290
/
!
!
i
330
370
410
450
Wavelength (nm) Fig. 23. Effect of heat treatment on the KI:Ag+ UV transmission spectrum. Dashed curve: aged crystal before heat treatment, showing the absorption produced by AgI aggregates. Dotdashed curve: after heating this sample to 200 C and quenching to room temperature, isolated Ag+ centers are produced. The crystal is now transparent beyond the 420 nm wavelength region. (After McWhirter and Sievers, unpublished).
178
A.J. Sievers and J.B. Page
Ch. 3
halide clusters which the modest heat treatment perfected by Hearon (1986) breaks up. Figure 24 shows the measured UV absorption spectrum of KI:Ag + for two different temperatures. Over this temperature range the center frequency of each of the three bands remains essentially fixed. A linear concentration dependence establishes that these three features labeled A, A' and C [using the nomenclature of Fussgiinger (1969)] in the figure are associated with isolated Ag + ions (Page et al. 1989). To display the three lines in one figure, three different Ag + concentrations are required, as described in the caption. The strength of the strong UV line C shown in fig. 24(a) is temperature independent, and is assigned to a delocalized charge transfer excitation in which an electron from a nearest neighbor anion occupies one of the empty Ag + excited states. The weak lines A and A' are assigned to the Alg(4d 10) -+ T2g(4d95s) and Alg(4d 10) --~ Eg(4d95s) transitions, respectively. They display nearly zero strength at 1.2 K but grow rapidly with increasing temperature. The change is much faster than can be obtained from the occupation number effect in the low-lying Tlu resonant mode, for example. This oscillator strength temperature dependent signature of A and A' does not match any of the cases displayed in fig. 21 but the increase in
Fig. 24. KI:Ag+ UV absorption spectra. Dashed line, T = 50 K; dot-dashed line, T = 1.2 K. Silver concentration: panel (1) 2 x 10-2 mole%, panel (2) 2 x 10-3 mole%, panel (3) 5 x 10-5 mole%. (After Page et al. 1989).
Unusual anharmonic local mode systems
w
1.0 .
i
179
,,,
o
m
It
Q_ 0
~. 0.5
~
r c-
\ (3
I
xU,..
tO
0.0 0
10 2O Temperature (K)
30
Fig. 25. Temperature dependence of Ag + on-center population, as determined by various measuring techniques. Dashed line, far IR resonant and gap mode strengths. Dotted line, dielectric constant data. Squares, UV absorption for the A' electronic state. Solid circles, Raman scattering intensity from the 16.1 cm -1 Eg mode. (After Page et al. 1989).
optical strength with temperature appears identical to that found earlier for the increase in the DC dielectric constant produced by the appearance of a permanent dipole moment with increasing temperature. This latter change was shown to be proportional to the population in the off-center configuration, Poff. By letting Pon 1 -Poff, both the UV and dielectric constant data can be compared directly with Pon determined from the temperature dependence of the Tlu resonant and gap mode strengths. These results are graphed in fig. 25. The three different experimental probes are seen to measure exactly the same rapid decrease in the population of the on-center population. -
-
2.1.10. Raman scattering from even-symmetry resonant modes
The polarized Raman intensities I((1,-1,0)(1,-1,0)),
I((1, 1,0)(1,-1,0)),
I(
for KI:Ag + have been reported for the 90 ~ scattering geometry (Page et al. 1989). The notation here identifies the polarization of the incident and scattered wave with respect to the crystal axes: ((incident)(scattered/). The results at three temperatures for the first two polarizations are shown in figs 26(a) and 26(b). No impurity-induced scattering was observed for the
180
Ch. 3
A.J. Sievers and J.B. Page ----T~T-~-'--'r----'-~T-------T---'--'T'-'-
280
I
-
(o) k
N~
-
\x 160
,...
A C/) ta C: :3 0 U
"-" 4 0 =,., 2 8 0 ..,. c/} C
I
"~
(b)
C: I--t
160
""\
:
".
- ................' o... ........
40
12 Raman
~ ~:~,-A.
20 shift (cm - l )
t,
"
28
Fig. 26. Temperature dependence of the polarized Raman spectra of a { l, 1, 0}-cut KI + 0.3% AgI crystal. The spectra, taken with 488 nm excitation, are displayed for two polarization geometries: (a) ( 1 , - 1 , 0 ) ( 1 , - 1 , 0 ) ( A l g + ( 1 / 4 ) E g + ( 1 / 2 ) T 2 g ) a n d (b) (1,1, 0) (1, - 1 , 0) ((3/4) Eg). The three temperatures shown are 6.3 K (dotted line), 12.9 K (solid line), and 25.2 K (dashed line). No ( 1 , - 1, 0)(0, 0, 1) (1/2 T2g) polarized lines were observed at any temperature. The resolution is 2.5 cm -1. (Page et al. 1989).
third orientation. The data are classified according to the even-parity representations Alg, Eg and T2g of the cubic group Oh, which yields for the polarized intensities the expressions I ( ( 1 , - 1 , 0 ) ( 1 , - 1 , 0 ) ) - Alg + (1/4) Eg + (1/2) T2g,
I((1, 1,0)(1,-1,0))
- (3/4) Eg,
I((1, -1, 0) (0, 0, 1)) = (1/2) ZEg. The Raman measurements presented in fig. 26(b) support the earlier low temperature Eg mode frequency assignment by Kirby (1971). On the other hand, neither of the previously proposed Alg and T2g higher frequency resonances are found. In addition, no pronounced Raman features are found in the gap region. A new discovery is the strong temperature dependence of
w
Unusual anharmonic local mode systems
181
the Raman scattering intensity of this 16.1 cm -1 Eg mode, which is shown for all of the temperatures in fig. 25. As temperature is increased, the scattering strength of this Eg mode decreases rapidly (see the solid circles in fig. 25), yet its center frequency stays nearly fixed. The temperature dependence of this Raman line is remarkably similar to that of the far IR active resonant and gap modes and the optical features discussed above, but it takes on added importance because the Eg mode involves no motion of the Ag + ion. Unpolarized Raman spectra at a number of temperatures between 6.9 and 25.7 K are shown in fig. 27. With increasing temperature up to 17.1 K, the 16.1 cm -1 Eg resonant mode Raman peak decreases in strength and is finally replaced by a second excited state feature at 12.2 cm-1. This excited state peak shows both Eg and Alg components and is consistent with the appearance of an A1 symmetry mode in a Cnv off-center configuration. At still higher temperatures this feature also disappears (or broadens) into a pronounced central peak having the same polarization properties, and likely produced by the off-center tunneling levels themselves. This polarization character of the quasielastic central peak is preserved up to 60 K, while at higher temperatures T2g symmetry scattering appears (Fleurent et al. 1991).
25.7K
A i.0 6.9
9.2K
IlK
15.2K
r O Ir
o
>,0.5
B
t/3 t--
C
5
a~"S--
ZO 35 Roman shift (cm-I)
Fig. 27. Unpolarized Raman spectra of a {100}-cut KI:Ag+ crystal versus temperature. The six different temperatures are given in the figure. With increasing temperature the 16.1 cm-1 Eg resonant mode is replaced by a lower frequency A1 symmetry mode appropriate to a C4v symmetry defect site which subsequently disappears in a quasielastic scattering peak at the highest temperature. (After Fleurent et al. 1991).
A.J. Sievers and J.B. Page
182
Ch. 3
2.2. Basic shell model description of the T = 0 K nearly unstable lattice dynamics
2.2.1. Local modes The Lifshitz harmonic Green's function method for computing general defect phonon properties has been described in detail (Maradudin et al. 1971; Bilz et al. 1984). For completeness, the application of this method to determine the frequencies, displacement patterns and IR spectra of localized modes arising from substitutional impurities is briefly reviewed here. A detailed discussion of the application of the Lifshitz method to the lowtemperature on-center dynamics of KI:Ag + is given by Sandusky et al. (1993a). The defect concentration is taken to be low enough so that the case of just a single impurity is required. For a system of N ions interacting via harmonic forces, the normal mode frequencies {w f} and displacements x ( f ) are determined by the 3N • 3N matrix eigenvalue equation
(4:' - w ~ M ) x ( f ) = O,
(2.18)
where ~ is the harmonic force constant matrix, M is the diagonal mass matrix and f = 1. . . . ,3N labels the modes. The force constant matrix is symmetric, and the modes are normalized such that the 0rthonormality relation is x ( f ) M x ( f ' ) = 5f f,. With a substitutional defect present, it is convenient to rewrite eq. (2.18) identically in terms of the host crystal harmonic Green's function matrix G0(w2) and the perturbing matrix C(w2), containing the impurity-induced force constant and mass changes. The result is +
x ( f ) = o.
(2.19)
This equation can be partitioned into two equations" one involving just components inside the "defect space", defined by the sites associated with nonzero elements of C, and a second equation which determines the mode displacements outside the defect space:
[III -k- GoiI(~)Cii(~o~)] xI(f) = 0
(2.20)
XR(f) = --GoRI(~)CII(OJ~)xI(f)"
(2.21)
and
w
Unusual anharmonic local mode systems
183
The subscripts I and R refer to components inside and outside the defect space, respectively. Equation (2.20) gives the determinental frequency condition: (2.22)
IIii + GoIi(w})Cii(w}) I - O.
For isoelectronic impurities, the defect space is usually small. Equation (2.22) then involves the determinant of a small matrix and is thus practicable, provided the defect-space elements of the unperturbed harmonic Green's function matrix can be computed. This is readily done for localized modes by direct summations involving the unperturbed host crystal phonons; for alkali-halides, these are well-known through phenomenological models (e.g. shell models) that account very well for the measured phonon dispersion curves. Once the local mode frequency is known, eq. (2.20) can be used to compute the defect-space displacement pattern, to within a normalization constant. The displacements outside the defect space may then be determined by eq. (2.21).
2.2.2. Infrared absorption In the long-wavelength limit, the interaction between an insulator obeying the Born-Oppenheimer approximation and an external field of monochromatic infrared radiation is
-M(u) . Ee -i~t,
(2.23)
where M(u) is the system's dipole moment for nuclear configuration u. If just the linear term MS(u) -- ~-,z# #~(l)u#(1) in the expansion of the ath component of the dipole moment in terms of the nuclear displacements is retained and a calculation similar to the one carried out by Klein (1968) is done, the resulting absorption coefficient a(w) for a cubic crystal with an impurity concentration Ci is given by 2
47rwCi ( n ~ + 2 ) ~(w)- cn(w) 3
~ I m [ G(W2+ ie)]/-~,
(2.24)
where n ~ is the high-frequency index of refraction, c is the speed of light in vacuum, G(w 2) is the harmonic Green's function matrix for the impurity crystal and the limit e --+ 0 + is understood. The quantity / ~ = {#~(l)} denotes the effective charges in this linear dipole moment approximation.
A.J. Sievers and J.B. Page
184
Ch. 3
In deriving eq. (2.24), it is assumed that the standard Lorentz local field correction holds and that the contribution from the electronic polarizability is adequately described by the high frequency dielectric constant coo. The index of refraction n(w) is taken to have its pure crystal value, owing to the assumption of low defect concentrations. Hence the integrated absorption strength Sf = f a(w)da~ for a single local mode f is n~+2
S.f = cn(w)
3
)
2
[~"~x(f)]2
(2.25)
Using the transformation properties of the dipole moment, it is straightforward to show that for an impurity crystal with Oh symmetry,/~ transforms under symmetry operations as the ath partner of irreducible representation Tlu. Hence, by a standard group-theoretic matrix element theorem, f ~ x ( f ) vanishes when x ( f ) is not an ath partner belonging to Tlu. In eq. (2.25), the sum implicit in ~ x ( f ) extends over the entire system, reflecting the fact that infrared radiation couples to all of the ions. For the case when the effective charges in the defect crystal are unperturbed from their host crystal values, Klein (1968) has shown how the absorption may be expressed in terms of just defect-space quantities.
2.2.3. Impurity model calculations The determination of the local mode (and resonant mode) displacement pattern and frequency requires knowledge of the perturbing matrix C(w 2) and the pure crystal Green's function elements. Two different methods have been employed to obtain the latter in the KI:Ag + defect mode calculations. The first is detailed in (Page 1974; Harley et al. 1971), and its present application is outlined here. The breathing shell model (Schrrder 1966) was used by Page et al. (1989) to compute the pure KI phonon frequencies {~ka} and complex plane wave displacement patterns {x(kj)} at 22,932 k vectors in the irreducible 1/48 element of the Brillouin zone, this being equivalent to one million vectors in the full zone. The imaginary parts of the unperturbed Green's function matrix Im Go(a;2) = 7r ~ x ( k j ) x +(kj) ~ (w~3 - to2) ka were approximated as histograms by dividing the pure KI phonon frequency range into 100 equally spaced "bins" and evaluating the sum over the modes
w
Unusual anharmonic local mode systems
185
whose frequencies fall within each bin. The real parts were then obtained by computing the Hilbert transforms of the imaginary parts. The defect model for a low-temperature on-center configuration of KI:Ag + consists of the mass change Am, longitudinal force constant changes 5 = - A ~ ( 0 0 0 , 100) between the defect and each of its six nearest neighbors, and longitudinal force constant changes 5' = - A ~ ( 1 0 0 , 200) between the defect's nearest neighbors and their adjacent fourth-nearest neighbors. Physically, 5 arises from the different binding of the impurity and is expected to be negative, consistent with the overall force-constant softening implied by the presence of the strong low-frequency IR impurity resonance at 17.3 cm -1. The force constant change 5' is postulated to arise from the expected defectinduced inward static relaxation of the nearest neighbors, also consistent with the overall force constant softening. Within this model, only the modes belonging to the Tlu, Eg and Alg irreducible representations of the Oh point group are perturbed. The necessary orthonormal symmetry basis vectors within the impurity space are shown in fig. 28. The two force constant changes (5, 5') were determined by requiring that the model reproduce the observed Tlu 17.3 cm -1 low-frequency resonant and 86.2 cm -1 gap mode IR frequencies. Both the resonant and gap mode frequencies are determined by the condition Re]l + Go(z)C(z)] = 0, where z = w2 + ie. For the gap mode, the Re in this equation is not necessary, since the imaginary parts of the Green's function vanish. Using this condition, 5 versus 5' curves are computed consistent with the 17.3 cm -1 and 86.2 cm -1 modes, with the results given in fig. 29. The crossing point of these two curves, denoted by a circle in the figure, gives the force constant changes for which the model reproduces both of these Tlu absorption peaks. The corresponding fractional force constant changes are d/kl - -0.563 and t~t/kl = --0.531, where kl -- 1.884 • 104 dyn/cm is the pure KI breathing shell model nearest-neighbor longitudinal overlap force constant. Having fixed the model's parameters, we next turn to its predictions. First, the model predicts the ratio of the IR absorption strengths for the 86.2 cm-1 and 17.3 cm -1 gap and resonant modes to be 1.4. This is in reasonable agreement with the observed value of ~ 3, considering that no local field or effective charge changes were included in the relative intensity calculation. Second, the mode predicts a low-frequency Eg resonance at 20.5 cm -1, in reasonable agreement with the observed Eg Raman peak at 16.1 cm -1. Figure 29 includes the 5 versus 5' curve computed for an Eg resonance at 16.1 cm -1, and one sees that this curve comes quite close to the circled (5, 5') point for this model. For comparison, fig. 29 also includes the (5, 5') curve computed for the case of a zero-frequency Tlu resonance (dashed curve). The force constant changes on this curve correspond to the defect lattice being unstable against Tlu displacements. As the force constants are
A.J. Sievers and J.B. Page
186
Ch. 3
2-1/2
~(Tlu x,1)
~(Tlu x,2) -
12-1/2
2 . 1 2 "1/2 0
o
~
~(Eg 1,1)
o
....
o
~(EQ2,1) z
6-1/2
~(Alg 1,1 ) Fig. 28. Symmetry-basis vectors on the defect and its nearest neighbors, used in the perturbed phonon model. For a basis vector ~(Fip, t), Fi labels the irreducible representation, p labels the partners within the representation, and t labels independent vectors for a given Fip. (After Sandusky et al. 1993a).
weakened, the resonant mode of Tlu symmetry is almost always the first to become unstable; hence only the portion of fig. 29 to the upper right of the dashed curve is physically reasonable. The close proximity of the fit point (circle) to the dashed curve means that this model corresponds to a nearly unstable defect/host system. Besides the Tlu (17.3 cm -1, fit) and Eg (20.5 cm -1, predicted) resonant modes, the model also predicts an Alg resonance at 37.3 cm -1. This mode is Raman-allowed, but has not been seen experimentally. However, since the Alg and Eg Raman strengths are determined by two independent electronic polarizability derivatives, the null Alg experiment could be used to place bounds on these derivatives.
w
Unusual anharmonic local mode systems - I .0
-0.5
187
o.o 0.0
..,
~~
I
I I jT, u zero freq.
T__!~_86.2 cm___[!
-
-0.5
-1.0 Fig. 29. Calculated fractional force constant changes for resonant modes and gap modes at fixed frequencies. The inset illustrates the force constant perturbations in our model; all other short range and Coulomb force constants are unperturbed. The fractional changes are given in units of the pure KI breathing shell model nearest neighbor longitudinal overlap force constant kl = 1.884 x 104 dyn/cm. The dashed curve is for a Tlu instability. (After Page et al. 1989).
A striking prediction of this model is the existence of three nearly degenerate gap modes" the three fold degenerate Tlu mode at 86.2 cm -1, a two fold degenerate Eg mode at 86.0 cm -1 and a nondegenerate Alg mode at 87.2 cm -1. These modes are found to have some very unusual properties. The first is the extent of the near-degeneracy- the frequencies are within about a wavenumber of each other. Second, they are essentially independent of the force constant change 6, as is evident from the curve for the 86.2 cm -1 gap mode in fig. 29. Third, ratios of the Raman strengths of the Alg and Eg gap modes to their low-frequency resonant mode counterparts are predicted to be negligible. Indeed, these modes have not been observed in Raman experiments. Detailed calculations of the mode displacement patterns reveals the origin of these unusual properties. All three gap modes are found to have displacement patterns which are strongly peaked on the impurity's fourthneighbor sites [(+200), (0 + 20), (00 + 2)], as shown in fig. 30(a)-(c). The displacements on these sites are a factor of 50 or more larger than those on the impurity or its six nearest neighbors. For comparison, Figure 30(d) also shows the displacement pattern for the 17.3 cm -1 Tlu resonant mode,
A.J. Sievers and J.B. Page
188
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Fig. 30. Calculated displacement patterns for different KI:Ag + modes. (a) 87.2 cm -1 Alg pocket gap mode. (b) 86.0 cm -1 Eg(1) pocket gap mode. (c) 86.2 cm -1 TluX pocket gap mode. (d) 17.3 cm -1 TluX resonant mode. Here Tlux denotes the Tlu partner which couples to z-polarized radiation, and Eg(1) denotes one of the two degenerate Eg partners. For our choice of partners (fig. 28), panels (a), (c), and (d) show displacements in the z - y plane, while (b) shows displacements in the y-z plane. Note that the displacement pattern for the resonant mode is peaked on the defect and its nearest neighbors, while the displacement patterns for the pocket gap modes, (a)-(c), are peaked on the fourth neighbor sites, away from the defect. The displacements for the different symmetries are not drawn to scale. (After Rosenberg et al. 1992).
w
Unusual anharmonic local mode systems
189
which is peaked at the impurity site and its nearest neighbors. The nearlydegenerate frequencies of these "pocket" gap modes reflect the fact that they are almost entirely determined by the local dynamics within each of the six pockets; the pockets are weakly coupled to produce the different symmetry modes. Moreover, the independence of these modes on the force constant change 5 is clear, since this force-constant change is seen to couple ions which have essentially no motion in these modes. Finally, the weak predicted Raman strengths for the Alg and Eg pocket modes is also explained by the negligible displacements on the impurity's nearest neighbors, since defect-induced first-order Raman scattering in alkali halides typically arises from the modulation of the electronic polarizability by the motion of the ions in the impurity's immediate vicinity. On the other hand, the Tlu pocket mode is readily observable with IR absorption, since this probe couples to all of the ions in the system. Unfortunately, the model is fit to the measured IR pocket mode frequency, so that the negligible Raman activity of the two even-parity pocket modes precludes a straightforward independent verification of the existence of these highly unusual modes.
2.2.4. Isotope mode splitting Owing to the negligible Raman activity of the Alg and Eg pocket gap modes, another test was needed to verify their existence. Fortunately, the naturally-occurring isotopic abundances of 7% 41K+ and 93% 39K+ in the host crystal provide just such a test. It has been demonstrated that the harmonic defect model predicts that the presence of one or more 41K+ ions at the impurity's fourth-neighbor sites strongly mixes the pocket gap modes of all three symmetry types, producing new IR-active "isotope" pocket gap modes which are experimentally observable (Sandusky et al. 1993a). To determine the frequencies and IR integrated absorption strengths for these isotope modes, nearly-degenerate perturbation theory is first applied to a single impurity/isotope combination (Sandusky et al. 1991, 1993a). This is done by expanding the isotope mode displacement patterns in terms of the six normalized unperturbed pocket gap mode patterns g,(i) = ~ , = 1 af'(i)X(ff) and substituting the expansion into eq. (2.18). Here i = 1,..., 6 labels the isotope modes. Multiplying the resulting equation on the left by ~(f) yields a 6 x 6 eigenvalue equation for the isotope-mode frequencies w~ and the expansion coefficients af(i)" 6
(W2f -- W2) af(i) = E w2f((f)Amx(f')aY '(i)" f'=l
(2.26)
190
A.J. Sievers and J.B. Page
Ch. 3
Here Am is the diagonal matrix containing the mass changes introduced by the 41K+ isotopes, and the {mos} are the six unperturbed pocket-gap-mode frequencies. A 41K isotope substitution changes the mass of the 41K+ ion, leaving its electronic structure and, hence the effective charges #~<~unchanged. Thus, the dipole moments for the isotope modes are produced exclusively by their Tlu components. When ff,(i) is substituted into eq. (2.25), the integrated z-polarized IR.absorption for the ith isotope mode, of frequency w~ produced by a single impurity/isotope combination in a crystal of volume V, is 2rr 2
2 (rzoo + 2 ) 2
si = cn(wi)V
3
[~=x(Tlux)]iaZ'ux(i)"
(2.27)
This expression is just the absorption for a single impurity with no isotopes present, multiplied by the Tlux fraction a2 uX(/) for the ith isotope mode and by an index of refraction correction n(w~ )ln(wO. Figure 31 shows the calculated splittlngs for a single Ag + impurity with a (200) 41K + isotopic substitution, which mixes the Alg, Eg2 and TluX pocket .
llu
No Isotope
87
Alg
_-
_
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.
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Fig. 31. Mixing of the three nearly degenerate Alg, Eg2 and Tlu x gap modes under a (200) 39K+ --+ 41K+ host-lattice isotopic substitution. The percentages give the strength sx of the z-polarized absorption per defect/(200) isotope combination, relative to a defect with no isotope substitution. Note that the percentages add up to slightly more than 100% because the absorption coefficient is not only proportional to the imaginary part of the dielectric function but is also inversely proportional to the host-crystal index of refraction, which is strongly frequency dependent in the gap mode region. (After Sandusky et al. 1991).
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Unusual anharmonic local mode systems o
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191
(a) 84.7 cm -1 isotope mode o
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(b) 86.1 cm -1 isotope m o d e Fig. 32. Computed displacements patterns for a (200) 39K+ -+ 41K+ host-lattice isotopic substitution. The impurity and the isotope are represented by a dark square and a dark circle, respectively. (a) 84.7 cm -1 isotope pocket gap mode and (b) 86.1 cm -1 isotope pocket gap mode. These are the isotope pocket gap modes having the largest and smallest frequency shifts relative to the unperturbed Tlu pocket gap mode. Note that the displacement pattern for the isotope mode with the largest frequency shift is strongly peaked on the isotope.
gap m o d e s to create three isotope modes. The calculated displacement patterns for the (200) isotope m o d e s with the largest and smallest frequency shifts from the unperturbed Tlu 86.2 cm -1 pocket gap m o d e are given in fig. 32. T h e s e isotope m o d e s are seen to be pocket gap modes with the displacements confined to a single pocket. It is not surprising that the isotope with a large displacement on the (200) 41K+ gives the largest frequency shift, while the isotope m o d e with a negligible displacement on this ion gives a very small frequency shift. As s h o w n in fig. 31, fourth-neighbor 41K+ substitutions can produce isotope m o d e s having frequency shifts larger than 1 cm -1 and IR absorption strengths c o m p a r a b l e to that p r o d u c e d by a defect with no isotopes present. In contrast, (110) (i.e. second-neighbor) and (400) 41K+ isotope substi-
A.J. Sievers and J.B. Page
192
Ch. 3
tutions are found to produce modes having negligible absorption strengths and/or frequency shifts of less than 0.1 cm -1, which would be very difficult to observe given the large width of the main mode. Since the displacements for the isotope mode are zero-order displacements in the perturbation approach, the Tlu fractions a2 ox(i) for a single isotope-impurity combination sum to unity. Hence, one sees from eq. (2.27) that provided the change with frequency of the index of refraction can be neglected, which is an excellent approximation for modes separated by less than 0.1 cm -1, the unperturbed mode strength obtained by including the weakly shifted modes is the same as if their contributions had been neglected from the outset. Similar arguments hold for the splitting of the pocket isotope modes, such as the 84.7 cm -1 (200) 41K+ mode, due to second-neighbor or (400) 41K+ isotopes- as long as the splitting produced by these substitutions is smaller than the experimental resolution, or these new modes have negligible absorption strengths compared with the fourth-neighbor pocket isotope modes, their contributions can be ignored. Accordingly, the second-neighbor and (400) 41K+ isotope substitutions were neglected in the isotope-induced IR absorption calculations (Sandusky et al. 1991, 1993a). However, in order to account for all 41K+ fourth-neighbor modes, the 41K+ fourth-neighbor isotope modes with frequency shifts of less than 0.1 cm -1 were included. The nearly-degenerate perturbation theory predicts the frequency shifts (eq. 2.26) and integrated IR absorption strengths (eq. 2.27) for isotope modes produced by a single impurity/isotope combination. However in a real KI:Ag + system, there is a distribution of impurity/isotope combinations. In a crystal with Nd impurities, the predicted x-polarized strength for the ith isotope mode produced by an impurity/isotope combination with 1 fourth neighbor isotopes and the Ni - 1 symmetry-equivalent modes produced by different isotope/impurity combinations, also with 1 41K+ fourth neighbors, is given by Si - NdNi(1 -
f)6-I fl si
(2.28)
where si is the x-polarized strength for the ith mode produced by a single isotope/impurity combination, as given by eq. (2.27). The computed shifts and strengths for 13 isotope/impurity configurations are tabulated in Sandusky et al. (1993a), which gives numerous additional details. The final result is that there are only two predicted isotope pocket gap mode IR peaks at frequencies resolvable from the main unperturbed line at 86.2 cm -1. These are an observably strong peak at 84.7 cm -1 and a much weaker one at 87.1 cm -1. The experimental isotope line is found at 84.5 cm -1 as discussed below.
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Unusual anharmonic local mode systems
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2.3. Pocket gap mode experiments
2.3.1. Isotope shift The absorption coefficient of KI+0.4 mole% AgI in the phonon gap region of KI for two different temperatures is shown in fig. 33 at a resolution of 0.1 cm -1. The strong impurity-induced feature at 86.2 cm -1 is the KI:Ag + gap mode corresponding to the low temperature on-center configuration of I
I
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I
75
80
85
I 90
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7 E 0 v t"-" 9 0
, m
q... 0 0 cO o m
k.. 0
~0
..0
0
70
Frequency
95
(cm -1)
Fig. 33. Impurity-induced absorption coefficient of KI + 0.4 mole% AgI. The restricted frequency interval covers the phonon gap region of KI. The resolution is 0.1 cm -1 The temperature for the upper spectrum is 1.6 K, and that for the lower one is 8.8 K. The strong mode at 86.2 cm -1 is the on-center KI:Ag + gap mode. This mode has lost about half its strength in the higher-temperature spectrum. The doublet at 76.8 and 77.1 cm -1 is due to C1- and the single peak at 82.9 cm -1 is due to Cs +. The weak temperature-dependent peak at 84.5 cm -1 is the Ag + isotope mode. Note that the KI:Ag + modes, which have a FWHM of 0.5 cm -1, are significantly broader than other KI gap modes, whose FWHM is ,-~ 0.14 cm -1. Additional temperature-induced changes are increased broadband absorption due to difference-band processes in the host KI crystal and the appearance of a KI:Ag + gap mode at 78.6 cm -1, corresponding to population of the off-center configuration of the Ag + defect in the KI lattice. (After Sandusky et al. 1993a).
A.J. Sievers and J.B. Page
194
Ch. 3
the defect system. Most of the weaker spectral features seen here and identified in the figure caption are associated with other unwanted monatomic impurities, present in either the host or dopant starting materials. But the strength of the weak line at 84.5 cm -1 also varies linearly with the Ag + concentration and hence is not due to pair modes. When the temperature is increased from 1.6 K to 8.8 K, the strengths of the strong Ag + gap mode and the neighboring satellite line at lower frequency are reduced in strength by a factor of two while the weak but sharp modes due to the other impurities remain unchanged. At the highest frequencies shown in the figure, the temperature change produces an increase in the host absorption coefficient due to intrinsic difference band processes (Love et al. 1989). The interplay between the temperature dependence of the gap mode strength and the underlying difference band absorption can be seen more easily in the 3-dimensional plot of fig. 34, which displays the results obtained for several temperatures from 1.6 K to 19 K. At the highest tempera12"
0
0 ~
B~
B7
~j
~o
8 frocloeoCY
Fig. 34. Temperature dependence of the absorption coefficient in the region of the KI:Ag+ pocket gap modes between 1.6 and 19 K. The resolution is 0.1 cm-1. Note that the two KI:Ag+ modes have nearly disappeared in the high-temperature spectrum, where two weak Rb+ gap mode peaks at 86.3 and 86.9 cm-1 have become visible. Note also the increase in the difference-band absorption with increasing temperature. (After Sandusky et al. 1993a).
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Unusual anharmonic local mode systems
195
ture shown, 19.0 K, the gap mode has nearly vanished; the remaining weak absorption peaks at 86.3 and 86.9 cm -I are due to a small concentration of naturally-occurring Rb + impurities. It is clear from these data that not only does the strength of the main mode disappear with increasing temperature, but also that the weak satellite line at 84.5 cm -~ disappears with a similar temperature dependence. At the same time that the difference band absorption increases in magnitude with increasing temperature, the weak features within and on the high frequency side of the main Ag + gap mode remain essentially temperature independent, but become easier to see with the disappearance of the Ag + gap mode. An identifiable property of the Ag + gap modes at 86.2 and 84.5 cm -1 is that their strengths become vanishingly small by ~ 25 K. This temperature dependence is associated with the depopulation of the Ag + on-center configuration. It is exhibited by all experimental probes of this defect system. In contrast, the KI:C1- doublet at 76.8 and 77.1 cm -1 and the KI:Cs + single peak at 82.9 cm -1, for example, have no temperature dependence in this restricted temperature region. There is another relevant feature generated by the Ag + defect in the 8.8 K spectrum. A weak broad gap mode is visible at 78.6 cm -1 which initially grows in strength with increasing temperature. Previously, this line was identified with a transition in the off-center Ag + configuration (Sievers and Greene 1984). The study by Sandusky et al. (1993a) shows that although the band initially grows with increasing temperature it then appears to stop growing for temperatures larger than about 12 K. It now is clear that the temperature dependence of the strength of this mode does not show the temperature dependence associated with the off-center configuration, which continues to grow in strength until about 25 K.
2.3.2. Temperature dependence of the isotope mode intensity To separate the temperature dependence of the gap mode spectrum from the two phonon difference band absorption a three step unfolding technique is used. First the data in the parts of the phonon gap region of KI where no impurity modes are present is used to fit a polynomial to the temperature dependent background absorption. Second, the background absorption is subtracted, leaving only the absorption peaks corresponding to the impurity modes. Third, the strengths of the KI:Ag + pocket gap modes are obtained by fitting the peaks to the sum of two Voigt functions corresponding to the unperturbed and isotope modes, respectively. Since the isotope mode is much weaker than the unperturbed mode (4% of the strength), it is necessary to limit the number of free parameters in the fits: the assumptions used are
196
A.J. Sievers and J.B. Page I
Ch. 3
'
Z
0 "0
I
0
I
'
I
10 Temperature (K)
20
Fig. 35. Temperature dependence of the strengths of the two ir-active modes produced by the Ag + center in the KI gap region. The solid circles are the data for the weak mode at 84.5 cm-1, produced by the 41K+ substitution on the fourth neighbor of the Ag + defect. The dashed line gives the temperature dependence of the unperturbed Ag + gap mode at 86.2 cm-1 (After Sandusky et al. 1991).
that the two Voigt functions have the same shape and that the separation of their center frequencies is temperature-independent. Thus, the remaining free parameters are the widths of the Gaussian and Lorentzian contributions to the lineshapes and the strengths of the two modes. The strength of the unperturbed gap mode obtained by this technique is in good agreement with the previous results, represented by the dashed line in fig. 35. The solid circles in this figure show the temperature dependence of the isotope mode as determined by this technique; within the experimental uncertainty, the temperature dependences of the unperturbed and isotope KI:Ag + pocket gap modes are clearly similar, and possibly identical. 2.3.3. Entire impurity induced spectrum at T = 0 K
It would be misleading to conclude from the success that of two-parameter defect model in describing the isotope effect that it can reproduce the complete impurity-induced far IR spectrum of KI:Ag + system. Figure 36 shows the measured low-temperature absorption spectrum of KI:Ag + versus frequency (curve A) over the frequency region below the reststrahl. In the same figure is presented the calculated impurity-induced absorption (curve B) in the acoustic spectrum generated by the two parameter model (8, St). The
w
Unusual anharmonic local mode systems
I
l
i
197
l
10
I
E o
1 c ~
o
0 o
c0. I o Q_ L_
0 o3 _K3
0.01
0
25 50 75 Frequency (cm -1)
100
Fig. 36. Impurity-induced absorption coefficient of KI:Ag + vs frequency. Curve A: Far IR absorption coefficient of KI:Ag + below the optical phonon region, at 1.7 K. The dominant features are the KI:Ag + resonant and gap modes at 17.3 and 86.2 c m - l , respectively, and the gap modes due to KI:C1- at 76.8 and 77.1 cm -1 and KI:Cs + at 82.9 cm -1 (C1- and Cs + are present as natural impurities). Additional weak features due to KI:Ag + are at 30, 44, 55.8, 63.6 and 84.5 c m - 1 ; all are associated with the on-center configuration of the Ag + impurity. The instrumental resolution is 0.1 cm-1. Curve B: the calculated absorption coefficient according to the two-parameter harmonic (5, 5 ~) model. The gap mode delta-function is not shown. Note that the structure in the acoustic spectrum for curve A is not associated with density of states features in curve B. (After Sandusky et al. 1993a).
resonant mode peak height was matched to the one in curve A. For clarity, the gap mode delta-function is not shown. Although this perturbed harmonic model gives a good account of the positions and the strengths of the resonant and gap modes, as well as the pocket mode isotope effect, it cannot at the same time account for the frequency dependence or strength of the broadband acoustic absorption. This could be due to neglected anharmonic or electronic deformation effects involving the Ag + ion, or to the neglect of the static relaxation effects beyond the defect's fourth-nearest neighbors.
A.J. Sievers and J.B. Page
198
Ch. 3
2.3.4. Anharmonicity studied by uniaxial stress and electric field measurements Given the anomalous thermal behavior of KI:Ag +, which falls well outside the harmonic approximation, it is of interest to explore this system's anharmonic interactions. Accordingly, uniaxial stress shifts and electric field measurements have been made. Note that for the pocket modes, these measurements should provide information about host lattice anharmonicity near the (200) family of ions, whereas such measurements for the resonant modes should be sensitive to anharmonicity near the defect. Because the stress and E-field induced frequency shifts are small, a "global" analysis is used to obtain the experimental shifts. The method consists of overlaying the shifted line at some given non-zero perturbation onto the corresponding unshifted line and varying the position and width of the line at non-zero perturbation until the area between the two curves is minimized.
J
O~I(SCO)
~2(C0 -- A(co))
peak O~1
peak O~2
d~o
(2.29)
as a function of the two variables s and Aw, where the two absorption lines O~1(03) and a2(w), are normalized to unit height. The width of the first line relative to that of the second is then equal to s and its frequency shift is Table 1 Measured values of the stress coupling coefficients of the IR-active gap and resonant modes of the KI:Ag +, KI:CI-, KI:e-(F-center), KI:Cs +, and KI:Rb + defect systems. Note that B/A nearly vanishes for the first two modes, which are pocket modes, and for the 55.8 cm-1 KI:Ag + mode in the acoustic spectrum. On the other hand, the KI:Ag + resonant mode at 17.3 cm-1 and the last six "standard" gap modes have much larger values of B/A. The stress coupling coefficients are given in units of cm-1/unit strain. The measurements were made between 1.7 and 3.5 K (After Rosenberg et al. 1992). Mode
Frequency (cm -1)
A
B
KI:Ag + KI:Ag + KI:Ag + KI:Ag +
86.2 84.5 55.8 17.3
186 188 143 390
4- 21 4- 22 4- 32 4- 90
KI:C1KI:CIKI:eKI:Cs + KI:Rb + KI:Rb +
76.8 77.1 82.7 82.9 86.3 86.9
102 101 143 148 147 143
4- 36 4- 36 + 55 + 56 + 55 4- 55
C
9 4- 6 7-4- 6 2 4- 9 510 4- 70 53 455 439 480-l69 468 +
10 10 18 15 14 14
B/A
- 7 4- 6 0 4- 6 - 1 6 4- 9 - 1 5 4- 10
0.05 0.04 0.01 1.31
+ 0.03 + 0.03 + 0.06 4- 0.35
-19 -20 -20 -20 -21 -20
0.52 0.54 0.27 0.54 0.47 0.48
+ 0.21 + 0.22 4- 0.16 4- 0.23 -I- 0.20 + 0.21
4- 10 4- 10 4- 15 4- 15 4- 15 -i- 15
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Unusual anharmonic local mode systems
199
Table 2 Measured and QD model predicted electric-field-induced frequency shifts for the Tlu pocket gap mode. The values in parentheses give the predicted minimum and maximum shifts produced by anharmonic parameters consistent with the uncertainties in the measured Tlu gap and resonant mode uniaxial stress coefficients (After Sandusky et al. 1994). EDC
EIR
[100]
[100]
[100]
[010]
[110]
[110]
[110]
[1-10]
Aw/AE2[IO-6 cm-1/(kV/cm) 2] Exp. QD Exp. QD Exp. QD Exp. QD
1.70 -t- 0.09 1.13 (0.67/1.67) -0.93 4- 0.05 -0.42 (-0.28/- 0.58) 0.08 -t- 0.36 0.32 (0.12/0.54) 0.20 + 0.29 0.39 (0.27/0.55)
related to Aw. The stress and electric field results for the different crystal orientations and polarizations are given in tables 1 and 2, respectively. Stress measurements have been carried out on a variety of different gap modes in KI. The behavior of these gap modes is qualitatively different from that of the KI:Ag + pocket gap modes. Inspection of the coupling coefficients collected in table 1 reveals similar behavior for all impurity gap modes listed, except for Ag +. This suggests that a small B/A ratio is a stress behavior signature of the pocket gap modes, which sets them apart from the other gap modes. Note that the B/A difference is also large between the KI:Ag + pocket gap mode and the 17.3 cm -1 resonant mode (see table 1). Interestingly, the B/A ratio for the sharp band mode peak at 55.8 cm -1 for KI:Ag + is very small, comparable to that for this system's pocket modes. These examples demonstrate the unique nature of the stress behavior of the KI:Ag + pocket gap mode, the related isotope mode and the 55.8 cm -1 band mode.
2.4. The (6, 6', 6") and the quadrupolar deformability models As detailed in (Rosenberg et al. 1992; Sandusky et al. 1994), quasiharmonic extensions of the (6, 6') model prove inadequate to explain the anharmonic shifts revealed by the stress and E-field experiments. The original (6, 6') model assumed that defect-induced inward relaxation of the silver ion's six nearest neighbors produces the force constant change 6'. The magnitude of its fit value is roughly half the pure KI nearest-neighbor overlap force constant, implying substantial relaxation. This suggests in turn that relaxation-induced force constant changes 6" = -Aq~xx(200, 300) should also be included. These changes could have a strong effect on the pocket
200
A.J. Sievers and J.B. Page
Ch. 3
gap modes, since their displacement patterns are so strongly peaked on the (200) family of ions. Recent work has shown how 8" can be included without adding any free parameters. By assuming that the introduction of the defect into the unrelaxed crystal produces radial forces on just the defect's six nearest neighbors and working within a linearized theory (i.e., small relaxations), one can use the pure crystal harmonic shell model Green's functions to compute the static displacements throughout the lattice, relative to those on the defect's six nearest neighbors. If these relative static relaxations are combined with the assumption of cubic anharmonicity arising from nearest-neighbor central potentials, then the necessary force constant change ~;" can be computed uniquely in terms of 8'. For KI this procedure yields, 8" - 0 . 6 8 ' with no adjustable parameters added to the (8, 8') model. For this new "relaxation" model (8, 8~, 8"), the fits to the measured IR Tlu resonant and gap mode frequencies again give three nearly degenerate Alg, Eg and Tlu pocket gap modes, which are almost identical in frequencies and displacement patterns to the pocket gap modes predicted by the (8, 8') model. Nevertheless, despite improvements with some experimental results and the fact that the (~, St, 8,,) model is well-motivated physically, it cannot be complete: it predicts a Raman-active Eg symmetry resonant mode at 26.3 cm -1, 10 cm -1 above the experimental frequency and 6 cm -1 above the (8, 8') model prediction. Furthermore, like the (8, 8~) model, it cannot provide a basis for a consistent explanation for both the measured stress and Stark shifts. A consistent explanation of the measured stress and electric field pocket gap mode shifts has been obtained by introducing a Ag + electronic quadrupolar deformability-induced harmonic force constant change A4~xx(100,- 100) = 2A~bxu(100, -010) = t~QD (Sandusky et al. 1994). Quantum mechanically, such force constant changes arise from virtual s-d electronic transitions and have been argued to be important for the Ag + ion (Fischer et al. 1972; Fischer 1974; Dorner et al. 1976; Kleppmann and Weber 1979; Bilz 1985; Jacobs 1990; Corish 1990; Kleppmann 1976). This QD model adds but a single free parameter to those included in the original (~, ~') model; ~" is still determined uniquely in terms of ~' as described above. The force constant changes d; and ~' are obtained by fitting the measured IR resonant mode and pocket gap mode peaks at 17.3 and 86.2 cm -1, as before, and t~QD is adjusted to reproduce the observed 16.1 cm -1 Eg Raman peak. Overall, the resulting predicted harmonic properties are in substantially better agreement with experiment than for the (~, ~') model, and now the Eg resonant mode frequency is (necessarily) correct. Moreover, the QD model provides a consistent explanation for the gap mode E-field and uniaxial pressure induced shifts, plus the large E-field mixing seen for the Tlu and Eg resonant modes. The stress and E-field experiments probe the anharmonicity in the vicinity of the defect. To model the measured stress shifts, a quasiharmonic
w
Unusual anharmonic local mode systems
201
Applied Stress
Purecrystal,strains!
i_ [
Harmonic defect model
i Defectcrystal strains
!Anharm~ .........
t
Li ~
Force constant ! changes ~9 ......
1Frequency
Normalized gap mode ! displacement patterns .......................
shifts
Fig. 37. Schematic diagram illustrating the procedure for calculating the stress coupling coefficients of the KI:Ag + impurity modes. Note that both the defect-crystal strains and impurity-mode displacements are determined from the perturbed harmonic shell model. The anharmonicity is only needed to determine the force-constant changes produced by the harmonic defect-crystal local strains. The electric-field Stark shift calculations follow an analogous procedure. (After Rosenberg et al. 1992).
theory is employed as follows. First, the perturbed harmonic force-constant model is used to compute the local strains induced by the applied stress. These strains are then combined with the assumption of nearest-neighbor cubic anharmonicity to yield the local force-constant changes. Finally, these are used in a perturbation theory calculation of the stress-induced frequency shifts. This procedure is schematically outlined in fig. 37 and is detailed in the appendix of Rosenberg et al. (1992). The E-field Stark shift calculations follow an analogous procedure, with the different symmetry of this perturbation being taken into account. Figure 38 shows measured and predicted pocket mode difference spectra for an 87 kV/cm applied static field and EIRIIEDc[100] probe-field geometry. The predicted spectrum in (a) is for the (~;,d;', ~") model, with the anharmonicity parameters obtained from measured stress shifts, while the prediction in (b) is for the QD model, again based on stress-fit anharmonicities. It is seen that the QD model accounts well for the measured difference spectrum, whereas the (5, ~',~") model is in marked disagreement-
A.J. Sievers and J.B. Page
202
2.0
0.0
v
Ch. 3
(a)
f '\ ""
.,,.
\~~l, \
',.i .,]
-2.0
E
!
~"-'"-'"
".9"~0.15
0.00
-0.15
,
s5
,
86
(0 (cm "1) Fig. 38. Experimental Stark effect pocket gap mode difference spectrum, compared with the predictions of two different models. The measured spectrum, Aa = a(EDc = 87 k V / c m ) a(EDc = 0) for EIRIIEDc [100], is represented by the solid curves. (a) Comparison with the (6, 6', 6") model spectrum (dot-dash line), which is more than an order of magnitude too large. (b) Comparison with the QD model spectrum (dashed line). The theoretical Stark shifts of table 2 are computed using anharmonicities fit to measured stress shifts. The predicted difference spectra shown here are generated from the theoretical shifts by using a Voigt lineshape determined from the measured zero-field lineshape for the IR pocket gap mode. (After Sandusky et al. 1994).
it predicts shifts which are nearly two orders of magnitude larger than the experimental shifts. Table 2 compares the measured static E-field induced pocket mode frequency shifts predicted by the QD model using the QD model stress-fit anharmonicities. The computed shifts of table 2 are determined from strengthweighted frequency averages ~ - ~ i S i ~ i / ~ i Si as a function of applied field strength, and the uncertainties in the predicted shifts arise from uncertainties in the stress measurements. The predicted and observed shifts for the EIRtiEDC [100] p r o b e - field geometry are seen to overlap, while the predicted shift for EIRIEDc [100] is half the observed shift. The predicted and observed shifts for EDC [ 110] also overlap, but the uncertainty in the experimental shifts for this configuration precludes strong conclusions. Using the same stress-fit anharmonicities, the QD model also predicts a relatively
w
Unusual anharmonic local mode systems
203
large E-field induced mixing for the low frequency Tlu and Eg resonant modes (frequency shifts ,-,, cm -1 at E = 100 kV/cm); these shifts are in excellent agreement with the experimental shifts (Kirby 1971). 2.5. Discussion and conclusions A comprehensive comparison of experiment and theory on KI:Ag + is collected in table 3. Since the theoretical work has necessarily focused on the T - 0 K results, much of the table is devoted to this limit. Ten different kinds of spectroscopic data are identified in the first column to compare with the three successively more refined harmonic shell models, labeled (a), (b) and (c). Column (a) identifies the two-parameter (3, 3') model; column (b) the two-parameter (3,/~, 3") model which includes relaxation without any additional parameters and column (c) the three-parameter QD model which includes electronic quadrupolar deformability in addition to (3, 3~, 3"). Although the simple (3, 3') model does a surprisingly good job in predicting the pocket modes and provides reasonable agreement with five of the ten experimental results listed, its quasiharmonic extension cannot account for the pocket mode stress and Stark electric field results listed as 11 through 14. The two-parameter (3, 3~, 3") model may be viewed as an intermediate model on the way to the QD model in that there is improved agreement with some of the eleven experimental items while at the same time the disagreement with the experimental Eg Raman resonant mode frequency increases. The QD model is a remarkable success. With its single additional parameter fit to the experimental Eg Raman resonant mode frequency, the QD model provides the same good overall agreement with the first ten experimental results as does the (3, 3~, 3") model, and its quasiharmonic extension describes both the stress and electric field measurements consistently. From the stress and Stark measurements (table 3, entries 11 through 15), there are 15 new pieces of spectroscopic data. When five of these from the stress measurements are used to fit the anharmonic QD model, the other 10 experimental results, which include the Stark electric field data, are predicted correctly. These successes of the QD model demonstrate that the silver ion possesses a significant electronic quadrupolar deformability in KI:Ag + and that this deformability plays an essential dynamical role. Conversely, the good agreement between the experimental data and the QD model also show that an anomalous potential energy anharmonicity does not appear to play a role in the description of the T = 0 K dynamics. Although the QD model provides an excellent description of most of the T = 0 K features, there are a few discrepancies that stand out and need to be considered. The most important ones are perhaps items 8 and 9 in table 3,
A.J. Sievers and J.B. Page
204
Ch. 3
Table 3 Qualitative Comparison of the KI:Ag + on-center experimental results with calculated results based on perturbed shell models. (a) the two parameter (6, 6t) model; (b) the two parameter (6, 6~, 6") model and (c) the three parameter (6, 8~, 6") + QD model. Experimental Results
SM Results
(a)
(b)
(c)
harmonic approx, harmonicapprox, harmonicapprox. fit to (6, 6') model fit to (6, 8', 6") 1. Tlu resonant and gap mode freqs (17.3 cm- l, model 86.2 cm- l)
fit to (6, 6', 6")+QD model
2. Relative Tlu resonant and gap mode strengths (,-~ 3)
fair agreement (1.4)
good agreement (3.0)
good agreement (3.0)
3. resonant mode isotope frequency shift (-0.14 4- 0.03)
poor agreement (-0.05)
good agreement (-0.12)
good agreement (-0.12)
4. gap mode isotope frequency shift ( - 1.7)
good agreement ( - 1.46)
good agreement (-1.63)
good agreement (-1.64)
5. relative gap mode isotope strength (0.04)
fair agreement (0.073)
fair agreement (0.074)
fair agreement (0.073)
6. Eg resonant mode (16.1 cm -1)
fair agreement (20.5 cm -1)
poor agreement (26.3 cm -1)
fit to QD model
7. Alg resonant mode not
Alg resonant mode Alg resonant mode Alg resonant mode (37.3 cm -1) (41.5 cm -1) (41.5 cm -1)
observed
poor agreement
poor agreement
poor agreement
9. weak absorption peaks poor agreement at 30, 44, 55.8, 63 cm -1
poor agreement
poor agreement
Predicts Alg and Eg pocket modes with negligible Raman strengths. (87.8 cm-1, 86.0 cm -1)
Predicts Alg and Eg pocket modes with negligible Raman strengths. (87.8 c m - 1, 86.0 cm -1)
8. magnitude of the broad acoustic Tlu absorption spectrum
10. Alg and Eg gap modes not observed in Raman.
Predicts Alg and Eg pocket modes with negligible Raman strengths. (87.2 cm -1 , 86.0 cm -1)
anharmonic QD model 11. pocket mode and 17.3 cm-1 resonant mode stress effect
five parameter fit to the QD-based quasiharmonic model
12. isotope pocket mode stress effect
good agreement
w
Unusual anharmonic local mode systems
205
Table 3 (continued)
13. pocket mode Stark effect
Experimental Results
SM Results data agrees with predictions of the QDbased quasiharmonic model with no additional parameters
14. isotope pocket mode Stark effect
good agreement
15. resonant mode Stark effect
good agreement
16. Large gap mode line width in comparison to other gap mode systems (0.5 cm -1) r (0.14 cm -1)
outside of the model
Experimental temperature dependences (with increasing temperature)
Temperature dependent properties outside the framework of the QD-based quasiharmonic model
17. disappearance of the Tlu and Eg resonant and Tlu pocket mode strengths 18. weak broad IR gap mode (78.6 cm -1) appears and then levels off 19. Raman resonant mode (Alg + Eg) (12.2 cm -1) appears and then disappears into a broad central peak 20. Temperature dependent pocket mode A and B stress coefficients. which refer to the poor agreement between the model and the magnitude and weak structure in the measured impurity-induced absorption in the acoustic region, shown in fig. 36. It is found that this disagreement is preserved for both the (~, ~', d;") and the QD models; hence these experimental properties appears to be outside of the scope of the current models. The ultrasharp feature at 55.8 cm -1, listed in item 9, may not be unique to Ag + but may simply be an impurity-induced density of states peak, since a similar feature has been seen for Na + impurities in KI (Ward et al. 1975). However, the feature is produced by the Ag + ion since like the resonant mode, it disappears as the sample temperature is increased. The three models described here do not reproduce such an ultrasharp feature in the acoustic spectrum. Item 16, the T = 0 K pocket gap mode linewidth is another property that stands out. Except for Ag + which has a gap mode width of 0.5 cm -1, the linewidth for all other point defects in KI is about 0.14 cm -1. This factor
206
A.J. Sievers and J.B. Page
Ch. 3
of three difference may be significant. The smaller of these two values could be a consequence of random DC strains in the crystal producing an inhomogeneously broadened gap mode band but, according to table 1, the ratio of the hydrostatic coupling coefficients of Ag + to the Cs + gap mode, which is closest in frequency, is equal to 1.26: a value much too small to explain this linewidth variation. Since the resulting QD model parameters now indicate that only standard anharmonicity parameters are associated with pocket modes, the large linewidth observed for KI:Ag + does not appear to be driven by an unusual potential energy anharmonicity. The last section in table 3 lists a number of observed temperature dependences (17 through 20), all of which are outside of the QD model. We wish to make the point here that the temperature dependence results appear to be outside the scope of many other phenomenological models as well. For example, applying Boltzmann statistics to the two configuration model (fig. 17) shows that to account for the rapid temperature dependence of the resonant mode strength (fig. 14), the number of levels in the off-center C2 configuration state must be at least 100 times that of the on-center C~ ground state configuration. What physical process for a single point ion in a lattice could generate this much entropy by 25 K? One conclusion is that these temperature-dependent experimental results are not only outside of current lattice dynamics theory but are difficult to understand even in a more general phenomenological framework. One conclusion which appears to be independent of the details of the temperature dependence is that the quantitative temperature-dependences of the strengths of the IR resonant and pocket modes provide important complementary information on the participation of the Ag + impurity's surrounding ions in this system's anomalous thermally-driven on-to off-center transition of the Ag + ion. Measurements and analysis have demonstrated that both the low-frequency IR resonant mode and the strong pocket IR gap mode disappear at identical rates with temperature in the range 0 K to ~ 25 K, even though the dynamics of each, according to all three perturbed shell models, involve ion motion in different regions of the defect vicinity. Since the mode frequencies are nearly temperature-independent over this temperature interval, it appears that these two nearly-harmonic modes simply monitor, in different spatial regions, the population of the on-center configuration; hence, the thermally-driven instability in this system involves the entire coupled defect/host system in the impurity region.
3. Intrinsic localized modes in perfect anharmonic lattices We now tum from the previous, experimentally-driven study of a fascinating anharmonic solid-state defect system, to the still purely theoretical
w
Unusual anharmonic local mode systems
207
problem of intrinsic localized modes in anharmonic lattices. As noted in the Introduction, our focus is on simple descriptive aspects of some of the interesting phenomena. We will begin with the simpler case of monatomic lattices with harmonic plus hard quartic nearest-neighbor interactions and will later consider the important generalizations of adding cubic anharmonicity or treating full realistic potential functions V(r). When ILMs occur in these monatomic systems, their frequencies are above the maximum frequency of the harmonic lattice, and we will describe their existence, stability, motion and interactions with plane-wave phonons. We will then consider diatomic lattices for the same interactions; in this case one has the interesting additional possibility that ILMs can exist in the frequency gap between the acoustic and optic phonon branches.
3.1. One-dimensional monatomic lattices
3.1.1. Heuristic illustration of vibrational localization We begin with a qualitative example to illustrate why localized modes might be expected to appear in a perfect but anharmonic lattice. For a onedimensional monatomic harmonic lattice with particles of mass m connected by nearest neighbor spring constant k2, the normal modes are described by an orthogonal set of homogeneous plane waves which are confined to a band of frequencies with a high frequency cutoff at Wm. The amplitude pattern of this highest-frequency mode involves every atom vibrating 7r out of phase with its neighbor and is shown in fig. 39(a). To estimate the rms amplitude of a particular particle in a plane wave harmonic mode ~o for a chain of N particles, we use the virial theorem. For a harmonic system the mean energy in a mode is equal to two times the mean kinetic energy, e.g., hw/2 - N[mwe(u2)] hence the rms amplitude at each site n in this plane wave mode, (u2) 1/e ~ (N)-l/z(h/2mw) 1/e, is quite small for a long chain. Next we form a localized wave packet at t -- 0 at a particular site with the appropriate linear superposition of this set of plane wave modes. This localized excitation in which only a few atoms, Nd, are displaced from their equilibrium position is shown in fig. 39(b). If this were a normal mode, the central atoms would have relatively large rms amplitudes, (U2) 1/2 ~ (Nd)-l/2(h/2mw) 1/2, independent of the number of atoms N in the long chain. However, this is not a normal mode for the harmonic system, and for t > 0 lattice dispersion insures that this localized excitation would not remain confined for long. For a similarly constructed one-dimensional anharmonic lattice with N particles, the plane wave spectrum derived in the small oscillation limit is
208
A.J. Sievers and J.B. Page (o) 9
&
9
COm
9
9
Ch. 3
9
9
....9
9
(b)
......
_i
[
t
i
L_ r-
t
Nd
~uj
--J --I
(c)
3~
Wm
/////////////////~ ///1////////// .
.
.
.
.
.
.
.
[
I
Fig. 39. A schematic representation of the frequencies and eigenvectors of plane wave and localized modes. (a) The eigenvector of the highest frequency plane wave mode at Wm. (b) A localized mode with frequency w~ involving a subset Nd of the N lattice sites. (c) The plane wave vibrational spectrum located between 0 and win, the anharmonic localized mode with frequency w], and the third harmonic of this local mode (dashed line). The local mode eigenvector A( . . . . 0, - 1/2, 1, - 1/2, 0 . . . . ) for the strongly localized (triatomic molecule) limit is also shown on the right.
necessarily similar to that found for the harmonic case. But when a wave packet is constructed at t = 0 to represent a localized disturbance in this chain, a different situation occurs from the harmonic example above. To be specific we assume that the potential energy anharmonicity is quartic with a positive coefficient, and we imagine that the wave packet ~is localized in stages. As the wave packet corresponding to fig. 39(b) is formed, the amplitude of the particle at the central site increases, but since the potential is no longer harmonic, so does its frequency of vibration. The quartic anharmonicity may be thought of as renormalizing the harmonic force constants within the packet to the higher values. According to Rayleigh's theorem the resulting frequency shift for each plane wave mode can be no larger than the frequency interval between the original lattice modes (Maradudin et al. 1971). As the amplitude pattern becomes more strongly localized around a particular site so that Nd --+ 1, the vibrational amplitude becomes
w
Unusual anharmonic local mode systems
209
larger, allowing the maximum harmonic lattice frequency win, which is not bounded from above, to rise above the plane wave spectrum, transforming to an inhomogeneous localized mode in the process. Given sufficient anharmonicity the end result is a true localized mode centered at the particle in question with frequency Wl as shown in fig. 39(c). The anharmonic system has evolved in such a way that both homogeneous and inhomogeneous modes exist simultaneously. Depending on the anharmonicity of the lattice, the eigenvectors shown in fig. 39 represent three different possibilities: Case (a) depicts the highest frequency ~m plane wave mode for the small anharmonicity limit. With increasing anharmonicity a localized solution with frequency a;~ > tom also exists above the plane wave spectrum, and its eigenvector is represented by (b). For this case there are roughly Na atoms involved in the localized vibration and since Na << N the amplitude of each atom in this mode is much larger than for the plane wave case. In the high frequency limit, Wl >> Wm, the local mode eigenvector takes on the particularly simple form of a vibrating triatomic molecule as shown in case (c). The spectrum shown in fig. 39(c) includes not only a plane wave spectrum of modes plus one localized mode but also an overtone of the local mode, represented by the dashed line. In principle, the anharmonic quartic potential produces a response at the third, fifth and higher harmonics, but because these frequencies are so much larger than both the plane wave and the local mode frequencies, it is reasonable to assume that the lattice cannot respond at these higher frequencies so that these nonlinear mixing terms can be ignored in the dynamical analysis. This approximation is called the rotating wave approximation (RWA). The original anharmonic lattice with the spectrum shown in fig. 39(c) is now described by a set of effective harmonic oscillators, both plane wave-like and localized.
3.1.2. Quantitative study of quadratic and hard quartic nearest-neighbor potentials (k2, k4) As noted in the Introduction, the prediction of Sievers and Takeno (1988) was based on this case, for which the potential energy function is given by ~2
v = -
2
(Un+l -n
k4 Z ( U n + l _
+ 7
Un)4
(3.1)
n
where k2 and k4 are the harmonic and quartic spring constants, respectively, and un is the longitudinal displacement of the nth particle from its equilibrium position. For a purely harmonic lattice, the solutions of the
210
A.J. Sievers and J.B. Page
Ch. 3
equations of motion are the familiar plane waves, with the dispersion relation w(k) = Odm sin(ka/2), where a:m =-- 2(k2/m) 1/2 is the maximum lattice frequency and a is the equilibrium lattice spacing. Writing the equation of motion for the nth particle, substituting the trial solution un(t) = A~,~ cos(a;t) and making the rotating wave approximation (RWA) by replacing the resulting cos3(wt) term by the first term (3/4)cos(cot) in its Fourier expansion, one obtains
--od2m~n
"-
k2(~n+l -- 2~n + ~,,+l) +
3k4A 2 4 [(~n+l -- ~n) 3 -- (~n -- ~n-1)3],
(3.2)
where m is the particle mass. This was the procedure followed by Sievers and Takeno (1988), and an approximate solution was then obtained via the use of lattice Green's function techniques. The result was that odd-parity ILMs with the approximate displacement pattern A(..., 0, - 1/2, 1, - 1/2, 0 . . . . ) given in w1.2 above can exist at any lattice site, with the frequency given by
(w) ~mm
3( ~
27A4 1+
16
'
(3.3)
where the anharmonicity parameter is A4 -- k 4 A 2 / k 2 9 This solution was argued to be a good approximation provided An >> 16/81. As we have noted, the molecular dynamics (MD) simulations of fig. 1 strikingly verify this solution. The value of A4 for the simulation in the lower panel was 1.63, well within the range of validity of eq. (3.3). A power spectrum of the MD displacements of fig. l(b) shows that the observed ILM frequency is within 2% of that predicted by this RWA equation. The theoretical arguments of Sievers and Takeno (1988) utilized a lattice Green's function formalism that is somewhat complicated to the uninitiated. Yet the ILM vibration appearing in fig. 1 is exceedingly simple. In the following subsection we give a simple and direct argument which shows that the tendency to localization in this case of strong quartic anharmonicity reflects a fundamental property of the underlying purely anharmonic system.
3.1.2.1. Asymptotic behavior; odd and even modes. To more simply understand the above ILM solution and its connection to the anharmonicity, we
w
Unusual anharmonic local mode systems
211
now focus directly on the equations of motion, for the hypothetical pure quartic case. Setting k2 - 0 in eq. (3.2) and rearranging, we have
w2 ---
3k4A 2
[(~n - ~n+l) 3 at- (~n -- ~n-1)3] 9
(3.4)
4m(n Let us now seek a localized odd-parity solution, centered at site n = 0. We thus require (0 = 1, ~-n = (n, and I~1 << I~11 for Inl > 1. Within these restrictions, the n = 0 and n = 1 versions of eq. (3.4) give
w2 =
3k4 A2
2(1 - (1)3,
n = 0,
(3.5)
4m and w2 ~
3k4A 2
[(~+(~1-1)3],
n=l.
(3.6)
4m~cl For a solution, these two frequencies should of course be the same, and we see that for ~1 = - 1 / 2 they become nearly equal"
w2k4A2(3) -
m
4 ~
,
n=0,
(3.7)
and k4A2 ( 3 ) 4 ( 60 2 ~
1+
1 )
n-
1
9
(3.8)
Next we proceed to the equation of motion for the particle at n = 2:
602
3k4 A2
[({2 -- ~3)3 -t- (~2 -- {1)3] 9
4m~2 In accord with our approximations, we set ~ 2 - (l ~,~ 1/2 and neglect ( 2 - (3 in comparison, obtaining w e ~ (3k4A2)/(32m(2). By equating this with the n = 0 expression for w 2 (eq. 3.7), we find 1
~2 ~ ~ . 54
(3.9)
A.J. Sievers and J.B. Page
212
Ch. 3
Thus 1~2] << I~l[ = 1/2. Finally, assuming I~n+ll << I ~ l for all n/> 2, we use eq. (3.4) to obtain a recursion relation for n/> 2
..~
.
(3.10)
Hence, all of the ~n's for In[/> 2 rapidly approach zero with increasing n, and it is seen that the odd-parity displacement A(..., 0, - 1/2, 1, - 1/2, 0,...) is indeed an approximate solution for the pure quartic case. The ILM frequency given by eq. (3.7) is the same as that given by the k2 = 0 version of eq. (3.3), and we find that this RWA frequency is within 2% of the exact frequency observed in MD simulations. When harmonic interactions are added, eq. (3.3) is recovered, and as noted earlier, it is a good approximation provided that the anharmonicity parameter does not become so weak that the homogeneous plane-wave harmonic solutions become dominant. Beyond the insight provided by this simple heuristic argument, it can easily be generalized to give an interesting exact result (Page 1990). Suppose that the nearest-neighbor pure quartic interaction is replaced by a nearestkr neighbor anharmonic interaction of arbitrary even-order: V = T ~,~(Un+lun) ~, where r = 4, 6 . . . . . The equations of motion are then
m~tn(~) ~ ]gr{ [Un+l(~)- Un(~)] r-1 -- [Un(~)- Un--l(~)] r-1 }.
(3.11)
Our focus here will be on the spatial behavior of the solutions, and for this purpose we could use the RWA for the time dependence, just as we did above for the pure quartic case. However, eqs (3.11) readily separate, giving rise to solutions periodic in time, for any even r (Kiselev 1990). Thus for the sake of generality, we briefly digress to bring in the exact, rather than the RWA frequency. For the trial solution un(t) = A~nf(t), it is straightforward to obtain an exact expression for the period T, from which the square of the frequency w = 27r/T is obtained as
W 2 B"k"A"-2[ rn~
(~ - ~+1
),.-1
+ (~n --
~-i
)r-l]
(3 12) ,
where the coefficient B~ is given by
)2
B,.-~
x/1-f ~
.
(3.13)
w3
Unusual anharmonic local mode systems
213
Notice that for the r = 4 pure quartic case, eq. (3.12) would go over to the RWA result eq. (3.4), provided/34 = 3/4. Indeed, evaluation of the elliptic integral appearing above gives B 4 = 0.718, and we again see that the RWA works very well for the pure quartic case. We now return to the question of the spatial behavior and focus on the (n's in the right-hand side of eq. (3.12). Following the preceding argument for eqs (3.5-3.10), we seek a localized odd-parity ILM, centered at site n - 0. Again neglecting (2 compared with (1, we find that the n = 0 and n = 1 versions of eqs (3.12) are nearly satisfied by (0 = 1 and (1 = - 1 / 2 :
co2__2krBrAr-2 ( 3 ) r-1 -
m
~
,
n-O,
(3.14)
and
w2 ~
2krBrAr-2(3)r-l[ (1) r-l] 1+
n-
1
(3.15)
These equations differ only by the factor 1 + (1/3) r - l , which approaches 1 in the asymptotic limit of large r. Turning to the n = 2 version of eq. (3.12), we again have ~2 - (1 .-~ 1/2 and neglect ~2 - (3 in comparison, obtaining w 2 ..~ (krB~A~-2)/(m(22~-l). Setting this equal to the n - 0 expression for w 2 (eq. 3.14), one finds (2 --~
1
2.3
,
(3.16)
r-1
which approaches zero in the limit of large r. As a final step, we again assume that I~n+xl << I~nl for all n >/ 2, and use eq. (3.4) to derive the general-r analog of the recursion relation eq. (3.10) for n t> 2:
,.+, ~n
..~
~n-1
.
(3.17)
Clearly, the (n's rapidly approach zero with increasing distance for fixed r, and they all vanish in the large r limit. Thus the odd parity ILM pattern A ( . . . , 0, - 1/2, 1, - 1/2, 0 , . . . ) is an asymptotically exact solution in the limit of increasing even anharmonic order (Page 1990). Even for the r = 4 "worst" case of a pure quartic system, this solution remains very a c c u r a t e - in the preceding reference, a more exact
A.J. Sievers and J.B. Page
214
Ch. 3
2.2 f
1.8
even ~
/i
odd
1.4
1.0
!
0
]
2
A4 Fig. 40. Computed odd- and even-parity ILM frequencies versus the quartic anharmonicity parameter A4. These curves were obtained by numerically solving the RWA equations of motion for all of the particles in a 40-particle monatomic linear chain, with periodic boundary conditions. Three of the corresponding odd-parity ILM displacement patterns are shown in the next figure.
calculation for the pure quartic case was found to correct the mode pattern only slightly, to A( . . . . 0, 0.02, -0.52, 1, -0.52, 0.02, 0 . . . . ). The simple odd-parity ILM pattern given by Sievers and Takeno (1988) therefore reflects a fundamental exact property of the underlying purely anharmonic system. This pattern is just that of a simple linear triatomic molecule of equal masses, and is in fact the most localized odd parity pattern which keeps the center of mass at rest. This leads naturally to the question of whether the most localized even parity displacement pattern, namely that of a linear diatomic molecule A( . . . . 0 , - 1, 1,0,...), might also be an asymptotically exact solution for the purely anharmonic system in the same limit of increasing evenorder anharmonicity. This was proven to be the case by Page (1990). For the pure quartic case this asymptotically exact mode pattern is corrected to A ( . . . , 0, 1/6, - 1, 1, - 1/6, 0 . . . . ), and the corresponding RWA frequency of the pure-quartic even-parity mode is given by w2 ~ (6kaA2/m)[1 +(7/12)3]. Again, this is readily verified by MD simulations. Even modes were discovered independently in numerical simulations by Burlakov et al. (1990a-d) and Bourbonnais and Maynard (1990). In the following section, we will see that the above asymptotic limit also gives simple insights into the stability properties of the odd and even ILMs. However, before moving to this topic, we return briefly to the harmonic plus quartic (k2, k4) case and note some additional simple aspects. As the quartic anharmonicity parameter An - kaA2/k2 decreases, the ILMs are expected to
w3
Unusual anharmonic local mode systems
215
(a) ~/O~m=1.79
(b) o)/O~m=l.
(c) o~/r
Fig. 41. Computed odd-parity ILM normalized displacement patterns {~n} as a function of the ILM frequency for a 40-particle (k2, k4) linear chain, with periodic boundary conditions. As for fig. 39, the RWA equations of motion were solved numerically for all 40 particles. In practice, one begins with an initial guess for the displacement pattern, and the routine converges to the correct pattern and frequency. The particle motion is longitudinal, but for clarity the displacements are plotted vertically. The displacement of the central particle is unity in each case. spatially broaden, and this is indeed found to be the case. For a fixed value of this parameter one can imagine moving out from the mode center until the displacements are so small that the anharmonic effects are negligible, and since the ILM frequency is necessarily above the maximum frequency Wm of the harmonic lattice, the amplitudes then decrease with distance just as for a localized impurity mode in the harmonic lattice. In one dimension this decrease is a simple exponential. This gives a straightforward numerical means for obtaining the mode displacement pattems: one simply solves the equations of motion (3.2) numerically for the particles having nonnegligible amplitudes and then applies the known harmonic-approximation analytic amplitude decrease for the particles beyond, as a boundary condition. A related approach is to apply periodic boundary conditions and simply solve the equations of motion numerically for all of the particles. These techniques have been used by a number of investigators (Bickham and Sievers 1991; Bickham et al. 1993; Kiselev et al. 1993; Kiselev et al. 1994b; Sandusky and Page 1994).
A.J. Sievers and J.B. Page
216
Ch. 3
6.0
5.0
1D
4.0
I
I
/
,I
,f
,,f./
/
2.0 1.0 0.0
, ! 1.0
2.0
3.0
4.0
A4 Fig. 42. Local mode frequency versus A4 as calculated by two different rotating wave approximations. The dashed curve follows from the single frequency rotating wave approximation while the solid curve includes an additional contribution from the third harmonic term. The more exact two frequency calculation produces a slight lowering of the ILM frequency over that produced by the simple RWA. (After Bickham and Sievers 1991). Figure 40 plots the computed odd- and even-parity ILM frequencies for a (k2, k4) lattice versus the quartic anharmonicity parameter An, and fig. 41 shows odd-parity ILM displacement patterns for three different values of this quantity. The ILM spatial broadening with decreasing anharmonicity is clearly apparent; in the purely harmonic limit (k4 = 0), both the odd and even ILMs broaden into the zone boundary phonon mode A ( . . . , 1 , - 1 , 1 , - 1 , . . . ) . Interestingly, we will see in a later section that when cubic anharmonicity is added, this spreading is largely suppressed. In developing these analytic local mode solutions, it is assumed that the system only responds at the "fundamental" frequency in the assumed c0s(wt ) solutions. Because of the quartic potential, response at 3w, 5co, etc. should also be present. A straightforward approach to investigate the influence of the next higher order term is to generalize the rotating wave approximation to u,~ = (1 - 3)~n cos(wt) + 3~n cos(3wt). When this trial solution is inserted back into the equations of motion, the result is that the ILM frequency is corrected to a new lower frequency value. Figure 42 shows the magnitude
w3
Unusual anharmonic local mode systems
217
of the shift on the odd mode solution for one, two and three dimensions. As might be expected, the correction term grows with increasing anharmonicity parameter but it remains a small contribution over the entire parameter range. This figure confirms the idea that the simple rotating wave approximation is a valid approximation for identifying ILMs in anharmonic systems.
3.1.2.2. Stability. An important question concerns the stability of the ILMs against infinitesimal perturbations. By returning briefly to the asymptotic limit discussed in the previous section, we can easily determine the basic stability properties of these modes. Subsequently, we will sketch some of the quantitative aspects of these properties, followed by a discussion of their consequences for ILM motion. Much of the following stability material follows from the study by Sandusky, Page and Schmidt (1992), which should be consulted for details. The material on moving ILMs derives from that reference and the study by Bickham et al. (1992). It was seen above that the odd- and even-parity displacement patterns A(..., 0, - 1/2, 1, - 1/2, 0,...) and A(..., 0, - 1, 1,0,...), are asymptotically exact for a purely anharmonic lattice in the limit of increasing even-order anharmonicity. In this limit the interparticle potential becomes that of a square well. For point masses the "repulsive" side of the well occurs when the particle collide, at Un+l - - U n --" - - a , and because of the reflection symmetry possessed by an even-order potential, the "attractive" side of the well occurs at un+l - u n -- +a. Thus the square well has width 2a, and the particles move completely freely until either of two situations arise: 1) they collide elastically when their separation is zero, or 2) they attract impulsively ("snap back") when their separation reaches 2a. This limiting behavior for strong even-order anharmonicity is intuitively clear, and it has been derived rigorously by Sandusky et al. (1992). In this limit, the asymptotically exact even- and odd-parity ILM patterns above require that the amplitudes have the fixed values A - a/2 and A = 2a/3, respectively. This is easily seen in fig. 43. More importantly, one also sees clearly that for the even mode the collision and "snap" occur at different instants, whereas for the odd ILM the central particle simultaneously collides with one of its nearest neighbors and "snaps back" due to its attraction to the other nearest neighbor. It is then easy to see that any perturbation which destroys the simultaneity of the collision and snap will destroy the coherence of the odd-parity mode pattern, rendering this mode unstable. On the other hand, since the collision and snap in the even parity ILM are not simultaneous, this mode is stable. The above simple picture of ILM instability in the asymptotic limit of high even-order anharmonicity carries over to the case of even- and odd-parity modes in (k2, k4) systems; however, this case is more complicated than the
218
A.J. Sievers and J.B. Page
Ch. 3
O _1 7
L n
Collision (repulsion)
2a _I
L
-A
"Snap" (attraction)
A
Even-parity mode A( .... 0,-1,1,0 .... )
Collision and snap simultaneous
21=1 I"
9
I_ I-
II
oi-.o -A/2
Odd-parity mode
-I _1 -I
~
9 A
-A/2
A( .... 0,-1/2,1,-1/2,0 .... )
Fig. 43. Collision (repulsion) and "snap" (attraction) for the even-parity ILM (top panel) and for the odd-parity ILM (bottom panel), in a monatomic lattice of point masses interacting via a nearest-neighbor anharmonic potential of even order, in the asymptotic limit of high order. In this limit, the potential becomes that of a square-well of width 2a and these mode patterns are exact. For the even-parity mode, the collision and snap do not occur at the same time, whereas for the odd-parity mode they do. This renders the odd-parity ILM unstable against any perturbation which destroys the simultaneity of the collision and snap. For the case of (k2, k4) lattices, the even- and odd-parity ILMs remain stable and unstable, respectively; moreover, the odd parity instability results in the ILM moving slowly from site to site, as discussed in the text. (After Sandusky et al. 1992).
above simple limiting case, and it requires careful analysis involving both analytic and numerical work. Briefly, one assumes a solution of the form
un(t) = A[~n + 5~ne~t] cos [wt + 6r
(3.18)
where 8~n and 8r are infinitesimal displacement and phase perturbations, respectively. Since M D simulations show that both the odd- and even-ILMs are generally stable over at least several periods, we assume that the above perturbations vary slowly in time with respect to a mode period. Performing an appropriate time-average (closely related to the RWA) and linearizing the equations resulting from the trial solution (eq. (3.18)), we arrive at a 4s • 4s eigenvalue problem to determine A, where s is the number of sites included in the unperturbed ILM displacement pattern {(n}. The ILM will
Unusual anharmonic local mode systems
w
219
0.18 o
0.14
= =,=,.
l-.
r 0.10 L_
i-.-
0.06
0
0.02 1.0
z~ I
I
I
2.0
3.0
4.0
5.0
(O/tOm Fig. 44. Instability growth rate vs. anharmonicity for odd-parity ILMs in a 21-particle harmonic plus quartic lattice, with periodic boundary conditions. The anharmonicity is measured by the ratio of the ILM frequency to the maximum frequency of the harmonic lattice. The solid curve gives theoretical predictions obtained from the stability analysis sketched in the text, and the triangles are growth rates measured in MD simulations for various values of the amplitude and the ratio k4/k 2. The dashed line gives the predicted growth rate for the pure quartic lattice. (After Sandusky et al. 1992). be unstable if one finds a perturbation (determined from the eigenfunction) of the displacements or phases (i.e. velocities) which is associated with an eigenvalue A having a positive real part, since such perturbations will grow exponentially in time. Sandusky et al. (1992) should be consulted for details. Numerically applying this analysis, we find that the odd-parity pure quartic ILM is always unstable against even-parity displacement or phase perturbations [e.g. ( . . . , - d ; a , 0, d;a. . . . )], whereas the even-parity ILM is always stable. These results agree with the more intuitively clear asymptotic limiting behavior discussed above. With harmonic interactions (k2) included, the odd-parity ILM instability growth rates are predicted to decrease with decreasing anharmonicity, and the even-parity ILM is predicted to be stable. Figure 44 shows the predicted odd-parity ILM instability growth rates as a function of anharmonicity for the (k2, k4) lattice, and these are compared with growth rates measured in MD simulations. The dashed line gives the pure quartic limit. The predicted and MD results are seen to be in very good agreement; as discussed by Sandusky et al. (1992), the ILM spatial broadening with lowering anharmonicity was not included in the growth-rate predictions in this figure, and when they are included the minor discrepancies at low anharmonicities are removed.
A.J. Sievers and J.B. Page
220
Ch. 3
2.5 1.5 t. .O, .
o r (D
0.5
-~ -0.5 ...=
ca. -1.5 -2.5 19420
i
I
19430
19440
19450
time (units of 2~(Or.) Fig. 45. Even-parity mode stability in a 20-particle pure quartic lattice, as seen in MD simulations after more than 32,000 oscillations. The initial displacement pattern is the pure quartic even-parity pattern ( . . . . 0, 1 / 6 , - 1, 1 , - 1/6, 0 . . . . ), centered at sites (-0.5, 0.5). For this run k4 and the amplitude are chosen so that w = 1.7Wm, where Wm= 1.0 (eV/~, 2 amu) 1/2 is a convenient frequency unit. The displacements are magnified, for clarity, and the displacements on the particles not shown are negligible. This mode is exceedingly stable, as predicted by the perturbation theory analysis. (After Sandusky et al. 1992).
As noted above, no instability is predicted for the even-parity ILM, and MD simulations have found this mode to be extremely stable: runs for the pure quartic and for the harmonic plus quartic case found no changes in the even-parity mode over more than 32,000 oscillations, as is illustrated in fig. 45 for the pure quartic case. As pointed by Sandusky et al. (1992), the even-parity ILM stability is further manifested by the fact that even when given t = 0 perturbation seeds which cause the odd-parity ILM to move after just tens of oscillations, the even mode was still found to persist unchanged for more than 32,000 oscillations (the maximum extent of the MD runs). Given that the odd-parity ILMs are unstable, the question arises as to how the instability manifests itself. In MD runs, it is observed that the instability does not destroy the ILMs, but rather causes them to move.
3.1.2.3. Translational motion. To find a traveling ILM, Bickham et al. (1992) substituted the trial solution un = A~n(t)cos(wt- kna) into the equations of motion for the different particles, where A is the maximum amplitude of the moving ILM in a lattice of spacing a, (n(t) is a slowly varying envelope function, and k and w are the wave vector and frequency, respectively. The resulting set of equations are then numerically solved by assuming that the
w
Unusual anharmonic local mode systems
2.5
9
m
9
i .......
9
I
"
I
221
9
O
2.0 E
:3 :3
1.5
"
1.0
o.o
,
,
.
.
.
0.2
.
.
~
,
.
.
e
o.3 0.4 0.5 ka
Fig. 46. Dispersion curve of the t : 0 odd-symmetry traveling ILM for four different anharmonicity values. The values from top to bottom are A4 = 2.5, 1.6, 0.9 and 0.4. The dashed curves identify solutions of the equations of motion using the localization condition in the text. The solid lines indicate regions where simulations of moving modes are successful. The open circles are determined from the simulated displacements by assuming a Gaussian envelope function. Qualitatively similar results have been found for the t = 0 even-symmetry traveling ILM, but the odd modes cover a larger region of w(k) space. (After Bickham et al. 1992). t = 0 solution has the form: ~-n = ~,~ = (-1)nA~I e x p [ - ( n - 1)Ka] for positive n and K . For given values of the wave vector k and the anharmonicity parameter A4 = k4A2/k2, the time-dependent equations for sites 0, 1 and 2 are numerically solved to obtain the normalized frequency (W/Wm), the relative amplitude at the nearest neighbor site ~1, and the decay constant K . It is found that smoothly moving ILMs of the assumed type are produced only in a restricted region of w(k) space, with this region becoming more restricted as the anharmonicity and the ILM frequency increase. Figure 46 shows the dispersion curve that is found when the trial solution is used to fit numerically the displacements as the wave packet associated with a t - 0 odd-symmetry ILM travels across several lattice sites. The solid curves identify regions of w(k) space where the excitation moves with a constant envelope velocity for at least 15 lattice sites. This velocity is typically smaller than 15% of the harmonic lattice sound velocity. The absence
222
A.J. Sievers and J.B. Page
Ch. 3
Fig. 47. Displacement vs. time as the vibrational excitation passes through a fixed lattice site. The solid curve gives the results. The group velocity of the envelope is 7.2% of the harmonic lattice sound velocity. The dashed curve represents the best fit of the excitation envelope to a Gaussian envelope function. For comparison the dotted curve represents a hyperbolic-secant-function envelope. (After Bickham et al. 1992).
of a solid line at small k values identifies that region where the mode either remains stationary or only moves a few lattice sites before coming to a stop. At larger k values, on the other side of the solid line, the mode moves but decelerates. Uniform motion is observed over a larger region w(k) for the t = 0 odd mode than for the t = 0 even mode, consistent with the idea that the odd mode has an intrinsic translational instability. The solid line in fig. 47 shows a typical simulation trace of displacement versus time as the vibrational excitation passes through a particular site. The dashed line represents a gaussian function best fit to the pattern. Note that a hyperbolic-secant-function best fit represented by the dotted curve does not agree with the simulated amplitude in the wings, while the Gaussian envelope matches fairly well over the entire interval. One conclusion is that the shape is clearly different from the hyperbolic secant function previously found for solitons in continuous systems. The preceding discussion is for a traveling ILM described by the trial solution un = A~(t)cos(wt-$n), where the phase Sn is equal to kna; that
w
U n u s u a l a n h a r m o n i c local m o d e s y s t e m s
223
Fig. 48. MD simulation of a traveling ILM in a 21-particle harmonic plus strong quartic lattice, with periodic boundary conditions. The anharmonicity parameter is A 4 --- 1.63, and the t = 0 displacements are given by the translationally unstable odd-parity ILM pattern A(.... 0,-1/2, 1,-1/2,0 .... ), with A = 0.1a. The mode frequency is W/Wm = 1.67. The ILM motion is produced by seeding the MD run with a small initial velocity perturbation /q = -/~l = 0.00718, in units of wma, corresponding to a relative phase perturbation of 6r - 6r = 6r - 6r = -0.086 rad. The speed of this traveling mode is 0.053 in units of wma, well below the harmonic sound speed of 0.5. The displacements are magnified, for clarity. (After Sandusky et al. 1992).
is for a constant phase difference between adjacent sites. There exists another type o e smoothly traveling ILM, for which the relative phases between adjacent sites is not constant (Sandusky et al. 1992). This type of traveling mode can exist for large values of the anharmonicity, and an example is shown in fig. 48. This mode moves with a speed approximately 1/10 of the harmonic sound speed, and as it slowly moves from site to site, its mode pattern alternates approximately between the odd- and even-ILM patterns. Figure 49 plots the phase difference between adjacent relative coordinates d~ -- Un - Un-1 for the moving mode of fig. 48 as a function of time, as the mode passes a single site. The phase difference is seen to be nonconstant. Moreover for this mode, nonconstant phase differences of the same magnitude are obtained between the adjacent displacements {un} themselves, as well as between the adjacent relative displacements {dn}. Depending on the strength of the anharmonicity and the initial conditions, at least two other sorts of ILM motion have been reported: (1) The ILM becomes trapped at a site (Bickham et al. 1992) and converts to a stable even-parity ILM (Sandusky et al. 1992). (2) The ILM oscillates between
224
A.J. Sievers and J.B. Page
Ch. 3
Fig. 49. Phase difference between the relative displacements d_4 = u_4 - u - 3 and d_ 3 = u_3 - u_2 for the traveling mode of fig. 47. The triangles give the relative phase as the traveling mode passes the n = - 3 particle. This is a type of traveling ILM where the phase is nonconstant. (After Sandusky et al. 1992).
Fig. 50. ILM oscillations between lattice sites, observed in MD simulations for a 21particle harmonic plus quartic lattice, with periodic boundary conditions. The anharmonicity is A4 = 1.25, and the initial displacement pattern is that of the odd-parity ILM for this anharmonicity, together with a small perturbation ( . . . . ~a, 0, - ~ a . . . . ), where da is 0.01% of the unperturbed mode amplitude. The dashed line is a guide that follows the mode's "center". Notice that this translationally unstable ILM does not move from its initial location until roughly 15 oscillations have occurred. The displacements are magnified, for clarity. (After Sandusky et al. 1992).
w
Unusual anharmonic local mode systems
225
Fig. 51. MD simulation of colliding ILMs in the 21-particle (k2,k4) lattice of fig. 48. The initial configuration produces two traveling ILMs of the same type as in that figure, and they are seen to reflect from the ends of the chain (for the symmetry of the initial configuration used here, the periodic boundary conditions are equivalent to free end boundary conditions). They then collide, producing ILMs traveling with much higher velocities. The displacements are magnified, for clarity. (After Sandusky and Page, unpublished).
adjacent sites (Sandusky et al. 1992). An example of this oscillatory behavior is shown in fig. 50. Finally, in passing we show in fig. 51 an MD simulation of two moving ILMs colliding. These are for a 21-particle (k2, k4) chain with periodic boundary conditions, and the symmetry of the initial conditions is such that the ILMs are symmetrically reflected inward from the ends. Upon collision it is seen that two ILMs emerge symmetrically, with much higher velocities. This is an isolated MD run, and it is not clear to what extent it is a special case. Nevertheless, this figure suffices to show clearly that traveling ILMs are not solitons in the usual sense.
3.1.2.4. The effect of a light mass defect. For the case when a light substitutional mass defect md < m is present at site n = 0, it is straightforward to generalize the arguments in w and show that in the asymptotic limit of increasing even-order anharmonicity, the exact odd-parity mode pattern A ( . . . , 0 , - 1/2, 1 , - 1 / 2 , 0,...) for the perfect lattice is replaced by
A.J. Sievers and J.B. Page
226
Ch. 3
(Kivshar 1991) A
md
md
)
. . . , 0 , - ~ m ' 1 , - ~,2m . . . .
(3.19)
Just as for the perfect lattice case, this remains a very good approximation for the case of pure quartic interactions. For the case of nearest-neighbor harmonic plus quartic potentials (k2, this mode pattern remains a good approximation provided that 16/81, just as for eq. (3.3), with the defect mode frequency given by
An>>
(Wd) 2 1 ( m ) ( ~
~
k4),
md)I l + ~ mm
3A4( r o d ) 2] 1+--~--l+2m
.
(3.20)
For md = m this reduces to eq. (3.3), and in the md --+ 0 limit of a very light mass, the defect mode pattern just becomes that of an Einstein oscillator A(..., 0, 1,0,...), as is intuitively clear, with the frequency going as
Bickhamet al. (1993) have studied the influence of a stationary light mass defect on the scattering of moving ILMs. Figure 52 shows the amplitude and frequency associated with the mass defect localized mode that is produced with the energy captured from a passing ILM. With defect masses in the range 0.33 < md/m < 0.5, the moving localized mode is partially captured and reflected. There is a peak in the amplitude of the mode localized at the defect when its frequency is near the vibrational frequency of the moving ILM, indicating a resonance effect. When the relative mass of the impurity increases to 0.5, the energy of the moving ILM is completely captured, except for some plane waves that are produced as the defect mode settles into its eigenvector. There is a dramatic increase of both the amplitude and frequency of the defect mode at this transition, as shown in the figure. As the mass increases further, the frequency of the defect mode decreases until it is nearly equal to the frequency of the moving localized mode at md/m = 0.91. Beyond this point, the impurity can support modes of the same frequency as the moving ILM and therefore becomes transparent. The key feature of this transfer process is evident by examining first the small defect mass limit. When the impurity is very light, it "adiabatically" follows the motion of its nearest neighbors and thus moves in phase with them at the incoming ILM frequency in the figure (dashed line). Therefore this defect region perfectly reflects moving ILMs in which adjacent particles vibrate 7r out of phase. Such reflection continues until the defect mass is sufficiently large that the impurity mode frequency approaches that of the moving ILM. The exact threshold at which the defect begins to capture energy depends on both the characteristic wave vector and anharmonicity of the moving ILM.
Unusual anharmonic local mode systems
w
0.30
227
..... 9
I
"
I
"
I
'
"13
,.,,0.20
E o0.10 (D ',I-,,
(D
"13
0.00
I
1.80
!
"
moving mode defect mode
......
1.60
'
E
a ~ a
.40 1.20 1.00
,
0.20
i
0.40
,
m
,
0.60
l
0.80
,
1.00
md/m Fig. 52. Simulation results for the frequency and amplitude of the anharmonic mass defect localized mode produced with captured energy from a passing ILM. (After Bickham et al. 1993).
3.1.2.5. Relation to anharmonic zone boundary mode stability. As noted above, both the odd and even (k2, k4) ILMs spatially broaden and merge with the maximum frequency Wm of the harmonic lattice as the anharmonicity parameter A4 is decreased to zero. That is, the ILMs just become the harmonic lattice zone boundary phonon mode (ZBM), with displacement pattem A(..., 1 , - 1, 1 , - 1. . . . ). An interesting "inverse" of this connection between the ILMs and the ZBM has also been investigated, in recent analytic and numerical studies (Burlakov and Kiselev 1991; Burlakov et al. 1990d; Burlakov et al. 1990b; Kivshar 1993; Sandusky et al. 1993b; Sandusky and Page 1994); the last of these is quite extensive. The upshot of this work is that the anharmonic ZBM can evolve into one or more ILMs and that an instability of the ZBM is the first step in this decay. The question of the stability of such an extended lattice phonon mode against decay into ILMs is of interest, since it appears to provide a possible avenue for creating ILMs via the application of external perturbations.
A.J. Sievers and J.B. Page
228
Ch. 3
In the rotating wave approximation, the anharmonic ZBM frequency for a (k2, k4) lattice is easily shown to be given by O32
~=
1 + 3A4.
(3.21)
By applying a suitable version of the stability analysis sketched in the preceding section, we find that the anharmonic ZBM in this lattice is indeed unstable (Sandusky et al. 1993b; Sandusky and Page 1994). For a given value of the anharmonicity A4, it is convenient to decompose the instability perturbations into spatial Fourier components, and one can readily predict the ZBM instability growth rates as a function of the perturbation wavevector kp. Figure 53(a) shows such predictions, together with measured rates from MD simulations; these are seen to agree well with the predicted rates. For different values of the anharmonicity (as measured by 60/60m), the maximum ZBM instability growth rate is found to occur at different perturbation wavevectors (kp)max, as shown in fig. 53(b). One sees that as the anharmonicity increases, (kp)max increases towards its largest value, which is that for the pure quartic lattice. Thus, larger anharmonicity favors shorter wavelength instability perturbations - the ZBM instability has introduced a new, anharmonicity-dependent, length scale into the system, namely the wavelength 27r/(kp)max associated with the fastest-growing instability perturbation. This ZBM instability length scale turns out to correlate precisely with the spatial extent of the ILMs for each value of the anharmonicity. Furthermore, when one goes beyond the perturbation instability analysis and uses MD simulations to observe the nonlinear time-evolution of the ZBM instability over finite times, it is found that the ZBM indeed evolves into a periodic array of ILM-like localized displacement patterns, as is strikingly evident from fig. 54. The characteristic width of these ILM-like structures decreases with increasing anharmonicity; it is again just the wavelength associated with the fastest-growing ZBM instability perturbation. Analogous ILM-related zone boundary mode instabilities are found for the more realistic case of lattices with both quartic and cubic anharmonicity (Sandusky and Page 1994).
3.1.3. (k2, k3, k4) nearest-neighbor potentials While the (k2, k4) lattices discussed in the previous sections have been fruitful for establishing some fundamental aspects of localized dynamics in periodic anharmonic lattices, it is also clear that this model leaves out one of the most important properties of the interparticle anharmonicity in real
w
Unusual anharmonic local mode systems
229
0.10
~
0.05
0.00 0.00
t-tll "1o
0.25 0.50 kpa (units of n radians)
0.50
o m
L _
0.25_
.-.
o
t-
/
• 0.00 E 1.0
%
1.'5 ~/[.0m
Fig. 53. Zone boundary mode instability and its relation to ILMs. Upper panel: predicted and measured ZBM instability growth rates in units of the anharmonic ZBM frequency w, as a function of the perturbation wavenumber kp, for a 40-particle (k2, k4) lattice with lattice constant a and periodic boundary conditions. The anharmonicity parameter is An -- 0.068, corresponding to a ZBM frequency of W/Wm= 1.1. The solid curve gives the stability analysis prediction, and the circles are measured in MD simulations. Lower panel: ZBM stability analysis prediction of the perturbation wavenumber (kp)max associated with the largest instability growth rate, as a function of the anharmonicity (measured by the ZBM frequency w). The dashed line gives the value for a purely quartic lattice. For a given value of the anharmonicity, the wavelength associated with (kp)max correlates accurately with the spatial extent of the ILMs for the same anharmonicity. (After Sandusky et al. 1993a, b).
systems, namely cubic anharmonicity. Typical interatomic potentials such as Lennard-Jones, Morse, and Born-Mayer plus Coulomb are repulsivedominated, and their Taylor-series expansions lead to a strong negative cubic anharmonic term (k3). Our preceding restriction of the anharmonicity to just even orders disproportionately weights the attractive component of the potential, as is clearly evident in the "square-well" limit of increasing order discussed earlier, and we have seen that it is in this limit that the most localized ILMs are asymptotically exact. Therefore it is important to
230
A.J. Sievers and J.B. Page
Ch. 3
Fig. 54. MD simulation showing the finite-time evolution of the predicted fastest-growing ZBM instability perturbation in a 40-particle (k2, k4) lattice with periodic boundary conditions. Here, the anharmonicity parameter is A4 -- 0.0067, corresponding to a ZBM frequency of W/Wm = 1.01. The top panel shows the t = 0 unperturbed ZBM pattern that is used. The fastest-growing perturbation for this case occurs at (kpa)max = 0.107r, and its pattern is shown in the second panel. The bottom panel shows the displacement pattern after a finite time has elapsed, and one sees that the ZBM has evolved into a periodic array of localized ILMlike patterns. The rate at which this evolution occurs depends upon the magnitude of the perturbation seed. The particle displacements are plotted vertically, and the relative vertical scales are indicated by the tic marks. (After Sandusky et al. 1993b; Sandusky and Page 1994).
reconsider the problem, with cubic anharmonicity included. We will first discuss ILMs in monatomic (k2, k3, k4) lattices and then generalize to the case of the above realistic analytic interatomic potentials. With nearest-neighbor cubic anharmonicity included, the potential energy of eq. (3.1) is replaced by
k2
k3
V = --2 ~ ( u n + l - u,~)2 + -~- ~(u,~+l - un) 3 n
n
k4 y ~ ( U n + l n
-
Un)4.
(3.22)
w
Unusual anharmonic local mode systems
231
The main qualitative feature introduced by k3 is to make the interparticle potential asymmetric, resulting in the particles' time-average displacements being nonzero and amplitude-dependent. Hence, amplitude- and site-dependent static distortions have to be added to the RWA trial solutions: Un(t) = A[~n cos(oat) q- An] ,
(3.23)
where the {An } are the static distortions, relative to the overall amplitude A. The {(n} give the dynamical mode pattern, as before. Within the RWA, Bickham et al. (1993)carried out detailed numerical studies of ILMs for the (k2, k3, k4) case and verified their results using MD simulations. Briefly, their technique consisted of implementing a standard nonlinear equation solver to obtain numerical solutions to the RWA equations for a restricted number of sites near the ILM center, and then applying the harmonic-approximation boundary condition of exponential decay for the particles beyond. Free-end boundary conditions were used. They found that by locally distorting the perfect lattice with the sign of the distortion determined by the sign of k3, stable ILM's could be generated even for large cubic anharmonicities but that independent of the sign of the cubic anharmonicity its effect was to decrease the frequency of the ILM. As the cubic anharmonicity increases, the eigenvector also becomes more localized until it resembles a triatomic molecule, beyond which the mode becomes unstable and decays. Bickham et al. (1993) should be consulted for details. In recent studies focused on the question of ZBM-stability/ILM-existence in (k2, k3, k4) lattices with periodic boundary conditions, Sandusky and Page (1994) employed a similar but slightly more computational approach, in that the RWA equations of motion were solved numerically for all of the particles, the number of which was typically taken as 40. As detailed there, the presence of sufficient cubic anharmonicity is found to stabilize the ZBM and simultaneously prevent the formation of ILMs. For the case of isolated ILMs, their results are similar to those of Bickham et al. (1993), when allowance is made for the different boundary conditions used. Figure 55 shows the RWA calculations by Bickham et al. (1993) for stationary even-parity modes. The ILM frequencies are shown as a function of the cubic anharmonicity parameter A3 = k3A/k2, for three different values of the quartic anharmonicity An in a 512-particle lattice with free ends. Also shown are the results of MD simulations, and they are seen to be in good agreement with the RWA predictions. The slight discrepancy observed for larger values of An arises from the higher harmonics neglected in the RWA. Figure 56 shows the results of calculations by Sandusky and Page (1994) of the RWA dynamic (a) and static (b) displacement patterns for an oddparity ILM in a 40-particle periodic boundary condition lattice with
A.J. Sievers and J.B. Page
232 , 0
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Fig. 55. Frequencies of even-parity ILMs in a 512-particle monatomic (k2, k3, k4) lattice with free ends. The dashed curves give the frequencies as a function of the cubic anharmonicity parameter A3, for four different values of the quartic parameter A4. The circles are the results measured in MD simulations. The results show that there is a limiting value of k 3 for a given value of k4. (After Bickham et al. 1993).
A3 = -2.1 and A4 = 1.1. The frequency is 1.1~m. Owing to the mode's symmetry, there is no static displacement on the central particle. However the static displacements on the remaining particles are nonzero, and they vary rapidly in the region where the dynamic displacements are large. Away from this region, the static displacements decrease linearly to zero at the boundaries, corresponding to a constant static strain away from the mode center. The magnitude of this strain decreases with increasing numbers of particles. The nature of the static displacements away from the mode center depends on the boundary conditions employed- for the free-end boundary conditions used by Bickham et al. (1993), these strains are in fact zero since the particles away from the mode center simply translate together, towards or away from the mode center, depending on the sign of k3. In the righthand column of fig. 57, we show the dynamical displacement patterns for odd-parity ILMs in a 40-particle periodic boundary condition lattice with fixed values of (k2, k3, k4), as the amplitude A is changed (Sandusky and
w
Unusual anharmonic local mode systems
233
(a)
....._ i,,,.--
Particle
(b)
b._ v
Particle Fig. 56. Predicted dynamical displacement pattern (a) and static displacement pattern (b) for an odd-parity ILM in a 40-particle (k2, k3, k4) lattice with periodic boundary conditions. The values of A3, A4 and the frequency are -2.1, 1.1 and 1.1 Wm, respectively. Notice that the static distortions away from the mode center decrease linearly to zero at the "supercell" boundaries, for this periodic boundary condition lattice. (After Sandusky and Page 1994). Page 1994). The lower pattern on the right corresponds to the same mode shown in fig. 56. The column on the left shows the results for ILMs of the same frequencies in a lattice that is identical, except that there is no cubic anharmonicity (k3 = 0). The interesting aspect here is that as the frequency is decreased, the presence of cubic anharmonicity is seen largely to quench the spreading out that occurs for the k3 -- 0 case. Eventually the mode rapidly broadens, merging with the zone boundary mode at a finite value of A3. It is straightforward to show for both monatomic (Sandusky and Page 1994) and diatomic lattices (Kiselev et al. 1994b) that the effect of the static displacements {An} is to renormalize the harmonic force constants k2 into site-dependent, effective harmonic force constants k~(n+l,n) -- k2 -k- 2k3A(An+l - A n ) + 3k4A2(An+l - An) 2
(3.24)
in one of the RWA equations of motion. As a result, the cubic anharmonicity is formally eliminated from the equation, and it is transformed into the pure
A.J. Sievers and J.B. Page
234
k =O
I m
Ch. 3
k ,0 "=
l
~
Particle
m=l
m---1
Fig. 57. Effect of cubic anharmonicity on ILM localization. The plots show the oddparity ILM dynamical displacement patterns for a 40-particle periodic boundary condition lattice having fixed values of (k2, k3, k4), as the ILM frequency is lowered by decreasing the amplitude A. The patterns on the left are for k3 = 0, while those on the right are all for the same nonzero value of k 3. The amplitude has been chosen to give the same frequencies for these cases. Notice that the k 3 # 0 ILM patterns remain localized as the frequencies decrease, while the k 3 = 0 ILM patterns spread out. (After Sandusky and Page 1994).
RWA eq. (3.20) above, except with k2 replaced by the site-dependent k~'s. This site-dependence is strong near the mode center, where the static displacements vary the most rapidly. Evidently, the dynamical displacements for the (k3 r 0) ILMs on the fight side of fig. 57 are formally like those of an impurity mode in a (kz, k4) lattice having defect-induced force constant weakening which enhances the mode's localization. (k2, k4)
3.1.3.1. ILM existence. Bickham et al. (1993) have generated an existence threshold curve for ILMs in the s a m e (k2,k3, k4) lattice used for fig. 55, with the results shown in fig. 58. The solid curve was obtained by fixing A3 and then finding the minimum value of An for which ILMs are supported- this is the point where the ILMs broaden into the zone boundary mode, and it occurs at the same point for the odd and even ILMs. The dashed curve above A4 ,~ 0.8 corresponds to the ILM dynamical displace-
w
Unusual anharmonic local mode systems 1.6
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A3-K3A/K2 Fig. 58. ILM existence in a 512-particle monatomic (k2, k3, k4) lattice with free ends. The solid curve is the existence threshold determined by the condition that the ILM spatially broadens into the zone boundary lattice phonon mode. The dashed curve corresponds to a high-localization triatomic-molecule limit. Area (d), lying between curves (b) and (c), designates the region of a continuum approximation. (After Bickham et al. 1993). ment patterns becoming more localized than the (k2, k4) asymptotic limit ( . . . , 0, - 1/2, 1, - 1/2, 0,...). At these points the ILMs were observed to be unstable in the MD runs of Bickham et al. (1993). The (A3, An) points along the dashed curve are very close to those where a double minimum appears in the (ke, k3, k4) interparticle potential (Sandusky and Page 1994).
3.1.3.2. ILM motion. Moving ILMs for (k2, k3, k4) lattices have been discussed by Bickham et al. (1993). The RWA was used within a moving envelope approximation characterized by a single wavevector, and numerical results were obtained for the same 512-particle free-end lattice considered in figs 55 and 58 above. For a fixed value of the quartic anharmonicity An, dispersion curves were predicted for several values of the cubic anharmonicity A3, as shown in fig. 59. Included in this figure are the corresponding results measured in MD simulations. The agreement is seen to be quite good, with the larger discrepancies occurring for larger values of the wavevector. The top curve is for the pure (k2, k4) lattice, and it reproduces the results given earlier by Bickham et al. (1992). Figure 60 shows the displacements as a function of time, for three particles in the above lattice as a moving ILM passes (Bickham et al. 1993). For
A.J. Sievers and J.B. Page
236 1.5
9
I
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.
.
.
i
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3
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-
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ko Fig. 59. Dispersion curves for moving ILMs for the (k2, k3, k4) lattice of fig 58. The quartic anharmonicity is fixed at A 4 = 1.6, and the lines are the predictions for different values of the cubic anharmonicity A3. The circles are obtained from MD simulations. Note that the agreement between the simulation and analytic results is better at smaller wave vectors and higher frequencies because the effect of the second derivative that were removed from the equations of motion is then minimized. (After Bickham et al. 1993). this free-end boundary condition case, all of the particles on one side of the mode center and well away from it have the same static displacement, while the static displacements of the corresponding particles on the other side are equal and opposite. This is manifested in the figure by the change in the particle offsets from their equilibrium positions as the mode passes. The group velocity of this particular excitation is approximately 15% of the harmonic lattice sound velocity. Note that there is a small increase in the static displacement at the n = 10 site after a time corresponding to three periods of the m a x i m u m plane-wave frequency. This is caused by a small amplitude supersonic pulse that propagates in both directions from the ILM at the start of the simulation and eventually reaches the free ends of the lattice. This pulse may be a long wavelength acoustic kink soliton that is also a solution of these equations.
3.1.4. Realistic potentials The previous studies of ILMs in (k2, k3, k4) lattices show that for a fixed value of the quartic anharmonicity parameter A4, both the even and odd
w
Unusual anharmonic local mode systems
237
Fig. 60. Displacements versus time for a moving ILM in the (k2, k3, k4) free-end lattice of fig. 58. The total displacements (static plus dynamic) are shown for three different sites, as the moving ILM passes. The group velocity of this particular moving excitation is 15% of the harmonic lattice sound velocity. The anharmonicity parameters are A3 = 0.8 and A4 = 0.4, and the wavevector is given by ka = 0.3, where a is the lattice constant. The three curves have been displaced vertically, for clarity. (After Bickham et al. 1993). versions of these modes exist, provided the magnitude of the cubic anharmonicity A3 does not exceed a threshold value. Moreover, it is not difficult to show that these conditions can be satisfied easily for (k2, k3,/ca) values obtained from the Taylor-series expansions of realistic interparticle potentials V(r), such as Lennard-Jones, Morse and B o r n - M a y e r plus Coulomb, about their m i n i m u m points. The question then naturally arises as to whether or not ILMs can exist in lattices in which the particles interact via these full potentials, which of course include all of the higher-order terms in the Taylor-series expansions. One-dimensional lattices with realistic potentials have been studied recently by Kiselev et al. (1993, 1994a, b) for the monatomic and diatomic cases and by Sandusky and Page (1994) for the monatomic case. The diatomic case will be discussed in w3.2. The work of Sandusky and Page concerns general questions about the interrelations between zone boundary phonon stability and ILM formation, and it includes the use of the full potentials listed in the previous paragraph. In some of the calculations, these potentials are restricted to nearest neighbors, while in others they are allowed to act out to sixth neighbors. With respect to the question of ILM existence,
238
A.J. Sievers and J.B. Page
Ch. 3
the basic finding of both the monatomic and the diatomic studies is that ILMs do not exist in either type of lattice with these realistic potentials. The presence of anharmonic terms beyond fourth order in the full potentials acts to suppress the presence of ILMs. However, the diatomic case gives a new possibility, namely that ILMs could occur with frequencies in the gap between the acoustic and optic phonon branches. That such "gap" ILMs exist is shown by Kiselev et al. (1993, 1994b), as discussed w3.2.
3.1.5. Mass defect anharmonic mode with realistic potentials Although ILMs do not exist above the top of the plane wave spectrum of a monatofnic lattice with realistic potentials, high frequency anharmonic impurity modes are still a possibility. Kiselev et al. (1994a, b) have investigated the properties of mass and force constant defect modes as a function of amplitude for four nearest neighbor potentials, which in order of increasing anharmonicity are: Toda, Born-Mayer plus Coulomb (BMC), LennardJones, and Morse. These potentials were adjusted to have the same k2 and k4 terms in their Taylor's series expansion. The light impurity mass defect parameter is e = 1 - md/m -----0.95, where md and m are the defect and the host particle masses. The form of the interaction potential between particles has the same value for all nearest neighbors, except for the bonds involving the impurity. The force constant defect parameter is taken to have the value 77 = 1 - V~'/V" = 0.87, where V~' and V" are proportional to the nearest-neighbor force constants between the impurity and the host lattice and between the particles of the host lattice, respectively. The frequency of the harmonic impurity mode in such a chain has the value 60harm/60m -- 1.125, where tom is the frequency of the top of the harmonic phonon band. For small amplitude vibrations a light impurity supports a localized vibration with a frequency higher than the plane wave spectrum, but with increasing amplitude the anharmonicity of the interaction potential becomes important. The soft attractive part of the standard two-body potentials makes this frequency decrease as the amplitude of the local mode vibration increases and the excitation transforms from a local mode to an in-band resonant mode. Figure 61 shows the frequency of an anharmonic impurity mode in a lattice with a mass and force constant defect for three different two-body potentials as a function of normalized amplitude A/a, where a is the lattice constant. The odd-parity mode is centered at the impurity site. The solid curves show the analytic results, and the points give the simulation results" squares (Toda), diamonds (BMC), and circles (Morse). The results for the Lennard-Jones interaction potential (not shown) are between the BMC and the Morse results. The conclusion is that the more anharmonic the two-body potential, the lower the anharmonic impurity mode frequency.
Unusual anharmonic local mode systems
w 1.2
........... .
239 _
I
.
I
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'
1.0 0.8 ~EO. 6 0.4 0.2 0
,
0.0
I
0.1
, .........
l
0.2
....................
i
I
0.3
i
0.4
Normalized amplitude Fig. 61. The frequency of an anharmonic impurity mode in a monatomic lattice with a force constant and mass defect for three different two-body potentials, as a function of the normalized amplitude A/a. The odd-parity mode is centered at the light impurity (mass defect parameter e = 0.95), which is bonded with its neighbors through the perturbed potential having the forceconstant defect parameter r/ = 0.87. Analytic results are shown as solid curves, while the simulation results are represented by squares (Toda), diamonds (Born-Mayer plus Coulomb), and circles (Morse). For each of the anharmonic potentials, k2 and k4 are the same. (After Kiselev et al. 1994b).
3.2. One-dimensional diatomic lattices
The study by Kiselev et al. (1993, 1994b) of diatomic one-dimensional lattices with standard full potential functions establishes that ILMs can occur in the gap between the optic and acoustic phonon branches. The study was basically numerical and focused on a specific diatomic lattice, namely one with 257 particles, free ends, and having the masses of lithium iodide. Three full potential functions V(r) were emphasized, namely the Toda potential (Toda 1989), the Morse potential and the BMC potential, and all three were restricted to act between just nearest neighbors. In addition, a nearest-neighbor (k2, k3, k4) potential was studied, with the spring constants obtained from the expansion of the BMC potential for LiI. The parameters in the remaining two potential functions were adjusted to reproduce the BMC values of k2 and k4. To obtain the stationary ILMs, an RWA approach based upon the trial solution eq. (3.23) was made, and the resulting equations were solved numerically.
A.J. Sievers and J.B. Page
240
Ch. 3
1-(a) 0
to
- - - - ' - -
-1 "
5. E
(b)
o
<
I~
0
-1
,
-8
~
m
A
.... i
-4
i
0
4
. .
1I
i
8
Particle number Fig. 62. Static and dynamic displacement patterns for ILMs in a one-dimensional diatomic lattice of 257 particles interacting via a nearest-neighbor Born-Mayer plus Coulomb potential. The masses and potential parameters are those of lithium iodide, and free-end boundary conditions are used. The circles and squares denote the light and heavy masses, respectively, with their open and solid versions giving the static and dynamic patterns. (a) Odd-parity gap ILM, with to = 0.74tom. This mode was found to be stable in MD runs. (b) Even-parity gap ILM, with to = 0.74tom. MD runs revealed this mode to be unstable. (c) Odd-parity ILM with to = 1.04tom, obtained by using just (k2, k3, k4), gotten by expanding the LiI potential used in (a) and (b). When the full LiI potential is used, no ILMs with to > tom are found. In all three panels, the overall amplitude is A/a = 0.233. (After Kiselev et al. 1993).
For the (k2, k3, k4) version of this diatomic lattice, an odd-parity ILM occurs, with a frequency W/Wm = 1.04, just above that of the maximum harmonic lattice frequency. Its static and dynamic displacement pattern are given in fig. 62 (c). Just as in the monatomic case discussed in w3.1.4, this high-frequency ILM disappears when any of the three full potentials is used instead of the (k2, k3, k4) potential. Again, no high-frequency ILMs have been found using the above full realistic potentials, for either the monatomic or diatomic cases; the higher-order anharmonic terms in the expansions of the full potentials suppress ILM solutions with frequencies above the phonon bands. However, both even- and odd-parity ILMs are found with frequencies in the phonon gap for either the (k2, k3, k4) case, or with the full potentials. (Note that for a diatomic periodic lattice, the midpoints between the particles' equilibrium sites are no longer symmetry centers, as they are for monatomic
w
Unusual anharmonic local mode systems
241
lattices. For the diatomic case, only the equilibrium sites are symmetry centers. Hence, even-parity modes for the diatomic case are qualitatively different than for the monatomic lattice; in particular they involve no motion of the central particle.) The odd-parity gap ILM is found to be stable, in the sense of persisting over many periods in MD simulations, whereas the even-parity gap ILM is found to be unstable. The displacements for the odd- and even-parity gap ILMs for the B M C potential are shown in panels (a) and (b) of fig. 62. Additional numerical tests show that odd-parity gap modes can be generated from the amplitude pattern for the nearest-neighbor potential cases even when second nearest-neighbor forces are included in the model. These findings suggest that anharmonic gap modes are also stable in the presence of long range forces (Kiselev et al. 1993). Figure 63 plots the frequencies of the odd-parity ILM gap mode for each of the three full potentials, as a function of the normalized amplitude of the 1.0 CO+ 0.8
0.6
~'~ 0.41-
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/f
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.
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.
,
0.1 0.2 0.3 Normalized amplitude
,
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.
Fig. 63. Frequencies of the odd-parity ILM gap mode in the diatomic lattice of fig. 62, for three different full potentials. The frequencies are plotted versus overall amplitude A/a, and w_ and w+ are the frequencies defining the gap between the acoustic and optic phonon branches in the harmonic crystal. The three frequency curves correspond to different nearest-neighbor potential functions: triangles (Toda), circles (Born-Mayer plus Coulomb), and squares (Morse). The inset shows these potentials, together with a Lennard-Jones and a (k2, k3, k4) potential. The Born-Mayer plus Coulomb potential is that appropriate to LiI, and the values of (k2, k 3, k4) were obtained by expanding this potential. The parameters for the remaining potentials were adjusted to reproduce these same values for k2 and kn. (After Kiselev et al. 1993).
A.J. Sievers and J.B. Page
242
Ch. 3
light particle A//a. The curves give the RWA results and the points give the results of MD simulations, and they are seen to be in good agreement with the RWA predictions for A/a < 0.2. When the normalized amplitude is increased to A/a ~ 0.05 the gap mode eigenvector becomes nearly that of a triatomic molecule. The discrepancies observed for larger amplitudes and "softer" potentials (e.g. Morse) indicate that the RWA is becoming less accurate. This is because the effective harmonic potential associated with the RWA looks very different from the effective anharmonic double well which forms at large ILM amplitudes (Kiselev et al. 1994b). The frequency spectrum for the displacement Of the light particle in the stable odd-parity ILM is given by the solid curve in fig. 64. It is obtained over a time interval of At = 1024(27r/o~m). In order of decreasing strength, the peaks are located at ~g, 3U)g, and 5Wg. This spectrum is remarkably similar to the dashed curve, obtained for the anharmonic ILM of the same amplitude in the same diatomic lattice with only the symmetric k2 and k4 potential terms, which is the form used in most earlier studies. Evidently, the
101
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O)I(JL)m Fig. 64. Power spectra of gap and local modes in a diatomic chain with 257 particles. The solid curve corresponds to the gap mode (a~g = 0.74 OJm), with the Born-Mayer plus Coulomb potential for LiI, and the dashed curve shows the ILM spectrum (wi = 1.24~0m) in the same lattice with the (k2, k4) potential from the Taylor's series expansion of the Born-Mayer plus Coulomb potential. The ordinate of the dashed curve has been shifted by a factor 20, for clarity. In each case the amplitude is A/a = 0.233. The results are remarkably similar: the resonant features occur at the same first, third and fifth orders with about the same strengths. The "noise" that appears in the solid curve near these frequencies is due to weak coupling between the gap mode and the plane wave spectrum. (After Kiselev et al. 1993).
w
Unusual anharmonic local mode systems
243
main function of the asymmetric potential terms in the Taylor's expansion of the two body potential is to contribute to the static distortion in the displacement patterns shown in fig. 62. Inspection of fig. 62(a) reveals that the displacement pattern of the stable odd-parity gap ILM for the LiI BMC potential is very much like that of an isolated Einstein oscillator- the odd-order (asymmetric) terms in the interparticle potential have caused the equilibrium positions of the particles on each side of the central particle to uniformly shift outwards, and the dynamical displacements are ~, 0 except at the central particle. This Einstein-oscillator-like dynamical behavior is not surprising in view of the fact that the central mass is so much lighter than its neighbors. Indeed, this mode is behaving qualitatively like that of the light mass defect anharmonic mode discussed earlier for monatomic (k2, k4) systems, with the additional feature that the harmonic force constants k2 on either side of the central particle have been renormalized to k~, due to the local static distortion produced in the ILM by k3. Gap ILMs in pure (k2, k4) diatomic systems have recently been discussed (Aoki et al. 1993; Chubykalo et al. 1993). It is now evident that ILM solutions can be obtained numerically for a perfect diatomic 1D chain incorporating a variety of realistic two body potentials in the sense of persisting over many periods in MD runs, while simulations demonstrate that the odd-parity gap modes are stable, but the even-parity modes are not. The incorporation of a realistic two-body potential into the lattice has brought out several important new features of ILMs: (1) a localized mode will not exist above the top of the plane wave spectrum for realistic potentials; (2) a long-lived odd-parity localized mode can exist in the gap between the optic and acoustic branches; (3) in the large amplitude limit, the triatomic molecule-like eigenvector previously observed for local modes in the hard quartic case is recovered for gap modes with realistic potentials and (4) for each of these potentials the gap mode is accompanied by a localized DC expansion of the lattice.
4.
Speculation
In the preceding two sections, we have discussed two very different sorts of unusual localized vibrational behavior in strongly anharmonic systems. First, we have seen that detailed experimental and theoretical studies of the impurity system KI:Ag + reveal a nearly unstable defect/host system at low temperatures, which exhibits a rapid, thermally-driven on --+ off-center transition between T = 0 K and T = 20 K. As detailed above, this transition is highly anomalous, in that it cannot be explained within any sort of a
244
A.J. Sievers and J.B. Page
Ch. 3
harmonic or weakly perturbed anharmonic f r a m e w o r k - thus the system is strongly anharmonic in an as yet undetermined way. At the same time, the T = 0 K on-center configuration and its associate dynamical properties as probed by extensive stress, E-field and host lattice isotope effect experiments are found to be consistent with a perturbed harmonic phonon model. Even though the on-center configuration is therefore apparently harmonic (or quasiharmonic under applied stress or E-fields), its associated dynamics are themselves very unusual" the impurity-host system is strongly coupled but nearly unstable against off-center displacements and it possesses a new type of localized impurity mode, the pocket gap modes, whose displacements are confined to sites away from the defects. Second, we have discussed some of the basic properties of an unusual new class of localized vibrational excitations (ILMs) which can occur in periodic lattices, provided there is sufficiently strong local anharmonicity present. This anharmonicity could arise in two ways: 1) from large values of the anharmonic coefficients in the expansion of the potential energy about the equilibrium configuration, or 2) from the presence of large amplitudes in a localized region of the lattice. The study of ILMs has been purely theoretical up to the present and is still at an early stage. The focus has been on simplified models, but they are becoming more realistic, as exemplified by the recent studies involving realistic potentials (Kiselev et al. 1993, 1994b; Sandusky et al. 1994). As the properties of ILMs become better established, we anticipate that there will be significant activity in experimentally verifying their existence in real solids. In the following we briefly mention some of the speculative ideas which have been suggested previously. One of these involves a link between ILMs and KI:Ag +, and it will be discussed first, in a separate subsection. 4.1. ILMs and anomalous defect properties
4.1.1. KI:Ag + One of the early proposals was that perhaps one or more ILMs are trapped at the Ag + site at low temperatures only to be released into the lattice at some finite temperature. When these ILMs are released into the lattice at an elevated temperature, they would leave behind a defect space with a different vibrational pattem and hence account for the change in the defect induced spectrum. Since a free ILM could move in any direction in the lattice, a large number of accessible states would exist; hence, there is a great deal of entropy associated with a change from a bound to this "free" configuration. The release of ILMs would be expected to give a strong temperature dependent signature.
w
Unusual anharmonic local mode systems
245
1.0
O.B I 0.6
I
\ I
\\ ,
~
',,f ~
0.4
0.2
-
\ l
0.2
\
0.4
0.6
0.8
kT/~-Fig. 65. Temperature dependence of the degree of association pon(T) for three different concentrations. The reduced temperature is kT/( where ff is the Gibbs free energy of association. The three different concentrations are (a) c = 1.2 x 10 -3, (b) c = 1.2 x 10 - 2 , and (c) c = 1.2 x 10-1. (After Lidiard 1957).
The temperature dependence can be calculated by making use of the similarity with the corresponding impurity-vacancy-complex activation problem. (See, for example, Lidiard 1957). Assume that an ILM complexes with the impurity. If the degree of association Pon is such that Cpon is the molar fraction of complexes, then application of the law of mass action to the process (impurity-ILM complex r unassociated impurity + unassociated ILM) gives = c exp (1
--port) 2
(,) k-T
(4.1) '
where ff is the Gibbs free energy of association. Curves showing Pon(T) as a function of temperature for three different impurity concentrations are presented in fig. 65. The temperature dependence of the lowest concentration curve looks remarkably similar to the data shown in fig. 14(a). Since both c and ( could be taken to be free parameters at this stage, a reasonable fit to the data could be generated for a single concentration, but fig. 65 indicates that the exact temperature dependence depends strongly on the concentration. What cannot be fit by this model is the experimental observation that the line strength temperature dependence observed for KI:Ag + is independent of Ag + concentration over a range of at least a factor of three; hence, the correct model cannot depend on concentration.
A.J. Sievers and J.B. Page
246
Ch. 3
4.1.2. MOssbauer recoilless fraction for Sn in Pb There are completely different kinds of measurements on Sn impurities in metallic Pb which show unusual temperature dependent behavior in the lattice vibrational spectrum. Mt~ssbauer recoilless fraction measurements as a function of temperature provide another way to investigate the dynamical coupling between an impurity and the lattice. For a simple Debye solid this quantity reduces to ~D
f = exp
3R 1+ 2k69D
~
ex - 1
'
(4.2)
where R is the recoil energy of the free nucleus and 690 is the Debye temperature of the solid (Frauenfelder 1963; Ashcroft and Mermin 1976). At temperature large compared to 69D the temperature dependent factor in the exponent is proportional to the temperature. (Note the similarity between eq. (4.2) and eq. (2.5). The measured temperature dependence of the recoilless fraction of ll9Sn in Pb (Haskel et al. 1993) is shown in fig. 66. For temperatures comparable to or slightly larger than the Debye 690 of Pb (88 K), the data give a linear temperature dependence for a semilog plot, as expected from eq. (4.2). The anomalous behavior appears at somewhat higher temperatures. For the lowest concentration the recoilless fraction disappears at temperatures above 120 K. For the 1% 119Sn in Pb, the recoilless fraction can be followed to higher temperatures beyond the knee at about 145 K. Similar temperature dependent results are found for the 2% 119Sn in Pb sample with the knee now at about 170 K. Finally, for the highest concentration sample the knee appears at about 200 K but is now somewhat rounded. These results demonstrate that the characteristic knee temperature at which the recoilless fraction rapidly decreases moves to higher temperature with increasing Sn concentration, with the resulting temperature dependence being a strong function of concentration. These results have been interpreted as evidence for a large number of low-lying states, a number much larger than can be derived from hopping or tunneling of the defect. The large number of states is attributed to "local melting" of about 30 atoms in the lattice near the defect, even though the temperature is far below the melting temperature of the alloy. The remarkable concentration dependence is not a natural feature of such a model. From w4.1.1 a concentration dependence of the temperature dependence would be a natural consequence if one or more ILMs are complexed to the Sn impurity at low temperatures but are released as the temperature is raised into the 150 K region. (See fig. 65). During this transformation
Unusual anharmonic local mode systems
w
247
A (/) .,I,-o~ e"
t,._ v
o
>, 0.10
.u. (/)
e"
o
t,,. .,0,(,.)
Or) 0.01
I00
150 Temperature (K)
200
Fig. 66. MOssbauer spectral intensity versus temperature for alloys of Sn in Pb. The different atomic % concentrations of l l9Sn for data taken with increasing temperature are as follows: crosses -0.5%; open squares -1%; solid diamonds -2%; open circles -3%. The z's show the data taken during cooling. The straight lines are fits to the experimental data using a standard anharmonic potential model for the Debye-Waller factor. (After Haskel et al. 1993). the Sn could move from an on-center site to an off-center site. The large entropy contribution would stem from the large number of states available to the ILMs and the effective knee activation temperature would increase with increasing Sn concentration in qualitative agreement with the results shown in fig. 66.
4.2. ILMs and other properties Section 3 focused on ILMs with frequencies outside the normal acoustic and optic phonon bands, since almost all of the ILM work up to now has concerned this case. However, Takeno and Sievers (1988) also theoretically discussed the possibility of low-frequency in-band ILMs for the case of monatomic (k2, k4) lattices having negative values of the quartic anharmonicity A4 - knA2/k2, arising from negative ka's. These "soft anharmonicity" in-band ILMs are reminiscent of defect-induced harmonic resonant modes, such as those occurring in the T = 0 K on-center configuration of KI:Ag +. Indeed, for the simple (k2, k4) lattice it was found that low-frequency resonant ILMs have frequencies and amplitude patterns behaving like those for a force constant defect in a harmonic lattice, with the role of the harmonic force-constant change parameter in the latter case being played by the anharmonicity parameter An. In the limit of low frequency, the resonant
248
A.J. Sievers and J.B. Page
Ch. 3
ILM amplitude pattern is confined to the central particle, in contrast to the behavior we have seen here for the high-frequency ILM in the hard quartic case, where the limiting displacement pattern is that of a simple triatomic molecule. On the other hand, the spectral width of the resonant ILM was found to be much greater than that for the corresponding harmonic forceconstant defect resonant mode, giving the resonant ILM a shorter lifetime against decay into the quasicontinuum of embedding band modes. All of the ILM work discussed here has been classical. Nevertheless, certain general aspects of the effects of ILMs on the equilibrium thermodynamics of lattices were discussed by Sievers and Takeno (1988) and by Takeno and Sievers (1988). The key point is that because ILMs can exist at any lattice site, they should give rise to a large configurational entropy and hence make an important contribution to the free energy for nonzero temperatures. Thus their equilibrium statistics should be like those for thermally-activated processes such as vacancy formation, being determined by the balance between the energy needed for ILM formation and the configurational entropy. For the case when the thermal equilibrium number n i l m of ILMs is much less than the number N of lattice sites, one has n i l m - - N exp(-/3f), where = 1/(kBT) and f is the work needed to create a single ILM in a lattice of fixed volume. These considerations apply to above-band ILMs, to gap ILMs and to in-band resonant ILMs, with the resonant case expected to have a smaller (but still positive) value of f (Takeno and Sievers 1988). Hence for each type of ILM, the picture which emerges is that the vibrational spectrum of a lattice at T = 0 K should be dominated by homogeneous plane-wave-like harmonic phonons (or anharmonically renormalized quasiharmonic phonons), but with increasing temperature ILMs are created and should be included in describing the dynamical properties. Interpreting the classical anharmonic potential functions as effective potentials for the quantum solids 3He and 4He, Sievers and Takeno (1988) and Takeno and Sievers (1988) speculated on the possible importance of ILMs for these highly anharmonic systems. 3He exhibits a low-temperature excess specific heat characteristic of thermally-activated processes, and this has been identified with vacancy production. X-ray measurements of the lattice parameter of both 3He and 4He in constant-volume cells have shown that there are indeed thermal vacancies produced. However, when these measurements and the specific heat results are interpreted in terms of localized vacancies, one obtains an unreasonably large energy for the production of a vacancy. An attractive mechanism in addition to thermal vacancies, is provided by thermally-activated ILM production. In this case the lowtemperature ILM formation energy should be relatively small, being the difference between zero-point energies of two types of vibrational modes,
w
Unusual anharmonic local mode systems
249
namely ILMs and phonons. Additional arguments, emphasizing the role of resonant ILMs, were advanced by Takeno and Sievers (1988). Another early application (Sievers and Takeno 1989) was to the problem of the observed anomalous low temperature specific heat of glasses. This application was based upon two main assumptions. 1) The disorder results in the presence of a substantial number of low-frequency resonant ILMs - these can be thought of as disorder-induced impurity ILMs. 2) The ILMs can move diffusively. Here in w3 we have seen that recent studies of moving ILMs in perfect lattices show that ILMs move very easily from site to site (Bickham et al. 1992; Sandusky et al. 1992); moreover, it is reasonable to assume that the glassy disorder would result in the motion being diffusive. Sievers and Takeno (1989) presented arguments leading from these assumptions to two main results. The first is that diffusive ILM motion produces a contribution to the low temperature specific heat which is linear in T, as is observed experimentally. The second is that the presence of stationary resonant ILMs contributes an excess specific heat term which is cubic in T, also as observed. For the simplified model calculations, the key parameter was the ratio of the resonant ILM frequency to the Debye frequency. Phenomenological fits of this parameter to the measured low-temperature linear specific heat term for 15 glassy solids yield similar values (0.063 to 0.12), even though the glasses range in type through covalent, van der Waals, ionic, metallic, inorganic and organic. Moreover, the resulting excess cubic specific heat in the model is of the correct order of magnitude. An obvious question is whether or not the atomic forces in real crystals are sufficiently strong to sustain ILMs at reasonable temperatures. Bickham and Sievers (1991) addressed this question in an approximate fashion in their work on high-frequency ILMs in (k2, k4) lattices having weak quartic force constants. As we have noted, strong anharmonicity An -- k4A2/k2 can still occur in such lattices, for sufficiently large amplitudes A. These authors obtained k4/k2 from lattice data for five alkali halides, with LiF being the most anharmonic (ka/k2 - - 5 . 5 0 ~-2). Then by combining the standard harmonic approximation expressions for the virial theorem and thermal mean square displacement, together with their results for the spatial width of the ILMs as a function of A4, they concluded that in thermal equilibrium, lattices with these force constants would have insufficient thermal energy to sustain ILMs at any temperature below the melting temperature. Despite the major approximations in these arguments, the results were sufficiently clear-cut that it was concluded to be very unlikely that high frequency ILMs could arise thermally in these (k2, k4) systems. This left open the possibility of thermally generated low-frequency resonant ILMs in such systems, but this question has not yet been addressed. Finally, these authors also suggested that crystals exhibiting soft-mode ferroelectric or anti-ferroelectric phase
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Ch. 3
transitions might be an appropriate candidate for ILMs - near such a phase transition the importance of the anharmonic potential energy terms should be strongly enhanced. Having shown that ILMs are not likely to be thermally generated in simple (k2, k4) lattices with force constants derived from alkali halide potentials, Bickham and Sievers (1991) then briefly considered the introduction of vibrational energy from a nonthermal source, e.g., a Mrssbauer recoil or an optical transition followed by local lattice relaxation. As an extreme case within their force constant parameters, they considered the vibrational energy relaxation of an F-center, which in LiF involves about 40 phonons of frequency hWm. The order-of-magnitude result was that for this case sufficient vibrational energy is available to produce ILMs. However, this would require a very rapid local relaxation of the optically excited center, and hence detailed model calculations are necessary before one can make definite predictions. The speculations summarized in the above paragraphs were carried out shortly after the ILM concept appeared in 1988, and they were therefore made in the context of simple (k2, k4) lattices. Owing to the novel possibility that perfect lattices could sustain localized vibrational excitations, and because of the inherent difficulties of dealing with nonlinear phenomena in general, most of the subsequent theoretical work has focused on detailed explorations of the properties of ILMs for this simple case. Although this work is still in a relatively early stage, we have seen in w3 that these studies have already revealed a rich variety of interesting phenomena. We have also pointed out, however, that the simple (k2, k4) case misses an important aspect of realistic potentials, namely cubic anharmonicity, whose implications for ILMs in real solids are only now being sorted out. Our recent analytic and numerical studies of the effects of including realistic values of k3 (Bickham et al. 1993; Sandusky et al. 1994) have shown that the qualitative features of ILMs in (k2, k4) lattices remain when k3 is included, with the primary additional feature being that the ILMs are accompanied by a strong, amplitude-dependent, local static distortion of the lattice. As discussed earlier, this distortion tends to enhance the ILM's spatial localization as the mode frequency is lowered. Moreover, recent studies using realistic interparticle full potentials show that ILMs above the maximum lattice frequency do not occur for the one-dimensional systems studied (Kiselev et al. 1993; Kiselev et al. 1994b; Sandusky et al. 1994), whereas ILMs do occur in the phonon frequency gap in diatomic lattices (Kiselev et al. 1993). Again, these gap ILMs appear to retain most of the qualitative features of ILMs in simple (k2, k4) lattices, with the main effect of the odd-order terms in the potential function just being to produce the local static distortion.
Unusual anharmonic local mode systems
251
Clearly, the study of ILMs is in its infancy, with a vast number of important open questions remaining. Nevertheless, the recent results using realistic interparticle potentials suggest that ILMs will be theoretically understood sufficiently well in the near future that sensible experimental investigations can be launched to establish their presence in real solids.
5.
Conclusion
The experimental and theoretical studies of the two strongly anharmonic systems discussed in this chapter have revealed a wealth of fascinating and unexpected new phenomena. While much understanding has been achieved, the important remaining unexplained questions should provide a fertile area for obtaining fundamental new insights into the dynamical behavior of condensed matter systems.
6. Acknowledgements During the preparation of this article A.J.S. was supported by NSF Grant DMR-9312381 and ARO Grant DAAL03-92-G-0369 and J. B. P. was supported by NSF Grant DMR-9014729 and the Alexander von Humboldt Foundation. We would also like to acknowledge the important contributions made by our many collaborators on this work, particularly R.W. Alexander, S.A. Bickham, B.P. Clayman, R.P. Devaty, H. Fleurent, L. Genzel, L.H. Greene, S.B. Hearon, A.M. Kahan, R.D. Kirby, S. Kiselev, J.T. McWhirter, C.M. Mungan, I.G. Nolt, M. Patterson, A.M. Rosenberg, T. R6ssler, K.W. Sandusky and S. Takeno.
Note added in proof Since the preparation of the manuscript for this chapter, the ILM-related literature has continued to grow. Listed below are some recent papers. 1. ILMs in perfect and/or defect (k2, k4) lattices are discussed in: Wallis, R.E, A. Franchini and V. Bortolani (1994),Phys. Rev. B 50, 9851; Flach, S. and C.R. Willis (1994), Phys. Rev. Lett. 72, 1777; Dusi, R. and M. Wagner (1995), Phys. Rev. B 51, 15847; Takeno, S. (1995), J. Phys. Soc. Jpn 64, 2380; Kovalev, S., F. Zhang and Y.S. Kivshar (1995), Phys. Rev. B 51, 3218. 2. Diatomic Toda lattices are discussed in: Aoki, M. amd S. Takeno (1995), J. Phys. Soc. Jpn 64, 809.
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A.J. Sievers and J.B. Page
Ch. 3
3. The counting of ILMs is discussed in: Kiselev, S.A., S. Bickham and A.J. Sievers (1995), Commun. in Condens Mat. Phys. 17, 135.
4. Surface or edge ILMs are discussed in: Takeno, S., K. Hori, K. Ohtsuka and S. Homma (1994), J. Phys. Soc. Jpn 63, 1295; Watanabe, T. and S. Takeno (1994), Phys. Soc. Jpn 63, 2028; Bonart, D., A.P. Mayer and U. Schr~der (1995), Phys. Rev. Lett. 75, 870; Bonart, D., A.P. Mayer and U. Schr/Sder (1995), Phys. Rev. B 51, 13739.
5. Exact RWA solutions for ILMs driven by external fields are discussed in: Roessler, T. and J.B. Page (1995), Phys. Lett. A (accepted).
6. General nonlinear dynamics/soliton approaches" Huang, G. (1995), Phys. Rev. B 51, 12347; Neuper, A., EG. Mertens and N. Flytzanis (1994), Z. Phys. B 95, 397.
7. On-site potentials include: Flach, S. (1994), Phys. Rev. E 50, 3134; Flach, S., K. Kladko and C.R. Willis (1994), Phys. Rev. E 50, 2293; Flach, S. (1995), Phys. Rev. E 51, 3579.
8. Some quantum mechanical aspects related to ILMs are discussed in: Kitamura, T. and S. Takeno (1994), Phys. Lett. A 190, 327; Kitamura, T. and S. Takeno (1995), Physica A 213, 539; Roessler, T. and J.B. Page (1995), Phys. Rev. B 51, 11382.
9. The application to self-localized magnons is considered by: Takeno, S. and K. Kawasaki (1994), J. Phys. Soc. Jpn 63, 1928; Wallis, R.E, D.L. Mills and A.D. Boardman (1995), Phys. Rev. B 52, R3828.
10. An overview of some properties of ILMs is given by: Page, J.B. (1995), Physica B (accepted); Bickham, S.R., S.A. Kiselev and A.J. Sievers (1995), in: Spectroscopy and Dynamics of Collective Excitations in Solids, Ed. by B. DiBartolo (Plenum Press, New York).
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Hearon, S.B. and A.J. Sievers (1984), Phys. Rev. B 30, 4853. Holland, U. and E Luty (1979), Phys. Rev. B 19, 4298. Hori, K. and S. Takeno (1992), J. Phys. Soc. Jpn 61, 2186. Jacobs, P.W.M. (1990), J. Imaging Sci. 34, 79. Kahan, A.M., M. Patterson and A.J. Sievers (1976), Phys. Rev. B 14, 5422. Kapphan, S. and E Luty (1972), Phys. Rev. B 6, 1537. Kirby, R.D. (1971), Phys. Rev. B 4, 3557. Kirby, R.D., A.E. Hughes and A.J. Sievers (1970), Phys. Rev. B 2, 481. Kiselev, S.A. (19.90), Phys. Lett. A 148, 95. Kiselev, S.A., S.R. Bickham and A.J. Sievers (1993), Phys. Rev. B 48, 13508. Kiselev, S.A., S.R. Bickham and A.J. Sievers (1994a), Phys. Lett. A 184, 255. Kiselev, S.A., S.R. Bickham and A.J. Sievers (1994b), Phys. Rev. B, 50, 9135. Kivshar, Y.S. (1991), Phys. Lett. A 161, 80. Kivshar, Y.S. (1993), Phys. Rev. E 48, 4132. Kivshar, Y.S. and D.K. Campbell (1993), Phys. Rev. E 48, 3077. Klein, M.V. (1968), in: Physics of Color Centers, Ed. by W.B. Fowler (Academic Press, New York), p. 430. Klein, M.V. (1990), in: Dynamical Properties of Solids, Vol. 6, Ed. by G.K. Horton and A.A. Maradudin (North-Holland, Amsterdam), p. 65. Kleppmann, W.G. (1976), J. Phys. C 9, 2285. Kleppmann, W.G. and W. Weber (1979), Phys. Rev. B 20, 1669. Kojima, K., M. Sakurai and T. Kojima (1968), J. Phys. Soc. Jpn 24, 815. Kosevich, A.M. and A.S. Kovalev (1974), Sov. Phys. - JETP 40, 891. Kosevich, Y.A. (1993a), Phys. Lett. A 173, 257. Kosevich, Y.A. (1993b), Phys. Rev. Lett. 71, 2058. Kosevich, Y.A. (1993c), Phys. Rev. B 47, 3138. Kremer, F. (1984), in: Infrared and Millimeter Waves (Academic Press, New York), chapter 4. Lidiard, A.B. (1957), in: Handbuch der Physik, Vol. 20, Ed. by S. Fliagge (Berlin), p. 246. Lifshitz, I.M. (1956), Nuovo Cim. Suppl. 3, 716. Love, S.P., W.P. Ambrose and A.J. Sievers (1989), Phys. Rev. B 39, 10352. Maradudin, A.A., E.W. Montroll, G.H. Weiss and I.P. Ipatova (1971), Theory of Lattice Dynamics in the Harmonic Approximation (Academic Press, New York); Suppl. 3. Narayanamurti, V. and R.O. Pohl (1970), Rev. Mod. Phys. 42, 201. Nolt, I.G. and A.J. Sievers (1968), Phys. Rev. 174, 1004. Page, J.B. (1974), Phys. Rev. B 10, 719. Page, J.B. (1990), Phys. Rev. B 41, 7835. Page, J.B., J.T. McWhirter, A.J. Sievers, H. Fleurent, A. Bouwen and D. Schoemaker (1989), Phys. Rev. Lett. 63, 1837. Patterson, M. (1973), MS Thesis, Cornell University. Pnevmatikos, S., N. Flytzanis and M. Remoissenet (1986), Phys. Rev. B 33, 2308. Poglitsch, A. (1984), PhD Thesis Max-Planck-Institut fur Festk~rperforschung, Stuttgart. Rosenberg, A., C.E. Mungan, A.J. Sievers, K.W. Sandusky and J.B. Page (1992), Phys. Rev. B 46, 11507. Sandusky, K.W. and J.B. Page (1994), Phys. Rev. B 50, 866. Sandusky, K.W., J.B. Page, A. Rosenberg, C.E. Mungan and A.J. Sievers (1991), Phys. Rev. Lett. 67, 871. Sandusky, K.W., J.B. Page and K.E. Schmidt (1992), Phys. Rev. B 46, 6161. Sandusky, K.W\, J.B. Page, A. Rosenberg and A.J. Sievers (1993a), Phys. Rev. B 47, 5731. Sandusky, K.W., J.B. Page and K.E. Schmidt (1993b), in: Dynamic Days Arizona Int. Conf. (unpublished).
Unusual a n h a r m o n i c local m o d e systems
255
Sandusky, K.W., A. Rosenberg, B.P. Clayman, J.B. Page and A.J. Sievers (1994), Europhys. Lett. 27, 401. Sch~ifer, G. (1960), J. Phys. Chem. Solids 12, 233. Schr6der, U. (1966), Solid State Commun. 4, 347. Sievers, A.J. (1964), Phys. Rev. Lett. 13, 310. Sievers, A.J. (1968), in: Localized Excitations in Solids, Ed. by R.E Wallis (Plenum, New York), p. 27. Sievers, A.J. (1969), in: Elementary Excitations in Solids, Ed. by A.A. Maradudin and G.F. Nardelli (Plenum, New York), p. 193. Sievers, A.J. and L.H. Greene (1984), Phys. Rev. Lett. 52, 1234. Sievers, A.J., L.H. Greene and S.B. Hearon (1984), in: 9th Int. Con.[. on Infrared and Millimeter Waves, J. Soc. Applied Physics, p. 144. Sievers, A.J., A.A. Maradudin and S.S. Jaswal (1965), Phys. Rev. 138, A272. Sievers, A.J. and S. Takeno (1965), Phys. Rev. 140, A1030. Sievers, A.J. and S. Takeno (1988), Phys. Rev. Lett. 61, 970. Sievers, A.J. and S. Takeno (1989), Phys. Rev. B 39, 3374. Smith, D.Y. (1985), in: Handbook of Optical Constants of Solids, Ed. by E. Palik (Academic Press, New York), p. 35. Stolen, R. and K. Dransfeld (1965), Phys. Rev. 139, A1295. Stoneham, A.M. (1975), Theory of Defects in Solids (Oxford University Press, Oxford). Takeno, S. (1990), J. Phys. Soc. Jpn 59, 3861. Takeno, S. (1992), J. Phys. Soc. Jpn 61, 2821. Takeno, S. and S. Homma (1991), J. Phys. Soc. Jpn 60, 731. Takeno, S. and K. Hori (1990), J. Phys. Soc. Jpn 59, 3037. Takeno, S. and K. Hori (1991), J. Phys. Soc. Jpn 60, 947. Takeno, S., K. Kisoda and A.J. Sievers (1988), Progr. Theor. Phys. Suppl. 94, 242. Takeno, S. and A.J. Sievers (1965), Phys. Rev. Lett. 15, 1020. Takeno, S. and A.J. Sievers (1988), Solid State Commun. 67, 1023. Toda, M. (1989), Theory of Nonlinear Lattices, 2nd edn (Springer, New York). Vegard, L. (1921), Z. Physik 5, 17. Visscher, W.M. (1963), 129, 28. Ward, R.W., B.P. Clayman and T. Timusk (1975), Canad. J. Phys. 53, 424. Weber, R. (1964), Phys. Lett. 12, 311. Weber, R. and P. Nette (1966), Phys. Lett. 20, 493. Wong, X. and F. Bridges (1992), Phys. Rev. B 46, 5122. Yoshimura, K. and S. Watanabe (1991), J. Phys. Soc. Jpn 60, 82. Zavt, G.S., M. Wagner and A. Ltitze (1993), Phys. Rev. E 47, 4108.
Unusual Anharmonic Local Mode Systems
A.J. SIEVERS
J.B. PAGE
Laboratory of Atomic and Solid State Physics and the Materials Science Center Cornell University Ithaca, NY 14853-2501 USA
Department of Physics and Astronomy Arizona State University Tempe, Arizona 85287-1504 USA
Dynamical Properties of Solids, edited by G.K. Horton and A.A. Maradudin
9 Elsevier Science B.V., 1995
137
Contents 1. Introduction
141
1.1. Impurity modes in crystals 141 1.2. Localized modes in perfect anharmonic lattices ,
143
Experimental and theoretical studies of a thermally anomalous nearly unstable impurity system: KI:Ag + 147 2.1. Initial experiments 147 2.2. Basic shell model description of the T = 0 K nearly unstable lattice dynamics 182 2.3. Pocket gap mode experiments 193 2.4. The (3, 3', 3") and the quadrupolar deformability models 199 2.5. Discussion and conclusions 203
,
Intrinsic localized modes in perfect anharmonic lattices 3.1. One-dimensional monatomic lattices 207 3.2. One-dimensional diatomic lattices 239
4. Speculation
243
4.1. ILMs and anomalous defect properties 4.2. ILMs and other properties 247 5. Conclusion
251
6. Acknowledgements Note added in proof References
251 251
252
139
244
206
We dedicate this chapter to LUDWIG GENZEL and the late HEINZ BILZ for their pioneering experimental and theoretical studies of numerous key aspects of lattice dynamics in crystals
1.
Introduction
1.1. Impurity modes in crystals The early far infrared measurements with grating instruments on the low temperature properties of defects in alkali halides produced a variety of sharp features associated with local modes (Sch~ifer 1960), gap modes (Sievers et al. 1965) and resonant modes (Sievers 1964; Weber 1964). The sharp resonant modes were somewhat of a surprise, even though the possibility of strongly coupled impurity-induced lattice resonances had been proposed in earlier theoretical work (Brout and Visscher 1962; Visscher 1963; Dawber and Elliott 1963). The observation of these low-lying sharp spectral features was quickly followed by more accurate measurements using the fourier transform IR technique (Sievers 1969) and somewhat later by Raman scattering studies. (Klein 1990 has recently reviewed this topic.) A variety of defect systems has been studied in detail, and much of that work has been reviewed from both the experimental (Barker and Sievers 1975; Bridges 1975) and theoretical (Stoneham 1975; Bilz et al. 1984) points of view. The findings are that most localized vibrational modes in defect-lattice systems behave in the expected fashion; that is, the modes identified with impurity motion can be interpreted from the experimental studies as harmonic or slightly anharmonic oscillators. However, for some systems the modes have anomalous properties that place them outside of the bounds of that interpretation. The vibrational properties of the light Li + impurity in KC1 or KBr are two defect-lattice systems that have received a great deal of attention over the years. KCI:Li + has been studied mainly because the impurity ion is off-center in the (111) directions so that low frequency tunneling states play a prominent role (Narayanamurti and Pohl 1970; Kirby et al. 1970; Devaty and Sievers 1979; Wong and Bridges 1992). On the other 141
142
A.J. Sievers and J.B. Page
Ch. 3
hand, an equal effort has gone into examining the resonant mode properties of KBr:Li + because now the impurity ion is barely stable at a normal lattice site (Sievers and Takeno 1965; Page 1974; Kahan et al. 1976). The near instability demonstrated by the single ion spectrum is fullfilled for Li + pairs in KBr which are found to be off-center and to tunnel coherently from site to site (Greene and Sievers 1982, 1985). Although these point defect systems have received ample attention in the literature and are thought to be understood there are other anomalous systems which have not played a central role since the paucity of available data did not allow their unusual features to be recognized as crucial. There are two "simple" heavy defect ion-lattice systems in particular, which display unusual vibrational properties: KI:Ag + and RbCI:Ag +. At low temperatures both of these systems show defectinduced lattice modes in the far IR. Early measurements showed that the Ag + impurity in KI was on-center at a K + lattice site, while the Ag + impurity in RbC1 was off-center in a (110) direction and exhibited tunneling between equivalent sites (Barker and Sievers 1975). The dynamics of this heavy impurity had been assumed to be similar to those found for the Li + impurity in KBr and KC1, respectively. In the last decade, paraelectric resonance studies of the tunneling modes in RbCI:Ag + as a function of hydrostatic pressure indicate unusual intensity dependences for some transitions. These have been interpreted as a signature for the existence of an on-center configuration nearby in energy (Bridges et al. 1983; Bridges and Chow 1985; Bridges and Jost 1988). For KI:Ag +, the early observation that the IR active resonant mode and the later observation that the entire T = 0 K impurity-induced spectrum disappear as the temperature is increased to 25 K has been shown to be associated with the Ag + ion moving from the on-center to an off-center position with increasing temperature (Sievers and Greene 1984). These results for KI:Ag + may indicate that at T = 0 K there is both an on-center configuration and an off-center one with nearly the same energy. The very unusual behavior of these two lattice-defect systems has called into question the underlying fundamentals of standard defect phonon theory. Where is the flexibility in the Lifshitz theory (Lifshitz 1956) for the existence of multiple elastic configurations? To address this fundamental lattice dynamics question, much theoretical and experimental effort has gone into reexamining in greater detail the T = 0 K properties of the simplest of the two systems, namely on-center KI:Ag +, which may have at least two low-lying configurations. The approach that has been used is first to apply the harmonic shell model in an attempt to explain the observed spectroscopic properties of this lattice defect system, excluding the temperature dependence. This has proved to be quite successful, and with the addition of some anharmonicity as described
w1
Unusual anharmonic local mode systems
143
here, it gives a quantitative description of the local dynamics of the KI:Ag + system at T = 0 K. The resulting comparison between theory and experiment in unprecedented detail has led to the prediction and verification of an unexpected new class of defect modes, called "pocket" gap modes (Sandusky et al. 1991). These defect modes have the unusual property that the maximum vibrational amplitude is not at the impurity but is localized at lattice sites well removed from the impurity. The experimental investigation of the isotope effect (Sandusky et al. 1991; Sandusky et al. 1993a), stress shifts (Rosenberg et al. 1992) and electric field Stark effect (Sandusky et al. 1994) provide sufficient experimental information to demonstrate that a dynamically-induced electronic deformability of the Ag + impurity plays an essential role in the dynamics. Upon analysis of much of the data, one finds that a convincing explanation of the T = 0 K experimental far IR and Raman defect-induced spectra can be made within the Lifshitz perturbed phonon framework, in terms of a slightly anharmonic shell model in which the coupled defect/host system is nearly unstable. In contrast, while a variety of phenomenological anharmonic models have been introduced to account for the observed temperature effects, it is still not possible to present a consistent dynamical picture of the temperaturedependent behavior of this system. One purpose of this review is illustrate how the temperature dependence models fail. The end result of these tightly woven arguments will be that the observed anomalous temperature dependence of the spectra appears to fall truly outside the bounds of current defect mode theory. A primary function of this review is to gather together all of the experimental and theoretical data on this particular on-center system, so that the reader can inspect, understand and appreciate what lies beyond the framework of current impurity dynamics theory. 1.2. Localized modes in perfect anharmonic lattices While it is not surprising that the loss of periodicity in defect crystals leads to localized vibrational phenomena, the standard description of the dynamics of defect-free periodic lattices in terms of plane wave phonons is so deeply ingrained that it was surprising to many researchers when it was argued theoretically in 1988 (Sievers and Takeno 1988; Takeno and Sievers 1988) that the presence of strong quartic anharmonicity in perfect lattices can also lead to localized vibrational modes, henceforth called "intrinsic localized modes" (ILMs). These papers studied the classical vibrational dynamics of a simple one-dimensional monatomic chain of particles interacting via nearestneighbor harmonic and quartic anharmonic springs, and they used a "rotating wave approximation" (RWA), in which just one frequency component was
144
A.J. Sievers and J.B. Page
Ch. 3
kept in the time dependence. For the case of sufficiently strong positive quartic anharmonicity, it was found that the lattice could sustain stationary localized vibrations having the approximate mode pattem A(..., 0 , - 1 / 2 , 1 , - 1 / 2 , 0 . . . . ), where A is the amplitude of the "central" atom. This pattem is odd under reflection in the central site and is "optic mode"-like, in that adjacent particles move 7r out of phase. As with all nonlinear vibrations, the ILM frequencies are amplitude dependent. The ILMs can be centered on any lattice site, giving rise to a configurational entropy analogous to that for vacancies, and some thermodynamic ramifications were explored in the preceding two references and by Sievers and Takeno (1989), along with speculations about the possible presence of ILMs in strongly anharmonic solids such as solid He and ferroelectrics. A provocative, but still unproven speculation was that these modes might be involved in the anomalous thermally-driven low temperature on-center --+ off-center transition of the nearly unstable KI:Ag + system, which is the other main subject of this chapter. It is straightforward to check the theoretical predictions using molecular dynamics simulations to solve the equations of motion numerically. This of course avoids the RWA and other approximations used by Sievers and Takeno (1988), and fig. 1 shows the results for a 21-particle monatomic chain with periodic boundary conditions and harmonic plus quartic anharmonic nearest-neighbor interactions. Each panel shows the time-evolution of the particle displacements for the same initial displacement pattem, namely that of the predicted ILM pattem given above, centered on the middle particle. The initial velocities were zero. The two panels give the subsequent displacements for the case of (a) purely harmonic interactions and (b) harmonic plus strong quartic anharmonic interactions. As expected, in (a) the initial displacements of the purely harmonic system spread into the lattice, since they are a linear combination of independently evolving homogeneous plane-wave normal modes. In sharp contrast, for the anharmonic system (b), the initial displacement pattem is seen to persist, and the theoretical predictions are strikingly verified. In the few years since the appearance of the 1988 Sievers and Takeno papers, there has been a rapidly increasing number of theoretical studies published related to ILMs. Interestingly, during the preparation of the present chapter, we discovered an earlier, brief paper (Dolgov 1986), which obtains the Sievers-Takeno solution and the analogous even-parity ILM solution, derived independently by Page (1990). The Dolgov paper has gone unnoticed by subsequent workers.
w
Unusual anharmonic local mode systems
145
(a) harmonic 0
o it} o
EL
9
.... J T
T - - _ -
....
-
5 0
r
tO
EL
----------x
10 t 0
~
5
-
~ 10
15
(b) anharmonic 10 rr
._o O
EL to
"s
5 0
-5
EL
-10 0
~i
1'0
15
time (units of 2~'(Om)
Fig. 1. Molecular dynamics simulations for a monatomic chain with (a) purely harmonic and (b) harmonic plus quartic anharmonic nearest-neighbor interactions. For both panels, the initial configuration is the odd-parity ILM displacement pattern A( . . . . 0, - 1/2, 1, - 1/2, 0 . . . . ), with the amplitude on the central particle being 0.1 of the equilibrium nearest-neighbor distance a. The initial velocities are zero. The masses are 39.995 amu, the lattice constant is 1 ,~ and k2 = 10 eV/,~2. The dashed lines in (a) are guides which delineate the spreading of the initial localized displacement pattern into the harmonic lattice. For (b) the value of the anharmonicity parameter A4 = k4A2/k2 is 1.63, well within the range when this ILM should be a valid solution. An implementation of the fifth-order Gear predictor-corrector method was used for the MD runs (Allen and Tildesley 1987), with the time step taken to be 1/180 of the period of the maximum harmonic frequency ~Om. In both panels, the particle displacements are magnified, for clarity.
T h e I L M p a p e r s p u b l i s h e d since 1988 r o u g h l y divide into t w o b r o a d and o v e r l a p p i n g c a t e g o r i e s . T h e first o f these i n v o l v e s the relation o f the n e w excitations to the g e n e r a l b e h a v i o r o f discrete n o n l i n e a r systems, with e m p h a s i s on c o n n e c t i o n s to soliton-like b e h a v i o r in lattices. T h e s e c o n d c a t e g o r y is m o r e tightly f o c u s e d on g e n e r a l i z a t i o n s and e x t e n s i o n s o f the 1988 papers, in o r d e r to d i s c o v e r the p r o p e r t i e s o f h i g h l y - l o c a l i z e d l a r g e - a m p l i t u d e I L M s and also to assess their likely i m p o r t a n c e for real solids. T h e general diffi-
146
A.J. Sievers and J.B. Page
Ch. 3
culty of dealing with complex nonlinear dynamical systems is reflected in the diversity of theoretical approaches used, which range from purely analytic studies to completely numerical molecular dynamics simulations. In our opinion the former should, whenever possible, be accompanied by the latter, since the unfamiliarity of the phenomena can easily lead to unwarranted approximations being made in analytical work. Computer studies of the dynamics of nonlinear lattices extend back to the pioneering work on one-dimensional chains by Fermi, Pasta and Ulam (1955), and they have been a vital adjunct to much of the subsequent work on solitons in anharmonic lattices. Discussions of both analytic and numerical aspects of soliton behavior in one-dimensional lattices with intersite cubic and quartic anharmonic interactions are found in Flytzanis et al. (1985) for the monatomic case and in Pnevmatikos et al. (1986) for the diatomic case. Of particular interest for ILMs is their relationship to lattice envelope solitons. Stationary lattice solitons were studied theoretically more than 20 years ago by Kosevich and Kovalev (1974) for anharmonic chains with cubic and quartic onsite and intersite anharmonicity. The emphasis was on onsite anharmonicity, and solutions were obtained for low-amplitude stationary envelope solitons whose spatial extent is broad compared with the lattice constant. Later MD simulations by Yoshimura and Watanabe (1991) for linear chains with quadratic plus quartic interactions showed that the stationary envelope soliton solutions account well for ILMs whose spatial widths are sufficiently broad, e.g., 10 or more lattice constants. However, as was subsequently emphasized by Kosevich (1993a), the stationary envelope soliton solutions do not describe the large-amplitude highly localized ILMs, such as that of fig. 1. Some recent papers dealing with general aspects of localized dynamics in discrete nonlinear systems and ILM-related soliton studies are: Flach and Willis (1993), Flach et al. (1993), Kivshar and Campbell (1993), Chubykalo and Kivshar (1993), Claude et al. (1993), Kosevich (1993b, c), and Cai et al. (1994). Some representative papers on ILM solutions in lattices with nearest-neighbor intersite anharmonicity and not already cited are: Takeno et al. (1988), Burlakov et al. (1990a, b, d), Kiselev (1990), Takeno (1990), Bickham and Sievers (1991), Takeno and Hori (1991), Takeno (1992), Fischer (1993), Aoki et al. (1993), Chubykalo et al. (1993), and Kiselev et al. (1993). Some representative studies discussing traveling ILMs are given in Takeno and Hori (1990), Bickham et al. (1992), Hori and Takeno (1992), Sandusky et al. (1993b), Bickham et al. (1993) and Kiselev et al. (1994b). A detailed study of the stability of ILMs was given by Sandusky et al. (1992), and the relationship between ILMs and the stability of homogeneous lattice phonon modes has been investigated by Burlakov et al. (1990c), Burlakov and Kiselev (1991), Dauxois and Peyrard (1993), Kivshar (1993),
w
Unusual anharmonic local mode systems
147
Sandusky et al. (1993b), and Sandusky and Page (1994). Energy transport and/or the effects of defects or disorder are discussed by Bourbonnais and Maynard (1990a, b), Takeno and Homma (1991), Kivshar (1991), Kiselev et al. (1994a, b) and Zavt et al. (1993). An important recent development has been the generation of stable ILMs in the presence of cubic anharmonicity (Bickham et al. 1993), and the subsequent inclusion of realistic potentials such as Lennard-Jones, Morse, or Born-Mayer plus Coulomb (Kiselev et al. 1993; Sandusky and Page 1994; Kiselev et al. 1994a, b). A primary effect of including the odd-order potential energy terms is that the ILMs are accompanied by localized DC lattice distortions. It is not our aim here to give a critical assessment of the rapidly growing literature in this area. Rather, we wish to describe some of the important phenomena and results as simply as possible, referring to the literature for details. We will emphasize the work of our own groups, and we will restrict our attention to one-dimensional monatomic or diatomic chains of particles interacting via quadratic, cubic and quartic springs, or via realistic potentials. Throughout, we will stress the phenomena themselves rather than the theoretical methods, which are well-described in the original papers.
0
Experimental and theoretical studies of a thermally anomalous nearly unstable impurity system: KI:Ag +
2.1. Initial experiments
2.1.1. Intrinsic temperature dependent absorption in KI A number of investigators have shown that the far infrared transmission of pure alkali halide crystals at frequencies below the reststrahlen region is controlled by two-phonon difference band energy-conserving absorption processes (Stolen and Dransfeld 1965; Eldridge and Kembry 1973; Eldridge and Staal 1977; Hardy and Karo 1982). The probability of absorption of a phonon is proportional to the phonon occupation number n~ and that of emission of a phonon is proportional to (1 + n~). For a two-phonon difference process in the absorption spectrum, the absorption of a photon is accompanied by the emission of one phonon (1 + n l) and the absorption of another phonon n2, giving a temperature-dependent factor (1 + n l)n2. The net absorption is obtained by correcting for spontaneous photon emission by subtracting the reverse process, namely (1 + n2)nl, so that the resulting temperature dependence is proportional to In2- nl I. Since thermal phonons are required to generate such an absorption process, pure crystals are extremely transparent at low temperatures and low frequencies, far from the
A.J. Sievers and J.B. Page
148
Ch. 3
reststrahlen band. However, since we are interested in examining the temperature dependence of defect induced absorption, these temperature-dependent properties of the pure crystal need to be quantified as well. As long as the sample does not have parallel sides, the absorption coefficient c~(w) is determined from the transmission expression It --
(1 =
-
R) 2
exp[-c~(w)/]
=
t(w)
Io
,
1 --
R 2
( 2 . 1 a)
exp[-2o~(w)/]
where the transmission coefficient t(w) is defined as the ratio of the transmitted to the incident intensity, R is the reflectivity coefficient and l is the sample length. When R is small and/or ~(w)l is large, the second term in the denominator can be ignored and eq. (2.1a) reduces to
t(w) ~ (1
-
R) 2
exp[-a(w)/].
(2.1b)
Figure 2 shows the temperature-dependent absorption coefficient a(w) for KI over a wide range by using three different ordinate scales versus frequency (Love et al. 1989). For the samples used here, the frequency and temperature-dependent absorption coefficient a(w, T) is obtained by dividing the spectra taken at the higher temperature by one at 4.2 K. From eq. (2.1b) this gives 1
c~(w, T) - c~(w,4.2 K) = -7 In[It(w, T)/It(w, 4.2 K)].
(2.2)
l
In fig. 2(a), (b) the absorption coefficient at 4.2 K is small enough on both of these scales that it can be ignored, hence these data give the temperature dependence of the difference band absorption coefficient directly. The observed spectral structure agrees with that predicted by early workers, including the weak difference band contribution at 11.3 cm -1 which stems from vertical transitions between the s163 symmetry branches in the frequency region where the dispersion curve slopes are nearly parallel. The sharp edge at 31 cm -1 in fig. 2(c) is due to relatively strong ZI(LA)-Z4(TO) two-phonon difference band transitions across the acousticoptic phonon gap.
2.1.2. Resonant mode IR absorption Because of the freezing out of the intrinsic difference band absorption processes for frequencies below the reststrahlen region in alkali halides at
w
Unusual anharmonic local mode systems i
i
i
i
I
I
. (0)
I
I
/82K
40-
-
.
/64K
20
~/47'
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,
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(b)
E
2
149
55K
5K
8
C
._~
-
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/
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e--
O .IQ
<
S
O
(c) 1.6-
82K
0.8-
0
20
40 Frequency
60
80
( c m -a)
Fig. 2. Temperature dependence of the absorption coefficient of pure KI at different temperatures. The sample temperature is varied from 4.2 to 82 K. The spectral resolution is 0.44 cm-1 for frequencies between 3 and 26 cm -1 , and is 0.35 cm -1 at higher frequencies. The spectral features are produced by two-phonon energy-conserving difference band processes. (After Love et al. 1989).
low temperatures, transmission measurements on impurity-doped crystals at p u m p e d liquid helium temperatures give directly the impurity induced absorption coefficient c~i(w). The Ag + ion was the first impurity to be studied at far infrared frequencies in these hosts below the reststrahen region (Sievers 1964; Weber 1964). A simple method in which the lattice-constant
A.J. Sievers and J.B. Page
150
Ch. 3
dependence of an impurity resonant mode can be seen is to plot the log of the observed mode frequency for a fixed impurity ion versus the log of the host lattice constant. This is presented for Ag + in fig. 2. The straight line through all but one of these frequencies indicates that this defect resonant mode obeys a Mollwo-Ivey-like rule (Hayes and Stoneham 1985) up to at least the large lattice constant region (Sievers 1968). This straight line implies that the dependence of the frequency upon the lattice constant can be written as
0 lnw 01na
3A = w '
(2.3)
where a is the lattice constant, w is the resonant frequency and A is the hydrostatic coupling coefficient. For five of the alkali halide hosts in fig. 3, the quantity 3A/w = 3; however, KI:Ag + is seen to give a much lower frequency than that projected from this straight line. The A coefficient of KI:Ag + can be estimated from uniaxial stress measurements, with the result that the quantity 3A/w in eq. 2.3, is equal to 68. This corresponds to a very different slope, as shown by the line at the KI:Ag + point in fig. 3. It is perhaps fortuitous that the resonant mode for KBr falls on the 3A/w = 3
! 601
~
I
I- oc,
I
,o F
I >'351
"-"
~,,~No I
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~ 3o
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. I
I
,
3.2s LATTICE CONSTANT(A) 3.00
,
t KI 3.50
Fig. 3. Center frequency of the Ag+ resonant mode for different alkali halide crystals as a function of the host lattice constant. For five different alkali halide lattices the resonant mode frequency can be fit to a straight line. The line centered at the KI:Ag+ point is from uniaxial stress measurements. (After Sievers 1968).
w
Unusual anharmonic local mode systems
151
line, since RbC1, which has nearly the same lattice constant, is an off-center system (Kirby et al. 1970). [The multiple absorption bands associated with excited tunneling transitions in RbCI:Ag + (Barker and Sievers 1975) are not included in fig. 3]. The resonant mode absorption bands associated with the Ag + ion, excluding KI, are found to be relatively temperature independent, at least for thermal energies up to the resonant mode energy. One possible reason that the KI:Ag + resonant mode frequency does not follow the 3A/w = 3 line in fig. 3 is that KI:Ag + might not be an on-center defect system. This possibility has been eliminated by studying the uniaxial stress dependence of this absorption line. A typical splitting for stress applied in the [100] direction for two different polarizations of the radiation is shown in fig. 4. The area under each of the three curves appears to be roughly the same. By making such measurements for uniaxial stress along the [100], [110] and [111] crystal directions, it is possible to distinguish between a frozen off-center system, a tunneling system or an on-center system (Nolt and Sievers 1968). From the number of components observed and the absence of any stress-induced dichroism, it can be concluded that the Ag + defect has either Oh or Td symmetry. In addition, these measurements provide experimental values for the three even-symmetry stress coupling coefficients between the resonant mode and the lattice: A(hydrostatic), B(tetragonal), and C(trigonal). 2
v
I
i
!
I
q
I
.--,.
.
E
~' [o~o]
I
'
I
K I : Ag
4-
(b)
[I 00]
Stress
Zero Stress
(J
---I I'--
r
._e .u_ ' r. - I
(o)
/
0
(.)
~' Doo]
\ \ 0 I
I 16
,
n
IB Frequency
20
J22
(cm"l)
Fig. 4. KI:Ag + absorption coefficient for a [100] stress of 3.3 kg/mm 2. The sample temperature is 1.5 K. The instrumental resolution is 0.3 cm -1. The polarizations are identified in the figure. The integrated line strength is 1.03, 1.23 and 1.1 4-0.1 cm -2 for curves (a), (b) and (c), respectively. (After Nolt and Sievers 1968).
A.J. Sievers and J.B. Page
152
Ch. 3
Electric field measurements on the IR active resonant mode confirmed that the defect had Oh symmetry but in addition produced surprises (Kirby 1971). The electric field induced absorption coefficient for KI:Ag + is shown in fig. 5. The negative Aa region identifies components associated with the zero-field spectrum that have changed. The complex difference spectrum in the resonant mode region arises from the Tlu mode at 17.3 cm -1 mixing with an even mode of Eg symmetry at 16.35 cm -1. These measurements also show another E-field induced line at 25 cm -1, which has been interpreted as an A lg symmetry mode. In addition, absorption features at larger frequencies, namely 30 and 44 cm -1, decrease in strength with applied Efield. The presence of the Eg symmetry mode was confirmed by Raman scattering studies, but no trace of the Alg mode was found. An example of the temperature-independent behavior found for the Ag + doped hosts that obey the Mollwo-Ivey-like rule is seen in fig. 6, which presents the spectrum of NaI:Ag + at 1.2 and 24.2 K (Greene 1984). At the higher temperature the resonant mode centered at 36.7 cm -1 appears on a two-phonon difference-band background spectrum. The resonant mode strength itself is expected to be temperature independent if the mode behaves as a harmonic oscillator. From the Kramers-Kronig dispersion relations, a general f sum rule for the absorption coefficient can be found (Smith 1985):
cx3
f0
71"~ dw c~i(w) = 2c = constant,
(2.4a)
where the exact expression for the plasma frequency squared or oscillator strength depends on the specific model employed. As long as the magnitude of the effective charge which produces the absorption coefficient is temperature independent and the integral is taken over the complete IR absorption spectrum, then the above sum rule holds. In particular, it applies to anharmonic as well as to harmonic oscillators. When anharmonicity is included the transition of interest may shift, broaden and change in strength, but the anharmonicity will also produce other lines in the spectrum (e.g. sidebands, overtones) which will ensure the constant sum rule (Bilz et al. 1984). A specific form of eq. (2.4a), appropriate to impurity-induced absorption over a frequency interval encompassing a line for the case of a host with dielectric constant e0 is (Sievers 1964)
dw Oq(W)
--
2c
=
CMQx/C~
3
'
(2.4b)
A.J. Sievers and J.B. Page
154 32 F
KBr:Li//'~
30~
/''o jw " ~ ' ~ l :cu "//~'~~
-'" 281 o ' E 26
~
P
~' 241-
," 18e~
I I
A
/o
_.I
i6~- ,,e~,~ 14~"~ 0
Ch. 3
~
~h--e-.-14
I , 2 Pressure,
I 3
,
I I 4 in kilobars
CsI : TI I 5
i
I 6
I
I 7
I
Fig. 7. Hydrostatic pressure-induced shift in the resonant mode frequency of four different resonant mode systems. The experimental uncertainties are +0.1 cm-] in the resonant mode frequency and -t-10% in the pressure. The sample temperature is 4.2 K. (After Patterson 1973). where e* is the effective charge and MQ is an effective mass associated with the mode. The fact that the resonant mode shown in fig. 6 is nearly temperature independent indicates that this defect system is not anharmonically coupled to other degrees of freedom, for if it were the area under the resonant mode transition would decrease with increasing temperature and the area difference would transfer into its sidebands such that the total area remained constant (Klein 1968; Barker and Sievers 1975). Hydrostatic pressure measurements up to 7 kbars on the resonant mode feature provide additional evidence that the simple absorption spectrum is not associated with a tunneling defect, since none of the complex spectral behavior previously found for the tunneling system KCI:Li + was observed here (Kahan et al. 1976). Figure 7 shows the hydrostatic frequency shifts observed for different on-center defect systems (Patterson 1973). For all of these systems only one IR active resonant mode is found at each pressure. One curious feature is that the effective resonant mode force constant for KI:Ag + varies nonlinearly with strain (for Aa//a up to 1.6%), while all other low lying resonant modes show a linear dependence over the same range. Figure 8 shows the resonant mode absorption spectrum c~(w) for a KI:Ag + sample at atmospheric pressure and at 1.9 kbars, respectively, as measured with respect to an empty pressure cell (Patterson 1973). Because of the different transmission charactistics of the empty cell, the ordinate gives the absorption coefficient in arbitrary units; however, since the absorption spectra at the two pressures were measured sequentially, the absorption strengths can be compared. Within experimental accuracy the area under the line
w
Unusual anharmonic local mode systems I
........
'
I
'
! 0.3 . - .
I
o.~
'
KI:Ag+ 175 kV/cm 4.2OK -'11"-
1
0.2
I
153
-
-
j
J
tJ
o
/'-"~_/~
'"
-o.i -0.2
!
IO
i
20
I
t
I
=
30 40 FREQUENCY ( cm-I)
~
-
50
Fig. 5. Electric-field-induced absorption coefficient for KI:Ag +. A negative Aa indicates a decrease in absorption with applied field, while a positive Aa indicates a field-induced increase. The complex line shape in the 17 cm -1 region indicates that two resonant modes of opposite parity are being mixed by the odd-parity electric field. Here Ei~llEdcll[100], and the instrumental resolution is 0.55 cm -1. (After Kirby 1971).
3.0
!
NaI +
A
3E v
0.2 %
I
AgI
..... 1 . 2 K
........ 24.2 K
O
..,./-
,.e...
,-" 2 . 0
.~. r
-
NI,-
O (..) C:
.9 .a,...
1.0
O. i,_ 0 U') ,JO
<:[ 0 ,, 20
I
30 Frequency
1
40
50
(cm -I)
Fig. 6. Temperature dependent Ag+-induced absorption in the resonant mode frequency region of NaI. The strength of the resonant mode centered at 36.7 cm -1 is relatively temperature independent, while the two phonon difference band absorption clearly increases between the two temperatures shown. The instrumental resolution is 0.52 cm -1. (After Greene 1984).
w
Unusual anharmonic local mode systems
r 0 "On O. t,. 0 r JO
9
155
~
9 00 4~ OIbO0000 O0 O0
O0
Oo~
Ollo
| OOO O
13
I, 15
1 7 3 c m -I
216cm -t
I 17
I 21
I 19
Frequency,
I 25
co
I 25
I 27
I 29
(cm -I)
Fig. 8. Strength of the KI:Ag + resonant mode at two different hydrostatic pressures. The pressures are 0 and 1.9 kbars, respectively. (After Patterson 1973).
appears to scale with frequency over this pressure range. This pressuredependent area is a surprising result since according to eq. (2.4a, b) for a single absorption line the area should remain fixed, independent of frequency (ignoring the small decrease in e0 with increased pressure). The uniaxial stress measurements, admittedly at lower pressures, did not appear to give such an effect. Although the eigenvector of the resonant mode would be expected to change with hydrostatic pressure, the magnitude of the effective charges in eq. (2.4b) are expected to remain fixed over this pressure range. Clearly, the possibility of this pressure-dependent effect should be examined with more experimental precision. 2.1.3. Anomalous temperature dependence of the resonant and gap modes
Figure 9 clearly indicates that the temperature dependence of the KI:Ag + resonant mode strength is qualitatively different from that observed for the other Ag + resonant mode systems. The strength of the mode decreases rapidly with increasing temperature until by 21.6 K it seems to have disappeared into a slightly larger background spectrum (Sievers and Greene 1984). This temperature dependence of the line strength is independent of the Ag + concentration. Notice that between 1.2 and 9.3 K where about half of the strength has been lost, the center frequency of the mode remains nearly unchanged.
A.J. Sievers and J.B. Page
156
I
1.0
I
I
Ch. 3 I
_ KI" AgI
E O.8 1.2 K
"5 0.6
--it-
U
~,
0.4
9.3 K
o
<~ 0.2
f2,.6 K ,,~..~.,.....~...~.~....,r I0
15 Frequency
20 (cm "l)
25
Fig. 9. Temperature dependence of the absorption spectrum of KI + 0.2 mole% AgI in the frequency region of the T]u resonant mode. The three temperatures are identified in the figure. At high temperatures the broad nonresonant absorption coefficient is about 3% of the low temperature resonant mode peak value. (After Sievers and Greene 1984). For emphasis it is helpful to contrast this unusual temperature dependence of the resonant mode strength with the temperature-independent properties of a related defect system. Each of the five temperature-independent Ag + induced resonant mode frequencies in fig. 3, when normalized to the appropriate host lattice Debye frequency, is much larger than observed for KI:Ag + (Wr/WD = 17.3/91 = 0.19). Perhaps below a certain value of this ratio a temperature-dependent strength occurs? The NaCI:Cu + resonant mode (Weber and Nette 1966) centered at 23.5 cm -1 provides a good test of this idea since now the ratio Wr/WD is 23.5/223 = 0.11. The temperature dependence of this mode's absorption is shown in fig. 10, where the temperature ranges from 4.2 K for curve (a) up to 77 K for curve (d). The resonant mode peak can easily be seen at each temperature. The center frequency and the linewidth are temperature dependent, as expected for a slightly anharmonic oscillator, but for at least the two lowest temperatures the strength appears to increase slightly with increasing temperature 8(21.5 K) /S(4.2 K) = 1.15. Because of the large width at still higher temperatures it is difficult to separate unambiguously the resonant mode and the two-phonon difference band absorption. At 42.6 K (curve c) the intrinsic two-phonon difference band absorption at the resonant mode center frequency is only 0.25 cm -1 (Love et al. 1989), compared to the resonant mode peak value of
w
Unusual anharmonic local mode systems 8
'i .........
157
I
._
89
caL , , "
:
:"lt
j
'G
~ u 0
9
.o
(b) 0
**
(a) .
!o
.
.
.
.
.
.
.
.
.
.
.
, i , ,
20
5o Frequency (crn-a)
|
40
Fig. 10. Temperature dependence of the absorption coefficient of NaCI+0.5 mole% CuC1 in the frequency region of the resonant mode. The four temperatures are (a) 4.2 K; (b) 21.5 K; (c) 42.6 K and (d) 77 K. At high temperatures the resonant mode resides on top of the intrinsic two phonon difference band absorption. The resolution is 0.43 cm-I . (After Sievers, unpublished). about 4 cm-1 but it is nevertheless clearly observable on the high frequency of the line in fig. 10. The unusual temperature-induced absorption strength change of the KI:Ag + resonant mode can be examined in a number of different ways to separate out the different contributions to the spectrum. One of these is to examine the strength change over AT jumps. Such changes can be seen with precision in fig. 11, where the absorption coefficient difference between neighboring temperatures as a function of frequency is presented (Greene 1984). The largest strength changes are seen to occur in the 6 K to 13 K region. Compared to the rapid resonant mode strength change with temperature, the line broadening and center frequency shift in this representation are again seen to be quite small. The broadening and shift also appear comparable to those measured in fig. 10 for the resonant mode of NaCI:Cu + at the two lowest temperatures. This strength change was first observed in 1965, with less precision (Takeno and Sievers 1965). In that work it was proposed that the resonant mode, through anharmonicity, is linearly coupled to other low-lying phonon
A.J. Sievers and J.B. Page
158
:(a)
Ch. 3
; all.2)-a(2.91
I I I
/,,%.~A
__1_
A -~'-
-]
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; ~
:
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9
:~
-
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l
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-
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a (10.9)'-a 3.7)ll (I .,,..,~
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-
A i!
IIII1~ ~ l l
Ii
. . . . . . . . . . . . . .
V
] i I
r
t I
~
(2(18.1)-Cl(21.6)
I
I
I
!
I
I
,
1
~5
20
I 0 - ~
I /~
L~-'"-,,~J_
-'
I
.
, I
g
, I
~o
.....
Frequency
I ....
--
I :
. v-
l
(cm -I)
Fig. 11. Absorption coefficient differences between adjacent temperatures, vs. frequency for KI:Ag +. The instrumental resolution is 0.47 cm -]. (After Greene 1984).
modes. Such a coupling could remove strength from the "zero phononlike" resonant mode transition without contributing to the temperature dependence of the center frequency and linewidth. The temperature-dependent strength data fit a Debye-Waller-like factor with a T 2 in the exponent. If the sidebands extended over a large enough frequency interval they would be negligible in the experimental measurement. The temperature dependence produced by such a Debye-Waller-like factor has the form
I(T) = exp
-
SO~ 2
1+ 4
~
JO
e :~ -- 1
'
(2.5)
where S'0 = 9W2c/4Mv2hw 3 contains an acoustic spectrum cutoff frequency We which is less than the Debye frequency coD. Here 0c = hwc/kB, v is
w
Unusual anharmonic local mode systems
159
the Debye sound velocity, and M is the average ion mass appropriate to the Debye modes. For Oh symmetry the effective strain coupling parameter A is determined by the three coupling coefficients A(hydrostatic), B(tetragonal), C(trigonal) for the even-parity symmetry types. Note that it is only at very low temperatures that the T 2 behavior contributes to the exponent. When the measured strain coupling coefficients from the uniaxial stress measurements are used to determine A, the value is an order of magnitude too small to agree with the temperature dependent strength data (Alexander et al. 1970). Besides linear coupling between a resonant mode and a Debye spectrum, Alexander et al. (1970) also attempted to fit the temperature dependence of the absorption strength with a linear coupling to a single even-parity resonant mode of frequency DE. In this case the intensity of the zero phonon line is given by I(T) = e-'YEIo(CE),
(2.6)
where 7E = SE(2~E + 1)
(2.7)
CE = SEcsch(hf2E/2kBT).
(2.8)
and
Because Sz is now of order unity instead of N -1, where N is the number of atoms as for extended lattice modes, one must include the modified Bessel function Io(Co) in the modulation analysis. One of the conclusions of the work of Alexander et al. (1970) was that eq. (2.6) provides a much better fit to the observed temperature dependence than does eq. (2.5), given the known strain coupling coefficients. This work, together with the experimental work of Kirby (1971) on the presence of both odd and even-parity resonant modes in the KI:Ag + spectrum, seemed to indicate that the vibrational properties of point defects were indeed well understood. But it was then discovered by Greene (1984) in the early 1980's that the reported good agreement between the linearly coupled odd and even mode analysis of the temperature-dependent spectrum was in fact due to an erroneous numerical calculation. No set of model parameters in eqs (2.6), (2.7) and (2.8) can produce agreement with the observed temperature dependence of the KI:Ag + resonant mode. In retrospect, it is now easy to see why the linear coupling model cannot explain the observed temperature dependence. For linear coupling to either extended band modes or to localized modes, the temperature
160
A.J. Sievers and J.B. Page
Ch. 3
dependence in the exponent comes from a (2n + 1) occupation factor. At temperatures large compared to the characteristic modulation frequency of the phonons or the low lying even mode this term must vary as kBT which, even in the exponent, is too slow to account for the rapid disappearance of the line strength by 25 K. When the E-field dependence of the resonant mode was investigated by Kirby (1971), other silver-induced features in the absorption spectrum were found, most notably a gap mode centered at 86.2 cm -l but also other much weaker but identifiable peaks located at 29.8, 44.4, 51.0, 55.9 and 63.6 cm -1. The interconnection between these different features remained unexplored until temperature-dependence measurements on the complete impurity-induced spectrum were made (Greene 1984; Sievers and Greene 1984). Some key features can be identified in the Ag+-induced absorption spectra shown in fig. 12. The solid curve in fig. 12(a) shows the impurity-induced absorption coefficient c~(w) versus frequency at 1.2 K. The assignment of the different features is the same as that given earlier by Kirby (1971). One surprising result is that the entire impurity-induced spectrum is very temperature dependent, as shown by the dotted absorption curve for 11 K in fig. 12(a). With increasing temperature the strength of the resonant mode, combination bands, and gap mode all decrease, while a broad absorption band associated with intrinsic two-phonon difference band processes increases at high temperatures. Precise measurements of the temperature dependence of the absorption coefficient in the low-frequency region around the resonant mode show that the decrease in the resonant mode strength is accompanied by a corresponding increase in a broad absorption which appears to be nearly frequency independent from the smallest wave number measured (3 cm -1) up to at least 25 cm -1. The magnitude of this non-resonant absorption is proportional to the low temperature strength of the IR active resonant mode over a factor 40 in Ag + concentration (Sievers and Greene 1984). There are other spectral features which appear at elevated temperatures, as can be seen in fig. 12(b), where the difference in absorption coefficient Ac~(w) between a given temperature and the 1.2 K reference temperature is plotted versus frequency. The solid curve is for T = 3.4 K, and the dashed one is for 10 K. A positive Aa(w) in this figure indicates that the sample absorbs more at high temperatures than at 1.2 K at that frequency. In addition to the nonresonant absorption (n) at low frequencies there are two additional new impurity-induced features which appear at elevated temperatures, namely a density of states peak (d) at 69 cm -1, and a gap mode (g) at 78.6 cm -1. Note that the KI phonon gap extends from 70 to 96 cm -1. The evolution of the excited state transitions with temperature can be seen more clearly in fig. 13, which gives absorption difference traces at a number of temperatures
A.J. Sievers and J.B. Page
162
0.6
----r
~
!
!
Ch. 3
r'""'T'-
q
/ 0.4 IE r
<1
0.2
-
s TK
30
J / !
50 70 Frequency (era-I)
Fig. 13. Temperature dependence study of the excited-state transitions for KI:Ag+. The ordinate shows the absorption coefficient difference between the temperature given in the figure and 1.2 K. The two features at 69 and 78.6 cm-1 are seen to grow with increasing temperature, as does the intrinsic two phonon difference band absorption. (After Greene 1984). (Greene 1984). All three of the high-temperature absorption processes (n, d and g) have similar temperature dependences: they are not observable at 1.2 K and grow in magnitude with increasing temperature. The common low-temperature behavior indicates that all of these absorptions may stem from the same elevated energy state. An important experimental finding is that the temperature dependence of the two strong modes, labeled r and g in fig. 12(a) are essentially the same. The strength I(T) of the IR active gap or resonant mode at a particular temperature T is given by eq. (2.4a, b), but with the frequency integration only across the line itself. The temperature dependence of the measured strength I(T) of each line is plotted in fig. 14(a). The growth of the excited state gap mode at 78.6 cm-1 is shown in fig. 14(b). The strength of this "hot" band is normalized to the low temperature strength of the gap mode at 86.2 cm -1. At first sight it might appear that these temperature dependences could follow from simple Boltzmann population effects for anharmonic oscillators, but this is not the case. One of the simplest explanations would be to ascribe the temperature dependence to the anharmonicity of the resonant mode itself, treated as an independent oscillator. If the 0 --+ 1 transition does not coincide with the 1 --+ 2 and higher transitions then only the 0 --+ 1 transition contributes to the unshifted resonant mode transition, and its intensity would decrease as the ground state is thermally depopulated (Alexander et al. 1970). Assuming that the anharmonicity is not too large
w
Unusual anharmonic local mode systems
161
Fig. 12. Temperature dependence of the Ag+-induced absorption spectrum in KI. The vertical dot-dashed lines divide the figure into three parts: The concentration in the center region (2 • 1018 Ag+/cm 3) is twice that of the two end regions. Assignment of the different features: r = resonant mode, c = combination band, d = density of states peak, g = gap mode, and n = nonresonant absorption. (a) The absorption coefficient vs. frequency for two temperatures: solid curve, 1.2 K; dashed curve, 10 K. (b) absorption coefficient difference between a given temperature and 1.2 K vs. frequency for two temperatures: solid curve, 3.4 K; dashed curve, 10 K. Note that the ordinate in the center region of (a) is expanded 5 times. (After Sievers and Greene 1984).
w
Unusual anharmonic local mode systems
163
Fig. 14. Normalized absorption strength vs. temperature for the Tlu resonant and gap modes in KI:Ag +. (a) Gap and resonant mode data for a number of Ag + concentrations. Solid and dashed curves are the predictions of the three dimensional double anharmonic oscillator model. The dotted curve, which is 1 minus the nonresonant growth curve data, is taken at a frequency of 4 cm -1. (b) Excited-state gap mode data. The solid curve shows the predicted temperature dependence of the excited-state strength for the same anharmonic model as used in (a). (After Sievers and Greene 1984). so that the h a r m o n i c oscillator partition functions can still be used, then for a t h r e e - d i m e n s i o n a l oscillator in w h i c h only o n e e x c i t e d state is i m p o r t a n t for e a c h polarization, o n e has
I(T)/I(O) - (1 - e - h l 2 r / k T ) 4
_ Zr4
(2.9)
w h e r e Zr is the partition function of a o n e - d i m e n s i o n a l oscillator. We n o w s u p p o s e that there is a t h r e e - d i m e n s i o n a l d o u b l e a n h a r m o n i c oscillator with e x c i t a t i o n n u m b e r s labeled (g,r). T h e t e m p e r a t u r e - d e p e n d e n t intensity o f two transitions are o f interest:
A.J. Sievers and J.B. Page
164
Ch. 3
(a) resonant mode (0, 0) ~ (0, 1) with intensity -,~ (Zg)-3(Zr) -4, (b) gap mode (0, 0) ~ (1,0) with intensity ~ (Zg)-n(Zr) -3. The dashed curve in fig. 14(a) gives the predicted temperature dependence for the gap mode and the solid curve gives the predicted dependence for the resonant mode. The solid line in fig. 14(b) shows the corresponding prediction for the excited state gap mode (0, 1) ~ (1, 1) transition. In each case the temperature dependence is not fast enough to fit the experimental data. Another missing feature required by this model at elevated temperatures is the (0, 1) ~ (0, 2) transition. Figure 15 shows that such a feature does not appear in the spectrum; instead the high temperature component consists of nonresonant absorption over the entire frequency region (Greene 1984). Here the difference between the absorption coefficient at the temperatures shown and that at 20.6 K is plotted versus temperature. Since the nonresonant absorption is not present at 1.2 K and the resonant mode absorption is essentially gone by 20.6 K, the lowest spectrum in fig. 15, cff l.2 K ) - c~(20.6 K), shows both the resonant mode and the nonresonant absorption components fully developed. 1
I
T
13.7 K
\
o
U
'
0
0 5
I0
15 20 Frequency (cm-I)
25
Fig. 15. Temperature-induced change in the absorption coefficient for KI:Ag+. Ac~ = c~(T) - c~(20.6 K) and T for each trace are given. Both the resonant mode and nonresonant absorption are evident in the lowest trace. The spectral resolution is 0.43 cm-1. (After Sievers et al. 1984).
Unusual anharmonic local mode systems
w
9
0.6
9
9
0..0~0
165 1.0
O0
A
o, 0.4
A
w, to to
/
A
e-
0.5 ~ , I--v
/"
_
i-
i:I
/"
0.2
,L" 0
-.,~-.~'~
0
I
I
I0
I
20
I
i
30
Temperoture (K) Fig. 16. Normalized nonresonant far IR absorption coefficient for KI:Ag + at 4 cm -1 vs. temperature (solid curve). The normalization factor is the high temperature value. Also shown (dots) is the measured temperature dependence of T A 1 defined later by eq. (2.15) in the text. This quantity is proportional to the population in the off-center configuration. The initial increase of the data with temperature follows an energy gap law with a gap of 24 K. (After Hearon and Sievers 1984).
In order to understand better the role of the nonresonant absorption, its contribution was examined at a frequency far removed from the resonant mode itself. A broad band millimeter wave spectrum with intensity maximum centered near 4 cm-] and a full width at half maximum of 4 cm-1 was used to measure the nonresonant absorption growth curves as a function of temperature. The experimental results normalized to the high temperature data are represented by the solid line shown in fig. 16 (the ordinate for this trace is on the right). By subtraction of the normalized growth curve from unity, the data can be compared directly with the temperature dependence of the resonant and gap mode strengths. These data are represented by the dotted curve in fig. 14 (a). It appears that the strength lost by the gap and resonant modes with increasing temperature is transferred to the nonresonant absorption (Sievers et al. 1984).
2.1.4. Two elastic configuration model When the unusual strong temperature dependent properties of KI:Ag + are contrasted with the temperature-independent properties of NaI:Ag +, it seems clear that the original Lifshitz approach (1956) would have qualitative difficulties in serving as a basis to explain these temperature dependent features,
A.J. Sievers and J.B. Page
166
I cz
Cl
I
[ i|
Ch. 3
ii
I
I
I
I I I d g
I
I I
I I I
i
i
I 0
I
Fig. 17. Schematic representation of the two configuration model. C1 identifies the on-center elastic configuration, with only the gap mode transition shown. C2 shows the higher energy (~) off-center elastic configuration with the excited state density of states and the gap mode transitions. One possible source for the relaxation time r is shown.
since the same impurity is surrounded by the same nearest neighbors in two alkali halides having the same symmetry. To describe in a phenomenological manner the temperature-dependent data, it was proposed that under certain circumstances a second elastic configuration could occur in addition to the ground state configuration of the defect-lattice system (Sievers and Greene 1984). Figure 17 illustrates such a possible two-configuration arrangement. Here C] identifies the ground state on-center configuration and C2 the offcenter one at a higher energy ~. A few of the assignments of the observed transitions are also shown. The rapid temperature dependence observed for KI:Ag + requires that the second configuration contain a low-lying multiplet with a large degeneracy in comparison to the on-center configuration, perhaps related to the displacive tunneling of the Ag + defect among many equivalent minima centered about a normal lattice site. This possibility is not so unusual, since a twelve-component spectrum with a tunneling splitting of ,,~ 0.1 cm -1 has been identified for the normal ground state configuration of Ag + in RbC1 (Kirby et al. 1970; Kapphan and Luty 1972; Holland and Luty 1979). As long as the thermal and electromagnetic energies are much larger than the tunnel splitting, the frequency dependence of the absorption coefficient for such an arrangement could have the simple Debye form considered below.
w
Unusual anharmonic local mode systems
167
The Clausius-Mossotti equation for a medium with intrinsic dielectric constant eo that contains Ni polarizable defects per cm 3 is (Hearon and Sievers 1984) A
~'- 1 ~"+ 2
=
eo + , 4 - 1 eo + ,~ + 2
=
eo- 1
47rNi~i
)
eo + 2
,
(2.10)
3
A
where ,4 is the defect-induced contribution to the complex dielectric constant and ~i is the dipolar polarizability of the defect. In the limit of a small offcenter concentration Ni, the impurity-induced dielectric constant becomes (Sievers et al. 1984)
Z~ = 47rNi~i ( e~ + 2 ) 2.
(2.11)
3
In the Debye approximation, the dielectric response at frequency w is
3 = A 1 -I--iA2 -- A0(1 - iw'r)- 1,
(2.12)
where
A0 = 4"n'Ni ~
3
'
(2.13)
p is the dipole moment, k is Boltzmann's constant, T is the temperature and ~- is the relaxation time. According to eq. (2.12) for the far IR limit (wT >> 1), the nonresonant absorption coefficient is
O~n
CV~
~
,
(2.14)
which is independent of frequency and is proportional to the number density of off-center ions. If the Debye model describes the nonresonant absorption data, then in addition to the far IR signature in the temperature-dependent absorption coefficient there should also be a radio frequency signature in the real part of the impurity-induced dielectric function.
A.J. Sievers and J.B. Page
168
Ch. 3
2.1.5. Radio frequency dielectric constant measurements Measurements of the real and imaginary parts of the dielectric function at 10 kHz as a function of temperature are shown in fig. 18 (Hearon and Sievers 1984). An important observed feature of these measurements is that only the real part of the dielectric function is temperature dependent; no temperature dependence is found for the imaginary part. Although e l ( T ) - ~1(1.4 K) shows a maximum at about 10 K, no corresponding loss peak is observed in e2(T). For ground state tunneling systems such as RbCI:Ag +, a peak in el (T) as a function of temperature is always accompanied by a concomitant peak in e2(T) as r(T) is swept through the wr = 1 condition with increasing temperature (Holland and Luty 1979); moreover, the radio frequency spectra shown in fig. 18 are independent of excitation frequency w. Both of these results indicate that wr << 1 over the entire temperature region of these KI:Ag + measurements. Equations (2.12) and (2.13) show that in this limit
A I = A oo(
Ni(T) T
and
A2=0,
0.04
(2.15)
I
z~
0.02 I
x~ x
F-x
X X
0.00 0
20
40
Temperature (K) Fig. 18. Temperature dependence of the impurity-induced dielectric constant for KI:Ag + as obtained from radio frequency measurements, el ( T ) - e1(1.4 K) is plotted vs. temperature for the single-crystal samples of 0.5 and 0.2 mole% AgI in KI in the melt (triangles and crosses respectively.) Also shown is the temperature dependence of e o ( T ) - e0(1.4 K) for pure KI (the z's). e 2 ( T ) is temperature independent and has the same value for all three samples (,-~ 1.5 x 10-3). The measuring frequency is 10 kHz. (After Hearon and Sievers 1984).
w
Unusual anharmonic local mode systems
169
so that the temperature dependence measurement of the real part of the dielectric function provides a direct determination of the population in the off-center configuration. The temperature dependence of the population in this second configuration is found as follows. The data in fig. 18 for the 0.5 mole% AgI are subtracted from those for pure KI, the result is multiplied by T and plotted vs. temperature, resulting in fig. 16. At high temperatures the data points in this figure indicate that the off-center population approaches a near constant value. Information about the reorientational relaxation time for Ag + in this offcenter state can be obtained by combining the experimental results for the dipole moment from the radio-frequency dielectric constant measurement with the measured temperature dependence of the nonresonant absorption coefficient c~n in the far IR spectral region. Inspection of fig. 16 shows that the far IR nonresonant absorption data follow Ni(T), while according to eq. (2.14) the data should vary as Ni(T)/[TT(T)]. The consequence is that TT(T) -= constant. These radio-frequency measurements appear to support the picture that at least two elastic configurations exist for this lattice defect system. At low temperature the Ag + ion is on-center with its associated on-center spectral features. When the temperature is increased, a second elastic configuration becomes populated. This configuration has an energy with respect to the on-center ground state of at least 24 K. The large increase in the defect contribution to the dielectric constant indicates that the Ag + ion is off-center in this second, higher-energy elastic state. Because the off-center population appears to approach a constant value at elevated temperatures, the available data seem to indicate that only these two elastic configurations are close by in energy. The large difference in the temperature-dependent properties of Ag + in the hosts NaI and KI may indicate that low-lying configurations can only occur when the impurity ion is smaller than the host ion it replaces (Na + < Ag + < K + < Rb+). This idea is supported by three different hydrostatic pressure measurements which have been used to transform off-center systems such as RbCI:Ag + (Holland and Luty 1979) and KCI:Li + (Devaty and Sievers 1980) to on-center ones. The almost discontinuous change observed in the impurity ion position at low temperatures and at finite temperatures would be a natural consequence if two different elastic configurations could coexist for a defect lattice system. The apparent transition would occur whenever the pressure-tuned ground states for the two configurations have approximately the same energy, so that direct configurational tunneling becomes possible. The detailed paraelectric resonance studies over a narrow hydrostatic pressure range for Ag + in the RbC1 host provide an experimental probe of the dynamics in this nearly degenerate two-configuration region. It has been
170
A.J. Sievers and J.B. Page
Ch. 3
shown that a modest hydrostatic pressure tunes the levels of what is interpreted as the on-center configuration into the same energy range as the off-center one, so that a microwave transition can be observed between these configurations (Bridges et al. 1983; Bridges and Chow 1985; Bridges and Jost 1988). It is noteworthy that the integrated strength of this transition decreases much more rapidly with increasing temperature than Boltzmann population effects predict (Bridges and Jost 1988). 2.1.6. Microwave absorption measurements
Absorption measurements on KI:Ag + as a function of temperature from 5 K to 300 K have been made at three different frequencies in the microwave region, in order to explore the off-center Debye model in more detail. According to eq. (2.14) the model predicts a very simple result for the absorption coefficient in the limit w~- >> 1. These absorption measurements were made with an oversized nonresonant cavity technique (Hearon 1986) which was developed at the Max Planck Institute in Stuttgart (Kremer 1984; Poglitsch 1984). Backward-wave oscillators are used to span the frequency range from 40 GHz to 160 GHz. In order to approach an isotropic field in the cavity, mode stirrers are used to vary the effective shape of the cavity as a function of time, so that a large number of modes interact with the sample. The sample is surrounded by an unsilvered fused quartz double-walled cold finger which hangs in the cavity, and it is cooled to low temperatures by a continuous flow of cryogenic helium gas. The temperature dependence of the absorption coefficient for samples of KI +0.2 mole% AgI and KI +0.03 mole% AgI are shown in fig. 19. The impurity-induced absorption coefficient is given by the difference between the two temperature dependent curves at 160 GHz and 80 GHz. (The low concentration sample was destroyed before the 40 GHz data set could be completed.) To obtain these results, the detector signal for the cavity with the sample is divided by that without the sample at the same temperature to determine the absorption coefficient for a fixed frequency. The estimated uncertainty in the data is represented by the cross-hatched region around each set of data. The qualitative temperature dependence of the impurityinduced nonresonant absorption (the difference between the two curves in a given panel) for the limit a~- >> 1 can be seen most clearly for the highest frequency data shown in fig. 19 (a). These data, which appear to be temperature independent over much of the temperature interval studied, indicate that Nip2/T~ - is essentially a constant for temperatures large enough to favor the off-center configuration. One possible conclusion is that T~- = constant, just as was found earlier by comparing the far IR and the radio frequency data.
w
Unusual anharmonic local mode systems
0.4-
171
o) f=160
0.2-
0
IE 0.2
-
t
b) f: 80 GHz
I
c) f=40GHz
v
E ~0.1
g := O
a= ,,~
0
0. IO
,/ / ,~/ ,,,'/',, / /j...~
0.0S
0 I
0
I00 200 Temperoture (K)
300
Fig. 19. Temperature dependence of the impurity-induced microwave absorption coefficient of KI:Ag + at three different frequencies. (a) 160 GHz (5.33 cm-]), (b) 80 GHz (2.67 cm - ] ) and (c) 40 GHz (1.33 cm-1). The upper curve in each panel is for KI + 0.2 mole% AgI and the lower ones in frames (a) and (b) are for KI + 0.03 mole% AgI. The estimated errors for the oversized microwave cavity technique are identified by the cross hatched regions in each frame. (After Hearon 1986).
A.J. Sievers and J.B. Page
172
Ch. 3
The lower frequency data in fig. 19 (b) and (c) show additional temperaturedependent structure at low temperatures, which probably indicates that both resonant and nonresonant absorption are being observed for the off-center configuration. It now appears clear that given sufficient nonresonant absorption data versus temperature for different frequencies, both the applicability of the Debye model and the temperature dependent behavior of the different factors in Nip2/TT could be isolated and explored, identifying important aspects of the mysterious off-center configuration.
2.1.7. Rotational motion in the off-center configuration Perhaps the simplest way to produce two configurations, one of which has a much larger effective degeneracy than the other, is to assume that the off-center configuration of the Ag + ion contains a large number of free rotor states. In this model we assume that the crystal field splitting of such free rotor states is negligible. In addition, we assume that the rotational moment of inertia I of the Ag + ion about the normal lattice site is large enough so that the spacing between the rotational levels is small compared to kT at all temperatures for which the upper configuration is populated. In this case the rotational levels are dense and the rotational motion can be treated by classical statistical mechanics. The rotational partition function becomes Zr ~
2IkT h2
•
T Tr
.
(2.16)
If we assume that the vibrational degrees of freedom are roughly the same in both configurations, then the on-center population for the two-configuration model displayed in fig. 17 is simply
Pon(T) =
l+
T -~-~ Trr e
.
(2.17)
The surprising result is that with both the energy gap ~ and the moment of inertia Tr treated as free parameters, it is still not possible to fit the data shown in fig. 14. When Tr is made sufficiently small (generating a high density of states) so that at 25 K a small value for Pon(T) can be produced, then the abrupt knee at about 5 K is missed. Apparently, even this limiting case of free rotor states without the quenching produced by crystalline electric field effects does not produce a dense enough set of levels to match the measured temperature dependence. A better fit to the data of fig. 14 is obtained simply by replacing (T/Tr) in eq. (2.17) by a constant degeneracy factor g t> 100.
w
173
Unusual anharmonic local mode systems
2.1.8. Ag + in KI alloys
To test rotor-like models, disorder has been introduced into the lattice to see if the temperature-dependent properties of the resonant mode are changed. Early work on Li + ions in KBr alloys (Clayman et al. 1967) has shown that a resonant mode can be shifted both to smaller and larger frequencies, since an alloy of two alkali halides has an average lattice constant intermediate to those of the two constituents (Havighurst et al. 1925). The dependence of the mode frequency and linewidth on lattice constant was found to be linear in Clayman's work. Since the average lattice constant (a) is calculated from the measured alloy concentration by the Vegard relation, namely (a) = al + ( a 2 - al)z where al and a2 are the lattice constants of the two components and z is the molar concentration of component 2 (Vegard 1921), the centroid frequency of the resonant mode is expected to be a linear function of the molar alloy concentration. Since A V / V -- 3Aa/(a), the frequency shift can be expressed as -Aw = A ( A V / V ) cm -1. Because RbI has a larger lattice constant than KI, alloying some RbI with KI:Ag + would be expected to decrease the resonant mode frequency, making the defect system more unstable. The far IR temperature dependent spectrum of a KI + 1 mole% RbI + 0.2 mole% Ag + sample is presented in fig. 20
1.0
m
A
"T i!
- o.s
.J // i
l 0
Z>v'"-7"~ C)"
I0
.
~
lyi'~" i
20
30
40
Frequency (cm "l) Fig. 20. Temperature dependence of the Ag+-induced resonant mode absorption coefficient for a KI alloy. The sample is KI + 1 mole% RbI + 0.2 mole% AgI. The solid line is for T = 1.2 K, the dot-dashed line is for T = 10 K and the dashed line is for T = 20 K. The center frequency of the resonant mode is shifted slightly to the red and one half of the low temperature resonant mode area is missing when compared to an unalloyed sample grown at the same time. (After Hearon 1986).
174
A.J. Sievers and J.B. Page
Ch. 3
(Hearon 1986). At 1.2 K (solid line) the resonant mode centroid is shifted to a smaller frequency and has lost half of its strength when compared to a KI + 0.2 mole% Ag + sample, grown at the same time. The dot-dashed trace shows the same spectrum for I0 K. The temperature-induced area loss is very similar to that found for the unalloyed crystal: compare this result with the 9.3 K trace shown in fig. 9. At the highest temperature of 20 K (dashed curve), the resonant mode peak has nearly vanished. These temperature dependent results appear very similar to those presented earlier for the unalloyed samples. As the concentration of RbI is increased from 1 mole% to 5%, the frequency shift is found to obey the relation -Aw = 300(AV/V) cm -1. For each alloy, the strength of the mode appears to have the same temperaturedependent properties as described above. Unfortunately, the alloy-induced shift of the resonant mode to smaller frequencies is accompanied by a rapid decrease in the low temperature line strength. As a result, for 3 mole% RbI the strength is about 1/20 of that for the unalloyed crystal, and by 5 mole% RbI, the strength ratio is a barely observable 1/50. Is this rapid change in the low temperature strength with increasing alloy lattice constant a consequence of the off-center configuration coming into resonance with the on-center one? Probably not. For the different alloy concentrations, the observed strength most likely stems from the Ag + centers that have unperturbed neighbors. If the Ag + centers are in a random distribution and it is the next-neighbor Rb + ions that destroy the mode, then the absorption strength should vary as ,-~ (1 - : c ) 12 where z is the alloy mole concentration. The rapid strength change observed here with increasing z could be interpreted as evidence for some degree of nonuniformity in the Ag + :Rb + distribution. When the crystal is grown, the Ag + ions may prefer to remain in a substitutional lattice site near the Rb + ions in the crystal. A remarkably similar strength change with alloy composition has been found previously for the KBr:Li + resonant mode. In this case expanding the alloy lattice by 5 mole% KI also causes the mode strength to disappear (Clayman and Sievers 1968). Although the strength of the Ag + resonant mode is strongly influenced by lattice disorder, its temperature dependent properties appear to remain unchanged, within the experimental uncertainties generated by the large strength change.
2.1.9. Temperature dependence of the Ag + electronic transitions in KI Another way that the temperature dependence of the position of the Ag + ion can be monitored is by making use of the known optical properties of this ion in alkali halide crystals. This behavior can be understood in
w2
Unusual anharmonic local mode systems
175
terms of the 4d 1~ --+ 4d95s parity-forbidden electronic transitions of the Ag + defect (Fussg~inger 1969; Kojima et al. 1968; Holland and Luty 1979) being made allowed by dynamic or static symmetry breaking. When the ion is a substitutional defect in a cubic crystal these electronic transitions are made somewhat allowed by vibronic coupling to odd-parity vibrational modes, or if the defect is off center, by the static odd-parity lattice distortion. In each case the larger the odd-parity contribution, the larger the line strength. A schematic representation of the temperature dependence of the oscillator strength of the transition for the different possibilities is given in fig. 21. Four cases as a function of temperature are shown: (A) gives the optical signature for an on-center defect coupled to a single Tlu symmetry harmonic vibration; (B) is for an on-center defect coupled to a single Tlu symmetry anharmonic vibration; (C) is for an off-center impurity with a shallow barrier and (D) is for a frozen off-center impurity with a large potential barrier. Little optical work has been done on KI:Ag + because this particular system has the unusual property of "aging" with time at room temperature much faster than for the other alkali halide hosts (Fussg~inger 1969). A systematic way to reverse the aging process and hence to reactivate the samples was found during the radio frequency studies of KI:Ag + by Hearon (1986). I
to-'
D
I
,
f
~
cons,
'
[x,] Id 2
td 3
,
0
I
|
200 Temperature (K) I00
Fig. 21. Schematic temperature-dependence signatures for the electronic oscillator strength of the Ag + ion coupled to different strength odd-parity distortions. (A) on-center ion coupled to a harmonic Ttu vibrational mode. (B) on-center ion coupled to an anharmonic Tlu vibrational mode. (C) off-center ion with a shallow potential barrier. (D) off-center ion with a large potential barrier. (After Holland and Luty 1979).
A.J. Sievers and J.B. Page
176
Ch. 3
o) ? Jan.
0.02 oo
OO
O Oo
O
O O O
O.OI
O O O
""
21 Jan. A&
& "~' a a t=
aa~ Aaat=a
A
,r
o ~ ~
.........................
b)
I
h-- 0 . 0 4
After ,txtxt•t="at=tbt= at=a ~ a
tx
0.02
t=
o 0
ooo
Before
oooooo,oooooooo 8 16 Temperature (K)
~ 24
Fig. 22. Aging and reactivation of isolated Ag + defect centers in KI. (a) The temperature dependent radio frequency dielectric constant versus temperature for an as-grown sample (circles) of KI + 0.2 mole% AgI and the aging observed in the same sample after it has been refrigerated for 14 days (triangles). The number of isolated defect centers has decreased by a factor two. (b) Radio frequency dielectric constant data (circles) for a KI + 0.5 mole% AgI sample maintained at room temperature for several months. The data for the same sample after heat treating at 200 C (triangles) shows that the isolated defect density is fully restored. (After Hearon 1986).
Such effects had been observed previously in the far IR, and samples were refrigerated between runs to slow the process. New crystals were always grown once previous samples had aged. Hearon (1986) found that the reactivation procedure for a Ag + doped KI crystal which has been stored at room temperature is first to heat it to 200 ~ in air and then cool it back down to room temperature in a few minutes. Examples of this aging and reactivation effect are shown in fig. 22. Here the temperature-dependent behavior of the dielectric constant is monitored to determine when the isolated Ag + defect state is present. Figure 22(a) shows the aging effect as measured via the radio frequency dielectric constant for a freshly grown sample
w
Unusual anharmonic local mode systems
177
of KI + 0.2 mole%AgI (circles) and for the same sample after it had been refrigerated for 14 days (triangles). The concentration of isolated centers has decreased by a factor 2. Figure 22(b) illustrates how the sample can be reactivated. A KI + 0.5 mole% AgI sample maintained at room temperature for several months is nearly devoid of isolated centers, as indicated by the circles. After cycling the sample to 200 ~ the temperature-dependent dielectric constant associated with isolated centers is fully restored (Hearon 1986). This aging effect is also quite pronounced in the UV absorption spectrum of the doped crystal. Figure 23 shows the spectrum of an aged KI:Ag + sample before (dashed) and after (dot-dashed) the heat treatment. Before heat treating, the UV spectrum has a strong edge near 420 nm that is in the same frequency interval as the band gap of pure AgI (Kleppmann 1976). After heat treatment, the AgI-like spectrum in the UV disappears, and the crystal becomes transparent. An Ag+-like absorption spectrum is now found at much higher frequencies. Apparently, at room temperature the Ag + ion is fairly mobile in the alkali halide host and tends to organize into silver 1.0
,
w
,..........
after before 0.8
.,,,..,,.v,
,,,-.--
9 - - . ~9 . , . ' . . ~ " ~- 9 - . - ~ , . ~ . , ~ . .
/.r ,,
I I ! I ! I I
! !
.s 0.6
! !
t
,..., 0 . 4 [.-,
/
i /
t
'--
/
i i
/ / /
!
0.2
,-. /
J
/
i
/
I /
t
i 0.0
I
/ /
......... 250
290
/
!
!
i
330
370
410
450
Wavelength (nm) Fig. 23. Effect of heat treatment on the KI:Ag+ UV transmission spectrum. Dashed curve: aged crystal before heat treatment, showing the absorption produced by AgI aggregates. Dotdashed curve: after heating this sample to 200 C and quenching to room temperature, isolated Ag+ centers are produced. The crystal is now transparent beyond the 420 nm wavelength region. (After McWhirter and Sievers, unpublished).
178
A.J. Sievers and J.B. Page
Ch. 3
halide clusters which the modest heat treatment perfected by Hearon (1986) breaks up. Figure 24 shows the measured UV absorption spectrum of KI:Ag + for two different temperatures. Over this temperature range the center frequency of each of the three bands remains essentially fixed. A linear concentration dependence establishes that these three features labeled A, A' and C [using the nomenclature of Fussgiinger (1969)] in the figure are associated with isolated Ag + ions (Page et al. 1989). To display the three lines in one figure, three different Ag + concentrations are required, as described in the caption. The strength of the strong UV line C shown in fig. 24(a) is temperature independent, and is assigned to a delocalized charge transfer excitation in which an electron from a nearest neighbor anion occupies one of the empty Ag + excited states. The weak lines A and A' are assigned to the Alg(4d 10) -+ T2g(4d95s) and Alg(4d 10) --~ Eg(4d95s) transitions, respectively. They display nearly zero strength at 1.2 K but grow rapidly with increasing temperature. The change is much faster than can be obtained from the occupation number effect in the low-lying Tlu resonant mode, for example. This oscillator strength temperature dependent signature of A and A' does not match any of the cases displayed in fig. 21 but the increase in
Fig. 24. KI:Ag+ UV absorption spectra. Dashed line, T = 50 K; dot-dashed line, T = 1.2 K. Silver concentration: panel (1) 2 x 10-2 mole%, panel (2) 2 x 10-3 mole%, panel (3) 5 x 10-5 mole%. (After Page et al. 1989).
Unusual anharmonic local mode systems
w
1.0 .
i
179
,,,
o
m
It
Q_ 0
~. 0.5
~
r c-
\ (3
I
xU,..
tO
0.0 0
10 2O Temperature (K)
30
Fig. 25. Temperature dependence of Ag + on-center population, as determined by various measuring techniques. Dashed line, far IR resonant and gap mode strengths. Dotted line, dielectric constant data. Squares, UV absorption for the A' electronic state. Solid circles, Raman scattering intensity from the 16.1 cm -1 Eg mode. (After Page et al. 1989).
optical strength with temperature appears identical to that found earlier for the increase in the DC dielectric constant produced by the appearance of a permanent dipole moment with increasing temperature. This latter change was shown to be proportional to the population in the off-center configuration, Poff. By letting Pon 1 -Poff, both the UV and dielectric constant data can be compared directly with Pon determined from the temperature dependence of the Tlu resonant and gap mode strengths. These results are graphed in fig. 25. The three different experimental probes are seen to measure exactly the same rapid decrease in the population of the on-center population. -
-
2.1.10. Raman scattering from even-symmetry resonant modes
The polarized Raman intensities I((1,-1,0)(1,-1,0)),
I((1, 1,0)(1,-1,0)),
I(
for KI:Ag + have been reported for the 90 ~ scattering geometry (Page et al. 1989). The notation here identifies the polarization of the incident and scattered wave with respect to the crystal axes: ((incident)(scattered/). The results at three temperatures for the first two polarizations are shown in figs 26(a) and 26(b). No impurity-induced scattering was observed for the
180
Ch. 3
A.J. Sievers and J.B. Page ----T~T-~-'--'r----'-~T-------T---'--'T'-'-
280
I
-
(o) k
N~
-
\x 160
,...
A C/) ta C: :3 0 U
"-" 4 0 =,., 2 8 0 ..,. c/} C
I
"~
(b)
C: I--t
160
""\
:
".
- ................' o... ........
40
12 Raman
~ ~:~,-A.
20 shift (cm - l )
t,
"
28
Fig. 26. Temperature dependence of the polarized Raman spectra of a { l, 1, 0}-cut KI + 0.3% AgI crystal. The spectra, taken with 488 nm excitation, are displayed for two polarization geometries: (a) ( 1 , - 1 , 0 ) ( 1 , - 1 , 0 ) ( A l g + ( 1 / 4 ) E g + ( 1 / 2 ) T 2 g ) a n d (b) (1,1, 0) (1, - 1 , 0) ((3/4) Eg). The three temperatures shown are 6.3 K (dotted line), 12.9 K (solid line), and 25.2 K (dashed line). No ( 1 , - 1, 0)(0, 0, 1) (1/2 T2g) polarized lines were observed at any temperature. The resolution is 2.5 cm -1. (Page et al. 1989).
third orientation. The data are classified according to the even-parity representations Alg, Eg and T2g of the cubic group Oh, which yields for the polarized intensities the expressions I ( ( 1 , - 1 , 0 ) ( 1 , - 1 , 0 ) ) - Alg + (1/4) Eg + (1/2) T2g,
I((1, 1,0)(1,-1,0))
- (3/4) Eg,
I((1, -1, 0) (0, 0, 1)) = (1/2) ZEg. The Raman measurements presented in fig. 26(b) support the earlier low temperature Eg mode frequency assignment by Kirby (1971). On the other hand, neither of the previously proposed Alg and T2g higher frequency resonances are found. In addition, no pronounced Raman features are found in the gap region. A new discovery is the strong temperature dependence of
w
Unusual anharmonic local mode systems
181
the Raman scattering intensity of this 16.1 cm -1 Eg mode, which is shown for all of the temperatures in fig. 25. As temperature is increased, the scattering strength of this Eg mode decreases rapidly (see the solid circles in fig. 25), yet its center frequency stays nearly fixed. The temperature dependence of this Raman line is remarkably similar to that of the far IR active resonant and gap modes and the optical features discussed above, but it takes on added importance because the Eg mode involves no motion of the Ag + ion. Unpolarized Raman spectra at a number of temperatures between 6.9 and 25.7 K are shown in fig. 27. With increasing temperature up to 17.1 K, the 16.1 cm -1 Eg resonant mode Raman peak decreases in strength and is finally replaced by a second excited state feature at 12.2 cm-1. This excited state peak shows both Eg and Alg components and is consistent with the appearance of an A1 symmetry mode in a Cnv off-center configuration. At still higher temperatures this feature also disappears (or broadens) into a pronounced central peak having the same polarization properties, and likely produced by the off-center tunneling levels themselves. This polarization character of the quasielastic central peak is preserved up to 60 K, while at higher temperatures T2g symmetry scattering appears (Fleurent et al. 1991).
25.7K
A i.0 6.9
9.2K
IlK
15.2K
r O Ir
o
>,0.5
B
t/3 t--
C
5
a~"S--
ZO 35 Roman shift (cm-I)
Fig. 27. Unpolarized Raman spectra of a {100}-cut KI:Ag+ crystal versus temperature. The six different temperatures are given in the figure. With increasing temperature the 16.1 cm-1 Eg resonant mode is replaced by a lower frequency A1 symmetry mode appropriate to a C4v symmetry defect site which subsequently disappears in a quasielastic scattering peak at the highest temperature. (After Fleurent et al. 1991).
A.J. Sievers and J.B. Page
182
Ch. 3
2.2. Basic shell model description of the T = 0 K nearly unstable lattice dynamics
2.2.1. Local modes The Lifshitz harmonic Green's function method for computing general defect phonon properties has been described in detail (Maradudin et al. 1971; Bilz et al. 1984). For completeness, the application of this method to determine the frequencies, displacement patterns and IR spectra of localized modes arising from substitutional impurities is briefly reviewed here. A detailed discussion of the application of the Lifshitz method to the lowtemperature on-center dynamics of KI:Ag + is given by Sandusky et al. (1993a). The defect concentration is taken to be low enough so that the case of just a single impurity is required. For a system of N ions interacting via harmonic forces, the normal mode frequencies {w f} and displacements x ( f ) are determined by the 3N • 3N matrix eigenvalue equation
(4:' - w ~ M ) x ( f ) = O,
(2.18)
where ~ is the harmonic force constant matrix, M is the diagonal mass matrix and f = 1. . . . ,3N labels the modes. The force constant matrix is symmetric, and the modes are normalized such that the 0rthonormality relation is x ( f ) M x ( f ' ) = 5f f,. With a substitutional defect present, it is convenient to rewrite eq. (2.18) identically in terms of the host crystal harmonic Green's function matrix G0(w2) and the perturbing matrix C(w2), containing the impurity-induced force constant and mass changes. The result is +
x ( f ) = o.
(2.19)
This equation can be partitioned into two equations" one involving just components inside the "defect space", defined by the sites associated with nonzero elements of C, and a second equation which determines the mode displacements outside the defect space:
[III -k- GoiI(~)Cii(~o~)] xI(f) = 0
(2.20)
XR(f) = --GoRI(~)CII(OJ~)xI(f)"
(2.21)
and
w
Unusual anharmonic local mode systems
183
The subscripts I and R refer to components inside and outside the defect space, respectively. Equation (2.20) gives the determinental frequency condition: (2.22)
IIii + GoIi(w})Cii(w}) I - O.
For isoelectronic impurities, the defect space is usually small. Equation (2.22) then involves the determinant of a small matrix and is thus practicable, provided the defect-space elements of the unperturbed harmonic Green's function matrix can be computed. This is readily done for localized modes by direct summations involving the unperturbed host crystal phonons; for alkali-halides, these are well-known through phenomenological models (e.g. shell models) that account very well for the measured phonon dispersion curves. Once the local mode frequency is known, eq. (2.20) can be used to compute the defect-space displacement pattern, to within a normalization constant. The displacements outside the defect space may then be determined by eq. (2.21).
2.2.2. Infrared absorption In the long-wavelength limit, the interaction between an insulator obeying the Born-Oppenheimer approximation and an external field of monochromatic infrared radiation is
-M(u) . Ee -i~t,
(2.23)
where M(u) is the system's dipole moment for nuclear configuration u. If just the linear term MS(u) -- ~-,z# #~(l)u#(1) in the expansion of the ath component of the dipole moment in terms of the nuclear displacements is retained and a calculation similar to the one carried out by Klein (1968) is done, the resulting absorption coefficient a(w) for a cubic crystal with an impurity concentration Ci is given by 2
47rwCi ( n ~ + 2 ) ~(w)- cn(w) 3
~ I m [ G(W2+ ie)]/-~,
(2.24)
where n ~ is the high-frequency index of refraction, c is the speed of light in vacuum, G(w 2) is the harmonic Green's function matrix for the impurity crystal and the limit e --+ 0 + is understood. The quantity / ~ = {#~(l)} denotes the effective charges in this linear dipole moment approximation.
A.J. Sievers and J.B. Page
184
Ch. 3
In deriving eq. (2.24), it is assumed that the standard Lorentz local field correction holds and that the contribution from the electronic polarizability is adequately described by the high frequency dielectric constant coo. The index of refraction n(w) is taken to have its pure crystal value, owing to the assumption of low defect concentrations. Hence the integrated absorption strength Sf = f a(w)da~ for a single local mode f is n~+2
S.f = cn(w)
3
)
2
[~"~x(f)]2
(2.25)
Using the transformation properties of the dipole moment, it is straightforward to show that for an impurity crystal with Oh symmetry,/~ transforms under symmetry operations as the ath partner of irreducible representation Tlu. Hence, by a standard group-theoretic matrix element theorem, f ~ x ( f ) vanishes when x ( f ) is not an ath partner belonging to Tlu. In eq. (2.25), the sum implicit in ~ x ( f ) extends over the entire system, reflecting the fact that infrared radiation couples to all of the ions. For the case when the effective charges in the defect crystal are unperturbed from their host crystal values, Klein (1968) has shown how the absorption may be expressed in terms of just defect-space quantities.
2.2.3. Impurity model calculations The determination of the local mode (and resonant mode) displacement pattern and frequency requires knowledge of the perturbing matrix C(w 2) and the pure crystal Green's function elements. Two different methods have been employed to obtain the latter in the KI:Ag + defect mode calculations. The first is detailed in (Page 1974; Harley et al. 1971), and its present application is outlined here. The breathing shell model (Schrrder 1966) was used by Page et al. (1989) to compute the pure KI phonon frequencies {~ka} and complex plane wave displacement patterns {x(kj)} at 22,932 k vectors in the irreducible 1/48 element of the Brillouin zone, this being equivalent to one million vectors in the full zone. The imaginary parts of the unperturbed Green's function matrix Im Go(a;2) = 7r ~ x ( k j ) x +(kj) ~ (w~3 - to2) ka were approximated as histograms by dividing the pure KI phonon frequency range into 100 equally spaced "bins" and evaluating the sum over the modes
w
Unusual anharmonic local mode systems
185
whose frequencies fall within each bin. The real parts were then obtained by computing the Hilbert transforms of the imaginary parts. The defect model for a low-temperature on-center configuration of KI:Ag + consists of the mass change Am, longitudinal force constant changes 5 = - A ~ ( 0 0 0 , 100) between the defect and each of its six nearest neighbors, and longitudinal force constant changes 5' = - A ~ ( 1 0 0 , 200) between the defect's nearest neighbors and their adjacent fourth-nearest neighbors. Physically, 5 arises from the different binding of the impurity and is expected to be negative, consistent with the overall force-constant softening implied by the presence of the strong low-frequency IR impurity resonance at 17.3 cm -1. The force constant change 5' is postulated to arise from the expected defectinduced inward static relaxation of the nearest neighbors, also consistent with the overall force constant softening. Within this model, only the modes belonging to the Tlu, Eg and Alg irreducible representations of the Oh point group are perturbed. The necessary orthonormal symmetry basis vectors within the impurity space are shown in fig. 28. The two force constant changes (5, 5') were determined by requiring that the model reproduce the observed Tlu 17.3 cm -1 low-frequency resonant and 86.2 cm -1 gap mode IR frequencies. Both the resonant and gap mode frequencies are determined by the condition Re]l + Go(z)C(z)] = 0, where z = w2 + ie. For the gap mode, the Re in this equation is not necessary, since the imaginary parts of the Green's function vanish. Using this condition, 5 versus 5' curves are computed consistent with the 17.3 cm -1 and 86.2 cm -1 modes, with the results given in fig. 29. The crossing point of these two curves, denoted by a circle in the figure, gives the force constant changes for which the model reproduces both of these Tlu absorption peaks. The corresponding fractional force constant changes are d/kl - -0.563 and t~t/kl = --0.531, where kl -- 1.884 • 104 dyn/cm is the pure KI breathing shell model nearest-neighbor longitudinal overlap force constant. Having fixed the model's parameters, we next turn to its predictions. First, the model predicts the ratio of the IR absorption strengths for the 86.2 cm-1 and 17.3 cm -1 gap and resonant modes to be 1.4. This is in reasonable agreement with the observed value of ~ 3, considering that no local field or effective charge changes were included in the relative intensity calculation. Second, the mode predicts a low-frequency Eg resonance at 20.5 cm -1, in reasonable agreement with the observed Eg Raman peak at 16.1 cm -1. Figure 29 includes the 5 versus 5' curve computed for an Eg resonance at 16.1 cm -1, and one sees that this curve comes quite close to the circled (5, 5') point for this model. For comparison, fig. 29 also includes the (5, 5') curve computed for the case of a zero-frequency Tlu resonance (dashed curve). The force constant changes on this curve correspond to the defect lattice being unstable against Tlu displacements. As the force constants are
A.J. Sievers and J.B. Page
186
Ch. 3
2-1/2
~(Tlu x,1)
~(Tlu x,2) -
12-1/2
2 . 1 2 "1/2 0
o
~
~(Eg 1,1)
o
....
o
~(EQ2,1) z
6-1/2
~(Alg 1,1 ) Fig. 28. Symmetry-basis vectors on the defect and its nearest neighbors, used in the perturbed phonon model. For a basis vector ~(Fip, t), Fi labels the irreducible representation, p labels the partners within the representation, and t labels independent vectors for a given Fip. (After Sandusky et al. 1993a).
weakened, the resonant mode of Tlu symmetry is almost always the first to become unstable; hence only the portion of fig. 29 to the upper right of the dashed curve is physically reasonable. The close proximity of the fit point (circle) to the dashed curve means that this model corresponds to a nearly unstable defect/host system. Besides the Tlu (17.3 cm -1, fit) and Eg (20.5 cm -1, predicted) resonant modes, the model also predicts an Alg resonance at 37.3 cm -1. This mode is Raman-allowed, but has not been seen experimentally. However, since the Alg and Eg Raman strengths are determined by two independent electronic polarizability derivatives, the null Alg experiment could be used to place bounds on these derivatives.
w
Unusual anharmonic local mode systems - I .0
-0.5
187
o.o 0.0
..,
~~
I
I I jT, u zero freq.
T__!~_86.2 cm___[!
-
-0.5
-1.0 Fig. 29. Calculated fractional force constant changes for resonant modes and gap modes at fixed frequencies. The inset illustrates the force constant perturbations in our model; all other short range and Coulomb force constants are unperturbed. The fractional changes are given in units of the pure KI breathing shell model nearest neighbor longitudinal overlap force constant kl = 1.884 x 104 dyn/cm. The dashed curve is for a Tlu instability. (After Page et al. 1989).
A striking prediction of this model is the existence of three nearly degenerate gap modes" the three fold degenerate Tlu mode at 86.2 cm -1, a two fold degenerate Eg mode at 86.0 cm -1 and a nondegenerate Alg mode at 87.2 cm -1. These modes are found to have some very unusual properties. The first is the extent of the near-degeneracy- the frequencies are within about a wavenumber of each other. Second, they are essentially independent of the force constant change 6, as is evident from the curve for the 86.2 cm -1 gap mode in fig. 29. Third, ratios of the Raman strengths of the Alg and Eg gap modes to their low-frequency resonant mode counterparts are predicted to be negligible. Indeed, these modes have not been observed in Raman experiments. Detailed calculations of the mode displacement patterns reveals the origin of these unusual properties. All three gap modes are found to have displacement patterns which are strongly peaked on the impurity's fourthneighbor sites [(+200), (0 + 20), (00 + 2)], as shown in fig. 30(a)-(c). The displacements on these sites are a factor of 50 or more larger than those on the impurity or its six nearest neighbors. For comparison, Figure 30(d) also shows the displacement pattern for the 17.3 cm -1 Tlu resonant mode,
A.J. Sievers and J.B. Page
188
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Fig. 30. Calculated displacement patterns for different KI:Ag + modes. (a) 87.2 cm -1 Alg pocket gap mode. (b) 86.0 cm -1 Eg(1) pocket gap mode. (c) 86.2 cm -1 TluX pocket gap mode. (d) 17.3 cm -1 TluX resonant mode. Here Tlux denotes the Tlu partner which couples to z-polarized radiation, and Eg(1) denotes one of the two degenerate Eg partners. For our choice of partners (fig. 28), panels (a), (c), and (d) show displacements in the z - y plane, while (b) shows displacements in the y-z plane. Note that the displacement pattern for the resonant mode is peaked on the defect and its nearest neighbors, while the displacement patterns for the pocket gap modes, (a)-(c), are peaked on the fourth neighbor sites, away from the defect. The displacements for the different symmetries are not drawn to scale. (After Rosenberg et al. 1992).
w
Unusual anharmonic local mode systems
189
which is peaked at the impurity site and its nearest neighbors. The nearlydegenerate frequencies of these "pocket" gap modes reflect the fact that they are almost entirely determined by the local dynamics within each of the six pockets; the pockets are weakly coupled to produce the different symmetry modes. Moreover, the independence of these modes on the force constant change 5 is clear, since this force-constant change is seen to couple ions which have essentially no motion in these modes. Finally, the weak predicted Raman strengths for the Alg and Eg pocket modes is also explained by the negligible displacements on the impurity's nearest neighbors, since defect-induced first-order Raman scattering in alkali halides typically arises from the modulation of the electronic polarizability by the motion of the ions in the impurity's immediate vicinity. On the other hand, the Tlu pocket mode is readily observable with IR absorption, since this probe couples to all of the ions in the system. Unfortunately, the model is fit to the measured IR pocket mode frequency, so that the negligible Raman activity of the two even-parity pocket modes precludes a straightforward independent verification of the existence of these highly unusual modes.
2.2.4. Isotope mode splitting Owing to the negligible Raman activity of the Alg and Eg pocket gap modes, another test was needed to verify their existence. Fortunately, the naturally-occurring isotopic abundances of 7% 41K+ and 93% 39K+ in the host crystal provide just such a test. It has been demonstrated that the harmonic defect model predicts that the presence of one or more 41K+ ions at the impurity's fourth-neighbor sites strongly mixes the pocket gap modes of all three symmetry types, producing new IR-active "isotope" pocket gap modes which are experimentally observable (Sandusky et al. 1993a). To determine the frequencies and IR integrated absorption strengths for these isotope modes, nearly-degenerate perturbation theory is first applied to a single impurity/isotope combination (Sandusky et al. 1991, 1993a). This is done by expanding the isotope mode displacement patterns in terms of the six normalized unperturbed pocket gap mode patterns g,(i) = ~ , = 1 af'(i)X(ff) and substituting the expansion into eq. (2.18). Here i = 1,..., 6 labels the isotope modes. Multiplying the resulting equation on the left by ~(f) yields a 6 x 6 eigenvalue equation for the isotope-mode frequencies w~ and the expansion coefficients af(i)" 6
(W2f -- W2) af(i) = E w2f((f)Amx(f')aY '(i)" f'=l
(2.26)
190
A.J. Sievers and J.B. Page
Ch. 3
Here Am is the diagonal matrix containing the mass changes introduced by the 41K+ isotopes, and the {mos} are the six unperturbed pocket-gap-mode frequencies. A 41K isotope substitution changes the mass of the 41K+ ion, leaving its electronic structure and, hence the effective charges #~<~unchanged. Thus, the dipole moments for the isotope modes are produced exclusively by their Tlu components. When ff,(i) is substituted into eq. (2.25), the integrated z-polarized IR.absorption for the ith isotope mode, of frequency w~ produced by a single impurity/isotope combination in a crystal of volume V, is 2rr 2
2 (rzoo + 2 ) 2
si = cn(wi)V
3
[~=x(Tlux)]iaZ'ux(i)"
(2.27)
This expression is just the absorption for a single impurity with no isotopes present, multiplied by the Tlux fraction a2 uX(/) for the ith isotope mode and by an index of refraction correction n(w~ )ln(wO. Figure 31 shows the calculated splittlngs for a single Ag + impurity with a (200) 41K + isotopic substitution, which mixes the Alg, Eg2 and TluX pocket .
llu
No Isotope
87
Alg
_-
_
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.
.
.
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-
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_
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_
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-(_1..3:_.
-
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(200) Isotope S,
\\
_. \ "
\
\\\ \\\
~\ %\ k\ It.
-
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....
\
.
.
.
.
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Fig. 31. Mixing of the three nearly degenerate Alg, Eg2 and Tlu x gap modes under a (200) 39K+ --+ 41K+ host-lattice isotopic substitution. The percentages give the strength sx of the z-polarized absorption per defect/(200) isotope combination, relative to a defect with no isotope substitution. Note that the percentages add up to slightly more than 100% because the absorption coefficient is not only proportional to the imaginary part of the dielectric function but is also inversely proportional to the host-crystal index of refraction, which is strongly frequency dependent in the gap mode region. (After Sandusky et al. 1991).
w
Unusual anharmonic local mode systems o
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191
(a) 84.7 cm -1 isotope mode o
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(b) 86.1 cm -1 isotope m o d e Fig. 32. Computed displacements patterns for a (200) 39K+ -+ 41K+ host-lattice isotopic substitution. The impurity and the isotope are represented by a dark square and a dark circle, respectively. (a) 84.7 cm -1 isotope pocket gap mode and (b) 86.1 cm -1 isotope pocket gap mode. These are the isotope pocket gap modes having the largest and smallest frequency shifts relative to the unperturbed Tlu pocket gap mode. Note that the displacement pattern for the isotope mode with the largest frequency shift is strongly peaked on the isotope.
gap m o d e s to create three isotope modes. The calculated displacement patterns for the (200) isotope m o d e s with the largest and smallest frequency shifts from the unperturbed Tlu 86.2 cm -1 pocket gap m o d e are given in fig. 32. T h e s e isotope m o d e s are seen to be pocket gap modes with the displacements confined to a single pocket. It is not surprising that the isotope with a large displacement on the (200) 41K+ gives the largest frequency shift, while the isotope m o d e with a negligible displacement on this ion gives a very small frequency shift. As s h o w n in fig. 31, fourth-neighbor 41K+ substitutions can produce isotope m o d e s having frequency shifts larger than 1 cm -1 and IR absorption strengths c o m p a r a b l e to that p r o d u c e d by a defect with no isotopes present. In contrast, (110) (i.e. second-neighbor) and (400) 41K+ isotope substi-
A.J. Sievers and J.B. Page
192
Ch. 3
tutions are found to produce modes having negligible absorption strengths and/or frequency shifts of less than 0.1 cm -1, which would be very difficult to observe given the large width of the main mode. Since the displacements for the isotope mode are zero-order displacements in the perturbation approach, the Tlu fractions a2 ox(i) for a single isotope-impurity combination sum to unity. Hence, one sees from eq. (2.27) that provided the change with frequency of the index of refraction can be neglected, which is an excellent approximation for modes separated by less than 0.1 cm -1, the unperturbed mode strength obtained by including the weakly shifted modes is the same as if their contributions had been neglected from the outset. Similar arguments hold for the splitting of the pocket isotope modes, such as the 84.7 cm -1 (200) 41K+ mode, due to second-neighbor or (400) 41K+ isotopes- as long as the splitting produced by these substitutions is smaller than the experimental resolution, or these new modes have negligible absorption strengths compared with the fourth-neighbor pocket isotope modes, their contributions can be ignored. Accordingly, the second-neighbor and (400) 41K+ isotope substitutions were neglected in the isotope-induced IR absorption calculations (Sandusky et al. 1991, 1993a). However, in order to account for all 41K+ fourth-neighbor modes, the 41K+ fourth-neighbor isotope modes with frequency shifts of less than 0.1 cm -1 were included. The nearly-degenerate perturbation theory predicts the frequency shifts (eq. 2.26) and integrated IR absorption strengths (eq. 2.27) for isotope modes produced by a single impurity/isotope combination. However in a real KI:Ag + system, there is a distribution of impurity/isotope combinations. In a crystal with Nd impurities, the predicted x-polarized strength for the ith isotope mode produced by an impurity/isotope combination with 1 fourth neighbor isotopes and the Ni - 1 symmetry-equivalent modes produced by different isotope/impurity combinations, also with 1 41K+ fourth neighbors, is given by Si - NdNi(1 -
f)6-I fl si
(2.28)
where si is the x-polarized strength for the ith mode produced by a single isotope/impurity combination, as given by eq. (2.27). The computed shifts and strengths for 13 isotope/impurity configurations are tabulated in Sandusky et al. (1993a), which gives numerous additional details. The final result is that there are only two predicted isotope pocket gap mode IR peaks at frequencies resolvable from the main unperturbed line at 86.2 cm -1. These are an observably strong peak at 84.7 cm -1 and a much weaker one at 87.1 cm -1. The experimental isotope line is found at 84.5 cm -1 as discussed below.
w
Unusual anharmonic local mode systems
193
2.3. Pocket gap mode experiments
2.3.1. Isotope shift The absorption coefficient of KI+0.4 mole% AgI in the phonon gap region of KI for two different temperatures is shown in fig. 33 at a resolution of 0.1 cm -1. The strong impurity-induced feature at 86.2 cm -1 is the KI:Ag + gap mode corresponding to the low temperature on-center configuration of I
I
I
I
75
80
85
I 90
>
7 E 0 v t"-" 9 0
, m
q... 0 0 cO o m
k.. 0
~0
..0
0
70
Frequency
95
(cm -1)
Fig. 33. Impurity-induced absorption coefficient of KI + 0.4 mole% AgI. The restricted frequency interval covers the phonon gap region of KI. The resolution is 0.1 cm -1 The temperature for the upper spectrum is 1.6 K, and that for the lower one is 8.8 K. The strong mode at 86.2 cm -1 is the on-center KI:Ag + gap mode. This mode has lost about half its strength in the higher-temperature spectrum. The doublet at 76.8 and 77.1 cm -1 is due to C1- and the single peak at 82.9 cm -1 is due to Cs +. The weak temperature-dependent peak at 84.5 cm -1 is the Ag + isotope mode. Note that the KI:Ag + modes, which have a FWHM of 0.5 cm -1, are significantly broader than other KI gap modes, whose FWHM is ,-~ 0.14 cm -1. Additional temperature-induced changes are increased broadband absorption due to difference-band processes in the host KI crystal and the appearance of a KI:Ag + gap mode at 78.6 cm -1, corresponding to population of the off-center configuration of the Ag + defect in the KI lattice. (After Sandusky et al. 1993a).
A.J. Sievers and J.B. Page
194
Ch. 3
the defect system. Most of the weaker spectral features seen here and identified in the figure caption are associated with other unwanted monatomic impurities, present in either the host or dopant starting materials. But the strength of the weak line at 84.5 cm -1 also varies linearly with the Ag + concentration and hence is not due to pair modes. When the temperature is increased from 1.6 K to 8.8 K, the strengths of the strong Ag + gap mode and the neighboring satellite line at lower frequency are reduced in strength by a factor of two while the weak but sharp modes due to the other impurities remain unchanged. At the highest frequencies shown in the figure, the temperature change produces an increase in the host absorption coefficient due to intrinsic difference band processes (Love et al. 1989). The interplay between the temperature dependence of the gap mode strength and the underlying difference band absorption can be seen more easily in the 3-dimensional plot of fig. 34, which displays the results obtained for several temperatures from 1.6 K to 19 K. At the highest tempera12"
0
0 ~
B~
B7
~j
~o
8 frocloeoCY
Fig. 34. Temperature dependence of the absorption coefficient in the region of the KI:Ag+ pocket gap modes between 1.6 and 19 K. The resolution is 0.1 cm-1. Note that the two KI:Ag+ modes have nearly disappeared in the high-temperature spectrum, where two weak Rb+ gap mode peaks at 86.3 and 86.9 cm-1 have become visible. Note also the increase in the difference-band absorption with increasing temperature. (After Sandusky et al. 1993a).
w
Unusual anharmonic local mode systems
195
ture shown, 19.0 K, the gap mode has nearly vanished; the remaining weak absorption peaks at 86.3 and 86.9 cm -I are due to a small concentration of naturally-occurring Rb + impurities. It is clear from these data that not only does the strength of the main mode disappear with increasing temperature, but also that the weak satellite line at 84.5 cm -~ disappears with a similar temperature dependence. At the same time that the difference band absorption increases in magnitude with increasing temperature, the weak features within and on the high frequency side of the main Ag + gap mode remain essentially temperature independent, but become easier to see with the disappearance of the Ag + gap mode. An identifiable property of the Ag + gap modes at 86.2 and 84.5 cm -1 is that their strengths become vanishingly small by ~ 25 K. This temperature dependence is associated with the depopulation of the Ag + on-center configuration. It is exhibited by all experimental probes of this defect system. In contrast, the KI:C1- doublet at 76.8 and 77.1 cm -1 and the KI:Cs + single peak at 82.9 cm -1, for example, have no temperature dependence in this restricted temperature region. There is another relevant feature generated by the Ag + defect in the 8.8 K spectrum. A weak broad gap mode is visible at 78.6 cm -1 which initially grows in strength with increasing temperature. Previously, this line was identified with a transition in the off-center Ag + configuration (Sievers and Greene 1984). The study by Sandusky et al. (1993a) shows that although the band initially grows with increasing temperature it then appears to stop growing for temperatures larger than about 12 K. It now is clear that the temperature dependence of the strength of this mode does not show the temperature dependence associated with the off-center configuration, which continues to grow in strength until about 25 K.
2.3.2. Temperature dependence of the isotope mode intensity To separate the temperature dependence of the gap mode spectrum from the two phonon difference band absorption a three step unfolding technique is used. First the data in the parts of the phonon gap region of KI where no impurity modes are present is used to fit a polynomial to the temperature dependent background absorption. Second, the background absorption is subtracted, leaving only the absorption peaks corresponding to the impurity modes. Third, the strengths of the KI:Ag + pocket gap modes are obtained by fitting the peaks to the sum of two Voigt functions corresponding to the unperturbed and isotope modes, respectively. Since the isotope mode is much weaker than the unperturbed mode (4% of the strength), it is necessary to limit the number of free parameters in the fits: the assumptions used are
196
A.J. Sievers and J.B. Page I
Ch. 3
'
Z
0 "0
I
0
I
'
I
10 Temperature (K)
20
Fig. 35. Temperature dependence of the strengths of the two ir-active modes produced by the Ag + center in the KI gap region. The solid circles are the data for the weak mode at 84.5 cm-1, produced by the 41K+ substitution on the fourth neighbor of the Ag + defect. The dashed line gives the temperature dependence of the unperturbed Ag + gap mode at 86.2 cm-1 (After Sandusky et al. 1991).
that the two Voigt functions have the same shape and that the separation of their center frequencies is temperature-independent. Thus, the remaining free parameters are the widths of the Gaussian and Lorentzian contributions to the lineshapes and the strengths of the two modes. The strength of the unperturbed gap mode obtained by this technique is in good agreement with the previous results, represented by the dashed line in fig. 35. The solid circles in this figure show the temperature dependence of the isotope mode as determined by this technique; within the experimental uncertainty, the temperature dependences of the unperturbed and isotope KI:Ag + pocket gap modes are clearly similar, and possibly identical. 2.3.3. Entire impurity induced spectrum at T = 0 K
It would be misleading to conclude from the success that of two-parameter defect model in describing the isotope effect that it can reproduce the complete impurity-induced far IR spectrum of KI:Ag + system. Figure 36 shows the measured low-temperature absorption spectrum of KI:Ag + versus frequency (curve A) over the frequency region below the reststrahl. In the same figure is presented the calculated impurity-induced absorption (curve B) in the acoustic spectrum generated by the two parameter model (8, St). The
w
Unusual anharmonic local mode systems
I
l
i
197
l
10
I
E o
1 c ~
o
0 o
c0. I o Q_ L_
0 o3 _K3
0.01
0
25 50 75 Frequency (cm -1)
100
Fig. 36. Impurity-induced absorption coefficient of KI:Ag + vs frequency. Curve A: Far IR absorption coefficient of KI:Ag + below the optical phonon region, at 1.7 K. The dominant features are the KI:Ag + resonant and gap modes at 17.3 and 86.2 c m - l , respectively, and the gap modes due to KI:C1- at 76.8 and 77.1 cm -1 and KI:Cs + at 82.9 cm -1 (C1- and Cs + are present as natural impurities). Additional weak features due to KI:Ag + are at 30, 44, 55.8, 63.6 and 84.5 c m - 1 ; all are associated with the on-center configuration of the Ag + impurity. The instrumental resolution is 0.1 cm-1. Curve B: the calculated absorption coefficient according to the two-parameter harmonic (5, 5 ~) model. The gap mode delta-function is not shown. Note that the structure in the acoustic spectrum for curve A is not associated with density of states features in curve B. (After Sandusky et al. 1993a).
resonant mode peak height was matched to the one in curve A. For clarity, the gap mode delta-function is not shown. Although this perturbed harmonic model gives a good account of the positions and the strengths of the resonant and gap modes, as well as the pocket mode isotope effect, it cannot at the same time account for the frequency dependence or strength of the broadband acoustic absorption. This could be due to neglected anharmonic or electronic deformation effects involving the Ag + ion, or to the neglect of the static relaxation effects beyond the defect's fourth-nearest neighbors.
A.J. Sievers and J.B. Page
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Ch. 3
2.3.4. Anharmonicity studied by uniaxial stress and electric field measurements Given the anomalous thermal behavior of KI:Ag +, which falls well outside the harmonic approximation, it is of interest to explore this system's anharmonic interactions. Accordingly, uniaxial stress shifts and electric field measurements have been made. Note that for the pocket modes, these measurements should provide information about host lattice anharmonicity near the (200) family of ions, whereas such measurements for the resonant modes should be sensitive to anharmonicity near the defect. Because the stress and E-field induced frequency shifts are small, a "global" analysis is used to obtain the experimental shifts. The method consists of overlaying the shifted line at some given non-zero perturbation onto the corresponding unshifted line and varying the position and width of the line at non-zero perturbation until the area between the two curves is minimized.
J
O~I(SCO)
~2(C0 -- A(co))
peak O~1
peak O~2
d~o
(2.29)
as a function of the two variables s and Aw, where the two absorption lines O~1(03) and a2(w), are normalized to unit height. The width of the first line relative to that of the second is then equal to s and its frequency shift is Table 1 Measured values of the stress coupling coefficients of the IR-active gap and resonant modes of the KI:Ag +, KI:CI-, KI:e-(F-center), KI:Cs +, and KI:Rb + defect systems. Note that B/A nearly vanishes for the first two modes, which are pocket modes, and for the 55.8 cm-1 KI:Ag + mode in the acoustic spectrum. On the other hand, the KI:Ag + resonant mode at 17.3 cm-1 and the last six "standard" gap modes have much larger values of B/A. The stress coupling coefficients are given in units of cm-1/unit strain. The measurements were made between 1.7 and 3.5 K (After Rosenberg et al. 1992). Mode
Frequency (cm -1)
A
B
KI:Ag + KI:Ag + KI:Ag + KI:Ag +
86.2 84.5 55.8 17.3
186 188 143 390
4- 21 4- 22 4- 32 4- 90
KI:C1KI:CIKI:eKI:Cs + KI:Rb + KI:Rb +
76.8 77.1 82.7 82.9 86.3 86.9
102 101 143 148 147 143
4- 36 4- 36 + 55 + 56 + 55 4- 55
C
9 4- 6 7-4- 6 2 4- 9 510 4- 70 53 455 439 480-l69 468 +
10 10 18 15 14 14
B/A
- 7 4- 6 0 4- 6 - 1 6 4- 9 - 1 5 4- 10
0.05 0.04 0.01 1.31
+ 0.03 + 0.03 + 0.06 4- 0.35
-19 -20 -20 -20 -21 -20
0.52 0.54 0.27 0.54 0.47 0.48
+ 0.21 + 0.22 4- 0.16 4- 0.23 -I- 0.20 + 0.21
4- 10 4- 10 4- 15 4- 15 4- 15 -i- 15
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Unusual anharmonic local mode systems
199
Table 2 Measured and QD model predicted electric-field-induced frequency shifts for the Tlu pocket gap mode. The values in parentheses give the predicted minimum and maximum shifts produced by anharmonic parameters consistent with the uncertainties in the measured Tlu gap and resonant mode uniaxial stress coefficients (After Sandusky et al. 1994). EDC
EIR
[100]
[100]
[100]
[010]
[110]
[110]
[110]
[1-10]
Aw/AE2[IO-6 cm-1/(kV/cm) 2] Exp. QD Exp. QD Exp. QD Exp. QD
1.70 -t- 0.09 1.13 (0.67/1.67) -0.93 4- 0.05 -0.42 (-0.28/- 0.58) 0.08 -t- 0.36 0.32 (0.12/0.54) 0.20 + 0.29 0.39 (0.27/0.55)
related to Aw. The stress and electric field results for the different crystal orientations and polarizations are given in tables 1 and 2, respectively. Stress measurements have been carried out on a variety of different gap modes in KI. The behavior of these gap modes is qualitatively different from that of the KI:Ag + pocket gap modes. Inspection of the coupling coefficients collected in table 1 reveals similar behavior for all impurity gap modes listed, except for Ag +. This suggests that a small B/A ratio is a stress behavior signature of the pocket gap modes, which sets them apart from the other gap modes. Note that the B/A difference is also large between the KI:Ag + pocket gap mode and the 17.3 cm -1 resonant mode (see table 1). Interestingly, the B/A ratio for the sharp band mode peak at 55.8 cm -1 for KI:Ag + is very small, comparable to that for this system's pocket modes. These examples demonstrate the unique nature of the stress behavior of the KI:Ag + pocket gap mode, the related isotope mode and the 55.8 cm -1 band mode.
2.4. The (6, 6', 6") and the quadrupolar deformability models As detailed in (Rosenberg et al. 1992; Sandusky et al. 1994), quasiharmonic extensions of the (6, 6') model prove inadequate to explain the anharmonic shifts revealed by the stress and E-field experiments. The original (6, 6') model assumed that defect-induced inward relaxation of the silver ion's six nearest neighbors produces the force constant change 6'. The magnitude of its fit value is roughly half the pure KI nearest-neighbor overlap force constant, implying substantial relaxation. This suggests in turn that relaxation-induced force constant changes 6" = -Aq~xx(200, 300) should also be included. These changes could have a strong effect on the pocket
200
A.J. Sievers and J.B. Page
Ch. 3
gap modes, since their displacement patterns are so strongly peaked on the (200) family of ions. Recent work has shown how 8" can be included without adding any free parameters. By assuming that the introduction of the defect into the unrelaxed crystal produces radial forces on just the defect's six nearest neighbors and working within a linearized theory (i.e., small relaxations), one can use the pure crystal harmonic shell model Green's functions to compute the static displacements throughout the lattice, relative to those on the defect's six nearest neighbors. If these relative static relaxations are combined with the assumption of cubic anharmonicity arising from nearest-neighbor central potentials, then the necessary force constant change ~;" can be computed uniquely in terms of 8'. For KI this procedure yields, 8" - 0 . 6 8 ' with no adjustable parameters added to the (8, 8') model. For this new "relaxation" model (8, 8~, 8"), the fits to the measured IR Tlu resonant and gap mode frequencies again give three nearly degenerate Alg, Eg and Tlu pocket gap modes, which are almost identical in frequencies and displacement patterns to the pocket gap modes predicted by the (8, 8') model. Nevertheless, despite improvements with some experimental results and the fact that the (~, St, 8,,) model is well-motivated physically, it cannot be complete: it predicts a Raman-active Eg symmetry resonant mode at 26.3 cm -1, 10 cm -1 above the experimental frequency and 6 cm -1 above the (8, 8') model prediction. Furthermore, like the (8, 8~) model, it cannot provide a basis for a consistent explanation for both the measured stress and Stark shifts. A consistent explanation of the measured stress and electric field pocket gap mode shifts has been obtained by introducing a Ag + electronic quadrupolar deformability-induced harmonic force constant change A4~xx(100,- 100) = 2A~bxu(100, -010) = t~QD (Sandusky et al. 1994). Quantum mechanically, such force constant changes arise from virtual s-d electronic transitions and have been argued to be important for the Ag + ion (Fischer et al. 1972; Fischer 1974; Dorner et al. 1976; Kleppmann and Weber 1979; Bilz 1985; Jacobs 1990; Corish 1990; Kleppmann 1976). This QD model adds but a single free parameter to those included in the original (~, ~') model; ~" is still determined uniquely in terms of ~' as described above. The force constant changes d; and ~' are obtained by fitting the measured IR resonant mode and pocket gap mode peaks at 17.3 and 86.2 cm -1, as before, and t~QD is adjusted to reproduce the observed 16.1 cm -1 Eg Raman peak. Overall, the resulting predicted harmonic properties are in substantially better agreement with experiment than for the (~, ~') model, and now the Eg resonant mode frequency is (necessarily) correct. Moreover, the QD model provides a consistent explanation for the gap mode E-field and uniaxial pressure induced shifts, plus the large E-field mixing seen for the Tlu and Eg resonant modes. The stress and E-field experiments probe the anharmonicity in the vicinity of the defect. To model the measured stress shifts, a quasiharmonic
w
Unusual anharmonic local mode systems
201
Applied Stress
Purecrystal,strains!
i_ [
Harmonic defect model
i Defectcrystal strains
!Anharm~ .........
t
Li ~
Force constant ! changes ~9 ......
1Frequency
Normalized gap mode ! displacement patterns .......................
shifts
Fig. 37. Schematic diagram illustrating the procedure for calculating the stress coupling coefficients of the KI:Ag + impurity modes. Note that both the defect-crystal strains and impurity-mode displacements are determined from the perturbed harmonic shell model. The anharmonicity is only needed to determine the force-constant changes produced by the harmonic defect-crystal local strains. The electric-field Stark shift calculations follow an analogous procedure. (After Rosenberg et al. 1992).
theory is employed as follows. First, the perturbed harmonic force-constant model is used to compute the local strains induced by the applied stress. These strains are then combined with the assumption of nearest-neighbor cubic anharmonicity to yield the local force-constant changes. Finally, these are used in a perturbation theory calculation of the stress-induced frequency shifts. This procedure is schematically outlined in fig. 37 and is detailed in the appendix of Rosenberg et al. (1992). The E-field Stark shift calculations follow an analogous procedure, with the different symmetry of this perturbation being taken into account. Figure 38 shows measured and predicted pocket mode difference spectra for an 87 kV/cm applied static field and EIRIIEDc[100] probe-field geometry. The predicted spectrum in (a) is for the (~;,d;', ~") model, with the anharmonicity parameters obtained from measured stress shifts, while the prediction in (b) is for the QD model, again based on stress-fit anharmonicities. It is seen that the QD model accounts well for the measured difference spectrum, whereas the (5, ~',~") model is in marked disagreement-
A.J. Sievers and J.B. Page
202
2.0
0.0
v
Ch. 3
(a)
f '\ ""
.,,.
\~~l, \
',.i .,]
-2.0
E
!
~"-'"-'"
".9"~0.15
0.00
-0.15
,
s5
,
86
(0 (cm "1) Fig. 38. Experimental Stark effect pocket gap mode difference spectrum, compared with the predictions of two different models. The measured spectrum, Aa = a(EDc = 87 k V / c m ) a(EDc = 0) for EIRIIEDc [100], is represented by the solid curves. (a) Comparison with the (6, 6', 6") model spectrum (dot-dash line), which is more than an order of magnitude too large. (b) Comparison with the QD model spectrum (dashed line). The theoretical Stark shifts of table 2 are computed using anharmonicities fit to measured stress shifts. The predicted difference spectra shown here are generated from the theoretical shifts by using a Voigt lineshape determined from the measured zero-field lineshape for the IR pocket gap mode. (After Sandusky et al. 1994).
it predicts shifts which are nearly two orders of magnitude larger than the experimental shifts. Table 2 compares the measured static E-field induced pocket mode frequency shifts predicted by the QD model using the QD model stress-fit anharmonicities. The computed shifts of table 2 are determined from strengthweighted frequency averages ~ - ~ i S i ~ i / ~ i Si as a function of applied field strength, and the uncertainties in the predicted shifts arise from uncertainties in the stress measurements. The predicted and observed shifts for the EIRtiEDC [100] p r o b e - field geometry are seen to overlap, while the predicted shift for EIRIEDc [100] is half the observed shift. The predicted and observed shifts for EDC [ 110] also overlap, but the uncertainty in the experimental shifts for this configuration precludes strong conclusions. Using the same stress-fit anharmonicities, the QD model also predicts a relatively
w
Unusual anharmonic local mode systems
203
large E-field induced mixing for the low frequency Tlu and Eg resonant modes (frequency shifts ,-,, cm -1 at E = 100 kV/cm); these shifts are in excellent agreement with the experimental shifts (Kirby 1971). 2.5. Discussion and conclusions A comprehensive comparison of experiment and theory on KI:Ag + is collected in table 3. Since the theoretical work has necessarily focused on the T - 0 K results, much of the table is devoted to this limit. Ten different kinds of spectroscopic data are identified in the first column to compare with the three successively more refined harmonic shell models, labeled (a), (b) and (c). Column (a) identifies the two-parameter (3, 3') model; column (b) the two-parameter (3,/~, 3") model which includes relaxation without any additional parameters and column (c) the three-parameter QD model which includes electronic quadrupolar deformability in addition to (3, 3~, 3"). Although the simple (3, 3') model does a surprisingly good job in predicting the pocket modes and provides reasonable agreement with five of the ten experimental results listed, its quasiharmonic extension cannot account for the pocket mode stress and Stark electric field results listed as 11 through 14. The two-parameter (3, 3~, 3") model may be viewed as an intermediate model on the way to the QD model in that there is improved agreement with some of the eleven experimental items while at the same time the disagreement with the experimental Eg Raman resonant mode frequency increases. The QD model is a remarkable success. With its single additional parameter fit to the experimental Eg Raman resonant mode frequency, the QD model provides the same good overall agreement with the first ten experimental results as does the (3, 3~, 3") model, and its quasiharmonic extension describes both the stress and electric field measurements consistently. From the stress and Stark measurements (table 3, entries 11 through 15), there are 15 new pieces of spectroscopic data. When five of these from the stress measurements are used to fit the anharmonic QD model, the other 10 experimental results, which include the Stark electric field data, are predicted correctly. These successes of the QD model demonstrate that the silver ion possesses a significant electronic quadrupolar deformability in KI:Ag + and that this deformability plays an essential dynamical role. Conversely, the good agreement between the experimental data and the QD model also show that an anomalous potential energy anharmonicity does not appear to play a role in the description of the T = 0 K dynamics. Although the QD model provides an excellent description of most of the T = 0 K features, there are a few discrepancies that stand out and need to be considered. The most important ones are perhaps items 8 and 9 in table 3,
A.J. Sievers and J.B. Page
204
Ch. 3
Table 3 Qualitative Comparison of the KI:Ag + on-center experimental results with calculated results based on perturbed shell models. (a) the two parameter (6, 6t) model; (b) the two parameter (6, 6~, 6") model and (c) the three parameter (6, 8~, 6") + QD model. Experimental Results
SM Results
(a)
(b)
(c)
harmonic approx, harmonicapprox, harmonicapprox. fit to (6, 6') model fit to (6, 8', 6") 1. Tlu resonant and gap mode freqs (17.3 cm- l, model 86.2 cm- l)
fit to (6, 6', 6")+QD model
2. Relative Tlu resonant and gap mode strengths (,-~ 3)
fair agreement (1.4)
good agreement (3.0)
good agreement (3.0)
3. resonant mode isotope frequency shift (-0.14 4- 0.03)
poor agreement (-0.05)
good agreement (-0.12)
good agreement (-0.12)
4. gap mode isotope frequency shift ( - 1.7)
good agreement ( - 1.46)
good agreement (-1.63)
good agreement (-1.64)
5. relative gap mode isotope strength (0.04)
fair agreement (0.073)
fair agreement (0.074)
fair agreement (0.073)
6. Eg resonant mode (16.1 cm -1)
fair agreement (20.5 cm -1)
poor agreement (26.3 cm -1)
fit to QD model
7. Alg resonant mode not
Alg resonant mode Alg resonant mode Alg resonant mode (37.3 cm -1) (41.5 cm -1) (41.5 cm -1)
observed
poor agreement
poor agreement
poor agreement
9. weak absorption peaks poor agreement at 30, 44, 55.8, 63 cm -1
poor agreement
poor agreement
Predicts Alg and Eg pocket modes with negligible Raman strengths. (87.8 cm-1, 86.0 cm -1)
Predicts Alg and Eg pocket modes with negligible Raman strengths. (87.8 c m - 1, 86.0 cm -1)
8. magnitude of the broad acoustic Tlu absorption spectrum
10. Alg and Eg gap modes not observed in Raman.
Predicts Alg and Eg pocket modes with negligible Raman strengths. (87.2 cm -1 , 86.0 cm -1)
anharmonic QD model 11. pocket mode and 17.3 cm-1 resonant mode stress effect
five parameter fit to the QD-based quasiharmonic model
12. isotope pocket mode stress effect
good agreement
w
Unusual anharmonic local mode systems
205
Table 3 (continued)
13. pocket mode Stark effect
Experimental Results
SM Results data agrees with predictions of the QDbased quasiharmonic model with no additional parameters
14. isotope pocket mode Stark effect
good agreement
15. resonant mode Stark effect
good agreement
16. Large gap mode line width in comparison to other gap mode systems (0.5 cm -1) r (0.14 cm -1)
outside of the model
Experimental temperature dependences (with increasing temperature)
Temperature dependent properties outside the framework of the QD-based quasiharmonic model
17. disappearance of the Tlu and Eg resonant and Tlu pocket mode strengths 18. weak broad IR gap mode (78.6 cm -1) appears and then levels off 19. Raman resonant mode (Alg + Eg) (12.2 cm -1) appears and then disappears into a broad central peak 20. Temperature dependent pocket mode A and B stress coefficients. which refer to the poor agreement between the model and the magnitude and weak structure in the measured impurity-induced absorption in the acoustic region, shown in fig. 36. It is found that this disagreement is preserved for both the (~, ~', d;") and the QD models; hence these experimental properties appears to be outside of the scope of the current models. The ultrasharp feature at 55.8 cm -1, listed in item 9, may not be unique to Ag + but may simply be an impurity-induced density of states peak, since a similar feature has been seen for Na + impurities in KI (Ward et al. 1975). However, the feature is produced by the Ag + ion since like the resonant mode, it disappears as the sample temperature is increased. The three models described here do not reproduce such an ultrasharp feature in the acoustic spectrum. Item 16, the T = 0 K pocket gap mode linewidth is another property that stands out. Except for Ag + which has a gap mode width of 0.5 cm -1, the linewidth for all other point defects in KI is about 0.14 cm -1. This factor
206
A.J. Sievers and J.B. Page
Ch. 3
of three difference may be significant. The smaller of these two values could be a consequence of random DC strains in the crystal producing an inhomogeneously broadened gap mode band but, according to table 1, the ratio of the hydrostatic coupling coefficients of Ag + to the Cs + gap mode, which is closest in frequency, is equal to 1.26: a value much too small to explain this linewidth variation. Since the resulting QD model parameters now indicate that only standard anharmonicity parameters are associated with pocket modes, the large linewidth observed for KI:Ag + does not appear to be driven by an unusual potential energy anharmonicity. The last section in table 3 lists a number of observed temperature dependences (17 through 20), all of which are outside of the QD model. We wish to make the point here that the temperature dependence results appear to be outside the scope of many other phenomenological models as well. For example, applying Boltzmann statistics to the two configuration model (fig. 17) shows that to account for the rapid temperature dependence of the resonant mode strength (fig. 14), the number of levels in the off-center C2 configuration state must be at least 100 times that of the on-center C~ ground state configuration. What physical process for a single point ion in a lattice could generate this much entropy by 25 K? One conclusion is that these temperature-dependent experimental results are not only outside of current lattice dynamics theory but are difficult to understand even in a more general phenomenological framework. One conclusion which appears to be independent of the details of the temperature dependence is that the quantitative temperature-dependences of the strengths of the IR resonant and pocket modes provide important complementary information on the participation of the Ag + impurity's surrounding ions in this system's anomalous thermally-driven on-to off-center transition of the Ag + ion. Measurements and analysis have demonstrated that both the low-frequency IR resonant mode and the strong pocket IR gap mode disappear at identical rates with temperature in the range 0 K to ~ 25 K, even though the dynamics of each, according to all three perturbed shell models, involve ion motion in different regions of the defect vicinity. Since the mode frequencies are nearly temperature-independent over this temperature interval, it appears that these two nearly-harmonic modes simply monitor, in different spatial regions, the population of the on-center configuration; hence, the thermally-driven instability in this system involves the entire coupled defect/host system in the impurity region.
3. Intrinsic localized modes in perfect anharmonic lattices We now tum from the previous, experimentally-driven study of a fascinating anharmonic solid-state defect system, to the still purely theoretical
w
Unusual anharmonic local mode systems
207
problem of intrinsic localized modes in anharmonic lattices. As noted in the Introduction, our focus is on simple descriptive aspects of some of the interesting phenomena. We will begin with the simpler case of monatomic lattices with harmonic plus hard quartic nearest-neighbor interactions and will later consider the important generalizations of adding cubic anharmonicity or treating full realistic potential functions V(r). When ILMs occur in these monatomic systems, their frequencies are above the maximum frequency of the harmonic lattice, and we will describe their existence, stability, motion and interactions with plane-wave phonons. We will then consider diatomic lattices for the same interactions; in this case one has the interesting additional possibility that ILMs can exist in the frequency gap between the acoustic and optic phonon branches.
3.1. One-dimensional monatomic lattices
3.1.1. Heuristic illustration of vibrational localization We begin with a qualitative example to illustrate why localized modes might be expected to appear in a perfect but anharmonic lattice. For a onedimensional monatomic harmonic lattice with particles of mass m connected by nearest neighbor spring constant k2, the normal modes are described by an orthogonal set of homogeneous plane waves which are confined to a band of frequencies with a high frequency cutoff at Wm. The amplitude pattern of this highest-frequency mode involves every atom vibrating 7r out of phase with its neighbor and is shown in fig. 39(a). To estimate the rms amplitude of a particular particle in a plane wave harmonic mode ~o for a chain of N particles, we use the virial theorem. For a harmonic system the mean energy in a mode is equal to two times the mean kinetic energy, e.g., hw/2 - N[mwe(u2)] hence the rms amplitude at each site n in this plane wave mode, (u2) 1/e ~ (N)-l/z(h/2mw) 1/e, is quite small for a long chain. Next we form a localized wave packet at t -- 0 at a particular site with the appropriate linear superposition of this set of plane wave modes. This localized excitation in which only a few atoms, Nd, are displaced from their equilibrium position is shown in fig. 39(b). If this were a normal mode, the central atoms would have relatively large rms amplitudes, (U2) 1/2 ~ (Nd)-l/2(h/2mw) 1/2, independent of the number of atoms N in the long chain. However, this is not a normal mode for the harmonic system, and for t > 0 lattice dispersion insures that this localized excitation would not remain confined for long. For a similarly constructed one-dimensional anharmonic lattice with N particles, the plane wave spectrum derived in the small oscillation limit is
208
A.J. Sievers and J.B. Page (o) 9
&
9
COm
9
9
Ch. 3
9
9
....9
9
(b)
......
_i
[
t
i
L_ r-
t
Nd
~uj
--J --I
(c)
3~
Wm
/////////////////~ ///1////////// .
.
.
.
.
.
.
.
[
I
Fig. 39. A schematic representation of the frequencies and eigenvectors of plane wave and localized modes. (a) The eigenvector of the highest frequency plane wave mode at Wm. (b) A localized mode with frequency w~ involving a subset Nd of the N lattice sites. (c) The plane wave vibrational spectrum located between 0 and win, the anharmonic localized mode with frequency w], and the third harmonic of this local mode (dashed line). The local mode eigenvector A( . . . . 0, - 1/2, 1, - 1/2, 0 . . . . ) for the strongly localized (triatomic molecule) limit is also shown on the right.
necessarily similar to that found for the harmonic case. But when a wave packet is constructed at t = 0 to represent a localized disturbance in this chain, a different situation occurs from the harmonic example above. To be specific we assume that the potential energy anharmonicity is quartic with a positive coefficient, and we imagine that the wave packet ~is localized in stages. As the wave packet corresponding to fig. 39(b) is formed, the amplitude of the particle at the central site increases, but since the potential is no longer harmonic, so does its frequency of vibration. The quartic anharmonicity may be thought of as renormalizing the harmonic force constants within the packet to the higher values. According to Rayleigh's theorem the resulting frequency shift for each plane wave mode can be no larger than the frequency interval between the original lattice modes (Maradudin et al. 1971). As the amplitude pattern becomes more strongly localized around a particular site so that Nd --+ 1, the vibrational amplitude becomes
w
Unusual anharmonic local mode systems
209
larger, allowing the maximum harmonic lattice frequency win, which is not bounded from above, to rise above the plane wave spectrum, transforming to an inhomogeneous localized mode in the process. Given sufficient anharmonicity the end result is a true localized mode centered at the particle in question with frequency Wl as shown in fig. 39(c). The anharmonic system has evolved in such a way that both homogeneous and inhomogeneous modes exist simultaneously. Depending on the anharmonicity of the lattice, the eigenvectors shown in fig. 39 represent three different possibilities: Case (a) depicts the highest frequency ~m plane wave mode for the small anharmonicity limit. With increasing anharmonicity a localized solution with frequency a;~ > tom also exists above the plane wave spectrum, and its eigenvector is represented by (b). For this case there are roughly Na atoms involved in the localized vibration and since Na << N the amplitude of each atom in this mode is much larger than for the plane wave case. In the high frequency limit, Wl >> Wm, the local mode eigenvector takes on the particularly simple form of a vibrating triatomic molecule as shown in case (c). The spectrum shown in fig. 39(c) includes not only a plane wave spectrum of modes plus one localized mode but also an overtone of the local mode, represented by the dashed line. In principle, the anharmonic quartic potential produces a response at the third, fifth and higher harmonics, but because these frequencies are so much larger than both the plane wave and the local mode frequencies, it is reasonable to assume that the lattice cannot respond at these higher frequencies so that these nonlinear mixing terms can be ignored in the dynamical analysis. This approximation is called the rotating wave approximation (RWA). The original anharmonic lattice with the spectrum shown in fig. 39(c) is now described by a set of effective harmonic oscillators, both plane wave-like and localized.
3.1.2. Quantitative study of quadratic and hard quartic nearest-neighbor potentials (k2, k4) As noted in the Introduction, the prediction of Sievers and Takeno (1988) was based on this case, for which the potential energy function is given by ~2
v = -
2
(Un+l -n
k4 Z ( U n + l _
+ 7
Un)4
(3.1)
n
where k2 and k4 are the harmonic and quartic spring constants, respectively, and un is the longitudinal displacement of the nth particle from its equilibrium position. For a purely harmonic lattice, the solutions of the
210
A.J. Sievers and J.B. Page
Ch. 3
equations of motion are the familiar plane waves, with the dispersion relation w(k) = Odm sin(ka/2), where a:m =-- 2(k2/m) 1/2 is the maximum lattice frequency and a is the equilibrium lattice spacing. Writing the equation of motion for the nth particle, substituting the trial solution un(t) = A~,~ cos(a;t) and making the rotating wave approximation (RWA) by replacing the resulting cos3(wt) term by the first term (3/4)cos(cot) in its Fourier expansion, one obtains
--od2m~n
"-
k2(~n+l -- 2~n + ~,,+l) +
3k4A 2 4 [(~n+l -- ~n) 3 -- (~n -- ~n-1)3],
(3.2)
where m is the particle mass. This was the procedure followed by Sievers and Takeno (1988), and an approximate solution was then obtained via the use of lattice Green's function techniques. The result was that odd-parity ILMs with the approximate displacement pattern A(..., 0, - 1/2, 1, - 1/2, 0 . . . . ) given in w1.2 above can exist at any lattice site, with the frequency given by
(w) ~mm
3( ~
27A4 1+
16
'
(3.3)
where the anharmonicity parameter is A4 -- k 4 A 2 / k 2 9 This solution was argued to be a good approximation provided An >> 16/81. As we have noted, the molecular dynamics (MD) simulations of fig. 1 strikingly verify this solution. The value of A4 for the simulation in the lower panel was 1.63, well within the range of validity of eq. (3.3). A power spectrum of the MD displacements of fig. l(b) shows that the observed ILM frequency is within 2% of that predicted by this RWA equation. The theoretical arguments of Sievers and Takeno (1988) utilized a lattice Green's function formalism that is somewhat complicated to the uninitiated. Yet the ILM vibration appearing in fig. 1 is exceedingly simple. In the following subsection we give a simple and direct argument which shows that the tendency to localization in this case of strong quartic anharmonicity reflects a fundamental property of the underlying purely anharmonic system.
3.1.2.1. Asymptotic behavior; odd and even modes. To more simply understand the above ILM solution and its connection to the anharmonicity, we
w
Unusual anharmonic local mode systems
211
now focus directly on the equations of motion, for the hypothetical pure quartic case. Setting k2 - 0 in eq. (3.2) and rearranging, we have
w2 ---
3k4A 2
[(~n - ~n+l) 3 at- (~n -- ~n-1)3] 9
(3.4)
4m(n Let us now seek a localized odd-parity solution, centered at site n = 0. We thus require (0 = 1, ~-n = (n, and I~1 << I~11 for Inl > 1. Within these restrictions, the n = 0 and n = 1 versions of eq. (3.4) give
w2 =
3k4 A2
2(1 - (1)3,
n = 0,
(3.5)
4m and w2 ~
3k4A 2
[(~+(~1-1)3],
n=l.
(3.6)
4m~cl For a solution, these two frequencies should of course be the same, and we see that for ~1 = - 1 / 2 they become nearly equal"
w2k4A2(3) -
m
4 ~
,
n=0,
(3.7)
and k4A2 ( 3 ) 4 ( 60 2 ~
1+
1 )
n-
1
9
(3.8)
Next we proceed to the equation of motion for the particle at n = 2:
602
3k4 A2
[({2 -- ~3)3 -t- (~2 -- {1)3] 9
4m~2 In accord with our approximations, we set ~ 2 - (l ~,~ 1/2 and neglect ( 2 - (3 in comparison, obtaining w e ~ (3k4A2)/(32m(2). By equating this with the n = 0 expression for w 2 (eq. 3.7), we find 1
~2 ~ ~ . 54
(3.9)
A.J. Sievers and J.B. Page
212
Ch. 3
Thus 1~2] << I~l[ = 1/2. Finally, assuming I~n+ll << I ~ l for all n/> 2, we use eq. (3.4) to obtain a recursion relation for n/> 2
..~
.
(3.10)
Hence, all of the ~n's for In[/> 2 rapidly approach zero with increasing n, and it is seen that the odd-parity displacement A(..., 0, - 1/2, 1, - 1/2, 0,...) is indeed an approximate solution for the pure quartic case. The ILM frequency given by eq. (3.7) is the same as that given by the k2 = 0 version of eq. (3.3), and we find that this RWA frequency is within 2% of the exact frequency observed in MD simulations. When harmonic interactions are added, eq. (3.3) is recovered, and as noted earlier, it is a good approximation provided that the anharmonicity parameter does not become so weak that the homogeneous plane-wave harmonic solutions become dominant. Beyond the insight provided by this simple heuristic argument, it can easily be generalized to give an interesting exact result (Page 1990). Suppose that the nearest-neighbor pure quartic interaction is replaced by a nearestkr neighbor anharmonic interaction of arbitrary even-order: V = T ~,~(Un+lun) ~, where r = 4, 6 . . . . . The equations of motion are then
m~tn(~) ~ ]gr{ [Un+l(~)- Un(~)] r-1 -- [Un(~)- Un--l(~)] r-1 }.
(3.11)
Our focus here will be on the spatial behavior of the solutions, and for this purpose we could use the RWA for the time dependence, just as we did above for the pure quartic case. However, eqs (3.11) readily separate, giving rise to solutions periodic in time, for any even r (Kiselev 1990). Thus for the sake of generality, we briefly digress to bring in the exact, rather than the RWA frequency. For the trial solution un(t) = A~nf(t), it is straightforward to obtain an exact expression for the period T, from which the square of the frequency w = 27r/T is obtained as
W 2 B"k"A"-2[ rn~
(~ - ~+1
),.-1
+ (~n --
~-i
)r-l]
(3 12) ,
where the coefficient B~ is given by
)2
B,.-~
x/1-f ~
.
(3.13)
w3
Unusual anharmonic local mode systems
213
Notice that for the r = 4 pure quartic case, eq. (3.12) would go over to the RWA result eq. (3.4), provided/34 = 3/4. Indeed, evaluation of the elliptic integral appearing above gives B 4 = 0.718, and we again see that the RWA works very well for the pure quartic case. We now return to the question of the spatial behavior and focus on the (n's in the right-hand side of eq. (3.12). Following the preceding argument for eqs (3.5-3.10), we seek a localized odd-parity ILM, centered at site n - 0. Again neglecting (2 compared with (1, we find that the n = 0 and n = 1 versions of eqs (3.12) are nearly satisfied by (0 = 1 and (1 = - 1 / 2 :
co2__2krBrAr-2 ( 3 ) r-1 -
m
~
,
n-O,
(3.14)
and
w2 ~
2krBrAr-2(3)r-l[ (1) r-l] 1+
n-
1
(3.15)
These equations differ only by the factor 1 + (1/3) r - l , which approaches 1 in the asymptotic limit of large r. Turning to the n = 2 version of eq. (3.12), we again have ~2 - (1 .-~ 1/2 and neglect ~2 - (3 in comparison, obtaining w 2 ..~ (krB~A~-2)/(m(22~-l). Setting this equal to the n - 0 expression for w 2 (eq. 3.14), one finds (2 --~
1
2.3
,
(3.16)
r-1
which approaches zero in the limit of large r. As a final step, we again assume that I~n+xl << I~nl for all n >/ 2, and use eq. (3.4) to derive the general-r analog of the recursion relation eq. (3.10) for n t> 2:
,.+, ~n
..~
~n-1
.
(3.17)
Clearly, the (n's rapidly approach zero with increasing distance for fixed r, and they all vanish in the large r limit. Thus the odd parity ILM pattern A ( . . . , 0, - 1/2, 1, - 1/2, 0 , . . . ) is an asymptotically exact solution in the limit of increasing even anharmonic order (Page 1990). Even for the r = 4 "worst" case of a pure quartic system, this solution remains very a c c u r a t e - in the preceding reference, a more exact
A.J. Sievers and J.B. Page
214
Ch. 3
2.2 f
1.8
even ~
/i
odd
1.4
1.0
!
0
]
2
A4 Fig. 40. Computed odd- and even-parity ILM frequencies versus the quartic anharmonicity parameter A4. These curves were obtained by numerically solving the RWA equations of motion for all of the particles in a 40-particle monatomic linear chain, with periodic boundary conditions. Three of the corresponding odd-parity ILM displacement patterns are shown in the next figure.
calculation for the pure quartic case was found to correct the mode pattern only slightly, to A( . . . . 0, 0.02, -0.52, 1, -0.52, 0.02, 0 . . . . ). The simple odd-parity ILM pattern given by Sievers and Takeno (1988) therefore reflects a fundamental exact property of the underlying purely anharmonic system. This pattern is just that of a simple linear triatomic molecule of equal masses, and is in fact the most localized odd parity pattern which keeps the center of mass at rest. This leads naturally to the question of whether the most localized even parity displacement pattern, namely that of a linear diatomic molecule A( . . . . 0 , - 1, 1,0,...), might also be an asymptotically exact solution for the purely anharmonic system in the same limit of increasing evenorder anharmonicity. This was proven to be the case by Page (1990). For the pure quartic case this asymptotically exact mode pattern is corrected to A ( . . . , 0, 1/6, - 1, 1, - 1/6, 0 . . . . ), and the corresponding RWA frequency of the pure-quartic even-parity mode is given by w2 ~ (6kaA2/m)[1 +(7/12)3]. Again, this is readily verified by MD simulations. Even modes were discovered independently in numerical simulations by Burlakov et al. (1990a-d) and Bourbonnais and Maynard (1990). In the following section, we will see that the above asymptotic limit also gives simple insights into the stability properties of the odd and even ILMs. However, before moving to this topic, we return briefly to the harmonic plus quartic (k2, k4) case and note some additional simple aspects. As the quartic anharmonicity parameter An - kaA2/k2 decreases, the ILMs are expected to
w3
Unusual anharmonic local mode systems
215
(a) ~/O~m=1.79
(b) o)/O~m=l.
(c) o~/r
Fig. 41. Computed odd-parity ILM normalized displacement patterns {~n} as a function of the ILM frequency for a 40-particle (k2, k4) linear chain, with periodic boundary conditions. As for fig. 39, the RWA equations of motion were solved numerically for all 40 particles. In practice, one begins with an initial guess for the displacement pattern, and the routine converges to the correct pattern and frequency. The particle motion is longitudinal, but for clarity the displacements are plotted vertically. The displacement of the central particle is unity in each case. spatially broaden, and this is indeed found to be the case. For a fixed value of this parameter one can imagine moving out from the mode center until the displacements are so small that the anharmonic effects are negligible, and since the ILM frequency is necessarily above the maximum frequency Wm of the harmonic lattice, the amplitudes then decrease with distance just as for a localized impurity mode in the harmonic lattice. In one dimension this decrease is a simple exponential. This gives a straightforward numerical means for obtaining the mode displacement pattems: one simply solves the equations of motion (3.2) numerically for the particles having nonnegligible amplitudes and then applies the known harmonic-approximation analytic amplitude decrease for the particles beyond, as a boundary condition. A related approach is to apply periodic boundary conditions and simply solve the equations of motion numerically for all of the particles. These techniques have been used by a number of investigators (Bickham and Sievers 1991; Bickham et al. 1993; Kiselev et al. 1993; Kiselev et al. 1994b; Sandusky and Page 1994).
A.J. Sievers and J.B. Page
216
Ch. 3
6.0
5.0
1D
4.0
I
I
/
,I
,f
,,f./
/
2.0 1.0 0.0
, ! 1.0
2.0
3.0
4.0
A4 Fig. 42. Local mode frequency versus A4 as calculated by two different rotating wave approximations. The dashed curve follows from the single frequency rotating wave approximation while the solid curve includes an additional contribution from the third harmonic term. The more exact two frequency calculation produces a slight lowering of the ILM frequency over that produced by the simple RWA. (After Bickham and Sievers 1991). Figure 40 plots the computed odd- and even-parity ILM frequencies for a (k2, k4) lattice versus the quartic anharmonicity parameter An, and fig. 41 shows odd-parity ILM displacement patterns for three different values of this quantity. The ILM spatial broadening with decreasing anharmonicity is clearly apparent; in the purely harmonic limit (k4 = 0), both the odd and even ILMs broaden into the zone boundary phonon mode A ( . . . , 1 , - 1 , 1 , - 1 , . . . ) . Interestingly, we will see in a later section that when cubic anharmonicity is added, this spreading is largely suppressed. In developing these analytic local mode solutions, it is assumed that the system only responds at the "fundamental" frequency in the assumed c0s(wt ) solutions. Because of the quartic potential, response at 3w, 5co, etc. should also be present. A straightforward approach to investigate the influence of the next higher order term is to generalize the rotating wave approximation to u,~ = (1 - 3)~n cos(wt) + 3~n cos(3wt). When this trial solution is inserted back into the equations of motion, the result is that the ILM frequency is corrected to a new lower frequency value. Figure 42 shows the magnitude
w3
Unusual anharmonic local mode systems
217
of the shift on the odd mode solution for one, two and three dimensions. As might be expected, the correction term grows with increasing anharmonicity parameter but it remains a small contribution over the entire parameter range. This figure confirms the idea that the simple rotating wave approximation is a valid approximation for identifying ILMs in anharmonic systems.
3.1.2.2. Stability. An important question concerns the stability of the ILMs against infinitesimal perturbations. By returning briefly to the asymptotic limit discussed in the previous section, we can easily determine the basic stability properties of these modes. Subsequently, we will sketch some of the quantitative aspects of these properties, followed by a discussion of their consequences for ILM motion. Much of the following stability material follows from the study by Sandusky, Page and Schmidt (1992), which should be consulted for details. The material on moving ILMs derives from that reference and the study by Bickham et al. (1992). It was seen above that the odd- and even-parity displacement patterns A(..., 0, - 1/2, 1, - 1/2, 0,...) and A(..., 0, - 1, 1,0,...), are asymptotically exact for a purely anharmonic lattice in the limit of increasing even-order anharmonicity. In this limit the interparticle potential becomes that of a square well. For point masses the "repulsive" side of the well occurs when the particle collide, at Un+l - - U n --" - - a , and because of the reflection symmetry possessed by an even-order potential, the "attractive" side of the well occurs at un+l - u n -- +a. Thus the square well has width 2a, and the particles move completely freely until either of two situations arise: 1) they collide elastically when their separation is zero, or 2) they attract impulsively ("snap back") when their separation reaches 2a. This limiting behavior for strong even-order anharmonicity is intuitively clear, and it has been derived rigorously by Sandusky et al. (1992). In this limit, the asymptotically exact even- and odd-parity ILM patterns above require that the amplitudes have the fixed values A - a/2 and A = 2a/3, respectively. This is easily seen in fig. 43. More importantly, one also sees clearly that for the even mode the collision and "snap" occur at different instants, whereas for the odd ILM the central particle simultaneously collides with one of its nearest neighbors and "snaps back" due to its attraction to the other nearest neighbor. It is then easy to see that any perturbation which destroys the simultaneity of the collision and snap will destroy the coherence of the odd-parity mode pattern, rendering this mode unstable. On the other hand, since the collision and snap in the even parity ILM are not simultaneous, this mode is stable. The above simple picture of ILM instability in the asymptotic limit of high even-order anharmonicity carries over to the case of even- and odd-parity modes in (k2, k4) systems; however, this case is more complicated than the
218
A.J. Sievers and J.B. Page
Ch. 3
O _1 7
L n
Collision (repulsion)
2a _I
L
-A
"Snap" (attraction)
A
Even-parity mode A( .... 0,-1,1,0 .... )
Collision and snap simultaneous
21=1 I"
9
I_ I-
II
oi-.o -A/2
Odd-parity mode
-I _1 -I
~
9 A
-A/2
A( .... 0,-1/2,1,-1/2,0 .... )
Fig. 43. Collision (repulsion) and "snap" (attraction) for the even-parity ILM (top panel) and for the odd-parity ILM (bottom panel), in a monatomic lattice of point masses interacting via a nearest-neighbor anharmonic potential of even order, in the asymptotic limit of high order. In this limit, the potential becomes that of a square-well of width 2a and these mode patterns are exact. For the even-parity mode, the collision and snap do not occur at the same time, whereas for the odd-parity mode they do. This renders the odd-parity ILM unstable against any perturbation which destroys the simultaneity of the collision and snap. For the case of (k2, k4) lattices, the even- and odd-parity ILMs remain stable and unstable, respectively; moreover, the odd parity instability results in the ILM moving slowly from site to site, as discussed in the text. (After Sandusky et al. 1992).
above simple limiting case, and it requires careful analysis involving both analytic and numerical work. Briefly, one assumes a solution of the form
un(t) = A[~n + 5~ne~t] cos [wt + 6r
(3.18)
where 8~n and 8r are infinitesimal displacement and phase perturbations, respectively. Since M D simulations show that both the odd- and even-ILMs are generally stable over at least several periods, we assume that the above perturbations vary slowly in time with respect to a mode period. Performing an appropriate time-average (closely related to the RWA) and linearizing the equations resulting from the trial solution (eq. (3.18)), we arrive at a 4s • 4s eigenvalue problem to determine A, where s is the number of sites included in the unperturbed ILM displacement pattern {(n}. The ILM will
Unusual anharmonic local mode systems
w
219
0.18 o
0.14
= =,=,.
l-.
r 0.10 L_
i-.-
0.06
0
0.02 1.0
z~ I
I
I
2.0
3.0
4.0
5.0
(O/tOm Fig. 44. Instability growth rate vs. anharmonicity for odd-parity ILMs in a 21-particle harmonic plus quartic lattice, with periodic boundary conditions. The anharmonicity is measured by the ratio of the ILM frequency to the maximum frequency of the harmonic lattice. The solid curve gives theoretical predictions obtained from the stability analysis sketched in the text, and the triangles are growth rates measured in MD simulations for various values of the amplitude and the ratio k4/k 2. The dashed line gives the predicted growth rate for the pure quartic lattice. (After Sandusky et al. 1992). be unstable if one finds a perturbation (determined from the eigenfunction) of the displacements or phases (i.e. velocities) which is associated with an eigenvalue A having a positive real part, since such perturbations will grow exponentially in time. Sandusky et al. (1992) should be consulted for details. Numerically applying this analysis, we find that the odd-parity pure quartic ILM is always unstable against even-parity displacement or phase perturbations [e.g. ( . . . , - d ; a , 0, d;a. . . . )], whereas the even-parity ILM is always stable. These results agree with the more intuitively clear asymptotic limiting behavior discussed above. With harmonic interactions (k2) included, the odd-parity ILM instability growth rates are predicted to decrease with decreasing anharmonicity, and the even-parity ILM is predicted to be stable. Figure 44 shows the predicted odd-parity ILM instability growth rates as a function of anharmonicity for the (k2, k4) lattice, and these are compared with growth rates measured in MD simulations. The dashed line gives the pure quartic limit. The predicted and MD results are seen to be in very good agreement; as discussed by Sandusky et al. (1992), the ILM spatial broadening with lowering anharmonicity was not included in the growth-rate predictions in this figure, and when they are included the minor discrepancies at low anharmonicities are removed.
A.J. Sievers and J.B. Page
220
Ch. 3
2.5 1.5 t. .O, .
o r (D
0.5
-~ -0.5 ...=
ca. -1.5 -2.5 19420
i
I
19430
19440
19450
time (units of 2~(Or.) Fig. 45. Even-parity mode stability in a 20-particle pure quartic lattice, as seen in MD simulations after more than 32,000 oscillations. The initial displacement pattern is the pure quartic even-parity pattern ( . . . . 0, 1 / 6 , - 1, 1 , - 1/6, 0 . . . . ), centered at sites (-0.5, 0.5). For this run k4 and the amplitude are chosen so that w = 1.7Wm, where Wm= 1.0 (eV/~, 2 amu) 1/2 is a convenient frequency unit. The displacements are magnified, for clarity, and the displacements on the particles not shown are negligible. This mode is exceedingly stable, as predicted by the perturbation theory analysis. (After Sandusky et al. 1992).
As noted above, no instability is predicted for the even-parity ILM, and MD simulations have found this mode to be extremely stable: runs for the pure quartic and for the harmonic plus quartic case found no changes in the even-parity mode over more than 32,000 oscillations, as is illustrated in fig. 45 for the pure quartic case. As pointed by Sandusky et al. (1992), the even-parity ILM stability is further manifested by the fact that even when given t = 0 perturbation seeds which cause the odd-parity ILM to move after just tens of oscillations, the even mode was still found to persist unchanged for more than 32,000 oscillations (the maximum extent of the MD runs). Given that the odd-parity ILMs are unstable, the question arises as to how the instability manifests itself. In MD runs, it is observed that the instability does not destroy the ILMs, but rather causes them to move.
3.1.2.3. Translational motion. To find a traveling ILM, Bickham et al. (1992) substituted the trial solution un = A~n(t)cos(wt- kna) into the equations of motion for the different particles, where A is the maximum amplitude of the moving ILM in a lattice of spacing a, (n(t) is a slowly varying envelope function, and k and w are the wave vector and frequency, respectively. The resulting set of equations are then numerically solved by assuming that the
w
Unusual anharmonic local mode systems
2.5
9
m
9
i .......
9
I
"
I
221
9
O
2.0 E
:3 :3
1.5
"
1.0
o.o
,
,
.
.
.
0.2
.
.
~
,
.
.
e
o.3 0.4 0.5 ka
Fig. 46. Dispersion curve of the t : 0 odd-symmetry traveling ILM for four different anharmonicity values. The values from top to bottom are A4 = 2.5, 1.6, 0.9 and 0.4. The dashed curves identify solutions of the equations of motion using the localization condition in the text. The solid lines indicate regions where simulations of moving modes are successful. The open circles are determined from the simulated displacements by assuming a Gaussian envelope function. Qualitatively similar results have been found for the t = 0 even-symmetry traveling ILM, but the odd modes cover a larger region of w(k) space. (After Bickham et al. 1992). t = 0 solution has the form: ~-n = ~,~ = (-1)nA~I e x p [ - ( n - 1)Ka] for positive n and K . For given values of the wave vector k and the anharmonicity parameter A4 = k4A2/k2, the time-dependent equations for sites 0, 1 and 2 are numerically solved to obtain the normalized frequency (W/Wm), the relative amplitude at the nearest neighbor site ~1, and the decay constant K . It is found that smoothly moving ILMs of the assumed type are produced only in a restricted region of w(k) space, with this region becoming more restricted as the anharmonicity and the ILM frequency increase. Figure 46 shows the dispersion curve that is found when the trial solution is used to fit numerically the displacements as the wave packet associated with a t - 0 odd-symmetry ILM travels across several lattice sites. The solid curves identify regions of w(k) space where the excitation moves with a constant envelope velocity for at least 15 lattice sites. This velocity is typically smaller than 15% of the harmonic lattice sound velocity. The absence
222
A.J. Sievers and J.B. Page
Ch. 3
Fig. 47. Displacement vs. time as the vibrational excitation passes through a fixed lattice site. The solid curve gives the results. The group velocity of the envelope is 7.2% of the harmonic lattice sound velocity. The dashed curve represents the best fit of the excitation envelope to a Gaussian envelope function. For comparison the dotted curve represents a hyperbolic-secant-function envelope. (After Bickham et al. 1992).
of a solid line at small k values identifies that region where the mode either remains stationary or only moves a few lattice sites before coming to a stop. At larger k values, on the other side of the solid line, the mode moves but decelerates. Uniform motion is observed over a larger region w(k) for the t = 0 odd mode than for the t = 0 even mode, consistent with the idea that the odd mode has an intrinsic translational instability. The solid line in fig. 47 shows a typical simulation trace of displacement versus time as the vibrational excitation passes through a particular site. The dashed line represents a gaussian function best fit to the pattern. Note that a hyperbolic-secant-function best fit represented by the dotted curve does not agree with the simulated amplitude in the wings, while the Gaussian envelope matches fairly well over the entire interval. One conclusion is that the shape is clearly different from the hyperbolic secant function previously found for solitons in continuous systems. The preceding discussion is for a traveling ILM described by the trial solution un = A~(t)cos(wt-$n), where the phase Sn is equal to kna; that
w
U n u s u a l a n h a r m o n i c local m o d e s y s t e m s
223
Fig. 48. MD simulation of a traveling ILM in a 21-particle harmonic plus strong quartic lattice, with periodic boundary conditions. The anharmonicity parameter is A 4 --- 1.63, and the t = 0 displacements are given by the translationally unstable odd-parity ILM pattern A(.... 0,-1/2, 1,-1/2,0 .... ), with A = 0.1a. The mode frequency is W/Wm = 1.67. The ILM motion is produced by seeding the MD run with a small initial velocity perturbation /q = -/~l = 0.00718, in units of wma, corresponding to a relative phase perturbation of 6r - 6r = 6r - 6r = -0.086 rad. The speed of this traveling mode is 0.053 in units of wma, well below the harmonic sound speed of 0.5. The displacements are magnified, for clarity. (After Sandusky et al. 1992).
is for a constant phase difference between adjacent sites. There exists another type o e smoothly traveling ILM, for which the relative phases between adjacent sites is not constant (Sandusky et al. 1992). This type of traveling mode can exist for large values of the anharmonicity, and an example is shown in fig. 48. This mode moves with a speed approximately 1/10 of the harmonic sound speed, and as it slowly moves from site to site, its mode pattern alternates approximately between the odd- and even-ILM patterns. Figure 49 plots the phase difference between adjacent relative coordinates d~ -- Un - Un-1 for the moving mode of fig. 48 as a function of time, as the mode passes a single site. The phase difference is seen to be nonconstant. Moreover for this mode, nonconstant phase differences of the same magnitude are obtained between the adjacent displacements {un} themselves, as well as between the adjacent relative displacements {dn}. Depending on the strength of the anharmonicity and the initial conditions, at least two other sorts of ILM motion have been reported: (1) The ILM becomes trapped at a site (Bickham et al. 1992) and converts to a stable even-parity ILM (Sandusky et al. 1992). (2) The ILM oscillates between
224
A.J. Sievers and J.B. Page
Ch. 3
Fig. 49. Phase difference between the relative displacements d_4 = u_4 - u - 3 and d_ 3 = u_3 - u_2 for the traveling mode of fig. 47. The triangles give the relative phase as the traveling mode passes the n = - 3 particle. This is a type of traveling ILM where the phase is nonconstant. (After Sandusky et al. 1992).
Fig. 50. ILM oscillations between lattice sites, observed in MD simulations for a 21particle harmonic plus quartic lattice, with periodic boundary conditions. The anharmonicity is A4 = 1.25, and the initial displacement pattern is that of the odd-parity ILM for this anharmonicity, together with a small perturbation ( . . . . ~a, 0, - ~ a . . . . ), where da is 0.01% of the unperturbed mode amplitude. The dashed line is a guide that follows the mode's "center". Notice that this translationally unstable ILM does not move from its initial location until roughly 15 oscillations have occurred. The displacements are magnified, for clarity. (After Sandusky et al. 1992).
w
Unusual anharmonic local mode systems
225
Fig. 51. MD simulation of colliding ILMs in the 21-particle (k2,k4) lattice of fig. 48. The initial configuration produces two traveling ILMs of the same type as in that figure, and they are seen to reflect from the ends of the chain (for the symmetry of the initial configuration used here, the periodic boundary conditions are equivalent to free end boundary conditions). They then collide, producing ILMs traveling with much higher velocities. The displacements are magnified, for clarity. (After Sandusky and Page, unpublished).
adjacent sites (Sandusky et al. 1992). An example of this oscillatory behavior is shown in fig. 50. Finally, in passing we show in fig. 51 an MD simulation of two moving ILMs colliding. These are for a 21-particle (k2, k4) chain with periodic boundary conditions, and the symmetry of the initial conditions is such that the ILMs are symmetrically reflected inward from the ends. Upon collision it is seen that two ILMs emerge symmetrically, with much higher velocities. This is an isolated MD run, and it is not clear to what extent it is a special case. Nevertheless, this figure suffices to show clearly that traveling ILMs are not solitons in the usual sense.
3.1.2.4. The effect of a light mass defect. For the case when a light substitutional mass defect md < m is present at site n = 0, it is straightforward to generalize the arguments in w and show that in the asymptotic limit of increasing even-order anharmonicity, the exact odd-parity mode pattern A ( . . . , 0 , - 1/2, 1 , - 1 / 2 , 0,...) for the perfect lattice is replaced by
A.J. Sievers and J.B. Page
226
Ch. 3
(Kivshar 1991) A
md
md
)
. . . , 0 , - ~ m ' 1 , - ~,2m . . . .
(3.19)
Just as for the perfect lattice case, this remains a very good approximation for the case of pure quartic interactions. For the case of nearest-neighbor harmonic plus quartic potentials (k2, this mode pattern remains a good approximation provided that 16/81, just as for eq. (3.3), with the defect mode frequency given by
An>>
(Wd) 2 1 ( m ) ( ~
~
k4),
md)I l + ~ mm
3A4( r o d ) 2] 1+--~--l+2m
.
(3.20)
For md = m this reduces to eq. (3.3), and in the md --+ 0 limit of a very light mass, the defect mode pattern just becomes that of an Einstein oscillator A(..., 0, 1,0,...), as is intuitively clear, with the frequency going as
Bickhamet al. (1993) have studied the influence of a stationary light mass defect on the scattering of moving ILMs. Figure 52 shows the amplitude and frequency associated with the mass defect localized mode that is produced with the energy captured from a passing ILM. With defect masses in the range 0.33 < md/m < 0.5, the moving localized mode is partially captured and reflected. There is a peak in the amplitude of the mode localized at the defect when its frequency is near the vibrational frequency of the moving ILM, indicating a resonance effect. When the relative mass of the impurity increases to 0.5, the energy of the moving ILM is completely captured, except for some plane waves that are produced as the defect mode settles into its eigenvector. There is a dramatic increase of both the amplitude and frequency of the defect mode at this transition, as shown in the figure. As the mass increases further, the frequency of the defect mode decreases until it is nearly equal to the frequency of the moving localized mode at md/m = 0.91. Beyond this point, the impurity can support modes of the same frequency as the moving ILM and therefore becomes transparent. The key feature of this transfer process is evident by examining first the small defect mass limit. When the impurity is very light, it "adiabatically" follows the motion of its nearest neighbors and thus moves in phase with them at the incoming ILM frequency in the figure (dashed line). Therefore this defect region perfectly reflects moving ILMs in which adjacent particles vibrate 7r out of phase. Such reflection continues until the defect mass is sufficiently large that the impurity mode frequency approaches that of the moving ILM. The exact threshold at which the defect begins to capture energy depends on both the characteristic wave vector and anharmonicity of the moving ILM.
Unusual anharmonic local mode systems
w
0.30
227
..... 9
I
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......
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a ~ a
.40 1.20 1.00
,
0.20
i
0.40
,
m
,
0.60
l
0.80
,
1.00
md/m Fig. 52. Simulation results for the frequency and amplitude of the anharmonic mass defect localized mode produced with captured energy from a passing ILM. (After Bickham et al. 1993).
3.1.2.5. Relation to anharmonic zone boundary mode stability. As noted above, both the odd and even (k2, k4) ILMs spatially broaden and merge with the maximum frequency Wm of the harmonic lattice as the anharmonicity parameter A4 is decreased to zero. That is, the ILMs just become the harmonic lattice zone boundary phonon mode (ZBM), with displacement pattem A(..., 1 , - 1, 1 , - 1. . . . ). An interesting "inverse" of this connection between the ILMs and the ZBM has also been investigated, in recent analytic and numerical studies (Burlakov and Kiselev 1991; Burlakov et al. 1990d; Burlakov et al. 1990b; Kivshar 1993; Sandusky et al. 1993b; Sandusky and Page 1994); the last of these is quite extensive. The upshot of this work is that the anharmonic ZBM can evolve into one or more ILMs and that an instability of the ZBM is the first step in this decay. The question of the stability of such an extended lattice phonon mode against decay into ILMs is of interest, since it appears to provide a possible avenue for creating ILMs via the application of external perturbations.
A.J. Sievers and J.B. Page
228
Ch. 3
In the rotating wave approximation, the anharmonic ZBM frequency for a (k2, k4) lattice is easily shown to be given by O32
~=
1 + 3A4.
(3.21)
By applying a suitable version of the stability analysis sketched in the preceding section, we find that the anharmonic ZBM in this lattice is indeed unstable (Sandusky et al. 1993b; Sandusky and Page 1994). For a given value of the anharmonicity A4, it is convenient to decompose the instability perturbations into spatial Fourier components, and one can readily predict the ZBM instability growth rates as a function of the perturbation wavevector kp. Figure 53(a) shows such predictions, together with measured rates from MD simulations; these are seen to agree well with the predicted rates. For different values of the anharmonicity (as measured by 60/60m), the maximum ZBM instability growth rate is found to occur at different perturbation wavevectors (kp)max, as shown in fig. 53(b). One sees that as the anharmonicity increases, (kp)max increases towards its largest value, which is that for the pure quartic lattice. Thus, larger anharmonicity favors shorter wavelength instability perturbations - the ZBM instability has introduced a new, anharmonicity-dependent, length scale into the system, namely the wavelength 27r/(kp)max associated with the fastest-growing instability perturbation. This ZBM instability length scale turns out to correlate precisely with the spatial extent of the ILMs for each value of the anharmonicity. Furthermore, when one goes beyond the perturbation instability analysis and uses MD simulations to observe the nonlinear time-evolution of the ZBM instability over finite times, it is found that the ZBM indeed evolves into a periodic array of ILM-like localized displacement patterns, as is strikingly evident from fig. 54. The characteristic width of these ILM-like structures decreases with increasing anharmonicity; it is again just the wavelength associated with the fastest-growing ZBM instability perturbation. Analogous ILM-related zone boundary mode instabilities are found for the more realistic case of lattices with both quartic and cubic anharmonicity (Sandusky and Page 1994).
3.1.3. (k2, k3, k4) nearest-neighbor potentials While the (k2, k4) lattices discussed in the previous sections have been fruitful for establishing some fundamental aspects of localized dynamics in periodic anharmonic lattices, it is also clear that this model leaves out one of the most important properties of the interparticle anharmonicity in real
w
Unusual anharmonic local mode systems
229
0.10
~
0.05
0.00 0.00
t-tll "1o
0.25 0.50 kpa (units of n radians)
0.50
o m
L _
0.25_
.-.
o
t-
/
• 0.00 E 1.0
%
1.'5 ~/[.0m
Fig. 53. Zone boundary mode instability and its relation to ILMs. Upper panel: predicted and measured ZBM instability growth rates in units of the anharmonic ZBM frequency w, as a function of the perturbation wavenumber kp, for a 40-particle (k2, k4) lattice with lattice constant a and periodic boundary conditions. The anharmonicity parameter is An -- 0.068, corresponding to a ZBM frequency of W/Wm= 1.1. The solid curve gives the stability analysis prediction, and the circles are measured in MD simulations. Lower panel: ZBM stability analysis prediction of the perturbation wavenumber (kp)max associated with the largest instability growth rate, as a function of the anharmonicity (measured by the ZBM frequency w). The dashed line gives the value for a purely quartic lattice. For a given value of the anharmonicity, the wavelength associated with (kp)max correlates accurately with the spatial extent of the ILMs for the same anharmonicity. (After Sandusky et al. 1993a, b).
systems, namely cubic anharmonicity. Typical interatomic potentials such as Lennard-Jones, Morse, and Born-Mayer plus Coulomb are repulsivedominated, and their Taylor-series expansions lead to a strong negative cubic anharmonic term (k3). Our preceding restriction of the anharmonicity to just even orders disproportionately weights the attractive component of the potential, as is clearly evident in the "square-well" limit of increasing order discussed earlier, and we have seen that it is in this limit that the most localized ILMs are asymptotically exact. Therefore it is important to
230
A.J. Sievers and J.B. Page
Ch. 3
Fig. 54. MD simulation showing the finite-time evolution of the predicted fastest-growing ZBM instability perturbation in a 40-particle (k2, k4) lattice with periodic boundary conditions. Here, the anharmonicity parameter is A4 -- 0.0067, corresponding to a ZBM frequency of W/Wm = 1.01. The top panel shows the t = 0 unperturbed ZBM pattern that is used. The fastest-growing perturbation for this case occurs at (kpa)max = 0.107r, and its pattern is shown in the second panel. The bottom panel shows the displacement pattern after a finite time has elapsed, and one sees that the ZBM has evolved into a periodic array of localized ILMlike patterns. The rate at which this evolution occurs depends upon the magnitude of the perturbation seed. The particle displacements are plotted vertically, and the relative vertical scales are indicated by the tic marks. (After Sandusky et al. 1993b; Sandusky and Page 1994).
reconsider the problem, with cubic anharmonicity included. We will first discuss ILMs in monatomic (k2, k3, k4) lattices and then generalize to the case of the above realistic analytic interatomic potentials. With nearest-neighbor cubic anharmonicity included, the potential energy of eq. (3.1) is replaced by
k2
k3
V = --2 ~ ( u n + l - u,~)2 + -~- ~(u,~+l - un) 3 n
n
k4 y ~ ( U n + l n
-
Un)4.
(3.22)
w
Unusual anharmonic local mode systems
231
The main qualitative feature introduced by k3 is to make the interparticle potential asymmetric, resulting in the particles' time-average displacements being nonzero and amplitude-dependent. Hence, amplitude- and site-dependent static distortions have to be added to the RWA trial solutions: Un(t) = A[~n cos(oat) q- An] ,
(3.23)
where the {An } are the static distortions, relative to the overall amplitude A. The {(n} give the dynamical mode pattern, as before. Within the RWA, Bickham et al. (1993)carried out detailed numerical studies of ILMs for the (k2, k3, k4) case and verified their results using MD simulations. Briefly, their technique consisted of implementing a standard nonlinear equation solver to obtain numerical solutions to the RWA equations for a restricted number of sites near the ILM center, and then applying the harmonic-approximation boundary condition of exponential decay for the particles beyond. Free-end boundary conditions were used. They found that by locally distorting the perfect lattice with the sign of the distortion determined by the sign of k3, stable ILM's could be generated even for large cubic anharmonicities but that independent of the sign of the cubic anharmonicity its effect was to decrease the frequency of the ILM. As the cubic anharmonicity increases, the eigenvector also becomes more localized until it resembles a triatomic molecule, beyond which the mode becomes unstable and decays. Bickham et al. (1993) should be consulted for details. In recent studies focused on the question of ZBM-stability/ILM-existence in (k2, k3, k4) lattices with periodic boundary conditions, Sandusky and Page (1994) employed a similar but slightly more computational approach, in that the RWA equations of motion were solved numerically for all of the particles, the number of which was typically taken as 40. As detailed there, the presence of sufficient cubic anharmonicity is found to stabilize the ZBM and simultaneously prevent the formation of ILMs. For the case of isolated ILMs, their results are similar to those of Bickham et al. (1993), when allowance is made for the different boundary conditions used. Figure 55 shows the RWA calculations by Bickham et al. (1993) for stationary even-parity modes. The ILM frequencies are shown as a function of the cubic anharmonicity parameter A3 = k3A/k2, for three different values of the quartic anharmonicity An in a 512-particle lattice with free ends. Also shown are the results of MD simulations, and they are seen to be in good agreement with the RWA predictions. The slight discrepancy observed for larger values of An arises from the higher harmonics neglected in the RWA. Figure 56 shows the results of calculations by Sandusky and Page (1994) of the RWA dynamic (a) and static (b) displacement patterns for an oddparity ILM in a 40-particle periodic boundary condition lattice with
A.J. Sievers and J.B. Page
232 , 0
.....
9
I
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'
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o --. o "-~ o o"-,.
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Ch. 3
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..... ---
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o\
o~..
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~
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I
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Fig. 55. Frequencies of even-parity ILMs in a 512-particle monatomic (k2, k3, k4) lattice with free ends. The dashed curves give the frequencies as a function of the cubic anharmonicity parameter A3, for four different values of the quartic parameter A4. The circles are the results measured in MD simulations. The results show that there is a limiting value of k 3 for a given value of k4. (After Bickham et al. 1993).
A3 = -2.1 and A4 = 1.1. The frequency is 1.1~m. Owing to the mode's symmetry, there is no static displacement on the central particle. However the static displacements on the remaining particles are nonzero, and they vary rapidly in the region where the dynamic displacements are large. Away from this region, the static displacements decrease linearly to zero at the boundaries, corresponding to a constant static strain away from the mode center. The magnitude of this strain decreases with increasing numbers of particles. The nature of the static displacements away from the mode center depends on the boundary conditions employed- for the free-end boundary conditions used by Bickham et al. (1993), these strains are in fact zero since the particles away from the mode center simply translate together, towards or away from the mode center, depending on the sign of k3. In the righthand column of fig. 57, we show the dynamical displacement patterns for odd-parity ILMs in a 40-particle periodic boundary condition lattice with fixed values of (k2, k3, k4), as the amplitude A is changed (Sandusky and
w
Unusual anharmonic local mode systems
233
(a)
....._ i,,,.--
Particle
(b)
b._ v
Particle Fig. 56. Predicted dynamical displacement pattern (a) and static displacement pattern (b) for an odd-parity ILM in a 40-particle (k2, k3, k4) lattice with periodic boundary conditions. The values of A3, A4 and the frequency are -2.1, 1.1 and 1.1 Wm, respectively. Notice that the static distortions away from the mode center decrease linearly to zero at the "supercell" boundaries, for this periodic boundary condition lattice. (After Sandusky and Page 1994). Page 1994). The lower pattern on the right corresponds to the same mode shown in fig. 56. The column on the left shows the results for ILMs of the same frequencies in a lattice that is identical, except that there is no cubic anharmonicity (k3 = 0). The interesting aspect here is that as the frequency is decreased, the presence of cubic anharmonicity is seen largely to quench the spreading out that occurs for the k3 -- 0 case. Eventually the mode rapidly broadens, merging with the zone boundary mode at a finite value of A3. It is straightforward to show for both monatomic (Sandusky and Page 1994) and diatomic lattices (Kiselev et al. 1994b) that the effect of the static displacements {An} is to renormalize the harmonic force constants k2 into site-dependent, effective harmonic force constants k~(n+l,n) -- k2 -k- 2k3A(An+l - A n ) + 3k4A2(An+l - An) 2
(3.24)
in one of the RWA equations of motion. As a result, the cubic anharmonicity is formally eliminated from the equation, and it is transformed into the pure
A.J. Sievers and J.B. Page
234
k =O
I m
Ch. 3
k ,0 "=
l
~
Particle
m=l
m---1
Fig. 57. Effect of cubic anharmonicity on ILM localization. The plots show the oddparity ILM dynamical displacement patterns for a 40-particle periodic boundary condition lattice having fixed values of (k2, k3, k4), as the ILM frequency is lowered by decreasing the amplitude A. The patterns on the left are for k3 = 0, while those on the right are all for the same nonzero value of k 3. The amplitude has been chosen to give the same frequencies for these cases. Notice that the k 3 # 0 ILM patterns remain localized as the frequencies decrease, while the k 3 = 0 ILM patterns spread out. (After Sandusky and Page 1994).
RWA eq. (3.20) above, except with k2 replaced by the site-dependent k~'s. This site-dependence is strong near the mode center, where the static displacements vary the most rapidly. Evidently, the dynamical displacements for the (k3 r 0) ILMs on the fight side of fig. 57 are formally like those of an impurity mode in a (kz, k4) lattice having defect-induced force constant weakening which enhances the mode's localization. (k2, k4)
3.1.3.1. ILM existence. Bickham et al. (1993) have generated an existence threshold curve for ILMs in the s a m e (k2,k3, k4) lattice used for fig. 55, with the results shown in fig. 58. The solid curve was obtained by fixing A3 and then finding the minimum value of An for which ILMs are supported- this is the point where the ILMs broaden into the zone boundary mode, and it occurs at the same point for the odd and even ILMs. The dashed curve above A4 ,~ 0.8 corresponds to the ILM dynamical displace-
w
Unusual anharmonic local mode systems 1.6
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I
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)
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b ...........
1.2
C
cxi L'Xl <~-0.8
I
ble Re
235
I
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I
II,d< 0.4 Unstable 0.0 -2.5
9
I
-1.5
~"~/k,~') ,
I
~
-0.5
I
0.5
Unstable ,
I
1.5
,
2.5
A3-K3A/K2 Fig. 58. ILM existence in a 512-particle monatomic (k2, k3, k4) lattice with free ends. The solid curve is the existence threshold determined by the condition that the ILM spatially broadens into the zone boundary lattice phonon mode. The dashed curve corresponds to a high-localization triatomic-molecule limit. Area (d), lying between curves (b) and (c), designates the region of a continuum approximation. (After Bickham et al. 1993). ment patterns becoming more localized than the (k2, k4) asymptotic limit ( . . . , 0, - 1/2, 1, - 1/2, 0,...). At these points the ILMs were observed to be unstable in the MD runs of Bickham et al. (1993). The (A3, An) points along the dashed curve are very close to those where a double minimum appears in the (ke, k3, k4) interparticle potential (Sandusky and Page 1994).
3.1.3.2. ILM motion. Moving ILMs for (k2, k3, k4) lattices have been discussed by Bickham et al. (1993). The RWA was used within a moving envelope approximation characterized by a single wavevector, and numerical results were obtained for the same 512-particle free-end lattice considered in figs 55 and 58 above. For a fixed value of the quartic anharmonicity An, dispersion curves were predicted for several values of the cubic anharmonicity A3, as shown in fig. 59. Included in this figure are the corresponding results measured in MD simulations. The agreement is seen to be quite good, with the larger discrepancies occurring for larger values of the wavevector. The top curve is for the pure (k2, k4) lattice, and it reproduces the results given earlier by Bickham et al. (1992). Figure 60 shows the displacements as a function of time, for three particles in the above lattice as a moving ILM passes (Bickham et al. 1993). For
A.J. Sievers and J.B. Page
236 1.5
9
I
9
I
9
I
o Simulation .
.
.
.
i
9
i
Results
/~=0.0
.
9
Ch. 3
-.~=-1.6
- -
........... b.~=-0.8 " - - - - - - - / ~ - - - 1.8 ..... .M=-1.2
1.4
E1.3
. .... . . . *b- ........ " ~ .i~.................... . . -*".... o " " . . ~ o.
3
0
:3
~
r. . . . .
1.2
.
.,.-
-
....
..........
0 ...........
.-,,
~-
"o--
" . . . . r_
g .......
o 0
1.1 0
o
0
~ 0 0
o0
,
0.0
I
0.1
,
I
0.2
'
i
0.3
.
~
0.4
.
,
0.5
9
0.6
ko Fig. 59. Dispersion curves for moving ILMs for the (k2, k3, k4) lattice of fig 58. The quartic anharmonicity is fixed at A 4 = 1.6, and the lines are the predictions for different values of the cubic anharmonicity A3. The circles are obtained from MD simulations. Note that the agreement between the simulation and analytic results is better at smaller wave vectors and higher frequencies because the effect of the second derivative that were removed from the equations of motion is then minimized. (After Bickham et al. 1993). this free-end boundary condition case, all of the particles on one side of the mode center and well away from it have the same static displacement, while the static displacements of the corresponding particles on the other side are equal and opposite. This is manifested in the figure by the change in the particle offsets from their equilibrium positions as the mode passes. The group velocity of this particular excitation is approximately 15% of the harmonic lattice sound velocity. Note that there is a small increase in the static displacement at the n = 10 site after a time corresponding to three periods of the m a x i m u m plane-wave frequency. This is caused by a small amplitude supersonic pulse that propagates in both directions from the ILM at the start of the simulation and eventually reaches the free ends of the lattice. This pulse may be a long wavelength acoustic kink soliton that is also a solution of these equations.
3.1.4. Realistic potentials The previous studies of ILMs in (k2, k3, k4) lattices show that for a fixed value of the quartic anharmonicity parameter A4, both the even and odd
w
Unusual anharmonic local mode systems
237
Fig. 60. Displacements versus time for a moving ILM in the (k2, k3, k4) free-end lattice of fig. 58. The total displacements (static plus dynamic) are shown for three different sites, as the moving ILM passes. The group velocity of this particular moving excitation is 15% of the harmonic lattice sound velocity. The anharmonicity parameters are A3 = 0.8 and A4 = 0.4, and the wavevector is given by ka = 0.3, where a is the lattice constant. The three curves have been displaced vertically, for clarity. (After Bickham et al. 1993). versions of these modes exist, provided the magnitude of the cubic anharmonicity A3 does not exceed a threshold value. Moreover, it is not difficult to show that these conditions can be satisfied easily for (k2, k3,/ca) values obtained from the Taylor-series expansions of realistic interparticle potentials V(r), such as Lennard-Jones, Morse and B o r n - M a y e r plus Coulomb, about their m i n i m u m points. The question then naturally arises as to whether or not ILMs can exist in lattices in which the particles interact via these full potentials, which of course include all of the higher-order terms in the Taylor-series expansions. One-dimensional lattices with realistic potentials have been studied recently by Kiselev et al. (1993, 1994a, b) for the monatomic and diatomic cases and by Sandusky and Page (1994) for the monatomic case. The diatomic case will be discussed in w3.2. The work of Sandusky and Page concerns general questions about the interrelations between zone boundary phonon stability and ILM formation, and it includes the use of the full potentials listed in the previous paragraph. In some of the calculations, these potentials are restricted to nearest neighbors, while in others they are allowed to act out to sixth neighbors. With respect to the question of ILM existence,
238
A.J. Sievers and J.B. Page
Ch. 3
the basic finding of both the monatomic and the diatomic studies is that ILMs do not exist in either type of lattice with these realistic potentials. The presence of anharmonic terms beyond fourth order in the full potentials acts to suppress the presence of ILMs. However, the diatomic case gives a new possibility, namely that ILMs could occur with frequencies in the gap between the acoustic and optic phonon branches. That such "gap" ILMs exist is shown by Kiselev et al. (1993, 1994b), as discussed w3.2.
3.1.5. Mass defect anharmonic mode with realistic potentials Although ILMs do not exist above the top of the plane wave spectrum of a monatofnic lattice with realistic potentials, high frequency anharmonic impurity modes are still a possibility. Kiselev et al. (1994a, b) have investigated the properties of mass and force constant defect modes as a function of amplitude for four nearest neighbor potentials, which in order of increasing anharmonicity are: Toda, Born-Mayer plus Coulomb (BMC), LennardJones, and Morse. These potentials were adjusted to have the same k2 and k4 terms in their Taylor's series expansion. The light impurity mass defect parameter is e = 1 - md/m -----0.95, where md and m are the defect and the host particle masses. The form of the interaction potential between particles has the same value for all nearest neighbors, except for the bonds involving the impurity. The force constant defect parameter is taken to have the value 77 = 1 - V~'/V" = 0.87, where V~' and V" are proportional to the nearest-neighbor force constants between the impurity and the host lattice and between the particles of the host lattice, respectively. The frequency of the harmonic impurity mode in such a chain has the value 60harm/60m -- 1.125, where tom is the frequency of the top of the harmonic phonon band. For small amplitude vibrations a light impurity supports a localized vibration with a frequency higher than the plane wave spectrum, but with increasing amplitude the anharmonicity of the interaction potential becomes important. The soft attractive part of the standard two-body potentials makes this frequency decrease as the amplitude of the local mode vibration increases and the excitation transforms from a local mode to an in-band resonant mode. Figure 61 shows the frequency of an anharmonic impurity mode in a lattice with a mass and force constant defect for three different two-body potentials as a function of normalized amplitude A/a, where a is the lattice constant. The odd-parity mode is centered at the impurity site. The solid curves show the analytic results, and the points give the simulation results" squares (Toda), diamonds (BMC), and circles (Morse). The results for the Lennard-Jones interaction potential (not shown) are between the BMC and the Morse results. The conclusion is that the more anharmonic the two-body potential, the lower the anharmonic impurity mode frequency.
Unusual anharmonic local mode systems
w 1.2
........... .
239 _
I
.
I
'
I
'
1.0 0.8 ~EO. 6 0.4 0.2 0
,
0.0
I
0.1
, .........
l
0.2
....................
i
I
0.3
i
0.4
Normalized amplitude Fig. 61. The frequency of an anharmonic impurity mode in a monatomic lattice with a force constant and mass defect for three different two-body potentials, as a function of the normalized amplitude A/a. The odd-parity mode is centered at the light impurity (mass defect parameter e = 0.95), which is bonded with its neighbors through the perturbed potential having the forceconstant defect parameter r/ = 0.87. Analytic results are shown as solid curves, while the simulation results are represented by squares (Toda), diamonds (Born-Mayer plus Coulomb), and circles (Morse). For each of the anharmonic potentials, k2 and k4 are the same. (After Kiselev et al. 1994b).
3.2. One-dimensional diatomic lattices
The study by Kiselev et al. (1993, 1994b) of diatomic one-dimensional lattices with standard full potential functions establishes that ILMs can occur in the gap between the optic and acoustic phonon branches. The study was basically numerical and focused on a specific diatomic lattice, namely one with 257 particles, free ends, and having the masses of lithium iodide. Three full potential functions V(r) were emphasized, namely the Toda potential (Toda 1989), the Morse potential and the BMC potential, and all three were restricted to act between just nearest neighbors. In addition, a nearest-neighbor (k2, k3, k4) potential was studied, with the spring constants obtained from the expansion of the BMC potential for LiI. The parameters in the remaining two potential functions were adjusted to reproduce the BMC values of k2 and k4. To obtain the stationary ILMs, an RWA approach based upon the trial solution eq. (3.23) was made, and the resulting equations were solved numerically.
A.J. Sievers and J.B. Page
240
Ch. 3
1-(a) 0
to
- - - - ' - -
-1 "
5. E
(b)
o
<
I~
0
-1
,
-8
~
m
A
.... i
-4
i
0
4
. .
1I
i
8
Particle number Fig. 62. Static and dynamic displacement patterns for ILMs in a one-dimensional diatomic lattice of 257 particles interacting via a nearest-neighbor Born-Mayer plus Coulomb potential. The masses and potential parameters are those of lithium iodide, and free-end boundary conditions are used. The circles and squares denote the light and heavy masses, respectively, with their open and solid versions giving the static and dynamic patterns. (a) Odd-parity gap ILM, with to = 0.74tom. This mode was found to be stable in MD runs. (b) Even-parity gap ILM, with to = 0.74tom. MD runs revealed this mode to be unstable. (c) Odd-parity ILM with to = 1.04tom, obtained by using just (k2, k3, k4), gotten by expanding the LiI potential used in (a) and (b). When the full LiI potential is used, no ILMs with to > tom are found. In all three panels, the overall amplitude is A/a = 0.233. (After Kiselev et al. 1993).
For the (k2, k3, k4) version of this diatomic lattice, an odd-parity ILM occurs, with a frequency W/Wm = 1.04, just above that of the maximum harmonic lattice frequency. Its static and dynamic displacement pattern are given in fig. 62 (c). Just as in the monatomic case discussed in w3.1.4, this high-frequency ILM disappears when any of the three full potentials is used instead of the (k2, k3, k4) potential. Again, no high-frequency ILMs have been found using the above full realistic potentials, for either the monatomic or diatomic cases; the higher-order anharmonic terms in the expansions of the full potentials suppress ILM solutions with frequencies above the phonon bands. However, both even- and odd-parity ILMs are found with frequencies in the phonon gap for either the (k2, k3, k4) case, or with the full potentials. (Note that for a diatomic periodic lattice, the midpoints between the particles' equilibrium sites are no longer symmetry centers, as they are for monatomic
w
Unusual anharmonic local mode systems
241
lattices. For the diatomic case, only the equilibrium sites are symmetry centers. Hence, even-parity modes for the diatomic case are qualitatively different than for the monatomic lattice; in particular they involve no motion of the central particle.) The odd-parity gap ILM is found to be stable, in the sense of persisting over many periods in MD simulations, whereas the even-parity gap ILM is found to be unstable. The displacements for the odd- and even-parity gap ILMs for the B M C potential are shown in panels (a) and (b) of fig. 62. Additional numerical tests show that odd-parity gap modes can be generated from the amplitude pattern for the nearest-neighbor potential cases even when second nearest-neighbor forces are included in the model. These findings suggest that anharmonic gap modes are also stable in the presence of long range forces (Kiselev et al. 1993). Figure 63 plots the frequencies of the odd-parity ILM gap mode for each of the three full potentials, as a function of the normalized amplitude of the 1.0 CO+ 0.8
0.6
~'~ 0.41-
~
I 0.2
/f
I o
|
0.5
1.0
Relative 0.0
!
0.0
_.,
.
,
.
1.5
2.0
displacement .,
.
,
0.1 0.2 0.3 Normalized amplitude
,
0.4
.
Fig. 63. Frequencies of the odd-parity ILM gap mode in the diatomic lattice of fig. 62, for three different full potentials. The frequencies are plotted versus overall amplitude A/a, and w_ and w+ are the frequencies defining the gap between the acoustic and optic phonon branches in the harmonic crystal. The three frequency curves correspond to different nearest-neighbor potential functions: triangles (Toda), circles (Born-Mayer plus Coulomb), and squares (Morse). The inset shows these potentials, together with a Lennard-Jones and a (k2, k3, k4) potential. The Born-Mayer plus Coulomb potential is that appropriate to LiI, and the values of (k2, k 3, k4) were obtained by expanding this potential. The parameters for the remaining potentials were adjusted to reproduce these same values for k2 and kn. (After Kiselev et al. 1993).
A.J. Sievers and J.B. Page
242
Ch. 3
light particle A//a. The curves give the RWA results and the points give the results of MD simulations, and they are seen to be in good agreement with the RWA predictions for A/a < 0.2. When the normalized amplitude is increased to A/a ~ 0.05 the gap mode eigenvector becomes nearly that of a triatomic molecule. The discrepancies observed for larger amplitudes and "softer" potentials (e.g. Morse) indicate that the RWA is becoming less accurate. This is because the effective harmonic potential associated with the RWA looks very different from the effective anharmonic double well which forms at large ILM amplitudes (Kiselev et al. 1994b). The frequency spectrum for the displacement Of the light particle in the stable odd-parity ILM is given by the solid curve in fig. 64. It is obtained over a time interval of At = 1024(27r/o~m). In order of decreasing strength, the peaks are located at ~g, 3U)g, and 5Wg. This spectrum is remarkably similar to the dashed curve, obtained for the anharmonic ILM of the same amplitude in the same diatomic lattice with only the symmetric k2 and k4 potential terms, which is the form used in most earlier studies. Evidently, the
101
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!
9
I
9
!
!
ffl
m =
10 o
10 -1 E
"•10-2
%
.
/
/
".,,
~....
0-3
~
0-4 !
0
2
4
6
8
O)I(JL)m Fig. 64. Power spectra of gap and local modes in a diatomic chain with 257 particles. The solid curve corresponds to the gap mode (a~g = 0.74 OJm), with the Born-Mayer plus Coulomb potential for LiI, and the dashed curve shows the ILM spectrum (wi = 1.24~0m) in the same lattice with the (k2, k4) potential from the Taylor's series expansion of the Born-Mayer plus Coulomb potential. The ordinate of the dashed curve has been shifted by a factor 20, for clarity. In each case the amplitude is A/a = 0.233. The results are remarkably similar: the resonant features occur at the same first, third and fifth orders with about the same strengths. The "noise" that appears in the solid curve near these frequencies is due to weak coupling between the gap mode and the plane wave spectrum. (After Kiselev et al. 1993).
w
Unusual anharmonic local mode systems
243
main function of the asymmetric potential terms in the Taylor's expansion of the two body potential is to contribute to the static distortion in the displacement patterns shown in fig. 62. Inspection of fig. 62(a) reveals that the displacement pattern of the stable odd-parity gap ILM for the LiI BMC potential is very much like that of an isolated Einstein oscillator- the odd-order (asymmetric) terms in the interparticle potential have caused the equilibrium positions of the particles on each side of the central particle to uniformly shift outwards, and the dynamical displacements are ~, 0 except at the central particle. This Einstein-oscillator-like dynamical behavior is not surprising in view of the fact that the central mass is so much lighter than its neighbors. Indeed, this mode is behaving qualitatively like that of the light mass defect anharmonic mode discussed earlier for monatomic (k2, k4) systems, with the additional feature that the harmonic force constants k2 on either side of the central particle have been renormalized to k~, due to the local static distortion produced in the ILM by k3. Gap ILMs in pure (k2, k4) diatomic systems have recently been discussed (Aoki et al. 1993; Chubykalo et al. 1993). It is now evident that ILM solutions can be obtained numerically for a perfect diatomic 1D chain incorporating a variety of realistic two body potentials in the sense of persisting over many periods in MD runs, while simulations demonstrate that the odd-parity gap modes are stable, but the even-parity modes are not. The incorporation of a realistic two-body potential into the lattice has brought out several important new features of ILMs: (1) a localized mode will not exist above the top of the plane wave spectrum for realistic potentials; (2) a long-lived odd-parity localized mode can exist in the gap between the optic and acoustic branches; (3) in the large amplitude limit, the triatomic molecule-like eigenvector previously observed for local modes in the hard quartic case is recovered for gap modes with realistic potentials and (4) for each of these potentials the gap mode is accompanied by a localized DC expansion of the lattice.
4.
Speculation
In the preceding two sections, we have discussed two very different sorts of unusual localized vibrational behavior in strongly anharmonic systems. First, we have seen that detailed experimental and theoretical studies of the impurity system KI:Ag + reveal a nearly unstable defect/host system at low temperatures, which exhibits a rapid, thermally-driven on --+ off-center transition between T = 0 K and T = 20 K. As detailed above, this transition is highly anomalous, in that it cannot be explained within any sort of a
244
A.J. Sievers and J.B. Page
Ch. 3
harmonic or weakly perturbed anharmonic f r a m e w o r k - thus the system is strongly anharmonic in an as yet undetermined way. At the same time, the T = 0 K on-center configuration and its associate dynamical properties as probed by extensive stress, E-field and host lattice isotope effect experiments are found to be consistent with a perturbed harmonic phonon model. Even though the on-center configuration is therefore apparently harmonic (or quasiharmonic under applied stress or E-fields), its associated dynamics are themselves very unusual" the impurity-host system is strongly coupled but nearly unstable against off-center displacements and it possesses a new type of localized impurity mode, the pocket gap modes, whose displacements are confined to sites away from the defects. Second, we have discussed some of the basic properties of an unusual new class of localized vibrational excitations (ILMs) which can occur in periodic lattices, provided there is sufficiently strong local anharmonicity present. This anharmonicity could arise in two ways: 1) from large values of the anharmonic coefficients in the expansion of the potential energy about the equilibrium configuration, or 2) from the presence of large amplitudes in a localized region of the lattice. The study of ILMs has been purely theoretical up to the present and is still at an early stage. The focus has been on simplified models, but they are becoming more realistic, as exemplified by the recent studies involving realistic potentials (Kiselev et al. 1993, 1994b; Sandusky et al. 1994). As the properties of ILMs become better established, we anticipate that there will be significant activity in experimentally verifying their existence in real solids. In the following we briefly mention some of the speculative ideas which have been suggested previously. One of these involves a link between ILMs and KI:Ag +, and it will be discussed first, in a separate subsection. 4.1. ILMs and anomalous defect properties
4.1.1. KI:Ag + One of the early proposals was that perhaps one or more ILMs are trapped at the Ag + site at low temperatures only to be released into the lattice at some finite temperature. When these ILMs are released into the lattice at an elevated temperature, they would leave behind a defect space with a different vibrational pattem and hence account for the change in the defect induced spectrum. Since a free ILM could move in any direction in the lattice, a large number of accessible states would exist; hence, there is a great deal of entropy associated with a change from a bound to this "free" configuration. The release of ILMs would be expected to give a strong temperature dependent signature.
w
Unusual anharmonic local mode systems
245
1.0
O.B I 0.6
I
\ I
\\ ,
~
',,f ~
0.4
0.2
-
\ l
0.2
\
0.4
0.6
0.8
kT/~-Fig. 65. Temperature dependence of the degree of association pon(T) for three different concentrations. The reduced temperature is kT/( where ff is the Gibbs free energy of association. The three different concentrations are (a) c = 1.2 x 10 -3, (b) c = 1.2 x 10 - 2 , and (c) c = 1.2 x 10-1. (After Lidiard 1957).
The temperature dependence can be calculated by making use of the similarity with the corresponding impurity-vacancy-complex activation problem. (See, for example, Lidiard 1957). Assume that an ILM complexes with the impurity. If the degree of association Pon is such that Cpon is the molar fraction of complexes, then application of the law of mass action to the process (impurity-ILM complex r unassociated impurity + unassociated ILM) gives = c exp (1
--port) 2
(,) k-T
(4.1) '
where ff is the Gibbs free energy of association. Curves showing Pon(T) as a function of temperature for three different impurity concentrations are presented in fig. 65. The temperature dependence of the lowest concentration curve looks remarkably similar to the data shown in fig. 14(a). Since both c and ( could be taken to be free parameters at this stage, a reasonable fit to the data could be generated for a single concentration, but fig. 65 indicates that the exact temperature dependence depends strongly on the concentration. What cannot be fit by this model is the experimental observation that the line strength temperature dependence observed for KI:Ag + is independent of Ag + concentration over a range of at least a factor of three; hence, the correct model cannot depend on concentration.
A.J. Sievers and J.B. Page
246
Ch. 3
4.1.2. MOssbauer recoilless fraction for Sn in Pb There are completely different kinds of measurements on Sn impurities in metallic Pb which show unusual temperature dependent behavior in the lattice vibrational spectrum. Mt~ssbauer recoilless fraction measurements as a function of temperature provide another way to investigate the dynamical coupling between an impurity and the lattice. For a simple Debye solid this quantity reduces to ~D
f = exp
3R 1+ 2k69D
~
ex - 1
'
(4.2)
where R is the recoil energy of the free nucleus and 690 is the Debye temperature of the solid (Frauenfelder 1963; Ashcroft and Mermin 1976). At temperature large compared to 69D the temperature dependent factor in the exponent is proportional to the temperature. (Note the similarity between eq. (4.2) and eq. (2.5). The measured temperature dependence of the recoilless fraction of ll9Sn in Pb (Haskel et al. 1993) is shown in fig. 66. For temperatures comparable to or slightly larger than the Debye 690 of Pb (88 K), the data give a linear temperature dependence for a semilog plot, as expected from eq. (4.2). The anomalous behavior appears at somewhat higher temperatures. For the lowest concentration the recoilless fraction disappears at temperatures above 120 K. For the 1% 119Sn in Pb, the recoilless fraction can be followed to higher temperatures beyond the knee at about 145 K. Similar temperature dependent results are found for the 2% 119Sn in Pb sample with the knee now at about 170 K. Finally, for the highest concentration sample the knee appears at about 200 K but is now somewhat rounded. These results demonstrate that the characteristic knee temperature at which the recoilless fraction rapidly decreases moves to higher temperature with increasing Sn concentration, with the resulting temperature dependence being a strong function of concentration. These results have been interpreted as evidence for a large number of low-lying states, a number much larger than can be derived from hopping or tunneling of the defect. The large number of states is attributed to "local melting" of about 30 atoms in the lattice near the defect, even though the temperature is far below the melting temperature of the alloy. The remarkable concentration dependence is not a natural feature of such a model. From w4.1.1 a concentration dependence of the temperature dependence would be a natural consequence if one or more ILMs are complexed to the Sn impurity at low temperatures but are released as the temperature is raised into the 150 K region. (See fig. 65). During this transformation
Unusual anharmonic local mode systems
w
247
A (/) .,I,-o~ e"
t,._ v
o
>, 0.10
.u. (/)
e"
o
t,,. .,0,(,.)
Or) 0.01
I00
150 Temperature (K)
200
Fig. 66. MOssbauer spectral intensity versus temperature for alloys of Sn in Pb. The different atomic % concentrations of l l9Sn for data taken with increasing temperature are as follows: crosses -0.5%; open squares -1%; solid diamonds -2%; open circles -3%. The z's show the data taken during cooling. The straight lines are fits to the experimental data using a standard anharmonic potential model for the Debye-Waller factor. (After Haskel et al. 1993). the Sn could move from an on-center site to an off-center site. The large entropy contribution would stem from the large number of states available to the ILMs and the effective knee activation temperature would increase with increasing Sn concentration in qualitative agreement with the results shown in fig. 66.
4.2. ILMs and other properties Section 3 focused on ILMs with frequencies outside the normal acoustic and optic phonon bands, since almost all of the ILM work up to now has concerned this case. However, Takeno and Sievers (1988) also theoretically discussed the possibility of low-frequency in-band ILMs for the case of monatomic (k2, k4) lattices having negative values of the quartic anharmonicity A4 - knA2/k2, arising from negative ka's. These "soft anharmonicity" in-band ILMs are reminiscent of defect-induced harmonic resonant modes, such as those occurring in the T = 0 K on-center configuration of KI:Ag +. Indeed, for the simple (k2, k4) lattice it was found that low-frequency resonant ILMs have frequencies and amplitude patterns behaving like those for a force constant defect in a harmonic lattice, with the role of the harmonic force-constant change parameter in the latter case being played by the anharmonicity parameter An. In the limit of low frequency, the resonant
248
A.J. Sievers and J.B. Page
Ch. 3
ILM amplitude pattern is confined to the central particle, in contrast to the behavior we have seen here for the high-frequency ILM in the hard quartic case, where the limiting displacement pattern is that of a simple triatomic molecule. On the other hand, the spectral width of the resonant ILM was found to be much greater than that for the corresponding harmonic forceconstant defect resonant mode, giving the resonant ILM a shorter lifetime against decay into the quasicontinuum of embedding band modes. All of the ILM work discussed here has been classical. Nevertheless, certain general aspects of the effects of ILMs on the equilibrium thermodynamics of lattices were discussed by Sievers and Takeno (1988) and by Takeno and Sievers (1988). The key point is that because ILMs can exist at any lattice site, they should give rise to a large configurational entropy and hence make an important contribution to the free energy for nonzero temperatures. Thus their equilibrium statistics should be like those for thermally-activated processes such as vacancy formation, being determined by the balance between the energy needed for ILM formation and the configurational entropy. For the case when the thermal equilibrium number n i l m of ILMs is much less than the number N of lattice sites, one has n i l m - - N exp(-/3f), where = 1/(kBT) and f is the work needed to create a single ILM in a lattice of fixed volume. These considerations apply to above-band ILMs, to gap ILMs and to in-band resonant ILMs, with the resonant case expected to have a smaller (but still positive) value of f (Takeno and Sievers 1988). Hence for each type of ILM, the picture which emerges is that the vibrational spectrum of a lattice at T = 0 K should be dominated by homogeneous plane-wave-like harmonic phonons (or anharmonically renormalized quasiharmonic phonons), but with increasing temperature ILMs are created and should be included in describing the dynamical properties. Interpreting the classical anharmonic potential functions as effective potentials for the quantum solids 3He and 4He, Sievers and Takeno (1988) and Takeno and Sievers (1988) speculated on the possible importance of ILMs for these highly anharmonic systems. 3He exhibits a low-temperature excess specific heat characteristic of thermally-activated processes, and this has been identified with vacancy production. X-ray measurements of the lattice parameter of both 3He and 4He in constant-volume cells have shown that there are indeed thermal vacancies produced. However, when these measurements and the specific heat results are interpreted in terms of localized vacancies, one obtains an unreasonably large energy for the production of a vacancy. An attractive mechanism in addition to thermal vacancies, is provided by thermally-activated ILM production. In this case the lowtemperature ILM formation energy should be relatively small, being the difference between zero-point energies of two types of vibrational modes,
w
Unusual anharmonic local mode systems
249
namely ILMs and phonons. Additional arguments, emphasizing the role of resonant ILMs, were advanced by Takeno and Sievers (1988). Another early application (Sievers and Takeno 1989) was to the problem of the observed anomalous low temperature specific heat of glasses. This application was based upon two main assumptions. 1) The disorder results in the presence of a substantial number of low-frequency resonant ILMs - these can be thought of as disorder-induced impurity ILMs. 2) The ILMs can move diffusively. Here in w3 we have seen that recent studies of moving ILMs in perfect lattices show that ILMs move very easily from site to site (Bickham et al. 1992; Sandusky et al. 1992); moreover, it is reasonable to assume that the glassy disorder would result in the motion being diffusive. Sievers and Takeno (1989) presented arguments leading from these assumptions to two main results. The first is that diffusive ILM motion produces a contribution to the low temperature specific heat which is linear in T, as is observed experimentally. The second is that the presence of stationary resonant ILMs contributes an excess specific heat term which is cubic in T, also as observed. For the simplified model calculations, the key parameter was the ratio of the resonant ILM frequency to the Debye frequency. Phenomenological fits of this parameter to the measured low-temperature linear specific heat term for 15 glassy solids yield similar values (0.063 to 0.12), even though the glasses range in type through covalent, van der Waals, ionic, metallic, inorganic and organic. Moreover, the resulting excess cubic specific heat in the model is of the correct order of magnitude. An obvious question is whether or not the atomic forces in real crystals are sufficiently strong to sustain ILMs at reasonable temperatures. Bickham and Sievers (1991) addressed this question in an approximate fashion in their work on high-frequency ILMs in (k2, k4) lattices having weak quartic force constants. As we have noted, strong anharmonicity An -- k4A2/k2 can still occur in such lattices, for sufficiently large amplitudes A. These authors obtained k4/k2 from lattice data for five alkali halides, with LiF being the most anharmonic (ka/k2 - - 5 . 5 0 ~-2). Then by combining the standard harmonic approximation expressions for the virial theorem and thermal mean square displacement, together with their results for the spatial width of the ILMs as a function of A4, they concluded that in thermal equilibrium, lattices with these force constants would have insufficient thermal energy to sustain ILMs at any temperature below the melting temperature. Despite the major approximations in these arguments, the results were sufficiently clear-cut that it was concluded to be very unlikely that high frequency ILMs could arise thermally in these (k2, k4) systems. This left open the possibility of thermally generated low-frequency resonant ILMs in such systems, but this question has not yet been addressed. Finally, these authors also suggested that crystals exhibiting soft-mode ferroelectric or anti-ferroelectric phase
250
A.J. Sievers and J.B. Page
Ch. 3
transitions might be an appropriate candidate for ILMs - near such a phase transition the importance of the anharmonic potential energy terms should be strongly enhanced. Having shown that ILMs are not likely to be thermally generated in simple (k2, k4) lattices with force constants derived from alkali halide potentials, Bickham and Sievers (1991) then briefly considered the introduction of vibrational energy from a nonthermal source, e.g., a Mrssbauer recoil or an optical transition followed by local lattice relaxation. As an extreme case within their force constant parameters, they considered the vibrational energy relaxation of an F-center, which in LiF involves about 40 phonons of frequency hWm. The order-of-magnitude result was that for this case sufficient vibrational energy is available to produce ILMs. However, this would require a very rapid local relaxation of the optically excited center, and hence detailed model calculations are necessary before one can make definite predictions. The speculations summarized in the above paragraphs were carried out shortly after the ILM concept appeared in 1988, and they were therefore made in the context of simple (k2, k4) lattices. Owing to the novel possibility that perfect lattices could sustain localized vibrational excitations, and because of the inherent difficulties of dealing with nonlinear phenomena in general, most of the subsequent theoretical work has focused on detailed explorations of the properties of ILMs for this simple case. Although this work is still in a relatively early stage, we have seen in w3 that these studies have already revealed a rich variety of interesting phenomena. We have also pointed out, however, that the simple (k2, k4) case misses an important aspect of realistic potentials, namely cubic anharmonicity, whose implications for ILMs in real solids are only now being sorted out. Our recent analytic and numerical studies of the effects of including realistic values of k3 (Bickham et al. 1993; Sandusky et al. 1994) have shown that the qualitative features of ILMs in (k2, k4) lattices remain when k3 is included, with the primary additional feature being that the ILMs are accompanied by a strong, amplitude-dependent, local static distortion of the lattice. As discussed earlier, this distortion tends to enhance the ILM's spatial localization as the mode frequency is lowered. Moreover, recent studies using realistic interparticle full potentials show that ILMs above the maximum lattice frequency do not occur for the one-dimensional systems studied (Kiselev et al. 1993; Kiselev et al. 1994b; Sandusky et al. 1994), whereas ILMs do occur in the phonon frequency gap in diatomic lattices (Kiselev et al. 1993). Again, these gap ILMs appear to retain most of the qualitative features of ILMs in simple (k2, k4) lattices, with the main effect of the odd-order terms in the potential function just being to produce the local static distortion.
Unusual anharmonic local mode systems
251
Clearly, the study of ILMs is in its infancy, with a vast number of important open questions remaining. Nevertheless, the recent results using realistic interparticle potentials suggest that ILMs will be theoretically understood sufficiently well in the near future that sensible experimental investigations can be launched to establish their presence in real solids.
5.
Conclusion
The experimental and theoretical studies of the two strongly anharmonic systems discussed in this chapter have revealed a wealth of fascinating and unexpected new phenomena. While much understanding has been achieved, the important remaining unexplained questions should provide a fertile area for obtaining fundamental new insights into the dynamical behavior of condensed matter systems.
6. Acknowledgements During the preparation of this article A.J.S. was supported by NSF Grant DMR-9312381 and ARO Grant DAAL03-92-G-0369 and J. B. P. was supported by NSF Grant DMR-9014729 and the Alexander von Humboldt Foundation. We would also like to acknowledge the important contributions made by our many collaborators on this work, particularly R.W. Alexander, S.A. Bickham, B.P. Clayman, R.P. Devaty, H. Fleurent, L. Genzel, L.H. Greene, S.B. Hearon, A.M. Kahan, R.D. Kirby, S. Kiselev, J.T. McWhirter, C.M. Mungan, I.G. Nolt, M. Patterson, A.M. Rosenberg, T. R6ssler, K.W. Sandusky and S. Takeno.
Note added in proof Since the preparation of the manuscript for this chapter, the ILM-related literature has continued to grow. Listed below are some recent papers. 1. ILMs in perfect and/or defect (k2, k4) lattices are discussed in: Wallis, R.E, A. Franchini and V. Bortolani (1994),Phys. Rev. B 50, 9851; Flach, S. and C.R. Willis (1994), Phys. Rev. Lett. 72, 1777; Dusi, R. and M. Wagner (1995), Phys. Rev. B 51, 15847; Takeno, S. (1995), J. Phys. Soc. Jpn 64, 2380; Kovalev, S., F. Zhang and Y.S. Kivshar (1995), Phys. Rev. B 51, 3218. 2. Diatomic Toda lattices are discussed in: Aoki, M. amd S. Takeno (1995), J. Phys. Soc. Jpn 64, 809.
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A.J. Sievers and J.B. Page
Ch. 3
3. The counting of ILMs is discussed in: Kiselev, S.A., S. Bickham and A.J. Sievers (1995), Commun. in Condens Mat. Phys. 17, 135.
4. Surface or edge ILMs are discussed in: Takeno, S., K. Hori, K. Ohtsuka and S. Homma (1994), J. Phys. Soc. Jpn 63, 1295; Watanabe, T. and S. Takeno (1994), Phys. Soc. Jpn 63, 2028; Bonart, D., A.P. Mayer and U. Schr~der (1995), Phys. Rev. Lett. 75, 870; Bonart, D., A.P. Mayer and U. Schr/Sder (1995), Phys. Rev. B 51, 13739.
5. Exact RWA solutions for ILMs driven by external fields are discussed in: Roessler, T. and J.B. Page (1995), Phys. Lett. A (accepted).
6. General nonlinear dynamics/soliton approaches" Huang, G. (1995), Phys. Rev. B 51, 12347; Neuper, A., EG. Mertens and N. Flytzanis (1994), Z. Phys. B 95, 397.
7. On-site potentials include: Flach, S. (1994), Phys. Rev. E 50, 3134; Flach, S., K. Kladko and C.R. Willis (1994), Phys. Rev. E 50, 2293; Flach, S. (1995), Phys. Rev. E 51, 3579.
8. Some quantum mechanical aspects related to ILMs are discussed in: Kitamura, T. and S. Takeno (1994), Phys. Lett. A 190, 327; Kitamura, T. and S. Takeno (1995), Physica A 213, 539; Roessler, T. and J.B. Page (1995), Phys. Rev. B 51, 11382.
9. The application to self-localized magnons is considered by: Takeno, S. and K. Kawasaki (1994), J. Phys. Soc. Jpn 63, 1928; Wallis, R.E, D.L. Mills and A.D. Boardman (1995), Phys. Rev. B 52, R3828.
10. An overview of some properties of ILMs is given by: Page, J.B. (1995), Physica B (accepted); Bickham, S.R., S.A. Kiselev and A.J. Sievers (1995), in: Spectroscopy and Dynamics of Collective Excitations in Solids, Ed. by B. DiBartolo (Plenum Press, New York).
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