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Anharmonic phonon decay in disordered systems
Physica B 271 (1999) 198 } 204
Anharmonic phonon decay in disordered systems Subir K. Bose!, Sean M. Kirkpatrick",1, W.M. Dennis",* !Department of Physics, University of Central Florida, Orlando, FL 32816, USA "Department of Physics and Astronomy, University of Georgia, Athens, GA 30602-2451, USA Received 4 June 1999; received in revised form 21 June 1999; accepted 14 July 1999
Abstract We investigate the e!ects of the relaxation of wave vector conservation due to broken translational symmetry on the anharmonic decay process. The relaxation of momentum conservation in the vicinity of a defect site is considered and its implications in various experimental methods of investigation are discussed. ( 1999 Elsevier Science B.V. All rights reserved. PACS: 63.20.Pw; 63.20.Ry Keywords: Anharmonic phonon decay; Disordered systems
1. Introduction Many optical techniques for the generation and detection of nonequilibrium acoustic phonons rely on the relaxation of wave vector conservation that occurs at a point defect. Examples of such techniques for phonon generation include spin lattice relaxation by both single phonon [1,2] and multiphonon emission [3,4] and defect-induced one phonon absorption (DIOPA) [5,6]. Phonon detection techniques that rely on an optically active defect for phonon detection include hot luminescence [1,2], vibronic sideband phonon spectroscopy (VSPS) [7] and phonon-induced coherence loss (PICOLO) [8]. * Corresponding author. Fax:#1-706-542-2492. E-mail address: [email protected] (W.M. Dennis) 1 Present address: Code 5640, Laser Physics Branch, US Naval Research Laboratory, Washington DC 20375, USA.
These optical techniques when applied to the study of phonon decay have been used to investigate the dynamics of nonequilibrium acoustic phonons with spectral, temporal and spatial resolution (see for example Ref. [9]). In particular, various combinations of these techniques have been used to investigate anharmonic decay in a variety of insulating crystals [10,11]. The results of these experimental investigations have been interpreted in terms of an isotropic model with linear dispersion in which wave vector conservation is strictly conserved. The above model for anharmonic three phonon decay was "rst introduced by Orbach and Vredevoe [12] and Klemens [13] and was shown to yield an u5 dependence of the phonon decay rate on frequency. The interaction of acoustic phonons with point defects can lead to both elastic and inelastic scattering processes. Elastic scattering from point defects leads to an u4 dependence of the phonon
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S.K. Bose et al. / Physica B 271 (1999) 198}204
lifetime [14], however, this power dependence will not manifest itself in those experiments which average over wave vector. Inelastic scattering processes can lead to non power-law frequency dependencies of the phonon lifetime [11,15] While the e!ect of point defects on phonon dynamics has been studied in detail both experimentally and theoretically, the interaction of acoustic phonons with a point defect has been treated as a separate process independent of anharmonic decay. The combination of elastic scattering with anharmonic decay leads to additional collective propagation modes such as quasidi!usion [16] and nonlocal heat transfer [17]. In this work, we consider the e!ect of broken translational symmetry on the three phonon anharmonic decay process itself. In particular we determine if the relaxation of wave vector conservation within the con"nes of an isotropic model with linear dispersion leads to a deviation from the u5 dependence which can be observed experimentally. Our approach in this paper is to consider the case where all interacting phonons are traveling wave modes. We point out that while experimental investigations have been performed to study the e!ect of increased disorder on phonon decay, the increased disorder may introduce additional decay mechanisms, such as inelastic scattering which may dominate the frequency dependence of the anharmonic decay. In addition, phonon transport e!ects can mask the frequency dependence of the phonon decay rate at low frequencies when the anharmonic decay time becomes longer than the time necessary for the phonons to escape from the interaction region [16]. The plan of this paper is as follows: in Section 2 a brief discussion of single-mode relaxation time (SMRT) theory of phonon lifetimes is presented. In Section 3, we discuss the relaxation of momentum conservation and indicate how the e!ects of disorder are introduced into the lifetime calculation. In Section 4, the phonon decay time is calculated for the case of traveling wave modes. Finally, in Section 5, we summarize our results and compare our theoretical predictions with current experimental results and discuss the implications for future experiments.
199
2. Single-mode relaxation time The theoretical treatment of a system of nonequilibrium phonons is fairly complicated. The approximation `single-mode relaxation timea provides a reasonable approximation in which one of the participating phonon modes is assumed to be away from the equilibrium, while the other modes are presumed to have their equilibrium values. Consider the decay of a phonon (qs) into two lower frequency phonons (q@s@) and (qAsA), where q stands for wave vector, and s is the branch index. In the single-mode relaxation time approximation, the mode (qs) is assumed to be nonequilibrium while the other modes are in thermal equilibrium. Denoting the occupation number of the mode by n , we write [18] qs 1 n " + , qs (eb u(qs)~Wqs !1)
(1)
Here, u(qs) is the frequency and b"1/k ¹; k is B B the Boltzmann constant, ¹ is the temperature and W denotes the deviation from the equilibrium qs distribution n6 . For small deviations, we have the qs linear approximation n "n6 #W n6 (n6 #1). qs qs qs qs qs
(2)
The collision term in the Boltzmann}Peierls equation in the relaxation time approximation can be written as [18] Rn (n !n6 ) qs "! qs qs , Rt q qs
(3)
where q is the decay time of the (qs) phonon. qs Following Ref. [18], Eqs. (2) and(3) yield an expression for q~1, i.e. qs
C
D
1 1 1 " + PM qAsA # PM q{s{,qAsA . qs,q{s{ q 2 qs n6 (n6 #1) qs qs qs q{s{,qAsA
(4)
PM qAsA and PM q{s{,qAsA are the equilibrium transition qs,q{s{ qs rates of the processes (qs)#(q@s@)P(qAsA) and (qs)P(q@s@)#(qAsA), respectively. The transition rates are calculated using the golden rule.
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3. Disorder and the breakdown of momentum conservation The interaction Hamiltonian for the spontaneous phonon decay due to three phonon processes can be written as [13,14] 1 H*/5" + + eir >(q~q{~qA)c(q, q@, qA)a as asA #h.c., q q{ q < q,q{,qA r (5) where the as (a ) are the appropriate creation (anniq q hilation) operators as de"ned in Ref. [13] and c(q, q@, qA) represents the strength of the anharmonic interaction. In a translationally invariant lattice, the sum of the phase factors over all atomic sites r, yields a Kronecker delta dq q qA G , signifying the ~ {~ , conservation of wave vector. We will restrict our discussion to Normal processes only and hence the reciprocal wave vector G can be set to zero. The Kronecker delta has its origin in the orthogonality of the plane wave eigenstates of the harmonic lattice. In a disordered system, the sum over two-phase functions instead obey the relation [19] + e*rj >(q~q{)"e*0S(q!q@), (6) j where ei0 is a phase factor and S(q!q@) is the `structure factora which embodies information about disorder in the system. In terms of the structure factor, the interaction Hamiltonian of the disordered crystal can therefore be written as N 0 H*/5" + ei S(q!k)c(q, q@, qA)a as asA , (7) q q{ q < q,q{,qA where k"q@#qA, and N is the number of atoms in the crystal. The above Hamiltonian will now be used to evaluate the phonon decay time for the spontaneous decay process.
the GruK neisen approximation and choose c(q, q@, qA) as [20] !i 2M c c(q, q@, qA)" uu@uA, JN J3 v6
(8)
where v6 is an average velocity of sound, c is the GruK neisen constant, and u"u(qs), u@"u(q@s@), and uA"u(qAsA). The transition rate, using the golden rule, can be written as
AB
ABA B
4M2 c 2 + 3 2p N 2 PM q{s{,qAsA" DS(q!k)D2 qs 3 v6 + < M ]uu@uAn6 (n6 #1)(n6 A A#1)d(u!u@!uA). (9) qs q{s{ qs The phonon decay rate is then given by
P
P
q~1" + A A dq@ d)@ dqA d)Aqq@3qA3 qs ss{s s{,sA (n6 #1)(n6 A A #1) qs ]DS(q!q@!qA)D2 q{s{ (n6 #1) qs ]d(u!u@!uA).
(10)
In deriving Eq. (10), we have used the isotropic continuum approximation, and replaced the summations over q and q@ by integrations. A A is ss{s a constant including the phonon speeds v , v and s s{ v A. s Also, in the low-temperature limit, (n6 #1)(n6 A A #1) q{s{ qs +1. (11) (n6 #1) qs The concept of the structure factor was introduced earlier by Morgan [21] and Morgan and Smith [22] to model Umklapp process in anharmonic phonon scattering in glassy solids in which the Normal processes were assumed to obey wave vector conservation. In the absence of experimental data on the structure factor, we assume a simple Lorentzian form, which also arises in the Ornstein}Zernicke theory of #uids [23], i.e.
4. Spontaneous decay rate
S(0) S(k)" , 1#f2k2
To evaluate the phonon decay rate by spontaneous anharmonic three phonon processes we use
where f is a correlation length. S(0) is the value of S(k) at k"0. From Eq. (10), we have, using
(12)
S.K. Bose et al. / Physica B 271 (1999) 198}204
201
Eqs. (11) and (12),
P
P
qq@3qA3 q~1J dq@ dqA dX@ dXA qs [1#f2(q!q@!qA)2]2 ]d(u!u@!uA).
(13)
We now evaluate Eq. (13) numerically [24] for the case of linear dispersion. For the numerical calculations we take v /v "v /v A to be 1.5 where s s{ s s v is the phase velocity of mode s. s As shown in Fig. 1, for small values of f/v , we s obtain a single-power-law behavior of q~1Ju8 qs while for large values of f/v we observe the exs pected power law of q~1Ju5. At intermediate qs values of f/v we observe a cross-over from a u8 bes havior at low frequencies to the expected u5 behavior at high frequencies. If the phonon decay rate as calculated from Eq. (13) is assumed to be of the form q~1Ju5f (uf/v ), qs s then a plot of u~5q~1 versus uf/v should collapse qs s the curves in Fig. 1 to a single curve as is shown in Fig. 2. Fig. 2. Plot of u~5q1 versus uf/v . Using this transformation, qs s the curves in Fig. 1. collapse to a single curve.
Fig. 1. Dependence of the phonon decay rate as a function of the mode frequency in Hz for a range of values of m/v . s
For comparison with experimental data which are usually plotted on a log}log scale and "t to a power law, we also have "t the frequency dependence of the phonon decay rate calculated from Eq. (13) to a power-law dependence. The power-law exponent (n) as a function of f/v is shown in Fig. 3. s We note that there is a physical lower limit to the values that f/v can take, i.e f/v '10~13 s, since s s this corresponds to a correlation length of the order of a few lattice spacings. We point out that Eq. (13) will not be valid for arbitrarily high defect concentrations, since under these conditions, the phonon density of states may deviate signi"cantly from the pure crystal, resulting in a modi"ed frequency dependence. We also note that Eq. (13) can be evaluated analytically for the case of collinear phonon decay. The details of this case are given in the appendix. Finally, we point out that while the form of S(k) in Eq. (12) has been physically motivated, we wish to emphasize that this distribution is not unique. Any distribution d (x) that can be used to f
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S.K. Bose et al. / Physica B 271 (1999) 198}204
Fig. 3. Dependence of the power-law exponent as a function of f/v . The power-law exponent crosses over from a value of 8 at s low values of f/v to 5 at high values of f/v . s s
approximate a delta function in the limit fPR and becomes constant as fP0 will also reproduce the limiting power-law behavior of the present calculation. The details of the cross over between the two power laws will indeed be distribution dependent, however, the exact form of the cross over behavior is not a central result of this paper.
5. Comparison with experimental data In this section we compare the results of Eq. (13) with experimental data in which the frequency dependence of the phonon lifetime has been investigated. In particular, we note the frequency dependence of the phonon decay rate at low frequencies. In an experiment performed by Will et al. [10] in LaF : Er3`, phonons were generated using a heat 3 pulse and detected on the Er3` impurity ion by magnetically tuning the splitting of the 4S (1) 3@2 state. The authors observed a u5 dependence at
higher frequencies converting to a frequency independent lifetime at lower frequencies. The exact value for the frequency independent lifetime was found to depend on the size of the interaction volume. However, in an experiment performed by Wang et al. [11], who used defect-induced one phonon absorption to generate monochromatic phonon distributions in YLF : Pr3` and measured the lifetimes of the decay products as a function of frequency, the decay products were found to exhibit an u5 dependence over the range from 30 to 100 cm~1. In another experiment by Wang et al. [25] in YAG : Pr3` the frequency dependence of the phonon decay rate was found to cross over to a value (5 at low frequencies. This behavior was attributed to the inelastic scattering of phonons o! low-energy electronic transitions. Finally, in what is perhaps the most complete study of phonon decay in disordered materials, Happek et al. [26] observed a ju5 frequency dependence in Ca Sr F . Here j is a parameter (1~x) x 2 which increased for increasing Sr concentrations up to x"0.05. In addition, these authors also observed, at low frequencies, a jun dependence in Ca La F , where n ranged from 5 to 3 as (1~x) x 2 x ranged from 0 to 0.025. They observed that as the concentration of defects increased, the frequency dependence of the phonon lifetime decreased. These authors suggested that the dominant dependence of the phonon lifetime at high concentrations might be explained by inelastic scattering processes and that it was unclear if the lattice anharmonicities could give rise to such a behavior. Our model predicts a frequency dependence that bends over to a u8 dependence at lower frequencies. To our knowledge a frequency dependence of where n'5 has not been observed to date. However, we point out that this frequency dependence should manifest itself at low phonon frequencies where the measured phonon lifetime in optical experiments is dominated by the time taken for the phonons to escape from the interaction volume or other decay processes such as inelastic scattering, making accurate measurements of the anharmonic decay rate di$cult. We conclude that under the conditions in which most optically detected phonon experiments are
S.K. Bose et al. / Physica B 271 (1999) 198}204
performed, the expected frequency dependence is u5. If deviations from this power law are obtained care should be taken to ascertain that the measured decay rate is due to anharmonic decay and not due to the additional mechanisms (inelastic scattering, di!usive escape) described above.
Acknowledgements We would like to thank Shubha Bose for her help in the computation. One of us (S.K.B.) would like to thank William Yen for his warm hospitality during the stay at the University of Georgia. We are indebted to Paul Klemens for valuable suggestions. We also wish to thank Michael Geller for many useful discussions. Finally, we wish to thank David Huber for many helpful discussions and comments and for suggesting the form of the universal curves shown in Fig. 2. In addition the authors would also like to acknowledge the support of the National Science Foundation grant DMR-9321052.
203
which evaluates to u8 q~1J . (A.3) qs [1#(f2/v2)u2(1!d)2]2 s For typical values of the correlation length, f (+ 100 As ), the phonon group velocity v (+3000 s ms~1), and for terahertz phonons, we "nd that (f2/v2)u2(1!d)2<1, indicating the dominant bes havior of the inverse lifetime to be u4 with a correction term of order u2. In Eq. (13) an additional term describing the positional correlation of the point defects may be introduced [27]. However, we do not consider such an e!ect here. Finally, we note that the analytical solution for the collinear case provides an independent test of the numerical integration procedure used for the noncollinear case. In the collinear case the numerical calculations indicate that q~1 exhibits an u8 qs dependence at low values of (f2/v2)u2(1!d)2 s crossing over to an u4 dependence in good agreement with the analytical expression given by Eq. (A.3).
References Appendix A To evaluate Eq. (13) we choose v /v and v /v A s s{ s s to be of order, but not equal to unity, i.e. v /v "v /v A "d where d is any suitable constant. s s{ s s The above expression is still a complicated function of the angles between q and qA and that between q and q@, q being chosen along the z-axis. To obtain an analytical expression, we consider collinear scattering. Note that this implies conservation of wave vector only for all phonons decaying along the same branch, or d"1. The above simpli"cation gives
PP
[1] J.I. Dijkhuis, A. van der Pal, H.W. de Wijn, Phys. Rev. Lett. 37 (1976) 1554. [2] R.S. Meltzer, J.E. Rives, Phys. Rev. Lett. 38 (1977) 421. [3] V.A. Benderskii, V.Kh. Brikenstein, V.L. Broude, A.G. Lavrushko, Solid State Commun. 15 (1974) 1235. [4] V.L. Broude, N.A. Vidmont, V.V. Korshunov, Y.B. Levinson, A.A. Maksimov, I.I. Tartakovskii, JETP Lett. 25 (1977) 261. [5] U. Happek, T.P. Lang, K.F. Renk, in: S. Hunklinger, W. Ludwig, G. Weiss (Eds.), Proceedings of the Third International Conference Phonon Physics, World Scienti"c, Singapore, 1990, p. 1427. [6] W.A. Tolbert, W.M. Dennis, W.M. Yen, Phys. Rev. Lett. 65 (1990) 607.
uu@3uA3d(u!u@!uA) du@ duA . [1#(f2/v2)(u2#d2u@2#d2uA2!2duu@!2duuA!2d2u@uA)]2 s
q~1J qs
(A.1)
Integrating over uA, we obtain
P
q~1J qs
u uu@3(u!u@)3 du@ , [1#(f2/v2)u2(1!d)2]2 0 s
(A.2)
[7] M.J. Colles, J.A. Giordmaine, Phys. Rev. Lett. 27 (1971) 670. [8] D.M. Boye, J.E. Rives, R.S. Meltzer, J. Lumin 45 (1990) 147.
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[9] K.F. Renk, in: W.E. Bron (Ed.), Nonequilibrium Phonon Dynamics, Ser. B 124, Plenum Press, New York, 1985 (Chapter. 3). [10] J.M. Will, W. Eisfeld, K.F. Renk, Appl. Phys. A 31 (1983) 191. [11] Xiao-jun Wang, W.M. Dennis, W.M. Yen, Phys. Rev. B 46 (1992) 8168. [12] R. Orbach, L.A. Vredevoe, Physics 1 Long Island City, New York, 1964, p. 91. [13] P.G. Klemens, J. Appl. Phys. 38 (1967) 4573. [14] P.G. Klemens, Proc. Phys. Soc. Lond. A 68 (1955) 1113. [15] A.A. Kaplyanski, S.A. Basun, in: W. Eisenmenger, A.A. Kaplyanski (Eds.), Nonequilibrium Phonons in Nonmetallic Crystals, North-Holland, Amsterdam, 1986 (chapter. 8). [16] T.E. Wilson, F.M. Lurie, W.E. Bron, Phys. Rev. B 30 (1984) 6103. [17] Y.B. Levinson, Solid. State Commun. 36 (1980) 73.
[18] G.P. Shrivastava, The Physics of Phonons, Adam Hilger, New York, 1990. [19] Z.M. Ziman, Models of Disorder, Cambridge University Press, Cambridge, 1979. [20] P.G. Klemens, Phys. Rev. 122 (1961) 443. [21] G.J. Morgan, J. Phys. C1 (1968) 347. [22] G.J. Morgan, D. Smith, J. Phys. C7 (1974) 649. [23] N.H. March, R.A. Street, M. Mosi (Eds.), Amorphous Solids and the Liquid State, Plenum Press, New York, 1985. [24] NAG #90 Manual, Release 2, Numerical Algorithms Group Limited, Oxford, 1995 (Chapter. 11). [25] Xiao-jun Wang, J. Ganem, W.M. Dennis, W.M. Yen, Phys. Rev. B 44 (1991) 900. [26] U. Happek, W.W. Fischer, J.A. Campbell, in: M. Meissner, R.O. Pohl (Eds.), Phonon Scattering in Condensed Matter VII, Springer, Berlin, 1992, p. 306 (Chapter. 6). [27] P.G. Klemens, in: V.D. Frechette (Ed.), non-crystalline Solids, Wiley, New York, 1960 (Chapter 20).
Anharmonic phonon decay in disordered systems Subir K. Bose!, Sean M. Kirkpatrick",1, W.M. Dennis",* !Department of Physics, University of Central Florida, Orlando, FL 32816, USA "Department of Physics and Astronomy, University of Georgia, Athens, GA 30602-2451, USA Received 4 June 1999; received in revised form 21 June 1999; accepted 14 July 1999
Abstract We investigate the e!ects of the relaxation of wave vector conservation due to broken translational symmetry on the anharmonic decay process. The relaxation of momentum conservation in the vicinity of a defect site is considered and its implications in various experimental methods of investigation are discussed. ( 1999 Elsevier Science B.V. All rights reserved. PACS: 63.20.Pw; 63.20.Ry Keywords: Anharmonic phonon decay; Disordered systems
1. Introduction Many optical techniques for the generation and detection of nonequilibrium acoustic phonons rely on the relaxation of wave vector conservation that occurs at a point defect. Examples of such techniques for phonon generation include spin lattice relaxation by both single phonon [1,2] and multiphonon emission [3,4] and defect-induced one phonon absorption (DIOPA) [5,6]. Phonon detection techniques that rely on an optically active defect for phonon detection include hot luminescence [1,2], vibronic sideband phonon spectroscopy (VSPS) [7] and phonon-induced coherence loss (PICOLO) [8]. * Corresponding author. Fax:#1-706-542-2492. E-mail address: [email protected] (W.M. Dennis) 1 Present address: Code 5640, Laser Physics Branch, US Naval Research Laboratory, Washington DC 20375, USA.
These optical techniques when applied to the study of phonon decay have been used to investigate the dynamics of nonequilibrium acoustic phonons with spectral, temporal and spatial resolution (see for example Ref. [9]). In particular, various combinations of these techniques have been used to investigate anharmonic decay in a variety of insulating crystals [10,11]. The results of these experimental investigations have been interpreted in terms of an isotropic model with linear dispersion in which wave vector conservation is strictly conserved. The above model for anharmonic three phonon decay was "rst introduced by Orbach and Vredevoe [12] and Klemens [13] and was shown to yield an u5 dependence of the phonon decay rate on frequency. The interaction of acoustic phonons with point defects can lead to both elastic and inelastic scattering processes. Elastic scattering from point defects leads to an u4 dependence of the phonon
0921-4526/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 2 2 4 - 0
S.K. Bose et al. / Physica B 271 (1999) 198}204
lifetime [14], however, this power dependence will not manifest itself in those experiments which average over wave vector. Inelastic scattering processes can lead to non power-law frequency dependencies of the phonon lifetime [11,15] While the e!ect of point defects on phonon dynamics has been studied in detail both experimentally and theoretically, the interaction of acoustic phonons with a point defect has been treated as a separate process independent of anharmonic decay. The combination of elastic scattering with anharmonic decay leads to additional collective propagation modes such as quasidi!usion [16] and nonlocal heat transfer [17]. In this work, we consider the e!ect of broken translational symmetry on the three phonon anharmonic decay process itself. In particular we determine if the relaxation of wave vector conservation within the con"nes of an isotropic model with linear dispersion leads to a deviation from the u5 dependence which can be observed experimentally. Our approach in this paper is to consider the case where all interacting phonons are traveling wave modes. We point out that while experimental investigations have been performed to study the e!ect of increased disorder on phonon decay, the increased disorder may introduce additional decay mechanisms, such as inelastic scattering which may dominate the frequency dependence of the anharmonic decay. In addition, phonon transport e!ects can mask the frequency dependence of the phonon decay rate at low frequencies when the anharmonic decay time becomes longer than the time necessary for the phonons to escape from the interaction region [16]. The plan of this paper is as follows: in Section 2 a brief discussion of single-mode relaxation time (SMRT) theory of phonon lifetimes is presented. In Section 3, we discuss the relaxation of momentum conservation and indicate how the e!ects of disorder are introduced into the lifetime calculation. In Section 4, the phonon decay time is calculated for the case of traveling wave modes. Finally, in Section 5, we summarize our results and compare our theoretical predictions with current experimental results and discuss the implications for future experiments.
199
2. Single-mode relaxation time The theoretical treatment of a system of nonequilibrium phonons is fairly complicated. The approximation `single-mode relaxation timea provides a reasonable approximation in which one of the participating phonon modes is assumed to be away from the equilibrium, while the other modes are presumed to have their equilibrium values. Consider the decay of a phonon (qs) into two lower frequency phonons (q@s@) and (qAsA), where q stands for wave vector, and s is the branch index. In the single-mode relaxation time approximation, the mode (qs) is assumed to be nonequilibrium while the other modes are in thermal equilibrium. Denoting the occupation number of the mode by n , we write [18] qs 1 n " + , qs (eb u(qs)~Wqs !1)
(1)
Here, u(qs) is the frequency and b"1/k ¹; k is B B the Boltzmann constant, ¹ is the temperature and W denotes the deviation from the equilibrium qs distribution n6 . For small deviations, we have the qs linear approximation n "n6 #W n6 (n6 #1). qs qs qs qs qs
(2)
The collision term in the Boltzmann}Peierls equation in the relaxation time approximation can be written as [18] Rn (n !n6 ) qs "! qs qs , Rt q qs
(3)
where q is the decay time of the (qs) phonon. qs Following Ref. [18], Eqs. (2) and(3) yield an expression for q~1, i.e. qs
C
D
1 1 1 " + PM qAsA # PM q{s{,qAsA . qs,q{s{ q 2 qs n6 (n6 #1) qs qs qs q{s{,qAsA
(4)
PM qAsA and PM q{s{,qAsA are the equilibrium transition qs,q{s{ qs rates of the processes (qs)#(q@s@)P(qAsA) and (qs)P(q@s@)#(qAsA), respectively. The transition rates are calculated using the golden rule.
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3. Disorder and the breakdown of momentum conservation The interaction Hamiltonian for the spontaneous phonon decay due to three phonon processes can be written as [13,14] 1 H*/5" + + eir >(q~q{~qA)c(q, q@, qA)a as asA #h.c., q q{ q < q,q{,qA r (5) where the as (a ) are the appropriate creation (anniq q hilation) operators as de"ned in Ref. [13] and c(q, q@, qA) represents the strength of the anharmonic interaction. In a translationally invariant lattice, the sum of the phase factors over all atomic sites r, yields a Kronecker delta dq q qA G , signifying the ~ {~ , conservation of wave vector. We will restrict our discussion to Normal processes only and hence the reciprocal wave vector G can be set to zero. The Kronecker delta has its origin in the orthogonality of the plane wave eigenstates of the harmonic lattice. In a disordered system, the sum over two-phase functions instead obey the relation [19] + e*rj >(q~q{)"e*0S(q!q@), (6) j where ei0 is a phase factor and S(q!q@) is the `structure factora which embodies information about disorder in the system. In terms of the structure factor, the interaction Hamiltonian of the disordered crystal can therefore be written as N 0 H*/5" + ei S(q!k)c(q, q@, qA)a as asA , (7) q q{ q < q,q{,qA where k"q@#qA, and N is the number of atoms in the crystal. The above Hamiltonian will now be used to evaluate the phonon decay time for the spontaneous decay process.
the GruK neisen approximation and choose c(q, q@, qA) as [20] !i 2M c c(q, q@, qA)" uu@uA, JN J3 v6
(8)
where v6 is an average velocity of sound, c is the GruK neisen constant, and u"u(qs), u@"u(q@s@), and uA"u(qAsA). The transition rate, using the golden rule, can be written as
AB
ABA B
4M2 c 2 + 3 2p N 2 PM q{s{,qAsA" DS(q!k)D2 qs 3 v6 + < M ]uu@uAn6 (n6 #1)(n6 A A#1)d(u!u@!uA). (9) qs q{s{ qs The phonon decay rate is then given by
P
P
q~1" + A A dq@ d)@ dqA d)Aqq@3qA3 qs ss{s s{,sA (n6 #1)(n6 A A #1) qs ]DS(q!q@!qA)D2 q{s{ (n6 #1) qs ]d(u!u@!uA).
(10)
In deriving Eq. (10), we have used the isotropic continuum approximation, and replaced the summations over q and q@ by integrations. A A is ss{s a constant including the phonon speeds v , v and s s{ v A. s Also, in the low-temperature limit, (n6 #1)(n6 A A #1) q{s{ qs +1. (11) (n6 #1) qs The concept of the structure factor was introduced earlier by Morgan [21] and Morgan and Smith [22] to model Umklapp process in anharmonic phonon scattering in glassy solids in which the Normal processes were assumed to obey wave vector conservation. In the absence of experimental data on the structure factor, we assume a simple Lorentzian form, which also arises in the Ornstein}Zernicke theory of #uids [23], i.e.
4. Spontaneous decay rate
S(0) S(k)" , 1#f2k2
To evaluate the phonon decay rate by spontaneous anharmonic three phonon processes we use
where f is a correlation length. S(0) is the value of S(k) at k"0. From Eq. (10), we have, using
(12)
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201
Eqs. (11) and (12),
P
P
qq@3qA3 q~1J dq@ dqA dX@ dXA qs [1#f2(q!q@!qA)2]2 ]d(u!u@!uA).
(13)
We now evaluate Eq. (13) numerically [24] for the case of linear dispersion. For the numerical calculations we take v /v "v /v A to be 1.5 where s s{ s s v is the phase velocity of mode s. s As shown in Fig. 1, for small values of f/v , we s obtain a single-power-law behavior of q~1Ju8 qs while for large values of f/v we observe the exs pected power law of q~1Ju5. At intermediate qs values of f/v we observe a cross-over from a u8 bes havior at low frequencies to the expected u5 behavior at high frequencies. If the phonon decay rate as calculated from Eq. (13) is assumed to be of the form q~1Ju5f (uf/v ), qs s then a plot of u~5q~1 versus uf/v should collapse qs s the curves in Fig. 1 to a single curve as is shown in Fig. 2. Fig. 2. Plot of u~5q1 versus uf/v . Using this transformation, qs s the curves in Fig. 1. collapse to a single curve.
Fig. 1. Dependence of the phonon decay rate as a function of the mode frequency in Hz for a range of values of m/v . s
For comparison with experimental data which are usually plotted on a log}log scale and "t to a power law, we also have "t the frequency dependence of the phonon decay rate calculated from Eq. (13) to a power-law dependence. The power-law exponent (n) as a function of f/v is shown in Fig. 3. s We note that there is a physical lower limit to the values that f/v can take, i.e f/v '10~13 s, since s s this corresponds to a correlation length of the order of a few lattice spacings. We point out that Eq. (13) will not be valid for arbitrarily high defect concentrations, since under these conditions, the phonon density of states may deviate signi"cantly from the pure crystal, resulting in a modi"ed frequency dependence. We also note that Eq. (13) can be evaluated analytically for the case of collinear phonon decay. The details of this case are given in the appendix. Finally, we point out that while the form of S(k) in Eq. (12) has been physically motivated, we wish to emphasize that this distribution is not unique. Any distribution d (x) that can be used to f
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Fig. 3. Dependence of the power-law exponent as a function of f/v . The power-law exponent crosses over from a value of 8 at s low values of f/v to 5 at high values of f/v . s s
approximate a delta function in the limit fPR and becomes constant as fP0 will also reproduce the limiting power-law behavior of the present calculation. The details of the cross over between the two power laws will indeed be distribution dependent, however, the exact form of the cross over behavior is not a central result of this paper.
5. Comparison with experimental data In this section we compare the results of Eq. (13) with experimental data in which the frequency dependence of the phonon lifetime has been investigated. In particular, we note the frequency dependence of the phonon decay rate at low frequencies. In an experiment performed by Will et al. [10] in LaF : Er3`, phonons were generated using a heat 3 pulse and detected on the Er3` impurity ion by magnetically tuning the splitting of the 4S (1) 3@2 state. The authors observed a u5 dependence at
higher frequencies converting to a frequency independent lifetime at lower frequencies. The exact value for the frequency independent lifetime was found to depend on the size of the interaction volume. However, in an experiment performed by Wang et al. [11], who used defect-induced one phonon absorption to generate monochromatic phonon distributions in YLF : Pr3` and measured the lifetimes of the decay products as a function of frequency, the decay products were found to exhibit an u5 dependence over the range from 30 to 100 cm~1. In another experiment by Wang et al. [25] in YAG : Pr3` the frequency dependence of the phonon decay rate was found to cross over to a value (5 at low frequencies. This behavior was attributed to the inelastic scattering of phonons o! low-energy electronic transitions. Finally, in what is perhaps the most complete study of phonon decay in disordered materials, Happek et al. [26] observed a ju5 frequency dependence in Ca Sr F . Here j is a parameter (1~x) x 2 which increased for increasing Sr concentrations up to x"0.05. In addition, these authors also observed, at low frequencies, a jun dependence in Ca La F , where n ranged from 5 to 3 as (1~x) x 2 x ranged from 0 to 0.025. They observed that as the concentration of defects increased, the frequency dependence of the phonon lifetime decreased. These authors suggested that the dominant dependence of the phonon lifetime at high concentrations might be explained by inelastic scattering processes and that it was unclear if the lattice anharmonicities could give rise to such a behavior. Our model predicts a frequency dependence that bends over to a u8 dependence at lower frequencies. To our knowledge a frequency dependence of where n'5 has not been observed to date. However, we point out that this frequency dependence should manifest itself at low phonon frequencies where the measured phonon lifetime in optical experiments is dominated by the time taken for the phonons to escape from the interaction volume or other decay processes such as inelastic scattering, making accurate measurements of the anharmonic decay rate di$cult. We conclude that under the conditions in which most optically detected phonon experiments are
S.K. Bose et al. / Physica B 271 (1999) 198}204
performed, the expected frequency dependence is u5. If deviations from this power law are obtained care should be taken to ascertain that the measured decay rate is due to anharmonic decay and not due to the additional mechanisms (inelastic scattering, di!usive escape) described above.
Acknowledgements We would like to thank Shubha Bose for her help in the computation. One of us (S.K.B.) would like to thank William Yen for his warm hospitality during the stay at the University of Georgia. We are indebted to Paul Klemens for valuable suggestions. We also wish to thank Michael Geller for many useful discussions. Finally, we wish to thank David Huber for many helpful discussions and comments and for suggesting the form of the universal curves shown in Fig. 2. In addition the authors would also like to acknowledge the support of the National Science Foundation grant DMR-9321052.
203
which evaluates to u8 q~1J . (A.3) qs [1#(f2/v2)u2(1!d)2]2 s For typical values of the correlation length, f (+ 100 As ), the phonon group velocity v (+3000 s ms~1), and for terahertz phonons, we "nd that (f2/v2)u2(1!d)2<1, indicating the dominant bes havior of the inverse lifetime to be u4 with a correction term of order u2. In Eq. (13) an additional term describing the positional correlation of the point defects may be introduced [27]. However, we do not consider such an e!ect here. Finally, we note that the analytical solution for the collinear case provides an independent test of the numerical integration procedure used for the noncollinear case. In the collinear case the numerical calculations indicate that q~1 exhibits an u8 qs dependence at low values of (f2/v2)u2(1!d)2 s crossing over to an u4 dependence in good agreement with the analytical expression given by Eq. (A.3).
References Appendix A To evaluate Eq. (13) we choose v /v and v /v A s s{ s s to be of order, but not equal to unity, i.e. v /v "v /v A "d where d is any suitable constant. s s{ s s The above expression is still a complicated function of the angles between q and qA and that between q and q@, q being chosen along the z-axis. To obtain an analytical expression, we consider collinear scattering. Note that this implies conservation of wave vector only for all phonons decaying along the same branch, or d"1. The above simpli"cation gives
PP
[1] J.I. Dijkhuis, A. van der Pal, H.W. de Wijn, Phys. Rev. Lett. 37 (1976) 1554. [2] R.S. Meltzer, J.E. Rives, Phys. Rev. Lett. 38 (1977) 421. [3] V.A. Benderskii, V.Kh. Brikenstein, V.L. Broude, A.G. Lavrushko, Solid State Commun. 15 (1974) 1235. [4] V.L. Broude, N.A. Vidmont, V.V. Korshunov, Y.B. Levinson, A.A. Maksimov, I.I. Tartakovskii, JETP Lett. 25 (1977) 261. [5] U. Happek, T.P. Lang, K.F. Renk, in: S. Hunklinger, W. Ludwig, G. Weiss (Eds.), Proceedings of the Third International Conference Phonon Physics, World Scienti"c, Singapore, 1990, p. 1427. [6] W.A. Tolbert, W.M. Dennis, W.M. Yen, Phys. Rev. Lett. 65 (1990) 607.
uu@3uA3d(u!u@!uA) du@ duA . [1#(f2/v2)(u2#d2u@2#d2uA2!2duu@!2duuA!2d2u@uA)]2 s
q~1J qs
(A.1)
Integrating over uA, we obtain
P
q~1J qs
u uu@3(u!u@)3 du@ , [1#(f2/v2)u2(1!d)2]2 0 s
(A.2)
[7] M.J. Colles, J.A. Giordmaine, Phys. Rev. Lett. 27 (1971) 670. [8] D.M. Boye, J.E. Rives, R.S. Meltzer, J. Lumin 45 (1990) 147.
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[9] K.F. Renk, in: W.E. Bron (Ed.), Nonequilibrium Phonon Dynamics, Ser. B 124, Plenum Press, New York, 1985 (Chapter. 3). [10] J.M. Will, W. Eisfeld, K.F. Renk, Appl. Phys. A 31 (1983) 191. [11] Xiao-jun Wang, W.M. Dennis, W.M. Yen, Phys. Rev. B 46 (1992) 8168. [12] R. Orbach, L.A. Vredevoe, Physics 1 Long Island City, New York, 1964, p. 91. [13] P.G. Klemens, J. Appl. Phys. 38 (1967) 4573. [14] P.G. Klemens, Proc. Phys. Soc. Lond. A 68 (1955) 1113. [15] A.A. Kaplyanski, S.A. Basun, in: W. Eisenmenger, A.A. Kaplyanski (Eds.), Nonequilibrium Phonons in Nonmetallic Crystals, North-Holland, Amsterdam, 1986 (chapter. 8). [16] T.E. Wilson, F.M. Lurie, W.E. Bron, Phys. Rev. B 30 (1984) 6103. [17] Y.B. Levinson, Solid. State Commun. 36 (1980) 73.
[18] G.P. Shrivastava, The Physics of Phonons, Adam Hilger, New York, 1990. [19] Z.M. Ziman, Models of Disorder, Cambridge University Press, Cambridge, 1979. [20] P.G. Klemens, Phys. Rev. 122 (1961) 443. [21] G.J. Morgan, J. Phys. C1 (1968) 347. [22] G.J. Morgan, D. Smith, J. Phys. C7 (1974) 649. [23] N.H. March, R.A. Street, M. Mosi (Eds.), Amorphous Solids and the Liquid State, Plenum Press, New York, 1985. [24] NAG #90 Manual, Release 2, Numerical Algorithms Group Limited, Oxford, 1995 (Chapter. 11). [25] Xiao-jun Wang, J. Ganem, W.M. Dennis, W.M. Yen, Phys. Rev. B 44 (1991) 900. [26] U. Happek, W.W. Fischer, J.A. Campbell, in: M. Meissner, R.O. Pohl (Eds.), Phonon Scattering in Condensed Matter VII, Springer, Berlin, 1992, p. 306 (Chapter. 6). [27] P.G. Klemens, in: V.D. Frechette (Ed.), non-crystalline Solids, Wiley, New York, 1960 (Chapter 20).