- Email: [email protected]
Calculation of the Huang-Rhys factor for F-centers
J. Phys.
Chenz. Solids
Pergamon
Printed in Great Britain.
Press 1965. Vol. 26, pp. 1837-1851.
CALCULATION
OF THE HUANG-RHYS
FACTOR
FOR F-CENTERS ANTHONY Physics Department,
M. LEMOS’
and JORDAN
J. MARKHAM
Illinois Institute of Technology, (Receiwed
Chicago,
Illinois
28 May 196.5)
Abstract-The Huang-Rhys factors for the absorption and emission of light by F-centers are derived using a modified tight binding approximation. Preparatory to this calculation, explicit representations for the discreet local phonon eigenfrequency spectra are determined as functions of electron state. Reduced forms for the diagonalizing unitary matrices are constructed and the corresponding normal coordinates determined. Parameter dependent forms for the Feynman integrals and the Huang-Rhys factors are then derived using a coulombic electron phonon interaction and the F-center wave functions developed by SMITH.The results of the calculation indicate that; (1) the Huang-Rhys factors for 1s a2p transitions depend on two distinct radial modes of vibration, each of which contributes significantly; (2) the forbidden s to s transitions depend only on the completely symmetric breathing mode and lead to small Huang-Rhys factors and consequently narrow band widths; (3) numerical calculations for the Huang-Rhys factors for several alkali halides indicate approximate agreement with the experimental values.
1. INTRODUCTION
THE PROBLEM of electron phonon interactions can be separated into three categories. The first is the case of the extremely weak interaction in which both the equilibrium configuration and the phonon eigenfrequencies are independent of electron state. This is the situation in metals and many semiconductors. In the second case, sometimes referred to as the linear approximation, the electron phonon interaction is considered to be weak enough to be approximated by a term which is linear in the displacement coordinates of the lattice ions. Here, while the equilibrium configuration is slightly sensitive to the electronic states, the phonon eigenfrequencies are completely independent of them. This is a reasonable approach in the case of extremely diffuse color centers. In the third case the electron phonon interaction is strong and local effects are important. We believe that most broad optical absorption bands, such as the F-band are due to this type of interaction. The problems related to strong electron phonon interactions can be approached by two means. * Present address: New York.
Adelphi University,
Garden City 1837
The first is semi-classical and phenomenological in which one uses configuration coordinate diagrams and introduces many factors from experimental results. Although this method is qualitatively useful, it does not have a firm theoretical foundation. In the second approach one uses rigorous theory. This method has not been applied to any realistic or even semi-realistic situations. If the model is made sufficiently simple and yet physically reasonable, one should be able to carry through the details of a rigorous calculation with a minimum of approximations. One objective of this paper is to show that this rigorous approach can be applied to a simplified model of the Fcenter to obtain Huang-Rhys factors(l*s) using simple electron wave functions.@) From our results it is shown that ; (1) more than one mode of vibration must be considered when transitions occur between s and p states. This means that there are at least two modes which interact strongly with the trapped electron and therefore the quantitative use of a single configuration coordinate is questionable. (2) Forbidden transitions between almost pure s states lead to narrow bands and perhaps in extreme cases to sharp lines as observed in ruby (see MCCLURE@))
1838
ANTHONY
M. LEMOS
and JORDAN
and the zero phonon lines observed by FITCHEN et aE.(s) (3) The phonon broadening results are of the right order of magnitude and predict the observed distinction between the narrower F-band in KC1 and the broader one in LiF. Several calculations have been made using the intermediate interaction where it is assumed that the trapped electron at the imperfection (F-center) is coupled to the longitudinal optical phonons of the perfect crystal. These calculations have only been partially successful. Theoretical and experimental evidence indicates that this assumption is incorrect for well localized centers. Our work indicates that despite several approximations, one can cast the calculations into a form which arises from pure quantum mechanical arguments. It suggests that with further work one will be able to replace the hypothesis of the configuration coordinate scheme with more rigorous concepts developed from the correct application of quantum mechanics. We first describe our modified tight binding model and determine the phonon eigenfrequency spectra as a function of electron state. A reduced set of normal coordinates, needed for the second portion of this paper, is determined. In the second part, we consider in detail the Huang-Rhys factors for the absorption and emission of light by F-centers. Explicit representations for the appropriate Feynman integrals are given for a variety of electronic transitions. Numerical calculations based on the SMITH@) parameters and empirical values of the phonon eigenfrequencies, are made. Finally, the physical significance and possible ramifications of our results are discussed.
J. MARKHAM
indicate that both the local and crystal modes contribute to the Huang-Rhys factor. Extending these considerations MCCOMBIE and co-workersol) CASSELMANand co-worker(ls) and MONTROLL(~~) have suggested that to a first approximation a localized imperfection can be treated as a molecule. LOFTYand GEBHARDT~~)have used such a model in interpreting their data. Therefore, in order to carry through a detuiled, rigorous calculation based on a semi-realistic model, we have associated a tightly bound seven particle system with the F-center. It might be surprising that a seven particle cluster was selected rather than only six which is used in the conventional tight binding approximation.(s) The seventh or core particle was introduced in an attempt to account for the effect of the exterior lattice in a simple fashion. For example, each of the six particles immediately surrounding the imperfection has a nearest neighbor in the outer shells which attracts it electrostatically but repels it very strongly due to the overlap, BORN-MAYER forces.(is) Therefore, one cannot ignore the rest of the lattice. Translational motion may be inhibited by allowing the mass of this particle to go to infinity or its influence may be eliminated entirely by letting the core interactions vanish. The seventh particle is not to be confused with the trapped electron whose wave function spreads outside the vacancy. This model seems to represent the simplest semi-realistic one which can be handled completely using the general theory of lattice vibrations. Figure 1 shows the equilibrium configurations as a function of electron state.
2. THE MODEL 3. PHONON EIGENFREQUENCIES We assume the DE BOER(~)model of the center In the harmonic approximation the Lagrangian and a modified tight binding approximation. for a system of interacting particles may be Since the charge distribution of the trapped written as electron is well localized in its ground and excited L = ~C~~~~~~-~Ccl(Vavjv)o:RaR*, states(T) local effects should be important. For (3.1) example, the calculation of WILLIAMS and HEBB@) where Rx is the displacement of the kth particle, which are based on local interactions shows first Va 0, is the 9th gradient dyad and the subscript order agreement with experiment. The analysis 0 indicates evaluation at the equilibrium conof BJo&s) indicates that if local modes arise as a figuration. The symbol “:” indicates the dyadic result of the imperfection, other modes are effecscalar product. The equation of motion for the tively excluded from the region. ROSENSTOCKand @h particle is KLICK(~@ recently considered the problem of local modes for the one dimensional case. They M,&+~,(V~Vjy)o~R~= 0 (34
CALCULATION
OF
THE
HUANG-RHYS
where k = 12, ,***, n. This set of equations solutions of the form
has
Rk = & eZWt where w is a phonon (3.2) and (3.3) yield
(3.3)
eigenfrequency.
C,[(vk:V3V)o-M,w26k~ll]‘Aj
Equations
= 0,
(3.4)
where 6 is the Kronecker delta and II is the identity dyad. In order to carry out an explicit determination of the phonon eigenfrequencies, ~2, we have assumed a Hooke’s potential function of the form
Y = Q‘;,
;[rtj-
l$j]?
Here rgj and l+f represent following vectors: rU = rc-rj;
FACTOR
FOR
F-CENTERS
1839
equilibrium position by about 16 per cent while the remaining four undergo a small inward displacement. Therefore we have associated a molecule of D4h symmetry with the excited state and one of 01, symmetry with the ground state. These are illustrated in Fig. 1. Changes in symmetry associated with such distortions seem to have been first pointed out by REDLICH and STRICKS.+~~)In carrying out the actual calculation we have represented the distortion by a perturbation parameter A. Our eigenfrequency results are correct to first order in A.
(3.5)
the magnitudes
1i* = Ii--lj,
of the (3.5a)
1
where rt and 16 are the instantaneous and equilibrium positions of the i+h particle. The prime in equation (3.5) means that i # j. The displacement vector, Rt, is defined in terms of these vectors by: Rz = ri-lg. The Kj+h gradient
(3.6)
dyad of the potential
(vkvjv)CI
=
8k&Lt,-
is
Lkj
(3.7)
where
Substitution
of equation
~,[Lk3-6k,(~tLZf-M~w211)]‘Aj
(3.7) into (3.4) yields =
0.
(3.9)
From this equation we see that the eigenfrequencies are strictly functions of the equilibrium configuration. This means that a calculation of the F-center phonon eigenfrequency spectra can be made without having any detailed knowledge of the electron wavefunctions. It is sufficient to know the configuration as a function of electron state. We now employ our model combined with the distortion calculations of WOOD and KORRINGA.(~@ Their results indicate that in the electronic ground state, the ions nearest the imperfection undergo a very small isotropic displacement. In the excited state the two ions located on the symmetry axis are displaced radially outward from their normal
FIG. 1. (a) Ground state equilibrium configuration (Oh symmetry). The six outer particles are equidistant from the core particle. (b) Excited state equilibrium configuration (Dm symmetry). Particles 5 and 6 are displaced radially outward along the z-axis while particles 14 are displaced radially inward along the x and y-axes.
1840
ANTHONY
M. LEMOS
and JORDAN
J. MARKHAM
Table 1. Phonon eigenfrequencies for the ground state (Oh symmetry) j
Group snecres 4k,+k;
Al,
M k,+k;
47
M k,
FZU
n/r 2k,
Fzs
x-
13 E
;I
l/2
&m(3mk,+mk;+2Mk;)+i[
($
- z)2+
A(2mk,+mk;+4Mk;)]
F lU
&(3mkg+mki+2Mkb)-i[
(2
- s)‘+ m
--$-(2mk0+mkL+4Mk;)]1’2
Flu
Since we have considered only central forces, the secular determinant is diagonalizable by a variety of techniques.(l*) We have employed the factoring method using external Cartesian coordinates. It has been assumed that the six outer particles are of identical mass, M, and are coupled to one another by coupling constants k,,%, depending on the electron state. The mass of the seventh or core particle is m and the constants coupling it to the six outer particles are k&. The results of the phonon eigenfrequency calculation are shown in Tables 1 and 2. A comparison of these tables reveals that the excited electron state has acted like a perturbation on the ground state equilibrium configuration. By reducing the symmetry of the ground state the eigenfrequency degeneracy has been partially removed. The resulting splitting of the phonon levels is illustrated in Fig. 2. This figure may be interpreted as an energy level diagram in which all oscillators are in the same quantum state, the various levels arising from the distribution of eigenfrequencies given in Table 1. The phasing results for those modes which are necessary for a determination of the Huang-Rhys factors are shown in Tables 3 and 4. The first
column represents the eigenfrequencies shown in Tables 1 and 2. The second column indicates the corresponding phase relationships. Here the A’s are defined by equation (3.3) and A,(k/j) refers to the displacement amplitude of the cc-component of the kth ion in the jth mode. Note that in Table 3 the phase relationships for the degenerate modes 2 and 3 are not unique. We have chosen this particular representation in order to conform to the results of Table 4. Figures 3 and 4 illustrate the phase relationships of Tables 3 and 4. 4. THE HUANGRHYS
PROBLEM
The Huang-Rhys factor was introduced by these authors(ls) as well as by PEKAFL(~@ Recently KLICK and SCHULMAN(~~) and MARKHAM(~) have generalized the factor and it is on this generalization that our calculation is based. The symbols S, and S, will be used for the factors associated with absorption and emission. The S-factor may be interpreted as either a measure of the phonon broadening of the F-bands or as the average number of phonons stored in the lattice immediately after an optical transition. To acquire some further insight into S, consider the one dimensional configuration coordinate
CALCULATION
OF THE
HUANG-RHYS
FACTOR
FOR
P-CENTERS
1841
Table 2. Fkonon ezjynfrequenciesfor tke excited state @Id&symmetry) Group species
j 4k,+ k;
1
M
2
k&k; M
.ke iii
3
k&k’ --.?+A!? M ke -+*Re M M
4 5
ke -M
6
*!! M
7
2ke X
8
2ke
9
M
10
M
1 [ (3ke z-
&(3mke+mk~+2Mk~+3mkeA)+Z
+
2h 4Mk;) f -A( MZm
>)2+--$-(Zmk8+mk;
1
l/2
9mke+ mk; - 6Mki)
&mke+mk~+2Mk~-3~~A)+~[
(2
- ~)2+~m~~mke+mk~ 112
+
13
4Mk~) - ~~A~9rn~+rnk~-6Mk~)
&[3mke+mk;+21Uk;+3m&A)-
i[ ($-
&
4Mk;) -f--A(9mke
(3mk,+mk;+2Mk;
Mzm
- T
z)‘+
sm(2mke+mk; l/2
2ke
+
I
$ rnk:- 6Mk;)
keA)- i[ (2
I
-s)‘+ m
&2mke+mki 112
+
4Mk;) -~~9~+~~-6Mk~)
1
1842
ANTHONY
M. LEMOS
and JORDAN
diagram illustrated in Fig. 5. Within the framework of the BORN-OPPENHEIMER(~~~~~) technique _&E,(p)and &(q) represent the potential energies of the lattice in the ground, g, and upper, U, electronic state. The horizontal lines designated vibrational levels, while the vertical lines represent electronic transitions. Therefore, Fig. 5 is
J. MARKHAM
phonon emissions take place and again the center relaxes to its ground state equilibrium coniigutation. The phonons stored in the lattice at b and d are S, and S,. Table 3. Radial phasing resultsfor the ground state --. Phase relationships
Eigenfrequency (0;)
Al(l) = A,(l/l)i
= -A&l)
As(l) = A,(l/l)j
= -A&l)
A#) ----_-__. <
= A,(l/l)k
= -A,#)
AT(l) = 0
-f
2
h(j) - -&(Uj)i=-h(j) I-
kg+k; M
h(j) = 4(2/j)j= -A.&i) As(j) = -tAxW) ~A~~2~~lk= --As(j) h(j) = 0,
---_--I
<
j = 2, 3.
MARKHAM@) has developed the following general forms for the Huang-Rhys factors:
S, = pi+%
(4.1)
and (4.2) FIG.2. Phonon eigenfrequency splitting for the simplest case, k’ = 0, kc = KS = k. Numbers to the right of each horizontal line represent the degeneracy. The six fold root is due to accidental degeneracy. The ground state degeneracies have been partially removed as a result of the reduced symmetry of the excited state.
interpreted as a center making a Franck-Condon transition frnm point a to b, corresponding to absorption, where it is left in an excited vibrational state which is inconsistent with the temperature of the crystal. A rapid readjustment process occurs whereby the center relaxes by means of phonon emissions to its excited state equilibrium configuration at point c. After roughly 10-a set a downward Franck-Condon transition occurs from point c to point d, where once again the center is left in an excited vibrational state, Subsequent
Here (4.3) where w&) and @j(g) are the jth phonon frequency of the excited and ground respectively. Sf is the Huang-Rhys factor sponding to the 1Y.hmode. The generalized may be interpreted as the weighted sum of phonons. The Sf’s are defined as follows:
and
eigenstates correfactor stored
CALCULATION
OF
THE
HUANG-RHYS
Table 4. Radial phasing results for the excited state Eigenfrequency
Phase relationships
(w,“)
Al(l)
= A,(l/l)i
M
A5(1)
=
A$)
k,+k;
M
a”
M
A2(2)
R
(4.7)
-A##)
Here the A,(k/j)‘s represent the elements of a unitary matrix, the significance of which has been discussed in the previous section, As a result of this transformation, equation (4.7) becomes
= 0
AI(~) = A,(1/2)i 2 w2=------
where 4(R) is the electron wave function and V is the potential energy of the system. This equation may be recast into a more convenient form by the introduction of the following transformation :
P
(1+&%(1/l) =
1843
F-CENTERS
(G)
= -Aa
k
FOR
%(k) = ~M,1/24(W)
= -Aa
AZ(l) = A,(l/l)j
2 4k?+k; or = -
FACTOR
=
= -AS(~)
-&(1/2) j=
-A42)
A5(2) = A6(2) = A7(2) = 0 AI(~) = A,(1/3)i
ke AZ(~) =&(1/3)j k+k; W3---fAke M M
= -As(S) = -A4
(3)
2_
As(S) = -2(1-A) A&/3)&
= -As(3)
A,(3) = 0
Here (4.4a) and (4.5a) where
AE = E,--E,.
(4.5b)
The l ~‘s defined by equations (4.4a) and (4.5a) may be determined by means of the FJXYNMANHELIXAN theorem.@4@) Thus
FIG. 3. (a) The & mode of the Oh group which corresponds to thej = 1 mode of Table 3. (b) The & mode of the Oh group, which corresponds to the j = 2,3 modes of Table 3.
1844
ANTHONY
M.
LEMOS
and JORDAN
J.
MARKHAM
This expression is completely general within the framework of the“adiabatic-h~monic”approximation. The potential energy of the system may be written as V = CrV~(r,R3)+~C2jVz(R~,R~), (4.9) where r and Rj represent the positions of the trapped electron and the jth ion respectively. A harmonic expansion of equation (4.9) would yield a form equivalent to equation (3.5). Since it is
a ~ONFIG~~TION
~05ROl~TE
One dimensional configuration coordinate diagram. The ordinate represents the total energy of the system. The vertical lines ab and cd correspond to the Fran&-Condon transitions while the lines 6c and da represent phonon emissions. q represents the configuration coordinate, I&(q) and E,(g) the energies of the ground and excited states, and Sa and Se the HuangRhys factors corresponding to optical absorption and emission. FIG.
5.
assumed that the electron wavefunctions form a complete orthonormal set, the second term of equation (4.10) does not contribute to (4.9). Therefore we define an effective potential as
Veff =
&V(r,Rj).
{4.10)
To carry out explicit calculations of the HuangRhys factars the following simplification has been made: e2 FIG. 4. (a) The first Al9 mode of the D4h group which corresponds to the j = 1 mode of TabIe 4. (6) The Bse mode of the D4h group which corresponds to the j = 2 mode of Table 4. (c) The second Alg mode of the I&, group which corresponds to the j = 3 mode of Table 4.
Veff = Cj-
I r--Rj I A more realistic treatment would be the actual charge distributions of the particles. However, we are motivated
(4.11) to include interacting _ _ here by an
CALCULATION
OF THE
HUANG-RHYS
attempt to obtain explicit expressions for the Huang-Rhys factor in a first approximation. We assume that equation (4.11) represents the interaction to such an approximation. However, before the Feynman integrals of equation (4.8) can be attacked, one more piece of information is necessary, namely the wave functions. The most reliable simple wave functions for F-centers seem to be those derived by SMITH,@) who following the initial suggestion of MOTT and GURNEY, assumed that the trapped electron has only two bound states, namely a 1s and a 2p state. The SMITH wave functions are: f.Dls = B(l +hr) exp(-Xr)
cos 0.
(4.13)
Here x3
B=
112
( ??I 1
(4.12a)
and p5
112
D=-,
F-CENTERS
1845
the lattice constant
A = [(X-a)s+JQ-zs]3~s,
(4.14a)
B = [Xs+ys+(x-u)s]s/s.
(4.14b)
and
Since the “molecule” has reflection symmetry about the planes at right angles to the x,y, and x axes, it follows that in every case &(1/j)
= &&(3/j),
(4.15)
&(2/j)
= +&(4/j),
(4.16)
&(5/j)
= +&(6/j).
(4.17)
and
Here the negative signs define the radial modes. From equation (4.14) we see that the E~‘Svanish for all non-radial modes. It should be noted that this result is independent of the coulombic electron-phonon interaction we have used. Any central force type interaction yields the same result. We restrict our attention to the reduced matrix scheme of Fig. 6, which defines the A,(K/j)‘s for the radial modes. There are only two non-vanishing l j’s, namely
(4.13a)
0 7r where X and /3 are adjustable
FOR
Here, a, represents
(4.12)
and @sp = Dr exp(-/3r)
FACTOR
parameters.
Is + 2p tra-nsitions Due to the symmetry of our ground state configuration a great number of the integrals of equation (4.8) vanish. In fact, we find that
(3M)l/a
(&+)
= &[e-X
1 1,
+7xss14x+14
-B&/A
-A,(3/j)+A,(2/j)-A,(4/j)
+Az’5ii)-az’6ii’1(g~~~g)
(uE~u)
(4’14)’
=
(:+2X3
-14
-
(4.20)
-&~-2f~(y4+4y3
+8y2+9y+9/2)+(y2_9/2)],
(4.21)
1846
ANTHONY
M.
LEMOS
and JORDAN
J. MARKHAM
and =
-&+U(2y6+4y5
+8y4+14y3+19ys+18y+9)-(y*+9)],
(4.22)
where X = 2Xa
(4.21a)
y =
(4.22a)
and pa.
2p -+ Is transitions The equation of cj(e) which corresponds emission is
,
,
-~,(3/j)
[~3+y2+((z-b)3]3’~
b = a’(l+A),
44 =
“[-+-+)((.f&) 2/M
d/6
x((&j;~U)-(U~~/r+)]~(4.25)
2v
3x
(4.24b)
where A is the perturbation parameter previously introduced. U’ is the distance from the center to one of the nearest neighbor ions in the xy plane and b is the distance from the center to either of the two ions on the z-axis. (Refer to Fig. 1.) Again, equations (4.15)-(4.17) apply and only the radial modes contribute to the l j(e)‘s. By substituting the appropriate values of the L&(/#‘S from Table 4 into equation (4.24) we obtain the following results.
-(UF[U))+$-(l+Y)
+(2/j)
lx
(4.24a)
and
where the (A’)‘s represent the elements of the unitary matrix for the excited state. Since the configuration symmetry has been reduced by the perturbation of the 2p wave function, the resultant expression is somewhat more complicated, namely
e2.
B’ =
to
El(e) = Ck,aM;1/2Ai(&!j)
(e) = -+$A,(W
Here
bz
4Y
0
0
FIG. 6. Reduced A-matrix for the ground state. This portionofthe matrix (refer Sections 3 and 4) shows those elements which correspond to the radial modes given in Table 3.
CALCULATION
2
3
OF
THE
HUANG-RHYS
lx
2Y
21
_-1 2
_;
&(l+y)
-&(l+?
&(l+y)
FIG. 7. Reduced
FACTOR
3x
)
FOR
4Y
52
62
12
0
0
-&(l+Y)
-~(l_~)
X-matrix for the excited state. This portion of the matrix (refer Sections those elements which correspond to the radial modes given in Table 4.
and
1847
F-CENTERS
+-$)
3 and 4) shows
and
es(e) = Z[~(l+~)((.$j+) 2/M 2/3
<“lYlu) are given in equations tively.
-(U~1U))-;(l-+)
(4.20) and (4.21) respec-
2s * 1s transitions x((&$)-(+$j+)].
(4.26)
Here = &[&(2os+4~5+
(+Jr+
+14w3+19er2+18a+9)-(w3+9)]
804 (4.27)
and ($j+)
= &[e-U(:+&3+7u2
+14u+14 where
> 1) -14
(4.28)
v = /!3b,
(4.27a)
u = 2Xb.
(4.25b)
and
The integrals
Although these transitions are forbidden, there are two basic reasons for considering them here. The first is that the lifetime of the excited state is of the order of 10-s set, whereas the expected lifetime of the order of 10-s sec. Although several author@-33) have considered this problem it has not been completely resolved. KLICK(~~) has suggested that this long lifetime might be the result of a forbidden 2s -+ 1s transition. In fact, MCCOMBIE and co-workers(lQ have considered only transitions between spherically symetric s states. The second reason for considering these transitions is that the narrow ruby R lines may correspond to forbidden transitions.@) Although there seems to be no recorded wave function for the 2s state of the F-center, we can make a reasonable guess as to its form. The Fcenter 2p wave function corresponds approximately to the hydrogenic 2p function. We therefore assume the following: @‘2.$= D(l -/+?-8’,
(BE+)
D = (+/3(8)3/3.
(4.28)
(4.28a)
1848
ANTHONY
M.
LEMOS
and JORDAN
Before proceeding further we should note that the perturbed equilibrium configuration is a function of the 2p charge distribution. Since the 2s state is spherically symmetric the distortion will no longer be present. Therefore we may determine the EJ’S for both the Is -+ 2s and 2s + 1s transitions from equation (14) by simply introducing a parameter c, such that
J. MARKHAM
is a function of the parameters associated with the electronic states. The results of any numerical calculation will depend critically on the parameters assumed. At the present time most of these are in a state of flux. For example, it is well known that the lattice surrounding the defect is distorted. Although GOURARY(~~) and WOOD and KORRINGA~~) have considered this problem, it remains unsolved. Also, the experimental spectrum is unknown. (4.29) of the local phonon eigenfrequencies Finally, the use of the simple, one parameter, a’(1 + A); (2s +- Is) Smith wave functions is questionable. At the There is only one-vanishing l3 term, ~1, which present time, work on improved, more general corresponds to the breathing mode. Here wave functions is being carried on by WOOD and JOY.(32) El = e2(6M)-112 [ We are now in a position to obtain numerical values for the Huang-Rhys factors corresponding to 1s ~2p and 1s ~22s transitions. Here we have used the ground and excited state parameters given by sMITH.c3) Since Smith concerned himself where with wave functions for absorption, the use of his parameters for the emission calculation is question(4.30a) c = [(~++p+~3~3/2; able. With regard to the phonon eigenfrequencies, we have associated the effective frequencies as determined by KONITZER and MAFZKHAM(~~~~) (K. & M), LUTY and GEBHARDT~~)(L. & G.), and (4.31) RUSSELLand KLICK(~~) (R. & K.) with the breathing mode. With conventional configuration co(.,,i~~%+ = &[e-x (:+2X3 ordinate techniques this seems to be a reasonable assumption. It should be noted that the same \ 7 (4.32) effective frequency has been used for both the +7X2+14X+14 absorption and emission calculation of NaCl. Thus the emission result for NaCl is probably v = /3c; (4.31a) correct only to an order of magnitude. In a recent and letter, KUHNERT and GEBHARDT@) have published new data on the effective frequencies. HOWx = 2hc. (4.32a) ever, the authors have decided to await the more detailed publication before using this new data. Discussion of calculations In carrying out the calculation we have used the Before presenting the numerical results, a few free ion masses and assumed that the ground state features of the calculations should be noted. We equilibrium distances between the core and outer find that the Huang-Rhys factors have the particles is given by the appropriate lattice constants. The results of the numerical calculation are following dependence shown in Table 5. The inequalities in this table enter as a result of the two frequencies involved in the calculation. We recall from Table 1 that
(m,,/3 CD29
- ( QIBI~/+]
The e4 term arises from the charges of the halogen ion and the electron, W, is the jth phonon eigenfrequency, Mis the free ion mass and f(pwp2p)
CALCULATION
OF THE
Table 5. Numerical Crystal Absorption
KC1 NaCl LiF
w X lOI set-l
HUANG-RHYS
results
for
Sl
FACTOR
1849
FOR F-CENTERS
the Huang-Rhys factor s
S3
S
exP
(1s + 2p transitions)
2~~2.96 (K. and M.) 2~ x4.4 (K. and M.) 2~~4.1 (R. and K.)
1
l<&
2
30
I-S< s<7*5
30
-5
l< s3<7
38
lO< Sa< 76
48< s< 114
116
0
2< s3< 18
2
30
0
3< Sa< 23
3
30
2
0
2
-
2
0
2
-
Emission (2p + 1s transitions)
KC1 NaCl
2n x4.25 (L. and G.) 2~ x4.4 (K. and M.)
(1 s * 2s transitions) KC1 2n x2.96 (K. and M.) NaCl 2n x4.4 (K. and M.)
Thus by letting K’ sweep from zero to infinity we arrive at the result 12
0’s
2.
In Table 5, Sr is associated with the breathing mode; Ss with the asymmetrical axial mode; S = Sl+ Ss; and Sexp is the approximate experimental value. The 1s e2s transitions have been included for the sake of completeness and indicate a smaller value. The most reliable theoretical results are the S factors corresponding to the 1s -+ 2p absorption. In spite of the numerous assumptions and simplifications we obtain qualitative agreement with experiment. This leads us to believe that our approach is valid as a first approximation. This is strengthened by our prediction regarding the rather narrow bands of KC1 and NaCl and the much broader band in LiF. One consequence of our results is that the Huang-Rhys factors for allowed transitions depend on at least two normal modes, each of which 6
contributes significantly. This means that there are at least two lattice modes which interact strongly with the trapped electron. Therefore, the conventional formula for the half width, which assumes a single frequency must be modified. This formula is
Hz = (8 In 2) (Rw)s S coth
(4.34)
where H represents the half-width of the F band at a temperature T, W, the effective phonon frequency, and S the corresponding Huang-Rhys factor. The modification indicated by our results is Hs = (8 In 2) 21 (tiu~,)~Sj coth where the summation extends over the two modes contributing to the Huang-Rhys factor. Such a generalization has been discussed by MARKHAM(~) and by CASSELMAN and MARKHAM.
1850 5. CONTUSIONS
ANTHONY
M.
LEMOS
and JORDAN
AND SUMMAR X DISCXJSSION ,. OF RESULTS
This work, as far as we have been able to determine, represents the first theoretical calculation of the Huang-Rhys factors (phonon broadening) based on the tight binding approximation. The bulk of theoretical and experimental work indicates that such an approximation is valid for well localized centers such as the F-center. Preparatory to this calculation the discrete local phonon eigenfrequency spectra corresponding to the ground and excited electron states has been determined. These results are applicable to any localized model of the F-center in the alkali halides which possesses oh and/or &h symmetry. Within the framework of the “adiabaticharmonic” approximation the secular determinant implicit in equation (3.9) is completely general and in no way restricted to the tight binding approximation or to a particular equilibrium configuration. This means that our calculations may be extended to include next nearest neighbor interactions and outer shells with a minimum of difficulty. Explicit forms for the Feynman integrals for various electronic transitions have been determined. (e.g. see equation (4.20) ). For allowed transitions it has been shown that the HuangRhys factors are dependent on two distinct radial modes, each of which contributes significantly to the phonon broadening of the F-bands. This means that there are two lattice modes which interact strongly with the trapped electron. There are two immediate consequences of this result: (1) Simple one dimensional configuration coordinate diagrams are not quantitatieretymeaningful. This scheme is based on the assumption of a single interacting mode for the tight binding appro~mation of the F-center. We have shown that this is theoretically incorrect. (2) The conventional formulation for the half-width must be modified. Numerical calculations of the Huang-Rhys factors for several alkali halides indicate approximate agreement with the experimental values. Our results predict the observed distinction between the rather narrow F-bands in KC1 and NaCl and the much broader band in LiF. Also it has been shown that the forbidden s to s transitions give rise to narrow bands and perhaps in the
J. MARKHAM
extreme case to sharp lines. In general, our work indicates that one may make completely quantum mechanical calculations using the tight binding approximation and obtain rough agreement with experiment, The various approximations and simplifications used in this development have been discussed in detail in the previous sections. One obvious continuation of this work would be the relaxation of some of these restrictions in the hope of obtaining a more realistic physical model. For example, our modified tight binding approximation could be extended to include second shell interactions. This would allow one to explore more diffuse centers and also to examine more closely the validity of our present model. On a more sophisticated level, the use of a more general electron-phonon interaction and the inclusion of polarization effects should be considered. At the present time, general wave functions for the excited states of the F-center are being developed by WOODand Jot. Using these wave functions it should be possible to investigate the problems of the F-center phonon broadening, especially for the case of emission, on a more reasonable level. Furthermore, if the KI and L bands do correspond to higher excited states of the F-center, it should be possible to determine their Huang-Rhys factors with a slight modification of our present formalism. In this connection one should be able to investigate phonon phenomena associated with centers which have a geometrical structure similar to the F-center, such as the U and KCl:Tl centers. One problem which can be attacked rigorously within the framework of our model is that of the effective mass of the center. This is an important concept which seems to have been introduced by W~LLI~~S(~~)in 1951. Since then, experimentalists seem to have adopted this concept without questioning its theoretical validity. REFERENCES
J. J., Reo. Mod. Phys. 31,956 (1959). 1. MARKHAM 2. DEXTER D. L., Solid State Physics (ads. SEITZ F. and TVRNBULL D.) Vol. 6. Academic Press, New York (1958). 3. S~vrr~a UT. A., “Energy Level Calculations for Fcenters and Positive Ions in Alkali Halides”, XnolIs Atomic Power Laboratory (Rept. KAPL1720, April 1,1957).
CALCULATION
OF
THE
HUANG-RHYS
4. MCCLURE D. S., Solid State Physics (eds. SEITZ F. and TURNBULL D.) Vol. 9. Academic Press, New York (1959). 5. FITCHEN D. B., SILSBEER. H., FULTON T. A. and WOLF E. L., Phys. Rev. Lett. 11, 275 (1963). 6. DE BOER J. H., Recuiel des Trav. Chem. d. PaysBas 56, 301 (1937). 7. GOURARY B. S., and ADRIAN F. J., Solid State Physics (eds. SEITZ F. and TURNBULLD.) Vol. 10. Academic Press, New York (1960). 8. WILLIAMS F. E. and HEBB M. H., Phys. Rev. 84, 1181 (1951). 9. BJORK R. L., Phys. Rev. 105,456 (1957). 10. ROSENSTOCKN. H. and KLICK C. C.. Phvs. Rev. 119, 1198 (1960). 11. MCCOMBIE C. W., MATTHEW J. A. and MURRAY A. M., J. Appl. Phys. Suppl. 33, 339 (1962). 12. CASSELMANT. N. and MARKHAM J. J., J. Phys. Chem. Solids 24, 669 (1963). 13. MONTROLL E. W., Bull. Am. Phys. Sot. 9,12 (1964). 14. LUTY F. and GEBHARDT W., 2. Physik 169, 475 (1962). 1.5. BORN M. and MAYER J. E., Z. Physik75,l (1932). 16. WOOD R. F. and KORRINGA J., Phys. Rev. 123, 1138 (1961). 17. REDLICK T. K. and STRICKS W., Wien Beriche IIb, 146,447 (1937). 18. WILSON E. B., DECIUS J. C. and CROSS P. C., Molecular Vibrations. McGraw-Hill, New York (1955). 19. HUANG K. and RHYS A., Proc. Roy. Sot. (Lond.) AZO4,406 (1950). 20. PEKAR S. I., J. Exptl. Theoret. Phys. (USSR) 20, 510 (1950). I
_
FACTOR
FOR
F-CENTERS
1851
21. KL.ICK C. C. and SCHULMAN J. H., Solid State Physics (eds. SEITZ F. and TURNSULL D.) Vol. 5. Academic Press, New York (1957). ’ 22. BORN M. and OPPENHEIMERJ. R., Ann. Physik 84, 457 (1927). Theory of 23. BORN M. and H~ANG K., Dynamical Crystal Lattices, Chap. 4. Clarendon Press, Oxford (1954). 24. FEYNMAN R. P., Phys. Rev. 56, 340 (1939). in die Q&ten Chemie, 25. HELLMAN H., Einfihrung P. 285. Teubner. Leimie (1937). N. F. and &JW~Y k: W.; Electronic Pro26. M&r cesses in Ionic Crystals. Clarendon Press, Oxford (1948). 27. FOWLER W. B. and DEXTER D. L., Phys. Rev. 128, 2154 (1962). 28. FOWLER W. B. and DEXTER D. L., Phys. Stat. Sol. 2, 821 (1962). 29. FOULER W. B:, Phys. Rev. 135, A1725 (1964). 30. KLICK C. C.. PATTERSOND. A. and KNOX R. S. (unpublished) 31. GOURARY B. S. and ADRIAN F. J., Phys. Rev. 105, 1180 (1957). 32. WOOD R. F. and JOY H. W., Phys. Rev. 136, A451 (1964). 33. KONITZERJ. D. and MARKHAM J. J., J. Chem. Phys. 32,843 (1960). 34. KONITZERJ. D. and MARKHAM J. J., J. Chem. Phys. 34, 1936 (1961). 3.5. RUSSELL G. A. and KLICK C. C.. Phys. Rev. 101. 1473 (1956). 36. GERHARDT W. and KUNHERT H.. , Phvs. < Lett. (Netherlands) 11, 1 (1964). 37. WILLIAMS F. E., Phys. Rev. 82, 281 (1951).
Note added in proof.-We are grateful to R. GILBEHTfor pointing out that there appears to be an intrinsic error in the work of SMITH. Although the forms of his wave functions may be correct, the parameters are not. Thus the values appearing in Table 5 should be considered as no more than qualitative at this stage. Work on a revised table has begun and should be published in the near future.
Chenz. Solids
Pergamon
Printed in Great Britain.
Press 1965. Vol. 26, pp. 1837-1851.
CALCULATION
OF THE HUANG-RHYS
FACTOR
FOR F-CENTERS ANTHONY Physics Department,
M. LEMOS’
and JORDAN
J. MARKHAM
Illinois Institute of Technology, (Receiwed
Chicago,
Illinois
28 May 196.5)
Abstract-The Huang-Rhys factors for the absorption and emission of light by F-centers are derived using a modified tight binding approximation. Preparatory to this calculation, explicit representations for the discreet local phonon eigenfrequency spectra are determined as functions of electron state. Reduced forms for the diagonalizing unitary matrices are constructed and the corresponding normal coordinates determined. Parameter dependent forms for the Feynman integrals and the Huang-Rhys factors are then derived using a coulombic electron phonon interaction and the F-center wave functions developed by SMITH.The results of the calculation indicate that; (1) the Huang-Rhys factors for 1s a2p transitions depend on two distinct radial modes of vibration, each of which contributes significantly; (2) the forbidden s to s transitions depend only on the completely symmetric breathing mode and lead to small Huang-Rhys factors and consequently narrow band widths; (3) numerical calculations for the Huang-Rhys factors for several alkali halides indicate approximate agreement with the experimental values.
1. INTRODUCTION
THE PROBLEM of electron phonon interactions can be separated into three categories. The first is the case of the extremely weak interaction in which both the equilibrium configuration and the phonon eigenfrequencies are independent of electron state. This is the situation in metals and many semiconductors. In the second case, sometimes referred to as the linear approximation, the electron phonon interaction is considered to be weak enough to be approximated by a term which is linear in the displacement coordinates of the lattice ions. Here, while the equilibrium configuration is slightly sensitive to the electronic states, the phonon eigenfrequencies are completely independent of them. This is a reasonable approach in the case of extremely diffuse color centers. In the third case the electron phonon interaction is strong and local effects are important. We believe that most broad optical absorption bands, such as the F-band are due to this type of interaction. The problems related to strong electron phonon interactions can be approached by two means. * Present address: New York.
Adelphi University,
Garden City 1837
The first is semi-classical and phenomenological in which one uses configuration coordinate diagrams and introduces many factors from experimental results. Although this method is qualitatively useful, it does not have a firm theoretical foundation. In the second approach one uses rigorous theory. This method has not been applied to any realistic or even semi-realistic situations. If the model is made sufficiently simple and yet physically reasonable, one should be able to carry through the details of a rigorous calculation with a minimum of approximations. One objective of this paper is to show that this rigorous approach can be applied to a simplified model of the Fcenter to obtain Huang-Rhys factors(l*s) using simple electron wave functions.@) From our results it is shown that ; (1) more than one mode of vibration must be considered when transitions occur between s and p states. This means that there are at least two modes which interact strongly with the trapped electron and therefore the quantitative use of a single configuration coordinate is questionable. (2) Forbidden transitions between almost pure s states lead to narrow bands and perhaps in extreme cases to sharp lines as observed in ruby (see MCCLURE@))
1838
ANTHONY
M. LEMOS
and JORDAN
and the zero phonon lines observed by FITCHEN et aE.(s) (3) The phonon broadening results are of the right order of magnitude and predict the observed distinction between the narrower F-band in KC1 and the broader one in LiF. Several calculations have been made using the intermediate interaction where it is assumed that the trapped electron at the imperfection (F-center) is coupled to the longitudinal optical phonons of the perfect crystal. These calculations have only been partially successful. Theoretical and experimental evidence indicates that this assumption is incorrect for well localized centers. Our work indicates that despite several approximations, one can cast the calculations into a form which arises from pure quantum mechanical arguments. It suggests that with further work one will be able to replace the hypothesis of the configuration coordinate scheme with more rigorous concepts developed from the correct application of quantum mechanics. We first describe our modified tight binding model and determine the phonon eigenfrequency spectra as a function of electron state. A reduced set of normal coordinates, needed for the second portion of this paper, is determined. In the second part, we consider in detail the Huang-Rhys factors for the absorption and emission of light by F-centers. Explicit representations for the appropriate Feynman integrals are given for a variety of electronic transitions. Numerical calculations based on the SMITH@) parameters and empirical values of the phonon eigenfrequencies, are made. Finally, the physical significance and possible ramifications of our results are discussed.
J. MARKHAM
indicate that both the local and crystal modes contribute to the Huang-Rhys factor. Extending these considerations MCCOMBIE and co-workersol) CASSELMANand co-worker(ls) and MONTROLL(~~) have suggested that to a first approximation a localized imperfection can be treated as a molecule. LOFTYand GEBHARDT~~)have used such a model in interpreting their data. Therefore, in order to carry through a detuiled, rigorous calculation based on a semi-realistic model, we have associated a tightly bound seven particle system with the F-center. It might be surprising that a seven particle cluster was selected rather than only six which is used in the conventional tight binding approximation.(s) The seventh or core particle was introduced in an attempt to account for the effect of the exterior lattice in a simple fashion. For example, each of the six particles immediately surrounding the imperfection has a nearest neighbor in the outer shells which attracts it electrostatically but repels it very strongly due to the overlap, BORN-MAYER forces.(is) Therefore, one cannot ignore the rest of the lattice. Translational motion may be inhibited by allowing the mass of this particle to go to infinity or its influence may be eliminated entirely by letting the core interactions vanish. The seventh particle is not to be confused with the trapped electron whose wave function spreads outside the vacancy. This model seems to represent the simplest semi-realistic one which can be handled completely using the general theory of lattice vibrations. Figure 1 shows the equilibrium configurations as a function of electron state.
2. THE MODEL 3. PHONON EIGENFREQUENCIES We assume the DE BOER(~)model of the center In the harmonic approximation the Lagrangian and a modified tight binding approximation. for a system of interacting particles may be Since the charge distribution of the trapped written as electron is well localized in its ground and excited L = ~C~~~~~~-~Ccl(Vavjv)o:RaR*, states(T) local effects should be important. For (3.1) example, the calculation of WILLIAMS and HEBB@) where Rx is the displacement of the kth particle, which are based on local interactions shows first Va 0, is the 9th gradient dyad and the subscript order agreement with experiment. The analysis 0 indicates evaluation at the equilibrium conof BJo&s) indicates that if local modes arise as a figuration. The symbol “:” indicates the dyadic result of the imperfection, other modes are effecscalar product. The equation of motion for the tively excluded from the region. ROSENSTOCKand @h particle is KLICK(~@ recently considered the problem of local modes for the one dimensional case. They M,&+~,(V~Vjy)o~R~= 0 (34
CALCULATION
OF
THE
HUANG-RHYS
where k = 12, ,***, n. This set of equations solutions of the form
has
Rk = & eZWt where w is a phonon (3.2) and (3.3) yield
(3.3)
eigenfrequency.
C,[(vk:V3V)o-M,w26k~ll]‘Aj
Equations
= 0,
(3.4)
where 6 is the Kronecker delta and II is the identity dyad. In order to carry out an explicit determination of the phonon eigenfrequencies, ~2, we have assumed a Hooke’s potential function of the form
Y = Q‘;,
;[rtj-
l$j]?
Here rgj and l+f represent following vectors: rU = rc-rj;
FACTOR
FOR
F-CENTERS
1839
equilibrium position by about 16 per cent while the remaining four undergo a small inward displacement. Therefore we have associated a molecule of D4h symmetry with the excited state and one of 01, symmetry with the ground state. These are illustrated in Fig. 1. Changes in symmetry associated with such distortions seem to have been first pointed out by REDLICH and STRICKS.+~~)In carrying out the actual calculation we have represented the distortion by a perturbation parameter A. Our eigenfrequency results are correct to first order in A.
(3.5)
the magnitudes
1i* = Ii--lj,
of the (3.5a)
1
where rt and 16 are the instantaneous and equilibrium positions of the i+h particle. The prime in equation (3.5) means that i # j. The displacement vector, Rt, is defined in terms of these vectors by: Rz = ri-lg. The Kj+h gradient
(3.6)
dyad of the potential
(vkvjv)CI
=
8k&Lt,-
is
Lkj
(3.7)
where
Substitution
of equation
~,[Lk3-6k,(~tLZf-M~w211)]‘Aj
(3.7) into (3.4) yields =
0.
(3.9)
From this equation we see that the eigenfrequencies are strictly functions of the equilibrium configuration. This means that a calculation of the F-center phonon eigenfrequency spectra can be made without having any detailed knowledge of the electron wavefunctions. It is sufficient to know the configuration as a function of electron state. We now employ our model combined with the distortion calculations of WOOD and KORRINGA.(~@ Their results indicate that in the electronic ground state, the ions nearest the imperfection undergo a very small isotropic displacement. In the excited state the two ions located on the symmetry axis are displaced radially outward from their normal
FIG. 1. (a) Ground state equilibrium configuration (Oh symmetry). The six outer particles are equidistant from the core particle. (b) Excited state equilibrium configuration (Dm symmetry). Particles 5 and 6 are displaced radially outward along the z-axis while particles 14 are displaced radially inward along the x and y-axes.
1840
ANTHONY
M. LEMOS
and JORDAN
J. MARKHAM
Table 1. Phonon eigenfrequencies for the ground state (Oh symmetry) j
Group snecres 4k,+k;
Al,
M k,+k;
47
M k,
FZU
n/r 2k,
Fzs
x-
13 E
;I
l/2
&m(3mk,+mk;+2Mk;)+i[
($
- z)2+
A(2mk,+mk;+4Mk;)]
F lU
&(3mkg+mki+2Mkb)-i[
(2
- s)‘+ m
--$-(2mk0+mkL+4Mk;)]1’2
Flu
Since we have considered only central forces, the secular determinant is diagonalizable by a variety of techniques.(l*) We have employed the factoring method using external Cartesian coordinates. It has been assumed that the six outer particles are of identical mass, M, and are coupled to one another by coupling constants k,,%, depending on the electron state. The mass of the seventh or core particle is m and the constants coupling it to the six outer particles are k&. The results of the phonon eigenfrequency calculation are shown in Tables 1 and 2. A comparison of these tables reveals that the excited electron state has acted like a perturbation on the ground state equilibrium configuration. By reducing the symmetry of the ground state the eigenfrequency degeneracy has been partially removed. The resulting splitting of the phonon levels is illustrated in Fig. 2. This figure may be interpreted as an energy level diagram in which all oscillators are in the same quantum state, the various levels arising from the distribution of eigenfrequencies given in Table 1. The phasing results for those modes which are necessary for a determination of the Huang-Rhys factors are shown in Tables 3 and 4. The first
column represents the eigenfrequencies shown in Tables 1 and 2. The second column indicates the corresponding phase relationships. Here the A’s are defined by equation (3.3) and A,(k/j) refers to the displacement amplitude of the cc-component of the kth ion in the jth mode. Note that in Table 3 the phase relationships for the degenerate modes 2 and 3 are not unique. We have chosen this particular representation in order to conform to the results of Table 4. Figures 3 and 4 illustrate the phase relationships of Tables 3 and 4. 4. THE HUANGRHYS
PROBLEM
The Huang-Rhys factor was introduced by these authors(ls) as well as by PEKAFL(~@ Recently KLICK and SCHULMAN(~~) and MARKHAM(~) have generalized the factor and it is on this generalization that our calculation is based. The symbols S, and S, will be used for the factors associated with absorption and emission. The S-factor may be interpreted as either a measure of the phonon broadening of the F-bands or as the average number of phonons stored in the lattice immediately after an optical transition. To acquire some further insight into S, consider the one dimensional configuration coordinate
CALCULATION
OF THE
HUANG-RHYS
FACTOR
FOR
P-CENTERS
1841
Table 2. Fkonon ezjynfrequenciesfor tke excited state @Id&symmetry) Group species
j 4k,+ k;
1
M
2
k&k; M
.ke iii
3
k&k’ --.?+A!? M ke -+*Re M M
4 5
ke -M
6
*!! M
7
2ke X
8
2ke
9
M
10
M
1 [ (3ke z-
&(3mke+mk~+2Mk~+3mkeA)+Z
+
2h 4Mk;) f -A( MZm
>)2+--$-(Zmk8+mk;
1
l/2
9mke+ mk; - 6Mki)
&mke+mk~+2Mk~-3~~A)+~[
(2
- ~)2+~m~~mke+mk~ 112
+
13
4Mk~) - ~~A~9rn~+rnk~-6Mk~)
&[3mke+mk;+21Uk;+3m&A)-
i[ ($-
&
4Mk;) -f--A(9mke
(3mk,+mk;+2Mk;
Mzm
- T
z)‘+
sm(2mke+mk; l/2
2ke
+
I
$ rnk:- 6Mk;)
keA)- i[ (2
I
-s)‘+ m
&2mke+mki 112
+
4Mk;) -~~9~+~~-6Mk~)
1
1842
ANTHONY
M. LEMOS
and JORDAN
diagram illustrated in Fig. 5. Within the framework of the BORN-OPPENHEIMER(~~~~~) technique _&E,(p)and &(q) represent the potential energies of the lattice in the ground, g, and upper, U, electronic state. The horizontal lines designated vibrational levels, while the vertical lines represent electronic transitions. Therefore, Fig. 5 is
J. MARKHAM
phonon emissions take place and again the center relaxes to its ground state equilibrium coniigutation. The phonons stored in the lattice at b and d are S, and S,. Table 3. Radial phasing resultsfor the ground state --. Phase relationships
Eigenfrequency (0;)
Al(l) = A,(l/l)i
= -A&l)
As(l) = A,(l/l)j
= -A&l)
A#) ----_-__. <
= A,(l/l)k
= -A,#)
AT(l) = 0
-f
2
h(j) - -&(Uj)i=-h(j) I-
kg+k; M
h(j) = 4(2/j)j= -A.&i) As(j) = -tAxW) ~A~~2~~lk= --As(j) h(j) = 0,
---_--I
<
j = 2, 3.
MARKHAM@) has developed the following general forms for the Huang-Rhys factors:
S, = pi+%
(4.1)
and (4.2) FIG.2. Phonon eigenfrequency splitting for the simplest case, k’ = 0, kc = KS = k. Numbers to the right of each horizontal line represent the degeneracy. The six fold root is due to accidental degeneracy. The ground state degeneracies have been partially removed as a result of the reduced symmetry of the excited state.
interpreted as a center making a Franck-Condon transition frnm point a to b, corresponding to absorption, where it is left in an excited vibrational state which is inconsistent with the temperature of the crystal. A rapid readjustment process occurs whereby the center relaxes by means of phonon emissions to its excited state equilibrium configuration at point c. After roughly 10-a set a downward Franck-Condon transition occurs from point c to point d, where once again the center is left in an excited vibrational state, Subsequent
Here (4.3) where w&) and @j(g) are the jth phonon frequency of the excited and ground respectively. Sf is the Huang-Rhys factor sponding to the 1Y.hmode. The generalized may be interpreted as the weighted sum of phonons. The Sf’s are defined as follows:
and
eigenstates correfactor stored
CALCULATION
OF
THE
HUANG-RHYS
Table 4. Radial phasing results for the excited state Eigenfrequency
Phase relationships
(w,“)
Al(l)
= A,(l/l)i
M
A5(1)
=
A$)
k,+k;
M
a”
M
A2(2)
R
(4.7)
-A##)
Here the A,(k/j)‘s represent the elements of a unitary matrix, the significance of which has been discussed in the previous section, As a result of this transformation, equation (4.7) becomes
= 0
AI(~) = A,(1/2)i 2 w2=------
where 4(R) is the electron wave function and V is the potential energy of the system. This equation may be recast into a more convenient form by the introduction of the following transformation :
P
(1+&%(1/l) =
1843
F-CENTERS
(G)
= -Aa
k
FOR
%(k) = ~M,1/24(W)
= -Aa
AZ(l) = A,(l/l)j
2 4k?+k; or = -
FACTOR
=
= -AS(~)
-&(1/2) j=
-A42)
A5(2) = A6(2) = A7(2) = 0 AI(~) = A,(1/3)i
ke AZ(~) =&(1/3)j k+k; W3---fAke M M
= -As(S) = -A4
(3)
2_
As(S) = -2(1-A) A&/3)&
= -As(3)
A,(3) = 0
Here (4.4a) and (4.5a) where
AE = E,--E,.
(4.5b)
The l ~‘s defined by equations (4.4a) and (4.5a) may be determined by means of the FJXYNMANHELIXAN theorem.@4@) Thus
FIG. 3. (a) The & mode of the Oh group which corresponds to thej = 1 mode of Table 3. (b) The & mode of the Oh group, which corresponds to the j = 2,3 modes of Table 3.
1844
ANTHONY
M.
LEMOS
and JORDAN
J.
MARKHAM
This expression is completely general within the framework of the“adiabatic-h~monic”approximation. The potential energy of the system may be written as V = CrV~(r,R3)+~C2jVz(R~,R~), (4.9) where r and Rj represent the positions of the trapped electron and the jth ion respectively. A harmonic expansion of equation (4.9) would yield a form equivalent to equation (3.5). Since it is
a ~ONFIG~~TION
~05ROl~TE
One dimensional configuration coordinate diagram. The ordinate represents the total energy of the system. The vertical lines ab and cd correspond to the Fran&-Condon transitions while the lines 6c and da represent phonon emissions. q represents the configuration coordinate, I&(q) and E,(g) the energies of the ground and excited states, and Sa and Se the HuangRhys factors corresponding to optical absorption and emission. FIG.
5.
assumed that the electron wavefunctions form a complete orthonormal set, the second term of equation (4.10) does not contribute to (4.9). Therefore we define an effective potential as
Veff =
&V(r,Rj).
{4.10)
To carry out explicit calculations of the HuangRhys factars the following simplification has been made: e2 FIG. 4. (a) The first Al9 mode of the D4h group which corresponds to the j = 1 mode of TabIe 4. (6) The Bse mode of the D4h group which corresponds to the j = 2 mode of Table 4. (c) The second Alg mode of the I&, group which corresponds to the j = 3 mode of Table 4.
Veff = Cj-
I r--Rj I A more realistic treatment would be the actual charge distributions of the particles. However, we are motivated
(4.11) to include interacting _ _ here by an
CALCULATION
OF THE
HUANG-RHYS
attempt to obtain explicit expressions for the Huang-Rhys factor in a first approximation. We assume that equation (4.11) represents the interaction to such an approximation. However, before the Feynman integrals of equation (4.8) can be attacked, one more piece of information is necessary, namely the wave functions. The most reliable simple wave functions for F-centers seem to be those derived by SMITH,@) who following the initial suggestion of MOTT and GURNEY, assumed that the trapped electron has only two bound states, namely a 1s and a 2p state. The SMITH wave functions are: f.Dls = B(l +hr) exp(-Xr)
cos 0.
(4.13)
Here x3
B=
112
( ??I 1
(4.12a)
and p5
112
D=-,
F-CENTERS
1845
the lattice constant
A = [(X-a)s+JQ-zs]3~s,
(4.14a)
B = [Xs+ys+(x-u)s]s/s.
(4.14b)
and
Since the “molecule” has reflection symmetry about the planes at right angles to the x,y, and x axes, it follows that in every case &(1/j)
= &&(3/j),
(4.15)
&(2/j)
= +&(4/j),
(4.16)
&(5/j)
= +&(6/j).
(4.17)
and
Here the negative signs define the radial modes. From equation (4.14) we see that the E~‘Svanish for all non-radial modes. It should be noted that this result is independent of the coulombic electron-phonon interaction we have used. Any central force type interaction yields the same result. We restrict our attention to the reduced matrix scheme of Fig. 6, which defines the A,(K/j)‘s for the radial modes. There are only two non-vanishing l j’s, namely
(4.13a)
0 7r where X and /3 are adjustable
FOR
Here, a, represents
(4.12)
and @sp = Dr exp(-/3r)
FACTOR
parameters.
Is + 2p tra-nsitions Due to the symmetry of our ground state configuration a great number of the integrals of equation (4.8) vanish. In fact, we find that
(3M)l/a
(&+)
= &[e-X
1 1,
+7xss14x+14
-B&/A
-A,(3/j)+A,(2/j)-A,(4/j)
+Az’5ii)-az’6ii’1(g~~~g)
(uE~u)
(4’14)’
=
(:+2X3
-14
-
(4.20)
-&~-2f~(y4+4y3
+8y2+9y+9/2)+(y2_9/2)],
(4.21)
1846
ANTHONY
M.
LEMOS
and JORDAN
J. MARKHAM
and =
-&+U(2y6+4y5
+8y4+14y3+19ys+18y+9)-(y*+9)],
(4.22)
where X = 2Xa
(4.21a)
y =
(4.22a)
and pa.
2p -+ Is transitions The equation of cj(e) which corresponds emission is
,
,
-~,(3/j)
[~3+y2+((z-b)3]3’~
b = a’(l+A),
44 =
“[-+-+)((.f&) 2/M
d/6
x((&j;~U)-(U~~/r+)]~(4.25)
2v
3x
(4.24b)
where A is the perturbation parameter previously introduced. U’ is the distance from the center to one of the nearest neighbor ions in the xy plane and b is the distance from the center to either of the two ions on the z-axis. (Refer to Fig. 1.) Again, equations (4.15)-(4.17) apply and only the radial modes contribute to the l j(e)‘s. By substituting the appropriate values of the L&(/#‘S from Table 4 into equation (4.24) we obtain the following results.
-(UF[U))+$-(l+Y)
+(2/j)
lx
(4.24a)
and
where the (A’)‘s represent the elements of the unitary matrix for the excited state. Since the configuration symmetry has been reduced by the perturbation of the 2p wave function, the resultant expression is somewhat more complicated, namely
e2.
B’ =
to
El(e) = Ck,aM;1/2Ai(&!j)
(e) = -+$A,(W
Here
bz
4Y
0
0
FIG. 6. Reduced A-matrix for the ground state. This portionofthe matrix (refer Sections 3 and 4) shows those elements which correspond to the radial modes given in Table 3.
CALCULATION
2
3
OF
THE
HUANG-RHYS
lx
2Y
21
_-1 2
_;
&(l+y)
-&(l+?
&(l+y)
FIG. 7. Reduced
FACTOR
3x
)
FOR
4Y
52
62
12
0
0
-&(l+Y)
-~(l_~)
X-matrix for the excited state. This portion of the matrix (refer Sections those elements which correspond to the radial modes given in Table 4.
and
1847
F-CENTERS
+-$)
3 and 4) shows
and
es(e) = Z[~(l+~)((.$j+) 2/M 2/3
<“lYlu) are given in equations tively.
-(U~1U))-;(l-+)
(4.20) and (4.21) respec-
2s * 1s transitions x((&$)-(+$j+)].
(4.26)
Here = &[&(2os+4~5+
(+Jr+
+14w3+19er2+18a+9)-(w3+9)]
804 (4.27)
and ($j+)
= &[e-U(:+&3+7u2
+14u+14 where
> 1) -14
(4.28)
v = /!3b,
(4.27a)
u = 2Xb.
(4.25b)
and
The integrals
Although these transitions are forbidden, there are two basic reasons for considering them here. The first is that the lifetime of the excited state is of the order of 10-s set, whereas the expected lifetime of the order of 10-s sec. Although several author@-33) have considered this problem it has not been completely resolved. KLICK(~~) has suggested that this long lifetime might be the result of a forbidden 2s -+ 1s transition. In fact, MCCOMBIE and co-workers(lQ have considered only transitions between spherically symetric s states. The second reason for considering these transitions is that the narrow ruby R lines may correspond to forbidden transitions.@) Although there seems to be no recorded wave function for the 2s state of the F-center, we can make a reasonable guess as to its form. The Fcenter 2p wave function corresponds approximately to the hydrogenic 2p function. We therefore assume the following: @‘2.$= D(l -/+?-8’,
(BE+)
D = (+/3(8)3/3.
(4.28)
(4.28a)
1848
ANTHONY
M.
LEMOS
and JORDAN
Before proceeding further we should note that the perturbed equilibrium configuration is a function of the 2p charge distribution. Since the 2s state is spherically symmetric the distortion will no longer be present. Therefore we may determine the EJ’S for both the Is -+ 2s and 2s + 1s transitions from equation (14) by simply introducing a parameter c, such that
J. MARKHAM
is a function of the parameters associated with the electronic states. The results of any numerical calculation will depend critically on the parameters assumed. At the present time most of these are in a state of flux. For example, it is well known that the lattice surrounding the defect is distorted. Although GOURARY(~~) and WOOD and KORRINGA~~) have considered this problem, it remains unsolved. Also, the experimental spectrum is unknown. (4.29) of the local phonon eigenfrequencies Finally, the use of the simple, one parameter, a’(1 + A); (2s +- Is) Smith wave functions is questionable. At the There is only one-vanishing l3 term, ~1, which present time, work on improved, more general corresponds to the breathing mode. Here wave functions is being carried on by WOOD and JOY.(32) El = e2(6M)-112 [ We are now in a position to obtain numerical values for the Huang-Rhys factors corresponding to 1s ~2p and 1s ~22s transitions. Here we have used the ground and excited state parameters given by sMITH.c3) Since Smith concerned himself where with wave functions for absorption, the use of his parameters for the emission calculation is question(4.30a) c = [(~++p+~3~3/2; able. With regard to the phonon eigenfrequencies, we have associated the effective frequencies as determined by KONITZER and MAFZKHAM(~~~~) (K. & M), LUTY and GEBHARDT~~)(L. & G.), and (4.31) RUSSELLand KLICK(~~) (R. & K.) with the breathing mode. With conventional configuration co(.,,i~~%+ = &[e-x (:+2X3 ordinate techniques this seems to be a reasonable assumption. It should be noted that the same \ 7 (4.32) effective frequency has been used for both the +7X2+14X+14 absorption and emission calculation of NaCl. Thus the emission result for NaCl is probably v = /3c; (4.31a) correct only to an order of magnitude. In a recent and letter, KUHNERT and GEBHARDT@) have published new data on the effective frequencies. HOWx = 2hc. (4.32a) ever, the authors have decided to await the more detailed publication before using this new data. Discussion of calculations In carrying out the calculation we have used the Before presenting the numerical results, a few free ion masses and assumed that the ground state features of the calculations should be noted. We equilibrium distances between the core and outer find that the Huang-Rhys factors have the particles is given by the appropriate lattice constants. The results of the numerical calculation are following dependence shown in Table 5. The inequalities in this table enter as a result of the two frequencies involved in the calculation. We recall from Table 1 that
(m,,/3 CD29
- ( QIBI~/+]
The e4 term arises from the charges of the halogen ion and the electron, W, is the jth phonon eigenfrequency, Mis the free ion mass and f(pwp2p)
CALCULATION
OF THE
Table 5. Numerical Crystal Absorption
KC1 NaCl LiF
w X lOI set-l
HUANG-RHYS
results
for
Sl
FACTOR
1849
FOR F-CENTERS
the Huang-Rhys factor s
S3
S
exP
(1s + 2p transitions)
2~~2.96 (K. and M.) 2~ x4.4 (K. and M.) 2~~4.1 (R. and K.)
1
l<&
2
30
I-S< s<7*5
30
-5
l< s3<7
38
lO< Sa< 76
48< s< 114
116
0
2< s3< 18
2
30
0
3< Sa< 23
3
30
2
0
2
-
2
0
2
-
Emission (2p + 1s transitions)
KC1 NaCl
2n x4.25 (L. and G.) 2~ x4.4 (K. and M.)
(1 s * 2s transitions) KC1 2n x2.96 (K. and M.) NaCl 2n x4.4 (K. and M.)
Thus by letting K’ sweep from zero to infinity we arrive at the result 12
0’s
2.
In Table 5, Sr is associated with the breathing mode; Ss with the asymmetrical axial mode; S = Sl+ Ss; and Sexp is the approximate experimental value. The 1s e2s transitions have been included for the sake of completeness and indicate a smaller value. The most reliable theoretical results are the S factors corresponding to the 1s -+ 2p absorption. In spite of the numerous assumptions and simplifications we obtain qualitative agreement with experiment. This leads us to believe that our approach is valid as a first approximation. This is strengthened by our prediction regarding the rather narrow bands of KC1 and NaCl and the much broader band in LiF. One consequence of our results is that the Huang-Rhys factors for allowed transitions depend on at least two normal modes, each of which 6
contributes significantly. This means that there are at least two lattice modes which interact strongly with the trapped electron. Therefore, the conventional formula for the half width, which assumes a single frequency must be modified. This formula is
Hz = (8 In 2) (Rw)s S coth
(4.34)
where H represents the half-width of the F band at a temperature T, W, the effective phonon frequency, and S the corresponding Huang-Rhys factor. The modification indicated by our results is Hs = (8 In 2) 21 (tiu~,)~Sj coth where the summation extends over the two modes contributing to the Huang-Rhys factor. Such a generalization has been discussed by MARKHAM(~) and by CASSELMAN and MARKHAM.
1850 5. CONTUSIONS
ANTHONY
M.
LEMOS
and JORDAN
AND SUMMAR X DISCXJSSION ,. OF RESULTS
This work, as far as we have been able to determine, represents the first theoretical calculation of the Huang-Rhys factors (phonon broadening) based on the tight binding approximation. The bulk of theoretical and experimental work indicates that such an approximation is valid for well localized centers such as the F-center. Preparatory to this calculation the discrete local phonon eigenfrequency spectra corresponding to the ground and excited electron states has been determined. These results are applicable to any localized model of the F-center in the alkali halides which possesses oh and/or &h symmetry. Within the framework of the “adiabaticharmonic” approximation the secular determinant implicit in equation (3.9) is completely general and in no way restricted to the tight binding approximation or to a particular equilibrium configuration. This means that our calculations may be extended to include next nearest neighbor interactions and outer shells with a minimum of difficulty. Explicit forms for the Feynman integrals for various electronic transitions have been determined. (e.g. see equation (4.20) ). For allowed transitions it has been shown that the HuangRhys factors are dependent on two distinct radial modes, each of which contributes significantly to the phonon broadening of the F-bands. This means that there are two lattice modes which interact strongly with the trapped electron. There are two immediate consequences of this result: (1) Simple one dimensional configuration coordinate diagrams are not quantitatieretymeaningful. This scheme is based on the assumption of a single interacting mode for the tight binding appro~mation of the F-center. We have shown that this is theoretically incorrect. (2) The conventional formulation for the half-width must be modified. Numerical calculations of the Huang-Rhys factors for several alkali halides indicate approximate agreement with the experimental values. Our results predict the observed distinction between the rather narrow F-bands in KC1 and NaCl and the much broader band in LiF. Also it has been shown that the forbidden s to s transitions give rise to narrow bands and perhaps in the
J. MARKHAM
extreme case to sharp lines. In general, our work indicates that one may make completely quantum mechanical calculations using the tight binding approximation and obtain rough agreement with experiment, The various approximations and simplifications used in this development have been discussed in detail in the previous sections. One obvious continuation of this work would be the relaxation of some of these restrictions in the hope of obtaining a more realistic physical model. For example, our modified tight binding approximation could be extended to include second shell interactions. This would allow one to explore more diffuse centers and also to examine more closely the validity of our present model. On a more sophisticated level, the use of a more general electron-phonon interaction and the inclusion of polarization effects should be considered. At the present time, general wave functions for the excited states of the F-center are being developed by WOODand Jot. Using these wave functions it should be possible to investigate the problems of the F-center phonon broadening, especially for the case of emission, on a more reasonable level. Furthermore, if the KI and L bands do correspond to higher excited states of the F-center, it should be possible to determine their Huang-Rhys factors with a slight modification of our present formalism. In this connection one should be able to investigate phonon phenomena associated with centers which have a geometrical structure similar to the F-center, such as the U and KCl:Tl centers. One problem which can be attacked rigorously within the framework of our model is that of the effective mass of the center. This is an important concept which seems to have been introduced by W~LLI~~S(~~)in 1951. Since then, experimentalists seem to have adopted this concept without questioning its theoretical validity. REFERENCES
J. J., Reo. Mod. Phys. 31,956 (1959). 1. MARKHAM 2. DEXTER D. L., Solid State Physics (ads. SEITZ F. and TVRNBULL D.) Vol. 6. Academic Press, New York (1958). 3. S~vrr~a UT. A., “Energy Level Calculations for Fcenters and Positive Ions in Alkali Halides”, XnolIs Atomic Power Laboratory (Rept. KAPL1720, April 1,1957).
CALCULATION
OF
THE
HUANG-RHYS
4. MCCLURE D. S., Solid State Physics (eds. SEITZ F. and TURNBULL D.) Vol. 9. Academic Press, New York (1959). 5. FITCHEN D. B., SILSBEER. H., FULTON T. A. and WOLF E. L., Phys. Rev. Lett. 11, 275 (1963). 6. DE BOER J. H., Recuiel des Trav. Chem. d. PaysBas 56, 301 (1937). 7. GOURARY B. S., and ADRIAN F. J., Solid State Physics (eds. SEITZ F. and TURNBULLD.) Vol. 10. Academic Press, New York (1960). 8. WILLIAMS F. E. and HEBB M. H., Phys. Rev. 84, 1181 (1951). 9. BJORK R. L., Phys. Rev. 105,456 (1957). 10. ROSENSTOCKN. H. and KLICK C. C.. Phvs. Rev. 119, 1198 (1960). 11. MCCOMBIE C. W., MATTHEW J. A. and MURRAY A. M., J. Appl. Phys. Suppl. 33, 339 (1962). 12. CASSELMANT. N. and MARKHAM J. J., J. Phys. Chem. Solids 24, 669 (1963). 13. MONTROLL E. W., Bull. Am. Phys. Sot. 9,12 (1964). 14. LUTY F. and GEBHARDT W., 2. Physik 169, 475 (1962). 1.5. BORN M. and MAYER J. E., Z. Physik75,l (1932). 16. WOOD R. F. and KORRINGA J., Phys. Rev. 123, 1138 (1961). 17. REDLICK T. K. and STRICKS W., Wien Beriche IIb, 146,447 (1937). 18. WILSON E. B., DECIUS J. C. and CROSS P. C., Molecular Vibrations. McGraw-Hill, New York (1955). 19. HUANG K. and RHYS A., Proc. Roy. Sot. (Lond.) AZO4,406 (1950). 20. PEKAR S. I., J. Exptl. Theoret. Phys. (USSR) 20, 510 (1950). I
_
FACTOR
FOR
F-CENTERS
1851
21. KL.ICK C. C. and SCHULMAN J. H., Solid State Physics (eds. SEITZ F. and TURNSULL D.) Vol. 5. Academic Press, New York (1957). ’ 22. BORN M. and OPPENHEIMERJ. R., Ann. Physik 84, 457 (1927). Theory of 23. BORN M. and H~ANG K., Dynamical Crystal Lattices, Chap. 4. Clarendon Press, Oxford (1954). 24. FEYNMAN R. P., Phys. Rev. 56, 340 (1939). in die Q&ten Chemie, 25. HELLMAN H., Einfihrung P. 285. Teubner. Leimie (1937). N. F. and &JW~Y k: W.; Electronic Pro26. M&r cesses in Ionic Crystals. Clarendon Press, Oxford (1948). 27. FOWLER W. B. and DEXTER D. L., Phys. Rev. 128, 2154 (1962). 28. FOWLER W. B. and DEXTER D. L., Phys. Stat. Sol. 2, 821 (1962). 29. FOULER W. B:, Phys. Rev. 135, A1725 (1964). 30. KLICK C. C.. PATTERSOND. A. and KNOX R. S. (unpublished) 31. GOURARY B. S. and ADRIAN F. J., Phys. Rev. 105, 1180 (1957). 32. WOOD R. F. and JOY H. W., Phys. Rev. 136, A451 (1964). 33. KONITZERJ. D. and MARKHAM J. J., J. Chem. Phys. 32,843 (1960). 34. KONITZERJ. D. and MARKHAM J. J., J. Chem. Phys. 34, 1936 (1961). 3.5. RUSSELL G. A. and KLICK C. C.. Phys. Rev. 101. 1473 (1956). 36. GERHARDT W. and KUNHERT H.. , Phvs. < Lett. (Netherlands) 11, 1 (1964). 37. WILLIAMS F. E., Phys. Rev. 82, 281 (1951).
Note added in proof.-We are grateful to R. GILBEHTfor pointing out that there appears to be an intrinsic error in the work of SMITH. Although the forms of his wave functions may be correct, the parameters are not. Thus the values appearing in Table 5 should be considered as no more than qualitative at this stage. Work on a revised table has begun and should be published in the near future.