P HY SIC AL 8 E VI E% 8 VOLUME 10, NUMBER 6 y 5 S E P I EM B E 8 1974.

Theory of multiphonon absorption due to nonlinear electric moments in crystals

Bernard BendowSolid State Sciences Laboratory, Air Force Cambridge Research Laboratories, Bedford, Massachusetts 01730

Stanford P. Yukon~Parke Mathematical Laboratories, Carlisle, Massachusetts 01741

See-Chen Yingf~

Department of Physics, Brown University, Providence, Rhode Island 02912(Received 10 April 1974)

%e formulate a correlation-function theory of the multiphonon absorption due to nonlinear electricmoments in anharmonic crystals, at frequencies co far above the reststrahl co„employing the method ofcumulants. In contrast to previous treatments, direct expansion of the moment and anharmonicity

potential in powers of displacements is avoided; we thus obtain expressions containing various classes ofphonon processes summed to infinite order. The frequency and temperature dependence of the

absorption coefficient a is calculated for various approximations and simplified limits, including: the

harmonic limit, for which computations with Debye and Einstein models are carried out; the quadraticanharmonic approximation to the cumulant, for which computations are carried. out in the

noninteracting-cell picture; and the single-particle model, within which "exact" computations are

pursued. The results indicate exponentiallike frequency dependences for a for all cases, with

enhancement and broadening of a for large anharmonicity, behavior similar to that predicted foranharmonicity absorption stemming from linear moments alone. Application of the single-particle model

to cubic diatomic crystals demonstrates significant effects of anharmonicity on cx, especially for

co/coo & 1. Moreover, for covalent crystals nonlinear moment contributions to a dominate, while for

highly ionic crystals the linear term can exceed or compete with the nonlinear ones for small tointermediate values of co/coo.

I. INTRODUCTION

Various papers have recently treated the prob-lem of multiphonon infrared absorption in the fre-quency region far above the reststrahl in crystals,where a large number (n ) 5) of phonons contributeto the absorption. This frequency region is of con-siderable interest in ir applications requiring highlytransparent components, such as fiber optics, inte-grated optics, and laser windows. The intrinsiclimiting absorption of many materials for the latterapplications will be determined by multiphonon pro-cesses. For the most part, existing work has beenconcentrated on anharmonicity absorption, in whichthe fundamental ir-active modes are excited by theincident light, and subsequently damp through an-harmonic interactions, conserving energy in theover-all process. However, it is well known thatmultiphonon absorption may proceed as well throughthe mediation of "higher-order" or "nonlinear"moments, where, in general, an arbitrary numberof phonons may be excited directly by the incidentlight. The nomenclatures higher order and nonlin-ear are often employed interchangeably in referringto terms in the moment greater than linear in lat-tice displacements u; (note, however, that we willconsider the linear response to the Photon field)We here adapt the correlation-function formalismof Bendow, Ying, and Yukon (hereafter, I) applied

previously to the anharmonicity absorption prob-lem. The effect of terms in the moment quadraticin u was considered within a noninteracting-cellmodel of a crystal by McGill et al. In the presentwork, we develop a formalism suitable for the fullcrystal problem, with an arbitrary functional formfor the moments, although specializations to vari-ous simplified models will be made. The latterwill be most suitable for the case of crystals pos-sessing linear moments arising from charge trans-fer, where the major contribution to the nonlinearmoment arises from axially symmetric distortionsof the electron clouds in the vicinity of the ions.On the other hand, for ir-inactive crystals withhomopolar bonds, such as Si, just the nonaxialcomponents of the distortion dipole moment surviveto contribute to absorption. '

Adopting an adiabatic-approximation viewpoint,the effective moment of the crystal may be takento be

1VI(~Its)) = QrI gq; r;+gQ&R

where lt) is the adiabatic electron wave function(ground state for fixed ionic positions) and (q;, r;)and (Q;, R;) are the charge and position vectors ofthe valence (charge-transfer) electrons and the ioncores, respectively. We now adopt the approachof Flytzanis' expounded in his treatment of higher-

10 2286

10 THEORY GF MULTIPHQNQN ABSORPTION DUE TQ 2287

order susceptibilities of semiconductors, whereM is taken as a sum of bond (i. e. , locaLized) mo-ments between pairs of atoms. The motivation andimplications of this approach are discussed in somedetail by the latter author. In the present instance,it implies that M may be expressed as

1M= —~ m„.(R„—R, ,r),lsl 's'

where (l, s) indicates the sth atom in the lth unitcell, and m„, is the bond dipole moment for thepair (ss'); terms with ls =l s are exciuded above.The function m{r), which contains contributionsboth from "static" and "distortion" charges, hasbeen discussed by various authors (Refs. 12 and

13}, to which the interested reader is referred.In addition to developing the formal theory of

higher-order moment absorption, the aims of thepresent paper are to investigate the frequency andtemperature dependence of the absorption, and toascertain the relative role of anharmonicity in theabsorption. The plan of the paper is as follows:The frequency- and temperature-dependent absorp-tion coefficient n((0, T) is expressed in terms of acumulant expansion involving displacement corre-lators in Sec. II, and various approximations to thegeneral results are indicated. Analytic results forsimplified models of the moments and the phononsare presented in Sec. III, while numerical compu-tations are pursued in Sec. IV. The results of thecalculations, including the (d and T dependence ofn, are discussed and compared with other work inSec. V.

the absorption coefficient for isotropic or cubiccrystals, which following Ref. 15 is given by

n((u, T) = Img [)( (k, (d)]„„,C tti((d y

(2. 3)

S$1 ~ $ S }t SS SSm. =——,e (m +m&), (2 5)

where R indicate equilibrium ion positions, and &,is a vector from the cell origin to the sth atom inthe unit cell. The calculation of n (or, equivalent-Ly, Imx) involves the evaiuation of

(M(t)M(0)) P exp[iq. QP( —R, .)$g'ls l's'r jr' j'

where Im indicates imaginary part, and p. , is thereal part of the refractive index, which generallyvaries negligibly in regimes of interest for thepresent problem. It is useful to express Im X interms of a correlator for M as

Imx =0 (M(t)M(0))„[n(0t)+ I]

(2. 4)n((d) —= (e "—1) ',

where P =k~T, with k~ Boltzmann's constant; wehave above set the photon wave vector k= 0, as willbe done henceforth.

Exploiting various analogies between the presentproblem and that investigated in I leads us to ex-press

M= ss' '&'~"l ~l»e'~ ~ls ~l's'qlslsse (1

II. MULTIPHONON ABSORPTION COEFFKIENT +iq' .(Rt —Rt.)]m" m"", *U, (2 6)The lattice contribution to the complex dielectric

susceptibility X may be written in standard tensorform as' (h =1)

)( (k, (d ) = —G„(M-„(t);Mg(0)}, (2 1)

where k and ~ are the wave vector and frequencyof the light, and G„ the Fourier transform of theretarded tensor Green's function (GF}

G (X(t); ii(0)}= —ie(t)([A(t), 5(0)]),

u =(2 )'Jrr 'u(r), r

(2. 2)

where () indicates therma! average, e is the Heav-iside function, and A(t) is A in the Heisenberg rep-resentation. For the present purposes we require

ir( ~ (utu(t&-utrur(t)i tur ~ (uty(0)-utr ~ (0)]U=-(e ' ' e

which we will express in terms of a cumulant ex-pansion below. For a single random variable,e. g. , the cumulant expansion is defined by

(e') = exp+ (a"/n!)(x"), ,n=1

(x).= (x), (x'), = (x') —(x)', etc.(2. '7)

It is straightforward to extend the above expansionto time-ordered operators, as detailed in Ref. 16,for exampie. For simpbcity, we consider aii (ls)to be sites of inversion symmetry, so that (u"„)= 0for n odd. Employing other standard properties ofcorrelation functions, one obtains directly

U(ls, l's ', rj, r'j ', q, q ', t, t) ) = e

lt(= u(q:q: [C (iso, iso)+ C(l s'0, l's'0) —2C(iso, l's'0)]+3iq:q:q: [C (iso, iso, l's'0)

+ C (iso, l's'0, 1's'0)]+ ~ ~ ~,

2288 BE ND0%', Y UKQ N) A ND YI NG 10

h2 =f, (Is -j r; I 's ' -j'r; q -q ),h, =q:q': [C {Istj r0)+ C (1's't j 'r'0) —C (l's'f j r0) —C {fstj 'r'0)] ——,iq: q': q': [C(lst, fst, jr0)

(2, 8)

—C(Let, Isi, rj '0)+ C (l's't, I's't, rj 0) —C(l s t, I's'f, rj''0) —2C(lst, I's t j r0)+ 2C (lst, I's't j 'r'0)]

+ ~ iq ':q:q: [C (Ist, rj 0, rj 0) + C {Ist, r'j '0, r'j '0) —C {I's't, rj 0, rj 0) —C (I's 't, r'j '0, r'j '0)

—2C(lst, rj 0, r'j '0)+ 2C {I's't, rj 0, rj''0)]+ ~ ~ ~,

F„=Q Qg„(A;)p„(R;, &u),tidal

gff f 8~ ff {2.11)

p„= d~'p„, (~-~' p, ('; p, =(2& -'

The computational and interpretational advantagesof the above series are summarized in I. Its appli-cation to the present problem will be illustrated inthe following sections.

It is evident from the above development that cal-

where C{f,s~t, , fzsztz, . . . , l~„t„) is the n-fold dis-placement correlator

C = {ug„,(t, )u,~,2(t~) ~ ~ u, , (t„)&, (2. 9)

and the notation ql .'q2. ~ - ~ . q„: C indicates that dotproducts are taken between each pair (q, , u. ..).s]The number of independent l, indices could be re-duced by invoking translational invariance, but thisis not useful at this point.

Various simplifying approximations are imme-diately suggested by Eq. (2. 8). If one retains justterms quadratic in u's in the cumulant expansion,results very similar formally to Eq. (2. 3) of I areobtained ("quadratic approximation"). An evensimpler result follows upon evaluation of U in theharmonic ensemble, in which case just quadraticcorrelators survive and, moreover, these are tobe evaluated in the harmonic ensemble ("harmonicapproximation"). More generally, one may con-ceive of a successive approximation scheme inwhich higher-order correlators are approximatedin terms of lower-order ones, but such procedurestend to become complicated and unwieldy.

Some practical considerations regarding evalua-tion of Imx are worthy of note. A useful procedurein many instances is the use of convolution expan-sions. Specifically, we here require the Fouriertransform P„of functionals of the form

8:(I)=P Z(It„D(a„f)), (2. 10)

where D is itself a (linear in the quadratic approxi-mation) functional of the correlators C, and & isobtained by carrying out the sums over q; in Eq.(2. 8). A convolution expansion in D yields (&u 40)

culation of n requires knowledge of C(t) or itstransform C„. Within the quadratic approximation,e. g. , we require the correlators'

C „,(&, t) = ~Z &"'", ""',gt2{AI„(tHI, ~ ( )&,

(2. 12)where f&'s are polarization vectors, M, 's masses,and (d s phonon frequencies; A»=bt, „+b», whereb's are creation-annihilation operators of phononsfrom branch y. Although one thus has a set ofbranch-coupled correlators {A„A„,&, it is often areasonable or necessary approximation to retainjust the diagonal terms y =y', in which case one re-quires

(2-„„(t)A~„(0)&„=2[n(hy(u) + 1]ImG (hr(u), (2. 13)

with 6 the standard anharmonic GF, a quantitywhich has been investigated extensively in the liter-ature. ' lf we reexpress G in terms of the complexself-energy II= g&+iH2, C„ for the above case be-comes

C„(R )=.—

(2. 14)where &ut =m;+ II,(i). For linear moments justterms linear in C arise; then for cu»~„-, , n-~2,a well known result. ~2 may itself be expressed interms of the C s, as detailed in I.

111. APPLICATION TO MODEL SYSTEMS

In this section we evaluate Imx within the qua-dratic approximation for various special cases,such as for Gaussian moments, and within simpli-fied lattice models, such as the independent-celland Einstein.

A. Gaussian moments

2 2If we choose m;=rnoe ', then the q integrations

in the quadratic approximation become straightfor-ward. ' For simplicity, we display results for theisotropic case where q ~ C ~ q' = q ~ q'C, with C sca-lar; invoking translational invariance one finallyobtains, after some algebra

8 (t) -=(M(t)M(0)&

THEORY OF MULTIPHQNQN ABSORPTION DUE TQ

2i21R1+s2R2 D~i ' ~2

momo „~ exp 2PVg Rg%3ss'rr ' 4a&a2 —D

stein model, for example,

C(t)- e "2'(n2+ 1)+e'"o'n2, (3. 5)

X (4a~ a2 -D2) 2+,ay =—2[C„(00)+C...,(00)] —C„.(Rq, 0)+R2,

a2 —= —,' [C„„(00)+ C„,„,{00)]—C„„,(%2, 0) + Ro,

D = C,„(--R2, t)+ C,.„,(R,, —R, —R„t)

—C,„.(R2- %2, t) —C,.„(-A~ —R2, t),Rg Rg+ Ks Kss

= R2+ K~ —K„a

(3. 1)

Kith the above results, numerical evaluation by themethod of convolutions becomes practicable. Ex-amples of such computations will be presented inSec. IV.

B. Einstein model

%e consider a generalized Einstein model wherespatial correlations between different cells are ne-glected, but temporal correlations, includingdamping, say, are retained. Then

C(Q, t) = C(t)&(R;), (3. 2)

mhere 4 is the Kronecker symbol, so that the sumover-sites may be carried out explicitly. In thequadratic approximation, for example, one obtains

where coo is the mode frequency and no its thermaloccupancy. 8'(&d) may be evaluated explicitly in thiscase, but me mill not display the somewhat lengthyresults here: Rather, the interested reader is re-ferred to Eq. (32) of I, where a similar result aris-ing in connection with anharmonic damping is dis-played. Essentially, in the present case each F;mill be replaced by a series of g-function peaks at(d =no)o, n integer, of meight

(1+1/no)"~I„[2q. C, ~ q n2t'(n2+1) t'], (3. 6)

where I„ is the modified Bessel function and C, isthe linear combination of C 's entering the corre-sponding E& term.

C. h&dependent-cell model

Another approach, similar in spirit to the Ein-stein, restricts moment interactions to atoms mith-in a single cell; however, the full crystalline spec-trum may be employed for C. This approximationis realized by choosing E =l', j=j' (but s 4s',rex'); in Eq. (3.1), e. g. , one would take R&=R2=0. In this instance it is useful to employ a some-what different formulation than that given above.Rather, me write the term corresponding to e"3 inU [see Eq. (2. 8)] in the form

p(t) = P m m", 2'.2U,qq'ss'rr ' exp d„~q'" = p „q q", (3. 7)

"[E2—E2 —Es]+ { Ei E2 —Ev Es Es

—Eg2+ Eii) + 2(E2+ E2+ E2+ E2)}, (3. 3)

x j&b (q —q'+ 4)[E, —E2 —E,]+Ni2, (q+ q'+ G) where the d's and P's are functionals of the corre-lators C, and depend on site indices and, in thenonisotropic case, on the directions of q and q'.Then, taking m;=m,

where sv; are the Debye-%aller-like terms corre-sponding to e "' and e & in Eq. (2. 8); G's are re-ciprocal-lattice vectors; and

E& = exp(q ~ [C,„(t)+C, ,„.(t)]. q'},E2= exp[q. C (t) q ]

E,= exp [q C,,„,{t)~ q'],E4= exp(-q. [C,„,(t)+ C...(t)]. q'},

mhere

mnss'rr ' P P (P&' )H„(~2' ),

iO'Ri +ig' 82

qq'mnss'r r'

(3. 8)

E, = exp [-q ~ C,„,(t). q'], H„(R) =g &'"'"m~iy„-ti" (3. 9)

F6=exp[-q C. ,(t). q'],E, = exp/q [C„(t)—C,„.(t)]. q'},

E, = exp/j. [C„(t)—C,.„(t)].q'},F2= exp(q ~ [C,.„.(t) —C,„.(t)] ~ q'},

(3 4)

( )1 s"m(R)Zn can (3. 10)

and the second equality obtains for the isotropiccase. In most cases of interest so, may be replacedby unity, except at very high temperatures; then

E,o= exp+. [C,...(t) —C,.„(t)] q },Eu = exp@ ~ [C,„(t)+ C,.„,{t)—C„,(t) —C,.„(t)].q'} .

For the conventional harmonic single-mode Ein-

and

mn mn

2290 BE NDOW, YUKON, AND YING 10

Up to this point, the results are fairly general. Theusefulness of the above form is, however, limitedto special circumstances, such as the independent-cell model. For a diatomic crystal in the lattercase one obtains simply

(3.12)

where m~"' indicates nth derivative of m, and

n = IT&,—Pre I; D is the quantity appearing in Eq. (3.1),

but evaluated for Ry —R~ = p and s =r = 1, s'= r' = 2.Thus we require just the sum over 0, of a series ofconvolutions p„weighted by (m'"') to calculate lmX.

The above result is particularly convenient forinvestigating the effect of anharmonicity on higher-order moment absorption; this is most practicablewhen a sufficiently simple form for C(t) or C„ob-tains. C„will be qualitatively similar as a func-tion of frequency to the anharmonicity absorptioncoefficient az arising from linear moments viadamping of the TO mode. ' At lower frequencies(&u/a&To-0-3), C is a complicated function reflect-ing selection rules and density-of-states effects.As an example of the influence of selection rules on

n itself, we note that the linear moment contribu-tion should contain only terms arising from k=0TO phonons, while higher-order moment contribu-tions in general involve all phonons in the zone.Nevertheless, it may be reasonable for qualitativepurposes to neglect structure in C„, since theprincipal contribution to absorption for &u/&oo» 1

results from higher convolutions, which tend tosmooth out any structure present in C„. In Sec. IVwe employ simplified models of this kind for in-vestigating the effects of anharmonicity on the fre-quency dependence of n.

D. Symmetry considerations

The proliferation of terms in n associated with

different sets of site indices may be reduced by

noting various symmetry relations. An obvious re-lation stemming from the definition of M is thatterms with ei thee s ~ s or r x are equivalent.Other symmetries are present in special circum-stances. In the quadratic approximation, whenever

C,~= C~, and C(R) = C(- 0), conditions which indeedhold for cases of interest here, then terms with

s r, s ~r or with s-x, s =r are equivalent.The demonstration of the latter follows directly,and will not be spelled out in detail here. It should

be remarked, however, that restrictions such asm" =0 for s =s and R, =8, imply that terms with

R~= 0 and/or Rs= 0 must be excluded for certainsets of site indices. For the case of diatomic crys-tals, for example, one finds that the 16 possibleterms are reduced to five different ones by appli-cation of the above rules. However, restrictionson %q or %3 accompany 12 of these. For the simpler

models to be considered in this paper we moreoverassume that nonvanishing contributions to m"arise only between different types of atoms (s Ws );thus all terms with either s =s or x=r vanish.Within the latter approximation only four equalterms remain, and it suffices, for example, tocompute just the configuration s =1, s =2, x=1,r = 2 to obtain n.

E. Single-particle model

The simplest model of all views the crystal as anassembly of identical noninteracting cells, with asingle particle in each which undergoes anharmonicmotion, and interacts with light via an electric mo-ment m. Then

Im)(„= —'(m(t)m(0))„[n(&u)+ 1]

—g (0(m (n) (n

(m (0)5(&u —m„), (3. 13)

n

thus resulting in a spectrum composed of lines at&u = ~„of strength l(0 Im In) I . Such a spectrum isa consequence of neglecting crystallinity, and onemust either interpret the peaks as the integratedabsorption in a given frequency interval, or elseaverage the predicted absorption over an appro-priate frequency distribution to simulate a crystal.Despite its interpretational drawbacks, the advan-tage of this model is its simplicity for numericalcalculations, especially for "exact" (nonperturba-tive) calculations which would be virtually impossi-ble in the full crystal case. Fortunately, variouscalculations of anharmonicity absorption based on

the single-particle model do appear to retrieve re-sults similar to those of more complicated models.This suggests that the model may be useful in thehigher-order moment case as well.

IV. NUMERICAL CALCULATIONS

In this section numerical calculations are carriedout at three increasing levels of accuracy {harmon-ic, quadratic anharmonic, exact) but with corre-spondingly decreasing sophistication in the modelsemployed. For example, full crystal computationsare feasible within the harmonic approximation,while exact computations require the simplificationsinherent in the single-particle model. But becauseof the complicated expression for 0. in the generalcase, it appears useful to calculate results for awide range of models and approximations, with an

eye to eliciting common features which could berepresentative of actual crystals.

A. Harmonic approximation

Gaussian moments; Debye and parabolic phonons

In the harmonic approximation, absorptien arisesexclusively from electric moments; n in this casewill be denoted a«. We consider a diatomic crys-

10 THEORY OF MULTIPHGNGN ABSORPTIQN DUE TQ 2291

tal, characterized by Gaussian moments m„.{q)= m~&e" o', restricted to pairs with s 4s . Calcu-lation of n« then requires evaluation of just a sin-gle term in the sum in f»»(t) of Eq. (3. 1), namely,that with s = 1, s'= 2, x = 1, x' = 2. Once C„has beenspecified, numerical computation by the method ofconvolutions is straightforward. We consider C 'sappropriate to a generalized Debye model mith dis-persion ~ =w&+sA;, and a parabolic model with&d = {k—ok /k„)A. C's for these cases are obtainedand displayed in Ref. 21, and we restrict the pres-ent discussion to just a brief mention of the princi-pal considerations involved: One chooses threebranches of identical dispersion, mith polarizationvectors characteristic of k=0 optic phonons, name-ly, f, = —(mo/m, )' f2. In dimensionless units (seeRef. 21) one obtains for the Debye case (»d &0)

9 f» f» sin[ko[(»d —»d, )/(1 —»d, )]R]2 (m»m ) kpR

x ' 2[»»(»o)+1]8(»o, &»d&1), (4. 1)o»(l —»d, )

mhere ko=12q, and 8 indicates the range where Cis nonzero. For &o negative, one employs C(- I»»» l)=e '"'C()&ol). Analogous results obtain for theparabolic ease as mell.

When application to rea, listic situations is de-sired, implicit as mell as explicit temperature de-pendences must be accounted for in C„. Mostimportantly, the phonon spectrum is temperaturedependent; the simplest approximation is to assumea uniform linear shift in m vs T for all k, as dis-cussed in Ref. 21, for example. The lattice con-stant [obscured by the nondimensionalization of Eq.(4. 1)] and possibly the parameters in m(»') and theinteratomic potential v(r), also depend on T, butthe effects on 0. are relatively minor compared tothe phonon shift.

The general behavior of O, E„may be predicted by

exploiting the close analogy between 0.«and theanharmonieity absorption coefficient a„, calculatedin detail in Ref. 21. One easily shows that foridentical m(r) and v(o ) the only differences betweenam„and ez, apart from constant prefactors, areas follows: (a) the coefficient 8"' 7/BD"" asso-ciated with D" in the convolution series for n„ isreplaced by 8"P/8D", and (b) the factor»d arisingfrom an extra propagator in a„ is replaced by afactor co in Q.E„. These differences have a second-ary effect, and one may directly predict the majorfeatures of nE„ from those of o.„described in Ref.21, say. Namely, one deduces (a) an exponential-like decrease in the envelope of e vs increasing ~,for all cases, (b) the appearance of structure in adue to narrow bandwidths and/or narrow peaks, inthe density of states, (c) a smoothing of a vs»o atlarge o», and (d) a very strong expbcit T dependence

2»» o» 3 " 1 S"m(r)

» t~o)" »"

}n(»d)+ 1»do (4. 2)

mhere v, is the volume per particle, and pp=)»»do»»o/5 is the dimensionless reduced mass. De-spite the model s artificiality, the simplicity of theresults for n make it convenient for a qualitativeanalysis.

We consider m(r) of the form m- e """, such assuggested in Ref. 27, for example; parameters inm may be determined by prescriptions to be de-scribed in Sec. IVC. In Fig. 2 we illustrate resultsfor aE„vs v for exemplary crystals (the curve ismade continuous by interpolation between integervalues of »o/»oo). Again, exponentiallike behaviorfor o. vs ~ is evident as in Fig. 1. The tempera-

20—

l6-C34UhJ

Q 12-

O 08-

KDIX)

O'~~P

0' I

2I I

4 6FREQUENCY

FIG. 1. LogIp of the absorption coefficient n vs di-mensionless frequency e/{dp, for the generalized Debyeand parabolic (P) models described in the text. The no-tation Dx indicates Debye with intercept x = ref/uDp Crys-tal parameters are chosen characteristic of KCl.

at large +, which can be substantially suppressedwhen typical T dependences of the phonon spectrumare included. Other dependences, such as on therange of m{r), may be similarly deduced; the in-terested reader is directed to Ref. 21. We herecarry out computations of n«at T = 0 K, as illus-trated in Fig. 1. The results may be seen to bevery similar to those for n„given in Ref. 21.

2. Independent-eel/, Einstein model

This model combines the independent-cell ap-proximation [Eq. (3.12)] with a C(f) characteristicof a single harmonic Einstein oscillator [see Eq.{3.5)]. One obtains directly (5, c = 1)

2292 BE NDOW, YUKON, AND YI NG 10

ture dependence of n is displayed in Fig. 3 for KClparameters, employing

~o(T) = (uo(1+aT), (4 6)

with a = —2. 't&&10 '/'K for KC1, a value estimatedfrom the Gruneisen approximation. The resultsindicate the suppression of what would have been avery strong temperature dependence at co»A@0 had

g been taken as zero.

B. Quadratic approximation

*-]0— K

The quadratic approximation is the simplest ve-hicle for incorporating anharmonicity effects in n,thereby improving on the harmonic results givenabove. There are a variety of indications, for ex-ample, that anharmonic damping of the TO modemay yield tQe dominant contribution to absorptionin ionic crystals; for such a case inclusion of an-harmonicity in e is essential. One requires an an-harmonic correlator C„which is available for se-lected modes and crystals, but is not available ingeneral (see Sec. II). As discussed in Sec. IIIC, itis useful to consider model C„'s of a fairly simpleform, which ignore any complex structure repre-sentative of real crystals, but satisfy various gen-eral physical requirements. We here employ theindependent-cell model, with the R3 dependence ofC„neglected as well. The absorption coefficienttakes the form (&u o0)

a{~)-~Q [m'"'(x)]'p„(~)/n! .fI=1

To consistently investigate effects of anharmonic-

ity on n, we require C's which satisfy appropriatesum rules. For optic vibrations within an isotropicsingle-cell model, the sum rules of Refs. 29 and

I I I I I I

3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7 5

FREQUE NC Y i')o

FIG. 3. Ln of the electric-moment absorption coeffi-cient nEM vs dimensionless frequency +/~o for Kcl, atvarious temperatures.

30 may be boiled down for our purposes to

d(d (d U((d),3m

2P 0

witha oo

{u(t)u(0)) = d(u e '"'C„,"0

C, = (g/g) sgn&o[n{&u)+ 1]U(~) .

(4. 4)

Ua =A Ice Ie ' (4. 6)

The origin, significance, and properties of U arediscussed in Ref. 30.

We now introduce a Gaussian model for C via thechoice

-10

-15—

w 2000

O -25—

-30—

-35—

Then Eq. (4. 4) yields

C„= (6v/p)I

&uIsgn~[n(~)+ 1]a'e ' "

(6v/p8)a e ', T-(6p/it)a &re ', T-0 .

For T-~ we obtain

G Cn a o)Q a 2 67t 2n

( )1/2 0 i 0p6

(4. 6)

{4. t)

I I I I I I I I

3.5 4,0 4.5 5.0 5.5 6.0 6 5 ?0 7.5 8.0FRFQUENC Y ~/&o

FIG. 2. Ln of the electric-moment absorption coeffi-cient n EM vs dimensionless frequency v jvo, for various

crystals. m(r) = mi(r) +mo(r) = m, r+ mo e (4 6)

for T-0 the p's are not expressible inclosed form.The anharmonicity parameter is p ', with large a '

implying large anharmonicity and broadened C„vsChoosing

THEORY OF MULTIPHONON ABSORPTION DUE TO

then in the high-temperature limit

(4. 9)At =—Co[mt(K)/mq(tt) —X„], A2 ——X Co,

where no is an appropriate constant. A.&//Co is ameasure of the static-charge linear moment con-tribution relative to that arising from distortiondipole moments; X measures the range of the dis-tortion moment. Figure 4 illustrates results ofcomputations for n at different values of a, forfixed sets of parameters Co/a, At/Co, and X . Asexpected, broader absorptions are associated withsmaller a's and larger X 's, i. e. , greater anhar-monicity and shorter-range moments.

A similar analysis follows for C„characteristicof a TO mode damped by Einstein oscillators. Fol-lowing Ref. 2, the independent-cell model with in-terionic potential v{t')- e """yields for T =0 'K,

1C„=Ct Z—(A4('t(td -neo), A4= Cent. „', (4. 10)

n=1

where CE is a dimensionless phonon parameter,Cs~ p,'. Application of Eq. {4.4) shows

DD

U 0UJO

1

oP

k- DCL O

tXt

ODDCV0

y t

D

0 ~ 00

DD

U

Ot

1 ~ 00 2.00I I

1

3.00 4.00FREQUENCY

6 00

C, = (3tt/2 it~, )A,' e "4,

with the final result for n

as(u)) =Q ct.;('t((d —j(uo),

(4. 11)o

DD

OC Ct

CQ

D

Q

Ej 4 APl

A, =-C, [mt(x)/m, (tt) —X„]', A~-=CtX~ .

(4. 12)

DDle

0.00 1 .00 2 00 3 CQ 4 00FREQUENCY

6 ~ 00

The interpretation of the various parameters fol-lows in direct analogy with the Gaussian case above;A4 plays a role similar to a ', for example. Re-sults of computations of nE vs + are illustrated in

Fig. 5, and are seen to display trends similar tothose for a~.

C. Exact calculations: Single-particle model

The single-particle model has been discussedabove in Sec. DIE. Although this model is highlyartificial because it neglects interactions betweencells (i. e. , crystallinity), it has neverthelessproven popular for a variety of quantum ' ' andclassical calculations because of its amenabilityto exact analyses. To calculate ImX we require thefunctions m and v, and the eigenstates of the Ham-

iltonia, n

DDD

DD

zW

4J4 ODo

t

0D„DCKo cn

CQ

ooDI ~

Vl CV

OI

DDlA

0 00k I I

1 I C

1 00 2 ~ 00 3 00 4 ~ 00 6 00 6 ~ 00FRE.QUENCH

H =p'/2 p+ U (~) . (4. 13)

Because we expect calculations within the modelto be at best qualitative in nature, it is sufficient tochoose rn's and e's of a fairly simple, but physical-ly reasonable form. Specifically, we characterize

FIG. 4. Log&0 of the absorption. coefficient o, vs dimen-sionless frequency +/~0, for a Gaussian. -model correla-tor. (a) A&/a =A2/a —-0.01, a =0.5, 0.35, 0.2 for curves1-3, respectively; {b) At/a =1.0, A2/a =0.1, a =0.5,0.35, 0.2 for curves 1-3, respectively; and (c) a =0.5,2=

Ag/a =A2/a =0.1, 0.2, 0.3, 0.4 for curves 1-4, re-spectively.

2294 BE NDO%, YUKON, AND YI NG

v(r)=vo(e '"—2e '"),m(r) = equ+ moeroe (4. 14)

u=—r —ro,where e& is the static charge. %e here specializeto T = O'K, where the absorption consists of a se-ries of 5-function lines of strengths

n„= (2v(u/ pc) l(o lm(r) ln) l' (4. 15)

at frequencies co = co„. A continuous-absorption co-efficient may be inferred by interpolation betweenpeaks, combined with the ansatz that the integratedabsorption in the nth-phonon regime equals the nth-peak strength.

The frequency dependence of 0. is especiallysimple in the harmonic limit. For m-e ~~", forexample, it is straightforward to show that

diatomic crystals by a Morse n and exponential dis-tortion moment, so that 50

LU

0.0—LL.LLLU

- 5.0—

0 - l0.0—CL.

KO

g) -I 5.0—~oIO -20,0

0.0

, 0.5)

05)

{0.~,0.0 5)l, l

2.0 4.0 6.0 S.O IO.O

F'RE Q UENCY + /co0

a-~y'"/n!, y'=2V, '(,', (4. 16)

n„,-n ((pn &'/2)" ', (4. 1V)

which is again exponentiallike for small $ and n.Anharmonicity is therefore dominant in determiningthe frequency dependence of n for small (&. High-er-order moment terms play a more significantrole, on the other harxi, for large $& and small $.Typical results for n vs co are illustrated in Fig. 6,

Cl

I—UJ

UO

LLIO

with $& measured in units of ao. This represents anexponentiallike dependence for small values of n andy. Anharmonicity provides further channels forabsorption; it is easy to show, for example, thatfor small $ in Eq. (4. 14), the linear moment termcontributes an absorption

FIG. 6. Logic of the absorption coefficient o.' vs dimen-sionless frequency ~/cuc for various choices of ] and ]~of the single-particle model described in the text.

and are observed to display the familiar exponen-tiallike dependence. To investigate the effects ofvariations in ( and (, in more detail, it is useful todisplay the results of exponential fits to n-vs-v cal-culations. This is done in Table I, employing then=3 and n= 6 values, for various choices of $ and

In obtaining these results we have kept ~0 andmo fixed (to values characteristic of ZnTe). Broad-ening and enhancement of e is evident as either $or $, is held fixed, with the other member allowedto increase. It is also evident that anharmonicityis relatively more important for smaller values ofEj, as mentioned previously.

Before proceeding with calculations representa-tive of crystals, it is necessary to determine theparameters entering Eq. (4. 14); we have recently

TABLE I. Exponential fits to ~ vs ~~ for various ~t: and

a{em ) =Ae~"I"c.

Cl0

en. .OVl

0.00 4 F 00L

8.00 12.00 15.00 20 ~ 00 24.00FREQUENC'/

FIG. 5. Logic of the absorption coefficient o' vs dimen-sionless frequency (d/cuc for an Einstein-model correlator;with ASA4e 4=A5A4e"4=0. 1, and A4=0. 2, 0.35, 0.5,0.65 for curves 1-4, respectively.

0, 020.10.63, 00.020.10.63.00.020.10.63.0

0.050.050.050.050.50.50. 50.55.05.05.05.0

7.6x 102.4x 103.Ox 103.2x 103.3x 101.2x 101.4x 105.0x 10l.4x 101.1x 103.2x 102. 0x 107

9.88.85.52. 97.47.05.02. 83.13.12.71.8

10 THEORY OF MU LTI PHONON ABSORPTION DUE TO 2295

TABLE II. Potential and moment parameters forrepresentative crystals.

Crystal

GaPGaAsGaSbAlSbZnSZnSeZnTeNaCI.

KC1KBrLiF

po (10 erg)

6.34.14.2

4. 73.43.64. 20.70.50.51.3

1.07l.251.131.081.311.191.041.191.301.251.26

—l.49—2.44

2 ~ 13—1.95—3.76—3.53—4.69

0.060.030. 040.02

(, (A.-')

0.640.540.520.540.520.500.440.720.860.781.16

indicated how phonon pressure-dependence datamay be employed to aid in this task. The pro-cedures utilized are outlined in the Appendix, andthe results tabulated in Table II for a variety ofrepresentative crystals. Employing these param-eters, one calculates the curves displayed in Figs.7 and 8 for representative semiconducting and ioniccrystals. Also, results for exponential fits be-tween n = 3 and n = 6 are listed in Table ID for avariety of semiconductors. Also of interest is therelative contribution of linear-vs-nonlinear termsin the moment to n, as well as the relative role ofanharmonieity in determining o. To elucidate theseeffects, Table IV lists, for ZnTe and NaCl param-eters, ratios of various matrix elements of m: ex-act to harmonic, linear term to full m, and linearterm to harmonic for the full m (the notation HA

indicates harmonic limit). Clearly, for the pres-ent values of the parameters, the harmonic ap-proximation provides very poor predictions for a.Moreover, the linear approximation to m is inade-quate for semiconductors, especially for u/&uo» l.For ionic crystals, on the other hand, the linea, rterm exceeds or competes with the nonlinear termsfor &o/u&o-l, although the latter always dominatefor sufficiently large values of &u/&uo. It is inter-esting to note that the value of the linear term inalkali halides is suppressed by a cancellation be-tween the e~ and m(y) contributions; moreover, the

e& and exponential terms interfere, causing thebend in n vs ~ in Fig. 8. An additional commentis in order regarding the absolute magnitude of npredicted by the single-particle model. Our cal-culations, based as they are on a single-cell mod-

el, should be enhanced by a factor approximatelyequal to N, , where N, is the coordination number.This is because in the nearest-neighbor (NN) ap-proximations each atom interacts with N, oppositelycharged atoms, rather than just the single oppositesign number within the unit cell accounted forabove.

V. DISCUSSION

ZUJ

LL

LL.

4JC)O

-5.0—Q.

-t 0.0—IX

CA

Cl P

a )50a pp

nS

Ga AsGcSbl, l, I

2.0 4.0 6.0 8.0 I Q. Q I 2.Q

FREQUENC~ aim

FIG. 7. Log~o of the absorption coefficient n vs dimen-sionless frequency u./~0 for representative semiconduc-tors, within the single-particle model.

In this section we outline and discuss some of theprincipal results of this paper. %'e also pursue acomparison with experiment and other theorieswhenever appli cable.

The present treatment provides a formulation ofmultiphonon absorption due to nonlinear momentsin the presence of anharmonicity, based on thebond-moment concept of Flytzanis. ' Rather thanexpanding the moment and potential directly interms of displacements, we have employed a cumu-lant expansion to express e as a functional of aseries in anharmonic correlators of the form(u,u2 ~ ~ ~ u„). This method provides a convenientstarting point for model calculations for crystals;especially simple are results for the harmoniclimit, and for the quadratic (anharmonic} approxi-mation to the cumulant. Applications of the for-malism were illustrated by numerical calculationsin various limits (see Sec. IV), for specific choicesof the moment, potential, and phonon spectrum. Asingle-particle model was required to make exactcalculations feasible, although the crystalline na-ture of the solid is suppressed in the model. In theother extreme of full crystal calculations, only theharmonic-approximation results are straightfor-ward, although use of a quadratic anharmonic ap-proximation to the cumulant is not overly difficultif sufficiently simple phonon models are employed.

An interesting question from a theory standpointis the effect of truncating the cumulant expansion;i.e. , does the cumulant series converge, and if sohow many terms must be retained in a given case.Intuitively convergence should result in cases ofinterest because (u )/ao« l. However, since the

2296 BE NDOW, YUKON, AND YING

DD

DD

~ ~

04

UJD

Ll C4. ,LUCD

D

I

CDC/0CQCC D

DCO ..

I

C)

FIG. 8. Log&{) of the ab-sorption coeff ic ient e vsdimensionless frequency{'j{0 for representativeionic crystals, within thes ingle-particle model.

D

I

DDD

I

D

P4I

0 ~ 00 2 ~ 00 4.00I I I

8.00 8.00 t0 00FREQUENCY m/m

I I

t2.00 14.00

final quantity required is the Fourier transform ofthe exponential of the series, such expectations arespeculative. Maradudin and Ambegaokar'~ calcu-lated the effect of cubic terms in the cumulant onthe neutron scattering cross section for materialssuch as lead, and found them to be small. Asidefrom the calculation being model dependent, the ap-proximations utilized by these authors are of ques-tionable validity for the present problem, where theresponse for &u/&up» 1 is required. Better resolu-tion of the question of the role of higher-ordercumulants must thus await future investigations.

Because the various numerical computations car-ried out in this paper all compromise on one oranother feature of an exact, full crystal calcula-tion, it seems most reasonable to emphasize com-mon trends in the results which may, in fact, be ofmore general validity. Among such observationsare the following:

{a) The absorption is exponentiallike in all eases,at least over frequency intervals spanning severalunits of &u/&uo, with most notable departures fromexponentiality for large anharmonicity.

(b) Increased anharmonieity tends to broaden n,and enhance n atro/&u, » 1. This broadening ap-pears to be significant for typical real crystals.

(c) For very small anharmonicity, the absorptiondue to higher moments is formally nearly equivalentto linear moment anharmonicity absorption, if onesimply replaces the potential z by the moment m .In particular, results of calculations for n„{&u, T)(Refs. 1-6, 22-25, and 32, e. g. ) carry over to thehigher-moment case with just minor modifications.

(d) All else remaining the same, n is enhancedsubstantially for m{r) and v(r) of progressivelyshorter range.

The above similarity in trends within differentmodels may be ascribed to both common featuresin the formalism, as well as in the models them-

Crystal

GaPGaAsGaSbAlSbZnSZnSeZnTe

6.6x 106.2 x.104.2x 104

5.8x 104

1.9x 1051.5x 1052.2x 10'

4. 014. 024. 203.953.704. 014.30

TABLE III. Exponential fits to o vs { for representativecrystals. g(cm ) =Ae

10 THEORY OF MUI TIPHGNGN ABSORPTION DUE TO 2297

TABLE IV. Ratios of matrix elements for Z.nTe andNaCl. A = (0 I (e&/e1/mo —2(&)u ln); P = (0 I e&/e I/mou+ 8 ~&"

I n).

Crystal

ZnTe(e, =0)

NaCl

(f.g/e = 1)

quantumNo. yg

1.00.560.400.310.250.22

1.01.8

11.3—2.6—1.1—0 7

Pm.

9.7 x 102. 06.62. Bx 101.6x 101.1x 10

1.01.50.45.44.6x 103.4x 10

1.01.12.69.04. lx 102.4x 10

1.02.75.01.4x 105.3 10'2.5x 10

selves. In the present approach, because of con-servation oi energy and the smallness of the dimen-sionless phonon parameter Co, the principal con-tribution in the n-phonon regime (cu/~0-n) arisesfrom n-phonon processes. If just j- and A,'-phononsplitting processes are retained in a. perturbationseries expansion for ImX, one finds that the con-tribution of n-phonon processes will vary roughlyas the square of vertices proportional to -(v Co)

'xn(j)m(k), where j+ k=n. Then, for physicallyreasonable choices fur v and m such as employedhere (exponentiallike, say), the weight of n-phononprocesses in ImX will be roughly -(Co/r, )", where

x, is a characteristic-length parameter for theshort-range portions of p and m {v- e " "&, m -e ""&,

with r, -r, -r„say). Thus the over-all decreasein o. with increasing n or + will be nearly exponen-tial. Pathological choices for v and m, or corre-lators calculated from them, ma.y be expected toyield results different from those obtained here, asdemonstrated previously for the anharmonieity ab-sorption problem.

Regarding temperature dependence, although we

have presented explicit computations only for theharmonic limit, it is easy to see that closely anal-ogous conclusions hold under more general condi-tions as well. The convolution expansion for thenoninteracting-cell model, for example, shows thatp„-C". For T —~, C -T; combined with the factor[n(u&)+ I] this leads to n„-7" . One can similarlyshow that anharmonicity tails from other p„againcontribute absorptions -T" . Strong departuresin n from T variation in the nth-phonon regimemay stem from either implicit variations in phonon

frequencies and related parameters, or from theexistence of widely separated peaks in the densityof states, which then lead to a superposition of Tterms with different m's.

The present treatment applies equally well for-mally in the limit of highly covalent crystals, al-though physically appropriate choices for m areless certain for this case. The vector characterof m, as manifest through angle-bending effects,may be especially crucial for elemental crystalssuch as Si, where, as pointed out by Flytzanis, "the component of the dipole moment along the di-rection of the bond vanishes. Thus, spec'al caremust be taken if one is to construct physicallymeaningful models for the moments in such ca-ses. ' For the crystals considered in the presentpaper, on the other hand, the neglect of vector ef-fects is not expected to materially affect the ~ or Tdependence of n, although it may affect the magni-tude of n. The reader is directed to Ref. 12 for aconsideration of some of the modifications neces-sary to account for the vector character of M.

In addition to the general conclusions describedabove, a variety of more specific and/or rnodel-dependent information emerges as well. %e hereturn attention to one particular case which is po-tentially of interest for actual crystals, namely,results of the single-particle calculations of Sec.IVC. Because of the gross oversimplifications ofthe model, as well as some serious uncertaintiesin the values of the parameters it requires, theresults must be viewed as suggestive rather thanconclusive. Among these are the following:

(a) Anharmonicity has a significant effect on mul-tiphonon absorption in typical crystals; for suchcases the harmonic approximation provides an ex-ceedingly poor picture of o. (u, T)

(b) Contributions due to nonlinear moment termsdominate n in covalent crystals. In ionic crystals,on the other hand, linear moment contributionsmay exceed or compete with nonlinear ones, es-pecially for lower values of w/u&0.

Mcoill et al. found that when just quadratic cor-rections to the linear term were employed, the ef-fects on o. for most alkali halides were minor. Thepresent work suggests that at least for sufficientlylarge w/&oo» 1 it will be necessary to retain higher-order terms in m, a.nd that these will affect at leastthe magnitude of 0, . If cancellation effects do notoccur {they in fact do for the present choice of pa-rameters) quadratic corrections may well be suffi-cient for lower values of &/coo. A method for eval-uating the net effect of moments vs anharmonicityfrom experimental spectra at different tempera-tures has been suggested recently by Hellwarth etal. , ' in which sum rules for o. are utilized. On

applying the method to LiF, they find that nonlinearmoments contribute only several per cent to theintegrated absorption. Presumably, the informa-tion extracted from the sum rule may be employedin relating the magnitudes of the linear to nonlinearterms of m, and thus provide a basis for prediction

2298 BE NDOW, YUKON, AND YING 10

of o. in the multiphonon regime.The similarity of the co and T dependence of n for

&u/ufo from both the linear and nonlinear terms in m

makes it especially difficult to distinguish the two

directly in experimental data. If cancellations oc-cur, a careful theory accounting for density-of-states effects and selection rules may enable theanalysis of spectra for ur/uo-2-3. Otherwise thelargest differences are anticipated for &u/&uo» 1, , aregime where n (typically &10 cm ) is often ei-ther unmeasureably small, or masked by impurityabsorption (due to residual impurities in what nor-mally would be considered ultrapure material).The various general properties of o. {&u, T) discussedabove have been extensively verified for a varietyof ionic crystals (see, for example, Refs. 38 and

39 for +, and 40 and 41 for T dependence. Unfor-tunately, no useful data appear to be available for&u/&uo» 1 for semiconductors or covalent crystals.A more useful evaluation of the theory, and the pos-sibility of more realistically assessing the roles oflinear vs nonlinear moments in covalent crystals,awaits the completion of such experiments.

APPENDIX: DETERMINATION OF PARAMETERSIN THE MOMENT AND POTENTIAL

We here describe a simplified set of prescrip-tions for determination of the parameters eo, $,mo, and $, of Eq. (4. 14). A total of four equationsare required, assuming all the other parametersare known. One relation for (vo, $) stems from thethermodynamic condition relating the compress-ibility to the potential, namely,

, 3 dnE(p11+ 2p12) 6 (A3)

in conjunction with a simplified model for the lat-tice susceptibility, namely,

X, =(e$'/pv, )(~'0 ~') '(A4)

Then use of HM's theoretical expressions lead di-rectly to

1 d(do K II I= —b, 2 [rov (ro) + 2v (r,) —2v (ro)/ro],o dI' 3pcoo

= —b~ ,'Kept@ (x-o)+ 2v2 (7p) —2m(ro)/ro],

where b, = 1 for rocksalt and —,' for zinc blende, and

52= 1/ef for rocksalt and —', e~r for zinc blende.Combined with the relation

ceeding in this fashion nevertheless, one obtainswith HM

e,*=e+ 2[m'(ro)+2m(ro)/ro] {rocksalt ionic),

e$3 [m'(ro) + 2m (ro)/r~) (A2)

(zinc-blende semiconductor),

where e~ is the Szigeti charge.HM calculate theoretical expressions, within the

nearest-neighbor approximation, for stress- inducedcharges in refractive index stemming from latticevibrations. We employ, in addition, the consisten-cy relation between the elasto-optic constants P;&and dn/dI' for cubic crystals, namely,

E = v,v "(rp) . (Al)(A6)

Another is forthcoming from the relation betweenthe ionic charges and the moment. FollowingHumphreys and Maradudin (HM), ~ we choose e, =e,the electronic charge, for ionic crystals, and e& = 0for covalent crystals, such as semiconductors.This is obviously a very crude approximation, ap-propriate just for qualitative investigations. Pro-

where e indicates electronic values, andy„o and

pro are mode Gruneisen parameters, Eqs. (Al ),(A2), and (A5) enable the determination of vo, $,mo, and $& for crystals where pressure data on TOand LG phonons are available, in addition to theother standard data. Calculations for representa-tive crystals are given in Table II of the text.

*Research supported by Air Force Cambridge ResearchLaboratories, AFSC, under Contract No. F19628-71-C-0142.

4A. P. Sloan Foundation Research Fellow.tSupported in part by NSF.M. Sparks and L. J. Sham, Solid State Commun. ~ll1452 (1972); Phys. Rev. B ~8 3037 (1973).

B. Bendow, S. C. Ying, and S. P. Yukon, Phys. Rev.B ~8 1679 (1973); B. Bendow, Phys. Rev. B 8, 5821(1973).

T. C. McGill, R. W. Helwarth, M. Mangir, and H. V.Winston, J. Phys. Chem. Solids 34, 2105 (1973).D. L. Mills and A. A. Maradudin, Phys. Rev. B ~8 1617(1973).

K. V. Namjoshi and S. S. Mitra, Phys. Rev. B 9, 815

(1974).6H. B. Rosenstock, J. Appl. Phys. 44, 4473 (1973):,

Phys. Rev. B 9, 1973 (1974).See, for example, the Proceedings of the 1972 and 1973Conferences on Electronic Materials, in J. Electron.Mater. 2 (1972); 3 (1973).

Both higher-order moments and anharmonicity have beenconsidered at frequencies near and below the fundamen-

tal, where contributions from just a few phonons (1-4)dominate the absorption. See, for example, J. S. Lan-ger, A. A. Maradudin, and R. F. Wallis, in LatticeDynamics, edited by R. F. Wallis (Pergamon, New

York, 1965).~M. Lax and E. Burstein, Phys. Rev. ~97 36 (1955); B.

Szigeti, in Lattice Dynamics, edited by R. F. Wallis

10 THEORY OF MU I TIPHONON ABSORPTION DUE TO

(Pergamon, New York, 1965).The advantages of correlation-function techniques fortreating higher-order moment absorption have beennoted previously by B. Bendow and S. C. Ying, Phys.I ett. A ~42 359 (1973).C. Flytzanis, Phys. Rev. Lett. ~29 772 (1972).C. Flytzanis, Phys. Rev. B 6, 1264 (1972).M. Born and K. Huang, 17yncmica/ Theory of CrystalLattices {Oxford U. P. , Oxford, England, 1954).For ~ ~ 0 a constant moment Mo may be added to Iwithout affecting the results. Choosing Mo as the nega-tive of the moment when the ions are in their equilibriumpositions provides an interpretation of X as the contribu-tion to the susceptibility due to vibrations of the ionsabout equilibrium.A. A. Maradudin, in 1962 Brandeis I ectures in Theo-retical Physics (Benjamin, New York, 1963), Vol. 2,Chap. VIII.A. A. Maradudin, in Solid State Physics, edited by F.Seitz axd D. Turnbull (Academic, New York, 1966),Vols. 18 and 19,

~Umklapp terms are omitted.See, for example, the papers by Cowley and Bilz, inPkonons, edited by R. %. Stevenson (Plenum, New

York, 1966).The simplicity of the present results is a consequenceof both the quadratic approximation and the Gaussianform for m~. When terms beyond quadratic in the cumu-lant expansion are retained, or if m, has a more com-plicated functional form, then a laborious numericalevaluation of the q integrals may be required.Of course, if one is willing to compute the p's numeri-cally then such simplifications are unnecessary.B. Bendow, Phys. Rev. B ~8 5821 (1973).B. Bendow, Appl. Phys. Lett. 23, 133 (1973).M. Sparks and L. J. Sham, Phys. Rev. Lett, . 31, 714{1973}.

24A. A. Maradudin and D. L. Mills, Phys. Rev. Lett.31, 718 (1973).

~T. C. McGil. l and H. V. Winston, Solid State Commun.13, 1459 (1973).B. Bendow and P. D. Gianino, Opt. Commun. 9, 306(1973).

YL. B. Humphreys and A. A. Maradudin, Phys. Rev. B6, 3868 {1972}.See, for example, S. S. Mitra, in Optical Properties ofSolids, edited by S. Nudelman and S. S. Mitra (Plenum,New York, 1969).P. C. Kwok, in Solid State Physics, edited by F. Seitzand D. Turnbull (Academic, New York, 1967), Vol. 20.

30A. A. Maradudin, K. W. Montroll, G. H. Weiss, and I.P. Ipatova, Theory of Lattice Dynamics on the Har-monic Approximation, 2nd ed. (Academic, New York,1971).Such a potential, neglecting long-range effects, is mostmeaningful for higher-order terms involving g»1 de-rivatives of v.S. P. Yukon and B. Bendow, Opt, Com. mun. 10, 53(1974).

3 B. Bendow, P. D. Gianino, Y. F. Tsay, and S. S.Mitra in Conference on IR Lase~ ~indo~ Materials1973, edited by B. Bendow and C. A. Pitha {AFRCL,Bedford, Mass. , 1974), Vol. I.

34A. A. Maradudin and V. Ambegaokar„Phys. Rev. 135,1071 {1964).

3~One serious deficiency is the use of the nn approxima-tion, since the predicted m and v are in some cases oflonger range than would be necessary to unambiguouslyjustify such a procedure. Among many other uncer-tainties are those related to the choice of the staticcharge and effective field, expecially for covalent crys-tals. We note that Flytzanis (Ref. 12) chooses eq je =1for III-IV s in his treatment, while we have here em-ployed the HM choice of ej/e = 0.

36R. W. Hellwarth, in Laser Induced Damage in OpticalMaterials, 1973, edited by A. J. Glass and A. H. Guen-ther (U. S. GPO, Washington, D. C. , 1973).

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