PHYSICAL REVIEW 8 VOLUME 47, NUMBER 11 15 MARCH 1993-I

Local-fieM correction for nonlinear optical coefficients

S. T. Chui and Hong-ru MaBartol Research Institute, University ofDelaware, Newark, Delaware 19716

R. V. Kasowski and William Y. HsuCentral Research and Development Department, E.I. du Pont de Xemours c% Company,

Experimenta) Station, 8'ilmington, Delaware 19898(Received 19 September 1991;revised manuscript received 30 March 1992}

We formulate the nonlinear optical coefficients in terms of the Matsubara Green's function and itsspectral representation. From this a formula for the ab initio calculation of the local-field effect is de-rived.

I. INTRODUCTION

Nonlinear optical materials are of obvious technologi-cal importance, yet there have been few ab initio calcula-tions in this area until recently. ' Pong and Shen com-puted the nonlinear optical coefficients of GaAs by theempirical pseudopotential technique. They found a non-linear optical coefficient smaller than the experimental re-sult by a factor of 20 in the low-frequency limit. They at-tributed this discrepancy to a local-field correction. Forionic systems, an estimate of this correction is([e(co)+2]/3) where e is the dielectric constant. Butthis correction factor may be inaccurate in strongly co-valently bonded material such as GaAs, where the elec-tronic bands exhibit strong dispersion. In recent ab initiocalculations of the nonlinear optical coefficients ofGaAs, ' reasonable agreement with the experimental re-sults was obtained and the local-field correction wasfound to be 3% with the present technique and 8% withthe local-density approximation (LDA), which is muchless than the factor of 20 based on simple estimates.

Some of the important nonlinear optical materials(KTiOPO4, LiNbO3) also exhibit ferroelectricity. The fer-roelectricity is believed to be due to a large local electricfield together with a large susceptibility of the individualions. It is thus important to understand the effect of thelocal field on the nonlinear optical properties and not rely

I

on the above dielectric constant estimate.Levine and co-workers' have discussed the calcula-

tion of the local-field correction for the nonlinearceefficient in a self-consistent LDA one-electron picturefor a scalar external potential. In this paper we formulatethe nonlinear optical coefficients in terms of the Matsu-bara Green's function and its spectral representation.From this a formula for the ab initio calculation of thelocal-field effect is derived. Gygi and Baldereschi sug-gest that in the LDA treatment of screening, the long-range Coulomb interaction is replaced by a short-rangeone and that may be the cause of the discrepancy betweentheory and experiment in the magnitude of the band gapfor insulators. The present approach to calculating thesusceptibilities does not require such a replacement.

II. LOCAL-FIELD EFFECTS IN LINEAR RESPONSE

Soon after the study of the random-phase approxima-tion for the electron gas, Ehrenreich and Cohen pro-posed a particularly simple way to rederive the resultfrom a self-consistent treatment. Adler' followed thesame line of reasoning and introduced the effect of a crys-tal lattice. He considered screening of an external ex-ponential periodic electromagnetic field so that theresponse involves wave vectors that are multiples of thereciprocal-lattice vectors Cx. The generalized dielectricconstant is given, in the random-phase approximation, by

4rre~ (j exp[i (G+q)q r] ~k ) ( k ~exp[i(G'+q) r] ~j )(f fk)—/G+qf /G +q/, .„&+E,E„—

The inverse of this matrix cannot be obtained analyticallyin general. In the tight-binding approximation, this in-version can be carried out in the following manner. Thesingle-particle wave functions in this approximation arewritten as ~f ) =+Rexp(ik R)pf(r —R), where pI isassumed not to be a function of k. The dielectricmatrix can then be written in terms of the Fourier trans-form of the products of P defined as g(k)= J d r P;(r)exp(ik r)P&(r). More precisely,

e(q+ G, q+ G') = l —g (G+ q)g *(G'+q)g(q),

where y(q)=4~e g, t, (f, ft, )/q (co+E, Eb), —where-a, b are state indices. This form is separable. Its inverseis given by

g (G+q)g*(G'+q)g(q)l —+ I

(G"+q) I'y(q)Qll

Adler exploited this fact by writing the inverse dielec-

47 6293

6294 CHUI, MA, KASOWSKI, AND HSU

tric constant first in a real-space representation and thenin a multipole expansion. In this way he showed that theinverse dielectric function possesses a Lorenz-Lorentzcorrection as

1/e '(0, 0) = 1+4lr(a —C, )/[1 —3lr(a —C, )/3],where a is the polarizability and C, is a self-polarizationcorrection. This makes a connection with the more tradi-tional understanding of the local-field correction.

III. MATSUBARA GREEN'S-FUNCTIONFORMULATION FOR THE NONLINEAR

COEFFICIENT

There has been some discussion of the calculation ofthe nonlinear optical coefficients. "" These have notbeen carried out in the language of modern many-bodytheory so that the actual computation would be cumber-some. In addition, these works do not allow for easy in-corporation, in a systematic manner of many-body effectssuch as the local-field correction or the excitonicCoulomb corrections in two-photon absorption. It isthus important to clarify the fundamental formalism fornonlinear response functions. More precisely, in analogy

I

q=0

(b)

FIG. 1. (a) The diagrams for the nonlinear optical coe%cient.(b) The diagram showing the screening of the external verticesthat lead to the local-field correction.

to the calculation of the linear-response coefficients, thenonlinear-response function y'„b defined as the ratio ofthe polarization to the second power of the electric field,(j, /(El, E, )) /iso for an exponential external field is, insecond-order perturbation theory, equal to

t(1/ill) f dt, f dt2expial(2t —t, t2)({[j,(—0),j„(t,)],j,(t2)] &c /co +a~b

for incoming frequencies co and outgoing frequency 2'.We add another term with the subscripts a and b inter-changed. Here the average is taken with respect to theexact ground-state wave function. After the commuta-tors of the current operators are expanded and the timeintegrals are carried out, this becomes identical in formto the standard formula" except that the single-particlestates in those formulas are now interpreted as exactmany-particle states.

To develop a systematic expansion, we first relate thisnonlinear-response function to the analytic continuationof a Matsubara Green's function g~ defined as a time-

ordered product

7

(c /iso )f dr, f dr2(Tj&(r, )j,(r2)j, (0))+a~b .0

Here P is the inverse temperature. The first-order term ofthis formula involves only one diagram illustrated in Fig.1(a) and corresponds to the eight diagrams introduced byYee and Gustafson. '

Indeed, carrying out the time integration and express-ing the intermediate states in terms of the exact eigen-states of the Hamiltonian, the Lehmann-Symanzik spec-tral representation of gM can be written as

f. g- f.—,( l &2+ &nn )( l &1+ l &2+ &gn ) (i CO2+ mn„. )( l m l

—mg„)(2)

al is the incoming frequency, and f„ is the occupation number of the state labeled by n. (j, ) „ is the matrix element ofthe ath component of the current matrix element between the states g and n. On setting ico2=2co and iso&= —co, thisform is identical to the nonlinear-response function first derived by Butcher and McLean, "

1x,".I, =, , Jim[(J. ).,(jb),.(J, ).. 1~ VA 4,g„

f. f, f.—f, —+a~b,

( 2' + co„„~) ( ci) +cog„) (2' +co„„)( co cog„)(3)

where V is the volume. The detailed identification is il-lustrated in Table I where we make an explicit connec-tion with formula (2.17) of Ref. 2 if we identify a with k,b with j, and c with i. Thus the analytic continuation of

I

the Matsubara Green's function indeed becomes the non-linear response function y

In writing this formula, we have followed Aspnes' andexploited time-reversal invariance. By explicitly display-

47 LOCAL-FIELD CORRECTION FOR NONLINEAR OPTICAL. . . 6295

ing the subtraction of the occupation number of thebands, the number of terms that needs to be summed andthe computation time is reduced.

At zero temperature, f can only be 0 or 1; whenI

f„Wf„, only one of the two terms of the above sum isnonzero, the denominator of the above formula is finitefor 2' smaller than the band gap. When f„=f„., theabove formula can be simplified as

(f„fg)(f—„fg)Im—[(j,)„g(jb )g„(j,)„„]/[(ei+~„)(ei ~ „)].

Hence the denominator remains finite.

IV. LOCAL-FIELD CORRECTION

The local-field correction results from polarization of the neighboring atoms. This polarization can be represented asa screening of the external field by the electrons which now form bands. The screening of an external electromagneticfield involves wave vectors that are multiples of the reciprocal-lattice vectors. This is represented by Fig. 1(b) where thewiggly line represents the bare electromagnetic coupling with the wave vector as indicated and the solid lines representelectron propagators for the intermediate states. For the nonlinear optical coeScient g, we follow an identical pro-cedure for the coupling to the external electromagnetic field indicated by the double solid line in Fig. 1(a).

The final result is summarized in Eqs. (12), (16), and (18). One can think of this correction as replacing the currentoperator in Eq. (3) by a screened one given by [1—+by, b(G, G')Rb, (G')] 'j(G), where

(nj~, exp[iG r]~g ) (g ~ jbexp[ i G'.r'] ~—n )(f„fg)—y,b(Cx, G')=pe /m A'V

ng- co+E„—E (4)

R,b(G=O) =0, R,b(G, co) = —4irG, Gb/G for G&0. We now describe our results in detail.The macroscopic field, which is the average of the microscopic field over a region large compared to the lattice space,

is in general different from the local field which actually acts on electrons. The microscopic field A(r) in the crystal isthe sum of applied field A'"' and the induced field A'":

A(r) = A'"'+ A'" =g A(G)exp(iG r),G

Aext( G ) Aext5Ci, o '

The applied field is slowly varying over the crystal unit cell, but the induced field is rapidly varying. The Fourier corn-ponents of the rapidly varying induced current, j(Cx), can be written in terms of susceptibilities E as a power series in

A(Cx) from straightforward perturbation theory as

j,(Cx)=gg K„(Cx, G')A, ( G') + g gK,',bt(G;G, , Gi)A, (G, )Ab(G2) . (6)G' a Gl, 02 ab

K(G, Cx') and IC' '(Cx,'Cx„Cx2) are derived in the Appendix and the result is

(p, )„,(k, k, G)(pb ),„(k,k —Cx')K,b(Cx, Cx', )= gq„„(k,k, G —G')f„5,b+ gg (f„f,), —

4mcV „z"" ' ' ' m2iricV „„k co —E„(k)+E„(k)

Kt2)(G G G 2 )e c ng» qgn» 1 2 n g

3 (p ) (k, k, Cx) (k, k, —Cx —G )(f f)—cab & I & 2& ab

Pl 'ACng

CO COng

e 5„q„(k,k, Cx —Cx, )(pb) „(k,k, —Cx2)(f„f)—2m fiC V ng k CO COng

e 5,b q„g(k, k, G —Gz)(p, )g„(k,k, —G, )(f„f)—2m'A'c'V

+I2m 'fi'c'V „g„(2~+~„„)(ei+~g„) (2~+~„„)(ei cog„)—

X Im[(p, )„g(pb ) „(p,)„„.]+a~i, (8)

where the matrix elements are

6296 CHUI, MA, KASO%SKI, AND HSU 47

p„„(k,k', G)= ——. d r [Vg„„(r)g„„(r)—g„k(r)VQ„ i, (r) ]exp( —i G r },2l

q„„.(k, k', G) =Id'r P„„(r)g„„(r)exp( i—G r)..

Our goal is to express the rapidly varying high Fouriercomponents in terms of the slowly varying macroscopicfields and currents at G=0.

The field in the crystal is related to the induced currentby Maxwell equations in the gauge ((}=0(E= —8, A/c). ' From the continuity equationi~V E=4eV.j; we have

G, Gbg = —4m.ab G2 2

and the prime indicates that the term with G=O is ex-cluded from the sum. Define R (G) =g(G)co, R (0)=0,

y (co) =cK/co

A, (G)= A, (0)5G 0+ g'g„(G }j,(G )/c,

where

(9) P(2) ic lC(2)( )CO

Substituting Eq. (9) into Eq. (6), we have

2CO

3

j,(G)=g y„(G,O)A, (0)+gg 0 (G, G')R,b(G')jb(G')+ g g . y'„b(G;G„G2)A, (Gi)Ab(G~) .a G' ab Xea GG ab

Combining terms proportional to the current j on both the left- and right-hand side of the equation, we have

2 3

ge,"~'(G, G')Jb(G') = gy, b(G, O) A„(0)+ 2 gy, 'b, '(G;G„G2)A„(G, ) A, (G~),b C C be

where

Kgb (G& G') =5,b5oc" XZ„(G—,G')R,b(G')

G,'Gb=5,b5GG +4~+y„( G, G')

GI2(12)

Note that t' ' is related but not equal to the dielectric constant, which is ordinarily defined to be a scalar in terms of the

density-response function. Multiplying by (e'") ', we get

CO3

Ja(G) X X( )ab (G&G )+bc(G &0)bc Ac(0)+ . p(e )ab (G&G )+bed(G &G &Gi2)bed Ac(G1) Ad(G2)

G' be 1c(13)

One can also express A(G) in terms of A(0). To first or-der in A from Eq. (6) we get

A(G) =(e"')-'(G,O) A(O), (15)

j,(G)=g+ IC„( GG') A, ( G') .G' a

Substituting this in Eq. (8), we get

(14) where

e',„'(G,G') =5,b5oG gR„(G )y—,b (G, G')

TABLE I. Translation table for the identification of Eq. (3)with corresponding terms in (2.17) of Ref. 2. The first and thelast columns map the corresponding terms in the sum that areequal. The middle column shows the renaming of the summa-tion labels that makes the identification explicit.

1 n'~g 32 n '+-+n 73 n ~g, n'~n, g ~n' 1

4 n ~n 5

G,G,=5,b5oG+4mg y,b(G, G') . (16)G2

In contrast to e'" where the G appears on the right, theG now appears on the left of y . Substituting Eq. (15)into (13) we finally arrive at the susceptibilities for boththe linear and the nonlinear response as

(17)G, e

and

LOCAL-FIELD CORRECTION FOR NONLINEAR OPTICAL. . . 6297

G&G&G&, c'a'b'(& ) (06/) y ~ b (G, , G2, G3) ~ b

a'a( 2&0)a'a( )b'b(63~0}b'b

(l8)

Define the Fourier transform of the current as

j(6)=—f1 r j(r)exp( —iG r),1

V

j(r)=gj(6)exp(iG r) .0

Equation (18) is the main result of the present paperCalculations using these results suggest that the local-field corrections are small for GaAs. Similar calculationsfor KTP are currently underway.

APPENDIX

The Hamiltonian of a solid system interacting withclassical electromagnetic field described by vector poten-tial A can be written as

H Ho+H

where

Ho =QE„(k )a„ka„k

Using

4(r) =gf„z(r}a„i,n, k

the current operator can be written as

where

X a„ka„k.A(G'),

j(G)= — g p „,(k, k', 6)a„„a„.„.mV ~

2

g'q„„(k,k', G —6')2mcV

and

n, k p„„(k,k', G)= ——.f d »[VS„'k(r)P„.k.(r )l

H, = fd r 0'(r) p A+Ap+ —A %(r).2mc C

Integrating by parts, we have

T

H, = d r 4 (r)A —VV(r}2mc l

—A. —%(r)VV (r)l

+ —A 4 (r)%(r)c

31.r—jr -ArC

Here, the current operator is

j(r) = . [0'(r)V'P (r) —'P (r)VV(r)]2ml

2

[A+t(r)+(r)] .2mc

Xexp( iG r—),q„„(k,k', 6)=fd'» g„k(r)P„„(r)exp( —iG r) .

One can divide the j into an ordinary and a diamagneticterm as

ji(6)= — Q p„,.(k k' 6)aagaa'kk, n'k'

2

j2(G+6')+2 g q„„(k,k', 6+6')a„ba„b

2mC nk n'k

Hi can be written as

H, = ——gj, (6).A( —6)1

+ g j2(G+6'}A( —G) A( —6'}G, G'

Standard linear-response theory gives us the response ofcurrent j(6) to the field A(G) to the order of A as

A(G~ai) =++&ca(6~6 ~~) ~ (6 ~aalu)+ g +&cab(G&GI&62&ai) a 61&p b 62&

G' a G)G~ ab

where

K(6,6', co) = f d&K(6, 6', r)exp(i'm),

(6;Gi 62 2') =fd&id&2& (GiGi 62 &i & 2)ex[&p~ & (i&+]2)

The K s are given by correlation functions ofj i and jz,

6298 CHUI, MA, KASOWSKI, AND HSU 47

K.„(G,G', t t—')= —&j,(G —G', t))5.b5(t —t')+ 8(t —t')& [j,.(G, t),j»(G', t')]),

c2i8(t t,—)5(t, —t, )

K,',b (G;G„G2, t t—„t t ~ )—= & [J&, (G, t),J i( —G, G—2, t, ) ] )5nbVc

8(t t—, )8(t, t, —)

& [[A,« t»A. «i ti)) jib(G2 t2)I &

Vc

8(t t, }—8(t, —t, )& [[A,« t»A. «2 t2)] Jlb(G1 tl)I ~

Vc

i 8(t —t, )5(t t, )5—„+ ' ' "&lj2(G Gl t) jib( G2 2)]~

Vc

i 8(t t, )5(—t —t, )5,„+ '

V' '"&[j(G—G t) J.( —G t )])

Vc

Since the response functions are retarded Green's functions of current operators, it can be calculated through an analyt-ic continuation of corresponding Matsubara functions. The latter is easy to evaluate and can take advantage of themodern many-body theoretic techniques. The anal results of K's are

p„,(k, k, G)p,„(k,k, —G')K(G, G', e)= gq„„(k,k, G —G')f„+ $g (f„f,), —

(p, )„„(k,k, G)q,„(k,k, —G, —Gz)(f„f„)—nu k CO COn„

e'5„ q„„(k,k, G —G, )(pb),„(k,k —G, )(f„ f,)—

(A1)

e'5, b q„„(k,k, G —Gi)(p, ),„(k,k, —Gi)(f„f,)—2m Pic V

e fn' fg fn fg 1~[ Pa n'g Pb gn Pc nn ]'+a~b .4m fi c V„„(2e~+ ')(~+eign ) '(2~+~nn')(~ ~gn )

Z. Levine and D. C. Allan, Phys. Rev. Lett. 66, 41 (1991);H.Zhong, Z. H. Levine, D. C. Allan, and J. W. Wilkins, ibid. 69,379 (1992).

Hong-ru Ma, S. T. Chui, R. V. Kasowski, and William Hsu,Opt. Commun. 85, 437 (1991).

3M.-Z. Huang and W. Y. Ching, Phys. Rev. B 45, 8738 (1992).4E. Ghahramani, D. J. Moss, and J. E. Sipe, Phys. Rev. B 43,

8990 (1991),and references therein.5C. Y. Fong and Y. R. Shen, Phys. Rev. B 12, 2325 (1976); see

also C. Willand, Opt. Commun. 57, 146 (1986).Y. R. Shen, Non-Linear Optics (Wiley, New York, 1984).

7Z. H. Levine and D. C. Allan, Phys. Rev. B 44, 12 781 (1991).

SF. Gygi and A. Baldereschi, Phys. Rev. Lett. 62, 2160 (1989).9H. Ehrenreich and M. Cohen, Phys. Rev. 115, 786 (1959).~ D. Adler, Phys. Rev. 126, 413 (1962).I P. N. Butcher and T. P. McLean, Proc. Phys. Soc. London 81,

219 (1963); Ma, Chui, Kaskowski, and Hsu (Ref. 2), Eq.(2.17).

~ T. K. Yee and T. K. Gustafson, Phys. Rev. A 18, 1597 (1978);see Huang and Ching (Ref. 3), Fig. 2.2.D. E. Aspnes, Phys. Rev. B 6, 4648 (1972)~

See, for example, J. D. Jackson, Classical E1ectrodynamics, 1sted. (Wiley, New York, 1972), p. 183, Eq. (6.53).