Original Papers phys. stat. sol. (b) 84, 417 (1977)

Subject classification: 14.3.4 and 20.1; 13.1; 22.2; 22.4

Institute of Physics, State Committee for Science and Technology, Hanoi

Interband Electronic Raman Scattering by a Three-Band Semiconductor in the Presence

of a Resonant Electromagnetic Field BY

NGUYEN VAN HIEU and NGUYEN AI VIET

The influence is investigated of a resonant electromagnetic field on the interband electronic Raman scattering of a three-band semiconductor with direct band gaps. Explicit expressions for the cross-sections are given. The form of the Raman spectra depends on the effective masses of electrons and holes.

Der EinfluB eines elektromagnetischen Resonanzfeldes auf die Interband-Raman-Streuung im direkten Dreiband-Halbleiter wird untersucht. Explizite Ausdrucke fur die Wirkungsquerschnitte werden angegeben. Die Form der Raman-Spektren ist abhangig von den effektiven Massen der Elektronen und Locher.

1. Introduction

In the presence of the intense electromagnetic field of a coherent light beam the energy bands of elementary excitation as well as the optical properties of semiconduc- tors must be changed. It was shown by many authors that due to the resonant inter- action of the electromagnetic field there arise a series of narrow band gaps in each energy band of the semiconductor. Now the optical phenomena are connected with the transitions of new elementary excitations. Galitskii et al. [ 11, Gorelavskii and Elesin [4], Elesin [5], Krokhin [GI, Klimontovich and Pogorelova [7], Perlin and Kovarskii [8], Yacobi [9], Becker et al. [lo], Ariutiunian et al. [ l l] , and the present authors [ 121 have studied the absorption spectra and the dielectric constant of several kinds of semiconductors in the presence of a resonant intense electromagnetic field. The Green function approach in [lo] permits to include both the dynamical Stark effect and the Franz-Keldysh effect. In this paper we consider the influence of the resonant intense laser field on the interband electronic Raman scattering by a three- band semiconductor of the same type as that studied in [8, 101. Our results can be applied to many Arrl-BV semiconductors. We suppose that the valence band “b” and both conduction bands “a” and “c” are parabolic with effective masses mb, ma, and m,, respectively, and have extrema a t p = 0. Denote by d a b , d,,, and d b c = d a b f d a c

tE r--- ,

s2

I

Fig. 1. Representation of energy bands

418 NGUYEN VAN HIEU and NGUYEN AI VIET

the distances of their extrema. By assumption the transitions a - b and a - c are allowed. We know that for this semiconductor the interband electronic Raman scatter- ing with the transition of electrons from the valence band to the conduction band “a” is forbidden, and that with the transition to the conduction band “c” is allowed. We show that together with the change of the forni of the cross-section for the allowed transition the resonant interaction of the intense electromagnetic field induces also some new Raman lines.

2. Matrix Element

We put the energy of the electron a t the top of the valence band to be zero. Then the energies of holes and electrons equal

Suppose that the angular frequency 9 of the radiation field equals the distance A,, = Abc - d a b between the extrema of the two conduction bands. Due to the reso- nant interaction of this field with the electrons there arise new elementary excitations with quasi-energies depending on the strength of the field. The Hamiltonian of the system of electrons and holes interacting with the resonant radiation field is

H, = z {-%(PI a;% + %(P) b;bP + EAP) C&P + 4 P ) [C&P ,-isat+ .;cp eiQtl> , (1) P

where u p , c p , and b, are the annihilation operators for electrons in the two bands and for holes, A(p) is proportional to the scalar product of the amplitude of the vector potential and the interband matrix element of the momentum -iv. By means of the . ” Gnitary transformation

int z (a$ap - b:bp -c,’cp)

U = e p

and a subsequent Bogolyubov transformation

1 aP = UPaP + VPYP , c p = - 7 1 p a p + u p y p >

6, = B, we rewrite the Hamiltonian in a t-independent diagonal form :

To study the electronic Raman scattering by new elementary excitations with anni- hilation operators a p , / I p , y p and quasi-energy spectra (5) we apply the perturbation theory to the interaction between this system of quasi-particles and a second electro-

Interband Electronic Raman Scattering by a Three-Band Semiconductor 419

magnetic field with the vector potential

+ h.c. (7 ) A(t) = El e - id + & e-W

Denote by nnn'(p) the matrix element of the momentum -iv between the states of the two bands n and n'. We have

+ e [&,naC(p) + c~a ,nca(p ) ] A - e [b'i&hCb(p) + cpb-,nbc(p)] A } . ( 8 )

where at small values of p

420 NGUYEN VAN HIEU and NGUYEN AI VIET

I n the case of a very weak resonant electromagnetic field A(p) 3 0 the matrix element (9a) becomes negligible, while the matrix element (9b) tends to the limit

(cbl f l 10) = - 2nie2 6 [ E b ( P ) + E,(P) + W2 - Wl] MbC(P) , (13)

which is the matrix element of the conventional allowed interband Raman scattering considered in many earlier papers [13 to 151.

3. Cross-Sections From equations (9) to (12) for the matrix elements it follows that the cross-section

can be divided into two parts:

do do, do- dw, dw, dw,

+-* _. = __

where the first part

is the new form of the conventional allowed interband Raman spectrum a t frequencies

gives a new Raman spectrum at higher frequencies

wz M w1- ( d a b - fz) = d b c + 252. In the limit of a very weak resonant radiation field the first part equals the conven- tional allowed Raman cross-section do,/do,, while the second part vanishes :

do, 'do, do- - " - 9 - - t o . dw, do, dw,

The explicit expressions of the cross-sections (15 a) and (15 b) depend on the relations between the effective masses ma, WL,,, mc. To formulate these relations we define new parameters p, WL, and Q :

Interband Electronic Raman Scattering by a Three-Band Semiconductor 421

For convenience, we introduce also some notations : Jf+(P) = J f P ( P ) 7 Jf-(P) = J f y Y P ) 7

N + @ ) = J f ! Y ( P ) 7 #-(PI = J f B Y ( P ) 7

"+ = "1 - (u2 f a) 7 = dab > A = IA(o)l,

4. Discussion The form of the scattering cross-sections is determined by the kinematical factors

f(x), g(x) and the matrix elements M&), N&), and is sensible to the values of the parameters p and e of the semiconductor. The factors f ( x ) and g(x) are functions of the frequency difference o1 - w2. They have poles if /el < 1, but they are smooth func-

Fig. 2. Dependencef(s) and g(z) on o+. f(z) for -1 < e < 0, g(z) for 0 < e < 1

I y 2 ' 2 x - *

422 NGUYEN VAN HIEU and NGUYEN Ar VIET: Interband Electronic Raman Scattering

tions if depend explicitly on the frequencies w1 and coZ. For some intervals of cool they have poles a t co2 == 52, SZ f 2A. In Fig. 2 we represent the dependence of f ( x+) and g ( x k ) on w+, and in Fig. 3 we plot the cross-sections in some special cases. From these results it follows that the experimental study of the interband Raman scattering in the presence of the resonant radiation field would provide useful information on the higher conduction band “c” of the semiconductor.

Our reasoning can be applied to many zincblende semiconductors A1I1-BV and AI1-BVr, for which the valence band “b” and the higher conduction band “c” are F15, while the lower conduction band ‘(a” is rl. Among these materials those with 52 < A would be more favourable, since for them the laser light with frequency SZ cannot excite the electron-hole pairs. An example of them may be GaP with

52 = 2.1 eV ,

> 1. The matrix elements M,(p) and N + ( p )

A = 2.8 eV . References

[l] V. M. GALITSKII, S. P. GORELAVSKII, and V. F. ELESIN, Zh. eksper. tcor. Fiz. 5i, 207 (1969). [2] V. D. BLAZHIN, Fiz. tverd. Tela l i , 2325 (1975). [3] Yu. I. BALRAREI and E. M. EPSTEIN, Fiz. tverd. Tela 15, 925 (1973); 17, 2312 (1975). [4] S. P. GORELAVSKII and V. I?. ELESIN, Zh. eksper. teor. Fiz. 10, 491 (1969). r5] V. F. ELESIN, Fiz. tverd. Tela 11, 1920 (1969). [6] 0. N. KROXHIN, Fiz. tverd. Tela 7, 2612 (1965). [7] Yu. L. KLIMONTOVICH and E. V. POCORELOVA, Zh. eksper. teor. Fiz. 51, 1722 (1966). [S] E. Yu. PERLIN and V. A. KOVARSKII, Fiz. tverd. Tela 12, 3105 (1970). [9] Y. YACOBI, Phys. Rev. B 1, 3105 (1970).

[lo] L. BICKER, 12. ENDERLEIN, and K. PEUKER, phys. stat. sol. (b) 60, 579 (1973). [ll] S. L. ARIUTIUNIAN, E. M. KAZARIAN, and G. P. MINASIAN, Fiz. tverd. Tela 18, 2568 (1976). [12] NGUYEN VAN HIEU and NGUYEN AT VIET, to be published. [13] S. S. JHA, Nuovo Cimento 63B, 331 (1969). [14] E. BURSTEIN, D. L. MILLS, and R. F. WALLIS, Phys. Rev. B 4, 2429 (1971). [15] S. SRIVASTAVA and K. ARYA, Phys. Rev. B 8, 667 (1973).

fRpceizied May 2, 1977,)