P H Y S I C S OF S E M I C O N D U C T O R S AND D I E L E C T R I C S

MULTIPHONON RENORMALIZATION OF THE SPECTRUM OF TWO-DIMENSIONAL

EXCITONS IN A MAGNETIC FIELD

A. D. Val' and N. V. Tkach UDC 537.226

The renormalization of the dispersion law for two-dimensional excitons in a mag- netic field is studied by the Green's-function method taking into account multi- phonon processes.

The experimental realization of two-dimensional systems [i, 2] has stimulated new the- oretical studies along these lines. One of the interesting problems, studied in detail in recent years, is the behavior of excitons in different two-dimensional and quasi-two-dimen- sional structures exposed to external fields. Thus in [3] the spectrum of Wannier-~ott ex- citons in strong magnetic fields was studied neglecting the phonon subsystem. In [4, 5] the renormalization of the spectrum of diamagnetic two-dimensional excitons by their interaction with optical phonons at T = 0 was studied. The specific nature of the calculation carried out in the last works cited makes it impossible to extrapolate the results obtained there to magnetic field intensities for which the energy corresponding to the cyclotron frequency is less than the excitonic rydberg. The purpose of this work is to study this problem.

Since a magnetic field alters the internal state of an exciton, this changes the bind- ing force between quasiparticles and phonons, intensifying or weakening it depending on the parameters of the crystal. We shall therefore study, by means of the Green's function method developed in [6], the renormalization of the quasiparticle spectrum by optical phonons with- out assuming that the interaction is weak in the sense of perturbation theory.

( heH < 2',~e~ i In studying a two-dimensional exciton in weak magnetic field I ~c h2~i j , we shall as-

sume that its energy levels are far apart so that we can neglect scattering between levels. The Hamiltonian of the system in the second quantization representation is derived from first principles. The analytical form of the coupling function is found by the same method as that employed in [7] for the three-dimensional case. Now, however, the interaction of an electron and a hole with two-dimensional optical phonons is taken into account [5], while the transi- tion to the second-quantized representation is carried out for wave functions of the two- dimensional exciton [8] in a magnetic field. As a result the Hamiltonian operator of the system has the typical form [9, I0]

(1)

Here

~n (~) = ~ - - 21~e~ h ~ i + ~n + h '~ ; '2m (2)

is the dispersion law for an exciton in a magnetic field (e is the gap width);

~n - 3 e~a'ex H~ (3) 64 ~c ~

i s t h e p u r e l y f i e l d - i n d u c e d s h i f t o f an e n e r g y l e v e l ( n e g l e c t i n g p h o n o n s ) .

The coupling function between the "magnetic" exciton and the phonons is represented in the form of two terms

(H; q) = ~ (q) + ?n (q), (4 )

Chernovitskii State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 3-7, April~ 1987. Original article submitted May 20, 1985.

0038-5697/87/3004-0267512.50 �9 1987 Plenum Publishing Corporation 267

m~

/,2

/,0

-28

I l 1 I

m'/~ I

,.o

Fig. I Fig. 2

Fig. i. m*/m (upper scale and curve i) and 5~h (lower scale and curve 2) versus the number n for H = 0.

Fig. 2. m~/m (upper scale and curve I) and A (lower scale, curve 2 neglecting phonons and curve 3 taking phonons into account) versus the intensity of the magnetic field H. For convenience the origins of the curves 2 and 3 are combined.

the first of which

- e : ~ -',) - --oo ( - 1)p ( 5 )

~P (q) = ('1+-:o) (1 + z ) == (1 -~ ,3pq2) :~j~

characterizes the interaction of an exciton with phonons without the field and, up to a fac- tor, is identical to that obtained in [ii], while the second term can be regarded as a cor- rection to the interaction owing to the field

The parameters

?H (q) = / ~ e ~ ~o -- ~oo dft: ~ ( - 1 )P (2~pq' -- 1 ) / .

- - (1 + ~ p q ~ ) S l ~

d = 2.2-lO-3(heo~aex/~ec)2; p = I; 2 = h; e; ,3e~n = (mh:~ae,,.'4ra) 2,

(6)

where a is the lattice constant and S is the "two-dimensional" volume (area) of the main region of the crystal, while the rest of the notation is standard, have been introduced in the formulas (5) and (6).

The optical phonons are assumed to be nondispersive:

~ ( q ) ___~Q. ( 7 )

To study the renormalization of the spectrum of a "magnetic" exciton by polarization phonons in the crystal it is necessary to find the mass operator corresponding to the Green's function. Such a mass operator was determined in [6] taking into account the arbitrary mag- nitude of the coupling with a Hamiltonian of the type (i). In our case it can be written in

the form I~(H: q,)I~A, (,c. q,)

M~'(~r = X h' X" I~(H; q2) l"A2(1r q,, q~.) ...i" ( 8 ) q, E~c_q~- .,

~E~_q,_q,-. . ._,~ I~(H, qJ I-~A,/'c, q, . . . . q , ) E H n

l = !

Here

_ = -- , ~ -- ._~ Oq t.

l = l l = l l = 1

The form of the coefficients A.(m, q,, q2 .... q.) is very cumbersome, and since they were de- determined exactly in [6] we shall not present them here. Since the functions I~(H, qn)! 2 have sharp maxima for the same values of qn = q0:M, this makes it possible, by virtue of the properties of the coefficients A n [6], to set

268

A. (to, q~, q~, . .q.) = n.

Then, by using the dispersion laws (2) and (7) and the coupling function (4) and introducing similar dimensionless parameters and variables

= . . 2ma-); :-.s = ( t im - - -: - - A n ) L; ". '~ L; ( L = .h:= ~ " '> e ~ o F = --- % -- %0 .

2aLl (1 -- %) (1 +-:o_J

K = - - . ~ K ; Q = : = q ; ~ : .~=,~;~ , .=~a ~, a

and also the function

2 t -D 2 ffS(Qn) = j ~ (-- I)" ( I --_~pQ,, / -~ (i = \ l - es-s ) j ,

we obtain the following expression in polar coordinates for the dimensionless full mass op- erator At (K, ~) = M H(K)/L

o r ,-.

:'v/(K; ~.)=F fH(Q,) ~n_K., Q~_2KQ, cos$_.:_ 0 O

7 2dQ,d~ --Fi,~ ,~t fH(Q2)L~_K~_QT_Q~_2K(Q~cosT,_TQ, c o s ~ ) - i Q , Q , cos~; - %)-~---. ...

0 o

�9 t 1 J"(Q,,) 0 0

ndQ,fl~. n

:n_K 2 "~(Q]+2KQ~cos?i) o ~ Q / Q / r ~ : - - 1=1 i > / = 1 (9)

To find the position of the bottom of the renormalized exciton band it is necessary to solve, according to the general theory of [9, I0], the dimensionless dispersion equation with the mass operator (9) for K = 0:

~ n - , ~ ( K = 0 , ~H) = O. (10)

The solution of this equation ~ h will determine (in units of L) the magnitude of the shift of the bottom of the band owing to the interaction of the "magnetic" exciton with phonons

= a ; p h -

The renormalized effective mass is given by the relation

m* m [ 1~ dM (~ = =M 4- K~' a (K 2)''' i x = o ] - '

Thus the new dispersion law for quasiparticles taking into account their interaction with phonons and the magnetic field will have the form

2:xe4 h ~ ~2

h2~L 2m*

w h e r e A = A H + A~h i s t h e t o t a l s h i f t o f t h e p u r e l y e x c i t o n i c l e v e l .

B e c a u s e o f t h e c o m p l e x i t y o f t h e a n a l y t i c a l e x p r e s s i o n s t h e c a l c u l a t i o n o f t h e mass op - e r a t o r (9), the solution of the dispersion equation (i0), and calculation of A and m* were carried out on an ES-I045 computer. For a model of a TIBr crystal with the parameters m e = 0.18; m h = 0.38; e0 = 35.1; e~ = 5.4; a = 3.97 ~; ~ = 116 cm -I the following results were obtained.

The shift A~h and the effective mass m* were calculated as a function of the number of

n-phonon processes taken into account. As an example Fig. 1 shows graphs of these dependences

269

for H = 0, where for convenience, instead of points, corresponding to specific values for different values of n, solid lines are drawn. One can see from the figure that both curves saturate for quite low values of n. Thus in the case under study the new quasiparticle spec- trum is completely formed by five phonon processes (for all values of H).

The field dependence of A and m * was calculated for values of n such that these quantities as a function of n were saturated. The results are presented in the form of graphs in Fig. 2.

One can see from Fig. 2 that the effective mass decreases smoothly as the field (H) in- creases relative to the value which is obtained owing to renormalization of the excitonic mass by virtue of phonons with H = 0. The magnetic dependence of the full shift in the bot- tom of the band is determined by the magnetic field directly (AH) and by the phonons (A~h). As the field increases (H) the shift AH is always into the short-wavelength region and fol- lows a quadratic law. The interaction of an exciton with virtual phonons shifts the bottom of the band into the long-wavelength side, but as H increases the absolute magnitude of this shift decreases.

Thus the resulting shift of the bottom of the exciton band occurs into the short-wave- length region relative to the value corresponding to the value renormalized by the exciton- phonon interaction in the absence of a field.

LITERATURE CITED

i. Zh. I. Alferov, A. T. Gorelenok, et al., Fiz. Tekh. Poluprovodn., 18, No. 7, 1230 (1984). 2. V. A. Volkov, D. V. Galchenkov, et al~, Fiz. Tekh. Poluprovodn., 19, No. 2, 333 (1985). 3. I. V. Lerner and Yu. E. Lozovik, Zh. Eksp. Teor. Fiz., 74, No. I, 274 (1978). 4. S. I. Beril and E. P. Pokatilov, Fiz. Tekh. Poluprovodn., 14, No. i, 37-42 (1980). 5. S. I. Beril, E. P. Pokatilov, and L. F. Chibotaru, Fiz. Tverd. Tela, 24, No. 3, 663-668

(1982). 6. N. V. Tkach, Teor. Mat. Fiz., 61, No. 3, 400-407 (1984). 7. A. A. Zinchenko, V. M. Nitsovich, and N. V. Tkach, Ukr. Fiz. Zh., 26, No. 8, 1287-1292

(1981). 8. M. Shinada and S. Sugano, J. Phys. Soc. 3pn., 21, No. i0, 1936-1946 (1966). 9. A. A. Abrikosov, L. P. Gor'kov, I. E. Dzyaloshinskii, et al., Methods of the Quantum

Theory of Fields in Statistical Physics [in Russian], Fizmatgiz, Moscow (1962). A. S. Davydov, Theory of Solids [in Russian], Nauka, Moscow (1976). O. Yu. Mikityuk, Fiz. Elektron., No. 23, 25-28 (1981).

i0. ii.

EFFECT OF SEMICONDUCTOR PARAMETERS ON SPECTRAL SENSITIVITY

OF A GALLIUM ARSENIDE-ELECTROLYTE CONTACT

S. N. Kravchenko and V. F. Musarova UDC 621.383.51:531.136

The dependence of spectral sensitivity of a gallium arsenide contact with a 0.01 N KCI solution in dimethylformamide upon impurity concentration in the semicon- ductor and applied anode bias is studied. It is known that a decrease in dopant level increases spectral sensitivity in the long-wave region. Experimental re- sults are compared with calculation and are found to agree satisfactorily.

Wide use of presently available regenerative solar cells which convert solar energy di- rectly into electrical current is hindered by the low efficiency or instability of these de- vices. These shortcomings are related to the absence of sufficient information on the physi- cal and photoelectrochemical processes upon which the action of these converters is based.

One of the most promising materials for semiconductor anodes in regenerative cells is gallium arsenide, since its forbidden band width corresponds well to the solar energy spec-

I. I. Mechnikov State University, Odessa. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 7-10, April, 1987. Original article submitted January 3, 1984; revision submitted May 31, 1985.

270 0038-5697/87/3004-0270512.50 �9 1987 Plenum Publishing Corporation