EXCITON-TWO-PHONON RABMAN SCATTERING OF LIGHT IN A QUANTUM WELL

The theory of exciton-two-phonon resonant Raman scattering of light in a quantum well has been developed. It is shown that in the case of exciton-two-phonon Raman scattering of light, in which two-dimensional excitons appear as intermediate states, it leads to a sharp increase in scattering (in( (cid:31) (cid:31) 0 3 2 0 (cid:30) / ln ) times (where α 0 – dimensionless Fröhlich constant of interaction of two-dimensional excitons with LO phonons, α 0 1  ) compared to the mechanism of electron-hole pairs. The amplification is due to the fact that in a quantum well, in which the energies of the electron and hole are dimensionally quantized, the process of direct creation (or direct annihilation) of an exciton and the actual emission of a second phonon by the exciton are possible. The scattering tensor at the maximum of the second phonon repetition peak is (cid:31) 0 1 (cid:30) .


Introduction
In recent years a great deal of interest has been devoted to the study and the engineering of high-quality devices of very low dimension, essentially quantum-well, wires, or quantum-dots (QDs) semiconductors (Xiao Z., Zhu J., He F., 1995).Because of their low -dimensionality of these systems exhibit many new physical effects (Mukkhopadhyay S., 1995) which are extremely interesting from the point of view of fundamental physics and also for their potential applications in microelectronic device technology.
Consequently much effort has lately gone into understanding and exploring the physical properties of these systems both theoretically and experimentally.These studies have been performed with the proposal of understanding Section 5. Physics the fascinating novel phenomena and of fabricating devices with new functions or to improve the performance of the existing devices (Wen Fang X. (2001).Excitons play a dominant role in their hysical properties; therefore, their stability is important for possible devices requiring this characteristic (Moussaouy A. El., Bria D., Nougauoi A., Charrour R., Bouhassoune M., 2003).In semiconductor quantum wells, the electron-phonon interaction is usually much less.
As is known, multiphonon resonant Raman scattering (MRRS) is observed during monochromatic irradiation of some polar semiconductors in the fundamental absorption region (Xiao Z., Zhu J., He F., 1995;Mukkhopadhyay S., 1995;Sood A. K., Menendez, J., Cardona M. & Ploog K., 1985;Meynadres M.H., Finkman E., Sturge M.D., Warlock J.M. & Tamargo M.C., 1987).The method (MRRRS) has been intensively used in recent years to obtain information about both vibrational modes and electronic states and features of electron-phonon and exciton-phonon interactions in systems of reduced dimensionality (heterostructures, quantum wells, wires and points).Secondary lines glow (phonon repetitions) is observed at frequencies Z Z Z , where ω l is the frequency of the exciting light; ω LO -fre- quency of bulk longitudinal (optical(LO) phonons; N is the scattering order, i.e. those.number of LO phonons backgrounds emitted during scattering.
Theoretical studies of MRRS processes in a bulk semiconductor have shown that two types of processes contribute to the scattering cross sections: scattering through intermediate states of free electron-hole (EDH) (Goltsev A.V., Lang I.G., Pavlov S.T. Bryzhina M. F., 1983) and through excitonic states (Korovin L. I., Pavlov S. T. & Eshpulatov B. E., 1990).
Scattering with the participation of free EHPs in two-dimensional systems was studied in (Korovin L. I., Pavlov S. T. & Eshpulatov B. E., 1991), where it was shown that the MRRS cross section is enhanced in D 0 1 times compared to the three-dimensional case.For phonon repetitions N ≥ 2.
This article examines two-phonon resonant Raman scattering (RRS) in a single quantum well in the case where the intermediate states are two-dimensional excitons.Inequality (1) ensures the two-dimensionality of the exciton.We will assume that the exciton, emitting phonons, remains in the 1S state all the time.

Statement of the problem and necessary relationships
In the case of the second phonon repetition, the scattering tensor is determined by the expression (Korovin L. I., Pavlov S. T. & Eshpulatov B. E., 1988).
. 2 (3) Scalar functions S 1 and S 2 represent oneand two-fold sums over quantum numbers of size quantization Where Green's function The expression for the Green's function (5) includes the function the lifetime of an exciton in the state n n e h ,K The function γ, which has the meaning of a mass operator, is not calculated further; it is only assumed that (6) If we calculate γ to first order from the coupling constant α 0 , then it is obvious that J D ~0.No 3-4.
Section 5. Physics , look like in S 1 and S 2 the Green's function G (n, n, 0,ω l ) corresponds to the direct production of an and G (n, u, 0, ω s ) corresponds to its direct annihilation.
These processes can only take place if K = 0 (the small impulse of the light wave is neglected).

G n n n
, , , Frequency dependence of the scattering tensor Let us first consider the case of scattering in the same zone, described by the function ) (real phonon emission is possible).Therefore, with sufficient accuracy we can assume that If the parameter is J Z / LO  1, then when integrating over the variable K, the contribution of the pole of the function G (n, n, Κ, ω l --ω LO ) becomes dominant,Therefore, with sufficient accuracy we can assume that Then in the frequency domain ( 13) 2 (frequency corresponding to direct annihilation), then G n n , , .0 см formula ( 16) is simplified and takes the form The frequency dependence of S 2 differs from the frequency dependence of S 1 in that This frequency interval includes the frequency real transition between bands nn and nn′ with emission of LO phonon.Replacing 7) with a δ -function and integrating over K, for S 2 we obtain the expression Where he part of formula ( 19), which depends on and from ( 7), ( 8) we obtain  E is the energy of the exciton band, K is the modulus of the exciton wave vector.
The considered excitonic mechanism of two-phonon RRS leads to a sharp increase in the scattering cross section (scattering tensor − / ln times.From this we can conclude that in a quasi-two-dimensional electron system, the mechanism of two-phonon RSRS is predominant.This conclusion seems justified specifically for two-phonon scattering, when the exciton appears only in the act of indirect creation (or indirect annihilation) and single emission of a LO -phonon.phonons, the question of the relationship between the contribution of the exciton mechanism and the EHP mechanism to scattering becomes more complicated.This is due to the fact that when a LO -phonon is emitted by a hot exciton, it can go into the EHP state and then phonons will be emitted by the electron and hole.Without exploring the relative roles of the two scattering mechanisms in this paper.We only note that the dependence of the scattering tensor on the coupling constant α_0 in the case of MFRRS with a purely exciton mechanism.Remains the same as in the case of two-phonon RSRS, since the appearance of an additional coupling constant in the numerator during the transition from N to N + 1 emitted phonons will be compensated by the appearance of the constant γ ~a0 in the denominator, which comes from the process of real emission of a phonon by an exciton .
The excitonic scattering mechanism in a bulk semiconductor is a 0 2 − times weaker than scattering in a quasi-two-dimensional system.The enhancement of two-phonon scattering compared to the bulk case is explained by the fact that in a quantum well in the frequency range corresponding to direct production or direct annihilation of an exciton, real phonon emission is possible, while in a bulk semiconductor two-phonon scattering consists of two indirect processes -creation and annihilation exciton.
Let us consider a single quantum well with infinite potential walls located between the z = 0 and z = d planes.Let us further assume that the relation , ε-dielectric constant of the quantum well material, e -electron charge, μ -reduced effective mass.
emission of a phonon by an exciton both for the case of scattering in the same zone (n = ′ n ) and for the case of transition to another zone (n ≠ ′ n ).Its square modulus is equal to and (17) are valid in the vicinity of the maximum of the Green's function, which lead to large values of S 1 and S S 2 has two strong maxi- ma: one at the frequency Z Z l l 1 (direct exciton production) and the other coinciding with the strong maximum of function S 1 .Exciton transitions for the cases n n z c are shown in (Fig. 1).

Figure 1 .
Figure 1.Scheme of exciton transitions in the case of taking into account only one (a) and two (б) excitonic zones