ivchenko_bookreg

phenomenon under consideration may be interpreted as both Rayleigh resonant scattering and resonant photoluminescence. Bearing this in mind they use sometimes the general term ‘resonant secondary emission’. Note that, under optical excitation by a non-monochromatic light with E0(ω1) being a smooth function of ω1 in the vicinity of ω0, we obtain for the radiation spectrum

In most cases, however, the di erence between the two phenomena can be justified. Indeed, in light scattering defined in the traditional way the excited states of a system are virtual, whereas in conventional photoluminescence the emission of the secondary photon is usually preceded by multiple transitions of the system between di erent real excited states.

In 1982, Hegarty et al. [6.2] reported for the first time a resonant enhancement of the Rayleigh scattering at the heavy-hole exciton transitions of GaAs/AlGaAs MQW structures. A systematic study of resonant Rayleigh scattering in semiconductor single QWs is presented in [6.3]. It is shown that, although the participation of propagating exciton states cannot be completely excluded, the main contribution to the resonant Rayleigh scattering comes from excitonic states localized (or confined) by 2D growth islands formed at the well interfaces during the growth process. A theory of steady-state scattering of light via 2D-excitons from a QW with rough interfaces has been developed in [6.4].

6.2 Light Scattering in Bulk Semiconductors

6.2.1 Scattering by Free Carriers

We define the di erential light-scattering cross-section as

where V is the emitting volume, the energy-flux density, J1, of the primary

¯

radiation is related with the mean number of photons Nq1 through

∆W is the scattering rate in the frequency region ∆ω2 and within a solid angle ∆Ω2 inside the medium of the dielectric constant æb. Note that the quantity d2σ/dωdΩ is defined in (6.13) as the scattering cross-section per unit volume and has the dimension cm−1s rather than cm2s. The spectral intensity, I(ω2), of the secondary radiation propagating in vacuum in a unit

292 6 Light Scattering

solid angle is connected with the intensity J10 incident on a semi-infinite crystal by the relation

Here, Ki is the absorption coe cient for the light of frequency ωi (i = 1, 2), R is the reflection coe cient, and the di erence between R(ω1) and R(ω2) is neglected. For the sake of simplicity, we consider the geometry of backscattering under normal incidence of the primary wave. While writing (6.15), we have taken into account that for radiation backscattered perpendicular to the surface the ratio of the solid angles dΩ20/dΩ2 in vacuum and in the crystal is equal to the squared refractive index æ(ω2).

In the limiting case of a rarefied plasma where the Coulomb interaction between electrons may be disregarded, one has

ks

where s is the electron spin index, r0 = e2/(m0c2) is the classical electron radius, m is the e ective mass, Ek = 2k2/(2m ) and fk is the equilibrium electron distribution function. The electron-photon interaction operator used in deriving (6.16) and written in the second-quantization representation has the following form

c†q , cq are the creation and annihilation operators for the photons and a†ks, aks are those for the electrons.

Substituting (6.14, 6.16) into (6.16) and neglecting the frequency dependence of æ, we come to

k

with the factor 2 accounting for spin degeneracy.

The scattering cross-section (6.18) can be expressed in terms of the electronic susceptibility

Indeed, using the relation

fk − fk+q = fk(1 − fk+q ) 1 − e− ω/kB T

valid for the equilibrium distribution and the identity

1

Im = πδ(ε)

ε − i Γ

we can rewrite (6.18) as

where

Nω = [exp ( ω/kB T ) − 1]−1 .

It follows from (6.19) that, in equilibrium and for the isotropic electron spectrum, the susceptibility χel(ω, q) is independent of the direction of the wave vector q. Note also that

Im {χel(−ω, q)} = −Im {χel(ω, q)} .

In Stokes scattering the transferred frequency ω is positive and Nω > 0, whereas, in the anti-Stokes process, ω < 0, Nω > 0 and 1 + Nω = −N|ω|. Therefore, the relation

(1 + N−ω )Im {χel(−ω, q)}

(1 + Nω )Im {χel(ω, q)}

defining the intensity ratio of the anti-Stokes and Stokes indexlight scattering!Stokes or anti-Stokesscattering lines is equal to exp (− ω/kB T ).

Since the operator (6.17) is proportional to the Fourier component of the electron-density operator

(6.20) gives the cross-section of light scattering by charge-density fluctuations in a rarefied plasma. With allowance made for Coulomb correlations, the

294 6 Light Scattering

di erential cross-section of scattering by charge-density fluctuations takes on the form

with the dielectric function

For simplicity, the contribution of optical phonons is here neglected and will be discussed in the next subsection. The expressions (6.18) and (6.22) di er in the factor æ2∞/|æ(ω, q)|2 accounting for the screening of the charge fluctuations appearing in the system. For a low-density plasma, |æel| æ∞ and this factor is close to unity. Equation (6.22) describes the scattering both from single-particle excitations with the transferred frequency ω = (Ek+q −Ek)/ and collective plasma oscillations, or plasmons, whose frequency satisfies the equation

Expression (6.17) for the operator Hel-phot describing light scattering by free electrons is valid provided the photon energy ωi is small compared to the energy separation Ec0 − El0 from the other bands l = c. If this condition is not met, one has to start from a more general expression

In accordance with the expression of the reciprocal e ective-mass tensor m−αβ1 in terms of the k · p theory, for ωi |Ec0 − El0| we obtain

γs s = δs s m0 e1αe2β .

αβ mαβ

In crystals of cubic symmetry,

mαβ = m δαβ , γs s = mm0 (e1 · e2)δs s

and (6.25) reduces to (6.17).

While deriving these equations we neglected the di erence between the frequencies ω1 and ω2 and included into the sum over l in (6.26) only the contributions of the upper valence bands Γ8 and Γ7. It was also assumed that the energies |Eg − ω1|, |Eg + ∆ − ω1| exceed the mean electron kinetic energy, the thermal energy kB T for the nondegenerate plasma and the Fermi energy EF for the degenerate electron gas. Substituting (6.27) into (6.25), we obtain

where the charge-density operator ρ is defined according to (6.21) and σq is the Fourier component of the electron-spin density operator

As follows from (6.25), the light can be scattered not only by fluctuations of the electron density, but also by those of the spin density as well. The first contribution to the cross-section is described by (6.22) where the ratio m0/m has to be replaced by the coe cient A. For the cross-section of spindependent scattering, we obtain

Since the spin-density fluctuations are not accompanied by violation of neutrality, they are not screened and, therefore, the di erential cross-section (6.29) does not contain the factor æ2∞/|æ(ω, q)|2. It should be mentioned that (6.29) includes both the contribution due to spin-flip scattering k, 1/2 → k+q, −1/2 or k, −1/2 → k+q, 1/2 which is proportional to |(e1 ×e2)+|2 and |(e1 × e2)−|2, respectively, and that due to spin-dependent spin-conserving

296 6 Light Scattering

scattering k, s → k + q, s (s = ±1/2) which is proportional to |(e1 × e2)z |2, where z is the spin quantization axis and

(e1 × e2)± = (e1 × e2)x ± i(e1 × e2)y .

In a classical magnetic field B z, the electron spin states are split and the transferred frequency in spin-flip scattering s → −s is given by

where g is the electron g factor. In a quantizing magnetic field, contributions to light scattering arise not only from the spin-flip processes, but also from carrier transitions between the Landau levels.

Besides the above two light-scattering mechanisms related to chargeand spin-density fluctuations, there exist others, in particular, scattering by energy fluctuations taking into account the nonparabolicity of the freecarrier spectrum, by mass fluctuations in a many-valley semiconductor with anisotropic e ective masses, by collective electron-hole plasma oscillations, and scattering involving carrier transitions between di erent subbands, e.g., between the heavy and light-hole subbands.

6.2.2 Scattering by Phonons

The main contribution to the phonon-assisted light scattering comes from the indirect interaction of photons with the lattice through the electron subsystem rather than from the direct photon-phonon interaction. Lattice vibrations produce in the medium a transient optical SL, and it is from the latter that the scattering occurs. Therefore, the e ciency of scattering by acoustic or optical phonons is inherently connected with the intensity of the corresponding fluctuations, δχαβ (r, t), of the medium susceptibility, see (6.9). As a result, the di erential scattering cross-section can be represented in the form

In such a semiphenomenological description, the fluctuation δχαβ can be expanded in the normal coordinates of lattice vibrations written in the secondquantization representation. Consequently, δχαβ is an operator acting on the wave function of the phonon subsystem, the angular brackets in (6.31) denoting the thermodynamic average of the operator product.

The operator δχαβ involved in the calculation of the cross-section of scattering by acoustic phonons is a linear combination of the deformation tensor components ulm, namely,

298 6 Light Scattering

where ul, El are the components of the displacement vector u and the electric field E induced by this displacement. Usually, the vector u is defined as the relative shift of the cation and anion sublattices multiplied by √ρ¯, where ρ¯ is the reduced-mass density

ρ = 2(Ma + Mb)/Ω0 is the density, Ma and Mc are the anion and cation masses, and Ω0 is the volume of the crystal primitive cell. In the secondquantization representation, the operators u and E can be written as

where Ω ≡ ΩLO, æ = æ0æ∞/(æ0 + æ∞), b†Q, bQ are the creation and annihilation operators for the longitudinal optical phonons. By using (6.31, 6.32, 6.36, 6.38) we come to

where δαβl is the unit antisymmetrical tensor of the third rank and x, y, z are the principal axes [100], [010], [001]. If the phonon damping Γ is taken into account, the function δ(ω − ΩLO) in (6.39) should be replaced by

where æ(ω) is the dielectric function