ivchenko_bookreg

matrix element of the operator pˆα calculated between the bulk Bloch functions, and ieν ,vν are the overlap integrals. We consider the backscattering geometry when the incident and scattered waves propagate in opposite directions, perpendicularly to the interface plane. Because of the wave-vector conservation, see (6.6), in this case the 2D wave vector k of an electron involved in the scattering process is conserved.

For the pair of conduction band Γ6 and valence band Γ7 in a semiconductor of Td symmetry, the components

ˆ

The matrices V for transitions from the heavyand light-hole states may be written in the simple form similar to (6.48) if the heavy-light hole mixing is ignored. In this case, one has

ˆ

The diagonal and o -diagonal components of the matrix V describe the scattering from spin-conserving and spin-flip intersubband excitations, respectively.

In symmetrical QWs, the parity of electron envelope functions is conserved. Therefore it is conserved as well for scattering by intersubband excitations, e.g., the scattering is allowed for the transitions e1 → e3 and forbidden for e1 → e2. In real conditions, a deviation from the selection rules in parity may be due to additional scattering of the light-excited electron-hole pairs by static defects, and hybridization of the heavy and light hole states with k = 0. Indeed, if hybridization is included, the combination e2αe1β Rαβ (ν s , νs) contains contributions of the type

transforming according to the representation B (or Γ2) of the D2d group, see (4.17). In asymmetrical QWs, particularly in one-side δ-doped QWs, the main reason for scattering from e2-e1 intersubband transitions is asymmetrical shape of the electron envelopes ϕeν (z), ϕvν (z) which allows ieν ,vν and ieν,vν to be simultaneously nonzero.

Equation (6.46) was derived in the single-particle approximation. In this case, the transferred photon energy ω coincides with the di erence between

302 6 Light Scattering

the single-particle energies Ei i = Ei − Ei. The inclusion of Coulomb interaction between electrons results in renormalization of the intersubband excitation energy. For the e1 → e2 transitions in a single QW, the renormalized energies, ECD and ESD, of the chargeand spin-density intersubband excitations can be written as

The parameters α21, β21 describe the depolarization and exchange-correlation (exciton) e ects, they have been introduced in Chap. 4, see (4.12). Note that the length L21 in (4.13) can be also presented in the form

L21 = dz dz ϕ2(z )ϕ1(z ) .

−∞ −∞

Note that as soon as the dispersion of the dielectric constant æ is taken into account the constant æb in (4.13) should be replaced by the function æ(ω) = æ∞ + æphon. As a result we obtain a mixed excitation of the intersubband plasmon and LO phonon satisfying the dispersion equation

where ω212 ,pl is defined by (4.14) with æb replaced by æ∞ and, for simplicity, the exchange-correlation correction is neglected. Equation (6.51) has two solutions

Figure 6.1 presents spectra of light scattering by intersubband chargeand spin-density excitations measured on a GaAs/AlGaAs SQW structure in parallel and crossed polarizations. The excitation-energy di erence, ECD −ESD, is seen to exceed 2 meV. In addition, the spectra of Fig. 6.1 show singleparticle intersubband excitations. In the normal-incidence backscattering geometry, the energy of these excitations coincides with the bare single-particle spacing E21. In the oblique-incidence backscattering geometry, the scattered wave vector equals to q = 2 sin θ(ω1/c) (θ is the incidence angle) and the energies of single-particle intersubband excitations cover a continuum bounded by E21 ± qvF , where vF is the Fermi velocity.

Raman scattering by collective and single-particle intersubband excitation has been observed not only in QW structures [6.5–6.13] but also in quantum

Fig. 6.1. Inelastic-light-scattering spectra of intersubband excitations of the highmobility 2D electron gas in a GaAs/AlGaAs QW. The peaks of spin-density excitations (SDE), charge-density excitations (CDE), and single-particle excitations (SPE) are shown. From [6.5].

wires [6.14–6.16]. Decca et al. [6.17] have studied inelastic light scattering in GaAs/AlGaAs double QWs and observed excitations associated with transitions between two lowest e1 subbands, symmetric and antisymmetric. They have uncovered a new aspect of electron-electron interactions in the 2D electron gas in the studied structure when both subbands are densely populated, i.e. when the ratio η = ∆SAS /EF reaches some critical value. Here EF is the Fermi energy and ∆SAS is the symmetric-antisymmetric splitting which can be easily varied by changing the thickness and Al content of the barrier. The excitation spectra in 0D systems can be revealed in light-scattering as well.

304 6 Light Scattering

For example, Lockwood et al. [6.17] have probed, by inelastic light scattering, shell structure and electronic excitations of many-electron QDs.

Not only interbut also intrasubband excitations contribute to the light scattering from electrons in 2D and 1D nanostructures. For 2D electrons, the dispersion of plasma oscillations exhibits a square-root behavior

where q is the 2D plasmon wave vector. In a periodic MQW structure, the plasmon frequency depends on the 3D wave vector Q and di ers from (6.53) in a structural factor [6.18]

where Q = Q2x + Q2y and d is the structure period. In the weak coupling

limit Q d 1 realized in thick-barrier MQWs or for plasmons with short

wavelengths, the interaction between plasmons in di erent wells may be neglected, S(Q) → 1 and ωplMQW(Q) → ωpl2D(Q ). In the strong coupling limit

Q d 1, the plasmons are characterized by a linear dispersion law (acoustical plasmons)

Of interest is also the particular case (Qz d)2 (Q d)2 1, where S(Q) ≈ 2/(Q d),

and the plasma oscillations regain the 3D character. Interaction of plasmons with polar optical phonons is included in (6.53,6.54) by replacing æ∞ with

æ∞ + æphon.

The intrasubband plasmons in a QWR which correspond to oscillations of the charge density in the direction of the wire exhibits an almost linear dispersion [6.19–6.21]. For a cylindrical wire of the radius R, the dispersion of the 1D plasmon frequency in the long-wavelength limit |q|R 1 can be presented analytically

where N1D is the 1D electron density, æb is the background dielectric constant and q is the plasmon wave vector.

6.4 Scattering by Folded Acoustic Phonons

The investigation of acoustic phonons in bulk semiconductors by light scattering spectroscopy is usually limited to the sub-meV range since the wave vector conservation only allows the observation of phonons which are very close to the Brillouin zone center, the Γ -point. In contrast to a bulk crystal, semiconductor SLs exhibit zone-folding of the bulk acoustic dispersion into minibands or minibranches, as shown in Chap. 2. Because of the artificial periodicity along the growth direction, each acoustic branch folds within the new Brillouin zone, |Qz | ≤ π/d, into a series of minibranches with forbidden minibands at the folding points Qz = 0, ±π/d. As a result, acoustic modes with energies of several meV appear at the superstructure Brillouin-zone center and can be easily detected in light scattering experiments.

To describe light scattering from folded acoustic phonons in terms of photoelastic mechanism, we expand, similarly to (6.32), the dielectric susceptibility fluctuation at frequency ω1 in components of the deformation tensor

In contrast to (6.32) written for a homogeneous semiconductor, here the elasto-optical, or photoelastic, coe cients Pαβλµ = ∂χαβ /∂uλµ depend on z changing stepwise at the SL interfaces and retaining their values within each layer [6.22]. One can conveniently expand the tensor P in a Fourier series

m

and the lattice displacement u in the eigenstates of folded acoustic phonons

where Gm = 2πm/d (m = 0, ±1...), and l = 0, ±1, ±2... is the folded-phonon branch index. Using (6.56, 6.57), we obtain

m

where q is the transferred wave vector. Bearing in mind to consider the light backscattering by longitudinal acoustic (LA) phonons in the parallel configuration z(xx)¯z or z(yy)¯z, we focus on the coe cient Pxxzz ≡ P12 that

306 6 Light Scattering

relates δχxx, δχyy with uzz . Its values in the layers A and B are labelled as PA and PB. Then the expansion (6.56) reduces to

For GaAs/AlGaAs-type SLs, the modulation of photo-elastic coe cients plays a more essential role than the mixing in (6.57) of the various space harmonics. This allows to start the description of light scattering from the approximation of straight-line dispersion valid for ε = 0, see (2.137). In this approximation, the correlation function for fluctuations δαβ in (6.31) can be evaluated by setting

ΩK = s¯|K|, bK and b†K are the annihilation and creation operators for a phonon with the 3D wave vector (0, 0, K). Notice that here K is the phonon wave vector in the extended-zone scheme and Q is that in the reduced-zone scheme. The second-order quantization allows to average correctly the square |δχ|2. We remind that

4 5 4 5

bK b†K = 1 + NΩK , b†K bK = NΩK ,

where NΩK is the phonon occupation number. Substituting the expansion (6.59) for P12 and equation (6.60) for uz (z) into (6.55) and taking into account that uzz = ∂uz /∂z we obtain

It follows then that the intensity of the light scattered from the l-th mode into the Stokes region of the spectrum can be written as

× (NΩK + 1)δ(ω1 − ω2 − ΩK )δq1,q2+Q .

Thus, in the scattering spectrum, one should observe doublets ±|l| with an |l|-independent splitting

where Ωl is ΩK for K related with l by (6.61).

The exact Stokes shift is obtained by changing ΩK in the δ-function by the exact dispersion ΩlQ of folded acoustic phonons, see (2.135). For small |Q| 2π/d, the deviation from the straight-line dispersion is described by

where Ωl0 = 2π|l|s/d¯ and ε is introduced in (2.136).

In (001)-grown SLs of the point symmetry D2d, the folded acousticphonon vibrations ±|l| at Q = 0 correspond to the irreducible representations A1 and B2 [6.23, 6.24]. For the above photoelastic mechanism, the mode A1 is scattering-active whereas the mode B2 is forbidden. Therefore, in the limit of small transferred wave vector q realized in forward Raman scattering, only one mode of the split doublet can be excited. At typical backscattering q-values the A1 and B2 components are mixed enough, the zone-center selection rules are relaxed, both branches ±|l| have nonzero scattering intensity well described by (6.63) and appear as doublets in the scattering spectrum. TA modes in this geometry are forbidden. Note that the doublet center-of- mass is independent of the excitation frequency ω1 or the scattering angle, in contrast to the Brillouin scattering from the lowest branch l = 0. Hence, the scattering by folded acoustic phonons with l = ±1, ±2... is not called Brillouin, but Raman scattering.

The first unambiguous Raman observation of phonon folding in an artificial semiconductor SL was reported by Colvard et al. [6.25]. Figure 6.2

with |l| = 1, 2, 3. In accordance with the selection rules, there is no scattering from folded LA phonons in the crossed geometry e2 e1. The peak at 160 cm−1 is attributed to 2TA scattering. Figure 6.3 shows Raman spectra in the acoustic frequency range measured in the z(yy)¯z configuration on hexagonal (wurtzite-lattice) GaN/Al0.28Ga0.72N SLs with three di erent periods [6.26]. Only the first folded doublet is observed for each sample. Here z is directed along the hexagonal c-axis. In agreement with theory, the doublet splitting is almost the same in all three cases while the shift of the doublet center- of-mass increases with decreasing the SL period. Forward and backward Raman scattering spectra are compared in [6.27]. Folded acoustic phonons in

308 6 Light Scattering

˚

A, T = 300 K. The inset sketches phonon dispersion by Rytov’s model. The arrows specify the peak positions derived from X-ray di raction data. From [6.25].

GaAs/AlAs SLs grown on non-(001)-oriented substrates have been studied by means of Raman scattering by Spitzer et al. [6.28]. A strong magneticfield enhancement of Raman scattering on folded LA phonons in a SL has been reported by Mirlin et al. [6.29]. Raman spectroscopy on folded acoustic phonons in BeTe/ZnSe SLs has been applied for determination of sound velocity in BeTe [6.30]. Localized folded acoustic phonon modes in SLs with structural defect layers has been analyzed in [6.31].

Raman-scattering spectroscopy of nanocrystals or QDs [6.32, 6.33] allows to reveal the vibrational characteristics of nanoparticles that are intermediate between the bulk and molecular states. A theory of Raman scattering by acoustic vibrations in spherical nanocrystals embedded in a dielectric matrix was developed by Goupalov and Merkulov [6.34]. Since the sound velocities and the densities of the studied semiconductor nanocrystals and silicate-glass matrix di er insignificantly, the acoustic phonon modes in a nanocrystal are not completely confined, and they can transfer into continuum phonons of the matrix. This gives rise to a substantial broadening of the corresponding Raman lines. For the spheroidal fully-symmetric phonons being purely longitudinal, the line shape is given by

Raman shift (cm-1)

Fig. 6.3. Raman spectra of GaN/Al0.28Ga0.72N SLs with di erent periods d = 61,

˚

128 and 238 A. λ1 = 488 nm, T = 300 K. From [6.26].

|ω|/c(in)LA , c(in)LA is the longitudinal sound velocity in the nanocrystal, the sign

± corresponds to the Stokes and anti-Stokes components, and the functions I1(x), I2(x) have been presented in [6.34]. The function I1 (qinR) has a broad peak at qinR π with the half maximum of the same order and reflects the fact that, in the simple model of interaction of quantum-confined carriers with bulk phonons, the interaction is optimal for phonon wave vectors q of the order of π/R. The function I2(qinR) has a series of narrow peaks contributed by di erent confined-phonon modes, the peak width being determined by the

310 6 Light Scattering

conversion of a confined phonon into a bulk acoustic phonon propagating in the matrix.

6.5 Scattering by Confined and Interface Optical Phonons

In optical spectroscopy, in addition to the propagating folded acoustical phonons, the confined optical phonons and interface phonon modes in semiconductor QWs and SLs have received considerable attention and are reasonably well understood.

In order to provide insight into the optical phonon structure of QWs or SLs, let us consider an isolated slab of the thickness a. Optical vibrations are confined within the slab and are equivalent to those in the infinite crystal whose wave vectors are given by ±mπ/a, m being an integer. In a multilayered structure, optical phonon modes are confined within individual layers, provided that the optical branches of the constituent materials do not overlap. Particularly, this is the case of GaAs/AlAs or GaSb/AlSb SLs. The series of phonons labelled by m are termed confined phonons.

In the normal-incidence backscattering from (001)-grown heterostructures, only LO phonons with zero in-plane wave vector are involved. Then, in the dielectric-continuum approximation, the displacement vector u and the induced electric field E = −Φ are parallel to the growth direction z and depend only on z. Three di erent models are usually employed for an analytical description of the confined optical modes [6.35]. In the slab model the electric field potential Φ(z) has nodes at the interfaces so that, for the A-like mode m confined in the layer A, Φm(z) and the LO displacement um(z) Ez (z), see (6.37), take the form

However, with such a choice of Φ(z), the field Ez is discontinuous at the boundary between the layers A and B and the displacement uz (z) becomes discontinuous as well. In the guided phonon model [6.36], the ionic displacements vanish at the interfaces. Therefore, the electric potential and vibrational amplitudes of the confined, or “guided”, modes are given by