ivchenko_bookreg

Huang and Zhu [6.37] proposed a relatively simple model (Huang-Zhu model) for which both electric potential and displacement have nodes at the interfaces. In this model which is now generally accepted, the scalar electric potential is given by

where µm and Cm are constants determined by the condition that Φm(z) and its derivative both vanish at the interfaces z = ±a/2. These conditions can be presented in the form [6.37]

An important point to stress is that m = 1 mode is excluded in (6.68) because this mode is associated with the interface mode considered in Chap. 2. For B-like confined optical modes, the thickness a is replaced by b and the point z = 0 is taken at the mid-point of the corresponding layer B.

The confined phonon frequency can be estimated from Ωm = ΩLO(qm), where Ω(q) is the bulk LO dispersion and qm = mπ/a is the phonon quantized wave vector. For small values of q, Ω(q) can be approximated by a parabolic function

q2

Ω(q) ≈ Ω(0) − ¯ , 2M

¯

where M is the parameter (the phonon e ective mass) describing the dispersion of longitudinal modes.

In the D2d point group, the confined LO phonons have the A1 or B2 symmetry. In the following we analyze the scattering selection rules for the phonon modes written in the form (6.67) or (6.68), which is equivalent. Accordingly, for the phonon modes with even m, the scalar electric potential Φm(z) has the A1 symmetry and the displacement envelope function um(z) transforms according to the representation B2, whereas, for phonons with odd m, Φm and um(z) correspond to the B2 and A1 representations.

Microscopically, light scattering by phonons in an undoped sample is described as a third-order process with the compound matrix element

312 6 Light Scattering

where Mn0 is the matrix element of the one-photon transition from the ground state of the structure, |0 , to the excited state |n which is an unbound electron-hole pair or exciton characterized by the excitation energy En and the damping rate Γn, Vn n is the matrix element of electron-phonon interaction including the interaction between a phonon with a hole, Ω is the phonon frequency. Note that in (6.69) only resonant contribution is taken into account. Before scattering occurs, the electronic subsystem is in the ground state |0 , there are N (ω1) incident photons, no photons with the frequency ω2 and N phonons with the frequency Ω and wave vector q. After scattering occurs, the electronic subsystem again finds itself in the ground state, the number of incident photons decreases by one, a photon ω2 is created and the number of phonons is increased by one in the Stokes Raman scattering or decreased by one in the anti-Stokes process. According to (6.69), in the first step of the three-step scattering process, the incident photon excites the electronic subsystem to the intermediate state |n . Then a phonon induces scattering from |n to another intermediate state |n . Finally, the electronic subsystem returns to the ground state with emission of the scattered photon.

Since Mn0 e1 ·jn0 and M0n e2 ·j0n where jn0 is the matrix element of the current-density operator, the scattered intensity is proportional to

I(e2, ω2|e1, ω1) |e2αRαβ e1β |2[(NΩ + 1)δ(ω − Ω) + NΩ δ(ω + Ω)] , (6.70)

where NΩ is the phonon occupation number and we introduced the scattering tensor, or Raman tensor, defined as

Fr¨ohlich interaction induced by the macroscopic electric fields. On the other

ˆ

hand, the short-range part, VDP , is the deformation-potential interaction, also found in nonpolar materials.

Linearly independent components of the Raman tensor Rαβ can be found by the method of invariants. The principal rule here reads as follows. Let

sentation Dµ of the system point group. Then, in light scattering induced by

ˆ

the perturbations Vi, the quantities R(Dµ, i) = e2αRαβ e1β should transform under point-symmetry operations according to the representation Dµ. In particular, for perturbations of the symmetry A1, B2 and E in a (001)-grown QW or SL we have

R(B2) = R2(e2xe1y + e2y e1x) ,

R(E, 1) = R3e2z e1y + R4e2y e1z , R(E, 2) = R3e2z e1x + R4e2xe1z ,

where R’s are scalar coe cients. To derive (6.72), we have to recall that as basis functions of the above three irreducible representations one may choose the functions x2 +y2 or z2 (representation A1), xy (B2), yz and xz (E), where x [100], y [010]. According to (6.72), the Raman tensors Rαβ (Dµ, i) can be represented in the following matrix form

For long-wavelength optical vibrations in bulk GaAs, the displacements have the F2 symmetry. Following the symmetry reduction Td → D2d, this representation of the group Td transforms to the representations B2 + E of the group D2d, compare Tables A.2 and A.5. Therefore, for the deformationpotential mechanism of electron-phonon interaction, the Raman tensors for phonons of the polarization u x, y, z in GaAs coincide with the tensors R(E, 1), R(E, 2) and R(B2), where one has to set R2 = R3 = R4. Taking into account the symmetry of confined LO modes, we conclude that the deformation-potential-induced scattering is allowed for the modes with odd m, i.e. with the even envelope um(z).

In bulk GaAs the Fr¨ohlich-interaction-induced scattering is “dipole-for- bidden” with the matrix element Vn n being proportional to the phonon wave vector q. In QWs and SLs, q is replaced by the much larger confinement wave vector qm = mπ/a. This enhancement of the Fr¨ohlich contribution allows to observe the Raman scattering by confined modes with even m, where the function Φm(z) is even and the function um(z) is odd. Under optical excitation far away from all interband resonances the deformation-potential term dominates (m odd), but under resonance photoexcitation the Fr¨ohlich term becomes stronger (m even).

Figure 6.4 shows Raman spectra for scattering by confined LO phonons measured on GaAs/AlAs SLs made up of 400 double layers with the layer

˚ ˚

thicknesses a = 20 A, b = 60 A. When excited under nonresonant conditions, the cross-sections of scattering by the phonons LO2l+1 and LO2l observed in the z(xy)¯z and z(xx)¯z polarizations, respectively, are comparable in order of magnitude. In accordance with theoretical predictions, under resonant

excitation, ω1,2 ≈ Ee01,hh1, one observes scattering by LO2l phonons. The presence of the same LO2l lines (though of a considerably lower intensity)

in the crossed geometry z(xy)¯z can be attributed to the influence of static

314 6 Light Scattering

frequency is due to the interface mode. From [6.38].

defects on the Fr¨ohlich interaction of carriers with optical phonons. Raman scattering from SLs and QWs has become an attractive alternative to inelastic neutron scattering for determining the bulk-phonon dispersion [6.39–6.42].

As compared to the confined optical phonons, observation of the interface modes was more di cult [6.43]. Nakayama et al. [6.44] have performed the first Raman study of the interface mode dispersion by changing the angle of incident light. A detailed investigation of interface modes was undertaken

¯

using micro-Raman backscattering from (010) and (110) surfaces of (001)- grown SLs [6.45]. Then it was realized that, when finite phonon dispersion is taken into account, the interface modes are partially mixed with nearby confined phonon modes [6.37, 6.46].

Most experiments on Raman scattering by confined and interface optical phonons were performed on heterostructures grown on (001)-oriented sub-

316 6 Light Scattering

QWs [6.48, 6.52, 6.53]. Sapega et al. [6.48] reported strong spin-flip-related Raman scattering in p-type GaAs/AlxGa1−xAs (001)-oriented MQWs in an external magnetic field. The experiment was performed in the backscattering Faraday configuration z(ση , σλ)¯z. This scattering is shown to be related to transitions within the magnetic-field-split ground state of the neutral acceptor into the ±3/2 sublevels and involves the angular-momentum reversal transitions +3/2 → −3/2 or −3/2 → +3/2 of a hole bound to an acceptor (peaks labelled H in Fig. 6.5). The e ect exhibits resonance behavior and the scattering e ciency is significant only within a narrow frequency region between the low-energy edge of the photoluminescence spectrum and the fundamental-absorption edge of the heterostructure. The polarization of the Stokes and anti-Stokes components is found to depend on the excitation energy.

˚ ˚

Fig. 6.5. Raman spectra of a Be-doped 46A/110A GaAs/AlxGa1−xAs MQWs measured in (σ−, σ+) (upper spectrum) and (σ+, σ−) (lower) configurations in a magnetic field B = 10 T and for excitation energy ω = 1.628 eV. H labels the hole spin-flip Raman peak, LE the localized-exciton angular-momentum-flip peak. [6.48]

Two pure limiting cases observed at the edges of the scattering resonance profile can be discriminated in experiment. When excited with circularly polarized light at the long-wavelength edge of the resonance profile, the backscattering line, which is also completely polarized, is observed in the z(σ+, σ−)¯z or z(σ−, σ+)¯z configurations (Fig. 6.5) and is not seen in z(σ+, σ+)¯z or z(σ−, σ−)¯z. Each of the (σ+, σ−), (σ−, σ+) spectra contains either the Stokes or anti-Stokes component in such a way that, if the z(σ−, σ+)¯z reveals a Stokes shift, then in the opposite geometry only the anti-Stokes component is present. Under circularly-polarized photoexcitation at the short-wavelength edge, one observes both the Stokes and anti-Stokes scattering lines simultaneously. These lines are circularly polarized with the polarization as high as 80%. The sign of the backscattered-light polarization coincides with that of specularly reflected light, thus implying that the scattering occurs predominantly in the z(σ+, σ+)¯z and z(σ−, σ−)¯z configurations. The intensity ratio of the Stokes and anti-Stokes components can be approximated with good precision by exp (| ω|/kB T ), where ω is the transferred photon frequency, ω1 − ω2, proportional to the magnetic field Bz . This scattering is observed only in p-type samples which confirms its interpretation as a process accompanied by a ±3/2 → 3/2 spin flip of the hole bound to an acceptor resulting from its exchange interaction with the hole in a photo-excited exciton. Thus, at least two di erent mechanisms are identified to contribute to the bound- hole-related spin-flip Raman scattering denoted as processes A and B and contributing, respectively, to the longand short-wavelength edges of the resonance profile. The process B involves three-particle complexes, A0LE, which can be considered as a localized exciton neighboring a neutral acceptor and weakly perturbed by it. The process A is associated with excitons bound to neutral acceptors, denoted A0X, acting as intermediate states and occurs due to acoustic-phonon-assisted spin-flip of an electron in such a complex.

We remind that the neutral-acceptor ground state of Γ8 symmetry is fourfold degenerate in a bulk GaAs crystal. Because of the confinement e ect of the barriers, this state splits into two close-lying levels Eh and El with the hole spin components ±3/2 and ±1/2. Under a magnetic field B parallel to z each of them splits into

where the hole energy is referred to the zero-field position of the h level and ∆C is the zero-field h-l splitting. In the first approximation, the values of g factors, gh and gl, coincide. In addition to the strongest ±3/2 → 3/2 Raman lines, two other acceptor-related lines are observed. Their polarization properties and magnetic-field behavior (Fig. 6.6) indicate that they originate from the ±3/2 → 1/2 interlevel transitions.

Now we describe in greater detail the scattering process B that occurs under resonant excitation of A0LE complexes. In this case a σ± photon excites an A0LE complex |s, j, m with s + j = ±1, where m = ±3/2 is the

318 6 Light Scattering

µ

∆

µ

∆

∆

Fig. 6.6. Magnetic-field dependence of the Raman shifts of −3/2 → +3/2, +3/2 →

(full circles) GaAs/AlxGa1−xAs MQW structures. ∆C denotes the “crystal-field” splitting of the acceptor ground state into two Kramers doublets at zero magnetic field. The inset shows the scheme of neutral acceptor energy levels at zero and nonzero z component of the magnetic field. [6.53]

initial spin of an acceptor-bound hole and s = ±1/2, j = ±3/2 are the spin indices of an electron and a hole in the photoexcited localized exciton. At the next stage the hole j in the exciton induces a flip m → −m of the bound hole as a result of a flip-stop-like exchange interaction described as [6.54]

the localized exciton (s, j) in the complex |s, j, −m annihilates, emitting a photon of the same circular polarization, but the neighboring bound hole has now the reversed spin −m and the secondary photon energy ω2 di ers from the initial energy ω1 by ±|∆A|, where ∆A = 3ghµB Bz with the minus and plus signs corresponding to Stokes and anti-Stokes scattering. Note that the interaction (6.77) flips the acceptor spin while leaving s and j invariant. Such an interaction becomes possible because the two holes in the complex A0LE have di erent centers of in-plane localization, ρA and ρLE . An existence of the in-plane direction ρA − ρLE reduces the symmetry of the complex and allows nonzero values of ∆hh. It is this reduction of symmetry that lifts the restrictions imposed by the conservation of the angular-momentum component m + M in the backscattering process B. Indeed, in this process values m1 + M1 and m2 + M2 di er by ±3 because the acceptor-bound hole changes its angular momentum by ∆m = ±3 and the photon angular momentum is unchanged, ∆M = 0.

As mentioned above, the A0X-mediated scattering process A can be considered as a double spin flip because it includes an acoustic-phonon-induced spin flip of an electron in the photoexcited complex A0X. The acousticphonon energy equals |ge µB Bz | and, therefore, the Raman-shift energy is given by (3gh −ge )µB Bz , i.e. it also depends on the electron g factor. The interpretation of the process A is confirmed by measurements of Raman spectra under tilted magnetic fields [6.53] in which case the selection rules allow the A0X-mediated backscattering without the assistance of an acoustic phonon and the Raman shift is given by ∆A = 3ghµB Bz .

Coming back to Fig. 6.5, we note that, besides the line H related to the spin flips of acceptor-bound holes, one can see a line with a smaller Raman shift, denoted as LE. It is assigned to Raman scattering by flipping the angular momentum, s + j, of a photoexcited localized exciton through interaction with acoustic phonons. This line is observed both in doped and undoped QWs.

The Raman shift of the interlevel scattering ±3/2 → 1/2 displays a strong dependence on the well width: ∆C = 7.3, 3.5 and 2 meV for the

˚

MQWs with the same barrier thickness b = 110 A and the QW thickness

˚

a = 46, 72 and 102 A, respectively. A value of ∆C measured on one structure increases monotonically with increasing ω1. This can be understood

320 6 Light Scattering

taking into consideration that the lower-energy excitation probes neutral acceptors with higher binding energy. Their states are less a ected by barrier confinement and, hence, the splitting ∆C is smaller.

Using Raman scattering in the absence of any applied magnetic field, Jusserand et al. [6.55–6.57] demonstrated spin splitting of the electron subbands in QW structures due to the inversion asymmetry (Chap. 2). They observed a double peak in the crossed-polarization scattering spectra of intrasubband excitations in an n-type modulation doped GaAs/AlxGa1−xAs SQW. Changing the angle of the sample surface normal with respect to the incident and scattered wave vectors allows one to study the scattering e ciency as a function of the transferred wave-vector in-plane component

q = q1 − q2 .

At small temperatures and q kF , kF being the Fermi wave vector, all involved electron states lie very close to the Fermi surface and the transferred photon frequency is given by

2

ω = m q kF cos (φk − φq ) ± ∆E(k, φk) ,

where φk, φq are the azimuth angles of k and q , ∆E(k, φk) is the spin splitting and the sign + (−) refers to the spin-down—spin-up (spin-up— spin-down) transitions. From an analysis of the Raman-scattering signal, Jusserand et al. obtained an overall understanding of the in-plane anisotropy of the electron spin splitting and showed that, in the studied samples, the Rashba contribution to the splitting induced by the asymmetric potential is of comparable magnitude to the bulk-inversion-asymmetry term.

6.7 Double Resonance in Raman Scattering

In optical spectroscopy the double resonance is defined as enhancement in the intensity of secondary emission under conditions where (i) the energies,ω1 and ω2, of the incident and secondary photons coincide with those of two interband excitations and (ii) the di erence (ω1 − ω2) equals to the energy of one or few optical phonons. This e ect which can be described in terms of both resonant Raman scattering and resonant photoluminescence has been observed in bulk materials as well as in QW structures. The role of the interband excitations is typically played by excitons. The enhancement

in the Raman scattering e ciency follows from (6.5) or (6.71) if di erences ω0 − ω, ω0 − (ω ± Ω) in (6.5) or En − ω1, En − ω1 ± Ω ≡ En − ω2 in (6.71) vanish simultaneously.

In bulk GaAs, the double Raman resonance was first realized by means of uniaxial stress, which splits the Γ8 valence band into the heavy-hole