ivchenko_bookreg

to 0 and, therefore, the electron g factor varies from gA to gB. Since in a GaAs/Al0.35Ga0.65As heterostructure the g factors gA and gB di er in sign, the electron g factor should change its sign at a certain well thickness. Moreover, gA and gB are close in absolute value and their contributions to the net g factor (5.94) remarkably cancel out within a wide range of the well thicknesses. In this case, the various corrections disregarded in (5.94) may play a noticeable role.

Another simplified method is based on the assumption of infinitely high barriers for both conduction and valence bands in which case the selection rules (2.152) allow nonzero matrix elements pcsν,vmν for the quantumconfined states with the same quantum number ν = ν only. Then the longitudinal and transverse g factors are given by

Ee1, En1 are the confinement energies in the lowest conduction (e1), heavyhole (hh1), light-hole (lh1) and spin-orbit split (so1) subbands. The equations for g , can be derived by using either the general equation (5.89) or equation (5.92) for g and the similar equation

for g . It is clear that the contribution to R , from the subband n1 is easily identified by the confinement energy En1 in the corresponding denominator.

and, therefore, the hh1 contribution to R is absent. Neglecting the hole confinement energies we obtain an isotropic electron g factor renormalized as compared with (5.92) due to the replacement of Eg by Eg + Ee1. If we take into account the confinement energies En1 in the corresponding denominators then we come to the following g-factor anisotropy

According to (5.95, 5.97) the relative anisotropy of the g factor is given by

242 5 Photoluminescence Spectroscopy

where

η = 1 − 3 Eg (Eg + ∆)m0 . 2 ∆|pcv |2

It follows then that in heterostructures the electron g factor should exhibit a remarkable anisotropy.

While deriving (5.95) we considered unstrained heterostructures with matched lattice constants. In structures with a noticeable lattice mismatch the strain-induced shifts, δEc and δEn, of the conduction and valence bands should be included into the calculation procedure. As a result in the simplified equations (5.95) the denominators Eg + Ee1 + En1 are changed by Eg + δEc + Ee1 + δEn + En1. Thus, the heavyand light-hole states are split due to both confinement and stress caused by the lattice mismatch between the compositional materials. In the CdTe/CdMgTe heterosystem, these effects act in the same direction pushing the light-hole states towards higher energies and increasing the g-factor anisotropy [5.54].

Detailed measurements have been performed on the transverse electron g factor g as a function of the well width a for GaAs/Al0.3Ga0.7As QW struc-

≈ ˚

tures [5.55–5.57] and showed the sign change of g at a 65 A. Moreover, a considerable di erence between g and g has been observed on A3B5 and A2B6 based heterostructures, namely in GaAs/AlGaAs, GaAs/AlAs, GaInAs/InP, and CdTe/CdMgTe, under optical orientation of free carriers in tilted magnetic fields [5.58–5.60], in Optically Detected Magnetic Resonance (ODMR) experiments [5.61, 5.62], by using quantum beat technique [5.63], and in resonant spin-flip Raman scattering [5.54, 5.64, 5.65].

where ηα;ss = e1, s|σα|e1, s and the magnetic-field induced perturbation

is given by

δH = −ec Avˆ ,

vˆ being the velocity operator. Note that, for a homogeneous magnetic field, the vector-potential is a linear function of the radius-vector r. The di erence between ηα;ss and σα,ss arises from an admixture in the state |e1, s of the opposite spin −s, see (2.44, 2.47). Numerical estimates show that this di erence is very small in all particular cases considered below and can be neglected.

In the Kane model, the velocity operator vˆ = −1∂H(k)/∂k is an 8 × 8 matrix with k-independent components. Using the explicit form for this matrix we obtain the Zeeman Hamiltonian in the form

This particular equation can be used while calculating the electron g factor in QWRs and QDs as well as the transverse g factor in QWs. Indeed, if the wave function is localized in the direction ζ the diagonal matrix element of the coordinate ζ is no more a functional and the matrix element of δH for B ζ can be calculated avoiding its transformation into the sum in (5.89). While calculating the longitudinal g factor in a QW one cannot find such a direction and needs other methods based on the application of (5.89), numerical calculation of the Landau levels and their spin splitting and so on [5.53].

It follows from the symmetry considerations that the electron g factor in a spherical QD or a cylindrical QWR is isotropic. Moreover, one can show that in these two particular cases (5.100) can be reduced to [5.53]

where d is the dimensionality of a nanostructure, d = 0 for a QD and d = 1 for a QWR, Vn is a volume of the sphere in the n-dimensional space: V2 = πR2

for a QD or dρf 2(ρ) for a QWR taken respectively over the A and B volumes. Note that the sum wA + wB slightly di ers from unity because of an admixture of the valence band states in the conduction-band wave function. An important point is that equation (5.101) describes as well the transverse g factor in a QW of the width a = 2R. In this case the dimensionality d = 2, the volume V3−d = 2R and f (R) ≡ f (a/2).

Figure 5.8 shows the dependence of the electron g factor on the radius R in QDs and QWRs calculated for the isomorphic nanostructure GaAs/AlGaAs and pseudomorphic structure Ga0.47In0.53As/InP. In addition, in the same graph, variation of the longitudinal and transverse g factors in a QW is presented. The parameters used in the calculation of the first heterosystem are as follows: Eg = 1.52 eV, ∆ = 0.34 eV, 2|pcv |2/m0 = 28.9 eV for bulk GaAs, and Eg = 1.94 eV, ∆ = 0.32 eV, 2|pcv |2/m0 = 26.7 eV for bulk GaAs/Al0.35Ga0.65As, the band o set ratio Vh:Vc = 2:3. The contribution of remote bands is taken into account by adding the constant ∆g = −0.12 to the Kane-model values of g. For the second heterosystem the following parameters were used: Eg = 0.813 eV, ∆ = 0.356 eV, 2|pcv |2/m0 = 25.5 eV for bulk Ga0.47In0.53As, and Eg = 1.423 eV, ∆ = 0.108 eV, 2|pcv |2/m0 = 20.4 eV for InP, the valence band o set Vh = 0.356 eV, the contribution from remote bands ∆g = −0.13. With increasing R the curves approach the bottom value of g in the bulk material A, in agreement with (5.101) where, for R → ∞, wA saturates to unity and values of wB, E and V3−d(R)f 2(R) tend to zero. The asymptotic behavior of g at large R is described by g(R) = gA(0) + ∆g +

(Rd/R)2 with the hierarchy R0 > R1 > R2 > R2, where R2 , characterize the convergence of g and g in QWs. In the opposite limit, R → 0, the

curves g(R) tend to the g factor value in the bulk barrier semiconductor,

gB(0) + ∆g = 0.57 in Al0.35Ga0.65As and 1.2 in InP. The relation gQD > grmQW R > gQW can be understood taking into account that the reduction

in dimensionality enhances the role of the electron spatial confinement. The estimation shows that the contribution of the term proportional to f 2(R) in (5.101) is not relatively small. In general, an approximate description of the dependence g(R) is not applicable in the simple form gA(E)wA + gB(E)wB + ∆g. Thus, Fig. 5.8 demonstrates main features of the e ect of dimensionality on the electron g factor.

Fig. 5.9. The electron g factor components gαα (α = x, y, z) in a GaAs/AlGaAs rectangular QWR vs. the length, 2b, of one side. The other side, 2a, is kept constant. The insert depicts the orientation of the coordinate system. The arrows indicate

˚

values of the transverse and longitudinal g factors in the 80 A-thick QW. [5.53]

In rectangular QWRs, the tensor gαβ is characterized by the three di erent diagonal components, gxx, gyy and gzz . The corresponding Kane-model electron wave functions were discussed in Chap. 2, see Fig. 2.3. Figure 5.9 shows the dependence of gαα (α = x, y, z) in GaAs/Al0.35Ga0.65As QWRs on one of the rectangular sizes while another size is kept constant and equal to

˚

2a = 80 A. With increasing b the components gyy and gzz converge on the

246 5 Photoluminescence Spectroscopy

˚

electron transverse g factor in a QW of the thickness 80 A and the component gxx approaches the QW value of g . In a square-shaped QWR, b = a, the two components gxx and gyy coincide as the symmetry predicts. Moreover, one can see from Fig. 5.9 that at the point b = a the anisotropy |gzz − gxx| is in fact quite small because, as it follows from (2.66), the function hz (x, y) in (2.65) vanishes at the four lines x = 0, y = 0, y = x, y = −x and its values are too suppressed to produce a significant contribution to g.

A theory of the Zeeman splitting of electron spin states in biased QW structures was developed in [5.52]. The electric-field induced in-plane anisotropy of the g factor, due to the nonzero o -diagonal components gxy , gyx, was predicted by Kalevich and Korenev [5.66] and observed by Hallstein et al. [5.67].

In heterostructures, the heavy-hole e ective g factor exhibits a strong anisotropy. Moreover, in the approximation of axial symmetry the in-plane hh1-hole g factor is zero at all. Kiselev and Moiseev developed a theory of the Zeeman spin splitting of size-quantized heavy holes in QWs and SLs in the multi-band envelope-function approximation, with a satisfactory agreement with experiment (see [5.68] and references therein). As far as in-plane magnetic fields are concerned, the transverse e ective g factor gh in the subband hh1 is nonzero due to the anisotropic parameter q in (2.31) and can

where as before the indices A and B refer to the well and barrier compositional materials, and wA,B is the probability to find a heavy hole in the quantum well or barrier. From comparison between theory and experiment on hole spin quantum beats in n-modulation-doped GaAs/AlGaAs QW structures a value of |gh | was found to be 0.04 ± 0.01 [5.69] which is smaller than the longitudinal g factor, gh , almost by two orders of magnitude.

5.3.4 Spin Quantum Beats in Photoluminescence

In order to elucidate the main idea of transient quantum beats let us first consider a general two-level quantum system with the initial state which is a linear superposition

ψ(t = 0) = C1|1 + C2|2

of the two eigenstates |1 and |2 with the eigenenergies E1, E2. The wave function of the unperturbed system varies in time as

one has for the signal registered by the detector

5.3 Optical Spin Orientation of Free Carriers 247

| | |2 | ¯ |2 | ¯ |2 ¯ ¯ −

D t = C1C1 + C2C2 + 2 Re C1C1 C2 C2 exp [i(E1 E2)t/ ] .

Thus, if ψ(t = 0) and |D are not the pure states |1 , |2 then the signal has an oscillating component with the oscillation period T = 2π /|E1 − E2|.

As an example we take

Then the squared scalar product | D|t |2 describing the measured signal is equal to 0.5{1 ± cos [(E2 − E1)t/ ]}.

Now we will illustrate the idea of quantum beats by the example of the transient Hanle e ect. We assume that the pulsed photoexcitation generates at t = 0 spin-polarized photoelectrons in the conduction band. We use the notations n0 ≡ n(t = 0) and s0 = (0, 0, s0z ) for the initial electron density and spin density, where s0z = æPc0n0/2 and Pc0, æ are introduced in (5.53). The time dependence of n and s in an external magnetic field B z can be derived by solving equation (5.55) where the derivatives dn/dt and ds/dt are added in the left-hand sides and the steady-state generation rates g, s˙ are set to zero. For B x, the solutions n(t), sx(t) are easily written: n(t) = n0 exp (−t/τ0), sx = 0. As for two other components of s, it is convenient to transform a pair of two equations for the real components sy , sz into one equation for the complex combination s+ = sz + isy

From this it follows that

sz (t) = s0z e−t/T cos ΩLt , sy (t) = −s0z e−t/T sin ΩLt .

The intensities of the PL circularly-polarized components are related to the total intensity I and the Stokes parameter Pc by I± = I(1 ± Pc)/2. Taking into account that, according to (5.53), Pc = æp = 2æsz /n, we obtain for the transient Hanle e ect

In order to analyze (5.105) in terms of the quantum beats we rewrite the spinor wave function ψ(t) of a photoelectron as a linear combination of the spin eigenstates |1 , |2 in the magnetic field B x which, in fact, are eigenstates of the Pauli matrix σx, namely

248 5 Photoluminescence Spectroscopy

We assume the initial electron spin to be polarized along z and neglect, for simplicity, the spin relaxation and decay due to the electron recombination. Then, the electron state is described by the time-dependent wave function

For interband optical transitions hh1 → e1 in a QW structure the factor æ in (5.105) equals −1. If the analyzer detects the circularly polarized photons,

we obtain D|t = e−iΩL t/2(1 eiΩL t)/2 or I± | D|t |2 = (1 cos ΩLt)/2 with the quantum-beat period 2π/ΩL. Allowance for the finite lifetime and spin relaxation leads to a multiplication of unity by e−t/τ0 and cos ΩLt by e−t/T in agreement with (5.105).

Note that, in a tilted magnetic field with Bz = 0, Bx = 0, equation (5.105)

where ΩL = Ω2 + Ω2 , Ω = g µB Bz , Ω = g µB Bx and the uniaxial anisotropy of the g factor is taken into account. Quantum beats in the exciton PL are mentioned in Sect. 5.5.

5.4 Hot Photoluminescence in Quantum-Well Structures

By the term hot photoluminescence of free carriers we mean the emission arising due to the radiative recombination of photoelectrons with their kinetic energies exceeding the average energy of the thermalized electrons. The emission occurs during the photoelectron energy relaxation to the conduction-band bottom and is usually related to band-to-acceptor transitions [5.70–5.73]. The hot-electron energy distribution is non-Maxwellian and, at low temperatures and low doping level, has a discrete character because of the consecutive emission of longitudinal optical phonons by the hot electrons.

The hot PL spectroscopy can be used to measure the valence subband dispersion in QW structures and SLs [5.73]. To simplify the picture let us ignore the in-plane anisotropy, or warping, of the QW subbands. Then to obtain the subband dispersion information, monochromatic photons are absorbed in a QW to generate hot electrons of energy Ee and hot holes of energy Eh. Most

of the hot electrons relax by phonon emission. However, a few of them immediately recombine radiatively at a neutral acceptor intentionally introduced as a dopant into the well and emit photons of the energy EP L. By measuring the hot PL energy for a given initial photon energy ω, values of Ee and Eh may be directly derived from the energy and wave-vector conservation laws

Here k = |k|, k is the electron wave vector, Eg is the QW band gap and Ea is the binding energy of a hole at the acceptor. The conduction-subband dispersion, Ee(k), is relatively simple and can be readily derived by using independent information. Therefore, the wave vector k is obtained from the first equation (5.107). According to (5.107), a value of Eh(k) equals to ω − Eg − Ee(k). By measuring the hot PL as a function of laser photon energy, the dispersion of the valence subbands can be mapped out.

The spin optical orientation for thermalized and hot electrons mostly differs quantitatively and, with some modifications, the principles formulated in Sect. 5.3.1 are valid for circular polarization of hot PL as well. A quite new phenomenon is the linear polarization of hot PL. It appears under optical excitation by linearly-polarized light and depends on the direction of the light-polarization unit vector e. It is due to the alignment of photoelectron momenta which, in turn, is related to the form of the valence-band wave functions in semiconductors with a degenerate valence band. First we explain this phenomenon for a bulk semiconductor with a zinc-blende lattice. For transitions from the heavyand light-hole subbands, hh and lh, to the conduction band c, the transition rate Wc,j (j = hh, lh) is a function of the angle θ between the electron wave vector k and the polarization unit vector e. Neglecting the warping of the valence band, one obtains, see, e.g., [5.74]

where P2(t) = (1/2)(3t2 − 1) is the second Legendre polynomial and αhh = −1, αlh = 1. Note that for α = −1 the transition rate is just proportional to sin2 θ. Thus, the momenta k of electrons excited from the heavy-hole subband are mostly normal to e and the initial distribution of electrons photoexcited from the light-hole subband is stretched along e. Recombination of the hot photoelectrons with the anisotropic distribution in the k-space leads to a linear polarization of the hot PL. In accordance with (5.108) the distribution function of the relaxing hot electrons is given by

¯

where f is the isotropic part of the distribution function and α is the parameter of anisotropy dependent on the electron kinetic energy Ee. The hot-PL intensity is determined by the integral

250 5 Photoluminescence Spectroscopy

I(ω ) [1 + αrP2(cos θ )]fkdΩ , (5.110)

where dΩ is an element of the solid angle, ω is the selected PL frequency, ω and k are related by Ee(k) + Eg − Ea = ω , and θ is the angle between k and the polarization unit vector, e , of the secondary radiation. The band- to-acceptor optical matrix element is proportional to the Fourier component, Ψa(k), of the spinor wave function of the bound-to-acceptor hole. At large k, the function Ψa(k) is formed by heavy-hole states and the anisotropy parameter αr is close to αhh = −1. Substituting fk from (5.109) into (5.110) and integrating over the solid angle one obtains

where χ is the angle between e and e . In the back-scattering geometry, the degree of linear polarization of the hot PL is given by

where I , I are the intensities of the radiation polarized parallel and perpendicular to the initial polarization e. For αr = −1 and αj = 1 we obtain

P= 1/7 and P = −3/9, respectively. Thus, the sign of polarization for the radiation of electrons excited from the heavy-hole subband coincides with that of the exciting light, and for the electrons excited from the light-hole subband, the PL polarization has the opposite sign.

In bulk semiconductors the linear polarization P weakly depends on the initial photoelectron kinetic energy. The main feature of the hot-electron PL in QWs, as compared with bulk crystal, is its strong dependence on the kinetic energy Ee. Values of P in the 2D structures vary from P = 0 at Ee = 0 up to

P= 0.5 at kinetic energies exceeding the confinement energy. This result can be understood taking into account that the valence-subband wave functions at kx = ky are pure states of heavy and light holes. Therefore, under optical

excitation at the band edge Eg,hhrmQW = Eg + Ee1 +Ehh1 or Eg,lhQW = Eg +Ee1 + Elh1, the optical transition rate is independent of the angle ϕ between e and

the 2D wave vector k , the anisotropic alignment in the k -space is absent and P = 0. However, at k = 0 each subband state is a hybrid of heavyand light-hole states and the alignment becomes possible. Similarly to (5.108), the angular dependence of the 2D-photoelectron distribution function can be presented as

In order to estimate the anisotropy parameter α2D for transitions hh1 → e1, we remind that, in a bulk semiconductor, the transition rate Wc,hh(k) is proportional to sin2 θ and, for e x, the latter can be rewritten as the ratio