ivchenko_bookreg

E21∆(2Eg + ∆)

Λ = 2Eg (Eg + ∆)(3Eg + 2∆) ,

written in the basis of spin states with sz = ±1/2. In order to perform a calculation taking into account all powers of βλµ one needs to use (8.23) and (8.24) in a straightforward way. As soon as we are interested in contributions to photocurrents linear in β we can set all β’s to zero except for one and proceed similarly to the Cs-symmetry case. For example, we retain the term βyx(ν)σy kx in (8.31) and disregard the term proportional to βxy(ν). The corresponding current is induced in the x-direction perpendicularly to the plane (y, z) of oblique incidence:

Then the eigenstates have a fixed spin component on the y axis and the spin split energies are determined by (8.25) where βν = βzx(ν) is changed by βyx(ν) and ± means spin states with sy = ±1/2. Since the component ez is present in (8.33) and the spin under z-polarized transitions is conserved, see (8.32), only spin-conserving processes (e1, sy ) → (e2, sy ) contribute to the circular photocurrent jx. From (8.32) one can find the corresponding matrix elements of the velocity operator

e2, sy |e · vˆsz sz |e1, sy = v21(ez + 2iΛsy ex)

and, hence,

where the term quadratic in Λ is neglected. The second consequence of the spin conservation is that, instead of (8.26), the wave vector component kx of electrons involved in the transitions is expressed not via the sum but via the di erence of β coe cients

An important conclusion is that the photocurrents (8.30) and (8.35) change their signs within the resonance absorption spectrum. If the inhomogeneous broadening has a Gaussian character, i.e., if

the inversion point lies at the photon energy

Usually τ1 > τ2 and the inversion point should be blue-shift relative to the

¯

resonance energy E21 determined from light transmission spectral measurements. However, for τ1 τ2 this blue shift is negligible. The sign inversion of the circular photocurrent in the resonant e2-e1 transition region has been recently observed in n-type GaAs/AlGaAs QW samples [8.12].

In contrast to electrons in the conduction band, the energy dispersion of holes in the valence band of QWs is essentially nonparabolic and intersubband absorption can involve simultaneously di erent pairs of subbands hhν and hhν . However, with some modifications of the theory and complications in calculations the intersubband circular CPE in p-doped samples can be considered in a way similar to that in n-QWs.

Now we turn to intrasubband optical transitions

(e1, k, s) + ω → (e1, k , s )

in the lowest electron subband e1. They are indirect in the k space, occur due to additional scattering by phonons or static imperfections and involve virtual intermediate states. This situation is realized in n-doped QWs for photon energies to be not high enough in order to excite direct intersubband transitions. The intrasubband photocurrent is given by the general equation (8.18) where the generation matrix is a sum of contributions due to the ingoing and outgoing electrons. The corresponding generation matrices have

374 8 Photogalvanic E ects

where the index n enumerates the intermediate states, Mn k;nk and Vn k ;nk are the matrix elements of the electron-photon and electron-phonon or electron-defect interaction, Ω is the phonon frequency, the sign ± corresponds to emission and absorption of phonons. For the scattering by static defects Ω is set to zero. An important point is that indirect transitions via intermediate states in the same subband do not contribute to the circular PGE. The effect appears if virtual processes involve intermediate states in other bands or subbands, n = e1. The same takes place for intrasubband optical orientation of electronic spins in the conduction band. The latter is discussed in more detail in the next subsection in connection with the spin-galvanic e ect.

8.2 Spin-Galvanic E ect

The mechanisms of the circular PGE discussed so far are linked with the asymmetry in the momentum distribution of carriers excited in optical transitions which are sensitive to the light’s circular polarization due to selection rules. Now we discuss an additional possibility to generate a photocurrent sensitive to the photon helicity [8.13, 8.14]. In a system of free carriers with non-equilibrium spin-state occupation but equilibrium energy distribution within each spin branch, the spin relaxation or Larmor precession in an external magnetic field can be accompanied by generation of an electric current. Phenomenologically, this linkage between an electric current and the total electronic spin s is described by

µ

The symmetry of the second-order pseudotensor Q coincides with that of the tensor γ describing the circular PGE, see (8.1). Similarly, its non-vanishing components can exist in non-centrosymmetric systems belonging to one of the gyrotropic classes. In (001)-oriented QWs of the C2v symmetry equation (8.41) reads

If the non-equilibrium spin is produced by optical orientation and the spin sµ is proportional to the degree of light circular polarization Pc the current generation can be reputed just as another mechanism of the circular PGE. However the non-equilibrium spin s can be achieved both by optical and non-optical methods, e.g., by electrical spin injection, and, in fact, (8.41) presents an independent e ect called the spin-galvanic e ect. Here we bear in mind spin-induced electric currents that appear under uniform distribution of the spin polarization in the 3D-, 2Dor 1D space, respectively, in a bulk semiconductor, a QW and a quantum wire. In this sense the spin-galvanic e ect di ers from surface currents induced by inhomogeneous spin orientation [8.15] and other phenomena where the spin current is caused by gradients

of potentials, concentrations, etc., like spin-voltaic e ect which occurs in inhomogeneous samples, e.g., the ‘paramagnetic metal-ferromagnetic’ junction or p-n junction.

Fig. 8.4. Current jx as a function of the magnetic field B for normally incident right-handed (open circles) and left-handed (filled circles) circularly polarized radiation at λ = 148 µm and radiation power 20 kW. Measurements are presented for an n-GaAs/AlGaAs single heterojunction at T = 4.2 K. Curves are fitted from (8.43) using the same value of the spin relaxation time τs and scaling of the jx value for both the solid and dashed curves. From [8.14].

Usually the circular photogalvanic and spin-galvanic e ects are observed simultaneously under illumination by circularly polarized light and do not allow experimental separation. However, they can be separated in time-resolved measurements. Indeed, after removal of light or under pulsed photoexcitation the circular photocurrent decays within the momentum relaxation time τp whereas the spin-galvanic current decays with the spin relaxation time. Next we consider a geometry of experiment under steady-state photoexcitation which allows to observe the spin-galvanic e ect and exclude the circular PGE [8.14]. The geometry is depicted in inset in Fiq. 8.4. The circularly polarized light is incident normally to the interface plane (001) of a QW, the light absorption yields a steady-state spin orientation s0z in the z direction proportional to the spin-generation rate s˙z . The symmetry of (001)-grown QWs forbids generation of a current proportional to the normal component of s. To obtain an in-plane component of the spins, necessary for the spingalvanic e ect, a magnetic field B x is applied. Due to Larmor precession a non-equilibrium spin polarization sy is induced,

where τs = √τs τs , τs , τs are the longitudinal and transverse electron spin relaxation times, ΩL is the Larmor frequency. The photocurrent measured in the x direction is depicted in Fig. 8.4 as a function of the magnetic field for two opposite circular polarizations of the light. In accordance with the phenomenological equations (8.42) and (8.43) the current jx exhibits nonmonotonous variation with the magnetic field. Comparison with theory allows to find a product gτs and the spin relaxation time if the electron g-factor is known.

There are two di erent microscopical mechanisms of the spin-galvanic effect, namely, kinetic and relaxational [8.13]. The experimental data of Fig. 8.4 can be understood in terms of the kinetic mechanism. It is inherently connected with the spin dependency of matrix elements, Mk s ,ks, of electron

scattering by impurities, other static defects and phonons. It is convenient to

× ˆ ˆ represent the 2 2 matrix Mk k as a linear combination of the unit matrix I

and Pauli matrices as follows

where Ak k = Akk , Bk k = Bkk due to hermiticity of the interaction and A−k ,−k = Akk , B−k ,−k = −Bkk due to the symmetry under time inversion. For the conduction subband e1 in a (001)-grown QW the electronphonon deformation-potential Hamiltonian has the form

+Ξcv Lcv {u12 [σ1(k2 + k2) − σ2(k1 + k1)] + σ3 [u31(k1 + k1) − u32(k2 + k2)]} .

Here k , k are the electron final and initial wave vectors, uij are the strain tensor components induced by acoustic vibrations, Ξc and Ξcv are the conduction-band and interband deformation potential constants, the indices 1, 2, 3 represent the principal axes [100], [010], [001],

One can readily find the scalar Ak k and pseudovector Bk k for this particular

¯

scattering mechanism. In the axes x [110], y [110], z [001] the first term in (8.45) remains invariant while the expression in braces in the second term of (8.45) becomes

12 (uxx−uyy ) σx(ky + ky ) − σy (kx + kx) +σz uzx(ky + ky ) + uzy (kx + kx) .

The spin-galvanic current observed in the geometry of Fig. 8.4 is caused by the asymmetric spin-flip scattering of spin-polarized electrons in the systems

Fig. 8.5. Microscopic origin of the spin-galvanic current in the presence of k-linear terms in the electron Hamiltonian. The σy kx term in the Hamiltonian splits the conduction band into two parabolas with the spin ±1/2 in the y direction. If one spin subband is preferentially occupied, asymmetric spin-flip scattering results in a current in the x direction. The rate of spin-flip scattering depends on the value of the initial and final k-vectors. There are four distinct spin-flip scattering events possible, indicated by the arrows. The transitions sketched by dashed arrows yield an asymmetric occupation of both subbands and hence a current flow. If, instead of the spin-down subband, the spin-up subband is preferentially occupied the current direction is reversed.

with k-linear contributions to the e ective Hamiltonian. Figure 8.5 illustrates the electron energy spectrum with the βyxσy kx term included. Spin orientation in the y direction causes an unbalanced population in the spin-down and spin-up branches. Spins oriented in the y direction are scattered along kx from the higher filled branch, say the spin-up or |1/2 y branch, to the less filled branch | − 1/2 y . The matrix elements for these spin-flip processes are proportional to the components Bk k,x and Bk k,z of the vector Bk k in (8.44).

Four di erent spin-flip scattering events are schematically sketched in Fig. 8.5 by arrows. Their probability rates depend on the values of the wave vectors of the initial and final states. Spin-flip transitions shown by solid arrows have the same rate. They preserve the symmetric distribution of carriers in the branches and, thus, do not yield a current. The two processes indicated by broken arrows are not equivalent and generate an asymmetric carrier dis-

378 8 Photogalvanic E ects

tribution around the branch minima in each spin branch. This asymmetric distribution results in a current flow along the x direction.

In considering the relaxational mechanism of the spin-galvanic e ect we can ignore spin-dependence of the scattering matrix elements but should

electron kinetic energy. We apply the spin density matrix formalism used in Chap. 5 while describing the electron spin relaxation and, in the same way, assume the following hierarchy of relaxation times to be fulfilled

where τp, τε, τs are, respectively, the electron momentum, energy and spin relaxation times, τ0 is the electron lifetime in case of the interband optical photoexcitation.

The conditions (8.46) allow a straightforward solution of the kinetic equation for the electron spin density matrix

tonians. The current is given by the standard equation (8.16). In what follows we derive a contribution to the current linear in the coe cients βλµ govern-

of thermal equilibrium the matrix ρk for electrons with non-equilibrium spin polarization S per particle can be written as

where f 0(E) is the Fermi-Dirac distribution function, {M N }s = (M N + N M )/2, Hk is the electron e ective Hamiltonian at zero magnetic field, i.e., the sum Ek0 +Hk(1) with Ek0 being 2k2/(2m ). Note that the total spin density s in (8.41) is a product of S and the electron density Ne. Practically, one should keep only zeroand first-order terms and replace (8.48) by

First of all we demonstrate that the distribution (8.49) leads to no electric current and the collision integral vanishes upon substitution of the matrix ρ0k. In order to demonstrate the absence of current we notice that the sum

j = e Tr vˆ(k)ρ0k

k

and also vanish. The second property, Qk{ρ0} ≡ 0, can be checked if we write the collision integral for any particular mechanism of electron scattering. Say, for elastic scattering with spin-independent scattering matrix elements Vk k one has

Here the curly brackets with the subscript s mean the antisymmetrization of matrices, Ni is the concentration of static defects and the δ-function is understood as

Substituting (8.49) into (8.50) and retaining zeroand first-order terms we obtain zero by using the identity

F (Ek0 ) − F (Ek0 ) ∂E∂ k0 δ Ek0 − Ek0 = −δ Ek0 − Ek0 ∂E∂ k0 F (Ek0 )

for any analytical function F (E).

We demonstrate the relaxational mechanism under conditions of interband optical orientation by circularly polarized light and then briefly discuss the intraband excitation. Since τε τ0 it is not necessary to study the relaxation of the energy of hot photoelectrons to the bottom of the conduction band, and the generation matrix Gk in (8.47) can be written in the form

380 8 Photogalvanic E ects

where g is the rate of optical excitation of electrons in the conduction band, n is their density and S0 is the average spin of a photocreated electron relaxing to the conduction-band bottom. As in the case with ρ0, one needs to expand (8.51) up to the first-order terms

We use the simple momentum-relaxation model and present the collision integral in the form

ρk − ρ0

Qk{ρ} = k . (8.52)

τp

While solving the kinetic equation (8.47) we write the spin density matrix as

and assume δρk to be a small correction. It is an odd function of k.

Under steady-state photoexcitation ρk is time-independent, and the first term in the left-hand side of (8.47) vanishes. After averaging the terms of

where ΩL is the Larmor vector frequency, and τs−1 is the spin relaxation rate including the contribution due to the Dyakonov-Perel’ mechanism related to the term (i/ )[Hk(1), ρk] in (8.47). These are the balance equations discussed in Chap. 5. Now we derive, from (8.47), an equation for δf = Tr{δρ/2}. It has the form

and, therefore, the correction δfk is given by

One can see that it is smaller than the last term in (8.49) by a factor of τp/τ0. However, this is the correction contributing to the current and for a nondegenerate gas of photoelectrons one obtains

or, according to (8.54),