ivchenko_bookreg

If S is varying in time and the characteristic time of variation is long as compared to τp, τε then (8.56) is generalized to

which can be just as well converted into (8.57).

In order to consider the relaxational mechanism under intraband optical orientation of electronic spins in n-doped QWs the lifetime τ0 in (8.47) should be set to infinity. The intraband generation matrix has the form

,'

0 H · ˙

Gk = 2 f ( k) , σ S , (8.59)

s

˙

where S is the spin generation rate per particle. Then we obtain for the correction δfk contributing to the current

˙

Under normal incidence of the light on a (001)-grown QW the vector S is directed along z and the current is zero. Thus, the relaxational mechanism makes no contribution to the spin-galvanic current in the set-up of Fig. 8.4 and the latter is completely related to the kinetic mechanism.

The radiation of the CO2 laser causes direct e2-e1 optical transitions in GaAs/AlGaAs MQWs. It can induce the resonant spin-galvanic current at normal incidence of radiation in the presence of an in-plane magnetic field, as this e ect was observed under intrasubband transitions (Fig. 8.4).

˙

Since the spin generation rate Sz K Kz , where K , Kz are defined in (4.21) the spectral behavior of the spin-galvanic current must coincide with the absorption spectrum. One can see from Fig. 4.5 that the wavelength dependence of the spin-galvanic e ect obtained between 9.2 µm and 10.6 µm, indeed, repeats the spectrum of the intersubband absorption.

8.3 Photon Drag E ect

In the calculation of the circular PGE, the photon momentum q was neglected. The photon drag e ect described by the tensor T in (8.1) is entirely owing to the existence of a photon momentum. Classically, the momentum carried by electromagnetic waves can manifest itself through a radiation pressure on a surface as described already by Maxwell. Light pressure e ects are

382 8 Photogalvanic E ects

found in several fields of physics, namely, astrophysics, atomic and molecular physics, solid-state physics, including metals and semiconductors, bulk materials and nanostructures. If the absorption in a solid is caused by free-carrier optical transitions, intraband, interband, bound-to-continuum etc., these carriers acquire a directed motion due to momentum transfer from photons to the particles. In bulk crystals the photon drag current was first observed experimentally in the microwave region [8.16] and in quantum transitions between the valence subbands in Ge excited by a CO2 laser [8.17, 8.18]. New features of the photon drag e ect in a 2D electron gas were predicted by Vasko [8.19], Luryi [8.20], Grinberg and Luryi [8.21] and observed experimentally by Wieck et al. [8.22]

A reasonable estimation of the light-induced drift of quantum-confined carriers can be obtained on the basis of the following simple argument applied for example to a QW. If η is the relative absorbance and q is the photon in-plane wave vector component then the rate of momentum transfer per particle per unit area is given by

ηI

F = q ωNe .

This is nothing but the drag force acting upon each electron. It gives rise to a drift velocity vd = (τp/m )F and a current density

The same estimation follows from the momentum and energy conservation

for the direct interband optical transition (e1, k) → (e2, k + q ). The kinetic energy 2(k + q)2/(2m ) has a linear-k term, 2(k · q)/m , which is reminiscent of the linear-k spin-dependent term in (8.8). One can see that the current (8.61) is obtained from the estimation (8.12) for the circular PGE as soon as the velocity, β/ , related to the electron linear-k dispersion is substituted by the velocity, q /m , of an electron with the momentum q . Continuing this analogy we can use (8.10), change βe by 2q /me, set βv to zero and Pc to −1, use the identity memh/µ = me + mh ≡ M and finally come to the interband photon drag current induced under normal incidence

The photon drag e ect induced by the e2-e1 resonant transitions can be found

where for the sake of brevity η21 stands for η21(e z). It is worth to note that q |ez |2 = q sin3 θ. In Fig. 8.6 the spectral response of the photon drag e ect in a modulation-doped GaAs/Al0.35Ga0.65As MQW structure is presented [8.23]. A Lorentzian absorption is taken for the fit using (8.64). The spectral line shape of the photon drag current yields a relaxation time ratio τ1/τ2 = 1.6.

Fig. 8.6. Photon drag spectrum at room temperature of the modulation-doped GaAs/AlxGa1−xAs MQW sample, measured at a micropulse intensity of 2.5 MW/cm2. For a better signal to noise ratio, an integration is made over the entire macropulse of FELIX (5.6 µs). Solid line is the best fit according to (8.64). From [8.23].

It is instructive to compare equations (8.63, 8.64) derived for particular mechanisms with the general symmetry considerations. For QWs of the symmetry C2v , the phenomenological description (8.1) gives for the photon drag e ect

=x,y,z

384 8 Photogalvanic E ects

In symmetrical QWs characterized by the symmetry D2d some of the above coe cients Tλµνη are interrelated, namely

Txxzz = Tyyzz , Txxxx = Tyyyy , Txxyy = Tyyxx , Txyxy = Tyxxy . (8.66)

The photon drag current (8.63) corresponds to equal Txxxx and Tyyyy while the photocurrent (8.64) is described by the coe cients Txxzz = Tyyzz . For more complicated band structures these relations can be violated and, moreover, all other coe cients in (8.65) become nonzero.

8.4 Linear Photogalvanic E ect

For the linear PGE, the phenomenological equation (8.1) in QWs of the C2v symmetry reduces to

jLP GE,x = χxxz (exez + ez ex) I , jLP GE,y = χyyz ey ez + ez ey I . (8.67)

In symmetrical QWs of the point-group D2d, the pair of coe cients are linearly dependent, χxxz = −χyyz .

Fig. 8.7. Photogalvanic current in a (113)-grown Si0.75Ge0.25(5 nm)/Si single QW normalized by the light power P as a function of the phase angle ϕ. The results are obtained under normal-incidence irradiation at λ = 280 µm at room temperature. The full line is fitted according to (8.70). Broken and dotted lines show jx sin 2ϕ and jx sin 2ϕ cos 2ϕ, respectively. From [8.9].

The Cs symmetry allows both circular and linear PGEs for normal incidence because in this case the tensors γ and χ have the additional nonzero components γxz , see (8.2), χxxy = χxyx, χyxx and χyyy . As a result, under normal incidence one has

jx = γxz Pc + χxxy exey + ey ex I , jy = χyxx|ex|2 + χyyy |ey |2 I . (8.68)

where χ± = (χyxx ± χyyy )/2 and α is the angle between the plane of polarization and x. Figure 8.7 presents the measured dependence of jx and jy as a function of the angle α and the fit to (8.69) for a p-type SiGe (113)-grown asymmetrical QW structure. In the experimental setup, where the laser light is linearly polarized along x and a λ/4 plate is placed between the laser and the sample, (8.68) takes the form

where ϕ is the angle between the initial plane of linear polarization and the optical axis of the polarizer. The circular and linear polarizations of the incident light vary with ϕ in accordance to Pc = cos ϕ, see (8.5), and Pl = sin ϕ. In Fig. 8.7 experimental data and a fit to these functions are presented for the same p-type SiGe (113)-grown QW structure.

The linear PGE was observed in some insulators as early as the 1950s, and possibly even earlier, but was correctly identified as a novel phenomenon only in 1974-75 [8.24, 8.25]. In semiconductors, the linear PGE was first observed on tellurium [8.26, 8.27] and then studied in detail on p-GaAs [8.28].

Microscopically, a current of the linear PGE consists of the so-called ballistic and shift contributions [8.29–8.32]. The first of them is described by the

Here the index n describes all quantum numbers characterizing the electron eigenstates, namely the band and subband labels, spin sublevel and wave vector k; the probability transition rate from the state n to n is given by Fermi’s golden rule

Mn n is the transition matrix element, vn and τp(n) are the electron velocity and momentum relaxation time in the state n, fn is the distribution function, or the occupation, of the state n. The energy En includes the photon or

386 8 Photogalvanic E ects

phonon energy in the initial or final state. Equation (8.71) is a contribution to the general expression for the current (8.13) of diagonal components of the electron density matrix, ρnn = fn, and of the velocity vnn ≡ vn. The ballistic current is nonzero only if one simultaneously includes in Mn n carrier interaction both with a photon and with another particle, a phonon, impurity or static defect, another electron or hole, including a geminate partner photocreated in the same photoabsorption process. In other words one needs to go beyond the Born approximation in calculating Mn n.

The second contribution to the linear PGE current comes from inclusion in (8.13) of the nondiagonal components ρnn and vnn with n = n. This current was shown [8.31] to originate from the shift of the wave packet’s center-of-mass in quantum transitions and can be written as

where Φn n is the phase of the transition matrix element, k and k are the wave vectors in the states n and n , Ωn is the diagonal matrix element of the coordinate

Ωn = i un kun dr ,

and un(r) is the Bloch periodical amplitude. In a steady-state regime, when the processes of generation, scattering and recombination are taken altogether into consideration, the contributions associated with Ωn vanish since they describe the static charge redistribution. The first term in the right-hand side of (8.74) can be rewritten as

This form is useful for practical calculations. It is worthy of note that a shift of an electron under the quantum transition has a physical tie to the wellknown Goos-H¨anchen e ect in classical optics [8.33]. In the latter, the totally reflected beam is spatially displaced with respect to a ray reflected by the geometrical interface between the two media. The physical connection becomes apparent when considering the totally reflected beam a superposition of plane waves and taking into account that the phase shift under reflection is a function of the angle of incidence [8.34, 8.35].

To illustrate, we estimate the linear PGE under the hh1 → e1 interband transitions induced by linearly polarized light normally incident on a [hhl]- grown QW (h = 0). The shifts (8.75) become nonzero if both even and odd terms in k are included into the interband matrix elements. For transitions

from the heavy-hole subband the optical matrix elements can be presented by

Mcv (k) ≡ Me1,±1/2,;hh1,±3/2(k) = [iM0(ex ± iey ) + Q(λµ±)kλeµ] . (8.76)

The first term on the right-hand side is the main contribution described by Table 2.2 while the complex coe cients Q(λµ±) describe linear-k corrections. They satisfy the conditions

Q(λµ−) = Q(+)λµ , Q(λµ±) = Q(µλ±) .

Now we can start from the interband matrix elements written in the principal

axes x0, y0, z0 and rewrite them in terms of x, y, z components. As a result, the ratios Q(+)λµ /M0 can be presented as

where the parameters P and Q are introduced in (3.210), and θ is the angle between the axes [hhl] and [001]. Then from (8.73) and (8.77) we come to the shift contribution to the interband photocurrent

Particularly, for QWs grown along [111] and characterized by the C3v symme-

for the C3v point group (Table A.4).

For the estimation of the ballistic photocurrent we should consider indirect interband transitions. Similarly to (8.40) the compound matrix element of the transition has the form

where γ → +0 and Vkc,vk are the scattering matrix elements in the conduction and valence bands. While calculating the square modulus of M ind, the energy denominators are transformed using the identity

&

Only the terms containing the product of the real part for one of the denominators in (8.79) and the imaginary term for the other denominator contribute to the current. Therefore, the e ect is due to the transitions where the energy conservation law holds not only between the initial and final states but also for one of the intermediate states as well. Formally it means that the

following part

πIm {Vkc kVkv kMcv (k)Mcv (k )}

of |M ind|2 contains a contribution antisymmetrical in k. Substituting this expression into (8.71) and taking into account the k-dependent correction to Mcv (k ) one can complete the calculation. The ballistic contribution to the photocurrent has the same order of magnitude as the shift contribution. Really, the antisymmetric part of |M ind|2 is proportional to the product V cV v while the momentum relaxation times in (8.71) are proportional to (V c)−2 or (V v )−2 so that the dependence of the ballistic current on the absolute values of electron-photon or electron-defect coupling constants disappears.

The linear PGE can also be induced in noncentrosymmetric SLs, i.e., in a saw-tooth SL, and MQW structures, i.e., in MQWs with asymmetric double wells, under illumination with unpolarized light [8.36–8.39]. The photocurrent is generated along the growth direction z because of the lack of reflection symmetry z → −z. Note that in MQWs the e ect has a threshold at the edge of transitions between quantized and continuum states, the so-called bound-to-continuum or above-barrier transitions.

8.5 Saturation of Photocurrents at High Light

Intensities

Here we discuss nonlinear behavior of the linear and circular PGEs, which will take place with increasing the light intensity due to saturation or bleaching of the absorption. Since the saturation e ect was observed on p-doped QW structures [8.40] we consider direct intersubband optical transitions from the heavy-hole subband hh1 to higher subbands, say the lh1 subband.

Spin sensitive bleaching can be analyzed in terms of the following simple model taking into account both optical excitation and nonradiative relaxation processes. The probability rates for direct optical transitions from the hh1 states with m = ±3/2 to higher subbands are denoted as W±. For linearly polarized light, W+ and W− are equal. For the circular polarization, right-handed, σ+, or left-handed, σ−, the rates W± are di erent but, due to time-inversion symmetry, satisfy the condition W+(σ±) = W−(σ ). The

photoexcited holes are assumed to loose their spin orientation in the course of energy relaxation to the bottom of the hh1 subband, due to rapid spin relaxation in hot states. Thus, spin orientation occurs only in this subband. If p+ and p− are the 2D densities of heavy holes occupying the subbands (hh1, +3/2) and (hh1, −3/2), respectively, then the rate equations for p± can be written as

The second terms on the left-hand side describe the spin relaxation trying to equalize the population in the (hh1, ±3/2) spin branches. The first terms on the right-hand side describe the removal of holes from the hh1 subband due to photoexcitation while the second terms characterize the relaxation of holes which come down to the (hh1, +3/2) and (hh1, −3/2) states with equal rates. If the laser-pulse duration is longer than any relaxation time, the time derivatives in (8.80) can be omitted and, instead of this equation, we have

where ρ is the hole spin polarization degree (p+ − p−)/(p+ + p−), η is a function of the light intensity I, the parameter ρ0 is defined as the ratio (W−−W+)/(W−+W+) for the σ+-polarized radiation of low enough intensity where η is constant and W± is proportional to I. The factors 1 ± ρ0Pc take into account the sensitivity of optical transitions to the circular polarization of light and spin of involved particle. The factors 1 ± ρ take into account that the transition probability rate depends on the occupation number of the initial state and, hence, on the hole spin polarization. Substitution of (8.82) into (8.81) leads to the linear equation for ρ

where ps is the hole density and we rewrote p+ − p− as psρ. The solution

Bleaching of absorption with increasing the intensity of linearly-polarized light is described phenomenologically by the function

Fig. 8.8. Photogalvanic current jx normalized by the intensity I as a function of I for circularly (curve 1) and linearly (curve 2) polarized radiation at T = 20 K. The inset shows the geometry of the experiment; eˆ indicates the direction of

¯

the incoming light. The current jx flows along [110] direction at normal incidence of radiation on p-type (113)A-grown GaAs/AlGaAs QWs. In order to obtain the circular PGE right or left circularly polarized light has been applied. To obtain the linear PGE linearly polarized radiation with the electric field vector E oriented at 45◦ to the x direction was used. The measurements are fitted to jx/I 1/(I +I/Is) with one parameter Is for each state of polarization (full line: circular, broken line: linear). From [8.40].

where η0 = η(I → 0) and Ise is the characteristic saturation intensity controlled by energy relaxation of the 2D hole gas. Since the photocurrent of linear PGE, jLPGE, induced by the linearly polarized light is proportional to ηI, one has

The circular current jCPGE induced by the circular polarized radiation is proportional to W+ − W− ρ. Substituting η(I) from (8.85) into (8.84) we find after some development