Microscopic origin of the photocurrent

In the following the microscopic origin of the observed photocurrents at nor-mal incidence will be discussed as well as the microscopic expressions for the relevant parameters χ1 andγ, limited to the case of unpolarized and circu-larly polarized light, respectively. The analysis of these two contributions to the ratchet effects is crucial for the understanding of the data summarized in Fig. 38. For this the classical Boltzmann equation for the electron distribution functionfk, has to be taken into account. Here k is the in-plane electron wave vector, F the sum of the time-dependent electric-field forceeE(t) = 2eRe[E₀exp(−iωt)]

of the light wave and the static force−dV(x)/dx, ω the light frequency,vk=

~k/m^∗ the electron velocity,e and m^∗ the electron charge and effective mass, Q^(p)k and Q^(ε)_k the collision terms responsible for the electron momentum and energy relaxation, respectively. The operatorQ^(p)k is taken in its simplest form (f^k−hf^ki)/τ, whereτ is the momentum relaxation time and the brackets imply an averaging over k directions. The operator Q^(ε)_k acts on the distribution function averaged over the directions of k and depends only on the modulus k =|k|. The assumptions for Eq. (60) are only valid for a weak and smooth potential, which satisfies the conditions |V(x)| ≪ εe and q ≡ 2π/a ≪ ke, whereke is the typical electron wave vector andεe the typical energy assumed to be larger than the photon energy~ω.

6.1.4.1 Polarization-independent photocurrent: The photocurrent contribu-tion, which constitutes the polarization-independent ratchet effect, is given by

the term includingχ1 in Eqs. (59) and describes the photocurrent proportional toD−C =χ1/√

2 in Eq. (58). It can be related to the heating of free carri-ers by an electromagnetic wave. In the case of high temperatures the kinetic Eq. (60) can be reduced to the macroscopic equations for the two-dimensional electron densityN(x), local nonequilibrium temperature Θ(x), current density jx and energy flux density iε,x(x) in x-direction. Under homogeneous optical excitation these equations have the solutions, given by

k_BΘ =k_BT +~ωGτ_ε, (61)

N(x)≡N(x,Θ) =N0e^−V(x)/kBΘ,

where G is the Drude absorption rate per particle, and N0 is x-independent.

For the general solution, shown in Eqs. (61), the current jx, and thus the carrier flow iε,x, are absent. Only if the generation rate G varies spatially, a nonzero jx arises. Such an inhomogeneous distribution of the radiation electric field is generated in the patterned samples due to near field effects, which occur because the distance between the surface pattern and the QW is smaller than the wavelength [43]. Resulting from this, the amplitude of a plane electromagnetic field shining through the superimposed grating be-comes a periodic function ofxwith the perioda. In addition an asymmetrical SL leads to a relative phase shift between the potential V(x) and the light intensity I(x), yielding a net current even for unpolarized radiation because in this case the product I(x)(dV /dx) averaged over the space does not van-ish. This spatial variation of G is described by the steady-state generation G(x) = G0+G1cos (qx+ϕG), which produces a stationary periodic electron temperature Θ(x)−Θ ≡ δΘ(x) = τε~ω[G(x)−G0] accompanied by a light-induced periodic correction of the space-oscillating contribution to the electron densityδN(x) ≈ −N0δΘ(x)/Θ. Microscopically the current is described by a sum of drift and diffusion terms,

j_x =µ on x, whereas the total j_x is x-independent. As the diffusion term averaged

6 PHOTOCURRENTS IN LATERAL STRUCTURED SAMPLES 95 over a period d vanishes and the average of the product N(x,Θ)dV(x)/dx is zero the current can be calculated as an averageµ[dV(x)/dx]δN(x).

The lateral potential can be modeled in the simple and periodic inx function V(x) = V1cos (qx+ϕV). Due to a phase shift between V(x) and the periodic electron temperature Θ(x) a current along x, given by Eq. (63), arises.

jx =χ1I =µN0~qζ^′G0

V1

2kBTωτε (63)

In this equation the asymmetry parameter ζ^′ = (G1/G0) sin (ϕV −ϕG), de-scribing the inhomogeneous photoexcitation, enters. For the derivation lead-ing to Eq. (63) a model function V(x) was used, which is similar to the one considered in Ref. [90] and is applicable to ratchets with a phase shift be-tween its sinusoidal potential and the temperature variation. Generally, such nonequilibrium asymmetric systems based on a periodic potentialV(x) and a periodic temperature profile Θ(x) are referred to as Seebeck ratchets [77]. As an important result Eq. (63) yields a polarization independent current, which increases with decreasing temperature T, as observed in the experiment (see Fig. 35) and is proportional to the energy relaxation timeτε.

6.1.4.2 Helicity-dependent photocurrent: The helicity-dependent photocur-rent which is described in Eqs. (59) by the parameter γ or in Eq. (57) by A, is driven by circularly polarized light and is therefore, called circular ratchet effect. Similar to the polarization-independent part this contribution is gener-ated in a lateral SL with a phase shift between the periodic potentialV(x) and the generation rate G(x). A photocurrent, which depends on the helicity at normal incidence of radiation (θ0 = 0^◦) is allowed by symmetry arguments, but only along they-direction. Circularly polarized radiation excludes theχ3-term in Eqs. (59), by which the remaining current is given byjy =±Iγ, with ± cor-responding to right- and left-handed circular polarized radiation, respectively.

More precisely, the resulting photocurrent along the direction of grooves reads j_y = 2e²τ

m^∗ Re{E_0y^∗ (x)δN_ω(x)}, (64)

whereδN_ω(x) is the electron density oscillation linear in the THz electric field E_0x. Considering the continuity equation −iωeδN_ω +dj_x,ω/dx = 0 and the

equation for the linear-response electric currentjx,ω(x) modulated in space the

This kind of directed current, which changes its sign upon switching the light helicity and thus is proportional to the circular polarizationP_circ, is based on two phase shifts relative to the periodic variation δN(x, t). One of them is a spatial phase shift between the potential V(x) and generation rate G(x), proportional to ϕ_V −ϕ_G. The second one is a temporal phase shift, given by arctan(ωτ) with respect to E_x(t). In comparison to the polarization-independent the helicity-dependent current is by a factor of 2ωτ_ε smaller but increases in the same way with decreasingT, also confirmed by the experiment.

Here the current is proportional to the momentum relaxation timeτ.