Coulomb-Induced Rashba Spin-Orbit Coupling in Semiconductor Quantum Wells

In the absence of an external field, the Rashba spin-orbit interaction (SOI) in a two-dimensional electron gas in a semiconductor quantum well arises entirely from the screened electrostatic potential of ionized donors. We adjust the wave functions of a quantum well so that electrons occupying the first (lowest) subband conserve their spin projection along the growth axis ($s_z$), while the electrons occupying the second subband precess due to Rashba SOI. Such a specially designed quantum well may be used as a spin relaxation trigger: electrons conserve $s_z$ when the applied voltage (or current) is lower than a certain threshold $V^{*}$; higher voltage switches on the Dyakonov-Perel spin relaxation.

Figure 1

The proposed device. The 2DEG is confined inside the quantum well grown along the $z$-([110]) direction; the current flows along the $x$ direction. The widths of the spacers separating the central zone (the well) from the doping layers on the two sides of the well are different.

Figure 2

The dependence of the SOI amplitudes for electrons in the first and second subband on the DDT-parameter $R$, see (8). The initial confinement potential, at $R=0$, is a square well, with parameters described in the caption of Fig. 3. Observe (i) that ${\alpha }_1=0$ at $R=0.82$ and (ii) strong dependence on the width of the well $a$. The curves are numerical [18] so that zeros are not exactly reached.

Figure 3

Confinement in the conduction and valence bands at $R=0.82$, see Fig. 2. The vertical walls correspond to the borders of the initial square well of width $a=400\AA$ [19] and depth $U_0=10\mathrm{meV}$. The left and right spacers have widths ${\delta }_L=0.1a$ and ${\delta }_R=0.04a$, respectively. Both doping layers have widths $w=0.06a$. The two lowest subbands are at energies ${ϵ}_1=4.5\mathrm{meV}$ and ${ϵ}_2=8.8\mathrm{meV}$ from the bottom of the original well [18].

Figure 4

The energy distribution function, $f_E({\tilde{x}}^{*})$ at a point ${\tilde{x}}^{*}$ inside a current-carrying sample [see Eq. (11)] at temperature $T=({ϵ}_2-{ϵ}_1)/30$. The dashed lines represent the edges of the conduction subbands, and the dotted line is the local electrochemical potential from Eq. (9). Because of the two-step form of the nonequilibrium distribution function some electrons have energy higher than the chemical potential $\mu ({\tilde{x}}^{*})$ (see the “dashed” area). These “hot” electrons came from the right; the “hottest” ones have energy $\sim {\mu }_R$ and originate from the right contact. Note that $f_E({\tilde{x}}^{*})\simeq 1$ for $E\lesssim {\mu }_L$ and $f_E({\tilde{x}}^{*})\simeq {\tilde{x}}^{*}$ for ${\mu }_L\lesssim E\lesssim {\mu }_R$; thus, the step in the distribution function $f_E({\tilde{x}}^{*})$ occurs precisely at $f_E({\tilde{x}}^{*})={\tilde{x}}^{*}$. By applying this rule one easily sees that the energy distributions at the edges of the sample, $f_E(0)$ and $f_E(1)$, are usual Fermi functions.

Figure 5

Population of the subbands in the sample, see Eq. (12). The width of the populated zone is controlled by the applied voltage $V$. For $V=1.5({ϵ}_2-{ϵ}_1)$, $n_2(x)$ approaches its maximal value at ${\tilde{x}}^{*}\equiv x^{*}/L=0.43$ [18].

Figure 6

The coordinate dependence of the SOI amplitudes inside the sample. Close to the right contact, the second subband is not populated (see Fig. 5); hence ${\alpha }_1=0$, and electrons do not experience spin-orbit scattering there [18]. The coordinate dependence of the SOI amplitudes inside the sample. Close to the right contact, the second subband is not populated (see Fig. 5); hence ${\alpha }_1=0$, and electrons do not experience spin-orbit scattering there [18].