Gate-controlled persistent spin helix state in (In,Ga)As quantum wells

In layered semiconductors with spin-orbit interaction (SOI) a persistent spin helix (PSH) state with suppressed spin relaxation is expected if the strengths of the Rashba and Dresselhaus SOI terms, $\alpha$ and $\beta$, are equal. Here we demonstrate gate control and detection of the PSH in two-dimensional electron systems with strong SOI including terms cubic in momentum. We consider strain-free InGaAs/InAlAs quantum wells and first determine a ratio $\alpha /\beta \simeq 1$ for nongated structures by measuring the spin-galvanic and circular photogalvanic effects. Upon gate tuning the Rashba SOI strength in a complementary magnetotransport experiment, we monitor the complete crossover from weak antilocalization via weak localization to weak antilocalization, where the emergence of weak localization reflects a PSH-type state. A corresponding numerical analysis reveals that such a PSH-type state indeed prevails even in presence of strong cubic SOI, however no longer at $\alpha =\beta$.

Figure 1

(a) Energy-band profile of conduction band with the Fermi energy at $E_{\mathrm{F}}=0$ in ${\mathrm{In}}_{0.53}{\mathrm{Ga}}_{0.47}\mathrm{As}/{\mathrm{In}}_{0.52}{\mathrm{Al}}_{0.48}\mathrm{As}$ 2DEG structures calculated by using a Poisson-Schrödinger equation solver. Black and gray lines correspond to 4 and 7 nm samples, respectively. Full lines denote the normalized electron probability density ${\left|\psi \right|}^2$. (b) Schematic cross section of the structures under study.

Figure 2

(a) Sketch of the experimental arrangement (top) and sample geometry used for measurements of the spin-galvanic effect. Here we used circularly polarized light at normal incidence with the magnetic field applied along the $x\parallel \left[100\right]$ direction. The photocurrent $J\left(\theta \right)$ is mapped by measuring successively signals from opposite contact pairs. (b) Azimuthal dependence of the SGE current $J$ measured in a 4 nm QW at room temperature, $\lambda =148$ $\mu$m, and $B_x=0.8$ T. The solid line shows the fit according to $J=J_{\mathrm{R}}\cos \theta +J_{\mathrm{D}}\sin \theta$ with the ratio of $J_{\mathrm{R}}/J_{\mathrm{D}}=0.98\pm 0.08$.

Figure 3

Magnetoconductance profiles (in units of $e^2/h$) measured at different gate voltages, i.e., carrier densities $N_s$, for (a) 4 and (b) 7 nm QWs at $T=$ 1.4 K. All curves in (a) and (b) are shifted positively (gray and green) and negatively (black and red) with respect to the blue curve, for which $\Delta \sigma =0$ at $B_z=0$ mT. For the 4 nm QW a clear WL dip occurs for a carrier density of $3.71\times {10}^{11}$ cm${}^{-2}$, which is absent for the 7 nm QW.

Figure 4

Numerical analysis of the PSH: Panels (a) and (b): Profiles of the average conductance correction $\delta g\left(\phi \right)=g\left(\phi \right)-g\left({\phi }_c\right)$ (in units of $e^2/h$) as function of flux $\phi$ (normalized by the flux quantum ${\phi }_0$) for disordered mesoscopic conductors with SOI ratios $\alpha /\beta =$ 0 to 4/3; see text. (a) Case of vanishing cubic SOI, ${\Gamma }_{\beta }=0$: Weak antilocalization is observed due to spin interference except for $\alpha /\beta =1$ when spin relaxation is suppressed by exact $SU\left(2\right)$ symmetry. (b) A finite cubic term ${\Gamma }_{\beta }=5/6$ leads to a weak localization signature at a shifted value $\alpha /\beta <1$, in agreement with the experimental observations in Fig. 3a. (c) Absence of weak localization for increased ${\Gamma }_{\beta }=16/5$ and larger ratios $\alpha /\beta$, consistent with the increase of the corresponding quantities when changing $L_W=$ 4 nm [Fig. 3a] to 7 nm [Fig. 3b]. Up to the lowest curves all data have been shifted by arbitrary offsets for the sake of clarity. (d) Magnetoconductance correction $\delta g_0=\delta g(\phi =0)$ as a function of the ratio $\beta /\alpha$ for fixed $\alpha$ and various values of ${\Gamma }_{\alpha }=k_{\mathrm{F}}^2\gamma /\alpha$; WL dips are linearly shifted with increasing ${\Gamma }_{\alpha }$ indicating a modified PSH condition in $\alpha /\beta$ space.