Control of Spin Helix Symmetry in Semiconductor Quantum Wells by Crystal Orientation

We investigate the possibility of spin-preserving symmetries due to the interplay of Rashba and Dresselhaus spin-orbit coupling in $n$-doped zinc-blende semiconductor quantum wells of general crystal orientation. It is shown that a conserved spin operator can be realized if and only if at least two growth direction Miller indices agree in modulus. The according spin-orbit field has in general both in-plane and out-of-plane components and is always perpendicular to the shift vector of the corresponding persistent spin helix. We also analyze higher-order effects arising from the Dresselhaus term, and the impact of our results on weak (anti)localization corrections.

Figure 1

(a) Global minimum ${\lambda }_{\min }$ (in terms of $Q_{\text{so}}=4m{\beta }^{(1)}/{\hslash }^2$) of the spectrum of the spin diffusion operator ${\mathrm{\Lambda }}_{\text{sd}}$ for the optimal ratio of Rashba and Dresselhaus coefficients $\alpha /{\beta }^{(1)}$ for different growth directions $[{\tilde{n}}_x,{\tilde{n}}_y,5]$ (${\tilde{n}}_x,{\tilde{n}}_y\in \mathbb{Z}$). The white lines emphasize the vanishing minima. (b) Ratio between the global minimum ${\lambda }_{\min }$ and the minimum $1/(D_{\mathit{e}}{\widehat{\tau }}_{\mathit{s},\min }^{(0)})$ found by considering the spectrum at $\mathbf{q}=0$ solely, which corresponds to the D’yakonov Perel’ spin-relaxation tensor, Eq. (4). Along the white lines, both minima vanish exactly due to the SU(2) symmetry.

Figure 2

Characteristic parameters in case of a persistent spin helix symmetry in dependence of the growth direction. Notice the degeneracy for $\alpha /{\beta }^{(1)}$ in the [001] direction.

Figure 3

2DEG grown along [113] for $c_e/c_{\varphi }=10^3$ and $c_e=1$ with $\alpha /{\beta }^{(1)}$ close to the SU(2) symmetry point. (a) Relative MC $\mathrm{\Delta }{\sigma }_R(B)=\mathrm{\Delta }\sigma (B)-\mathrm{\Delta }\sigma (0)$ for different values of $\epsilon$. For compactness, we restrict the plots for $\epsilon <0(\epsilon >0)$ to negative(positive) magnetic fields. The colored lines correspond to exact numerical calculations, the black dotted lines to the approximate expression, Eq. (11). Gray dashed lines show the trend of the minima $\mathrm{\Delta }\sigma (B_{\min })$ in dependence of $\epsilon$. (b) The respective MC minimum as a function of $\epsilon$. Red solid lines correspond to exact numerical calculations, black dotted and green dashed lines to approximate formulas.