Signatures of spin-preserving symmetries in two-dimensional hole gases

We investigate ramifications of the persistent spin helix symmetry in two-dimensional hole gases in the conductance of disordered mesoscopic systems. To this end we extend previous models by going beyond the axial approximation for III-V semiconductors. For heavy-hole subbands we identify an exact spin-preserving symmetry analogous to the electronic case by analyzing the crossover from weak antilocalization to weak localization and spin transmission as a function of extrinsic spin-orbit interaction strength.

Figure 1

Fermi surface for the different spin directions obtained from Eq. (1) (black and blue (gray) contours) and the corresponding direction of the effective spin-orbit field, ${\mathit{\Omega }}_{2\mathrm{DHG}}$, illustrated by arrows. The SOI parameters establish a persistent spin helix for holes with uniaxial spin orientation corresponding to $\eta \equiv {\lambda }_{\mathrm{R}}/{\lambda }_{\mathrm{D}}=+1$ (a) and $\eta =-1$ (b). In both cases the Luttinger parameters are $\overline{\gamma }=-\delta$, i.e., ${\gamma }_3=0$.

Figure 2

Signatures of spin-preserving symmetries in weak localization of a two-dimensional hole gas. (a) Disorder-averaged magnetoconductance correction $\langle T\left(\varphi \right)\rangle -\langle T\left({\varphi }_{\max }\right)\rangle$ as a function of flux $\varphi$ (in units of ${\varphi }_0=h/e$; ${\varphi }_{\max }/{\varphi }_0=3.1$) for spin-orbit coupling ratios $\eta \equiv {\lambda }_{\mathrm{R}}/{\lambda }_{\mathrm{D}}=-1.65,-0.017,-1.33,-0.67,-1$ (from top to bottom). (b) Conductance correction $\langle T\left(0\right)\rangle -\langle T\left({\varphi }_{\max }\right)\rangle$ as a function of $\eta$. Negative magnetoconductance reflects suppression of spin relaxation close to $\eta =-1$. System parameters: disorder average over 1000 impurity configurations for a scattering region of aspect ratio (length $L$ to width $W$) 200:80 unit cells with periodic boundary conditions in transverse direction. Quantum transmission computed for $k_{\mathrm{F}}W/\pi =13$ hole states per spin supported in the leads, elastic mean free path $l=0.04W$, ${\gamma }_3=0$, and fixed Dresselhaus spin precession length $k_{\mathrm{D}}W\approx 1$, defined below Eq. (18).

Figure 3

Top: ratio of disorder averaged diagonal transmission over total transmission $\langle T_{\mathrm{D}}\rangle /\langle T\rangle =\langle T_{\mathrm{D}}\rangle /\langle T_{\mathrm{D}}+T_{\mathrm{OD}}\rangle$ as a function of $\eta$ for a scattering region with 150:80 aspect ratio for fixed Dresselhaus spin precession length $k_{\mathrm{D}}^{-1}\approx {\left(k_F^2{\lambda }_D\right)}^{-1}=1.3W$ and Luttinger parameters ${\gamma }_2=1$ and ${\gamma }_3=0.25$ for which no exact spin-preserving symmetry can be established. The peak of $\langle T_{\mathrm{D}}\rangle /\langle T\rangle$ at $\eta =1$ (indicated by the arrow), coincides with the maximum of the diabatic transition probability of Eq. (18). Average transmission shown includes 1000 disorder configurations. With respect to the eigenbasis of $\eta =-1$, we obtain a curve that coincides with the mirror image of the shown plot, displaying a maximum at $\eta =-1$. Bottom: amplitude of the magnetoconductance correction in which no WAL-WL-WAL transition is observable in the vicinity of the symmetry point of $\left|\eta \right|=1$ due to insufficient spin randomization in the regime $\left|\eta \right|<1.19$. This example indicates that in a sweep of the Rashba SOI, the point where Rashba and Dresselhaus SOI are balanced displays a clear signal in the diagonal transmission, while it may not be detectable in the form of a WAL-WL-WAL transition.