Symmetry breaking of the persistent spin helix in quantum transport

We exploit the high-symmetry persistent spin helix state obtained for similar Rashba and linear Dresselhaus interactions in a quantum well to revisit the weak localization problem within a perturbative approach in a Landau level formulation. We define the small parameter of the theory as the deviation from the symmetry state introduced by the mismatch of the linear terms and by the strength of the cubic Dresselhaus term. In the vicinity of the helix state, the SO field becomes uniaxial, offering a natural direction of spin quantization, thus defining the $z$ axis within the 2D plane. In contrast to previous theories, this reveals a full decoupling of the Cooperon triplet scattering modes as well as decoupled Landau levels, to lowest order in the small parameter. This makes it possible to derive a closed-form expression for the weak localization magnetoconductivity, thus providing a new paradigm of localization in the weakly-broken spin symmetry regime. We perform quantum transport experiments in GaAs quantum wells, finding very good agreement with the new theory. We present a reliable two-step method to extract the SO and transport parameters from fits of the new expression, obtaining excellent agreement with recent experiments. This is an important step towards engineering and controlling the spin-orbit interaction as a powerful resource in emerging quantum technologies.

Figure 1

Cooperon terms around the PSH symmetry $\alpha =\beta$, with singlet $\left(S\right)$ and triplet $(T_0,T_{\pm })$ states all in the same, generic Landau level $|n\rangle$. (a) Energy of the Cooperon eigenstates as functions of $k_x$, the Cooperon momentum along $\widehat{x}$ which fixes the center of the orbit. The $S$ and $T_0$ states are located at $k_x$ and become degenerate at $\alpha =\beta$ and ${\beta }_3=0$. $T_{+}$ and $T_{-}$ are degenerate, but since the orbits are separated by $2Q_{+}$ there is no coupling between them, giving WL. Note there is no coupling to other Landau levels. (b) Energies of the eigenstates in one Landau level as a function of the ratio of $\alpha /\beta$. The full curves correspond to the case where the cubic term is zero and all states are degenerate at $\alpha =\beta$ and WL is observed. The dashed lines correspond to the states, when the cubic term is strong, then $S$ and $T_0$ state are not degenerate at $\alpha =\beta$ giving WAL even at $\alpha =\beta$.

Figure 2

Magnetoconductance curves in the regime close to the spin helix symmetry as given by Eq. (5). The black traces correspond to $\alpha =\beta$. Left panel: Spin orbit coupling causes a quench of the WL (dip less deep) before WAL appears (dashed red traces). Right panel: For a strong cubic term, WAL appears even at $\alpha =\beta$ and is defined by the WAL trace with the WAL minima closest to $B_{\perp }=0$.

Figure 3

Density map with symmetry points (purple triangles) as a function of top $V_T$ and back gate $V_B$ voltage for data set #2 (see SM [70] for details). Along contours of constant density (labeled in units of ${10}^{15}{\mathrm{m}}^{-2})$, $B_{SO-}$ is changing as a function of detuning $\delta E_z$, while $B_{SO3}$ is constant. The gray circles indicate the measured gate configurations. The triangles correspond to the approximate position where $\alpha =\beta$ and the purple line corresponds to a plot of the calculated PSH condition from the extracted SO parameters. Equation (5) is valid everywhere between the red dashed lines.

Figure 4

(a) Measured conductivity traces (black) around the symmetry point and fits (red) using Eq. (5) with the respective density labeled at each trace in units of ${10}^{15}{\mathrm{m}}^{-2}$. The measured traces have been symmetrized in $B_{\perp }$ for fitting. The gray dashed curves correspond to the fit range obeying $B_{\perp }\ll B_{\text{tr}}$. (b) Upper panel: extracted $B_{\mathrm{SO}3}$ values versus density. The red dashed curve is a quadratic fit to $B_{\mathrm{SO}3}$. The blue markers and curve correspond to the extracted and calculated $B_{\phi }$ (see main text). Middle panel: extracted ${\tau }_3/{\tau }_1$ using the later to be determined $\gamma$ for each individual value of $B_{\mathrm{SO}3}$ from the upper panel. The red dashed line is the average of ${\tau }_3/{\tau }_1$. Lower panel: coherence time from the extracted $B_{\phi }$ for the respective density and mobility. The red dashed curve is a fit to the data assuming Nyquist dephasing. For the two lowest densities 6.0 and 6.5 (left of the dashed vertical line), the extracted values of $B_{\mathrm{SO}3}$ and $B_{\phi }$ are only bounds, see text.

Figure 5

Measured traces away from $\alpha \approx \beta$. (a) Fits (green) to the conductivity traces (black) for constant density $n=9.25\times {10}^{15}{\mathrm{m}}^{-2}$ in Fig. 3. The gray dashed lines indicate the range for the diffusive approximation. Each curve is labeled with its detuning value $\delta E_z$. (b) Extracted values of $B_{\mathrm{SO}-}$ versus the detuning $\delta E_z$ for all densities (arranged vertically and labeled in units of ${10}^{15}{\mathrm{m}}^{-2}$ for each $B_{\mathrm{SO}-}$ curve). The error bars correspond to the error on the fit parameter (i.e., one standard deviation). The data in the gray shaded area is included in the fit.

Figure 6

SO parameters from five measurements obtained on two samples. Measurements 1 and 3 correspond to Hall bar no. I and measurements 2, 4, and 5 correspond to Hall bar no. II, where measurement 5 is from another cool down, details in SM [70] Sec. II. The blue lines correspond to the values obtained in a previous work [23], the red lines correspond to their average, and the red dashed lines correspond to the standard deviation of the mean. (a) Dresselhaus coefficient $\gamma$ (b) offset ${\alpha }_0$ of the Rashba parameter, (c) ${\alpha }_1$ of the Rashba parameter and (d) average scattering time ratio ${\tau }_3/{\tau }_1$.