Closed-Form Weak Localization Magnetoconductivity in Quantum Wells with Arbitrary Rashba and Dresselhaus Spin-Orbit Interactions

We derive a closed-form expression for the weak localization (WL) corrections to the magnetoconductivity of a 2D electron system with arbitrary Rashba $\alpha$ and Dresselhaus $\beta$ (linear) and ${\beta }_3$ (cubic) spin-orbit interaction couplings, in a perpendicular magnetic field geometry. In a system of reference with an in-plane $\widehat{\mathbf{z}}$ axis chosen as the high spin-symmetry direction at $\alpha =\beta$, we formulate a new algorithm to calculate the three independent contributions that lead to WL. The antilocalization is counterbalanced by the term associated with the spin relaxation along $\widehat{\mathbf{z}}$, dependent only on $\alpha -\beta$. The other term is generated by two identical scattering modes characterized by spin-relaxation rates which are explicit functions of the orientation of the scattered momentum. Excellent agreement is found with data from GaAs quantum wells, where, in particular, our theory correctly captures the shift of the minima of the WL curves as a function of $\alpha /\beta$. This suggests that the anisotropy of the effective spin-relaxation rates is fundamental to understanding the effect of the spin-orbit coupling in transport.

Figure 1

Experimentally determined magnetoconductivity corrections $\mathrm{\Delta }\sigma (B)-\mathrm{\Delta }\sigma (0)$, in $e^2/2\pi \hslash$ units, plotted as a function of the magnetic field $B$ (mT), are compared with the theoretical result in Eq. (19) (continuous line) for $\alpha /\beta =0.05$, 0.29, 0.37, 0.44.

Figure 2

Magnetoconductivity corrections are plotted as a function of the ratio of the linear SOI couplings $\alpha /\beta$. In the inset, we track the magnetic field value $B_{\min }$ at which the magnetoconductivity reaches a minimum as a function of $\alpha /\beta$.