Spin Accumulation in Diffusive Conductors with Rashba and Dresselhaus Spin-Orbit Interaction

We calculate the electrically induced spin accumulation in diffusive systems due to both Rashba (with strength $\alpha )$ and Dresselhaus (with strength $\beta )$ spin-orbit interaction. Using a diffusion equation approach we find that magnetoelectric effects disappear and that there is thus no spin accumulation when both interactions have the same strength, $\alpha =\pm \beta$. In thermodynamically large systems, the finite spin accumulation predicted by Chaplik, Entin and Magarill, [Physica E {\bf 13}, 744 (2002)] and by Trushin and Schliemann [Phys. Rev. B {\bf 75}, 155323 (2007)] is recovered an infinitesimally small distance away from the singular point $\alpha =\pm \beta$. We show however that the singularity is broadened and that the suppression of spin accumulation becomes physically relevant (i) in finite-sized systems of size $L$, (ii) in the presence of a cubic Dresselhaus interaction of strength $\gamma$, or (iii) for finite frequency measurements. We obtain the parametric range over which the magnetoelectric effect is suppressed in these three instances as (i) $\mathrm{\mid }\alpha \mathrm{\mid }-\mathrm{\mid }\beta \mathrm{\mid }\lesssim 1\mathrm{/}mL$, (ii)$|\alpha|-|\beta| \lesssim \gamma p_{\rm F}^2$,and(iii)$|\alpha|-|\beta| \lesssiM \sqrt{\omega/m p_{\rm F}\ell}$with$\ell$theelasticmeanfreepathand$p_{\rm F}$ the Fermi momentum. We attribute the absence of spin accumulation close to $\alpha =\pm \beta$ to the underlying U (1) symmetry. We illustrate and confirm our predictions numerically.