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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 $α)$ and Dresselhaus (with strength $β)$ 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, $α=\pm β$. 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 $α=\pm β$. 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 $γ$, or (iii) for finite frequency measurements. We obtain the parametric range over which the magnetoelectric effect is suppressed in these three instances as (i) $|α|-|β| \lesssim 1/mL$, (ii)$|α|-|β| \lesssim γp_{\rm F}^2$, and (iii) $|α|-|β| \lesssiM \sqrt{ω/m p_{\rm F}\ell}$ with $\ell$ the elastic mean free path and $p_{\rm F}$ the Fermi momentum. We attribute the absence of spin accumulation close to $α=\pm β$ to the underlying U (1) symmetry. We illustrate and confirm our predictions numerically.