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 13, 744 (2002)] and by Trushin and Schliemann [Phys. Rev. B 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) $\left|\alpha \right|-\left|\beta \right|\lesssim 1/mL$, (ii) $\left|\alpha \right|-\left|\beta \right|\lesssim \gamma p_{\text{F}}^2$, and (iii) $\left|\alpha \right|-\left|\beta \right|\lesssim \sqrt{\omega /m{p}_{\text{F}}\ell }$ with $\ell$ the elastic mean-free path and $p_{\text{F}}$ 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.

Figure 1

Upper panel: Spin polarization $S_{x,\left(\alpha \right)}/S_{x,\left(\alpha =0\right)}$ for $\mathbf{E}\parallel \left[110\right]$ (dashed) and $S_{y,\left(\alpha \right)}/S_{y,\left(\alpha =0\right)}$ (solid line) for $\mathbf{E}\parallel \left[1\overline{1}0\right]$ as a function of Rashba SOI $\alpha /\beta$ for ${\xi }_{\beta }=2\beta p_{\text{F}}\tau =0.1$ and ${\xi }_{\gamma }=0.02$. Lower panel: $\alpha$ dependence of the normalized spin polarization $S_{y,\left(\alpha ,\gamma \right)}/S_{y,\left(\alpha ,\gamma =0\right)}$ for $\mathbf{E}\parallel \left[110\right]$, ${\xi }_{\beta }=2\beta p_{\text{F}}\tau =0.1$, and ${\xi }_{\gamma }=0.0,0.01,0.02,0.03$.

Figure 2

(Color online) Normalized spin accumulation $S_y/S_{2\text{DEG}}$ as a function of $\alpha /\beta$ for fixed $\beta /2a=t_{\text{D}}=0.15t$ (giving ${\ell }_{\text{so}}^{\text{D}}\approx 21a$), $U=2t$ (giving $\ell \approx 8.5a$), and Fermi energy $E_{\text{F}}=0.5t$, for different linear system size $W=L=70a$ (red squares), $150a$ (blue diamonds), and $310a$ (gray circles). Data are averaged over 5000 disorder configurations. The solid lines are the theoretical prediction, Eq. (16), with renormalized bulk spin accumulation and system size, $S_{2\text{DEG}}\to {\delta }_{\text{fit}}{S}_{2\text{DEG}}$ and $L\to L_{\text{fit}}$ with ${\delta }_{\text{fit}}\approx 0.84$, $L_{\text{fit}}\approx 39.3a$ for $L=70a$, ${\delta }_{\text{fit}}\approx 0.93$, $L_{\text{fit}}\approx 69.7a$ for $L=150a$, and ${\delta }_{\text{fit}}\approx 0.93$, $L_{\text{fit}}\approx 117.1a$ for $L=310a$. The electric current is in the direction $\widehat{x}\parallel \left[1\overline{1}0\right]$.

Figure 3

(Color online) Disorder-averaged normalized spin accumulation $⟨S_y⟩/S_{y;2\text{DEG}}$, with $S_{y;2\text{DEG}}=\alpha \tau \left(dn/dx\right)$, as a function of (a) the mean-free path $\ell$ (for fixed width $W=50a$ and (b) the width $W$ of the wire (for fixed $U=2t$, $\ell \approx 8.5a$). The electric current is in the direction $\widehat{x}\parallel \left[100\right]$. Different data sets correspond to different values of $\beta /\alpha =n/15$, $n=10$ (black circles), 11 (red), 12 (green), 13 (dark blue), and 14 (light blue). In both panels, other parameters are fixed at $t_{\text{R}}=0.15t$, $E_{\text{F}}=0.5t$, and $L=40a$ and data have been averaged over 3000 disorder configurations.