Mesoscopic Fluctuations in the Spin-Electric Susceptibility due to Rashba Spin-Orbit Interaction

We investigate mesoscopic fluctuations in the spin polarization generated by a static electric field and by Rashba spin-orbit interaction in a disordered 2D electron gas. In a diagrammatic approach we find that the out-of-plane polarization—while being zero for self-averaging systems—exhibits large sample-to-sample fluctuations which are shown to be well within experimental reach. We evaluate the disorder-averaged variance of the susceptibility and find its dependence on magnetic field, spin-orbit interaction, dephasing, and chemical potential difference.

Figure 1

Disordered two-dimensional electron gas with spin-orbit interaction and electric contacts at $x=0$ and $x=L$. Mesoscopic fluctuations of the magnetoelectrical spin susceptibility ${\chi }^{31}$ allow for a finite out-of-plane polarization $S^3$ in response to an in-plane electric field $\mathbf{E}$.

Figure 2

Dominant diagrams (for $1/p_{F}l\ll 1$) leading to the variance $\overline{\delta {\chi }^{ij}\delta {\chi }^{km}}$ given in Eq. (3), see also Fig. 3. The upper diagrams (a) and (b) contain the Hikami boxes (HBs) $V_{La}$, $V_{Ra}$ (with solid arrows) and two diffusons ($D$) and $V_{Lb}$, $V_{Rb}$ (with dashed arrows) and two Cooperons ($C$), respectively. The lower diagrams contain the HBs $V_{Lc}$, $V_{Rc}$ (solid arrows, two diffusons) and $V_{Ld}$, $V_{Rd}$ (dashed arrows, two Cooperons).

Figure 3

General Hikami box (HB): shaded area represents the correction to the “empty” box $V_{(a)}$ by $V_{(b)}$, $V_{(c)}$ containing a single impurity line (dashed). The vertices are denoted by ${\kappa }^i\in \{{\sigma }^{\mu },v_1,v_2\}$, and $R/A$ stands for the averaged $G_{E_F,b}^{R/A}$.

Figure 4

Example of a path along which an initially $y$-polarized spin rotates exclusively into the positive out-of-plane direction (circled dots). The spin directions are denoted by small arrows at the left of each path segment and result from precession about the Rashba spin-orbit field ${\mathbf{B}}_R$ that is generated by the propagation along the path. The length of the horizontal segment is chosen half the spin-orbit length ${\lambda }_{SO}/2=1/4m\alpha$ so that the spin rotates about ${\mathbf{B}}_R$ by an angle $\pi$. An interfering path is shown (dashed line) generating the same spin rotations.

Figure 5

The relative variance of the out-of-plane spin susceptibility $\overline{(\delta {\chi }^{31}{)}^2}/{\overline{\delta {\chi }^{21}}}^2$ is shown as a function of the in-plane magnetic field $2b\tau$ for $x=0.1$, $L_{\varphi }=100l$, and sample sizes $L=100\pi l$ (solid line) and $L=200\pi l$ (dashed line), respectively. The low field approximation, Eq. (7), (curved dotted line) and half of the $b=0$ value (straight dotted line) are shown for $L=\pi l100$. Inset: Variance as a function of $\Delta E$ for $x=0.1$, $L_{\varphi }=100l$, and $L=100\pi l$ (solid line) and $L=200\pi l$ (dashed line). The suppression of $\overline{\delta {\chi }^{31}}$ with increasing $b$ and $\Delta E$ is described by Eqs. (7, 8).