Quantum kinetic equations and anomalous nonequilibrium Cooper-pair spin accumulation in Rashba wires with Zeeman splitting

We derive the theoretical and numerical framework for investigating nonequilibrium properties of spin-orbit coupled wires with Zeeman splitting proximized by a superconductor in the nonlinear diffusive regime. We demonstrate that the anisotropic behavior of triplet Cooper pairs in this system leads to novel spin accumulation profiles tunable by the magnetic field and strength of applied voltage bias. This paves the way for enhanced manipulation of superconducting spintronic devices, and it enables further investigation of nonequilibrium effects in proximity-coupled superconducting structures more generally.

Figure 1

The model: nanowire with intrinsic spin-orbit coupling and Zeeman splitting, proximized by an $s$-wave superconductor and a voltage bias $V_{\mathrm{in}}$ applied via a bulk normal metal. The magnetization exchange field $\mathit{h}$ is purely in-plane.

Figure 2

Total spin accumulation $M_{\tau }$ along unit vectors $\tau$ for a nanowire with SOC and Zeeman splitting as a function of the applied voltage bias. Two field orientations are shown: $\mathit{h}=\left|\mathit{h}\right|(1,0,0)$ in the upper half, and $\mathit{h}=\left|\mathit{h}\right|/\sqrt{2}(1,1,0)$ in the lower half. Two field strengths are displayed for each orientation: $\left|\mathit{h}\right|=0.5\mathrm{\Delta }$ in the first row, and $\left|\mathit{h}\right|=\mathrm{\Delta }$ in the second row. The first column represents spin accumulation along the field (as indicated on the coordinate schematic), while the others show perpendicular components. The nanowire has length $L/{\xi }_S=0.8$ and the SOC is of pure Rashba type with coefficient $\alpha =5/L$. Spin accumulation amplitudes are given in units of $2M_0$.

Figure 3

(a) Total spin accumulation along the field $\mathit{h}=0.5\mathrm{\Delta }/\sqrt{2}(1,1,0)$ for increasing SOC strength at zero bias, in units of $2M_0$. The inset shows a linearly increasing spatial frequency of oscillation, derived by fitting the numerical curves to a sinusoid via least squares. Real and imaginary parts of the component wave vectors with increasing Rashba coupling $\alpha$ for the analytic weak proximity solution at field orientation $\left(b\right)\theta =0$ [Eqs. (A3)] and (c) $\theta =\pi /4$ [Eqs. (A4)]. The nanowire has length $L/{\xi }_S=0.8$.

Figure 4

Normalized charge conductance as a function of applied voltage bias for exchange-field orientations $\theta =0$ and $\pi /4$, where $\mathit{h}=\left|\mathit{h}\right|(cos\theta ,sin\theta ,0)$. The nanowire has length $L/{\xi }_S=0.8$, exchange-field strength $\left|\mathit{h}\right|=0.5\mathrm{\Delta }$, and the SOC is of pure Rashba type with coefficient $\alpha =5/L$. The inset shows the local density of states $D\left(\epsilon \right)$ in the middle of the sample.

Figure 5

Spin accumulation $M_{\tau }^{\prime }$, without contribution from the equilibrium component of the Green's function, along unit vectors $\tau$ for a nanowire with SOC and Zeeman splitting as a function of the applied voltage bias. All system parameters are as in Fig. 2.

Figure 6

Components of the $d$-vector at zero bias, for exchange-field orientations $\theta =0$ and $\pi /4$. The field strength is $\left|\mathit{h}\right|=0.5\mathrm{\Delta }$, and similarly all other parameters are as given in Fig. 2.

Figure 7

Local density of states $D\left(\epsilon \right)$ in the middle of the sample for field strengths $\left|\mathit{h}\right|=0.5\mathrm{\Delta }$ and $\left|\mathit{h}\right|=\mathrm{\Delta }$, and rotations $\theta =0$ and $\theta =\pi /4$.

Figure 8

Total spin accumulation $M_{\tau }$ and isolated nonequilibrium component $M_{\tau }^{\prime }$ along unit vector $\tau$, for exchange field aligned along the nanowire, $\mathit{h}=0.5\mathrm{\Delta }\widehat{\mathit{z}}$, i.e., parallel with the SO gauge field.