Long-Range Spin-Triplet Helix in Proximity Induced Superconductivity in Spin-Orbit-Coupled Systems

We study proximity induced triplet superconductivity in a spin-orbit-coupled system, and show that the $\mathbf{d}$ vector of the induced triplet superconductivity undergoes precession that can be controlled by varying the relative strengths of Rashba and Dresselhaus spin-orbit couplings. In particular, a long-range spin-triplet helix is predicted when these two spin-orbit couplings have equal strengths. We also study the Josephson junction geometry and show that a transition between 0 and $\pi$ junctions can be induced by controlling the spin-orbit coupling with a gate voltage. An experimental setup is proposed to verify these effects. Conversely, the observation of these effects can serve as a direct confirmation of triplet superconductivity.

Figure 1

Energy dispersion and Fermi surfaces are shown for (a) normal metals, (b) ferromagnets, (c) a 2DEG with Rashba SOC and (d) a 2DEG with equal strengths of Rashba and Dresselhaus SOCs. The possible forms of spin states of Cooper pairs, including singlet and triplet pairs, are also illustrated in the figures. For $k_y\ne 0$, $k_x=0$, the gap between two spin bands in (c) is $|2\alpha k_y|$.

Figure 2

A schematic plot of a SC-FM-SOC junction. Energy dispersions for different regions are shown above the junction structure. The colors in the dispersion relation represent different spin indices and the solid lines (dashed lines) denote electron (hole) bands. $k_{1(2),f}$ and $k_{3(4),f}$ are the Fermi momenta of different spin bands for SOC and FM regions, respectively. Different propagation or reflection processes are denoted by $T_{\text{fm}(\text{soc})}^{\text{in}(\text{out})}$ or $R_{\mathrm{ad}}$.

Figure 3

The magnetization direction (the green arrows) and the effective magnetic field direction of SOC (the purple arrow) are shown (a) for a 0 junction and (b) a $\pi$ junction. The red arrows reveal the spatial distribution of the $\mathbf{d}$ vector. The phases of SCs at two sides are taken to be $\varphi /2$ and $-\varphi /2$. The color in (c) and (d) shows the spectral function of the SC-FM-SOC-FM-SC junction (logarithmic plot) as a junction of the relative phase $\varphi$ for a 0 junction and $\pi$ junction, respectively. The black lines are the Andreev levels from analytical calculations. (f) shows the current-phase relation for the 0 and $\pi$ junction.

Figure 4

The spatial dependence of the $\mathbf{d}$ vector of triplet pairs (the red arrows) and the corresponding effective magnetic field (the purple arrows) of the SOC are shown for (a) $\alpha =\beta$ ($\pi$ junction) and (b) $\alpha =-\beta$ (0 junction). TSC means triplet superconductor. (c) The proposed 2D SC-FM-SOC-FM-SC structure for an electronic-tunable Josephson junction.