General framework for transport in spin-orbit-coupled superconducting heterostructures: Nonuniform spin-orbit coupling and spin-orbit-active interfaces

Electronic spin-orbit coupling (SOC) is essential for various newly discovered phenomena in condensed-matter systems. In particular, one-dimensional topological heterostructures with SOC have been widely investigated in both theory and experiment for their distinct transport signatures indicating the presence of emergent Majorana fermions. However, a general framework for the SOC-affected transport in superconducting heterostructures, especially with the consideration of interfacial effects, has not been developed even regardless of the topological aspects. We hereby provide one for an effectively one-dimensional superconductor-normal heterostructure with nonuniform magnitude and direction of both Rashba and Dresselhaus SOC as well as a spin-orbit-active interface. We extend the Blonder-Tinkham-Klapwijk treatment to analyze the current-voltage relation and obtain a rich range of transport behaviors. Our work provides a basis for characterizing fundamental physics arising from spin-orbit interactions in heterostructures and its implications for topological systems, spintronic applications, and a whole variety of experimental setups.

Figure 1

Illustration of an effective 1D N-S system with SOC and a spin-orbit-active interface (I). The vectors denote directions of Rashba ${\alpha }_{\mathrm{N}\left(\mathrm{S}\right)}{\sigma }_y$ and Dresselhaus ${\beta }_{\mathrm{N}\left(\mathrm{S}\right)}{\sigma }_x$ on the N (S) side, as well as interfacial Rashba $Z_{\alpha }{\sigma }_y$, Dresselhaus $Z_{\alpha }{\sigma }_x$, and Zeeman $Z_h\widehat{n}·\vec{\sigma }$ components. In general, the spin basis in each region can be different. Insets: schematic energy spectra with two isospin branches (solid and dashed curves, respectively) on the corresponding sides. The solid circles denote possible outgoing states corresponding to an incoming electron (empty circle) from the N side.

Figure 2

(a) [(b)] Zero-bias NDC (scaled by colors in bar graph) in the $Z_0\text{−}{Z}_{\gamma }$ $\left(Z_h\right)$ plane given $Z_h\left(Z_{\gamma }\right)=0$. Data are for $\Delta =0.002,\delta {\gamma }^2=0$, and $T=0$.

Figure 3

(a)–(c) Interfacial phase effects on NDC for a general spin-orbit-active interface described by Eq. (14) with $Z_0=0,0.66$, and $1.5$, respectively. Curves with diamonds, triangles, squares, circles, and crosses correspond to ${\zeta }^{\prime }=0,0.25\pi ,0.5\pi ,0.75\pi$, and $\pi$, respectively. The other relevant parameters are set as $Z_1=Z_3=0,Z_2=Z_4=0.5,\Delta =0.002,\delta {\gamma }^2=0$, and $T=0.15\Delta$. (d)–(f) Charge imbalance effects on NDC at cases of ${\zeta }^{\prime }=0,0.65\pi$, and $\pi$, respectively. Curves with diamonds, triangles, squares, circles, and crosses correspond to $m\delta {\gamma }^2=-0.975,-0.5$, 0, $0.5$, and $0.975$, respectively. The other parameters are the same as (a)–(c) except $Z_0=0$ and $\delta {\gamma }^2$ varies.

Figure 4

(a) Zero-bias NDC vs $T$ at various ${\zeta }^{\prime }$ [convention and setting are the same as Fig. 3]. (b) Zero-bias NDC vs $T$ at various $\delta {\gamma }^2$ [same as Fig. 3].