Josephson diode effect via a non-equilibrium Rashba system

Abstract

A non-equilibrium state in a Rashba system under an in-plane magnetic field is identified as the origin of the Josephson diode effect. This state is induced by a current bias--necessary for measuring the current-voltage characteristics--which shifts the Fermi momentum away from equilibrium. This essential mechanism has been overlooked in previous studies. This oversight stems from the implicit assumption that the equilibrium-based formulations are sufficient to describe Josephson effect. We formulate the Josephson coupling via the non-equilibrium Rashba system under current bias using a tunneling Hamiltonian, where the Rashba system is modeled as one-dimensional. When the magnetic field is applied perpendicular to the current, the Josephson coupling becomes asymmetric, giving rise to the diode effect. The magnitude and sign of this effect depend on the distance between the superconducting electrodes $d$, the in-plane magnetic field, and the spin-orbit coupling strength. Our results clarify the microscopic origin of the Josephson diode effect, which can be optimized by tuning $d$.

Figures (6)

• Figure 1: (a) The device geometry of the Josephson junction through the Rashba system (M), which is supposed to be one-dimensional. The two SCs (SC$_{\rm R}$ and SC$_{\rm L}$) are separated by the Rashba system with distance $d$. An external magnetic field $h_y$ is applied in the direction perpendicular to the applied current $I_B$. The signs of $I_B$ and $h_y$ are defined by the Cartesian coordinate system shown in this panel. (b) The schematics of the current-voltage curve of the Josephson junction. Due to the Josephson diode effect, the amplitude of critical current $I_c$ in the positive branch ($I_c^+$) colored by red is different from that in the negative one ($I_c^-$) colored by blue. The broken line is the curve without magnetic field, i.e., $h_y=0$. For $h_y>0$ ($h_y<0$), the curve is shifted up (down) from that with $h_y=0$. The shift is reversed by reversing $h_y$.
• Figure 2: Diagrams contributing to the Josephson coupling. The solid lines represent the Green's function in each region. Two contributions are shown: non-spin flip (left) and spin flip (right). The Josephson diode effect comes from the process in the right panel. This term appears only when the spin-orbit interaction $\alpha_R$ is there.
• Figure 3: The black solid lines indicate the linearized dispersion relation for M in the absence of $\alpha_R$ and $h_y$. In the left panel (a) $I_B=0$, the Fermi energy $\varepsilon_F$ is shown by thin line (blue) and is common to both of left and right branches. In the right panel (b) with $I_B\ne 0$, the Fermi energy in the left branch is different from that in the right branch shown by thin liens (red).
• Figure 4: The $d$-dependenced of the assymmetry ratio $Q\equiv (I_c^+-I_c^-)/(I_c^++I_c^-)$ is plotted with $\alpha_R=10$ meVÅ for $h_y=0.01$ T (blue), 0.1 T (green), and 0.2 T (red).
• Figure 5: $Q$ is plotted as a function of $\alpha_R$ with $h_y=0.1$ T for $d=50$ nm (blue), 100 nm (green), and 150 nm (red).