Effects of spin-orbit coupling and spatial symmetries on the Josephson current in SNS junctions

Abstract

We present an analysis of the symmetries of the interference pattern of critical currents through a two-dimensional superconductor-semiconductor-superconductor junction, taking into account Rashba and Dresselhaus spin-orbit interaction, an arbitrarily oriented magnetic field, disorder, and structural asymmetries. We relate the symmetries of the pattern to the absence or presence of symmetries in the Hamiltonian, which provides a qualitative connection between easily measurable quantities and the spin-orbit coupling and other symmetries of the junction. We support our analysis with numerical calculations of the Josephson current based on a perturbative expansion up to eighth order in tunnel coupling between the normal region and the superconductors.Semiconductors with strong spin-orbit interaction (SOI) attracted a lot of attention in recent years. The prospect of manipulating electron spin efficiently with electric fields instead of magnetic fields makes SOI an attractive ingredient for spintronic applications [1, 2], as well as spin-based quantum computing [3,4]. Furthermore, several concrete proposals were put forward on how to create topological states of matter in hybrid structures relying on semiconductors with strong SOI: One-or twodimensional semiconductors proximitized by an s-wave superconductor can behave as a p-wave topological superconductor [5][6][7][8]. Two-dimensional semiconductor heterostructures can acquire an "inverted band structure" and enter a (topological) quantum spin Hall state [9,10]. The notion that such topological systems can host nonAbelian quasiparticles and the prospect of using these particles for topologically protected quantum computing [11] sparked an intense activity of research and fueled the interest in semiconductors with strong SOI.In most lower-dimensional semiconductor structures, the electric fields contributing to SOI have two important contributions: (i) a so-called Dresselhaus field resulting from the lack of inversion symmetry of the crystal structure and (ii) a Rashba field due to asymmetries in the applied confining potential. Although the underlying mechanisms are thus well understood, it still remains a challenge to determine the absolute and relative strength of both contributions in a given sample [12,13].Investigating the DC Josephson current through a superconductor-semiconductor-superconductor junction in the presence of an applied magnetic field has been proposed as a way to acquire information about SOI in the semiconductor [14][15][16]. Indeed, SOI can make the current depend anisotropically on the field [16] or produce an anomalous supercurrent (a current at zero phase difference) [14,[17][18][19]. These effects depend on the orientiation and type (Rashba or Dresselhaus) of the SOI and as such could therefore be used to determine or parametrize the SOI in a given sample [20].Previous models produced (semi-)analytic results for the Josephson current as a function of SOI parameters, e.g. for strictly one-dimensional wires [16,17], for quasi-one-dimension...

Gate controlled anomalous phase shift in Al/InAs Josephson junctions

“…This condition is protected by parity and time-reversal symmetries. However, the combined presence of spinorbit coupling and magnetic field breaks these symmetries [1] and can lead to a finite supercurrent even when the phase difference is zero [2,3]. This is the so called anomalous Josephson effectthe hallmark effect of superconducting spintronics -and can be characterized by the corresponding anomalous phase shift (φ 0 ) [4,5].…”

“…This condition is protected by parity and time-reversal symmetries. However the presence of spin-orbit coupling along with the application of an in-plane magnetic field can break these symmetries [1]. This allows an anomalous phase (φ 0 ), which means that with no current flowing there can be a non-zero phase across the junction or, conversely, at zero phase a current can flow.…”

Detecting topological superconductivity with φ0 Josephson junctions

“…1(b), we again find a finite phase shift which results from the singlet-triplet mixing of the doubly occupied QD states. When the spin-orbit vector Ω and the magnetic field are orthogonal, the system is invariant under a composition of time reversal and mirroring in the plane perpendicular to Ω, under which the superconducting phase goes to opposite itself; because the energy must be invariant under this symmetry, there can be no terms that are odd in the superconducting phase difference in the Hamiltonian and thus no non-trivial phase offset [25,38]. However, unlike the ground state of the SC leads, the ground states of the TS leads transform nontrivially under the above transformations and we thus anticipate a nonzero phase shift.…”

Tunable- φ Josephson junction with a quantum anomalous Hall insulator

“…The CPR in such ϕ junction J = J 0 sin(θ − ϕ) suggests that the current flows even at the zero phase difference [10][11][12]. Breaking time-reversal symmetry in X is a necessary condition to realize the ϕ junction.…”