Chiral current-phase relation of topological Josephson junctions: A signature of the 4π -periodic Josephson effect

Abstract

The 4π -periodic Josephson effect is an indicator of Majorana zero modes and a ground-state degeneracy which are central to topological quantum computation. However, the observability of a 4π -periodic Josephson current-phase relation (CPR) is hindered by the necessity to fix the fermionic parity. As an alternative to a 4π -periodic CPR, this paper proposes a chiral CPR for the 4π -periodic Josephson effect. This is a CPR of the form J(φ) ∝ C | sin(φ/2)|, describing a unidirectional supercurrent with the chirality C = ±1. Its non-analytic dependence on the Josephson phase difference φ translates into the 4π -periodic CPR J(φ) ∝ sin(φ/2). The proposal requires a spin-polarized topological Josephson junction which is modeled here as a short link between spinsplit superconducting channels at the edge of a two-dimensional topological insulator. In this case, C coincides with the Chern number of the occupied spin band of the topological insulator. The paper details three scenarios of achieving a chiral CPR: by Zeeman-like splitting only, by Zeeman splitting combined with bias currents, and by an external out-of-plane magnetic field.

Transport in two-dimensional topological materials: recent developments in experiment and theory

“…More recent theories have revisited the fractional JE in the context of MZMs in hybrid structures combining conventional superconductors with topological insulators or semiconductor nanowires [440,448,494,495,496,497,498,499,500,501,502,503,504,505,506,507,455,508,509,510,457,511,512,513,514,515,516,517,518,519,520,521,522,523,524]. Theoretically, the 4π -periodic CPR (69) has been proposed for ferromagnetic weak links in quantum spin-Hall insulator/superconductor structures under assumption of the local fermion-parity conservation [440].…”

“…This example illustrates the fractional JE which is characterized by a subharmonic CPR with the frequency 1/2. References [466,467] proposed that CPR (69) could be observed in voltage-biased JJs where the phase difference evolves with time as φ(t) = 2eU t/ , producing the current [440,448,494,495,496,497,498,499,500,501,502,503,504,505,506,507,455,508,509,510,457,511,512,513,514,515,516,517,518,519,520,521,522,523,524]. Theoretically, the 4π -periodic CPR (69) has been proposed for ferromagnetic weak links in quantum spin-Hall insulator/superconductor structures under assumption of the local fermion-parity conservation [440].…”

“…A recent example is the chiral CPR proposed in Ref. [524] for 2DTI-based JJs. This is a CPR of the form describing a unidirectional supercurrent with the chirality C at T = 0.…”

“…An external magnetic field can also modify the shape of the equilibrium CPR J(φ), exposing the hidden fractional JE despite the conventional 2π-periodicity in φ. A recent example is the chiral CPR proposed in [538] for 2DTI-based JJs. This is a CPR of the form…”

Zero-frequency supercurrent susceptibility signatures of trivial and topological zero-energy states in nanowire junctions

“…A promising approach toward the detection of topological ZESs is using equilibrium phase-biased transport, see e.g. [52][53][54][55][56][57][58][59][60][61][62][63][64][65]. Here, due to a phase difference between two superconductors, a supercurrent flows across the junction, producing the Josephson effect [66].…”