Theory of reentrant superconductivity in Corbino Josephson junctions

TL;DR

The paper addresses how Corbino Josephson junctions, realized on the surface of a three-dimensional topological insulator, respond to threaded magnetic flux and how topology modifies the current-phase relation. It combines a geometry-based phase-evolution framework for conventional JJs with a Majorana-edge-mode description to study topological Corbino JJs, including a discretized Majorana ladder to compute spectra and currents. The main findings are that circular junctions behave similarly in topological and conventional cases, but non-circular geometries exhibit reentrant superconductivity with a period set by the number of corners, halved in the topological case due to a sin(2φ) component; this halving offers a potential experimental marker of topological superconductivity in TI–SC hybrids. Overall, the work provides a geometry-tuned, experimentally accessible signature of Majorana-dominated transport and clarifies how edge Majorana modes reshape the current-phase relation in Corbino JJs.

Abstract

Josephson junctions made of conventional superconductors display Fraunhofer-like oscillations of the critical current as a function of the threaded magnetic flux. When the superconductors are deposited on the surface of a three-dimensional topological insulator, this pattern is slightly modified due to the presence of chiral Majorana modes. Here we calculate the critical current of a Corbino Josephson junction, where the fluxoid becomes quantized and the superconducting phase has an integer winding. We discover that circular junctions exhibit similar behavior in both topologically trivial and non-trivial scenarios, while non-circular junctions demonstrate a remarkable distinction. Using a simple analytical model, we show that these non-circular junctions exhibit reentrant superconductivity with a period related to their number of corners, and numerically we find that this period is halved in the topological case. The period halving may help establish the existence of topological superconductivity in hybrid topological insulator-superconductor junctions.

Figures (3)

• Figure 1: (a) Illustration of a Corbino Josephson junction. Blue: normal region (taken to be either a standard metal or a three-dimensional topological insulator). Gray: superconducting pads. The out-of-plane magnetic field $\mathbf{B}$ penetrates the normal region between the superconductors. (b) Phase gradient as a function of the coordinate $\theta$. For a circular junction the gradient is constant, whereas the square junction shows kinks due to the corners. The simplified single-harmony model of Eq. \ref{['eq:single_harmony']} captures this kinks structure. Inset: top view of the junction with the curves $r_{\rm in}\left(\theta\right)$ and $r_{\rm out}\left(\theta\right)$ annotated; these are the inputs to the exact phase evolution in Eq. \ref{['eq:general_phase_evolution']}.
• Figure 2: Critical current as a function of the number of vortices $n_{\rm v}$ for (a) a circular Corbino Josephson junction, (b) a conventional square junction, and (c) a topological square junction. The circular junction behaves identically for the conventional and topological cases: any $n_{\rm v}>0$ completely destroys superconductivity. In the square junction, the critical current is nonzero when $n_{\rm v}$ is a multiple of four, with the topological junction having nonzero $I_c$ also when $n_{\rm v}$ is a multiple of two. All values are normalized by their respective zero-flux critical current $I_c\left(0\right)$. For clarity of the logarithmic scale, we cut off the signals from below at $10^{-3}$.
• Figure 3: (a) Majorana ladder corresponding to the variant of the Grover--Sheng--Vishwanath Hamiltonian grover_emergent_2014li_coupled_2020 we adopt, Eq. \ref{['eq:H_ladder']}. When $t_1=2t_2$ two Majorana modes of opposite chirality are localized each on a different chain. (b) Two such ladders arranged in a closed geometry to resemble the inner and outer modes of a Corbino JJ on a 3DTI. The Josephson coupling between the inner and outer modes (dashed red lines) depends on the local phase difference $\phi\left(\theta\right)$.