Current Flow in Topological Insulator Josephson Junctions due to Imperfections

TL;DR

The paper addresses the origin of nonzero Josephson currents in Corbino topological Josephson junctions carrying an integer number of flux quanta. In the atomic limit with nonoverlapping CdGM states, it shows that y-dependent imperfections, such as width inhomogeneities $W(y)$, lift flux-quantization zeros and generate a finite current described by $I(\varphi_0) = -\frac{\pi}{\Phi_0} \sum_k \sum_{n} \tanh(\beta E_{n,k}/2) \partial E_{n,k}/\partial \varphi_0$, while the zero mode remains currentless and excited CdGM levels contribute progressively larger currents as $I_n \propto \sqrt{n}$. The low-energy spectrum follows $E_{n,k}=\pm \omega_{Bk}\sqrt{n}$ with $\omega_{Bk}$ given by a combination of system parameters, and currents exhibit clear dependence on the perturbation gradients $\gamma_k$. The authors also predict microwave spectroscopy signatures with resonances at $\Omega_{n,k}=\omega_{Bk}(\sqrt{n}+\sqrt{n-1})$, providing concrete experimental tests to probe the CdGM spectrum and the presence of Majorana modes in long topological Josephson junctions.

Abstract

Recent experiments on planar superconductor-topological insulator-superconductor (S-TI-S) junctions, e.g., in Corbino geometry, have reported low-temperature nonzero Josephson currents in states with integer fluxoid (flux) induced in the junction by a perpendicular magnetic field. This effect was discussed in connection with Majorana zero modes localized in Josephson vortices of such junctions. Here, we provide an explanation for this phenomenon, attributing it to imperfections. We focus on the ``atomic" limit in which the low-energy bound states of different vortices do not overlap. In this limit, we can associate the nonvanishing critical current with the irregularities, e.g., in the junction's width. The low-temperature contribution to the current is provided by the bound states with low but nonzero energy. We also propose clear experimental tests based on microwave spectroscopy, revealing distinctive selection rules for vortex transitions.

Figures (8)

• Figure 1: Left panel: Planar Josephson junction; right panel: Corbino geometry Josephson junction.
• Figure 2: A cartoon showing a chain of vortices at distance $\ell_B$ from each other. The low-energy CdGM states are localized at distance $\lambda_B\ll \ell_B$.
• Figure 3: Spectral flow of a single topological Josephson vortex. The left panel shows the evolution of the spectrum with dimensionless magnetic field $\frac{2\pi \xi}{\ell_{B}}\propto B$. The right panel shows the evolution of the vortex spectrum with the chemical potential of the central region, $\mu_{N}$. Note that the spectrum condenses around the Fermi energy for the values of $\mu_{N}$ in resonance with the effective cavity modes $\mu_{N}=v\frac{\pi m}{W},\ m\in\mathbb{Z}$.
• Figure 4: Current profiles of the individual CdGM states. The upper panels correspond to the unperturbed states. The lower panels show the corrections (with $\gamma_k/\omega_{Bk}=0.05$) due to the gradient of the width $W(y)$.
• Figure 5: The schematic depiction of the allowed transitions in vortex atoms. In particular, by driving the system at frequencies $\Omega=\omega_{B}(\sqrt{n}+\sqrt{n-1}),\ n=1, 2, \dots$ it is possible to break Cooper pairs in the ground state to populate pairs of neighboring excited states at energies $\omega_{B}\sqrt{n}$ and $\omega_{B}\sqrt{n-1}$.