Even-odd flux quanta effect in the Fraunhofer oscillations of an edge-channel Josephson junction

Abstract

We calculate the beating of h/2e and h/e periodic oscillations of the flux-dependent critical supercurrent I c ( ) through a quantum spin-Hall insulator between two superconducting electrodes. A conducting pathway along the superconductor connects the helical edge channels via a nonhelical channel, allowing an electron incident on the superconductor along one edge to be Andreev reflected along the opposite edge. In the limit of small Andreev reflection probability the resulting even-odd effect is described by I c ∝ | cos(e / ) + f |, with |f | 1 proportional to the probability for phase-coherent interedge transmission. Because the sign of f depends on microscopic details, a sample-dependent inversion of the alternation of large and small peaks is a distinctive feature of the beating mechanism for the even-odd effect. Superconductor-normal-metal-superconductor junctions with edge channel conduction in the normal region are governed by the interplay of charge e and charge 2e transport: Charge can only enter or exit the superconductor in units of 2e, but in the normal region this Cooper pair can be split over opposite edges, when an electron incident on the normalsuperconductor (NS) interface along one edge is Andreev reflected as a hole along the opposite edge.For quantum Hall edge channels this mechanism produces Fraunhofer oscillations (oscillations of the critical current with enclosed flux ) having a fundamental period of h/e, twice the usual periodicity [1]. These are chiral edge channels, so Andreev reflection along the edge of incidence is forbidden and only the circulating path of Fig. 1(a) contributes to the supercurrent. When the edge channels allow for propagation in both directions, the critical current includes the usual h/2e-periodic contributions from Andreev reflection along a single edge, and further h/e periodic contributions from circulating paths without charge transfer [ Fig. 1(b)].Here we investigate this beating of h/e and h/2e periodic contributions to the Fraunhofer oscillations. We are motivated by recent work on proximity induced superconductivity in quantum spin-Hall (QSH) insulators [2][3][4][5][6][7][8], which in one series of experiments [4] showed Fraunhofer oscillations with an even-odd effect: Large peaks in the critical current at even multiples of h/2e alternate with smaller peaks at odd multiples.The QSH insulator has helical edge channels (with direction of motion tied to the spin), so we consider that case in what follows (although the beating mechanism for the even-odd effect does not rely on helicity). Following Ref.[6] we assume that the superconductors dope the contacted QSH insulator, locally pushing the Fermi level in the conduction band. The broad conducting pathway that appears along the NS interface will be gapped by the superconducting proximity effect, but a narrow gapless channel may remain because superconductivity only becomes effective at some penetration length ξ 0 from the NS interface. (Reference [4] estimates ξ 0 240 nm, comparable to the estimated width o...

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