Hybrid graphene-superconductor devices have attracted much attention since the early days of graphene research. So far, these studies have been limited to the case of diffusive transport through graphene with poorly defined and modest-quality graphene/superconductor interfaces, usually combined with small critical magnetic fields of the superconducting electrodes. Here, we report graphene-based Josephson junctions with one-dimensional edge contacts of molybdenum rhenium. The contacts exhibit a well-defined, transparent interface to the graphene, have a critical magnetic field of 8 T at 4 K, and the graphene has a high quality due to its encapsulation in hexagonal boron nitride. This allows us to study and exploit graphene Josephson junctions in a new regime, characterized by ballistic transport. We find that the critical current oscillates with the carrier density due to phase-coherent interference of the electrons and holes that carry the supercurrent caused by the formation of a Fabry-Pérot cavity. Furthermore, relatively large supercurrents are observed over unprecedented long distances of up to 1.5 μm. Finally, in the quantum Hall regime we observe broken symmetry states while the contacts remain superconducting. These achievements open up new avenues to exploit the Dirac nature of graphene in interaction with the superconducting state.

The band structure of a type-II Weyl semimetal has pairs of electron and hole pockets that coexist over a range of energies and touch at a topologically protected conical point. We identify signatures of this Weyl point in the magnetic quantum oscillations of the density of states, observable in thermodynamic properties. Tunneling between the electron and hole pockets in a magnetic field is the momentum space counterpart of Klein tunneling at a p-n junction in real space. This magnetic breakdown happens at a characteristic field strength that vanishes when the Fermi level approaches the Weyl point. The topological distinction between connected and disconnected pairs of type-II Weyl cones can be distinguished by the qualitatively different dependence of the quantum oscillations on the direction of the magnetic field. DOI: 10.1103/PhysRevLett.116.236401 Weyl semimetals provide a condensed matter realization of massless relativistic fermions [1]. Their spectrum features a diabolo-shaped surface in energy-momentum space that separates helical electronlike states (moving in the direction of the momentum) from holelike states (moving opposite to the momentum) [2]. These "Weyl cones" are the three-dimensional analogue of the two-dimensional Dirac cones in graphene. The third spatial dimension provides a topological protection, by which the conical point (Weyl point) cannot be opened up unless two Weyl cones of opposite helicity are brought together in momentum space [3].Although the Weyl point cannot be locally removed, the cones can be tilted and may even tip over [4][5][6][7][8][9][10][11][12]. For the relativistic Weyl cone such a distortion is forbidden by particle-hole symmetry, but that is not a fundamental symmetry in condensed matter. While in graphene the high symmetry of the honeycomb lattice keeps the cone upright, strain providing only a weak tilt [13], the tilting can be strong in 3D Weyl semimetals. This leads to a natural division of Weyl cones into two topologically distinct types [9]. In type I the cone is only weakly tilted so that the electronlike states and holelike states occupy separate energy ranges, above or below the Weyl point. In type II the cone has tipped over so that electron and hole states coexist in energy. Many experimental realizations of a type-II Weyl semimetal have recently been reported [14][15][16][17][18][19][20][21].In a magnetic field the coexisting electron and hole pockets of a type-II Weyl semimetal are coupled by tunneling through the Weyl point (Fig. 1). Here we investigate how this process, a momentum space manifestation of Klein tunneling [22], affects the magnetic quantum oscillations of the density of states (de Haas-van Alphen effect), providing a unique thermodynamic signature of the topologically protected band structure (an alternative to proposed transport signatures [9,[23][24][25]). Because the quantum oscillations are governed by extremal cross sections of the Fermi surface, one might wonder whether some symmetry is required to align the extremal cross sect...

The LaAlO 3 =SrTiO 3 interface hosts a two-dimensional electron system that is unusually sensitive to the application of an in-plane magnetic field. Low-temperature experiments have revealed a giant negative magnetoresistance (dropping by 70%), attributed to a magnetic-field induced transition between interacting phases of conduction electrons with Kondo-screened magnetic impurities. Here we report on experiments over a broad temperature range, showing the persistence of the magnetoresistance up to the 20 K rangeindicative of a single-particle mechanism. Motivated by a striking correspondence between the temperature and carrier density dependence of our magnetoresistance measurements we propose an alternative explanation. Working in the framework of semiclassical Boltzmann transport theory we demonstrate that the combination of spin-orbit coupling and scattering from finite-range impurities can explain the observed magnitude of the negative magnetoresistance, as well as the temperature and electron density dependence. The mobile electrons at the LaAlO 3 =SrTiO 3 (LAO=STO) interface [1] display an exotic combination of superconductivity [2,3] and magnetic order [4][5][6][7]. The onset of superconductivity at sub-Kelvin temperatures appears in an interval of electron densities where the effect of Rashba spin-orbit coupling on the band structure at the Fermi level is strongest [8,9], but whether this correlation implies causation remains unclear.Transport experiments above the superconducting transition temperature have revealed a very large ("giant") drop in the sheet resistance of the LAO=STO interface upon application of a parallel magnetic field [10][11][12][13]. An explanation has been proposed [13,14] in terms of the Kondo effect: Variation of the electron density or magnetic field drives a quantum phase transition between a highresistance correlated electronic phase with screened magnetic impurities and a low-resistance phase of polarized impurity moments. The relevance of spin-orbit coupling for magnetotransport is widely appreciated [10,[14][15][16][17][18][19], but it was generally believed to be too weak an effect to provide a single-particle explanation of the giant magnetoresistance.In this work we provide experimental data (combining magnetic field, gate voltage, and temperature profiles for the resistance of the LAO=STO interface) and theoretical calculations that support an explanation fully within the single-particle context of Boltzmann transport. The key ingredients are the combination of spin-orbit coupling, band anisotropy, and finite-range electrostatic impurity scattering.The thermal insensitivity of the giant magnetoresistance [10,11], in combination with a striking correspondence that we have observed between the gate voltage and temperature dependence of the effect, are features that are difficult to reconcile with the thermally fragile Kondo interpretationbut fit naturally in the semiclassical Boltzmann description.We first present the experimental data and then turn to the theoretical de...

It was pointed out by Tewari and Sau that chiral symmetry (H → −H if e ↔ h) of the Hamiltonian of electron-hole (e-h) excitations in an N -mode superconducting wire is associated with a topological quantum number Q ∈ Z (symmetry class BDI). Here we show that Q = Tr r he equals the trace of the matrix of Andreev reflection amplitudes, providing a link with the electrical conductance G. We derive G = (2e 2 /h)|Q| for |Q| = N, N − 1, and more generally provide a Q-dependent upper and lower bound on G. We calculate the probability distribution P (G) for chaotic scattering, in the circular ensemble of random-matrix theory, to obtain the Q-dependence of weak localization and mesoscopic conductance fluctuations. We investigate the effects of chiral symmetry breaking by spin-orbit coupling of the transverse momentum (causing a class BDI-to-D crossover), in a model of a disordered semiconductor nanowire with induced superconductivity. For wire widths less than the spin-orbit coupling length, the conductance as a function of chemical potential can show a sequence of 2e 2 /h steps -insensitive to disorder.

A two-dimensional superconductor with spin-triplet p-wave pairing supports chiral or helical Majorana edge modes with a quantized (length L-independent) thermal conductance. Sufficiently strong anisotropy removes both chirality and helicity, doubling the conductance in the clean system and imposing a super-Ohmic L 1/ decay in the presence of disorder. We explain the absence of localization in the framework of the Kitaev Hamiltonian, contrasting the edge modes of the two-dimensional system with the one-dimensional Kitaev chain. While the disordered Kitaev chain has a log-normal conductance distribution peaked at an exponentially small value, the Kitaev edge has a bimodal distribution with a second peak near the conductance quantum. Shot noise provides an alternative, purely electrical method of detection of these charge-neutral edge modes.