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We consider a two-dimensional electron gas with strong spin-orbit coupling contacted by two superconducting leads, forming a Josephson junction. We show that in the presence of an in-plane Zeeman field the quasi-one-dimensional region between the two superconductors can support a topological superconducting phase hosting Majorana bound states at its ends. We study the phase diagram of the system as a function of the Zeeman field and the phase difference between the two superconductors (treated as an externally controlled parameter). Remarkably, at a phase difference of $pi$, the topological phase is obtained for almost any value of the Zeeman field and chemical potential. In a setup where the phase is not controlled externally, we find that the system undergoes a first-order topological phase transition when the Zeeman field is varied. At the transition, the phase difference in the ground state changes abruptly from a value close to zero, at which the system is trivial, to a value close to $pi$, at which the system is topological. The critical current through the junction exhibits a sharp minimum at the critical Zeeman field, and is therefore a natural diagnostic of the transition. We point out that in presence of a symmetry under a modified mirror reflection followed by time reversal, the system belongs to a higher symmetry class and the phase diagram as a function of the phase difference and the Zeeman field becomes richer.
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Majorana zero modes are quasiparticle states localized at the boundaries of topological superconductors that are expected to be ideal building blocks for fault-tolerant quantum computing. Several observations of zero-bias conductance peaks measured in tunneling spectroscopy above a critical magnetic field have been reported as experimental indications of Majorana zero modes in superconductor/semiconductor nanowires. On the other hand, two dimensional systems offer the alternative approach to confine Ma jorana channels within planar Josephson junctions, in which the phase difference {phi} between the superconducting leads represents an additional tuning knob predicted to drive the system into the topological phase at lower magnetic fields. Here, we report the observation of phase-dependent zero-bias conductance peaks measured by tunneling spectroscopy at the end of Josephson junctions realized on a InAs/Al heterostructure. Biasing the junction to {phi} ~ {pi} significantly reduces the critical field at which the zero-bias peak appears, with respect to {phi} = 0. The phase and magnetic field dependence of the zero-energy states is consistent with a model of Majorana zero modes in finite-size Josephson junctions. Besides providing experimental evidence of phase-tuned topological superconductivity, our devices are compatible with superconducting quantum electrodynamics architectures and scalable to complex geometries needed for topological quantum computing.
We theoretically study topological planar Josephson junctions (JJs) formed from spin-orbit-coupled two-dimensional electron gases (2DEGs) proximitized by two superconductors and subjected to an in-plane magnetic field $B_parallel$. Compared to previous studies of topological superconductivity in these junctions, here we consider the case where the superconducting leads are narrower than the superconducting coherence length. In this limit the system may be viewed as a proximitized multiband wire, with an additional knob being the phase difference $phi$ between the superconducting leads. A combination of mirror and time-reversal symmetry may put the system into the class BDI. Breaking this symmetry changes the symmetry class to class D. The class D phase diagram depends strongly on $B_{parallel}$ and chemical potential, with a weaker dependence on $phi$ for JJs with narrower superconducting leads. In contrast, the BDI phase diagram depends strongly on both $B_parallel$ and $phi$. Interestingly, the BDI phase diagram has a fan-shaped region with phase boundaries which move away from $phi = pi$ linearly with $B_parallel$. The number of distinct phases in the fan increases with increasing chemical potential. We study the dependence of the JJs critical current on $B_parallel$, and find that minima in the critical current indicate first-order phase transitions in the junction only when the spin-orbit coupling strength is small. In contrast to the case of a JJ with wide leads, in the narrow case these transitions are not accompanied by a change in the JJs topological index. Our results, calculated using realistic experimental parameters, provide guidelines for present and future searches for topological superconductivity in JJs with narrow leads, and are particularly relevant to recent experiments [A. Fornieri et al., Nature (London) 569, 89 (2019)].
We show that the time reversal symmetry inevitably breaks in a superconducting Josephson junction formed by two superconductors with different pairing symmetries dubbed as i-Josephson junction. While the leading conventional Josephson coupling vanishes in such an i-Josephson junction, the second order coupling from tunneling always generates chiral superconductivity orders with broken time reversal symmetry. Josephson frequency in the i-junction is doubled, namely $omega = 4eV /h$. The result provides a way to engineer topological superconductivity such as the d + id -wave superconducting state characterized by a nonzero Chern number.
Topological superconductivity in quasi-one-dimensional systems is a novel phase of matter with possible implications for quantum computation. Despite years of effort, a definitive signature of this phase in experiments is still debated. A major cause of this ambiguity is the side effects of applying a magnetic field: induced in-gap states, vortices, and alignment issues. Here we propose a planar semiconductor-superconductor heterostructure as a platform for realizing topological superconductivity without applying a magnetic field to the 2D electron gas hosting the topological state. Time-reversal symmetry is broken only by phase-biasing the proximitizing superconductors, which can be achieved using extremely small fluxes or bias currents far from the quasi-one-dimensional channel. Our platform is based on interference between this phase biasing and the phase arising from strong spin-orbit coupling in closed electron trajectories. The principle is demonstrated analytically using a simple model, and then shown numerically for realistic devices. We show a robust topological phase diagram, as well as explicit wavefunctions of Majorana zero modes. We discuss experimental issues regarding the practical implementation of our proposal, establishing it as an accessible scheme with contemporary experimental techniques.
The phase-dependent bound states (Andreev levels) of a Josephson junction can cross at the Fermi level, if the superconducting ground state switches between even and odd fermion parity. The level crossing is topologically protected, in the absence of time-reversal and spin-rotation symmetry, irrespective of whether the superconductor itself is topologically trivial or not. We develop a statistical theory of these topological transitions in an N-mode quantum-dot Josephson junction, by associating the Andreev level crossings with the real eigenvalues of a random non-Hermitian matrix. The number of topological transitions in a 2pi phase interval scales as sqrt(N) and their spacing distribution is a hybrid of the Wigner and Poisson distributions of random-matrix theory.
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