Controlled Finite Momentum Pairing and Spatially Varying Order Parameter in Proximitized HgTe Quantum Wells

Conventional $s$-wave superconductivity is understood to arise from singlet pairing of electrons with opposite Fermi momenta, forming Cooper pairs whose net momentum is zero [1]. Several recent studies have focused on structures where such conventional $s$-wave superconductors are coupled to systems with an unusual configuration of electronic spin and momentum at the Fermi surface. Under these conditions, the nature of the paired state can be modified and the system may even undergo a topological phase transition [2, 3]. Here we present measurements and theoretical calculations of several HgTe quantum wells coupled to either aluminum or niobium superconductors and subject to a magnetic field in the plane of the quantum well. By studying the oscillatory response of Josephson interference to the magnitude of the in-plane magnetic field, we find that the induced pairing within the quantum well is spatially varying. Cooper pairs acquire a tunable momentum that grows with magnetic field strength, directly reflecting the response of the spin-dependent Fermi surfaces to the in-plane magnetic field. In addition, in the regime of high electron density, nodes in the induced superconductivity evolve with the electron density in agreement with our model based on the Hamiltonian of Bernevig, Hughes, and Zhang [4]. This agreement allows us to quantitatively extract the value of $\tilde{g}/v_{F}$, where $\tilde{g}$ is the effective g-factor and $v_{F}$ is the Fermi velocity. However, at low density our measurements do not agree with our model in detail. Our new understanding of the interplay between spin physics and superconductivity introduces a way to spatially engineer the order parameter, as well as a general framework within which to investigate electronic spin texture at the Fermi surface of materials.