Transverse shift in Andreev reflection

Abstract

An incoming electron is reflected back as a hole at a normal-metal-superconductor interface, a process known as Andreev reflection. We predict that there exists a universal transverse shift in this process due to the effect of spin-orbit coupling in the normal metal. Particularly, using both the scattering approach and the argument of angular momentum conservation, we demonstrate that the shifts are pronounced for lightly-doped Weyl semimetals, and are opposite for incoming electrons with different chirality, generating a chirality-dependent Hall effect for the reflected holes. The predicted shift is not limited to Weyl systems, but exists for a general three-dimensional spin-orbitcoupled metal interfaced with a superconductor.Spin-orbit coupling (SOC) underlies many topics that are at the frontier of current research. Intuitively, under SOC, the change of a particle's spin polarization will tend to alter its orbital motion. The effect is particularly pronounced when particles are scattered at certain interfaces. For example, when reflected at an interface, a circularly-polarized light beam acquires a transverse shift normal to its plane of incidence, known as ImbertFedorov shift [1][2][3][4][5][6], due to the intrinsic SOC of light [7]. Recently, an analogous transverse shift is predicted for Weyl electrons [8,9], the fermionic cousin of photons with strong SOC in so-called Weyl semimetals [10][11][12][13][14][15][16][17][18][19], when they are scattered by electrostatic potentials or velocity gradients. This discovery has generated great interest [20,21], and hints at possible universality of such effect in spin-orbit-coupled systems.There is an intriguing scattering process unique for the normal-metal-superconductor (NS) interface-Andreev reflection [22], in which an incoming electron excitation from the normal metal at energy ε above the Fermi level E F is reflected back as a hole excitation with energy ε below E F [23]. The process conserves energy and momentum but not charge: the missing charge of (−2e) is absorbed as a Cooper pair at Fermi level into the superconductor. For excitation energies below the superconducting gap, electrons cannot penetrate into the superconudctor, and Andreev reflection becomes the dominating mechanism for transport through the NS interface. A natural question arises: Is there a transverse shift associated with Andreev reflection? This question is not trivial, since the incoming and outgoing particles possess distinct identities with opposite charges. To our knowledge, it has not been posed or studied before.In this work, we answer the above question in the affirmative. We predict that a transverse shift generally exists in Andreev reflection between a three-dimensional spin-orbit-coupled metal and a conventional superconductor. We explicitly demonstrate the effect for two examples: a lightly-doped Weyl semimetal and a spin-orbitcoupled metal without any band-crossing. The result is derived via the quantum mechanical scattering approach, and in special cases can be exactly veri...

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