Topological Bardeen-Cooper-Schrieffer theory of superconducting quantum rings

TL;DR

This paper develops a mathematical model for the topology of the Fermi surface in quantum rings, revealing topological transitions that influence superconductivity, and derives how the critical temperature varies with geometric parameters.

Contribution

It introduces a novel exactly solvable model for the Fermi surface topology in nanorings, linking topological transitions to superconducting properties.

Findings

Two topological transitions in the Fermi surface upon shrinking the ring.

Non-monotonic variation of $T_c$ with confinement size, showing a maximum.

Monotonic increase of $T_c$ in the special case of a square toroid.

Abstract

Quantum rings have emerged as a playground for quantum mechanics and topological physics, with promising technological applications. Experimentally realizable quantum rings, albeit at the scale of a few nanometers, are 3D nanostructures. Surprisingly, no theories exist for the topology of the Fermi sea of quantum rings, and a microscopic theory of superconductivity in nanorings is also missing. In this paper, we remedy this situation by developing a mathematical model for the topology of the Fermi sea and Fermi surface, which features non-trivial hole pockets of electronic states forbidden by quantum confinement, as a function of the geometric parameters of the nanoring. The exactly solvable mathematical model features two topological transitions in the Fermi surface upon shrinking the nanoring size either, first, vertically (along its axis of revolution) and, then, in the plane orthogonal to it, or the other way round. These two topological transitions are reflected in a kink and in a characteristic discontinuity, respectively, in the electronic density of states (DOS) of the quantum ring, which is also computed. Also, closed-form expressions for the Fermi energy as a function of the geometric parameters of the ring are provided. These, along with the DOS, are then used to derive BCS equations for the superconducting critical temperature of nanorings as a function of the geometric parameters of the ring. The $T_c$ varies non-monotonically with the dominant confinement size and exhibits a prominent maximum, whereas it is a monotonically increasing function of the other, non-dominant, length scale. For the special case of a perfect square toroid (where the two length-scales coincide), the $T_c$ increases monotonically with increasing the confinement size, and in this case, there is just one topological transition.