Topological Bardeen-Cooper-Schrieffer theory of superconducting quantum rings

TL;DR

This work develops an exact analytical framework for metallic nanorings to capture how 3D confinement reshapes the Fermi sea topology. By modeling the ring as a hollow cylinder, it reveals four hole-pocket regions in $k$-space that evolve with geometry, producing two topological transitions: from a trivial Fermi surface ($\pi_1(S^2)=0$) to a nontrivial $\mathbb{Z}$ surface, and then to $\mathbb{Z}_6$. These transitions imprint distinct features in the DOS (a kink and a jump) and enable closed-form expressions for the Fermi energy; these feed into a BCS-based calculation of the superconducting critical temperature $T_c$, which shows non-monotonic dependence on the dominant confinement length and monotonic behavior with the secondary length, with a special square toroid case ($D=\mathcal{L}$) yielding a single transition. Overall, the paper demonstrates how nanoring geometry can tune electronic structure and superconductivity, offering a route to geometry-driven control of $T_c$ in superconducting quantum rings.

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.

Figures (15)

• Figure 1: 2D section of a thin film of thickness $L$, confined along $z$ direction and infinite along the $y$ and $x$ directions. A free electron (quantum plane wave) is assumed to have a maximum wavelength equal to the length of the medium in the direction of motion, which can be expressed as a function of the angle $\theta$, thanks to the cylindrical symmetry, as $\lambda_{max} = L/ \cos{\theta}$. This leads to a cutoff in the accessible values of the wavevector $k$. Adapted with permission from Ref. Phillips_2021.
• Figure 2: The allowed momentum space for phonon plane waves propagating in a confined sample with the thin film geometry sketched in Fig. (\ref{['fig:travaglino1']}). The two inner spheres represent the set of forbidden states in k-space (hole pockets), while the outer sphere is the Fermi sphere for the bulk material. The volume of available states in k-space is represented by the volume of the outer Fermi sphere minus the volumes of the two smaller spheres which represent states that are not available due to confinement. Adapted with permission from Ref. Phillips_2021.
• Figure 3: Schematic section in real space of the confined ring sample. (a) Projection of the confinement in xy space; a ring of inner radius $a$ and outer radius $b$. (b) Projection of the confinement in the xz plane: a rectangle with dimensions $b$ and $D$. The inner light blue rectangle with dimensions $a$ and $D$ depicts the area in the xy plane where the inner circle is located.
• Figure 4: Schematic section in real space of the confined ring sample. (a) Projection of the confinement in xy space; the red line represents the path with maximum length. (b) Projection of the confinement in xz space; the red line represents the path with maximum length. $\theta$ is the angle between the red line and the z-axis.
• Figure 5: Rendering of the geometry of the different regions of the k-space. The yellow sphere is the Fermi sphere, while the orange and the green spheres are the regions in which are located the hole pockets of forbidden states. This is not to scale.