Landrò Elena, Fomin Vladimir M, Zaccone Alessio
Department of Physics "A. Pontremoli", University of Milan, Via Celoria 16, 20133 Milan, Italy.
Department of Science and High Technology, Insubria University, Via Valleggio 11, Como, 22100 Italy.
Eur Phys J B. 2025;98(1):7. doi: 10.1140/epjb/s10051-024-00851-9. Epub 2025 Jan 15.
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 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 increases monotonically with increasing the confinement size, and in this case, there is just one topological transition.
量子环已成为量子力学和拓扑物理学的一个研究领域，具有广阔的技术应用前景。实验上可实现的量子环，尽管尺寸在几纳米量级，但却是三维纳米结构。令人惊讶的是，目前尚无关于量子环费米面拓扑的理论，纳米环中超导性的微观理论也缺失。在本文中，我们通过建立一个关于费米面和费米表面拓扑的数学模型来弥补这一情况，该模型表明，由于量子限制，电子态存在非平凡的空穴口袋，它是纳米环几何参数的函数。这个精确可解的数学模型表明，在缩小纳米环尺寸时，费米表面会出现两个拓扑转变，要么首先沿垂直方向（沿其旋转轴），然后在与之正交的平面内，要么反之。这两个拓扑转变分别反映在量子环电子态密度（DOS）的一个扭结和一个特征不连续上，同时我们也计算了电子态密度。此外，还给出了作为环几何参数函数的费米能量的封闭形式表达式。然后，利用这些表达式以及电子态密度来推导纳米环超导临界温度作为环几何参数函数的BCS方程。超导临界温度随主导限制尺寸非单调变化，并呈现出一个显著的最大值，而它是另一个非主导长度尺度的单调递增函数。对于完美方形环面的特殊情况（其中两个长度尺度重合），超导临界温度随限制尺寸的增加而单调增加，在这种情况下，只有一个拓扑转变。
