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arXiv:2104.07850 · Europhysics Letters (EPL)

Distinct properties of vortex bound states driven by temperature

Xinwei Fan, Xiaoyu Chen, Huan Yang, Hai‐Hu Wen

Published Nov 1, 2021• 4 citations
arXivHTMLPDF

Abstract

Abstract We investigate the behavior of vortex bound states in the quantum limit by self-consistently solving the Bogoliubov-de Gennes equation. We find that the energies of the vortex bound states deviates from the analytical result with the half-integer angular momentum μ at very low temperature. Specifically, the energy ratio for the first three orders is closer to 1: 2: 3 than to 1: 3: 5 in the extreme quantum limit . The local density of states reveals a Friedel-like behavior associated with that of the pair potential, which will be smoothed out by the thermal effect above the quantum limit. Our studies show that the vortex bound states can exhibit very distinct features in different temperature regimes, which provides a comprehensive understanding and should stimulate more experimental efforts for verification.

DOI
10.1209/0295-5075/ac3927

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Abstract We investigate the behavior of vortex bound states in the quantum limit by self-consistently solving the Bogoliubov-de Gennes equation. We find that the energies of the vortex bound states deviates from the analytical result with the half-integer angular momentum μ at very low temperature. Specifically, the energy ratio for the first three orders is closer to 1: 2: 3 than to 1: 3: 5 in the extreme quantum limit . The local density of states reveals a Friedel-like behavior associated with that of the pair potential, which will be smoothed out by the thermal effect above the quantum limit. Our studies show that the vortex bound states can exhibit very distinct features in different temperature regimes, which provides a comprehensive understanding and should stimulate more experimental efforts for verification.
Focusing on a quantum-limit behavior, we study a single vortex in a clean sss-wave type-II superconductor by self-consistently solving the Bogoliubov--de Gennes equation. The discrete energy levels of the vortex bound states in the quantum limit are discussed. The vortex core radius shrinks monotonically up to an atomic-scale length on lowering the temperature TTT, and the shrinkage stops to saturate at a lower TTT. The pair potential, supercurrent, and local density of states around the vortex exhibit Friedel-like oscillations. The local density of states has particle-hole asymmetry induced by the vortex. These are potentially observed directly by scanning tunneling microscopy.
The temperature dependence of an isolated quantum vortex, embedded in an otherwise homogeneous fermionic superfluid of infinite extent, is determined via the Bogoliubov--de Gennes (BdG) equations across the BCS-BEC crossover. Emphasis is given to the BCS side of this crossover, where it is physically relevant to extend this study up to the critical temperature for the loss of the superfluid phase, such that the size of the vortex increases without bound. To this end, two techniques are introduced. The first one solves the BdG equations with ``free boundary conditions,'' which allows one to determine with high accuracy how the vortex profile matches its asymptotic value at a large distance from the center, thus avoiding a common practice of constraining the vortex in a cylinder with infinite walls. The second one improves on the regularization procedure of the self-consistent gap equation when the interparticle interaction is of the contact type, and permits us to considerably reduce the time needed for its numerical integration by drawing elements from the derivation of the Gross-Pitaevskii equation for composite bosons starting from the BdG equations.
A theoretical study on discrete vortex bound states is carried out near a vortex core in the presence of a van Hove singularity (VHS) near the Fermi level by solving Bogoliubov–de Gennes (BdG) equations. When the VHS lies exactly at the Fermi level and also at the middle of the band, a zero-energy state and other higher-energy states whose energy ratios follow integer numbers emerge. These discrete vortex bound state peaks undergo a splitting behavior when the VHS or Fermi level moves away from the middle of the band. Such splitting behavior will eventually lead to a new arrangement of quantized vortex core states whose energy ratios follow half-odd-integer numbers.
The spatial distribution of vortex bound states is often anisotropic, which is correlated with the underlying property of materials. In this work, we examine the effects of Fermi surface anisotropy on vortex bound states. The large-scale calculation of vortex bound states is introduced in the presence of fourfold or twofold Fermi surface by solving the Bogoliubov–de Gennes (BdG) equations. Two kinds of quasiparticles' behaviors can be extracted from the local density of states (LDOS) around a vortex. The angle-dependent quasiparticles will move from high energy to low energy when the angle varies from curvature maxima to minima of the Fermi surface, while the angle-independent quasiparticles tend to stay at a relatively higher energy. In addition, the weight of angle-dependent quasiparticles can be enhanced by the increasing anisotropy degree of Fermi surface.
A full quantum-mechanical treatment of the Bogoliubov-de Gennes equation for a single vortex in a d -wave superconductor is presented. First, we find low-energy states extended in four diagonal directions, which have no counterpart in a vortex of s -wave superconductors. The four-fold symmetry is due to " quantum effect ", which is enhanced when p F ξ is small. On the other hand, when p F ξ is large, the four-fold symmetry is suppressed. Second, for p F ξ ∼ 1, a peak with a large energy gap E 0 ∼ Δ is found in the density of states, which is due to the formation of the lowest bound states.