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arXiv:cond-mat/9801166 · Physical Review Letters

Low-Lying Quasiparticle Excitations around a Vortex Core in Quantum Limit

Nobuhiko Hayashi, Tomoya Isoshima, Masanori Ichioka, K. Machida

Published Mar 30, 1998• 193 citations
arXivHTMLPDF

Abstract

Focusing on a quantum-limit behavior, we study a single vortex in a clean $s$-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 $T$, and the shrinkage stops to saturate at a lower $T$. 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.

DOI
10.1103/physrevlett.80.2921

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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.
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.
Making use of the Bogoliubov-de Gennes equation, we study the quasi-particle spectrum and the vortex core structure of a single vortex in quasi 2D s-wave superconductors for small pFξ0p_Fξ_0pF​ξ0​, where pFp_FpF​ is the Fermi momentum and ξ0=vF/Δ0ξ_0=v_F/Δ_0ξ0​=vF​/Δ0​ is the coherence length(ℏ=1\hbar=1ℏ=1). In particular we find that the number of bound states decreases rapidly for decreasing pFξ0p_Fξ_0pF​ξ0​. Also for pFξ0∼1p_Fξ_0\sim 1pF​ξ0​∼1, the Kramer-Pesch effect stops around T/Tc≃0.3T/T_c\simeq 0.3T/Tc​≃0.3.
The quasiparticle resonant states around a single nonmagnetic impurity with unitary scattering in a d-wave superconductor is studied by solving the Bogoliubov--de Gennes equations based on a $t\ensuremath{-}J$ model. Both the spatial variation of the order parameter and the local density of states (LDOS) around the impurity have been investigated. We find that (i) a particle-hole symmetric system has a single symmetric zero-energy peak in the LDOS regardless of the size of the superconducting coherence length ${\ensuremath{\xi}}_{0};$ (ii) for the particle-hole asymmetric case, an asymmetric splitting of the zero-energy peak is intrinsic to a system with a small value of ${k}_{F}{\ensuremath{\xi}}_{0}.$
We study several aspects of the structure of vortices in conventional s-wave type-II superconductors. It is well known that there are low-energy quasiparticles bound to the core of a vortex. We show that under certain conditions, these quasiparticles form a degenerate Fermi gas with a finite density of states at the Fermi energy. In three-dimensional superconductors, the result is a one-dimensional Fermi gas of quasiparticles bound to the core of a vortex line. As is usual in one-dimensional systems, interactions between the quasiparticles lead to Luttinger liquid behavior. This may be probed through scanning tunneling microscopy into the vortex core. We further suggest that a Peierls-type instability in the shape of the vortex line may develop due to interactions between the quasiparticle gas and fluctuations in the vortex line shape.
In order to clarify the structure of a singly quantized vortex in a superfluid fermion gas near the Feshbach resonance, we numerically solve the generalized Bogoliubov-de Gennes equation in the boson-fermion model. The superfluid gap, which contains contributions from both condensed fermion pairs and condensed bosons, is self-consistently determined, and the quasiparticle excitation levels bound in the vortex core are explicitly shown. We find that the boson condensate contributes to enhance the matter density depletion and the discreteness of localized quasiparticle spectrum inside the core. It is predicted that the matter density depletion and the discrete core levels are detectable in the vicinity of the BCS-Bose-Einstein condensation crossover point.
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.