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Progress of Theoretical Physics

Quasi-Particle Spectrum around a Single Vortex

Masaru Kato, Kazumi Maki

Published May 25, 2000• 6 citations
OpenAlex

Abstract

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 pFξ0, where pF is the Fermi momentum and ξ0 = vF/∆0 is the coherence length (¯h = 1). During our numerical calculation the particle number is conserved for each pFξ0. In particular, we find that there are only 1 or 2 bound states for pFξ0 = 1. Also, for pFξ0 = 1, the Kramer-Pesch effect ceases to exist at around T/Tc ≃ 0.3. There are renewed interests in vortex structure since the discovery of high Tc cuprate superconductors. It is possible that the superconductivity of high Tc cuprates can be characterized as d-wave superconductivity 1), 2) and is very close to the quantum

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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 pFξ0, where pF is the Fermi momentum and ξ0=vF/Δ0 is the coherence length (ħ=1). During our numerical calculation, the particle number is conserved for each pFξ0. In particular, we find that there are only 1 or 2 bound states for pFξ0=1. Also, for pFξ0=1, the Kramer-Pesch effect ceases to exist at around T/Tc≃0.3.

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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 pFξ0, where pF is the Fermi momentum and ξ0 = vF/∆0 is the coherence length (¯h = 1). During our numerical calculation the particle number is conserved for each pFξ0. In particular, we find that there are only 1 or 2 bound states for pFξ0 = 1. Also, for pFξ0 = 1, the Kramer-Pesch effect ceases to exist at around T/Tc ≃ 0.3. There are renewed interests in vortex structure since the discovery of high Tc cuprate superconductors. It is possible that the superconductivity of high Tc cuprates can be characterized as d-wave superconductivity 1), 2) and is very close to the quantum
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 pFξ0, where pF is the Fermi momentum and ξ0=vF/Δ0 is the coherence length (ħ=1). During our numerical calculation, the particle number is conserved for each pFξ0. In particular, we find that there are only 1 or 2 bound states for pFξ0=1. Also, for pFξ0=1, the Kramer-Pesch effect ceases to exist at around T/Tc≃0.3.
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.
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.
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.
In order to understand what kind of information can be extracted from the vortex-core structure in superfluid fermion-atom gases we make a systematic study for the core structure of a singly quantized vortex in the range of broad Feshbach resonance, solving numerically the generalized Bogoliubov--de Gennes equation derived from the fermion-boson model. Numerical calculations reveal that the vortex core can be well characterized by three length scales from the spatial variation of the fermionic and bosonic gap functions. We also predict that the profile of matter depletion and the quasiparticle spectrum show distinctive differences between the BCS and Bose-Einstein Condensate regimes.
Self-consistent Bogoliubov-de Gennes calculations have long shown that solving the Poisson equation inside a superconducting vortex turns a one-signed charge depletion into a modulation that alternates in sign with period π/kFπ/k_Fπ/kF​. We give an elementary account of that result. Taking the Caroli-de Gennes-Matricon bound states in a step-like gap, we show that the normalization of the bound-state spinor is nearly independent of angular momentum, which collapses the mode sum into closed form. Inside the core the vortex winding removes one Bessel channel from a completeness sum, so the density vanishes on the vortex line and carries Friedel-like oscillations of wavevector 2kF2k_F2kF​; outside it the sum gives a 1/r1/r1/r envelope decaying over a coherence length, with a residual ripple. The bound-state charge does not integrate to zero, so neutrality obliges the extended states to compensate it exactly. That compensation is complete at long wavelength but fails at the diameter of the Fermi circle, and what survives is a sign-alternating 2kF2k_F2kF​ modulation reduced only by 4kF2/(4kF2+kTF2)4k_F^2/(4k_F^2+k_{TF}^2)4kF2​/(4kF2​+kTF2​), a factor lying between one-half and three-quarters for any metal. The oscillation is therefore not a delicate effect but a consequence of neutrality and the inefficiency of screening at large momentum transfer: in the screened total the smooth terms cancel and only the ripple is left.