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arXiv:cond-mat/9706118 · Europhysics Letters (EPL)

Low-lying excitations around a single vortex in a d -wave superconductor

Y Morita, Mahito Kohmoto, Kazumi Maki

Published Oct 15, 1997• 26 citations
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Abstract

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.

DOI
10.1209/epl/i1997-00446-x

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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.
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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 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.
Starting with a microscopic Hamiltonian, we derive the Schr\"odinger equation of a superconductor with a vortex. The analysis describes the many-body state of a superconductor as a delocalized superposition of Bogoliubov-de Gennes mean-field vortex states. Taking a suitable one-band extended Hubbard model as the microscopic Hamiltonian of electrons in high-Tc{T}_{c}Tc​ compounds, solutions to the vortex Schr\"odinger equation are computed. We compute the vortex mass. Tuning the Hamiltonian parameters, we obtain solutions for the case of an sss-wave order parameter and for the ddd-wave case and consider consequences of order parameter symmetry for vortex directional motion.
Within numerically exact solutions of the Bogoliubov-de Gennes equations, we demonstrate that discrepancies between predicted low-energy quasiparticle properties in disordered 2D d-wave superconductors occur because of the unanticipated importance of disorder model details and normal state particle-hole symmetry. For the realistic case, which is best described by a binary alloy model without particle-hole symmetry, we predict density-of-state suppression below an energy scale which appears to be correlated with the corresponding single impurity resonance.