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 quantumMaking 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

Progress of Theoretical PhysicsMay 25, 20006 citations

Quasi-particle spectrum around a single vortex in s-wave superconductors

Masaru Kato, Kazumi Maki

arXiv:cond-mat/9810187AIP conference proceedingsJan 1, 19990 citations

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

Y Morita, Mahito Kohmoto, Kazumi Maki

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.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.

arXiv:cond-mat/9706118Europhysics Letters (EPL)Oct 15, 199726 citations

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

Nobuhiko Hayashi, Tomoya Isoshima, Masanori Ichioka +1 more

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.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.

arXiv:cond-mat/9801166Physical Review LettersMar 30, 1998193 citations

Microscopic structure of a quantized vortex core in atomic Fermi gases

Masahiko Machida, Yoji Ohashi, T. Koyama

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.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.

Physical Review AAug 25, 20066 citations

Analytical Charge Density Profile of Vortex Core in Weak-Coupling Superconductor

Chi-Ken Lu

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 $π/k_F$ . 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 $2k_F$ ; outside it the sum gives a $1/r$ 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 $2k_F$ modulation reduced only by $4k_F^2/(4k_F^2+k_{TF}^2)$ , 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.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 $π/k_F$. 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 $2k_F$; outside it the sum gives a $1/r$ 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 $2k_F$ modulation reduced only by $4k_F^2/(4k_F^2+k_{TF}^2)$, 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.

arXiv:2608.00226Jul 31, 20260 citations

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

p_Fξ_0

, where

p_F

is the Fermi momentum and

ξ_0=v_F/Δ_0

is the coherence length(

\hbar=1

). In particular we find that the number of bound states decreases rapidly for decreasing

p_Fξ_0

. Also for

p_Fξ_0\sim 1

, the Kramer-Pesch effect stops around

T/T_c\simeq 0.3

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.

π/k_F

. 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

2k_F

; outside it the sum gives a

1/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

2k_F

modulation reduced only by

4k_F^2/(4k_F^2+k_{TF}^2)

, 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.