Application of Bogoliubov-de Gennes equations to vortices in Hubbard superconductors
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Application of Bogoliubov-de Gennes equations to vortices in Hubbard superconductors Chumin Wang1, César G. Galván1 and Luis A. Pérez2 1 2
Instituto de Investigaciones en Materiales, Universidad Nacional Autónoma de México, D.F., México Instituto de Física, Universidad Nacional Autónoma de México, D.F., México
ABSTRACT In this work, the formation of d-wave superconducting magnetic vortex is studied within the Bogoliubov-de Gennes formalism and the generalized Hubbard model, which leads to 2N2 coupled self-consistent equations for a supercell of N×N atoms. These equations determine the spatial variation of the superconducting gap as a function of the electron concentration and electron-electron interactions. The results show that the superconducting states induced by the correlated hopping ( ∆t3 ) are more sensitive to the presence of magnetic field than those induced by attractive nearest-neighbor interaction (V). Furthermore, we calculate the electronic specific heat as a function of the temperature for a given applied magnetic field, whose behavior has a qualitative agreement with experimental data. INTRODUCTION In type-II superconductors there are two critical magnetic field values and, when the applied magnetic-field strength is between these two values, vortices with quantum magnetic fluxes appear in these superconductors [1]. The Bogoliubov-de Gennes formalism [2] provides a microscopic description of the vortex formation, vortex symmetry and interaction between vortices. On the other hand, the generalized Hubbard model, which includes firstneighbor (∆t) and second-neighbor (∆t3) correlated hopping interactions besides the onsite (U) and nearest-neighbor (V) electron-electron interactions, has been used to investigate the anisotropic superconductivity [3]. In particular, the d-wave superconductivity observed in high-Tc ceramic superconductors can be studied by means of a negative V [4] or a positive ∆t3 [5] within the generalized Hubbard model. In this work, we report a comparative study of the d-wave superconducting states originated from V and ∆t3 under an applied magnetic field within the Bogoliubov-de Gennes formalism. This study was carried out by using the supercell technique, which leads to 2N2 coupled self-consistent equations for a supercell of N×N atoms. These equations allow determining the spatial variation of the superconducting gap as a function of the electron concentration and electron-electron interactions. THE MODEL Let us consider a single-band Hubbard Hamiltonian with U =∆t =0 on a square lattice given by [3] V (1) = Hˆ t ∑ cˆm† ,σ cˆn ,σ + t ′ ∑ cˆm† ,σ cˆn ,σ + ∑ nˆm nˆn + ∆t3 ∑ cˆm† ,σ cˆn ,σ nˆ l 2 < m , n > ,σ > ,σ > ,σ , < m ,l > , < n ,l >
where cˆm† ,σ ( cˆm ,σ ) is the creation (annihilation) operator with spin σ = ↑ or ↓ at site m,
nˆm,σ = cˆm† ,σ cˆm,σ , nˆm = nˆm,↑ + nˆm,↓ , < m, n > and > respectively denote first- and secondneighbor sites. For a singlet superconductor with uniform electron charge and spin densities
ρ
( nˆl ,↑ nˆl ,↓ ) in an external magnetic field (B =
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