Self-consistent Bogoliubov–de Gennes theory of the vortex lattice state in a two-dimensional strongly type-II superconductor at high magnetic fields

Abstract

A self-consistent Bogoliubov deGennes theory of the vortex lattice state in a 2D strong type-II superconductor at high magnetic fields reveals a novel quantum mixed state around the semiclassical Hc2, characterized by a well-defined Landau-Bloch band structure in the quasi-particle spectrum and suppressed order-parameter amplitude, which sharply crossover into the well-known semiclassical (Helfand-Werthamer) results upon decresing magnetic field. Application to the 2D superconducting state observed recently on the surface of the topological insulator Sb2Te3, accounts well for the experimental data, revealing a strong type-II superconductor, with unusualy low carrier density and very small cyclotron mass, which can be realized only in the strong coupling superconductor limit.PACS numbers: 74.25.Ha, 74.25.Uv The recent discoveries of surface and interface superconductivity with exceptionally high superconducting (SC) transition temperatures in several material struc- [3] promise to stimulate fundamental studies of the phenomenon of strong type-II superconductivity in two-dimensional (2D) and quasi 2D electron systems, particularly under high magnetic fields.[4] In a pure strong type-II superconductor under a uniform magnetic field the quasi particle spectrum is gapless in a broad field range below the upper critical field H c2 [5], [6], [4], where scattering of quasi particles by the vortex lattice interferes with the Landau quantization of the electron motion perpendicular to the magnetic field to form magnetic (Landau) Bloch's bands. In pure 2D, or quasi 2D, strong type-II SC systems, such as that realized in the multilayer system of the organic charge transfer salt κ − (ET ) 2 Cu (SCN ) 2 [7], under a magnetic field perpendicular to the easy conducting plane, the underlying normal electron spectrum is fully quantized and the effect of the vortex lattice is very pronounced. 2D vortex lattices can realize in such strongly layered electronic systems due to the presence of weak crystalline disorder [8], [9], where pinning of a few flux lines provide support for the entire vortex lattice against melting under an increasing magnetic field up to the irreversibility line [10]. Of special interest in the present paper is the unique situation of the 2D superconductivity realized in surface states of topological insulators, e.g. Due to the suppressed energy dispersion along the magnetic field direction, characterizing the 2D electron system, and the particle-hole symmetry inherent to the SC state, the quasi particle spectrum exhibits peculiar features that are missing in the 3D case. For example, at discrete magnetic field values where the chemical potential is located in the middle of a Landau band, so that the underlying normal state spectrum satisfies particle-hole symmetry, the calculated quasi-particle density of states shows a linear, Dirac-like energy dependence, which reflects topological singularities at the vortex lattice cores [4], [14]. Both the enhanced Landau quantization effect and the lucid reflection ...

2D massless Dirac Fermi gas model of superconductivity in the surface state of a topological insulator at high magnetic fields

“…[5], is then comparatively studied with respect to both models. It is found that, similar to the well known solution of the linearized self consistency equation, derived by Helfand and Werthamer for the standard model [11], [9] the desired solution of the corresponding integral equation in the 2D Weyl model is greatly facilitated by initially finding analytical solutions of the eigenvalue equation for the SC order parameter. Furthermore, the calculated H − T phase diagram for the Weyl model in the semiclassical limit (i.e.…”

“…Our calculations for Sb 2 Te 3 show that the SC helical surface state reported in [5] was in the moderate semiclassical range (n F ≥ 10), so justifying the mapping with the standard model. They reveal a very unusual, strong type-II superconductivity at low carrier density and small cyclotron effective mass, m * = 0.065m e , which can be realized only in the strong coupling (λ ∼ 1) superconductor limit [9]. Further reduction of the carrier density in such a system could yield an effective cyclotron energy comparable to or larger than the Fermi energy, LL filling factors smaller than unity, and cutoff energy larger than the chemical potential, resulting in significant deviations from the predictions of the standard model.…”

“…Using a simple s-wave BCS model, similar to the standard model described in Sec.3, with the experimentally observed dHvA frequency, F 0 = 36.5 T (implying n F 0 (H) = F0 H ), and cyclotron mass m * 0 = 0.065m e , it was shown in [9] that such an unusual SC state can exist only in the strong coupling superconductor limit. In particular, the zero field limit of the self-consistent order parameter amplitude, ∆ SC (n F 0 → ∞) → ω D / sinh (1/λ 0 ), calculated in [9], was found, for λ 0 = 1 and ω D = 0.25E F 0 , to basically agree with the spatially average SC energy gap, derived from the STS measurements (i.e. 13 meV) [5], whereas the LL filling factor, calculated at the semiclassical H c2 ( n F 0 ≈ 14), was found to agree with the experimentally determined field of the resistivity onset downshift H R (∼ 2.5 T, n F 0 ∼ 14) [5].…”

“…0.065 in Sb 2 Te 3 , see also [8]) of the free electron mass m e , resulting in a dramatic enhancement of the cyclotron frequency, ω c = eH/m * c, and the corresponding Landau level (LL) energy spacing. In a recent paper [9] we have exploited a standard electron gas model, with a quadratic energy-momentum dispersion and an effective band mass m * = 0.065m e , in a systematic investigation of the quasi-particle states and the SC pair-potential in the vortex lattice state of this system under high perpendicular magnetic fields, by solving self-consistently the corresponding Bogoliubov de Gennes (BdG) equations. The results account reasonably well for the 2D SC state observed on the surface of Sb 2 Te 3 under magnetic fields of up to 3 T [5], revealing a strong type-II superconductivity at unusually low carrier density and small cyclotron effective mass, which can be realized only in the strong coupling (λ ∼ 1) superconductor limit.…”

“…It should be noted, however, that the use of the standard LL spectrum, arising from a parabolic bandstructure, in the self-consistent BdG theory, presented in Ref. [9], has been done heuristically, without actual derivation from the effective 2D Weyl Hamiltonian describing the helical surface states observed in these topological insulators [10], [5]. Such a derivation is particularly necessary for the spin-momentum locked model under study here, since SC pairing involves certain spinorbital correlations.…”