Temperature Dependence of the Upper Critical Field in Disordered Hubbard Model with Attraction
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, DISORDER, AND PHASE TRANSITION IN CONDENSED SYSTEM
Temperature Dependence of the Upper Critical Field in Disordered Hubbard Model with Attraction1 E. Z. Kuchinskiia,*, N. A. Kuleevaa, and M. V. Sadovskiia,b,** a
Institute for Electrophysics, Russian Academy of Sciences, Ural Branch, Yekaterinburg, 620016 Russia Institute for Metal Physics, Russian Academy of Sciences, Ural Branch, Yekaterinburg, 620108 Russia * e-mail: [email protected] ** e-mail: [email protected]
b Mikheev
Received August 30, 2017
Abstract—We study disorder effects upon the temperature behavior of the upper critical magnetic field in an attractive Hubbard model within the generalized DMFT+Σ approach. We consider the wide range of attraction potentials U—from the weak coupling limit, where superconductivity is described by BCS model, up to the strong coupling limit, where superconducting transition is related to Bose–Einstein condensation (BEC) of compact Cooper pairs, formed at temperatures significantly higher than superconducting transition temperature, as well as the wide range of disorder—from weak to strong, when the system is in the vicinity of Anderson transition. The growth of coupling strength leads to the rapid growth of Hc2(T), especially at low temperatures. In BEC limit and in the region of BCS–BEC crossover Hc2(T), dependence becomes practically linear. Disordering also leads to the general growth of Hc2(T). In BCS limit of weak coupling increasing disorder lead both to the growth of the slope of the upper critical field in the vicinity of the transition point and to the increase of Hc2(T) in the low temperature region. In the limit of strong disorder in the vicinity of the Anderson transition localization corrections lead to the additional growth of Hc2(T) at low temperatures, so that the Hc2(T) dependence becomes concave. In BCS–BEC crossover region and in BEC limit disorder only slightly influences the slope of the upper critical field close to Tc. However, in the low temperature region Hc2(T) may significantly grow with disorder in the vicinity of the Anderson transition, where localization corrections notably increase Hc2 (T = 0) also making Hc2(T) dependence concave. DOI: 10.1134/S1063776117120159
INTRODUCTION The studies of disorder influence on superconductivity have a rather long history. In pioneer papers by Abrikosov and Gor’kov [1–4] they analyzed the limit of weak disorder (pFl ≫ 1, where pF is the Fermi momentum and l is the mean free path) and weak coupling superconductivity, which is well described by BCS theory. The well-known “Anderson theorem” on the critical temperature Tc of superconductors with “normal” (nonmagnetic) disorder [5, 6] is usually also attributed to this limit. The generalization of the theory of “dirty” superconductors for the case of strong enough disorder (pFl ~ 1) (and up to the region of Anderson transition) was done in [7–10], where superconductivity was also analyzed in the weak coupling limit. Most dramatically, the effects of disordering are reflected in the behavior of the upp
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