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Superconductivity in disordered materials

Fabio Siringo

Published Dec 1, 1997• 0 citations
OpenAlex

Abstract

— The real-space single-particle Green’s function formalism is extended to the attractive disordered Hubbard Hamiltonian by matrix Nambu notation, thus providing a general approach to the superconductive instability of a disordered system in quantum mean-field approximation. Configurational averages may be performed by a recent method, already employed for the repulsive model, thus allowing for spatial fluctuations of the local order parameter. The method is shown to be a natural way of investigating the validity limit of the Anderson theorem. An exact analytical solution is provided for a Bethe lattice with no diagonal disorder and infinitely large connectivity.

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— The real-space single-particle Green’s function formalism is extended to the attractive disordered Hubbard Hamiltonian by matrix Nambu notation, thus providing a general approach to the superconductive instability of a disordered system in quantum mean-field approximation. Configurational averages may be performed by a recent method, already employed for the repulsive model, thus allowing for spatial fluctuations of the local order parameter. The method is shown to be a natural way of investigating the validity limit of the Anderson theorem. An exact analytical solution is provided for a Bethe lattice with no diagonal disorder and infinitely large connectivity.
We present a real-space approach to study the effect of disorder on superconductivity. The method is based on augmented space formalism that goes beyond mean-field approximations for configuration averaging and effectively deals with the influence of configuration fluctuations of the neighborhood of an atom. In the regime of validity of Anderson's theorem our results for sss- and ddd-wave dirty superconductors have excellent agreement with existing results. The formalism is extended and tested for random negative UUU Hubbard model. Having verified the reliability of our method we use it to study environment-dependent inhomogeneous randomness in disordered superconducting systems.
We study the normal (nonsuperconducting) phase of the attractive Hubbard model within the dynamical mean field theory (DMFT) using the numerical renormalization group (NRG) as an impurity solver. A wide range of attractive potentials U is considered, from the weak-coupling limit, where superconducting instability is well described by the BCS approximation, to the strong-coupling region, where the superconducting transition is described by Bose condensation of compact Cooper pairs, which are formed at temperatures much exceeding the superconducting transition temperature. We calculate the density of states, the spectral density, and the optical conductivity in the normal phase for this wide range of U, including the disorder effects. We also present the results on superconducting instability of the normal state dependence on the attraction strength U and the degree of disorder. The disorder influence on the critical temperature T c is rather weak, suggesting in fact the validity of Anderson’s theorem, with the account of the general widening of the conduction band due to disorder.
To explore correlated electrons in the presence of local and nonlocal disorder, the Blackman-Esterling-Berk method for averaging over off-diagonal disorder is implemented into dynamical mean-field theory using tensor notation. The impurity model combining disorder and correlations is solved using the recently developed fork tensor-product state solver, which allows one to calculate the single particle spectral functions on the real-frequency axis. In the absence of off-diagonal hopping, we establish exact bounds of the spectral function of the noninteracting Bethe lattice with coordination number ZZZ. In the presence of interaction, the Mott insulating paramagnetic phase of the one-band Hubbard model is computed at zero temperature in alloys with site- and off-diagonal disorder. When the Hubbard UUU parameter is increased, transitions from an alloy band insulator through a correlated metal into a Mott insulating phase are found to take place.
Using the generalized DMFT+Σ approach, we study the influence of disorder on single-particle properties of the normal phase and the superconducting transition temperature in the attractive Hubbard model. A wide range of attractive potentials U is studied, from the weak coupling region, where both the instability of the normal phase and superconductivity are well described by the BCS model, to the strong-coupling region, where the superconducting transition is due to Bose-Einstein condensation (BEC) of compact Cooper pairs, formed at temperatures much higher than the superconducting transition temperature. We study two typical models of the conduction band with semi-elliptic and flat densities of states, respectively appropriate for three-dimensional and two-dimensional systems. For the semi-elliptic density of states, the disorder influence on all single-particle properties (e.g., density of states) is universal for an arbitrary strength of electronic correlations and disorder and is due to only the general disorder widening of the conduction band. In the case of a flat density of states, universality is absent in the general case, but still the disorder influence is mainly due to band widening, and the universal behavior is restored for large enough disorder. Using the combination of DMFT+Σ and Nozieres-Schmitt-Rink approximations, we study the disorder influence on the superconducting transition temperature T c for a range of characteristic values of U and disorder, including the BCS-BEC crossover region and the limit of strong-coupling. Disorder can either suppress T c (in the weak-coupling region) or significantly increase T c (in the strong-coupling region). However, in all cases, the generalized Anderson theorem is valid and all changes of the superconducting critical temperature are essentially due to only the general disorder widening of the conduction band.
We study the normal (non-superconducting) phase of attractive Hubbard model within dynamical mean field theory (DMFT) using numerical renormalization group (NRG) as impurity solver. Wide range of attractive potentials UUU is considered, from the weak-coupling limit, where superconducting instability is well described by BCS approximation, up to the strong-coupling region, where superconducting transition is described by Bose-condensation of compact Cooper pairs, which are formed at temperatures much exceeding superconducting transition temperature. We calculate density of states, spectral density and optical conductivity in the normal phase for this wide range of UUU, including the disorder effects. Also we present the results on superconducting instability of the normal state dependence on the attraction strength UUU and the degree of disorder. Disorder influence on the critical temperature TcT_cTc​ is rather weak, suggesting in fact the validity of Anderson theorem, with the account of the general widening of the conduction band due to disorder.
I review the quantum theory of the electron moving in a random environment. First, the quantum mechanics of individual particles scattered on a random potential is discussed. The quantum-mechanical description is extended to many-body systems by using many-body Green functions. The many-body approach is used to derive the coherent-potential approximation and to show how it fits into the dynamical mean-field theory. The generating functional of the coherent-potential approximation is obtained in an analytic form from the limit to infinite dimensions of the general many-body description of non-interacting electrons in random lattices. The analytic generating functional of the mean-field description of random systems is extended to the Falicov-Kimball model with thermally equilibrated scattering lattice potential. The many-body Green functions are then used to calculate transport properties. The electrical conductivity of the coherent-potential approximation is derived from the two-particle Green function calculated in infinite spatial dimensions. Finally, a perturbation theory for the vertex corrections to the mean-field conductivity is introduced. In particular, it is shown how to make the expansion beyond the local approximations conserving.