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Physical Review B

Augmented space recursion study of the effect of disorder on superconductivity

Shreemoyee Ganguly, A. Venkatasubramanian, Kartick Tarafder, Indra Dasgupta, Abhijit Mookerjee

Published Jun 22, 2009• 5 citations
Publisher

Abstract

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 $s$- and $d$-wave dirty superconductors have excellent agreement with existing results. The formalism is extended and tested for random negative $U$ Hubbard model. Having verified the reliability of our method we use it to study environment-dependent inhomogeneous randomness in disordered superconducting systems.

DOI
10.1103/physrevb.79.224204

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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.
— 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 vector recursion based approach to study the effect of disorder on superconductivity in a system modeled by the two-band attractive Hubbard model. We use the augmented space formalism for the disorder averaging. In the presence of only intraband pairing in a two-band disordered system with disorder in either or both bands, our calculations reveal that the gap survives in the quasiparticle spectrum; similar to single band systems. However, for interband pairing the gap in the quasiparticle spectrum ceases to exist beyond a critical value of the disorder strength. In the presence of both interband and intraband pairing interaction, depending on the relative magnitude of the pairing strength, only a particular kind of pairing is possible for a half filled two-band system.
We study the effect of disorder symmetry in a disordered s-wave superconductor. We begin with an attractive Hubbard model on a two-dimensional square lattice in the presence of diagonal- and off-diagonal disorder. The model is studied numerically using Bogoliubov-deGennes mean-field approach. We find that the off-diagonal disorder is much more efficient in destroying superconducting order. A detailed analysis brings out very distinct qualitative pictures in the two cases: diagonal disorder leads to formation of small scale isolated superconducting islands separated by large region of normal metal, whereas off-diagonal disorder leads to a connected network of superconducting region. Naively, it seems that the connected network of superconducting regions should have a large average value for superconducting order parameter, but this does not hold true for large disorder, as we demonstrate explicitly. Our qualititive picture is supported by real-space data on local order parameters.
For High-Tc superconductors the fluctuation regime is much wider than conventional superconductors due to the short coherence length and high transition temperature. AC measurements can probe the conductivity at a wide range of frequencies, and therefore provide a test of the scaling properties predicted from theory. The measured critical exponents and width of the fluctuation peak are sample dependent, which suggests that the sample inhomogeneities may play an important role in the critical region. In this thesis we study the effects of disorder on the fluctuation conductivity in the critical regime of zero-field normal-superconducting transition by the time-dependent Ginzburg-Landau theory. We set up a discretized model of the superconductor and calculate the two-dimensional and three-dimensional scaling function without disorder above Tc. The result of the discretized model deviates slightly from previous theoretical studies of the continuous model, which can be explained by a finite short wavelength cutoff in Ginzburg-Landau theory. Our results agree well with other theoretical investigations on cutoff effects from an analytical approach. Disorder in a superconductor is modeled by a distribution of Tc's at the lattice sites. We add random disorder to a two-dimensional lattice and calculate the scaling functions averaged over 1000 disorder configurations. When disorder is weak, the scaling functions are increased when T is close Tc . At strong disorder anomalous behavior may occur at the region close to Tc.
The effect of random on-site disorder on s-wave (BCS) superconductors described by a two dimensional negative-U Hubbard model is studied using Bogoliubov-de Gennes (BdG) method. The mean field pairing amplitudes become spatially inhomogeneous at large values of disorder where the system breaks up into superconducting islands with large pairing amplitudes, separated by insulating strips. The amplitude fluctuations are correctly accounted for via BdG calculations, however it misses phase fluctuations which are inherent to low dimensions. The phase fluctuations affect superconducting properties strongly, and the effect is more pronounced in the limit of large disorder. We provide a close estimate of the actual transition temperature, Tcby incorporating phase fluctuations about the inhomogeneous BdG state. This is obtained by relating the jump in renormalized Ds(obtained from a self consistent Harmonic approximation on a phase-only Hamiltonian) at the Kosterlitz-Thouless (KT) transition and the KT transition temperature, TKTobtained from Ds(T) = (2/π)TKTas temperature tends to TKTfrom below. This yields opening of a large region sandwiched between [Formula: see text] (obtained by the vanishing of [Formula: see text]) and TKTwhere there is no phase coherence between the pairs, however amplitude correlations continue to exist, reminiscent of a pseudogap phase in high-Tcsuperconductors which is marked by short ranged preformed pairs without any definite phase relation between them. Further, the appearance of the superconducting islands from a homogeneous phase indicates the evolution of a system consisting of large and overlapping pairs to one that contains short and tightly bound pairs - a scenario termed as BCS-BEC crossover. We have investigated in details the crossover phenomenon as a function of disorder and confirmed it's existence at small values of electron concentration, however, absent at larger densities.
I discuss the mean-field theory of superconductivity in a strongly disordered system of fermions with short-range attraction. It is argued that in this limit the effective theory at low energies is equivalent to the disordered Bose–Hubbard model, and I consider both the infinite-range and the "nearest-neighbor" hopping of bosons between the localized states. In the infinite-range case the mean-field theory is exact, and the superconducting gap is uniform in space, while in the latter case, the gap becomes highly non-uniform in space, but, surprisingly, is finite everywhere at T < T MF . I find that the mean-field transition temperature T MF > 0 always, and argue that the superconductor–insulator transition at T = 0 in models with net attraction between fermions is in the universality class of "dirty-bosons".