We investigate the effect of white-noise disorder on the local phase stiffness and thermodynamic properties of a two-dimensional
s-wave superconductor. Starting from a local attractive model and using path-integral formalism, we derive an effective action by decoupling the superconducting order parameter into amplitude and phase components in a gauge-invariant manner. Perturbative techniques are applied to the phase fluctuation sector to derive an effective phase-only XY model for disordered superconducting systems. Solving the saddle-point Green's function using Bogoliubov-de Gennes theory, we calculate the distributions of nearest-neighbor couplings for various disorder strengths. A single-peak distribution is observed for low disorder strength, which becomes bimodal with one peak at negative couplings as the disorder strength increases. The local phase stiffness remains randomly distributed throughout the lattice and shows no correlation with pairing amplitudes. The temperature dependence of the superfluid stiffness (
Js) is studied using Monte Carlo simulations. At strong disorder and low temperatures,
Js increases with increasing temperature, exhibiting anomalous behavior that may indicate the onset of a glassy transition. Additionally, calculations of the Edwards-Anderson order parameter in this disorder regime suggest the emergence of a phase-glass state at very low temperatures.
We study a model of a composite system constructed from a ``pairing layer'' of disconnected attractive-
U Hubbard sites that is coupled by single-particle tunneling, ${t}_{\ensuremath{\perp}}$, to a disordered metallic layer. For small interlayer tunneling the system is described by an effective long-range
XY phase model whose critical temperature,
Tc, is essentially insensitive to the disorder and is exponentially suppressed by quantum fluctuations.
Tc reaches a maximum for intermediate values of ${t}_{\ensuremath{\perp}}$, which we calculate using a combination of mean-field, classical, and quantum Monte Carlo methods. The maximal
Tc scales as a fraction of the zero-temperature gap of the attractive sites when
U is smaller than the metallic bandwidth, and is bounded by the maximal
Tc of the two-dimensional attractive Hubbard model for large
U. Our results indicate that a thin, rather than a thick, metallic coating is better suited for the enhancement of
Tc at the surface of a phase fluctuating superconductor.
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.
The effect of nonmagnetic impurities on 2D
s-wave superconductors is studied beyond the weak disorder regime. Within the Bogoliubov--de Gennes (BdG) framework, the local pairing amplitude develops a broad distribution with significant weight near zero with increasing disorder. Surprisingly, the density of states continues to show a finite spectral gap. The persistence of the spectral gap at large disorder is shown to arise from the breakup of the system into superconducting ``islands.'' Superfluid density and off-diagonal correlations show a substantial reduction at high disorder. A simple analysis of phase fluctuations about the highly inhomogeneous BdG state is shown to lead to a transition to a nonsuperconducting state.
The finite-temperature phase diagram for the two-dimensional attractive fermion Hubbard model with spin-dependent disorder is considered within Bogoliubov--de Gennes mean-field theory. Three types of disorder are studied. In the first, only one species is coupled to a random site energy; in the second, the two species both move in random site energy landscapes which are of the same amplitude but different realizations; and finally, in the third, the disorder is in the hopping rather than the site energy. For all three cases, we find that, unlike the case of spin-symmetric randomness, where the energy gap and average order parameter do not vanish as the disorder strength increases, a critical disorder strength exists separating distinct phases. In fact, the energy gap and the average order parameter vanish at distinct transitions,
Vcgap and
Vcop, allowing for a gapless superconducting (gSC) phase. The gSC phase becomes smaller with increasing temperature until it vanishes at a temperature
T∗.
We show, using a Bogoliubov-de Gennes (BdG) mean field theory, that the local pairing amplitude Δ(r) becomes highly inhomogeneous with increasing disorder in an s-wave superconductor. The probability distribution P(Δ) is peaked about the BCS value at low disorder, but with increasing disorder, progressively develops into a broad distribution with significant build up of weight near Δ≈0. At high disorder, the system is found to form superconducting "islands" separated by a non-superconducting sea. Surprisingly, a finite energy gap persists into the highly disordered state in spite of many sites having negligible pairing amplitude and is understood in detail within the BdG framework. Once the pairing amplitude becomes inhomogeneous, the role of quantum phase fluctuations becomes crucial in driving a superconductor-insulator transition at a critical disorder. The insulator is unusual as it has a finite gap for all disorder strengths in marked contrast to the Anderson insulator in non-interacting systems. We treat the phase fluctuations within a self consistent harmonic approximation and obtain the superfluid stiffness as a function of disorder, which agrees well with our earlier quantum Monte Carlo studies.