We study the effect of disorder on the superconductor-insulator transition in an inhomogeneous d-wave superconductor using the kernel polynomial method. As the Bogoliubov-de Gennes equations of the two-dimensional square lattice are solved self-consistently for the cases with more than 100000 unit cells, it is possible to observe the spatial fluctuations of the superconducting order parameters at the nanoscale. We find that strong spatial fluctuation of the superconducting order parameters can be introduced by disorder, and some superconducting specific order parameters are even enhanced. Moreover, we find that some isolated superconducting “islands” can survive the strong disorder, giving a boson insulator with some localized Cooper pairs. Our numerical calculations predict the existence of two sequential transitions with the increasing disorder strength: a d-wave to s-wave superconductor transition, and then an s-wave superconductor to insulator transition. The possibility of the appearance of a metallic phase between the superconducting and insulating phases is excluded by performing the lattice-size scaling of the generalized inverse participation ratio. In addition, we also discuss the effect of disorder on the optical conductivity of the d-wave superconductors.
We study the effect of disorder on the superconductor-insulator transition in an inhomogeneous d-wave superconductor using the kernel polynomial method. As the Bogoliubov-de Gennes equations of the two-dimensional square lattice are solved self-consistently for the cases with more than 100000 unit cells, it is possible to observe the spatial fluctuations of the superconducting order parameters at the nanoscale. We find that strong spatial fluctuation of the superconducting order parameters can be introduced by disorder, and some superconducting specific order parameters are even enhanced. Moreover, we find that some isolated superconducting “islands” can survive the strong disorder, giving a boson insulator with some localized Cooper pairs. Our numerical calculations predict the existence of two sequential transitions with the increasing disorder strength: a d-wave to s-wave superconductor transition, and then an s-wave superconductor to insulator transition. The possibility of the appearance of a metallic phase between the superconducting and insulating phases is excluded by performing the lattice-size scaling of the generalized inverse participation ratio. In addition, we also discuss the effect of disorder on the optical conductivity of the d-wave superconductors.
We study the superconductor-insulator transition (SIT) in d-wave superconducting ultrathin films. By means of the kernel polynomial method, the Bogoliubov-de Gennes equations are solved for square lattices with up to 360×360 unit cells self-consistently, making it possible to observe fully the nanoscale spatial fluctuations of the superconducting order parameters and discriminate accurately the localized quasiparticle states from the extended ones by the lattice-size scaling of the generalized inverse participation ratio. It is shown that Anderson localization can not entirely inhibit the occurrence of the local superconductivity in strongly-disordered d-wave superconductors. Separated by an insulating 'sea' completely, a few isolated superconducting 'islands' with significant enhancement of the local superconducting order parameters can survive across the SIT. The disorder-driven SIT, therefore, is a transition from a d-wave superconductor to a Bose insulator which consists of localized Cooper pairs. Unlike an s-wave superconductor which presents a robust single-particle gap across the SIT, the optical conductivity of a d-wave superconductor reveals a gapless insulating phase, where the SIT can be detected by observing the disappearance of the Drude weight with the increasing disorder.
Competition between superconductivity and disorder plays an essential role in understanding the metal-insulator transition. Based on the Bogoliubov-de Gennes framework, we studied an 2D s-wave fermionic optical lattice system with both spin- orbit coupling and disorder are presented. We find that, with the increase of the strength of disorder, the mean superconducting order parameter will vanish while the energy gap will persist, which indicates that the system undergoes a transition from a superconducting state to a gapped insulating state. This can be confirmed by calculating the inverse participation ratio. We also find that, if the strength of disorder is small, the superconducting order parameter and the energy gap will decrease if we increase the strength of spin-orbit coupling and Zeeman field. In the large disorder limits, the increase of the strength of spin- orbit coupling will increase the mean superconducting order parameter. This phenomenon shows that the system is more insensitive to disorder if the spin-orbit coupling is presented. Numerical computing also shows that the whole system breaks up into several superconducting islands instead of being superconductive.
The kernel polynomial method, as an approach to deriving highly-accurate spectral properties of large sparse matrices, is extended to investigate the effects of disorder in two-dimensional inhomogeneous superconductors. Being able to solve the Bogoliubov-de Gennes equations for quite large square lattices self-consistently, we gain new microscopic insights and make intuitive observations of the localization of low-energy quasiparticles on the nanometer scale. We also compare the behaviors of the optical conductivities of superconductors with different pairing symmetries. We find that the Drude weight turns out to be finite only for the weakly-disordered d -wave superconductor while obvious suppression of the optical conductivity in the infrared range can be observed in the disordered s -wave superconductor.
Competition between superconductivity and disorder plays an essential role in understanding the metal-insulator transition. Based on the Bogoliubov-de Gennes framework, we studied an 2D s-wave fermionic optical lattice system with both spin- orbit coupling and disorder are presented. We find that, with the increase of the strength of disorder, the mean superconducting order parameter will vanish while the energy gap will persist, which indicates that the system undergoes a transition from a superconducting state to a gapped insulating state. This can be confirmed by calculating the inverse participation ratio. We also find that, if the strength of disorder is small, the superconducting order parameter and the energy gap will decrease if we increase the strength of spin-orbit coupling and Zeeman field. In the large disorder limits, the increase of the strength of spin- orbit coupling will increase the mean superconducting order parameter. This phenomenon shows that the system is more insensitive to disorder if the spin-orbit coupling is presented. Numerical computing also shows that the whole system breaks up into several superconducting islands instead of being superconductive.
We study a simple model of a two-dimensional s-wave superconductor in the presence of a random potential as a function of disorder strength. We first use the Bogoliubov--de Gennes (BdG) approach to show that, with increasing disorder, the pairing amplitude becomes spatially inhomogeneous, and the system cannot be described within conventional approaches for studying disordered superconductors that assume a uniform order parameter. In the high-disorder regime, we find that the system breaks up into superconducting islands, with large pairing amplitude, separated by an insulating sea. We show that this inhomogeneity has important implications for the physical properties of this system, such as superfluid density and the density of states. We find that a finite spectral gap persists in the density of states, even in the weak-coupling regime, for all values of disorder, and we provide a detailed understanding of this remarkable result. We next generalize Anderson's idea of the pairing of exact eigenstates to include an inhomogeneous pairing amplitude, and show that it is able to qualitatively capture many of the nontrivial features of the full BdG analysis. Finally, we study the transition to a gapped insulating state driven by quantum phase fluctuations about the inhomogeneous superconducting state.
In this article we study superconductor-insulator transitions within the general framework of an attractive Hubbard model. This is a well-defined model of s-wave superconductivity which permits different tuning parameters (disorder and field). Furthermore, it allows a comparison of various analytical and computational approaches in order to gain a complete understanding of the various effects of amplitude and phase fluctuations. We present a systematic pedagogical approach, aiming to equip the lay reader with enough apparatus to be able to understand the numerical calculations, reproduce some of the simpler results, and be able to tackle future problems related to inhomogeneous phases. We go into considerable detail on mean-field theory (MFT) and the Bogoliubov-de Gennes (BdG) approach, as these are a first line of attack which can capture much of the physics, but we also outline cases where this fails to capture phase fluctuations and more sophisticated Quantum Monte Carlo (QMC) calculations are necessary. We discuss the behavior of many observables, including densities of states, superfluid stiffness, and dynamical conductivity, for the disorder-tuned superconductor-insulator transition. We also discuss SITs tuned by parallel magnetic field, which are quite different due to pairbreaking.