We present the effect of disorder on the optical conductivity of two-dimensional inhomogeneous superconductors by applying the kernel polynomial method to solve the Bogoliubov-de Gennes equations. By means of the lattice size scaling of the generalized inverse participation ratio, we find that the localization length of the quasiparticle decreases significantly with the increase of the disorder strength. Meanwhile, the weak disorder can readily restrain the Drude weight, while the superconducting gap has the tendency to suppress the low-energy optical conductivity. We also employ the Lanczos exact diagonalization method to study the competition between the on-site repulsive interactions and disorder. It is shown that the screening effect of repulsive interactions significantly enhances the Drude weight in the normal phase.
The strong electron correlations are widely accepted as the key to solve the fundamentally important problems of the high-temperature superconductors [
In this paper, we study an effective tight-binding model of the inhomogeneous superconductor on a square lattice [
where are the electronic annihilation (creation) operators at sites with spin (or), denote the hopping integrals between nearest neighbor (NN) sites, present the on-site disorder energies. The superconducting order parameters, [
where represent the NN attractive interactions, is the Fermi-Dirac distribution function, and indicate the off-diagonal Green’s function of NN sites and.
We introduce the kernel polynomial method (KPM) [
With
where represent the scaled energies within the interval [–1, 1], and denote the Chebyshev polynomials of the first kind, and we introduce the Lorentz kernel to overcome the Gibbs oscillation.
We define the on-site disorder energies as random variables distributed uniformly between and, where can be regarded as the strength of disorder. In
Neglecting the intervalley scatterings, the interactions can introduce metal-insulator transition (MIT) in the two dimensional (2D) electron systems with a large number of degenerated valleys [
where denote the local density of states (LDOS) at sites. We find that the dependence of is a good approximation for the lattice size scaling of GIPR of the 2D inhomogeneous systems. As shown in
The optical conductivity can be calculated by [
where, and is the coordination number. In
To study the competition between the on-site repulsive interactions and disorder, we employ the Lanczos method [
where is the on-site repulsive interactions. We fix the disorder strength to study the effect of the on-site repulsive interactions on the optical conductivity. As shown in
In summary, the localization effect of disorder has been investigated by applying the scaling of generalized inverse participation ratio. We find that the off-diagonal superconducting order has the delocalization effect, while the on-site repulsive interactions can suppress significantly the localization of quasiparticles by screening strongly the disorder potential. The foregoing solutions can be demonstrated by observing the evolution of Drude weight.
The work was supported by the NSFC of China, under Grant No. 10974018 and 11174036, and the National Basic Research Program of China (Grant Nos. 2011CBA 00108).