Localized superconductors

Abstract

We study the effects of Anderson localization on superconductivity by using a Bardeen-Cooper-Schrieffer (BCS)-type trial wave function which pairs electrons in exact time-reversed eigenstates of the single-particle Hamiltonian. Within this approximation, and neglecting localization effects on the effective Coulomb repulsion and the electron-phonon coupling, we find that superconductivity persists below the mobility edge. In fact, Anderson's theorem is valid in the localized phase as long as \ensuremath{\rho}${\ensuremath{\Delta}}_{0}$${L}^{d}$>1 (\ensuremath{\rho} is the density of states averaged over \ifmmode\pm\else\textpm\fi{}${\ensuremath{\Delta}}_{0}$ of the Fermi energy, ${\ensuremath{\Delta}}_{0}$ the BCS gap parameter, and L the localization length). Hence the gap order parameter \ensuremath{\Delta}(r) remains uniform in space at the BCS value ${\ensuremath{\Delta}}_{0}$. The superfluid density and response to electromagnetic perturbations, however, show marked differences from the ``dirty superconductor'' regime. For \ensuremath{\rho}${\ensuremath{\Delta}}_{0}$${L}^{d}$1, \ensuremath{\Delta}(r) fluctuates spatially and eventually drops to zero. In the limit when states are site localized, the system crosses over into the ``Anderson negative-U glass.'' Considerations beyond the trial wave-function approximation will speed up the destruction of superconductivity. The superconductor formed from localized states has the property that its quasiparticle excitations are also localized. Such excitations can be probed by observing the normal current in a tunneling junction.