Model for vortex pinning in a two-dimensional inhomogeneousd-wave superconductor

Abstract

We study a model for the pinning of vortices in a two-dimensional, inhomogeneous, type-II superconductor in its mixed state. The model is based on a Ginzburg-Landau (GL) free energy functional whose coefficients are determined by the mean-field transition temperature ${T}_{c0}$ and the zero-temperature penetration depth $\ensuremath{\lambda}(0)$. We find that if (i) ${T}_{c0}$ and $\ensuremath{\lambda}(0)$ are functions of position and (ii) ${\ensuremath{\lambda}}^{2}(0)\ensuremath{\propto}{T}_{c0}^{y}$ with $yg0$, then vortices tend to be pinned by regions where ${T}_{c0}$ and therefore the magnitude of the superconducting order parameter $\ensuremath{\Delta}$ are large. This behavior is in contrast to the usual picture of pinning in type-II superconductors, where pinning occurs in the small-gap regions. We also compute the local density of states of a model BCS Hamiltonian with $d$-wave symmetry, in which the pairing field $\ensuremath{\Delta}$ is obtained from the Monte Carlo simulations of a GL free energy. Several features observed in scanning tunneling spectroscopy measurements on $\mathrm{Y}{\mathrm{Ba}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{6+x}$ and ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}\mathrm{Ca}{\mathrm{Cu}}_{2}{\mathrm{O}}_{8+x}$ are well reproduced by our model: far from vortex cores, the local density of states spectra have a small gap and sharp coherence peaks, while near the vortex cores, they have a larger gap with low, broad peaks. Additionally, also in agreement with experiment, the spectra near the core do not exhibit a zero-energy peak which is, however, observed in other theoretical studies.