Microscopic derivation of magnetic-flux-density profiles, magnetization hysteresis loops, and critical currents in strongly pinned superconductors

We present a microscopic derivation, without electrodynamical assumptions, of B(x,y,H(t)), M(H(t)), and ${\mathit{J}}_{\mathit{c}}$(H(t)), in agreement with experiments on strongly pinned superconductors, for a range of values of the density and strength of the pinning sites. We numerically solve the overdamped equations of motion of these flux-gradient-driven vortices which can be temporarily trapped at pinning centers. The field is increased (decreased) by the addition (removal) of flux lines at the sample boundary, and complete hysteresis loops can be achieved by using flux lines with opposite orientation. The pinning force per unit volume we obtain for strongly pinned vortices, ${\mathit{J}}_{\mathit{c}}$B∼${\mathit{n}}_{\mathit{p}}$ ${\mathit{f}}_{\mathit{p}}^{1.6}$, interpolates between the following two extreme situations: very strongly pinned independent vortices, where ${\mathit{J}}_{\mathit{c}}$B∼${\mathit{n}}_{\mathit{P}}$ ${\mathit{f}}_{\mathit{p}}$, and the two-dimensional Larkin-Ovchinikov collective-pinning theory for weakly pinned straight vortices, where ${\mathit{J}}_{\mathit{c}}$B∼${\mathit{n}}_{\mathit{P}}$ ${\mathit{f}}_{\mathit{p}}^2$. Here, ${\mathit{n}}_{\mathit{p}}$ and ${\mathit{f}}_{\mathit{p}}$ are the density and maximum force of the pinning sites.