Two-dimensional collective pinning and vortex-lattice melting in a-${\mathrm{Nb}}_{1\mathrm{-}\mathit{x}}$${\mathrm{Ge}}_{\mathit{x}}$ films

Current-voltage (I-V) and ac-resistivity measurements in the mixed state of amorphous ${\mathrm{Nb}}_{1\mathrm{-}\mathit{x}}$ ${\mathrm{Ge}}_{\mathit{x}}$ (x≊0.3) films in perpendicular magnetic fields are extensively analyzed for evidence of different vortex-lattice (VL) phases. In the low-temperature-field (T-B) regime flux flow is suppressed due to the presence of weak pinning centers giving rise to collective pinning of straight vortices. Various criteria for the depinning current are discussed and the impact on the collective pinning analysis is investigated. In the high T-B regime, just below the mean-field transition line ${\mathit{B}}_{\mathit{c}2}$(T), linear I-V behavior shows evidence of uniform flux flow. The flux-flow resistivity follows the theoretical predictions providing a means to determine the upper critical field. It turns out to be distinctly larger than the field ${\mathit{B}}_0$ at which the pinning disappears or the field ${\mathit{B}}_{\mathit{p}}$ where the pinning force starts to decrease sharply. In the narrow regime between ${\mathit{B}}_{\mathit{p}}$ and ${\mathit{B}}_0$ the ac resistivity ${\mathrm{\rho }}_{\mathrm{ac}}$ rises steeply towards the dc flux-flow resistivity. Either ${\mathit{B}}_{\mathit{p}}$ or ${\mathit{B}}_0$ coincides with the VL melting field ${\mathit{B}}_{\mathit{m}}$(T) as was predicted by the theory for two-dimensional melting. Two scenarios are further investigated: one in which ${\mathit{B}}_{\mathit{p}}$ is taken to be ${\mathit{B}}_{\mathit{m}}$, so that the steep rise in ${\mathrm{\rho }}_{\mathrm{ac}}$ is caused by inhomogeneous flow of the viscous vortex liquid in presence of disorder. In the other ${\mathit{B}}_0$ is identified with ${\mathit{B}}_{\mathit{m}}$ and the behavior of ${\mathrm{\rho }}_{\mathrm{ac}}$ is supposed to be evidence of thermally activated hopping of VL defects. The latter scenario is in better agreement with the experiments.