Critical currents and vortex-unbinding transitions in quench-condensed ultrathin films of Bismuth and Tin

We have investigated the I-V characteristics of strongly disordered ultra-thin films of {\it Bi} and {\it Sn} produced by quench-condensation. Our results show that both these sytems can be visualized as strongly disordered arrays of Josephson junctions. The experimentally observed I-V characteristics of these films is hysteretic, when the injected current is ramped from zero to critical current and back. These are remarkably similar to the hysteretic I-V of an underdamped single junction. We show by computer simulations that hysteresis can persist in a very strongly disordered array. It is also possible to estimate the individual junction parameters ($R$, $C$ and $I_c$) from the experimental I-Vs of the film using this model. The films studied are in a regime where the Josephson-coupling energy is larger than the charging energy. We find that a simple relation $I_c(T)=I_c(0)(1-(T/T_c)^4)$ describes the temperature dependence of the critical current quite accurately for films with sheet resistance $\sim$ 500$\Omega$ or lower. We also find evidence of a vortex-unbindi