Dimensional Crossover for Intrinsic dc Josephson Effect in B${\mathrm{i}}_2$S${\mathrm{r}}_2$CaC${\mathrm{u}}_2$${\mathrm{O}}_8$ 2212 Single Crystal Whiskers

The variations of the critical current $I_c$ across the layers as a function of the parallel magnetic field $H$ have been studied on small area stacked junctions fabricated from perfect single crystal B${\mathrm{i}}_2$S${\mathrm{r}}_2$CaC${\mathrm{u}}_2$ ${\mathrm{O}}_8$ whiskers. Intrinsic dc Josephson effect has been clearly demonstrated on structures with in-plane size $L$ less than $20\mu \mathrm{m}$. With $L$ increases, a dimensional crossover to monotonic size independent behavior of $I_c\left(H\right)$ has been observed. $I_c\left(H\right)$ decrease in this region is proportional to $\sqrt{H}$ in accordance with the Fistul-Giuliani theory.