Magnetic field-tuned superconductor-insulator transition in quenched-condensed ultrathin Be films

We quenched condensed ultrathin Be films onto glass slides inside a dilution refrigerator with the substrates held near 10 K. The films were first tuned from insulating to superconducting by increasing the thickness in fine steps. The thickness-tuned transition occurs at a normal state sheet resistance R$_{N} \quad \sim $ 13 k-Ohm measured at 10 K. For superconducting films of various thickness, the field-tuned superconductor-insulator transition was then investigated. Remarkably, the critical resistance of the field-tuned transition was found to be R$_{C}$ = h/4e$^{2}$, independent of the thickness for films of R$_{N}$ ranging from 11 to 6 k-Ohm, critical temperature T$_{C}$ ranging from 1.3 to 5.2 K, and critical field B$_{C}$ ranging from 1 to 8 T. This result is a strong evidence for a duality quantum phase transition from a vortex glass in the superconducting state to a Bose glass in the field-induced insulating state. For thicker films with R$_{N} \quad <$ 5 k-Ohm, the critical resistance no longer remained at h/4e$^{2}$ but was nearly equal to R$_{N}$. This observation suggested that these thick films were no longer in the vicinity of the quantum critical point.