Electrostatic and parallel-magnetic-field tuned two-dimensional superconductor-insulator transitions

The two-dimensional superconductor-insulator transition in disordered ultrathin amorphous bismuth films has been tuned both by electrostatic electron doping using the electric field effect and by the application of parallel magnetic fields. Electrostatic doping was carried out in zero and nonzero magnetic fields, and magnetic tuning was conducted at multiple strengths of electrostatically induced superconductivity. The various transitions were analyzed using finite size scaling to determine their critical exponent products. For the electrostatically tuned transition, the exponent product nu z=0.7 +/- 0.1, using data from intermediate temperatures down to 60 mK. Here nu is the correlation length exponent and z is the dynamical critical exponent. In the case of electrostatically tuned transitions in field, and the field-tuned transitions at various values of electrostatically induced superconductivity, scaling was successful with nu z=0.65 +/- 0.1 from intermediate temperatures down to about 100 or 150 mK. The parallel critical magnetic field, B-c, increased with electron transfer as (Delta n-Delta n(c))(0.33), and the critical resistance decreased linearly with Delta n. However, at lower temperatures, in the insulating regime, the resistance became larger than expected from extrapolation of its temperature dependence at higher temperatures, and scaling failed. These observations imply that although the electrostatic and parallel magnetic-field-tuned superconductor-insulator transitions would appear to belong to the same universality class and to be delineated by a robust phase boundary that can be crossed either by tuning Delta n or B, in the case of the field-tuned transition at the lowest temperatures, some different type of physical behavior turns on in the insulating regime.