Magnetic-Field Dependence of Microwave Absorption and Energy Gap in Superconducting Films

Abstract

We have measured the change in absorption of microwave radiation by superconducting indium, tin, and lead films as a function of magnetic field. Experiments have been conducted on films 145-700 \AA{} thick using 8-mm- and 4-mm-wavelength microwave radiation, primarily with the magnetic field parallel to the plane of the film, but also as a function of angle between the magnetic field and the plane of the film. We observe a second-order phase change in the magnetic field, as is to be expected for films of this thickness according to the theories of Ginzburg and Landau and of Nambu and Tuan. The angular dependence of the critical field found in our measurements is in good agreement with the theory of one of the authors, which predicts an angular dependence given by the formula $(\frac{{H}_{c}sin\ensuremath{\theta}}{{H}_{c\ensuremath{\perp}}})+{(\frac{{H}_{c}cos\ensuremath{\theta}}{{H}_{c\mathrm{II}}})}^{2}=1$.We have attempted to infer the magnetic-field dependence of the effective energy gap $\ensuremath{\Delta}$ from our absorption measurements in a parallel magnetic field. This inference was made by using the calculations of Miller, based on the theory of Mattis and Bardeen. The magnetic-field dependence of $\ensuremath{\Delta}$ which we found is not in agreement with previous experiment and theory, as $\ensuremath{\Delta}$ decreases much more rapidly in low magnetic fields in our experiments, nor is it completely consistent for the two frequencies used. Thus, it appears from our experimental absorption measurements that the effect of a magnetic field on a superconducting film cannot be adequately described simply by introducing a field-dependent energy-gap parameter.