Magnetic Field Dependence of the Superconducting Energy Gap in Ginzburg-Landau Theory with Application to Al

A theoretical calculation is given of the magnetic field dependence of the superconducting energy gap, using the Ginzburg-Landau theory. In addition to depending upon the size of the specimen, the field dependence of the energy gap depends quite strongly on the nature of the boundary conditions. For the usual case with the magnetic field equal on opposite sides of a film, the calculations show that for a ratio of thickness, d, to penetration depth, λ, less than √5, the energy gap goes smoothly to zero as the critical field is approached—a second-order phase transition. When d/λ > √5, the energy gap approaches a finite value as the critical field is approached—a first-order phase transition. Energy-gap measurements for aluminum agree very well with these calculations. When one changes the boundary conditions so that the magnetic field is constrained to be zero on one side of the film, the theory predicts a very different behavior. For this case, at all thicknesses, the energy gap approaches a finite value as the critical field is approached—a first-order phase transition. An experiment involving cylindrical films is proposed to test this latter case. It is shown that in this proposed experiment these boundary conditions are appropriate for predicting the current dependence of the energy gap.