Calculation of TC in a normal-superconductor bilayer using the microscopic-based Usadel theory

The Usadel equations give a theory of superconductivity, valid in the diffusive limit, that is a generalization of the microscopic equations of the BCS theory. Because the theory is expressed in a tractable and physical form, even experimentalists can analytically and numerically calculate detailed properties of superconductors in physically relevant geometries. Here, we describe the Usadel equations and review their solution in the case of predicting the transition temperature TC of a thin normal-superconductor bilayer. We also extend this calculation for thicker bilayers to show the dependence on the resistivity of the films. These results, which show a dependence on both the interface resistance and heat capacity of the films, provide important guidance on fabricating bilayers with reproducible transition temperatures.