Temperature dependence of the superheating field in type-II superconductors

The expulsion of an applied magnetic field is a hallmark characteristic of superconductivity. For a sufficiently large external field, the superconducting state transitions to a normal metal (type-I) or a flux-lattice state (type-II) at a field $H_{c1}$. The superconducting state is metastable and persists up to a field above $H_{c1}$, the so-called superheating field. We numerically solve the semi-classical equations of Eilenberger for the anomalous Green's functions, order parameter, and vector potential for a clean superconductor in an external magnetic field. We use a linear stability analysis to explore the local stability of the free energy to two-dimensional fluctuations, mapping the stability onto an eigenvalue problem of a linear operator. We systematically calculate the dependence of the superheating field on both temperature and the Ginzburg-Landau parameter $\kappa$. We compare our results with the analogous calculation for Ginzburg-Landau theory, which is valid only near the critical temperature, and to experimental measurements.