Superheating field of superconductors within Ginzburg-Landau theory

We study the superheating field of a bulk superconductor within Ginzburg-Landau theory, which is valid near the critical temperature. We calculate, as functions of the Ginzburg-Landau parameter $\kappa$, the superheating field $\Hsh$ and the critical momentum $k_c$ characterizing the wavelength of the instability of the Meissner state to flux penetration. By mapping the two-dimensional linear stability theory into a one-dimensional eigenfunction problem for an ordinary differential equation, we solve the problem numerically. We demonstrate agreement between the numerics and analytics, and show convergence to the known results at both small and large $\kappa$. We discuss the implications of the results for superconducting RF cavities used in particle accelerators.