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 $H_{\mathrm{sh}}$ 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.

Figure 1

Solving for the 1D Ginzburg-Landau superheating field. By fixing value of the order parameter at the surface, we implicitly define the applied magnetic field $H_a(A)$ at the surface. This definition produces a branch of unstable solutions, but guarantees that our equations will have a solution for all guesses of $A$. The “nose” of the curve occurs at the $H_{\mathrm{sh}}^{1\text{D}}$, the superheating field ignoring two-dimensional fluctuations, and is the largest $H_a$ for which a nontrivial solution to Eq. (2) can be found. This example was calculated for $\kappa =1$, and $H_{\mathrm{sh}}^{1\text{D}}\approx 0.9$.

Figure 2

Numerically calculated $H_{\mathrm{sh}}$ and corresponding analytical approximations [Eqs. (10) and (11)] versus the Ginzburg-Landau parameter $\kappa$.

Figure 3

Comparison between the numerical and asymptotic critical momentum $k_c$, Eq. (13). The approximate behavior of $k_c$ near ${\kappa }_c$ (dot-dashed) is given in Eq. (12).

Figure 4

Profile of the order parameter at $H_{\mathrm{sh}}$ for $\kappa =50$ together with analytic approximations given in Eqs. (A2) and (A15).

Figure 5

Numerical profile (solid line) of the critical perturbation $\delta {\mathrm{q̃}}_y$ determining $H_{\mathrm{sh}}$ for $\kappa =50$ compared to the large-$\kappa$ perturbative result (dashed), Eq. (B22).

Figure 6

The wavelength of the critical perturbation ($2\pi /k_c$) and the Abrikosov vortex spacing ($a$) calculated at the superheating field both vanish at large $\kappa$, although the former diminishes much more quickly.