Effects of anisotropy and disorder-mediated nucleation of vortices on the superheating field of superconductors

We provide a theory for the effects of disorder and materials anisotropy on the maximum parallel surface field $H_{\mathrm{sh}$ that a superconductor can sustain, important for accelerating cavities in current particle accelerators. (Current niobium cavities routinely operate above $H_{c1}$, in a metastable regime susceptible to vortex penetration). Dirt is discussed in an 'instanton' calculation of disorder-mediated vortex nucleation. The increased susceptibility to dirt due to the smaller coherence lengths in new materials (Nb$_3$Sn, NbN, MgB$_2$) is swamped by much stronger effects of the distance from the pure $H_{\mathrm{sh}$: Nb$_3$Sn should be as reliable at 0.92 Tesla as Nb at typical operating fields of 0.18 Tesla, according to a crude estimate. The effects of anisotropy in layered materials is calculated within Ginzburg-Landau theory, applicable near the critical temperature. For high-$\kappa$ materials like MgB$_2$, the anisotropy is negligible near $T_c$; we speculate about possible large anisotropies at lower temperatures. We briefly review current experimental development of Nb$_3$Sn cavities.