RF Superconductivity and the Superheating Field H sh James P. Sethna, Gianluigi Catelani, and Mark Transtrum Superconducting RF cavity Lower losses Limited by maximum of H(t) in cycle Each superconducting material has maximum possible H sh Radio Frequency cavity Oscillating E(t) to accelerate particle bunches Maxwell implies oscillating H(t) Best shaped cavities: E/H = 36 MV/(m G)

Metastability and Nucleation Raindrops: the Liquid-Gas Transition “Superheating” like 110% humidity Gas phase metastable for T c > T > T sp, spinodal temperature Metastable energy barrier B droplet nucleation R 2 surface tension cost R 3 bulk energy gain Unstable spontaneous separation at T sp linear stability theory sinusoidal threshold  ~  exp(i k·z) lowers energy k TcTc T sp

Superconductors and magnetic fields What’s the superheating field? Type II (Nb and Nb 3 Sn) RF cavity operating conditions already above H c1 Vortex nucleation slower than RF frequency (GHz)   Type I (Pb) Type II superconductors  Magnetic flux lattice H > H c1 Coherence length: Decay of  Penetration depth: Decay of  Energy cost Energy gain Can we calculate the phase diagram for H sh ?

Why a superheating field? Metastability threshold and H sh Why is there a barrier to vortex penetration? How to calculate H sh ?    Costly core  enters first; gain from field  later Theories of superheating field Line nucleation H sh ~H c /  discouraging, but wrong Ginsburg-Landau theory H sh ~ H c Eilenberger equations H sh = 0.84 H c Eliashberg equations (hard!) Field where barrier vanishes Linear stability analysis determines nucleation mechanism: vortex array Barrier

Theories of superconductivity Validity versus complexity Ginzburg-Landau (GL)  (r), H(r) order parameters Spatial dependence OK Valid only near T c RF cavity operating conditions Ginzburg-Landau valid Bardeen Cooper Schrieffer (BCS) theory Pairing k, -k within vibration energy Excellent for traditional superconductors H c1 (T), H c2 (T) done H sh (T) hard (spatial dependence) n kFkF ħdħd

Theories of superconductivity Validity versus complexity Eilenberger Equations Valid at all temperatures Assumes  (r), H(r) vary slowly Green’s function f, g Vortex core collapse?? Eliashberg equations Needs electronic structure Never done before for H sh Ginzburg- Landau Underestimate for H sh MgB 2 Nb 3 Sn Nb at 2K Eilenberger equation results

Theories of superconductivity Validity versus complexity Bogoliubov-deGennes equations Pairs all k, -k Local equations for quasiparticle eigenstates We solved for vortex core states, predicted split peak Sum over all quasiparticle states to get self-consistent  (r), H(r) n kFkF Shore et al.Hess et al. Quasiparticle density of states at different distances from vortex center Experiment verified our theoretical prediction of split peak away from vortex center