Two Major Open Physics Issues in RF Superconductivity H. Padamsee & J. Sethna

1. Two Major Open Physics Issues in RF SuperconductivityH. Padamsee & J. Sethna • What is the RF critical magnetic field? Is it • Hc1, Hc, Hsh? • How does it depend on temperature? • How does it depend on Ginzburg-Landau parameter • (k = l/x)? • High-Field Q-slope (next session) • Why does the RF surface resistance of niobium increase sharply at high RF magnetic field?

2. Quick Review of DC Critical Magnetic fields Internal magnetic field Normal State Meissner State External field Hc

3. Type I and Type II SC Ginzburg and Landau treated the surface energy associated with a normal/SC phase boundary. Qualitatively, the free energy per unit volume increases by µ0 Ha2l/2 over the penetration depth (L  due to the diamagnetism; work is done to exclude the magnetic flux and falls by µ0 Hc2x0 /2 over the coherence length due to the increase of the super-electron density. If the coherence length is smaller than the penetration depth, there is a negative surface energy and it is energetically favorable at equilibrium to have normal/superconducting boundaries… Type II. If x0 > L, there is a positive surface energy and the formation of normal/SC regions is not favorable, Type I

4. Magnetization vs External FieldHc1, Hc2, Hc Boundary between Type I and Type II defined by G-L parameter k = l/x G-L theory relates Hc1, Hc2 and Hc to k, over a restricted range of k. Hsh is the maximum permissible value of the applied field, which satisfies GL equations. Metastability allows Hsh > Hc1 (or Hc)  = Hc2/√2Hc

5. Measuring Hc1 and Hc of Nb, 4.2 K from Magnetization CurvesGives questionable results for Hc1 and Hc due to flux pinning, which depends on state of sample DESY

6. Hc1 and Hc are also difficult to measure because of hysteresis (flux pinning)…Rarely (if ever) do you get a reversible curve

7. Attempted Magnetization Measurements on NON-Reversible Nb (Saito)

8. Attempted Determination of k (T)  = Hc2/√2Hc

9. Superheating Field Hsh is defined as the maximum permissible value of the applied field, which satisfies GL equations. Matricon and Saint-James solved GL equations numerically for the one-dimensional case where half of the space is occupied by a superconductor.

10. Hsh in RF Fields • In RF, fields change rapidly, within nanoseconds. • If the time it takes to nucleate fluxoids is long compared to the rf period (10 - 9 s) • There is a tendency for the meta-stable superconducting state to persist up to Hsh > Hc1. • T. Yogi Measured Hsh > Hc1 for Alloys Sn-In and In-Bi over a range of kappa values to show no discontinuity across Type I and Type II

11. T. Yogi’s Results Nb

12. T. Hays Measured the RF Critical Field for Superconductors: Nb and Nb3Sn Using High Pulse Power We plan to repeat these measurements with large grain and single crystal cavities (if funded)

13. Heuristic Arguments to determine Hsh In the process of phase transition, a boundary must be nucleated. In a Type I superconductor, the positive surface energy suggests that, in dc fields, the Meissner state can persist metastably beyond the thermodynamic critical field, up to the superheating field, Hsh. At this field, the surface energy per unit area vanishes: For Type II superconductors, it is also possible for the Meissner state to persist meta-stably above Hc1, How far above Hc1 ? is the open question

14. Saito extended the energy balance argument to other dimensional forms of nucleation such as a line nucleation (vortex nucleation). The diamagnetic energy is given by and the condensation energy is Balancing the two contributions, the superheating field is Is line nucleation the proper model for the RF critical field, i.e. the superheating field?

15. Issues with this Energy Balance Approach for H > Hc1 • As an energy-balance argument, the vortex nucleation model gives an upper bound on the equilibrium critical field for vortex penetration, which is related to Hc1. • Nothing in the energy balance argument discusses meta-stability, which is the key aspect for Hsh • The line nucleation model is useful in the context of nucleation on in-homogeneities on the scale of the coherence length, but not as a fundamental limit for uniform, flat, pure superconductors.

16. Temperature Dependence • Saito determines k (T) = Hc2 (T) /√2Hc (T) and Hc(T) from DC magnetic field data • This is questionable for two reasons • Superheating is a prediction from the Abrikosov, G-L theory, which is a perturbation theory. Therefore it is valid for T ~ Tc or D ~ 0 • Non-local effects important at lower temperatures have been shown to introduce large qualitative changes in vortex behavior. • Hysteresis (pinning) in magnetization gives unreliable answers for Hc1 and Hc • Final problem, Saito introduces Hsh-rf = √2 Hsh-dc ?? • This is incorrect for phase transition field

17. 1700 – 1750 Oe Best

18. Problem • If Hrf (0) = 1800 Oe, • According to Saito, Hsh (dc) at zero temperature = 1270 Oe • which is << Hc1 (known to be 1740 – 1900 Oe) !!

19. Experimentally Peak RF Magnetic Fields Are Rising ! Cavities material and surfaces are getting better !

20. How to correctly calculate Hsh? • Field where barrier vanishes • Linear stability analysis will also determine vortex array • At large k and T~Tc, 1-D analysis gives Hsh = 0.745 Hc (as discussed) • At lower T, we need the Eilenberger equations • (Non-local, Green’s functions, …)

21. Fundamental limits to Hsh in NiobiumWhat can theory tell us? Why a superheating field? How to calculate Hsh? • Field where barrier vanishes • Linear stability analysis • determines nucleation • mechanism: vortex array • At large kandT~Tc, 1-D analysis • Hsh = 0.745 Hc James P. Sethna, Gianluigi Catelani x • “Line nucleation” wrong • Yogi, Saito Hsh~Hc /kdiscouraging • Via “energy balance”, no barrier calculation • Does not work for large k, • Hsh < Hc1 = Hc ln(k)/(√2 k) • Correct balance theory gives Hc1not Hsh Barrier L> x Costly core x enters first; gain from field L later

22. Which Theory? If we trusted Ginsburg-Landau far below Tc, things looks good: Niobium: Hsh ~ 2400 Oe → 63 MV/m Nb3Sn: Hsh ~ 4000 Oe → 110 MV/m But Ginsburg-Landau is only valid near Tc ; RF cavities at 2K << Tc for Nb • At lower T, we need the • Eilenberger equations • (Non-local, Green’s functions, …) • Known corrections from G-L: • small effects on Hc1, Hc2 • huge effect on vortex core • (collapse from x to 1/kF) • Unknown so far: • Effect on Hsh • Work proposed & in progress… James P. Sethna, Gianluigi Catelani

23. Eilenberger/Gorkov for Hsh in Nb • Equations of motion for the (anomalous) Green’s functions f(w,n,x) and g(w,n,x) • Self-consistent equation for gap D • Maxwell equation for H from current • Constraint on the Green’s function James P. Sethna, Gianluigi Catelani Matsubara frequencies w and Fermi wavevector direction n Solve 1D ODE for uniform, superconducting state Find 2D instability threshold Hsh(functional eigenvalue crosses zero)