Two Major Open Physics Issues in RF Superconductivity

H. 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

• ()?

• High-Field Q-slope (next session)– Why does the RF surface resistance of niobium

increase sharply at high RF magnetic field?

Hc

Quick Review of DC Critical

Magnetic fields

Meissner State

Normal State

External field

Inte

rnal

mag

netic

fie

ld

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 Hc2 0 /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.

Type I and Type II SC

If 0 > L, there is a positive surface energy and the formation of normal/SC regions is not favorable, Type I

Magnetization vs External FieldHc1, Hc2, Hc

Boundary between Type I and Type II defined by G-L parameter

G-L theory relates Hc1, Hc2 and Hc to , over a restricted range of .

= Hc2/√2Hc

Hsh is the maximum permissible value of the applied field, which satisfies GL equations. Metastability allows Hsh > Hc1 (or Hc)

Measuring Hc1 and Hc of Nb, 4.2 K from Magnetization Curves

Gives questionable results for Hc1 and Hc due to flux pinning, which depends on state of sample

DESY

Hc1 and Hc are also difficult to measure because of hysteresis (flux pinning)…

Rarely (if ever) do you get a reversible curve

Attempted Magnetization Measurements on NON-Reversible Nb (Saito)

Attempted Determination of

= Hc2/√2Hc

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.

Superheating Field

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

T. Yogi’s Results

Nb

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)

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

Heuristic Arguments to determine Hsh

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?

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.

Temperature Dependence• Saito determines = 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 ~ 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

1700 – 1750 Oe Best

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) !!

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

How to correctly calculate Hsh?

• Field where barrier vanishes • Linear stability analysis will also determine

vortex array• At large 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, …)

•

Why a superheating field?

Fundamental limits to Hsh in NiobiumWhat can theory tell us?

James P. Sethna, Gianluigi CatelaniHow to calculate Hsh?

Costly core enters first; gain from field later

“Line nucleation” wrong

• Yogi, Saito Hsh~Hc / discouraging• Via “energy balance”, no barrier calculation• Does not work for large

Hsh < Hc1 = Hc ln/(√2 ) • Correct balance theory gives Hc1 not Hsh

• Field where barrier vanishes • Linear stability analysis determines nucleation mechanism: vortex array• At large andT~Tc, 1-D analysis

Hsh = 0.745 Hc

Bar

rier

Which Theory?James P. Sethna, Gianluigi Catelani

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 theEilenberger equations

(Non-local, Green’s functions, …)

Known corrections from G-L:• small effects on Hc1, Hc2

• huge effect on vortex core (collapse from to 1/kF)

Unknown so far:• Effect on Hsh

Work proposed & in progress…

Eilenberger/Gorkov for Hsh in NbJames P. Sethna, Gianluigi Catelani

• Equations of motion for the (anomalous) Green’s functions f(,n,x) and g(,n,x)

• Self-consistent equation for gap • Maxwell equation for H from current• Constraint on the Green’s function

Matsubara frequencies and Fermi wavevector direction nSolve 1D ODE for uniform, superconducting stateFind 2D instability threshold Hsh (functional eigenvalue crosses zero)