RF Losses due to incomplete Meissner – Ochsenfeld effect:

Difference between Bulk Nb and Nb/CU

Legnaro National Laboratories ISTITUTO NAZIONALE DI FISICA NUCLEARE

and University of Padua

Enzo Palmieri

Because of the• Large dimensions of a cavity • Mesoscopic effect • Policristallinity of the SC (Grain Boundaries)• Pin-holes and defects in the film• Large Demagnetization factor due to steps and protrusions however present• Unavoidably slight inhomogheneities of the superconductor

However small, Incomplete Meissner-Ochsenfeld effect will be always present due to

Partially Unscreened Earth Magnetic Flux Trapping

After W. Weingarten 1986 CERN

Residual Resistance versus the magnetic field left during cooling

500 MHz 4,2 K

Two mechanisms of dissipation:

– The RF dissipation of the vortex normal core

– The energy damping due to fluxons dynamical flow

In case of trapped earth magnetic field H << HC1

Vortex density is so low, that there is no Abrikosov Lattice

Vortexes are single insulated flux quanta oscillating under RF

Bardeen and Stephen - Phys Rev 140 A1197 (1965) but also

Kim, Hempstead and Strnad - Phys Rev 139 A1163 (1965)

have shown that

The well known equation of the damped forced oscillator can be used for describing the oscillation of a vortex from and to a pinning center

If u is the displacement of a single flux line respect to the pinning center

Where

M is the effective Mass of the Vortex per unit lenght

h is the flow Viscosity

K is the elastic constant of the linearized pinning force in the approximation of small displacements

Fo is the vortex quantum

J is the current density induced by rf fields

For single non interacting vortexes,

the pinning constant

the viscosity

being s the low temperature conductivity before SC transition

Kim, Hempstead and Strnad - Phys Rev 139 A1163 (1965) A. Schmid,W. Hauger, J. Low Temp. Phys., 11,667, (1973)

for Frequencies lower than the electron collision frequency << 1/w t

The effective mass M has no appreciable dynamical effect

Then the motion equation get simplified into:

Then setting and

Since

We arrive to the Ohm law

From which the flux flow conductivity is found

Discussing the oscillatory motion of fluxoids in type II superconductros, De Gennes and Matricon introduced the notion of a depinning frequency

• below which ( << w w0) the motion is largely inhibited by pinning to crystal lattice defects

• above which ( >> w w0) pinning is relatively uneffective

The depinning frequency w0:

depends on the Residual Resistivity Ratio b

indeed

Shown by Eileberger (Phys Rev 153, 584 (1967) that the max difference between K and K1 is less than 9%

and mean free path

Attention must be paied to JC that indeed is not

the expression found by Bardeen for a thin film

that is the depairing current

but the depinning current that is definitely smaller

To estimate the depinning current let’s use Larkin-

Ovchinikov Expressions ( JLTP, 34, P409 (1979) for the

pinning force exerted by a grain boundary of

thickness t parallel to the vortex

being •n the density of states

•g1 the deviation of the electron phonon interaction constant

•t the density od states

Bulk Niobium Niobium film

Free to move Pinned by Grain Boundaries

Vortex

Vortex

Low depinning frequency w0 High depinning frequency w0

Zn 1 i

n 1 i

n

s1-is2 in place of sn

Tinkham demostrated that the Surface impedance Zn

for a normal metal in the normal regime can be extended to

SC by simply substituting the Mattis and Bardeen complex

conductivity at the place of s in the Zn formula

sn = 1 / rn = dc conductivity at T

d = skin depth

1 2

n

f E f E g E dE

,

2 11 2

n

f E g E dE

Zn 1 i

n 1 i

n

sf in place of sn

Analogously, we introduce sf in change of sn for

calculating the residual term due to the vortex flow

sn = 1 / rn = dc conductivity at T

d = skin depth

Hence the penetration depth becomes

or better written

Hence the Surface Resistance becomes

or better written

Whose real part reduces to

/ w w0

For >> w w0, practically all vortexes are depinned and Rf reaches his saturation value

For << w w0, the flux flow losses decrease as w3/2

since less and less vortexes have enough energy to overcome the pinning attraction.

After W. Weingarten 1986 CERN

Residual Resistance versus the magnetic field left during cooling

500 MHz 4,2 K

Since the critical parameter is the ratio / w w0 the Flux

flow losses increase both when working at higher

frequancies w and when increasing the RRR value b, i.e.

decreasing the depinning frequency.

In Conclusion:

If for low frequency RF Structures, thin films coated cavities

do not require magnetic screenings, it is not said that at

higher frequencies, thin films will pin the vortexes, being the

ratio / w w0 the critical parameter