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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