Motivation

1
Motivation
The critical-state model is a very powerful tool
to characterize the irreversible magnetic
behavior of superconductors Geometries where
analytical results are available
subject of the present work
2
The Bean model
Flux penetrated region J Jc const
Flux free region B 0
Cylinder
Ba
Thin disk
Ba
3
Thin disk with Jc(B)
We derive three coupled integral equations
The flux and current density distributions, B(r)
and J(r), for a given Ba are found by numerical
solution
4
B(r) and J(r) - partial penetration
Profiles for fixed Ba and a, but with different
Jc0 and B0
Jc(B) Jc0 exp(-B/B0)
Bean model

• The B-dependence of Jc leads to
• non-uniform J(r) at altrltR
• cusped peak in J(r) at ra
• steeper B(r) near ra
• reduced peak in B(r) at rR

5
B(r) and J(r) - decreasing Ba
Field descent after a large Ba J Jc
everywhere
Jc(B) Jc0 / (1B/Bc)
Bc ?0 Jc0 /2
The shape of B(r) strongly depends on Ba
6
Flux penetration depth
Jc(B) Jc0 exp (-B/B0)
Bean model limit (B0? ? ) a/R
1/cosh(Ba/Bc)
The flux penetration depth, a, depends strongly
on the applied field Ba as well as on the value
of B0 .
7
Low-field behavior of a(Ba)
All the models show the same asymptotic behavior

• Bean model a /R 1 - (Ba/Bc)2
• Jc(B) leads to renormalization of Bc
• Bc Bceff ? Bc (1-?
  Bc /B0)

8
Comparison vs ? geometry
At small applied fields
In bulk samples B(r) is also small everywhere
?Jc ? Jc(B0) everywhere. ? B-dependence of
Jc can be ignored
Ba
In thin samples B(r) is divergent at the
edge ? near the edge Jc ? Jc(B0) ?
B-dependence of Jc is essential for any Ba
Ba
9
Magnetization
(Bean model)
10
AC-susceptibility
Parametric plot
11
Low-Field Behavior
with Bc ? Bceff Bc ( 1 - ? Bc / B0
) ? 0.42 exponential model ? 0.36
Kim model
12
High-Field Behavior of ?
Geometry-independent result
13
Comparison with Experiment
Stoppard et al., Physica C 241 375 (1995)
Herzog et al., Phys. Rev. B 56 2871 (1997)
14
Flux-Creep Simulations
15
Conclusions
The financial support from the Norwegian Research
Council and NATO is gratefully acknowledged. One
of the authors (Y.M.G) receives support from
Centre for Advanced Studies, Norway.