Nucleation of Vortices in Superconductors in Confined Geometries

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Nucleation of Vortices in Superconductors in
Confined Geometries

• W.M. Wu, M.B. Sobnack and F.V. Kusmartsev
• Department of Physics
• Loughborough University, U.K.
• July 2007

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• Nucleation of vortices and anti-vortices
• Characteristics of system
• Nucleation of vortices
• Physical boundary conditions
• Characteristics of vortex interaction

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• Geim paramagnetic Meissner effect
• Chibotaru and Melnikov anti-vortices,
  multi-quanta-vortices
• Schweigert multi-vortex state ? giant vortex
• Okayasu no giant vortex

A.K. Geim et al., Nature (London) 408,784
(2000). L.F. Chibotaru et al., Nature (London)
408,833 (2000). A.S. Melnikov et al., Phys. Rev.
B 65, 140501 (2002). V.A. Schweigert et al.,
Phys. Rev. Lett. 81, 2783 (1998). S. Okayasu et
al., IEEE 15 (2), 696 (2005).
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Grigorieva et al., Phys. Rev. Lett. 96, 077005
(2006)
Total flux LF0
Applied H
Baelus et al. predictions different from
observations Phys. Rev. B 69, 0645061 (2004)
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• Theories at T 0K
• Experiments at finite T ? 0K
• This study extension of previous work to include
• multi-rings and finite temperatures

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Model
H Hk ??Aapp
R
HHc1
d
Local field B H
R lt ?2/d ?, d ltlt rc
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T 0K
H gt Hc1 Vortices penetrate
Flux Fv qF0 , F0 hc/2e
H lt Hc1 Meissner effect
js
js
js -(c/4??2)A
js -(c/4??2)(A-Av)
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Boundary condition normal component of js
vanishes
Method of images
ri (R2/r)ri
image anti-vortex
ri
Fi (r) qF0 /2?r
Fi
-Fi
Fi qF0
Av ?Fi (r-ri) - Fi (r-r'i)?
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LF0
N2 vortices qF0
H
r1
r2
r1 lt r2
N1 vortices qF0
L gt 0 vortex L lt 0 anti-vortex
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T 0 K
Gibbs free Energy
zi ri/R
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a
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Finite temperature T ? 0K
Gibbs free energy
SEntropy
Dimensionless Gibbs free energy
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• Minimise g(L,N1,N2,t) with respect to z1, z2
• Grigorieva Nb
• R 1.5nm, ?0 100nm
• Tc 9.1K, tc 0.7
• T 1.8K, t 0.14

(L, N1) a central vortex of flux LF0 at centre,
N1 vortices (F0) on ring z1 (L,N1,N2) a central
vortex, N1 vortices on z1 and N2 on z2
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Results t 0 (T 0K)
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Results t 0.14 (T 1.8K)
H60 Oe ? h20.5
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Vortex Configurations with 9?0
(0,2,7)
(1,8)
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Total flux 9?0
(L,N1,N2)(0,2,7) at t 0.14
(L,N)(1,8) at t 0
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Vortex Configurations with 10?0
(1,9)
- - (0,3,7)
H 60 Oe ? h 20.5
(0,2,8)
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Total flux 10?0
(L,N1,N2)(0,2,8) t 0.14
(L,N)(1,9) t 0
(L,N1,N2)(0,3,7) t 0.14
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Conclusions and Remarks

• Modified theory to include temperature
• Results at t 0.14 in very good agreement with
  experiments of Grigorieva her group
• Extension to gt 2 rings/shells
• Underlying physics mechanisms