1. 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

2.  Nucleation of vortices and anti-vortices 1.Characteristics of system 2.Nucleation of vortices 3.Physical boundary conditions 4.Characteristics of vortex interaction

3.  Geim: paramagnetic Meissner effect  Chibotaru and Mel’nikov: 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. Mel’nikov 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).

4. Total flux = LΦ 0 Grigorieva et al., Phys. Rev. Lett. 96, 077005 (2006) Applied H Baelus et al.: predictions different from observations [Phys. Rev. B 69, 0645061 (2004)]

5.  Theories at T = 0K  Experiments at finite T ≠ 0K This study: extension of previous work to include multi-rings and finite temperatures

6. Model H = Hk =  A app d R < λ 2 /d = Λ, d << r c H~H c1 R Local field B ~ H

7. T = 0K H < H c1 : Meissner effect H > H c1 : Vortices penetrate Flux Φ v = qΦ 0, Φ 0 = hc/2e HH j s = -(c/4  2 )A j s = -(c/4  2 )(A-A v ) jsjs jsjs

8. Method of images riri r’ i = (R 2 /r)r i Boundary condition: normal component of j s vanishes image anti-vortex Φ i = qΦ 0 Φ i (r)= qΦ 0 /2  r A v =  [Φ i (r-r i ) - Φ i (r-r' i )]θ ΦiΦi -Φi-Φi

9. H r1r1 r2r2 L > 0 vortex L < 0 anti-vortex r 1 < r 2 LΦ0LΦ0 N 1 vortices qΦ 0 N 2 vortices qΦ 0

10. T = 0 K Gibbs free Energy z i = r i /R

12. α

13. Finite temperature T ≠ 0K Gibbs free energyS=Entropy Dimensionless Gibbs free energy:

14.  Minimise g(L,N 1,N 2,t) with respect to z 1, z 2  Grigorieva: Nb R ~ 1.5nm, 0 ~ 100nm T c ~ 9.1K, t c ~ 0.7 T ~ 1.8K, t ~ 0.14 (L, N 1 ): a central vortex of flux LΦ 0 at centre, N 1 vortices (Φ 0 ) on ring z 1 (L,N 1,N 2 ): a central vortex, N 1 vortices on z 1 and N 2 on z 2

15. Results: t = 0 (T = 0K)

17. Results: t = 0.14 (T = 1.8K) H=60 Oe  h=20.5

19. Vortex Configurations with 9  0 – (0,2,7) * * (1,8)

20. Total flux = 9  0 (L,N 1,N 2 )=(0,2,7) at t = 0.14 (L,N)=(1,8) at t = 0

21. Vortex Configurations with 10  0 – (1,9) * * (0,2,8) - - (0,3,7) H = 60 Oe  h = 20.5

22. Total flux = 10  0 (L,N 1,N 2 )=(0,3,7) t = 0.14 (L,N 1,N 2 )=(0,2,8) t = 0.14 (L,N)=(1,9) t = 0

23. 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 > 2 rings/shells  Underlying physics mechanisms