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
Nucleation of vortices and anti-vortices
1. Characteristics of system
2. Nucleation of vortices
3. Physical boundary conditions
4. Characteristics of vortex interaction
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).
Total flux = LΦ0
Grigorieva et al., Phys. Rev. Lett. 96, 077005 (2006)
Applied H
Baelus et al.: predictions different from observations[Phys. Rev. B 69, 0645061 (2004)]
Theories at T = 0K
Experiments at finite T ≠ 0K
This study: extension of previous work to include
multi-rings and finite temperatures
Model
H = Hk = Aapp
d
R < λ2/d = Λ, d << rc
H~Hc1
R
Local field B ~ H
T = 0K
H < Hc1: Meissner effect
H > Hc1: Vortices penetrate
Flux Φv = qΦ0 , Φ0 = hc/2e
H
rxBHBG 3222 d)()(8
1
H
js = -(c/42)A js = -(c/42)(A-Av)
js
js
Method of images
ri
r’i = (R2/r)ri
Boundary condition: normal component of js vanishes
image anti-vortex
Φi = qΦ0
Φi (r)= qΦ0 /2r
Av = [Φi (r-ri) - Φi (r-r'i)]θ
Φi -Φi
Hr1
r2
L > 0 vortex L < 0 anti-vortex r1 < r2
LΦ0
N1 vortices qΦ0
N2 vortices qΦ0
T = 0 K
02 / RHh
),()0,(')0,('
ln2ln2ln4
),,(
211221
2211212
2
NNgNgNgLh
zqLNzqLNr
RNq
hNNLg
c
Gibbs free Energy
zi = ri/R
Gd
tLNg 20
2)(16),,(
1
12
4222
2222
)/(sin4
)/2cos(21ln
2)1(
)1ln(ln)1(ln)0,('
iiii
ii
iiiiiii
N
n
c
Nn
zNnzqNzqhN
zqNzqNNr
RqNNg
1
1
1
1
222
2 1
212121
212121
2112 ))//(2cos(2
))//(2cos(21ln),(
N
m
N
n NmNnzzzz
NmNnzzzzqNNg
α
Finite temperature T ≠ 0K
TSTGG )0(
)lnlnlnln
ln2ln2(),,(),,,(
2121
2121
NNzz
r
RtNNLgtNNLg
c
Gibbs free energy S=Entropy
220
/)(16 dTktB
Dimensionless Gibbs free energy:
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 LΦ0 at centre, N1 vortices (Φ0) on ring z1
(L,N1,N2): a central vortex, N1 vortices on z1 and N2 on z2
Results: t = 0 (T = 0K)
Results: t = 0.14 (T = 1.8K)
H=60 Oe h=20.5
Vortex Configurations with 90
– (0,2,7)
* * (1,8)
Total flux = 90
(L,N1,N2)=(0,2,7) at t = 0.14
(L,N)=(1,8) at t = 0
Vortex Configurations with 100
– (1,9)
* * (0,2,8)
- - (0,3,7)
H = 60 Oe h = 20.5
Total flux = 100
(L,N1,N2)=(0,3,7)t = 0.14
(L,N1,N2)=(0,2,8)t = 0.14
(L,N)=(1,9)t = 0
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