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