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I. Grigorieva, L. Vinnikov, A. Geim(Manchester)
V. Oboznov, S. Dubonos (Chernogolovka)
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• Vortices in small superconductors (size R ~ ξ,λ) expected to behave similar to electrons in artificial atoms, i.e. obey specific rules for shell filling, exhibit magic numbers, etc.
• In confined geometries, superconducting wave function must obey boundary conditions which determine total vorticity L
• Vortex states are further influenced by vortex interactions with screening currents (for R > λ)
• Numerical studies of vortex states exist but so far no direct observations
• We present direct observations of vortex states in small superconducting dots by magnetic decoration
Motivation
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Starting Nb films: λ(0) 90 nm; ξ(0) 15 nm; Hc2(0) 1.5 T;
6; Tc=9.1 K; thickness d = 150 nm > ξ, λ
Vortex structure in a macroscopic Nb film. External field Hext = 80 Oe
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200 µm
20 µm
5 µm
Each structure contained circular disks, squares and triangles of four different sizes: 1µm; 2 µm; 3µm; 5 µm
Over 500 dots decorated in each experiment (same field, temperature, decoration conditions)
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field-cooling in perpendicular magnetic field
external magnetic field varying between 20 and 160 Oe, i.e., H/Hc2 = 0.002 – 0.016, where Hc2(3.5 K) 1 T;
experimental details
H decoration captures snapshotsof vortex states at T 3.5 K =0.4Tc;
thickness of all nanostructured samples d = 150 nm > ξ, λ
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L = 9
L = 25 (3,8,14)
L = 94
despite strong pinning, confinement has dominating effect on vortex states:
well defined shell structures observed for L 35 in circular disks;
a variety of states with triangular / square symmetries observed for L 15 for triangular and square dots
for larger L (L>30-35), vortex arrangements are less well defined and for L > 50 become disordered, similar to macroscopic films
due to many different combinations of Hext
values and dot sizes, almost all possible vorticities between L=0 and L 50 were observed (L = 0,1,2,3,4,5,6,…)
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for all values of vorticity L, external filed (total flux) required for nucleation of L vorticies significantly exceeds corresponding field for a macroscopic film
0
10
20
30
40
0 5 10 15 20 25
(0)(1)(2)(3)(4)(5)(1,5)(1,6)(1,7)(1,8)(2,7)(2,8)(3,7)(3,8)(4,10)(1,5,11) (1,6,12)(1,6,13)(1,6,14)
L = /0
L
/ 0
Vorticity vs field
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B.J. Baelus and F.M. Peeters, Phys. Rev. B 65, 104515, 2002
experiment, disk size R 100ξ
nucleation of the first vortex requires magnetic flux corresponding to over 30
states with small vorticities are stable overappreciable field intervals, e.g. for a 2µm disk, H 20 Oe for transition to L=1; H 10 Oefor transition to L=2
numerical study, R = 6ξ
0
1
2
3
4
0 5 10 15 20 25
L
(-
0L)/
L 0
Vorticity vs field
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0
10
20
30
40
frequency
(2,7) (2,8) (3,3,3)(1,8) (3,7)
0
20
40
60
fre
qu
en
cy
(2,8)(2,7) (3,7)(3,3,3) (1,8)
at least two or three different states observed in every experiment in dots of nominally the same size
2 m dots, Hext= 80 Oe
3 m dots, Hext= 60 Oe
Multiplicity of vortex arrangements
variations in dot sizes, shape irregularities lead to variations in flux up to 0
small differences in energy of different states with same L implied
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21
22
23
7 8 9 10 11 12 13
(1,8)(2,8)(3,7)(3,8)
L
/ 0 0.50
Multiplicity of vortex arrangements
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0
5
10
15
20
0 4 8 12
L
/ 0
(0)(1)
(2)(3)
(5)(1,5)
(6)
(4)
(1,6)
(1,7) (1,8)
(2,7)
(2,8)
(3,7) (3,8)
Evolution of vortex states
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Comparison with theory
B.J. Baelus, L.R.E. Cabral,F.M. Peeters, Phys.Rev.B69, 064506 (2004)
observed vortex states in good agreement with numerical simulations
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Magic numbers
we are able to
identify magic numbers (maximum numbers of vortices in each shell before the next shell nucleates) identify shell filling rules
…
L = 5
L = 6
L = 7 L = 8
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…after that new vortices appear in either the first or second shell:
L=11 (3,8)L=10 (3,7)
L=10 (2,8)
L=9 (2,7)
Magic numbers
… and this continues until the total vorticity reaches L=14 (L1=4; L2=10)
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Magic numbers… third shell appears at L>14 in the form of one vortex in the centre …
L=17 (1,5,11) L=18 (1,6,11)… after that additional vortices nucleate in either first, second or third shell until L3 reaches 16…
L=22 (2,7,13) L=24 (3,7,14)
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Magic numbers
… fourth shell appears at L3>18 in the form of one vortex in the centre, and so on …
L=35 (1,5,11,18)
rules of shell filling similar to electrons in artificial atoms (V.M. Bedanov and F.M. Peeters, Phys. Rev.B 49,667, 1994)
magic numbers: one shell L1=6 two shells L2=10 three shells L3=18 …..
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Vortex states in triangular dots
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Vortex states in square dots
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Conclusions
direct observations of multiple vortex states in confined geometry
low-vorticity states (L<4) are stable over surprisingly large intervals of magnetic field
well defined shell structures in circular geometry
magic numbers for vortex shell filling
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L=20 (1,6,13)Hext=160 Oe
L=18 (1,6,11)Hext = 30 Oe
L=21 (1,7,13)Hext=160 Oe
vortex configurations do not change with increasing external field