I. Grigorieva, L. Vinnikov, A. Geim (Manchester) V. Oboznov, S. Dubonos (Chernogolovka)

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

Starting Nb films: λ(0)  90 nm; ξ(0)  15 nm; H c2 (0)  1.5 T;   6; T c =9.1 K; thickness d = 150 nm > ξ, λ Vortex structure in a macroscopic Nb film. External field H ext = 80 Oe

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)

field-cooling in perpendicular magnetic field external magnetic field varying between 20 and 160 Oe, i.e., H/H c2 = – 0.016, where H c2 (3.5 K)  1 T; experimental details H decoration captures snapshots of vortex states at T  3.5 K =0.4T c ; thickness of all nanostructured samples d = 150 nm > ξ, λ

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 H ext values and dot sizes, almost all possible vorticities between L=0 and L  50 were observed (L = 0,1,2,3,4,5,6,…)

for all values of vorticity L, external filed (total flux) required for nucleation of L vorticies significantly exceeds corresponding field for a macroscopic film Vorticity vs field

B.J. Baelus and F.M. Peeters, Phys. Rev. B 65, , 2002 experiment, disk size R  100ξ nucleation of the first vortex requires magnetic flux corresponding to over 3  0 states with small vorticities are stable over appreciable field intervals, e.g. for a 2µm disk,  H  20 Oe for transition to L=1;  H  10 Oe for transition to L=2 numerical study, R = 6ξ Vorticity vs field

(2,7) (2,8)(3,3,3)(1,8)(3,7) (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, H ext = 80 Oe  3  m dots, H ext = 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

 0.5  0 Multiplicity of vortex arrangements

(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

Comparison with theory B.J. Baelus, L.R.E. Cabral, F.M. Peeters, Phys.Rev.B 69, (2004) observed vortex states in good agreement with numerical simulations

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

…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 (L 1 =4; L 2 =10)

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 L 3 reaches 16… L=22 (2,7,13) L=24 (3,7,14)

Magic numbers … fourth shell appears at L 3 >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  L 1 =6 two shells  L 2 =10 three shells  L 3 =18 …..

Vortex states in triangular dots

Vortex states in square dots

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

L=20 (1,6,13) H ext =160 Oe L=18 (1,6,11) H ext = 30 Oe L=21 (1,7,13) H ext =160 Oe vortex configurations do not change with increasing external field