Vortex matter in nanostructured and hybrid superconductors François Peeters University of Antwerp In collaboration with: B. Baelus, M. Miloševic V.A. Schweigert (Russian Academy of Sciences, Novosibirsk) D. Vodolazov (Russian Academy of Sciences, Nizhny Novgorod) P. Deo (Bose Institute, Calcutta) A.K. Geim (University of Nijmegen, The Netherlands) (since 1: University of Manchester, UK)
Common wisdom about superconductivity # Superconducting phase boundary: is a material property independent of the size of the superconductor # Two general classes of superconductors # Quantization of flux in units of Φ =hc/e
S/N boundary: mesoscopic effects film S/N boundary depends on the size of the sample. Decreasing the size of the superconductor enhances the superconducting state
Magnetization T =.4K Al-disks d =.7-.15 µm λ() = 7 nm ξ() =.15 µm κ = λ/ξ =.8 < 1/ -4π M (a. u.) 3 1 x1 x.5 x1.3 x1.5 µm.5 µm Type I or type II behavior is no longer.75 µm only determined by the material. The size of the sample is very important and can turn a type I superconductor into a type II superconductor. r = 1. µm A.K. Geim, I.V. Grigorieva, S.V. Dubonos, J.G.S. Lok, J.C. Maan, A.E. Filippov, and F.M. Peeters, Nature 39, 59 (1997) 5 1 H (Gauss)
Fractional and negative vortices Al-disks R = 7.5 µm d =.1 µm T =.5 K Φ (φ ) 3 δφ 1 5.5 H (G) 6 φ Vortex entry no longer leads to an increase of flux by a quantized unit. The amount of flux can even be negative with increasing H. 4 6 8 H (Gauss) A.K. Geim, S.V. Dubonos, I.V. Grigorieva, K.S. Novoselov, F.M. Peetrs, and V.A. Schweigert, Nature 47, 55 ()
Ginzburg-Landau formalism condensation energy (<) interaction energy (>) α r Gs = Gn + dr r β r 4 Ψ ( r) + Ψ( r) 1 * + Ψ m* internal magnetic field urr rr rota( r) = h( r) * r urr r r e 1 () r ih Ar () Ψ () r + (() hr H) c 8π kinetic energy operator for Cooper pairs: m*=m, e*=e applied magnetic field α = h mξ o T (1 ) T o, length scales: ξ =h / m* α coherence length λ = c m β 4 e π α penetration length
G s Ψ, r A Ginzburg Landau equations δ G s δψ = 1 m e i ur r h A Ψ = αψ βψ Ψ c δ δ G s r A = ur ur ur 4π r A = j c r ur ur ur j eh im ( * *) 4 e = Ψ Ψ Ψ Ψ Ψ mc A Boundary conditions: # At the superconducting/insulating interface: # Magnetic field far away from disk = applied field: e ih A Ψ= c r r n ur 1 r Ar = Ho ρ eφ
Superconducting Normal phase boundary E Electrons e, m, E H H Cooper pairs e*=e, m*=m, E -α 1-T/T o S N T Box Ψ Electrons Ψ(R)= R Ψ Cooper pairs (-ih -e*a/)ψ n = R
electrons Cooper pairs A.K. Geim, I.V. Grigorieva, S.V. Dubonos, J.G.S. Lok, J.C. Maan, A.E. Filippov, and F.M. Peeters, Nature 39, 59 (1997) R=.55µm d=.8µm ξ()=.µm
Circular symmetry: Giant vortex states r Ψ ( r) = F( ρ, ze ) ilϕ -M free energy..1. -.1. 5 6 7 8 3 4 -. -.4 -.6 -.8-1. L= 1 L= 1 3 4 5 6 7 R/ξ=3.5 d/ξ=.18 κ=.8 8 equilibrium F( ρ) vorticity (for thin disks and cylinders) Magnetic field: A ϕ (ρ,z) [Ψ, A j ] [1D, D] - problem..5 1. 1.5. H/H c
Density of superconducting condensate density Cooper pairs 1..8.6.4. L=1 L= L= L=3 L=4 L=5 L=6 L=7 L=8....4.6.8 distance in µm -magnetization (Gauss).3..1 magnetic field (Gauss) 1 1 8 6 4 Magnetization r.6 1.5 Ψ.5 ( r) = F( ρ, ze ) ilϕ 1..4.8 Magnetic field distribution 119.57G 111.5G 13.43G 87.8G 79.1G 71.14G 63.7G 46.9G 38.85G L=8 L=7 L=6 L=5 L=4 L=3 L= L=1 L= R=.8µm d=.134µ m ξ()=.183 µm λ()=.7µm T=.4K..5 1. 1.5. ρ L (ρ ) distance in µm.75.5.5. 4 6 8 1 1 magnetic field (Gauss).
Experiment theory -M (Gauss).3.5..15.1 T=.4K R=.5 µm d=.134 µm 1.5 1.5 1..75.5.5.5..6. 1.5 1 3 4 5 6 7 Superconducting dots H (Gauss) Two-dimensional electron gas -magnetization (Gauss).5.4.3..1 Total magnetic field (Gauss) 1 8 6 4 Averaging over detector DETECTOR DISK -.3 1 3 4 5 6 7 Magnetic field (Gauss).. 4 6 8 1 1 - -1 1 3 Distance along a line through the disc center in µm -Magnetization (Gauss).15.1.9.6.3. magnetic field (Gauss) T =.4K R =.8 µm d =.134 µm 1..8.75.5.5
Larger disks Small disks: symmetry of disk determines the symmetry of superconducting density r Ψ ( r) = F ( ρ, z) e ilϕ Larger disks: competition between symmetry boundary of sample and Abrikosov lattice [Ψ, A] [D, 3D] - problem
Giant - multi-vortex transitions F/F -M/H c (x 1-3 ). -. -.4 -.6 -.8-1. 1 8 6 4 R/ξ = 4. d/ξ =.1 κ =.8 L = 1 3 4..5 1. 1.5. 5 H /H c 6 7 8 9 11 1 y/ξ y/ξ 4 H=.55H c - -4 H=.75H c 4 - -4 Magnetic field distribution -4-4 x/ξ H=.65H c H=.8H c -4-4 x/ξ
Multiple-small-tunnel junctions The idea: Lead 1 (N) I 1 small tunnel junction superconductor ξ 4 Al, R=75nm, d=33nm Apply a fixed current I 1 = 1pA and measure the voltage as a function of the field. Decrease of the energy gap by the supercurrent underneath the junction (Bardeen, 196): ( J S ) = () 1 7 J J S C y/ξ - -4-4 - 4 x/ξ Axial symmetry: V A = V B = V C = V D -4-4 x/ξ Non-Axial symmetry: V A V B V C V D A. Kanda et al (4)
Multi-vortex transitions
Giant - multi-vortex states. L=6 L=7-4π M 35 4 45 5 H (Gauss) A.K. Geim, S.V. Dubonos and J.J. Palacios, Phys. Rev. Lett. 84, 1796 ()
Vortex states (R=6ξ) L=6-14: one vortex in center + (L-1) vortices in a ring Are 9 vortices on a ring stable? L=1-14: two vortices in center + (L-) vortices in a ring 6 4 L = 11 y/ξ - -4-6 -6-4 - 4 6 x/ξ L=1-14: three vortices in center + (L-3) vortices in a ring
Free energy F/F L 3 vortices on an outer ring + 3 vortices on an inner. ring for L = 1 14.. -.1 -. -.3 -.4 -.5 -.6 -.7 -.8 -.9-1. R = 6.ξ L vortices on a ring -. L - 1 vortices on a ring + 1 vortex in the center -.5 Giant vortex state with vorticity L L - vortices on a outer ring + vortices on a inner ring L - 3 vortices on a outer ring + 3 vortices on a inner ring...4.6.8 1. 1. 1.4 1.6 1.8 H /H c F/F -.5 -.1 -.15 R = 6.ξ.7.8.9 1. 1.1 H /H c 1 13 14
Larger disks (L = 16) Three different vortex configurations are stable: y/ξ 15 1 5-5 -1-15 - - -15-1 -5 5 1 15 x/ξ y/ξ 15 1 5-5 -1-15 - - -15-1 -5 5 1 15 x/ξ y/ξ 15 1 5-5 -1-15 - - -15-1 -5 5 1 15 x/ξ (4,1) (5,11) (1,5,1) R =? H = 1.H c Lowest energy B.J. Baelus, L.R.E. Cabral, and F.M. Peeters, Phys. Rev. B 69, 9451 (4)
Disks with R 5? High fields: triangular lattice in the center Cooper-pair density L = 3 Voronoi construction L = 3 L. R. E. Cabral et al, PRB (4)
T =.4K Summary: size dependence 3 x1.5 µm Only Meissner state Only axially symmetric states (giant vortex states) Multivortex states: one vortex shell -4π M (a. u.) Multivortex states: several vortex shells 1 x. 5 x1. 3 x 1 Circular vortex shells near the boundary + Triangular vortex lattice in the center 5 1 H (Gauss).5 µm.75µm r = 1.µm Small disks Large disks
Entrance and exit of vortices Bean-Livingston surface barrier -.17 R=4.8ξ, d= H/H c =.86 free energy -.18.78.8.84.8 r Ψ, A -.19 [18 18, (768 768) 6] 1 3 4 5 x/ξ
Fractional and negative vortices
Fractional and negative vortices Al-disks R = 7.5 µm d =.1 µm T =.5 K Φ (φ ) 1 3 5.5 δφ H (G) Φ (φ ) 5 4 3 6 (a) (b) φ µm 1 (c) 4 6 15 8 3 45 H (Gauss) H (Gauss) A.K. Geim, S.V. Dubonos, I.V. Grigorieva, K.S. Novoselev, F.M. Peeters, and V.A. Schweigert, Nature (London) 47, 55 ().
Surface roughness -. -.3 -.4 -.5 -.5 -.6 -.7 -.8 -.9-1. (d) (e) (e)..1..3 (d) (c) (a) Superconducting density free energy -.6 -.7 -.8 -.9 (a) (a) (b) (c) (b) (d) (b) (a) (b) -1. (e) (c)..1..3.4 H/H c Vortex exit Vortex entrance
Ring structures Sub-micron Hall probe Detector area Multiple flux entry/exit (Ørsted Laboratory, Copenhagen) R 5.5ξ w 1.3ξ D.Y. Vodolazov, F.M. Peeters, S.V. Dubonovs, A.K. Geim, Phys. Rev. B 67, 5456 (3) S. Pedersen, G. R. Koford, J. C. Hollingbery, C. B. Sorensen, and P. E. Lindelof, Phys. Rev. B 64, 145 (1).
Time-dependent Ginzburg-Landau equations ψ ( ) ( u + iϕψ = i A ψ+ψ ψ 1) +χ t ( i A) ( ( * )) ϕ= div Re ψ ψ
Comparison with experiment R 5.5 ξ w 1.3ξ
Conclusions Ginzburg-landau theory is able to describe the superconducting state of mesoscopic superconductors (also deep inside the superconducting state) Depending on the radius of the disk: type I, type II, multi-type I behavior Geometry (i.e. size) determines the properties Giant multi-vortex transition (second-order) Multi-vortex configurations (first order) Metastability: Hysteresis Fractional and negative flux entrance Multi-vortex entry/exit in ring structures
University of Antwerp Condensed Matter Theory Group Nanostructured ferromagnets + superconducting film D.S. Golubovic et al, Phys. Rev. B 68, 1753 (3) J.I. Martín et al, Phys. Rev. B 6, 911 () J.E. Villegas et al, Science 3, 1188 (3) Ferromagnets create inhomogeneous magnetic fields with <H>= will locally destroy superconductivity pinning centra <H> vortex/antivortex pairs can be created
University of Antwerp Condensed Matter Theory Group I II III IV V Single magnetic dot on top of a SC film Φ + /Φ 4 6 8 1 1 -.96 VI VII VIII F/F -.97 -.98 -.99-1. I II III IV VII VIII d/ξ=.5 d d /ξ=.5 R d /ξ=1. κ=1. 5 1 15 5 3 V m/(h C ξ 3 ) VI
University of Antwerp Condensed Matter Theory Group Larger magnetic disk favoring the multivortex state h mz /H c,5, 1,5 1,,5, -,5 1 3 4 5 6 7 r / x d d /ξ=1. R d /ξ = 1. 4. R /ξ d = 4. 1.
University of Antwerp Condensed Matter Theory Group 5 y/ξ -5-5 5 x/ξ 5 y/ξ -5-5 5 x/ξ The equilibrium vortex phase diagram: Solid lines: transitions between different vortex configurations for different size (R d ) and magnetic moment (m) of the dot. Dashed lines: denote the formation of a new ring of anti-vortices. N is the number of the anti-vortices involved. Shaded area: the multi-vortex state under the dot is energetically favorable (giant vortex splits into individual vortices).
University of Antwerp Condensed Matter Theory Group Transition between successive N-states: the baby vortex-antivortex pair F/F 4 6 8 1 1 -.96 -.97 -.98 -.99-1. IV III Φ + /Φ IV III II I VIII VII VI V 8 1 1 14 16 18 5 1 15 5 3 m/m IV III m/m
University of Antwerp Condensed Matter Theory Group Normal state N=1 N= N=3 N=4 Φ + /Φ =1.65 Φ + /Φ =1.63 N= Φ + /Φ =1.9 Φ + /Φ =1.7
University of Antwerp Condensed Matter Theory Group Normal state N=1 N= N= N=3 N=4 single dot-like behavior
Fractional states +3 + Unit cell - Appear only as metastable states, in the considered range of parameters - Significantly influenced by the Ginzburg-Landau parameter κ enhanced stability in strong type II superconductors
University of Antwerp Condensed Matter Theory Group Magnetic-field-enhanced critical current L / ξ = 6.5 a / ξ =. D / ξ =. l / ξ =.1 d / ξ =. κ = 1.
University of Antwerp Condensed Matter Theory Group Magnetic-field-induced superconductivity M. Lange et al., Phys. Rev. Lett. 9, 1976 (3) H/H 1 =1 φ + / φ =.8 H/H 1 = H/H 1 =
Conclusions Single FM dot on top of a SC film - Vortex/anti-vortex molecule - Anti-vortex ring structure Lattice of FM dots: Vortex/anti-vortex lattice Perpendicular magnetization Magnetic field enhanced critical current Magnetic field induced superconductivity New vortex/anti-vortex lattice structures: each of them may have a different unit cell
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