Vortex matter in nanostructured and hybrid superconductors

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1 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)

2 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

3 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

4 Magnetization T =.4K Al-disks d = 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)

5 Fractional and negative vortices Al-disks R = 7.5 µm d =.1 µm T =.5 K Φ (φ ) 3 δφ 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 H (Gauss) A.K. Geim, S.V. Dubonos, I.V. Grigorieva, K.S. Novoselov, F.M. Peetrs, and V.A. Schweigert, Nature 47, 55 ()

6 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

7 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

8 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

9 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

10 Circular symmetry: Giant vortex states r Ψ ( r) = F( ρ, ze ) ilϕ -M free energy L= 1 L= 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 H/H c

11 Density of superconducting condensate density Cooper pairs L=1 L= L= L=3 L=4 L=5 L=6 L=7 L= distance in µm -magnetization (Gauss).3..1 magnetic field (Gauss) Magnetization r Ψ.5 ( r) = F( ρ, ze ) ilϕ Magnetic field distribution G 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 ρ L (ρ ) distance in µm 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 -.

12 Experiment theory -M (Gauss) T=.4K R=.5 µm d=.134 µm Superconducting dots H (Gauss) Two-dimensional electron gas -magnetization (Gauss) Total magnetic field (Gauss) Averaging over detector DETECTOR DISK Magnetic field (Gauss) Distance along a line through the disc center in µm -Magnetization (Gauss) magnetic field (Gauss) T =.4K R =.8 µm d =.134 µm

13 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

14 Giant - multi-vortex transitions F/F -M/H c (x 1-3 ) R/ξ = 4. d/ξ =.1 κ =.8 L = H /H c y/ξ y/ξ 4 H=.55H c - -4 H=.75H c 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.

15 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 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)

16 Multi-vortex transitions

17 Giant - multi-vortex states. L=6 L=7-4π M H (Gauss) A.K. Geim, S.V. Dubonos and J.J. Palacios, Phys. Rev. Lett. 84, 1796 ()

18 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 x/ξ L=1-14: three vortices in center + (L-3) vortices in a ring

19 Free energy F/F L 3 vortices on an outer ring + 3 vortices on an inner. ring for L = 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 H /H c F/F R = 6.ξ H /H c

20 Larger disks (L = 16) Three different vortex configurations are stable: y/ξ x/ξ y/ξ x/ξ y/ξ 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)

21 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)

22 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

23 Entrance and exit of vortices Bean-Livingston surface barrier -.17 R=4.8ξ, d= H/H c =.86 free energy r Ψ, A -.19 [18 18, ( ) 6] x/ξ

24 Fractional and negative vortices

Fractional and negative vortices Al-disks R =

5 δφ H (G) Φ (φ ) 5 4 3 6 (a) (b) φ µm 1 (c) 4

25 Fractional and negative vortices Al-disks R = 7.5 µm d =.1 µm T =.5 K Φ (φ ) δφ H (G) Φ (φ ) (a) (b) φ µm 1 (c) 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 ().

26 Surface roughness (d) (e) (e) (d) (c) (a) Superconducting density free energy (a) (a) (b) (c) (b) (d) (b) (a) (b) -1. (e) (c) H/H c Vortex exit Vortex entrance

27 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).

28 Time-dependent Ginzburg-Landau equations ψ ( ) ( u + iϕψ = i A ψ+ψ ψ 1) +χ t ( i A) ( ( * )) ϕ= div Re ψ ψ

29 Comparison with experiment R 5.5 ξ w 1.3ξ

30 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

Golubovic et al, Phys. Rev. B 68, 1753 (3) J.I.

Villegas et al, Science 3, 1188 (3) Ferromagnets create

31 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

1 -.96 VI VII VIII F/F -.97 -.98 -.99-1.

32 University of Antwerp Condensed Matter Theory Group I II III IV V Single magnetic dot on top of a SC film Φ + /Φ VI VII VIII F/F I II III IV VII VIII d/ξ=.5 d d /ξ=.5 R d /ξ=1. κ= V m/(h C ξ 3 ) VI

33 University of Antwerp Condensed Matter Theory Group Larger magnetic disk favoring the multivortex state h mz /H c,5, 1,5 1,,5, -, r / x d d /ξ=1. R d /ξ = 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

34 University of Antwerp Condensed Matter Theory Group 5 y/ξ x/ξ 5 y/ξ 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).

vortex-antivortex pair F/F 4 6 8 1 1 -.96 -.97 -.98 -.99-1.

35 University of Antwerp Condensed Matter Theory Group Transition between successive N-states: the baby vortex-antivortex pair F/F IV III Φ + /Φ IV III II I VIII VII VI V m/m IV III m/m

36 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

37 University of Antwerp Condensed Matter Theory Group Normal state N=1 N= N= N=3 N=4 single dot-like behavior

38 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

39 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.

40 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 =

41 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

42 THE END

University of Antwerp Condensed Matter Theory Group Vortices in superconductors IV. Hybrid systems

University of Antwerp Condensed Matter Theory Group Vortices in superconductors IV. Hybrid systems Vortices in superconductors IV. Hybrid systems François Peeters Magnetic impurities T c decreases with increasing impurity density Origin: exchange interaction between electron and impurity: Γ(r i -r e