Superconductors and Vortices at Radio Frequency Magnetic Fields (Ernst Helmut Brandt - 50') Speaker: Ernst Helmut Brandt - Max Planck Institute for Metals Research, D-70506 Stuttgart, Germany | Duration: 50 min. Abstract After an introduction to superconductivity and Abrikosov vortices, the statics and dynamics of pinned and unpinned vortices in bulk and thin film superconductors is presented. Particular interesting is the case of Niobium, which has a Ginzburg-Landau parameter near 0.71, the boundary between type-I and type-II superconductors. This causes the appearance of a so called type-II/1 state in which the vortex lattice forms round or lamellar domains that are surrounded by ideally superconducting Meissner state. This state has been observed by decoration experiments and by small-angle neutron scattering. Also considered are the ac losses caused at the surface of clean superconductors, in particular Niobium, in the Meissner state, when no vortices have yet penetrated. The linear ac response is then xpressed by a complex resistivity or complex magnetic penetration depth, or by a surface impedance. At higher amplitudes, several effects can make the response nonlinear and increase the ac losses. In particular, at sharp edges or scratches of a rough surface the magnetic field is strongly enhanced by demagnetization effects and the induced current may reach its depairing limit, leading to the nucleation of short vortex segments. Strong ac losses appear when such vortex segments oscillate. In high-quality microwave cavities the nucleation of vortices has thus to be avoided. Once nucleated, some vortices may remain in the superconductor even when the applied magnetic field goes through zero. This phenomenon of flux-trapping is caused by weak pinning in the bulk or by surface pinning.

2. Superconductivity Zero DC resistivity Kamerlingh-Onnes 1911 Nobel prize 1913 Perfect diamagnetism Meissner 1933 T c ->

3. YBa 2 Cu 3 O 7- δ Bi 2 Sr 2 CaCu 2 O 8 39K Jan 2001 MgB 2 Discovery of superconductors Liquid He 4.2K ->

7. Alexei Abrikosov Vitalii Ginzburg Anthony Leggett Physics Nobel Prize 2003 Lev Landau 10 Dec 2003 Stockholm

8. Grigorii Volovik Richard Klemm Boris Shklovskii George Crabtree Ernst Helmut Brandt Boris Altshuler Lev Gor'kov David Bishop Alexei Abrikosov David Nelson Michael Tinkham Phil W. Anderson Valerii Vinokur Igor' Dzyaloshinskii David Khmel'nitskii Abrikosov‘s 70th Birthday Symposium, 6 Nov 1998 in Argonne

9. Abrikosov‘s 80th Birthday Symposium, 8 Nov 2008 in Argonne Tony Leggett Alexei Abrikosov

10. Decoration of flux-line lattice U.Essmann, H.Träuble 1968 MPI MF Nb , T = 4 K disk 1mm thick, 4 mm ø B a = 985 G, a =170 nm D.Bishop, P.Gammel 1987 AT&T Bell Labs YBCO , T = 77 K Ba = 20 G, a = 1200 nm similar: L.Ya.Vinnikov, ISSP Moscow G.J.Dolan, IBM NY electron microscope

11. Type-I supercond. Tantalum disk 33 μ m thick, 4 mm diameter, B a = 58 mT, T=1.2 K Type-II supercond. Niobium disk 40 μ m thick, 4 mm diameter, B a = 74 mT, T=1.2 K Optical microscope, looks like Type-I Same Niobium disk but Electron microscope shows vortices 0.1 mm 0.1 mm 1 μ m Essmann 1968 and Review: EHB + U.Essmann, phys.stat.sol.b 144, 13 (1987)

12. Decoration of a square disk 5 x 5 x 1 mm 3 of high-purity polycrystalline Nb , T=1.2 K, in increasing B a =1100 Gauss. Fingers of vortex lattice penetrate . When the edge barrier is overcome, single vortices or droplets of vortex lattice jump to the center. (U.Essmann)

13. Vortex-vortex interaction, schematic originates when Fourier trans. deviates from V(k) ~ 1/(1+k 2 λ 2 ) and for BCS from Eilenberger method London, GL repulsion attraction

14. jump B 0 EHB, Phys. Lett. 51A, 39 (1975); phys. stat. sol.(b) 77, 105 (1976) H c2 -> κ 1 (T) slope -> κ 2 (T) H c1 -> κ 3 (T)

15. Auer Auer and Ullmaier, PRB 7, 136 (1973) with many refs. and phase diagram TaN N << 1 cylinder  = 0.665 T C = 4.38 K Domains with vortex lattice Type II / 1 vortex attraction B 0 vortex lattice Type II / 2 vortex repulsion

16. examples: Nb, TaN, PbIn, PbTl B a - M Theor.  –T phase diagram : Ulf Klein, JLTP 69, 1 (1987) Exp.al.: Auer+Ullmaier 1973 sphere long cylinder

17. Isolated vortex (B = 0) Vortex lattice: B = B 0 and 4B 0 vortex spacing: a = 4 λ and 2 λ Bulk superconductor Ginzburg-Landau theory EHB, PRL 78, 2208 (1997) Abrikosov solution near B c2 : stream lines = contours of | ψ |2 and B

18. Magnetization curves of Type-II superconductors Shear modulus c 66 (B, κ ) of triangular vortex lattice c 66 -M Ginzburg-Landau theory EHB, PRL 78, 2208 (1997) B C1 B C2

19. Isolated vortex in film London theory Carneiro+EHB, PRB (2000) Vortex lattice in film Ginzburg-Landau theory EHB, PRB 71, 14521 (2005) bulk film sc film vac

20. Magnetic field lines in films of thicknesses d / λ = 4, 2, 1, 0.5 for B/B c2 =0.04, κ =1.4 4 λ λ 2 λ λ /2

21. Pearl vortex in an infinite thin film 1. Vortex in ideal screening thin infinite film ( London depth = 0 ) 2. Vortex in infinite thin film with 2D penetration depth > d film vortex Magnetic field Circulating sheet current J(r) Force between two vortices Interaction potential = -V´(r) 3D 2D

22. EHB, PRB 79, 13526 (2009) J.Pearl, APL 5, 65 (1964) exact Pearl potential analytic approximation:

23. Interaction of one vortex with a vortex pair = stream function g of this vortex pair = inverse matrix K ij for fixed index j EHB, PRB 2005 peak: ~ ln(2.27 Λ / r)

24. Vortex-vortex interaction for one vortex in center of square film : numerical V num divided by Pearl potential V Pearl for infinite film V/V = 1 V/V = 0

25. Pinning of flux lines Types of pins: ● preciptates: Ti in NbTi -> best sc wires ● point defects, dislocations, grain boundaries ● YBa 2 Cu 3 O 7- δ : twin boundaries, CuO 2 layers, oxygen vacancies Experiment: ● critical current density j c = max. loss-free j ● irreversible magnetization curves ● ac resistivity and susceptibility Theory: ● summation of random pinning forces -> maximum volume pinning force j c B ● thermally activated depinning ● electromagnetic response ● width ~ j c H H c2 - M ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Lorentz force B х j -> -> FL pin

26. magnetization force 20 Jan 1989

27. Levitation of YBCO superconductor above and below magnets at 77 K 5 cm Levitation Suspension FeNd magnets YBCO

28. Importance of geometry Bean model parallel geometry long cylinder or slab Bean model perpendicular geometry thin disk or strip analytical solution: Mikheenko + Kuzovlev 1993: disk EHB+Indenbom+Forkl 1993: strip B a j J J B a J c B J B a B a r r B B j j j c r r r r B a

29. equation of motion for current density: EHB, PRB (1996) Long bar A ║J║E║z Thick disk A ║J║E║ φ Example integrate over time invert matrix! sc as nonlinear conductor approx.: B= μ 0 H, H c1 =0 J x B a , y z J r B a B a -M

30. Flux penetration into disk in increasing field field- and current-free core ideal screening Meissner state + + + _ _ _ 0 B a

31. Same disk in decreasing magnetic field B a no more flux- and current-free zone _ _ + + + + _ _ _ + + _ + _ remanent state B a =0 B a

32. YBCO film 0.8 μ m, 50 K increasing field Magneto-optics Indenbom + Schuster 1995 Theory EHB PRB 1995 Thin sc rectangle in perpendicular field stream lines of current contours of mag. induction ideal Meissner state B = 0 B = 0 Bean state | J | = const

33. Thin films and platelets in perp. mag. field, SQUIDs EHB, PRB 2005 2D penetr. depth Λ = λ 2 /d

34. Vortex pair in thin films with slit and hole current stream lines

35. Dissipation by moving vortices (Free flux flow. Hall effect and pinning disregarded) Lorentz force on vortex: Lorentz force density: Vortex velocity: Induced electric field: Flux-flow resistivity: Where does dissipation come from? 1. Electric field induced by vortex motion inside and outside the normal core Bardeen + Stephen, PR 140, A1197 (1965) 2. Relaxation of order parameter near vortex core in motion, time Tinkham, PRL 13, 804 (1964) ( both terms are ~ equal ) 3. Computation from time-dep. GL theory: Hu + Thompson, PRB 6, 110 (1972) B c2 B Exper. and L+O B+S Is comparable to normal resistvity -> dissipation is very large !

36. Note: Vortex motion is crucial for dissipation. Rigidly pinned vortices do not dissipate energy. However, elastically pinned vortices in a RF field can oscillate: Force balance on vortex: Lorentz force J x B RF (u = vortex displacement . At frequencies the viscose drag force dominates, pinning becomes negligible, and dissipation occurs. Gittleman and Rosenblum, PRL 16, 734 (1968) E x | Ψ | 2 order parameter moving vortex core relaxing order parameter v v

37. Penetration of vortices, Ginzburg-Landau Theory Lower critical field: Thermodyn. critical field: Upper critical field: Good fit to numerics: Vortex magnetic field: Modified Bessel fct: Vortex core radius: Vortex self energy: Vortex interaction:

38. Penetration of first vortex 1. Vortex parallel to planar surface: Bean + Livingston, PRL 12, 14 (1964) Gibbs free energy of one vortex in supercond. half space in applied field B a Interaction with image Interaction with field B a G( ∞ ) Penetration field: This holds for large κ . For small κ < 2 numerics is needed. numerics required ! H c H c1

39. 2. Vortex half-loop penetrates: Self energy: Interaction with H a : Surface current: Penetration field: 3. Penetration at corners: Self energy: Interaction with H a : Surface current: Penetration field: for 90 o H a 4. Similar: Rough surface, H p << H c vortex half loop image vortex super- conductor vacuum R vacuum H a sc R H a vortices

40. Bar 2a X 2a in perpendicular H a tilted by 45 o H a Field lines near corner λ = a / 10 current density j(x,y) log j(x,y) x/a y/a y/a y/a x/a x/a λ large j(,y)

41. 5. Ideal diamagnet, corner with angle α : H ~ 1/ r 1/3 Near corner of angle α the magnetic field diverges as H ~ 1/ r β , β = ( π – α )/(2 π - α ) H ~ 1/ r 1/2 cylinder sphere ellipsoid rectangle a 2a b 2b H/H a = 2 H/H a = 3 H/H a ≈ (a/b) 1/2 H/H a = a/b Magnetic field H at the equator of: (strip or disk) b << a b << a Large thin film in tilted mag. field: perpendicular component penetrates in form of a vortex lattice H a vacuum H a sc r α α = π α = 0

42. Irreversible magnetization of pin-free superconductors due to geometrical edge barrier for flux penetration Magnetic field lines in pin-free superconducting slab and strip EHB, PRB 60, 11939 (1999) b/a=2 flux-free core flux-free zone b/a=0.3 b/a=0.3 b/a=2 Magn. curves of pin-free disks + cylinders ellipsoid is reversible!

43. Radio frequency response of superconductors DC currents in superconductors are loss-free ( if no vortices have penetrated ), but AC currents have losses ~ ω 2 since the acceleration of Cooper pairs generates an electric field E ~ ω that moves the normal electrons (= excitations, quasiparticles ). 1. Two-Fluid Model ( M.Tinkham, Superconductivity, 1996, p.37 ) Eq. of motion for both normal and superconducting electrons: total current density: super currents: normal currents: complex conductivity:

44. dissipative part: inductive part: London equation: Normal conductors: parallel R and L: crossover frequency: power dissipated/vol : London depth λ skin depth power dissipated/area: general skin depth: absorbed / incid. power:

45. 2. Microscopic theory ( Abrikosov, Gorkov, Khalatnikov 1959 Mattis, Bardeen 1958; Kulik 1998 ) Dissipative part: Inductive part: Quality factor: For computation of strong coupling + pure superconductors (bulk Nb) see R. Brinkmann, K. Scharnberg et al., TESLA-Report 200-07, March 2000: Nb at 2K: R s = 20 n Ω at 1.3 GHz, ≈ 1 μΩ at 100 - 600 GHz, but sharp step to 15 m Ω at f = 2 Δ /h = 750 GHz (pair breaking), above this R s ≈ 15 m Ω ≈ const When purity incr., l ↑, σ 1 ↑ but λ ↓

47. 10 Dec 2003 Stockholm Princess Madeleine Alexei Abrikosov