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Electrodynamics of Superconductors exposed to high frequency fields Ernst Helmut Brandt

Electrodynamics of Superconductors exposed to high frequency fields Ernst Helmut Brandt Max Planck Institute for Metals Research, Stuttgart. Superconductivity Radio frequency response of ideal superconductors two-fluid model, microscopic theory Abrikosov vortices

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Electrodynamics of Superconductors exposed to high frequency fields Ernst Helmut Brandt

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  1. Electrodynamics of Superconductors exposed to high frequency fieldsErnst Helmut Brandt Max Planck Institute for Metals Research, Stuttgart • Superconductivity • Radio frequency response of ideal superconductors • two-fluid model, microscopic theory • Abrikosov vortices • Dissipation by moving vortices • Penetration of vortices "Thin films applied to Superconducting RF:Pushing the limits of RF Superconductivity"Legnaro National Laboratories of the ISTITUTO NAZIONALE DI FISICA NUCLEARE in Legnaro (Padova) ITALY, October 9-12, 2006

  2. Superconductivity Tc→ Zero DC resistivity Kamerlingh-Onnes 1911 Nobel prize 1913 Perfect diamagnetism Meissner 1933

  3. Discovery of superconductors Bi2Sr2CaCu2O8 YBa2Cu3O7-δ 39K Jan 2001MgB2 Liquid He 4.2K →

  4. Radio frequency response of superconductors DC currents in superconductors are loss-free (if no vortices have penetrated), but AC currents have losses ~ω2 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:

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

  6. When purity incr., l↑, σ1↑ but λ↓ 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: Rs= 20 nΩ at 1.3 GHz, ≈ 1 μΩ at 100 - 600 GHz, but sharp step to 15 mΩ at f = 2Δ/h = 750 GHz (pair breaking), above this Rs ≈15 mΩ≈ const

  7. Vortices: Phenomenological Theories • 1911 Superconductivity discovered in Leiden by Kamerlingh-Onnes • 1957 Microscopic explanation by Bardeen, Cooper, Schrieffer: BCS • 1935 Phenomenological theory by Fritz + Heinz London: • London equation: λ = London penetration depth • Ginzburg-Landau theory:ξ = supercond. coherence length, • ψ = GL function ~ gap function • GL parameter: κ= λ(T) /ξ(T) ~ const • Type-I scs: κ≤ 0.71, NS-wall energy> 0 • Type-II scs:κ≥ 0.71, NS-wall energy < 0: unstable ! !

  8. Abrikosov finds solution ψ(x,y) with periodic zeros = lattice • of vortices (flux lines, fluxons) with quantized magnetic flux: • flux quantum Φo = h / 2e = 2*10-15 T m2 • Nobel prize in physics 2003 to V.L.Ginzburg and A.A.Abrikosovfor this magnetic field lines flux lines currents

  9. Abrikosov 28 Sept 2003 • Abrikosov finds solution ψ(x,y) with periodic zeros = lattice • of vortices (flux lines, fluxons) with quantized magnetic flux: • flux quantum Φo = h / 2e = 2*10-15 T m2 • Nobel prize in physics 2003 to V.L.Ginzburg and A.A.Abrikosovfor this

  10. Physics Nobel Prize 2003 Landau Alexei Abrikosov Vitalii Ginzburg Anthony Leggett

  11. 10 Dec 2003 Stockholm Princess Madeleine Alexei Abrikosov

  12. Decoration of flux-line lattice U.Essmann, H.Träuble 1968 MPI MF Nb, T = 4 K disk 1mm thick, 4 mm ø Ba= 985 G, a =170 nm electron microscope 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

  13. Isolated vortex (B = 0) Vortex lattice: B = B0 and 4B0 vortex spacing:a = 4λ and 2λ Bulk superconductor Ginzburg-Landau theory EHB, PRL 78, 2208 (1997) Abrikosov solution near Bc2: stream lines = contours of |ψ|2 and B

  14. c66 BC1 -M BC2 Magnetization curves of Type-II superconductors Shear modulus c66(B, κ ) of triangular vortex lattice Ginzburg-Landau theory EHB, PRL 78, 2208 (1997)

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

  16. Magnetic field lines in films of thicknesses d / λ = 4, 2, 1, 0.5 for B/Bc2=0.04, κ=1.4 4λ 2λ λ/2 λ

  17. ● ● pin ● ● ● ● ● ● ● ● → ● ● ● ● ● ● ● ● FL ● ● ● ● ● ● ● ● ● ● ● ● ● Lorentz force B х j → -M H Hc2 Pinning of flux lines Types of pins: ●preciptates: Ti in NbTi → best sc wires ●point defects, dislocations, grain boundaries ● YBa2Cu3O7- δ:twin boundaries, CuO2 layers, oxygen vacancies Experiment: ●critical current density jc = max. loss-free j ● irreversible magnetization curves ● ac resistivity and susceptibility Theory: ● summation of random pinning forces → maximum volume pinning force jcB ● thermally activated depinning ●electromagnetic response width ~ jc ●

  18. 20 Jan 1989 magnetization force

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

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

  21. Ba Ba, y J J r x z -M Ba Example sc as nonlinear conductor approx.: B=μ0H, Hc1=0 Long bar A ║J║E║z Thick disk A ║J║E║φ invert matrix! equation of motion for current density: EHB, PRB (1996) integrate over time

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

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

  24. tail to scale d/2w = 1/25 θ = 45° Ha + _ 0 tail _ stretched along z tail + tail + 0 _ Bean critical state of thin sc strip in oblique mag. field 3 scenarios of increasing Hax, Haz Mikitik, EHB, Indenbom, PRB 70, 14520 (2004)

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

  26. Λ=λ2/d Thin films and platelets in perp. mag. field, SQUIDs EHB, PRB 2005 2D penetr. depth

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

  28. Is comparable to normal resistvity → dissipation is very large ! B+S Exper. and L+O B Bc2 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)

  29. v 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 forceJ x BRF (u = vortex displacement . At frequencies the viscose drag force dominates, pinning becomes negligible, and dissipation occurs. Gittleman and Rosenblum, PRL 16, 734 (1968) E order parameter |Ψ|2 v x relaxing order parameter moving vortex core

  30. 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:

  31. Penetration of first vortex 1. Vortex parallel to planar surface:Bean + Livingston, PRL 12, 14 (1964) Interaction with field Ba Interaction with image G(∞) Gibbs free energy of one vortex in supercond. half space in applied field Ba Penetration field: Hc1 Hc This holds for large κ. For small κ < 2 numerics is needed. numerics required!

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

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

  34. Ha r vacuum sc α α = π α = 0 5. Ideal diamagnet, corner with angle α: Near corner of angle α the magnetic field diverges as H ~ 1/ rβ, β = (π – α)/(2π - α) H ~ 1/ r1/3 Magnetic field H at the equator of: cylinder sphere ellipsoid rectangle H/Ha = 2 H ~ 1/ r1/2 H/Ha = 3 b Ha H/Ha = a/b a b << a (strip or disk) Large thin film in tilted mag. field: perpendicular component penetrates in form of a vortex lattice 2b H/Ha≈ (a/b)1/2 2a b << a

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

  36. Summary • Two-fluid model qualitatively explains RF losses in ideal superconductors • BCS theory shows that „normal electrons“ means „excitations = quasiparticles“ • Their concentration and thus the losses are very small at low T • Extremely pure Nb is not optimal, since dissipation~ σ1 ~ lincreases • If the sc contains vortices, the vortices move anddissipate very much energy, • almost as if normal conducting, but reduced by a factor B/Bc2 ≤ 1 • Into sc with planar surface, vortices penetrate via a barrier at Hp ≈ Hc > Hc1 • But at sharp corners vortices penetrate much more easily, at Hp << Hc1 • Vortex nucleation occurs in an extremely short time, • More in discussion sessions ( 2Δ/h =750 MHz )

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