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Published on October 16, 2007

Author: FunSchool


Slide1:  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 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 Superconductivity:  Superconductivity Zero DC resistivity Kamerlingh-Onnes 1911 Nobel prize 1913 Perfect diamagnetism Meissner 1933 Tc → Slide4:  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: Slide5:  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: Slide6:  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 Slide7:  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 ! Vortices: Phenomenological Theories ! Slide8:  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.Abrikosov for this magnetic field lines flux lines currents Slide9:  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.Abrikosov for this Slide10:  Alexei Abrikosov Vitalii Ginzburg Anthony Leggett Physics Nobel Prize 2003 Landau Slide11:  10 Dec 2003 Stockholm Princess Madeleine Alexei Abrikosov Slide12:  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 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 Slide13:  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 Slide14:  Magnetization curves of Type-II superconductors Shear modulus c66(B, κ ) of triangular vortex lattice c66 -M Ginzburg-Landau theory EHB, PRL 78, 2208 (1997) BC1 BC2 Slide15:  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 vac Slide16:  Magnetic field lines in films of thicknesses d / λ = 4, 2, 1, 0.5 for B/Bc2=0.04, κ=1.4 4λ λ 2λ λ/2 Slide17:  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 Slide18:  magnetization force 20 Jan 1989 Slide19:  Levitation of YBCO superconductor above and below magnets at 77 K 5 cm Levitation Suspension FeNd magnets YBCO Slide20:  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 Ba j Slide21:  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=μ0H, Hc1=0 Slide22:  Flux penetration into disk in increasing field field- and current-free core ideal screening Meissner state + + + _ _ _ 0 Slide23:  Same disk in decreasing magnetic field Ba no more flux- and current-free zone _ _ + + + + _ _ _ + + _ + _ remanent state Ba=0 Slide24:  Bean critical state of thin sc strip in oblique mag. field 3 scenarios of increasing Hax, Haz Mikitik, EHB, Indenbom, PRB 70, 14520 (2004) to scale d/2w = 1/25 stretched along z Ha tail tail tail tail + + _ _ 0 0 + _ θ = 45° Slide25:  YBCO film 0.8 μm, 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 Slide26:  Thin films and platelets in perp. mag. field, SQUIDs EHB, PRB 2005 2D penetr. depth Slide27:  Vortex pair in thin films with slit and hole current stream lines Slide28:  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) Slide29:  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 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 x |Ψ|2 order parameter moving vortex core relaxing order parameter v Slide30:  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: Slide31:  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 Ba Interaction with image Interaction with field Ba G(∞) Penetration field: This holds for large κ. For small κ < 2 numerics is needed. numerics required! Hc Hc1 Slide32:  2. Vortex half-loop penetrates: Self energy: Interaction with Ha: Surface current: Penetration field: 3. Penetration at corners: Self energy: Interaction with Ha: Surface current: Penetration field: for 90o Ha 4. Similar: Rough surface, Hp << Hc Slide33:  Bar 2a X 2a in perpendicular Ha tilted by 45o Ha 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) Slide34:  5. Ideal diamagnet, corner with angle α : H ~ 1/ r1/3 Near corner of angle α the magnetic field diverges as H ~ 1/ rβ, β = (π – α)/(2π - α) H ~ 1/ r1/2 cylinder sphere ellipsoid rectangle a 2a b 2b H/Ha = 2 H/Ha = 3 H/Ha ≈ (a/b)1/2 H/Ha = 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 Ha Slide35:  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 Slide36:  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 ~ l increases If the sc contains vortices, the vortices move and dissipate 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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