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

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 →

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:

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:

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 , March 2000: Nb at 2K: R s = 20 nΩ at 1.3 GHz, ≈ 1 μΩ at GHz, but sharp step to 15 mΩ at f = 2Δ/h = 750 GHz (pair breaking), above this R s ≈ 15 mΩ ≈ const When purity incr., l↑, σ 1 ↑ but λ↓

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 1952 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 !

1957 Abrikosov finds solution ψ(x,y) with periodic zeros = lattice of vortices (flux lines, fluxons) with quantized magnetic flux: flux quantum Φ o = h / 2e = 2* T m 2 Nobel prize in physics 2003 to V.L.Ginzburg and A.A.Abrikosov for this magnetic field lines flux lines currents

1957 Abrikosov finds solution ψ(x,y) with periodic zeros = lattice of vortices (flux lines, fluxons) with quantized magnetic flux: flux quantum Φ o = h / 2e = 2* T m 2 Nobel prize in physics 2003 to V.L.Ginzburg and A.A.Abrikosov for this Abrikosov 28 Sept 2003

Alexei Abrikosov Vitalii Ginzburg Anthony Leggett Physics Nobel Prize 2003 Landau

10 Dec 2003 Stockholm Princess Madeleine Alexei Abrikosov

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

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

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) BC1BC1 B C2

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

Magnetic field lines in films of thicknesses d / λ = 4, 2, 1, 0.5 for B/B c2 =0.04, κ=1.4 4λ4λ λ 2λ2λ λ/2

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 ● H H c2 -M-M width ~ j c ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● Lorentz force B х j → → FL pin

magnetization force 20 Jan 1989

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

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 BaBa j J J BaBa JcJc B J BaBa BaBa r r BB j j jcjc r rr r BaBa

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

Flux penetration into disk in increasing field BaBa field- and current-free core ideal screening Meissner state _ _ _ 0

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

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

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

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

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

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 !

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) v E x |Ψ|2|Ψ|2 order parameter moving vortex core relaxing order parameter v

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:

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! HcHc H c1

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

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

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)/(2π - α) vacuum HaHa sc r α α = π H ~ 1/ r 1/2 α = 0 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 Large thin film in tilted mag. field: perpendicular component penetrates in form of a vortex lattice H a

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, (1999) b/a=2 flux-free core flux-free zone b/a=0.3b/a=2 Magn. curves of pin-free disks + cylinders ellipsoid is reversible! b/a=0.3

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 ~ σ 1 ~ 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/B c2 ≤ 1 Into sc with planar surface, vortices penetrate via a barrier at H p ≈ H c > H c1 But at sharp corners vortices penetrate much more easily, at H p << H c1 Vortex nucleation occurs in an extremely short time, More in discussion sessions ( 2Δ/h = 750 MHz )