Superconductors and Vortices at

Radio Frequency Magnetic Fields

Ernst Helmut Brandt Max Planck Institute for Metals Research, Stuttgart

• Superconductivity and Abrikosov vortices

• Vortex attraction in Niobium

• Perpendicular vortices in thin films

• Pinning and dissipation by moving vortices

• Penetration of first vortices

• Vortex-free superconductors at radio frequencies:

two-fluid model and microscopic BCS theory

"Thin films and new ideas for pushing the limits of RF Superconductivity" Legnaro National Laboratories of the ISTITUTO NAZIONALE DI FISICA NUCLEARE

in Legnaro (Padova) ITALY, October 4-6, 2010

Superconductivity

Zero DC resistivityKamerlingh-Onnes 1911Nobel prize 1913

Perfect diamagnetismMeissner 1933

Tc →

YBa2Cu3O7-δ

Bi2Sr2CaCu2O8

39K Jan 2001 MgB2

Discovery ofsuperconductors

Liquid He 4.2K →

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 Φo = h / 2e = 2*10-15 T m2

Nobel prize in physics 2003 to V.L.Ginzburg and A.A.Abrikosov for this

magneticfield 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 Φo = h / 2e = 2*10-15 T m2

Nobel prize in physics 2003 to V.L.Ginzburg and A.A.Abrikosov for this Abrikosov28 Sept 2003

Alexei Abrikosov Vitalii Ginzburg Anthony Leggett

Phy

sics

Nob

el P

rize

200

3

Lev Landau

10 Dec 2003 Stockholm

Grigorii Volovik

RichardKlemm

Boris Shklovskii

GeorgeCrabtree

Ernst HelmutBrandt

BorisAltshuler

LevGor'kov

DavidBishop

Alexei Abrikosov

DavidNelson

MichaelTinkham

Phil W. Anderson

ValeriiVinokur

Igor'Dzyaloshinskii

DavidKhmel'nitskii

Abrikosov‘s 70th Birthday Symposium, 6 Nov 1998 in Argonne

Abrikosov‘s 80th Birthday Symposium, 8 Nov 2008 in Argonne

Tony LeggettAlexei Abrikosov

Decoration of flux-line lattice

U.Essmann,H.Träuble 1968 MPI MFNb, T = 4 Kdisk 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 MoscowG.J.Dolan, IBM NY

electron microscope

Type-I supercond. Tantalum disk33 μm thick,4 mm diameter,Ba= 58 mT, T=1.2 K

Type-II supercond. Niobium disk 40 μm thick,4 mm diameter,Ba= 74 mT, T=1.2 K

Optical microscope,looks like Type-I

Same Niobium disk butElectron microscopeshows vortices

0.1 mm

0.1 mm

1 μmEssmann 1968 andReview: EHB + U.Essmann,phys.stat.sol.b 144, 13 (1987)

Decoration of a square disk 5 x 5 x 1 mm3 of high-purity polycrystalline Nb, T=1.2 K, in

increasing Ba =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)

Vortex-vortex interaction, schematic

originates when Fourier trans.

deviates from V(k)~1/(1+k2λ2)

and for BCS from Eilenberger

method

London, GL

repulsion

attraction

jump B0

EHB, Phys. Lett. 51A, 39 (1975); phys. stat. sol.(b) 77, 105 (1976)

Hc2→κ1(T)

slope→κ2(T)

Hc1→κ3(T)

Auer

Auer and Ullmaier, PRB 7, 136 (1973) with many refs. and phase diagram

TaN N << 1 cylinder

= 0.665 TC = 4.38 K

Domains withvortex latticeType II / 1vortex attraction

B0

vortex latticeType II / 2vortex repulsion

spherelong cylinder examples:

Nb, TaN, PbIn, PbTl

Ba

- M

Theor. –T phase diagram:Ulf Klein, JLTP 69, 1 (1987)Exp.al.: Auer+Ullmaier 1973

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

Magnetization curves ofType-II superconductors

Shear modulus c66(B, κ )

of triangular vortex lattice

c66

-M

Ginzburg-Landau theory EHB, PRL 78, 2208 (1997)

BC1

BC2

Isolated vortex in film

London theoryCarneiro+EHB, PRB (2000)

Vortex lattice in film

Ginzburg-Landau theoryEHB, PRB 71, 14521 (2005)

bulk film

sc film

vac

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

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

exact Pearl potential

analytic approximation:

EHB, PRB 79, 13526 (2009)

J.Pearl, APL 5, 65 (1964)

Interaction of one vortex with a vortex pair = stream function g of this vortex pair

= inverse matrix Kij for fixed index j

EHB, PRB2005

peak:~ ln(2.27Λ / r)

Vortex-vortex interaction for one vortex in center of square film:

numerical Vnum divided by Pearl potential VPearl for infinite film

V/V = 1

V/V = 0

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

H Hc2

-M

width ~ jc

●

●

●

●

●

●

●

● ●

●

● ●

●

●

● ●

●

●

● ●

● ●

●

● ● ● ●

●

●

●

●

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 modelparallel geometrylong cylinder or slab

Bean modelperpendicular geometrythin disk or strip

analytical solution:Mikheenko + Kuzovlev 1993: diskEHB+Indenbom+Forkl 1993: strip

Ba

j

JJ

Ba

Jc

B

J

Ba Ba

r

r

B B

jjjc

r

r r

r

Ba

equation of motionfor current density:EHB, PRB (1996)

J

x

Ba, y

z

J

r

Ba

Long bar

A ║J║E║z

Thick disk

A ║J║E║φ

Example

integrateover time

invert matrix!

Ba

-M

sc as nonlinear conductorapprox.: B=μ0H, Hc1=0

Flux penetration into disk in increasing fieldBa

field- andcurrent-free core

ideal screeningMeissner state

+

+

+ _

_

_

0

Same disk in decreasing magnetic fieldBa

Ba

no more flux- and current-free zone

_

_

+

+++

_

__

+ +_

+ _

remanent state Ba=0

YBCO film0.8 μm, 50 Kincreasing fieldMagneto-opticsIndenbom +Schuster 1995

TheoryEHBPRB 1995

Thin sc rectangle in perpendicular field

stream lines of current

contours ofmag. induction

ideal Meissner

state B = 0

B = 0

Bean state| J | = const

Λ=λ2/d

Thin films and platelets in perp. mag. field, SQUIDs

EHB, PRB2005

2D 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)

Bc2

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 BRF

(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|2orderparameter

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 Ba

Interactionwith image

Interactionwith field Ba

G(∞)

Penetration field:

This holds for large κ.

For small κ < 2 numerics is needed.

numerics required !

Hc

Hc1

2. Vortex half-loop penetrates:

Self energy:

Interaction with Ha:

Surface current:

Penetration field:

vortex half loop

imagevortex

super-conductorvacuum

R

3. Penetration at corners:

Self energy:

Interaction with Ha:

Surface current:

Penetration field:

for 90o

Ha

vacuum

Ha

sc

R

4. Similar: Rough surface, Hp << HcHa

vortices

Bar 2a X 2a in perpendicularHa tilted by 45oHa

Field linesnear corner λ = a / 10

current density j(x,y)

log j(x,y)

x/ay/a

y/a

y/a

x/a

x/a

λ

large j(,y)

5. Ideal diamagnet, corner with angle α :

H ~ 1/ r1/3Near corner of angle α the magnetic field

diverges as H ~ 1/ rβ, β = (π – α)/(2π - α)

vacuum

Ha

sc

r

αα = π

H ~ 1/ r1/2

α = 0

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 tiltedmag. field: perpendicularcomponent penetrates in form of a vortex lattice

Ha

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=2

Magn. curves of pin-free disks + cylinders

ellipsoid isreversible!

b/a=0.3

Radio frequency response of superconductors

DC currents in superconductors are loss-free (if no vortices have penetrated), butAC currents have losses ~ ω2 since the acceleration of Cooper pairs generates anelectric 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 normaland 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:

Londondepth 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) seeR. 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

When purity incr., l↑, σ1↑ but λ↓

Summary

• Vortices in superconductors predicted by Abrikosov are usually repulsive

but can be attractive in Nb, forming clusters, lamellae, chains

• If the sc contains vortices, the vortices move and dissipate much energy,

almost as a normal conductor, 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

( pair breaking at 2Δ/h = 750 MHz )

• In scs with no vortices, the two-fluid model qualitatively explains RF losses

• BCS theory shows that the excitations are „quasiparticles“, Cooper pairs

• Their concentration and thus the losses are very small at low T

• Extremely pure Nb is not optimal, since dissipation ~ σ1 ~ l (path) increases

10 Dec 2003 Stockholm Princess Madeleine Alexei Abrikosov

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

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 )