electrodynamics of superconductors exposed to high frequency fields ernst helmut brandt max planck...
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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
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Superconductivity
Zero DC resistivityKamerlingh-Onnes 1911Nobel prize 1913
Perfect diamagnetismMeissner 1933
Tc →
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YBa2Cu3O7-δ
Bi2Sr2CaCu2O8
39K Jan 2001 MgB2
Discovery ofsuperconductors
Liquid He 4.2K →
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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:
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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:
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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↓
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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
!
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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
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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
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Alexei Abrikosov Vitalii Ginzburg Anthony Leggett
Phy
sics
Nob
el P
rize
2003
Landau
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10 Dec 2003 Stockholm Princess Madeleine Alexei Abrikosov
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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
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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
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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
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Isolated vortex in film
London theoryCarneiro+EHB, PRB (2000)
Vortex lattice in film
Ginzburg-Landau theoryEHB, PRB 71, 14521 (2005)
bulk film
sc
vac
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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
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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
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magnetization
force
20 Jan 1989
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Levitation of YBCO superconductor
above and below magnets at 77 K
5 cm
Levitation Suspension
FeNd magnets
YBCO
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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
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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
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Flux penetration into disk in increasing fieldBa
field- andcurrent-free core
ideal screeningMeissner state
+
+
+ _
_
_
0
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Same disk in decreasing magnetic fieldBa
Ba
no more flux- and current-free zone
_
_
+
+++
_
__
+ +_
+ _
remanent state Ba=0
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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°
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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
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Λ=λ2/d
Thin films and platelets in perp. mag. field, SQUIDs
EHB, PRB2005
2D penetr.depth
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Vortex pair in thin films with slit and hole current stream lines
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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 !
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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
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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:
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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
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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
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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)
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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
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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
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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 )