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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 - PowerPoint PPT Presentation
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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
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 →
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↓
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
2003
Landau
10 Dec 2003 Stockholm Princess Madeleine Alexei 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
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
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
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
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°
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
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 )