Transcript Folie 1 - LNL-INFN

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
Tc →
Zero DC resistivity
Kamerlingh-Onnes 1911
Nobel prize 1913
Perfect diamagnetism
Meissner 1933
Discovery of
superconductors
Bi2Sr2CaCu2O8
YBa2Cu3O7-δ
39K Jan 2001 MgB2
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 ~ 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:
dissipative part:
inductive part:
London equation:
London
depth λ
Normal conductors:
skin depth
parallel R and L:
crossover frequency:
power dissipated/vol:
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:
When purity incr., l↑, σ1↑ but λ↓
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
Vortices: Phenomenological Theories
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:
Type-II scs:
κ ≤ 0.71, NS-wall energy > 0
κ ≥ 0.71, NS-wall energy < 0: unstable !
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
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 Φo = h / 2e = 2*10-15 T m2
Nobel prize in physics 2003 to V.L.Ginzburg and A.A.Abrikosov for this
Abrikosov
28 Sept 2003
Physics Nobel Prize 2003
Landau
Alexei Abrikosov
Vitalii Ginzburg
Anthony Leggett
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
Ba= 985 G, a =170 nm
electron microscope
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
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
c66
-M
BC
1
BC2
Magnetization curves of
Type-II superconductors
Shear modulus c66(B, κ )
of triangular vortex lattice
Ginzburg-Landau theory
EHB, PRL 78, 2208 (1997)
bulk
vac
sc
Isolated vortex in film
London theory
Carneiro+EHB, PRB (2000)
Vortex lattice in film
Ginzburg-Landau theory
EHB, PRB 71, 14521 (2005)
film
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
-M
Theory:
● summation of random pinning forces
→ maximum volume pinning force jcB
● thermally activated depinning
● electromagnetic response
● ● pin
●
●
●
● ●
●
●
●
●
●
→
●
●
●
●
●
FL
●
●
●
● ●
●
●
●
●
●
●
●
●
●
Lorentz force B х j →
width ~ jc
H
Hc2
20 Jan 1989
magnetization
force
Levitation of YBCO superconductor
above and below magnets at 77 K
Levitation
5 cm
YBCO
FeNd magnets
Suspension
Importance of geometry
Ba
j
Bean model
Ba
parallel geometry
long cylinder or slab
B
r
jc
Bean model
perpendicular geometry
thin disk or strip
B
J
analytical solution:
Mikheenko + Kuzovlev 1993: disk
EHB+Indenbom+Forkl 1993: strip
j
r
j
r
r
J
Ba
Jc
r
J
B
Ba
Ba
r
Example
Ba, y
sc as nonlinear conductor
J
Ba
J
approx.: B=μ0H, Hc1=0
z
x
Long bar
A ║J║E║z
Thick disk
A ║J║E║φ
r
-M
Ba
invert matrix!
equation of motion
for current density:
EHB, PRB (1996)
integrate
over time
Flux penetration into disk in increasing field
Ba
ideal screening
Meissner state
+
field- and
current-free
core
+
_
0
+ _
_
Same disk in decreasing magnetic field
Ba
no more flux- and current-free zone
_
+
_
+
_
remanent state Ba=0
+ _
+
Ba
_
+
_
+
_
+
to scale
d/2w = 1/25
θ = 45°
tail
Ha
+0_
tail
stretched along z
tail
tail
+
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)
_
_
+
Thin sc rectangle in perpendicular field
stream lines
of current
contours of
mag. induction
ideal Meissner
state B = 0
B=0
| J | = const
Bean state
Theory
EHB
PRB 1995
YBCO film
0.8 μm, 50 K
increasing field
Magneto-optics
Indenbom +
Schuster 1995
Thin films and platelets in perp. mag. field, SQUIDs
EHB,
PRB
2005
Λ=λ2/d
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:
Is comparable to normal resistvity
→ dissipation is very large !
B+S
Where does dissipation come from?
Exper.
and L+O
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
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
|Ψ|2
order
parameter
v
v
x
moving vortex core
relaxing order parameter
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)
Interaction
with field Ba
Interaction
with image
G(∞)
Gibbs free energy of one
vortex in supercond. half
space in applied field Ba
Penetration field:
Hc1
Hc
This holds for large κ.
For small κ < 2
numerics is needed.
numerics required!
2. Vortex half-loop penetrates:
superconductor
vacuum
Self energy:
Ha
Interaction with Ha:
R
Surface current:
image
vortex
Penetration field:
vortex
half loop
3. Penetration at corners:
Ha
Self energy:
vacuum
Interaction with Ha:
Surface current:
R
for 90o
sc
Penetration field:
4. Similar: Rough surface, Hp << Hc
Ha
vortices
Ha
y/a
Bar 2a X 2a in
perpendicular
Ha tilted by 45o
large
j(,y)
log j(x,y)
y/a
Field lines
near corner
λ = a / 10
x/a
x/a
current density
j(x,y)
λ
y/a
x/a
5. Ideal diamagnet, corner with angle α :
Near corner of angle α the magnetic field
diverges as H ~ 1/ rβ, β = (π – α)/(2π - α)
Magnetic field H at the equator of:
cylinder
Ha
vacuum
sc
H ~ 1/ r1/3
α
α=π
H/Ha = 2
sphere
r
H ~ 1/ r1/2
H/Ha = 3
α=0
H/Ha = a/b
Ha
ellipsoid
b
a
b << a
rectangle
(strip or disk)
H/Ha ≈ (a/b)1/2
2b
2a
b << a
Large thin film in tilted
mag. field: perpendicular
component penetrates
in form of a vortex lattice
Irreversible magnetization of pin-free superconductors
due to geometrical edge barrier for flux penetration
b/a=2
Magn. curves of pin-free disks + cylinders
flux-free
core
b/a=0.3
b/a=2
ellipsoid is
reversible!
flux-free
zone
b/a=0.3
Magnetic field lines in pin-free
superconducting slab and strip
EHB, PRB 60, 11939 (1999)
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 )