Carmine SENATORE
Superconductivity and its applications
Lecture 4
http://supra.unige.ch
Group of Applied SuperconductivityDepartment of Quantum Matter Physics
University of Geneva, Switzerland
Type-II superconductors: Mixed state and quantized vortices
• Magnetic flux penetrates beyond Hc1
1
2E 0
wall energy
• Being the wall energy negative, the system prefers to maximize the walls
• The entering flux is fractionated in
vortices, each one carrying a flux
quantum hc
e 0
2
Previously, in lecture 3
Previously, in lecture 3From the Ginzburg-Landau equations
In a type-II superconductor, the critical fields are related to the characteristic lengths
cc
HH ln ln
0
1 24 2
c cH H
0
2 22
2
The structure of the vortex lattice
r(r ) tanh
2h(r ) ln .
r
02
0 122
From the solution of the linearized 1st G-L equation at Hc2
n
nqx
H0
2
n nC C and
Interaction between vortices
vortex h
0
1 08
and in the case of 2 vortices
vortices h r h r h r h r
02 1 1 1 2 2 1 2 2
8
h r h r
0 01 1 1 22
8 4
vortex interaction 12
In lecture 3, we found
Interaction between vortices
The force of vortex 1 on vortex 2 is
The obvious generalization to an arbitrary array is
s ˆf J zc
0
Js is the total supercurrent due to all other vortices Jarray + any
transport current Jext at the vortex core position.
Obviously, at equilibrium
array ˆJ zc
0 0
interaction ˆf J r zc
02 1 2
Vortex motion and dissipation: Flux Flow
Let’s focus on the effects of a transport current Jext
On a single vortex
extJc
ˆf z
0
On the vortex lattice
v ext vF f zn f J nc
0 ˆ
ext
BJ
c
Therefore, vortices tend to move transverse to Jext . If v is their velocity
vE B
c DISSIPATION !!
Vortex motion and dissipation: Flux Flow
Bardeen and Stephen [PR 140 (1965) A1197] showed that the vortex
velocity v is dumped by a viscous drag term
ext LJ vc
0
And ff
EB
J c
0
2
vE B
c
The following form is predicted for
c
n
B
c
0 2
2
And thus ff n
c
B
B
2
Normal fraction in the
SC, occupied by the
vortex cores
condensation condensation magG G defect G vortex G
condensation magG G defect G
Vortex-defect interaction
vortex defect
Force to extract the vortex from the defect pf U r
Defects are impurities, grain boundaries and any spatial inhomogeneity, whose
size is comparable with and
extf Jc
0
Vortex-defect interaction
Force exerted from Jext
pf
Pinning Force exerted
from defects
cJc
0
Jc is the critical current density
pf f v 0 0If then and
pf f v 0 0then andIf
Only superconductors with defects are truly superconducting ( = 0 ) !!
Critical state: the Bean model
BF J
c H J
c
4+
z
yx
BFor an infinite slab in parallel field
JBF
c
dBB
dx
1
4c
P
J BF
c
In the critical state pF Fc
dBJ
dx c
4
N.B. The critical current density Jc is different from the depairing current Jd
Depairing current * cs d d
Hn m v J
c
22 2 2
2
1 2
2 8c
d
HJ c
4
vc
dnJ
dx c
0
4
C.P. Bean, Phys. Rev. Lett. 8 (1962) 250
Critical state: the Bean model
*cB J a
c
4
Critical state: Pinning strength
The critical current density Jc depends on the type, size and distribution of the
pinning centers
Critical state: Reversing field
The shaded area
corresponds to the sample
magnetization
Critical state: Hysteresis loop
*B B0
1
2
*B B0
5
4
*B B0 3
cM J a
Critical state: Hysteresis loop
Critical state: Hysteresis and losses
*
B
B 0
Q MdB B 20
Superconducting wires are multifilamentary. WHY ?
Bi2223
Bi2212
MgB2
Nb3Sn
NbTi
Superconducting wires are multifilamentary. WHY ?
cM J d d is the filament size
cM nJ d
d is the filament size
n is the number of filaments
M
B
With the subdivision of the superconducting layer in filaments, hysteretic
losses are reduced but the critical current density Jc is unchanged
Extracting Jc from magnetization
Knowing that it is possible to calculate the
expression of M for a given, constant JV
M r J dVV
1
2
Sphere c
MJ
R
3
Slab in perpendicular field
Slab in parallel field
Cylinder in parallel field
Cylinder in perpendicular field
c
MJ
d
2
c
MJ
b b a
3
3
c
MJ
R
3
2
c
MJ
R
3
8
ab
Field dependence of Jc
-10 -8 -6 -4 -2 0 2 4 6 8 10-3
-2
-1
0
1
2
3
m [
emu
]
B [Tesla]
T = 7.5 K
Nb3Sn
M M B c c J J B
Field dependence of Jc
0 1 2 3 4 5 65
10
100
@4.2K
M
[e
mu/c
m3]
B [Tesla]
LHC dipole NbTi strand
0 1 2 3 4 5 6 7 80.5
1
10
50
Jc [
kA
/mm
2]
B [Tesla]
Reference, transport measurement
LHC dipole NbTi strand
@4.2K
c
MJ
R
3
8
-6 -4 -2 0 2 4 6-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
m [
emu
]
B [Tesla]
LHC dipole NbTi strand
@4.2K
Temperature dependence of Jc
-10 -8 -6 -4 -2 0 2 4 6 8 10-3
-2
-1
0
1
2
3
m [
emu
]
B [Tesla]
T = 7.5 KNb3Sn
-10 -8 -6 -4 -2 0 2 4 6 8 10-3
-2
-1
0
1
2
3
10 K
m [
emu
]
B [Tesla]
T = 7.5 KNb3Sn
-10 -8 -6 -4 -2 0 2 4 6 8 10-3
-2
-1
0
1
2
3
12.5 K10 K
m [
emu
]
B [Tesla]
T = 7.5 KNb3Sn
-10 -8 -6 -4 -2 0 2 4 6 8 10-3
-2
-1
0
1
2
3
15 K12.5 K
10 K
m [
emu
]
B [Tesla]
T = 7.5 KNb3Sn
M M B,T c c J J B,T
E-J relation for a type-II superconductor
E
J0
Jc
pinning: Bean Model
E
J0
Jc
ff n
c
B
B
2
Bean Model + flux flow
E-J relation for a superconductor in the mixed state
E
J
0
ff n
c
B
B
2
w/o pinning: flux flow
The effect of thermal excitation was not yet included
Beyond Bean: Thermally activated flux creep
BU/ k TR e 0
In the presence of an external current
B B BU / k T U / k T U / k TR R R e e e
0
Thermally activated flux creep
B B BU / k T U / k T U / k TR R R e e e
0
U U J BU U / k TR R e
0
In a first approximation
where
The B-profile relaxes with t
Thermally activated flux creep
BU U J / k TR R e
0
E B
x t
+
In the infinite slab – parallel field geometry
E vB
2) J is constant in space
Thermally activated flux creep: Anderson-Kim model
The solution for J is Bk T tJ t J ln
U t
0
0 0
1
In the Anderson-Kim model where U U B,T0 0
JU U
J
0
0
1
P.W. Anderson and Y.B. Kim, RMP 36 (1964) 39
BH
0
1) B << Hd 2
Thermally activated flux creep: Anderson-Kim model
Bk T tJ t J ln
U t
0
0 0
1
The relaxation rate S is directly proportional to T and inversely proportional to U0
BdM k TS
M d lnt U
0 0
1
M J
The current density J and the magnetization M both relax with the logarithm of time
Measuring S as a function of B and T, we have an experimental access to the pinning energy
U U B,T0 0
Magnetic relaxation experiments
1 10 100 1000 10000650
700
750
800
850
900
@ 10K, 3T
MgB2
MgB2 + 10wt% SiC
MgB1.9
C0.1
J c [A
/mm
2]
time [s]
M
B
Typical values of the activation energy U0 span in the range 10-100 meV
(~100-1000 K)
Bibliography
Fosshein & SudbøSuperconductivity: Physics and ApplicationsChapter 8
TinkhamIntroduction to SuperconductivityChapter 5