Carmine SENATORE
Superconductivity and its applications
Lecture 2
Département de Physique de la Matière QuantiqueUniversité de Genève
Previously, in lecture 1
If it is a superconductor, then…
temperature
resi
stiv
ity
Tc
0
1
T < Tc
H < Hc
2 3
Type-I and Type-II superconductorsPreviously, in lecture 1
H
-4πM
Hc0 H < Hc
Type-IHc 10-3 – 10-2 T
H
-4πM
Hc20 Hc1H < Hc1 Hc1 <H < Hc2
Type-IIHc2 10 – 102 T
Previously, in lecture 1
2
8c
S N
HG G
cH Hif1)
S NS S2) higher order in the superconducting state
3) From the 1st London equation * 2
*24 L
S
m c
n e
with
penetration depth
h
z0
insulator superconductor
H L
z
h(z) He
Landau theory of the order-disorder transition
The order parameter is a quantity which is zero in one phase
(usually above the critical point) and non-zero in the other
Example: para- / ferro- magnetic transition
order parameter( )
( )(0)
M Tm T
M
1i
i
MV
where
T > TCuriem = 0
T < TCuriem 0
Landau theory of the order-disorder transition
The Landau expansion for the free energy density close to TCurie
2 40
1( , ) ( , )
2F r T F r T m m
At the equilibrium 0F
m
32 2 0m m
2 0m m
Two solutions: 0m 2m
and
Landau theory of the order-disorder transition
and are function of the temperature
20 1 2
1( ) ( ) ( ) ...
2Curie CurieT T T T T
20 1 2
1( ) ( ) ( ) ...
2Curie CurieT T T T T
From the minimization conditions
1 CurieT T
0
Two solutions: 0m 2m
and
Landau theory of the order-disorder transition
2 m
Temperature dependence of the order parameter close to TCurie
112
1 2
0Curiem T T
Ginzburg-Landau Theory of Superconductivity
In the G-L theory ( )sn r where ( ) ( ) ir r e
2 4( , ) ( , )
2S NF r T F r T
In the London model ns does not depend on the position
The superfluid density ns is the order parameter
2( )r
Free energy expansion in the zero-field case
2 2
*2m
rigidity of the order parameter
Ginzburg-Landau Theory of Superconductivity
2 4( , ) ( , )
2S NF r T F r T
Free energy expansion in the zero-field case
2 2
*2m
Free energy expansion for B ≠ 0
qp p A
c B A In electrodynamics where
*ei i a
c For the momentum operator
2* 2
2 4
*
1( , ) ( , )
2 82S N
e hF r T F r T i a
cm
Ginzburg-Landau Theory of Superconductivity2
* 22 4
*
1( , ) ( , )
2 82S N
e hF r T F r T i a
cm
2* 2
2 4
*
1( ) ( , )
2 82S N
V
e hF T dr F r T i a
cm
* * * , , , ,S S SF F a a F a
*
V
SF dr a
Energy density
( ) ( , )S S
V
F T dr F r T Energy
Minimization procedure variations of , *, and a
0SF
*0SF
0SF
a
And set
The two Ginzburg-Landau equations
0SF
*0SF
0SF
a
2*
2
*
1 0
2
ei a
cm
1st G-L equation
* *2
2* ** *2
e eJ a
m i m c
2nd G-L equation
Zero-field case deep inside superconductor2
* 22 4
*
1( , ) ( , )
2 82S N
e hF r T F r T i a
cm
2 4
2S NF F
becomes
2*
2
*
1 0
2
ei a
cm
And the 1st G-L equation
2 0 becomes
The energy density
2
0 Two solutions: and
Zero-field case deep inside superconductor
On the sign of and
2 1 1 2
4
3
2
1
α > 0 andβ < 0
2 1 1 2
12
10
8
6
4
2
α ≤ 0 and β < 0
FS FS
2 1 1 2
2
4
6
8
10
12
2 1 1 2
1
2
3
4
α 0 and β > 0 α < 0 andβ > 0
FS FS
Zero-field case deep inside superconductor
2
112
1 2
0cT T
2 4
2S NF F
21
2
The solution for the superconducting state
1 cT T 0 Choosing and
and
Zero-field case deep inside superconductor
2 4
2S NF F
21
2
Relation between , and Hc
2
8c
S N
HF F
and
22
2 8cH
Thus
Zero-field case near superconductor boundary2
*2
*
1 0
2
ei a
cm
The 1st G-L equation
becomes 22
*
1 0
2i
m
We define f
2
where
In one dimension2 2
3* 2
02
d ff f
m dx
22 2
2(1 ) 0
d ff f
dx with
22
*2m
0 0f x with the boundary condition
Zero-field case near superconductor boundary
22 2
2(1 ) 0
d ff f
dx with
22
*
1( )
1 /2 ( ) c
TT Tm T
tanh2
xf
tanh2
x
The solution is
x0
insulator superconductor
The coherence length is*2m
H 0 case near superconductor boundary
The 2nd G-L equation * *2
2* ** *2
e eJ a
m i m c
*22
*
eJ a
m c
*22
*
eh
m c
4 h J
c
2 2 0h h with* 2
2 *24
m c
e
If is small *2
2
*
eJ a
m c see C.P. Poole, Superconductivity (2007), pp. 150-154
H 0 case near superconductor boundary
2 2 0h h with* 2
2 *24
m c
e
h
x0
insulator superconductor
H ( )x
h x He
* 2
*24L
S
m c
n e
The London penetration depth is
Flux exclusion & retention : the resistenceless circuitThe total magnetic flux threading a closed resistanceless circuit cannot change so long as the circuit remains resistanceless.
1. The circuit is cooled below Tc in an applied field Ba . The magnetic flux in the circuit is = A Ba
2. The value of Ba is changed
adB dIA RI L
dt dt ( ) (t) constantaLI t AB
Flux quantization
The 2nd G-L equation * *2
2* ** *2
e eJ a
m i m c
2* * 2 ( )i r ( )( ) i rr e
* *2
2 2
* *2
2
e eJ i a
m i m c
*
2 **2
m c ca J
ee
*
*2 2 *
m Jdl cadl dl
e e
Flux quantization
*
*2 2 *
m Jdl cadl dl
e e
S
adl a ndS
( )S
S
h ndS h is a single-value function
2dl n
*
*2 2 *( )S
m Jdl hch n
e e
*( )S
hch n
e 0n
Calculation of the domain-wall energy
<<
At a domain wall
magnetic energy is gained
condensation energy is lost
2
18
cHE A
2
28
cHE A
2
8cH
E A
0 2
08
cHE A
<<
Calculation of the domain-wall energy
Ginzburg-Landau parameter
1
2 0E
Type-I superconductor
1
2 0E
Type-II superconductor
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