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Superconductivity
Electrical resistance
r
𝑇c …
critical temperature
maα Tc  const.
pure metal
a is a material constant
𝑇c
metal with impurities
0.1 K
(isotopic shift of the critical
temperature)
𝑇
1
Superconductivity
Heike Kamerlingh Onnes
1913 Nobel prize in physics
The superconductivity was
discovered in 1911 by Heike
Kamerlingh Onnes at the Leiden
University. At 4.2 K (-296°C), he
observed a disappearance of
resistivity in mercury. His
experiments were made possible by
the condensation of helium (1908).
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Superconductivity
Superconducting elements
Al
Cd
Ga
Hg
In
Ir
La
Mo
Nb
Os
Pb
Re
T [K]
1.19
0.56
1.09
4.00
3.40
0.14
5.00
0.92
9.13
0.65
7.19
1.70
Ru
Sn
Ta
Tc
Th
Ti
Tl
U
V
Zn
Zr
T [K]
0.49
3.72
4.48
8.22
1.37
0.39
2.39
0.68
5.30
0.87
0.55
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Isotopic Shift
maα Tc  const.
Material T [K]
Zn
Cd
Sn
Hg
Pb
Tl
0.87
0.56
3.72
4.00
7.19
2.39
a
0.45±0.05
0.32±0.07
0.47±0.02
0.50±0.03
0.49±0.02
0.61±0.10
Material
T [K]
a
Ru
Os
Mo
Nb3Sn
Mo3Ir
Zr
0.49
0.65
0.92
18
0.00±0.05
0.15±0.05
0.33
0.08±0.02
0.33±0.03
0.00±0.05
0.55
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Superconductivity
Temperature dependence of
the critical magnetic field
Superconductor in
a magnetic field
Hc
 T2 
H c  H 0 1  2 
 Tc 
normal state

 
B  µ0 ( H  M ); 0  4 10 7 Vs /( Am)
Superconductor: Meissner effect



B  0  H  M
  M H  1
superconducting state
Tc
T
Otherwise:   -10-6
5
Meissner-Ochsenfeld effect
6
Magnetic levitation train
7
Superconductor in a magnetic field
External field:
Inner field:
Magnetization:


Be  µ0 H e



B  µ0 ( H e  M )  0


M  H e
Be
 
 
Be2
W   MdB    BdB  
2
0
0
Be
Work per unit of
volume
(magnetization direction of a superconductor is
opposite to the magnetic field direction)
Energy of a superconductor within an magnetic field
is higher than without an magnetic field
This is caused by the “superconducting” electrons
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Transition between normal and
superconducting state
Thermodynamic consideration
G  U  TS
Be2
G U 
 TS
2
Be2 
1

Tc  U  G 
S
2 
Be  2U  G  Tc S 
𝐺 … Gibbs free energy
𝑈 … enthalpy
𝑇 … temperature
𝑆 … entropy
𝐵e … external magnetic field
𝑇 < 𝑇c: 𝑈 (and 𝑆) small for SC state, therefore the SC state is stable
𝑇 > 𝑇c: 𝑆 bigger in normal state (less order), therefore the normal
state is stable
𝐵 > 0: free Gibbs energy is smaller, if 𝑆 is bigger (normal state)
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Superconductivity
Material
𝑇 [K]
NbC
14
NbN
16
Nb3Al
18
Nb3Ge
23
Nb3Sn
18
SiV3
17
La2-xBaxCuO4
30
MgB2
40
YBa2Cu3O7-d
110
S.L. Bud’ko and P.C. Canfield: Temperature-dependent Hc2
anisotropy in MgB2 as inferred from measurements on
polycrystals, Phys. Rev. B 65 (2002) 212501.
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Crystal structures of
La2-xBaxCuO4 and YBa2Cu3O7-x
La2-xBaxCuO4
Space group: Bmab
Lattice parameters:
a = 5.33915(9) Å
b = 5.35882(9) Å
c = 13.2414(2) Å
YBa2Cu3O7-x
Space group: Pmmm
Lattice parameters:
a = 3.856(2) Å
b = 3.870(2) Å
c = 11.666(3) Å
ab
a/2 < c/3 < a
a  b  c/3
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Superconductivity
Type I superconductors
Type II superconductors
• Transition to normal state
after exceeding 𝐻c
• Superconductivity decreases
gradually between 𝐻c1 und 𝐻c2
• Transition to normal state after
exceeding 𝐻c2
−𝑀
−𝑀
superconducting
normal
state
𝐻c
𝐻
𝐻c1 𝐻c
𝐻c2
𝐻
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Theories of Superconductivity
Superelectrons :
• No scattering
• Entropy of the system is zero (the
system is perfectly ordered)
• Large coherence length
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London Theory (Meissner Effect)
 

1 
A; rot A  B
London: j  
2
µ0 λL


Ohm: j  E
London:
Maxwell:
Meissner
effect:

1 
rot j  
B
2
µ0 λL





E
rot B  µ0 j  µ0 0
 rot B  µ0 j
(static conditions)
t




Solution:
rot rot B  grad div B  B  µ0 rot j

0


 x
B
B x   B0 exp  
 B   2
λL
 λL 




B
𝜆L … London penetration depth
x
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Consequences of the London Theory
𝜆L describes the penetration depth of the magnetic field into a
material. Inside the material at a distance 𝜆L to the surface the
intensity of the magnetic field falls to 1/e of its original value.
An external magnetic field 𝐵e penetrates completely
homogeneous a thin layer, if the thickness is much smaller
than 𝜆L . In such a layer, the Meissner effect is not complete.
The induced field (in the material) is smaller than 𝐵e , therefore
the critical magnetic field, which is oriented parallel to the thin
layers is very high.
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Coherence Length
The distance in which the width of the energy gap, in a spatial
variable magnetic field, doesn’t change essentially.
London:
 
1  
j r   
Ar 
2
µ0 λL
 
 
rot Ar   Br 
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BCS Theory of Superconductivity
J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 106 (1957) 162.
J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 108 (1957) 1175.
1. Interactions between electrons can cause a ground state, which is
separated from the electronically excited states by an energy gap.
However: there are also superconductors without an energy gap!
𝐸
𝐸
17
BCS Theory of Superconductivity
2. The energy gap is caused by the interaction between electrons
via lattice vibrations (phonons). One electron distort the crystal
lattice, another electron “sees” this and assimilate his energy to
this state in a way, which reduces the own energy. That’s how
the interaction between electrons via lattice vibrations work.
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BCS Theory of Superconductivity
3. The BCS theory delivers the London penetration depth for the
magnetic field and the coherence length. Thereby the Meissner
effect is explained.
London:
 
 
 
1  
j r   
Ar ; rot Ar   Br 
2
µ0 λL
Meissner:

 B
 x
B  2  B x   B0  exp  
λL
 λL 
Coherence length:
0 
vF
πE g
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