Macroscopic theory of
superconductivity
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Zero resistance
Meissner effect
Specific heat
Intermediate state
London equations
Penetration depth
Resistance
Experimental facts: there are metals and metallic compounds
with the following properties:
¾ Below certain critical temperature
Examples:
W – 0.012K
Al – 1.2K
α-Hg – 4.15K
Nb – 9.26K
Tc
the resistance vanishes.
Kamerlingh-Onnes, 1914
MgB2 - 39K
YBa2Cu3O7− x- 95K
R
T
Tc
If currents are generated in a superconductor, they do not decay.
Meissner effect
¾ Superconductors are ideal diamagnets. Weak magnetic fields
not penetrate superconductors (Meissner effect). Fields
greater than H destroy superconductivity.
Meissner effect
Magnetic field disappears in the bulk of a superconductor
c
Examples:
W :1G = 10−4 T α − Hg : 411G
Al : 99G
Nb :1980G
The external field generates surface currents that screen the field.
Must be a characteristic scale of the magnetic field decay:
Penetration depth
Phenomenologically:
δ (T ) =
1 − (T / Tc )
δ
B
δ (0)
4
S
N
Phase transition:
¾Zero magnetic field – second order (at the transition point, there is no
Meissner effect, and phases are the same)
¾ Finite magnetic field – first order phase transition
Specific heat
Experimentally: activation benaviour in the superconducting phase
c(T ) ∝ exp(−∆ / kBT ), T
Meissner effect
Consider a metallic ellipsoid in an external magnetic field
Tc
H
At the transition point: jump of the specific heat
Conclusion: new state of electrons in a superconductor, different
from the normal metal
Avenues to proceed:
¾ Phenomenological theory: London equations, Ginzburg-Landau
equations
¾ Microscopic theory: Bardeen – Cooper - Schrieffer
Hin
A tiny bit of electrodynamics
Outside:
H = B, M = 0
Inside:
Hin = H − 4π nM
H – magnetic field
B – magnetic induction
M – magnetization
0 < n < 1 – demagnetization factor
(geometry dependent)
Bin = Hin + 4π M
1
Meissner effect
Intermediate state
Consider now a superconducting ellipsoid in an external magnetic field
Inside:
H in = H − 4π nM
H c (1 − n) < H < H c
Coexistence of normal and superconducting states?
Bin = H in + 4π M = 0
Hin = H /(1 − n) > H
H > Hc
N
H H (1 − n)
Hin = H c ⇒ B = − c
n
n
- superconducting state
H c (1 − n) < H < H c
S
Solution: “Intermediate state”
- normal state
H < H c (1 − n)
N
- a problem!
Does not work: the field
penetrates the normal parts,
but then the field must decay away
from S – no normal part is possible
The field in the whole superconductor
equals H c
B
Normal and superconducting domains coexist
S
London equations
∂ m
j=E
∂t ne2
No resistance: The electrons are accelerated by the field
1 ∂H
∇× E = −
c ∂t
∇×
∂ m
j=E
∂t ne2
d ∂
= + v∇
dt ∂t
∂⎛
m
1 ⎞
⎜ ∇× 2 j + H ⎟ = 0
∂t ⎝
ne
c ⎠
m
1
j+ H =0
ne2
c
N
H
Penetration depth
Phenomenological theory based on the experimental facts:
Zero resistance and Meissner effect.
dv
d m
m = eE , j = nev ⇒
j=E
dt
dt ne2
I
related to
Fermi velocity
∇×
Maxwell equation:
m
1
j+ H =0
ne2
c
∇× H =
H
S
4π
j
c
∆H = δ −2 H
δ=
mc 2
4π ne2
- penetration depth
Decay of the magnetic field in the bulk of S:
H exp(− x / δ )
(also the current)
related to the
current
Estimates:
δ ∼ 3 ⋅10−6 m
-reasonably agrees with experiments.
Should we substitute density of all electrons???
2