Download Ginzburg-Landau theory Free energy Ginzburg

Ginzburg-Landau theory
Ginzburg-Landau equations
•
•
•
•
•
•
•
•
•
BCS: microscopic, very powerful – but too complicated
Order parameter
Free energy
Ginzburg-Landau equations
Critical field of a film
Type-II superconductors
Upper critical field
Lower critical field
Vortex structure
Vortex motion. Pinning
London equations: describe a very special situation
Applicability range: close to the transition temperature.
No restrictions concerning the nature of (conventional) superconductors.
Idea: order parameter is a complex number – wave function of Cooper pairs.
Ψ(r )
Free energy – functional of the order parameter.
Ginzburg-Landau equations
τ = (T − Tc ) / Tc
2e ⎞
H ⎪⎫
⎛
⎬
⎜ −i ∇ − A ⎟ Ψ +
c ⎠
8π ⎪⎭
⎝
2
2
Minimizing the free energy:
2
⎧⎪
b 4 1 ⎛
2e ⎞
H 2 ⎫⎪
2
Fs = Fn + ∫ dV ⎨aτ Ψ + Ψ +
⎬
⎜ −i ∇ − A ⎟ Ψ +
2
4
m
c
8π ⎭⎪
⎝
⎠
⎪⎩
With respect to
Expansion at
the critical
point
Energy of the
normal state
Without
fluctuations:
⎧−aτ / b T < Tc
2
Ψ0 = ⎨
T > Tc
⎩ 0
δ
Way out: to develop a theory based on Landau theory
of second order phase transitions.
Free energy
b 4 1
2
⎪⎧
Fs = Fn + ∫ dV ⎨aτ Ψ + Ψ +
2
4m
⎪⎩
ξ
Ψ*
2
Energy of
magnetic
field
Fluctuation
contribution
(“kinetic
energy of
Cooper
pairs”)
aτΨ + b Ψ Ψ +
2
Boundary condition:
Ginzburg-Landau equations
Minimizing the free energy:
⎧⎪
b 4 1 ⎛
2e ⎞
H 2 ⎫⎪
2
Fs = Fn + ∫ dV ⎨aτ Ψ + Ψ +
⎬
⎜ −i ∇ − A ⎟ Ψ +
m
c
2
4
8π ⎪⎭
⎝
⎠
⎩⎪
1 ⎛
2e ⎞
⎜ −i ∇ − A ⎟ Ψ = 0
4m ⎝
c ⎠
2e ⎞
⎛
n ⎜ −i ∇ − A ⎟ Ψ = 0
c ⎠
⎝
Rescaling GL equations
Ψ → Ψ / Ψ 0 , r → r / δ , H → H / H 0 , H 0 = 2 2π aτ b3/ 2
2
With respect to
∇× H =
4π
j
c
The rescaled equations contain only one constant!
κ = 2eH 0δ 2 / c
(−iκ −1∇ − A)2 Ψ − Ψ + Ψ Ψ = 0
2
A
j =−
“Maxwell equation”
(
)
ie
2e2 2
Ψ*∇Ψ − Ψ∇Ψ* −
Ψ A
mc
2m
(
)
n −iκ −1∇ − A Ψ = 0 at the surface
∇×∇× A = −
(
)
i
2
Ψ*∇Ψ − Ψ∇Ψ* − Ψ A
2κ
1
Critical field of a film
Steps of the calculation:
- Write the energy difference between S and N phases;
- Find at which field the energies are the same (depends on
- Solve GL equations and find the order parameter;
- Find the critical field
Results: Thick films,
Thin films
δ
d
H c = H cb (1 + δ / d )
δ
d
Critical field of a film
Let us now look at the details.
Ψ)
H cb = H 0 / 2
H c = 2 6 H cbδ / d
Hc
Thick films
d
κ 1 give Ψ ≈ const
δ Ψ 2 ≈ 1 First-order phase transition
Thin films
d
δ
GL equations for
Fs − Fn
Ψ=0
Second-order phase transition
At certain thickness the transition order changes
d
Fs − Fn
δ
d
δ
dc = 5δ
H
H cb
H cb
Ψ
δ /d
Critical field of a film
Type-I superconductors: 1st order phase transition in the bulk
is realized by splitting into N and S layers
Type II superconductors: does not work!
Interface energy: negative
Ψ
2
Phase transition must be of the second order and occurs in the interval
of fields
Optimal thickness of a layer
Intermediate state!
H cb
2
Type-II superconductors
H > Hc2
Interface energy: positive
Critical field of one layer is higher
H
H < H c1
(Upper critical field)
Even a small piece of superconductor is unstable
Normal metal
(Lower critical field)
Fully developed Meissner effect
Superconductor
H c1 < H < H c 2
Mixed state
No Meissner effect
Formation of infinitely thin layers is preferable
First-order phase transition is shifted to infinitely strong fields!
Upper critical field
Two possible mechanisms:
- Spins of two electrons in a Cooper pair become parallel – too strong fields
- Larmor radius of a Cooper pair becomes shorter than ξ
Larmor radius:
p⊥
rL =
cp⊥
c
<
eH eH ξ
Lower critical field
What happens below
Persistent currents (no decay): vortices!
Quantum vortices:
-component of momentum of the pair transverse to the
magnetic field
ξ < rL ⇒ H < H c 2 ∼
c
eξ 2
Hc 2 ?
Magnetic field penetrates the superconductor
H
v
Bohr-Sommerfeld quantization:
ρ
2π n =
∫ pdq
= 2m ∫ vdl = 2mv ⋅ 2πρ
H c 2 = κ H cb > H cb
v = n / 2mρ
- quantized velocity
n=1: energetically the most favorable
ξ
ρ
δ
2
Lower critical field
v = n / 2mρ ξ
Vortex energy (per unit length):
E=
δ
ρ
δ
H
ns
πn
δ πn
2mvs2 ⋅ 2πρ d ρ = s ln = s ln κ
2 ∫ξ
4m
ξ
4m
2
Isolated vortex
2
What is the magnetic flux in a vortex?
Ψ
v
Vortex magnerization (per unit length):
ns 2e
πn e
ρ vs ⋅ 2πρ d ρ = s δ 2
2 c ∫ξ
2mc
H
Lower critical field: the first vortex is formed
H
ξ
δ
Ψ = Ψeiθ
Magnetic energy: -MH
δ
M=
Vortex
core
ρ
Current:
ρ
j =−
)
2e2 ⎞
2⎛e
j = Ψ ⎜ ∇θ −
A⎟
mc ⎠
⎝m
Far from the vortex core: no current
Let us integrate over a contour around the vortex center.
H cb
H c1 = E / M = (c / eδ 2 ) ln κ ∼ H cb ln κ / κ 2 < H cb
1/ 2
κ
∫ ∇θ dl =
2e
c
∫ Adl =
Vortex lattice
2eΦ
c
Φ = nΦ s , Φ s ≡ π c / e
Form a triangular vortex lattice
-superconducting
flux quantum
Vortex motion
Close to lower critical field – sparse vortices
I
If current is passed through a
superconductor in a mixed state:
Lorentz force acts at the core
electrons
a
Lattice period: determined by H
a
(
ie
2e2 2
Ψ*∇Ψ − Ψ∇Ψ* −
Ψ A
2m
mc
F ∝ j×H
Per plaquette: half a vortex
Vortex motion
transverse to the current
Φs
3 2
aH
=
2
4
Total flux per plaquette:
Hc 2
: vortices overlap
Motion of magnetic flux with a constant velocity
Electric field
a ∼ ξ ⇒ Hc 2 ∼ Φs / ξ 2
H c1 ∼ Φ s / δ ; H cb ∼ Φ s / δξ
2
Resistance!!
1
E = H ×v
c
ρ ∼ ρN H / Hc2
Like if produced by vortex cores.
Pinning
Does it mean Type-II superconductors have resistance down to
very low fields?
No, vortices do not move (at weak currents)
Pinning:
Is more favorable
than
It costs energy to depin a vortex.
3