Chapter 9 Ginzburg-Landau theory

Chapter 9 Ginzburg-Landau theory

The limit of London theory The London equation

∇×J = −

B

µ0 λ 2 The London theory is plausible when

1. The penetration depth is the dominant length scale coherent length λ l mean free path λ ξ0 2. The field is small and can be treated as a perturbation 3. ns is nearly constant everywhere The coherent length should be included in a new theory

Ginzburg-Landau theory 1. A macroscopic theory 2. A phenomenological theory 3. A quantum theory

London theory is classical

Introduction of pseudo wave function Ψ (r ) Ψ (r )

2

is the local density of superconducting electrons

Ψ (r ) = ns2 (r ) 2

The free energy density The difference of free energy density for normal state and superconducting state can be written as powers of Ψ 2 and ∇Ψ 2

potential energy

Kinetic energy

Ginzburg-Landau free energy density at zero field

β

1 gs = gn + α Ψ + Ψ + ∇Ψ * 2 2m i 2

2nd order phase transition

2

4

Quantum mechanics

2nd order phase transition Potential energy

U =α Ψ + 2

β 2

Ψ

4

A reasonable theory is bounded, i. e. U ( Ψ → ∞ ) → ∞

β >0 Classical solutions

U

U

Ψ=0

α > 0 Single well

Ψ

Ψ

α < 0 double well

Spontaneous symmetry breaking U The phase symmetry of the ground state wave function is broken

Ψ = Ψ eiϕ

Ψ

Ψ = Ψ ∞2 = − 2

α >0

α =0

Ψ=0

Critical point

Normal state Ψ

2

α β α 0

α =0

α Tc

T = Tc

T < Tc

α = α ′ ( t − 1)

Near the critical point,

If β is regular near Tc then

α′ Ψ = − ( t − 1) βc

The London penetration depth is 1 2

2

λL2 =

⎛1⎞ 1 λL ∝ ⎜ ⎟ ∝ 1 n ⎝ s⎠ (1 − t ) 2

α t=1 T t= Tc

m µ0 ns e2 Consistent with the observation λL (T ) 1 = λL (0) 1 − t 4 12

(

)

t

Magnetic field contribution at non zero field, there are two modifications

p → p − e* A

The vector potential

B = ∇× A 1 ∆g = µ0 H 2 2 ∆g ( H a ) = − µ0

For perfect diamagnetism

Ha

∫ MdH 0

a

The canonical momentum The first modification is to include the hamiltonian of a charged particle in a magnetic field

E = −∇φ −

∂ A ∂t

B = ∇× A t

For a charged paticle,

mv(t ) = mv (0) + q ∫ Εdt 0

= mv (0) − qA

mv(t ) + qA = mv(0)

is conserved in the magnetic field

The canonical momentum is chosen as

p canonical = mv + qA

1 2 1 2 m v = p − q A ( ) The kinetic energy is canonical 2 2m

Gauge transformation A → A ′ = A + ∇χ ∂ ′ φ →φ =φ − χ ∂t

E = −∇φ −

∂ A ∂t

B = ∇× A

The physics is unchanged The phase of the particle wave function will be changed by a phase factor ⎛ ie ⎞ Ψ (r ) → Ψ ′(r ) = Ψ (r ) exp ⎜ χ ⎟ ⎝ ⎠ ⎧ 1 2 ⎛ ie ⎞ ⎫ ′ ′ ′ − Ψ = − ∇ − Ψ ( ) exp χ p A r A e i e ( ) ( )⎨ H= p − eA ) + U ( ⎜ ⎟⎬ ⎝ ⎠⎭ ⎩ 2m ⎛ ie ⎞ H Ψ = H ′Ψ′ = exp ⎜ χ ⎟ {( −i ∇ − eA′ ) Ψ + ( ∇χ ) Ψ} ⎝ ⎠ ⎛ ie ⎞ = exp ⎜ χ ⎟ ( −i ∇ − eA ) Ψ ⎝ ⎠ Comment: not all theory are gauge-invariant, the theory keeps gauge-invariance is called a gauge theory

The meaning of |Ψ|2 Energy density 2

1 ⎛ 1 ⎛ * ⎞ * e A ϕ e AΨ ∇ − Ψ = ∇ Ψ + Ψ ∇ − ⎜ ⎟ ⎜ * * 2m ⎝ i 2m ⎝ i ⎠

Real part

Im. part

1 = 2m*

{

⎞ iϕ ⎟e ⎠

2

with Ψ = Ψ eiϕ 2

( ∇ Ψ ) + ( ∇ϕ − e A ) Ψ 2

*

2

2

}

•The first term arises when the number density ns has a nonzero gradient, for example near the N-S boundary (the length scale is coherent length ξ, and in type I SC, ξ