Chapter 9 Ginzburg-Landau theory

The limit of London theoryThe London equation 2

0µ λ∇× = −

BJ

The London theory is plausible when1. The penetration depth is the dominant length scale

mean free path coherent length

2. The field is small and can be treated as a perturbation3. ns is nearly constant everywhere

lλ 0λ ξ

The coherent length should be included in a new theory

Ginzburg-Landau theory

1. A macroscopic theory2. A phenomenological theory3. A quantum theory London theory is classical

Introduction of pseudo wave function ( )Ψ r

2( )Ψ r is the local density of superconducting electrons

2 2( ) ( )snΨ =r r

The free energy densityThe difference of free energy density for normal state and superconducting state can be written as powers of and2Ψ 2∇Ψ

potential energy Kinetic energy

22 4

*

12 2s ng g

m iβα= + Ψ + Ψ + ∇Ψ

Quantum mechanics

Ginzburg-Landau free energy density at zero field

2nd order phase transition

2nd order phase transition2 4

2U βα= Ψ + Ψ

A reasonable theory is bounded, i. e.

Potential energy

( )U Ψ → ∞ → ∞

0β >Classical solutions

Ψ

U U

0Ψ =Ψ

Single well 0α < double well 0α >

Spontaneous symmetry breaking

Ψ

U

The phase symmetry of the ground state wave function is broken

ie ϕΨ = Ψ

2 2 αβ∞Ψ = Ψ = −

0α > 0α = 0α <

0Ψ = 0Ψ ≠superconducting stateCritical pointNormal state

2Ψ density of superconducting electrons

The meaning of α0α =The superconducting critical point is α

t=1 t0α = 0α <0α >

cT T<cT T=cT T>

( )1tα α′= −Near the critical point,

c

TtT

=( )2 1

c

tαβ

′Ψ = − −If β is regular near Tc then

22

0L

s

mn e

λµ

=The London penetration depth is

( )

12

12

1 1

1L

sn tλ

⎛ ⎞∝ ∝⎜ ⎟

⎝ ⎠ −( )

14 2

( ) 1(0) 1

L

L

T

t

λλ

=−

Consistent with the observation

Magnetic field contributionat non zero field, there are two modifications

*e→ −p p A The vector potential= ∇×B A

20

12

g Hµ∆ = For perfect diamagnetism

( ) 00

aH

a ag H MdHµ∆ = − ∫

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

tφ ∂

= −∇ −∂

= ∇×

E A

B A

0

( ) (0)

(0)

t

m t m q dt

m q

= +

= −

∫v v Ε

v A

For a charged paticle,

( ) (0)m t q m+ =v A v is conserved in the magnetic field

canonical m q= +p v AThe canonical momentum is chosen as

( )22canonical

1 12 2

m qm

= −v p AThe kinetic energy is

Gauge transformationχ′→ = + ∇A A A

tφ φ φ χ∂′→ = −

∂

tφ ∂

= −∇ −∂

= ∇×

E A

B A

The physics is unchanged

The phase of the particle wave function will be changed by a phase factor

( ) ( ) ( ) exp ie χ⎛ ⎞′Ψ → Ψ = Ψ ⎜ ⎟⎝ ⎠

r r r

( )212

H e Um

= − +p A( ) ( )

( ) ( )

( )

( ) exp

exp

exp

iee i e

ie i e

ie i e

χ

χ χ

χ

⎧ ⎫⎛ ⎞′ ′ ′− Ψ = − ∇ − Ψ⎨ ⎬⎜ ⎟⎝ ⎠⎩ ⎭

⎛ ⎞ ′= − ∇ − Ψ + ∇ Ψ⎜ ⎟⎝ ⎠⎛ ⎞= − ∇ − Ψ⎜ ⎟⎝ ⎠

p A r A

A

A

H H ′ ′Ψ = Ψ

Comment: not all theory are gauge-invariant, the theory keeps gauge-invariance is called a gauge theory

The meaning of |Ψ|2Energy density

2 2* *

* *

1 12 2

ie A e A em i m i

ϕϕ⎛ ⎞ ⎛ ⎞∇ − Ψ = ∇ Ψ + Ψ ∇ − Ψ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

ie ϕΨ = Ψwith

( ) ( ) 22 22 **

12

e Am

ϕ= ∇ Ψ + ∇ − Ψ

Real partIm. part

•The first term arises when the number density ns has a non-zero gradient, for example near the N-S boundary

•The second term is the kinetic term associated with the supercurrent. If the phase is constant of position, it gives

2 2* 2

*2e A

mΨ

=

(the length scale is coherent length ξ, and in type I SC, ξ<<λ)

Penetration near the N-S boundaryNear the surface, the magnetic induction is z

( ) ( )0 xz zB x B e λ∼ for x<0

x

y Ha

super

B(x)= ∇×B A

( ) yz

AB x

x∂

= −∂

Choose the gauge

( ) ( )y zA x B xλ=

We found the energy density2 22 2* 2 * 2 2

* *2 2e A e B

m mλΨ Ψ

=

normal

2

02BµShould be equal to the field energy density

*2

2*0 S

me n

λµ

=*

22 2*

0

me

λµ

=Ψ

From London’s theory

Kinetic energy density= * 212S Sn m v⎛ ⎞

⎜ ⎟⎝ ⎠

The supercurrent velocity= * *

*

S Sm e

eϕ

= −

= ∇ −

v p A

AFor 0ϕ∇ =

2** 2

*

12 2S S S

en m v n

m⎛ ⎞ =⎜ ⎟⎝ ⎠

A While in GL theory, the energy density 2 2* 2

*2e A

mΨ

=

2Sn = Ψ

The meaning of the wavefunction Ψ

GL theory and London theoryIn bulk superconductors 20

2 4

2

2

2

2

s n Cg g Hµ

βα

αβ

∞ ∞

− = −

= Ψ + Ψ

= −

In previous discussion, we have(in bulk)2

Sn αβ∞

−= Ψ =

*2

2*0 S

me n

λµ

= we have22 2 * 2 2

0 0*

C C

S

H e Hn m

µ µ λα − −= =with

43 * 4 20

2*Ce H

mµ λβ =

The temperature dependences near critical pointNear the critical point

4

11 t

λ ∝− C

TtT

=

*2 4

2* 20

1 1Smn t t

e µ λ∞Ψ = = ∝ − −

( ) ( ) ( )441 1 1 1 1 4 4 4 1t tε ε ε− = − − − − = −

0ε →

( )( )20 1C CH H t−

( ) ( )2 22

2 24

1 21

1 4C

tH t

tε

α λε

−∝ ∝ −

−

( )( )

( )( )

2 224 2

2 24

1 2constant of

41C

tH t

t

εβ λ

ε

−∝ ∝ =

−

Parameters in GL theory can be determined by λ(T) and HC(T)

GL differential eqnsThe solution for minimizing gs in absence of the field, boundary and current is

∞Ψ = Ψ

In general cases, the wavefunction can be written as

( )Ψ = Ψ r

By variational method 0SV

g dVδ =∫2

2 **

1 02

em i

α β ⎛ ⎞Ψ + Ψ Ψ + ∇ − Ψ =⎜ ⎟⎝ ⎠

A

( )

( )

2* *2* *

* *

*2 2* *

*

2

S

e em i m

e e em

ϕ

= Ψ ∇Ψ − Ψ∇Ψ − Ψ

= ∇ − Ψ = Ψ

J A

A v

We have (1st eq)

(2nd eq)

Derivation for GL eqns0S

V

g dVδ =∫

( )1 2 3, ,∇Ψ ≡ ∂ Ψ ∂ Ψ ∂ Ψgs is a function of Ψ and

With boundary conditions, i.e 0 or 0Ω Ω

Ψ = ∇Ψ =

( )0S S

ii i

g g∂ ∂− ∂ =

∂Ψ ∂ ∂ Ψ∑ Euler-Lagrange eq.

2 22 4 *

0*0

12 2 2s ng g e

m iβα µ

µ⎛ ⎞= + Ψ + Ψ + ∇ − Ψ + − ⋅⎜ ⎟⎝ ⎠

BA M H

*2 *

* *2Sg e e

m iα β∂ − ⎛ ⎞= Ψ + Ψ Ψ + ⋅ ∇ − Ψ⎜ ⎟∂Ψ ⎝ ⎠

A A

( )* *0S S

ii i

g g∂ ∂− ∂ =

∂Ψ ∂ ∂ Ψ∑

( )*

**

22 *

*

12

12

Si i i i

i ii

g e Am i i

em i i

∂ − ⎛ ⎞ ⎛ ⎞∂ = ∂ ∂ − Ψ⎜ ⎟ ⎜ ⎟∂ ∂ Ψ ⎝ ⎠ ⎝ ⎠

⎧ ⎫− ⎪ ⎪⎛ ⎞= ∇ Ψ − ⋅∇Ψ⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭

∑ ∑

A

2*2 * 2 *

* *

222 2 * * 2

*

22 *

*

102 2

1 22

12

e e em i m i i

e em i i

em i

α β

α β

α β

⎧ ⎫− ⎪ ⎪⎛ ⎞ ⎛ ⎞= Ψ + Ψ Ψ + ⋅ ∇ − Ψ + ∇ Ψ − ⋅∇Ψ⎨ ⎬⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭

⎧ ⎫⎪ ⎪⎛ ⎞= Ψ + Ψ Ψ + ∇ Ψ − ⋅∇Ψ + Ψ⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭

⎛ ⎞= Ψ + Ψ Ψ + ∇ − Ψ⎜ ⎟⎝ ⎠

A A A

A A

A

First GL equation

0SV

g dVδ =∫gs is a function of Α and ∇× =A Β

With boundary conditions, i.e 0 or 0j iAΩ Ω

= ∂ =A

( )0S S

iij i j

g gA A

∂ ∂− ∂ =

∂ ∂ ∂∑ Euler-Lagrange eq.

2 22 4 *

0*0

12 2 2s ng g e

m iβα µ

µ⎛ ⎞= + Ψ + Ψ + ∇ − Ψ + − ⋅⎜ ⎟⎝ ⎠

BA M H

( )

** * * *

*

2**2* *

* *

2

2

S

j jj

j

j

g e e eA m i i

e Aem i m

⎧ ⎫∂ − ⎪ ⎪⎛ ⎞ ⎛ ⎞= Ψ ∇ − Ψ + Ψ − ∇ − Ψ⎨ ⎬⎜ ⎟ ⎜ ⎟∂ ⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭

− ⎛ ⎞= Ψ ∇Ψ − Ψ∇Ψ + Ψ⎜ ⎟⎝ ⎠

A A

( )( )( )

( )( ) ( )

( ) ( )

( )

0

0

0 0

0

12

121 1

1

m n q rSi lmn lqr i

i i lmnqri j i j

lmn lqr i mi nj q r qi rj m ni lmnqr

lmn lij i m n ijl i li lmn i

j

A AgA A

A A

A

ε εµ

ε ε δ δ δ δµ

ε ε εµ µ

µ

∂ ∂ ∂∂∂ = ∂

∂ ∂ ∂ ∂

= ∂ ∂ + ∂

= ∂ ∂ = ∂ ∇×

−= ∇×∇×

∑ ∑ ∑

∑ ∑

∑∑ ∑ A

A

( ) ( )( )2lmn lqr m n q r

lmnqrA Aε ε∇× = ∂ ∂∑A

( )

( )

2* *2* *

* *0

2* *2* *

* *

102

2

e em i m

e em i m

µ− ⎛ ⎞= Ψ ∇Ψ − Ψ∇Ψ + Ψ + ∇×∇×⎜ ⎟

⎝ ⎠

= Ψ ∇Ψ − Ψ∇Ψ − Ψ

A A

AJ

0 0

1 1µ µ

∇×∇× = ∇× =A Β J

second GL equation

Boundary conditionsThe GL eqns are derived by assuming boundary conditions that

* 0ei Ω

⎛ ⎞∇ − Ψ =⎜ ⎟⎝ ⎠

A and 0Ω

=J

These are true for a SC-insulator boundary, but not correct for an N-SC boundary

The boundary condition for N-SC is derived by de Gennes using a microscopic theory:

* iei bΩ

⎛ ⎞∇ − Ψ = Ψ⎜ ⎟⎝ ⎠

ASC normal

b

Thus the wavefunction will “leak” into the normal region with a characteristic length, b. This is called proximity effect.

GL coherent lengthAt zero field, H=0

0=J * * 0Ψ ∇Ψ − Ψ∇Ψ = and

Superconducting phase is constant of position0ϕ∇ =

f∞

Ψ≡

Ψ2

2 2* 0

2mα βΨ + Ψ Ψ − ∇ Ψ =

(GL eq 1)

αβ∞Ψ ≡ −In 1D system

2 22 3

* 2 02

d f f fm dx

α β ∞− + + Ψ =

2 23

* 2 02

d f f fm dxα

− + − =

Dimension=[L2]

22

*2mξ

α=A length scale can be defined

22 3

2 0d f f fdx

ξ− + − =

2 11 t

ξ ∝−

since 1 tα ∝ −

Consider the situation that f~1 (deep in the SC)We can expand the GL eq: 1 0f g g= −

( ) ( )2

322 1 1 0d g g g

dxξ− + + − + =

2 2 0g gξ ′′− − = ( ) 2xg x e ξ±

Exact solution( )

22 2

2 1 0d f f fdx

ξ− + − =

2xuξ

=tanhu u

u u

e ef ue e

−

−

−= =

+

2

1cosh

dfdu u

=2

2 2

tanh2cosh

d f udu u

= −with

The solution

When u is large( )( )

( )( )

2

2

2 2

2 2

1tanh

1

1 1

1 2 1 2

u u

u u

u u

u x

e ef u

e e

e e

e e ξ

−

−

− −

− −

−= =

+

= − − +

− = −

f∞

Ψ≡

Ψ

xξ

Slope=12

1

tanh2xf u uξ

= =When u is small

Dimensionless GL parameter22 * 2 2

0*

Ce Hm

µ λα −=

22

*2mξ

α=

0*

0 02 2 2C Ce H Hξ

µ λ πµ λΦ

= =

0 *

he

Φ = The fluxoid

20

0

2

4 2

2 2

1 11 1

CH

tt t

πµ λλκξ

= =Φ

−∝ =

− +12

κ <12

κ > type 1 SCWhen type 2 SC

London penetration depth(GL eq 2) ( )

*2*

*

e em

ϕ= ∇ − ΨJ A

0ϕ∇ = for2*

2*

em

= − ΨAJ

2 2* *2 2

* * 20

1e em m µ λ

∇× = − Ψ ∇× = − Ψ = −J A B B

0µ∇× =B J

If

From Ampere’s law( ) 2 2

0µ ∇× = ∇×∇× = ∇ ∇ ⋅ − ∇ = −∇J B B B B

0∇ ⋅ =B

22

1λ

∇ =B B

x

We get the London eq.

ξ

00

x xz

AB B e eλ λ

λ− −= = −

( ) 02

0

xy

AJ x e λ

µ λ−=

Ba

J, A

x ξ

x

λ

ξ

Jy(x)

Bz(x)

0x

zB B e λ−=

( ) 0x

yJ x J e λ−=

tanh2xξ∞Ψ = Ψ 0 ξ λ

Type 2 SC

Effect of small Ψ

Effect of small Ψ

normal

SC

Two length scales, ξ and λ

|Ψ|