SUPERCONDUCTIVITY

OVERVIEW OF EXPERIMENTAL FACTSEARLY MODELS

GINZBURG-LANDAU THEORYBCS THEORY

Jean Delayen

Thomas Jefferson National Accelerator FacilityOld Dominion University

USPAS June 2008 U. Maryland

Historical Overview

Perfect Conductivity

Unexpected result

Expectation was the opposite: everything should become an isolator at 0T Æ

Kamerlingh Onnes and van der Waals in Leiden with the helium 'liquefactor' (1908)

Perfect Conductivity Persistent current experiments on rings have measured

1510s

n

ss

>

Perfect conductivity is not superconductivity

Superconductivity is a phase transition

A perfect conductor has an infinite relaxation time L/R

Resistivity < 10-23 Ω.cm

Decay time > 105 years

Perfect Diamagnetism (Meissner & Ochsenfeld 1933)

0Bt

∂ =∂

0B =

Perfect conductor Superconductor

Penetration Depth in Thin Films

Very thin films

Very thick films

Critical Field (Type I)

2

( ) (0) 1c cc

TH T HT

È ˘Ê ˆÍ ˙- Á ˜Ë ¯Í ˙Î ˚

Superconductivity is destroyed by the application of a magnetic field

Type I or “soft” superconductors

Critical Field (Type II or “hard” superconductors)

Expulsion of the magnetic field is complete up to Hc1, and partial up to Hc2

Between Hc1 and Hc2 the field penetrates in the form if quantized vortices or fluxoids

0 epf =

Thermodynamic Properties

Entropy Specific Heat

Energy Free Energy

Thermodynamic Properties

( ) 1

2

When phase transition at is of order latent heat

At transition is of order no latent heat jump in specific heat

stc c

ndc

es

T T H H T

T T

C

< = fi

= fi

3

( ) 3 ( )

( )

( )

electronic specific heat reasonable fit to experimental data

c en c

en

es

T C T

C T T

C T T

ga

=

ª

∼

Thermodynamic Properties

3 3

2 200

3

3

: ( ) ( )

(0) 0

3 3

( ) ( )

( ) ( )

At The entropy is continuous

Recall: and

For

c c

c s c n c

T T

esc c

s nc c

c s n

T S T S T

S CST T

T T Tdt dt CT T T T

T TS T S TT T

T T S T S T

a g ga g

g g

=

∂= =∂

fi = Æ = =

= =

< <

ÚÚ

superconducting state is more ordered than normal state

A better fit for the electron specific heat in superconducting state is

9, 1.5 with for cbT

Tes c cC a T e a b T Tg

-= ª ª

Energy Difference Between Normal and Superconducting State

( )4 4 2 22

( ) ( )3( ) ( ) ( ) ( )4 2

Εnergy is continuousc

n c s c

T

n s es en c cTc

U T U T

U T U T C C dt T T T TTg g

=

- = - = - - -Ú

( ) ( )2

210 0 04 8

at cn s c

HT U U Tgp

= - = =2

8is thecondensationenergycH

p2

0,8

at is the free energy differencecHTp

π

( ) ( )222

2( ) 1 18 4c

n s n c cc

H T TF U U T S S TT

gp

È ˘Ê ˆÍ ˙= D = - - - = - Á ˜Ë ¯Í ˙Î ˚

( )21

2( ) 2 1c cc

TH T TT

pgÈ ˘Ê ˆÍ ˙= - Á ˜Ë ¯Í ˙Î ˚

The quadratic dependence of critical field on T is related to the cubic dependence of specific heat

Isotope Effect (Maxwell 1950)

The critical temperature and the critical field at 0K are dependent on the mass of the isotope

(0) with 0.5c cT H M a a-∼ ∼

Energy Gap (1950s)

At very low temperature the specific heat exhibits an exponential behavior

Electromagnetic absorption shows a threshold

Tunneling between 2 superconductors separated by a thin oxide film shows the presence of a gap

/ 1.5 with cbT Tsc e b-μ

Two Fundamental Lengths• London penetration depth λ

– Distance over which magnetic fields decay in superconductors

• Pippard coherence length ξ– Distance over which the superconducting state decays

Two Types of Superconductors

• London superconductors (Type II)– λ>> ξ– Impure metals– Alloys– Local electrodynamics

• Pippard superconductors (Type I)– ξ >> λ– Pure metals– Nonlocal electrodynamics

Material Parameters for Some Superconductors

Phenomenological Models (1930s to 1950s)

Phenomenological model:Purely descriptiveEverything behaves as though…..

A finite fraction of the electrons form some kind of condensate that behaves as a macroscopic system (similar to superfluidity)

At 0K, condensation is complete

At Tc the condensate disappears

Two Fluid Model – Gorter and Casimir

( )

( ) ( ) ( )

( )

( )

( )

1/2

2

2

1- :

( ) = 112

14

gives =

fractionof "normal"electronsfractionof "condensed"electrons (zero entropy)

Assume: free energy

independent of temperature

Minimizationof

c

n s

n

s c

C

T T xx

F T x f T x f T

f T T

f T T

TF T xT

g

b g

< =

+ -

= -

= - =-

ÊË

( ) ( ) ( )

C

4

41/2

3

2

( ) 1 1

T3T

n sC

es

TF T x f T x f TT

C

b

g

ˆÁ ˜

È ˘Ê ˆÍ ˙fi = + - = - + Á ˜Ë ¯Í ˙Î ˚

fi =

Two Fluid Model – Gorter and Casimir

( ) ( ) ( )

( )

41/2

22

2

2

( ) 1 1

( ) 22

8

1

Superconducting state:

Normal state:

Recall difference in free energy between normal and

superconducting state

n sC

nC

c

C

TF T x f T x f TT

TF T f T TT

H

TT

b

g b

p

b

È ˘Ê ˆÍ ˙= + - = - + Á ˜Ë ¯Í ˙Î ˚

Ê ˆ= = - = - Á ˜Ë ¯

=

Ê ˆ= - Á ˜Ë ¯

( )2 2

1(0)

c

c C

H T TH T

È ˘ Ê ˆÍ ˙ fi = - Á ˜Ë ¯Í ˙Î ˚

The Gorter-Casimir model is an “ad hoc” model (there is no physical basis for the assumed expression for the free energy) but provides a fairly accurate representation of experimental results

Model of F & H London (1935)

Proposed a 2-fluid model with a normal fluid and superfluid components

ns : density of the superfluid component of velocity vsnn : density of the normal component of velocity vn

2

superelectrons are accelerated by

superelectrons

normal electrons

s s

s s

n n

m eE Et

J en

J n e Et m

J E

u

u

s

∂ = -∂= -

∂=

∂

=

Model of F & H London (1935)

2

2 2

2

0 = Constant

= 0

Maxwell:

F&H London postulated:

s s

s ss s

ss

J n e Et m

BEt

m mJ B J Bt n e n e

m J Bn e

∂=

∂

∂—¥ = -∂

Ê ˆ∂fi —¥ + = fi —¥ +Á ˜∂ Ë ¯

— ¥ +

Model of F & H London (1935)combine with 0 sB = Jm—¥

( ) [ ]

22 0

12

20

- 0

exp /

s

o L

Ls

n eB B

m

B x B x

mn e

m

l

lm

— =

= -

È ˘= Í ˙Î ˚

The magnetic field, and the current, decay exponentially over a distance λ (a few 10s of nm)

( ) ( )

12

20

4

14 2

1

10

1

From Gorter and Casimir two-fluid model

Ls

sC

L L

C

mn e

TnT

T

TT

lm

l l

È ˘= Í ˙Î ˚

È ˘Ê ˆÍ ˙μ - Á ˜Ë ¯Í ˙Î ˚

=È ˘Ê ˆÍ ˙- Á ˜Ë ¯Í ˙Î ˚

Model of F & H London (1935)

Model of F & H London (1935)

2

0

2

0, 0

1

London Equation:

choose on sample surface (London gauge)

Note: Local relationship between and

s

n

s

s

BJ H

A H

A A

J A

J A

lm

l

—¥ = - = -

—¥ =

— = =

= -

i

Penetration Depth in Thin Films

Very thin films

Very thick films

Quantum Mechanical Basis for London Equation

( ) ( ) ( )

( )

( ) ( ) ( )

( )

2* * *

1

0

2

2

0 0 ,In zero field

Assume is "rigid", ie the field has no effect on wave function

n n n n nn

e eJ r A r r r dr drmi mc

A J r

r eJ r A r

mer n

y y y y y y d

y y

y

r

r

Ï ¸È ˘= — - — - - -Ì ˝Î ˚Ó ˛

= = =

= -

=

ÂÚ

Pippard’s Extension of London’s ModelObservations:

-Penetration depth increased with reduced mean free path

- Hc and Tc did not change

- Need for a positive surface energy over 10-4 cm to explain existence of normal and superconducting phase in intermediate state

Non-local modification of London equation

( ) ( )4

0

0

1-

34

1 1 1

Local:

Non local:

R

J Ac

R R A r eJ r d

c R

x

l

s upx l

x x

-

=

È ˘¢Î ˚=-

= +

Úi

London and Pippard KernelsApply Fourier transform to relationship between

( ) ( ) ( ) ( ) ( ):4

and cJ r A r J k K k A kp

= -

( ) ( )2

2

2

ln 1Specular: Diffuse:eff eff

o

o

dkK k k K k

dkk

pl lp

•

•= =

+ È ˘+Í ˙

Î ˚

ÚÚ

Effective penetration depth

London Electrodynamics

Linear London equations

together with Maxwell equations

describe the electrodynamics of superconductors at all T if:– The superfluid density ns is spatially uniform– The current density Js is small

22 2

0

1 0sJ E H Ht l m l

∂= - — - =

∂

0sHH J Et

m ∂—¥ = —¥ = -∂

Ginzburg-Landau Theory

• Many important phenomena in superconductivity occur because ns is not uniform– Interfaces between normal and superconductors– Trapped flux– Intermediate state

• London model does not provide an explanation for the surface energy (which can be positive or negative)

• GL is a generalization of the London model but it still retain the local approximation of the electrodynamics

Ginzburg-Landau Theory

• Ginzburg-Landau theory is a particular case of Landau’s theory of second order phase transition

• Formulated in 1950, before BCS

• Masterpiece of physical intuition

• Grounded in thermodynamics

• Even after BCS it still is very fruitful in analyzing the behavior of superconductors and is still one of the most widely used theory of superconductivity

Ginzburg-Landau Theory

• Theory of second order phase transition is based on an order parameter which is zero above the transition temperature and non-zero below

• For superconductors, GL use a complex order parameter Ψ(r) such that |Ψ(r)|2 represents the density of superelectrons

• The Ginzburg-Landau theory is valid close to Tc

Ginzburg-Landau Equation for Free Energy

• Assume that Ψ(r) is small and varies slowly in space

• Expand the free energy in powers of Ψ(r) and its derivative

2 22 4

01

2 2 8ne hf f

m i cba y y y

p

*

*

Ê ˆ= + + + — - +Á ˜Ë ¯

A

Field-Free Uniform Case

Near Tc we must have

At the minimum

2 40 2nf f ba y y- = +

0f fn- 0f fn-

•

0 > 0<

0 ( ) ( 1)t tb a a> = -¢

2 22

0 (1 )8 2

and cn c

Hf f H ta yp b

- = - = - fi μ -

2 ayb• = -

Field-Free Uniform Case

At the minimum

2 40 2nf f ba y y- = +

20 ( ) ( 1) (1 ) t t tb a a y•> = - fi μ -¢

22

0 2 8(1 )

(definition of )cn c

c

Hf f H

H t

ab p

- = - = -

fi μ -

2 ayb• = -

It is consistent with correlating |Ψ(r)|2 with the density of superelectrons

2 (1 ) cnear Tsn tl-μ μ -

which is consistent with ( )20 1c cH H t= -

Field-Free Uniform Case

Identify the order parameter with the density of superelectrons

222

22 2

22

2 22 42

2 4

( )(0)1 1 ( )( ) ( ) (0)

( )1 ( )2 8

( ) ( )( ) ( )( )4 (0) 4 (0)

since

and

Ls

L L

c

c cL L

L L

T TnT T n

H TT

H T H TT Tn T n

l al l b

ab p

l la bp l p l

Y= Y fi = = -

Y

=

= - =

∼

Field-Free Nonuniform Case

Equation of motion in the absence of electromagnetic field

221 ( ) 02

Tm

y a y b y y*- — + + =

Look at solutions close to the constant one2 ( )where Tay y d y

b• •= + = -

To first order:21 0

4 ( )m Td d

a* — - =

Which leads to 2 / ( )r Te xd -ª

Field-Free Nonuniform Case

2 / ( )2

(0)1 2( )( ) ( )2 ( )

where r T L

c L

ne Tm H T Tm T

x lpd xla

-**

ª = =

is the Ginzburg-Landau coherence length.

It is different from, but related to, the Pippard coherence length. ( )0

1/22( )

1T

t

xx-

GL parameter:( )( )( )

L TTT

lkx

=

( ) ( )

( )

Both and diverge as but their ratio remains finite

is almost constant over the whole temperature range

L cT T T T

T

l x

k

Æ

2 Fundamental Lengths

London penetration depth: length over which magnetic field decay

Coherence length: scale of spatial variation of the order parameter (superconducting electron density)

1/2

2( )2

cL

c

TmTe T Tbla

*Ê ˆ= Á ˜ -¢Ë ¯

1/22

( )4

c

c

TTm T T

xa*

Ê ˆ= Á ˜ -¢Ë ¯

The critical field is directly related to those 2 parameters

0( )2 2 ( ) ( )c

L

H TT Tf

x l=

Surface Energy

2 2

2

2

18

:8

:8

Energy that can be gained by letting the fields penetrate

Energy lost by "damaging" superconductor

c

c

H H

H

H

s x lp

lpxp

È ˘-Î ˚

Surface Energy

2 2

c

Interface is stable if >0If >0

Superconducting up to H where superconductivity is destroyed globally

If >> <0 for

Advantageous to create small areas of normal state with large

cH H

sx l s

xl x sl

>>

>

1 :2

1 :2

area to volume ratio quantized fluxoids

More exact calculation (from Ginzburg-Landau):

= Type I

= Type II

lkxlkx

Æ

<

>

2 218 cH Hs x lpÈ ˘-Î ˚

Magnetization Curves

Intermediate State

Vortex lines in Pb.98In.02

At the center of each vortex is a normal region of flux h/2e

Critical Fields

2

2

12

2

1 (ln .008)2

Type I Thermodynamic critical field

Superheating critical field

Field at which surface energy is 0

Type II Thermodynamic critical field

(for 1)

c

csh

c

c c

cc

c

c

HHH

H

H H

HHH

H

k

k

k kk

=

+

Even though it is more energetically favorable for a type I superconductor to revert to the normal state at Hc, the surface energy is still positive up to a superheating field Hsh>Hc → metastable superheating region in which the material may remain superconducting for short times.

Superheating Field

0.9

1

Ginsburg-Landau:

for <<1

1.2 for 0.75 for 1

csh

c

c

HH

HH

kk

kk >>

∼

∼ ∼∼

The exact nature of the rf critical field of superconductors is still an open question

Material Parameters for Some Superconductors

BCS

• What needed to be explained and what were the clues?

– Energy gap (exponential dependence of specific heat)

– Isotope effect (the lattice is involved)

– Meissner effect

Cooper Pairs

Assumption: Phonon-mediated attraction between electron of equal and opposite momenta located within of Fermi surface

Moving electron distorts lattice and leaves behind a trail of positive charge that attracts another electron moving in opposite direction

Fermi ground state is unstable

Electron pairs can form bound states of lower energy

Bose condensation of overlappingCooper pairs into a coherentSuperconducting state

Dw

Cooper PairsOne electron moving through the lattice attracts the positive ions.

Because of their inertia the maximum displacement will take place

behind.

BCS

The size of the Cooper pairs is much larger than their spacing

They form a coherent state

BCS and BEC

BCS Theory

( )

0 , 1 ,-

, : ( ,- )

0 1

:states where pairs ( ) are unoccupied, occupied

probabilites that pair is unoccupied, occupied

BCS ground state

Assume interaction between pairs and

q q

q q

q qq qq

qk

q q

a b q q

a b

q k

V

Y = +

=

P

0q k if and

otherwiseD DV x w x w- £ £

=

BCS

• Hamiltonian

• Ground state wave function

destroys an electron of momentum creates an electron of momentum

number of electrons of momentum

k k qk q q k kk qk

k

q

k k k

n V c c c c

c k

c k

n c c k

e * *- -

*

*

= +

=

 ÂH

( ) 0q q q qqa b c c f* *

-Y = +P

BCS

• The BCS model is an extremely simplified model of reality– The Coulomb interaction between single electrons is

ignored– Only the term representing the scattering of pairs is

retained– The interaction term is assumed to be constant over a

thin layer at the Fermi surface and 0 everywhere else– The Fermi surface is assumed to be spherical

• Nevertheless, the BCS results (which include only a very few adjustable parameters) are amazingly close to the real world

BCS

( )

( )

2

2 20

10

0

0, 1 0

1, 0 0

2 1

21sinh0

Is there a state of lower energy than the normal state?

forfor

yes:

where

q q q

q q q

qq

q

VDD

a b

a b

b

e

V

r

xx

x

x

w w

r

-

= = <

= = >

= -+ D

D =È ˘Í ˙Î ˚

BCS

( )( )

11.14 exp

0 1.76

c DF

c

kTVN E

kT

wÈ ˘

= -Í ˙Î ˚

D =

Critical temperature

Coherence length (the size of the Cooper pairs)

0 .18 F

ckTux =

BCS Condensation Energy

( ) 20

20 0

0

0

4

02

8/ 10

/ 10F

Condensation energy: s n

F

VE E

HN

k K

k K

r

e p

e

D- = -

Ê ˆD- D =Á ˜Ë ¯

D

BCS Energy GapAt finite temperature:

Implicit equation for the temperature dependence of the gap:

( )( )( )

1 22 2

1 22 20

tanh / 210

D kTd

V

w ee

r e

È ˘+ DÍ ˙Î ˚=+ DÚ

BCS Excited States

0

2 22k

Energy of excited states:

ke x= + D

BCS Specific Heat

10

Specific heat

exp for < ces

TC T

kTDÊ ˆ-Á ˜Ë ¯

Electrodynamics and Surface Impedance in BCS Model

( )

[ ] ( )

0

4

,

, ,

There is, at present, no model for superconducting sur

is treated as a small pe

face resistance at high rf

rturbation

similar to Pippar

field

ex

ex i i

ex

rf c

Rl

H H it

eH A r t pmc

HH H

R R A I R T eJ dr

R

ff f

w-

∂+ =∂

=

<<

◊μ

Â

Ú( ) ( ) ( )( )

40 0 :

d's model

Meissner effect

cJ k K k A k

Kp

= -

π

Penetration Depth

( ) 2

4

2 ( )

c

c

specular

1Represented accurately by near

1-

dk dkK k k

TTT

lp

l

=+

Ê ˆÁ ˜Ë ¯

Ú

∼

Surface Resistance

( )( )

4

32 2

2

:1

:2

exp

- kT

2

Temperature dependence

close to dominated by change in

for dominated by density of excited states e

kTFrequency dependence

is a good approximation

c

c

s

tT tt

TT

ART

l

w

w

D

--

- <

DÊ ˆ-Á ˜Ë ¯

∼

∼

Surface Resistance

Surface Resistance

Surface Resistance