1 Superconductivity

8/8/2019 1 Superconductivity

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SUPERCONDUCTIVITY

OVERVIEW OF EXPERIMENTAL FACTSEARLY MODELS

GINZBURG-LANDAU THEORY

BCS THEORY

Jean Delayen

Thomas Jefferson National Accelerator Facility

Old Dominion University

USPAS June 2008 U. Maryland


2/67

Historical Overview


3/67

Perfect Conductivity

Unexpected result

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

Kamerlingh Onnes and van der Waals in

Leiden with the helium 'liquefactor' (1908)


4/67

Perfect Conductivity

Persistent current experiments on rings have measured

1510s

n

s

s>

Perfect conductivity is not superconductivity

Superconductivity is a phase transition

A perfect conductor has an infinite relaxation time L/R

Resistivity < 10-23 .cm

Decay time > 105 years


5/67

Perfect Diamagnetism (Meissner & Ochsenfeld 1933)

0B

t

= 0B =

Perfect conductor Superconductor


6/67

Penetration Depth in Thin Films

Very thin films

Very thick films


7/67

Critical Field (Type I)

2

( ) (0) 1c cc

T H T H

T

-

Superconductivity is destroyed by the application of a magnetic field

Type I or soft superconductors


8/67

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 e

pf =


9/67

Thermodynamic Properties

Entropy Specific Heat

Energy Free Energy


10/67


( ) 1

2

When phase transition at is of order latent heat

At transition is of order no latent heat

jump in specific heat

st

c c

nd

c

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

g

a

=


11/67


3 3

2 200

3

3

: ( ) ( )

(0) 0

33

( ) ( )

( ) ( )

At The entropy is continuous

Recall: and

For

c c

c s c n c

T T

es

c c

s n

c c

c s n

T S T S T

S CS

T TT T T

dt dt C T T T T

T TS T S T T T

T T S T S T

a g ga g

g g

=

= =

fi = = =

= =

< >

Impure metals

Alloys

Local electrodynamics

Pippard superconductors (Type I)

>>

Pure metals Nonlocal electrodynamics


17/67

Material Parameters for Some Superconductors


18/67

Phenomenological Models (1930s to 1950s)

Phenomenological model:

Purely descriptive

Everything behaves as though..

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

At 0K, condensation is complete

At Tc the condensate disappears


19/67

Two Fluid Model Gorter and Casimir

( )

( ) ( ) ( )

( )

( )

( )

1/2

2

2

1- :

( ) = 1

1

2

1

4

gives =

fractionof "normal"electrons

fractionof "condensed"electrons (zero entropy)

Assume: free energy

independent of temperature

Minimizationof

c

n s

n

s c

C

T T x

x

F T x f T x f T

f T T

f T T

TF T x

T

g

b g

< =

+ -

= -

= - = -

( ) ( ) ( )

C

4

4

1/2

3

2

( ) 1 1

T

3 T

n s

C

es

TF T x f T x f T

T

C

b

g

fi = + - = - +

fi =


20/67

Two Fluid Model Gorter and Casimir

( ) ( ) ( )

( )

4

1/2

2

2

2

2

( ) 1 1

( ) 22

8

1

Superconducting state:

Normal state:

Recall difference in free energy between normal and

superconducting state

n s

C

nC

c

C

TF T x f T x f T T

TF T f T T

T

H

T

T

b

g

b

p

b

= + - = - +

= = - = -

=

= -

( )2

2

1(0)

c

c C

H T T

H T

fi = - 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


21/67

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 E t

J en

J n e Et m

J E

u

u

s

= -

= -

=

=


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2

2 2

2

0 = Constant

= 0

Maxwell:

F&H London postulated:

s s

s s

s s

s

s

J n eE

t m

BE

t

m m J B J B

t n e n e

mJ B

n e

=

= -

fi + = fi +

+


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combine with 0 sB = Jm

( ) [ ]

22 0

1

2

2

0

- 0

exp /

s

o L

L

s

n eB B

m

B x B x

m

n e

m

l

lm

=

= -

=

The magnetic field, and the current, decay

exponentially over a distance (a few 10s of nm)


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( ) ( )

1

2

2

0

4

1

4 2

1

10

1

From Gorter and Casimir two-fluid model

L

s

s

C

L L

C

m

n e

Tn

T

T

T

T

lm

l l

=

-

=

-



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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


26/67

Penetration Depth in Thin Films

Very thin films

Very thick films


27/67

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 n

n

e e J r A r r r dr dr

mi mc

A J r

r e J r A r

m e

r n

y y y y y y d

y y

y

r

r

= - - - -

= = =

= -

=

f


28/67

Pippards Extension of Londons 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 toexplain existence of normal and superconducting phase in

intermediate state

Non-local modification of London equation

( ) ( )40

0

1-

3

4

1 1 1

Local:

Non local:

R

J Ac

R R A r e J r d

c R

x

l

s upx l

x x

-

=

= -

= +

i


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London and Pippard KernelsApply Fourier transform to relationship between

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

andc

J r A r J k K k A k p

= -

( ) ( )22

2

ln 1

Specular: Diffuse:eff eff o

o

dk

K k k K kdk

k

pl l

p

= =

+ +

Effective penetration depth


30/67

London Electrodynamics

Linear London equations

together with Maxwell equations

describe the electrodynamics of superconductors at all T if: The superfluid density nsis spatially uniform

The current density Js is small

2

2 2

0

10s

J EH H

t l m l

= - - =

0s

H H J E t

m = = -


31/67

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


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Ginzburg-Landau theory is a particular case ofLandaus 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 thebehavior of superconductors and is still one of themost widely used theory of superconductivity

G


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Theory of second order phase transition is based onan order parameter which is zero above the transition

temperature and non-zero below

For superconductors, GL use a complex order

parameter(r) such that |(r)|2

represents thedensity of superelectrons

The Ginzburg-Landau theory is valid close toTc


34/67

Ginzburg-Landau Equation for Free Energy

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

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

derivative

2

22 4

0

1

2 2 8n

e hf fm i c

ba y y y p

**

= + + + - + A

Fi ld F U if C


35/67

Field-Free Uniform Case

Near Tc we must have

At the minimum

2 4

02

nf f ba y y- = +0

f fn

-0

f fn

-

0> 0 = -

2 22

0(1 )

8 2

andcn c

H f f H t

ay

p b

- = - = - fi -

2 ay

b = -

Fi ld F U if C


36/67


At the minimum

2 4

02

nf f ba y y- = +

20 ( ) ( 1) (1 )t t t b a a y > = - fi -

22

0

2 8(1 )

(definition of )cn c

c

H f f H

H t

a

b p

- = - = -

fi -

2 ayb

= -

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

superelectrons

2 (1 ) cnear Tsn tl- -

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

Fi ld F U if C


37/67


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

TTn

T T n

H TT

H T H T T Tn T n

l al l b

ab p

l la bp l p l

Y= Y fi = = -Y

=

= - =

Fi ld F N if C


38/67

Field-Free Nonuniform Case

Equation of motion in the absence of electromagnetic field

221 ( ) 0

2

T

m

y a y b y y *- + + =

Look at solutions close to the constant one2 ( )

whereTa

y y d y

b = + = -

To first order:21 0

4 ( )m Td d

a* - =

Which leads to 2 / ( )r Texd -

Field Free Non niform Case


39/67

Field-Free Nonuniform Case

2 / ( )

2

(0)1 2( )

( ) ( )2 ( )where

r T L

c L

ne T

m H T T m T

x lpd xla

-**

= =

is the Ginzburg-Landau coherence length.

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

0

1/22

( )1

Tt

xx

-

GL parameter:( )

( )( )

LT

TT

lk

x=

( ) ( )

( )

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

2 Fundamental Lengths


40/67

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

c

L

c

TmT

e T T

bl

a

* = -

1/22

( ) 4c

c

TT m T Tx a*

= -

The critical field is directly related to those 2 parameters

0( )2 2 ( ) ( )

c

L

H TT T

f

x l=


41/67

Surface Energy

2 2

2

2

1

8

: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

l

p

x

p

-

S f1


42/67

Surface Energy

2 2

c

Interface is stable if >0

If >0

Superconducting up to H where superconductivity is destroyed globally

If >> >

>

1

:2

1:

2

area to volume ratio

quantized fluxoids

More exact calculation (from Ginzburg-Landau):

= Type I

= Type II

l

k x

lk

x

2 21

8c

H Hs x lp

-

Magnetization Curves


43/67

Magnetization Curves

Intermediate State


44/67

Intermediate State

Vortex lines in Pb.98In.02

At the center of each vortex is a

normal region of flux h/2e

C iti l Fi ld


45/67

Critical Fields

2

2

1

2

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

c

sh

c

c c

c

c

c

c

H

HH

H

H H

HH

H

H

k

k

k k

k

=

+

Even though it is more energetically favorable for a type I superconductor torevert 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


46/67

Superheating Field

0.9

1

Ginsburg-Landau:

for

The exact nature of the rf critical

field of superconductors is stillan open question


47/67

Material Parameters for Some Superconductors

BCS


48/67

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


49/67

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 overlapping

Cooper pairs into a coherent

Superconducting state

Dw

Cooper Pairs


50/67

Cooper Pairs

One electron moving through the lattice attracts the positive ions.Because of their inertia the maximum displacement will take place

behind.

BCS


51/67

BCS

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

They form a coherent state

BCS and BEC


52/67

BCS and BEC

BCS Theor


53/67

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

0

q kif and

otherwise

D DV x w x w -

=

BCS


54/67

BCS

Hamiltonian

Ground state wave function

destroys an electron of momentumcreates an electron of momentum

number of electrons of momentum

k k qk q q k k

k qk

k

q

k k k

n V c c c c

c k

c k

n c c k

e * *- -

*

*

= +

=

H

( )0q q q q

q

a b c c f* *-Y = +P

BCS


55/67

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 veryfew adjustable parameters) are amazingly close to the real

world

BCS


56/67

BCS

( )

( )

2

2 2

0

1

0

0

0, 1 0

1, 0 0

2 1

21

sinh

0

Is there a state of lower energy than

the normal state?

for

for

yes:

where

q q q

q q q

q

q

q

VDD

a b

a b

b

e

V

r

x

x

x

x

ww

r

-

= =

= -

+ D

D =

BCS


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BCS

( )( )

11.14 exp

0 1.76

c D

F

c

kT

VN E

kT

w

= -

D =

Critical temperature

Coherence length (the size of the Cooper pairs)

0 .18F

ckT

ux =

BCS Condensation Energy


58/67

BCS Condensation Energy

( ) 20

2

0 00

0

4

0

2

8

/ 10

/ 10F

Condensation energy: s n

F

V

E E

HN

k K

k K

r

e p

e

D

- = - D

- D =

D

BCS Energy Gap


59/67

BCS Energy Gap

At finite temperature:Implicit equation for the temperature dependence of the gap:

( )

( )

( )

1 22 2

1 22 20

tanh / 21

0

D kT

dV

w e

er e

+ D = + D

BCS Excited States


60/67

BCS Excited States

0

2 22k

Energy of excited states:

ke x= + D

BCS Specific Heat


61/67

BCS Specific Heat

10

Specific heat

exp for < ces

TC T

kT

D -

Electrodynamics and Surface Impedancei BCS M d l


62/67

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

R

l

H H i t

e H A r t p

mc

HH H

R R A I R T e J dr

R

f

f f

w-

+ =

=


63/67

Penetration Depth

( ) 2

4

2( )

c

c

specular

1Represented accurately by near

1-

dkdk

K k k

T

T

T

lp

l

= +

Surface Resistance


64/67

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

kT

Frequency dependenceis a good approximation

c

c

s

tT t

t

TT

AR

T

l

w

w

D

--

-

1 Superconductivity

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