1 PHY 712 Electrodynamics 9-9:50 AM MWF Olin 103 Plan for Lecture 34: Special Topics in Electrodynamics: Electromagnetic aspects of superconductivity

PHY 712 Spring 2015 -- Lecture 34 1

PHY 712 Electrodynamics9-9:50 AM MWF Olin 103

Plan for Lecture 34:

Special Topics in Electrodynamics:

Electromagnetic aspects of superconductivity

London equations

Tunneling between two superconductors

04/20/2015

PHY 712 Spring 2015 -- Lecture 34 204/20/2015


London model of superconducting state

0

2

/

Recall Drude model of conductivity in "normal" materials

(

)

for

t

dm e mdt

et e

m

nene t

m

v vE

Ev v

J v J E E

2

Suppose London model of conductivity in superconducting materials

From Maxwell's equations:

4 1 1

0;

dm edt

d e d d nene

dt m dt dt m

t tc c c

vE

v E J

E

E

B

v

BJ

E


London model, continued

2

22

2 2

2

London model of conductivity in superconducting materials

From Maxwell's equations:

4 1 1

4 1

4

d d nene

dt dt m

c c c

c

t t

t

t

c

c

J v

E BB E

BB B

B

E

J

J

J 3

2

2 32

2

2 3

3

3

32

2 2

22 2

2

2 222

1

4 1

4 1

0 wi1

1

h4

t LL

c

mc c

mc c

m

t t

ne

t t

ne

t t t

t t

c

c ne

B

B BE

B B B

B


London model – continued

2

22 2

2 2 2

2

2

2

2

0 with

0

London model of conductivity in superconducting materials

1 1

4

For slowly varying solution:

1 for

LL

L

d d nene

dt dt m

t t ne

mc

c

t

B

B

J v E

B

/

/

( , )ˆ :

( , ) (0, )

( , ) (0, )London leap:

L

L

z

xz z

xz z

B x t

t t

B x t B te

t t

B x t B t e

z

2x/2

2

Consistent results for current densit

( ) ( ) B (0)e

y:

4 1ˆ L

y y L zL

J Jc

nex x

mc

B JBJ y


Behavior of superconducting material – exclusion of magnetic field according to the London model

22

2

/

/

4

Vector po

Penetration len

tential for

gth for superconductor:

( , ) (0, )

0 :

ˆ ( ) ( ) ( 0)

L

L

L

xz z

xy y L z

ne

B x t B t e

A

m

A x B e

c

x

A

A y2

x/

2

( ) B (0)e Current

0 or

de

=0

nsity: Ly L z

nex

mc

ne ne em

mc m

J

c

J A v A

x

lL

7Typically, 10L m


Magnetization field

0

,

Gibbs fr

Treating Lond

ee energy ass

on current in terms of cor

ociated with magnetization

responding magnetization

for superconductor

field :

=

:

1( ) ( 0) ( ) (0

4

For

)8

4

a

L

H

S a S SG H G H dHM H G

x

M

B H

H

M

M

2

2

Gibbs free energy associated with magnetization for normal conductor:

( ) ( 0)

Condition at phase boundary between normal and superconducting states:

1( ) (0) ( ) (0)

8

(0) (0

N N

N N C C

N

a

a

C S S

S

H

G H G H

G H G G H G H

G G

2

2 2

1)

81

( ) ( ) 8for

0 for

C

C a CS a N a

a C

a

H

H H HH G

H

HHG

H

(At T=0K)


Magnetization field (for “type I” superconductor at T=0K)

H

B

HC

-4pM

HC

GS-GN

HC

H

H

Inside superconductor

=0= +4 for < CH HMB H


Temperature dependence of critical field2

( ) (0) 1c cc

T HHT

T

From PR 108, 1175 (1957)

Bardeen, Cooper, and Schrieffer, “Theory of Superconductivity”

2/( ( ) )FN E VcT e

k

characteristic phonon energy

density of electron states at EF

attraction potential between electron pairs


Type I elemental superconductorshttp://wuphys.wustl.edu/~jss/NewPeriodicTable.pdf

http://wuphys.wustl.edu/~jss/NewPeriodicTable.pdf

http://wuphys.wustl.edu/~jss/NewPeriodicTable.pdf


Quantization of current flux associated with the superconducting state (Ref: Ashcroft and Mermin, Solid State Physics)

=0

Now suppose that the

From the London equations for the interior of t

current carrier is a pair of electrons charact

he supercondu

erized

by a wavefunction of

cto

th

r:

em

c v A

e form ie

2

2* *

22

The quantum mechanical current associated with the electron pair is

2=

2

2 =

e e

mi mc

e e

m mc

j A

A


Quantization of current flux associated with the superconducting state -- continued

dl

Suppose a superconducting material has a cylindrical void. Evaluate the integral of the current in a closed path within the superconductor containing the void.

22

0

magnetic flux

for some integer

Quantization

2

of flux in the void

0

2

: 2

d d

d d

e e

m m

d

d n n

hc

c

n ne

j l A

l a BA

l

A a

l

Such “vortex” fields can exist within type II superconductors.


Josephson junction -- tunneling current between two superconductors (Ref. Teplitz, Electromagnetism (1982))

d

Bz

x


Josephson junction -- continued

Bz

Supercon left Supercon rightJunction

d

( /2)/0

0( /2)/

0

/ 2

( ) / 2 / 2

/ 2

L

L

x d

zx d

B e x d

x B d x d

B e x d

B



( /2)/0

0

( /2)/0

/ 2

( ) / 2

2

/ /2

/ 2

2

/L

L

x dL L

x dL

y

L

d

A x

B e x d

x B d x d

B e x dd

Ay

Supercon left Supercon rightJunction

d


Josephson junction -- continuedd

x

L R

0

0

Quantum mechanical model of tunnelling current

Let denote a wavefunction for a Cooper pair on left

Let denote a wavefunction for a Cooper pair on rightR

LiL

iR R

R

R

L

LL L

e

e

i E

i

t

t

R R LE Coupling parameter



202 20 0 0 0

202 20 0 0 0

Solving for wavefunctions

1

2

1

2

R L

R L

L

L L L LiL

R LR

R R

R

R iR

t

ii E e

ii E e

t

t t

2 20 0

2( ) sin

cos

cos

R

R

R R L

L L R LR L R

LL R LR

L L RLR

L

LRR

n nn n

t t

n

t n

n

n n

E

E

t n



2 ( ) si4

Tunneling current:

If = and in absense of magnetic field, ( )

n

(0)

LT L R LR

L LR LR L

R R

eJ

En n

ne n nt

Et t

x

L R

JL JR

JT

0

Relationship between superconductor currents and

and tunneling current. Within the superconductor, denote the

generalize current operator acting on pair wavefunction

1 2ˆ

2

L

i

RJ J

ei

m c

e

v A **2ˆ ˆ with current

2

eJ

v v

20

20

2 2

2

2 2

2R R

L L L

R

e eJ

m c

e eJ

m c

A

A



x

L R

JL JR

JT

20

20

2

2

2

2 2

2

2 2

2

2 2

2

R

L L L L L

R

L

R R R

RRL

e eJ

m c

e eJ

m

en

c

e e

c c

en

m m

v

v

A

Av

Av

A

4

Tunneling current:

Need to evaluate in presence of magnetic field

2 ( ) sinLT L R LR

LR

ne n n

eJ

t


2 2

2 2 R

RL

L

m

c

m e e

c

v vA A

0

0

Recall that for / 2

fo

ˆ 0 and

ˆ 0r an / 2d

L L

LR

x d

x d

B

B

v A y

v A y


x

L R

JL JR

JT

4Tunneling current: ( ) sinT L R LR

eJ n n

00

Integrating the difference of the phase angles alon

)

:

2

(

g

LR LR L y

y

B d



x

L R

JL JR

JT

/2

/2

00

0 0

00

4Tunneling current density:

Integrating current density throughout width of superconductors

sin

sin

/ ))

/

(sin

2(2 ) and

(

where 2

T L LR

T T T LR

w

L

w

en

h dy

cB w d

e

J

w

I J hwJ

00

Integrating the difference of the phase angles along

2

:

(

2)

LR LR L de

cB y

y



IT

/F F0

Note: This very sensitive “SQUID” technology has been used in scanning probe techniques. See for example, J. R. Kirtley, Rep. Prog. Physics 73, 126501 (2010).

SQUID =superconducting quantum interference device


Scanning SQIUD microscopy Ref. J. R. Kirtley, Rep. Prog. Phys. 73 126501 (2010)


1 PHY 712 Electrodynamics 9-9:50 AM MWF Olin 103 Plan for Lecture 34: Special Topics in Electrodynamics: Electromagnetic aspects of superconductivity

Documents

josephson junction

continued itff0note

continued xlrjljrjt

superconducting state

magnetization field

london model xll

quantization of current

solid state physics