PHY 712 Electrodynamics 10-10:50 AM MWF Olin 107 Plan for Lecture 32:

PHY 712 Spring 2014 -- Lecture 32 1

PHY 712 Electrodynamics10-10:50 AM MWF Olin 107

Plan for Lecture 32:Special Topics in Electrodynamics:

Electromagnetic aspects of superconductivity -- continued

04/14/2014

PHY 712 Spring 2014 -- Lecture 32 204/14/2014

PHY 712 Spring 2014 -- Lecture 32 304/14/2014

PHY 712 Spring 2014 -- Lecture 32 404/14/2014

PHY 712 Spring 2014 -- Lecture 32 504/14/2014

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

neB x t B t e

A

m

A x B e

c

x

AA y

2x/

2

( ) B (0)e Current

0 or

de

=0

nsity: Ly L z

nexmc

ne ne emmc m

J

c

J A v A

xL

7Typically, 10L m

PHY 712 Spring 2014 -- Lecture 32 604/14/2014

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

4For

)8

4

a

L

H

S a S SG H G H dHM H G

x

MB H

HM

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)8

1( ) ( ) 8

for

0 for

C

C a CS a N a

a C

a

H

H H HH G

H

HHG

H

PHY 712 Spring 2014 -- Lecture 32 704/14/2014

Magnetization field (for “type I” superconductor)

H

B

HC-4M

HCGS-GN

HC

H

H

PHY 712 Spring 2014 -- Lecture 32 804/14/2014

Examples of type I superconductors http://hyperphysics.phy-astr.gsu.edu/hbase/tables/supcon.html#c1

PHY 712 Spring 2014 -- Lecture 32 904/14/2014

PHY 712 Spring 2014 -- Lecture 32 1004/14/2014

Josephson junction -- tunneling current between two superconductors

d

Bz

x

PHY 712 Spring 2014 -- Lecture 32 1104/14/2014

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 dx B d x d

B e x dB

PHY 712 Spring 2014 -- Lecture 32 1204/14/2014

Josephson junction -- continued

( /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

PHY 712 Spring 2014 -- Lecture 32 1304/14/2014

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

PHY 712 Spring 2014 -- Lecture 32 1404/14/2014

Josephson junction -- continued

202 20 0 0 0

202 20 0 0 0

Solving for wavefunctions

12

12

R L

R L

LL L L L

iL

R LR

R R

R

R iR

tii 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 n n nt t

nt n

n

n n

E

Et n

PHY 712 Spring 2014 -- Lecture 32 1504/14/2014

Josephson junction -- continued

2 ( ) si4Tunneling 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 JRJT

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

eim c

e

v A **2 ˆ ˆ with current 2eJ

v v

20

20

2 222 22R R

L L L

R

e eJm ce eJm c

A

A

PHY 712 Spring 2014 -- Lecture 32 1604/14/2014

Josephson junction -- continued

x

L R

JL JRJT

20

20

2

2

2

2 222 22

2 2 2

R

L L L L L

R

L

R R R

RRL

e eJm ce eJm

en

ce ec c

en

m m

v

v

A

AvAv

A

4Tunneling current:

Need to evaluate in presence of magnetic field

2 ( ) sinLT L R LR

LR

ne n neJt

PHY 712 Spring 2014 -- Lecture 32 1704/14/2014

2 2 2 2 RR

LL

mc

m e ec

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

Josephson junction -- continued

x

L R

JL JRJT

4Tunneling current: ( ) sinT L R LReJ n n

00

Integrating the difference of the phase angles alon

)

:

2

(

g

LR LR L y

y

B d

PHY 712 Spring 2014 -- Lecture 32 1804/14/2014

Josephson junction -- continued

x

L R

JL JRJT

/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 LRw

L

w

e n

h dy

cB w de

J

w

I J hwJ

00

Integrating the difference of the phase angles along

2

:

(

2 )

LR LR L decB y

y