PHY 712 Spring 2014 -- Lecture 33 1

PHY 712 Electrodynamics10-10:50 AM MWF Olin 107

Plan for Lecture 33:Special Topics in Electrodynamics:1. Electromagnetic aspects of

superconductivity – continued2. Review – reflection and refraction

04/21/2014

PHY 712 Spring 2014 -- Lecture 33 204/21/2014

PHY 712 Spring 2014 -- Lecture 33 304/21/2014

Josephson junction -- tunneling current between two superconductors

d

Bz

x

PHY 712 Spring 2014 -- Lecture 33 404/21/2014

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 33 504/21/2014

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 33 604/21/2014

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 33 704/21/2014

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 33 804/21/2014

Josephson junction -- continued

02 ( ) si4Tunneling current:

If = and in absense of magnetic field, ( )

n si

)

n

(0

LT L R LR T LR

L LR L

RR LR

eJ J

En

ne n

n t

nt

E 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

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 33 904/21/2014

Josephson junction -- continued

x

L R

JL JR

JT

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

0Tunneling current:

Need to evaluate in presence of magnetic f

2

ie d

s n

l

iLT T LR

LR

nJ et

J

PHY 712 Spring 2014 -- Lecture 33 1004/21/2014

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 JR

JT

0Tunneling current: sin TT LRJ J

00

Integrating the difference of the phase angles alon

)

:

2

(

g

LR LR L y

y

B d

PHY 712 Spring 2014 -- Lecture 33 1104/21/2014

Josephson junction -- continued

x

L R

JL JR

JT

/2

/2

0

00

0 0

00

Tunneling current density: Integrating current density throughout width and height :

sin(sin(

wher

sin

/ ))/2(2 ) ande

2

T T LR

T T Tw

LR

L

w

J Jw h

I J hwh dy

cw

J

B de

00

Integrating the difference of the phase angles along

2

:

(

2 )

LR LR L decB y

y

PHY 712 Spring 2014 -- Lecture 33 1204/21/2014

Josephson junction -- continued

IT

/0

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

PHY 712 Spring 2014 -- Lecture 33 1304/21/2014

Review of reflection and refraction Consider the normal incidence case; 3 media

n1 n2 n3

E0

E1R

E2T

E2R

E3T

PHY 712 Spring 2014 -- Lecture 33 1404/21/2014

Review of reflection and refraction -- continued Consider the normal incidence case; 3 media

n1 n2 n3

E0

E1R

E2T

E2R

E3T

Note that in this steady-state formulation, we must match the tangential components of the E and H fields at each boundary

d

( / )( )

( / )( ) ( / )( )

0

Each plane wave component has the form:

ˆ( , )=

ˆ ˆ( , )= in our case

j

j j

x

x

i c n ctj j

i c n ct i c n ctj j j j xj

j

t E

n E n

e

Ee e

ct

c

E r y

H r z z

PHY 712 Spring 2014 -- Lecture 33 1504/21/2014

Review of reflection and refraction -- continued Consider the normal incidence case; 3 media

n1 n2 n3

E0

E1R

E2T

E2R

E3T

d

0 1 2 2

110

22

3

2 22

2 2 3

2 3

Matching equations:

R R

R R

i iR

i iR

E E E

E

En En

e e En E e E e E

E

E E

n

E

2

Here:n dc

PHY 712 Spring 2014 -- Lecture 33 1604/21/2014

Review of reflection and refraction -- continued Consider the normal incidence case; 3 media

n1 n2 n3

E0

E1R

E2T

E2R

E3T

d

1 1 1

2 2 22 2 2

2 2 2

3 3 2 3

2 2 22 2

2 2

1 1

2

2 2 2

3 3 2 3

After some algebra:

1 1 1 1 2 1 1 cos(2

1 1 1 1 1

)

2

n n n n n nn n n n n n

n n n n nn n n n n

R =1

2

2

)1 cos(2nn

PHY 712 Spring 2014 -- Lecture 33 1704/21/2014

Review of reflection and refraction -- continued

n1 n2 n3

E0

E1R

E2T

E2R

E3T

d1 1 1

2 2

1

2 2 22 2 2

2 2 2

3 3 2 3

2 2 22 2 2

2 2 2

3 3

1 1

2 3 22

1 1 1 1 2 1 1 co )s(2

1 1 1 1 2 1 1

n n n n n nn n n n n n

n n n n n nn n n n n n

R =cos(2 )

2 2 22 2 2

2 2 2

3 3 2 3

1 1 1

2 2

2 22 2 1

Condition for zero ref

) 0

2 4) 1

lectance:

1 1 1 1 2 1 1 cos(2

cos(2 (2 1) (2 1) also 4

n n n n n nn n n n n n

n d n d n n nc d

3n