4/18/2018 - Wake Forest...

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PHY 712 Electrodynamics9-9:50 AM MWF Olin 105

Plan for Lecture 36:

Special Topics in Electrodynamics:

Electromagnetic aspects of superconductivity

• London equations

• Brief mention of quantum mechanism

• Tunneling between two superconductors

04/18/2018 PHY 712 Spring 2018 -- Lecture 36

204/18/2018 PHY 712 Spring 2018 -- Lecture 36

04/18/2018 PHY 712 Spring 2018 -- Lecture 36 3

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Special topic: Electromagnetic properties of superconductors

Ref:D. Teplitz, editor, Electromagnetism – paths to research,Plenum Press (1982); Chapter 1 written by Brian Schwartz

and Sonia Frota-Pessoa

History:1908 H. Kamerlingh Onnes successfully liquified He1911 H. Kamerlingh Onnes discovered that Hg at 4.2 K has vanishing resistance1957 Theory of superconductivity by Bardeen, Cooper, and Schrieffer


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

l

l

l

l

A

A y2

x/

2

( ) B (0)e Current

0 or

de

=0

nsity: L

y L z

nex

mc

ne ne em

mc m

J

c

ll

J A v A

x

lL

7Typically, 10L ml


Behavior of magnetic field lines near superconductor

normalstate:

superconducting state:

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2) )2 2/ ( ((0) (0) 2 ( )( )

8F VC

S FN

NEH

G N EG e

characteristic phonon energy

density of electron states at EF

attraction potential between electron pairs


Temperature dependence of critical field2

( ) (0) 1c c

c

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

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Type I superconductors:


Type II superconductors


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

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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

F

F

F

j l A

l a BA

l

A a

l

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



Crystal structure of one of the high temperature superconductors

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Some details of single vortex in type II superconductor

22 2

2 2

2 002 2

London equation without vortices:

4 1

4

Equation for field with single quantum of vortex along -

where

ˆ ˆ ˆ ( )

axis:

2

1

L

L

L L

mc

ne

hx y

c

c

z

e

l

l

l l

F F

J B B

B B z r r x y

002

ˆSolution: ( )=2 LL

rK

ll

F

B r z

2

02 2

2

02 2

0

00

0 2 ' '

Sin

Check:

1 1For 0 0

1 1For

ce K ( ) ln

2

L L

r

L L

u

dr r

d d rr K

dr r dr

d d rr K

dr r

u

r

u

d

l l

l l


002

ˆ( )=2 LL

rK

ll

FB r z

Scanning probe images of vortices in YBCO at 22 K


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

d

Bz

x

Mechanism for vortex detection --

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Josephson junction -- continued

Bz

Supercon left Supercon rightJunction

d

( /2)/0

0

( /2)/0

/ 2

( ) / 2 / 2

/ 2

L

L

x d

z

x d

B e x d

x B d x d

B e x d

B

l

l



( /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

l

l

l l

l l

Ay


d

B A


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

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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 L

iL

R LR

R R

R

R i

R

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

LR

R

n nn n

t t

n

t n

n

n n

E

E

t n

Note that

1LRL RE

tE



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

generalized current operator acting on pair wavefunction

2ˆ ˆ

2

L R

i

J

e

J

eJ

v v * 1 2ˆ with

2

ei

m c

v A

20

20

2 2

2

2 2

2R R

L L L

R

e eJ

m c

e eJ

m c

A

A



4Tunneling current:

If = = and in absense of magnetic field, ( ) (0)

Constant Josephson tunneling current for 0

2

) i

( n

sLT L R LR

L LR L

R

R

R LR

L

eJ

En n n

n

t t

n nt

E

EE

e

4 J (0)

Oscillatory Josephson tunneling current for 2

4 2 J (0

s

)

in

n

si

R L

T LR

T LR

en

E eV

e eVn

E

t

Method for precise measurement of /e

x

L R

JL JR

JT

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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)/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

l

l

l l

l l

Ay


d

B A


0

0

Recall that for / 2

fo

ˆ 0 and

ˆ 0r an / 2d

L L

LR

x d

x d

B

B

l

l

v A y

v A y


x

L R

JL JR

JT

Tunneling current:

( ) in4

sT L R LR

enJ n

0

2 2

0

2 2

Integrating the difference of the phase angles along :

, ,0 , ,0

2 2 ) (

d d d dRLR L L

LR L

R

y

y y

eB d yc

l

Bz

2

2 2

2

2 2R R

L L

e

m mc

e

m mc

v

v

A

A

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x

L R

JL JR

JT

0

/

00 0

0

2

/2

0

0

4 2Tunneling current density: sin

Integrating current density throughout width of supercondu r

sin (2 )

(2

(2 )

cto s

= cos2

T L LR T LR L

T T

TLR L

w

w

L

e eJ J B

w

I J

wJ ewB

eB

n d yc

w dy

cdc

l

ll

00

00 0 0

cos

Define:

) (2 )

2(2 ) and (2 )

2

LR L

L L

ew

w

B

ew

c

B

d d

cB d d

e c

l

l l

F

FF

F

00

Integrating the difference of

the phase

(2 )

angles along :

2LR LR L d y

c

y

eB l

ww



x

L R

JL JR

JT

0 000 0

0

02 0

0

/2

/2

0

Integrating current density throughout width of superconductors

= cos cos2

sin( = sin(

where

(2 ) (2 )(2 )

/ ))

/

w

T T

TLR L LR L

L

T LR

w

w dy

d dc cd

c

w

I J

wJ ew ewB B

eB

w J

l ll

F F

F F

00

2(2 ) and

2L

cB w d

e

lF F

00

Integrating the difference of

the phase

(2 )

angles along :

2LR LR L d y

c

y

eB l

ww



x

L R

JL JR

JT

/2 02 0

0 0

00

/2

4Tunneling current density:

Integrating current density throughout width of superconductors

sin

sin

/ ))

/

2(2 ) an

(sin(

d er2

wh e

T L L

w

w

R

T T T LR

L

eJ

w

n

I w dy

cB w

w

d

J J

e

l

F F

F F

F F

ww

Bz

d

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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 microscopyRef. J. R. Kirtley, Rep. Prog. Phys. 73 126501 (2010)


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pg. 481

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