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4/18/2018
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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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04/18/2018 PHY 712 Spring 2018 -- Lecture 36 4
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
04/18/2018 PHY 712 Spring 2018 -- Lecture 36 5
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
04/18/2018 PHY 712 Spring 2018 -- Lecture 36 6
Behavior of magnetic field lines near superconductor
normalstate:
superconducting state:
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04/18/2018 PHY 712 Spring 2018 -- Lecture 36 7
04/18/2018 PHY 712 Spring 2018 -- Lecture 36 8
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
04/18/2018 PHY 712 Spring 2018 -- Lecture 36 9
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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04/18/2018 PHY 712 Spring 2018 -- Lecture 36 10
Type I superconductors:
04/18/2018 PHY 712 Spring 2018 -- Lecture 36 11
Type II superconductors
04/18/2018 PHY 712 Spring 2018 -- Lecture 36 12
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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04/18/2018 PHY 712 Spring 2018 -- Lecture 36 13
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.
04/18/2018 PHY 712 Spring 2018 -- Lecture 36 14
04/18/2018 PHY 712 Spring 2018 -- Lecture 36 15
Crystal structure of one of the high temperature superconductors
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04/18/2018 PHY 712 Spring 2018 -- Lecture 36 16
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
04/18/2018 PHY 712 Spring 2018 -- Lecture 36 17
002
ˆ( )=2 LL
rK
ll
FB r z
Scanning probe images of vortices in YBCO at 22 K
04/18/2018 PHY 712 Spring 2018 -- Lecture 36 18
Josephson junction -- tunneling current between two superconductors (Ref. Teplitz, Electromagnetism (1982))
d
Bz
x
Mechanism for vortex detection --
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04/18/2018 PHY 712 Spring 2018 -- Lecture 36 19
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
04/18/2018 PHY 712 Spring 2018 -- Lecture 36 20
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
l
l
l l
l l
Ay
Supercon left Supercon rightJunction
d
B A
04/18/2018 PHY 712 Spring 2018 -- Lecture 36 21
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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04/18/2018 PHY 712 Spring 2018 -- Lecture 36 22
Josephson junction -- continued
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
04/18/2018 PHY 712 Spring 2018 -- Lecture 36 23
Josephson junction -- continued
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
04/18/2018 PHY 712 Spring 2018 -- Lecture 36 24
Josephson junction -- continued
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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04/18/2018 PHY 712 Spring 2018 -- Lecture 36 25
Josephson junction -- continued
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
04/18/2018 PHY 712 Spring 2018 -- Lecture 36 26
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
l
l
l l
l l
Ay
Supercon left Supercon rightJunction
d
B A
04/18/2018 PHY 712 Spring 2018 -- Lecture 36 27
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
Josephson junction -- continued
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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04/18/2018 PHY 712 Spring 2018 -- Lecture 36 28
Josephson junction -- continued
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
04/18/2018 PHY 712 Spring 2018 -- Lecture 36 29
Josephson junction -- continued
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
04/18/2018 PHY 712 Spring 2018 -- Lecture 36 30
Josephson junction -- continued
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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04/18/2018 PHY 712 Spring 2018 -- Lecture 36 31
Josephson junction -- continued
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
04/18/2018 PHY 712 Spring 2018 -- Lecture 36 32
Scanning SQIUD microscopyRef. J. R. Kirtley, Rep. Prog. Phys. 73 126501 (2010)
04/18/2018 PHY 712 Spring 2018 -- Lecture 36 33
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04/18/2018 PHY 712 Spring 2018 -- Lecture 36 34
pg. 481
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