1 phy 712 electrodynamics 9-9:50 am mwf olin 103 plan for lecture 34: special topics in...
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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
PHY 712 Spring 2015 -- Lecture 34 304/20/2015
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
PHY 712 Spring 2015 -- Lecture 34 404/20/2015
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
PHY 712 Spring 2015 -- Lecture 34 504/20/2015
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
PHY 712 Spring 2015 -- Lecture 34 604/20/2015
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
PHY 712 Spring 2015 -- Lecture 34 704/20/2015
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)
PHY 712 Spring 2015 -- Lecture 34 804/20/2015
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
PHY 712 Spring 2015 -- Lecture 34 904/20/2015
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
PHY 712 Spring 2015 -- Lecture 34 1004/20/2015
Type I elemental superconductorshttp://wuphys.wustl.edu/~jss/NewPeriodicTable.pdf
PHY 712 Spring 2015 -- Lecture 34 1104/20/2015
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
PHY 712 Spring 2015 -- Lecture 34 1204/20/2015
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.
PHY 712 Spring 2015 -- Lecture 34 1304/20/2015
Josephson junction -- tunneling current between two superconductors (Ref. Teplitz, Electromagnetism (1982))
d
Bz
x
PHY 712 Spring 2015 -- Lecture 34 1404/20/2015
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
PHY 712 Spring 2015 -- Lecture 34 1504/20/2015
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 2015 -- Lecture 34 1604/20/2015
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
PHY 712 Spring 2015 -- Lecture 34 1704/20/2015
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 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
PHY 712 Spring 2015 -- Lecture 34 1804/20/2015
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
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
PHY 712 Spring 2015 -- Lecture 34 1904/20/2015
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
PHY 712 Spring 2015 -- Lecture 34 2004/20/2015
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
Josephson junction -- continued
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
PHY 712 Spring 2015 -- Lecture 34 2104/20/2015
Josephson junction -- continued
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
PHY 712 Spring 2015 -- Lecture 34 2204/20/2015
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
PHY 712 Spring 2015 -- Lecture 34 2304/20/2015
Scanning SQIUD microscopy Ref. J. R. Kirtley, Rep. Prog. Phys. 73 126501 (2010)
PHY 712 Spring 2015 -- Lecture 34 2404/20/2015