phy 712 electrodynamics 10-10:50 am mwf olin 107 plan for lecture 32:
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PHY 712 Electrodynamics 10-10:50 AM MWF Olin 107 Plan for Lecture 32: Special Topics in Electrodynamics: Electromagnetic aspects of superconductivity -- continued. Behavior of superconducting material – exclusion of magnetic field according to the London model. l L. x. - PowerPoint PPT PresentationTRANSCRIPT
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