Purdue University
Purdue ePubs
Computer Science Technical ReportsDepartment of Computer Science1993
A SemiLinear Elliptic PDE Model for the StaticSolution of Josephson Junctions
J. G. CaputoN. FlytzanisE. A. Vavalis
Report Number:
93049
is document has been made available through Purdue ePubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu foradditional information.
Caputo, J. G.; Flytzanis, N.; and Vavalis, E. A., "A SemiLinear Elliptic PDE Model for the Static Solution of Josephson Junctions"(1993).
Computer Science Technical Reports.
Paper 1063.hp://docs.lib.purdue.edu/cstech/1063
A
SemiLinear
Elliptic
PDE
Model
for
the
StaticSolution
of
JosephsonJunctions
J.G.Caputo!,
N.
Flytzanis
2
and
E.A.Vavalis
3
CSDTR93049
July
1993
A
SEMILINEARELLIPTIC
PDE
MODEL
FOR
THE
STATIC
SOLUTION
OFJOSEPHSON
JUNCTIONS
J.
G.
CAPUTO·,
N.
FLYTZANISt
AND
E.A.
VAVALJS~
Abstract.
Inthis
study
we
deriveasemilinearEllipticPartialDifferentialEquation
(PDE)
problem
that
modelsthe
static
(zerovoltage)behavior
of
aJosephsonwindowjUllction.
Iterativemethods
forsolvingthisproblemareproposed,analyzed
and
theirconvergenceanalysisispresented.
The
preliminary
computational
results
that
aregiven,show
the
modelingpower
of
our
approach
and
exhibititscomputational
efficiency.
1.
Introduction.
Josephsonjunctiondeviceshavebeenextensivelyusedin
many
applications
likeverysensitive
magnetometers,
high
frequencyoscillators,fastswitches
etc.
To
study
and
analyzesuch
devices,differential
equationproblems
that
model
them
have
been
widelyused.
Inthispaper
we
present
asemilinearelliptic
PDE
problem
whicheffectively
and
accurately
models
the
static
behavior
of
a2dimensional
Josephson
window
junction.
The
existence
of
the
solutions
and
the
analysisof
their
smoothness
andstability
are
not
addressedhere.Efficient
and
stablenumerical
methods
areproposed
to
solve
this
PDE
problem
and
asoftware
infrastructure
that
can
be
usedasa
simulationengine
for
Josephson
junction
devicesis
presented.
The
rest
of
the
paper
isorganizedasfollows.
In
Section2we
present
the
derivation
of
thePDE
problem
that
models
the
Josephson
junction.
A
brief
discussion
about
the
underlying
physics
and
alistofapplicationsbasedon
the
Josephson
junctionare
alsogiven.
In
Section3
numericalalgorithms
forsolving
the
derived
PDE
problem
arcpresented,
their
convergenceanalysisisdiscussed
andcertainimplementation
issues
are
addressed.InSection4,
preliminarynumericalexperiments
that
confirm
the
convergence
properties
of
the
method,
aswell
its
efficiency,
arepresented.
A
list
of
plotsconcerning
the
characteristics
of
the
computed
solutionsfordifferent
junction
geometries
and
boundary
conditions
are
alsogiven.
Ourpreliminary
conclusions
arepresented
in
Section5
togetherwithour
future
researchplans.2.
Derivation
of
the
Josephson
junction
PDE
model.
AJosephson
junction
isaweak
linkbetween
two
superconductors
allowing[or
the
couplingof
electron
pairs.
Josephson
[6]
showed
that
the
electrodynamics
of
such
adevice
aredescribed
not
by
currentand
voltage
but
by
their
integrals,charge
and
phase
whichheshowed
to
beproportionalto
the
phase
of
the
macroscopicwavefunctionsof
the
electronpairs
inthe
twosuperconductors.
He
showed
thatthe
current
of
pairs
across
the
link
dependedon
the
sineof
that
phase
difference.
These
tworelationsallow
the
device
to
act
asafrequencyvoltageconvertorleading
to
applicationslikehighfrequencyoscillators,
fast
switches
or
the
definitionofavoltage
standard
((14]'[7]'
[2]).
Another
effectis
•Laboratoirede
Mathematiques,
Instilut
deSciencesAppliquees,BPS,76131MontSaintAignanCedex,France.
(caputo@lmi.insaTOuen.fr)
t
Physics
Department,
University
or
Crete,Heraklion,Greece.
(/lytzani@minos.cc.uch.gr)
I
Purdue
University,
Computer
Science
Department,West
Lafayette,IN47907,USA.
(mav@cs.purd11e.edu)
Work
supported
in
part
byNationalScience
Foundationgrant
CCR
9202536.
1
I
i
I
FIG.
1.
AwindowJosephsonjunction.
:
,.'
..

._.
..
~
...
_._
...
_
.
_._.;
..
,
the
existence
ofa
current
of
pairsin
the
absenceof
voltage.
This
current
can
be
modulated
intoan
interference
pattern
by
an
externalmagnetic
field
makingthese
devicesverysensitive
magnetometers
[2].
For
anisolatedjunction
theseeffects
remain
small
energetically
andan
idea
toincrease
theenergy
output
isto
have
thelll
drive
an
electromagnetic
cavity.In
practice
this
is
done
by
makingthetop
andbottom
superconducting
layers
much
larger
than
the
junction
area.
This
design
leads
toverywell
controlled
specifications
and
thejunctions
do
notdeterioratewithtime.
A
view
of
sucha
device
is
presentedin
Figure
2.
In
this
workwe
model
such
type
of
junctionand
find
thestatic
(zerovoltage)solutions.
The
properties
of
these
solutions
can
be
verified
experimentally
and
from
adifferent
point
of
viewthese
static
solutionsprovide
adapted
initial
conditionsfor
the
timedependentproblem.
The
governing
equations
of
suchadevice
are
Maxwell's
equationstogether
with
the
Josephson
equationsmentioned
above.Designissues
and
the
fact
that
the
thickness
of
the
weak
link
(oxidelayer)isvery
small,
force
the
fieldsto
be
in
aplane.One
can
then
model
eachsuperconductinglayer
by
anarray
of
inductancesand
their
couplingbya
nonlinearelement
givenby
the
Josephson
equations
ill
parallel
witharesistor
representing
thecurrent
of
normal
electrons
and
a
capacity
[7].
Writing
the
equations
for
the
phase
difFerence
between
thetopand
the
bottom
layer,
and
assuming
a
perfect
symmetry
for
the
parameters
we
get
the
modelpresented
in
Figure
2.
The
Kirchofflawsfor
thisarray
can
be
condensed
into
the
following
discrete
evolution
equation
for
the
phase
(fl,
(integral
of
the
voltage)
at
eachllodei
of
the
array:
(I)
L
neMesl
neighbors
<Pi<Pi
C··
<PO.
(<Pi)
1'"
L
..
+
(fli+
L
.\2
sm

=
i·
j
l]
,A
!PO
Thisequation
expresses
the
fact
that
the
algebraic
sum
of
all
currents
iszero
at
agivennode.
The
first
term
corresponds
to
thecurrent
in
the
iuductances
(surface
current),
2
FIG.2.
Thcequivalentcircuitmodel.
the
secondis
the
ernrentin
the
capacitanceand
thethird
is
the
current
of
pairs
whichexistsonlyin
the
junction
region.
The
last
term
If
xl
is
an
external
currentappliedsymmetrically
to
the
boundary
of
the
array
and
it
can
alsodescribe
the
effect
ofanexternalmagnetic
fieldifappliedantisyrnetrically
[9].
The
physical
parameters
are,
L
ij
the
inductance
in
branch
ij
,
C
the
capacitance
at
node
i,
A
the
Josephsonlength,
L;
theinductancein
eachbranch,
<Po
thequantum
offlux.
The
time
independent
problem
givesrisetoadiscretesemilinearellipticequationwhichreducesin
the
continuum
limit
to:
1
1..
(2)\7(L(
)\7u)«x,y)"
(
)sm(u)=O
III
il,
x,y
A
L
x,y
where
((x,
y)
is
the
indicator
functionof
n
J.
i.e.
()
_{l
if(x,y)Eilj
t:
x,y

o
otherwise
au
an
=f(x,y)
on
ail,
and
where
:n
denotesthe
outward
normalderivative.Whenconsideringadevicesuch
as
the
onein
Figure
1
with
a
totalarea
nand
ajunetion
area
n
J
the
function
L
assumesa
constant
value
in
n
andanotherconstant
value
in
njn
J.
Then
equation
(2)
reduces
to
the
boundary
value
problem
whichcan
be
expressedformally
as
(3)\7([<
+
k(l
<)]\7u)

«x,y)
;,
sin(u)
=
0
in
il,
au
an
=
f(x,y)
on
oil,
(4)
where
k
=
LLin
and
where
Lin
and
Lout
are
the
inductances
inside
and
outside
the
".,
junction
n
J
respectively.
This
PDE
problem
isequivalent
tothe
compositesystem:
.LlUin
=
sin(Uin)
>.'
3
,
,
I
r