Simulation of Nonlinear Dynamics Effects for Josephson Junctions

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15 Νοε 2013 (πριν από 3 χρόνια και 4 μήνες)

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Simulation of
Nonlinear Dynamics
Effects for
Josephson Junctions


MOSTAFA BORHANI, M. HADI VARAHRAM

Electrical Engineering

Department

Sharif University of Technology

Tehran, Azadi St.

postal code:
11365
-
9363

Islamic
Republic

of
Iran



Abstract
:

The purpose

of this paper is to examine the behavior of Josephson Junctions using nonlinear
methods. A brief historical explanation of superconductivity is provided for the reader, and then no time is
wasted as the concept of Cooper pairs is described with some mathe
matical derivation. Then, using multiple
references, a brief formulation of the concepts behind Josephson Junctions is laid out, preparing the reader for
a mathematical model that describes the current vs. voltage behavior of the junction. After this is c
omplete,
the equations are transformed into dimensionless form and the techniques of analyzing a two
-
dimensional,
coupled, nonlinear system are employed.
Whenever possible, plots and diagrams are used to explain the
significance of presented data. Finally,

conclusions are drawn from the nonlinear analysis, and a connection is
made between the harmonic nature of the Josephson Junction and it’s forced, damped pendulum analog.


Key
-
Words:

Josephson Junctions
,
Nonlinear Effects
,
Cooper pairs
, M
athematical mode
l
, Superconductivity


1
Introduction


Superconductors hold a lot of promise for many
applications, and the hope for the discovery of
superconducting materials that exhibit
superconductivity at higher temperatures has pulled
at the imaginations of physicis
ts since the January,
1986 discovery of the phenomenon at 30 K. The
implications of room temperature superconductors
are infinite. Power transmission from power
stations with zero
-
resistance power lines could not
only minimize cost of electricity, but al
so cut down
on fossil fuel consumption, while trains levitating on
superconducting rails could finally ensure cheap and
efficient mass transit. While neither of the
aforementioned technologies exist, the
many
reason
s

for further
investigation of

supercond
uct
ivity

are readily apparent
.

Prior to 1986, however,
high temperature
superconductors were not yet observed, and the
theoretical study of the materials led to more
intricate, detailed discoveries about the nature of
superconducting media.
The theoretica
l model of
Cooper pairs

coupled with fundamental quantum
mechanical framework led to a breakthrough by an
eccentric 22
-
year old by the name of Brian
Josephson.

In 1962,
Brian
Josephson, a graduate student at the
time, suggested a simple model where two
sup
erconductors, separated by some non
-
superconducting media with no voltage between the
two, would have a current pass from one
superconductor to the other. This behavior is
classically impossible, though the process of
tunneling

in quantum mechanics illustr
ated that such
behavior might be possible

in the non
-
classical
regime
. About one year later, the effect that
Josephson had suggested was observed
experimentally and it was called
The Josephson
Effect
.


2

Cooper Pairs


The Josephson Effect cannot be studied

in any
complete sense without a
basic

understanding of
Cooper pairs.
Heuristically, Cooper pairs can be
described by imagining that two electrons, which
are fermions (spin ½ particles), pair up such that
their spins are anti
-
parallel, and the resulting pa
ir
acts like a single, spin
-
less particle. The pair can
then be modeled like an indistinguishable spin
-
zero
particle, i.e., like a boson. The Cooper pairs in the
superconductor will adopt the same phase to
minimize th
e energy of the superconductor, and
eac
h pair can be modeled as a single particle with a
single wave
-
function.

As in most models for conductors, one can assume
that the electrons exist in a “Fermi sea”, where all
states less than the Fermi energy (E
F
) are filled

[1]
.
With the states filled, i
f any more electrons are to be
added they will have energy greater than the Fermi
energy, and these electrons will interact with one
another via some potential. The total Hamiltonian
for the added pair is

)
(
2
1
0
r
r
V
H
H




w
here H
0

is the Hamiltonia
n of the pair
assuming no
electron
-
electron interaction.

)
(
2
2
0
r
V
m
p
H



The perturbing
potential
)
(
2
1
r
r
V


represents the
fact that the electrons do interact with one another.


The Schrodinger equation then yields



E
H


for energy E with eigenstate

. Solutions of the
Schrodinger equation are then of the form



k
jkR
k
e
a


where
R

is the relative coordinate
2
1
r
r

. This
wavefunction is stationary state solution. It is a set
of wavefunctions for the C
ooper pair, and it implies
that the pair can be treated as a single particle. This
wavefunction, like any other wavefunction, is
distributed in space such that quantum mechanical
tunneling
through a Josephson Junction is both
possible and observable.


3

Ma
thematical Foundation of the

Josephson Junction Model


A Josephson junction, as described earlier, is
constructed by placing two superconductors in close
proximity to one another, and coupling the two with
a non
-
superconductor or a superconductor that is n
ot
at its superconducting temperature. The junction is
connected in series with a DC current source so that
constant current is driven through the junction.

The spacing between the two superconducting
electrodes (labeled as
d

in the figure below) is the
governing physical characteristic of the junction that
limits tunneling. If
d

~ 10
-
5

cm

or less, the net
current
I

flowing through the junction

contains a
component called the
supercurrent

[2]
.

The
supercurrent is not a function of the voltage
V

across
th
e gap, but is instead a function of the phase
difference of the Cooper pair wavefunctions, given
by

2
1






This supercurrent is periodic with period 2


and can
be simplified to a sinusoidal function in the simplest
cases,
and it will be sh
own that


Sin
I
I
c
s


(
1
)


1

2
Superconductor
1
Superconductor
2
Weak coupling
d

Fig.

1.
A Josephson junction

The constant
I
c

is called the
critical current
, and its
value is completely dependent on the dimensions

(shape and structure) of the junction.

It turns out that
this critical current is the value required for voltage
to be developed across the junction. For currents
below
I
c
, there is no voltage, and for currents above
I
c
, a voltage is observed

[3]
. The si
gnificance of this
is that current still flows through the junction below
I
c
, but since there is no observable voltage, there is
essentially no resistance (the entire junction behaves
like a superconductor). Justification of this can be
seen through the f
amiliar Ohm’s Law

(
IR
V

).

For nonzero
I
, but zero voltage
V
,
R

must be zero.
When the current in the junction exceeds
I
c
, a
constant phase difference can no longer be
maintained across the junction and a voltage
develops.

The voltage is

then proportional to the
change in phase, given by

dt
d
e
h
V

2




(
2
)

Th
e voltage and supercurrent relations are

found by
recognizing that the boundary problem of the
wavefunction at the junc
tion interface can be
modeled with the coupled equations

[1]

2
1
1
2



K
eV
dt
d
jh




1
2
2
2



K
eV
dt
d
jh



Here,
K

is

a coupling constant.
T
he wavefunctions
are conveniently expressed in terms of their pair
densities

i
j
i
i
e



2
/
1

.

so that sub
stitution of the new wavefunction form
into the coupled equations yields





Sin
K
h
dt
d
2
/
1
2
1
1
)
(
2


(
3
)





Sin
K
h
dt
d
2
/
1
2
1
2
)
(
2



(
4
)

h
eV
Cos
h
K
dt
d
2
)
(
2
/
1
1
2
1








(
5
)

h
eV
Cos
h
K
dt
d
2
)
(
2
/
1
2
1
2








(
6
)

From these equations
one

see
s

that the changing
density of Cooper pairs varies with the sine of the
phase difference, and that the rate of decrease of
pair density in one superconductor is the negative o
f
that in the other (from equations (
3
) and (
4
)). The
first two equations yield the supercurrent
relationship

by
equation

(
1
).

Subtracting (
5
) from (
6
), and setting
2
1



, the
expression for the voltage

is equation (
2
)
.


4

A
Model of the

Josephson Junction

The supercurrent relation is only the current
that
arises due to the movement of
the electron pairs.
The total current also contains a “displacement”
current as well as an “ordinary” current. A simple
circuit model can be constructed w
here one
represents the displacement current with a capacitor
and the ordinary current with a resistor,
and each is

placed in a circuit like the one shown below.

R
C

S
S
: Superconductivity Element

Fig.

2.
A simple circuit model

Following the Kirchoff conventions

of adding
current and voltage, the voltage drop across each
branch must be equal to some voltage
V
. The sum
of these currents and the supercurrent must be equal
to the
total

current
I


I
Sin
I
R
V
dt
dV
C
c






(7)

Su
bstituting the phase
-
varying voltage (
2
) into the
above equation, we find

I
Sin
I
dt
d
eR
h
dt
d
e
hC
c






2
2
2
2


(8)

According to Strogatz, the critical current
I
c

for a
Josephson Junction is usually between 10
-
6

and 10
-
3

A, and a typical volta
ge is on the order of 10
-
3

V.
The typical oscillation frequency is on the order of
10
11

Hz, and a reasonable length scale is on the
order of 10
-
6

m.


5

Dimensionless Analysis and
Trajectories in the


y

Plane

O
ne can define dimensionless
parameters for (8) in
the following way:


t
hC
eI
c
2
/
1
)
(






c
B
I
I
I

,
2
/
1
2
)
2
(
C
R
eI
h
c



so that the dimensionless form of (8) becomes

I
Sin






'
'
'



(9)

and we have a simplified, dimensionless, second
order non
linear differential equation in


. Here,
I
B


is
the bias current in the circuit.

This second order, nonlinear differential equation
can be decomposed into a two dimensional

system,
making analysis more practical. If one makes the
substitution
'


y
,
one changes the single, second
order equation to a pair of first order equations, i.e.,

y

'




(10)

y
Sin
I
y





'


(11)

A pair of differential equations like the set of
equations (10) and (11) begs for a nonlinear
dynamics approach. The first goal is to find the
fixed points of the

system, i.e., the points where
'
'
0
y




(simultaneously). The fixed points will
give an indication as to how the trajectory of the
system behaves.

The resulting
Jacobean

for the system formed by
(10) and (11) is










-
cos
-
1
0
A
.

Th
e trace of
A

yields the sum of the eigenvalues to
be






2
1

which is less than zero. In
addition, the product of the
eigenvalues is the
determinant of
A
, namely



Cos

2
1
. So we have









2
1
)
(
A
Trace




Cos
A
Det


2
1
)
(

The eigenvalues of the
Jacobean

matrix say a great
deal about the fixed points of a system and its
stability.
Fixed points represent some type of node
in the phase space trajectory of the system. The
fixed
point may be a
saddle point
,
stable node
,
unstable node
,
center, spiral,
or
star node
. There are
other possibilities, but in general the aforementioned
cases are sufficient for this discussion.
Examples of
each type of fixed point are included below.


Tab
le

1
:
Examples of each type of fixed point

1


and
2


Fixed points

Both negative

unstable node

Both complex

“center” or “spiral”

One is negative

saddle point

One is zero

line of fixed points

Both are zero

Plane

of fixed points

Equal

star node


Star node
Stable node
Degenerate node
Saddle point
Spiral
Center

Fig.

3.

Some type of node in the phase space trajectory of
the system

It is now necessary to examine the possible
combinations of eigenvalues that may reveal the
nature of the fixed points. W
e must consider many
cases. The nature of the problem changes drastically
for I >1, since Det(
A
) would imply that one of the
eigenvalues is complex. If I<1, then
one of the
eigenvalues can be negative, both can be negative,
or both can be positive. In the

case that I = 1, one or
both of the eigenvalues can be zero. Refer to the
chart below for a breakdown of the various
scenarios.


Table

2
:
breakdown of the various scenarios


2
1




2
1




Classification

+

-

unstable node ( I < 1)

-

+

saddle point (I < 1)

-

-

saddle point (I < 1)

Real

Complex

Center or spiral (I > 1)

Complex

Complex

Center or spiral (I >1)

Zero

+

Line of fixed points (I = 1)

Zero

-

Line of fi
xed points (I = 1)

Zero

0

Plane of fixed points (I = 1)


-10
-8
-6
-4
-2
0
2
4
6
8
10
-10
-8
-6
-4
-2
0
2
4
6
8
10
y



Fig.

4.
Plot for

= 0, I = 0 (no damping, no current).


0
1
2
3
4
5
6
7
8
9
10
-10
-8
-6
-4
-2
0
2
4
6
8
10
y



Fig.

5.
Plot for


= 0, I = 0.5 (no da
mping,
I
c

> I
B

).

0
1
2
3
4
5
6
7
8
9
10
-10
-8
-6
-4
-2
0
2
4
6
8
10
y


Fig
. 6.
Plot for


= 0, I = 1 (no damping,
I
c

=
I
B
).


0
1
2
3
4
5
6
7
8
9
10
-10
-8
-6
-4
-2
0
2
4
6
8
10
y


Fig. 7.
Plot for


= 0, I = 2 (no damping,
I
c

< I
B

).


0
10
20
30
40
50
60
70
80
90
100
-10
-8
-6
-4
-2
0
2
4
6
8
10
y


Fig. 8.
Plot for


= 1, I = 2 (weak damping,
I
c

< I
B

).

0
10
20
30
40
50
60
70
80
90
100
-10
-8
-6
-4
-2
0
2
4
6
8
10
y


Fig. 9.
Plot for


=10 , I = 2 (heavy damping,
I
c

< I
B

).

6

Conclusions


One can see that for I >1 there seems to be a stable

limit cycle.
Plots
8

and
9

display this result clearly.
In order to explain this, one must c
onsider the
nullcline
. (
A nullcline is the function that arises
by computing
0
'

y
.)


)
(
1


Sin
I
y



This sinusoidal function is observ
ed most clearly in
plot
9

(for weak damping).
A
ll trajectories (any
initial conditions) lead to an eventual steady state
along this nullcline. This means that for any phase
difference


and any y (voltage), as long as
I
c

< I
B
,
there
will be a stable oscillatory voltage across the
junction.

For small damping (small

) and I <1, the junction
is operating in the zero
-
voltage state. As I is
increased, nothing happens until the bias current (
I
B
)
overcomes the critical current (
I
c
), when the time
-
averaged voltage begins to grow. Its amplitude
depends on the magnitude of I

and the damping
factor

. If I is then decreased slowly, the stable
cycle remains even for I <1, until the current reaches
the critical current when the voltage drops to zero.
This behavior leads to a hysteretic current
-
voltage
curve, as seen below.


I
<V>
I
c
1


Fig. 10.
Hysteresis in the Josephson Junction.


Experimental verification of this type of behavior
has been recorded by Zimmerman

[4]
, who, as
mentioned before, designed a mechanical analog of
the Josephson
junction
.

The data from

this model
shows a jump to zero rotation rate at the bifurcation.






References
:

[1] Theodore van Duzer and Charles W. Turner.
Principles of Superconductive Devices and
Circuits
. 2
nd

edition. 1999, Prentice Hall.


[2] Likharev, Konstantin K.
Dynamics

of Josephson
Junctions and Circuits
. 1986. Gordon and
Breach Science Publishers.

[
3
]
Strogatz, Steven H.
Nonlinear Dynamics and
Chaos
. 1994, Perseus Books.

[4] D.B. Sullivan and J.E. Zimmerman.
Mechanical
Analogs of Time Dependent Josephson
Phenomena
. Ame
rican Journal of Physics, Vol.
39. December, 1971, pg. 1504.

[
5
]
Pines, David.
Understanding High Temperature
Superconductivity: Progress and Prospects.
June, 1997
.

[
6
]
Feynman, Richard.
The Feynman Lectures on
Physics
. Vol. III. 1965, California Insti
tute of
Technology.

[
7
]

ODE Software for Matlab
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Department of Mathematics.

[
8
] François Alouges and Virginie Bonnaillie,
Analyse numérique de la supraconductivité:
Numerical analysis of superconductivity


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ume 337,
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-
548

[
9
] M. Borhani, S.B. Rafe, M.H. Varahram,
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th

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-
351.

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lication of
Superconductivity

in Electrical Enginnering
,
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th

Iranian Student Conference in electrical
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)
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[11]
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Paulo Rodrigues, “
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DISTRIBUTED SYSTEMS (SEPADS 2004),
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200.pdf

[12] Davion Hill, “A Nonlinear Dynamics Approach
to Josephson Junctions”,
Condensed Matter
Physics,
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[13] M. Borhani, B.
Sc
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Electrical Engineering
Department,
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