Download Nonlinear Dynamics of Josephson Junctions

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 mathematical 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 complete,
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, Mathematical model, 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 physicists 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 also 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
reasons
for
further
investigation
of
superconductivity 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 theoretical 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
superconductors, separated by some nonsuperconducting 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 illustrated 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 pair
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 the energy of the superconductor, and
each 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 (EF) are filled [1].
With the states filled, if 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
H  H 0  V (r1  r2 )
where H0 is the Hamiltonian of the pair assuming no
electron-electron interaction.
p2
H0 
 V (r )
2m
The perturbing potential V ( r1  r2 ) represents the
fact that the electrons do interact with one another.
The Schrodinger equation then yields
H  E
for energy E with eigenstate . Solutions of the
Schrodinger equation are then of the form
  a k e jkR

k
where R is the relative coordinate r1  r2 . This
wavefunction is stationary state solution. It is a set
of wavefunctions for the Cooper 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 Mathematical 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 not
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
the gap, but is instead a function of the phase
difference of the Cooper pair wavefunctions, given
by   1   2
This supercurrent is periodic with period 2 and can
be simplified to a sinusoidal function in the simplest
cases, and it will be shown that
I s  I c Sin
(1)
1
d
Superconductor1
Weak coupling
2
Fig. 1. A Josephson junction
Superconductor 2
The constant Ic 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 Ic, there is no voltage, and for currents above
Ic, a voltage is observed [3]. The significance of this
is that current still flows through the junction below
Ic, 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 familiar Ohm’s Law ( V  IR ).
For nonzero I, but zero voltage V, R must be zero.
When the current in the junction exceeds Ic, 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
h d
(2)
V
2e dt
The voltage and supercurrent relations are found by
recognizing that the boundary problem of the
wavefunction at the junction interface can be
modeled with the coupled equations [1]
d
eV
jh 1    1  K 2
dt
2
d 2 eV
jh

 2  K 1
dt
2
Here, K is a coupling constant. The wavefunctions
are conveniently expressed in terms of their pair
densities  i   i1 / 2 e ji .
so that substitution of the new wavefunction form
into the coupled equations yields
d1 2
(3)
 K ( 1  2 )1 / 2 Sin
dt h
d 2
2
(4)
  K ( 1  2 )1 / 2 Sin
dt
h
d1
K 
eV
(5)
  ( 2 )1 / 2 Cos 
dt
h 1
2h
d 2
K 
eV
(6)
  ( 1 )1 / 2 Cos 
dt
h 2
2h
From these equations one sees 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 of
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 1   2 , the
expression for the voltage is equation (2).
4 A Model of the Josephson Junction
'  y
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 where 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.
(11)
y  I  Sin  y
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
1
 0
(10) and (11) is A  
.
- cos -  
The trace of A yields the sum of the eigenvalues to
be 1   2   which is less than zero. In
addition, the product of the eigenvalues is the
determinant of A , namely 1 2  Cos  . So we have
Trace ( A)  1   2  
Det ( A)  1 2  Cos 
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.

S
R
C
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
dV V
(7)
C
  I c Sin  I
dt R
Substituting the phase-varying voltage (2) into the
above equation, we find
hC d 2
h d
(8)

 I c Sin  I
2
2e dt
2eR dt
According to Strogatz, the critical current Ic for a
Josephson Junction is usually between 10-6 and 10-3
A, and a typical voltage is on the order of 10-3 V.
The typical oscillation frequency is on the order of
1011 Hz, and a reasonable length scale is on the
order of 10-6 m.
5
Dimensionless
Analysis
Trajectories in the y Plane
and
One can define dimensionless parameters for (8) in
the following way:
eI
I
h
)1 / 2 
 ( c )1 / 2 t  I  B ,   (
2
hC
Ic
2eI c R C
so that the dimensionless form of (8) becomes
 ''   '  Sin  I  (9)
and we have a simplified, dimensionless, second
order nonlinear differential equation in . Here, IB 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.,
(10)
'
Table 1: Examples of each type of fixed point
1 and  2
Both negative
Both complex
One is negative
One is zero
Both are zero
Equal
Fixed points
unstable node
“center” or “spiral”
saddle point
line of fixed points
Plane of fixed points
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
10
8
6
4
2
0
y
It is now necessary to examine the possible
combinations of eigenvalues that may reveal the
nature of the fixed points. We 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.
-2
-4
-6
-8
-10
0
1
2
3
4
5

6
7
8
9
10
Fig. 6. Plot for  = 0, I = 1 (no damping, Ic = IB).
Table 2: breakdown of the various scenarios
10
8
6
4
2
y
1  2 
1   2
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 fixed points (I = 1)
Zero
0
Plane of fixed points (I = 1)
0
-2
-4
-6
-8
-10
10
0
8
2
3
4
5

6
7
8
9
10
Fig. 7. Plot for  = 0, I = 2 (no damping, Ic < IB ).
6
4
y
1
10
2
8
0
6
-2
4
-4
2
0
y
-6
-8
-2
-10
-4
-10
-8
-6
-4
-2
0

2
4
6
8
-6
10
-8
Fig. 4. Plot for  = 0, I = 0 (no damping, no current).
-10
0
y
10
20
30
40
50

60
70
80
90
100
Fig. 8. Plot for  = 1, I = 2 (weak damping, Ic < IB ).
10
8
10
6
8
4
6
2
4
0
2
y
-2
0
-4
-2
-6
-4
-8
-6
-10
-8
0
1
2
3
4
5

6
7
8
9
10
-10
0
Fig. 5.Plot for  = 0, I = 0.5 (no damping, Ic > IB ).
10
20
30
40
50

60
70
80
90
100
Fig. 9. Plot for  =10 , I = 2 (heavy damping, Ic < IB ).
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 consider the
nullcline. (A nullcline is the function that arises
by computing y '  0 .)
1
y  ( I  Sin )

This sinusoidal function is observed most clearly in
plot 9 (for weak damping). All 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 Ic < IB,
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 (IB)
overcomes the critical current (Ic), when the timeaveraged 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.
<V>
Ic
1
I
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. 2nd edition. 1999, Prentice Hall.
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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
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Phenomena. American Journal of Physics, Vol.
39. December, 1971, pg. 1504.
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June, 1997.
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