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 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 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

where H0 is the Hamiltonian of the pair assuming no electron-electron interaction.

The perturbing potential represents the fact that the electrons do interact with one another. The Schrodinger equation then yields

for energy E with eigenstate Y. Solutions of the Schrodinger equation are then of the form

where is the relative coordinate . 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

This supercurrent is periodic with period 2p and can be simplified to a sinusoidal function in the simplest cases, and it will be shown that

(1)

Fig. 1. A Josephson junction

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 ().

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

(2)

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]

Here, K is a coupling constant. The wavefunctions are conveniently expressed in terms of their pair densities .

so that substitution of the new wavefunction form into the coupled equations yields

(3)

(4)

(5)

(6)

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 , 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 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.

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

(7)

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

(8)

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 and Trajectories in the f-y~```` Plane

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

, ,

so that the dimensionless form of (8) becomes

(9)

and we have a simplified, dimensionless, second order nonlinear differential equation in f. 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, one changes the single, second order equation to a pair of first order equations, i.e.,

(10)

(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 (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 .

The trace of yields the sum of the eigenvalues to be which is less than zero. In addition, the product of the eigenvalues is the determinant of, namely . So we have

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.

Table 1: Examples of each type of fixed point

and / 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

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. 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.

Table 2: breakdown of the various scenarios

/ / 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)

Fig. 4. Plot for = 0, I = 0 (no damping, no current).

Fig. 5.Plot for = 0, I = 0.5 (no damping, Ic > IB ).

Fig. 6. Plot for = 0, I = 1 (no damping, Ic = IB).

Fig. 7. Plot for = 0, I = 2 (no damping, Ic < IB ).

Fig. 8. Plot for = 1, I = 2 (weak damping, Ic < IB ).

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 .)

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 a) 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 time-averaged voltage begins to grow. Its amplitude depends on the magnitude of I and the damping factor a. 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.

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.

[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. American 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 Institute of Technology.

[7] ODE Software for Matlab. Rice University Department of Mathematics.

[8] François Alouges and Virginie Bonnaillie, Analyse numérique de la supraconductivité: Numerical analysis of superconductivity ,Comptes Rendus Mathematique, Volume 337, Issue 8, 15 October 2003, Pages 543-548