Josephson Junction Description

The Josephson junction device has unique behavior which complicates simulation with a SPICE-type simulator. Central is the idea of phase, which is a quantum-mechanical concept and is generally invisible in the non-quantum world. However with superconductivity, and with Josephson junctions in particular, phase becomes not only observable, but a critical parameter describing these devices and the circuits that contain them.

Without going into the detailed physics, one can accept that phase is an angle which applies to any superconductor. The angle is a fixed value anywhere on the superconductor, unless current is flowing. Flowing current produces magnetic flux, and magnetic flux produces a change in phase. One can express this as follows:

LI=flux= (/2)

Here, *L*
is the inductance,
is the magnetic flux quantum
(Planck's constant divided by twice the electron charge) and
is
the phase difference across the inductor. The supercurrent flowing in
a Josephson junction is given by

I=I_{c}sin()

where *I*_{c}
is the junction critical current, and
is the phase
difference across the junction. The junction phase is proportional to
the time integral of junction voltage:

= (2/)V(t)dt

The important consequence is that the sum of the phase differences around any loop consisting of Josephson junctions and inductors must be a multiple of 2 . This is due to the requirement that the superconducting wave function be continuous around the loop. Further, if the loop phase is not zero, it implies that a persistent current is flowing around the loop, and that the magnetic flux through the loop is a multiple of the flux quantum .

We therefor observe that in a circuit containing loops of Josephson junctions and inductors, which includes about all useful circuits:

- The DC voltage across each Junction or inductor is zero.
- The DC current applied to the circuit splits in such a manner as to satisfy the phase relations above.

Without any built-in concept of phase, it would appear to be impossible to find the DC operating point of a circuit containing Josephson junctions and inductors with a SPICE simulator. However, there are ways to accomplish this.

The time-honored approach, used successfully for many years, is to
skip DC analysis entirely. One generally is interested only in
transient analysis, describing the time evolution of the circuit under
stimulus, and a DC analysis would only be necessary to find the
initial values of circuit voltages and currents. Instead of a DC
analysis, what is done is every voltage and current source starts at
zero voltage or current, and ramps to the final value in a few
picoseconds. The transient analysis is performed using the ``use
initial conditions'' (```uic`'') option, where there is no DC
operating point analysis, and transient analysis starts immediately
with any supplied initial condicions (which are not generally given in
this case). By ramping up from zero, the phase condition around
junction/inductor loops is satisfied via Kirchhoff's voltage law.
Actually, this ensures that the loop phase is constant, but it is zero
as we started from zero. Initially, there is no ``trapped flux'' in
the inductor/junction loops, so assumption of zero phase is correct.
Thus the prescription is to ramp up all sources from zero, use the
`uic` option of transient analysis, and wait for any transients
caused by the ramping sources to die away before starting the ``real''
simulation. The ramping-up effectively replaces the DC operating
point analysis.

The second approach is to use phase-mode DC analysis (see
2.7.3.1), which is used in * WRspice* transparently when
Josephson junctions are present. This applies to explicit DC analysis
as well as operating point analysis. Further, this enables AC and
similar small-signal analysis with Josephson junctions in

See the Josephson junction model description for more information.