Long Josephson junction explained
. This definition is not strict.
, which is only a function of time, but not of coordinates i.e. the Josephson junction is assumed to be point-like in space. In contrast, in a
long Josephson junction the Josephson phase can be a function of one or two spatial coordinates, i.e.,
or
.
Simple model: the sine-Gordon equation
The simplest and the most frequently used model which describes the dynamics of the Josephson phase
in LJJ is the so-called perturbed
sine-Gordon equation. For the case of 1D LJJ it looks like:
where subscripts
and
denote partial derivatives with respect to
and
,
is the Josephson penetration depth,
is the Josephson plasma frequency,
is the so-called characteristic frequency and
is the bias current density
normalized to the critical current density
. In the above equation, the r.h.s. is considered as perturbation.
and time is normalized to the inverse plasma frequency
. The parameter
is the dimensionless damping parameter (
is McCumber-Stewart parameter), and, finally,
is a normalized bias current.
Important solutions
- Small amplitude plasma waves.
\phi(x,t)=A\exp[i(kx-\omegat)]
Here
,
and
are the normalized coordinate, normalized time and normalized velocity. The physical velocity
is normalized to the so-called Swihart velocity
, which represent a typical unit of velocity and equal to the unit of space
divided by unit of time
.
[2] Notes and References
- M. Tinkham, Introduction to superconductivity, 2nd ed., Dover New York (1996).
- J. C. Swihart . 1961 . Field Solution for a Thin-Film Superconducting Strip Transmission Line . J. Appl. Phys. . 32 . 3 . 461–469 . 10.1063/1.1736025. 1961JAP....32..461S .
