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In superconductivity, a long Josephson junction (LJJ) is a Josephson junction which has one or more dimensions longer than the Josephson penetration depth . This definition is not strict.
In terms of underlying model a short Josephson junction is characterized by the Josephson phase , 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
editThe 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.
Usually for theoretical studies one uses normalized sine-Gordon equation:
where spatial coordinate is normalized to the Josephson penetration depth 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
edit- Small amplitude plasma waves.
- Soliton (aka fluxon, Josephson vortex):[1]
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]
References
edit- ^ 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. Bibcode:1961JAP....32..461S. doi:10.1063/1.1736025.