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Long Josephson junction

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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 λ J {\displaystyle \lambda _{J}} . This definition is not strict.

In terms of underlying model a short Josephson junction is characterized by the Josephson phase ϕ ( t ) {\displaystyle \phi (t)} , 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., ϕ ( x , t ) {\displaystyle \phi (x,t)} or ϕ ( x , y , t ) {\displaystyle \phi (x,y,t)} .

Simple model: the sine-Gordon equation

The simplest and the most frequently used model which describes the dynamics of the Josephson phase ϕ {\displaystyle \phi } in LJJ is the so-called perturbed sine-Gordon equation. For the case of 1D LJJ it looks like:

λ J 2 ϕ x x − ω p − 2 ϕ t t − sin ⁡ ( ϕ ) = ω c − 1 ϕ t − j / j c , {\displaystyle \lambda _{J}^{2}\phi _{xx}-\omega _{p}^{-2}\phi _{tt}-\sin(\phi )=\omega _{c}^{-1}\phi _{t}-j/j_{c},}

where subscripts x {\displaystyle x} and t {\displaystyle t} denote partial derivatives with respect to x {\displaystyle x} and t {\displaystyle t} , λ J {\displaystyle \lambda _{J}} is the Josephson penetration depth, ω p {\displaystyle \omega _{p}} is the Josephson plasma frequency, ω c {\displaystyle \omega _{c}} is the so-called characteristic frequency and j / j c {\displaystyle j/j_{c}} is the bias current density j {\displaystyle j} normalized to the critical current density j c {\displaystyle j_{c}} . In the above equation, the r.h.s. is considered as perturbation.

Usually for theoretical studies one uses normalized sine-Gordon equation:

ϕ x x − ϕ t t − sin ⁡ ( ϕ ) = α ϕ t − γ , {\displaystyle \phi _{xx}-\phi _{tt}-\sin(\phi )=\alpha \phi _{t}-\gamma ,}

where spatial coordinate is normalized to the Josephson penetration depth λ J {\displaystyle \lambda _{J}} and time is normalized to the inverse plasma frequency ω p − 1 {\displaystyle \omega _{p}^{-1}} . The parameter α = 1 / β c {\displaystyle \alpha =1/{\sqrt {\beta _{c}}}} is the dimensionless damping parameter ( β c {\displaystyle \beta _{c}} is McCumber-Stewart parameter), and, finally, γ = j / j c {\displaystyle \gamma =j/j_{c}} is a normalized bias current.

Important solutions

  • Small amplitude plasma waves. ϕ ( x , t ) = A exp ⁡ [ i ( k x − ω t ) ] {\displaystyle \phi (x,t)=A\exp}
  • Soliton (aka fluxon, Josephson vortex):
ϕ ( x , t ) = 4 arctan ⁡ exp ⁡ ( ± x − u t 1 − u 2 ) {\displaystyle \phi (x,t)=4\arctan \exp \left(\pm {\frac {x-ut}{\sqrt {1-u^{2}}}}\right)}

Here x {\displaystyle x} , t {\displaystyle t} and u = v / c 0 {\displaystyle u=v/c_{0}} are the normalized coordinate, normalized time and normalized velocity. The physical velocity v {\displaystyle v} is normalized to the so-called Swihart velocity c 0 = λ J ω p {\displaystyle c_{0}=\lambda _{J}\omega _{p}} , which represent a typical unit of velocity and equal to the unit of space λ J {\displaystyle \lambda _{J}} divided by unit of time ω p − 1 {\displaystyle \omega _{p}^{-1}} .

References

  1. M. Tinkham, Introduction to superconductivity, 2nd ed., Dover New York (1996).
  2. 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.
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