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 \lambda_J. This definition is not strict.

In terms of underlying model a short Josephson junction is characterized by the Josephson phase \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., \phi(x,t) or \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 \phi in LJJ is the so-called perturbed sine-Gordon equation. For the case of 1D LJJ it looks like:

{{center|

\lambda_J^2\phi_{xx}-\omega_p^{-2}\phi_{tt}-\sin(\phi)

=\omega_c^{-1}\phi_t - j/j_c,

}}

where subscripts x and t denote partial derivatives with respect to x and t, \lambda_J is the Josephson penetration depth, \omega_p is the Josephson plasma frequency, \omega_c is the so-called characteristic frequency and j/j_c is the bias current density j normalized to the critical current density j_c. In the above equation, the r.h.s. is considered as perturbation.

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

{{center|

\phi_{xx}-\phi_{tt}-\sin(\phi)=\alpha\phi_t - \gamma,

}}

where spatial coordinate is normalized to the Josephson penetration depth \lambda_J and time is normalized to the inverse plasma frequency \omega_p^{-1}. The parameter \alpha=1/\sqrt{\beta_c} is the dimensionless damping parameter (\beta_c is McCumber-Stewart parameter), and, finally, \gamma=j/j_c is a normalized bias current.

=Important solutions=

  • Small amplitude plasma waves. \phi(x,t)=A\exp[i(kx-\omega t)]
  • Soliton (aka fluxon, Josephson vortex):M. Tinkham, Introduction to superconductivity, 2nd ed., Dover New York (1996).

{{center|

\phi(x,t)=4\arctan\exp\left(\pm\frac{x-ut}{\sqrt{1-u^2}}\right)

}}

Here x, t and u=v/c_0 are the normalized coordinate, normalized time and normalized velocity. The physical velocity v is normalized to the so-called Swihart velocity c_0=\lambda_J\omega_p, which represent a typical unit of velocity and equal to the unit of space \lambda_J divided by unit of time \omega_p^{-1}.{{cite journal

| author = J. C. Swihart

| year = 1961

| title = Field Solution for a Thin-Film Superconducting Strip Transmission Line

| journal = J. Appl. Phys.

| volume = 32

| issue = 3

| pages = 461–469

| doi = 10.1063/1.1736025|bibcode = 1961JAP....32..461S }}

References

{{reflist}}

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Category:Superconductivity

Category:Josephson effect