Fluxon Coupling in Dual Thin Films

Abstract

The coupling force for an isolated fluxon which extends through two thin films is calculated in the thin-film approximation. When the distance between centers of the fluxon in the two films is $R$, the force of attraction is ${F}_{c}=(\frac{{\ensuremath{\varphi}}_{0}}{{\ensuremath{\mu}}_{0}})[\frac{{d}_{1}{d}_{2}}{({d}_{1}{\ensuremath{\lambda}}_{2}^{2}+{d}_{2}{\ensuremath{\lambda}}_{1}^{2})}]{A}_{\frac{d}{{\ensuremath{\lambda}}^{2}}}(R)$; ${A}_{\frac{d}{{\ensuremath{\lambda}}^{2}}}(R)$ is the vector potential describing the magnetic field distribution of a fluxon in a film for which $\frac{d}{{\ensuremath{\lambda}}^{2}}$ has the effective value $\frac{({d}_{1}{\ensuremath{\lambda}}_{2}^{2}+{d}_{2}{\ensuremath{\lambda}}_{1}^{2})}{{\ensuremath{\lambda}}_{1}^{2}{\ensuremath{\lambda}}_{2}^{2}}$. The film thicknesses are ${d}_{1}$ and ${d}_{2}$, the penetration depths are ${\ensuremath{\lambda}}_{1}$ and ${\ensuremath{\lambda}}_{2}$, and ${\ensuremath{\varphi}}_{0}$ is the flux quantum. The result is obtained by using a simple superposition principle, and is valid when ${d}_{1}\ensuremath{\ll}{\ensuremath{\lambda}}_{1}$, ${d}_{2}\ensuremath{\ll}{\ensuremath{\lambda}}_{2}$, the fluxon core radii are much less than $\frac{2{\ensuremath{\lambda}}^{2}}{d}$, and the separation between the films is small compared to ${\ensuremath{\lambda}}_{1}$ or ${\ensuremath{\lambda}}_{2}$.