Fluxon Coupling in Dual Thin Films

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 $F_c=\left(\frac{{\phi }_0}{{\mu }_0}\right)\left[\frac{d_1{d}_2}{(d_1{\lambda }_2^2+d_2{\lambda }_1^2)}\right]A_{\frac{d}{{\lambda }^2}}\mathrm{}\left(R\right)\mathrm{}$; $A_{\frac{d}{{\lambda }^2}}\mathrm{}\left(R\right)\mathrm{}$ is the vector potential describing the magnetic field distribution of a fluxon in a film for which $\frac{d}{{\lambda }^2}$ has the effective value $\frac{(d_1{\lambda }_2^2+d_2{\lambda }_1^2)}{{\lambda }_1^2{\lambda }_2^2}$. The film thicknesses are $d_1$ and $d_2$, the penetration depths are ${\lambda }_1$ and ${\lambda }_2$, and ${\phi }_0$ is the flux quantum. The result is obtained by using a simple superposition principle, and is valid when $d_1\ll {\lambda }_1$, $d_2\ll {\lambda }_2$, the fluxon core radii are much less than $\frac{2{\lambda }^2}{d}$, and the separation between the films is small compared to ${\lambda }_1$ or ${\lambda }_2$.