Ferromagnetic film on a superconducting substrate

We study the equilibrium domain structure and magnetic flux around a ferromagnetic film with perpendicular magnetization ${M}_{0}$ on a superconducting (SC) substrate. At $4\ensuremath{\pi}{M}_{0}l{H}_{c1}$ the SC is in the Meissner state and the equilibrium domain width in the film, l, scales as $(l/4\ensuremath{\pi}{\ensuremath{\lambda}}_{L}{)=(l}_{N}/4\ensuremath{\pi}{\ensuremath{\lambda}}_{L}{)}^{2/3}$ with the domain width on a normal (nonsuperconducting) substrate, ${l}_{N}/4\ensuremath{\pi}{\ensuremath{\lambda}}_{L}\ensuremath{\gg}1;$ ${\ensuremath{\lambda}}_{L}$ being the London penetration length.