Skyrmion-vortex hybrid and spin wave solutions in superconducting ferromagnets

Abstract: The coexistence of Ferromagnetism and superconductivity in so called ferromagnetic superconductors is an intriguing phenomenon which may lead to novel physical effects as well as applications. Here in this work we have explored the interplay of topological excitations, namely vortices and skyrmions, in ferromagnetic superconductors using a field theoretic description of such systems. In particular, numerical solutions for the continuous spin field compatible to a given vortex profile are determined in absence and presence of a Dzyaloshinskii-Moriya interaction (DMI) term. The solutions show that the spin configuration is like a skyrmion but intertwined with the vortex structure -- the radius of the the skyrmion-like solution depends on the penetration depth and also the polarity of the skyrmion depends on the sign of the winding number. Thus our solution describes a topological structure: namely a skyrmion-vortex composite. We have also determined the spin wave solutions in such systems in presence and absence of a vortex. In absence of vortex frequency and wave vector satisfy a cubic equation which leads to various interesting features. In particular, we have shown that in the low frequency regime the minimum in dispersion relation shifts from $k=0$ to a non zero $k$ value depending on the parameters. We also discuss the nature of spin wave dispersion in the $\omega \sim \Tilde{m}$ regime which shows a similar pattern in the dispersion curve. The group velocity of the spin wave would change it's sign across such a minimum which is unique to FMSC. Also, the spin wave modes around the local minimum looks like roton mode in superfluid and hence called a magnetic roton. In presence of a vortex, the spin wave amplitude is shown to vary spatially such that the profile looks like that of a N\'eel Skyrmion. Possible experimental signature of both solutions are also discussed.