Superluminal propagation of composite collective modes in superconductor-ferromagnet heterostructures

Superconductor/ferromagnet/superconductor Josephson junctions are paradigmatic systems for studying the delicate interplay of superconductivity and magnetism via proximity effects as well as their composite excitations. Here, we analyze the collective modes (CM) in such a heterostructure by taking into account the interplay between the demagnetization field ${H}_{\mathrm{dem}}$ and the field caused by the anisotropy of the ferromagnet ${H}_{\mathrm{an}}$, which was previously neglected. It turns out that the spectrum of composite collective modes, $\ensuremath{\omega}(k)$, has a qualitatively different form in the case of ${H}_{\mathrm{dem}}<{H}_{\mathrm{an}}$ and of ${H}_{\mathrm{dem}}>{H}_{\mathrm{an}}$. In the first case, the dependence $\ensuremath{\omega}(k)$ has the same form as in previous studies, whereas in the second case, the spectrum looks completely different. In particular, for moderate or weak anisotropy in ferromagnet, the group velocity of collective modes demonstrates inflection point where the group velocity becomes infinite and is superluminal. Furthermore, this point separates purely real and complex-conjugate solutions for the collective modes and is also an exception point. We show that the difference of the CMs spectra can be revealed by the Fiske experiment, i.e., by measuring the $I\text{\ensuremath{-}}V$ characteristics in the presence of magnetic field and voltage.