Title: Direct Evidence of Spontaneous Abrikosov Vortex State in Ferromagnetic Superconductor EuFe$_2$(As$_{1-x}$P$_x$)$_2$ with $x=0.21$,
Abstract: Using low-temperature Magnetic Force Microscopy (MFM) we provide direct
experimental evidence for spontaneous vortex phase (SVP) formation in
EuFe$_2$(As$_{0.79}$P$_{0.21}$)$_2$ single crystal with the superconducting
$T^{\rm 0}_{\rm SC}=23.6$~K and ferromagnetic $T_{\rm FM}\sim17.7$~K transition
temperatures. Spontaneous vortex-antivortex (V-AV) pairs are imaged in the
vicinity of $T_{\rm FM}$. Also, upon cooling cycle near $T_{\rm FM}$ we observe
the first-order transition from the short period domain structure, which
appears in the Meissner state, into the long period domain structure with
spontaneous vortices. It is the first experimental observation of this scenario
in the ferromagnetic superconductors. Low-temperature phase is characterized by
much larger domains in V-AV state and peculiar branched striped structures at
the surface, which are typical for uniaxial ferromagnets with perpendicular
magnetic anisotropy (PMA). The domain wall parameters at various temperatures
are estimated.