  Rare earth free alloys are in focus of permanent magnet research since the
accessibility of the elements needed for nowadays conventional magnets is
limited. Tetragonally strained iron-cobalt (Fe-Co) has attracted large interest
as promising candidate due to theoretical calculations. In experiments,
however, the applied strain quickly relaxes with increasing film thickness and
hampers stabilization of a strong magnetocrystalline anisotropy. In our study
we show that already 2 at% of carbon substantially reduce the lattice
relaxation leading to the formation of a spontaneously strained phase with 3 %
tetragonal distortion. In these strained
(Fe$_{0.4}$Co$_{0.6}$)$_{0.98}$C$_{0.02}$ films, a magnetocrystalline
anisotropy above 0.4 MJ/m$^3$ is observed while the large polarization of 2.1 T
is maintained. Compared to binary Fe-Co this is a remarkable improvement of the
intrinsic magnetic properties. In this paper, we relate our experimental work
to theoretical studies of strained Fe-Co-C and find a very good agreement.
