  A series of Fe$_x$Rh$_{100-x}$ ($30 \leq x \leq 57$) films were epitaxially
grown using magnetron sputtering, and were systematically studied by
magnetization-, electrical resistivity-, and Hall resistivity measurements.
After optimizing the growth conditions, phase-pure Fe$_{x}$Rh$_{100-x}$ films
were obtained, and their magnetic phase diagram was revisited. The
ferromagnetic (FM) to antiferromagnetic (AFM) transition is limited at narrow
Fe-contents with $48 \leq x \leq 54$ in the bulk Fe$_x$Rh$_{100-x}$ alloys. By
contrast, the FM-AFM transition in the Fe$_x$Rh$_{100-x}$ films is extended to
cover a much wider $x$ range between 33 % and 53 %, whose critical temperature
slightly decreases as increasing the Fe-content. The resistivity jump and
magnetization drop at the FM-AFM transition are much more significant in the
Fe$_x$Rh$_{100-x}$ films with $\sim$50 % Fe-content than in the Fe-deficient
films, the latter have a large amount of paramagnetic phase. The
magnetoresistivity (MR) is rather weak and positive in the AFM state, while it
becomes negative when the FM phase shows up, and a giant MR appears in the
mixed FM- and AFM states. The Hall resistivity is dominated by the ordinary
Hall effect in the AFM state, while in the mixed state or high-temperature FM
state, the anomalous Hall effect takes over. The absence of topological Hall
resistivity in Fe$_{x}$Rh$_{100-x}$ films with various Fe-contents implies that
the previously observed topological Hall effect is most likely extrinsic. We
propose that the anomalous Hall effect caused by the FM iron moments at the
interfaces nicely explains the hump-like anomaly in the Hall resistivity. Our
systematic investigations may offer valuable insights into the spintronics
based on iron-rhodium alloys.
