Epitaxial thin films of the substitutionally alloyed half-Heusler series CoTi$_{1-x}$Fe$_x$Sb were grown by molecular beam epitaxy on InAlAs/InP(001) substrates for concentrations 0.0$leq$x$leq$1.0. The influence of Fe on the structural, electronic, and magnetic properties was studied and compared to that expected from density functional theory. The films are epitaxial and single crystalline, as measured by reflection high-energy electron diffraction and X-ray diffraction. Using in-situ X-ray photoelectron spectroscopy, only small changes in the valence band are detected for x$leq$0.5. For films with x$geq$0.05, ferromagnetism is observed in SQUID magnetometry with a saturation magnetization that scales linearly with Fe content. A dramatic decrease in the magnetic moment per formula unit occurs when the Fe is substitutionally alloyed on the Co site indicating a strong dependence on the magnetic moment with site occupancy. A crossover from both in-plane and out-of-plane magnetic moments to only in-plane moment occurs for higher concentrations of Fe. Ferromagnetic resonance indicates a transition from weak to strong interaction with a reduction in inhomogeneous broadening as Fe content is increased. Temperature-dependent transport reveals a semiconductor to metal transition with thermally activated behavior for x$leq$0.5. Anomalous Hall effect and large negative magnetoresistance (up to -18.5% at 100 kOe for x=0.3) are observed for higher Fe content films. Evidence of superparamagnetism for x=0.3 and x=0.2 suggests for moderate levels of Fe, demixing of the CoTi$_{1-x}$Fe$_x$Sb films into Fe rich and Fe deficient regions may be present. Atom probe tomography is used to examine the Fe distribution in a x=0.3 film. Statistical analysis reveals a nonhomogeneous distribution of Fe atoms throughout the film, which is used to explain the observed magnetic and electrical behavior.
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