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Epitaxial films of the B20-structure alloy Fe$_{1-y}$Co$_y$Ge were grown bymolecular beam epitaxy on Si (111) substrates. The magnetization variedsmoothly from the bulk-like values of one Bohr magneton per Fe atom for FeGe tozero for non-magnetic CoGe. The chiral lattice structure leads to aDzyaloshinskii-Moriya interaction (DMI), and the films' helical magnetic groundstate was confirmed using polarized neutron reflectometry measurements. Thepitch of the spin helix, measured by this method, varies with Co content $y$and diverges at $y \sim 0.45$. This indicates a zero-crossing of the DMI, whichwe reproduced in calculations using first principle methods. We also measuredthe longitudinal and Hall resistivity of our films as a function of magneticfield, temperature, and Co content $y$. The Hall resistivity is expected tocontain contributions from the ordinary, anomalous, and topological Halleffects. Both the anomalous and topological Hall resistivities show peaksaround $y \sim 0.5$. Our first principles calculations show a peak in thetopological Hall constant at this value of $y$, related to the strongspin-polarisation predicted for intermediate values of $y$. Half-metallicity ispredicted for $y = 0.6$, consistent with the experimentally observed linearmagnetoresistance at this composition. Whilst it is possible to reconciletheory with experiment for the various Hall effects for FeGe, the largetopological Hall resistivities for $y \sim 0.5$ are much larger then expectedwhen the very small emergent fields associated with the divergence in the DMIare taken into account.

Helical magnetic structure and the anomalous and topological Hall effects in epitaxial B20 Fe$_{1-y}$Co$_y$Ge films