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We report on the first example of epitaxial Mn$_{3 + \delta}$Ge thin filmswith a cubic $L1_2$ structure. The films are found to exhibit frustratedferromagnetism with an average magnetization corresponding to0.98$~\pm~$0.06$~\mu_B$/Mn, far larger than the parasitic ferromagnetism inhexagonal Mn$_3$Ge and the partially compensated ferrimagnetism in tetragonalMn$_3$Ge. The Hall conductivity is the largest reported for the kagome magnetswith a low temperature value of $\sigma_{xy} = 1587~$S/cm. Density functionalcalculations predict that a chiral antiferromagnetic structure is lower inenergy than a ferromagnetic configuration in an ordered stoichiometric crystal.However, chemical disorder driven by the excess Mn in our films explains why afrustrated 120$^\circ$ spin structure is not observed. Comparisons between themagnetization and the Hall resistivity indicate that a non-coplanar spinstructure contributes the Hall signal. Anisotropic magnetoresistance and planarHall effect with hysteresis up to 14 T provides further insights into thismaterial.

Giant anomalous Hall effect in epitaxial Mn$_{3.2}$Ge films with a cubic kagome structure