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

We report on the first example of epitaxial Mn$_{3 + \delta}$Ge thin films with a cubic $L1_2$ structure. The films are found to exhibit frustrated ferromagnetism with an average magnetization corresponding to 0.98$~\pm~$0.06$~\mu_B$/Mn, far larger than the parasitic ferromagnetism in hexagonal Mn$_3$Ge and the partially compensated ferrimagnetism in tetragonal Mn$_3$Ge. The Hall conductivity is the largest reported for the kagome magnets with a low temperature value of $\sigma_{xy} = 1587~$S/cm. Density functional calculations predict that a chiral antiferromagnetic structure is lower in energy than a ferromagnetic configuration in an ordered stoichiometric crystal. However, chemical disorder driven by the excess Mn in our films explains why a frustrated 120$^\circ$ spin structure is not observed. Comparisons between the magnetization and the Hall resistivity indicate that a non-coplanar spin structure contributes the Hall signal. Anisotropic magnetoresistance and planar Hall effect with hysteresis up to 14 T provides further insights into this material.