The composition-dependent behavior of the Dzyaloshinskii-Moriya interaction (DMI), the spin-orbit torque (SOT), as well as anomalous and spin Hall conductivities of Mn$_{1-x}$Fe$_x$Ge alloys have been investigated by first-principles calculations using the relativistic multiple scattering Korringa-Kohn-Rostoker (KKR) formalism. The $D_{rm xx}$ component of the DMI exhibits a strong dependence on the Fe concentration, changing sign at $x approx 0.85$ in line with previous theoretical calculations as well as with experimental results demonstrating the change of spin helicity at $x approx 0.8$. A corresponding behavior with a sign change at $x approx 0.5$ is predicted also for the Fermi sea contribution to the SOT, as this is closely related to the DMI. In the case of anomalous and spin Hall effects it is shown that the calculated Fermi sea contributions are rather small and the composition-dependent behavior of these effects are determined mainly by the electronic states at the Fermi level. The spin-orbit-induced scattering mechanisms responsible for both these effects suggest a common origin of the minimum of the AHE and the sign change of the SHE conductivities.
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