Vanishing contribution of electron-phonon scattering to the anomalous Hall effect due to phonon softening

The intricate interplay between electronic and lattice degrees of freedom poses significant theoretical and experimental challenges in investigating the influence of electron-phonon scattering (EPS) on the anomalous Hall effect (AHE). In this study, by synergizing experimental measurements with theoretical calculations in $\mathrm{MnA}{\mathrm{l}}_{1\ensuremath{-}x}{\mathrm{Ga}}_{x}\mathrm{Ge}$, we observed an enhancement in the anomalous Hall conductivity (AHC) with a peak value of $922\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Omega}}}^{\ensuremath{-}1}\phantom{\rule{0.16em}{0ex}}\mathrm{c}{\mathrm{m}}^{\ensuremath{-}1}$. Notably, we found that EPS plays a pivotal role in contributing to AHE in pristine MnAlGe, whereas this contribution diminishes gradually and ultimately disappears in Ga-doped systems. Raman spectroscopy revealed that the ${A}_{1g}$ and ${B}_{2g}$ modes undergo a shift towards lower frequencies in Ga-doped samples. Furthermore, Brillouin-light-scattering measurements demonstrated a marked reduction in the energy of the acoustic phonon branches. This phonon softening behavior is attributed to the weakened hybridization of the Al-Ge bond. Our findings illuminate a potential link between the impact of EPS on AHE and phonon softening.