Dzyaloshinskii-Moriya Interaction and Hall Effects in the Skyrmion Phase of ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Ge}$

We carry out density functional theory calculations which demonstrate that the electron dynamics in the Skyrmion phase of Fe-rich ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Ge}$ alloys is governed by Berry phase physics. We observe that the magnitude of the Dzyaloshinskii-Moriya interaction directly related to the mixed space-momentum Berry phases, changes sign and magnitude with concentration $x$ in direct correlation with the data of Shibata et al. [Nat. Nanotechnol. 8, 723 (2013)]. The computed anomalous and topological Hall effects in FeGe are also in good agreement with available experiments. We further develop a simple tight-binding model able to explain these findings. Finally, we show that the adiabatic Berry phase picture is violated in the Mn-rich limit of the alloys.

Figure 1

Strength of the DMI as a function of concentration $x$ in ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Ge}$ alloys. The total value of the DMI (filled squares) is decomposed into the contributions coming from the transition metal (red dots) and Ge (blue dots). The inset depicts the crystal structure of the studied B20 compound, with light gray and dark gray spheres representing the transition-metal (TM) and Ge atoms, respectively.

Figure 2

Bottom: Strength of the DMI as a function of electron occupation computed from a simple tight-binding model of a finite trimer (structure shown in inset). Essential is the breaking of local inversion symmetry of the bond between the transition-metal (light gray) and Ge (dark gray). The direction of the left (${\mathbf{S}}_1$) and right (${\mathbf{S}}_2$) spin lies in the $xy$ plane, and the DMI vector is pointing out of plane. Top: schematic evolution of the $d$ states at the Fermi energy in the tight-binding model with electron occupation.

Figure 3

Computed anomalous Hall conductivity ${\sigma }_{xy}$ and the absolute value of topological Hall constant $R_{yx}^{\text{top}}$ as a function of concentration $x$ in ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Ge}$ alloys. The thin line marks the critical concentration $x_c=0.8$.