Composition-dependent magnetic response properties of ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Ge}$ alloys

The composition-dependent behavior of the Dzyaloshinskii–Moriya interaction (DMI), the spin-orbit torque (SOT), as well as anomalous and spin Hall conductivities of ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Ge}$ alloys have been investigated by first-principles calculations using the relativistic multiple scattering Korringa–Kohn–Rostoker (KKR) formalism. The $D_{\mathrm{xx}}$ 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, because 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 anomalous Hall effect and the sign change of the spin Hall effect conductivities.

Figure 1

(a) Results for $D_{xx}\left(x\right)$ in ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Ge}$ calculated by using Eq. (3) (circles) and for $D_{xx}\left(x\right)$ (diamonds) and $D_{yy}\left(x\right)$ (empty squares) calculated by using using Eq. (2) in comparison with the results of other calculations from Ref. [12] (filled squares) and Ref. [30] (triangles). (b) The element-resolved Dzyaloshinskii–Moriya interaction in ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Ge}{D}_{xx}^{\text{Mn}}$ (triangles up) and $D_{xx}^{\text{Fe}}$ (triangles down). The total $D_{xx}\left(x\right)$ function is again shown as circles as in panel (a).

Figure 2

The spin- and element-resolved DOS on (a) Mn and (b) Fe atoms in ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Ge}$ for $x=0.1$, 0.5, and 0.9.

Figure 3

The spin-resolved Bloch spectral function in ${\mathrm{Mn}}_{0.9}{\mathrm{Fe}}_{0.1}\mathrm{Ge}$ and ${\mathrm{Mn}}_{0.1}{\mathrm{Fe}}_{0.9}\mathrm{Ge}$ for (a), (b) minority- and (c), (d) majority-spin states, respectively.

Figure 4

Spin magnetic moments of Mn (circles) and Fe (squares) in ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Ge}$ alloy, and the average magnetic moment per site (diamonds). Experimental results of Kanazawa et al. [34] are shown as green triangles.

Figure 5

(a) Total torkance per unit cell (solid squares) as well as its Fermi surface (empty circles) and Fermi sea (filled circles) contributions in ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Ge}$ calculated via the Kubo–Bastin formalism [Eq. (4)]. (b) Comparison of Fermi-sea contribution to the torkance calculated via Eq. (4) (circles) with results obtained using the expressions (5) (triangles) and (6) (squares).

Figure 6

Energy dependence of the DMI parameter $D_{xx}\left(E\right)$ (upper panel) and the torkance $t_{xx}\left(E\right)$ (lower panel) in ${\mathrm{Mn}}_{0.7}{\mathrm{Fe}}_{0.3}\mathrm{Ge}$ (circles) and ${\mathrm{Mn}}_{0.1}{\mathrm{Fe}}_{0.9}\mathrm{Ge}$ (squares). Integrated up to the Fermi energy $E_{\mathrm{F}}$, these functions give $t_{xx}$ and $D_{xx}$.

Figure 7

(a) Anomalous Hall conductivity calculated for ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Ge}$ via the CPA–Kubo–Bastin formalism (circles), compared with calculations using the Berry-curvature approach and the virtual crystal approximation [12] (triangles), and with low-temperature experimental data (squares) [34]. (b) Anomalous Hall coefficient calculated via the Kubo–Bastin equation (circles) compared with experimental data at 50 K (squares) [34].

Figure 8

(a) Anomalous and (b) spin Hall conductivities ${\sigma }_{xy}$ and ${\sigma }_{xy}^z$, respectively, as functions of $x$ in ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Ge}$ calculated via the Kubo–Bastin formalism. Fermi- surface contributions are given in red, those from the Fermi sea in blue, and their sum in black. The superscripts 0 and 1 indicate on- and off-site terms. Results for the latter are shown excluding (NV) and including vertex corrections (VC).

Figure 9

Comparison of all nonzero torkance tensor elements as functions of $x$ in ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Ge}$ calculated via the Kubo–Bastin formalism. The diagonal elements $t_{xx}$ and $t_{yy}$ (left $y$ scale) are given as black squares and red circles, respectively, the off-diagonal torkances (right $y$ scale) $t_{xy}$ and $-t_{yx}$ are given as blue up- and green down-facing triangles.