Topology analysis for anomalous Hall effect in the noncollinear antiferromagnetic states of ${\mathrm{Mn}}_{3}A\mathrm{N}$ $(A=\mathrm{Ni}, \mathrm{Cu}, \mathrm{Zn}, \mathrm{Ga}, \mathrm{Ge}, \mathrm{Pd}, \mathrm{In}, \mathrm{Sn}, \mathrm{Ir}, \mathrm{Pt})$

We investigate topological features of electronic structures which produce large anomalous Hall effect in the noncollinear antiferromagnetic metallic states of antiperovskite manganese nitrides by first-principles calculations. We first predict the stable magnetic structures of these compounds to be noncollinear antiferromagnetic structures characterized by either $T_{1g}$ or $T_{2g}$ irreducible representation by evaluating the total energy for all of the magnetic structures classified according to the symmetry and multipole moments. The topology analysis is next performed for the Wannier tight-binding models obtained from the first-principles band structures. Our results reveal the small Berry curvature induced through the coupling between occupied and unoccupied states with the spin-orbit coupling, which is widely spread around the Fermi surface in the Brillouin zone, dominantly contributes after the $\mathit{k}$-space integration to the anomalous Hall conductivity, while the local divergent Berry curvature around Weyl points has a rather small contribution to the anomalous Hall conductivity.

Figure 1

Energetically inequivalent magnetic structures of ${\mathrm{Mn}}_{3}A\mathrm{N}$ classified according to the multipole moments following Ref. [20]. The green, yellow, and blue balls indicate Mn, $A$, and N atoms, respectively. Arrows on Mn atoms indicate the magnetic moments.

Figure 2

Energy bands from the first-principles calculations (red) and from Wannier interpolation (green) of (a) ${\mathrm{Mn}}_3\mathrm{NiN}$, (b) ${\mathrm{Mn}}_3\mathrm{GeN}$, and (c) ${\mathrm{Mn}}_3\mathrm{IrN}$ along high-symmetry points in the first Brillouin zone of a simple cubic shown in Fig. 3.

Figure 3

The first Brillouin zone (black) corresponding to the crystal primitive unit cell with the high-symmetry points. The hexagonal plane (green line) shows minimum periodicity in the (111) plane for the simple cubic Brillouin zone with the center point at $\mathrm{\Gamma }$. The orange rectangle is the region used to plot the Berry curvature in Fig. 5.

Figure 4

Total energy as the function of lattice constants for different magnetic configurations in ${\mathrm{Mn}}_3\mathrm{GaN}$. The equilibrium total energy of the $(M_x^{\alpha },M_y^{\alpha },M_z^{\alpha })=\left(111\right)$ magnetic structure is chosen as the origin of total energy. The values are fitted to Birch-Murnaghan's equation of state [31] by the least-square method.

Figure 5

The [111] Berry curvature component after taking band summation, ${\mathrm{\Omega }}_{\mathrm{sum}}^{111}\equiv \frac{1}{\sqrt{3}}({\mathrm{\Omega }}_{yz,\mathrm{sum}}+{\mathrm{\Omega }}_{zx,\mathrm{sum}}+{\mathrm{\Omega }}_{xy,\mathrm{sum}})$, on (111) plane centered at $\mathrm{\Gamma }$, shown in Fig. 3, for ${\mathrm{Mn}}_3\mathrm{GeN}$ with (a) the MTQ and (b) the MO configuration, respectively.

Figure 6

Number of Weyl points around the Fermi level (black boxes and blue boxes) with the calculated AH conductivity (red dots) for the series of ${\mathrm{Mn}}_{3}A\mathrm{N}$.

Figure 7

Band structure in (a) ${\mathrm{Mn}}_3\mathrm{GeN}$ and (b) ${\mathrm{Mn}}_3\mathrm{PtN}$ along high-symmetry lines in the first Brillouin zone of a simple cubic shown in Fig. 3.

Figure 8

(a) Band structure and (b) Berry curvature along the [111] direction, ${\vec{k}}_{111}$, having Weyl points near Fermi energy that produce the positive Berry curvature after taking band summation in ${\mathrm{Mn}}_3\mathrm{SnN}$. Each panel shows an interval $0.109({\AA }^{-1}$) along ${\vec{k}}_{111}$ with Weyl point at the middle of the line. The relative energies with respect to the Weyl points are written in red, the blue number $+1$ and $-1$ indicate the chiralities of the Weyl points. The value “$-0$ meV” indicates the Weyl point within the energy range of −1 $\text{meV}<E<0$ meV. The coordinates of these Weyl points in the reciprocal space from left to right are ${\mathrm{W}}_1=(-0.06,-0.34,-0.34)$, ${\mathrm{W}}_2=(-0.04,0.34,0.34)$, ${\mathrm{W}}_3=(-0.05,0.44,-0.16)$, ${\mathrm{W}}_4=(-0.05,-0.16,0.44)$, ${\mathrm{W}}_5=(-0.16,-0.05,0.44)$, ${\mathrm{W}}_6=(-0.34,-0.34,0.03),$ and ${\mathrm{W}}_7=(-0.15,0.47,0.05)$, respectively.

Figure 9

The bar chart showing contribution of the Berry curvature to the resultant AH conductivity of ${\mathrm{Mn}}_{3}A\mathrm{N}$ with the $A$ elements having (a) small and (b) large SOC. The horizontal axis is the absolute intensity of the Berry curvature. The contribution is also shown for ${\mathrm{Mn}}_3\mathrm{Ir}$, which shows the same magnetic alignment on Mn atoms in ${\mathrm{Mn}}_{3}A\mathrm{N}$.

Figure 10

The Berry curvature integrated on the (111) hexagonal area as shown in Fig. 3 with its center changing from $\mathrm{\Gamma }$ to $R$ for ${\mathrm{Mn}}_{3}A\mathrm{N}$ ($A=\text{Ni}$, Pd, Pt).

Figure 11

(a)–(c) Distribution of the Berry curvature after taking band summation, ${\mathrm{\Omega }}_{\mathrm{sum}}^{111}$, and Fermi surfaces on the BZ plane as shown in Fig. 3 with its center point of $R$. (d)–(f) The band structure around Fermi energy and (g)–(i) Berry curvature ${\mathrm{\Omega }}_{\mathrm{sum}}^{111}$ on the $G_1\text{−}{G}_2$ line shown in (a)–(c), respectively.