Large anomalous Nernst and spin Nernst effects in the noncollinear antiferromagnets ${\mathrm{Mn}}_{3}X$ ($X=\mathrm{Sn},\mathrm{Ge},\mathrm{Ga}$)

Noncollinear antiferromagnets have recently been attracting considerable interest partly due to recent surprising discoveries of the anomalous Hall effect (AHE) in them and partly because they have promising applications in antiferromagnetic spintronics. Here we study the anomalous Nernst effect (ANE), a phenomenon having the same origin as the AHE, and also the spin Nernst effect (SNE) as well as AHE and the spin Hall effect (SHE) in noncollinear antiferromagnetic ${\mathrm{Mn}}_{3}X$ ($X=\mathrm{Sn}$, Ge, Ga) within the Berry phase formalism based on ab initio relativistic band structure calculations. For comparison, we also calculate the anomalous Nernst conductivity (ANC) and anomalous Hall conductivity (AHC) of ferromagnetic iron as well as the spin Nernst conductivity (SNC) of platinum metal. Remarkably, the calculated ANC at room temperature (300 K) for all three alloys is huge, being 10–40 times larger than that of iron. Moreover, the calculated SNC for ${\mathrm{Mn}}_3\mathrm{Sn}$ and ${\mathrm{Mn}}_3\mathrm{Ga}$ is also larger, being about five times larger than that of platinum. This suggests that these antiferromagnets would be useful materials for thermoelectronic devices and spin caloritronic devices. The calculated ANC of ${\mathrm{Mn}}_3\mathrm{Sn}$ and iron are in reasonably good agreement with the very recent experiments. The calculated SNC of platinum also agrees with the very recent experiments in both sign and magnitude. The calculated thermoelectric and thermomagnetic properties are analyzed in terms of the band structures as well as the energy-dependent AHC, ANC, SNC, and spin Hall conductivity via the Mott relations.

Figure 1

(a) Layered hexagonal ($D_{6h}^4$) structure of ${\mathrm{Mn}}_{3}X$ ($X=\mathrm{Sn}$, Ge, Ga) with (b) the associated hexagonal Brillouin zone (BZ). (c) Type A and (d) type B antiferromagnetic configurations considered in this paper. Both magnetic structures have an orthorhombic symmetry and thus their irreducible BZ wedge (IBZW) [i.e., the trapzian prism indicated by the blue dashed lines in (b)] is three times larger than the hexagonal IBZW [i.e., the triangle prism indicated by the blue dashed lines in (b)]. The vertical [horizontal] black dashed line in (c) [(d)] denotes the mirror plane.

Figure 2

Berry curvature $[{\mathrm{\Omega }}^x\left(\mathbf{k}\right),{\mathrm{\Omega }}^y\left(\mathbf{k}\right),{\mathrm{\Omega }}^z\left(\mathbf{k}\right)]$ (in units of ${\AA }^2$) of configuration A ${\mathrm{Mn}}_3\mathrm{Sn}$. (a) (${\mathrm{\Omega }}^x,{\mathrm{\Omega }}^y$) on the $k_x{k}_y$ ($k_z=0$) plane and (b) (${\mathrm{\Omega }}^x,{\mathrm{\Omega }}^z$) on the $k_x{k}_z$ ($k_y=0$) plane.

Figure 3

${\mathrm{Mn}}_3\mathrm{Sn}$. (a) Relativistic band structure in magnetic structure A. (b) Anomalous Hall conductivity (AHC) (${\sigma }_{yz}^A$) and spin Hall conductivity (SHC) (${\sigma }_{xy}^z$) as well as (c) anomalous Nernst conductivity (ANC) (${\alpha }_{yz}^A$) and spin Nernst conductivity (SNC) (${\alpha }_{xy}^z$) as a function of energy. The Nernst conductivities were calculated at $T=300$ K. The Fermi level is at the zero energy. Note that in (b) [(c)], the unit for the SHC [SNC] should be ($\hslash /e$) S/cm [($\hslash /e$) A/m K].

Figure 4

${\mathrm{Mn}}_3\mathrm{Ge}$. (a) Relativistic band structure in the A magnetic structure. (b) Anomalous Hall conductivity (AHC) (${\sigma }_{zx}^A$) and spin Hall conductivity (SHC) (${\sigma }_{xy}^z$) as well as (c) anomalous Nernst conductivity (ANC) (${\alpha }_{zx}^A$) and spin Nernst conductivity (SNC) (${\alpha }_{xy}^z$) as a function of energy. The Nernst conductivities were calculated at $T=300$ K. The Fermi level is at the zero energy. Note that in (b) [(c)], the unit for the SHC [SNC] should be ($\hslash /e$) S/cm [($\hslash /e$) A/m  K].

Figure 5

${\mathrm{Mn}}_3\mathrm{Ga}$. (a) Relativistic band structure in the A magnetic structure. (b) Anomalous Hall conductivity (AHC) (${\sigma }_{zx}^A$) and spin Hall conductivity (SHC) (${\sigma }_{xy}^z$) as well as (c) anomalous Nernst conductivity (ANC) (${\alpha }_{zx}^A$) and spin Nernst conductivity (SNC) (${\alpha }_{xy}^z$) as a function of energy. The Nernst conductivities were calculated at $T=300$ K. The Fermi level is at the zero energy. Note that in (b) [(c)], the unit for the SHC [SNC] should be ($\hslash /e$) S/cm [($\hslash /e$) A/m K].

Figure 6

(a) Anomalous Nernst conductivity (ANC) (${\alpha }^A$) and (b) spin Nernst conductivity (SNC) (${\alpha }_{xy}^S$) as a function of temperature. The values displayed on the vertical dashed line (a) and (b) are the ANC and SNC at 100 K calculated using the Mott relations [Eqs. (5) and (6)] and the energy derivatives of AHC and SHC listed in Table 2, respectively.