Spin-order dependent anomalous Hall effect and magneto-optical effect in the noncollinear antiferromagnets ${\mathrm{Mn}}_{3}X\mathrm{N}$ with $X=\mathrm{Ga}$, Zn, Ag, or Ni

The anomalous Hall effect (AHE) and the magneto-optical effect (MOE) are two prominent manifestations of time-reversal symmetry breaking in magnetic materials. Noncollinear antiferromagnets (AFMs) have recently attracted a lot of attention owing to the potential emergence of exotic spin orders on geometrically frustrated lattices, which can be characterized by corresponding spin chiralities. By performing first-principles density functional calculations together with group-theory analysis and tight-binding modeling, here we systematically study the spin-order dependent AHE and MOE in representative noncollinear AFMs ${\mathrm{Mn}}_{3}X\mathrm{N}(X=\mathrm{Ga}$, Zn, Ag, and Ni). The symmetry-related tensor shape of the intrinsic anomalous Hall conductivity (IAHC) for different spin orders is determined by analyzing the relevant magnetic point groups. We show that while only the $xy$ component of the IAHC tensor is nonzero for right-handed spin chirality, all other elements—${\sigma }_{xy},{\sigma }_{yz}$, and ${\sigma }_{zx}$—are nonvanishing for a state with left-handed spin chirality owing to lowering of the symmetry. Our tight-binding arguments reveal that the magnitude of IAHC relies on the details of the band structure and that ${\sigma }_{xy}$ is periodically modulated as the spin rotates in-plane. The IAHC obtained from first principles is found to be rather large, e.g., it amounts to 359 S/cm in ${\mathrm{Mn}}_3\mathrm{AgN}$, which is comparable to other well-known noncollinear AFMs such as ${\mathrm{Mn}}_3\mathrm{Ir}$ and ${\mathrm{Mn}}_3\mathrm{Ge}$. We evaluate also the magnetic anisotropy energy and find that the evolution of spin order is related to the number of valence electrons in the $X$ ion. Interestingly, the left-handed spin chirality could exist in ${\mathrm{Mn}}_{3}X\mathrm{N}$ with some particular spin configurations. By extending our analysis to finite frequencies, we calculate the optical isotropy $[{\sigma }_{xx}\left(\omega \right)\approx {\sigma }_{yy}\left(\omega \right)\approx {\sigma }_{zz}\left(\omega \right)]$ and the magneto-optical anisotropy $[{\sigma }_{xy}\left(\omega \right)\ne {\sigma }_{yz}\left(\omega \right)\ne {\sigma }_{zx}\left(\omega \right)]$ of ${\mathrm{Mn}}_{3}X\mathrm{N}$. Similar to the IAHC, the magneto-optical Kerr and Faraday spectra depend strongly on the spin order. The Kerr rotation angles in ${\mathrm{Mn}}_{3}X\mathrm{N}$ are in the range of $0.3^{\circ }\sim 0.4^{\circ }$, which is large and comparable to other noncollinear AFMs like ${\mathrm{Mn}}_3\mathrm{Pt}$ and ${\mathrm{Mn}}_3\mathrm{Sn}$. Our finding of large AHE and MOE in ${\mathrm{Mn}}_{3}X\mathrm{N}$ suggests that these materials present an excellent antiferromagnetic platform for realizing novel spintronics and magneto-optical devices. We argue that the spin-order dependent AHE and MOE are indispensable in detecting complex spin structures in noncollinear AFMs.

Figure 1

(a) Right-handed $\left(\kappa =+1\right)$ and (b) left-handed $\left(\kappa =-1\right)$ vector spin chiralities in coplanar noncollinear spin systems. The open arrows indicate the clockwise rotation of spin with a uniform angle, which results in a different spin configuration with the same spin chirality. (c) The crystal and magnetic structures of ${\mathrm{Mn}}_{3}X\mathrm{N}(X=\mathrm{Ga}$, Zn, Ag, and Ni). The purple, green, and blue balls represent Mn, $X$, and N atoms, respectively. The spin magnetic moments originate mainly from Mn atoms, while the spin polarization of $X$ and N atoms is negligible. The spins on three Mn sublattices $({\mathrm{Mn}}_1,{\mathrm{Mn}}_2$, and ${\mathrm{Mn}}_3)$ are indicated by red arrows that are aligned within the (111) plane (here, the right-handed spin chirality is shown as an example). The angles between neighboring spins are always ${120}^{\circ }$, while the spins can simultaneously rotate within the (111) plane that is characterized by an azimuthal angle $\theta$ away from the diagonals of the face. (d) The (111) plane of ${\mathrm{Mn}}_{3}X\mathrm{N}$, which can be regarded as a kagome lattice of Mn atoms. The dotted lines mark the two-dimensional unit cell. (e)–(g) The R1, R2, and R3 phases with the right-handed spin chirality. There are one three-fold rotation axis $[C_3$, which is along the [111] direction $(z$ axis)], three two-fold rotation axes $(C_2^{\left(1\right)},C_2^{\left(2\right)}$, and $C_2^{\left(3\right)})$, and three mirror planes $(M^{\left(1\right)},M^{\left(2\right)}$, and $M^{\left(3\right)})$ in the R1 phase; only $C_3$ axis is preserved in the R2 phase; the time-reversal symmetry $T$ has to be combined with a two-fold rotation and mirror symmetries in the R3 phase. (h)–(j) The L1, L2, and L3 phases with the left-handed spin chirality. There are one two-fold rotation axis $\left(C_2\right)$ and one mirror plane $\left(M\right)$ in the L1 phase; the time-reversal symmetry $T$ is combined with two-fold rotation and mirror symmetries in both the L2 and the L3 phases.

Figure 2

(a) Band structures of the kagome lattice with the spin orders $\kappa =+1$ and $\theta =0^{\circ },{30}^{\circ },{90}^{\circ }$. (b) Band structures of the kagome lattice with the spin orders $\kappa =-1$ and $\theta =0^{\circ },{15}^{\circ },{30}^{\circ }$. (c) IAHC of the kagome lattice as a function of $\theta$ for $\kappa =\pm 1$ states for the three positions of the Fermi energy $E_F$ at 1.8 eV (top panel), 0 eV (middle panel), and $-1.8\mathrm{eV}$ (bottom panel). The curves of the $\kappa =-1$ state (green lines) are scaled by a factor of 10. (d)–(g) Berry curvature ${\mathrm{\Omega }}_{xy}\left(\mathit{k}\right)$ with $\kappa =+1$ and $\theta =0^{\circ },{30}^{\circ },{90}^{\circ },{270}^{\circ }$ at $E_F=-1.8\mathrm{eV}$. (h)–(k) Berry curvature ${\mathrm{\Omega }}_{xy}\left(\mathit{k}\right)$ with $\kappa =-1$ and $\theta =0^{\circ },{15}^{\circ },{30}^{\circ },{90}^{\circ }$ at $E_F=-1.8\mathrm{eV}$. Dotted lines in panels (d)–(k) indicate the first Brillouin zone.

Figure 3

The first-principles band structures of (a,b) ${\mathrm{Mn}}_3\mathrm{ZnN}$ and (c,d) ${\mathrm{Mn}}_3\mathrm{NiN}$ for different spin orders $(\theta =0^{\circ },{30}^{\circ }$, and ${90}^{\circ }$ for the right-handed state with $\kappa =+1$, and $\theta =0^{\circ },{15}^{\circ }$, and ${30}^{\circ }$ for the left-handed state of opposite spin chirality). The $k$-path lies within the (111) plane.

Figure 4

(a) Magnetic anisotropy energy of ${\mathrm{Mn}}_{3}X\mathrm{N}(X=\mathrm{Ga}$, Zn, Ag, and Ni) as a function of the azimuthal angle $\theta$. The results for left-handed spin chirality $\left(\kappa =-1\right)$ are only shown in ${\mathrm{Mn}}_3\mathrm{ZnN}$ and ${\mathrm{Mn}}_3\mathrm{NiN}$ as the representatives. The solid and dotted lines are expressed by $\text{MAE}\left(\theta \right)=K_{\text{eff}}{sin}^2\left(\theta \right)$ and $\text{MAE}=K_{\text{eff}}/2$, respectively. (b), (c) The IAHC of ${\mathrm{Mn}}_3\mathrm{ZnN}$ and ${\mathrm{Mn}}_3\mathrm{NiN}$ as a function of the azimuthal angle $\theta$ for both right-handed $\left(\kappa =+1\right)$ and left-handed $\left(\kappa =-1\right)$ spin chiralities. The solid and dotted lines are the polynomial fits to the data.

Figure 5

Energy dependence of the optical conductivity in ${\mathrm{Mn}}_3\mathrm{NiN}$. (a,b) Real and imaginary parts of ${\sigma }_{xx}$ for the $\kappa =+1$ state. Characteristic peaks and valleys are marked by black arrows. (c,d) Real and imaginary parts of ${\sigma }_{xy}$ for the $\kappa =+1$ state. (e,f) The real and imaginary parts of ${\sigma }_{xx}$ for the $\kappa =-1$ state. (g,h), (i,j), (k,l) The real and imaginary parts of ${\sigma }_{xy},{\sigma }_{yz}$, and ${\sigma }_{zx}$ for the $\kappa =-1$ state.

Figure 6

Magneto-optical spectra of ${\mathrm{Mn}}_{3}X\mathrm{N}$ for $X=$ (a) Ga, (b) Zn, (c) Ag, and (d) Ni in the $\kappa =+1$ spin configuration. The panels from left to right show Kerr rotation angle ${\theta }_K^z$, Kerr ellipticity ${\varepsilon }_K^z$, Faraday rotation angle ${\theta }_F^z$, and Faraday ellipticity ${\varepsilon }_F^z$, respectively.

Figure 7

The magneto-optical spectra of ${\mathrm{Mn}}_3\mathrm{NiN}$ for $\kappa =-1$ state: (a) ${\varphi }_{K,F}^x$, (b) ${\varphi }_{K,F}^y$, and (c) ${\varphi }_{K,F}^z$. The panels from left to right are Kerr rotation angle, Kerr ellipticity, Faraday rotation angle, and Faraday ellipticity, respectively.