Spin-order dependent anomalous Hall effect and magneto-optical effect in the noncollinear antiferromagnets Mn3XN with X=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 ${\mathrm{Mn}}_{3}X\mathrm{N}\phantom{\rule{4pt}{0ex}}(X=\mathrm{Ga}$, 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---${\ensuremath{\sigma}}_{xy},{\ensuremath{\sigma}}_{yz}$, and ${\ensuremath{\sigma}}_{zx}$---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 ${\ensuremath{\sigma}}_{xy}$ 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 $[{\ensuremath{\sigma}}_{xx}(\ensuremath{\omega})\ensuremath{\approx}{\ensuremath{\sigma}}_{yy}(\ensuremath{\omega})\ensuremath{\approx}{\ensuremath{\sigma}}_{zz}(\ensuremath{\omega})]$ and the magneto-optical anisotropy $[{\ensuremath{\sigma}}_{xy}(\ensuremath{\omega})\ensuremath{\ne}{\ensuremath{\sigma}}_{yz}(\ensuremath{\omega})\ensuremath{\ne}{\ensuremath{\sigma}}_{zx}(\ensuremath{\omega})]$ 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 $0.{3}^{\ensuremath{\circ}}\ensuremath{\sim}0.{4}^{\ensuremath{\circ}}$, 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.