Ferroic Order for Anisotropic Magnetic Dipole Term in Collinear Antiferromagnets of $(t_{2g}{)}^4$ System

We present the possibility of x-ray magnetic circular dichroism on ${\mathrm{RuO}}_2$ with collinear antiferromagnetism (AFM). Given that the crystal symmetry breaks the time reversal symmetry when the antiparallel spin aligns along the [100] direction, the expectation vector of the anisotropic magnetic dipole operator $⟨\mathit{t}⟩$ remains uncanceled along the [010] direction. Our Letter reveals that the magnetic dipole ($T_z$) term in the x-ray magnetic circular dichroism is induced by the residual $⟨\mathit{t}⟩$. Because the features of the magnetic moment can be detected via absorption measurements even in the AFM, this technique will be useful for determining the magnetic phase, the Van Vleck–type paramagnet or the excitonic AFM in $(t_{2g}{)}^4$ system.

Figure 1

(a) Crystal and magnetic structure with the Néel vector aligned along the $[\overline{1}00]$ direction for the rutile structure. (b) Definition of local coordinates of ${\mathrm{RuO}}_6$ cluster, and (c) electronic configuration for an intermediate spin state $(t_{2g}{)}^4$ without the spin-orbit interaction (SOI). Two approximation states, i.e., (d) $jj$ coupling and (e) $LS$ coupling, are shown for the electronic state with SOI.

Figure 2

Anisotropic magnetic dipole operator $⟨\mathit{t}⟩$ for the collinear antiferromagnet rutile, (a), (b) $d_{3z^2-r^2}$, (c) $d_{x^2-y^2}$, (d) $d_{xy}$, (e) $d_{yz}$, and (f) $d_{zx}$ orbitals. The Néel vector is set to [100] for (b) and $[\overline{1}00]$ for the other frames.

Figure 3

(a) Phase diagram and $⟨T_z⟩$ values for the $(t_{2g}{)}^4$ model. The inset shows the direction of $h_{\mathrm{MF}}$ on the ${\mathrm{RuO}}_6$ cluster under ${\mathit{h}}_{\mathrm{MF}}\parallel \mathrm{QA}$. (b) $⟨T_z⟩$ value of the ${\mathrm{Ru}}^{4+}$ model. $⟨T_z⟩$ values are indicated by colored symbols, and the black dotted line indicates the $⟨T_z⟩$ behavior of the $(t_{2g}{)}^4$ model where $g_J{h}_{\mathrm{MF}}/\mathrm{\Delta }=0.0118$ when $\lambda /\mathrm{\Delta }=0.1$. ${\zeta }_{4d}$ is the spin-orbit coupling constant for the $4d$ orbital, where the ${\zeta }_{4d}$ value of ${\mathrm{Ru}}^{4+}$ ion (${\zeta }_{4d}^{{\mathrm{Ru}}^{4+}}$) is 0.161 eV.

Figure 4

X-ray absorption spectroscopy (upper result) and XMCD (lower result) spectra around absorption Ru-$L$ edges where (a) $\mathit{k}\parallel [010]$ and (b) $\mathit{k}\parallel [100]$: the right (left) structures denote $L_{\mathrm{II}}$ ($L_{\mathrm{III}}$).