Fragility of the magnetic order in the prototypical altermagnet ${\mathrm{RuO}}_2$

Altermagnetism is a topic that has lately been gaining attention and the ${\mathrm{RuO}}_2$ compound is among one of the most studied altermagnetic candidates. However, the survey of available literature on ${\mathrm{RuO}}_2$ properties suggests that there is no consensus about the magnetism of this material. By performing density functional theory (DFT) calculations, we show that the electronic properties of stoichiometric ${\mathrm{RuO}}_2$ are described in terms of a Hubbard $U$, within $\mathrm{DFT}+U$, smaller than the value required to have magnetism. We further argue that Ru vacancies can actually aid the formation of a magnetic state in ${\mathrm{RuO}}_2$. This in turn suggests that a characterization of the amount of Ru vacancies in experimental samples might help the resolution of the controversy between the different experimental results.

Figure 1

Crystal structure of ${\mathrm{RuO}}_2$: Ru atoms are shown in red and blue (different colors denote different spin orientations), and O atoms are shown in teal.

Figure 2

Total energy (left) and local magnetization at the Ru site (right) as a function of $U_{\mathrm{eff}}=U-J$ are explored in two sets of calculations. In one set (“increasing $U_{\mathrm{eff}}$”), denoted with $\times$ (energy results) and $\square$ symbols (magnetization results) the calculations were done starting from $U=0$ eV and progressively increasing $U$ in each subsequent calculation. In the other set (“decreasing $U_{\mathrm{eff}}$”), denoted with $+$ and $*$ symbols, the direction of the calculation was reversed. The calculations are without SOC contributions.

Figure 3

Magnetic anisotropy as a function of the angle in degrees, when rotating the Neel vector in the indicated planes.

Figure 4

Projected nonmagnetic density of states onto Ru $d$ orbitals: blue, orange, cyan, red, and teal are used to depict $z^2, x^2-y^2, xy, xz$, and $yz$ orbitals, respectively. The coordinate system is aligned with Ru-O bonds.

Figure 5

Spin-polarized total density of states for majority spin as a function of $U_{\mathrm{eff}}$ (in eV). The color corresponds to various values of $U_{\mathrm{eff}}$, where solid lines in blue shades correspond to the nonmagnetic ground state and dashed lines in red shades correspond to the AM order. Note that here the calculations were done using wien2k; thus, the value of $U_{\mathrm{eff}}$ is not directly comparable with other calculations where vasp was employed due to the differences in implementations.

Figure 6

Dependence of the local atomic magnetization $m$ on hole doping (an undoped case corresponds to 56 electrons per cell, i.e., per two formula units) and the effective Hubbard parameter $U_{\mathrm{eff}}=U-J$. Isolines for $m=0.05$, 0.10, and 0.15 ${\mu }_B$ are depicted by white lines (with no attempts to smooth the plotted lines).