Magnetism of Ba${}_4$Ru${}_3$O${}_{10}$ revealed by density functional calculations: Structural trimers behaving as coupled magnetic dimers

Density-functional calculations have been carried out to investigate the origin of the peculiar magnetic properties of Ba${}_4$Ru${}_3$O${}_{10}$. These calculations suggest that, under the influence of Ru-Ru covalent bonding, structural trimers behave in this compound as strong $S=1$ antiferromagnetic dimers ordering below $T_N$ under the influence of weaker intertrimer couplings.

Figure 1

(a) Atomic structure of Ba${}_4$Ru${}_3$O${}_{10}$. (b) Local system of coordinates employed to decompose Ru-$4d$ LDOS. (c) Structure of a corrugated layer observed along the $b$ axis. Dominant magnetic couplings occurring between Ru(2) ions are indicated in (a) and (c) and circles highlight the structural trimers.

Figure 2

(a) Nonmagnetic DOS of Ba${}_4$Ru${}_3$O${}_{10}$ calculated in GGA. (b) Detail of Ru(1) and Ru(2) paramagnetic LDOS. (c) and (d) Ru(1) and Ru(2)-$4d$ paramagnetic LDOS decomposed on the real $t_{2g}^0$, $t_{2g}^{+}$, $t_{2g}^{-}$, $e_g^{+}$, $e_g^{-}$ orbitals and with the $z$ axis along the metal-metal bond direction within the trimer.

Figure 3

(a) DOS in Ba${}_4$Ru${}_3$O${}_{10}$ calculated in GGA+U for the ground-state AFM order. (b) Corresponding $4d$-LDOS of Ru(1) and (c) of one of the Ru(2) of a trimer.

Figure 4

(a) Results of the least-squares fitting procedure employed to calculate magnetic couplings in GGA: only the low-energy configurations shown in blue have been used, the configurations shown in green have been discarded. The energy of the ground-state AFM configuration is used as the reference energy. (b) Detailed presentation of the eight low-energy configurations considered for the evaluation of magnetic couplings. (c) and (d) The results of the same procedure applied to GGA+U. Red (blue) Ru(2)O${}_6$ coordination octahedra correspond to up (down) arrangements of the Ru(2) magnetic moments. According to the convention used in Eq. (1), AFM couplings are positive.

Figure 5

Schematic of the kinetic superexchange mechanism proposed to explain the AFM coupling within the trimers.

Figure 6

Magnetic susceptibility of Ba${}_4$Ru${}_3$O${}_{10}$ recorded up to 350 K fitted using Eq. (4).

Figure 7

(a) Dependence of energy on the direction of the Ru(2) magnetic moments in the $ab$ and $bc$ planes calculated for the ground-state AFM order in GGA+U+SO. (b) Rietveld refinement of the neutron powder diffraction pattern recorded at 1.8 K (Ref. 12). Red circles, black lines, and green lines represent the observed, calculated, and difference patterns, respectively. The expected positions of the Bragg reflections are shown as vertical bars: the upper series corresponds to the nuclear phase, while the lower series marks the magnetic contribution. The inset shows an enlargement of the (020)/(002) doublet (marked by an arrow), with the blue line corresponding to magnetic contribution to the total intensity.