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

From a simple ionic picture, the only magnetically active ions in this compound are the three Ru$\sp{4+}$ atoms which form trimers of faced shared RuO$_6$ octahedral. The Ru atom in the middle of the trimer (named Ru(1)) is cristallographically inequivalent to the ones at the corners (named Ru(2)). A na\"ive analysis of the magnetic properties of this compound compatible with the expected low spin magnetic configuration of the Ru ions would predict a complicate magnetic order at low temperature involving the Ru(1) and Ru(2) ions and a high temperature susceptibility corresponding to three S=1 ions per unit cell. In spite of that, we demonstrate in this work, from density functional calculations, that under the influence of Ru-Ru covalent bonding, the structural trimers behave in an extended range of temperature from 0 to 600K, as strong ($S=1$) antiferromagnetic dimers. Our calculations of the effective exchange interactions show a strong intra-dimer interaction and a weaker inter-dimer one which explains the antiferromagnetic order observed below $T_N = 105K$ and the magnetic susceptibility in the intermediate and high temperature range (from $T_N$=105K up to 612~K).