Chain metallicity and antiferro-paramagnetism competition in underdoped YBa${}_2$Cu${}_3$O${}_{6+x}$: a first principles description

We describe from advanced first principles calculations the energetics of oxygen doping and its relation to insulator-metal transitions in underdoped YBa${}_2$Cu${}_3$O${}_{6+x}$. We find a strong tendency of doping oxygens to order into non-magnetic Cu${}^{1+}$O${}_x$ chains at any $x$. Ordering produces one-dimensional metallic bands, while configurations with non-aligned oxygens are insulating. The Cu${}^{2+}$O${}_2$ planes remain insulating and antiferromagnetic up to a threshold between $x$=0.25 and 0.5, above which a paramagnetic normal-metal state prevails. The in-plane antiferro-paramagnetic competition depends on $x$, but only weakly on the ordering state of the chains.