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Electron-phonon mediated superconductivity in La$_6$Ni$_5$O$_{12}$ nickel oxides

Nickel oxide superconductors offer an alternative playground for understanding the formation of Cooper pairs in correlated materials such as the famous cuprates. By studying the La$_{n+1}$Ni$_n$O$_{2n+2}$ phase diagram on the basis of hybrid and spin-polarized density functional theory simulations, we reveal the existence of charge and bond ordered (CBO) insulating phases that are quenched by doping effects, ultimately resulting in a metallic phase at the n=5 member. Nevertheless, the phonons associated with the CBO identified in the phase diagram remain sufficiently large to mediate Cooper pairs in La$_6$Ni$_5$O$_{12}$, yielding a computed critical temperature between T$_c$=11-19K consistent with the 13K observed experimentally in Nd$_6$Ni$_5$O$_{12}$. Thus, in order to identify the superconducting mechanism, extracting the relevant instabilities in the doping phase diagram of superconductors appears critical.