The origin and scarcity of breathing pyrochlore lattices in spinel oxides

Breathing pyrochlores are a unique class of materials characterized by a three-dimensional lattice of corner-sharing tetrahedra. However, unlike conventional pyrochlores where all tetrahedra are identical in size, the breathing pyrochlore lattice is composed of alternating large and small tetrahedra. Experimental realizations of the breathing pyrochlore lattice are rare but they do occur in $A$-site ordered spinels, as in the prototype materials LiGaCr$_4$O$_8$ and LiInCr$_4$O$_8$. In this work, we demonstrate that Cr cannot be straightforwardly substituted with other magnetic transition metals while retaining the breathing pyrochlore structure. To explain this observation, we perform density functional theory (DFT) calculations to investigate the formation and stability of LiGaCr$_4$O$_8$ and LiInCr$_4$O$_8$, focusing on the energy scales associated with $A$ and $B$-site orderings as well as the magnetic exchange interactions of Cr ions. We identify the strong octahedral site preference of Cr$^{3+}$ as a key factor in protecting the structural integrity of the pyrochlore sublattice, which in turn enables the breathing distortion to proceed. We also demonstrate that the charge order between the $A$ (Li) and $A'$ sites (Ga or In) is maintained at all temperatures up to decomposition. Furthermore, while the magnetic exchange interactions constitute a relatively small energy scale and therefore do not play a fundamental role in the structural stability of LiGaCr$_4$O$_8$ and LiInCr$_4$O$_8$, magnetism may play a critical role in setting the magnitude of the tetrahedral distortion. Ultimately, we conclude that the scarcity of breathing pyrochlores is a consequence of the stringent requirement on site ordering for their formation.