  Self-organizing nanocheckerboards have been experimentally fabricated in
Mn-based spinels, but have not yet been explained with first principles. Using
density-functional-theory, we explain the phase diagram of the
$\mathrm{ZnMn_xGa_{2-x}O_4}$ system and the origin of nanocheckerboards. We
predict total phase separation at zero temperature, then show the combination
of kinetics, thermodynamics, and Jahn-Teller physics that generates the
system's observed behavior. We find the {011} surfaces are strongly-preferred
energetically, which mandates checkerboard ordering by purely geometrical
considerations.
