  Molybdenum borides were studied theoretically using first-principles
calculations, empirical total energy model and global optimization techniques
to determine stable crystal structures. Our calculations reveal the structures
of known Mo-B phases, attaining close agreement with experiment. Following our
developed lattice model, we describe in detail the crystal structure of
boron-rich $MoB_x$ phases with 3<x<9 as the hexagonal $P6_3/mmc$-$MoB_3$
structure with Mo atoms partially replaced by triangular boron units. The most
energetically stable arrangement of these $B_3$ units corresponds to their
uniform distribution in the bulk of the crystal structure, which leads to the
formation of a disordered nonstoichiometric phase, with ordering arising at
compositions close to x=5 due to a strong repulsive interaction between
neighboring $B_3$ units. The most energetically favorable structures of $MoB_x$
correspond to the compositions 4<x<5, with $MoB_5$ being the boron-richest
stable phase. The estimated hardness of $MoB_5$ is 37-39 GPa, suggesting that
the boron-rich phases are potentially superhard.
