  The absence of both spin freezing and of a static Jahn-Teller effect have
lead to the proposition that Ba$_3$CuSb$_2$O$_9$ is a quantum spin-orbital
liquid. However, theoretical understanding of the microscopic origin of this
behavior has been hampered by a lack of consensus on the lattice structure. Cu
ions have been proposed to realize either a triangular lattice, a short-range
ordered honeycomb lattice or a disordered lattice with stripelike correlations.
Here we analyze the stability of idealized versions of these arrangements using
density functional theory. We find stripe order of Cu ions to be energetically
favored, hinting towards stripelike local Cu-Cu arrangements, while long-range
order is presumably hindered due to disorder effects. Furthermore, we find
evidence of significant interlayer interactions between Cu-Sb dumbbells, which
affects the out-of-plane arrangement. Analysis of the relaxed crystal
structures, electronic properties and tight-binding parameters provides clues
as to the nature of the Jahn-Teller distortions.
