  Bi$_2$Se$_3$ is a topological insulator and it is often doped with Te to
compensate the $n$-type carriers due to Se vacancies. Different doping patterns
of Te would influence the transport characteristics of the surface states. We
study the Te atom distribution in Bi$_2$Se$_3$ with different Te
concentrations, using first-principles density-functional-theory calculations.
We show that Te prefer the outer layer, until the composition becomes
Bi$_2$SeTe$_2$ and the outer layer is full of Te. And for a fixed ratio of Se
and Te atoms within a single layer, we find that the structures with less
number of adjacent Te-Se pairs tend to have lower energies. This might result
in the formation of local Te clusters under low annealing temperatures.
