Observation of Coulomb repulsion between Cu intercalants in Cu$_{\mathrm{x}}$Bi$_{2}$Se$_{3}$

Using scanning tunneling microscopy and \textit{ab initio }simulations, we have identified several configurations for Cu-dopants in Cu$_{\mathrm{x}}$Bi$_{2}$Se$_{3}$, with Cu intercalants being the most abundant. Through statistical analysis, we show strong short-range repulsive interactions between Cu intercalants. At intermediate range (\textgreater 5nm), the pair distribution function shows oscillatory structure along the \textless 1 0 -1\textgreater\ directions which appears to be influenced by different diffusion barriers along the \textless 1 0 -1\textgreater\ and \textless 2 -1 -1\textgreater\ directions.