Anomalous behavior of the $E_u^1$ infrared-active phonon mode in a Bi${}_{2-x}$Sr${}_x$Se${}_3$ crystal

We have studied spectral evolution of the $E_u^1$ phonon line of a topological insulator Bi${}_{2-x}$Sr${}_x$Se${}_3$ with temperature. Unlike the Raman-active phonons, the $E_u^1$ mode demonstrates softening upon cooling the crystal, and the corresponding spectral line acquires a pronounced Fano-like shape at temperatures $T\lesssim 100$ K. We interpret the latter effect as a signature of specific coupling of the bulk infrared active phonons to surface Dirac electrons. Using coherent resonant excitation of the $E_u^1$ mode as a surface sensitive tool, we have detected softening of the surface counterpart of the bulk $E_u^1$ phonon mode upon strontium doping. This observation can be an evidence of enhanced electron-phonon interaction at the surface of the Bi${}_{2-x}$Sr${}_x$Se${}_3$ crystal.