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High-Resolution Faraday Rotation and Electron-Phonon Coupling in Surface States of the Bulk-Insulating Topological Insulator Cu$_{0.02}$Bi$_2$Se$_3$

We have utilized time-domain magneto-terahertz spectroscopy to investigate the low frequency optical response of topological insulator Cu$_{0.02}$Bi$_2$Se$_3$ and Bi$_2$Se$_3$ films. With both field and frequency dependence, such experiments give sufficient information to measure the mobility and carrier density of multiple conduction channels simultaneously. We observe sharp cyclotron resonances (CRs) in both materials. The small amount of Cu incorporated into the Cu$_{0.02}$Bi$_2$Se$_3$ induces a true bulk insulator with only a \textit{single} type of conduction with total sheet carrier density $\sim4.9\times10^{12}/$cm$^{2}$ and mobility as high as 4000 cm$^{2}/$V$\cdot$s. This is consistent with conduction from two virtually identical topological surface states (TSSs) on top and bottom of the film with a chemical potential $\sim$145 meV above the Dirac point and in the bulk gap. The CR broadens at high fields, an effect that we attribute to an electron-phonon interaction. This assignment is supported by an extended Drude model analysis of the zero field Drude conductance. In contrast, in normal Bi$_2$Se$_3$ films two conduction channels were observed and we developed a self-consistent analysis method to distinguish the dominant TSSs and coexisting trivial bulk/2DEG states. Our high-resolution Faraday rotation spectroscopy on Cu$_{0.02}$Bi$_2$Se$_3$ paves the way for the observation of quantized Faraday rotation under experimentally achievable conditions to push chemical potential in the lowest Landau Level.