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

We have utilized time-domain magnetoterahertz spectroscopy to investigate the low-frequency optical response of the topological insulator ${\mathrm{Cu}}_{0.02}{\mathrm{Bi}}_2{\mathrm{Se}}_3$ and ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ 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 ${\mathrm{Cu}}_{0.02}{\mathrm{Bi}}_2{\mathrm{Se}}_3$ induces a true bulk insulator with only a single type of conduction with a total sheet carrier density of $\sim 4.9\times {10}^{12}/{\mathrm{cm}}^2$ and mobility as high as $4000{\mathrm{cm}}^2/\mathrm{V}·\mathrm{s}$. This is consistent with conduction from two virtually identical topological surface states (TSSs) on the top and bottom of the film with a chemical potential $\sim 145\mathrm{meV}$ 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 ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ films, two conduction channels were observed, and we developed a self-consistent analysis method to distinguish the dominant TSSs and coexisting trivial bulk or two-dimensional electron gas states. Our high-resolution Faraday rotation spectroscopy on ${\mathrm{Cu}}_{0.02}{\mathrm{Bi}}_2{\mathrm{Se}}_3$ paves the way for the observation of quantized Faraday rotation under experimentally achievable conditions to push the chemical potential in the lowest Landau level.

Figure 1

Data summary for 64 quintuple layer (QL) ${\mathrm{Cu}}_{0.02}{\mathrm{Bi}}_2{\mathrm{Se}}_3$ (sample1). (a) Real sheet conductance of 64 QL ${\mathrm{Cu}}_{0.02}{\mathrm{Bi}}_2{\mathrm{Se}}_3$ and pure ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ films at 5 K. The inset is the Drude conductance $G_D$ of 64 QL ${\mathrm{Cu}}_{0.02}{\mathrm{Bi}}_2{\mathrm{Se}}_3$ after subtracting the phonon and ${\epsilon }_{\infty }$ contributions. The arrow indicates the deviation from pure Lorentzian form that arises from electron-phonon coupling. (b) Field-dependent conductance at 5 K. Arrows are guides to the eye for cyclotron frequencies. (c) Real and (d) imaginary parts of the complex Faraday rotation data at different fields at 5 K. (e) Cyclotron frequency versus field. The solid line is a linear fit. The inset is a cartoon indicating $E_F\sim 145\mathrm{meV}$ above the Dirac point (75 meV below the conduction band minimum). (f) Complex Faraday angle with fits at 7 T. The solid curve is a fit with a field-dependent scattering rate. The dashed curve is a fit using the zero-field scattering rate. Arrows are guides to the eye for cyclotron frequency.

Figure 2

(a) Real conductance of 100 QL ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ comparing single- and two-component Drude fits as detailed in the text. The inset enlarges the low-frequency regime. (b) Field-dependent conductance at 5 K. (c) The real and (d) imaginary parts of the complex Faraday rotation data at different fields at 5 K. (e) The cyclotron resonance frequency versus field. The inset is a cartoon indicating $E_F\sim 350\mathrm{meV}$ above the Dirac point. (f) Complex Faraday rotation data with one- and two-component Drude fits at 7 T. The dashed and solid curves are one- and two-Drude fits, respectively.

Figure 3

(a) The scattering rate as a function of cyclotron frequency (red). The fully renormalized scattering rate by mass through extended Drude analysis as a function of frequency (green). The black arrow indicates $\beta$ surface phonon frequency at the Kohn anomaly. (b) The renormalized mass as a function of frequency. The error bars express the uncertainty in ${\omega }_{pD}$.