Sum-Rule Constraints on the Surface State Conductance of Topological Insulators

We report the Drude oscillator strength $D$ and the magnitude of the bulk band gap $E_g$ of the epitaxially grown, topological insulator $(\mathrm{Bi},\mathrm{Sb}{)}_2{\mathrm{Te}}_3$. The magnitude of $E_g$, in conjunction with the model independent $f$-sum rule, allows us to establish an upper bound for the magnitude of $D$ expected in a typical Dirac-like system composed of linear bands. The experimentally observed $D$ is found to be at or below this theoretical upper bound, demonstrating the effectiveness of alloying in eliminating bulk charge carriers. Moreover, direct comparison of the measured $D$ to magnetoresistance measurements of the same sample supports assignment of the observed low-energy conduction to topological surface states.

Figure 1

Panels (a) and (b) show the band structure of a model TI, where the SS is composed of simple linear bands. The bulk band structure is also indicated by the dotted blue lines, consisting of two hole bands and an electron band. In (a) $E_F$ is at the Dirac point, while in (b) $E_F$ is 0.20 eV below the Dirac point, where the arrows in both panels schematically show the allowed transitions. In (a), interband transitions are allowed at all energies, giving rise to the conductance spectra shown in (c). When $E_F$ is shifted below the Dirac point, transitions cannot occur from the depopulated region. Thus, transitions are suppressed below $2|E_F|$, giving rise to the conductance spectra shown in (d).

Figure 2

(a) Temperature dependent spectrum of the conductance of $(\mathrm{Bi},\mathrm{Sb}{)}_2{\mathrm{Te}}_3$ obtained from terahertz time-domain spectroscopy and FTIR. The inset shows (symmetrized) dc magnetoresistivity data for this film. The “cusp” at the low-$B$ field in the 2 K data is indicative of weak antilocalization. (b) Drude-Lorentz model fit (4) (gray) of the 6 K experimental conductance (blue). Individual Drude or Lorentzian oscillators are displayed as thin black lines. (c) Temperature dependence of the Drude oscillator strength $D$ (black points) corresponding to the left axis, and square root of the charge carrier density $n$ (open red squares) extracted from Hall effect measurements, corresponding to the right axis. The upper bounds for the Drude response of the linear SS ($D_{\text{LB},\max }^s\pm 10\%$), and more realistic estimate of the SS contribution including hexagonal warping ($D_{\text{BST},\max }^s$) in $(\mathrm{Bi},\mathrm{Sb}{)}_2{\mathrm{Te}}_3$, are indicated by the light gray and dark gray bars, respectively. (d) Temperature dependence of the free carrier scattering rate 1/$\tau$ from optics (black points, left axis) and charge mobility $\mu$ from Hall effect measurements (open red squares, right axis).