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 of the epitaxial topological insulator alloy (Bi,Sb)$_2$Te$_3$. The bulk band gap is used in conjunction with f-sum rules to establish an upper bound for the D expected in a typical Dirac like system composed of linear bands. We expand our result from the linear band model to include both hexagonal warping and electron-hole asymmetry, as is typical in topological insulator systems. The corresponding maximum value of D arising from Dirac bands in this more complex system is also determined. The observed D is found to be close to this upper bound, demonstrating the effectiveness of alloying in eliminating bulk charge carriers. Moreover, Hall effect parameters and the weak anti-localization observed in transport on the same sample support assignment of the low-energy conduction to topological surface states.