Systematic Size-Dependence of Electrical Resistivity Profiles in Bi$_{0.9}$Sb$_{0.1}$ Crystals

Recently, Fu and Kane[1] predicted that surface states with a odd-$Z_{2}$ topopological signature exist in the alloy Bi$_{1-x}$Sb$_{x}$ over a broad rangeof $x$. Hsieh \textit{et al}. [2] have confirmed by ARPES that, in Bi$_{0.9}$Sb$_{0.1}$, an odd number of surface states cross the chemical potential. To investigate the surface-state transport properties, we have examined how the resistivity-temperature (\textit{$\rho $-T}) profiles change as the transverse widths of crystals are varied from 20 to 1000 $\mu $m. In large crystals, the bulk conductance dominates, whereas in the small sample limit, surface conductance may be observable. Measurements on a large number of crystals reveal a reproducible systematic change in the $\rho $-T profiles. We find that the conductance at 4 K converges to a well-defined value when expressed as sheet conductance per square$ G$. The value is roughly 1000 times the quantum of conductance $e^{2}$/$h$. We discuss the interpretation of these results in the context of surface state conduction. 1. L. Fu and C. L. Kane, Phys. Rev. B 76, 045302 (2007). 2. D. Hsieh et al., Nature 452, 970 (2008).