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We introduce a new resistance measurement method that is useful incharacterizing materials with both surface and bulk conduction, such asthree-dimensional topological insulators. The transport geometry for this newresistance measurement configuration consists of one current lead as a closedloop that fully encloses the other current lead on the surface, and two voltageleads that are both placed outside the loop. We show that in the limit wherethe transport is dominated by the surface conductivity of the material, thefour-terminal resistance measured from such a transport geometry isproportional to $\sigma_b/\sigma_s^2$, where $\sigma_b$ and $\sigma_s$ are thebulk and surface conductivities of the material, respectively. We call this newtype of measurement \textit{inverted resistance measurement}, as the resistancescales inversely with the bulk resistivity. We discuss possible implementationsof this new method by performing numerical calculations on different geometriesand introduce strategies to extract the bulk and surface conductivities. Wealso demonstrate inverted resistance measurements on SmB$_6$, a topologicalKondo insulator, using both single-sided and coaxially-aligned double-sidedCorbino disk transport geometries. Using this new method, we are able tomeasure the bulk conductivity, even at low temperatures, where the bulkconduction is much smaller than the surface conduction in this material.

A New Method for Characterizing Bulk and Surface Conductivities of Three-Dimensional Topological Insulators: Inverted Resistance Measurements