Magnetic Field Signatures of Topological States in 3D Time-Reversal Invariant Insulators

While the topological behavior of Bi$_2$Se$_3$ has been identified experimentally\footnote{Y.L. Chen et al., \emph{Science} {\bf 325}, 178 (2009).} \footnote{P. Roushan et al., \emph{Nature} {\bf 460}, 7259 (2010).}, characterization by electron transport has been difficult due to high bulk transport caused by inadvertent doping of the crystal.\footnote{N. P. Butch et al., \emph{Phys. Rev. B.} {\bf 81}, 24 (2010).} We perform self-consistent quantum transport calculations to show that patterned surfaces offer a unique environment in which the system may be characterized by resultant magnetic field distributions. We compare doped and undoped Bi$_2$Se$_3$ samples with normal metals to show a qualitative difference in current flow around the patterned surface. We find that the surface to bulk conductance ratio can be inferred from the magnetic field in patterned systems due to the spatial separation of bulk and surface currents created by the corrugation, which applies even in heavily doped systems. The magnetic field is sufficiently large so as to be observed using ultracold atom microscopy.