Many promising building blocks of future electronic technology - including non-stoichiometric compounds, strongly correlated oxides, and strained or patterned films - are inhomogeneous on the nanometer length scale. Exploiting the inhomogeneity of such materials to design next-generation nanodevices requires a band structure probe with nanoscale spatial resolution. To address this demand, we report the first simultaneous observation and quantitative reconciliation of two candidate probes - Landau level spectroscopy and quasiparticle interference imaging - which we employ here to reconstruct the multi-component surface state band structure of the topological semimetal antimony(Sb). We thus establish the technique of band structure tunneling microscopy (BSTM), whose unique advantages include nanoscale access to non-rigid band structure deformation, empty state dispersion, and magnetic field dependent states. We use BSTM to elucidate the relationship between bulk conductivity and surface state robustness in topological materials, and to quantify essential metrics for spintronics applications.
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