  Nanowires with a GaSb core and an InAs shell (and the inverted structure) are
interesting for studies of electron-hole hybridization and interaction effects
due to the bulk broken band-gap alignment at the material interface. We have
used eight-band $\mathbf{k\cdot p}$ theory together with the envelope function
approximation to calculate the band structure of such nanowires. For a fixed
core radius, as a function of shell thickness the band structure changes from
metallic (for a thick shell) to semiconducting (for a thin shell) with a gap
induced by quantum confinement. For intermediate shell thickness, a different
gapped band structure can appear, where the gap is induced by hybridization
between the valence band in GaSb and the conduction band in InAs. To establish
a relationship between the nanowire band structures and signatures in
thermoelectrical measurements, we use the calculated energy dispersions as
input to the Boltzmann equation and to ballistic transport equations to study
the diffusive limit and the ballistic limit, respectively. Our theoretical
results provide a guide for experiments, showing how thermoelectric
measurements in a gated setup can be used to distinguish between different
types of band gaps, or tune the system into a regime with few electrons and few
holes, which can be of interest for studies of exciton physics.
