Extracting band structure characteristics of GaSb/InAs core-shell nanowires from thermoelectric properties

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 $\mathbf{k}·\mathbf{p}$ 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.

Figure 1

(a) The GaSb/InAs bulk band alignment with conduction-band edges in blue, valence-band edges in green, and split-off band edges in pink. (b) A cylindrical core-shell nanowire with a GaSb core radius $R_C$ and an InAs shell thickness $t_S$. (c) A setup for transport measurement of a nanowire. Electrodes are depicted in blue and red, with applied source voltage $V$. The nanowire is subject to a gate voltage $V_G$ and a heat gradient $\mathrm{\Delta }T$.

Figure 2

Band structure for a GaSb/InAs nanowire with core radius $R_C=20$ nm and shell thickness (a) $t_S=8.5$ nm, (b) $t_S=7.8$ nm, (c) $t_S=7.62$ nm, and (d) $t_S=7.5$ nm. In (b), a hybridization gap of 1.1 meV opens up.

Figure 3

Band structure for an InAs/GaSb core-shell nanowire with core radius $R_C=20$ nm and shell thickness (a) $t_S=2.7$ nm, (b) $t_S=2.1$ nm (c) $t_S=2$ nm, and (d) $t_S=1.9$ nm.

Figure 4

(a)–(d) Conductivity (red solid line) and conductance (blue dashed line) calculated in the diffusive and the ballistic limit, respectively, for the GaSb/InAs $\left[111\right]$ nanowires with core radius $R_C=20$ nm. (e)–(h) Seebeck coefficient calculated in the diffusive limit (red solid line) and the ballistic limit (blue dashed line) for the same wires. (a)–(d) [(e)–(h)] correspond to the nanowire dimensions in Figs. 2–2.

Figure 5

(a)–(d) Conductivity (red solid line) and conductance (blue dashed line) calculated in the diffusive and the ballistic limit, respectively, for the InAs/GaSb $\left[111\right]$ nanowires with core radius $R_C=20$ nm. (e)–(h) Seebeck coefficient calculated in the diffusive limit (red solid line) and the ballistic limit (blue dashed line) for the same wires. (a)–(d) [(e)–(h)] correspond to the nanowire dimensions in Figs. 3–3.