Band alignment and scattering considerations for enhancing the thermoelectric power factor of complex materials: The case of Co-based half-Heusler alloys

Producing high band and valley degeneracy through aligning of conducting electronic bands is an effective strategy to improve the thermoelectric performance of complex band-structure materials. Half-Heuslers, an emerging thermoelectric material group, has complex band structures with multiple bands that can be aligned through band engineering approaches, giving us an opportunity to improve their power factor. Theoretical calculations to identify the outcome of band engineering usually employ detailed density functional theory for band-structure calculations, but the transport calculations are kept simplistic using the constant relaxation time approximation due to the complications involved with detailed scattering physics. In this work, going beyond the constant relaxation time approximation, we perform an investigation of the benefits of band alignment in improving the thermoelectric power factor under different density of states dependent scattering scenarios. As a test case we consider the Co-based p-type half-Heuslers TiCoSb, NbCoSn, and ZrCoSb. First, using simplified effective mass models combined with Boltzmann transport, we investigate the conditions of band alignment that are beneficial to the thermoelectric power factor under three different carrier scattering scenarios: (i) the usual constant relaxation time approximation, (ii) intraband scattering restricted to the current valley with the scattering rates proportional to the density of states as dictated by Fermi's golden rule, and (iii) both intra- and interband scattering across all available valleys, with the rates determined by the total density of states at the relevant energies. We demonstrate that the band-alignment outcome differs significantly depending on the scattering details. Next, using the density functional theory calculated band structures of the half-Heuslers we study their power factor behavior under strain induced band alignment. We show that strain can improve the power factor of half-Heuslers, but the outcome heavily depends on the curvatures of the bands involved, the specifics of the carrier scattering mechanisms, and the initial band separation. Importantly, we also demonstrate that band alignment is not always beneficial to the power factor. In addition, we show that the band structure itself can undergo changes as the bands are aligned in practice, which further affect the band alignment optimization. Our work illustrates the importance of going beyond the constant relaxation time approximation, as well as understanding how the band structure of each material behaves when considering band alignment.