Understanding how to optimize electronic band structures for thermoelectrics is a topic of long-standing interest in the community. Prior models have been limited to simplified bands and/or scattering models. In this study, we apply more rigorous scattering treatments to more realistic model band structures - upward-parabolic bands that inflect to an inverted parabolic behavior - including cases of multiple bands. In contrast to common descriptors (e.g., quality factor and complexity factor), the degree to which multiple pockets improve thermoelectric performance is bounded by interband scattering and the relative shapes of the bands. We establish that extremely anisotropic `flat-and-dispersive bands, although best-performing in theory, may not represent a promising design strategy in practice. Critically, we determine optimum bandwidth, dependent on temperature and lattice thermal conductivity, from perfect transport cutoffs that can in theory significantly boost $zT$ beyond the values attainable through intrinsic band structures alone. Our analysis should be widely useful as the thermoelectric research community eyes $zT>3$.
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