  Materials with ring-shaped electronic bands are promising thermoelectric
candidates, since their unusual dispersion shape is predicted to give large
power factors. While previous calculations of these materials have relied on
the assumption of a constant mean-free-path or relaxation time, recent
first-principles modeling of electron-phonon scattering suggests that the
scattering rates may be better approximated by the electron density-of-states
(so-called DOS scattering model). In this work, we use density functional
theory to investigate single and double quintuple-layer Bi$_2$Te$_3$,
Bi$_2$Se$_3$ and Sb$_2$Te$_3$, with a focus on understanding how the three
aforementioned scattering approximations impact thermoelectric performance --
emphasis is placed on the DOS scattering model. The single quintuple-layer
materials possess two ring-shaped valence band maxima that provide an abrupt
increase in conducting channels, which benefits the power factor. Additionally,
below the band edge a ring-shaped minimum, located between the two maxima, is
found to further enhance the thermoelectric performance but only with the DOS
scattering model. This comes from a sharp drop in the DOS, and thus scattering,
just below the ring-shaped minimum. An analytic octic dispersion model is
introduced and shown to qualitatively capture the observed features. The double
quintuple-layer materials display notably worse thermoelectric properties,
since their dispersions are significantly modified compared to the single
quintuple-layer case. The benefits of ring-shaped bands are sensitive to the
alignment of the two ring maxima and to the degree of ring anisotropy. Overall,
single quintuple-layer Bi$_2$Te$_3$ and Bi$_2$Se$_3$ are most promising, with
the DOS scattering model giving the highest power factors.
