The valence band of a variety of few-layer, two-dimensional materials consists of a ring of states in the Brillouin zone. The energy-momentum relation has the form of a `Mexican hat or a Rashba dispersion. The two-dimensional density of states is singular at or near the band edge, and the band-edge density of modes turns on nearly abruptly as a step function. The large band-edge density of modes enhances the Seebeck coefficient, the power factor, and the thermoelectric figure of merit ZT. Electronic and thermoelectric properties are determined from ab initio calculations for few-layer III-VI materials GaS, GaSe, InS, InSe, for Bi$_{2}$Se$_{3}$, for monolayer Bi, and for bilayer graphene as a function of vertical field. The effect of interlayer coupling on these properties in few-layer III-VI materials and Bi$_{2}$Se$_{3}$ is described. Analytical models provide insight into the layer dependent trends that are relatively consistent for all of these few-layer materials. Vertically biased bilayer graphene could serve as an experimental test-bed for measuring these effects.
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