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We study the relationship between the shape of the electronic band structureand the thermoelectric properties. In order to study the band shape dependenceof the thermoelectric properties generally, we first adopt models with bandstructures having the dispersion $E({\bf k}) \sim |{\bf k}|^n$ with $n = 2, 4$and 6. We consider one, two- and three dimensional systems, and calculate thethermoelectric properties using the Boltzmann equation approach within theconstant quasi-particle lifetime approximation. $n = 2$ corresponds to theusual parabolic band structure, while the band shape for $n = 4, 6$ has a flatportion at the band edge, so that the density of states diverges at the bottomof the band. We call this kind of band structure the "pudding mold type band".$n \ge 4$ belong to the pudding mold type band, but since the density of statesdiverges even for $n= 2$ in one dimensional system, this is also categorized asthe pudding mold type. Due to the large density of states and the rapid changeof the group velocity around the band edge, the spectral conductivity of thepudding mold type band structures becomes larger than that of the usualparabolic band structures. It is found that the pudding mold type band has acoexistence of large Seebeck coefficient and large electric conductivity, andsmall Lorenz number in the Wiedemann--Franz law due to the specific band shape.We also find that the low dimensionality of the band structure can contributeto large electronic conductivity and hence a small Lorenz number. We concludethat the pudding mold type band, especially in low dimensional systems, canenhance not only the power factor but also the dimensionless figure of meritdue to stronger violation of the Wiedemann--Franz law.

Enhanced power factor and reduced Lorenz number in the Wiedemann--Franz law due to pudding mold type band structures