The role of dimensionality on the electronic performance of thermoelectricdevices is clarified using the Landauer formalism, which shows that thethermoelectric coefficients are related to the transmission, T(E), and how theconducing channels, M(E), are distributed in energy. The Landauer formalismapplies from the ballistic to diffusive limits and provides a clear way tocompare performance in different dimensions. It also provides a physicalinterpretation of the "transport distribution," a quantity that arises in theBoltzmann transport equation approach. Quantitative comparison ofthermoelectric coefficients in one, two, and three dimension shows that thechannels may be utilized more effectively in lower-dimensions. To realize theadvantage of lower dimensionality, however, the packing density must be veryhigh, so the thicknesses of the quantum wells or wires must be small. Thepotential benefits of engineering M(E) into a delta-function are alsoinvestigated. When compared to a bulk semiconductor, we find the potential for~50 % improvement in performance. The shape of M(E) improves as dimensionalitydecreases, but lower dimensionality itself does not guarantee betterperformance because it is controlled by both the shape and the magnitude ofM(E). The benefits of engineering the shape of M(E) appear to be modest, butapproaches to increase the magnitude of M(E) could pay large dividends.
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