The role of dimensionality on the electronic performance of thermoelectric
devices is clarified using the Landauer formalism, which shows that the
thermoelectric coefficients are related to the transmission, T(E), and how the
conducing channels, M(E), are distributed in energy. The Landauer formalism
applies from the ballistic to diffusive limits and provides a clear way to
compare performance in different dimensions. It also provides a physical
interpretation of the "transport distribution," a quantity that arises in the
Boltzmann transport equation approach. Quantitative comparison of
thermoelectric coefficients in one, two, and three dimension shows that the
channels may be utilized more effectively in lower-dimensions. To realize the
advantage of lower dimensionality, however, the packing density must be very
high, so the thicknesses of the quantum wells or wires must be small. The
potential benefits of engineering M(E) into a delta-function are also
investigated. When compared to a bulk semiconductor, we find the potential for
~50 % improvement in performance. The shape of M(E) improves as dimensionality
decreases, but lower dimensionality itself does not guarantee better
performance because it is controlled by both the shape and the magnitude of
M(E). The benefits of engineering the shape of M(E) appear to be modest, but
approaches to increase the magnitude of M(E) could pay large dividends.Comment: 23 pages, 5 figure