Size effect in thermoelectric power factor of nondegenerate and degenerate low-dimensional semiconductors

Low-dimensional materials have been known to give high thermoelectric (TE) performance by reducing the confinement length of the materials. Recently, we have shown that the TE power factor of low-dimensional semiconductors depends not only on the confinement length, but also on the thermal de Broglie wavelength of electrons or holes [Phys. Rev. Lett. 117, 036602 (2016)], in which the calculation was performed by assuming the semiconductors to be nondegenerate, i.e., we approximated the Fermi energy to lie only within the energy band gap, or in other words, the low doping approximation. Now, in this work, we generalize the previous results considering the degenerate case, in which the Fermi energy can exist in the valence or conduction bands, thus enabling a full consideration of heavy doping. An analytical formula for the TE power factor is derived to describe the size effect in the power factor of the low-dimensional semiconductors. We find that for both nondegenerate and degenerate cases, the TE power factor is enhanced in one- and two-dimensional semiconductors when the confinement length is smaller than the thermal de Broglie wavelength of the semiconductors, with Fermi energy around top (bottom) of valence (conduction) band for the p-type (n-type) semiconductors.