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We performed the first-principles calculation on common thermoelectricsemiconductors $\rm Bi_2Te_3$, $\rm Bi_2Se_3$, $\rm SiGe$, and $\rm PbTe$ inbulk three-dimension (3D) and two-dimension (2D). We found that miniaturizationof materials does not generally increase the thermoelectric figure of merit($ZT$) according to the Hicks and Dresselhaus (HD) theory. For example, $ZT$values of 2D $ \rm PbTe$ (0.32) and 2D $ \rm SiGe$ (0.04) are smaller thantheir 3D counterparts (0.49 and 0.09, respectively). Meanwhile, the $ZT$ valuesof 2D $\rm Bi_2Te_3$ (0.57) and 2D $\rm Bi_2Se_3$ (0.43) are larger than thebulks (0.54 and 0.18, respectively), which agree with HD theory. The HD theorybreakdown occurs because the band gap and band flatness of the materials changeupon dimensional reduction. We found that flat bands give a larger electricalconductivity ($\sigma$) and electronic thermal conductivity ($\kappa_{el}$) in3D materials, and smaller values in 2D materials. In all cases, maximum $ZT$values increase proportionally with the band gap and saturate for the band gapabove $10\ k_BT$. The 2D $Bi_2Te_3$ and $Bi_2Se_3$ obtain a higher $ZT$ due tothe flat corrugated bands and narrow peaks in their DOS. Meanwhile, the 2D PbTeviolates HD theory due to the flatter bands it exhibits, while 2D SiGepossesses a small gap Dirac-cone band.

Non-universal Scaling of Thermoelectric Efficiency in 3D and 2D Thermoelectric Semiconductors