  We performed the first-principles calculation on common thermoelectric
semiconductors $\rm Bi_2Te_3$, $\rm Bi_2Se_3$, $\rm SiGe$, and $\rm PbTe$ in
bulk three-dimension (3D) and two-dimension (2D). We found that miniaturization
of 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 than
their 3D counterparts (0.49 and 0.09, respectively). Meanwhile, the $ZT$ values
of 2D $\rm Bi_2Te_3$ (0.57) and 2D $\rm Bi_2Se_3$ (0.43) are larger than the
bulks (0.54 and 0.18, respectively), which agree with HD theory. The HD theory
breakdown occurs because the band gap and band flatness of the materials change
upon dimensional reduction. We found that flat bands give a larger electrical
conductivity ($\sigma$) and electronic thermal conductivity ($\kappa_{el}$) in
3D materials, and smaller values in 2D materials. In all cases, maximum $ZT$
values increase proportionally with the band gap and saturate for the band gap
above $10\ k_BT$. The 2D $Bi_2Te_3$ and $Bi_2Se_3$ obtain a higher $ZT$ due to
the flat corrugated bands and narrow peaks in their DOS. Meanwhile, the 2D PbTe
violates HD theory due to the flatter bands it exhibits, while 2D SiGe
possesses a small gap Dirac-cone band.
