Why is the Peltier / Seebeck Effect's efficiency so low in practical devices? Usually solid state devices are able to attain excellent efficiency. However, currently existing Peltier coolers and Seebeck generators have horrible efficiency -- far worse than Carnot, and also far worse than gas or vapor compression machinery full of moving parts. 
What gives? What would be needed to make these devices perform within a few perfect of Carnot?
 A: Here is why.
To be efficient in this context requires two contradictory properties: 1) that the thermoelectric material have low thermal conductivity (so the temperature difference across the junction is not thermally short-circuited) while at the same time having high electrical conductivity (so the cooling effect is not swamped out by I^2R losses in the junction). 
To the extent that these properties are both affected by things like electron mobility, you can't make one low and the other high in the same chunk of material. 
You can finesse the problem somewhat with exotic materials from obscure regions of the periodic table, which is why solid-state refrigerator junctions are made from things like balonium-doped canthavium/unobtainium alloys. This is a materials science joke; you may laugh now.
A: In order to reach Carnot's efficiency, the materials that makes up the TEG must have a ZT value of infinity. That is so because the efficiency of a TEG can be expressed as the Carnot efficiency multiplied by a term that equals 1 when $ZT$ is infinite. (see https://en.wikipedia.org/wiki/Thermoelectric_materials#Device_efficiency for example.)
However $ZT = \frac{\sigma S^2T}{\kappa}$ where $\kappa$ is the thermal conductivity, $\sigma$ is the electrical conductivity, $S$ is the Seebeck coefficient and $T$ is the absolute temperature.
This value cannot reach infinity.
A good candidate material must therefore possess a good electrical conductivity, a low thermal conductivity, and a high Seebeck coefficient. In practice metals are discarted because they somehow obey to Wiedeman-Franz law which states that a good electrical conductor is also a good thermal conductor, and this holds true for most metals. Insulator have high values of $S$ but extremely low values for $\sigma$ and so they aren't good candidates either. The best candidates are heavily doped semiconductors, which fall in between metals and insulators (note that there's a theoretical optimum doping concentration value). For semiconductors it is possible to split the thermal conductivity into a lattice contribution and into an electronic contribution: $\kappa = \kappa_l + \kappa_e$. While $\kappa_e$ satisfies reasonably well Wiedeman-Franz law, the part $\kappa_l$ is independent of it. This means that if one reduces the lattice contribution to the thermal conductivity, one increases the ZT coefficient. In practice such a reduction is accomplished through a variety of ways, such as creating very complex materials (with very large chemical formulae) with complex  crystal structure geometry, impacting on phonons propagation. Many researchers are nowadays working on engineering materials with very low $\kappa_l$.
If we turn to superconductors, they unfortunately have a vanishing Seebeck coefficient, so they are totally unable to generate power out of a temperature difference through the Seebeck effect. I.e. they cannot be used as materials to generate power in a TEG or to cool down things in a Peltier module.
Note that there is a theoretical minimum value of $\kappa$ which can be attained, which is roughly the one of a glass. Since $S$ is also limited (cannot be arbitrarily big), which is many times related to the electrical conductivity through Mott formula and since we've already seen that superconductors (where one could be lead to think that $\sigma$ is infinity while it would be better to say that $\rho$ vanishes), there is no hope left for $ZT$ to reach infinity.
This only explains why thermoelectric materials lead to heat engines that do not match Carnot efficiency. 
Now the saddest part: The ZT value of commercial TEGs is around 0.7 near room temperature (the material is Bi2Te3, which is the same material used since the 1970's despite many decades of improvements!). In the lab they've found a material with a ZT of above 2, but such a material is so chemically unstable and fragile that it cannot be used in the making of a TEG or Peltier module. That's the real reason why thermoelectric engines have a so low efficiency. There is, thus far, no material having a high ZT despite insane efforts since decades. Even DFT (density functional theory) applied to all the elements of the periodic table were used to predict good candidates and the best candidates had miserable ZT values compared to the efficiency one gets from solar panels and many heat engines.
