If there is no limit to $ZT$, why are there no materials with $ZT>3$? It is well understood that the thermoelectric performance of a material (described by the parameter $ZT $ ) has no transport related upper theoretical bound. This means a material can exist that can thermoelectrically convert energy with Carnot limit efficiency. It is also well known that there no materials currently with $ZT>3$.
Is it possible to theoretically 'design' a material with  ZT $ \rightarrow \infty$?

http://cvining.com/system/files/articles/vining/Vining-ICT92-1992.pdf
 A: The thermoelectric figure-of-merit $zT$ is strengthened by electrical conductivity $\sigma$ as well as the Seebeck coefficient $S$ that both aid the motion of charges into an electrical current, and is weakened by thermal conductivity $\kappa$ that seeks to shrink the difference in temperature $T$, which is the driving factor for current generation, by passing the thermal energy to the colder end. These are all combined in the definition:
$$zT=\frac{\sigma S^2}{\kappa}T$$
Now, reaching $zT\to \infty$ for a finite temperature state requires either $\sigma\to \infty$ or $S\to \infty$ or $\kappa\to 0$. Let's look into the details of each of these.

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*Theoretically, $\sigma\to \infty$ is an often used idealisation in electronics work but in practice it depends tightly on the charge-carrier mobility $\mu$ and concentration $n$ typically simplified to the following with $e$ being the individual charge-carrier charge: $$\sigma=en\mu$$ Carrier concentration and charge is material-specific, so only mobility is of interest. It depends on the practical carrier scattering mechanisms within the material. Reaching $\sigma\to \infty$ thus requires no charge scattering which requires an idealised material structure with no electrical resistance. This might be found theoretically in certain super-conductors but we typically in turn require excessive energy input to reach such super-conductivity. $\sigma\to \infty$ might thus be energy excessive which would defeat the purpose of the thermoelectric element as an energy generator.


*$S$ is a material property which basically represents the entropy "transport" along with moving charge-carriers on a per-charge basis. It has a theoretical maximum (for materials with a small hole/electron mobility ratio) at: $$S=\pm\frac{E_g}{2eT}$$ where $E_g$ is the energy band gab of the thermoelectric semiconductive material. Reaching a larger $S$ value is a tough research discipline where we have to engineer a new semiconductive material. The issue with this is that, while a larger band gab increases $S$ this also drastically reduces the free charge-carrier concentration needed for high electrical conductivity above. I wouldn't count on this factor as ever reaching far high, although it does have quite an influence in the $zT$ figure-of-merit since it is squared.


*Finally, the $\kappa$ is where most research is focused, including my own during my theses at DTU in Denmark a few years back. The full thermal conductivity of a material typically has two main contributions: $$\kappa=\kappa_e+\kappa_{ph}$$ One is from thermal energy carried along with the charge-carrier, the other is from phonon transport - small vibration "chunks" or "packages" - that move through the lattic or structure of the material itself. The former might be basically impossible to reduce to zero, since it is tightly bound to the electric conductivity via the Wiedeman-Franz' law: $$\kappa_e=LT\sigma$$ where $L$ is the Lorentz' factor (a constant). It typically comes out to constitute one third of the total $\kappa$ in modern thermoelectric materials. The latter, the phonon conductivity, is where we most likely have good chances of reducing the value significantly. Phonon movement depends on proper "paths" within the lattice at the atomic scale; the more perfect the crystal the better the phonon transport. The more disruptions, the lower. I was working on sintering processes where thermoelectric pieces were produced by pressing-without-melting the dust particles into flat crystalline structures that constituted a highly layered final thermoelectric leg. With fine electrical but poor thermal conductivity perpendicular between layers, low $\kappa_{ph}$ was achieved without sacrificing the $\sigma$. But note, even with low $\kappa_{ph}$, the $\kappa_{e}$ still constitutes quite a portion of the total $\kappa$, so reaching $\kappa\to 0$ is not really realistic.
All in all it doesn't seem to become theoretically possible to achieve something like $zT\to\infty$. Still, keep in mind that this measure is only an on-the-fly indicator. What is actually relevant is the final efficiency $\eta$ which also depends on operating temperature range. If you can reach better than something around $\eta>\;\sim10\%$ you are not far from a commercially interesting level; a level that solar panels were at some 30 years back. With more research this will hopefully increase further decade by decade.
