my question is about whether it's possible in principle
The answer is yes.
and whether anyone tried it.
The answer is by all chances, no.
So, how come?
The effect
The thermoelectric effect for electricity generation (called the Seebeck effect) is the phenomenon that a voltage is generated at a temperature different across the ends of a conductor:
$$V=S\Delta T$$
where $S$ is the Seebeck coefficient, a material propety. All we need is a temperature difference - that is, a hot and a cold source. Said in other words, any hot source and any cold source. So sure, if you have a hot source which nuclear certainly is, and water cooling which is a usual method for nuclear power plants, you just need a material that has good properties in this temperature range.
The materials
But here comes the problem. The search for and research in thermoelectric materials is the brake in this field. We simply do not have good enough materials at the moment. The topic still feels new, though discovered some hundreds of years ago, and the best materials at present are still those that were found in the 1950's. We are improving and improving, but the efficiencies are simply too low compared to any other source.
Efficiency
I do not know the typical efficiency of a nuclear power plant. But a diesel engine as an example is at about 40% and is one of the most efficient practical engines existing and in use today.
Now, the maximum efficiency of thermoelectric devices is given as:
$$\mu_{max} =\frac{T_C-T_H}{T_H}\cdot \frac{\sqrt{1+ZT}-1}{\sqrt{1+ZT}+T_H/T_C}=\mu_{Carnot}\cdot\mu_{r}$$
where $T_H$ and $T_C$ are hot and cold end temperatures, and the Carnot efficiency is $\mu_{Carnot}=\frac{T_C-T_H}{T_H}$. The $\mu_r$, called the reduced efficiency or conversion efficiency, is maybe realisticly at around $10\%$, while the Carnot could be at maybe $60\%$ - multiply them together, and the maximum efficiency $\mu_{max}$ is simply too low.
Figure of merit
It all comes down to the so called figure of merit $ZT$, given as:
$$ZT=\frac{S^2\sigma}{\kappa}$$
For present state-of-the-art materials it is only at around 1 to 1½. I am myself working with the alloy Bismuth Telluride, the best material at the moment in the lower temperature range around room temperature to 100-200 degress C, of which the higest $zT$ achieved is $zT \sim 1.45$ It is usually stated that it needs to reach around 3-4 for a material to be usable in industry. See the graph of the $\mu_r$ below showing the value for different $ZT$s at varying temperature.
Source: Rowe, D. M.: ”Termoelectrics Handbook - Macro to nano”, Taylor & Francis Group, 2006.
The problem is mainly the issue of reducing the thermal conductivity $\kappa$. This is mainly material science and a material problem, we need to overcome - but I fully agree that this physical phenomenon must have a huge potential at some point.
So, I do not know if anyone ever tried putting it into a nuclear power plant. But I really don't think so. Whenever better materials are found, it will take years for them to be integrated into largescale plants.
