I have read quite a lot on Peltier elements but I still can't figure out why they work exactly.
Consider the following (well-made) animated diagram of a Peltier element:
I think the reason why electrons conduct heat is random vibration energy: since temperature is measuring how much an atom or molecule randomly moves around, move that atom or molecule from one place to another and heat flow is generated. I don't know why electrons (or atoms or molecules in fact) randomly move around when they're hot in the first place, but if this applies to atoms or molecules I don't see any reason why electrons wouldn't be affected (considering they are a particule).
Now, I'm having a hard time understanding why holes would conduct heat. Holes are not a particule, they are merely an empty slot for an electron that went elsewhere. So a flow of holes is always in reality a flow of electrons in the opposite direction, right? Then why wouldn't the right hand side conduct heat in the wrong direction, resulting in zero net heat transferred from the cold side to the hot side? It must be because of semiconductors, because from that diagram I can't see why a PN junction is necessary... But why/how?