How do electrons carry thermal energy in Peltier coolers?
What happens is that the electrons need to gain energy to jump the energy gap at the metal-semiconductor junction.


The valence band is shown in the shaded area. In conductors the conductance band and the valence band overlap. In semiconductors there is an energy gap between the conductance and valance bands, as shown in the figure. A conducting electron must gain energy as it transitions from the conductance band in the conductor to the conductance band in the semiconductor and must lose energy vice versa. This energy is the thermal energy, hence heating or cooling. A similar argument applies for p-type semiconductors.