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I have just begun reading about thermoelectricity. However I’ve been struggling to find a satisfying explanation for the the thermoelectric effects, specifically the Seebeck and Peltier effect, The different explanations that I’ve come across are as follows:

  1. H Julian Goldsmid, 'The Thermoelectric and Related Effects', in Introduction to Thermoelectricity (Springer, 2016), pp. 1-7.

    Nowadays, we understand that electric current is carried through a conductor by means of electrons that can possess different energies in different materials. When a current passes from one material to another, the energy transported by the electrons is altered, the difference appearing as heating or cooling at the junction, that is as the Peltier effect. Likewise, when the junction is heated,electrons are enabled to pass from the material in which the electrons have the lower energy into that in which their energy is higher,giving rise to an electromotive force.

  2. How does the Seebeck Effect create a voltage difference from a temperature differential? by Steven J Greenfield on The Educational Blog

    Voltage Induced Across a Conductor In a Thermal Gradient

    Albert E. Seaver & Brian P. Seaver

    Nesperd Engineering

    Abstract—When a bar of electrically conductive solid material is thermally insulated except at its ends and is then heated to a temperature Th at one end while it is held to a cooler temperature Tc at the other end, a linear thermal gradient ∇T develops along the length of the bar. If the bar has a free-electron charge density ρ then, since the hot end of the bar has expanded relative to the cold end, the free-electron charge density at the hot end will be lower than at the cold end due to this expansion. As a result, a charge density gradient ∇ρ is set up along the bar; and some of the free-electrons diffuse from the cooler temperature end towards the hotter temperature end of the bar. This diffusion results in a diffusion charge flux (current density) JD which is controlled by the diffusion coefficient D. On the other hand, the movement of some free electrons to the hotter end results in an excess of free-electrons near the hotter end; and, since the counter-ions of the solid cannot move, an excess of positive charge occurs near the cooler end.

  3. Seebeck coefficient - Wikipedia ….As before, the high-energy carriers diffuse away from the hot end, and produce entropy by drifting towards the cold end of the device. However, there is a competing process: at the same time low-energy carriers are drawn back towards the hot end of the device. Though these processes both generate entropy, they work against each other in terms of charge current, and so a net current only occurs if one of these drifts is stronger than the other. The net current depends literally on how conductive high-energy carriers are, compared to low-energy carriers.

Of course there may be a flaw in my understanding here and all three explanations may be pointing to the same thing, but as far as I can tell they describe completely different things. So basically my question is, should the Seebeck effect be attributed to

  1. The difference in intrinsic electron energy between dissimilar conductors
  2. Thermal expansion of the hot end or
  3. Competing thermal drifts
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  • $\begingroup$ There are more. For metals, the derivative of the density of states at the Fermi level is important. For semiconductors (as in commercial Peltier elements), the charge carriers (electrons, holes) can be treated as a classical ideal gas. $\endgroup$ – Pieter Jan 3 at 9:26
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There are a lot of questions asked, I will try to clear some doubts.

The Seebeck effect can indeed occur in a single material, under which a temperature gradient is established. This will give rise to a potential difference, creating a steady-state (not an equilibrium state) in which the electric field inside the material arising from the non-homogeneous electronic distribution cancels out the electronic "diffusion" due to the temperature gradient. However this non-homogeneous electronic distribution does not arise from thermal expansion. One could reach it by solving the Boltzman transport equation, for example.

To reemphasize, the Seebeck effect has nothing to do with thermal expansion, which is in most cases completely negligible. Some materials have a vanishing Seebeck coefficient at a given temperature, while they have a non vanishing coefficient of thermal expansion. And possibly almost vice-versa.

As is, the explanation 1 is correct, Wikipedia (explanation 3) is also mostly correct, while explanation 2 is dead wrong.

By the way, the three thermoelectric effects are the same manifestation of the same physical processes. Once one obtains any thermoelectric coefficient (e.g. the Seebeck coefficient of a material), one obtains all the other ones via Thomson relations.

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