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Two materials are usually necessary to manifest the Seebeck effect. However, the Seebeck effect occurs when we apply a temperature gradient to a material, and this material produces a potential difference. The usual reason to justify the need of two materials it is to break the symmetry.

Let's imagine a metal bar. If I apply a temperature gradient at the bar extremities, it will induce a flux of electrons, that is, an electric current, and a potential difference. After some time, the electrons will be in "balance", they will just vibrate in the extremities of the bar, but there will be no flux of these electrons. To break this symmetry, we join the extremities of a metal bar made of a different material to the extremities of the previous bar. Since the bars are different, we break the symmetry, and so we obtain a flux of electrons. But what happens if, instead of using a different material, we connect a resistance or a lamp (for example) to the first bar? Should it not break the symmetry? This way we just need one material to produce the Seebeck effect.

My doubt is, historically, the Seebeck effect it has been observed with two materials, but in reality, it can occur in a single material. I do not understand this concept. Why does sometimes it needs two materials, and other times just need one?

Honestly, the fact of using only one material makes much more sense when connecting any type of electronics. But if so, can all materials be considered thermoelectric? Since the application of a thermal gradient produces oscillation of electrons, and apart from the case of wood, which burns, there should be charge flow in any type of metal or semiconductor due to an increase in temperature. I am really confused, and if someone can help me, I would appreciate it!

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Yes, the Seebeck effect happens in pretty much all materials except superconductors. It is just very hard to observe in a very poor conductor such as dry wood which has a resistivity $\gtrsim 20$ orders of magnitude larger than common metals. The Seebeck Effect could also be made very close to zero in a carefully constructed alloy of metals with positive and negative Seebeck coefficients, e.g. see Table 1.1 of Macía-Barbers Thermoelectric Materials and Advances.

Just adding a lamp won't break the symmetry as long as the lamp is also made of the same material as the rest of the circuit. Essentially all we've done is made a longer wire and the potential drop will still only depend on the temperature difference between two locations, which will be the same going in either direction and hence no current will flow. (If the lamp is made from a different material, than we are back to the normal two-component thermoelectric situation.)

As noted in the answer to "Seebeck effect and the need for two conductors", however, you can see Seebeck currents in a single conductor if the thickness of the conductor changes at the heated/cooled point. This happens because thermal and electrical conductivities change when a sample becomes very thin, and the Seebeck coefficient depends on these conductivities.

In metals, for example, thermal energy can be conducted by both electrons and phonons, but only electrons carry electric charge. The Seebeck coefficient depends how the phonons and electrons scatter from the lattice, defects and impurities, each other, and from any surfaces or interfaces. If the thickness of the sample is comparable to the bulk mean free path of the electrons (or phonons), the scattering from surfaces and interfaces starts to dominate and the Seebeck coefficient will change from the bulk value. Because of this size-dependence of the Seebeck coefficient, Single-Metal Nanoscale Thermocouples can be constructed.

One can expect also differences in Seebeck coefficient for materials that are otherwise identical except for their microcrystalline structure. (Although I guess this means they are no longer a "homogeneous material".) Substantial differences in Seebeck Coefficient are indeed observed in materials depending on how they were annealed, e.g. a factor of two in ZnO between annealing at $500$ °C and $800$ °C.

Because it is very hard to make a completely homogeneous material, changes in the Seebeck coefficient can be observed along the length of a single gold nanowire, so hitting the right spot with a laser can generate a tiny current.

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