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Based on this question, I was trying to understand what happens with the electrons when there is a gradient of temperature between the ends of a metal rod.

I put the contacts of a voltimeter at the two ends of an iron rod. It was initially 0V as expected.

One of the ends was then heated by an incandescent lamp, while the other was kept cold using a type of artificial ice at approximately $0^\circ$C.

After some time, the voltimeter measured 0.1mV, with the positive polarity at the hot end. It remains in this value even after removing the hot and cold sources, and only several minutes later returns to $0V$, after the thermal gradient went away.

My tentative explanation:

It is related to band structure. Iron, as the other metals has a conduction band only partially filled. The electrons fill all the states of lower energy, leaving empty states above the energy of the higher occupied states (Fermi level).

That is the picture before we consider thermal energy. Above $0 K$ the boundary between the last occupied state and the next empty state blurs. The electrons follow Fermi-Dirac statistics, and some of them are above the Fermi level, leaving empty states of lower energy. Increasing temperature makes that drift also increase, with more electrons having higher energy and momentum.

If one part of a metal rod is hotter than the other, more electrons tends to flow to the cold region than the opposite, because there are more ocuppied states with higher momentum in the hot region. But once in colder places, the FD statistic is not the same, and they make a transition to lower available states, liberating thermal energy. That is the reason for the thermal flow in metals.

There is a limitation in the velocity of that process however. When the electrons spread from the hotter end, it creates a (hot) region depleted of electrons, and a cold region negative charged. The resultant polarity (that was measured between the ends of the iron rod), avoids that the process can be so fast as the convection for gases for example.

Problems with the explanation

My doubt is that I could not find that effect in the web, (except here, but with the opposite polarity!)

On the other hand there is a lot of reference to Seebeck effect, that needs 2 different metals. Is it what I measured (copper from the voltimeter contacts and the iron bar)?

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