I'm trying to understand the phonon-electron interaction in superconductors. I'm a high school student doing a project on superconductors and I can't find understandable information about it. Almost everything about the topic is quite above my physics knowledge. I wonder if someone can explain the mechanisms behind this interaction :)
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2$\begingroup$ Could you perhaps let us know what you understand so far? $\endgroup$– Codename 47Commented Dec 7, 2021 at 15:43
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$\begingroup$ I understand that phonons are the quantized descriptions of lattice vibrations, and I understanf how coopers pairs are formed. an electrons distors the lattice of the metal and an area gets more positive dense, which attracs another electron. the presence of this positive dense area overcomes the repulsion of the two electrons and their join forming a cooper pair. I read that " the positive region of the lattice is part of what is called a phonon, a colletive motion of multiple ions with the same frequency " so what I think I understand is that energy in form of heat enters the material $\endgroup$– ShelyyCommented Dec 7, 2021 at 16:36
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$\begingroup$ it causes the lattices to vibrate, and these vibrations are the descriptions of phonons so the formation of the cooper pairs are the result of this electron-phonon interaction. ?? $\endgroup$– ShelyyCommented Dec 7, 2021 at 16:38
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1 Answer
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The picture you describe in the comments corresponds to that described on the Wikipedia article for BCS theory, which is our current best theory for how superconductivity works. In the overview, they state that
Roughly speaking the picture is the following:
An electron moving through a conductor will attract nearby positive charges in the lattice. This deformation of the lattice causes another electron, with opposite spin, to move into the region of higher positive charge density. The two electrons then become correlated. Because there are a lot of such electron pairs in a superconductor, these pairs overlap very strongly and form a highly collective condensate.
In this "condensed" state, the breaking of one pair will change the energy of the entire condensate - not just a single electron, or a single pair. Thus, the energy required to break any single pair is related to the energy required to break all of the pairs (or more than just two electrons). Because the pairing increases this energy barrier, kicks from oscillating atoms in the conductor (which are small at sufficiently low temperatures) are not enough to affect the condensate as a whole, or any individual "member pair" within the condensate. Thus the electrons stay paired together and resist all kicks, and the electron flow as a whole (the current through the superconductor) will not experience resistance. Thus, the collective behavior of the condensate is a crucial ingredient necessary for superconductivity.
This is only a rough picture, in particular since the electrons in question act as approximately free electrons, and thus their position is not well defined. Any description of an electron attracting "nearby" ions is therefore a simplified picture, because "nearby" is ill-defined. But it is true that electrons interact with the phonons, and that this causes a small attraction between the electrons, which are otherwise repulsed by each other.
As mentioned, this small attraction allows for collective behavior at low temperatures. A relatively large disruption is needed to disrupt this collective behavior, which leads to a superconducting state, where the electrons move uninterrupted through the material (corresponding to zero resistance).
To get any more accurate than this, math is needed. At your level, you might be able to look at these lecture notes and follow these steps
The origin of the attraction. In section 8.3, they state that "electrons can interact with one another by exchanging phonons", i.e. one electron can cause a lattice vibration which is then absorbed by another. In section 8.4, they then make a model where electrons are both attracted (by phonon interaction) and repulsed (by Coulomb interaction, i.e. due to having the same charge). The plot shows the energy cost for electrons moving closer to each other as a function of a parameter $\nu$. Since for some values of $\nu$ the graph is negative, under these circumstances electrons can actually gain energy by moving closer together! This corresponds to attraction, where the phonon interaction surpasses the Coulomb interaction.
The energy gap. The energy of the electrons is given in Eq. (10.29) as approximately $E_k = \sqrt{\epsilon_k^2 + |\Delta|^2}$, where $k$ is the momentum, $\epsilon_k = \hbar k^2/2m$ is the kinetic energy, and $\Delta$ is an energy presence due to the attractive interaction, which we have assumed to be constant in $k$. You can plot $E_k$ as a function of $k$ to see for yourself that the presence of $\Delta$ causes a gap in the energy to appear, i.e. no energy states are available below $\Delta$. Inside the gap, the electrons can exist as Cooper pairs, but to exist as single electrons, they need to be lifted to a state above the gap, which requires a lot of energy (about $\Delta$).
It is very understandable if most of this is way over your head, but hopefully it helps a bit.
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$\begingroup$ Thanks a lot for answering and sorry for the late comment. It took a while to understand this but I think that I have a clearer view of what a electron-phonon-electron interaction is! $\endgroup$– ShelyyCommented Jan 25, 2022 at 19:33
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$\begingroup$ Happy to help! If you feel like this answer was helpful, you can mark it as the correct answer. $\endgroup$ Commented Jan 26, 2022 at 13:56