Talk:Electronic band structure

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covalent and molecular bonding

This article should make it obvious that it (mostly) applies to covalently bonded crystalline solids. (Though non-crystalline covalent solids also have band structure.) It mostly does not apply to molecular solids, where individual molecules are held together by weaker bonds, with little electron overlap between them. Gah4 (talk) 21:54, 10 October 2020 (UTC)

full

Does it say in the article, or explain well enough, that full bands don't contribute to conduction? I wanted to link to it, but I don't see it. Gah4 (talk) 16:29, 1 February 2021 (UTC)

Pauli principle

There is some sort of confusion about the meaning of the Pauli principle here. I added a request for a citation near the invocation of the Pauli principle. The statement about the Pauli principle is now gone from the main text, but still exists in the figure caption. I already edited the Pauli principle out once in December, but this change was reverted, with a message "This is an elementary explanation for general readers. It should include that the underlying reason for the splitting of the energy levels is the Pauli exclusion principle". But the point is not about elementary vs. technical, it is about factually correct statements and clear concepts. If you invoke the principle, I would ask you to at least provide a textbook reference in which it is used in the calculation of the band structure.

How I view the role of the Pauli principle technically: the crystal is a many-body system, but with the simplifying assumption no interactions, we can describe the many-body states in terms of single-particle states. The band structure calculation then reduces to an application of Bloch's theorem for a single-particle Schrödinger equation. Thus, there is no Pauli principle at this point, only hybridization.

Now, to describe the filling of the bands, we have to revert back to many-body description, and here we encounter the Pauli principle. It tells us that our terminology makes sense; the states indeed become "filled" and we cannot put a second electron to the same state. I think it will benefit even the elementary reader if these two things, the formation of the band structure and how the electrons occupy it, are presented as separate concepts and the principle is invoked only for the latter one. -Jähmefyysikko (talk) 06:22, 5 February 2022 (UTC)

I think that agrees with why I took it out. Without Pauli there are no atomic orbitals to make atoms, much less molecules. If you believe in LCAO, which sometimes I don't, you make molecular orbitals out of atomic orbitals. (In theory you can use any basis, but that isn't the reason for LCAO.) But otherwise, molecular orbital theory solves Schrodinger around the nuclei of the molecule, and band theory in a crystal lattice. Atomic orbitals already satisfy Pauli, and so do linear combinations of them. It is actually delocalization that creates the bonds, so we should credit Heisenberg instead of Pauli. Gah4 (talk) 06:59, 5 February 2022 (UTC)

Thanks, although I don't fully agree with all the above statements. The stability of the atom (or its electron configuration) requires Pauli principle, but the existence of atomic orbitals does not, because atomic orbital is a single-electron concept, just like the band structure. For this reason, I don't understand what do you mean by "atomic orbitals already satisfy Pauli". The Pauli principle only comes into play as we consider how the orbitals are occupied. Could you clarify? -Jähmefyysikko (talk) 11:50, 5 February 2022 (UTC)

OK, yes, which atomic (or molecular) orbitals are occupied. I suppose that seemed obvious enough not to say it, but yes. There are no electrons in the unoccupied orbitals! Otherwise, Pauli comes from Fermi–Dirac statistics. Otherwise, if you look at the molecular orbitals for straight chain hydrocarbons of increasing length, as with bands, the energy of the top and botton MO stays about the same, but more and more fill in the region between. Or consider the Discrete Fourier transform of a time series, as the number of samples increases and the sample rate stays constants. The range of frequencies stays the same, but more and more fill in, such that the frequency resolution increases. I am pretty sure that Pauli doesn't have much to do with the DFT, though the reasoning is the same. Bandwidth depends on atomic spacing, and state spacing on the number of atoms, for the same reason as for DFT. Gah4 (talk) 14:47, 5 February 2022 (UTC)

Exactly, thanks for the clarification. Jähmefyysikko (talk) 15:23, 5 February 2022 (UTC)

When two identical atoms join to form a molecule, their atomic orbitals overlap.

The article says: When two identical atoms join to form a molecule, their atomic orbitals overlap. This at least needs to allow for compound semiconductors. It works for any crystal made of identical or not atoms. And for many non-crystalline materials, too. Gah4 (talk) 19:32, 18 July 2023 (UTC)

Pauli principle, pt. 2

I removed the statement about Pauli principle. It was supported by two references (Holgate and Van Zeghbroeck). The first one is a semi-popularized textbook and the second one is self-published. Stardard textbooks (Kittel, Ashcroft & Mermin, Oxford Solid State basics) do not invoke Pauli Principle when describing the band formation. Jähmefyysikko (talk) 09:00, 23 September 2023 (UTC)

Without the Pauli principle, which is what makes fermions do what they do, all electrons are in the 1s state in atoms, and one fat band in metals. Well, I suspect that there wouldn't be metals. However, since the Pauli principle is implied by electron energy levels in atoms, putting atoms together doesn't need to add it. It is already assumed. I suppose I should read Ashcroft and Mermin, sitting on the shelf not so far away. Gah4 (talk) 22:03, 23 September 2023 (UTC)

I agree that Pauli principle is important. It would even be useful to emphasize more the section Electronic_band_structure#Filling of bands which discusses it. It is just that the energy level splitting is not explained by it. To illustrate this, one may consider a situation with only one electron in the diatomic molecule (or a lattice). Since there is only one electron, Pauli principle does not play any role, but there is still energy level splitting (or band structure). Jähmefyysikko (talk) 09:52, 24 September 2023 (UTC)

I think so. I used to do group seminars about this, but haven't thought in so much detail. Last year I was remembering the book Bonds and Bands in Semiconductors I used to work with chemistry people, so explaining the connection between band theory and molecular orbital theory made sense. Maybe I will even see what it says about this one. Gah4 (talk) 10:52, 24 September 2023 (UTC)

I added the phrase mentioning the Pauli exclusion principle to the caption of the graph, which was removed by Jähmefyysikko.
My version:

When far apart (right side of graph) all the atoms have discrete valence orbitals p and s with the same energies. However, when the atoms come closer (left side), their electron orbitals begin to spatially overlap. The Pauli Exclusion Principle prohibits them from having the same energy, so the orbitals hybridize into N molecular orbitals each with a different energy, where N is the number of atoms in the crystal.

Jähmefyysikkos version:

When far apart (right side of graph) all the atoms have discrete valence orbitals p and s with the same energies. However, when the atoms come closer (left side), their electron orbitals begin to spatially overlap, so the orbitals hybridize into N molecular orbitals each with a different energy, where N is the number of atoms in the crystal.

Which is better? --Chetvorno^TALK 05:19, 8 November 2023 (UTC)

I am not a solid state physicist, just an engineer, but many physics books do include the Pauli principle in explanations of band structure. Supporting sources are given below. The first two are almost word for word similar to my version:

• "Fig. 41-2a shows two isolated copper atoms. [...] If we bring the atoms closer together, their wavefunctions begin to overlap, starting with the outer electrons. We then have a single two-atom system with 58 electrons, not two independent atoms. The Pauli exclusion principle requires that each of these electrons occupy a different quantum state. In fact, 58 levels are available, because each energy level of the isolated atom splits into two levels of the two-atom system. [...] If we bring up more atoms, we gradually assemble a lattice of solid copper. For N atoms, each level in the isolated copper atom must split into N levels in the solid. Thus the individual energy levels in the solid form energy bands..." Halliday, Resnick, Walker, Fundamentals of Physics, p.1254
• "According to the Pauli exclusion principle, each state can be occupied by at most two electrons of opposite spin. [...] If one considers a thought experiment in which several initially isolated atoms are gradually brought closer together, their interaction will lead to a splitting of their energy levels. If a very large number of atoms are involved, as in the case of a real solid, then the energy levels will lie on a quasi-continuous scale and one therefore speaks of energy bands." Ibach, Luth, Solid State Physics, p.2
• "According to the Pauli exclusion principle, no two electrons in an atom can share one identical quantum state. In fact this discreteness requirement can be extended beyond an isolated atomic system. In a crystalline solid, for example, discrete energy levels are created due to the covalent bonding of the atoms in the crystal lattice. These allowed energy levels are lumped into two energy bands, the valence band and conduction band" Cai, Shaalev, Optical Metamaterials: Fundamentals and Applications, p.12
• "The semiclassical dynamics of electrons in energy bands is then considered; together with the Pauli exclusion principle for occupation of states, it gives a qualitative distinction between metals, semiconductors, and insulators." (p.2) "Consider an assembly of particles at thermal equilibrium moving independently in a volume. If the particles obey the Pauli exclusion principle, the occupation probability for a one particle quantum state is given by the Fermi-Dirac distribution..." (p.112) Grosso, Paravicini, Solid state physics
• "As the assembly of atoms condenses, the degenerate, sharp atomic states interact and are broadened in energy to a band. (The Pauli exclusion principle prohibits two particles in the same system from occupying the same energy state.)" Lionel Kimerling, "Bands and bonds", Course notes: Photonic materials and devices, MIT

--Chetvorno^TALK 05:19, 8 November 2023 (UTC)

Thanks for engaging in discussion. The above references do invoke the Pauli principle, but I don't see a clear claim that the level splitting itself would be a result of that principle. Rather, it is only the filling of the bands that is explained by the principle. This is non-interacting physics, so the many-body problem splits neatly into two separate stages: First, one uses the single-particle Hamiltonian to determine the energy levels (the splitting appears here). Second, the filling of those levels is determined by the Fermi-Dirac distribution (Pauli exclusion is here).

Often, the hybridization is explained by referring to "overlapping wavefunctions", and this may be source of confusion, as it makes it look like the electrons are somehow interacting. But this is actually just a description of mathematics of the perturbation theory as applied to the tight binding method, and is still single-particle physics, with no Pauli exclusion.

Further evidence that the fermionic Pauli exclusion is not the cause of the level splitting is given by the fact that similar splitting and band formation also happens in bosonic or classical systems.

It is also incorrect to say that The Pauli Exclusion Principle prohibits them from having the same energy. The Pauli principle forbids two electrons from occupying the same quantum state. But two states can be degenerate and have the same energy, and still be simultaneously occupied. Jähmefyysikko (talk) 08:48, 8 November 2023 (UTC)

Okay, I will accept your division of the explanation into the splitting of levels and the filling of levels, although a number of the above sources do seem to say the splitting is due to the Pauli principle. But the name of this article is Electronic band structure, so by definition we are talking about many-particle physics and nondegenerate levels. I drop my support for my original wording above, but the explanation needs to include the Pauli principle as the reason for the distribution of electrons, as virtually every elementary explanation of bands does. --Chetvorno^TALK 17:31, 8 November 2023 (UTC)

How about this:

When far apart (right side of graph) all the atoms have discrete valence orbitals p and s with the same energies. However, when the atoms come closer (left side), their electron orbitals begin to spatially overlap, so the orbitals hybridize into N molecular orbitals each with a different energy, where N is the number of atoms in the crystal. Since N is such a large number, the orbitals are very close in energy, forming bands. The Pauli Exclusion Principle limits the number of electrons in any one orbital to two, so the electrons are distributed within the bands' orbitals, beginning with the lowest energy. --Chetvorno^TALK 17:56, 8 November 2023 (UTC)

Ok, I added Pauli principle to the caption (it would be a bit better if the image also showed the filling of the bands). The caption was getting too long, so I also tried reducing some detail. What do you think? Jähmefyysikko (talk) 06:01, 9 November 2023 (UTC)

References