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For some quick background, I work as a high voltage electrician at a major metropolitan utility. In training apprentices, we use books that are often very poor in explaining fundamental electrical concepts. I'd like to try to improve the instruction that we provide new students, starting with some basic band theory. So here's my main question: is it correct to describe the valence electrons in an insulator as being bound to specific atoms? In reading different sources, I've encountered answers that seem to disagree with each other, though I am unsure if they actually represent equivalent descriptions of the same thing. In volume 3 of the Feynman Lectures it says:

Of course, in a real crystal there are already millions of electrons. But most of them (nearly all for an insulating crystal) take up positions in some pattern of motion each around its own atom—and everything is quite stationary.

(https://www.feynmanlectures.caltech.edu/III_13.html#Ch13-S1)

Hyperphysics similarly says:

"Conductor" implies that the outer electrons of the atoms are loosely bound and free to move through the material. Most atoms hold on to their electrons tightly and are insulators.

(http://hyperphysics.phy-astr.gsu.edu/hbase/electric/conins.html#c2)

Here at Physics Stack Exchange, user @anna v writes:

One can also visualize the electrons energy of the conduction band as solutions on the whole lattice, whereas in the valence band the electrons are tied to individual atoms , that is why it is called "valence".

(What makes electrons in the conduction band conductive?)

In contrast (maybe!), I have read many answers that emphasize that both the valence and conduction bands are delocalized in a regular lattice. Essentially, the Heisenberg Uncertainty Principle demands that if the momentum of an electron is known with precision (i.e., delta p=0) its wavefunction spreads across the entire lattice. In such a description, insulators don't allow current because because all proximate momentum states are filled (with electrons moving in opposite directions--zero net drift velocity), and it would take a lot of energy to cross the bandgap.

Is this a fundamentally different description then the Feynman one quoted above? And how can the picture of delocalized insulator electrons accommodate bulk polarization of insulators in electric fields?

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  • $\begingroup$ I’m going with no, not at all. Even in an insulator the top valence band is well described by Bloch functions and an $E$ vs $k$ band diagram. As one example, diamond used to be called an insulator, and is now a wide band gap semiconductor. The valence band did not change, regardless of what we call it. $\endgroup$
    – Jon Custer
    Commented Sep 12, 2022 at 20:50
  • $\begingroup$ Are Bloch functions the only useful description of electronic states in band theory? What about Wannier functions? physics.stackexchange.com/a/77598/345557 $\endgroup$
    – Dave
    Commented Sep 13, 2022 at 13:30
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    $\begingroup$ Is it correct? It's a model, a product of human imagination. Models in physics are never correct when you push deep enough into them. But models can be useful. Is the bound electron concept useful? Absolutely! $\endgroup$
    – John Doty
    Commented Sep 13, 2022 at 15:43
  • $\begingroup$ @JonCuster Also I have come across work by Raffaele Resta that builds on Kohn's picture of localized states in insulators. Unfortunately, I don't have the technical knowledge to analyze the mathematical reasoning presented. $\endgroup$
    – Dave
    Commented Sep 13, 2022 at 18:06
  • $\begingroup$ @JohnDoty I think that's a long the lines of what I'm unsure about. What models or mental pictures are at least reasonable to present? There's always some degree of imperfection and simplification associated with models of reality. It's also confusing that people with significant physics literacy and work experience have such adamantly different takes on what seems like a pretty basic question. $\endgroup$
    – Dave
    Commented Sep 13, 2022 at 18:24

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Valence electrons are outermost (closest to zero) in energy on a binding energy scale. Valence electrons in atoms form the covalent, ionic, or metallic bonds in molecules or bulk substances. In a single covalent bond between two atoms, valence electrons that make (two) molecular orbitals are shared between the two atoms. In metallic bonds, valence electrons are shared between all atoms in the metal.

For bonding in metals, at absolute zero, all electrons are in bound states in the equivalent of a highest occupied molecular orbital (HOMO) band. The topmost occupied level in the HOMO band defines what is called the Fermi energy. The occupied portion of the HOMO band below the Fermi energy is called the valence band. The bound states in the valence band are shared by all atoms collectively. The electrons in the valence band are said to be delocalized throughout the valence band. They have no specific position that defines their location. By comparison, the electrons in metals at higher binding core level energy states are spatially localized about individual atoms. This is not about the depth of the potential well on any given atom, rather it is that core level electrons are at binding energies that are factors of 10x or more above binding energies for valence electrons. Core level electrons are essentially entirely localized about the metal atom. Since valence electrons are delocalized, they have no sense of a specific place to go when we apply a voltage field--essentially, they are already "everywhere" spatially. Since valence electrons are bound, they can only be displaced by a voltage field, they cannot be moved out of position. Think of voltage field shifting an entire delocalized valence band cloud of electrons rather than of a voltage field causing any one or more specific electrons to travel from point A to point B. The former is polarization (leading to capacitance), the latter is current (leading to conduction).

Any free (unoccupied) levels above the Fermi energy constitute what is called the conduction band. No electrons occupy these states at absolute zero. No electrons are "free" from their bound, valence band states; they are all below the Fermi energy. At absolute zero in temperature, since no electrons are free to travel through the conduction band, metals are electrical insulators.

As we raise temperature above absolute zero, electrons can move above the Fermi energy. This is a thermal excitation process; it is not induced by voltage. Electrons in the conduction band region are less delocalized, and, because they are unbound in the shared metallic bonds, they are also free to travel anywhere in the metal. Specifically, they are free to carry current when we apply a voltage field.

The picture is not about whether electrons are tightly or loosely bound. The distinction is about whether electrons are bound and delocalized in the valence band or whether they are unbound and more localized in the conduction band. The former cannot carry current; they latter can respond to a voltage field and travel (carry current) from one location to another.

An insulator is nothing more than a system where the portion that we call the valence band is completely full and the portion that we call the conduction band is separated by an energy gap. The conduction band sits closer to the zero point of energy while the valence band lies below it at higher binding energy. This is also the distinction between a metal and a semiconductor; the former has no band gap, while the latter has a band gap. The band gap in semiconductors is smaller than that in insulators. Electrons from the (full) valence band can be thermally excited to hop over the band gap in semiconductors. Those electrons that reach the conduction band in semiconductors can carry current (the details about holes in semiconductors are the subject of a different discussion). The band gap in insulators is too large for thermal excitation to cause any electrons from the (delocalized yet bound) valence band states to reach the (localized yet unbound) conduction band states.

Finally, metals behave as insulators at absolute zero in temperature because all electrons are in the bound states in the valence band. Even though phonons are frozen out and therefore resistance becomes zero, no free electrons occupy the available states to carry current.

All told, for the same temperature in a given volume of a solid, insulators have no "free" electrons (electrons in the conduction band) because the band gap is too large, semiconductors have some "free" electrons because the band gap exists but can be overcome by thermal motion, and metals have more "free" electrons because no band gap exists.

To close on one note in your question, polarization is the displacement of charge remaining tied to a fixed point, conduction is the movement of charge from point A to point B. When we relax a voltage field, charges displaced by polarization return back to the fixed point, but charges that have moved to point B by conduction do not return back to point A. Charges that cause polarization do not contribute to current, and charges that cause current do not contribute to polarization.

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  • $\begingroup$ Thank you for taking the time to post an answer. There are a number of points that are unclear. You mention greater localization at higher binding energy--I assume that means deeper in the potential well. But then you describe conduction band electrons as being localized in contrast to the valence band. I don't think I've ever seen such a description before... $\endgroup$
    – Dave
    Commented Sep 13, 2022 at 13:52
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    $\begingroup$ And it seems unclear that metals are necessarily insulators at absolute zero. Phonons go away, so wouldn't that lower resistance? Plus aren't metals defined (at least in part) by the fact that there are available states at 0K (the Fermi energy lies in the valence band)? $\endgroup$
    – Dave
    Commented Sep 13, 2022 at 13:59
  • $\begingroup$ I've updated to your questions. I will hedge and say conductions electrons are less delocalized. Phonons go away, but no free electrons occupy the available states, so metals behave as insulators. $\endgroup$ Commented Sep 13, 2022 at 15:38
  • $\begingroup$ Regarding polarization, in metals an external electric field causes conduction electrons to accumulate on the surface in such a way as to cancel out the field within the conductor body (the electrostatic case). If we describe polarization in insulators as the shifting of the delocalized electron cloud, how is that distinct from what happens in metals? $\endgroup$
    – Dave
    Commented Sep 13, 2022 at 18:14
  • $\begingroup$ All materials have atoms (or ions) with electron clouds. The electron clouds can be shifted relative to the atom or ion nucleus. This is electronic polarization. Metals have a free electron "sea". This free electron sea can be shifted en masse relative to the lattice positions of the nuclei. This is free electron or plasmon polarization. $\endgroup$ Commented Sep 14, 2022 at 2:28

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