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I am currently studying Diode Lasers and Photonic Integrated Circuits, Second Edition, by Larry A. Coldren, Scott W. Corzine, Milan L. Masanovic. Chapter 1.2 ENERGY LEVELS AND BANDS IN SOLIDS says the following:

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On the other hand, in a covalently bonded solid like the semiconductor materials we use to make diode lasers, the uppermost energy levels of individual constituent atoms each broaden into bands of levels as the bonds are formed to make the solid. This phenomenon is illustrated in Fig. 1.4. The reason for the splitting can be realized most easily by first considering a single covalent bond. When two atoms are in close proximity, the outer valence electron of one atom can arrange itself into a low-energy bonding (symmetric) charge distribution concentrated between the two nuclei, or into a high-energy antibonding (antisymmetric) distribution devoid of charge between the two nuclei. In other words, the isolated energy level of the electron is now split into two levels due to the two ways the electron can arrange itself around the two atoms.$^1$ In a covalent bond, the electrons of the two atoms both occupy the lower energy bonding level (provided they have opposite spin), whereas the higher energy antibonding level remains empty.
If another atom is brought in line with the first two, a new charge distribution becomes possible that is neither completely bonding nor antibonding. Hence, a third energy level is formed between the two extremes. When $N$ atoms are covalently bonded into a linear chain, $N$ energy levels distributed between the lowest-energy bonding state and the highest-energy antibonding state appear, forming a band of energies. In our linear chain of atoms, spin degeneracy allows all $N$ electrons to fall into the lower half of the energy band, leaving the upper half of the band empty. However in a three-dimensional crystal, the number of energy levels is more generally equated with the number of unit cells, not the number of atoms. In typical semiconductor crystals, there are two atoms per primitive unit cell. Thus, the first atom fills the lower half of the energy band (as with the linear chain), whereas the second atom fills the upper half, such that the energy band is entirely full.
The semiconductor valence band is formed by the multiple splitting of the highest occupied atomic energy level of the constituent atoms. In semiconductors, the valence band is by definition entirely filled with no external excitation at $T = 0 \text{ K}$. Likewise, the next higher-lying atomic level splits apart into the conduction band, which is entirely empty in semiconductors without any excitation. When thermal or other energy is added to the system, electrons in the valence band may be excited into the conduction band analogous to how electrons in isolated atoms can be excited to the next higher energy level of the atom. In the solid then, this excitation creates holes (missing electrons) in the valence band as well as electrons in the conduction band, and both can contribute to conduction.

$^1$ The energy level splitting is often incorrectly attributed to the Pauli exclusion principle, which forbids electrons from occupying the same energy state (and thus forces the split, as the argument goes). In actuality, the splitting is a fundamental phenomenon associated with solutions to the wave equation involving two coupled systems and applies equally to probability, electromagnetic, or any other kind of waves. It has nothing to do with the Pauli exclusion principle.

Parts of this explanation seem logically incoherent to me. The authors say that

In a covalent bond, the electrons of the two atoms both occupy the lower energy bonding level (provided they have opposite spin), whereas the higher energy antibonding level remains empty.

But they then say that

The semiconductor valence band is formed by the multiple splitting of the highest occupied atomic energy level of the constituent atoms.

But we know that the bonds in a semiconductor are covalent bonds, and the authors say in the first quote that the electrons of the covalent bond both occupy the lower energy bonding level (provided they have opposite spin), so how does it make sense to then say that the semiconductor valence band is formed by the multiple splitting of the highest occupied atomic energy level of the constituent atoms?

Furthermore, the authors also say that

Likewise, the next higher-lying atomic level splits apart into the conduction band, which is entirely empty in semiconductors without any excitation.

But they just said that the semiconductor valence band is formed by the multiple splitting of the highest occupied atomic energy level of the constituent atoms, so how does it then make sense to talk about a higher-lying atomic level? After all, if it is the highest occupied atomic energy level, then, logically, there is no higher-lying level!

What am I misunderstanding/confusing with these?

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    $\begingroup$ Are you confused about the author’s use of the word “occupied” in this context? $\endgroup$
    – J. Murray
    Nov 7, 2021 at 1:42
  • $\begingroup$ @J.Murray Hmm, that might be it. What do you think the confusion might be (with regards to "occupied")? $\endgroup$ Nov 7, 2021 at 3:16
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    $\begingroup$ well, I’m not sure. But it makes perfectly good sense to talk about an energy level which is higher than the highest occupied every level, if the higher energy level is unoccupied. $\endgroup$
    – J. Murray
    Nov 7, 2021 at 3:39
  • $\begingroup$ @J.Murray Ohhh, ok, I think I see what it's saying. So the highest occupied atomic energy level splits apart into the valence band, and then the next higher atomic energy level, which is unoccupied, splits into the unoccupied conduction band? $\endgroup$ Nov 7, 2021 at 4:12
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    $\begingroup$ Yes, that's right - at least, that's the point the author is trying to convey. In reality it is a somewhat oversimplified description of the formation of energy bands, but it's good enough for an intuitive picture. $\endgroup$
    – J. Murray
    Nov 7, 2021 at 20:26

1 Answer 1

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The picture the author is trying to convey is that if you build a covalently-bonded crystal lattice out of atoms, then the discrete energy levels which characterize the isolated atoms split apart into continuous energy bands. The highest occupied level in the isolated atom becomes the valence band, and the next-highest level (which is unoccupied) becomes the (empty) conduction band. This is a bit of an oversimplification, but it provides a reasonable intuitive picture which can be refined with more detail later.

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