One can understand the formation of energy bands from the Kronig-Penny model which assumes a periodic potential. But I heard that even if the potential is aperiodic, for example in amorphous substances (glass, plastic) there also exist bands. If not periodicity what is the fundamental physics that causes band formation?
[Physics] Can amorphous solids have energy bands
amorphous-solidscondensed-matterelectronic-band-theoryglasssolid-state-physics
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The deep insight of Anderson is that the difference between insulators and conductors is not the energy spectrum. In fact the entire picture we are taught in introductory courses is highly misleading. [Note: Everything I am going to talk about will be about single particle effects, so no interaction.]
First lets just remember the introductory picture. We have a perfect crystal, so we get energy bands. We fill those bands up with electrons. In the case when a band is partially filled we get a conductor. In the case when all of our bands are completely occupied, so that the Fermi level lies in the gap, we get an insulator.
Now that problems: finite conductivity is entirely dependent on impurities. In the absence of impurities momentum is completely conserved. If I give the carriers any momentum, they will never lose it. Therefore a finite current can never dissipate, which is the same as saying the resistance is zero. Since there will all always be some carriers at any non-zero temperature, in the absence of impurities all materials will be "perfect conductors".
So it is clear that to make any sense we need to add impurities. However if we add impurities the nice energy band picture disappears. Since we just added random stuff to our Hamiltonian there is no reason we shouldn't be able to to find a state of any energy if we look hard enough. Obviously there will be more states in what used to be the bands, but there will also be states in the gap. In short the bands will blur together.
But if the bands blur together then there is no longer any notion of a gap - so what could possibly separate insulators and conductors? It is not the electronic energy spectrum, it is the electronic wavefunction themselves. Since there is no longer translational symmetry these are not restricted to the Bloch form. There are two main possibilities:
1) The wavefunctions near the Fermi level are extended, i.e. their magnitude is roughly constant over the entire system, like a plane wave. This is a conductor.
2) The wavefunctions near the Fermi level are localized, i.e. their magnitude decays roughly exponentially as you go out from some point. This is an insulator.
This is what actually distinguishes insulators and conductors. Going back to the band gap classification of materials - why does it basically work? The reason is if one adds disorder to a perfect crystal, the states that are added in the gap and near the band edges are usually localized states, so thinking about the gaps leads to the correct answer. But this is not the direct physical mechanism.
If you consider a single energy level in an atom, e.g. the $1s$ state of hydrogen, then if you bring two atoms together the level will split into two states (often called the bonding and anti-bonding orbitals). If you bring in a third atom the two states will split again giving four states, and so on. Grouping $n$ atoms will split the original distinct level into $2^{n-1}$ levels.
The number of atoms doesn't have to be very big before the the spacing between all these levels is far less than $kT$, and at that point we have a band i.e. effectively we have a continuous range of energies available to electrons.
So the bands, and therefore the band gaps, arise due to bringing many atoms together, not just two.
Best Answer
In general, one can understand emergence of band structure starting from two extreme scenarios, both not contradicting formation of broadened levels/bands in amorphous solids:
A nice real-world example of existence of electron bands in an aperiodic solid is amorphous SiO2, i.e. the major component of common glass. It is transparent to visible light, because corresponding photon energies do not suffice to create electron-hole pairs by overcoming the band gap energy of amorphous SiO2. Photons with energies from the ultraviolet spectrum, however, have enough energy to excite electrons from the valence band to the conduction band of glass (i.e. the photons are absorbed), explaining why glass is (mostly) opaque to ultraviolet light. (Here is an abstract of a medical paper, that summarizes how efficient different types of glass are at blocking UV light.)
Regarding a citation about band structure in amorphous solids: the following article discusses the optical band gap dependence of soda lime borosilicate glass (measured band gap $\Rightarrow$ experimental evidence for electronic energy bands) as a function of TiO2 dopant concentration: Ruengsri, Kaewkhao, Limsuwan, Procedia Engineering 32, 772 (2012) (open access).
Figures 3 and 4 of the following article: Roth, "Tight-Binding Models of Amorphous Systems: Liquid Metals", Phys. Rev. B 7, 4321 (1973) (behind pay wall) show density of states for a (tight-binding) model of an amorphous solid: the broad density of states reveal bands (i.e. there are no individual discrete "spikes" in the density of states indicative of isolated states).