Question about Electronic Band Structure

[TJonline]

TL;DR Summary

N level (~continuous) energy bands formed from the discrete energy levels of individual atoms when joined in macroscopic lattices of N atoms (N being one jillion)

According to this Wikipedia entry:

https://en.wikipedia.org/wiki/Electronic_band_structure

the discrete energy levels of an isolated atom split into N levels in a macroscopic crystalline lattice (N being the number of atoms in the lattice) forming essentially continuous bands from the discrete levels. I can understand why a molecular bond between two atoms could cause a discrete level to split into two hybrid levels. But I don't understand how a macroscopic lattice could split into N levels (N being enormous), as though ALL of the atoms of the macroscopic material were somehow bonded individually no matter how many intervening atoms. Can someone enlighten me?

 

[bob012345]

Summary:: N level (~continuous) energy bands formed from the discrete energy levels of individual atoms when joined in macroscopic lattices of N atoms (N being one jillion)

According to this Wikipedia entry:

https://en.wikipedia.org/wiki/Electronic_band_structure

the discrete energy levels of an isolated atom split into N levels in a macroscopic crystalline lattice (N being the number of atoms in the lattice) forming essentially continuous bands from the discrete levels. I can understand why a molecular bond between two atoms could cause a discrete level to split into two hybrid levels. But I don't understand how a macroscopic lattice could split into N levels (N being enormous), as though ALL of the atoms of the macroscopic material were somehow bonded individually no matter how many intervening atoms. Can someone enlighten me?

It is similar to and an extension of multiple square wells in Quantum Mechanics. When two wells are separate they do not interact but when they are close they do and the energy levels split. When you bring multiple atoms together they interact. *Kronig and Penney first showed how band structure developed using Quantum Mechanics and is based on the work of Bloch where the model solves one electron wavefunction in a periodic potential which represents the effect all the other atoms. You can see how the band structure comes out of the math in the first three pages of this simplified model. See figure #2.

* https://web.physics.utah.edu/~lebohec/P5510/References/Kronig_Penney_periodic_potential.pdf

 

[TJonline]

Kronig and Penney's math is quite beyond me and I'm amazed that they were so proficient with QM way back in 1930. The dilemma for me remains that in a MACROSCOPIC lattice, all N jillion atoms are NOT close, yet ALL N atoms evidently have an equal influence on the transition from discrete orbitals to the N-level band structure. I guess I'll have to just chalk it up to quantum weirdness, recall that electrons in a sense take all possible paths even around Jupiter according to Feynman as to the double slit experiment, be satisfied that you at least can make sense of the math, and just take it as a given. I'm going to bed with a headache now. Lol. Thanks.

 

[bob012345]

Kronig and Penney's math is quite beyond me and I'm amazed that they were so proficient with QM way back in 1930. The dilemma for me remains that in a MACROSCOPIC lattice, all N jillion atoms are NOT close, yet ALL N atoms evidently have an equal influence on the transition from discrete orbitals to the N-level band structure. I guess I'll have to just chalk it up to quantum weirdness, recall that electrons in a sense take all possible paths even around Jupiter according to Feynman as to the double slit experiment, be satisfied that you at least can make sense of the math, and just take it as a given. I'm going to bed with a headache now. Lol. Thanks.

I think you may be taking the model too literally. Taking $N→∞$ is a mathematical simplification but because the number of atoms has been so large in typical transistors, it has been historically reasonable. I believe using a periodic potential as Bloch proposed in 1928 is tantamount to assuming an infinite crystal. Obviously, if you have a quantum dot in isolation you have a reasonably finite piece of semiconductor material and the band structure is no longer continuous as seen in the chart below. Modern transistors have only on the order of perhaps tens of thousands of atoms. Every energy level contributing to the band structure in a real transistor has a spread and when thousands overlap you get an experimental continuum even if you only included the few thousand nearest atoms but that assumption is less true as devices approach their ultimate nanoscale limits such as in quantum dot devices where there may be only tens to hundreds of atoms involved and the energy spectrum becomes more discreet. I believe if one does the actual computation of solving the Shrodinger equation, which we can now do by computer, with increasing orders of magnitude of atoms starting from ten onwards one would get very close to the continuous band structure after only a few orders of magnitude. The infinite crystal is a mathematical simplification.

from https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5688190/