Band theory and qualitative character of bands

[learn.steadfast]

I'm interested in the the band theory of solids for semiconductor modeling.
I haven't solved the Schrödinger equation for multiple atoms, but I'd like to know some details that experienced physicists might already know.

Several texts show how the bands develop from molecular coupling of orbitals; often the description is based on hydrogen since that makes math simpler in ignoring lots of un-necessary electrons. The orbital energies, whether filled with actual electrons or not, split. A typical diagram in undergrad courses is the following. As the atoms are brought close, bands form from distinct orbitals. The number of energy levels inside each band is proportional to the number of atoms being condensed into a solid.

As n atoms are brought closer to each other, each orbital will split into 'n' energy levels.
My understanding is that half of the energy levels are bonding, and half are anti-bonding; at least with a hydrogen molecule that is proven to be the case. With only two levels, it's trivial that one energy level is the "bonding" state, and the other the "anti-bonding"; But I'm not sure what happens when more than two hydrogen atoms are brought together with the same spacing as the previous hypothetical molecule; Would the "top" most energy value be the same as before, or would the "anti-bonding" energy change in value, and can it be approximated from knowledge of the original anti-bonding value? eg: does splitting purely "interpolate" the space between fixed bonding and anti-bonding energies at a given atomic spacing; or does the band's height also change vs.. number of atoms in a "crystal" of an arbitrary but fixed atomic spacing?

I ask, because I think the same qualitative properties should exist in real semiconductors and would help modeling of semiconductors for electronics.

I am not able to solve Schrödinger's sufficiently to get a realistic answer for larger atoms or crystals. Normally, orbitals increase in energy with the inverse square law of their orbit number. $$ E \varpropto { R_h \over n^2 } $$ But, conventional wisdom is that the spacing of the n energy levels inside each band is equal, and I'm not completely convinced of why this should be so.

Models, like Krog & Penny, don't really deal with how bands migrate as groups up and down in energy, or how the spacing of energy levels would change vs. atomic separation. Is the energy spacing inside the band really uniform as would be predicted by simplistic models?

In the above graph, the energy levels inside the "bands" is not traced out.
How do they split in detail? at first, (tracing from right side of the graphs to left), both S and P bands seems to grow equally above and below the original energy level of their respective orbitals (which suggests symmetry, and likely linearity of splitting); but at a certain distance, the bands do opposite things. All 2p orbitals go up in energy, but the S orbitals all go down in energy.

What I would like is reasonable and simple models that give (even if crude) the statistics of how individual bands morph vs. atomic spacing.

For example, the m'th energy level inside a 3S band of n atoms; is it even crudely predictable as to where it will be relative to the band's edge or center? (linear or quadratic, etc.) Do all S orbitals go down the same amount of energy for the same change in atomic spacing, or is it inversely proportional to the square of the orbital number, etc.

For another example: It's clear from the diagram, that the 2S band gets taller vs. 'a' spacing, until a spot near a0, but then starts shrinking again. But I have no idea why, or what form of equation would qualitatively describe the S band's difference in height at any give atomic spacing 'a', it's center, or the absolute position of it's top or bottom.

Thanks.

 

[DeathbyGreen]

It might help you get more responses if you condensed your question a little bit. Could you paraphrase the bullet points or what you're asking? The band structure (if I understood your question) doesn't depend on the number of atoms present, within a continuum model apprixmation. A lattice will have (depending on the atoms present) s or p (or any orbital label) bands which consist of hybridized orbitals of neighboring atoms. Electrons residing in either the p or s bands can only take those energy values in the band structure; because of the lattice nature we describe the bands using Bloch waves which represent the lattice periodicity.

 

[ZapperZ]

I'm interested in the the band theory of solids for semiconductor modeling.
I haven't solved the Schrödinger equation for multiple atoms,

I will tell you right off the bat that no one can solve a multi-atom Schrödinger equation. That is why we have band structure calculations using approximations, such as mean-field approximations, and techniques such as density-functional theory (DFT).

Several texts show how the bands develop from molecular coupling of orbitals; often the description is based on hydrogen since that makes math simpler in ignoring lots of un-necessary electrons. The orbital energies, whether filled with actual electrons or not, split. A typical diagram in undergrad courses is the following. As the atoms are brought close, bands form from distinct orbitals. The number of energy levels inside each band is proportional to the number of atoms being condensed into a solid.

View attachment 209123
As n atoms are brought closer to each other, each orbital will split into 'n' energy levels.
My understanding is that half of the energy levels are bonding, and half are anti-bonding; at least with a hydrogen molecule that is proven to be the case. With only two levels, it's trivial that one energy level is the "bonding" state, and the other the "anti-bonding"; But I'm not sure what happens when more than two hydrogen atoms are brought together with the same spacing as the previous hypothetical molecule; Would the "top" most energy value be the same as before, or would the "anti-bonding" energy change in value, and can it be approximated from knowledge of the original anti-bonding value? eg: does splitting purely "interpolate" the space between fixed bonding and anti-bonding energies at a given atomic spacing; or does the band's height also change vs.. number of atoms in a "crystal" of an arbitrary but fixed atomic spacing?

I ask, because I think the same qualitative properties should exist in real semiconductors and would help modeling of semiconductors for electronics.

I am not able to solve Schrödinger's sufficiently to get a realistic answer for larger atoms or crystals. Normally, orbitals increase in energy with the inverse square law of their orbit number. $$ E \varpropto { R_h \over n^2 } $$ But, conventional wisdom is that the spacing of the n energy levels inside each band is equal, and I'm not completely convinced of why this should be so.

Models, like Krog & Penny, don't really deal with how bands migrate as groups up and down in energy, or how the spacing of energy levels would change vs. atomic separation. Is the energy spacing inside the band really uniform as would be predicted by simplistic models?

In the above graph, the energy levels inside the "bands" is not traced out.
How do they split in detail? at first, (tracing from right side of the graphs to left), both S and P bands seems to grow equally above and below the original energy level of their respective orbitals (which suggests symmetry, and likely linearity of splitting); but at a certain distance, the bands do opposite things. All 2p orbitals go up in energy, but the S orbitals all go down in energy.

What I would like is reasonable and simple models that give (even if crude) the statistics of how individual bands morph vs. atomic spacing.

For example, the m'th energy level inside a 3S band of n atoms; is it even crudely predictable as to where it will be relative to the band's edge or center? (linear or quadratic, etc.) Do all S orbitals go down the same amount of energy for the same change in atomic spacing, or is it inversely proportional to the square of the orbital number, etc.

For another example: It's clear from the diagram, that the 2S band gets taller vs. 'a' spacing, until a spot near a0, but then starts shrinking again. But I have no idea why, or what form of equation would qualitatively describe the S band's difference in height at any give atomic spacing 'a', it's center, or the absolute position of it's top or bottom.

Thanks.

Maybe what you need to start with is to look at something "simpler", such as the tight-binding approximation. After all, you want the band structure of a solid, not a "molecule".

Zz.

 

[learn.steadfast]

DeathbyGreen:

Could you paraphrase the bullet points or what you're asking?

I'm approaching the problem from the "tight binding" theory. I would love to bullet point the issues, but I'm not sure how to do so; yet. I think discussing the theory, and highlighting individual points in lectures is probably the best I can do; I'll devote this post to a single point and answer your objection.

A "continuum approximation" is merely an assumption that the number of (available) stationary states for electrons is so large, that it divides up the band into discrete energy states of negligibly small distance apart, therefore, we can replace discreet mathematics with calculus approximations that assume continuous distributions. In that case, summations can be replaced with integration. None the less, energy levels in reality are quantized ... they aren't continuous.

The band theory of solids is derived from a quantum approach by taking a discreet number of atoms, computing the molecular oribitals for N atoms (eg: hybridized orbitals), and as a side effect, show that for N atoms there are N hybridized orbitals. This is a consequence of the (a)symmetry requirements for fermions and the Pauli exclusion principle.

In the following youtube video, the teacher shows how to do three hydrogen atoms for simplicity, and then switches to lithium atoms for more than three atoms ..., and then gives the "oh by the way" comment that a realistic number of atoms per cubic centimeter in a solid is on the order of 10**23 or so; therefore, the number of energy levels (quantum stationary states) divides the energy band into discrete steps that are extremely small.



For another example; here's a teacher that approaches band theory from a "free electron model" first, and then the "tight binding model" second, to arrive at the same general conclusions. At about 17 to 18 minutes into the video, he discusses generically how the models differ. (really if you want to understand free electron model, you need to go back one lecture) But, at around 24 minutes into the lecture, the instructor begins drawing the discrete split energy levels of different energy orbitals due to hybridizing of 100 atoms. He is just free-hand drawing them, so his graphs don't really show the mathematical density of the energy values inside a "band" but only a crude conceptual idea. He does show, though, that the onset of splitting happens at different atomic spacing for different starting point atomic orbitals. See 28 minutes into the video. The different onset of splitting is because the overlap of the orbitals happens at different atomic spacings.



When you say:

Electrons residing in either the p or s bands can only take those energy values in the band structure; because of the lattice nature we describe the bands using Bloch waves which represent the lattice periodicity.

That's one of the major points I am interested in. What is the "distrubution" or "density" of the splitting energy levels vs Energy and atomic spacing, and how do we know for certain?

It's clear, for example, that all three hybrid orbitals in the last youtube lecture will eventually split into 100 energy levels each. Yet, it's not clear what's going on internal to each band; eg: how far apart each stationary state is in energy. Clearly: The total band height being split is "thicker/taller" as the graph is traced more leftward. Therefore, it's obvious that the spacing between each available stationary energy level must be increasing as we go to the left, because a fixed number of 100 levels is dividing an increasing range of values (vertical distance inside a band). However, it's not clear how that distance is divided up into 100 levels. (linear/quadratic/etc.)

1. How do the stationary energy states in a given band, distribute themselves between the top of the band and the bottom if given a fixed atomic spacing ?
2. As an example; Does anyone know of a plot of three hydrogen atom's hybrid orbital energies numerically solved/simulated/etc. that shows how the splitting of Energy levels spoken of in the first youtube video (@ 9 minutes 47 seconds) would actually vary with atomic spacing? ( The author of the video said his drawing was not to scale. )

 

[learn.steadfast]

I will tell you right off the bat that no one can solve a multi-atom Schrödinger equation. That is why we have band structure calculations using approximations, such as mean-field approximations, and techniques such as density-functional theory (DFT).

Yes, I'm sure. But there are approximate solutions; and that's all I meant.
I seem to recall reading about hartree-fock variational methods and preturbation theory in undergraduate courses a long time ago.

Maybe what you need to start with is to look at something "simpler", such as the tight-binding approximation. After all, you want the band structure of a solid, not a "molecule".
Zz.

Umm, yeah. See the videoes in the previous post. I sort of thought I was starting with the tight binding model.
It's used for as little as three atoms... :)