Semiconductors - Energy Band Clarification

[Marcin H]

Homework Statement

This isn't really a problem, but I wanted some clarification on this topic. In an energy band like the one I drew below, what determines the size of the conduction band and valence band? Etop - Ec and or Ev - Ebot? Is that just a property of the material? Also, how do electrons choose where to go inside the conduction band? What I mean is if an electron gets enough energy to go from the valence band to the conduction band, how does it decide to stay towards the bottom of the band vs the middle or close to the top?

Energy bands come from a bunch of atoms together and we are looking at top view I guess (the valence shell of each atom together) and that is what forms our valence band and conduction band, correct? But in the band diagram of an atom I thought we can only be at the valence band and not above or below or between, but strictly at the band. There are no energy states inbetween the bands I mean. You can only be at one or the other. Is my understanding of this incorrect?EDIT***

Also, for a big picture view of everything, how does the energy diagram correspond to our physical material? For example, if we have a chunck or block of intrinsic silicon, where is this energy band with respect to that chunck of silicon? I don't know if my question is making sense, but I guess another way to put it, where is the valence band and conduction band in our physical chunk of material?

Edit 2***

I am struggling to relate the energy band diagram to the physical object itself. They are 2 different ideas correct? Energy band and physical chunk of material? Does the valence band and conduction band have anything to do with the atoms valence shell and conduction shell? Separately I understand the 2 for the most part. Relating them together, I get completely lost. I understand properties of the energy band diagram and how electrons and holes move around in it and stuff, but what is physically happening in our material. If a electron gets enough energy to jump from the valence band to the conduction band then what is physically happenign in our block of material? What is happening in a chunk of intrinisc silicon for example: https://imgur.com/a/0UFz2

Homework Equations

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The Attempt at a Solution

 

[Drakkith]

In an energy band like the one I drew below, what determines the size of the conduction band and valence band? Etop - Ec and or Ev - Ebot? Is that just a property of the material?

The "size" of the valence and conduction bands is probably a complicated function of the number of and electronic configuration of the atoms, but I don't know the details. You could probably say it's an inherent property of the material.

Also, how do electrons choose where to go inside the conduction band? What I mean is if an electron gets enough energy to go from the valence band to the conduction band, how does it decide to stay towards the bottom of the band vs the middle or close to the top?

It's mostly a matter of how much energy the electron acquires and where it starts. If it acquires enough energy to jump the bandgap, then it will end up in whatever state it had the energy to jump to from its location in the valence band. In other words, if an electron acquires 2.5 eV and jumps across the bandgap, it will end up in whatever state corresponds to 2.5 eV above its previous state in the valence band.

Energy bands come from a bunch of atoms together and we are looking at top view I guess (the valence shell of each atom together) and that is what forms our valence band and conduction band, correct? But in the band diagram of an atom I thought we can only be at the valence band and not above or below or between, but strictly at the band. There are no energy states inbetween the bands I mean. You can only be at one or the other. Is my understanding of this incorrect?

Energy bands come from combining enormous numbers of orbitals. For example, the valence band comes from the combination of something like 10^²⁰ valence orbitals. Since no electrons can have the same quantum numbers, this means that each orbital has to have a slightly different energy. The difference in energy between two adjacent orbitals in the bands (adjacent in energy terms, not in a physical sense) turns out to be on the order of 10^^-22 eV, a supremely tiny number. A number so small that we can model the valence and conduction bands as being continuous, even though they really aren't.

Also, for a big picture view of everything, how does the energy diagram correspond to our physical material? For example, if we have a chunck or block of intrinsic silicon, where is this energy band with respect to that chunck of silicon? I don't know if my question is making sense, but I guess another way to put it, where is the valence band and conduction band in our physical chunk of material?

The electrons exist throughout the material. Energy band diagrams represent the energy that the electrons in the valence and conduction orbitals can have. The diagrams don't represent the physical location of the electrons.

 

[Marcin H]

Since no electrons can have the same quantum numbers, this means that each orbital has to have a slightly different energy.

Can you explain this a bit further. What do you mean by quantum number? What is a quantum number? Also, if I am understanding this correctly you are saying that if we have 1 silicon atom with an orbital or energy level at 7ev then another silicon atom that bonds with this one can't have an orbital with 7ev?

Are you saying no X amount of electrons can have the same energy level? Or when bonding no X amount of electrons can have the same energy level?

The electrons exist throughout the material. Energy band diagrams represent the energy that the electrons in the valence and conduction orbitals can have. The diagrams don't represent the physical location of the electrons.

but are the valence bands and conductance bands physically in the materiall? I feel like I'm really close to understanding this, but something is still bothering me about this. if we have an electron moving from the valence band to the conduction band, how does that relate to what is physically happening in the material? I am this stage where I feel like I'm understanding this, but then think of something and it all stops making sense again, lol.

Edit* should I be thinking about this like we are in two different domains? Like the space domain and the energy domain or something?

 

[Drakkith]

Can you explain this a bit further. What do you mean by quantum number? What is a quantum number?

In this context, a quantum number is one of several quantities which give a valid solution to the schrodinger wave equation for an atom (specifically a hydrogen atom, but we can use its as a reasonable approximation for semiconductor physics at this level). Per the Pauli exclusion principle, no two electrons can exist in the same state. The particular set of quantum numbers determines which state an electron exists in. Thus, no two electrons can have the same quantum numbers. Furthermore, since an "orbital" is literally defined by the quantum numbers of the electron occupying it, different electrons must occupy different orbitals at all times. I'm counting electrons with opposite spins as being in different orbitals, though you'll usually read that electrons can exist within the same orbital if they have opposite spins. Either way it's the same result. Any two electrons must have unique combinations of quantum numbers.

In addition, each unique set of quantum numbers results in a different energy level for the electron with that set of numbers. So different orbitals also have different energies. This also means that when you try to force two atoms very close together, close enough that they will bond together if possible, their individual orbitals cannot remain the same since this would mean that they have the same quantum numbers. Instead the electrons are forced to occupy different orbitals with different energies. These new orbitals are molecular orbitals.

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

Also, if I am understanding this correctly you are saying that if we have 1 silicon atom with an orbital or energy level at 7ev then another silicon atom that bonds with this one can't have an orbital with 7ev?

It's likely that neither will have an orbital with an energy of 7 eV. The two atomic orbitals at 7 eV will likely combine into molecular orbitals with energies just above or below 7 eV if I remember my solid state chemistry class correctly. Unfortunately my textbook from my class doesn't actually appear to say anything on the matter...

but are the valence bands and conductance bands physically in the materiall? I feel like I'm really close to understanding this, but something is still bothering me about this. if we have an electron moving from the valence band to the conduction band, how does that relate to what is physically happening in the material?

See below.

Edit* should I be thinking about this like we are in two different domains? Like the space domain and the energy domain or something?

Pretty much, yes. The valence and conduction bands exist everywhere in the material and aren't physically separated from each other. This makes sense given that the valence and conduction bands are just the combination of huge numbers of valence and higher atomic orbitals. All of these orbitals overlap each other in terms of where you might find an electron from any particular orbital. In an atom, a 1s electron (the "closest" to the nucleus) still has a small chance of being found waaaaaay out where you would likely find a 5f electron, and a 5f electron as a small chance of being found near to the nucleus where you'd likely find a 1s electron. The same is true when you get to molecules and bulk metals and semiconductors. You can find any particular electron anywhere within the material. Certain locations just have a larger or smaller probability of finding the electron there. So in essence all of these orbitals and energy levels all overlap each other somewhat physically.

 

[Marcin H]

Thanks!