What determines the energy levels in the conduction band of solids?

[RaduAndrei]

Consider the figure from section "Electrical conductivity":https://en.wikipedia.org/wiki/Valence_and_conduction_bands

I understand that below the Fermi level, we have bands of allowed energy separated by band gaps. Then, we have the Fermi level. Then we have the conduction band.

My question. Above the Fermi level I see band gaps. Is this the case? I thought that the free electrons can have any energy that they want.

A second question. I know that below the Fermi level, the energies of the electrons are quantified. Thus, we say "continuous" bands. But actually these bands contain discrete energies very close one to another. Is it the same for the conduction band? Or when the electron is found in the conduction band, can it have any energy that he wants? Or its energy is still quantified as a free electron?

 

[DrDu]

My question. Above the Fermi level I see band gaps. Is this the case? I thought that the free electrons can have any energy that they want.

If an electron gets lifted above the Fermi level, e.g. by absorption of light, it is not more free than an electron below the Fermi level. It still sees the periodic potential of the cores and the other electrons. Hence there are still bands and band gaps.

 

[RaduAndrei]

Thanks for the answer.

My physics teacher said in my course: " An electron found in the valence band is a bound electron that has its energy quantified, while an electron found in the conduction band is a free electron which has a continuous energy".

But on google, I seem to always find that the energy of a free electron is also quantified, and not a continuous value. Maybe he got it wrong.

 

[RaduAndrei]

So, inside the solid, the energy of an electron is like so.

We have bands separated by band gaps. And these bands contain discrete energies very close to one another. Hence, we say "continuous" but they are actually discrete. And two particular bands of interest are those situated near the Fermi level. If an electron receives the ionization energy then he goes to the conduction band and is a free electron. Above the conduction band there is a band gap. Above this band gap there is another band and so on. Right? But until when?

I imagine that if an electron receives sufficient large energy then he can jump out of the solid maybe.

 

[RaduAndrei]

I think what my physics teacher meant is like so. As the energy of the electron increases, the height of the band gaps decreases in comparison to the height of the bands. So at sufficient large energy, the height of the band gaps is so small that we can say we have a single continuous conduction band.

 

[DrDu]

I imagine that if an electron receives sufficient large energy then he can jump out of the solid maybe.

Yes, and only this process is considered ionisation. But there are electronic exitations at lower energy which don't let to ionisation and were electrons end up in excited bands.

 

[RaduAndrei]

Yes, and only this process is considered ionisation. But there are electronic exitations at lower energy which don't let to ionisation and were electrons end up in excited bands.

I thought ionization means when the electron leaves its atom, but not the entire solid.

 

[RaduAndrei]

Or what you mean is that maybe the band theory considers that the solid is infinite. And the electron energy is negative. And when the separation from the atom goes to infinity, the energy of the electron goes to 0. And at 0, we have ionization.

 

[DrDu]

Yes exactly. Maybe things become clearer if you consider a hypothetical solid made up of -say- hydrogen atoms which are at large distance from each other (i.e. the lattice constant is very large). Then the atoms hardly interact and the bands are very narrow, centered at the disrecte energy levels of isolated hydrogen atoms. The Fermi level lies within the lowest band formed from the 1s states. However, there are many more empty bands before you reach the ionization continuum at 13.6 eV above the ground state level.

 

[RaduAndrei]

Yes exactly. Maybe things become clearer if you consider a hypothetical solid made up of -say- hydrogen atoms which are at large distance from each other (i.e. the lattice constant is very large). Then the atoms hardly interact and the bands are very narrow, centered at the disrecte energy levels of isolated hydrogen atoms.

I understood this.

The Fermi level lies within the lowest band formed from the 1s states. However, there are many more empty bands before you reach the ionization continuum at 13.6 eV above the ground state level.

The Fermi level lies within the lowest band meaning that it lies right at the height of the lowest band. Right at the last discrete level of the band. Right? I know that the Fermi level is the last level occupied with electrons at 0K.

And "many more empty bands". We have a band gap, followed by the lowest band, followed by a band gap, followed by a band gap that extends to infinity. There are no bands above the second band, right?

 

[DrDu]

No, not the last discrete level, rather somewhere in the center as each hydrogen orbital can take up up to 2 electrons but each hydrogen only contributes one.

 

[RaduAndrei]

No, not the last discrete level, rather somewhere in the center as each hydrogen orbital can take up up to 2 electrons but each hydrogen only contributes one.

Aa, ok. If for example I have N hydrogen atoms, then the first level will split into N levels. And the N electrons will go in the first N/2 levels. So the band is half empty. So the Fermi level is at the middle. I see.

And I think there are still bands above the second band. Not only an infinite band gap. Because the hydrogen atom does not have an electron found in the 3rd energy level, it does not mean that it can't go there. Right?

 

[RaduAndrei]

I mean. The electron does not care from which network he is part of. He just has a quantified energy. The low levels are separated by large heights and the quantified energy is obvious. But as the energy of the electron increases, the differences between the energy levels decreases and we can say that the energy of the electron is continuous.

For example, for hydrogen we have -13.6, -3.4, -1.5, -0.85, -0.54... (eV)
So the differences between the levels decreases until the basic quantum energy unit which is h*frequency.

So, every atom is characterized by a frequency.

But now I can ask. What happens when an electron goes from one atom (Si) to another different atom (Ge). Its energy suddenly updates so it can have other values of energy levels? Is the energy content of an electron imposed by the atom from which he is part of?

 

[DrDu]

Aa, ok.

And I think there are still bands above the second band. Not only an infinite band gap. Because the hydrogen atom does not have an electron found in the 3rd energy level, it does not mean that it can't go there. Right?

Yes, of course

 

[RaduAndrei]

I mean. The electron does not care from which network he is part of. He just has a quantified energy. The low levels are separated by large heights and the quantified energy is obvious. But as the energy of the electron increases, the differences between the energy levels decreases and we can say that the energy of the electron is continuous.

For example, for hydrogen we have -13.6, -3.4, -1.5, -0.85, -0.54... (eV)
So the differences between the levels decreases until the basic quantum energy unit which is h*frequency.

So, every atom is characterized by a frequency.

But now I can ask. What happens when an electron goes from one atom (Si) to another different atom (Ge). Its energy suddenly updates so it can have other values of energy levels? Is the energy content of an electron imposed by the atom from which he is part of?

I mean, who imposes an electron what quantified energy it can have? The atoms? Maybe it has not been discovered yet. We just know that they behave like this.

Anyway, here lies the problem that some atoms can't make bonds with all the other atoms and such, only with some of them. I still have much to learn.

 

[DrDu]

Of course it has been discovered. There are all kinds of programs out there to calculate the band structure ab initio. Just have a look at you favourite solid state physics book.

 

[RaduAndrei]

Ok, thanks for the answers.

 

[RaduAndrei]

I have another question.
Consider the figure from the image.

In the book it says:

" Solids are conductors in the following two cases:
a) the band with the highest allowed energy, called valence band (BV), is not fully occupied with electrons (see figure 4.3.a). In this case, the valence band is the same as the conduction band.
b) the valence band overlaps with the next band, completely empty, called conduction band (see figure 4.3.b)"

So, in case a) I can see what happens. For example, sodium described by 1s2 2s2 2p6 3s1. We will have 3 fully occupied bands followed by a band occupied only half of it. So, even an weak electric field applied to the crystal can excite the electrons in the other half of the band. So sodium is conductive.

But I don't understand the case b). How can it overlap?

I think they make a distinction between metals and semimetals but don't say this specifically.
And the overlapping refers to phonons, momentum and such.

 

[nasu]

About overlapping, you can think what happens as you bring the atoms closer and closer. Every atomic energy level splits into closely spaced levels, which form the bands.
If the widths of two neighboring bands becomes larger (in energy) than the distance between initial levels they will overlap.
See this diagram, for example:
http://ecee.colorado.edu/~bart/book/book/chapter2/gif/fig2_3_2.gif

 

[RaduAndrei]

I see. Another question. (I might ask questions and maybe it would be better to go through a whole book, but I just want to get a correct picture of what happens and be done with it and move further into transistors and such. There are textbooks that treat this subject of valence and conduction bands too lightly for my taste. And to go through a whole book of solid state physics to really understand what happens...well, maybe in the future. At the moment, I don't have time).

So my question.

Consider two hydrogen atoms (so each has one electron) very far apart so that they don't interact. Thus, the two electrons have the same 4 quantum numbers: 1, q1, q2, up. Where 1 is the number of the energy level, up is the spin and q1, q2 the other quantum numbers.

When I approach them, the first level must split into two levels so that the Pauli exclusion principle (which says that every electron must have its own unique set of quantum number) is respected. Thus, the two electrons will have the quantum numbers: 1,q1,q2,up and 1,q1,q2,down. Thus the two electrons go into the first energy level, each with a different spin.

So in approaching the atoms, the quantum numbers must change. And what changes is the energy level and the spin.

But can't only the spin change without the energy level getting split? So if initially both electrons had the quantum numbers 1,q1,q2,up, then after approaching them, one of the electrons just changes his spin to "down" so that the new quantum numbers of the electrons are: 1,q1,q2,up and 1,q1,q2,down.

Why the energy level must split? Pauli exclusion principle will still be respected if just one of the electrons changes its spin without splitting the energy level.

 

[RaduAndrei]

I think that the energy level splits and the two electrons go into the lowest level because a H2 molecule must have less energy than the energy of the two separate hydrogen atoms in order for the two atoms to make the covalent bond. So changing just the spin of the electron does not lower the energy of the system and, thus, the molecule will maybe have a higher energy than the energy of the two separate hydrogen atoms. Hence, the molecule will not form in the first place.
Changing the spin just changes the direction but not the energy content.