# Are the electrons in the conduction band completely free?

• I
I'm having trouble to understand why it's said that electrons in the conductor band are free while electrons in the valence band are not.
I know by the Schrodinger equations that the trajectory of an electron inside a specific band and with a specific energy level is a probability. From what I understand, there is a probability (even though it's really really low), that the H-electron of a H-tom in Earth appears in the moon in some moment and then comes back to us. So, by this understanding, I would say that even the electrons in the valence band could indeed travel an undefined distance far from the nucleus. And this to me is conductivity.

So my questions is, what really defines the conduction band and differentiate it from the valence band? Why in the conduction band we would see much more conductivity, if in the valence band we already see some degree of conductivity.

By what I searched it's said that the conduction band is more conductive because the electrons are free. But in that sense, what are free electrons? As I said, an electron in any atom could go to the moon and come back, this is free enough for me. What is defined to be this freedom of electrons in the conduction band that the electrons in the valence band does not have? And how do this affect conductibility?

Does it have anything to do with a negative and positive total energy? For example, we know by Kepler law that if 2 bodies have negative total energy (potential + kinetic) the motion will be an ellipse, and the bodies will never be able to go to infinity (they will be "tied"). But if the total energy is positive the motion will be a hyperbola. The bodies will be able to distance themselves how much they want. Is the conduction band defined like that? A band in witch the potential electrical energy + kinetic energy of the electrons are positive?

Drakkith
Staff Emeritus
If I remember my solid state physics class correctly, and I admit it's been a while, then the difference between a 'free' electron that is in the conduction band vs a 'bound' electron in the valence band is that the conduction band has a huge number of close energy levels that are unoccupied that the electron can easily transition to. These transitions can be driven by anything that adds or removes energy from an electron, such as an electric field, thermal energy, collisions with other electrons or ions, etc.

The valence band also has a huge number of closely spaced energy levels, but these are almost entirely occupied by electrons. Hence an electron in the valence band simply has nowhere to go if only a small amount of energy is added or subtracted from it.

See here for an explanation of the underlying band structure theory: https://en.wikipedia.org/wiki/Electronic_band_structure

By what I searched it's said that the conduction band is more conductive because the electrons are free. But in that sense, what are free electrons? As I said, an electron in any atom could go to the moon and come back, this is free enough for me. What is defined to be this freedom of electrons in the conduction band that the electrons in the valence band does not have?

The availability of very closely spaced empty energy levels (aka electronic states). The conduction band has these, the valence band does not, as the latter is made up of states that are already filled.

PeterDonis
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2020 Award
I would say that even the electrons in the valence band could indeed travel an undefined distance far from the nucleus. And this to me is conductivity.

No, it isn't conductivity.

When you say "the electrons in the valence band could indeed travel an undefined distance far from the nucleus", that's not correct. A correct statement would be "if you were to make a position measurement on an electron in the material, there is a small but nonzero probability of the result being a position an undefined distance far from the nucleus". But that statement is only meaningful if you are measuring the electron's position.

When materials conduct electricity, nobody is measuring the positions of the electrons. As a rough approximation, the observable involved is the energy of the electrons, and "conductivity" means "if you make some extra energy available inside the material by putting an electrical potential across it, there will be lots of electrons that can each take up a little bit of that energy and transfer it from one side of the material to the other". Whether or not that can happen has nothing to do with the probability of measuring the position of an electron to be very far from the nucleus. It has to do with what energy states are available to the electrons.

Dale and vanhees71
PeterDonis
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2020 Award
what are free electrons?

Electrons that are in states that are not bound to a particular atom. Again, that has nothing to do with the probability of measuring an electron's position to be very far from the nucleus of some atom. It has to do with the overall wave function of the electron--whether that wave function is centered on a particular nucleus (valence electron, bound) or has fairly uniform amplitude all throughout the material (conduction electron, free).

vanhees71
@PeterDonis What would mean to say that a given electron is bounded?

I thin k you tried to answer that when you said " It has to do with the overall wave function of the electron--whether that wave function is centered on a particular nucleus (valence electron, bound) or has fairly uniform amplitude all throughout the material (conduction electron, free)."

Do you know where I could thing the graph of a conduction band wave function? Also, your above explanation does not have a limiting point. That is, the wave function of the valence band also goes to infinity, so to say that the conductive band has a more uniform wave function for me does not completely differentiate the two

PeterDonis
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2020 Award
Do you know where I could thing the graph of a conduction band wave function?

Unfortunately I don't have a handy source, no.

the wave function of the valence band also goes to infinity, so to say that the conductive band has a more uniform wave function for me does not completely differentiate the two

Why not? I'm simply describing two very different kinds of spatial dependence of the amplitude: fairly uniform across the entire material (i.e., over a distance that is large compared to the size of a single atom), vs. centered on one particular atom (i.e., much, much larger amplitude around that one atom than anywhere else).

Unfortunately I don't have a handy source, no.
Why not? I'm simply describing two very different kinds of spatial dependence of the amplitude: fairly uniform across the entire material (i.e., over a distance that is large compared to the size of a single atom), vs. centered on one particular atom (i.e., much, much larger amplitude around that one atom than anywhere else).

So you are saying that even from the last valence band to the first conduction band we would have that big change in the graph? Even in metals that have the valence and conduction band almost tied together?

The band structure of a metal has nothing to do with the spatial distribution of the charges. Your typical electron in a metal will be totally delocalized throughout that material and not have much variation on a macroscopic scale.

PeterDonis
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2020 Award
So you are saying that even from the last valence band to the first conduction band we would have that big change in the graph? Even in metals that have the valence and conduction band almost tied together?

Yes. Even a small difference in energy can make a big difference in the spatial wave function, if it's the difference between a bound state energy (less than the potential energy due to the nucleus of a single atom) and a free state energy (greater than the potential energy due to the nucleus of a single atom).

vanhees71
Gold Member
@PeterDonis What would mean to say that a given electron is bounded?
It means that you need some finite minimal energy to set it free from the atom/molecule it is bound to. Take a single atom. There the electron is in a bound state, and it takes the binding energy to "push" it into a scattering state, which describes a situation where the electron is moving away from the nucleus more or less freely.

It's not too different in a medium too. Only here you have to take into account the fermionic nature of the electron because there are of course many electrons around. In a conductor there are some electrons which are not bound to a specific atom anymore and there are enough free states for them to move across the entire medium when you apply an external electric field. They are not totally free though and get scattered, which leads effectively to friction and this makes the conductivity of the material as a macroscopic consequence of this microscopic picture.

ZapperZ
Staff Emeritus
By what I searched it's said that the conduction band is more conductive because the electrons are free. But in that sense, what are free electrons? As I said, an electron in any atom could go to the moon and come back, this is free enough for me. What is defined to be this freedom of electrons in the conduction band that the electrons in the valence band does not have? And how do this affect conductibility?

Does it have anything to do with a negative and positive total energy? For example, we know by Kepler law that if 2 bodies have negative total energy (potential + kinetic) the motion will be an ellipse, and the bodies will never be able to go to infinity (they will be "tied"). But if the total energy is positive the motion will be a hyperbola. The bodies will be able to distance themselves how much they want. Is the conduction band defined like that? A band in witch the potential electrical energy + kinetic energy of the electrons are positive?

Here's a simplistic illustration. The figure on the left is an example of a "bound" state, while the figure on the right is an example of a "free" state where the particle has enough energy to NOT be bounded to the potential.

In a conductor, it is a bit more complicated, because now you have a periodic potential that can be approximated to look like this:

This is the Bloch periodic potential, and when you solve for the wave function, you get the Bloch wave function. Notice that the wave function is a superposition of plane waves! This looks nothing like the wavefunction that you solve for a bound electron in a central potential.

This would have been more appropriate to be asked in the Condensed Matter Physics forum, especially when you are asking about the conduction band wavefunction.

Zz.

Dale, etotheipi and vanhees71
PeterDonis
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2020 Award
This would have been more appropriate to be asked in the Condensed Matter Physics forum

Good point. The thread has now been moved there.

vanhees71