# I Why energy band diagram is flat?

1. Aug 1, 2015

### persia77

if in a solid there arent no states with same energy then why energy band vs location diagram is flat?

2. Aug 1, 2015

### NaOH

The energy states are spread throughout the whole solid. So each horizontal line represents the same state.

3. Aug 2, 2015

### persia77

but pauli exclusion principle states that there are not same states in a solid

4. Aug 2, 2015

### NaOH

The same energy state is extended throughout the solid, in other words, the electron wave is spread throughout the whole solid.

5. Aug 2, 2015

### persia77

if you are correct then two electron can occupy same energy state but pauli exclusion principle states that they cannot
is it correct?

6. Aug 2, 2015

### NaOH

I did not mean to imply that two electrons occupy the same state. I am saying that the electron is spread throughout the whole solid. However, as we move along the crystal and we are still talking about the same electron (and thus state), the line is diagrammatically shown to be horizontal. The electrons still occupy states as allowed by pauli's exclusion principle.

Last edited: Aug 2, 2015
7. Aug 2, 2015

### persia77

if you are correct then a electron dosnt have position in solid and electron move through the solid but in valance band electron is not free and dont move

8. Aug 2, 2015

### NaOH

To understand this you need another piece of knowledge which is not found in your typical band diagram -- that of (crystal) momentum.

Associated with each energy state is a certain (crystal) momentum. This momentum is what allows the net movement of electrons, which we measure as current. When we apply an electric field, we can describe it as shifting all the electron states with some momentum. In metals, in general, this happens without any huge problems because the shifting is done by occupying the conduction band states.

In insulators, things behave slightly differently. Now, when we apply the electric field, we are not able to shift the electron states! Why? Because of the band gap -- there are no states to be shifted to.

In semiconductors, we get to access some of these states, but usually it is more than we need.

9. Aug 2, 2015

### ZapperZ

Staff Emeritus
This is the full band structure of copper:

You'll notice that it isn't "flat", especially the conduction band.

As NaOH is trying to tell you, your band diagram is a "cartoon" picture simply to indicate where the available states are. It is really a reduced picture where the momentum k has been integrated out, leaving only the available energy states. This means that the electrons in the bands are characterized by not just energy, but also momentum (E, k). So while they may occupy the same energy, there is degeneracy in the form of differing momentum, resulting in a band dispersion. This does not result in an overall "flat" curves.

Zz.

10. Aug 2, 2015

### persia77

i say about valance(bonding) electron not free electron
valance electron is fixed not moving
you say there are same energy state in solid ,it is contradicted with pauli exclusion principle

11. Aug 2, 2015

### persia77

i say about real space not k space
please show real space diagram

12. Aug 2, 2015

### ZapperZ

Staff Emeritus
WHY? That makes no sense. The band structure is DEFINED as E vs k. Your "energy band diagram" is a compressed version of this!

Don't you care to know what the FULL description looks like?

Zz.

13. Aug 2, 2015

### ZapperZ

Staff Emeritus
Look at the valence band structure of copper! There's a lot of curving lines there! And again, there are degeneracies!

Zz.

14. Aug 2, 2015

### NaOH

Consider an Oxygen molecule. The outermost electrons are moving. Electrons don't simply stop because they participate in chemical bonds.

A crystal can be very crudely described as a very very large molecule. The electrons here then are shared (and thus moving) throughout this whole collection of atoms.

To put it very simply, each electron belongs has one state belonging to some energy. But since this electron is shared by all, at different location, if you are looking at the same electron it will have the same energy (therefore horizontal line).

Pauli's principle is not violated because the each energy state only has one electron, but as we plot energy vs. location we draw a horizontal line because it is still the same energy state -- it is still the same electron.

So each energy state also has a momentum associated with it. When the valance band is full, the electrons are still moving all over, but there is no net movement because electrons that travel left are balanced by those that travel right. It is by going to an empty band that the electrons are able to occupy energy states that have momentum that goes, say, to the right!

But this is a very gross simplification. As ZapperZ noted there are degeneracies, which means to say that different states have the same energy. I don't think you need to worry about this, because the main point is that each state you see is for the whole crystal, and when an electron is in that state, no other electron can occupy that state.

Last edited: Aug 2, 2015
15. Aug 2, 2015

### persia77

if they move then what is diffence between them and free electron?
why we can define effective mass only for free electron?

16. Aug 2, 2015

### NaOH

The difference between them and free electron is effective mass. In solids, you must understand that bulk properties are usually not the result of each single electron minding their own business but rather the effect of the collection of all their properties.

Since the electrons are still moving in the valance band, the question is why the valance band does not conduct. The reason is that collectively, there is no net movement of electrons.

As for what effective mass is, it is a model we use to describe the motion of electron in solids because it is not entirely free, it is bonded to atoms, but in many ways it is like a free electron since it moves throughout the whole crystal.

17. Aug 2, 2015

### persia77

please show me a text book that say your statement

18. Aug 2, 2015

### NaOH

Introduction of Solid State Physics, charles kittel.

Look under bloch functions

19. Aug 2, 2015

### persia77

thanks
but its not mentioned there

20. Aug 2, 2015

### NaOH

I can't help you any further. What I can say is that the book mentions the form of wavefunction for travelling waves, for another the Bloch function is a (travelling) plane wave.