Understanding Band Theory: A Basic Guide

[Maddie1609]

Hi!

I am having difficulty grasping what band theory actually is, especially what the bands are and the energy gap between them. Are they probabilities? Shared electrons between atoms such as in metallic bonding?

What is the difference between the conductor band and the valence band? I think I read somewhere that the conductor band is the outermost band, but shouldn't that be the valence band as it comprises of valence electrons?

And finally, does someone have photos or videos for visualization purposes?

I'm learning it on a very basic level by the way.

 

[moriheru]

I am not really familiar with the field I study QM and not solid state physics. But as far as I have there are several types of bands, the forbiden band and the allowed band, where the allowed band is the allowed energy range of a free electron. I don't know how far you are in the theory or how far your mathematical skills are, but if you have any questions I can try and help...

 

[Maddie1609]

I am not really familiar with the field I study QM and not solid state physics. But as far as I have there are several types of bands, the forbiden band and the allowed band, where the allowed band is the allowed energy range of a free electron. I don't know how far you are in the theory or how far your mathematical skills are, but if you have any questions I can try and help...

Thanks! Do you know why in solids there are bands instead of energy levels? I think I read something in the likes of; atoms in a solid interact with each other which broadens the energy levels to make room for two sets of electron. Does this have any merit to it?

Edit: not far at all, I'm actually just learning about semiconductors, conductors and insulators, but I have a hard time conseptualizing it without knowing more about energy bands.

 

[marcusl]

A band is the collection of allowed energy levels for electrons in a crystal. If you think of energy levels of electrons in hydrogen--the simplest atom--they are quantized. The levels of two hydrogen atoms far apart are identical, furthermore, but if you bring them near each other, interactions split each level a tiny bit into two closely spaced levels, one above and one below the original level. The energy diagram for the two-atom system, therefore, consists of level doublets that are available to the two electrons in the system. The gaps between allowed levels are there as before.

Now move to a more complex atom in a crystal. A crystal has 10^22 atoms per cubic centimeter, so each discrete energy level of an atom is smeared into a huge number of closely spaced levels due to its interactions with many neighbors. There are so many levels spaced so close together that they appear as a continuum that is called a band. Electrons occupy these bands, that is, they have energies within one band or another. The band gaps are what remains of the disallowed energies between the discrete levels of a single atom.

There are, in general, many filled bands corresponding to the filled shells familiar from chemistry. Only the highest occupied band is of interest for electric conduction, and it is called the valence band. At low temperature, electrons in a semiconductor drop into the lowest available level and are bound to the crystal atoms. At higher temperatures, or with the addition of an electric potential, some valence electrons acquire enough energy to jump into the vacant band above (called the conduction band) where they are free to roam through the crystal. Thus conductivity in a pure semiconductor rises with temperature, which is opposite to the behavior of metals (metals have electrons occupying the conduction band even at 0K).

As for materials, I have to believe that the web will have lots of tutorials so look around. Come back if you have more questions.

 

[moriheru]

1.In QM you have discrete energy levels that, for example if you are talking about harmonic oscillators the energy levels are half integers multiplied by omega and placnkcconstant bar. That is for a single atom. SO what happens if we have several atoms and even wmore and even more, well the energy levels will become continuos, so we speak of bands...

(as I said I am not familiar with the field so pleace don't qoute me but I am sure)

2. You said something about probabilitys earlyer. Yes one does use probabilitys in Band theory, that is one uses the wavefunction of the elctrons to derive the bands.

 

[Maddie1609]

A band is the collection of allowed energy levels for electrons in a crystal. If you think of energy levels of electrons in hydrogen--the simplest atom--they are quantized. The levels of two hydrogen atoms far apart are identical, furthermore, but if you bring them near each other, interactions split each level a tiny bit into two closely spaced levels, one above and one below the original level. The energy diagram for the two-atom system, therefore, consists of level doublets that are available to the two electrons in the system. The gaps between allowed levels are there as before.

Now move to a more complex atom in a crystal. A crystal has 10^22 atoms per cubic centimeter, so each discrete energy level of an atom is smeared into a huge number of closely spaced levels due to its interactions with many neighbors. There are so many levels spaced so close together that they appear as a continuum that is called a band. Electrons occupy these bands, that is, they have energies within one band or another. The band gaps are what remains of the disallowed energies between the discrete levels of a single atom.

There are, in general, many filled bands corresponding to the filled shells familiar from chemistry. Only the highest occupied band is of interest for electric conduction, and it is called the valence band. At low temperature, electrons in a semiconductor drop into the lowest available level and are bound to the crystal atoms. At higher temperatures, or with the addition of an electric potential, some valence electrons acquire enough energy to jump into the vacant band above (called the conduction band) where they are free to roam through the crystal. Thus conductivity in a pure semiconductor rises with temperature, which is opposite to the behavior of metals (metals have electrons occupying the conduction band even at 0K).

As for materials, I have to believe that the web will have lots of tutorials so look around. Come back if you have more questions.

Thank you so much, that was perfect! Exactly what I've been searching for for hours :-)