# Help Understanding Energy Band Diagram

I come across the attached diagram in electrical engineering books quite often and, not being a physics text, they often fail to explain it. I tried to do my homework by spending a couple of hours on google trying to figure out the answer to my question myself, but all I can find online are summaries lacking any real meat or explanations practically identical to my text's.

Some help with my diagram:
I am copying a "diagram of a cross section of the total energy surface" of an SI atom. The red lines are total energy. The +4 on the x-axis is supposed to be the nucleaus of the SI atom plus 10 inner electrons. I left out the 3s2 layer purposely to keep the diagram cleaner.

My questions are:
I get the y-axis but what is the x-axis susposed to be? Is it the radius of the orbital circumference, the diameter of the orbital circumference, or something else entirely? My guess is the later because I cannot figure out how either can pass through zero on the x-axis for an outer shell. I guess what I really want to know is, how was the cut made through the total energy surface? Are all the electrons in the same surface, but occuping different points on that surface. Or do they actually have a unique surface? I think the "unique energy level" criteria (Pauli Exclusion Principle) could be satisfied either way. I dunno, perhaps my real problem is with visualizing the energy surface...

This diagram is supposed to provide "deep insight" into the operation of semiconductor devices so I really would like to understand it.

es

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ZapperZ
Staff Emeritus
es said:
I come across the attached diagram in electrical engineering books quite often and, not being a physics text, they often fail to explain it. I tried to do my homework by spending a couple of hours on google trying to figure out the answer to my question myself, but all I can find online are summaries lacking any real meat or explanations practically identical to my text's.

Some help with my diagram:
I am copying a "diagram of a cross section of the total energy surface" of an SI atom. The red lines are total energy. The +4 on the x-axis is supposed to be the nucleaus of the SI atom plus 10 inner electrons. I left out the 3s2 layer purposely to keep the diagram cleaner.

My questions are:
I get the y-axis but what is the x-axis susposed to be? Is it the radius of the orbital circumference, the diameter of the orbital circumference, or something else entirely? My guess is the later because I cannot figure out how either can pass through zero on the x-axis for an outer shell. I guess what I really want to know is, how was the cut made through the total energy surface? Are all the electrons in the same surface, but occuping different points on that surface. Or do they actually have a unique surface? I think the "unique energy level" criteria (Pauli Exclusion Principle) could be satisfied either way. I dunno, perhaps my real problem is with visualizing the energy surface...

This diagram is supposed to provide "deep insight" into the operation of semiconductor devices so I really would like to understand it.

es

First of all, you need to be "less sloppy" on your notation. I spent several minutes trying to figure out what is "SI atom". Please note that atomic symbols are often case-sensitive. So silicon is "Si", not "SI".

Secondly, a "band diagram" is not about the atom itself. The atom has lost its individuality when it forms a solid. So the bands are really the "hybridization" of all the orbitals that have overlapped and form a continuum of states for the solid.

Thirdly, in a crystal structure, the ions that make up the solid form a regular pattern or location in space. So where these ions are located are usually the potential energy well. The red line is trying to show that, but doing it very poorly, because it (i) ignores that there are other repeated potential well and (ii) it seems to show that a charge carrier in that semiconductor as always confined unless it is above the vacuum state.

It is a poorly drawn sketch.

Zz.

P.S. The y-axis is energy, sometime called binding energy.

Gokul43201
Staff Emeritus
Gold Member
es said:
This diagram is supposed to provide "deep insight" into the operation of semiconductor devices so I really would like to understand it.
Looking at that picture, I can not disagree more. I strongly recommend you abandon that diagram and get a fundamental understanding of semiconductor physics from a regular solid-state physics textbook.

Wow, I had no idea things were so badly depicted. Perhaps that's why I was so confused.

I did know that as atoms are added to a crystal, electrons states split due to the Pauli exclusion principle and that in a real solid, due to the large number of atoms, the bands become a continuum. So I previously assumed the diagram was just trying to show the simplified case where the number of atoms in the solid was one. I guess this is an invalid case for a band diagram.

Also, sorry about the notation confusion.

I was looking to understand the physics that underlies the operation of a MOSFET but perhaps it would be better to just study solid-state physics first. Can you recommend a textbook suitable for self study? I do have a good working knowledge of linear algebra, calculus and differential equations.

Thanks for the help.

Kittel's Introduction to solid state physics is a fairly standard introductory text. Your mathematical skills should be enough but you should also know a bit of quantum mechanics.