# Effective mass versus bandgap

1. Sep 28, 2008

### Siggy

Hello-

I feel that this is a "graduate-level" question, so I hope that this question is not found to be insultingly easy.

Can someone help me gain an understanding of why bandgap and electron effective mass in semiconductors are proportional to one another (e.g. trend of C, Si, Ge, alpha-Sn shows decreasing bandgap and decreasing electron effective mass)? Or connect me to appropriate links or books with a good explanation? A combo of rigor and intuition would be best, though at this point, I would settle for the intuitive part.

Regards,
Siggy

2. Sep 28, 2008

### Manchot

If you're talking about a direct-gap material, you can use k.p perturbation theory to solve for the periodic part of the Bloch function (for small k values). This will give you an expression for the E-k relation, which in turn gives you the effective mass.

Roughly speaking, the result is because of Wigner's "no-crossing" theorem. As you vary the strength of a potential continuously, even if it looks like two distinct energy levels will intersect with each other (i.e., become degenerate), they never will. Instead, they "repel" each other and continue on their merry way.

When you're talking about two strongly coupled energy bands, they will also repel each other (with the k value representing the thing you are continuously varying). How quickly they repel each other depends on how close they start together (i.e., the energy gap). So, if two bands are very close to each other at k=0, they will quickly separate for nonzero k values, resulting in a small effective mass. If they are energetically very far apart from each other, they are essentially uncoupled and won't even see each other, and will have an effective mass equal to the free electron mass.

3. Sep 29, 2008

### granpa

http://en.wikipedia.org/wiki/Excitons#Subtypes

Excitons can be treated in two limiting cases, which depend on the properties of the material in question. In semiconductors, the dielectric constant is generally large, and as a result, screening tends to reduce the Coulomb interaction between electrons and holes. The result is a Mott-Wannier exciton, which has a radius much larger than the lattice spacing. As a result, the effect of the lattice potential can be incorporated into the effective masses of the electron and hole, and because of the lower masses and the screened Coulomb interaction, the binding energy is usually much less than a hydrogen atom, typically on the order of 0.1 eV.

4. Sep 29, 2008

### granpa

that article makes it sound as though one can consider either the mass or the charge to change. (the charge being screened by the movement of bound charges in the dielectric).

I dont really understand how changing the mass is equivalent to changing the charge though.

Last edited: Sep 30, 2008
5. Sep 30, 2008

### Minich

Dear Siggy.
This question is very simple, but i had never seen it cleared enough.
Especially when it is mixed with Fermi energy and definitions of "zero" energy and "zero" momentum in the p-E plane of electron-quasiparticle.

I try to explain this topic.

1. For free electron. There are orbitals with momentum p and energy E. For each E we have two orbitals: p and -p.
$$E=p^2/(2m)$$

2. In crystal lattice there is no free electron. Electron is believed to become a quasiparticle, ie it occupies the orbital with exact energy E(!) and averaged momentum of operator p=<p>.

E exact because orbital is exact solution of Shroedinger equation in crystal potential.
p could not be exact eigenvalue of operator p because operator p does not commute with crystal potential.

Orbital is the linear combination of eigenfunctions of operator p.

3. In crystal we can childly assume that when electron move along lattice it is partly reflected back (bragg reflection) and then again re-reflected and so on. The reflection is especially big if
a) initial pi and reflected pr are connected:
|pi-pr|=const=invers lattice vector (conservation of momentum)
b) Ei and Er are very close.

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