Nearly free electron model: band gap

E-U_0 \right ]c_{\frac{\pi}{a}}+ U_{-\frac{2\pi}{a}}c_{-\frac{\pi}{a}}##Solving for the energies at these points, we can determine the band gap between the second and third bands. I hope this helps you in your calculation. If you are still having trouble, please don't hesitate to ask for help. Good luck!
  • #1
Telemachus
835
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I was trying to determine the bandgap in the nearly free electron model. I'm having trouble to determine the band gap bewteen the second and the third band. Its a one dimensional problem.

So, the central equation reads:

##\displaystyle \left [ \frac{\hbar}{2m} (k-G)^2-E \right ]c_{k-G}+ \sum_{K'} U_{G-K'} c_{k-K'}##

So, I want to determine the band gap which would be at the point ##k=2{\pi}{a}## in the extended diagram (the band gap between the second and the third band).

And my ##K'## I think that should be: ##K'=\frac{2\pi}{a},\frac{4\pi}{a}##. When I carry on the whole calculation I get something that don't make any sense to me, because I don't find any degeneracy on the energy, and I think that's wrong.

For ##G=\frac{2\pi}{a}## I get:

##\displaystyle \left [ \frac{\hbar}{2m} (k-\frac{2\pi}{a})^2-E-U_0 \right ]c_{k-\frac{2\pi}{a}}+ U_{\frac{2\pi}{a}}c_{k-\frac{4\pi}{a}}##

And for ##G=\frac{4\pi}{a}##:

##\displaystyle \left [ \frac{\hbar}{2m} (k-\frac{4\pi}{a})^2-E-U_0 \right ]c_{k-\frac{4\pi}{a}}+ U_{-\frac{2\pi}{a}}c_{k-\frac{2\pi}{a}}##

I don't know what I'm doing wrong.

Thanks in advance.
 
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  • #2


Hello,

Thank you for sharing your problem in determining the band gap in the nearly free electron model. The band gap is an important concept in solid state physics and is related to the energy difference between the highest occupied energy band and the lowest unoccupied energy band.

In the nearly free electron model, the central equation you have written is correct and represents the Schrödinger equation for the electron in a periodic potential. However, in order to determine the band gap between the second and third bands, you need to consider the Brillouin zone of the one-dimensional problem, which in this case is a line segment from ##k=0## to ##k=\frac{2\pi}{a}##.

In this Brillouin zone, the allowed values of ##k## are given by ##k_n = n\frac{\pi}{a}##, where ##n## is an integer. So for the band gap between the second and third bands, we need to consider the points ##k_2=\frac{2\pi}{a}## and ##k_3=\frac{3\pi}{a}##.

Now, for ##G=\frac{2\pi}{a}##, the central equation becomes:

##\displaystyle \left [ \frac{\hbar}{2m} (k-\frac{2\pi}{a})^2-E-U_0 \right ]c_{k-\frac{2\pi}{a}}+ U_{\frac{2\pi}{a}}c_{k-\frac{4\pi}{a}}##

Substituting ##k=k_2=\frac{2\pi}{a}##, we get:

##\displaystyle \left [ \frac{\hbar}{2m} (0)^2-E-U_0 \right ]c_{0}+ U_{\frac{2\pi}{a}}c_{-\frac{2\pi}{a}}##

Similarly, for ##G=\frac{4\pi}{a}##, the central equation becomes:

##\displaystyle \left [ \frac{\hbar}{2m} (k-\frac{4\pi}{a})^2-E-U_0 \right ]c_{k-\frac{4\pi}{a}}+ U_{-\frac{2\pi}{a}}c_{k-\frac{2\pi}{a}}##

Substituting ##
 

Related to Nearly free electron model: band gap

What is the Nearly Free Electron Model for the band gap?

The Nearly Free Electron Model is a simplified theoretical model used to describe the behavior of electrons in solids. It assumes that the electrons in a solid experience a periodic potential due to the arrangement of atoms, but also interact weakly with each other and the lattice. This model helps to explain the existence and behavior of band gaps in solids.

What is a band gap and how does it relate to the Nearly Free Electron Model?

A band gap is an energy range in a solid material where no electron states exist. In the Nearly Free Electron Model, band gaps arise due to the periodic potential experienced by electrons, which causes some energy levels to be forbidden. This model can predict the size and location of band gaps in materials.

What factors affect the size of the band gap according to the Nearly Free Electron Model?

The size of the band gap in a material depends on several factors, including the strength of the periodic potential, the distance between atoms in the lattice, and the strength of electron-electron interactions. These factors can be adjusted in the Nearly Free Electron Model to study their effects on the band gap.

How does the Nearly Free Electron Model explain the conductivity of materials?

The Nearly Free Electron Model can be used to understand the conductivity of materials by considering the energy levels and band structure of the material. If the band gap is small or non-existent, electrons can easily move between states and the material will conduct electricity. However, if the band gap is large, electrons will not have enough energy to move and the material will be an insulator.

What are the limitations of the Nearly Free Electron Model?

The Nearly Free Electron Model is a simplified theoretical model and does not fully capture the complexities of real materials. It assumes a perfectly periodic potential and does not account for effects such as impurities or defects in the material. It also does not take into account the quantum mechanical nature of electrons and their interactions with the lattice. Therefore, while the model can provide valuable insights, it may not always accurately predict the behavior of real materials.

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