Nearly free electron model: band gap

[Telemachus]

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.

 

[mark726]

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 ##