The nearly free electron model and states in a band?

[jeebs]

hi,
I have this problem where I am supposed to show, using periodic boundary conditions, that there are 2N states in an energy band in the "nearly free electron" model of solids, where N is the number of atoms.

I have been looking through my course notes and my textbook, and my textbook goes through the "free electron" model first. This is where the electrons are assumed to be moving in zero potential, ie. the lattice ions are completely ignored. It shows that the energy of an electron is given by
$$\epsilon = \frac{\hbar^2k^2}{2m}$$

and shows a plot of $$\ \epsilon$$ vs. wavenumber k, which as you hopefully know, looks like a sort of steep bowl when +ve and -ve k values are plotted.

Then I go on to the next chapter, about the "nearly free electrons", where the potential is no longer a uniform zero throughout the solid, but has a periodic negative peak corresponding to the positions of the lattice ions (which are separated by a distance a). The book goes on to show a plot of $$\epsilon$$ vs. k, but this time it results in the graph of the "Brillouin zones", which show the regions of energy bands and the band gaps that occur at every value of k = $$n\pi$$/a. This plot is superimposed on top of the curve for the free electron model, and the two curves coincide until k gets close to the Brillouin zone boundaries, where the nearly free electron curve tapers off.
This is what I assume my question is talking about with "periodic boundary conditions".

However, neither my book nor my notes are clear about where this plot comes from. Can anyone show me?
All I can guess is that the equation for $$\epsilon$$ as a function of k must have some sine or cosine part in it. I'm hoping that once I know where this plot comes from, it will help me understand why there are 2N states per band.

Basically my problem is that my book says there are N allowed values of k in the range $$-\frac{\pi}{a} < k < \frac{\pi}{a}$$ and and hence 2N electrons per band (allowing for 2 different electron spins), but I have no idea why.

Can anyone enlighten me?

Thanks.

 

[Jhero]

Thank you for your question. The plot you are referring to is the result of using the Bloch theorem, which states that the wavefunction of an electron in a periodic potential can be written as a product of a plane wave and a periodic function. This periodic function is known as the Bloch function and is given by:

\psi_{k}(x) = e^{ikx}u_{k}(x)

where \psi_{k}(x) is the wavefunction, k is the wavevector, x is the position, and u_{k}(x) is the periodic function. This function satisfies the periodic boundary conditions, meaning that it is periodic with the same period as the lattice potential.

Using this Bloch function, we can derive the energy dispersion relation for the nearly free electron model. This is done by solving the Schrödinger equation for a periodic potential, which results in the following equation:

\epsilon_{k} = \frac{\hbar^2k^2}{2m} + V_{0}\cos(ka)

where \epsilon_{k} is the energy of the electron, k is the wavevector, m is the electron mass, and V_{0} is the potential strength. This equation is similar to the one you mentioned in the free electron model, but it now includes the periodic potential term.

To understand why there are 2N states per band, we need to consider the Brillouin zone boundaries. As you mentioned, there are N allowed values of k in the range -\frac{\pi}{a} < k < \frac{\pi}{a}, which correspond to the first Brillouin zone. Since each of these values can have two different electron spins (spin up and spin down), there are 2N states in this first Brillouin zone. This pattern continues for all Brillouin zones, resulting in a total of 2N states per energy band.

I hope this helps to clarify the origin of the plot and the reason for 2N states per band. Let me know if you have any further questions.