dispersion relation « sidk.info| Sidharth Kshatriya

The purpose of this essay is the revise, clarify and elucidate concepts in Dispersion and Band Structure.

A dispersion relation is a relation between Energy and Momentum. For a free particle $E = \frac{p^2}{2m}$ . The dispersion relation is generally between $E$ and $k$ . (Note that $p=\hbar k$ . This always holds. It comes from De Broglie’s equation $\lambda=\frac{h}{p} \Rightarrow \frac{2 \pi}{\lambda} = k = \frac{p}{\hbar}$ . For a crystal, using the nearly free electron model, the relation is a (modified) parabola (just the free particle) with some “breaks” in it. For diagram see here (look at the Extended Zone Scheme part of the diagram). The breaks occur at $k=\frac{\pi}{a},\frac{2\pi}{a}$ etc. Why? For $k=\frac{\pi}{a} \Rightarrow a = \frac{\lambda}{2}$ . This is a standing wave situation where the group velocity of the electron wave becomes zero (left traveling reflected wave superimposes on right traveling incoming wave. The atoms represent the “walls” for the reflection). So its not possible to have $k=\frac{\pi}{a}$ and a break occurs at that point. Now the $a = \lambda$ is a similar situation, so the $k=\frac{\pi}{a}$ point also has a break in the parabola and so on. Note that the reduced zone scheme is just a compact way of representing the band structures. Depending on what curve you are, you can read off the energy but to get the right $k$ you need to add right number of $2\pi/a$ terms.