# Direct and Indrect band semiconductor

There's a fundamental concept about direct and indirect band gap semiconductor that I am not clear about Given below is the image of the direct and indirect band gaps from Kittel

The y axis represents energy. But I am a bit confused about the x axis. What that K is actually? All that I can make out is that its some kind of wave vector.

Below is an image from wikipedia

It shows the x axis as momentum. The accompanying description states this as crystal momentum. On looking up the related pg again from wikipedia

I would like to know what exactly this value on the x axis represents? Why is it taken on the y axis? And what does the variation of energy with k actually signifies?

• For a free particle, $E = mv^2/2 = p^2/(2m)$. Now if you take the de-Broglie wavelength and from that the wavenumber $k$ you get $E = \hbar^2k^2/2$ as the dispersion relation for the free particle. This gets modified in a crystal as explained in the answer below. – Pieter Dec 31 '17 at 12:02

## 1 Answer

In these bandstructure diagrams, the abscissas are the wavevectors $k$ and the ordinates $E$ are the energies of the electrons in the crystal which are represented by Bloch wave functions $$\psi(x) = u(x)\exp {(ikx)}$$ where $u(x$) has the lattice periodicity and the wave vector $k$ has arbitrary values within the Brillouin zone. The so-called crystal momentum $p=\hbar k$ plays a role in the conservation of momentum in cases like photon absorption and emission. The two band diagrams illustrate this for the case of an indirect semiconductor where the CB minimum is not at the same k vector as the VB maximum and for a direct semiconductor where the CB minimum is at the same k as the VB maximum. Because photon momentum $p=\hbar \omega/c$ is very small, the change in crystal momentum for a transition of an electron from the VB to the CB due to photon absorption/emission is very small and the transition in the band diagram is practically vertical. In the direct semiconductor a direct transition of an electron from the VB maximum to the CB minimum by photon absorption is possible from the VB maximum to the CB minimum. As shown in the graph for the indirect semiconductor, such a transition is only possible with aid of an additional phonon of large crystal momentum which provides the change in crystal momentum from VB maximum to CB minimum. Thus the photon absorption at the band gap energy is less likely (weaker) in the indirect semiconductor than in the direct semiconductor.