# Density of States

1. Nov 3, 2009

### fk08

Hello,

is it possible to roughly estimate the density of states just looking at the band structure E(k) of a solid?

Thanks

2. Nov 3, 2009

### ZapperZ

Staff Emeritus
Nope. The density of states require that you sum over all k values at a particular energy as well. E(k) is simply the energy at a particular k value.

Zz.

3. Nov 3, 2009

### kanato

There are some tricks for guessing what the density of states would look like, for instance, if the bands are very wide (small effective mass and large velocity) then the density of states would be small, and if there are flat parts in the bands then there might be a van Hove singularity at that energy. But not every feature of the DOS can be accounted for in this way, so even as a qualitative analysis tool this is not very good. You really need to do the sum over k.

4. Nov 3, 2009

### Gerenuk

Yes, it is possible, of course. At least for 1D. For higher dimensions it depends on how well you are at thinking in 3D or 4D

Each k-value gives two states. So you pick an energy E with a small interval +/- dE around it and check where this little horizontal "energy strip" intersects the bands. The width of the intersections and hence the slope of band at that energy determine the number of states.

Basically the flatter the slope, the higher the contribution to the density of state from that part of the band.

5. Nov 3, 2009

### kanato

Oh, one thing that could be done, if you have an idea of what the lattice looks like is to construct a tight binding model and fit the tight binding parameters to band structure and then use the tight binding model to compute the density of states. Depending on the bands and the lattice this may be rather non-trivial.

6. Nov 3, 2009

### sokrates

It is possible if you have a closed form expression for E(k), including all dimensions involved.

E(k) is your dispersion relation that connects the discrete "states" (k) to their individual "energies" (E).

You first sum the discrete k-states up to an energy in k-space (which is usually done by converting the sum to an integral, because you are summing over a very large number of states) - then you write the sum in terms of E using your E(k) relation, if it is in closed form, and take the derivative. That's it.

Let me give an example:
In 2D, you switch to polar coordinates and draw a circle (corresponding to any energy--say E) and you write the total number of states under this circle as a function of k:

$$N_{total} =~ \frac{\pi |\vec{k}|^2}{(2\pi/L_x) (2\pi/L_y)}$$

where the denominator is the "area" a single state occupies. So total area/ area by a single state gives the total number of states for a given energy. I am assuming periodic boundary conditions here, i.e, the spacing between the states is 2pi/L.

Then you write N_total as a function of energy, if you you have a parabolic band this will be:

$$N_{total} (E) = \frac{m^*E}{\pi \hbar^2}$$

where I assumed:

$$E = \frac{\hbar^2 |\vec{k}|^2}{2m^*}$$

Note that k is the wave"vector" here. Its components are kx and ky.

Now you know the TOTAL number of states up to any given energy E... If you take the derivative with respect to E, you know the density of states for a 2D material, which is independent of energy (for a parabolic band).

$$D(E) = \frac{m^*}{\pi \hbar^2}$$

Last edited: Nov 3, 2009