Estimating Density of States from Band Structure E(k)

In summary, to estimate the density of states of a material, you need to sum over all k values at a particular energy. Zz.
  • #1
fk08
31
0
Hello,

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

Thanks
 
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  • #2
fk08 said:
Hello,

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

Thanks

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
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
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 :smile:

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
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
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:

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

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:

[tex] N_{total} (E) = \frac{m^*E}{\pi \hbar^2}[/tex]

where I assumed:

[tex] E = \frac{\hbar^2 |\vec{k}|^2}{2m^*} [/tex]

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).

[tex] D(E) = \frac{m^*}{\pi \hbar^2}[/tex]
 
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1. What is the density of states?

The density of states is a concept used in solid state physics to describe the number of available energy states per unit volume in a material. It is a fundamental property that helps us understand the electronic and thermal properties of materials.

2. How is the density of states related to the band structure?

The density of states is directly related to the band structure of a material. In a solid, the energy levels available to electrons form bands, and the density of states describes the number of states within a given energy range. The shape and width of the bands in the band structure determine the density of states at different energies.

3. What is the relationship between the density of states and the Fermi energy?

The Fermi energy is the highest energy level occupied by electrons at absolute zero temperature. The density of states at the Fermi energy is related to the number of electrons in the system, as it represents the total number of states available for electrons to occupy. Therefore, the density of states at the Fermi energy is a crucial parameter in understanding the electronic properties of a material.

4. How is the density of states calculated from the band structure?

The density of states can be calculated by integrating the density of states function over all energy levels. This function is derived from the band structure by taking into account the energy levels and degeneracies of the bands. There are also various methods for approximating the density of states, such as the Gaussian broadening method and the tetrahedron method.

5. Why is estimating the density of states important?

The density of states is a crucial parameter in understanding the electronic and thermal properties of a material. It provides valuable information about the number of electrons, their energy distribution, and the available energy levels for them to occupy. Estimating the density of states is essential for predicting the behavior of materials in different conditions and for designing new materials with specific properties.

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