Band structure and Density of states (DOS)

Question

Can someone explain how these two plots are related?

How are the peaks in the right are associated with the left figure?

Answer 1

The dos graph is on right. the band structure is on the left.

The DOS (right) is the density of the lines of the band structure for a specific energy. So there's no line that pass through -1 so there's no DOS there. At -0.5, there a almost flat line. On the dos, you can see a clear spike at that value.

That's how both graph are related.

Now let's further the explanation step by step.

1. There is 1 kpt for every 2*4π^3/V in the k-space. This is similar to a free electron enclosed in a volume (The most basic solution of Schrodinger equation).

2. For every energy, you can draw a fermi surface in k space. It is an isosurface of energy. For a free electron, the Fermi surface is a sphere. Every point on the surface of this sphere has an energy ε.

3. Take 2 energy : ε and ε + dε. You can find the volume between these 2 sphere. By that mean, you can also find the number of kpts between these 2 spheres. If dε is infinitesimal, you have the DOS.

I need to revise what the electronic band structure (left) exactly means. For now, I can say it is where the k point are located in the DOS. In other words, it is in which direction of k-space the surface ε+dε has expanded a lot compared to the surface ε.

Edit: You can only think of the electronic band structure with a sphere as a trivial case since a sphere can not grow more in a direction, it must grow equally in every direction. You will need to think of a Fermi Surface instead.

Answer 2

If you have dispersions of the form $\epsilon_\mu(k)$ where $\mu$ is the band index and $k$ is momentum, the DOS is given by:

$$ \rho(\omega) = \sum_\mu \int \frac{dk}{(2\pi)^d} \delta ( \omega- \epsilon_\mu(k)) $$

Where $d$ is the spatial dimension. It should be pretty self-explanatory. For every value of $\omega$ you "sum" over all possible states that contribute to that value. In a translational invariant systems you count states by counting momenta (with a certain factor of $2\pi$).

Answer 3

as you can from the plot many different k values correspond to the same energy near about -0.75eV for example. While in the plot the number of k values appears infinite, in reality they are actually spaced by 2Pi/a, where a is the lattice constant. This means there is a discrete number of k values at a specific energy. ie (DOS/eV)*(energy of the state) = Number of electrons at that energy

Answer 4

A density functional theory (DFT) method was used to obtain the energy of bandgaps. These energies are determined by calculating the microscopic average potential in each solid sample. The average potential may change with the pure and doped sample and that may affect the bandgap and DOS of the solids. e. The valance bands correspond to the values below zero energy and conduction bands are the one above zero.

Answer 5

Gamma, X, S... are some k-points (wavenumbers), connected among them, defining a path of highly symmetric points in the first Brillouin of a crystal.