Electron Density in Bulk Materials

ELECTRON DENSITY IN BULK MATERIALE

The bulk material is a collection of atoms having properties that are from individual atoms.

The minute size of nanomaterials gives them unique electronic properties. One of the major ways in which small‒volume materials differ from bulk materials is the number of available energy states. In a bulk material, the states within each energy sublevel are so close that they blend into a band.

The total number of electron states, N, with energies upto E, can be determined based on quantum mechanics using the following equation

...(1)

Here, we represent the volume as V [V being the characteristic dimension of the solid], m is the mass of an electron and h is the Plancks constant.

The number of energy states per unit volume is given by

 ...(2)

If we differentiate the above equation with respect E, we get a relation for density of states Z(E).

Density of states

Density of states is defined as number of available electron energy states per unit volume, per unit energy

i.e., Z(E)= dn / dE    

       ...(3)

We can write equation (2) as

The most important thing to note about the density of states equation for a bulk material is that it is proportional to the square root of energy.

Z(E) ∝ √E         ...(5)

The specific electrical properties of semiconductors and superconductors will depend on density of states.

The most relevant application of density of states is that it provides information about nanomaterials.

The existance of does not mean that it is occupied all the time. The Fermi function gives us the probability of occupation by a free electron in a given energy state.

i.e., Fermi function F(E) = 1 / [e^(E‒Er)/KBT +1]

...(6)

Multiplying fermi function F(E) with density of states Z(E) will give us the number of free electrons per unit volume per unit energy

...(7)

To be specific, the number of free electrons per unit volume

...(8)

In a conductor at T = OK, [Hint: put F(E)= 1] or electron density in a conductor at OK is given by the number of free electrons

   ...(9)