Now, how do we calculate density of states then?

Well, we need to know something about the energy band.

And here, we adopt something called a parabolic energy band approximation.

Now what is a parabolic energy band?

Energy band that is a parabola in the e

versus k energy band relationship.

So typical parabolic energy band is written like this,

E energy is proportional to k squared, and

the curvature is inversely proportional to the effective mass.

Now this is an exact expression for free space electron.

And it is also a good approximation in semiconductor near the band edge,

that is, at the bottom of the conduction band.

At the top of the valence band,

the energy band very much should look like a parabola.

And we can use this approximation by appropriately choosing

the value of the effective mass for each band.

Now, once you adopt this parabolic energy band structure,

then you can define a sphere with a radius given by this.

What is this?

This is the k value corresponding to energy E.

So for a given energy E here, if you plot

all the k values that corresponds to the same energy E,

this equation will tell you that that will form a sphere.

And the sphere radius is given by this.

And this is pictorially shown here, okay?

So there is a sphere, section one-eighth of a sphere in the first octet.

In this sphere, the surface of this sphere defines

all the k values corresponding to energy E.

Now, then we can look at the volume occupied by this sphere,

one eighth of a sphere.

And that sphere represents the k-space volume

corresponding to the energy range from zero to E.

And you can easily calculate the volume.

The volume of a sphere is 4pi over 3 times r cubed.

And we're taking only the positive values of k here, so we take one-eighth of it.

This is the volume in k-space corresponding

to energy range from zero to E.