Q:In a three dimensional semiconductor crystal with a perfect lattice. Assume the distance between atoms is ‘b’ center meter. The reciprocal lattice is what we call k-space. It’s Fourier transform of original lattice. We solve the Schrödinger equation of electron wave in original lattice for one unit cube with side length b and obtain the electron wave mode. We can represent all solutions with the k-space with each node of the k-space as one solution i.e. one mode. (note: all these solutions are solved for one single unit in the lattice). This problem is asking you to derive the mode density in this crystal. We will do this step-by-step as the following:

a). Assume the electrical potential distribution is infinite between atoms. This results in a periodic solution with periodic unit (π/b, π/b, π/b) in kx, ky and kz directions of k-space. Write done the volume of the unit cell in the k-space. (5 points)

b). Imagine in k-space, there is a spherical shell, the radius of the inner sphere is k, the thickness of the shell is the differential dk, write down the volume of the spherical shell. (5 points)

c). How many cells in the spherical shell in total? Due to k can only be positive, how many cells in the first octave? Each k point can correspond to two modes for electron spin up and spin down. So how many modes in the first octave in k-space. (5 points)

d). if we related the results to the density of modes, we can equal it to ρ(k)*dk, where the ρ(k) is the density of the modes. Please find the ρ(k) based on c) result. (5 points)

e). Based on d) result, please normalized it to unit volume mode density. (5 points)

f). Based on e), convert unit volume normalized ρ(k) to ρ(ν) by using ρ(k)dk=ρ(ν)dν relationship. It’s more convenient to use frequency ν instead of k in the real practice. (5 points)