Problem $3$:

You are given a semiconducting crystal with the following energy diagram.

Calculate the intrinsic number of charge carriers at room temperature.

You have two different dopant options to add to the material to make an n$-$type semiconductor. Dopant $1$ has an energy of $0\mathrm{.}1$ eV below the conduction band and a solubility limit in the semiconductor of $0\mathrm{.}001\%.$ Dopant $2$ has an energy of $0\mathrm{.}2$ eV below the conduction band and a solubility limit in the semiconductor of $0\mathrm{.}01\%.$

a$)$ Calculate the extrinsic carrier concentration for these dopant options at room temperature. Which dopant would be the better choice for achieving a higher conductivity?

b$)$ Is there a temperature at which you would change your choice of dopant?