To Calculate the self consistent equation for Mean field in a Doped Semiconductor

1. The problem statement, all variables and given/known data Magnesium atom is introduced into a Copper crystal where electrons are free to move in a fermi sphere until equilibrium is reached. Assuming the mean field (slowly varying) description applies with potential energy eψ(r) the energy given by ε(r,p) where the dist of electrons is given by fermi function f(r,p) for change in eletronic density due to impurity we have δn®. Mean field is related to its source (charge distribution) through poisson equation -Δψ(r) . Solve for the self consistent equation by Fourier Transform

2. Relevant equations

ε(r,p)=ε(p)+eψ(r)

f(r,p)=1/( e^(ε(p)+eψ(r)-εf/kt) +1 )

δn®=2∫dp/(2∏h)^3 (f(r,p)-f(p)

-Δψ(r)=e/ε (δn®-δ(r) )