Show that the Fermi energy or Fermi potential is spatially constant in the n-type Si slice with linearly varying donor dopant impurity concentration as given in given problem.

Problem:-

An n-type SI slice of a thickness L Is inhomogeneously doped with phosphorus donor whose concentration profile is given by N_DD(x) N₀ + (N_L-N₀)(x/L).

What is the formulae for the electric potential difference between the front and the tack surfaces when the sample is at thermal and electric equilibria regardless of how the mobility and diffusivity varies with position?

What is the formulae for the equilibrium electric field at a plane x from the front surface for a constant diffusivity and mobility? How is the formulae modified if diffusivity and mobility vary with impurity ion concentration?