(a) If Ef equals Ec, calculate the Fermi probability of a state being occupied at E=E+kT. (b) If Ef equals Ev, determine the probability of a state being empty at E=E₁-kT. (c) Find the probability that an energy level is occupied by an electron when the state is positioned above the Fermi level by 5kT. (d) Determine the probability that an energy level is occupied by an electron when the state is positioned above the Fermi level by 10kT. Utilize Fermi-Dirac statistics with a temperature of T=300K.

The 2DEG in (iii) is patterned to produce a clean, quasi-1D channel. The current I through the channel is = Nev, where N = the number of electrons, e the electronic charge and = the electrons' group velocity. The number of electrons N(ɛ) = f(ɛ, µ)g(ɛ), where f (ɛ, u) =Fermi-Dirac distribution = 1 and g(ɛ) density of states = dn/dɛ. 1+exp() kBT (a). Write down the dispersion relation for free electrons of mass m. What is their group velocity v? (b). Find an expression for g(ɛ) involving the group velocity. Leave your answer in terms of v.

The Fermi energy for copper is EF = 7.00 eV. For copper at T = 1020 K, (a) find the energy of the energy level whose probability of being occupied by an electron is 0.911. For this energy, evaluate (b) the density of states N(E) and (c) the density of occupied states No(E).

A silicon p-n junction (ni = 1010 cm3, Na = 1017 cm-3 and Nd = 4 x 1016 cm 3) is biased with an applied voltage Va = -5 V. Calculate the built-in potential, the depletion layer width and the depletion capacitance. Take the temperature as 27°C.