Properties of the two-dimensional electron gas consider a two-dimensional electron gas (2DEG) with twofold spin degeneracy but no valley degeneracy. 

(a) Show that the number of orbital’s per unit energy is given by: D(ε)= m/πh²

(b) Show that the sheet density is related to the Fermi wave vector by: n_s = k²_F/2π. 

(c) Show that, in the Drude model, the sheet resistance, i.e., the resistance of a square segment of the ZDEG can be written as: R_s = (h/e²) (kFℓ) where ℓ = v_μτ is the mean free path.