Question

Please show all steps and explain your reasoning in detail for parts e, f, & g. Ignore a - d. Thank you.

Consider a "2D electron gas". Ne electrons of mass m * confined to a square ofarea A = L2 with zero potential. Take the lowest level to be energy zero. a. What is the total number of states N with energy less than E including spin degeneracy? b. Deduce the energy density of states, defined by D(E) c. Find the Fermi energy EF (here, equal to the chemical potential at temperature T = 0. d. Find the total energy Etot at T = 0. l. 1 dN A dE Find the 2D spreading pressure, P =-dEtot/dA. Find the 2D bulk modulus, B =-Ad Write down the expression relating Ne to D(E) and the Fermi-Dirac occupancy n (E) at finite T. Can dP it be solved to determine μ?

Answer 1

E_tot = sigma_k epsilon_k n_k = sigma_k epsilon 1/e^B(epsilon - mu)_+1 = (2s + 1) A/(2 pi)^2 integral epsilon k dk d theta/e_beta(-mu()+1 where epsilon = h^2 k^2/2m At T = 1, epsilon < mu (otherwise 0) => E_tot = (2s + 1) A/(2 pi)^2 integral pi^2 k^2/2n k d theta dk/ = (2s + 1) A/(2 pi)^2 h^2/2m integral^2 pi_0 d theta integral KF_0 k^3 dk = 2 A/(2 pi)^4 h^2/2m times 2 pi times K^3_F/3 E_tot = 2A h^2/4 pi m k^3_F/3 = 4 h^2 k^3_F/6 pi m we have, N = sigma n_k = (2s + 1) A/(2 pi)^2 integral k dt d theta = (2s + 1) A/(2 pi)^2 times 2 pi k^2_F/2 s = 1/2 N = A/2 pi K^2_F => K_F = (2 pi M/A)^1/2 => E_tot = A h^2/6 pi m (2 pi N/A)^3/2 = (2 pi N)^3/2 h^2/6 pi m A^-1/2 Pressure, p = -d E_tot/dA = -(2 pi N)^3/2 h^2/6 pi m times