(a) In the T = 0 approximation, the total number of particles that corresponds to a Fermi energy E_F is given by

Calulate the area density of the particles needed to have a Fermi energy of E_F = 100 meV for a two-dimensional isotropic electron gas with effective mass m = m₀, where m₀ is the free electron mass.

(b) Suppose that we produce the two-dimensional electron gas of part (a) in a quantum well with thickness 10 nm and infinitely high barriers. Then the total volume of a quantum well with area 1 mm² is 10⁻⁵ mm³. Compare the total number of electrons needed in this system to get a Fermi energy of 100 meV, and the total number of electrons needed for a three-dimensional isotropic electron gas with the same total volume.

EF = D(E)dE. - Lote JO N= (2.8.8)