Two-Dimensional Fermi Gas

Consider a noninteracting nonrelativistic gas of N spin-1/2 fermions at T = 0 in a box of area A.

a) Find the Fermi energy.

b) Show that the total energy is given by

c) Qualitatively discuss the behavior of the heat capacity of this system at low temperatures.

SOLUTION

a) At T = 0 the noninteracting fermions will be distributed among the available states so that the total energy is a minimum. The number of quantum states available to a fermion confined to a box of area A with momentum between p and p + dp is given by

(1)

where the multiplicity g = 2s + 1 and the spin s = 1/2. The N fermions at T = 0 fill all the states of momentum from 0 to p_F. We can therefore calculate this maximum momentum p_F from

(2)

The Fermi energy ε_F for this nonrelativistic gas is simply

(3)

Using (2) and (3) we obtain

(4)

(5)

For s = 1/2 and g = 2, (5) becomes

(6)

b) The total energy of the gas

(7)

Substituting p_F from (2) into (7) gives

(8)