Obtain an asymptotic form, in the limit T << TT , for the specific heat of a gas of N free fermions adsorbed on a surface of area A, at a given temperature T

For the two-dimensional gas of free fermions, we write

and

with the Fermi-Dirac distribution,

Discarding exponentially small corrections (in the limit βε_F →∞), we have

Note that exp (x) / [exp (x) + 1]² is an even function of x, and that

Also, we have

From these results, we write a low-temperature expression for the internal energy (except for vanishingly exponential correc- tions),

where

which leads to a linear dependence of the specific heat with temperature as in the three-dimensional case.