Gradient corrections to the kinetic energy density functional of a two-dimensional Fermi gas at finite temperature

We examine the leading order semiclassical gradient corrections to the non-interacting kinetic energy density functional of a two dimensional Fermi gas by applying the extended Thomas-Fermi theory at finite temperature. We find a non-zero von Weizs"acker-like gradient correction, which in the high-temperature limit, goes over to the familiar functional form $({\mathrm{\hslash }}^2\mathrm{/}24m)(\mathrm{\nabla }\rho {)}^2\mathrm{/}\rho$. Our work provides a theoretical justification for the inclusion of gradient corrections in applications of density-functional theory to inhomogeneous two-dimensional Fermi systems at any {\em finite} temperature.