Spin normalization in two-dimensional free-electron state counting ($g(ϵ)=g_{s}Am/(2\pi {\hslash }^2)$)

Each allowed wavevector occupies area $(2\pi {)}^2/A$ under periodic boundary conditions, for each independent spin state. For parabolic dispersion, the number below energy $ϵ$ is $g_{s}Amϵ/(2\pi {\hslash }^2)$. Thus the often-quoted coefficient $Am/(2\pi {\hslash }^2)$ is per spin. Unpolarized electrons have $g_s=2$; the same convention must be used in particle counting and the Fermi energy. A linear Dirac dispersion has a different energy dependence.