Surface-electron and bulk-phonon heat-capacity crossover ($C_{\mathrm{surf}}/C_{\mathrm{bulk}}\propto A/(NT)$)

A neutral two-dimensional linear electronic spectrum has heat capacity proportional to $A{T}^2$, while three-dimensional acoustic phonons have Debye heat capacity proportional to $N{T}^3/{\Theta }_D^3$. Thus their ratio scales as $A{k}_B^2{\Theta }_D^3/({\hslash }^2{v}^{2}NT)$. At fixed crystal composition and shape, $A/N\propto 1/L$ for linear size $L$, favoring the electronic surface contribution in smaller or colder specimens. The continuum laws require thermally populated wavelengths small compared with sample size; their formal low-temperature crossover need not lie inside that regime.