Thermoelectronic emission from monolayer graphene with temperature dependent work functions

For the first time we have derived an equation for the temperature (T) dependent work function (W(T)) that will be important for modeling thermoelectronic current density (J) and energy distribution of emitted electrons specially, from nano-materials. The equation containing terms up to fifth power of T gives a modified Richardson-Dushman (MRDE) equation that fits excellently well the experimental data of J vs T for suspended graphene. It provides a unique technique for accurate determination of W$_{\mathrm{0}}$, Fermi energy, E$_{\mathrm{F0}}$ at 0 K and surface density of charge carriers, n$_{\mathrm{s}}$ of graphene. The corresponding values obtained for suspended graphene are: W$_{\mathrm{0}} \quad =$ 4.42 \underline {}$+$ 0.01 eV, E$_{\mathrm{F0}} \quad =$ 0.166 \underline {}$+$ 0.002 eV; n$_{\mathrm{s}} \quad =$ 2.34x10$^{\mathrm{12}}$ cm$^{\mathrm{-2}}$. The model gives --ve thermal expansion coefficient of graphene (-8x10$^{\mathrm{-6}}$ /K) which has been experimentally confirmed. The equations are expected to hold for carbon nanotubes.