Role of work function distribution on field emission effects

Field emission effect is the emission of electrons from a cold metallic surface in the presence of an electric field. The emission current exponentially depends on the work function of the metallic surface. In this work we consider the role of work function distribution on the field emission current. The work function distribution can arise due to nano-scale inhomogeneities of the surface as well as for collection of nano-particles with size distribution. We consider both Gaussian distribution as well as log-normal distribution. For Gaussian distribution, the field emission current, $J_{\text{av}}$, averaged over work function distribution shows Gaussian dependence, $J_{\text{av}}\propto \exp (\alpha {\sigma }^2)$, where $\sigma$ is the width of the work function distribution and $\alpha$ is a fitting parameter. For log-normal distribution, $J_{\text{av}}$ shows compressed exponential behaviour, $J_{\text{av}}\propto \exp (\gamma {\sigma }^n)$, where the exponent $n>1$ is a non-universal parameter. We also study in detail field emission current for various electric field strength applied to systems with high density, characterised by Fermi energy, $E_F\gg \mathrm{\Phi }$, $\mathrm{\Phi }$ being the work function of the system as well as systems with low density characterised by $E_F\ll \mathrm{\Phi }$.