  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,
$J_{\textrm{av}}$, averaged over work function distribution shows Gaussian
dependence, $J_{\textrm{av}}\propto \exp(\alpha\sigma^{2})$, where $\sigma$ is
the width of the work function distribution and $\alpha$ is a fitting
parameter. For log-normal distribution, $J_{\textrm{av}}$ shows compressed
exponential behaviour, $J_{\textrm{av}}\propto \exp(\gamma\sigma^{n})$, 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, $E_{F} \gg \Phi$, $\Phi$
being the work function of the system as well as systems with low density
characterised by $E_{F} \ll \Phi$.
