Why is the local work function so much lesser than Work Function? I have been trying to understand the difference between the Work function of a metal and the Local Work Function. 
I did some experiments to find the Local Work function of Graphite using an STM and found values ~0.3-0.7 eV. However, the work function of Graphite is ~4eV. Literature values indicate that my experimental results are correct. 
Some research leads me to these definitions on a website:
Work Function:
The work function corresponds to the minimum energy necessary to
extract an electron from the metal
Local Work function:
The local work function, which is defined as the energy required to take an electron from the Fermi level to a specified position outside the metal.
I frankly can't see a difference between these two definitions. Admittedly, my concepts in Solid State Physics are shaky. What is the difference between the two work functions? Can someone clarify, without assuming I know too much about Fermi levels.
 A: Well, the local work function depends heavily on the surface topography, e.g. dense step concentration causes dramatic reduction of the local work function. a good read can be found here.
Look at it this way: Just as the energy levels in the bulk are determined by the lattice's potential energy (which is determined by the structure and dopants in the bulk), the energy on the surface is determined by the surface's potential - which is also determined by the surface structures and possible contaminants. 
Electrically non-neutral local spots also bend the energy lines in phase-space (called band bending).
The same material can have very different local work function values depending on it's surface structure.
A: Maybe you have had water layer on your sample? Presence of water (or other liquid*) is known to dramatically lower the workfunction. For liquid layer the potential barrier height is kind of oscillatory with the layer thickness (i.e. tip-sample distance in this case, as for STM water would form a water neck between the two) which should show characteristic waviness of the exponential I(z) curve. If your exponential curve goes up and down on the rise, then it is strong indication of having a water (liquid) layer.
I obtained workfunction of ca. 0.25 eV on HOPG while measuring in air instead of UHV due to water on my sample.
The surface roughness, as mentioned above, also lowers workfunction, but I guess you performed a broad scan of the surface going into I(z) spectroscopy, to check for flatness?
*Or only polar liquids? Would have to check for that.
