The tunneling potential for field emission from nanotips

In the quasi-planar approximation of field emission, the potential energy due to an external electrostatic field $E_0$ is expressed as $-e \gamma E_0 \Delta s$ where $\Delta s$ is the perpendicular distance from the emission site and $\gamma$ is the local field enhancement factor on the surface of the emitter. We show that for curved emitter tips, the current density can be accurately computed if terms involving $(\Delta s/R_2)^2$ and $(\Delta s/R_2)^3$ are incorporated in the potential where $R_2$ is the second (smaller) principle radius of curvature. The result is established analytically for the hemiellipsoid and hyperboloid emitters and it is found that for sharply curved emitters, the expansion coefficients are equal and coincide with that of a sphere. The expansion seems to be applicable to generic emitters as demonstrated numerically for an emitter with a conical base and quadratic tip. The correction terms in the potential are adequate for $R_a \gtrapprox 2$ nm for local field strengths of $5$ V/nm or higher. The result can also be used for nano-tipped emitter arrays or even a randomly placed bunch of sharp emitters.