Space charge effects in field emission: Three dimensional theory

Field emitters rely on high gradients on microfabricated structures to enable substantial levels of emitted current, which can in turn act to reduce the field at the emission site of a single emitter. An account of that effect is obtained in the following steps: a model of the emitter geometry based on the point charge model that allows for the determination of the apex radius and field enhancement factor for arbitrarily sharp emitter structures is given, followed by an analytical formula for the calculation of total current from such a structure and then by a model of the effect of emitted current in suppressing the field at the emission site. Predictions of the impact of space charge on the emitted current are made and compared to findings of Barbour [Phys. Rev. 92, 45 (1953)] for sharpened structures with varying work function. A discussion is given of the method to combine single tip three-dimensional results with a study of space charge on field emission in one dimension.