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The density of emission current by a rough surface with fractal structure is analyzed. A possible experimental setup is modeled by an irregular emitting surface, iteratively generated by a fractional Brownian motion (FBM) algorithm with roughness (or Hurst) exponent $H=0.3$ and a far away plane representing the anode. The boundaries are held at a fixed potential difference and the Laplace's equation, with lateral periodic conditions, was numerically solved. The solution for the potential leads to the evaluation of the local field and, subsequently, the current density evaluated by using the Fowler-Nordheim's approach. The results hint at a strong local dependence of the electric field with small variations in the roughness of the irregular boundary. The value of the turn-on electric field is found to be ~8×10⁶ V/cm. Finally, Fowler-plot curves make possible to discuss the connection between the field amplification factor and geometric properties of the emitter surfaces.
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Assis et al. (2008) studied this question.
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