The Fowler-Nordheim law gives the current density extracted from a surface under strong fields, by treating the emission of electrons from a metal-vacuum interface in the presence of an electric field normal to the surface as a quantum mechanical tunneling process. Child’s law predicts the maximum transmitted current density by considering the space charge effect. When the electric field becomes high enough, the emitted current density will be limited by Child’s law. This work analyzes the transition of the transmitted current density from the Fowler-Nordheim law to Child’s law space charge limit using a one-dimensional particle-in-cell code. Also studied is the response of the emission model to strong electric fields near the transition point. We find the transition without geometrical effort is smooth and much slower than reported previously [J. P. Barbour, W. W. Dolan, J. K. Trolan, E. E. Martin, and W. P. Dyke, Phys. Rev. 92, 45 (1953)]. We analyze the effects of geometric field enhancement and work function on the transition. Using our previous model for effective field enhancement [Y. Feng and J. P. Verboncoeur, Phys. Plasmas 12, 103301 (2005)], we find the geometric effect dominates, and enhancement β>10 can accelerate the approach to the space charge limit at practical electric field. A damped oscillation near the local plasma frequency is observed in the transient system response.

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The local field enhancement factor β is often introduced in the Fowler–Nordheim equation to represent the geometrical effects at the surface of the cathode, where β(s)=En(s)∕E0 for macroscopic applied field E0. Local variation of β determines the local normal surface electric field, En(s), resulting in local dependence of injection current by the Fowler–Nordheim law. In computational models, it is impractical to determine the time-dependent local surface field each time step on a microscopic space scale. Effective β is introduced in this paper which allows us to study the emission properties at a macroscopic scale. Microscopic (subgrid) local effective β is calculated only at the initial time step, and then the effective β can be recomputed for different surface electrical field through this model. The model allows reduction of dimensionality as well as the ability to include subgrid effects. The model is demonstrated on fundamental cases and compared to a calculation with a mesh fine enough to resolve the geometric features.