Transition from Fowler-Nordheim field emission to space charge limited current density

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.