Fitting nonlinear FN plots of field emission strips with a self-consistent parallel plane model

Abstract

A self-consistent parallel plane (SCPP) model has been developed for fitting nonlinear Fowler-Nordheim (FN) plots of field emission strips. The saturation behavior of field emission strips is investigated by using a finite-difference time-domain (FDTD) particle-in-cell (PIC) simulation. The field emission process is described quantum mechanically by the Fowler-Nordheim equation. Space-charge effects are automatically included in the FDTD PIC simulations. The simulation results have confirmed that the saturation is caused by the space charge effects. Within the framework of an effective work function approximation and energy conservation, the FN and Poisson's equations are solved self-consistently in the SCPP model. It is demonstrated that the nonlinear FN plots of the field emission strips can be fitted by the SCPP model very well. The electric properties of the field emission strips can be characterized by the effective work function and the effective emission area. In addition, the field emission of Spindt-type cathodes may be possibly fitted with the SCPP model as well as the field emission strips.The FDTD PIC simulations were performed using the two-dimensional (2D) MAGIC code (ATK Mission Research, VA, US), which has been demonstrated to be capable of well describing space charge effects [1][2][3]. In the MAGIC simulations of the model, the interaction between charged particles and electromagnetic fields is considered by solving the Maxwell's equations and the Lorentz force equation, and the space-charge effects are automatically included. Boundary conditions including conductors and field emission of the cathode surface are used. The SYMMETRY PERIODIC boundary conditions are employed to describe the cell-to-cell relationships. Figure 1 shows a unit cell of the array of field emission strips, where L is the length of the unit cell, h and w are the height and width of the strip, respectively. PBC stands for the periodic boundary conditions we employed, and d is the distance between the cathode and the anode. Figure 2 shows the J-V curves of the field emission strips with several heights of h = (0.1, 0.2, …, 0.5) µm. Figure 3 shows the corresponding FN plots of the J-V curves. The MAGIC simulation results are marked with symbols. The simulation results have confirmed that the saturation is caused by the space charge effects. The saturation is achieved at a current density level close to that predicted by Barbour et al. [4] and the FN equation become nonlinear.In summary, the saturation behaviour has been investigated by the FDTD PIC simulations. We have confirmed that the nonlinear FN plots are caused by the space charge effects. It is also found that the saturation IV characteristics can be fitted with the self-consistent parallel plane model very well [5,6]. The details of the MAGIC simulation and the fitting procedures of the SCPP model will be presented and discussed.