Numerical analysis in focusing characteristics of the three-dimensional trapezoidal electrodes for carbon nanotube field emitters

In this study, the focusing characteristics of three-dimensional trapezoidal focusing electrodes for carbon nanotube field emitters are analyzed by the combination of the finite-difference time-domain and particle-in-cell methods. To investigate the divergences of the electrons, the three-dimensional trajectories of the electron beams are simulated by evaluating the electrons’ positions and momentums. The divergence angles of electrons in the emitters with three-dimensional focusing electrodes are proved to be smaller than those with the conventional triode structure or with the planar focusing electrodes. Characteristics of the 4 μm-thick three-dimensional geometries (trapezoidal, rectangular and inversed trapezoidal) are compared. The 1 V biased three-dimensional trapezoidal focusing electrode is optimal in regulating the divergence of the beams and forming uniform electron spots on the anode. The average divergence angle of the electrons is limited to 0.1309 rad on the anode plane while the maximum divergence angle is 0.3236 rad.