Insulating funnels, called electron hop funnels, use secondary electron emission and an electric field created by an electrode to transport current through the device. The surface charge along the funnel wall self-adjusts to get unity-gain transmission. Electron hop funnels allow increased control over the spatial and energy uniformity of the transmitted beam from a field emitter array. Measurements performed on the relationship between transmission through the device and the electrode voltage has shown hysteresis. To better understand the origin of the hysteresis, simulations have been performed using the particle trajectory code Lorentz 2E. The simulations reveal two very important mechanisms that define the transmission through the funnel. The simulations also show that hysteresis is a fundamental characteristic of hop funnels

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Enhanced output current density of an active-matrix high-efficiency electron emission device array with 13.75 μ m pixels Electron hop funnels provide a method to integrate field emission arrays into microwave vacuum electron devices, to protect the arrays, and to provide a method to study the secondary electron characteristics of dielectrics. A hop funnel is a dielectric material with an electrode, known as the hop electrode, placed around the narrow end (exit) of the funnel to control the current transmitted through the device. Current is transmitted through the funnel via electron-hopping transport. This work investigates a hysteresis observed in the current-voltage characteristic of the device. The experimental results showing the observed hysteresis will be presented. This work will demonstrate that charging on the bottom of the hop funnel is not the fundamental cause of the hysteresis.

While prior theoretical studies of multi-dimensional space-charge limited current (SCLC) assumed emission from a small patch on infinite electrodes, none have considered emission from an entire finite electrode. In this paper, we apply variational calculus (VC) and conformal mapping, which have previously been used to derive analytic solutions for SCLC density (SCLCD) for nonplanar one-dimensional geometries, to obtain mathematical relationships for any multi-dimensional macroscopic diode with finite cathode and anode. We first derive a universal mathematical relationship between space-charge limited potential and vacuum potential for any diode and apply this technique to determine SCLCD for an eccentric spherical diode. We then apply VC and the Schwartz–Christoffel transformation to derive an exact equation for SCLCD in a general two-dimensional planar geometry with emission from a finite emitter. Particle-in-cell simulations using VSim agreed within 4%–13% for a range of ratios of emitter width to gap distance using the thinnest electrodes practical for the memory constraints of our hardware, with the difference partially attributed to the theory's assumption of infinitesimally thin electrodes. After generalizing this approach to determine SCLCD for any orthogonal diode as a function of only the vacuum capacitance and vacuum potential, we derive an analytical formulation of the three-dimensional Child–Langmuir law for finite parallel rectangular and disk geometries. These results demonstrate the utility for calculating SCLCD for any diode geometry using vacuum capacitance and vacuum potential, which are readily obtainable for many diode geometries, to guide experiment and simulation development.

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