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.

Crossed-field space-charge limited current (CF-SCLC) represents the maximum stable current that can be produced in crossed-field devices (CFDs); prominent CFDs include magnetrons and cyclotrons, critical technologies in vacuum electronics. While planar crossed-field geometries have the most developed theoretical models, cylindrical geometries are far more common and useful in practical applications. CFDs are characterized by the strength of the externally applied magnetic field $B$ , orthogonal (crossed) to the electric field; this is often normalized to the magnetic insulation condition, the Hull cutoff field $B_H$ . We apply variational calculus to derive a new theoretical model for CF-SCLC for planar and cylindrical devices below and above $B_H$ . The variational model offers a concise derivation for planar results without transforming to the time domain and gives the first analytic results from first principles for cylindrical CF-SCLC. We implement a fully three-dimensional simulation in CST Particle Studio which, in addition to additional derived simple theoretical models, explains the often overlooked experimental current scaling $\propto {\left[1-{\left(B/B_H\right)}^2\right]}^{1/2}$ which decreases to zero current as $B\to B_H^{-}$ . These additional simple models reduce the maximum mismatch magnitude between theory and experiments or simulations by up to 68% compared to the variational model, with the most improvement at the critical limit $B\to B_H^{-}$ . Justification for the variational model and future applications are discussed.

Silicon field emitter arrays (Si FEAs) are being explored as an electron source for vacuum channel transistors for high temperature electronics. Arrays of 1000 × 1000 silicon tip based gated field emitters were studied by measuring their electrical characteristics up to 40 V of DC gate bias with a 1.3 mA emission current at different temperatures from 25 to 400 °C. At ∼350 °C, residual gas analyzer measurements show that water desorption and carbon dioxide partial pressures increase significantly, the gate to emitter leakage current decreases by more than ten times, and the collector current increases by more than ten times. These improvements remained after heat-treatment but were then lost once the device was exposed to the atmosphere for several days. The improvements could be recovered upon additional baking suggesting that adsorbates (primarily water) on the surface affected field emission and surface leakage. It was also found that after heat-treatment, the electrical characteristics of the devices exhibited <3% variation in collector current at 40 V, which (without exposure to the atmosphere) can be termed as a weak temperature dependence. These results suggest that Si FEAs could be viable as a high temperature transistor.

A complementary vacuum field emission device structure is proposed, and its operation is analyzed by multiphysics simulation. A freely moving double-clamped cantilever, which balances the electrostatic attraction force and elastic restoration force, is used as the source electrode while the drain electrode is fixed. The electron-emitting cathode is formed on the source for the n-type device and on the drain for the p-type. Thus, complementary current–voltage characteristics are obtained only with Fowler–Nordheim tunneling electron transport. The impact of various design parameters such as vacuum gap, cantilever beam thickness, gate width, and tip offset on the drain and gate leakage currents is investigated. Inverter logic operation is verified successfully using complementary devices.

Lateral field emission devices have been characterized and degradation tested for >1000 h to study stability and reliability. Two types of planar device structures, diode and bowtie, were studied. These nanoscale devices have 10–20 nm tip to tip or tip to collector dimensions with the tips fabricated from Au/Ti. Typical currents of 2–6 nA at 6 V were measured. The devices were placed on lifetime tests in a vacuum of 1000 h. Seven total devices were tested with one failing at 300 h. and three of the devices showed <5% degradation in current until 1400 h when testing was stopped, and three other devices showed a sudden drop of ≈20% ranging from 700 to 900 h. Optical microscope images of one of the devices that failed catastrophically at 350 h show physical arc damage where the bond pad narrows to the emitter trace. Scanning electron microscope images of a bowtie part that completed 1400 h of operation showed no obvious erosion or damage to the tips.

A magnetron concept using field emission cathodes has been modeled with the Air Force Research Laboratory particle-in-cell code ICEPIC and the two-dimensional particle trajectory simulation Lorentz2E. In this approach, field emitters are used to provide a distributed cathode in place of a traditional thermionic cathode. The emitters are placed below the interaction space in a shielded structure. The cathode is comprised of facet plates with slits to protect the emitters. Simulation of an L-band rising sun magnetron shows that the faceted magnetron will oscillate using both five and ten facet cathodes. The startup times are very similar to that of a cylindrical cathode magnetron. The electron trajectories of the shielded slit structure have been modeled, and the results indicate that electrons can be injected through the slits and into the interaction space using lateral edge emitters and a pusher electrode design.