We perform direct numerical simulations of quasi-static magnetohydrodynamic turbulence, and compute various energy transfers including the ring-to-ring and conical energy transfers, and the energy fluxes of the perpendicular and parallel components of the velocity field. We show that the rings with higher polar angles transfer energy to ones with lower polar angles. For large interaction parameters, the dominant energy transfer takes place near the equator (polar angle θ ≈ π 2 ). The energy transfers are local both in wavenumbers and angles. The energy flux of the perpendicular component is predominantly from higher to lower wavenumbers (inverse cascade of energy), while that of the parallel component is from lower to higher wavenumbers (forward cascade of energy). Our results are consistent with earlier results, which indicate quasi two-dimensionalization of quasi-static magnetohydrodynamic (MHD) flows at high interaction parameters.

The Child–Langmuir law relates the voltage applied across a planar diode to the saturation value JCL of current density that can be transmitted through it in case the injection velocity of electrons is zero. The Child–Langmuir current density JCL is, at the same time: (i) the maximum current density that can be transmitted through a planar diode, (ii) the current density below which the flow is steady and unidirectional in the long time limit, and (iii) the average transmitted current density for any value of injected current density above JCL. Existing generalizations of Child–Langmuir law to nonzero velocities of injection are based on the characteristics (i) and (ii) of JCL. This paper generalizes the law to nonzero velocities of injection based on the characteristic (iii) by deriving an analytical expression for the saturation value of current density. The analytical expression for the saturation current density is found to be well supported by numerical computations. A reason behind preferring the saturation property of the Child–Langmuir current density as the basis for its generalization is the importance of that property in numerical simulations of high current diode devices.

The field enhancement factor at the emitter tip and its variation in a close neighbourhood determines the emitter current in a Fowler-Nordheim like formulation. For an axially symmetric emitter with a smooth tip, it is shown that the variation can be accounted by a cosθ˜ factor in appropriately defined normalized co-ordinates. This is shown analytically for a hemiellipsoidal emitter and confirmed numerically for other emitter shapes with locally quadratic tips.

Modeling high aspect ratio field emitter arrays is a computational challenge due to the enormity of the resources involved. The line charge model (LCM) provides an alternate semi-analytical tool that has been used to model both infinite as well as finite sized arrays. It is shown that the linearly varying charge density used in the LCM generically mimics ellipsoidal emitters rather than a Cylindrical-Post-with-an-Ellipsoidal-Tip (CPET) that is typical of nanowires. Furthermore, generalizing the charge density beyond the linear regime allows for modeling shapes that are closer to a CPET. Emitters with a fixed base radius and a fixed apex radius are studied with a view to understanding the effect of nonlinearity on the tip enhancement factor and the emitter current in each case. Furthermore, an infinite square array of the CPET emitters is studied using the nonlinear line charge model, each having a height h=1500 μm and a base radius b=1.5 μm. It is found that for moderate external field strengths (0.3−0.4 V/μm), the array current density falls sharply for lattice spacings smaller than 43h. Beyond this value, the maximal array current density can be observed over a range of lattice spacings and falls gradually thereafter.

Diodes used in most high power devices are inherently open. It is shown that under such circumstances, there is a loss of electromagnetic radiation leading to a lower critical current as compared to closed diodes. The power loss can be incorporated in the standard Child-Langmuir framework by introducing an effective potential. The modified Child-Langmuir law can be used to predict the maximum power loss for a given plate separation and potential difference as well as the maximum transmitted current for this power loss. The effectiveness of the theory is tested numerically.

The current-voltage data of a gated metallic nanotipped pyramidal emitter are analyzed using recent advances in field emission theory such as curvature corrections to the tunneling potential and the generalized cosine law of local electrostatic field variation near the emitter apex. It is first shown numerically that the cosine law holds for gated emitters. The theory is then subjected to an experimental test [C. Lee, S. Tsujino, and R. J. Dwayne Miller, Appl. Phys. Lett. 113, 013505 (2018)] where minor uncertainties in the pyramid base length Lb and the apex radius of curvature Ra exist. It is found using comsol multiphysics that the best-fit theoretical prediction for the emission current corresponds to Lb≃1.275 μm and Ra≃5.41 nm, both of which are within their respective uncertainties. The errors for the best-fit curve follow a known pattern with a change in the applied field, with higher field strengths (>5 V/nm) reporting less than 10% error.

Drift space is a region free from externally applied fields. It is an important part of many devices involving charged particle beams. The space charge effect imposes a limit on the current that can be transported through a drift space. A reasonable estimate of the space charge limited current density (J c SCL ) in closed drift tubes can be obtained from electrostatic considerations despite the fact that charge particle transport results in electromagnetic radiation. We deal here with the situation where the drift tube is open to electromagnetic radiation, for instance due to the presence of a dielectric window. In such cases, electromagnetic radiation leaks out of the window which results in a decrease in the average kinetic energy of electrons. If the injected current density is much lower than J c SCL , power loss does not result in a change in the transmitted current. As the injected current density is increased, power loss increases and at a critical value lower than J c SCL , reflection of electrons begins to occur. We also show that the lowering of the space charge limited current on account of power loss can be incorporated in the standard electrostatic picture by introducing an effective voltage for the collection plate.

Multi-stage cathodes are promising candidates for field emission due to the multiplicative effect in local field predicted by the Schottky conjecture and its recently corrected counterpart [Biswas, J. Vac. Sci. Technol. B 38, 023208 (2020)]. Due to the large variation in length scales even in a 2-stage compound structure consisting of a macroscopic base and a microscopic protrusion, the simulation methodology of a gated field emitting compound diode needs to be revisited. As part of this strategy, the authors investigate the variation of local field on the surface of a compound emitter near its apex and find that the generalized cosine law continues to hold locally near the tip of a multi-scale gated cathode. This is used to emit electrons with appropriate distributions in position and velocity components with a knowledge of only the electric field at the apex. The distributions are consistent with contemporary free-electron field emission model and follow from the joint distribution of launch angle, total energy, and normal energy. For a compound geometry with local field enhancement by a factor of around 1000, a hybrid model is used where the vacuum field calculated using COMSOL is imported into the Particle-In-Cell code PASUPAT, where the emission module is implemented. Space charge effects are incorporated in a multi-scale adaptation of PASUPAT using a truncated geometry with “open electrostatic boundary” condition. The space charge field, combined with the vacuum field, is used for particle-emission and tracking.