Beyond the Child–Langmuir law: A review of recent results on multidimensional space-charge-limited flow

Abstract

A simple physical derivation of Child-Langmuir space-charge-limited emission using vacuum capacitance Am.Space-charge-limited ͑SCL͒ flows in diodes have been an area of active research since the pioneering work of Child and Langmuir in the early part of the last century. Indeed, the scaling of current density with the voltage to the 3/2's power is one of the best-known limits in the fields of non-neutral plasma physics, accelerator physics, sheath physics, vacuum electronics, and high power microwaves. In the past five years, there has been renewed interest in the physics and characteristics of SCL emission in physically realizable configurations. This research has focused on characterizing the current and current density enhancement possible from two-and three-dimensional geometries, such as field-emitting arrays. In 1996, computational efforts led to the development of a scaling law that described the increased current drawn due to two-dimensional effects. Recently, this scaling has been analytically derived from first principles. In parallel efforts, computational work has characterized the edge enhancement of the current density, leading to a better understanding of the physics of explosive emission cathodes. In this paper, the analytic and computational extensions to the one-dimensional Child-Langmuir law will be reviewed, the accuracy of SCL emission algorithms will be assessed, and the experimental implications of multidimensional SCL flows will be discussed.

Space-charge limited surface currents between two semi-infinite planar electrodes embedded in a uniform dielectric medium

“…Langmuir [2] also considered cylindrical electrodes, and noted that the V 3/2 dependence in (1) does not depend on the shape of the electrodes, using a scaling argument. Only recently the Child-Langmuir law was extended to electron emission over a finite patch on a planar cathode [11,12,13]. Whereas the geometry of many practical devices is indeed one-dimensional, there are exceptions, as just mentioned.…”

Power loss in open cavity diodes and a modified Child-Langmuir law

“…Generalizations for non-zero initial velocity can be found in [4,5,6]. In two dimensions, analytical and numerical studies [7,8,9] indicate that the critical current is somewhat higher than the Child-Langmuir expression and is given by [7,9] …”

Space charge effects in field emission: One dimensional theory

“…Nevertheless, there have been several theoretical efforts [11][12][13][14][15][16]: in particular, the treatment of the 1D field emission / space charge problem (Ref. [17] and references therein), the usage of Particle-in-Cell codes to treat field emission (Ref.…”