Discrete Electron Emission

TL;DR

The paper addresses space-charge effects at mesoscopic scales where electrons must be treated as discrete point charges rather than a continuum. It develops simple discrete-space-charge models for point-, line-, and sheet-emitter configurations, and validates them against molecular-dynamics simulations to derive scaling laws for space-charge limited emission. A key result is the identification of a critical length $\xi_* = \sqrt{\frac{q}{2\pi\varepsilon_0 E_0}}$ that governs emission spacing and current, with distinct scalings: $I \propto E_0^{3/4}$ for point emitters, $I \propto E_0^{5/4}$ for line emitters, and the conventional $I \propto E_0^{3/2}$ for large-area emitters. The work clarifies how discreteness and emitter geometry influence spacing and current, providing predictive scaling laws and benchmarks for nano-structured cathodes and single-electron emission regimes, extending beyond continuum PIC models.

Abstract

Analysis of space-charge effects on electron emission typically makes some assumption of continuity and smoothness, whether this is continuity of charge as in the classical derivation of the Child-Langmuir current, or the mean-field approximation used in particle-in-cell simulations. However, when studying the physics of electron emission and propagation at the mesoscale it becomes necessary to consider the discrete nature of electronic charge to account for the space-charge effect of each individual point charge. In this paper we give an extensive analysis of some previous work on the distribution of electrons under space-charge limited conditions. We examine the spacing of electrons as they are emitted from a planar surface, We present simplified models for analysis of such conditions to derive scaling laws for emission and compare them to computer simulations.

Figures (10)

• Figure 1: Surface field, $\overline{E}_{2mD}$, from Eq. ( \ref{['eq:SheetEmitterNorm']}) as a function of elevation for a discrete sheet of particles. $\overline{l}=\sqrt{2\pi}$. $\overline{E}_{m2D}\approx1-\overline{\xi}\frac{16.5172}{\overline{l}^3}$ shown as dashed line.
• Figure 2: The surface field, $\overline{E}_{m1D}$, from Eq. ( \ref{['eq:StringEmitterNorm']}) as a function of $\overline{\xi}$, for discrete string of particles
• Figure 3: The magnitude of the surface electric field on the emitting ring at two instances in time. $D = 1000$ nm, $V = 200$ V. The bottom picture shows the situation after 62.5 fs have elapsed from the top picture is taken. Note the signature effects of newly emitted electrons.
• Figure 4: Distribution of separation between adjacent electrons emitted from a ring. Boxplot shows first to third quartile with line at the median. Whiskers extend to 1.5 times the inter-quartile range. Diamonds show statistical outliers. Blue line shows pitch spacing $=1.834\xi_*$ and red line shows $R_*=\sqrt{2}\xi_*$ .
• Figure 5: Distribution of distance between adjacent electrons, emitted from the ring, for three different values of applied field.