Kevin L. Jensen

Electron emission theory and its application: Fowler–Nordheim equation and beyond

In this article, we examine the Fowler–Nordheim (FN) equation for field emission using pedagogical models to introduce and illuminate its origins, limitations, extensions, and application to multidimensional structures. The analyses of modern electron sources generally invoke either the FN equation or the Richardson–Laue–Dushman equation (thermionic emission) to interpret experimental data. These equations have ranges of validity that are increasingly challenged by operating conditions. The present article shall therefore have several aims. An introduction to and review of the FN equation shall be presented. Extensions to account for many body and other effects, shall be motivated by accessible models, and a generalized thermal-field emission methodology developed to account for low work function, high fields, photoexcitation, and other conditions in which the incident electron energy is near the barrier maximum. An account of effects such as resonance, which are not generally part of the standard emissio...

An analytical solution for microtip field emission current and effective emission area

Classical field emission theory relates the material work function and applied electric field at the emitter surface to tunneling current density, based upon the one-dimensional, planar Fowler–Nordheim equation. The extension of this theory to nonplanar, tip-anode geometries is complicated by the spatial variation of the electric field and resultant current density along the microtip surface. To relate this spatially dependent electric field to the applied voltage in this tip-to-plane geometry, experimenters have adopted empirically determined field enhancement factors (i.e., F=βV) and emission area terms to explain quantitative measurements of microtip current. In this work, the nature of the field enhancement and area terms are defined analytically by employing an exact three-dimensional electric field solution in prolate spheroidal coordinates and examining the relationship between total tip current and current density. The analytical and numerical results based on this model provide a physical context...

General formulation of thermal, field, and photoinduced electron emission

The canonical equations of electron emission are the Richardson-Laue-Dushman equation for thermionic emission, the Fowler-Nordheim equation for field emission, and the Fowler-Dubridge equation for photoemission. They are widely used and experimentally well vetted for the determination of current density from electron sources. While all three may be obtained from a general underlying formulation, they are treated as distinct as their domains of applicability do not overlap. Here, a tractable analytical model is given that not only devolves to the canonical equations, but also and more importantly is effective in the transition regions when the emission currents from tunneling, thermal emission, and/or photoemission become comparable but where the canonical equations are inapplicable. The resulting thermal-field-photoemission equation has application to emerging advanced electron sources as well as for emission models required by beam simulation codes.

The methodology of simulating particle trajectories through tunneling structures using a Wigner distribution approach

The authors introduce a trajectory methodology to describe elementary space- and time-dependent events in a tunneling process in the resonant tunneling diode (RTD). A methodology for constructing quantum particle trajectories is presented. The trajectories for a RTD are presented and their behavior, as a function of scattering and self-consistency, is shown to be consistent with the steady-state current-voltage/quantum well electron density characteristics of the RTD, and with the response of the RTD to a sudden bias switch. The trajectories also exhibit a conservation-of-energy-like behavior. The trajectory formulation is thus shown to be potentially useful for incorporating into a multidimensional particle Monte Carlo simulation of quantum-based devices in which the tunneling region is small compared to the dimensions of the device. >

Emittance of a field emission electron source

An analytical formula of the emittance of a field emitter is given. In contrast to thermal and photoemission, such a formula contains complexity due to the multidimensional nature of the source. A formulation of emittance is given for one- and three-dimensional (3D) field emitters. The 3D formulation makes use of the point charge model of a unit cell emitter coupled with a trajectory analysis to follow electrons to an evaluation plane where emittance is determined. The single tip theory is extended to an array and the resulting theory predicts the emittance of a Spindt-type square array of emitters 0.2cm on a side producing 2000A∕cm2 is 23mmmrad. Theory compares favorably with experimental measurements in the literature from ungated and gated sources. The impacts of several complications are estimated: the effects of a gate for modulating the emitter; the influence of space charge within the unit cell on the beam; and constraints imposed by modulation frequency, emitter dimensions, and rise/fall time requ...

Numerical simulation of field emission and tunneling: A comparison of the Wigner function and transmission coefficient approaches

Quantum transport through one‐dimensional potential barriers is usually analyzed using either the transmission coefficient (TC) or the Wigner distribution function (WDF) approach. Fast, accurate, and efficient numerical algorithms are developed for each and are compared for (a) calculating current‐field relationships for field‐emission potentials with silicon parameters (and current‐voltage relationships for resonant tunneling diodes), (b) their ability to accommodate scattering, self‐consistency, and time dependence, and for (c) the behavior of their ‘‘particle trajectory’’ interpretations. In making the comparisons, the concern will be on the ability of each method to be incorporated into a larger ensemble‐particle Monte Carlo simulation; it is argued that, in this regard, the WDF approach has significant advantages. Since the TC calculations rely on the Airy function approach, a detailed comparison of this method is made with the widely used Wentzel–Kramers–Brillouin and Fowler–Nordheim approaches for ...

Exchange-correlation, dipole, and image charge potentials for electron sources: Temperature and field variation of the barrier height

Potential barrier profiles for large applied fields and/or high temperature are developed for the study of field and thermionic emission electron sources intended for radio frequency power tube applications. The numerical implementation provides a fast and flexible method to obtain the barriers which govern current density, and yet allows for complications such as nanoprotrusions, adsorbates, “internal” field emission, the sputtering of low work function emission sites, and so on. The model consists of (i) a modified form of the Wigner Lattice expansion of the electron ground state energy to evaluate the exchange and correlation potential, (ii) a simplified form of the ionic core potential to correct the “Jellium” model, (iii) a triangular representation of the barrier with a single adjustable parameter which enables both the solution of Schrodinger’s equation in terms of Airy functions and thus an exact evaluation of the electron density near the barrier, and (iv) a numerical integration of Poisson’s equ...

Simulation of resonant tunneling structures: Origin of the I–V hysteresis and plateau-like structure

Hysteresis and plateau-like behavior of the I–V curves of a double-barrier resonant tunneling structure are simulated in the negative differential resistance region. Our simulation results show that the creation of an emitter quantum well after the current passes its maximum value is the key point in understanding the origin of the I–V plateau-like structure. It is demonstrated that the plateau-like behavior of the I–V curves is produced by the coupling between the energy level in the emitter quantum well and that in the main quantum well. The hysteresis is a manifestation of the above-mentioned energy level coupling, the accumulation and distribution of electrons in the emitter, and the coupling between the energy level in the quantum well and the conduction band edge or the three-dimensional continuum states in the emitter. The effects of the structural parameters on the bistability of the I–V curves of resonant tunneling devices are discussed. The creation and disappearance mechanism of the emitter qua...

A photoemission model for low work function coated metal surfaces and its experimental validation

Photocathodes are a critical component many linear accelerator based light sources. The development of a custom-engineered photocathode based on low work function coatings requires an experimentally validated photoemission model that accounts the complexity of the emission process. We have developed a time-dependent model accounting for the effects of laser heating and thermal propagation on photoemission. It accounts for surface conditions (coating, field enhancement, and reflectivity), laser parameters (duration, intensity, and wavelength), and material characteristics (reflectivity, laser penetration depth, and scattering rates) to predict current distribution and quantum efficiency (QE) as a function of wavelength. The model is validated by (i) experimental measurements of the QE of cesiated surfaces, (ii) the QE and performance of commercial dispenser cathodes (B, M, and scandate), and (iii) comparison to QE values reported in the literature for bare metals and B-type dispenser cathodes, all for vari...

New results in the theory of Fowler–Nordheim plots and the modelling of hemi-ellipsoidal emitters

This paper reports further progress in understanding the theory of emission-area extraction from Fowler-Nordheim plots, and reports some useful interim results derived by modelling field electron emission from hemi-ellipsoidal emitters. The mathematical nature of the relationship between a new approach to emission-area extraction, recently proposed, and older approaches is demonstrated. The new approach is extended to cover field dependence in emission area. Preliminary results are reported from an investigation into the effects of making erroneous assumptions about the tunnelling barrier seen by the electron and the absence of field dependence in emission area. If wrong theoretical assumptions are made, then emission area can be overpredicted by a factor of as much as 10 or 20. On the other hand, if correct theoretical assumptions are made, then the extracted emission area is close to an emission area derived directly from the model calculations. The problematical nature of the concept of emission area, when emission area is a function of field, is pointed out.