Fowler-Nordheim Field Emission Research Articles

A data-driven model for the field emission from broad-area electrodes

Electron emission from cathodes in high field gradients is a quantum tunneling effect. The 1928 Fowler–Nordheim field emission (FE) equation and the 1956 Murphy–Good FE equation have traditionally been key in describing cold field emissions, offering estimates for emitters for almost a century. Nevertheless, applying FE theory in practice is often constrained by the lack of data on the distribution and geometry of the emission sites. Predictions become more challenging with an uneven electric field distribution at the cathode surface. Consequently, FE formulations are frequently calibrated using current–voltage data after test, limiting their efficacy as true predictive models.This study develops an alternative model for field emission using a data-driven predictive approach based on (1) vast experimental data, (2) electrostatic simulations of the cathode surface, and (3) detailed material and geometry properties, which together overcome these limitations. The objective of this work is to develop and harness this comprehensive dataset to train a machine learning model capable of providing precise predictions of the cathode current in order to further the understanding and application of field emission phenomena. More than 259 h of experimental data have been processed to train and benchmark some of the well-known machine learning models. After two stages of optimization, a coefficient of determination >98% is achieved in the prediction total field emission current using ensemble models.

Influence of frequency on the partial discharge characteristics of biaxially oriented polypropylene film under nanosecond pulse voltage

Partial discharge (PD) is one of the key factors leading to premature insulation failure of film capacitors used in pulse power devices. To study the effect of frequency on the PD characteristics of biaxially oriented polypropylene (BOPP) film, a sphere-plate electrode was used to simulate defects of the air gap between the film layers. PD experiments of the BOPP film were carried out under 1, 10, 100, 500 Hz, and 1 kHz with a rising edge of 65 ns and a pulse width of 300 ns. The results show that the repetitive partial discharge inception voltage (RPDIV), the number of discharges, the rising stage discharge (RSD) amplitude, and the falling stage discharge (FSD) time-lag, all show an increase and then decrease with increasing pulse frequencies and reach a maximum value at a pulse frequency of 100 Hz. Moreover, the FSD discharge amplitude and the RSD discharge time-lag increase with an increasing pulse frequency. From the perspective of initial electron generation, the quantitative mathematical relationship between PD characteristics (RPDIV, PD amplitude, PD number, and PD time-lag) and frequency was established based on the Fowler–Nordheim field emission effect for the first time, which can reflect the influence of frequency on the memory effect and the development process of PD. The results can provide an experimental basis and a mechanistic explanation for the evaluation of PD characteristics for film capacitors under nanosecond pulse voltage.

Corona Discharge Behaviors with Presence of Floating Electrode in Air Insulation using Numerical Model Analysis

This paper deals with numerical analysis method for analyzing the partial discharge behaviors in terms of corona discharge in the presence of a floating electrode under lightning impulse voltage. To analyze the corona discharge behaviors at 0.5 mm short electrode gap system as basic research work between main conductor and floating electrode, we considered fully coupled numerical model based on the charge continuity equations for positively charged ion, negatively charged ion and free electrons. Poisson’s equation is coupled with hydro-dynamic drift-diffusion equations derived from the three kinds of charge continuity equation. Fowler-Nordheim field emission is assumed that the electron emission by electric field is used as boundary conditions for main conductor and floating electrode. As a results, it is found that the floating electrode creates a non-uniform electric field distribution with a large field enhancement, causing ionization to initiate corona discharge. In other words, floating electrode is early corona discharge inception by collecting the induced charge developed by high voltage electrode. This result is expected to be applicable to various corona discharge generating systems as well as insulation design of power apparatus with presence defects such as a floating metal particle under lightning impulse voltage.

Study of dielectric breakdown in liquid xenon with XeBrA: The xenon breakdown apparatus.

Maintaining the electric fields necessary for the current generation of noble liquid time projection chambers (TPCs), with drift lengths exceeding 1m, requires a large negative voltage applied to their cathode. Delivering such high voltage is associated with an elevated risk of electrostatic discharge and electroluminescence, which would be detrimental to the performance of the experiment. The Xenon Breakdown Apparatus (XeBrA) is a 5-l, high voltage test chamber built to investigate the contributing factors to electrical breakdown in noble liquids. In this work, we present the main findings after conducting scans over stressed electrode areas, surface finish, pressure, and high voltage ramp speed in the medium of liquid xenon. Area scaling and surface finish were observed to be the dominant factors affecting breakdown, whereas no significant changes were observed with varying pressure or ramp speed. A general rise in both the anode current and photon rate was observed in the last 30s, leading up to a breakdown, with a marked increase in the last couple of seconds. In addition, the position of breakdowns was reconstructed with a system of high-speed cameras and a moderate correlation with the Fowler-Nordheim field emission model was found. Tentative evidence for bubble nucleation being the originating mechanism of breakdown in the liquid was also observed. We deem the results presented in this work to be of particular interest for the design of future, large TPCs, and practical recommendations are provided.

FIB fabrication and electron emission characteristics of overhanging tungsten and platinum nanostructures

The recent advancement in nanofabrication technology has enabled realization of vacuum electron devices with nanogap spacing. These devices have advantages over semiconductor counterparts due to possible operation under hard conditions and some unique functionalities. The challenging task, however, lies with the full understanding of their current–voltage (I–V) characteristics, resulting from various electron emission mechanisms. The reliable extraction of device parameters is, therefore, vital for its potential applications. An attempt has, therefore, been made here to fabricate two three-dimensional overhanging electrodes of tungsten and platinum with a nanoscale gap of 70–100 nm on glass substrates using chemical vapor deposition and focused ion beam milling. Their (I–V) characteristics measured in situ at ∼10−6 mbar are shown to follow the Child–Langmuir law and Fowler–Nordheim field emission at low and high-bias conditions, respectively. The extraction of parameters with a simple procedure suggested earlier yields an effective emission area of ∼3510 Å2, work function of ∼2.5 eV, and field enhancement factor (β) of ∼ 1.8 for tungsten; the values for platinum are 12.5 Å2, 3.0 eV, and 5.0, respectively. The higher β in the case of platinum can be attributed to the formation of a comparatively rough emitter surface with some fine protrusions. The nanostructure gives a current spike at high voltages, which marks its transition to an explosive emission state, breakdown, and dispersion of spherical metal particles over the substrate.

Linkage of electron emission and breakdown mechanism theories from quantum scales to Paschen's law

Numerous applications such as micro- and nanoelectromechanical systems, microplasmas, and directed energy increasingly drive device miniaturization to nanoscale and from vacuum to atmospheric pressure. This wide range of operating conditions and relevant mechanisms complicates the derivation of a single scaling law for electron emission and gas breakdown; therefore, theoretical studies often unify two or three mechanisms piecemeal. This study defines a common set of scaling parameters across the range of dominant mechanisms to derive a theory that links electron emission and breakdown mechanism theories from quantum scales to Paschen's law and yields asymptotic solutions for quantum space-charge limited emission (QSCL), classical space-charge limited emission (CSCL), space-charge limited emission with collisions (MG), Fowler–Nordheim field emission (FN), field emission driven gas breakdown, and classical gas breakdown defined by Paschen's law (PL). These non-dimensionalized equations are universal (true for any gas) across all regimes except for PL, which contains a single, material-dependent parameter. This approach reproduces various nexuses corresponding to the transitions across multiple mechanisms, such as QSCL to CSCL, CSCL to FN, CSCL to MG to FN, and field emission-driven breakdown as described by FN to PL, using a single non-dimensionalization scheme to facilitate experimental designs concerned with crossing these regimes. Furthermore, we demonstrate the conditions for more complicated nexuses, such as matching QSCL, CSCL, MG, and FN. This provides valuable information to experimentalists concerning regimes where slight perturbations in conditions may alter the electron emission mechanism and to theorists concerning the applicability of the asymptotic solutions or reduced nexus theories.

Conductor Surface Roughness-dependent Gas Conduction Process for HVDC GIL—Part I: Simulation

Charges due to ionization at conductors at low electric field serves as the main source of charge carriers that is responsible for surface charge enhancement, especially for hetero-polar charges on the convex surface side of spacers inside DC gas-insulated transmission lines (GILs). In this paper, simulation results of charge generation processes from gas side, including natural ionization, micro-discharge and field emission, are presented. The results show that for the case where the electric field strength is -6.548 kV/mm and the gas pressure is 0.4 Mpa, when conductor roughness Ra is lower than 24.16 μm, natural radiation ionization is the main cause of the dark current. When Ra is between 24.16 μm and 188.59 μm, the dark current is dominated by micro-discharge. The Fowler-Nordheim (F-N) field emission occurs when Ra exceeds 188.59 μm. Considering the roughness of the conductor in the actual GIL, the increase in the dark current due to local roughness is mainly due to micro-discharge rather than F-N field emission. The results in this paper provide an important part for further determining of conductor surface conditions in high voltage direct current (HVDC) GILs and serve as an important reference for completion of the surface charging model of spacers under DC.

A study of DC electrical breakdown in liquid helium through analysis of the empirical breakdown field distributions

We report results from a study on electrical breakdown in liquid helium using near-uniform-field stainless steel electrodes with a stressed area of ∼0.7cm2. The distribution of the breakdown field is obtained for temperatures between 1.7 K and 4.0 K, pressures between the saturated vapor pressure and 626 Torr, and with electrodes of different surface polishes. A data-based approach for determining the electrode-surface-area scaling of the breakdown field is presented. The dependence of the breakdown probability on the field strength as extracted from the breakdown field distribution data is used to show that breakdown is a surface phenomenon closely correlated with Fowler–Nordheim field emission from asperities on the cathode. We show that the results from this analysis provide an explanation for the supposed electrode gap-size effect and also allow for a determination of the breakdown-field distribution for arbitrary shaped electrodes. Most importantly, the analysis method presented in this work can be extended to other noble liquids to explore the dependencies for electrical breakdown in those media.

A reformulated general thermal-field emission equation

Theoretical models of thermal, field, and thermal-field emission of electrons generally require the evaluation of the Gamow, or tunneling, factor. A rapid means to do so is developed using shape factor methods for general potentials, but, in particular, for the image charge (or Schottky-Nordheim) barrier from which the Fowler-Nordheim (field emission) and Richardson-Laue-Dushman (thermal emission) equations are derived. The shape factor method provides greater accuracy than the elliptical integral functions that the electron emission equations conventionally rely on and eliminates the need to use them. The mild overprediction of the field emission current by the Fowler-Nordheim equation at very high fields is corrected. Undesired behaviors associated with prior versions of the general Thermal-Field equation in the thermal-field regime are eliminated by properly identifying the maximum of the current density integrand and accounting for its shape. The methodology to rapidly identify the current density maximum in general is demonstrated for various combinations of field, temperature, and work function, and the improvements demonstrated. The utility of the reformulated methodology for the simulation of electron beams and the ability to characterize cathode configurations of technological interest are discussed.

Solution of the time dependent Schrödinger equation leading to Fowler-Nordheim field emission

We solve the time-dependent Schrödinger equation describing the emission of electrons from a metal surface by an external electric field E, turned on at t=0. Starting with a wave function ψ(x,0), representing a generalized eigenfunction when E=0, we find ψ(x,t) and show that it approaches, as t→∞, the Fowler-Nordheim tunneling wavefunction ψE. The deviation of ψ from ψE decays asymptotically as a power law t−32. The time scales involved for typical metals and fields of several V/nm are of the order of femtoseconds. We plot the short-time evolution of the current and density.

Field emission and explosive electron emission process in focused ion beam fabricated platinum and tungsten three-dimensional overhanging nanostructure

Abstract Three-dimensional platinum and tungsten overhanging nanogap (∼70 nm) electrodes are fabricated on a glass substrate using focused ion beam milling and chemical vapour deposition processes. Current-voltage (I-V) characteristics of the devices measured at a pressure of ∼10−6 mbar shows space-charge emission followed by the Fowler-Nordheim (F-N) field emission. After the F-N emission, the system enters into an explosive emission process, at a higher voltage generating a huge current. We observe a sharp and abrupt rise in the emission current which marks the transition from the F-N emission to the explosive emission state. The explosive emission process is destructive in nature and yields micro-/nano-size spherical metal particles. The chemical compositions and the size-distribution of such particles are performed.

Self-consistent modeling of laminar electrohydrodynamic plumes from ultra-sharp needles in cyclohexane

This paper presents a self-consistent model of electrohydrodynamic (EHD) laminar plumes produced by electron injection from ultra-sharp needle tips in cyclohexane. Since the density of electrons injected into the liquid is well described by the Fowler-Nordheim field emission theory, the injection law is not assumed. Furthermore, the generation of electrons in cyclohexane and their conversion into negative ions is included in the analysis. Detailed steady-state characteristics of EHD plumes under weak injection and space-charge limited injection are studied. It is found that the plume characteristics far from both electrodes and under weak injection can be accurately described with an asymptotic simplified solution proposed by Vazquez et al. [“Dynamics of electrohydrodynamic laminar plumes: Scaling analysis and integral model,” Phys. Fluids 12, 2809 (2000)] when the correct longitudinal electric field distribution and liquid velocity radial profile are used as input. However, this asymptotic solution deviates from the self-consistently calculated plume parameters under space-charge limited injection since it neglects the radial variations of the electric field produced by a high-density charged core. In addition, no significant differences in the model estimates of the plume are found when the simulations are obtained either with the finite element method or with a diffusion-free particle method. It is shown that the model also enables the calculation of the current-voltage characteristic of EHD laminar plumes produced by electron field emission, with good agreement with measured values reported in the literature.

Large area fast-AFM scanning with active “Quattro” cantilever arrays

In this work, the fabrication and operation of an active parallel cantilever device integrating four self-sensing and self-actuating probes in an array is presented. The so called “Quattro” cantilever system is controlled by a multichannel field programmable gate array (FPGA) controller. The integrated cantilever devices are fabricated on the basis of a silicon-on-insulator wafer using surface micromachining and gas chopping plasma-etching processes [I. W. Rangelow, J. Vac. Sci. Technol., A 21, 1550 (2003)]. The unique design of the active cantilever probes provides both patterning and readout capabilities [Kaestner et al., J. Micro-Nanolithogr. MEMS 14, 031202 (2015)]. The thermomechanical actuation allows the individually operation of each cantilever in static and dynamic modes. This enables a simultaneous atomic force microscopy operation of all cantilevers in an array, while the piezoresistive read-out of the cantilever bending routinely ensures atomic resolution at a high imaging speed. The scanning probe lithography capabilities of the active cantilevers are based on the utilization of a Fowler–Nordheim field emission process of low-energy electrons (20–50 eV) for direct writing maskless lithography. The cantilever in the Quattro active cantilever array have a pitch of 125 μm (tip-to-tip distance), which allows an image size of 0.5 × 0.2 mm to be acquired within a single scan with 0.2 nm resolution in the vertical direction. Using parallel imaging, an effective scanning speed of 5.6 mm/s is achieved. The multichannel, scalable controller architecture allows four FPGA channels to scan and collect data simultaneously. A data buffer of 128 Mbits for a single frame of 4096 × 1024 pixels is applied. The designed data transfer system allows a packet size of 128 pixels to be transmitted within less than 10 μs, respectively. Thus, the entire image frame is transferred in less than 280 ms, which exceeds the required throughput in the practical cases like critical dimension-metrology and inspection. In this article, the authors are presenting the concept of the system, which combines imaging, metrology, and lithography capabilities with a low-cost of ownership. In this context, the authors are investigating the throughput capability, reproducibility, resolution, and positioning accuracy of the Quattro active cantilever system.

Field emission characteristics of a small number of carbon fiber emitters

This paper reports an experiment that studies the emission characteristics of small number of field emitters. The experiment consists of nine carbon fibers in a square configuration. Experimental results show that the emission characteristics depend strongly on the separation between each emitter, providing evidence of the electric field screening effects. Our results indicate that as the separation between the emitters decreases, the emission current for a given voltage also decreases. The authors compare the experimental results to four carbon fiber emitters in a linear and square configurations as well as to two carbon fiber emitters in a paired array. Voltage-current traces show that the turn-on voltage is always larger for the nine carbon fiber emitters as compared to the two and four emitters in linear configurations, and approximately identical to the four emitters in a square configuration. The observations and analysis reported here, based on Fowler-Nordheim field emission theory, suggest the electric field screening effect depends critically on the number of emitters, the separation between them, and their overall geometric configuration.

Field emission driven direct current argon discharges and electrical breakdown mechanism across micron scale gaps

In this paper results of the experimental and theoretical studies of the field emission driven direct current argon microdischarges for the gaps between 1 μm and 100 μm are presented and discussed. The breakdown voltage curves and Volt-Ampere characteristics proved to be a fertile basis providing better understanding of the breakdown phenomena in microgaps. Based on the measured breakdown voltage curves, the effective yields have been estimated confirming that the secondary electron emission due to high electric field generated in microgaps depends primarily on the electric field leading directly to the violation of the Paschen’s law. Experimental data are supported by the theoretical predictions that suggest departure from the scaling law and a flattening of the Paschen curves at higher pressures confirming that Townsend phenomenology breaks down when field emission becomes the key mechanism leading to the breakdown. Field emission of electrons from the cathode, the space charge effects in the breakdown and distinction between the Fowler-Nordheim field emission and the space charge limited current density are also analyzed. Images and Volt-Ampere characteristics recorded at the electrode gap size of 20 μm indicate the existence of a discharge region similar to arc at the pressure of around 200 Torr has been observed.

The particle production at the event horizon of a black hole as gravitational Fowler-Nordheim emission in uniformly accelerated frame, in the non-relativistic scenario

In the conventional scenario, the Hawking radiation is believed to be a tunneling process at the event horizon of the black hole. In the quantum field theoretic approach the Schwinger’s mechanism is generally used to give an explanation of this tunneling process. It is the decay of quantum vacuum into particle anti-particle pairs near the black hole surface. However, in a reference frame undergoing a uniform accelerated motion in an otherwise flat Minkowski space-time geometry, in the non-relativistic approximation, the particle production near the event horizon of a black hole may be treated as a kind of Fowler-Nordheim field emission, which is the typical electron emission process from a metal surface under the action of an external electrostatic field. This type of emission from metal surface is allowed even at extremely low temperature. It has been noticed that in one-dimensional scenario, the Schrödinger equation satisfied by the created particle (anti-particle) near the event horizon, can be reduced to a differential form which is exactly identical with that obeyed by an electron immediately after the emission from the metal surface under the action of a strong electrostatic field. The mechanism of particle production near the event horizon of a black hole is therefore identified with Schwinger process in relativistic quantum field theory, whereas in the non-relativistic scenario it may be interpreted as Fowler-Nordheim emission process, when observed from a uniformly accelerated frame.

Application of the general thermal field model to simulate the behaviour of nanoscale Cu field emitters

Strong field electron emission from a nanoscale tip can cause a temperature rise at the tip apex due to Joule heating. This becomes particularly important when the current value grows rapidly, as in the pre-breakdown (the electrostatic discharge) condition, which may occur near metal surfaces operating under high electric fields. The high temperatures introduce uncertainties in calculations of the current values when using the Fowler–Nordheim equation, since the thermionic component in such conditions cannot be neglected. In this paper, we analyze the field electron emission currents as the function of the applied electric field, given by both the conventional Fowler–Nordheim field emission and the recently developed generalized thermal field emission formalisms. We also compare the results in two limits: discrete (atomistic simulations) and continuum (finite element calculations). The discrepancies of both implementations and their effect on final results are discussed. In both approaches, the electric field, electron emission currents, and Joule heating processes are simulated concurrently and self-consistently. We show that the conventional Fowler–Nordheim equation results in significant underestimation of electron emission currents. We also show that Fowler–Nordheim plots used to estimate the field enhancement factor may lead to significant overestimation of this parameter especially in the range of relatively low electric fields.

Structural tuning of nanogaps using electromigration induced by field emission current with bipolar biasing

We report a new method for fabrication of Ni nanogaps based on electromigration induced by a field emission current. This method is called “activation” and is demonstrated here using a current source with alternately reversing polarities. The activation procedure with alternating current bias, in which the current source polarity alternates between positive and negative bias conditions, is performed with planar Ni nanogaps defined on SiO2/Si substrates at room temperature. During negative biasing, a Fowler-Nordheim field emission current flows from the source (cathode) to the drain (anode) electrode. The Ni atoms at the tip of the drain electrode are thus activated and then migrate across the gap from the drain to the source electrode. In contrast, in the positive bias case, the field emission current moves the activated atoms from the source to the drain electrode. These two procedures are repeated until the tunnel resistance of the nanogaps is successively reduced from 100 TΩ to 48 kΩ. Scanning electron microscopy and atomic force microscopy studies showed that the gap separation narrowed from approximately 95 nm to less than 10 nm because of the Ni atoms that accumulated at the tips of both the source and drain electrodes. These results show that the alternately biased activation process, which is a newly proposed atom transfer technique, can successfully control the tunnel resistance of the Ni nanogaps and is a suitable method for formation of ultrasmall nanogap structures.

Field-emission–driven direct current hydrogen discharges in microgaps

This letter contains results of the experimental and theoretical studies of the direct current hydrogen discharges generated between two plane-parallel tungsten electrodes separated from 1 μm to 100 μm. Breakdown voltage curves and volt-ampere characteristics of the discharges are presented and discussed in order to investigate fundamental processes leading to the breakdown in microgaps. Our experimental results are in accordance with the theoretical predictions which state that Townsend phenomenology breaks down when field emission becomes the key mechanism affecting the breakdown and deforming the left-hand side of the breakdown voltage curves. Based on the measured breakdown voltage curves, the dependence of the effective yield on the electric field in microgaps has been determined. Space charge effects are also estimated and distinction between Fowler-Nordheim field emission and space charge limited current density is considered.

Fowler-Nordheim Field Emission from Carbon Nanotubes Under Intense Electric Field

In this paper, we study the Fowler-Nordheim field emission (FNFE) from carbon nanotubes on the basis of a newly formulated electron dispersion law by considering the fact that the intense electric field needed for FNFE changes the band structure in a fundamental way. It has been found that the field emitted current increases with increasing electric field in oscillatory manner due to the appearance of van Hove singularities and exhibits spikes for particular values of the electric field where the singularity occurs. The numerical values of the field emitted current in all the cases vary widely and the determined by the chiral indices and the diameter in the respective cases. The results of this paper find three applications in the fields of nanoscience and technology.