Extraction of emission parameters for large-area field emitters, using a technically complete Fowler-Nordheim-type equation
Richard G. Forbes
备注：
36 pages, including two figures, and list of acronyms (to be electronic supplementary materiaL). Version scientifically equivalent to corrected proof for publication in Nanotechnology. Some minor typographical errors have been corrected
In papers on cold field electron emission from large area field emitters (LAFEs), it has become widespread practice to publish a misleading Fowler-Nordheim-type (FN-type) equation. This equation over-predicts the LAFE-average current density by a large highly-variable factor thought to usually lie between 1000 and 1000 000 000. This equation, although often referenced to FN's 1928 paper, is a simplified equation used in undergraduate teaching, does not apply unmodified to LAFEs, and does not appear in the 1928 paper. Technological LAFE papers often do not cite any theoretical work more recent than 1928, and often do not comment on the discrepancy between theory and experiment. This usage has occurred widely, in several high-profile American and UK applied-science journals, and in various other places. It does not inhibit practical LAFE development, but can give a misleading impression of potential LAFE performance to non-experts. This paper shows how the misleading equation can be replaced by a conceptually complete FN-type equation that uses three high-level correction factors. One of these, or a combination of two of them, may be useful as an additional measure of LAFE quality; this paper describes how to estimate factor values using experimental data. Suggestions are made for improved engineering practice in reporting LAFE results. Some of these should help to prevent situations arising whereby an equation appearing in high-profile applied-science journals is used to support statements that an engineering regulatory body might deem to involve professional negligence.
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As the title implies the article describes the possibility of taking into account the relativistic correction to the field current density of the field emission of electrons from the metal. The article provides the reader with some analytic generalization of the Fowler-Nordheim equation with the relativistic correction. The relativistic correction to the Fowler-Nordheim equation makes it possible to take into account the influence of the relativism on the field emission current. It is especially noted that the consideration of this correction is necessary in the case of sufficiently strong ele…
Generalized Fowler-Nordheim field-induced vertical electron emission model for two-dimensional materials
目前，场致电子发射的理论描述仍然主要局限于近一个世纪前发展起来的经典Fowler-Nordheim（FN）框架。对于新兴的二维材料，FN模型的许多基本假设由于它们的降维和奇异的电子性质而变得无效。在这项工作中，我们开发的解析和半解析模型的场致垂直电子发射从二维材料的表面，明确考虑到降维，非抛物线能谱，非守恒的面内电子动量，有限温度和空间电荷限制的影响。我们表明，传统的FN法律不再适用于二维材料。本文提出的修正垂直场发射模型与实验结果有更好的一致性。有趣的是，一个新的高场制度的\emph{saturated surface field emission}出现由于二维材料的降维。这种饱和场致发射效应的一个显著后果是在三维体材料中通常在高场下预期的空间电荷限制电流的缺乏。
Field emission: the theoretical link between voltage loss, reduction in field enhancement factor, and Fowler-Nordheim-plot saturation
对于大面积场致电子发射体，当单个柱状发射体具有足够的电阻，并且通过它的电流足够大时，则沿其发生电压损失。这封信提供了一个简单的分析和概念论证，即这种电压损失与后顶点处的场增强因子（FEF）的降低直接且不可分割地联系在一起。E.米努岛Groening，K. B. K. Teo，S. H.达拉尔湖Gangloff，J. -施内尔湖胡丹斯基岛Y. Y.但是，P. Vincent，P. Legagneux，G. A. J. Amaratunga和W. I. Milne [Nano Lett. 5，2135（2005）]，通过拟合到来自拉普拉斯解算器的数值结果。这封信通过使用“浮动球体”模型，解析地推导出了相同的公式。分析证明更清楚地揭示了潜在的物理学，并表明这种效应是一种普遍现象，与发射体最突出部分的表面电荷大小减少有关。电压依赖性FEF降低是Fowler-Nordheim图中“饱和”的原因之一。另一种是由于测量电路电阻引起的分压器效应。提出了这两种效应的综合理论。两者一起或单独都会导致饱和。实验上，如果发生饱和但电压损失很小（比如< 20 V），则饱和更可能是由于FEF降低而不是分压。在这种情况下，可能需要修改紧密间隔的发射器之间的静电相互作用（“屏蔽”）的现有处理。其他假定的饱和原因也存在，所以目前的理论是片面的。它的扩展似乎是可能的，并可能导致更普遍的物理理解FN图饱和的原因。
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The Fowler-Nordheim tunneling current formula has been widely used in the design of devices based on metal/insulator (metal/semiconductor) heterojunctions with triangle potential barriers, such as the flash memory. Here we adopt the model that was used to derive the Landauer formula at finite temperature, the nearly-free electron approximation to describe the electronic states in semi-infinite metal electrode and the Wentzel-Kramers-Brillouin (WKB) approximation to describe the transmission coefficient, and derive a tunneling current formula for metal/insulator heterojunctions under large bias…
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This paper focuses on one small but significant part of a long-term project that aims to improve the interpretation of measured field electron emission (FE) current-voltage [I_m(V_m)] data, and (later) the formulation of FE theory. A new form of FE I_m(V_m) data-analysis plot - the so-called "Murphy-Good (MG) plot" - has recently been introduced, within the general framework of the prevailing "smooth planar metal-like emitter" methodology. This new plot form, and the reasons for its introduction, are discussed. It is shown that the MG plot can perform all the functions that the traditional (90…
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Both single emitters and large area field emitters (LAFE) are generally characterized using the slope and intercept of a Murphy-Good (or Fowler-Nordheim) plot which are used to extract the field enhancement factor and the emission area. Using a shielding model that has been developed recently for a LAFE, the validity of the underlying assumption is investigated. It is found that in case of a LAFE, the slope has contributions from the enhancement factor {\it as well as} the rate at which the effective number of super-emitters changes with the applied field. As a consequence, the emission area i…
Field electron emission (FE) has relevance in many different technological contexts. However, many related technological papers use a physically defective elementary FE equation for local emission current density (LECD). This equation takes the tunneling barrier as exactly triangular, as in the original FE theory of 90 years ago. More than 60 years ago, it was shown that the so-called Schottky-Nordheim (SN) barrier, which includes an image-potential-energy term (that models exchange-and-correlation effects) is better physics. For a metal-like emitter with work-function 4.5 eV, the SN-barrier-r…
Field emission tips with apex radius of curvature below 100nm are not adequately described by the standard theoretical models based on the Fowler-Nordheim and Murphy-Good formalisms. This is due to the breakdown of the `constant electric field' assumption within the tunneling region leading to substantial errors in current predictions. A uniformly applicable curvature-corrected field emission theory requires that the tunneling potential be approximately universal irrespective of the emitter shape. Using the line charge model, it is established analytically that smooth generic emitter tips appr…
This Comment suggests that technological field electron emission (FE) papers, such as the paper under discussion [P. Serbun et al.,, Rev. Sci. Instrum. 91, 083906 (2020)], should use FE theory based on the 1956 work of Murphy and Good (MG), rather than a simplified version of FE theory based on the original 1928 work of Fowler and Nordheim (FN). Use of the 1928 theory is common practice in technological FE literature, but the MG treatment is known to be better physics than the FN treatment, which contains identifiable errors. The MG treatment predicts significantly higher emission current dens…
In field electron emission (FE) studies, it is important to check and analyse the quality and validity of experimental current-voltage data, which is usually plotted in one of a small number of standard forms. These include the so-called Fowler-Nordheim (FN), Millikan-Lauritsen (ML) and Murphy-Good (MG) plots. The Field Emission Orthodoxy Test is a simple quantitative test that aims to check for the reasonableness of the values of the parameter "scaled field" that can be extracted from these plots. This is done in order to establish whether characterization parameters extracted from the plot w…
A model is constructed to describe the thermal-field emission of electrons from a three-dimensional ($3$D) topological semimetal hosting Dirac/Weyl node(s). The traditional thermal-field electron emission model is generalised to accommodate the $3$D non-parabolic energy band structures in the topological Dirac/Weyl semimetals, such as cadmium arsenide (\ch{Cd3As2}), sodium bismuthide (\ch{Na3Bi}), tantalum arsenide (\ch{TaAs}) and tantalum phosphide (\ch{TaP}). Due to the unique Dirac cone band structure, an unusual dual-peak feature is observed in the total energy distribution (TED) spectrum.…
Electrostatic shielding is an important consideration for large area field emitters (LAFE) and results in a distribution of field enhancement factors even when the constituent emitters are identical. Ideally, the mean and variance together with the nature of the distribution should characterize a LAFE. In practice however, it is generally characterized by an effective field enhancement factor obtained from a linear fit to a Fowler-Nordheim plot of the $\text{I V}$ data. An alternate characterization is proposed here based on the observation that for a dense packing of emitters, shielding is la…
We develop analytical models of optical-field electron tunneling from the edge and surface of two-dimensional (2D) materials, including the effects of reduced dimensionality, non-parabolic energy dispersion, band anisotropy, quasi-time dependent tunneling and emission dynamics indueced by the laser field. We discover a universal scaling between the tunneling current density $J$ and the laser electric field $F$: In($J/|F|^β)\propto1/|F|$ with $β= 3 / 2$ in the edge emission and $β= 1$ in the vertical surface emission, which both are distinctive from the traditional Fowler-Nordheim (FN) model of…
Measured field electron emission (FE) current-voltage Im(Vm) data are traditionally analysed via Fowler-Nordheim (FN) plots, as ln{Im/(Vm)**2} vs 1/Vm. These have been used since 1929, because in 1928 FN predicted they would be linear. In the 1950s, a mistake in FN's thinking was found. Corrected theory by Murphy and Good (MG) made theoretical FN plots slightly curved. This causes difficulties when attempting to extract precise values of emission characterization parameters from straight lines fitted to experimental FN plots. Improved mathematical understanding, from 2006 onwards, has now enab…
In field electron emission (FE) studies, it is important to check and analyse the quality and validity of results experimentally obtained from samples, using suitably plotted current-voltage [I(V)] measurements. For the traditional plotting method, the Fowler-Nordheim (FN) plot, there exists a so-called "orthodoxy test" that can be applied to the FN plot, in order to check whether the FE device/system generating the results is "ideal". If it is not ideal, then emitter characterization parameters deduced from the FN plot are likely to be spurious. A new form of FE I(V) data plot, the so-called …
Measurements of current-voltage characteristics from ideal carbon nanotube (CNT) field electron emitters of small apex radius have shown that these emitters can exhibit a linear Fowler-Nordheim (FN) plot [e.g., Dean and Chalamala, Appl. Phys. Lett., 76, 375, 2000]. From such a plot, a constant (voltage-independent) characteristic field enhancement factor (FEF) can be deduced. Over fifteen years later, this experimental result has not yet been convincingly retrieved from first-principles electronic structure calculations, or more generally from quantum mechanics (QM). On the contrary, several Q…
This article proposes that we should think differently about predicting and interpreting measured field electron emission (FE) current-voltage [I(V)] characteristics. It is commonly assumed that I(V) data interpretation is a problem in emission physics and related electrostatics. Many experimentalists then apply Fowler-Nordheim plot methodology, developed in 1929. However, with modern emitting materials this 90-year-old interpretation methodology often fails (maybe in nearly 50% of cases), and yields spurious values for characterization parameters, particularly field enhancement factors. This …
This paper presents a new methodology for making comparisons between the theory of field electron emission (FE) and experiment, and is intended as a "demonstration of concept". This methodology is based on the value of the exponent kappa that describes the power to which voltage is raised in the pre-exponential of a mathematical equation that describes (for an electronically ideal FE system) the dependence of measured emission current on measured voltage. The aim is to use experimental exponent-values kappa^expt in an attempt to decide between two alternative FE theories, for both of which all…
Experimental Fowler-Nordheim plots taken from orthodoxly behaving carbon nanotube (CNT) field electron emitters are known to be linear. This shows that, for such emitters, there exists a characteristic field enhancement factor (FEF) that is constant for a range of applied voltages and applied macroscopic fields $F_\text{M}$. A constant FEF of this kind can be evaluated for classical CNT emitter models by finite-element and other methods, but (apparently contrary to experiment) several past quantum-mechanical (QM) CNT calculations find FEF-values that vary with $F_\text{M}$. A common feature of…
