Fowler-Nordheim Theory Research Articles

Discharge characteristics of the planar microscale gap electrodes with various geometry structures in the atmosphere environment

The discharge of microscale gaps for MEMS systems is a serious issue. To reveal the dark discharge characteristics of the microscale gap, electrodes with various gap feature sizes were prepared, and the DC current–voltage characteristics for the electrodes were implemented in atmospheric air. The results showed minor differences in dark discharge current on a nanoampere scale among electrodes with varying gap sizes of micro-protrusions. The current formation mechanism was consistent with the electron avalanche theory. Morphological characterization identified the presence of non-ideal thin-layer trailings at the electrode edges. To evaluate the effect of trailings on the dark discharge characteristics of the microscale gap structures, the electrodes with varying trailing lengths were fabricated. It showed that the current increased exponentially in the presence of long thin-layer trailing. In this case, the breakdown is most pronounced with an effective gap distance of 30 μm and a trailing length of 14.34 μm for the electrodes. For this structure, the dark discharge current was 198 nA at 36 V bias voltage, while the discharge current increased to 50 µA when the bias was increased to 37 V, and stabilized at 100 µA when the bias voltage was higher than 39 V. The rapid increase in current caused by trailing originates from the field emission as judged by the Fowler-Nordheim theory. Simulation results showed that the field strength can reach up to 3.76 × 109 V/m at a thin layer trailing length of 3.5 μm and a bias voltage of 200 V, which matches the conditions for the field emission. By calculating the effective secondary electron emission coefficient, it was found that thin-layer trailing significantly intensifies the ion-enhanced effect, further lowering the threshold for field emission and resulting in a rapid increase in microscale gap current at lower voltages.

High performance field emission cathode based on the diamond nanowires prepared by nanocrystalline diamond films annealed in air

This paper reports the field emission (FE) characteristics of a diamond nanowires (DNWs) array. The nanocrystalline diamond (NCD) film was deposited on silicon by microwave plasma chemical vapor deposition (MPCVD) and then annealed in air forming DNWs and hydrogenated at last. A high-field flat-plate emission test structure with a 1.03 μm gap between anode and cathode was prepared and the electrical properties proved it feasible. The FE performance of DNWs array was measured in a vacuum test system and that of NCDs film as a comparison. Finally, their FE parameters were analyzed and extracted based on the Fowler–Nordheim (F-N) theory. The results show that transforming NCDs film into DNWs array can improve the FE characteristics greatly. The turn-on field is as low as 1.36 V μm−1 dropping by one order of magnitude, while the field enhancement factor and FE current density are up to 156.68 and 484.75 mA cm−2 respectively rising both by two orders of magnitude. This excellent FE performance stems from the characteristics of large aspect ratio, very small tip radius and high density of DNWs.

Characteristics of Field-Emission-Induced Breakdown With Conical Microprotrusion

Micro-protrusions on plane electrode surface of vacuum interrupter is the main cause of field emission induced breakdown. The micro-protrusions are difficult to observe because of the ceramic insulations and floating shield, and they will be destroyed by breakdown events. This paper aims to reveal the characteristics of field emission induced breakdown with conical micro-protrusion. The conical micro-protrusions are reconstructed based on Fowler-Nordheim theory with field emission current and applied voltage from breakdown experiment. The conical micro-protrusions temperature characteristics are obtained taking joule heating effect into consideration, which could help to confirm the validity of the reconstructed micro-protrusions. The relationship between conical micro-protrusions geometrical parameters and field emission characteristics are analyzed. The results would clarify effects of conical micro-protrusion on plane electrode surface on field emission induced breakdown.

Generalizing Fowler–Nordheim Tunneling Theory for an Arbitrary Power Law Barrier

Herein, the canonical Fowler–Nordheim theory is extended by computing the zero‐temperature transmission probability for the more general case of a barrier described by a fractional power law. An exact analytical formula is derived, written in terms of Gauss hypergeometric functions, that fully capture the transmission probability for this generalized problem, including screened interaction with the image potential. First, the quality of approximation against the so far most advanced formulation of Fowler–Nordheim, where the transmission is given in terms of elliptic integrals, is benchmarked. In the following, as the barrier is given by a power law, in detail, the dependence of the transmission probability on the exponent of the power law is analyzed. The formalism is compared with results of numerical calculations and its possible experimental relevance is discussed. Finally, it is discussed how the presented solution can be linked in some specific cases with an exact quantum‐mechanical solution of the quantum well problem.

Breakdown of plane-wave-based theories of field emission from a subnanometer-radius tip

We present a combined experimental and theoretical study of field electron emission from an ultrasharp carbon tip with a subnanometer radius formed by a field-emission-induced structural modification of a diamond needle-like microcrystallite. The tip dimensions were measured by high-resolution transmission electron microscopy and used to calculate the potential distribution outside the emitter. Substitution of the calculated field into the Fowler-Nordheim (FN) theory and the more refined theory based on the Simmons equation overestimate the field emission current by six and five orders, respectively. We show that such a strong discrepancy is due to the incorrect use of plane wavefunctions to describe electrons inside a subnanometer emitter. We present a theory based on more realistic non-planar wavefunctions, which takes into account the effect of localization and one-dimensional density of states inside the emitter. These modifications of the FN theory allowed us to obtain a good order-of-magnitude agreement with the experimental values of the field emission current. The presented results are of great practical importance, since field emitters with nanometer radius are intensively studied experimentally and are the basis of state-of-the-art vacuum nanoelectronic devices.

Gap distance dependence on field emission at the nanogap between silicon cleavage surfaces

In this study, we developed a model of field emission at a vacuum nanogap composed of semiconductor electrodes and conducted numerical calculations to evaluate its field emission characteristics. The current at the silicon nanogap was also measured, and the results were compared with the calculated results. Large-area vacuum nanogaps are expected to be used for highly efficient room-temperature thermionic power generation owing to their good thermal insulation, and their electrical conduction of tunneling current is sufficient for practical use. Because vacuum nanogaps should be parallel and smooth with a large area, we proposed and examined the use of cleavage to fabricate nanogaps in single-crystal materials. A nanogap with a nominal area of 26.5 μm2 was fabricated in a microelectromechanical device to evaluate its properties, and the model was used for the analysis. The calculated current–voltage characteristics of the semiconductor nanogap of both n-type and p-type silicon were affected by the bandgap and the presence of carriers, i.e., electrons and holes. Moreover, the Fowler–Nordheim plot was confirmed to be valid for semiconductor nanogaps based on the calculated results. We proposed an effective work function that compensates for the discrepancy between semiconductor nanogaps and the Fowler–Nordheim theory and enables a more accurate analysis. Field-emission currents flowing through a 1.1 nm nanogap were measured and matched with the calculated results by adjusting the field enhancement factor and emission area. The model developed and confirmed experimentally in this work will be helpful in examining field emissions at semiconductor nanogaps.

Strong field driven extreme nonlinear photoemission from individual single-walled carbon nanotubes

Photoemission from nanostructures under strong fields has attracted significant interests in materials science and optoelectronic applications. Here, we report the extreme nonlinear photoemission in single-walled carbon nanotubes (SWCNTs) and its underlying mechanisms related to the microscopic carrier excitation and the sequential emission of occupied-state electrons. We show that the characteristic photoemission responses, e.g., nonlinear slope, of individual SWCNTs strongly depend on the unique electronic structures near the Fermi level, i.e., localization of the emitting states and their tunneling probabilities. The transition of photoemission mechanisms giving rise to the extreme nonlinear behavior was elucidated using Kohn-Sham potentials and the extended Fowler-Nordheim theory. This paper provides insights into photoemission dynamics of SWCNTs on the ultimate atomic length scale and attosecond time scale, will hopefully help design materials for tunable optoelectronic responses.

Quantitative Analysis on the Field Strength in the Addressable Gated ZnO Nanowire Field Emitter Arrays: Model and Experiment

Addressable gated field emitter arrays (FEAs) have important applications in vacuum microelectronic devices. To fabricate high-performance device, a comprehensive understanding on the field emission characteristics of gated FEAs is necessary, which requires a quantitative analysis method. In this work, a general model based on Fowler-Nordheim (FN) theory has been established to fulfill this blank. It is found that nonlinear FN plot with positive and negative slopes could occur in gated FEAs, which its turning point is related to the proportion of anode and gate field. This provides a way for obtaining the field strengths applied by the anode and gate structures. Besides, the transconductance of the gated FEAs increases exponentially with the total surface field, which indicates that both the anode and gate field need to be increased for achieving a high transconductance device. As a demonstration, the addressable gated ZnO nanowire FEAs using multimicrosize pattern with radius as small as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2.5 ~\mu \text{m}$ </tex-math></inline-formula> have been designed and fabricated. The relationship between the transconductance and anode voltage can be well fitted by using the model, which deduces the proportion of anode and gate field in the device to be about 3:1.

Copper Nanowire Arrays: Growth and Properties

Cu nanowire arrays of three different diameters have been synthesized via template assisted electrode position technique. Morphological, structural, optical, electrical and field emission properties were examined for these three nanowire arrays. Morphological study reveals the nanowire arrays are of uniform diameter throughout the length and of desired dimensions. Structural study shows the face centred cubic structure of nanowires. The crystallite size has been calculated using the Debye-Scherrer relation and micro strain has been calculated using Williamson-Hall analysis. The surface plasmon resonance absorption peak shifts towards red end with increase in diameter of nanowires. I-V characteristics show the ohmic behaviour. Electron field emission properties of Cu nanowire arrays have been studied using Fowler Nordheim theory. The field emission parameters like maximum emission current density, turn on field and field enhancement factor have been calculated for nanowire arrays of different diameters. Cu nanowire arrays of diameter 50 nm exhibit better field emission properties, indicating that thin nanowire arrays have great potential to be used as field emitters.

Strongly anisotropic field emission from highly aligned carbon nanotube films

The field electron emission properties of carbon nanotube (CNT) films composed of densely packed and highly aligned CNTs were investigated. The CNT films were produced by a continuous film casting process and are spooled into long lengths with the CNTs aligned lengthwise in the film. The anisotropic nature of the CNT film morphology was confirmed by performing specific conductivity measurements in directions both parallel and perpendicular to the aligned CNT microstructure. Field emission experiments were performed on 5 and 10 mm wide films that were mechanically cut into small samples and then vertically mounted so that the emission occurred from the film edge. The films were mounted with the aligned CNT microstructure oriented either parallel or perpendicular to the direction of the applied electric field. The highest emission currents were produced by films mounted in the parallel alignment configuration. Additional experiments were performed on films that were folded, which eliminated surface irregularities at the film edge due to the cutting process. SEM imaging performed at the ridge of the folded film before and after field emission (FE) experiments showed that films mounted in the parallel alignment configuration had minimal surface damage after FE, while films mounted in the perpendicular alignment configuration showed substantial damage. The effective emission area and field enhancement factor were extracted from the FE data using the orthodox Fowler–Nordheim theory. Folded CNT film cathodes mounted in the parallel alignment configuration produced the highest emission currents, while demonstrating a larger emission area and lower field enhancement factor.

The Parameters of the Field Emission Model and the Fabrication of Zinc Oxide Nanorod Arrays/Graphene Film

A large-scale growth of zinc oxide (ZnO) nanorod arrays on graphene sheets was fabricated by a hydrothermal technique, and the Fowler–Nordheim theory was used to build a model to describe the properties of the arrays’ field emission. The results indicated that the morphological characteristics of the ZnO nanorods grown on the graphene sheets can be easily tuned by varying the reaction time and concentrations of the reaction solution. The regular ordered ZnO nanorods arrays on the graphene sheets were obtained under the appropriate experimental conditions. Further, this composite cathode was demonstrated to possess excellent field emission properties due to the outstanding mechanical and electrical properties of graphene. The field emission current density of the composite cathode reached 1,448 μA cm–2 at the electric field of 16.5 V μm–1. The key parameter, field enhancement factor, reached 6,366, while the pure graphene cathode field is about 1,660. These specific nanorod arrays with enhancement of the field emission properties would be useful to sensor or modulator units for accessing networks.

Effects of Antimony Deposition on Field-Emission Current Density of Ge/Si

Abstract To estimate the field-emission current density of a Ge/Si heterosystem, 20-nm germanium/silicon (100) samples were grown by molecular beam epitaxy. The surface of one sample was covered with a layer of antimony, which was removed in vacuum prior to the samples being measured. A second sample of Ge/Si was exposed to room air in the absence of antimony. The current–voltage characteristics of both samples obtained by scanning tunneling microscopy (STM) were discovered to be in agreement with classical Fowler–Nordheim theory. The density of emission current from Ge nanocrystal exceeds the density of emission current from the wetting layer of Ge/Si. The density of emission current of pure Ge nanocrystal is less than the density of emission current of Ge nanocrystal with adsorption layers.

Design and optimization of a nano-antenna hybrid structure for solar energy harvesting application

A novel hybrid structure with high responsivity and efficiency is proposed based on an L-shaped frame nano-antenna (LSFNA) array for solar energy harvesting application. So, two types of LSFNAs are designed and optimized to enhance the harvesting characteristics of traditional simple electric dipole nano-antenna (SEDNA). The LSFNA geometrical dimensions are optimized to have the best values for the required input impedance at three resonance wavelengths of λ res = 10 μm, 15 μm, and 20 μm. Then the LSFNAs with three different sizes are modeled like a planar spiral-shaped array (PSSA). Also, a fractal bowtie nano-antenna is connected with the PSSA in the array gap. This proposed hybrid structure consists of two main elements: (I) Three different sizes of the LSFNAs with two different material types are designed based on the thin-film metal–insulator–metal diodes that are a proper method for infrared energy harvesting. (II) The PSSA gap is designed based on the electron field emission proposed by the Fowler–Nordheim theory for the array rectification. Finally, the proposed device is analyzed. The results show that the PSSA not only has an averaged 3-time enhancement in the harvesting characteristics (such as return loss, harvesting efficiency, etc.) than the previously proposed structures but also is a multi-resonance wide-band device. Furthermore, the proposed antenna takes up less space in the electronic circuit and has an easy implementation process.

Extreme lightwave electron field emission from a nanotip

We report on sub-cycle terahertz light-field emission of electrons from tungsten nanotips under extreme conditions corresponding to a Keldysh parameter $\gamma_K\approx10^{-4}$. Local peak THz fields up to 40~GV/m are achieved at the apex of an illuminated nanotip, causing sub-cycle cold-field electron emission and acceleration in the quasi-static field. By simultaneous measurement of the electron bunch charge and energy distribution, we perform a quantitative test of quasi-static Fowler-Nordheim tunnelling theory under field conditions that completely suppress the tunnel barrier. Very high bunch charges of $\sim10^6$ electrons/pulse are observed, reaching maximum energies of 3.5~keV after acceleration in the local field. The energy distribution and emission current show good agreement with Fowler-Nordheim theory even in this extreme field regime. Extending this model to the single-shot regime under these conditions predicts peak electron distributions with a spectral purity of $10^{-4}$. THz field-induced reshaping and sharpening of the nanotip is observed, reducing the tip radius from 120~nm to 35~nm over roughly $10^9$ THz shots. These results indicate THz-driven nanotips in the extreme field limit are promising electron sources for ultrafast electron diffraction and microscopy.

Characterization of InSb nanopillars for field emission applications

A piezoelectrically driven metallic nanoprobe is installed inside a scanning electron microscope to perform local characterization of the field emission properties of InSb nanopillars. The tip-shaped anode can be precisely positioned at sub-micron distances from the emitters to collect electrons from areas as small as 1μm2 under the application of an external bias up to 100 V. Current-voltage characteristics are measured for cathode-anode separation down to 500 nm and are analyzed in the framework of the Fowler-Nordheim theory. We give estimation of performance parameters such as the field enhancement factor and the turn-on field and their dependence on the cathode-anode separation distance. We demonstrate the time stability of the emitted current for several minutes. Finally, we perform a finite element electrostatic simulation to calculate the electric field in proximity of the nanopillars and we evaluate the effective emitting area as well as the screening effect due to presence of other pillars in close vicinity. We show that InSb nanopillars are very stable emitters that allow current density as high as 104 A/cm2 and excellent time stability, crucial characteristics to envisage device exploitation.

An infrared energy harvesting device using planar cross bowtie nanoantenna arrays and diode-less rectification based on electron field emission

ABSTRACT In this paper, we present a rectification process through electron emission from sharp tips under a high electric field of 109–1010 V/m based on the Fowler–Nordheim theory. A large planar array of 104–105 nanoantennas is proposed to achieve the required electric field for electron emission in the nanoantenna gap. Spatial coherence of the incident wave is a prohibitive issue in using large antenna arrays, and it is overcome by choosing proper size for the array. The design starts with a single-element nanoantenna, for which, five different types are considered, namely dipole, spiral, cross bowtie, sinuses and square spiral. Full-wave simulations show that cross bowtie has the highest field enhancement. Moreover, two array configurations, i.e. series and parallel feed networks, are considered and their field enhancement are compared. Moreover, field enhancement in large array configurations is predicted using proposed equations based on curve fitting.

Iron oxide-coated MWCNTs nanohybrid field emitters: a potential cold cathode for next-generation electron sources

In this paper, we have reported that the field emission properties of multiwall carbon nanotubes (MWCNTs) were significantly increased by decorating their surface by iron oxide nanoparticles. MWCNTs were prepared on silicon substrate through low-pressure chemical vapour deposition using acetylene as source gas. The iron oxide nanoparticles were grown on the surface of MWCNTs by thermal evaporation technique. Modified surface morphologies of the prepared films were characterized through field emission scanning electron microscope (FESEM), Raman spectroscopy and X-ray diffraction. A significant change in current density, stability and turn-on voltage has been observed in iron oxide-coated MWCNT films. Decoration of iron oxide nanoparticles reduces turn-on voltage from 4 to 3.4 V/μm, while current density increases from 10.15 to 33.26 mA/cm2. For practical MWCNT-based field emission devices, it is necessary to improve the emission current density and stability. The excellent field emission parameters are obtained and calculated which make them useful for high-performance field emission-based devices. The mechanism of emission of electrons is well described by the Fowler Nordheim theory.

Tower-Like ZnO Nanorod Bundles Grown on Freestanding Diamond Wafers for Electron Field Emission Improvement

Tower-like ZnO nanorod (ZNR) bundles were grown on freestanding diamond (FSD) wafers by a low-temperature hydrothermal method without any catalysts, and the corresponding electron field emission property was investigated. The structural characterizations disclosed that these tower-like bundles were composed of single crystalline ZNRs with hexagonal wurtzite structure. They grew perpendicularly to each crystal planes of the diamond grains. Different hydrothermal reaction times (e.g., 1.5, 3, 6, 8 h) were carried out to investigate the morphology features of ZNRs grown on the growth surface of FSD wafers. The ZNR bundles with a growth time of 6 h showed the highest crystallinity and aspect ratio. Such ZNR bundles/FSD hybrid exhibited a high emission current density of 0.2 mA at an applied field of 11.8 V/μm and a reduced turn-on field of 6.8 V/μm. The field enhancement factor β was calculated to be ~4979 based on the Fowler-Nordheim theory. The geometry and structure of polycrystalline diamond were responsible for the formation of ZNR bundles and the high β-value.

Field emission from AlGaN nanowires with low turn-on field

We fabricate AlGaN nanowires by molecular beam epitaxy and we investigate their field emission properties by means of an experimental setup using nano-manipulated tungsten tips as electrodes, inside a scanning electron microscope. The tip-shaped anode gives access to local properties, and allows collecting electrons emitted from areas as small as 1 µm2. The field emission characteristics are analysed in the framework of Fowler–Nordheim theory and we find a field enhancement factor as high as β = 556 and a minimum turn-on field = 17 V µm−1 for a cathode–anode separation distance = 500 nm. We show that for increasing separation distance, increases up to about 35 V µm−1 and β decreases to ∼100 at = 1600 nm. We also demonstrate the time stability of the field emission current from AlGaN nanowires for several minutes. Finally, we explain the observation of modified slope of the Fowler–Nordheim plots at low fields in terms of non-homogeneous field enhancement factors due to the presence of protruding emitters.

Electric field emission characteristics of a triode-structured metallic edge emitter from geometric conditions

Spindt-tip type cold electron emission has potential in various applications; however, it has a limited current capacity that prevents the emitters from being used for higher current sources, and the fabrication of the emitters is not easy. In this study, we investigate the electric field emission characteristics of a triode-structured metallic edge emitter from geometric conditions. We perform a systematic simulation based on Fowler-Nordheim theory to analyze the feasibility of using the proposed emitter. The dependences of the emission current, electron transmittance, and region of the triode mode on specific geometrical parameters, including the anode-gate vertical distance, gate-cathode vertical distance, gate-cathode horizontal distance, anode voltage, and gate voltage, are analyzed. The results show that the electric field for a narrow gate-cathode vertical distance is superior to the electric field of the gate electrode for a narrow gate-cathode horizontal distance, while the electric field for a wide gate-cathode vertical gap is more dominant than the electric field of the anode electrode for a wide gate-cathode horizontal gap. Moreover, the electron transmittance gradually increases with a low gate voltage, wide anode-gate vertical distance, and high anode voltage, and the region of the triode mode also increases with a high gate voltage, wide anode-gate vertical distance, and low anode voltage. Consequently, our results appear to be effective for an optimization in an experiment and are expected to improve present electric field emission techniques.