With the continual miniaturization of electronic devices, there is an urgent need to understand the electron emission and the mechanism of electrical breakdown at nanoscale. For a nanogap, the complete process of the electrical breakdown includes the nano-protrusion growth, electron emission and thermal runaway of the nano-protrusion, and plasma formation. This review summarizes recent theories, experiments, and advanced atomistic simulation related to this breakdown process. First, the electron emission mechanisms in nanogaps and their transitions between different mechanisms are emphatically discussed, such as the effects of image potential (of different electrode's configurations), anode screening, electron space-charge potential, and electron exchange-correlation potential. The corresponding experimental results on electron emission and electrical breakdown are discussed for fixed nanogaps on substrate and adjustable nanogaps, including space-charge effects, electrode deformation, and electrical breakdown characteristics. Advanced atomistic simulations about the nano-protrusion growth and the nanoelectrode or nano-protrusion thermal runaway under high electric field are discussed. Finally, we conclude and outline the key challenges for and perspectives on future theoretical, experimental, and atomistic simulation studies of nanoscale electrical breakdown processes.

The physical mechanisms of electrical breakdown are vitally important for the performance and lifetime of electronic devices and electrical equipment. The continuing trend of miniaturizing electronic devices and electromechanical devices causes the electrical insulation of microelectromechanical systems (MEMS), nanoelectromechanical systems (NEMS), vacuum micro/nano electronic devices, and molecular devices to encounter more complicated and severe operating environments, i.e., ultrahigh electromagnetic field and local high temperature in micro- or nanogaps.1–5 Conversely, the microplasmas generated by the electrical breakdown may be used in various applications such as electric propulsion,6 nanomaterial synthesis,7 and medical treatment.8 Thus, understanding the electrical breakdown mechanisms at microscale and nanoscale is essential for the proper design and operation of such applications.

The breakdown phenomenon in the gas environment is a transition process of the gaseous gap from the insulative state to the conductive state, leading to the insulation failure and equipment fault. The classical theory of characterizing gas breakdown is Townsend avalanche and the Paschen's law (PL), which provides the breakdown voltage as a function of gas pressure p and gap distance d in the operating regime pd < 200 Torr cm. However, as the gap spacing decreases to less than ∼5 μm, field emission (FE) will dominate the breakdown over the Townsend avalanche,9–18 where the collision ionization can barely occur in such a small gap due to the relatively large electron mean free path λ (∼500 nm) in ambient air, implying that gas breakdown in nanogaps resembles vacuum breakdown.11,19–23 Instead of exhibiting the “u-shape” with pd characteristic of PL, breakdown in the FE-driven regime instead decreases linearly with decreasing gap distance, as demonstrated theoretically24 and experimentally.25 

In general, vacuum breakdown is initiated by thermal instability at the cathode or anode due to the electron emission current, which heats the protrusion itself or transfers kinetic energy to the anode, which will cause the metal atom evaporation.26–31 Due to the evaporated atoms, the gap pressure increases and the mean free path of electrons may become less than the gap distance, where the avalanche ionization will occur and cause the plasma discharge.32 Thus, the dynamics of electron emission and the electrode's properties will determine the dielectric strength and insulation performance of the gap.33,34 However, the electron emission process and the breakdown mechanism at nanoscale are more complicated, and the traditional electron emission theories developed decades ago may not be accurate, where the quantum effects should be considered.35 The electrode surface effects at nanoscale play a more important effect because of the comparable size of gap and surface roughness,36 and even direct field ionization rather than collision ionization may dominate the ionization process for plasma discharge.37,38

It is of interest to understand the dynamics of electron emission and electrical breakdown process in a nanogap. First, when a voltage is applied on the nanogap, the electrons from the electrode may undergo tunneling through the potential barrier under a high electric field, which is commonly described by the Fowler–Nordheim (FN) equation. However, the traditional FN equation could fail to accurately describe this electron emission process for the nanogap and nanoelectrodes, where the image-charge potential (including anode screening), exchange-correlation effects, and space-charge Coulomb interaction in the nanogap must be considered35,39 and modified for nanoelectrodes.40 Qualitatively, electron emission at nanoscale follows three regimes: direct tunneling, field emission, and space-charge limited.41–43 During the breakdown process, electron emission may transit from pure field emission to thermal assisted field emission due to Joule heating and the Nottingham effect.44,45 Second, the surface effects of the electrodes are expected to play a more dominant role in the breakdown characteristics in a nanogap because of the comparable nanometer size between gap spacing and surface roughness36,46 and also demonstrated for microscale gaps at atmospheric pressure when crater formation in the cathode changed the effective gap distance.47 In addition to initial nano-protrusions due to surface roughness, the size of the nano-protrusions can become larger (under high electric field) due to field-induced biased surface atom diffusion48–53 or plastic deformation due to Maxwell stress.54–60 During the breakdown process, the electron emission from the nano-protrusion can produce more Joule heating and Nottingham effect at nanoscale, where the size of these nano-protrusion is comparable to the electron mean free path, and the resistivity of the nano-protrusions increases due to the electron scattering at surfaces and grain boundaries.61 The enhanced heating would partially melt the nano-protrusion, and the field-induced force could further elongate it and enhance its sharpness. This would provide positive feedback to the thermal runaway process, causing the final evaporation of large fractions of the nano-protrusion, which become the neutral atoms for plasma discharge.62,63 The entire breakdown process at nanoscale is illustrated in Fig. 1. Hence, new insight of electron emission and electrical breakdown at nanoscale must be developed, such as the distinctive electron emission/transport process, significant surface effects, nano-protrusions or nanoelectrodes deformation under high electric field and high temperature, and the strong coupling between electron emission, local overheating, and the morphology deformation of the nano-protrusions or nanoelectrodes.

FIG. 1.

The illustration of the breakdown process at nanoscale.

FIG. 1.

The illustration of the breakdown process at nanoscale.

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This review highlights the recent studies on the theoretical modeling, experimental measurements, and atomistic simulation on the electron emission and electrical breakdown at nanoscale. Section II summarizes the fundamental theories of electron emission under high electric field, such as Fowler–Nordheim (FN) equation, Child–Langmuir (CL) law, and their modern revisions, and emphatically discusses the theoretical progress at nanoscale. Section III describes the experimental findings in electron emission and electrical breakdown at nanoscale with an emphasis on the electrode morphology evolution, space-charge effects, and electrical breakdown characteristics. Section IV describes the research progress in atomistic simulation of the surface morphology evolution of metal nanoelectrodes or nano-protrusions under high electric field. Finally, concluding remarks and future works are given in Sec. V.

Electron emission models can be classified as thermionic emission (TE), photoemission, field emission (FE), and space-charge limited (SCL) emission.64,65 Among them, the field electron emission is the primary mechanism to initiate electrical breakdown in nanogaps, which is described by the Fowler–Nordheim (FN) law at low current density and it is limited by the space-charge limited emission or the Child–Langmuir (CL) law at high current density. The FN equation describes the field emitting current density JFN from a metal–vacuum interface as a quantum mechanical tunneling process.66,67 The simplest one-dimensional (1D) FN equation for a planar emitting surface is
J F N = A F N E 0 2 Φ W F t 2 ( y ) exp [ B F N v ( y ) Φ W F 3 / 2 E 0 ] ,
(1)
where AFN = 1.54 × 10−6 A eV V−2 and BFN = 6.83 × 107 eV−3/2 V cm−1 are constants, t2(y) and v(y) are Nordheim parameters (where y = 3.79 × 10−4 E01/2WF), ФWF is the work function, and E0 is the applied electric field in V/cm.68,69

For sharp emitters, it is a common practice to assume that the surface electric field E0 is enhanced by a factor of β in Eq. (1): E = βE0 = βV/d. However, this may be not correct because of the drastically different tunneling barrier when the radii of emitter tips are less than 10 nm,70 where the deviation of the barrier shape and of the interaction between an electron and image charge near the tip apex must be considered.71–74 Biswas and Ramachandran found that for a hyperboloid surface, the image potential could be approximated by a sphere image potential.73 Kyritsakis et al. generalized the FN equation for emitters with radii less than 10 nm, where the barrier potential was revised by adding a second-order term or even the cubic term to the usual formula.40,75,76 Patterson and Akinwande derived the field emission equations by considering the quantum confinement (QC) effects near to the emitting surface due to nanoscale field emitters and explored the competing effects of a reduced electron supply due to quantum confinement and increased electron transmission probability from geometry-dependent local field enhancement as emitter dimensions decrease.77,78 Additionally, a fractional FN equation that considers field emission from a random and rough surface was derived analytically.79 

For a 1D planar gap of spacing D with an applied voltage Vg, the 1D Child–Langmuir (CL) law predicts the maximum transmitted space-charged limited current density JCL given by
J C L = 4 2 9 ε 0 e m V g 3 / 2 D 2 ,
(2)
where ε0 is the free space permittivity, and e and m are the charge and mass of the electron, respectively.80,81 Detailed reviews of the CL law can be found elsewhere.36,65 For a nanotip or rough electrode, the SCL electron emission has been studied.82–87 By assuming an unlimited source of electron emission, the emitter could supply sufficient electrons (even at low voltage) to drive the electric field at the cathode to be zero, which is consistent with the original derivation of the 1D CL law. Based on this condition, Ang et al. reported a revised scaling of JSCL ∝ Vg3/2/Dm for the space-charge limited current of a tip, where m (= 1.1–1.2) depends on the emission area and radius of the tip.88 Moreover, Singh et al. established an extension of CL law of JSCL ∝ γaVg3/2/D2 by considering the field enhancement factor γa of the curved emitter.89 Harsha and Garner presented an analytical solution for current density from a sharp tip in the tip-to-plate configuration by using the variational calculus methods90 and derived an exact solution for SCL current that is true for any multidimensional geometry as long as the equation for the electric potential in vacuum is known.91 For a rough cathode, a closed-form fractional-dimensional generalization of CL law is derived in classical regime, where the roughness of the cathode is modeled as a “slab of fractal dimension” with a parameter α (≤1) determined by the box-counting method for a given image of the cathode's roughness.92 The analytical CL scaling for a planar gap was also extended to cylindrical and spherical setting,93,94 including nonzero initial velocity.95,96

When the electric field is sufficiently strong, the emitted current density will be limited by the buildup of space-charge in the gap, leading to a transition from the FN equation to the CL law. One may derive equations from first principles to unify these mechanisms by starting from the electron force law in what is referred to as nexus theory, which provides important signposts for the conditions where one can use a standard electron emission equation (e.g., FN or CL) directly or one must use a more complicated exact equation.25 Lau et al. derived universal curves (true for any electrode material) unifying FN and CL.97 Forbes proposed a simple dimensionless equation to examine space-charge effects on field emission in parallel-plane geometry.98 Feng and Verboncoeur studied the response of the surface electric field and the injected current density for three different applied fields by using a 1D particle-in-cell (PIC) code and found that increasing the applied field enhanced the space-charge effect and may even contribute to the reverse of electric field because of a few damped oscillations before the equilibrium.99 Lin considered the formation of cathode plasma and surface properties within the framework of the effective work function approximation and investigated ions effects at anode on the steady state of electron emission by using a self-consistent approach. They found that negative space-charge from the electrons significantly reduced the surface electric field, while positive space-charge due to the upstream ion current could enhance the FE current on the contrary.100 

It is naturally of interest to study the electrical breakdown including various effects such as the geometrical properties of the electrode, heating of the electrode, and other different electron emission mechanisms.44,101–105 Feng et al. modified the FN-SCL theory with consideration of injection velocity and geometric effects by solving the energy conservation equation, FN equation, and Poisson's equation simultaneously and found that either initial velocity or geometric effects would make the actual transmitted current density closer to the space-charge limited current.106 Chen et al. investigated the space-charge effects of field emission in nanotriodes with different geometries and work functions by numerically solving the coupled FN equation and Poisson's equation and found that the field emitters with lower work function are likely to trigger the space-charge effects, and scaling the emitter radius and gate aperture of FE nanotriodes to smaller dimensions could overcome the space-charge effects.107 Zhu and Ang examined the space-charge affects for a single hyperbolic tip by iteratively solving the Poisson equation and particle trajectories and found that the classical CL law failed to predict the maximum current density from a sharp tip, and for a practical tip with nonuniform work function, the maximum current density may be not located at the apex.88 Kyritsakis et al. developed a three-dimensional theoretical model, describing the scaling laws of space-charge limited emission at high electric fields in the weakly space-charge regime, which can be applicable for any geometry by using a geometry-specific correction factor.108 There were some studies related to the space-charge effects in field emitter arrays.109,110 With the enhanced field emission, higher temperature of the emitters (beyond room temperature) due to Joule heating and Nottingham effect indicated that thermionic contribution cannot be ignored, and Jensen et al. proposed a general thermal field (GTF) equation that successfully bridges field emission and the thermionic emission (TE).44 TE is generally represented by the Richardson–Laue–Dushman (RLD) equation, elucidating the process wherein electrons acquire sufficient thermal energy to overcome potential barriers.111 TE dominates the electron emission at high temperature and low electric field, while FE dominates the electron emission at low temperature and high electric field, and the GTF model should be employed during the transition regimes, where the contributions of the FE and TE are comparable. This GTF model was later extended to include the geometric effects by using a modified potential barrier45 and space-charge.112 

When the gap distance decreases to a few nanometers, comparable to the electron de Broglie wavelength, quantum effects cannot be neglected for electron emission behavior. In 1963, Simmons studied tunneling in a nanogap by considering anode screening in terms of the image charge potential and the emission process from both metal electrodes, but this model assumed the classical image charge potential and neglected the quantum effects of the space-charge field and the electron exchange-correlation potential.113,114 Later, Lau et al. examined the effect of the space-charge on electron emission in nanogaps by using the mean field theory and solving the Poisson and Schrödinger equations self-consistently to show that the classical CL law could be exceeded by a large amount due to electron tunneling.115 Ang et al. further considered the exchange-correlation interaction and presented a consistent and exact 1D quantum CL law (JQCL) in a nanogap with a scaling of JQCL ∝ Vg1/2 and JQCL ∝ D4 (Refs. 116 and 117), which differs from the classical CL law of JCL ∝ Vg3/2 and JCL ∝ D2 from Eq. (2). Figures 2(a) and 2(b) show the scaling of the quantum CL with respect to D and Vg, respectively. Such quantum enhancement of CL law can also be realized in ultrafast short pulse regime.118 Reference 119 reviews the quantum CL law and its other properties, such as bipolar flow, quantum gap capacitance, and transit time.

FIG. 2.

(a) The normalized quantum CL law μQ as a function of the normalized gap spacing λ at different normalized gap voltage Φg = 10–2–102 (solid lines: top to bottom), Φg ≫ 1 (dashed line), and classical limit (dash-dotted line). (b) The value μQVg3/2 as a function of gap voltage Vg at various gap spacing D = 1, 10, and 100 nm (solid line: top to bottom). The dashed lines are results without exchange-correlation and the short-dashed line is the classical CL law. Reproduced with permission from Ang et al., Phys. Rev. Lett. 91, 208303 (2003). Copyright 2003 American Physical Society.116 

FIG. 2.

(a) The normalized quantum CL law μQ as a function of the normalized gap spacing λ at different normalized gap voltage Φg = 10–2–102 (solid lines: top to bottom), Φg ≫ 1 (dashed line), and classical limit (dash-dotted line). (b) The value μQVg3/2 as a function of gap voltage Vg at various gap spacing D = 1, 10, and 100 nm (solid line: top to bottom). The dashed lines are results without exchange-correlation and the short-dashed line is the classical CL law. Reproduced with permission from Ang et al., Phys. Rev. Lett. 91, 208303 (2003). Copyright 2003 American Physical Society.116 

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Similarly, the transitions between different electron emission mechanisms in a nanogap and other geometrical settings have been studied. Zhang et al. proposed a self-consistent model to characterize quantum tunneling current in similar and dissimilar metal–insulator–metal nanoscale junctions by including the effects of anode screening, space-charge, exchange-correlation potential, and current emission from the anode and solving the coupled Schrödinger and Poisson equations self-consistently and revealed the general scaling for quantum tunneling current and its dependence on the bias voltage, the gap distance, and material properties; they also found that the J-V curve spans three regimes: direct tunneling, field emission, and space-charge limited, as shown in Fig. 3.41–43 However, the image potential in this model is the classical image charge potential, which may underestimate the potential barrier. Alternatively, Koh and Ang adopted a modified Thomas–Fermi free electron model for computing image charge potential, which eliminates the singularities at the metal–vacuum interfaces, and presented a 1D modified Thomas–Fermi approximation (TFA) quantum model to predict the field emission behavior in a nanogap.35 Similarly, based on this method, Li et al. further studied the field electron emission characteristics with different nanogap spacing and electric field strength and divided them into four emission regimes including the direct tunneling regime (DTR), field emission regime (FER), quantum regime (QR), and space-charge limited regime (SCLR), which are shown in Fig. 4.39 Specifically, QR is characterized by the dominance of exchange-correlation potential in the total space-charge potential profile. The exchange-correlation potential is always negative, which can effectively reduce the electron tunneling potential height, leading to a higher emission current density than that without considering such a phenomenon in the classical CL law. For SCLR, the electron tunneling process is determined mainly by the space-charge Coulomb potential. The FER refers to the classic electron emission process that is reliably described by FN equation. For DTR, the space-charge density is small due to the low applied electric field value. In this regime, both space-charge Coulomb potential and exchange-correlation potential are negligible, and the overall electron tunneling energy profile is largely controlled by image charge potential. In addition, in terms of the temperature effect, Koh and Ang found that when the space-charge is not important, the thermionic emission at high temperature will increase the emission current at low voltage, whereas when the space-charge becomes significant, the emission current will converge in the space-charge-limited regime, which is independent of temperature.120 

FIG. 3.

Normalized (in terms of CL law) current density γ as a function of applied gap voltage Vg, for Au-vacuum-Au system separated by a 1 nm gap based on the self-consistent model. JV curves span three regimes: direct tunneling, field emission, and space-charge-limited. Reproduced with permission from Zhang and Lau, J. Plasma Phys. 82, 595820505 (2016). Copyright 2016 Cambridge University Press.43 

FIG. 3.

Normalized (in terms of CL law) current density γ as a function of applied gap voltage Vg, for Au-vacuum-Au system separated by a 1 nm gap based on the self-consistent model. JV curves span three regimes: direct tunneling, field emission, and space-charge-limited. Reproduced with permission from Zhang and Lau, J. Plasma Phys. 82, 595820505 (2016). Copyright 2016 Cambridge University Press.43 

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FIG. 4.

Four different electron emission regimes QR (quantum regime), SCLR (space-charge limited regime), DTR (direct tunneling regime), and FER (field emission regime) are divided when D = 1–10 nm and F = 1–100 V/nm for the quantum model. Reproduced with permission from Li et al., Front. Phys. 11, 1223704 (2023). Copyright 2023 authors, licensed under a Creative Commons Attribution (CC BY) 4.0 International License.39 

FIG. 4.

Four different electron emission regimes QR (quantum regime), SCLR (space-charge limited regime), DTR (direct tunneling regime), and FER (field emission regime) are divided when D = 1–10 nm and F = 1–100 V/nm for the quantum model. Reproduced with permission from Li et al., Front. Phys. 11, 1223704 (2023). Copyright 2023 authors, licensed under a Creative Commons Attribution (CC BY) 4.0 International License.39 

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Further studies are needed to examine the Wentzel–Kramers–Brillouin (WKB) approximation, the effects of electrode geometry, and the effect of rough surface.43 It may be worthwhile to note that the transition from field emission to SCL emission vanishes when the field emission from 2D materials becomes non-FN type emission predicted in a recent paper121 for a nanogap. In addition, the connection between the electron emission theories and the practical electron emission and breakdown behavior needs to be bridged, although it is difficult due to the complicated environment during breakdown. For example, the inevitable nano-protrusions and surface contamination on the electrode surface induce a non-uniform work function that leads to non-uniform electron emission and space-charge density.88 Nanoelectrode deformation during electron emission also complicates the breakdown process,122–124 motivating some recent experiments (see Sec. III) at nanoscale to compare with the relevant theories.

Due to the restriction of observation, fabrication, and displacement capabilities, experimental investigation of electron emission and electrical breakdown across nanogaps is a great challenge compared to macroscale studies, requiring new experimental techniques to explore the underlying principles. To establish a nanogap, two typical methods are employed in the published works with the corresponding setup diagrams, as shown in Fig. 5. The first method is to fabricate a fixed nanogap on a substrate with representative methods including mechanical break junctions,125 electromigration,126 electron beam lithography,127–129 electron beam annealing,20 electrochemical narrowing,130 and focused ion beam (FIB) technology;131 however, in such cases, it is difficult to control the electrode structure effects and precise electrode spacing and it is critical to account for the effects of the substrate. Alternatively, one can set up an adjustable gap between electrodes using a nanomanipulator in an electron microscope, such as the scanning electron microscope (SEM) and the transmission electron microscope (TEM).132 For these adjustable gaps, the electrodes are mounted onto a nanomanipulator and a sample stage. With the aid of the nanomanipulator, the electrodes could be adjusted in precise increments and SEM could be used to determine the gap distance, which could be as small as 10 nm. Hirata,133 Peschot,23 Ziemann,134–136 and Meng137–144 have studied nanogap breakdown using this technique.

FIG. 5.

Diagrams showing the setup for (a) fixed and (b) adjustable nanogap configurations.

FIG. 5.

Diagrams showing the setup for (a) fixed and (b) adjustable nanogap configurations.

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The surface condition (adsorbed gases,145,146 water vapor,146–148 oxide layer,149 etc.) and geometry of electrodes used in the experiment will have a great influence on the nanogap electron emission and electrical breakdown, which must be well controlled for more accurate results. For the first method, Bhattacharya et al. found that the water vapor present in the gas is the major reason for the increase in the work function, the degradation of the field emission current, and the increase in the leakage current and proposed that the ultraviolet (UV) light cleaning can desorb the water vapor, restore the field emission current and decrease the leakage current.146–148 For the second method, Meng et al. established an electrode fabrication technique, which could obtain a metal nanotip by using FIB milling and a hemisphere electrode by combining electrochemical etch and Joule melting.142  Figure 6 shows the as-fabricated molybdenum needle electrode with a tip radius of 15 nm and the as-fabricated hemisphere molybdenum electrode with a radius of 20 μm. Generally, there are two main advantages to this method. One is that the last process in the fabrication is conducted inside the high vacuum environment, so the metal electrode will not be oxidized during the test, which avoids the influence of oxide layers. The other advantage is that the fabrication process is controllable and visible, so the prepared tip and hemisphere are both in regular shapes.

FIG. 6.

(a) The as-fabricated needle electrode with a tip radius of 15 nm. (b) The as-fabricated hemisphere electrode with a radius of 20 μm. Reproduced with permission from Meng et al., 28th International Symposium on Discharges and Electrical Insulation in Vacuum (2018), Vol. 1, pp. 15–18. Copyright 2018 IEEE.137 

FIG. 6.

(a) The as-fabricated needle electrode with a tip radius of 15 nm. (b) The as-fabricated hemisphere electrode with a radius of 20 μm. Reproduced with permission from Meng et al., 28th International Symposium on Discharges and Electrical Insulation in Vacuum (2018), Vol. 1, pp. 15–18. Copyright 2018 IEEE.137 

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Fixed nanogaps on a substrate are always relevant to the nanoscale photonic and electronic devices.150 Interestingly, reducing the gap distance to nanoscale provides a solution for applying field emission devices in integrated circuits, where the advantages of both vacuum and solid state devices can be achieved together, such as ballistic transport in a nanogap, stability at harsh conditions, high frequency/power output, and circuit integration,21,22 further motivating electron emission and breakdown at nanoscale. So far, many studies have examined field emission to improve device performance;68,151,152 however, fewer studies have assessed breakdown characteristics. In this review, we will focus on electrode deformation, space-charge effects during field emission, and electrical breakdown characteristics at nanoscale.

For nanogap devices, the electrode stability determines the performance and lifetime of these devices, and the interaction between the electric field and electrode surface plays a critical role in electrical breakdown at nanoscale. There are three main kinds of electrode deformation in the experiments, including field assisted diffusion,149 electromigration,153 and local evaporation.22 For field assisted diffusion, Gupta and Willis achieved the sub-nanometer increment in fabricating nanometer spaced electrodes by using atomic layer deposition, and the electrode properties were characterized by field emission and metal–vacuum–metal tunneling. They found that the electrode spacing could be reduced by temperature and voltage manipulation. Higher voltage could lead to field assisted diffusion of surface atoms, and higher temperature would lead to thermal expansion, thus decreasing the gap. After exposing the copper electrode to ambient air, the barrier height of copper electrode may be reduced due to the oxide layer and air contamination on the copper surface.149 In terms of the electromigration, Tomoda et al. fabricated the nanogap electrodes based on field-emission-induced electromigration, where the tunnel resistance was completely determined by the preset current, regardless of the initial gap separation.153 For local evaporation, Driskill-Smith et al. fabricated a device consisting of multiple emitter tips of radii about 1 nm within an extractor electrode aperture of diameter 50 nm that was successfully operated at atmospheric pressure with an ultra-low turn-on voltage of about 7.5 V. During the measurement, they found a sudden fall in current, which could be attributed to the destruction of the dominant nanopillar source of electrons at that particular voltage.154 Han et al. tested the performance of a gate-insulated vacuum channel transistor fabricated by the standard silicon semiconductor processing and found that after a single measurement, the sharp tips ruptured due to local evaporation of the cathode.22 Jones et al. demonstrated a paradigm for CMOS compatible, low voltage (<10 V), robust devices, which were operational at atmospheric pressures and could be independently gated on a single integrated chip. They also suggested that the current instability observed during the test was most likely caused by changes in the emitter due to current-induced heating.21 

Except for the electrode morphology changes, when the electric field is high enough, the emitted current density would be limited by space-charge. Brimley et al. studied the field emission characteristics from Ir/IrO2 tips separated by gaps below 100 nm in air. They found that the space-charge limited emission came before the FN emission in Fig. 7(a), which was explained that the CL/QCL comes from sites near the tip apex under low-potential, and the subsequently observed FN emission comes from other sites further from the gap center. The noticeable morphology changes before and after field emission were observed from the electron micrographs of the devices in Fig. 7(b).155 Singh and Kumar also demonstrated that the I–V curve followed modified CL relation, classical CL law, and FN tunneling, respectively, by testing a copper nanotip-based diode with a 120 nm gap, as shown in Figs. 7(c) and 7(d). The spherical nanoparticles in Fig. 7(e) were formed over the substrate due to the local evaporation of electrode material under enormous current flows through the nanogap.156 However, the CL law was observed before (i.e., for lower currents) the FN field emission, which needs further investigation and verification. On the contrary, the following studies show that the CL law appeared after (i.e., for higher currents) the FN equation. Bhattacharjee et al. carried out experiments to study the transition from field emission to space-charge-limited flows of aluminum electrodes in different nanogaps (30, 50, 70 nm), as can be seen in Fig. 8(a). The results in Figs. 8(b) and 8(c) showed that the I–V characteristics were governed by the FN field emission at low voltages and by space-charge effects at high voltages, and when the gap was 50 nm, a transition from the QCL law to CL law took place at around 10 V.157,158 Using nexus theory, Loveless et al. determined the transition from the QCL law to CL law to occur ∼100 nm.159 They further demonstrated the importance of accounting for collisions to examine conditions for the transitions between the QCL law, CL law, FN equation, and the Mott–Gurney law for collision SCL current, while also linking these conditions to FE-driven breakdown using a common nondimensionalization scheme. Pescini et al. fabricated a free-standing silicon nanostructure, a nanoscale lateral field emission triode, as shown in Fig. 8(d). During the measurement, they observed a saturation of the emitted current for the highest applied voltages due to space-charge effects, see Fig. 8(e).19 Except for the single device measurement above, there are also some studies on the space-charge effects of field emission arrays. Using lateral-type polysilicon field emission arrays fabricated by using local oxidation of the silicon process and conventional photolithography, Lee et al. analyzed the I–V characteristics for the prepared array, which consisted of 100 field emission tips with an average interelectrode spacing of 23 nm. They observed good linearity of FN plots to confirm field emission with deviations at high current due to space-charge or the destruction of some of the tips.160 

FIG. 7.

Space-charge limited emission occurs at lower currents than field emission (FN): (a) I–V curves from device 10-2; (b) electron micrographs from before and after field emission measurements of the device 10–2 (d = 100 nm) in Ref. 155; (c) current–voltage (I–V) characteristics of a copper nanotip based diode as measured at a pressure of 10–6 mbar; (d) scanning electron micrograph of the FIB fabricated copper nanotip based diode structure with a gap of 120 nm; and (e) scanning electron micrograph of the spherical nano-particles of copper formed over the substrate due to local evaporation of the electrode under high current. (a) and (b) Reproduced with permission from Brimley et al., J. Appl. Phys. 109, 094510 (2011). Copyright 2011 American Institute of Physics.155 (c)–(e) Reproduced with permission from Singh and Kumar, J. Appl. Phys. 113, 053303 (2013). Copyright 2013 American Institute of Physics.156 

FIG. 7.

Space-charge limited emission occurs at lower currents than field emission (FN): (a) I–V curves from device 10-2; (b) electron micrographs from before and after field emission measurements of the device 10–2 (d = 100 nm) in Ref. 155; (c) current–voltage (I–V) characteristics of a copper nanotip based diode as measured at a pressure of 10–6 mbar; (d) scanning electron micrograph of the FIB fabricated copper nanotip based diode structure with a gap of 120 nm; and (e) scanning electron micrograph of the spherical nano-particles of copper formed over the substrate due to local evaporation of the electrode under high current. (a) and (b) Reproduced with permission from Brimley et al., J. Appl. Phys. 109, 094510 (2011). Copyright 2011 American Institute of Physics.155 (c)–(e) Reproduced with permission from Singh and Kumar, J. Appl. Phys. 113, 053303 (2013). Copyright 2013 American Institute of Physics.156 

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FIG. 8.

Space-charge limited emission occurs at higher currents than field emission (FN): (a) SEM image of the aluminum electrodes with 70 nm gap; and (b) current voltage I–V characteristics for 30, 50, and 70 nm nanogaps. Inset shows magnified view of the current voltage I–V characteristics for the 30 and 70 nm samples with a FN fit at lower voltages. (c) A plot of ln I vs ln V, where current is in pA and voltage is in volts. The 50 nm sample shows a transition from the QCL law (slope of 0.5) to the CL law (slope of 1.5) at around 10 V.158 (d) Scanning electron micrograph of the free-standing silicon nanostructure, and the aerial view of the device with the contacts marked as source (S), drain (D), gate 1 (G1), and gate 2 (G2). (e) ID vs VDS for different gate voltages VG1D is shown (G2 is grounded). (a)–(c) Reproduced with permission from Bhattacharjee and Chowdhury, Appl. Phys. Lett. 95, 061501 (2009). Copyright 2009 American Institute of Physics.158 (d)–(e) Reproduced with permission from Pescini et al., Adv. Mater. 13, 1780–1783 (2001). Copyright 2001 John Wiley & Sons, Inc.19 

FIG. 8.

Space-charge limited emission occurs at higher currents than field emission (FN): (a) SEM image of the aluminum electrodes with 70 nm gap; and (b) current voltage I–V characteristics for 30, 50, and 70 nm nanogaps. Inset shows magnified view of the current voltage I–V characteristics for the 30 and 70 nm samples with a FN fit at lower voltages. (c) A plot of ln I vs ln V, where current is in pA and voltage is in volts. The 50 nm sample shows a transition from the QCL law (slope of 0.5) to the CL law (slope of 1.5) at around 10 V.158 (d) Scanning electron micrograph of the free-standing silicon nanostructure, and the aerial view of the device with the contacts marked as source (S), drain (D), gate 1 (G1), and gate 2 (G2). (e) ID vs VDS for different gate voltages VG1D is shown (G2 is grounded). (a)–(c) Reproduced with permission from Bhattacharjee and Chowdhury, Appl. Phys. Lett. 95, 061501 (2009). Copyright 2009 American Institute of Physics.158 (d)–(e) Reproduced with permission from Pescini et al., Adv. Mater. 13, 1780–1783 (2001). Copyright 2001 John Wiley & Sons, Inc.19 

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More recently, to investigate whether the actual transition from FN equation to space-charge effects exists in the entire electrical breakdown process at nanoscale, Wang et al. tested the electrical breakdown and electron emission for various nanogap distances and electrode aspect ratios with an anode protrusion in atmospheric air, where the electrodes used titanium and gold, as shown in Figs. 9(a) and 9(b). For the gap distance deff > 200 nm, the breakdown voltage Vb in Fig. 9(e) decreases linearly with the deff decreases, and the protrusion width has a weak effect on the Vb. For deff < 200 nm, Vb decreases less rapidly with decreasing deff. This may correspond to a combination of quantum effects159 and/or a rapid increase in the field enhancement factor for smaller gaps. The SEM images before and after breakdown are shown in Figs. 9(c) and 9(d). Interestingly, in terms of the transition process in the electron emission mechanism before breakdown, one of the experimental FN plots is shown in Fig. 9(f), which shows that the FN curve flattens before the breakdown, indicating the presence of the space-charge effects.25,161

FIG. 9.

(a) Top view of the device schematic; (b) side view of the device schematic; (c) and (d) SEM images of a representative device before and after breakdown; (e) breakdown voltage as a function of deff for different protrusion width 2a; and (f) Fowler–Nordheim (FN) plots for deff = 400 nm and 2a = 25 nm. Reproduced with permission from Wang et al., Appl. Phys. Lett. 120, 124103 (2022). Copyright 2022 Authors published under an exclusive license by AIP Publishing.161 

FIG. 9.

(a) Top view of the device schematic; (b) side view of the device schematic; (c) and (d) SEM images of a representative device before and after breakdown; (e) breakdown voltage as a function of deff for different protrusion width 2a; and (f) Fowler–Nordheim (FN) plots for deff = 400 nm and 2a = 25 nm. Reproduced with permission from Wang et al., Appl. Phys. Lett. 120, 124103 (2022). Copyright 2022 Authors published under an exclusive license by AIP Publishing.161 

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The electron emission and electrical breakdown characteristics studied by the adjustable nanogap method are more systematic and intrinsic than the fixed nanogap method, and this method can realize the in situ record of the electrode morphology evolution during electron emission, before and after breakdown. Here, we summarize studies of electron emission and the electrode morphology evolution during field emission, and the influences of electrode geometries, electrode materials, and gap distance on the electrical breakdown behavior and possible mechanisms.

Understanding the electron emission characteristics in the nanogap is important for figuring out the electrical breakdown process at nanoscale. Cabrera et al. studied the I–V characteristics of a tip-planar structure, where a sharp metallic tip (5–30 nm radius) placed at a variable distance d from a planar collector, ranging from a few nm to a few mm, as shown in Fig. 10(a). The scaling behaviors of the I–V curves and U-d curves were observed, and the different scaling variables were obtained for different distances. The I–V curves in 3–300 nm gaps and their scaling behaviors were shown in Fig. 10(b), where R(d) ∝ d−λ, λ ≈ 0.22, and additionally, they found that when the gap distance is less than 10 nm, the power-law exponent λ tends to a larger value, as can be seen in Fig. 10(c).162 Recently, Li et al. studied the influence of the cathode radius (R = 2, 24, 220 nm) of a tungsten nanotip (cathode) on the electron emission characteristics in vacuum nanogaps from 2 to 25 nm based on an in situ TEM electrical measurement system.132 Furthermore, the relationship between electron emission and electrode evolution (electrode reliability) plays a critical role in the electrical breakdown process. Huang et al. studied the reliability of the Si nano-cathode. Representative SEM images of deformed individual Si nano-tip apex in sequence following the increase in the electric field (d = 500 nm) shown in Figs. 10(d)–10(h) revealed that the apex changed its tip shape into a nano-whisker-like structure. The selected area electron diffraction (SAED) images in Figs. 10(i)–10(k) showed that a crystalline Si tip apex deformed to an amorphous structure. The field emission current in Fig. 10(l) showed the poor uniformity due to the cathode instability, and the breakdown events are extremely prone to occur when the current reaches several nanoamps. For a 100 nm gap, the clear deformation resembling a 500 nm gap occurred, as shown in Figs. 10(m)–10(p). For a 50 nm gap, the nano-whisker due to cathode deformation would contact the anode and caused a breakdown. This low-field deformation of the crystalline-to-amorphous phase transformation of the Si lattice could occur due to the strong electrostatic force exerting on the electrons in the surface lattices, and the increased inner stress and the electron density due to the arsenic-dopant in the Si surface lattice.124 Except for the morphology changes of cathode, the anode material may transfer to the cathode due to the anodic temperature rise in nanogaps during electron emission.163,164

FIG. 10.

(a) Schematic view of the experimental setup. (b) The I–V curves in 3–300 nm and their scaling behaviors [R(d)∝d−λ, λ ≈ 0.22]. (c) Summary of V-d curves at 0.2 nA in a log10 V vs log10 d plot, d from 3 to 2000 nm. The curves have been shifted along the vertical scale relative to each other for clarity.162 (d)–(h) The typical SEM images illustrating the deformed individual Si tip emitter in sequence following the increase in the applied field. The cathode-to-anode separation is 500 nm. (i) The typical TEM image of a Si nano-apex with a whisker on top; the inset is the corresponding EDX spectra of the apex. (j) The typical SAED image of the nano-whisker. (k) The typical SAED image of the bulk of the Si tip. (l) The typical field emission I–E characteristics of the individual tips in the 500 nm cathode-to-anode separation tests. The inset is the corresponding FN plots. (m)–(o) The typical SEM images showing the deformed Si tip in sequence following the increase in the applied field. The cathode-to-anode separation is 100 nm. (p) The typical field emission I–E characteristics and the corresponding FN plots of the three tested tips in the 100 nm cathode-to-anode separation tests. (a)–(c) Reproduced with permission from Cabrera et al., Phys. Rev. B 87, 115436 (2013). Copyright 2013 American Physical Society.162 (d) and (e) Reproduced with permission from Huang et al., Sci. Rep. 5, 10631 (2015). Copyright 2015 Authors licensed under a Creative Commons Attribution (CC BY) 4.0 International License.124 

FIG. 10.

(a) Schematic view of the experimental setup. (b) The I–V curves in 3–300 nm and their scaling behaviors [R(d)∝d−λ, λ ≈ 0.22]. (c) Summary of V-d curves at 0.2 nA in a log10 V vs log10 d plot, d from 3 to 2000 nm. The curves have been shifted along the vertical scale relative to each other for clarity.162 (d)–(h) The typical SEM images illustrating the deformed individual Si tip emitter in sequence following the increase in the applied field. The cathode-to-anode separation is 500 nm. (i) The typical TEM image of a Si nano-apex with a whisker on top; the inset is the corresponding EDX spectra of the apex. (j) The typical SAED image of the nano-whisker. (k) The typical SAED image of the bulk of the Si tip. (l) The typical field emission I–E characteristics of the individual tips in the 500 nm cathode-to-anode separation tests. The inset is the corresponding FN plots. (m)–(o) The typical SEM images showing the deformed Si tip in sequence following the increase in the applied field. The cathode-to-anode separation is 100 nm. (p) The typical field emission I–E characteristics and the corresponding FN plots of the three tested tips in the 100 nm cathode-to-anode separation tests. (a)–(c) Reproduced with permission from Cabrera et al., Phys. Rev. B 87, 115436 (2013). Copyright 2013 American Physical Society.162 (d) and (e) Reproduced with permission from Huang et al., Sci. Rep. 5, 10631 (2015). Copyright 2015 Authors licensed under a Creative Commons Attribution (CC BY) 4.0 International License.124 

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Some studies have reported the intrinsic electrical breakdown mechanism in nanogaps. Our group previously established a vacuum nanogap breakdown test system, which consisted of a FIB system that was integrated with a high-resolution SEM, a three-dimensional nanometer manipulator, a Keithley electrometer (Model 6517B) for the dc voltage supply and weak current measurement, and a current monitor to detect breakdown current. We used this system to systematically study the influences of the electrode geometry, electrode material, gap separation, and injected voltage waveform on breakdown.137–144 For sphere–sphere and needle–sphere electrodes, the I–V curves and the corresponding FN plots are shown in Figs. 11(a) and 11(b). The needle–sphere electrodes showed the obvious field emission process and lower breakdown voltages than the sphere–sphere electrodes, where collective field electron emission rather than continuous field electron emission initiated the vacuum breakdown at nanoscale under a uniform electric field. The movement of the tip of the needle cathode to the anode surface after breakdown under nonuniform electric field indicated the cathode-initiated breakdown for the needle-to-sphere electrodes. The breakdown voltage for a pulsed waveform with the pulse duration of 200 ns and the rising time of 30 ns was about 4–5 times higher than for a dc waveform. The I–V curves for Mo and W cathodes demonstrated similar trends, and the evaporation of metal atomic vapor from the cathode, which depends on the thermal properties of the cathode materials, played a key role in breakdown. Figure 11(c) compares the average breakdown voltage142 as a function of gap distance to other results (Hirata et al.133 and Peschot et al.23). As the gap decreases, the breakdown voltage decreases and reaches about 40 V in a 30 nm gap for tungsten nanotip (cathode)-stainless steel plane (anode) structure.133 Moreover, the SEM images of the typical electrode morphology of the sphere–sphere electrodes at small/large current breakdown mode and the needle–sphere electrodes before and after breakdown are shown in Figs. 11(d)–11(g). It was hypothesized that for the needle–sphere electrodes, the field emission dominated the initial stage and then thermionic emission took over until the nanogap was filled with massive ions and electrons. Alternatively, it was conjectured that for the sphere–sphere electrodes, electron emission would turn on simultaneously from a large area rather than some random protrusions once the applied electric field reaches a critical value, and the subsequent collision of the substantial emission electrons with the anode surface would evaporate and ionize the anode, leading to breakdown.142 

FIG. 11.

(a) The current-voltage curves for different electrode geometries; (b) the corresponding FN plots;143 (c) the average breakdown thresholds as a function of electrode geometries, electrode materials and gap distances;23,133,142 (d) SEM image of small current mode for sphere-sphere electrodes; (e) SEM image of large current mode for sphere–sphere electrodes; (f) SEM image before electrical breakdown for needle-sphere electrodes; and (g) SEM image after electrical breakdown for needle-sphere electrodes. (a) and (b) Reproduced with permission from Meng et al., IEEE International Conference on Dielectrics (ICD) (2016), 1159–1162. Copyright 2016 IEEE.143 (d)–(g) Reproduced with permission from Meng et al., IEEE Trans. Dielectr. Electr. Insul. 21, 1950–1956 (2014). Copyright 2014 IEEE.142 

FIG. 11.

(a) The current-voltage curves for different electrode geometries; (b) the corresponding FN plots;143 (c) the average breakdown thresholds as a function of electrode geometries, electrode materials and gap distances;23,133,142 (d) SEM image of small current mode for sphere-sphere electrodes; (e) SEM image of large current mode for sphere–sphere electrodes; (f) SEM image before electrical breakdown for needle-sphere electrodes; and (g) SEM image after electrical breakdown for needle-sphere electrodes. (a) and (b) Reproduced with permission from Meng et al., IEEE International Conference on Dielectrics (ICD) (2016), 1159–1162. Copyright 2016 IEEE.143 (d)–(g) Reproduced with permission from Meng et al., IEEE Trans. Dielectr. Electr. Insul. 21, 1950–1956 (2014). Copyright 2014 IEEE.142 

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As discussed above, there are some studies on electron emission and electrical breakdown behavior based on the fixed nanogaps on substrates and adjustable nanogaps. The studies using fixed nanogaps are more inclined to make for improving the performance of devices directly. The studies of adjustable nanogaps are helpful to understand the intrinsic properties of electrical breakdown at nanoscale. However, there is still a lot to be done in the experiments. First of all, it is experimentally difficult to achieve a pure nano-electrode. For instance, an oxide layer would coat the electrode surface once it is exposed to air and the absence of ultrahigh vacuum sample chambers in SEMs and FIB-SEMs would deposit a carbon layer on the electrode,165–168 which could cause significant effects on electron emission and breakdown characteristics at nanoscale due to the comparable size of the surface contaminants and gap. Thus, there is an urgent need for in situ preparation and electrical measurement of pure electrodes in a higher vacuum environment, such as in TEM.132 Second, for the electrode morphology evolution during electron emission process, the reports focus mainly on the Si electrode.124 Studies must be performed on other electrode materials to elucidate the interaction between electric field and electrodes (e.g., electrode materials, electrode geometries) during the electron emission process. Even the interaction between electric field and other electrode surface properties, such as the grain boundaries or dislocations, must be examined to determine the origin of breakdown from the microscopic scale of electrode materials. Third, the FN equation is typically used to connect experiments and electron emission theories. Aside from recent studies applying nexus theory to breakdown,161,169 the FN theory is common for analyzing the electron emission behavior until now, but there is an urgent anticipation for observing additional features and phenomenon, such as CL/QCL, before electrical breakdown at nanoscale.159 Finally, it is conjectured that electron emission under high electric field dominates the first stage of electrical breakdown mechanism at nanoscale. The electrode that vaporizes the atoms first depends on which electrode reaches thermal criticality first, which depends on the electrode geometry and electrode materials. The evaporated atoms then expand and initiate a metallic plasma; however, the formation of the metallic plasma due to collision ionization or direct ionization in nanogaps due to the strong electric field requires further study. Hence, these all motivate a more systematic study of electron emission and electrical breakdown characteristics from 1 nm to ∼5 μm gap with different electrode materials, electrode geometries, surface conditions, and test environments. This requires developing a more comprehensive in situ test platform, such as high-resolution electron microscopes to in situ investigate the dynamic process of electrode morphology evolution during electron emission process and optical diagnostics to characterize the dynamic process of nanoscale breakdown. These diagnostics must then be combined with a more general model or electrical mechanism at nanoscale to better guide and optimize device design and operation.

In systems with nanoelectrodes or nano-protrusions, the surface-to-volume ratio increases significantly, and surface effects start playing a more significant role in the processes leading to electrical breakdowns. To understand the onset of the phenomenon at nanoscale, it is important to include high electric field effects on solid surfaces in the theoretical models. Although SEM and TEM imaging of surface evolution under high electric fields could provide in situ observations of deformation and structural phase transitions in the electrodes, the experimental results discussed in Sec. III are inclined to demonstrate only comparative study of surface morphology of electrodes before and after electrical breakdown. Whether the morphological evolution during electron emission can be observed depends on the electrode material, electrode size, and electrode structure. Until now, only the growth of a nano-whisker-like structure at the Si apex was observed during electron emission.124 In addition, the evolution of electrode morphology during breakdown cannot be followed during experiments because of the short (nanosecond) timescales of the entire breakdown process after the electrode enters thermal runaway. Instead, atomic dynamics on a metal surface under high electric fields can be followed using computer simulations that combine classical molecular dynamics with the effects of electric fields.62,170 Since the motion of atoms in these simulations is determined simultaneously by interatomic interactions and electrostatic interactions with the external electric field, this method is a suitable alternative for studying the mechanisms of electrical breakdown onset at nanoscale.

It is generally reported that for an electrical breakdown event to occur, an extremely large geometric field enhancement factor (of the order of several hundreds) is needed to explain high electron emission currents measured prior to the event.171 This enhancement factor is assumed to appear due to localized sharp protrusions on the metal surface, which have not been observed experimentally, especially for the metal surface after prior conditioning, so it is believed that the sharp surface features may appear on metal surfaces spontaneously under high electric field.172 In terms of growth mechanisms of such protrusions, field-induced biased surface diffusion48–53 and plastic deformations due to high electric field-induced stress54–60 have been proposed. Djurabekova and her research team at the University of Helsinki have carried out many computer simulations to elucidate the possible mechanisms of nano-protrusion growth. For instance, their Kinetic Monte Carlo (KMC) model, which takes into account the electric field effects on migration energy barriers, showed that an applied electric field can induce a bias in surface diffusion of atoms toward enhancing the growth of the nanotips in high electric fields and elevated temperatures.51,173 Figure 12(a) shows the simulation results for a high anode electric field (∼72 GV/m) and a substrate temperature of 3000 K. This group's molecular dynamic (MD) simulations also showed the possible mechanisms of initiation of growth of a nano-protrusion in the presence of a near-surface void56,174 or grain boundaries.60 In the presence of a near-surface void, the applied high electric field can exert sufficient high electric field-induced stress (Maxwell stress175–177) on the charged surface atoms to cause the plastic deformation above the void. This process shifts the material above the void toward the surface to create a small surface protrusion [cf. Fig. 12(b)]. Then the enhancement of both electric field and plastic deformation continues until the maximum shear stress on the void surface is reached, where the dislocations nucleating will lead to the quick growth of the nano-protrusion in a self-reinforcing manner.56 Furthermore, for a Cu polycrystalline metal surface, they revealed the possibility of the initiation and growth of nano-protrusions along the grain boundary under the Maxwell stress in another MD simulation study, as shown in Fig. 12(c).60 More recently, Xiao et al. demonstrated the formation of nano-protrusions along the grain boundary on a planar metal electrode made of equal-molar W-Mo alloy under the Maxwell stress.59 

FIG. 12.

(a) The growth process of a W nanotip at different stages in an applied electric field of 50 GV/m (initial local field 72 GV/m) at 3000 K, starting from a hemispherical asperity. (b) The growth of the nano-protrusion on the Cu surface in the presence of a near surface void under the high electric field. (c) The cross section (A) and a perspective view (B) of a formed surface protrusion along the grain boundary under the high electric field-induced stress. (a) Reproduced with permission from Jansson et al., Nanotechnology 31, 355301 (2020). Copyright 2020 IOP Publishing Ltd.51 (b) Reproduced with permission from Pohjonen et al., J. Appl. Phys. 114, 033519 (2013). Copyright 2013 AIP Publishing LLC.56 (c) Reproduced with permission from Kuppart et al., Micromachines 12, 1178 (2021). Copyright 2021 Authors licensed under a Creative Commons Attribution (CC BY) 4.0 International License.60 

FIG. 12.

(a) The growth process of a W nanotip at different stages in an applied electric field of 50 GV/m (initial local field 72 GV/m) at 3000 K, starting from a hemispherical asperity. (b) The growth of the nano-protrusion on the Cu surface in the presence of a near surface void under the high electric field. (c) The cross section (A) and a perspective view (B) of a formed surface protrusion along the grain boundary under the high electric field-induced stress. (a) Reproduced with permission from Jansson et al., Nanotechnology 31, 355301 (2020). Copyright 2020 IOP Publishing Ltd.51 (b) Reproduced with permission from Pohjonen et al., J. Appl. Phys. 114, 033519 (2013). Copyright 2013 AIP Publishing LLC.56 (c) Reproduced with permission from Kuppart et al., Micromachines 12, 1178 (2021). Copyright 2021 Authors licensed under a Creative Commons Attribution (CC BY) 4.0 International License.60 

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To simulate the dynamic evolution of nanoelectrode surface or nano-protrusions under the high electric field and understand the underlying mechanisms of vacuum arcing, Djurabekova and her research team have developed a set of simulation programs in the past decade. First, they developed a hybrid electrodynamics–molecular dynamics (ED-MD) model (HELMOD code) based on the classical MD code PARCAS that considers the electron emission process and corresponding Joule heating.170,174,178–180 In Ref. 174, they simulated (i) thermal and field-enhanced evaporation, (ii) only field-enhanced evaporation, and (iii) only thermal evaporation. They observed no evaporation for the only thermal evaporation simulation. The critical electric field for the only field-enhanced evaporation simulation was higher than the simulation that considered both thermal and field-enhanced evaporation, where the electric field force increased the distance between atoms and weakened the bonds, and the increased temperature could give extra energy to atoms for breaking. Additionally, they used this code to calculate the critical temperature of spontaneous reorientation, defined as the temperature where the number of atoms forming the dislocation increased sharply, for Cu nanotips with different diameters (1.6–2.8 nm) and heights (6–14 nm) under various electric fields (0.4–1.5 GV/m). They found that the stabilizing effect of an external applied electric field was an order of magnitude greater than the destabilization caused by the Joule heating from the field emission current.179 

However, the aforementioned method does not include the Nottingham effect and the generalized potential barrier for emitters, so they further developed a general algorithm and realized it in a Fortran 2003 computational tool, named “General Tool for Electron Emission Calculations—GETELEC,” which provides an easy-to-use and accurate tool for estimating electron emission under various conditions and shapes of emitters.181 In this algorithm, the current density J and the Nottingham heat deposited per unit area PN were given by
J = Z S k B T D ( E ) log ( 1 + exp ( E / k B T ) ) d E
(3)
and
P N = Z S E 1 + exp ( E / k B T ) E D ( ε ) d ε dE ,
(4)
where E is the electron's energy, D(E) is the transmission coefficient, kB is the Boltzmann constant, T is the temperature, and Z S = e m / 2 π 2 3 is the Sommerfeld current constant. GETELEC was implemented and tested with the HELMOD code. Figure 13 shows total deposited heating power along the tip as calculated in mode (A) (crosses) and in mode (B) (circles), where the mode (A) considers both Joule and Nottingham effects as calculated by GETELEC, while the mode (B) only considers Joule heat. The results show that the total current and the Joule heat are overestimated by several orders of magnitude when only Joule heating is considered, and the Nottingham effect can make the apex heat up to the same temperature with ten times lower current.181 
FIG. 13.

Total deposited heating power along the tip as calculated in mode (A) (crosses) and in mode (B) (circles). The Joule component of mode (A) calculation is also shown (red squares). Mode (A) considers both Joule and Nottingham effects, while mode (B) only considers Joule heating. Reproduced with permission from Kyritsakis and Djurabekova, Comput. Mater. Sci. 128, 15–21 (2017). Copyright 2016 Elsevier B.V.181 

FIG. 13.

Total deposited heating power along the tip as calculated in mode (A) (crosses) and in mode (B) (circles). The Joule component of mode (A) calculation is also shown (red squares). Mode (A) considers both Joule and Nottingham effects, while mode (B) only considers Joule heating. Reproduced with permission from Kyritsakis and Djurabekova, Comput. Mater. Sci. 128, 15–21 (2017). Copyright 2016 Elsevier B.V.181 

Close modal

Although the aforementioned works could assess the behavior of a metal surface with small-scale features under high electric field, using the finite difference method (FDM) with a structured static mesh limited the simulation model, simulation time, and simulation size due to the high computational cost. To improve computational efficiency, scalability, and tolerance with respect to the crystallographic structure of studied materials, they proposed a simulation code called FEMOCS, which couples atomistic simulations with a finite element solver.182,183 Later, they combined HELMOD, FEMOCS, and GETELEC to simulate the field-assisted thermal evaporation from Cu nanotips of different initial geometries under different electric fields and found that the aspect ratio of nanotips has a significant influence on the efficiency of thermal evaporation of Cu atoms. Higher aspect ratios reduced the critical evaporation electric field necessary to initiate field-assisted thermal evaporation from the tip heated by the electron emission processes.63 Furthermore, the effect of a space-charge (SC) model was investigated in these simulations. Initially, only a simplified 1D model was added to the multiphysics simulations.184 Although it provided insightful results, the 1D model was insufficient to capture fully the space-charge effect on the thermal runaway process. To decipher the plasma ignition mechanism more accurately, they used the PIC method to allow for a full 3D calculation of the space-charge effects. This final step allowed the coupling of the evolution process of the metal surface under high electric field with the evolution process of the vaporized and ionized material that leads to plasma ignition in the vacuum, and the final simulation flow chart of this ED-MD-PIC multiscale-multiphysics model is shown in Fig. 14(a).62 Based on the model, they simulated the runaway process of the Cu nanotips with R = 3 nm, h = 93 nm and different widths (17 and 54 nm) under the same field emission current, as shown in Fig. 14(b). The time evolution results of averaged nanotip heights and corresponding some characteristic excerpts are shown in Figs. 14(c) and 14(d), indicating that the thermal runaway of the thin emitters shows a cyclic process, and the thin emitters with a high neutral evaporation rate have a higher probability to ignite self-sustainable plasma than the wide emitters. Moreover, based on the same method, Xiao et al. performed this ED-MD-PIC multiscale-multiphysics simulations for the refractory metal such as W and Mo, and the structural evolution and thermal runaway process of them are compared to the Cu. Figures 14(e) and 14(f) show the geometry and internal crystal structure evolution and atomic evaporation process of these metals and found that for the refractory metals (W, Mo), the structural thermal process shows the rapid growth of small protrusions and the following sharpening and thinning at the apex, while for Cu, the evolution process is caused by the ejection of large droplets generated by recrystallization and necking of the molten region at the apex, indicating that the melting and boiling points play a vital role in the breakdown process.185 More recently, Xiao et al. systematically studied nine different metal nano-emitters, including body centered cubic (BCC) structures (Mo, W, and V), face centered cubic (FCC) structures (Cu, Au, and Al), and hexagonal structures (HCP; Ti, Zr, and Zn), which show a high diversity in crystal structure, work function, melting point, and boiling point. They found that the nanotips could deform under a low electric field, and two mechanisms are suggested. The tilting of the nanotips in the bulk crystalline region far away from the apex region can be explained by the reversible elastic distortions of atomic registries away from their equilibrium configurations, while the tilting at the apex region can be attributed to the movement of partial dislocations at the apex. The ED-MD-PIC simulations indicated that those dislocations are initiated either at the surface or in the interior of the nanotips. After the initiation, the partial dislocations could quickly propagate through the nanotip, directly causing the bending behaviors of the nanotips under electric field. Furthermore, these simulations showed that the initial crystallographic orientation of the nanotip also plays a significant role in the bending or tilting phenomena in molecular dynamics simulations. For example, three distinct initial crystallographic orientations for Cu nanotips have been studied, including [001]⟨001⟩, [110]⟨110⟩, and ⟨111⟩[111] in Ref. 186. The tilting of nanotips at the apex only occurs for the first two initial orientations, but not for the third case. Otherwise, the thermal runaway deformation of the nanotips under high electric field can also be classified into two types, where the soft metal nanotip with low melting points (Al, Cu, Au, and Zn) during thermal runaway behaves the elongation, thinning, and necking of the molten region, while the refractory metal nanotip (Ti, Zr, Mo, and W) during thermal runaway behaves the elongation, thinning, and sharpening. Meanwhile, soft metals show a higher neutral evaporation rate than refractory metals. Reference 186 also emphasized the importance of correlating the electrical pre-breakdown characteristics of metal nanotips to their microscopic atomic structures (crystal structures, coordination numbers, lattice constants, crystallographic plane spacing, average bond length, etc.) and macroscopic thermophysical parameters (work function, electrical conductivity, thermal conductivity, boiling point, melting point, cohesive energy of bulk crystal, and evaporation enthalpy). Many of those structural and thermophysical properties are available from experimental studies and literature. In addition, they established a multi-variable linear regression model to predict the critical evaporation electric field, where using the boiling point, atomic coordination number of liquid metals, and equilibrium lattice parameter as the independent variables gives the best prediction accuracy.186 Meanwhile, neither lattice types nor work function shows obvious correlation with the critical pre-breakdown E-field value for metal nanotips. Notably, even for the proposed multi-variable models, the melting point or boiling point plays the decisive role in determining the critical E-field value of metal nanotips.186 The metals having higher values for those two physical properties may lead to large E-field pre-breakdown strength under the thermal runaway mechanism. So far, the ED-MD-PIC simulations are mainly applied to conical metal nanotips. In addition to thermophysical properties of metals, the size and geometry of nanotip are also expected to influence both thermal runaway mechanism and pre-breakdown strength, requiring further investigations in the future.

FIG. 14.

(a) Simulation flow chart of the ED-MD-PIC multiscale-multiphysics model.62 (b) Boundary regions and geometry of the simulation domain, which consists of an atomistic region and an extension region. (c) Time evolution of the averaged nanotip heights (left) and (d) some characteristic excerpts from the simulation (right). Labels (a)–(h) on the left correspond to the frames on the right side. Error bars show the variation of the data between parallel runs in a form of standard error of the mean.62 (e) and (f) Geometry and internal crystal structure evolution during thermal runaway and atomic evaporation process of W, Mo, and Cu nanotips (R0 = 1, H0 = 100 nm) under critical electric field. The color coding corresponds to the profiles of atomic coordination number analysis (CNA) for different metal structures. (a)–(d) Reproduced with permission from Veske et al., Phys. Rev. E 101, 053307 (2020). Copyright 2020 Authors licensed under a Creative Commons Attribution (CC BY) 4.0 International License.62 (e)–(g) Reproduced with permission from Gao et al., J. Phys. D 55, 335201 (2022). Copyright 2022 IOP Publishing Ltd.185 

FIG. 14.

(a) Simulation flow chart of the ED-MD-PIC multiscale-multiphysics model.62 (b) Boundary regions and geometry of the simulation domain, which consists of an atomistic region and an extension region. (c) Time evolution of the averaged nanotip heights (left) and (d) some characteristic excerpts from the simulation (right). Labels (a)–(h) on the left correspond to the frames on the right side. Error bars show the variation of the data between parallel runs in a form of standard error of the mean.62 (e) and (f) Geometry and internal crystal structure evolution during thermal runaway and atomic evaporation process of W, Mo, and Cu nanotips (R0 = 1, H0 = 100 nm) under critical electric field. The color coding corresponds to the profiles of atomic coordination number analysis (CNA) for different metal structures. (a)–(d) Reproduced with permission from Veske et al., Phys. Rev. E 101, 053307 (2020). Copyright 2020 Authors licensed under a Creative Commons Attribution (CC BY) 4.0 International License.62 (e)–(g) Reproduced with permission from Gao et al., J. Phys. D 55, 335201 (2022). Copyright 2022 IOP Publishing Ltd.185 

Close modal

Interestingly, the mechanical properties of metals such as bulk modulus and Young's modulus have not been included as the possible descriptors for developing the predictive multi-variable model in Ref. 186. Certainly, the elastic–plastic deformation behavior of metal nanotips is directly correlated with the mechanical moduli under the thermal and electrical stresses. Microscopically, the mechanical module measures the chemical bonding strength among atoms in atomic structure. The strong chemical bonding mechanism is expected to be more resistant to structural plastic deformation and thermal runaway for metal nanotips under E-field and heats. The findings of Ref. 186 clearly support the idea that the refractory metals such as Mo (BCC), W (BCC), Ti (HCP), and Zr (HCP) exhibited much higher pre-breakdown electric field strength than malleable metals including Al (FCC), Cu (FCC), Au (FCC), and Zn (HCP). The results may suggest that materials with higher mechanical stiffness could sustain larger E-field pre-breakdown strength before thermal runaway occurring in general.

The likely dependence of ED-MD-PIC simulation results on the employed inter-atomic potential forms is another important aspect concerning all previous studies. Typically, the embedded atomic potentials (EAM), which are carefully trained for atomic structures, thermodynamic properties, elastic constants, and phase transition properties, are used for all MD simulations. Generally, those EAM potentials could provide reliable descriptions for structural and thermophysical properties of metal nanotips at finite temperature, compared to those of experimental results. However, it is indeed that few of the available EAM potentials are properly fitted to describe the evaporation process under intense heating process. In addition to the interatomic potentials, the occurring of thermal runways in nanotips is crucially related to the resistive heating process and the associated thermal conduction mechanism. Those calculations require the knowledge of both the size and temperature dependencies of electrical resistivity and total thermal conductivity for nanotips. However, such dependencies are scarcely available for the two parameters in the literature, even though the current ED-MD-PIC simulation results could explain some characteristic experimental observations for electrical breakdown of nanotips including the deformation, melting, and transferring of cathode materials to anode in nanogaps.140,144

Although the ED-MD-PIC model can efficiently simulate the structural deformation, phase transition, and thermal evaporation of the nanotips under high electric field, there are still some limitations. For example, the space-charge Coulomb potential and the space-charge exchange-correlation potential, which are important for electron emission in nanogaps as discussed in Sec. II, are not considered in the standard ED-MD-PIC methodology using the FEMOCS code. More recently, Xiao et al. have dramatically improved the algorithm for the current ED-MD-PIC method, forming the new computational tool called “Field Emission coupled Molecular Dynamics” or “FEcMD.”59 First, the space-charge potential and exchange-correlation potential are considered to determine the electron tunneling barrier profile with the WKBJ model, which is calculated based on the obtained space-charge density distribution from the PIC simulation. In addition, the image charge potential is modified based on the free electron Thomas–Fermi approximation (TFA) with the random phase approximation (RPA), which eliminates the singularity at the interface between the electrode and vacuum.35,39 They found that the exchange-correlation effects could overwhelm the space-charge potential for the nanogap in the quantum electron field emission regime (see Fig. 4), which could lower the electron emission barrier and increase the emission current.59 Second, the two-temperature thermal conduction model is implemented, which can decouple the electron and phonon temperatures and help elucidate the heating process for nanotips under both RF and static electric fields. Although the two-temperature thermal conduction model allows the phonon and electron temperature evolve separately, the two subsystems are coupled by the electron–phonon coupling constant, which crucially determines the thermal energy exchange rate between them. The two-temperature model in FEcMD code shows that when the electric field is weakly time-dependent or the time to breakdown is sufficiently long, the conventional one-temperature model can serve as a valid approximation. Otherwise, if the electron emission period approaches the characteristic relaxation time of the electron–phonon coupling mechanism, the nanotips enter into a highly non-equilibrium condition. A large difference between phonon and electron temperature profiles is obtained for nanotips, leading to a prominent high frequency electronic heat conduction mechanism. Finally, all numerical algorithms relevant to MD simulations in the FEcMD software have been completely rewritten and now fully support three widely used interatomic potentials [i.e., the Lennard–Jones pair potential (LJ), embedded atomic method (EAM) potentials, and moment tensor potentials (MTP)] for both alloys and pure metals, improving the versatility and flexibility of the methodology.

As discussed above, the current multiscale-multiphysics model can realize the formation of the nano-protrusions from a defective metal surface under high electric field-induced stress, and the subsequent morphology evolution and thermal runaway of the nanotips under intense electron emission. This model has been extended to the nanogap, which provides an in-depth understanding of the interaction between the high electric field, electron emission, and metal surfaces from the atomic scale, and is significant for the breakdown process at nanoscale. However, some challenging problems remain unsolved for the current multiscale-multiphysics model. First, the simulated deformation behavior relies heavily on the accuracy of the physics and geometry used in the MD simulation, which may suffer from the accuracy of empirical or semiempirical interatomic potentials and electrode geometry.187–190 Meanwhile, the interatomic potentials used in the simulation may be inaccurate for the structural relaxations in melting and vaporization at very high temperature.186 Although the utilization of machine learning potentials such as moment tensor potentials (MTP) for ED-MD-PIC simulations could achieve the level of accuracy approaching that of first-principles calculations, the whole workflow for training and validating of MTPs for metals experiencing multiple simultaneous phase transitions including crystalline solid, liquid, and vapor during the thermal runaway is an unsolved issue currently. In addition, the current ED-MD-PIC methodology only supports the electrodynamics and atomic structure evolution for metal nanotips. For semiconductors and dielectrics, there is lack of appropriate interatomic potentials to describe the phase transitions and structural damages induced by a high electric field and strong heating process. Otherwise, in great contrast to that of metals, the space-charge dynamics is rather complicated in dielectrics, and which involves various mechanisms including the charge injection, diffusion and drifting, and the trapping process. It is relatively straightforward to develop the ED-PIC based on finite element method to solve the electrodynamics, heat balance equations, and space-charge dynamics for nano- or micro-protrusions consisting of dielectric materials, as demonstrated in Ref. 191. Nevertheless, a comprehensive understanding of various mechanisms for the formation of space-charge in dielectrics and a full knowledge of all related physical parameters are the fundamental challenges for employing such methodology to dielectric nanotips. For dielectric nanotips, full quantum mechanical calculations may be eventually required to reliably capture the electron polarization and chemical bond breaking process under the high E-field. Thus, there is an urgent need to develop and use more accurate interatomic potentials (i.e., machine learning potentials) or more advanced theoretical framework to study the thermal runaway process of the metal and dielectric nanotips in the future. Second, neither the increase in the electrical resistivity due to the increased electron scattering at the line and plane defects, such as dislocations, stacking faults, and twin boundaries, nor the electron wind force exerted on the atoms are included.192 Finally, it is worth noting that in the current model, the PIC simulation only includes electrons, ignoring the presence of electron–neutral interactions and ions, which must be included in future simulations to extend the model to include the initiation and subsequent evolution of the plasma.38 Notably, while PIC-MCC simulations can describe the collisions among electrons, ions, and neutrals, it has not been interfaced to ED-MD simulations to provide a unified simulation workflow that includes electron emission and thermal runway to electric breakdown and plasma formation.

This work reviews the progress on theories, experiments, and atomistic simulations to characterize electron emission and electrical breakdown at nanoscale. The electrical breakdown process across nanogaps begins with the growth of field driven nano-protrusion, followed by electron emission enhancement, nano-protrusion thermal runaway, and initiation of a metallic plasma. The electron emission theory across nanogaps could provide the theoretical support for electron tunneling, but will still be limited by the calculation approximation, such as the WKB approximation for the barrier transmission coefficient, which does not involve the interaction between the material surface atoms and the electric field. The experiments at nanoscale can provide insight into electron emission, electrode morphology evolution, and electrode deformation before and after breakdown, but the thermal runaway process or breakdown process are almost impossible to capture. Atomistic simulations can demonstrate the nano-protrusion deformation under electric field-induced stress or the thermal runaway process under intense electron emission based on the latest electron emission theory and advanced computational methods; however, they are limited by some simulation parameters, such as interatomic potentials. Hence, theory, experiments, and atomistic simulations examining electrical breakdown at nanoscale all require substantial future studies. Characterizing this behavior requires a more general theory or model for a nanogap that simultaneously addresses electron emission, nano-protrusion morphology evolution, and plasma formation. Developing such a theory requires a concerted combination of theory, simulation, and experiment to guide and optimize device design and operation.

Interestingly, a growing number of experimental studies focus on the physics of electron emission and electronic devices from graphene and other 2D materials, which are expected to exhibit different current–voltage characteristics in the carrier transport because of (i) electrostatics and electrodynamics due to reduced dimensionality and (ii) non-parabolic energy–momentum dispersion relation of the transport carriers.121,193–195 Thus, we anticipate extensive future studies on two-dimensional material related devices.196–201 It would be interesting to use the thermal-field emission from electrodes based on 2D materials and topological materials202–204 to study electrical breakdown in the nanogaps discussed in this paper.

In this paper, we have not discussed the physics and stability of the nonrelativistic beam–plasma interaction for the interelectrode gap (IEG) plasma that was studied in a spacing of a few mm to 1.5 cm.205,206 The analysis of the sheath layer in nanoscale and anisotropic IEG plasmas may be important for nanoscale discharge physics discussed in this paper. Other than the quantum emission models developed for 2D materials,196–201 quantum confinement effects near to the emitting surface due to nanoscale field emitters have been developed.77,78 These emission models77,78,196–201 can be combined with the gas discharge models in a nanogap with SCLC effects to have a better consistent treatment.

It is important to note that we have focused our review on metallic field emitters that are valid under FN law. It is known that FN law is not valid for nonmetallic nano-materials. For example, field emission from carbon-based nano-island films207 requires hot-electron emission mechanism supplemented with thermoelectric phenomena caused by the phonon drag phenomenon near the boundary between sp2 and sp3 carbon nanodomains. The surface's deformation of carbon-based cathodes has resulted low-threshold electron emission,208,209 where the FN law is no longer valid and qualitative model based the idea of surface reconstruction and the formation of centers with negative correlation energies has been proposed. Thus, a methodology of comparing various field emission models with experiments is by having a general “k” in the pre-exponential term in the model to determine the best agreement.210 Detailed studies in linking such nonmetallic field emitter for nanogap discharge are critical. Finally, the effects of nano-protrusion growth at the emitter surface are similar to the dewetting effect taking place within the thin-film covered field emitters reported in earlier works.211,212

The work is supported by National Natural Science Foundation of China (51977169) and the Fundamental Research Funds for the Central Universities (xtr062023001). L.K.A. is supported by the A*STAR AME IRG (A2083c0057).

The authors have no conflicts to disclose.

Yimeng Li: Writing – original draft (lead). Lay Kee Ang: Writing – review & editing (equal). Bing Xiao: Writing – review & editing (equal). Flyura Djurabekova: Writing – review & editing (equal). Yonghong Cheng: Conceptualization (equal). Guodong Meng: Conceptualization (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

1.
S. S.
Baturin
,
T.
Nikhar
, and
S. V.
Baryshev
,
J. Phys. D
52
(
32
),
325301
(
2019
).
2.
L.
Boodhoo
,
L.
Crudgington
,
H. M. H.
Chong
,
Y.
Tsuchiya
,
Z.
Moktadir
,
T.
Hasegawa
, and
H.
Mizuta
,
Microelectron. Eng.
145
,
66
70
(
2015
).
3.
C. B.
Ru
,
Y. H.
Ye
,
C. L.
Wang
,
P.
Zhu
,
R. Q.
Shen
,
Y.
Hu
, and
L. Z.
Wu
,
Appl. Mech. Mater.
490–491
,
1042
1046
(
2014
).
5.
S.
Banerjee
,
P. Y.
Wong
, and
P.
Zhang
,
J. Phys. D
53
(
35
),
355301
(
2020
).
6.
O. O.
Baranov
,
S.
Xu
,
L.
Xu
,
S.
Huang
,
J. W. M.
Lim
,
U.
Cvelbar
,
I.
Levchenko
, and
K.
Bazaka
,
IEEE Trans. Plasma Sci.
46
(
2
),
230
238
(
2018
).
7.
W. H.
Chiang
,
D.
Mariotti
,
R. M.
Sankaran
,
J. G.
Eden
, and
K.
Ostrikov
,
Adv. Mater.
32
(
18
),
1905508
(
2019
).
8.
A. P.
Papadakis
,
S.
Rossides
, and
A. C.
Metaxas
,
Open Appl. Phys. J.
4
(
1
),
45
63
(
2011
).
9.
R. S.
Dhariwal
,
M. F.
Ahmad
, and
M. P. Y.
Desmulliez
, “MEMS design, fabrication, characterization, and packaging,”
Proc. SPIE
4407
,
172
179
(
2001
).
10.
R. S.
Dhariwal
,
J. M.
Torres
, and
M. P. Y.
Desmulliez
, in
IEE Proc. Sci. Meas. Technol.
147
,
261
265
(
2000
).
11.
P. G.
Slade
and
E. D.
Taylor
,
IEEE Trans. Compon. Packag. Technol.
25
(
3
),
390
396
(
2002
).
12.
A.
Wallash
and
L.
Levit
, “Reliability, testing, and characterization of MEMS/MOEMS II,”
Proc. SPIE
4980
,
87
96
(
2003
).
13.
J. M.
Torres
and
R. S.
Dhariwal
,
Microsyst. Technol.
6
(
1
),
6
10
(
1999
).
14.
L. H.
Germer
,
J. Appl. Phys.
30
(
1
),
46
51
(
1959
).
15.
S.
Matejcik
,
B.
Radjenovic
,
M.
Klas
, and
M.
Radmilovic-Radjenovic
,
Eur. Phys. J. D
69
(
11
),
251
(
2015
).
16.
G. D.
Meng
,
X. Y.
Gao
,
A. M.
Loveless
,
C. Y.
Dong
,
D. J.
Zhang
,
K. J.
Wang
,
B. W.
Zhu
,
Y. H.
Cheng
, and
A. L.
Garner
,
Phys. Plasmas
25
(
8
),
082116
(
2018
).
17.
Y.
Fu
,
P.
Zhang
,
J. P.
Verboncoeur
, and
X.
Wang
,
Plasma Res. Express
2
(
1
),
013001
(
2020
).
18.
A. L.
Garner
,
A. M.
Loveless
,
J. N.
Dahal
, and
A.
Venkattraman
,
IEEE Trans. Plasma Sci.
48
(
4
),
808
824
(
2020
).
19.
L.
Pescini
,
A.
Tilke
,
R. H.
Blick
,
H.
Lorenz
,
J. P.
Kotthaus
,
W.
Eberhardt
, and
D.
Kern
,
Adv. Mater.
13
(
23
),
1780
1783
(
2001
).
20.
C. T.
Lu
,
S.
Johnson
,
S. P.
Lansley
,
R. J.
Blaikie
, and
A.
Markwitz
,
J. Vac. Sci. Technol. B
23
(
4
),
1445
1449
(
2005
).
21.
W. M.
Jones
,
D.
Lukin
, and
A.
Scherer
,
Appl. Phys. Lett.
110
(
26
),
263101
(
2017
).
22.
J. W.
Han
,
J. S.
Oh
, and
M.
Meyyappan
,
Appl. Phys. Lett.
100
(
21
),
213505
(
2012
).
23.
A.
Peschot
,
N.
Bonifaci
,
O.
Lesaint
,
C.
Valadares
, and
C.
Poulain
,
Appl. Phys. Lett.
105
(
12
),
123109
(
2014
).
24.
A. L.
Garner
,
A. M.
Loveless
,
A. M.
Darr
, and
H.
Wang
,
Gas Discharge and Electron Emission for Microscale and Smaller Gaps
(
Springer
,
2023
).
25.
A. L.
Garner
,
G. D.
Meng
,
Y. Y.
Fu
,
A. M.
Loveless
,
R. S.
Brayfield
, and
A. M.
Darr
,
J. Appl. Phys.
128
(
21
),
210903
(
2020
).
26.
P.
Osmokrović
,
D.
Ilić
,
K.
Stanković
,
M.
Vujisić
, and
B.
Lončar
,
Acta Phys. Pol. A
118
(
4
),
585
588
(
2010
).
27.
P. A.
Chatterton
,
Proc. Phys. Soc. London
88
(
1
),
231
(
1966
).
28.
D. K.
Davies
and
M. A.
Biondi
,
J. Appl. Phys.
42
(
8
),
3089
3107
(
1971
).
29.
D. W.
Williams
and
W. T.
Williams
,
J. Phys. D
5
(
2
),
280
(
1972
).
30.
F. M.
Charbonnier
,
C. J.
Bennette
, and
L. W.
Swanson
,
J. Appl. Phys.
38
(
2
),
627
633
(
1967
).
31.
T.
Utsumi
,
J. Appl. Phys.
38
(
7
),
2989
2997
(
1967
).
32.
I.
Beilis
,
Plasma and Spot Phenomena in Electrical Arcs
(
Springer Nature
,
2020
).
33.
D.
Ilic
,
K.
Stankovic
,
M.
Vujisic
, and
P.
Osmokrovic
,
Radiat. Eff. Defects Solids
166
(
2
),
137
149
(
2011
).
34.
K.
Tsuruta
,
IEEE Trans. Electr. Insul.
EI-18
(
3
),
204
208
(
1983
).
35.
W. S.
Koh
and
L. K.
Ang
,
Nanotechnology
19
(
23
),
235402
(
2008
).
36.
P.
Zhang
,
Y. S.
Ang
,
A. L.
Garner
,
A.
Valfells
,
J. W.
Luginsland
, and
L. K.
Ang
,
J. Appl. Phys.
129
(
10
),
100902
(
2021
).
37.
S.
Calatroni
, in
8th International Workshop on Mechanisms of Vacuum Arcs
(
2019
).
38.
R.
Koitermaa
,
A.
Kyritsakis
,
T.
Tiirats
,
V.
Zadin
, and
F.
Djurabekova
,
Heating and Plasma Processes
, arXiv:2402.08404.
39.
N.
Li
,
K.
Wu
,
Y. H.
Cheng
, and
B.
Xiao
,
Front. Phys.
11
,
1223704
(
2023
).
40.
A.
Kyritsakis
and
J. P.
Xanthakis
,
Proc. R. Soc. A
471
(
2174
),
20140811
(
2015
).
41.
S.
Banerjee
and
P.
Zhang
,
AIP Adv.
9
(
8
),
085302
(
2019
).
42.
43.
P.
Zhang
and
Y. Y.
Lau
,
J. Plasma Phys.
82
,
595820505
(
2016
).
44.
K. L.
Jensen
,
Y. Y.
Lau
,
D. W.
Feldman
, and
P. G.
O'Shea
,
Phys. Rev. Spec. Top.-Accel. Beams
11
(
8
),
081001
(
2008
).
45.
A.
Kyritsakis
and
J. P.
Xanthakis
,
J. Appl. Phys.
119
(
4
),
045303
(
2016
).
46.
P. G.
Muzykov
,
X. Y.
Ma
,
D. I.
Cherednichenko
, and
T. S.
Sudarshan
,
J. Appl. Phys.
85
(
12
),
8400
8404
(
1999
).
47.
R. S.
Brayfield
,
A. J.
Fairbanks
,
A. M.
Loveless
,
S. J.
Gao
,
A.
Dhanabal
,
W. H.
Li
,
C.
Darr
,
W. Z.
Wu
, and
A. L.
Garner
,
J. Appl. Phys.
125
(
20
),
203302
(
2019
).
48.
T. T.
Tsong
and
G.
Kellogg
,
Phys. Rev. B
12
(
4
),
1343
1353
(
1975
).
49.
G. L.
Kellogg
,
Phys. Rev. Lett.
70
(
11
),
1631
1634
(
1993
).
50.
G. L.
Kellogg
,
Appl. Surf. Sci.
76
(
1–4
),
115
121
(
1994
).
51.
V.
Jansson
,
E.
Baibuz
,
A.
Kyritsakis
,
S.
Vigonski
,
V.
Zadin
,
S.
Parviainen
,
A.
Aabloo
, and
F.
Djurabekova
,
Nanotechnology
31
(
35
),
355301
(
2020
).
52.
T. X.
Phuoc
and
R. H.
Chen
,
Combust. Flame
159
(
1
),
416
419
(
2012
).
53.
A.
Kyritsakis
,
E.
Baibuz
,
V.
Jansson
, and
F.
Djurabekova
,
Phys. Rev. B
99
(
20
),
205418
(
2019
).
54.
E. Z.
Engelberg
,
J.
Paszkiewicz
,
R.
Peacock
,
S.
Lachmann
,
Y.
Ashkenazy
, and
W.
Wuensch
,
Phys. Rev. Accel. Beams
23
(
12
),
123501
(
2020
).
55.
E. Z.
Engelberg
,
Y.
Ashkenazy
, and
M.
Assaf
,
Phys. Rev. Lett.
120
(
12
),
124801
(
2018
).
56.
A. S.
Pohjonen
,
S.
Parviainen
,
T.
Muranaka
, and
F.
Djurabekova
,
J. Appl. Phys.
114
(
3
),
033519
(
2013
).
57.
A. S.
Pohjonen
,
F.
Djurabekova
,
A.
Kuronen
,
S. P.
Fitzgerald
, and
K.
Nordlund
,
Philos. Mag.
92
(
32
),
3994
4010
(
2012
).
58.
A. S.
Pohjonen
,
F.
Djurabekova
,
K.
Nordlund
,
A.
Kuronen
, and
S. P.
Fitzgerald
,
J. Appl. Phys.
110
(
2
),
023509
(
2011
).
59.
N.
Li
,
X.
Gao
,
X.
Feng
,
K.
Wu
,
Y.
Cheng
, and
B.
Xiao
, arXiv:2310.04751 (
2023
).
60.
K.
Kuppart
,
S.
Vigonski
,
A.
Aabloo
,
Y.
Wang
,
F.
Djurabekova
,
A.
Kyritsakis
, and
V.
Zadin
,
Micromachines
12
(
10
),
1178
(
2021
).
61.
H. J.
Han
,
S.
Kumar
,
G. T.
Jin
,
X. Y.
Ji
,
J. L.
Hart
,
D. J.
Hynek
,
Q. P.
Sam
,
V.
Hasse
,
C.
Felser
,
D. G.
Cahill
,
R.
Sundararaman
, and
J. J.
Cha
,
Adv. Mater.
35
(
13
),
2208965
(
2023
).
62.
M.
Veske
,
A.
Kyritsakis
,
F.
Djurabekova
,
K. N.
Sjobak
,
A.
Aabloo
, and
V.
Zadin
,
Phys. Rev. E
101
(
5
),
053307
(
2020
).
63.
X. Y.
Gao
,
A.
Kyritsakis
,
M.
Veske
,
W. J.
Sun
,
B.
Xiao
,
G. D.
Meng
,
Y. H.
Cheng
, and
F.
Djurabekova
,
J. Phys. D
53
(
36
),
365202
(
2020
).
64.
S.
Nirantar
,
T.
Ahmed
,
M.
Bhaskaran
,
J. W.
Han
,
S.
Walia
, and
S.
Sriram
,
Adv. Intell. Syst.
1
(
4
),
1900039
(
2019
).
65.
P.
Zhang
,
A.
Valfells
,
L. K.
Ang
,
J. W.
Luginsland
, and
Y. Y.
Lau
,
Appl. Phys. Rev.
4
(
1
),
011304
(
2017
).
66.
L. W.
Nordhlim
,
Proc. R. Soc. London A
121
,
626
639
(
1928
).
67.
R. H.
Fowler
and
L.
Nordheim
,
Proc. R. Soc. London A
119
,
173
181
(
1928
).
68.
N.
Egorov
and
E.
Sheshin
,
Field Emission Electronics
(
Springer
,
2017
).
69.
N. F.
George
,
Field Emission in Vacuum Microelectronics
(
George
,
2005
).
70.
P. H.
Cutler
,
J.
He
,
N. M.
Miskovsky
,
T. E.
Sullivan
, and
B.
Weiss
,
J. Vac. Sci. Technol. B
11
(
2
),
387
391
(
1993
).
71.
G. N.
Fursey
and
D. V.
Glazanov
,
J. Vac. Sci. Technol. B
16
(
2
),
910
915
(
1998
).
72.
J.
He
,
P.
Cutler
,
N.
Miskovsky
,
T.
Feuchtwang
,
T.
Sullivan
, and
M.
Chung
,
Surf. Sci.
246
(
1–3
),
348
364
(
1991
).
73.
D.
Biswas
and
R.
Ramachandran
,
Phys. Plasmas
24
(
7
),
079901
(
2017
).
74.
K. L.
Jensen
,
D. A.
Shiffler
,
J. R.
Harris
,
I. M.
Rittersdorf
, and
J. J.
Petillo
,
J. Vac. Sci. Technol. B
35
(
2
),
02C101
(
2017
).
75.
A.
Kyritsakis
,
J. P.
Xanthakis
, and
D.
Pescia
,
Proc. R. Soc. A
470
(
2166
),
20130795
(
2014
).
76.
K. L.
Jensen
,
D. A.
Shiffler
,
M.
Peckerar
,
J. R.
Harris
, and
J. J.
Petillo
,
J. Appl. Phys.
122
(
6
),
064501
(
2017
).
77.
A. A.
Patterson
and
A. I.
Akinwande
,
J. Appl. Phys.
117
(
17
),
174311
(
2015
).
78.
A. A.
Patterson
and
A. I.
Akinwande
,
J. Vac. Sci. Technol. B
38
(
2
),
023206
(
2020
).
79.
M.
Zubair
,
Y. S.
Ang
, and
L. K.
Ang
,
IEEE Trans. Electron Devices
65
(
6
),
2089
2095
(
2018
).
80.
C. D.
Child
,
Phys. Rev.
32
(
5
),
0492
0511
(
1911
).
81.
I.
Langmuir
,
Phys. Rev.
2
(
6
),
450
486
(
1913
).
82.
K. L.
Jensen
,
D. A.
Shiffler
,
I. M.
Rittersdorf
,
J. L.
Lebowitz
,
J. R.
Harris
,
Y. Y.
Lau
,
J. J.
Petillo
,
W.
Tang
, and
J. W.
Luginsland
,
J. Appl. Phys.
117
(
19
), 194902 (
2015
).
83.
S.
Sun
and
L. K.
Ang
,
Phys. Plasmas
19
(
3
),
033107
(
2012
).
84.
K. L.
Jensen
,
D. A.
Shiffler
,
J. J.
Petillo
,
Z. G.
Pan
, and
J. W.
Luginsland
,
Phys. Rev. Spec. Top.-Accel. Beams
17
(
4
),
043402
(
2014
).
85.
Y. Y.
Lau
,
J. Appl. Phys.
61
(
1
),
36
44
(
1987
).
86.
S.
Sun
and
L. K.
Ang
,
J. Appl. Phys.
113
(
14
),
144902
(
2013
).
87.
K.
Torfason
,
A.
Valfells
, and
A.
Manolescu
,
Phys. Plasmas
23
(
12
),
123119
(
2016
).
88.
Y. B.
Zhu
and
L. K.
Ang
,
Phys. Plasmas
22
(
5
),
052106
(
2015
).
89.
G.
Singh
,
R.
Kumar
, and
D.
Biswas
,
Phys. Plasmas
27
(
10
),
104501
(
2020
).
90.
N. R. S.
Harsha
and
A. L.
Garner
,
IEEE Trans. Electron Devices
68
(
12
),
6525
6531
(
2021
).
91.
N. R. S.
Harsha
,
M.
Pearlman
,
J.
Browning
, and
A. L.
Garner
,
Phys. Plasmas
28
(
12
),
122103
(
2021
).
92.
M.
Zubair
and
L. K.
Ang
,
Phys. Plasmas
23
(
7
),
072118
(
2016
).
93.
Y. B.
Zhu
,
P.
Zhang
,
A.
Valfells
,
L. K.
Ang
, and
Y. Y.
Lau
,
Phys. Rev. Lett.
110
(
26
),
265007
(
2013
).
94.
A. L.
Garner
,
A. M.
Darr
, and
N. R. S.
Harsha
,
IEEE Trans. Plasma Sci.
50
(
9
),
2528
2540
(
2022
).
95.
N. R. S.
Harsha
,
J. M.
Halpern
,
A. M.
Darr
, and
A. L.
Garner
,
Phys. Rev. E
106
(
6
),
L063201
(
2022
).
96.
X. J.
Zhu
,
N. R. S.
Harsha
, and
A. L.
Garner
,
J. Appl. Phys.
134
(
11
),
113301
(
2023
).
97.
Y. Y.
Lau
,
Y. F.
Liu
, and
R. K.
Parker
,
Phys. Plasmas
1
(
6
),
2082
2085
(
1994
).
98.
R. G.
Forbes
,
J. Appl. Phys.
104
(
8
),
084303
(
2008
).
99.
Y.
Feng
and
J. P.
Verboncoeur
,
Phys. Plasmas
13
(
7
),
073105
(
2006
).
100.
M. C.
Lin
,
J. Vac. Sci. Technol. B
23
(
2
),
636
639
(
2005
).
101.
W. B.
Nottingham
,
Phys. Rev.
49
(
1
),
78
97
(
1936
).
102.
F. M.
Charbonnier
,
R. W.
Strayer
,
L. W.
Swanson
, and
E. E.
Martin
,
Phys. Rev. Lett.
13
(
13
),
397
(
1964
).
103.
K. L.
Jensen
,
Mod. Dev. Vac. Electron Sources
135
,
345
385
(
2020
).
104.
J.
Norem
,
V.
Wu
,
A.
Moretti
,
M.
Popovic
,
Z.
Qian
,
L.
Ducas
,
Y.
Torun
, and
N.
Solomey
,
Phys. Rev. Spec. Top.-Accel. Beams
6
(
8
),
089901
(
2003
).
105.
M. G.
Ancona
,
J. Vac. Sci. Technol. B
13
(
6
),
2206
2214
(
1995
).
106.
Y.
Feng
,
J. P.
Verboncoeur
, and
M. C.
Lin
,
Phys. Plasmas
15
(
4
),
043301
(
2008
).
107.
P. Y.
Chen
,
T. C.
Cheng
,
J. H.
Tsai
, and
Y. L.
Shao
,
Nanotechnology
20
(
40
),
405202
(
2009
).
108.
A.
Kyritsakis
,
M.
Veske
, and
F.
Djurabekova
,
New J. Phys.
23
(
6
),
063003
(
2021
).
109.
K. L.
Jensen
,
J. Appl. Phys.
107
(
7
),
079902
(
2010
).
110.
K. L.
Jensen
,
J. Vac. Sci. Technol. B
29
(
2
),
02b101
(
2011
).
111.
O. W.
Richardson
and
A. F. A.
Young
,
Proc. R. Soc. London A
107
(
743
),
377
410
(
1925
).
112.
A. M.
Darr
,
C. R.
Darr
, and
A. L.
Garner
,
Phys. Rev. Res.
2
(
3
),
033137
(
2020
).
113.
J. G.
Simmons
,
J. Appl. Phys.
34
(
6
),
1793
1803
(
1963
).
114.
J. G.
Simmons
,
J. Appl. Phys.
35
(
8
),
2472
2481
(
1964
).
115.
Y. Y.
Lau
,
D.
Chernin
,
D. G.
Colombant
, and
P. T.
Ho
,
Phys. Rev. Lett.
66
(
11
),
1446
1449
(
1991
).
116.
L. K.
Ang
,
T. J. T.
Kwan
, and
Y. Y.
Lau
,
Phys. Rev. Lett.
91
(
20
),
208303
(
2003
).
117.
L. K.
Ang
,
Y. Y.
Lau
, and
T.
Kwan
,
IEEE Trans. Plasma Sci.
32
(
21
),
410
412
(
2004
).
118.
L. K.
Ang
and
P.
Zhang
,
Phys. Rev. Lett.
98
(
16
),
164802
(
2007
).
119.
L. K.
Ang
,
W. S.
Koh
,
Y. Y.
Lau
, and
T. J. T.
Kwan
,
Phys. Plasmas
13
(
5
),
056701
(
2006
).
120.
W. S.
Koh
and
L. K.
Ang
,
Appl. Phys. Lett.
89
(
18
),
183107
(
2006
).
121.
C.
Chua
,
C. Y.
Kee
,
Y. S.
Ang
, and
L. K.
Ang
,
Phys. Rev. Appl.
16
(
6
),
064025
(
2021
).
122.
K.
Yeong
and
J.
Thong
,
J. Appl. Phys.
100
(
11
),
114325
(
2006
).
123.
K. S.
Yeong
and
J. T. L.
Thong
,
J. Appl. Phys.
99
(
10
),
104903
(
2006
).
124.
Y. F.
Huang
,
Z. X.
Deng
,
W. L.
Wang
,
C. L.
Liang
,
J. C.
She
,
S. Z.
Deng
, and
N. S.
Xu
,
Sci. Rep.
5
,
10631
(
2015
).
125.
M. A.
Reed
,
C.
Zhou
,
C. J.
Muller
,
T. P.
Burgin
, and
J. M.
Tour
,
Science
278
(
5336
),
252
254
(
1997
).
126.
H.
Park
,
A. K. L.
Lim
,
A. P.
Alivisatos
,
J.
Park
, and
P. L.
McEuen
,
Appl. Phys. Lett.
75
(
2
),
301
303
(
1999
).
127.
Y. V.
Kervennic
,
H. S. J.
Van der Zant
,
A. F.
Morpurgo
,
L.
Gurevich
, and
L. P.
Kouwenhoven
,
Appl. Phys. Lett.
80
(
2
),
321
323
(
2002
).
128.
P.
Steinmann
and
J. M. R.
Weaver
,
J. Vac. Sci. Technol. B
22
(
6
),
3178
3181
(
2004
).
129.
P.
Kumar
and
K.
Sangeeth
,
Nanosci. Nanotechnol. Lett.
1
(
3
),
194
198
(
2009
).
130.
A. F.
Morpurgo
,
C. M.
Marcus
, and
D. B.
Robinson
,
Appl. Phys. Lett.
74
(
14
),
2084
2086
(
1999
).
131.
N. S.
Rajput
,
A. K.
Singh
, and
H. C.
Verma
,
J. Appl. Phys.
112
(
2
),
024310
(
2012
).
132.
Y.
Li
,
F.
Zhan
,
J.
Tang
,
Y.
Cheng
, and
G.
Meng
, in
30th International Symposium on Discharges and Electrical Insulation in Vacuum (ISDEIV)
(
IEEE
,
2023
), pp.
1
4
.
133.
Y.
Hirata
,
K.
Ozaki
,
U.
Ikeda
, and
M.
Mizoshiri
,
Thin Solid Films
515
(
9
),
4247
4250
(
2007
).
134.
T.
Muranaka
,
T.
Blom
,
K.
Leifer
, and
V.
Ziemann
,
Nucl. Instrum. Methods Phys. Res. Sect. A
657
(
1
),
122
125
(
2011
).
135.
J.
Ögren
,
S. H. M.
Jafri
,
K.
Leifer
, and
V. G.
Ziemann
, “
Surface characterization and field emission measurements of copper samples inside a scanning electron microscope
”, in
Proc. 7th Int. Particle Accelerator Conf. (IPAC'16), Busan, Korea, May 2016
(
IPAC
,
2016
), pp.
283
285
.
136.
T.
Muranaka
,
T.
Blom
,
K.
Leifer
et al., “
European coordination for accelerator research and development CON
,” (
2011
); available at https://cds.cern.ch/record/1407219 EuCARD-CON-
2011
-022.
137.
G. D.
Meng
,
Y. H.
Cheng
,
C. Y.
Dong
,
X. Y.
Gao
,
K. J.
Wang
, and
D. J.
Zhang
, in
28th International Symposium on Discharges and Electrical Insulation in Vacuum
(
IEEE
,
2018
), pp.
15
18
.
138.
G. D.
Meng
,
Y. S.
Cheng
,
L.
Chen
,
Y.
Chen
, and
K.
Wu
, in
IEEE International Conference on Solid Dielectrics (ICSD)
(
IEEE
,
2013
), pp.
662
665
.
139.
G. D.
Meng
,
Y. H.
Cheng
,
C. Y.
Dong
, and
K.
Wu
, in
Proceedings of 2014 International Symposium on Electrical Insulating Materials
(
IEEE
,
2014
), pp.
417
420
.
140.
C.
Men
,
Y. H.
Cheng
,
B. W.
Zhu
,
W. J.
Song
,
C. Y.
Dong
, and
G. D.
Meng
, in
International Conference on Condition Monitoring and Diagnosis (CMD
) (
IEEE
,
2016
), pp.
222
225
.
141.
C. Y.
Dong
,
Y. H.
Cheng
,
C.
Men
,
L.
Chen
,
B. W.
Zhu
, and
G. D.
Meng
, in
IEEE International Conference on Dielectrics (ICD)
(
IEEE
,
2016
), Vol. 1–2, pp.
1139
1142
.
142.
G. D.
Meng
,
Y. H.
Cheng
,
K.
Wu
, and
L.
Chen
,
IEEE Trans. Dielectr. Electr. Insul.
21
(
4
),
1950
1956
(
2014
).
143.
G. D.
Meng
,
Y. H.
Cheng
,
C. Y.
Dong
,
L.
Chen
,
B. W.
Zhu
, and
C.
Men
, in
IEEE International Conference on Dielectrics (ICD)
(
IEEE
,
2016
), pp.
1159
1162
.
144.
G. D.
Meng
,
Y. H.
Cheng
,
C. Y.
Dong
,
X. Y.
Gao
,
D. J.
Zhang
,
K. J.
Wang
, and
C.
Men
, in
IEEE 2nd International Conference on Dielectrics (ICD)
(
IEEE
,
2018
), pp.
1
4
.
145.
S.
Itoh
,
T.
Niiyama
, and
M.
Yokoyama
,
J. Vac. Sci. Technol. B
11
(
3
),
647
650
(
1993
).
146.
R.
Bhattacharya
,
M.
Cannon
,
G.
Rughoobur
,
N.
Karaulac
,
W.
Chern
,
R. F.
Asadi
,
Z.
Tao
,
B. E.
Gnade
,
A. I.
Akinwande
, and
J.
Browning
,
J. Vac. Sci. Technol. B
41
(
5
),
053201
(
2023
).
147.
R.
Bhattacharya
,
M.
Cannon
,
R.
Bhattacharjee
,
G.
Rughoobur
,
N.
Karaulac
,
W.
Chern
,
A. I.
Akinwande
, and
J.
Browning
,
J. Vac. Sci. Technol. B
40
(
1
),
010601
(
2022
).
148.
R.
Bhattacharya
,
M.
Turchetti
,
M. T.
Yeung
,
P. D.
Keathley
,
K. K.
Berggren
, and
J.
Browning
,
J. Vac. Sci. Technol. B
41
(
6
),
063202
(
2023
).
149.
R.
Gupta
and
B. G.
Willis
,
Appl. Phys. Lett.
90
(
25
),
253102
(
2007
).
150.
S.
Luo
,
B. H.
Hoff
,
S. A.
Maier
, and
J. C.
deMello
, “
Scalable Fabrication of Metallic Nanogaps at the Sub-10 nm Level
Adv. Sci.
2021
,
8
,
2102756
.
151.
S. J.
Heo
,
J. H.
Shin
,
B. O.
Jun
, and
J. E.
Jang
,
ACS Nano
17
(
19
),
18792
18804
(
2023
).
152.
S.
Nirantar
,
B.
Patil
,
D. C.
Tripathi
,
N.
Sethu
,
R. V.
Narayanan
,
J.
Tian
,
M.
Bhaskaran
,
S.
Walia
, and
S.
Sriram
,
Small
18
(
47
),
2203234
(
2022
).
153.
Y.
Tomoda
,
K.
Takahashi
,
M.
Hanada
,
W.
Kume
, and
J.
Shirakashi
,
J. Vac. Sci. Technol. B
27
(
2
),
813
816
(
2009
).
154.
A. A. G.
DriskillSmith
,
D. G.
Hasko
, and
H.
Ahmed
,
Appl. Phys. Lett.
71
(
21
),
3159
3161
(
1997
).
155.
S.
Brimley
,
M. S.
Miller
, and
M. J.
Hagmann
,
J. Appl. Phys.
109
(
9
),
094510
(
2011
).
156.
A. K.
Singh
and
J.
Kumar
,
J. Appl. Phys.
113
(
5
),
053303
(
2013
).
157.
S.
Bhattacharjee
,
A.
Vartak
, and
V.
Mukherjee
,
Appl. Phys. Lett.
92
(
19
),
191503
(
2008
).
158.
S.
Bhattacharjee
and
T.
Chowdhury
,
Appl. Phys. Lett.
95
(
6
),
061501
(
2009
).
159.
A. M.
Loveless
,
A. M.
Darr
, and
A. L.
Garner
,
Phys. Plasmas
28
(
4
),
042110
(
2021
).
160.
H. I.
Lee
,
S. S.
Park
,
D. I.
Park
,
S. H.
Hahm
,
J. H.
Lee
, and
J. H.
Lee
,
J. Vac. Sci. Technol. B
16
(
2
),
762
764
(
1998
).
161.
H. X.
Wang
,
R. S.
Brayfield
,
A. M.
Loveless
,
A. M.
Darr
, and
A. L.
Garner
,
Appl. Phys. Lett.
120
(
12
),
124103
(
2022
).
162.
H.
Cabrera
,
D. A.
Zanin
,
L. G.
De Pietro
,
T.
Michaels
,
P.
Thalmann
,
U.
Ramsperger
,
A.
Vindigni
,
D.
Pescia
,
A.
Kyritsakis
,
J. P.
Xanthakis
,
F. X.
Li
, and
A.
Abanov
,
Phys. Rev B
87
(
11
),
115436
(
2013
).
163.
M.
Vincent
,
S. W.
Rowe
,
C.
Poulain
,
D.
Mariolle
,
L.
Chiesi
,
F.
Houzé
, and
J.
Delamare
,
Appl. Phys. Lett.
97
(
26
),
263503
(
2010
).
164.
C.
Poulain
,
A.
Peschot
,
M.
Vincent
, and
N.
Bonifaci
, in
IEEE 57th Holm Conference on Electrical Contacts (Holm)
(
IEEE
,
2011
), pp.
1
7
.
165.
D. A.
Zanin
,
H.
Cabrera
,
L. G.
De Pietro
,
M.
Pikulski
,
M.
Goldmann
,
U.
Ramsperger
,
D.
Pescia
, and
J. P.
Xanthakis
, “Fundamental aspects of near-field emission scanning electron microscopy,” in
Advances in Imaging and Electron Physics
(Elsevier,
2012
), Vol.
170
, pp.
227
258
.
166.
N. T. H.
Farr
,
G. M.
Hughes
, and
C.
Rodenburg
,
Materials
14
(
11
),
3034
(
2021
).
167.
X. L.
Wei
,
Y.
Liu
,
Q.
Chen
, and
L. M.
Peng
,
Nanotechnology
19
(
35
),
355304
(
2008
).
168.
R. K.
Hart
,
T. F.
Kassner
, and
J. K.
Maurin
,
Philos. Mag.
21
(
171
),
453
467
(
1970
).
169.
H. X.
Wang
,
A. M.
Loveless
,
A. M.
Darr
, and
A. L.
Garner
,
J. Vac. Sci. Technol. B
40
(
6
),
062805
(
2022
).
170.
F.
Djurabekova
,
S.
Parviainen
,
A.
Pohjonen
, and
K.
Nordlund
,
Phys. Rev E
83
(
2
),
026704
(
2011
).
171.
A.
Descoeudres
,
Y.
Levinsen
,
S.
Calatroni
,
M.
Taborelli
, and
W.
Wuensch
,
Phys. Rev. Spec. Top.-Accel. Beams
12
(
9
),
092001
(
2009
).
172.
V.
Zadin
,
A.
Pohjonen
,
A.
Aabloo
,
K.
Nordlund
, and
F.
Djurabekova
,
Phys. Rev. Spec. Top.-Accel. Beams
17
(
10
),
103501
(
2014
).
173.
V.
Jansson
,
E.
Baibuz
,
A.
Kyritsakis
, and
F.
Djurabekova
, in
International Vacuum Nanoelectronics Conference
(
2017
).
174.
S.
Parviainen
,
F.
Djurabekova
,
A.
Pohjonen
, and
K.
Nordlund
,
Nucl. Instrum. Methods Phys. Res. Sect. B
269
(
14
),
1748
1751
(
2011
).
175.
D. J.
Griffiths
,
Introduction to Electrodynamics
(
American Association of Physics Teachers
,
2005
).
176.
F. W.
Hehl
and
Y. N.
Obukhov
,
Foundations of Classical Electrodynamics: Charge, Flux, and Metric
(
Springer Science & Business Media
,
2003
).
177.
J. D.
Jackson
,
Classical Electrodynamics
(
American Association of Physics Teachers
,
1999
).
178.
S.
Parviainen
,
F.
Djurabekova
,
H.
Timko
, and
K.
Nordlund
,
Comput. Mater. Sci.
50
(
7
),
2075
2079
(
2011
).
179.
M.
Veske
,
S.
Parviainen
,
V.
Zadin
,
A.
Aabloo
, and
F.
Djurabekova
,
J. Phys. D
49
(
21
),
215301
(
2016
).
180.
K.
Eimre
,
S.
Parviainen
,
A.
Aabloo
,
F.
Djurabekova
, and
V.
Zadin
,
J. Appl. Phys.
118
(
3
),
033303
(
2015
).
181.
A.
Kyritsakis
and
F.
Djurabekova
,
Comput. Mater. Sci.
128
,
15
21
(
2017
).
182.
M.
Veske
,
A.
Kyritsakis
,
F.
Djurabekova
,
R.
Aare
,
K.
Eimre
, and
V.
Zadin
, in
29th International Vacuum Nanoelectronics Conference (IVNC)
(
IEEE
,
2016
), pp.
1
2
.
183.
M.
Veske
,
A.
Kyritsakis
,
K.
Eimre
,
V.
Zadin
,
A.
Aabloo
, and
F.
Djurabekova
,
J. Comput. Phys.
367
,
279
294
(
2018
).
184.
A.
Kyritsakis
,
M.
Veske
,
K.
Eimre
,
V.
Zadin
, and
F.
Djurabekova
,
J. Phys. D
51
(
22
),
225203
(
2018
).
185.
X. Y.
Gao
,
N.
Li
,
A.
Kyritsakis
,
M.
Veske
,
C. Y.
Dong
,
K.
Wu
,
B.
Xiao
,
F.
Djurabekova
, and
Y. H.
Cheng
,
J. Phys. D
55
(
33
),
335201
(
2022
).
186.
X. Y.
Gao
,
N.
Li
,
Z. F.
Song
,
K.
Wu
,
Y. H.
Cheng
, and
B.
Xiao
,
J. Phys. D
56
(
26
),
265203
(
2023
).
187.
C.
Deng
and
F.
Sansoz
,
Acta Mater.
57
(
20
),
6090
6101
(
2009
).
188.
H.
Zheng
,
A. J.
Cao
,
C. R.
Weinberger
,
J. Y.
Huang
,
K.
Du
,
J. B.
Wang
,
Y. Y.
Ma
,
Y. N.
Xia
, and
S. X.
Mao
,
Nat. Commun.
1
,
144
(
2010
).
189.
S. X.
Zheng
and
S. X.
Mao
,
Extreme Mech. Lett.
45
,
101284
(
2021
).
190.
J. W.
Wang
,
F.
Sansoz
,
J. Y.
Huang
,
Y.
Liu
,
S. H.
Sun
,
Z.
Zhang
, and
S. X.
Mao
,
Nat. Commun.
4
,
1742
(
2013
).
191.
S. B.
Cárceles
,
A.
Kyritsakis
,
V.
Zadin
,
A.
Mavalankar
, and
I.
Underwood
, in
International Vacuum Nanoelectronics Conference
(
2023
).
192.
M.
Biesuz
,
T.
Saunders
,
D. Y.
Ke
,
M. J.
Reece
,
C. F.
Hu
, and
S.
Grasso
,
J. Mater. Sci. Technol.
69
,
239
272
(
2021
).
193.
Y. S.
Ang
,
S. J.
Liang
, and
L. K.
Ang
,
MRS Bull.
42
(
7
),
505
510
(
2017
).
194.
Y. S.
Ang
,
L. M.
Cao
, and
L. K.
Ang
,
Infomat
3
(
5
),
502
535
(
2021
).
195.
Y. S.
Ang
,
Y. Y.
Chen
,
C.
Tan
, and
L. K.
Ang
,
Phys. Rev. Appl.
12
(
1
),
014057
(
2019
).
196.
Z. S.
Wu
,
S. F.
Pei
,
W. C.
Ren
,
D. M.
Tang
,
L. B.
Gao
,
B. L.
Liu
,
F.
Li
,
C.
Liu
, and
H. M.
Cheng
,
Adv. Mater.
21
(
17
),
1756
1760
(
2009
).
197.
G. T.
Wu
,
X. L.
Wei
,
Z. Y.
Zhang
,
Q.
Chen
, and
L. M.
Peng
,
Adv. Funct. Mater.
25
(
37
),
5972
5978
(
2015
).
198.
G. T.
Wu
,
X. L.
Wei
,
S.
Gao
,
Q.
Chen
, and
L. M.
Peng
,
Nat. Commun.
7
,
11513
(
2016
).
199.
X. L.
Wei
,
Y.
Bando
, and
D.
Golberg
,
ACS Nano
6
(
1
),
705
711
(
2012
).
200.
S.
Kumar
,
G. S.
Duesberg
,
R.
Pratap
, and
S.
Raghavan
,
Appl. Phys. Lett.
105
(
10
),
103107
(
2014
).
201.
A.
Di Bartolomeo
,
F.
Giubileo
,
L.
Iemmo
,
F.
Romeo
,
S.
Russo
,
S.
Unal
,
M.
Passacantando
,
V.
Grossi
, and
A. M.
Cucolo
,
Appl. Phys. Lett.
109
(
2
),
023510
(
2016
).
202.
L. K.
Ang
,
Y. S.
Ang
, and
C. H.
Lee
,
Phys. Plasmas
30
(
3
),
033103
(
2023
).
203.
W. J.
Chan
,
C.
Chua
,
Y. S.
Ang
, and
L. K.
Ang
,
IEEE Trans. Plasma Sci.
51
(
7
),
1656
1670
(
2023
).
204.
W. J.
Chan
,
Y. S.
Ang
, and
L. K.
Ang
,
Phys. Rev. B
104
(
24
),
245420
(
2021
).
205.
V. S.
Sukhomlinov
,
A. S.
Mustafaev
,
H.
Koubaji
,
N. A.
Timofeev
, and
A.
Zaitsev
,
J. Phys. Soc. Jpn.
92
(
4
),
044501
(
2023
).
206.
V. S.
Sukhomlinov
,
A. S.
Mustafaev
,
H.
Koubaji
,
N. A.
Timofeev
,
O. G. M.
Hiller
, and
G.
Zissis
,
Phys. Plasmas
29
(
9
),
093103
(
2022
).
207.
A.
Andronov
,
E.
Budylina
,
P.
Shkitun
,
P.
Gabdullin
,
N.
Gnuchev
,
O.
Kvashenkina
, and
A.
Arkhipov
,
J. Vac. Sci. Technol. B
36
(
2
),
02c108
(
2018
).
208.
R.
Smerdov
and
A.
Mustafaev
,
J. Appl. Phys.
134
(
11
),
114903
(
2023
).
209.
G. N.
Fursey
,
M. A.
Polyakov
,
N. T.
Bagraev
,
I. I.
Zakirov
,
A. V.
Nashchekin
, and
V. N.
Bocharov
,
J. Surf. Investig.
13
(
5
),
814
824
(
2019
).
210.
S. V.
Filippov
,
A. G.
Kolosko
,
E. O.
Popov
, and
R. G.
Forbes
,
R. Soc. Open Sci.
9
(
11
),
220748
(
2022
).
211.
F.
Ruffino
and
M. G.
Grimaldi
,
Phys. Status Solidi A
212
(
8
),
1662
1684
(
2015
).
212.
F.
Niekiel
,
P.
Schweizer
,
S. M.
Kraschewski
,
B.
Butz
, and
E.
Spiecker
,
Acta Mater.
90
,
118
132
(
2015
).