Space Charge Limited Emission Research Articles

Advanced space charge limited field emission cathodes.

Several novel field emission cathodes are constructed by attaching a variety of materials to a steel substrate using multiple methods of bonding. These cathodes are tested using a pulsed relativistic (350-550kV) vircator in a space charge limited operating regime. The viability of these cathodes as advanced or alternative components in high power microwave systems is evaluated. The currents measured from the cathodes are compared to values predicted from a 3D space charge limited flow model. The performance of the cathode is determined by how the current deviates from the model, as well as the consistency and durability.

Efficient kinetic particle simulations of space charge limited emission in magnetically insulated transmission lines using reduced physics models

Efficient kinetic particle simulations of space charge limited emission in magnetically insulated transmission lines using reduced physics models

Transitional behaviors of fractional dimensional models of field-induced and space-charge limited emission from rough surfaces

Electron emission is an important physical phenomenon in both vacuum and solid-state devices. Previously, the effects of surface roughness or interface irregularity have been incorporated in the fractional Fowler–Nordheim (FNα) equation for field emission, the fractional Child–Langmuir (CLα) law for space-charge limited emission (SCLE) in vacuum, and the fractional Mott–Gurney (MGα) law for SCLE in the presence of carrier collisions using a fractional dimensional approach, where the fractional dimensional parameter (0<α≤1) defines the degree of roughness at electrode surface or irregularity of the gap medium. Here, the transitions between FNα, CLα, and MGα regimes have been investigated for electron emission from rough surfaces. It is shown that the fractional second-order or third-order nexuses, where at least two or three of the FNα, CLα, and MGα regimes converge, respectively, have a strong dependence on the surface conditions as captured by the parameter α. This work provides an important insight into the mechanism of electron emission in electron emitters where surface roughness is inevitable due to practical experimental fabrication and/or emitter surface degradation over time. The model shall serve as a versatile analytical tool to characterize the roughness by fitting the model to the experimental measurements.

Transitions between field emission and vacuum breakdown in nanoscale gaps

The continuing reduction in device size motivates a more fundamental understanding of breakdown and electron emission for nanoscale gaps. While prior experiments have separately studied breakdown and electron emission in vacuum gaps, no study has comprehensively examined the transitions between these mechanisms. In this study, we measure the current-voltage (I−V) curves for electrodes with different emitter widths for 20–800 nm gaps at vacuum (∼1 μTorr) to measure breakdown voltage and assess electron emission behavior. The breakdown voltage Vb increases linearly with increasing gap distance from ∼15 V at 20 nm to ∼220 V at 300 nm and remains nearly constant for larger gaps; Vb does not depend strongly on the emitter width. Breakdown can proceed directly from the field emission regime. Nexus theory, which predicts transitions between space-charge limited current (SCLC) and field emission (FE), shows that the experimental conditions are in the Fowler–Nordheim regime and within a factor of 0.7 to the FE-SCLC transition. We also present the results of electrode damage by emission current-induced heating to explain the flattening of Vb at larger gaps that was absent in previous experiments for similar gap distances at atmospheric pressure.

The transition from thermionic to space charge limited emission for needle-like surfaces—A particle-in-cell simulation study

We perform particle-in-cell simulations of thermionic emission from hemi-ellipsoidal shaped needle emitters. The Schottky corrected Richardson–Dushman equation is used to compute the emitted current from the curved emitter surface. It is shown that the transmitted thermionic current increases with cathode temperature before eventually settling to the enhanced space charge limited (SCL) current. It is further shown that the critical temperature at which transition to the saturation region occurs increases with the height of the emitter, leading to a softening in the transition. This may be attributed to the non-uniformity in the cathode work function as in planar diodes. The SCL current is found to increase linearly with the electrostatic apex field enhancement factor as predicted recently [Singh et al., Phys. Plasmas 27, 104501 (2020)].

Numerical simulation of a relativistic magnetron using a fluid electron model

An approach to numerically modeling relativistic magnetrons, in which the electrons are represented with a relativistic fluid, is described. A principal effect in the operation of a magnetron is space-charge-limited (SCL) emission of electrons from the cathode. We have developed an approximate SCL emission boundary condition for the fluid electron model. This boundary condition prescribes the flux of electrons as a function of the normal component of the electric field on the boundary. We show the results of a benchmarking activity that applies the fluid SCL boundary condition to the one-dimensional Child–Langmuir diode problem and a canonical two-dimensional diode problem. Simulation results for a two-dimensional A6 magnetron are then presented. Computed bunching of the electron cloud occurs and coincides with significant microwave power generation. Numerical convergence of the solution is considered. Sharp gradients in the solution quantities at the diocotron resonance, spanning an interval of three to four grid cells in the most well-resolved case, are present and likely affect convergence.

Ampere-class bright field emission cathode operated at 100 MV/m

High-current bright sources are needed to power the next generation of compact rf and microwave systems. A major requirement is that such sources could be sustainably operated at high frequencies, well above 1 GHz, and high gradients, well above $100\text{ }\text{ }\mathrm{MV}/\mathrm{m}$. Field emission sources offer simplicity and scalability in a high-frequency era of the injector design, but the output rf cycle charge and high-gradient operation remain a great and largely unaddressed challenge. Here, a field emission cathode based on ultra-nano-crystalline diamond, an efficient planar field emission material, was tested at $100\text{ }\text{ }\mathrm{MV}/\mathrm{m}$ in an $L$-band injector. A very high charge of 38 pC per rf cycle was demonstrated (300 nC per rf pulse corresponding to an rf pulse current of 120 mA). This operating condition revealed a two-dimensional space charge limited emission where the one-dimensional Child-Langmuir limit was surpassed. An injector brightness of ${10}^{14}\text{ }\text{ }\mathrm{A}/(\mathrm{rad}\text{ }\text{ }\mathrm{m}{)}^{2}$ was estimated for the given operating conditions.

Incorporating photoemission into the theoretical unification of electron emission and space-charge limited current

Photon emitters are becoming increasingly important due to their ability to generate high brightness, low emittance, and spatiotemporally coherent electron bunches for multiple applications; however, these emitters rarely produce electrons solely due to photoemission. Often, photon emitters are prone to undesired thermionic emission; alternatively, some devices intentionally leverage field and thermionic emission to increase output current. Regardless, attempting to extract higher currents from these devices raises concerns about space-charge buildup. While theories have examined the transitions between many of these mechanisms, none have used a common framework to unify photo-, thermionic, field, and space-charge limited emission simultaneously, typically represented individually by the Fowler–Dubridge (FD), Richardson–Laue–Dushman (RLD), Fowler–Nordheim (FN), and Child–Langmuir (CL) equations, respectively. This paper derives an exact solution unifying these mechanisms and reports conditions where emission bypasses RLD to directly transition from FD to FN based on asymptotically matching the three models at a nexus point. Furthermore, we provide a step-by-step approach for developing nexus phase space plots exhibiting the operating conditions for transitions among FD, RLD, FN, CL, Mott–Gurney for space-charge limited current with collisions, and Ohm's law for an external resistor. We demonstrate the utility of nexus plots for assessing the applicability of the simple well-known theories based on a single mechanism or the necessity to use more complicated solutions combining multiple mechanisms. As such, nexus theory provides a simple framework for guiding theorists in model development, simulation experts in algorithm development and selection, and experimentalists in device design.

Low‐Energy Oxygen Plasma Injection of 2D Bi2Se3 Realizes Highly Controllable Resistive Random Access Memory

AbstractResistive random access memory (RRAM) based on ultrathin 2D materials is considered to be a very feasible solution for future data storage and neuromorphic computing technologies. However, controllability and stability are the problems that need to be solved for practical applications. Here, by introducing a damage‐less ion implantation technology using ultralow‐energy plasma, the transport mechanisms of space charge limited current and Schottky emission are successfully realized and controlled in RRAM based on 2D Bi2Se3 nanosheets. The memristors exhibit stable resistive switching behavior with a high resistive switching ratio (>104), excellent cycling endurances (300 cycles), and great retention performance (>104 s). The reliability and controllability of Bi2Se3 memory endowed by oxygen plasma injection demonstrate the great potential of this ultralow‐energy ion implantation technology in the application of 2D RRAM.

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

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

Space-Charge Effects in the Field-Assisted Thermionic Emission from Nonuniform Cathodes

We use computational simulations to study the electron emission and propagation in planar vacuum diodes. We show how space-charge affects thermionic emission from cathodes with two different values of work function that form a checkerboard pattern of finite extent on the cathode surface. We confirm that, for intermediate cathode temperature, the local current density from low work function regions can exceed the space-charge limit for the entire cathode. As the cathode temperature rises space-charge effects lead to homogeneous current density from the interior of the emitting area and a higher current density from its periphery. We show how beam emittance and brightness are affected, and show that the operating temperature for optimal brightness is such that it corresponds to the transition region between source-limited and space-charge limited emission. Finally we show how beam current and beam quality depend on how fine grained the structure of the cathode is.

Transitions between electron emission and gas breakdown mechanisms across length and pressure scales

This Perspective outlines theoretical, simulation, and experimental studies linking electron emission and gas breakdown. Many studies have investigated field emission-driven microscale gas breakdown, including recent reviews [Garner et al., IEEE Trans. Plasma Sci. 48, 808–824 (2020); Fu et al., Plasma Res. Express 2, 013001 (2020)]. This Perspective focuses on generalizing field emission-driven microscale gas breakdown to consider the contribution of other forms of electron emission, specifically thermionic and space-charge limited. Recent theoretical studies have unified thermionic, field, and space-charge limited emission with and without collisions to derive “nexuses” where the individual solutions match, indicating transitions in the mechanisms. Reducing device size to nanoscale at atmospheric pressure leads to a transition from field emission to space-charge limited emission for nitrogen at ∼250 nm. This Perspective summarizes the derivation of these nexuses and future extensions. We next describe simulation and theoretical studies for field emission-driven microscale gas breakdown and highlight how the nexus theory may be integrated to account for temperature, space-charge, and pulse parameters. Finally, we summarize the development of optical techniques to assess microscale gas breakdown and recent nanoscale experiments at atmospheric pressure that suggest that space-charge may begin to contribute to field emission prior to gas breakdown. We highlight the combination of theory, simulation, and experiment to link electron emission and gas breakdown mechanisms across length, pressure, and temperature scales for applications that include vacuum electronics, pulsed power, and medicine.

Theoretical analysis of the transition from field emission to space-charge-limited emission in liquids and gases

Discharge formation and breakdown in liquids have critical implications for water purification, medicine, and combustion, thereby motivating characterization of liquid behavior under high voltages. While several experiments with dielectric liquids have demonstrated that current transitions from Fowler–Nordheim (FN) scaling for field emission to Mott–Gurney (MG) scaling for space-charge-limited emission (SCLE) with increasing voltage, there is no common model linking electron emission from the liquid to gas phase. We show that applying a theory unifying FN, MG, and the Child–Langmuir law (CL) for SCLE at vacuum as a function of electron mobility and gap distance [Darr et al., Appl. Phys. Lett. 114, 014103 (2019)] yields excellent agreement with dielectric liquid emission experiments [Dotoku et al., J. Chem. Phys. 69, 1121 (1978)]. Specifically, current follows FN scaling at lower voltages before space charge contributions dominate, although none of the dielectric liquids considered achieve MG scaling in the experimental regime of interest. Considering a higher mobility representative of a vapor in the theory demonstrates the feasibility of achieving CL scaling for the gaps of the size considered experimentally at reasonably achievable applied voltages. Increasing the gap distance by an order of magnitude eliminates the contribution of space charge; decreasing gap distance by an order of magnitude causes a transition to MG. The implications of these results on electron emission in liquids and during a phase change to vapor and gas will be discussed.

Extinguishment of hot cathode discharges by space-charge and surface magnetic effects

Emission characteristics of a directly heated, rod-shaped LaB6 hot cathode has been studied in a multi-dipole confinement device. It is found that contrary to predictions from the conventional space-charge limited emission theory, emission current is not simply limited to a maximum value but eventually extinguished by space-charge related effects. In addition, directional preference of electron emission towards the positive lead of the applied heating voltage can be visibly observed. Results suggest that thermionically emitted electrons trapped in the virtual cathode are pumped towards the high voltage lead of the applied heating, and that the magnetic effects for filament heating currents exceeding 50 A may be the underlying mechanism of extinguishment; that is, the gyromotion of surface emitted electrons for sufficiently great heating currents may prevent electrons from escaping the virtual cathode. Geometrical effects are also briefly discussed.

Eulerian analysis of the electron flow emitted from an infinitely sharp electrified cone

The axisymmetric steady-state electron flow emitted from an electrified semi-infinite perfectly-conducting filament in vacuum is theoretically analyzed using an Eulerian model. The high symmetry allows for a solution in terms of self-similar functions, with radial dependence described by power-laws, leading to an eigenvalue problem for the angular dependence, which is numerically solved.Our results show a unique eigenvalue compatible with the boundary conditions, leading to a r−1.6308 dependence for the electron number density, r0.3692 for the electric potential, r0.1846 for the velocity field. Hence, for an infinitely thin filament, space charge limited emission is not compatible with our boundary conditions.

A new simple algorithm for space charge limited emission

Many high power electronic devices operate in a regime where the current they draw is limited by the self-fields of the particles. This space charge limited current poses particular challenges for numerical modeling where common techniques like over-emission or Gauss' Law are computationally inefficient or produce nonphysical effects. In this paper, we show an algorithm using the value of the electric field in front of the surface instead of attempting to zero the field at the surface, making the algorithm particularly well suited to both electromagnetic and parallel implementations of the particle-in-cell algorithm. We show how the algorithm is self-consistent within the framework of finite difference (for both electrostatics and electromagnetics). We show several 1D and 2D benchmarks against both theory and previous computational results. Finally, we show the application in 3D to high power microwave generation in a 13 GHz magnetically insulated line oscillator.

Oxygen annealing effect on resistive switching characteristics of multilayer CeO2/Al/CeO2 resistive random-access memory

Cerium oxide-based memristors have been extensively studied because of their compatibility with CMOS technology. Yet, inconsistency of resistive switching parameters is one of the main contests in development of nonvolatile memory for commercialization. Owing to filamentary nature of the resistive switching devices, variability of the resistive switching characteristics can be reduced by doping, where conductive filaments can easily grow due to reduction in the formation energy of oxygen vacancies. In this work, multilayer CeO2/Al/CeO2 films were prepared through radio-frequency (rf) sputtering at room temperature to study the effect of oxygen annealing on the resistive switching characteristics. Device with CeO2/Al/CeO2 multilayer structure after annealing exhibits reduction of defects and improved switching endurance, good data retention, and uniformity in operational parameters. The resistive switching characteristics have been simulated using space charge limited conduction and Schottky emission at high field region of the high resistance state, which is well fitted by linear curve fitting analysis. Improvement in the switching characteristics revealed that Al charge trapping layer has diffused into the CeO2 matrix, which might have resulted in lower density of oxygen vacancies due to oxygen annealing. Experimental I–V analysis indicate that oxygen annealing is an effective approach to enhance the switching characteristics of RRAM devices.

New variational calculus equation calculates space-charge limited current for any geometry

Rather than directly solving Poisson’s equation, researchers apply variational calculus to calculate space charge limited emission current density for cylindrical and spherical geometries and beyond.

A coordinate system invariant formulation for space-charge limited current in vacuum

While space-charge limited emission current density Jcr is calculated exactly for one-dimensional (1D) planar geometry, 1D cylindrical and spherical geometries require approximations such as the Langmuir-Blodgett (LB) equations or nonphysical assumptions. Using variational calculus (VC), we derive a differential equation from first principles to calculate Jcr for any geometry. This yields exact, closed-form analytical solutions for 1D coaxial cylindrical and concentric spherical geometries that approach LB for sufficiently close cathode (Rc) and anode (Ra) radii. VC agrees better with simulations in cylindrical geometry than LB at Rc/Ra = 0.5. The analytical VC solutions also demonstrate the asymptotic behavior for Jcr. For cylindrical geometry, Jcr ∝ 1/Rc2 as Rc/Ra approaches zero or infinity. For spherical geometry, Jcr ∝ 1/Rc2 as Rc/Ra → 0 and Jcr ∝ Ra2/Rc4 as Rc/Ra → ∞.

Ion current losses in the convolute and inner magnetically insulated transmission line of the Z machine

We introduce a 1D planar static model to elucidate the underlying mechanism of large ion current losses in the vacuum convolute and the inner magnetically insulated transmission line (MITL) of the $Z$ machine. We consider $\mathbf{E}\ifmmode\times\else\texttimes\fi{}\mathbf{B}$ electron flow, parallel to the electrodes, and ion motion across the vacuum gap, for given voltage $V$, gap distance $d$, anode magnetic field ${B}_{a}$, and vacuum electron current $\mathrm{\ensuremath{\Delta}}I$. This model has been introduced and solved before by Desjarlais [Phys. Rev. Lett. 59, 2295 (1987)] for the applied magnetic field ion diode. Here we apply it to convolute and inner MITL ion losses of $Z$, relaxing the fix magnetic flux condition of that reference. In the absence of ions we show that the electron vacuum flow must be close to the anode if its current exceeds the value given by the local flow impedance, implying high electric fields there. We then introduce space charge limited ion emission from the anode, neglecting the magnetic force on ions. We obtain the solution of the steady state equations for two special cases: (a) when both the electric potential and the electric field are zero inside the gap, and there is a layer of electrons not carrying current that neutralizes the ion charge between the virtual and the electrode cathode, making that region electric field free, and (b) when the electric field is zero inside the gap, but the potential is not, and zero electron charge between that point and the physical cathode. For case (a) we obtain an ion current density which we conjecture is the maximum attainable for any electron charge distribution in the electron current carrying layer, given $V,d,{B}_{a},\mathrm{\ensuremath{\Delta}}I$ an ion species. We obtain the enhancement factor for both cases with respect to the ion-only Child-Langmuir ion current density, and show that it can be significantly larger than that of the electron saturated flow case. Furthermore, imposing electron current conservation as the flow enters the inner MITL from the four outer MITLs, we recover the well-known dependence ${j}_{\mathrm{ion}}\ensuremath{\sim}{V}^{3/2}/{d}^{2}$, where voltage and gap are taken near the joining point of those outer MITLs. The implications and limitations of the proposed model are discussed.