Space Charge Limited Current Density Research Articles

Two-dimensional model of space-limited current in the weakly collisional regime for an inhomogenous medium

This short report describes the phenomenon of space-charge-limited (SCL) current transport between ballistic and collision-dominated regimes for an inhomogeneous medium with a finite emission area. This intermediate regime can be considered as a weakly collisional regime where the collisional mean free path is comparable to the length of the medium. The SCL current density is calculated as a function of the degree of collision, inhomogeneity of the medium, and the geometrical properties of the emitting area. The inhomogeneity of the medium is characterized by a parameter of (0<α≤1), where α = 1 denotes a perfect homogenous medium. The calculated SCL current density is enhanced by finite emission area effects by a factor of 1+F×G, where F measures the mean position of the electrons in the medium, and G is a geometrical correction factor due to finite emission area. The enhancement is found to be higher in the collisional regime as compared to the ballistic regime. A higher inhomogeneity (with smaller α) also increases the enhancement. Smooth transition between the fully ballistic SCL transport (Child–Langmuir model) and the collision-dominated SCL transport (Mott–Gurney model) is demonstrated and verified, respectively, by the particle-in-cell code and the device simulator. This model is useful for the characterization of high-current SCL transport where the non-ideal conditions (such as inhomogenous medium and weakly collisional regime) cannot be described by the existing SCL models.

Coordinate System Invariant Formulation of Fractional‐Dimensional Child‐Langmuir Law for a Rough Cathode

AbstractThis work presents an exact closed‐form analytical model of the space charge limited (SCL) current density for planar, cylindrical, and spherical geometries in fractional dimensional spaces (Fα) where α represents the fractional dimensions having a range 0 < α ⩽ 1. Decreasing value of α shows an increasing degree of surface roughness. This analytical model approaches Langmuir–Blodgett equations for sufficiently close anode (Ra) and cathode (Rc) radii for integer dimensions (α = 1). To calculate SCL current density for any geometry, variational calculus (VC) is used to derive the differential equations in fractional (α) dimensions. The effect of electrode radius ratios on space charge limited current is investigated here considering the surface roughness.

Space-charge-limited current density for nonplanar diodes with monoenergetic emission using Lie-point symmetries.

Understanding space-charge-limited current density (SCLCD) is fundamentally and practically important for characterizing many high-power and high-current vacuum devices. Despite this, no analytic equations for SCLCD with nonzero monoenergetic initial velocity have been derived for nonplanar diodes from first principles. Obtaining analytic equations for SCLCD for nonplanar geometries is often complicated by the nonlinearity of the problem and over constrained boundary conditions. In this Letter, we use the canonical coordinates obtained by identifying Lie-point symmetries to linearize the governing differential equations to derive SCLCD for any orthogonal diode. Using this method, we derive exact analytic equations for SCLCD with a monoenergetic injection velocity for one-dimensional cylindrical, spherical, tip-to-tip (t-t), and tip-to-plate (t-p) diodes. We specifically demonstrate that the correction factor from zero initial velocity to monoenergetic emission depends only on the initial kinetic and electric potential energies and not on the diode geometry and that SCLCD is universal when plotted as a function of the canonical gap size. We also show that SCLCD for a t-p diode is a factor of four larger than a t-t diode independent of injection velocity. The results reduce to previously derived results for zero initial velocity using variational calculus and conformal mapping.

Elucidation of inter/intra molecular H-bonding mediated charge transportation mechanism in water swelled, hydrophilic organic PVA-Co3O4 nanocomposite films

In this article, we have explored the water molecules mediated charge transportation kinetics in Polyvinyl alcohol (PVA)-Cobalt Oxide (Co3O4) flexible nanocomposite films that rely heavily on the comprehension of surface and interface chemistry. Understanding the chemical interactions at the aqueous hydrophilic PVA-Co3O4 interface is essential in recognising the H-bonding revolution for complex nanocomposite films. The amount of water intake with time evolution has been monitored by the contact angle and wettability measurements. The chemical interactions and the revolution of the weak dangling bonds have been studied by Fourier transform infrared spectroscopy (FTIR), Raman and photoluminance (PL) spectra of the water-swollen hydrophilic PVA-Co3O4 nanocomposite films. The dissociation of water molecules following the fragmentation of inter/intra-molecular dangling bonds is correlated with the water diffusion kinetics and saturation in the sorption capacity of water-swollen PVA-Co3O4 nanocomposite films. The trap-states development due to the interfacial charge accumulation, after absorption of water molecules, are favourable for the trap-assisted space charge limited current conduction (TA-SCLC) mechanism and increase current density ∼ 102 times as compared to the dehydrated nanocomposite films. Meticulous tuning of the strong interfacial interactions and the manifestations of fragmented H+ ions have been identified as the primary craftsman for enhancing the dielectric constant up to eight times in the presence of 30 μL deionized water. Due to the disintegration of the free water molecules at the measurement voltage, the ac conductivity (charge transport) changes from the Jonscher's power law type variation to the jump relaxation model (JRM). The origin and accumulation of H+ ions at the grain boundary interfaces have been analyzed with impedance spectroscopic (IS) studies. An electrical equivalent circuit model is proposed for the water-mediated charge transport mechanism in swelled, hydrophilic, organic nanocomposite films.

A Tutorial on Calculating Space-Charge-Limited Current Density for General Geometries and Multiple Dimensions

Space-charge-limited current (SCLC) remains a critical issue across material phases (vacuum, gas, liquids, and solids) and timescales. The solution for one-dimensional (1D), planar SCLC density (SCLCD), referred to as the Child–Langmuir (CL) law, remains the common benchmark for these calculations; however, practical devices are neither 1D nor planar. This article reviews recent progress on developing approaches to derive SCLCD for nonplanar and multidimensional diodes. First, we summarize the application of variational calculus (VC) to derive 1D SCLCD based on minimizing the current in the gap. We also describe the nuances of applying Poisson’s equation to 1D nonplanar diodes compared to three-dimensional (3D) geometries. We next discuss the application of conformal mapping (CM) to extend 1D, planar SCLCD to more complicated 1D geometries, including a detailed justification of the application of CM for solving Poisson’s equation. We then discuss the application of vacuum capacitance and a generalized relationship between the electric potential in vacuum and space-charge-limited (SCL) diodes to derive exact analytic solutions for SCLCD in two-dimensional (2D) and 3D diodes with emission from the full cathode. We further relate the SCLCD for 3D planar square and disk electrodes across a wide range of electrode sizes. We conclude by discussing future theoretical and practical extensions of these approaches.

A multi-dimensional Child–Langmuir law for any diode geometry

While prior theoretical studies of multi-dimensional space-charge limited current (SCLC) assumed emission from a small patch on infinite electrodes, none have considered emission from an entire finite electrode. In this paper, we apply variational calculus (VC) and conformal mapping, which have previously been used to derive analytic solutions for SCLC density (SCLCD) for nonplanar one-dimensional geometries, to obtain mathematical relationships for any multi-dimensional macroscopic diode with finite cathode and anode. We first derive a universal mathematical relationship between space-charge limited potential and vacuum potential for any diode and apply this technique to determine SCLCD for an eccentric spherical diode. We then apply VC and the Schwartz–Christoffel transformation to derive an exact equation for SCLCD in a general two-dimensional planar geometry with emission from a finite emitter. Particle-in-cell simulations using VSim agreed within 4%–13% for a range of ratios of emitter width to gap distance using the thinnest electrodes practical for the memory constraints of our hardware, with the difference partially attributed to the theory's assumption of infinitesimally thin electrodes. After generalizing this approach to determine SCLCD for any orthogonal diode as a function of only the vacuum capacitance and vacuum potential, we derive an analytical formulation of the three-dimensional Child–Langmuir law for finite parallel rectangular and disk geometries. These results demonstrate the utility for calculating SCLCD for any diode geometry using vacuum capacitance and vacuum potential, which are readily obtainable for many diode geometries, to guide experiment and simulation development.

Analytic Solutions for Space-Charge-Limited Current Density From a Sharp Tip

While analytic equations for space-charge limited current density (SCLCD) have been derived for planar and nonplanar geometries, the SCLCD for emission from a sharp tip has not been derived. In this article, we use variational calculus (VC) to derive an exact analytic equation for SCLCD for a 1-D tip-to-tip geometry, represented by hyperboloids in the prolate spheroidal coordinates, and recover the SCLCD for a tip-to-plate geometry as a special case. We then consider circles in the extended Poincaré disk, which is the stereographic projection of hyperboloids onto a plane, as conformal transformations to derive the SCLCD for a misaligned tip-to-tip geometry, where the axes of rotation of the hyperboloids are displaced by a small distance. This mapping technique is also applied to study the effect of a small angle tilt in a tilted tip-to-tip geometry, where the axes of rotation of the hyperboloids meet at an angle.

Space-Charge-Limited Current Injection Into Free Space and Trap-Filled Solid

We present a model of the space charge limited (SCL) electron injection into a gap combined with trap-filled dielectric solid and vacuum or a gap combined with two different trap-filled dielectric solid. An optimized calculation method is proposed to calculate the SCL current density J inside the gap under different gap voltages V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g</sub> . The n scaling of J- V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> varies in a different range of V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g</sub> , which is described as the Ohm regime, trap-limited regime, and SCL regime. It is found that the n scaling is dependent on electron mobility and relative length scale between two media, which indicates that traps in the dielectric have a profound impact on SCL electron transport properties.

Space charge limited current for bipolar flow with uniform initial velocity

The characteristics of space charge limited (SCL) bipolar flow in a planar ion diode with uniform initial velocity are studied in terms of the potential distribution. As a negative potential minimum occurs, a general expression for the limiting current density as a function of the normalized initial kinetic energy of electron αe and ion αi, normalized potential minimum αm, and ratio of ion current density to electron current density q is shown to be given by Je=JCL[Γ(αe,αi,αm,q)]2 in the form of a definite integral, where the function Γ is independent of the gap voltage and gap spacing of the diode, and JCL is the classical Child–Langmuir law. According to the expression, the SCL current density can be found by traversing all the values of the potential minimum. We also proposed a separation method to numerically obtain the spatial profile of the electrostatic potential as a negative potential minimum develops in a steady state. The theoretical and numerical results are compared with some special cases that have been derived previously and also verified by the extrapolation method in the presence of ions.

Space charge limited current with distributed velocity of initial electrons in planar diode

The research on the space-charge limited current is of great importance for developing insight into the high current diodes. The Child-Langmuir law presents one-dimensional (1D) space-charge limited current density by considering a full electrostatic solution. Jaffe investigated the 1D space-charge limited current density modified by the uniform initial velocity of electrons emitted from the cathode by using the equation of motion for the individual electron. Luginsland et al. presented the two-dimensional (2D) space-charge limited current density by using the particle-in-cell (PIC) codes neglecting initial velocity distribution of the emitted electrons. In this Letter, the effects of initial electrons with the uniform energy distribution and thermal equilibrium distribution on the space charge limited current density are studied by using the full electromagnetic PIC method; the 2D modification factors to the space charge limited current density for the two cases are presented and discussed.

High-power vacuum terahertz photomixer and integrated circuits based on microscale phototubes.

Technologies and industrials in long-distance communication, detection, and imaging applications are still in great need of higher-output-power terahertz sources. This paper proposes two kinds of microscale vacuum phototube based high-power terahertz source: vacuum photomixer and terahertz integrated circuit. The principle of photomixer based on photoemission and field-assisted photoemission is demonstrated. Its capability of producing radiation power beyond 1 mW is estimated based on theoretical analysis and experimental evidence. Simulation and theoretical analysis have shown that the fundamental THz photodiode devices can operate with a space-charge limited current density of 4496 A/cm2 at 60 V, and the amplifier circuits are calculated to have a gain performance of around 10 dB. The two photoemission-based roadmaps have the potential to be developed from an emerging and interdisciplinary field to more promising future directions of THz science and technology.

A Numerical Approach for Space Charge Limited Bipolar Flow in Cylindrical diodes

In this paper, a numerical iteration approach to resolve the space charge limited (SCL) bipolar flow problem in cylindrical geometries has been developed. Such an approach is basis for the simultaneous determination of the unknown current densities and the potential distribution. We employed this method to study the characteristics of the SCL bipolar flow. By considering a cylindrical geometry with a cathode radius Rc and an anode radius Ra, the enhancement over the classical Langmuir-Blodgett (LB) law is investigated as a function of Rc/Ra. It is found that for the bipolar flow model, the SCL current density can be given by F×JLB, where F and JLB represent the enhancement factor on account of the influence of ions and the LB law, respectively. The enhancement factor F follows a Rc/Ra scaling and gradually converges to a constant with increasing Rc/Ra. The planar bipolar flow solution is recovered in the condition where the values of Rc and Ra are much greater than that of the gap spacing.

The electric field at the hole-injecting metal/organic interface controls the bias dependence of the current–voltage hole mobility

Based upon the room temperature current–voltage data of some published organic diode structures the unique phenomenon of the decreasing hole mobility, μ, with the increasing applied electric field, E a, is interpreted. The measurable quantity, the hole drift mobility μ d is formulated in terms of E a and the electric field at the hole injecting metal/organic interface, E int, dependent algebraic function multiplied by the intrinsic hole mobility, μ max that is organic morphology dependent but E a independent scaling factor. On account that the intrinsic mobility, μ max, is uncoupled from both E a and E int it is shown that the origin of the negative field hole mobility effect occurs due to E int, that is a linear function of E a. The bias and the space distribution of the internal organic electric field, E, as well as the free hole density, p, for poly(3-hexylthiophene) is calculated in detail. Depending on the organic layer morphology the internal electric field may exhibit, at the particular value of E a, a deep well in the vicinity of the hole injecting metal/organic interface. Then the strong peak of the free hole density exists there the effect of which is spreading some 10 nm into the organic. If E int happens to be E a independent constant, then from the resulting space charge limited current density, the increasing hole drift mobility, μ d, with the increasing applied electric field, E a, is deduced. The published current–voltage data of two distinct metal-substituted phthalocyanine thin films provide an additional confirmation of the described formalism.

Enhanced space charge limited current for curved electron emitters

The maximum current that can be transported across a vacuum diode is limited by forces arising due to space charge. In a planar thermionic or explosive emission diode, the space charge limited current density from an emitting patch is given by the Child–Langmuir law JCL∼Vg3/2/D2, where Vg is the potential difference across the diode and D is the separation between the anode and the cathode. We show here, analytically using the nonlinear line charge model, that for a curved emitter in a planar diode configuration, the limiting current obeys the scaling relationship JSCL∼γaVg3/2/D2, where γa is the apex field enhancement factor of the curved emitter. For an emitter with a large height (h) to the apex radius of curvature (Ra) ratio, the limiting current far exceeds the planar value. The result is verified using the particle-in-cell code PASUPAT for two curved emitter shapes.

ISIS Penning source extraction studies reveal the 3-dimensional Child-Langmuir effect.

The standard 1X ISIS negative Penning surface plasma source has reliably produced an H- beam for ISIS operations for 35 years. In order to meet the 60 mA, 2 ms, and 50 Hz beam current and duty cycle required for the front end test stand (Letchford et al., in Proceedings of IPAC2015, Richmond, VA, USA, 2015), a 2X scaled source has been developed [Faircloth et al., AIP Conf. Proc. 2052, 050004 (2018)]. The 2X source has a plasma chamber twice the linear dimensions of the 1X source. This paper investigates the comparison between different emission areas (plasma electrode aperture dimensions) for both the 1X and 2X sources. Slit and circular extraction schemes are studied. A 3D Child-Langmuir relationship is observed where the space charge limited current density depends on the aspect ratio of the extraction aperture.

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 → ∞.

Space Charge Effects on Ion Mobility Spectrometry.

We study the space charge limited maximal current density j″ of mobility-selected ions that can be transmitted in ion mobility spectrometry (IMS). Theory and experiments focus on differential mobility analyzers (DMAs), but are readily generalizable to other IMS devices. Repulsion between the ions in the cloud leads to beam spreading, with significant broadening once the ion number density n becomes comparable to the space charge saturation limit nsat = εoEo/(eΔ). Δ is the distance traversed by the ions in the direction of an externally imposed electric field Eo, and e is the charge on each ion. For ions of electrical mobility Z, j″ is then limited below j″sat = ZEoensat = ZεoEo2/Δ. A theory including diffusion and space charge effects is developed that reduces to Burgers' exactly solvable equation. The theory is tested in experiments with room temperature electrosprays (ES) of 100mM [ethyl3N+-formate-] in methanol. This spray produces primarily a single ionic species at very high initial concentration n, which may be tuned above or below nsat by varying the distance from the ES emitter to the inlet slit of the DMA. Mobility-selected ion densities n > 3.108 ions/cm3 are achieved, with n~nsat, and with drastically broadened mobility peak shapes having the approximate top hat-predicted shapes. However, the largest n values approaching nsat are not quantitatively measurable because the densest sprays do not fill the outlet slit length. Graphical Abstract .

A comparative study of the co-precipitate synthesis of barium-calcium-aluminates impregnants for dispenser cathodes by using different precipitants

Abstract In order to optimize the synthesis process of barium-calcium-aluminate (BCA) for the application of impregnated dispenser cathodes, the influence of two precipitants, ammonium carbonate (AC) and ammonium hydrogen carbonate (AHC), on the chemical composition, phase transformation, thermal behaviors and morphology of the precursors was systematically investigated by means of XRD, TG-DSC, FESEM, and EDS mapping analysis. The results revealed that AC precipitant facilities to form NH4Al (OH)2CO3 (AACH) phase and barium calcium carbonate in the precursor, while the product precipitated by AHC comprises amorphous γ-AlOOH phase and the carbonate. Both precursors precipitated by the AC and AHC undergo various stages of phase transformations and finally converted to an emission-active material Ba3CaAl2O7 phase after sintering at 1200 °C. DC emission current values have been measured and the maximum space charge limited current densities are 8.0 ± 0.3 and 6.5 ± 0.4 A/cm2 at 1130 °Cb for the testing cathodes impregnated with the AC and AHC precursor aluminates, respectively. The difference in the emission capacity originates from the barium content in the aluminates prepared with different precipitants.

Space-charge effect of the time-varying electron injection in a diode: Classical and relativistic regimes

Conventionally, space-charge (SC) limited current density is defined as the maximal cur- rent density allowed to traverse a diode under a DC voltage when a time-invariant flow is injected from the cathode. In this work, we study the SC limited current density under a time-varying injection for both classical and relativistic regimes and determine the maxi- mal amount of limited current density under certain conditions. Our simulations show that it is unlikely that a time-varying injection emitted from cathode exceeds the known SC limits, in either classical or relativistic regime.