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

Harsha, N. R. Sree; Garner, Allen L.

doi:10.1109/ted.2021.3122393

Cited by 13 publications

(3 citation statements)

References 48 publications

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“…[32][33][34][35] The proposed fractional VC approach could be applied to t-t and t-p geometries by carefully observing the impact of electrode surface roughness on electrode work function. [36] ∑ k=1…”

Section: Discussionmentioning

confidence: 99%

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

Zubair

Fatima

Raheem

et al. 2023

Advanced Physics Research

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This 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 (F 𝜶 ) 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 (R a ) and cathode (R c ) 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.

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Section: Discussionmentioning

confidence: 99%

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

Zubair

Fatima

Raheem

et al. 2023

Advanced Physics Research

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“…One challenge is that many practical devices are nonplanar [17], which motivated past extensions of SCLC to concentric cylindrical and spherical geometries [28][29][30][31][32][33][34][35][36][37][38][39][40]. Previous studies derived SCLC for non-planar diodes by applying variational calculus (VC) [41,42] and conformal mapping (CM) [43,44]. The first VC study extremized the current across the gap and obtained analytic solutions for concentric cylinders and spheres; a subsequent study applied VC to derive SCLC for a crossed-field diode with orthogonal electric and magnetic fields [45].…”

Section: Introductionmentioning

confidence: 99%

“…The first VC study extremized the current across the gap and obtained analytic solutions for concentric cylinders and spheres; a subsequent study applied VC to derive SCLC for a crossed-field diode with orthogonal electric and magnetic fields [45]. The VC cylindrical solution was then verified using CM, which used maps to translate the planar SCLC solution to more complicated geometries [43,44]. These methods have been extended for SCLC in general geometries and multiple dimensions [46].…”

Section: Introductionmentioning

confidence: 99%

A coordinate system invariant formulation for space-charge limited current with nonzero injection velocity

Halpern¹,

Darr²,

Harsha³

et al. 2022

Plasma Sources Sci. Technol.

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Recent studies have extended the classical space-charge limited current (SCLC) solution in a non-magnetic, planar diode with zero injection velocity to other geometries using variational calculus (VC). We further extend VC to solve for SCLC with a non-relativistic, monoenergetic injection velocity from first principles for nonplanar diodes. By extremizing either the current or a functional of the electric field (and not its derivative), we demonstrate that VC can find either the bifurcation or the SCLC solution, respectively. The bifurcation solution is characterized by the onset of particle reflection, resulting in a singularity in the derivative of the electric field at the virtual cathode, physically analogous to the singularity at the cathode in SCLC for zero injection velocity. Alternatively, using VC to extremize a functional of the potential and its gradient (electric field) yields the maximum current SCLC result. We then derive the SCLC solutions in cylindrical and spherical diodes; additionally, we derive a method to determine SCLC numerically and the bifurcation solution exactly for any orthogonal geometry. Implications for the potential profile and virtual cathode are discussed, especially the behavior for other geometries.

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

et al. 2021

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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.

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

Cited by 13 publications

References 48 publications

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

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

A coordinate system invariant formulation for space-charge limited current with nonzero injection velocity

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

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