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

Sree Harsha, N. R.; Pearlman, Marcus; Browning, Jim; Garner, Allen L.

doi:10.1063/5.0071018

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

Journal Article · Fri Dec 03 00:00:00 EST 2021 · Physics of Plasmas

DOI:https://doi.org/10.1063/5.0071018· OSTI ID:1978980

^[1]; Pearlman, Marcus ^[2];

^[2];

^[1]

Purdue University, West Lafayette, IN (United States)
Boise State University, ID (United States)

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.

View Accepted Manuscript (DOE)

View Accepted Manuscript (Publisher)

Cite

Export

Save

Research Organization:: Purdue Univ., West Lafayette, IN (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Fusion Energy Sciences (FES); US Air Force Office of Scientific Research (AFOSR)

Grant/Contract Number:: SC0018156; FA9550-18-1-0218; FA9550-19-1-0101

OSTI ID:: 1978980

Alternate ID(s):: OSTI ID: 1833876

Journal Information:: Physics of Plasmas, Vol. 28, Issue 12; ISSN 1070-664X

Publisher:: American Institute of Physics (AIP)Copyright Statement

Country of Publication:: United States

Language:: English

References (63)

Novel Scaling Laws for the Langmuir-Blodgett Solutions in Cylindrical and Spherical Diodes Zhu, Y. B.; Zhang, P.; Valfells, A. Physical Review Letters, Vol. 110, Issue 26 https://doi.org/10.1103/PhysRevLett.110.265007	journal	June 2013
Space-Charge Effects in the Field-Assisted Thermionic Emission from Nonuniform Cathodes Sitek, Anna; Torfason, Kristinn; Manolescu, Andrei Physical Review Applied, Vol. 15, Issue 1 https://doi.org/10.1103/PhysRevApplied.15.014040	journal	January 2021
Enhanced space charge limited current for curved electron emitters Singh, Gaurav; Kumar, Raghwendra; Biswas, Debabrata Physics of Plasmas, Vol. 27, Issue 10 https://doi.org/10.1063/5.0028007	journal	October 2020
A simple physical derivation of Child–Langmuir space-charge-limited emission using vacuum capacitance Umstattd, R. J.; Carr, C. G.; Frenzen, C. L. American Journal of Physics, Vol. 73, Issue 2 https://doi.org/10.1119/1.1781664	journal	February 2005
Space‐charge‐limited current and capacitance in double‐junction diodes Grinberg, A. A.; Luryi, Serge Journal of Applied Physics, Vol. 61, Issue 3 https://doi.org/10.1063/1.338165	journal	February 1987
Inertial confinement fusion: Review and perspective Johnson, T. H. Proceedings of the IEEE, Vol. 72, Issue 5 https://doi.org/10.1109/PROC.1984.12897	journal	January 1984
High power microwave generation by a coaxial virtual cathode oscillator Jiang, W.; Woolverton, K.; Dickens, J. IEEE Transactions on Plasma Science, Vol. 27, Issue 5 https://doi.org/10.1109/27.799836	journal	January 1999
Currents Limited by Space Charge between Concentric Spheres Langmuir, Irving; Blodgett, Katharine B. Physical Review, Vol. 24, Issue 1 https://doi.org/10.1103/PhysRev.24.49	journal	July 1924
Applying Conformal Mapping to Derive Analytical Solutions of Space-Charge-Limited Current Density for Various Geometries Harsha, N. R. Sree; Garner, Allen L. IEEE Transactions on Electron Devices, Vol. 68, Issue 1 https://doi.org/10.1109/TED.2020.3038619	journal	January 2021
Effect of space charge limited current on performance of organic field-effect transistors Trukhanov, Vasily A.; Bruevich, Vladimir V.; Paraschuk, Dmitry Yu. Synthetic Metals, Vol. 246 https://doi.org/10.1016/j.synthmet.2018.11.002	journal	December 2018
Response to “Comment on ‘A coordinate system invariant formulation for space-charge limited current in vacuum’” [Appl. Phys. Lett. 118 , 266101 (2021)] Darr, Adam M.; Harsha, N. R. Sree; Garner, Allen L. Applied Physics Letters, Vol. 118, Issue 26 https://doi.org/10.1063/5.0057754	journal	June 2021
Edge Corrections for Strip and Disc Capacitors Wintle, H. J.; Kurylowicz, S. IEEE Transactions on Instrumentation and Measurement, Vol. IM-34, Issue 1 https://doi.org/10.1109/TIM.1985.4315253	journal	March 1985
Space-charge-limited current in a film Grinberg, A. A.; Luryi, S.; Pinto, M. R. IEEE Transactions on Electron Devices, Vol. 36, Issue 6 https://doi.org/10.1109/16.24363	journal	June 1989
Space-charge-limited current in cylindrical diodes with finite-length emitter Kostov, K. G.; Barroso, J. J. Physics of Plasmas, Vol. 9, Issue 3 https://doi.org/10.1063/1.1446876	journal	March 2002
Numerical solution of Fredholm integral equations of the second kind by using integral mean value theorem Avazzadeh, Z.; Heydari, M.; Loghmani, G. B. Applied Mathematical Modelling, Vol. 35, Issue 5 https://doi.org/10.1016/j.apm.2010.11.056	journal	May 2011
Space-Charge Limited Current From a Finite Emitter in Nano- and Microdiodes Gunnarsson, Johannes Bergur; Torfason, Kristinn; Manolescu, Andrei IEEE Transactions on Electron Devices, Vol. 68, Issue 1 https://doi.org/10.1109/TED.2020.3037280	journal	January 2021
Space-charge and trap-filling effects in organic thin film field-effect transistors Koehler, Marlus; Biaggio, Ivan Physical Review B, Vol. 70, Issue 4 https://doi.org/10.1103/PhysRevB.70.045314	journal	July 2004
The Effect of Space Charge and Residual Gases on Thermionic Currents in High Vacuum Langmuir, Irving Physical Review, Vol. 2, Issue 6 https://doi.org/10.1103/PhysRev.2.450	journal	December 1913
Multidimensional short-pulse space-charge-limited flow Koh, W. S.; Ang, L. K.; Kwan, T. J. T. Physics of Plasmas, Vol. 13, Issue 6 https://doi.org/10.1063/1.2208086	journal	June 2006
Space–charge limited current in nanodiodes: Ballistic, collisional, and dynamical effects Zhang, Peng; Ang, Yee Sin; Garner, Allen L. Journal of Applied Physics, Vol. 129, Issue 10 https://doi.org/10.1063/5.0042355	journal	March 2021
Modeling space-charge-limited currents in organic semiconductors: Extracting trap density and mobility Dacuña, Javier; Salleo, Alberto Physical Review B, Vol. 84, Issue 19 https://doi.org/10.1103/PhysRevB.84.195209	journal	November 2011
A new simple algorithm for space charge limited emission Stoltz, P. H.; Luginsland, J. W.; Chap, A. Physics of Plasmas, Vol. 27, Issue 9 https://doi.org/10.1063/5.0020781	journal	September 2020
Two-Dimensional Space-Charge-Limited Emission: Beam-Edge Characteristics and Applications Umstattd, R. J.; Luginsland, J. W. Physical Review Letters, Vol. 87, Issue 14 https://doi.org/10.1103/PhysRevLett.87.145002	journal	September 2001
Approximate Solutions of the Space‐Charge Problem for Some Unusual Electrode Geometries Ivey, Henry F. Journal of Applied Physics, Vol. 24, Issue 12 https://doi.org/10.1063/1.1721200	journal	December 1953
Currents Limited by Space Charge between Coaxial Cylinders Langmuir, Irving; Blodgett, Katharine B. Physical Review, Vol. 22, Issue 4 https://doi.org/10.1103/PhysRev.22.347	journal	October 1923
Space charge limited two-dimensional electron flow in a rectangular geometry Rokhlenko, A.; Lebowitz, J. L. Journal of Applied Physics, Vol. 102, Issue 2 https://doi.org/10.1063/1.2757718	journal	July 2007
On the Space‐Charge‐Limited Current between Nonsymmetrical Electrodes O'Neill, George D. Journal of Applied Physics, Vol. 26, Issue 8 https://doi.org/10.1063/1.1722130	journal	August 1955
Three-dimensional Child–Langmuir law for uniform hot electron emission Koh, W. S.; Ang, L. K.; Kwan, T. J. T. Physics of Plasmas, Vol. 12, Issue 5 https://doi.org/10.1063/1.1913612	journal	May 2005
Onset of space charge limited current for field emission from a single sharp tip Sun, S.; Ang, L. K. Physics of Plasmas, Vol. 19, Issue 3 https://doi.org/10.1063/1.3695090	journal	March 2012
Space charge limited electron flow in two dimensions without magnetic field Rokhlenko, A.; Lebowitz, J. L. Journal of Applied Physics, Vol. 110, Issue 3 https://doi.org/10.1063/1.3622152	journal	August 2011
Space-Charge-Limited Current Injection Into Free Space and Trap-Filled Solid Zhu, Ying Bin; Geng, Kuiwei; Cheng, Zheng Shan IEEE Transactions on Plasma Science, Vol. 49, Issue 7 https://doi.org/10.1109/TPS.2021.3084461	journal	July 2021
Two-dimensional analytical Mott-Gurney law for a trap-filled solid Chandra, W.; Ang, L. K.; Pey, K. L. Applied Physics Letters, Vol. 90, Issue 15 https://doi.org/10.1063/1.2721382	journal	April 2007
The circular disk parallel plate capacitor Carlson, G. T.; Illman, B. L. American Journal of Physics, Vol. 62, Issue 12 https://doi.org/10.1119/1.17668	journal	December 1994
100 years of the physics of diodes Zhang, Peng; Valfells, Ágúst; Ang, L. K. Applied Physics Reviews, Vol. 4, Issue 1 https://doi.org/10.1063/1.4978231	journal	March 2017
Space-Charge-Limited 2D Electron Flow between Two Flat Electrodes in a Strong Magnetic Field Rokhlenko, A.; Lebowitz, J. L. Physical Review Letters, Vol. 91, Issue 8 https://doi.org/10.1103/PhysRevLett.91.085002	journal	August 2003
A Set of Benchmark Tests for Validation of 3-D Particle in Cell Methods O'Connor, Scott; Crawford, Zane D.; Verboncoeur, John P. IEEE Transactions on Plasma Science, Vol. 49, Issue 5 https://doi.org/10.1109/TPS.2021.3072353	journal	May 2021
Simple Theory for the Two-Dimensional Child-Langmuir Law Lau, Y. Y. Physical Review Letters, Vol. 87, Issue 27 https://doi.org/10.1103/PhysRevLett.87.278301	journal	December 2001
Inertial‐Electrostatic Confinement of Ionized Fusion Gases Hirsch, Robert L. Journal of Applied Physics, Vol. 38, Issue 11 https://doi.org/10.1063/1.1709162	journal	October 1967
Non-uniform space charge limited current injection into a nano contact solid Zhu, Y. B.; Ang, L. K. Scientific Reports, Vol. 5, Issue 1 https://doi.org/10.1038/srep09173	journal	March 2015
Discharge From Hot Cao Child, C. D. Physical Review (Series I), Vol. 32, Issue 5 https://doi.org/10.1103/PhysRevSeriesI.32.492	journal	May 1911
Analytical results for the capacitance of a circular plate capacitor Reichert, Benjamin; Ristivojevic, Zoran Physical Review Research, Vol. 2, Issue 1 https://doi.org/10.1103/PhysRevResearch.2.013289	journal	March 2020
Two-Dimensional Child-Langmuir Law Luginsland, J. W.; Lau, Y. Y.; Gilgenbach, R. M. Physical Review Letters, Vol. 77, Issue 22 https://doi.org/10.1103/PhysRevLett.77.4668	journal	November 1996
Analytic Solutions for Space-Charge-Limited Current Density From a Sharp Tip Harsha, N. R. Sree; Garner, Allen L. IEEE Transactions on Electron Devices, Vol. 68, Issue 12 https://doi.org/10.1109/TED.2021.3122393	journal	December 2021
Current flow and pair creation at low altitude in rotation-powered pulsars’ force-free magnetospheres: space charge limited flow Timokhin, A. N.; Arons, J. Monthly Notices of the Royal Astronomical Society, Vol. 429, Issue 1 https://doi.org/10.1093/mnras/sts298	journal	December 2012
Effects of emitting surfaces and trapped ions on the sheath physics and current flow in multidimensional plasma systems Johnson, G. R.; Campanell, M. D. Plasma Sources Science and Technology, Vol. 30, Issue 1 https://doi.org/10.1088/1361-6595/abcc7b	journal	January 2021
Solutions of Laplace's equation with simple boundary conditions, and their applications for capacitors with multiple symmetries Morales, Mayckol; Diaz, Rodolfo A.; Herrera, William J. Journal of Electrostatics, Vol. 78 https://doi.org/10.1016/j.elstat.2015.09.006	journal	December 2015
Two-dimensional relativistic space charge limited current flow in the drift space Liu, Y. L.; Chen, S. H.; Koh, W. S. Physics of Plasmas, Vol. 21, Issue 4 https://doi.org/10.1063/1.4869732	journal	April 2014
Space charge limited current for bipolar flow with uniform initial velocity Liao, Mei Yan; Yao, Ruo He; Zhu, Ying Bin Physics of Plasmas, Vol. 28, Issue 6 https://doi.org/10.1063/5.0034995	journal	June 2021
Approximate analytical solutions for the space-charge-limited current in one-dimensional and two-dimensional cylindrical diodes Chen, Xupeng; Dickens, James; Hatfield, L. L. Physics of Plasmas, Vol. 11, Issue 6 https://doi.org/10.1063/1.1743309	journal	June 2004
Accurate 3-D Capacitance of Parallel Plates From 2-D Analytical Superposition Matick, Richard E.; Ruehli, Albert E. IEEE Transactions on Components, Packaging and Manufacturing Technology, Vol. 3, Issue 2 https://doi.org/10.1109/TCPMT.2012.2223817	journal	February 2013
Space-charge limited thermionic sheaths in magnetized fusion plasmas Komm, M.; Ratynskaia, S.; Tolias, P. Nuclear Fusion, Vol. 60, Issue 5 https://doi.org/10.1088/1741-4326/ab7c27	journal	March 2020
A coordinate system invariant formulation for space-charge limited current in vacuum Darr, Adam M.; Garner, Allen L. Applied Physics Letters, Vol. 115, Issue 5 https://doi.org/10.1063/1.5115261	journal	July 2019
Two-dimensional space-charge-limited flows in a crossed-field gap Koh, W. S.; Ang, L. K. Applied Physics Letters, Vol. 90, Issue 14 https://doi.org/10.1063/1.2720710	journal	April 2007
Beyond the Child–Langmuir law: A review of recent results on multidimensional space-charge-limited flow Luginsland, J. W.; Lau, Y. Y.; Umstattd, R. J. Physics of Plasmas, Vol. 9, Issue 5 https://doi.org/10.1063/1.1459453	journal	May 2002
Two-dimensional electromagnetic Child–Langmuir law of a short-pulse electron flow Chen, S. H.; Tai, L. C.; Liu, Y. L. Physics of Plasmas, Vol. 18, Issue 2 https://doi.org/10.1063/1.3553451	journal	February 2011
Space-Charge Limited Photocurrent Mihailetchi, V. D.; Wildeman, J.; Blom, P. W. M. Physical Review Letters, Vol. 94, Issue 12 https://doi.org/10.1103/PhysRevLett.94.126602	journal	April 2005
Space charge limited current emission for a sharp tip Zhu, Y. B.; Ang, L. K. Physics of Plasmas, Vol. 22, Issue 5 https://doi.org/10.1063/1.4919936	journal	May 2015
A Tutorial on Theoretical and Computational Techniques for Gas Breakdown in Microscale Gaps Garner, Allen L.; Loveless, Amanda M.; Dahal, Jiba Nath IEEE Transactions on Plasma Science, Vol. 48, Issue 4 https://doi.org/10.1109/TPS.2020.2979707	journal	April 2020
Breakdown of a Space Charge Limited Regime of a Sheath in a Weakly Collisional Plasma Bounded by Walls with Secondary Electron Emission Sydorenko, D.; Kaganovich, I.; Raitses, Y. Physical Review Letters, Vol. 103, Issue 14 https://doi.org/10.1103/PhysRevLett.103.145004	journal	October 2009
Space-charge limited current in planar and cylindrical crossed-field diodes using variational calculus Darr, Adam M.; Bhattacharya, Ranajoy; Browning, Jim Physics of Plasmas, Vol. 28, Issue 8 https://doi.org/10.1063/5.0054307	journal	August 2021
Impact of Nonuniform Thermionic Emission on the Transition Behavior Between Temperature-and Space-Charge-Limited Emission Chen, Dongzheng; Jacobs, Ryan; Morgan, Dane IEEE Transactions on Electron Devices, Vol. 68, Issue 7 https://doi.org/10.1109/TED.2021.3079876	journal	July 2021
The Capacitance of a Parallel-Plate Capacitor by the Schwartz-Christoffel Transformation Palmer, Harlan B. Transactions of the American Institute of Electrical Engineers, Vol. 56, Issue 3 https://doi.org/10.1109/T-AIEE.1937.5057547	journal	March 1937
Space-charge limited current with a finite injection velocity revisited Lafleur, T. Plasma Sources Science and Technology, Vol. 29, Issue 6 https://doi.org/10.1088/1361-6595/ab9069	journal	June 2020

Similar Records

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

Journal Article · Thu Dec 03 00:00:00 EST 2020 · Journal of Applied Physics · OSTI ID:1978980

Garner, Allen L.; Meng, Guodong; Fu, Yangyang; +3 more

CROSSED-FIELD THERMIONIC CONVERSION. Scientific Report No. 10

Technical Report · Wed Mar 01 00:00:00 EST 1961 · OSTI ID:1978980

Peters, P H

Quantitative modeling of ICRF antennas with integrated time domain RF sheath and plasma physics

Journal Article · Wed Feb 12 00:00:00 EST 2014 · AIP Conference Proceedings · OSTI ID:1978980

Smithe, David N.; D'Ippolito, Daniel A.; Myra, James R.

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
computational electromagnetics
electrostatics
conformal differential geometry
integral equations
integral calculus
mathematical modeling
calculus of variations
theoretical plasma physics
particle-in-cell method
space charge dominated beams

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

Citation Formats

References (63)

Similar Records

Related Subjects