3:30pm  3:50pmAnalytical Methods: 1
Comparison of ParticleinCell and Continuum Simulations for RF Microscale Gas Breakdown
A. M. Loveless^{1}, V. Ayyaswamy^{2}, S. Mahajan^{1}, A. Semnani^{3}, A. L. Garner^{1}
^{1}Purdue University, United States of America; ^{2}University of California Merced, United States of America; ^{3}University of Toledo, United States of America
Understanding and accurately characterizing electron emission and gas breakdown is necessary with increasing device miniaturization. For DC voltages, Paschen’s law, which is based on the Townsend avalanche criterion, is commonly used to predict gas breakdown; however, for microscale gaps, the resulting strong electric fields at breakdown induce the release of additional electrons by field emission (FE), which considers the enhanced surface electric field due to the decreased potential barrier at the cathode [1]. Accurately predicting breakdown under these conditions requires combining field emission and Townsend avalanche. Similarly, field emission also contributes to breakdown for microscale gaps under RF and microwave fields, motivating theoretical studies and particleincell (PIC) simulations [2] to account for this behavior. While effective for gaps below ~10 microns at atmospheric pressure, PIC is not computationally efficient for larger gaps due to the computational expense encountered with additional particles. Thus, this study compares RF breakdown simulations using PDP1, a 1D/3v (onedimensional in space, threedimensional in velocity) PIC code, to continuum simulations using SOMAFOAM [3], a finite volume framework to simulate lowtemperature plasmas. The results between PDP1 and SOMAFOAM will be compared to each other and theory for various frequencies, pressures, and gap distances, particularly to assess scaling laws between these parameters in different operational regimes. The computational efficiency of the two methods and assessment to theory and experiment will be discussed.
[1] A. L. Garner, A. M. Loveless, J. N. Dahal, and A. Venkattraman, “A tutorial on theoretical and computational techniques for gas breakdown in microscale gaps,” IEEE Trans. Plasma Sci., vol. 48, pp. 808824, 2020.
[2] M. U. Lee, J. Lee, J. K. Lee, and G. S. Yun, “Extended scaling and Paschen law for microsized radiofrequency plasma breakdown," Plasma Sources Sci. Technol., vol. 26, art. no. 034003, 2017.
[3] A. K. Verma and A. Venkattraman, “SOMAFOAM: An OpenFOAM based solver for continuum simulations of lowtemperature plasmas,” Comp. Phys. Comm., vol. 263, art. no. 107855, 2021.
Work supported by the Office of Naval Research under Grant Number N000142112441.
3:50pm  4:30pmAnalytical Methods: 2
CrossedField Nexus Theory: Incorporating Collisions, Field Emission, Thermionic Emission, and SpaceCharge
L. I. Breen^{1}, A. M. Loveless^{1}, A. M. Darr^{1}, K. L. Cartwright^{2}, A. L. Garner^{1}
^{1}Purdue University, West Lafayette, IN 47906 USA; ^{2}Sandia National Laboratories, Albuquerque, NM
Understanding electron emission is vital for characterizing diode performance for numerous applications, including directed energy systems, thermionic converters, timeresolved electron microscopy, and xray systems. The “nexus theory” formulation may be used to predict the physical conditions where multiple electron emission mechanisms, such as thermionic, field, and spacecharge limited emission, may need to be solved jointly [1]. Once nexus theory identifies such a regime, one can derive exact equations from first principles that couple the relevant physics to assess behavior. The exact model should recover the standard equations for the individual emission mechanisms under appropriate asymptotic limits [1]. Operating conditions near where these asymptotic solutions match require more complicated equations coupling the relevant mechanisms; regimes farther away from these intersections may use the simpler, wellknown solutions.
A common diode design in high power applications incorporates an external magnetic field perpendicular to the electric field induced by the applied voltage. Electron trajectories in these crossedfield diodes may either cross the gap if the magnetic field is below a limiting value known as the Hull cutoff or be turned back to the cathode for magnetic fields exceeding the Hull cutoff. Above the Hull cutoff, the diode is magnetically insulated. Much as the ChildLangmuir equation characterizes planar spacecharge limited current (SCLC), similar equations may be derived for the limiting current in crossedfield diodes under nonmagnetically insulated [2] and magnetically insulated conditions [3]. These conditions do not strongly depend on the specific electron emission mechanism, but rather define the maximum current that may be emitted into the gap based on geometry and boundary conditions.
This presentation highlights our application of nexus theory to crossedfield diodes. We unify thermionic and field emission with the limiting current in a crossedfield diode by introducing the generalized thermalfield emission current density equation, as was previously derived for nonmagnetic diodes [1]. We will next introduce collisions into the derivation of the limiting current of crossedfield diodes [2,3] to derive a collision limiting current for a crossedfield diode, equivalent to a MottGurney law for nonmagnetic SCLC with collisions. The implications of the transitions between these mechanisms under various conditions and the respective limits on device operation will be discussed.
1. A. M. Darr, C. R. Darr, and A. L. Garner, “Theoretical assessment of transitions across thermionic, field, and spacechargelimited emission,” Phys. Rev. Res., vol. 2, 2020, Art. no. 033137.
2. Y. Y. Lau, P. J. Christenson, and D. Chernin, “Limiting current in a crossedfield gap,” Phys. Plasmas, vol. 5, pp. 44864489, 1993.
3. P. J. Christenson and Y. Y. Lau, “Transition to turbulence in a crossed‐field gap,” Phys. Plasmas, vol. 12, pp. 37253727, 1994.
4:30pm  4:50pmAnalytical Methods: 3
Novel techniques for deriving the spacecharge limited current for nonplanar diodes
N. R. Sree Harsha, A. M. Darr, J. M. Halpern, A. L. Garner
Purdue University, United States of America
Spacechargelimited current (SCLC) is the maximum current that can flow in the steadystate operation of the diode. Characterizing SCLC is critical for understanding the behavior of various devices, including highpower vacuum devices, organic fieldeffect transistors, quantum diodes, nin or pip diodes, and photovoltaic devices [1]. The SCLC in a onedimensional (1D) planar diode was derived independently by Child and Langmuir over a century ago [1]. Recently, we applied variational calculus (VC) and conformal mapping (CM) to derive analytic solutions to SCLC for nonplanar diode geometries [2].
In this presentation, we review the application of VC and CM to obtain analytic solutions for SCLC for nonplanar diodes. The analytic solutions for SCLC in any orthogonal coordinate system can be obtained using VC by extremizing the total current in the gap [2]. While VC is a powerful technique to solve for SCLC, the calculations become tedious for diodes exhibiting curvilinear flow. For such geometries, we have applied CM to transform the curvilinear flow into a rectilinear flow, thereby obtaining analytic SCLC solutions [2]. We extend VC to obtain a mathematical relationship between vacuum potential and spacechargelimited potential in any orthogonal geometry [3]. The exact solutions for SCLC in twodimensional and threedimensional planar diodes with finite emitters are presented [3]. We also apply Lie point symmetries to derive SCLC with nonzero injection velocity in nonplanar diode geometries and describe how similar solutions may be obtained using VC. The practical importance of this flexibility and a comparison between these mathematically powerful techniques will be discussed.
[1] P. Zhang, Y. S. Ang, A. L. Garner, Á. Valfells, J. W. Luginsland, and L. K. Ang, “Space–charge limited current in nanodiodes: Ballistic, collisional, and dynamical effects,” J. Appl. Phys., vol. 129, no. 10, Mar. 2021, Art. no. 100902.
[2] A. L. Garner, A. M. Darr, and N. R. Sree Harsha, “Calculating spacecharge limited current density for general geometries and multiple dimensions,” IEEE Trans. Plasma Sci., submitted.
[3] N. R. S. Harsha, M. Pearlman, J. Browning, and A. L. Garner, “A multidimensional Child–Langmuir law for any diode geometry,” Phys. Plasmas, vol. 28, no.12, Dec. 2021, Art. no. 122103.
________________________________
This material is based upon work supported by the Air Force Office of Scientific Research under award number FA95501910101 and a Purdue Doctoral Fellowship.
4:50pm  5:10pmAnalytical Methods: 4
Assessment of Techniques for Determining SpaceCharge Limited Current for Nonplanar Crossedfield Diodes
H. Wang, N. R. Sree Harsha, A. M. Darr, A. L. Garner
Purdue University
The maximum stable current that can flow in a diode, known as the spacecharge limited current (SCLC), is essential for numerous applications, including nano vacuum transistors, electric thrusters, and timeresolved electron microscopy. Recently, several general approaches for deriving analytic solutions for nonplanar and multidimensional diodes have been developed [1]. Crossedfield diodes (CFDs), where an external magnetic field B is applied perpendicular to the electric field, may also be characterized by a maximum current that depends on whether an emitted electron crosses the gap or turns back to the cathode [2]. Unlike nonmagnetic SCLC, the spacecharge limit does not characterize the maximum current in a CFD, which is instead determined by electron flow stability [2]. These initial studies derived solutions for the limiting current that were only valid for planar diodes [2], which are not representative of typical crossedfield devices.
This presentation assesses various approaches to derive the limiting current for nonplanar diodes. We first describe the derivation of the SCLC in both magnetically insulated and noninsulated CFDs by using the EulerLagrange equation for planar and concentric cylinder diodes [3]. While this approach may, in principle, be extended to any general geometry, the actual mathematical application is daunting. Thus, we also apply conformal mapping, which was used to derive the mapping of the spacecharge limited potential from a given geometry to the standard planar geometry, to obtain SCLC for concentric cylinders [1]. We next apply Lie point symmetries, which may be considered as a generalization of conformal mapping, to derive SCLC in other complicated geometries, including concentric spheres, which are not amenable to conformal mapping [1]. An overall assessment and comparison of the SCLC using these different techniques will be discussed, as will the extension of conformal mapping and Lie point symmetries to more complicated geometries.
[1] A. L. Garner, A. M. Darr, and N. R. Sree Harsha, “Calculating spacecharge limited current density for general geometries and multiple dimensions,” IEEE Trans. Plasma Sci., submitted. [2] P. J. Christenson, “Equilibrium, stability, and turbulence in cycloidal electron flows in crossed electric and magnetic fields,” Ph.D. dissertation, Department of Nuclear Engineering and Radiological Sciences, University of Michigan, 1996. [3] A. M. Darr, R. Bhattacharya, J. Browning, and A. L. Garner, “Spacecharge limited current in planar and cylindrical crossedfield diodes using variational calculus,” Phys. Plasmas, vol. 28, no. 8, 2021, Art. No. 082110.
5:10pm  5:30pmAnalytical Methods: 5
Optimization of a Set of ElectronNeutral Collision Cross Sections in Fluorinated Nitrile (C4F7N)
M. Flynn, A. Neuber, J. Stephens
Texas Tech University, United States of America
Plasma fluid models for highvoltage gaseous discharges rely on transport coefficients which are often calculated with an electron swarm kinetic model (e.g. Monte Carlo, Boltzmann equation). These calculations, however, require the input of a set of electronneutral cross sections which are not well known for many gases. C_{4}F_{7}N (i.e. 3M™ Novec™ 4710) is one such gas. Owing to its short atmospheric lifespan and large dielectric strength, C_{4}F_{7}N has received recent attention as an insulating gas with significantly reduced global warming potential, when compared to SF_{6}.
This report details the progress made in the development of a complete and selfconsistent set of cross sections for electron swarms in C_{4}F_{7}N. MultiBolt, a multiterm Boltzmann equation solver, is utilized to optimize elastic and inelastic cross sections for the calculation of swarm parameters, which are compared with available literature. The cross section optimization procedure and considerations for the Boltzmann model will be discussed.
SNL is managed and operated by NTESS under DOE NNSA contract DENA0003525
