Conference Agenda

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Session Overview
Analytical Methods
Wednesday, 22/June/2022:
3:30pm - 5:30pm

Session Chair: Amanda M. Loveless, Purdue University
Location: 301B

Oral Session

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3:30pm - 3:50pm
Analytical Methods: 1

Comparison of Particle-in-Cell and Continuum Simulations for RF Microscale Gas Breakdown

A. M. Loveless1, V. Ayyaswamy2, S. Mahajan1, A. Semnani3, A. L. Garner1

1Purdue University, United States of America; 2University of California Merced, United States of America; 3University 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 particle-in-cell (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 (one-dimensional in space, three-dimensional in velocity) PIC code, to continuum simulations using SOMAFOAM [3], a finite volume framework to simulate low-temperature 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. 808-824, 2020.

[2] M. U. Lee, J. Lee, J. K. Lee, and G. S. Yun, “Extended scaling and Paschen law for micro-sized 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 low-temperature plasmas,” Comp. Phys. Comm., vol. 263, art. no. 107855, 2021.

Work supported by the Office of Naval Research under Grant Number N00014-21-1-2441.

3:50pm - 4:30pm
Analytical Methods: 2

Crossed-Field Nexus Theory: Incorporating Collisions, Field Emission, Thermionic Emission, and Space-Charge

L. I. Breen1, A. M. Loveless1, A. M. Darr1, K. L. Cartwright2, A. L. Garner1

1Purdue University, West Lafayette, IN 47906 USA; 2Sandia National Laboratories, Albuquerque, NM

Understanding electron emission is vital for characterizing diode performance for numerous applications, including directed energy systems, thermionic converters, time-resolved electron microscopy, and x-ray systems. The “nexus theory” formulation may be used to predict the physical conditions where multiple electron emission mechanisms, such as thermionic, field, and space-charge 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, well-known 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 crossed-field 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 Child-Langmuir equation characterizes planar space-charge limited current (SCLC), similar equations may be derived for the limiting current in crossed-field diodes under non-magnetically 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 crossed-field diodes. We unify thermionic and field emission with the limiting current in a crossed-field diode by introducing the generalized thermal-field emission current density equation, as was previously derived for non-magnetic diodes [1]. We will next introduce collisions into the derivation of the limiting current of crossed-field diodes [2,3] to derive a collision limiting current for a crossed-field diode, equivalent to a Mott-Gurney law for non-magnetic 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 space-charge-limited emission,” Phys. Rev. Res., vol. 2, 2020, Art. no. 033137.

2. Y. Y. Lau, P. J. Christenson, and D. Chernin, “Limiting current in a crossed-field gap,” Phys. Plasmas, vol. 5, pp. 4486-4489, 1993.

3. P. J. Christenson and Y. Y. Lau, “Transition to turbulence in a crossed‐field gap,” Phys. Plasmas, vol. 12, pp. 3725-3727, 1994.

4:30pm - 4:50pm
Analytical Methods: 3

Novel techniques for deriving the space-charge limited current for nonplanar diodes

N. R. Sree Harsha, A. M. Darr, J. M. Halpern, A. L. Garner

Purdue University, United States of America

Space-charge-limited current (SCLC) is the maximum current that can flow in the steady-state operation of the diode. Characterizing SCLC is critical for understanding the behavior of various devices, including high-power vacuum devices, organic field-effect transistors, quantum diodes, n-i-n or p-i-p diodes, and photovoltaic devices [1]. The SCLC in a one-dimensional (1-D) 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 space-charge-limited potential in any orthogonal geometry [3]. The exact solutions for SCLC in two-dimensional and three-dimensional 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 space-charge 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 multi-dimensional 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 FA9550-19-1-0101 and a Purdue Doctoral Fellowship.

4:50pm - 5:10pm
Analytical Methods: 4

Assessment of Techniques for Determining Space-Charge Limited Current for Non-planar Crossed-field 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 space-charge limited current (SCLC), is essential for numerous applications, including nano vacuum transistors, electric thrusters, and time-resolved electron microscopy. Recently, several general approaches for deriving analytic solutions for non-planar and multidimensional diodes have been developed [1]. Crossed-field 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 non-magnetic SCLC, the space-charge 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 crossed-field devices.

This presentation assesses various approaches to derive the limiting current for non-planar diodes. We first describe the derivation of the SCLC in both magnetically insulated and non-insulated CFDs by using the Euler-Lagrange 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 space-charge 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 space-charge 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, “Space-charge limited current in planar and cylindrical crossed-field diodes using variational calculus,” Phys. Plasmas, vol. 28, no. 8, 2021, Art. No. 082110.

5:10pm - 5:30pm
Analytical Methods: 5

Optimization of a Set of Electron-Neutral Collision Cross Sections in Fluorinated Nitrile (C4F7N)

M. Flynn, A. Neuber, J. Stephens

Texas Tech University, United States of America

Plasma fluid models for high-voltage 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 electron-neutral cross sections which are not well known for many gases. C4F7N (i.e. 3M™ Novec™ 4710) is one such gas. Owing to its short atmospheric lifespan and large dielectric strength, C4F7N has received recent attention as an insulating gas with significantly reduced global warming potential, when compared to SF6.

This report details the progress made in the development of a complete and self-consistent set of cross sections for electron swarms in C4F7N. MultiBolt, a multi-term 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 DE-NA0003525

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