Explorations of Space-Charge Limits in Parallel-Plate Diodes and Associated Techniques for Automation

Space-charge limited flow is a topic of much interest and varied application./nWe extend existing understanding of space-charge limits by simulations,/nand develop new tools and techniques for doing these simulations along the way./nThe Child-Langmuir limit is a simple analytic solution for space-charge limited/ncurrent density in a one-dimensional diode. It has been previously extended to two dimensions/nby numerical calculation in planar geometries. By considering an axisymmetric cylindrical system with axial emission from a circular cathode of finite radius $r$ and outer drift tube $R > r$/nand gap length $L$, we further examine the space charge limit in two dimensions./nWe simulate a two-dimensional axisymmetric parallel plate diode of various aspect ratios ($r/L$),/nand develop a scaling law for the measured two-dimensional space-charge limit (2DSCL)/nrelative to the Child-Langmuir limit as a function of the aspect ratio of the diode./nThese simulations are done with a large ($100 T$) longitudinal magnetic field to restrict electron/nmotion to 1D, with the two-dimensional particle-in-cell simulation code OOPIC./nWe find a scaling law that is a monotonically decreasing function of this aspect ratio,/nand the one-dimensional result is recovered in the limit as $r >> L$./nThe result is in good agreement with prior results in planar geometry,/nwhere the emission area is proportional to the cathode width./nWe find a weak contribution from the effects of the drift tube for current at the beam edge,/nand a strong contribution of high current-density ``wings'' at the outer-edge of the beam,/nwith a very large relative contribution when the beam is narrow./nMechanisms for enhancing current beyond the Child-Langmuir limit remain a matter of great importance./nWe analyze the enhancement effects of upstream ion injection on the transmitted current in a one-dimensional parallel plate diode./nElectrons are field-emitted at the cathode,/nand ions are injected at a controlled current from the anode./nAn analytic solution is derived for maximizing the electron current throughput in terms of the ion current./nThis analysis accounts for various energy regimes, from classical to fully relativistic./nThe analytical result is then confirmed by simulation of the diode in each energy regime./nThe simulation approach involved iteratively testing injected ion current,/nand treating the measured transmitted electron current as a feedback mechanism./nThe feedback loop was automated, allowing for a single simulation to locate the optimized current./nBy tuning the injected ion current, we are able to optimize the transmitted electron current./nThis tuning of the ion current is automated by the integration of a high-level Python interface,/nwrapping the C++ particle-in-cell simulation code OOPD1./nIn this particular system, analysis showed that simulation runtime would be a function of transit time, and thus ion mass./nBy experimenting with reduced ion mass, we were able to significantly reduce simulation times,/nwhile recovering the same physical results./nField-limited emission is an approach for using Gauss's law to to satisfy the space charge limit for emitting current in particle-in-cell simulations./nWe find that simple field-limited emission models make several assumptions,/nwhich introduce small, systematic errors in the system./nWe make a thorough analysis of each assumption,/nand ultimately develop and test a new emission scheme that accounts for each./nThe first correction we make is to allow for a non-zero surface field at the boundary./nSince traditional field-emission schemes only aim to balance Gauss's law at the surface,/na zero surface field is an assumed condition./nBut for many systems, this is not appropriate,/nso the addition of a \emph{target} surface field is made./nThe next correction is to account for nonzero initial velocity,/nwhich, if neglected, results in a systematic underestimation of the current,/ndue to assuming that all emitted charge will be weighted to the boundary,/nwhen in fact it will be weighted as a fraction strictly less than unity,/ndepending on the distance across the initial cell the particle travels in its initial fractional timestep./nA correction is made to the scheme, to use the actual particle weight to adjust the target emission./nThe final analyses involve geometric terms,/nanalyzing the effects of cylindrical coordinates,/nand taking particular care to analyze the center of a cylindrical beam,/nas well as the outer edge of the beam, in Cartesian coordinates./nWe find that balancing Gauss's law at the edge of the beam is not the correct behavior,/nand that it is important to resolve the profile of the emitted current,/nin order to avoid systematic errors./nA thorough analysis is done of the assumptions made in prior implementations,/nand corrections are introduced for cylindrical geometry,/nnon-zero injection velocity, and non-zero surface field./nParticular care is taken to determine special conditions for the outermost node,/nwhere we find that forcing a balance of Gauss's law would be incorrect./nThe new emission scheme is tested in a two-dimensional periodic simulation,/nto demonstrate that the Jaffe limit for a one-dimensional diode with finite initial velocity is recovered./nWe also extend the iterative scheme developed earlier,/nand apply it to determine a scaling law for the Child-Langmuir limit/nin an axisymmetric planar diode, with finite initial velocity./nWe find that the new scheme reproduces prior results,/nand in significantly less computation time due to no longer needing to overinject,/nand leads to rapid convergence of the surface field,/nusing our new algorithmic optimization wrapper to seek the local limiting current along an emitter.

Advisor: Lieberman, Michael
Verboncoeur, John