THE KINETIC SIMULATION IN VACUUM ELECTRONICS: UNCOVERING THE FUNDAMENTAL NATURE OF NON-MAXWELLIAN DISTRIBUTION FUNCTION EFFECTS Текст научной статьи по специальности «Физика»

i l St. Petersburg Polytechnic University Journal. Physics and Mathematics. 2022 Vol. 15, No. 3.3 Научно-технические ведомости СПбГПУ. Физико-математические науки. 15 (3.3) 2022

Conference materials UDC 533.9.02

DOI: https://doi.org/10.18721/JPM.153.314

The kinetic simulation in vacuum electronics: uncovering the fundamental nature of non-Maxwellian distribution function effects

V. Yu. Kozhevnikov 1H, A. V. Kozyrev \ A. O. Kokovin \ N. S. Semeniuk 1 1 Institute of High Current Electronics SB RAS, Tomsk, Russia H Vasily.Y.Kozhevnikov@ieee.org

Abstract. This paper presents main theoretical results obtained in vacuum explosive-emission electronics by using of the numerical methods aimed to the direct solution of Vlasov Poisson kinetic equations. It was shown that computational physical kinetics makes it possible to explain a number of important physical laws occurring in vacuum diodes unlike widely used Particle-in-Cell or hydrodynamic simulation. The kinetic approach makes it possible to take into account nonequilibrium non-Maxwellian effects mainly associated with the "tails" of distribution functions. The advantages of kinetic simulation are shown on two highly relevant problems of vacuum electronics are considered in details.

Keywords: vacuum electronics, physical kinetics, numerical simulation

Funding: The work was carried out within the framework of the state assignment of the Ministry of Science and Higher Education of the Russian Federation on the topics FWRM 2021 0007, FWRM-2021-0014.

Citation: Kozhevnikov V. Yu., Kozyrev A. V., Kokovin A. O., Semeniuk N. S., The kinetic simulation in vacuum electronics: uncovering the fundamental nature of non-Maxwellian distribution function effects, St. Petersburg State Polytechnical University Journal. Physics and Mathematics, 15 (3.3) (2022) 76-81. DOI: https://doi.org/10.18721/JPM.153.314

This is an open access article under the CC BY-NC 4.0 license (https://creativecommons. org/licenses/by-nc/4.0/)

Материалы конференции УДК 533.9.02

DOI: https://doi.org/10.18721/JPM.153.314

Кинетическое моделирование в вакуумной электронике: раскрытие фундаментальной природы эффектов немаксвелловской функции распределения

В. Ю. Кожевников 1Н, А. В. Козырев 1, А. О. Коковин 1, Н.С. Семенюк 1 1 Институт сильноточной электроники СО РАН, Томск, Россия н Vasily.Y.Kozhevnikov@ieee.org

Аннотация. В статье представлены основные теоретические результаты, полученные в вакуумной взрывоэмиссионной электронике с использованием численных методов решения системы кинетических уравнений Пуассона-Власова. Было показано, что именно вычислительная физическая кинетика позволяет объяснить ряд важных физических закономерностей, происходящих в вакуумных диодах. Кинетический подход согласованно учитывает неравновесные немаксвелловские эффекты, определяемые т.н. «хвостами» функций распределения. Подробно показаны преимущества кинетического моделирования на примере двух актуальных задач вакуумной электроники.

Ключевые слова: вакуумная электроника, физическая кинетика, численное моделирование

Финансирование: Работа выполнена в рамках Государственных заданий «Разряды в газах высокого и низкого давления и применения низкотемпературной газоразрядной

© Kozhevnikov V. Yu., Kozyrev A. V., Kokovin A. O., Semeniuk N. S., 2022. Published by Peter the Great St.Petersburg Polytechnic University.

плазмы в электрофизических устройствах и технологиях» (код темы FWRM 2021 0007) и «Высокоинтенсивные источники спонтанного и индуцированного излучения оптического диапазона спектра и их применение для решения научных и технологических задач» (код темы FWRM 2021 0014).

Ссылка при цитировании: Кожевников В. Ю., Козырев А. В., Коковин А. О., Семенюк Н. С., Кинетическое моделирование в вакуумной электронике: раскрытие фундаментальной природы эффектов немаксвелловской функции распределения // Научно-технические ведомости СПбГПУ. Физико-математические науки. 2022. Т. 15. № 3.3. С. 76-81. DOI: https://doi.org/10.18721/JPM.153.314

Статья открытого доступа, распространяемая по лицензии CC BY-NC 4.0 (https:// creativecommons.org/licenses/by-nc/4.0/)

Introduction

For decades, vacuum electronics has remained an area of plasma physics attracting a great interest which is explained by its wide practical application [1]. From a theoretical point of view, vacuum electronics is a self-consistent description of space-charge limited (SCL) flows of charged particles. The SCL can be clearly explained by the example of a simple two-electrode system (vacuum diode) to which a certain voltage is applied. Even if the charges injection from cathode is assumed to be unlimited, the total current through the diode is always limited to

. _ 4S0 ¡2qU3/2

hcL 9 \ m D2 ' (1)

w.r.t. the voltage U applied to the infinite parallel plates located at a distance D from each other (m is the electron rest mass, q is the electron charge, s0 is the vacuum dielectric permittivity). The expression of (1) is called the Child's law [2]. It involves many assumptions, e.g. zero initial electron velocities, electrons only emission and a steady-state approximation, however the Child's law provides an ideal theoretical understanding of the SCL phenomenon and the tendency limiting the maximum current that can be carried through the vacuum device. Subsequently [3], Child's law was significantly corrected to relax the approximations used in (1).

In the beginning of the computational plasma physics era, the particle-in-cell (PIC) method was introduced for studying many transient processes in plasma [4]. Another more fundamental and much more computationally difficult principle is based on simultaneous solution of Boltzmann equations together with the electromagnetic field equations. The primary objects of kinetic numerical simulation are the electron and ion distribution functions (EDF and IDF), which provide more information about the system of charged particles than the characteristics of macroparticles in PIC approach. As an example, a pair of one-dimensional (1D) problems of non-stationary current flow in planar vacuum diodes are considered below. The 1D makes it possible to exclude from consideration geometric factors, which are usually referred to for the phenomenological explanation of complex phenomena. Thus, physical kinetics is used in this paper to simulate problems that are simple in terms of computational performance, but at the same time, the methods reveal non-trivial features that can only be described using a more fundamental approach.

Materials and Methods

Let us denote f the EDF and f the single-charge IDF. Thus, in 1D formulation the EDF and IDF depend on spatial variable x, momentum px, and time variable t. Both distribution functions obey collisionless kinetic (Vlasov) [5] equations

^L + LL f - qEx J _ о dt m dx x dp

e (2) J + Л J + qEx J _ о dt mi dx dpx

where me and m. are rest masses of electron and ion, respectively.

© Кожевников В. Ю., Козырев А. В., Коковин А. О., Семенюк Н. С., 2022. Издатель: Санкт-Петербургский политехнический университет Петра Великого.

Neglecting the magnetic field influence, the system of Vlasov equations (2) describes the electrons and ion motion in the electric field Ex. The electric field equation has to be solved mutually with (2) in order to obtain a self-consistent solution, so we complement (2) with 1D Poisson's equation for electric potential ^

f? = -±{ni-ne), ex =-d£, ,(x,t)=J{x,px,t)dpx, (3)

where e0 is dielectric permittivity of vacuum, ne and n. are electron and ion number densities, respectively.

The cathode is maintained at the point of x = 0 with the electric potential ^ = 0, and the anode is at a distance D in the point x = D with the electric potential U(t). Thus, the Poisson's equation (3) can be solved in quadratures in the form of the following expressions:

Ex (x, t) = -Ujt) -q J[ne (x ', t)- n, (x ', t)]dx' + -q- J J[n, (x ', t)- n, (x ', t)]dx' dx,

D -0 0 -0D 0 0 D x x x'

9( x, t ) = U (t)x + JJ[ne (x ', t)-n (x', t)] dx 'dx - q JJ[ ne (x '', t)- nt (x '', t)] dx "dx'.

D -0D 0 0 -0 0 0

The initial conditions for the equations (2)-(4) are chosen to be zero fe = f. = 0, U = 0. The system of equations (2)-(4) is solved numerically using semi-Lagrangian methods [5] at a rectangular uniform phase-space grid (x, px).

The "Anomalous" Electrons in Nanosecond Vacuum Diode

One of the novel physical phenomena regarding vacuum diodes became the effect of fast electrons generation with energies e exceeding the maximum voltage Umax applied to the diode (multiplied by an elementary charge q) e > qUmax. This group of electrons are also known as electrons with the "anomalously high" energies. Depending on experimental conditions fairly large groups of electrons with "anomalously" high kinetic energies can exist.

To elucidate the conditions for the "anomalous electrons" appearance, we posed the problem of a non-stationary current flow in planar vacuum diode with predominant electron emission from the cathode during the applying to it a voltage pulse U(t) with a short leading edge trlee and a constant amplitude of Umax. The term "short leading edge" suggests that the front duration is comparable to the electrons time of flight on their way from the cathode to the anode. By

setting the cathode electron emission condition with the boundary condition at x = 0 for the

2

_ _ Px

EDF fe (x = 0,px,t) = n0 / ^ 2nmW0e 2mW", where Wg is the electron temperature, n0 is emission electron number density, and f. = 0, for short anode voltage rise time we obtain the relaxation oscillation of the collector current (Fig. 1) as it was shown in [6]. No oscillations are observed at the large lengths of the anode voltage rise time, and in the case of intermediate values (~ 0.5 ns), the oscillations have a lower amplitude and decay more intensively than at short rise times (below or equal to 0.1 ns). The establishment of a stationary current flow leads to that the collector current density becomes equal to jSCL, regardless of the duration of the anode voltage pulse duration.

The collector current peak arises when the electron beam reaches the anode. The relaxation oscillations buildup is caused by the irregularity of charge inflow and outflow into the interelectrode gap. However, the computational physical kinetics allows to look at the process of the peak collector current density formation from the more fundamental point of view. Fig. 2 shows the comparison of the EDF density plots corresponding to the shortest anode voltage rise time (triee = 0.1 ns) in the case of stationary current flow (a) and at the time point corresponding to the maximum collector current density (b).

The electron beam reaches anode at this time point (Fig. 2, b) have mean energy exceeding qU by more that 20 percent with respect to steady-state value equal exactly to qU . If the

max max

rise time is relatively longer (here ~2 ns), the electron beam mean energy doesn't override qUmax value during the whole voltage pulse duration. Its EDF is similar to Fig. 2, a. The electrons with "anomalously high" energies form anode current density relaxation oscillations peaks.

Fig. 1. The collector current density given for different risetimes (D = 1 cm, If = 2 kV)

v 7 max '

2 4 6 8

K-coorSinsle. mm

Fig. 2. The EDF density plots corresponding to a steady-state (t = 5 ns) and to the maximum of the collector current (t = 770 ps)

The "Anomalous" Ion Transport in Nanosecond Vacuum Diode

Another problem of vacuum electronics is connected to an explosive emission generating a dense quasi-neutral cathode plasma [7]. Explosive cathode plasma consists of electrons and a group of multiple-charged metal ions located in a thin near-cathode sheath. In various experiments it was convincingly shown that the critical role in the vacuum breakdown plays the existence of directed ion flows from cathode to anode [8]. Numerous experiments indicate that ions not only move towards the anode, but also their kinetic energies are exceeding the typical vacuum discharge combustion voltage (multiplied by q), e.g. see [9]. These ions are called the "anomalous", and their transfer processes in vacuum diodes are called the "anomalous ion transport", due to an ability to travel from cathode to anode. Here we give the simplest self-consistent explanation to this phenomenon from a standpoint of physical kinetics. The simplified case of a cathode plasma composition consisting of electrons and single-charged metal (carbon) ions is considered in planar 1D case here.

At t = 0 emission plasma enter the vacuum diode from the cathode as a nonequilibrium (T = 3 eV, Te = 1 eV) quasi-neutral plasma with average number density of ne « n. = n0. The regular plasma flow from cathode into the gap is modeled by simple boundary conditions

fe,t (x = 0Px,f) = no ^2%mqTeie

2mqTe i

Fig. 3 shows the EDF dynamics fe at the initial stage of vacuum breakdown. The very first frame shows that the EDF looks similar to the case of stationary current flow in a vacuum diode (Fig. 2). Fig. 4 illustrates the phenomenon of the "anomalous ions acceleration" as is. Fast electron plasma component displacement is accompanied by slower ion dynamics also moving towards the anode. Fig. 5 shows the true electrodynamic cause of the "anomalous" ion acceleration. Starting from the first time points, electrons move towards the anode forming a region of negative electric potential near the cathode ("virtual cathode"). This forces ions to accelerate towards the anode. During the ion motion between the physical and the "virtual" cathode the displacement of a negative potential region occurs, and the cathode plasma starts to fill the near-cathode space. This process continues until the plasma fills the entire diode. These later stages of a vacuum breakdown are shown in details in [10] both for two- and multi-component cathode plasma.

The ion plasma component motion occurs at velocities that are significantly higher than the average value corresponding to a "virtual cathode" depth. One can say that "anomalous" ion acceleration is a motion of ions having an "anomalously" high energies. The IDF "tail" prolongs from near to thermal energies of several electron-volts up to 100 eV, while the instant electric potential minimum doesn't exceed —25 V. The reason of why ions are moving in the "anomalous transport" mode acquire higher energies is in the nonstationary influence of the electric field due to the formation and subsequent displacement of the "virtual cathode" region towards anode. The ions turn into the continuous electrodynamic acceleration regime. This physical mechanism is similar to the formation of runaway electron beams [11].

p

Fig. 3. The EDF f density plots computed for selected time steps. In arbitrary units normalized to

boundary maximal value

200

150

g? 100 5

2 50

50 ns 100 ns 150 ns 200 ns

i jf J L/

i J e a i-coordmate, mm

10

2 A 6 B K-DQCfdina1e, mm

2 <! 6 B s-coondinata, mm

2 4 6 fl x-ooofdinale. mm

Fig. 4. The IDF f density plots computed for selected time steps. Plot is given in arbitrary units; the absolute values have to be multiplied by 1039 s/(kg-m4)

¡-oooranele

Fig. 5. The electrostatic potential evolution for selected time steps

The results of this paper convincingly show key advantages of computational physical kinetics with respect to the fundamental problems of vacuum electronics.

REFERENCES

1. Beck A. H., Marton L., Handbook of Vacuum Physics, Vol. 2: Physical Electronics. Physics Today, 23.5 (1970).

2. Child C. D., Discharge from Hot CaO, Physical Review (Series I), 32 (5) (1911) 492-511.

3. Ragan-Kelly B., Explorations of Space-Charge Limits in Parallel-Plate Diodes and Associated Techniques for Automation, PhD Thesis, UCLA, Berkley, 2013.

4. Birdsall C. K., Langdon A. B., Plasma Physics via Computer Simulation, 1991.

5. Kozhevnikov V. Y., Kozyrev A. V., Semeniuk N. S., Modeling of Space Charge Effects in Intense Electron Beams: Kinetic Equation Method Versus PIC Method, IEEE Transactions on Plasma Science, 45(10) (2017) 2762-2766.

6. Airapetov A. Sh., Feoktistov A. L., Current limit in a non-stationary flat diode, Kratkie Soobshchenia po Fizike, 2 (18) (1990) 18-20.

7. Litvinov E. A., Mesyats G. A., Proskurovskiï D. I., Field Emission and Explosive Electron Emission Processes in Vacuum Discharges, Soviet Physics Uspekhi, 26 (2) (1983) 138-159.

8. Krinberg I. A., Three modes of plasma expansion in a vacuum arc. The Proceedings of XXI International Symposium on Discharges and Electrical Insulation in Vacuum (ISDEIV), Yalta, Russia, (2004) 178-181.

9. Oks E. M., Savkin K. P., Yushkov G. Y., Nikolaev A. G., Anders A., Brown I. G., Measurement of total ion current from vacuum arc plasma sources. Review of Scientific Instruments, 77(3) (2006)

10. Kozhevnikov V., Kozyrev A., Kokovin A., Semeniuk N., The Electrodynamic Mechanism of Collisionless Multicomponent Plasma Expansion in Vacuum Discharges: From Estimates to Kinetic Theory. Energies, 14 (22) (2021) 7608.

11. Zubarev N. M., Kozhevnikov V. Y., Kozyrev A. V., Mesyats G. A., Semeniuk N. S., Sharypov K., Shunailov S. A., Yalandin M. I., Mechanism and dynamics of picosecond radial breakdown of a gas-filled coaxial line. Plasma Sources Science and Technology, 29 (12) (2020) 125008.

03B504.

THE AUTHORS

KOZHEVNIKOV Vasily Yu.

Vasily.Y.Kozhevnikov@ieee.org ORCID: 0000-0001-7499-0578

KOKOVIN Aleksandr O.

kokovin.alexandr@mail.ru ORCID: 0000-0003-2068-7674

KOZYREV Andrey V.

kozyrev@to.hcei.tsc.ru ORCID: 0000-0002-7078-7991

SEMENIUK Natalia S.

viliiskoeozero@yandex.ru ORCID: 0000-0002-5972-2839

Received 04.05.2022. Approved after reviewing 05.07.2022. Accepted 05.07.2022.

© Peter the Great St. Petersburg Polytechnic University, 2022