Electron Boltzmann Equation Research Articles

Analysis of parameters of coaxial dielectric barrier discharges in argon flow at atmospheric pressure

This work deals with the numerical and experimental investigation of atmospheric pressure dielectric barrier discharges (DBDs). In the experiment, the current and voltage of DBD sustained in an argon flow in coaxial discharge cell are measured. Numerical models are based on the drift-diffusion theory of gas discharges. Different modelling approaches, where the electron kinetics is determined on the basis of the Maxwellian electron energy distribution function (EEDF), vs models with more realistic EEDF obtained from the solution of the electron Boltzmann equation are implemented. The effect of energy loss due to heating of the gas is considered.

On the electron energy distribution function in the high power impulse magnetron sputtering discharge

We apply the ionization region model (IRM) and the Orsay Boltzmann equation for electrons coupled with ionization and excited states kinetics (OBELIX) model to study the electron kinetics of a high power impulse magnetron sputtering (HiPIMS) discharge. In the IRM the bulk (cold) electrons are assumed to exhibit a Maxwellian energy distribution and the secondary (hot) electrons, emitted from the target surface upon ion bombardment, are treated as a high energy tail, while in the OBELIX the electron energy distribution is calculated self-consistently using an isotropic Boltzmann equation. The two models are merged in the sense that the output from the IRM is used as an input for OBELIX. The temporal evolutions of the particle densities are found to agree very well between the two models. Furthermore, a very good agreement is demonstrated between the bi-Maxwellian electron energy distribution assumed by the IRM and the electron energy distribution calculated by the OBELIX model. It can therefore be concluded that assuming a bi-Maxwellian electron energy distribution, constituting a cold bulk electron group and a hot secondary electron group, is a good approximation for modeling the HiPIMS discharge.

Transition characteristics and electron kinetics in microhollow cathode discharges

We demonstrate the transition characteristics and electron kinetics of microdischarges in a microgap with a cathode having microstructures using two-dimensional particle-in-cell/Monte Carlo collision (2d3v) simulations. It is found that the microdischarge is inside the hollow cavity at lower pressures, forming microhollow cathode discharges (MHCDs), while the dominant discharge moves outside the cavity at higher pressures, locating above the cathode rectangular protrusion tip. The spatial distributions of the microdischarge parameters (e.g., electric potential and electron density) are presented by capturing the transition characteristics. The electron kinetics of the microdischarges are examined based on the moment analysis of the electron Boltzmann equation. The collisional and non-collisional components of the electron power absorption are decomposed, which were seldomly investigated for microdischarges previously. The results provide fundamental insights into MHCD formation with a structured electrode at varying pressure conditions, which could be beneficial for the design and fabrication of microplasma devices in practical applications.

Decomposition of сarbon tetrachloride under the action of a dielectric barrier discharge of atmospheric pressure in an oxygen atmosphere

In this study, the process of decomposition of carbon tetrachloride (CCl4) vapor in oxygen DBD at atmospheric pressure and its kinetic regularities have been studied. In the course of the experiments, it was shown that the efficiency of the decomposition of carbon tetrachloride in DBD can reach 100%. Depending on the conditions of the experiments, the effective rate constants were equal to (0.16–0.59) s−1, and the decomposition energy yields were (0.001–0.012) molecules per 100 eV of the inputed energy. The main decomposition products were CO2 and Cl2 molecules. The formation of a solid on the internal electrode of the reactor was also found. The substance contains atoms of carbon, oxygen, chlorine (C:O:Cl) = 1:0.38:0.01, as well as hydrogen atoms. The substance also contains functional groups –CH, –CH2, –OH and dimers of carboxylic (chlorocarboxylic) acids. Based on the solution of the Boltzmann equation for electrons, it is shown that for the compositions of a gas containing O2 molecules, ССl4, and decay products, the kinetic and transport characteristics of electrons are the same as in a pure oxygen discharge. Using the kinetic characteristics of electrons and the reaction rate constants the mechanisms of reactions leading to the found reaction products are proposed. It was shown that the primary reaction of destruction is the reaction of dissociation of CCl4 by electron impact, leading to the formation of CCl3• and Cl and the reaction with the O (1D) atom, as a result of which CCl3• and ClO• are formed.

Helium and oxygen excited states densities in a He-air RF-driven atmospheric pressure plasma jet

The volumetric photon emission from a pulsed, RF-driven atmospheric pressure plasma jet was monitored from ultraviolet to the near infrared part of the spectrum. When a small admixture of dry air, as low as ∼0.01%, is added to the helium flow, the helium line intensities decrease, while those of oxygen increase dramatically. To understand the emission trends, a non-equilibrium, collisional-radiative model is developed to estimate He and O excited state densities as a function of relative air concentration. The model is based on a numerical solution of the electron Boltzmann equation for the electron energy distribution function self-consistently coupled to the balance equations for helium and air species. The spectral analyses reveal a good agreement between experiments and modeling, with the latter indicating that a complex interplay between electron kinetics and plasma chemistry is responsible for the substantial changes in emission profiles when He is diluted with a small amount of air.

Comparing Kinetic and MEP Model of Charge Transport in Graphene

Graphene has attracted the attention of several researchers because of its peculiar features. In particular, the study of charge transport in graphene is challenging for future electron devices. Usually, the physical description of electron flow in graphene given by the semiclassical Boltzmann equation is considered to be a good one. However, due to the computational complexity, its use in simulation tools is not practical and, as already done for traditional semiconductors such as Si or GaAs, simpler models are warranted. Here we will assess the validity of a class of hydrodynamical models based on the maximum entropy principle (MEP), by comparing, in the case of suspended monolayer graphene, the direct solution of the semiclassical Boltzmann equation for electrons, obtained by employing a discontinuous Galerkin approach, with the MEP distribution function. A reasonable agreement is observed.

Analysis of the multicomponent relativistic Boltzmann equation for electron scattering in big bang nucleosynthesis

Big-bang nucleosynthesis (BBN) is valuable as a means to constrain the physics of the early universe and it is the only probe of the radiation-dominated epoch. A fundamental assumption in BBN is that the nuclear velocity distributions obey Maxwell-Boltzmann (MB) statistics as they do in stars. Specifically, the BBN epoch is characterized by a dilute baryon plasma for which the velocity distribution of nuclei is mainly determined by the dominant Coulomb elastic scattering with mildly relativistic electrons. One must therefore deduce the momentum distribution for reacting nuclei from the multicomponent relativistic Boltzmann equation. However, the full multicomponent relativistic Boltzmann equation has only recently been analyzed and its solution has only been worked out in special cases. Moreover, a variety of schemes have been proposed that introduce nonthermal components into the BBN environment which can alter the thermal distribution of reacting nuclei. Here, we construct the relativistic Boltzmann equation in the context of BBN. We also derive a Langevin model and perform relativistic Monte-Carlo simulations which clarify the baryon distribution during BBN and can be used to analyze any relaxation from a nonthermal injection. We show by these analyses that the thermalization process leads to a nuclear distribution function that remains very close to MB statistics even during the most relativistic environment relevant to BBN. Hence, the predictions of standard BBN remain unchanged.

Third-order transport coefficient tensor of electron swarms in noble gases

In this work we extend a multi term solution of the Boltzmann equation for electrons in neutral gases to consider the third-order transport coefficient tensor. Calculations of the third-order transport coefficients have been carried out for electrons in noble gases, including helium (He), neon (Ne), argon (Ar), krypton (Kr) and xenon (Xe) as a function of the reduced electric field, E/n0 (where E is the electric field while n0 is the gas number density). Three fundamental issues are considered: (i) the correlation between the longitudinal component of the third-order transport tensor and the longitudinal component of the diffusion tensor, (ii) the influence of the third-order transport coefficients on the spatial profile of electron swarm, and (iii) the errors associated with the two term approximation for calculating the third-order transport coefficients for electron swarms in noble gases. It is found that a very strong correlation exists between the longitudinal components of the third-order transport coefficient tensor and diffusion tensor for the higher values of E/n0. The effects of the third-order transport coefficients on the spatial profile of electron swarms are the most pronounced for noble gases with the Ramsauer-Townsend minimum in the cross sections for elastic scattering. The largest errors of two term approximation are observed in the off-diagonal elements of the third-order transport coefficient tensor in Ar, Kr and Xe for the higher values of E/n0.

Electron swarm parameters in C2H2, C2H4 and C2H6: measurements and kinetic calculations

This work presents swarm parameters of electrons (the bulk drift velocity, the bulk longitudinal component of the diffusion tensor, and the effective ionization frequency) in C2Hn, with n = 2, 4, and 6, measured in a scanning drift tube apparatus under time-of-flight conditions over a wide range of the reduced electric field, 1 Td ≤ E/N ≤ 1790 Td (1 Td = 10−21 V m2). The effective steady-state Townsend ionization coefficient is also derived from the experimental data. A kinetic simulation of the experimental drift cell allows estimating the uncertainties introduced in the data acquisition procedure and provides a correction factor to each of the measured swarm parameters. These parameters are compared to results of previous experimental studies, as well as to results of various kinetic swarm calculations: solutions of the electron Boltzmann equation under different approximations (multiterm and density gradient expansions) and Monte Carlo simulations. The experimental data are consistent with most of the swarm parameters obtained in earlier studies. In the case of C2H2, the swarm calculations show that the thermally excited vibrational population should not be neglected, in particular, in the fitting of cross sections to swarm results.

Kinetics versus thermodynamics on CO2 dissociation in high temperature microwave discharges

Recent experiments by Groen et al (2019 Plasma Sources Sci. Technol. 28 075016) on CO2 activation at high translational temperature (3500 < Tg < 5500 K) are analyzed with a self-consistent code describing the vibrational kinetics of reacting CO2 together with CO and O2 and the corresponding electron Boltzmann equation. The kinetics results are compared with the corresponding thermodynamics values. This comparison shows that the major components of the mixture, i.e. CO and O can be described by the thermodynamic approach, while the other minor components present large deviation from equilibrium. A qualitative agreement is found between experimental and theoretical values of electron density, E/N and electron temperature in both diffuse and contracted regimes. The theoretical vibrational distributions as well as the corresponding electron energy distribution functions present non-equilibrium effects even at the considered high translational temperature.

Deep learning for solving the Boltzmann equation of electrons in weakly ionized plasma

A novel direct numerical method to calculate the electron velocity distribution function (EVDF) in hydrodynamic equilibrium under a uniform DC electric field is presented. In the present method, an artificial feedforward neural network learns the EVDF governed by both the Boltzmann equation and boundary conditions. The present method dost not require the expansion of the EVDF in the Legendre polynomials and the discretization of both the EVDF and the Boltzmann equation. As a benchmark, the EVDF in Reid’s ramp model gas and Ar gas was calculated by the present method, and then the validity of the present method was demonstrated by comparing electron energy distributions and electron transport coefficients deduced from the EVDF with those calculated by Monte Carlo simulation.

Simulation of bipolar charge transport in graphene on h-BN

Purpose The purpose of this paper is to simulate charge transport in monolayer graphene on a substrate made of hexagonal boron nitride (h-BN). This choice is motivated by the fact that h-BN is one of the most promising substrates on account of the reduced degradation of the velocity due to the remote impurities. Design/methodology/approach The semiclassical Boltzmann equations for electrons in the monolayer graphene are numerically solved by an approach based on a discontinuous Galerkin (DG) method. Both the conduction and valence bands are included, and the inter-band scatterings are taken into account as well. Findings The importance of the inter-band scatterings is accurately evaluated for several values of the Fermi energy, addressing the issue related to the validity of neglecting the generation-recombination terms. It is found out that the inclusion of the inter-band scatterings produces sizable variations in the average values, like the current density, at zero Fermi energy, whereas, as expected, the effect of the inter-band scattering becomes negligible by increasing the absolute value of the Fermi energy. Research limitations/implications The correct evaluation of the influence of the inter-band scatterings on the electronic performances is deeply important not only from a theoretical point of view but also for the applications. In particular, it will be shown that the time necessary to reach the steady state is greatly affected by the inter-band scatterings, with not negligible consequences on the switching on/off processes of realistic devices. As a limitation of the present work, the proposed approach refers to the spatially homogeneous case. For the simulation of electron devices, non-homogenous numerical solutions are required. This last case will be tackled in a forthcoming paper. Originality/value As observed in Majorana et al. (2019), the use of a Direct Simulation Monte Carlo (DSMC) approach, which properly describes the inter-band scatterings, is computationally very expensive because the valence band is highly populated and a huge number of particles is needed. Even by simulating holes instead of electrons does not overcome the problem because there is a certain degree of ambiguity in the generation and recombination terms of electron-hole pairs. The DG approach, used in this paper, does not suffer from the previous drawbacks and requires a reasonable computing effort.

Time dependent selfconsistent electron energy distribution functions during nano-second repetitive discharges in reacting N2/H2 mixtures

Electron energy distribution function, vibrational and electronic excited states are self-consistently solved to investigate the formation of ammonia under nano-second repetitive pulsed discharges. The inclusion of transitions starting from electronically and vibrationally excited states of N2 and H2 molecules in the electron Boltzmann equation is discussed to rationalize the evolution of the electron energy and level distributions, as well as of chemical composition. The results are presented focusing on the dependence of the vibrational distributions of both N2 and H2 molecules and eedf on the applied reduced electric field. The relevance of using complete sets of electron impact cross sections, including transitions from vibrationally excited states has been investigated, comparing with the ground state model.

A kinetic model for the electrostatic spark discharge in atmospheric-pressure air

This paper presents a 0-D kinetic model of electrostatic spark discharges consisting of the time-dependent Boltzmann equation of electrons, a discharge-circuit equation, and heavy particles’ kinetic equations to investigate the energy-transfer mechanisms from the electrostatic energy given to the energy of gases by the spark discharge. In this report, the model is applied to the discharges in atmospheric-pressure air under optimum conditions corresponding for the minimum ignition energies of the typical flammable gases, hydrogen, ethylene and propane, in three types of explosion groups for gases.

1D Modeling of the Microwave Discharge in Liquid n-Heptane Including Production of Carbonaceous Particles

Microwave plasma in the liquid is initiated inside a gas bubble formed at the end of the electrode-antenna, through which microwave energy is introduced into the liquid. A 1D model a set of gas phase kinetic reactions describes the evolution processes of ionization, heat transfer and formation of gas and solid products inside the plasma bubble. The code is based on joint solution of the Boltzmann equation for free electrons of the plasma, a simplified equation for the microwave field, the heat conduction equation, the balance equation for the electron density and the balance equations for the weight fraction for all gas and solid products of n-heptane pyrolysis. The Joule heat released in the plasma is expended on the evaporation of liquid n-heptane into the bubble and the decomposition of the n-heptane molecules. The model includes both the description of gas phase processes and formation of solid carbon-containing particles. The growth mechanism for generation of solid particles describes simultaneous processes of the initial nucleation, surface growth and coagulation of soot particles. The results of calculations are compared with known experimental results.

Boltzmann equation studies on electron swarm parameters in Townsend breakdown of copper vapor plasma using independently assessed electron-collision cross sections

Electron transport coefficients in copper vapor plasma are calculated both by two-term expansion of electron Boltzmann equation Bolsig+ and tracking the random motion of electrons using Monte Carlo collision code METHES based upon recently evaluated cross section sets. The copper atoms are evaporated from hot electrode during the post-arc phase of vacuum circuit breakers, in which Townsend breakdown between electrode gaps is probable. The electron energy probability function, electron mean energy, flux/transport mobility and diffusion coefficients, as well as Townsend ionization coefficients are shown in reduced fields 10∼1000 Td at a typical vapor temperature 2000 K. The validity of two-term approximation is checked by comparison to well benchmarked METHES code. If the electrode temperature varies between 1500∼2500 K, the influence of vapor temperature on ionization coefficients is about 5% at 200.4 Td, and drops to 0.5% at 493 Td according to Bolsig+ results. Similar to classic gas discharge theory, the Paschen curve is proposed for Townsend breakdown of metal vapor. Using the calculated ionization coefficient and a constant secondary electron yield, the Paschen minimum is determined to be 106∼122 V at a critical value of the product of vapor density and gap length (4.7∼5.7)×1019 m-2. A satisfactory agreement was found with the previously measured ignition voltage between vacuum interrupter contacts after the arcing.

Improved mobility models for charge transport in graphene

Abstract Charge transport in graphene is crucial for the design of a new generation of nanoscale electron devices. A reasonable model is represented by the semiclassical Boltzmann equations for electrons in the valence and conduction bands. As shown by Romano et al. (J. Comput. Phys., 2015), the discontinuous Galerkin methods are a viable way to tackle the problem of the numerical integration of these equations, even if efficient DSMC with a proper inclusion of the Pauli principle have been also devised. One of the advantages of the solutions obtained with deterministic approach is of course the absence of statistical noise. This fact is crucial for an accurate estimation of the low field mobility as proved by Majorana et al. (J. Math. Industry, 2016) in the case of a unipolar charge transport in a suspended graphene sheet under a constant electric field. The mobility expressions are essential for the drift-diffusion equations which constitute the most adopted models for charge transport in CAD. Here the analysis by Majorana et al. (J. Math. Industry, 2016) is improved in two ways: by including the charge transport both in the valence and conduction bands; by taking into account the presence of an oxide as substrate for the graphene sheet. New models of mobility are obtained and, in particular, relevant improvements of the low field mobility are achieved.

Particle-in-cell simulation of ultrafast hot-carrier transport in Fe/Au heterostructures

We describe a theoretical approach for spin-polarized hot-electron transport, as it occurs after excitation by ultrafast optical pulses in heterostructures formed by ferromagnetic and normal metals. We formulate a spin-dependent particle-in-cell model that solves the Boltzmann equation for excited electrons. It includes lifetimes and transmission coefficients as parameters, which can be taken from ab-initio calculations or experiment, and can be easily extended to multilayer systems. This approach is capable of describing electron transport in the ballistic, super-diffusive and diffusive regime including secondary-carrier generation. We apply the model to optically excited carriers in Fe/Au bilayers and Fe/Au/Fe spin-valve structures. We gain microscopic insight into the hot-electron transport dynamics probed in recent experiments on spin-valves. We find contributions to the demagnetization dynamics induced in Fe/Au/Fe trilayers regardless of the parallel or antiparallel magnetic alignment of the Fe layers.

Kinetic mechanisms in air plasmas

A self-consistent kinetic model is developed to study air (N2–20%O2) plasma discharges in single pulses, their afterglows, and in repetitively pulsed discharges produced at low-pressures (133, 210 and 400 Pa) in a 1 cm inner radius tube. Since the pulse duration ranges from ∼0.3 up to ∼10 ms, the model includes the interplay of the vibrational kinetics of N2 and chemical kinetics involving neutral, excited and charged species, which are coupled to the electron Boltzmann equation during the active plasma phase. A satisfactory agreement is found between modelling simulations and different experimental results. A comprehensive and detailed analysis of the modelling calculations is carried out, providing a general overview and insight of the most important mechanisms that take place in air plasmas at low-pressures.

Non-equilibrium kinetics of the ground and excited states in N2–O2 under nanosecond discharge conditions: extended scheme and comparison with available experimental observations

A numerical 0D kinetic model is developed here to simulate the detailed kinetics of the ground and excited electronic states of atmospheric gases occurring under streamer discharge conditions. The model is based on an extended kinetic scheme that involves state-to-state vibrational kinetics of the ground electronic states of N2, O2 and NO diatomics, including e–V, V–V and V–T energy transfers, and contains about 104 processes for more than 500 tracked state-specific states (including the higher excited and autoionising states of NI/OI atomic species), and also for state-non-specific species. This scheme takes into account the most important radiative processes occurring in N2 and N2–O2 mixtures that are of diagnostic interest, including the extreme ultraviolet and vacuum ultraviolet radiation produced by many excited and autoionising states of NI/OI. The dependence of the rates of electron-impact processes on the reduced electric field E/N is found by solving the Boltzmann equation for electrons in the two-term approximation using the Boltzmann equation solver Open Source library, BOLOS.We examine several test cases: pure nitrogen at 50 and 200 Torr and synthetic air at 10, 50 and 760 Torr, at a gas temperature of 300 K. The numerical results for pure nitrogen were compared with experimental results recently obtained by laser induced fluorescence for metastables. Good qualitative agreement was observed between numerical simulations and experiments at vibronic levels v = 2–5, giving important feedback on the appropriateness of the kinetic scheme. Numerical results were also used to find several general spectrometric signatures that might be helpful in the development of diagnostic procedures.