Self-consistent Charge Transport Research Articles

Self-consistent charge transport model with ionization for the alphie plasma thruster

The Alternative Low-Power Hybrid Ion Engine (alphie) is a new technology for space propulsion based on plasma. Its distinct characteristic is the counterflow of charges (ions and electrons) passing through its two-grid system. This means that electrons coming from an external cathode are accelerated toward the ionization chamber, in which a neutral gas (typically Ar) is injected. The strong magnetic field therein confines these electrons, which ionize and exchange energy with the propellant gas. Thus, the operation of alphie is strongly affected by the electrons coming from the external cathode and their collisions with the neutral atoms. This work studies the counterflow employing a particle-in-cell simulation of ions and electrons passing through a single hole as a function of the electron cathode currents (Ice) and potential drops between grids (VAC). Transparency of the grid system to ions and electrons and the ion current extracted by the grid system are studied under sweeps of these two parameters. The number of ionization events by each high-energy electron entering the ionization chamber is evaluated using a physical model based on the gas density and the cross section for ionization. These new ions are then extracted by the same electric field that accelerates the electrons inward. Thus, simulations are self-consistent, since the ionizing electron flow from the external cathode drives the ion outflow at the exit section of the two-grid system. The electrical transparency of the two-grid system to ions and electrons, related to the axial charge currents, is also studied under sweeps of aforementioned operation parameters. This new way to deal with ionizations can be useful to study other plasma thrusters in which electrons for ionization come from an external cathode without modeling the complex structure of the ionization chamber.

BORIS – Micromagnetic, Spin Transport and Multiscale Atomistic Software for Modelling Magnetic Information Storage

A brief review of BORIS is given here, together with a review of recent works using this software, including applications to modelling magnetic hard-disk-drive read heads, ultrafast magnetization processes, computation of thermodynamic equilibrium states using Monte Carlo algorithms, and modelling skyrmions as information carriers. BORIS is a state-of-the-art multi-physics and multi-scale research software designed to solve three-dimensional magnetization dynamics problems, coupled with a self-consistent charge and spin transport solver, heat flow solver with temperature-dependent material parameters, and elastodynamics solver including thermoelastic and magnetoelastic/magnetostriction effects, in arbitrary multi-layered structures and shapes. Both micromagnetic and atomistic models are implemented, also allowing multi-scale modelling where computational spaces may be configured with multiple simultaneous micromagnetic and atomistic discretization regions. The software allows multi-GPU computations on any number of GPUs in parallel, in order to accelerate simulations and allow for larger problem sizes compared to single-GPU computations – this is the first magnetization dynamics software to allow multi-GPU computations, enabling large problems encompassing billions of cells to be simulated with unprecedented performance.

Simulation of positive streamers in atmospheric air by a macroscopic model with a new branching criterion

Streamers are the initial stage of electric breakdown in a long air gap. The modelling of streamers in atmospheric air can lay the foundation for the modelling of long sparks and help the insulation design in ultra-high voltage projects. In this paper, a 3D macroscopic model is built to describe the development of positive streamers. This model is based on a discharge tree model, first implemented by Luque and Ebert, which describes the self-consistent charge transport in streamers. The modified model considers the multiple initial streamers, streamer width variation after branching and improved streamer head mobility. More importantly, a new practical branching criterion of positive streamers is proposed. A higher branching possibility is shown if the average photoionization rate ahead of a streamer decreases to a certain critical value. In atmospheric air this critical value is estimated to be 4.5×1019 cm−3 S−1 from experimental results. We performed a simulation on the development of streamers with tens of centimeters. The calculated results of streamer branching ratio agrees well with the experimental data. The influences of the critical photoionization rate and channel electrical conductance are discussed.

Growing discharge trees with self-consistent charge transport: the collective dynamics of streamers

We introduce the generic structure of a growth model for branched discharge trees that consistently combines a finite channel conductivity with the physical law of charge conservation. It is applicable, e.g., to streamer coronas near tip or wire electrodes and ahead of lightning leaders, to leaders themselves and to the complex breakdown structures of sprite discharges high above thunderclouds. Then we implement and solve the simplest model for positive streamers in ambient air with self-consistent charge transport. We demonstrate that charge conservation contradicts the common assumption of dielectric breakdown models that the electric fields inside all streamers are equal to the so-called stability field and we even find cases of local field inversion. We also find that, counter-intuitively, the inner branches of a positive-streamer tree are negatively charged, which provides a natural explanation for the observed reconnections of streamers in laboratory experiments and in sprites. Our simulations show the structure of an overall ‘streamer of streamers’ that we name collective streamer front, and predict effective streamer branching angles, the charge structure within streamer trees and streamer reconnection.

Secondary electron emission and self-consistent charge transport in semi-insulating samples

Electron beam induced self-consistent charge transport and secondary electron emission (SEE) in insulators are described by means of an electron-hole flight-drift model (FDM) now extended by a certain intrinsic conductivity (c) and are implemented by an iterative computer simulation. Ballistic secondary electrons (SE) and holes, their attenuation to drifting charge carriers, and their recombination, trapping, and field- and temperature-dependent detrapping are included. As a main result the time dependent “true” secondary electron emission rate δ(t) released from the target material and based on ballistic electrons and the spatial distributions of currents j(x,t), charges ρ(x,t), field F(x,t), and potential V(x,t) are obtained where V0 = V(0,t) presents the surface potential. The intrinsic electronic conductivity limits the charging process and leads to a conduction sample current to the support. In that case the steady-state total SE yield will be fixed below the unit: i.e., σ = η + δ < 1.

Fast electron beam charge injection and switching in dielectrics

AbstractBasic investigations of secondary electrons (SE) relaxation and attenuation are made by means of Monte Carlo simulations using ballistic electron scattering and interactions with optical and acoustic phonons as well as impact ionization of valence band electrons. Then the electron beam induced selfconsistent charge transport and secondary electron emission in insulators are described by means of an electron‐hole flight‐drift model (FDM). Ballistic secondary electrons and holes, their attenuation and drift, as well as their recombination, trapping, and field‐ and temperature‐dependent Poole‐Frenkel detrapping are included. Whereas the initial switching‐on of the secondary electron emission proceeds over milli‐seconds due to long‐lasting selfconsistent charging, the switching‐off process occurs much faster, even over femto‐seconds.Thus a rapid electron beam switching becomes possible with formation of ultra‐short electron beam pulses offering an application in stroboscopic electron microscopy and spectroscopy. (© 2011 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Time-dependent start-up and decay of secondary electron emission in dielectrics

Electron beam induced selfconsistent charge transport and secondary electron emission in insulators are described by means of an electron-hole flight-drift model implemented by an iterative computer simulation. Ballistic secondary electrons and holes, their attenuation and drift, as well as their recombination, trapping, and field-dependent and temperature-dependent detrapping are included. As a main result the time dependent secondary electron emission rate σ(t) and the spatial distributions of currents j(x,t), charges ρ(x,t), field F(x,t), and potential V(x,t) are obtained. Whereas the switching-on of the secondary electron emission proceeds over milliseconds due to selfconsistent charging, the switching-off process occurs much faster, even over femtoseconds.

Ultrafast relaxation of electrons and selfconsistent electrical charging in insulators

Abstract Low energy electron scattering in insulators is described by a Monte Carlo program based on acoustic and optical phonon scattering as well as on impact ionization of valence band electrons, especially aimed to the scattering of ballistic and drifting electrons in wide gap dielectrics. There is a rapid relaxation of excited electrons within the conduction band of wide gap insulators like SiO 2 and Al 2 O 3 over femtoseconds. The field-dependent transport and trapping parameters allow us to model the selfconsistent charge transport and charging-up of insulating samples during electron irradiation. The resulting spatial and time-dependent distributions of all currents j ( x ,t), charges ρ ( x ,t), the electric field F ( x ,t) and the potential V ( x ,t) are obtained and compared with experimental measurements.

Electron beam probing of silica surface layers on alumina

The electron beam induced self-consistent charge injection and transport in a layered insulator SiO 2 –Al 2 O 3 is described by means of an electron–hole flight–drift model FDM and an iterative computer simulation. Thermal and field-enhanced detrapping are included by the Poole–Frenkel effect. The surface layer with a modified electric surface conductivity is included which describes the surface leakage currents. Furthermore, it will lead to particular charge incorporation at the interface between the surface layer and the bulk substrate. As a main result the time-dependent spatial distributions of currents j ( x , t ), charges ρ ( x , t ), field F ( x , t ), and potential V ( x , t ) are obtained. The spatial charge distribution with depth shows a quadro-polar plus–minus–plus–minus structure in nanometer dimension.

Injection of charge nanostructures into insulators

The electron beam induced selfconsistent charge transport in insulators is described by means of an electron-hole flight–drift model FDM and an iterative computer simulation. Ballistic secondary electrons and holes, their attenuation and drift, as well as their recombination, trapping, and detrapping are included. Thermal and field-enhanced detrapping are described by the Poole–Frenkel effect. As a main result the time dependent secondary electron emission rate σ ( t ) and the spatial distributions of currents j ( x , t ) , charges σ ( x , t ) , field F ( x , t ) , and potential V ( x , t ) are obtained. The spatial charge distributions with depth show a quadropolar plus–minus–plus–minus structure in nanometer dimensions.

Electron beam charging of insulators with surface layer and leakage currents

The electron beam induced self-consistent charge transport in layered insulators (here, bulk alumina covered by a thin silica layer) is described by means of an electron-hole flight-drift model and an iterative computer simulation. Ballistic secondary electrons and holes, their attenuation and drift, as well as their recombination, trapping, and detrapping are included. Thermal and field-enhanced detrapping are described by the Poole–Frenkel effect. Furthermore, an additional surface layer with a modified electric surface conductivity is included which describes the surface leakage currents and will lead to particular charge incorporation at the interface between the surface layer and the bulk substrate. As a main result, the time-dependent secondary electron emission rate σ(t) and the spatial distributions of currents j(x,t), charges ρ(x,t), field F(x,t), and potential V(x,t) are obtained. For bulk full insulating samples, the time-dependent distributions approach the final stationary state with j(x,t)=const=0 and σ=1. In the case of a measurable surface leakage current, the steady stationary state is reached for σ&amp;lt;1. First measurements are extended to the sample current measurement including nonstationary components of charge incorporation and polarization as well as dc components of leakage currents.

Injection and selfconsistent charge transport in bulk insulators

The new flight-drift model (FDM) of selfconsistent electron transport and electrical charge storage in wide-gap insulators reflects a more realistic simulation of these processes in dielectric and insulating materials than the former mainly ballistic model. Thus, electron-hole creation, their ballistic flight, followed by field-drift transport, and finally trapping in localized states and/or recombination are taken into account. The experimentally accessable quantities of field assisted secondary electron emission σ as well as the resulting surface potential V 0 due to internal current j ( x , t ) , charge ρ ( x , t ) , field F ( x , t ) , and potential V ( x , t ) distributions are obtained. The calculations are performed for bulk Al 2O 3 ceramics with open and metal-coated and grounded surfaces.

Relationships between dielectric breakdown resistance and charge transport in alumina materials—Effects of the microstructure

Dielectric breakdown is the main cause of insulator degradation. Breakdown strength strongly depends on materials microstructure (grain size, grain boundaries nature,…) [Liebault, J., Vallayer, J., Goeuriot, D., Tréheux D., Thévenot, F., How the trapping of charges can explain the dielectrics breakdown performance of alumina ceramics, J. Eur. Ceram. Soc., 2001, 21, 389–397; Si Ahmed, A., Kansy, J., Zarbout, K., Moya, G., Liebault, J., Goeuriot, D., Microstructural origin of the dielectric breakdown strength in alumina: a study by positron lifetime spectroscopy, J. Eur. Ceram. Soc., 2005, 25, 2813–2816]. The experimental study of these materials behaviour towards charge injection was performed by the scanning electron microscopy mirror effect (SEMME) method. It allows to measure the amount of injected charges finally trapped in the insulator. In order to explain the experimental results, we developed an iterative computer simulation of the self-consistent charge transport in bulk alumina samples during electron beam irradiation, based on a new flight-drift model (FDM). Ballistic and drift electron and hole transport as well as their recombination, trapping and detrapping (due to temperature and electric field) are taken into account. As a main result the time dependent secondary electron emission rate and the spatial distributions of currents, charges, field and potential are obtained. The analysis of these two kinds of results allowed us to identify the effect of the microstructure on the behaviour of the injected charges in the insulator and then to propose some mechanisms depending on the temperature leading to a good dielectric breakdown resistance. Indeed, at room temperature a huge localisation of charges limits further injection into the sample that permits to delay breakdown. On the other hand, when the temperature increases, the efficiency of the charge spreading behaviour is improved, leading to a higher dielectric breakdown strength.

Electron beam charging of insulators: A self-consistent flight-drift model

Electron beam irradiation and the self-consistent charge transport in bulk insulating samples are described by means of a new flight-drift model and an iterative computer simulation. Ballistic secondary electron and hole transport is followed by electron and hole drifts, their possible recombination and/or trapping in shallow and deep traps. The trap capture cross sections are the Poole-Frenkel-type temperature and field dependent. As a main result the spatial distributions of currents j(x,t), charges ρ(x,t), the field F(x,t), and the potential slope V(x,t) are obtained in a self-consistent procedure as well as the time-dependent secondary electron emission rate σ(t) and the surface potential V0(t). For bulk insulating samples the time-dependent distributions approach the final stationary state with j(x,t)=const=0 and σ=1. Especially for low electron beam energies E0&amp;lt;4keV the incorporation of mainly positive charges can be controlled by the potential VG of a vacuum grid in front of the target surface. For high beam energies E0=10, 20, and 30keV high negative surface potentials V0=−4, −14, and −24kV are obtained, respectively. Besides open nonconductive samples also positive ion-covered samples and targets with a conducting and grounded layer (metal or carbon) on the surface have been considered as used in environmental scanning electron microscopy and common SEM in order to prevent charging. Indeed, the potential distributions V(x) are considerably small in magnitude and do not affect the incident electron beam neither by retarding field effects in front of the surface nor within the bulk insulating sample. Thus the spatial scattering and excitation distributions are almost not affected.

Selfconsistent electrical charging in insulators

A Monte Carlo program based on acoustic and optical phonon scattering as well as on impact ionisation of valence band electrons has advantages in description of very low energy electron scattering (in eV and meV regions) and is aimed especially to wide gap dielectrics and insulators. Thus, the rapid relaxation and the ballistic transport of excited electrons within the conduction band of a wide gap insulator occurs over femtoseconds. The field-dependent transport and trapping parameters allow us to model the selfconsistent charge transport and charging-up of SiO 2 thin layers as well as bulk Al 2O 3 samples during electron bombardment. The resulting distributions of currents, charges and electric fields within these samples explain, e.g. the phenomena of field-enhanced and field-blocked secondary electron emission. In order to prove the accessible quantity of the surface charging-up potential we have chosen the X-ray bremsstrahlung (BS) spectra, i.e. the shift of the short wavelength threshold due to the negative surface potential V 0 and respective retarding of the incident electron beam. This effect is demonstrated for a 3 mm bulk Al 2O 3 sample and E 0 = 30 keV electron beam irradiation resulting in a huge negative surface potential of V 0 exp = − 17 kV .

Monte Carlo Modeling of Electron Scattering in Nonconductive Specimens

Very low energy electrons in a solid should behave like Bloch electrons and will interact with perturbations of the atomic lattice, that is, with phonons. So we use the acoustic phonon scattering for replacing the elastic binary encounter approximation of the Mott scattering for electrons with low energies E < 100 eV. For ballistic electrons (1 eV < E < Eg) and higher energies up to 1 keV we determined the acoustic phonon scattering and the impact ionization rate by means of the "backscattering-versus-range" proof and respective eta(E0) - R(E0) diagrams. Electron trajectories demonstrate the relatively short range of primary electrons (PE) with energies E > 50 eV due to strong impact ionization losses (cascading) and the much greater range of secondary electrons (SE) with E < 50 eV, finally as a consequence of less effective phonon losses. The field-dependent transport parameters allow us to model the self-consistent charge transport and charging-up of insulating SiO2 layers during electron bombardment maintained by the current components of primary electrons jPE, secondary electrons jSE, and associated ballistic holes jBH, as well as by Fowler-Nordheim field injection jFN from the substrate. The resulting distributions of currents j(x, t), charges p(x, t), electric fields F(x, t), and the potential V(x, t) across the dielectric layer explain the phenomena of field-enhanced and field-blocked secondary electron emission with rates delta >or<or=1.

Secondary electron emission and self-consistent charge transport and storage in bulk insulators: Application to alumina

The self-consistent charge transport in bulk alumina samples during electron beam irradiation is described by means of an iterative computer simulation. Ballistic electron and hole transport as well as their recombination and trapping are included. As a main result the time-dependent secondary electron emission rate σ(t) and the spatial distributions of currents j(x,t), charges ρ(x,t), the field F(x,t), and the potential slope V(x,t) are obtained. For bulk insulating samples, the time-dependent distributions approach the final stationary state with j(x,t)=const=0 and σ=1. Especially for low electron beam energies E0=1 keV, the incorporation of charges can be controlled by the potential VG of a vacuum electrode in front of the target surface. Finally, for high electron beam energies, the real negative surface potential V0&amp;lt;0 is measured by x-ray bremsstrahlung spectra and the shift of the short wavelength edge. For the initial beam energy E0=30 keV, the experimental value V0=−16 kV is still in good agreement with our simulations.

Self-consistent electrical charging of insulating layers and metal-insulator-semiconductor structures

By means of a computer simulation the self-consistent charge transport with the current densities j(x,t), the respective charges ρ(x,t), field strengths F(x,t), and potential distributions V(x,t) in SiO2 layers are obtained as a function of the insulator depth x and the injection time t. The SiO2 layers are considered as open layers on silicon substrate or they are embedded in metal-oxide-semiconductor (MOS) structures. The given currents of primary electrons, the field-dependent ballistic currents of secondary electrons and holes as well as the Fowler–Nordheim injection of electrons from the substrate into the dielectric layer are taken into account. This method allows a defined charge storage and the explanation of complicated emission, charging-up, and breakdown processes within insulating layers during electron bombardment and/or high-field charge injection from adjacent electrodes, e.g., in MOS structures.

Applicability of the High Field Model: An Analytical Study Via Asymptotic Parameters Defining Domain Decomposition

In this paper, we present a mesoscopic-macroscopic model of self-consistent charge transport. It is based upon an asymptotic expansion of solutions of the Boltzmann Transport Equation (BTE). We identify three dimensionless parameters from the BTE. These parameters are, respectively, the quotient of reference scales for drift and thermal velocities, the scaled mean free path, and the scaled Debye length. Such parameters induce domain dependent macroscopic approximations. Particular focus is placed upon the so-called high field model, defined by the regime where drift velocity dominates thermal velocity. This model incorporates kinetic transition layers, linking mesoscopic to macroscopic states. Reference scalings are defined by the background doping levels and distinct, experimentally measured mobility expressions, as well as locally determined ranges for the electric fields. The mobilities reflect a coarse substitute for reference scales of scattering mechanisms. See [9] for elaboration.The high field approximation is a formally derived modification of the augmented drift-diffusion model originally introduced by Thornber some fifteen years ago [25]. We are able to compare our approach with the earlier kinetic approach of Baranger and Wilkins [5] and the macroscopic approach of Kan, Ravaioli and Kerkhoven [20].