Self-consistent Electric Field Research Articles

Nonlinear Charge Transport and Excitable Phenomena in Semiconductor Superlattices.

Semiconductor superlattices are periodic nanostructures consisting of epitaxially grown quantum wells and barriers. For thick barriers, the quantum wells are weakly coupled and the main transport mechanism is a sequential resonant tunneling of electrons between wells. We review quantum transport in these materials, and the rate equations for electron densities, currents, and the self-consistent electric potential or field. Depending on superlattice configuration, doping density, temperature, voltage bias, and other parameters, superlattices behave as excitable systems, and can respond to abrupt dc bias changes by large transients involving charge density waves before arriving at a stable stationary state. For other parameters, the superlattices may have self-sustained oscillations of the current through them. These oscillations are due to repeated triggering and recycling of charge density waves, and can be periodic in time, quasiperiodic, and chaotic. Modifying the superlattice configuration, it is possible to attain robust chaos due to wave dynamics. External noise of appropriate strength can generate time-periodic current oscillations when the superlattice is in a stable stationary state without noise, which is called the coherence resonance. In turn, these oscillations can resonate with a periodic signal in the presence of sufficient noise, thereby displaying a stochastic resonance. These properties can be exploited to design and build many devices. Here, we describe detectors of weak signals by using coherence and stochastic resonance and fast generators of true random sequences useful for safe communications and storage.

Data-driven solutions of ill-posed inverse problems arising from doping reconstruction in semiconductors

ABSTRACT The non-destructive estimation of doping concentrations in semiconductor devices is of paramount importance for many applications ranging from crystal growth to defect and inhomogeneity detection. A number of technologies (such as LBIC, EBIC and LPS) have been developed which allow the detection of doping variations via photovoltaic effects. The idea is to illuminate the sample at several positions and detect the resulting voltage drop or current at the contacts. We model a general class of such photovoltaic technologies by ill-posed global and local inverse problems based on a drift-diffusion system which describes charge transport in a self-consistent electrical field. The doping profile is included as a parametric field. To numerically solve a physically relevant local inverse problem, we present three approaches, based on least squares, multilayer perceptrons, and residual neural networks. Our data-driven methods reconstruct the doping profile for a given spatially varying voltage signal induced by a laser scan along the sample's surface. The methods are trained on synthetic data sets which are generated by finite volume solutions of the forward problem. While the linear least square method yields an average absolute error around 10%, the nonlinear networks roughly halve this error to 5%.

The kinetic theory of cathode plasma expansion in a spatially non-uniform geometric configuration of a vacuum diode

This paper presents a theoretical explanation for the occurrence of anomalous ion acceleration in vacuum diodes, leading to the expansion of cathode plasma towards the anode, a characteristic phenomenon of vacuum breakdown. The explanation is derived from first principles based on equations of collisionless physical kinetics, using the example of an axisymmetric vacuum diode. The proposed theoretical interpretation convincingly demonstrates that the primary mechanism behind the anomalous ion acceleration of cathode plasma is the collisionless motion of ions in a self-consistent electric field.

Electron plasma diagnostics in ELTRAP by electron cyclotron resonance heating method.

Electron cyclotron resonance heating method of Particle-in-Cell code was used to analyze heating phenomena, axial kinetic energy, and self-consistent electric field of confined electron plasma in ELTRAP device by hydrogen and helium background gases. The electromagnetic simulations were performed at a constant power of 3.8 V for different RF drives (0.5 GHz- 8 GHz), as well as for 1 GHz constant frequency at these varying amplitudes (1 V-3.8 V). The impacts of axial and radial temperatures were found maximum at 1.8 V and 5 GHz as compared to other amplitudes and frequencies for both background gases. These effects are higher at varying radio frequencies due to more ionization and secondary electrons production and maximum recorded radial temperature for hydrogen background gas was 170.41 eV. The axial kinetic energy impacts were found more effective in the outer radial part (between 0.03 and 0.04 meters) of the ELTRAP device due to applied VRF through C8 electrode. The self-consistent electric field was found higher for helium background gas at 5 GHz RF than other amplitudes and radio frequencies. The excitation and ionization rates were found to be higher along the radial direction (r-axis) than the axial direction (z-axis) in helium background gas as compared to hydrogen background gas. The current studies are advantageous for nuclear physics applications, beam physics, microelectronics, coherent radiation devices and also in magnetrons.

On the Existence of Global Compactly Supported Weak Solutions of the Vlasov–Poisson System with an External Magnetic Field

We consider the first mixed problem for the Vlasov–Poisson system with an external magnetic field in a domain with piecewise smooth boundary. This problem describes the kinetics of a two-component high-temperature plasma under the influence of a self-consistent electric field and an external magnetic field. The existence of global weak solutions is proved. In the case of a cylindrical domain, sufficient conditions are obtained for the existence of global weak solutions with supports in a strictly internal cylinder; this corresponds to the confinement of high-temperature plasma in a mirror trap.

Kinetic Model of Vacuum Plasma Expansion in a Cylindrical Gap

Results of a theoretical description of collisionless kinetics of radial expansion of two-component (electron–ion) plasma in the one-dimensional cylindrical formulation of the problem are presented. The electric-field mechanism of supersonic expansion of the plasma flame due to the motion of the electron–ion ensemble and self-consistent electric field in the diode with the potential difference applied to it is demonstrated. The spatiotemporal evolution of the ion energy distribution function, electric potential, and rate of expansion of the emission boundary of the plasma flame is shown. The calculated rates of flame expansion at the copper cathode (~1.5 × 106 cm/s) well agree with the experimental data.

Kinetic Model of Vacuum Plasma Expansion in a Cylindrical Gap

Results of a theoretical description of collisionless kinetics of radial expansion of two-component (electron–ion) plasma in the one-dimensional cylindrical formulation of the problem are presented. The electric-field mechanism of supersonic expansion of the plasma flame due to the motion of the electron–ion ensemble and self-consistent electric field in the diode with the potential difference applied to it is demonstrated. The spatiotemporal evolution of the ion energy distribution function, electric potential, and rate of expansion of the emission boundary of the plasma flame is shown. The calculated rates of flame expansion at the copper cathode (~1.5 × 106 cm/s) well agree with the experimental data.

Unambiguous coupling between acoustic and electromagnetic emissions in plastically deformed crystals

An evident coupling between acoustic (AE) and electromagnetic (EME) emissions has been proved experimentally during plastic deformation of LiF ionic monocrystals under uniaxial compression with simultaneous recording of both AE and EME. The strong correlation between AE and EME demonstrate clearly that the observed EME is caused by dynamical dislocations and charged vacancies in the ionic lattice during work hardening. The theoretical interpretation proposed to explain the observable EME is based on the well-known Stepanov effect that means sweeping-up the charged vacancies of a preferable sign by gliding edge dislocations and formation of charged Cottrell clouds. During work hardening dislocation pile-ups are formed, and a certain nonequilibrium charge density is accumulated at their heads, resulting to the dynamic electric polarization of the deformed crystal. As the external loading increases, a locked dislocation pile-up bursts through the stoppers and quickly loses its bound charge. The relaxation of this charge produces intrinsic polarization currents generating electric pulses strongly correlated with dynamic dislocation process during plastic deformation. To build the theoretical model, it is assumed that the relaxation current can be described as an athermic viscous motion of vacancies under the kinetic friction force ∼Bυ (B is the friction coefficient and υ is the vacancy velocity) in a self-consistent electric field determined by the distribution of the total charge density. The electrical signal generated by an acting slip system has been calculated. By comparing the calculated and experimentally measured electric signal patterns, the friction coefficient for the linear chain of vacancies (the analogue of an edge dislocation extra-plane) in LiF has been estimated to be B≃ 0.9⋅10–5 g cm–1⋅s–1. This value is in accordance with the corresponding coefficient for dislocations in ionic lattices.

Initial kinetics of electrons, ions and electric field in planar vacuum diode with plasma cathode

The paper presents the results of a 1D1V theoretical study of the initial stage of plasma expansion into a planar vacuum gap 1 cm long with an applied voltage of 500–5000 V. Based on the collisionless kinetic and Maxwell equations, the motion of a two-component electron–ion plasma in a self-consistent electric field is described. The fundamental mechanism of ‘superthermal’ velocities of the emission edge of the plasma and anode-directed velocities of positive ions is demonstrated in detail. The general regularities of this process are established for variations in the number density of the initial plasma and the applied voltage.

Stability of the planar rarefaction wave to three-dimensional full compressible Navier–Stokes–Poisson system

A full compressible Navier–Stokes–Poisson system models the motion of viscous ions under the effect of variable temperature and plays important roles in the study of self-gravitational viscous gaseous stars and in simulations of charged particles in semiconductor devices and plasmas physics. We establish the time-asymptotic nonlinear stability of a planar rarefaction wave to the initial value problem of a three-dimensional full compressible Navier–Stokes–Poisson equation when the initial data are a small perturbation of the planar rarefaction wave. The proof is given by a delicate energy method, which involves highly non-trivial a priori bounds due to the effect of the self-consistent electric field. This appears as the first result on the nonlinear stability of wave patterns to the full compressible Navier–Stokes–Poisson system in multi-dimensions.

Simulations of Dynamical Electronic Vortices in Charge and Spin Density Waves

Charge and spin density waves are typical symmetry broken states of quasi one-dimensional electronic systems. They demonstrate such common features of all incommensurate electronic crystals as a spectacular non-linear conduction by means of the collective sliding and susceptibility to the electric field. These phenomena ultimately require for emergence of static and transient topological defects: there are dislocations as space vortices and space-time vortices known as phase slip centers, i.e., a kind of instantons. Dislocations are statically built-in under a transverse electric field; their sweeping provides a conversion among the normal carriers and condensate which ensures the onset of the collective sliding. A special realization in a high magnetic field, when the density wave is driven by the Hall voltage, originated by quantized normal carriers, reveals the dynamic vorticity serving to annihilate compensating normal and collective currents. Spin density waves, with their rich multiplicative order parameter, bring to life complex objects with half-integer topologically bound vorticities in charge and spin degrees of freedom. We present the basic concepts and modelling results of the stationary states and their transient dynamics involving vorticity. The models take into account multiple fields in their mutual non-linear interactions: the complex order parameter, the self-consistent electric field, and the reaction of normal carriers. We explore the traditional time-dependent Ginzburg–Landau approach and introduce its generalization allowing the treatment of intrinsic normal carriers. The main insights and illustrations come from numerical solutions to partial differential equations for the dissipative dynamics of one and two space dimensions.

Parametric investigation of azimuthal instabilities and electron transport in a radial-azimuthal E × B plasma configuration

Partially magnetized low-temperature plasmas (LTP) in an E × B configuration, where the applied magnetic field is perpendicular to the self-consistent electric field, have become increasingly relevant in industrial applications. Hall thrusters, a type of electrostatic plasma propulsion, are one of the main LTP technologies whose advancement is hindered by the not-fully-understood underlying physics of operation, particularly, with respect to the plasma instabilities and the associated electron cross field transport. The development of Hall thrusters with unconventional magnetic field topologies has imposed further questions regarding the instabilities' characteristics and the electrons' dynamics in these modern cross field configurations. Accordingly, we present in this effort a detailed parametric study of the influence of three factors on the plasma processes in the radial-azimuthal coordinates of a Hall thruster, namely, the magnetic field gradient, secondary electron emission, and plasma number density. The studies are carried out using the reduced-order particle-in-cell code developed by the authors. The setup of the radial-azimuthal simulations largely follows a well-defined benchmark case from the literature in which the magnetic field is oriented along the radius, and a constant axial electric field is applied perpendicular to the simulation plane. The salient finding from our investigations is that, in the studied cases corresponding to elevated plasma densities, a long-wavelength azimuthal mode with the frequency of about 1 MHz is developed. Moreover, in the presence of strong magnetic field gradients, this mode results from an inverse energy cascade and induces a significant electron cross field transport as well as a notable heating of the ions.

Azimuthal structures and turbulent transport in Penning discharge

Azimuthal structures in cylindrical Penning discharge are studied with a 2D3V radial–azimuthal PIC/MCC model with the axial magnetic field. The discharge is self-consistently supported by ionization due to the axial injection of electrons. It is shown that the steady-state discharge can be supported in two different regimes with different types of observed azimuthal structures. The transition between the regimes is controlled by the mechanism of the energy input to the discharge. In the first regime (low energy of the injected electrons), with the pronounced m = 1 spoke activity, the power input is dominated by the energy absorption due to the radial current and self-consistent electric field. In the other regime (higher energy of the injected electrons), with prevalent small-scale m > 1 spiral structures, and the lower values of the anomalous transport, the total energy deposited to the discharge is lower and is mostly due to the direct input of the kinetic energy from the axial electron beam. We show that the large (m = 1) spoke and small-scale structures occur as a result of Simon–Hoh and lower hybrid instabilities driven by the electric field, density gradient, and collisions.

Calculation of the ion current to a conducting cylinder in a supersonic flow of a collisionless plasma

The diagnostics of low-temperature plasma flows using cylindrical probes is based on the classical Langmuir relation for the ion current to a thin, in comparison with the Debye length, cylinder. The aim of this work is to study the applicability of the Langmuir relation for a cylinder whose radius exceeds the Debye length. The interaction of a conducting cylinder with a rarefied plasma flow was simulated numerically. The cylinder had a negative potential with respect to the plasma. Free molecular flow around the cylinder was simulated on the basis of a two-dimensional system of the Vlasov–Poisson equations. The electron-repulsing local equilibrium self-consistent electric field was calculated using the Poisson–Boltzmann model in the approximation of local equilibrium electrons and taking into account an electron sink on the cylinder surface in the central field approximation. The Vlasov equations for ions and the Poisson–Boltzmann equations for the self-consistent electric field were solved on nested grids by a finite-difference relaxation method with splitting by physical processes and using the method of characteristics. The reliability of the calculated results was confirmed by the solution of known model problems and a comparison with the results of other authors and the results of solving identical physical problems with the use of different mathematical models and methods. The ion current to a cylinder placed transversely to a plasma flow was calculated as a function of the cylinder potential, the ion velocity ratio, and the ratio of the characteristic dimension of the cylinder to the Debye length. From the calculated results, numerical estimates were obtained for the range of applicability of the classical Langmuir relation for the ion current to a cylinder whose radius exceeds the Debye length. The results obtained may be used in the diagnostics of supersonic flows of a low-temperature rarefied plasma.

MPI Parallelization of Numerical Simulations for Pulsed Vacuum Arc Plasma Plumes Based on a Hybrid DSMC/PIC Algorithm

The transport characteristics of the unsteady flow field in rarefied plasma plumes is crucial for a pulsed vacuum arc in which the particle distribution varies from 1016 to 1022 m−3. The direct simulation Monte Carlo (DSMC) method and particle-in-cell (PIC) method are generally combined to study this kind of flow field. The DSMC method simulates the motion of neutral particles, while the PIC method simulates the motion of charged ions. A hybrid DSMC/PIC algorithm is investigated here to determine the unsteady axisymmetric flow characteristics of vacuum arc plasma plume expansion. Numerical simulations are found to be consistent with the experiments performed in the plasma mass and energy analyzer (EQP). The electric field is solved by Poisson’s equation, which is usually computationally expensive. The compressed sparse row (CSR) format is used to store the huge diluted matrix and PETSc library to solve Poisson’s equation through parallel calculations. Double weight factors and two timesteps under two grid sets are investigated using the hybrid DSMC/PIC algorithm. The fine PIC grid is nested in the coarse DSMC grid. Therefore, METIS is used to divide the much smaller coarse DSMC grid when dynamic load imbalances arise. Two parameters are employed to evaluate and distribute the computational load of each process. Due to the self-adaption of the dynamic-load-balancing parameters, millions of grids and more than 150 million particles are employed to predict the transport characteristics of the rarefied plasma plume. Atomic Ti and Ti2+ are injected into the small cylinders. The comparative analysis shows that the diffusion rate of Ti2+ is faster than that of atomic Ti under the electric field, especially in the z-direction. The fully diffuse reflection wall model is adopted, showing that neutral particles accumulate on the wall, while charged ions do not—due to their self-consistent electric field. The maximum acceleration ratio is about 17.94.

Modeling of a 43 GHz microplasma within a photonic crystal

The electron density (ne) and argon metastable density (1s5) of a 43GHz microplasma are obtained using the zero-dimensional plasma kinetics solver (ZDPlasKin®) for an incident power of 160 and 1000mW and argon pressures of 10–600Torr (1.3×103 to 8.0×104Pa). These simulations are compared with previously published experimental data. To determine the self-consistent electric field in the modeled plasma, the three-dimensional millimeter wave fields are computed as a function of electron density using ANSYS EM19.2, HFSS®. This electromagnetic field model is coupled to ZDPlasKin such that any increase in the simulated plasma density correctly attenuates the simulated electric field within the microplasma. The electron density is found to be sensitive to argon gas temperature, so a two-zone temperature model was needed to obtain agreement with experimental measurements. The temperature in the central core of the microplasma was determined by a previous experimental study. That temperature was used as an input to the model for the simulation of volume recombination losses. The outer regions of the microplasma are assumed to be in equilibrium with the walls (300K). This second temperature was used in the model to determine diffusion losses. The modeled electron and metastable densities are of the order 1020 and 1018m−3, respectively. This is in good agreement with those measured experimentally as long as the two-zone temperature model is used. In the absence of a hot gas core, the modeled three-body recombination rates are excessive and the simulation severely under-predicts the electron density and over-estimates the metastable density. We conclude that the millimeter wave microplasma has a hot core (2500K at 600Torr) that rarifies the argon gas and effectively reduces three-body recombination. This allows one to achieve high electron density on the order of 1020m−3 with only 100mW of wave power.

The RAM-SCB model and its applications to advance space weather forecasting

Energetic particle fluxes that are part of the Earth’s ring current and radiation belts can intensify significantly during space weather events like geomagnetic storms and could cause severe damage to satellite-based technologies. Understanding the physical processes that control their dynamics and improving our capability for their prediction is thus extremely important. In the context of space weather applications and user needs, this paper provides a brief description of our kinetic ring current-atmosphere interactions model with self-consistent magnetic field (RAM-SCB) and its further extension to implement a self-consistent electric (E) field. Specific examples that demonstrate RAM-SCB capabilities and limitations to reproduce the near-Earth space weather environment are given. The current status of RAM-SCB is assessed and plans for its further improvement are discussed.

Nonlinear electron transport in miniband superlattice driven by dual terahertz fields and a transverse magnetic field

Time-dependent electron current response of GaAs-based miniband superlattice under dual ac electric fields and a magnetic field is studied using balance equation approach. The space charge-induced self-consistent electric field is taken into account in the model. The miniband superlattice operates in the diffusive regime without electric field domain formation. Electron current displays very complicated oscillating behavior with the influence of external fields. The effect of dissipation on nonlinear electron transport is carefully studied based on Poincaré bifurcation diagram and power spectrum. The exhibition of complicate nonlinear oscillation in superlattice is attributed to the nonlinearity induced by self-consistent field and interaction between external radiation and internal cooperative oscillating mode relative to Bloch oscillation and cyclotron oscillation.

Entropy and convergence analysis for two finite volume schemes for a Nernst\u2013Planck\u2013Poisson system with ion volume constraints

In this paper, we consider a drift-diffusion system with cross-coupling through the chemical potentials comprising a model for the motion of finite size ions in liquid electrolytes. The drift term is due to the self-consistent electric field maintained by the ions and described by a Poisson equation. We design two finite volume schemes based on different formulations of the fluxes. We also provide a stability analysis of these schemes and an existence result for the corresponding discrete solutions. A convergence proof is proposed for non-degenerate solutions. Numerical experiments show the behavior of these schemes.

Contact-PIC numerical methods for simulating Vlasov–Poisson–Fokker–Planck problem

This paper develops a numerical discretization for solving the Vlasov–Poisson–Fokker–Planck system, which is a model for describing collisional plasma in the presence of a self-consistent electric field. This numerical discretization is established in the particle-in-cell framework in which the dynamical behavior of charged particles follows a stochastic differential equation. We study the geometry structure of the corresponding stochastic differential equation, and construct contact methods to numerically solve it. We also analyze the time accuracy order of the proposed numerical schemes. Numerical experiments are conducted to validate our methods.