Boltzmann-Poisson System Research Articles

Ion-acoustic waves in weakly ionized plasmas with charge-exchange collisions

Ion-neutral charge-exchange collisions in plasmas of laboratory, space, and astrophysical origins are fundamental to understanding wave dissipation and wave generation phenomena. This paper implements a charge-exchange collision operator in the Boltzmann–Poisson system equations for a weakly ionized plasma. When considering an electric field perturbation, the governing kinetic equations provide significant results concerning the plasma conductivity and the dielectric function, appearing in simple, sensible forms. The present analysis reveals a backward wave propagation phenomenon at maximum conductivity when the wavenumber of the plasma wave is smaller than the reciprocal of the ion-neutral collisions mean free path. In addition, it is shown that ion-neutral coupling resulting from charge-exchange collisions enhances ion-acoustic waves below and beyond the ion plasma frequency and leads to the onset of a fundamental instability that overcomes Landau damping under certain circumstances. The collisionless model is recovered as a limiting case, i.e., in the asymptotic limit of a long mean free path.

Continuum kinetic investigation of the impact of bias potentials in the current saturation regime on sheath formation

In this work, we examine sheath formation in the presence of bias potentials in the current saturation regime for pulsed power fusion experiments. It is important to understand how the particle and heat fluxes at the wall may impact the wall material and affect electrode degradation. Simulations are performed using the 1X-1V Boltzmann–Poisson system for a proton–electron plasma in the presence of bias potentials ranging from 0 to 10 kV. The results indicate that the sheath near the anode remains generally the same as that of a classical sheath without the presence of a bias potential. However, the sheath near the cathode becomes more prominent with a larger potential drop, a significant decrease in the electron density, and larger sheath lengths. The spatially constant current density increases to a saturation value with the increasing bias potential. For high bias potentials, the material choice needs to consider that the anode faces significantly larger particle and heat fluxes compared to the cathode. In general, the results trend with theory with differences attributed to the simplified assumptions in the theory and the kinetic effects considered in the simulations. Due to the significant computational cost of a well resolved 1X-2V simulation, only one such simulation is performed for the 5 kV case showing higher current.

Stochastic Galerkin methods for the Boltzmann-Poisson system

We study uncertainty quantification for a Boltzmann-Poisson system that models electron transport in semiconductors and the physical collision mechanisms over the charges, using the stochastic Galerkin method in order to handle the randomness associated with the problem. In this study we choose first as a source of uncertainty the phonon energy, taking it as a random variable, as its value influences the energy jump appearing in the collision integral for electron-phonon scattering. Then we choose the lattice temperature as a random variable, since it defines the value of the collision operator terms in the case of electron-phonon scattering by being a parameter of the phonon distribution. Finally, we present our numerical simulations for the latter case. We calculate then with our stochastic Discontinuous Galerkin methods the uncertainty in kinetic moments such as density, mean energy, current, etc. associated to a possible physical temperature variation (assumed to follow a uniform distribution) in the lattice environment, as this uncertainty in the temperature is propagated into the electron PDF. Our mathematical and computational results let us predict then in a real world problem setting the impact that possible variations in the lab conditions (such as temperature) or limitations in the mathematical model (such as assumption of a constant phonon energy) will have over the uncertainty in the behavior of electronic devices.

A fast implicit solver for semiconductor models in one space dimension

Several different approaches are proposed for solving fully implicit discretizations of a simplified Boltzmann-Poisson system with a linear relaxation-type collision kernel. This system models the evolution of free electrons in semiconductor devices under a low-density assumption. At each implicit time step, the discretized system is formulated as a fixed-point problem, which can then be solved with a variety of methods. A key algorithmic component in all the approaches considered here is a recently developed sweeping algorithm for Vlasov-Poisson systems. A synthetic acceleration scheme has been implemented to accelerate the convergence of iterative solvers by using the solution to a drift-diffusion equation as a preconditioner. The performance of four iterative solvers and their accelerated variants has been compared on problems modeling semiconductor devices with various electron mean-free-path.

Polarization and relaxation effects induced by two-scale homogenization

Motivated by the two-scale convergence method introduced by G. Nguetseng and G. Allaire, we study the polarization and relaxation effects induced by two-scale homogenization. By virtue of Boltzmann–Poisson system, we characterize the microscopic properties of electrons in dielectric materials. The homogenized equations are obtained which describe the mean behaviors. By the homogenized equation, we get the polarization and relaxation effects which lead to the modified Gauss law; moreover, the dielectric function and the conductivity are also obtained.

Diffusion limit of a Boltzmann–Poisson system with nonlinear equilibrium state

The diffusion approximation for a Boltzmann–Poisson system is studied. Nonlinear relaxation type collision operator is considered. A relative entropy is used to prove useful [Formula: see text]-estimates for the weak solutions of the scaled Boltzmann equation (coupled to Poisson) and to prove the convergence of the solution toward the solution of a nonlinear diffusion equation coupled to Poisson. In one dimension, a hybrid Hilbert expansion and the contraction property of the operator allow to exhibit a convergence rate.

A stochastic asymptotic-preserving scheme for the bipolar semiconductor Boltzmann-Poisson system with random inputs and diffusive scalings

In this paper, we study the bipolar Boltzmann-Poisson model, both for the deterministic system and the system with uncertainties, with asymptotic behavior leading to the drift diffusion-Poisson system as the Knudsen number goes to zero. The random inputs can arise from collision kernels, doping profile and initial data. We adopt a generalized polynomial chaos approach based stochastic Galerkin (gPC-SG) method. Sensitivity analysis is conducted using hypocoercivity theory for both the analytical solution and the gPC solution for a simpler model that ignores the electric field, and it gives their convergence toward the global Maxwellian exponentially in time. A formal proof of the stochastic asymptotic-preserving (s-AP) property and a uniform spectral convergence with error decaying exponentially in time in the random space of the scheme is given. Numerical experiments are conducted to validate the accuracy, efficiency and asymptotic properties of the proposed method.

Galerkin methods for Boltzmann–Poisson transport with reflection conditions on rough boundaries

We consider in this paper the mathematical and numerical modeling of reflective boundary conditions (BC) associated to Boltzmann–Poisson systems, including diffusive reflection in addition to specularity, in the context of electron transport in semiconductor device modeling at nano scales, and their implementation in Discontinuous Galerkin (DG) schemes. We study these BC on the physical boundaries of the device and develop a numerical approximation to model an insulating boundary condition, or equivalently, a pointwise zero flux mathematical condition for the electron transport equation. Such condition balances the incident and reflective momentum flux at the microscopic level, pointwise at the boundary, in the case of a more general mixed reflection with momentum dependant specularity probability p(k→). We compare the computational prediction of physical observables given by the numerical implementation of these different reflection conditions in our DG scheme for BP models, and observe that the diffusive condition influences the kinetic moments over the whole domain in position space.

A brief survey of the discontinuous Galerkin method for the Boltzmann-Poisson equations

We are interested in the deterministic computation of the transients for the Boltzmann-Poisson system describing electron transport in semiconductor devices. The main difficulty of such computation arises from the very high dimensions of the model, making it necessary to use relatively coarse meshes and hence requiring the numerical solver to be stable and to have good resolution under coarse meshes. In this paper we give a brief survey of the discontinuous Galerkin (DG) method, which is a finite element method using discontinuous piecewise polynomials as basis functions and numerical fluxes based on upwinding for stability, for solving the Boltzmann-Poisson system. In many situations, the deterministic DG solver can produce accurate solutions with equal or less CPU time than the traditional DSMC (Direct Simulation Monte Carlo) solvers. In order to make the presentation more concise and to highlight the main ideas of the algorithm, we use a simplified model to describe the details of the DG method. Sample simulation results on the full Boltzmann-Poisson system are also given.

Recovering doping profiles in semiconductor devices with the Boltzmann–Poisson model

We investigate numerically an inverse problem related to the Boltzmann–Poisson system of equations for transport of electrons in semiconductor devices. The objective of the (ill-posed) inverse problem is to recover the doping profile of a device, presented as a source function in the mathematical model, from its current–voltage characteristics. To reduce the degree of ill-posedness of the inverse problem, we proposed to parameterize the unknown doping profile function to limit the number of unknowns in the inverse problem. We showed by numerical examples that the reconstruction of a few low moments of the doping profile is possible when relatively accurate time-dependent or time-independent measurements are available, even though the later reconstruction is less accurate than the former. We also compare reconstructions from the Boltzmann–Poisson (BP) model to those from the classical drift–diffusion-Poisson (DDP) model, assuming that measurements are generated with the BP model. We show that the two type of reconstructions can be significantly different in regimes where drift–diffusion-Poisson equation fails to model the physics accurately. However, when noise presented in measured data is high, no difference in the reconstructions can be observed.

On the Diffusion Limit of a Semiconductor Boltzmann–Poisson System Without Micro-Reversible Process

This paper deals with the diffusion approximation of a semiconductor Boltzmann–Poisson system. The statistics of collisions we are considering here, is the Fermi–Dirac operator with the Pauli exclusion term and without the detailed balance principle. Our study generalizes, the result of Goudon and Mellet [11], to the multi-dimensional case.

A discontinuous Galerkin solver for Boltzmann–Poisson systems in nano devices

In this paper, we present results of a discontinuous Galerkin (DG) scheme applied to deterministic computations of the transients for the Boltzmann–Poisson system describing electron transport in semiconductor devices. The collisional term models optical-phonon interactions which become dominant under strong energetic conditions corresponding to nano-scale active regions under applied bias. The proposed numerical technique is a finite element method using discontinuous piecewise polynomials as basis functions on unstructured meshes. It is applied to simulate hot electron transport in bulk silicon, in a silicon n + – n – n + diode and in a double gated 12 nm MOSFET. Additionally, the obtained results are compared to those of a high order WENO scheme simulation and DSMC (Discrete Simulation Monte Carlo) solvers.

Kinetic effects on the transport properties of nanostructured devices investigated by deterministic solutions of the Boltzmann-Poisson system

Deterministic numerical methods developed for solving Boltzmann-Poisson systems of carriers and phonons are applied to determine the transport properties of sub-micron semiconductor devices and bulk graphene. Kinetic effects on the far-from-equilibrium electron transport in a silicon npn-structure are studied by comparing the solution of the Boltzmann equation with corresponding maximum entropy distributions, which serve as reference for a hydrodynamical description. An indium phosphide n+-n-n+ diode is considered to investigate the impact of nonequilibrium polar optical phonons on the electron transport. Remarkable transport properties due to the exotic band structure of graphene are investigated on a kinetic level. The numerical studies based on a direct determination of the distribution functions of electrons and phonons demonstrate the significant influence of nonequilibrium phonons on the carrier transport in sub-micron electronic building blocks.

Efficient deterministic parallel simulation of 2D semiconductor devices based on WENO-Boltzmann schemes

A flexible parallel deterministic solver of the Boltzmann–Poisson system for 2D semiconductor device simulation on computer clusters is presented. The simulator is obtained by parallelizing a previously proposed numerical scheme based on high order finite difference weighted essentially non-oscillatory (WENO) schemes. Although the underlying numerical scheme presents important advantages over direct simulation Monte Carlo methods, this scheme imposes very high demands of computing power. Due to this, the parallelization of the different calculation phases in the numerical scheme has been tackled. The data subdomain which demands most of the computational workload has been suitably distributed among the processors and several parallel design decisions has been taken in order to achieve good performance. Moreover, the resultant parallel application can be easily adjusted to simulate a wide range of devices and could be easily used by engineers without mathematical background about the underlying numerical scheme. The parallel algorithm has been implemented in C++ augmented with calls to MPI functions and functions of optimized linear algebra libraries. Several experiments have been performed by simulating particular MOSFET and DG-MOSFET devices on a SMP cluster in order to show its efficiency.

Discontinuous Galerkin solver for Boltzmann-Poisson transients

We present results of a discontinuous Galerkin scheme applied to deterministic computations of the transients for the Boltzmann-Poisson system describing electron transport in semiconductor devices. The collisional term models optical-phonon interactions which become dominant under strong energetic conditions corresponding to nano-scale active regions under applied bias. The proposed numerical technique is a finite element method using discontinuous piecewise polynomials as basis functions on unstructured meshes. It is applied to simulate hot electron transport in bulk silicon, in a silicon n +-n-n + diode and in a double gated 12 nm MOSFET. Additionally, the obtained results are compared to those of a high order WENO scheme simulation.

DIFFUSION AND HOMOGENIZATION APPROXIMATION FOR SEMICONDUCTOR BOLTZMANN–POISSON SYSTEM

In this paper, the diffusion and homogenization approximation of the Boltzmann–Poisson system in the presence of a spatially oscillating electrostatic potential is studied. By analyzing the relative entropy, the uniform energy estimate for a well-prepared boundary data is proved. An averaging lemma and two-scale convergence techniques are used to prove rigorously the convergence of the scaled Boltzmann equation (coupled to Poisson) to a homogenized drift–diffusion-Poisson system.

A Two-Surface Problem of the Electron Flow in a Semiconductor on the Basis of Kinetic Theory

A steady flow of electrons in a semiconductor between two parallel plane Ohmic contacts is studied on the basis of the semiconductor Boltzmann equation, assuming a relaxation-time collision term, and the Poisson equation for the electrostatic potential. A systematic asymptotic analysis of the Boltzmann–Poisson system for small Knudsen numbers (scaled mean free paths) is carried out in the case where the Debye length is of the same order as the distance between the contacts and where the applied potential is of the same order as the thermal potential. A system of drift-diffusion-type equations and their boundary conditions is obtained up to second order in the Knudsen number. A numerical comparison is made between the obtained system and the original Boltzmann–Poisson system.

Diffusion Limit of a Semiconductor Boltzmann–Poisson System

The paper deals with the diffusion limit of the initial‐boundary value problem for the multidimensional semiconductor Boltzmann–Poisson system. Here, we generalize the one‐dimensional results obtained in [5] to the case of several dimensions using global renormalized solutions. The method of moments and a velocity averaging lemma are used to prove the convergence of the renormalized solutions to the semiconductor Boltzmann–Poisson system towards a global weak solution of the drift‐diffusion‐Poisson model.

DSMC versus WENO-BTE: A double gate MOSFET example

Time-dependent solutions to the Boltzmann-Poisson system in two spatial dimensions and three-dimensional velocity space are obtained by using a recently developed high order WENO scheme. The collision operator of the Boltzmann equation models the scattering processes between electrons and phonons which are assumed to be in thermal equilibrium. In this paper, the deterministic numerical solutions for a double gate silicon MOSFET are compared with Monte Carlo simulations. The main aim of this investigation is to show how direct solutions of the Boltzmann transport equation coupled with the Poisson equation can, through comparisons, suggest improvements of the DSMC algorithms such as, in particular, the charge assignment to the mesh, the treatment of the boundary conditions and the free flight duration.

Efficient time integration of the Boltzmann-Poisson system applied to semiconductor device simulation

This paper presents new efficient methods for performing the time integration of finite-difference WENO approximations of the Boltzmann-Poisson system applied to semiconductor device simulation. The developed methods are based on local time-step schemes for hyperbolic conservation laws which permit the use of different time increments at different positions in space. A strategy to dynamically adapt the space-time grid according to the actual stability criteria imposed by the CFL-condition is proposed. The resulting numerical schemes are used to simulate the electron transport in a n+−n−n+ silicon diode and in a silicon MESFET. Several numerical tests and comparisons with computations performed with TVD Runge-Kutta type algorithms prove the efficiency of the presented time integration methods.