Semiconductor Boltzmann Equation Research Articles

Effects of the Nature of Boundaries Conditions and Their Truncation Errors on the Distribution of Minority Carriers in Silicon Solar Cell

In this work, the effects of boundaries conditions and truncation errors in the distribution of minority carriers in the semiconductor are studied. It is a one-dimensional digital study of a polycrystalline silicon solar cell under polychromatic illumination in a dynamic state. Starting from the Boltzmann equation of semiconductors, the author establishes the general equation of particle transport. Assumptions made on the latter allow it to give the equation of distribution of minority carriers in a general way in its case to be studied. This dimensioned distribution equation reveals the parameters of influences on the distribution of carriers. It obtains a partial derivative equation for the carrier distribution function. The boundary conditions are then discretized to order one and then to order two. By considering boundary conditions and the nature of the carriers, the author numerically resolves the discretized general equation by assessing the influence of the nature of the boundary conditions and truncation errors and the influence of the discretization step on the density of the charge carriers by setting certain parameters and varying others. The work ends with a conclusion and logical follow-up to this work.

Derivation of a hydrodynamic heat equation from the phonon Boltzmann equation for general semiconductors

We present a formalism to solve the phonon Boltzmann transport equation (BTE) for finite Knudsen numbers that supplies a hydrodynamic heat transport equation similar to the Navier-Stokes equation for general semiconductors. This generalization of Fourier's law applies in general cases, from systems dominated by momentum-preserving normal collisions, as is well known, to kinetic materials dominated by resistive collisions, where it captures nonlocal effects. The key feature of our framework is that the macrostate is described in terms of the heat flux and its first derivatives. We obtain explicit expressions for the nonequilibrium phonon distribution and for the geometry-independent macroscopic parameters as a function of phonon properties that can be calculated from first principles. Ab initio model predictions are found to agree with a wide range of experiments in silicon. In contrast to approaches directly based on the BTE, the hydrodynamic equation can be solved in arbitrary geometries, thus providing a powerful tool for nanoscale heat modeling at a low computational cost.

Semiconductor Boltzmann-Dirac-Benney equation with a BGK-type collision operator: Existence of solutions vs. ill-posedness

A semiconductor Boltzmann equation with a non-linear BGK-type collision operator is analyzed for a cloud of ultracold atoms in an optical lattice: \begin{document} $ \partial _t f + \nabla _pe(p)·\nabla _x f - \nabla _x n_f·\nabla _p f = n_f(1- n_f)(\mathcal{F}_f-f),\;\;\;\; x∈\mathbb{R}^d, p∈\mathbb{T}^d, t>0. $ \end{document} This system contains an interaction potential \begin{document}$n_f(x,t): = ∈t_{\mathbb{T}^d}f(x,p,t)dp$\end{document} being significantly more singular than the Coulomb potential, which is used in the Vlasov-Poisson system. This causes major structural difficulties in the analysis. Furthermore, \begin{document}$e(p) = -\sum_{i = 1}^d$\end{document} \begin{document}$\cos(2π p_i)$\end{document} is the dispersion relation and \begin{document}$\mathcal{F}_f$\end{document} denotes the Fermi-Dirac equilibrium distribution, which depends non-linearly on \begin{document}$f$\end{document} in this context. In a dilute plasma—without collisions (r.h.s \begin{document}$. = 0$\end{document} )—this system is closely related to the Vlasov-Dirac-Benney equation. It is shown for analytic initial data that the semiconductor Boltzmann equation possesses a local, analytic solution. Here, we exploit the techniques of Mouhout and Villani by using Gevrey-type norms which vary over time. In addition, it is proved that this equation is locally ill-posed in Sobolev spaces close to some Fermi-Dirac equilibrium distribution functions.

The backward Monte Carlo method for semiconductor device simulation

A backward Monte Carlo method for the numerical solution of the semiconductor Boltzmann equation is presented. The method is particularly suited to simulate rare events. The general theory of the backward Monte Carlo method is described, and several estimators for the contact current are derived from that theory. The transition probabilities for the construction of the backward trajectories are chosen so as to satisfy the principle of detailed balance. This property guarantees stability of the numerical method and allows for a clear physical interpretation of the estimators. A symmetric sampling method which generates wave vectors always in pairs symmetric to the origin can be shown to yield zero current exactly as thermal equilibrium is approached. The properties of the different estimators are evaluated by simulation of an n-channel MOSFET. Quantities varying over many orders of magnitude can be resolved with ease. Such quantities are the drain current in the sub-threshold region, the high-energy tail of the carrier distribution function, and the so-called acceleration integral which varies over 30 orders in the example shown.

Uniform spectral convergence of the stochastic Galerkin method for the linear semiconductor Boltzmann equation with random inputs and diffusive scaling

In this paper, we study the generalized polynomial chaos (gPC) based stochastic Galerkin method for the linear semiconductor Boltzmann equation under diffusive scaling and with random inputs from an anisotropic collision kernel and the random initial condition. While the numerical scheme and the proof of uniform-in-Knudsen-number regularity of the distribution function in the random space has been introduced in [15], the main goal of this paper is to first obtain a sharper estimate on the regularity of the solution-an exponential decay towards its local equilibrium, which then lead to the uniform spectral convergence of the stochastic Galerkin method for the problem under study.

An Asymptotic-Preserving Stochastic Galerkin Method for the Semiconductor Boltzmann Equation with Random Inputs and Diffusive Scalings

In this paper, we develop a generalized polynomial chaos approach based stochastic Galerkin (gPC-SG) method for the linear semiconductor Boltzmann equation with random inputs and diffusive scalings. The random inputs are due to uncertainties in the collision kernel or initial data. We study the regularity (uniform in the Knudsen number) of the solution in the random space and prove the spectral accuracy of the gPC-SG method. We then use the asymptotic-preserving framework for the deterministic counterpart developed in [S. Jin and L. Pareschi, J. Comput. Phys., 161 (2000), pp. 312--330] to come up with the stochastic asymptotic-preserving gPC-SG method for the problem under study which is efficient in the diffusive regime. Numerical experiments are conducted to validate the accuracy and asymptotic properties of the method.

Extended Hydrodynamic Models and Multigrid Solver of a Silicon Diode Simulation

AbstractExtended hydrodynamic models for carrier transport are derived from the semiconductor Boltzmann equation with relaxation time approximation of the scattering term, by using the globally hyperbolic moment method and the moment-dependent relaxation time. Incorporating the microscopic relaxation time and the applied voltage bias, a formula is proposed to determine the relaxation time for each moment equation, which sets different relaxation rates for different moments such that higher moments damp faster. The resulting models would give more satisfactory results of macroscopic quantities of interest with a high-order convergence to those of the underlying Boltzmann equation as the involved moments increase, in comparison to the corresponding moment models using a single relaxation time. In order to simulate the steady states efficiently, a multigrid solver is developed for the derived moment models. Numerical simulations of an n+-n-n+ silicon diode are carried out to demonstrate the validation of the presented moment models, and the robustness and efficiency of the designed multigrid solver.

An asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions: A splitting approach

We present a new asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions --- a leading-order elastic collision together with a lower-order interparticle collision. When the mean free path is small, numerically solving this equation is prohibitively expensive due to the stiff collision terms. Furthermore, since the equilibrium solution is a (zero-momentum) Fermi-Dirac distribution resulting from joint action of both collisions, the simple BGK penalization designed for the one-scale collision [10] cannot capture the correct energy-transport limit. This problem was addressed in [13], where a thresholded BGK penalization was introduced. Here we propose an alternative based on a splitting approach. It has the advantage of treating the collisions at different scales separately, hence is free of choosing threshold and easier to implement. Formal asymptotic analysis and numerical results validate the efficiency and accuracy of the proposed scheme.

An asymptotic-preserving scheme for the semiconductor Boltzmann equation toward the energy-transport limit

We design an asymptotic-preserving scheme for the semiconductor Boltzmann equation which leads to an energy-transport system for electron mass and energy as mean free path goes to zero. As opposed to the classical drift-diffusion limit where the stiff collisions are all in one scale, new difficulties arise in the two-scale stiff collision terms because the simple BGK penalization [15] fails to drive the solution to the correct limit. We propose to set up a spatially dependent threshold on the penalization of the stiffer collision operator such that the evolution of the solution resembles a Hilbert expansion at the continuous level. Formal asymptotic analysis and numerical results confirm the efficiency and accuracy of our scheme.

Implicit-Explicit Runge-Kutta Schemes for the Boltzmann-Poisson System for Semiconductors

AbstractIn this paper we develop a class of Implicit-Explicit Runge-Kutta schemes for solving the multi-scale semiconductor Boltzmann equation. The relevant scale which characterizes this kind of problems is the diffusive scaling. This means that, in the limit of zero mean free path, the system is governed by a drift-diffusion equation. Our aim is to develop a method which accurately works for the different regimes encountered in general semiconductor simulations: the kinetic, the intermediate and the diffusive one. Moreover, we want to overcome the restrictive time step conditions of standard time integration techniques when applied to the solution of this kind of phenomena without any deterioration in the accuracy. As a result, we obtain high order time and space discretization schemes which do not suffer from the usual parabolic stiffness in the diffusive limit. We show different numerical results which permit to appreciate the performances of the proposed schemes.

Finite difference scheme for semiconductor Boltzmann equation with nonlinear collision operator

Finite difference scheme for semiconductor Boltzmann equation with nonlinear collision operator

Asymptotic-Preserving Numerical Schemes for the Semiconductor Boltzmann Equation Efficient in the High Field Regime

We present asymptotic-preserving numerical schemes for the semiconductor Boltzmann equation efficient in the high field regime. A major challenge in this regime is that there may be no explicit expression of the local equilibrium which is the main component of classical asymptotic-preserving schemes. Inspired by [F. Filbet and S. Jin, J. Comput. Phys., 229 (2010), pp. 7625--7648] and [F. Filbet, J. Hu, and S. Jin, Math. Model. Numer. Anal., 46 (2012), pp. 443--463], our idea is to penalize the stiff collision term with a classical Bhatnagar--Gross--Krook operator---which is not the local equilibrium in the high field limit---while treating the stiff force term implicitly with the spectral method. These schemes, despite being implicit, can be inverted easily with a stability independent of the physically small parameter. We design these schemes for both nondegenerate and degenerate cases and show their asymptotic properties. We present several numerical examples to validate the efficiency, accuracy, and asymptotic properties of these schemes.

Asymptotic Preserving Schemes for Semiconductor Boltzmann Equation in the Diffusive Regime

Asymptotic Preserving Schemes for Semiconductor Boltzmann Equation in the Diffusive Regime

Calculation of the internal electric field within doped semiconductors

A detailed model for the calculation of the internal potential and electric field profile within doped semiconductors is developed from a first-principles approach and presented in this paper. The model utilizes Poisson's equation and basic Boltzmann statistics to develop a standard nonlinear Poisson–Boltzmann equation (NPBE) for doped semiconductors. The resultant NPBE links the internal electrostatic potential within the doped semiconductor to the doping concentration profile of the semiconductor device under consideration. The NPBE is solved by the application of numerical methods, is general in formulation, supporting multiple simultaneous dopant configurations, and may be applied to any semiconductor type. Calculated results of the electric field profile for various semiconductor dopant structures derived using the model are additionally presented in this paper. The electric field results predicted by the model are shown to be in excellent agreement with those found by other methods. The model may be expanded to accommodate effects involving internal substrate electron–hole pair generation (gemination) caused by photo-ionization for application to and the modeling of solar cell device structures.

Diffusive semiconductor moment equations using Fermi–Dirac statistics

Diffusive moment equations with an arbitrary number of moments are formally derived from the semiconductor Boltzmann equation employing a moment method and a Chapman–Enskog expansion. The moment equations are closed by employing a generalized Fermi–Dirac distribution function obtained from entropy maximization. The current densities allow for a drift-diffusion-type formulation or a “symmetrized” formulation, using dual-entropy variables from nonequilibrium thermodynamics. Furthermore, drift-diffusion and new energy-transport equations based on Fermi–Dirac statistics are obtained and their degeneracy limit is studied.

Energy transport in semiconductor devices

The modelling, analysis and numerical approximation of energy-transport models for semiconductor devices are reviewed. The derivation of the partial differential equations from the semiconductor Boltzmann equation is sketched. Furthermore, the main ideas for the analytical treatment of the equations, employing thermodynamic principles, are given. A new result is the proof of the weak sequential stability of approximate solutions to some time-dependent energy-transport equations with physical transport coefficients. The discretization of the stationary model using mixed finite elements is explained, and some numerical results in two and three space dimensions are presented. Finally, energy-transport models with lattice heating or quantum corrections are reviewed.

Variational formulation of the steady Boltzmann equation for semiconductors and applications

Variational formulation of the steady Boltzmann equation for semiconductors and applications

THE HIGH FIELD ASYMPTOTICS FOR A FERMIONIC BOLTZMANN EQUATION: ENTROPY SOLUTIONS AND KINETIC SHOCK PROFILES

The high field approximation of a fermionic Boltzmann equation of semiconductors is performed after the formation of shocks. By employing a new entropy, whose dissipation measures the departure from the high field equilibrium, convergence towards the entropic solution of the limiting conservation law is proven. The entropy is also used to construct kinetic shock profiles for entropic shocks and to prove non-existence of non-entropic shock profiles.

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.

Adaptive energy discretization of the semiconductor Boltzmann equation

A new deterministic method is presented for solving the semiconductor transport equation of electrons in one‐dimensional silicon device structures. The developed numerical scheme provides a dynamical adaption of the energy discretization to speed up the computation. The adaptive discretization is realized by a time‐ and space‐dependent scaling of the energy variable. Several numerical tests are performed for the bulk case and for a one‐dimensional diode structure. The obtained results prove the efficiency of the proposed method to be significantly increased compared to schemes using fixed energy grids.