Deterministic kinetic solvers for charged particle transport in semiconductor devices

Cáceres, María J.; Carrillo, José A.; Gamba, Irene M.; Majorana, Armando; Shu, Chi-Wang

doi:10.1007/978-0-8176-4554-0_7

Cited by 10 publications

(14 citation statements)

References 43 publications

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“…Furthermore, a deterministic simulation has several advantages over direct Monte-Carlo simulations due to the results being noise-free, the accurate resolution of transients, and the possibility of obtaining all macroscopic quantities of interest directly from suitable moments of the pdf, ρ(t, v, g). A similar strategy was adopted for obtaining highly accurate resolution of the charged particle transport in semiconductors (see [14,15,10] and references therein). One of the possible drawbacks of the deterministic simulation of the Fokker-Planck equation (3) is its computational cost.…”

Section: Introductionmentioning

confidence: 99%

“…The accurate approximation of the v and g derivatives allows us to use explicit time integrators with a fairly moderate CFL condition. The derivation of this scheme follows similar ideas as in work done in Boltzmann-Poisson kinetic systems for semiconductors [14,15,10]. Also, splitting schemes were developed for plasmas and semiconductor modelling [17,16], which may be applied to the present situation for further computational savings.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A numerical solver for a nonlinear Fokker–Planck equation representation of neuronal network dynamics

Cáceres¹,

Carrillo²,

Tao³

2011

Journal of Computational Physics

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Abstract. To describe the collective behavior of large ensembles of neurons in neuronal network, a kinetic theory description was developed in [13,12], where a macroscopic representation of the network dynamics was directly derived from the microscopic dynamics of individual neurons, which are modeled by conductance-based, linear, integrate-and-fire point neurons. A diffusion approximation then led to a nonlinear Fokker-Planck equation for the probability density function of neuronal membrane potentials and synaptic conductances. In this work, we propose a deterministic numerical scheme for a Fokker-Planck model of an excitatory-only network. Our numerical solver allows us to obtain the time evolution of probability distribution functions, and thus, the evolution of all possible macroscopic quantities that are given by suitable moments of the probability density function. We show that this deterministic scheme is capable of capturing the bistability of stationary states observed in Monte Carlo simulations. Moreover, the transient behavior of the firing rates computed from the Fokker-Planck equation is analyzed in this bistable situation, where a bifurcation scenario, of asynchronous convergence towards stationary states, periodic synchronous solutions or damped oscillatory convergence towards stationary states, can be uncovered by increasing the strength of the excitatory coupling. Finally, the computation of moments of the probability distribution allows us to validate the applicability of a moment closure assumption used in [13] to further simplify the kinetic theory.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A numerical solver for a nonlinear Fokker–Planck equation representation of neuronal network dynamics

Cáceres¹,

Carrillo²,

Tao³

2011

Journal of Computational Physics

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“…Very recently, deterministic solvers to the BoltzmannPoisson system (1) for two-dimensional devices were proposed [1][2][3]7]. These methods provide accurate results which, in general, agree well with those obtained from Monte Carlo (DSMC) simulations, often at a fractional computational time.…”

Section: Methodsmentioning

confidence: 62%

“…Time-dependent solutions of the Boltzmann-Poisson system contain all the information on the evolution of the carrier distribution. In (1), f represents the electron probability density function (pdf ) in phase space k at the physical location x and time t. E is the electric field and ε is the energy-band function. Physical constants and q are the Planck constant divided by 2π and the positive electric charge, respectively.…”

Section: Introductionmentioning

confidence: 99%

Discontinuous Galerkin solver for Boltzmann-Poisson transients

et al. 2008

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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 nanoscale 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.

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“…Due to the mathematical difficulties that arise by this interaction, operators as well as the large number of unknowns, the numerical approximation of such kinetic transport models is highly demanding from a computational standpoint. Traditional computational schemes for this type of linear transport developed several decades ago were based in Discrete Simulations Monte Carlo (DSMC) [3,39] or more recently deterministic approaches initiated by [26,37] and more recently extended WENO finite difference based schemes [6,7,8,9,5].…”

Section: Introductionmentioning

confidence: 99%

Positivity-preserving discontinuous Galerkin schemes for linear Vlasov-Boltzmann transport equations

2012

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Abstract. We develop a high-order positivity-preserving discontinuous Galerkin (DG) scheme for linear Vlasov-Boltzmann transport equations (Vlasov-BTE) under the action of quadratically confined electrostatic potentials. The solutions of such BTEs are positive probability distribution functions and it is very challenging to have a mass-conservative, high-order accurate scheme that preserves positivity of the numerical solutions in high dimensions. Our work extends the maximum-principle-satisfying scheme for scalar conservation laws in a recent work by X. Zhang and C.-W. Shu to include the linear Boltzmann collision term. The DG schemes we developed conserve mass and preserve the positivity of the solution without sacrificing accuracy. A discussion of the standard semi-discrete DG schemes for the BTE are included as a foundation for the stability and error estimates for this new scheme. Numerical results of the relaxation models are provided to validate the method.

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Deterministic kinetic solvers for charged particle transport in semiconductor devices

Cited by 10 publications

References 43 publications

A numerical solver for a nonlinear Fokker–Planck equation representation of neuronal network dynamics

A numerical solver for a nonlinear Fokker–Planck equation representation of neuronal network dynamics

Discontinuous Galerkin solver for Boltzmann-Poisson transients

Positivity-preserving discontinuous Galerkin schemes for linear Vlasov-Boltzmann transport equations

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