A Class of Stochastic Algorithms for the Wigner Equation

Communications in Applied and Industrial Mathematics

2017

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The Wigner equation represents a promising model for the simulation of electronic nanodevices, which allows the comprehension and prediction of quantum mechanical phenomena in terms of quasi-distribution functions. During these years, a Monte Carlo technique for the solution of this kinetic equation has been developed, based on the generation and annihilation of signed particles. This technique can be deeply understood in terms of the theory of pure jump processes with a general state space, producing a class of stochastic algorithms. One of these algorithms has been validated successfully by numerical experiments on a benchmark test case. Keywords:Nanostructures, Wigner transport equation, Direct simulation Monte Carlo AMS subject classification: 82D80, 82S30, 65M75The Wigner equation is a full quantum transport model able to capture the relevant physics in next generation semiconductor devices. It is well known that the pure state Wigner equation is an equivalent phase-space reformulation of the Schrödinger equation. At the same time the Wigner equation can be augmented by a Boltzmann-like collision operator accounting for the process of decoherence. However, this equation has represented a numerically daunting task and it has raised more problems than solutions. A numerical treatment of the Wigner equation can be dealt with deterministic schemes [1][2][3][4], Direct Simulation Monte Carlo [5][6][7][8], and it is often used to derive reduced transport models, such as quantum-hydrodynamic models [9][10][11][12].Among the several particle Monte Carlo methods developed during these years (see [13] for a review), we have focused in the so called Signed Monte Carlo method [14], where the Wigner potential is treated as a scat-

Section: Discussionmentioning

confidence: 99%

“…The creation process can be better understood in terms of the Markov jump process theory [15,16]. We consider a particle system…”

Section: Markov Jump Process Theorymentioning

confidence: 99%

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A benchmark study of the Signed-particle Monte Carlo algorithm for the Wigner equation

Communications in Applied and Industrial Mathematics

2017

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“…In this framework, we can then construct a kinetic equation in which the distribution function evolves in time under the streaming motion of external forces and spatial gradient, and the randomizing influence of nearly point-like (in space-time) scattering events. For devices with a characteristic length of a few tens of nanometers, the transport of electrons along the axis of the wire can be considered semiclassical within a good approximation; otherwise, a quantum-kinetic approach must be used [32]. The distribution function for the electrons in a quantum wire, with linear expansion in x-direction, depends on the x-direction in real space, the wave vector in x-direction k x and the time t, i.e.,…”

Section: Kinetic and Hydrodynamic Modelmentioning

confidence: 99%

A Hydrodynamic Model for Silicon Nanowires Based on the Maximum Entropy Principle

Castiglione

2016

Entropy

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Silicon nanowires (SiNW) are quasi-one-dimensional structures in which the electrons are spatially confined in two directions, and they are free to move along the axis of the wire. The spatial confinement is governed by the Schrödinger-Poisson system, which must be coupled to the transport in the free motion direction. For devices with the characteristic length of a few tens of nanometers, the transport of the electrons along the axis of the wire can be considered semiclassical, and it can be dealt with by the multi-sub-band Boltzmann transport equations (MBTE). By taking the moments of the MBTE, a hydrodynamic model has been formulated, where explicit closure relations for the fluxes and production terms (i.e., the moments on the collisional operator) are obtained by means of the maximum entropy principle of extended thermodynamics, including the scattering of electrons with phonons, impurities and surface roughness scattering. Numerical results are shown for a SiNW transistor.

“…To solve the BBP equations is not an easy task also from the numerical point of view, because they form a set of partial integrodifferential equations. Particle-based solvers [1][2][3][4][5][6] of the BBP system can be proposed but with a huge computational effort. For engineering purposes, one has to introduce hydrodynamic models, which are obtained by taking the moments of the BBP equations and by using a suitable truncation procedure.…”

Section: Introductionmentioning

confidence: 99%

A hierarchy of hydrodynamic models for silicon carbide semiconductors

Communications in Applied and Industrial Mathematics

Stefano

2017

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The electro-thermal transport in silicon carbide semiconductors can be described by an extended hydrodynamic model, obtained by taking moments from kinetic equations, and using the Maximum Entropy Principle. By performing appropriate scaling, one can obtain reduced transport models such as the Energy transport and the drift-diffusion ones, where the transport coefficients are explicitly determined.