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A class of stochastic algorithms for the numerical treatment of the Wigner equation is introduced. The algorithms are derived using the theory of pure jump processes with a general state space. The class contains several new algorithms as well as some of the algorithms previously considered in the literature. The approximation error and the efficiency of the algorithms are analyzed. Numerical experiments are performed in a benchmark test case, where certain advantages of the new class of algorithms are demonstrated.

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PurposeThe purpose of this paper is to set up a consistent off‐equilibrium thermodynamic theory to deal with the self‐heating of electronic nano‐devices.Design/methodology/approachFrom the Bloch‐Boltzmann‐Peierls kinetic equations for the coupled system formed by electrons and phonons, an extended hydrodynamic model (HM) has been obtained on the basis of the maximum entropy principle. An electrothermal Monte Carlo (ETMC) simulator has been developed to check the above thermodynamic model.FindingsA 1D n+−n−n+ silicon diode has been simulated by using the extended HM and the ETMC simulator, confirming the general behaviour.Research limitations/implicationsThe paper's analysis is limited to the 1D case. Future researches will also consider 2D realistic devices.Originality/valueThe non‐equilibrium character of electrons and phonons has been taken into account. In previous works, this methodology was used only for equilibrium phonons.

Abstract. The paper studies the convergence behavior of Monte Carlo schemes for semiconductors.A detailed analysis of the systematic error with respect to numerical parameters is performed. Different sources of systematic error are pointed out and illustrated in a spatially one-dimensional test case. The error with respect to the number of simulation particles occurs during the calculation of the internal electric field. The time step error, which is related to the splitting of transport and electric field calculations, vanishes sufficiently fast. The error due to the approximation of the trajectories of particles depends on the ODE solver used in the algorithm. It is negligible compared to the other sources of time step error, when a second order Runge-Kutta solver is used. The error related to the approximate scattering mechanism is the most significant source of error with respect to the time step.Mathematics Subject Classification. 82D37, 65C05.

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-

Thermoelectric effects in bulk silicon are investigated by using a hydrodynamic model for the electron-phonon system, derived in the framework of extended thermodynamics. This model consists of a set of balance equations, where the higher order moments and the production terms are completely determined without any fitting procedure. If the system is in local thermal equilibrium, the thermopower and Peltier coefficients have been obtained and the phonon-drag contribution has been recovered. The model allows us to define and evaluate the Peltier coefficient when the system is out of thermal equilibrium.

In this paper, a hydrodynamic model coupling electron and phonon transport in silicon semiconductors has been formulated to describe off-equilibrium phenomena. Closure relations for the higher order moments and the production terms have been obtained on the basis of the maximum entropy principle of extended thermodynamics. Applications to bulk silicon are presented.