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We propose an extended hydrodynamical model for plasmas, based on the moments of the electron distribution function which satisfies the Fokker–Planck–Landau (FPL) transport equation. The equations for the moments can be obtained by multiplying the FPL equation by the corresponding weight functions and integrating over the velocity space. The moments are decomposed in their convective and non–convective parts and closure relations for the fluxes and production terms can be obtained by using the maximum entropy distribution function, which depends on Lagrangian multipliers. These latter can be expressed in terms of the state variables by imposing the constraints that the maximum entropy distribution function reproduces the moments chosen as state variables. In particular, we will concentrate on the 13-moment system. As a first application, we treat the case of the relaxation towards equilibrium of a homogeneous plasma with a temperature anisotropy, showing that the results are in good agreement with those obtained by means of the Kogan solution of the kinetic equation.

Graphene has attracted the attention of several researchers because of its peculiar features. In particular, the study of charge transport in graphene is challenging for future electron devices. Us...

The present work aims at formulating quantum energy-transport and drift- diffusion equations for charge transport in graphene from a quantum hydrodynamic model proposed in [1], obtained from the Wigner-Boltzmann equation via the mo- ment method. In analogy with the semiclassical case, we are confident that the energy- transport and drift-diffusion models have mathematical properties which allow an easier numerical treatment.

In this chapter a short overview will be given regarding the physics of semiconductors and some properties of the Boltzmann equation that is the starting point for describing the semiclassical transport of charge carriers.

In this chapter we investigate the formal properties of the hydrodynamical model for semiconductors based on MEP. We will prove that it forms a hyperbolic system in the physically relevant region of the space of the dependent variables. Such a property is a consequence of the general theory developed in Chap. 3 when applied to the complete non linear model but since we have performed an expansion of the original non linear MEP distribution function, the hyperbolicity have to be checked.

In this chapter the ideas presented in the previous sections will be employed to get a mathematical model for the simulation of a Double Gate MOSFET (hereafter DG-MOSFET). This model is based on the Schrodinger–Poisson system coupled to a set of energy-transport equations, one for each subband, arising from the moment systems associated to the equations ( 1.28) with closure relations obtained by MEP.

The aim is to simulate the DG-MOSFET of first figure in this chapter with the model presented in Chap. 7 consisting of the Schrodinger–Poisson block ( 1.25), ( 1.27) coupled to the energy-transport equations ( 7.30), ( 7.31).

In this chapter MEP is applied to close the moment equations for electrons in silicon semiconductors. In our model we consider the electrons distributed in the six X-valleys assumed as equivalent. The approximation given by Kane will be used as dispersion relation.

The last years have witnessed a great interest in 2D-materials for their promising applications. The most investigated one is graphene, but lately also the single-layer transition metal dichalcogenides (TMDCs), like molybdenum disulphite, tungsten diselenite, and black phosphorus have received a certain attention.

Thermal effects are playing a crucial role for the design of electron nanoscale devices. The present contribution deals with charge and phonon transport under an applied external electric field in a suspended monolayer of graphene. A major question is represented by the phonon-phonon collision operator involving in general a three particle scattering mechanism. To model the phonon-phonon interactions a relaxation time approximation is employed. This requires the introduction of a local equilibrium phonon temperature whose definition is still a matter of debate for a general non equilibrium situation. Here, two different approaches are presented and discussed.

The description of a physical system requires the knowledge of some information, for example the prediction of the motion of a point particle in classical mechanics requires the knowledge of its initial position and momentum besides, of course, the system of forces acting upon it. In the case of a great quantity of particles, i.e. of the order of the Avogadro’s number (6.022 × 1023), the information for a detailed description of the motion of every particle are practically not available; therefore distribution functions and statistical methods are introduced in order to describe the behavior of complex systems.

MEP can be used for solving the closure problem related to the moment systems associated to the electron transport equations. Here the case of 3D electron gas is considered. Lower dimensional electron gases will be treated in the next chapters.

Based on the considerations of the previous chapters, a natural way to get a quantum macroscopic model is to use MEP in a quantum framework to close the moment system arising from the Wigner equation.

To study the electron transport in a some tens of nanometers long channel of a metal oxide field effect transistor, in order to reduce the computational cost of simulations, it can be convenient to divide the electrons into a 2D and a 3D population. Near the silicon/oxide interface the two populations coexist, while in the remaining part of the device only the 3D component needs to be considered because quantum effects are negligible there. The major issue is the description of the scattering mechanisms between the 2D and the 3D electron populations, due to interactions of electrons with nonpolar optical phonons and interface modes. Here, we propose a rigorous treatment of these collisions based on an approach similar to that used in Fischetti and Laux (Phys Rev B 48:2244–2274, 1993), in the context of a Monte Carlo simulation. We also consider all the other main scatterings, which are those with acoustic phonons, surface roughness, and impurities.

Thermal effects in monolayer graphene under an electron flow are investigated with a Monte Carlo (MC) analysis. The crystal heating is described by simulating the phonon dynamics. In particular, the balance equations for the energy of acoustic, Γ optical, and K phonons are deduced. They contain a heating source which is evaluated by counting the emission and absorption processes during the MC simulation of electron flow. The rate of temperature rise is obtained for several Fermi energies and applied electric voltages.

We present a hierarchy of macroscopic models which generalize Cattaneo equation and can be used to describe heat conduction in semiconductor materials. In particular, from these models we derive a new formula for the lattice thermal conductivity which is a able to reproduce the results achieved by means of the celebrated Callaway formula and to give some insights into the dynamical thermal conductivity of semiconductor materials. The models make use of a set of macroscopic state variables for the acoustic phonons, that are moments of their occupation numbers. The evolution equations for these variables are obtained starting from the Boltzmann–Peierls transport equations, and are closed by means of the maximum entropy principle. All the main interactions of phonons among themselves, with isotopes and boundaries are taken into account. Numerical results are shown for the cases of silicon and germanium.

In this work, we propose a hierarchy of macroscopic models for the description of phonon transport in graphene. They are devised starting from the phonon Boltzmann transport equation by using the moments method and the maximum entropy principle. The state variables are the phonon energy densities and the fluxes of integer powers of the energy. We underline that the full energy dispersion relations are used. We test the validity of the models by analysing the steady-state and dynamical thermal conductivity, and the heat transport in zigzag and armchair graphene nanoribbons under a fixed temperature gradient.

The minisymposium was concerned with the mathematical modeling and simulation of charge transport in graphene and other 2D materials and in structures, like double gate MOSFETs, nano-ribbons and nano-wires, where the presence of confinement effects allows for the formal description of the carrier flow as that of a two dimensional or one dimensional electron gas.

We propose, for a finite crystal, that the entire system be expressed in terms of the phase space, including momentum, configuration position and spin. This is to be done both classically and quantum mechanically. We illustrate this with a silicon crystal. Then, quantum mechanically measuring with a wire containing electrons, we obtain a theoretically good approximation for an electron in one semiconductor. As well, we outline what has to be done for any crystal.

In this paper, the low and high field mobilities of graphene on substrates are studied, by means of deterministic solutions, obtained using a Discontinuous Galerkin (DG) numerical scheme, of the semiclassical Boltzmann equation for charge transport in graphene. It is shown that there is a strong dependence on the distance between the impurities and the graphene layer with significant changes both in the low and high field mobility curves. We remark that the use of a DG scheme avoids the intrinsic noise typical of the Direct Monte Carlo Simulation (DSMC) results and allows to evaluate the low field mobility with considerable accuracy, making less ambiguous the comparison with experimental measurements.

A hydrodynamical model for the charge and the heat transport in graphene is presented. The state variables are moments of the electron, hole and phonon distribution functions, and their evolution equations are derived from the respective Boltzmann equations by integration. The closure of the system is obtained by means of the maximum entropy principle and all the main scattering mechanisms are taken into account. Numerical simulations are presented in the case of a suspended graphene monolayer.

PurposeThis paper intends to present a hydrodynamical model which describes the hole motion in silicon and couples holes and electrons.Design/methodology/approachThe model is based on the moment method and the closure of the system of moment equations is obtained by using the maximum entropy principle (hereafter MEP). The heavy, light and split‐off valence bands are considered. The first two are described by taking into account their warped shape, while for the split‐off band a parabolic approximation is used.FindingsThe model for holes is coupled with an analogous one for electrons, so obtaining a complete description of charge transport in silicon. Numerical simulations are performed both for bulk silicon and a p‐n junction.Research limitations/implicationsThe model uses a linear approximation of the maximum entropy distribution in order to close the system of moment equations. Furthermore, the non‐parabolicity of the heavy and light bands is neglected. This implies an approximation on the high field results. This issue is under current investigation.Practical implicationsThe paper improves the previous hydrodynamical models on holes and furnishes a complete model which couples electrons and holes. It can be useful in simulations of bipolar devices.Originality/valueThe results of the paper are new since a better approximation of the band structure is used and a description of both electron and hole behavior is present, therefore the results are of a certain relevance for the theory of charge transport in semiconductors.

Recently [1], a procedure has been proposed in order to close the set of the moment equations of relativistic radiative fluid dynamics. In particular explicit expressions for the moments of the bremsstrahlung and Thomson scattering source terms have been given. In this work, as anticipated in [1], we shall treat in a systematic way Comptonization and double Compton scattering too. Numerical results relative to the Compton cooling of hot electrons are shown.

In this paper we present a hydrodynamical model for the charge and the heat transport in graphene. The macroscopic variables are moments of the electron, hole and phonon distribution functions, and their evolution equations are derived from the Boltzmann equations by integration. The system of equations is closed by means of the maximum entropy principle and all the main scattering mechanisms are taken into account. Numerical simulations are presented in the case of a suspended graphene monolayer.

A nanoscale double-gate MOSFET is simulated by using a subband model based on the maximum entropy principle (MEP).

We present the first macroscopical model for charge transport in compound semiconductors to make use of analytic ellipsoidal approximations for the energy dispersion relationships in the neighbours of the lowest minima of the conduction bands. The model considers the main scattering mechanisms charges undergo in polar semiconductors, that is the acoustic, polar optical, intervalley non-polar optical phonon interactions and the ionized impurity scattering. Simulations are shown for the cases of bulk 4H and 6H-SiC.

In this article we present a fluid dynamical model for the charge transport in semiconductors based on a recently found asymptotic solution of the semiconductor Boltzmann transport equation (BTE) for strong fields.

In this article, we present a theoretical foundation for tail electron hydrodynamical models (TEHM) in semiconductors.

We present a macroscopic model for electron transport in silicon, in which the state variables are moments of the electron distribution functions. The evolution equations are obtained from the Boltzmann transport equations and are closed by means of the maximum entropy principle (MEP). Analytic approximations are used for the electron and phonon dispersion relations. All the main scattering mechanisms electrons undergo in silicon are taken into account.

A nanoscale double-gate MOSFET is simulated with an energy-transport subband model for semiconductors formulated starting from the moment system derived from the Schrodinger–Poisson–Boltzmann equations. The system is closed on the basis of the maximum entropy principle and includes scattering of electrons with acoustic and non-polar optical phonons. The proposed expression of the entropy combines quantum effects and semiclassical transport by weighting the contribution of each subband with the square modulus of the envelope functions arising from the Schrodinger–Poisson subsystem. The simulations show that the model is able to capture the relevant confining and transport features and assess the robustness of the numerical scheme.

In this paper we present a hydrodynamical model which, in principle, is able to describe charge transport in a generic compound semiconductor. The model makes use of an analytic approximation for the conduction bands. Energy dispersion relationships in the neighbors (valleys) of the lowest minima are, in fact, taken to be spherical, nonparabolic. The model considers the main scattering mechanisms in polar semiconductors, that is the acoustic, polar optical, intervalley non-polar optical phonon interactions and the ionized impurity scattering. Simulations are shown for the cases of bulk GaN and SiC.

A hydrodynamic subband model for semiconductors has been formulated in (Mascali and Romano, IL NUOVO CIMENTO 33C:155163, 2010) by closing the moment system derived from the Schrodinger-Poisson-Boltzmann equations on the basis of the maximum entropy principle (MEP). Explicit closure relations for the fluxes and the production terms are obtained taking into account scattering of electrons with acoustic and non-polar optical phonons, as well as surface scattering. Here a suitable numerical scheme is presented for the above model together with simulations of a nanoscale silicon diode.

An energy-transport subband model is formulated with the use of the Maximum Entropy Principle. Numerical simulations of a quantum diode show the feasibility of the model.

In this paper we present exact closures of the 8-moment and the 9-moment models for the charge transport in silicon semiconductors based on the maximum entropy principle. The validity of these models is assessed by numerical simulations of an n-n-n device. The results are compared with those obtained from the numerical solution of the Boltzmann Transport Equation both by Monte Carlo method and directly by a finite difference scheme.

We address the problem of coupling a system of network equations corresponding to an electric circuit with a detailed model for a device connected to the circuit. The device is modeled by an hydrodynamic model based on the maximum entropy principle, which results in a hyperbolic system of partial differential equations. We perform a numerical simulation with an oscillator coupled with an n-n-n channel.