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\title{Applications of a non-parabolic Tail Electron Hydrodynamical Model

to  Silicon devices}

\author{Angelo Marcello Anile\thanks{e-mail anile@dmi.unict.it}~

and Giovanni Mascali\thanks{e-mail

mascali@dmi.unict.it} \\

{\em Dipartimento di Matematica ed Informatica,}\\

{\em Universit\`a di Catania,}

{\em Viale A. Doria 6, 95125 Catania, Italy}}

\date{}

\maketitle

Computer aided design of modern  semiconductor

devices requires increasingly accurate physical

models which are able to

describe high-field phenomena such as hot electron effects, impact ionization,

thermal self-heating, etc. Such phenomena play an important role

in the degradation and breakdown of devices. \\

Using Monte Carlo methods (MC) is extremely CPU

intensive and therefore not practical.

On the other hand traditional hydrodynamical models for carrier transport

in semiconductors

cannot cope with hot electrons because they deal only with average values

over the whole carrier population.\\

Several considerations suggest that electrons

having energies respectively lower and higher than a

suitable threshold energy may be described by

two distinct thermal

distributions at different temperatures \cite{LaFis}.

This justifies

the introduction of new fluid dynamical

models, \cite{Yao,Scro}, where two well-defined subpopulations of electrons

are considered, each subpopulation being described by the relative

macroscopic quantities.\\

Recentely \cite{AniMas}, a theoretical foundation

for these models has been given by utilizing the moment method and a closure

by which one can obtain both the constitutive fluxes and the production terms,

appearing in the moment equations,

as functions of the fundamental hydrodynamical variables, without
resorting to MC.

However, for the sake of simplicity, a parabolic band approximation
has been used

also for the highly energetic electrons. Here we propose an improvement

of the model attained by using the Kane dispersion relation.

Starting from the Boltzmann equation and introducing a threshold energy

$ \epsilon_{thr}$, (usually the threshold energy of impact ionization),

we obtain a macroscopic model where the electron flow is described

by the following fundamental variables:

number density, average velocity,

energy and energy flux of electrons having energy less

than and greater

than $\epsilon_{thr}$ (zone $1$ and $2$ electrons respectively),

\begin{eqnarray}

&{}&\left\{ \begin{array}{ll}n_1=\int_{\tilde{\Delta}} f\,d {\bf k}\quad

\quad\quad

n_1\,{\bf u_1}=\int_{\tilde{\Delta}}{\bf v} f\,d{\bf k} \\

n_1\,W_1=\int_{\tilde{\Delta}}\epsilon f\,d{\bf k}\quad\quad

n_1\,{\bf S_1}=\int_{\tilde{\Delta}}\epsilon {\bf v} f\,d{\bf k}

\label{eqDF.2}

\end{array}\right.\\

&{}&\left\{ \begin{array}{ll}n_2=\int_{\Delta} f\,d {\bf k}\quad\quad\quad

n_2\,{\bf u_2}=\int_{\Delta}{\bf v} f\,d{\bf k}\\

n_2\,W_2=\int_{\Delta}\epsilon f\,d{\bf k}\quad\quad

n_2\,{\bf S_2}=\int_{{\Delta}}\epsilon {\bf v} f\,d{\bf k}\,.

\label{eqDF.3}

\end{array}\right.

\end{eqnarray}

$f=f({\bf x},t,{\bf k})$ is the electron distribution function and

${\bf k}$ the electron wave vector. $\epsilon({\bf k})$

is the electron energy in the conduction band.

$\Delta=\left\{{\bf k}:\epsilon({\bf k})\geq \epsilon_{thr}\right\}$ and

$\tilde\Delta=\Re^3-\Delta$.\\

The necessary closure relations for fluxes and production terms

are found by means of the maximum entropy

principle \cite{Mul}. \\

In order to test the model, we consider applications to bulk silicon and

1-dimensional electron diodes.

















\begin{thebibliography}{50}



\bibitem{LaFis}

S.~Laux, M.~Fischetti, Transport models for advanced device simulation-truth or

consequences, Proceedings of the Bipolar-BiCMOS technology Meeting,
October 1-3,

Minneapolis, MN, (1995).



\bibitem{Yao}

C.~-S.~Yao, J.~-G.~Ahn, Y.~-J.~Park, H.~-S.~Min and R.~W.~Dutton,

Formulation of a Tail electron Hydrodynamic model based on Monte Carlo

Simulations, IEEE Electron Dev. Lett.,{\bf 16}, 26-29, (1995).



\bibitem{Scro}

P.~Scrobohaci,T.~-W.~Tang, Modeling of the hot electron subpopulation

and its application to the impact ionization in silicon submicron

devices: part I, IEEE Trans. Electron Devices, {\bf 41}, 1197-1205 , (1996).



\bibitem{AniMas}

A.~M.~Anile, G.~Mascali,

{\sl Theoretical foundations for tail electron hydrodynamical models

in semiconductors}, Appl.~Math.~Lett., {\bf 14}, 245-252, (2001).



\bibitem{Mul}

I.~M\"uller and T.~Ruggeri  (1998) Rational Extended Thermodynamics.

Springer-Verlag, Berlin

\end{thebibliography}







\end{document}

We shall consider the two electron subpopulations to be described by

two different distribution functions and by maximizing the electron entropy

with respect to them, we shall find the