Three-Dimensional Implementation of a Unified Transport Model
This paper describes a unified transport model which self-consistently accounts for thermal effects and hot-carrier phenomena. Such result is achieved by including the energy balance equations for electrons, holes and lattice. The model has been incorpora
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Edited by S. Selberherr, H. Stippel, E. Strasser - September 1993
Three-Dimensional Implementation of a Unified Transport Model A. Pierantoni, A. Liuzzo, P. Ciampolini t , and G. Baccarani Dipartimento di Elettronica, Informatica e Sistemistica, Universita di Bologna Viale Risorgimento 2, 1-40136 Bologna, ITALY hstituto di Elettronica, Universita di Perugia 1-06131 Santa Lucia-Canetola, Perugia, ITALY
Abstract This paper describes a unified transport model which self-consistently accounts for thermal effects and hot-carrier phenomena. Such result is achieved by including the energy balance equations for electrons, holes and lattice. The model has been incorporated into the three-dimensional device simulator HFIELDS3D and its numerical efficiency is tested by simulating both unipolar and bipolar devices.
1. The physical model The behavior of sub micron devices is influenced by complex interactions among nonstationary phenomena, device self-heating, as well as fringing and proximity effects. In principle, therefore, such effects have to be self-consistently described within a unified framework: at present, however, only a few attempts to exploit such a model have been reported, mainly because of the computational burden associated with it. For this reason, Szeto and Reif [1 J, as well as Benvenuti et 801. [2], limited themselves to one-dimensional analyses, whereas Katayama and Toyabe [3J developed a threedimensional, finite-difference discretization scheme. We present the inclusion of a generalized transport model into the more versatile environment provided by HFIELDS-3D, and compare the efficiency of some solution schemes. The model originates from the three BTEs which describe the dynamics of three interacting subsystems: namely electrons, holes and phonons. Energy-balance equations are obtained by taking the moments of order two of the BTEs:
. ~
awn,p
~ ~
~+dIVSn,p=F·Jn,p+
(awn,p) ~
(1) coll
aWL d'IV S~L_--+ at
(OWL) at
(2)
--
coll
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A. Pierantoni et al.: Three-Dimensional Implementation of a Unified Transport Model
In the above equations, W represents the energy density and defined as follows:
-
Sn,p
(5 )-kBTn,p q - In,p
= -Kn,p grad Tn,p =f "2 -
while the current densities
S the
energy flow,
S
(3)
SL = -KL grad TL
(4)
J are given by:
- = qPn [kBTn In -q- gradn
+ n grad (kBTn)] -q- - t.p
(5)
(6) The collision terms, which account for the interactions among different subsystems, are described according to the relaxation-time approximation and fulfill the following relationship:
+ (aWn) + (aWp) (OWL) at at at coll
coil
= EG U
(7)
coli
Finally, the model is completed by Poisson's and current-continuity equations. Spatial discretization is carried out by the Box Integration Method (in analogy with the scheme presented in [4]) applied to the hybrid mesh suggested by Conti et a1. [5].
2. Simulation results In this section, the simulation of two simple devices is discussed: namely, the MOSFET shown in Fig. 1 and the BJT shown in Fig. 2 are considered. In both cases, comp
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