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VLSI DESIGN
1998, Vol. 8, Nos. (14), pp. 515520
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(C) 1998 OPA (Overseas Publishers Association) N.V.
Published by license under
the Gordon and Breach Science
Publishers imprint.
Printed in India.
Calibration of a One Dimensional Hydrodynamic
Simulator with Monte Carlo Data
O. MUSCATO a,,, S. RINAUDOband P. FALSAPERLA
Dipartimento di Matematica, Viale Andrea Doria, 95125 Catania (Italy)"
bSGSTHOMSON Microelectronics, Stradale Primo Sole 50, 95121 Catania (ltaly)
In this paper we use the code Exemplar for matching a hydrodynamic 1D, time
dependent simulator and the transport coefficients obtained by the Monte Carlo
simulator Damocles. This code is based on the Least Square method and it does not
require any a priori knowledge about the simulator (analytical form of the equations
etc.). The stationary electron flow in a one dimensional n+nn+submicron silicon
diode is simulated.
Keywords." TCAD, VLSI, BTE, transport theory, fluid mechanics, electronic devices
1. INTRODUCTION
Electronic transport in semiconductors can be
described by hydrodynamic models (hereafter
HM), obtained by taking the moments of the
Boltzmann transport equation (hereafter BTE):
the resulting mathematical model consists of an
infinite hierarchy of Partial Differential Equations
expressing balance laws for the particle number n,
velocity,total energy E, deviatoric stress tensor
(0"), energy flux
coupled with the Poisson equation.
Recently Anile and Muscato [1] presented an
Extended Hydrodynamic model where the closure
of the moment hierarchy is obtained by exploiting
the entropy principle:
(or heat flux h ) and so on,
this system, which is
hyperbolic, consists of 13 scalar equations in the
13 unknowns
determines the description of the stress and of
the heat flux, at variance with the other models.
Such a model has been tested successfully with
Monte Carlo (hereafter MC) simulations in Silicon
[2]. The Left H and Side of the balance equations,
called production terms, represent the average rate
of change of carrier total energy Qw, momentum
Qp, energyflux Qs and stress Q</j> due to the
scattering of carriers with the lattice. Usually they
are approximated with ad hoe empirical formula
[3] or as relaxation terms [4]: this last approxima
tion leads to a serious inconsistency with one of
the fundamental principles of Linear Irreversible
Thermodynamics, the Onsager Reciprocity Princi
(n,,E,( ij),) and completely
*Corresponding author.
515
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O. MUSCATO et al.
pie. In order to tackle this problem, we expanded
the electron distribution function in Hermite
polynomials around the state of global thermal
equilibrium (limiting ourselves to the first two
polynomials), where the electrons lie close to the
bottom of the conduction band. By using a well
known procedure due to Grad [5] we obtained the
following equations:
Qw
(1)
"rw
Qp
aPoo +aPl
+aP2
nff
+ bP+bPoo+blo
kBTo
(2)
Qs
ag+ asl
+
kBTonff
++
+
(3)
where Tois the room temperature, E0the lattice
total energy, noa reference impurity concentration
(1018cm3).
Since in
distribution function
is quasiisotropic
deviatoric stress tensor/isnegligible and by this
method we cannot extract the corresponding
production term (unless we consider higher order
Hermite polynomials).
For the sake of simplicity we modeled the stress
production as a relaxation term:
this approximation the
[2] the
Q<ij>
<ij>
(4)
We should emphasize that:
the coefficients appearing in Eqs. (14) are not
fitting parameters but rather will be extracted by
MC data obtained by the Damocles code [6], in
the case of parabolic spherical band approxima
tion, in order to be consistent with the Anile and
Muscato model, obtained under these restric
tions;
these coefficients are not functions of the
positions in the device as some authors claim.
With such coefficients we obtained a closed set of
hydro equations which has been solved by an
adequate numerical scheme. Since the Monte
Carlo method gives a stochastic solution of the
BTE, the results are noisy: how does the hydro
solution change for small variations of the para
meters Tw, T, a0, al, a2, b0, bl, b29. In order to
answer this question the simulator Exemplar,
developed by one of us [7], has been used.
The plan of the paper is the following: in section
2 we discuss the optimization problem; in section 3
we simulate the usual n+nn+diode with the
Damocles code and with our Extended hydrody
namic model. We discuss the range of validity of
the optimization procedure for various biases and
conclusion are drawn.
2. THE OPTIMIZATION PROBLEM
The optimization problem is well known: defining
the residual function
N
R()
wi(f(xi,)
i=1
yi)2
(5)
where Y is the vector of the parameters, f(xi, ) is
the analytical description of the event computed in
xi(modeled by the vector parameter ),Yithe real
event value obtained inxiandwian appropriate
weight factor. In a mathematical language the
problem of the least squares method could be
expressed in the form:
min{R(7)fi7 e Rn}
(6)
where n is the dimension of the vector 7 and 7 the
solution of the system, generally, nonlinear
VR() 0
(7)
and R" the n dimensional space of the real R.
Ifwe know the analytical equation off(x, 7), we
could evaluate and resolve the system vR()
0,
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CALIBRATION OF A ONE DIMENSIONAL SIMULATOR
517
using one of many methods proposed
Marquardt). However, if we do not know the
analytical form off(x, E), it is not possible to find
the solution of the problem (6) finding the solution
of the system (7). This happens when we want to
determine a set of parameters to model a general
device with any kind of simulator: we run the
simulator obtaining the macroscopic quantities
but not the derivatives of the Residual function. In
this case we can resort numerical methods as the
simplex method and Powell’s method, which
minimize the Residual function, without having
to know its derivatives.
The Exemplar code runs the simulator to
calculate the function f(xi, ) (i= 1,...,N) then
compute and optimize the Residual function with
the Simplex or Powell method (according the user
choice) and the procedure restarts until a tolerance
of the Residual function is obtained.
The proposed algorithm can be used with any
simulators which can read their input from files
and store their output in columnformatted ASCII
files. It was implemented in a software program
with a friendly user interface based on XWindow,
which resolves the problem of the optimization
well, but, since during the optimization it runs the
simulator, it needs a large computational time.
(e.g.
50
MC
Eq.(2)
"700
0.05
0.10.15
0.2
0.25
(micron)
0.30.35
0.4
O.
0.5
FIGURE
vs. distance: Monte Carlo data (with ***) and the fitting formula
Eq. (2) (with ooo).
Average rate of change of carrier momentum Qp
25
MC
Eq.(3)
Volt bias
.300
0.05
0’.1 0.’15
0.2 0.25
(micron)
0’.3
0.35 0.4
0.450.5
3. SIMULATION OF A ONE DIMENSIONAL
n+nn+SILICON DIODE
FIGURE 2
vs. distance: Monte Carlo data (with ***) and the fitting formula
Eq. (3) (with ooo).
Average rate of change of carrier energy flux Qs
The n+nn+diode consists of two n+regions
0.1 gmlong doped to a density ofN
while the central n region is 0.3 gm wide, with a
doping density of N
Ohmic boundary conditions, To
of applied bias; simulations refer to the stationary
regime.
In Figures
and 2 we plot the MC data for Qv
and Qs and the fitting with eqs. (2) and (3) (with
ooo): we see that our functional form fits well the
data. We observe that the MC data are noisy
especially near the contacts (this run needed
1018 cm3,
1016 cm3. We consider
300 K, and
V
month of CPU in a IBM Risc 6000 590) and
consequently the fitting coefficients are not unique.
By using these coefficients the Extended Hydro
dynamic model is solved by using a simulator
based on the splitting method between relaxation
and convection (Tadmor scheme) [8]. Then we run
the Exemplar code in order to find the best
coefficients which fit well the MC data. In Figures
3, 4 and 5 we compare the MC data (with ***), the
hydro data without optimization (with ooo), the
hydro data optimized with Exemplar (with xxx).
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O. MUSCATO et al.
1.4
0.4
(micron)
Monte Carlo
Hydro
Exemplar
Volt bias
x
0.4
0.45
0.5
FIGURE 3
using the hydrodynamic simulator without optimization (ooo),
the hydro data optimized with the Exemplar code (xxx) and
Monte Carlo data (***), with
Average drift velocity vs. distance computed by
Volt bias.
0.15
0.05
0.1
Monte Carlo
Hydro
Exemplar
Volt bias
(micron)
FIGURE 5
hydrodynamic simulator without optimization (ooo), the hydro
data optimized with the Exemplar code (xxx) and Monte Carlo
data (***), with
Volt bias.
Heatflux vs. distance computed by using the
Monte Carlo
Hydro
Exemplar
Volt bias
0.25
0.2
0.1
0.05
oi5
o.
o/15
0.2
o.,
(micron)
ot
o.,
o:,
o:;.
o.
FIGURE 4
using the hydrodynamic simulator without optimization (ooo),
the hydro data optimized with the Exemplar code (xxx) and
Monte Carlo data (***), with
Average electron energy vs. distance computed by
Volt bias.
The optimized coefficients (shown in Tab. I)
differ from the previous ones at maximum by 40%.
We notice that the well known ’spurious’ peak in
the velocity curve Figure 3 is lowered and the
energy and heat flux curves Figures 4 and 5 are
closer to the MC data. However this ’spurious’
peak cannot be eliminated by any small change in
the parameters and therefore its persistence calls
for a more radical examination of the basic
assumptions of the model.
A crucial question to be addressed is what is the
range of validity of the optimization procedure? In
order to answer this question we simulate the
device with the coefficients of Table I (obtained
with
Volt) for various biases. The result is that in
the range 0.8 / 1.2 Volt the hydro simulations fit
well the respective MC data (see Figs. 6, 7 and 8 in
case of 1.2V):
for higher biases the hydro
simulations diverge considerably with respect to
the MC data.
0,4
Monte Carlo
Hydro
1.2 Volt bias
(micron)
0.05 0.1 0.15
0.2
0.3
0.35
0.4
0.45
0.5
FIGURE 6
using the hydrodynamic simulator with transport coefficients of
Table
(obtainedwith
Volt) (ooo) and Monte Carlo data (***),
with 1.2 Volt bias.
Average drift velocity vs. distance computed by
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CALIBRATION OF A ONE DIMENSIONAL SIMULATOR
519
TABLE
Transport coefficients (in psec), channel 0.3 tm,
Volt bias
a0
a
a2
b0
1.6078
21.7514
b
b2
Qp
Qs

0.1376
5.7677
2.3042
27.7312
1.3978
24.1639
0.1930
2.6697
0.00769
3.5269
7w
0.4094 (in psec.)
0.0712 (in psec.)
0.4
0.35
0.25
0,2
0.15
0.05
Monte Carlo,
Hydro
1.2 Volt bias
ooO
ooO
o
o
O0
0.;5
0:1
0.’15 0:2
0.5
(micron)
0.3
0.35
0.4
0.45
0.5
FIGURE 7
using the hydrodynamic simulator with transport coefficients of
Table
(obtained with
Volt) (ooo) and Monte Carlo data
(***), with 1.2 Volt bias.
Average electron energy vs. distance computed by
0,15
0"05
0.05
0.1
Monte Carlo
Hydro
1.2 Volt bias
0.05
0.1
0.15
0.2
0.25
(micron)
0.35
0.4
0.45
0.5
FIGURE 8
hydrodynamic simulator with transport coefficients of Table
(obtained with
Volt) (ooo) and Monte Carlo data (***), with
1.2 Volt bias.
Heat flux vs. distance computed by using the
Finally we say that the modeling of the
production terms and the closure of the hydro
model are crucial points. They should be obtained
by using the physics (e.g., an entropy principle,
expansions of the distribution function). The
transport coefficients can be obtained either by
experiments or by MC simulations, and their
validity is restricted to a neighborhood.
We are trying to improve our model and to
simulate 2D devices: work along this line is in
progress and will presented elsewhere.
Acknowledgements
This work has been supported by the C.N.R.
Progetto Speciale Modelli Matematici per
conduttori 1996, the MURST project 40% and
60% 1996.
semi
References
[1] Anile, A. M. and Muscato, O. (1995). Phys. Rev. B., 51,
16728.
[2] Anile, A. M. and Muscato, O. (1996). Cont. Mech Therm.,
1,1.
[3] Tang, T. W., Ramaswamy, S. and Nam, J. (1993). IEEE
Trans. Elec. Dev., 40, 1469.
[4] Baccarani, G. and Wordemann, M. E. (1982). Solid State
Elec., 29, 970.
[5] Grad, H. (1958). In Thermodynamics ofgases edited by
S. Flugee, Handerbruch der Physik 12, SpringerVerlag,
Berlin.
[6] Laux, S. E., Fischetti, M. V. and Frank, D. J. (1990). IBM
J. Res. Develop., 34.
[7] Rinaudo, S. (1995). A General algorithm to calibrate any
kindofsimultators without any knowledge about analytical
form ofthe implemented models, SGSTHOMSON Micro
electronics, internal report.
[8] Falsaperla, P. and Trovato, M. (1997). Preprint University
of Catania.
Authors’ Biographies
Orazio Muscato is Assistant Professor of Theore
tical Mechanics at Catania University. His re
search interests include mathematical models for
semiconductors, Monte Carlo simulations, wave
propagation.