1333
IEEE
TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL.
40,
NO.
7,
JULY
1992
0
Lintarandmlinear
trequencydomaln
(Ha- Balance)
analysis
Phy
sics-Based Electron Device Modelling and
Computer-Aided
MMIC
Design
Circuit Analysis
G
=
WY)
Fabio Filicori, Giovanni Ghione, Member,
IEEE,
and Carlo
U.
Naldi, Senior Member,
IEEE
(Invited
Paper)
Abstract-The paper provides an overview on the state of the
art and future trends in physics-based electron device model-
ling for the computer-aided design of monolithic microwave
IC’s.
After a review of the main physics-based approaches to
microwave modelling, special emphasis is placed on innovative
developments relevant to circuit-oriented device performance
assessment, such as efficient physics-based noise and para-
metric sensitivity analysis. The use of state-of-the-art physics-
based analytical or numerical models for circuit analysis is dis-
cussed, with particular attention to the role of intermediate be-
havioural models in linking multidimensional device simulators
with circuit analysis tools. Finally, the model requirements for
yield-driven
MMIC
design are discussed, with the aim of point-
ing out the advantages of physics-based statistical device
modelling; the possible use of computationally efficient ap-
proaches based on device sensitivity analysis for yield optimi-
zation is also considered.
I.
INTRODUCTION
N THE traditional approach to the design of hybrid or
I
monolithic microwave IC’s (MMIC’s) the circuit is
built around packaged or foundry devices which are
modelled by behavioral electrical models (e.g., equiva-
lent circuits), characterized through standard or on-chip
measurements performed on manufactured prototypes.
Circuit optimization is performed in the space of the elec-
trical or geometrical parameters of the passive elements.
Conversely, in the physics-based approach to MMIC de-
sign, optimization also involves the technological param-
eters
of
the active devices. This requires that the active
devices be characterized through physical models, which
provide the link between the physical and process input
data and the electrical performances within the framework
of
an integrated CAD environment (see Fig.
1)
whose
main steps are:
Process modelling, relating the process parameters
a
(i.e., intrinsic semiconductor characteristics, control pa-
rameters for epitaxial or ion-implantation processes, ge-
ometry of photolithographic masks..
.)
to the correspond-
ing physical parameters
0
(e.g., activated doping profile,
Manuscript received September
16,
1991;
revised March
I,
1992.
This
work was partially supported by EEC in the framework of the ESPRIT
255
and
5018
COSMIC projects. Partial support has
also
been
provided from
the Italian National Research Council (CNR).
F. Filicori is with the Universita di Ferrara, Istituto di Ingegneria, Via
Scandiana
21,
44100 Ferrara, Italy, and with the Centro di Studio fer
I’In-
terazione Operatore-Calcolatone Facolta di Ingegneria, Viale Risorgi-
mento
2,
40136 Bologna, Italy.
G. Ghione and C.
U.
Naldi are with the Dipartimento di Elettronica.
Politecnico di Torino, Corso Duca degli Abruzzi
24,
Torino. Italy.
IEEE Log Number
9200768.
a
Prooess
parameters
’
p
Physicaldevice
Physics-based
0
Anatytiipmcess
models
0
20
numerical
models
I
0
quasi-20
and
20
Device Modelling
numerical
and
analylii models
I
I
y
Electrical device
Circuit-oriented
behavioral
0
Direcl
link
0
Intermediate
equivalent
cirmks
0
Intermedie
black-box
mathematical
models
’TGCi,cuitpebrmance
Fig.
1.
Functional flow chart for process, device and circuit modelling.
actual gate length, recessed gate depth, surface or sub-
strate state density..
.)
which characterize the manufac-
tured semiconductor device.
Physics-based device modelling (PBDM), relating the
physical parameters
0
to
the electrical parameters
y
(i.e.,
frequency-dependent S-parameters, dc characteristics, RF
transconductance, junction capacitances, noise parame-
ters..
.)
of a given semiconductor device.
Circuit analysis finally providing the link between the
electrical device parameters
y
and the corresponding cir-
cuit performance
G.
There are several reasons for adopting a physics-based
approach to MMIC design. In a performance-driven de-
sign, PBDMs allow the designer to tailor, at least up to a
certain extent, the active devices
so
as to further improve
the circuit response. However, the physics-based ap-
proach has special advantages in yield-driven MMIC de-
sign, where the electrical device parameters must be char-
acterized statistically. In fact, while the physical
parameters deriving from the manufacturing process are
either practically uncorrelated or subject to simple corre-
lations, the statistics
of
electrical device parameters are
affected by complex correlations introduced by the device
physics. The cumbersome and expensive characterization
of
many manufactured device prototypes can be avoided
0018-9480/92$03.00
0
1992 IEEE
1334
IEEE
TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 40, NO. 7. JULY
1992
if the statistics of the electrical parameters is derived, by
means of PBDM’s, from the physical parameters whose
experimental statistical characterization and Monte Carlo
simulation is easier.
In this perspective, the process and device physics-
based models menticned above should not only yield the
nominal (expected) values for the physical and electrical
device parameters, respectively, but also the self and joint
probability distributions of the deviations
A
0
and
Ay
be-
tween actual and expected values. In the case of small
physical parameter changes the electrical parameters can
be statistically characterized through the first order ap-
proximation
Ay
=
SJ
Ab,
where
SJ
is the device small-
change sensitivity.
The above considerations suggest that PBDMs, tradi-
tionally a tool for device design only, should also play an
important role in physics-based MMIC design. However,
while the computer algorithms for circuit analysis
[73],
[87], [77], [63], [38], [83], [98]
have now reached suffi-
cient maturity to enable MMIC optimization even on me-
dium-power workstations, the physical device models, on
which the accuracy and efficiency of performance predic-
tion ultimately depends, still involve considerable com-
putational problems.
In fact, a model able to provide complete device per-
formance prediction (dc characteristics, bias-dependent
small-signal ac parameters, large-signal response, noise,
temperature dependence) in terms of physical parameters
alone must be based on fundamental semiconductor equa-
tions. Unfortunately, even for the relatively simple drift-
diffusion model, accurate and general-purpose algorithms
for the solution of the PBDM equations require the nu-
merical treatment of sets of partial differential equations
over a two- or three-dimensional domain. As a conse-
quence, numerical physics-based models are computa-
tionally intensive and therefore unsuitable for direct in-
clusion into CAD tools for circuit analysis and
optimization.
The implementation of physical models can be simpli-
fied and made more efficient (but possibly less accurate)
by taking advantage of the specific structure of microwave
FET’s; this leads to the so-called quasi-2D numerical or
analytical models. However, although recently proposed
numerical quasi-2D models can be used for simple large-
signal circuit analyses
[88],
only analytical models are
directly compatible with optimization-driven circuit anal-
ysis algorithms based on frequency-domain harmonic-
balance
(HB)
techniques. The use of numerical physics-
based device models for circuit analysis, as discussed in
Sec. V, is possible only through “off line” device simu-
lation and indirect linking with circuit analysis algorithms
by means of intermediate behavioral models.
Although the above remarks seem to suggest that only
analytical models are really suitable for physics-based
MMIC design, this cannot be considered a final conclu-
sion in view of the accuracy requirements posed by this
task. In fact, success in performing physics-based MMIC
design obviously depends on the accuracy achieved by the
physical models used to this aim. However, accuracy re-
quirements for PBDM’s are difficult to establish a priori,
and should be properly understood. While both fully 2D
and simplified models can be highly accurate in reproduc-
ing the electrical characteristics of a particular device, as
repeatedly shown in the literature, this goal is often
achieved by properly adjusting the values of some of the
physical input parameters around initial estimates (model
tuning). This procedure may ultimately turn the physical
model into an almost behavioral model, whose so-called
physical input parameters actually depend on the real
physical parameters; consequently, the excellent agree-
ment shown for a specific device does not guarantee that
the physical model is able to accurately reproduce the
variations in the electrical characteristics caused by vari-
ations in the physical input parameters. Simplified imple-
mentations of physical models are expected to suffer from
this limitation more than fully 2D numerical implemen-
tations.
As discussed in Sec. VI, physics-based performance
and above all yield optimization requires a model able not
only to accurately simulate the electrical behaviour of a
device, but also to closely reproduce the effect of small
variations of its physical parameters with respect to the
nominal values. This conclusion can be intuitively under-
stood when considering that physics-based MMIC opti-
mization normally starts from a circuit which has already
been performance-optimized around “standard” foundry
devices. Now, according to the more or less critical per-
formance requirements and to the maturity of the tech-
nology, several situations may arise. If the tolerance
ranges for circuit performance are not critical, further op-
timization is probably useless. The same can be said of
yield optimization if the technology is poor; in this case,
in fact, the spread in the physical parameters is
so
large
that little can be achieved by design centering, and yield
improvement becomes mainly a technological issue. On
the other hand, the design of high-performance circuits
using a mature technology offers good possibilities in
terms both of performance and yield physics-based opti-
mization. In either case, however, we expect that, owing
to the tight performance tolerances and to the low spread
of
the physical parameters, performance or yield optimi-
zation can be achieved by means of small variations in the
physical parameters around the “standard” values of an
initial performance-optimized design. This leads to the
conclusion that, in order to achieve a practically mean-
ingful design, the PBDM must provide a highly accurate
estimate of the device sensitivity to physical parameters
variations with respect to a nominal condition to which
the model has been somehow fitted.
These remarks suggest that physics-based performance
or yield optimization, up to now camed out through an-
alytical models only, could also take advantage of more
complex and potentially more accurate physical models
run “off line” with respect to the circuit optimizer, when
FILICORI
et
al.:
PHYSICS-BASED ELECTRON DEVICE MODELLING
1335
these are able to provide a good estimate of the device
sensitivity. Such a possibility is offered by the efficient
sensitivity analysis techniques presented
in
Sec. IV.
Therefore, while the efforts towards achieving more and
more accurate analytical models are certainly worthwhile,
the use of the more computationally intensive numerical
PBDMs is possible for circuit analysis and yield optimi-
zation by the proper use of intermediate behavioural
modelling and device sensitivity analysis.
The paper is structured as follows. A comprehensive
review of the physics-based modelling of GaAs devices
for MMIC’s is presented in Sec. 11, with special emphasis
on MESFET’s. Section I11 covers a less conventional
topic, i.e., physics-based noise modelling, whose impor-
tance both in microwave device design and in physics-
based performanc prediction hardly needs to be stressed;
some recent developments introduced by the authors in
the domain of two-dimensional noise modelling of GaAs
FET’s are included. Section IV covers the problem of
physics-based
device sensitivity
analysis and also includes
some new material recently developed by the authors.
Section V deals with the issue of physics-based circuit
analysis carried out either directly through analytical
PBDM’s or indirectly through intermediate behavioral
models consisting either of large-signal equivalent cir-
cuits or of black-box mathematical models, for which
some innovative developments are presented. Finally Sec.
VI is devoted to a discussion of physics-based MMIC per-
formance and yield optimization by means of state-of-the-
art analytical PBDMs, and also to some possible devel-
opments concerning yield optimization through “off line”
multidimensional numerical PBDM’s.
11. PHYSICS-BASED MODELS
A.
Process Modelling
Process modelling is an important but critical step in
MMIC CAD. In fact, the practical characterization of the
GaAS process requires extensive measurements on a spe-
cific set of technological facilities; the resulting data can
be strongly process-dependent and have limited general
validity. An even more demanding task is the statistical
characterization of the physical parameters deriving from
a given process.
For
these reasons, efforts toward a com-
prehensive GaAs process modelling are comparatively
rare; an excellent example is found in the work by Anholt
et al.
[3],
[4],
[6]. From the statistical data reported in
[6]
it can be inferred that improvements in technology
have now made the standard deviation of the physical pa-
rameters (doping profiles, etch depths, and
so
on) reason-
ably low, i.e., of the order of less that
10%
(see Table I1
in [6]). This is important in view of physics-based yield
optimization, since whenever the technological uniform-
ity is poor, realistic yield improvement is more dependent
on progress in technology than on design centering. On
the other hand, a good process uniformity and repeatabil-
ity makes yield optimization through design centering
meaningful and worth doing.
B.
An
Overview
on
the Basic Semiconductor Device
Models
Most available physics-based models for GaAs FET’s
are based on the drift-diffusion picture of camer trans-
port, in which the carrier drift velocity
U
is a function of
the local electric
E
field through the static field-velocity
curve, and the diffusivity
D
follows the equilibrium Ein-
stein relationship. Since the device dimensions are typi-
cally much smaller than the operating wavelength, the
electric potential and the charge density can be related
through Poisson equation. For bipolar transport, the drift-
diffusion model reads:
(3)
where
n
is the electron density, p is the hole density,
$J
the electric potential,
=
-V$J
the electric field,
Ni
and
NA
the ionized acceptor and donor densities,
R
the net
recombination rate. The model becomes slightly more
complex in heterostructure FETs, since space-dependent
bandgaps and semiconductor affinities must be allowed
for.
The drift-diffusion model is already a heavy approxi-
mation when compared
to
other, more complete descrip-
tions of camer transport. Although quantum effects are
globally significant to the operation of many high-fre-
quency or optical devices, quantum models (i.e., the
Schrodinger equation in the effective mass approxima-
tion) can often be applied locally. A typical example is
provided by the high electron mobility transistor
(HEMT), in which carriers are mostly confined in a quan-
tized-system, the so-called two-dimensional electron gas
(2DEG). The sheet density of the
2DEG
can be separately
characterized from a quantum standpoint and the resulting
model can be easily interfaced to non-quantum transport
models.
Semiclassical
transport models deal with camers as
classical particles, whose motion properties (effective
mass and interactions with lattice impurities, phonons,
etc.) derive from quantum models. The fundamental
semiclassical model for semiconductor transport is the
Boltzmann equation
[61] which directly yields the time-
and space-dependent
momentum distribution function
of
carriers in the phase space, and therefore provides full
information on both low- and high-energy phenomena.
The only technique currently able to cope with this model
without resorting to drastic approximations is the Mon-
tecarlo simulation method [61], [81], which is still too
1336
IEEE TRANSACTIONS
ON
MICROWAVE THEORY AND TECHNIQUES. VOL.
40,
NO.
7.
JULY
1992
computationally intensive to enable device design and op-
timization, let alone circuit-oriented CAD. From Boltz-
mann equation the so-called
hydrodynamic
transport
models can be derived, whose unknowns are the
central
moments
of the carrier distribution, which correspond to
the average parameters (average density, average energy,
average momentum, and
so
forth) of the carriers, consid-
ered collectively as a
carrier gas.
Hydrodynamic models
are sets of partial differential equations which express, in
divergence form, the conservation of the central moments
of the carrier distributions [18], [55]; in the case of uni-
polar transport, a widely accepted choice leads to a set of
three equations for each equivalent minimum of the con-
duction band, corresponding to a particle continuity equa-
tion (the current continuity equation), an energy transport
equation, and a (vector) momentum transport equation.
Several simplifications have been proposed to reduce
the computational complexity of the full hydrodynamic
model. Firstly, by approximately averaging the transport
equations of all the equivalent minima, the
single electron
gas
transport model is obtained
[
181,
which has been re-
cently exploited for multidimensional device simulation
[37]. However, further approximation are often intro-
duced, mainly to avoid the explicit solution of the mo-
mentum transport equation. By neglecting space and time
variations in the momentum equation one obtains the
so-
called
energy transport models
(see e.g., [108]), which
can be further simplified by neglecting the kinetic vs. the
thermal electron energy
of
the carriers
(temperature
models,
see e.g., [29]), or the electron heat flow in the
energy transport equation
[
1081. Although the above ap-
proximations are meant to trade off accuracy for compu-
tational efficiency, the errors introduced thereby are dif-
ficult to control and the simplified hydrodynamic models
yield results which may be as different from each other as
from the drift-diffusion model [37]. Drift-diffusion models
can be finally considered as hydrodynamic models
in
which both the energy and the momentum equation are
approximated with their steady-state, space-independent
expressions. For a more detailed discussion the reader can
refer e.g., to [84].
The numerical treatment of hydrodynamic or drift-dif-
fusion device models requires discretization and solution
algorithms [103, 801. Discretization can be carried out
through finite-differences or finite-elements techniques by
means of special schemes, like the so-called Scharfetter-
Gummel scheme
[
1021. After discretization, the time-do-
main physical model becomes a large, sparse system
of
coupled ordinary non-linear differential equations whose
unknowns are, for instance, the charge density, average
energy and electric potential at the discretization nodes.
The solution step requires this system to be analyzed in
the several possible operating conditions of the device. In
the
dc problem
all time derivatives are set to zero and the
resulting nonlinear system is solved through Newton
linearization; in the
ac small-signal
problem, device anal-
ysis is better carried out in the frequency domain, by
means of numerical techniques analogous to those usually
adopted for small-signal circuit analysis. Finally, the
large-signal analysis
with periodic or arbitrary (transient)
excitation, requires the differential system to be solved
through time-stepping algorithms, since harmonic-bal-
ance analysis in the framework of numerical device sim-
ulation would be too computationally intensive. Thus,
large-signal multidimensional models have been mainly
exploited for transient simulation. For the purpose of
complete performance prediction, other less conventional
kinds of device analysis should also be considered, like
noise and parametric sensitivity analysis. These will be
separately discussed in Section I11 and
IV,
respectively.
C.
Quasi-Two Dimensional and Analytical Physics-
Based Models
Owing to their computational intensity, exact, multi-
dimensional implementations of transport models cannot
be directly included into circuit analysis and optimization
algorithms; however, proper approximations enable
greater computational efficiency in the analysis
of
specific
devices. In particular, the cross-field structure of micro-
wave FET’s, in which the channel current and the gate
control mechanism are orthogonal (see Fig. 2), suggests
an approximate spatial decoupling which is exploited in
the so-called
quasi-2D
implementations of transport
models. In most quasi-2D models the gate charge control
is treated according to a 1D quasi-equilibrium approxi-
mation along
y,
while the analysis of channel current is
reduced to a 1D continuity equation along
x.
The solution
of the two decoupled 1D models can either be numerical
or analytical; in its simplest form, the gate control model
is based on the depletion approximation and the channel
model is based on a two-zone (ohmic and velocity-satu-
rated) channel approximation (Fig. 2), which ultimately
reduces, for constant mobility, to Shockley’s JFET model.
According to the different possible levels of approxima-
tion made, several classes of models have been derived,
with widely different complexity and accuracy. Repre-
sentative examples are:
1. Quasi-2D energy-transport models [25],
[
1091, [88],
based on an approximate 1D version of the energy and
momentum transport equations; the charge control mech-
anism is either analytical or implicit. The computational
intensity is not negligible, since the 1D solution for the
transport model is performed numerically.
2. Quasi-2D models with numerical charge control and
two-zone channel approximation
[
1011, [91]. Such models
use an accurate quasi-equilibrium numerical model for
charge control
[
1011, [91], which can provide detailed in-
sight into the static behavior of substrate impurities and
traps. Since the charge control model can be separately
solved and the results stored as a look-up table, the com-
putational burden is limited.
3.
Quasi-2D models with analytical charge control and
two-zone channel model. Since charge control is based on
the abrupt depletion approximation
(n
=
0
under the gate
and
n
=
No
(
y)
in the conducting channel), which is poor
FILICORI
et
al.
:
PHYSICS-BASED ELECTRON
DEVICE
MODELLING
1337
Source
Ca
te
Drab
adopted [92]. While analytical 2D models [68] seem to
Fig.
2.
FET
structure
and
cross-field
control;
two-region
channel
approx-
imation.
for rapidly varying (e.g., implanted) profiles, transition
functions have been introduced in an attempt to better ap-
proximate
n(y)
[90]. Examples of early models allowing
for doping profiles
of
increasing generality
(constant,
Gaussian, arbitrary) are those proposed by Puce1
et
al.
[93],
Shur [105]. de Santis
[34]
and Higgins [56]; recent
refinements allow the treatment of complex velocity-field
curves [27]. State-of-the-art examples of MESFET models
are the GATES simulator
[4]
and the SIMTEC simulator
[90],
which
also
provides an empirical treatment of non-
stationary effects through a gate-length dependent satu-
ration velocity. Short-gate geometrical and non-stationary
effects were
also
introducted in [57].
4.
Analytical 2D model. A fully 2D approximate ana-
lytical treatment of the drift-diffusion model was first sug-
gested in 1976 by Yamaguchi and Kodera
[
1171, who pro-
pose an accurate parametrized approximation of the
channel mobile charge, based on results from 2D simu-
lation. The potential is derived as the superposition of
a
Laplacian component (obtained through Fourier expan-
sion) and a Poissonian component (evaluated by neglect-
ing the potential curvature along the channel). From the
electric field and the approximate charge distribution the
current density can finally be obtained. Yamaguchi's
treatment was extended to buffered devices by Bonjour
et
al.
[20]. In 1981 Madjar and Rosenbaum [76] proposed a
full large-signal analytical model obtained by integrating
a dc Yamaguchi-like model with the quasi-static capaci-
tance matrix derived from a self-consistent charge distri-
bution. A state-of-the-art example of an analytical 2D
model is the
TEFLON
large-signal MESFET simulator
developed by Trew
et
al.
[68].
The classification attempted above is not exhaustive and
only aims at outlining some basic trends in quasi-2D FET
modelling.
HEMT
models have been omitted for brevity,
since the quantum effects included in the charge control
mechanism bring about further complexities and lead to
an impressive variety of possible analytical models (see
[32]
for an overview).
Analytical quasi-2D models are not always completely
suitable for describing state-of-the-art MESFET's, since
the two-zone channel approximation becomes unsatisfac-
tory in the presence
of
geometrical short-gate effects (i.e.,
when
L/u
Z
5,
where
L
is the gate length and
a
the
equivalent channel thickness); this leads to
a
poor esti-
mate
of
the output resistance, unless special models are
provide
a
satisfactory model for the dc characteristics,
some problems are still open in the modelling of dynamic
(small- or large-signal) behavior. In fact, the small-signal
capacitance model is based on quasi-static approxima-
tions, and
ad
hoc
assumptions must be introduced to es-
timate those small-signal elements which cannot be de-
rived from dc current-voltage or charge-voltage
characteristics
(e.g.,
the intrinsic resistance
R,
or the gate
delay
7).
Moreover, no physics-based description is avail-
able for the static or dynamic behavior of substrate and
surface trapping effects, which play an important role in
the low-frequency dispersion of the transconductance and
output conductance, although several empirical or pa-
rametrized models have been proposed
[
151,
[70],
[72].
From the standpoint of computational intensity, quasi-
2D numerical models are typically one order of magni-
tude faster than full 2D models, which typically require a
few minutes CPU per working point on a medium-size
workstation. This, however, is not enough to directly in-
clude them in circuit simulators. On the other hand, an-
alytical PBDM's, while being slower than the behavioral
models to quasi-2D and analytical models). The choice of
ent the only physics-based models fast enough to be di-
rectly incorporated into circuit simulators.
D.
Discussion
The above overview has outlined physical models of
decreasing intrinsic complexity (from the Boltzmann
equation down to drift-diffusion models) and then of de-
creasingly complex implementation (from 2D numerical
models to quasi-2D and analytical models). The choice of
a
simpler model or implementation is often considered as
a way to trade off accuracy in favour of computational
efficiency, but,
as
a
matter of fact, several examples can
be found in literature of very good matching between ex-
tremely simple models and experiments; on the other
hand, complex models sometimes seem to yield predic-
tions which are quantitatively inaccurate when compared
to experiments.
In fact, most of the microscopic information provided
by complex models may be redundant or second-order in
modelling the operation of a particular device. For in-
stance, high-energy carrier distribution tails in MES-
FET's,
as
accurately modelled by Boltzmann-Monte
Carlo models, are only relevant to the breakdown behav-
iour of the device. This leads to the rather obvious con-
clusion that only those features which are relevant to the
operation
of
the device should be accurately modelled.
A first point is the need to include into the model non-
stationary transport effects. The inadequacy of the drift-
diffusion approximation to model submicron devices has
been discussed in several papers, see e.g., [108], [37]
among the most recent ones. According to
[
1081
the main
effects of non-stationary transport are:
1)
the equivalent
saturation velocity of the carriers increases due to spatial
overshoot effects; 2) electron heating makes the electron