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