
Citation 
 Permanent Link:
 https://ufdc.ufl.edu/UF00098282/00001
Material Information
 Title:
 Equivalentcircuit modeling of the largesignal transient response of fourterminal MOS field effect transistors /
 Creator:
 Arreola, José Ignacio, 1950
 Publication Date:
 1978
 Copyright Date:
 1978
 Language:
 English
 Physical Description:
 vii, 132 leaves : ill. ; 28 cm.
Subjects
 Subjects / Keywords:
 Approximation ( jstor )
Drains ( jstor ) Electric current ( jstor ) Electric potential ( jstor ) Electronics ( jstor ) Electrons ( jstor ) Engineering ( jstor ) Equivalent circuits ( jstor ) Modeling ( jstor ) Narrative devices ( jstor ) Dissertations, Academic  Electrical Engineering  UF Electrical Engineering thesis Ph. D Fieldeffect transistors ( lcsh )
 Genre:
 bibliography ( marcgt )
nonfiction ( marcgt )
Notes
 Thesis:
 ThesisUniversity of Florida.
 Bibliography:
 Bibliography: leaves 128131.
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by José Ignacio Arreola.
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 University of Florida
 Holding Location:
 University of Florida
 Rights Management:
 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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04075260 ( OCLC ) AAX4400 ( NOTIS )

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Full Text 
EQUIVALENTCIRCUIT MODELING OF THE
LARGESIGNAL TRANSIENT RESPONSE
OF FOURTERMINAL MOS FIELD EFFECT TRANSISTORS
By
JOSE IGNACIO ARREOLA
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1978
to
IGNACIO
and
CELIA,
my pvaent
ACKNOWLEDGMENTS
I am deeply indebted to Prof. Fredrik A. Lindholm
for his contribution to this work and for his continued
guidance, support and encouragement. I also wish to
thank Dr. D. R. MacQuigg for his help in doing experimental
measurements and for many interesting discussions. I would
like to express my appreciation to Prof. A. D. Sutherland
for allowing me to study the results of his twodimen
sional calculations which broadened my understanding of the
MOSFET.
The financial support of Consejo Nacional de Ciencia
y Tecnologia (Mexico) throughout this work is gratefully
acknowledged. I must also thank Mrs. Vita Zamorano for her
careful typing of the manuscript. Finally, I owe a special
debt of gratidude to my wife, Patricia, for her forbearance
and encouragement.
iii
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS . . . . . . . . . . iii
ABSTRACT . . .. . . . . . . . . vii
CHAPTER
I INTRODUCTION . . . . . . . . 1
II A NONLINEAR INDEFINITE ADMITTANCE MATRIX FOR
MODELING ELECTRONIC DEVICES . . . . 4
2.1 Introduction . . . . . . 4
2.2 Indefinite Admittance Matrix . . . 5
2.3 Extension for Nonlinear Electronic Devices 7
2.4 Conclusions . . .. . . . 13
III EQUIVALENTCIRCUIT MODEL FOR THE FOURTERMINAL
MOSFET . . . . . . . . . 16
3.1 Examples of Engineering Needs for a Model
for the LargeSignal Transient Response 16
3.1.1 Reasons for the Poor Modeling of
the Transient Substrate Current by
Existing MOSFET Models . . .. 17
3.2 Problems Involved in Modeling of Four
Terminals Devices . . . . .... 19
3.3 EquivalentCircuit for the Intrinsic
MOSFET .. . . . . . . . 23
3.3.1 Transport Current . .. ... 23
3.3.2 Charging Currents . . . .. 24
3.4 Special Considerations . . .. . 29
3.5 Modeling of the Extrinsic Components .34
3.6 Relation to Existing Models . . ... .35
IV STEADYSTATE MOSFET THEORY MERGING WEAK,
MODERATE AND STRONG INVERSION . .. . 38
4.1 Introduction . .. . . . . 38
4.2 Fundamentals . . . . . . .. 40
4.2.1 Drain Current . . . . . 40
4.2.2 Charge Components . . . .. 42
4.2.3 Surface Potential . . . . 47
CHAPTER
IV (continued)
4.3 Drain Current and Charge Components in a
Model Merging Weak, Moderate and Strong
Inversion . . . . . . . . 51
4.3.1 Drain Current . . . 51
4.3.2 Charge Components . . . .. 57
4.3.3 Limits for the Strong, Weak, and
Moderately Inverted Portions of
the Channel . . . .. .. . 59
4.4 Results and Evaluation of the Model . 62
4.5 Conclusions . . . . . . . 74
V FUNCTIONAL DEPENDENCIES FOR THE ELEMENTS IN THE
FOURTERIINAL EQUIVALENTCIRCUIT . . .. 76
5.1
5.2
5.3
Introduction . . . . . . .
SourceDrain Current Source . . . .
Capacitances . . . . . . .
5.3.1 Expression for the Capacitances
5.3.2 Physical Interpretation of the
Results for the Capacitances .
5.3.3 An Engineering Approximation for
the Functional Dependencies of the
Intrinsic Substrate Capacitances
C and CDB
SB. DB
5.3.4 Engineering Importance of the In
trinsic Substrate Capacitances CS
and CDB . . . . .
5.4 Transcapacitors . . . . .
5.4.1 Expressions for the Trans
capacitors . . . . .
5.4.2 Engineering Importance of the
Transcapacitance Elements .
5.4.3 Transcapacitances in a Three
Terminal EquivalentCircuit
5.5 Conclusions . . . . . . .
VI SCOPE AND FUTURE WORK . . . . .
APPENDIX
A PROPERTIES OF QUASIFERMI POTENTIALS . .
B APPROXIMATED EXPRESSION FOR THE DIFFUSION/
DRIFT RATIO IN THE MOSFET . . . .
C COMPUTER SUBPROGRAM TO CALCULATE THE VALUE
THE ELEMENTS IN THE EQUIVALENTCIRCUIT .
97
. . 98
S 98
S 00
S 106
S. 109
. 111
S. 115
S. 119
OF
. 123
Page
Page
LIST OF REFERENCES . . . .
BIOGRAPHICAL SKETCH . . .
. 128
. 132
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida
in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy
EQUIVALENTCIRCUIT MODELING OF THE
LARGESIGNAL TRANSIENT RESPONSE
OF FOURTERMINAL MOS FIELD EFFECT TRANSISTORS
By
Jose Ignacio Arreola
March 1978
Chairman: Fredrik A. Lindholm
Major Department: Electrical Engineering
An approach is proposed that yields equivalentcircuit
models for the largesignal transient response for all
electronic devices described by chargecontrol. The ap
proach is applied to derive an improved equivalentcircuit
model for the fourterminal MOSFET. It is suggested that
the model proposed gives a better description of the physics
internal to the device than was previously available.
A static characterization of current and charges in
the MOSFET is also proposed that unifies the descriptions
of the weak, moderate and strong inversion modes of opera
tion. Predictions of this characterization agree better with
experimental results than previous work of similar complexity.
The static characterization of current and charges is used
to derive functional dependencies for the equivalentcircuit
components in terms of applied voltages and physical makeup
of the MOSFET.
vii
CHAPTER I
INTRODUCTION
Computer simulations of MOSFET digital circuits can
disagree severely with measured performance. A particular
case of such a disagreement, which results in suboptimal
circuit design, is the poor simulation of transient cur
rents flowing in a substrate terminal of MOS field effect
transistors [1]. The sources of such disagreements are
either in the computer programs in use or in the inadequacies
of existing largesignal equivalentcircuit models for the
fourterminal MOSFET.
The purpose of this dissertation is to derive an im
proved equivalentcircuit model for the fourterminal MOSFET.
Improvements are made in the following aspects of the
equivalentcircuit model:
(a) the representation of capacitive effects in a four
terminal device;
(b) the characterization of the dc steadystate
currents and charges;
(c) the inclusion, in principle, of two and three
dimensional effects present, for example, in
shortchannel MOSFETs.
As will be seen, all of these improvements are inter
related and result from basing the derivation of the
equivalentcircuit model on the internal physics that deter
mines the operation of the MOSFET.
We begin in Chapter III by proposing an approach that
yields equivalentcircuit models for the largesignal
transient response of all electronic devices described by
charge control [24]. The relation of this approach to
the indefinite admittance matrix of circuit theory offers
advantages in the modeling of devices having more than three
terminals.
Chapter III starts by discussing the problems arising
from the fourterminal nature of the MOSFET. Such problems
were apparently not previously recognized. For the in
trinsic part of the device (see Fig. 3.1), we apply the
systematic approach developed in Chapter II. This approach,
whose power is emphasized because of the four terminals of
the MOSFET, yields a general description of the device that
offers improvements (a) and (c) listed earlier.
To define fully the equivalentcircuit model of Chapter
III, one needs a suitable description of the dc steadystate
behavior. Extensive work has been done in the past to
characterize operation in the dc steadystate; however, none
of this work is completely suitable for the purposes of
equivalentcircuit modeling. In Chapter IV, a new model for
the dc steadystate behavior is derived that unifies the
description of the full range of operation of the device 
from weak to strong inversion and from cutoff to saturation.
The model avoids discontinuities in the characteristics
present in all previous characterizations of similar com
plexity, and shows good agreement with experimental results.
The new model also improves the characterization of the
charges in the device.
In Chapter V we derive, using the results of Chapter IV,
the functional dependence of each circuit element in the
equivalentcircuit developed in Chapter III. In Chapter V
we also assess the engineering importance of the improvements
introduced in the equivalentcircuit model for the MOSFET and
propose possible simplifications of the model.
Chapter VI treats possibilities for future research.
CHAPTER II
A NONLINEAR INDEFINITE ADMITTANCE MATRIX
FOR MODELING ELECTRONIC DEVICES
2.1 Introduction
This chapter describes a new approach for developing
equivalentcircuit models of electronic devices. The
models developed by this approach represent the large
signal (hence nonlinear) response to transient excitation.
The approach applies to all devices whose operation is
described by the principles of charge control [2z], in
cluding, therefore, field effect transistors of various
kinds, bipolar transistors, and certain electron tubes.
The models yielded by the approach are compact, com
posed of few circuit elements. As a result of their com
pactness, the models are meant to be useful in the computer
aided analysis of electronic circuits. This intended use
contrasts with that intended for equivalentcircuit models
[5] containing many circuit elements, which pertain chiefly
to detailed studies of the physics underlying electronic
device behavior.
The approach to be described applies independently of
the number of device terminals. Indeed, the greater that
number, the more the power of the approach is disclosed.
The approach applies also independently of multidimensional
spatial dependence that may be present in the boundary
value problem describing the device. This generality is
needed, for example, in modeling the MOS field effect tran
sistor (MOSFET), because the substrate terminal constitutes
a fourth terminal through which sizable transient currents
flow in some circuit applications, and because shortchannel
devices give rise to multidimensional effects.
Models of fourterminal devices [6,7] and models that
include multidimensional effects [8] have been proposed
earlier. But this previous work has not focused on laying
down systematic procedures for developing models, which is
the aim of this chapter.
Systematic procedures exist for modeling the linear
response of multiterminal circuits subjected to smallsignal
excitation. These procedures are linked to the indefinite
admittance matrix (IAM), which we first shall review and
then exploit to model the nonlinear response of multi
terminal electronic devices to largesignal excitation.
2.2 Indefinite Admittance Matrix
Consider a lumped electrical network which has n ter
minals. Let an additional external node be the common
reference. From the standpoint of its behavior at the ter
minals, the network, if linear, may be described by a set
of equations as follows:
I = yV (2.1)
The required linearity is assured for any network operating
under smallsignal conditions. The matrix elements of y are
I.
yjk V (2.2)
k vi=0, i k
where I and V correspond to the current and voltage at the
terminals.
The matrix y defined in (2.1) and (2.2) is called the
indefinite admittance matrix [9,10], and its elements satisfy
the following property imposed by Kirchhoff's laws
I Yjk = Yjk = 0 (2.3)
j k
that is, the elements in any row or any column sum to zero.
As will be seen, our development of largesignal models
for electronic devices will make use of two special cases
of the IAM. In the first case, the matrix y is symmetric
and has one of the following forms:
d
y = a = b y = c/dt (2.4)
dt
Here a, b, and c are real symmetric matrices, and each matrix
element corresponds to a single lumped resistor or capacitor
or inductor connected between each pair of the n terminals.
In the second case, the matrix y is nonsymmetric, but is
the sum of two indefinite admittance matrices: a symmetric
matrix, like (2.4), and a residual nonsymmetric matrix, each
element of which corresponds to a controlled current source
placed between each pair of terminals. In this second case,
then, the circuit representation of the IAM results from
connecting the network corresponding to the symmetric matrix
in parallel with that corresponding to the nonsymmetric
matrix. In general, summing of indefinite admittance
matrices corresponds to connecting their circuit representa
tions in parallel.
2.3 Extension for Nonlinear Electronic Devices
Consider an electronic device having n terminals. The
modeling begins by specifying the physical mechanisms
relevant to the operation of the device. For many devices,
only three such mechanisms, at most, are relevant: the
transport of charged carriers between terminals; the net
recombination of charged carriers within the device; the
accumulation of these carriers within the device. Thus the
current i flowing at any terminal J is the sum of three
components: a transport current (i ) T a recombination cur
rent (i )R, and a charging current (i )C. That is
i = (i )T + (i )R + (iJ)C (2.5)
We now characterize these components.
The transport mechanism consists of the injection of a
charged carrier in one terminal, followed by its transport
across the device until it reaches any of the other terminals,
where it recombines at a surface with a carrier of opposite
charge. The recombination mechanism differs from the
transport mechanism only in that the carriers recombine
within the bulk of the device instead of at the terminals.
Therefore, both mechanisms can be characterized by the same
form
(i )T, = (i K)T R (2.6)
K J
Here iJK represents the current due to the charged carriers
injected from terminal J, which recombine, at a surface or
in the bulk, with oppositecharged carriers injected from
terminal K. From this characterization, it follows that
(iJK)T,R satisfies the following properties:
JK = JJ = 0 (2.7)
These properties allow transport and net recombination to be
represented by controlled current sources connected between
pairs of terminals. The value of the current source between
terminals J and K is iJK.
The last mechanism to be considered is the accumulation
of mobile carriers within the device, which requires the
charging current
dqj
(ij)c = T (2.8)
As Fig. 2.1 illustrates, dqJ is the part of the total charge
accumulated within the device in time dt that is supplied
i = (iJ) T,R + (i)C
(ij)c
dqj
dt
Fig. 2.1 The charging current (i ) at terminal J
produces the accumulated charge dq .
from terminal J. The charge accumulation expressed in (2.8)
is a mechanism basic to any electronic device that operates
by charge control [24].
Now, using (2.6) and (2.8), we may rewrite (2.5) as
dqj
ij = (i K) + (2 9)
K~ J JK T,R dt
Although (2.9) is valid, it does not correspond to a con
venient network. To get a convenient network representation,
we apply one additional constraint which costs small loss in
generality in that it holds for all chargecontrol devices
[24]. We apply the constraint that the overall device
under study is charge neutral. Or, more exactly and less
demanding, we assume the device accumulates no net overall
charge as time passes. This constraint of overall charge
neutrality requires a communication of the flux lines among
the terminals to occur that maintains charge neutrality by
coulomb forces and by drift and diffusion currents. The
requisite overall neutrality may result either from
neutrality occurring at each macroscopic point, as in a
transistor base, or from a balancing of charges that are
separated, as on the gate and in the channel of a MOSFET.
As a result of overall neutrality, the current at any
terminal J becomes the sum of the currents flowing out of all
of the other terminals
iJ = i (2.10)
KJK
This global counterpart of the Kirchhoff currentnode law
implies for the charging currents of (2.8) that
(i )c (iK)C (2.11)
KXJ
which means that a charging current entering one terminal
flows, in its entirity, out of all of the other terminals.
Hence, as is true also for the transport and recombination
mechanisms, charge accumulation can be represented by a
controlled current source connected between each pair of
terminals.
For a model to be useful in circuit analysis, the
elements of the model must all be specified as functions of
the terminal currents and voltages. To do this, we now make
use of the principles of charge control [24] and of the
closely allied quasistatic approximation [6,7,11].
For the transport and recombination mechanisms, charge
control gives directly
(iJK) T,R = JK /tK. (2.12)
Here qJK is the charge of the carriers that contribute to
the current flowing between terminals J and K. The recombina
tion time tJK is the time constant associated with that cur
rent: a transit time if the mechanism being described is
transport, a lifetime if it is recombination. Then, to
produce the desired functional dependence, a quasistatic
approximation [6,7,11] is used that specifies each
(iJK)T,R as a function of the instantaneous voltages at
the device terminals.
This characterization of (iJK)T,R, combined with the
properties expressed in (2.7), can be manipulated to de
scribe transport and recombination by an IAM, like a in
(2.4). Because iK = iKJ the matrix is symmetric.
There are two network representations of transport and
recombination described by this matrix. As noted before,
just below (2.7), one of these consists of controlled
current sources connected between pairs of terminals.
Another network representation consists entirely of non
linear resistors, RJK = (V VK)/iJK.
Similar procedures are applied to model charge ac
cumulation. To the charging current defined in (2.8) a
quasistatic approximation is applied [6,7,11],
specifying the functional dependence of qJ on the terminal
voltages and enabling thereby the employment of the chain
rule of differentiation. The resulting characterization of
(i )C describes charge accumulation by a matrix that has
the form of b in (2.4), a matrix whose elements are
aqj
bJK = V (2.13)
bJK V K dVO=0, IK
Matrix b also satisfies the key properties of the indefinite
admittance matrix that are given in (2.3). For a general
nterminal electronic device, this matrix describing charge
accumulation is nonsymmetric, and is therefore the sum of a
symmetric and a residual nonsymmetric part. The symmetric
part corresponds to an allcapacitor network; the network
representation of the residual nonsymmetric matrix consists
of controlled current sources.
2.4 Conclusions
From the properties of the IAM it follows that the
threebranch circuit of Fig. 2.2 serves as a building block
for model generation. Connecting a circuit of this form
between each terminal pair yields the general network
representation for an nterminal electronic device. For
any particular device of interest, certain of the circuit
elements may vanish. In a MOSFET, for instance, no trans
port or recombination currents flow to the gate, and the
corresponding circuit elements will be absent.
Any equivalentcircuit model generated by this approach
can be regarded in two ways: either as a product of the
building block of Fig. 2.2 or as a circuit described by a
matrix which obeys the key properties of the IAM. Descrip
tion by the IAM treats all terminals equally in that none is
singled out as the reference node; the advantages of this will
show up plainly in the modeling of a fourterminal device,
such as the MOSFET.
From Fig. 2.2 notice that the mobile charge accumulation
within a general nterminal electronic device is not rep
resented by the flow of displacement currents in an all
capacitor model. The residual nonsymmetric matrix, and
'q J= J/t K
dv
3 qK/ ) dtK
(8q /vK qK/VJ) dt
Fig. 2.2 General equivalentcircuit between each pair
of terminals of an nterminal electronic device.
the corresponding transcapacitance current source of Fig.
2.2, provides the needed correction. This correction has
practical engineering consequences in certain MOSFET cir
cuits although a discussion of that is postponed for a
later chapter.
To use the approach set forth here in modeling any
particular device requires that the static dependence on
the terminal voltages be specified for the currents and
charges defined in (2.12) and (2.13). This requires that
a physical model for the device be chosen to describe the
dc steady state. For the MOSFET this has been done, and
the corresponding equivalentcircuit model is derived in
the following chapters.
CHAPTER III
EQUIVALENTCIRCUIT MODEL FOR
THE FOURTERMINAL MOSFET
The main contribution of this chapter is the deriva
tion of an equivalentcircuit model for the fourterminal
MOSFET by use of the method described in Chapter II.
The resulting model is intended to represent with good
accuracy the largesignal transient currents flowing through
each of the four terminals of the device, including the
substrate terminal.
3.1 Examples of Engineering Needs for a Model
for the LargeSignal Transient Response
In many digital integratedcircuit applications of the
MOSFET, the substrate terminal of each device is connected
to a power supply. This connection serves at least two pur
poses: it provides a means to control the threshold voltage
of the device, and it enables a good layout of the circuit
[12,13]. In a largescale integrated circuit, the large
transient current flowing through a power supply can result
in poor voltage regulation and poor circuit performance un
less both the circuit and the power supply are properly de
signed. An optimum design of a circuit will provide the
maximum density of components on the chip consistent with
the requirement that the voltage regulation of each power
supply remains acceptable.
To design circuits using computer aids therefore requires
that one has available a set of equivalentcircuit models
for the MOSFET that adequately represent the transient
currents flowing through the terminals in response to large
signal excitation of the devices. According to engineers
involved in such designs, such models are not now available
[1]. This absence of accurate models forces the engineer
to suboptimal designs, by which we mean less densely packed
circuits than those that could be designed if accurate enough
device models were available.
3.1.1 Reasons for the Poor Modeling of the Transient Sub
strate Current by Existing MOSFET Models
The substrate current during transients arises from
capacitive effects in two regions of the device (Fig. 3.1):
the depletion region around the source and drain islands
(extrinsic substrate capacitances); and the depletion region
underneath the inversion channel (intrinsic substrate
capacitance). In general, however, the substrate current
is modeled as arising only from the pn junction (extrinsic)
capacitances around the source and drain islands. These
capacitances have the form [13]:
C.
S= o (3.1)
j [ VB n
18
GATE
SOURCE
N+
x
DRAIN
__
N+
intrinsic region
I
extrinsic region
SUBSTRATE
Fig. 3.1 An nchannel enhancement MOSFET divided into
intrinsic and extrinsic parts.
IK_ ........... I
where V is the applied junction voltage, dB is the built
in potential and n is an exponential factor. The maximum
value of these capacitances, given by Cjo, is estimated
for typical doping concentrations to be in the order of
108 F/cm2 [13]. As we shall see in Chapter V this is also
the order of magnitude of the intrinsic substrate
capacitances. Because the area of the channel and the
area of the source and drain islands are in many circuits
comparable, the inclusion of the intrinsic capacitive
effects to model the substrate current is essential. More
over, in new fabrication technologies such as silicon on
saphire [14] considerable reduction of the substrate ex
trinsic capacitances can be achieved. These reductions
can also be achieved by employing special circuit techniques
in the conventional technology [15]. In both these cases,
the intrinsic effects are dominant and must be included in
the modeling.
3.2 Problems Involved in Modeling
of FourTerminal Devices
The modeling of the intrinsic effects of the four
terminal MOSFET presents special problems not previously
considered. To lead into these problems, consider first a
twoterminal device. As is shown in Fig. 3.2(a), we apply
a small voltage dV. The figure illustrates that there is
only a single path of communication between the terminals.
That is, there is only one way the flux lines can link be
(a)
3
dQ
11 2
dQ 4
dQ1 / I do2
 I\7~
Fig. 3.2 Illustration of the paths of communication
between terminals in a twoand fourterminal
device.
device.
tween the terminals and thus there is no uniqueness in
the charge that flows at the terminals. The charge that
flows at each terminal is dQ. A nonuniqueness does occur,
however, in devices with more terminals. Consider now a
fourterminal device. From Fig. 3.2(b) one sees that there
are six paths of communication of the flux lines among the
terminals in a general fourterminal device. Thus, suppose
one applies a small voltage between any two terminals while
appropriately terminating the other terminals so that
charge can flow through them. Then one must account properly
for the apportionment of the charges amongst the terminals.
Of the total charge dQ that flows, what will be the charges
dQ1, dQ2, dQ3 and dQ4 flowing at each of the four terminals?
There is a second related problem. One way of seeing
this problem is to suppose that within the box of Fig. 3.1(b),
for the time being, is an allcapacitor network. Then apply
a small voltage between terminals 1 and 3,having shorted the
other terminals to an arbitrary reference. In response, a
certain amount of charge flows at terminal 4. Now inter
change the roles of terminals 3 and 4. That is, apply the
small voltage at terminal 4 and measure the amount of charge
flowing past terminal 3. The result of this experiment is
that one finds exactly the same amount of charge as before.
That is a property of a reciprocal network, of which an all
capacitor configuration is an example [16].
Now if one does the same experiment with a MOSFET one
finds that this reciprocity does not apply, as we shall prove
in Section 5.4. The reason is that terminal 3 represents
the gate and terminal 4 represents the substrate; and the
gate and substrate are highly different physical struc
tures. This asymmetry in physical structure introduces
a nonreciprocity in the network properties not present in
an allcapacitor network. To account for this asymmetry,
therefore, one should expect that the network representa
tion for a MOSFET must contain elements describing mobile
charge accumulations in addition to capacitors.
To manage these problems one requires a systematic
approach. In Chapter II we have developed a methodology
that permits one to obtain a lumped network representation
of multiterminal electronic devices obeying the principles
of charge control whose largesignal transient behavior
depends on three physical mechanisms: mobile charge trans
port, net recombination within the device and mobile charge
accumulation. The result is the equivalentcircuit of
Fig. 2.2, which applies between any pair of terminals and
is the basic building block from which an equivalentcircuit
can be constructed for the overall multiterminal device.
The currents representing transport and net recombination
flow in the current source iJK. The charging current re
presenting mobile charge accumulation flow through the
capacitor CJK = qK /9Vj and through the controlled current
source characterized by dJK = 9qj/ vK qK/D3Vj'
To apply this methodology to the MOSFET, one needs
only to describe the components of charge accumulation dqJ
J
in each region and the transport and recombination flow in
terms of the physics underlying the device behavior. We
will now apply this methodology to the MOSFET.
3.3 EquivalentCircuit for the Intrinsic MOSFET
For concreteness, consider the enhancementmode n
channel MOSFET illustrated in Fig. 3.1. A central idea in
the equivalentcircuit modeling is to resolve the electronic
device under study into two parts [11]: an intrinsic part
where the basic mechanisms responsible for the operation of
the device occur, and an extrinsic part which depends on
the details of the device structure. For the particular
MOSFET under consideration this is done in Fig. 3.1.
The behavior of the intrinsic region in the MOSFET is
described by charge control [24], and thus an equivalent
circuit of its operation can be obtained by applying the
methodology described in Chapter II.
3.3.1 Transport Current
At normal operating voltages and temperatures the
leakage current in the insulated gate is negligible and the
recombination/generation rate in the channel and in the sub
strate can be neglected. Charge transport occurs, there
fore, only along the highly conductive inversion channel
induced at the semiconductor surface. This transport
mechanism is represented in the equivalent circuit as a con
trolled current source iSD connected between source and
drain. Its explicit functional dependence in terms of the
physical makeup and the terminal voltages is obtained by
using a quasistatic approximation to extrapolate the
steadystate functional dependence of the drain current
ID. This will be considered in Chapter V.
3.3.2 Charging Currents
If we neglect recombination and generation, the cur
rent flowing in the substrate terminal iB is solely a
charging current, that is, current that changes the number
of holes and electrons stored in the intrinsic device.
Thus, if during time dt a change dqB occurs in the hole
charge stored in the substrate, then
dqB
i dt (3.2)
B dt
Similarly, neglecting any leakage current in the
insulator, the current flowing in the gate iG is only a
charging current. If this current changes the charge of
the metal gate by dqG in time dt, then
dqG
i (3.3)
G dt
The current flowing at the source terminal consists of
two components. The first component changes the electron
charge stored in the channel by an amount dqS in time dt.
The second component arises from electrons that, flowing in
from the source, pass through the channel and then out of
the drain terminal. Thus,
dqS
S dt + SD
Similarly, on the drain side,
dq
D d
D dt
ISD
(3.4)
(3.5)
The total change of charge in the inversion channel dqN is
then
dqS dqD dqN
S + iD = + dt dt (3.6)
If we apply a quasistatic approximation [6,11] and
then use the chain rule of differentiation, equations (3.2)
through (3.5) can be expressed in the following matrix form:
aqs
DvS
DqD
DV
DvS
aqB
vS
Dqs
VD
D
aVD
q G
avD
DqB
DvD
q9S
VG
qD
DVG
qG
DvG
DqB
DVG
aVB
DqG
DvB
DqB
DvB
SD
SD
0
0
o
o)
(3.7)
Here the dot notation designates time derivatives.
By applying the constraint that the overall intrinsic
device is charge neutral, one can prove as is done in
Chapter II [17,18] that the first matrix in (3.7) satisfies
the properties of the indefinite admittance matrix of net
work theory [9]. That is, the sum of all the elements in
any row or column is equal to zero.
The matrix description of (3.7) together with the
building block of Fig. 2.2 yields, therefore, the general
largesignal equivalentcircuit for the intrinsic four
terminal MOSFET. This network representation is shown in
Fig. 3.3 and its elements are defined in Table 1.
Elements in addition to capacitors that represent
charging currents appear in the circuit of Fig. 3.3. These
transcapacitors would be zero only if the matrix in (3.7)
were symmetric. That is, if dqJ/3vK = qK /3vJ forall J
and K. The physical structure of the MOSFET, however, is
nonsymmetric and hence one should expect that the elements
dJK are in general nonzero. This is the case, indeed, as
it will be shown in Chapter IV where we calculate the func
tional dependencies of these elements in terms of the ap
plied voltages and the device makeup.
The transcapacitive elements in the network representa
tion can be also seen as related to error terms yielded by
an ideal allcapacitor model. In this sense, we will study
and assess their importance in Chapter V.
In the circuit of Fig. 3.3, the capacitive effects be
tween source and drain are represented by a capacitor CSD
and a controlled current source characterized by dSD In
the theory of operation of the MOSFET based on the gradual
case [19], it has been shown [20] that there are no capaci
tive effects between source and drain. In this work, we
will consider this to be the case and therefore we will
assume 3qS/3vD = 3qD/vS = 0, implying
27
Iz m
41
> 4
4 I,
C H
roo
P
(9 0
c~ co
co
U) U)o
H P c
m I c~ Cd
u(
u I a d
3 0
(4 a Cd
rCd
U) (5I
(5 IIc
U) U)
Definitions for the elements of the general
equivalentcircuit for the MOSFET.
CAPACITANCES
S 9G
avS
D9B
C 
SB vS
3qD
SDD
CSD =v
S
DqG
C 
DG DvD
cqB
DB DvD
8qB
CGB G
G
TRANSCAPACITANCES
d 
SG D G
SqS
SB vB
q9S
SD D
SqD
DG v
G
qG
vvS
B
8vS
^D
dD
d 
DB aVB
SqG
dGB B
B
Table 1
aqG
vD
qB
3vD
avG
^G
CSG
CSD = dSD = (3.8)
In more detailed characterizations of the device 
for example, the ones including channel length modulation
[21] and twodimensional effects in short channel devices
[8] the drain voltage directly influences the charging
of the channel and capacitive effects between source and
drain as modeled by Fig. 3.3 may need to be included.
3.4 Special Considerations
For the equivalent circuit model in Fig. 3.3 to be
useful in circuit analysis we require that all the elements,
current sources and capacitors, must be specified as func
tions of the terminal current and voltages. In doing this,
as indicated in Chapter II, we will use the quasistatic
approximation [6,111, which is based on the steady state
operation of the MOSFET. A particular detailed model for
steadystate operation is considered in Chapter IV and the
functional dependencies for the elements of Table 1 will
be derived in Chapter V from this model. However, before
approaching these problems, we must give special considera
tion to two charge components that are not described in the
conventional steadystate characterization of the device:
the contributions from the source and the drain, dqS and dqD,
to the total charging of the channel. To gain physical in
sight as to how dqS and dqD contribute to the charging of
the channel, consider the following.
If we apply a change in the gate voltage, a change
of the charge in the channel dqN will occur. The electrons
necessary to supply this additional charge are injected
into the channel by charging currents flowing in from the
source and drain, that is
SdqS dqD dqN
is + i + dt d (3.9)
S D dt dt dt
The contributions of dqs and dqD to dqN are, in general,
unequal and depend, as we shall see, on the operating con
ditions of the device.
Figure 3.4 shows a simplified energy band diagram at
the surface of an Nchannel MOSFET under various operating
conditions determined by the magnitude of VD. Consider
first the case when VD = 0 and AVG is applied. Because the
barrier height that the electrons have to overcome in both
sides of the channel is equal (Fig. 3.1(b)), we expect that
charging currents flowing into the source and drain ends will
be equal,
[dq dqD
dt dt (3. 10
Now apply a small VD>0 and change the voltage by AVG. As in
the previous case, electrons are injected from both sides of
the channel. The electric field produced by the application
of VD, however, will present an additional barrier height
for the electrons injected from the drain side (Fig. 3.4(c)).
Thus we expect the charging current in the source to be larger
EFn
(a) Equilibrium
(b) V = 0, VG 0
(c) VD small, VG 0
(c) VD small, VG f 0
VD large, VG 7 0
Fig. 3.4 Energy band diagram at the surface of a MOSFET
under the effect of applied drain and gate voltage.
I
than the charging current in the drain. That is,
[dqS dq D
> dt (3.11)
SVVD >0 VD>0
For larger values of VD the device will be eventually driven
into saturation. The high electric field produced near the
drain will impede charging of the channel from that end
(Fig. 3.4(d)). Hence, the additional electrons required
when AVG is applied will be supplied mainly from the source
end. That is,
dq
dqD] 0 (3.12)
dt SATURATION
A similar argument can be employed to explain the contribu
tions of dqS and dqD to the charging of the channel due to
changes in the substrate voltage.
From the above discussion we can define an apportionment
function X such that the source and drain charging currents
can be expressed as
[= dqS] N (3.13)
d t = D d t V / D
dJVs,VD V ,V
S SD S D
and
dq dq
SVSV t fl (3.14)
VSVD VSVD
The apportioning function A takes values from 1/2 to 1 be
tween the conditions of V DS = 0 and saturation.
A convenient expression for X results from combining
its definition in (3.13) with the indefinite admittance
matrix that characterize the charging currents in (3.7).
Using chain rule differentiation in (3.13),
qS q S FN G N
G + v B =B B (3.15)
G B L GB
This equation must remain valid for any value of vG and vB.
Thus
aqS/ VG + aqS/aVB
= /v + /(3.16)
qN /3VG + qN/B
By using the properties of the indefinite admittance matrix,
the numerator and denominator of (3.16) can be rewritten as:
qS 3qS 3q qS 3S
+ = + j
G 3B S avD
aqG aqB rvD Sqv
q + + (3.17)
9VS vS S Dv DJ
and
+ + I
vG vB vS vD)
_qG B qG B
+ + + (3.18)
vS vS D vD
Substituting (3.17) and (3.18) into (3.16) and, using the
definitions in Table 1, we obtain,
1
C + C
SDG DB
1 SG
CSG + CSB
(3.19)
Here, we have used the assumption that no direct capacitive
effects exist between source and drain (aqS/avD = SqD/vs = 0).
Equation (3.19) can now be used to obtain the functional
dependencies of equivalentcircuit elements involving dq and
dqD directly from an extrapolation of the steadystate be
havior of the device. From Table 1, these elements are,
d q = q
SG vG vS
aqD aqG
d D q G
DG BvG V D
G D
dSB
N
qN + C
DG
vG
= dSG
dDB = dSB
dGB = dSG + dDG (3.24)
Equations (3.22)(3.24) have been simplified by direct
application of the properties of the indefinite admittance
matrix.
3.5 Modeling of the Extrinsic Components
The extrinsic components depend on details of the
fabrication of a specific type of MOSFET. In many cases,
(3.20)
(3.21)
(3.22)
(3.23)
the extrinsic part can be modeled by inspection of the
geometry of the device. Elements commonly found are:
overlapping capacitances due to the overlap of the gate
oxide over the source and drain islands; bonding
capacitances resulting from metalization over areas where
the oxide is relatively thick; PN junction capacitances
arising from sourcesubstrate and drainsubstrate dif
fusions; and resistance components due to finite resis
tivity at the source, drain and substrate. In general,
these elements are distributed capacitances and resistors
but can be transformed to lumped elements by applying a
quasistatic approximation. Lindholm [11] gives the de
tails of the general approach for modeling extrinsic ef
fects in a fourterminal MOSFET. For particular devices,
the details of the extrinsic modeling have been worked
out in the literature [7,22].
3.6 Relation to Existing Models
A wide variety of equivalentcircuit models of dif
ferent complexity and accuracy have been advanced for the
MOSFET [7,11,20,22,23]. The general development of these
models follows a partially heuristic and partially sys
tematic approach that consists in interpreting in circuit
form the different terms of the equations describing the
device physics. The definitions of the elements in these
circuit models depend on the particular approximations of
the physical model involved.
In contrast, the equivalentcircuit of Fig. 3.3 and
the definitions of its elements in Table 1, having been
developed from a methodology based on fundamentals, are
quite general. For example, the new network representa
tion can take into account two and threedimensional ef
fects such as those in the shortchannel MOSFET. To use
the model one needs only compact analytical descriptions
of these effects in physical models for the dc steady
state. Such descriptions, we anticipate, will appear in
the future. Indeed, as new physical models for dc be
havior appear, such as the one presented in the next chap
ter, the equivalentcircuit developed here is designed to
make immediate use of them to yield new and better network
representations of the largesignal transient response of
the MOSFET.
Most of the past work in equivalentcircuit modeling
of the intrinsic MOSFET neglects the effect of charging
currents flowing into the substrate terminal. Among the
models that consider these effects, the treatment of Cob
bold [6] is the most detailed. His model, derived for
smallsignal applications, involves an equivalentcircuit
in which the charging effects are represented by four
capacitors (sourcegate, sourcesubstrate, draingate and
drainsubstrate) and a controlled current source (gate
substrate). As can be observed in the general equivalent
circuit between any two terminals shown in Fig. 2.2, the
representation by a capacitor alone of charging currents
between two terminals requires certain specific conditions
related to symmetry and apportionment of charge in the
device to be satisfied. For example, if the terminals are
the source and the gate a singlecapacitor representation
between these terminals would require 9qS/ G = 9qG/9vS'
Because of the physical asymmetry of the MOSFET, these
requirements are, in general, not satisfied. This problem
was apparently not recognized by any of the previous workers
in the field.
CHAPTER IV
STEADYSTATE MOSFET THEORY MERGING
WEAK, MODERATE AND STRONG INVERSION
4.1 Introduction
In Chapter III we have developed a circuit representa
tion for the transient behavior of the intrinsic four
terminal enhancementmode MOSFET. Each circuit element in
this representation depends on the constants of physical
makeup of the MOSFET and on the voltages at the terminals
of the intrinsic device in a way that is determined by the
static model chosen to represent the current and the in
version, substrate and gate charges. To complete the
modeling, therefore, one must choose a static model for
this current and these charges that is general enough to be
suited to whatever circuit application is under considera
tion. None of the static models previously developed are
suitable for this purpose, for reasons that will be soon
discussed. Thus the purpose of this chapter is to develop
a model that has the properties required.
One necessary property of the static model is that it
represents the entire range of operation to be encountered
in various circuit applications, including the cutoff,
triode, and saturation, including operation in weak, moderate
and strong inversion, and including fourterminal operation.
The model of Pao and Sah [24] comes nearestto this ideal.
It covers in a continuous form the entire range of opera
tion. However, its mathematical detail makes it inconvenient
for computeraided circuit analysis, and it does not include
the substrate charge and the influence of the substrate
terminal.
The Pao and Sah model has provided the basis for other
modeling treatments. Swanson and Meindl [25], and Masuhara
et al. [26] have presented simplified versions covering
the entire range of operation. Their approach consists in
a piecewise combination of models for the limits of weak and
strong inversion. This approach introduces discontinuities
in the slopes of the characteristics for moderate inversions,
which are computationally undesirable. These models, fur
thermore, do not include charge components and the influence
of the substrate terminal.
Following a different line of reasoning ElMansey and
Boothroyd [27] have derived an alternative to the Pao and
Sah model. Their work includes charge components and four
terminal operation. However it also is mathematically more
complicated than is desirable for computeraided circuit
design.
The goal of this chapter is to develop a model that
includes:
(a) fourterminal operation;
(b) cutoff, triode and saturation regions;
(c) weak, moderate and strong inversion;
(d) current and total charges.
The model, furthermore, should avoid the discontinuities
of a piecewise description while maintaining enough
mathematical simplicity for computeraided circuit analysis.
In Section 4.2 a review of the general fundamental of
MOSFET operation are presented. A discussion, in Section
4.2.3, of the relation between the surface potential and
the quasiFermi level for electrons sets the basis of our
approach. In Section 4.3 expressions for the drain current
and the total charge components are derived. To assess the
validity of our approach, the predictions of our model for
the drain current are compared against experimental data in
Section 4.4. In the last section we include a discussion of
the limitations of the model.
4.2 Fundamentals
4.2.1 Drain Current
In an nchannel MOSFET, illustrated by Fig. 3.1, the
steadystate drain current density JD(x,y) is essentially
the electron current density in the inversion channel [24]:
dN
J (xy) = J (x,y) = qp NE + q D
D n n y n dy
dV
= qn N (4.1)
where V = VNVP is defined as the difference between the
quasiFermi potential for electrons V and the quasiFermi
potential for holes Vp. Because there is no significant
hole current flowing in the device [28] Vp is nearly constant
41
and coincides with the bulk Fermi potential,
(F = kT/q ,n NAA/ni. The voltage V is referred to as
the "channel voltage" [20], and at the boundaries of the
channel, y=0 and y=L, it has the values V(0) = V and
V(L) = VD. These and other properties of V will be derived
in Appendix A.
The total drain current is obtained by using the
gradual channel approximation [19]:
dV
= Z JD(x,y)dx = Zyn Q (4.2)
Here Z is the channel width, n is an effective mobility,
and Qn is the electron charge per unit area in the inversion
channel defined by
Qn = q N dx (4.3)
0
The differential equation in (4.2) is solved by integrating
along the channel
Z Co D Qn
I= L V S C 0dV (4.4)
where L is the effective channel length and C is the oxide
capacitance per unit area.
The effects due to mobility reduction and channel
length modulation have been studied in detail by different
authors [21,29,30]. They could be included in this work by
appropriately modifying pn and L.
4.2.2 Charge Components
For the purposes of equivalentcircuit modeling it is
convenient to divide the charge distribution within the in
trinsic device in three components: charge associated with
the gate, charge in the bulk and charge in the inversion
channel.
In the charge associated with the gate we include:
the actual charge in the metallic gate (CoV o), the fixed
charge in the oxide Q and the charge due to surface
states at the oxidesemiconductor interface Q ss Inspec
tion of the energy band diagram of Fig. 4.1 shows that this
effective gate charge Q per unit area can be expressed as
Q Q Q
Qg ox+ ss (4. 5)
C G 4MS S + C C
0 0 0
where MS (= (mXsq(ECEI) (F) is by definition the metal
semiconductor work function, S is is the surface potential
and VG is the applied gate voltage. In this work we will
assume that the charge in the surface states Q is in
ss
dependent of voltage. It has been demonstrated, however,
that when the device is operating under low voltage condi
tions [31] the voltage dependence of Qss becomes important
ss
in determining the relation between surface potential and
external applied voltages. A typical characterization of
Qss is given by [32]:
Qss = CSS (SV)
(4.6)
43
XS
VE /q
^N I
EE /q
.. Epp/q OpF
ErFn /EV/q
EF/q 
Sx
Oxide Semiconductor
Fig. 4.1 Energy band diagram under nonequilibrium condi
tions. All voltages are referred to the substrate.
Note that qVn = EFn, qVp = EFp, and qVI = EI.
where NSS, representing the surface state density per unit
area, is used as a parameter to obtain improved fit with
experiment. Typical values for NSS are on the order of
10 2 1
1x10 cm eV [25,31,32]. The work presented here can be
modified to include this effect.
The charge in the bulk consists mainly of ionized
atoms and mobile majority carriers. In a psubstrate devic
the bulk charge Qb per unit area can be approximated by
Qb = o q(PAA)dx
0
(4.7)
To solve this integral equation, one can change the variable
of integration to the potential V (x) by using the solution
to Poisson's equation for the electric field. This proce
dure requires numerical integration of (4.7). In the present
analysis we will obtain an analytic solution by assuming
that, because of its "spikelike" distribution [33], the
mobile electron charge in the channel has a negligible ef
fect on the potential distribution. Although this approxima
tion is only strictly valid under depletion or weak inversion
conditions, it serves also as a good approximation under
strong inversion conditions because the major contribution
to Qb in strong inversion comes from the uncompensated and
ionized impurities in the depletion layer [34].
Using the approach described above, we obtain
e
Qb kTf sV IS 2
C K VB + ( e 1
K V kT (4.8)
Here,
K 2qESNAA 1/2
=  AA(4.9)
02
is a constant that depends on fabrication parameters. The
exponential term in (4.8) results from integrating the
contribution to the charge density of the mobile holes in
the substrate P/NAA = exp[B(SVB)]. This yields
PS/NAA = exp[B(SVB)] where PS is the density of holes
at the surface. For the regions of interest, depletion to
strong inversion, this exponential term can be neglected.
The charge in the inversion channel, defined by (4.3),
the charge in the gate, and the bulk charge are all related
through a onedimensional Gauss' law, which requires
Q + Q + Qb = 0 (4.10)
The total charge components are obtained by integrating
Q g Qb and Q along the channel:
Total gate charge,
L
QG = Z Q 9 dy (4.11)
46
Total substrate charge,
L
QB = Z Qb dy (4.12)
Total inversion charge,
QN = (QG+Q) (4.13)
Or, alternatively, we may change the variable of integration
to the channel voltage V by using (4.2),
VD
G ZL QgQn dV
V
ZL
QB L QbQn dv (4.14)
D VS
VD
ZL 2
Q V Q dV
N I* n
D fV S
S
where
I
I* = (4.15)
D Zn C
n o
L
is a normalized drain current. The dimensions of I* are:
(volts)2. Similar expressions have been obtained by Cobbold
[7 ] by assuming drift only. In contrast, (4.14) includes
the effect of drift and diffusion which, as we shall see,
is necessary in obtaining a model for the complete operating
range of the MOSFET.
4.2.3 Surface Potential
The complete characterization of the charge components
per unit area Q and Qb requires the functional relation
between the surface potential S and the applied voltages.
This relation is established by applying Gauss Law,
ignoring the ycomponent of electric field, which requires
that the effective charge in the gate be the source of the
xdirected electric field in the semiconductor. That is,
Qg = SEx x=0 = KCo F(,'V'VB'F) (4.16)
The function F(SV,VB, F) is the normalized electric field
at the surface obtained from the solution to Poisson's equa
tion. This solution has been worked out by several authors
for the case when VB = 0 [24]. If extended now to the case
when a bias voltage VB is applied, we find that
pF v^ fkT 1/2 (SVB) B 1
F(SVVB'F) = e + kT/q
8 ( SV2(F) s^VB B2mF 8 (VVB+2(F)] 1/2
e e e (4.17)
kT/q
For the usual substrate doping, VVB+2 F is always much larger
than kT/q. Furthermore, if we neglect the majority carrier
concentration at the surface (PS<
approximation in both the depletion and inversion modes, one
can show from (4.17) that (4.16) reduces to
V = V kT + (+SV 2 F) 1 1 2 (4.18)
VG S = K IS VB q e 1 (4.18)
where
Q Q
= VG MS + + C (4.19)
0 0
The solution of the integral equations defining the current
(4.4) and the charge components (4.14), in which the variable
of integration is the channel voltage V, requires the func
tional relation between S and V. This relationship, how
ever, has not been found in closed form and hence the pos
sibility of direct integration of (4.4) and (4.5) is excluded.
A numerical integration can be performed [24] but, because
of the large computer times involved, we will look for an
approximation that will yield an analytic solution.
Let us consider some important characteristics of the
functional relation between S and V that will set the basis
for our approach. Figure 4.2 shows the solution for S ob
tained from (4.18) for a specific device having x0 = 2000A
15 3
and NAA = lx10 cm Figure 4.2 shows that, for values of
VG for which CS(O) is below 2F', S is nearly independent of V.
For VG such that S(0) > 2 F' S increases almost linearly
with V provided as it is shown below, that drift dominates
in determining the channel current. For V greater than a
certain critical voltage, however, diffusion begins to dominate
and dip/dV 0. This characteristic behavior can be explained
by studying the relative importance of the drift and diffu
sion components along the channel [24]:
49
I I
3
/ i^ ,
P
o 2
2 F
I I
1 2
V (volts)
Fig. 4.2 Surface potential 4s as function of channel voltage V.
S
Dn dN/dy 1 d4S/dV NA P NAA
(4.20)
pn NE dEs /dV N N
n y S S S
which is derived in Appendix B. For V' such that OS(0) < 2 F,
the channel is weakly inverted (<
and (4.20) implies that d s/dV 0. When V is such that
0S(0) > 2 F the channel near the source is strongly inverted
(NS(0) >> NAA); then near the source, (4.20) implies that
drift dominates and thus dip/dV  1. As we move toward the
drain, the electron concentration decreases, the channel be
comes weakly inverted and there again diffusion dominates
and dOS/dV  0. The channel voltage for which the channel
becomes weakly inverted corresponds approximately in the
strong inversion theory [7 ] to the pinchoff voltage. At
higher gate voltages, the channel remains strongly inverted
in its entire length and drift is the main mechanism. In
the strong inversion theory this corresponds to nonsaturated
operation.
The behavior of cS as described above has been used to
establish two approximations often used in characterizing
MOSFET behavior: the strong inversion and the weak inversion
approximations. In the strong inversion approximation, which
is applied when N (0)>> NAA, the surface potential is assumed
to be related to the channel voltage by S = V+2 F [ 7 .
Because then diS/dV = 1, this assumption is equivalent to
neglecting contributions due to diffusion mechanisms near the
drain. In the weak inversion approximation, which is applied
when NS(0) < NAA the surface potential is assumed to be in
dependent of voltage iS = iS(0) [31]. Then diS/dV = 0,
and, therefore, drift mechanisms near the source are neglected.
Although these two approximations produce satisfactory agree
ment with experiment in the strong and weak inversion limits,
they fail for moderate inversion (NS=NAA) where neither of
the criteria used in strong or weak inversion can be applied.
In the following section we will relax the strong and
weak inversion approximations by using the basic properties
of dS/dV. As shown in the previous discussion, these pro
perties relate to the degree of inversion in the channel.
As we shall see, the resulting model not only will merge the
operation in the strong and weak inversion modes, but also
will provide a firstorder approximation for moderate in
version.
4.3 Drain Current and Charge Components in
a Model Merging Weak, Moderate, and
Strong Inversion
4.3.1 Drain Current
In Section 4.2 we found that the drain current could be
expressed as
I VD
I* = Q' dV (4.21)
Z C VC
I L
Here, and in the rest of the chapter, the notation Q' is used
to designate a charge component divided by the oxide capacitance
per unit area C The dimensions of 0' are volts. By
using the condition of charge neutrality Qn = (Q +Q)'
(4.21) can be rewritten as
V VD
I Qg dV + Qb dV (4.22)
S S
In equation (4.22) a very convenient change of variables
can be introduced by noting from (4.5) and (4.8) that
dQ' dlpd
dg d d S dV
dy dy (V) dV dy
(4.23)
dQ' a L ,vuB 1/2 2 d. dv
b d kT ] K S dV
dy dy sVB q 2Qb dV dy
Thus,
dQ' 20'/K
dV = 9 = b(4.24)
(dp /dV) (d S/dV)
Substituting (4.24) in the expression for the current, we ob
tain
Q (VD) Qb(VD) 2 2
SQg dQg 2Qb2/K2 dQO
I* = d + 2 /K db (4.25)
D (dS/dV) + (d/dV)
Qg(VS) b(Vs)
Figure 4.3 shows the elements constituting the integrands
in (4.25) for a specified device operating in the pinchoff
mode. This represents the most general case because the
channel is strongly inverted at the source and becomes
strong
inversion
moderate weak
inversion inversion
0
Qb
vI vC
Qn
____sM
0
1.5.
/ F
1
1 V (volts) 2
Fig. 4.3 Components of charge per unit area and surface
potential as functions of the channel voltage V.
weakly inverted toward the drain. As discussed in Section
4.2.3, d S/dV has almost constant values along the channel;
in the strongly inverted portion d S/dV = 1 while in the
weakly inverted portion dPs/dV = 0. In the transition
between strongly and weakly inverted regions, where the
channel is moderately inverted, d S/dV is not constant.
However, because this represents a small portion of the
characteristic (PS vs. V), we will assume in a firstorder
approximation that S is there linearly related to V with
the value for the slope dpS/dV lying between 0 and 1.
Our approach will consist then in dividing the channel
into three regions by defining appropriate limits V1 and V2
as shown in Figure 4.3. Below V1 the channel will be assumed
to be strongly inverted with d S/dV = SS, a constant.
Above V2 we will consider the channel to be weakly inverted
with d s/dV = S a constant. In the transition region the
channel will be assumed to be moderately inverted with
d~s/dV = SM, also a constant. These approximations allow us
to write the expression for the current as the sum of the
contributions in each region. Furthermore, because dis/dV
is assumed constant in each case, it can be taken out of the
integrals which can then be directly evaluated. If we define
a function, F related to ID by
Q (Vb) Q (Vb)
FI(V,Vb) = O' dQ' + J 20 /K2d Q
',2b 3 Vb
g +2 2 b
2 3 2 (4.26)
JV V
a a
then the expression for the drain current becomes
F (VsV ) F (V1,V2) F (V2 VD)
I* = + + (4.27)
D SS SM SW
The three components of (4.27) result from carrying out the
details of the integration indicated in (4.21). Here, if
we let V1 = VD and SS = 1, (4.27) reduces to the conventional
expression (obtained by using the strong inversion approx
imation) for the drain current of a device operating in the
triode mode.
In computing the drain current from (4.27), a numerical
problem could occur in evaluating the term corresponding to
the weakly inverted channel because SW is very small. To
avoid this problem an alternate form for this term can be
obtained as follows. The channel charge Qn was defined in
(4.3) as
Q = q N(x,y)dx (4.3)
Taking derivatives on both sides with respect to y yields
Taking derivatives on both sides with respect to y yields
do
n dN
y = q (x,y)dx (4.28)
S0
but, because N = n. exp[B(V VN)] = n. exp[B(V VVp)], it
follows from the gradual approximation [19] that
dN = N i
dy kT/q
dSs dV
dV j dy
(4.29)
Substituting (4.29) in (4.28), using the definition of
Q and reordering the terms, we obtain
dQ'
dV kT/q n 43
Qn (ld/dV) (4.30)
From (4.30) the contribution to the drain current from the
weakly inverted channel can then be alternatively written as
D
I* = 
DW2
O' dV
n
kT D)
q (
Q' (V )
dQ'
( n
(1dis/dV)
But since we are assuming that dis/dV has a constant value
SW in this region, we finally obtain:
(4.32)
kT
DW q
(4.31)
Here, if we let SW = 0 and V2 = VS, (4.32) reduces to the
conventional expression for the drain current of a device
operating in weak inversion [31].
4.3.2 Charge Components
The procedure to calculate the total charge components
is entirely analogous to the one presented for the drain
current. Combining (4.10) and (4.14) and using the change
of variable indicated in (4.24), we obtain for the total
charge components
V
ZL D
G I*~
D DVS
S
ZL
I*
D
2
(Qg +QgQ b)dV
Q' (VD)
Q (VD
QgVs)
O'2 do'
g g +
(1di /dV)
Qb(VD) 2 2/2 2d
( b dv (4.33)
(1d /dV)s
b(VS)
ZL
B I*VS
D V
S
ZL
I*
*D
(02 + Q'OQ)dV
 gb
D) 2Q3/K2dQ
+ S
S (1dS/dV)
Qb(VS)
b(VD) 2 2
Q(VD) 2Q'Q /K2 dQ'
gb b
I (1dis/dV)
Q (VS)
Again, if the channel is divided in three regions and we
assume that dis/dV is constant in each region, the charges
can be obtained by direct integration. Let us define func
tions FOG and FQB such that,
QG QBo
and
(4.34)
0' (Vb) Q2 (Vb)
F (Va 2 dO' + 2Q' b2/K2 dQb
QG a b fg gb
O' (Va) QO(Va)
g 3 2 f 2 b l ^
= + 2 QQ' + 2 (4.35)
3 2gbg 5
3K K
L a
and
Qb(Vb) Q(Vb)
FQB (VaVb) = 2Qb3/K2 dQ' + 2Q'Q'2/K2 dQ
QB af b bgb
Qb(Va) Qb(Va)
Q 4 + 27 O'O2 + j (4.36)
22 b 32 b K5 Va
2K 3K f
a
where the second integral in (4.35) and (4.36) was evaluated
using integration by parts with u = Q' and dv = Q'2doa .
Then the total charges can be expressed as
FZL FQG(V ,V ) FQG (V ,V2) FOG (V2 VD
Q =  Gs1 + "+ (4.37)
G I* S + s S
D S M W
_ZL FQB(VSV) FQB(VlV2) QB(V2VD (4.38)
B I* S S S
D S M + W
As in the case of the drain current, to avoid numerical
problems due to the smallness of SW, an alternative expression
can be obtained for the contribution of the weakly inverted
portions of the channel. Using (4.30) directly in (4.14),
we obtain
Q' (V )
Q, = ZL kT g dO'
GW ID ) q (1ds /dV) n
Q (V2)
2 3 D
ZL kT/q g 2 b V
I* (1S 3 K2 (4.39)
SV2
Here we have used integration by parts with u = O' and
"g
dv = dQb. A similar expression results for the bulk charge
in weak inversion,
Q' (VD)
nD
SZL kT b
B I q (ldis/dV) Qn
n 2
Q(V,)
03 02 "VD
ZL kT/q 2 (b b
S (4 40)
I (1S U) 3 K2 2
D JK
V2
4.3.3 Limits for the Strong, Weak, and Moderately Inverted
Portions of the Channel
The threeregion piecewiselinear approximation employed
in Sections 4.3.1 and 4.3.2 to obtain expressions for the
current and charges uses two parameters: (1) the limits V1
and V2 that divide the strong, moderate, and weakly inverted
portions of the channel; and (2) the approximate values at
the slope disg/dV (S S S ) in each of the three regions.
These parameters will be now defined in terms of the applied
external voltages.
In Section 4.2.3 we concluded that dIs/dV could be
considered as a measure of the level of inversion along
the channel. Here we will show that it is also the ratio
of the contribution of the drift current to the total
current. In (4.20) we indicated that
I 1 d Sp/dV
= (4.20)
DRIFT diS/dV
Thus, rearranging terms we obtain
DRIFT dS (4 4
= (4.41)
DRIFT DIFF
We will use this property of diS/dV to define quantitatively
the voltages V1 and V2 as follows.
In the strongly inverted regions we previously observed
that diS/dV is close to unity and drift dominates while in
the weakly inverted regions diffusion dominates with dis/dV
being close to zero. Thus we will define the transition
region corresponding to moderate inversion as the region in
which both drift and diffusion are comparable. More
specifically, we will define V1 as the channel voltage at
which the drift current constitutes 80% of the total current
and V2 as the channel voltage for which the drift component
is 20% of the total current. This specification of V1 and
V2 provides, approximately, the best leastsquares fit
between the piecewise linear approximation and the si versus
V characteristic. Based on these definitions we can now
obtain expressions for V1 and V2 by solving
dis
dV A
dV
(4.42)
where A has the value A = 0.80 when solving for V1 and A = 0.20
when solving for V2. Differentiation of both sides of (4.18)
yields
dS
dV
e (iSV24F)
e
20'
+ 1 + e
K2
K"
= A
Combining (4.43) and (4.18) and using the definition of Q ,
we find that
2V 2
kT x K2
V1V2 = V' 2 n + V
G1 22F q 2 2 x
A'K
where
k2 + T 1 /2
Vx =K GVB 4 q 2
(A' K)
(4.43)
(4.44)
kT/q
+ A
Here, A' = (1A)/A. Hence, A' = 1/4 when calculating V1 and
A' = 4 when calculating V2. Equation (4.44) applies only
when VS < VI, V2 < VD. The complete functional dependencies
for V1 and V2 are given by
Vl
V1 = Vs
SVD
v2D
V2 = VS
VD
from (4.44) if V < V1 < VD
if V1 < VS
if V1 > VD
from (4.44) if VS < V2 < VD
if V2 < VS
if V2 > VD
(4.45)
B (SV2F)p
Using the functional dependencies for V1 and V2 given
by (4.45), we now can solve (4.18) to obtain 4s at the
limits V1 and V2. The surface potential at those points
can be used to define the approximate slopes dis/dV in each
region, which constitute the second parameter at our three
region piecewise linear approximation. They are,
s=(V ) S(Vs)
sS V VS
(S(V2) IS(V1)
SM = V2 (4.46)
SV 2 V
2 1
SS(VD) s(V2
W V V
Here, to calculate is(VS), ls(Vl)' ls(V2) and S(VD)'
one needs to solve (4.18) numerically. This process does not
require much computer time. We used the NewtonRaphson
method [35] to calculate the solution and found that less
than five iterations were necessary to achieve convergence.
4.4 Results and Evaluation of the Model
Table 2 summarizes the results of the model merging
weak, moderate and strong inversion. In Figures 4.4 through
4.7 we illustrate the characteristics for the drain current
and the total charge components obtained from the proposed
model. Notice that the curves in these characteristics and
their slopes are continuous throughout the entire range of
63
Table 2 Drain Current and Total Charge Components
DRAIN CURRENT
ID FI(VS,V ) FI(V1,V2)
= I* = + + I*
AnCoZ/L D SS M DW
TOTAL CHARGE COMPONENTS
1 FQG(Vs,Vl)
1 o S
1 FOB (VSV 1
I* S
DL S
FOB(VIV2)
S.M
FOB (VlV2)
S 
M
QN + QG + QB = 0
QG
ZLCo
QB
ZLC
o
+ GW
ZLCo
+ "BW
ZLC
o
__
(Continued)
FOR THE CURRENT:
FI (Va ,Vb)
I* = kT/q
DW 1SW
W
g2 2Q vb
2 3 K2
a
V
D
[(nl
FOR THE CHARGES:
F G(Va ,Vb)
Q= ;
3
FB(VaVb) =
QGW 1
ZLC I*
o D
21<2
v b
+ 2 ( 'Q3 + 25/K5)
V
a
SV
v
K2 g3 + 2Qb /K )
3Ka
a
kT/q
1SW
o_2 o3v D
+ Q + 2
S2 g b 3 K2 V?
QBW 1 kT/q
ZLC I* 1SW
0 D WV
V
2 '3 Q2' D
2 bb
3 K2 2
V,
Table 2
0.2 B
41
0
/
0.1 
/
extrapolated
/ threshold
1 2
Vt (volts)
Fig. 4.4 Calculated squareroot dependence of the drain
current on gate voltage.
102
" 15
IH 10
1 1.5 2
V' (volts)
Fig. 4.5 Calculated drain current as function of gate
voltage for three doping concentrations
(x = 2000A).
o
(N
1)
il
0
H
0.1
0.3 0.4
VD (volts)
Fig. 4.6 Calculated drain current
and moderate inversion.
characteristics in weak
0.2
0.5
1 2 3 4 5
V' (volts)
Fig. 4.7 Calculated charge components as function of
gate voltage.
operation. This feature results from including the
transition region for moderate inversion, which is not
included in previous work treating weak [31] and strong [7
inversion.
Figure 4.4 shows that for strong inversion the func
tional relation between the drain current and the gate voltage
follows a square law [20], while in weak inversion this rela
tion is exponential, as shown in Figure 4.5. This behavior
agrees qualitatively with previous models for the extremes
of strong and weak inversions.
In Figure 4.6 the drain current is shown as a function
of the drain voltage for weak and moderate inversion. The
inclusion of drift and diffusion in our model has produced
a smooth transition into saturation. The necessity of in
cluding diffusion to produce this smooth transition was first
recognized by Pao and Sah [24].
The total charge components are shown in Figure 4.7 as
functions of the gate voltage. Notice that the inversion
charge increases exponentially at low gate voltages. The
relationship between the charge components and the terminal
voltages has apparently not been established previously for
weak and moderate inversion. As is demonstrated in the next
chapter, these relationships provide a basis for characteriza
tion of the device capacitances and the displacement currents.
In assessing the validity of our modeling approach and
the accuracy of the expressions developed for the current
and charges, we compare the results of our model against
results from previous theoretical treatments. Figure 4.8
shows experimental data for the square root of the drain
current against gate voltage obtained in a commercial
15 3 0
device (4007) having NAA = 3x10 cm and x = 1000A.
AA o
In this figure we also show the calculated characteristics
obtained from the model just derived. Since QSS and other
fabrication parameters are not accurately known for this
device, the calculated and the observed characteristics were
matched using the value of the voltage and current at the extra
polated threshold voltage. Good agreement between experiment
and theory is observed. We also show in Figure 4.8 theo
retical characteristics obtained from a model using the
strong inversion approximation [7 ]. The discrepancy at
low gate voltages between this model and the experimental
data arise because the strong inversion approximation
assumes that an abrupt transition between depletion and in
version occurs when the surface potential at the source is
equal to 2pF. This results in a discontinuity in the slope
of the characteristics at the boundary between cutoff and
saturation. A discrepancy also exists at high gate voltages.
This arises because the surface potential, which in the strong
inversion approximation is assumed independent of gate voltage,
is in fact a logarithmic function of VG. As one can show from
equation (4.18) for VG > VT, this function can be approximated
by
Vb2 2 Vy K(2
kT G VG F B(24FV
S(O) = 2F KT log G 2 B) (4.47)
SqIK kT/q
71
/i
30
Cc /
'/
15 3
 N A= 3xl0 cm
x = 1000 /
20 
V = 2v
V = Ov
/
10
/
S Experiment
/ Our model
I  Strong inversion
I! model [ 7 ]
Extrapolated
S*/ threshold
1 1.5 2 2.5
VG (volts)
Fig. 4.8 Experimental values for the drain current compared
with values calculated using our model and using
a model for strong inversion.
14 3 /
N =7x10 cm /
105 AA / /
x = 1470A
o /
VD = 2v /
6
V = Ov
cn B
1 7 /I x
S Experiment [26]
108 Our model
S Strong inversion
model [7 ]
I  Weak inversion
9
10 model [31]
i I / I
10
0.5 0 0.5
VG (volts)
Fig. 4.9 Experimental values for the drain current
compared with values calculated using our
model, using a model for strong inversion
and using a model for weak inversion.
73
VG = Iv
104
VG = 0.3v
G o
F 5 VG = 0.lv a
S10 >
VG = Ov "
106 G = 0.1
0
107
107
V, = 0.2v
8
10 
V, = 0.3v
9
10 
I I I I I
0.1 0.2 0.3 0.4 0.5
VD (volts)
Fig. 4.10 Experimental values for the drain current
compared with values calculated using our
model and using a previous model
(NA = 7x1014, x = 1470A).
AA o
Experiment [26]
Our model
 Previous model [26]
The proposed new model includes implicitly this dependence
of S in VG.
Figures 4.9 and 4.10, which compare the predictions of
our model with experimental data from the literature [26],
show excellent agreement. Because information was available
only for the doping concentration NAA and oxide thickness x
in this device, the calculated and the experimental charac
teristics were matched using the value of the gate voltage
and drain current at the extrapolated threshold. In Figure
4.9 we show for comparison previous models obtained for weak
inversion [31] and for strong inversion [7]. In Figure 4.10
we compare our model against a recently developed model for
the entire range of operation [26]. Although this model shows
good agreement with experiment in the weak and strong inver
sion limits, it fails for gate voltages near the transition
region (VG ~ 0.lv). Furthermore notice the discontinuities
in the slope of the characteristics which our model avoids.
4.5 Conclusions
The major achievement of this chapter is the analytical
description given in Table 2 that unifies weak, moderate and
strong inversion and covers the cutoff, triode and saturation
modes of operation. This description has the following
properties:
(1) It includes the effects of substrate bias
which enables the representation of four
terminal properties of the MOSFET.
(2) It includes the charges in the gate, channel
and substrate regions as well as the drain
current. These charges provide the basis for
modeling capacitive effects.
(3) It consists of simple expressions having
continuous derivatives with respect to the
terminal voltages. This helps make the
description useful for computeraided
circuit analysis.
The model developed here is subject to the limitations
of the onedimensional gradual channel approximation which
become severe in MOSFET structures with short channel lengths.
Other limitations arise from the idealizations used in Sec
tion 4.2: effective channel length, field independent
mobility and effective charge in surface states. A number
of publications in the technical literature deal with more
detailed descriptions of these parameters and also with
shortchannel effects. As explained in Section 4.2, our
model has enough flexibility to incorporate these descrip
tions.
CHAPTER V
FUNCTIONAL DEPENDENCIES FOR THE ELEMENTS IN
THE LARGESIGNAL FOURTERMINAL EQUIVALENTCIRCUIT
5.1 Introduction
In Chapter III we developed an equivalentcircuit
representation for the transient response of the MOSFET.
By employing the results of Chapter IV, the functional
dependencies of each element in this equivalentcircuit
will be now derived in terms of the applied voltages and
the fabrication parameters of the device. The main ap
proximation used in deriving such dependencies is a quasi
static approximation through which, as discussed in Chap
terms II and III, one extends the knowledge of the dc
steadystate behavior of the device to describe its large
signal transient response.
The equivalentcircuit for the intrinsic MOSFET
derived in Chapter III is shown in Fig. 3.3. The definition
for each element in the circuit is given in Table 1.
Three types of elements are present: a current source be
tween drain and source representing charge transport, and
capacitors and transcapacitors connected between each node
representing charge accumulation within the device. In Sec
tions 5.2 through 5.4, the functional dependence of each of
these elements is derived. The resulting mathematical
expressions are valid for the entire range of operation
of the MOSFET, and include the effect of the substrate
terminal. Such expressions are new.
This chapter also provides the first detailed dis
cussion of the intrinsic capacitive effects of the sub
strate and the transcapacitive effects due to the non
symmetry of the fourterminal MOSFET. In Sections 5.3.4
and 5.4.2 we discuss the engineering importance of these
two effects. Under certain conditions determined by the
particular circuit environment in which the device is used
the equivalent network representation can be simplified.
An example is discussed in Section 5.4.3.
5.2 SourceDrain Current Source
Through the use of a quasistatic approximation, as
discussed in Chapter III, the functional dependence of the
nonlinear sourcedrain current source can be determined by
extrapolating the static characteristics of the drain cur
rent found in Section 4.3.1. Thus,
iSD = ID(VS,'DG',B) (5.1)
which has the same functional dependencies on the terminal
voltage as those describing the dc steadystate.
5.3 Capacitances
5.3.1 Expressions for the Capacitances
The capacitors in the equivalentcircuit are defined
in Table 1 as the partial derivatives with respect to
voltage of the time varying total charge components qG, qB'
q As in the case of the transport current iSD a quasi
static approximation allows us to write
qG = QG(vSvD' G'B
qB = QN(VS'D'VG'VB) (5.2)
qN = (qG + )
One can anticipate that a partial differentiation of (5.2)
with respect to the voltages would lead to very complicated
expressions. But we will now show that because of the sys
tematic approach used in Chapter III to define the circuit
elements, one can find simple expressions for the functional
dependencies of the capacitors.
From Table 1 the capacitors connected to the source
are
SQG
C = (5.3)
SG av
and
aQ
C B (5.4)
SB av
We can use (4.14) to rewrite CSG
SG
79
C vD
CSG _VD
C osG l i 1 (
C Ss Vs Q' Q' dv (5.5)
ZLC I0*Dv g n
S
where Q' denotes a charge per unit area normalized by the
oxide capacitance C (the dimensions of Q' are volts).
Using chain rule differentiation and the fundamental theorem
of integral calculus,
C Q I*
SG 1 G D
SLC I Dv Qg(V) Qn (Vs) (5.6)
o D S
But since
DI* VD
avD Qn dv = Q(vs) (5.7)
S vs vS
we finally obtain
C Q'(0) O
SG n Gj Q;(O)(
ZLCo I Z Qg(0) (5.8)
0 D
where Qn(0) and Q (0) are the normalized and gate charge
per unit area, given by (4.5) and (4.8), evaluated at the
source end (y=0). Similarly
C Q'() QO
SB Q0 (0) (5.9)
ZLC I* cZL b
For the capacitances connected to the drain, the ap
proach is the same except that
qI* D D
D D
aD. Q' dv = Q'(L) (5.10)
Qy n n
vD vD v
Thus, we obtain
CDG (L) [ Q (L)] (5.11)
ZLC I*
and
CDB Q( QQ(L) (5.12)
ZLC ID
o D
where Q'(L), Q'(L) and Q'(L) are the normalized channel,
gate and substrate charge evaluated at the drain end (y=L).
The gatesubstrate capacitance is defined in Table 1
as
qB QB
CB 3 vG (5.13)
GB 3vG av
Substituting (4.14), which gives the functional relation
for QB' and applying the chain rule for differentiation
yields
C 3I1* D
GB 1 D I V ]
ZLCB I QB vG + Q dv d) (5.14)
ZLC0 I B DvG VG )v b n
The expression for CGB is more complicated than those for
CSG, CDG, CSB, and CDB. To find this expression we take
v
the partial derivatives, aI*/3vG and 9/9vG (D QbQ dv),
using (4.27) and (4.36). The procedure is straightforward,
and the results follow:
81
CGB 1 DFI (v DF (vl' 2) DFI(v2'D)
C + + +
ZLC I* SS S+ S
o D L S M W
DFQB (S' 1 DFQB(V1'V2) DFQB(2',VD)
DFQB (s'V + +  (5.15)
S S Sw (5
Here we have defined the functions DF and DFQB as:
I Q V
Q
DFn (v vb Q + (5.16)
I ag K2 B (:sV24F)
1 + e v
1 + 2Q + e va
DFQ(vl vb) 12 n (5.17)
L 2Q' a
An alternative form for the contributions of the weakly in
verted portions of the channel results from taking partial
derivatives with respect to vG in (4.32) and (4.40). This
yields,
I* DF (v v )1 + K2 /Q D
DW I 2 kT/q K2/ VD(5
vG SW 1Sw 1 + K2/2Q'
and
DFQB (v2'D)_ kT/ + K /Q (5.19)
SW 1S W 1 + K /2Q1
The results for the capacitances are summarized in Table 3.
Figure 5.1 illustrates the functional dependencies of the
Functional dependencies for the capacitors.
Q (0)
I* (QG/ZL Q (0))
D
Q'(L)
I* (Q/ZL Q'(L))
D g
Qn(0)
I* (Q'/ZL Qb(0))
D
CSG
o
CDG
ZLCo
CSB
ZLC
o
CDB
ZLC
o
1
Ir IQ
(Q;/ZL Q(0))
VG DVG
v(L) (
Iv(O) b dv
(*) given by (5.15) through (5.17).
Q'(L)
n
I*
D
CGB
ZLCo
_
Table 3
0
*C
o / /
C) >
0 > >
C J ri oC
II II II / \ I m e
< 0 QM\
> >
C(
rrq
S u
0_
m
a) u
IC
o Cd
o n u
Cd d
0
S O
0ce "L
^^Z ? s^Tu UT ous^oedB
rl
capacitances in a specific device. In contrast with results
obtained from models using the strong inversion approxima
tion [7,36], these curves present smooth transitions between
the different regions of operation: cutoff, saturation and
nonsaturation. A physically based discussion about the main
features of these characteristics is given in the next section.
5.3.2 Physical Interpretation of the Results for the
Capacitances
Consider first the capacitances connected to the source
and drain nodes in the equivalentcircuit. These capacitances
are directly related to the apportionment between the currents
charging the channel from the source island and from the drain
island. To observe how this apportionment occurs, let us
consider the total capacitance at the source CSS given by
3QN
C = C + C = v (5.20)
SS SG SB Dv
and the total capacitance at the drain CDD given by
aQ
CDD = CDG + CDB (5.21)
As we shall see, the functional dependence of these capacit
ances shown in Fig. 5.2 has the form to be expected from the
discussion of the charge apportionment in Section 3.3. In
cutoff there is no charging of the channel and both CSS and
CDD are equal to zero. As the gate voltage is increased,
the channel is turned on in an exponential form (see Fig. 4.5)
causing an abrupt change in CSS. At higher gate voltages,
(1)
u
4)
+ I H
C0
u O
4U)
U U)
U U
o \ 
U 0m
(In
C) 7
U \
S\04
H ^
+ ^^ V
while the device is in the saturation region, QN increases
almost linearly with gate voltage and hence CSS is nearly
constant. In the saturation region, because there is no
charging of the channel from the drain end, CDD = 0.
Further increase of the gate voltage drives the device into
nonsaturation. Here the channel opens gradually into the
drain allowing thereafter an increasing contribution of
the drain end to the charging of the channel while the con
tribution from the source decreases. Thus in this region,
as shown in Fig. 5.2, CSS decreases while CDD increases.
For very large gate voltages the charging of the channel
will tend to occur equally from the drain than from the
source. When this happens the values of CSS and CDD tend
to one another as shown in Fig. 5.2.
A measure of the apportionment of the contributions
of the drain and source islands to the charging of the
channel is given by the apportionment function \ defined
in Chapter III as
1 1
C + CC (5.22)
+ DG DB DD
1+ 1+
SG + CSB SS
This function is used in the next section to obtain expres
sions for the transcapacitances. Its functional dependence
for a particular device is shown in Fig. 5.3. In saturation,
CDD = 0 and X = 1, while in nonsaturation the values of CDD
and CSS approach one another and X tends to 1/2.
1
0.9
0.8
0.7
0.6
0.5
2 4 6
V (volts)
Fig. 5.3 Apportionment function X for the device described
in Fig. 5.1.
Note from Fig. 5.1 the similarity between the charac
teristics of the substrate capacitances CSB and CDB and the
characteristics of the gate capacitances CSG and CDG. This
similarity, which also can be observed in the expressions
defining these capacitances, will be used in the next sec
tion to obtain an engineering approximation for CSB and CDB'
Consider now the gate substrate capacitance
CGB = QB/3VG. This capacitance is related to the control
of the gate over the substrate charge. In cutoff, where
VG is not large enough to turn on the channel, this capacit
ance is equal to the capacitance of a (twoterminal) MOS
capacitance [37]. As VG increases, an inversion channel
starts forming at the surface of the semiconductor and more
field lines emanating from the gate will terminate in the
inversion channel. Thus, CGB will decrease as shown in
Fig. 5.1. For larger gate voltages, where a strong inverted
channel is formed over the entire length of the intrinsic
device, the gate will exert even less control over the sub
strate charge and CGB decreases at a faster rate reaching
eventually a zero value as illustrated in Fig. 5.1.
Figures 5.4 and 5.5 show the total gate capacitance CGG
and the total substrate capacitance CBB together with their
components,
C GG GN + (5.23)
CBB VB ___ +  5.4
G G CG
C 3Q  '= 1 + + (5.24)
BB BvB 3vB B
89
U
c)
C)
H
(1)
4J
0
P
u)
C4
0
o
oU dH
I0(0 rd
O>p
00
> 0
o ard
rp
> *
urt
0
O>
tn
rnr jo s;Tun uT cnu2;T dd2
1
0 > 1/ roP 1
>o(Q \\/4 
0*
DqZ 9 Sq~n UT OU~qopdp
0
In
4,
C)
0
a4
E
0
D o
(1)
U)
4,
H $
oo
& >
o
> (
41
0 \
DZ o Tuno uT u;TDd
u >
\0 in
\ EV s
\^<
^/^ ^s, ^
/ ^ rl
u sQ)
In the cutoff region there is no inversion channel and
CGG and CBB are equal. Their functional dependency is
that of an MOS capacitance [37]. In the saturation region,
the gate charge depends almost linearly on the gate voltage
(Fig. 4.7), and CGG shows a constant value of about 2/3 ZLC
as predicted by strong inversion theory [20]. In this re
gion, as VG is increased, the surface potential increases
producing a widening of the depletion layer; consequently
CBB decreases as shown in Fig. 5.5. At the onset of the
nonsaturation region CGG abruptly rises due to the increase
of electron concentration over the entire channel length.
For even larger gate voltages CGG approaches the value of
the total oxide capacitance. In this region, CBB attains
a constant value because the substrate charge becomes in
dependent of gate voltage. This constant value cannot be
clearly determined from the expressions of the substrate
capacitances just found. In the next section, however,
we discuss an approximation for the substrate capacitance
that permits a good estimation of their values for engineering
purposes.
The main features of the functional dependencies for
the gate capacitances in the MOSFET have been predicted by
previous authors [7,20] using simplified models. Our results
agree qualitatively with these predictions, giving additionally
a detailed and continuous description for these capacitances
and also for the substrate capacitances.
5.3.3 An Engineering Approximation for the Functional
Dependencies of the Intrinsic Substrate Capacitances
C and C
SB DB
The functional dependencies for the substrate capacit
ances CSB and CDB were derived in Section 5.3.1. Figure 5.1
shows these functional dependencies together with the func
tional dependencies for the gate capacitances and the gate
bulk capacitance. We pointed out previously the similarity
between the functional dependencies of the gate and substrate
capacitances appearing in this figure. From an engineering
point of view, this similarity is advantageous because it
suggests the existence of relations of the form:
CSB = S CSG
(5.25)
CDB = CD CDG
where aS and cD may be simple functions of the voltages.
Such relations would allow considerable simplification in
the computation of the substrate capacitances. In recent
engineering applications [1], CSB and CDB are modeled to a
first order approximation as
CSB = a CSG
(5.26)
CDB = CDG
with a being a constant. Because expressions for CSB and
CDB were not previously available this approximation has not
been verified. With the functional dependencies for CSB and
CDB made available in the previous section we can now study
this engineering approximation. Figure 5.6 shows aS and
aD, defined in (5.25), as functions of the applied voltages.
Notice that although in the nonsaturation region aS and aD
are practically independent of the gate voltage they are
in general not constant.
Using the functional dependencies for CSB and CDB given
in Table 3 we will now derive an improved approximation
for aS and aD that shows a better functional dependence on
the applied voltages while remaining a simple function of
the voltages.
Consider first
CSB Q /ZL Qb(0)
SB _B (5.27)
S CSG QG/ZL Q (0) 5 )
Substituting the expression for Q;, QC and ID given in (4.4)
and (4.14) aS can be rewritten as
fD
Q i(v) (Q (v) Q vs)) dv
vS
aS = VD (5.28)
Qn(v) (Qa(v) Q'(vs)) dv
The integrals in (5.28) can be approximated by a series solu
tion using the trapezoidal rule for the integration. A
numerical comparison between the exact solution and the
series solution shows that by taking only the first term in
this series we can obtain an approximation that is both sim
ple and accurate:

Full Text 
PAGE 1
EQUIVALENTCIRCUIT MODELING OF THE LARGESIGNAL TRANSIENT RESPONSE OF FOURTERMINAL MOS FIELD EFFECT TRANSISTORS By JOSE IGNACIO ARREOLA A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1978
PAGE 2
to JGNACIO and CELIA, my pcuie,nti)
PAGE 3
ACKNOWLEDGMENTS I am deeply indebted to Prof. Fredrik A. Lindholn for his contribution to this work and for his continued guidance, support and encouragement. I also wish to thank Dr. D. R. MacQuigg for his help in doing experimental measurements and for many interesting discussions. I would like to express my appreciation to Prof. A. D. Sutherland for allowing me to study the results of his twodimensional calculations which broadened my understanding of the MOSFET. The financial support of Consejo Nacional de Ciencia y Tecnologia (Mexico) throughout this work is gratefully acknowledged, I must also thank Mrs. Vita Zamorano for her careful typing of the manuscript. Finally, I owe a special debt of gratidude to my wife, Patricia, for her forbearance and encouragement. 2 11
PAGE 4
TABLE OF CONTENTS Page ACKN0V7LEDGMENTS iii ABSTRACT vii CHAPTER I INTRODUCTION 1 II A NONLINEAR INDEFINITE ADMITTANCE MATRIX FOR MODELING ELECTRONIC DEVICES 4 2.1 Introduction 4 2.2 Indefinite Admittance Matrix 5 2.3 Extension for Nonlinear Electronic Devices 7 2.4 Conclusions 13 III EQUIVALENTCIRCUIT MODEL FOR THE FOURTERMINAL MOSFET 16 3.1 Examples of Engineering Needs for a Model for the LargeSignal Transient Response . 16 3.1.1 Reasons for the Poor Modeling of the Transient Substrate Current by Existing MOSFET Models 17 3.2 Problems Involved in Modeling of Four Terminals Devices 19 3.3 EquivalentCircuit for the Intrinsic MOSFET 2 3 3.3.1 Transport Current 23 3.3.2 Charging Currents 24 3.4 Special Considerations 29 3.5 Modeling of the Extrinsic Components . . 34 3.6 Relation to Existing Models 35 IV STEADYSTATE MOSFET THEORY MERGING WEAK, MODERATE AND STRONG INVERSION 38 4.1 Introduction 38 4.2 Fundamentals 40 4.2.1 Drain Current 40 4.2.2 Charge Components 4 2 4.2.3 Surface Potential 47 iv
PAGE 5
CHAPTER IV (continued) Page 4.3 Drain Current and Charge Components in a Model Merging VJeak, Moderate and Strong Inversion 51 4.3.1 Drain Current 51 4.3.2 Charge Components 57 4.3.3 Limits for the Strong, Weak, and Moderately Inverted Portions of the Channel 59 4.4 Results and Evaluation of the Model ... 62 4.5 Conclusions 74 V FUNCTIONAL DEPENDENCIES FOR THE ELEMENTS IN THE FOURTERMINAL EQUIVALENTCIRCUIT 76 5.1 Introduction 76 5.2 SourceDrain Current Source 77 5.3 Capacitances 78 5.3.1 Expression for the Capacitances . 79 5.3.2 Physical Interpretation of the Results for the Capacitances ... 84 5.3.3 An Engineering Approximation for the Functional Dependencies of the Intrinsic Substrate Capacitances Cgg and Cj^g 92 5.3.4 Engineering Importance of the Intrinsic Substrate Capacitances C ^"d Cj33 SB g^ 5.4 Transcapacitors 98 5.4.1 Expressions for the Transcapacitors 98 5.4.2 Engineering Importance of the Transcapacitance Elements .... 100 5.4.3 Transcapacitances in a ThreeTerminal EquivalentCircuit . . . 106 5.5 Conclusions 109 VI SCOPE AND FUTURE WORK Ill APPENDIX A PROPERTIES OF OUASIFERflI POTENTIALS 115 B APPROXIMATED EXPRESSION FOR THE DIFFUSION/ DRIFT RATIO IN THE MOSFET 119 C COMPUTER SUBPROGRAM TO CALCULATE THE VALUE OF THE ELEMENTS IN THE EQUIVALENTCIRCUIT .... 123 V
PAGE 6
Page LIST OF REFERENCES 128 BIOGRAPHICAL SKETCH 13 2 VI
PAGE 7
Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy EQUIVALENTCIRCUIT MODELING OF THE LARGESIGNAL TRANSIENT RESPONSE OF FOURTERMINAL MOS FIELD EFFECT TRANSISTORS By Jose Ignacio Arreola March 1978 Chairman: Fredrik A. Lindholm Major Department: Electrical Engineering An approach is proposed that yields equivalentcircuit models for the largesignal transient response for all electronic devices described by chargecontrol. The approach is applied to derive an improved equivalentcircuit model for the fourterminal MOSFET. It is suggested that the model proposed gives a better description of the physics internal to the device than was previously available. A static characterization of current and charges in the MOSFET is also proposed that unifies the descriptions of the weak, moderate and strong inversion modes of operation. Predictions of this characterization agree better with experimental results than previous work of similar complexity. The static characterization of current and charges is used to derive functional dependencies for the equivalentcircuit components in terms of applied voltages and physical makeup of the MOSFET. Vil
PAGE 8
CHAPTER I INTRODUCTION Computer simulations of MOSFET digital circuits can disagree severely with measured performance. A particular case of such a disagreement, which results in suboptimal circuit design, is the poor simulation of transient currents flowing in a substrate terminal of MOS field effect transistors [1] . The sources of such disagreements are either in the computer programs in use or in the inadequacies of existing largesignal equivalentcircuit models for the fourterminal MOSFET. The purpose of this dissertation is to derive an improved equivalentcircuit model for the fourterminal MOSFET. Improvements are made in the following aspects of the equivalentcircuit model: (a) the representation of capacitive effects in a fourterminal device; (b) the characterization of the dc steadystate currents and charges; (c) the inclusion, in principle, of twoand threedimensional effects present, for example, in shortchannel MOSFETs.
PAGE 9
As will be seen, all of these improvements are interrelated and result from basing the derivation of the equivalentcircuit model on the internal physics that determines the operation of the MOSFET. We begin in Chapter III by proposing an approach that yields equivalentcircuit models for the largesignal transient response of all electronic devices described by charge control [24] . The relation of this approach to the indefinite admittance matrix of circuit theory offers advantages in the modeling of devices having more than three terminals. Chapter III starts by discussing the problems arising from the fourterminal nature of the MOSFET. Such problems were apparently not previously recognized. For the intrinsic part of the device (see Fig. 3.1), we apply the systematic approach developed in Chapter II. This approach, whose power is emphasized because of the four terminals of the MOSFET, yields a general description of the device that offers improvements (a) and (c) listed earlier. To define fully the equivalentcircuit model of Chapter III, one needs a suitable description of the dc steadystate behavior. Extensive work has been done in the past to characterize operation in the dc steadystate; however, none of this work is completely suitable for the purposes of equivalentcircuit modeling. In Chapter IV, a new model for the dc steadystate behavior is derived that unifies the description of the full range of operation of the device 
PAGE 10
from weak to strong inversion and from cutoff to saturation. The model avoids discontinuities in the characteristics present in all previous characterizations of similar complexity, and shows good agreement with experimental results. The new model also improves the characterization of the charges in the device. In Chapter V we derive, using the results of Chapter IV, the functional dependence of each circuit element in the equivalentcircuit developed in Chapter III. In Chapter V we also assess the engineering importance of the improvements introduced in the equivalentcircuit model for the MOSFET and propose possible simplifications of the model. Chapter VI treats possibilities for future research.
PAGE 11
CHAPTER II A NONLINEAR INDEFINITE ADMITTANCE MATRIX FOR MODELING ELECTRONIC DEVICES 2.1 Introducti on This chapter describes a new approach for developing equivalentcircuit models of electronic devices. The models developed by this approach represent the largesignal (hence nonlinear) response to transient excitation. The approach applies to all devices whose operation is described by the principles of charge control [2^], including, therefore, field effect transistors of various kinds, bipolar transistors, and certain electron tubes. The models yielded by the approach are compact, composed of few circuit elements. As a result of their compactness, the models are meant to be useful in the computeraided analysis of electronic circuits. This intended use contrasts with that intended for equivalentcircuit models [5] containing many circuit elements, which pertain chiefly to detailed studies of the physics underlying electronicdevice behavior. The approach to be described applies independently of the number of device terminals. Indeed, the greater that number, the more the power of the approach is disclosed.
PAGE 12
The approach applies also independently of multidimensional spatial dependence that may be present in the boundaryvalue problem describing the device. This generality is needed, for example, in modeling the MOS field effect transistor (MOSFET) , because the substrate terminal constitutes a fourth terminal through which sizable transient currents flow in some circuit applications, and because shortchannel devices give rise to multidimensional effects. Models of fourterminal devices [6,7] and models that include multidimensional effects [8] have been proposed earlier. But this previous work has not focused on laying down systematic procedures for developing models, which is the aim of this chapter. Systematic procedures exist for modeling the linear response of multiterminal circuits subjected to smallsignal excitation. These procedures are linked to the indefinite admittance matrix (lAM), which we first shall review and then exploit to model the nonlinear response of multiterminal electronic devices to largesignal excitation. 2 . 2 Indefinite Admittance Matrix Consider a lumped electrical network which has n terminals. Let an additional external node be the common reference. From the standpoint of its behavior at the terminals, the network, if linear, may be described by a set of equations as follows:
PAGE 13
I yv . (2.1) The required linearity is assured for any network operating under smallsignal conditions. The matrix elements of y are 'i^'k (2.2) v^=0, iT^k where 1_ and V correspond to the current and voltage at the terminals. The matrix y defined in (2.1) and (2.2) is called the indefinite admittance matrix [9,10] , and its elements satisfy the following property imposed by Kirchhoff's laws I ^jk = I ^jk = (2.3) that is, the elements in any row or any column sum to zero. As will be seen, our development of largesignal models for electronic devices will make use of two special cases of the lAM. In the first case, the matrix y is symmetric and has one of the following forms: y=a y=b^ y= c/dt . (2.4) Here a, b, and c are real symmetric matrices, and each matrix element corresponds to a single lumped resistor or capacitor or inductor connected between each pair of the n terminals. In the second case, the matrix y is nonsymmetric , but is the sum of two indefinite admittance matrices: a symmetric
PAGE 14
matrix, like (2.4), and a residual nonsymmetric matrix, each element of v/hich corresponds to a controlled current source placed between each pair of terminals. In this second case, then, the circuit representation of the lAM results from connecting the network corresponding to the symmetric matrix in parallel with that corresponding to the nonsymmetric matrix. In general, summing of indefinite admittance matrices corresponds to connecting their circuit representations in parallel. 2 . 3 Extension for Nonlinear Electronic Devices Consider an electronic device having n terminals. The modeling begins by specifying the physical mechanisms relevant to the operation of the device. For many devices, only three such mechanisms, at most, are relevant: the transport of charged carriers between terminals; the net recombination of charged carriers within the device; the accumulation of these carriers within the device. Thus the current i flowing at any terminal J is the sum of three components: a transport current (i ) , a recombination curJ T rent (i ) , and a charging current (i ) . That is J K J Lij (ij^T + (ij)R ^ (ij)c (2.5) We now characterize these components. The transport mechanism consists of the injection of a charged carrier in one terminal, followed by its transport
PAGE 15
8 across the device until it reaches any of the other terminals, where it recombines at a surface with a carrier of opposite charge. The recombination mechanism differs from the transport mechanism only in that the carriers recombine within the bulk of the device instead of at the terminals. Therefore, both mechanisms can be characterized by the same form ^\l^T,R = ^Ij (ijK)T,R Â• (2.6) Here i^^^ represents the current due to the charged carriers injected from terminal J, which recombine, at a surface or in the bulk, with oppositecharged carriers injected from terminal K. From this characterization, it follows that (ijp.),p j^ satisfies the following properties: ^JK ^ ^KJ ijj = Â• (2.7) These properties allow transport and net recombination to be represented by controlled current sources connected between pairs of terminals. The value of the current source between terminals J and K is ijj^. The last mechanism to be considered is the accumulation of mobile carriers within the device, which requires the charging current (^j^c = arÂ• (2.8) As Fig. 2.1 illustrates, dq^ is the part of the total charge accumulated within the device in time dt that is supplied
PAGE 16
Fig. 2.1 The charging current (ij) st terminal J produces the accumulated charge dq .
PAGE 17
10 from terminal J, The charge accumulation expressed in (2.8) is a mechanism basic to any electronic device that operates by charge control [24]. Now, using (2.6) and (2.8), we may rewrite (2.5) as dq 'j = ^Ij (^Jk)t,R ^ dFÂ• (2.9) Although (2.9) is valid, it does not correspond to a convenient network. To get a convenient network representation, we apply one additional constraint which costs small loss in generality in that it holds for all chargecontrol devices [24]. We apply the constraint that the overall device under study is charge neutral. Or, more exactly and less demanding, we assume the device accumulates no net overall charge as time passes. This constraint of overall charge neutrality requires a communication of the flux lines among the terminals to occur that maintains charge neutrality by coulomb forces and by drift and diffusion currents. The requisite overall neutrality may result either from neutrality occurring at each macroscopic point, as in a transistor base, or from a balancing of charges that are separated, as on the gate and in the channel of a MOSFET. As a result of overall neutrality, the current at any terminal J becomes the sum of the currents flowing out of all of the other terminals h ^l iK Â• (2.io:
PAGE 18
11 This global counterpart of the Kirchhoff currentnode law implies for the charging currents of (2.8) that (ij)c = J._^ (iK)c (2.11) which means that a charging current entering one terminal flows, in its entirity, out of all of the other terminals. Hence, as is true also for the transport and recombination mechanisms, charge accumulation can be represented by a controlled current source connected between each pair of terminals . For a model to be useful in circuit analysis, the elements of the model must all be specified as functions of the terminal currents and voltages. To do this, we now make use of the principles of charge control [24] and of the closely allied quasistatic approximation [6,7,11]. For the transport and recombination mechanisms, charge control gives directly (ijK^,R = ^jr/^jk Â• (2.12; Here q is the charge of the carriers that contribute to the current flowing between terminals J and K. The recombination time tjT, is the time constant associated with that current: a transit time if the mechanism being described is transport, a lifetime if it is recombination. Then, to produce the desired functional dependence, a quasistatic approximation [6,7,11] is used that specifies each
PAGE 19
12 ^^JK^T R ^^ ^ function of the instantaneous voltages at the device terminals. This characterization of (ij^)^ ^, combined with the properties expressed in (2.7), can be manipulated to describe transport and recombination by an lAM, like a in (2.4). Because i^^^ = ~^kj' ^^^ matrix is symmetric. There are two network representations of transport and recombination described by this matrix. As noted before, just below (2.7), one of these consists of controlled current sources connected between pairs of terminals. Another network representation consists entirely of nonlinear resistors, R^^ = (v^v^^) /i^^. Similar procedures are applied to model charge accumulation. To the charging current defined in (2.8) a quasistatic approximation is applied [6,7,11], specifying the functional dependence of q^ on the terminal voltages and enabling thereby the employment of the chain rule of differentiation. The resulting characterization of ^"'"J^C ^sscribes charge accumulation by a matrix that has the form of b in (2.4), a matrix whose elements are 8qj (2.13) dVj=0, IT^K VK 9V K Matrix b also satisfies the key properties of the indefinite admittance matrix that are given in (2.3). For a general nterminal electronic device, this matrix describing charge accumulation is nonsymmetric , and is therefore the sura of a
PAGE 20
13 symmetric and a residual nonsymmetric part. The symmetric part corresponds to an allcapacitor network; the network representation of the residual nonsymmetric matrix consists of controlled current sources. 2 . 4 Conclusions From the properties of the JAM it follows that the threebranch circuit of Fig. 2.2 serves as a building block for model generation. Connecting a circuit of this form between each terminal pair yields the general network representation for an nterminal electronic device. For any particular device of interest, certain of the circuit elements may vanish. In a MOSFET, for instance, no transport or recombination currents flow to the gate, and the corresponding circuit elements will be absent. Any equivalentcircuit model generated by this approach can be regarded in two ways: either as a product of the building block of Fig. 2.2 or as a circuit described by a matrix which obeys the key properties of the lAM. Description by the lAM treats all terminals equally in that none is singled out as the reference node; the advantages of this will show up plainly in the modeling of a fourterminal device, such as the MOSFET. From Fig. 2.2 notice that the mobile charge accumulation within a general nterminal electronic device is not represented by the flow of displacement currents in an all capacitor model. The residual nonsymmetric matrix, and
PAGE 21
14 'JK = ^Jk/^JK (3qj/9v^ 3qK/9Vj) ^ Fig. 2.2 General equivalentcircuit between each pair of terminals of an nterminal electronic device
PAGE 22
15 the corresponding transcapacitance current source of Fig. 2.2, provides the needed correction. This correction has practical engineering consequences in certain MOSFET circuits although a discussion of that is postponed for a later chapter. To use the approach set forth here in modeling any particular device requires that the static dependence on the terminal voltages be specified for the currents and charges defined in (2.12) and (2.13). This requires that a physical model for the device be chosen to describe the dc steady state. For the MOSFET this has been done, and the corresponding equivalentcircuit model is derived in the following chapters.
PAGE 23
CHAPTER III EQUIVALENTCIRCUIT MODEL FOR THE FOURTERMINAL MOSFET The main contribution of this chapter is the derivation of an equivalentcircuit model for the fourterminal MOSFET by use of the method described in Chapter II. The resulting model is intended to represent v/ith good accuracy the largesignal transient currents flovzing through each of the four terminals of the device, including the substrate terminal. 3 . 1 Examples of Engineering Needs for a Model for the LargeSignal Transient Response In many digital integratedcircuit applications of the MOSFET, the substrate terminal of each device is connected to a power supply. This connection serves at least two purposes: it provides a means to control the threshold voltage of the device, and it enables a good layout of the circuit [12,13]. In a largescale integrated circuit, the large transient current flowing through a power supply can result in poor voltage regulation and poor circuit performance unless both the circuit and the power supply are properly designed. An optimum design of a circuit will provide the maximum density of components on the chip consistent v/ith 16
PAGE 24
17 the requirement that the voltage regulation of each power supply remains acceptable. To design circuits using computer aids therefore requires that one has available a set of equivalent~circuit models for the MOSFET that adequately represent the transient currents flowing through the terminals in response to largesignal excitation of the devices. According to engineers involved in such designs, such models are not now available [1]. This absence of accurate models forces the engineer to suboptimal designs, by v;hich we mean less densely packed circuits than those that could be designed if accurate enough device models were available. 3.1.1 Reasons for the Poor Modeling of the Transient Substrate Current by Existing MOSFET Models The substrate current during transients arises from capacitive effects in two regions of the device (Fig. 3.1) : the depletion region around the source and drain islands ( extrinsic substrate capacitances) ; and the depletion region underneath the inversion channel ( intrinsic substrate capacitance). In general, however, the substrate current is modeled as arising only from the pn junction (extrinsic) capacitances around the source and drain islands. These capacitances have the form [13] : C. = . ''j".^ (3.1) D 1 f"
PAGE 25
GATE SOURCE I) irn'n r n N iiiiiinn/iimn /i})ii > )}j })>)))>)> i ) i >>>> > )TTi j A DRAIN t )}l):\}llltr X N intrinsic region extrinsic region \}!> > !ii)>}}> )>)> mr SUBSTRATE Fig. 3.1 An nchannel enhancement MOSFET divided into intrinsic and extrinsic parts.
PAGE 26
19 where V is the applied junction voltage,
PAGE 27
20 dQ dQ + odv (a) dO, (b) Fig. 3.2 Illustration of the paths of cominunication between terminals in a twoand fourterminal device .
PAGE 28
21 tween the terminals and thus there is no uniqueness in the charge that flows at the terminals. The charge that flows at each terminal is do. A nonuniqueness does occur, however, in devices with more terminals. Consider now a fourterminal device. From Fig. 3.2(b) one sees that there are six paths of communication of the flux lines among the terminals in a general fourterminal device. Thus, suppose one applies a small voltage between any two terminals while appropriately terminating the other terminals so that charge can flow through them. Then one must account properly for the apportionment of the charges amongst the terminals. Of the total charge dQ that flows, what will be the charges dQ_f '^^2' ^'^3 ^^*^ ^^04 flowing at each of the four terminals? There is a second related problem. One way of seeing this problem is to suppose that v/ithin the box of Fig. 3.1(b) , for the time being, is an a 11capacitor network. Then apply a small voltage between terminals 1 and 3, having shorted the other terminals to an arbitrary reference. In response, a certain amount of charge flov/s at terminal 4. Now interchange the roles of terminals 3 and 4. That is, apply the small voltage at terminal 4 and measure the amount of charge flowing past terminal 3. The result of this experiment is that one finds exactly the same amount of charge as before. That is a property of a reciprocal network, of which an allcapacitor configuration is an example [16], Now if one does the same experiment with a MOSFET one finds that this reciprocity does not apply, as we shall prove
PAGE 29
22 in Section 5.4. The reason is that terminal 3 represents the gate and terminal 4 represents the substrate; and the gate and substrate are highly different physical structures. This asymmetry in physical structure introduces a nonreciprocity in the network properties not present in an allcapacitor network. To account for this asymmetry, therefore, one should expect that the netv/ork representation for a MOSFET must contain elements describing mobile charge accumulations in addition to capacitors. To manage these problems one requires a systematic approach. In Chapter II we have developed a methodology that permits one to obtain a lumped network representation of multiterminal electronic devices obeying the principles of charge control whose largesignal transient behavior depends on three physical mechanisms: mobile charge transport, net recombination within the device and mobile charge accumulation. The result is the equivalentcircuit of Fig, 2.2, which applies between any pair of terminals and is the basic building block from which an equivalentcircuit can be constructed for the overall multiterminal device. The currents representing transport and net recombination flow in the current source i . The charging current representing mobile charge accumulation flow through the capacitor C^j, = dq^/'dVj and through the controlled current source characterized by djj^ = 9qj/3vj^ 3qj^/9Vj. To apply this methodology to the MOSFET, one needs only to describe the components of charge accumulation dq.
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23 in each region and the transport and recombination flow in terms of the physics underlying the device behavior. We will now apply this methodology to the MOSFET. 3. 3 EquivalentCircuit for the Intrinsic MOSFET For concreteness, consider the enhancementmode nchannel MOSFET illustrated in Fig. 3.1. A central idea in the equivalentcircuit modeling is to resolve the electronic device under study into two parts [11] : an intrinsic part where the basic mechanisms responsible for the operation of the device occur, and an extrinsic part which depends on the details of the device structure. For the particular MOSFET under consideration this is done in Fig. 3.1. The behavior of the intrinsic region in the MOSFET is described by charge control [24] , and thus an equivalent circuit of its operation can be obtained by applying the methodology described in Chapter II. 3.3.1 Transport Current At normal operating voltages and temperatures the leakage current in the insulated gate is negligible and the recombination/generation rate in the channel and in the substrate can be neglected. Charge transport occurs, therefore, only along the highly conductive inversion channel induced at the semiconductor surface. This transport mechanism is represented in the equivalent circuit as a controlled current source i ^ connected between source and drain. Its explicit functional dependence in terms of the
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24 physical makeup and the terminal voltages is obtained by using a quasistatic approximation to extrapolate the steadystate functional dependence of the drain current 1^. This will be considered in Chapter V. 3.3.2 Charging Currents If we neglect recombination and generation, the current flowing in the substrate terminal i is solely a charging current, that is, current that changes the number of holes and electrons stored in the intrinsic device. Thus, if during time dt a change dq^ occurs in the hole charge stored in the substrate, then ^^B ^B dtÂ• (3.2) Similarly, neglecting any leakage current in the insulator, the current flowing in the gate i is only a charging current. If this current changes the charge of the metal gate by dq^ in time dt, then ^G d^ Â• (3.3) The current flowing at the source terminal consists of two components. The first component changes the electron charge stored in the channel by an amount dq in time dt. The second component arises from electrons that, flowing in from the source, pass through the channel and then out of the drain terminal. Thus,
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25 ^S " dF + 1 SD (3.4: similarly, on the drain side. dq 1_ = D D dt "SD (3.5) The total change of charge in the inversion channel dq is then ic + 1 dqg ^ dq^ dq^ + S "^ ^D ^t~ ^ dt ~ dt (3.6) If we apply a quasistatic approximation [6,11] and then use the chain rule of differentiation, equations (3.2) through (3.5) can be expressed in the following matrix form: 'D "B 3v^ dq D 3^S gvg 3v^ 9qq 9qc 3v D 9q D 9v D sq^ 8v D 3q B 3V D 3v, 3q D 3v, 3q, 3v, aq B 3v, 3qc 3v B 3q D 3v B 3q. 9v B 3q B 3v B V, V D V, V B + 1 ;d SD (3.7; Here the dot notation designates time derivatives. By applying the constraint that the overall intrinsic device is charge neutral, one can prove as is done in Chapter II [17,18] that the first matrix in (3.7) satisfies the properties of the indefinite admittance matrix of network theory [9] . That is, the sum of all the elements in any row or column is equal to zero.
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26 The matrix description of (3.7) together with the building block of Fig. 2.2 yields, therefore, the general largesignal equivalentcircuit for the intrinsic fourterminal MOSFET. This network representation is shown in Fig. 3.3 and its elements are defined in Table 1, Elements in addition to capacitors that represent charging currents appear in the circuit of Fig. 3.3. These transcapacitors would be zero only if the matrix in (3.7) were symmetric. That is, if dq^Sv^ = dq^/dv^ for all J and K. The physical structure of the MOSFET, however, is nonsymmetric and hence one should expect that the elements ^JK ^^^ ^^ general nonzero. This is the case, indeed, as it will be shown in Chapter IV where we calculate the functional dependencies of these elements in terms of the applied voltages and the device makeup. The transcapacitive elements in the network representation can be also seen as related to error terms yielded by an ideal allcapacitor model. In this sense, we will study and assess their importance in Chapter V. In the circuit of Fig. 3.3, the capadtive effects between source and drain are represented by a capacitor C and a controlled current source characterized by d . In the theory of operation of the MOSFET based on the gradual case [19], it has been shown [20] that there are no capacitive effects between source and drain. In this work, we will consider this to be the case and therefore we will assume aqg/'dVp = dq^/^v^ = 0, implying
PAGE 34
27 Eh M Cm O s o H 0) C Â•H ^ +J C H Q) O U U I rH to > H 0) n3 C O 00 Â•H
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28 Table 1 Definitions for the elements of the general equivalentcircuit for the MOSFET. CAPACITANCES ^SG =
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29 ^SD ^ Â° ' ^SD = Â° Â• (3.8) In more detailed characterizations of the device for example, the ones including channel length modulation [21] and twodimensional effects in short channel devices [8] the drain voltage directly influences the charging of the channel and capacitive effects between source and drain as modeled by Fig. 3.3 may need to be included. 3.4 Special Considerations For the equivalent circuit model in Fig. 3.3 to be useful in circuit analysis we require that all the elements, current sources and capacitors, must be specified as functions of the terminal current and voltages. In doing this, as indicated in Chapter II, we will use the quasistatic approximation [6,11], which is based on the steady state operation of the MOSFET. A particular detailed model for steadystate operation is considered in Chapter IV and the functional dependencies for the elements of Table 1 will be derived in Chapter V from this model. However, before approaching these problems, v/e must give special consideration to two charge components that are not described in the conventional steadystate characterization of the device: the contributions from the source and the drain, dqÂ„ and dq_ , to the total charging of the channel. To gain physical insight as to how dqÂ„ and dq^^ contribute to the charging of the channel, consider the following.
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30 If we apply a change in the gate voltage, a change of the charge in the channel dq^^ will occur. The electrons necessary to supply this additional charge are injected into the channel by charging currents flowing in fron the source and drain, that is ^s ^ ^D = dF dq dq dq + D 'N dt dt (3.9) The contributions of dq^ and dq^^ to dq^^ are, in general, unequal and depend, as we shall see, on the operating conditions of the device. Figure 3.4 shows a simplified energy band diagram at the surface of an Nchannel MOSFET under various operating conditions determined by the magnitude of V . Consider first the case when V^^ = and AV^ is applied. Because the barrier height that the electrons have to overcome in both sides of the channel is equal (Fig. 3.1(b)), we expect that charging currents flowing into the source and drain ends will be equal. dq, dt V^=0 dq D dt (3.10) V^=0 Now apply a small Vj^>0 and change the voltage by AV . As in the previous case, electrons are injected from both sides of the channel. The electric field produced by the application of V^, however, will present an additional barrier height for the electrons injected from the drain side (Fig. 3.4(c)). Thus we expect the charging current in the source to be larger
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31 r N + HI 1 1 1 1 1 II 1 1 1 1 inti 1 1 It I I } I J ) II 1 1 I II 1 1 1 1 1 1 I ,,,,,,, , , V D li' "'i I " N E Fn E (a) Equilibri um E. V \ (b) v^ = 0, Vg ^ (c) V^ small, Vg 7^ \ \ Fig. 3.4 Energy band diagram at the surface of a MOSFET under the effect of applied drain and gate voltage.
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32 than the charging current in the drain. That is. dq, dt vo dq D dt Vo (3.11) For larger values of V^^ the device will be eventually driven into saturation. The high electric field produced near the drain will impede charging of the channel from that end (Fig. 3.4(d)). Hence, the additional electrons required when AV^ is applied will be supplied mainly from the source end. That is. ^%^ dt SATURATION (3.12) A similar argument can be employed to explain the contributions of dqg and dq^^ to the charging of the channel due to changes in the substrate voltage. From the above discussion we can define an apportionment function X such that the source and drain charging currents can be expressed as dqg, dt ^^V ^S'^D dt V , V S' D and ^%. dt = (1X) ^S'^^D ^%V dt \ J ^S'^D (3.13) (3.14) The apportioning function A takes values from 1/2 to 1 between the conditions of V DS and saturation,
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33 A convenient expression for X results from combining its definition in (3.13) with the indefinite admittance matrix that characterize the charging currents in (3.7). Using chain rule differentiation in (3.13), Â— 2. v^ + Â— 5. V = A dv^ G 3Vg B Â— ^G ^ 8Vr ^B l3^G B (3.15) This equation must remain valid for any value of Vq and Vo, Thus A = 3qs/8Vg + 8qg/avg 3q^/3v^ + ^V^^B (3.16) By using the properties of the indefinite admittance matrix, the numerator and denominator of (3.16) can be rewritten as: and 3q, 3qr dv 3q B 3v, S ^ ^%^ 3v aq 3v, n aq + B 3v, DJ + 3q D 9^Sl 3v, 3v D 3q N 3q N 3v, 3v B 3q N 3q N 3v, 3v D 3q( 3v^ 3q + B 3qr 3q B 3v, 3v D 3v D (3.17) (3.18) Substituting (3.17) and (3.18) into (3.16) and, using the definitions in Table 1, we obtain,
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34 ^ = c +r Â— Â• (3.19) , DG ^DB C + C SG ^SB Here, we have used the assumption that no direct capacitive effects exist between source and drain Oqe/3v = 3q /dv ^ 0) Equation (3.19) can now be used to obtain the functional dependencies of equivalentcircuit elements involving dq and dq^^ directly from an extrapolation of the steadystate behavior of the device. From Table 1, these elements are, 3qg aq^. 3qj^ ^SG = ^ 3Vr ^ ^ + Cg^^ (3.20) DG 9vÂ„ 3v^ ^ 9v^ ^ ^DG (3.21) dgB = dg^ (3.22) ^DB ^ ^SB (3.23) ^GB = ^SG + ^DG (324) Equations ( 3 . 22) ( 3 . 24 ) have been simplified by direct application of the properties of the indefinite admittance matrix. 3. 5 Modeling of the Extrinsic Components The extrinsic components depend on details of the fabrication of a specific type of MOSFET. In many cases,
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35 the extrinsic part can be modeled by inspection of the geometry of the device. Elements commonly found are: overlapping capacitances due to the overlap of the gate oxide over the source and drain islands; bonding capacitances resulting from metalization over areas where the oxide is relatively thick; PN junction capacitances arising from sourcesubstrate and drainsubstrate diffusions; and resistance components due to finite resistivity at the source, drain and substrate. In general, these elements are distributed capacitances and resistors but can be transformed to lumped elements by applying a quasistatic approximation. Lindholm [11] gives the details of the general approach for modeling extrinsic effects in a fourterminal MOSFET. For particular devices, the details of the extrinsic modeling have been worked out in the literature [7,22]. 3 . 6 Relation to Existing Models A v^ide variety of equivalentcircuit models of different complexity and accuracy have been advanced for the MOSFET [7,11,20,22,23]. The general development of these models follows a partially heuristic and partially systematic approach that consists in interpreting in circuit form the different terms of the equations describing the device physics. The definitions of the elements in these circuit models depend on the particular approximations of the physical model involved.
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36 In contrast, the equivalentcircuit of Fig. 3.3 and the definitions of its elements in Table 1, having been developed from a methodology based on fundamentals, are quite general. For example, the new network representation can take into account two and threedimensional effects such as those in the shortchannel MOSFET . To use the model one needs only compact analytical descriptions of these effects in physical models for the dc steadystate. Such descriptions, we anticipate, will appear in the future. Indeed, as new physical models for dc behavior appear, such as the one presented in the next chapter, the equivalentcircuit developed here is designed to make immediate use of them to yield new and better network representations of the largesignal transient response of the MOSFET. Most of the past v/ork in equivalentcircuit modeling of the intrinsic MOSFET neglects the effect of charging currents flowing into the substrate terminal. Among the models that consider these effects, the treatment of Cobbold [6] is the most detailed. His model, derived for smallsignal applications, involves an equivalentcircuit in which the charging effects are represented by four capacitors (sourcegate, sourcesubstrate, draingate and drainsubstrate) and a controlled current source (gatesubstrate) . As can be observed in the general equivalentcircuit between any two terminals shown in Fig. 2.2, the representation by a capacitor alone of charging currents
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37 between two terminals requires certain specific conditions related to symmetry and apportionment of charge in the device to be satisfied. For example, if the terminals are the source and the gate a singlecapacitor representation between these terminals would require 3qÂ„/3v^ = 9q_/3vÂ„. Because of the physical asymmetry of the MOSFET, these requirements are, in general, not satisfied. This problem was apparently not recognized by any of the previous workers in the field.
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CHAPTER IV STEADYSTATE MOSFET THEORY MERGING WEAK, MODERATE AND STRONG INVERSION 4 . 1 Introduction In Chapter III we have developed a circuit representation for the transient behavior of the intrinsic fourterminal enhancementmode MOSFET. Each circuit element in this representation depends on the constants of physical makeup of the MOSFET and on the voltages at the terminals of the intrinsic device in a way that is determined by the static model chosen to represent the current and the inversion, substrate and gate charges. To complete the modeling, therefore, one must choose a static model for this current and these charges that is general enough to be suited to whatever circuit application is under consideration. None of the static models previously developed are suitable for this purpose, for reasons that will be soon discussed. Thus the purpose of this chapter is to develop a model that has the properties required. One necessary property of the static model is that it represents the entire range of operation to be encountered in various circuit applications, including the cutoff, triode, and saturation, including operation in weak, moderate and strong inversion, and including fourterminal operation.
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39 The model of Pao and Sah [24] comes nearest to this ideal. It covers in a continuous form the entire range of operation. However, its mathematical detail makes it inconvenient for computeraided circuit analysis, and it does not include the substrate charge and the influence of the substrate terminal. The Pao and Sah model has provided the basis for other modeling treatments. Swanson and Meindl [25], and Masuhara et al. [26] have presented simplified versions covering the entire range of operation. Their approach consists in a piecewise combination of models for the limits of weak and strong inversion. This approach introduces discontinuities in the slopes of the characteristics for moderate inversions, which are computationally undesirable. These models, furthermore, do not include charge components and the influence of the substrate terminal. Following a different line of reasoning ElMansey and Boothroyd [2 7] have derived an alternative to the Pao and Sah model. Their work includes charge components and fourterminal operation. However it also is mathematically more complicated than is desirable for computeraided circuit design. The goal of this chapter is to develop a model that includes : (a) fourterminal operation; (b) cutoff, triode and saturation regions; (c) weak, moderate and strong inversion; (d) current and total charges.
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40 The model, furthermore, should avoid the discontinuities of a piecewise description while maintaining enough mathematical simplicity for computeraided circuit analysis. In Section 4.2 a review of the general fundamental of MOSFET operation are presented. A discussion, in Section 4.2.3, of the relation between the surface potential and the quasiFermi level for electrons sets the basis of our approach. In Section 4.3 expressions for the drain current and the total charge components are derived. To assess the validity of our approach, the predictions of our model for the drain current are compared against experimental data in Section 4.4. In the last section vie include a discussion of the limitations of the model. 4. 2 Fundamentals 4.2.1 Drain Current In an nchannel MOSFET, illustrated by Fig. 3.1, the steadystate drain current density J (x,y) is essentially the electron current density in the inversion channel [24]: J (x,y) :. J (x,y) = qy NE + q D ^ '' n' ny^ndy = q^n N ay (4.1) where V = V^Vp is defined as the difference between the quasiFermi potential for electrons V^^ and the quasiFermi potential for holes Vp . Because there is no significant hole current flowing in the device [28] V is nearly constant
PAGE 48
41 and coincides with the bulk Fermi potential, (j)p = kT/q In N^^/n . . The voltage V is referred to as the "channel voltage" [ 20] , and at the boundaries of the channel, y=0 and y=L, it has the values V(0) = V and V(L) Vjj. These and other properties of V will be derived in Appendix A. The total drain current is obtained by using the gradual channel approximation [19] : ^D = ^ JÂ„(x,y)dx = Zy Q ^ D ' ^ ' ^n n dy (4.2: Here Z is the channel width, y is an effective mobility, and Q is the electron charge per unit area in the inversion channel defined by Qj^ = q N dx . (4.3) ^ The differential equation in (4.2) is solved by integrating along the channel ^D = fZy^C no (4.4) where L is the effective channel length and C is the oxide o capacitance per unit area. The effects due to mobility reduction and channel length modulation have been studied in detail by different authors [21,29,30]. They could be included in this work by appropriately modifying y and L,
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42 42.2 Charge Components For the purposes of equivalentcircuit modeling it is convenient to divide the charge distribution within the intrinsic device in three components: charge associated with the gate, charge in the bulk and charge in the inversion channel. In the charge associated with the gate we include: the actual charge in the metallic gate (C V J , the fixed charge in the oxide Q^^^, and the charge due to surface states at the oxidesemiconductor interface Q . inspection of the energy band diagram of Fig. 4.1 shows that this effective gate charge Q^ per unit area can be expressed as QQ ox , Â•o ^S CÂ„ + C Cr= ^G *MS ^S ^ r^ + C^ (4.5; where 0^g (= ^^x^q{E^E^) <^^) is by definition the metalsemiconductor work function, ^g is is the surface potential and V^ is the applied gate voltage. m this work we will assume that the charge in the surface states Q is in'ss dependent of voltage. It has been demonstrated, however, that when the device is operating under low voltage conditions [31] the voltage dependence of Q^^ becomes important in determining the relation between surface potential and external applied voltages. A typical characterization of Qgg is given by [32] : Qss = ^"SS ^"^s"^) (4.6)
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43 E^/q Ej/q ^Wl_ 1^ E^q Oxide Semiconductor Fig. 4.1 Energy band diagram under nonequilibrium conditions. All voltages are referred to the substrate Note that qV = E ^ n Fn' ^^P ^ Epp, and qVj = E^,
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44 where N^^, representing the surface state density per unit area, is used as a parameter to obtain improved fit with experiment. Typical values for N^g are on the order of 1x10 cm eV [25,31,32]. The work presented here can be modified to include this effect. The charge in the bulk consists mainly of ionized atoms and mobile majority carriers. In a psubstrate device the bulk charge Qj^ per unit area can be approximated by Qb q(PNz^)dx (4.7) To solve this integral equation, one can change the variable of integration to the potential V^ (x) by using the solution to Poisson's equation for the electric field. This procedure requires numerical integration of (4.7). In the present analysis we will obtain an analytic solution by assuming that, because of its "spikelike" distribution [33], the mobile electron charge in the channel has a negligible effect on the potential distribution. Although this approximation is only strictly valid under depletion or weak inversion conditions, it serves also as a good approximation under strong inversion conditions because the major contribution to Qj_^ in strong inversion comes from the uncompensated and ionized impurities in the depletion layer [34]. Using the approach described above, we obtain
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45 K *sVb ^ f 8(*3Vb) e 1 1/2 Here , K lit V Â•S B q 1/2 (4.8: K 2^^sNaa11/2 C o (4.9: is a constant that depends on fabrication parameters. The exponential term in (4.8) results from integrating the contribution to the charge density of the mobile holes in the substrate P/N^^^ exp [6 (ijgVg) ] . This yields Pg/N^. = exp [3 (ijJg'Vg) ] where Pg is the density of holes at the surface. For the regions of interest, depletion to strong inversion, this exponential term can be neglected. The charge in the inversion channel, defined by (4.3) , the charge in the gate, and the bulk charge are all related through a onedimensional Gauss' law, which requires Q + Q + 0, = n g b (4.10) The total charge components are obtained by integrating Q , Q, and Q along the channel: g' b n ^ Total gate charge. Qg = 2 QÂ„ dy ^ (4.11)
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46 Total substrate charge. Qb = Z Qh dy (4.12) Total inversion charge. % = (VQb) (4.13) Or, alternatively, we may change the variable of integration to the channel voltage V by using (4.2), V Qb ZL Â•ZL D V, Q Q dV g n V D Qn = ZL I* D ^ V, .^D QbQn dV V, Q^ dV n (4.14) where I* D "D Zy C n o (4.15) is a normalized drain current. The dimensions of I* are: (volts) . Similar expressions have been obtained by Cobbold [ 7 ] by assuming drift only. In contrast, (4.14) includes the effect of drift and diffusion which, as we shall see, is necessary in obtaining a model for the complete operating range of the MOSFET.
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47 4.2.3 Surface Potential The complete characterization of the charge components per unit area and Q^^ requires the functional relation between the surface potential ip^ and the applied voltages. This relation is established by applying Gauss Law, ignoring the ycomponent of electric field, which requires that the effective charge in the gate be the source of the xdirected electric field in the semiconductor. That is. Qg = ^S^x x=0 KC^ F(^g,V,Vg,4)p) (4.16: The function F (t) v, V^ , (}>Â„) is the normalized electric field at the surface obtained from the solution to Poisson's equation. This solution has been worked out by several authors for the case when Vg = [24]. if extended now to the case when a bias voltage VÂ„ is applied, we find that F(iJ;3,V,Vg,c})p; kT q 1/2 kT/q 'S B + + e 3 (i;gV2<,p) i(;gVg 62((,j, 6(VVg+24,^; kT/q e 1/2 (4.17) For the usual substrate doping, VV^+2(p is always much larger than kT/q. Furthermore, if we neglect the majority carrier concentration at the surface (Pg<
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48 where Q Q \7 ' Â— T7 J, 1 ox s s The solution of the integral equations defining the current (4.4) and the charge components (4.14), in which the variable of integration is the channel voltage V, requires the functional relation between ()g and V. This relationship, however, has not been found in closed form and hence the possibility of direct integration of (4.4) and (4.5) is excluded. A numerical integration can be performed [24] but, because of the large computer times involved, we will look for an approximation that will yield an analytic solution. Let us consider some important characteristics of the functional relation between ijj^ and V that will set the basis for our approach. Figure 4.2 shows the solution for ijj obtained from (4.18) for a specific device having x = 2000A 15 3 ^'^ ^AA " 1^10 cm . Figure 4.2 shows that, for values of V^ for which !jg(0) is below 2(j)p, ^^ is nearly independent of V. For V^ such that ^^(0) > 2(^^, ^^ increases almost linearly with V provided as it is shown below, that drift dominates in determining the channel current. For V greater than a certain critical voltage, however, diffusion begins to dominate and dijg/dV > 0. This characteristic behavior can be explained by studying the relative importance of the drift and diffusion components along the channel [24] : a
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49 m P O 2 > en
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50 Â°n ^^/^y ^ ^ ^^s/^^ ^AA Ps V ^n ^ ^y ^i>s/<3i'^ ^ N^ " N^ (4.20) which is derived in Appendix B. For V' such that i, (0) < 26 , the channel is weakly inverted (Ng< 2({)p the channel near the source is strongly inverted (Ng(0) >> N^^) ; then near the source, (4.20) implies that drift dominates and thus di,,./dv ^1. As we move toward the drain, the electron concentration decreases, the channel becomes weakly inverted and there again diffusion dominates and d^g/dV > 0. The channel voltage for which the channel becomes weakly inverted corresponds approximately in the strong inversion theory [ 7 ] to the pinchoff voltage. At higher gate voltages, the channel remains strongly inverted in its entire length and drift is the main mechanism. In the strong inversion theory this corresponds to nonsaturated operation. The behavior of ijg as described above has been used to establish two approximations often used in characterizing MOSFET behavior: the strong inversion and the weak inversion approximations. In the strong inversion approximation, which is applied when Ng(0)>> N^, the surface potential is assumed to be related to the channel voltage by .u ^ = \/+2 [71. Because then dij^^/dV = 1, this assumption is equivalent to neglecting contributions due to diffusion mechanisms near the drain. in the weak inversion approximation, which is applied when Ng(0) < Uj^ the surface potential is assumed to be in
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51 dependent of voltage if^g = xjj^iO) [31]. Then dij; /dV = 0, and, therefore, drift mechanisms near the source are neglected Although these two approximations produce satisfactory agreement with experiment in the strong and weak inversion limits, they fail for moderate inversion (N =^N ) where neither of the criteria used in strong or weak inversion can be applied. In the following section we will relax the strong and weak inversion approximations by using the basic properties of difj /dV. As shown in the previous discussion, these properties relate to the degree of inversion in the channel. As we shall see, the resulting model not only will merge the operation in the strong and weak inversion modes, but also will provide a firstorder approximation for moderate inversion. 4 . 3 Drain Current and Charge Components in a Model iMerging Weak, Moderate, and Strong Inversion 4.3.1 Drain Current In Section 4.2 we found that the drain current could be expressed as c Z u C ' '^ n o 0' dV . (4.21) ^S " Here, and in the rest of the chapter, the notation O' is used to designate a charge component divided by the oxide capacitance
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52 per unit area C . The dimensions of 0' are volts. By using the condition of charge neutrality (4.21) can be rev/ritten as n ^V^b)' I* D V. Q^ dV + V, V, Qb d^ (4.22) In equation (4.22) a very convenient change of variables can be introduced by noting from (4.5) and (4.8) that dQ' g dy dy d7 ^V'^s) = di) S dV dV dy d__ dv K ^S B q 1/2' T.2 di> S dV 20^:^ dV dy (4.23) Thus, dV dQ' (dijg/dV) 20'/K^ (dijjg/dV) (4.24) Substituting (4.24) in the expression for the current, we obtain I* D Q'(Vd) Qg(^s) O do Â—2 ^_ + (d;;Â„/dV) Qb^^D) "b^^s) (dij^g/dV) (4.25) Figure 4.3 shows the elements constituting the integrands in (4.25) for a specified device operating in the pinchoff mode. This represents the most general case because the channel is strongly inverted at the source and becomes
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53 strong Â• inversion moderate inversion v;eak inversion W Fig. 4.3 Components of charge per unit area and surface potential as functions of the channel voltage V,
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54 weakly inverted toward the drain. As discussed in Section 4.2.3, d^g/dV has almost constant values along the channel; in the strongly inverted portion dijg/dV ^ 1 while in the weakly inverted portion dij^ is there linearly related to V with the value for the slope d^^g/dV lying between and 1. Our approach will consist then in dividing the channel into three regions by defining appropriate limits V and V as shown in Figure 4.3. Below V^ the channel will be assumed to be strongly inverted with di>^/dV = Sg, a constant. Above V^ we will consider the channel to be weakly inverted with dil^g/dV S^^^, a constant. In the transition region the channel will be assumed to be moderately inverted with di^g/dV = S^, also a constant. These approximations allow us to write the expression for the current as the sum of the contributions in each region. Furthermore, because di^g/dV is assumed constant in each case, it can be taken out of the integrals which can then be directly evaluated. If we define a function, F^, related to I by
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55 lab Q' (V, ) g b O' (V ) Â•g' a' O' dQ' + g ^g Qb^^b^ b a "b b Q ,2 V V Â•* a b ^.3 ^ 4 V b V ^ a (4.26; then the expression for the drain current becomes ^D = Â— s; ^ Â— ^^ Â— ^ Â— s^ Â— 'M 'W (4.27; The three components of (4.27) result from carrying out the details of the integration indicated in (4.21). Here, if we let V, V^ and SÂ„ = 1, (4.27) reduces to the conventional expression (obtained by using the strong inversion approximation) for the drain current of a device operating in the triode mode. In computing the drain current from (4.27) , a numerical problem could occur in evaluating the term corresponding to the weakly inverted channel because S is very small. To avoid this problem an alternate form for this term can be obtained as follows. The channel charge was defined in 'n (4.3) as Qn <5 N (x,y) dx (4.3) Taking derivatives on both sides with respect to y yields
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56 do ^oo ^ ^ J ^ (4.28) but, because N = n^ exp [ 3 (V^V^) ] = n. exp [ 3 (V VV ) ] , it follov?s from the gradual approximation [19] that dN ^ dy N Si dV 1 _ ^ uv kT/q [ dV J dy (4.29) Substituting (4.29) in (4.28), using the definition of Q and reordering the terms, we obtain dV ^ _ kT/q ^Â«; Q^ (ld,;g/dV) (4.30) From (4.30) the contribution to the drain current from the weakly inverted channel can then be alternatively written as V r D 2 kr q QA^^dI q;(v,) do' n (ldifg/dV) (4.31) But since v/e are assuming that dijj^/dv has a constant value S^j in this region, we finally obtain: tV DVl kT q; q (is,,) D (4.32) V,
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57 Here, if we let S^,^ = and ^2 = V<^, (4.3 2) reduces to the conventional expression for the drain current of a device operating in weak inversion [ 31] . 4.3.2 Charge Components The procedure to calculate the total charge components is entirely analogous to the one presented for the drain current. Combining (4.10) and (4.14) and using the change of variable indicated in (4.24) , we obtain for the total charge components Â°h ZL IS r^D V, (Q+ Q^Q^)dv ZL g D Qq(Vs) o'^ do' "g 'g (ld,J;g/dV) + q; (V ) Â°^ 2Q'0'^/K^ do' "g "b "b (ld,jj^^/dV) Qb(Vs) , (4.33) and Ob ZL f"" 2 ZL Qb(^D^ Qb^^s) (ldiJ;g/dV) Â«b(^D) Qi^^s) 2Q'0'^/K^dQ' g 'b b (ldi;g/dV) (4.34 Again, if the channel is divided in three regions and v;e assume that d([)^/dV is constant in each region, the charges can be obtained by direct integration. Let us define functions FÂ„Â„ and F^,, such that,
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58 V^^a'^b) = O' (V. ) g b Q' (V ) g a o' ^ dn' + ^b^^b) ^b^^a) 2Q'Q'2/k2 doQ .3 3K' 2 2Q' g b j^5 5^ iv b V " a and ^QB^^a'^^ 2b(\) Qb^^a) 2Q,3/j^2 ^ , ^ b 'b ^b^^) Qi(^a) 2Q'Q'2/k2 dO' g b ' b (4.35) 2K ,4 9 ^b q: ' + 3K 20 ,5 O'O'' gb K^ V, ^a (4.36) where the second integral in (4.35) and (4.36) was evaluated using integration by parts with u = Q* and dv = Q'^dO'. g b "b' Then the total charges can be expressed as 0' = Â— ^B I* D ^QG^^S'^: ^ !WVV ^ V(^2%: n w >qb1VV_ ^ ^QB^^l'^2) ^ V(V2^d)" M W (4.37) (4.38) As in the case of the drain current, to avoid numerical problems due to the smallness of S^^, an alternative expression can be obtained for the contribution of the weakly inverted portions of the channel. Using (4.30) directly in (4.14), we obtain
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59 Q' GW
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60 In Section 4,2.3 we concluded that di/j /dV could be considered as a measure of the level of inversion along the channel. Here we will show that it is also the ratio of the contribution of the drift current to the total current. In (4.20) we indicated that ^DIFF ^ d^c/^^ Idrift ~ d'J^s/^v ^^'^^ Thus, rearranging terms we obtain '''DRIFT _ ^"^s I 4^T ~ dV~ (4.41) DRIFT ' DIFF We will use this property of d(;^^/dV to define quantitatively the voltages V^ and y^ as follows. In the strongly inverted regions we previously observed that di;g/dV is close to unity and drift dominates while in the weakly inverted regions diffusion dominates with dij; /dV being close to zero. Thus we will define the transition region corresponding to moderate inversion as the region in which both drift and diffusion are comparable. More specifically, we will define V^ as the channel voltage at which the drift current constitutes 80% of the total current and V^ as the channel voltage for which the drift component is 20% of the total current. This specification of V and ^2 provides, approximately, the best leastsquares fit between the piecewise linear approximation and the \h versus ^s V characteristic. Based on these definitions we can now obtain expressions for V^ and y^ by solving
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61 dv" = A (4.42) where A has the value A = 0.80 when solving for V, and A = 0.20 when solving for V^ . Differentiation of both sides of (4.18) yields dij< B(i]jgV24,p) dV = A 20' Â•'g 9 + 1 + e {il>^V2(^p) (4.43; Combining (4.43) and (4.18) and usinq the definition of O , g' v;e find that ?\7 ? V ,V = V' 24)^ Â— Â£n Â— A5+ ^ V 1 2 G F q ^.^2 2 where (4.44) V^ = K V' V + Â— + Â— ^G ^B ^ 4 ^ q f 1 (A'Kl 1 1/2 + kT/q Here, A' = (1A)/A. Hence, A' = 1/4 when calculating V, and A' = 4 when calculating V . Equation (4.44) applies only when Vg < V, , V2 < V . The complete functional dependencies for V and V^ are given by V, V from (4.44) if Vg < V, < V ^s if ^1 < ^S ^D if ^1 > ^D V^ f VÂ„ from (4.44) if VÂ„ < V^ < V_ Z b ^ U 1 V3 if V2 < Vg V D If V2 > V^ (4.45)
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62 Using the functional dependencies for V, and V2 given by (4.45), we nov/ can solve (4.18) to obtain ^ at the limits V^ and V^. The surface potential at those points can be used to define the approximate slopes diiÂ„/dv in each S region, which constitute the second parameter at our threeregion piecewise linear approximation. They are. 3 _ '^S(^) ^s(^s) ' ^ \s 1^5(^2) ^aj^l) ^M ~ V ^^~V^ (4.46) 2 1 ^ ^ ^s^^^ ^5(^2) w V^ V, Here, to calculate i^s(V,,), (i'g(V^), i>^{V^) and 4's(V^), one needs to solve (4.18) numerically. This process does not require much computer time. We used the NewtonRaphson method [35] to calculate the solution and found that less than five iterations were necessary to achieve convergence. 4. 4 Results and Evaluation of the Model Table 2 summarizes the results of the model merging weak, moderate and strong inversion. In Figures 4.4 through 4.7 we illustrate the characteristics for the drain current and the total charge components obtained from the proposed model. Notice that the curves in these characteristics and their slopes are continuous throughout the entire range of
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63 Table 2 Drain Current and Total Charge Components DRAIN CURRENT "D y C Z/L n o = I* D + . 4T* TOTAL CHARGE COMPONENTS 'G 1 ^LCo Ig !WVV_^ V^^^2^' 'M + GW ZLC o ZLC B ^ 1 I* o D WVV , %^^^2' ri o ZLC O Qn ^ Qg ^ Qb = Â°
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64 Table 2 (Continued) FOR THE CURRENT; ^I^^a'^) = 2 3 ^2 rV, V i a I* DW kT/q 1S V r T D FOR THE CHARGES V^^a'^b! Q . 3 rV, 3 3k2 ^g"b V ^OB^^a'^b) Q .4 2K 3K ,2 g b ,V, V I a GV7 ^ _ J^ kT/q ZLC I* 1S o D W O' ^ O' ^ tV D V, BW ^ J_ kT/q ZLC I* 1S O D ^VJ 2 5^ ^ 3 j,2 2 iV V.
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65 N AA X o V D 0.2 1^B m P H O > *Q 0.1 1 inl5 3 1x10 cm 2000A Iv Ov extrapolated threshold I V' (volts) Fig. 4.4 Calculated squareroot dependence of the drain current on gate voltage.
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66 v^ (volts: Fig. 4.5 Calculated drain current as function of gate voltage for three doping concentrations (x = 2000A) .
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67 CN m IÂ— I o > *Q 10 2 10 10 0.1 ^i = ^ V= 1.6 V= 1.4 N AA X o V B 1 inl^ 3 1x10 cm 2000A Ov V^ = 1.2 J. 0.2 0.3 V D 0.4 (volts) 0.5 Fig. 4.6 Calculated drain current characteristics in weak and moderate inversion.
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68 P H O > U N \ o 1x10 cm 2000A Iv Ov V^ (volts) Fig. 4.7 Calculated charge components as function of gate voltage.
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69 operation. This feature results from including the transition region for moderate inversion, v/hich is not included in previous work treating weak [31] and strong [7 ] inversion. Figure 4.4 shows that for strong inversion the functional relation between the drain current and the gate voltage follows a square law [20], while in weak inversion this relation is exponential, as shov/n in Figure 4.5. This behavior agrees qualitatively with previous models for the extremes of strong and weak inversions. In Figure 4.6 the drain current is shov/n as a function of the drain voltage for weak and moderate inversion. The inclusion of drift and diffusion in our model has produced a smooth transition into saturation. The necessity of including diffusion to produce this smooth transition was first recognized by Pao and Sah [24]. The total charge components are shown in Figure 4.7 as functions of the gate voltage. Notice that the inversion charge increases exponentially at low gate voltages. The relationship between the charge components and the terminal voltages has apparently not been established previously for weak and moderate inversion. As is demonstrated in the next chapter, these relationships provide a basis for characterization of the device capacitances and the displacement currents. In assessing the validity of our modeling approach and the accuracy of the expressions developed for the current and charges, we compare the results of our model against
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70 results from previous theoretical treatments. Figure 4.8 shows experimental data for the square root of the drain current against gate voltage obtained in a commercial device (4007) having N^. = 3x10^^ cm~^ and x = lOOOA. AA O In this figure we also show the calculated characteristics obtained from the model just derived. Since and other fabrication parameters are not accurately known for this device, the calculated and the observed characteristics v/ere matched using the value of the voltage and current at the extrapolated threshold voltage. Good agreement between experiment and theory is observed. We also show in Figure 4.8 theoretical characteristics obtained from a model using the strong inversion approximation [7 ]. The discrepancy at low gate voltages betv/een this model and the experimental data arise because the strong inversion approximation assumes that an abrupt transition between depletion and inversion occurs when the surface potential at the source is equal to 2({)p. This results in a discontinuity in the slope of the characteristics at the boundary between cutoff and saturation. A discrepancy also exists at high gate voltages. This arises because the surface potential, which in the strong inversion approximation is assumed independent of gate voltage, is in fact a logarithmic function of V_ . As one can show from equation (4.18) for V^ > V^ , this function can be approximated by T kT ij;e(0) ^ 2
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71 30 CN 71. k^ 20 10 N AA X o V D V B Â• Experiment Our model Strong inversion model [ 7 ] extrapolated threshold I L 1.5 2.5 V^ (volts) Fig. 4.8 Experimental values for the drain current compared with values calculated using our model and using a model for strong inversion.
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72 10 10 10 5 6 HÂ° 10"^ 10 10 9 10ii N AA X o V. V B 7 inl4 3 7x10 cm 1470A 2v Ov Â• Experiment [26] Our model Strong inversion model [ 7 ] VJeak inversion model [31] 0.5 0.5 V ^ (volts) Fig. 4.9 Experimental values for the drain current compared with values calculated using our model, using a model for strong inversion and using a model for weak inversion.
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73 H 10 to E <Â« 10 10"Â° 
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74 The proposed new model includes implicitly this dependence of ij;g in Vq. Figures 4.9 and 4.10, which compare the predictions of our model with experimental data from the literature [26] , show excellent agreement. Because information was available only for the doping concentration IJ . and oxide thickness x AA O in this device, the calculated and the experimental characteristics were matched using the value of the gate voltage and drain current at the extrapolated threshold. In Figure 4.9 we show for comparison previous models obtained for weak inversion [31] and for strong inversion [7]. in Figure 4.10 we compare our model against a recently developed model for the entire range of operation [26]. Although this model shows good agreement with experiment in the weak and strong inversion limits, it fails for gate voltages near the transition region (V^ . 0.lv). Furthermore notice the discontinuities in the slope of the characteristics which our model avoids. 4 . 5 Conclusions The major achievement of this chapter is the analytical description given in Table 2 that unifies weak, moderate and strong inversion and covers the cutoff, triode and saturation modes of operation. This description has the following properties: (1) It includes the effects of substrate bias which enables the representation of fourterminal properties of the MOSFET.
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75 (2) It includes the charges in the gate, channel and substrate regions as well as the drain current. These charges provide the basis for modeling capacitive effects. (3) It consists of simple expressions having continuous derivatives v/ith respect to the terminal voltages. This helps make the description useful for computeraided circuit analysis. The model developed here is subject to the limitations of the onedimensional gradual channel approximation v/hich become severe in MOSFET structures with short channel lengths Other limitations arise from the idealizations used in Section 4.2: effective channel length, field independent mobility and effective charge in surface states. A number of publications in the technical literature deal with more detailed descriptions of these parameters and also with shortchannel effects. As explained in Section 4.2, our model has enough flexibility to incorporate these descriptions .
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CHAPTER V FUNCTIONAL DEPENDENCIES FOR THE ELEMENTS IN THE LARGESIGNAL FOURTERMINAL EQUIVALENTCIRCUIT 5 . 1 Introduction In Chapter III we developed an equivalentcircuit representation for the transient response of the MOSFET. By employing the results of Chapter IV, the functional dependencies of each element in this equivalentcircuit will be now derived in terms of the applied voltages and the fabrication parameters of the device. The main approximation used in deriving such dependencies is a quasistatic approximation through which, as discussed in Chapterms II and III, one extends the knowledge of the dc steadystate behavior of the device to describe its largesignal transient response. The equivalentcircuit for the intrinsic MOSFET derived in Chapter III is shown in Fig. 3.3. The definition for each element in the circuit is given in Table 1. Three types of elements are present: a current source between drain and source representing charge transport, and capacitors and transcapacitors connected between each node representing charge accumulation within the device. In Sections 5.2 through 5.4, the functional dependence of each of 76
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77 these elements is derived. The resulting mathematical expressions are valid for the entire range of operation of the MOSFET, and include the effect of the substrate terminal. Such expressions are new. This chapter also provides the first detailed discussion of the intrinsic capacitive effects of the substrate and the transcapacitive effects due to the nonsymmetry of the fourterminal MOSFET. In Sections 5.3.4 and 5.4.2 we discuss the engineering importance of these two effects. Under certain conditions determined by the particular circuit environment in which the device is used the equivalent network representation can be simplified. An example is discussed in Section 5.4.3. 5 . 2 SourceDrain Current Source Through the use of a quasistatic approximation, as discussed in Chapter III, the functional dependence of the nonlinear sourcedrain current source can be determined by extrapolating the static characteristics of the drain current found in Section 4.3.1. Thus, ^SD = ^D^^S'^D'^G'^B^ ' (^i: which has the same functional dependencies on the terminal voltage as those describing the dc steadystate.
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78 5.3 Capacitances 53.1 Expressions for the Capacitan ices The capacitors in the equivalentcircuit are defined in Table 1 as the partial derivatives with respect to voltage of the time varying total charge components q , q , G B IN" As in the case of the transport current i SD a quasistatic approximation allows us to write ^G = Qg^^S'^D'^G'^B^ ^B = Qn^^S'^D'^G'^B^ (5.2) One can anticipate that a partial differentiation of (5.2) with respect to the voltages would lead to very complicated expressions. But we will now show that because of the systematic approach used in Chapter III to define the circuit elements, one can find simple expressions for the functional dependencies of the capacitors. From Table 1 the capacitors connected to the source are 3Qg 'SG 3v, (5.3) and 8Q B 'SB dVc (5.4) We can use (4.14) to rewrite C SG
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79 V SG ZLC 'O 8v, I* D V, Q' 0' dv g n (5.5) where Q' denotes a charge per unit area normalized by the oxide capacitance C (the dimensions of Q' are volts) . Using chain rule differentiation and the fundamental theorem of integral calculus, SG SLC o iÂ£ Q' 31* ZT 9^Qg(s)Qn(^s) (5.6) But since 3v^ 3v, /d V, q; dv q;(s^ (5.7) we finally obtain SG ZLC o Q;(0) rQl "D zr Qg(o) (5.8) where Q' (0) and Q' (0) are the normalized and gate charge per unit area, given by (4.5) and (4.8), evaluated at the source end (y=0) . Similarly SB ZLC q:(o) n I* D Qb z!%^'^ (5.9) For the capacitances connected to the drain, the approach is the same except that
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80 3^D 3v D 3v D V, D V, Q' dv = 0' (L) n n (5.10) Thus, we obtain and DG ZLC Qg Qg(L) (5.11) DB ZLC o Qb Q'(L) (5.12) where Q^(L), Qg(L) and Q^(L) are the normalized channel, gate and substrate charge evaluated at the drain end (y=L) . The gatesubstrate capacitance is defined in Table 1 as C ^^B 3Qb 'GB Sv, 3v^ Â• (5.13) Substituting (4.14), which gives the functional relation for Qg, and applying the chain rule for differentiation yields 'GB ZLC o T* 3v, G jv. QkO' dv b n (5.14) The expression for C^^ is more complicated than those for 'GB ^SG' "^DG' ^cn. and C SB' 'DB' To find this expression we take v D the partial derivatives, 3l*/3vÂ„ and 3/3v^ (/ " Q^O dv) , u G G 'Â•' vÂ„ b n ' ' using (4.27) and (4.36). The procedure is straightforward, and the results follow:
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GB ZLC o ^D Qb DFj(v3,v^; DF^(v^,v^) DFj(v2,Vj^) M W + ^V^^S'^l^ P^QB^^l^2^ Â°V^^2'^d)", + + = 1 (5.1b, ri w Here we have defined the functions DF and DF as I QB DF^(v^,Vj^) = 1 V Q' + g q; 1 + K' 2Q' g 1 + e 6 ((j)sv2<)p) b V (5.16) Â°V^^a'^b) 2 ^
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82 Table 3 Functional dependencies for the capacit ors SG ZLC o Q' (0) Â—^ (QyZL Q^(0)) DG ZLC. q;(l) ~Tf(Qy^L Q'(L)) YY~ (Qg/ZL Q^(0)) ZLC I* '^B CD DB ZLC o q;(l) I* (Q^/ZL Q^(0)) c GB ZLC o
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83 41 o o > _ fC rH 3 u rH H in Cn Â•H Cm
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84 capacitances in a specific device. In contrast with results obtained from models using the strong inversion approximation [7,36], these curves present smooth transitions between the different regions of operation: cutoff, saturation and nonsaturation. A physically based discussion about the main features of these characteristics is given in the next section. 5.3.2 Physical Interpretation of the Results for the Capacitances Consider first the capacitances connected to the source and drain nodes in the equivalentcircuit. These capacitances are directly related to the apportionment between the currents charging the channel from the source island and from the drain island. To observe how this apportionment occurs, let us consider the total capacitance at the source C given by ^SS = ^SG ^ ^SB = ^ (520) and the total capacitance at the drain C given by %D = ^DG ^ C^B = 3^ (5.21) As we shall see, the functional dependence of these capacitances shown in Fig. 5.2 has the form to be expected from the discussion of the charge apportionment in Section 3.3. In cutoff there is no charging of the channel and both C and ""DD ^^^ equal to zero. As the gate voltage is increased, the channel is turned on in an exponential form (see Fig. 4.5) causing an abrupt change in C^^. At higher gate voltages,
PAGE 92
85 o oil JO srtTun ux seouBi^TOPdEo
PAGE 93
86 while the device is in the saturation region, Q increases almost linearly with gate voltage and hence C is nearly constant. In the saturation region, because there is no charging of the channel from the drain end, C^q = 0. Further increase of the gate voltage drives the device into nonsaturation. Here the channel opens gradually into the drain allowing thereafter an increasing contribution of the drain end to the charging of the channel while the contribution from the source decreases. Thus in this region, as shown in Fig. 5.2, C decreases while C^^ increases. For very large gate voltages the charging of the channel will tend to occur equally from the drain than from the source. When this happens the values of Cgg and Cpj^ tend to one another as shown in Fig, 5.2. A measure of the apportionment of the contributions of the drain and source islands to the charging of the channel is given by the apportionment function X defined in Chapter III as X = = . (5,22) 1 + DG DB DP c + c c SG ^SB ^SS This function is used in the next section to obtain expressions for the transcapacitances. Its functional dependence for a particular device is shown in Fig. 5.3. In saturation, ^DD ~ Â° ^^^ A = 1, while in nonsaturation the values of C and Cg2 approach one another and A tends to 1/2.
PAGE 94
87
PAGE 95
Note from Fig. 5.1 the similarity between the characteristics of the substrate capacitances Cgg and C^g and the characteristics of the gate capacitances CÂ„Â„ and C^^. This similarity, which also can be observed in the expressions defining these capacitances, will be used in the next section to obtain an engineering approximation for CÂ„Â„ and C^^^. SB DB Consider now the gate substrate capacitance Cgg = 3Qg/3v^. This capacitance is related to the control of the gate over the substrate charge. In cutoff, where Vg is not large enough to turn on the channel, this capacitance is equal to the capacitance of a (twoterminal) MOS capacitance [37]. As V^^ increases, an inversion channel starts forming at the surface of the semiconductor and more field lines emanating from the gate will terminate in the inversion channel. Thus, CÂ„^ will decrease as shown in Fig. 5.1. For larger gate voltages, where a strong inverted channel is formed over the entire length of the intrinsic device, the gate will exert even less control over the substrate charge and C^^ decreases at a faster rate reaching eventually a zero value as illustrated in Fig. 5.1. Figures 5.4 and 5.5 show the total gate capacitance CÂ„^ and the total substrate capacitance CÂ„_ together with their DSD components, 'GG 3v, .30^ 903 9"g 3^g (5.23) 30 B BE 3v B 3Qn ^ 3^r3v B 3v B (5.24)
PAGE 96
89 P Â•{ o > > 0) o > +J H O m m p c c o g o u +J Â•H c m o c (d p H u (0 o rO Â•'^ Â•H o u ^ p rH u in o OTZ JO sq.Tun UT aoueq^TOPdeo IT) Â•H
PAGE 97
90 0) ii p ri O > Â• o OTZ 50 st^Tun UT aoue:^TopdeD u o m en +J C cu c o a E o o w c u c m p Â•H u u p LT)
PAGE 98
91 In the cutoff region there is no inversion channel and C^^ and Cgg are equal. Their functional dependency is that of an MOS capacitance [37]. In the saturation region, the gate charge depends almost linearly on the gate voltage (Fig. 4.7), and C shows a constant value of about 2/3 ZLC as predicted by strong inversion theory [20] . In this region, as V is increased, the surface potential increases producing a widening of the depletion layer; consequently Cgg decreases as shown in Fig. 5.5. At the onset of the nonsaturation region C^^ abruptly rises due to the increase of electron concentration over the entire channel length. For even larger gate voltages C^_ approaches the value of the total oxide capacitance. In this region, CÂ„_, attains BB a constant value because the substrate charge becomes independent of gate voltage. This constant value cannot be clearly determined from the expressions of the substrate capacitances just found. In the next section, hov\7ever, we discuss an approximation for the substrate capacitance that permits a good estimation of their values for engineering purposes. The main features of the functional dependencies for the gate capacitances in the MOSFET have been predicted by previous authors [7,20] using simplified models. Our results agree qualitatively with these predictions, giving additionally a detailed and continuous description for these capacitances and also for the substrate capacitances.
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92 53.3 An Engineering Approximation for the Funct ional "De pendencies of tne Intrinsic Substrate Capacitances The functional dependencies for the substrate capacitances Cgg and Cj^g were derived in Section 5.3.1. Figure 5.1 shows these functional dependencies together with the functional dependencies for the gate capacitances and the gatebulk capacitance. We pointed out previously the similarity between the functional dependencies of the gate and substrate capacitances appearing in this figure. From an engineering point of view, this similarity is advantageous because it suggests the existence of relations of the form: SB "S ^SG ^DB "d ^DG (5.25) where a^ and a^ may be simple functions of the voltages. Such relations would allow considerable simplification in the computation of the substrate capacitances. In recent engineering applications [1], C^^ and C^^ are modeled to a first order approximation as ^SB Â« *^SG ^DB " ^DG (5.26) with a being a constant. Because expressions for C and SB 'DB we re not previously available this approximation has not been verified. With the functional deoendencies for Coo and ^DB ^^'^^ available in the previous section v.'g can now study
PAGE 100
93 this engineering approximation. Figure 5.6 shows a^ and a j~), defined in (5.25), as functions of the applied voltages Notice that although in the nonsaturation region Â«Â„ and a D are practically independent of the gate voltage they are in general not constant. Using the functional dependencies for CÂ„Â„ and C^^ given in Table 3 we will now derive an improved approximation for ag and aj^ that shov;s a better functional dependence on the applied voltages while remaining a simple function of the voltages. Consider first ac ^SB Q^^^ Qb^Â°) S C SG Q'/ZL Q^(o; (5.27) Substituting the expression for Q ' , 0' and I^ given in (4.4) B G D and (4.14) a^ ^^n be rewritten as D V, Q;(v) (Q^(v) Q^(Vg)) dv a, ^v D V, Qj!i(v) (Q^(v) Q'g(v^) ) dv (5.28) The integrals in (5.28) can be approximated by a series solution using the trapezoidal rule for the integration. A numerical comparison between the exact solution and the series solution shows that by taking only the first term in this series we can obtain an approximation that is both simple and accurate:
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94 V^ (volts) "s'%
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95 a 2 ~ Qg(^D^ Qg(^s) (5.29) If we repeat the procedure for aÂ„ we find that a^ = ag = a (5.30) We can express a and a in terms of the external voltages by employing the expressions for Q and Q, given in Section 4.2. If we use the strong inversion approximation ijjg = v+2({)p in (4.5) and (4.8) these expressions become Qg = ^o^^G ^ 2<}>f) (5.31) Qb = KCo(v + 2(f,p Vg) 1/2 (5.32) Substituting in (5.30) we obtain a = K V DS L (V DS + 2<^F ^BS^'^' (2*p V33)^/25.33) 2 1/2 where K = (2qegN^^/CQ) . In saturation v g must be sub2 stituted by the saturation voltage: v (= v' 2,^ + K /2 DSS K(v^ Vg + K^/4)^/^) Figure 5.6 shows the approximation in (5.33) as a function of voltage. In Fig. 5.7 we compare the functional dependencies for C^^ and C^Â„ obtained from Section 5.3.1 with the one obtained by using (5.25) and (5.33). In both figures, good agreement is shown between the approximation and the more detailed expressions.
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96 o p IÂ— I o > > Â• C Â• > in r\ o c > o c H H V u c 0) o en H p 1+! ra o e Â•H c o O !h H a CO Q. CO fC 0) in 0) ax: X p QJ c DIZ go sq;Tun ut aoupt^Toedeo in P ri T3 P fd p o IÂ— I C U H W P3 :3 Q u CD 13 Â•p a 3 pa U CO u x: +j PQH Q5 Â• U Q Â•r! 0) fO fO c a rO mg coo C/2 u u a in Â•H
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97 5.3.4 Engineering Importance of the Intrinsic Substrate Capacitances C ^ p and C p^^ From a simplified theory of MOSFET operation [20] the approximated value of the gate capacitances C_,_, and C^Â„ are SG DG 'SG 2/3 ZLC in saturation o 1/2 ZLC in nonsaturation :5.34) 'DG 1/2 ZLC o in saturation in nonsaturation (5.35) where ZLC is the total oxide capacitance. The simplified theory does not include the intrinsic substrate capacitances, However, an excellent estimation of their value can be obtained by using the approximation discussed in Section 5.3.3, From (5.25) we obtain C SB 2a/3 ZLC^ a/2 ZLC o (5.36) C DB a/2 ZLC (5.37) with a defined in (5.33) as a = ^ (2qÂ£gn^^) o (^DS^^^F^BS^^''^ (2*F^BS)''''' V DS (5.38: To study the importance of the substrate capacitances we will consider a particular device with N AA 1x10 cm "^ and
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98 x^ = 2000A. The functional dependency of a in the terminal voltages for this case is shown in Fig. 5.6. Notice that the value of a in this example is about one half which implies values of C^^ and C^g of about one half the magnitude Â°^ ""SD ^^'^ "^DG" However, because the value of a is directly related to the square root of the substrate doping N and to the oxide thickness (equation (5.38)), the relative value of Cgg and C^^ will also depend on these parameters. For instance, if the doping concentration is changed from 15 15 1x10 to 4x10 in our example the functional dependence of a shown in Fig, 5.7 would be shifted upwards by a factor of two. This would result in values of C^d and C^^^ of about the same magnitude of CÂ„Â„ and C^^. For even larger values of N^^ or x^ the values of C^^ and C^^^ would exceed the values of C^^ and C^^^. Thus, for the doping concentrations considered in this example, the values of C Â„ and C SB DB exceed the values of CÂ„_ and C^^. oCj DG Notice from equations (5.36) through (5.37) that C SB ^""^ ^DB ^^^ proportional to aC^, but, because a is directly related to l/C^, Cgg and C^^g are independent of C^ . This contrasts with Cg^ and C^^ which directly depend on C . 5. 4 Transcapacitors 54.1 Expressions for the Transcapacitors In Chapter III the transcapacitances were defined (equations ( 3. 20) (3 . 24) ) as:
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99 9Q SG DG GB ^SB = ^ 3v N + C SG 9Q ^DB = ^1^) 9v '^SG ^ ^DG N + C. 'DG (5.39) To transform these definitions into functional dependencies of the voltages v/e require only an expression for 9Qj^/9vq in terms of the terminal voltages. Taking derivatives with respect to v^ in (4.14) which defines 0,,, we obtain ^Qn
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100 The complete expression for 9 Q /3 v is then N G 3Vn I* % ' ^^I^^S'^1^ Â°^I^''l'^2^ '^^I^''2'''Dh ^S Sm % M W (5.43) Here, the function DF^{v^,v^) was previously defined by (5.16). The functional dependencies for the transcapacitances are illustrated in Fig. 5.8 for a specific device. In this figure notice that the value of the transcapacitors is about one order of magnitude smaller than the value of the capacitors (Fig. 5.1) except when the device is operating in weak inversion (between cutoff and saturation) . In contrast with the current flowing in the capacitors, the current flowing in the transcapacitors is not determined by the voltage across their terminals. Thus, the relative value of the transcapacitors with respect to the capacitors is not enough to assess their importance in the equivalentcircuit. In the next section we will consider the engineering importance of the transcapacitors in the overall equivalentcircuit. 54.2 Engineering Importance of the Transcapacitance Elements The derivation of the equivalentcircuit for the MOSFET in Chapter III demonstrated the need for circuit elements in addition to capacitors for the network representation of charging currents. This need arises from the basic asymmetry of the physical structure of the device. The
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101 0.1 o CJ m o Ui p Â•H c c H 0) u c p H o u m (d 0.1 V' (volts) Fig. 5.8 Calculated values for the trans capacitances .
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102 additional circuit elements can be regarded as correction terms to an allcapacitive network representation. Thus, we can study the importance of the transcapacitance elements in the overall netv/ork representation by considering, at each terminal, the ratio of the charging currents flowing through transcapacitors to the charging currents flowing through capacitors. From an analysis of the equivalentcircuit of Fig. 3.3 and using the properties of the indefinite admittance matrix describing this network, we obtain the following expressions for the relative importance of the transcapacitive currents: (is^T
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103 basic building block of Chapter II (Fig. 2.2) was used to construct the complete equivalentcircuit for the MOSFET . In equations (5.44) through (5.47) notice that the relative importance of the transcapacitors depends on the particular circuit environment in which the device is used. For example, if the particular circuit environment IS such that Vq = Vg the contribution of the transcapacitors to the total charging currents at any terminal would be zero. A contrasting example is a circuit environment such that v^ 7^ vÂ„ and Cj hi ^DG^G = ^DB^B ^G "d Â• ^n ^ Â— r TTT^ = ^T + ^i ^^ (5.48) Â° ^DG "*" ^DB 1 + Â«D 1 + Â«D ^ where a^ = Cj^g/C^^^. From equation (5.45) one can see that in this case the charging currents flowing through the transcapacitors would be the main contribution to the total charging current at the drain. To illustrate the practical importance of the transcapacitors, consider the input device of the inverter circuit shown in Fig. 5.9. In this figure there are also shown qualitative sketches of the waveforms of a particular applied excitation and the response of the circuit. If we apply (5.44)(5.47) , TY^=^ (5.49) ^ S' C ^SG
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104 4V . 1 GG DD ^BB >
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105 ^^D^ ^V^ DG 'DG V. DD V. V + 1 + C DD DB V, (5.50) GB V C DD DG V + (CgQ+C[~,(^+CQg) (5.51) (^B^C :5.52) An inspection of Figs. 5.1 and 5.8 shows that the maximum values for these ratios will occur in the vicinity of the transition between cutoff and saturation and between saturation and nonsaturation. As an example consider V /V = 1. In the vicinity of the transition between cutoff and saturation (V' 1.5v), we obtain (^D^C 0.36 (^B^C (i G^T (^G^C 0.20 (5.53) = while, in the vicinity of the transition between saturation and nonsaturation (V' :^ 3v) , we obtain G 0.06 0.05 (^B^C ^ 0.04 = (5.54)
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106 These numbers represent approximately the maximum percentage of error involved when evaluating the charging currents at each terminal using an allcapacitive network. Thus for this particular example the neglect of the transcapacitances would result in error of about 36% in calculating the charging current flowing in the source and errors of about 20% and 4% in calculating the charging currents flowing in the gate and in the drain. Although the example used for illustrating purposes is simple, it indicates the potential engineering importance of the transcapacitors for calculating rise times and other related behavior. The use of the transcapacitors adds complexity to the programs in computer aided design. Detailed studies involving computer simulations in largescale integration are necessary to assess the practical engineering importance of these elements. This, however, lies beyond the scope of this work. 5.4,3. Transcapacitances in a ThreeTerminal EquivalentCircui"t ~ ~~ ~~~ Among the circuit environments in which the MOSFET is used, one in which the source and the substrate are connected together as shown in Fig. 5.10(a) is often found in practical applications. In this case we will show that transcapacitive effects are not necessary to model charging currents in the MOSFET and therefore the equivalentcircuit reduces to the result [11] for a threeterminal device shown in Fig, 5.10(b). In Section 3,2.2 we proved that charging currents in the MOSFET can be represented by an indefinite admittance
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107
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108 matrix (equation (3.7)). The interconnection of source and substrate imposes two constraints in the circuit: a constraint on the currents (i becomes iÂ„+iÂ„) , and a contraint on the voltages (Vg = Vg) . Consider these constraints applied to the indefinite admittance matrix in equation (3.7). The constraint on the currents requires adding the first and fourth rows. The constraint on the voltages requires adding the first and the fourth columns. The matrix resulting is then, ^S + ^B "D 3qc 3q + B 3q. dq. 3v, + 3v B 3v D 3q D 3q D 9q D 3Vc 3v 3q 3v, + B G 3v D 3qc 3v3 3Vr 3qQ 3q( 3v B 3v D 3v, Vc V D V, + SD 1. "SD (5.55) This matrix still satisfies the properties of the indefinite admittance matrix. Furthermore, if we neglect capacitive effects between source and drain (3qe/3v = 3q /3vÂ„ + 3qi)/3Vg = 0), one can see that this matrix is symmetric. That is. 3qg 3qQ aq^ avg 9Vj3 (5.56)
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109 Therefore, the network representation of this symmetric matrix consists only of capacitors [16]. One of these capacitors , SG 3q^ ^ a^G 3Vg avg, (5.57; is connected betv/een source and gate, and the other, 'DG (5.58) is connected between drain and gate, as shown in Fig. 5.10(b) 5.5 Conclusions In this chapter we derived the functional dependencies for the elements in the equivalentcircuit developed in Chapter III: a currentsource representing charge transport, and capacitors and transcapacitors representing mobile charge accumulation. The main achievement of the chapter is the derivation of functional dependencies for these circuit elements that are valid and continuous in the entire range of operation of the device. This constrast with similar previous work [11,20,28] which applies only for the strong inversion operation of a threeterminal MOSFET. Another result of this chapter is an approximation of engineering importance that permits the calculation of substrate capacitances directly from knowledge of the gate capacitances. The approximation is both simple and accurate.
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110 The importance of the transcapacitances was discussed from an engineering point of view. If the transcapacitances are neglected, they can be a potential source of error in certain circuit applications. Work is needed to assess the practical importance of the new equivalentcircuit model in computeraided MOSFET circuit analysis. As a step in this direction Appendix C gives a computer subprogram that uses the results of this chapter to calculate the circuit elements as functions of the external voltages.
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CHAPTER VI SCOPE AND FUTURE WORK In this study we have proposed a methodology for developing models for the largesignal transient response of nterminal electronic devices, and have applied it to a particular device, the fourterminal MOSFET. The principal contributions achieved are summarized in this chapter together with recommendations for future research. In Chapter II we presented a systematic modeling approach that applies to nterminal electronic devices obeying the principle of charge control [2,3]. It is based on an extension of the indefinite admittance matrix from network theory. The approach is especially useful when modeling devices with three or more terminals. Its power is emphasized in Chapter III where we applied it to the fourterminal iMOSFET, The methodology considers the physical mechanisms commonly involved in the operation of electronic devices: the transport, net recombination, and accumulation of mobile charge. If analogous mechanisms occur in other systems, for example, in the chemical, physical, societal, and biological sciences [38] , then to these systems the methodology could be also applied. Such extensions are beyond the objectives of the present work. Ill
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112 Chapter III presents a new largesignal equivalentcircuit model for the transient response of the MOSFET (Fig. 3.3) that differs in a fundamental sense from those developed by the more intuitive schemes previously used. It includes tv/o effects not fully included in previous work: (1) the apportionment of charge amongst the four terminals, and (2) the properties resulting from the asymmetry of the physical structure of the MOSFET. The equivalentcircuit model applies generally, subject only to the validity of the quasistatic approximation. In principle it can include twoand threedimensional effects such as those occurring in shortchannel MOSFETs. Static descriptions for these effects have been already advanced [8] and more can be expected in the future. Work toward applying such descriptions is in order. Chapter IV describes the development of a dc static characterization for the behavior of the MOSFET. Its results offer advantages to computeraided circuit design over previous static characterizations. The dc steadystate description covers the entire range of operation: weak to strong inversion, cutoff to saturation. Â• It includes the effect of the substrate and the substrate terminal. It contains compact expressions for current and charges with continuous derivatives with respect to the terminal voltages. The static characterization is used in Chapter V to obtain the functional dependencies for the equivalentcircuit elements. The results apply to the entire range of operation
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113 of the MOSFET. Good agreement with sone experimental results is shown; a more comprehensive experimental confirmation is in order, in particular, with respect to the capacitors and transcapacitors. Research is also necessary to investigate the practical value of the new equivalentcircuit in computeraided circuit design. A central approximation on which the results of our study depend is the quasistatic approximation (QSA) . This approximation underlies not only our work but nearly all the models in common use for either bipolar transistors or MOSFETs in computeraided circuit analysis. It is widely used because it yields compact network representations. Recent v/ork by Frazer and Lindholm [39] has indicated, however, that under certain circumstances the QSA, for some of the MOSFET models now widely used in computeraided design, fails to be consistent with itself. This lack of selfconsistency can produce errors in computing system parameters such as the turnoff propagation delay time. By including in our nev; model a selfconsistency test proposed by Lindholm and Frazer [40] , one can detect under which particular applications the QSA is violated. The violations are related to the transit time across various regions of the device, and therefore to the thickness of the device region under consideration. Thus an approach for eliminating the inadequacies of the model in these situations is to divide the intrinsic region into subregions and model each of them using the approach of Chapter II [41] . The
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114 complete equivalent circuit is then the series connection of the models of each region. The size and number of these regions could be determined by the magnitude of the violation of the QSA. More research is needed to assess the suitability of this approach. Another approach for eliminating the inadequacies of the QSA results from abandoning this approximation in the modeling process. Such an approach, based on approximating Shockley's six basic differential equations [42] by finite differences, has been proposed by Sah [43] but has never been applied to the MOSFET. Future work could include applying this approach to the MOSFET.
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APPENDIX A PROPERTIES OF QUASIFERMI POTENTIALS The objective of this appendix is to explore some properties of the quasiFermi potentials in semiconductor materials. These properties are used in the analysis of MOSFET behavior presented in Chapter IV. Consider a long piece of sem.iconductor material and apply a voltage V across its terminals as shown in Fig. A1. C X V Fig. A1 This system must satisfy Faraday's law expressed mathematically by E'd2. = (A1) or t Ed2, + ;) Eda = . semiconductor (A2) external circuit 115
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116 In the semiconductor, E =^ E^ = dV /dx, where V is the electrostatic potential. Thus, Faraday's law requires rh V Mx L dV, d3r ^^ = ^l(L) V^(0) (A3) When studying semiconductor devices a very important tool of analysis results from defining electron and hole quasiFermi potentials V and V as follows [42]; ^^I q^" HT 'N V. VkT Â„ P ^ ^^ ^ (A4) Here, N and P are the electron and hole concentrations. Vie wish to find the functional relation between the applied external voltage and the electron and hole quasiFermi potentials. From (A4) V,(x) = V^(x) + ^ in ^(^) n (A5) Substituting this equation in (A3) we obtain for electrons ^a = V^(^) V,(0) ^f ^nl^ (A6) If we impose the constraint that the contacts between the semiconductor and the external circuit are ohmic , then at the boundaries x0 and x=L the electron concentrations must have their equilibrium values
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117 N(L) = N^(L) E N(0) = N^(0) E (A7) Because (A6) must hold in equilibrium, when V^ = and the quasiFermi levels for electron and holes coincide with the Fermi level which is position independent, we obtain which implies (A8) Ng(L) = Ng(0: (A9) That is, the equilibrium concentration of electrons is equal at both ohmic contacts. Notice that this interesting result is not restricted to homogeneous material. For holes, one can show that a similar result is obtained. If we now substitute (A7) and (A9) in (A6) we obtain V, = V^(L) V.,(0) 'N N (A10) Analogously, for holes ^a " ^P^^) ' ^P^Â°^ (A11) The relationships in (A10) and (A11) are very useful to establish boundary conditions when analyzing the physics of a semiconductor device as will be shown for the particular case of the MOSFET. In the analysis presented in Chapter IV of the nchannel MOSFET we defined a "channel voltage" related to the quasi
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118 Fermi potentials by ^ = ^N ^P (A12) Because there is no significant hole current in this device, Vp can be considered constant. If we denote the position of the ohmic contacts in the source and drain islands by 0' and L' , then (A10) requires % " ^q = ^N^^') ^M^O') = ^(L) V(O') (A13) or D S N" ' N V(O') V + c V(LM=V, .c . '^"' where c is an arbitrary constant which for convenience we will set equal to zero. The source and drain islands are heavily doped and the quasiFermi levels there are independent of position. In these regions, therefore, V is also independent of position and at the boundaries of the channel, y=0 and y=L (Fig. 3.2), we have V(0) Vg (A15) V(L) = V^ These are the boundary conditions for V used in Chapter I V,
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APPENDIX B APPROXIMATED EXPRESSION FOR THE DIFFUSION/DRIFT RATIO IN THE MOSFET In this appendix we will discuss an analytical justification to the expression for the diffusion/drift ratio given in equation (4.20): ^n ^''/^ _ 1 ^^s/^^ ^AA^S ^AA y^ N E^ d^g/dV N3 N3 Â• (^20) We will begin by proving the first part of this equation. The ratio of the diffusion to the drift component along the channel in a MOSFET is ^DIFF Â°n dN/dy '^DRIFT ^n ^ ^y VJe can express the electron concentration N in terms of the electrostatic potential V and the quasiFermi potential for electrons Vj, by using B(VjV ) N = n^e ". (B2; Inspection of the energy band diagram in Fig. 4.1 shows that VjVp = 4j(j)p. Because V = ^^'^P ' ^^ follows that V V i)V()p. Thus N can be rewritten as 119
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120 N = n . e Biiiv^^) (B3) Taking derivatives with respect to y in this equation we obtain, dN N dy q/kT 'djjj_ _ dV" dy dy (B4) From the gradual chaanel approximation [19], ^ = dij ;(x,y) ^ d4,(0,y) ^ ^ dy dy dy dy (B5) Substituting (B5) and (B4) in (B1) , using Einstein's relation (D/y = kT/q) , and simplifying common terms we finally obtain D^dN/dy dV/dy dij)g/dy 1 dij^^/dV ^n ^^ ^Y di;g/dy d\l)^/dV (B6) which proves the first part of (4.20). To prove the second part of (4.20) we will use the relationship between the surface potential ij; and the applied voltages obtained from Gauss' law in Section 4.2.3: ^i ^'S = ^ ^'S ^B + f ^ 6(iJ>gV2(j)p) ^nl/2 ^ ' 1 (4.18)
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121 Taking partial derivatives with respect to y in both sides of (4.18) we obtain d^3 ^6(iPsV2^j,) dV~ " 2Q^^ Bi^V2^F) ^^"^^ V 3+ I + e Rearranging terms. 1 diijÂ„/dV 2QVk2 + 1 b q dijjg/dV B(ii;gV2<,p) e (B8) The exponential term in (B9) is related, as we shall see, to the electron concentration at the surface. From (B4), 6 (<^cV())p) Ng = n^e ^ (B9) because N^^/n^ = e ''^ , the exponential term in (B9) is therefore equal to N /N . We can then write 1 d,^ /dV N d4.s/dV = (2Q/k2.1) ^ (B10) The term (2Q'/K^ + 1) (2 (V^^j^) /K^ + l) varies along the g (j o channel within the same order of magnitude (Fig. 4.3) while N changes by several orders of magnitude. Therefore, the behavior of the diffusion/drift ratio is strongly dominated by changes in NÂ„ and thus, for purposes of illustrating the main features of this ratio, we can take
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122 1 dj.g/dV N, _^^_^___ AA (B11) This justifies the second part of (4.20)
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APPENDIX C COMPUTER SUBPROGRAM TO CALCULATE THE VALUE OF THE ELEMENTS IN THE EQUIVALENTCIRCUIT The subroutine that we list in this appendix employs the results of Chapter V. The input to the subroutine is the voltage at the terminals: VÂ„ , V , V_ , V^ . The output b U (j B is the value of the elements in the equivalent circuit: the current source I , the capacitors CÂ„Â„, CÂ„_, C_Â„, C^^ , U oCj DO DG DB C^Â„ and the transcapacitors denoted by TÂ„^, TÂ„^, T_^, T_^ GB ^ ^ SG SB DG DB and T . The parameters BK = K (defined in equation (4.9)) , Go TK = kT/q, and TFIF = 2 kT/qJln ^^tka/^j. '^^^^ ^^ entered in a COMMON block. 123
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124 * * * * 10 SUBROUTINE EQCKT ( VS , VD , VG , V3 , IE , +ID,CSG,CSB,CDG,CDB,CGB,TSG,TSB,TDG,TDB,TGB) COMMON BK,TK,TFIF REAL LAf4BDA, LIM, LIMl , LIM2 , IDS , IDM, IDVJ, ID CHARGE PER UNIT AREA QG(U)=VGU QB (U) =BK*SQRT (UVBTK) QN{U) =(OG(U)+QB(U) ) DEFINE FUNCTIONS USED IN SUBROUTINE G(U)=QG(U) **2/2. B(U)=2*0B(U) **3/(3.*BK**2) GG(U)=OG(U) **3/3. BB(U)=QB(U)**4/(2*BK**2) GB(U)=2*(QG(U)*QB(U)**3+2*OB(U)**5/(5*BK**2))/(3*BK**2) DSG(U,V)=1./(1+.5*BK**2* {1+EXP( (UVTFIF) /TK) )/OG(U) ) DSI(ABK) =TK*ALOG((2./ABK)*(TK/ABK+SQRT(VGVB+BK**2/4 + + TKMTK/ABK**21.))) ) VX(LIM) =VGTFIFLIM+BK**2/2.+ BK*SQRT ( VGVB+BK**2/4 . +TK* (EXP (LIM/TK) 1) ) MINIMUM VALUE OF GATE VOLTAGE FOR ONSET OF INVERSION N=NI ; PSISVB=FIF IE=2 IF (VGVB .LT. TFIF/2.+BK*SQRT(TFIF/2.)) RETURN ESTABLISH LIMITS FOR WEAK MODERATE STRONG INVERSION LIM1=DSI (BK/4. ) LIM2=DSI(4. *BK) V1=VX(LIM1) V2=VX(LIM2) VDSS=VX(0.0) COMPUTE SURFACE POTENTIAL U AT THE LIMITS AND APPROXIMATE THE SLOPES IN THE THREE REGIONS ESTIMATION FOR US TSI=V2+TFIF FG=(VG**2VGBK**2* (TFIFVB) )/(BK**2*TK) IF (FG .GT. 0.0) TSI=TFIF+TK*ALOG(FG) ASSIGN 1 TO KK GO TO 51 US=SIS IF (VD .GT. VS) GO TO 5 ID=0. GO TO 3 5 V=VD ESTIMATION FOR UD TSI=V2+TFIF IF (V2 .GT. VD) TSI=VD+US ASSIGN 10 TO KK GO TO 51 UD=SIS SW0.
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125 SM=0.5 SS=1. IF (VI .GT. VS) GO TO 12 V1=VS U1=US GO TO 18 12 IF (VI .LT. VD) GO TO 14 V1=VD U1=UD GO TO 16 14 V=V1 ASSIGN 15 TO KK GO TO 50 15 U1=SIS 16 SS=(U1US)/(V1VS) 18 IF (V2 .GT. VS) GO TO 20 V2=VS U2=US GO TO 2 6 20 IF (V2 .LT. VD) GO TO 22 V2= VD U2==UD GO TO 2 8 22 V=V2 ASSIGN 25 TO KK GO TO 50 25 U2=SIS 26 SW=(UDU2)/(VDV2) 28 IF (V2 .NE. VI) SM= (U2U1 ) / (V2V1 ) DRAIN CURRENT IDS=(G(U1)+B(U1)G(US)B(US) ) /SS IDM=(G(U2)+B(U2)G(U1)B(U1) ) /SM IDW=TK* (QN(UD)QN(U2) )/(lSW) ID=IDS+IDM+IDW IF (ID .GT. 0.0) GO TO 40 35 IE=1 DQGSI=BK**2* (1+EXP(USVSTFIF)/TK)EXP( (VBUS)/TK) )/(2*QG(US) ) CGB=DQGSI/ (l.+DQGSI) ALL THE ELEMENTS IN EQCKT^O BUT ID AND CGB RETURN 40 IE=0 TOTAL CHARGE COMPONENTS SGNS= (GG (Ul) +GB (Ul) GG (US) GB (US) ) /SS SGNM= (GG (U2) +GB (U2 ) GG (Ul) GB (Ul) ) /SM SGNVJ=TK* (G (UD) B (UD) QG (UD) *QB (UD) + G(U2)+B(U2)+QG(U2) *QB(U2) )/(lSW) QGG= (SGNS+SGNM+SGNW) /ID SBNS= (BB (Ul) +GB (Ul) BB (US) GB (US) ) /SS SBNM= (BB (U2) +GB (U2 ) BB (Ul ) GB (Ul) ) /SM SBNW=TK* (B(UD)QB(UD) **2/2 . B (U2 ) +QB (U2 ) **2/2. )/(lSW) QBB= (SBNS+SBNM+SBNW) /ID QNN=(QGG+QBB) IF (QGG .GT. QG(US) .OR. QBB . GT . QB(US)) GO TO 35
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126 TRANSCONDUCTANCES GS=QN(US) GD=QN(UD) GMS=0(QG(U1)+QN(U1)*DSG(U1,VL)QG(US)QN(US)*DSG(US,VS) ) /SS GMM=(QG(U2)+QN(U2)*DSG(U2,V2)QG(U1)QN(U1)*DSG(U1,V1) ) /SM GMW=TK* ( (1.5*BK**2/QE(UD) ) *DSG(UD,VD) + (1.5*BK**2/QB(U2) )*DSG(U2,V2) )/(lSV\J) GM=GMS+Gr4M+GMW GMBÂ— (GS+GD+GM) INTEGRAL OF (D/DVG)QB*QN DV DBNGS=(B(U1)QN(U1) *QB (Ul ) *DSG (Ul , VI ) + B(US)+QN(US) *QB(US) *DSG(US,VS) ) /SS DBNGM=(B(U2)QN(U2) *QB(U2) *DSG(U2,V2) + B(U1)+QN(U1)*QB(U1)*DSG(U1,V1) ) /SM DBNGW=TK* ( (QB(UD)BK**2/2. )*DSG(UD,VD) + (QB(U2)BK**2/2.)*DSG(U2,V2) )/(lSW) DBNG=DBNGS+DBNGM+DBNGW VARIATION OF TOTAL INVERSION CHARGE QNN WITH VG DNNGS=2*IDS+(QN(U1) **2*DSG (Ul , VI ) ON (US) **2*DSG (US , VS) ) /SS DNNGM=2*IDM+(QN(U2) **2*DSG (U2 , V2 ) ON (Ul ) ** 2*DSG (Ul , VI ) ) /SM DNNGW= TK*(QN(UD)*(1(1BK**2/(2*QB(UD) ) )*DSG(UD,VD) ) + QN(U2)*(1(1BK**2/(2*QB(U2) ) )*DSG(U2,V2) ))/(lSW) DNNG=DNNGS+DNNGM+DNNGW DQNG=(QNN*GM+DNNG) /ID CAPACITANCES CSG=QN(US) * (QGGQG(US) )/lD CSB=QN(US) * (QBBQB(US) ) /ID CDG=QN(UD) * (QGGQG(UD) ) /ID CDB=QN (UD) * (QBBQB (UD) ) /ID CGB= (QBB*GM+DBNG) /ID APPORTIONING FUNCTION LAMBDA LAMBDA=1 . / (1+ (CDG+CDB) / (CSG+CSB) ) TRANSCAPACITANCES TSG=LAMBDA*DQNG+CSG TDG= (1LAMBDA) *DQNG+CDG TSB=TSG TDB=TDG TGB=TSG+TDG RETURN ROUTINE TO CALCULATE SURFACE POTENTIAL AT GIVEN V 50 TSI=V+US 51 DO 55 1=1,100 E= (TSIVTFIF) /TK IF (E .LT. 100.) E=100. QS=BK*SQRT (TSIVB+TK* (EXP (E) 1) ) SI=TSI(VGTSI+QS)/(.5*BK**2*(1+EXP(E) )/QSl) IF (ABS(SITSI) .LE. l.E06) GO TO 60
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127 55 TSI=SI 60 SIS=SI GO TO KK, (1,10,15,25) END
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LIST OF REFERENCES [1] W.W. Lattin, Program Manager of INTEL CORPORATION, private communication [2] E.O. Johnson and A. Rose, "Simple general analysis of amplifier devices with emitter, control, and collector functions," Proc. IRE , vol. 47, pp. 407418, Mar. 1959. [3] R.D. Middlebrook, "A modern approach to semiconductor and vacuum device theory," lEE Proc. , vol. 106B, suppl. 17, pp. 887902, Mar. 1960. [4] P.E. Gray, D. DeWitt, A.R. Boothroyd, and J.F. Gibbons, Physical Electronics and Circuit Models of Transistors . New York: Wiley, 1964. [5] C.T. Sah, "Equivalent circuit models in semiconductor transport for thermal, optical, augerimpact, and tunneling recombinationgenerationtrapping processes," Phys. Status Solidi (a) , vol. 7, pp. 541559, 1971. [6] F.A. Lindholm and P.R. Gray, "Largesignal and smallsignal models for arbitrarilydoped fourterminal fieldeffect transistors," IEEE Trans. El ectron Devices, vol. ED13, pp. 819829, Dec. 1966. [7] R.S.C. Cobbold, Theory and Applications of FieldEffect Transistors . New York: WileyInterscience , 1970. [8] a) H.C. Poon, L.D. Yau, R.L. Johnson, and P. Beecham, "D.C. model for shortchannel IGFET's," in 1973 Int. Electron Device Meet. Dig. , p. 156. b) L.D. Yau, "A simple theory to predict the threshold voltage of short channel IGFET's," SolidState Electron. vol. 17, p. 1059, 1974. c) Y.A. ElMansy and A.R. Boothroyd, "A simple twodimensional saturation model for short channel IGFET's for CAD applications," in 1974 Int. Electron Device Meet. Dig. , p. 35. [9] J. Shekel, "Matrix analysis of multiterminal transducers," Proc. IRE , vol. 42, pp. 840847, May 1954. 128
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129 [10] L.A. Zadeh, Multipole analysis of active networks," IRE Trans. Circuit Theory , vol. CT4, pp. 97105, Sept. 1957. [11] F.A. Lindholm, "Unified modeling of fieldeffect devices," IEEE J. SolidState Circuits, vol. SC6, Aug. 1971. [12] D.J. Hamilton and W.G. Howard, Basic Integrated Circuit Design . New York: McGraw Hill, 1975. [13] W.M. Penney and L. Lau, MPS Integrated Circuits . New York: Van Nostrand Reinhold Co., 1972. [14] a) E.C. Ross and C.W. Mudler, "Extremely low capacitance silicon film MOS transistors," IEEE Trans. Electron Devices , vol. ED13, p. 379, March 1966. b) Y.A. ElMansy, D. Michael Caughey, "Characterization of silicononsaphire IGFET transistors," IEEE Trans. Electron Devices , vol. ED24, pp. 11481153, Sep. 1977. [15] Richard D. Pashley and Gary A. McCormic, INTEL Corp., "A 70ys Ik MOS RAM," 1976 lEE International SolidState Circuits Conference, pp. 138139, Feb. 1976. [16] W.R. Smythe, Static and Dynamic Electricity , New York: McGrawHill, 1950. [17] J.I. Arreola and F.A. Lindholm, "A nonlinear indefinite admittance matrix for modeling electronic devices," IEEE Trans. Electron Devices , vol. ED24, pp. 765767, June 1977. [18] J.I. Arreola, "A methodology for the lumped network representation of displacement and transport currents in electronic devices, with applications to MOS transistors," Master's Thesis, University of Florida, 1975. [19] W. Shockley, "A unipolar fieldeffect transistor," Proc . IRE , vol. 40, pp. 13651376, Nov. 1952. [20] C.T. Sah, "Characteristics of the metaloxidesemiconductor transistors," IEEE Trans. Electron Devices , vol. ED11, pp. 324345, July 1964. [21] D. FrohnanBetchkowsky and A.S. Grove," Conductance of MOS transistors in saturation," IEEE Trans. Electron Devices , vol. ED16, pp. 108113, Jan. 1969. [22] J.E. Meyer, "MOS models and circuit simulation," RCA Review, vol. 32, pp. 4363, March 1971.
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130 [23] R.H. Crawford, MQSFET in Circuit Desi gn. New YorkMcGrawHill, 1967T ~ [24] H.C. Pao and C.T. Sah, "Effects of diffusion current on characteristics of netaloxide (insulator) semiconductor transistors," Solid State E lectron. . vol. 9, pp. 927937, 1966. " [25] R.M. Swanson and J.D. Meindl, "Ionimplanted complementary MOS transistors in lowvoltage circuits," IEEE J. Solidstate Circuits , vol. SC7, pp. 146153, Apr. 1972. [26] T. Masuhara, J. Etoh, M. Nagata, "A precise MOSFET model for lowvoltage circuits," IEEE Trans. Electron Devices , vol. ED21, pp. 363371, June 1974. [27] V.A. ElMansy, A.R. Boothroyd, "A new approach to the theory and modeling of insulatedgate fieldeffect transistors," IEEE Trans. Electron Devices, vol. ED24, pp. 24125 3, March 1977. " [28] D.J. Hamilton, F.A. Lindholm, A.H. Marshak, Principles and Applications of Semiconductor Modeling , Chapter II , New York: Holt, Rinehart and Winston, 1971. [29] V.G.K. Reddi and C.T. Sah, "Source to drain resistance beyond pinchoff in metaloxidesemiconductor transistors (MOST)," IEEE Trans. Electron Devices, vol. ED12 pp. 139141, Mar. 1965~ ' [30] D. FrohmanBentchkowsky, "On the effect of mobility variations on MOS device characteristics," Proc . IEEE vol. 56, pp. 217218, Feb. 1968. "' [31] M.B. Barron, "Low level currents in insulated gate field effect transistors," Solid State Electr., vol. 15. pp. 293302, 1972. [32] R.J. Van Overstraeten, G. DeClerck, G.L. Bronx, "Inadequacy of the classical theory of the MOS transistor operating in weak inversion," IEEE Trans. Elec tron Devices , vol. ED20, pp. 11501153, Dec. 1973. [33] H.K.J. Ihantola and J.L. Moll, "Design theorv of a surface fieldeffect transistor," Solid St ate Electronics, vol. 7, pp. 423430, June 1964. ~~ [34] C.T. Sah and H.C. Pao, "The effects of fixed bulk charge on the characteristics of metaloxidesemiconductor transistors," IEEE Trans. Electron Devices, vol. ED13 pp. 393409, April 1966.
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131 [35] F.B. Hildebrand, Introduction to Numerical A nalysis, New York: McGrawHill, 1974 . [36] F.A. Lindholm, and J.I. Arreola, "Equivalent circuit studies," Chapter III (Part I), Technical Report of the Electron Device Research Center to ARPA, U.S. Army, Aug. 1977. [37] C.T. Sah, "Theory of MOS capacitor," Solid State Electronic Lab., University of Illinois, Urbana, Report No. 1, Dec. 14, 1964. [38] a) R.G. Weigert, "A general ecological model and its use in simulating algalfly energetics in a thermal spring community," Insects: Studies in Population Management (P.W. Geier, et al., eds.) vol. 1, 1973. b) S. Harris, The Boltzmann Equation , Chapter 11, New Yorl^: Holt, Rinehart and Winston, Inc., 1971. c) W.F. Ames, Nonlinear Partial Differential Equations in Engineering , New Yor]<; : Academic Press, 1965, Nonlinear Ordinary Differential Equations in Transport Processes , Chapter 1, New York: Academic Press, 1968. [39] D.L. Frazer, Jr. , and F.A. Lindholm, "Violations of the quasistatic approximation in largesignal MOSFET models," to be published. [40] F.A. Lindholm and D.L. Frazer, Jr.," A selfconsistency test for device models in transient computer simulation of largesignal circuits," to be published. [41] J.L. Clemens and F.A. Lindholm, "Systematic modeling of MOS transistors," unpublished work, 1965. [42] W. Shockley, "The theory of pn junctions in semiconductors and pn junction transistors," Bell Syst. Tech. J^, vol. 28, pp. 435489, 1949. [43] C.T. Sah, "The equivalent circuit model in solid state electronics III, conduction and displacement currents," Solid State Electronics, vol. 13, pp. 15471575, 1970.
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BIOGRAPHICAL SKETCH Jose Ignacio Arreola V7as born in Mexico City, Mexico, on January 10, 1950. He received the degree of Ingeniero Mecanico Electricista from the Universidad iberoamericana in Mexico City in 1973. He worked for Institute Nacional de Astrofisica, Optica y Electronica located in Puebla, Mexico, for one year. Jose Ignacio has been a Fellow from Consejo Nacional de Ciencia y Tecnologia at the University of Florida since June, 1974 He received the degree of Master of Science, major in Electrical Engineering, in August 1975. 132
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Fredrik A. Lindholm, Chairman Professor of Electrical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Alan D. Sutherland Professor of Electrical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. David R. MacOuiqq /'./' Assistant Professor of / Electrical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. ct]/\J Earnest D. Adams Professor of Physics
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This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. March 1978 iLJ^^dJ, \M^^ Dean, College of Engineering Dean, Graduate School
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AM23 78. 10.60.

