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
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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:
