Equivalent-circuit modeling of the large-signal transient response of four-terminal MOS field effect transistors /

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EQUIVALENT-CIRCUIT MODELING OF THE
LARGE-SIGNAL TRANSIENT RESPONSE
OF FOUR-TERMINAL 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 pva-ent















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 two-dimen-

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 EQUIVALENT-CIRCUIT MODEL FOR THE FOUR-TERMINAL
MOSFET . . . . . . . . . 16

3.1 Examples of Engineering Needs for a Model
for the Large-Signal 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 Equivalent-Circuit 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 STEADY-STATE 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
FOUR-TERIINAL EQUIVALENT-CIRCUIT . . .. 76


5.1
5.2
5.3


Introduction . . . . . . .
Source-Drain 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 Equivalent-Circuit
5.5 Conclusions . . . . . . .

VI SCOPE AND FUTURE WORK . . . . .

APPENDIX

A PROPERTIES OF QUASI-FERMI POTENTIALS . .

B APPROXIMATED EXPRESSION FOR THE DIFFUSION/
DRIFT RATIO IN THE MOSFET . . . .

C COMPUTER SUBPROGRAM TO CALCULATE THE VALUE
THE ELEMENTS IN THE EQUIVALENT-CIRCUIT .


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



EQUIVALENT-CIRCUIT MODELING OF THE
LARGE-SIGNAL TRANSIENT RESPONSE
OF FOUR-TERMINAL MOS FIELD EFFECT TRANSISTORS

By

Jose Ignacio Arreola

March 1978

Chairman: Fredrik A. Lindholm
Major Department: Electrical Engineering

An approach is proposed that yields equivalent-circuit

models for the large-signal transient response for all

electronic devices described by charge-control. The ap-

proach is applied to derive an improved equivalent-circuit

model for the four-terminal 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 equivalent-circuit

components in terms of applied voltages and physical make-up

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 large-signal equivalent-circuit models for the

four-terminal MOSFET.

The purpose of this dissertation is to derive an im-

proved equivalent-circuit model for the four-terminal MOSFET.

Improvements are made in the following aspects of the

equivalent-circuit model:

(a) the representation of capacitive effects in a four-

terminal device;

(b) the characterization of the dc steady-state

currents and charges;

(c) the inclusion, in principle, of two- and three-

dimensional effects present, for example, in

short-channel MOSFETs.









As will be seen, all of these improvements are inter-

related and result from basing the derivation of the

equivalent-circuit model on the internal physics that deter-

mines the operation of the MOSFET.

We begin in Chapter III by proposing an approach that

yields equivalent-circuit models for the large-signal

transient response of all electronic devices described by

charge control [2-4]. 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 four-terminal 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 equivalent-circuit model of Chapter

III, one needs a suitable description of the dc steady-state

behavior. Extensive work has been done in the past to

characterize operation in the dc steady-state; however, none

of this work is completely suitable for the purposes of

equivalent-circuit modeling. In Chapter IV, a new model for

the dc steady-state behavior is derived that unifies the

description of the full range of operation of the device -









from weak to strong inversion and from cut-off 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

equivalent-circuit developed in Chapter III. In Chapter V

we also assess the engineering importance of the improvements

introduced in the equivalent-circuit 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

equivalent-circuit 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 [2-z], 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 equivalent-circuit 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 short-channel

devices give rise to multidimensional effects.

Models of four-terminal 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 small-signal

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 large-signal 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 small-signal 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 large-signal 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 opposite-charged 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 [2-4].

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 charge-control devices

[2-4]. 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 current-node 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 [2-4] and of the

closely allied quasi-static 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 quasi-static

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

quasi-static 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

n-terminal 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 all-capacitor 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

three-branch 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 n-terminal 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 equivalent-circuit 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 four-terminal device,

such as the MOSFET.

From Fig. 2.2 notice that the mobile charge accumulation

within a general n-terminal 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 equivalent-circuit between each pair
of terminals of an n-terminal 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 equivalent-circuit model is derived in

the following chapters.















CHAPTER III

EQUIVALENT-CIRCUIT MODEL FOR
THE FOUR-TERMINAL MOSFET



The main contribution of this chapter is the deriva-

tion of an equivalent-circuit model for the four-terminal

MOSFET by use of the method described in Chapter II.

The resulting model is intended to represent with good

accuracy the large-signal 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 Large-Signal Transient Response


In many digital integrated-circuit 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 lay-out of the circuit

[12,13]. In a large-scale 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 equivalent-circuit 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 p-n 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 n-channel 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

10-8 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 Four-Terminal 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

two-terminal 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 two-and four-terminal
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

four-terminal 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 four-terminal 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 all-capacitor 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 all-capacitor 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 large-signal transient behavior

depends on three physical mechanisms: mobile charge trans-

port, net recombination within the device and mobile charge

accumulation. The result is the equivalent-circuit of

Fig. 2.2, which applies between any pair of terminals and

is the basic building block from which an equivalent-circuit

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 Equivalent-Circuit for the Intrinsic MOSFET


For concreteness, consider the enhancement-mode n-

channel MOSFET illustrated in Fig. 3.1. A central idea in

the equivalent-circuit 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 [2-4], 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 make-up and the terminal voltages is obtained by

using a quasi-static approximation to extrapolate the

steady-state 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 quasi-static 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

large-signal equivalent-circuit 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 make-up.

The transcapacitive elements in the network representa-

tion can be also seen as related to error terms yielded by

an ideal all-capacitor 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




















4-1
> 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
equivalent-circuit 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 two-dimensional 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 quasi-static

approximation [6,111, which is based on the steady state

operation of the MOSFET. A particular detailed model for

steady-state 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 steady-state 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 N-channel 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 equivalent-circuit elements involving dq and

dqD directly from an extrapolation of the steady-state 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; P-N junction capacitances

arising from source-substrate and drain-substrate 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

quasi-static approximation. Lindholm [11] gives the de-

tails of the general approach for modeling extrinsic ef-

fects in a four-terminal 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 equivalent-circuit 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 equivalent-circuit 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 three-dimensional ef-

fects such as those in the short-channel 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 equivalent-circuit developed here is designed to

make immediate use of them to yield new and better network

representations of the large-signal transient response of

the MOSFET.

Most of the past work in equivalent-circuit 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

small-signal applications, involves an equivalent-circuit

in which the charging effects are represented by four

capacitors (source-gate, source-substrate, drain-gate and

drain-substrate) 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 single-capacitor 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

STEADY-STATE 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 enhancement-mode MOSFET. Each circuit element in

this representation depends on the constants of physical

make-up 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 cut-off,

triode, and saturation, including operation in weak, moderate

and strong inversion, and including four-terminal 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 computer-aided 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 El-Mansey 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 computer-aided circuit

design.

The goal of this chapter is to develop a model that

includes:

(a) four-terminal operation;

(b) cut-off, 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 computer-aided 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 quasi-Fermi 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 n-channel MOSFET, illustrated by Fig. 3.1, the

steady-state 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 = VN-VP is defined as the difference between the

quasi-Fermi potential for electrons V and the quasi-Fermi

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 equivalent-circuit 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 oxide-semiconductor 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 (= (m-Xs-q(EC-EI) (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 (S-V)


(4.6)





43













XS
VE /q




^N -I
EE /q
.. Epp/q OpF

Er-Fn /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 p-substrate 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 "spike-like" 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(S-VB)]. This yields

PS/NAA = exp[-B(S-VB)] 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 one-dimensional 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 y-component of electric field, which requires

that the effective charge in the gate be the source of the

x-directed 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 -(S-VB) B 1
F(SVVB'F) = e + kT/q-


8 ( S-V-2(F) s^-VB -B2mF -8 (V-VB+2(F)] 1/2
e e e (4.17)
kT/q

For the usual substrate doping, V-VB+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 + -(+S-V- 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 pinch-off 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 first-order 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 s-VB 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
S-Qg 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 pinch-off

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 first-order

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 -V-Vp)], 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 (l-d/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
(1-dis/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 +
(1-di /dV)


Qb(VD) 2 2/2 2d
( -b dv (4.33)
(1-d /dV)s
b(VS)


ZL
B I*VS
D V
S


ZL
I*
*D


(02 + Q'OQ)dV
-- g-b


D) 2Q3/K2dQ

+ S
S (1-dS/dV)
Qb(VS)


b(VD) 2 2
Q(VD) 2Q'Q /K2 dQ'
gb b
I (1-dis/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 2-gbg 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 (1-ds /dV) -n
Q (V2)


2 3 D
ZL kT/q -g 2 b V
I* (1-S 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 (l-dis/dV) Qn
n 2
Q(V,)


03 02 "VD
ZL kT/q 2 (b -b
S- (4 40)
I (1-S U) 3 K2 2
D JK
V2


4.3.3 Limits for the Strong, Weak, and Moderately Inverted
Portions of the Channel

The three-region piecewise-linear 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 least-squares 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 (iS-V-24F)
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 G-VB 4 q 2-
(A' K)


(4.43)


(4.44)


kT/q
+ A


Here, A' = (1-A)/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 (S-V-2F)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 Newton-Raphson

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 1-SW
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

S-V
v

K2 g3 + 2Qb /K )
3Ka
-a


kT/q
1-SW


o_2 o3v D
+ Q + 2
S2 g b 3 K2 V?


QBW 1 kT/q
ZLC I* 1-SW
0 D WV


V
2 '3 Q2' D
2 bb
3 K2 2
V,


Table 2






















0.2 B


4-1
0






/
0.1 -


/



extrapolated
/ threshold

1 2
Vt (volts)




Fig. 4.4 Calculated square-root dependence of the drain
current on gate voltage.



















10-2


" 1-5
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)
i-l
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 cut-off 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(24F-V
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 /-
10-5 AA / /
x = 1470A
o /

VD = 2v /
-6
V = Ov
cn B


1 -7- /I x



S Experiment [26]
10-8 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

10-4

VG = 0.3v
G o

F -5 VG = 0.lv a
S10 >

VG = Ov "


10-6 G = -0.1
0
10-7


10-7
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 cut-off, 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 computer-aided

circuit analysis.

The model developed here is subject to the limitations

of the one-dimensional 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

short-channel 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 LARGE-SIGNAL FOUR-TERMINAL EQUIVALENT-CIRCUIT



5.1 Introduction


In Chapter III we developed an equivalent-circuit

representation for the transient response of the MOSFET.

By employing the results of Chapter IV, the functional

dependencies of each element in this equivalent-circuit

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

steady-state behavior of the device to describe its large-

signal transient response.

The equivalent-circuit 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 four-terminal 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 Source-Drain Current Source


Through the use of a quasi-static approximation, as

discussed in Chapter III, the functional dependence of the

nonlinear source-drain 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 steady-state.









5.3 Capacitances


5.3.1 Expressions for the Capacitances

The capacitors in the equivalent-circuit 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 gate-substrate 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 (:s-V-24F)
1 + e v





1 + 2Q + e va
DFQ(vl vb) 1-2 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 1-Sw 1 + K2/2Q'


and

DFQB (v2'D)_ kT/ + K /Q (5.19)
SW 1-S 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 r-i 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






-r-l









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: cut-off, 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 equivalent-circuit. 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

cut-off 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 cut-off, where

VG is not large enough to turn on the channel, this capacit-

ance is equal to the capacitance of a (two-terminal) 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/ ro-P 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 cut-off 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:

PAGE 1

EQUIVALENT-CIRCUIT MODELING OF THE LARGE-SIGNAL TRANSIENT RESPONSE OF FOUR-TERMINAL 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 two-dimensional 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 EQUIVALENT-CIRCUIT MODEL FOR THE FOUR-TERMINAL MOSFET 16 3.1 Examples of Engineering Needs for a Model for the Large-Signal 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 Equivalent-Circuit 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 STEADY-STATE 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 FOUR-TERMINAL EQUIVALENT-CIRCUIT 76 5.1 Introduction 76 5.2 Source-Drain 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 Equivalent-Circuit . . . 106 5.5 Conclusions 109 VI SCOPE AND FUTURE WORK Ill APPENDIX A PROPERTIES OF OUASI-FERf-lI 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 EQUIVALENT-CIRCUIT .... 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 EQUIVALENT-CIRCUIT MODELING OF THE LARGE-SIGNAL TRANSIENT RESPONSE OF FOUR-TERMINAL MOS FIELD EFFECT TRANSISTORS By Jose Ignacio Arreola March 1978 Chairman: Fredrik A. Lindholm Major Department: Electrical Engineering An approach is proposed that yields equivalent-circuit models for the large-signal transient response for all electronic devices described by charge-control. The approach is applied to derive an improved equivalent-circuit 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 equivalent-circuit components in terms of applied voltages and physical make-up 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 large-signal equivalent-circuit models for the four-terminal MOSFET. The purpose of this dissertation is to derive an improved equivalent-circuit model for the fourterminal MOSFET. Improvements are made in the following aspects of the equivalent-circuit model: (a) the representation of capacitive effects in a fourterminal device; (b) the characterization of the dc steady-state currents and charges; (c) the inclusion, in principle, of twoand threedimensional effects present, for example, in short-channel MOSFETs.

PAGE 9

As will be seen, all of these improvements are interrelated and result from basing the derivation of the equivalent-circuit model on the internal physics that determines the operation of the MOSFET. We begin in Chapter III by proposing an approach that yields equivalent-circuit models for the large-signal transient response of all electronic devices described by charge control [2-4] . 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 equivalent-circuit model of Chapter III, one needs a suitable description of the dc steady-state behavior. Extensive work has been done in the past to characterize operation in the dc steady-state; however, none of this work is completely suitable for the purposes of equivalent-circuit modeling. In Chapter IV, a new model for the dc steady-state 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 cut-off 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 equivalent-circuit developed in Chapter III. In Chapter V we also assess the engineering importance of the improvements introduced in the equivalent-circuit 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 equivalent-circuit 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 equivalent-circuit 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 short-channel devices give rise to multidimensional effects. Models of four-terminal 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 small-signal 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 large-signal 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 small-signal 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 large-signal 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 • (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 opposite-charged carriers injected from terminal K. From this characterization, it follows that (ijp.),p j^ satisfies the following properties: ^JK ^ -^KJ ijj = • (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 = ar• (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 [2-4]. Now, using (2.6) and (2.8), we may rewrite (2.5) as dq 'j = ^Ij (^Jk)t,R ^ dF• (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 charge-control devices [2-4]. 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 • (2.io:

PAGE 18

11 This global counterpart of the Kirchhoff current-node 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 [2-4] and of the closely allied quasi-static approximation [6,7,11]. For the transport and recombination mechanisms, charge control gives directly (ijK^,R = ^jr/^jk • (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 quasi-static 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 quasi-static 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 all-capacitor 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 three-branch 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 n-terminal 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 equivalent-circuit 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 four-terminal device, such as the MOSFET. From Fig. 2.2 notice that the mobile charge accumulation within a general n-terminal 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 equivalent-circuit between each pair of terminals of an n-terminal 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 equivalent-circuit model is derived in the following chapters.

PAGE 23

CHAPTER III EQUIVALENT-CIRCUIT MODEL FOR THE FOUR-TERMINAL MOSFET The main contribution of this chapter is the derivation of an equivalent-circuit model for the four-terminal MOSFET by use of the method described in Chapter II. The resulting model is intended to represent v/ith good accuracy the large-signal transient currents flov-zing through each of the four terminals of the device, including the substrate terminal. 3 . 1 Examples of Engineering Needs for a Model for the Large-Signal Transient Response In many digital integrated-circuit 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 lay-out of the circuit [12,13]. In a large-scale 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 p-n 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 n-channel 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 four-terminal 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 four-terminal 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 four-terminal 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 all-capacitor 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 large-signal transient behavior depends on three physical mechanisms: mobile charge transport, net recombination within the device and mobile charge accumulation. The result is the equivalent-circuit of Fig, 2.2, which applies between any pair of terminals and is the basic building block from which an equivalent-circuit 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 Equivalent-Circuit for the Intrinsic MOSFET For concreteness, consider the enhancement-mode nchannel MOSFET illustrated in Fig. 3.1. A central idea in the equivalent-circuit 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 [2-4] , 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 make-up and the terminal voltages is obtained by using a quasi-static approximation to extrapolate the steady-state 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 dt• (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^ • (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 quasi-static 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.

PAGE 33

26 The matrix description of (3.7) together with the building block of Fig. 2.2 yields, therefore, the general large-signal equivalent-circuit 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 make-up. The transcapacitive elements in the network representation can be also seen as related to error terms yielded by an ideal all-capacitor 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 •H ^ +J C -H Q) O U U I rH to > H 0) n3 C O 00 •H

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28 Table 1 Definitions for the elements of the general equivalent-circuit for the MOSFET. CAPACITANCES ^SG =

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29 ^SD ^ ° ' ^SD = ° • (3.8) In more detailed characterizations of the device for example, the ones including channel length modulation [21] and two-dimensional 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 quasi-static approximation [6,11], which is based on the steady state operation of the MOSFET. A particular detailed model for steady-state 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 steady-state characterization of the device: the contributions from the source and the drain, dq„ and dq_ , to the total charging of the channel. To gain physical insight as to how dq„ 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 N-channel 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

PAGE 38

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.

PAGE 39

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 = (1-X) ^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,

PAGE 40

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), — 2. v^ + — 5. V = A dv^ G 3Vg B — ^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 — • (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 equivalent-circuit elements involving dq and dq^^ directly from an extrapolation of the steady-state behavior of the device. From Table 1, these elements are, 3qg aq^. 3qj^ ^SG = ^ 3Vr ^ ^ + Cg^^ (3.20) DG 9v„ 3v^ ^ 9v^ ^ ^DG (3.21) dgB = -dg^ (3.22) ^DB ^ -^SB (3.23) ^GB = ^SG + ^DG (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,

PAGE 42

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; P-N junction capacitances arising from source-substrate and drain-substrate 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 quasi-static approximation. Lindholm [11] gives the details of the general approach for modeling extrinsic effects in a four-terminal 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.

PAGE 43

36 In contrast, the equivalent-circuit 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 three-dimensional effects such as those in the short-channel 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 equivalent-circuit developed here is designed to make immediate use of them to yield new and better network representations of the large-signal transient response of the MOSFET. Most of the past v/ork in equivalent-circuit 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 small-signal applications, involves an equivalent-circuit in which the charging effects are represented by four capacitors (source-gate, source-substrate, drain-gate and drain-substrate) 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

PAGE 44

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 single-capacitor representation between these terminals would require 3q„/3v^ = 9q_/3v„. 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.

PAGE 45

CHAPTER IV STEADY-STATE 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 enhancement-mode MOSFET. Each circuit element in this representation depends on the constants of physical make-up 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 cut-off, triode, and saturation, including operation in weak, moderate and strong inversion, and including four-terminal operation.

PAGE 46

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 computer-aided 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 El-Mansey 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 computer-aided circuit design. The goal of this chapter is to develop a model that includes : (a) four-terminal operation; (b) cut-off, triode and saturation regions; (c) weak, moderate and strong inversion; (d) current and total charges.

PAGE 47

40 The model, furthermore, should avoid the discontinuities of a piecewise description while maintaining enough mathematical simplicity for computer-aided 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 quasi-Fermi 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 n-channel MOSFET, illustrated by Fig. 3.1, the steady-state 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 quasi-Fermi potential for electrons V^^ and the quasi-Fermi 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) Vj-j. 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 = -^ J„(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 4-2.2 Charge Components For the purposes of equivalent-circuit 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 oxide-semiconductor 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 , •o -^S C„ + 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)

PAGE 50

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^,

PAGE 51

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 p-substrate device the bulk charge Qj^ per unit area can be approximated by Qb q(P-Nz^)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 "spike-like" 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 *s-Vb ^ f -8(*3-Vb) e -1 1/2 Here , -K lit -V •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 (i|jg-Vg) ] . 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 one-dimensional 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 Q„ 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 •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 y-component of electric field, which requires that the effective charge in the gate be the source of the x-directed 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^ , (}>„) 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 V„ is applied, we find that F(iJ;3,V,Vg,c})p; kT q 1/2 kT/q 'S B + + e 3 (i|;g-V-2<|,p) i(;g-Vg -62((,j, -6(V-Vg+24,^; kT/q e 1/2 (4.17) For the usual substrate doping, V-V^+2(p is always much larger than kT/q. Furthermore, if we neglect the majority carrier concentration at the surface (Pg<
PAGE 55

48 where Q Q \7 ' — 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 di|jg/dV -> 0. This characteristic behavior can be explained by studying the relative importance of the drift and diffusion components along the channel [24] : a

PAGE 56

49 m -P O 2 > en

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50 °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 pinch-off 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 i|jg 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-

PAGE 58

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 first-order 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' (di|jg/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 —2 ^_ + (d;|;„/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 pinch-off mode. This represents the most general case because the channel is strongly inverted at the source and becomes

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53 strong • 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 di|jg/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 ) •g' a' O' dQ' + -g ^g Qb^^b^ b a "b -b -Q ,2 V V •* a b ^.3 ^ --4 V b V -^ a (4.26; then the expression for the drain current becomes ^D = — s; ^ — ^^ — ^ — s^ — '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 S„ = 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 -V-V ) ] , 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 ^«; Q^ (l-d,|;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 (l-dif-g/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 (i-s,,) 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 °h ZL IS r^D V, (Q+ Q^Q^)dv ZL g D Qq(Vs) -o'^ do' "g 'g (l-d,J;g/dV) + q; (V ) °^ 2Q'0'^/K^ do' "g "b "b (l-d,jj^^/dV) Qb(Vs) , (4.33) and Ob ZL f"" 2 ZL Qb(^D^ Qb^^s) (l-diJ;g/dV) «b(^D) Qi^^s) 2Q'0'^/K^dQ' g 'b b (l-di|;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 F„„ 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 do-Q .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'' -g-b 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' = — ^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 least-squares 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 di|j< B(i]jg-V-24,p) dV = A 20' •'g 9 + 1 + e {il>^-V-2(^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)^ — £n — A5+ ^ V 1 2 G F q ^.^2 2 where (4.44) V^ = K V' V + — + — ^G ^B ^ 4 ^ q f 1 (A'Kl 1 1/2 + kT/q Here, A' = (1-A)/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 V„ from (4.44) if V„ < 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 di|i„/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 Newton-Raphson 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 = °

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64 Table 2 (Continued) FOR THE CURRENT; ^I^^a'^) = 2 3 ^2 -rV, V -i a I* DW kT/q 1-S 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* 1-S o D W O' ^ O' ^ tV D V, BW ^ J_ kT/q ZLC I* 1-S 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 square-root 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 I— 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 cut-off 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 • 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 H° 10-"^ 10 10 -9 10-ii N AA X o V. V B -7 inl4 -3 7x10 cm 1470A 2v Ov • 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 <« 10 10"° -

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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 cut-off, 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 computer-aided circuit analysis. The model developed here is subject to the limitations of the one-dimensional 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 short-channel 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 LARGE-SIGNAL FOUR-TERMINAL EQUIVALENT-CIRCUIT 5 . 1 Introduction In Chapter III we developed an equivalent-circuit representation for the transient response of the MOSFET. By employing the results of Chapter IV, the functional dependencies of each element in this equivalent-circuit 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 steady-state behavior of the device to describe its largesignal transient response. The equivalent-circuit 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 Source-Drain Current Source Through the use of a quasi-static approximation, as discussed in Chapter III, the functional dependence of the nonlinear source-drain 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 steady-state.

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78 5.3 Capacitances 5-3.1 Expressions for the Capacitan ices The capacitors in the equivalent-circuit 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 i£ 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 gate-substrate capacitance is defined in Table 1 as C ^^B 3Qb 'GB Sv, 3v^ • (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*/3v„ and 3/3v^ (/ " Q^O dv) , u G G '•' v„ 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^ °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)s-v-2<|)p) b V (5.16) °V^^a'^b) 2 ^

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82 Table 3 Functional dependencies for the capacit ors SG ZLC o Q' (0) —^ (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 4-1 o o > _ fC rH 3 u rH H in Cn •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: cut-off, 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 equivalent-circuit. 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 = ^ (5-20) 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 cut-off 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 ~ ° ^^^ 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 C„„ 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 C„„ 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 cut-off, 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, C„^ 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 C„^ and the total substrate capacitance C„_ 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 •-{ o > > 0) o > +J H O m m -p c c o g o u +J •H c m o c (d -p -H u (0 o rO •'^ •H o u ^ p rH u in o OTZ JO sq.Tun UT aoueq^TOPdeo IT) •H

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90 0) -i-i -p r-i O > • 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 •H u u -p LT)

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91 In the cut-off 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, C„_, 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 5-3.3 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 « *^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

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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 «„ and a D are practically independent of the gate voltage they are in general not constant. Using the functional dependencies for C„„ 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^°) 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 a„ 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^„ 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 I— I o > > • C • > in r-\ o c > o c -H -H V u c 0) o en -H -p 1+-! ra o e •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 I— I C U H W P3 :3 Q u CD 13 •p a 3 pa U CO u x: +j PQH Q5 • U Q •r! 0) fO fO c a rO mg coo C/2 u u a in •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^„ 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 = ^ (2q£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 °^ ""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 C„„ 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 „ and C SB DB exceed the values of C„_ 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 5-4.1 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^ °^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 cut-off 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 equivalent-circuit. In the next section we will consider the engineering importance of the transcapacitors in the overall equivalent-circuit. 5-4.2 Engineering Importance of the Transcapacitance Elements The derivation of the equivalent-circuit 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 •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 all-capacitive 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 equivalent-circuit 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 equivalent-circuit 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^ v„ and Cj hi ^DG^G = ^DB^B ^G "d • ^n ^ — r TTT^ = ^T + ^-i ^^ (5.48) ° ^DG "*" ^DB 1 + «D 1 + «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 cut-off and saturation and between saturation and nonsaturation. As an example consider V /V = 1. In the vicinity of the transition between cut-off 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 all-capacitive 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 large-scale 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 Three-Terminal 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 equivalent-circuit reduces to the result [11] for a three-terminal 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 i„+i„) , 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 /3v„ + 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 equivalent-circuit developed in Chapter III: a current-source 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 three-terminal 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 equivalent-circuit model in computer-aided 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 large-signal 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 n-terminal 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 large-signal 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 equivalent-circuit model applies generally, subject only to the validity of the quasi-static approximation. In principle it can include twoand three-dimensional effects such as those occurring in short-channel 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 computer-aided circuit design over previous static characterizations. The dc steady-state description covers the entire range of operation: weak to strong inversion, cut-off to saturation. • 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 equivalent-circuit 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 computer-aided circuit design. A central approximation on which the results of our study depend is the quasi-static approximation (QSA) . This approximation underlies not only our work but nearly all the models in common use for either bipolar transistors or MOSFETs in computer-aided 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 computer-aided design, fails to be consistent with itself. This lack of selfconsistency can produce errors in computing system parameters such as the turn-off propagation delay time. By including in our nev; model a self-consistency 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 QUASI-FERMI POTENTIALS The objective of this appendix is to explore some properties of the quasi-Fermi 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. A-1. C X V Fig. A-1 This system must satisfy Faraday's law expressed mathematically by E'd2. = (A-1) or t E-d2, + ;) E-da = . 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) (A-3) When studying semiconductor devices a very important tool of analysis results from defining electron and hole quasi-Fermi potentials V and V as follows [42]; ^^I q^" HT 'N V. VkT „ P ^ ^^ ^ (A-4) 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 quasi-Fermi potentials. From (A-4) V,(x) = V^(x) + ^ in ^(^) n (A-5) Substituting this equation in (A-3) we obtain for electrons ^a = V^(^) V,(0) ^f ^n|l^ (A-6) If we impose the constraint that the contacts between the semiconductor and the external circuit are ohmic , then at the boundaries x-0 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 (A-7) Because (A-6) must hold in equilibrium, when V^ = and the quasi-Fermi levels for electron and holes coincide with the Fermi level which is position independent, we obtain which implies (A-8) Ng(L) = Ng(0: (A-9) 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 (A-7) and (A-9) in (A-6) we obtain V, = V^(L) V.,(0) 'N N (A-10) Analogously, for holes ^a " ^P^^) ' ^P^°^ (A-11) The relationships in (A-10) and (A-11) 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 n-channel MOSFET we defined a "channel voltage" related to the quasi-

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118 Fermi potentials by ^ = ^N ^P (A-12) 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 (A-10) requires % " ^q = ^N^^') ^M^O') = ^(L-) V(O') (A-13) 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 quasi-Fermi 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 (A-15) 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 • (^-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 °n dN/dy '^DRIFT ^n ^ ^y VJe can express the electron concentration N in terms of the electrostatic potential V and the quasi-Fermi potential for electrons Vj, by using B(Vj-V ) N = n^e ". (B-2; Inspection of the energy band diagram in Fig. 4.1 shows that Vj-Vp = 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 Biii-v-^^) (B-3) Taking derivatives with respect to y in this equation we obtain, dN N dy q/kT 'djjj_ _ dV" dy dy (B-4) From the gradual chaanel approximation [19], ^ = dij ;(x,y) ^ d4,(0,y) ^ ^ dy dy dy dy (B-5) Substituting (B-5) and (B-4) in (B-1) , 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 (B-6) 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>g-V-2(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(iPs-V-2^j,) dV~ " 2Q^^ Bi^-V-2^F) ^^"^^ V 3+ I + e Rearranging terms. 1 diij„/dV 2QVk2 + 1 b q dijjg/dV B(ii;g-V-2<|,p) e (B-8) The exponential term in (B-9) is related, as we shall see, to the electron concentration at the surface. From (B-4), 6 (<|^c-V-())p) Ng = n^e ^ (B-9) because N^^/n^ = e ''^ , the exponential term in (B-9) is therefore equal to N /N . We can then write 1 d,^ /dV N d4.s/dV = (2Q-/k2.1) ^ (B-10) 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 N„ 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 (B-11) 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 EQUIVALENT-CIRCUIT 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: V„ , 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 C„„, C„_, C_„, C^^ , U oCj DO DG DB C^„ and the transcapacitors denoted by T„^, T„^, 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)=VG-U QB (U) =-BK*SQRT (U-VB-TK) 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( (U-V-TFIF) /TK) )/OG(U) ) DSI(ABK) =TK*ALOG((2./ABK)*(TK/ABK+SQRT(VG-VB+BK**2/4 + + TKMTK/ABK**2-1.))) ) VX(LIM) =VG-TFIF-LIM+BK**2/2.+ BK*SQRT ( VG-VB+BK**2/4 . +TK* (EXP (LIM/TK) -1) ) MINIMUM VALUE OF GATE VOLTAGE FOR ONSET OF INVERSION N=NI ; PSIS-VB=FIF IE=2 IF (VG-VB .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**2-VG-BK**2* (TFIF-VB) )/(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 SW-0.

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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=(U1-US)/(V1-VS) 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=(UD-U2)/(VD-V2) 28 IF (V2 .NE. VI) SM= (U2-U1 ) / (V2-V1 ) 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) )/(l-SW) ID=IDS+IDM+IDW IF (ID .GT. 0.0) GO TO 40 35 IE=1 DQGSI=BK**2* (1+EXP(US-VS-TFIF)/TK)-EXP( (VB-US)/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) )/(l-SW) 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. )/(l-SW) 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) )/(l-SV\J) GM=GMS+Gr4M+GMW GMB— (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) )/(l-SW) 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-(1-BK**2/(2*QB(UD) ) )*DSG(UD,VD) ) + -QN(U2)*(1-(1-BK**2/(2*QB(U2) ) )*DSG(U2,V2) ))/(l-SW) DNNG=DNNGS+DNNGM+DNNGW DQNG=(QNN*GM+DNNG) /ID CAPACITANCES CSG=QN(US) * (QGG-QG(US) )/lD CSB=QN(US) * (QBB-QB(US) ) /ID CDG=-QN(UD) * (QGG-QG(UD) ) /ID CDB=-QN (UD) * (QBB-QB (UD) ) /ID CGB= (QBB*GM+DBNG) /ID APPORTIONING FUNCTION LAMBDA LAMBDA=1 . / (1+ (CDG+CDB) / (CSG+CSB) ) TRANSCAPACITANCES TSG=LAMBDA*DQNG+CSG TDG= (1-LAMBDA) *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= (TSI-V-TFIF) /TK IF (E .LT. -100.) E=-100. QS=-BK*SQRT (TSI-VB+TK* (EXP (E) -1) ) SI=TSI-(VG-TSI+QS)/(.5*BK**2*(1+EXP(E) )/QS-l) IF (ABS(SI-TSI) .LE. l.E-06) 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. 407-418, Mar. 1959. [3] R.D. Middlebrook, "A modern approach to semiconductor and vacuum device theory," lEE Proc. , vol. 106B, suppl. 17, pp. 887-902, 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, auger-impact, and tunneling recombination-generation-trapping processes," Phys. Status Solidi (a) , vol. 7, pp. 541-559, 1971. [6] F.A. Lindholm and P.R. Gray, "Large-signal and smallsignal models for arbitrarily-doped four-terminal fieldeffect transistors," IEEE Trans. El ectron Devices, vol. ED-13, pp. 819-829, Dec. 1966. [7] R.S.C. Cobbold, Theory and Applications of Field-Effect Transistors . New York: Wiley-Interscience , 1970. [8] a) H.C. Poon, L.D. Yau, R.L. Johnson, and P. Beecham, "D.C. model for short-channel 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," Solid-State Electron. vol. 17, p. 1059, 1974. c) Y.A. El-Mansy 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 multi-terminal transducers," Proc. IRE , vol. 42, pp. 840-847, May 1954. 128

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129 [10] L.A. Zadeh, Multipole analysis of active networks," IRE Trans. Circuit Theory , vol. CT-4, pp. 97-105, Sept. 1957. [11] F.A. Lindholm, "Unified modeling of field-effect devices," IEEE J. SolidState Circuits, vol. SC-6, 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. ED-13, p. 379, March 1966. b) Y.A. El-Mansy, D. Michael Caughey, "Characterization of silicon-on-saphire IGFET transistors," IEEE Trans. Electron Devices , vol. ED-24, pp. 1148-1153, Sep. 1977. [15] Richard D. Pashley and Gary A. McCormic, INTEL Corp., "A 70-ys Ik MOS RAM," 1976 lEE International SolidState Circuits Conference, pp. 138-139, Feb. 1976. [16] W.R. Smythe, Static and Dynamic Electricity , New York: McGraw-Hill, 1950. [17] J.I. Arreola and F.A. Lindholm, "A nonlinear indefinite admittance matrix for modeling electronic devices," IEEE Trans. Electron Devices , vol. ED-24, pp. 765-767, 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 field-effect transistor," Proc . IRE , vol. 40, pp. 1365-1376, Nov. 1952. [20] C.T. Sah, "Characteristics of the metal-oxide-semiconductor transistors," IEEE Trans. Electron Devices , vol. ED-11, pp. 324-345, July 1964. [21] D. Frohnan-Betchkowsky and A.S. Grove," Conductance of MOS transistors in saturation," IEEE Trans. Electron Devices , vol. ED-16, pp. 108-113, Jan. 1969. [22] J.E. Meyer, "MOS models and circuit simulation," RCA Review, vol. 32, pp. 43-63, March 1971.

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130 [23] R.H. Crawford, MQSFET in Circuit Desi gn. New YorkMcGraw-Hill, 1967T ~ [24] H.C. Pao and C.T. Sah, "Effects of diffusion current on characteristics of netal-oxide (insulator) -semiconductor transistors," Solid State E lectron. . vol. 9, pp. 927-937, 1966. " [25] R.M. Swanson and J.D. Meindl, "Ion-implanted complementary MOS transistors in low-voltage circuits," IEEE J. Solid-state Circuits , vol. SC-7, pp. 146-153, Apr. 1972. [26] T. Masuhara, J. Etoh, M. Nagata, "A precise MOSFET model for low-voltage circuits," IEEE Trans. Electron Devices , vol. ED-21, pp. 363-371, June 1974. [27] V.A. El-Mansy, A.R. Boothroyd, "A new approach to the theory and modeling of insulated-gate field-effect transistors," IEEE Trans. Electron Devices, vol. ED-24, pp. 241-25 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 pinch-off in metal-oxide-semiconductor transistors (MOST)," IEEE Trans. Electron Devices, vol. ED-12 pp. 139-141, Mar. 1965~ ' [30] D. Frohman-Bentchkowsky, "On the effect of mobility variations on MOS device characteristics," Proc . IEEE vol. 56, pp. 217-218, Feb. 1968. "' [31] M.B. Barron, "Low level currents in insulated gate field effect transistors," Solid State Electr., vol. 15. pp. 293-302, 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. ED-20, pp. 1150-1153, Dec. 1973. [33] H.K.J. Ihantola and J.L. Moll, "Design theorv of a surface field-effect transistor," Solid St ate Electronics, vol. 7, pp. 423-430, June 1964. ~~ [34] C.T. Sah and H.C. Pao, "The effects of fixed bulk charge on the characteristics of metal-oxide-semiconductor transistors," IEEE Trans. Electron Devices, vol. ED-13 pp. 393-409, April 1966.

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131 [35] F.B. Hildebrand, Introduction to Numerical A nalysis, New York: McGraw-Hill, 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 algal-fly 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 quasi-static approximation in large-signal MOSFET models," to be published. [40] F.A. Lindholm and D.L. Frazer, Jr.," A self-consistency test for device models in transient computer simulation of large-signal 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 p-n junctions in semiconductors and p-n junction transistors," Bell Syst. Tech. J^, vol. 28, pp. 435-489, 1949. [43] C.T. Sah, "The equivalent circuit model in solid state electronics III, conduction and displacement currents," Solid State Electronics, vol. 13, pp. 1547-1575, 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.