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PHYSICAL MODELS OF MOSFET DEVICES

October 1975

Prepared by

Electron Device Research Center Cniversity of Florida

Department of Electrical Engineering DOC Gainesville, Florida 326.11~-n~1 nr?rrT

EDRC R~:::,\:=" EDRC-75-/ If:~~~\~~~ \\\ DAAG39-74-C-0193 U UGQ:J-;....=lU ____ ........ JJ

A

u.s. Army Materie' Command

HARRY DIAMOND LABORATORIES

Adelphi, Maryland 20783

This research was sponsored by the Defense Nuclear Agency under Subtask Z99QAXTB029, Work Unit 43, "TREE Effort MEI/LSI." AMCMS Code: 691000.22.10807, DNA HIPR: 74-523.

~o fOIl FUBliC RHEAS(, OtSTRI8U1OH lH.. .. rn.o

PHYSICAL MODELS OF MOSFET DEVICI:;S

David P. Kennedy rrcdrik A. Lindholm

Contents

L"t-L1l,tcr I. - Thc MOS Field-Effect Transistor by R. F. Motta and D. P. Kennedy

1.0 Introduction 5

6 6

1.1 Traditional Theory of MOSFET Operation 1.1.1 Mathematical Development 1.1.2 Electric Current Continuity 1.1.3 Drift and Diffusion 1.1.4 Other Limitations

12 16 18

i.2 Extensions of Traditional MOSFET Theory 1.2.1 One-Dimensional Drift-and-Diffusion 1.2.2 Weak Inversion Operation

20 21 25 33 35

1.2.3 Channel Length Modulation 1.2.4 Models for Short-Channel Structures

1.3 Overview of Present Research 58 1.3.1 Continuity of Source-Drain Electric Current 60 1.3.2 The Weak Inversion Mode of Operation 62 i.3.3 Short-Channel MOSFET Structures 62

1.4 ~o-Dimensional Mechanisms in MOSFE~ Operation 63 ~:4.l Electric Current Saturation 66

1.4.2 Channel T~rmination in a MOSFET 69 1.4.3 Velocity Saturation in a MOSFET 72 1.4.4 Depletion Charge Distribution 75

1.5 A Modified Theory for MOSFET Operation 83 1.5.1 A Physical, Interpretation of the Separa-

tion Parameter A(V ) 86 1.5.2 The Volt-Ampere Chara8teristics of a MOSFET 87 1.5.3 The Saturation Mode of Operation 91 1.5.4 Calculation of Q. in a MOSFET 92 1.5.5 Calculation of A~oin a MOSFET 97

o 1.6 Conclusions 99

Ch,lptcr II - Inversion Layer Studies for MOSFET Operation by C. T. Hsing and D. P. Kennedy

2.1 Introduction

2.2 Solution of the Schroedinger Equation for a MOS Structure

2. 3 Inversion Layer Corrier Distribution in a MOSFET

2.4 Discl.;::;sion

~ • 5 Conclusions

3

101

103

109

III

115

Chapte r I I I - Lumped Network Represen ta t ~()n v f ~os

Transistors by J. I. Arreo ia ana F. /, :,1 n:J:.() i;1

~.l Introduction

3.2 Current-Flow Equatio- for an r~-Terll'in,ll Dj'VJ(~"

3.3 A Quasi-Static Appr~~imation

3.4 Lumped Network RepresentC1tion of a Semlcondur:tnr Devic0

3.4.1 Simplified Capacitive Representation (d jk = d

kj)

3.4.2 General Capacitive Representation Cd jk 1 d kj )

3.5 Lumped Network Representation of a MOS Transistor 3.5.1 Evaluation of Model Parameters

3.5.1.1 Transport Current Models 3.5.1.2 Displacement Current Model Elements

3.6 Conclusions

Literature Cited

List of Figures

List of Symbols

Appendix A Derivation of Solution Equations for t h!' Present Theory of MOSFET Operation

Appendix B Electric Current Continuity in the Present Theory of MOSFET Operation

Distrlbution ~i~t

4

! t.'

1 ..

J, (J ~ f )

if,

I ~ ':

1 'j 7

Chapter I

'I'll<' NOS Fie1d-Eff~'rt Transistor

R. 1'. Motta and D. P. Kennedy

1.0 Introduction

Thl! traditional theory of MOSFET operation 1S based upon ;1;,

early treatment by Ihant01a [1]* and Ihantola and Hall [2]. Tills

trcatml!nt utilized a series of simplifying asswr.ptions and approx:­

mations similar to those found in Shockley's theory of the unipol~r

transistor [3J. The purpose of these simplification~ was to

achieve mathematical tractability in analyzing this semiconduc~or

structure. For the device fabrication technology at that time,

th is trea tmen t produced calcula ted resul ts in sat is factory ag ret'­

ment with experiment.

Following the Ihantola-Moll work, other researchers further

developed and refined this theory of MOSFET operation. The

physical mechanisms associated with this device were studied in

detail, each investigation yielding an improved agreement between

theory and experiment. At the present time, numerous textbooks

[4-8] offer discussions on this topic. In addition, the techni­

cal literature contains hundreds of papers covering a wide range

of specialized studies of the physical and electrical properties

of a MOSFET [9,10].

Concurrent with improvements in MOSFET theory, rapid

cha!I(]es took place in fabrication techniques for this semi­

conductor device. For example, integrated-circuit technologies

were developed whereby high packing-density logic structures

(c.g., CMOS) were used extensively in the design of complex

electronic systems. These technologies utilized new and

refined photolithographic methods to fabricate very short­

chrtnnel MOSFET structures. In addition, TTL compatibility

was achieved with the development of low voltage MOSFET

--------------* Due to the number of:; Ii terature references cited and the nature of portions of this report Ca review of current literature results), all references will be listed at the end of this report as per the table of contents.

5

these technologi c<\l adviHlcf'S camp d:'1 1 :71!,'):- t d ",'

the traditional theory of !-1OSFET oper,l~ I .\

results that are in poor a'jrel';1('nt · ... ·i·h

ThlS sItuation irlltiated new studies into the phi'~-ll~:,ll

::-•• chanisms at f-10SFET operation (11-30). These studips C'l,':,r!','

s;J()' .... that tilC simplifying assumptions and apprOXifTldtion'; of

t ::adl tional theory are not always appl icable to the ,1:'1,,1':'5 l'~

!' r:1odern MOSFET devices .• This inapplicabilit'/ DeCnln('s

:hHti(:ularly evident when dealing \,rith the wea].: in-.;pr:;ion

:7,0dc' of :·10SFET opera tion and/or very short-channC'l s t nwt u res.

Some of these new studies developed and utill zed t .... 'o­

olme:'1sional computer solutions to obtain insight into MnSfET

(,peration [24-30]. These studies have aided in undcrst..lndiny

limltations of the traditional MOSFET theory. Further::-.orc: ,

1;1 many cases such studics have also provlded a basis for the

development of new and improved MOSFET models [11,13,17, 18,24J.

In a recent investigation, ri70rous computer calc~lations

showed that two-dimensional mechanisms can contribute in ~

~iqnlficant fashion to MOSFET operation [251. Resultino frn"1

this investigation, a model was proposed to account for these

two-dimensional mechanisms [24]. An inportant task un(lC'rtak~n

1": ~he present study is J~di~ional development and rpfin0rn0~t

u: t n i s mode 1 •

The ultiMate 10al <if the present rnsearr.h is to obtdin

1 :-l<jon)'lS mathemJtlcal Model for the physical nechanisfTls

nf ,) .'WSFET, with an attempt thereilftpr to inclucIe thE'

"f fC('Lei of ri.ldlJt ion.

The purpose of this report is two-fold: 1) to q i vc ;w

~'l-depth critical review of existing ~10SFE'T' theories; ,lnd

2) to discuss progress in the present research towards

development of an improved M0SFET model. In .• ddition, thiS

rc~port discusses a rigorous two-dimensl.-)n11 comput,;[ ~;()111t l(lll

for :10SFET operation, utiliz(·d throuqhout tilt' fJrl'spnt n";!',trch.

6

~ 1 • 1 . l

1.1 Traditional Theory of MOSFET Operdtion

This section presents from a fundamental viewpoint a

qll,} l i t.a t ive assessment of tradi tiona I HOSFET theory.

[) 1 SL~U~S ions are pres('n ted to estab Ii sh some importan t

\ n,ld,-,quacies of this theory, inadequacies relevant to the

present task of model development. Following these dis­

cussions, later sections of this report deal with proposed

l~odifications of MOSFET theory which offer improved agree­

ment with experiment. For consistency, an n-channel

enhancement MOSFET will be assumed throughout this report.

1.1.1 Mathematical Development

The Ihantola-t1ol1 treatment of MOSFET opera tion [1,2 J contains simplifying assumptions and approximations similar

to those used by Shockley in his junction-FET theory [3J.

In a one-dimensional analysis of this structure, Ihantola

and Moll proposed that souTce-drain electric current in a

MOSFET is attributable to drift mechanisms alone. Further,

as in Shockley's gradual channel approximation, they suggest

that electric current saturation in a HOSFET is a consequence

of channel "pinch-off" similar to that obtained in a JFET

structure.

The main elements of this theory result from a one­

dimensional application of Gauss's law. It is assumed that

the total electrostatic charge (OT) within this semiconductor

structure (i.e., both inversion charge and depletion charge)

is quantitatively determined by the gate-innuced electric

field at the oxide-semiconductor interface (Fig. 1.1):

~ c E (x,O) = -OT soy ( 1.1)

For simplicity, in Eq. (1.1) we neglect electrostatic charges

that may reside at the semiconductor surface, i.e., surface

states.

For additional simplification, we now also neglect:

1) the difference in work function that may exist between the

7

- OXIDE

SOURCE ISLAND

Figure 1.1

I . I. !

r-GATE ELECTRODE

L ------11 ....

P-TYPE SUBSTRATE

DRAIN ISLAND

Illustrative Model of a MOSFET.

8

§l.l.l

y,lt.t: (~lc:ct r-nde tlncl the semiconductor material, and 2)

01ectrostntic charges that may reside within the gate oxide.

i, e " oxide states. Based upon these simplifying assumptions

contlnuity of electric field at the oxide-semiconductor

interface requires that

E ys E (x,O) y

(1. 2)

In Eq. (1.2), VG is the gate voltage an~ Vs represents the

electrostatic potential at this interface, hereafter called

the surface potential. Throughout this report, all voltages

dre assumed to be referenced to the charge neutral regions

of the semiconductor substrate.

Substituting Eg. {I ?} into Eg. (1.1) results in the

following expression for the total gate-induced electrostatic

charge:

Q = T

In one spatial dimension, the surface potential (V ) within s this structure arises in conjunction with an electrostdtic

(1. 3)

charge (OT) due to unneutralized impurity ions and mobile

electrons. From a one-dimensional solution of Poisson's

equation in the semiconductor substrate, mobile electrons

accumulate in an extrenely thin layer at the oxide-semicon­

ductor interface [31J. When the applied gate voltage is

sufficiently positive, the density of electrons near the

semiconductor surface can exceed the density of holes (majorjty

carriers in the bulk), thereby forming an inversion layer.

Thus, the mobile electrons within such a layer constitute an

inversion charge (Qi)' The corresponding distribution of

unneutralized impurity ions is basically eguivalent to the

depletion charge (Qo) in a one-sided abr~pt p-n junction

[32J. Fig. 1.2 gualitatively illustrateg such one-dimensional

distributions of Qi and QD in a MOSFET.

9

0\ o .....

......... LI...J o ->< o ........

I (SEMICONDUCTOR)

UHNEUTRAL I ZED IMPURITY IONS (COMP'RISING Qo)

o~--~~~----------------­\ , \

" , " " c

Figure 1.2 One-Dimensional Distributions of Qi and QD in a MOSFET.

10

I . I. i

::fl. • .1. • .1.

Rigornus c~!culations of the electrostatic potential

'!l~~ributlnn throushout the s~bstrate provide justification

:or 1n lr.',portClnt simplifying assumption ~n tro.clitional !10SFF.T

S0"cificdlly, it is assuned that the surf?(~(>

pot·.::ntial (V s ) · ..... oulLl be unchanged if 0.11 nobile '.'lectrc;ns

~ere removod from the gate-induced space-charge reglon.

Based upnn t~is asswnption, the depletion rharqe (OD' is

determined from abrupt p-n junction theory [32J:

(1. 4)

At ~he time this theory was developed, MOSFET devices were

n::Jrr:;:11Iy oper-:lted under conditions of very "stron-:j inversion";

such operation corresponds to a surface density of elpctrons

(n ) substantially in excess of the substrate impurIty con-s nn~t-"'aj-ion (~I ) "- ..... '--......... i-... A • Under these conditions, exact calculations

fr');'l Poisson's equation show that the mobile electrons ::on-

tribute about 5-7 kT/q to the surface potentia! IV ). Thero­s

fore, Eg. (1.2) is a reasonable approximation for conditions

of strong inversion.

The total electrostatic charge (OT' wlth~n this

structure is comprised of the inversion charge (0,) and l

the depletion charge (OD)'

(1. 5)

Therefore, by substituting Egs. (1.3) and (1.4) into Eg. (1.5)

and then solving for 0., we obtain l

-K, ( 0. (V )::: 1 0 (V - V , .. /2K f': gtl

AV-

1 s t G s so s ox (1. 6)

In the traditional theory of MOSFET operation, it 15

next assuI'llpd that inversion layer electrons are transported

hetween the source and drain solely by the mechanism of dr1ft.

';herefore, a~ any location along the oxide-semiconductor

1::'

~. 1 . 1 . :2

interface, th~ source-drain electric current can be cxpresserl

as

I = uWOl.' (Va) dVs o CiX

where Q. (V ) is given by Eq. (1.6). In Eq. (1. 7), W is tr . .e 1 s

.... idth of a MOSFET and w represents the drift mobility of

inversion layer carriers. Finally, assuming 10 is every .... here

constant bet .... een the sourCQ and drain, integrating both sides

of Eq. (1. 7) give s

(L ID I dx

10 ./

rVD+2 ¢'F

= uW I

j 2~F Q. (V ) dV

1 S S

.... here ¢F is the Fermi potential given by

Implied by Eq. (1.8) is an assumption that the mobility of

(1. "7)

(1.8)

(1.9)

these inversion layer electrons is constant bet .... een the source

and drain. Further, the ~~sumed bounds for V (V = 2¢F at s s the source and Vs = Vo + 2~F at the drain) are justifiable

approxUnations based on rigorous calculations for strong

inversion operation of this semiconductor structure. Thus,

in the traditional theory, Eq. (1.8) in conjuction .... ith

Eq. (1. 6) is used to establish the vol t-dlnpere characte ri s tics

of a Mo..C;P'ET.

1.1.2 Electric Current Continuity

Contained in the Ihantoia-Moll treatment of MOSFET

oper;!tion is an assu!nption of electric current continui ty

.... ithin the source-dr~in inversion layer, thus

12

§l.1.2

== 0

From Eq. (1.10) , in conju~~tion with Eq. (1. 7' , 1oo;e obtain

( I

d 2V (UD

I dO. (::5; ~ Qi s + 1 = 0

dx :: UW

dx 2 dV

t s

therefore

d 2V 1

dQ. (:f s l.

dx 2 :: - --

Q. dV l. S

Next we show that the right-hand side of Eq. (1.12) has a

nonzero value.

(1.10)

(1.11)

(1.12)

Let us consider, individually, each term on the right­

hand side of Eq. (1.12). From Eq. (1. 7), if IO + 0 it is

evident that Qi(Vs ) + 0 and dVs/dx + O. Further, because a

well-defined difference of potential exists between the source

and drain of a MOSFET (2~F~VS~D+2~F)' Eq. (1.6) implies that

dQ./dV + O. Clearly, from the foregoing considerations we must 1 s 2 2

conclude that d V /dx + 0 in Eq. (1.12). s Equation (1.12) could be interpreted as a one-dimensional

Poisson equation,

= -

where p (x) =

1

Ie: E: S 0

p (x)

dQ. l.

dV s

or, L . ..:.:t:ead, it could be interpreted as a two-dimensional

form of Laplace's equation:

(1.13)

(1.14)

1.1. :)

+ • o ! ;

We now outline the ba!:1.s for thest! two different interpretduor ..

and their consequences from a physical point of view.

~n general, the potential distribution within the

inversion and depletion regions uf a MOSFET is given by

Poisson's equation:

= - p(x) +

~ E: S 0

The complete solution of this differential equation is com-

(1.1f.)

posed of t~o parts: a particular solution for the ir.h~moqeneous

part and a general solution for the homogeneous part. (The

homogeneous part of Eq. (1.16) is Laplace's equation, E4. (1.15) \.

From Gauss' law [Eq. (1.1)], it is evident that the

Ihantola-Moll Theory for MOSFET operation requires all SCt1 rces

of electrostatic charge within the semiconductor to yield ar.

electric field component normal to the oxide-semiconductor

interface. Therefore, we must conclude that this theory pro-

vides no ~ources of electrostatic charge capable of satisfying

Eq. (1.13); hence, from this point of view, Eq. (1.12) should

represent a two-dimensional form of Laplace's equation, Eg. (1.15).

It can be shown readily that the potential distribution

within any two-dimensional electrtcnl conductor satisfies

Laplace's equation only when the strtlcture is of homogeneous

electrical conductivity. It is our aim to show that the

electrical conductivity of a MOSFET inversion layer is not

homogeneous (i.e., spatially constant) between its source and

drain and, therefore, that Eq. (1.12) is not a two-dimensional

form of Laplace's equation. Thereby, we reveal an inconsistency

in the lhantola-Moll model, and a fundamental problem in the • traditional theory of MOSFET operation.

14

§1.1.2

In one B~atial cimension, Poisson's equation within

the inversion laye::- of an n-channel MOSFET has the form:

., c""V -2 cy

q =

• Co

(N - P + n) A

... ·here t!1e hole anc electron densities are assumed to be

~deq~ately described by the Boltzmann distribution:

r -, I

I

(1. 1'7)

1- q P = n. 811P (V-f

F)

1 (1.15)

I ~

n • n i exp

kT

r q

1;;-...

(V-¢ ) I nj

!

0.19 )

:\!ter substituting Eqs. (1.18) and (1.19) into 'oisson's

equatlcn, Eq. (1.17), a single integration yields (for n.»NA':

q, f Cs Eo E2 N V

ns • kT 1-;;; ys - 1\ s

Eq~ation (1.20) establishes the inversion l~yer electron

density at the oxide-semiconductor interface (n ). 5

Thus, Eq. (1.20) shows that the magnitude of ns

(1.20)

undergoes changes with a variation of either the gate-induced

electric field (E ) or surface potential (V ), and, therefore, YS s

varies spatially (with x). For this reason, We musE consider

the inversion layer to be an inhomogeneous electrical conductor,

so that Laplace's equation is inapplicable to the problem at hand.

From the foregoing discussion of traditional MOSPET

theory come the following observations: 1) this theor} assumes

thdt all sources of electrostatic charge within the semicon­

ductor contribute to the electric field component (Ey) directed

15

:; 1 . 1 . 3

perpendicular to the oxide-semiconductor interface; and 2)

the electric field component (E } producing a source-drain x electric current canrot be described by Laplace's equation.

In short, electric cu.·rent continuity is realized only if

electrostatic charges in the vicinity of this inversion

layer contribute to Ex. Thereby, we must conclude that a

nonzero divergence of electric current between the source

and drain is implicit in the traditional theory of MOSPET

operation.

1.1.3 Drift and Diffusion

~he traditional theory of MOSFET operation contains an

assumption that electric current in the source-drain channel

arises from transport of mobile electrons, due entirely to

drift mechanisms. This assumption implies that thermal dif­

fusion has a negligible influence on the volt-ampere charac­

teristics of this semiconductor device. It is our aim herein

to show that difficulties arise in this drift model, when ,

attempting to explain electric current saturation.

Traditional MOSFET theory attributes the onset of

electric current saturation to a "pinch-off" mechanism,

similar to that proposed by Shockley in his treatment of th2

JFET. Briefly. it is presumed that pinch-off produces a

zero ~ensity ot inversion charge (Oi) in the vicinity of the

drain junction, thereby terminating the source-drain channel.

Fi<jure 1.3 shows a graphical illustration of the calcu­

latee pinch-off condition, using Eq. (1.6). Clearly, from

Fig. 1.3, a well-defined channel pinch-off i~ predicted at

approxtmately Vs - 3.0 volts. Further, in comparison to the

inversion charge density at the source [0. (2¢p) 1, Q. is shar~-l. ~

ly reduced in the :immediate vicinity of this pinch-off point

(Fig. 1.3).

Experimental volt-ampere characteristics of a MOSPET drl'

continuous between the triode region and the satuation region.

16

,,,,",

w t.!:) 0::: c:::( ::t:: U

z:. 0

(/) 0::: W :.;:.-z

0 w N

-.l c:::(

~ 0 ::z:

LL -&

N

.... 0 , ...-.

(/)

> '-" ...... a

(SOURCE)

10':'1

10-2

1.0 1.5

Figure 1.3

.;J..a.. .........

2.0 3.0 3.5 4.0

Calculated Inversion Charge (Oi) in a HOSFBT;

t ox = 1000 AO, NA • 2 x l015 cm-3, VG - 4.3 volts.

17

n.1. 4

Thereby, Fig. 1.3, in conjunction with Eq. (l. 7), revedls

a difficulty when attempting to apply traditional MOSFET

theory to the saturation mode of operation: because the

inversion charge (Q.) unde goes a drastic decrease in the 1

vicinity of this pinch-off point, ~q. (1.7) could only pre-

dict saturation mode current (1 0 ) if the maynitude of the

source-drain electric field (E = -dV /dx) becomes extremely x s large in that vicinity. This necessity for an extremely

large magnitude of Ex places in doubt the .~lidity of Eq. (1.7).

Finally, consistent with this one-dimensional t~Qatment

of MOSFET operation, it was shown (Section 1.1.2) that a

change in Qi (Vs ) mnst be accompained by a variation of in­

version carrier density. In this theory, the dramatic de­

crease in Q. (V ) associated with channel pinch-off implies 1 s

that a substantial gradient of inversion carrier density

would exist within the source-drain channel. Therefore, in

the presence of this gradient, thermal diffusion must be

taken into account in any rigorous calculation of the source­

drain electric current. In contrast, the traditional theory

of MOSFET operation neglects diffusion and, thus, does not

provid~ a consistent explanation of electric current satura­

tion.

1.1.4 Other Limitation_

For the device technology existing at the time traditional

MOSFET theory was developed, this theory produced satisfactory

agreement with experiment. However, soon thereafter, rapid

changes took place in MOSFET fabrication techniques. These

~hanges resulted in situations for which this theory sometimes

proved t') be inadequate. Herein, we will discuss two stIch

situations, to illustrate limitations of this traditional

MOSFET theory.

In this theory, the triode-mode volt-ampere characteristics

of a MOSFET are determined by substituting Eq. (1.6) into

10

H.1.4

l:q. (loB) (",~)(i, thereafter, integrating the result; thus, for

VD~)2~F' we obtain

where

c ox

i j V V -( G 0 '~

r 3/2 • \ (Vo + 2¢F)

1(,(

C ox :: 1 0

t ox

3C ox

\

Further, this theory defines a "threshold voltage" ('IT) to

a~proximate the value of gate voltage at which channel

conductance approaches zero:

= V2K s£oqNA (2~F)

C ox

(1. 21)

(1. 21)

(1.23)

In engineering practice, an experimental method is often

used to determine a threshold or "turn-on" voltage. First,

from measurements, the square root of saturated drain current

is plotted versus gate voltage. Then, making a linear extra­

polation of this plot, the voltage axis intercept establishes

this threshold voltage ,[ 81. Henceforth, we wi 11 refer to this

definition as the "extrapolated thresh~ld voltage."

Advances in device fabrication technology led to the de­

sigr of low leakage MOSFET circuits. Thereby, interest be­

came directed towards the weak inversion mode of MOSFET

operation (ns~NA) which i~ obtained roughly for gate voltages

less than the extrapolated threshold voltage. Experiment

19

< I . ~)

showed that nearly six orders of magnitude of sourc0-draj~

current in a MOSFET are controlled by gate voltages b~low

thil" extrapolated thrr>shold voltage [3Jj. This range of d

MOSFET volt-ampere ch...:t'acteristics has been named the "Sub­

threshold region." Further, oy curve fitting experimental

plots 0f MOSFET drain current versus gate voltage in this

regio~, this current was shown to be proportional to

exp (qVG/mkT) (where m is an empirical constant) [34 J. From

such results quantitative and qualitative discrepancies

were demonstrated between t radi t lonal MOSFET theory r Eq. (J. 21) 1

and experiment.

Similarly, progress in device fabrication techniques led

to the development of very short-channel MOSFET structures.

For these devices, the traditional definition [Eq. (1. :21) 1 of

threshold voltage proved to be inapplicable. From Eq. (1.23) I

VT is a function of substrate doping (NA

) and oxide thickness

(t ). In contrast, experiments showed that the extrapolated ox threshold voltage of a short-channel MOSFET is also dependent

on metallurgical channel length and applied drain voltage

[30, 35J.

1.2 Extensions of Traditional MOSFET Theory

MOSFET operat10n in the weak inversion mode is critical to

the performance ot circuits requiring small values of source

drain cUt"n::nt: for example, CMOS and MOSFET memory cells.

An example of its importance is found in MOS rremory cells,

where weak inversion produces source-drain leakage and, thereby,

considerably influences the refershing time of the system [36J.

Paralleling the development of low-threshold MOSFET devicl:s,

an equal effort was directed toward the development of hlgh­

speed structures. To attain this increased spend, there has

been a continuous reduction of source-drain channel length;

in fact, some devices have been fabricated with a channel

length as small as 0.5 urn. [38].

20

H.2.1

Clearly, it is hiljhly subjective to classify MOSFET tran­

sistors as either having a "long-channel" or a "short-channel".

For this reason, there is no well defined boundary between

these two types of device. Nevertheless, a survey of the

technical literature [17,22] shows that some device engineers

believe this boundary lies in the vicinity of 5-8 ~m. In con­

trast with this oversimplified view, other authors believe

thdt the short channel boundary is related to the sourc(= and

drain space charge layer widths, relative to the source and

drain separation distance [18,21,35].

The purpose of this section is to present theoretical

studies of MOSFET operation that have been developed either to

attain better agreement between theory and experiment or,

instead, to refine physical concepts that are clearly inadequate

in the Ihantol~ treatment of this problem. It is shown here

that these studtes often represent extensions of the Ihantola

solution and, as a consequence, retain many shortcomings of

this work.

1.2.1 One-Dimensional Drift- and-Diffusion

Traditional MOSFET theory assumes that conduction in the

source-drain channel is due entirely to transport of mobile

electrons by drift mechanisms. Section 1.1.3 showed that

difficulties arise in this model when attempting to explain

('le~tric current saturation. Briefly, it was shown that any

rigorous calculation of electric current saturation in a i

MOSFF:T must take thermal diffusion into account.

The influence of thermal diffusion on the volt-ampere

characteristics of a MOSFET was investigated by Pao and Sah [39J.

III their analysis, the inversion layer source-drain electric

('urn'nt density is given by

a~

jx = qun an x

21

(1.24)

,; 1 , :> ,

where ¢ is the electron qUnsi-Fermi potenti<:ll. Equ.ltlon (1.::'4) n

impli<:itly contains both 0'··'-t and diffusion mechanisms of

Cdrrler transport [40]. From Eq. (1.24) / the toted ;;;oun:I.-dldl!l

electric current (at any location along the oxide-sE'mic.)!,Ju".tUt

interface) is obtain"d by integrating this expression over 1.111'

channel cross section:

f a~ T qlJW n

dy. ( J. •• ! ) I = n --·0 ;Ix

0

In Eq. (1.25) / y. is the depth beneath the oxide-semlconduct:Jt, 1

interface at which n=n i . t'·ssurning 10 is everywhere const.ant

between the source and drain, integrating both sides o~ Eq. (1.25) gives

= 9.l:.!! L

L y. 5 51 o 0

If we now assume no electric currp.nt exists in rt direction 1-'(,'_

pendlcular to the oxide-semiconductor interface (iy we have fl~ /fl =0 thus,

4 Dn ·; .

n y

= ~!! L

o 0

n dyd~ . n

The inner inteS'ral of Eq. (1.27) represents tht, lnV('rSlOn

charge in a MOSFET, y.

o i • -q S 1 n (y) d Y •

o

22

( 1 . 2 k)

§1.2.1

A change of variable in Eq. 1.28 (setting Ey = -dV/dy)

Ylelds the relation v (x)

s

Q. = -g j n(V,Qn) dv. 1 E; (V,.p n)

tF

(1.29)

'fhereforc, after substituting Eg. 1.29 into Eq. 1.27 we cbtain

,jf1 expression for the source-drai.n electric current wilhin this

semiconductor device:

"'~H~re n (v, 1 ) is given by the Boltzmann distribution for n electrons in this region of the structure, [Eq. (1.19)].

(1.30 )

A rlnorous calculation of 10

, using Eq. (1.30), would require

<1 two-dimensional solution of Poisson's equation, Eq. (1.16).

Tn simplify this problem, Pao and Sah treated the case where

x J V/'Jy ; thus, they neglect (j2v/ax2 in Eg. (1.16) { 1

~n(j, thereby, reduce this equation to one spatial dimension.

Tr,t' lnclgnitude of the electric field component Ey(V,$n) is

,hereby obtained ~y a direct integration of the one-dimensional

,'olsson's equation for this semiconductor structure, Eq. (1.17):

(1.)1)

is the extrinsic Debye length,

0.32 )

23

Setting in Eg. (1.31) V=V we obtain the electric field s

within ~ ~~SFET, due to tb· ate electrode, E Thus, after ys

substi tt.ting one-dimensional form of Gauss' law[ (Eq. (1.2) 1 into Eg. 1.31 w~en (V=V ) we obtain

s

v -G

,~ (KSfC\ (kT) fp-sv s + BV -1 C Lo I g s

ox I L

+ [.2:.]' [e s n -1] n· ') lqV - ¢) ] ~ NA

where C =K.£ It (the gate to semiconductor electrical ox 1 0 ox

(1. 33)

capacitance). Equation (l 30) in conjunction with Eqs. (1. ~l)

and (1.33) can be solved numerically, to determine theoretical

volt-ampere characterictis of a MOSFET.

Tile foregoing derivation makes clear two central postulates

in the Pao-Sah theory. First, from a one-dimensional Poisson

equation [Eq. (1.17)], all sources of electrostatic charge within

the semiconductor are rpquired to produce ~n electric field

component [E , given by Eq. (1. 31)] normal to the oxide-semi-ys

conductor interface. Second, from Eq. (1.26), this theory

assurne~ electric current continaity within the source-drain

The.se postulates also form the basis of the

of MOSFET operation (Section 1.1.1). Thus,

inversion layer.

traditional theory

the Pau-Sah theory can be viewed as an extension of traditional

MOSFET theory, Lh_ essential difference being that the former

takes into account thermal diffusion. It is our aim herei.n to

show that (as with traditional MOSFET theory) the Pao-Sah

theory introduces difficulties when considered from a funda­

mental viewpoint.

The Pao-Sah theory of MOSFET operation neglects the term

a2V/3x 2 in Poisson's equation [Eq. (1.16)], and, therE,fore,

it is implied that Ex is constant between the source and drain.

If E is, indeed, constant between the source and drain of a MO~FET, V (x)· would be a linear potential dish ibution.

s Therefore, a further impllcation arising from this treatment

24

91.2.2

~s that distributions of Q (x) calculated from Eg. (1.29) are 1

limited to solutlons resulting from a constant gradient of

V (x). s

Based upon this treatment of MOSFET operation, Pao and Sah

sho~ed numerical calculations of E (x) for a device operating xs

:i'':,lr electric current satL:ration. These calculations established

a substantial variation ~f E between the source and drain. xs

Because it is implicit in this theory that E (x) = const., we xS

ha\/c an important inconsistency. This lncon5istency places

in doubt the applicability of this model to the saturation mode

of MOSFET operation.

1.2.2 Weak Inversion Operation

Several publications treating the weak inversion mode of

MOSFET operation have recently appeared in the technlcal litera­

ture [11-16]. A review of these publications shows that all of

these studies offer a modification of the Pao-Sah theory,

~lth ~n aim toward obtaining better agreement between experi­

ment and theory. It is important to note that these studies

provlde no modification altering the basic concepts (and diffi­

culties) associated with the Pao-Sah model (see Section 1.2.1):

electric current continuity is assumed within the source-drain

channel, yet there are fundamental reasons why this condition

0: electric current continuity is not attained.

A. M. B. Barron [refs. 11,26]

This author derive(~ a modified form of Eg. (1. 33), by

taking into account a surface-state density (Q ) at the oxide-55 semiconductor

v :: V G s

6nterfac~ (" E:) r 5S i2 s a (kT) -8Vs C--+C-L--le ox ox D q

25

+ av -1 s

(I.34i

Applying numerical nethods to this equation, Barron '.',i!culdt:r'l:

the potential distribution v, I" ); this calculation l.>d (C'1

conclllsion that for conditions of weak inversion V is 'H'C;f:r.t 1,,1

constant for O~~n~Vo' i.e. between the

therefore, proposed the following weak

source and drClin. He,

inversion Clppro:<im,ltl rl:

v =V 9 0

(1 • 35)

where V represents a MOSFET surface potential at the source. o

This potential can be calculated through a numerical solution

of Eg. (1.34), setting V =V and ~ =0. son

Barron also modified Eg. (1.36) by assuming a 1 O\ .. ·c [ lunlt

V=O rather than V=~F:

Vo Vs

L L auW ""'---

L

n(V'~n)

E (V, ~ ) y n

dV d¢ n

(1. 36)

This choice of lower limit corresponds to n=nf/NA (rather than

n=ni

) .

.l\ssuming V =V in Eq. 1.36, Barron obtained an aPIJroximLlte s 0

closed-form solution for the source-drain electric current.

This solution provided important information on the weak inver-

sion mode of operation: it showed that the transition into ·-·lcctrlC

r"lrrent satL:.ration is exponential, with drain voltage,

I ( 1 - -qvo/kT) Th' l'" 1 . t . O~ ~ e . IS cone uSl0n IS In qua lta Ive ayreement

with both laCora~Jr-" measurements and a two-dimensional computer

solution for the case of weak inversion. Quantitative agree-

ment with experiment was realized only by an empirical adJust­

ment of Qss

in Eq. (1. 34).

B. Swanson and Meindl (ref. 12J

These authors derived a model for MOSFET operation

applicable for moderately weak inversion (V >20kT/q) through s

strong inversion. In this model, invers ion charge (Q.) 1 S 1

obtained as the difference between the total electrostatic

26

§1.2.2

charge (QT) and a depletion charge (QO):

Qi = QT - Qo (1. 37)

From d one-dimensional application of Gauss' law [Eq. (1.1)]

In ,~onjunction with Eg. (1.31) I the total charge is given by

:::: I( <: E S 0 ys

(n.) 2

+ N:

BV -1 s

(1.38)

Further, for Vs > 20kT/g, it was shown that Eq. (1.38) is well

approximated by

2

(1.39)

Substituting Eq. (1.39) and Eg. (1.4) (the depletion approxima­

tion fo!: QO) into Eg. (1.37) yields

Q. = 1

+ ·I2K c gNAll-s 0 s

27

B(V -+ ) e s n

(1. 40)

; 1. 2.2

Equation (1.40) can be used to calculate 10

, provideJ that

VS(~n) is determined.

By a mathe~atically complicated method, Swanson and Meindl

darived approximate relations for VS(.n)' applicable resp0c­

tively to the weak inv rsion and strong inversion modes of

operation. Finally, these aut\.. ..... rs obtained expressions for

the source-drain electric current (10 ) by a three-region

approximation: 1) strong inversion only, 2) mixed inversion

(strong inversion near source, weak inversion near drain) ,

and 3) weak inversion only. For weak inversion operation

their model shows that 10 depends exponentially on gate voltage.

Purther, this weak inversion model also shows an exponential

transition into electric current saturation as (l_e-qVo/kT),

in qualitative agreement with Barron's result [11].

C. Troutman and Chakravarti Iref. 13)

These autho_s developed a model for MOSFET operation thilt

is applicable from th~ onset of weak inversion (Vo=~F) to the

onset of strong inversion (Vo=2~F). It was asserted that the

surface space charge of a MOSFET operated in the weak inversion

mode consists primarily of inunobile charge and, therefore, the

surface potential (V ) should be independent of the electron s

~uasi-Fermi level (t). Thus, for a given gate voltaye, this n

treatment assumes

= V = b. ItS 0 F (1. 41)

where 1~b~2. Further, these authors calculated V through a o numerical solution of Eq. (1.33), with V =V and ¢ =0:

son

= V o

.'2""" + -­

C ox

28

BV o

-1

(1. 42)

§l.2.2

T!1t' "b~lnd-IH?ndinq par,Hoeter" (b) t~iin be found from b=Vo/,tF

•

l's I nq I:q. (1.41) for V , Troutman and Chakravarti obtai ned s

d cl.)st'd-fl1rm .lpproxim~te solution of Eq. (1. 36) I for sourc€'-

dr.llll L'l,'C'tric current (IO) in the weak inversion mode. This

solutlon showed that TO depends exponentially on the band­

bending parameter (bl. Moreover, in this solution, the transi­

tion into electric current saturation is exponential as -gV r/'<T

(l-e ), in gualitative agreement with the weak inversion

models of Barron [11] and of Swanson and Meindl [12]. Finally,

Troutman and Chakravarti showed good agreement between theory

and experiment throughout the subthreshold region.

o. Kat~o and Itoh (ref. 14]

These authors d~rived an approximate solution of

Eg. (}. 361, by first reyersing the order of integration and

then dividing the resulting equation into two parts, as follows:

I ~ ~ !) L

Vs(Vn) ,~

+ I I

)

V 0

o

Vo (

J

n(V'·n) E (V,. )

y n

n(V'·n)

d¢ dV n

Ey(V'+n) d. dV n

~ n (V s)

(1.43)

In Eg. (1.43), Vs(VD

) represents the surface potential at the

pinch-off point. The potentials Vo and VS(Vo ) are obtained

by approximate solutions of Eq. (1.33), setting, =0 and. =Vo ' n n

respectively. Finally, after making extensive mathematical

approximations of Eg. (1.45), these authors obtained a c10sed­

form approximate solution for IO' applicable to MOSFET operation

for dny degree of inversion.

q. 2.2

E. EI-Mansy [ref. 161

A novel modification of the Pao-Sah treatment was

developed by E l-Mansy [16]; this treatment d if fe rs sign i hcan tl'/

from those previously discussed. This author does not rely on

cla~sical Boltzmann di~tributions [Eq. (1.18) and (1.19)J to

de8cribe the distribution of inversion layer carriers perpen­

dicular to the oxide-semiconductor interface. EI-Mansy com­

pared calculated electron distribution$ from classical and

quantum-mechanical treatments of an inversion layer; the more

rigorous quantum-mechanical solution [42] predicted a much more

uniform distribution of el~~trons than the classical solution.

From this comparison he hypothesized that this difference could

be largely responsible for discrepancies between previous MOSFET

theories and experiment. For this reason, he proposed to

alleviate these discrepancies by using an ad hoc recta:gular

approximation for the electron distribution, asserting that

rigorous quantum-mechanical calculations of an inversion layer

can be more accurately approximated by a rectangular elect_on

distribut:on than by a classical Boltzmann distribution.

An important contribution of the EI-Mansy research is an

interpretation he gave concerning calculated differences bl~tween

the inversion charge based upon classical electrostatics and

from quantum mechanics. He suggested that the source-drain

channel mobility ~_ have been using is a reduced mobility

to "chplain-away" an inaccurate evaluation of the inversion

charge.

To simplify his analysi~, EI-Mansy approximated the net

charge distribution (due to both inversion charge and depletion

charge) by an equivalent constant charge. He argued that this

approximation is reasonable in strong inversion, because then

the net charge is comprised mainly of inversion layer electrons

(which, by assumption, have a constant distribution). Similarly,

for weak inversion, he asserted that the net charge distribution

is approximately constant because it is comprised mainly of the

30

H.2.2

depletion charge which (he also assumed) has a constant distri­

bution. In the transition region between weak and strong inver­

sion, it was assumed that the error resulting from this constant

charge approximation would be negligible.

The main elements of this theory follow from an assumed

prod~ct solution for electrostatic potential within the source­

drain inversion layer:

kT V(x,y) = - g(x) fey) - ~ q F

(1.44)

Chooslng f (0) = 1 in Eq. (1.44) and, in addition, requiring

that v(x,o) = Vs + ~F' he obtained

V(x,y) = k~ f(y)Vs

(1.45)

To evaluate the function fey) in Eq. (1.45), this author

assumed: 1) the space-charge layer beneath the gate oxide can

be rep~esented by an equivalent constant charge distribution of

thickness y ; a~~ 2) the electrostatic potential distribution s within this space-charge layer is adequately described by a one-

dimensional Poisson equation (d2V/dy2 = constant). From two

successive integrations of this Poisson equation,

(1.46)

The thickness of this space-charge layer (y ) is determined s from a requirement for continuity of gate-induced electric

field:

(1. 47)

y=o

31

',1 .2. 2

Thus, by substituting Eq. (1.46) into Eq. (1.47) and, there­

after, solving for y , he obtained s

Ys = 2V s

1(1 [VG-V:;:.l

K t J s ox

(1. 4 R)

By substituting Eq. (1.48) into the depletion approximation

(QD = -qNAYs)' the depletion charge can be written

(l.49)

EI-Mansy's expression for inversion charge is obtained by sub­

stituting Eq. (1.3) and Eq. (1.49) into Eq. (1.37):

Q. ~

K.C ~ 0

= ---t ox +

K. ~

K S

2V s v -v G s

tox

(1.50)

J EI-Mansy obtained an expression for surface potential (Vs )

in terms of terminal voltages, by a procedure in which he

"matched" his chu.:".-"'l region model to boundary condi tions at

the source and drain junctions. A complicated implicit rela­

tion for Vs resulted from this procedure.

Finally, he showed that the source-drain electric current

could be evaluated from

(1.51)

o

32

~1.2.3

Eq.'dt Ion (1.~']) .)!,pears similar to the Pao-Sah treatment of

MOSFE't I)peration, except that r. in Eg. (1.51) takes the place

of ;. • n

lie d(>scribes ' .• 1S an "effective potentinl" in his

(rectangular charge distribution) model which is analogous to

the electron gUnsi-Fermi level (t , in the (classical) Pao-Sah n

model. Using Eq. (1.51), in conjunction with Eg. (1.50), he

dctcr."1iner. the s()urce-rirain el~ctric current. Cslng this model,

F.J-Mansy showed good agreement between theory and experiment

for MOSFET operation, from weak inversion through strong inver-

510n.

1.2.3 r;'han_nel Length Modulation

Channel length modulation is a mechanism initially proposed

by rhantola [1] to explain the finite drain conductance of a

MOSFE~ when operating in electric current saturation. This

proposed mechanism is based upon several simplifying assUQp­

tlons (see Fig. 1.4):

1. an inversion charge (Qi) of zero exists at the pinch­

off point in a source-drain channel:

2. a virtual drain for minority carriers bounds the

inversion layer at this pinch-off point:

3. an increase df drain voltage (Vo' produces an increased

distance between the pinch-off point and the drain: and

4. if ISA'f designates the drain current at the onset of

current saturation, when operating deep in saturation

this drain current is given by

L I :: o L-llL ISAT I (1.52)

Where i'lL is the drain to pinch-off point distance.

Reddi and Sah [43] viewed drain junction depletion layer

widening, (and the consequences of this widening upon IO) as

analogous to the Early effect in a bipolar trans~stor [44].

33

u.. u.. e

I ....... ::::t: z: u­z 0 - 0... 0...

c..!)

>

Figure 1.4

Z L..I..J e I-- « (.I') 0:: 0:: 0:: I.I..J I-

~>- (.I')

zS (Xl =:;:) - V')

L..I..J 0...

~ I

0...

Illustrative Model tor Channel-Length Modulation.

34

~1.2.3

§1.2.4

For this reason, it was assumed the pinch-off pOl.nt to drain

junction distance could be calculated using the t.heory of an

abrupt asymmetrical p-n junction. In their calculations, it

was assumed the pinch-off point was located at the space-charge

layer edge and, therefore,

(1.53)

where VSAT = Vo when lSAT = 10 ,

In the Reddi-Sah analysis, VSAT

is calculated using

Eg. (1.6), and assuming Qi = 0 when Va = VSAT + 2~F;

where C = K.C It . ox 1 0 ox

-v:-:-2VGC2 1 + ox

K E: qNA s a

(1. 54)

Using this channel shrinkage model for MOSFET operation in

the saturation mode, Reddi and Sah showed reasonable agreement

with experiment for substrate impurity concentrations on the

order of l016 cm-3. However, it was subsequently shown that this

model can be in gross disagreement with experiment. In particu­

lar, this model tends to overestimate the output conductanc~ of

more lightly doped devices [45].

1.2.4 Models for Short-Channel Structure~

Recently, a substantial number of publications treating

short-channel MOSFET structures have appeared in the technical

literature [13,16-23,45]. Similar to the situation with recent

weak inversion studien, these short-channel studies have retained

an inherent weakness from elementary MOSFET theory: they

possess no constraint to assure source-drain electric current

continuity.

35

,1. /..1

A review of recent ly publ i shed short -chZlnne 1 tllcor ll~S

shows thZlt each treatment att-ributes short-channel eff'1c:::; to

one or more of the following mechanisms: 1) channel' n(jth

modulZltion, 2) ~arrier velocity saturation, and 3) twc-dlmen­

sional electric field:. Among these recent studies, tht': '~drllt"

publications dealt pr.J..1arily with channel length modulat10n,

thereby seeking to explain finite output conduct~ncc 111 ~hc

saturation mode of MOSFET operation. Subsequently, other

studies also took carrier velocity saturation into account.

in an effort to obtain improved agreement with experimtwt.

With insight from rigorous computer calculat ions of MOSFf-:T

operation, later studies fr~used attention on the influence 01

two-dimensional electric field distributions within this semI­

conductor structure. It was suggested that these electr'ic

field distributions are responsible for threshold volta(je

dependency on such parameters as channel length and dr,! 1n

voltage.

In each of these short-channel studies, model development

was based upon heuristic approximations. Typically, these

developments utilized empirically defined functions, and

obtained agreement wi th experiment through the use of on" or

more adjustable parameters.

A. Frohman-Bentchkowsky and Grove [ref. 45J

These authors modified the Reddi-Sah treatment of channel

length rr.odulation (Section 1.2.3). It was assumed that, between

the inve=sion layer and the drain, the average electric f1l'lJ

parallel to the oxide-semiconductor interface (E ) is adeyudtely x approximated by

E x (l. 55)

This field was attributed to the superp~sition of electric

fields (see Fig. 1.5): 1) Ell arising from acceptor ions

within the drain depletion layer, 2) E I the x-axis component x2

36

CHARGE-NEUTRAL REGION OF P-TYPE SUBSTRATE

I

GAlE-bRAIN DEPLETION REGION

I I

GATE ELECTRODE

~ ~ ____ ~f--------~ I

!'ISURE 1.'; Two-Dimensional Electric Fields in the Gdte­Draln Depletion Region of a MOSrET (after f~ohm~n-Bentchkowsky and Grove [45]).

37

of fringinq field £) which arises from the drain-gate potentLcll

dlfference, and 1) £ , the v-'lxis component of fringinq fi(Jld x-, E, which arisl's frum the gate to pinch-off point potenti;ll

difference (V -'J ) G 'SAT .

E x

Thus,

(1. ::16)

substituting Eg. (1.56) into Eg. (1.55) and, thereafter,

solving for AL gives

VO-VSAT f\L :::: E

El + E + x2 x3

(1. 57)

From abrupt junction ~heory, E} was approximated by

(1. 58)

The fields E and E were given by the approximations x2 Xl

(1.S9i))

(1. 59b)

where III and a2 are adjustable parameters. Thus, substituting

Eqs. (1. 58) and (1.59) into Eg. (1.57),

[~NA _ 6L = (V -V) (VO VSAT ) o SAT 2K E: s 0

- -1

+ + (1. 60)

In this model, Eq. (1.60) is used in conjunction with

Eg. (1.52) to calculate the saturation mode volt-ampere charac­

teristics of a MOSFET. Using al = 0.2 and a2 = 0.6, these

38

~ 1. 2 . ..j

r€'f. 17

,1 I . \.1 ( ~ r~ r". \.'! I

jrl~l<'~- rr;(il)\l:~y W:thl:'1 thL' lnverSlon layer:

ho]('s,

v In

C J o

" o

x I, I', ( 'I ~ 1 XS Cj

1/1

., , ~()r clectl'ons,

1.1.61)

dnd l:: c

( 1. C 2)

dr.d low field mobility withln this inversion layer.

':'his medt· 1 tli:ylects diffusion, therefore, the source­

drain ,,·:!",:tric current is det.err:1ined by substituting Eq. (1.61;

in t J Eq. (1. 7 )

hole:;, so th:ll

rdV /0 s x

r dV /d ~ I s X

+1 ! c

11 1/

t

)

( 1 . 63)

in Eq. (1. 6 3) . For this case, dssu~ing

0), t r.e se

, u t t . ()! ~ ,'; ,) • IJ I ,d E (J . (1. 6 3) b Y s epa rat 1 () n ~) f va ria b 1 e s, 1. '-='. ,

::11<,: '.; ('J ) L S

fur' triode-IT:Jde operation, the y 0 b t a .1, ned

(1. (4)

39

where IDO

1S the source-draill current given by vlemt!rltdr':'

, "

theory [41]. In d p-type subst- ~~r:?, the inversion la'lvr ('elrCI(',

are electrrms, S0 that 1 = 2 in Eg. (1.63). This CdSf' ",1~1not

be solved by ::epar\ition of variables and, instc3d, Armstr{J~;{: inr!

Magowan obtained a numer cal solution.

At the onset of eleccric currpnt saturatioll, ISAT .s

determined from Eg. (1.64) with VD

::: VSAT

: t~e resulting

expression for ISAT is then uSed in conjunction with the

preceding Frohman-Bentchkowsky and Grove mode 1 tr) ca lcu 1 ate

MOSFET volt-ampere characteristics in the saturat10n mode,

Using th1S model, Armstrong and Mago .... an showed reasonablt>

agreement between theory and ~ xperiment for a short-chanr:," 1

MOSFET (L =3um) with substrate doping of 2.7 x 10 1 'cm

C. D. P. Smith [ref. 18J

Smith modif1ed elementary MOSFET theory [41)J by taklnc; lnt"

account channel length modulation. In addition, he lmpllcitl':,

accounted for the influence of two-dimens1onal electrIc :lelJ

distribut10ns on the volt-ampere characteristics of a short

channel structure. Briefly, the inherent multi-dimenslon:tlit]

I)f these fields was accounted for by redefining threshold

voltage.

This author's treatment of channel leIlgth modulatIon IS

based upon Insight gained from a rigorous computer solutIon [or

MOSFET operation ~/~]. For saturation mode operation, this

Solut1on sho\"'ed: 1) at some point along the oxide-semJ-

conductor 1nterface Vs = VG and, therefore, the gate-induced

electrIc fIeld (E ) is zero; and 2) there 1S an equipotL'l1tlal ys

line perpendicular to this interface at that part:icular locd-

tion (see Fig. l.ha).

Smith argued that no inversion layer can exist alon0 thIS

Interface between x == !,L and the drain, because E reverse~, 1'5

di;:-ection in this region. Fur':her, rigorous computer calculd-

tions [28J showed that the inversion charge diminishes ,Ibruptl,;'

~o

-r IH'{U

..,)" L

',',: \

FrC;UHE l.h

L

,,1. 2.4

EQU I POTENT I AL -\ EyS == rJ (V == VG)

,/c(x) ,.)

SURFACE SPACE CHARGE LAYER DRAIN

I Y ,--L,L NTIAL

CHARGE I == VD) NEUTRAL I

I A -- - ------- ---B SUBSTRATE WD I

( a) VD

I

Vs(x) -----

(b)

---IoID

/ '/

/

:, L 0

,I. Ch,u-,Icteristlc Equipotentla1 (V = VG) for Saturation-Mode Operation of a MilSPET.

b. Electrostatlc Potential Distribution Associated with Fig. 1.6a.

G u

I-<C -..J t- .cc:

~~ I- W U I­WO -..J a.. W

(after Smith [18J).

41

near thIS point, Ther,~forL>, Smith propos(·d ~o L1Sc' tl:;

account for ,:hannel length '". 'ulation,

SufficIe:'tly far fro!11 the gate elt~ctrode, the p()t,'r~t 11 i

cllstribution ,iCIOSS the dra~n depletion layer IS adl'qlJ.lt.

approximated by one-d'menSlonal abrupt p··n )UnL-tlon tik()r-':',

Thus (alona 1 in.e A-B 1'1 Flq.

[V B (x) J is g 1 ve n by

'Ja) this potentl,ll (jlstrIL)utl"ll

where

v (l-x/W ), o D

f). -. -- S 0

~ qNA

( ; , (. ') )

( 1 . ~,II )

In Fig. 1.6b, Vs'x) and VB(x) are superimposed; at the ~olnt

of intersection Vs(tL) = VG so that

Substituting Ega, (1.66) and (1.67) into Eq. (1.65) and, there'

after solving for ~L yields

( 1 , h 8)

For long-channel structures, Eg. (1.68) could tw us,,] 111

conjunction "ilth Eg. (1.52) to calculate soun't'-c:r,llil l I·'."!!

currc.-nt (11)) In thl' saturation mode. However, for sfl(·rr-

chan.nel st::uctures, Smith argued that two-dimensiofl.ll "l.··t r"

fIelds withIn the semiconductor matt.'rla1 WIll modify the

threshold voltage and, therefore, also ID' Specific.Jl';.',

because the depletIon charge beneath the gate will bt, f',ln 1 d: ly

supported by t.he drain-induced electric field, he hypottl,'!;[z<.',J

that threshold voltage would depend on VO'

42

\1.2.4

i'n lmpJl"ltl',' d('coclnt ~or ttllS short-chi.lnnel effect, Smith

r')f'l),"'C: t i1<' :<.! !·" .... lnq .1f'proXlmatl0n; he modi: l~·d LD from

;.t", ... ·t.lr\· t~l' (.ry ~4hl i,,; introduci'lq <If I "cff"ctiVO thrc:>hcdd

( 1. 69)

\/:T 1I],l!" dl,tL~rmined by rcqcl1r1ng cont1nuIty (in rnaqnltudc ami

"lo~)f'1 or the 'Jclt-am~'ere characterIstics, between the :rIode:

.!:~.j sdtur . .1tIon reqions; thus,

V T

v -v G T 2V

G

WD

\ --i

L I

{2 ,,' : ;~ ~ N-A

( 2 't, F )

C OX

(1. 70)

(1. 71 )

(1 . 72)

and ..... here W in Eg. (1. 71) is given by Eg. (1.66). Equation D

(1.72) can be recognized as a threshold voltage from elementary

MOSFET theory [46J. Smith showed that this model reduces to that of elementary

MOSFET theory in the limit of long channel lengths, small drain

':oltages, and gate voltages near VT

[Eq. (1.72)]. For hIgher

gatc dnd drain voltages, this model predicts significantly

Irt'dt"[ draIn currents than elementary theory. Further, as

sho ..... n 1n Fig. 1.7, this model predicts greater saturation-mode

output conductance than the model of Frohman-Bentchkowsky and

Grove [45], but less than the model of Heddi and Sah [43J. fInally, Smith showed good agreement between his short-channel

model and ex?eriment for channel lengths of 5 to 30 um and sub-

strate impurity concentration~ of lOiS to 10 16 em-I. However,

43

w 15 u z <r:: I--<....> ;:::)

0 z 0 U

I--;:::) 10 CL I-- >-;:::)

0 til 0

W ..c: 0 E 0 :l

:E '--' I

Z 0

I--<r:: ex::: 5 ;:::) I--<r:: (/)

o o

FIGURE l. 7

2

I ... j

4 6 8 10

Calculated Saturation-Mode Output Conductance (substrate impurit~ concentration 3 x lO 1 . cm - j, c han n ell e n q t h 4. ;.,) ;

(A) Reddi-Sah Model [43J, (8) Smith Model [18], (C) Frohrnan- Ben tchko .... sk v ,'Hlel (; ro\',

Mode 1 [45]; . (after Smith [laj).

44

for more heavily doped substrates this model can deviate signifi­

cantly from experiment; specifically, calculated saturation-mode

output conductance showed noticeable deviation from device data

at large gate voltages.

D. Hoeneisen and Mead [ref. 19]

Through modifications of elementary MOSFET theory [41J,

these authors also proposed a model for short-channel struc­

tures. Thits model takes channel length modulation into account,

with the aid of an adjustable parameter. In addition, they

treat carrier velocity saturation in the source-drain channel,

by means of an empirical relation for carrier drift velocity.

Neglecting diffusion, the source-drain current was written

in the form

(1. 73)

where veff is an "effective carrier velocity." Assuming

dIO/dx = 0 between the source and drain, an integratiQn of

Eg . (1. 73) give s

W XI-X2 (1. 74)

In Eg. (1.74), Xl and X2 represent boundaries near the source

and drain, respectively, between which this one-dimensional

treatment is presumed applicable. Therefore, XI-X2 represents

an active channel length, whereby channel length modulation is

taken into account. By a change of variables, Eq. (1.74) was

written

I .. o

45

(1.75)

§1.2.4

To account for carrier velocity saturation, these authors

~8ed an empirical relation for veff

:

= v (dV Idx) m s

{:: + d:~} (1. 76)

In Eq. (1.76), vm is the maximum carrier velocity and Wo is

the low-field mobility within an inversion layer. Equation

(1.7S)p in conjunction with Eq. (1.76) for veff and Eq. (1.6)

for Q. (V), was used to derive an expression for the source-1. s

dr.ain electric current.

Based on this approximation method, Hoeneisen and Mead

obtained a good fit to experimental volt-ampere characteristics

of MOSFET structures with channel lengths as short as 4lJrn. How­

ever, in eSSence, this fit was obtained by using Lc as an

adjustable parameter.

E. A. Pop a [ref. 20)

This author proposed a theory for the saturation mode of

MOSFET operation. His theory takes into account channel length

modulation, wherein the pinch-off region by the drain is

treated like an abrupt p-n junction. In addition, this theory

accounts for a nonzero density of mobile carriers within this

region, and for c~=rier velocity saturation. However, calcu­

lation of the output conductance requires an iterative solu­

tion; moreover, agreement with experiment was obtained by using

the drain junction depth as an adjustable parameter.

F. H. S. Lee [ref. 21]

Based on heuriatic geometric approximations for the dis­

tribution of depletion charge, Lee derived a threshold voltage

applicable to short-channel MOSFET structures. To approximate

rigorously calculated depletion charge distributions in a

46

~1.2.4

',,', , '\"',\, '7( 27J 1,1 ). f.. \ "-. ), I this dutnor proposed a plecvwis·.'-r,'ct.c.lflgular

:t,\~:l:Tlatl<.m, shown in Fig. 1.8. In this flgur,-: Ws ' We ~lnd

""!' ,-,-present, respectively, source, Chdtl!lel, Lind draln dvplt.'-

,! ],!')','I \<.'1,1t.li:;;:-; IS thl' source/dra"\ 1~~ldf]d lClnctlon dt: It:; J

l'l.i l~; dli .Hllustablt." parampter, tor lateral extL'nSlon of the'

.1:~·(-" .H,d draIn dt~pletion layers.

d:; : r~~ abrupt. 1 uncti on theory .

... I.lths, th" SUrfdC(' fJ()u~ntial (Vs ) was piecewise dfJproximatt'd

'c. f () I low 5 :

v s

(0 x yW ) S

(.W < x' L-YW ) S D

K" ... \' " D b i

x L;

(1. - -, j ,"-

( 1 . 7 -,'- i

1:1 I:q. (1./"), Vbi

IS the source/drain built-:rl pot'>:ltial [c;;,

I· 15 a band-bending pdrameter, and K is an adJustable.' pdrameu:r.

The approximation given by Eg. (1. 77b) was based on Barron's

('omputer solution for MOSFET operation[26]; thiS dpproximatlon

·'''.is presUJTIed valid for we,3k inversion operation, wherein b 2

signifles the onset of strong inversion. From the thf:oory of

() n c - sid (' dab r up t P - n j un c t ion s, inc 0 n j un c t ion \.,' i ": h E q . (1. 7 7 )

till' depletion layer widths were given by

( 1. 7 fLJ)

(1.78b)

,21< • ,J s 0 liuN . ~ . A

( 1 .7 He)

47

....... ----L

-"Y (I.! +1.1 ) • S IlO

',' de r-----, , , - --1

I ______ .J

I , I I I

l ________ 1 __ _ LHAR(lE -NEUTRAL SUBSTRATE

,l'Ct'wtSL'RectanCluL, ,'\pprO:':lmatlClf\ I', ,

:·10SFET ~)t'pl\'tl(>n ('r' ::01' Distr-d)ut :,,: I 1ft l' r ! ,t • e [H).)

48

, I. 'i

'i 1 . 2.4

!"'\"fl',, hy "'·II.1tlTlQ the total charqc on the' Cjiltt:: ci<'ctrr)dL'

lil" ('hdrd" COil! lined in th0Sf' depIctIon t<1Yl:rs, Lei' cl('r-,v(·d

This ('xpre:';sion ('an

: H \,: I , f I \ . i 1

I' -, ,., I

1 C L

ox (1. 79)

.... ',\.,.:',. \'p, :s :1 trddltion.,d (long channel) threshold vnltaql' 1

~I" ,-])~ and f is an algebraic function of substratE:

'1::"11,:' r'()!l,'cntrdt Ion (NA), drain voltage (Vo )' sOClrce/dr()lI1

11::d ]IF,,·t t0n depth (x,), and two adjustable parameter:o (. )

iI:': /-.). 'J'h(' adjustable parameters, y and X, must be d(~tpr'mJned

:" fltt in') Eg. (J. 79) to lneasurements of thr('c~hold voltac'l!.

:,';Ifl'l r:11j; pmpiricdl relation [Eq. (1.79)], Le(,' inv0stigat::d

! I,,' '!":)(''',':C[lC,' of thr('>sholrl voltaqe on d1annel lenqth, sub-

1;rl1'lJrlty '':Ollc(!ntrdtion, oxide thicknc:ss, sClurce/d Ll: n

.:'11ndiur-Iction depth, and drain voltage. I~ short, Lee fClU~d

(.,It t'II, threshold voltage decreased w:'th decreasing chan.::l

n'] t h. Sppctfically, factors that enhance this decrease of

1) increased drain voltage. 2nd

:n I ~Ic'rf'd::(>tl source/drdin island junction depth. Lee's c('sults

f'JC thiS r hccshold voltaqe showed significant decrementc'd

b"havic:r- ~nl- channel lengths less than about 2;:m.

(~. ':'routman and ChDkravar" --------"'----iudt:;cl b,/ a rigorolls two-dimensional computer snlut ~():: fn:

"l(JSFLT op"ratlon [251, tl.<..!se authors modified their w(~ak invor­

,on thc'ory (Se~tion 1.2.2C) to make it applicable to short-

This modification relies on an adjustabl~

'I: H'l(·ter thdt is evaluated using this rigorous solution.

Prom rigorous two-dimensional calculations, the source-

.,' ; 1 :: (. 1 e' . t r i r fie 1 d (E ) qoes to zero Clt a "oint (llonq the xs r

JXl,jV-Sl!mlconJuctor interidc..:. Between t.his l·Olr.t .lnd till'

E IS directed towdrds the drain; conversely, between ){s

t ~ I • E is directed towzlrd~, the S()~.Jrc:e, xs

49

I . .' .

Thus, at this ~'.~ro fi~ld point, the surf.:lct' put',·nt.!.l!

~xhibits a minil1um.

{ '/ s

point <.I5.:l localized potcnti.:l1 b~lrri('r or "vir!lL!! SIIlI'"

For long-clhllH1cl structures, the 1'routman-('!ld,',rd'/ .r:

weak inversion th,·\.)["y gives the [0110win'1 (~xfJr.:s~,I"r f· r

SOurl'L'!-drd i n eh"'ctr ic currPllt:

\\ kT\ I 0 = -L ,-- I . J V . q I 10

I..

In Eg. (1. 80), Q. is the inversion ch.:lrge for zero dP;;J I,'d 10

drain bias. An expression for Q. , <lS .J : unct 101, ol d i),,',.,; 10

bendinq par;lmeter, was glven in tht!ir .... (~ak in'Jcrslor: t : ••. ·'11·/·.

These :1uthors iH'Oposed a modification of Eq. (1.80) tu "U"I,d

~ts applicabil ty to short-channel structures.

This modification was based UpOl! ,Ill <lppro: ... irr.all'J: :":

Q. • 10

It was presumed that Q. depends uniquelv on V .It th.' 10 • S

r.oint where E r- Xl> :::: O. Further, they assumed that thl' surfd''-''

potenti.:ll at this point (Vo ) could be adequately .tt.lfJrc)~:un.lt.'d

by

v o

.... here " is an adjustable parameter.

(1. H J \

Equa t ion (1. 8 1) 1 S ~; I 1~1 J l.t;'

to Cq. (1. 41) fro:n their long-channel model, (';':Cl'!Jt tid!

I:.:g. (l.81) contains the term VO/6 to account! ()r ShUl'! -,'11 Illll.·1

effects. This approximation was used to obta j I; il meld I I I "d

(~xpression for Q, and, thereby, for 10 ln Eg. (1.80!. 10

Troutman and Chakravarti showed 'Jood <lqr(:('rnent i;":\o,'l(':,

this model ane cxppriment for two MOSFET structures, · .... lth

l(.'ll'jll'.-' "f Kill! "r'·; ? m, respectlvely. ii· .''';.' Vt ~', ! () \)b' 1." • 1

,lO I'" .••••• I. I ill ,'/I,; 1 r: ,.'.1 I parameter I' '.'1

\ •• '" I

'. ()

'·!l\L!I'~.'d t.\, l!Il1<iniJ riqornus lwo-<ilm.'nSl()lldl ('c\mput';.'r c:"lcui,)-

'11~:. '!'hc:orofnro, thl' !'I~.I'·tic'1lit.y t)f usintj t!ll:::; ['lo(h·1 for

Ii . II. C. Poon l't .11. ret. 2}J

In t's";C'ncC',

:,)nnUll WdS d<'l-lvod to account for the inh'.'reLr.ly two-dimcr.­

';~()!1,J d:stribution of electric fields in such 3tructur,·s.

Brit·fly, :t WilS n:cognized that dCplt..;tlOfl C:l,jr ;\~ with!.:i this

:.;,'mi·'(lliductor mtlterial must be !;h"rl'd b.~n,·.·\'" the qate' ,]!;d

drdl!1 .md, therefore, only part of this dL'plellO:1 dElrtlC' coo­

t.tth~lt/·S to tIl(' gate-induced electric field.

'I'll .Jccount for gatL-dr .. lin shuring (If tillS d,-~pletl<.r 'h.!n:c,

In this figure, \-J_ represents the \odth of the yatl' '-

d"plpt lotI luycr ilnd x, j s the source/drdin island jun-:-t lO!1 J

rddiG:L In .Jddition, the dr.Jin junction depletion lay,·r ..... ·ld':.t;

j:; approximated as WC' At the onset of strong inversion

(V 2.tJF), Wc was approximated from abrupt Junction tt.cory:

i'2K ',I 0

~ qNA W =

C (1. 82)

'!'!wrt':tJ':, from Fig. 1. Q , depletion charqe electrostatical Ly

t i,": t.) tho g.:lte-el,'ct.rodc WilS ,lpproximated hy an cquivalC'nt

':ld:"qC rt!sidin-J within,} trapezoidal dop1etior~ rL!9iun of the

:,,;,'H.: conduct or subs t I"a te.

.. 1\ •

Tne tc.ta1 ch.lrq(· electrostatically associated ..... ·lth th.>

,I.' l'!pc'troclp i~ (]ivl'n by Gauss' law:

Q1' :;.. - C ( V - V ) ox G s

of stronq 1 n \'0 r s ion, \' s " ,-, F

Eq. (1. H 3 ; Thus, bast~d upon this trapezoidal dcp1etlon layer

51

Thus, , t this . I \) It·IJ ;)0 I nt t hl' . , ~; i

'I'r' • ~ . I ~l . I : I , l (' h . I i-; r I ': ,1 r t I

p()lrl! ,I,; .. lOC.,[l.c.'ci :"d":ltLll t).ll-rl' r' 1[' VI!

1-'''1 lor l-,·il.I:·,:,,·1 :itru"tllr-(';';, tb,' Trf)llt'mol,'-

i)

h

L 1-('

_ .. v . [)

, I

i" ["1, (1. '10\ Q IS th.· !nVerSI')n ,'hdrq,· f'J! /('1(, d.: !, ':

I' '

. 1S. :\r l'~:pr,:ss l\.n fur Q ! (

,uthor-c; l"c)!)o3eJ cl moJiflc"tllln c,t :'1, h(),

I. '0

r twas p ,e s umed t hd( () i 0 dep,'nds llfl I q U(· 1:' 1,:'1 \.

W:lerl' E " x.>

v

o. Fur the' r ,

()

15 d~ ad)'Jstablc pardmetcr.

t ()

contains te rm v " 0'

,ffeces.

'xpn~ssion for

Troutman

'.' .1nc1, thereby, for 10

,Inc Cha k ra vart i showL'·..!

" f, .J .. \ ':n c:: ! : ) t lmL'Ur 1 t ,. ·'·,f

:'I(

, t· i I

:'jUdt lOll 1.1. r: I,

lnl ".]1_,1 , : .t

In lrl Le;. 1,1.2,",

'luud .Iq!'··.·!'!<·:lt

. ,. j .

r

r"

: j 'f' I , , >1 11 (11 m, resp{'ct : Vt· I ) '.'. ,t,

: I tit il pdramet.l:'l

, , ()

J t

: ,J ,.

, ' . I ..

.1 '

I ~ \ j

' ... ' t

I. .·1

> ... I .... ' i 1:1 .:, ,ff:: ~ J t •• r "1

, J , J'- '~j ( I ( : I

" : f : 11" ) t_ (.; j t,

" I ;,

1'\'

" ( ') r~ .. ~ I t . , ~ ',."r;

I: • r t ru~~t 'I

" ,I !l It \ ~ : '..... . ~ t ~:

;-'I' ! t . 'r- , ! ,~ ,. ;)#' I r >cj I),' t \', <

: !"r i' Iii I I! ; j I· ,t I,'" \ 'j ) (!-

. , :' 1', '.'cl I,' 1 i' • t rw 1 ....

.1 ',''''1;:.· ~ ( , ! 11, \ \ - \ ~ r \. t 1 ! I ;,,-,

~ 1 t > l. '( ) !'1. > t 1" 1 ~ ~ I. 1 : :r ~ , ~ (l~.,

1 ~ L' t.l r L S \ ' ~ 1 t S t ~ 1 t' \', 1 1..1 t ~ 1 () f t h L) ") t t

. .; 1 ,I:: [: ) u r: " •

, U: 1 L' t 1 ():~

2 F) 11

:,! , ' ~ , )r~',

(: .• t't·-t l,'\~~ t~()>c!" Wd~-) '1~)!--'r()Xlrn~lt(': t~', '-.}!'l

J 1 n ' ... 1 t l~

I) 11"

,t d 1

; . t! ).11' t '!1 1)', (~,) U S S I 1,'1 t,.,' :

( 1 "']'

t " ~' :-, " ( ~

, ' v

c;

':. • l s

. ..;tr()~'I,j l:I\'t'! , ' ( ,

,; \ dl';,;l,.:·, ;':'

5 I

...... -----L

CH~RGE-NEUTRAL SUBSTRATE

:,',(;[,1;;: 1.'1 Ceunlc'll'lL· /\IJIJroximatiull to ,I ;,10:,/.'1.':' Dl'pl.'tlon ('tLlrqe Distlibutl():1 (,1 f t (. r POOl! ft' t ill. [.) :! J ) .

1'1 Ii

l

:'1 ) 1 \' I '1 :

( ,

. [, . l'

';' Ill' ' :-;!I' - I ,: :

. [) (1 \ '" \ '"~ : U t\ i'

" '.' i (';' ,.) " '1' 1 ~ .

-, ' + 2. - 'i ~~" (2 : I' )

S (' 1\ -

':"F C ox

u 1 -­

L

I.-d 1.

1. L. Y

F,om Eq. (l.2G) and Fig. 1.9, these ,'luthors <JDtalr.cd ,1:1 <.'x:;r.:';"­

Slon for threshold voltage, which could be written

V,""", ;. L ..

I ".,--------... '~2K C qN (2<P)

J _~_o __ lL_!:._ c ox

(1. 8 7 )

Fo!' channel lengths much greater than the sour<..:c/dr'lin

r.lulu;; (1.> Xj), f(N",L,X

j) goes to unity and. the/'pEon:·,

f,q. (l.B7) reduces to elementary MOSFET theory LE'1. (1.72)].

:_',ll'~:ul"tiOllS utilizing Eq. (1.87) showed that this th:-l'sholJ

voltc!yt.,! decreases significant-ly for channel ~er1'!ths 0:1 til'-:

ordel' of 2:.m or less; furthermol:.-e, this beh.'lvior is cnhanc(:(~

by ir:crvasinq sourc··/dr~iin island radius. Thcs ... ' charact r-

istics ,)(jre(~ qu.:.iitatively with theoretical and experiment.)}

result;, reported c,lsewhere [21,30,35J. Usin9 Eq. (1.87).

Poon ct. al showed reasonable aqreement between thpcrl' ,1nd

('Xpel' iln(~nt for channel lenqths from 7. 4\lm down to 1.4 .. m ar,d

substratt.: impurity concentration of 1.6 x 10' l em

')1

, 1 , :2 , ,1

1. El-Hansv -~-~ ~~

'J\lk i nq a nove 1 approac fl to ch..lnn~' 1 1eng th mod:J1 d t 1 fll!,

El-~1clnsy extended the applicability of his ~10SFE'!' mod"l !()

hort-chdnnL'l ,;tructures .

• 1 drcd:1 re-lion by an ,1rbitr;\t'ily defined bound.Jl":',

.k-fined this boundary by

';I~V/"JvZ ----'''-- ;: 1 0 . :;:V/;'x

( ; . H', !

Thl~ channtcl region is that part of the source-drdin ':~dn;ll'l

bet .... ·een this boundary dnd H~e source: similarly, th., dr'lin

re,non lies between this boundary and the drain.

prl.';;llIned the .:lppli'cability of a one-dimensionill pnissc):; ";; It lUl:

[Ui. (1.17)] in this channel regior.; therefore, the <.'1·'(,': r,,­

steHle potential in this region is given by Eg. (1.4fi) (::')[:1

the El-Mansy long-channel model). By substitutinq Eq. (1.4 1,)

into Ey. (1.89) and evaluating the derivatives, r;l-Mdn~;'/

obtained an implicit equation for the surfcice potenti.ll V, .It 51

this boundary. (To determine V , this equation must bl: sol'.',,-,u S I

numerically.) From Eg. (1.48), the width Yl of the sur:,\('·'

s~aC0 charge region at this boundary is qiven by

= 2V

SI K, -V - V

1 G 51

ie t 5 ox I

..J

lid'/ing established these boundary conditions

( 1. 9U I

(V clnd y:), 51

E I-Hans)" treated the drain reg ion as a vol urne obey i :1q Cd u:;,;'

law,

( 1, q! )

Thus, without requiring a detailed knowledge of chJrqc Lii~tr 1-

but lon, the total charge co.ltained within this region ,'.II: 0,'

S4

.. " -'- --

1. ! . ..:

" iY.., <; ttl i s Jl!)i)r()t!\~l:, l 111 :-; tL"t !l()! ::;l()~;(' t "( * I (1

n 1 ('

! i dt ':1\; \ l . ').' ) 1 n t ( (1. 91) '11 Vt.':';

;.: . I ~ • d::' -rl (N \ • n ) rl,,'d:, .

0 ( ) . . 'j ~ J

j.

",:: 1 . . E1 ) dy ';' ;I'LiS ~.: l' \:'

~ ... s 0 .j

X l' W K r (E s 0 2

0 l:.) dx .. (l .94)

t! E:. 1.(14) I trw electric field in the chan;." nl~U': .. :-;'.l:)-

!) )

• r IJ ! 'I'

c,

I', .9'.'

,.--- \I = · s

- OX ! 1)[ - q ;-11 CO:lLJlJf f fJ p

I ~ r l :. f /\, C L

ir x

IV I I Y ~ 1 I 1 J' 1 (3} l _______ ~~ ______ __ _

D(J r, /" i.I' 14

P-TYPI:

CHARGE-NEUTRAL SUBSTRATl

!'IC',j~l 1.10 \'dllssi.ln-Surfdc~' B()undlrl('~', (I -I,):,

AI-nund ,111 Aroitr,lrily lJC·!ll ... ·d DI·.I;I

Hc'q ion (aft(.>I' LI-M,ll,!,y [J (,1).

56

! n Eq

.it t hc'

F

1 : i:

dV x 1 -

1 -

'I .... V. i

• t

'. 1 .1.4

2 ( 1. (ii.)

f' ,,~.

, 1 • J I 1

(1.9(1), Er is the surface sourcc-c:rd.n I.·lvc~:. ::l .. ., I

bound.lry bc"t' .... et·n the channel regi0n .lnd tilt.: .•. : .•...

1.1 Flom Eqs. (1.96) dnd (1.97), tt,,,, :lr • :':'v::JI

( I . 44

K s

IS evaluated as follows:

ii: 1 - £1) dy ~) J

- WK ,i" I •

oS 0 )

: ,

;-:;U!JS'ltut l!l<j E.15. (1.93), (l.95) and (lo9S) Int . ~! •

(IV S

I dx

v • 1 3 + v: rX

. c eV. - V(') dx J ox (, .>

C

;,rt('r suiJstantJ.al In.tthematical manipuLltion of ::'!. (:.'1 4

c "

ox I (VD

t, ·.2 J(V • I --" L' . -r ). - v } \' I' I'

I' .) S - (; ~ 'D ' S,

" 1 • 1 .'

• qN'Y I (V - V ) ks'oYl

I E;;n ) - E-

1\ D SI () c: • J

57

I.

In Eq. (1.100), ESD represC'nts 1-: Jt the dr<1::: JU:l,-ti'Jfl. ~ X5

EI-~,lnsy _I~;sumed ncqligiblC' diffusion in this drdlll r. :l(,n,

so th.It dr-lin current is (,J\l!ll by E4. (1. 7).

formuLl. ,\ssUml:1Q 2('rO div('rqc~nce of source-drain curr,-rt,

El-,.1ans'/ ~)btJinC'd an cxpression for drJin current.

expreSSIon in conjunction W~_.I Eq. (1.100), he established .til

.~xprcss 1 on for output conduct.1nce wh ich produced qood dCj I '-'I.·rn' -;: t,

wIth experiment. This agreement was obtained over a widc·

range of .1pplied Jrain bias for: 1) a silicon gate deVl('!.!

".;lth a ch:H'.f1cl Ipngth of 10 IJm and substrate lmpurit/ ','OfiCl'n­

tration of 10: 'cm- 3; and 2) three metal gate devices Wlttl

channel lenyths of 70,50, and (;.6 urn, and substrate imptHlt

,: 0 nee n t ~- J t 1 0 n 0 f 5 x 1 0 I .., 3 • 2 x 1 0 I ", and 1. 3 x 1 0 1 • em - ,

1.3 Overview of ~resent Research

St'l:tions 1.1 and 1.2 of this report present importa:lt

aspects of ~OSFET theory that form a basis for this resc.!rc!J,

It '"as shown in Section 1.1 that traditional theory for ~1()SIT':

operation contains a multitude of simplifications and apprC)Xl­

~ations that are clearly inapplicable in many situations.

~Je\'t'rthc'less, this theory offered great mathematical simplli"t·;,

and, In .lddition, lt agreed with experiment for a period o!

In time, and with many improvements in devi,'"

fab:'i(,l· :on technology, it became increasingly eJcLlr thcll

lrcHlltl.,n,d theory required improvements. Speci f ic.III y, f)()(,r

,1'lrec:nL'nt was ::lttilined bet .... een theory and experimt.'nt for S~lurt

Ch.1n:lt'1 structurc's and/or in the weak inversion mode of ('I-"'rd-

r_ ior:.

This situation was recognized by many workers, ~nd d mult 1-

tude of <llternate theories were proposed to improve thls situ .. -­

tior.; cha!1nel length modulation, the incorporation of dlfful;i(lll

me(:han j s:ns I etc. we re proposed as an "upda te" 0 f the t r- dJ I t I (' I!. i 1

~O;;FET the-ory, as a means to obtain better aqre('m~'nt \0,'1 t h

58

J,

,' •• 1\ i. J. ;:1,-':\ t • l:"'~ . ! I ' 1>'. r I f I '-h' 1 '.' l! ~ ~ ~_ ' .•

,( ~;l'liL !t,-Sl ~lrl"t t~ff()rt:

'" ;:,,,j:: j '.ltions of tradttional theory (:i1.21.

I f .1 i: t 10',.1,

: ':11' 1 : l' 1 t 1 nth i s t tH.' 0 r y .

:. '\" 1 "t \ .s ..

I\~'" .j 101' ,I1S to ;litcr this p,)ttern of rt.·sedrch, r;:.· Pl"'SL':H

d'~ IS Jlr','cted to"'ilrd dn elimination <..1f heuristic :n.!t·:ods.

·'rn:JJtl'r pI'OsraM was available for the two-dimcnslo:131

,: .. 11·::'IS of MOSFET O!.H'rdtion.

·qU.l: 10'1'; solved by this computer program repres'..:l1t o,,·:;(:r.!l

·.··1 .11lSrr.S dssociated with hole-electron transl'Jo!'r ll~ ,-,;,::1"::-

,·!.!,:ct\)r !:1dteri,;tl; tlJ(~s,,·' equations are der:.vt·d ,'!")I.l 3rd H1-

I~' ;.::J'51<..'5. ilnc1 thc", ·in' not unique to the de'.'!,'" ·.;:.d\~!,

Bhort channel st.rlJ~·-

did ..... e.lf: :nvcrSi(l!l operation. Thus, th 1 S CO;;1PU t,,:-

,,;,dTT' ...... 1~, :lSl'c! to s~t:dy !'hysical mechdnisrns of Mn:-;F'ET ,);"t' ,­

I; tn .. t cOllld not be' dl,tt'rmincd throUClh 1.,bor,Jt<.)r' eX~'crl-

.1' 1:1..! cornpu".ltion'1! ~;,:nsc, we could "·;t'c" l::~'l,ie ~,:lC'

';l:~',)r dfld .Iccur.ltely c~valuCltc the domi!1ant ml~L'unl~;:;lS

,.: :\l"!;/')lIy its '.'h~(·tric,ll charClcteristi,:s.

59

-'-"~--~-- ---

I . ;.

into MO;-'f"!:', o,>"l" , t ,(

Thc purpose' (.r" !~'i:; ,.. '.': .":

dlmC:!lS.lt'n,ll C'~~lL"ulatlons, Jnd to dis(·us~ lrnportdnt S~l()! t, ,,;!;:.,

II: ft.IJ:tltHLll th,.n ':' wherl" mOdlfi(·.Jtlnn~; .~lrl· np'·(·';" ...... ·.· t·.

,I:· .• lt:~ our- c(),ils :'or this r',J<,'drch.

1. 3. Curn·:,t ----~---------

It ..... as pn'vlously shown (:il.l.2) that sourr-.·-Jr ,:1. ('1··," r

'urrcnt conttnutty 15 not redlized in the tr .. HlitJ·)I .. l: ~"urtherrn()re, it was st10wn tha t 1 d· .' .. ,"

::.Odl:~e(: .1:1<3 lmproved til.s trdditiondl theory (':] . .!.!' ..•.

: LI<.."u:q SUltdDlc import<lnce upon this DdSIC phYS1C,,1

:' .. :::t. Itl shol't, most availdDle mathemati,,'al tr','dtm"r:':

··~()SFE7 ()tJeLH,i..()~ .)ssume that electriC' current C():ltl:H;;t·,'

:'\·.dL.:eG. wIthout imposing this characteristic dS d ; .. : .... I': .. ··:,t.

i\ detailed study of traditional theory (.Jnd m()d;ll~' .. ,:I .

. ), -:his thvoryl reveals an important defic-ipn,:y.

t:it.·!'e 1S :lO mechdnism (either implied or statcd) .... ·tH'rl'b',

el~ctric current continuity can be attained.

t:'V.:ltments havt! insufficient degrees of fre-:tlC>n'. whereb/ ':h'

req~lrement of electric current continuity could l)(.: imtJ,-,.;t·d

'-1pC':. t !l(, approximat lng mathematical eguations, 'r h 1. S !J lj r' t ~ ( . t l 1 !:

.j S~c<.."1 of the problem was addressed in .J prev ious pub I 1,' . t ,( .

O~ th:s topic LL~J.

~'():·l'" study of the two-dimensional computt.:r suluti·>!'"

,·lv ... :t ,-_. current continuity is, inde('d, rC.:Jllzed 111 ·.t1L· 111\····-

dO;) ;d':t~r of tl MOSFET, although this necessary situ .. t ;u'\

I t tel i nL'd through mechanisms not prev ious 1 y d: sCUSSt'd 1 ' .

Namcly, the total source-Oral!! ,_,l"l:t I,

currcllt (ID

) could be described as a sum of a drift c:ump()ll";:~

(Idrift) ilnd a diffusion component (I diff ),

60

,L 1

,Jrdl!l jU:h'l:,'!l 10 aru;es pl'imarlly from dr1ft, d!lJ till' ci:: :u­,>l.m L'pmpr.!:.'nt 15 neyliqible. In regions near tt.·.· ,jralfl

~u!lction trc,movc'cl from the drain SpaCf:>-Chd!"t;e lc1yt"-rl t ~ ~ . - ....

dr'dl'l C'uI'rL'nt drlse!'> primarily from diffu~;ion, .... ,lth ,j :;tltr

:;lal1tidl r,~duclion of drift mc:ch,"lnisms.

From this computer solution it became obViOUS ';:.1' 'i::fu­

,~l()n 15 .Hl lmpfH·tant mechanisnl associated w1th ('r:,,~,:.:' ~:J:,SiJ rr

:11 tlh' inversion layer of a HOSF'?:". It was ,lIsa ... '"

, lcctric '.:urrent continuity is not accurately,;,:;;.; ,:.;: .. "c! if; ,,!~.:

t:1f'('!'Y .1r MOSF'ET op(~ratlon without explicitl';, in:.:l'~:::-.,; i:l ~I,J

<h·()!·), it source-drain electric curre!1t arislny frc·:n :J):i. :,;"

"nJ difrd:;ion.

In d previous study this aspect of MOSFET th('(',!; ";,1.',

"dd u:-sscd [24], and a modi f ied theory was pr0tJ0scd t:. j t J !-':,,,:' ;,

:;ou:-C'c-drain electric current continuity in thIS S"!"~l"():~ :'J('tCI"

.lev 1 C", It was :hown that electric current cont 1l1Ud.y "r15(-S

thrc~uqh two-dimensional mechanisl:lsth.lt are funcar.1L'nLl. tc'

~OSfET operation. Through these two-dlmens lona 1 me,'lld!l, :';:::s

tddjtion,ll degree of freedom is Icali7.ed in thl' (JPCL1tc .. ()~

this semiconductor device and, thereby, electrh- Cl.:r"'_':'.t ,")n­

lnuily is ~ttained.

Furthermore, this new theory of MOSFET operatlO!l .... .:!,::.

reduced to a simplified one-dimensional system of l'qu~tlnns

tt1a t are sui table for engineer ing ?urposes. It was shown that

this simpllfied theol:v is adequate for enY~.leet"lng purposes

whcr applied to long-channel MOSFET structures (channel len,;ti~

-' lO"m), but that this theory fails for other impol'clnt SltU:l­

lior~s. Namely, this theory fails to adequately describe the:

volt-ampere characteristics of short-channel MOSF'ET struclUft',;

(ell,lnr""l length, .:: 7.0 wm). Further, this tiw,ny Lllls tu

d\l('qUd te 1 y descr i be the vol t-ampere character 1 st 1 CS 0 t ..I

61

l. ~. 3

MOSFET structure in its weak inversion mode of operdti()rl,

regardless of channel length.

Thus, we have the m:.u. purpose and direction of the presellt

research effort: to identify inadequacies ~ssociated with

this previous theory of MOSFET operation th.:lt n'nder It i f1.lppJ 1-

cab Ie to these fore r .2ntioned s i tua t ions.

1.3.2 The Weak ~nversion ~ode of Operation

Two-dimensional computer studies of weuk invers i on hav.,

established a situation not considered in traditional theory or

~OSFET operation. Namely, these studies show that 111 ":C'dK lrlVt'r­

Slon nearly all source-drain electri~ current is ~ttribut~bl~

to the diffusion of inversion layer carriers. With this

i nSlght, we can say that this tradi tional theory should I:ot b •.

expected to agree with weak inversion experiments.

The engineering theory resulting from this two-dimensioTl')}

study approximates this transition to diffusion, in the lim~t

of weak inversion. Thereby, this engineering theory is In

agreement with two-dimensional calculations of MOSFET opcra­

tlon. Nonetheless, quantitative disagreement is found bet .... ccn

the calculated volt-ampere charccteristics based upon this

newly developed engineering theory and two-dimensional ~omputer

calculations. The reason for this disagrec:rlent was unknGwn,

and a solution for this problem represents an important goal

for the present research effort.

1.3.3 Short-Channel MOSFET Structures

Comput~r calculations for MOSFET operation have also

identified an important two-dimensional mechanism producinq

threshold modulation in short channel structures. Specifically,

ion sharing between the gate and drain produces a two-dimer.­

slonal electric-field distribution that cannot be adequately

described on an elementary one-dimensional basis. Although ion

sharing exists in all MOSFET structures, in short channel

s t ruct ures thi s mechanism is encountered near the source i Ull,'­

t ion to the extent required for a sign if icant mod i f i C,1 t ion 0 f

the threshold voltage.

62

.: 1 .4.0

!, simplified view of this two-dimensional mech.lnism can :w ~)btdlnt'(~ frolTl Eq. (1.6). In this expression it is assumed that

~~. i!; C0n~tdnt and, the'refore, Q. (V ) is uniyuely defined dt t\ 1 s

" ,11':"!1 10c~lti0n l.Jy (VG-Vsl. This is not so in (j two-dim(~:-,-

In two spatial dimcrtsic r,:-"

tJCCO!l1CS .~vl>ryv.'here a variable, and its magnltude dc('r'_,c;'·::;

Wl th ? decrease of channel length and/or an increase of <lea it.

\'01 LHlC. A~ a consequence, the specific maqnltude of 0 (V ; j S

~lcfincd ,1S threshold can be <ltttlined at a reduced gate voltaqc

.. ,·hen (' i ther the channel length is decreased or the drair.

\'oltagc is increased.

ThIlS, a second goal for this research is to modify the

foremel~tioned engineering theory to adequat.!ly dpproximat·_'

thi:-; two-dimensional mechanism. Thereby, it is believed th:.:~

~'ngineerlng theory will adequately describe the volt-ampclf'

('~ldracteristics of a short-channel MOSFET.

1.4 Two-Dimensional Hechanisms in .HOSFET Operatlon

Because the gate electrode and the drain junction are

qeometric~lly perpendicular, impurity ions within the del,lc­

tion regions of a MOSFET must be shared. For these regions,

Po i sson' s equa tion has tr.e form

-qN A

K (: S 0

(1.102)

where the x-axis and the y-axis are in the direction from

sou r,'t'-ci ra in, and the di reetion from surf ace-subs t ra te ,

p;spC'ct ively. Since E := -grad (V), this expression can also

b(~ wri ttC'n in the form:

dE + ~.

dy 0( ,-

S 0

(1.103)

Tile :;p"ciflC manner.' in which ions are shared bt·tw"l'n the' q,ltl'

il:ld drclln is unknown. However, fOl- steady-state opL'r<1tion,

63

Ion den~itll'S t: and"J C.ln b(' d"flnt-d sllch Uul x \'

dE

.. _. , 1\

! . '\ -

"J x

s () x 'j dx

: 1. f!.:,

dE bl N

s ,.. ,--y ( 1 i I,

Y q dy

F~om this interpretatIon, ion-sharinq becomes -l!1 lfllf)('rtd[;-

mechdnis!n: ions N thilt are electrostdtlc~lll'/ dS~;()Cl"~t-'! -', x . che dr"iln junction do not contribute to the q.llc d·'pl.·t til'

charge. Similarly, ions N that are electrosLltic','lll':- dS,;',­y

·liltt.'d wIth the gate electrode do not contributt· trl tlw c!:,;'

spac':'-Chtl rqe 1 aye r·. For this reuSOIl, til(' substratv clu;Jll\:

:n Eq. (1.6) (tradition3l MOSFET lh(!ory) L'~l!1not be- ('U1\:'10.,:,,:

(I constant.

Fiqur,- 1.11 qualitatively establishes the dcqn:l' u~· idt·

Lind draIn interaction in it MOSFET. In thIs i llustrut Ion ',,'.

show contours of constant source-drain electric field ,:orn;)(')["'!I:

Ex' From Gauss' lel""', the magnitude of this electric flt·l:

;'omponent is directly proportional to the inteqrated subs',r.!!,

l'lectrosLltic charge contributino to E. ThIs illustr.,tlClr. x

... nO'. .. ·s t:1.1C .) :lon-negliqible ilmount of thf' substratt' l'!t·('trt;-

stelt1\.: c!'>arq(> contributes to E and, therefore, this ('har I" x

cunnot contribute Lv the electric field component UtJOIl Whl"~l

t r3d: t lonill ~10SFET theory is based, Eq. (1.1).

From Fig. 1.11, it is evident th.lt Eq. (1.6) is most

accurdte in the vicinity of the source junct.ion. Tilrouqh()llt

othpr regions of thIs structure d non-neqliglble amount of

substrate electrostatic charqe contributes to the x-dxi~

component of electric field and, therefore, this charqp rllust

be subtracted from NA in Eg. (1.6), Thereby, two-dirn('nSlnr'.!1

mechdn 1 sms become an important part of MOSFET opera t lOr!.

64

...... " - - - - - _lSf)O V I U1 - ..... ""-, -- ""-" 2000 VIC: ,

\ '- ..... L;IJO V / cr, - -- - _ " - -' , 'Jr, V /,' -,.- - - - - - ...... \ ')t,,; /LII --__ -_ ' \ ---- " '\ ---- ...... , ' \

(~----------------------~-~- ", ' -...... ,,\ \ ..... ,\ \

\\

I 1 f ~ l !" E I. 1 ;

SU:)S 1 RA Tl

- -

\\ ~ ~

i ~ • • !

(',ll"U;,lt",: l'ont()UI'; ,,' ,',,: "t \~,'

:~{)llr"t\-1)r,'1r: 1·:1 'l'\.t il' I·'lt 1 ... 1 .~~1

I "1();-~rT';' II, , 11) I:i I ~;. x : I ,';;'

- -

<] .4. I

'!'hrollqh this two-dimc.'nsional mechanism we obt<lin the: addi­

tional deqree of flo"l'dom needed to establish a continuuLJ:;

sourct,-drdin l'lectric current. It is th(~ purpose of this

section to rC'v!t" .... t:"),,v two-dimension<ll aspL'cts of MOSFf':T

ot>eLltion thclt ,lrl' H1f'f)rLlnt to the research task ilt h,llvl.

1 • .t • 1 E 1 c c t ric Cur 1°,' 11 l S d t U r <l t i () n

,\ccordinq to troddilion,ll 1v10SFE'1' theory, clect:.ril: currc'llt

:':;,ltur,1tion le; .tttl'ibut.lbl,' tel <l ch<lnnel pinch-off lIlCChdfll:;rn [1].

Implied by this till'or,:: is that current sdtur<ltion is d;:;so,·idt(·d

' ... :ith two different ell'ctrostatic potential distributions; 1.';.

!) one dist.ribution between the source' and a pinch-off pu I nl,

.,!;c! :::) another betw('en thiF Dinch-off point and the dr.] ill.

':':'is concept of MOSFET operation has undergonf' siqnificant.

f:":()c!ification by V<lrious <luthors [17,18,43,45]. However, litth,

:)!lyslc.d insight is qained from these theories.

Another viewpoint attributes electric current saturdtiun

to constriction of the source-drain channel in the vicinity 0f

':1 pinch-off point. Pao dnd Sah viewed this constriction ClS d

"bottleneck" in which diffusion is an important mechanism of

carrier transport [39J. This viewpoint was corroborated by

t· .... o-d imens iona 1 numl~ rica 1 solutions of MOSFET opera tion, eel l.cu­

Idt.ed by Barron [26]" and by Vandorpe and Xuong [29J. In \.:on­

[rast, another two-dimensional solution of this MOSFET pr~blem

show;- :,0 ev idence to support this tradi tional concept of cl

plnch-')ff :nechanlSlt. [24,25J.

from ~ two-dimensional solution, Fig. 1.12 shows calcu­

lat.·,d i nVC'rsion carrier distributions in a MOSFET biased in

the saturation mode. This figure shows a notable modific:ltion

of ch.:ll1nel configuration in the vicinity of the drain. /low­

~ver, in contrast to the traditional pinch-off concept, signi­

ficant inversion carriers reside in this drain region. further,

an associated calculation of the mobile carrier flux di~~tribll­

tion (Fiy. 1.13) shows that saturation mode operation pr()duc~'~~

an irlcrease of channel width. These illustrations place in

question the concept of channel pinch-off.

66

1"\ L. U ........ (1)

0::: r.:= n:: DC .-:: r)

t..D l.O .....-t .....-t

0 0 .-; .....-t

x x 1"\ .-;

o

LJ\ LJ\ .....-t .....-t

0 G .--; .-;

x X

r'<'l ~-1

C'J

o

.::::::r

.....-t 0 .....-t

X

~r-,

NIV~O

.::::::r

o

3J~nOS

LJ\

C)

(UlI\) 3JNV1SIO

67

to

(:J

1. 4. I

ex:;

C

.::::r

c:::'

N

0

--l .......... x

:; c

E - '.') .- -0: ~.

~ ~

:> E-~ ~

L:.. '1) .::. r 3f

'J,

:.: '-'

:::: ~' :.-::: ,J '-

\..,

> . ... -: ~. .. c. ~

r: '. ~

C ." ~

~

"

'J .. .. :> , .

+.

" ~

lIL\NNL L

GATE :- t Rr1 I riA T ION

o ------------------

0.1

0.2

......... 0.3 '" E eX> :l

'-'

LLJ

0,4 ~ LLJ

~ Z U U z: ~ <r. c:::r:: => c:::r.:-f- a a (/) (/)

a

0.5

0.6

.;;..

0.7 () 0.2 0.4 O.G O ~) . ;) ~ • lJ

x/L l](~i.;RE 1. 13 CalclJli'lted mobilr, carrier flux dlstrlblj' :0:, 1i; ,j "11 Tf'';' ,:.'~"":

d ,~unstai1t carrie:- mntJi l' "':' 1\' _ -= < ,; V~ ,', I', j.

1. <1 .2

, ., • 't •.

(V,,-V) chanqt's pOLlr-it·,. ,It. ~.omc llnl!]! (J S

It'Vi'r~,dl 1;1 the direction of the gatc-induc"d I.'j.:ctric fIeld

ii' ':Jl. 1.14) r3f:>tween lhlS point ~1nd the ~)ourCt~, the ga'::.(·-

1:1(ju('c'd fit lel produces rtn accumu1,'ltion of inversion carric:rs

(lor;., the' scmj,"·I!.lt.ctor sur£'ace--thereby L,r-mlnq d conducti'.'(

,.'{)tlVcrSt!Jy, between this point anJ tile driJin, the:

'l,tt:"-lndllced elect'rlC field forces mobile carriers away from

(This component of electric field terrnindtes

, ) 11 1 (j Ii sin the d r a i n i slit n d • ) Between this channel termination

~)()il1t .'1n(~ the drclln, a negligible density of inversion carriers

:·.··sid(·~; .It the semiconductor surface (Fiq. 1.12).

Fiqun~s LIS dnd 1.16 show rigorous twc,-dimenslonal '::.llcu-

1.1t j()[;S of ('jl'l.·trnst.atic potential within d 1'-1OSFET substrdte,

F lq. 1.1 r) "hows t:l)(~ potential distribut 10n corresponding to d

t r i ,).1,· mod.: u j il S <-.'ond i t i on. Co n vcr scI y, i n r.' i 'J, 1. 1 G t his

:-.10SFr:T lS bldS('d w(·ll into saturation mode op'.::rdtion. Far from

th·· 'LIt.' (:h:ctrodc:, reqions of the drain depletIon layer exhibit

,I; '('t',ll,I! distribution that is well approxim.lted byille-

dim"'ISlon,ll .1iJ;"upL p-n junction tht:ory. In (:ontLlst, near the

(:-.~,:,,-~,.,rt,i(·')nd\Jclnr interface, depletion charq('s ·Ir,.' shi1red

L. : y" .'11 t ll·.' 'Jato ('lL'ctrode and the drain iU!ls~.lon.

r." 1(\~1~; ttl(> c'quipotential contours are par<111el to neithl:r the

;:\1 ... · l. J. ,'trod,: nul' the dr,lin junction.

:;I,i)~)t..lntjal dif:cr(!nces can be observed between ttlt·

TiH..'se d 1 f-

fl' id: III trJod .. operdlJOn, this field

Inc 0 n t r as t, f II r ~; d t U rat ion mod e

69

SOURCE

GATE ELECTRODE

OXIDf.

E-FIELD

P

SUBSTRATE

(VG - Vs) REVI:RQ',

POLAR ITY HF Rf

DRAIN

FIGLJH[ 1. 14 Vualitative Illustration of Gate-Induced

-~ .. -- - - --~.......... -.....- -----...

E 1 e c t ric - Fie 1 d 0 r i en tat ion 1 nth e S ,} t u r' .I t I url­Mode.

70

, I . ~)

(, 1 II . )

:J

() . [) E

...w ,....)

c:r:

I . ()

GA I [

- i) 5 ...

o 1.0 2.0 ( • IT! )

3.0

< 0:::

5.0

('ret,k! I.lr, l:,dculated potentic11 distrlbut 1(;:1

o lor]

l.n a MOSf'ET(NA = 2 x 10L Cltomsicm~; V D = 1. 0 vo 1 t ; V" = 3. 0 vo 1 t 5 )

2.0 1.5 1.0

).0 iI.O S.O

2.0 3.0 4.0 5.0 ( . rn )

!'!I,;t:HI: l.lb ",li"uLltl'd potenti.1l cll:;tributJOll III

d rl(JSPET (NA = 2 x 10 1 ' d toms/em' ; VI) 6.0 volts; Vt ; 3.0 volts).

71

, , , • 0-

• <i.

put t'nt i .11 equa 1 s qa te vol tagt~.

ti,tl IlnL' p,lrallel to th(' dr,injunction <It this l)()lr~t of ri':';

revl'rs.:ll. C10'.1rly, the potential difference bt't· .... C·l·l' r;~I~ dr,lll

.lnd this l'quipotcnti,:ll line (VO-VC) is less th.J.1l :11(' "f,pl !(",i

dL1ln ':nlt<lqt>; thercf )re, this point of channel tvrlllillclt 1(11)

r.1ust alw.\ys reside within this drain dt'pletion Idy,'r.

t.'mphasi:~,-,d that this point of channel termination Cclrrlc'S n(l

llllplic.ltlon of .1 pinch-off or channel constricti()n 1llt.'clldnIS!lI.

Within both the inversion channel and the adj.1cent dr.llr:

depletion layer, a substantial source-drain electric f il It!

forces mobile carriers towards the drain junction.

1.4.3 Veloci~aturation in a MOSFET

Rigorous two-dimensional calculations of MOSFET opvr.ltion

show t~at two fundamentally different mechanisms are cap~blc

of producing electric current saturation. In a long-channel

MOSFET electric current saturation is a consequence of th,'

potential distribution in the drain depletion layer betwccn

channel termination and the drain junction. In contrast,

electric current saturation in a short channel MOSFET can

result from velocity saturation within the source-drain channel.

Computer calculations show that velocity saturation hiS

a negligible influence on the volt-ampere characteristics of

a long-channel MOSFET; this situation is seen through a com­

parison of Figs. 1.17 and 1.18. In a long-channel MOSFET the

electric fields are sufficient to produce velocity saturation

on ly wi th in the dra in space-charge layer -- a reg ion where t hl'

mobile carrier trajectory is two-dimensional, Fig. 1.13.

This two-dimensional trajectory is a consequence of a

large electric field component E forcing carriers toward the x

drain island, and a small electric field component E forcing y

carriers away from the oxide-semiconductor interface. The

relative magnitude of these electric field components imply

velocity saturation in a direction parallel to the oxide-scnli­

~onductor interface (x-axis), and little (or no) velocit\'

72

7

6

5

- 4 Ii u

....... l1li :1 ......-

~ 3

-2

1

0

3.5

3.0

2.5

2.0

0 1 2 3 4 5 VD(VOLTI) .

rlauJS 1.17 Calculated volt·..,.re char.oteri.tio. of • MO.'." ••• uain, a channel left,\h (L) of 10 ~. (con.tant carrier mobi1!t;).

73

11.4.1

6

E v -: ..

:::: ..........

0

t 1. 4. 3

9 x 103 r---......,--....... --...... --...... --.....----.

3

7

6

r-

)

4

o.

J

2

1

o

(VOLTS)

3.0

2.5

2.0

o 1 2 3 4 Vn (VOLTS)

FIGURE 1. 18 Calculated volt-ampere characteristics of .:l MOSFET, ussuming a channel length (L) of 10.0 pm (fh;ld-dependent cLlrri('r mobility) .

74

-l . 4

, 'J .\ 'w' l t ~ I \ J t • 1 t

I;, ::

;! !, ~j :; i,1 t l "( \ - ~'! l, t ! '.; t. I,

';"'u:, \l.~l()"lti :~aturdtl():l prouuccs littlL' ;or nUl ,.'ndfi·". 1:1 ':,.

\".], -,lmpC'I't' l'Ildrd'~t(~rlst ics of a lonlj-chdnnL'J r-1()SF1:'!',

\"t:ntr .. i:~tirlq WJth this l()nq-ch.~nnc\) ~;itULlti()n, ·.'·f) .. :t,;,

:.; "t "r d L 1 , ) n , ) f m L) b i 1 t! C, 1 r r i e r s has apr 0 " 011 n din flu ': r, '. ' :: ',J:1 t: .' ' '.'l!jl-lnpt·r,· .'hill·d(~tel·l!,ti('s of <t vee:, sho!'t-chdlU11':~ \~(,:~l':-;':'

1;'{)f1lP.Jlt' Fiqs. 1.20 ,1nu 1.21). Here WI.! find thdt VC1,;clt/

,;.ltlJ"n 10n in il short-,'hannel MOSFET produc('s a !OV':l:r out:Jut

"Olldu,'LiHV'P i.n rt'C: tor.s of electric current SiltULltl0:1. I tl

~dd:tlC>:I, vl~loclt':' satllratior: produces electric curnc·nt. S.ltU­

r.lt ie)l' .. tt d urail' '.'oJ Llqe substdntially 1m·ler tha:l would be

;)r.dlvtl'd from thf:! tr<1l!itlonal theory cf MOSFET operation,

A study of these short-channel calculations has rendil:'

"St.tUllShcd the source of these changes. N~mely, 1" :1 short-

'."ll.I:H)t'l. ~-lOSFET VL'lOClty saturat1(ln of mobill~ c~rriers ~s foun;:

v.'lthi:1 t!lC sourcL>-dr;1111 :'nV(~rsion layer, a region hc:lvinq a

;J ro found ill f 1 u(~nce UpOl1 the to td 1 dra incur rt~n t . ,\s a CVISc'­

qtl(-ill"', an increasc'd l,lcctric field within til.:.' inv(~rsion layi~r

i::u .. to an increas('d drdin volta'1el h~s ,~ illini::lum influence­

UpUll lIll: drall1 Curr(~nt of a very short-channel stru\.'ture.

,4.4

l<eqiolls of carr ier depl(;tion in a MOSFET play two lmport.!1~t

r-r,j.,s: 1) they insulate the source-drain channel from ci hicl:lly

;'unductlV(' substrate; dnti 2) they siqnific,lntly influcnce c,ho

t~l('t.t()st<ltic potential distrlbution. E 1 cmen t ary ~10SFET then ry

tr"lt~, illl depletion layc·r calculations on a one-dlmensional

udsi:;. At best, this one-dimensional treatment may provide ;1

[C'dsonable approximation for long-channel structures--howevcr,

whc;n clpplied to short-channel srructurcs, it results in serLOUS

err(. r-;.;.

75

o N

o

Nl~dU

..::::r o

(Wrl) 3JN\llSIG

76

L1I

o

3J~nOS

U)

o

1. 4 .4

~,

'--

--;

oc 0

~ lD

(-. ,J

--.J

"-x

..::::r 0

" .-.

. ~ .-c,

'F. " --::

-r--~,

:;-, •. ' (; :::

-' ~,

..

:-. f":

,. :-. >-. _. -- .. ~) .. l~ ~. v

..: c. ,->-. '.

:; ~ ~,

;... ,--." - '-~

3.6

3.2

2.8

0.8

0.4

o o 1

FIGURE 1.2 0

§1.4.4

2 4 5 6

Calculated volt-ampere characteristics of a short-channel (L = 1.0 um)MOSFET (constant carrier mobility).

77

2.0 x 105

1.8

1.6

1.4

1.2

t: 1.0 :.; , "-

ro

3: 0.8 .......... 0

O.G

0.4

0.2

o o 1

FIGURE 1.21

1 . 4. 4

3.5

2.5

2.0

1.5

2 G

Calculated volt-ampere characteristics of a short-channel (L = 1. 0 ~m) !o1()~;Ft:T (Field dependent carrier mobility).

18

.. '.I. I MOSFET deplt·tlC)11

31.4.4

li"lY'I:r t'Xhiblt

;'(':~ Ii "[('ct r"st.lt ic interactions; for ex,)mpl(~, cl n..'Cjlon

. I' .,!,jch \If:;\"ut J'alized impurity ions must be sh,'ln'd

: " ~ \-. 1 • t ' f t; ~l t .' ,t n I.! J r a in. Arising from these interactJons

Another interpretation ()f thl-; ~:iitu,:j-

, 1 : > t :11t thl'sf' lnteractions result in ,) reduction ut

lons available to each depletion lcly,.'r--

., t ~. J .' sit u ,H j O:l i s e q l!..L ~~: len! to are d u c t ion 0 f 3 U b s t rat e

:'u j! lustrate th.ls lon-sharinq mechanism, Fig. 1.22 shows

.-d:mpn:;llmal calculations of a MOSFET depletion layer. In

',;.' ,'"l,~ulations, the depletion layer edge was arbitrarily

1:1 .. ·d tIl be the locus at which majority carrier denslty equals I,

t ;;'.' ubstrate impurlty concentration. These fiqut'es show

")!.'"dl" ()f constant (1000 V/cm) sour<.~e-drain elc'C::l'ic fl'~)ld

.j ','IlSUlllt appl:ed (late voltage. As the drclin voltage is

i"~-!'(}dS,'d tfL'om a volts to 3 volts), Fig. 1.22 qu,:llita.tively

: .. c. c r It c'l" ,ll'l expans i on of the elra i n dep let ion 1 aye r towa rds

From I:'ig. 1.22, this expansion is most pronounced

: :1" vl,-inity of the oxide-semiconductor interface, where a

'-lL~t,.lntldl part of tbl! substratt'? elcctrost.:::tic charqc produces

!·'ct~J;C field perpendicular to the oxide-semiconductor

Thu~, the ~xpansion shown in Fig. 1.22 is equiva-

" t ~~ ,I reduced 01ectrostatic charge and, i1cncc, iJ reduced

J:::"': r'lf unnt'utr'llized impurity ions available to the c.;ate

For t:li.S reason, the g.1.te depletion layer ~ehaves

, r,'qlDrl r.lf vdriable impurity ion density, due to draln

,:, L:,t 1(d1a1 calculations confirm that this "[(~dch-thrullgh"

"", ;:'1 LS dt'pendent on qatt> voltage (Fiqs. 1.23 .lnd 1.24)

,I! ,>:cc: drain voltaqe, Fiqs. 1.23 and 1.24 shell" th;tt c1t1

r",'cl,'c] Clate volt;l(je :)roduces an increased de9rvt: of orulll

Also shown in these illustrdtions

79

(a)

------

f'ICURE 1. 22

OXIDE

- .... _-

(b)

-- ...... -............

(d)

Calculated Contour of Constant (1000 V/cm) Source-drain Electric Field (broken line) in a ').0 I,m MUSFET

for V(; = 2.0 volts and

(a) Vo == 0

(b) Vo == 0.5 volts

(c) Vo 1.0 volts

(d) Vo = 3.0 volts

(Cross-hatched areas represent charge-neutral substrate.)

80

I . 'I. ,1

J.)

I ..

1~ v

w u a::::

(/")

x

-., "-

" \

~AT~ OXIDE "\.~., S\S\ S :;:~\SS\ \SSS\\ \\$S s~

- j

( I I \ (t\ ;' I

(R) \

\ \ \ ,...,...,7\ X X X :x 7'7*', , <>"'"'-

o

I \..- -'" 4...-'L J<. '-.0. J<.. A ~ .. ~ __________ ~

FIGURE 1.23 CaJeu ' atcd Cont.ours of ConS::d(.:: r,(;\;rce-LJr,lir: E:,,(·tr: F 1 e ] d 1- j I a 5. \) .' n r·:OSfFT for 'II, C, \:;, .1.1, \. 1

V 'J.O 'Jolls; l]

. ~. ) E X

o ( 53 0 '.1 '~C ( , d e p 1 '-' t 1 Q n - ] Zl Y C 1 (' d q '.') .

Hi E X

100u \'/l:m.

.c.

.c.

CXl l'V

,------1 ...... X GATE

y

." - ... -- ..... .---,.... ""'-----.. ..... _....... ......, ..... .......

....... ,.. ...... .......

" . ............ (B' \ .. ....... J . ....... \ . ..... VD = 5.0

w U 0::: => o U)

(A)

,/VD = 0 ....................... r , I , I /Jf(~ V 0 ___ "

OXIDE

FIGURE 1.24 Calculated Contours of Constont Source-Drain Ele~trlc Field in a 5.0 ~,m MOSFET for V,. 2 'Jolt:.s.

u

l\ ) E x

o (SOl:XCC: depletlon-l,-:,:,er edC'Jc)

8) E 1000 om. x

< 0::: P

...:.."

~ 1. ')

This ~dge is defined as

tl\~ locus of zero sourc(~-drain electri.c field.

rOt' .:1 'liven I-J.:1te voltage (VG

::: 2 volts) I Fig. 1. 24 st,ows

! l. I ~, ~ () U r c e d ( ~ P 1 e t i () n 1 i1 Y ere d g e f () r t w () d iff ere n t val u e ~; 0 f

dtall, \'{~lL.l<Je (VD

::c 0 volts and Vo 0= ') 'Jolts, respectLvely).

J ! \ t! '<l~; rig u H:, i tis a p par e n t t hat rea c h - t h r () u q h () f the d r a i n

,'.,'pl!'tlon layer has produced a contractior. of the SO'Jrce

d(:plet iOll layer. At this time, the influence of reach-through

();~ sourC(:-drcl i n electrica 1 current is not clear 1 y unde rs tood.

1.~) /\ M0dified . .::r.Q~2..~L for MOSFET Operation

1n the inversion layer of a MOSFET, both drift and diffu­

:;iOl1 contribute to the source-drain electric current:

(1.106)

I!, thi s expression, Ie represents the total electric current

p~r~ll('l to the oxide-semiconductor interface.

If we neglect recombination-generation mechanisms within

this semiconductor device, the source-drain electric current

,(Eq. IJ. LOC)l must be constant at any location between the I J

:;('U'(.'e dnd drain. Unlike other previously outlined studies of

this structure, in the present analysis we require electric

eu!: f."'.'n t con [.1 nui ty; thus

o qWIJ O. 1 (1.107)

:\ :.;ubstLlntial degree of mathemati.cal simplification is r ali.zed

h'/ the chain rule of differentiation,

" fld

d'Q , 1 dO. 1

dV ~! x s

dO, 1

dV s

d/V S ----

dx:'

81

dV s

dx

d?Q, 1

+ dV

s

(1.108)

dV S

( l . 109) x

and a direct sUbstitution of Eq. (1.108) and (1.109) lilt"

Eq. (1.107) yields

d.'V S

dV 2 S

~ r dQ. I i __ 1. !Q dV i ., J L

kT g

dQ. -1

1 dV-

S _J

(1.110)

Equation (1.110) introduces a fundamental reli1ti.on that

must be satisfied between V (x) and Q. (V ) in order to outLlill s 1 s

electric current continuity in a MOSFET. The traditional

one-dimensional theory of this semiconductor structure imlJ1ics

that Q. (V ) is uniquely esta~lished by a one-dimensional form 1 s

of Gauss' law (Eq. 1.6) and, therefore, cannot satisfy Eg. (1.110)

Two-dimensional computer calculations show that this impl~cation

from traditional theory is incorrect. The invers10n layer

charge Q. (V ) does, indeed, satisfy Gauss' law, althouCjL it 1 S

satisfies a two-dimensional form of this law: and this two-

dimensional form contains the additional degree of freedom

needed to also satisfy Eq. (1.110).

For this reason, it is unnecessary to undertake a

rigorous two-dimensional analysis of the electrostatic potential

distribution in a MOSFET. We need only recognize that Q. (V ) 1 s

LuSt satisfy Eq. (1.110) in order to attain electric current

continuity within the inversion layer, and that the additional

degree of freedom nt:::eded to satisfy Eq. (1.110) resulls from a

readjustment of two-dimensional electrostatic interactions

between the gate and drain. Gauss' law is always satisfied

within this semiconductor device, although the manner in vlhich

Gauss' law is satisfied is, in reality, of little concern to

the problem under consideration.

A physically meaningful modification of Eq. (l.110) is

realized by introducing into this expression the separation

84

',) d I ,1 ml' • l' r ,; V ) Y 1 e 1 din g s

\ (V ) s

Therefore,

dV s

dx

dV s

dx

1 d '0, r~ T 1

q dV ;, L S

\ -2

) -1

kT dO i

q dV s '

_ A (V ) : ___ s ::: 0 l,dV \)2 S i dx

\ ,

]

dO. .-3. 1 kT dV s

_(3)~(V)Q' kT s 1

§ 1. 5

( 1. 11 1.1)

(1.111b)

(1.1l2a)

o (l.1l2b)

FrOlfl ;"'PF":lL;... , ..... , eqs. (i.112) hove the so:'ution equations

Q. (V ) 1 s

where

x = x o 1

E o J

'I s

V o

r

exp [n(O]d~

qV /kT t -qv /kT s 0 - [~ 1 ::: O. e e

- Q io \ 10 kT

V

r s

'V 0

D ([, )

exp [ _'l5. kT

:::

E,

-J v o

85

"'I

l d' l + D ( E,) i " (

j .J

A (n) dn

(1.1l3a)

dQi )

l

I dV s J .V

0

(l.1l3b)

(1.114)

31.').1

These solution equations provide the foundation for new mathe­

matical expressions for thf' "olt-ampere characte::-istics of a

MOSFET.

Through algebraic manipulations, it can be readily proved

that if, indeed, a solution is found for A(V ), the magnitudes s of V (xl and Q. (V ) will always yield electric current 20ntinuity s ~ s between the source and drain. This proof is shown in Appendix B.

1.5.1 A Physical Interpretation of the Separation Pdrameter'C'l:sl

The physical significance of this separation parameter

A (V ) (§1.5) becomes clear from the derivations given 1n s Appendix B. Therein, it is shown that the source-drain electric

current is constant at any location within this semiconductor

device if Eq. (1.112) is satisfied. Further, from Appendix B

we have

\ (V ' sl d dV s (

d Q . /dx ) ] log e dV:/dX (1. 115)

From Eq. (1.115), it is evident the separation parameter

\ (V s ' produces a modification of the proportions of drif~

current and diffusion current necessary to yield constant

source-drain electric current at all locations within this

semiconductor device.

From our two-dimensional computer solution for this

problem it was found that A(V s ) always attains a magnitude cf

q/kT in th'c limit of weak inversion. From Eq. (1.115), t-his

limit implies that ~ll source-drain electric current is

attributable to diffusion. Furthermo.ce, from this computer

calculation it was found that A(V ) can be adequately approxi-s

mated by the first two terms of a Taylor series expansion of

Eg. (1.115) about a location near the source junction:

A (V s) = AO l-A(V -V ) s 0

86

(1.116)

§1.5.2

\· .. he re'\ ~ (Vo)' Because \ (V ) approaches thi s same 1 imi tat

o s the point of channel termination (where Vs == Ve ) we have

l\ ::

kT. 1- -- I A

, q J 0 (V,.. - V )

\J 0

(1.117)

Thcr-efore, substituting Eq. (1.117) into Eq. (1.116) I we obtain

,\ (V ) = s

lVG-V P­o 0

(V -v ) +[kTl(V -V ).,\ G s qi s 0 0

..

1.5.2 The Volt-Ampere Characteristics of a MOSFET

(1.118)

From the solution equations for Q. (V ) and V (x) [Eq. (1.113) J 1. s s

in conjunction with the qualitative form for .,\{Vs

) [Eq. (1.116)J

we can readily develop an expression for the vo1t-ampe;

characteristics of a MOSFET. From Eq. (1.113b) we obtain

for (,;. (V ) 1 S

Q. (V ) 1. 5

where

A o

Q. ) 1 - (1 + kT 10 I q

A ~, ) z

o 0

. exp

z

L' o

(Z ) o

\ +~

A '

:.L (V -V _1:.) kT 5 0 A

'" _9.... 1 kT A

87

Z ) -o

\ o f(l + A' (1.119)

(1.120a)

(1.12Gb)

:; 1.5.2

Although Eq. (1.119) has an inherently complicated forml

throughout the range of var':'d; .es encountered in this boundary

value problem ~his expression can be adequately approximated

by the relation

Qi IV51 ~ Qio [1-AIV 5 -Vo l] ,\

o A

In a similar manner, from Eg. (1.113a) we obtain

= V o

A 'I

1 If7.\ : + A)E xj 0

o J (

.J

(1.121)

(1.122)

where E is the inversion layer electric field at the source o

end of the structure:

E o

= _ dV s I dx

(1.123) x=\)

Because this system of equations has been designed to

yield a divergence free electric current, the magnitude of

this current,

kT q (-.!. dQ i) "J

O. dx ' ~

(1.124)

~an be calculated at any location within the source-drain

channel. Selecting for this calculation the source end of this

structure, Eq. (1.124) has the form

10 -W~O. [ dV5 kT (L d

Qi ) 1 = ~o dx q °i dx J

x=O

( kT 1 (:~:) I = w~o. E ) 1 (1.125)

~o 0 \ q °io ) I I I , V

0

88

I·'r(lrl f'i. (1.121) we 011taln thE' rc1dtion

d() . 1

J\'

\I .::V S 0

·1 , h, ¢

(1.126)

FlllL:lC'f, from Eq. (1.122) we have, upon recognizing V::;=VD

\ ... ' ! ~'. j 1 : .. :. I"

I: ()

o +A

A .. ( I

)

(1.127)

Th'l~:;, aft,}r substitut.ing Eq. (1.126) and Eg. (1.127) into

E>;. (J. 1. 2 S), we h.:lve an express ion for the vo 1 t-ampe re

,." ~ i Jr'wtcrJ sties of a MOSFET I

I i'i:. (1 .

r' .., I

k1' J ~ I 10 i 1 + ( \ '+';\Ti,- I -II

'U I g 0 ()

L.

I t should be noted that Eg.

1

,\ +A)

[ -~\ - l-A(VD-VO}~ r

)

(1.128) contains only

(1.128)

two unknown

p,.lrdmet(·rs; Q, and).,. 10 0

Both Of these parameters arise at

(h(' ~.;ourc(' t'IH.i of this structure where two-dimen::iona1 mechanisms

:1:" ;:Iln;r-Iurn; thus, these parameters can be eva1u;ited on a one-

Befon~ determininq these parameters, we first consider

,J!: .1I'p1i('ation of Eq. (1.128) to the weak inversion mode of

')tH.'cdllOfl. From Eg. (1.117), as "'0+ g/kT in weak inversion

t~l( !',jrdlfl .. :ter A becomes small and, therefore, Eg. (1.128) has

IIII;l ( 1" ) .)

'1/1-: 'I I'

::: 2WDQ,

10 I - c (1.129)

·: 1 . ' .. .:

Clearly, Eq. (1.129) shows an exponential saturation of till,'

source-drain electric current wi th an increase of dr.J i tl va 1 t,J<jl'.

ThUS, from the .:oncept of electric current continuity \·:ithl!1

the inversion layer of a MOSFET, we have an expc('ssion frn

current saturation in ,leak inversion that is ill sUDstdnt I,ll

agreement with both experimental observation und the ptl'\'ir)u~;

studies of Barron [11] and Swanson and Meindl [12].

Further insight is gained from Eq. (1.129) if we COnSJdVI

the drain junction as a minority carrier (electron) sink --

like the collector junction of a bipolar transistor.

Boltzmann statistics, the tnrm

1 - e

-q(V -v )/kT o 0

,\"sum i, nq

(1.130)

in Eq. (1.129) is identical to the minority carrier sir'f.

offered by a reverse-biased collector junction.

when Vo is large we have from Eq. (1.129)

In addition,

2WOQ. 1::= 10 ( 1 . 1 31 ) o L

Clearly, in this situation TO is a consequence of minoritv

carrier diffusion from a source of magnitude Q. to an ideal 10

sink that is located a distance L from this source.

Thus, Eq. (1.L~8) contains all the qualitative require-. ments for the weak inversion mode of operation. First, this

expression shows that electric current saturation exhibits

an exponential form in weak inversion. Second, in weak invt'r-

sion Eg. (1.128) predicts that all source-diain electric

current arisLs from diffusion.

90

51.5.3

.'-:\,3 TI)(' ~aturLlti_C2r2 Mode of Operation

It: ,'.hould t;(. r,\cognized thLlt Eg. (1. '1.28), and its cxten-

is applicable only when

;, \.; .j known quantity. This situation exists when Vo < VG'

::1 ... \l,-,~tric current saturation (when Vo :;:. VG

), the source-drain

I ~: 'J • • t s i O!1 1.1 Y • ~ r In i:li n t a ins a tot a 1 vol tag e 0 f (VG-V ) across its J 0

,-nl I rl' ]ctlqth, . .11<,houCJh the length of this inversion layer

dn incre.:..se of VI . )

':rhus, in electric current

.dUc,lt.ion E,!. (.l.128j has the torn:

I) 1 + kT,\

q 0

.j

, 1 -

I

l-A(V,-V ) Go,

,\ +1\ o

",;h .. n' Lc (V D) is a 'JoltLlge-dependent channel length.

I,

> , , (1.i32j

Assuming the substrate region of this MOSFET has 6 homo­

q('·tl.~ous impur j t.:y a tom densi ty, we apply to this calcula tion

t he d,:pl(~tion layer theory of an abrupt asymmetrical p-n j unc­

ti0Il. From this theory, the drain junction space-charge layer

,·xtensinn into the substrate is given by

(1.133)

wll,-, r" N A represents the subs trate impur i ty i on dens i ty.

Further, from this depletion layer theory, we have () voltLlge

drop of VO-VG across a distance of l'lL, where

'I'll.,:;, if we subtract this distance (I'.L) from the total

:iuur<'e-drain chanr.cl length (L), vIe obtain

91

(1.134)

L - I W D

w' , [) l.

qN" I". ('; -v ) D (;

lenqth Clcross which we lu'/e u total vo 1 L i.Hll' f ,.

fJ. \" .. ( ,

Eq. (1.132) in con)unctio, with Eq. (l.l),)) ,·,;t.J!,ll:;tll'S t:;,

drain current when VD

. VG

.

1.5.4 Calculations of Q. in a MOSFET 10

To obtain source-drain electric current HI ,I "1()SfT'f', ;t

has been shown that C). (V ) must satisfy d rpldtl,l: ,li!!'I", 1 S •

from that given by a one-dimen' ional form of Cdt!>;,' lei ..... ·.

relation is given by Eq. (1.111b) and it dlff('r~; slqnifl ',':1 I'.

from both the

,lnd Moll [Eq.

and Soh [Eq.

elementary one-dimensional form C;lVI.'r', by 1:1.::!:"\'

(1.6)], and the more riqorous form 'lIVl'n lJ': ,'j"

(1.29)J. Before Eq. (1.113b) ancl, hVflCt' , i:j.

(1.128) can be used for a quantitative evalu.lti,)n of tllr' '.(.!t-

:3mpere characteristics of a MOEFET, it is r.eC1·SSilry t:o (',Il"ttl.lt"

a magnitude for Q. -- the vClllle of Q. at the ~;ource end (It 10 1

thi~ semiconductor structure (wl-}ere V = V ). s 0

From previous discussions, it '''''''IS shown that elc'ctro­

stutie interactions between the gat(~ and dri'lin <1[,C' at :1

'lllni'"'1um near the source junction. for this n~.:t~;on, it IS

:)rl.'sumed that a. one--dimensionll for.m of Cau~;s' law is d 1,[>11-

'.:.:able in U)is regiuI. ,{' the structure. Furthermore, tlr,:' llille'rellt

!,;im[)licity of the Ihantola-Moll form for Gauss' lal-' [1,\. (l.i,)

c,ffers many computational advantages, jet then' .11'(> nurnl'['utJ:-;

i nd ica t ion::> tha t th i s form is inappl iCilb Ie to t-.he W("j k 1 :1V,_ ['

'~~on mode of MOSFET operation. Specifically, for wedk },1\"'1--

,;ion (where Vs <: - 2~F) the Ihantola-Moll equ.1tion yield:,

totally incorrect values for Q .. 1

For purposes of illustration, Fig. 1.25 show!, il compal-I-

son bet\oJeen the calculated values of Q, 1 usinq the Ih,lnt()i.I-1'1n) 1

92

_.J

:i 1. ~. 4

fl(llSSON'S ErJIJAT!ON

A T RAD IT I Otil\L APP~OX Irl/\ T InN l Eq. (1. 6) ]

l~]-' 0 REVISED APPRnXJr1.~TIOr1 [Eq. (l.139)J

r; • J .G5

t·' I, ;/: I.

,7 .75

A

• A

A

A

4

A

A

() : )

2~F = .i75 VOLTS

.85 .9

(JI\(> DlrlC'nS10nal Calculations of lr.version "h.lcq,> ((lj) 111 i,n ,'105 Structure; NA =

i I) 'M

93

.95

, J. :1

equation [Eq. (1.f))] suitably modified to elimindllC: t.:h ;.It,

voltage VG ' "nd using the rigorous solution firsl ,;;""r' hy

Pao and Sah [Eq. (1.29)J. In the Ihantola-f'.loll equ,lLl':' w,'

replace the term reprE'sentinq the gate-induced eJ'-.:ctric fl' lu

by an exact one-dimenf: onal solution for P()is~;on's '-.~qu:;t ie,ll:

i(.C 1 0

t OX

(V.-V) =-12 (, s

-SV s e + Bv s

"V ,. s - 1

Therefore, the Ih3.ntc,la-Moll expression ha:". the ron:;

Q. (V ) 1 S

+ f<V ,~ S

. ;·v ! c: ie ---1

+ {~~t qNAV !3 0 :.

, ' .

From Fig. 1.25, the IhantoJa-M·")ll (trad.;tit)llcll) ~l[JjJL':-:

matio;\ i~; shown to yield values of Q: tha:: ,.;.r::: ,1C}'-·ql""'· olll,;,

for st:ro'1g InVe::SlOTl modes of f'.10SFET O'.)f.-Lltiull.

~hrcqhold voltage of n MC~~ET (V s approxilTll~ion yield; a C:. of zero, anJ for wenk jllvl~rsion

1

(VS

< 2~F) this 2pproxim~tion yields n

Q .• 1

( 11'

(1.137)

The source of this error has be~1 identified: 1 t \...\ r! I,':;

Lrom an application of the depletion :pproximnt [on In ('\,,-11',1,;"

ting the depletiOl: charge (Qo) in a ~OSFE'l' [Eq. (1.·1) J. Th i

deplE:tion approximation over-cstil1ates (sli(~\..,tlv) the t,'.t.l'

depletion ch<:rge, due to an elimination 0 moLile ,'ill'!'l, r" 11\

the li.athematical modFl.

94

§1.5.4

for stronq inversion conditions (when QT » Qo); however,

under weak inversion conditions QT

= Qo in [Eq. (1.5)] and,

therefore, a small error in the calculatrd value of Qo produces

~ large error in the calculated value of Q .. 1

It should be noted that this source of error also exists

within the Xennedy theory of MOSFET operation [24] and, there­

fore, this theory is clearly inapplicable for weak inversion

calculations. In their theoretical studies Swanson and Meindl

[12J also used Eq. (1.6) for calculating Q., although Lhey J.

introduced a compensating approximation. After writing

Eg. (1.6) in the form shown in Eq. (1.137) they eliminated

from this expression the terms arising from substrate majority

carrIers and, thereby, minimized this source of error.

A substantial increase in accuracy is obtained through a

modification of the Ihantola-Moll equation for Q.. This J.

improvement arises by replacing the traditional depletion

charge term (obtained from the depletion layer theory of an

abrupt asymmetrical p-n junction) by an approximation derived

from an exact solution of Poisson's equation [Eq. (1.138)].

As in the Ihantola-Moll theory [lJ, for the depletion

charge calcul.ation we assume the voltage V is attributable s only to ionized impurity atoms. Implicit in this assumption

is an inversion layer width that is small, relative to the

depletj,n layer width. Thus, from this assumption we can

neglect the contribution of inversion carriers in Eq. (1.138)

ond obtaIn an approximate expression for the depletion charge:

(

K £ I ) [ -I3V ] ~ Qo{Vs' = -12~! \kT e s + BV - 1 Lo I q s, J

(1.138)

Thus

Q. (VG

, V ) I S

K.£ 1 0

- ---(V -V ) t G s ox

95

I ~ I

BV - 1 i

S J (1.139)

1 _ 'j _ ·1

From Fig. 1.2:>, this revised approximation ('qu.1t Ir,:) fur

the inversion charge in a MOs!-~s:;l' yields results th;)t ;H(, in

satisfactory agreement with a rigorous solution of thl~ r)rnLl··~.

Clearly, this new approximation equation offl'rs adequ.!rc- aor(',·­

ment throughout the rar Ie from extremely weak i nvers ion tr)

strong inv~rsion.

To assess the adequacy of using this one-dimensional model

for calculating Q. in a MOSFET, direct comparisions have been ~o

made between Eq. (1.139) and a two-dimensional solution of thiS

problem. A numerical integration of the inversion l;)ycr carri(~r

distribution was used in our computer mod~l to establish this

inversion layer charge.

parison.

Following are the results of this co~-

._-

Qio (two-dimensional

°io [Eq

Condition calculation) ( ! . '. ) ]

(coul. /ern 2) (caul _ /cm' )

Strong Inversion 1.9 x 10- 9 1.9 x

Threshold Operation 3.83 x 10-1 0 4. 33

Weak Inversion 1. 44 x 10- 11

I 4.6 x

In view of interpolation errors associated with this numerical

integration technique, this degree of agreement is considcrcri

satisfactory.

It should be noted that by utilizing Eq. (1.139) in this

theory of MOSFET operation we encounter an additional cornpli

cation. Namely, this theory yields the magnitude of 0. , anc: 10

hence ID

, explicitly in terms

'J G' FrO~r.;E(~~) «(~~:3)6(~Tw)er ~::: V = V + ~2 -- -- -- <e

G SKi LD q l

of V and implicitly in terms of s

( . 2 [ SV n~ s

+ BV - I + -I e -s '. NAJ

I , " !

1 minor algebraic campii-Clearly, this problem represents on Y a

cation.

96

( 1. 1"; 1 )

1. 5.5

1.5.5 Calculation of A in a MOSFET

In the present MOSFET theory, source-drain electric current

is qiven by Eq. (1.128). This equation contains two unknown

parameters (Q. and AO) which must be evaluated in order to ~o

calculate this current. In Section 1.5.4 we gave an expression

for Qio Eq. (1.139) , which we derived from a rigorous one­

dimensional solution of Poisson's equ~tion. Herein we will

d('rive an expression for A , the remaining parameter needed o

for calculating the source-drain electric current.

To obtain an expression for A , we will utilize a general o

relation for A (Vs ) given by Eq. (l.lllb). In addition, since

A = A (V ) represents the magnitude of A(V ) at the source end o 0 s

of this seriliconductor structure, we will use Eq. (1.139) to

approximate Q. (V ) in this vicinity. A substantial degree of ~ s

mathematical simplification results in we neglect the term -8V s e in this equation; thereby

Q. (V ) ~ - I( i £ 0 (V _ V ) + 1{2 (K S e: 0) (k T) [s V _ I ] ~ 1 S t G s L!) q s ox .

(1.142)

We can neglect this term when

V s » (kT/qL [1-eXPC-qVs/kT)], (1.143)

which is clearly the situation in all cases of practical

interest.

From Eq.

we have

dQ. 1

dV = s

K.( 1 0

t ox

(1.142), by differentiating with respect to V , s

1 + --

-v2 (KSEO)(kT)~[v _ \ LD \ q s

(1.144)

97

" 1 " ') " ')

:: (1.145)

substituting Eqs. (1.144) and (1.145) into Eq. (1.1l1b) and,

thereafter, evaluating the result for V = V , we obtain s 0

, ~ , ' 'I

A o

= Cox £T) Vo- q- + 12 Cd l+~-ql ,Va - -~ I ( q.~( kT) -.!. l (kT'f kT\-1jl ~

lc [V -V + kT]l~)~{v - kT)~_ {2 C [V -~t'~T)l '

( 1. 146)

where

ox G a q kT s q d s 2 q !I

C OX

K.t: 1 0

- -t-OX

J)

(1.147)

(1.148)

For V o 3

» i(kT/q), Eq. (1.146) reduces to the following a~proxi-

mation:

f­o

::: r

C ox

C IV -V + ox; G 0

'-

(1.149)

kTl-fV - l{kT)] [2K E. qN / (V -." 0 2 q so A 0 'J l

Calculations using Eq. (1.149) showed that this expression

adequately appr~ximates A for any degree of inversion. In o

particular, such calculations showed this e>:pression predicts

A ~ q/kT for conditions of weak inversion, in agreement with o

rigorous two-dimensional calculations of MOSFET operation.

§ 1. 6

1.£1 Conclusions

A detailed review of the technical litErature has shown

c111 lmp0rtant shortcoming in our theory of MOSFET operation.

Spocifically, workers in this field have assumed continulty

of inversion layer electric current within their mAthematical

models, without making electric current continuity a funda­

m0ntal requirement. As a consequence, most available theories

of MOSFET operation predict a totally unphysical situation:

they predict different magnitudes of electric current at the

source junction and at the drain junction.

A detailed two-dimensional computer solution for this

problem shows that electric current continuity does, indeed,

exist in the inversion layer of a MOSFET -- as it must, from

a physical point of view. Further, this computer solution

shows that both drift and diffusion of inversion layer carriers

are important components of the associated transport process.

The relative importance of drift and diffusion differs between

the limits of strong and weak inversion modes of operation.

In the strong inversion mode of operation, near the source

junction, most (but not all) inversion layer electric current

arises from the mechanism of drift. The ratio of drift

current and diffusion current undergoes a continuous change,

with an increase of distance from the source junction, although

their sum remains constant. Thereby, electric current conti­

nui ty is iTldirltained in this semiconductor device.

In the weak inversion mode of operation, mobile carrier

transport from the source to the drain is almost entirely a

consequence of diffusion. In this fashion, MOSFET operation

exhibits many similarities to the mechanisms encountered in a

bilJolar transistor. The source junction injects carriers

Into the inversion layer, similar to the role of an emitter

Junction. These carriers diffuse along the oxide-semiconductor

interface and, eventually, reach the reverse-biased drain

junctioll. Hence, the drain is like the reverse-biased collectc­

of a bipolar transistor.

99

· J • r,

Implied by this situation 1S a necessity to includ, :)()ttl

drift and diffusion in any r.':),-~()U3 theory for MOSF'ET 0f)VrdtlUll.

This necessi ty ~;hows the inherent two-dimens iona 1 n.:l tup n:- ~ !:i'

mechanisms encountered in this device. On a one-dimensl()nc.ll

basis, most authors uti 'ize Gauss' law to establish the total

invers ion layer charge through r ' _ '- the entire source-d r din in VI i -­

s ion layer. I t is shown that this inversion charge dis t r i bu­

tion cannot satisfy a req~irement of electric current contirlulty

and, in addition, a one-dimensional form of Gauss' law -- tll"~'

are insufficient degrees of freedom. On a one-dimensi~n~l

basis this inversion charge rlistribution is determi ned ci thl'r

by Gauss' law or by a require.nent for electric current con t \­

nuity; if one requirement is satisfied, the othe-,- is not.

A study of two-dimensional computer solutions l;.lS ·~:sl.J

blished that everywhere \Jithin a MOSFET the inverslon cLdr;t'

distribution satisfies bo'::' the requirement of electric' curn·~~t

continuity and Gauss' law. 'A fundamental difference found J"

this computer solution is that the inversion layer chdrge

satisfies Gauss' law on a two-dimensional basis; thcrl_.'uy trw

structure can exhibit an additional degree of freedom.

It is thus recognized that two-dimensioned elcctro:ot.i~

interactions between the gate and drain alw.:lYs '-dke f-.'lac'c', lfl

order to maintain electric curr~nt continuity within clfl lll'J,'r-­

sion layer. 'As:> cons~·quence, one can mathem.:ltically anal',//'l'

this structure on a one-dimensional basis by imposing .:l rvqul L('­

ment of ~lectric current continuity.

It is shown that this approach to the MOSPET probl(~m yields

results that are physically reasonable and, in .:lddition, .:lrc in

agreemen t wi th experiment. First, thi s approach to the prob} c'm

shows that true channel pinch-cff does not take place in a MOSPET.

Second, this approach yields an exponential type of electric'

current saturation, with an increase of drain voltage. l\ddi-

t ional agreement is obtained wi th the two-dimensional comj.Ju\.e t­

solution: in weak inversion, all source-drain electric current

is attributable to the mechanism of diffusion within the illvor­

sion layer.

100

Chapter II

Inversion Layer Studies f0r MOSFET Operation

C. T. Hsing and D. P. Kennedy

'J 2 1

A quantum mechanical analysis is presented for the mobile

carrier distribution in the inversion layer of a MOSFET. This

analysis is b~sed upon a one-dimensional solution of Schroe­

dinqer's equation, in conjunction with an assumed constant

(>lectros ta t ic potential gradien t. Included is the mechanism 0 f

tunneling from the semiconductor into the oxide. It is shown

that a non-zero density of mobile carriers resides at the oxide­

semiconductor interface. It is also shown that the point of o 0

maximum mobile carrier density lies about 15 A to 25 A from this

interface; this distance is dependent upon the magnitude of semi­

conductor doping and the gate induced electric field.

2.1 Introduction

Conventional thecry of MOSFET operation [l,2J is based upon

~n assumed constant carrier mobility within the source-drain

inversion layer. This approximation describes MOSFE~ operation

only through a limited range of gate voltage. It has been experi­

mentally established that the inversion layer carrier mobility

exhi~its a large change, with a change of applied gate voltage,

throughout the normal range of device operation. A theory for

this phenomenon was first fO!"mulated by J. R. Schrieffer [3J; he

proposed that diffuse (random) scattering of the oxide and semi­

conductor interface was the basic source of this difficulty.

Following his work, other researchers proposed theories to

explain this change of mobility with gate voltage [4,5J. How­

ever, none were found to b~ in satisfactory agreement with

experiment [6].

The reduction of inversion layer carrier mobility is know~

to result from scattering at the oxide-semiconductor interface,

yet the details of this scattering mechanism are not adequately

underst~0d. It is evident that this scattering process is inti-

101

---- ---'--~.~---.-

.1

mately related to the average djstance betwee:1 these invvr:; I 'JI

layer carriers and the silicon surface: a deCretlse of f r is d I s­

tance should produce a decrease of .... arr ier mobi 1 i ty. For' hi,;

reason, a first step tow<""rd attaining a theory for the mobl 1 i Ly

of these inversion layer 'arriers is to accurately establj ;;11 t~ci r

distribution, relative to the oxide-semiconductor interface. It

is toward this goal the present research has been directed.

In a MOS structure the inversion layer carriers are bounded

within a potential well; on one side the~e is the oxide and s0mi­

conductor interface, and on the other side a large substrate

electric field. For this reaF'n, it was postulated that these

~nversion layer carriers would exhibit a quantized energy dis­

tribution, as in most problems of this type [7]. Arter many

years of research, proof was obtained for this qUantization ill

the form of Shubneknov-de Hass oscillations [8J. Thereby, it was

established that mechanisms other than traditional electrostatics

determine the carrier distribution in the inversion layer of a

MOSFET.

In traditional theory of MOSFET operation it was initially

assumed that the inversion layer carrier distribution could be

established from solutions of Poisson's equation. This proof

of ~ quantization in energy clearly established that a true eval­

uation of this carrier distribution required a simUltaneous soll'­

tion of both Poisson'_ equation and Schroedinger's equation [9 / 1C J. To date, all available solu~ions for this problem have utilized

a simplifying approximation that is unwarranted from a physical

point of view, and which is inadequate for surface scattering

calculations. Namely, it is assumed the oxide-semiconductor

potential barrier is exceedingly large and, thereby, all eigen­

functions are zero at thi~ location. Because tunneling has,

indeed, been observed into the gate oxide [llJ, a zero eigen­

function bou~dary is inconsistent with experiment.

In the present investigation we aim toward a rigorous numeri­

cal solution of this Schroedinger-Poisson problem. Clearly, all

numerical solutions of this type require a first "guess"; the

quality of this supposition will significantly influence the

102

L. L

,:0l'1\t,uter tirle' rf'q'cl~rcd to iittain d (:omplcte solution of thL'

i'roblem.

Til.: pre-sent '",ork is direc~cd r.oward obt;linlnr: dr. 0xact

dr1dl)'tlcal solution for Schroedinger's equatio:l, !,dSt·d Uf)0; "l

fil',;t post-ulale for the ir~version loyer potenti,al dist.riL)-l~ j0i,:

v..'t: ~lS;;\lmL~ it has d ::or,stant gr3uient. prom this soluti~n, we

can thl'reafter introduce into Poisson'::; equation tLi,: caiculated

in'1(>r~;ion lc1ycr carrier distribution, and obtain Lin lr.1[>roved

(:"timd!." of the inversion layer potential distribution.

f,:lshin:1, a Picard iteration between Poisson's ecu3l1c)!1 and

Schroedinger's equation can be used to obtain an (,('curate numcr;­

cal evaluation of the inversion layer carrier distribution.

The following discussions outline the solution of Schroe­

din<Jer's equation for a constant substrate potential qradier.t.

This solution is compared with the classical solution for an

idnntical MOS problem. In addition, comparisons are made between

tv/o quantum mechanical solutions, assuming differ~nt values of

substrate potential gradient.

2.2 SolutIon of the Schroedinger Equation for a MOS Structure

Fig. 2.1 illustrates the mathematical model used in this

analysis. We assu.!ne an n-channel MOS structure in one spatial

rlimension that is bounded (at x ::= O) by a Si0 2 insulator. For

convenience, this model is divided into two regions:

Region I -- the semiconductor material (x < O)

Region II -- the Si02 insulator (x > 0).

Furthermore, a potential barrier is assumed at tr.e interf')cE:

between these two reqioTls; the magnitude of this barrier (E B) is

tdken from published measurements [12].

In this mathematical model v.:: also assume Regions I and II

contain electric fields £: and c of constant nlagnitude. The s 0

magnitudes of [ and [ are r0~dc2d by a requirement of electric s 0

flux continuity at the semico~ctuctor and oxide interface (x = 0).

Thus K r. ::= K L , where K and K represent the relative dielec-S s 0 0 s 0

tric constants of the semiconductor and oxide, respectively.

103

Region I (Si)

-ec x _____ s

Figure 2.1

r: B

E. ~

Reqion II (SiO~)

E - c x B 0

Simplified energy diagram at the interface of Si -Si0 2 •

104

x

In Fig. 2.1 all potential energies are measured with res~ect

to the conduction band edge of the semiconductor. For this rea­

son, the Fermi level (EF in Fig. 2.l) is a negativ.c: quantit,/,

wherens the energy eigenvalues (E.) are positive qu~ntltlcs. 1

Thus, separate forms of Schroedinger IS equa t ion mus t b,-,

C;pt~C if ied for the two reg ions of this semiconductor st ruc:tlJ rc.

'T'!wrP-ilfter, the solutions of these two equation:::; can be matched

,'. ~ilei~i)ounoary (x 8):

Re'Jion I (x 0)

ReSlon II (x 0)

d2

". "'Ii --~

dx + ~ (E + e E x) nL 1 s

o

2m (E. - EB + e Ex) ~ . == 0 . ~ 1 0 III

( 2 . 1 )

(2.2)

In Eq. (2.1) and Eq. (2.2), m represents the effective mass of

electrons in a direction perpendicular to the semiconductor­

oxiJe interface.

Substantial simplification of these equations is obtained

through a change of variables. For this reason, we assume the

following:

where

,;( . :::;

1

8 i =

and

'.

[x + __ E i. )

': eEs J (t

-[x + E i -. En]

cEo

[

2meEs 11/3

h2

J

12meE 11/3 [-h20

(2. 3)

(2.4)

(2.5)

Kg (2.6)

(2. 7)

(2.8)

105

.L

In troduc 1ng these expres s i. )ns

obtain

. ) Eq. (2. 1 \ c1 tid !:q. ( 2. :::) WE:

x u

x o

I rl ~ • • T

+ :: .. .' i . I I o

It can be shown (13J that solutions for Eq. ( .(l) and

Eg. (2.10) have the +:or:n:

.} 1 / 3 ( (, i) + b i .] - 1 /i ( " J )

(::. 10:

(2. Jl)

(2.12,

where JI/J and J-I

/ 3 are Bessel functions of the fj~st kind and

of orders 1/3 and -1/3, respectively. Similarly, K?/3 is a

modified Bessel function of the second kind, 2nd of (,rder 1/3.

The p<lrameters T'i and t'i are given by

1 ; , i

2 6,3/2 3 1

(2.13)

In Egs. (2.11) and (2.12), the terms at b, and c <lrbitrary con­

stants used to satisfy the particular boundary value problem

under consideration.

The relative magnitUdes of a and b (Eg. (2.11)) are reacily

established using Eq. (2.3) in conjunction with Fig. 2.1. Bec')l!se

x < 0 in Region I, the variable a, becomes negative when - 1

X < -E./eE and, thereby, the Bessel functions of Eg. (2.10) have 1 s

a negative argument yielding

( 2 . 14)

where In(ai

) is a modified Bessel function of the first kind.

106

!l2.2

Insight is gained by considering an approximation fOL

II

/3

(ai

) and J_1/3

(ai

) that is val~d when 0i is very larg~ [l~J:

It;. I e ~

/ 2n ll;il ( 2. 1') )

Eq. (2.15) is unbounded when a. = 00. Clearly, no valid solution 1.

for this problem can be unbounded; this situation is avoided in

Eg. (2.11) by setting a ::: b, thus

aMi [J l/3 (~i) + J- 1/3 ((,i) 1 ::: aAi (-'J. )

1.

where Ai(m) is the Airy function [15J.

(2.16 )

Similarly, Eq. (2.12) can be written in terms of the Airy

function:

(2.17)

Thereby, the two solutions for this problem are obtained in one

Cop.unon functional relation.

Next we establish the arbitrary constants (a and b) associa­

ted with Eqs. (2.16) and (2.17). This is readily accomplished

by assuming continuity for both the magnitudes and derivatives

of~I and ¢rr at the semiconductor-oxide boundary (x = 0):

(2. 18)

. d~~1 !x=O

(2.19 )

x=O

107

~ 2.2

Substituting Eq. (2.16) anti Eo

the equality

'2.17: into Eg. (2.18) yields

aAi(-a .) = cAi(B .) o~ o~

(2.20 )

where a . an.d £3 • designate the magnitl1'1eS of a. and 13. at o~ o~ ~ ~

x = 0, respectively. Similarly, after substituting Eg. (2.16)

an.d Eq. (2.17) into Eq. (2.19), we obtain

, ' aAl(-a .}K = -cAi(B .)K B o~ a o~

(2.21)

where Ai(m) represents the first derivative of the Airy function.

FrOOl Eq. {2.20) and Eq. (2.21), we obtain the relation

, K 8 ' Ai (8 .) Ai (-ex .) + - Ai (-a .) Ai (8 . \ = f)

01 o~ v 01 OJ (2.22 )

'1

and this relation establishes the energy eigenval 11es (E.) for ~

this boundary value problem.

Thus Eq. (2.l6) and Eq. (2.l7) represent solutions for this

problem when x :5. 0 and x .:. 0, respectively. Furthermore, these

solutions are continuous at x = a (Egs. (2.18) and (2.19» when

the individual eigenvaluc= (Ei

) satisfy the equality of Eq. (2.22).

In addition, the arbitrary constant of Eq. (2.16) is (from

Eq . ( 2 . 20» given by

c. = a. 1 ~

Ai(-a .) o~

Ai (C .) o~

(2.23)

Therefore, thz eigenfunctions of this problem are given ~y

x < 0

x > 0

WIi(x) = aiAi[-ui(X)]

Wrri (x) = a. 1.

Ai(-a.) [ 1 Ai (Bo~) Ai f\ (x) .

01 j

(2.24)

(2.25 )

where a. is determined from the normalizins rQquir~nont 1

2 I a· I 1

(' I )_co

2 I ~ i (x) I dy = 1 (2.26 )

108

2.3 Inversion LJyer Carrier Distribution in aMOS Structure*

III section 2.2 we established the energy eigenvalues (E.) 1-arid the assoc i a ted eigenfunction solutions for Schroed1-nge r IS

"l!uation ['J!. (x)]. From these solutions, we can obtain the: spettial 1

rrobahility for inversion layer carriers l~. (x) 12 at each allowed 1

energy level (E. ) • 1

This probability distribut.ion, in conjunction

with the den~ity of carriers (N.) at each energy leve~ (E.), yields 1 1

the spatial distribution of these particular energeti~ carriers ~. i 'x) I;' ... 1 I • 1 \ ~ • After summing the carrier distributions from all

enerqy eigenvalues, we obtain an expression for the total distri­

but inn of inversion layer carriers within this semiconductor

rlt'V i ce:

n (x) :::

.x;

r

L i:::l

N. II ~'.(x) i 2 11'

(2.27 )

~~ this section we derive a rigorous mathematical expression tor

this inversion layer carrier distribution.

The density of states in an inversion layer can be written

as:

(2.28 )

where Dl IE) is the density of states in a plane parallel to the

oxide-silicon interface, and D2

(E) is the density of states in a

direction perpendicular to this interface.

First we consider directions parallel to the oxide-semi­

conductor interface. The total number of states per unit area

in this plan residing between E and E+dE is given by

Dl (E)dE = dr(E) = 2~ . dk (2.29 )

The electron energy in this direction is

(2.30 )

* The' methods used here are from Gn~r~ingers and Talley, ref. [16].

109

.,2.. 1

where m' is the effective electron density-of-sla"(;~> m.:Js::, 1"

l:le y-z djrection. Thus, from Eg. (2.29), in coniuncti r ,; v.dl:

Eg. (2.30), we have

D (E) = dr(E) 1 dE

= d r ( ) /dk: = dl:./dk

It 211 m' ( 2 . 31 ) '" n./m

The tot~l density of states perpendicular to the oxide­

se~iconductor interface D2 (E) can be written as

n v

H(F:-E.) 1

IL. 32)

where n is the degeneracy factor. This factor depends upon v

the crystal orientation, with respec~ to the semiconductor sur-

face. In Eg.

by:

H (t:) = 1

= 0

(2.32), H(E-E.) is a step function, and is dpfined ~

• > ') (? . 1 i

< 0

From Egs. (2.28), (2.31), and (2.37.) I the density of st ~tl'S

for each energy eigenvalue within the inversion layer of aMOS

st:Jcture is given by

D. (E\ '"l

rm·"" 1

l~J utE - E.) 1

In Eg. (2.34), a factor two has been introduced for spin

degeneracy.

(2. 34)

~he total number of electrons at each energy level E. is 1

therefore given by

(" N. = J D. (E)f(E)dE ~ 0 1

(2. 35)

where feE) is the probability an electron occupies the energy

level E. Assuming Fermi-Dirac statistics, feE) is given by

110

f (E) (2. 36)

where Ep is the Permi energy. After substituting into Eq. (2.35)

thl- expressions given in Eq. (2.34) and Eq. (2.36), we obtain the

carrier density at each energy eigenvalue:

r ," _ {m I k Tn v ) { ~l' N == 'I D, (E) f (E ) dE - 2 ; log e 1 + ex p ( E P

1 Jo 1 . ~h ) , E. ) /kTJ~ .~

1 )

(2.37 )

!laving calculated the carrier density N., Eq. (2.27) yields 1

the density distribution of these carriers within the inversion

layer of a MOS structure:

n (xl "" m'n kT v

2 nn

2.4 Discussion

(2. 38)

Fig. 2.2 illustrates the calculated inversion layer energy

levels for two different values of electric field at the semi­

conductor surface: ~ = 1.25 x 105 v/crn and ~ = 3.4 x 10 4 v/crn. s s These two calculations establish a general trend that has thus

fnr been verified in this analysis: an increased level of energy

quantization is realized with an increase of electric field. Prom

a practical point of view, implied here is an increase level of

quantum mechanical mechanisms with an increase of electric field.

Thus, in strong inversion we can assume the inversion layer car­

rier distribution will be poorly described by traaitional MOS

thpory. As the gate voltage is reduced and the structure enters

weak inversion, these quantum mechanical mechanisms should tend

to disappear. Thus, it is suggested tha~ in the normal range of

MOSFCT operation, the inversion layer carrier distributior. will

change between those regions where quantum mechanical mechanisms

are significant (strong inversion), and those regions where

traditional electrostntic mechanisms dominate (weak inversion).

111

~

> C,i ~

..;..

> CJ ~

.... w

2.4

.200

,175

.150

,125

.100

,075

.050 ,025

,000 () 100 200 30g 4()f) sr)n

X ---> (A)

,175

,150

,125

.100 -- .----~

.075

.050

,025

.0Of) 0 100 200 300 400 500

o

X > (A)

FIGURE 2.2 Calculated elect:ron energy levels in '_he surface potential well for two different potential gradients (surface electron fields) .

112

~ L. q

This type of situation is generally c0~=i5tent with other

well knOH! solutions of Schroedinger's equation for electrons

in a potential well. Under conditions of w~ak inv~rsion the sub-

strat~ electric field is small and, hence, the interface potential

well is very wid2; little energy quantization is therefore obsc~­

ved. Contrasting with this weak j,version case, under strong

inversion the substrate electric field is very large and, hence,

the potential well is very narrow. A narrow potential well tends

to produce a large separation between the energy eigenvalues of

Schroedinger's e~uation and, therefore, quantum mechanical

me~hanisms tend to dominate.

Fig. 2.3 pres~nts a comparison between the inversion layer

mobkle carrier distribution derived from quantum mechanical and

from traditiona~ solutions of this problem. This illustration

shows an important difference between these two solutions. The

traditional solution yields a maximum carrier density at the oxide

and semiconductor interface, whereas the quantum mechanical solu-a a

tion places a maximum density at abo~t 15 A to 25 A from this

interface. Qualitatively, the important difference between these

sol ut ions is quantum ,llechanica;' reflection a t the s i licon-Si0 2

bO'Jndary.

Ir. this semiconductor structure, large coulomb forces attract

inversion layer carriers toward the semiconductor and oxide inter­

face. Upon reaching this interface, these ene~getic electrons

~ither reflect back into the semiconductor mate-ial ~r, instead,

tunnel into the oxide. This reflection mechanism produces a

maxlmum carrier density at some location rem0~ed from the reflect­

Ing surface (or potential barrier). This process is similar to

standing waves that are obtained in electromagnetic systems. It

is emphasized that if the semiconductor and oxide potential bar­

rier is infinitely large, no tunneling takes place and all carriers

are reflected from the interface. Thereby, an infinite barrier

potential produces a situation where no mobile carriers reside

at the interface surface.

113

18

17

5 16

x

13

12 Q 100

FIGURE 2.3

f s

1.25 x 11):) volts/cm

classical solution

quantum mechanical ~~ol u tion

200 300 40r) 500 70r) 800 o

X CA)

Comparison between quantum mechanical solution and classical solution of the inv~rsion 13'/cr carrier distribution.

114

~ L • :.>

The quant.um mechanical solution shown in Fig. 2.3 must be

recognized ~s only a first-order approximation for the inversion

carrier distribution in a MOS structure. This solution is based

upon a constant potential gradient throughout the inversion region

whereas, in a rigorously accurate solution, this situation would

not exist. Therefore, the relative widths of these calculated

inversion layers (quantum mechanical and classicl1) must be viewed

a3 qualitative, rather than quantitative. A more accurate calcu­

lation of this inversion layer will ~e obtained after subst.itu­

ting the electron distribution (Fig. 2.J) into Poisson's equation,

and thereby calculating a better approximation for the associated

electrostatic potential distribution. Thereafter, usir.g this

improved potential distribution, a new solution of Schroedinger's

~quation will yield a more accurate representation of this inver­

sion layer carrier distribution.

Nevertheless, this approximate quantum mechanical solution

yields important qualitative information concerning the overall

shape of the inversion layer. For example, Fig. 2.4 sho",'s that

a substantial reduction of gate voltage (and, hence, electric

field) produ~es a large increase of inversion layer width. It is

expected that an exact solution for this small electric field

situation (weak inversion) will closely approach the classical

electrostatic solution for this problem.

The foregoing calculations are based upon assumptions similar

to those made by other workers in this field [In]. Assumed here

is a silicon MOS structure containing an impurity atom density

(NA

) of 4 x 1015 atoms/em J . The silicon-oxide surface is taken

to be in the (100) directLon. In addition, we have assumed for

the effective electron mass (m) a value of 0.916 m , and for the o density-of-states mass (m') (in the y-z direction) a value of

0.19 m . o

2.5 Conclusion

A first trial solution for Schroedinger's equation has shown

an important deficiency in previously published solutions of this

probler .. [10,16]. Specifically, by assuming an inf ini te poten tial

115

18

17

~r;

If. ~ u

'-../

1 ,... -. )

,......., X -'

- .1_ ~t ~-

~ (..::--

, -, .1.)

12

11 sn lr)O

FIGURE 2.4

ISO 2()O 3S0 /:;- ,\ .!.J )

Quantum mechanical solutions of inversion carrier nistribution with different surface electric fields (fs ).

( A ) E s = 1. 2 5 x 10 5 V / em

(8) Es = 3.4 x 104

V/em

116

h~rr10r at the silicon-oxide interface, other workers have con­

cluded that no inversion layer carriers reside at this location.

Thl' !=lresent solution implies that such a conclusion is inaccurate.

It is shown that a non-zero carrier density exists at this inter­

face surface and, in addition, a finite tunneling current is

present from the semiconductor into the oxide.

Unanswered questions reMain concerning the consequences of

this mobile carrier tunn~ling. Clearly, tunneling cannot take

place for an indefinite period of d~vice operation without

altering the interface barrier height. It is suspected that this

particular problem should be given serious consideration for

MOSFET operation in u radiation type of enviro~~ent. Oxide damage

by radiatio~ could, indeed, produce trapping sites for these

tunneling carriers and, thereby, result in a significant shift of

MOSFET threshold voltage.

Because research on MOSFET operation is now terminat-cd under

this contract, no further studies will be undertaken to qllar.ti­

tatively establish the consequences of radiation on the operation

of MOSFET structures. ·levertheless, we expect to complete this

quantum mechanical analysis of the MOSFET inversion layer and to

apply conclusions drawn from this effort to other aspects of

device theory.

117

PFECf:.DHG PAGE j~llOT .tt'IU-lliD __ .. 'J......... . 7"''';;.0'.' ~

Chapter III

LUMPED tlET\\"IJRK REPRESENTATION OF MOS TRANSISTORS

by J. I. ARREOLA and F. A. LINDHOLM

1.1 Int~oduction

53.1

As an ~id to ~he design of MOS circuits by comouter, t~e

existing network representations (or equivalent-circuit

Models) for the MOS transistor fall far short of bei~g ide~l

in several respects. Among these deficienc:es a~e:

(a) the models fail to accurately represent thE

current flow during transients;

(b) the models fail to ~ccurately represent t!le

behavior of short-channel devices and their

behavior in weak inversion.

The practical consequence of these deficiencies is a~ 1n­

ability to design desired circuits by computer simulaticn.

Hence, either one desig~s using Many empirical iterations,

which is enormously costly, or one contrives circuits and

device configurations to avoid those aspects of device

behavior (such as "a" and "b" above) that are poorly mode!~d"

This resulting conservative design yields worse circuit per­

formance them present fabrication technology can prov ide.

The deficiencies of the existing models can be traced

to the way in which they were developed. First, as with

other devices, the interconnection of capacitors, resistol"S,

and controlled sources used to represent an MOS transistor

evolved in a largely heuristic and intuitive manner. Second,

it evolved with an emphasis on the discrete MOS transistor,

which, in contrast to its integrated-circuit counterpart, can

re~sonably be treated as a three-terMinal device. This

emphasis led to a fundamental misrepresentation of the dis-

118

placement currents in an i~~egrAted-circuit Mrs transist,or

(a four-terminal device) I which contributed to defi;~i"3I1c'/

(a) named abc'Je. Third, as Cobbold [lJ h,ls pointed out, f::I,C

attempts to model t~e displacement currents ill a four terffiindl

MOS transistor were ')ased on Lindholm and Gray's '..;orr: [2J (;r

the four-terfTlinal junction field-effect transistor. This '"c,,'K

proved to be inapproprL1te for the MOS transistor. Finelli')

the piecewise one-dimensional analysis of Shockley [3J, ~s

adapted first by Ihantola and Moll [4J to the MOS trano'islor,

undergirded the development of alJ existina network rc~resellt~-

tions. For short-channe' devices and for operation 1n weak

inversion, a piecewise one-dimensional model no lonqer hnlds.

In the present work, ",:e hav'e tried t.o bull'] ,1 fi riT.

foundation for network representations that adcquatei~

account for the bahavior of a four-terminal device, ,'1(1 that

are general enough to enable the inclu~ion of the mulri­

dimensional effects present in short-channel devices, or in

devices operating in weak inversion. Our object has heen tCl

der i ve the network represen ta t ions wi th ani n imurr, numb,,' r (1 f

approximations and a minimum loss jn generality. This con­

trasts with the earlier work of Lindholm [5J, which 30uqnt

to tie the existing network representations to the underlying

physics by clarifyins the approximations involved.

In Section ~.2 below, we set up the equations for

current flow that describe ~ny n-termin~l device. In Section

3.3, using the quasi-static approximation [1,2,5J. we trans­

form this description into relations between current d:!C

voltage. Section 3.4.0 gives several forms of netwe,d: rerre­

sentation of this description. In Sectlon 3.5.0, we apply

the methodology developed to the four-terminal MOS tran3istor.

The result, shown in Figure 3.5, differs from existing net­

work representations by the presence of a controlled sourC0

in parallel with each cu?acitor. One can view t he me.," i f1(j

119

3.2

as heinq necessary in the full

;1(' t'.;ork reprc sen ta t ion 0 f the r·10S trans i stor \ ::0 .... 1 it hi n the

,: ,1 1 i d i t v () f the q \J a s i - s t a tj c d P pro x i mat ion); 0 r a s r e 1 ate d t e

errors inherent in all-capacitor models for the displacement

~urr0nts.

3. 2 tio:1s for the Current Flow in an n-Terminal De'Jice

The major objective of this study is to find a networK

representation of the physjca 1 mechanisms relevar.t to the

operation of MO~ transistors. The network representation

;;ou'lht will describe the device response to lLlrge-signal und

limc:-varying excitation, with the smLlll-siqnal or dc res~on~~'~

b01ng obtainLlble as a special caSE.

Ay network representftion, we mean an interconnection ('~

circult elements between circuit nodes. Ecch node of the

network rt'presentatlon wi 11 correspond to an int~jnsic ,=-ont.:..:~ct

o f tilt' C 1 e c t r or i i c d n vic e, w h L c h 'N i 11 bed e fin c· c:l ~; It 0 r t 1 Y .

By th" physical mecr.dnisms relevant to the: oper,]tion '.);

t!l(' ~1ns tr:l~~sistor, WI' m,",lr1 tht~ follo'winq m0chanisms (,CCUrrlnc;

,l~ ar)(~ within a cc:rt:,lin sur-fuce S to be dc:finc:d :::hortly. t,t

this surfac'2 tile mair, phyc~lcal mechunism to be considered is:

(a) the flo~ of mobile charge cLlrriers across the

surfLlce.

WIthin the surfdce, the m<;in physiccll !'lc:chanlsms to be consi.dered

,I re :

(h) the tra:1sport of mobile cr«:rqc: curric.'rs c:ntjre:ly

across the volume bounded by the surface; and

(CI t.he dCCL1~~1.jtion of mobile charge carriers .... ,lithin

the V01U1110 boundcd by the surface.

Thus, in this study, bec.Juse of our concentr<ltlon on the ,"10S

translstor, we ilnorc various other phenomena t.hat are imrJorLlnt

in other devices, includinq: magnetic induction; and inter-

b~nd, band-bound, or bound-bound transitions in a semiconductor

(fnr '~xample, reCOml.H.lation, generation, tunnelinq, or trappinq

f)rO(~essc:s)

120

,2

In general, the bc'havicr of an (~lt;ctconi( __ ,:·"/1"(' 'N! 1:

depend on a complicated m' " ':-c c diff(':rent !,hl'i\()ml'il.l, cerUllr'

ones being r-esent in some r~qions of thc' dl'vi('(~ butd)~;('nt

from others.

device, it is commor to divide the device Into int'-in:';l;' and

extrinsic parts, as h~s beer' me [51, for ,'xc1mrJle, f"l til"

MOS transistor. This study will concentr,lt,· on the lilt cinsic

part which is defined as beinq bounded by t'w ,.;urfac(· ,',

referred to earlier. It is convenient to definr, tt,l,' Ir:t:rir.s1c

part so that the surface E' er'{:loses nn nc·t ('fur';"

so that it obeys the condition of o\"('r<.:111 c:fldrq,: nt·u t 1',1]1[':'.

In general, there w}! 1 exist certain purt ions

S. (j~cl,2 ... n) of the sur:ace S across which an appn'cidb](' 1

flow of mobile charge carriers will occur. l\cross tn"

remainder of th2 surface, the flow of mobile cLarcll,: ';,lrriL:cs

will be insignificant. Thus, in qeneral,

\ S. < S· L l-j

Let the subsurfaces S. be called the J

defined so that at each point on s. ,

( '1 • 1 )

intrinsic contacts,

the component of the J

conduction current density JC

normal + 0 S. 'N ill be a p p I Ii,) b 1 I: , 1

which we denote by the statement: J c not'ma 1 t o. MOS

trdnsistor, there are four such intcinfic contacts 151: the'

actual m2tal cu,,_.,~j- of the ga:-'C:i the surfaces parallel t.)

and just inside the facitl(j portions of the sourc(' ,md drain

is land- subs trate junctions; and the" plane furmed by t h(~

depletion region in the substrate nearest the ac:ual substCJtl

contact.

The current flow i. out 0- anv intrinsjc contact S IS J > 1

carried by mobi12 carriers and is qivcn by

i. J

This corresponds to

- -{S. JC'dS'

J physical mechanism

121

(a)

( 3 .2)

named previously.

§ 3.2

The mobile charge carriers flowing across co~tact S J

can be divided into two components, corresponding tc physical

mechanisms (b) and (c) occurring within the surface S. In

part, the mobile charge carriers will be entirely transport 0 d

across the volume bounded by the surface S; and, in part, th~y

will accumulate within this volume. Thus,

i. = J

i + T.

)

dQ. -...J.. dt

( 3 • 3)

Here, iT. represents transport; and dQ/dt r£~resents accumu10-) )

tion. We define iT. and dQj/dt as the components corresponding )

respectively to transport and accumulati'.' of mobile charge

carriers within S contributed by the current i. at contact S. J J

The component iT.' being produced by the transpurt of )

mobile charge carriers across the volume, can be expressed by

i = T. )

iT ' jk

(3.4)

where iT jk

Sj resulting

is that part of the transport current iT, at contact

from the

contact Sk' From the

iT jk

and

i T .. ))

mobile charge carriers

way i Tjk

is d,-:=fined

= -i Tkj

= 0 .

)

transported from

(3.5)

(3.6)

Because surface S encloses no net charge, it follows that:

I j

dQ. <? = 0 (3.7)

From the continuity equation integrated over the volume

bounded by S,

122

) . !

dQ. ~ =:; 0

i dt ( 3 • 8)

Thus,

(J. 9)

the last property being a qlnhal co~nterpart of Kirchhoff's

current node law.

As wlll be seen, the properties expressed in (3.4)

through (3.9) will prove important in deriving 3 network

representation of devi ce b8havior. They j .ave a 11 been

obtained in the foregoing by physical reasoning.

Alternativ0ly, (3.7 and (3.9; can be "'crived in a

straightforward manner by ;:'ombininr; two of Maxwell's equations

and

aD V'xH=J +-C 3t

V'D = p

(3.10 )

(3.11)

with the definitions of S, Sj' and ij

. (Here Il is the magnecic

field intensity, D is the electric flux displacement, and 0

is the net charge density.) From this derivation one can

identify thp terms dQj/dt corresponding to the acct....nulat'Qn of

mobile charge carriers as being associatf~d with the displace­

ment current.

As later oaqes will demonstrate, the result,

dO. i. = L iT +-Tt.

J k;ij jk (3. 12)

will constitute the basis for our development of network

representations for the MOS transistor. Its physi~al ~eaning

is that the current i. at contact S. may contribute tn current , J at the other contact.s, as implied by physical mechanism (b)

named above, and that it may contribute to the accumulation of

123

§ 3 . 3

mobil~ charge c~rl·iers w~thin the intrinsic dEvice, as

implied by physical mechanism (c) named above. As will be

illustrated later for the MOS transistor, the description of

t.he transport terms iT and the accumulation terms dQ.;ldt jk J

must be derived from consideration of the physics pertinent

to device operation.

I~ physical mechanisms apart from (b) and (c) contribute

significantly to d~vice behavior, a generalization of (3.12)

will be needed. In this generalization, additional terms

will need to be added to (3.12), one for each additional

physical mechanism considered.

3.3 A Quasi-Static Approximation

In general, the transport and displncement terms in (3.10l

will depend not only on the tErminal voltages vl,v2 .. v n

but also on its derivatives with respect to time dVI/dt,

dV2/dt ... d2vl/dt2, etc. As a first-orde~ approximation,

assume that the functional dependencies of the charge and the

transport current are the same as in the dc steady state [5J.

That is,

I = I (v l ,v2 ••• v n ) tjk tjk

v ) n

where I and Q. are the functional dependences in the dc tjk )

steady state.

With (3.13) and (3.14), Eq.

i. = ) L

k,tj

124

(3.12) A

dQ. -=::J.. dt

becomes

(3.13)

( 3.14)

( 3.15)

Applicc,tion of the chai: rule f':H di fft:rc~ntii)tioil J::

dQ j gives dt

It is

dQ. -.-.:.l dt

convenient

(lQ. -=.1 (lV

k

~

i. k

to

t. ::

;)0, --l v . (lv, k

y.

( 3 . 16)

define 1n (3.1(,)

y

d'k 1 ' (Ll7)

in which

terminal

and has

d jk r.orresponds t, the ciisplacement of charC!c in

and

a)

j due to a varia t i lin in the vo] tage on termina 1 k I

the following properties:

Since the device is neutrally ~harged

(3.18)

b} If all the terminals are incrementally shorted,

no displacement currents will flow. In other terms I if

VI :: V/ :: .... v oj 0, n

then

1. -= 0 :: vI y. d

jk 0

1 k

and

'i d - 0 k jk - . ( 3. 19)

Eq. (3.19) can also be prove(~ by contradiction; for if it

were not true, then Eq. (3.16) would be invalid in the

special case: vI:: V7 ::

125

§ 3. 4

vI i th the definition given in (3.17), Eq. (3.1S) becomes ,.

i . -- ! I + L djk 'ilk· (3.20 )

J k,j tjk k

3.4 Lumped Network Representation of a Semiconductor Device

~

Eq. (j .20) . -r~.

The re/rr::'sentatlon :or terminal j is shnwn in

Figure 3.1. In most npplications, however, it is more con-

vcnienl to use a network representation with elements inter­

connecting the nodes.

j

. !.l 't II d jk\

1 (i.,v.) J J

d ..

1 JJ l .. i jk ¥o;)

-- -

Figure 3.1 Representation for terminal j of equivalent circuit.

A circuit with the desired characteristics requires symmetry

among the ele~ents connecting any two nodes. Eg. (3.20) can

be rewritten to satisfy this condition by adding and subtracting

\. djk

Cr. to I. J k,i that d kj =

k

i . J

the .th J

0,

I , k,j I

equation. Collecting terms and noting

1

Itjk + (djK~k - dkj~j)J (3.21 )

The branch connecting nodes j and k in the resultant circuit

is shown in Figure 3.2.

126

3. 4 .1

, ,--------; -"-------.....

J----I I--__ K

d jk \ -dkj V j

Figure 3.2 Branch connecting nodes j and k of ecjt:IvaTe-ri t circuit.

3.4.1 Simplified Capacitive Representation

In the previous representation, a simplification occurs

in the branch jk whenever djk

:::: dkj

. The current gen,'rator

(djk~k - dkj~j) becomes a capacit0~ Cjk o~ value

In this case the current in node j IS gIven by:

i. = )

1

I + C. k (.:. . -':'k) I

tjk J J J

(3.22 )

Figure 3.2 ~~~ws the corresponding representation for

the bra:lch ;k.

r--------4 -}-------..

j ---

Figure 3.3 Representation-for branch jk of equivalent circuit.

127

( 3.23)

§3.4.2

This kind of simplification applies to any two-termir.al device,

and for certain other cases as determined by Eq. (3.18)

together with Eq. (3.19).

3.4.2 General Cap~citive Representation (d' k ~ d 1,.)

J 1\.J

For multi terminal devices the condition d jk = dkj

is

not always true. However, a capacitive representation is

obtained by defining:

(3.24)

The difference in displacements djk will represent (in the

circuit) the deviation of the branch jk with respect to a

totally capacitive representation.

The following properties of djk are derived from defIni­

tion (3.24), and from Eqs. (3.18) and (3.19):

d ~ . = 0 ( a) JJ

djk = -dkj (b) (3.25 )

L djk = L djk = 0 (c) j k

The use of definition (3.24) in the equation for the

current flow in terminal j (equation 3.21) leads to

i. = k~jl I + djkVk + dk j (v k - v j ) ] J tjk (3.26 )

whic.l in the branch jk corresponds to a current gE":nerator of

value djk~k' and a capacitor Cjk of value

Cjk = -dkj ,

as is shown in Figure 3.4.

128

(3.27)

:: ].4.2

,... 1

tjle

r- ....

j Ie

Cjle

Figure 3.4 Equivalent circuit between nodes j and k.

129

,f

§3. 5.0

3.5.0 Lumped-Netw\n'k Representation of an MOS Transistor

The conclusions of the foregoing sections apply generally

to any n-terminal device subject only to the constraint of

overall charge neutrality_ The application of these conclu

sions to finding a network representation for a specific devic~f

however, will typically require approximations, as we shall

now see.

Consider an MOS transistor. Let terminals I and 2

represent the source and drain contacts, and terminals 3 and

4 the contacts of the gate and the substrate, respectively.

In normal operation, the gate and substrate voltages are

controlled to induce a conduction channel between the source

and drain regions. Under these conditions, the main assump­

tions made with respect to the currents flowing in the device

are:

1. Recombination currents are neglected.

2. Transport currents at terminals 3 and 4 are neglected,

This is due to the presence of the insulator in the gate,

and to the reverse bias voltage in the substrate.

3. Displacement currents in the channel are neglected (this

assumption is not valid for three-dimensional models).

This is because the channel forms a direct conduction

path between the drain and the source.

With these assumptions, a network representation is

obtained by using the representation found in Section 3.4.2.

Figure 3.5 shows the resultant circuit.

3.5.1 Eval tion of Model Parameters

3.5.1.1 Transport Current Model-Elements

The current generator 112 is the functional dependence

of the dt".:;in ,-~urrent in dc steady state It is determined from

the dc analysis of the device.

130

GATE

-----+--------------~~--_+--~_. ~------_r----~----? SOURCE

I C12 I , I I

"'-T- ---'I----·r- J , I

: - : C24 L.... ____ ,'--.\ _____ ,;.

4

SUBSTRATE OR BULK

'- '

DB AI};

(TId' v ! ?4 4 I

------ Hepresentation for threp dimenRional modp1s

Figure 3.5 Gen~ra1 Network Repreoentation for the MOS Model.

131

, ' 1 f c om il r t • 1 d t i 0 1\ l , \, 1 r,

..cluiv,l1('nt to the fundamental charge control c'quatlCJn

Qt (-)

T

t jk

when-- Qt and T are the total charge in transit arid the t:ansit

time between terminals j and k.

3.5.1.2 nt-Current Model Elements

a) C'i1pacitors.

The value of the capacitors is directly obtained from

(1.27) and ( 3.17) I

C 13 :lO)

== - --dVl

C14

d0 4 = - --

dVl ( 3.2 d ;

C34

d03 == - dV 4

b) Current generators.

The current generators are evaluated crom ( 3.24) and (3. Pi,

di) :::: dOl dO)

d 2) dC 2 d0 3

dV1 == -- -dV dV3 oV ) 2

dOl d04 d 24

a0 2 dQ 4

d' == - == dV 4 ( 3 . 2') )

14 dV4 ~vl v l

d 34

dQ3 ()Q4

== dV) dV 4

132

useful,

diJ 1 Ji4 0 (a)

d2J

+ d 24 0 (b)

( 3 . j () )

diJ + d23 -d 4J d 34 (c)

di4 + d24 -dj4 d 43

, (d)

As Eqs. (3.28) and (3 .• :9) imply, the eV<1luiltlon of t~1('

displacement circuit elements reyuires knowlcdqe of thv . . ,

functional dependencies of 0], 0;", 01, 04' ::;, nce nc t r,lns-

port currents flow in terminals 3 and 4, the charqes Q ,ind

Q" are found, respectively, by evaluating the chanj" '~)n ~.hc

metal in the gate, and the charge accumu 1 a ted in the d,'p Ie ted

region in t!le substrate.

In contrast, both Q) and 0; contribut~ to maintaining the

total charge in the channel. The contributil):ls 0: OJ and Q:

to this m,::;.intl.'nance are unequal, however, beca'..!se of the 0 1 rllc

drop along the channel produced by the transport current.

Hence the identification of Q} and 02' presents a major

problem in eva111ating the elements of the circuit that are

defined i, terms of t/lese charges.

As one approach to this problem, let us consider two

limiting cases:

a) The device is turned off, Vi - VI = O.

Under these circumstances the distribution of the

charge in the channt 1 is symmetr ic, and the di sp laceme n t of

charge in terminals 1 and 2 is equal,

( 3 . 31 )

133

Thils, in terms of the circuit elements,

d I! :.::: d 23 ,

.1 nd from (3. 30 c)

h) The devicf

voltage) .

d j 1 == d;' -l :=

IS in saturation, v? - Vj

S3.S.1.2

( 3. 32)

(3.33)

= v (pinch-off p

In pinch-off operation the channel has a termination

point near terminal 2, and tile charge displaced at the drain

becomes zero,

( 3. 34)

In this case

d 23 = 0, (3. 35)

and from ( 3 . 30 c )

(3.36 )

From the two extreme conditions just consi~nred, we can

sC 0 that the current generators d1 3 , d 23 in the model can be

expressed as

( 3 . 37)

( ]. 38)

where f (v.)-vl) is a function of (v2-vl) that satisfies the

conditions

1 for v2-vl = v . P

( 3. 39)

A suitable approximation needs to be found for ir :mediate

voltages.

Th0 current generators dl 4 and d 24 are found directly

from (3.30a) and (3. 30b),

d!4 = -di3 (3.40 )

(3.41 )

134

; . (,

3.6 Conclusions

Although during tlH.' r- __ r~0d 01 this ccntract ,v(' 1,,1V(' L:1id

a major part )f the foundations for the netwurk n"r)r"C,('nLl~' i 0fl

of the MOS traLsistor, our work at this stage is in(;omplctL'.

Laboratory and compu "'r experiments are needed to ilssess the

degree to which 0i:= new mode' 0vercomes the deficiencies of

its predecessors. Effort is needed to imbed the devicl'

physics fur short-channel d~vices and for weak inversion into

our model. The quasi-static approximation needs careful

examination for its contrib~tions to inadequate representation.

Finally, the consequences in circuit design of the revised

models need to be assessed, methodology needs to be developed

to enable use of the simplest model for each device in a

circuit, consistent with obtaining a good simulation of

circuit response.

135

LITERATURE CITEO

CHAPTER I

1. II .•. ,T. Thantola, Stanfo~rl E:lectronics Laborator'l, Techili,,]1 ; , '! : ",',.." .; /l 0 1 - J ( 1 9 G 1) •

2. I!. K. J. IhantoL.1 and J. L. Moll, Solid-State le'le,'croniCs,

7, 423 (1964).

1. IV. Shockley, !'roccedin9s of the I,R.I'" 40, 13fi5 (1952). =

4. R. H. Cruwford, "MOSFET in Circuit [)esign," McGraw-rilll Company, N.Y. (1967).

). l\. S. Grove, "Phys ics and Technology 0 f Semiconductor Devices," John Wiley and Sons, Inc., N.·i. (1967).

6. S. M. ,Szc, "Physics of Semiconductor Devices," Wjley-Ir.te;­science, N.Y. (1969).

7. O. J. Hamilton, F. A. Lindholm, and A. H. Marshak, "p r lnc iples and Appl ications of Semiconductor Dev i C~! Modeling," Holt, Rinehart and Winston, Ir.c., N.Y. (1971)

8. W. M. Penn(~,/ (ed.), "MOS Integrated Circuits," Van NostL~nd Rpinhold Company, N.Y. (1972).

9. H. F. Storm, IEEE Transactions on Electron Devices,

~D.:li, 710 (1967); also EO-16, 957 (1969).

10. E. S. Schlegel, IEEE Transactions on El~ctron Devices,

ED-14, 728 (1967); also EO-IS 951 (1968).

11. M. B. Barron, Solid-State Electronics, 15, 293 (1972). =

12. 1< .. ~1. Swanson and J. O. Meindl, IEEE Journ'11 of Solid­

SC.:Jtr' Circuits, SC-7, 146 (1972). ==-

1). R. R. Troutman and S. N. Chakravarti, IEEE Transaction

on L' J r (' u itT h L' 0 r lJ, CT- 2 0, 659 (1973).

>L If. Kutto and Y. Itoh, Solid-State Electronics, 17, 1283 ( 1974) .

lS. T. Masuhara, J. Etoh, and M. Nagata, IEEE Tr~nsactions

on Fc'lectron Devices, EO-21, 363 (1974).

lL, Y. A. EI-Mansy, "Modeling of Insulated-Gate Field-Effect ';'1 l:lsistors," Ph.D. Thesis, Carleton University (1974).

136

17. G. A. Armstronq and J. A. f-1agowan, f:lectronic's i., t./,'r.':,

~, 313 (1970).

18. D. P. Smith, Stanford Electronics Laborato["II, Tt'elillica'

R c po r t No. 48:1 5 - 4 ( 1971) •

19. B. lIoeneisen and ~. A. Mead, IEEE Transactions on FlcctruI.

Devices, ED-19, 3.2 (1972).

20. A. Popa, IEEE Transactions on Electron Devices, ED-19, 774 (1972).

21. H. S. Lee, Solid-State Electronics, 16, 1407 (1973). =

22. H. C. Poon, L. D. Yau, R. L. Johnston, and D. Beecham, International Electron Devices Meeting, WashinqtoIl, D.C.,

156 (1973).

23. L. D. Yau, Solid-State Electronics, l2., 1059 (1974).

24. D. P. Kennedy and P. C. Murley, IBM JOurnal of RI':;'.'uIC!;

and Development, t7, 2 (1972).

25. D. P. Kennedy, Mathematical Simulation of the Effe(:ts ul Ionizing Radiation on Semiconductors, Final Report

AFCRL-72-0257, (1972).

2 6 . M . B. Bar ron, S tan for dEl e c t ron i c s Lab 0 rat 0 1 i e 5, Tn: f) n 1 (" a I

Report No. 5501-1 (1969).

27. G. A. Armstrong, J. A. Magowan, and M. D. Ryan, Elee rr'!iI{_·.~ Letters, ~, 406 (1969).

~8. J. E. Schroeder and R. S. Huller, IEEE Transaction,' on f:l':ctron Devic(:s, ED-15, 954 (1968).

29. D. Vandorpe a .. ~ N. H. Xuong, Electronics Letters, 7, 47 (:S17l).

30. F. H. De La Moneda, IEEE Transactions on Circuit T'It.'Olll,

CT- 2 0, 666 ( 1 9 7 3) .

31. C. G. B. Garrett and W. H. Brattain, Physicai RCV1~W, 99

376 (1955).

32. W. Shockley, Bell System Technical Journal, 28, 435 (1949).

33. W. M. Gosney, IEEE Transactions on Electron Device:;,

ED-19, 213 (1972).

34. R. A. Stuart and W. Eccleston, Electronics Letters,

225 (1972).

137

8 I

35. R. H. Dennard, F. H. Gaensslen, H. N. Yu, V. L. Rideout, E. Bassous and A. R. LeBlanc, IEEE Journal of Solid-State Circuits, SC-9, 256 (1974). -

36. R. J. Van Overstraeten, G. Declerk, and G. L. Broux, IEEE Transactions on Electron Devices, ED-20, 1150 (1973)

37. R. R. Troutman, IEEE Journal of Solid-State Circuits, SC- 9, 55 (1974). =

38. R. H. Dennard, F. H. Gaensslen, H. N. Yu, V. L. Rideout, E. Bassous and A. R. LeBlanc, International Electron Devices Meeting, Washington, D.C., 152 (1973).

39. H. C. Pao and C. T. Sah, Solid-State Electronics, 9, 927 (1966) .

40. W. Shockley, "Electrons and Holes in Semiconductors," D. Van Nostrand Company, Inc., N. Y. (1950).

41. C. T. Sah and H. C. Pao, IEEE Transactions on Electron Devices, ED-13, 393 (1966).

42. A. P. Gnadinger and H. E. Talley, Solid-State Electronics, 13, 1301 (1970).

43. V. G. K. Reddi and C. T. Sah, IEEE Transactions on Electron Devices, EO-12, 139 (1965).

44. J. M. Early, Proceedings of the IRE, .!2., 1401 (1952).

45. D. Frohman-Bentchkowsky and A. S. Grove, IEEE Trans­actions on Electron Devices, ED-16, 108 (1969).

46. C. T. Sah, IEEE Transactions on Electron DevicES, ED-11 , 324 (1964).

CHAPTER II

1. H.K.J. Ihantola and J. L. Moll, solid State Electronics, 2, 423 (1964).

2. R.S.C. Cobbold, "Theory and Application of Field-Effect Transistors," Wiley-Interscience, N.Y. (1970).

3. J. R. Schrieffer, Phys. Rev., ~Q, 641 (1955).

4. R. F. Greene, D. R. Frankl and J. Zemel, Phys. Rev., lId, 967 (1960).

5. R. F. Pierret and C. r. Sah, solid State Electronics, ~, 279 (1968).

138

----~- -----~-.,....--- ~---

6. N.S.J. Murphy, F. Berz. ~~d I. Flinn, Solid State E10c­tronics, g. 775 (1969).

7. J. R. Schri~ffer, "Semiconductor Surface Physics," (edited by R. A. Kingston), Univ. of Pennesylvania Press (1957) .

8. A. B. Fowler, F. p. Fang, W. E. Howard and P. J. Stiles, Phys. Rev. Letters, 16, 901 (1966).

9. F. Stern, Phys. Rev. Letters, 21, 1687 (1968).

10. F. Stern, Phys. Rev. B, 2' 4891 (1972).

11. M. A. Green, F. D. King, and J. Shewchun, solid State Electronics, 17, 551 .'974).

12. R. Williams, Phys. Rev., 140, A569 (1965).

13. F. B. Hildebrand, "Advanced Calculus for Applications," 156, Prentice-Hall (1962~.

14. F. B. Hildebrand, "Advanced Calculus for Arplications," lSI, Prentice-Hall (1962).

15. "Handbook of Mathematical Functions," (edited by M. Abramowitz and I. A. Stegun), U.S. GPO, Washington, D.C. (1964).

16. A. P. Gnidinger and H. E. Talley, Solid State Electronics, 13, 1301 (1970).

CHAPTER III

1. R.S.C. Cobbold, 'Theory and Applications of Field-Effect Transistors," Wiley-Interscience, New York, 1970.

2. F. A. Lindholm and P. R. Gray, IEEE Trans. Electronic Devices, ED-~, 819-829, December 1966.

3. W. Shockley, Proc. IRE, vol. 40, Nov. 1952, pp. 1365-1376.

4. H.K.J. Ihantola and J. L. Moll, Solid State Electronics, ~, 423-430, June 1964.

5. F. A. Lindholm, IEEE Journal of Solid State Devices, Vol. SC-6, No.4, 250, August, 1971.

139

Figure

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

1. 1 C)

LIST OF FIGURES

Illustrative Model of a MOSFET.

One-Dimensional Distributions of Q. and QD

in a MOSFET. 1

Calculated Inversion Charge (Q.) in a MOSFET; o 1

tox = 1000 A, NA -= 2 x 101Scm- 1, VG = 4.3 volts.

Illustrative Model for Channel-Length Modulation.

Two-Dimensional Electric Fields in the Gate-Drain Depletion Region of a MOSFET.

(a) Characteristic Equipotential (V = VG ) for Saturation-Mode Operation of a MOSFET.

(b) Electr0static Potential Distribution Associated with Fig. 1.6a.

Calculdted Saturation-Mode Output Conductance (substrate impurity concentration -

3 x 101~cm-3, channel length = 411m); (A) Reddi-Sah Model. (~) Smith Hodel. (C) Frohman-Bentchkowsky and Grove M0del.

Piecewise Rectangular Approximation tv a MOSFET Depletion Charge Distribution.

Geometric Approximation to a MOSFET Depletion Charge Distribution.

Gaussian-Surface Boundaries, (1) - (4), Around an Arbitrarily Defined Drain Region.

8

10

17

34

37

41

44

48

52

56

1.11 Calculated Contours of Constant Source-Drain Electric r~ield in a MOSFET (L = 1011m, NA = 2 x l015 cm-3). 65

1.12 Calculated inversion carrier distribution in a MOSFET assuming a constant carrier mobility (VG = 3.0

volts; VD = 6.0 volts). 67

1.13 Calculated ~obile carrier flux assumin(J a constant carrier volts; VD = 6.0 volts).

distribution in a MOSFET

1.14

1.15

mobility (VG = 3.0

Qualitative Illustration of Gate-Induced Electric­Field Orientation in the Saturation-Mode.

Calculated potential distribution in a MOSFET (N

A = 2 x 1015 atoms/cm 2

j VD

= 1.0 volt;

VG = 3.0 volts).

140

68

70

71

Figure

1.16

1.17

1.18

1.19

1. 20

1. 21

1. 22

1. 23

1. 24

1. 25

Calculated notential distribution in a MOSFET (N

A - 2 ~ 101 s atoms/cm 2

; VD

= 6.0 volts;

VG :: 3.0 volts.

Calculated volt-ampere characteristics of a MOSFET, assuming a channel length (L) of 10 um (constant carrier mobility).

Calculated volt-ampere characterist1cs of a MOSFET, assuming a channel length (L) of 10.0 um (field-dependent carrier mobility) .

Calculated inversion c~rrier distribution in a MOSFET, assuming a _ "eld dependent carrier mobility (V

G = 3.0 '"olts; VD :: 6.0 volts).

Calculated volt-ampere characteristics of a short­channel (L = 1.0 um) MOSFET (constant carrier mobility).

Calculated volt-ampere characteristics of a short­channel (L = 1.0 um) MOSFET (Field dependent carrier mobility).

Calculated Contour of Constant (1000 V/cm) Source­Drain Electric Field (broken line) in a 5.0 urn MOSFET for VG ~ 2.0 volts and

(a) VD

:: 0

(b) VD

= 0.5 volts

(c) VD

= 1.0 volts

(d) Vo = 3.0 volts

71

73

74

76

77

78

80

(Cross-hat. .... :.ed areas represent charge-neutral subst ra te)

Calculated Contours of Constant Source-Drain Electric Field in a 5.0 urn MOSFET for VG = 1.0 volt and VD = 5.0 volts; 81

(A) E = 0 (source depletion-layer edge); x

(B) Ex = 1000 V/cm.

Calculated Contours of Constant Source-Drain Electric Field in a 5.0 um MOSFET for V = 2 volts. 82 (A) E = 0 (source depletion-fayer edge) i

x (B) Ex = 1000 v/cm.

One-Dimensional CalcUlations of Inversion Charge (Qi) in a MOS Structure; NA = 5 x 1016 crn-3. 91

141

Figure

2. 1

2.2

2.3

2.4

Simplified energy diagram at the interface of Si-SiOz.

Calculated electron energy levels in the surface po!:ential well for two different potential gradients (surface electron fields).

Comparison between qu~ntum mechanical solution and classical solution of the inversion layer carrier distribution.

Quantum mechanical solutions of inversion carrier distribution with different surface electric fields «( ).

s (A) (s = 1.25 X 10 5 V/cm

(B) ( = 3.4 X 10 4 V/cm S

104

112

114

116

3.1 Representation for terminal j of equivalent circuit. 126

3.2

3.3

3.4

3.5

Branch connecting nodes j and k of equivalent circuit.

Representation for branch jk of equivalent circuit.

Equivalent circuit between nodes j and k.

General Network Representation for the MOS Model.

142

127

127

129

131

ntECWlf'('; .?J.GB; ):J.ANK-;40T J! IUv.:D - __ _ _ _ .f-.............. ~ ... _-",~""; __ . _____ ~_

b

c OX

D

.. D

LIST OF SYMBOLS (Chapter I)

band bending parameter (dimensionless)

capacitance of gate oxide 2 (farads/m )

electron diffusivity in a MOSFET inversion layer

electric flux density 2 (coul. /m )

2 (m /sec)

E magnitude of E at source-end of a MOSFET channel (volts/m) o xs

E source-drain electric field (volts/m) x

E E along oxide-semiconductor int~rface (volts/m) xS x

E ga te- induced electric field (vol ts/m) y

E E at oxide-semiconductor interface (volts/m) ys y

gd output conductance of a M()SFET (mhos)

1D source-drain electric current in a MOSFET (amps)

diffusion component of ID (amps)

Idrift drift component of ID (amps)

kT/q

L

magnitude of ID at the onset of channel pinch-off (amps\

source-drain electron current density 2 (amps/m )

Boltzmann's constant (8.62 x 10- 5 ev/oK)

thermal voltage (.0259 volts at T = 300 0 K)

scurce-drain distance in a MOSFET (m)

extrinsic Debye length in semiconductor substrate of a MOSFET (m)

L voltage-dependent channel length of a MOSFET (m) c

n

n. 1

n s

electron density in semiconductor material (m- 3 )

intrinsic carrier concentration in semiconductor material

magnitude of n along oxide-semiconductor interfcce (m- 3 )

-3 (m )

NA acceptor impurity ion density in semiconductor substrate (m- 3 )

N x equivalent density of substrate ions electrostatically

associated with drain (m-3)

143

---~,..,...----~_ ... , -- -""""b...._

: J Y

l'

T

cquiv~lcnl density of substrate ions electrostatically associated with gate ( -3, m I

hole ,jt~nsity in semiconuuctor material (m- 3 )

electronic charge (coulombs)

depletion charge in semiconductor substrate

inversion charge in semiconductor substrate

2 (coul. 1m )

(coul./m2)

~agnitude of Q. at the source-end of a MOSFET channel (coul./m2

) 1

surface state charge at the oxide-semiconductor inter­face (coul. In.2)

total electrostatic charge within the semiconductor substrate (coul./m2 )

t t ( OK) empera ure

t qate oxide thickness :m) ox

v m

'J eff

v

maximum (scattering limited) electron drift velocity in ~emiconductor material (m/sec)

effective electron drift velocity (m/sec)

electrostatic potential (volts)

Vb i bui 1 t- in potentia I of a semiconductor p-n junction (vol ts)

Vo applied drain-source biasing voltage (volts)

V G applied gate-source biasing vol tage (volts)

Vo magnitude of Vs at source-end of a MOSFET channel (volts)

V s

surface potential in a MOSFET (electrostatic potential at oxide-xemiconductor interface) -- (volts)

magnitude of Vo at the onset of channel pinch-off (volts)

VT gate threshold voltage from elementary MOSFET theory (volts)

VTE effective threshold voltage of a short-channel MOSFET (volts)

w width of a MOSFET (m)

Wo drain depletion layer width (m)

x distance from source towards drain, parallel to oxide­semiconductor interface (m)

144

x source/drain island junction depth (m) 1

,. 1

y. 1

\1,

o

distance from semiconductor surface into substrate, perpendicular to oxide-semiconductor interface (m)

magnitude of y at which n = n. -- (m) 1

1 -1 (kT/q)- (volts)

channel shortening (L-L ) associated with "pinch-off" or with channel terffiination (m)

permittivity of free space (8.854 x 10-12 farads/m)

, . rf'lative dielectric constant of gate oxide (dimensionless) 1

~ o

o

relative dielectric constant of semiconductor substrate

Kennedy model separation parameter -1

(volts )

(dimen­sionless;

magnitude of ~ at source-end of a MOSFET channel -1

(volts )

electron drift mobility in a MOSFET inversion layer (m 2/volt-s0cl

magnitude of ~ at low electric fields 2 (m /volt-sec)

electrostatic charge density within semiconductor sub­strate (m- J )

1, equi librium Fermi potential in semiconduct0r substra te (vo 1 ts) F

,1 electron quasi-Fermi potential in semiconductor substrate (vol ts ~ n

145

1. S.

Since

and,

APPENDIX A

Derivation 9f Solution Equations for

the Present Theory of MOSFET Operation

Herein we derive Egs. (1.1l3a) and (1.113b) from Section

r,. Derivation of Eg. (l.113a)

From Eg. (1.112a), we have

E -dVs/dx, xs

dE xs

<:fX

therefore,

dE xs

s

Eg. (A-l) can

A (V ) E s xs (:~ S1

= ,\ (V ) dV . s s

o (A-l)

be :-ewritten

::= 0 (A-2)

(A- 3)

Equation (A-3) can be integrated along the oxide-semi­

conductor interface, from the source end of the channel to

any arbitrary point within the channel:

E (V) V rs

5( __ 1 )dE =J s "(n)dn E F: xs

o xs V o

(A-4 )

The integral on the left-hand side of Eg. (A-4) can be evaluated

din: .. ctly giving

f I

)

E (V) xs s

E o

dE xs = log [E (V) IE 1 e xs s 0 i

.J

146

(A-S)

solving for E (V) yields x,; s

(fVS

E (V) = E c.<p < XS sOl

I V , 0 ,

Making the substit~tion E (V) xs s

( V

aV 'rs s -E eX~l-vo -dx -0

and, therefore

V

1 _fs dx = exp

E 0

V 0

Integrating Eg. (A-8) along

f1ce, from the source end of the

within the channel

x V r E, 1 ( S

, ,

ex I

Jv "'" exp [ E

X 0 ~V

0 0 0

we obtain: V t,

(1\- G)

-dV /dx in Eg. S

( A - 6) '1 i \',~ S

I q r)) dn \ (1\-7 )

I

I ;.. (r))dr) ( dV (A-8 )

S

I

)

the oxide-semiconductor in '.e r-

channel to any arbitrary point

l .\ (q) dr) i d~, (1\- g)

.J

( s (

, (OldC]dr 1 I

x := X exp , 0 E ) J

0 V L V

0 0

(A-l 0)

117

!"ur' :-,Iir:p] i(·it~'/ "i notation, we now define

"U)

[,

-J \(rlldn

V o

(i\ 1 1 )

Thus, substitutinq Ey. (A-ll) into Eg. (A-I0), we obtain

V

x x o 1

E o

rs

(A-I L)

w hi c h 1 S i de n tic a 1 to Eq. ( 1 . 11 3 a), g i v c: n inS 12 c t ion I. 5 .

B. Derivation of Eg. (l.l13b)

From Eq. (1.112b) , we have

d'O 3l dO, L 1 + (V ) 1 ,\ (V ) 0, 0 (A-13 ) I'

dV -dV

s kT J kT S 1 l S

S

Equation (A-I3) can be greatly simplified if ~e introduce the

substitution

gV /kT 0, (V ) = flV)e s

1 s S (A-l4)

Differentiating Eg. (A-14) with respect to V yields s

dQ df gV /kT ( q) qv /kT 1 s

\-_ f(V)e s (Iv- :=;

dV e + kT s s s

rj:'O, rj.'f qV /kT (---'l.)df qVs/kT )'. qv /kT - 1 s 3 " s e + 2 kT dV 12 + (kT f(Vs)e . clV

-, dV "

s s s

148

(A-IS)

(A-16 )

Thereby, substituting Eqs. (A- ] 4 ) - (A-16) into Ey. ( ,\ 1 3) I

dLf [ A (V s)

-+ ~ ! df

0 (/\-17)

dV 2 kT J

dV s s

An additional deg l ~e of simplification results if W(~

assume

df dV

5

-qVs/kT = g(V )e s

Differentiating Eq. (A-18) with respect to V yields s

=~ dV s

-qV /kT e s

(A-18)

(A-19 )

substituting Eqs. (A-18) and (A-19) into Eq. (A-17) yiclus

~~ + A. (V s) g (V s) = 0 (A-20) s

and, therefore,

~ = - A (V )dV g(V ) s s s

(A-21)

Equation (A-21) can be integrated along the oxide-semi­

conductor interface, from the source end of the channel to any

arbitrary point within the channel:

V -J SA 'n)dn (A-22 )

V o

149

!.'n11j.Jlinrl lhc int('qr.ll on the left-hand side of Efj. (1\-22),

',N( ~ 1,1 '/t ,

q(V ) s

~.9. -" q q(V )

o

(1\-23)

Thus, substituting Eq. (A-23) into Eg. (A-22l and, thereafter,

solving for g(V ) yields s

,(oldO ~ )

(

df 1 gV /kT g(V ) = -- e s

s dV s

and, therefore,

g(V ) o = (~~ s) .

V o

gV /kT e 0

(A-24 )

(A-25 )

(A-26 )

Thus, substituting Egs. (A-25) and (A-26) intv Eg. (A-24) and,

thereafter, solving for df/dV : s

df dV

s

150

exp -J" V

o

A (f])df] (A- 27 1

From Eg. (A-27), we have

).(rj)dll dV . s

(1\-28)

Integrating Eg. (A-28) along the oxide-semiconductor

interface, from the source end of the channel to any arbitrary

point within the channel

f (V )

, r s

J ( 1 gV /kT

df = ~~s ve a t ). (rll dll ide (A-29)

f (V ) o

we obtain

f (V ) = f (V ) s 0

From Eg.

o V

o

V

(;~sL gV /kT .Js + 0 e

V 0

(A-l4) , we have

-gV /kT f (V : - Q. (V ) e s

S 1 S

0

exp [-gf kT

and, therefore, defil;ing Qio ::: Qi

(Vo

) , we have

I

J

(

( I

J V

0

-gV /kT -gV /kT f (V ) = Q. (V ) e 0 Q e 0

o 1. 0 = io

.., i

,\ (ll)dr1id[. (A-30)

J

(A-3l)

(A- 32)

Differentiating Eg. (A-3l) with respect to V and, thereafter, s evaluating this derivative at V = V yields

S 0

(~) = [_L + (dO i) J -gV o/kT dV kT Qio dV e

S V s V

(A- 33)

o 0

151

Substituting Eqs. (A-31 ) - (A- 33) into

after,

Q. (V ) 1 S

solving for 0. (V ), we 1 s

=::

qV /kT J -qV /kT Q. C S Le 0 10

V s

("

I )

V o

r exp l ~-J kT

V o

obtain

[_L - kT

Eq. (A-30) and, there-

1 I dQ i \ 1 d\T) J °io s V 0

(A-34)

F ina lly, substi tuting Eq. (A-l1) into Eq. (A- 34), we obtain

Q. (V ) 1 S

qV /kT { -qV /kT s 0 = Q. e e

10

V s

_[9- _ kT

r .J exp [ - ~ + n(~)] v

o

1 lIdO. I,

11 __ I

dV I Sl V •

oj

(A-35)

which is identical to Eq. (l.ll3b), given in Section 1.5.

152

-~ .----

!·.PPENDIX B

Electric Current Continuity in th2

Prcs~nl Theory of MOSFET Operation

In tnis appendix, it is our aim to show that th0 present

MOSFET theory (Section 1.5) fulfills a requirement for contI­

nuity of source-drain electric current. We will show that

solutions [V (x) and O. (V)] that satisfy Eqs. (1.112a) and SIS

(1.112b) also satisfy this requirement, regardless of the

functional relation we may assume for the separation parameter

[, (V )]. s

In addition, we will show that A (V ) adjusts the s

proportions of drift current and diffusion current necessary

to maintain constant the total source-drain electrIc current.

From Eq. (1.107), we have

dID [ d:V dO. dV

(k~) d 2 0. 1 s 1 S __ 1 I (B-1 ) -- :-:: qW~ O. -- + dx . dx-dx I dx L dV' J s

By introducing the following substitutions, from the chain

rule of differentiation,

dO· dO. :dV ) 1 1 I S . dx == dV I dx

S

(B-2 )

d 2 0. dO. d 2 V d 2 0. (:~sr 1 1 S 1

dV + --dx 2

5 dx' dV 2

S

(B- 3)

Eq. (B-1 ) can be written

dID dO. 1 d 2 V r dO i d 2 0. dV qW

LWOi 0

1 5 qW 0 1 5

dx = - dV J + I LJ dV 5

-; dx j

S dx 2 dV 2

L 5

153

(B-4 )

fUlther, by the chain rule of differentiation,

and, therefore,

dV . ~I

dx I

dV s . dx

-1

) Substituting Lg. (8-4) into Eg. (8-6), we obtain

+ gW

d2

V s (dV s)-l

dx 2 dx

d2

Q. ] D __ 1

dV 2 S

(n-'j)

(8-(;)

(8-7)

From Egs. (l.112a) and (l.112b), respectively, the present

theory of MOSFET operation requires

d"V (dV s \' s A (V ) 0 - crx) = dx' s

d'Q. r 1 dQ.

l~; A (V s) Qi 1 (V ) :L ,_1 -- + -

dV 2 S riV s L J ;:

Equation (8-8) can be rewritten

(dV s 1 -1 =

dx I

dV \ (V ) __ s

s dx

(8-b)

0 = . (8-9)

(8-10)

Introducing Eg. (8-10) as a substitution in Eg. (8-7), we

obtain

154

diD dV d:Q. r .., dQ. I I

q\oJO s 1 + A (V ) L 1

dV-=: dx - ; dV J

\ s kT I dV I s L J s I.. s

(8-11)

I

Cledrly, in Eq. (B- 11) , thl' term in brackets . is identIcal

to the left-hand side of Eq. (8-9). Thus, we have the fo11ow1:1<:;

situation: if Q. (V ) and V (x) satisfy the requirements of 1 s, s

the present theory ~qs. (B-8) and (8-9)J dIO/dVs (and hence

dlo/dx) is zero -- regardless of the funct10nal relatIon we

assume for A(VS).

To show how >(V ) adjusts proportions of drift d:ld diffu­s sion components of I

D, we begin by a maT1ip'..!l::!tiun of E::q. (l.ll.lb):

.\ (V ) s

From Eq. ( B- 2) ,

I I

i--.9. : kT L

dQ. 1

dV s

2 I '. -1 ' d Q. I dQ.. ; __ 1 __ 1 I I dV 2 \ dV J !

=

s s J

dQ. 1

dx

dQ. 1 -- -dV s

1 -1

Q. 1 I

J

and, therefore, Eq. (8-12) can be rewritten

r d 2 Q. ( :~: r1

J I

.\ (V ) ~ -'1 _ 1 s LkT dV 2

s

(~T! d~~ [(k~) dQ.

dV 1 l. Qi d~ dx

155

(8-12)

(B-13 )

-1

(8-14)

~he total source-dr~ n electric current is given by

where

w

dO. 1

- dx - Q 1

dV 1 s I x I

J

is the dIffusion comfoonent of 10

; thus,

dO i [ '. k T) d<.. I - 1- --dx I q dx

dV s ] -1 - Q -

i dx

SubstitutInQ Eq. (8- .. 7) into ';"'q. (8-14) , we obta~n

and,

: (V ) s

theref0re,

. (V ) s

Idiff '"

D

Fi;'ally, substitutincj

',' ) -,\J S -

~d.l.;.f

o

- dLQ. l ~- 1

kT dV

d dV s

2

s (:~:) -1 ]

Eq. (8-13) into Eq (8-19),

r q d I dQ I/dx ) ] l~f - dV

s lOge \ dV s 7(f)(

(A-l S)

(8-16)

(8- 1 7)

(8-18 )

(8·1 q)

we have

(8-20)

Equation (8-20) implcitly shows how A(V_) adjusts the prnpor-::>

tions of drift and diffusion components of the total source­

drain electric current.

1')6