4‘”?—

 

 

Trans
develc
This s
the Sp
risetc
induce
exist;j
Conside
jUnczim

ABSTRACT

CURRENT SATURATION MECHANISM IN
FIE LD-EFFE CT TRANSISTORS

BY

Sukhbir Singh

It is proposed that current saturation in Field-Effect
Transistors is due to the presence of Space—charge layers which
develop inside the channel normal to the direction of current flow.
This space charge is induced by the gate field. It is shown that
the Spatial variation of the majority carriers in the channel gives
rise to High-Low junctions inside the channel region and that these
induced junctions are responsible for a large diffusion current to
exist inside the channel.

An idealized model of a junction, field-effect transistor is
considered. The two gates are assumed to be two abrupt p-n
junctions placed symmetrically with respect to the center of the
channel. It is also assumed that the drain and source contacts
are ohmic and that the source contact overlaps the gates. Only

the case in which a drain voltage is specified is considered.

 

[276

7.9 1'

Fun?

doubl

 

 

SUKHBIR SINGH

The device is analyzed in two Space variables with the
majority carriers as the sole current carriers. It is shown that
the drain current flow is restricted to a very narrow region around
the center of the channel and that the channel width never reduces to
zero--even for moderately large fields. Two electric double layers
are found to be present in the channel; these double layers are
responsible for the observed current saturation phenomenon.
Channel current control is found to be analogous to the control of
collector current by the collector field in a bipolar transistor.

The existence of a diffusion mechanism in the channel and the
absence of velocity saturation are two important conclusions.
Furthermore, it is concluded that the presence of an electric

double layer is a fundamental property of field -effect devices.

CURRENT SATURATION MECHANISM IN
FIE LD-EFFECT TRANSISTORS

BY

Sukhbir Singh

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOC TOR OF PHILOSOPHY

Department of Electrical Engineering
and Systems Science

1970

if.

45; 7.2 VI

ACKNOWLEDGEMENTS

The author wishes to express his gratitude to Dr. David
Fisher for providing excellent direction to the research reported
in this thesis. The author is also thankful to Dr. H. E. Koenig
for his interest in the author's doctoral studies in his capacity of
chairman of the department. Sincere thanks are due to Dr. Dubes,
Dr. Hedges, Dr. Bong Ho, and Dr. N. Hills for their guidance
and willing participation in the author's guidance committee.

I would also like to thank the National Science Foundation

for providing funds for computational work on CDC 6500 computer.

 

.4

 

TABLE OF CONTENTS

Chapter
1 INTRODUCTION
1.1 Background
1.2 Basic FET Amplifier
1.3 FET Characteristics
1.4 Objectives and Significance of present work

2 EXISTING THEORIES OF CURRENT SATURATION

NNNNN
m-hLNND—I

»a
E

CNQNLNCNUJLNLNUJM
OGDVO‘U'l-hUJND-J

4 THE

4.1
4.2
4.3

Shockley's Model

Shockley-Prim Modification
Kennedy's Model

Goldberg's Mbdel

Gradual Channel Nonlinear Mobility

PROPOSED CURRENT SATURATION MODEL

FET Geometry

Assumptions

Mathematical Model

Space Charge in the Channel
Mechanism.of Drain Current Saturation
Solution of Transport Equation
NUmerical Method

Solution of Poisson's Equation
Iterative Scheme

RESULTS

The Results of the Present Analysis
Advantages of Present Theory
Disadvantages Over the Other Theories

5 SUMMARY AND CONCLUSIONS

AP P END IX

REFERENCES

High-Low Junctions in Semiconductors

Page
11

ll
12
15
'17

20

21
27
30
33
35

37

38
38
40
42
54
58
61
66
68

71

72
91
92

93
95
97

LIST OF SYMBOLS

width of half channel

length of channel

permittivity of semiconductor

mobile electron density

intrinsic carrier density

mobile hole density

Spatial variables

electron mobility

hole mobility

diffusion constant for electrons
tlll'l'lmlun nmntuni l'ur lmlna

electric field

drain or channel current

electric current density due to electrons
electric current density due to holes
Boltzman‘s constant

Debye shielding distance or Debye length

concentration of impuritie s

LIST OF SYMBOLS (cont.)

electronic charge

charge density

dimensionless electrostatic potential
applied drain voltage

channel width

quasi-Fermi level for electrons
quasi-Fermi level for holes

ele ctrostatic potential

Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.

Fig.

10.
11.
12.
13.
14.
15.
16.
17.

18.

LIST OF ILLUSTRATIONS
Schematic diagram of an n-channel JFET
FET Amplifier; A pictorial representation
FET characteristics
JFET model; space charge regions
Gradual channel approximation
Channel length modulation by space charge
Small current amplification device
Impurity profile in a SCAD
Electrostatic potential variation in a p-n-p structure
JFET model of present work
Gate-channel-gate structure on a p-n-p device
Potential and Fermi level diagrams of a p-n junction
Effect of bias on transition region width
Computed values of transition layer width
Gate space charge in a symmetrical channel

Lateral field and potential variation in channel

Majority carrier density in a partially depleted channel

Conductive channel analogous to a p-n-p bipolar

transistor

Page
13

16
18

23

28
31

31
34

39
39
46
49

52
52
56

56

Fig.

19.

20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.

LIST OF ILLUSTRATIONS (cont.)

Coordinate system and device structure parameters
used in solution of Poisson's equation

Block diagram of iterative scheme

Electrostatic potential in the channel
Electrostatic potential in the channel

Electron density in the channel

Electron density in the channel

Channel width

Electron mobility in the channel

Electrostatic potential in the channel

Electron density in the channel

Electrostatic potential in the channel

Electron density in the channel

Potential in the channel for various channel widths
Electron density in the channel

Drain current versus drain voltage

Electron concentration in a high-low junction

1*)

67

70
76
77
78
79
80
81
82
83
84
85
86
87
90

i)()

CHAPTER 1

INTRODUCTION

1.1 Background

 

The Field-Effect Transistor (FET) is a unipolar device.
The electric current which flows through the device is due to the
majority carriers. Current control is obtained by changing the
dimensions of the channel and/or by changing the carrier con-
centration in the channel.

There are basically two types of field-effect transistors:
the Junction Field-Effect Transistors (JFET) and the Insulated
Gate Field-Effect Transistors (IGFET). The latter is sometimes
called the 'metal-oxide-semiconductor' (MOS) transistor. Their
principle of operation (current controlled by an electric field) is
very similar to the JFET; the primary difference being the manner
in which the control element is made. In the junction FET, re-
verse biased p-n junctions are used to change the dimensions of
the conducting channel. In the surface FET, both the carrier

concentration and dimensions of the channel are controlled by an

12

electric field at the surface. In this thesis, only the case of
JFET'S is considered.

Field-effect devices are attractive both as active and pas-
sive components because of the readiness with which they can be
integrated into large-scale monolithic circuitry. They are also
easier to manufacture by evaporation techniques and can be made
from less refined materials than can the bipolar transistors. The
most notable advantages of field-effect transistors are their high
input impedance and their 'self-biasing' feature. Absence of any
charge storage mechanisms makes them especially useful for

switching purpose 8 .

1. 2 Basic FET Amplifier

 

A field-effect transistor is essentially a voltage controlled
resistor. The 'resistor' consists of a region of n-type material
sandwiched between two regions of p-type material. In Figure.
1.1(a) the electrical contacts S and D at the two ends of the 're-
sistor' are called the Source and Drain respectively. The p-type
regions, called the Gate (31 and Gate (32, are generally connected
together. When the gates have a reverse or no bias with respect
to the source terminal, the Space charge of p-n junctions lies
partially in the n-type region. The dimensions of the conducting

region, therefore, depend upon the bias on the gates. The length

b of the conducting region between the two gates is called the

l3

 

U
WEN _,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    

 

 

 

<+++++++++++
_. b I
s {A I D
o n-Type Channel 2a (a) V =(); \I : O
u #1 .5 d g
F Cd
3 +++++++jf++ ;
m g\ NP

2 \_p-Type gate
V V
g _ d
+1" ‘6 ”IF
V
Id
0» V' > O ; V S 0
I) — d
S n VV n
+ +
+ + + + + +
+++++f+ ++ ++
s\ \\P
Gz

 

 

 

 

 

 

 

 

 

 

 

 

 

figmeldL

Schematic diagram of an n-channel junction FET under various
bias conditions. The approximate shape of space charge regions
in the channel is shown. (a) Zero bias on all electrodes;

(b) Negative gate-source voltage and positive drain source
voltage; (c) Saturation condition for negative gate voltage.

y

ij
1 "
l‘

4.1—1!

14

channel. The width, w, is smallest between the gates and there-
foredetermines the resistance of the channel. In an undepleted
channel, w : 2a, and the resistance is the lowest.

When a voltage is applied between the source and the drain,
current flows in the channel. Also the reverse bias on the gate-
channel junctions gradually increases from S to D and Space charge
expands into the channel as depicted in Figure 1. 1(b). The channel
has the largest width near the source end, where the voltage drop
is low, and has the smallest width near the drain end, where the
voltage drop is the largest. The p-n junction can thus be used to
control the dimensions and hence the resistance of the channel.

As the reverse bias on the gates increases, a condition is reached
where the space charge extends completely across the channel and
the width of the channel reduces to zero [see Figure 1. l(c)]. The
reverse bias needed to achieve this condition is called the Pinch-
Off voltage Vp. The pinch-off can also be achieved, with zero
gate voltage, by increasing the drain voltage till the channel width
becomes zero.

In typical devices the pinch-off voltage varies from one to
ten volts and the resistance of the channel may vary from as little
as 100 ohms with zero bias to as high as many Megohms near the
pinch-off condition.

The above explanation was in terms of an FET with an

n-type channel; FET's are also fabricated with p-type channel

15

and n-type gates. These two FET's are complementary.

To optimize the working of the device, it is so designed
that the Space charge of p-n junctions expands primarily into the
channel. This is done by choosing the channel to have a higher
resistance than the gates. Under these conditions the Space
charge lies entirely in the channel.

Figure 1. 2 shows schematically an FET as an amplifica-
tion device. Amplification action occurs because of the fact that
the width of the space charge depends upon vgs--the signal voltage.
A changing channel current causes a changing voltage drop across
R --the load resistance. Amplification is obtained because the

d

change in v S--the output voltage, can be many times larger than

d
the change in v .
gs

In normal operation of an FET the gates are always re-

verse biased. The reverse current of the gates, and hence the

power drawn from the source is negligible. The input impedance

is, therefore, very large.

1. 3 FET Characteristics

 

The important characteristics of the FET are shown in
Figure 1. 3. Figure 1. 3(a) is the drain or output characteristics.
At low drain-source voltage, for fixed bias, the current increases
in proportion to voltage. The portions of the curve where the

slope is not zero correspond to the condition in which the channel

 

16

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 1.2.

An FET Amplifier; A pictorial representation.

H E
I)
I . +\
+ +~+ + +
Rg +++ +++ 1
C32 +-+-+ + + (31 Q
+ A++ n ++“\
P++ ++p vds
+-+ +-+
+ +
+ +
v
gs
~v .1-
_:;HV
S -F ds
le - '
ll ‘ —v
V'
g

17

between the drain and source acts like a controlled resistor.

This region is sometimes referred to as the triode region. In
this region of operation, volt-ampere characteristics can be de-
rived from the Gradual Channel Approximation Model [1]. At
higher drain voltages the current begins to bend over indicating
an increased resistance of the channel. When Vd becomes greater
than Vp, the curve levels off and the current becomes essentially
independent of the drain voltage. The portion of the curve where

the slope is zero (V greater than Vp) correSponds to the condition

d
of pinch-off, and the current is said to be saturated. The tran-
sistor in this region is a voltage-controlled, constant-current
device. It is in this region that it is normally operated as an
amplifier. A. sharp increase in current at higher voltages indi-
cates the onset of avalanche breakdown. Figure 1. 3(b) illustrates
a typical family of output characteristics and shows the kind of
control exercised by the gate voltage on the drain current.

Figure 1. 3(c) is the gate-drain transfer characteristic
for the condition when the channel is pinched-off. The unique

shape of these characteristics makes FET'S very suitable as

'remote control' devices .

1.4 Objectives of the Present Work

 

In the above simplified model of the FET, the behavior of

the device, in terms of its drain current, is the same as that

18

I

I) d 111
Constant

/current ‘

(Saturation region} Triode I Vgs = 0 j

R egion

   
  

Saturation —.
region

 

«.— Constant
resistance
(Triode region)
1 l l

0 1.0 2.0(VDS/VP) 0 0.5 1.0 2.0 3.0 (v /vp)

 

 

(a) (b)

Figure 1.3. The output family of characteristics. (a) Variation
of drain current with drain voltage. (b) Drain

current versus drain voltage with different values
of gate bias.

 

 

p gs

Figure 1.3. (c) Drain voltage-gate bias transfer characteristics

for Vds = VP'

19

predicted by the Gradual Channel Approximation theory [1].
According to this theory the channel width decreases in propor~
tion to the square root of the drain voltage. Thus for drain
voltages equal to or greater than the pinch-off voltage, the channel
is pinched off and has a zero width--at least over some portion of
its length. In other words, the channel is incapable of sustaining
a flow of current. Similarly, at a fixed drain voltage, the space
charge regions of the gate junctions eXpand with increasing gate
bias and eventually extend completely across the channel. A
large negative bias is thus able to make the drain current zero.
This predicted value of drain current is contrary to the observed
effects in the actual devices, i. e., even for a pinched off channel
a large current continues to flow and is independent of the drain
voltage.

The present work endeavors to explain:
1. The phenomenon responsible for the saturation of drain

current.

2. The mechanism which tends to maintain the drain current at

a near constant value with drain voltage.

3. The non-zero slope of the drain-current versus drain-

voltage characteristics .

CHAPTER 2

EXISTING THEORIES OF CURRENT SATURATION

The analysis of the unipolar transistor was first accom-
plished by Shockley in 1952 [l]. The keypoint in his analysis was
the so-called 'Gradual Channel Approximation, ' whereby it was
assumed that the rate of change of channel width was much smaller
than the actual width of the channel. Space charge regions of the
gates were assumed to be completely depleted of the majority
carriers. Shockley further assumed that the transition from the
depleted to the neutral channel was abrupt, and, accordingly,
that electrical neutrality existed inside the channel. For drain
voltages larger than the pinch-off voltage, the channel was assumed
to be pinched off; any increase in drain voltage would be absorbed
by the expanding gate-channel boundary. Under these conditions
Shockley's model predicted the current to be confined to an
I infinitely thin plane near the center of the channel and that the
current was maintained at the saturation level as a result of the
channel length modulation by the drain voltage.

Numerous subsequent elaborations of Shockley's model
have been in the direction of interpreting the saturation

20

f.

21

phenomenon while staying within the Gradual Channel Approxima-
tion. The most significant extensions of Shockley's original theory
are: (1) carrier mobility dependence on electrical field [3-7],
(2) different geometries and impurity distributions, and (3) the
effect of temperature. and charge injection [8-11].

Kennedy [11] used the special geometry of the SCAD
(Small Current Amplification Device) to establish the presence of
High-Low junctions [12] in the channel. This prevents the chan-
nel from choking itself due to its own current. Kennedy has further
shown [13], by an extensive numerical analysis of the device,
that a high-low junction is also present in a uniformly d0ped
channel provided the mobility of the carriers is taken to be depen-
dent upon the electric field. By an approximate analytical solution
of Poisson's equation, Goldman [14] predicted the presence of an
inversion layer inside the channel. This inversion layer should
give rise to increased minority carriers which are responsible

for maintaining a large current even in the pinched off channel.

2.1 Shockley's Model

 

In an idealized p-n-p structure, two abrupt and reverse
biased p-n junctions are used to modulate the width of a uniformly
dOped n-type channel. If p-type regions are dOped much more
heavily than the n-type channel, the space charge regions can be

assumed to lie entirely inside the n-type channel. The Space charge

22

within the channel can be adequately described by Poisson's

equation:

2

a V a V
———Z + —-Z = - Q(x,y)/k, (1)
8x 3y

-3
where Q(x, y) is the charge density (coulombs - cm ) in the chan-
nel and k is the permittivity of the material.

When the drain and source are at the same potential,

Equation (1) reduces to one dimension,

7 = - -—— = -Q<y)/k. (2)

where E = - g1;- is the potential gradient. As shown in Figure 2.1(a),

the boundary conditions are

and

q and Nd are defined as the electronic charge and the donor im-

purity density in atoms - cm-3 respectively. Equation (2) is
readily solved and yields

N‘H

am... 141(2)
P

23

 

 

 

 

 

 

 

N
Y
a.——-—n
T I
O—————_._____._.________b _____ _ I O
Source space charge VD

0 W I _[d Drain

 

\ /
+/ j W 2a 1:3 n-channel n \/

n L
_ — j space charge
L
I.

 

 

 

 

 

 

 

Figure 2.1. JFET model; space charge regions are shown for the case
of zero drain voltage.

24

2
qua
2k

 

where VI) = is the voltage necessary to extend the Space
charge completely across the channel. Vp is called the pinch-off
voltage.

When the drain is raised to a positive potential, the current
flows and causes a potential change along the x-direction. Since
the gate electrodes are equipotential, the reverse voltage between
the gate and the channel varies with x; hence the channel width
varies. The channel slope, variation of w with x, can be obtained

by solving Equation (1). This equation is nonlinear and cannot be

solved conveniently in terms of known mathematical functions.
2

2
8x

dimensional approximation can be used for V(x, y), with the channel

 

However, if is very small compared to Q(x, y), then a one-

width still being described by Equation (4). The condition for

2V 2
2.. «2.1.

2 2
3x 3y

is termed as the Gradual Channel Approximation [1] . This approxi-
mation means that the voltage at any point in the channel is deter-
mined by the voltage on the gates only and is independent of the

drain voltage and drain current.

The ohmic flow of current in the channel is given by

Idx
0' 2 flat -d(V)]

 

dV

 

:}< ,
o- is the conductivity of the channel material.

25

Equations (4) and (5) are then combined to integrate V(x) with
re Spect to x. The constant of integration is so chosen that d = a

when x = O. The resulting shape of channel is

(iat- [eh- (5)1}
I 6 Z 3 ,

where s : d/a and

I = 8 u kV z/a2

o o p
Figure 2.1(b) shows the shape of the channel at pinch off. Varia-
tion of drain current with drain voltage, below the pinch-off
condition, is shown in Figure 2.1(c).

As the drain voltage is increased from zero volts, the
current increases linearly at first, and then finally tends to level
off as Vd becomes comparable to VP. For Vd greater than Vp,
the model is no longer adequate to describe the large and almost
constant flow of current in the channel.

The gradual channel model is quite adequate for drain
voltages less than the pinch-off voltage; however, it fails to explain
the behavior of the FET beyond pinch off. In this region (Vd
greater than VP) some of the approximations are no longer valid.

The most notable assumptions are:

 

I P z is the thickness of the channel and uo is the mobility of
the carriers.

d/a

Normalized Channel Width:

'0

I
p...

H

Normalized drain current :

H
.

OWC‘fiNON-PO‘GDC

26

 

 

 

 

 

 

 

 

_ Vg
_ IT I = 0
P
_ (b)
L j l l
0 ° 2 ~4 . 6 .8 1.0
Normalized Channel Length: x/b
1.0 _
V
lv—gl = 0
8 P
6 ,.
(C)
.4 "
.2 _
0 ‘ ‘ 1 1 1
.2 .4 .6 .8 1.0
Vd
Normalized drain voltage: V—
P

Figure 2.1. The gradual channel approximation; (b) shape of channel,
(c) drain current.

27

l. The longitudinal electric field due to the drain electrode and
carrier flow in the channel is ignored. For short devices
(b/a << 10) this cannot be justified; near the drain contact the

longitudinal field is large and comparable to the laterial field.

2. Equation (6) predicts that at pinch-off an infinite current den-
sity may exist. Under this condition the gradual channel approxi-
mation is badly violated. It can be shown that the gradual
channel approximation is not valid for distances less than a/ 2

from the drain end [2].

3. The assumption that abrupt termination of space charge on the
neutral channel exists, eliminates any mechanism by which a

space charge can terminate on a neutral semiconductor.

4. Due to the back biasing of the gates, Shockley's theory implies
that the space charge regions should completely penetrate
across the channel and reduce the drain current to zero. This,
however, does not occur in practice. Thus field-effect theory,

based on complete depletion, is not correct.

5. Near the drain end, where extremely large fields are pn-mlic-Iml,
no account is taken of the nonlineurity ol’ Iln- lnulullly “I' II”-

charge car riers.

6. Because of the large amount of the charge present in the channel,
the mathematical analysis should be by the use of Poisson's

equation rather than the Laplace equation.

2. 2 Shockley-Prim Modification

 

Shockley and Prim [1-2] extended the above theory further
by including the effect of expansion of the depletion region towards the
source contact. When the drain voltage is larger than the pinch-off

voltage, the depletion eXpands as shown in Figure 2. 2. The point

I:

 

28

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I) y
b
GATE
=””1“”H”IHHHHHIHHHIHHHHHWHHIHHHHHHII”WNW“”lllmliiiim”””“lmim”
s a #— Xa vs
0 E l W \\ % wD: X
s 1 /' \ e
8 E a EXPOP E 7
m E E :4
8 ? E i
to ”Hill!!!” “IHIHHHHHI”IHIHHHHHHHI”HIHHHIHIHHH HIHHHHHIHH “HIHHHHHIHHHH Q
GATE
.— a/Z ‘1
x : 0

Figure 2.2. Channel length modulation by gate space charge.

 

29

x : O, the extrapolated pinchuoff point (expop), is defined as the
point where 3 would reduce to zero if the channel were extrapolated
beyond the range of its validity.

On the right of the expop the electrons, instead of flowing
over a finite width of the channel, can be assumed to be constrained
to the y = 0 plane. This shift does not effect the total potential
much because the field of the electrons is much smaller than the
field due to the gate Space charge and the drain electrode voltage.
The only field responsible for moving the carriers is Ex--the field
due to the drain electrode. Towards the right of the expop and at
the y :: 0 plane, the charge due to the electrons is so small that

the channel can best be described by the Laplace equation

 

2 2
8 V 3 V
2 + 2 ~ 0 . (7)
8x 8y
solution of Equation (7) is matched at x = -a/ 2 to the field due to
the electrons [Z] to give the width of the depletion region xd; one
finds that
a 2 I
x = -— 1n(V -V )( o) - (0.65-0.641nf) ,
d 11' d p 2
IVp

where f is the scaling factor used to match the solution of Equation
(7) and Equation (6). The value of f has been estimated [1-2] to be

about 0. 254.

30

l‘here are several flaws in this theory: The electric
fields encountered in this formulation are within the range of non-
linear carrier mobilities and yet the nonlinearities were not taken
into account. Also, the model does not give a satisfactory explana-
tion of the manner in which the current flows through the space
charge region of the gate channel. The saturation conductance of
the channel is zero and is far less than the measured value in the

actual transistor S .

2. 3 Kennedy's Model

 

In his report on the 'mathematical investigation of semi-
conductor devices' [11] Kennedy explains very successfully the
current saturation in a SCAD (Small Current Amplification Device).
Figure 2. 3 shows the geometry of the SCAD under investigation.

The channel lies between Specially shaped p-type gates symmetri-
cally placed with respect to the center of the channel. The source
contact is diffused with an impurity gradient of the type shown in
Figure 2.4. The gates are so shaped that the channel has a narrow
neck in the region where the impurity gradient changes rapidly. The
two dimensional Poisson's equation was solved numerically to obtain
the charge density in the channel. A High-Low junction [12] was
shown to be present at the neck of the channel where the depletion
layer theory would predict a channel pinch-off. The High-Low

junction arises as a consequence of the spatial distribution of the

31

SOURCE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Gate 3_,_ M Gate

junction ___ junction
p-type 4-!- p'type

 

8‘ \
\

n-type channel

Figure 2.3. Small Current Amplification Device [11].

l

atoms/cm

21

19

 

 

 

Figure 2.4. Impurity profile at the center of the SCAD.

32

majority carriers to form an accumulation layer adjacent to the
depletion layer. The space charge of this accurnulation-depletion
layer is equivalent to an electric double layer.

With appropriate biasing voltages, the expansion of gate-
channel depletion layers into the channel is blocked by the presence
of the electric double layers and hence the channel width cannot be
completely reduced to zero. With certain dimensions of the channel
the accumulation layer seems to enhance the electric conduction.
Various combinations of channel dimensions and bias voltages were
tried to confirm that it is an electric double layer that is responsi-
ble for the current saturation.

Kennedy proposes the presence of such a double layer, as
the cause of current saturation, in all field-effect devices. In
uniformly doped channels, it is further proposed, that an electric
double layer is present in the drain current path, because of the
redistribution of the voltage, after the pinch off. For a near zero
drain current, the charge in the channel, obtained by the integration
of the longitudinal field, consists of an equal number of positive and
negative charges. Due to the partial depletion of the channel, posi-
tive charges are balanced by an adjacent layer of accumulated
negative charges. This, Kennedy surmises, gives rise to an
electric double layer. However, a near zero value of drain current
is necessary for the above hypothesis to hold. This is not always

found to be the case.

‘
I
a . we 'w -

 

33

2.4 Goldberg's Model

 

Goldberg [l4] predicts the presence of an inversion layer
in the channel as a result of an approximate solution of one-dimen-
sional Poisson's equation. For a p-n-p structure, if the width of
the n region is made extremely small--either through metallurgy
or by reverse biasing the p-n junctions, then an inversion layer is
shown to be present in the n region.

As Shown in Figure 2. 5, U is the bulk electrostatic poten—

b
tial, in equilibrium, in the n region. AS the width of the n region
is reduced U passes through a maximum in passing from one p
region to the other. If the n region is made so narrow that U never
reaches its bulk value then the majority carrier density reduces at
this point. The corresponding minority carrier density increases,
Goldberg proposes, thereby giving rise to an inversion layer.

However, it is well known that with such narrow junctions,
the Fermi levels can be assumed to be constant and continuous
across the p-n junction and hence, before an inversion can take
place, the junction is completely depleted.

Actually an inversion layer, in the channel, is not needed to
cause a saturation. Even a little depletion, at any point in the

channel in the path of current, can give rise to a high-low junction

and can thus modify the current flow.

Positive
‘ potential

 

in

34

 

Id" T A II
I
,1" “b \

 

 

_-——

 

 

l

l

T

l

I /
T

l

l
I”! IN
I I I |
l l I |

 

 

 

 

 

 

 

 

 

Figure 2.5.

 

 

Electrostatic potential variation in a p—n-p structure;
(a) wide n-region, (b) narrow n-region.

35

2. 5 Gradual Channel Nonlinear Mobility

 

Various attempts have been made to explain the saturation
phenomenon by incorporating into the above model the phenomenon
of field dependence of the charge carrier mobility. For field values
as large as those near the drain contact, on the basis of gradual
channel model, the carrier mobilities are in the nonlinear range.

From experimental data there are two critical fields E and EC

cl 2

with E less than E . For fields below E , the mobility is
c1 c2 cl

constant. For fields between Ec and EC the mobility is propor-

1 2

tional to the square root of the field. For fields above E 2' the
c

mobility is inversely proportional to the field. For Silicon the

values of these fields are approximately:

n-type p-type
EC1 2.5 KV/cm 7 KV/cm
Ecz 15.0 KV/cm 20 KV/cm

Reduced mobility at increased fields requires still larger
fields to maintain the current at the same level and the velocity
quickly saturates to its maximum value. With the velocity of
carriers saturated at its maximum value, Grebne and Ghandhi [4]
postulate a minimum residual thickness of the channel near the
pinch-off point in order to estimate the longitudinal field. This field
is then matched with the field obtained from Shockley-Prim formu-

lation; the result is the saturated drain current.

36

Dacey and Ross [8-9] considered the effect of temperature
on the mobilities to explain the output conductance of the field-
effect transistor.

All the above modifications start with the gradual channel
approximation as the basic model; therefore, the explanation of
saturation mechanism is avoided. At pinch off almost all the drain
voltage drop is across the depleted region; also, in Spite of the
large space charge present in the channel, the Space charge current

is ignored.

CHAPTER 3

THE PROPOSED CURRENT SATURATION MODEL

The works of Shockley [l ], Sah [3] and Ghandhi [4] assume
that the transition between the conductive channel and the depletion
region takes place abruptly, and accordingly, electric neutrality
exists in the channel. This assumption is correct for channel
widths much larger than the transition region width which is of the
order of 5 to 10 Extrinsic Debye Lengths in Silicon [11]. With a
uniform impurity doping of the order of 1015 atoms/cm3 the ex-
trinsic Debye length in silicon is of the order of 4 x 10"2 microns
and with no voltage applied, the depletion layer of a p-n junction,
has a width of the order of 0.8 microns; that is, about 20 Debye
lengths. Thus with channel widths of the order of two microns the
usual assumption of abrupt termination of Space charge on the
neutral semiconductor is no longer valid. Again under normal
operation of the field-effect transistor, when the channel is near
'pinched off," the transition regions occupy nearly the whole width
of the channel.

With the drain voltage near the pinch-off value and the
channel narrowed at the drain end of the gates, the width of

37

38

transition regions becomes comparable to the residual width of the
channel. When the two transition regions meet at the center of the
channel, the channel passes from its neutral state to quasi-neutral
state. This partial depletion of the channel gives rise to a Spatial
variation of the majority carriers. As a consequence, double
layers are formed normal to the current flow; these “modify the
current in a manner similar to the emitting and collecting junctions

of a bipolar transistor.

3.1 Geometry of the Model

 

The device is schematically represented by two abrupt p-n
junctions placed symmetrically with respect to the center of the
channel [See Figure 3.1]. Uniform impurity doping of Na atoms/
cm3 in the gates and of Nd atoms/cm3 in the channel is assumed.
The source and drain contacts are placed at x = O and x = b res-
pectively. The gate metallurgical junctions M1 and M2 are placed
at y = 0 and y = 2a respectively. In order to simplify the mathe-
matical formulation, the gate contacts are taken to be overlapping
the source contact. Also, in order to make the analysis tractable

several assumptions are made. These are outlined in the next

section.

3. 2 Assumptions

 

l. The gates are metallic (degenerate) so that the depletion region

is entirely in the channel, i. e., Na >> Nd.

39

 

 

 

 

 

 

 

 

 

  

fl"

 

M

Figure 3.1. JFET model.

 

 

 

v.

1

G../////e//////aa,

Figure 3.2. Section at xx' of Figure 3.1.

3.

40

The potential in the absence of applied gate voltage at the
metallurgical junctions (at y = 0 and y : 2a) is taken to be as

zero. This also is taken as the boundary condition.

The donor impurity atoms in the channel are completely ionized.
Also, the Space charge region of the gate channel junctions is

completely depleted of the majority carriers.

The channel current is only due to the majority carriers. The
recombination and generation in the gate transition region and

the channel is ignored.

The current flow in the channel is entirely due to the longitudi-

nal component of the field, such that,

Ex(x: Y) : Ex(X).

The reverse current of the gate junctions is neglected.

The boundary between the Space charge of the gate and the
channel is taken as abrupt. Also, the charge density in the
channel, at right angles to the current flow, is approximated
to have. a constant value. which will give the same current as
would be obtained by the. integration o." current density in the

actual channel.

The two cases of majority carrier mobility are considered:
(1) mobility independent of the electric field, and (2) mobility

related to the electric field as formulated in Section 2. 5.

Mathematical Model

 

The analysis of a field-effect transistor involves the know-

ledge of the majority carrier distribution. This is obtained by

solving Simultaneously Poisson's equation, the Continuity equation,

and the Transport equation; subject to constraints imposed by

41

the geometrical configuration of the device and the external bias
voltages.

In this work, a simplified form of the current transport
equation, combined with the Poisson's equation and the continuity
equation for the majority carriers, is taken as the mathematical
description of the device.

Equation (1) is the Poisson's equation which relates the
divergence of the electric field [E(x, y) = - grad V(x, y)] to the
charge density arising out of the mobile charge carriers, p(x, y)
and n(x, y) (holes and electrons, respectively) and immobile im-
purity atoms N(x,y). Equations (2) and (3) are the Transport equa-
tions and express the current density, in terms of electron and hole
concentration, mobilities un and up and diffusion constants Dn and
DP. Equations (4) and (5) are the Continuity equations and are in
terms of the current densities due to electrons and holes; RH and

R are recombination and generation currents. Equation (6) is the

total current.

div grad Maw) = - E [N(x. y) - n(x.y) + ptx.y)] (1)
JP<X.Y) = -qu grad p(X.Y) + quth.y) p(X. y) grad ‘i‘(X.Y)
(Z)
Jn(x, y) = an grad n(x, y) + q un(x,y) n(x, y) grad \Jy(x, y)

(3)

I
I
:U

div Inesy) (4)

l
I
71

42

Jttxw) = Inesv) + Jp(X.Y) (6)

In addition to the above, Einstein's equations, are assumed valid:

Dn : Vt un(x, y)
D = V u (x, )
p t p y
where Vt is the thermal potential (Vt = 55-; T is the absolute

temperature of the device).

The analytical solution of these equations in terms of known
mathematical functions is a difficult task; therefore, some approxi-
mation techniques are employed.

Since the field-effect transistor is a majority carrier device,
the contribution, to the total current, of the minority carriers (in
this case holes) is negligible. Therefore, this set of equations is
further simplified by ignoring the presence of minority carriers
and the absence of recombination and generation. The system then

reduces to:

. div grad ¢(x,y) : - "E [N(x, y) - n(x, y)] (7)

Jn(x, y) = q Dn grad n(x, y) + q un(x, y) n(x, y) grad Mx, y)
(8)

div Jn(x,y) = 0 (9)
Equations (8) and (9) can be further simplified to one-dimensional
form as is explained in the following sections.

3. 4 Space Charis in the Channel.

 

Along any x-plane the gates and the channel form a p-n-p

structure (see Figure 3. 2). For a sufficiently wide n-region,

43

space charge effects at M1 and M2 do not cause a disturbance at C;
hence, the two junctions can be analyzed independent of each
other. However, for a narrow n-region, Space charge at M1 is

not independent of that at M therefore, the boundary condition

23
at C must be chosen very carefully.
Consider the case when the drain and source are at the

same potential so that a one-dimensional analysis can be under-

taken. The junction at M1 is described by the Poissons's equation

2
d .
--‘:,'f- = -E [N(y) +p(y) - um] (10)
dv
Nd , 0 E y 3 2a,
where N(y) =
Na 3 Y E 0'

and Nd and Na are the Donor and Acceptor impurity density levels.

Since the compensation is not relevant to our analysis, we will

assume Na .—. 0 in the n-region and Nd :- 0 in the p-region.
Introducing the normalized potential U m]: we haw: m the
n-region
U = 1n n n.
n ( n/ 1t
and
U = 1n n n,
p < p/ 1).

where o, = l/Vt .
In the n-region,

slit
dv

-n+Nd]

N

.9.
-k[p
__q_

k

[ni up (-0) - “1 exth) + Nd].

44

or alternately,

 

 

2 2n qz N
dU : i exp(U)-eXP(-u) - __(_i_
2 KTk 2 Zn.
dy 1
l Nd
= —' [Sinh(U) - —“ . (12)
L2 2n,
d 1
KTk . . . . . .
where Ld : is the 1ntr1n51c Debye Length (33. 8 microns in
Zn.q
Silicon). 1

For large distances from the junctions (neutral region) the

charge density from Equation (12) is zero and we can write

Nd
Slnh(Ub) = 3;."

9

1

where Ub is the height of the potential barrier. In the quasi-

neutral region, the potential can be writen as U = U + 6 , where

b

6 is the small deviation from the equilibrium potential. Equation

(12) can then be reduced to the linear form as

———= —— (13)

where L is defined as the Extrinsic Debye Length, and

de
[ KTk ] ”Z
L :
de

2 ‘ 2
2q ni J1 + (Nd/Zni)

 

 

In a reasonably intrinsic material, Lde reduces to

I"d
Lde = N /2 )2
( d “i

 

45

Equation (13) shows that in a region where the departure
from neutrality is small, the potential varies exponentially with
position.

If the potential were exponentially dependent upon position,
for large distances from neutrality, we would expect a distance of
5 to 6 Debye Lengths in order to change from complete neutrality
to complete depletion.

Similarly, in the p-region,

2 N
d
J = -—1—— [Sinh(U) +4] (14)
2 2 2n.
dy Ld 1

Equations (12) and (14) can be integrated once under the conditions

 

 

 

sill _ ill _
dy = -m- dy ya... - 0 (Ma)
Ul = U and U = U (15b)
y: +00 n y: —m p
to yield
N
(g[-1 2 [coshU-coshU -(U-U )——d- (16a)
dy Z n n 2n.
L 1
d
2 N
(it-J- = -2—[coshU-coshU +(U-U )A] . (16b)
dy L2 p p 2ni
d

Under reverse bias conditions, the Fermi levels (hp and ¢n
are assumed constant and parallel and are separated by a potential
V across the junction (Figure 3. 3). In effect, this assumption
means that in the transition regions the spatial variation of n and p

is dependent only on LP. This will result in the free carriers being

 
 
 
 
 

Transition
region

 

Figure 3.3.

 

46

\

Metallur gical Junction

 

t\/
l"

p-region

transition
region (a)

 
 
 
  

carriers density

1 atoms/cm

 

(b)

/\
l“...

 

Electrostatic potential variation across a p-n junction;
(3.) reverse biased junction, (b) carrier density levels

in the space charge regions.

47

slightly underestimated. However, in these regions where'they
are minority carriers, they do not contribute significantly to the

total space charge .

¢ + Cb
If an average Fermi level 4) = -—-2—2 is defined, then the
new carrier concentration is
n = ni exp (Un)
p = niexm-Un +aV). (17)
and Equation (16a) becomes
2
dU 2.
(Ti— : ——2— e)\:p(----zY-)[cosh((U)-coshUn +—)]
Y L
d N
+--‘-i—(Un POLE-U) . (18)

ni

Equation (18) is correct as long as condition (17) is valid.
However, Equation (17) does not describe actual strength of the

holes. This, as assumed earlier, does not cause any significant

error because even in the bulk region, the concentration of holes

is many orders of magnitude less than the electron concentration.

Let (U - o. V/Z) = Ud. The the field in the transition region

is
dU 1 Nd ] 1/2
a? : EE-[exP (U) +exp-(Ud +QV) - Zni Ud (19)

Even for moderately low negative bias (U is negative in most of

d

the Space charge region), the term exp -(Ud + (1V) can be ignored,

and hence, the field in the transition region is

48

dU N I”

l d
“g L [exp(Ud)+--r-1-'—Ud (20)
d 1
Thus, the width of the transition region (see Figure 3.4) is indepen-
dent of the bias. Also, the Spatial variation of the electrostatic
potential remains the same and is more or less logarithmic.

The boundary conditions at the junction between the n and

p regions are

U' = UI = U
y:-0 Y=+O O
and
d d
.9- - .—U
Yy3-O Y Y=+O

 

 

Combining Equations (15a), (15b) and (15c), the potential at the
junction is

.. T
cos h(UP) cos h(Un) + (Nd Ln + N UP)/ 2ni

 

 

A
U = (21)
2
0 (ND + NA)fni
for an unsymmetrical junction.
For an infinitely thin n-region, the boundary condition for
the space charge is defined by
z—U = 0
Y Y = + 00
and
Q(Y)‘x a C = Nd - ni exp (Un) +ni eXp (-Un +aV) (23)

without a serious error. The corresponding equation in the p side

is

49

 

 

 

 

 

 

 

 

Transition
re/gion f
it I I _
n Un U _Uno '
¢n —_ — ‘_ — ‘4’ — Ud = O
o. V/Z l l
U20 1 JUd = -(1 U/Z
I ' . ct) = 0
1 I
L 1‘ ‘ ‘ “a
l _| *4”
P
l | 1 W
I y 2 a l
Depletion
region
Figure 3.4. Effect of bias on the width of a transition region

(Equation 29).

50

Z 2 V
(d—U) = -— exp (- -a—~)[coshU - cos hU
dy L2 2 p
d N aV
4-2;:(U +—Z—- Up) (24)

1

Note: The far side in the p-region is the gate contact where q)
n
and ¢ coincide.
P

Variation of bulk potential U in the p-region can be shown

b
to be defined by the equation

 

N N
- 2
coshU - e°V/ .coshm 9% .im .21) .41“, _2_V_
p n 2 Zni n 2 Zni p 2
U = .(25)
b N
(Ens
2n. 2n.
1 1

The computed values of U are sketched in Figure 3. 5 for various

b
bias voltages.

For junction, field-effect transistors the electrostatic
potential gradient passes through zero at the center of the channel.
We always have a potential maxima at this point. If the. channel
width is reduced either through metallurgy or by increased reverse
bias, then, for a fixed distance between the gate junctions, the
potential maxima is lowered. Figure 3.6 depicts modified potential
at the channel center. The corresponding majority carrier density
is now lower than its value in the neutral region. The lowered
potential maxima in the channel gives rise to two interesting
situations:

1. For an extremely narrow n-region and the junctions in thermal

. . . Z . .
equilibrium (pn = ni ), the electrostatic potential meets the

(J;

 

51

iuob

Eocmm uh

OOZvCAmo <>rcmm on mrmoqmomdbjo uO4m2i>r

wvbom Olbmom meOZ on b biz cczojoz wow

<>m_ocm m_>m <Ord>omm .

Zbamabrn 9.1502

0020» omzmi. 2a" _.m x 5.... oz;

$3.56 omm<m 523:. {ouromm x as o:

.2 41m

 

 

 

 

 

Imob 1.0.0 IO.O

ZMdbrrchODP LCZOjOZ
IT ZOmEDCNmD toamzjbr C

 

_o.o

l5.0

I0.0

5.0

0.0

IN EXTRINSIC DEBYE LENGTHS

DISTANCE

52

 

 

max no

 

U :0 I

d. d \ \ (pn

eXpan ing \ / \ / expanding
space charge Space charge

\
W \‘ i

 

 

 

 

Figure 3.6. Gate space charge in a symmetrical channel.

 

 

\
Gate Gae
Lateral Lateral .
electro , j 7 electrostatic
s tati c \ 2 V ,__ 5 fie ld
Pote nti a1 ' " “ "’ E

 

conductive
channel
width

 

Figure 3.7. ‘Variation of electrostatic potential and field normal
to the flow of channel current.

S3

Fermi level and the channel becomes intrinsic. For an n-
region width of the order of O. 1 micron in Silicon, the electron
density falls below the equilibrium value at the cost of increased
hole concentration. This gives rise to an inversion layer in
the channel.

2. In case of reverse bias on the gates the hole electron product
is given by up : niZ [exp -(aV)]. For narrow n regions, when
the electrostatic potential meets the Fermi level, the density
of holes is many orders of magnitude lower than the electron
concentration; hence, before inversion can occur, the channel
is already depleted. Thus, in the case of an FET, the inver-
sion is not the cause of current saturation, as is predicted by
Goldberg.

Again, as shown by Equation (20) and Figure 3. 5, the deple-
tion layer does not terminate abruptly on the neutral region and the
depletion of channel is a continuous process. The channel current
instead of suddenly going to zero, as predicted by complete deple-
tion theory, is now governed by the behavior of the space charge
of the partially depleted channel. The nature of this space charge
is investigated in the next section.

Figure 3. 7 shows quantitatively the variation of lateral
field in a partially depleted channel. The field is of such a polarity

that the electrons are not swept away by the gates but are rather

54

restricted to a narrow region around the center of the channel.
The drain current is still confined to the central part of the channel
where the conductivity is the highest. Away from the center, the
electron density decreases almost exponentially, and for all
practical purposes, it can be taken as constant at some effective
value neff over the width of the channel. For the rest of the
analysis, the lateral variation in the electron density and the
electrostatic potential is ignored. Thus, the Equations (8) and (9)
reduce to a one-dimensional form:

Jn(x) : an grad n(x) + q un(x)n(x) grad LIJ(X) (Z6)
and

div Jn(x) : 0 . (27)
Equations (26) and (27) along with Equation (7) describe completely

the behavior of the FET.

3. 5 Mechanism of Drain Current Saturation.

 

With increasing reverse bias on the gates (either through.
applied voltage at the gates or as a consequence of the drain
current flow), the neutral channel narrows to such an extent that
its width is comparable to the width of the transition regions of the
Space charge of the gate channel junctions. The transition region
of one space charge instead of terminating on the neutral channel
now encounters the transition region of the second Space charge as
a boundary. A further increase in the reverse bias results in the

'partial depletion' (section 3. 4) of the channel at its narrowest

55

point (electron density falls below its equilibrium value, but is

larger than the intrinsic carrier density). The uncompensated

impurity centers at this point give rise to a positive space charge.

The conductive channel is no longer neutral and has a Spatial

variation of the majority carriers, at the y : 0 plane, as depicted

in Figure 3. 8.

The positive Space charge in the cmductive channel modifies
the flow of the drain current in a very significant manner. For a
qualitative explanation the following terms are defined:

1. 'Gradual Channel' - The part of the neutral conductive channel
between the source and the 'pinched-off' region (S - A). The
current flow and the dimensions of this part of the channel are
described by Shockley's gradual channel approximation [1].

2. 'Pinched-off Region' - That part of the channel where the
carrier density is below its equilibrium value.

3. n- - The carrier density in the pinched-off region. Its value
is solely determined by the bias on the gate junction.

Because of its higher resistivity the pinched-off part of the
channel absorbes a larger part of the drain voltage than would the
same length of neutral channel. The carriers are now subjected to
an increased drain field and move with a greater velocity.

The current flow in the gradual channel is purely due to
drift field and is governed by Ohm's law. However, in the 'pinched-
off' part of the channel, the current has a large diffusion component.

The diffusion density gradient in the region AB acts like an injecting

56

V

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4-“ g Gate
lllllllllllllllllllllllllllfialllllllllll G) G
.(
e 9 9 9369 ed as o G
Source 5 m D Drain
\/ ’ @—@—~
@ 9 Q g
9 9 e
V lllllllllllllllllllllllll llllllll llllllllllllllIlllllllllllllllllllllllllllll
Li-LE’ Gate
4::
—B\_C/c
0 t x
LE

Figure 3.8. Majority carrier density in a "partially" depleted

channel for Vd Vb.

reverse
biased

forward
biased

Gr adual

———t

 

 

 

 

 

Figure 3.9. Conductive channel analogous to a n-p-n bipolar transistor.

S7

junction. The injection is further aided by the drain field which

'forward biases' this junction. In the region CD the diffusion den--

sity gradient is in the opposite direction and impedes the flow of

carriers towards the drain contact. The drain contact field acts
as a reverse bias on this 'junction'. Thus, the carriers 'injected'
by the forward biased 'junction' AB are accelerated by the field in
the region BC and are then 'Swept away' by the 'reverse biased

junction' CD. The current is thus analogous to that of a n-p-n

transistor as pictured in Figure 3. 9.

The three salient characteristics of the device at this

point are:

1. An increase in drain voltage is absorbed, in most part, across
the 'junction' CD. The control of the drain voltage on the
current is thus reduced.

2. The current 'injected' by the 'junction' AB is determined by
its diffusion potential and is lirriited by the resistance of the
gradual channel. This resistance is a function of its length,
Vg and V . The 'injected' current, therefore, stays constant,
for a fixed V and with changing drain voltage.

3. The redistribution of majority carriers in the channel gives
rise to two High-Low junctions (Appendix A). The high-low
junction at AB causes a voltage minima and therefore a zero

field at point E. This is equivalent to a virtual cathode being

58

present at E, and the device behaves as a temperature satura-
ted cathode current device.

1"or Vd > Vp the length of 'pinched-off' channel (n- region)
increases because of expanding Space charge towards the source
contact. This effects the current in the following ways:

1. Increased resistance of the gradual channel tends to reduce the
injected current.

2. Current increases due to the increased field across the
'pinched-off’ channel.

3. A virtual cathode at E and the Space-charge density in the
'pinched-off' region are precisely the conditions for a Space
charge limited current (the current is proportional to E3/Z[2].
However, any increase in the field also increases the length of
the n- region. The net effect is a slight increase in the drain

current with drain voltage.

The current saturation in the above model is thus a conse-
quence of the Space charge in the channel. The mobility dependence
on the field and space charge limited current effects only appear
as second order effects. The gates control the current by reducing
the dimensions of the gradual channel and thus decreasing the in-

jected current available to the 'junction' AB.

3. 6 Solution of Transport Equation.

 

If the transport equation (26) is combined with equation (27),

we obtain

59

dw [Jn(x)] 0

din [q Un(x)n(x) grad 111(x) + an grad n(x)] .
(28)

Substituting for n(x) from the Boltzman's equation

n(X) = r1i eXPuW - ct) .
we get

div [qUn(x)n(x) grad ¢(x)] = 0.
The drain current at any section of the channel is constant and is
obtained by the integration of Equation (29) over the width of the
conductive channel. The drain current is

.W(X)

I = 23 Jn(x)dy
o
.W(X)
: 2‘8 qUn(x)n(x) grad ¢(x) dy
o
.W(X)
: Zq ni grad ¢(x) exp [-n¢(x)]\ Un(x) exp [atll(x)] rly
Z q ”i grad (Mir) eXp [— ut()(x)] V(x) , (Ht)
W(X)
where F(x) = 5 Un(x) exp [a41(x)] dy (31)
o

Rewriting Equation (30) in terms of ¢(x) and integrating with respect

to x yields,

 

 

I
exp [-a¢(x)] = 2:11.5‘ F(X) . + C ,
1

where C is the integration constant. Taking the limits of inte-
n

gration as x and b and expressing cn in terms of the boundary

60

conditions at the point b we have

 

 

I d
est-aw] = nieXP[-%(b)] +2; (x F;
i

Substituting from the Boltzman's equation the values of ¢(x) and

¢(b) the electron density in terms of the applied drain voltage is

given by

 

 

b .
I d
n(x) = exp[a¢(x)] [n(b) exp [-a¢<b>] Mahdi F03] (32)

If the electron density is evaluated at the x : 0, then from Equation
(32) we obtain an expression for the drain current as a function of
the applied drain voltage and the boundary conditions for the elec-

tron density:

ZqN
I : d 11(0) ‘ n(b) exp [’04) (b)] . (33)

(1 Sb dx
F(x)
0

The only approximation in the solution of the transport equation

 

 

 

was in ignoring (section 3. 5) the lateral component of current in
the transition regions between the gate and the channel.

The set of two boundary conditions n(O) and n(b) at the
external contacts 0 and b is in general given by the relationships
involving the currents at the external contacts. Assuming the
simplest form of the contacts of the ohmic type, the boundary

conditions are:

61

and (32.1)

n(b)

ll
:3

where nN is the electron equilibrium density at the external con-
tacts of the n-type material. An equivalent definition requires

charge neutrality at the contacts:

i-mLE

n(O) - p(O ) - Nd(0) : 0
and

n(b) -p<b> - N (b) = o.

d

 

Since the Fermi levels for the holes and electrons coincide at the

l
\1

2

contacts, the hole electron product is given by np = n, and hence
i

the boundary conditions can be written in terms of the doping

concentrations as:

 

 

 

N (O) 2 N (0)
n(0)=nN=/ d2 +1 + d2
for N(O) :0 ,
where n(b) = nN. (32.2)

3. 7 Numerical Method

 

In order to explain the first and second order effects re-
sponsible for current saturation in the FET, knowledge of the
detailed picture of the distribution of the carriers and voltage in
the channel is necessary. However, because of the nonlinear
nature of Poisson's equation [Equation (7)], an analytical solution

is impossible and, therefore, recourse is taken to the numerical

62

methods of solution. Of the various numerical methods of solu-
tion the choice is limited by the accuracy, a guarantee of con-
vergence, the total time of computation and, most important of all,
the mesh Size. A large number of mesh points indicates the
difficulty in a method in actual computation in that it requires a
large storage space and large computation time.

The most straight-forward numerical method of solution is
via the relaxation techniques. Through a judicious selection of the
mesh size and the relaxation factor, the convergence of this scheme
is more or less assured. However, 10 to 15 iterations are needed
for a fourth place decimal accuracy. Another disadvantage of this
technique is that it requires a storage space for 2(N x M) words
per variable (where N and M is the number of node points in the
x and y direction, reSpectively).

For a reasonable amount of accuracy the mesh size Should
be less than or at least comparable to the extrinsic Debye Length--
the characteristic length of decay of voltage in Equation (20). Also,
an FET, in saturation, should be analyzed for drain voltages of the
order of t to 10 times the pinch-off voltage. In the solution of the
tranSport equation (Equation 32), the size of the numbers range
between exp (4OVD) and exp (-40 Vd). Thus, if Vd = 5 Vp the largest

number is exp (200 VP). So that these numbers can be handled

efficiently in a large sized computer, the value of Vp should not

 

63

exceed 2 volts. From equation (13) the extrinsic Debye Length Lde and
the pinch-off voltage Vba are related by the equation

a/Lde = 2Vp/Vt25 8017p,
where a is the half—channel width. Thus from the point of view of the
size of numbers which can be safely handled in a computer we have

a/Lde g 160.

As was assumed in section 3.2, if the transition between complete

depletion and neutrality is taken to be abrupt then

Length of transition region < 0.1.
HaIfPChannel width

 

The length of the transition region has been determined in Equation (13) ‘

 

and therefore,

SLde/a < 0.1
or

a/Lde > 50.

In short devices (devices in which the gradual channel approximation
does not apply), let

b/a = 2.0.

If a = SOLde then: N = 101 and M = 101.
Thus for a numerical solution via relaxation techniques, the storage
space needed is of the order of 2(101 x 101) words per variable.

In this work a truncated Fourier's series is used to construct
the required function; since the function can be constructed in one
step--the required number of storage spaces is therefore, only
N'x:Nt= 101 x 101. Advantage is taken of the symmetry of the device

structure and thus we need only N x M/Z storage spaces. The required

 

64

number of storage spaces can be further reduced if the device length
is divided into sections. The required functions are then constructed
in one section at a time. In fact the entire computer program was
developed for IBM 1800 computer which has storage space of about 2500
real variable words.

3.8 Solution of Poisson's Equation

 

For any drain voltage Va and gate voltage V8, the shape of con-
ducting channel is a function of x (Figure 3.10). If V(x,y) is the

potential in the rectangular region x = 0, x = b, y = 0, and y = 2a,

 

the net charge density in the channel is defined by the equation

2 2 ‘

37‘; + 337‘; = —Q(x,)')/k (34) ~
where Q(x,y) = -q(Na-n) -qNa; 0 s y S Y1(x)
= -q(Nd-n); Yltx) s y 3 ms

= -q(Na-n) -qNa; Y2(X) s y S Za,
where Nd is s the donor density in the channel and n is the density
of the mobile majority carriers. The boundary conditions on V(x,y) are

V(x,0) = V(x,2a) = Vb(x),

V(b,y) = V6, and

V(0,y) = 0.

Let L = 32 + 32 = L .
SEZ' 3;2' aplaCian,

Let F(x,y) satisfy LF=0 and
F=Vb on boundary.
Let V - F = G such that G is zero on the boundary. Therefore,

LU

LV - LF

LV; because LF = 0.

65

 

Therefore,
F(b,a) = 4vd 3:? 1 Sin Nn/z N = 1,3,5,....
N=1 N
(38)
= 4Vd fl = Vd; [y f 0 and y # 23].
T ° I

The effect of this term on the channel current (b/a greater than 10)
can be ignored. Hewever, for smaller channel length to width ratios,
this effect is very significant.
The solution of LG = 0 with G = 0 on the boundary can again be
represented by a two-dimensional Fourier's series:
G(x,y) = 33° °z° Sin M Sin wa/b , (39)
M=l N=l ZEX"

where AMN is the coefficient determined by the boundary conditions

such that
AMN = _64 YR 1 ,7; 51112er . 5111an
W3 2 12 NTT T T
b M+(Z.a:’1)
b
g n(x) Sin.Mn Sin.Mw W(x) , Sin Nnx
-% (1 :— 1“? "5“”
O
= _64 V 1 2
ml w "N“
W b M2+(_2£N_)2 Tr
b

 

)
_1 81 n(x) Sin.Mn Sin.Mn W(x) Sin wa dx
5' i 2_'° 4—' a ° —IT_

0

for all values of N and for all odd values of M. (40)

 

g
a
t

66

Thus LF = 0 with F = Vb on the boundary and LG = Q(x,y) with V = 0
on the boundary is the solution of Equation (7).

The solution of LF = 0 is a two-dimensional Fourier series

F(x,y)= 35’ mm SinhN ,
N=1aNS 25"' '2E[

where a is determined by the boundary conditions. The potential

N ,
F

in the channel due to applied voltages is:

F(x,y) = 4vd i? Sinh ch/za Sin NTry/Za
—-- N=l Sifih N6572a °
(35)

 

b
+ 2 i9 Sinh /b Sin ch/b I Vb(x) . Sin NiTx . dx
5 N=l 1 2a L ' o ’5

where V(x,Za) = V(x,0) = Vb(x)
(36)

V8 for 0 S x SgL.

For gL 5 x 5 L; Vb(x) is given by the solution of one dimensional
Poisson's equation

2
LLi” = 0 ‘ (17)
0x2

such that Vb(x = L) = Vd'
The first term in Equation (35) is convergent for all odd values of N.
Atx=a,

l Sinh wa/Za Sin Nn/Z.
l N Sinh wa/2a

 

F(a,x) = 4Vd

'IT

ll M8

The term Sinh wa/Za is much smaller than unity and is almost zero
Sinh Nnb/Za

 

(for b/Za greater than 10) for all values of x except x = b, where

Sinh Nnx/Za

3155 N5572a x=b = 1°

67

 

 

g! 7i Vb(x)

///////€///// . “i
S _ E‘3W_ _ _ _ G) 1:

(V=0) —’ y//7/{W///////:IZ

0 III Illllll llllllllllllllllllllllIllllllllllflflllllfllflfll
,L. . \ J.

—"l|’

 

 

 

 

 

 

 

\
mm

Figure 3.10. Coordinate system and device structure parameters
used in the solution of Poisson's equation.

68

Equation (39) is the voltage due to space charge effects of the
gates and channel current. The factor N(x) represents the effect of
charge in the channel both due to the applied field and the channel
current. In the region of the partially depleted channel, this com-
ponent decreases; consequently, it tends to reduce the total voltage
in the channel. (The second term of Equation (40) equals the first
term for W(x)=2a). Any applied voltage on the drain contact increases
the drain current until the reverse bias reduces the channel width
to a point where a further reduction tends to reduce the voltage at

that point. For large channel length to width ratios (b/a>>10) the

 

terms of Equation (40) decay very rapidly with increasing values of __
M. For computational purposes the Fourier's series has been truncated

and terms up to and including M=N=21 are considered. The step size

taken was 0.05 normalized to half channel width.

3.9 Iteration Scheme

 

The six basic Equations (1) to (6) have been rearranged and
approximated to set of three equations (7), (26) and (27). Equations
(26) and (27) combined with the boundary conditions (32.2) define the
electron concentration in Equation (32). The solution of Poisson's
Equation (7), subject to the constraints of the applied voltage and
geometrical boundaries, is given by Equations (35) to (37) and (40).

In this new formulation of the problem, the electrostatic potential
'V(x,y), the electron density n(x) and the channel width W(x) are
chosen as the independent quantities. These three quantities represent

the unknowns of the reduced equations.

69

An iterative scheme is now used to cope with the various non—
linearities of the problem. The complete iteration scheme is shown
in Figure 3.11. The applied drain contact voltage Vd is specified.
F(x,y)--the component of channel voltage due to Vd, is computed from
Equation (35). A trial channel width Wj(x) and a trial electron
distribution nj(x) are chosen. To start the first cycle of the main

iteration loop, Gj+1(x,y)--the component of channel voltage, due to

. ' I '-' '3‘-

the gate bias and channel current, is computed from Equation (40),
in terms of Wj(x) and nj(x). Total voltage in the channel Vj+1(x,y)

is the sum of Gj+1(x,y) and F(x,y). Wj(x) and Vj+1(x,y) are used to

 

compute, from Equation (32), the new value of electron density-- t
nj+1(x). Poisson's equation with Vj+l(x,y) is now used to obtain

the charge density and hence the modified channel width Wj+1(x).

Solution of Poisson's Equation (34) with nj+1(x) and Wj+1(x) yields

an improved potential distribution. Iteration is continued until a

stable value of drain current [Equation (33)] is obtained.

Different initial values of Wj+1(x) and nj+1(x)-~other than
those shown in Figure 3.11, can be chosen to hasten the convergence
of the iteration scheme. The computed electron density nj+1(x) is
tested for any unrealistic values (n(x) orders of magnitude larger
than the impurity density Nd). In such a case a secondary iteration
100p is used to scale down the voltage Gj+1(x,y).

With this method complete freedom is available in the choice of
impurity distribution, the carrier boundary conditions at the external
contacts, and the dependence of mobility on the electric field and
doping. If applied voltage Vd is specified, the method solves for
total current and all the quantities of interest in the interior of

;.vice as a function of position.

70

 

Specify
VD
Y
F(x) = f(V

[Eq. 35]

 

 

 

D)

 

 

 

 

 

V,j+l : Fj+l + C;j+l

 

 

 

 

Start

 

 

Trial W(x)=2a,1‘21:0. o

n(x)=l. o, SCALE:1. o

 

 

1)

 

 

1

 

j

 

 

i A.
I

05H = {(wj, nj)

 

 

 

 

j+1

[Eq 40]

 

 

 

 

 

.+ .
n : f(VJ 1,WJ) =-

 

 

[Eq. 32]

 

 

 

 

 

 

 

 

wJ = f(VJ+1)
I

.+ . . .

de l : f(VJ+1, WJ+1,nJ+1)
[Eq. 33]

 

 

 

 

SCALE : O. 9XSCAL

 

 

 

 

1!

 

 

 

 

 

 

Figure 3.11. Block diagram of iterative scheme.

 

Iii-F cm )_-'_ui-_\'B. '. an,“

CHAPTER 4

RESULTS

Solutions obtained with the numerical procedure outlined in the
previous chapter are presented for two special structures: (1) an
n-channel FET with the gate length equal to a fraction of the channel
length and (2) Shockley's model (gate and the channel of equal lengths;
the gates overlap the source and the drain contacts). Distributions
of electrostatic potential, mobile carrier density and the channel

width are shown as functions of position throughout the interior of

 

the device. The terminal property in the form of drain current versus

tin-n2.

drain voltage is illustrated. “Exact" and conventional approximate
analytical results are compared and discrepencies are exposed.

In the previous chapter a numerical method of solution of the
two-dimensional, one-carrier, transport equation and Poisson's equation
describing the behavior of the FET has been described in detail.
Although the method of solution and numerical technique allow for an
arbitrary doping profile, boundary conditions at the external contacts,
and mobility dependencies, the results for a special type of idealized
structure are presented as an example of the numerical technique. The
analysis is restricted to "short” structures (b/a<<10) so that the
effects of drain field are accentuated. The choice of abrupt dOping
p ifiles has two motivations:

(a) the achievement of an analysis for the numerically worst case

(abrupt boundary conditions).

(b) the comparison between "exact" and approximate analytical solutions
are available only for idealized structures.

Motivation (b) also justifies the selection of appropriate constant

values for the carrier mobilities.
71

72

In addition to the above assumptions, external contacts of ohmic
type are specified for the device under consideration. The physical
parameters characterizing the structure are listed in Table 4.1.

For efficient arithmetic relevant quantities are expressed in
dimensionless form; the set of normalizing constants is chosen with
the criterion of achieving the highest simplification in the relations
of interest. The list of normalizing factors is given in Table 4.2.

Calculations were performed on the CDC 6500 system. The output
data was punched on cards and displayed in graphical form with the
help of HYPLOT routine on the IBM 1800 system. The computation time
for one value of drain voltage is of the order of 18 seconds.

4.1 The Results of the Present Analysis

 

For various applied voltages the electron distribution, channel
width, electrostatic potential and carrier mobilities are shown in
Figures 4.1 to 4.6 for the structure I; the same set of quantities is
shown in Figures 4.7 and 4.8 for the structure III and in Figures 4.9
to 4.10 for the structure 11. Figures 4.11 and 4.12 illustrate the
variation of electrostatic potential and carrier densities, for
Vd = VP, for channels of various initial widths. The drain current
versus drain voltage characteristics is shown in Figure 4.13, for
the three structures, for Vd > VP. For Vd < VP the drain current
is calculated only for the structure I. For all the above cases
there was no significant variation in the general shape and size of
the channel width, for Vd > VP and therefore only one curve is shown

(Figure 4.5). For all the above results the carrier mobility was

assumed constant with the electrostatic field.

 

73

 

 

 

Material: Silicon (relative permittivity 12)
Temperature: 300°K
Intrinsic carrier density: 1.5 x 1010 cm'3
Doping: n-chaJmel, N(x)=Nd=1.0 x 1016 cm‘3
gates, N(x)=Né=Metallic
Mobility: electrons, 1306.0 cmz/volt-sec.
Parameter Structure
I II III
channel length: 80 Lde 80 Lde . 80 Lde
channel width: 20 Lde 40 Lde 20 Ld0
Gate length: 80 Lde 80 L3; 60 Lde

 

Pinch-off voltage: 1.293V*** 5.172V 1.293V

 

Table 4.1. The physical parameters characterizing the FET structure,
analyzed under steady-state conditions for various applied

voltages.

 

* Unit of length in Extrinsic Debye Lengths.
** Voltage distribution on the drain contact is sinusoidal in the
y-direction.

*** Unit of voltage in Volts.

74

 

umoaoucfl mo moflpfiucmsc one now monunsm noflummflfimsnoz mo umfiq .N.e pomH

 

 

 

 

 

 

 

mofipfimcop
i nouqooom was
SHOH x o.H pz a: -2 .z .nocoe .xufiasmsfl we:
pass emmmo.o s> ea Hm>sa aesoa-ams:e
Hmwwcouoa
sao> emmNo.o o> s> Anastase poo scamsooae
oaoa x o.H m2 c xuwmcop amassed
eu\sao> S.m~s as m ensue Unsouofim
u omwufio>
“Hos ewmwo.o > e> measaev sadness
u HmfiecoHOQ
sacs ewmmo.o e\axn_> Unsssmospusae
0
Eu m-oH x assne.o ezs\xs>i es s apes: assemeu-eass
m >.x moumcfipaoou cowuwmoa
mbn<>.q<uHmmz§z. Homzaw
Cunhzmso
L a 4m 20353394 Han“ irmoz 29.530me

 

 

75

The weakness of the gradual channel theory is evident in
Figures 4.3, 4.4, 4.10 and 4.12. The carrier density distribution
has little resemblance for the first order idealization (neutral
channel); even for structures with very wide channels the effect
of drain contact field and the current space charge cannot be
ignored.

For narrow channels (structures I and III), the channel is
more or less depleted near the drain end. This kind of situation

exists in modern devices with extremely low pinch-off voltages. ‘

 

Figure 4.2 illustrates the progressive depletion of the channel
with increasing drain voltage. The carrier density passes through
a minimum near the drain contact. For higher drain voltages (Vd>Vp)
this minima lies and stays near the drain contact; also, it has a
higher minimum value than in the case of lower drain voltages. This
is due to the contribution of the space charge of increasing drain
current (Figure 4.4). The minimum value of carrier density in the
channel is dependent upon the device struture and drain voltages
(see Figures 4.3, 4.4, 4.8, 4.10 and 4.12).

In the Figures 4.3, 4.8 and 4.10 the effect of drain voltage
on the carrier density minimum should be noted; the minima moves
towards the source contact with increasing drain contact field.

This effect confirms Shockley's theory [1] of travelling pinch-Off
point towards the source.

The carrier density minima near the drain contact (Figure 4.4)
gives rise to a high-low junction. The confirmation for the presence
of a second high-low junction is made by the magnitude of diffusion
current in the channel; or, in other words, the gradient of the

carrier density in the channel. Towards the left of the carrier

 

76

. . . was...
. I
d o. a . p. (or... .. . . . . .0. . Huh. '

.cofiuflmoq mo cofluocsm m we Hoccmco one Cw Hmflucouoa ofiumumoauooam ”H esoposnum .H.e

n\x oocmumflp ponwfimEhoz

o.H m. . n. o. m. a. m. N. .M.

w
11a n 1w n 1111i1:.i~.

tilt)
\I

Q .t-

 

> no u e>
e> 3. u e>
a> ms u e>
a> no u e>
a> me u e>

N

 

SQ’O/(X)A {etiuaiod 3119150113913 pazitewioN

O
,_¢

77

>H

>m

>v

>0

o.H

 

.cofluwmoq Mo cofiuocsm m we Hoccmco one cw HmfiucoHOQ ”Monumonpuofim ”H manuospum .N.e .wflm

n\x mucoumwp nmnflflmeaoz

no 3. 3. 4.0 ms 2 no as
I a s1 n n

 

1
‘i
8'2/(X)A [etiuaiod 3:19150113919 pezttewioN

 

78

.H oasuosnum ”cofiufimom mo :oflpocsm m mm Hoccwcu may no poucoo may cw xufimcop coauooam

n\x oocwumfip pouflfimsaoz

o; as as We out. 3 so no as

T 11111q11qn1d1_n1

q
\\1z

 

 

.3. .mE

H5 o.o
4 n

 

N

o.H

PN/(x)u Aitsuap uoxioata pezttewioN

 

.coflufimoa :oHuUQSw m we Housmao mo aoudoo ogu :fl xuwmcop conuoofim ”H manposapm .e.e .mflm

n\x oocmpmwp poNflHmEnoz

on. as ms is so we so 2 2 no as
14 A a q 1 ed. a a a q a a A n din).

 

79

 

 

 

0N

PN/(x)u Airsuap uoxioeta pazttewi

80

 

a ..
IIHE

.coHuMmOQ mo :oHpoczw m we gone: Hoccsau ”H oasuosapm

n\x oocmumflp pomfiamenoz

.3. a:

 

 

 

N

V
BZ/(XM H1PIM Isuueqo paztremioN

o.~

81

 

 

.coflufimoq mo :ofiuocsw m we xpfiflflnoe cospoofim .o.e .mfiu
o\x museumfip poNflHmEHoz
5.0 o.o m.o v.0 m.o N.o H.o 9.0
.41 at .141 at us a _. 1w) .1 .q at a .1 a,

 

 

 

 

N.
s.
e.
seamN u a m.
meson u s

 

o.~

5'9021/(x)n Aittrqom uoxioeta pszttewioN

82

>H

 
 

[3:51 .‘u’

.cofiuwmoq mo

:ofiuocsm m we Hoccmco one cw Hmwucouoa owumumopuommm woufifimenoz ”HHH manpospum .n.e .ufim

p\x mandamus emusnsnhsz

o.H m.o m.o 5.0 o.o m.o q.m m.o ~.o H.o

sin a T. « dintd_tqt

  

o.o

 

8'2/(X)n {etiueiod paztIemioN

83

 

vilrvr . . . I. «in

 

 

 

 

.cOfluflmoa cofiuocsw m we Hoccmno co noucoo one ca Swanson conuooam ”HHH esoposaum .w.v .mHm
, D\X mugpmww VONMHNEOZ
on as as as es 3. so 3. NS no as
nannnninWIwanTn. 1 a
i
LN.
L
L e.
L o.
i
.iw.
i
54

 

PN/(x)u Airsuap uollbela PGZIIBmJON

84

.coflufimoa mo
:ofipocsw m we Hocemco we noucoo one cw HmflpcopOQ owumumoauoofim ”HH onsuosnum .m.e .mfim

as. 858% Essaanoz
no 2 NS no as

 

 

$3..
>
1
¢

> Nna.m u

241'5/(x)A [etiuaiod ot1915011oate pazttewioN

 

 

8S

 

. II‘1? a1. ... E

.nofipfimom mo
coHuo::m m we noncono mo nopnoo one an xuwmcoe nonuuoum ”HH on:uo:hpm .oa.e .mwm

n\x oonoumwp noanmEpoz

 

on ad 1 (We 1 he who m6 ed m.“ NS H5 o.o
Wtqqnq1111.nnn1<«l qa1_
L
a i N.
£26" >
L
l we

 

 

 

PN/(x)u Airsusp uoxioata pazttemioN

86

 

.Hoccmnu 0:» :fl mopw Hmfiucouom exp :0 mnuvwz.am::m:u HmwprM acoummmflw mo uuowmm .HH.¢ .wflm

fix 853me wmufimfiaoz

 

 

 

 

 

 

 

04 m5 2 2 05 m5 Yo 2 N5 2. o.o

Tfi4.fi41!4uidfi1afi.fiu4-i-“1....‘6aW
L m
TL.
2
9
1 N. p
9
TL
9
L n
l
m
038nm L q. m
1.
TL-
L D
d
a w
>82" >nw> L o. m
ocqmmum m.
L .rL
M
L m. m
mfomum H
1 ”L

.54

87

 

.Hmccmnu mcu :« coflusnfihumww achuumfio esp co mcpwwz Hoczwao Hmwuficfi ucohowwww mo puommm .NH.¢ .mflm

n\x muqmpmflw woufiamshoz

 

 

 

 

 

 

o.H m.o w.o n.o o.o m.o v.0 m.o N.o H.o o.o

11d . q 1‘ q 1%q4. « . q a . _ 1~
L
1N.
1v.
L
Lo.
L
La.
L
r:

pn/(x)u Anxsuap nalnaete pazgtemon

88

density minima, it can be seen that any change in drain voltage or
the position of pinch-off point does not alter the magnitude of
diffusion current by any significant amount. This is precisely

the property of a forward biased injecting junction. Near the
source end of the channel, the current is predominently of the
diffusion type. Hewever, at higher drain voltages the drain cur-
rent is completely of drift type near the drain contact. Again the
channel is never completely depleted of the mobile carriers.

Figure 4.5 shows the channel shape with drain voltage. One
observes that the channel width never reduces to zero; this is
contrary to the gradual channel approximation theory. For low
voltages the current control is through the channel width modula-
tion. At higher drain voltages, the channel shape remains unchanged
but the carrier density reduces. This is equivalent to channel
length modulation. The shape and width of the channel drain end is
almost independent of the drain voltage. This is different from the
predictions of the gradual channel approximation theory whereby the
channel width is always determined by the gate field.

Figure 4.6 shows the carrier mobility with position (the zero
value of mobility near the source contact is due to an error in the
computer program; actual value can be obtained by extrapolating the
curves). For the calculation of the results shown in Figures 4.1
to 4.12 the above effects are not taken into account. Therefore,
the velocity saturation of carriers is not a factor in the saturation

of drain current.

‘.. "‘Q-

 

89

Figure 4.13 shows the drain current variation with drain voltage.
The method of analysis confirms the predictions of the gradual channel
approximation theory below the pinch-off point. For Vd>vp non-zero
conductance is self-evident; also, the conductance is larger than
that predicted by the earlier models considered and is more near the
values observed for the practical transistors. The near linear varia-
tion of drain current, with drain voltage, is indicated by the denomi-
nator of Equation (32), chapter 3. This factor is only influenced by

the potential variation near the source; the contribution of the

field near the drain contact being insignificant.

{MWLGE zm- .
. .

 

a>\ U> nommfiocw Emup pouflmguoc

 

 

 

 

 

 

 

 

 

o .m o .v o .m o .m. o A
_ _ _ q _
owning game mumue> “confine Edna 2 .w. muswfim
a a
mfio> mom 4 u > .H enduoduum
A; LT Olul “IO.
3 L Q1|l + (1| D
o. Ila} a
{[01 33> on A n > “H: ousuospum
0“
0
9
a

 

33> ~26 u

 

 

m

o

.o

.N

we 39.13de 1U811U3 UIEJp

91

4.2 Advantages of Present Theory

 

The attempts at explaining the saturation current in FET's [1-12]

centered around the classical model of the gradual channel approximation.

This approximation, as explained earlier, is very satisfactory for
Vd<Vp but fails in the saturation region. Moreover, the precondition
for this approximation to hold, in a long device (b/a>>10), is seldom
met in practice.

The major drawback in this approximation is in the fact that the
space charge effects of the channel current and drain voltage are
ignored. The "feedback" mechanism present due to this space charge
controls the channel width (reduction of current reduces the voltage
drop; hence, the bias on the gate junctions is lowered).

In this analysis the current saturation is shown to be due to
physical phenomena--space charge and applied accelerating fields
rather than the geometry of the device. No voltage component in the
channel is ignored. This is made possible by the solution of the
two-dimensional Poisson's equation.

The numerical method adopted here has the advantage of a large
saving in the storage space required in the computer and hence the
cxnmmatation cost; the storage space requirement is only one—fourth
of that needed for the relaxation techniques. The method is par-
ticularly suitable for small computers which are handicapped due to

the memory space.

gear-Ln“! . ..... ._-__n -

 

w-

92

4.3 Disadvantages of Present Theory

 

The present method consists of the analytical solution of the
Poisson's equation for the applied voltages; this is convenient only
for simple geometry. For more complicated geometry and boundary
conditions the solution is very involved and a large number of addi-
tional calculations is required for the construction of the voltage Pu
function. .

The solution of the transport equation, for the carrier density,

in the form used here, requires the use of very large and very small j

 

numbers. This problem is especially critical for Vd>2Vp. The Fourier's E
series is not suitable for constructing the step-type drain voltage I
boundary; an infinite number of terms is needed to do this near the
gate end of the channel. Use of truncated Fourier's series results
in the superimposed ripples on the voltage distribution.
The channel width W(x) does not appear as a by-product of the
solution but its judicious initial selection is essential for the
faster progress of the iteration.
Saving in the storage space is at the cost of increased compu—
tation time. Also, the large number of mathematical functions needed

is a drawback. However, by clever programming these functions need

be called only once.

93

CHAPTER 5

SUMMARY AND CONCLUSIONS

A research program was undertaken with the objective of dev-
eloping a more accurate model to explain current saturation in field-
effect transistors. In particular, we wished to uncover the principal
phenomena responsible for current saturation in field—effect tran-
sistors. Also, we wanted to develop a more exact model for the
mechanism which tends to maintain the drain current at a mean constant
value with drain voltage as well as the mechanism responsible for
the non-zero slope of the drain-current versus drain-voltage charac-
teristic.

The field-effect transistor selected for analysis was schematically
represented by two abrupt p-n junctions placed symmetrically with
respect to the center of the channel. In order to simplify the mathe-
matical formulation, the gate contacts were taken to overlap the source
contact, and the gate and channel were assumed to be uniformly doped.

Numerical solutions to Poisson's equation were obtained subject
to certain restraints (See Sections 3.3-3.5) and the results were
analyzed. The major conclusions drawn from the analysis are:

l. The partial depletion of the channel (uniformly distributed
impurity density) and the consequent spatial gradient of the
majority current carriers is responsible for the current saturation.
This is in striking contrast to the models of Kennedy [11] and Kim
and Yang [16]. In their models the property of spatial distribution
of the carriers was deliberately introduced by selecting special

impurity density profiles in the channel.

 

94

Dependence of carrier mobility on the electrostatic field does

not seem to play any significant part in the saturation of cur-
rent. All the results presented in Section 4.1 have been obtained
under the assumption that the carrier mobility is independent of
the electric field. No accumulation of carriers has been observed
in the channel and the electric field increases continuously from
the source towards the drain.

The channel cannot be regarded as neutral even for voltages less
than the pinch-off value. With increasing drain voltage, the
high-low junction near the drain electrode remains more or less
fixed in position [Figures 4.3, 4.4 and 4.8]. The position of
second high-low junction is a function of drain voltage and drain
current. The slight increase in drain current, with increasing
drain voltage, is due to the increase of diffusion gradient across
this "junction" [Figures 4.4 and 4.8]. The movement of the
carrier density minima with the increase of drain voltage, in

the partially depleted region, towards the source is clearly
noticed in these figures.

Near the source contact the channel shape resembles the shape
predicted by the gradual channel approximation theory. However,
near the drain contact the channel width never reduces to zero
and reaches a minimum value of the order of 2 Debye lengths. For
narrow channels and voltages greater than the pinch-off value

the channel shape is invarient with the drain voltage.

 

95

APPENDIX

HIGH-LOW JUNCTIONS IN SEMICONDUCTORS

High-low junctions arise due to a discontinuity in the impurity
concentration in a semiconductor. Figure A1 shows a junction between
a heavily doped material n+ and a lightly doped material n. The
difference in concentration across the x=0 plane sets up a carrier [I
concentration gradient. The electrons diffuse into the n~region
causing an accumulation of electrons on the right side of the x=0 i

plane. Equilibrium is established when the accumulated charges are

 

balanced by the fixed positive charges of the impurity atoms.

An accumulation layer adjacent to a positive space charge gives
rise to an electric double layer. This layer impedes the flow of
current through it and is thus equivalent to a junction. Such
layers can arise in devices which have n+ type contacts. Contacts
obtained by diffusing similar type impurity into a substrate are

examples of this behavior.

96

 

t‘&:’~"u' ' - -

 

 

 

Figure A. Electron concentration in a high-low junction.

[1]

[2]
I3]
[4]
[5]
[6]
[7]

[8]
[9]

[10]

[11]

[12]

97
REFERENCES

Shockley W., "Unipolar Field-Effect Transistor", Proc. Inst.
Radio Engrs., vol. 40, pp. 1365, 1952.

Prim R.C. and Shockley W., "Joining Solutions at the Pinch-
Off Point in the Field—Effect Transistor”, IRE Trans.
Electronic Devices, vol. 4, pp. 1, 1953.

wu S.Y. and Sah C.T., "Current Saturation and Drain Conductance
of Junction-Gate Field—Effect Transistors”, Solid State
Electronics, vol. 10, pp. 593, 1967.

Grebne A.B. and Ghandhi S.K., "General Theory for the Pinched-
Operation of the Junction-Gate Field-Effect Transistor",
Solid State Electronics, vol. 12, pp. 573, 1969.

Grosvaley J., Motsh C. and Tribes R., "Physical Phenomenon for
Field-Effect Devices", Solid State Electronics, vol. 6, pp.
65, 1963.

Ryabinkin Yu S., "Field-Effect Transistor Based on Field-Effect
Theory", Radio Engineering and Electronics Physics", pp.
433, March 1968.

Hauser J.R., "Characteristics of Junction Field-Effect Devices
with Small Channel-to-Width Ratio", Solid State Electronics,
vol. 10, pp. 577, 1967.

Dacey G.C. and Ross I.M;, "The Field-Effect Transistor", Bell
System Tech. Journal, vol. 34, pp. 1149, 1955.

Dacey G.C. and Ross I.M., "Unipolar Field-Effect Transistor",
Proc. Inst. Radio Engrs., vol. 41, pp. 970, 1963.

Burger R.M. and Donovon R.P., "Fundamentals of Silicon
Integrated Device Technology vol. II”, Prentice-Hall,
New Jersey, 1968. .

Kennedy D.P., ”Mathematical Investigation of Semiconductor
Devices", Final Report (11), IBM, New York, April 1, 1967.

Lade R.W. and Jordan.A.G., "On the Characteristics of High-Low
Junctions Devices", Journal<1f.Electronics and Controls, vol.
13, pp. 23, 1963.

 

[13]

[14]

[15]

[16]

98

Kennedy D.P. and O'Brien R.R., "Computer Aided Two-Dimensional
Analysis of the Junction Field-Effect Transistor", IBM J.
Res. Develop., pp. 95, March 1970.

Goldberg Colman, "Space Charge Regions in Semiconductors",
Solid State Electronics, vol. 7, pp. 593, 1964.

Mari A. De, "An.Accurate NUmerical Steady-State One-Dimensional
Solution of the p-n Junctions”, Solid State Electronics, vol.
11, pp. 33, 1968.

Choong-Ki Kim and Yang E.S., "An Analysis of Current Saturation
Mechanism of Junction Field-Effect Transistors”, IEEE Trans.
on Electron Device, vol. ED-17, pp. 120, 1970.

 

flag lint.» bit-Ru. 24AM.“ h. .m . .. u‘, h