close

Вход

Забыли?

вход по аккаунту

?

Small signal AC model for the velocity-saturated MODFET and the prediction of the microwave characteristics of MODFETs

код для вставкиСкачать
INFORMATION TO USERS
This manuscript has been reproduced from the microfilm master. UMI
films the text directly from the original or copy submitted. Thus, some
thesis and dissertation copies are in typewriter face, while others may
b e from any type of computer printer.
T he quality of this reproduction is dependent upon the quality of the
copy submitted. Broken or indistinct print, colored or poor quality
illustrations and photographs, print bleedthrough, substandard margins,
and improper alignment can adversely affect reproduction.
In the unlikely event that the author did not send UMI a complete
manuscript and there are missing pages, these will be noted. Also, if
unauthorized copyright material had to be removed, a note will indicate
th e deletion.
Oversize materials (e.g., maps, drawings, charts) are reproduced by
sectioning the original, beginning at the upper left-hand corner and
continuing from left to right in equal sections with small overlaps. Each
original is also photographed in one exposure and is included in
reduced form at the back of the book.
Photographs included in the original manuscript have been reproduced
xerographically in this copy. H igher quality 6" x 9” black and white
photographic prints are available for any photographs or illustrations
appearing in this copy for an additional charge. Contact U M I directly
to order.
University M icrofilm s International
A Bell & H ow ell Information C o m p a n y
3 0 0 North Z eeb R oad . A n n Arbor. Ml 4 8 1 0 6 -1 3 4 6 USA
3 1 3 /7 6 1 -4 7 0 0
8 0 0 /5 2 1 -0 6 0 0
O rder N um ber 9201684
Small sig n a l AC m odel fo r the velo city -satu rated M O D FET a n d
the p re d ic tio n of th e m icrow ave characteristics o f M O DFETs
Kang, Sung Choon, Ph.D.
The Ohio State University, 1991
UMI
300 N. Zeeb Rd.
Ann Arbor, MI 48106
S m a l l S i g n a l AC M o d e l f o r t h e
V e l o c i t y - S a t u r a t e d MODFET a n d T h e
P r e d i c t i o n o f t h e m ic r o w a v e
CHARACTERISTICS OF MODFETS
D IS S E R T A T IO N
Presented in Partial Fulfillment of th e Requirem ents for
the D egree D octor of Philosophy in the G raduate
School of T h e Ohio S ta te U niversity
By
Sung Choon Kang, B .S .E .E ., M .S.E.E.,
s|e s|c >|c s|c sfc
The Ohio State U niversity
1991
D issertation Com mittee:
Professor Patrick Roblin
Professor Steve Bibyk
Approved by
%. L-.l
Adviser
Professor Furrukh Khan
D epartm ent of Electrical
Engineering
A cknow ledgem ents
I wish to express m y sincere appreciation and gratitude to m y advisor, Professor
Patrick Roblin for his advice, guidance and financial support throught my M aster’s
and P h.D . research at T he Ohio State University.
Furthermore, I wish to thank Professor Steve B ibyk and Professor Furrukh S.
Khan for reading m y dissertation and providing constructive criticism . I also express
m y gratitude to Dr. Hardis Morko$ in University of Illinois at Urbana-Cham paign
for providing the m easured data for m y research work.
Finally, I wish to express m y gratitude to my wife, Kyungsook, m y son, Sangwoo,
and m y daughter, Yousun, for their help and understanding. Especially, I owe to my
parents and m y relatives in Korea who have supported m y Ph.D . studies.
V it a
June 18, 1955 ................................................... Born — ChoonChun, Korea
1973-1977 ...........................................................B .S .E .E ., The Seoul National University,
Seoul, Korea
1977-1979 ...........................................................M ilitary Service
1980-1985 ...........................................................Manufacturing Engineer,
G old Star Cable C o.,
Seoul, Korea
1985-1988 ...........................................................M .S .E .E ., The O hio State University,
C olum bus, Ohio
1986-1991 ...........................................................G raduate Research Associate,
D epartm ent of E lectrical Engineering,
T h e Ohio State University,
Colum bus, Ohio
PUBLICATIO NS
1. P. Roblin, S.C . Kang, and H. Morkog, “Analytic Solution of th e VelocitySaturated M O S F E T /M O D F E T Wave Equation and Its Application to the Pre­
diction of th e Microwave Characteristics o f M O D FET’s ,” IEEE Trans. Electron
D evices, vol. ED-37, No. 7, p p .1608-1622, 1990
2. P. Roblin, S.C . Kang, and H. Morkog, “Microwave Characteristics of the MODF E T and th e Velocity-Saturated M O SFET W ave-Equation,” Proceedings of the
1990 International Sym posium on C ircuit and System s, vol.2, p p .1501-1504,
M ay 1990
3. P. Roblin, S.C . Kang, A. K etterson, and H. Morkog, “Analysis of M O D FET
Microwave C haracteristics,” IEEE Trans. Electron D evices, vol. ED-34, N o.9,
p p .1919-1928, 1987
FIELDS OF STU D Y
M ajor Field : Electrical Engineering
Studies in M icroeletronics
: Professor Patrick Roblin
Studies in Com puter Engineering
: Professor Fusun Ozguner
Studies in Physics
: Professor Thom as Lemberger
T able of C ontents
AC K N O W LED G EM EN TS
.................................................................................................
ii
V I T A ............................................................................................................................................
iii
LIST OF T A B L E S ...................................................................................................................
viii
LIST OF F I G U R E S ...............................................................................................................
ix
LIST OF S Y M B O L S ...............................................................................................................
xv
CHAPTER
I
I N T R O D U C T I O N ........................................................................................................
1
B a c k g r o u n d .................................................................................................... ... .
Problem S ta te m e n t............................................................................................
Structure of D isse r ta tio n ..................................................................................
1
3
6
TH E V ELO C ITY -SA TU R A TED M O D FET W AVE EQ UATIO N A N D
ITS SO LUTION ...........................................................................................................
8
1.1
1.2
1.3
II
2.1
2.2
2.3
III
PAGE
Derivation of th e W a v e -E q u a tio n ................................................................
E xact Solution of the Velocity-Saturated M O D FET wave-equation
Y-Param eters w ithin the Frequency Power-Series approxim ation . .
8
15
17
EQ UIVALENT CIRCUIT R E P R E S E N T A T IO N ...............................................
26
3.1
3.2
3.3
3.4
26
27
37
48
Introduction .......................................................................................................
T h e First Order Equivalent Circuit for th e GCA r e g io n .....................
T he O ptim al Second-Order Equivalent C ir c u it.......................................
T he V elocity-Saturated M O D FET Equivalent C ir c u it .........................
IV
PR ED IC TIO N OF T H E MICROWAVE CH AR AC TERISTIC S OF M ODF E T ’S .................................................................................................................................
4.1
4.2
4.3
V
55
E xtraction of param eters for the ac m o d e l .............................................
Com parison of the m easured and calculated d a t a ...............................
D is c u s sio n ..............................................................................................................
55
57
72
UNILATERAL P O W E R GAIN R ESO NAN CES AND f T-f\iAX O R D E R ­
ING .....................................................................................................................................
77
5.1
5.2
5.3
Introduction ......................................................................................................
Long and Short Channel M ode and the ac-current G a i n .................
Unilateral power gain of th e wave-equation m o d e l .................................
77
79
82
V I C O N C L U S IO N .................................................................................................................
88
6.1
6.2
C o n c l u s i o n ..........................................................................................................
Future Work .......................................................................................................
88
90
A P PE N D IC E S
A
Frequency Power-Series Solution for the Velocity-Saturated M O D FE T
wave e q u a t io n .................................................................................................................
A .l
A .2
A .3
Calculation of Vg c ( x = L ) ..............................................................................
Calculation of Vq C{ X s ) .................................................................................
Power-Series Solution of W a v e -E q u a tio n .................................................
A .3.1
Calculation of v q ...................................................................................
A .3.2
Calculation of v \ .................................................................................
A .3.3
Calculation of v 2 .................................................................................
A .3.4
Calculation of ig and i d ...................................................................
A .3.5
Calculation of V12and K2 2 ..................................................................
A .3 .6 Calculation of Yu and Y2 1
91
91
92
93
97
100
106
I ll
112
114
B
Exact Solution for V elocity-Saturated M O D FET Wave E q u a tio n ......................116
C
The Fourth Order Frequency Power-Series Solution in th e GCA R egion .
vi
127
D
D evelopm ent of Equivalent Circuit for the velocity-saturated M O D FET .
131
D .l Equivalent circuit for the saturation r e g io n ...............................................
D .2 Calculation of th e Y-param eters for the two region m o d e l ...............
131
136
B I B L I O G R A P H Y ...................................................................................................................
139
vii
L ist o f T a b l e s
PAGE
D evice parameters for calculating the intrinsic Y -p a r a m e te r s ...............
25
D evice parameters for the dc characteristics o f the A lG aA s/G aA s M O D ­
F E T (2045) and th e G aA lA s/InG aA s/G aA s pseudom orphic M O D FET
(2379) . .
...............................................................................................................
58
Microwave parasitics for the A lG aA s/G aA s M O D FET (2045) and the
G aA lA s/In G aA s/G aA s pseudom orphic M O D FE T (2379)
69
D eviation of calculated S param eters from th e m easured data for all
bias conditions for the device 2045 ( ds = 1500 A, C g d = 50 fF, and
C o s = 50 f F ) ...........................................................................................................
70
D eviation of calculated S param eters from the m easured data for all
bias conditions for the device 2379 ( da = 500 A, C g d = 70 fF, and
C g s — 50 f F ) ............................................................................................................
71
D eviation of calculated S param eters from th e measured data for device
2045 .............................................................................................................................
74
D eviation of calculated S param eters from th e measured data for device
2379 .............................................................................................................................
75
D evice parameters for the intrinsic short-channel M O D F E T ...............
81
L ist o f F ig u r e s
FIG U R E
PA G E
1
T he equivalent circuit of an extrinsic M O D F E T .........................................
4
2
Equivalent circuit for th e velocity-saturated intrinsic M O D FE T pro­
posed by Rohdin [ 2 1 ] .............................................................................................
5
3
Idealized representation of th e intrinsic M O D F E T .....................................
9
4
E xact solution (solid lines) and frequency power-series solu tion (dotted
lines) for m agnitude o f Y u ..................................................................................
21
E xact solution (solid lines) and frequency power-series solution (dotted
lines) for phase of Y u .............................................................................................
21
E xact solution (solid lines) and frequency power-series solu tion (dotted
lines) for m agnitude o f Y u ..................................................................................
22
E xact solution (solid lines) and frequency power-series solution (dotted
lines) for phase of Y u .............................................................................................
22
E xact solution (solid lines) and frequency power-series solution (dotted
lines) for m agnitude o f > 2 1 ..................................................................................
23
E xact solution (solid lines) and frequency power-series solution (dotted
lines) for phase of Y n .............................................................................................
23
E xact solution (solid lines) and frequency power-series solution (dotted
lines) for m agnitude o f Y 2 2 ..................................................................................
24
E xact solution (solid lines) and frequency power-series solution (dotted
lines) for phase of Y 2 2 .............................................................................................
24
A pproxim ate sm all-signal equivalent-circuits for the intrinsic MOD­
F E T in GCA region..................................................................................................
28
5
6
7
8
9
10
11
12
13
Comparison of the am plitude of Y \ \ / g o ............................................................
30
14
Comparison of the phase o f Y n / g o ......................................................................
30
15
Comparison of the am plitude of Y u / g o ............................................................
31
16
Comparison of the phase o i Y u / g o ......................................................................
31
17
Comparison of the am plitude of Y n / g o ............................................................
32
18
Comparison of the phase of Y u / g o ......................................................................
32
19
Comparison of the am plitude of Y i i / g o ............................................................
33
20
Comparison of the phase of T^/flp.....................................................................
33
21
Plot of f 5 %(Yij)/fo for i i i
asa function of the biasing param eter k. .
35
22
Plot of fs%(Yij)lfo for Y \ 2
asa function of the biasing param eter k. .
35
23
Plot of fs%(Yij)/fo for I 21
asa function of the biasing param eter k. .
36
24
Plot of fs%(Yij) / f 0 for Y 2 2
asa function of the biasing param eter k. .
36
25
Two different optimal second order sm all-signal equivalent circuits for
the intrinsic M ODFET in GCA r e g i o n .........................................................
38
26
Comparison of the m agnitude of Y n / g o for k = 0 . 6 5 ................................
40
27
Comparison of the phase of Yn/go f ° r k = 0 .6 5 ...........................................
40
28
Comparison of the m agnitude of Y n / g o for k = 0 . 6 5 ................................
41
29
Comparison of the phase of Yu/go for k = 0 .6 5 ...........................................
41
30
Comparison of the m agnitude of Y n / g o for k = 0 . 6 5 ................................
42
31
Comparison of the phase o f F21/50 for & = 0 - 6 5 ...........................................
42
32
Comparison of the m agnitude of Y 2 2 /go for k = 0 . 6 5 ................................
43
33
Comparison of the phase of
43
34
Plot of fs%{Yij)/fo for T ii as a function of the biasing param eter k.
.
45
35
Plot of fs%{Yij)/fo for T12 as a function of the biasing param eter k.
.
45
for k = 0.65 ............................................
x
36
Plot of h % ( Y i j ) l f 0 for Y21 as a function o f the biasing parameter k.
.
46
37
Plot of fs%(Yij)lfo for F22 as a function o f the biasing parameter k.
.
46
A pproxim ate second-order sm all-signal equivalent-circuits for th e in­
trinsic M O D F E T .....................................................................................................
47
First-order non-quasi-static equivalent circuit for the velocity-saturated
M O D FE T w ave-eq u ation ......................................................................................
49
38
39
40
Comparison of th e am plitude o f Y\\ for
Vd s
41
Comparison of the phase of Y u for
= 3 F and
42
Comparison of the am plitude of F12 for
43
Comparison of the phase of Y 1 2 for
Vd s
Vd s
Vd s
= 3 F and
Vg s
— 3 F and
= 3 F and
44
Comparison of th e am plitude of F2i for
Vd s
45
Comparison of th e phase of Y<i\ for
= 3 F and
46
Comparison of the am plitude o f >22 for
Vd s
47
Comparison of th e phase of I 22 for
= 3 F and
48
Vg s
= OF.
.
.
= OF...................
51
= OF..................
51
52
= OF...................
52
— OF.
. .
53
— OF...................
53
Measured (solid lines) and calculated (dotted lines) IV characteristics
of the A lG aA s/G aA s M O D FE T 2045 .............................................................
59
Measured (solid lines) and calculated (d otted lines) IV characteristics
of the pseudom orphic G aA lA s/In G aA s/G aA s M O D FE T 2379 . . . .
59
50
Equivalent circuit for the extrinsic M O D FE T
............................................
60
51
Measured (solid lines) and calculated (d otted lines) scattering param e­
ters for VGS = -08 V and VDS = 0.5 V for th e A lG aA s/G aA s M O D FE T
2045.................................................................................................................................
61
Measured (solid lines) and calculated (dotted lines) unilateral pow er
gain for V g s = -08 V and V d s = 0.5 V for th e A lG aA s/G aA s M O D FE T
2045.................................................................................................................................
61
49
52
xi
Vd s
Vg s
= 3 F and
Vg s
Vgs
Vg s
= OF.
.
50
.
Vg s
= OF.
50
. .
Vd s
= 3 F and
Vgs
53
54
55
56
57
58
59
60
61
62
Measured (solid lines) and calculated (dotted lines) scattering param­
eters for V g s = -08 V and V d s = 0.75 V for th e A lG aA s/G aA s MOD­
FE T 2045......................................................................................................................
62
M easured (solid lines) and calculated (dotted lines) unilateral power
gain for V g s = -08 V and V d s = 0.75 V for th e A lG aA s/G aA s MOD­
FE T 2045......................................................................................................................
62
M easured (solid lines) and calculated (dotted lines) scattering parame­
ters for V g s = -08 V and V d s = 1.0 V for the A lG aA s/G aA s M ODFET
2045.................................................................................................................................
63
M easured (solid lines) and calculated (dotted lines) unilateral power
gain for V g s = -08 V and V d s = 1.0 V for the A lG aA s/G aA s M ODFET
2045.................................................................................................................................
63
M easured (solid lines) and calculated (dotted lines) scattering parame­
ters for V q s = -08 V and V d s = 3.0 V for the A lG aA s/G aA s M ODFET
2045.................................................................................................................................
64
M easured (solid lines) and calculated (dotted lines) unilateral power
gain for V g s = -08 V and V d s = 3.0 V for the A lG aA s/G aA s M ODFET
2045.................................................................................................................................
64
M easured (solid lines) and calculated (dotted lines) scattering parame­
ters for V d s = 3 V and V g s = - 0.15 V for th e G aA lA s/InG aA s/G aA s
pseudom orphic M O D FE T 2379...........................................................................
65
Measured (solid lines) and calculated (dotted lines) unilateral power
gain for V d s = 3 V and V g s = -0.15 V for th e G aA lA s/InG aA s/G aA s
pseudom orphic M O D FE T 2379............................................................................
65
Measured (solid lines) and calculated (dotted lines) scattering param­
eters for V d s = 3 V and V g s — -0.08 V for th e G aA lA s/InG aA s/G aA s
pseudom orphic M O D FE T 2379...........................................................................
66
Measured (solid lines) and calculated (dotted lines) unilateral power
gain for V d s = 3 V and Vgs = -0.08 V for th e G aA lA s/InG aA s/G aA s
pseudom orphic M O D FE T 2379...........................................................................
66
xii
63
64
65
66
67
68
M easured (solid lines) and calculated (dotted lines) scattering param­
eters for Vd s = 3 V and Vgs = 0.25 V for th e G aA lA s/InG aA s/G aA s
pseudom orphic M O D FET 2379............................................................................
67
M easured (solid lines) and calculated (dotted lines) unilateral power
gain for Vd s = 3 V and Vgs = 0.25 V for th e G aA lA s/In G aA s/G aA s
pseudom orphic M O D FET 2379............................................................................
67
Measured (solid lines) and calculated (dotted lines) scattering param­
eters for Vd s = 3 V and Vg s = 0.56 V for th e G aA lA s/InG aA s/G aA s
pseudom orphic M O D FET 2379............................................................................
68
Measured (solid lines) and calculated (dotted lines) unilateral power
gain for Vd s = 3 V and Vgs = 0.56 V for the G aA lA s/In G aA s/G aA s
pseudom orphic M O D FET 2379............................................................................
68
Current and transconductance weight functions w \ (a ) (plain line) and
iw2( a _1) (dashed line) plotted versus a and a -1 respectively...................
79
Variation of th e unity current gain cutoff frequency f j versus gate
length Lg plotted versus a = E cLg/( V gs — Vt ) in log scale for an
intrinsic M O D FE T with Vgs = 0, 0.1, and 0.2 V and Vd s = 1 V. . .
80
69
Exam ple of th e unilaterization of a two port device by loss-less feedback. 82
70
M agnitude of th e unilateral power gain versus frequency for an intrinsic
M O D FET (V gs = 0 V and Vd s = 1 V) w ith a gate len gth of 3 /z
(dashed-dotted line), 1 /z (dashed line), and 0.3 fi (plain lin e )................
83
Equivalent circuit for the extrinsic M O D FET . C gs and C g d are the
fringe capacitors of the gate..................................................................................
85
Unilateral power gain versus frequency for a 0.3 /z extrinsic M O D FET
w ith parasitics resistances R s = R g = R d = 0-01 ^ (plain lin e ), 0.1 D
(dashed line), 1 fI (dotted dashed line), and 5 f! (dashed dashed line).
85
Unilateral power gain and short circuit current gain (plain line) ver­
sus frequency for a 0.3 /z extrinsic M O D FET using two different gate
resistances R g = 5 fl (dashed line) and 25 D (d otted dashed line). . .
87
List of calvn p r o g r a m ...............................................................................................
128
71
72
73
74
xiii
75
List of calix p r o g r a m ...............................................................................................
129
76
Equivalent circuit for the saturation r e g io n ....................................................
135
xiv
L ist o f S y m b o l s
Cg
: the gate capacitance per unit area
Cgs
'•the fringe capacitors between gate and source
Cgd
'■the fringe capacitors between gate and drain
ds
fMAX
: the channel width in th e saturation region
'• m axim um oscillation frequency
E c : the critical electric field to attain th e peak velocity
ei
: the dielectric constant for the channel material
62
: the average dielectric constant for th e high-bandgap region
I(x,t)
: total current in the channel
Idc(x)
: the dc channel current
i( x )
: the ac channel current
q
: electron charge
Lg
: the gate length
I
n s( x , t)
: the len gth of the saturation region
: two dim ensional electron density in the channel
H : the channel m obility
vs
: saturation velocity o f electron
ts
:
vs
V t
— tim e delay due to saturation region
v3
: the saturation v elocity of electrons
■ the threshold voltage
Vc s( x)
: DC channel to source voltage at th e position x
vcs(x)
: total channel to source voltage at th e position x
vg c
{x )
: total gate to channel voltage at th e position x
V g c (x )
•’ DC gate to channel voltage at th e position x
VgC( x )
: AC gate to channel voltage at th e position x
Vg s
VgS
Vd s
• DC applied voltage between the g a te and the source
: AC applied voltage between the g a te and the source
’ DC applied voltage between the drain and th e source
xv
Vds
k
'•AC applied voltage between th e drain and the source
= Vd s / V gs — Vr
xs
for unsaturated device
: instantaneous p osition of the G C A /satu ration boundary
Xs
: dc position of th e G C A /satu ration boundary
xs
: ac m otion of the G C A /satu ration boundary
Wg
: the gate width
(Vgs ~ VT)
u °
■
t1
r 2 --------
xvi
C H A PT E R I
IN T R O D U C TIO N
1.1
B a c k g ro u n d
T h e recently developed M O D FET is a prom ising device for both microwave and
m illim eter-w ave applications and high-speed digital circuits. It has dem onstrated re­
m arkable high-speed performances [1]. Propagation delays as small as 12 ps and below
10 ps have been obtained in ring oscillator m easurem ents at 300° K and 11° K respec­
tiv e ly [2], [3]. Very low noise figures have been m easured at microwave frequencies
(0.4 db with 14 db gain at 10GHz at 11°K [4]). Its low noise figure at high frequencies
m akes it attractive in microwave applications. M O D FET s based on novel com pound
sem iconductors are showing very prom ising results (see [5] for a review). Recently th e
G E Electronics Laboratory reported 0.15 fi gate length InA lA s/In G aA s/In P latticem atched M O D FETs w ith m axim um frequency of oscillation as high as 405 GHz [6].
In support of the developm ent of the M O D FET technology , there has been a
strong m odeling effort reported in th e literature. Indeed since the first dc model re­
ported by D elagebeaudeuf [7] a large number of dc m odels for the M O D FETs have
been published. Som e authors have in addition derived sm all-signal m odels for th e
M O D FE T using the quasi-static form ula Cgs = d Q / d V gs and Cgd = d Q / d V gd where
1
Q is the charge stored in th e channel. So far, however, there have been fewer re­
ported attem p ts to compare the microwave performance predicted from these models
w ith the published m icrowave data available for the M O D FETs. Yeager and Dutton
[8] reported a large-signal m odel which they compared with reasonable success to
scattering parameters m easured at 4 GHz. A num erical m icrowave m odel based on
a transm ission line circuit m odel was reported [9]. T h e sim ulation results obtained
w ith their numerical m odel show a good agreement w ith the scattering parameters of
a 0.3 m icron gate length M O D FE T m easured at a single bias point.
An analytic microwave m odel including distributed effects for the unsaturated
M O D FE T [10] was recently reported. It perm itted us to reasonably reproduce si­
m ultaneously the dc characteristics and microwave perform ance of an unsaturated
M O D FE T using a unique set of device param eters.
This analytic ac m odel was
however lim ited to the linear regime up to the edge of saturation.
Since the M O D FET wave equation has the same form as the three term inal MOSF E T ’s, it is useful to investigate the previous work on the M O SFE T wave equation.
T he wave-equation for th e unsaturated M O SFET was derived independently by Can­
dler and Jordan [11], Geurst [12] and Hauser [13]. Geurst derived an exact solution of
th e three term inal M O SFET wave-equation in term s of Stokes’ functions [12]. Burns
[14] and Treleaven and Trofimenkoff [15] derived independently an exact solution in
term s of B essel functions. These exact solutions are not however analytic per se, as
th ey involve Stokes or B essel functions which much be num erically generated. Ap­
proxim ate equivalent circuits were derived by both Burns [14] and Treleaven and
3
TrofimenkofF [15] for the case of the M O SFET operated in pinch-off. For this m od e
of operation th ey reported approxim ated analytic expressions for Y u and Y2\ (in
pinch-off Y X 2 = Y 2 2 = 0). A n alternative procedure based on an iterative scheme was
introduced b y Van Nielen [16] to obtain accurate approxim ate results of the M O S­
F E T wave equation. T his iterative solution was used by Bagheri and Tsividis [17]
and Bagheri [18] for deriving the sm all-signal Y-param eters of th e long-channel four
term inal M O SF E T and three-term inal M O D FET, respectively. M ore recently using
a frequency pow er series first introduced by Van der Ziel and Ero [19] for the junction
FE T , Van der Ziel and W u [20] solved the M O D FET w ave-equation and calculated
Y\\ in term s o f a frequency power series for the unsaturated M O D FE T . Roblin and
Kang [10] continued their calculation and derived th e remaining Y parameters Y 1 2 ,
Y2\ and Y22- T he frequency power series has the advantage of being analytic and
holding up to high frequencies.
To sum m arize there ex ist both an exact solution and an analytic frequency pow er
series solution of the M O D FE T (and M O SFET) wave-equation. T h e M ODFET (and
M OSFET) wave-equation applies however only to th e unsaturated M OSFET up to
th e edge of saturation or to long-channel devices operated in pinch-off.
However
saturation in a M OSFET or M O DFET results from velocity saturation and not pinchoff for subm icron gate length.
1.2
P r o b le m S ta te m e n t
In order to analyze the microwave characteristics of th e velocity-saturated M O D FET,
an equivalent circuit is often used to fit the device perform ance m easured at various
4
Lg
Rg
C dg
Cgs ^
Rd
Ld
V
m
in
m —
Source
Figure 1: The equivalent circuit of an extrinsic M O D FET
frequencies. A typical equivalent circuit is shown in Figure 1. This approach does not
perm it to determ ine the elem ent values from the device parameters and to predict
its bias dependence. The elem ent values of the equivalent circuit are generally valid
only over the frequency range for which the parameter extraction is performed, so
that attem p ts to extrapolate the response of the circuit beyond this frequency range
can produce m isleading results.
The sm all-signal ac m odels quoted in previous section are directly derived from
their dc m odels using the quasi-static approximation.
Since these m odels cannot
account directly for the propagation delay across the channel and for distributed
effects such as the effective channel charging resistances of the device capacitances,
these m ight not successfully sim ulate the observed frequency and bias dependence
of the M O D FE T characteristics. A high frequency ac m odel should account for the
5
g'dd
C'dd
G
S
S
Figure 2: Equivalent circuit for the velocity-saturated intrinsic M O D FET proposed
by Rohdin [21]
propagation delay and th e distributed effects to predict the correct high frequency
dependence o f the M O D F E T characteristics.
Recently Rohdin [21] extended th e unsaturated m odel [10] to th e saturated MOD­
FET with th e aid of a drain resistor and capacitor in parallel to represent th e charac­
teristics of th e saturation region. T here is however no system atic way to predict the
values of th e drain conductance and capacitance from the d ev ice parameters or the
bias conditions. Consequently, their values are fitted so as to o b ta in a good agreement
with the m easured data.
In order to account for these saturation effects in the m icrowave characteristics
of the velocity-saturated M O D FET , th e wave equation in saturation region should
be derived and solved.
T h e equivalent circuit based on wave equation w ill gives
better representation of th e velocity-saturated M O D FET . It w ill give the basis for
6
th e developm ent of a large signal m odel for short channel M O D FETs.
1.3
S tr u c tu r e o f D is s e r ta tio n
In Chapter II th e wave equation of velocity-saturated M O D FE T including both
velocity-saturation and channel length m odulation effects will be derived, using a
sim ple transport picture. T h e intrinsic M O D FET is divided into two regions sepa­
rated by a floating boundary.
The w ave equation holding in these regions will be
derived. B oth an exact solu tion using B essel function as reported in [14] and an ap­
proxim ate frequency power-series solution are derived. The comparison betw een two
m ethod will b e m ade to estim ate the validity of th e analytic frequency power-series
solution.
In Chapter III an equivalent circuit representation of the velocity-saturated MOD­
F E T will be introduced. F irst a sim ple RC equivalent circuit for the GCA region (or
long channel M O D FET ) w ill be derived based on th e frequency power-series solution
of the GCA w ave equation.
An optim al second-order equivalent circuit developed
using fourth order frequency power-series solution w ill also be derived. An equivalent
circuit for the saturation region based on the exact solution in th e saturation region,
will also be developed. T h e total equivalent circuit for the velocity saturated MOD­
FE T will be then constructed by com bining the circuits for the GCA region and the
saturation region.
In Chapter IV we will discuss the integration of this ac-m odel with a dc-model.
We then com pare the frequency and bias dependence o f the scattering param eters cal­
culated from this microwave m odel with the scattering parameters of a A lG aA s/G aA s
7
M O D FET and G aA lA s/In G aA s/G aA s pseudom orphic M O D FE T m easured for sev­
eral bias conditions. In order to introduce som e physical insights, a discussion on the
significance of some of th e physical param eters will be given.
In Chapter V the exact solution of the wave equation of velocity-saturated M OD­
FE T will b e used to analyze the m icrowave characteristics of intrinsic M O D FETs.
The dependence of th e unilateral gain, / x and
/
m a x
dependence upon gate length,
parasitics resistance and capacitance w ill then be discussed.
Chapter V I concludes this dissertation by discussing the future developm ent re­
quired in order to im prove the ac m odel and develop a large signal m odel.
The appendices give th e detailed calculation of the frequency power-series solution
(Appendix A ), exact solution (A ppendix B) and equivalent circuit for th e saturated
M O D FET (A ppendix D ). Also included is M acsym a program to calculate the fourthorder frequency power-series solution of the wave equation.
C H A PTER II
THE VELOCITY-SATURATED M O D FET
WAVE EQUATION AND IT S SOLUTION
2.1
D e r iv a tio n o f th e W a v e-E q u a tio n
T h e distributed ac-m odel for the saturated M O D FE T is based o n a simple b u t well
founded transport picture which assum es that transport is taking place in the 2 DEG
channel and relies on th e following electron velocity-field relation
ve =
—
fiE
for E < E c
v a = nEc
for E > E c
(2.1)
where va is an effective saturation velocity, v, ty p ica lly corresponds to the p e a k ve­
locity of th e stationary velocity-field relation of the m aterial constituting the channel
[22]. V elocity overshoot over the stationary velocity is partially im plied in t h e as­
sumption th a t the electron
velocity rem ains the p ea k velocity v a for channel fields
beyond th e critical field E c.Velocity overshoot above the effective saturation v elo city
is neglected since it has a minimal im p a ct on the d c characteristics for m oderately
sub-micron gate-length M ODFETs. Indeed it have recently b een shown in [22] for
a 0.5 fi M O D F E T that a hydrodynam ic model developed by W id iger and H ess [23],
allowing th e velocity to overshoot u p to three tim e s the peak stationary velocity,
9
Gate
O
Jl
GCA Region
V
S ource
O—
Saturation Region
d
Drain
^ ds
2 DEG channel
Xs
—O
Lg
Figure 3: Idealized representation of th e intrinsic M O D FET
predicted the same drain current and transconductance as a sim ple analytic m odel
using th e peak stationary velocity for effective saturation velocity. The hydrodynam ic
model predicts however a transport induced degradation in th e drain conductance not
accounted for by th e analytic m odel. Therefore th e sim ple transport m odel selected
here is m ore appropriate than th e stationary velocity-field relation used in [9] for
micron and m oderately sub-micron gate length F E T ’s .
For th e purpose o f analysis th e intrinsic M O D FE T is divided into two regions,
the so-called gradual-channel approximation (G C A ) and saturation regions, as was
done in a previously reported dc m odel for th e saturated M O D FET [24]. As shown
in Figure 3, the G CA and saturation regions are located betw een the gate and th e
2DEG channel on th e source and drain side respectively. In the G CA region th e
gradual channel approximation holds and the 2 D E G concentration ns is controlled
by the gate to channel potential
qns ( x , t) = Cg [v©s(<) - vc s ( x , t ) - VT]
(2.2)
where Ca is the 2DEG gate capacitance. In the satu ration region tw o dim ensional field
effects dom inate and th e GCA approximation breaks down. T h e channel potential
v c s can th en be approxim ately ob tain ed by solving the Poisson Equation a lo n g the
2DEG channel
<PvCS( x , t )
— s ^ —
q n s ( x , t)
nrt_ ^
= - s ^ r = m x ’i)
where /? = 1/ e i v sW gd a.
ON
( 2 -3 )
This sim p le model has th e advantage over the G rebene
Ghandhi m odel [24] o f predicting a larger drain conductance.
However a detailed
analysis of two-dim ensional field effects in the satu ration region [22] reveals th a t the
charge distribution in th e channel o n ly partially account for th e d c drain conductance
go (additional contributions to gD appear to be transport and traps related). As it
will be seen in Chapter IV, it is preferable for th e microwave m o d e l to use a physical
value for th e channel w idth da even though an artificial closer fit of the m easured dc
drain conductance g o can be achieved with unphysically large channel openings ds.
Following the pioneering work o f Grebene and Ghandhi it is assumed as in dc
model [24] that the boundary betw een the GCA a n d saturation regions occurs when
the channel field d v c s / d x reaches t h e critical field E c. Consequently the G C A region
includes th e portion o f the channel where the electron velocity has not yet reached
saturation and the saturation region includes the portion of th e channel in w h ich the
velocity saturation is taking place.
Let us now establish the wave equation w hich applies in each region. The wave
equation for the GCA region was derived in [13], [12] and, [11]. The relationship
between th e ac current and voltage in the GCA region is [20]
*'(*) =
\9 (Vac(x))vac{x)]
(2.4)
T he wave equation obtained for th e GCA region is [20]
[9{v Gc{x))vgc(x)\ = j u C g WgVgc(x)
(2.5)
using the function g(VGc(x)) = fJtWgCg(VGc(x) — Vr)- The dc potential Vg c {%) — Vr
is given by (see A ppendix B)
Vg c { x ) - VT = (VGS - VT)^I 1 + (Ps - 2ks) ^ ~
(2.6)
with ks = Vc s (X s )/(V g s —Vt ) and V cs(A s) the dc channel to source potential across
the entire GCA region.
The channel current in the saturation region can be expressed by
I ( x , t ) = Idc + i ( x ) e jut = qWgn s (x, t)v a
(2.7)
and the continuity equation in the channel
dijx^t) =
dx
w d v sn s ( x , t ) _
9
dx
dn s ( x , t ) _
9
dt
1 dl{x,t)
vs
dt
Extracting the ac part from Equation (2.8) and retaining the first order terms yields
12
In th e saturation region the ac current is related to th e ac voltage by the Poisson
Equation (2.3). D ecom posing Equation (2.3) into dc and ac parts yields the following
relationship between the ac voltage vgc(x) and current i
= - m
(2 .io )
Equations (2.9) and (2.10) m ake up th e wave equation for the saturation region.
T h e solution of the w ave equation across the entire channel requires a set of
boundary conditions to b e enforced at x = 0 and x = Lg and at th e boundary
betw een the GCA and saturation region. The boundary conditions to be used at
x = 0 and x = Lg for the com m on source configuration are
Uflc(0)
U^c(T^)
=
vgs
— Vgs
(2-11)
Vds
(2.12)
The continuity of the 2D E G carrier concentration, channel electric field and channel
potential, electron velocity and current are enforced at the G C A /satu ration boundary.
These are naturally enforced by the continuity of the ac voltage vgc and ac current i
at th e boundary.
N ote that according to saturation picture, the channel electric field at the floating
boundary between the G C A and saturation region is the dc (constant) critical field
E c. T he ac channel field is therefore null at the boundary, and the GC A /satu ration
boundary m ust m ove when ac voltages are applied at th e device term inals so as to
m aintain a zero ac channel field. In th e sm all signal analysis the total (dc + ac)
position of the GC A /satu ration boundary is written
13
x s { t ) = X s + x aeiwt
(2.13)
where X s is the dc position and x a th e ac m otion o f the boundary. Let us now derive
th e relationship between th e ac m otion x a of th e G C A /satu ration boundary and the
GCA ac field v'gc. The total (dc + ac) channel field at th e floating boundary xs is
th e spatial derivative of th e total potential vgc at this boundary
v ’ gc ( x s )
= V c c ( x s ) + v'gc(xs )e:,wt
(2.14)
T he ac electric field at th e floating boundary is th en , neglecting second order terms
V ( * s ) = VZc ( X s ) x a + v'gc( X s )
(2.15)
Setting th e ac electric field at th e floating boundary to zero yields th e boundary
m otion x a as a function o f v'„
gc
Xa~
V a d X s f 90^
(2 '16)
where one can easily calculate Vq C{ X s ) to be given by: (see Appendix A .2 for detail
calculation)
k H i-\k .y v c s-v T
Vcc(Xs) - - — S I J ,
X2—
(217)
T he solution of the wave equation across th e entire intrinsic M O D FET relies on
the continuity of the ac voltage and ac current at th e floating boundary. It is therefore
necessary to calculate th e ac voltage at the floating boundary and account for the
m otion o f th e G C A /saturation boundary. Let us derive th e modified G C A channel
potential obtained at th e floating boundary. T h e total (dc + ac) channel potential
at the floating boundary is given by
14
vg c ( x s , t)
= VGc { x s ) + vgc(xs )e3wt
(2.18)
w here vgc is th e ac potential obtained by solving Equation (2.5). Expanding Equation
(2.18) with a Taylor series around the dc boundary position X s for small variations
x s o f the boundary position yields the ac voltage vgc( x s ) at the floating boundary x s
(second order term s are neglected)
vflC(x s)
=
VqC(X s ) x 3 + vgc( X s )
(2.19)
=
—Ecxa + v gc( X s )
(2.20)
T h e potential drop across the saturation region is also m odified by the m otion of
th e boundary w hich m odulates the w idth of the saturation region. Integrating the
Poisson equation
f iVGC^ x } ) = _ j3I{x^
ax*
=
^ + ,(x)eM )
(2>21)
across the tim e varying saturation region yields th e ac potential at x = L g (see
A ppendix A .l for detail calculation)
vgc(Lg) = ( E c + Phcl) Xs + Vgc{xs ) + A vgc(l)
(2.22)
w here we introduced I = (L g — X s ) the dc width of th e saturation region and where
A vgc(l) is the ac potential vgc(x) obtained by solving th e Poisson Equation (2.10) for
a fixed saturation region w id th I, and zero ac potential vgc( X s ) = 0 and zero ac field
v'gc( X s ) = 0 at X s - Substituting Equation (2.20) into Equation ( 2 .22) gives
Vgc(Lg) = f ll d jx s + A Vgc(l) + vgc( X s )
(2.23)
15
One observes that th e contribution of the m otion x a of the G C A /saturation boundary
is to add the ac potential term /3IdJxa.
F in ally note th at the ac current at the floating boundary x s is to first order equal
to the ac current at the fixed boundary X s - T his originates in th e fact that the dc
current Idc is continuous (constant) along th e channel.
2.2
E x a ct S o lu tio n o f t h e V e lo c ity -S a tu r a te d M O D F E T w a v eeq u a tio n
It has been shown by Burns [14] that the voltage-w ave solution of the wave-equation
(2.5) can be expressed in term s of the m odified B essel functions I± 2 / 3 (Y ) (see A p ­
pendix B for detail calculation)
v ( x , u ) = C J 2 / 3 (Y ) + C 2 I . 2 / 3 ( Y )
(2.24)
The current-wave is then derived from Equation (2.4) to be
i(x,u>) = G'doaV S ' P ^ 4 [ C J . 1 / 3 ( Y ) + C 2 I 1/3( Y )]
(2.25)
where Y is a variable defined by Y = 4 /3 y / S ' ( P ) 3^4, P is a position variable defined
by P = 1 — (2ks — k 3 ) x / X a, S' is the norm alized frequency S' = jui/ujok with u>ok =
p{Vgs ~ Vt ){2ks - k l Y / X j , and G'dos = (2A:S - kl)Gdos with Gdos in Section 2.3. T h e
wave-equation in th e saturation region can be readily derived. The current-wave is
obtained by integrating Equation (2.9)
*(*) = i ( X a)e~j ^ {x~x,)
and th e voltage-wave by integrating Equation (2.10)
(2.26)
16
A <W O = \ p 0
)
<(*.)
1 - l]
(2.27)
The unknown coefficients Ci and C 2 are obtained from th e boundary conditions (2.11)
and (2 . 12),
Ygc{Lg) =
kgc(O) =
^ l lC l + A \ 2 C 2 = Vgs — Vds
(2.28)
A 2 \C\ + >122^2 = Vgs
(2.29)
where th e coefficients A,j are evaluated using Equations (2.22) and (2.24)
An
=
h / z m + G ^ J S 'P 'J ' | g ( ^ ) 3 [e- i * ' - l ] + , - ^ l
/ -------
- p l d J
ks
An
=
x
2
[ i _ i / 3( x aj+ h / 3 ( Y ,) \
s ) ^ >u*'
/ - 2/3 (U ) + G i„ .V S ;P ' ' 1 ^ 0 ) ! [ e - > S ' - l ] + j / 3 ^ ;
- p l d J — — 7------, r -------l-/i/3(Es) + i _ 5/3(r s)J
k, (1 - \ k s) Kut
A 21
=
I 2/3
A 22
=
I-2 /3
using A = j4iiA22 — ^ 12^ 21 , Ys — i / ^ ' / S ' ( P s)3^4, and Ps = (1 — ks)2. The unknown
coefficients C 1 and C 2 are then derived from the system of equations (2.29) to be
Cr
O2
=
A 22 — ^12
~v gs
^22.
^
v ds
A
A 2i
A n — A 2i
—
^
Vgs + ^ Vds
(2.30)
(2.31)
F in a lly the ac current flowing into the gate ig and th e ac current flowing into the
drain id are given in terms of th e applied gate to source voltage Vds and drain to
17
source voltage vgs by
id
ig
= i(L g) = i { X s) e - j % ‘
(2.32)
=
(2.33)
G'doay/S' •P.1/ V ' £ , [C'1/ - 1/3(F .) + C 2 I 1 / 3 (YS)\
= i ( 0 ) - i ( L g)
(2.34)
=
G'doaVS> [CaJ_1/3 Q x / S 7) + C 2 I 1 / 3 ( j ^ ) ]
(2-35)
-
G'doaV S ' P ^ 4 e - j ^ l[ C , I . l / 3 (Ys) + C 2 h , 3 (Ya)}
(2.36)
T hese currents hold for arbitrary large frequencies. N ote that the m odified Bessel
functions can be num erically calculated using the expansion [25]
/y\ n
^ = (
7
00
( —) 2j
) Sm ^TTT)
^
The drain and gate currents obtained for th e saturated M O D FE T m odel reduce for
/ = 0 , X s = Lg and V c s(A s) = Vd s to the drain and gate currents of the unsaturated
M O D FET s.
2.3
Y -P a r a m e te r s w ith in th e F req u en cy P o w er-S eries ap­
p ro x im a tio n
The ex a ct solution derived in th e previous section was obtained in term s of modified
B essel functions. Since these functions m ust be generated num erically it is convenient
to use instead a frequency power series solution which usually holds up to very high
frequencies. This frequency power series solution have directly derived from the waveequation using th e m ethod proposed by Ziel and Ero [19]. T he detailed calculation
is described in A ppendix A.
18
The frequency power series yields th e ac current flowing into the gate ig and the
ac current flow ing into the drain id in term s of th e applied gate to source voltage Vda
and drain to source voltage vga.
ig = Y \ \ v ga + Y\ 2 Vda
(2.38)
id = Y 2 1 vga 4- Y2 2 Vda
(2.39)
T h e Y coefficients calculated are the com m on source Y-param eters of the intrinsic
M O D FET. Port 1 is defined between gate and source and port 2 between drain and
source.
Before givin g the calculated Y-param eters let us first define the following terms;
1
A ( k a)
B ( k a)
6 (l
C ( k a)
5-5*.
D ( k a)
*-**'
(I-**.)*
i _ l l 4. i p
E ( k a)
6
6"'» ^
M
J
F ( k a)
Lt
24
30
*
* .) 3
J_p
30 a ' 180
*
(i
G(ka)
H ( k a)
i
_ i t
_1
{i - f a ) 4
i t
I J_p
24
120
4 . -1-k 2
20 a ^
80 * ~
72 »
180 *
(1 - P . ) 5
l_p
1440 *
19
Ri
=
Rgc
=
( 1 - f c , ) 2 IdcPIXs
k]{\ - \ k , f
Vout
■'Os
GdOs
Ry
=
E cX s
(1 - ka)Vout
Ra
=
(1 — ka)R{ + -jjj-Gdoa( l — ka)
Rb
=
1 + (1 — ka)RiRy
Rd
=
1
Ra + Rb
G dO s
1 + @^-GdOs(l — ka) + i? ,(l — ka)( 1 + R y)
nC gW g(VGs - V T )
=
Xs
Vcs(Xs)
ka =
Vgs ~ Vt
C os
Cg W g X s
=
The Y u and Yu param eters for th e saturated M O D FE T are:
~
[~GgaTa + Cga + Ega]
F ii
=
cu2
Y2 1
=
Gga- u
G ga
=
GdoaRd {ka + (1 — ka) R y R i )
C ga
—
R a c G a s D
—
H gs
=
R g c G g s E
— R gcCoaB — R
Ega
—
R gcG gaR d [ R a D + R b E ] — C o aR d [ R a A + R b C ] -------- — G gaG d a
Fga
—
R g c R j
^ l l l - C
G
—
a,T a + H ga + F ga
T2
—Cgara -f- H,gt + ju> [—Ggara + Cga]
where
-
<-gc'~’gs-L'
C o s A — ■
E‘-'gs
,
\sOaJi-
g CE g a D
— Fga
f3l2r
[ R g c G ga ( R
a F
+
R b H
)
—
C q s
(
R
a B
+
R
b G
)
E „ ( R AD + R BE)\ + ^ - G i , G t . - ^ - G i , C 3,
20
Gds is defined in the next section.
T he Yi2 and
for th e saturated M O D FET are:
G dl
JU) [ G dsr s
2 ~ 2 ^Ts ~~ CdsTs + Hda + Fds
Y12
—
W
Y2 2
=
Gds —w2 ^
Gds
=
G dOs R d (1
—
ks)
Cds
=
R g cG d sD
—
E ds
H ds
=
R 2gcG d s F
—
R g c E d sD — Fds
Eds
=
R gcR dG ds
Fds
—
z
l l - C d s T a + Hd,
+
ju> [—GdaTa +
Cds
"I” Eds\
Cds]
where
P I2-
[Ra D + R b E) - ^ ~ G 2da
[RycGj., (R^F
s /-i2
24
~ trj,
Jds —
R g H ) — E^t, ( R a D + R b E)\
P l\
G dsC ds
N ote that th e Y-param eters derived for the saturated M O DFET reduce for / = 0,
X s = L g and V cspC s) = V d s to the Y-param eters reported in [10] for th e unsaturated
M O D FET , excep t, however, for the term f?22(^), w h ich as pointed out by [26] is
incorrect. T h e correct term R 2 2 (k) is
9
R
( k )
22( }
=
^
^(45
180^
720 ^
(l-£*)6
1 6 0 +
14400 k4)
(2.40)
The correction introduced is small.
It is necessary to assess the frequency range of valid ity of the frequency powerexpansion solution. For th is purpose th e frequency dependence of th e Y parameters
21
MAGNITUDE OF Yll
xlOE-3
350
Y
I
N
M
H
0
POWER./
250
150
1
10
100
FREQUENCY (GHz)
Figure 4: Exact solution (solid lines) and frequency power-series solution (dotted
lines) for m agnitude of Yn
PHASE OF Y l l
POWER
80
EXACT
70
60
1
10
FREQUENCY (GHz)
100
Figure 5: Exact solution (solid lines) and frequency power-series solution (dotted
lines) for phase o f Yn
22
MAGNITUDE OF Y12
xlOE-3
3 .5
POWER /
1 .5
EXAC'
0 .5
1
10
100
FREQUENCY (GHz)
Figure 6: Exact solution (solid lines) and frequency power-series solution (dotted
lines) for m agnitude of Yn
PHASE OF Y12
-110
D
E
G
R
E
E
-1 3 0
POWER
-1 5 0
EXACT
-1 7 0
FREQUENCY (GHz)
100
Figure 7: Exact solution (solid lines) and frequency power-series solution (dotted
lines) for phase of Yu
23
MAGNITUDE OF Y21
xlOE-3
160
Y
I
N
M
H
0
140
POWER
j
120
EXACT
100
1
100
10
FREQUENCY (GHz)
Figure 8: E xact solution (solid lines) and frequency power-series solution (d otted
lines) for m agnitude of Y2i
PHASE OF Y21
-20
D
E
G
R
E
E
-4 0
POWER',
-6 0
-8 0
-100
FREQUENCY (GHz)
100
Figure 9: E xact solution (solid lines) and frequency power-series solution (dotted
lines) for phase of Y2i
24
MAGNITUDE OF Yll
xlOE-3
350
POWER./
Y
I
N
M
H
0
250
150
1
100
10
FREQUENCY (GHz)
Figure 10: E xact solution (solid lines) an d frequency power-series solution (dotted
lines) for m agnitude of Y22
PHASE OF Y l l
POWER
80
EXACT
70
60
1
10
FREQUENCY (GHz)
100
Figure 11: E xact solution (solid lines) an d frequency power-series solution (dotted
lines) for phase of Y22
25
Table 1: D evice parameters for calculating the intrinsic Y-param eters
Parameters
l
3
W9
/*
vs
VT
d
ds
Cl
^2
value
gate length (fim)
gate width (fim)
mobility ( cm / V . s e c )
Saturation velocity ( m / s e c )
threshold voltage (V )
gate to channel spacing (A)
channel width in saturation (A)
channel dielectric constant
gate dielectric constant
2
1
290
4400
3.45 x 105
-0.3
430
1500
13.1 e0
12.2 e0
are com pared for a fixed bias of both the analytic and exact solutions in Figure 4 11.
T h e device param eters used for th is comparison are shown in Table 1. T he intrinsic
Y-param eters are calculated for
Vg s
= OV and
Vd s
= 3V . One observes that the
analytic solution com pare to the exact solution for frequencies up to 40 GHz. This
frequency is near th e extrinsic f max as will be found in Chapter IV.
C H A PT E R III
EQUIVALENT CIRCUIT R EPRESENTATIO N
3.1
In tr o d u c tio n
The M O D FE T wave-equation adm its an exact small-signal solution in the frequency
dom ain in term s of th e m odified Bessel functions.
However Bessel functions are
difficult to generate and do not perm it the developm ent of a large-signal m odel. The
approxim ate analytic solutions are preferred for CAD applications since they are fast
and perm it the developm ent of equivalent circuits useful for tim e domain analysis.
The equivalent circuit for the velocity-saturated M O DFET can be directly derived
from the frequency power-series solution using a first-order RC topology for the entire
M O D FET. However th is does not provide a physical representation of the circuit so
that it is desirable to derive the equivalent circuits for each region and cascade them
together to obtain a m ore physical equivalent circuit.
Two m ethods, the iterative and power-series m ethods, have been used to obtain
approxim ate solutions o f the M O D FET wave equation in th e G CA region. The smallsignal Y parameters obtained by an iteration [18] of order two adm it a frequency
power-series expansion valid up to power two. These iterative Y-param eters hold for
higher frequencies and have the advantage of providing a m ore graceful degradation
26
27
outside their frequency range of validity, compared to th e Y-param eters obtained
by the frequency power-series of order two [18]. The iterative procedure yields for
M O D FET up to first order, sm all-signal Y-param eters of th e following form
Y ii = 9ij
1 + jujbij
(3.1)
This equation suggests the first-order RC topology for th e equivalent circuit m odel
in GCA region. However the circuit directly derived from th e iterative m ethod does
not im prove the total performance of the circuit, so that it is needed to derive the
circuit from the second-order power-series solution using th e first-order RC topology.
Based on the first-order RC topology, the second-order RC topology can be derived
from the fourth-order frequency power-series expansion.
In order to develop the equivalent circuit for the saturation region th e Poisson’s
equation and wave equation have to be solved. The solution will account for the
drain delay and th e potential drop in the drain region. A s seen in Chapter II this
equation can be solved exactly and this is used to develop th e equivalent circuit of
the saturation region. Combining the equivalent circuit o f th e GCA and saturation
regions give the com plete equivalent circuit.
3.2
T h e F irst O rder E q u iv a len t C ircu it for th e G C A region
A sim ple RC equivalent circuit m odel will be introduced to provide a graceful degra­
dation of th e Y-param eters for frequencies u> larger than u>o. T he RC m odel selected
consists of th e DC (u; = 0) sm all-signal param eters gij shunted by a capacitor C,j in
28
g o-
-O
R11
©
©
j < o C 12
C11
1 + j C 0R12C 12
s
±
Vds
d
R 22
9d
9m +
C 22
j ( 0 C 2i
1 +jiaR2iC 2i
O -
Figure 12: A pproxim ate sm all-signal equivalent-circuits for th e intrinsic M O D FET
in G CA region.
series with a charging resistor i?,j. The resulting intrinsic Y-param eters are
ju C n
=
Y12
=
Y
-*21
v
-—
_
22
1 + juR nC n
jwCii
1 + ju}Ri2Ci2
9a m
4+
,
9d +
1 + j w R 2\C2\
j u C 22
l + i a ; i ? 22C22
T h e associated equivalent-circuit for th e intrinsic M O D FET is shown in Figure
12. For frequencies to < < 1/ (RijC\j) these Y-param eters adm it the frequency powerseries
Y j — 9ij + j u C i j + u>2 RijCfj
(3.2)
We can now readily identify th e resistors and capacitors to be
^
On
n
_
=
_
9o(Vg s )1'u(k)
w0
--------------------
9 o ( V Gs ) I i 2( k )
D_
tin =
D
_
Ru
go(VGS)Ih(k)
O i2 — ------------------------------------------- J1 1 2 = —
w0
R 12
9 o ( V Gs ) ^ 2 ( k )
29
n
_
9 o(V g s )^2
l(&)
d
Cl1 "
^
n,
_ 9o(VGs)^22{k)
C-22 — --------------------
_
7^21
R n - ~g.(Vo s m * )
n
II22
4122 —
fl'0(V"G5)222( fe)
where
11, i JL J . 2 ____ 7 _ i 3
6
' 80
240
J
12
^ n (A :)
=
Tx x{ k )
=
IM5
(1-5*)
1 - k + \k2
(i n 12(k)
(1 - fc)(l - I k)
2 i a(fc) =
2(1 - I*)*
^ 2l(*) =
=
72.22 (&)
=
X22(*)
=
w
(! ~ * ) ( £ - £ p k + jgfc2 - 3Igfe3)
(1 - ! * ) «
=
J 2i ( *)
i___ 1_ L4
' 360
J
9a
i n ~ 4an'v
JL
I _
3L , U 2
9
(1
43 p _ I p . J L p ___ 1 Jf5
«ftA’ * iaaftf''
i«nnA'
(1 - !2* ) •
1
L i.3
a "■
20
(1 - I*)®
-
* )(&
-
if e * +
-
JL fc* +
(1 - | * ) 6
( ! - * ) ( ! - j * + ^ * 2)
(1 - 12'*)*
The time-constants r,j = RijCij appearing in the small-signal Y-parameters are
then given by
m
-
r i2
-
1
-
t 2
r22 -
D ^
11
11
_ 1 60 — 120& + 81 A:2 —21 A:3 + 2k4
- ^
15^2 _ k^ 6 _ 6k + k2j
D
^ 2C 12 - -
21 21 22
22
1 30 - 41* + 16P - 2*3
^
_ k)
1 600 — 1440* + 1290*2 — 540*3 + 110A:4 — 9 * 5
^
3 Q ^2
_ k^ 3Q _ A5k + 20fc2 _ ^
1 320 - 560& + 340P - 90P + 9*4
- ^
30^2 _ fc^ 2() _ l5k + 3jfc2^
MAGNITUDE OF Yll
12
POWER
EXAC'
0> < v <
8
— B
4
0 ^=
0.1
1
F/FO
10
100
^
igure 13: Comparison of the amplitude of iii/gro-
PHASE OF Y l l
MPWOKO
70
50
\ ^ \ POWER
30
10 L
0.1
1
F /F 0
10
100
Figure 14: Comparison o f the phase o f Yn /g0.
31
MAGNITUDE OF Y12
4
EXAC' ?
POWER /
3
— B
2
1
O1^
0. 1
1
F/FO
10
100
Figure 15: Comparison of the am plitude of Yu/go-
PHASE OF Y12
-110
R
-1 3 0
EXACT
\ \
-1 5 0
-1 7 0
F/F0
p 8 wer
100
Figure 16: Com parison of the ph ase of Y^/go.
32
MAGNITUDE OF Y21
4
POWER
3
EXAC'
I
2
1
0
100
F/FO
Figure 17: Comparison o f the am plitude of Yix/go-
PHASE OF Y21
-40
-80
-120
EXACT
-160
0.1
w
1
F/FO
10
100
Figure 18: Comparison of the phase of Y2i/g0.
MAGNITUDE OF Y22
4
POWER/
EXAC1
OK-^K
3
2
1
0
100
F/FO
igure 19: Comparison of the amplitude of Yn/go.
PHASE OF Y22
MMptJCDWO
45
EXACT
35
25
\ \\
15
\
1
F/FO
10
V POWE t
100
Figure 20: Comparison of the phase of Y^/go-
34
The m agnitude and phase of each Y-param eters are shown in Figure 13 - 20 for
k = 0.65, obtained with the RC equivalent-circuit (dashed-dotted line, E Q ), the exact
solution (plain line, E X A C T ), the frequency power-series (dashed line, P O W E R ), and
the second-order iterative Y-param eters derived in [18] (dashed line, B ). A s can be
seen in figures, the first-order equivalent shows th e graceful degradation.
In order to establish the range of validity of the RC circuit representation for all
bias conditions, the frequency fs%(Yij) for each parameter Yij is calculated. An error
Err(Y{j) of 5 % is obtained between the exact B essel solution and the approxim ate
results for th e frequency fs%(Yij). T he error E r r (Y ij) is
\Yij(exact) — Yij(approximate)\
.
\Yij(exact)\
For the sake o f comparison /s% (Y j)//o for each Yij param eter are plotted in Figure
21-24 as a function of th e biasing param eter k for the frequency power-series model
(dashed line, P O W E R ),the second-order iterative results [18] (dashed line, B2), the
first-order iterative results [18] (dashed line, B l ) and the sim p le RC circuit represen­
tation of th e frequency power-series m odel (dashed-dotted line, EQ). O ne observes
that the sim ple RC representation of the frequency power series holds for all bias
conditions up to a higher frequency than both th e frequency power series and the it­
erative results. On the sam e curve we have also plotted the u n ity current gain cut-off
frequency
fr /fo
(dashed line, FT) and the m axim um frequency of oscillation
f max/fo
(plain line, F M A X ), (frequency at which the unilateral gain is one [27]). B oth f x and
/max are calculated using th e exact Bessel solution.
All approxim ate sm all-signal m odels except th e first-order iterative m odel hold
35
Yll ERROR 5 PERCENT
FMAX
6
4
2
POWER
FT
B1
0
0
0.2
0 .4
0.6
0.8
1
Figure 21: Plot of fs%{Yij)/fo for Y u as a function of the biasing param eter k.
Y12 ERROR 5 PERCENT
FMAX
6
4
2
POWER
FT
B1
0
0
0. 2
0 .4
0.6
0.8
1
Figure 22: Plot of f 5 %(Yij)/fo for Yu as a function o f the biasing param eter k.
36
Y21 ERROR 5 PERCENT
10
FMAX
8
F
6
(
0
4
2
U .
POWER
B1
FT
0
Figure 23: P lot of fs%(Yij)/fo for Y2 1 as a function o f the biasing param eter k.
Y22 ERROR 5 PERCENT
6
FMAX
4
2
POWER
FT
0
B1
Figure 24: P lot of fs%{Yij)/fo for Y22 as a function o f the biasing param eter k.
37
for frequencies larger than the cut-off frequency f o for all bias conditions. The RC
circu it representation holds for frequencies larger than the m axim um frequency of
oscillation f max for k sm aller than ~ 0.9. For k larger than ~ 0.9, / 5% is however
sm aller than fmax• Note th a t both th e exact and the approxim ate m odels predict an
in fin ite m axim um frequency of oscillation at k = 1. O bviously in the extrinsic device
th e unavoidable source, drain and gate resistances and drain output conductance
w ill limit fmax to a finite value. The infinite f max predicted for the intrinsic FET is
nonetheless an indication o f the lim ited validity of the long-channel m odel. Indeed
e v e n in long channel devices the drain current saturation ultim ately results from
v elo city saturation and not pinch-off so that we always have k < 1 in the unsaturated
p a rt of the channel.
To conclude note that th e norm alization frequency f 0 is bias dependent. For gate
voltages approaching the threshold voltage, the norm alization frequency f 0 is small
a n d none of th e se so-called high-frequency approxim ate m odels can account for the
distributed effects arising even at low frequencies.
3 .3
T h e O p tim a l S eco n d -O rd er E q u iv a len t C ircu it
T h e simple R C equivalent circuit shown in Figure 12 is valid when the frequency
considered is sm all enough so that the unsaturated M O D FE T behaves like a lumped
d ev ice. At high-frequencies transm ission line-effects becom e im portant and a secondord er equivalent circuit becom es desirable.
T he topology of the optim al second-order equivalent circuit will be based on
th e second-order RC topology obtained by rewriting the second-order iterative Y-
38
Ri
o------- A V -------- ------ A V ------^ - Ci
Yij ‘ 9|g
=- c2
O
(a)
Ci
°
C2
+
lf"
Yij - 9ig
Ri
(b)
Figure 25: Tw o different optim al second order sm all-signal equivalent circuits for
the intrinsic M O D FET in GCA region
param eters under the form
juciij -|- ( j u f b i j
Yij
— 9ij d-
(3.4)
1 + jujcij + ( j u f d i j
A ctually tw o different second-order RC equivalent circuits can be used to im plem ent
this equation, as is seen in Figure 25.
(a) is preferable over (b) as its topology
physically im plem ents the distributed effects of the channel.
Let us now evaluate each elem ent in th e equivalent circuit (a).
First Equation
(3.4) is rew ritten in term s of the tim e constants Tuj, r 2 ij, and T3,j.
Yij — 9ij
d" jw C ij
1+
j ^ i j
(3.5)
d- ( i^ r ) 2r3<j
where
Cij
TU j
— fljj — C \ij "f" Cij2
Cli jC 2 i j R 2 ij
,
= U}0 b ij = U o ~
W ij T C^2ij
T2ij
= WoCij
T3ij
= ^ Q ^ ij =
W0 =
= U0(C 2ijR 2 ij
W
qC \ij
+
C iijR iij
+
C 2i j R l i j )
Cj2ij R l i j R2ij
0 r
(Vgs - V j)
27T/o = p
yx-----Ll
In order to extract th e value o f t u j , r 2 ij, and Th
Equation
(3.5) is expanded in a
fourth-order frequency power series. The denom inator of Equation (3.5) is obtained
- T V,- „ N2-2 = 1 - b “ r20- +
l + 3 —0TKj + \3—0) T3ij
“o
+ [ j ^ r 2ij + ( ; ^ ) 2r32t, ] 2 - \ j ^ r 2ij + ( A
Wo
Wo
Wo
o
27& ]3
(3.6)
Wo
Substituting Equation (3.6) in Equation (3.5) and neglecting the fifth- and higher
order term s in w gives the fourth-order frequency-power series.
Yu = Sij + i - F i j - U — f S i j + 0 - ) % - - ( j - ) ' D h
Wo
Wo
where th e coefficients, tu j, r 2 ij, and
Wo
(3.7)
Wo
are given in term s of Fij, Sij, T,j, and Dij by
MAGNITUDE OF Yll
ITER
POWER
12
8
4
0
10
0
20
F/FO
30
40
Figure 26: Comparison of the m agnitude of F n/<7o for k = 0.65
PHASE OF Y l l
ITER
POWER
-10
F/FO
F igu re 27: Comparison of th e phase of Yn /g0 for k = 0.65
MAGNITUDE OF Y12
5
ITER
4
POWER
3
2
1
0
0
10
20
F/FO
30
40
Figure 28: Comparison o f th e m agnitude o f Y u / g o for k — 0.65
PHASE OF Y12
D
E
G
R
E
E
POWER
-4 0
-8 0
-120
EXACT
-1 6 0
ITER
-200
F/FO
Figure 29: Comparison o f th e phase of Y^/go for k = 0.65
MAGNITUDE OF Y21
5
ITER
4
POWER
EXACT
3
2
1
0
0
10
20
F/FO
30
40
Figure 30: Comparison of th e m agnitude of F21/50 for k = 0.65
PHASE OF Y21
D
E
G
R
E
E
-4 0
-8 0
-1 2 0
ITER
-1 6 0
-2 0 0
F/FO
Figure 31: Comparison o f the phase o f Y2i/go for k = 0.65
MAGNITUDE OF Y22
5
IT E R
POWER
4
EXACT
3
2
1
0
0
20
10
F /F O
30
40
Figure 32: Comparison of the m agnitude of Y 2 2 / 9 0 for k = 0.65
PHASE OF Y22
140
100
D
E
G
R
E
E
EXACT
IT E R
0WER
-2 0
-6 0
-100
20
F /F O
Figure 33: Comparison of th e phase of Y2 2 /go for k = 0.65
44
T h e coefficients F,j, Sij, Tij and D{j can be obtained from the M O D FET waveequation using th e m ethod developed by Ziel [20].
The procedure used and the
obtained F,j, Sij , TtJ- and Dij coefficients for each Y param eters Yij are given in
A ppendix C. N ote that these parameters are all dependent on the norm alized bias
param eter k = v v*‘V t ■ The elem ents of the optim al second-order RC circuit can now
be obtained by inverting the system of Equations (3.5) so as to express R u j , R 2 ij,
Cuj, and C 2 ij in term s of rltj, r 2 ij, and r3tj.
R \ i j
—
ik
T\ij
_________ (Cj jl ~3j j ~ T \ j j T 2i j ) 2_______
r lij(CfjTlj - CijTujTxj + Tfc)
C
U j
k
=
C i j T 3 ij
C 2 ij
=
C j j T3 ij
~
~
T l i j T2 ij
C i j T l i j T 2 ij
+
^ • j T3i j ~ T U j T 2ij
T h e m agnitude and phase of each Y-param eters are p lotted in Figure 26 - 33
to dem onstrate th e graceful degradation. The Y-param eters are obtained w ith the
second-order equivalent circuit (dashed-dotted line, EQ ), th e exact solution (plain
line, E X A C T ), the frequency power series (dashed line, P O W E R ), and fourth order
iterative solution (dashed dashed line, ITE R ). A s can be seen in figures, th e secondorder equivalent circuit exhibits a more graceful degradation compared to th e fourthorder iterative solution and power series solution. The frequency range of validity of
this equivalent circuit can be evaluated by calculating the frequency /s%(Y'j) for each
param eter Yij for which a 5% error is obtained. Figure 34 - 37 show /s%(Tij) for each
param eter Yij. O ne observes that the circuit holds to a much higher frequency than
45
Yll ERROR 5 PERCENT
60
50
40
30
20
IT E R 4tH
10
0
IT E R 2nd
l****"-**"T * ^ —-fc-» -i—AU
1—
POWER
K
F igure 34: Plot o f / 5% (Kj)//o for Y\\ as a fu n ction of the biasing param eter k.
Y12 ERROR 5 PERCENT
EQ
IT E R 4 th
_ITER 2nd
~*1 ' 1
— •*—
POWER
K
Figure 35: Plot o f /s%(VIj)//o for Yu as a function of the biasing param eter k.
46
Y21 ERROR 5 PERCENT
80
60
F
/
F
0
40
ITER 4 th
20
IT E R 2nd
0
POWER
Figure 36: Plot of /s% (Y j) /fo for Y2\ as a function of the biasing param eter k.
Y22 ERROR 5 PERCENT
40
30
F
/
0
F
20
ITER 4 th
10
0
POWER
4 th
0
0.2
0 .4
0.6
0.8
1
Figure 37: Plot of / 5 % (Y j)//o for Y22 as a function of the biasing param eter k.
47
Gate
Drain
111
R 211
^T '
Cm
Source
122
Source
Figure 38: Approxim ate second-order sm all-signal equivalent-circuits for the
intrinsic M O D FET
any other m odel for all k values except for >2i- /s%(F2i ) / fo for Y 2 1 is the sam e as
th a t of first-order RC equivalent circuits at k = 0.9.
As m entioned above tw o different topologies for th e second-order equivalent circuit
are possible (see Figure 2 5 ). The transfer functions of both circuits for sm all-signal
analysis in th e frequency dom ain are th e sam e, even though different RC elem ents
are used. For th e same b ias condition, th e values o f R iy and C iy in circuit (b) are
th e same as th at of Ry a n d Cy in th e optim al first-order RC circuit. This m eans
th a t circuit (b ) extends th e frequency range by adding the R2ij, C2y circuit to the
in itial RC circuit. However in circuit (a) th e sum of capacitors, Cu j + C^y, give the
capacitor C y o f the optim al first-order R C circuit. T h e small-signal analysis does
n ot differentiate between (a ) and (b), how ever circuit (a) will be preferable to circuit
(b) since circuit (a) is a m ore physical representation o f the distributed channel for
large-signal analysis. T he resulting equivalent circuit for long gate length device is
48
show n in Figure 38.
3 .4
T h e V e lo c ity -S a tu r a te d M O D F E T E q u iv a len t C ircu it
T h e sm all-signal model presented above for th e intrinsic M O D FET holds only for
th e region of th e channel for w h ich the gradual channel approxim ation (G C A) holds.
H owever in saturation it b eco m es necessary to account for the contribution of the
b u ilt-in potential. A more co m p le x equivalent circuit results in which the equivalent
circu it introduced for the M O D F E T wave-equation is now just a subcircuit.
L et us dem onstrate this approach for th e velocity-saturated M O D FE T waveequation.
In th is conventional M O DFET m odel the F E T channel is divided into
th e GCA and saturation regions of length X s = Lg — £ and £ respectively. In the
saturation region the electron velocity is assum ed to saturate (to a value v s) while
th e GCA is failing. The channel potential in the saturation region is then assumed
to b e supported uniquely by t h e electron distribution in th e channel.
T h e derivation of the equivalent circuit for saturation region is given in Appendix
D , which is based on the exact solution. B y combining tw o equivalent circuit, one of
G C A region and the other o f saturation region, one can obtain Figure 39.
T h e total Y-parameters Y i j( s ) in terms o f the Y-param eters of the GCA region
Yij(g) of reduced gate length X s = Lg — £, are obtained as follows
Yn (sat)
=
Yn (s ) + V M S , +
Ynisat)
=
Yn{g)l, +
1 + r » (j)& (« )
~
(J )(1 -
~ U ^)Yn(g))
- * < « ) * ■ ( ,) )
49
>d
I
Jy.
——o—'V'vAr——
-°d
Rdd
R od
CD CD
X
go
YsVd. * v . ©
Y^)Vd6
Cgg
Y,a(®)
Cdd
6
6
>d6
■JOT,
icoCnd
1 +jtORg(JCg(j
Y2i(co) = g m +
foCda
1 +jmRdgCdg
Figure 39: First-order non-quasi-static equivalent circuit for the velocity-saturated
M O D F E T wave-equation
* 22(sat)
=
-, v e"Jur'
1 -f Y2 2 {g)Zs {ui)
where t , = v , /£ is th e transit tim e of the saturation region, Z,(u) an im pedance
specified below and
7. = 1 - S , =
6
, and y s tw o constants given by
x + p lDc(A
w ith
A
=
B
=
2X 5(1 - ks)
( 2 k, - k 2 )(Vas - VT)
4 X 5(1 - k , f
G dosi^k,
N ote th a t k, =
— k f ) 2(VGs
~ Yt )
Vc s ( X s ) /( V g s — Vt )
and Gdos = ^ C gWg(VGs — V t ) / X s are
values
used for k and the drain conductance gd respectively, in th e G CA Y-Param eters Yij(g)
given in section 3.2.
50
MAGNITUDE OF Yll
xlOE-3
600
I
EQUI
POWER
Y
I
N
400
M
H
0
200
F/FO
Figure 40: Comparison of the am plitude of
Yu
for
Vd s
= 3 F and
Vg s
= 0V.
PHASE OF Y l l
90
70
50
EXACT
EQUI
30
10
POWER
0
2
4
F/FO
6
8
Figure 41: Comparison of the phase of F n for Vds = 3 F and Vgs = OF.
51
MAGNITUDE OF Y12
xlOE-3
20
/ POWER
16
EXACT
EQUI
12
8
4
0
0
2
4
F/FO
8
6
Figure 42: Comparison of the am plitude of Yi2 for
Vds =
10
3 V an d
Vg s = O V .
PHASE OF Y12
300
200
D
E
G
R
E
E
POWER
100
EXACT
-100
F/FO
Figure 43: Comparison of the phase of Y12 for Vds = 3F and Vgs = 0V .
52
MAGNITUDE OF Y21
200
160
I
POWER
Y
N
120
M
H
0
EXACT
EQUI
F/FO
Figure 44: Comparison o f the am plitude of F2i for
Vd s
= 3F and
Vg s =
OF.
PHASE OF Y21
400
300
D
E
G
R
E
E
POWER
200
100
EQUI
EXACT
-100
F/FO
Figure 45: Comparison o f the phase o f Y2 1 for Vds = 3 F and Vgs = OF.
53
xlOE-3
MAGNITUDE OF Y22
2U
/ POWER
EXACT
EQUI
F/FO
Figure 46: Comparison o f the am plitude of K22 for Vd s = 3 V and Vgs = OV.
PHASE OF Y22
400
POWER
300
G
R
EXACT
200
EQUI
100
F/FO
Figure 47: Com parison of the phase of F22 for Vbs = 3V and Vgs = OV.
54
T h e im pedance Z a(u) is approxim ated by a first order RC network providing th e
correct second-order frequency power-series expansion
= £.1 + i1 r+ ?j u rC aRDa 2
(3-9)
with
p i DCt B - \ p e
R si
=
R s2
=
„
_
3(1 + P I d c U )
3 (1 + p i p p i A )
~
8
1 + PI dc ^A
2pe2
9
Ta
pi2
using P = 1l t xv aWgda.
T h e resulting equivalent-circuit provides an optim al first-order non-quasi-static
equivalent-circuit adm itting th e correct second-order frequency power expansion as
well as a graceful degradation. T his is dem onstrated in Figure 40 - 47 for an intrinsic
M O D FE T with th e parameters given in Table 1 in Chapter II and for an intrinsic
bias o f
Vd s
=
3
V and
Vg s
= OV. The phase and am plitude of Y-param eters versus
frequency calculated using this first-order RC equivalent-circuit (dashed-dotted line,
E Q U I), the exact solution (plain line, E X A C T ), and th e frequency power-series ap­
proxim ation (dashed line, PO W E R ) are compared in figures. The optim al first-order
RC m odel (EQ U I) is seen to hold to a m uch higher frequency than th e frequency
power-series approximation (P O W E R ).
C H A PTER IV
PR E D IC T IO N OF THE MICROWAVE
C H ARACTERISTICS OF M O D FET’S
4.1
E x tr a c tio n o f p a ra m eters for t h e ac m o d e l
The intrinsic ac model developed relies on the m aterial, device and bias param eters
ei,
L g, W g ,
(i, v„, Cg,
V t, V g c {X s), I
or X s ,
VG , I d c ,
which sh ou ld all be obtained
directly from the dc-m odel. As was m entioned in Chapter II, it is necessary to use an
accurate dc m odel to m o d e l the m icrow ave performance. The dc m odel used is th e dc
m odel recently reported [24], except for th e saturation voltage 14 =
Vd s
—Vcs(Xs)
which in accordance w ith the proposed saturation picture is derived from Equation
(2.3) to be
14 = I d c X l H ^ d ' W g V , ) - ECX S
(4.1)
This dc m od el provides tw o features: a field dependent m obility (see Equation (4.3)
below) and a non-linear charge control
Tis = nso [(1
(1 —
- a )) + atanh
Vq c - V
gm'
14
which perm its one to o b ta in an im proved fit of th e dc-characteristics (nso,
(4.2)
Vg m , a ,
and 14 are used as fittin g parameters). Since the ac-m odel introduced in C hapter II
55
56
relies on a constant m obility /z (in th e GCA region), constant gate capacitance Cg,
and constant threshold voltage Vt an extraction theory is required. Following the
approach described in reference [10] th e m obility is given by,
** = 1 + E ( 0 )/E !
where Ei = E cf (/j E c/ v a — 1). N ote th at the m obility used by th e ac-m odel is the
chordal m obility and not th e differential m obility (see Gunn [28]).
In reference [10] the threshold voltage was calculated using
VT = V g c ( X s ) x
14
'
^
L(l-°)
( V c c ( . X s ) - Vc m
V
cosh2 ^ G c i X s ^ G M ^
(4.4)
and the 2D E G capacitance using
= „n,„(.1 - ° ) ^
Vi
F g c (0) — Vgm
V!
(4.5)
T hese expressions are only valid for sm all gate-to-channel voltages. At large gate-tochannel voltages the high-bandgap m aterial between the gate and th e 2DEG channel
is no longer fully depleted.
This leads to the saturation of th e 2DEG concentra­
tion in Equation (4.2), an effect reflected in turn in Equations (4.4) and (4.5) by
th e reduction of both the threshold voltage and th e 2DEG capacitance.
However
due to the large RC constant of the depleted parasitic M ESFET channel, th e charge
distribution (electrons and ionized donors) in the high-bandgap m aterial does have
tim e to respond at high-frequencies. T h e 2DEG capacitance at large gate-to-channel
voltages is then lim ited by the m axim um 2DEG capacitance Cgmax =
+ Ad) ,
57
where d is the gate to channel spacing and A d is a constant arising solely from the
variation of th e Fermi level w ith the 2DEG concentration [29]. T he gate-to-channel
voltage
Vgco
a t which this ta k e s place is sim ply obtained from C*3(Vgco) =
C a m ax-
For gate-to-channel voltage larger than Vgco the threshold voltage selected is then
given by Vrmax = Vt(Vgco). T h e use of th e m axim um 2DEG capacitance and m ax­
im um threshold voltage at m icrow ave frequencies for large gate-to-channel voltages
supports the n otion that the R F transconductance gxf (R F) can be larger than the dc
transconductance gM{dc). A greatly improved fit of the scattering param eters results
from this choice.
4 .2
C o m p arison o f t h e m ea su red and c a lc u la te d d a ta
In order to te s t this m icrowave model, th e theory is applied to tw o one-micron
g a te length devices; an nAlo. 2 sGaO' 7 5 A s / iA lo . 2 sGao. 7 5 A s / G a A s (35 0 /3 0 /1 0 ,0 0 0 A )
M O D FET (d ev ice 2045) and an nAl.i 5 Gao,s 5 A s /iA lo .i 5 Gao,s 5 A s / I n o . 2 Gao,&A s / G a A s
(3 5 0 /3 0 /1 5 0 /1 0 ,0 0 0 A) pseudom orphic M O D FE T (device 2379). T h e data are ob­
tained from U niversity of Illinois at Urbana-Champaign.
The device parameters u sed for fitting th e IV characteristics of th e devices are
shown in Table 2. The com parison of the calculated and m easured IV characteristics
are shown in Figures 48 and 49. The device parameters for device 2045 are the sam e
as reported in [10] except for ds , R a, and Rd- A smaller value for d„ (closer to the
equilibrium channel width, see [22]) is used. T h e nonlinear source and drain resistance
m odel reported in [30] is used for R s and RdThe equivalent circuit used for the extrinsic m odel is shown in Figure 50. Dis-
58
Table 2: Device param eters for th e dc characteristics of th e A lG aA s/G aA s
M O D FE T (2045) a n d th e G aA lA s/In G aA s/G aA s pseudom orphic M O D FE T (2379)
Parameters
Lg
wg
V
va
Ec
n so
Vj
V gm
gate length (fim)
gate width (fim)
m obility (cm / V . s e c )
Saturation velocity ( m / s e c )
Critical Field ( K V/cm )
( cm -2 )
2
(V )
(V )
a
d
Ad
ds
Rs
Rd
C\
C2
gate to channel spacing (A)
see [29] (A)
channel width in saturation (A)
low field source resistance (fl)
low field drain resistance (fl)
channel dielectric constant
channel dielectric constant
2045
2379
1
1
290
290
4400
5600
3.45 x 105
3.5 x 105
10.9
31.2
1.02 x 1012
1.0 x 1012
0.3
0.19
0.15
-0.12
0.5
0.5
380
380
50
50
1500
500
3
2.7
6
2.7
13.1 eo
12.44 e0
12.2 eo
12.65 e0
59
I V CHARACTERISTICS
x lO E -3
VG=0.7
VG=0.5
30
— VG=0.3
" VG=0.1
VOLTAGE
Figure 48: Measured (solid lines) and calculated (dotted lines) IV characteristics of
th e A lG aA s/G aA s M O D FET 2045
IV CHARACTERISTICS
x lO E -3
VG=0.7
60
VG=0.5
VG=0•3
40
VG =0.1
20
V G = -0 .1
V G = -0 .3
0
VOLTAGE
Figure 49: Measured (solid lines) and calculated (dotted lines) IV characteristics of
the pseudom orphic G aA lA s/In G aA s/G aA s M O D FET 2379
60
CGd
Gate
Drain
GS
Source
Figure 50: Equivalent circuit for th e extrinsic M O D FET
tributed effects along th e gate w idth are not included, as they were found to be sm all
for the gate w idth and frequencies considered. T h e value of the parasitics used for
calculating th e extrinsic Y parameters are shown in Table 3. The gate to source
and gate to drain
Cgd
Cgs
fringe capacitances were estim ated to be 50-70 fF (see [21]).
T h e bond inductance at the gate, source and drain term inals and gate resistance R g
are obtained from deem bedding. T he source and drain resistances used are obtained
from the dc model.
The scattering param eters were m easured from 2 GHz to 18.4 GHz for 5 bias
conditions for device 2045 and 13 bias conditions for device 2379.
The extrinsic Y param eters, including parasitics, were calculated from the ac-
61
BIAS CONDITION
VGS = 0.08
VDS = 0 . 5
S12 x 10
Figure 51: M easured (solid lin es) and calculated (dotted lines) scattering param eters
for Vqs = -08 V and Y d s — 0-5 V for the A lG aA s/G aA s M O D FE T 2045.
UNILATERAL GAIN
30
VGS = 0 .0 8
VDS = 0 .5
A
I
N
(dB)
10
0
1
-
10
FREQUENCY (GHz)
100
Figure 52: M easured (solid lin es) and calculated (dotted lines) unilateral power gain
for VGS = .08 V and Yds = 0.5 V for the A lG aA s/G aA s M O D FET 2045.
62
BIAS CONDITION
Figure 53: Measured (solid lines) and calculated (d otted lines) scattering param eters
for VGS = .08 V and VDS = 0.75 V for the A lG aA s/G aA s M O D FE T 2045.
u n il a t e r a l g a in
30
VGS = 0 .0 8
VDS = 0 .7 5
20
10
0
1
10
FREQUENCY (GHz )
100
Figure 54: Measured (solid lines) and calculated (d otted lines) unilateral power gain
for V g s = -08 V and Vbs = 0.75 V for the A lG aA s/G aA s M O D FE T 2045.
63
90
b ia s
condition
VGS = 0.08
-]50
F ig u r e 55: M easured (solid lin e s ) and calculated (dotted lin e s ) scattering parameters
for VGS = .0 8 V and VDs = 1-0 V for t h e A lG aA s/G aA s M O D FET 2045.
u n il a t e r a l
g a in
30
VGs = 0 .0 8
VDS = 1
20
10
0
1
10
FREQUENCY (GHz )
100
F ig u r e 56: M easured (solid lin e s ) and calcu lated (dotted lin e s ) unilateral power gain
for Vgs = .0 8 V and Vps — 1-0 V for t h e A lG aA s/G aA s M O D FET 2045.
64
BIAS CONDITION
Figure 57: M easured (solid lines) and calculated (dotted lines) scattering parameters
for Vgs = -08 V and VDS = 3.0 V for the A lG aA s/G aA s M O D FE T 2045.
UNILATERAL GAIN
30
VGS = 0 . 0 8
VDS = 3
20
10
0
1
10
FREQUENCY (GHz)
100
Figure 58: M easured (solid lines) and calculated (dotted lines) unilateral power gain
for V g s = -08 V and V d s = 3.0 V for the A lG aA s/G aA s M O D FE T 2045.
65
BIAS CONDITION
Figure 59: M easured (solid lines) and calculated (dotted lines) scattering
param eters for V d s — 3 V and V g s = - 0.15 V for th e G aA lA s/InG aA s/G aA s
pseudom orphic M O D FE T 2379.
UNILATERAL GAIN
30
VGS = - 0 . 1 5
VDS = 3
G
A
I
N
20
<dB)
0
1
10
FREQUENCY (GHz)
100
Figure 60: Measured (solid lines) and calculated (dotted lines) unilateral power gain
for V d s = 3 V and V g s — -0.15 V for th e G aA lA s/In G aA s/G aA s pseudom orphic
M O D FE T 2379.
66
BIAS CONDITION
Figure 61: M easured (solid lines) and calculated (d o tte d lines) scattering
param eters for V d s = 3 V and V g s = -0.08 V for th e G aA lA s/InG aA s/G aA s
pseudomorphic M O D FE T 2379.
UNILATERAL GAIN
30
VGS = - 0 .0 8
G
A
I
N
(dB)
VDS = 3
20
10
0
1
10
FREQUENCY (GHz)
100
Figure 62: M easured (solid lines) and calculated (dotted lin es) unilateral power gain
for V d s = 3 V and V g s = -0.08 V for th e G aA lA s/In G aA s/G aA s pseudomorphic
M O DFET 2379.
67
BIAS CONDITION
Figure 63: M easured (solid lines) and calculated (d o tted lines) scattering
param eters for V d s = 3 V and Vgs = 0.25 V for th e G aA lA s/InG aA s/G aA s
pseudom orphic M ODFET 2379.
UNILATERAL GAIN
30
VGS = 0 .2 5
G
A
I
N
(dB)
VDS = 3
20
10
0
1
10
FREQUENCY (GHz)
100
Figure 64: Measured (solid lines) and calculated (dotted lines) unilateral power gain
for V d s = 3 V and V g s = 0.25 V for th e G aA lA s/InG aA s/G aA s pseudom orphic
M O D F E T 2379.
68
BIAS CONDITION
F igure 65: M easured (solid lines) and calculated (d otted lines) scattering
param eters for V d s = 3 V and V g s = 0.56 V for the G aA lA s/In G aA s/G aA s
pseudom orphic M O D FE T 2379.
UNILATERAL GAIN
30
VGS = 0 . 5 6
VDS = 3
20
G
A
I
N
(dB)
10
0
1
10
FREQUENCY (GHz)
Figure 66: Measured (solid lines) and calculated (dotted lin es) unilateral power gain
for V d s = 3 V and V g s = 0.56 V for the G aA lA s/In G aA s/G aA s pseudom orphic
M O D FET 2379.
69
T able 3: M icrowave parasitics for the A lG aA s/G aA s M O D FE T (2045) and the
G aA lA s/In G aA s/G aA s pseudom orphic M O D F E T (2379)
2045
Parameters
2379
C GS
Gate to source parasitic capacitance (fF )
50
70
C
Gate to drain parasitic capacitance (fF )
50
50
RG
Gate resistance (Q)
4.16
5.4
LG
Gate bond inductance (nH )
0.35
0.3
L5
Source bond inductance (nH )
0.08
0.07
LD
Drain bond inductance (nH )
0.33
0.36
gd
m odel for the sam e bias conditions and then converted to scattering parameters for
comparison with th e measured data.
T h e scattering param eters and unilateral power gain versus frequency for four bias
conditions are show n in Figures 51 - 58 for device 2045 and Figures 59 - 66 for device
2379. The scattering parameters for
Vg s
= -08 V and
Vd s
= 0.5 V, 0.75 V , 1 V, and
3 V are shown in Figures 51, 53, 55, 57 respectively for the A lG aA s/G aA s M ODFET.
The scattering param eters for Vds = 3 V and Vgs = -0.15 V , -0.08 V , 0.25 V, and
0.56 V are shown in Figures 59, 61, 63, 65 respectively for th e G aA lA s/InG aA s/G aA s
pseudomorphic M O D FE T . In th e polar and Sm ith chart p lots, the solid line is used
to represent the m easured data and the d otted line is used for the calculated data.
Figures 52, 54, 56, 58, 60, 62, 64, and 66 show the unilateral power gain given by
[31]
70
Table 4: D eviation of calculated S param eters from th e m easured d a ta for all bias
conditions for the device 2045 ( d a = 1500 A , C g d = 50 fF, and C g s = 50 fF)
Vi.
AS„
<1
0.08
0.25
0.09709
0.08
0.5
0.08
^
Vg.
a s 22
A f7(d B )
0.12828
0.04286
0.08892
2.41553
0.12094
0.41329
0.03103
0.12960
0.60579
0.75
0.14212
0.61521
0.03263
0.21576
0.61067
0.08
1.0
0.12278
0.61916
0.04603
0.25717
1.45705
0.08
3.0
0.30447
1.32780
0.07536
0.28605
1.16160
rH
cs
A 5 i2
__________IF21 —F12 1_________
, . gx
“ 4 (R e (y n )R e(F22) - R e ( y 12)R e(F 21))
1
'
In these figures, the solid lin e represents the value o f U obtained from the m easured
data and th e dotted line th e value of U calculated from the m icrowave model.
To estim a te the perform ance of th e m odel the error obtained for all m easured
bias conditions is sum m arized in Table 4 and Table 5 for devices 2045 and 2379
respectively. Since scattering parameters are norm alized quantities, th e error used is
1 N
A Sij = "T7 ) ^ | Sijdaia (cj,')
-'V i=l
Sijcalculatedi^i) |
(4.7)
As can be seen in Figures 52, 54, 56, 58, 60, 62, 64, and 66 th e calculated and
measured unilateral gain plotted on a log scale differ by a few dB over the entire
frequency range. The error used for th e unilateral power gain is therefore
1 N
AUdB =
I l0log(Udata(Ui)) ~ 10log(Ucalculated^)) \
(4.8)
71
Table 5: D eviation of calculated S param eters from th e measured d a ta for all bias
conditions for the device 2379 ( d s = 500 A, C g d = 70 fF, and C g s = 50 fF)
vg.
Vu
A 5U
<1
0.07
0.25
0.10488
0.07
0.5
0.07
A522
AC7(dB)
0.09403
0.06810
0.12519
3.25594
0.13381
1.01989
0.06220
0.54647
3.10049
0.75
0.14001
0.93923
0.03592
0.43403
1.10848
0.07
1.0
0.14074
0.67644
0.03820
0.32584
2.06435
0.07
2.0
0.15359
0.62508
0.05199
0.26670
3.08550
-0.15
0.5
0.14178
0.70650
0.03433
0.34632
1.24863
-0.08
0.5
0.12351
0.83907
0.04749
0.43257
0.99085
0.25
0.5
0.11836
0.16675
0.05722
0.10181
4.08520
0.5
0.5
0.13181
0.06170
0.06089
0.15007
0.87395
-0.08
3.0
0.14849
0.67419
0.04942
0.26523
2.96939
-0.15
3.0
0.15725
0.72022
0.04232
0.27461
2.60278
0.25
3.0
0.12164
0.89009
0.06550
0.24644
1.71173
0.56
3.0
0.09435
1.42184
0.07672
0.27800
1.19325
i-M
a s 12
72
The error averaged o v er all bias con d ition s is given in Tables 4 and 5 for devices
2045 and 2379, respectively. A minimum average error on the unilateral gain of 1.25
dB and 2.17 dB is obtained for all bias conditions for devices 2045 and 2379 respec­
tively. N ote that an error o f + 2 dB in th e unilateral gain corresponds to an error in the
calculated f max given by f m a x { d a t a ) / f max(theory) = 1.26. This error is reasonable
considering that apart fro m ds and C q S an d C g d , all o f the m odel param eters are
extracted from the dc m easurem ent or th e microwave deem bedding. A m ain source of
error originates from the d c drain conductance and transconductance, which are only
approxim ately fitted w ith th e present extraction techniques. This em phasizes the
im portance of an accurate d c model. It is believed th a t the error obtain ed could be
reduced w ith an improved d c parameter extraction technique. A dditional substantial
error m ight also originate from calibration and deem bedding.
4.3
D iscu ssio n
It is now relevant to elab orate on the physical insights gained from th e ac-model
reported for the saturated M ODFET. F ir st let us ju stify the need for an ac-m odel for
th e saturated MODFET.
As m entioned the p reviou s solutions for the wave-equation were only valid up
to the edge of saturation which occurs w hen the channel field is th e critical field
E c. It is important to n o te that one ca n n o t infer th e performance of the device in
saturation from the perform ance of the d e v ic e at the ed g e of saturation. Indeed once
th e device is driven deeper in saturation, one observes a substantial increase of the
transconductance, a stron g reduction o f th e drain conductance and a reduction of
73
th e effective channel length. These effects drastically modify, and in fact im prove the
device performance.
In a recent paper [21], Rohdin reported an ac-m odel for the saturated M O D FE T
(shown in Figure 2 in Chapter I). In his m odel the GCA region is described b y an
equivalent circuit based on the Y-param eters derived for the unsaturated M O D F E T
[10]. To account for the saturated region, Rohdin introduces in series in the drain
term inal, a conductance gdd> lim iting th e output conductance. A capacitance C d d ' in
parallel with gdd' is introduced to account for the charge m odulation in the saturation
region. U sing his model Rohdin develops a m ethod perm itting one to extract from
th e measured Y-param eters the source resistance, fringe capacitances, gate len gth
and effective saturation velocity. The physical equivalent circuit m odel proposed by
Rohdin em phasizes the im portance of th e saturation region.
Interestingly the GCA region in a saturated M O D FE T does not behave, at highfrequencies, as the GCA region of th e unsaturated M O D FET. This is verified by
developing a m odel relying on the Y-param eters derived for the unsaturated M O D ­
F E T (using for gate length: X s ) and th e exact solution of the wave-equation in the
saturation region. The continuity of th e total current and voltage was enforced at
th e floating boundary. T his approach failed com pletely, and th e S-parameters cal­
culated were uncorrelated to the m easured ones, for one cannot use the unsaturated
Y-param eters in the GCA region of th e saturated M O D FET. Indeed these u n sa tu ­
rated Y-param eters are derived with th e boundary conditions
Vgco(Xs-)
=
vgc0( X s + , t )
(4.9)
74
Table 6: D eviation of calculated S parameters from the m easured data for device
2045
<ASu>
<AS21)
(A S 12)
(A S 22)
(A f/)(d B )
d s = 500
0 .1 5 2 2
0 .7 4 1 8
0 .0 4 8 4 8
0 .2 5 7 4
1 .8 8 9 8
ds =
0 .1 5 7 4
0 .6 2 0 7
0 .0 4 5 5 8
0 .1 9 5 5
1 .2 5 0 1
0 .1 6 7 3
0 .5 8 0 9
0 .0 4 4 2 9
0 .1 8 9 7
1 .5 2 6 0
0 .3 5 4 9
1 .2 7 3 8
0 .0 6 1 9 9
0 .3 7 8 3
3 .0 4 2 6
ds =
A
1500 A
2500 A
Cgs — C g d ,
v g c i ( X
s
—)
=
—
0
V g C2 ( X s
—) = 0
( 4 .1 0 )
which are incorrect for the saturated M O D FET. Instead it m ust be assumed, as is
done in A ppendix A , that each voltage com ponent is continuous: wflCo,i,2(-Xs—) =
u3co,i,2(-^5 + ) . Note th a t these correct boundary conditions are naturally enforced by
a circuit im plem entation such as th e one proposed by Rohdin [21].
Let us now consider in more d etail the im pact of ds upon th e scattering parameters.
ds is in ten d ed to represent the effective channel w idth in the saturation region. T he
best fit of th e IV characteristics at all bias conditions was obtained for d„ = 2 5 0 0 A
and ds = 1 5 0 0 A on devices 2 0 4 5 and 2 3 7 9 respectively.
In Tables 6 and 7 it is
compared th e error on the scattering parameters and unilateral gain averaged over
all bias p oin ts obtained for different values of ds. It is noted from Tables 6 and 7 that
a reduced error results from using a value of da = 1 5 0 0 A and da = 5 0 0 A which is
closer t o t h e equilibrium channel w idth (see [2 2 ]). This supports the concept that th e
RF extrinsic drain conductance g o ( R F ) can be sm aller than the dc extrinsic drain
75
Table 7: D eviation of calculated S param eters from the m easured data for device
2379
A
da — 500 A
ds = 1000 A
ds = 1500 A
ds = 300
C g S
=
C g d
=
0
<ASU >
(A S 21)
<ASia)
(A S22)
(A C /)(dB )
0.1323
0.7153
0.05459
0.3133
2.3031
0.1316
0.6796
0.05310
0.2918
2.1762
0.1315
0.6114
0.05080
0.2577
2.206
0.1324
0.5654
0.04975
0.2390
2.3746
0.2531
1.8118
0.09147
0.6197
5.2966
conductance gD{dc). T h e difference betw een the dc drain conductance <jfn(dc) and RF
drain conductance g o ( R F ) is thought to originate in som e devices from traps present
in the buffer [32].
It is p oin ted out in S ection 4.1 th at the RF transconductance g \ i { R F ) was larger
in this device than the dc transconductance gM{dc). This ac-m odel does not account
in itself for either the large charging tim e constant of th e buffer trap s or the large RC
tim e constant in the w ide bandgap m aterial. These effects must therefore be reflected
in the choice m ade for th e parameters o f the ac-m odel. For exam p le for a device with
negligible g a te leakage (perfect insulated gate) the intrinsic drain conductance Gds and
th e intrinsic transconductance
G ga
in th e ac-model should be related to the extrinsic
R F drain conductance g n ( R F ) and extrinsic RF transconductance gM(RF) by Chou
76
and Antoniadis relationships inverted [33]
1 +
Gds ( R a 4 - R d ) + G gaR „ = 9 d ^R F )
(4 ,U )
1 +
G da{ R a + R d)
(4 '12)
+
G gaR s
=
9 m ^R F )
Finally let us n o te the im pact o f the parasitics on the microwave characteristics.
Tables 6 and 7 show that the error in the unilateral power gain increases
by 2-3 dB
when we set C g s = C g d = 0. Therefore parasitics play an im portant role
prediction of the d ev ice performance.
in the
C H A PTE R V
UNILATERAL PO W ER G AIN R ESO NANCES
A N D j T “ f m a x O R D ER IN G
5.1
In tr o d u c tio n
In the velocity-saturated model th e FE T is divided into two regions [34], a gradualchannel region (GCA) and a saturation region. In the gradual-channel region [35] the
carrier concentration is determ ined by
0
q n a( V G c )
=
for V g c ^ V r
C 9 {V gc — V r)
q n SM A x
for
for V t < V g c
V g sm a x
<
<
V g sm a x
V sc
Constant mobility fi, g a te capacitance Cg and threshold voltage Vj are assumed in
this region. In the saturation region, the Poisson equation is solved in th e direction
parallel to the channel, assuming a constant channel width d3. T he velocity of the
electron is assumed to have saturated to vB in this region. T h e boundary between the
GCA region and the saturation region is the position in the channel where th e channel
electric field reaches th e critical field E c =
This sim ple m odel can handle both
long and short channel FE T s. In th e long-channel m ode the transconductance varies
linearly w ith gate voltage. In the short-channel m ode the transconductance saturates
78
to a constant value for large gate voltages. The sw itch from the lon g channel m ode
to the short channel mode is determ ined b y the ratio a = ( E cLg) / ( V a s — Vr). Indeed
one can easily verify for th e velocity-saturated M O D FET m odel that th e ratio o f th e
drain current at the onset o f saturation Idc(sat) (when th e channel field at the drain
is equal to th e critical field E c) by th e drain current in pinch-off I dc(pinch) for th e
sam e V g s voltage is given b y
Idc(sat)
I
/
■
= 2a
h s ----------w -
Idcipmch)
'' 1 + 4 2- 1 *
=
Wl(a )
(5 J )
Sim ilarly the ratio of the transconductance gm(sat) at th e onset of saturation by th e
m axim um transconductance gm,MAx{sai) is
9m( s a t )
_
a-> _
=
9m,MAx(sat)
^ 1 + (a -l)2
(5 .2)
The weight functions W i ( a ) (plain lin e) and w2(a - 1 ) (dashed lin e) are plotted in
Figure 67 versus a and a -1 respectively. From the tangential dashed-dotted lines
shown in Figure 67 it is seen that the transition from long- to short-channel m od e
occurs for a = 1 / 2 for the saturation current ratio
and a -1 = 1 for th e saturation
transconductance ratio W2 - N ote that th is criterion is o n ly applicable in the range o f
validity of th e GCA approximation. A G C A region is always expected in the case of
high-aspect ratio (Lg/ d ) F E T where d is the gate to channel spacing. Small aspect
ratio FETs, where two-dim ensional field effects are im portant over th e entire gated
channel are not considered here. The G C A approximation will also fail in conventional
high-aspect ratio FET when th e Vgs voltage reaches V g s m a x and th e channel charge
n s saturates to tism a x - In a M O D FET th is occurs w hen th e parasitic M ESFET turns
on [36]. The ratio a is therefore more correctly defined by
79
1
0.8
N
E
I
G
H
T
0.6
0 .4
0. 2
0
0
2I
4
6
ALPHA AND 1 /ALPHA
8
10
F igu re 67: Current and transconductance w eight functions tw i(a) (plain lin e) and
w2(a _1) (dashed line) p lo tte d versus a and a -1 respectively.
a
E c^g
CL
(5.3)
m i n \ y GSM A X , I'gs] - V t '
As a consequence it is not possible i n practice to turn on th e short-channel m o d e in
a long gate-length F E T (e.g., lo ^ ).
5.2
Long a n d Short C h a n n e l M o d e and th e ac-cu rren t G a in
The high-frequency small-signal characteristics o f the velocity-saturated M O D F E T
model w ill be stu d ied in both th e short and lo n g channel m ode.
O bviously the
transport picture u p o n which this ac-m odel is based will in practice break dow n for
frequencies corresponding to the e n e r g y relaxation tim e ( ~ 1 T H z) and the m om entum
relaxation time ( ~ 10 THz).
80
ALPHA VS FT
1000
100
(GHz)
L g**2
0.01
0.001
0. 01
JU1L
ALPHA
100
1000
Figure 68: Variation of the unity current gain cutoff frequency / y versus g a te length
Lg plotted versus a = E cL 9/( V g s — Vr) in log scale for an intrinsic M O D FE T with
V g s = 0, 0.1, and 0.2 V and V d s = 1 V .
T he exact solution derived in Chapter II is used to calculate th e unity current-gain
cutoff frequency /y ( m t ) of the intrinsic M O D FE T versus gate length Lg for th e gate
to source voltages
Vg s
= 0 , 0.1, and 0.2 V and a drain to source voltage of
Vos
= 1 V.
T h e M O D FET param eters used are given in Table 8 . These unilateral current-gain
cutoff frequencies f r i i n t ) are plotted versus a in log scale in Figure 68. / r is defined
here as the frequency at which we have
,
. \
21 T
|y 2 i(<*r)| _ M ^ r ) |
|!/ii(w t)|
\z-n{u}T)\
.
f r i i n t ) increases with shrinking gate length ( a ). The increase of /y (in t ) switches
from the 1/ L 2g law of the long channel FE T to th e 1 / L g law expected forth e
short
channel FE T. This results from the saturation o f the transconductance du e to the
81
Table 8 : Device parameters for th e intrinsic short-channel M O DFET
value
Parameters
gate width (/zm)
threshold voltage (F )
mobility ( cm 2jW .sec)
saturation velocity ( m / s e c )
gate to channel spacing (A)
channel width in saturation (A)
channel dielectric constant
gate dielectric constant
w ,
VT
/*
Vs
d
ds
^1
^2
290
-0.3
5600
1.85 x 10s
430
500
13.1 c0
12.2 e0
velocity saturation in the FE T channel.
N ote that the switch from the long to short channel m ode occurs for a betw een 1
and 2 as predicted in the previous section. For the saturation velocity and m obility
of Table 1 and V d s = 1 V the corner point a = 1 corresponds to a gate length of 1
and 1.66 /z for
Vg s
= 0 and 0.2 V respectively.
If the effective saturation velocity v s were to increase with decreasing gate length
L g one would have in th e sub-micron regim e an 1 /
l aw with 1 < 7 < 2.
Rohdin [21] has dem onstrated that despite the expected occurrence of velocity
overshoot th e effective saturation velocity is essentially independent of gate length
for M O D FE T s with gate length varying from 0.9 to 0.3 /z. H is analysis is based
on the system atic reverse modeling of large number of FETs on different wafers. A
82
o — [jx
N = -^ 2 -
_
X ]2 p 22
X 22P12
Figure 69: Exam ple of th e unilaterization of a two port device b y loss-less feedback.
constant saturation velocity of 1.8510s m /se c is used in this analysis.
5.3
U n ila te r a l p o w e r ga in o f th e w a v e -e q u a tio n m odel
T h e analysis of the high-frequency performance of a device is typ ically done using
th e unilateral power gain U derived by Mason [27].
if
_
___________ I gai ~
I2___________
4 [Re(j/n)Re(?/22) - R e(y i2)R e(y2i)]
I^21-
zu
(k ^
I2
4 [Re(2n )R e (222) - R e(z12)R e(22i)]
U is th e m axim um available power gain (MAG) of a device once it has been
unilaterized (?/i2 = z 12 = 0) using feedback techniques. Figure 69 shows a possible
feedback circuit (proposed by Mason him self) to unilaterize a three-term inal device.
T h e m axim um frequency of oscillation / max is defined as the frequency at w hich
U is unity. /
m a x
is often referred to as the frequency at which a three-port device
switches from active to passive. The importance o f U and / m
a x
for characterizing
a device hinges on their invariance upon loss-less coupling (feedback and loading).
83
GAIN VS FREQUENCY
40
20
■"s:
V
P;:
G
A
I
N
is
0
(dB)
-2 0
-4 0
-6 0
10
100
FREQUENCY (GHz)
1000
10000
Figure 70: M agnitude o f the unilateral power gain versus frequency for an intrinsic
M O D F E T ( V g s = 0 V and V o s = 1 V ) with a gate length of 3 /x (dashed-dotted
lin e), 1 n (dashed line), and 0.3 // (plain line).
However because the feedback network required to unilaterize a device could only be
achieved at a single frequency with loss-less passive com ponents, / m a x is a narrow­
band figure of merit. A narrow-band figure of m erit is useful in classifying transistors
for the design of tuned amplifiers and oscillators. / m a x is therefore used as a R F or
m icrowave figure of m erit. This is in contrast w ith f r which is a broad-band figure
of m erit and is therefore more relevant for classifying transistors for the design of
broad-band and large-signal circuits.
Figure 70 shows th e m agnitude o f th e intrinsic unilateral gain calculated using the
solution o f the velocity-saturated wave-equation for a 3, 1 and 0.3 fi gate length FET
with
Vg s
— 0V and
Vd s
=
IV .
For sm all frequencies one observes th e usual 20 dB per
84
decade decrease of th e intrinsic Unilateral power gain. In all th ese FETs, on e observes
at large-frequencies a periodic divergence of th e intrinsic \U\ and alternate regions of
positive U and negative U.
/„ = (n +
T h e resonant frequencies are approxim ately given by
for positive integer n where t s ( V g s , V d s ) = £ / v a is th e transit-tim e
through the saturation region o f bias-dependent length £ ( V g s , V d s ) -
Clearly these
resonances are associated with the saturation region and occur for frequencies for
which th e phase of the drain current phaser e x p ( —ju>Ta) is approxim ately ± 7r. The
appearance of the negative U regions are also correlated w ith th e periodic development
of a negative output ac-resistance.
T h e unilateral power gain resonances could be an artifact of the FET m odel which
assum es the existence of a constant velocity saturation region indirectly controlled by
the gate. The possibility of steady-state gain at frequencies above the 20 dB /decade
extrapolated f M A x {i n t ) is not however in contradiction w ith th e principle o f operation
of an F E T . An F E T is a transit tim e device, and its switching speed is therefore limited
by th e length of its gate. However the am plification of a steady-state signal does not
convey any inform ation. For such a steady-state application, an F E T is therefore
not necessarily transit-tim e lim ited. Note th at for the 0.3 fi intrinsic M O D F E T , the
first resonant frequency f 0 occurs before the 20 dB /decade extrapolated fMAx{int).
A resonance can also be predicted by the approxim ate solu tion of the wave-equation
based upon the frequency-power series.
In practical devices, lossy parasitics (see Figure 71) w ill prevent the observation
of th e unilateral power gain resonances. Figure 72 show th e im pact of a source, gate
85
C gd
Drain
Gate
GS
Source
Figure 71: Equivalent circuit for th e extrinsic M O D FET. C g s and C g d are the
fringe capacitors o f the gate.
GAIN VS FREQUENCY
A
(dB)
-2 0
-4 0
.0
100
FREQUENCY (GHz)
1000
10000
Figure 72: Unilateral power gain versus frequency for a 0.3 fi extrinsic M O D FE T
w ith parasitics resistances R s = R a = R D = 0.01 fi (plain lin e), 0.1 fi (dashed line),
1 0 (dotted dashed line), and 5 fi (dashed dashed line).
86
and drain resistances
power gain of the 0.3
Rs = Rg
=
[/, M O D FET
Rd
= 0.01, 0 .1, 1 and 5 fl upon the Unilateral
in the presence of th e parasitics capacitors
Cgd
=
C g s = 50 fF (see Figure 71) and w ith bias Vgs = 0 V and V o s = 1 V. For large
enough resistances the unilateral power gain exhibits a switch from the 20 dB to
the 40 dB drop per decade approxim ately at the resonant frequency
f 0.
This is not
w ithout resem blance with the perform ance of the equivalent circuit reported by Steer
and Trew [37]. N ote however that here the corner frequency
f 0
is above the extrinsic
f M A x ( e x t)- Up to now a 40 dB per decade decrease of the unilateral power gain has
not been experim entally observed/reported in M O D FE T s (or M O SFET s).
It is noted that f a can be smaller than the intrinsic fMAx(int) (extrapolated w ith
a 20 dB per decade slope) in a 0.3 fi M O D FET. However for realistic lossy parasitics
the extrinsic
/ m
a x
(ext) is much sm aller than
f 0
(see Figure72 ) and the unilateral
power gain calculated from the extrinsic velocity-saturated M O D FE T wave-equation
will exh ib it a 20 dB per decade decrease in the entire active range.
Let us now address the issue of th e ordering of f o and
/
m a x
• It is shown in Figure
73 th at by the use of a sufficiently large gate resistance in presence of the parasitics
resistance R s = R d = 2 0 and parasitics capacitance C g s = C g d = 50 fF w ith
bias V g s = 0 V and V d s = 1 V, it is possible to reduce the extrinsic unilateral power
gain U while m aintaining a constant extrinsic /x ( e x f ) until
/
m a x
(ext ) is smaller than
f x ( e x t ) . Large gate resistances are indeed a problem in subm icron gate F E T ’s and
m ushroom T- and L- shape gates are used to circum vent it.
A s it was seen, lossy parasitics (gate, source and drain resistances combined w ith
87
GAIN VS FREQUENCY
40
20
G
A
I
N
0
(dB)
-20
-4 0
1
10
100
FREQUENCY (GHz)
1000
10000
Figure 73: U nilateral pow er gain and short circuit current gain (plain line) versus
frequency for a 0.3 fi extrinsic M O D FE T using two different gate resistances R q =
5 fi (dashed line) and 25 ft (dotted dashed line).
th e parasitics capacitors C g s and C g d ) play a dom inant role in shaping the highfrequency characteristics of a short-channel device (e.g ., 12 dB drop of U per octave).
C H A PT E R VI
CONCLUSION
6.1
C o n c lu sio n
The velocity-saturated M O D FET wave-equation was derived. This ideal model is
based on a piece-w ise linear charge-control m o d el and velo city field relation and ac­
counts for velocity saturation, and channel le n g th narrowing. T h e exact solution was
obtained in term s o f Bessel functions and a n a ly tic expressions for the Y parameters
in term s of a frequency power series.
A sim ple RC equivalent circuit developed from the frequency power-series Yparam eters was presented for th e unsaturated intrinsic M O D FE T . T h is first-order
RC equivalent circuit was found to hold to higher frequencies than th e frequency
power-series from w hich it is derived or the m ore com plicated second-order iterative
Y-param eters reported by [18]. Like the iterative Y-param eters this RC equivalent cir­
cuit features a graceful degradation of the sm all-signal param eters at high frequencies.
A lthough quite sim ple the RC equivalent c ircu it selected departs from conventional
equivalent circuit m odels which usually rely o n a transm ission line or RC delay for
the drain transconductance and a C or series R C feedback elem en t betw een the drain
and g a te and an inductor in series with the drain conductance.
88
89
In order to increase th e frequency range of validity an optim al second-order RC
equivalent circuit which ad m its a fourth-order frequency power-series solution was
developed. It was dem onstrated that th is equivalent circuit exhibits a graceful degra­
dation and holds to much higher frequencies than the first-order RC equivalent circuit,
and even fourth-order iterative Y param eters of the M O D FE T w ave equation.
This non-quasi-static sm all-signal equivalent circuit m odel was th en extended to
th e short-channel velocity-saturated M O D FE T wave-equation. T h e resulting equiva­
lent circuit provided a graceful degradation of the sm all-signal Y-param eters at high
frequencies.
To apply th is ideal M O D FE T ac-m odel to real M O D FE T d evices a param eter
extraction technique was proposed. T h e resulting m icrowave m od el perm itted one
to reasonably predict the microwave characteristics (scattering param eters and u n i­
lateral power gain versus frequency for different bias) of a one m icron gate len gth
A lG aA s/G aA s M O D FET and G aA lA s/InG aA s/G aA s M O D FET. T he param eters
used by the intrinsic ac-m odel were all obtained from th e fit of th e IV characteristics
alone. The m icrowave parasitic elem ents were either measured ( R g, Lq , Ls, L d ) or
estim ated
( C g s , C g d )-
T h e merit of th is analytic ac m odel is therefore its capacity
to predict th e microwave performance from the dc characteristics.
Using th e exact solution o f M O D FET wave equation, the high-frequency charac­
teristics of th e saturated-velocity M O D F E T wave-equation was studied. This nonquasi-static sm all-signal m od el presents som e novel features (e.g., unilateral pow er
gain resonances). The observation or u se of these non-quasi-static effects in th e e x ­
90
trinsic M O D FE T seems how ever to require unrealistically sm all source and drain
resistances assum ing this canonic ac-m odel is applicable to real devices.
6 .2
F u tu re W ork
N ew m odels are required to evaluate the device parasitics and im prove the extraction
techniques for the gate capacitance Cg and the threshold voltage Vr-
Finally it
m ight be possible to develop an extraction techniques for the m obility y and the
effective saturation velocity v s, perm itting to apply this sim ple ac model to sub­
m icron M O D FE T s for w hich velocity overshoot and undershoot have a strong effect
on device performance.
The large-signal analysis is beyond th e purpose of the dissertation. However a
large-signal m odel based on th e proposed first-order equivalent circuit was reported
[38] recently for the long channel M O D FE T . However a large-signal m odel for the
velocity-saturated M O D FET has not yet been developed. Even though the analysis
for the saturation region is n o t as sim ple as for the GCA region, it should be possible to
develop a large signal m odel from the sm all signal equivalent circuit m odel developed
in this dissertation for th e velocity-saturated M O D FET.
A ppendix A
Frequency Power-Series Solution for the
V elocity-Saturated M ODFET wave equation
A .l
C a l c u l a t i o n o f Vg c ( x = L g)
T h e Poisson’s equation in the channel is given in (2.3). For convenience the equation
is repeated below
d?V G C
dx 2
= —f3I(x,t) = —f3(Idc +
(A .l)
For sim plicity a new variable x' — x — X s is introduced, which make ( A .l) as
d2VGC
= - / ? / ( * ' , t) = - f 3 ( I dc + iejut)
dx12
(A .2)
T h e boundary conditions are
0) = i>Gc(a:s)
v'Gc ( x ' = 0) = - E c
vgc{x' =
L et’s start w ith integrating (A .2) regarding to x'.
d v Gc
=
a [ X'(T , .NJ , dvGc
< ^ + ‘) ^ + - S r
f x'
—Phcx' — E c — (3 J i(x)dx
91
(A .3)
v g c (Lg)is
obtain ed by integration from 0 to £
VGc{Lg)
=
rLg—XS
fLg—XS rx1
u g c (z s ) — E c(Lg — x s ) — ft I
Idcx'dx' — ft
/ i(x) dx
=
v g c (x s
—
ft J
Jo
) — E c(Lg
—x s )
Jo
Jo
— f3-Idc(L g — x s ) 2
fLg-xs rx'
J i(x)dxdx.
(A .4)
Retaining th e first order term s gives
v Gc ( L g )
«
Vg c ( X s ) -
E J -
f t l- I dce
+ E cx aejwt + vgc(xs ) e ^ + A v gJ e jwi + f tl dc£xaejujt
(A .5)
where I — Lg — X s and
x'
n
i{x)dxdx'
.
The ac poten tial at x = L g is then:
vge(Lg) = E cx aejut + vgc( x s )ejujt + A v gc(£)ejut + f t l dc£xaejut
A .2
(A .6)
C a lc u la tio n o f Vq C( X s )
T he dc poten tial Vgc ~ Vr is given in Equation (2.6). For convenience the equation
is rewritten as below
Vgc ( x ) - V t = (Vgs - VT)^1 + (k*a - 2 ka)^ ~
Differentiating both side of the above equation gives
(A.7)
93
d2
x
x=xs
V out j
—
dx
x=Xg
d
Vout
—
M
i
i _ i M
dx
M 1 - |fca)
Xs
(i +
( t ; _
2 t.) -£ -)-■ /*
A5
A5
x=Xs
Vout
* ? (i-& )2
x-=Xs
Voutl 1 +
fcs2 - 2fcs] - 3/2
Vou(^ ( l - ifc s) 2
(A.8)
^ s ( l ~ k3)3
where w e use
Vout
=
v° c(X s) = “
A .3
Vg s
— hr-
x j ( i - k, Y
(A .9)
P o w e r -S e r ie s S o lu tio n o f W a v e-E q u a tio n
The w ave equation in th e GCA an d saturation regions can be solved using the fre­
quency power series an alysis which w as first introduced by Ziel and Wu [20]. Since we
are interested in the ran ge of frequency up to f max w e will use an expansion up to the
second order terms. T h e frequency power-series solu tion up to th e fourth order will
be shown in Appendix C without detailed calculation. The ac voltage and current,
VgC(x), i ( x ) , and x, are expanded in Taylor series in powers of ju> up to the second
order
i(x)
=
iQ + juji-L + (j u ) 2i 2
(A .10a)
94
xs
=
x a0 + j u x ai + ( j u ) 2x a2
(A .10c)
Substituting vgc( x ) and i(x) in to Equations (2.5), (2.9) and (2.10) and equating th e
power of ju> yields for the G CA region
- j ^ [ g ( V Gc ( x ) ) v gc0(x)} =
0
(A.11a)
- ^ [ g ( V G c ( x ) ) v gcl(x)] =
WgCgvgd0(x)
( A .lib )
■ ^ \ 9 (v o c ( x ) ) v gc2(x)] =
WgCgvgci ( x )
(A .11c)
and for the saturation region,
di0{x)
dx
dx
di2(x)
=
0
v.s
1 .
_
dx
(A .12)
vs
and
d2A vsc0(a;)
dx2
= - 0 i o{x)
(A .13)
Introducing th e new variable x' = x — X s in Equation (A . 12) gives
|
- v
d%\
2o
dx1
di2
vs
ii
dx1
v2
(A .14)
95
and
f^AUgcO
=
Z
g
r —
0
K
<A-i5)
iP&Vgcl
„.- d ^ ~ = - ^ ’2
Solving th e Equation (A .14) yields
io
=
Ci
i\
— — - x -+- C 2
Vs
(A .16)
— *0 —
^ 72:
<r/2 — X
<r' -L
n
?2 —
+ C3
2 u2
us
where C \ and C2 are arbitrary co n sta n t. Substituting Equation (A .16) in to Equation
(A. 15) a n d integrating from 0 to i g iv es the p o ten tia l across th e saturation boundary.
Augco =
A u g Cl =
xn
^ * ' 3 - ^ 2 x '2
§Vs
2
(A.17a)
(A.17b)
(AJ7c»
where w e used the boundary con d ition s
AUgcO,l,2(AS) = 0
= 0
NoW th e boundary conditions are needed in order to solve th e above wave equation.
The boundary conditions used at x = 0 and x = L g are
Ugco(O) ~
Ugs
(A.18a)
96
VgcO ^Lg)
wf l d ( 0 )
—
Vgg
~
^ 302 ( 0 ) — v g c l { L g ) — v gc2 { L g ) — 0
Vds
(A.18b)
(A.18c)
And th e following boundary conditions are used at the G C A /satu ration boundary:
• Each com ponent of th e ac current is continuous at the boundary
[5(V fcc(A s))usc0,i,2(As)] = io,i,2( A s + )
(A.19)
• Each com ponent of th e ac channel electric field in the saturation region is set to
zero at th e boundary Av'gdax2 = 0.
• Each com ponent of th e boundary m otion is calculated using
x,0,l,2
v „ ^ x ^ V gc0ih2( X s )
(A.20)
• Each com ponent of th e ac voltage is continuous at the boundary so th a t we have
^flcO,l,2(Ls ) = PIdJXsO,l,2 + AvgcO, 1,2(0 + UscO,1,2(As)
(A .21)
Before th e calculation of the channel voltage for entire region, let us define the
following simpler notation.
Vout
— Vgs — Vr
L
= lg
V
= VGc ( x )
V
Vs
=
Vq
c {x
)
= Vgc ( A s )
97
And it will be helpful for understanding the calculation that the following term s are
calculated first.
g(V)
=
G doa( l - y ) X s
(A.22)
SW
=
G doa( l - k3) X s
(A .23)
g'(Va)
=
- Gd
o
a
(A .24)
We are interested in th e ac current in terms o f vga and v da in order to calculate the
intrinsic Y param eters. However th e boundary conditions are given in form of ac
voltages so that the ac channel voltages have to be calculated before th e ac current
can be obtained.
A .3.1
C a lc u la tio n o f t>o
T he first order ac voltage
vq
for th e entire channel can be obtained by solving the
Equation (A .11a) for th e GCA region and Equation (A .17a) for the saturation region
w ith the boundary conditions (A .18a), (A .18b) and G C A /saturation boundary con­
ditions. Integrating Equation (A. 11a) gives th e channel voltage in th e GCA region
for Vo in term s of x
g ( V ) v0{ x ) = d \ x + d2
where di and d2 are arbitrary constant.
(A .25)
d2 can be obtained from th e boundary
condition (A.18a) w ith Equation (A .25)
= vgaG doaX s
Substituting Equation (A .26) into Equation (A .25) yields
(A .26)
98
X *-f- VggGd OSX s
«■<*) -
; (v )
<A -27>
d\ will b e determ ined later on w h en the wave equation is solved in the saturation
region.
The calculation o f A v 0 in the saturation region (Equation (A .17a)) is obtained by
solving th e Poisson equation (A .13) for a fixed saturation region width £, and zero ac
potential vgc( X s ) = 0 and zero ac field v'gc( X s ) = 0. The relation between di and C\
can be obtained by settin g the ac current continuous at G C A /saturation boundary.
The ac current at th e GC A /satu ration boundary is
* o P k ) = ~ -j^[g(V)vo(x)]\x=xs = ~di
and th e ac current in the saturation region is C i so that w e can find C\ = —d\.
v0(x = Lg) can be found in E quation (A.21) from the zero order term w hich is
u0(L3) = fHdJx o + A v 0(l) + wo(As )
(A .28)
The value o f Ci (= —d \ ) can be found from this equation using th e boundary condition
(A.18b).
In order to obtain C \ , we have to express th e above equation in terms of C\ . Let
us start w ith the calculation ^ ( A s - ) at the GC A /satu ration boundary
v0( a :5 )
=
d ! X s
+
V g y G ,J0 S X
S
Gdoa{ 1 —ks) X s
~ C \ X s + VgsGdosXs
G d o s ( l - k a) X s
.
1
yj
a:0 can b e obtained by substituting Equation (A .9) into Equation (A .20) and extract­
ing the zero order com ponent
99
*o =
X J(1
,/ v ,
-k .)3
(A .30)
7oT, -----r r ^
vo ( * s )
V ^ A f tl - ifcfl)2
„
Vq( X s ) is obtained by differentiating Equation (A .27) at G C A /saturation boundary
and expressing it in term s o f d\.
• to )
=
"j" VgsQdos^-S
5
s(v)
X
d
=
x=Xs
d 'T x
[g{v)\
VggGdosXs
.9(V)\
x= Xs
g ( V ) - x g ' jV )
d
-
*
~
d' T x
.
4 ~ VgSGdosXs
92(V)
x=Xs
x=Xs
g 'iv )
9 2( V )
(A .31)
x=Xs
Substituting Equation (A .24) into Equation (A .31) yields
Ci
tiX s) = -
Gdos{ 1 — ks) X s
1+
E CX S
(1
ka^V0Ut
+
v'gs
{
E cX s
(i - k s y x s vout
(A .32)
From Equation (A .30) and Equation ( A .32) we obtain
(1 - K ) 2Ci
{ ^ f ) G doak ] { \ - \ k sf
x0 =
(1 - k a) 2v'g s
( ^
E CX S
1+
(1 - ks )V0Ut
E CX S
(A .33)
( i - I M 2 v out( i - ka)
The next step is to calculate Xvo(£) in Equation (A.17a).
Auo(^) =
E ie
(A .34)
Substituting Equation (A .33), (A .34), (A .29) in to Equation (A .28) gives
vo(Lg)
=
(1 - h f C r f h J
-
i+
{ ^ f ) G d0s k i { \ - \ k ay
+
(1
- k a) 2Vga/3I dc£
(3 ? )* 2 0 Ci
E CX S
(1 - ka)Vout
PCi
E cX s
Vout(l — ka)
V,'gs
' G doa( l - k a) + ( I - k 3)
— Vga
Vda
(A .35)
100
B y solving E quation (A .35) C \ will be known in term s o f vgs and Vds - For sim plicity
we will introduce the new sym bols i2, and R y which are defined in Section 2.3.
< L g)
=
- ^ [ i + R y] + R iR vvg. - f ¥ - C 1
^
I* dot
G doa{Cl
1 -
ks) + h
(1
-
ks
)
= vso ~ v *s
(A .36)
M anipulating th e above equation gives
Gdos\.(<k a "I"
1 +
~
k a) ) v ga
^G doai 1
—
k a)
G dosRd[(ka + R i R y (
+
1 -
(1 —
“ I" ( 1
^ s )U (is ]
k a) R i ( l
k a) ) v ga
+
-f
(1 -
Ry)
fc s ) u d s ]
where
8£2
R d = 1 + ~2~Gdoa( 1 — ka) + (1
—
k a) R i ( 1 + R y)
C a lc u la tio n o f v\
A .3.2
We shall now integrate the wave equation ( A .lib ) in order to get i>i in the G C A
region
i]
=
C , w sv„(x)
=
C , W ~ C ' X Jrg V" ° * “ X s
( A .37)
Before to do so let us introduce some new variables which will sim plify th e integration.
T he channel current is given by
Ido
=
fxC g W g
[F g s -
F t -
Vcs]
Using Equation (A .38) dx can be expressed in terms o f V c s(z) as follow s
( A .3 8 )
101
dx
=
[VG5 -
(A.39)
- VC 5( X )] rfVc s (a:)
Id c
Integrating Equation (A .39) from 0 to x yields
x
=
» C gWg
[(V o s -V iJ V o s W -i^ * )]
Ii C , W , ( V g
s
-V
Vcs(x)
t?
V g s — Vt
Id c
V
-V t)
Vc7s(
s(x)
2 \V g s
Let us introduce the new variable y
Vcs(x)
^
Vg s — Vt
so that we can write
x =
fiCgW g (VG S - V T)2
(A.40)
Integrating Equation (A .38) from 0 to X s gives
he = ^ ^ [ ( V
gs-
V t )V c s ( X s ) - \ v 3 s ( X s )\
fiCgWg (VG S - V T ) 2
Vcs(Xs)
1 ( Vcs(Xs)\
Xs
VGs — V t
2 \ VGs — V t
fiCgWg (VG S -
V t )2
Xs
(ka
2 kfj
)
(A.41)
where we introduced the new variable
.
Vcs(Xs )
5
Vos - Vt
Replacing Equation (A .41) into Equation (A .40) we obtain an expression related
x to y
x =
xX s
(
1
2\
(l- it ,) !* " ? ')
102
Let us differentiate the above equation by y in order to obtain the relation between
dx and dy
— Xs
dy
k3 ( l -
M_
1
so that dx can be expressed in term s o f dy
( A
' 4 2 )
L et’s start integration o f the wave equation (A .37) using these new variables
=
fx
C a W a Jo
C\X -J- VgSGfi()S^Cs
1 ; ? r au3“ J dx
W )
'9 ” 9 L
___ 1
X sC l - y)
fy r X s ( y - h I 2)
fvr As[y~ *y )
1
X s ^ ~ y) .h.
0 U 1l ks ( l1 - I * ,) Gd0, ( l - y ) X s ks( 1 - \ k 3) ay
-
- r w
1
+
GdOsXsVg
n tnr ffV
y GdOsXsVgs
j ,
° ’ W ' Lo G
M, ( l -—y y) )X s dy + e '
G^osCi
CosCi
Ci
fpy (
11 2\ i
G d o s k K 1 -- = \rk r3yy /Jo0 V
I*
G
q
3G \
Gd0skj(l - \ k 3y
Coav
C0.i>„,
~ 22 y J) dy +
G
g d om
sU
;v j
- ±i fkc .)) I
1 -
dy + e'
Co3Vg3
!
G
o 3V g 3
+
7
;—
TTi—
^ y + ei
y j
\ 2 y ~ 66 V
GdoM 1 - \rkr 3)
/I
2
1 3^
Integrating once more yields
g(V)vi(x)
G q3C i________ t x
_ _____
/ I
2
1 3\
G«k>.*2(1 - i M 2 A) \2^
~Gdosk23( l - l k 3y l
,)
£ y^
+
k fi%
=
- o J ^ o - L y
( 2^ "
6 y3)
dy+
f
j
-
6^ J
Co3Vg3
fX J
1
1
+ Jb,(l - iJb.) X V + ClX+ 62
* .( / -
\ k 3) dy
+e>
( ^ 2“ F 3+ k
) dy
- U a)2
.) Jo '
C 0aX ssCx
C1
G doak 3( 1
VgSC oaX s
'
f l
1 4
Qy
3
[6y
1 3l I
r1 2
+ k i ( i - \ K Y l2y “ 5*1+
62
.
1 sl
30 .
,
= 0 because Uo(0) = 0 from boundary condition (A .18c).
ui(x)
g(Y)
C oaX s C
C\\
fl 3
G dosk3( 1 - \Uk s)
s ) 3 V6y
VgaCoaX s
f 1 2
*2U - 5*-)
1 4
6
j_ 5)
30
V
1 3^ .
- F J + 6' 1
J
(A .43)
(A .44)
AC potential at X 5 , where y = ka, is
« i(* s ) =
r
^-s)
coac1 E + vgsC oaC +
G do
Ci
(A .45)
Unknown variable e\ will be obtain by applying the rem aining boundary condition
to Equation (A . 2 1 ) for the first order term. For convenience Equation ( A .21) will be
rew ritten for th e first order term .
ux(Ls ) = f3Idclxi + Aux(^) + ni(A's)
(A.46)
T he other boundary condition v \ { L g) = 0 w ill be used to obtain ex w ith Equation
(A .46).
Equation (A .46) m ust b e rewritten in terms of ex through th e following
steps. First the relation betw een ex and C 2 , which is used in saturation region, will
be established through GC A /satu ration boundary condition, ac current continuity,
the ac current in G CA region at X s is
iiC Xs)
=
—^ [ 9 ( V ) v i ( x ) ] \ x=xs
104
And ac current in saturation region at X s { x ' = 0) is
i i( x ' = 0) = C 2
C i can be expressed in term s of
C2 =
Cos“D
G dlos
l
C \
Vg g C Q g A
from the above equations
(A .47)
ei
where A and D are defined in Chapter II section 3. N ext step is to rew rite Aui(£) in
term s o f e\. From Equation (A. 17b)
a „ iM = E x e - E i p
6va
(A .48)
2
Sub stitu ting Equation (A .47) into Equation (A .48) yields
C o
D C \
V g g C o s A
■G dog
e\
(A.49)
Last step is calculation x\ in term s of ei. Substituting Equation (A .9) into Equation
(A.20) and extracting the first order term yields
Xl
* 1(1 ~ k* f _ ' , v ,
v outk]{ 1- \ k ay l( 5)
(A .50)
In order to get xi in term s of e i, we m ust calculate ac electric field, tq (X s ), in term s
of ei- ac electric field in GCA region is differentiating Equation (A .44) by x,
1
» .( * » )
=
C o sC l
D + vgsCosA + ei
Gdc
C oaX s C1
g '( V s )
E + vgsC 0SX s C + e \ X s
9 2{Vs)
G (los
1
CoaC x
D +
+ e\
U
-Y vgaCoaA
VgS
GdosXs{ 1 — ka)
^dos
g(V.)
Q
E CX S
( J d o a V out
Gl,X}(
C ,„ X sC ,
1 - *.)2
Gd
CoaC x
G dosXs( 1 — ka)
+
E 4* VgSC oaX s C + e i X s
( D + E R y)
G doa
vgaC oa(A + C R y) + e i ( l -f f?y)]
105
where A , C , D , and E are defined in Chapter II. Now x x can be expressed in term s
of ei
(1 - K f
Xi
C oaC x ,
=
( D
+
vgaC oa( A + C R
+ e i(l +
y )
+
E R y )
Gdos
Gjo, f e ) *J(1 - 1kay
(A .51)
R y ) ]
R eplacing Equation (A .51), (A .49), (A .45) into Equation (A .46) and applying
boundary condition (A .18c), v x(Lg) = 0.
0
Ci
'^ ■ ( D
=
Gdos{ 1
ka)
V a a C o a
■
[ R i (
C’do*(l
ei
ka)
GdosiX
P C i*
6u.
ka)
1 - ka) (A +
\cDD C \
R i ( D
GL(1 - M
vgaC 03
R i (
ks)
ei
Gdosi, 1
C R y )
+
C ]
vgaCoaA
Ci
■Gdo
cxc0
+
'-r dos
[1 + iZ«(l + R y ) { 1 — &s)]
2
C^dos(l
+ E R , ) ( 1 - k .) + ^ - E
'Jrdos
+
E R y ) { \
1 - ka)(A +
PI2
— ka) •+• E H— — G<i0» ( l — ka)D
C R y )
+
C
1 + /? ,(! + i?y)(l — ka) +
^
3£2
+ ^ - G doa{ 1 - ka)A
8£2
Gdosi)- + ka)
^ PC X£2
H
7 T'a
)
where i?, and i?y are defined in Chapter II. M ultiplying Gdos(1 — ks) to both sides to
make calculation easy gives
0 =
_ C
i C
£a
3£2
Ri( 1 - ka) D + (1 + 7 ^ ( 1 - ka) ) E + ^ - G d o s { 1 - ka)D
Gdos
~\'VgaC oa
+ e i
Rd +
Ri{ 1 - k2)A + (1 + J f c i ^ l - ka) ) C +
— Ts G d 0 J( l —
k a)
3£2
Gdoa( 1 - h ) A
106
e\ can be obtained by solving the above equation, which is
ei
=
Rd
3£2
ClCos
R i { 1 - k . ) D + R b E + ^ - G doa( 1 - ka) D
Gdos
f3£2
vgsCoa
{JCxl 2
6
— GosRd
R i (
1 — ka)A -f- R b C + — ^ 05(1 - ka)A
'TaJ£dGdos{t1
Ci
kg']
[ R a D + R g E ] — v ga[ R A A + R b C ]
G do
n r
“
n
As
7 s-ti'cl'C* dos\*-
(A .52)
J
where RA and R g are g iven in Chapter II. Equation (A .47) and (A .52) gives
Ci
J3
(A .53)
Vgs A ~ Ci
■Gdoa
Now ui is known for en tir e gate len gth.
A .3.3
C a lc u la tio n o f t;2
The wave equation for th e second order ac potential in GCA region is
<P
dx2
[g(v)v2] = c aw avi
(A .54)
Substituting Equation (A .4 4 ) into Equation (A .54) and integrating both side gives
_
"
r c‘w’
Jo
()
9 0
f
c°-c 'Xs
- \k,y
_CosVgaXs__ ( \ 2 _ 1 3^ i
k 2a (l - \ k a)2 \ 2 y
3 /
IK*
ViM
6s T 30*J
+ fl
C l , C^ 1i ________ fX 1
3
^ 4 _L ^
A
G d0akf( 1 - ±fcs)3 Jo g { V ) U 22 “ 6 y + 30y J
J
X
107
+
i m
^ l l w
) &
- \ &
x + c ‘w‘ C w ) i x + h
C l C11
i ______
_____ 1____ l y 3 _ 1 4 +
*\ x s ( l
y) ,
\ k a)3 Jo
Vo GU>.XS
6 y + 3300 y )/ ka(
U 11 - ±ka)
U a) y
G d0ak3( l - ±ka)3
G dos Xs (1
{l - «)
y) \ 6
r
.
4.
(
±
-
Gqsvga
ry
1
f - v 2 _ i , / 3^
~ y ) j„
Zy ) ka( \ - \ k a) y
k * { l - \ k ay" J Jo
o Gd0aX s ( 1l - y) \ 2 y
r* w
fv
1
X s { y - \ y 2) X s ( l - y )
fl V o G W M l - y ) M l - W
M l - W ^
_______ G qsC i_____ / 4
1^ 5
1 6A
G i)A 4( l - | M 4 V24y
.
30y
GftgVgs_____ (1 3
Gdo.fcJU - !&s )3 U y
180y J
1 4\ _________Cps6!_____ / I 2
12y )
G d0sk*(l - |fcs)2 V2y
^ 3^ , r
6y J + *
Integrating again gives
g ( V ) v 2{x)
C 20aC
" lx
r
1=
" *^* .)/ o
Gjo.*J(1 - \ k , ) * L
+
G ^ { C - \k,Y
4
! s
+
Coaex
fy
G
- \ k a)3 Jo
” d0ak3(l
.................
( e y3
G jo4 ,4(1 - J*,)'
CosXset
G d0ak3( l - \ k a)
6
^
12y>) dx
(?3
ft* + h
fy
( 1«<3
^
+ m
ye) k J h w ) dy
' - , ( i - 1 * .) '
f 1v 2
\2
C l C xX s
rv 1
G 2d0M 1 - \ k a)3 J 0 \24
C%aVgaX S
*
* - 3 0 1' + l 8 0 I' ) ‘fe
±ka)* J 0 V24 y , -
+
+
m
f Vf l
Gfogs
fy
G
" d0sk3(l - \ k a)3 Jo
'
+
1
(
f (f2- ?3)<>*+
CICX
G 2d0sk<a(l -
/
""
.
' W i - 1W
108
C^XsCr
——y 5 — —y G H— — y 7 ------ — y 8]
.120
80*
180
1440 J
G l s k K i - l k ay
+
+
v9s C l X s
1
777 y
Gd0sK{ 1 - \ks)A L24
4
-
1
20
1
5
77772/ +
7^
ri
1 .4 , 1
G doak*( 1 - \ k a)3 * v ~ 6 V + 3 0 y
C psXse i
+ f\x + f 2
Since boundary condition (A. 18c) gives v<i(x = 0) = 0, f 2 is zero.
v 2(x)
C tX sd,
=
9 ( V ) [ G 2doak l ( l - 1 ksy
+
vaaC l X s
[— y 4
G dosk i( 1 - \| kks)
sY [24
+
CoaX s £ 1
G doak*( 1 +
.120V
- —
8 0 V + 180y
y5 + —
20V
72
1440y
y 6l
J
The second order ac potential at x = X s , where y = ka, is given as
v 2( X s )
=
+
C t X s C , „ . vaaC
lX s
H + 9^ os'4 - G
G d0
do3
G doa( 1 — ka) X s
C oaX s e 1
E + fiX s
G dos
1
+ CoseiE
C oalCi H +
G j oa “ 1 G ,os ~ 1 G d0
+
^
(A .55)
where H , G , and £ are given in Chapter II. The other boundary condition gives ac
potential at drain side zero, so that we can calculate the f \ .
T h e second order ac
potential at drain side can be obtained from Equation (A .21) by extracting second
order term s
v 2(\jg) = /3Idclx 2 + A v 2(£) + v2( X s )
(A .56)
A ^ (^ ) is given in Equation (A. 17c)
A
v
2(£) =
24ug
6u,
(A .57)
109
ac current is continuous at G C A /satu ration boundary so that
i2(X s )
=
- f a \ g ( V ) v a(ar)]|*=jrff
c20A r
v9sClD
G doa
=
C3 =
Coa&1
G doa
G d,
D -h
(A .58)
( ' = 0)
12 2
w here jF, B, and D are given in Chapter II. From Equation (2.16) and Equation (A .9)
*2 =
* 1 ( 1 - k a)3
ToTi
i T ^ 2( * s )
v outk i ( i - \ k ay
(A .59)
77
1
*2 ( Xs )
C2 £ ± F +
G ldos
9 ( V S)
9'{Vs)
9 ( V a)
XJ
—rn
l^dos
+ /x
Gdo
I
Gdos
^ S ^ g s^ o s
I
+
Gdos( l — fcj)A s
G d oVso u^t
^
=
T ^ T T -T
I-*dos
B +
Gdoa
dos
I
f
\r
h + h Xs
^ 1 D + /,
G doa
[C 2, A s d1 / f + ^ , X s
G l sX U l - k3)2
+
n
^Jdoa
rc|,d x „ . uflsC 2s
+
^ S ^ O S ^ l jp
H + — n ------- G +
G
G doa
doa
l X s E + /x A s
“ ’
yy-
\ l ¥ 1 ( F + n » H ) + ' ^ ( B + R llG)
fc ajJiS I ^d o s
doa
+ £ z * ( D + R yE) + f 1( l + R y)]
&doa
J
(A .60)
Substituting E quation (A .60) into Equation (A .59) yields
(1 - ka)2
x2 =
^ ( F
+ RyH)
doa
+
v„*c l ( B + HyG) + G ^ . ( D + R , E ) + M 1 + R , )
G doa
G doa
(A .61)
110
Substitu ting Equation (A .55), (A .57), and (A .61) into Equation (A .56)
0
= V2(Lg ) =
+
R
i ___
C-s£ ± ( F + R . H ) +
+ R,G )
Gdoa
&' doa
{* d o a
dos
P C 1 / A i P C 2/!3
P C 3i
1
C osC i j j
Gdoai^ 1
^s) _
^
VgSCos q
G^a
CcoSe i ^ ^ ^
Gdoa
(A .62)
Gdoa
From Equation (A .58) we can obtain C 2 and introduce it in Equation (A .62)
0
= V2(L g) =
R i _________
G doa
+
+
Goa^l
VgaCl,
( F + R SH ) +
'- 'd
os
r doa
(D + R y E ) + / l ( l + Ry) | —
/ ^ 2r 2Cx
24
Gdoa
p e c 2Ts
^ c r F+^ c i B+c ^ D+fi
p t2 r
+ 2
6
G doa
l
+
G doa (1
& s)
+ RyG)
U dos
Gdo
dos
G doa
G do
G do
pi2T2C\ pec2ra
4-
24
1
6
\Cldx
2-7dos(l
*.)
p e
i? i(l — ka) ( F + R y H ) 4— — Gd04( l — ks) F 4* H
dos
pi2
i? ,(l — ks) ( B + R y G ) 4— — Gdos(l — ks) B 4- G
4G doa
4-
Gos^l
R i ( l - ks) ( D + R yE ) 4- ^ - G dos{ 1 - ks)D 4- E
G dos
+ /l
P i2
■R,(l — &s) ( l 4- Ry) 4— — Gdoa{ 1 — ks) 4-1
= 0
p£2r2Ci pe2c2r3
4-
fi
=
6
24
1
C ld,
G dos
\
s-v2
[Ra F + R b H] +
GdoaiP
ks)
4-
[Ra D 4- R b E) 4- f i / R d ] = 0
Rd
Gd,
t-*doa
\ P t 2r 2C x
24
Gdo
[Ra B 4- RbG]
J
G d o s il -
ks) -
G doai 1-
ks)
Ill
[r a F + R b H] ^dos
Ra B + R b G\ ^dos
[Ra D + R BE\
{^dos
~ 2 4 ~ Gdos(1 “ ks)Rd " ^ 6 ^
^
Gdos(1 “ *s) +
[CosCl [Ra F + R b H] - vgaCos[RA B + R BG\ - E 1[RAD + R BE]
LG,dos
'-3
Cl,Cl
— -3=^— r
Wos
A .3 .4
v„Cl
Cm
33— i f - -3;— i f - i i
<-*(*05
'GTrfoa
C a lcu la tio n o f ig and id
The ac current flow ing into the g a te can be obtained by subtracting i( x = Lg) from
i(x = 0). We assum e that the ac channel current is flowing from drain to source.
The ac current flow ing into the drain is i(x = L g). i(x = 0) can be obtained from
the current equation o f the G CA region and i ( x = Lg) from the current equation
of the saturation region. The relation between ac current and ac p otential in GCA
region is given by E quation (2.4). Rewriting th is equation in term s of th e frequency
com ponents at x = 0 yields
*o(0)
=
~ j ^ [ g ( V ) v 0]\x=0 =
*i(0)
=
~ - ^ l 9 ( V ) v i]U=o = - e i
*2(0 )
=
~ ^ 9 { V ) v2]\x= o =
—di = Ci
(A .63)
(A .64)
-/l
(A .65)
The ac current in th e saturation region is given by Equation (A . 16) so th a t i(x = Lg)
can b e obtained by substituting x' = 1 into E quation (A .16).
*0 {Eg)
=
Ci
i i{ L g)
= ~~CiTa + C2
(A .66)
(A.67)
112
i 2( La)
=
\C
it2
.-
C 2t. + C s
(A .68)
where rs = l / v 3.
As shown in Equation (A. 10a) th e ac current consists of th e zero, first and second
order term s. In order to get the to ta l gate and drain current we have to combine
them using to Equation (A.10a).
T h e ac current will be calculated up to second
order term s
*(0)
=
Ci - j u e i - (ju>)2/ i
(A .69)
i ( L g)
=
C 1 - M C 1ra - C 2) + (ju:)2 ( ^ C 1T2 - C 2Ta + C 3^
(A .70)
The gate current is then given by
ig =
jw [-e
1 + CiTa - C 2] + ( j u ) 2 - f i - ~ ^ Ta + C 2Ta - C 3
(A .71)
and th e drain current by Equation (A .70).
A .3.5
C a lcu la tio n o f Yi2 a n d Y22
Fj2 and Y22 are obtained from ig/vda and id/vda respectively w ith vga = 0. For vga = 0
we m ust introduce th e new constants C[, C'2, C'3, e^, and f [ calculated in Equation
(A .69), ( A .70), and (A .71).
C\
— G dos (1
n' _
G2
^3
t
—
_
ks)RdVda
GoaC\ n
D
I* dos
ei
^ — y — U —h
CoaR d C i fr> n ! D C,1
^
4" -Rb-®]
$ C xt 2_ D ^
/i
a TsRdGdos{ 1
0
f N
&s)
113
f'l
=
+
- 2£ Cl Gdos( 1 - ka) R d - ^ I l R dG dos{ 1 - ka)
14
0
GoaRd
[Ra F + R b H ] — Eda[RAD + R b E ]
Gdoa
Gdo
\coacx
■
Substituting th e above equations in to Equation (A .71) and Equation (A .70) and
dividing it by v ds gives Yx2 and Y22
Y12 =
y 22
ig
j u [ - e x' + C [ t s - C ' 2] + (jus)2 { - f ' x - l C ' r f + C'2t s - C ' 3]
-2 - =
Vds
Vds
—
j w [ —Eds
GdaTs ~ C da]
=
U)
=
id
C[ + M - C ' x T a + C'2) + (.j u ) 2[\C [r * - C'2t s + C'3]
— =
+
Fds + ^
+
( jw ) 2 [—
T, ~ CdsTs + H ds
Vds
F ds ~ G d a / ^ T 2
ju)[Eda
+
C dsTs
Gds^a "h Cds]
Vds
— Gds + j b j ( —GdsTs + Cda) + ( i w)2[G?ds/ 2 r 2 — Cd„rs + H ds]
=
Gds - a;2 [ ^ r 2 - C d. r . + Hds] - jw[G'lJ.r . - C ds\
where
9 1 — G dos{\
Vds
C"2 _ CosGds
Gds
Cda
Vds
r
dos
^S^Ed
-
Eds
t
e.
Eds
Vds
C 0s E d C di
Ordos
[Ra D + R b E] - ^ 0
tsG 2
ds
Cs _ C l G ds F _ CosEda D _
Vds
Gdoa
Gdoa
Hds
Fds
I L —- Pn al s sr ids - S f/•>- rTs '~>da'Jdi
r r
24
CoaRd C 0aGda
(R a F + R b H ) — Eda{R AD + i?B ^)j
Gdoa L Gdo
Vds
+
—
Hda\
114
C a lc u la tio n o f Yn and Y21
A .3.6
F n and i 2i is the ac gate and drain current divided by vga w hen Vda = 0.
Cj
=
GdosRd(ka “i
Ry (1
^gs
= cDLG,Ci - D
dos
c; =
C ‘-C \ F _
0 1 ,
— Go$Rd
_ 9 ^ 1 D _ fl
Gdos
Gdos
cx [Ra D 4- R b E] — v93[R a A + -KbC]!
r
G dos
pe
C \ T s R d G dos ( 1 -
fl
=
+
^
f
C oaR d
G dos
—
'
C d o s il -
r c osG i
.
Gdo
k .)
k a)R d -
^
^
T
a R d G dos{ 1 -
K )
[Ra F + R b H] — vgaC oa[RAB + R b G]
E\ [R a D + R b E]]
Substituting the above equations into Equation (A .71) and Equation (A .70) and
dividing it b y vgs gives I n and I 21
Y11
=
ia _ M ~ e i + Cx-T. - <%] + ( j uQ2[ - / ; - \ C l r t + (% t . - C l]
— =
ugs
ug s
— j u [ —E ga + GgaTs — Cga] + (juj)2[—Fga — Gas/ 2 r 2 + CgaTa —H g, s i
y ja
G gaTa -f- C g3]
--
u>
=
id
c ; + M - c y . + c i ) + c h 2[ ± c ; v 2 - c ; r . + c i \
— = ------------------------------------------------------------------------------
-
C ,.r . +
j i o [ E ga
H ,gs
Jgs
ug s
=
Gga + j u j ( - G gaTa + Cga) + C?w)2[ - G sar 2 - CgaTa + Htgsi
=
G ga-u>>
~ C , . t. + U „
ju>[GgaTa
C ga]
where
Ggs
C
—- = GdosRd{ks + R i R y ( l — ks))
=
v gs
c gs
=
=
D - C oaA - E as
Vgs
Egs
=
^
& dos
= ^ J £ R d[RAD + R BE ] - C 0SR d[RAA + R BC]
Vgs
U gs
~
~
^ d03
Cl
C lG g.
ft,
~
Vgs
r
i2
CjTd o s
Cl
f l _ pPTg
93
24
C0SEgS
r i
{-*doS
n
dos
9S
f3PTS
-
vgs ~
93 ds
+
^ ^ R d[RAF + R b H ] - ^ - R d[RAB + R BG}
^dos
C 0SR dEgS
Gdoa
6
33ds
'- 'd o s
[Ra D + R b E]
A p p en d ix B
E xact S olution for V elo city -S a tu ra ted M O D F E T
W ave E q u a tio n
The exact solution of the M O D FE T wave equation is based on the original calculation
of Burn [14] for the M O SFE T in pinch off (k = 1). His wave equation can be m odified
to hold for th e unsaturated regime.
Before proceeding, we need to derive the channel voltage. The dc channel potential
Vc(x ) in th e GCA region is obtained from the current equation. T he dc current in
GCA region is
IDC(x) = fiC(Vas - V T - V c s ( x ) ) dVcdSx{x)
Since th e dc current Idc is independent of x, we have
(VGs - V t - V c s ( x ) ) d ^G
J ^
= constant
Integrating both sides from 0 to a: and m anipulating with boundary conditions,
Fcs(O) = 0 and V cs(A 5 ) = V 'D , yields
117
It is convenient to introduce the dc gate to channel voltage,
V a c ( x ) = Vg s - V
=
t
-V
V g c ( x ),
c s (x )
defined as
(B .l)
(Vos - VT ) J l - ( 2 k . - k * ) ± -
(B.2)
w here ks = vJ j i VTThe equation is derived for total voltage from continuity equation and current
equation. The tim e dependent current equation can then be rewritten
I ( x , t ) = - f i C v G c ( x , t ) - V^ * , t ^
(B.3)
Vac(x, t ) =
(B.4)
w here
V g c {x )
+ vgc{ x , t)
T h e continuity equation becom es
d l j x , t)
dx
^ d vG cjx, Q
dt
/g
D ifferentiating Equation (B .3 ) on both sides with respect to x and substitutin g in
E quation (B .5) yields an equation for
t
¥
/
2
x
^
vgc
(x ^ ) :
2 d v GC( x ,t )
=
~
a t
^
(B.6)
For small signal analysis, th e above equation will be decomposed into a dc part
a n d a sm all-signal ac part. It is assumed th at the second order term (such as VgC( x , t ))
o f sm all-signal ac part is negligible. Using th is assum ption VQC( x ,t) can be rewritten
approxim ately
v G c ( X >0 »
VG c ( x ) + 2 V G c ( x ) V g e( x , t )
118
Substituting Equation (B .2) in the above equation and sim plifying yields
vh o M
*
(Vos - VT )2 ( l - (2fcs - Ar2) | - )
+
2{Vgs - VT) ^ 1 - (2ka - k j ) j - v ge( x , t )
=
(Vgs - Vt )2P + 2{Vo s - VT) y / P v gc( x , t)
(B .7)
where
P = 1 - (2fca - A:2)
X ,
The new variable P is introduced to sim plify calculation. The relation betw een
dx and d P has to be calculated before differentiating Equation (B .7).
X5
dP
2 ka — k2
dx =
The procedure to differentiate Equation (B .7) is as follows
d2
dx2
dP2
2 (2k, ^ ? ( V e S - V T ) ^
2
=
2
=
2
=
d2
r (2 * . - * . 2)i
v 2GC( x ,t )
[l/Gcfa.O]
p V ic M ]
d v 9C
,
(2 ka - k2)2(VGS - V t ) d
X2
dP
2y / P v + y / P d P
(2ks - k2)2(VGS - VT)
X2
(2 ks - k2)2(VGS - VT)
d2v,
Vp dP2
X 2
1
V+
4 /3 3 /2
dv gc
2y / P d P
+
1 dvgc
y /P d P
+
1
dv gc
2 y /P d P
+ VP
1
4 p 3 /2 V9c
d2i gc
dP2
(B .8)
Replacing Equation (B .8) in Equation (B .6) yields
d v ge( x , t )
n(VG S - V T)(2k3 - k 2s ) 2
dt
X2
/~pd2Vgc(x i P) . 1 d V g / x , t ) _____ 1
dP2
y/P
dP
4 P 3 /2 ^ C
\ r p d?Vgc(x,t)
1 dvgc( x , t )
V
dP2
+ y /P
dP
1
4 p 3 /2
'
’
'
(B.9)
119
w here
OJ0k
li{VGs - V T ) { 2 k a - k l f
^
Laplace transform of Equation (B.9) is
+ p d ^ t )
_ (1 +
$ ,p , /a)v(S i4) = „
(B 10)
w h ere S' = S/to0kEquation (B .1 0 ) is the s space representation of the M O D F E T wave equation and
can b e verified t o be equivalent to Equation (2.5). Equation (B.10) can be derived
from Equation (2 .5 ) by substituting g (V o c (x )) = gW gCg (VGc(x ) ~ V r ) into Equation
(2 .5 ) and differentiating. This is a modified B essel’s differential equation so that one
can find an a n alytic solution [25]. The com p lete solution is w ritten as
v,c(P,S) =
( ^ ( P ) 3' 4) + C2/_ 2/3 ( ] V ? ( P ) 3/4)
(B .ll)
w h ere C\ and C2 are arbitrary constants. T h e boundary conditions w ill define C\
and C 2 .
T h e ac current will be obtained decom posing Equation (B .3) into its dc and ac
com ponents w hile neglecting second order terms:
i(P .S )
where
=
- ^ C ^ ( t , « ( P , 5 ) V b c (P ))
=
- /* C ( - (2* X * ; ) )
=
G 'doa
- ^ ) ^ ( V V ' P ) = ffdQ, - ± ; ( v V P )
(B.12)
120
r ,
y C gW a{VGS -
V t ) (2 k a - k j )
^dos
E quation (B.12) has to be expanded in term s of C\ and C 2 in order to apply the
boundary conditions. It sim plifies th e calculation to introduce the new variable Y
Y
=
%
dP
=
J S ' P - 11*
Note th a t the m odified B essel function has the following properties
=
j f A + .M + V l M )
First dvgC/ d P will be expanded in terms of C \ and C 2
d v ,c
_
dYdv^ = ^
dP
p - l / t _ d [ c lh /! s { Y ) + C i h / d Y ) ]
dP d Y
= V s ;p - 1/4
dY
C i , ,
,
,
,, ,,,
,
C i,
1 (B.13)
Substituting Equation (B .13) and ( B . l l ) in Equation (B.12) gives
i(p,s) = g;„ [ i p - 1/a( c ,/2/3(y) + C2I-„3(Y))
+
=
V S ' P 1/4 [ ^ ( / s „ ( Y ) + /_ ,/3 (K )) +
Y pl/2^ T T f ~ l,-'ls{Y) -
+
+
y (/i/3 (n
+ /-5 /3 (V ))
/s/3<y)1
- A /s d ')]
y f S ' P XlA
S a V s ! p > / 4 [ C l7_1/3( y ) _ C ,/„ „ !> ') - C 2I - 5I3( Y ) + C 2/ 1/3(V)
121
+
C XI 5/3( Y ) + C XI . X/3(Y ) + C 2I 1/3(Y ) + C 2I - 5/3(Y)]
=
G'doay fS 'P xl A[CxI - X/z{Y ) + C 2I x/3(Y)}
Now the ac voltage and current in the G CA region can be obtained in term s of C x
and C2
vge( P ,S )
=
C xI2/3( Y ) + C 2U f 3( Y )
(B.14)
i ( P ,S )
=
G’dotV S ' P 1' 4[C1I - x/3( Y ) + C 2I x, 3( Y )\
(B.15)
Solving th e continuity and P oisson’s equations in th e saturation region gives th e
ac voltage and th e ac current. Integrating th e continuity equation (2.9) in chapter II
gives the ac current
i(x') — i0 e ~ ^ x
(B.16)
Since the ac current is continuous at the G C A /saturation boundary, w e have
i'o =
=
i(P , S )U = x s
G ^ V S ’P ^ I C J ^ Y . ) + C 2I x/3(Y .)]
(B .17)
where
Ps
=
Ya =
(1 — k s ) 2
| ^ P S3/4
T h e channel potential can b e approxim ately obtained by solving th e Poisson equa­
tion along the 2DEG channel.
(2.10) and integrating gives
Substituting Equation (B .16) in P oisson equation
122
n ( a : ') =
—fiio
e~ j ~>x
+
ax'
+
b
Since t h e ac electric fie ld is zero at th e G C A /saturation boundary, ^ j \ x= x s — 0?
the unknov/n constant
=
a
is
jfiio u>
(B .18)
The ch a n n el voltage in th e saturation region is
vK (x ') = P i o f c Y e - * * * + M o - x ' + b
\W /
ui
(B .1 9 )
Now t h e a c potential a n d ac current for the entire region are know n if C\ and C 2
are known. A s mentioned in Chapter II, the boundary conditions are wflc(®) — v gs
and vgc(Lg ) = vgs —vds. T h e channel p o ten tia l at th e drain end is show n in E quation
(2.23) in C hapter II. uflc( 0 ) can be d irectly obtained from the E quation (B .14) by
setting a: = 0 , P = 1 and Y = 4 \/5 '/3 .
M O ) = C iI V3( ~ V f ' ) + C2/_ 2/3( | v ^ ) = U3S
(B .2 0 )
Let us calcu late vgc(Lg) in term s of C i a n d C2. First th e voltage drop, Au(£), in th e
saturation region is the difference of v ( x ' = i) and vgc(x' = 0).
Augc(^)
=
#o0
)
e-*™*+ j / 3 i 0^ £ - p i 0 ( ^
=
/3i0 0 ) 2 [e-j %e ~ l ] + j P i ^ e
x a was given in Equation (2 .1 6 ) in C hapter II,
X, —
1
,
- .// . -- rV„c( j £ s )
V c c i X s f 3'
(B .2 1 )
123
Vq C is calculated in Appendix A .2:
K ( l — 2^®) Vgs — Vr
(1 - ksf
X?
Vg c ~
The calculation of v' ( X s ) in term s of Ci and C 2 will give us
/ .
,
V*c'
S)
dVgc
=
d P d V dVgQ
= ~dx~dP~dY
2 ks —
Xs
+
y
{ /i/3«
)
+
/- 5 /3 ( X ) }
so that
.
=
L---------- ^
*8(1 - j k . ) Vas_vT
(1-fc.)4
X?
. A
Xs
-----------
x
+ w
y /S 'p -U *
1'-)} + y U i / a t n ) + / - 5 / 3( n ) } ]
(1 - K f X . J S ' P ; ' ! *
- k , { i - \ k , ) {va s - v Tf ' { , -''°(Y‘] + h '*(Y‘))
+
c 2{ /1/3(y .) + /_ s/3(v;)}]
(B .22)
Substituting Equation (B .22), (B .21), and (B .14) into Equation (2.23) in Chapter II
and m anipulating yields
Ugc(-kg)
=
f i I d cE x s
-
^
+
C2{hi3,{Vs) + -f-5/3(K )}] +
+
j f - i
+
C i l 2 / 3(Y.) + C 2I-2/s(Ys)
” }■
X v g C( £ )
“ f*
Vgc(Xs')
( - ( 1 7 k/ f X i f f ^ ~ 1/4) [ c . M
\
ks (1 - j k sJ Vout
W
+ W i'.) )
[ e J>'*< - l ]
[<Zo. J S ' P } /4{ C i I - i /3{Y .) + C 2h /3(Ys)}}
(B.23)
124
T w o equations (B .20) and (B .23) are used to obtain C i and C2. For sim plicity two
equations m ay be rewritten as
Vgc{Lg)
v gc( 0)
— A n C \ + A 1 2 C 2 = Vgs — Vds
=
A 21 C 1
+
(B.24)
A 22 C 2 — vgs
where
An
=
h M Y . ) + a ’^ V s ' p ; / '
-
PhJ-
,(1 - k . f X s V S !P ;
0
u /3(y.)
/4
ks ( l - Ik s) Vc
r OUt
A 12 =
[ e - W - l] +
■ [ / - . / , « ) + / ./a « ) ]
/_ 2 /3( n ) + ^ osv ^ P 1/4 ^ 0 ) 2 [ e - ^ - l ] + i ^
h , 3 ( Y 5)
,( 1 - k s ) 3X s V & P s - 1/4
- fild J 7--------- :-----X-----k s (1 - I k s )
o u t
vc
(fv^7)
A 21
=
^2/3
A 22
=
1 -2 /3
From Equation (B.24) it is obvious that
Cx
A22(Vga
Vds)
A
A 22 — A 12
A
c2 =
A n v as
122
35
-
Aj2Vgs
A
'Vds
A 2\{VgS - v ds)
A\x — A 21
A 21
'Vgs T A Vds
A
' A
where A = ^4iii422 — •'4i2-d.2i*
For the calculation of Y u and F2i Vds have to be set to zero so that C \ and C2 will
125
be m odified as follow s
n <
(A 2 2 — A 1 2 )
^1
—
r > _
—
^2
V9*
A
( A n - A 2i )
^
v gs
The gate and drain current w ill b e expressed in terms of C [ and C1'2 as
Id
— i{x' = £) = ioe
G,1„ ' J S ' P t l l e - 1% ‘ [ C [ U i 3(Y,) + CS/1/ 3(y.)]
ta
—
i(x = 0 ) — i(x' = £)
=
g^
v s
7 [ c ; / _ 1/3 ( | v ^ ) + c ; / , /3 ( ^ ) ]
- G'to.T/S'PyU-WlCll-^Y.) + C M Y.)]
T he Txi and
v1 2 1
param eters for th e saturated M ODFET are
— —
Zd—
Vgs
Xu
V
9
— ----
—
ug s
=
G '^ y /S '
~
P } /4e - j %e(ClgsI - l /3 (Ys) + C 2 gsh ,z { Y s))\
C i„ I-
where
C \ gs
(A 22 _ A 12)
A
( A n — A 21)
C%g s
A
,/3 ( | v ^ ) + C 2„ I 1/3 ( i s ? )
126
vgs is set to zero to calculate Y12 and Y22 and we m ust introduce the new constant
C'i and C"
n"
Cl
a 22
— -^ V ds
=
si"
-d-21
C 2
—
^
v ds
The gate current and drain current are then
id
=
i[x' = t) = i o e ~ ^ e
=
G'dosVS' P y 4e - ^ e[ C ';i-1/3(Ys) + c"/1/3(ys)]
i(x = 0) — i(x' = t )
In
=
G L .V S ' c ; i . m
+c ; i„ 3
-
G '^ V S 'P ^ e -^ C U -^ Y ,) + c ; h ,3(y,)]
+
The Yi2 and Y22 param eters for the saturated M O D FET are
1d
y22
Vds
=
Y t 2
G'doay /S 'P } f 4e - i %e[CldaI - 1/3(Y.) + C 2dsI 1/3(Ya)]
=
^ds
G'dn,VS' C u s l - n z ( 3 ^
^dos
)
+ C 2dsI 1/3 ( - v ^ )
P.1/ 4e - J' * <[C'lli, / _ 1/ 3( y ,) + C 2ds l i /3(.YS)]}
where
r
lds
G2ds
-
—
^ 22
A
i2l
A p p en d ix C
T h e F ourth Order F req uency P ow er-S eries
S olu tion in th e G C A R eg io n
The fourth-order power-frequency solution of the w ave equation can be obtained by
expanding th e Equation (6a ),(6b ),(6c) in [10] up to v 4(x) terms
d2
dxM V a c ( x
-
Vt ) vq( x )
=
0
(C .la )
-
V r ) v i(s )
=
v 0(x)
(C .lb )
- VT ) v 2(x)
=
Vx(x)
(C .lc )
- VT )v3(x)
=
V2( x )
(C .ld )
[h (V g c { x - VT )v 4(x )
=
v 3(x)
(C .le )
^ H V a c(x
dx2
The current in the channel is obtained from
- ^ gWg- [ v ( x ) { V a c { x ) - VT)]
(C .2)
with
u (s )
=
v0(x) + j u jv i(x ) + (j w ) 2v 2(x ) + (ju>)3v 3( x ) + ( j u ) 4v 4(x)
(C .3a)
i (x )
=
i0(x) + j(x>h{x) + (j u ) 2i 2( x ) + ( j u ) 3i 3(x) + ( ju ) * i4(x)
(C .3b)
i l l and i 2i are determ ined by calculating ig and i d for Vds = 0 w ith the boundary
conditions no(0) = v0(L g) = u3s;v„(0) = vn(Lg) = 0 for n ± 0.
127
Ygd and Ydd are
128
calvn (fx, vdsn) := b lock ([i,m ,n,c,gdos,k,tm p,cl,c2],
fx:in tegrate(fx*C *2*(l-y)/k /(2-k ),y),
fx :in tegrate(fx*L *2*(l-y)/k /(2-k ),y) + C l*L *(2*y-y**2)/(2*k -k**2) + C2,
fx :fx /g d o s /L /(l-y ),
m :last(first(solve(ev(fx,y= 0),c 2))),
fx:ev(fx,c2= m ) ,
n :last(first(so lv e(ev (fx ,y = k )-fv d sn ,cl))),
fx :e v (fx ,c l= n ),
return(factor(fx))) $
Figure 74: List of calvn program
determ ined by calcu latin g ig and i d for vga = 0 w ith the boundary conditions vo(0) =
0,u o(L fl) = - v da, v n (0) = v n(Lg) = 0 for n ^ 0. T h e gate and drain currents (ig and
id) are obtained from th e channel current i(x ) using
id
=
ig =
i{x = L g)
(C.4)
i{x = L g) — i(x = 0)
(C.5)
Due the tedious nature of this calculation the sym bolic m anipulator M ACSYM A was
used. Two programs are used to calculated th e ac voltage and current, which are
calvn and calix fu n ction , calvn function solves th e wave equation and calix function
calculates the ac current from th e ac voltage for each com ponent. The programs are
shown in Figure 74 and 75. In calvn function vdsn is a boundary condition for ac
voltage.
T he calculations gives:
Y n = juFgg -
( j u ) 2 Sgg + ( j u f T g g ~ { j u J ^ D g g
129
calix (fx) : = block([tmp,id,k,m ,n],
ix: -k * (2 -k )* d iff(fx * g d o s,L * (l-y ),y )/2 /L /(l-y ),
return(factor(ix))) $
Figure 75: List of calix program
w ith
f
- ZO09r
fag
—
l'33
^ 0 3(2 - b)2
6
6 fc
+
fc 2
_ _ 60 - 12* + 81b 2 - 21b3 + 2 b4
~
4 5 (2 - b)5
77560
^ n - oo
226801b + 267301b2 - 156601b3 + 48071b4 - 7571b5 + 49b6
— r*
—
^0
14175(2 - lb)8
=
T*99
"
2 C q ------------------------------— --------------------------------- -------------
A 20196°0 - 80784001b + 135735601b2 - 124462801b3 + 6793245b4
° \\
233875(2
I
-— lb)
/vy11
33
22674901b5 - 4574111b6 + 51646b7 - 2527b8) \
2338875(2 - lb)11
J
Y12 = ju}Fgd - ( j u ) 2Sgd + (ju>)3Tgd - (j u ) 4D 3d with
3 - 41b + lb2
F g d
=
- 2
C
0
3(2 — b) 2
(1 - lb)(30 — 411b + 161b2 — 2*3)
S g d
=
T g d
=
n
_
3d ~
2 C 0
45(2 - lb)5
(1 - lb)(3780 - 8970b -f 79201b2 - 32471b3 + 637lb4 - 491b5 + 491b6)
— C o
14175(2 - lb)8
( ( * - * ) ( 1009800 - 3407580b + 4702830b2 -34307651b3
°V
2338875(2 - b ) U
1432920b4 - 346531b5 + 45486b6 - 2527b7) \
2338875(2 - b )11
J
130
= ju F dg - (ju>fSdg + ( j u f T dg - ( j u f D dg with
—2Co
30 - 45k + 20k2 - 3 k3
15(2 - k f
1
£
105
II
>21
II
to
DO - 1440A: + 1290A:2 - 540A;3 + 110A;4 - 9k5
II
—46*0
225(2 - k)6
(415800 - 1405800A: + 1945350k 2 - 1424610A:3
779625(2 - k f
V
599005A:4 - 146755A;5 + 19705A;6 - 1134fc7\
779625(2 - k f
+
J
(20196000 - 88387200k + 166148400A2 - 175707600A:3
° \\
dg
+
+
>22
iiu
j 'ia i
—
11694375(2
- ay
A;)12
115222050A;4 - 48760380A:5 + 13410670A:6 - 2330920 A;7
11694375(2 - k)12
234250A:8 - 10449A:9\
11694375(2 - k f 2 )
= ju F dd - ( j u f S d d + (ju j)3Tdd - (j u f D d d with
(1 - fc)(20 - 15* + 3A;2)
Fdd
~
2Co
15(2 - k f ----------
^ (1 - A:)(320 - 560A: + 340A:2 - 90k3 + 9A:4)
dd ~
0
0^/1
Tu
=
225(2 - k f
i \ ( 10560 - 290400A; + 315920A:2 - 175175A:3
779625(2 - k f -------------------
53165A:4 — 8505A;5 + 567k6 \
+
n
D dd =
779625(2 - k f
)
/10137600 - 38016000A: + 60064000A:2 - 52256000A3
C Q( l - k ) ^ -------------------------_ _ _ _ _ _ -----------------------27500560A:4 - 9035640A:5 + 1825520A:6 - 208980A:7 + 10449A8\
+
11694375(2 - A:)11
J
A p p en d ix D
D ev elo p m en t o f E quivalent C ircu it for th e
v elo city -sa tu ra ted M O D F E T
D .l
E q u iv a len t circu it for t h e sa tu r a tio n reg io n
T h e equivalent circuit for saturation region is based on the exact solution of the
velocity-saturated M O D FE T wave equation so that th e procedure for solving the
wave equation will be repeated. The exact solution for th e GCA wave equation is
given in A ppendix B.
vgs( Y ,S )
=
C 1/ 2/ 3( F ) + C'2/ _ 2/ 3( F )
(D .la )
i(Y, S )
=
G ' ^ P ^ V S ' i C . I ^ Y ) + C 2I 1/3( Y ))
(D .lb )
where
Y
=
\ V S iPa/A
O
P
=
l - ( 2 ka - k l ) ^
It is assum ed that vgc(Y3, S ) = vg3 — v'da and i = i'd at boundary.
Now the
developm ent of th e equivalent circuit for the saturation region requires to derive the
relationship betw een v'ds and i d. Let us start with the voltage and current in the GCA
131
132
region. At th e boundary x = X s we have
v (X .)
=
C J y s i Y , ) + C 2I-2I3(Y.) = vga - v'ds
(D.2a)
i(X .)
=
G'doaP t /4y / S ' { C 1I _ l/3(Ys) + C2I1/3(Ya)) = i'd
(D.2b)
The ac current and ac voltage in the saturation region is given by Equation (B.16)
and (B.19) in Appendix B . In these equations z'o can be replaced i'd directly. For
convenience th e se equations are rewritten
i(x')
=
i'de~^~>x'
(D.3a)
v(x')
=
p i'd ( ^ ) \ - ^ + j p i ^ x ' + b
\(jJ /
U}
(D.3b)
The boundary condition at the drain side is vgc{ty
v9• —Via- T h e gate to channel
voltage vgc at drain side is given by Equation (2.23)(see Chapter II).
Vac{Lg) = p i d j x s + A v gc(£) + Vgc( X a)
(D.4)
In order to derive the equivalent circuit Equation (D.4) has to be expressed in
term s of vda and i d. x a is g iv en in Equation (2.16) in Chapter II
X' = " V a c ( X s ) V^ Xs>>
(D-5)
where Vqc ( X s ) is given in Equation (2.17) in Chapter II. N ote that the derivative of
th e Bessel functions verify
=
/ „ + ,( * ) + ^ / „ M
(D .6)
133
Using th ese properties one can o b tain v'gc( X s )
' (V \
V“ {X S)
d P d Y dv,gc
dx d P d Y lx=Xv s
<lVgc
=
v
X=Xs
2 k a - k2
Xs
S S ’P 3~1/4
[c7i { /-i/s(y .) - ^ r h / s ( Y s)
+ C 2 { l 1/3(YS) + -_ ^2/ I3 .U 2/ 3(F s )
2 ks - k]
Xs
3 Ys
|
V ^ P 5- 1/4 [ { C r l ^ / s i Y ) + C2I 1/3(Ys )}
{C xIm
{Ys) +
(D.8)
C 2 / _ 2 /3 ( F s ) } ]
Substituting Equation (D .2a) and (D .2b) into Equation (D.8) yields
vK ( X s )
=
G ^ P ^ V S ' ( Z k . - k*)
2 ks - k2 / 7^ p _ i /4 2 uss —
+
Xs
V iF ‘
G'dosX s P } /2
+
i'd
G '^ X s il-k .)
i i V s ’P?'*
2&s - k2.
IX sPs
+
(Vgs ~ Vjs)
2ka- k 2
2Xs ( l - k , ) * '
( Vgs ~
(D.9)
V'd s )
VqC( X s ) was given in Equation (2.17) in Chapter II so that x s can be obtain ed in
terms o f i d and vds.
4 X |( 1 - ks)3
Xs
"
(2 ka - k 2) 2Vout
,
G'dosX s ( l - k s)
2k° ~ k s
r..
+ 7T^~T,
7T z{.vgs
2X 5(1 — ks )2
2X 5(1 - k.)
4 X 5 (1 - h f
(Vgs — Vds)
-i'd +
G'd0, ( 2 k s - k2)2Vout d ’ (2k3 - k 2)Vout
Vds)
(D.10)
where Vout = Vqs — Vt - For convenience the new constant A and B are introduced
A
=
B
=
^ X s(l ~ ks)
(2k3 - k 2)V0Ut
4 X 5 (1 - h ) 2
G'dos(2ks - k ])W out
134
A sv 3C(f) is given in Equation (2.27) in Chapter II. Now Equation (D.4) can be rewrit­
ten in term s of v da and id.
Vgc^^g)
— Pldtfi^s “i"^SVgc(^) 4" ^(A s) = Vga
Vda
=
p I dJ [ - B i ' d 4 A ( v ga - «$,)] + Pi'i ( 5 ) 2 [e_ j^
4
jPi'd
~ !]
i 4- Vga - v'da
( D .ll)
From th e above equation one can easily derive th e relation betw een i d and v ds.
v ds
=
/3ld<j*AVgS
Vds
4
1 4 Pldc^A
1 4 fildcf-A
Pi'd
1 4 P IdM
(D.12)
It is difficult to develop the equivalent circuit for th e saturation region from th e above
equation so that th e exponential term is expanded in power-series up to fourth order
term.
Vds
PldP'-A.Vgg
Vds
4
1 4 pidc^A
1 4 Pldc^A
Pi'd
i dc£ B ~ ( ^ y { \ - j - i
\(jJ J I
v.
1 4 pj-dJLA
/3Id<£Avga
Vds
4
1 4 pidc^A
1 4 pidcf-A
f3 P
D I a J B + \ p e - j w r . S ^ . - (o,r.)2 ^
24
6
1 4 pidJ-A
(D.13)
where rs = £ /v a T h e drain current is obtained from Equation (D .3a) by settin g x' = I.
id = *'dt J“' T5
(D.14)
One can easily ob tain ed the g a te current in th e saturation region from th e difference
of th e channel current at drain side and the G C A /satu ration boundary.
135
S
Figure 76: Equivalent circuit for the saturation region
i , = i'd ( l -
(D .15)
T h e above three equations gives the equivalent circuit for the saturation region
w hich is shown in Figure 76. Comparing Figure 76 w ith Equation (D .13) one can
easily find 7 s and S3
7s
=
1+ p ia A
fiIdc£A
(D ,16)
.
=
T + Jh M
(D '17)
Let us calculate th e value of each elem ent in the im pedance part.
T h e following
equation can be found from equivalent circuit
v da = 7sVds + S.vg, -
( R ai + i .1 •'
) i'd
V
+ JwCs J
(D .18)
In order to com pare Equation (D.18) w ith Equation (D.13) we should expand
th e denom inator in power-series up to the second order using the assum ption that
136
C s R s2 «
OJ.
v ds
~
7 sVds + 6avga - [ if o + R a2 - ju>CaR 2a2 - u>2C 2R%2] i'd
(D .19)
Com paring Equation (D.13) and (D.19) and equating th e term of sam e order gives
'Idc£B +
1 + ftldc^A
r,,/^ 2
=
<D-21)
6(1
Solving the above equations yields
p i dci B
Rsl
~
R a2 =
-
l + 0 I dc£A
(D>23)
2/y
3(1 + p I dc£A)
(D .24)
c■ =
N ow
( ° - 25)
is
— ^av da -t- £>avga
Z ai d
(D .26)
where
Z a = i?si H— j/is2 + jio C a
D .2
C a lc u la tio n o f th e Y -p a ra m eter s for th e tw o reg io n
m odel
T h e Y-param eters of the velocity saturated M O D FET , Yij(sat), w ill be obtained
from the equivalent circuit in term s of the Y-param eters of the GCA region Yij(g) of
137
reduced gate length X s = Lg —£. The gate and drain currents are easily obtained by
inspection from th e equivalent circuit (see Figure 39)
ig
= Ygg{g)vga -\-Ygd{g)v,d a-\-i,d{ \ - e ~ 3WT‘ )
(D.27)
id
= i'de~3wr‘
(D.28)
i'd
- Ydg(g)vga + Ydd(g )v da
(D.29)
where v'da is given by Equation (D .26).
Substituting E quation (D.26) in to (D.29)
yields
id
= Ydg(g')Vga
~
Ydd(g')('JaVda
8aVga
Y>aid)
Ydg^g)vga -)- Ydd{g)^av da + Ydd(g')8avga
Ydd(g } Z ai d
(D.30 )
From Equation (D .30) one can easily obtain i d in terms of vgs and vda as follows
. Ydg(g) + Ydd(g)6a
"
i +
,
Ydd{g)^a
v>‘ + i + Y M z . Vdl
( D3l)
Replacing Equation (D .31) in (D .28) gives
*=
+ (t W
m
) ^
(a32)
Substituting Equation (D .26) into (D.27) yields
ig
= Ygg(g)vgs + Ygd(g)(~jav ds + 6avga - Z si d) + i d ( l - e _ja,Ts)
= ( Ygg(9 ) + Ygd{g)8a)vga + Ygd(g)l a v da + i'd { \ - e - i “T‘ - Z aYgd{g ))
Substituting Equation (D .31) into (D.33) gives
ig
= (Ygg(g) + Ygd(g)8a)vga + Y g ^ g ^ v ^ + ( l - e~3“T’ - Z aYgd(g ))
(D.33)
138
Yd3 (g)
x
+
Ydd(g)&,
1 + Ydd( g ) Z s
Y d d (g )is
^
3‘ + 1 + Ydd(g)Za Vds)
99
M +rM (. +
^ “ e’,"T‘ “ Z,Y‘d(s)^
v.'gs
v d,
Ys i \
(D .34)
By definition Equation (D .34) and (D .32) defined th e total Y-param eters of th e
velocity-saturated M O D FET.
M
^ Yi
3^
c- ,w . _ z sF5d(5 ))
+ Y
_ Z. sy3(i(5 ))
y , + T ¥ 0 ^ ( l - e -e-iu-r.
^ - Z
Y ,M
-
YM
Y
( o\
Yd9{S)
—
“
( Ydgjg) + Ydd(g)Ss \
j WTt
\ 1 + Ydd( g )Z s ) e
Y“ {s)
“
(
t
W
m
) ■ '* "
t
I1
r .M
BIBLIOGRAPHY
[1] H. Morkog and P. Solomon, “T he HEMT: A super fast transistor,” I E E E Spec­
trum, vol. 21, pp. 28-35, 1984.
[2] C. P. Lee, S. J. Lee, D. L. Miller, and R. J. Anderson, “U ltra high speed digital
integrated circuits using G aA s/A lG aA s high electron m obility transistors,” Proc.
IEEE G a A s I C Symposium, pp. 162-165, 1983.
[3] R. H. H endel, S. S. Pei, C. W . Tu, B . J. Rom an, N. Shah, and R. Dingle, “Realiza­
tion of sub 10 picosecond switching tim es in selectively doped (A l,G a)A s/G aA s
heterostructure transistors,” IE E E IE D M Tech. Dig., pp. 857-858, 1984.
[4] N. T. Linh, M. Laviron, P. D elescluse, P. N. Tung, D. D elagebeaudeuf, D . Diamand, and J. Chevrier, “Low noise performance of two dim ensional electron
gas F E T s,” Proceedings o f the 10th Cornell Conference on Advanced concepts in
High-Speed Sem iconductor Devices and Circuits, vol. 7, pp. 187-192, 1985.
[5] U. K. M ishra, A. S. Brown, M. J. Delaney, P. T . Greiling, and C. F. K rum m , “The
A lInA s-G alnA s H EM T for m icrowave and m illim eter-wave applications,” IEEE
Transactions on Microwave Theory and Techniques, vol. 37, no. 9, pp. 1279-1285,
1989.
[6] A. J. Tessm er, P. C. Chao, K. H. G. Duh, P. H o, M. Y. K ao, S. M. J. Liu, P. M.
Sm ith, J. M. Ballingall, A. A. Jabra, and T. H. Yu, “Very high perform ance 0.15
p m gate-length In A lA s/In G aA s/In P lattice-m atched H E M T s,” Proceedings of
the 12th Cornell Conference on Advanced concepts in High-Speed Semiconductor
Devices and Circuits, 1989.
[7] D. D elagebeaudeuf and N . Linh, “M etal-(n) A lG aA s-G aA s tw o-dim ensional elec­
tron gas F E T ,” IEEE Transactions on Electron Devices, vol. ED-29, pp. 955-960,
1982.
[8] H. R. Yeager and R. W . D utton, “Circuit sim ulation m odels for high electron
m obility transistor (H E M T s),” IE E E Transactions on Electron Devices, vol. ED33, p p .
6 8 2 -6 9 2 ,
1986.
[9] D. H. H uang and H. C. Lin, “Dc and transm ission line m odels for a high electron
m obility transistor,” IE E E Transactions on M icrowave Theory and Techniques,
vol. 37, no. 9, pp. 1361-1370, 1989.
139
140
10] P. Roblin, S. Kang, A . K etterson, and H. Morkoc, “Analysis of M O D FET m i­
crowave characteristics,” IEEE Transactions on Electron Devices, vol. ED -34,
pp. 1919-1928, 1987.
11] D . B. Candler and A. G. Jordan, “A sm all-signal analysis of th e insulated-gate
field-effect transistor,” International Journal o f Electronics, vol. 19, pp. 181-196,
August 1965.
12] J. A. G eurst, “Calculation of high-frequency characteristics o f thin-film transis­
tors,” Solid-State Electronics, vol. 8 , pp. 88-90, January 1965.
13] J. R. Hauser, “Sm all-signal properties of field-effect devices,” I E E E Transactions
on Electron Devices, pp. 605-618, 1965.
14] J. R. Burns, “High-frequency characteristics of the insulated gate field-effect
transistor,” R C A Review, vol. 28, pp. 385-418, Septem ber 1967.
15] D . H. Treleaven and F . N . Trofimenkoff, “M O SFET equivalent circuit at pinchoff,” Proceedings o f the IEEE, vol. 54, pp. 1223-1224, Septem ber 1966.
16] J. V. N ielen, “A sim ple and accurate approxim ation to th e high-frequency charac­
teristics o f insulated-gate field-effect transistors,” Solid-State Electronics, vol. 12,
pp. 826-829, 1969.
17] M . Bagheri and Y. T sivid is, “A sm all signal dc-to-high-frequency nonquasistatic
m odel for th e four- term inal M O SFET valid in all regions of operation,” IE E E
Transactions on Electron Devices, vol. ED-32, pp. 2383-2391, N ovem ber 1985.
18] M . Bagheri, “An im proved M O D FET microwave analysis,” I E E E Transactions
on Electron Devices, vol. ED-35, no. 7, p. 1147, 1988.
19] A . V. D. Ziel and J. W . Ero, “Sm all-signal high-frequency theory of field-effect
transistors,” IEEE Transactions on Electron Devices, vol. E D -11, pp. 128-135,
April 1964.
20] V . Ziel and E. N. W u, “High-frequency adm ittance of high electron m obility
transistors(H E M T s),” Solid-State Electronics, vol. 26, pp. 753-754, 1983.
21] H. Rohdin, “Reverse m odeling of E /D logic sub-micron M O D FE T s and predic­
tion of m axim um extrinsic M O D FET current cutoff frequency,” I E E E Transac­
tions on Electron Devices, vol. E D -37, pp. 920-934, 1990.
22] P. Roblin, H. Rohdin, C. J. Hung, and S. W . Chiu, “C apacitance-voltage anal­
ysis and current m odeling of pulse-doped M O D F E T ’s,” IE E E Transactions on
Electron Devices, vol. E D -36, pp. 2394-2404, 1989.
23] D. J. W idiger, Two-Dimensional Simulation o f the High-Electron M obility Tran­
sistor. P hD thesis, U niversity of Illinois, 1984.
141
[24] H. Rohdin and P. R oblin, “A M O D F E T dc m o d el with im proved pinchoff and
saturation characteristics,” IEEE Transactions on Electron Devices, pp. 664-672,
1986.
[25] H. B. D w ight, Tables o f Integrals and Other M athem atical Data.
Publishing Co. Inc., 1961.
M acm illan
[26] M. Bagheri, “An im proved M O D F E T microwave analysis,” I E E E Transactions
on Electron Devices, vol. ED-35, p . 1147, 1988.
[27] S. J. M ason, “Power gain in feedback amplifiers,” IRE Transactions on Circuit
Theory, vol. CT-1, pp. 20-25, June 1954.
[28] J. B. Gunn, “Transport of electrons in a strong built-in electric field,” Journal
o f Applied Physics, vol. 39, no. 10, pp. 4602-4604, 1968.
[29] M . J. Moloney, F. P onse, and H. Morkoc, “G ate capacitance-voltage character­
istics of M O D FE T ’s: Its effect o n transconductance,” I E E E Transactions on
Electron Devices, vol. ED-32, no. 9 , pp. 1675-1684, 1985.
[30] P. Roblin, L. Rice, and H. M orkoc, “Nonlinear parisitics in M O D FE T ’s and
M O D FE T I-V characteristics,” I E E E Transactions on Electron Devices, vol. ED35, no. 8, pp. 1207-1214, 1988.
[31] P. Wolf, “Microwave properties o f schottky-barrier field-effect transistor,” I B M
Journal o f Research and Development, vol. 9, p p. 125-141, 1970.
[32] J. B. Kuang, P. J. Tasker, G. W . Wang, Y. K . Chen, L. F . Eastman, O. A.
A ina, H. Hier, and A. Fathim ulla, “Kink effect in subm icrom eter-gate M BEgrown InA lA s/InG aA s heterojunction M E SFE T s,” IEEE Electron Device Let­
ters, vol. EDL-9, pp. 630-632, 1988.
[33] S. Y. Chou and D. A. Antoniadis, “Relationship between m easured and intrinsic
transconductances o f F E T ’s,” I E E E Transactions on Electron Devices, vol. ED34, pp. 448-450, February 1987.
[34] A. B. Grebene and S. K. Ghandhi, “General th eory for pinched operation o f the
junction-gate fet,” Solid-State Electronics, vol. 12, p. 573, 1969.
[35] W . Shockley, “A unipolar ’field-effect’ transistor,” Proceedings o f IRE, vol. 40,
p. 1365, 1952.
[36] K. Lee, M. S. Shur, T . J. Drum m ond, and H. Morkog, “Parasitic M E SF E T
in (A l,G a)A s/G aA s m odulation d op ed FE T’s and M ODFET characterization,”
IE E E Transactions on Electron D evices, vol. E D -31, pp. 2 9 -3 5 , January 1984.
[37] M. B. Steer and R. J. Trew, “High-frequency lim its of m illim eter-wave tran sis­
tors,” IE E E Electron Device Letters, vol. EDL-7, November 1986.
[38] P. Roblin, S. Kang, and W . Liou, “Improved sm all-signal equivalent circuit m odel
and large-signal state equations for th e M O SF E T /M O D F E T wave eq uation ,”
IE E E Transactions on Electron Devices, vol. E D -38, June 1991.
Документ
Категория
Без категории
Просмотров
0
Размер файла
4 210 Кб
Теги
sdewsdweddes
1/--страниц
Пожаловаться на содержимое документа