A Modified EKV Model for Circuit Simulation

Abstract- A revised model for the current-voltage (I-V) characteristics based on the
current EKV model is presented. A EKV model uses a linear combination of to
logarithmic functions to interpolate beteen to regions of operation to generate a
single-piece model hich allos continuous derivatives ith respect to the e!ternal bias
voltages. "his model is rather convenient for circuit simulation for its simplicit# and
continuous nature. $oever% the original EKV model does not have the re&uired
precision compared to the simplified% s#mmetrical to-piece model% hich is commonl#
used in S'(CE. "hus% a modified EKV model has been developed to obtain the precision
re&uired as ell as preserving the original advantages of such a model.
(. ()"*+,-C"(+)
"o simulate M+S transistors in a circuit% e need a model hich can generate results in a
reasonable amount of time hile the accurac# is ma!imi.ed. Although the complete
charge-sheet model can produce the best accurac#% it is rather computational intensive
attribute to the fact that it contains a high-order of pol#nomial as ell as implicit
e!pressions for the surface potentials. And% to be sure% the latter re&uires iterative
techni&ues to be evaluated% hich is completel# unsuitable for circuit simulation.
"ill no% one of the most popular S'(CE M+S/E" 0evel-1 model is the
Simplified S#mmetric Model 234 5 264. "his model as named suggested is simple and
reasonabl# accurate ithin a certain range of operation% namel#% the ea7 and strong
inversion. 8et% this model completel# ignores the moderate inversion have e not
stretched the boundar# beteen strong and moderate inversions. 9ecause of the nature of
this model% a discontinuit# occurs hen e go from the ea7 inversion into the strong
inversion. (n addition% even inside the strong inversion% there are to e&uations re&uired
to describe saturation and non-saturation regions. $ence% in this regard% this multi-piece
model does not reall# simplif# the matter for it complicates the derivatives of the current
ith respect to the e!ternal biases.
A EKV model solves the problem. "he initiative of an EKV model is not to
improve the accurac# but to be able to collapse the simple model discussed above into a
single-piece hich has a continuous derivative ith each of the e!ternall# applied bias
voltage% and most important of all% it describes the moderate inversion. "o do so% it
interpolates beteen the strong and ea7 inversion to appro!imate hat is happening in
the complete charge-sheet model. $ence% it is rather a mathematical effort as opposed to
ph#sics to arrive such a model. )evertheless% the present EKV model% though simple% is
biased either toard the strong inversion% depending on some parameters. 9ut if e can
fi! this problem b# mathematical means% an EKV model is reasonabl# suitable for circuit
simulation if the precision e!pectation does not e!ceed that from the s#mmetrical model.
(n the folloing section% the revised EKV model is presented% and compared to
the complete charge-sheet model as ell as the simplified s#mmetrical model. (n section
(((% the results and precision issues from the modified EKV model ill be discussed.
((. "he Model
"he anal#tical model presented in this paper is based on the assumptions used to derive
the simplified s#mmetric model. "his includes the assumptions: first of all% a gradual
channel appro!imation is considered. Second% once velocit# saturation occurs near the
drain end of the channel% further increase in the drain current is onl# due to channel
length modulation. )e!t% an uniform substrate doping concentration. 0ast% the
negligence of the gate current in order to ma7e the model simple.
Equation 1
here
Co! +!ide Capacitance per unit area
; ;idth of the channel
0 0ength of the channel
t
thermal voltage
n< a slope function given b#
P
V
n
< = >
3 <
+
+

23.34
here
) 3 ) )(tanh( 1 ( > = + +
HB
V Vgb t f 23.>4
= )
6 >
( <
>
>

+ +

Vfb Vgb V
P
23.14
= = + + Vfb V
X
23.64
) 3 ) (tanh(
>
3

MB
V Vgb
23.?4
here
=
an interpolating surface potential function beteen
f >
and
t f @ > +
f
intrinsic fermi potential of the substrate
V$9 the boundar# beteen the strong and moderate inversion in Vgb
VM9 the boundar# beteen the moderate and ea7 inversion in Vgb
V<' the pinch-off voltage function at surface potential
=
VA an analogous function to V"=% but ith a d#namic surface potential

a dumm# function interpolating beteen > to ->

channel length modulation coefficient

'

1
1
]
1

+
1
1
]
1

+ +

>
< >
<
>
< >
<
>
3 ln( ) 3 ln( ) < ( >
t n
Vdb n V Vgb
t n
Vsb n V Vgb
EKV
X X
e e n t Cox
L
W
Ids


Equation 2
here
f > =
2>.34
+n the other hand% hen the value of Vgb is large% i.e. in the strong inversion region%
e&uation 3 corresponds to the simplified s#mmetric model in the strong inversion given
b#:
( ) ( )
1
]
1

) (
>
> >
=
Vsb Vdb
n
Vsb Vdb V Vgb Cox
L
W
Idsn
T s

Equation 3
here
t f @ > = +
21.34
$ence% beteen the strong and ea7 inversion% e can see that the EKV model must
provide an interpolation solution to the moderate inversion for e&uation 3 is continuous
both in itself as ell as its derivative correspondent.
(n the original EKV model% e&uations 23.34 5 23.?4 are simplified reduced to their
correspondent e!pressions hen the surface potential is either in the strong 21.34 or ea7
inversion 2>.34. (n addition% beta function is reduced to = at the strong inversion and to 53
for the ea7 inversion. "he techni&ue used here is the h#perbolic function of tangent
hich gives the folloing characteristics:
8(!) B tanh(!)
9# choosing different shifting for ! and #% e can interpolate beteen an# to points
ith a reasonable sharp edge% i.e. appro!imate a step function. "hus% the interpolating

'

1
1
]
1

1
1
]
1

t n
nVdb V Vgb
t n
nVsb V Vgb
sw
T T
e e n t Cox
L
W
Ids


> > >
= =
) 3 (
3
3
8 ! ( )
3= 3= !
3= ? = ? 3=
3
=
3
functions from 23.34 to 23.?4 are used in order to produce the correct simplification hen
(dsEKV is in either the strong or ea7 inversion.
(((. *esult and ,iscussion
(n the folloing% several aspects of modified EKV model is presented against either the
complete charge-sheet model or the simplified s#mmetric model or both. More
specificall#% the discussion is divided into three parts: 3) I-V curve >) derivative of I vs. V
1) accurac# evaluation. Each of hich contains characteristic plots against all e!ternal
bias voltages% Vgb% Vdb% and Vsb. "he purpose of these plots is to demonstrate the
accurac# of the modified EKV model.
1) I-V Characteristics
a) Ids vs. Vgb
) !t"ong Inve"son
) Mode"ate Inve"son
@.>@=1=> 3=
6
.
1.=C?@33 3=
?
.
(ds Vgb vdb , vsb , 3.? , 3.? , ( )
(ds
EKV
Vgb vdb , vsb , ( )
>.DDD>3D 3.1C>>3D Vgb
3.> 3.6 3.@ 3.C > >.> >.6 >.@ >.C 1
3 3=
?
3 3=
6
3 3=
1
) Wea# Inve"son
b) Ids vs. Vdb
$) Ids vs. Vsb
1.>@E>16 3=
?
.
?.1?316C 3=
C
.
(ds Vgb vdb , vsb , 3.> , 3.> , ( )
(ds
EKV
Vgb vdb , vsb , ( )
3.1C>>3D =.EC>>3D Vgb
=.E =.C =.D 3 3.3 3.> 3.1 3.6
3 3=
C
3 3=
E
3 3=
@
3 3=
?
3 3=
6
=.======3
>.D3=6>3 3=
3?
.
(ds Vgb vdb , vsb , =.? , =.? , ( )
(ds
EKV
Vgb vdb , vsb , ( )
=.EC>>33 =.>3?C>3 Vgb
=.> =.1 =.6 =.? =.@ =.E =.C
3 3=
3?
3 3=
36
3 3=
31
3 3=
3>
3 3=
33
3 3=
3=
3 3=
D
3 3=
C
3 3=
E
%) &e"vatve of I vs. V
a) I'ds vs. Vgb
) !t"ong Inve"son
) Mode"ate Inve"son
) Wea# Inve"son
b) I'ds vs. Vdb
$) I'ds vs. Vsb
@.>@?>=6 3=
6
.
=
(ds vgb Vdb , vsb , 1 , 1 , ( )
(ds
EKV
vgb Vdb , vsb , ( )
1 =.1 Vdb
= =.? 3 3.? > >.? 1
=
3 3=
6
> 3=
6
1 3=
6
6 3=
6
? 3=
6
@ 3=
6
E 3=
6
@.>@?>=6 3=
6
.
6.>D?>=> 3=
>=
.
(ds vgb vdb , Vsb , 1 , 1 , ( )
(ds
EKV
vgb vdb , Vsb , ( )
1 =.1 Vsb
= =.? 3 3.? > >.? 1
3 3=
6
=
3 3=
6
> 3=
6
1 3=
6
6 3=
6
? 3=
6
@ 3=
6
E 3=
6
() Accurac# Evaluation
)) E""o" of Modfed EKV Mode*+ $o,-a"ed to t.e $o,-*ete $.a"ge-s.eet ,ode*
a) E""o" of !/,,et"$ Mode*+ $o,-a"ed to t.e $o,-*ete $.a"ge-s.eet ,ode*