1997’
IEEE
In ema ional Symposium
on
Ci cui s
and
Sys ems,
June
9-12,1997,
H-Kong
--
Misma ch Dis ance Te m Compensa ion in[ Cen oid Con igu a ions
wi h Nonze o-A ea De ices
E.S.
Ka hunen, F ancisco V. Fe niindez and
A,.
Rod iguez Viizquez
Dep . o Analog and Mixed-Signal In eg a ed Ci cui Design, TMSE-CNM
Edi .
CICA,
A da. Reina Me cedes s/n, E,-41012 Se illa, SPAIN
Tel.: +34
5
4239923. FAX: +34
5
4231832, E-mail: [email p o ec ed]
Abs ac
This pape p esen s an analy ical app oach
o
dis-
ance e m compensa ion in misma ch models o in e-
g a ed de ices. Fi s ly, he condi ions ha minimize
pa ame e misma ch a e examined unde he assump ion
o
ze o-a ea de ices. The analy ical de elopmen s a e
illus a ed using cen oid con igu a ions. Then, de ia-
ions om he p e ious app oach due o he nonze o
de ice a eas a e s udied and e alua ed.
1.
In oduc ion
The endency in in eg a ed ci cui design owa ds
he use
o
submic on echnologies has mo i a ed
imp o emen s in he modeling
o
he second o de e ec s
o he
MOS
ansis o and o he andom a ia ions in he
ab ica ion p ocesses, which a e c i ical in he design
o
high pe o mance analog ci cui s
[
11-[7].
Va iabili y phenomena o elec ical cha ac e is ics
in in eg a ed ci cui s can be classi ied in o wo g oups:
in e -die a iabili y and in a-die misma ch. The i s one
accoun s o di e ences die- o-die o wa e - o-wa e ,
while he second is due
o
he exis ence o pa ame e luc-
ua ions in he wa e . We will ocus on in a-die misma ch
as
i
is he main esponsible o he de ia ions in analog
ci cui beha io .
The misma ch e ec on ci cui pe o mance has
been examined by many au ho s, and di e en misma ch
models ha e been p oposed ecen ly [3]-[9J. Sec ion
2
will b ie ly desc ibe he mos widely known in a-die
misma ch model: he Pelg om’s model. In sec ion 3 we
will ocus on he second e m o his model, which
accoun s o he misma ch due o he dis ance be ween
dle ices. An analy ical echnique o compensa e his mis-
ma ch is in oduced and applied
o
se e al cen oid con-
igu a ions. Finally, sec ion 4 examines he e ec o
aking in o accoun de ice a eas on he dis ance e m and
he magni ude
o
his pe u ba ion is e alua ed.
2.
Misma ch modelling
In 1988, Pelg om p oposed
a
model which has
become a e e ence o misma ch e ec s on analog in e-
0-7803-3583-X/97
$10.00
01997
IEEE
1644
g a ed ci cui s
[3].
Fo
wo samples
o
a
de ice wi h
equal
W.
L
and
a
sepa a ion dis ance
Ol2,
his model
associa es
a
s ochas ic a iable wi h he di e ence
be ween he alues o he same pa ame e in each ansis-
o (i. e. h eshold ol age). This s ochas ic a iable can
be ep esen ed by i s s anda d de ia ion,
A
aiid
S
being echnology-dependen i ing cons an s.
The i s e m in
(1)
includes he in luence
o
an-
sis o sizes and shapes ( hese do no appea explici ly
as
ec angula shapes a e assumed).
I
can be assumed ha
each ansis o pa ame e has
a
nominal alue and
a
supe imposed whi e noise. Associa ing Laplace ans-
o ms o he ansis o con igu a ion, i can be conside ed
as
a
noise il e ing p ocess. Then, his e m in
(1)
appea s
as
a
esul o noise powe no elimina ed in he il e ing
p ocess by he con igu a ion. The e o e, he con igu a-
ion in oduces
a
se o a iables in (l), namely, sizes and
shapes.
The second e m in (1) ep esen s he in luence o
he dis ance be ween ansis o s
DI2,
and is
a
conse-
quence o he exis ence o g adien s in he wa e . The
alue o any ansis o elec ical pa ame e
P
is ep e-
sen ed by
a
unc ion o dis ance in he wa e ,
P=Kd,
whe e:
d
is he dis ance om he poin whe e
P
is e alu-
a ed
o
he poin known
as
pe u ba ion cen e . This pe -
u ba ion cen e is he poin in he wa e whe e he
pa ame e akes i s maximudminimum alue. The e-
o e, he adial dependence o
a
pa ame e can be mod-
eled
as
a
cone whose e ex is he pe u ba ion cen e
[4].
P
P
3.
Misma ch Reduc ion Techniques
‘The physical in e p e a ion
o
each e m in (1) sug-
ges s di e len echniques
o
educe hem. The a ea e m
can only be educed by inc easing de ice a eas
[8]-[9].
Fo
sinall
de ices, his is he dominan e m and he dis-
ance e m can be neglec ed. Howe e , o high-pe o -
mance design, in which a eas a e ela i ely la ge,
dis ance e m plays an impo an ole. In o de o educe
his in luence, pa i ioning echniques and cen oid con-
igu a ions a e commonly used, al hough his is only
based in heu is ic conside a ions. In he ollowing, educ-
ion echniques a e analyzed using an analy ical app oach
which allows o ind ou con igu a ions wi h in e es ing
s a is ical p ope ies.
3.1.
De ice pa i ioning
Le us assume ha he pe u ba ion cen e is he e -
e ence poin o e e y dis ance. The e o e, he new se o
a iables
di
is he dis ance om each ansis o o he
pe u ba ion cen e (de ice sepa a ion
Ol2
can be easily
ob ained
as
he di e ence ec o ), as illus a ed in Fig.
1.
Ku ba ion cen e
Figu e
1:
Dis ances
o
pe u bacion cen e.
Gi en wo de ices,
ou
echnique pa i ions one o
bo h
o
hem in o a se o subdi isions
(M
and
N
espec-
i ely) and ies o combine hem, gene a ing di e en
con igu a ions, o imp o e he s a is ical pa ame e s. This
is illus a ed o he con en ional cen oid con igu a ion
in Fig.
2(b)
whe e one o he de ices in Fig. 2(a) has been
pa i ioned in o ou pieces. The g adien e ec compen-
sa ion due o he ou subdi isions seems o make he
pa ame e alue mo e uni o m. In his way we could
ob ain a educ ion o he s anda d de ia ion compa ed
wi h he one in he o iginal con igu a ion.
nu
Figu e
2:
(a) Basic S uc u e; (b) Cen oid s uc u e.
3.2.
Ma hema ical
model
A ma hema ical model mus be de eloped o e alu-
a e he in luence on misma ch o he pa i ions pe o med
on he ansis o s. Pa ame e misma ch is an s ochas ic
a iable, hence he unique way o ob ain his model is
using i s i s - and second-o de s a is ics: mean and s an-
da d de ia ion.
We will i s assume ze o-a ea ansis o s placed in
hei geome ic cen es. This is easonable as we a e
y-
ing o educe he dis ance e m o Pelg om's model, and
hence in a i s app oxima ion we a e in e es ed only in
he dis ance be ween he pa i ions bu no in hei a eas.
Assuming a adial dependence o he pa ame e
P
,
P=
P
0
+K,d
(2)
and a e aging dis ances om each pa i ion o he pe u -
ba ion cen e, he pa ame e a ia ion is
N
M
(3)
j=l
j=]
whe e
K
is he g adien slope on he wa e ,
di
and
d.,
a e
he dis ances om each pa i ion o he pe u ba ion cen-
e, and,
M
and
N
a e he numbe o pa i ions in each
de ice.
Du ing he design o a ci cui he ela i e posi ion o he
con igu a ion wi h espec o he pe u ba ion cen e is
unknown,
so
we mus calcula e misma ch as an a e age
o
AP
o e e y possible con igu a ion posi ion wi h
espec o he pe u ba ion cen e. The mean is,
J
27[/
N
and he s anda d de ia ion is gi en by,
2
27c
N
M
0
i=l
j=1
IS^
=
&I
[A
di-G
1
dj]'I
da-E
[AP]
(5)
a
This ma hema ical model can be used o compa e
di e en con igu a ions. I equi es he e alua ion o (4)
and
(3,
wha can be done by exp essing dis ances o he
pe u ba ion cen e as a unc ion o he symme ical cen-
e o he con igu a ion.
3.3.
E alua ion o di e en cen oid s uc u es
Fig.
3
shows se e al cen oid con igu a ions and
he e alua ion o hei s a is ical p ope ies using equa-
ions
(4)
and
(5).
In each case, dis ances om ansis o s
o pe u ba ion cen e,
d,
and
d,
we e exp essed as a
unc ion o he dis ance
R
om he con igu a ion cen e
o he pe u ba ion cen e , as shown in Fig.
4.
Assuming
ze o-a ea de ices, dis ances can be exp essed as:
-,
I is impossible o sol e (4)-(6) analy ically. Bu , expand-
ing
di
and
dj
in Taylo se ies a ound
R
and unca ing
i app op ia ely, hese in eg als can be sol ed analy ically
wi h high accu acy (e o is
<0.1%).
T unca ion o Tay-
lo
se ies is a easonable app oxima ion, since
<(
R
.
1645
on
2
K
AP=-
4R
_.
I
do
I
'I
0
ilP
=
0
32
R'
'xp
I
'V
U.
0
FP
=
0
2
o=
32
R
0%
TP
=
0
2
K2/
(J
=-
2
32
R
2
AP=-
4R
-
K
Figu e
3:
Mean and de ia ion o di e en s uc u es.
ype u ba ion cen e
Figu e
4:
Illus a ing dis ances
o
he pe u ba ion cen e .
The esul s in Fig.
3
a e e y in e es ing o design-
e s. Fo ins ance, he de ailed analy ical s udy
o
he con-
en ional cen oid s uc u e shows ha al hough i
p esen s qui e low s anda d de ia ion, he mean is no
ze o; on he con a y, i is in e sely p opo ional o he
dis ance o pe u ba ion cen e. This is disad an ageous
compa ed o ins ance o ci cula s uc u es. The p ice o
pay in hese is
a
la ge layou a ea.
4.
A ea in luence
on
dis ance e m
In he model shown in sec ion
3,
he dis ances om
each ansis o o he pe u ba ion cen e ha e been mea-
su ed om he geome ic cen e
o
each pa i ion.
In
his
sec ion he in luence
o
conside ing non-ze o a ea pa i-
ions
is
s udied.
I
mus be no iced
ha
we a e ocusing
only
on
he dis ance e m o Pelg om's model. This will
make clea ha he e exis a co ela ion be ween he mis-
ma ch e m due o he dis ance be ween ansis o pa i-
ions and hei sizes.
In o de o unde s and his in luence, conside an
isola ed de ice, as illus a ed in Fig.
5.
I
he ansis o is
conside ed
a
poin loca ed in i s geome ical cen e,
much in o ma ion is being
los .
A
mo e accu a e calcula-
ion is
10
a e age ou he dis ance o he pe u ba ion cen-
e :
D
--
-!--.~JA/.X
22
+y
dA
1
--
WL (7)
A
Then, he ze o-a ea model equa ions can be
used
subs i-
u ing
d
by an a e age dis ance. The new mean is,
and a co esponding exp ession can be ob ained o he
s anda d de ia ion. The in e p e a ion o hese equa ions
has a clea geome ical meaning
as
shown in
Fig.
6.
Fi s ,
o
each angula posi ion
o
he con igu a ion o he pe -
u ba ion cen e
a,
an a e age dis ance o each pa i ion
mus be calcula ed. Then, his p e ious esul mus be
a e aged o each possible ini ial posi ion o a pa i ion
wi h espec o he Pe u ba ion cen e
p
.
L
-
I.
Figu e
5:
Illus a ing a ea in luence.
This echnique is
oo
complex
as
an a e age dis ance
o
each pa i ion mus be calcula ed o each angula posi-
ion
a.
In he ze o-a ea model, ajec o ies desc ibed by
1646
Figu e
6:
Geome ic
in e p e a ion
o
(8)
pa i ion cen e s a e pe ec ly ci cula (pa i ion cen e is
cons an and equal o he geome ical cen e). Howe e ,
in he nonze o-a ea model, hese ajec o ies a e no ci -
cula , because he a e age dis ance o each pa i ion
depends on he alue o
a.
The esolu ion complexi y
o
equa ions in ze o-a ea
model was high bu , assuming he pa i ion cen e s
dependen on
a,
he equa ion esolu ion
is
e en mo e
di icul .
To
a oid his p oblem, an equi alen p-inde-
penden poin can be conside ed. Wi h his app oach, he
a ea in eg als a e elimina ed and he model complexi y
is
signi ican ly educed. This poin is ob ained a e aging all
he a ea o each pa i ion o each
p
alue. This gi es:
2
0A
which can be used in he equa ions o he ze o-a ea mod-
els (4)-(5).
5.
Pe u ba ion magni ude
I
is
also impossible o sol e equa ion
(9)
in gene al,
bu he displacemen o he pa i ion equi alen cen e
wi h espec o i s geome ical cen e can be a alua ed o
some simple case, i.e.
p
=
-
,
shown in Fig.
5.
Assum-
2
ing
D>>
W,
expanding equa ion
(7)
in
a
Taylo se ies, and
in eg a ing o e he gi en a ea, an app oxima ed analy i-
cal esul can be ob ained.
A
mo emen o he pa i ion
cen e wi h espec
o
he geome ical cen e
is
obse ed,
which o he case
in
Fig.
5
is&.
In he gene al case, we should a e age his mo e-
men o each alue o
p.
The p oblem can be sol ed
nume ically, using enough
p
samples o ob ain an accu-
a e alue o he displacemen . The calcula ed alue is
a ound
0.0427
.
Taking in o accoun p e ious esul s, nonze o-a ea
de ices cause a loss o symme y in he con igu a ion.
Assuming ha we a e wo king wi h
a
wa e diame e
o
10
cen ime e s, ha he pe u ba ion cen e is on he
7
WL
wa e , and ha we ha e ypical alues
o
de ice sizes
(50ym),
he assump ion
D
>>
,
W,
L
can be conside ed
conse a i e, and hence he app oxima ions in he mod-
els a e alid.
E alua ing he pa i i ion cen e mo emen using
he p e ious nume ical esul , and he alues o he di -
e en a iables in ol ed, he ob ained a e age displace-
men
is
a ound 0.02y.m. Conside he dis ance alues
used
in
sec ion
3,
which a e
he
dis ances om he pa i-
ion geome ical cen e s
o
he con igu a ion cen e ,
which a e hal a ansis o size a leas (a ound
25pm).
The e o e, he dis ance e o using 25pm ins ead
o
al-
ues a ound 25.02
pm
is smalle han
0.01%.
So,
his
mo emen can be neglec ed.
Bu , i is he in luence on he pa ame e misma ch
he eally signi ican in o ma ion. The e alua ion o he
mean and s anda d de ia ion o each s uc u e using
nonze o and ze o-a ea models gi es
a
ela i e e o
be ween he esul s p o ided by each model a ound 0.5%
in qui e ex eme cases (assuming squa e de ices wi h
a
200ym
side and
lR=Q.
1).
6.
Conclusion.
A misma ch e alua ion echnique o pa i ioned
s uc u es has been de eloped. Fi s ly, ansis o pa am-
e e s ha e been s a is ically cha ac e ized assuming ze o-
a ea de ices. Then, he a ea in luence has been exam-
ined. These a eas cause a ansis o cen e mo emen , bu
i
is
e i ied ha hei in luence on ze o-a ea esul s can
be neglec ed, assuming ypical dis ance alues.
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B.
Shyu, G. C. Temes, and K.
Yao,
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948-955, Dec. 1984.
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o
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