Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l.
5, N
o
. 5
,
O
c
tob
e
r
201
5, p
p
. 1
003
~101
1
I
S
SN
: 208
8-8
7
0
8
1
003
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJECE
Effect of
Devi
ce Variabl
e
s on
Surface Potential and Threshold
Voltagei
n DG-GNRFET
Baharak Meh
r
del
*
,
Az
lan Abdul Az
iz
*
, M
a
hdi
ar H
o
ssei
n
Gh
adiri**
* School of
Ph
y
s
ics, Univ
ersiti S
a
ins Malay
s
ia
** El
ectr
i
c
a
l
and
El
ectron
i
c
Engi
neering
,
Engi
ne
e
r
ing Cam
pus, U
n
iversiti
Sains
Mala
y
s
ia
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Apr 8, 2015
Rev
i
sed
Ju
l 7
,
2
015
Accepte
d
J
u
l 23, 2015
In this paper we present four simple
analy
t
ical threshold voltag
e
model for
short- channel and length of saturation
velocity region (LVSR) effect th
at
takes in
to accou
n
t the built – in
potential of th
e source and
drain chann
e
l
junction
,
th
e surface po
tent
ial
an
d the surfac
e
el
e
c
tri
c
fie
l
d effe
ct
on double –
gate gr
aphen
e
n
a
noribbon tr
ansi
stors. Four esta
blished m
odels
for surface
potenti
al
, la
tera
l
elec
tri
c
fie
l
d,
LVS
R
and threshold voltag
e
are presented
.
Thes
e m
odels
are bas
e
d on the eas
y ana
l
ytic
al s
o
lution
of the two
dimensional potential distrib
u
tion in
the gr
ap
hene and Poisson equation
which can b
e
us
ed to obta
i
n s
u
rface po
tent
ial
,
l
a
t
e
ral
ele
c
tri
c
fi
eld
,
LVS
R
and
threshold voltag
e
. These models
give a
clos
ed form
s
o
lution of
the s
u
rface
potenti
al
and
ele
c
tri
cal
fi
eld
dis
t
ri
bution as a function
of structur
al
parameters
and
drain bias. M
o
st of
anal
yt
i
c
al out
com
e
s
ar
e s
hown to
correlate with o
u
tcomes acquir
e
d b
y
Matlab simulation and th
e end model
applicability
to
the published silic
on base dev
i
ces is demonstrated.
Keyword:
D
oub
le g
a
te t
r
an
sistor
Gra
p
hene
na
n
o
r
i
b
bo
n t
r
a
n
si
st
or
Leng
th
o
f
saturatio
n
Surface potenti
a
l
Th
r
e
sho
l
d vo
ltag
e
Copyright ©
201
5 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
B
a
hara
k M
e
hr
del
,
Scho
o
l
of
Ph
ysics,
Un
i
v
ersiti Sains Malaysia,
1
118
00
, Pu
lau
Pin
a
ng
, Malaysia.
Em
a
il: b
a
h
a
ri
meh
r
@g
m
a
il.c
o
m
1.
INTRODUCTION
Inc
r
easing de
mand for hi
gh spee
d effici
ency in
m
e
m
o
ry
an
d l
ogi
c
appl
i
cations
has created a
co
n
tinuo
us tend
en
cy
for sm
a
ller d
e
v
i
ce sizes. CMOS te
c
h
n
o
l
o
gy
h
a
s s
h
i
f
t
e
d t
o
t
h
e s
ubm
i
c
ron
st
ru
ct
ure
t
o
achieve large density as well as highe
r efficiency inte
gra
t
ed circuits, so the sh
ort cha
nnel effects can be
great
er
.
Am
ong t
h
e i
m
port
a
nt
fact
o
r
s
whi
c
h
det
e
rm
i
n
e sho
r
t
-
cha
nnel
e
ffec
t
s are t
h
e
deca
y
of
de
vi
ce t
h
r
e
sh
ol
d
voltage
by using re
duce
d cha
nnel length [1]. In order to
proceed
with this type of
scaling factor equality, the
ch
ann
e
l leng
t
h
regard
ing
silico
n
MOSFETs
as esti
m
a
ted
by ITRS
will p
r
o
b
a
b
l
y n
e
ed
t
o
b
e
scaled
t
o
satisfy
t
h
e
nee
d
s of n
e
xt
-
g
ene
r
at
i
on tech
no
log
i
es.
Non
e
th
eless, th
ere are sev
e
ral q
u
e
stion
related
to
th
e act
io
n
of
silico
n
d
e
v
i
ces un
d
e
r ex
cessi
v
e
scaling
.
Acco
rd
ing
l
y, so
me n
e
w m
a
terial
b
a
sed
d
ev
ice
has b
e
en
p
r
op
o
s
ed
; for
exam
ple, nanowire FETs
, carbon nanoribbon FE
T [2-4].
Recent experi
mental studi
es have disc
usse
d the
feasib
ility o
f
man
u
f
act
u
r
i
n
g graph
e
n
e
n
a
no
ri
b
bon
tran
sist
o
r
s
[5
,
6
]
. The m
a
j
o
rity o
f
scien
tists h
a
v
e
b
eco
m
e
i
n
t
e
rest
ed i
n
t
h
i
s
area a
nd
prese
n
t
e
d
va
ri
ous t
y
pe
s o
f
GNR tra
n
sistor features
an
d ap
p
licatio
ns
[7-14
]
.
Howe
ver, there is an abse
nc
e of
researc
h
i
n
m
odelling
those feat
ures cl
ose to the
drain junction, which is
k
now
n
br
eak
d
o
w
n
vo
ltag
e
.
Wo
o et
al
an
d G
u
o an
d
W
u
[
1
5
,
1
6
]
ha
s creat
ed s
h
o
r
t
-
cha
nnel
t
h
re
shol
d v
o
l
t
a
ge
desi
g
n
s
by
resol
v
i
n
g t
h
e
t
w
o
-
di
m
e
nsi
o
n
a
l
Poi
sso
n eq
uat
i
o
n
.
Im
am
et
al
[17]
desi
gne
d t
h
e t
h
res
h
ol
d v
o
l
t
a
ge by
fo
rm
ali
z
i
ng t
h
e t
w
o di
m
e
nsional
P
o
i
sso
n f
o
rm
ul
a as a pai
r
of t
h
e o
n
e di
m
e
nsi
onal
(
1
D
)
Poi
s
s
on f
o
rm
ul
a an
d
two dim
e
nsional
Laplace form
ula.
For
t
h
e thres
hol
d vo
ltage a
n
e
x
pone
ntial functi
on of L
was
estimated.
Lately, Ban
n
a
[18
]
, app
lied
th
e qu
asi two
di
m
e
n
s
io
n
a
l m
e
th
od
and
stated
th
at th
e th
resh
o
l
d
vo
ltag
e
m
o
d
e
l
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
JECE Vo
l. 5
,
N
o
. 5
,
O
c
tob
e
r
20
15
:
100
3
–
10
11
1
004
gave
a
q
u
i
t
e
sim
i
l
a
r de
pen
d
e
n
ce
on
L t
o
t
h
at
st
at
ed
by
I
m
a
m
et al. [17]. In t
h
is
pa
per, a
n
ea
sy analytical
m
e
t
hod
fo
r t
h
r
e
sh
ol
d
vol
t
a
ge
wi
t
h
t
h
e s
h
ort
– cha
n
nel
,
co
m
p
l
e
t
e
l
y
depl
et
ed D
G
-
GNR
F
ET can
be ex
p
l
ai
ned.
In
ad
d
ition
,
an an
alytical so
l
u
tio
n
is based
o
n
the 2D
po
ten
tial d
i
strib
u
t
io
n
s
in
the g
r
ap
h
e
n
e
film
. Su
itab
l
e
bo
u
nda
ry
co
nd
i
t
i
ons can b
e
set
for
uni
f
o
rm
do
pi
n
g
i
n
t
h
e
gra
p
hene
fi
lm
. Acco
rdi
ngl
y
,
t
h
i
s
st
udy
prese
n
t
s
an
unc
om
plicated design for the surface
pote
ntial and electrical field distri
bution of double
- gate
graphe
ne
nan
o
r
i
b
bo
n fi
e
l
d effect
t
r
a
n
si
st
or.
Du
ri
n
g
t
h
e fol
l
o
wi
n
g
sectio
n
,
th
e su
rface p
o
t
en
tial as
well as th
e th
resho
l
d
v
o
ltag
e
fo
r
short ch
an
n
e
l
o
f
GNR will b
e
d
e
fin
e
d
first. To
be ab
le to
sim
p
lify th
e an
alysis, we
do
no
t assu
m
e
interface c
h
arges. In a
d
dition, carrier
acc
umulation or
inve
rsion
has
bee
n
ignore
d in the
gra
p
hene
fil
m
burie
d
oxi
de. M
o
re
o
v
e
r,
we
want
t
o
expl
ai
n
t
h
e
be
h
a
vi
o
u
r
o
f
D
G
-
GNR
F
ET cl
o
s
e t
o
t
h
e
d
r
ai
n
j
unct
i
o
n as
wel
l
as t
h
e
brea
k
d
o
w
n
vol
t
a
ge i
n
c
o
m
p
ar
i
s
on
wi
t
h
t
h
e si
l
i
c
on
base t
r
an
si
st
or.
2.
R
E
SEARC
H M
ETHOD
The Proposed
Model
for
LVSR
(Sur
face P
o
te
ntial
)
and
the
Sh
ort –Ch
a
nnel Solu
tion
The velocity-s
aturation-regi
on le
ngth
of F
ETs as well a
s
the widt
h of the drai
n re
gi
on i
n
which
carrier
velocit
y
saturation a
nd im
p
act ionization takes
place can be m
o
re
significant
varia
b
les for s
h
ort –
chan
nel
de
vi
ce
s i
n
na
noscal
e
t
r
ansi
st
o
r
s. T
h
e LV
SR
co
nt
rol
s
t
h
e
hot
-el
ect
ro
n ge
nerat
i
on
, su
bst
r
at
e c
u
r
r
ent
,
t
h
e
drai
n
brea
k
d
o
w
n
vol
t
a
ge
and
drai
n
c
u
r
r
e
nt
i
n
t
h
e
d
r
ai
n
regi
on
[
1
9-
2
4
]
.
At
t
h
e
FET
,
w
h
e
n
t
h
e
use
d
drai
n
v
o
ltag
e
is greater th
an th
e sat
u
ration
d
r
ai
n
vo
ltag
e
, t
h
e electric field
clo
s
e to
th
e
d
r
ai
n
reg
i
o
n
isg
r
eater th
an
the critical field powe
r whic
h leads to
car
r
i
er vel
o
ci
t
y
sat
u
rat
i
o
n [
25]
.
A st
an
dar
d
sc
hem
a
ti
c cross-
sect
i
on
fr
om
dou
bl
e ga
t
e
GNR
FE
T i
s
dem
onst
r
at
ed i
n
fi
gu
re 1
.
Wh
ere t
h
e
oxi
de t
h
i
c
k
n
ess
of t
h
e
fr
ont
a
nd
bac
k
gat
e
s
i
s
m
e
nt
i
oned
b
y
t
with
a d
i
electric co
n
s
tan
t
o
f
.
The L, W,
and
are t
h
e l
e
ngt
h,
wi
dt
h
,
t
h
i
c
k
n
ess
and
di
el
ect
ri
c con
s
t
a
nt
of t
h
e GNR
res
p
ec
t
i
v
el
y
.
In ge
ne
ral
,
fo
r anal
y
z
i
ng t
h
e
pot
e
n
t
i
a
l
di
st
ri
but
i
o
n
i
n
t
h
e
g
r
aph
e
n
e
, it is
n
ecessary to
solv
e th
e
Po
isson equ
a
tio
n first: [26
]
Fi
gu
re
1.
Sc
he
m
a
t
i
c
cross sec
t
i
on
of
a
Do
u
b
l
e
Gat
e
GNR
F
E
T
,
,
0
,
0
(1
)
Whe
r
e
Ψ
x,
y
m
u
st
be t
h
e
pot
e
n
t
i
a
l
any
w
here
o
f
(
x
, y
)
t
h
ro
u
g
h
o
u
t
t
h
e G
N
R
,
t
h
e
el
ect
ri
c char
ge
am
ount
i
s
q,
th
e dop
ing
co
ncen
tration
o
f
GNR is
N
.
In
fact,
th
e b
u
ilt-
i
n
p
o
t
en
tial
in
GNR with
a ban
d
g
a
p
is n
o
t
zero.
Howev
e
r, in this wo
rk
,
we
h
a
v
e
in
cl
u
d
ed the bu
ilt-in
p
o
t
en
tial o
f
th
e
sou
r
ce and
d
r
ain ch
an
n
e
l
jun
c
tio
n,
for
Eq. (1),
whe
r
e
b
y it is
n
ecessary to
determ
i
n
e t
h
e
bounda
ry conditions a
s
,
Ψ
0,0
V
and
Ψ
0
,,L
=
V
V
, the
bu
ilt-in
po
ten
tial an
d the so
ur
ce-drain
v
o
ltag
e
are represen
ted b
y
V
and
V
re
spectivel
y. Sinc
e t
h
e
electr
i
c f
l
ux
acr
o
ss th
e top
,
d
o
wn
G
N
R
and
o
x
i
d
e
bo
undar
y
can b
e
con
s
tan
t
, th
e
po
ten
tial f
u
n
c
tion h
a
s t
o
satisfy
[2
7]
.
,
|
=
,
(2
)
and
,
|
=
,
(3
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Ef
f
ect
of
Devi
c
e
Vari
abl
es
o
n
Surf
a
ce P
o
t
e
nt
i
a
l
a
n
d
T
h
res
h
ol
d
Vol
t
a
gei
n
DG
-GN
R
FET
(
B
a
har
ak Mehr
d
el
)
1
005
W
h
er
e
V
=
,V
=
,
is th
e
g
a
te-so
u
rce fo
r
fron
t and
is th
e
g
a
te-so
u
rce fo
r
back flat ba
nd volta
ge. T
h
e
flat band voltage is
fo
r t
h
e
G
N
R
by
usi
n
g a
ba
nd
ga
p
[2
8]
.
ln
⁄
,
wh
erev
erth
e
Ferm
i v
e
lo
city of t
h
e
g
r
aph
e
n
e
is
~
10
m/s. By fo
llowing th
e sam
e
m
e
t
hod a
n
al
y
s
i
s
as i
n
R
e
fs
, [
2
6,
2
9
]
t
o
s
o
l
v
e
Eq.
(
1
)
,
we ca
n
dec
o
m
pose
Ψx,
y
into
two
p
a
rts, such
th
at
Ψ
x,
y
V
x
+ U(
x, y)
(4
)
V
x
can
be t
h
e
o
n
e
di
m
e
nsi
on s
o
l
u
t
i
on
fr
om
t
h
e Poi
sso
n e
q
uat
i
o
n:
=
(5
)
Eq. (
5
) i
s
usual
l
y
used fo
r t
h
e
l
o
n
g
– cha
n
nel
effect
s. The e
a
si
est
sol
u
t
i
on
t
h
at
com
put
es fo
r t
w
o
di
m
e
nsi
onal
sho
r
t
c
h
a
nnel
effect
s
i
s
U(
x,
y
)
f
o
r
det
e
rm
ini
n
g
Vx
as
pr
ovi
ded
by
E
q
.
(
4
)
,
U(
x,
y
)
w
h
i
c
h m
u
st
f
u
l
f
i
l
t
h
e
Laplace e
quati
on:
Ux,
y
Ux,
y
0
(6
)
In
E
q
.
(
4
),
t
h
e
bo
u
nda
ry
c
o
n
d
i
t
i
ons
of
Ψ
x,
y
can
a
l
so be di
vi
ded
i
n
t
o
t
w
o
part
s sui
t
a
bl
e fo
r
t
h
e
sol
u
t
i
o
n
of
Eq.
(5
) an
d (
6
). H
o
weve
r,
b
y
breaki
ng
u
p
Eq.
(2
) an
d (
3
) t
h
e b
o
u
n
d
ary
con
d
i
t
i
ons a
r
e
usual
l
y
di
spl
a
y
e
d as
fo
llows:
|
=
(7
)
and
|
=
(8
)
whe
r
e be
fo
re
V
and
V
, was
e
x
pressed, it can al
so
be
stated
V0
=
V
,
wh
ich
is a
fro
n
t
surface
p
o
t
en
tial
o
b
t
ain
e
d
b
y
reso
lv
ing
Eq
. (5
), u
s
ing
bou
nd
ary co
n
d
ition
s
in Eq
. (7
), and
Eq
. (8
), and
wh
ere th
e d
e
v
i
ce facto
r
s
can be di
spl
a
y
e
d
al
o
n
g
wi
t
h
bi
as
c
o
n
d
i
t
i
ons
l
i
k
e:
[2
7]
2
V
V
1
q
N
t
1
1
(9
)
We ca
n a
ppl
y
t
h
e ze
ro
gat
e
bi
as co
n
d
i
t
i
on,
V
V
0
, s
o
fr
om
, Eq.
(9)
we ca
n
w
r
ite
qN
t
2
(1
0)
On
t
h
e
o
t
h
e
r han
d
, if t
h
e
d
e
vice is at g
a
te bias co
nd
ition
s
b
y
u
s
i
n
g
V
=
V
V
fr
o
m
Eq. (9
), it c
a
n
be
stated that:
2
V
q
N
t
(1
1)
In
Eq
.
(6
) t
h
e
b
o
u
n
d
ary
c
o
ndi
t
i
onsca
n
be st
at
ed as;
U
0,
0
V
V
and
U
0,
L
V
V
V
,
|
=
,
(1
2)
A
nd
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE Vo
l. 5
,
N
o
. 5
,
O
c
tob
e
r
20
15
:
100
3
–
10
11
1
006
,
|
=
,
(1
3)
with
th
e prev
io
u
s
bo
und
ary
co
nd
itio
ns, th
e resu
lts fo
r Eq
. (6
) can
b
e
foun
d
fro
m
th
e tech
n
i
qu
e for th
e
di
ffe
re
nt
i
a
t
i
on
of
pa
ram
e
t
e
rs. The s
o
l
u
t
i
o
n
o
b
t
a
i
n
ed
at
x =
0 ca
n
be e
x
p
r
e
ssed
usi
n
g
t
h
e
seri
es [
2
9]
U
0,
y
A
ex
p
λ
y
B
ex
p
λ
y
(1
4)
w
h
er
e
A
V
V
V
ex
p
λ
L
V
V
e
x
p
2
λ
L
1e
x
p
2
λ
L
(1
5)
B
A
(1
6)
In
add
itio
n,
λ
is a p
a
ram
e
ter that d
e
p
e
nd
s
o
n
tech
no
log
y
,
wh
ich
it can
be
d
e
scri
b
e
d as
bein
g
t
h
e so
lu
tio
n to
th
e equ
a
tion
:
t
λ
2
t
λ
t
λ
(1
7)
The value
of
t
is sm
al
l (
a
r
ound
1
0
-9
), s
o
we c
a
n a
p
p
r
oxi
m
a
te t
a
n
(
t
λ
to
t
λ
. Accordi
n
g
to[
2
7]
,
λ
is
λ
1
t
1
2
(1
8)
We can approa
ch
Ψ
(0, y) the
surface
pote
n
tial dist
ribution of the just
initial ter
m
n=1 from
the series i
n
Eq.
(14
)
. Th
e m
a
i
n
reaso
n
is that th
e
resu
lts
wh
ich
ca
n
b
e
sign
ifican
t
for
d
e
sign
ing
t
h
e th
resho
l
d vo
ltag
e
p
r
od
u
c
ed
fro
m
th
e su
rface po
ten
tial are lo
west wh
en
th
ey h
a
p
p
e
n
to
beh
a
pp
en
s clo
s
e to
th
e cen
tre in
th
e
channel (
y
≅L
2
⁄
[26]
.
U
0,
y
+(
(1
9)
Accord
ing
l
y, fo
r th
e sho
r
t chan
n
e
l, th
e
surface po
ten
tial can
b
e
d
e
scri
b
e
d
as
Ψ
0,
y
sinh
sinh
sinh
sinh
(2
0)
wi
t
h
a m
i
nim
u
m
gi
ven by
∝
∝
ex
p
L
2
(2
1)
We can d
e
termin
e
∝
,
∝
, a
n
d t
h
e
val
u
e
of
V
as th
e short – chan
n
e
l
thresho
l
d
vo
ltag
e
.
W
i
t
h
t
h
e l
o
west
surface potentials
Ψ
equals
2∅
a
s
ar
gu
ed
in [2
6
]
.
T
h
e
r
efo
r
e
,
E
q
.
(21) can be
indicated as:
2
∅
∝
∝
ex
p
L
2
(2
2)
The value
of
extracted from
Eq.
(
2
2)
, by
m
eans
o
f
des
c
ri
bes t
h
e t
h
re
shol
d
v
o
l
t
a
ge
i
n
t
h
e
sh
ort
-
chan
nel
gra
p
he
ne devi
ce.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Ef
f
ect
of
Devi
c
e
Vari
abl
es
o
n
Surf
a
ce P
o
t
e
nt
i
a
l
a
n
d
T
h
res
h
ol
d
Vol
t
a
gei
n
DG
-GN
R
FET
(
B
a
har
ak Mehr
d
el
)
1
007
∅
2∅
L
2
2
V
q
N
t
(2
3)
The pa
rticular
lateral electric
a
l field across
the semi
conductor surface ca
n be acq
uire
d from
di
fferenti
a
ting
Eq
. (2
0)
.
0,
,
=
(2
4)
We take
the
1st term
(n = 1) from
the seri
es
of E
q
.
(14), as a s
u
rface
pote
n
tial distri
bution
U
0,
y
[2
3]
.
Acco
r
d
i
n
g
t
o
[
27]
fr
om
Eq. (
1
4
)
ca
n be o
b
t
a
i
n
ed Eq
.
(
2
5)
.
U
0,
y
(2
5)
In Eq.
(4), t
h
e
surface
potential along y ca
n
be
displayed as
Ψ
0,
y
(2
6)
More
ove
r, t
h
e lateral electric
a
l field across
the cha
nnel
can
b
e
easily ob
tain
ed
b
y
th
e deriv
a
tion
of Eq. (26),
ove
r y
.
E (0
,
y
) =
(2
7)
Add
itio
n
a
lly,
by tak
i
n
g
y = L
,
Ψ
0,
y
. T
h
ere
f
ore, for
we ca
n state
that
L
ln
(2
8)
wh
ich
can
b
e
so
lv
ed
nu
m
e
ri
cally.
In
Eq.
(2
8),
is d
r
ai
n
satu
ration
vo
ltag
e
an
d
i
s
t
h
e
l
e
ngt
h
of
t
h
e
saturation velocity
region. The
rela
tions
hi
p
betwee
n the
s
u
rface
pote
n
tial,
electrical field and t
h
e le
ngth
of
satu
ration
reg
i
o
n
with
,
,
and
L i
s
s
h
o
w
n i
n
t
h
e
pr
o
pose
d
e
q
uat
i
o
n
s
.
3.
RESULTS
A
N
D
DI
SC
US
S
I
ON
In t
h
e f
o
l
l
o
wi
ng sect
i
o
n, by
usi
ng t
h
e a
b
o
v
e pr
oce
d
u
r
e,
t
h
e t
h
res
hol
d vol
t
a
ge ca
n be
com
put
ed i
n
devi
ces
of
20
nm
l
e
ngt
h
a
n
d
se
veral
th
ickn
esses.
In
figure2, t
h
e
d
e
si
g
n
estim
a
t
ed
thresh
o
l
d vo
ltag
e
h
a
s
b
een d
i
sp
layed as th
e
fun
c
tion
o
f
t
h
e ch
annel len
g
t
h an
d
can
also b
e
com
p
ared
with
silico
n
b
a
se
d
e
v
i
ces, i
n
ref [30]. Figure 2, s
h
ows the
surface
potential across the
chan
nel length in a thres
hol
d situation ext
r
acted
fr
om
Eq. (
1
7)
,
as wel
l
as
Eq
.
(2
0
)
pl
ot
t
e
d.
The
pr
o
x
i
m
i
t
y
bet
w
ee
n t
h
e
s
o
urce
an
d
drai
n
i
n
t
h
e
sh
o
r
t
c
h
an
nel
devices
caus
e
d the s
u
rface
potential to
vary from
V
. I
n
ref
[2
6]
t
h
e e
s
t
i
m
a
t
i
on
of
t
h
e t
a
nge
nt
fu
nct
i
o
n
i
n
t
λ
by
t
λ
is b
a
sed
on
v
e
ry t
h
in
sili
co
n film
s. In
ad
d
ition
,
t
h
is esti
m
a
tio
n
was ap
p
lied b
e
cau
s
e th
e
GNR th
ickn
ess is v
e
ry th
in
i
f
ou
r
figu
reis
co
m
p
ared
with th
e th
resho
l
d
v
o
ltag
e
i
n
ref
[31
]
. The sim
u
latio
n
m
odel used matches all the pres
um
ptions in the anal
ysis. Qua
n
tum
effects do not take
into account
effects
whe
n
t
h
ey
be
g
i
n t
o
pl
ay
a very
im
port
a
nt
rol
e
i
n
G
N
R
t
h
i
n
ner t
h
a
n
5
n
m
. In fi
g
u
re
3,
t
h
e beha
vi
o
r
f
o
r t
h
e
t
h
res
hol
d vol
t
a
ge vers
us
c
h
a
n
nel
l
e
n
g
t
h
L w
i
t
h
di
f
f
ere
n
t
th
ick
n
e
ss is sh
own.
During
th
i
s
p
a
rt, th
e pro
f
ile
of this s
u
rface
electrical field as
wel
l
as t
h
e pot
ent
i
a
l
cha
nge
di
spl
a
y
e
d
and t
h
e
res
u
l
t
s
of va
ri
o
u
s va
r
i
abl
e
s
(f
or i
n
st
ance,
d
r
ai
n-
so
urce
v
o
l
t
a
ge, o
x
i
d
e t
h
i
c
kne
ss, c
h
an
ne
l
l
e
ngt
h a
nd
d
o
p
i
n
g co
nce
n
t
r
a
t
i
on o
n
t
h
e l
e
n
g
t
h
of
sat
u
rat
i
o
n re
gi
on
) ca
n be a
n
al
y
zed near t
o
t
h
e d
r
ai
n
reg
i
on.
The
p
r
o
p
o
se
d m
odel
can be
co
nfi
r
m
e
d by
com
p
aring the
com
puted val
u
es
with
th
e su
gg
ested
sam
p
le,
as well
as si
m
u
latio
n
s
fo
r a
Si b
a
sed
d
e
vice,
in
ref [
30]
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE Vo
l. 5
,
N
o
. 5
,
O
c
tob
e
r
20
15
:
100
3
–
10
11
1
008
Fi
gu
re
2.
Th
res
hol
d
vol
t
a
g
e
al
on
g t
h
e c
h
an
ne
l
l
e
ngt
h
an
d rep
r
esen
ts
d
a
ta fro
m
th
e
Matlab
si
m
u
latio
n
with
L=2
0
nm
from
Eq
. (
1
7)
an
d E
q
.
(
2
0
)
Fi
gu
re
3.
Th
res
hol
d
vol
t
a
g
e
v
e
rsus
c
h
an
nel
l
e
ngt
h L
fo
r diffe
re
nt
t
OX
th
ickn
ess
o
f
th
e
DG -
GNRFET
m
odel
and
re
pr
esent
s
dat
a
f
r
o
m
t
h
e M
a
t
l
a
b
si
m
u
latio
n
w
i
t
h
L=20
n
m
f
r
o
m
Eq
. (
17)
an
d Eq. (20
)
A fi
n
e
settlemen
t
can b
e
ob
t
a
in
ed
am
o
n
g
si
m
u
latio
n
ou
tco
m
es p
l
u
s
th
e
sam
p
le with
variou
s
d
o
p
i
ng
conce
n
t
r
at
i
o
ns
and
oxi
de t
h
i
c
kne
sses
al
o
n
g
wi
t
h
t
h
e i
n
t
e
r
v
al
o
f
t
h
e
drai
n.
The
su
rface
po
t
e
nt
i
a
l
fo
r
Ψ
(0
, y
)
=
. Fi
gure
4, s
h
ows t
h
e
varia
n
c
e
of s
u
rface
potential acros
s t
h
e c
h
a
nnel
wi
t
h
a
va
ri
et
y
o
f
d
r
ai
n
bi
ases.
It
c
a
n
be observe
d
, t
h
at as the drai
n bias is incre
a
sed the
su
rface p
o
t
en
tial aro
und
th
e dr
ain side increases
if it
co
n
tinu
e
s to
b
e
con
s
tan
t
in the sou
r
ce area,
wh
ich
sh
ows t
h
e reliab
ility o
f
our su
ppo
sed bo
und
ary co
nd
itio
n
s
.
Figure
5, indic
a
tes the cha
n
ge of s
u
rface
potential acr
oss t
h
e na
nori
bbon
channel
for va
rious
oxide thic
kne
ss
wi
t
h
drai
n
bi
as
,
V
=1
.5
V and
ch
ann
e
l leng
th
L=15
and
N
=
11
0
cm
. It
can
be dem
onstrated that
whe
n
the oxi
d
e thic
kness
dec
r
eases
the l
o
we
st
pot
e
ntial close t
o
the s
o
urce si
de
increa
ses; howev
er, th
e
op
po
site
phe
n
o
m
e
non
o
ccurs
cl
ose t
o
t
h
e d
r
ai
n
d
u
e t
o
t
h
e
fact
t
h
at
t
h
e
oxi
de t
h
i
c
kne
ss
decrease
s
o
x
i
d
e ca
paci
t
a
nce
increases. This boosts t
h
e surface c
h
ar
ge a
n
d the
surface
potential for t
h
e
corrected bias conditions.
Hence,
on t
h
e d
r
ai
n si
de by
red
u
ci
n
g
of t
h
e o
x
i
d
e t
h
i
c
k
n
ess,
the
oxide capacitance rises t
ogether with the off-state
cu
rren
t
wh
ich
in
creases. As a
resu
lt of th
is, th
e po
ten
tial
i
n
t
h
e drai
n si
de i
s
red
u
ce
d. Fr
o
m
fi
gure
6
, i
t
can b
e
shown that as t
h
e c
h
annel lengths a
r
e
reduc
e
d from
L=
20
nm
to L=15
nm
, the surface
pote
n
tial is similar to
fig
u
re
4 f
o
r
V
1
.
5
V at each side
of the
de
vice. Howe
ver,
t
h
e
lowest potent
ial
m
oves in an upward
di
rect
i
o
n as c
h
annel
l
e
ngt
hs a
r
e di
m
i
ni
shed.
Thi
s
t
a
kes
pl
ac
e beca
use o
f
t
h
e ext
e
nsi
on
o
f
t
h
e de
pl
et
i
on
r
e
gi
o
n
below the gat
e
at the surfac
e
. Figure 7, s
h
ows
a strong correlation am
ong the sim
u
lation outcom
es and
sam
p
l
e
s used i
n
va
ri
o
u
s
do
pi
ng c
o
ncent
r
at
i
ons
, wi
t
h
t
= 5
nm
and di
st
an
ces fr
om
t
h
e drai
n i
n
do
u
b
l
e
gat
e
GNRFET.
When the surface
pote
n
tial
m
odel was confi
r
m
e
d
the particula
r
LVSR m
odel was also proven to
b
e
efficien
t
since it is an
i
d
eal m
e
th
o
d
o
f
the surface
p
o
t
en
tial
Ψ
0,
y
V
,
[2
7]
.
F
u
rtherm
ore,
the
sur
f
ace
p
o
t
en
tial d
i
ffers in
th
e lo
cation
of th
e ch
an
nel fo
r
v
a
riou
s
p
eak
do
p
i
n
g
co
n
c
en
tration
s
. It can
b
e
stated
th
at
whe
n
e
fficient
carrie
r
c
once
n
tration ac
ross
the c
h
an
nel i
n
crease
s
, t
h
e
surface
potent
ial rises. T
h
e
exact
out
c
o
m
e
s are a
s
di
spl
a
y
e
d
f
o
r
V
=1
.5
V
.
Fi
g
u
r
e
8,
show
s t
h
e field
d
i
st
r
i
bu
tion
ov
er th
e
n
a
no
r
i
b
bon
sur
f
a
ce
to
th
e v
a
riou
s field ox
id
e layer th
ickn
esses.
As exp
l
ai
ne
d
bef
o
re,fi
g
u
r
e
5
,s
ho
ws
t
h
e
p
o
t
e
nt
i
a
l
di
st
ri
b
u
t
i
o
n
acros
s t
h
e cha
nnel
l
e
n
g
t
h
f
o
r
t
h
e vari
o
u
s fi
el
d oxi
de t
h
i
c
k
n
e
sses.
As a resu
lt, th
ere is si
m
i
lar effect o
f
th
e
fro
n
t
in
terface o
x
i
d
e
layer th
ickn
esses
o
n
th
e field
as well as p
o
t
en
t
i
al d
i
strib
u
tion to
th
e silico
n
b
a
sed
devi
ces
. As s
h
ow
n i
n
fi
g
u
re
8, t
h
e
op
p
o
si
t
e
phe
n
o
m
e
non
was o
b
ser
v
e
d
, beca
use t
h
e
hi
g
h
est
fi
el
d u
s
ual
l
y
o
ccurs in th
e
p
n
junction interfa
ce in which
the avala
n
che
breakdown take
s
place. T
h
e
fie
l
d
oxi
de layer
t
h
i
c
kne
ss ef
fec
t
s t
h
e brea
kd
o
w
n
vol
t
a
g
e
. I
f
i
t
i
s
t
h
i
c
k en
ou
gh t
o
ca
usea
br
eakd
o
w
n
vol
t
a
ge w
h
en t
h
e v
o
l
t
a
ge
becom
e
s wea
k
er [32].
Figure 9, exhibit the analytical
outcom
e
s
of the re
sults of the el
ectric
field distri
bution ext
r
acte
d
f
r
o
m
Eq
. (
2
4
)
f
o
r
n
a
n
o
r
i
bbon
tr
ansisto
r
s
with
v
a
r
i
o
u
s
dop
ing
con
c
en
tr
atio
n
s
N
. A reas
ona
ble eval
uation
am
ong
st
t
h
e a
n
al
y
t
i
cal
and
n
u
m
e
ri
cal
fi
nal
out
c
o
m
e
s
m
i
g
h
t
u
s
ual
l
y
be
d
i
scove
red
.
T
h
e
di
f
f
ere
n
ces
be
t
w
e
e
n
t
h
e t
w
o
co
ul
d
be as
a
res
u
l
t
of
t
h
e
i
n
fl
ue
nc
e f
r
om
t
h
e s
p
a
ce cha
r
ge
regi
on
i
n
t
o
t
h
e
p
B
a
se re
gi
o
n
an
d
n
Drai
n di
ff
usi
o
n re
gi
o
n
,
w
h
i
c
h i
s
vi
si
bl
e f
r
o
m
t
h
e fi
el
d at
x=0
an
d
x=L.
Fi
gu
re
9, s
h
o
w
s t
h
at
t
h
e el
ect
ri
cal
field distri
buti
on ac
ross the
GNR surf
ace for the
va
rious
doping, c
o
m
p
ared
to t
h
e silicon
film surfac
e
with
vari
ous
dopi
ng. It can be clea
rly sh
own t
h
at there exists two electrical
field pea
k
s across the silicon surface,
one
ha
p
p
en
s i
n
t
h
e
p
n
jun
c
tio
n in
terface an
d n
e
x
t
on
t
h
e
n
n
junction i
n
terface.
Close to t
h
e
drain re
gi
on
dem
onst
r
at
e exact
l
y
t
h
e sam
e
beha
vi
o
r
i
n
t
h
eSi
base
d de
vice. As can
be seen, the m
a
gnitude of the electric
field dec
r
eases
with as the
N
increase
s
[32].
The s
ubst
r
ate doping c
o
n
c
entratio
n
isth
e main
factor in
t
h
e
opt
i
m
i
zati
on o
f
t
h
e G
N
R
t
r
a
n
si
st
or an
d
defi
nes t
h
e am
pl
i
t
ude a
n
d t
h
e
po
si
t
i
on o
f
t
h
e m
a
xi
m
u
m
peak el
ect
ri
c
field.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Ef
f
ect
of
Devi
c
e
Vari
abl
es
o
n
Surf
a
ce P
o
t
e
nt
i
a
l
a
n
d
T
h
res
h
ol
d
Vol
t
a
gei
n
DG
-GN
R
FET
(
B
a
har
ak Mehr
d
el
)
1
009
Fi
gu
re
3.
S
u
r
f
a
ce p
o
t
e
nt
i
a
l
ve
rsus
ch
an
nel
l
e
ngt
h
curves with va
rious drai
n
biases, L=
15nm
and oxide
th
ick
n
e
ss t
OX
=10 nm
Fi
gu
re
4.
S
u
r
f
a
ce p
o
t
e
nt
i
a
l curves
across
the
channel
wi
t
h
vari
ous
o
x
i
d
e t
h
i
c
k
n
esse
s, L=
15
nm
an
d
V
ds
=1
.5
V
Fi
gu
re
5.
S
u
r
f
a
ce p
o
t
e
nt
i
a
l curves
across
the
channel
wi
t
h
vari
ous
c
h
an
nel
l
e
n
g
t
h
s,
t
OX
=5 nm
, V
ds
=1
.5
V
Fi
gu
re
6.
S
u
r
f
a
ce p
o
t
e
nt
i
a
l curves
across
the
channel
wi
t
h
vari
ous
p
r
ofi
l
e
do
pi
n
g
c
o
ncent
r
at
i
ons
, L
= 1
5
nm
and Vd
s=1
.
5 V
Fi
gu
re
7.
El
ect
ri
cal
fi
el
d
di
st
r
i
but
i
o
ns ac
ro
ss
t
h
e
lateral d
i
rection
for
n
a
noribbo
n with v
a
rious fron
t
in
terface ox
id
e layer th
ickn
esses
Fi
gu
re
8.
El
ect
ri
cal
fi
el
d
di
st
r
i
but
i
o
n ac
ros
s
t
h
e
nanoribbon
s
u
rface with vari
ous dopi
ng
N
d
4.
CO
NCL
USI
O
N
From
this work, a
n
uncom
p
licated
analytical approach has bee
n
fo
u
n
d
t
o
obt
ai
n sa
m
p
l
e
s of t
h
e
thres
hol
d voltage for short channel double gate nano
ribbon FET whic
h induce
d
the s
u
rface potential effects
t
h
at
are
us
ual
l
y
descri
bed
an
d acc
ou
nt
ed
f
o
r i
n
o
u
r m
ode
l. Gen
e
rally, th
ere are an
alytical
m
o
d
e
ls to
fi
n
d
th
e
surface potenti
a
l and electrical fiel
d along with LVSR from
DG-GNR tr
ansistors for
the saturation
regi
on
whi
c
h
was i
n
v
e
st
i
g
at
ed
usi
n
g
t
h
e rec
o
m
m
ende
d m
odel
.
Ad
di
t
i
onal
l
y
,
by
usi
n
g
t
h
e i
n
t
r
od
uce
d
m
odel
s
, t
h
e
con
s
eq
ue
nces of de
vi
ce va
ri
abl
e
s;
fo
r i
n
st
ance na
no
ri
b
b
on t
h
i
c
kne
ss,
do
pi
n
g
co
nce
n
t
r
at
i
on an
d ch
anne
l
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE Vo
l. 5
,
N
o
. 5
,
O
c
tob
e
r
20
15
:
100
3
–
10
11
1
010
len
g
t
h was analyzed
and
com
p
ared
to
t
h
e silico
n
b
a
sed d
e
v
i
ces an
d
was
p
r
ov
en to b
eclose to the
m
. As
expecte
d
, as a result of sm
a
l
l-scale geom
etry for the pr
oduct, the large lateral electrical f
i
eld was discovere
d
to
b
e
clo
s
e t
o
th
e drain
ju
n
c
t
i
o
n
,
b
e
co
m
i
n
g
a po
in
t
o
f
trust an
d reliab
ility o
f
su
ch
d
e
vices. In
ad
d
ition
,
t
h
e
connection am
ong t
h
e c
r
itical dopi
ng
con
c
en
tration
aroun
d th
e drift
reg
i
on
al
o
n
g
with
t
h
e th
ick
n
e
ss was
descri
bed
.
M
o
st
of t
h
e a
n
al
y
t
i
cal
out
com
e
s were
bei
n
g
sho
w
n co
rrel
a
t
e
wi
t
h
t
h
e o
u
t
c
om
es acqui
re
d by
Matlab
si
m
u
lat
i
o
n
.
To
in
crease th
e b
r
eakdow
n
p
r
op
erties
for d
e
v
i
ces,
h
o
w
e
v
e
r g
e
tting
p
e
rfect surface field
di
st
ri
b
u
t
i
o
n
a
n
d t
h
e
cri
t
i
cal
d
opi
ng
co
nce
n
t
r
at
i
on
has si
gni
f
i
cant
val
u
e [
3
2
]
.
ACKNOWLE
DGE
M
ENTS
The a
u
t
h
ors
w
oul
d l
i
k
e t
o
ac
kn
o
w
l
e
d
g
e t
h
e
Sch
ool
of
P
h
y
s
i
c
and
Sc
ho
ol
of
El
ect
ri
cal
an
d El
ect
r
oni
c
Eng
i
n
eeri
n
g,
Un
i
v
ersitiSains Malaysia, fo
r th
eir supp
ort
an
d
con
t
ribu
tion
to
th
is st
ud
y. Sp
ecial th
an
ks to
Dr.
Yoon Tiem
Leong
for
his a
dvice and encouragem
ent
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