Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
8,
No.
2,
April
2018,
pp.
900
–
907
ISSN:
2088-8708
900
I
ns
t
it
u
t
e
o
f
A
d
v
a
nce
d
Eng
ine
e
r
i
ng
a
nd
S
cie
nce
w
w
w
.
i
a
e
s
j
o
u
r
n
a
l
.
c
o
m
Simulation
and
Modeling
of
Silicon
Based
Single
Electr
on
T
ransistor
Malik
Ashter
Mehdy
,
Mariagrazia
Graziano,
and
Gianluca
Piccinini
Department
of
Electronics
and
T
elecommunications,
Politecnico
di
T
orino,
Italy
Article
Inf
o
Article
history:
Recei
v
ed:
Jul
29,
2017
Re
vised:
Dec
29,
2017
Accepted:
Jan
27,
2018
K
eyw
ord:
Single
Electron
T
ransistor
TCAD
Simulations
NEGF
Coulomb
Blockade
T
ransmission
Spectrum
ABSTRA
CT
In
this
w
ork,
we
simulated
and
modeled
silicon
quantum
dot
based
single
electron
transistor
(SET).
W
e
simulated
the
de
vice
using
non-equilibrium
Green’
s
function
(NEGF)
formal-
ism
in
transport
direction
coupled
with
Schrodinger
equation
in
transv
erse
directions.
The
characteristics
of
SET
such
as
Coulomb
blockade
and
Coulomb
diamonds
were
observ
ed.
W
e
also
present
a
ne
w
ef
ficient
model
to
calculate
the
current
v
oltage
(
IV)
characteristics
of
the
SET
.
The
IV
characteristic
achie
v
ed
from
the
model
are
v
ery
similar
to
those
from
simulations
both
in
shape
and
magnitude.
The
proposed
model
is
capable
of
reproducing
the
Coul
omb
diamond
diagr
am
in
good
agreement
with
the
simulati
ons.
The
model,
which
is
based
on
transmission
spectrum,
is
simple,
ef
ficient
and
pro
vides
insights
on
the
ph
ysics
of
the
de
vice.
The
transmission
spectrum
at
equilibrium
is
achie
v
ed
from
simulations
and
gi
v
en
as
input
to
the
model.
The
model
then
calculates
the
e
v
olv
ed
transmission
spectra
at
non-equilibrium
conditions
and
e
v
aluates
the
current
using
Landauers
formula.
Copyright
c
2018
Institute
of
Advanced
Engineering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
Malik
Ashter
Mehdy
Department
of
Electronics
and
T
elecommunications
Politecnico
di
T
orino,
corso
Duca
de
gli
Abruzzi,
24,
T
orino,
Italy
malik.mehdy@polito.it
1.
INTR
ODUCTION
The
CMOS
technology
is
at
the
v
er
ge
of
the
miniaturization
and
it
needs
to
be
replaced
by
the
ne
w
tech-
nologies
which
should
o
v
ercome
its
limitations.
One
of
the
most
promising
technologies
is
the
one
based
on
single
electron
tunneling
phenomenon.
If
a
metallic
or
semiconducting
is
land
is
placed
between
the
electrodes
separated
by
the
tunnel
junctions
wi
th
v
ery
small
capacitance,
the
single
electron
phenomenon
can
be
observ
ed.
When
an
electron
tunnels
into
the
island
it
raises
t
he
electrostatic
potential
of
island
stopping
the
tunneling
of
the
follo
wing
electrons
until
e
xternal
potential
is
applied.
This
phenomenon
is
also
kno
wn
as
Coulomb
blockade.
If
the
island
size
is
v
ery
small
(so
that
the
capacitance
is
v
ery
small),
tunneling
of
single
electrons
can
be
controlled
by
the
e
xternal
electrode
potentials.
Hence
the
de
vices
based
on
single
electron
phenomenon
ha
v
e
the
attrib
utes
of
small
size
and
lo
w
po
wer
consumption.
Ov
er
the
years,
man
y
dif
ferent
schemes
for
digital
logic
using
single
electron
transistor
ha
v
e
been
presented
[1–5].
In
one
scheme,
the
l
ogic
states
are
represe
n
t
ed
by
the
single
electrons
located
at
tin
y
metallic
electrodes
[1].
While
such
a
scheme
can
pro
vide
the
huge
benefits
in
terms
of
high
speed
and
v
ery
lo
w
po
wer
consumption,
its
practical
implementation
f
aces
serious
challenges.
Firstly
,
the
y
should
be
operated
at
temperature
much
less
than
the
critical
temperature,
otherwise
a
thermally
induced
tunneling
e
v
ent
can
change
the
entire
calculation.
Secondly
,
the
output
of
one
logic
g
ate
(single
electron)
should
be
capable
of
char
ging
the
input
of
follo
wing
g
ates.
This
requires
the
stray
capacitance
of
interconnections
to
be
k
ept
ne
gligible
and
complicated
design
rules
for
dif
ferent
number
of
subsequent
g
ates
[2].
The
de
vices
based
on
silicon
are
v
ery
important
part
of
the
today’
s
digital
w
orld
[6–8].
Single
electron
transistors
(SET)
based
on
silicon
ha
v
e
been
pre
viously
studied
in
[9–11].
In
order
to
operate
the
transistor
at
room
temperature,
the
size
of
the
is
land
should
be
less
than
5nm
[12].
The
nano-lithograph
y
techniques
no
w
a
days
allo
w
to
define
such
small
structures
with
only
a
fe
w
v
ariations.
In
this
w
ork,
we
simulate
and
model
the
SET
based
on
J
ournal
Homepage:
http://iaescor
e
.com/journals/inde
x.php/IJECE
I
ns
t
it
u
t
e
o
f
A
d
v
a
nce
d
Eng
ine
e
r
i
ng
a
nd
S
cie
nce
w
w
w
.
i
a
e
s
j
o
u
r
n
a
l
.
c
o
m
,
DOI:
10.11591/ijece.v8i2.pp900-907
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
901
a
)
H
e
i
g
h
t
W
i
d
t
h
L
e
n
g
t
h
b
)
C
o
n
t
a
c
t
C
o
n
t
a
c
t
Si
l
i
con
Insulator
Ai
r
Si
Island
T
unnel
Junction
T
unnel
Junction
Figure
1.
a)
The
three-dimensional
structure
of
the
de
vice
which
is
simulated
in
Atlas
TCAD
b)
Cross
section
of
the
same
de
vice
as
seen
from
the
sides.
Dif
ferent
materials
and
the
dimensional
parameters
are
highlighted
and
drain
and
source
contacts
are
mentioned.
The
g
ate
contact
is
all
around
the
de
vice.
silicon
quantum
dot
with
a
size
of
1nm.
W
e
present
tw
o
no
v
el
contrib
utions
in
this
w
ork.
i)
W
e
simulate
Si
based
SET
,
for
the
first
time,
using
Atlas
TCAD
[13]
with
non-equilibrium
Green’
s
function
(NEGF)
technique.
The
results
are
comparable
to
the
e
xperimental
results
with
similar
de
vice
s.
NEGF
formalism
is
a
rob
ust
technique
for
the
quantum
le
v
el
simulations.
Anantram
et.
al
used
NEGF
technique
to
model
nanode
vices
[14].
The
transport
through
quantum
dot
weakly
coupled
with
the
contacts
is
studied
using
NEGF
technique
in
[15].
ii)
W
e
propose
transmission
spectrum
based
model
for
the
e
v
aluation
of
SET
characteristics.
The
proposed
model,
which
is
the
e
xtended
v
ersion
of
[16],
uses
transmission
spectrum
to
calculate
the
current.
It
does
not
need
an
y
comple
x
initial
set
up
and
is
v
ery
ef
ficient,
simple,
suf
ficiently
accurate
and
fle
xible
to
implement
in
dif
ferent
conditions.
In
comparison,
Monte
Carlo
based
models
[17]
are
accurate
b
ut
v
ery
time
consuming.
Macromodel
ing
based
models
[18]
[19]
are
ef
ficient
b
ut
pro
vide
a
little
ph
ysical
insights
on
the
de
vice
beha
vior
.
Analyt
ical
models
based
on
ph
ysics
[20]
and
master
e
qu
a
tion
[21]
solv
e
this
problem
b
ut
the
y
still
require
a
comple
x
procedure
to
calculate
tunnel
junction
resistances
and
capacitances
[22].
These
models
require
the
tunnel
junction
resistances
and
capacitances
as
input
while
the
proposed
model
requires
transmission
spectrum
as
input.
The
transmission
spectrum
pro
vides
insights
on
the
ph
ysics
of
de
vice
especially
about
the
density
of
states
of
island
which
lacks
in
case
of
other
models.
W
e
can
easily
change
the
model
for
the
v
ariations
in
contact
coupling,
g
ate
coupling
and
density
of
state
v
ari
ations
due
to
dif
ferent
f
abricated
size
of
island.
From
the
transmission
spectrum
current
is
calculated
usi
n
g
Landauers
formula.
The
rest
of
the
paper
is
or
g
anized
as
follo
ws
.
In
section
II
the
e
xplanation
of
t
h
e
de
vice
and
simulations
are
described.
In
section
III
the
details
of
proposed
model
are
gi
v
en.
In
section
IV
the
results
are
discussed
and
the
section
V
concludes
the
paper
.
2.
RESEARCH
METHODOLOGY
W
e
simulated
the
single
electron
transistor
(SET)
based
on
silicon
island
and
the
drain
and
source
contacts
made
of
highly
doped
silicon
(Fig.
1).
The
cross
section
of
the
island
(height
into
width)
is
1x1nm
and
the
length
is
2nm.
The
tunnel
junctions
are
created
by
the
g
ap
between
the
island
and
contacts.
These
g
aps
pro
vide
the
confinement
in
the
transport
direction
and
decouple
the
island
from
electrodes.
The
silicon
island
is
w
arped
by
the
oxide
through
which
g
ate
contact
controls
the
conduction
in
the
island.
The
de
vice
w
orks
as
the
single
e
lectron
transistor
and
sho
ws
the
characteristics
of
SET
such
as
Coulomb
diamonds
as
sho
wn
in
results
section.
W
e
simulated
the
de
vice
in
Atlas
TCAD
using
non-equilibrium
Green’
s
function
(NEGF)
formalism
i
n
transport
direction
coupled
with
Schrodinger
equation
in
transv
erse
direction.
The
simulator
uses
mode
space
technique
instead
of
solving
2d
or
3d
problem.
First
solving
the
Schrodinger
equat
ion
eigen
v
alues
and
eigen
v
ectors
are
calculated
in
each
slice
of
de
vice.
Then
a
transport
equation
is
solv
ed
for
electrons
mo
ving
in
lo
west
sub-bands
which
are
occupied.
This
reduces
the
size
of
the
problem
[13].
But
still
the
simulations
are
v
ery
time
consuming
and
there
is
a
need
for
an
ef
ficient
model.
From
the
simulations,
we
can
achie
v
e
the
transmission
spectrum
(TS)
at
equilibrium
which
i
s
fed
to
the
model
to
compute
the
IV
characteristics.
The
IV
characteristics
are
also
calculated
from
the
simulations
to
compare
with
those
from
the
model.
Simulation
and
Modeling
of
Silicon
Based
Single
...
(Malik
Ashter
Mehdy)
Evaluation Warning : The document was created with Spire.PDF for Python.
902
ISSN:
2088-8708
Figure
2.
The
flo
w
chart
of
the
model:
at
equilibrium,
the
transmission
spectrum
is
constructed
using
parameters
from
simulations,
while
at
non-equilibrium
conditions,
shift
is
calculated
by
the
self-consistent
loop.
3.
THE
PR
OPOSED
MODEL
The
propose
d
model
uses
transmission
spectrum
(TS)
to
calculate
the
characteristics
of
single
electron
tran-
sistors.
In
our
pre
vious
w
orks,
we
ha
v
e
implemented
earlier
v
ersions
of
t
he
model
to
molecular
de
vices
[23]
and
silicon
nanocrystals
[16].
The
current
model
is
implemented
on
SET
and
the
dif
ference
from
[23]
is
the
implemen-
tation
of
tw
o
spectra
for
dif
ferent
spin
as
e
xplained
in
[16].
In
this
w
ork,
we
e
xtend
the
model
presented
in
[16]
to
implement
the
g
ate
v
oltage
in
order
to
use
the
silicon
nanocrystal
as
SET
.
Further
e
xplanation
on
the
modeling
of
g
ate
v
oltage
can
be
found
in
the
follo
wing
description
and
in
the
results
section.
The
model
tak
es
as
input
the
transmission
spectrum
at
equilibrium
from
the
simulations.
From
the
transmission
spectrum,
current
i
s
calculated
using
Landauers
formula.
The
e
v
olution
in
transmission
spectrum
with
applied
v
oltage
is
calculated
by
the
model
using
self-consistent
field
technique.
By
e
v
aluati
n
g
the
TS
at
all
v
oltages
(
V
ds
;
V
g
)
and
using
the
Landauers
formula,
IV
characteristics
of
the
de
vice
can
be
calculated
[24].
W
e
calculate
the
TS
at
equilibrium
using
Atlas
TCAD.
The
TS
parameters
i.e.
position,
amplit
ud
e
and
width
of
peaks
are
gi
v
en
as
input
to
the
model.
The
model
uses
these
parameters
to
construct
the
TS
at
equilibrium.
The
shape
of
the
peaks
is
defined
by
the
Lorentzian
function.
D
0
=
k
0
N
or
m
(
1
2
(
E
)
2
+
2
)
(1)
Here
D
0
is
the
transmission
peak
defined
o
v
er
the
range
of
ener
gies
E
,
is
the
position
of
the
corresponding
peak
and
is
the
sum
of
drain
and
source
rate
constants
1
,
2
.
The
width
(or
broadening)
of
the
transmission
peaks
is
defined
by
.
After
normalizing
the
Lorentzian
function,
it
is
multiplied
by
the
amplit
ud
e
of
the
corresponding
peak
k
0
.
Other
peaks
are
calculated
similarly
and
then
TS
is
constructed
by
adding
all
the
transmission
peaks.
The
same
procedure
is
repeated
for
both
spins
and
tw
o
spectra
are
achie
v
ed.
Although
at
this
point
both
the
spectra
are
similar
,
the
dif
ference
arises
when
the
shift
in
spectra
is
calculated
as
e
xplained
belo
w
.
When
the
v
oltage
is
applied,
transmission
spectrum
shifts
on
the
ener
gy
axis.
The
shift
is
caused
by
the
e
xternal
potential
and
because
of
the
change
in
number
of
electrons.
At
equilibrium,
the
Fermi
le
v
el
of
island
is
IJECE
V
ol.
8,
No.
2,
April
2018:
900
–
907
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
903
aligned
with
the
drain
and
source
chemical
potential.
When
the
drain
to
source
v
oltage
(
V
ds
)
is
applied
the
drain
and
source
chemical
potential
split
and
the
separation
is
so
called
bias
windo
w
.
If
there
is
a
transmis
sion
peak
inside
or
at
the
edge
of
the
bias
windo
w
then
the
number
of
electrons
(
N
)
at
the
island
change.
If
the
transmission
peak
corresponds
to
the
spin
up
electrons
the
change
in
number
of
electrons
are
calculated
using
follo
wing
formula.
N
U
P
=D
N
=
1
f
1
U
P
=D
N
(
)
+
2
f
2
U
P
=D
N
(
)
1
+
2
(2)
Here
f
1
U
P
=D
N
(
)
and
f
2
U
P
=D
N
(
)
are
the
drain
and
source
Fermi
functions
with
respect
to
the
corresponding
spin
up/do
wn
transmiss
ion
peaks
(
)
If
the
number
of
electrons
increase
in
the
island
the
ener
gy
of
the
system
increases
so
the
transmission
spectrum
shifts
to
w
ards
higher
ener
gies
and
vice
v
ersa.
The
o
v
erall
shift
is
the
result
of
both
i.e.
change
in
number
of
electrons
and
the
applied
drain
to
source
bias.
The
po
s
iti
v
e
g
ate
v
oltage
lo
wers
the
ener
gy
of
the
island
so
it
shifts
the
transmission
spectrum
to
w
ards
lo
w
ener
gies.
The
o
v
erall
shift
(
U
)
can
be
calculated
using:
U
U
P
=D
N
=
U
L
+
U
0
N
D
N
=U
P
(3)
where
U
0
is
the
single
electron
char
ging
ener
gy
,
N
D
N
=U
P
is
the
change
in
number
of
spin
do
wn/up
electrons
and
U
L
=
d
V
ds
+
g
V
g
is
Laplace
potential.
The
ef
fect
of
the
e
xternal
v
oltages
is
modeled
by
the
coupling
coef
ficients
d
;
g
.
Equation
2
and
3
are
solv
ed
self-consistently
to
calculate
the
shift
in
tra
nsmission
spectrum
with
respect
to
the
applied
v
oltage.
The
shift
in
both
the
spectra
for
spin
up
and
spin
do
wn
is
calculated.
The
shift
in
the
spin
up
transmission
spectrum
is
calculated
from
t
he
change
in
number
of
spin
do
wn
electrons
as
spin
up
electron
feels
the
potential
due
to
spin
do
wn
electron
and
vice
v
ersa
[24].
When
a
posi
ti
v
e
g
ate
v
oltage
is
applied
and
a
lo
west
unoccupied
molecular
le
v
el
(LUMO)
goes
bel
o
w
the
drain
and
source
chemical
potentials,
the
number
of
electrons
increase
on
the
island.
This
increase
in
number
of
electrons
causes
the
rate
constant
(
)
to
decrease
so
the
width
of
the
transmission
peaks
also
reduces.
In
the
current
v
ersion
of
the
model,
the
change
in
width
is
calculated
using
fitting
parameters.
Another
ef
fect
of
the
increase
in
number
of
electrons
on
the
island
is
to
reduce
the
g
ate
coupling
(
g
).
W
ith
the
increase
in
number
of
electrons
on
the
island
the
amount
of
shift
in
transmission
spectra
due
to
increase
in
g
ate
v
oltage
reduces.
W
e
implement
this
-2
-1
0
1
2
3
Energy (eV)
0
1
2
3
Model
Si
mul
ations
-2
-1
0
1
2
3
0
1
2
3
Model
Si
mul
ations
(b)
-2
-1
0
1
2
3
Energy (eV)
0
1
2
3
Model
Si
mul
ations
-2
-1
0
1
2
3
Ener
gy (eV)
0
1
2
3
Model
Si
mul
ations
T
r
anmissi
on
-2
-1
0
1
2
3
Energy (eV)
0
1
2
3
Model
Si
mul
ations
(a)
(d)
(c)
E
f
=0
HOMO-LU
MO Gap
(e)
Figure
3.
T
ransmission
spectra
at
dif
ferent
V
g
and
V
ds
achie
v
ed
from
simulations
and
model.
The
bias
windo
w
is
sho
wn
as
the
dashed
lines.
a)
V
g
=
0
:
0
V
,
V
ds
=
0
:
0
V
b)
V
g
=
1
:
0
V
,
V
ds
=
0
:
0
V
c)
V
g
=
1
:
0
V
,
V
ds
=
1
:
0
V
d)
V
g
=
1
:
5
V
,
V
ds
=
1
:
0
V
e)
V
g
=
2
:
5
V
,
V
ds
=
1
:
0
V
.
Simulation
and
Modeling
of
Silicon
Based
Single
...
(Malik
Ashter
Mehdy)
Evaluation Warning : The document was created with Spire.PDF for Python.
904
ISSN:
2088-8708
ef
fect
in
the
model
by
linear
fitting
of
g
with
respect
to
the
change
in
number
of
electrons
(
N
)
.
After
e
v
aluating
the
transmission
spectra
e
v
olv
ed
with
the
applied
v
oltage
we
can
find
the
current
using
Landauers
formula.
I
U
P
=D
N
=
q
2
h
Z
+
1
1
T
U
P
=D
N
(
E
)[
f
1
(
E
)
f
2
(
E
)]
dE
(4)
Here
q
is
the
electron
char
ge,
h
is
plank’
s
constant
and
T
(
E
)
is
the
transmission
spectrum.
The
dif
ference
between
the
source
and
drain
Fermi
functions
models
the
ef
fect
of
bias
windo
w
so
that
only
the
part
of
transmission
spectrum
which
is
inside
the
bias
windo
w
is
inte
grated.
The
total
current
is
calculated
by
the
sum
of
spin
up
and
spin
do
wn
currents.
4.
RESUL
TS
AND
DISCUSSION
Single
electron
transistor
based
on
silicon
quantum
dot
of
1nm
cross
section
w
as
simulated
in
Atlas
TCAD
using
NEGF
formalism
coupled
with
Schrodinger
equation.
The
IV
characteristics
of
the
de
vice
were
calculated
from
simulations
as
well
as
from
the
model.
In
Fig.
3
the
transmission
spec
tra
(TS)
from
simulati
ons
and
model
are
sho
wn
for
dif
ferent
v
alues
of
V
g
and
V
ds
.
The
lo
west
unoccupied
molecular
le
v
els
(LUMO)
are
sho
wn
while
the
highest
occupied
molecular
le
v
els
(HOMO)
are
not
sho
wn
because
the
y
do
not
contrib
ute
to
the
conduction
in
this
case
as
the
y
are
f
ar
belo
w
the
Fermi
ener
gy
le
v
el
(
E
f
=
0
).
The
HOMO-LUMO
g
ap
and
the
Fermi
ener
gy
le
v
el
are
also
mentioned
in
the
Fig.
3
(a).
The
LUMO
le
v
els
are
relati
v
ely
closer
to
Fermi
le
v
el
and
by
applying
a
positi
v
e
g
ate
v
oltage
we
can
shift
them
near
Fermi
le
v
el
(Fig.
3
(b)).
Then
at
a
small
v
alue
of
V
ds
LUMO1
enters
inside
the
bias
windo
w
and
contrib
utes
to
current
(Fig.
3
(c,d,e)).
The
comparison
between
the
IV
characteristics
is
sho
wn
in
the
Fig.
4.
Drain
current
is
plotted
as
a
function
of
drain
to
source
bias
for
dif
ferent
v
alues
of
g
ate
v
oltage.
At
lo
w
g
ate
v
oltages
(i.e.
less
than
1
V)
there
are
no
transmission
peaks
inside
the
bias
windo
w
s
o
a
v
ery
small
current
flo
ws
through
de
vice
because
of
tunneling
of
electrons
with
v
ery
lo
w
probability
.
The
model
pro
vides
acceptable
estimation
of
the
current
as
sho
wn
in
the
figure
in
case
of
V
g
=
0
:
7
;
0
:
8
;
0
:
9
;
1
:
0
V
.
The
first
peak
starts
to
enter
in
the
bias
windo
w
at
V
g
=
1
V
as
sho
wn
in
the
Fig.
3
(c).
0.8V
1.2V
1.6V
0.7V
1.4V
2.1V
1.1V
1.3V
1.5V
1.0V
1.7V
2.3V
(a)
(b)
(c)
(d)
2.4V
0.9V
Figure
4.
Comparison
of
IV
characteristics
of
the
de
vice
at
v
arious
g
ate
v
oltages
between
the
model
and
simulations.
Dif
ferent
cur
v
es
are
for
dif
ferent
g
ate
v
oltages
mentioned
by
the
curv
es.
At
lo
wer
g
ate
v
olt
ages,
less
than
1.0V
,
there
are
no
peaks
inside
the
bias
windo
w
.
First
peak
enters
in
the
bias
windo
w
at
V
g
=
1
:
1
V
so
a
high
current
flo
ws.
As
the
g
ate
v
oltage
increases
other
peaks
also
contrib
ute
to
the
current.
IJECE
V
ol.
8,
No.
2,
April
2018:
900
–
907
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
905
-1
-0.5
0
0.5
1
Drain
to
Source
Voltage(V)
-20
-10
0
10
20
Drain Current(
µ
A)
Simulations
Model
Vg=1.8V
0
0.5
1
Drain to Source Voltage(V)
0
10
20
30
40
50
Drain Current(
A)
Simulations
Model w/o Spin
Model w Spin
Figure
5.
a)
IV
characteristics
of
the
simulated
de
vice
at
g
ate
v
oltage
equal
to
1.8V
and
for
both
positi
v
e
and
ne
g
ati
v
e
drain
to
source
bias.
b)
IV
characteristics
obtained
from
the
pre
vious
model
without
considering
the
spin
and
from
the
current
model
with
the
dif
ferent
spectra
for
spin
at
V
g
=
2
:
0
V
.
The
current
is
modeled
v
ery
ef
fecti
v
ely
at
the
g
ate
v
oltages
higher
than
1V
(Fig.
4)
because
the
bias
windo
w
,
for
the
selected
range
of
V
ds
,
includes
the
transmission
peaks.
As
the
model
is
transmission
spectrum
based
so
the
results
are
better
when
there
is
a
transmission
peak
inside
the
bias
windo
w
.
F
or
the
sak
e
of
clarity
,
the
dif
ferent
curv
es
sho
wn
here
are
selected
in
such
a
w
ay
that
the
y
do
not
o
v
erlap
each
other
.
In
Fig.
5
(a)
the
IV
characteristics
for
both
positi
v
e
and
ne
g
ati
v
e
drain
to
source
bias
are
sho
wn
for
V
g
=
2
:
0
V
.
This
sho
ws
that
our
model
is
v
alid
for
bidirectional
electron
flo
w
.
And
in
the
Fig.
5
(b)
the
results
from
the
pre
vious
model
without
considering
the
spin
(a)
(b)
(c)
Figure
6.
The
current
plotted
as
a
funct
ion
of
V
ds
and
V
g
.
The
shade
of
gre
yscale
represents
the
drain
current
v
alue
based
on
the
scale
sho
wn
in
the
figure.
The
three
figures
are
obtained
from
a)
simulations
b)
model
c)
model
using
refined
set
of
applied
v
oltages.
Simulation
and
Modeling
of
Silicon
Based
Single
...
(Malik
Ashter
Mehdy)
Evaluation Warning : The document was created with Spire.PDF for Python.
906
ISSN:
2088-8708
of
electron
(Model
w/o
Spin)
is
compared
with
the
proposed
model
(Model
w
Spin)
under
the
same
conditions
and
parameters.
W
e
can
see
that
the
proposed
model
pro
vides
v
ery
accurate
results
compared
t
o
the
pre
vious
v
ersion
of
the
model
in
[23].
The
current
as
a
function
of
V
ds
and
V
g
is
plotted
on
grayscale
in
Fig.
6.
The
magnitude
of
the
current
is
represented
by
the
shade
of
the
gray
according
to
the
scale
sho
wn
in
the
figure.
The
dark
re
gions
represent
the
zero
or
v
ery
small
current
while
t
he
current
of
higher
magnitudes
is
represented
by
the
light
re
gions.
The
coulomb
blockade
diamonds
which
are
unique
characteristics
of
SET
can
be
seen
i
n
the
form
of
dark
re
gions.
One
lar
ge
blockade
diamond
and
one
small
one
can
be
observ
ed
in
the
figure.
The
model
successfully
predicts
the
same
beha
vior
as
obtained
from
simulations.
The
resolution
of
the
plots
in
Fig.
6
(a,b)
is
not
good
and
it
w
ould
tak
e
high
computational
ef
fort
to
increase
it
in
case
of
simulations.
W
e
ha
v
e
sho
wn
the
high
resolution
achie
v
ed
from
the
model
in
Fig.
6
(c).
The
time
required
for
t
he
calculations
in
Fig.
6
(a,b)
is
6346s
by
the
TCAD
simulations
and
1.57s
by
the
model.
The
model
tak
es
128s
for
the
calculation
with
high
resolution
plot
(210
V
ds
points
and
260
V
g
points)
Fig.
6
(c).
All
the
simulations
were
performed
on
the
same
system
with
the
Intel
core
i5
1.8
GHz
processor
and
6
GB
of
RAM.
The
proposed
model
pro
vides
suf
ficiently
accurate
results
while
maintaining
the
ef
ficie
nc
y
.
While
other
compact
models
based
on
macromodeling
are
also
ef
ficient,
the
proposed
model
captures
the
ph
ysical
information
of
de
vic
e
with
more
details.
5.
CONCLUSIONS
In
this
w
ork,
we
simulated
and
modeled
the
single
electron
transistor
based
on
silicon
quantum
dot.
W
e
simulated
the
de
vice
in
Atlas
TCAD
using
NEGF
formalism
in
t
ransport
direction
coupled
with
Schrodinger
equation
in
transv
erse
direction.
The
SET
characteristics
such
as
Coulomb
blockade
and
Coulomb
diamonds
were
observ
ed
in
the
results
obtained
from
the
simulations
and
the
model.
The
IV
characteristics
of
the
SET
were
obtained
from
the
model
ha
ving
a
good
agreement
with
the
simulations.
The
proposed
model
is
simple,
ef
ficient,
fle
xible
and
pro
vides
insights
on
the
ph
ysics
of
the
de
vice.
In
the
future,
we
will
include
in
the
model
the
ef
fect
of
the
change
in
number
of
electrons
on
the
width
of
the
transmission
peaks.
W
e
will
also
use
the
de
vice
for
circuit
le
v
el
implementation.
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BIOGRAPHIES
OF
A
UTHORS
Malik
Ashter
Mehdy
recie
v
ed
his
B.E.
Electronics
de
gree
from
Federal
Urdu
Uni
v
ersi
ty
,
Islam-
abad,
P
akistan
in
2011
and
M.S.
de
gree
in
Nanote
chnology
from
Politecnico
di
T
orino,
Italy
in
2014.
He
is
currently
a
PhD
student
in
Department
of
Electronics
and
T
elecommunication
at
Po-
litecnico
di
T
orino,Italy
.
His
research
interests
include,
simulation
and
modeling
of
silicon
based
single
electron
transistor
.
Mariagrazia
Graziano
recei
v
ed
the
DrEng
and
the
PhD
de
grees
in
Electronics
Engineering
from
the
Politecnico
di
T
orino,
Italy
,
in
1997
and
2001,
respecti
v
ely
.
Since
2002,
she
is
a
researcher
and
since
2005,
an
assistant
professor
at
the
Politecnico
di
T
orino.
Since
2008,
she
is
adjunct
f
aculty
at
the
Uni
v
ersity
of
Illinois
at
Chicago
and
since
2014
she
is
a
Marie-Skodo
wska-Curie
Intra-European
Fello
w
at
the
London
Centre
for
Nanotechnology
.
Her
re
search
interests
include
design
of
CMOS
and
be
yond
CMOS
de
vices,
ci
rcuits
and
architectures.
She
is
author
and
coauthor
of
more
than
120
published
w
orks.
She
is
a
Member
of
the
IEEE
since
2007.
Further
info
on
her
homepage:
http://www
.det.polito.it/personale/scheda/(nominati
v
o)/mariagrazia.graziano
Gianluca
Piccinini
is
a
Full
Professor
since
2006
at
the
Department
of
Electronics
of
Po-
litecnico
di
T
orino,
Italy
,
where
he
teaches
electron
de
vices
and
inte
grated
system
technol-
ogy
.
He
recei
v
ed
the
Dr
.
Ing
and
the
Ph.D.
de
grees
in
electronics
engineering
in
1986
and
1990
respecti
v
ely
.
His
research
acti
vities
started
at
the
end
of
the
1980s,
were
initially
fo-
cused
on
VLSI
architecture
for
artificial
intelligence
and
mo
v
ed,
during
the
1990s,
to
w
ard
the
ph
ysical
design
of
VLSI
systems
for
high
rat
e
and
high-speed
transmission
and
coding
algo-
rithms.
His
current
interests
in
v
olv
e
the
introduction
of
ne
w
technologies
as
molecular
elec-
tronics
in
inte
grated
systems
where
he
studies
transport,
adv
anced
microf
abrication
and
self-
assembly
technologies
in
molecular
scale
systems.
He
is
author
and
co-author
of
more
than
100
published
w
orks
and
is
the
holder
of
one
international
patent.
Further
info
on
his
homepage:
http://www
.det.polito.it/personale/scheda/(nominati
v
o)/gianluca.piccinini
Simulation
and
Modeling
of
Silicon
Based
Single
...
(Malik
Ashter
Mehdy)
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