Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l.
5, N
o
. 2
,
A
p
r
il
201
5, p
p
.
24
3
~
25
0
I
S
SN
: 208
8-8
7
0
8
2
43
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
Three-Dimension
a
l Devices T
r
ans
p
ort
Simulation Lifetim
e
and
Relaxation Semiconductor
Nouar
S
o
u
a
d
Fadila*,
S.
Mans
ouri
**,
M.
Amr
a
ni
**, P.
Marie
***
,
A. Massoum*
*
El
ectro
technics
Departm
e
nt
, F
a
cult
y
of
Techno
l
o
g
y
, Univ
ersi
t
y
Djilal
i
Liab
ès of
Sidi bel
Abbès 2
2000 Algeri
a
**
Electron
i
cs D
e
partment, Faculty
of
Technolog
y
,
University
Djilali Liab
ès
of Sidi bel Abbès 220
00 Algeria
***
C
I
MAP UMR
6
252CN
R
S-
ENS
I
CAEN
-C
EA-U
CBN,
6
Bou
l
ev
ar
d
du
Mar
e
chal Ju
in, 1
425
0
C
a
en
C
e
dex
F
r
ance
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Nov 8, 2014
Rev
i
sed
D
ec 27
, 20
14
Accepte
d
Ja
n 20, 2015
Our work is
to creat
e a thr
ee-d
i
m
e
ns
ional S
i
m
u
lator (3D) us
ed f
o
r the s
t
ud
y
of the components to low geo
m
etr
y
of d
e
sign
, and to d
e
termine in
the
volum
e structure
the poten
tia
l d
i
stributions and
densities of fr
ee
carri
ers in
bias voltage giv
e
n b
y
solving
the s
y
stem of P
o
isson and two
continuities
equations. The initial versio
n can
simulate components of lifetim
e
semiconductor.
In this stud
y
,
w
e
ma
ke a comparison between
the lif
etime
and relax
a
tion
semiconductor
in
the condu
ction
mode. In
order
to
create
a
larger Sim
u
la
tor
,
we
'll
perform
a ca
lcul
ation b
y
var
y
ing
am bipolar lifetime
way
to move fr
om lifetime semiconduc
tor to relaxation semicon
ductor. We
consider th
e case corresponding
at two
diff
erent
values of diffusion lifetim
e
τ
0
which is corr
esponding to a
measured lif
etime in curren
t
tr
ansport. Th
e
method of resolution consists to
linear
iz
ation of
the equations transport
b
y
the fin
i
te diff
erences method. The algo
rithm for
solving linear
and strong
ly
coupled
equatio
ns deduced
from the ph
y
s
ical model is
th
e Newto
n
-Raphson.
However, in
or
der to
allow
a
bett
er
conv
erg
e
nce and
consequently
an
improvement of
time
com
puting
3D,
a m
e
thod
c
o
m
b
ined, in
corp
orating
the
Newton algor
ith
m and the Gummel met
hod was
develo
ped. PIN diodes ar
e
used for test of
the simulation model.
Keyword:
Gumm
el algorithm
Li
fet
i
m
e sem
i
conduct
o
r
Newt
on al
gori
t
h
m
R
e
l
a
xat
i
on sem
i
conduct
o
r
Three
-
di
m
e
nsional
si
m
u
l
a
ti
on
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
:
A. M
a
ss
o
u
m
,
Electrotechnic
s
De
partm
e
nt
, Faculty of technol
ogy, ,
Un
i
v
ersity Dj
ilali Liab
ès
o
f
Sid
i
b
e
l
Ab
b
è
s,
2
200
0 A
l
g
e
r
i
a.
Em
a
il: ah
m
a
ss
o
u
m
@yah
oo
.fr
1.
INTRODUCTION
To i
m
prove t
h
e SIM
3
D
so
ft
ware
devel
ope
d i
n
o
u
r
l
a
bo
ra
t
o
ry
, ha
ve cr
ea
t
e
d a
m
o
re ge
n
e
ral
t
ool
t
o
cal
cul
a
t
e
pot
en
t
i
a
l
di
st
ri
but
i
o
ns an
d co
nce
n
t
r
at
i
ons
of f
r
ee
carri
ers
,
by
a num
eri
cal
sol
u
t
i
on o
f
eq
uat
i
o
ns o
f
trans
p
o
r
t f
o
r
I
I
I
-
V sem
i
cond
ucto
rs.
Wo
rth
τ
0
, conside
r
abl
y
higher t
h
an
the
dielectric re
laxation tim
e
τ
rd (
τ
rd
<<
τ
0
), th
e results can
b
e
an
alyzed
using
simp
le inj
ect
i
o
n a
n
d
ass
u
m
i
ng a
com
m
on uni
t
hol
ders
i
n
e
x
c
e
ss l
i
f
e
expect
a
n
cy
wh
i
c
h i
s
const
a
nt
fo
r el
ect
ro
ns and
hol
es t
h
r
o
u
gh t
h
e
ν
reg
i
o
n
th
is typ
e
o
f
semico
n
d
u
c
tor will b
e
called a sem
i
c
o
nductor to life accordi
n
g to
the term
inology of Va
n R
o
os
broec
k
.
In the
contra
ry case
or
τ
0
is
m
u
ch lowe
r t
h
an t
h
e
dielectric relaxation ti
me
τ
rd
(
τ
rd
>>
τ
0
)
[1
],
th
e ef
f
e
cts of
space cha
r
ge is
very im
portant by any and lifetim
e
s of
excess carriers
vary greatly
from
one
poi
nt to anothe
r
alo
n
g
th
e structu
r
e. Th
is type o
f
sem
i
co
n
d
u
c
tor is called
se
m
i
co
n
d
u
c
tor to
relax
a
tio
n. Th
is is th
e case
of
GaAs. Th
e al
go
rith
m
b
e
st com
p
l
y
with
resolu
tio
n
no
n
linea
r eq
u
a
tion
s
i
n
p
a
rtial d
i
fferentials arisin
g
from
th
e
phy
si
cal
m
odel
,
est
a
bl
i
s
hed i
s
t
h
at
of Ne
wt
on
, t
h
i
s
m
e
t
h
o
d
h
o
we
ver c
o
n
v
er
ges ra
pi
dl
y
i
f
t
h
e ent
e
red
val
u
e
s
are p
r
o
p
erl
y
c
hos
en
, we
ha
ve t
h
er
ef
ore a
d
apt
e
d a m
e
t
h
od c
o
m
b
i
n
ed,
whi
c
h b
r
i
n
gs
bot
h t
h
e m
e
tho
d
of
Gum
m
el
and Newt
on
, t
h
us
creat
i
ng a
l
i
n
k b
e
t
w
ee
n t
h
e two
al
g
o
rithm
sa
t th
e en
d for a
redu
ctio
n
of
com
put
at
i
on t
i
m
e
and a
bet
t
e
r
co
nve
r
g
ence
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Th
ree-Di
mensio
na
l Devices
Tra
n
spo
r
t S
i
mu
l
a
tio
n
Lifetime a
n
d
Rela
xa
tio
n
…
(No
u
a
r
So
ua
d
Fa
d
ila
)
24
4
We
prese
n
t i
n
the
order s
u
itable m
a
the
m
atical
m
odel, as
well as t
h
e
physical m
odel, num
e
rica
l
m
o
d
e
l fo
llowed
b
y
m
o
d
e
lin
g
resu
lts and
th
ei
r in
terpretatio
n.
2.
THREE-
DI
M
E
NSIO
N
A
L
REPRESE
N
T
A
TIO
N
O
F
T
H
E PH
YSI
C
AL EQ
UATI
O
NS
[
2]
-[
3
]
In the stationary case and
for t
h
e a
n
alysis of
a ho
m
ogene
ou
s st
ru
ct
ure
i
n
t
h
ree
di
m
e
nsi
o
n
s
, t
h
e
B
a
si
c
(Fish and
con
tin
u
ity) equ
a
tio
ns tak
e
t
h
e fo
llowing
form
[
4
]-[
5
]
U
z
j
y
j
x
j
q
U
z
j
y
j
x
j
q
n
N
N
p
n
q
z
y
x
p
p
p
n
n
n
r
A
D
.
1
.
1
.
.
2
2
2
2
2
2
(1
)
with
1
.
1
.
1
.
1
.
p
p
ne
n
n
pe
p
n
p
n
p
U
n
U
U
z
p
y
p
x
p
p
D
q
z
y
x
p
p
q
p
j
z
n
y
n
x
n
n
D
q
z
y
x
n
n
q
n
j
.
.
.
.
.
.
.
.
.
.
(2
)
with
r
o
.
Th
e
d
i
electric p
e
rm
i
ttiv
it
y o
f
th
e sem
i
co
n
ducto
r
o :
Th
e p
e
rm
i
ttiv
ity o
f
v
a
cuum
r
:
Relativ
e to
th
e sem
i
co
n
d
u
c
to
r
p
e
rm
ittiv
it
y
: Electro
statiq
u
e
po
ten
tiel
q: elem
entary charge =
1
.6.10
-19
C
p
et
n
:
free
ho
l
e
s and
electrons d
e
n
s
ities
N
D
+
et N
A
-
:
donors a
n
d acce
ptors i
onize
d
de
nsities
n
r
: The c
h
arge
trapped on a
de
ep ce
ntre.
Whe
r
e
the
r
e
a
r
e n deep
ce
nters be
replace
d nr by
n
i
ri
n
1
j
n
et j
p
: vect
or current densitie
s of electrons
and holes
n
et
p
:
m
obil
ities of electrons and
holes.
D
n
et
D
p
: Dif
f
u
s
ion
o
f
électr
o
ns a
n
d
holes c
o
nstants
q
T
K
p
p
D
et
q
T
K
n
n
D
.
.
(3
)
W
ith
K : BOLTZM
ANN c
o
nstant
T : absolue
te
m
p
erature.
n
et
p
:
m
obil
ities of electrons and
holes
W
ill
be carrying loads in a m
a
terial with a l
o
w
density
N
R
c
e
ntre c
o
m
b
ining the
cha
r
ge
bom
b
space
(-e
nr) is l
o
w com
p
ared with the space
of
free
ca
rriers
an
d d
o
n
o
r
s or
io
ni
zed Acce
ptor c
h
arge.
Recom
b
ination plays an im
portant role, it
de
p
e
nd
s on
th
e
valu
es
o
f
th
e
p
a
ra
m
e
ter
s
τ
ne
and
τ
pe
, the
dielectric relaxation tim
e
τ
rd
and
di
ff
usio
n
τ
o life are im
port
a
nt to a
n
alyze t
h
e beha
vio
r
u
n
d
er
co
n
d
itions of
no
n
-
bala
nce
of
a gi
ven
sem
i
con
d
u
cto
r
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
088
-87
08
IJEC
E V
o
l. 5, No
. 2, A
p
ril 20
15
:
24
3 – 2
5
0
2
45
(4
)
(5
)
These tw
o ch
aracteristic tim
e
s allow def
i
ning t
h
e two
ty
pes of sem
i
con
duct
o
rs: s
e
m
i
cond
uct
o
r
lifetim
e or rela
xation
sem
i
conduct
o
r
has
d
u
r
ation
of
life
τ
rd
<<
τ
0
Sem
i
cond
ucto
r a
relaxati
on
τ
rd
<<
τ
0
3.
N
U
M
E
RICAL M
O
D
EL
First,
we
were interested to ac
hieve a
non-uniform
m
e
sh that esta
blishes from
the application of
a
calculation ba
s
e
d o
n
a ge
om
etric series of c
onsta
nt reas
on
(a) wit
h
a vari
able size
m
e
sh. This m
e
sh type can
be coa
r
se in
neutral areas a
n
d fine i
n
the
neighbor
ing re
gions of the P + inte
rface [6]-[7]. T
h
e electrical
param
e
ters calculation at the
r
m
odyna
m
i
c equilibrium
is
de
rive
d
by a
pplying th
e
m
e
thod
of Gum
m
e
l
m
e
thod
also called
de
coupled m
e
thod,
[8]-[9]. Gum
m
e
l
m
e
thod
consists of a
successi
ve re
solution
of t
h
e three
sy
stem
s couple
d
n
eq
uatio
ns i
n
N
u
n
k
n
o
w
ns
.
Each sy
stem
of eq
uatio
ns is
d
e
dicated to
det
e
rm
ine the value o
f
a sing
le typ
e
o
f
unk
nown categ
o
r
y
.
Fo
r ex
am
p
l
e th
e Poisso
n equ
a
tion pr
ov
id
es th
e
v
a
lu
es fo
r
potential
base
d
on
the c
once
n
tratio
ns
o
f
N
a
n
d
P.
The
gene
ral
prin
cip
l
e of
the
G
u
m
m
e
l
m
e
thod is
as f
o
llow
s
:
From
an estimated
initial sol
u
tion (
0
, N
0
, P
0
), t
h
e e
quati
on
F
(
,N,
P
) =0
u
nkn
own
is firstly
resol
v
ed. The
values
of t
h
us determ
ined will
be carried forward i
n
syste
m
s of equations F
n
et F
p
. The
equatio
n F
n
(
, N, P) = 0 is thus u
p
d
ated a
nd re
sol
v
ed to
tu
r
n
th
e un
kno
wn
N. This update and resolution
process is re
pe
ated alternately for
F
, F
n
, F
p
up
to f
u
ll
co
nv
erge
nce of
the
sy
stem
,
whe
r
e:
p
p
p
p
n
n
n
n
r
A
D
f
U
z
j
y
j
x
j
q
f
U
z
j
y
j
x
j
q
f
n
N
N
p
n
q
z
y
x
0
.
1
0
.
1
0
.
.
2
2
2
2
2
2
(6
)
Gumm
el algorithm
is represente
d by t
h
e
following
flowchart:
Following the
electrical parameters
at therm
odyna
m
i
c equilibrium
cal
culations, these values will be
inj
ected into t
h
e Newton
al
gorithm
to calculate the sam
e
param
e
te
rs u
nde
r
polarizati
o
n
,
[
1
0]
-[
11]
-
[
12]
, is
addi
ng t
o
each tim
e
one not
for t
h
e po
larizat
ion of 1KT
/q (Figure 2). Th
e
application of
Newt
on's m
e
th
od i
n
num
erical sim
u
lation o
f
de
vi
ces
leads to solve si
m
u
ltaneously
, Fn, Fp.
This is equi
valent to calculate
, N,
P as a
sol
u
tion
of a system
to 3N eq
uations
a
t
each poi
nt of the
net
w
ork
of three-dim
e
nsional
disc
retization.
Three
sy
stem
s
of
eq
uatio
ns
discretized
a
r
e
gr
ou
pe
d int
o
a
single
sy
stem
:
P
N
P
N
P
N
F
,
,
F
,
,
F
,
,
p
n
The sta
g
e
of linearization e
x
tende
d t
o
three
syste
m
s F
, F
n
, F
p
lead
s to
so
lv
e the
followi
ng system
:
.
,
whe
r
e:
: The
com
p
lete Jacobian
m
a
trix of
the system;
: The c
o
rrection
vector
This m
e
thod a
l
so called m
e
tho
d
c
o
u
p
led it
to re
du
ce c
o
n
s
idera
b
ly
the tim
e
of calculation, the
fix
was to deal with three-dim
e
nsional m
a
trices
or eac
h ele
m
ent of t
h
e m
a
trix
is a squa
re
m
a
trix of dim
e
nsio
n
3
*3
p
n
p
p
p
n
n
n
P
F
N
F
F
P
F
N
F
F
P
F
N
F
F
U
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN:
208
8-8
7
0
8
Three
-
Di
mensional Devices
Transp
ort Simul
a
tion
Lifetime and Rela
xa
tio
n … (No
u
a
r
So
ua
d Fa
d
ila
)
24
6
Figu
re
1.
R
e
sol
u
tion
by
t
h
e
de
cou
p
led
m
e
thod c
h
art
Figu
re
2.
O
r
ga
nizational st
ruc
t
ure
of
res
o
luti
on
by
the c
o
uple
d
m
e
thod
Th
us wa
s crea
ted a ne
w m
e
tho
d
c
o
m
b
ined
the G
u
m
m
el m
e
thod a
n
d N
e
wto
n
'
s
m
e
thod u
s
in
g n
o
n
-
uni
form
m
e
sh rather tight are
a
s with loa
d
s
of space a
n
d m
o
re coa
r
se in
ne
utral areas,
wit
h
a consi
d
era
b
le gain
in tim
e
per
f
o
r
m
a
nce that tu
r
n
s l
o
n
g
t
h
ree
-
d
i
m
e
nsional c
o
m
putation.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
088
-87
08
IJEC
E V
o
l. 5, No
. 2, A
p
ril 20
15
:
24
3 – 2
5
0
2
47
Figu
re 3.
Str
u
c
t
ure use
d
The structure has bee
n
studi
e
d is a structure p
ν
,
ν
layer
is slightly N, and
we treat the case
of
semiconductors to life and to relax, t
h
ese
oppo
site be
ha
viors a
r
e
obtaine
d by
a
d
justin
g
settings
τ
ne
et
τ
pe
, w
e
will assu
m
e
t
h
at th
e bo
undar
y
co
nd
itio
ns ar
e su
ch
th
at
th
e car
r
i
er
co
n
c
en
tr
ation
s
ar
e attach
ed
t
o
th
eir
therm
odynam
i
c equilibri
um
values, and we will use
two types of
boun
dary
conditions of Dirichl
e
t
on
u
nkn
own
p
l
an
s.
4.
RESULTS AND INTE
RPRETATION
Digital sim
u
lat
i
on
software is
written i
n
C++
,
DE
V C++
.
Therm
odynam
i
c
eq
uilibrium
study is an
essential step
for a fi
rst sim
u
lation
of a test s
t
ructur
e.
Furtherm
ore, the
sensitivity of Ne
w
t
on'
s m
e
thod to the
initial values
will lead us t
o
use re
sults from
the therm
odyna
mic equilibrium
, as initial
values.
In addition the
therm
odynam
i
c Equilibri
um
State can
gi
ve
us a first
vision of the distri
but
ions of
potential and free carriers
thr
o
u
g
h
a junct
i
on [
13]
Figure
4. Profile potential dist
ribution at
therm
odynam
i
c equilibrium
Figu
re
5.
Pr
o
f
ile h
o
les
di
stribution at therm
o
dynam
ic
equilibrium
Therm
odynam
i
c equilibrium
si
m
u
lation results are ob
tained by application
of the algorithm called
gum
m
e
l as decou
p
led m
e
thod
results fo
r the SC to life is
t
h
e sam
e
for th
e sc to relaxation a
nd they
co
m
p
ly
with the
p
h
y
s
ical param
e
ters g
i
ven t
o
the
p
re
gio
n
=
3.
1
0
14
cm
- 3
and the
re
gio
n
ν
N
d
=
1
.
5.
10
11
cm
-
3
to
better
see it will represent these resu
lts followi
ng the
x-axis
only
It is visible that space charge zone e
x
tends
m
a
in
ly
through the largest
st
ructure through the less
doped structure N m
a
y has th
e consequence
that conductio
n regim
e
will
be controlled
by an effect of contact
P
ν
,
or
dif
f
u
sio
n
pote
n
tial is 2
0
K
T
\q c
o
nf
or
m
e
d o
f
a
n
aly
t
ical calculation:
vd=l
o
g
(
(
N
A
*N
d
)
/
(n
i*n
i
))
,
[14]
By digital application found
V
d
=20.03KT/q
P
+
1µm
m
100
µm
10
µ
m
2µ
m
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN:
208
8-8
7
0
8
Three
-
Di
mensional Devices
Transp
ort Simul
a
tion
Lifetime and Rela
xa
tio
n … (No
u
a
r
So
ua
d Fa
d
ila
)
24
8
Figure
6. Profile holders
distribu
tion at t
h
erm
ody
nam
i
c equilibrium
The a
ppea
r
a
n
c
e
s of
curve
s
as well as the
orders
of
m
a
gnitude
o
f
di
ffe
ren
t
conce
n
tratio
n
s
determ
ined
by calculating
3D are
physically correct.
Figure
7. Profile of the
distri
bution potential
for a
pola
r
ization
o
f
10
K
T
\q
fo
r a
lifetim
e SC
Figu
re
8.
Pr
o
f
ile o
f
the
de
nsity
distrib
u
tio
n
of
electro
ns
f
o
r
a
po
lar
i
zatio
n of
10
KT\q
for a lifetim
e SC
Figu
re
9.
Pr
o
f
ile o
f
the
de
nsity
distrib
u
tio
n
of
the
holes
f
o
r
a
pol
arization
o
f
10
KT\q
f
o
r
lifetim
e SC
Figu
re 1
0
. Pr
of
ile
of
t
h
e den
s
ity
distributi
o
n
of
the
electrons
f
o
r
a pola
r
ization o
f
10
K
T
\q fo
r relaxation
SC
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
088
-87
08
IJEC
E V
o
l. 5, No
. 2, A
p
ril 20
15
:
24
3 – 2
5
0
2
49
Figu
re.
1
1
Pr
of
ile of t
h
e
den
s
ity
distributi
o
n
of
electrons
f
o
r
a pola
r
ization o
f
10
K
T
\q SC
Figu
re 1
2
. Pr
of
ile
of
t
h
e den
s
ity
distributi
o
n
fo
r
a
relaxation
of the holes for a
po
larization of
10
KT\q for a
relaxation SC
Plots a
r
e
give
n accordi
n
g to
m
e
sh
point
and
not
by le
ngth; the
P
+
regi
o
n
is disc
retized
o
n
1
5
poi
nts
an
d r
e
g
i
on
N
o
n
135
po
in
ts, th
us allowing
to better presenting
th
e s
p
ac
e
c
h
ar
g
e
zo
ne
.
The
plot a
r
e giv
e
n
according to m
e
sh points
a
n
d not
by the length, P
+
re
gio
n
is discretized
on
15
p
o
ints an
d t
h
e N
o
n
1
35
p
o
ints,
allowing
better present a
r
ea
of space
cha
r
ge
.
In
o
r
de
r t
o
e
n
able com
p
aris
ons
b
e
twee
n
d
i
ffere
nt
distrib
u
tion
cu
r
v
es
o
f
f
r
ee ca
rrie
r
s
in di
ffe
rent
polarizations, and the ability to
draw i
n
the sa
m
e
gr
aph, we chose
to
dr
aw in
x- axis
Figu
re 1
3
. Pr
of
ile
of
t
h
e den
s
ity
distributi
o
n
of
electrons for a
lifetim
e SC
Figu
re 1
4
. Pr
of
ile
of
t
h
e den
s
ity
distributi
o
n
of
the
holes for a lifet
im
e SC
Figu
re 1
5
. Pr
of
ile
of
t
h
e den
s
ity
distributi
o
n
of
electrons for a
relaxation SC
Figu
re 1
6
. Pr
of
ile
of
t
h
e den
s
ity
distributi
o
n
of
the
holes
f
o
r
a rela
xation
SC
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN:
208
8-8
7
0
8
Three
-
Di
mensional Devices
Transp
ort Simul
a
tion
Lifetime and Rela
xa
tio
n … (No
u
a
r
So
ua
d Fa
d
ila
)
25
0
5.
INTERPRET
ATION OF
RESULTS
All results fo
un
d u
n
d
er p
o
l
a
rization are
obtaine
d
by
applicatio
n of the Ne
wto
n
algo
rithm
.
The
str
u
ctur
e is equ
i
pp
ed with only o
n
e
contact inj
ecting.
For t
h
e lifetim
e se
m
i
cond
ucto
r u
nde
r
polarization note
a re
duction in the
widt
h
of
the area
of space cha
r
ge
from
equilibrium
therm
odynam
i
cs but wi
dely m
o
re
extensi
v
e tha
n
that o
f
the
rela
xation
sem
i
conduct
o
r
.
The e
x
tensi
o
n of t
h
e area
of space in t
h
e region
ν
c
h
a
r
ge
deri
ves esse
nt
ially
to free carriers
a
n
d
io
n
i
zed don
or
,
n
o
t
e th
at thr
oug
hou
t th
e
structure
for a lifetime se
m
i
conductor
∆
n =
∆
p, t
h
e t
r
end is towards the
ne
utra
lization
of the
space cha
r
ge
, t
h
e
valu
es of
n
a
n
d p
i
n
neut
ral zone
s
increase c
h
ec
ki
ng the e
q
uation
For
relaxati
on
sem
i
cond
ucto
r
fo
r lo
w p
o
lari
zation, t
h
e val
u
es
of
n an
d
p
in ne
utral zo
n
e
s kee
p
s the
sam
e
values as that of therm
odyna
m
i
c equilibrium
,
for voltages lower than 20
KT\
q
the conduction is
co
n
t
r
o
lled b
y
th
e eff
ect of
co
n
t
act if
µn
> µp
,
th
e
f
r
o
n
t
o
f
r
e
co
m
b
in
ati
o
n o
c
cu
rs in
neig
hb
orh
ood
of
the
contact P the right of the
front of recom
b
ination we
ob
ser
v
ed de
pletion of h
o
les.
F
o
r v
o
ltages
in or
de
r
to 2
0
KT\q is reached alm
o
st fla
t
bands re
gim
e
, there is
the sa
m
e
re
m
a
rks and an inc
r
ea
se in the density of
electrons a
n
d holes in P si
de
and
ν
si
de re
spectively
,
u
n
like the relaxati
on sem
i
cond
uc
tor,
whe
r
e we
note a
fr
ont o
f
rec
o
m
b
inatio
n ov
er n
ear
the
P regi
o
n
..
6.
CO
NCL
USI
O
N
The
pu
rp
ose
of
o
u
r st
udy
i
s
the com
p
ari
s
on
o
f
tw
o se
m
i
cond
ucto
rs
havi
ng t
h
e sa
m
e
electrical
param
e
ters exc
e
pt the
relaxati
on
tim
e
dielectric, f
o
r
the lifet
im
e sem
i
condu
ctor
nt
=
pt
=10
-8
s and a
relaxation
sem
i
cond
ucto
r
nt
=
pt
=1
0
-11
s with a
rd
=4
.8
10
-9
s. F
o
r the
ν
Z
o
ne t
h
is a
d
justm
e
nt leads t
o
a
di
ffe
rent
b
e
h
a
v
i
or
fo
r th
e two sem
i
c
o
ndu
ctor
s typ
e
s.
We also
note a great
difference
bet
w
ee
n t
h
e sem
i
conduct
o
r
Relaxation and the insulators
whose
free-
carrier concentrati
o
n is
negligi
b
le.
REFERE
NC
ES
[1]
R. Ardebili, J
.
C. Nathalie. Stud
y b
y
num
er
ica
l
si
m
u
lation of t
r
an
sport in sem
i
co
n
ductors
in th
e pr
es
ence
of c
e
nter
deep.
Thesis Ph
D Centre M
ontp
e
llier electronics
, (1992)
[2]
H. Mathi
e
u.
Physics
o
f
semico
nductors and
electron
i
c
compone
nts Courses and exer
cises
cor
r
ected.
6th edition
Masson
, Paris 2
009.
[3]
F.S. Nouar. Three-dimensiona
l modeling of transport in PN junctions in
the p
r
es
ence of deep
centers
. Th
es
is
of
Univers
i
t
y
of
Dj
ilal
i
Liab
ès
,
Sid
i
Bel Abbès, 200
2
[4]
J.D. Chatel
in. Di
spositifs of sem
i
c
onductor
.
Edition Georgi
, 1979
[5]
R. M
e
nezl
a. CL
AC 3D program three-dim
e
nsion
a
l resolution of Poisson’s equation
.
Thesis Phd, Ecole Centrale
de
Ly
on
, no
85-05,
1990
[6]
Jean-Pierre Corr
iou
.
Numerical
methods and optimization
:
th
eor
y
and pr
actice fo
r the
engin
eer
.
É
d
ition Paris
te
c et
doc
,
2010
[7]
M. Kemp, C.G.
Tannous, M. Meunier
.
Am
orphous silicon d
e
vice sim
u
lation b
y
an adapt
e
d Gum
m
e
l m
e
thod.
I
E
EE
transation on
electron de
vices, Pergamon press
,
vol 27, no 4
,
pp
.319-328, 1987
[8]
O.
E.
Akcasu
.
C
onvergence pro
p
rieties of N
e
wt
on’s methode f
o
r the solution
of the semicon
ductor
transport
equat
i
ons and hybrid solut
i
on te
chniques for m
u
ltidim
ensi
ona
l sim
u
lation of VLSI devices Solid state el
ectron
i
cs
.
Pergamon press
, vol 27
.
no 4
,
pp
349-328,1987
.
[9]
M. Khadraoui.
Use of coupled
and decoup
led
methods for th
ree-dimensional simulation of
d
e
vices at junction P
N
.
Thesis, Universit
y
Dji
llal
i
Liabès
, Sidi B
e
l Abb
è
,
1998
[10]
A. Resfa. 3D mo
deling of
this breakdown b
y
avalanche in
th
e stru
ctures of the PN junction of GaAs
insulating
semi
junction
presen
ted centers deep.
Thesis, Universit
y
of
Djila
li
Liabès
, sidi bel Abbès, 2005
[11]
C.S. Raff
erty
, M.R.
Pinto
and W. Dutton.
I
t
erativ
e methods in semi
conductors device simulation
. IEEE transation
on el
ectr
o
n
dev
i
ces
, vo
l
ED-32 n
o
100, pp. 2018-
2027, Octob
e
r 1
985
[12]
Bouabddellah Badra. Simulatio
n
numer
ical of conduction in the volume of
components to semiconductors to
relax
a
tion
.
T
h
es
i
s
PhD Facu
lt
y of
engin
eer
s
c
ienc
e
, Univ
ersity
S
i
d
i
Bel Abbès Alg
e
ria, 2004
[13]
Claude Delan
n
o
y
. C++ for
C pro
g
rammers
.
Edit
i
on Eyrol
l
es
, paris 2007
[14]
A. Bench
i
heb
.
Modélisation
d’
un tr
ansistor b
i
p
o
lair
e de puissance.
Thèse de magister université de Constan
tin
e
,
1996
Evaluation Warning : The document was created with Spire.PDF for Python.