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International Workshop of Computational Electronics Purdue University, 26th of October 2004
Treatment of Point Defects in Nanowire Treatment of Point Defects in Nanowire MOSFETs Using the Nonequilibrium Green’s MOSFETs Using the Nonequilibrium Green’s
Function Formalism Function Formalism
M. Bescond, J.L. Autran*, N. Cavassilas, D. Munteanu, and M. Lannoo
Laboratoire Matériaux et Microélectronique de Provence UMR CNRS 6137 - Marseille/Toulon (France) - www.l2mp.fr
* Also Institut Universitaire de France (IUF)
OutlineOutline Introduction
MOSFETs downscaling: statistical fluctuations of doping impurity positions
3D Quantum simulation of point defects in nanowire transistors Nonequilibrium Green Function formalism: Mode-Space approach Treatment of point defects
Results: influence of the impurity location and type Energy subbands Transverse modes Current characteristics
Conclusions and perspectives
Dimensions of nanowire MOSFETsVG
VDVS
L
DRAINSOURCE CHANNEL
VG
Gates
GatesOxide
W
XZ
Y
VG
VDVS
L
DRAINSOURCE CHANNEL
VG
Gates
GatesOxide
W
XZ
Y
XZ
Y
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
a) b)
VG
VDVS
L
DRAINSOURCE CHANNEL
VG
Gates
GatesOxide
W
XZ
Y
VG
VDVS
L
DRAINSOURCE CHANNEL
VG
Gates
GatesOxide
W
XZ
Y
XZ
Y
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
a) b)
WSiVG
VDVS
L
DRAINSOURCE CHANNEL
VG
Gates
GatesOxide
W
XZ
Y
VG
VDVS
L
DRAINSOURCE CHANNEL
VG
Gates
GatesOxide
W
XZ
Y
XZ
Y
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
a) b)
VG
VDVS
L
DRAINSOURCE CHANNEL
VG
Gates
GatesOxide
W
XZ
Y
VG
VDVS
L
DRAINSOURCE CHANNEL
VG
Gates
GatesOxide
W
XZ
Y
XZ
Y
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
a) b)
WSi
p-type channel region: Volume = LWSiTSi=1053=150 nm3
If doping concentration=1019 cm-3 1.5 impurity on average.
Discrete distribution and statistical location.
Effect of the impurity type and location.
Source and drain region: continuous doping of 1020 cm-3.
Dimensions: L=8 nm, WSi=3 nm, and TSi=3 nm, TOX=1 nm.
Channel region: discrete doping of 1019 cm-3, with 1 impurity on average.
Nonequilibrium Green’s function Nonequilibrium Green’s function formalismformalism
Point Defect Treatment
The 3D Mode Space Approach*
The 3D Schrödinger = 2D (confinement) + 1D ( transport)
VG
VDVS
L
DRAINSOURCE CHANNEL
VG
Gates
GatesEOT
W
VG
VDVS
L
DRAINSOURCE CHANNEL
VG
Gates
GatesEOT
W
2D (confinement)
1D (transport)
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
Si TSi
SiO2
TOX
i,nψ
ith eigenstate of the nth atomic plan
Hypothesis: n,i is constant along the transport axis.
* J. Wang et al. J. Appl. Phys. 96, 2192 (2004).
3 2
3
2
11
YX
Z
3 2
3
2
11
3 2
3
2
11
YX
Z
YX
Z
3 2
3
2
11
YX
Z
3 2
3
2
11
3 2
3
2
11
YX
Z
YX
Z
Y
X
Z
Y
X
Z
The 3D Mode Space Approach Electron distribution along subbands (valley (010)):
Simplified tight-binding approach: cubic lattice with ax, ay, az.
- 1 orbital/atom: : position z=laz, y=may, x=nax.
- Interactions between first neighbors.
1st
2nd
3rd
y
0 5 10 15 20-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
EFD
EFS
L=8 nmV
DS=0.4 V
VG=0 VP
oten
tial e
nerg
y (e
V)
X (nm) 1st
2nd
3rd
y
0 5 10 15 20-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
EFD
EFS
L=8 nmV
DS=0.4 V
VG=0 VP
oten
tial e
nerg
y (e
V)
X (nm)
i,nψ
i,nε
( )εGi
: Transverse eigenstate
: Transverse eigenvalue
: 1D Green function
For each subband i:
i=1
i=2
i=3
n,m,l
Point defect description
Treated as a macroscopic variation
Included in the self-consistent mode-space approach without coupling the
electron subbands
Treated as a localized variation = Chemical structure
Included in the real space approach based on the Dyson equation.
[G=(I-G0V)-1G0]
Point Defect = On-site Potential + Coulomb Tail
Energy
Impurity
r
erV
Si
Lr
Coul
c
0
/
4
1
dU
Treatment of on-site potential After achieving self-consistence including the Coulomb Tail, the device is subdivided at the point defect location:
Source Drain
S1 S2
X
Y
Z
Y
Source Drain
S1 S2
X
Y
Z
Y
Calculation of the Green’s functions of the surfaces S1 and S2:
( ) ( ) ',',',,=',',',, 111,','1
,,1111111111 1111∑ nmlψψεGψψnmlnmlεGnml ininii
ininS
( ) ( ) ',',',,=',',',, 222,','2
,,2222222222 2222∑ nmlψψεGψψnmlnmlεGnml ininii
ininS
Treatment of on-site potential Ud is then included using the Dyson equation:
2
10
'0
0
S
SS
G
GG
0
0
21
12
H
HV
2
1-
22 -' SSS GUGIG
DS ffVNVNTrd
eI 1121
0221211
0111
4
00 Im
1SGN
100
VVGGI SS
Intra-atomic potential matrix
Retarded Green function of the uncoupled system:
Calculation of the current*:
* M. Bescond et al., Solid-State Electron. 48, 567 (2004).
ResultsResults
Influence of point defect
Simulation results Electronic subbands: Effect of the Coulombic potential (valley (010))
Source Drain
1 nmX
Y
Z
Y
Source DrainSource Drain
1 nmX
Y
Z
Y
Subband profile is affected by the impurity. Subbands are still independant: justification of the mode-space approach. Acceptor impurity increases the channel barrier.
Simulation results Evolution the 1st confinement eigenstate (valley (010)):
Highest variations of the eigenstate with centered impurity.
Scalar product : weak variations.95.0≈ψψcent,1free,1
Defect free Centered defect
2
cent,1ψ
2
free,1ψ
z
Defect in the corner
2
nocent,1ψ
Simulation results First subband profile and current characteristics:
Defect in the corner: weak influence on the subband profile. Defect free: highest current. Centered defect: lowest current. Defect in the corner: intermediate behavior: current decrease of 50%. Variation of the subthreshold slope.
5 10 15-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
Defect free Centered defect Non centered defectF
irst s
ubba
nd p
rofil
e (e
V)
Source-drain axis (nm)
ZZ
ZZ
5 10 15-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
Defect free Centered defect Non centered defectF
irst s
ubba
nd p
rofil
e (e
V)
Source-drain axis (nm)
ZZ
ZZ
5 10 15-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
Defect free Centered defect Non centered defectF
irst s
ubba
nd p
rofil
e (e
V)
Source-drain axis (nm)
ZZ
ZZ
0.0 0.2 0.4 0.6 0.8
10-11
10-9
10-7
10-5
I D (
A)
VG (V)
Defect free Defect in the center Defect in the corner
VG=0 V VDS=0.4 V
Ud=2 eV
Simulation results Influence of Coulomb potential:
On-site potential defect does not affect the total current. Coulombic potential has the most significant impact. Electrons can be transmitted through the unperturbated neighboring atoms.
VDS=0.4 V
Conclusion
Modeling of electron-ion interaction based on the NEGF formalism.
Study of the effect the acceptor impurity in terms of physical properties.
Centered impurity involves a significant degradation of the current.
Not only a shift of the current but rather a subthreshold slope variation.
The Coulomb potential has a prevalent rule compared to the on-site potential of the impurity.
Treatment of donor impurities in source and drain.
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