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c Bandsues.
odeling6 Crolles
nd H. Jaouen
Q&TPS
Strained Silicon, ElectroniStructure and Related Is
STMicroelectronics, Q&TPS, Device M850 rue Jean Monnet, BP 16, F-3892
CEDEX, France
D. Rideau, F. Gilibert, M. Minondo, C. Tavernier a
D. RIDEAU
2/35
®
8 APRIL 2005 MOS-AK STRASBOURG
Q&TPS
OUTLOOK (1/4)
Current & Capacitances
STRAIN Matrix
INPUT
DESIRED VALUES
What to do?
Strain matrix
Ansys ...
?
D. RIDEAU
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®
8 APRIL 2005 MOS-AK STRASBOURG
Q&TPS
Electronic Structure upon Strain
Empirical: TB KP EPMAbinitio: LDA RPA GW
Dispersion relation and Gaps
OUTLOOK (2/4)
D. RIDEAU
4/35
®
8 APRIL 2005 MOS-AK STRASBOURG
ng Times: “Fermi Golden rule”“modified” GR Algorithm
Q&TPS
Electronic Structure upon Strain
Empirical: TB KP EPM
Integration over Brillouin Zone
ScatteriDensity of States
Carrier DensityMean Carrier Energy
Linear Response Theory
GR Algorithm
Fermi Dirac Statistics
Kubo-Greenwood formula
Mobility
Abinitio: LDA RPA GW
Dispersion relation and Gaps
G. Gilat and J. Raubenheimer, PR 144, 390 (1966)
OUTLOOK (3/4)
D. RIDEAU
5/35
®
8 APRIL 2005 MOS-AK STRASBOURG
ng Times: “Fermi Golden rule”
MC
“modified” GR Algorithm
Mobility
µαI
Current & Capacitances
Q&TPS
Electronic Structure upon Strain
Empirical: TB KP EPM
Integration over Brillouin Zone
ScatteriDensity of States
Carrier DensityMean Carrier Energy
Linear Response Theory
Poisson Schrodinger
GR Algorithm
Fermi Dirac Statistics
Kubo-Greenwood formula
Current & Capacitances
Mobility
Semiconductor Equation
Abinitio: LDA RPA GW
Dispersion relation and Gaps
G. Gilat and J. Raubenheimer, PR 144, 390 (1966)
Compact Models
OUTLOOK (4/4)
D. RIDEAU
6/35
®
8 APRIL 2005 MOS-AK STRASBOURG
ethod
Q&TPS
Review for Electronic Band Structure
Ab initio vs Empirical methods
A strain example: Si on SiGe buffer
Lower Dimension Aproximation
6X6 KP and Effective mass Hamiltonian
1Review for Electronic Band Structure M
D. RIDEAU
7/35
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8 APRIL 2005 MOS-AK STRASBOURG
Block Functions
itting parameters
in DamoclesY.M. Niquet et al, PRB 62 5109 (2000) and
(includes SO)
C. Tserbak et al, PRB 47 7104 (1993)
KPEPM
eigenvalues
od
Q&TPS
Troullier-Martins psp from fhi98PPHartwingsen psp including SO
Schrodinger Equation
Wave function BasisPlane WavesGaussian-orbital
Matrix ElementsEvaluation
FAb initio
Kohn-Sham Scheme GW correction
HKS Ec= Vion VHF+ + … Σ ψnk⋅ r3
d∫+
C.Hartwingsen et al, PRB, 58 3641 (1998)
EPM “Best” TB “Best” in
KP (UTOX)
DFT + LDA
TB
Self-consistentevaluation
Simple eigenvalueproblem
Electronic Band Structure, Overlap integral...
Hψnk Enkψnk=
Review for Electronic Band Structure Meth
D. RIDEAU
8/35
®
8 APRIL 2005 MOS-AK STRASBOURG
K G
ABINIT V4.4.3
3641 (1998))
Q&TPS
L G X W K’,U L W X
−10
−8
−6
−4
−2
0
2
4
6E
NE
RG
Y (
eV)
WAVE VECTOR
GWLDA KSS
PSP: Hartwingsen psp (C.Hartwingsen et al, Phys. Rev. B, 58
Ab Initio: vs
D. RIDEAU
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®
8 APRIL 2005 MOS-AK STRASBOURG
K G
HYS. REV. B 14, 556 (1976)
Q&TPS
L G X W K’,U L W X
−10
−8
−6
−4
−2
0
2
4
6
EN
ER
GY
(eV
)
WAVE VECTOR
EPM (local)
“NON LOCAL EFFECT”
UTOX AFTER J.R. CHELIKOWSKY AND M.L. COHEN, P
D. RIDEAU
10/35
®
8 APRIL 2005 MOS-AK STRASBOURG
KPGW
K G
Q&TPS
KPUTOX
L G X W K’,U L W X
−10
−8
−6
−4
−2
0
2
4
6
EN
ER
GY
(eV
)
WAVE VECTOR
D. RIDEAU
11/35
®
8 APRIL 2005 MOS-AK STRASBOURG
40
(110)
ass Hamiltonian
Structure
Q&TPS
0 20
k(108m−1)
−20 0 20−900
−800
−700
−600
−500
−400
−300
−200
−100
0
100
(111) k(108m−1) (100)
Ek (m
eV)
Valence Bands: KP and Effective M
KPMASS
Effective mass approximation for Electronic Band
D. RIDEAU
12/35
®
8 APRIL 2005 MOS-AK STRASBOURG
Mass Hamiltonian
Q&TPS
Hole “curvature mass” for Effective
0 20 40 60 800.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
wafer in−plane orientation (deg.)
Hol
e C
ondu
ctio
n m
ass
(m0
units
)
hhlhsh
D. RIDEAU
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®
8 APRIL 2005 MOS-AK STRASBOURG
e Mass Hamiltonian
Q&TPS
−10 −5 0 5 10
50
100
150
200
250
300
(100) k(108 m−1) (001)
Ek
(me
V)
∆2∆4
Conduction Bands: KP and Effectiv
MASSKP
D. RIDEAU
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8 APRIL 2005 MOS-AK STRASBOURG
0 0.1 0.2 0.3 0.40
02
04
06
08
01
12
14
16
x
Si
ε = Strain
Q&TPS
0.0
0.0
0.0
0.0
0.
0.0
0.0
0.0
exx
SiGe lattice larger than Si lattice
STRAIN (STUDIED CASE)
D. RIDEAU
15/35
®
8 APRIL 2005 MOS-AK STRASBOURG
G
EQUIVALENT VALLEY
Q&TPS
L G X,Z G Y,X−3
−2
−1
0
1
2
3
4
ENER
GY
(eV)
RELAXED SILICON
D. RIDEAU
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®
8 APRIL 2005 MOS-AK STRASBOURG
G
Q&TPS
L G X,Z G Y,X−3
−2
−1
0
1
2
3
4
ENER
GY
(eV)
2% TENSILE
D. RIDEAU
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®
8 APRIL 2005 MOS-AK STRASBOURG
AIN
CONDUCTION BANDS
VALENCE BANDS
Q&TPS
ENERGY SHIFT VS . STR
0.97 0.98 0.99 1 1.01 1.02 1.03−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
a||/a
0
EN
ER
GY
(eV
)∆2
∆4
LH
HH
SH
D. RIDEAU
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®
8 APRIL 2005 MOS-AK STRASBOURG
1% Compressive
0 20 40
k(108m−1) (110)
−20 0 20−900
−800
−700
−600
−500
−400
−300
−200
−100
0
100
(111) k(108m−1) (100)
Ek
(meV
)
Q&TPSVALENCE BANDS
1% Tensile
0 20 40
k(108m−1) (110)
−20 0 20−900
−800
−700
−600
−500
−400
−300
−200
−100
0
100
(111) k(108m−1) (100)
Ek
(meV
)
0 20 40
k(108m−1) (110)
−20 0 20−900
−800
−700
−600
−500
−400
−300
−200
−100
0
100
(111) k(108m−1) (100)
Ek
(meV
)
Relaxed
D. RIDEAU
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®
8 APRIL 2005 MOS-AK STRASBOURG
ZONE
Q&TPS
Density Of States and DOS masses
Scattering Rates
Carrier Density
1
2
3
INTEGRATION OVER BRILLOUIN
Electronic Structure
Empirical: TB KP EPMAbinitio: LDA RPA GW
Dispersion relation and Gaps
D. RIDEAU
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®
8 APRIL 2005 MOS-AK STRASBOURG
Q&TPS
DENSITY OF STATES
ρE E( ) δk BZ∈
∑ E En k( )–[ ]n∑=
W
L
K
X
U
1/48 1/8
G. Gilat and J. Raubenheimer, PR 144, 390 (1966)
DENSITY OF STATES
SYMMETRIES
Electronic Structure
Empirical: TB KP EPMAbinitio: LDA RPA GW
Dispersion relation and Gaps
INTEGRATION
D. RIDEAU
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®
8 APRIL 2005 MOS-AK STRASBOURG
−4 −2 0 2 4 6E (eV)
Q&TPS
−12 −10 −8 −60
1
2
3
4
5
g E (
1022
cm
−3 e
V−
1 )
DENSITY OF STATES
−12 −10 −8 −6 −4 −2 0 2 4 60
1
2
3
4
5
g E (
1022
cm
−3 e
V−
1 )
E (eV)
EPM (LINES)KP (LINES)
GW (DASHED LINES)GW (DASHED LINES)
D. RIDEAU
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8 APRIL 2005 MOS-AK STRASBOURG
EFF MASSES)
CTRONS
Q&TPS
DENSITY OF STATES (FB VS.
ELEHOLES
D. RIDEAU
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®
8 APRIL 2005 MOS-AK STRASBOURG
ture masses and valence band
EPMd
1.170.21
0.916
4.270.315
1.3867
Q&TPS
MASSES
Table 1 Experimental and theoretical band gap, conduction band curvaLuttinger parameters for Silicon.
Exp.a
k.p GWc
(eV) 1.170 1.17 1.1
(mo)0.191 0.194 0.191
(mo)0.916 0.916 0.921
4.27 4.27b 4.27 0.32 0.315b 0.315
1.458 1.386b 1.386
Eg
mt
ml
γ1
γ2
γ3
a Ref.[11]; b Fit for the 6-level k.p;c with ABINIT V4.3.3 [3]; d Ref. [17].
D. RIDEAU
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8 APRIL 2005 MOS-AK STRASBOURG
ALENCE BANDS)
0 200 400 600 8000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
msh
(D
OS
)
T (K)
x=0x=0.1x=0.2x=0.3
Q&TPS
DOS MASSES INSI/SIGE (V
0 200 400 600 8000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
m(D
OS
)
T (K)
x=0x=0.1x=0.2x=0.3
0 200 400 600 8000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
mlh
(D
OS
)
T (K)
x=0x=0.1x=0.2x=0.3
KP (UTOX)
D. RIDEAU
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®
8 APRIL 2005 MOS-AK STRASBOURG
IN :
.1 0.2 0.3 0.4 0.5−E
F (eV)
Q&TPS
CARRIER DENSITY VS. STRA
−0.2 −0.1 0 010
15
1016
1017
1018
1019
1020
1021
1022
EV
p (/
cm3)
FB 1.5% tensileOB 1.5% tensileFB relaxed SiOB relaxed Si
−0.2 −0.1 0 0.1 0.2 0.3 0.4 0.510
15
1016
1017
1018
1019
1020
1021
1022
EF−E
C (eV)
n (/c
m3)
FB 1.5% tensileOB 1.5% tensileFB relaxed SiOB relaxed Si
UTOX AFTER M. V. FISCHETTI ET AL . IN DAMOCLES
D. RIDEAU
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®
8 APRIL 2005 MOS-AK STRASBOURG
ENTS)
0 0.5 1 1.5VG (V)
PMOS
x=0.2
relaxed
Q&TPS
CAPACITANCE (MEASUREM
−1.5 −1 −0.5 0 0.5 1
0.006
0.008
0.01
0.012
0.014
0.016
0.018
VG (V)
C (
µF/m
m2)
−1 −0.5
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
C (
µF/m
m2)
NMOS
D. RIDEAU
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®
8 APRIL 2005 MOS-AK STRASBOURG
D CURVES)
0 0.5 1 1.5VG (V)
Q&TPS
−1.5 −1 −0.5 0 0.5 1
0.006
0.008
0.01
0.012
0.014
0.016
0.018
VG (V)
C (
µF/m
m2)
CAPACITANCE (SIMULATE
Charge Sheet Model Density Gradient
x=0.2
relaxed
−1 −0.50.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
C (
µF/m
m2)
D. RIDEAU
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®
8 APRIL 2005 MOS-AK STRASBOURG
iile
Q&TPS
CURRENT (LOW FIELDS )
1 1.2 1.4 1.6 1.8 2 2.2
10−11
10−10
10−9
10−8
10−7
10−6
10−5
VG−VFB(V)
ID(A
/µm
2)
UTOX relaxed SUTOX 1.5% tens
Recommended