BSIM: Industry Standard Compact MOSFET Modelshome.iitk.ac.in/~chauhan/ESSDERC2012_YSChauhan.pdf ·...
61
Y. S. Chauhan 1,2 , S. Venugopalan 2 , M. A. Karim 2 , S. Khandelwal 2 , N. Paydavosi 2 , P. Thakur 2 , A. Niknejad 2 and C. Hu 2 1 IIT Kanpur, India 2 UC Berkeley, USA BSIM: Industry Standard Compact MOSFET Models Sept. 19 th , 2012 ESSDERC Bordeaux, France
Transcript of BSIM: Industry Standard Compact MOSFET Modelshome.iitk.ac.in/~chauhan/ESSDERC2012_YSChauhan.pdf ·...
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Y. S. Chauhan1,2, S. Venugopalan2, M. A. Karim2, S. Khandelwal2, N. Paydavosi2, P.
Thakur2, A. Niknejad2 and C. Hu21IIT Kanpur, India2UC Berkeley, USA
BSIM: Industry Standard Compact MOSFET Models
Sept. 19th, 2012
ESSDERC Bordeaux, France
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SPICE Transistor Modeling
2
Simulation Time ~ 10s per DC data point No complex numerical method
allowed
Accuracy requirements ~ 1% RMS Error after fitting
Excellent Convergence
Example: BSIM-CMG 5,000 lines of VA code
50+ parameters
Open-source softwareimplemented in major EDA tools
Medium of information exchange
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Compact MOSFET Modeling Approaches Threshold Voltage based Models (e.g. BSIM3,
BSIM4) Fully Analytical solution (easy to implement) Fast Currents expressed as functions of Voltages
Different equations for Sub-threshold and above-threshold Linear/saturation regions Use interpolation function to get smooth current
3
( ) 21
2 gs
WI C V V V VLds ox th ds ds
=
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Compact MOSFET Modeling Approaches Surface Potential based Models (e.g. PSP, HiSim,
BSIM-CMG, BSIM-IMG)
Implicit equation is solved either iteratively or analytically Might be slower than threshold voltage based models
Charge based Models (e.g. BSIM5, BSIM6, EKV) Solve for charge instead of surface potential No iterations Faster than Surface Potential based approach with similar
accuracy in charge/current4
SVV
V
tV
toxSsiox
siSFBG
t
S
t
CHF
t
S
eeVeVCsignQCQVV +
+
==
+
11)(,2
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Outline
History of BSIM Models
BSIM6 Model
BSIM-CMG Model
BSIM-IMG Model
5
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History of BSIM Models BSIM1 Model
Defined as an engineering model (vs a purely physical model) Focus on the implementation in circuit simulator Only a fast demonstration for DC simulation (no attention on
derivatives)
BSIM2 Model A semi-physical (semi-empirical) model Improvement over BSIM1 to include better fitting to output
resistance and other first order derivatives Huge attention placed on parameter extraction methodology Still being used in many companies as internal model due to its
fitting ability and simple parameter extraction6
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History of BSIM Models BSIM3 Model
Starts as a simple physical model with very few parameters First time Continuous I-V and derivatives for fast convergence The 3rd version (BSIM3v3) becomes industry standard Need to fit many technologies from different foundries many
new parameters added Example Need to have fitting parameters Due to difficult to describe structural detail
7Source: Mansun Chan
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History of BSIM Models BSIM4 model
Started as model for statistical simulation Priority on physical effects (gate current, mobility models etc.) Added gate and body resistance networks to emphasize accuracy
on RF simulation Industry standard in 2000 and most widely used model by
semiconductor industry Provides better fitting with more number of parameters
BSIM-SOI Parallel work went on SOI modeling PD/FD/DD Real device effects same as BSIM3/BSIM4 Industry standard in 2002
8
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BSIM Family of Compact Device Models
9
BSIM: Berkeley Short-channel IGFET Model
1990 20102000 20051995
BSIM1,2 BSIM3
BSIM4
BSIMSOI
BSIM-MG
BSIM5 BSIM6
Bulk MOSFET
Silicon on Insulator
MOSFET
Multi-Gate MOSFET
New
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BSIM6: Charge based MOSFET model BSIM6 Next BSIM Bulk MOSFET model
Charge based core derived from Poissons solution Real device effects (SCE, CLM etc.) from BSIM4 Parameter names matched to BSIM4
Physical Capacitance model Short channel CVVelocity saturation & CLM
Symmetry Currents & derivatives are symmetric @ VDS=0 Capacitances & derivatives are symmetric @VDS=0 Provide accurate results in Harmonic Distortion simulation
Continuous in all regions of operations Better Statistical Modeling using physical parameters
10
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Physics of BSIM6 Model Gauss Law
Poissons solution for long channel MOSFET
Bulk charge density is given by
Combining these, we have
11
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Physics of BSIM6 Model Defining Pinch-off potential P = S, when
Qi=0
For S> few Vt, we have
Inversion Charge linearization12
P is evaluated from implicit equation
nq is the slope factor
*Ref.: Tsividis book & J.M. Sallese et al., Solid State Electronics
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Physics of BSIM6 Model Using linearization approach and normalization
No approximation to solve the charge equation compared to other models.
Solved the charge equation analytically13
chfpipiiqq
i vqqqnn
q =+
+++
2222
22ln)ln(
ACM/EKV/BSIM5 ignored the circled term
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Drain current with velocity saturation Drain current
Mobility model (ensures symmetry)
Using charge linearization & normalization
14
+
=+=
dxdQV
dxdQWIII iTSidiffdriftD
( )LvV
VCL
Wn
IiVCn
QqnCQ
sat
tvc
toxvq
Dd
Toxq
iSPq
ox
i
2,2
,2
,2
==
==
+
+
=dx
dQVdx
dQW
dxd
v
I iTSiS
sat
v
vD 2
1
( ) ( )( )[ ]
++
++=
2
22
1121
dsc
ddssd
qq
qqqqi
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Normalized Qi-VG & derivatives
15
qi vs VG1st derivative
2nd derivative
3rd derivative
Red Numerical Surf. Pot. model
Blue BSIM6 model
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Normalized IDS-VGS & derivatives
16
1st derivative
IDS vs VG
2nd derivative 3rd derivativeRed Numerical Surf. Pot. model
Blue BSIM6 model
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Mobility Model Mobility model has been adopted from BSIM4
17
( ) UCS
bs
is
EUeffbsx
eff
qq
UDEVUCUA
U
+
+++=
121
1
0
MV/cm
where
BSIM4
BSIM6
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Saturation Voltage Vdsat Vds to Vdsat BSIM4 formulation causes asymmetry
New Vdsat evaluation:
18
eff
teffsc LVSAT
V
=
2
)1(2112
1 2
sc
sscdsat
q
qqKSATIVq++
+=
+
==
122
ln22q
qdsatqdsatdsatfp
t
dsatdsat n
nqnqq
VVv
DELTADELTA
sdsat
ds
dsdseff
VVV
VV
+
=/1
1
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CV Model
19
Physical Capacitance Model Poly-depletion & Quantum Mechanical Effect Channel Length Modulation Velocity Saturation Effect
Charge conservation
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Junction capacitance model BSIM4 junction capacitance model gave
asymmetry Updated diode junction capacitance model for
AC symmetry
20
Transition point is at Vj=0
Transition point is at Vj=0.9V (pushed to strong forward bias)
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Junction capacitance model
U.C. Berkeley- 21
Symmetry problem using old Qj New model is infinitely differentiable @ VDS=0
BSIM6
BSIM4
BSIM6
BSIM4
BSIM6
BSIM4
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IDS-VX Gummel Symmetry
22
XI
X
X
dVdI
2
2
X
X
dVId
3
3
X
X
dVId
4
4
X
X
dVId
5
5
X
X
dVId 6
6
X
X
dVId
7
7
X
X
dVId
IX vs VX (VD=VX & VS=-VX)
All derivatives are continuous at VDS=0
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AC Symmetry test (C. McAndrew, IEEE TED, 2006)
23
Capacitance & derivatives are symmetric
Capacitances and derivatives are continuous at VDS=0
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Validation on Measured Data (Large device)
24
IDVG @ VDS=50mV
IDVG @ VDS=50mV
gmVG @ VDS=50mV
IDVG in saturation
IDVG in saturation gmVG in saturation
IDVD gdsVD IBVG for different VDS
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Validation on Measured Data (Short device)
25
IDVG @ VDS=50mV
IDVG @ VDS=50mV gmVG @ VDS=50mV
IDVG in saturation
IDVG in saturation
gmVG in saturation
IDVD gdsVD
IBVG for different VDS
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Harmonic Balance Simulation BSIM6 gives correct slope for all harmonics
26
Slope=1
Slope=2Slope=3
Slope=4 Slope=5
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BSIM6 model summary Rapid development: Released BSIM6.0.0-beta8 in Aug. 2012 Charge based physical compact model
Physical effects & Parameter names matched to BSIM4 Smooth charge/current/capacitance & derivatives
Symmetric and continuous around VDS=0 Fulfills Gummel symmetry and AC symmetry Shows accurate slope for harmonic balance simulation
BSIM4s extraction methodology can be easily used for BSIM6 fast deployment & lower cost
Under standardization review in CMC
27
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Outline
History of BSIM Models
BSIM6 Model
BSIM-CMG Model
BSIM-IMG Model
28
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Why next generation transistors?
MOSFET becomes resistor at small L.
Gate
Source Drain
OxideCg
Cd
C.Hu, Modern Semicon. Devices for ICs 2010, Pearson
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Making Oxide Thin is Not Enough
Gate
Source Drain
Leakage Path
Gate cannot control the leakage current paths that are far from the gate.
C.Hu, Modern Semicon. Devices for ICs 2010, Pearson
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One Way to Eliminate Si Far from Gate
31
Thin body controlled By multiple gates.
Gate
Gate
Source Drain
FinFET body is a thin Fin.
N. Lindert et al., DRC paper II.A.6, 2001
Gate Length
Sour
ce
Dra
in
Fin WidthFin Height
Gate Length
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New MOSFET Structures: Demonstration
32
K. Cheng et al. IEDM 2009 - (IBM / ST)
Y. Choi et al. IEEE EDL 2000- (UC Berkeley)X. Huang et al. IEDM 1999 (UC Berkeley)
C.C. Wu et al. IEDM 2010 (TSMC)
FinFET UTBSOI
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Challenges in developing a new model New Physics
Fully depleted channel Quantum confinement etc. When to include them?
Support Multiple Device architectures Inertia with BSIM4 Large user base
Familiarity with the parameters
Convergence Model behavior in extreme cases Balance Physics and Flexibility Balance Speed and Accuracy
33
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Multi-Gate Compact Model: BSIM-MG
34
Fin
BOX
Gat
e 1
Gat
e 2
BOX
p-subP+ back-gate
BSIM-IMG
BSIM-CMG
UTBSOI
BG-ETSOI
FinFETs on Bulk and SOI Substrates
Vertical Fin IMG
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BSIM-CMG Core Models Four device architectures
Three core models Intrinsic Double Gate Core (Y. Taur et al., IEEE EDL, 2004) Perturbation based DG Core for high-doping Cylindrical Gate Core
Bulk and SOI Substrate35
Double Gate Double Gate / Trigate / FinFET
Quadruple Gate Cylindrical Gate / Nanowire FET
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Surface Potential Core Double Gate Solution of Poissons equation and Gausss Law. Poissons equation inside the body can be written
as (Vch is channel potential)
Body doping complicates the solution of the Poissons equation.
Perturbation approach is used to solve this problem.
36
NANA
M. Dunga et al., IEEE TED, Vol. 53, No. 9, 2006
M. Dunga et al., VLSI 2007
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Core Drain Current Model No charge sheet approximation
37
0.0 0.5 1.0 1.50
500
1m
Vg = 0.9V
Vg = 1.2V
Vg = 1.5V
Na = 3e18cm-3
Drai
n Cu
rrent
(A)
Drain Voltage (V)0.0 0.5 1.0 1.50
500
1m
Drai
n Cu
rrent
(A)
Gate Voltage (V)
Na = 3e18 cm-3 Vd = 0.1 Vd = 0.2 Vd = 0.4 Vd = 0.6
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Core Capacitance Model Model inherently exhibits symmetry
Cij = Cji @ Vds= 0 V
Model matches TCAD data (No parameter used) Accurate short channel behavior
38Symbols: TCAD Results; Lines: Model
0.5 1.0 1.50.0
0.5
1.0
Cdg
Csg
Cgg
Na = 3e18cm-3Vds = 1.5V
Symbols : TCADLines : Model
Norm
alize
d Ca
paci
tanc
e
Gate Voltage (V)0.0 0.5 1.0 1.5
0.0
0.5
1.0
ModelSymmetry
Cgs
CgdCdg
Csg
Cgg
Na = 3e18Vg = 1.5V
Symbols : TCADLines : Model
Norm
alize
d Ca
paci
tanc
e
Drain Voltage (V)
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Short Channel (2D) Effects Quasi-2D analysis
Analytical expressions model
39
Characteristic Length
Symbols: TCAD Results; Lines: Model
Auth and Plummer, IEEE EDL, 2007
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Quantum Mechanical Effects Predictive model for confinement induced Vth
shift due to band splitting present in the model
Effective Width model that accounts for reduction in width for a triple/ quadruple/ surround gate structure
40
Width reduction due to structural confinement of inversion charge. (Dotted lines represent the effective width perimeter)
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FinFET Cfr Modeling TCAD Verification
41
Cfg: fin gate Ccg=Ccg1+Ccg2+Ccg3: contact gate
Both Cfg and Ccg agree well with 2D numerical simulations
0 10 20 30 40 50 60 70 800.16
0.18
0.20
0.22
0.24
0.26 Lext=20nm Lext=25nm Lext=30nm Lext=35nm Lext=40nm
Cfg
(fF)
Wg (nm)
20 25 30 35 40 45 500.06
0.09
0.12
0.15
Ccg
(fF)
Wg (nm)
Tc = 20nm --> 40nmin steps of 2nm
Increasing Tc
Fin
Epi-Si
Gate
Cfg
Ccg2
Ccg1Wg
Lext
Tc
Ccg3
0 25 50 75 1000.0
0.1
0.2
0.3
0.4
0.5
Tc (nm)
Ccg1
(fF)
Increasing Lext
Lext = 15nm --> 40nmin steps of 5nmTc = Wg + ToxTox = 1nm
Wfin=1um
-
Real Device Effects
42
Core SPE
I-V C-V
Short Channel Effects
GIDL Current
Channel Length Modulation and DIBL
Mobility Degradation
Direct tunneling gate current
Overlap capacitances
Temperature Effects
Velocity Saturation
S/D Resistance/ Parasitic Resistance
Fringe Capacitances
Noise models
Impact Ionization current
Quantum Effects
-
Bulk FinFET Fitting Bulk FinFETs are fabricated by TSMC TFIN=25nm, HFIN=27.5nm, EOT=2.42nm
43
Drain Current vs. Vgs (Lg = 50nm) Drain Current vs. Vds (Lg = 50 nm) Output Conductance (Lg = 50nm)
0.3 0.6 0.9 1.20
10
20
30
40
50Vds = 1.2V
Vds = 50mV
Vds = 1.2V
Vds = 50mV
Lg = 50nm
Drai
n Cu
rrent
(A)
Gate Voltage (V)
1p
1n
1
1m
0.0 0.3 0.6 0.9 1.20
10
20
30
40
50
Lg = 50nm
Vgs = 0.4V
Vgs = 0.8V
Vgs = 0.6V
Vgs = 1.0V
Vgs = 1.2V
Drain
Cur
rent
(A)
Drain Voltage (V)0.0 0.3 0.6 0.9 1.2
10n
1
100 Vgs = 1.2V --> 0.2V (in steps of 0.2V)
Lg = 50nm
Out
put C
ondu
ctan
ce (S
)
Drain Voltage (V)
0.0 0.3 0.6 0.9 1.21p
1n
1
1m
Vgs = 1.2V --> 0.4V (in steps of 0.2V)
Lg = 0.97m
Out
put C
ondu
ctan
ce (S
)
Drain Voltage (V)0.0 0.3 0.6 0.9 1.20
200n
400n
600n
800n
L = 0.97m
Vds = 1.2V
Vds = 50mV
Tran
scon
duct
ance
, gm (S
)
Gate Voltage (V)
0
2
4
6
8
10
0.0 0.3 0.6 0.9 1.21p
10p
100pVds = 1.2V
Lg = 50nm
Subs
trate
Cur
rent
(A)
Gate Voltage (V)
Transconductance (Lg = 0.97m) Output Conductance (Lg = 0.97 m) Substrate Current (Lg = 50nm)
Symbols: DataLines: Model
-
Asymmetric Vertical Nanowire fittiing Lg=120nm, D=80nm, Tox=3nm
44
Transconductance Output Conductance Temperature Dependence
Drain Current vs. VdsDrain Current vs. Vgs
Symbols: DataLines: Model
-
Symmetry / Continuity Tests
45
Model passes both DC and AC Symmetry Tests
Drain Current Capacitances (Cgg and Csd)C.C.McAndrew, IEEE TED, Vol. 53, No. 9, 2006
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BSIM-CMG Summary BSIM-CMG 106.0.0 is industry standard
production level model standardized in March 2012 Available in major EDA tools
Released BSIM-CMG 106.1.0 in Sept. 2012 Physical, Scalable Core Models for multiple
device architectures Supports both SOI and Bulk Substrates Many Real Device Effects captured
Validated on Hardware Data from different technologies
46
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Outline
History of BSIM Models
BSIM6 Model
BSIM-CMG Model
BSIM-IMG Model
47
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Device Structure & BSIM-IMG Asymmetric structure
Different Gate Work-functions Allows dissimilar Gate Potentials Different Oxide thickness and Material !
Captures important features Vth tuning through Back-Gate Multi-Vth technology
48
Source Drainx
yNA
Vfg
Tsi
Tox1
Tox2
Back Gate
Front Gate
Vs Vd
Vbg
-
Computationally Efficient Core Efficient Non-iterative Surface Potential calculation Surface potential needs to solved at least twice -
Source and Drain side Obtain s / Qis and d / Qid
49
0 2 4 6 8 100
20
40
60
80
100
IterativeMethod B
Cum
ulat
ive P
erce
ntag
e
Computational Time (s)
IterativeMethod A
BSIM-IMG s
FAST
S. Khandelwal et al., "BSIM-IMG: A Compact Model for Ultra-Thin Body SOI MOSFETs with Back-Gate Control", IEEE TED, Aug. 2012.
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Results: Surface-Potential
50
Comparison with Numerical Solution Absolute Error (
-
Volume Inversion Preserves Important Property like Volume
Inversion In sub-threshold (Low field), the charge density Qi is
proportional to the body thickness Tsi
51
-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.01E-121E-111E-101E-91E-81E-71E-61E-51E-41E-30.01
-1.0 -0.5 0.012345
Q i R
atio
sf (V)
Tsi = 5nm Tsi = 10nm Tsi = 20nm
Char
ge D
ensit
y, Q
i (cou
l/m2 )
Front Surface Potential, sf (V)
Toxb = 10m- - - - - - -
- - - - - - -- - - - - - -
Tsi
-
Drain Current Model
52
( ) ( )
+
+= dinvsinvssds
dinvsinvds QQq
kTQQL
WI ,,,1,1,,
2
2
2
222
ssiinv
ssi
EQE
+
= Qinv: inversion carrier density Es2: back-side electric field s1: front-side surface potential
Drift Diffusion
0.0 0.2 0.4 0.6 0.8 1.00
50
100
150
200
250 Charge sheet This Work TCAD
Vfg = 0.2v, 0.4v, 0.6v, 0.8v, 1.0v
Drai
n Cu
rrent
(A)
Drain Voltage (V)
-0.5 0.0 0.5 1.0-15
-10
-5
0
5
10
15
Charge-sheet Model This Work
Erro
r Rel
ative
to T
CAD
(%)
Front Gate Voltage (V)
No Charge-sheet Approximation Very high accuracy
-
Length Dependent Model Capacitive coupling ratio
53
0.01 0.1 1
0.18
0.20
0.22
0.24
Gam
ma
()
Gate Length (m)
TCAD Model
Vds = 1V
degraded at short channel
Tsi=8nm
Tbox=4nm
Front Gate
Back Gate
Source / Drain
Cox1
Csi
Cox2
Cd1(Leff)
Cd2(Leff)
-0.2 0.0 0.2 0.4 0.6 0.8 1.010-9
10-810-710-610-510-410-310-2
Vbg = 10v, 15v, 20v, 25v
Cross: MeasurementsLines: BSIM-IMG
W=50 x 0.5mL = 50 nm
Drai
n Cu
rrent
(A)
Gate Voltage (V)
Captures Vbg effect in I-V
-
QM Effect: Inv. Charge Centroid Model
54
0
10
20
30
40
50
60
0 0.5 1 1.5Gate Voltage (V)
Cgto
tal (
fF/
m^2
)
SCHRED_ClasBSIM-IMGSCHRED_QMBSIM-IMG Fitted
EOTphys = 0.65nmTinv = 1.13nm EOTeff = 0.95nm
Vdd = 0.9 V ; Vfb = 28 mV
TBOX = 140nm; Tsi = 6nm; Nsub = 1e16 cm-3
BOX
p-sub
Vgs
Chart2
0.00000390590.4689450.00000132780.444168
0.00002612940.4742640.00000883890.449232
0.0001769820.4791910.00005983540.453511
0.001198990.4851510.0004051970.457666
0.008116450.5009330.002743150.465
0.05463580.5819750.01853010.495712
0.3547591.045650.1233710.671869
1.927093.03790.7539141.57635
6.728347.758793.335644.39063
14.56514.74828.717939.09701
22.598822.380214.756214.43
29.116828.985219.244919.1532
34.051534.007522.305322.782
37.702137.661324.54425.4253
40.403540.321326.284327.3441
42.43342.29427.680528.761
43.989943.79228.822429.8323
45.210144.956529.771830.6617
46.185645.881530.569831.3182
46.979246.630431.247931.8481
47.634447.247231.830332.2833
48.182747.762532.335532.646
48.646548.198832.778132.9525
49.042848.572333.169233.2144
49.384448.895333.517733.4405
49.68149.177133.830933.6375
49.940449.42534.114333.8106
50.168649.644634.380433.9637
50.370649.840334.617434.1
50.550450.015934.828234.2222
50.711150.174135.030934.3322
50.855450.317435.2234.4318
50.985550.447835.397334.5223
51.103350.566835.573334.6049
51.210450.67635.730334.6806
51.30850.776535.8734.7503
51.397250.869236.021534.8145
51.479150.95536.156234.8739
51.554551.034636.285234.929
51.62451.108736.41934.9803
SCHRED_Clas
BSIM-IMG
SCHRED_QM
BSIM-IMG Fitted
Gate Voltage (V)
Cgtotal (fF/mm^2)
Device Parameters
MKS UnitsRegular Units
Tox11.00E-09
Tox21.40E-07
epsrox13.9
epssi11.7
epsrox23.9
eps08.85E-12
Tsivary
Cox13.45E-023.45E+00uF/cm^2
Cox22.47E-042.47E-02uF/cm^2
Csivary
Vbg0
Na1.00E+16
WF14.56
WF24.97E+00
Ea0.05
Ed5.00E-02
Results
QuantumTsiCtot @ -1.9vCtot @ 0.9vCtot @ 1.9vTinv @ -1.9vTinv @ 0.9vTinv @ 1.9vTcen @ -1.5vTcen @ 0.9vTcen @ 1.9vDist @ 0.9vDist. @ 1.9vQi @ 0.9vQi @ 1.5vVtsi
UnitsnmuF/cm^2uF/cm^2uF/cm^2nmnmnmnmnmnmnmnm
42.522.272.601.371.521.331.111.560.981.641.265.14E+121.42E+130.560
52.532.262.601.361.531.331.091.580.981.881.365.23E+121.43E+130.554
62.522.252.601.371.531.331.111.600.982.081.435.27E+121.43E+130.550
72.522.242.601.371.541.331.111.620.982.231.475.29E+121.43E+130.548
82.522.232.601.371.551.331.111.650.982.341.55.29E+121.43E+130.548
92.522.232.601.371.551.331.111.650.982.431.525.30E+121.43E+130.547
102.522.232.601.371.551.331.111.650.982.511.545.30E+121.43E+130.547
152.522.232.601.371.551.331.111.650.982.71.585.30E+121.43E+130.547
ClassicalTsiCtot @ -1.9vCtot @ 0.9vCtot @ 1.9vTinv @ -1.9vTinv @ 0.9vTinv @ 1.9vTcen @ -1.5vTcen @ 0.9vTcen @ 1.9vDist @ 0.9vDist. @ 1.9vQi @ 0.9vQi @ 1.5vVtsi
nmuF/cm^2uF/cm^2uF/cm^2nmnmnmnmnmnmnmnm
4convergenceissue
52.812.782.981.231.241.160.690.730.481.060.6816.93E+121.77E+130.514
62.812.772.981.231.251.160.690.740.481.130.7076.94E+121.77E+130.513
72.812.772.981.231.251.160.690.740.481.20.7276.94E+121.77E+130.513
82.812.772.981.231.251.160.690.740.481.250.7446.94E+121.77E+130.513
92.812.772.981.231.251.160.690.740.481.30.7596.94E+121.77E+130.513
102.812.772.981.231.251.160.690.740.481.340.7716.95E+121.77E+130.512
152.812.772.981.231.251.160.690.740.481.480.8516.95E+121.77E+130.512
TsiDel_VthSqr_Well
447.60325.625
542.02416.4
637.83411.3888888889
735.7258.3673469388
835.7256.40625
934.6675.0617283951
1034.6674.1
1534.6671.8222222222
Results
Vth shift
Tsi (nm)
Delta Vtsi (mV)
CV_Curves
Tsi=4nTsi=5nTsi=6nTsi=7n
0-0.046-0.037-0.0284.17E-087.82E-081.27E-071.81E-07
0.050.0040.0130.0222.77E-075.18E-078.39E-071.20E-06
0.10.0540.0630.0721.88E-063.50E-065.66E-068.04E-06
0.150.1040.1130.1221.27E-052.36E-053.81E-055.41E-05
0.20.1540.1630.1728.59E-050.0001596030.0002568530.000363442
0.250.2040.2130.2220.0005812610.001076190.001726230.00243404
0.30.2540.2630.2720.003910990.007185290.01141030.0159074
0.350.3040.3130.3220.02546330.04513360.06848470.0910076
0.40.3540.3630.3720.1399610.2180470.2911620.347655
0.450.4040.4130.4220.4893060.6297550.7287920.787959
0.50.4540.4630.4721.018141.134231.191551.21198
0.550.5040.5130.5221.467351.50991.515351.50465
0.60.5540.5630.5721.743731.742551.723991.70281
0.650.6040.6130.6221.912691.895181.869871.84804
0.70.6540.6630.6722.026582.004891.979841.9609
0.750.7040.7130.7222.110222.089392.066932.05141
0.80.7540.7630.7722.175452.156972.137792.1256
0.850.8040.8130.8222.228442.212542.19652.1871
0.90.8540.8630.8722.272732.259762.24642.23857
0.950.9040.9130.9222.310522.29982.28872.2825
10.9540.9630.9722.343692.334112.324872.32054
1.051.0041.0131.0222.372372.364372.356592.35315
1.11.0541.0631.0722.397332.391492.38442.38128
1.151.1041.1131.1222.420032.415122.409412.4064
1.21.1541.1631.1722.441052.435772.431272.42868
1.251.2041.2131.2222.459482.454822.450432.44861
1.31.2541.2631.2722.475652.472062.468122.46711
1.351.3041.3131.3222.491062.488372.484172.48333
1.41.3541.3631.3722.50592.502692.499432.49752
1.451.4041.4131.4222.518912.515242.512822.51106
1.51.4541.4631.4722.530292.527442.524532.52352
Tcen@Vdd (QM)
Tcen@Vdd (Clas)
Tsi (nm)
Tcen (nm)
CV_Curves
Tsi=4n
Tsi=5n
Tsi=6n
Gate Voltage (V)
Capacitance (fF)
Classical Qi-Tsi
Vg=0.9Tsi = 5nmTsi = 7nmTsi = 10nm
-1.93E-159.42E+191.13E+19-2.19E+041.93E-159.37E+191.12E+192.17E+04-1.93E-159.34E+191.12E+19-2.17E+04
0.127.51E+198.58E+181.03E+180.127.46E+198.64E+181.04E+180.127.43E+198.70E+181.04E+18
0.23426.11E+196.64E+181.56E+180.23596.04E+196.76E+181.59E+180.23716.00E+196.86E+181.63E+18
0.3435.06E+195.24E+181.80E+180.34784.97E+195.37E+181.87E+180.35144.91E+195.48E+181.93E+18
0.44654.27E+194.21E+181.88E+180.45594.15E+194.33E+181.98E+180.4634.08E+194.45E+182.06E+18
0.54513.65E+193.43E+181.87E+180.56023.52E+193.55E+181.99E+180.57193.44E+193.65E+182.09E+18
0.63893.17E+192.84E+181.81E+180.6613.03E+192.95E+181.95E+180.67822.93E+193.04E+182.06E+18
0.72832.79E+192.37E+181.73E+180.75842.63E+192.47E+181.88E+180.78192.52E+192.55E+182.00E+18
0.81332.49E+192.01E+181.64E+180.85242.32E+192.10E+181.79E+180.88312.20E+192.17E+181.92E+18
0.89432.24E+191.72E+181.54E+180.94312.06E+191.81E+181.71E+180.98191.93E+191.86E+181.83E+18
0.97132.03E+191.50E+181.45E+181.0311.84E+191.55E+181.60E+181.0781.72E+191.61E+181.74E+18
1.0451.86E+191.30E+181.36E+181.1151.67E+191.37E+181.52E+181.1721.54E+191.41E+181.66E+18
1.1151.71E+191.13E+181.26E+181.1971.52E+191.20E+181.43E+181.2641.38E+191.24E+181.57E+18
1.1811.59E+191.00E+181.18E+181.2761.39E+191.06E+181.35E+181.3541.25E+191.10E+181.49E+18
1.2441.49E+199.07E+171.13E+181.3521.28E+199.48E+171.28E+181.4421.14E+199.73E+171.40E+18
1.3051.40E+197.95E+171.04E+181.4261.19E+198.43E+171.20E+181.5271.05E+198.72E+171.33E+18
1.3621.32E+197.24E+179.86E+171.4971.11E+197.63E+171.14E+181.619.68E+187.94E+171.28E+18
1.4171.25E+196.49E+179.19E+171.5661.04E+196.83E+171.07E+181.6928.97E+187.09E+171.20E+18
1.4691.19E+195.81E+178.54E+171.6329.73E+186.23E+171.02E+181.7718.35E+186.51E+171.15E+18
1.5181.13E+195.32E+178.08E+171.6969.18E+185.69E+179.66E+171.8497.80E+185.85E+171.08E+18
1.5651.08E+194.88E+177.64E+171.7588.70E+185.22E+179.17E+171.9247.31E+185.41E+171.04E+18
1.611.04E+194.48E+177.22E+171.8188.26E+184.71E+178.56E+171.9986.88E+184.95E+179.90E+17
1.6531.00E+194.12E+176.81E+171.8757.87E+184.41E+178.27E+172.076.49E+184.55E+179.41E+17Average Distance
1.6949.70E+183.69E+176.24E+171.9317.52E+184.06E+177.84E+172.146.15E+184.24E+179.08E+17TsiSpreadsheetSchred
1.7329.39E+183.47E+176.02E+171.9857.21E+183.75E+177.44E+172.2095.83E+183.91E+178.63E+174nm0.98nm1.06
1.7699.11E+183.19E+175.64E+172.0376.92E+183.46E+177.05E+172.2765.55E+183.61E+178.21E+177nm1.11nm1.2
1.8048.86E+183.01E+175.43E+172.0876.66E+183.20E+176.67E+172.3415.29E+183.39E+177.93E+1710nm1.24nm1.34
1.8388.63E+182.67E+174.91E+172.1356.42E+183.02E+176.45E+172.4055.06E+183.14E+177.55E+17
1.8698.41E+182.61E+174.88E+172.1826.21E+182.79E+176.09E+172.4674.85E+182.96E+177.30E+17
1.98.22E+182.30E+174.37E+172.2276.01E+182.64E+175.89E+172.5284.65E+182.79E+177.06E+17
1.9288.04E+182.25E+174.34E+172.2715.83E+182.45E+175.56E+172.5884.47E+182.55E+176.60E+17Schred Manual
1.9567.88E+182.05E+174.01E+172.3135.66E+182.32E+175.37E+172.6454.31E+182.46E+176.50E+17The electron average distance is calculated assuming quantum mechanical chargedescription of the electrons. If a semiclassical model for the electron charge descriptionwere used, the average distance would be about three times larger
1.9827.73E+181.93E+173.83E+172.3545.50E+182.15E+175.05E+172.7024.16E+182.29E+176.18E+17
2.0077.59E+181.82E+173.66E+172.3935.36E+182.04E+174.88E+172.7574.02E+182.17E+175.98E+17
2.0317.47E+181.64E+173.34E+172.4315.23E+181.93E+174.70E+172.8113.89E+182.06E+175.79E+17
2.0537.35E+181.62E+173.32E+172.4685.11E+181.79E+174.41E+172.8643.77E+181.92E+175.50E+17
2.0757.24E+181.45E+173.00E+172.5034.99E+181.70E+174.25E+172.9153.65E+181.83E+175.32E+17
2.0957.14E+181.36E+172.84E+172.5374.88E+181.61E+174.09E+172.9653.55E+181.74E+175.16E+17
2.1147.04E+181.34E+172.83E+172.574.78E+181.53E+173.94E+173.0143.45E+181.62E+174.89E+17
2.1336.95E+181.25E+172.67E+172.6024.69E+181.45E+173.78E+173.0613.36E+181.58E+174.83E+17
2.1516.87E+181.10E+172.36E+172.6334.60E+181.38E+173.64E+173.1083.27E+181.47E+174.58E+17
2.1676.79E+181.09E+172.36E+172.6634.52E+181.27E+173.37E+173.1533.19E+181.44E+174.53E+17
2.1836.72E+181.01E+172.20E+172.6914.45E+181.25E+173.35E+173.1983.12E+181.34E+174.29E+17
2.1986.66E+189.98E+162.19E+172.7194.38E+181.18E+173.21E+173.2413.05E+181.28E+174.15E+17
2.2136.59E+189.23E+162.04E+172.7464.31E+181.12E+173.08E+173.2832.98E+181.22E+174.01E+17
2.2276.53E+188.50E+161.89E+172.7724.24E+181.06E+172.94E+173.3242.92E+181.20E+173.98E+17
2.246.48E+187.78E+161.74E+172.7974.18E+181.00E+172.81E+173.3652.86E+181.11E+173.75E+17
2.2526.43E+187.71E+161.74E+172.8214.13E+189.49E+162.68E+173.4042.80E+181.06E+173.62E+17
2.2646.38E+187.66E+161.73E+172.8444.07E+189.37E+162.66E+173.4422.75E+181.04E+173.59E+17
2.2766.33E+186.33E+161.44E+172.8674.02E+188.45E+162.42E+173.482.70E+189.71E+163.38E+17
2.2866.29E+186.92E+161.58E+172.8883.98E+188.35E+162.41E+173.5162.65E+189.54E+163.35E+17
2.2976.25E+185.63E+161.29E+172.9093.93E+188.25E+162.40E+173.5522.60E+188.86E+163.15E+17
2.3066.21E+186.21E+161.43E+172.933.89E+187.39E+162.16E+173.5862.56E+188.71E+163.12E+17
2.3166.18E+184.94E+161.14E+172.9493.85E+187.31E+162.16E+173.622.52E+188.32E+163.01E+17
2.3246.14E+185.53E+161.29E+172.9683.81E+186.86E+162.03E+173.6532.48E+188.19E+162.99E+17
2.3336.11E+184.89E+161.14E+172.9863.77E+186.79E+162.03E+173.6862.45E+187.58E+162.79E+17
2.3416.08E+184.87E+161.14E+173.0043.74E+186.35E+161.91E+173.7172.41E+187.47E+162.78E+17
2.3496.05E+184.24E+169.96E+163.0213.70E+185.93E+161.79E+173.7482.38E+187.13E+162.67E+17
2.3566.03E+184.22E+169.94E+163.0373.67E+185.88E+161.78E+173.7782.34E+187.03E+162.66E+17
2.3636.00E+183.60E+168.51E+163.0533.64E+185.46E+161.67E+173.8082.31E+186.48E+162.47E+17
2.3695.98E+184.19E+169.92E+163.0683.61E+185.42E+161.66E+173.8362.28E+186.40E+162.45E+17
2.3765.96E+183.57E+168.49E+163.0833.59E+185.38E+161.66E+173.8642.26E+186.09E+162.35E+17
2.3825.94E+182.97E+167.07E+163.0983.56E+184.63E+161.43E+173.8912.23E+186.02E+162.34E+17
2.3875.92E+183.55E+168.47E+163.1113.54E+184.95E+161.54E+173.9182.20E+185.73E+162.24E+17
2.3935.90E+182.95E+167.05E+163.1253.51E+184.57E+161.43E+173.9442.18E+185.45E+162.15E+17
2.3985.88E+182.94E+167.05E+163.1383.49E+184.19E+161.31E+173.9692.16E+185.39E+162.14E+17
2.4035.86E+182.34E+165.63E+163.153.47E+184.16E+161.31E+173.9942.13E+185.12E+162.04E+17
2.4075.85E+182.92E+167.03E+163.1623.45E+184.14E+161.31E+174.0182.11E+185.07E+162.04E+17
2.4125.83E+182.33E+165.62E+163.1743.43E+183.77E+161.20E+174.0422.09E+184.81E+161.94E+17
2.4165.82E+182.33E+165.62E+163.1853.41E+183.75E+161.19E+174.0652.07E+184.55E+161.85E+17
2.425.80E+182.32E+165.62E+163.1963.39E+183.39E+161.08E+174.0872.05E+184.51E+161.84E+17
2.4245.79E+182.32E+165.61E+163.2063.37E+183.37E+161.08E+174.1092.03E+184.47E+161.84E+17
2.4285.78E+181.73E+164.21E+163.2163.36E+183.36E+161.08E+174.1312.01E+184.03E+161.66E+17
2.4315.76E+181.73E+164.20E+163.2263.34E+183.01E+169.71E+164.1512.00E+184.19E+161.74E+17
2.4345.75E+182.30E+165.60E+163.2353.33E+182.99E+169.69E+164.1721.98E+183.96E+161.65E+17
2.4385.74E+181.72E+164.20E+163.2443.31E+182.98E+169.67E+164.1921.97E+183.73E+161.57E+17
2.4415.73E+181.15E+162.80E+163.2533.30E+182.97E+169.66E+164.2111.95E+183.71E+161.56E+17
2.4435.72E+181.72E+164.19E+163.2623.29E+182.63E+168.58E+164.231.94E+183.48E+161.47E+17
2.4465.71E+181.71E+164.19E+163.273.27E+182.62E+168.56E+164.2481.92E+183.65E+161.55E+17
2.4495.71E+181.14E+162.79E+163.2783.26E+182.28E+167.48E+164.2671.91E+183.24E+161.38E+17
2.4515.70E+181.14E+162.79E+163.2853.25E+182.60E+168.54E+164.2841.89E+183.22E+161.38E+17
2.4535.69E+181.71E+164.19E+163.2933.24E+182.27E+167.46E+164.3011.88E+183.20E+161.38E+17
2.4565.68E+181.14E+162.79E+163.33.23E+182.26E+167.46E+164.3181.87E+182.99E+161.29E+17
2.4585.68E+181.14E+162.79E+163.3073.22E+181.93E+166.38E+164.3341.86E+182.97E+161.29E+17
2.465.67E+181.13E+162.79E+163.3133.21E+182.24E+167.44E+164.351.85E+182.95E+161.28E+17
2.4625.66E+181.13E+162.79E+163.323.20E+181.92E+166.37E+164.3661.84E+182.75E+161.20E+17
2.4645.66E+185.66E+151.39E+163.3263.19E+181.91E+166.36E+164.3811.82E+182.74E+161.20E+17
2.4655.65E+181.13E+162.79E+163.3323.18E+181.91E+166.36E+164.3961.81E+182.72E+161.20E+17
2.4675.65E+181.13E+162.79E+163.3383.17E+181.59E+165.29E+164.4111.80E+182.53E+161.11E+17
2.4695.64E+185.64E+151.39E+163.3433.16E+181.58E+165.29E+164.4251.79E+182.51E+161.11E+17
2.475.64E+181.13E+162.78E+163.3483.15E+181.89E+166.34E+164.4391.79E+182.32E+161.03E+17
2.4725.63E+185.63E+151.39E+163.3543.15E+181.57E+165.28E+164.4521.78E+182.31E+161.03E+17
2.4735.63E+185.63E+151.39E+163.3593.14E+181.57E+165.27E+164.4651.77E+182.30E+161.03E+17
2.4745.62E+185.62E+151.39E+163.3643.13E+181.25E+164.22E+164.4781.76E+182.29E+161.02E+17
2.4755.62E+181.12E+162.78E+163.3683.13E+181.56E+165.27E+164.4911.75E+182.10E+169.43E+16
2.4775.62E+185.62E+151.39E+163.3733.12E+181.25E+164.21E+164.5031.74E+182.09E+169.42E+16
2.4785.61E+185.61E+151.39E+163.3773.11E+181.25E+164.21E+164.5151.74E+181.91E+168.62E+16
2.4795.61E+185.61E+151.39E+163.3813.11E+181.24E+164.20E+164.5261.73E+182.07E+169.38E+16
2.485.60E+185.60E+151.39E+163.3853.10E+181.24E+164.20E+164.5381.72E+181.89E+168.59E+16
2.4815.60E+185.60E+151.39E+163.3893.10E+181.24E+164.20E+164.5491.71E+181.88E+168.57E+16
2.4825.60E+185.60E+151.39E+163.3933.09E+181.24E+164.20E+164.561.71E+181.71E+167.78E+16
2.4835.60E+180.00E+000.00E+003.3973.09E+189.26E+153.15E+164.571.70E+181.87E+168.55E+16
2.4835.59E+185.59E+151.39E+163.43.08E+181.23E+164.19E+164.5811.69E+181.69E+167.76E+16
2.4845.59E+185.59E+151.39E+163.4043.08E+189.23E+153.14E+164.5911.69E+181.69E+167.75E+16
2.4855.59E+185.59E+151.39E+163.4073.07E+189.22E+153.14E+164.6011.68E+181.51E+166.96E+16
2.4865.59E+180.00E+000.00E+003.413.07E+189.20E+153.14E+164.611.68E+181.68E+167.72E+16
2.4865.58E+185.58E+151.39E+163.4133.06E+189.19E+153.14E+164.621.67E+181.50E+166.94E+16
2.4875.58E+185.58E+151.39E+163.4163.06E+189.18E+153.14E+164.6291.66E+181.50E+166.93E+16
2.4885.58E+180.00E+000.00E+003.4193.06E+189.17E+153.13E+164.6381.66E+181.33E+166.16E+16
2.4885.58E+185.58E+151.39E+163.4223.05E+189.15E+153.13E+164.6461.65E+181.49E+166.92E+16
2.4895.58E+180.00E+000.00E+003.4253.05E+186.10E+152.09E+164.6551.65E+181.32E+166.14E+16
2.4895.57E+185.57E+151.39E+163.4273.04E+189.13E+153.13E+164.6631.64E+181.32E+166.13E+16
2.495.57E+180.00E+000.00E+003.433.04E+186.08E+152.09E+164.6711.64E+181.31E+166.13E+16
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