2007 年計算數學研討會 中山大學
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2007 年計算數學研討會 中山大學. Ren-Chuen Chen 陳仁純 高雄師範大學 O. Voskoboynikov 霍斯科 交通大學. Modeling and Simulation of Classical and Quantum Computer Devices. Jen-Hao Chen 陳人豪 交通大學 Jinn-Liang Liu 劉晉良 高雄大學. Part 1. Classical Computer. Microprocessor. Microchips. MOSFET. MOSFET - PowerPoint PPT PresentationTRANSCRIPT
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2007 年計算數學研討會中山大學
Ren-Chuen Chen 陳仁純
高雄師範大學
O. Voskoboynikov霍斯科
交通大學
Modeling and Simulation of Classical and Quantum Computer Devices
Jen-Hao Chen陳人豪交通大學
Jinn-Liang Liu 劉晉良高雄大學
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Part 1. Classical Computer
Microchips Microprocessor
MOSFET
1or 0Either :Bit
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MOSFET (Metal Oxide Semiconductor Field Effect Transistor)
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Semiconductor
A semiconductor is a material that can behave as a conductor or an insulator depending on what is done to it. We can control the amount of curre
nt that can pass through a semiconductor.
Kingfisher Science Encyclopedia
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Czochralski Crystal Growth
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12 吋矽晶圓
Sand Ingot Wafer Doping IC
Silicon IngotGold Ingots
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Silicon Crystal
-
Si Si Si
Si
SiSi
Si
Si
Si
Shared electrons
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Doping Impurities (n-Type)
Electron
-
Si Si Si
Si
SiSi
Si
Si
As
Extra
Valence band, Ev
Eg = 1.1 eV
Conducting band, Ec
Ed ~ 0.05 eV
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Valence band, Ev
Eg = 1.1 eV
Conducting band, Ec
Ea ~ 0.05 eV
Electron
-
Si Si Si
Si
SiSi
Si
Si
B
Hole
Doping Impurities (p-Type)
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S. Roy and A. Asenov, Science 2005
3D, 30nm x 30nm
2003 L = 4 nm Research2005 L = 45 nm Production2018 L = 7 nm Production
MOSFET (Metal Oxide
Semiconductor Field Effect Transistor)
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Gate Length: 90 nm (2005 In Production) (Device Size) 65 nm (2006 In Production) 34 nm (This Talk)
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Device SizesVs.Models
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n+ n+
p-
interfacelayer
junctionlayer
junctionlayer
gate contactsource contact drain contact
bulk contact
BC D
I J
E
A F
B’ E’
C’ D’
R.-C. Chen and J.-L. Liu, JCP 2005L=IJ=34nm
Quantum Corrected Energy Transport Model
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Doping Concentration
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Physical Models
Drift diffusion model (3 PDEs)
Energy transport model
(5 PDEs)
Hydrodynamic Model
(7 PDEs)
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Energy Transport Model
(2.5) ),(
(2.4) ),(
(2.3) ,
(2.2) ,
(2.1) ),(
0
0
p
ppp
n
nnn
p
n
DAS
p
n
R
R
NNpnq
EJS
EJS
J
J
• the electrostatic potential • n the electron concentration• p the hole concentration• J the current density• S the energy flux• E the electric field• R the generation-
recombination rate
nqDnq nnn J )()(
)(),(
00
2
TpTn
i
nnpp
nnpqpnR
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Auxiliary Relationships
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Density Gradient Theory (Bohm Quantum Potentials)
,2
,2
qp
qn
p
pb
n
nb
p
n
pqDpq
nqDnq
pqppp
nqnnn
)(
)(
J
J
Constant sPlanck' ,12/ *2 qmb nn
)10( ,12/ 34*2 Oqmb pp
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New Variables
nnp
Self-Adjoint Formulation
expexp 2n
T
qni
T
qnni u
Vn
Vnn
2expexp pT
qpi
T
qppi v
Vn
Vnp
i
nTqn unV
ln
2
i
pTqp vnV
ln
2
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Self-Adjoint DGET Model
(3.26) ,
(3.25) ,
(3.24) ,
(3.23) ,
(3.22) ,
(3.21) ,
(3.20) ,
p
n
pp
n
p
n
p
R
R
Z
Z
R
R
F
G
G
J
J
n
n
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Adaptive Algorithm
SolveSolve
Initial meshInitial mesh
Error > TOLError > TOL
Error EstimationError Estimation RefinementRefinement
Yes
Post-ProcessPost-Process
No
PreprocessingPreprocessing
Gummel outer iterationGummel outer iteration
Solve Poisson Eq.Solve Poisson Eq.
SolveSolve pnvu ,,,
Error > TOLError > TOL
pn gg ,
Yes
No
(3.26) ,
(3.25) ,
(3.24) ,
(3.23) ,
(3.22) ,
(3.21) ,
(3.20) ,
p
n
pp
n
p
n
p
R
R
Z
Z
R
R
F
G
G
J
J
n
n
Finite Element MethodMonotone IterationExponential Fitting
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The Final Adaptive Mesh
0 20 40 60 80 100
0
20
40
60
80
100
Transverse Distance (nm)
Dep
th (
nm)
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Electron Concentration
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Electron Temperature
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Hole Quantum Potential
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Electron Current Density
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Drain Current for MOSFET
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
3
3.5
4
4.5
VDS
(V)
I DS (
mA
/ m
)
ETDGDGET
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Conclusion
New Model (DGET, Self-Adjointness)
Better Approximation
Global Convergence (Monotone Iterative Method)
Efficiency
Easy Implementation