Quantum Hydrodynamic Modeling, Numerical Methods, and Applications Semiconductor Transistors Jinn-Liang Liu 劉劉劉 National Hsinchu Univ. of E du. 劉劉劉劉劉劉 Jan. 25-27, 2010 劉劉劉劉劉劉劉劉劉劉劉劉劉劉劉劉劉
Jan 08, 2016
Quantum Hydrodynamic Modeling, Numerical Methods, and Applications
Semiconductor Transistors
Jinn-Liang Liu
劉晉良National Hsinchu Univ. of Edu.
新竹教育大學Jan. 25-27, 2010
交通大學數學建模與科學計算研究中心
Classical Computer
Microchips Microprocessor
MOSFET
1or 0Either :Bit
Transistor
The most important invention of the 20th century?
A transistor is an electronic device used as a switch or to amplify an electric current or voltage.
1930 First Transistor Patent Filed by J. E. Lilienfeld in 1926
The First Transistor Invented at Bell Labs in 1947
The original version of the paper was rejected for publication by Physical Review on the referee's unimaginative assertion that it was 'too speculative' and involved 'no new physics.'
Received his Ph.D. at University of Tokyo in 1959, Esaki was awarded the Nobel Prize in 1973 for research conducted around 1958 on electron quantum tunneling (Esaki Diode).
假設 20歲年輕人之創造力是 100%、辨別力是 0%, 70 歲老年人創造力是 0%、辨別力是 100%,人生分歧點是 45歲。分析諾貝爾獎得主獲獎事由和年齡關聯性,會發現得獎人年齡大多集中於 35歲至 39歲時,而我於 44歲發明人造量子結構。
MOSFET (Metal Oxide Semiconductor Field Effect Transistor)
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
Czochralski Crystal Growth
12 吋矽晶圓
Sand Ingot Wafer Doping IC
Silicon IngotGold Ingots
Silicon Crystal
-
Si Si Si
Si
SiSi
Si
Si
Si
Shared electrons
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
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)
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)
Gate Length: 90 nm (2005 In Production) (Device Size) 65 nm (2006 In Production) 34 nm (This Talk)
Device SizesVs.Models
n+ n+
p-
interfacelayer
junctionlayer
junctionlayer
gate contactsource contact drain contact
bulk contact
BC D
I J
E
A F
B’ E’
C’ D’
L=IJ=34nm
Quantum Corrected Energy Transport Model (Chen & Liu, JCP 2005)
Self-Adjoint Energy Transport Model (Chen & Liu, JCP 2003)
Doping Concentration
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
• electrostatic potential • n electron density• p hole density• J current density• S energy flux• E electric field• R generation-
recombination rate nqDnq nnn J
)()(
)(),(
00
2
TpTn
i
nnpp
nnpqpnR
nnnBnnn TqTk /JS
Auxiliary Relationships
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
Bohm’s Quantum Potential
,2
,2
*
2
qp
*
2
qn
p
p
qm
n
n
qm
p
n
pqDpq
nqDnq
pqppp
nqnnn
)(
)(
J
J
)O(10
Constant sPlanck' 34-
nRn in PDEorder fourth a J
Self-Adjoint QCET Model
(3.7) ,
(3.6) ,
(3.5) ,
(3.4) ,
(3.3) ,
(3.2) ,
(3.1) ,
p
n
pp
n
p
n
p
R
R
Z
Z
R
R
F
G
G
J
J
n
n
Singularly Perturbed QCET Model (Liu, Lee, & Chen, 2009 Preprint)
Dimensionless Scaling
• Nano devices extremely singular• Boundary layer• Junction layer• Quantum potential layer
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
(3.7) ,
(3.6) ,
(3.5) ,
(3.4) ,
(3.3) ,
(3.2) ,
(3.1) ,
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
The Final Adaptive Mesh
0 20 40 60 80 100
0
20
40
60
80
100
Transverse Distance (nm)
Dep
th (
nm)
Electron Temperature
Hole Quantum Potential
Electron Current Density
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
models QCET and ETby predicted difference 60%
Alternative Future MOS
MOS Scaling Challenges
1. Technology Scaling Parasitic Effects: Leakage, Capacitance, Risistance
2. Power Limits: End of Voltage Scaling3. Band-Structure Engineering4. Scattering: e-insulator, e-imp, e-ph, e-e 5. Dopant Atom Fluctuations6. Non-Equilibrium Electron & Phonon Distrb.7. Long Range Coulomb Interactions8. Full-Band Bias-Induced Quantization9. Phonon Transport Models10.Automatic Multi-Scale Computing
MOS Simulation Challenges
Conclusion
Self-Adjoint QCET Model: More AdvancedTechnology Scaling Challenges in Physics and
EngineeringMuti-Scaling Modeling and Numerical MethodsHigh-Performance Architecture, Algorithms, an
d Coding