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交通大學數學建模與科學計算研究中心
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
Classical Computer
Bit : Either 0 or 1
Microprocessor
Microchips
MOSFET
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
current that can pass through a semiconductor.
Kingfisher Science Encyclopedia
Czochralski Crystal Growth
Sand
Gold Ingots
Ingot
Wafer
Silicon Ingot
Doping
IC
12吋矽晶圓
Silicon Crystal
Shared electrons
Si
Si
Si
Si
Si
Si
Si
Si
-
Si
Doping Impurities (n-Type)
Si
Si
Si
As
Si
Si
Si
Si
-
Conducting band, Ec
Extra
Electron
Ed ~ 0.05 eV
Eg = 1.1 eV
Si
Valence band, Ev
Doping Impurities (p-Type)
Si
Si
Conducting band,
Ec
Si
Hole
Si
B
Eg = 1.1 eV
Si
Ea ~ 0.05 eV
Si
Si
-
Si
Electron
Valence band, Ev
S. Roy and A. Asenov, Science 2005
MOSFET
(Metal Oxide
Semiconductor
Field Effect
Transistor)
2003 L = 4 nm Research
2005 L = 45 nm Production
2018 L = 7 nm Production
3D, 30nm x 30nm
Gate Length: 90 nm (2005 In Production)
(Device Size) 65 nm (2006 In Production)
34 nm (This Talk)
Device Sizes
Vs.
Models
Self-Adjoint Energy Transport Model (Chen & Liu, JCP 2003)
Quantum Corrected Energy Transport Model (Chen & Liu, JCP 2005)
L=IJ=34nm
gate contact
source contact
drain contact
I
J
E
B
n+
C
C’
D’
D
n+
B’
E’
interface
layer
junction
layer
junction
layer
pA
F
bulk contact
Doping Concentration
Energy Transport Model
 
q
S
(n  p  N A  N D ),
(2.1)
  J n  R,
(2.2)
  J p   R,
(2.3)
 n  0
  S n  J n  E  n(
), (2.4)
 n
 p  0
  S p  J p  E  p(
), (2.5)
 p
J n  qn n  qDnn
S n  J n n  kBTn  / q   nTn
•
•
•
•
•
•
•
electrostatic potential
n electron density
p hole density
J current density
S energy flux
E electric field
R generationrecombination rate
q(np  ni )
R(n, p)  0
 n ( p  pT )   0p (n  nT )
2
Auxiliary Relationships
Self-Adjoint Formulation
    n  qn 
   qn 
  ni exp 
u   n2
n  ni exp 
VT


 VT 
     p  qp 
    qp 
  ni exp 
v   p2
p  ni exp 
VT


 VT 
p
n  n
 qn
  n2
 VT ln 
 uni
 qp
  p2
 VT ln 
 vni

New Variables

  


 


Bohm’s Quantum Potential
qn
qp
 n 

,
 n 
2   p 


,
*
2m p q 
p 


2

2mn* q
  Planck' s Constant
 O(10 )
-34
J n  q n n(  qn )  qDnn
J p  q p p(  qp )  qD pp
  J n  R a fourth order PDE in n
Self-Adjoint QCET Model
  F ,
(3.1)
  J n  R , (3.2)
  J p   R , (3.3)
 n  Z n ,
(3.4)
 p  Z p ,
(3.5)
  G n  Rn , (3.6)
  G p  R p , (3.7)
Singularly Perturbed QCET Model
(Liu, Lee, & Chen, 2009 Preprint)
Dimensionless Scaling
• Nano devices extremely singular
• Boundary layer
• Junction layer
• Quantum potential layer
Adaptive Algorithm
Initial
Initialmesh
mesh
Preprocessing
Preprocessing
  F ,
  Jn  R ,
Gummel
Gummelouter
outeriteration
iteration
  J p   R , (3.3)
Solve
SolvePoisson
PoissonEq.
Eq.
Solve
Solve
u , v,  n ,  p
Yes
Error
Error>>TOL
TOL
Solve
Solve g n , g p
Error
Error>>TOL
TOL
No
Post-Process
Post-Process
 n  Z n ,
(3.4)
 p  Z p ,
(3.5)
  G n  Rn , (3.6)
No
Error
ErrorEstimation
Estimation
(3.1)
(3.2)
Refinement
Refinement
Yes
  G p  R p , (3.7)
Finite Element Method
Monotone Iteration
Exponential Fitting
The Final Adaptive Mesh
100
80
Depth (nm)
60
40
20
0
0
20
40
60
Transverse Distance (nm)
80
100
Electron Temperature
Hole Quantum Potential
Electron Current Density
Drain Current for MOSFET
4.5
ET
DG
DGET
4
3.5
2.5
2
I
DS
(mA/  m)
3
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
VDS (V)
0.6
0.7
0.8
0.9
1
60% difference predicted by ET and QCET models
Alternative Future MOS
MOS Scaling Challenges
1. Technology Scaling Parasitic Effects:
Leakage, Capacitance, Risistance
2. Power Limits: End of Voltage Scaling
3. Band-Structure Engineering
4. Scattering: e-insulator, e-imp, e-ph, e-e
5. Dopant Atom Fluctuations
6. Non-Equilibrium Electron & Phonon Distrb.
7. Long Range Coulomb Interactions
8. Full-Band Bias-Induced Quantization
9. Phonon Transport Models
10.Automatic Multi-Scale Computing
MOS Simulation Challenges
Conclusion
Self-Adjoint QCET Model: More Advanced
Technology Scaling Challenges in Physics and
Engineering
Muti-Scaling Modeling and Numerical Methods
High-Performance Architecture, Algorithms,
and Coding
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