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Jinn-Liang Liu 劉晉良高雄大學
Quantum Hydrodynamic ModelsBased on the
Maximum Entropy Principle
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
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
Model Hierarchy (D. Vasileska, 2006)
Model Hierarchy (D. Vasileska, 2006)
Model Hierarchy (A. Jüngel, 2000)
Classical Models
[ ]
),( re temperatu,),(ity mean veloc , :models cMacroscopi ),,(function on distributi :models cMicroscopi
),',(' , to' from raten transitio:),',(
operatorcollision :')'1()',,()1('),',()(
Eq.) (Boltzmann )(
in density number electron :),,(),(
in eunit volumper electrons ofnumber :),,(
3D) (2D, )( 0
Eq.) )(Transport Liouville (1D 0
1),,( electron single a ofdensity y probabilit :),,(
txTtxVntvxftvxffvvvvxs
dvffvvxsffvvxsfC
fCfEmqfv
tf
dxdvtvxftxn
dxdvtvxf
qqEvmFfEmqfv
tf
dtdv
vf
dtdx
xf
tf
dtdf
dxdvtvxftvxf
veffx
xvx
=
−−−=
=∇⋅−∇⋅+∂∂
=
∇=−===∇⋅−∇⋅+∂∂
=∂∂
+∂∂
+∂∂
=
=
∫
∫
∫∫
φ&
Semi-Classical Models for Semiconductors
( )
densityacceptor : density,donor :
Eq.)(Poisson , ,
Eq.)Boltzmann Classical-(Semi )(
band conductionenergy : elocity,electron v :1
densityelectron :),,(),(
in eunit volumper electrons ofnumber :),,(
Eq.)Transport Classical-(Semi 0
Eq.)Transport (Classical 0
zoneBrillouin first Constant, sPlanck' reduced ,
1),,( electron single a offunction on distributi :),,(
AD
xAD
kx
k
B
kx
vx
B
NN
EnNNq
fCfEqfvtf
v
dktkxftxn
dxdktkxf
fEqfvtf
fEmqfv
tf
Bkmvp
dkdxtkxftkxf
φε
φ
εε
∇=−−=∆−
=∇⋅−∇⋅+∂∂
∇=
=
=∇⋅−∇⋅+∂∂
=∇⋅−∇⋅+∂∂
====
=
∫
∫∫
η
η
η
ηη
Moment Method
.)( , ,1)( take we,)(
)()()(
have we,over gintegratin and )(function moment aby SCBE thegMultiplyin
(SCBE) )(
kkkfdkkM
Qdkfdkk
kqEfdkkvkxt
MB
k
fCfEqfvtf
B
BBj
j
B
i
i
kx
εψψ
ψψψ
ψ
ψ
ψ
η
η
==
=∂∂
−∂∂
+∂
∂
=∇⋅−∇⋅+∂∂
∫
∫∫∫
Conservation Eqs.
energy average :)(1 ,)()(
momentum average :1 ,)()(
velocityaverage :1 ,0)(
dkkfn
WnCEnqVx
nSt
nW
dkkfn
PnCnqEx
nUt
nP
dkfvn
Vx
nVtn
BWi
ii
i
B
iiiP
i
j
iji
B
ii
i
i
∫
∫
∫
==+∂
∂+
∂∂
==+∂
∂+
∂∂
==∂
∂+
∂∂
ε
η
Closure Problem of Conservation Eqs.
,)()(
,)()( ,0)(
Wi
ii
i
iP
i
j
iji
i
i
nCEnqVxnS
tnW
nCnqEx
nUt
nPx
nVtn
=+∂
∂+
∂∂
=+∂
∂+
∂∂
=∂
∂+
∂∂
.,, of in terms expressed becannot ,,,, :problem Closure WVnCSCUP Wii
Piji
.,, : variableslFundamenta WVn
Maximum Entropy Principle (MEP) (Postulate)
( )
∫∫
=
−−=
B AA
B
A
dkfkM
dkfffkfH
fM
ME
B
ME
)( that sconstraint under the
log)( functionalentropy the
maximizes that function on distributi a exists then thereknown, are moments ofnumber certain a If
ψ
Legendre Tansform of MEP
( )( )
( )
etc. ,32 , :solved problem Closure
),,(Inversion
1exp1exp ,,,
,,1 ,,, ,0)(
smultiplier Lagrange : ,log)(
)()()( Maximize
*
BBME
B
ME
ijijii
iAA
ii
AAiA
AiA
A
B
B AAA
WUVmP
WVn
vkk
f
kWVnMdf
fL
dkfffkfH
dkfkMfHfL
δ
λλλψλλλ
ψ
ψ
ε
ε
εε
==
Λ=Λ
++
−=
Λ
−==Λ
===
Λ−−=
−Λ−=
⇒∫
∫
η
Quantum Hydrodynamic Models
A Quantum Energy Transport ModelChen & Liu, JCP 2005
Adaptive Algorithm
S o lv eS o lv e
I n it ia l m e s hI n it ia l m e s h
E r r o r > T O LE r r o r > T O L
E r r o r E s t im a t io nE r r o r E s t im a t io n R e f in e m e n tR e f in e m e n t
Y e s
P o s t-P r o c e s sP o s t-P r o c e s s
N o
P r e p r o c e s s in gP r e p r o c e s s in g
G u m m e l o u te r ite r a t io nG u m m e l o u te r ite r a t io n
S o lv e P o is s o n E q .S o lv e P o is s o n E q .
S o lv eS o lv e pnvu ζζ ,,,
E r r o r > T O LE r r o r > T O L
pn gg ,
Y e s
N o
)( )(
)()(
),( ),(
)(
p
n
pp
nn
gRgR
ZZ
vuRvuR
F
pp
n
p
n
p
=⋅∇−=⋅∇−
=∆−=∆−
−=⋅∇−=⋅∇−
=∆−
GG
JJ
n
n
ςςςς
φφ
n+ n+
p-
interfacelayer
junctionlayer
junctionlayer
gate contactsource contact drain contact
bulk contact
BC D
I JE
A F
B’ E’
C’ D’
The Final Adaptive Mesh
0 20 40 60 80 100
0
20
40
60
80
100
Transverse Distance (nm)
Dep
th (
nm)
Electron Concentration
Electron Temperature
Hole Quantum Potential
Electron Current Density (DGET)
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
/ µ m
)
ETDGDGET
Conclusion
New QET Model by MEPGlobal, Optimal Convergence, Stable Solution
Monotone Parameters + Grid Sizes + ExponentialFitting + Adaptive + Automatic to Treat Singularly Perturbed Nature => Boundary, Interior, and Quantum Potential LayersOpen Problems: Existence, Multiple Solutions,Uniqueness, Asymptotic, 3D, High Fields, Tunneling,Multi-scale Modeling, Atomic-scale Variations,Robust-Rapid-Accurate Simulation …