quantum hydrodynamic models based on the maximum entropy ... · semiconductor a semiconductor is a...
TRANSCRIPT
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Jinn-Liang Liu 劉晉良高雄大學
Quantum Hydrodynamic ModelsBased on the
Maximum Entropy Principle
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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 current that can pass through a semiconductor.
Kingfisher Science Encyclopedia
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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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Model Hierarchy (D. Vasileska, 2006)
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Model Hierarchy (D. Vasileska, 2006)
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Model Hierarchy (A. Jüngel, 2000)
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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&
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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
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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
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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
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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
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Quantum Hydrodynamic Models
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A Quantum Energy Transport ModelChen & Liu, JCP 2005
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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
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n+ n+
p-
interfacelayer
junctionlayer
junctionlayer
gate contactsource contact drain contact
bulk contact
BC D
I JE
A F
B’ E’
C’ D’
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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 (DGET)
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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
/ µ m
)
ETDGDGET
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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 …