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 …