2007 年計算數學研討會中山大學

Ren-Chuen Chen 陳仁純

高雄師範大學

O. Voskoboynikov霍斯科

交通大學

Modeling and Simulation of Classical and Quantum Computer Devices

Jen-Hao Chen陳人豪交通大學

Jinn-Liang Liu 劉晉良高雄大學

Part 1. Classical Computer

Microchips Microprocessor

MOSFET

1or 0Either :Bit

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’

R.-C. Chen and J.-L. Liu, JCP 2005L=IJ=34nm

Quantum Corrected Energy Transport Model

Doping Concentration

Physical Models

Drift diffusion model (3 PDEs)

Energy transport model

(5 PDEs)

Hydrodynamic Model

(7 PDEs)

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

• the electrostatic potential • n the electron concentration• p the hole concentration• J the current density• S the energy flux• E the electric field• R the generation-

recombination rate

nqDnq nnn J )()(

)(),(

00

2

TpTn

i

nnpp

nnpqpnR

Auxiliary Relationships

Density Gradient Theory (Bohm Quantum Potentials)

,2

,2

qp

qn

p

pb

n

nb

p

n

pqDpq

nqDnq

pqppp

nqnnn

)(

)(

J

J

Constant sPlanck' ,12/ *2 qmb nn

)10( ,12/ 34*2 Oqmb pp

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

Self-Adjoint DGET Model

(3.26) ,

(3.25) ,

(3.24) ,

(3.23) ,

(3.22) ,

(3.21) ,

(3.20) ,

p

n

pp

n

p

n

p

R

R

Z

Z

R

R

F

G

G

J

J

n

n

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

(3.26) ,

(3.25) ,

(3.24) ,

(3.23) ,

(3.22) ,

(3.21) ,

(3.20) ,

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 Concentration

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

Conclusion

New Model (DGET, Self-Adjointness)

Better Approximation

Global Convergence (Monotone Iterative Method)

Efficiency

Easy Implementation