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

交通大學數學建模與科學計算研究中心

Classical Computer

Microchips Microprocessor

MOSFET

1or 0Either :Bit

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 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’

L=IJ=34nm

Quantum Corrected Energy Transport Model (Chen & Liu, JCP 2005)

Self-Adjoint Energy Transport Model (Chen & Liu, JCP 2003)

Doping Concentration

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

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

recombination rate nqDnq nnn J

)()(

)(),(

00

2

TpTn

i

nnpp

nnpqpnR

nnnBnnn TqTk /JS

Auxiliary Relationships

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

Bohm’s Quantum Potential

,2

,2

*

2

qp

*

2

qn

p

p

qm

n

n

qm

p

n

pqDpq

nqDnq

pqppp

nqnnn

)(

)(

J

J

)O(10

Constant sPlanck' 34-

nRn in PDEorder fourth a J

Self-Adjoint QCET Model

(3.7) ,

(3.6) ,

(3.5) ,

(3.4) ,

(3.3) ,

(3.2) ,

(3.1) ,

p

n

pp

n

p

n

p

R

R

Z

Z

R

R

F

G

G

J

J

n

n

Singularly Perturbed QCET Model (Liu, Lee, & Chen, 2009 Preprint)

Dimensionless Scaling

• Nano devices extremely singular• Boundary layer• Junction layer• Quantum potential layer

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

(3.7) ,

(3.6) ,

(3.5) ,

(3.4) ,

(3.3) ,

(3.2) ,

(3.1) ,

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 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

models QCET and ETby predicted difference 60%

Alternative Future MOS

MOS Scaling Challenges

1. Technology Scaling Parasitic Effects: Leakage, Capacitance, Risistance

2. Power Limits: End of Voltage Scaling3. Band-Structure Engineering4. Scattering: e-insulator, e-imp, e-ph, e-e 5. Dopant Atom Fluctuations6. Non-Equilibrium Electron & Phonon Distrb.7. Long Range Coulomb Interactions8. Full-Band Bias-Induced Quantization9. Phonon Transport Models10.Automatic Multi-Scale Computing

MOS Simulation Challenges

Conclusion

Self-Adjoint QCET Model: More AdvancedTechnology Scaling Challenges in Physics and

EngineeringMuti-Scaling Modeling and Numerical MethodsHigh-Performance Architecture, Algorithms, an

d Coding