(phd dissertation defense) theoretical and numerical investigations on crystalline solid material

Theoretical and Numerical Investigations on Crystalline Solid Material and Their Application in the Simulation of Laser-Assisted Nano-Imprinting

Ph.D. Candidate:Di-Bao Wang （王地寶）Committee Members: Cheng-Kuo. Sung , Yeau-Ren Jeng, Jong-Shinn Wu, Yetmen Wang（宋震國教授）（鄭友仁教授）（吳宗信教授）（王逸民博士） Tei-Chen Chen, Yung-Chun Lee, Yu-Neng Jeng, Chin-Hsiang Cheng, Fei-Bin Hsiao （陳鐵城教授）（李永春教授）（鄭育能教授）（鄭金祥教授）（蕭飛賓教授） Jan 18, 2008

Ph.D. Dissertation Defense Nat’l Cheng Kung Univ.

晶狀固體材料之理論與數值研究與其在雷射輔助壓印模擬之應用

contents

Introduction Thermal Motion in Crystalline Solid Improvement of THK (time-history-kernel) Method Development of ABL (absorbing boundary layer) Method Acceleration of Neighbor List Updating Application: Laser Assisted Nano Imprinting Concluding Remarks Perspectives

Introduction Thermal Motion in Crystalline Solid Improvement of THK Method Development of ABL Method Acceleration of Neighbor List Updating Application: Laser Assisted Nano Imprinting Concluding Remarks Perspectives

Introduction

• Nanotechnology (science and fabrication)1) Nano-: one angstrom ~ one micrometer2) Key players: electrons and atoms/molecules3) Physical models: QM (quantum mechanics), NM

(Newtonian mechanics), SM (statistical mechanics)

http://www.cineca.it/sap/area/chimica.htmhttp://www.wikipedia.org

http://upload.wikimedia.org/wikipedia/en/4/41/NanoCartriangle.jpg

Introduction

• Nanomechanics1) Based on: QM & NM & SM & SP (solid-state physics)2) Focusing on: mechanical behavior of atomic-scale

system under external loading (P,T, etc.)3) Linked to : continuum mechanics

www.chemsoc.org/.../ezine/2003/trebin_apr03.htm

Introduction

• Computational Nanomechanics1) Based on: nanomechanics & MD & MC2) Applied for: complex nanomechanical phenomena and d

esign of nanodevices3) Linked to : computational continuum mechanics

www.chemsoc.org/.../ezine/2003/trebin_apr03.htm

Introduction

• Computational Requirements of Nanomechanics:

① domain truncation of simulated system with physics-compatible thermostat boundary without spurious wave reflection on boundaries

② fast searching of neighboring atoms (neighbor-list updating)③ proper information-exchange algorithms for multiscale modeling

Environment Domain

∞System Domain∞

∞

Introduction

• About “Isothermal/Non-reflecting B.C.”

W. E and Z. Huang, Matching Conditions in Atomistic-Continuum Modeling of Materials, Phys. Rev. Lett. 87,135501 (2001).

=> simple parametric model, but lacks linking to lattice physics

E. G. Karpov et al, A Green's function approach to deriving non-reflecting boundary conditions in molecular dynamics simulations , Vol. 62, 9, pp.1250~1262, Int’l J. Num. Meth. Eng (2004).

=> complete formulation & enormous computation, behaves well in some cases.

Shaofan Li et al, Perfectly Matched Multiscale Simulations For Discrete Lattice Systems, Phys. Rev. B., 74, 045418 (2006)

=> simple formulation, but hard to optimize. Not thermalized so far.

Introduction

• About “Fast List-Updating” …

Z. Yao, J.-S. Wang, G.-R. Liu, and M. Cheng, “Improved neighbor list algorithm in molecular simulations using cell decomposition and data sorting method”, Computer Physics Communications 161, pp.27-35 (2004)

=> no analytical evaluation and therefore optimization restricted

D.R. Mason, “Faster neighbour list generation using a novel lattice vector representation”, Computer Physics Communications 170, pp.31-41 (2005)

=> restricted to special lattice system

G. Sutmann and V. Stegailov, “Optimization of Neighbor List Technique In Liquid Matter Simulations”, Journal of Molecular Liquids, 125, pp.197-203 (2006)

=> complete discussion but Verlet Cell-linked List is not studied

Introduction

• Research Motivation

1) study & development of isothermally non-reflecting boundary conditions for nanomechanical & nano-fluidic systems

2) rigorous modeling/simulation of nanoscale fabrication process

3) optimization of neighbor list-updating algorithms

4) determination of optimized scheme for related problems

www.nalux.co.jp/glass_e.htm nanomolding.yonsei.ac.kr/research/4_7.htm

Introduction

• Research Objectives

1) Analyze/modify the thermal motion in crystalline solid from the theory of solid-state physics

2) Improve the derivation/computation of the time-history kernel through operation of complex functions

3) Develop a general formulation to re-construct/optimize the equation of motion in absorbing boundary layer

4) Propose an optimized list-updating technique to accelerate the computation in MD simulation

5) Implement the improved/developed isothermally non-reflecting boundary condition to study the nano imprinting problem

• Introduction• Thermal Motion in Crystalline Solid • Improvement of THK Method• Development of ABL Method• Acceleration of Neighbor List Updating• Application: Laser Assisted Nano Imprinting• Concluding Remarks• Perspectives

Thermal Motion in Crystalline Solid

• Material and Lattice Structure: => 2-D hexagonal atomic lattice

Associate Cell(first-nearest

neighbor)xn &1

yn &2

Unit Cell

nt vectordisplaceme atom ...

ectorposition v atom ... ˆ)(ˆ)(

vectorsite lattice ... ˆˆindex site lattice ... ),(

21

21

nru

etyetxr

ebneannnnn

nn

yxn

yx


• Present Assumption of Inter-atomic Interaction:

Associate Cell(first-nearest

neighbor)

1) Pair-wise interaction potential

2) First-nearest neighbor interaction

3) Small displacements about equilibrium points

22',',

2

'

'''

~

)()(~)(~

equusnsnnn

extn

nnnnn

uuUK

tftuKtuM

• Equation of Motion after Harmonic Approximation:

Liu, W.K et al, “Nano Mechanics and Materials: theory, multiscale methods and applications,”Wiley, Hoboken, NJ, 2006


• Normal-Mode Solution/ Lattice Standing Waves (thermal motion)

shiftphase vector- wavenormal p

frequency normal vector onpolarizati d

amplitude normal A

nptidAtu

:as solutionmode-normal the Consider

tuKtuM

:loading external without

n

nnnn

::::

:

exp)(

0)(~)(~'

'


• Dispersion Relationship

22

122

211

2112

sinsin3coscos2)2cos(

coscos2)2cos(3

ppppp

pppkm

:problem eigenvaluethe solving by


• Thermal Motion in Crystalline Solid

• Thermal Energy in Crystalline Solid

physics thermal lstatisticaby determinedbe to ...

:modes all over summation

p

pppppn

A

nptidAtR

exp)(

ps

ps

pspspspspsps

onsmodes/phon all over mationenergy sum HH

nmode/phono each ofengergy NAPEKEH

,,

,2

,2

,,,, 21


• Classical Approach (Gibbs canonical distribution):

• Quantum-mechanical Approach (Bose-Einstein distribution):

psps

BpsBps N

TkATkH

,2

,

2,,

2

by E.G. Karpov et al, 2006

psBpsps

B

B

ps

psps

Bps

pspsps

ps

B

psps

pspsps

TkN

TkTkN

TkA

nN

A

Tk

n

nH

,,

2,

2,

,2

,

2,

,,,

2,

,,

,,,

212112

212

1exp

121

… by this dissertation


• Comparisons of thermal amplitude for each approach:

T = 10 K T = 1000 K

Nanomechanics of Solids

• Lattice Green Function

1211

''0 '

'''

)(~̂~)(~

where

Solution Homog. Solution Particular

)()()()(~)(

0)0(,0)0(:I.C.

)()(~)(~

pKMsFLtG

tutudftGtu

uu

tftuKtuM

npts

nnn

n

t

nnn

nn

nn

nnnn


• Lattice Time-History-Kernel

a dynamic boundary condition inversely solve displacements at the boundary as in an infinite lattice

1

011

'

0'

,','),'(

~̂~̂)(~)(~

)()()()(~)(

211

1

2121211

GGFLtt

tRdRuttu

nnptsnnn

n

t

nnnnnnnnn


• Verification of THK Method


• Application of THK to Computational Nanomechanics => acts as an isothermally non-reflecting B.C. => helps the lattice to achieve relaxation


• Comparison between THK and FBC + Thermal Layer => during lattice relaxation


• Comparison between THK and FBC + Thermal Layer => under external forcing


• Comments on the Present THK Method

1) The most rigorous and systematic approach so far in published literatures2) Massive computation requirement of time convolution (0 -> t)3) Corner-effect is not studied and solved so far.

2)1( isn computatio the time,simulation for whole

)()()(1

0

nnO

nOgfdgtfn

k

kknt

Improvement of THK Method

• Inverse Laplace Transform by Crump’s Method

max

max

max

1

::

)sin()(Im)exp(2)(

t/1.5ak

t/freq. of number total A

time maximum t

ttiaatt

k

A

kkk

converge easier than method of Laguerre polynomial

controllable error

access to exponential form


• Recursive Algorithm for THK Time-Convolution

kt

iazttiaB

dtttwtdtttwt

tt

THK of form complextzBt

k

kk

kk

tt

A

kkk

max

max

00

1

/)(Im2

')'()'(Im')'()'(

)(Im)(

exp)(

tnUDtnwCtnUtzD

tzzBC

tnU

dtttnwt

kbkkkb

kk

kk

kk

A

kkb

tn

)1()1()()exp(

1)exp(

)(

')'()'(

,,

1,

0

reduced to recursive operation


• Performance of Recursive Algorithm

NAOtnUdtttnwt

NOtwdttwtttNu

A

kkb

tn

N

n

nnNtN

b

2)(')'()'(

')'()'()(

1,0

2

10

2)1(:

2)1(

~logln

:

2:

22

2

NNnIntegratio Direct

NN :off)-cut (with nIntegratio Direct

NNNN

NIFFTFFT

NAAlgorithm Recursive

cc

N

j

N

jb

r

tj

tjutjC

1

1

)(

)()(

• Absorbing Boundary Layer1) A finite-thickness artificial boundary2) Re-construct the equation of motion3) Reduce reflectivity and enhance attenuation

Development of ABL Method


• General Formulation to have the solution with decaying terms in time or space

upSKuMSDF

pSp

pS

upKuMtuKtuM

npt

nnnn

ˆ~~̂ˆ~

~),(

0ˆ~̂ˆ~0)(~)(~

2*211

*

2

''

:region ABL in motion of Eqn.

:Mapping General

:lattice original in motion of Eqn.


• ω-mapping Formulation ….. only for frequency

nnn

nnnn

nnnn

uMuMuKuM

:Region ABL in Motionof Eqn.

ISi

S :Mapping-

upKuMtuKtuM

:Lattice Original in Motionof Eqn.

~~2~~

~~

0,1

0ˆ~̂ˆ~0)(~)(~

2**

'''

**

*

2

''


• p-mapping Formulation ….. only for wave-number

nKKtuKtuM

ipS

IS

upKuMtuKtuM

nnn

nnn

ss

ss

nnnn

exp~'~,0)(~)(~

0,1

~

0ˆ~̂ˆ~0)(~)(~

''

*

2

''

:Region ABL in Motion of Eqn.

:Mapping-p

:Lattice Original in Motion of Eqn.


• 1-D Investigation of ω-mapping

ppp

Mk :Dispersion

pkM

pkMe :nAttenuatio

ntpniAtu

uuukMuuMuM


n

nnnnnn

2

22*22

2**

112**

sincos1

2sin4

14sin

2sin

221

expexp)()(

)2(2


• 1-D Investigation of ω-mapping (cont.)

iipkZiipkZ

iipkZ

iRZZZZR

/1)exp()exp(/1)exp(

/1)exp(

)exp(

22

11

11

12

21

?


• 1-D Investigation of ω-mapping (cont.)Li, S., Liu, X., Agrawal, A. and To, A.C., “Perfectly Matched Multiscale Simulations for Discrete Lattice Systems,” Phys. Rev. B., 74, 045418, 2006


• 1-D Investigation of p-mapping

2sin4

0expexp)()(

0,)(2)(

22

11

pMk :Dispersion

:nAttenuationtpniAtu

ukekuukeuM


n

nnnn


• 1-D Investigation of p-mapping (cont.)

inherent diverging characteristics in the reverse direction !


• Extensive ABL Methods to reduce the reflectivity to approach the real response of unbounded systems (or THK method)


• Extensive ABL Methods (cont.) Ordinary ABL method vs. Langevin ABL method

nn

nnnn

nn

nnnn

uMuKuM

: ABL Langevin of Motionof Eqn.

uMuMuKuM

: ABLOrdinary of Motionof Eqn.

~2~~

~~2~~

*

'''

2**

'''


• Extensive ABL Methods (cont.) Langevin ABL vs. Gradual-damping Langevin ABL

nnn

nnnn

nn

nnnn

uMuKuM

: ABL Langevin-G of Motionof Eqn.

uMuKuM

: ABL Langevin of Motionof Eqn.

~2~~

~2~~

*

'''

*

'''

• Extensive ABL Methods (cont.) G-Langevin ABL vs. Pole-Shifted G-Langevin ABL

')'(exp)'(~

~ˆ~~

~~~

0

*

*

'''

*

'''

dttttuM

uMuKuM

: ABL Langevin-PSG of Motionof Eqn.

uMuKuM

: ABL Langevin-G of Motionof Eqn.

t

sfnsfn

nnn

nnnn

nnn

nnnn

dttttutt

sfn 0

)'(exp)'()(

)()()1( 21 tjtjutj n

isf

n*

1



• Extensive ABL Methods (cont.) PSG-Langevin ABL vs. Gradually Pole-Shifted Langevin ABL

')'(exp)'(~

~ˆ~~

')'(exp)'(~

~ˆ~~

0 ,,*

*

'''

0

*

*

'''

dttttuM

uMuKuM

: ABL Langevin-GPS of Motionof Eqn.

dttttuM

uMuKuM

: ABL Langevin-PSG of Motionof Eqn.

t

nsfnnsfn

nnn

nnnn

t

sfnsfn

nnn

nnnn

Interim Conclusions

• The recursive-THK is the most suitable scheme for nanomechanical system with simple geometry for the exact response & cheaper computation.

• For general problems in nanomechanical system, the GPS-Langevin ABL method is recommended for its flexibility in geometry and simplicity in formulation.

Verlet Radius Determination

interval updating-list : ,)1( max

kNktVkRR CV

JETkR

ETkkRR

C

BCV

200

2

1

0

10.... 10)1(1

108)1(1

)/(10810 2

0

2max sm

mTkVV B

(sec)10 10 3

0

030

EmRtt C

tVk max)1(

Neighbor-List Updating Acceleration

nintegratiofor time:summationfor time:

ncalculatio forcefor time:ncalculatio distancefor time:

factorreduction :atoms gneighborinidentify to time:

atoms ofnumber total: simulation MDin on counsumpti time:

i

s

f

r

n

MD

isfrnMD

N

N

Neighbor-List Updating AccelerationComputation

Estimation

submitted to Computer Physics Communication

Algorithm GVCL

VR

VRVR

2:Number Dividing-Cell dC

36/25)6/5(

ratioreduction 2

=> Generalized Verlet Cell-linked List

2/R V


GVCLComputation Time

isfrs

LsjrdhdcGVCLMD

NNN

NNNNCNNk

'

''3,

)12(1

NCNC

CN d

d

dd

,1

12 '3

'

If Cd = 1, back to algorithm VCL


It can be shown that,

(1) Optimal list-updating interval, k, may exit !!

(2) Optimal cell-dividing number, Cd, exits !!

Numerical Validation For Effects of Neighbor-Holding Number and Cell-Dividing Number. All Results Are Normaliz

ed By The Case With (k,Cd) = (1,1). (a) T = 10 K, (b) T = 300 K

VCLVCL

40 % reduction

25 % reduction

(10,3) , opt dCk (5,3) , opt dCk

Neighbor-List Updating AccelerationParameter

Optimization

Interim Conclusions The rigorously defined Verlet radius takes into account the system t

emperature T and list-updating interval k An optimization domain of (N,N”) is derived for each MD algorithm, i

ndicating algorithm VCL is the most efficient for systems in large-scale systems

List-updating interval, k, should not be chosen arbitrarily and may owns an optimized value for Verlet-list-related algorithms, depending on physical conditions of molecule system

Algorithm GVCL is predicted to reduce the computation time by 25%~40%, compared with VCL. This is verified by numerical tests with optimized cell-dividing times Cd and list-updating interval k.


repetitive domain

mold

substrateIsothermally Non-reflecting

Boundary Condition

Displacement Control

Periodic Boundary C

onditionPerio

dic

Bou

ndar

y C

ondi

tion

System Description

Laser Assisted Nanoimprinting

Governing Equation

ninteractio nickel-copper :ninteractio nickel-nickel :

ninteractiocopper -copper :

)()(20

12

2

00

NiCu

NiNi

CuCu

jiij

RRRRij

n

jii

iji

rrR

eeE

dtrdm

ijij


Initial Conditions

K) (zero 0distant)-(equi 0

condition initial

ntdisplaceme atom :coordinate lattice :

i

i

i

i

iii

vu

uX

uXr

iu

x

y

iX


Boundary Conditions -- PBC

MDS Domain

),( minmin yx

),( maxmax yx

minxxif

)( minmax xxxx

maxxxif

)( maxmin xxxx


Domain Truncation Technique

=> lattice time-history-kernel


function sGreen' lattice :~kernel-history- timelattice :)(~

ntdisplaceme random thermally:)(

~̂~̂)(~)(~

)()()()(~)(

10

11'

0'

,','),'(

211

1

2121211

G

t

tR

GGFLtt

tRdRuttu

n

nnptsnnn

n

t

nnnnnnnnn

Mold Translation

molding

molding

demolding

demolding

holding

holding

lattice relaxation

lattice relaxation


Lattice Relaxation


Modeling of Laser Heating


Modeling of Laser Heatingcell ain absorptedenergy :E

cells within ies which varcell, ain atoms ofnumber :/tommolecule/a afor absorptedenergy :

c

c

NNEe

i

ii

iyixiyix

iyixiyix

ii

ee

eee

eVVVV

eVVVV

eee

/1

21

21

21''

21

'

2

2,

2,

2,

2,

2,

2,

2,

2,

Assume that each component of velocity is scaled by the same factor α.



Process Overview

Process Overview (cont.)


temperat ure

force

Force & Temperature


heating

moldingdemolding

Isothermal THK functions well during whole process

Substrate stress has high correlation with substrate temperature

Stress & Temperature


Both average/maximum force of substrate peaks when mold travels to the lowest position

Maximum stress is more relevant to moldingdemolding process

Average stress is closely related to substrate temperature

After molding-demolding-cooling process, substrate stress backs to its original value … no residual stress exists

Effect of Mold-Holding Interval


If this duration is not long enough, the substrate surface will be too sticky

and the trench feature cannot be successfully transferred by the mold.

Without Sufficient Holding Interval With Sufficient Holding Interval

Effect of Oxidation/ Parting-Agent


The inter-atomic potential between mold atoms and substrate atoms is

lowered artificially to model oxidation or parting agent coated on mold

surface.

The molded feature is sharper with oxidation or coated parting agent.

Without Parting Agent With Perfect Parting Agent

the real case shall be between these two

Interim Conclusion:

• THK functions well in this heating-indentation combined simulation.

• The maximum/average of substrate stress both peak at the end of molding process but they are closely related to molding process and substrate temperature respectively.

• The duration of holding process and effect of surface situation of mold is shown to be important control parameters in LADI process.

• According to this study, no residual stress exists after LADI process


Concluding Remarks

With the analogy of phonons to normal modes, the thermal amplitude for all temperature range is derived, which covers the result obtained by previous researchers above room temperature.

Based on the Crump’s method to generate and express the kernel function in a complex exponential form, the computational cost of THK time-convolution can be reduced from O(N^2) down to O(N)

It is shown that the gradually-damping pole-shifted Langevin ABL method (GPS-Langevin ABL) can efficiently avoid the low-frequency reflection in comparison with ordinary ABL methods.

The proposed GVCL algorithm, based on a rigorous definition of Verlet radius, can reduce the MD computation by 25% ~ 40%.

A two-dimensional MD simulation with isothermally non-reflecting boundary condition is successfully performed for the investigation of laser-assisted nano imprinting fabrication

Perspectives

• Extend the THK-related algorithm to polyatomic system.

• Develop an isothermal-ABL for computational nanomechanics.

• Parallel-processing version for GVCL list-updating.

• Multiscale simulation for nanoimprinting: MD + isothermal non-reflecting BC + continuum mechanics

• Develop THK/ABL in meso-dynamicsA. Strachan and B. L. Holian, “Energy Exchange between Mesoparticles and Their Internal Degrees of Freedom,” 94, 014301, PRL, 2005

Acknowledgment

• Prof. Fei-Bin Hsiao and all committee members

• Prof. Wing Kam Liu’s group in Northwestern University

• Prof. Eduard G. Karpov in University of Tennessee

• Prof. Shaofan Li in UC Berkeley

• Prof. W. E in Princeton Univ.

thanks~

(phd dissertation defense) theoretical and numerical investigations on crystalline solid material

Engineering