(phd dissertation defense) theoretical and numerical investigations on crystalline solid material
TRANSCRIPT
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
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
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etyetxr
ebneannnnn
nn
yxn
yx
Thermal Motion in Crystalline Solid
• 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
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'''
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)()(~)(~
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extn
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• Equation of Motion after Harmonic Approximation:
Liu, W.K et al, “Nano Mechanics and Materials: theory, multiscale methods and applications,”Wiley, Hoboken, NJ, 2006
Thermal Motion in Crystalline Solid
• Normal-Mode Solution/ Lattice Standing Waves (thermal motion)
shiftphase vector- wavenormal p
frequency normal vector onpolarizati d
amplitude normal A
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Thermal Motion in Crystalline Solid
• Dispersion Relationship
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Thermal Motion in Crystalline Solid
• Thermal Motion in Crystalline Solid
• Thermal Energy in Crystalline Solid
physics thermal lstatisticaby determinedbe to ...
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Thermal Motion in Crystalline Solid
• Classical Approach (Gibbs canonical distribution):
• Quantum-mechanical Approach (Bose-Einstein distribution):
psps
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,2
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by E.G. Karpov et al, 2006
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Thermal Motion in Crystalline Solid
• Comparisons of thermal amplitude for each approach:
T = 10 K T = 1000 K
• 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
Nanomechanics of Solids
• Lattice Green Function
1211
''0 '
'''
)(~̂~)(~
where
Solution Homog. Solution Particular
)()()()(~)(
0)0(,0)0(:I.C.
)()(~)(~
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tutudftGtu
uu
tftuKtuM
npts
nnn
n
t
nnn
nn
nn
nnnn
Nanomechanics of Solids
• 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
Nanomechanics of Solids
• Verification of THK Method
Nanomechanics of Solids
• Application of THK to Computational Nanomechanics => acts as an isothermally non-reflecting B.C. => helps the lattice to achieve relaxation
Nanomechanics of Solids
• Comparison between THK and FBC + Thermal Layer => during lattice relaxation
Nanomechanics of Solids
• Comparison between THK and FBC + Thermal Layer => under external forcing
Nanomechanics of Solids
• 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
Improvement of THK Method
• Recursive Algorithm for THK Time-Convolution
kt
iazttiaB
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THK of form complextzBt
k
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)(Im)(
exp)(
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)1()1()()exp(
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reduced to recursive operation
Improvement of THK Method
• Performance of Recursive Algorithm
NAOtnUdtttnwt
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)(
)()(
• 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
• Absorbing Boundary Layer1) A finite-thickness artificial boundary2) Re-construct the equation of motion3) Reduce reflectivity and enhance attenuation
Development of ABL Method
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.
Development of ABL Method
• 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
''
Development of ABL Method
• 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.
Development of ABL Method
• 1-D Investigation of ω-mapping
ppp
Mk :Dispersion
pkM
pkMe :nAttenuatio
ntpniAtu
uuukMuuMuM
:Region ABL in Motionof Eqn.
n
nnnnnn
2
22*22
2**
112**
sincos1
2sin4
14sin
2sin
221
expexp)()(
)2(2
Development of ABL Method
• 1-D Investigation of ω-mapping (cont.)
iipkZiipkZ
iipkZ
iRZZZZR
/1)exp()exp(/1)exp(
/1)exp(
)exp(
22
11
11
12
21
?
Development of ABL Method
• 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
Development of ABL Method
• 1-D Investigation of p-mapping
2sin4
0expexp)()(
0,)(2)(
22
11
pMk :Dispersion
:nAttenuationtpniAtu
ukekuukeuM
:Region ABL in Motionof Eqn.
n
nnnn
Development of ABL Method
• 1-D Investigation of p-mapping (cont.)
inherent diverging characteristics in the reverse direction !
Development of ABL Method
• Extensive ABL Methods to reduce the reflectivity to approach the real response of unbounded systems (or THK method)
Development of ABL 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**
'''
Development of ABL Method
• 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
Development of ABL Method
Development of ABL Method
• 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.
• 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
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
Neighbor-List Updating Acceleration
GVCLComputation Time
isfrs
LsjrdhdcGVCLMD
NNN
NNNNCNNk
'
''3,
)12(1
NCNC
CN d
d
dd
,1
12 '3
'
If Cd = 1, back to algorithm VCL
Neighbor-List Updating Acceleration
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.
Neighbor-List Updating Acceleration
• 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
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
Laser Assisted Nanoimprinting
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
Laser Assisted Nanoimprinting
Boundary Conditions -- PBC
MDS Domain
),( minmin yx
),( maxmax yx
minxxif
)( minmax xxxx
maxxxif
)( maxmin xxxx
Laser Assisted Nanoimprinting
Domain Truncation Technique
=> lattice time-history-kernel
Laser Assisted Nanoimprinting
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
Laser Assisted Nanoimprinting
Lattice Relaxation
Laser Assisted Nanoimprinting
Modeling of Laser Heating
Laser Assisted Nanoimprinting
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.
Laser Assisted Nanoimprinting
Laser Assisted Nanoimprinting
Process Overview
Process Overview (cont.)
Laser Assisted Nanoimprinting
temperat ure
force
Force & Temperature
Laser Assisted Nanoimprinting
heating
moldingdemolding
Isothermal THK functions well during whole process
Substrate stress has high correlation with substrate temperature
Stress & Temperature
Laser Assisted Nanoimprinting
Both average/maximum force of substrate peaks when mold travels to the lowest position
Maximum stress is more relevant to molding- demolding 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
Laser Assisted Nanoimprinting
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
Laser Assisted Nanoimprinting
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
Laser Assisted Nanoimprinting
• 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
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
• 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
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~