Some examples in theory and computation in nano-science

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Some examples in theory and computation in
nano-science
Sanjay V. Khare  Department of Physics and
Astonomy University of Toledo Ohio
43606 http//www.physics.utoledo.edu/khare/
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Outline

• About nano-science and technology
• Length Scales and Techniques
• My lines of research, some examples
• Dislocation driven surface dynamics
• Medium range order in a-Si
• Pt-Ru and Pt clusters on carbon, structure and
  electronic properties
• Summary
• The Future

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The scale of things Sub-nanometer to
MacroNatural
Manmade
Adapted from http//www.sc.doe.gov/bes/
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What Happens at the Nanoscale?

• Surfaces/interfaces between materials often
  exhibit different properties (geometric,
  electronic, and magnetic structure, reactivity,
  ) from bulk due to broken symmetry  and/or lower
  dimensionality.
• New surface and interface properties are the
  origin of new technological developments
• high-density magnetic recording, phase-change
  recording, catalysis, lab-on-a-chip devices,
  and biomedical applications (gene therapy, drug
  delivery and discovery).

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What is Nanotechnology?

• Activities at the level of atoms and molecules
  that have
• applications in the real world.
• Nanotechnology encompasses all technical fields.
• Key questions
• How can we synthesize a system?
• What are the properties (measured and
  calculated)?
• How can we take advantage of them?

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Why the excitement now?
Convergence of device technology, physical
instrumentation, chemical synthesis, biological
assays, theory and computation.
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General theme of my research

• Static
• Energetic, thermodynamic, electronic, and
  structural properties related to materials
  phenomena.
• Dynamic
• Near equilibrium and non-equilibrium mass
  transport mechanisms at surfaces.
• Techniques
• Use of appropriate theoretical and computational
  techniques.
• Touch with reality
• Direct contact with experiments through
  explanations, predictions, and direction for
  future experimental work.

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Theoretical Techniques and Length Scales

• 10 100 nm and above Continuum equations, FEM
  simulations, numerically solve PDEs, empirical
  relations.
• 1-10 nm Monte Carlo Simulations, Molecular
  Dynamics, empirical potentials.  
• lt 1 nm Ab initio theory, fully quantum
  mechanical.
• Integrate appropriate and most important science
  from lower to higher scale.

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Large length scale 100 nm
Length scale 100 nm Materials Metals,
semiconductors, metal nitrides (Ag, Pt, Si, Ge,
TiN) Phenomenon Energetics, dynamics,
fluctuations of steps, islands Techniques
Analytical, Numerical solutions to PDEs, Monte
Carlo
Example Length scale gt 100 nm Materials
surface of TiN(111) Phenomenon Dislocation
driven surface dynamics Techniques Analytical
model
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Low energy electron micrographs of decay of two
dimensional (2D) TiN islands on TiN(111)
Rate of area change dA/dt exp(-Ea/kT), Ea
activation energy for atom detachment from step
to terrace
Ta 1280 ?C
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Rate island area change dA/dt vs. temperature T
Ea 2.5 eV
Measured Ea is in agreement with detachment
limited step-curvature driven surface transport
S. Kodambaka, V. Petrova, S.V. Khare, D. Gall,
A. Rockett, I. Petrov, and J.E. Greene, Phys.
Rev. Lett. 89, 176102 (2002).
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Low energy electron micrographs of growth of
spirals and loops of TiN on TiN(111)
T/Tm 0.5
Spiral
2D Loop
2D Loop schematic
T 1415 oC
T 1380 oC
field of view 1.0 ?m
field of view 2.5 ?m
treal 200 s tmovie 21 s
treal 90 s tmovie 9 s
Not BCF growth structures
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TiN(111) spiral step growth
t 0 s
15 s

• near-equilibrium
• shape-preserving
• period ?
• ? (2p/?) exp(-Ed/kT),
• is thermally-activated
• absence of applied stress net mass change by
  deposition/evaporation.

? 47 s
47 s
31 s
T 1415 oC
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? versus T for spirals
Activation energy for island decay Ea 2.5 eV
Activation energy for spiral or loop growth Ed
4.5 eV Activation energy for sublimation
Eevaporation 10 eV Ea ltlt Ed ltlt
Eevaporation Spiral ( loop) nucleation and
growth MUST be due to bulk mass transport !!
? is thermally-activated
? (10-2 rad/s)
Ed 4.5?0.2 eV
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Modeling dislocation-driven spiral growth
Assumptions

• driving force bulk dislocation line energy
  minimization
• ? surface spiral step formation via bulk point
  defect transport

• dislocation cores emit/absorb point defects at a
  rate R(T).

C - point defect concentration (1/Å2) Ds -
surface diffusivity (Å2/s) ks -
attachment/detachment rate (Å/s) ? - area/TiN
(Å2)
? constant growth rate dA/dt
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Modeling dislocation-driven spiral growth
Analytical model, two key assumptions (1)
driving force bulk dislocation line energy
minimization ? surface spiral step formation
via bulk point defect transport (2) dislocation
cores emit/absorb point defects at a constant
rate R(T).
R(T)

• Results of model consistent with observations
• Loop or spiral growth rate dA/dt and ? are
  constant
• Both are thermally activated
• (3) Activation energy Ed corresponds to facile
  point defect
• migration along bulk dislocation cores.

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Spirals Summary

• TiN(111) step dynamics and the effect of
  surface-terminated dislocations were studied
  using LEEM (1200-1500 oC).
• Spiral step growth kinetics qualitatively
  quantitatively different from 2D TiN(111) island
  decay.
• Mechanism facile bulk point defect migration
  along the dislocations (Ed 4.50.2 eV).
• Dislocation Driven Surface Dynamics on Solids,
• S. Kodambaka, S. V. Khare, W. Sweich, K. Ohmori,
  I. Petrov,
• and J. E. Greene, Nature, 429, 49 (2004).
• Available at http//www.physics.utoledo.edu/khar
  e/pubs/

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Intermediate length scale 10 nm
Length scale 10 nm Materials amorphous
semiconductors, disordered metal alloys, silica,
(a-Si, a-SiO2, a-Al92Sm8) Phenomenon Structural
properties, order-disorder transition,
Techniques Monte Carlo, Molecular dynamics,
Image simulation
Example Length scale 10 nm Materials
a-Si Phenomenon Structural properties of a-Si
Techniques Monte Carlo simulation, image
simulation Motivation Solar cells, medium range
order
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Measuring MRO by Fluctuation Transmission
Electron Microscopy
(low variance V(k) in I(k))
(high variance V(k) in I(k))
P. M. Voyles, Ph.D. Thesis, UIUC (2000).
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Typical Variance Data
Courtesy, Nittala et al.
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Medium range order (MRO) everywhere

• All materials observed to date, a-Si, a-Ge,
  a-HfO2, a-Al92Sm8,
• a-Ge2Sb2Te5 show medium range order.
• Hypothesis PC grains are frozen-in subcritical
  crystal nuclei

Data for a-Si from Voyles et al.
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Para-crystalline (p-Si) model of a-Si
c-Si nano-crystals
continuous random network (CRN) matrix


Grains are randomly Orientated and highly
strained gt Material is diffraction amorphous.
CRN nano p-Si
c-Si
p-Si has medium range order (MRO)
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Algorithm to make p-Si models

• First place grains of bulk terminated c-Si in a
  fixed volume V. Atoms in these grains are called
  grain atoms.
• Then randomly distribute atoms in the remaining
  volume to create a correct density of a-Si. These
  atoms are matrix atoms.
• Connect all matrix atoms in a perfect 4-fold
  random network.
• Sew the grain surfaces to the matrix such that
  the (grains matrix) form a perfectly 4-fold
  coordinated network.

At this stage of construction Note bonds can be
un-physically large. Bonds are just nearest
neighbor tables not chemical bonds!
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Modified WWW Dynamics

• Do bond switches similar to WWW method to lower
  the energy.
• Use Monte Carlo probability.
• Use Keating potential for relaxation and bond
  switches.
• After all moves are exhausted anneal at kT
  0.2-0.3 eV.
• Go back to step 1 till no more convergence can be
  achieved.

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Order in Crystalline Si
crystalline Si
Crystalline Si Each atom has 4 bonds and bond
angles are fixed. There is short range order and
long range order
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Continuous random network (CRN) of Si
continuous random network (CRN) matrix
CRN Each atom has 4 nearest neighbors but bond
angles vary. There exists short-range order. But
no long range order.
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Change in peak heights ratio with substrate Ts
MRO increases smoothly with Ts.
Voyles, Gerbi, Treacy, Gibson, Abelson, PRL 86,
5514 (2001)
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Questions for theory and modeling
General How does the structure of the
disordered material affect the V(k)
data? Specific for today When is the
second peak higher than the first?
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Grain alignment increases second peak
Non-aligned grains Aligned grains
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CRN reduction increases second peak
Big matrix
Small CRN matrix same grain size
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Effect of crystallite shape on relative peak
heights
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Synopsis of a-Si modeling

• Large aligned fraction of paracrystalline grains
  give a higher second peak.
• Similar questions such as dependence of V(k) on
  grain size distribution can be explained by
  detailed modeling.
• Evidence from atomistic simulations of
  fluctuation electron microscopy for preferred
  local orientations in amorphous silicon, S. V.
  Khare, S. M. Nakhmanson, P. M. Voyles, P.
  Keblinski, and J. R. Abelson, App. Phys. Lett.
  (85, 745 (2004).
• Available at http//www.physics.utoledo.edu/khar
  e/pubs/

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Small length scale 1 nm
Length scale 1 nm Materials Metals,
semi-conductors (Ag, Pt, Si, Ge, Pt-Ru clusters,
Graphite) Phenomenon Energetics, structural
and electronic properties Techniques Ab initio,
molecular dynamics, Image simulation
Example Length scale 1 nm Materials Pt-Ru and
Pt clusters on carbon Phenomenon Structural and
electronic properties Techniques Ab initio
method Motivation Fuel cells, adsorbate
substrate interaction
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Motivation and Conclusions

• Pure Pt is used extensively as a catalyst
• Pt-Ru alloys are used a catalysts at the anode in
  fuel cells in the oxidation reaction
• 2CO O2 2CO2
• Ru prevents Pt from being poisoned.
• Model system to study binay nano-cluster
  properties
• Existing experiments at UIUC

• Close-packing geometry preferred by the clusters
• Pt segregates on top of Ru
• Novel substrate mediated effects influence the
  structure
• Nanoassemblies are supported for functional
  devices.
• Supports add (semi-infinite) periodicity and
  affect properties.

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Supported nano-cluster production

• Pure Pt clusters were deposited on various
  graphitic C surfaces by a similar process

• PtRu5C(CO) 16 clusters were deposited on various
  graphitic C surfaces

Topology of both pure Pt and Pt-Ru clusters were
then studied using various probes such as STEM,
EXAFS, XANES. The structures exhibit a raft like
shape
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Chemistry of inter-metallic nano-cluster
deposition
Nucleation and growth of bimetallic nanoparticles
PtRu5n from the cluster precursors PtRu5C(CO)5
as observed by EXAFS, occurring on C substrate.
Pt atoms segregate from the core at 400-500 K
to the surface at 700 K. Experiment C. W.
Hills et al., Langmuir 15, 690 (1999) M. S.
Nashner et al., J. Am. Chem. Soc. 120, 8093
(1998) 119, 7760 (1997) A. I. Frenkel et al.,
J. Phys. Chem B 105, 12689 (2001).
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Features of the nano-clusters
Pt

• Self-organized nano-clustering on carbon,
• cluster size 1.0 - 2.0 nm

(2) Cube-octohedral fcc(111)
stacking (3) Magic sizes 10,
37, 92 atoms
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Pt goes on top and bulk bond lengths
Pt92
Pt6Ru31
(4) In Pt-Ru clusters Pt goes to the top layer
(5) Even small 37 and 92 atom clusters show bond
lengths equal to that in the bulk metals, on
inert graphitic substrate!
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Surprise about bulk bond lengths

• Average bond lengths in clusters from the
  experiment are 99 - 100.
• In 37 free atom cluster only 8 atoms are fully
  coordinated.
• In 92 free atom cluster only 20 of atoms are
  fully coordinated.

Substrate carbon must be playing a significant
role!
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Theoretical line of attack

• Must do ab initio to get structure reliably!
• Do Pt/Ru and Ru/Pt complete surfaces with full
  coverage and clusters
• Cannot do large clusters on graphite with ab
  initio
• Do large clusters in vacuum only
• Do small ones on graphite and vacuum
• Compare results in vacuum against results on
  graphite for small clusters
• Compare with experiment

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Some checks on our ab initio method
Table of lattice constants in Å.
C (Honeycomb Graphite)
Bulk Pt
Bulk Ru
2.45
3.92
3.78
Experiment (E)
2.45
3.91
3.76
Theory (T)
100
99.74
99.36
T/E
Ab initio theory reproduces bond distances very
well!
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Pt on top of Ru always wins theoretically

• Simulated cube-octohedral nanocluster of Pt6Ru31
  with Pt on top is stable
• Simulated cube-octohedral nanocluster of Pt6Ru31
  with Pt in the middle breaks cube-octohedral
  symmetry and is unstable
• Theoretically Pt on top wins over Pt sub-surface
  by
• 0.31 eV/(surface atom) for hcp(111) Ru
  surface.
• Theoretically Pt on top wins over Pt sub-surface
  by
• 0.48 eV/(surface atom) for fcc(111) Ru
  surface.

Pt sub-surface Pt on top
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Pt6Ru31 neighbour shell distances (Å)
Theory PtRu simulated in vacuum Expt. From
fits to EXAFS data on C Percentages are
comparisons with bulk values
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Pt92 neighbour shell distances (Å)
Theory Pt92 simulated in vacuum Expt. From
fits to EXAFS data Percentages are comparisons
with bulk values
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Small clusters in vacuum and on C
Average bond lengths in Å from ab initio theory
dimer
trimer
capped trimer
capped 10-atom
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Bulk-like Bonds A Substrate-Mediated Effect
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Relative scales Substrate versus Ru
Honeycomb structure of graphene
Substrate length scales lt adsorbate
scales Effect of substrate is not just
geometric Lengths not in simple ratios, hence
adsorbate clusters are incommensurate Subtle
electronic effect due to graphene p electrons
C-C distance ( ) 1.42 Å , Center to Center(
) 2.45 Å Ru dimer on C ( ) 2.54 Å Ru
bulk bond length 2.66 Å
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Theory Enhances Understanding

• Nano-assemblies are always substrate-supported
• Substrate mediated effect
• Properties highly affected by support
• For metallic nano-clusters on carbon,
  bond-lengths and
• distributions agree with experiment once support
  is included
• Theory yields fundamental insight
• Location and electronic properties can be
  analyzed atom by atom
• Not always possible with simple experiment
• Experimental data is only simulated to fit with
  measured signal
• Ab initio methods are reliable for structural
  and electronic properties!
• S. V. Khare, D. D. Johnson et al., (In
  preparation).

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Future theory for catalytic nano-clusters

• Obtain molecular orbital picture of the bonding.
• Study catalysis on Pt-Ru surfaces.
• Investigate other alloy systems which are being
  discovered such
• as ceria, tungsten oxide, alumina and others.
• Predict new useful catalytic materials.

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Collaborators

• Senior
• Theorists
  Experimentalists
• D. D. Johnson (UIUC)
  J. E. Greene (UIUC)
  I. Petrov (UIUC)
• J. R. Abelson (UIUC)
• A. A. Rockett (UIUC)
• 
  R. G. Nuzzo (UIUC)
• Colleagues and Students
• V. Chirita (U. of Linkoping, Sweden) S.
  Kodambaka (UIUC)
• P. Keblinski (RPI) P. M. Voyles
  (Wisconsin)
• S. Nakhmanson (NCSU) K. Ohmori (UIUC)
• W. Swiech (UIUC)
• 
  K. Ohmori (UIUC)

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Institutional Support
Department of Materials Science and Engineering
and Frederick Seitz Materials Research Lab
University of Illinois at Urbana-Champaign
Illinois 61801 USA Support NSF, DARPA
Program, DOE, and ONR.
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Exciting future for synergy between theoretical
modeling and experiments

• Combination of appropriate theoretical tools for
  the right length scale and close contact with
  experimentalists is mutually fruitful!

Thank you!
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Electronic Density Plot Free Dimer
Z0.125 Å
Z0.500 Å
Z1.000 Å
Z0.250 Å
Z0.625 Å
Different Z slices
Z0.375 Å
Z0.750 Å
Free Ru2 bond length 1.9 Å
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Electronic Density Plot Dimer on C
Ru dimer on C slice through Z 0.80 Å
Jahn-Teller distortion Ru dimer ion cores are
not at symmetric hexagon centers. A single Ru
adatom favors hexagon center not side.
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Dimer is canted not parallel to graphite
Ru dimer on C slice through Z0.89 Å
Bottom Ru ion cores is closer to carbon
surface. Ru dimer asymmetrically placed in
hexagon and canted.
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Electronic Density Plot Trimer on C
Ru trimer on C slice through Z0.18 Å
Close to graphite plane
Ru trimer ion cores are not at symmetric hexagon
centers.
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Charge Difference Isosurface of Planar Ru
Trimer relative to unsupported trimer
2 e/A3 isosurface red charge
deficit yellow charge gain
From the bottom
Courtesy of Lin Lin Wang and D.D. Johnson (UIUC)

• Symmetry of the charge distribution matches the
  symmetry of the substrate - lowering energy. As
  will all 3-fold and 6-fold symmetric clusters.
• Hence cub-octahedral stacking occurs on layers
  that have such symmetry, such a 7-atom layer,

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Pt6Ru31 Bond Lengths (Å) per n.n. Shell

• 99 (94-99) of bulk value in experiment
  (theory).
• No 2nd n.n. bond for Pt-Pt with Pt atop
  position!
• Graphite only important for atoms near graphite
  surface.

For Pt92 cluster (5 shells) 99 in experiment,
96-99 in theory
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Ru trimer is planar, unlike dimer
Slice through trimer atoms
Z1.77 Å
Average distance from C-graphite remains same as
dimer.
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Ab initio method details

• LDA, Ceperley-Alder exchange-correlation
  functional as parameterized by Perdew and Zunger
• Used the VASP code with generalized ultra-soft
  Vanderbilt pseudo-potentials and plane wave basis
  set
• 14 Å cubic cell in vacuum with (4x4) graphite
  surface cell, 7 layers of vacuum
• 18 Ry. energy cut-off with G point sampling in
  the irreducible Brillouin zone
• Forces converged till lt 0.03 eV/ Å
• Used RISC/6000 and DEC alpha machines at UIUC

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Self-organized Pt and PtRu nano-assemblies on
carbon
Nucleation and growth of bimetallic nanoparticles
PtRu5n from the cluster precursors PtRu5C(CO)5
as observed by EXAFS, occurring on C substrate.
Pt atoms segregate from the core at 400-500 K
to the surface at 700 K.
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Embedded Atom Method (EAM) details

• Classical potential between atoms made up of a
  pair potential and an embedding function
• EAM analytical functional for fcc metals from
  R.A. Johnson, PRB 39,12554(1989)
• EAM potential is well fitted to cohesive energy,
  bulk modulus, vacancy formation energy and other
  properties
• Forces converged till lt 0.03 eV/ Å
• The potential also yields good surface properties
  such as the diffusion barrier on Pt(111)

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Three areas of my research
Length scale 100 nm Materials metals,
semiconductors, metal nitrides (Ag, Pt, Si, Ge,
TiN) Phenomenon Energetics, dynamics,
fluctuations of steps, islands Techniques
Analytical, Numerical solutions to PDEs, Monte
Carlo Length scale 10 nm Materials amorphous
semiconductors, disordered metal alloys, silica,
(a-Si, a-SiO2, a-Al92Sm8) Phenomenon Structural
properties, order-disorder transition,
Techniques Monte Carlo, Molecular dynamics,
Image simulation Length scale 1 nm Materials
Metals, semi-conductors (Ag, Pt, Si, Ge, Pt-Ru
clusters, Graphite) Phenomenon Energetics,
structural and electronic properties
Techniques Ab initio, molecular dynamics, Image
simulation
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Density Functional Theory (DFT)
Synonyms DFT Ab initio First Principles

• Hohenberg Kohn Theorems (1964)
• The external potential of a quantum many body
  system is uniquely determined by the r(r), so the
  total energy is a unique functional of the
  particle density E Er(r).
• The density that minimizes the energy is the
  ground state density and the energy is the ground
  state energy,
• MinEr(r) E0

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Kohn Sham Theory (1965)

• The ground state density of the interacting
  system of particles can be calculated as the
  ground state density of non-interacting particles
  moving in an effective potential veff r(r).

Coulomb potential of nuclei
Exchange correlation potential
Hartree electrostatic potential
is universal!
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Practical Algorithm

• Effective Schrodinger equation for non-interactng
  electrons

• Implementation
• Guess an initial charge density for N electrons
• 2. Calculate all the contributions to the
  effective potential
• 3. Solve the Schrodinger equation and find N
  electron states
• 4. Fill the eigenstates with electrons starting
  from the bottom
• Calculate the new charge density
• Calculate all the contributions to the effective
  potential and iterate until the charge density
  and effective potential are self-consistent.
• Then calculate total energy.

67
Value of ab initio method

• Powerful predictive tool to calculate properties
  of materials
• Fully first principles gt
• (1) no fitting parameters, use only fundamental
  constants (e, h, me, c) as input
• (2) Fully quantum mechanical for electrons
• Thousands of materials properties calculated to
  date
• Used by biochemists, drug designers, geologists,
  materials scientists, and even astrophysicists!
• Evolved into different varieties for ease of
  applications
• Awarded chemistry Nobel Prize to W. Kohn and H.
  Pople 1998

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What is it good for?
Pros Very good at predicting structural
properties (1) Lattice constant good to
1-10 (2) Bulk modulus good to 1-10 (3) Very
robust relative energy ordering between
structures (4) Good pressure induced phase
changes Good band structures, electronic
properties Good phonon spectra Good chemical
reaction and bonding pathways Cons Computational
ly intensive, Si band gap is wrong Excited
electronic states difficult
69
Schematic of FEM measurement
70
FEM measures medium range order MRO

• Long standing problem Lack of a technique for
  direct measurement of Medium Range Order (MRO).
• Diffraction is only sensitive to the 2- body
  correlation function g2(r1,2).
• 3- and 4-body correlation functions, g3(r12,r13)
  and g4(r12,r13,r14) carry MRO statistics.

dihedral angle f
1
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Basis for present work

• Keblinski et al. Quench from the melt
• Paracrystallites give
  V(k) with peaks
• S. Nakhmanson et al. Modified WWW dynamics
• Fit one
  data set for V(k)
• Studied structural, vibrational, and electronic
  properties.
• Review N. Mousseau et al. Phil. Mag. B 82, 171
  (2002).
• _________________________________________________
• Present work Follow Nakhmanson et al. make
  family of models.

72
12 p-Si models 1 CRN model
All models made of exactly 1000 Si atoms

• All models have similar pair-distribution
  function g2(r).
• All models have bond-angle distribution peaked
  at 109o 10o.
• All models have double peaked dihedral angle
  distribution at 60o
• and 180o.

73
43 of c-Si differing number of grains
74
12 of c-Si differing number of grains
75
Single grain variance differing of c-Si
76
Two grain variance differing of c-Si data
77
Four grain variance differing of c-Si
78
Effect of strain on CRN
79
Strain effect on single grain data
80
Strain effect on two grain data