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/
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
The scale of things: Sub-nanometer to Macro
Natural
Adapted from http://www.sc.doe.gov/bes/
Manmade
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).
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?
Why the excitement now?
Convergence of device technology, physical instrumentation,
chemical synthesis, biological assays, theory and computation.
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.
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.
• < 1 nm Ab initio theory, fully quantum mechanical.
• Integrate appropriate and most important science from lower to
higher scale.
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: > 100 nm
Materials: surface of TiN(111)
Phenomenon: Dislocation driven surface dynamics
Techniques: Analytical model
Low energy electron micrographs of decay of
two dimensional (2D) TiN islands on TiN(111)
treal = 12 min
tmovie = 17 sec
Rate of area change
dA/dt ~ exp(-Ea/kT),
4x4 m2
Ta = 1280 C
Ea = activation
energy for atom
detachment from
step to terrace
Rate island area change dA/dt vs. temperature T
T (K)
10
3
10
2
10
1
10
0
2
dA/dt (Å /s)
1700 1615 1530 1235 1170 1105
Ea ~ 2.5 eV
6.9 7.2 7.5
9.5 10.0 10.5
-1
1/kT (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).
Low energy electron micrographs of growth
of spirals and loops of TiN on TiN(111)
Spiral
T = 1415 oC
T/Tm ~ 0.5
field of view: 2.5 m
treal = 90 s; tmovie = 9 s
2D Loop
T = 1380 oC
field of view: 1.0 m
treal = 200 s; tmovie = 21 s
2D Loop
schematic
Not BCF growth
structures
TiN(111) spiral step growth
t=0s
15 s
• near-equilibrium
• shape-preserving
31 s
 = 47 s
• period = 
47 s
•  = (2p/) ~ exp(-Ed/kT),
is thermally-activated
• absence of applied
stress & net mass change
by deposition/evaporation.
T = 1415 oC
 versus T for spirals
 is thermally-activated
T (K)
-2
 (10
(10-2 rad/s)
rad/s)
1720
1680
1640
10
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
1
6.8
6.9
7.0
-1
1/kBT (eV )
Ed = 4.50.2 eV
7.1
Ea << Ed << Eevaporation
Spiral (& loop) nucleation and
growth
MUST be due to bulk mass
transport !!
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).
rloop
At steady state:  2Ci (r)  0
rcore
R
B.C.s:  DsC(r) 
2π rcore r  r
core
 k s [C(rloop ) - Ceq
loop ]
r  rloop
Step velocity:
drloop
dt
 Ωk s [C(rloop ) - C
eq
loop
ΩR  1 
]=


2π  rloop 
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
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)
rloop
rcore
Results of model consistent with observations:
(1) Loop or spiral growth rate dA/dt and  are constant
(2) Both are thermally activated
(3) Activation energy Ed corresponds to facile point defect
migration along bulk dislocation cores.
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 = 4.5±0.2 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/~khare/pubs/
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
Measuring MRO by Fluctuation Transmission
Electron Microscopy

incident electrons
near
random
I 2 k , Q 
I k , Q 
far from
B
ordered clusters
(low variance V(k) in I(k))
V (k , Q ) 
B
2
(high variance V(k) in I(k))
 1  f  g 2 r12 , g 3 r12 , r13 , g 4 r12 , r13 , r14 
P. M. Voyles, Ph.D. Thesis, UIUC (2000).
Typical Variance Data
• 15 Å image resolution
• peaks at a-Si diffraction
maxima
Courtesy, Nittala et al.
• average of 8-10 V(k)
traces
• error bars: one s
mean
Medium range order (MRO) everywhere
3.5
RMS no H
RMS 15 at. % H
RMS 20 at. % H
3.0
0.08
2.5
0.06
440 °C  0.2: polycrystalline
350 °C: crystals
300 °C: no crystals
250 °C
200 °C
 0.2
V(k)
2.0
0.04
1.5
0.02
1.0
0.00
0.5
0.3
0.4
0.5
0.6
-1
k (Å )
0.0
0.3
0.4
0.5
-1
0.6
0.7
k (Å )
• 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.
0.7
Para-crystalline (p-Si) model of a-Si
c-Si nano-crystals
continuous random
network (CRN) matrix
Grains are randomly
Orientated and highly strained
==> Material is diffraction
amorphous.
+
=
CRN + nano =
c-Si
p-Si has medium range order (MRO)
p-Si
Algorithm to make p-Si models
1. First place grains of bulk terminated c-Si in a fixed volume V.
Atoms in these grains are called grain atoms.
2. Then randomly distribute atoms in the remaining volume to
create a correct density of a-Si. These atoms are matrix atoms.
3. Connect all matrix atoms in a perfect 4-fold random network.
4. 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!
Modified WWW Dynamics
1. Do bond switches similar to WWW method to lower the energy.
2. Use Monte Carlo probability.
3. Use Keating potential for relaxation and bond switches.
2 2


3α
3β 
R0 
2
2 2
E K  ij R ij  R 0   2 ijk  R ij  R ik  
16
8R 0 
3
4. After all moves are exhausted anneal at kT = 0.2-0.3 eV.
5. Go back to step 1 till no more convergence can be achieved.
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
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.
Change in peak heights ratio with substrate Ts
0.08
V(k)
0.06
440 °C  0.2: polycrystalline
350 °C: crystals
300 °C: no crystals
250 °C
200 °C
 0.2
0.04
0.02
0.00
0.3
0.4
0.5
0.6
0.7
-1
k (Å )
MRO increases smoothly with Ts.
Voyles, Gerbi, Treacy, Gibson, Abelson, PRL 86, 5514 (2001)
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?
Grain alignment increases second peak
Non-aligned
grains
Aligned grains
CRN reduction increases second peak
Big matrix
Small CRN
matrix
same grain size
Effect of crystallite shape on relative peak
heights
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/~khare/pubs/
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
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.
Supported nano-cluster production
PtRu5C(CO) 16 clusters were
deposited on various
graphitic C surfaces
Pure Pt clusters were deposited
on various graphitic C
surfaces by a similar process
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
Chemistry of inter-metallic nano-cluster
deposition
CO
CO
H2 673 K
CO
CO
H2, 673 K
H2, 673 K
Carbon Black
Carbon Black
CO
4
7
3
H2
PtRu5C(CO)16
CO
K
H2 473 K
PtRu5C(CO)16
673 K
H2
[PtRu5]/C
[PtRu5]/C
Nucleation and growth of bimetallic nanoparticles [PtRu5]n 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.
Scheme 1
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).
Features of the nano-clusters
Pt
Ru
Pt
Graphitic carbon support
(1) Self-organized nano-clustering on carbon,
cluster size 1.0 - 2.0 nm
10
37
1
2
92
3
185
326
4
5
525
6
Cluster order, L
(2) Cube-octohedral fcc(111) stacking
(3) Magic sizes: 10, 37, 92 … atoms
Pt goes on top and bulk bond lengths
Pt
Ru
Pt6Ru31
Pt92
Graphitic carbon support
(4) In Pt-Ru clusters Pt goes to the top layer
10
37
92
18
(5) Even small 37 and 92 atom clusters show bond lengths equal
to that in the bulk metals, on “inert” graphitic substrate!
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.
10
37
92
Substrate carbon must be playing a significant role!
185
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
Some checks on our ab initio method
Table of lattice constants in Å.
Bulk
Bulk
C
Ru
Pt
(Honeycomb
Graphite)
Experiment (E) 3.78
3.92
2.45
Theory (T)
3.76
3.91
T/E %
99.36% 99.74%
2.45
100%
Ab initio theory reproduces bond distances very well!
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
Pt6Ru31 neighbour shell distances (Å)
Expt.
NN shell
Theory Expt.
Pt-Pt
Theory Expt.
Pt-Ru
Theory
Ru-Ru
1st
2.69
2.70 2.70
2.62 2.67
2.52
2nd
3.78
N/A 3.79
3.71 3.78
3.53
3rd
4.66
4.67 4.70
4.50 4.68
4.41
4th
5.38
5.30 5.40
5.05 5.42
5.12
% of
bulk
Both ~ 97%
Expt. ~99%
Expt. ~ 100+%
Theory 93-96% Theory ~ 94-96%
Theory: PtRu simulated in vacuum
Expt.: From fits to EXAFS data on C
Percentages are comparisons with bulk values
Pt92 neighbour shell distances (Å)
NN
shell
Pt92
Expt. [ % of bulk]
Theory [% of bulk]
1st
2nd
2.76[99.57%]
3.91[99.74%]
2.71[97.77%]
3.81[97.19%]
3rd
4.78[99.56%]
4.67[97.27%]
4th
5th
5.52[99.57%]
6.18[99.71%]
5.34[96.33%]
5.96[96.16%]
Theory: Pt92 simulated in vacuum
Expt.: From fits to EXAFS data
Percentages are comparisons with bulk values
Small clusters in vacuum and on C
Average bond lengths in Å from ab initio theory
# of
Ru on C Ru in
Pt on C Pt in
atoms
vacuum
vacuum
2
2.55
1.90
2.43
2.29
dimer
3
2.54
2.24
2.52
2.44
trimer
4
2.48
2.33
2.59
2.55
capped
trimer
10
2.50
2.43
2.65
2.60
capped
10-atom
bulk
2.66
2.77
Bulk-like Bonds: A Substrate-Mediated Effect
Relative scales: Substrate versus Ru
Honeycomb structure of graphene
Substrate length scales <
adsorbate scales
Effect of substrate is not just
geometric
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
Lengths not in simple ratios,
hence adsorbate clusters
are incommensurate
Subtle electronic effect due
to graphene p electrons
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).
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.
Collaborators
Senior
Theorists
D. D. Johnson (UIUC)
Experimentalists
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)
P. Keblinski (RPI)
S. Nakhmanson (NCSU)
S. Kodambaka (UIUC)
P. M. Voyles (Wisconsin)
K. Ohmori (UIUC)
W. Swiech (UIUC)
K. Ohmori (UIUC)
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.
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!
Electronic Density Plot: Free Dimer
Z=0.125 Å
Z=0.500 Å
Z=0.250 Å
Z=0.625 Å
Z=0.375 Å
Z=0.750 Å
Free Ru2 bond length = 1.9 Å
Z=1.000 Å
Different Z slices
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.
Dimer is canted – not parallel to graphite
Ru dimer on C
slice through
Z=0.89 Å
Bottom Ru ion cores is closer to carbon surface.
Ru dimer asymmetrically placed in hexagon and canted.
Electronic Density Plot: Trimer on C
Ru trimer on C
slice through
Z=0.18 Å
Close to graphite plane
Ru trimer ion cores are not at symmetric hexagon centers.
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, …
Pt6Ru31 Bond Lengths (Å) per n.n. Shell
6
5
4
Distance
3
(A)
2
6
1
5
0
4
stance
3
(A)
2
Experiment
Theory
6
1st
5
3rd
4th
4
Pt-Ru Shells
Distance
3
Experiment
(A)
Theory
2
5
2nd
1
4
Distance
3
(A)
Experiment
2
Theory
1
1
1st
2nd
3rd
4th
0
Expe
Theo
0
1st
Pt-Pt Shells
2nd
3rd
2nd
3rd
4th
Pt-Ru Shells
H2 673 K
CO
Ru-Ru Shells
1st
4th
CO
0
6
• 99+% (94-99%) of bulk value in experimentH(theory).
, 673 K
CO
H2, 673 K
2CO
Carbon Black
Carbon Black
• No 2nd n.n. bond for Pt-Pt with
Pt atop position!
4
7
3
K
673 K
• Graphite only important for
atoms
near
graphite
surface.
H2 473
K
H
2
PtRu5C(CO)16
PtRu5C(CO)16
H2
[PtRu5]/C
[PtRu5]/C
For Pt92 cluster (5 shells): 99+% in experiment, 96-99% in theory
Ru trimer is planar, unlike dimer
Slice through
trimer atoms
Z=1.77 Å
Average distance from C-graphite remains same as dimer.
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 < 0.03 eV/ Å
• Used RISC/6000 and DEC alpha machines at UIUC
Self-organized Pt and PtRu nano-assemblies
on carbon
Pt
Ru
support
Nucleation and growth of bimetallic nanoparticles [PtRu5]n 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.
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 < 0.03 eV/ Å
• The potential also yields good surface properties such as the
diffusion barrier on Pt(111)
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
Density Functional Theory (DFT)
Synonyms: DFT = Ab initio = First Principles
Hohenberg Kohn Theorems (1964)
(1)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 = E[rr)].
(2) The density that minimizes the energy is the ground state
density and the energy is the ground state energy,
Min{E[rr)]} = E0
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)].
  


 veff [ r (r )]  n (r )   n n (r ),
 2m

2
2
N
r (r )    n (r )
2
n 1
 r (r )  3
veff [ r (r )]  vnuc. (r )   
d r   v xc [ r (r )]
 r  r  
Coulomb
potential of
nuclei
Hartree electrostatic
potential
E xc [ r (r )]
vxc [ r (r )] 
,
r(r )
Exc [ r (r )]
Exchange
correlation
potential
is universal!
Practical Algorithm
Effective Schrodinger equation for non-interactng electrons
  


 veff [ r (r )]  n (r )   n n (r ),
 2m

2
2
N
r (r )    n (r ) ,
2
n 1
Implementation:
1. 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
5. Calculate the new charge density
6. Calculate all the contributions to the effective potential and
iterate until the charge density and effective potential are selfN
consistent.
E[ r (r )]   n
7. Then calculate total energy.

n 1
Value of ab initio method
• Powerful predictive tool to calculate properties of materials
•
Fully first principles ==>
– (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
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
Computationally intensive, Si band gap is wrong
Excited electronic states difficult
Schematic of FEM measurement
FEM measures medium range order MRO
Long standing problem: Lack of a
technique for direct measurement
of Medium Range Order (MRO).
4
3
 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.
2
1
dihedral angle
φ
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.
12 p-Si models + 1 CRN model
All models made of exactly 1000 Si atoms
% of c-Si
# of c-Si
Total # of
atoms
grains
models
43%
32%
21%
1 or 2 or 4
1 or 2 or 4
1 or 2 or 4
3
3
3
11%
0%
1 or 2 or 4
0
3
1(CRN)
• 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 60
and 180o.
o
43% of c-Si differing number of grains
12% of c-Si differing number of grains
Single grain variance differing % of c-Si
Two grain variance differing % of c-Si data
Four grain variance differing % of c-Si
Effect of strain on CRN
Strain effect on single grain data
Strain effect on two grain data