Approaches%20for%20Nanomaterials%20Modeling

1
Approaches for Nanomaterials Modeling

• Scott Dunham
• Professor, Electrical Engineering
• Adjunct Professor, Materials Science
  Engineering
• Adjunct Professor, Physics
• University of Washington

2
Introduction

• Modeling and simulation provides powerful tool
  for process/device development.
• In semiconductor industry, this is called
  Technology Computer Aided Design.
• Essential to rapid advancement of technology.
• Crucible for development of new approaches.
• Can we build on that foundation to efficiently
  develop understanding and tools for
  Nanotechnology?
• Apply modeling at several levels
• First principles (DFT) calculations of physical
  and electronic structure to guide.
• Empirical atomistic simulations molecular
  dynamics (MD)
• Mesoscale models to span time/spatial scales
  (e.g., MC)
• Continuum simulation of functional systems.

3
VLSI Technology CAD
Current semiconductor processes/devices designed
via technology computer aided design (TCAD)
1. 30 min, 1000C, O2 2. CVD Nitride, 800C, 20
min 3. Implant 2keV, 2?1015 As 4.

• Similar tools for design of other nano systems
  would be extremely powerful.

4
Ab-initio (DFT) Modeling Approach
Expt. Effect
Behavior
Validation Predictions
Critical Parameters
Model
DFT
Ab-initio Method Density Functional Theory (DFT)
Parameters
Verify Mechanism
5
Modeling Hierarchy

• accessible time scale within one day of
  calculation

Parameter Interaction DFT Quantum mechanics MD Empirical potentials KLMC Migration barriers Continuum Reaction kinetics
Number of atoms 100 104 106 108
Length scale 1 nm 10 nm 25 nm 100 nm
Time scale psec nsec msec sec
6
Modeling Hierarchy
Induced strains, binding/migration energies.
Configuration energies and transition rates.
Nanoscale Behavior
Calibration/testing of empirical potentials
Behavior during regrowth, atomistic vs. global
stress
Configuration energies and transition rates
7
Multi-electron Systems

• Hamiltonian (KE e-/e- e-/Vext)
• Hartree-Fockbuild wave function from Slater
  determinants
• The good
• Exact exchange
• The bad
• Correlation neglected
• Basis set scales factorially Nk!/(Nk-N)!(N!)

8
DFT Density Functional Theory

• Problem
• For more than a couple of electrons, direct
  solution of Schrödinger Equation intractible
  ?(r1,r2,,rn).
• Solution Hohenberg-Kohn Theorem
• There exists a functional for the ground state
  energy of the many electron problem in terms of
  the electron density En(r).
• Caveat
• No one knows what it is, but we can make guess
• Hamiltonian
• Functional

P. Hohenberg and W. Kohn, Phys. Rev. 136, B864
(1964)
9
Hohenberg-Kohn Theorem

• Theorem
• There is a variational functional
  for the ground state energy of the many electron
  problem in which the varied quantity is the
  electron density.
• Hamiltonian
• N particle density
• Universal functional

P. Hohenberg and W. Kohn,Phys. Rev. 136, B864
(1964)
10
Density Functional Theory

• Kohn-Sham functional
• with
• Different exchange functionals
• Local Density Approx. (LDA)
• Local Spin Density Approx. (LSD)
• Generalized Gradient Approx. (GGA)
  

Walter Kohn
W. Kohn and L.J. Sham, Phys. Rev. 140, A1133
(1965)
11
Implementation of DFT in VASP

• VASP features
• Plane wave basis
• Ultra-soft Vanderbilt type pseudopotentials
• QM molecular dynamics (MD)
• VASP parameters
• Exchange functional (LDA, GGA, )
• Supercell size (typically 64 Si atom cell)
• Energy cut-off (size of plane waves basis)
• k-point sampling (Monkhorst-Pack)

12
Sample Applications of DFT

• Idea Minimize electronic energy of given atomic
  structure
• Applications
• Atomic Structure (a)
• Formation energies (b)
• Transitions (c)
• Band structure (d)
• Charge distributions (e)

(a)
(e)
(b) (c)
(d)
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Sample Applications of DFT

• Idea Minimize energy of given atomic structure
• Applications
• Formation energies (a)
• Transitions (b)
• Band structure (c)
• Elastic properties (talk)

(a) (b)
(c)
14
DFT Only as good as its results

• Cohesive energy
• J.P. Perdew et al., Phys. Rev. Lett. 77,
  3865 (1996)
• Silicon properties

Method Li2 C2H2 20 simple molecules (mean absolute error)
Experiment 1.04 eV 17.56 eV -
Theoretical errors Hartree-Fock LDA GGA (PW91) -0.91 eV -0.04 eV -0.17 eV -4.81 eV 2.39 eV 0.43 eV 3.09 eV 1.36 eV 0.35 eV
Property Experiment LDA GGA
Lattice constant Bulk modulus Band gap 5.43 Å 102 GPa 1.17 eV 5.39 Å 96 GPa 0.46 eV 5.45 Å 88 GPa 0.63 eV
15
Predictions of DFT

• Atomization energy
• J.P. Perdew et al., Phys. Rev. Lett. 77,
  3865 (1996)
• Silicon properties

Method Li2 C2H2 20 simple molecules (mean absolute error)
Experiment 1.04 eV 17.56 eV -
Theoretical errors Hartree-Fock LDA GGA (PW91) -0.91 eV -0.04 eV -0.17 eV -4.81 eV 2.39 eV 0.43 eV 3.09 eV 1.36 eV 0.35 eV
Property Experiment LDA GGA
Lattice constant Bulk modulus Band gap 5.43 Å 102 GPa 1.17 eV 5.39 Å 96 GPa 0.46 eV 5.45 Å 88 GPa 0.63 eV
16
Elastic Properties of Silicon

• Lattice constant Hydrostatic
• Elastic properties
• Uniaxial

Method bSi Å
Experiment 5.43
DFT (LDA) 5.39
DFT (GGA) 5.45
GGA
Method C11 GPa C12 GPa
DFT (LDA) 156 66
DFT (GGA) 155 55
Literature 167 65
GGA
Method K GPa Y GPa ?
DFT (LDA) 96 117 0.297
DFT (GGA) 88 126 0.262
Literature 102 131 0.266
17
Behavior of F Implanted in Si

• Potential Advantages
• F retards B and P diffusion
• F enhances B activation (Huang et al.)
• Mysterious F behavior
• Exhibits anomalous diffusion
• Retards/enhances B, P diffusion
• Experiment
• 30keV F implant ? anneal

Data from Jeng et al.
18
Fluorine Reference Structure

• Lowest energy structure of single F
• F in bond-centered interstitial site (0.18eV
  preference over tetrahedral site)
• Diffusion barrier of ? highly
  mobile

19
Charge State of F in Si

• Lowest energy structure of single F
• F in bond-centered interstitial site (p-type
  material)
• F- in tetrahedral interstitial site (n-type
  material)
• Diffusion barrier of ?
  highly mobile

20
FnVm Clusters

• Idea Fluorine decoration of vacancies ? immobile
  clusters
• FnVm clusters are formed via decoration of
  dangling Si bonds
• Ab-initio binding energies
• Reference Fi, V or V2
• Results
• FnVm clusters have large binding
• energies

21
Charge States Analysis FnVm
Idea Fluorine decoration of vacancies ? immobile
clusters For mid gap Fermi level dominant
clusters are uncharged
Reference Fi, V and V2
22
Extended Fluorine Continuum Model

• Formation
• Dissociation
• Diffusion of Mobile F, I, V
• Defect Model Boundary Conditions
• Extended defect model including In, Vn, and 311
  defects
• Thin oxide layer on surface (20 Å) (segregation
  diffusion of Fi)

FnV dissoc. ?E eV
n1 n2 n3 n4 4.78 2.53 0.58 0.04
FnV2 dissoc. ?E eV
n4 n5 n6 2.81 0.99 -0.81
23
Fluorine Redistribution

• Simulation Experiment
• Fluorine diffusion mechanism
• Fast diffusing Fi get trapped in V
• Release Fi via I
• F decoration of V leads to F dissolving
• from deeper regions (I excess) and
• accumulation near surface (V excess)

I rich
V rich
24
F Effect on B and P

• Possible effects on dopant redistribution
• Direct interaction via B-F and P-F binding
• Indirect interaction via point-defect
  modifications
• Ab-initio calculation
• No significant binding energies
• ? indirect mechanism
• F alters local point-defect concentration
• Prediction
• F model should explain effect of F on B and P
  comprehensively

25
Simplified Fluorine Model
F dose time evolution Interaction
Mechanism 30min anneal at 650C After
20s, F3V and F6V2 Dissolution reaction Key
parameter ? a/c interface ?
F profile depth
26
Fluorine Diffusion

• Experiment
• 20keV 3x1015cm-2 F
• implant
• 1050C spike anneal
• No other dopants

27
F Effect on Phosphorus

• Source/Drain
  Pocket

28
F Effect on Boron

• Source/Drain
  Pocket

29
Summary Fluorine Model

• Tasks
• Identified F diffusion mechanism
• Developed simplified F model to understand F
  effect on P and B
• Modeled range of experimental data (Texas
  Instruments)
• Simplified F model
• Comprehensive treatment of amorphous and
  sub-amorphous conditions
• a/c depth and F profile are key parameters to
  understand F effect
• Results
• F diffusion can be understood via FnVm clusters
• F affects P and B diffusion indirectly via
  modification of local point-defect concentrations

30
Peptide-Surface Interactions

• Apply hierarchical approaches to
  understanding/modeling peptide binding to
  inorganic surfaces
• First step is to explore via DFT calculations
• Interesting problem is specific binding to
  different noble metals (Au, Ag, Pt).
• List of strong/weak binding
• trimers and quadramers (Oren)
• Picked Arginine (R) as first
• to explore

Strong Weak
RRS GGP
RWR PNG
VRS PTP
SRWR GPNG
WIRR NGGP
WWSR TGPP
31
Peptide-Surface Interactions

• Structure from MD (Oren)
• Distinctive 3N structures
• Explore via DFT

32
Peptide-Surface Interactions

• Truncated arginine with O acceptor.
• 111 Au surface (upper layers free)no H2O
• VASP-GGA structure minimization (limited)

33
Peptide-Metal Interactions

• Explore basis for peptide binding to metals.
• Charge distribution with and without arginine.
• Fractional charge transferred to O (0.2e-)
• Small induced charge on metal (exploring further)

34
Peptide-Metal Interactions

• 3D Charge distribution for peptide/surface system

35
MD Simulation
Initial Setup Stillinger-Weber or Tersoff
Potential
5 TC layer
1 static layer
4 x 4 x 13 cells
Ion Implantation (1 keV)
36
Recrystallization
1200K for 0.5 ns
37
MD Results Regrowth of SiAs

• As Diffusion in a-Si
• Enhanced relative to c-Si.
• As Bonding
• As coordination close to 3 in amorphous
• Changes to 4 in crystalline Si.
• V Incorporation
• No vacancies for low As concentrations
• Grown-in AsnV clusters at high CAs.

38
Molecular Dynamics

• MD widely and effectively used for organic
  molecules and inorganic materials.
• A key challenge is accurate (transferable)
  interatomic potentials
• A solution is use of DFT for calibration.

39
Empirical Potential Optimization

• Data set for training includes
• Lattice constant, structure
• Elastic properties (stiffness tensor)
• Point defect formation energies
• Configurations from high T ab-initio MD
• Match to both energies and forces.
• Start with pure Si and then add impurities
  one-by-one

40
Time Scale Issue

• Systems evolves slowly because there are local
  metastable states with long lifetime (high
  barriers relative to kT).
• Also need to follow atomic vibrations (fs)
• Can speed up by only
• considering only
• transitions.

41
Temperature Accelerated Dynamics

• TAD (Sorenson and Voter) speeds up MD by running
  system at higher T to find transitions.
• Multiple high T runs to identify possible
  transitions.
• Actual transition chosen based on lower T under
  study.
• Need enough to ensure finding low barrier process
  which dominates at lower T.
• Acceleration factors of 107 or more possible
  (depends on ?E and T).

42
Dimer Method

• Create dimer of system in configurational space
  near energy minimum.
• Minimize energy of dimer keeping center fixed.
• Finds lowest curvature direction (Voter 1997).
• Invert force component along dimer to define
  effective force.
• Minimize effective force.

Mirror Plane
Henkelman and Jónsson, JCP 111, 7010 (1999)
43
Adaptive Kinetic Monte Carlo
Henkelman Jonsson., JCP. 115, 9657 (2001).
44
Need for Mesoscale Models

• Some problems are too complex to connect DFT
  directly to continuum.
• High C, alloys, discrete effects, peptide binding
• MD suffers from time scale dilemma need to
  follow atomic vibrations (t10-100fs)
• Need a scalable atomistic approach.
• Apply Transition State Theory.
• Only follow major transitions

45
Kinetic Lattice Monte Carlo (KLMC)

• With crystal lattice, there is countable set of
  transitions
• Energies/hop rates from DFT
• Much faster than MD because
• Only consider defects
• Only consider transitions
• Develop discrete model for energy vs.
  configuration
• Base hop rates vary with ?E

46
KLMC Simulations
Set up crystal lattice structure
(10-50nm)3 Defects (dopant and point defects)
initialized - based on equilibrium
concentration - or imported from implant
simulation - or user-defined
47
Kinetic Lattice Monte Carlo Simulations
KLMC Simulations
Simulations include B, As, I, V, Bi, Asi and
interactions between them. Hop/exchange rate
determined by change of system energy due to the
event.
Energy depends on configuration with numbers from
ab-initio calculation (interactions up to 9NN).
48
Kinetic Lattice Monte Carlo Simulations
KLMC Simulations

• Calculate rates of all possible processes.
• At each step, Choose a process at random,
  weighted by relative rates.
• Increment time by the inverse sum of the rates.

Perform the chosen process and recalculate rates
if necessary. Repeat until conditions satisfied.
49
High Concentration As Diffusion

• DFT shows long-range As/V binding
• Possible configurations too numerous for simple
  analysis.
• Can use Kinetic Lattice Monte Carlo (KLMC)
  simulation.

50
Acceleration of KLMC Simulations
Problem Once a cluster is formed, the system
can spend a long time just making transitions
within a small group of states.
51
Acceleration of KLMC Simulations
A solution is to consider the group of states as
a single effective state. States inside the group
are near local equilibrium.
52
Acceleration of KLMC Simulations
Acceleration of KLMC Simulations
Comparison of time that a vacancy is free as a
function of doping concentration via simulations
and analytic function
Both simulations with/without acceleration
mechanism agree with the analytic prediction, but
acceleration saves orders of magnitude in CPU
time.
53
High Concentration Arsenic Diffusion
Equilibrium vacancy concentration increased
significantly since the formation energy is
lowered due to presence of multiple arsenic
atoms. At high concentration, vacancy likely
interacts with multiple dopant atoms. The barrier
is lowered due to attraction of nearby dopant
atoms.
54
High Concentration As Diffusion

• For moderate doping, DAs / (n/ni) Diffusion with
  I-, V-
• Above 1020 cm-3, As diffusion increases very
  rapidly

55
3D Atomistic Device Simulation
1/4 of 40nm MOSFET (MC implant and anneal)
56
Evolution of Population

• Evolution of size distribution critical
  (nucleation) but challenging for continuum
  simulation.
• Evolution of size distribution---behavior depends
  on size

S.T. Dunham, J. Electrochem. Soc. (1995)
57
Full Kinetic Precipitation Model (FKPM)
What if n is large (expensive)?
S.T. Dunham, J. Electrochem. Soc. (1995)
58
Reduced Moment-based Precipitation Model
Normalization
I. Clejan et al., J. Appl. Phys. (1995) A.H.
Gencer et al., J. Appl. Phys. (1997)
59
Comparison to Experiments
B B

• Experiment from Cowern et al.
• 40keV Si ion implantation with a dose of
  2x1013cm-2 over buried B epi- layers
• Interstitial supersaturation were extrapolated
  from boron profile during the anneal at 600, 700,
  and 800oC
• Simulation results
• Applied FKPM and RKPM-DFA
• Good agreements for both FKPM and RKPM-DFA to the
  experimental data

Epi-
0.9 µm
1.3 µm
2.5 µm
N.E.B Cowern et al., J. Appl. Phys. (1999) This
work was in conjunction with Chen-Luen Shih.
60
Comparison between FKPM and RKPM-DFA

• Time evolution of m0, m1,and CI at 700oC

• Time evolution of average size
• of 311 defects at different T

m1
m0
CI
61
Summary

• Advancement of nanotechnology is pushing the
  limits of understanding and controlling
  materials.
• Future challenges in nanotechnology require
  utilization of full set of tools in the modeling
  hierarchy (QM to continuum).
• Increasing opportunities remain as
  computers/tools and understanding/needs advance.

62
Challenges/Opportunities

• Complementary set of strengths/limitations
• DFT fundamental, but small systems, time scales
• KLMC scalable, but limited to predefined
  transitions
• MD for disordered systems, but limited time scale
• New, more efficient methods for long time scale
  dynamics, structure optimization.
• Meso/nanoscale systems the most difficult
• Materials/devices via model-based design.
• Optimized composition, structure, strain.
• Bio/nano (organic/inorganic) interfaces.
• Efficient empirical potentials to include
  electrostatic (organic), covalent/metallic
  (inorganic) bonding, and charge transfer.