Approaches for Nanomaterials Modeling

Scott DunhamUniversity of Washington

Nanotechnology ModelingEE539, Winter 2008

Nanotechnology Modeling Lab

Approaches for Nanomaterials Modeling

Scott Dunham

Professor, Electrical EngineeringAdjunct Professor, Materials Science & Engineering

Adjunct Professor, PhysicsUniversity of Washington




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.




VLSI Technology CAD

ProcessSimulator

Device Simulator

ProcessSchedule

DeviceStructure

ElectricalCharacteristics

Current semiconductor processes/devices designed via technology computer aided design (TCAD)

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

1. 30 min, 1000C, O2

2. CVD Nitride, 800C, 20 min

3. Implant 2keV, 21015 As

4. …




Ab-initio (DFT) Modeling Approach

Model

Expt. Effect

DFT

Validation&

Predictions Critical Parameters

Parameters

Behavior

Verify Mechanism

Ab-initio Method:Density Functional Theory (DFT)




Modeling Hierarchy

* accessible time scale within one day of calculation

ParameterInteraction

DFTQuantum

mechanics

MD Empirical potentials

KLMCMigration barriers

ContinuumReaction kinetics

Number of atoms 100 104 106 108

Length scale 1 nm 10 nm 25 nm 100 nm

Time scale* ≈ psec ≈ nsec ≈ msec ≈ sec




Modeling Hierarchy

Configuration energies and transition rates.

Calibration/testing of empirical potentials

Configuration energies and transition rates

Nanoscale Behavior

Behavior during regrowth, atomistic vs. global stress

Induced strains, binding/migration energies.




Multi-electron Systems

Hamiltonian (KE + e-/e- + e-/Vext):

Hartree-Fock—build wave function from Slater determinants:

The good: Exact exchange

The bad: Correlation neglected Basis set scales factorially [Nk!/(Nk-N)!(N!)]




DFT: Density Functional Theory

Problem: For more than a couple of electrons, direct solution of Schrödinger Equation intractible: (r1,r2,…,rn).

Solution: Hohenberg-Kohn TheoremThere exists a functional for the ground state energy of the many electron problem in terms of the electron density: E[n(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)




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)




Density Functional TheoryKohn-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)




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)

Calculation converged

Guess:

Electronic IterationSelf-consistentKS equations:

Ionic IterationDetermine ionic forcesIonic movement

Arrangement of atoms




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) …

(b) (c) (d)

(a) (e)




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)




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-FockLDAGGA (PW91)

-0.91 eV-0.04 eV-0.17 eV

-4.81 eV2.39 eV0.43 eV

3.09 eV1.36 eV0.35 eV

Property Experiment LDA GGA

Lattice constantBulk modulusBand gap

5.43 Å102 GPa1.17 eV

5.39 Å 96 GPa0.46 eV

5.45 Å88 GPa0.63 eV




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-FockLDAGGA (PW91)

-0.91 eV-0.04 eV-0.17 eV

-4.81 eV2.39 eV0.43 eV

3.09 eV1.36 eV0.35 eV

Property Experiment LDA GGA

Lattice constantBulk modulusBand gap

5.43 Å102 GPa1.17 eV

5.39 Å 96 GPa0.46 eV

5.45 Å88 GPa0.63 eV




Elastic Properties of Silicon

Lattice constant: Hydrostatic:

Elastic properties: Uniaxial:

Method K [GPa] Y [GPa]

DFT (LDA) 96 117 0.297

DFT (GGA) 88 126 0.262

Literature 102 131 0.266

Method bSi [Å]

Experiment 5.43

DFT (LDA) 5.39

DFT (GGA) 5.45

Method C11 [GPa] C12 [GPa]

DFT (LDA) 156 66

DFT (GGA) 155 55

Literature 167 65

GGA

GGA




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.




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




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




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




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




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]

n=1n=2n=3n=4

+4.78+2.53+0.58+0.04

FnV2 dissoc. E [eV]

n=4n=5n=6

+2.81+0.99-0.81




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 richV rich




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




Simplified Fluorine Model

F dose time evolution: Interaction Mechanism:

30min anneal at 650°C

After 20s, F3V and F6V2

Dissolution reaction: Key parameter: a/c interface F profile depth




Fluorine Diffusion

Experiment: 20keV 3x1015cm-2 F

implant 1050°C spike anneal

No other dopants




F Effect on Phosphorus

Source/Drain Pocket




F Effect on Boron

Source/Drain Pocket




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




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





Structure from MD (Oren)Distinctive 3N structuresExplore via DFT





Truncated arginine with O acceptor.111 Au surface (upper layers free)…no H2O

VASP-GGA structure minimization (limited)




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…)




Peptide-Metal Interactions

3D Charge distribution for peptide/surface system




MD Simulation

5 TC layer

1 static layer

4 x 4 x 13 cells

Initial Setup Stillinger-Weber or Tersoff Potential

Ion Implantation (1 keV)




Recrystallization

1200K for 0.5 ns




MD Results: Regrowth of Si:As

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.




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.

DFT

MDOptimizer

Error

Configs.

Fi, E

Fi, E

Parameters




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




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.




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 ΔE and T).




High

Potential

Low

Potential

Lowest

Curvature

Direction

Dimer

Image

Force with Component

Along Dimer Inverted

True Force

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)

Dimer Method




1. Find saddle points using dimer method.

2. Calculate transition rates.3. Use random number to

choose transitions.4. Advance system.5. Increment time.6. Repeat 1 to 5, until t = tmax.

Henkelman & Jonsson., JCP. 115, 9657 (2001).

Adaptive Kinetic Monte Carlo




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 (t~10-100fs)

Need a scalable atomistic approach.

Apply Transition State Theory.

Only follow major transitions

H

Tk

H

BN

j

j

N

i

i

exp13

3




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

Tk

EE

B

fi

2exp0




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

KLMC SimulationsKLMC Simulations




Tk

EE

B

fi

2exp0

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).

Kinetic Lattice Monte Carlo SimulationsKinetic Lattice Monte Carlo SimulationsKLMC SimulationsKLMC Simulations




Kinetic Lattice Monte Carlo SimulationsKinetic Lattice Monte Carlo Simulations

1

1

4

1

N

m jmjt

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.

KLMC SimulationsKLMC Simulations




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.




Problem: Once a cluster is formed, the system can spend a long time just making transitions within a small group of states.

state0

state1

state2

state3

~ eV

r01

r10

r23

r12

r21

r32

Acceleration of KLMC SimulationsAcceleration of KLMC Simulations




state0

state1

state2

state3

~eV

r0k

rk0

rk3

r12

r21

r3k

state K

A solution is to consider the group of states as a single effective state.

States inside the group are near local equilibrium.

kBkB

kBk

kk

TkETkE

jkTkEEjkpj

/exp[

)(/][exp)()(

Acceleration of KLMC SimulationsAcceleration 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.

Acceleration of KLMC SimulationsAcceleration of KLMC SimulationsAcceleration of KLMC SimulationsAcceleration of KLMC Simulations




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.

1

10

100

1000

10000

100000

1E+16 1E+17 1E+18 1E+19

Ar seni c Concent r at i on ( cm- 3)

Norm

aliz

ed V

acan

cy C

once

ntra

tion

(Cv

/Cv0

)High Concentration Arsenic DiffusionHigh Concentration Arsenic Diffusion




High Concentration As Diffusion

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

t

xD

6

2




1/4 of 40nm MOSFET (MC implant and anneal)

3D Atomistic Device Simulation




Evolution of Population

S.T. Dunham, J. Electrochem. Soc. (1995)

Evolution of size distribution critical (nucleation) but challenging for continuum simulation.

Evolution of size distribution---behavior depends on size




Full Kinetic Precipitation Model (FKPM)

S.T. Dunham, J. Electrochem. Soc. (1995)

What if n is large (expensive)?

“RKPM”




Reduced Moment-based Precipitation Model

I. Clejan et al., J. Appl. Phys. (1995)A.H. Gencer et al., J. Appl. Phys. (1997)

Normalization




Comparison to Experiments

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

40keVSi ion implantation

2x1013cm-2

B B

0.9 μm

1.3 μm2.5 μm

Epi-

N.E.B Cowern et al., J. Appl. Phys. (1999)This work was in conjunction with Chen-Luen Shih.




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




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.




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 difficultMaterials/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.

Documents

Approaches for Nanomaterials Modeling