Multiscale Materials Modeling

Scott DunhamUniversity of Washington

Nanotechnology ModelingEE539, Winter 2008

Nanotechnology Modeling Lab

Multiscale Materials Modeling

Scott Dunham

Professor, Electrical EngineeringAdjunct Professor, Materials Science &

EngineeringAdjunct Professor, PhysicsUniversity of Washington




Structure Density Functional Theory (DFT) Molecular Dynamics (MD) Kinetic Monte Carlo (kMC) Continuum

Transport Tunneling Conductance Quantization Non-equilibrium Green’s Functions (NEGF)

Outline




TCAD

ProcessSimulator

Device Simulator

ProcessSchedule

DeviceStructure

ElectricalCharacteristics

Current technology often designed via the aid of technology computer aided design (TCAD) tools

Complex trade-offs between design choices.

Many effects unmeasurable except as device behavior

Pushing the limits of materials understanding

Solution: hierarchical modeling (atomistic => continuum)




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




Ab-initio (DFT) Modeling Approach

Model

Expt. Effect

DFT

Validation&

Predictions Critical Parameters

Parameters

Behavior

Verify Mechanism

Ab-initio Method:Density Functional Theory (DFT)




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!)]




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)




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




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 energy of given atomic structure

Applications:

Formation energies (a)

Transitions (b)

Band structure (c)

Elastic properties (talk)

…(a) (b) (c)




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




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




Kinetic Lattice Monte Carlo (KLMC)

Some problems are too complex to connect DFT directly to continuum.

Need a scalable atomistic approach.

Possible solution is KLMC.

Energies/hop rates from DFT

Much faster than MD because:

Only consider defects

Only consider transitions

Tk

EE

B

fi

2exp0




V

Vacancy Mechanism Interstitial Mechanism

Si

Dopant

Fundamental processes are point defect hop/exchanges.

Kinetic Lattice Monte Carlo SimulationsKinetic Lattice Monte Carlo Simulations

Vacancy must move to at least 3NN distance from the dopant to complete one step of dopant diffusion in a diamond structure.




Tk

EE

B

fi

2exp0

Simulations include As, I, V, Asi and interactions between them.

Hop/exchange rate determined by change of system energy due to the event.

Energy depends on configuration and interactions between defects with numbers from ab-initio calculation (interactions up to 9NN).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.

Kinetic Lattice Monte Carlo Simulations




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

3D Atomistic Device Simulation




Summary

DFT (QM) is an extremely powerful tool for: Finding reaction mechanisms Addressing experimentally difficult to access phenomena Foundation of modeling hierarchy

Limited in system size and timescale:Need to think carefully about how to apply most effectively to nanoscale systems.




Conclusions

Advancement of semiconductor technology is pushing the limits of understanding and controlling materials (still 15 year horizon). Future challenges in VLSI technology will require

utilization of full set of tools in the modeling hierarchy (QM to continuum).

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

Increasing opportunities remain as computers/ tools and understanding/needs advance.




Acknowledgements

Contributions: Milan Diebel (Intel) Pavel Fastenko (AMD) Zudian Qin (Synopsys) Joo Chul Yoon (UW) Srini Chakravarthi (Texas Instruments) G. Henkelman (UT-Austin) C.-L. Shih (UW)

Involved Collaborations: Texas Instruments SiTD, Dallas Hannes Jónsson (University of Washington)

Computing Cluster Donation by Intel

Research Funded by SRC

Multiscale Materials Modeling

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