Multiscale Materials Modeling

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Multiscale Materials Modeling

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

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Outline

• Structure
• Density Functional Theory (DFT)
• Molecular Dynamics (MD)
• Kinetic Monte Carlo (kMC)
• Continuum
• Transport
• Tunneling
• Conductance Quantization
• Non-equilibrium Greens Functions (NEGF)

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TCAD
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 gt
  continuum)

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

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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)
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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)
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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
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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)

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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)
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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
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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)
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Recrystallization
1200K for 0.5 ns
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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

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Kinetic Lattice Monte Carlo Simulations
Fundamental processes are point defect
hop/exchanges.
Vacancy must move to at least 3NN distance from
the dopant to complete one step of dopant
diffusion in a diamond structure.
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Kinetic Lattice Monte Carlo Simulations

• 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.

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3D Atomistic Device Simulation
1/4 of 40nm MOSFET (MC implant and anneal)
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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.

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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.

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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