1. Multiscale Materials Modeling
Scott Dunham
Professor, Electrical Engineering
Adjunct Professor, Materials Science & Engineering
Adjunct Professor, Physics
University of Washington

2. Outline Structure Transport Density Functional Theory (DFT)
Molecular Dynamics (MD)
Kinetic Monte Carlo (kMC)
Continuum
Transport
Tunneling
Conductance Quantization
Non-equilibrium Green’s Functions (NEGF)

3. TCAD
Current technology often designed via the aid of technology computer aided design (TCAD) tools
Process
Simulator
Device
Schedule
Structure
Electrical
Characteristics
Complex trade-offs between design choices.
Many effects unmeasurable except as device behavior
Pushing the limits of materials understanding
Solution: hierarchical modeling (atomistic => continuum)

4. MD Empirical potentials
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

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

6. 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!)]

7. Hohenberg-Kohn 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)

8. 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)

9. 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

10. Implementation of DFT in VASP
Calculation converged
Guess:
Electronic Iteration
Self-consistent
KS equations:
Ionic Iteration
Determine ionic forces
Ionic movement
Arrangement of atoms
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)

11. 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)

12. 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

13. 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)

14. Recrystallization
1200K for 0.5 ns

15. 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

16. 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.

17. 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.

18. 3D Atomistic Device Simulation
1/4 of 40nm MOSFET (MC implant and anneal)

19. 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.

20. 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.

21. 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