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

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)

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)

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

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

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 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 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)

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

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)

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 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

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)

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

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.

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.

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

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