Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab Approaches for Nanomaterials Modeling Scott Dunham Professor, Electrical Engineering Adjunct Professor, Materials Science & Engineering Adjunct Professor, Physics University of Washington

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 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.

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab VLSI Technology CAD Process Simulator Device Simulator Process Schedule Device Structure Electrical Characteristics Current semiconductor processes/devices designed via technology computer aided design (TCAD)  Similar tools for design of other nano systems would be extremely powerful min, 1000C, O 2 2. CVD Nitride, 800C, 20 min 3. Implant 2keV, 210 15 As 4. …

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab Ab-initio (DFT) Modeling Approach Model Expt. Effect DFT Validation & Predictions Critical Parameters Behavior Verify Mechanism Ab-initio Method: Density Functional Theory (DFT)

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 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 Length scale1 nm10 nm25 nm100 nm Time scale* ≈ psec≈ nsec≈ msec ≈ sec

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 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.

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab Multi-electron Systems Hamiltonian (KE + e - /e - + e - /V ext ): Hartree-Fock—build wave function from Slater determinants: The good: Exact exchange The bad: Correlation neglected Basis set scales factorially [N k !/(N k -N)!(N!)]

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab DFT: Density Functional Theory Problem: For more than a couple of electrons, direct solution of Schrödinger Equation intractible: (r 1,r 2,…,r n ). Solution: Hohenberg-Kohn Theorem There 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)

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 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)

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 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)

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 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 Iteration Self-consistent KS equations: Ionic Iteration  Determine ionic forces  Ionic movement Arrangement of atoms

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 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)

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 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)

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab DFT: Only as good as its results Cohesive energy: J.P. Perdew et al., Phys. Rev. Lett. 77, 3865 (1996) Silicon properties: MethodLi 2 C2H2C2H2 20 simple molecules (mean absolute error) Experiment1.04 eV17.56 eV- Theoretical errors: Hartree-Fock LDA GGA (PW91) eV eV eV eV 2.39 eV 0.43 eV 3.09 eV 1.36 eV 0.35 eV PropertyExperimentLDAGGA 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

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab Predictions of DFT Atomization energy: J.P. Perdew et al., Phys. Rev. Lett. 77, 3865 (1996) Silicon properties: MethodLi 2 C2H2C2H2 20 simple molecules (mean absolute error) Experiment1.04 eV17.56 eV- Theoretical errors: Hartree-Fock LDA GGA (PW91) eV eV eV eV 2.39 eV 0.43 eV 3.09 eV 1.36 eV 0.35 eV PropertyExperimentLDAGGA 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

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab Elastic Properties of Silicon Lattice constant: Hydrostatic: Elastic properties: Uniaxial: MethodK [GPa]Y [GPa] DFT (LDA) DFT (GGA) Literature Methodb Si [Å] Experiment5.43 DFT (LDA)5.39 DFT (GGA)5.45 MethodC 11 [GPa]C 12 [GPa] DFT (LDA)15666 DFT (GGA)15555 Literature16765 GGA

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 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.

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 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

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 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

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab F n V m Clusters Idea: Fluorine decoration of vacancies  immobile clusters F n V m clusters are formed via decoration of dangling Si bonds Ab-initio binding energies: Reference: F i, V or V 2 Results:  F n V m clusters have large binding energies

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab Charge States Analysis F n V m Idea: Fluorine decoration of vacancies  immobile clusters For mid gap Fermi level dominant clusters are uncharged Reference: F i, V and V 2

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab Extended Fluorine Continuum Model Formation: Dissociation: Diffusion of Mobile F, I, V Defect Model & Boundary Conditions:  Extended defect model including I n, V n, and {311} defects  Thin oxide layer on surface (20 Å) (segregation & diffusion of F i ) F n V dissoc. E [eV] n=1 n=2 n=3 n= F n V 2 dissoc. E [eV] n=4 n=5 n=

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab Fluorine Redistribution Simulation:Experiment: Fluorine diffusion mechanism: Fast diffusing F i get trapped in V Release F i via I F decoration of V leads to F dissolving from deeper regions (I excess) and accumulation near surface (V excess) I rich V rich

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 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

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab Simplified Fluorine Model F dose time evolution:Interaction Mechanism: 30min anneal at 650°C After 20s, F 3 V and F 6 V 2 Dissolution reaction:Key parameter: a/c interface F profile depth

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab Fluorine Diffusion Experiment: 20keV 3x10 15 cm -2 F implant 1050°C spike anneal No other dopants

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab F Effect on Phosphorus Source/Drain Pocket

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab F Effect on Boron Source/Drain Pocket

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 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 F n V m clusters F affects P and B diffusion indirectly via modification of local point-defect concentrations

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 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 StrongWeak RRSGGP RWRPNG VRSPTP SRWRGPNG WIRRNGGP WWSRTGPP

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab Peptide-Surface Interactions  Structure from MD (Oren)  Distinctive 3N structures  Explore via DFT

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab Peptide-Surface Interactions  Truncated arginine with O acceptor.  111 Au surface (upper layers free)…no H 2 O  VASP-GGA structure minimization (limited)

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 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…)

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab Peptide-Metal Interactions  3D Charge distribution for peptide/surface system

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 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)

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab Recrystallization 1200K for 0.5 ns

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 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 As n V clusters at high C As.

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 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 MD Optimizer Error Configs. F i, E Parameters

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 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

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 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.

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 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 10 7 or more possible (depends on ΔE and T).

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 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

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 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 = t max. Henkelman & Jonsson., JCP. 115, 9657 (2001). Adaptive Kinetic Monte Carlo

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 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

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 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

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 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 Simulations

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab Simulations include B, As, I, V, B i, As i 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 Simulations KLMC Simulations

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab Kinetic Lattice Monte Carlo Simulations 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 Simulations

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 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.

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab Problem: Once a cluster is formed, the system can spend a long time just making transitions within a small group of states. Acceleration of KLMC Simulations

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab A solution is to consider the group of states as a single effective state. States inside the group are near local equilibrium. Acceleration of KLMC Simulations

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 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 Simulations

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 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. High Concentration Arsenic Diffusion

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab High Concentration As Diffusion  For moderate doping, D As / (n/n i ) [Diffusion with I -, V - ]  Above cm -3, As diffusion increases very rapidly

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 1/4 of 40nm MOSFET (MC implant and anneal) 3D Atomistic Device Simulation

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 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

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab Full Kinetic Precipitation Model (FKPM) S.T. Dunham, J. Electrochem. Soc. (1995) What if n is large (expensive)? “RKPM”

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab Reduced Moment-based Precipitation Model I. Clejan et al., J. Appl. Phys. (1995) A.H. Gencer et al., J. Appl. Phys. (1997) Normalization

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab Comparison to Experiments Experiment from Cowern et al.  40keV Si ion implantation with a dose of 2x10 13 cm -2 over buried B epi- layers  Interstitial supersaturation were extrapolated from boron profile during the anneal at 600, 700, and 800 o C Simulation results  Applied FKPM and RKPM-DFA  Good agreements for both FKPM and RKPM-DFA to the experimental data 40keV Si ion implantation 2x10 13 cm -2 B 0.9 μm 1.3 μm 2.5 μm Epi- N.E.B Cowern et al., J. Appl. Phys. (1999) This work was in conjunction with Chen-Luen Shih.

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab Comparison between FKPM and RKPM-DFA  Time evolution of m 0, m 1,and C I at 700 o C  Time evolution of average size of {311} defects at different T m1m1 m0m0 CICI

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 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.

Scott Dunham University of Washington Nanotechnology Modeling EE539, Winter 2008 Nanotechnology Modeling Lab 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 difficult  Materials/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.