Electronic-structure calculations at macroscopic scales
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1 Electronic-structure calculations at macroscopic scales M. Ortiz California Institute of Technology In collaboration with: K. Bhattacharya, V. Gavini (Caltech), J. Knap (LLNL) BAMC, Bristol, March, 2007
2 Predicting Properties of Matter from Electronic Structure The quantum mechanics of electrons and ions lies at the foundation of a large part of low-energy physics, chemistry and biology The Born-Oppenheimer approximation: Decouples the electronic and nuclear motion, electrons respond instantaneously to any change in nuclear coordinates Time-independent Schrödinger equation for an isolated N-electron atomic or molecular system:
3 Quantum mechanics and material properties
4 Perfect crystals Periodic systems Many fundamental properties of materials can be elucidated by considering perfect crystal order: Excitation spectra and optical properties Equations of state, elastic moduli, phase diagrams Piezoelectric, ferroelectric, magnetic properties Bloch s theorem: Representation of the wave function in terms of plane waves Fourier transform Pseudopotentials: Core electrons are replaced by an effective potential Pseudopotentials
5 Defective crystals Supercells Some macroscopic properties of materials are mediated by lattice defects: Vacancies: Creep, spall, prismatic loops, radiation ageing Dislocations: Metal plasticity Domain walls, grain boundaries, free surfaces, interfaces The core structure of these defects can be elucidated at the atomic level Electronic structure calculations? Lattice defects break the translational symmetry of the crystal Nonperiodic calculations Recover periodic case by considering periodic distributions of defects Supercells
6 Defective crystals Supercells Electronic structure of the 30 o partial dislocation in silicon (Csányi, Ismail-Beigi and Arias, Phys. Rev. Let. 80 (1998) 3984). Ab initio study of screw dislocations in Mo and Ta (Ismail-Beigi and Arias, Phys. Rev. Let. 84 (2000) 1499).
7 Defective crystals The chasm Because of computational cost, supercells limited to small sizes Exceedingly large defect concentrations Often the objective is to predict bulk properties of defects: Vacancies: cell size ~ 100 nm Dislocation cores: cell size ~ 100 nm Domain walls: cell size ~ 1 μm Grain boundaries: cell size ~ 20 μm Small-cell calculations lead to discrepancies with experimental measurements! How can bulk properties of defects (>> million atom computational cells) be predicted from electronic structure calculations?
8 Density functional theory
9 Orbital-Free Density functional theory
10 Orbital-Free Density functional theory
11 OFDFT Coarse-graining Defective lattices, well-separated defects: Away from defects, atoms see the electron density of a uniformly distorted periodic lattice: Cauchy-Born electron density + slowly varying modulation (Blanc, Le Bris and Lions, ARMA, 2002) Only near defect cores the electron density and the electrostatic potential deviate significantly from those of a periodic lattice All-atom OFDFT calculations are inherently wasteful! Objective: Model reduction such that: The coarse graining is unstructured and adaptive OFDFT is the sole input to the model (no spurious physics) Reduced solutions all-atom OFDFT in fully-resolved limit General approach (QC-OFDFT): Derive a real space, nonperiodic, formulation of OFDFT Effect a quasi-continuum (QC) model reduction
12 OFDFT Real space formulation Fourier transform periodicity Fast-multiple structure nonlocal! pseudopotentials
13 OFDFT Real space formulation
14 OFDFT FE approximation
15 OFDFT FE approximation 4-node tetrahedral finite-elements Second-order 4-point quadrature rule Optimal mesh gradation (a priori) Dirichlet boundary conditions on electrostatic potential and electron-density Penalty method to enforce constraint Nested conjugate gradients for solving the electrostatic potential, electron-density and atomic positions (configurational-force equilibrium) Parallel implementation with domain decomposition Freudenthal s algorithm for uniform subdivision of mesh
16 Convergence test Hydrogen atom Energy of hydrogen atom as a function of number of subdivisions of initial mesh
17 Example Aluminum clusters TF-λW + LDA, λ = 1/6 Heine-Abarenkov pseudopotential Simulations of 1x1x1 3x3x3 5x5x5 9x9x9 clusters Equilibrium configurations are determined
18 Example Aluminum clusters mid-plane face Contours of electron density in 3x3x3 aluminum cluster
19 Example Aluminum clusters mid-plane face Contours of electron density in 5x5x5 aluminum cluster
20 Example Aluminum clusters Property DFT-FE KS-LDA a Experiments b Lattice parameter (a.u.) Cohesive energy (ev) Bulk modulus (Gpa) a/ Goodwin et al. (1990), Gaudion et al. (2002) b/ Brewer (1997), Gschneider (1964) Binding energy/atom (ev) Bulk modulus (GPa) n -1/3 n -1/3
21 OFDFT Coarse-graining Real-space formulation and finite-element approximation Nonperiodic, unstructured, OFDFT calculations However, calculations are still expensive: 9x9x9 cluster = 3730 atoms required 10,000 CPU hours! Objective: Model reduction such that: The coarse graining is unstructured and adaptive OFDFT is the sole input to the model (no spurious physics) Reduced solutions all-atom OFDFT in fully-resolved limit General approach (QC-OFDFT): Derive a real space, nonperiodic, formulation of OFDFT Effect a quasi-continuum 1 (QC) model reduction Challenge: Subatomic oscillations and lattice scale modulations of electron density and electrostatic potential 1 Tadmor, Ortiz and Phillips, Phil. Mag., A73 (1996) 1529.
22 QC/OFDFT Multigrid hierarchy coarse resolution, nuclei in interpolated positions atomic resolution, nuclei in arbitrary positions nuclei Coarse grid
23 QC/OFDFT Multigrid hierarchy coarse resolution, slowlyvarying correction 1 predictor corrector subatomic resolution, rapidlyvarying correction nuclei 1 Blanc, LeBris, Lions, ARMA, 164 (2002) Fago el al., Phys. Rev., B70 (2004) (R) Intermediate grid
24 QC/OFDFT Multigrid hierarchy nucleus nuclei fine grid
25 QC/OFDFT Attributes The overall complexity of the method is set by the size of the intermediate mesh (interpolation of ρ h, φ h ) All approximations are numerical: interpolation of fields, numerical quadrature No spurious physics is introduced: OFDFT is the sole input to the model A converged solution obtained by this scheme is a solution of OFDFT Coarse graining is seamless, unstructured, adaptive: no periodicity, no interfaces Fully-resolved OFDFT and continuum finite elasticity are obtained as extreme limits Million-atom OFDFT calculations possible at no significant loss of accuracy
26 QC/OFDFT convergence Al vacancy Good test case: Both electronic core and long-range electrostatic, elastic effects are important Vacancy formation energies are used as benchmarks for orbital free kinetic energy functionals TFW + LDA, Heine-Abarenkov pseudopotentialdirchlet boundary conditions on correction fields: all fields approach bulk values at the boundary Sample sizes nominally containing 4, 32, 256, 2048, 16348, and one million atoms Million-atom sample: ~1500 representative atoms and ~450,000 elements in the electronic-mesh Optimal mesh gradation h(r) ~ r 6/5 (a priori)
27 QC/OFDFT convergence Al vacancy (100) plane 4% of nuclei accounted for in calculation at no loss of accuracy! Convergence of QC reduction
28 QC/OFDFT convergence Al vacancy (100) plane E f = 0.66 ev Triftshauser, Phys Rev, B12 (1975) ,000,000 atoms required to approach bulk conditions! Convergence with material sample size
29 QC/OFDFT convergence Al vacancy QC reduction converges rapidly: 16,384-atom sample: ~200 representative atoms required for ostensibly converged vacancy formation energy. 1,000,000-atom sample: ~1,017 representative atoms and ~ 450,000 electron-density nodes give vacancy formation energy within ~0.01 ev of converged value Vacancies have long-range elastic field and convergence with respect to sample size is slow: ~1,000,000 atom sample required to attain singlevacancy formation energy! What can we learn from large cell sizes? Case study 1: Di-vacancies in aluminum Case studey 2: Prismatic loops in aluminum
30 Case study 1 Di-vacancies in Al Di-vacancy along <100> Di-vacancy along <110> Core electronic structure
31 Case study 1 Di-vacancies in Al repulsive attractive Binding energy vs. material sample size
32 Case study 1 Di-vacancies in Al Calculations evince a strong cell-size effect: binding energy changes from repulsive at large concentrations to attractive at bulk concentrations Sample sizes containing > 1,000,000 atoms must be used in order to approach bulk conditions Di-vacancy binding energies are computed to be: ev for <110> di-vacancy; ev for <100> di-vacancy Agreement with experimental values: -0.2 to -0.3 ev (Ehrhart et al., 1991; Hehenkamp, 1994) Small-cell size values consistent with previous DFT calculations (Carling et al., 2000; Uesugi et. al, 2003) : ev for <110> di-vacancy; ev for <100> di-vacancy No discrepancy between theory and experiment, only strong vacancy-concentration effect!
33 Case study 2 Prismatic loops in Al Prismatic dislocation loops formed by condensation of vacancies in quenched aluminum Kulhmann-Wilsdorff and Kuhlmann, J. Appl. Phys., 31 (1960) 516. Prismatic dislocation loops formed by condensation of vacancies in quenched Al-05%Mg Takamura and Greensfield, J. Appl. Phys., 33 (1961) 247. Prismatic dislocation loops also in irradiated materials Loops smaller than 50 nm undetectable: Nucleation mechanism? Vacancy condensation?
34 Case study 2 Prismatic loops in Al stable unstable (100) plane Quad-vacancy binding energy vs. material sample size
35 Case study 2 Prismatic loops in Al (111) (001) Non-collapsed configuration Binding energy = ev (111) 1/2<110> prismatic loop Binding energy = ev Stability of hepta-vacancy
36 Case study 2 Prismatic loops in Al Growth of planar vacancy clusters is predicted to be energetically favorable for sufficiently small concentrations Elucidation of relevant conditions requires large cell-size calculations Vacancy clustering and subsequent collapse is a possible mechanism for formation of prismatic dislocation loops Prismatic loops as small as those formed from heptavacancies are stable!
37 Concluding remarks Models must be predictive of material behavior, especially under extreme difficult-to-test conditions This requires: physics-based models (Kohn-Sham DFT!) Approximations that do not compromise the physics and that introduce controllable errors and the possibility of convergence QC/OFDFT: Real-space formulation of OFDFT Unstructured, adaptive, multi-grid finite-element discretization Numerical quadrature Behavior of material samples may change radically with size (concentration): Need electronic structure calculations at macroscopic scales!
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