Quantum Monte Carlo Benchmarks Density Functionals: Si Defects

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1 Quantum Monte Carlo Benchmarks Density Functionals: Si Defects K P Driver, W D Parker, R G Hennig, J W Wilkins (OSU) C J Umrigar (Cornell), R Martin, E Batista, B Uberuaga (LANL), J Heyd, G Scuseria (Rice) Supported by the DOE and NSF Computation: OSC, NERSC, NCSA Jacob s Ladder of E XC approximations in Density Functional Theory (DFT) [1]: H DF T = T + V Hartree + [n, ] Experiment Heaven of Chemical Accuracy Quantum Monte Carlo () dependence Many body wavefunction density + density gradient + some exact exchange () (mixture of PBE and HF) [2] density + density gradient + orbital kinetic energy (TPSS) density + density gradient (PW91, PBE) cements charge density [1] J P Perdew et al Climbing the Density Functional Ladder Phys Rev Lett 91, (23) [2] J Heyd and G Scuseria, J Chem Phys 12, 7274 (24)

2 Silicon is Important Silicon is the cornerstone of microelectronics Formation and diffusion of dopant impurities is crucial Ion implantation-induced interstitials nucleate extended defects Experiments do not agree on interstitial energetics Band Gap [ev] nts XC Energy [ev] Bulk Silicon Band Gap m Experiment : Williamson et al, Phys Rev B, 57 (1998) Energy difference [mev/atom] m Diamond to β tin Phase Energy Difference : R Hennig; See Alfe et al, Phys Rev B, 7 (24) Use to identify accurate density functionals for theoretical investigations

3 Quantum Monte Carlo Method Trial Wave Function and Jastrow Factor Ψ T = (Slater determinant) (Jastrow factor) Density-Functional Calculations J L Martin s CPW2, VASP, and Gaussian Variational Monte Carlo Optimize Jastrow by energy minimization [3] Diffusion Monte Carlo Stochastic method of solving the many-body Schrödinger equation Projects out the ground state 64-atom cell: 6, cpu hours and 4 GB per processor Approximations* Controlled Statistical (increase MC steps) Finite-size (larger systems) Time-step (smaller time step) Population control (more walkers) Grid-size (decrease grid spacing) Uncontrolled Pseudopotential** Pseudopotential-locality Fixed node *See W Parker, Session U27 **See R Hennig, Session U27 [3] Umrigar and Filippi, Phys Rev Lett (25)

4 Accuracy of Functionals for Single Interstitials 6 Neutral Defects 16 atom cells cements Formation energy [ev] m m m X H T Single interstitial results agree with earlier by Leung et al [4] Observe systematic improvement of density functionals, and is accurate [4] W K Leung, R J Needs, G Rajagopal, S Itoh and SIhara, Phys Rev Lett 83, 2351 (1999)

5 Accuracy of Functionals for Diffusion Barriers Energy [mev] Neutral Defects 64 atom cells ments X X-H H T T-X X Reaction coordinate Lowest barrier from X to H defect is similar in and DFT Neutral T defect formation energy and barrier are higher in

6 Accuracy of Functionals for Di and Tri-Interstitials g replacements Formation Energy (ev) nts XC I a Di-interstitial Formation Energies I 2 a I a 2 Defect Type I 2 b I b 3 I 3 a I 3 b Defect Type Tri-interstitial Formation Energies In di-interstitials, and display ladder trend In tri-interstitials, improves for less distorted defects I b 3 I 3 c

7 Conclusions can accurately benchmark density functionals Perdew s Jacobs Ladder of functionals shows systematic improvement in representative silicon calculations hybrid functional closely reproduces -validation of functionals allows for accurate DFT beyond eg: larger systems or dynamics 6 Neutral Defects 16 atom cells Formation energy [ev] m m m X H T