Modelling of Defects DFT and complementary methods

1. Modelling of Defects DFT and complementary methods Tor Svendsen Bjørheim PhD fellow, FERMiO, Department of Chemistry University of Oslo NorFERM-2008, 3rd -7th of October 2008

2. Outline • Background / Introduction • DFT • Theory • DFT calculations in practice • Case studies • Supplementary Methods • Nudged Elastic Band • Molecular Dynamics • Monte Carlo approach • Summary

3. Introduction to defect modeling • Experimental techniques • Time consuming • Expensive • Inconclusive results • Supplement with computational studies • Condensed systems • Mutually interacting particles • Described by the full Hamiltonian: • Increasingly complex with larger number of electrons (3N variables) • Need simplifications! • Hartree-Fock • DFT

4. DFT - Density Functional Theory

5. Density Functional Theory • Ab initio ground state theory • Basic variable electron density, n(r) • n(r) depends only on the three spatial variables • Hohenberg-Kohn theorems (1964): • For a system of interacting particles in an external potential, the external potential and hence the total energy is a unique functional of n(r) • The ground state energy can be obtained variationally; the density that minimizes the total energy is the exact ground state density • Problem: kinetic energy of the electrons… • Practical use: Kohn-Sham approach (1965) • Introduce a reference system of non-interacting particles (with the same n(r)) • Kinetic energy easily determined: Functional = Function of a function

6. Total energy: • Exc: • Need to determine the orbitals of the reference system • Kohn-Sham equations: • Kohn-Sham potential: • Depends on the electron density  Can not be solved directly • Self consistent solutions using iterative schemes • Kohn-Sham approach • In principle exact • Do not know the form of EXC  Approximate • Modern DFT: new and improved EXC

7. Exchange-Correlation Functionals • Simplest approach: Local Density Approximations - LDA • Assume Exc equal to Exc in a homogeneous electron gas: • Locally constant electron density • Systems with slowly varying electron densities • Inadequate for systems with quickly varying electron densities • Improvements: Generelized-Gradient Approximations (GGA) • Include gradients of the electron density at each point • E.g. GGA-PW91 and GGA-PBE • Problems: • Band gaps and Van der Waals forces…..

8. DFT calculations in practice • Real solids ~1023 atoms • Huge number of wave functions.. • Need further simplifications! • Popular approach: plane waves • Periodically repeating unit cell • Bloch’s theorem: • Finite number of wave functions over an infinite number of k-points in the 1. Brillouin Zone! • K-point sampling • Electronic states at a finite number of k-points • Finite number of wave functions at a finite number of k-points in the 1. Brillouin Zone! • K-mesh needs to be chosen carefully

9. Defects in solids Solids with one or more defects Aperiodic systems Bloch’s theorem can not be applied Can not use plane wave basis sets Introduce the concept of a periodically repeating supercell Supercell method Defects in a ’box’ consisting of n unit cells Define the supercell with the defect as the new unit cell Periodic boundaries 3D periodic ordering of defects Typical size: <500 atoms Surfaces/interfaces & molecules

10. Alternative approaches • Finite-Cluster approach • Defects in finite atomic clusters • No interaction with defects in neighboring unit cells • Avoid surface effects: large clusters • Green’s Function Embedding Technique • Purely mathematical • Defect regions embedded in known DFT Green’s function of bulk • Ideal for studies of isolated defects (in theory) • Numerically challenging…..

11. Structural studies • Structural studies • Defect positions • Locate global minimum • E.g. proton positions • Local arrangement around isolated defects • E.g. local displacements in ferroelectrics

12. Thermodynamics • Formation energy of isolated point defects: • Defects that change the composition: • Total energies and a set of chemical potentials • E.g. isolated protonic defects & oxygen vacancies: • Formation: • Formation energy: • Atmospheric conditions • Thermodynamic tables • - Supercell size (unit cells) Effective defect concentration Fermi level:

13. Formation energies not directly comparable with experimental results • DFT = ground state = 0 K! • Chemical potential of the electrons • Combine to e.g. hydration: • E.g. hydration of AZrO3 perovskites • Increasing stability in orthorhombic perovskites • ∆Hhydr(DFT) reproduce experimental trends

14. Defect levels • Defect levels • Transition between charge states: Ef(q/q’) • Experimentally: DLTS • Determine the preferred charge state of defects • In supercell calculations: • Determine ΔEf for all charge states • ΔEf for all Fermi level positions • Most stable: charge state with lowest ΔEf

15. Hydrogen in semiconductors • Possibilities: • Protons, neutral hydrogen & hydride ions • Transition levels ε(0/-), ε(+/0), ε(+/-) • ZnO [2] • ε(+/-) above CBM • H: shallow donor • Only stable • LaNbO4 [3] • ε(+/-) 1.96 eV below CBM • favorable at high Fermi levels • n-type LaNbO4: dominated conductivity? [2] C.G. Van de Walle and J. Neugebauer, Nature (2003), 423 [3] A. Kuwabara, Private Communication, (2008)

16. Overall dominating defect (concentration-wise) Need to calculate all possible defects LaNbO4 [3] Interstitial oxygen dominates at high Fermi levels (0 K!) [3] A. Kuwabara, Private Communication, (2008)

17. When DFT fails… • In defect calculations: band gaps • Heavily underestimated • Both LDA/GGA functionals • Affects: • Defect levels • Defect formation energies • Correction: scissor operation! • Shift the conduction band • Align Eg(DFT) and Eg(EXP) • Donor type defects follow CB • Correct formation energies • Alternative: hybrid or semi-empirical functionals

18. Complementary methods

19. Nudged Elastic Band Method • Method for studying transition states: • Saddle points • Minimum Energy Paths (MEP) • Know initial and final states • Local/global minima • Obtained by total energy calculations (DFT) • Reaction path • Divided into ’images’ (< 20) • Image = ‘ snapshots’ between initial and final states • Equidistant images - connected by ’springs’ • Optimize each image (DFT+forces)  MEP • Chosen images not saddle points  interpolate

20. Proton transport in LaPO4 [5] • Proton transport in LaPO4 using NEB+DFT (VASP) • Saddle points: • Jump rates: • Determine dominating transport mechanism • Diffusion and conductivity: • Proton transport processes: • Rotation around oxygen • Oscillatory motion • Intra-tetrahedral jumps • Inter-tetrahedral jumps [5] R. Yu and L. C. De Jonghe, J. Phys. Chem. C111 (2007) 11003

21. Intratetrahedral jumps: • High energy barrier • Intertetrahedral jumps: • Two tetrahedrons • Lower energy barrier • Intertetrahedral jumps: • Three tetrahedrons • Even lower energy barrier • Experimentally: 0.8-0.9 eV

22. Molecular Dynamics • Used to simulate time evolution of classical many-particle systems • Obey the laws of classical mechanics • Condensed systems: • Classical particles moving under influence of an interaction potential, V(R1,…,RN) • Forces on the ions: • MD algorithms: • Discretize equation of motion • Trajectories: stepwise update positions and velocities

23. Interionic interactions: • Model potential vs. first-principles • Model potential • Parameterized to fit experimental or first principles data • Advantages • Possible to treat large systems • Long time evolution • Disadvantages • Inaccurate potentials  Poor force representation • First-principles • First-principles electronic structure calculation (e.g. DFT) at each ionic step • Advantages: • Accurate forces • Realistic dynamic description • Disadvatages • Computationally demanding • Small systems (~100 ions) • Short time periods (~ps) Good statistical accuracy Poor statistical accuracy..

24. Monte-Carlo approach • Loosely described: • Statistical simulation methods • Conventional methods (MD): • Discretize equations describing the physical process • E.g. equations of motion • Monte-Carlo approach • Simulate the physical process directly • No need to solve the underlying equations • Requirement: process described by probability distribution functions (PDF) • Average results over the number of observations • Proton diffusion • Jump frequency and PDF:

25. Summary Many different computational approaches • Systematic trends • Fundamental processes • Predict defect properties of real materials • Combine different methods

26. Acknowledgements Akihide Kuwabara Espen Flage-Larsen Svein Stølen Truls Norby Colleagues at the Solid State Electrochemistry group in Oslo Thank You !!

27. Thank You!