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

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

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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

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DFT - Density Functional Theory

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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

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Total energy: E xc : 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 E XC Approximate Modern DFT: new and improved E XC

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Simplest approach: Local Density Approximations - LDA Assume E xc equal to E xc 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….. Exchange-Correlation Functionals

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DFT calculations in practice Real solids ~10 23 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

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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

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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….. Alternative approaches

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

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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: Effective defect concentration Supercell size (unit cells) Atmospheric conditions - Thermodynamic tables - Fermi level:

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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 AZrO 3 perovskites Increasing stability in orthorhombic perovskites ∆H hydr (DFT) reproduce experimental trends (eV) (kJ/mol) CaZrO SrZrO PbZrO BaZrO

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

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Possibilities: Protons, neutral hydrogen & hydride ions Transition levels ε(0/-), ε(+/0), ε(+/-) ZnO [2] ε(+/-) above CBM H: shallow donor Only stable LaNbO 4 [3] ε(+/-) 1.96 eV below CBM favorable at high Fermi levels n-type LaNbO 4 : dominated conductivity? [2] C.G. Van de Walle and J. Neugebauer, Nature (2003), 423 [3] A. Kuwabara, Private Communication, (2008) Hydrogen in semiconductors

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Overall dominating defect (concentration-wise) Need to calculate all possible defects LaNbO 4 [3] Interstitial oxygen dominates at high Fermi levels (0 K!) [3] A. Kuwabara, Private Communication, (2008)

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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 E g (DFT) and E g (EXP) Donor type defects follow CB Correct formation energies Alternative: hybrid or semi-empirical functionals / eV BaZrO PbZrO ZnO1.93.4

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Complementary methods

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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

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Proton transport in LaPO 4 [5] Proton transport in LaPO 4 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. C 111 (2007) 11003

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Intratetrahedral jumps: High energy barrier Intertetrahedral jumps: Two tetrahedrons Lower energy barrier Intertetrahedral jumps: Three tetrahedrons Even lower energy barrier Experimentally: eV

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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(R 1,…,R N ) Forces on the ions: MD algorithms: Discretize equation of motion Trajectories: stepwise update positions and velocities

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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..

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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:

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

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

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Thank You!

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