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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.

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Presentation on theme: "Modelling of Defects DFT and complementary methods Tor Svendsen Bjørheim PhD fellow, FERMiO, Department of Chemistry University of Oslo NorFERM-2008, 3."— Presentation transcript:

1 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

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: 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

7 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

8 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

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 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

11 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: Effective defect concentration Supercell size (unit cells) Atmospheric conditions - Thermodynamic tables - 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 AZrO 3 perovskites  Increasing stability in orthorhombic perovskites  ∆H hydr (DFT) reproduce experimental trends (eV) (kJ/mol) CaZrO SrZrO PbZrO BaZrO

14 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

15 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

16 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)

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 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

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 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

21 Intratetrahedral jumps:  High energy barrier Intertetrahedral jumps:  Two tetrahedrons  Lower energy barrier Intertetrahedral jumps:  Three tetrahedrons  Even lower energy barrier Experimentally: 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(R 1,…,R N )  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!

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