Presentation on theme: "Modelling of Defects DFT and complementary methods"— Presentation transcript:
1 Modelling of Defects DFT and complementary methods Tor Svendsen BjørheimPhD fellow, FERMiO, Department of ChemistryUniversity of OsloNorFERM-2008, 3rd -7th of October 2008
2 Outline Background / Introduction DFT Supplementary Methods Summary TheoryDFT calculations in practiceCase studiesSupplementary MethodsNudged Elastic BandMolecular DynamicsMonte Carlo approachSummary
3 Introduction to defect modeling Experimental techniquesTime consumingExpensiveInconclusive resultsSupplement with computational studiesCondensed systemsMutually interacting particlesDescribed by the full Hamiltonian:Increasingly complex with larger number of electrons (3N variables)Need simplifications!Hartree-FockDFT
5 Density Functional Theory Ab initio ground state theoryBasic variable electron density, n(r)n(r) depends only on the three spatial variablesHohenberg-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 densityProblem: 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 Need to determine the orbitals of the reference system Total energy:Exc:Need to determine the orbitals of the reference systemKohn-Sham equations:Kohn-Sham potential:Depends on the electron density Can not be solved directlySelf consistent solutions using iterative schemesKohn-Sham approachIn principle exactDo not know the form of EXC ApproximateModern DFT: new and improved EXC
7 Exchange-Correlation Functionals Simplest approach: Local Density Approximations - LDAAssume Exc equal to Exc in a homogeneous electron gas:Locally constant electron densitySystems with slowly varying electron densitiesInadequate for systems with quickly varying electron densitiesImprovements: Generelized-Gradient Approximations (GGA)Include gradients of the electron density at each pointE.g. GGA-PW91 and GGA-PBEProblems:Band gaps and Van der Waals forces…..
8 DFT calculations in practice Real solids ~1023 atomsHuge number of wave functions..Need further simplifications!Popular approach: plane wavesPeriodically repeating unit cellBloch’s theorem:Finite number of wave functions over an infinite number of k-points in the 1. Brillouin Zone!K-point samplingElectronic states at a finite number of k-pointsFinite 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 Supercell method Aperiodic systemsBloch’s theorem can not be appliedCan not use plane wave basis setsIntroduce the concept of a periodically repeating supercellSupercell methodDefects in a ’box’ consisting of n unit cellsDefine the supercell with the defect as the new unit cellPeriodic boundaries3D periodic ordering of defectsTypical size: <500 atomsSurfaces/interfaces & molecules
10 Alternative approaches Finite-Cluster approachDefects in finite atomic clustersNo interaction with defects in neighboring unit cellsAvoid surface effects: large clustersGreen’s Function Embedding TechniquePurely mathematicalDefect regions embedded in known DFT Green’s function of bulkIdeal for studies of isolated defects (in theory)Numerically challenging…..
11 Structural studies Structural studies Defect positions Locate global minimumE.g. proton positionsLocal arrangement around isolated defectsE.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 potentialsE.g. isolated protonic defects & oxygen vacancies:Formation:Formation energy:Atmospheric conditionsThermodynamic tables-Supercell size (unit cells)Effective defect concentrationFermi level:
13 Formation energies not directly comparable with experimental results DFT = ground state = 0 K!Chemical potential of the electronsCombine to e.g. hydration:E.g. hydration of AZrO3 perovskitesIncreasing stability in orthorhombic perovskites∆Hhydr(DFT) reproduce experimental trends(eV)(kJ/mol)CaZrO3-1.790.46-147SrZrO3-1.540.69-120-106.1PbZrO3-1.320.65-75-103BaZrO3-1.220.76-66-80
14 Defect levels Defect levels Transition between charge states: Ef(q/q’) Experimentally: DLTSDetermine the preferred charge state of defectsIn supercell calculations:Determine ΔEf for all charge statesΔEf for all Fermi level positionsMost stable: charge state with lowest ΔEf
15 Hydrogen in semiconductors Possibilities:Protons, neutral hydrogen & hydride ionsTransition levels ε(0/-), ε(+/0), ε(+/-)ZnO ε(+/-) above CBMH: shallow donorOnly stableLaNbO4 ε(+/-) 1.96 eV below CBMfavorable at high Fermi levelsn-type LaNbO4: dominated conductivity? C.G. Van de Walle and J. Neugebauer, Nature (2003), 423 A. Kuwabara, Private Communication, (2008)
16 Overall dominating defect (concentration-wise) Need to calculate all possible defectsLaNbO4 Interstitial oxygen dominates at high Fermi levels (0 K!) A. Kuwabara, Private Communication, (2008)
17 When DFT fails… / eV In defect calculations: band gaps Heavily underestimatedBoth LDA/GGA functionalsAffects:Defect levelsDefect formation energiesCorrection: scissor operation!Shift the conduction bandAlign Eg(DFT) and Eg(EXP)Donor type defects follow CBCorrect formation energiesAlternative: hybrid or semi-empirical functionals/ eVBaZrO33.25.3PbZrO33.03.7ZnO1.93.4
19 Nudged Elastic Band Method Method for studying transition states:Saddle pointsMinimum Energy Paths (MEP)Know initial and final statesLocal/global minimaObtained by total energy calculations (DFT)Reaction pathDivided into ’images’ (< 20)Image = ‘ snapshots’ between initial and final statesEquidistant images - connected by ’springs’Optimize each image (DFT+forces) MEPChosen images not saddle points interpolate
20 Proton transport in LaPO4  Proton transport in LaPO4 using NEB+DFT (VASP)Saddle points:Jump rates:Determine dominating transport mechanismDiffusion and conductivity:Proton transport processes:Rotation around oxygenOscillatory motionIntra-tetrahedral jumpsInter-tetrahedral jumps R. Yu and L. C. De Jonghe, J. Phys. Chem. C 111 (2007) 11003
21 Intratetrahedral jumps: High energy barrierIntertetrahedral jumps:Two tetrahedronsLower energy barrierIntertetrahedral jumps:Three tetrahedronsEven lower energy barrierExperimentally: eV
22 Molecular DynamicsUsed to simulate time evolution of classical many-particle systemsObey the laws of classical mechanicsCondensed systems:Classical particles moving under influence of an interaction potential, V(R1,…,RN)Forces on the ions:MD algorithms:Discretize equation of motionTrajectories: stepwise update positions and velocities
23 Interionic interactions: Model potential vs. first-principlesModel potentialParameterized to fit experimental or first principles dataAdvantagesPossible to treat large systemsLong time evolutionDisadvantagesInaccurate potentials Poor force representationFirst-principlesFirst-principles electronic structure calculation (e.g. DFT) at each ionic stepAdvantages:Accurate forcesRealistic dynamic descriptionDisadvatagesComputationally demandingSmall systems (~100 ions)Short time periods (~ps)Good statistical accuracyPoor statistical accuracy..
24 Monte-Carlo approach Proton diffusion Loosely described: Statistical simulation methodsConventional methods (MD):Discretize equations describing the physical processE.g. equations of motionMonte-Carlo approachSimulate the physical process directlyNo need to solve the underlying equationsRequirement: process described by probability distribution functions (PDF)Average results over the number of observationsProton diffusionJump frequency and PDF:
25 Summary Many different computational approaches Systematic trends Fundamental processesPredict defect properties of real materialsCombine different methods
26 Colleagues at the Solid State Electrochemistry group in Oslo AcknowledgementsAkihide KuwabaraEspen Flage-LarsenSvein StølenTruls NorbyColleagues at the Solid State Electrochemistry group in OsloThank You !!