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 Supplementary Methods Summary
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. Need to determine the orbitals of the reference system
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 Supercell method
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
(eV)
(kJ/mol)
CaZrO3
-1.79
0.46
-147
SrZrO3
-1.54
0.69
-120
-106.1
PbZrO3
-1.32
0.65
-75
-103
BaZrO3
-1.22
0.76
-66
-80

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… / eV 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
/ eV
BaZrO3
3.2
5.3
PbZrO3
3.0
3.7
ZnO
1.9
3.4

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

27. Thank You!