2.6 Modelling Solid State Defects
2.6.1 Mott-Littleton Methodology
In the Mott-Littleton methodology[274] the defect energy is calculated by splitting the system into two regions; in the first region the ions are fully and explicitly affected by the defect, which sits at the centre of the region, whereas for region 2, which extends to infinity, only the displacements of the ions, caused by the presence of the defect, are considered.
The total energy is then calculated using the following equation:
E= E1(x) + E12(x, y) + E2(y) (2.84)
where E1is the energy of region 1, dependent on the coordinates, x, of the ions in
the region, E12 is the interaction between regions 1 and 2, and E2 is the energy of
region 2, dependent on the displacements, y, of the ions in this region.
Therefore, in order to determine the defect energy, the energy of the system needs to be minimised with respect to the positions x and displacements y. However, the displacements in region 2 are themselves dependent on the positions of the ions in region 1, making the energy minimisation difficult. Therefore, in order to simplify the minimisation process, the forces on the ions in region 1 are required to be zero, instead of the energy of the region being at a minimum. Additionally, the ions in region 2 need to be at equilibrium to minimise the energy.
In practice, it turns out that atomistic interactions with region 1 need to be consid- ered only for a small component of region 2. Consequently, region 2 is split into two
sections; region 2a and region 2b. In region 2a the interactions with ions in region 1 are calculated explicitly, and the positions of the ions displaced by the forces acting on them from the electrostatic field of the defect. Whereas for region 2b, which extends to infinity, it is assumed only polarisation of the ions changes as a result of the defect, and is treated as an electrostatic potential surrounding the lattice with all ions remaining in a fixed position. Therefore, the energy associated with region 2b is the polarisation of the ions in region 2b by the electrostatic field of the defect. When calculating the defect energy, an iterative approach is used to find the config- uration in which all the forces in region 1 are zero and the displacements in region 2a are at equilibrium. The final defect energy is then taken as the difference be- tween the total energy of the defective lattice and the perfect lattice. Importantly, calculated energies of vacancies and interstitials are with respect to the added or removed ion at infinite distance.
Figure 2.1: A visual representation of the Mott-Littleton methodology for calculating de- fect energies.
2.6.2
Supercell Method
Whereas in the Mott-Littleton method an isolated defect, or defect cluster, is placed at the centre of a lattice which is then modelled to infinity, the supercell method places a defect in a supercell, which is then subject to periodic boundary conditions.[275–277]
The main challenge associated with this method is that, instead of an isolated de- fect at the centre of a lattice, as in Mott-Littleton,[274] a periodic array of defects is generated. Therefore the potential interactions between defects in neighbouring supercells needs to be addressed. Interactions that need considering include: elec- trostatic, magnetic and elastic. All interactions will become small at large enough supercell sizes; however, the large computational requirements for such supercell sizes mean that it is impractical to employ such a method to remove these errors. Therefore alternative methods are employed to minimise their effect on the final result.
The effect of elastic and magnetic interactions between defects in neighbouring supercells tend to be neglected in solid state systems; elastic interactions, which arise when the defect distorts the surrounding lattice, are assumed to be negligi- ble at the supercell size used in this work, discussed further in Chapter 3, while there is currently no scheme that systematically corrects for magnetic interactions in supercells.[278] The effect of electrostatic interactions on defect formation en- ergies, on the other hand, has been addressed, and the methods used to correct for these interactions are outlined below.
Electrostatic interactions occur when the defect species present in the supercell cause the supercell to have an overall non-zero net charge. The presence of a net charge causes several issues including; the Ewald summation - used to determine the Coulombic contribution to the energies - diverging,[279, 280] defects interact- ing with images of themselves - due to the slow decay of the Coulomb potential of the charged defect - and the defect changing the electrostatic potential of the super- cell compared to the defect-free (pure) supercell, which in turn effects the formation energy of the defect.
The first of these is tackled by compensating for the net charge by introducing a uniform background charge density which effectively sets the average electrostatic potential of the supercell to zero.[278] The interaction of defects with images of themselves and the potential alignment issue are corrected after calculations have been performed with correction schemes.
An image charge correction scheme was initially proposed by Makov and Payne[280] who suggested a correction scheme in a linear supercell dimension L, where L is equal to the cube root of the supercell volume.[281]
∆EMP=
q2αM
2εL +
2πqQr
3εL3 (2.85)
where α is the lattice dependent Madelung constant, ε is the static dielectric con- stant of the pure supercell and Qris the second radial moment of the electron density
difference between the defective supercell and pure supercell.
However, a number of issues were raised on whether this charge correction was appropriate and if it lead to improved results.[282–285] Leading to Lany and Zunger[281] suggesting a slightly altered correction:
∆EMP= (1 + f )
q2αM
2εL (2.86)
based on the proportionality of the first and third order terms in equation 2.85
∆EMP3 = f ∆EMP1 (2.87) where ∆EMP1 and ∆EMP3 refer to the first and second terms in equation 2.85 re- spectively, and f is a proportionality factor which Lany and Zunger found equals −0.35.[281]
A key aspect of this correction scheme being more well regarded than the original scheme proposed by Makov and Payne[280] was the inclusion of the potential align- ment correction as these two corrections are not independent and therefore needed to be treated consistently.[278]
electrostatic interactions. To correct for the defect changing the electrostatic poten- tial of the supercell, the average electrostatic potential of the defective supercell is shifted so it is in line with the electrostatic potential of the pure bulk material:
∆EPA(D, q) = q(VD,qr −VHr) (2.88)
where D refers to the defect species in charge state q, and Vrare the reference poten- tials of the charged defect and the host, H. The reference potentials are determined from atomic-sphere-averaged electrostatic potentials at atomic sites far away from the defect.[281]
2.7
Computer Codes
Two main computer codes have been used throughout the work presented in this thesis, the Vienna ab initio simulation package (VASP),[289, 290, 290] and the general utility lattice program (GULP).[291] Here we give a brief overview of each code.
2.7.1
VASP
VASP is an atomic scale materials modelling program that can use either DFT or the HF approximation to compute an approximate solution the many-body Schr¨odinger equation. In our work, VASP is used for DFT calculations. VASP uses plane wave basis sets to express quantities such as the one-electron orbitals, with the inter- actions between electrons and ions described using either pseudopotentials or the projector-augmented-wave method, as described in Section 2.3.4. The implemen- tation of DFT and DFT+U is well known to be accurate and reliable in VASP, and is therefore well suited to our work, which considers both DFT and DFT+U when choosing modelling parameters, which will be discussed in Chapter 3.
2.7.2
GULP
For our interatomic potential-based calculations, the GULP code[291] was used. GULP is a force field-based code for modelling condensed phases and has a wide variety of capabilities, while being designed to be as easy to use as possible. The
variety of tasks GULP can perform, along with the ease of implementation, made this code well suited to the interatomic potential-based calculations performed for this thesis.
2.8
Computer Hardware
Calculations have been run on several clusters throughout this project, including three based at UCL and two national systems. In this section we will briefly cover how each was used and their architecture.
2.8.1
UCL-based
Three UCL hosted clusters have been used; Faraday, Legion and Grace. Faraday is a small, shared research cluster for the UCL Chemistry department, it has 240 cores and was therefore used for our interatomic potential-based calculations, which are computationally inexpensive compared to DFT. Legion and Grace are both larger clusters, Legion contains 8 different types of node and is a mixed use clus- ter whereas Grace is designed for parallel workloads, having 680 identical nodes available each with 16 cores (2×8 core Intel Xeon processors). Legion was used initially, and our work establishing a DFT-based modelling approach for LaFeO3 (Chapter 3) was performed on Legion, along with defect calculations using DFT (Chapter 4). Grace was used for our work on dopants (Chapter 6) and the Nudged Elastic band calculations for oxide ion migration (Chapter 5). Grace was also used for establishing a number of parameters for our surface model of LaFeO3(Chapter
7).
2.8.2
National
Two national high performance computing (HPC) facilitates have been used to run calculations; Archer and the Hartree Centre. Archer is the UK National Supercom- puting Service, provided by EPSRC, NERC, EPCC, Cray Inc. and The University of Edinburgh, it is designed for calculations that require large numbers of cores working in parallel. Archer consists of the Cray XC30 MPP supercomputer, and has 4920 compute nodes, each with 24 cores (2×12 core Ivy Bridge series proces- sors). Time to run calculations on Archer was requested and granted through my
membership of the UK’s HPC Materials Chemistry Consortium (MCC), funded by the EPSRC (Grant number EP/L000202). The Hartree Centre is part of the Science and Technology Facilities Council and provides a range of HPC facilities. Through the partnership of The Hartree Centre with the EPSRC project ’Energy Materi- als: Computational Solutions’ (EP/K016288/1) I was provided access to Iden and Napier. Iden has 84 nodes each of which has 24 cores (2×12 core Intel Xeon pro- cessors) while Napier has 360 nodes with 24 cores - of the same design as Iden. Both Archer and the computing facilities at the Hartree Centre were used to run surface relaxation calculations along with defect calculations on the (001) surface of LaFeO3(Chapter 7).
2.9
Concluding Remarks
In this chapter the theoretical background to the computational techniques used in the following work have been outlined, covering the general topic of modelling solid state materials, as well as the more specific subject of studying defective systems. In the next chapter, more details will be given regarding the software and parameters used in the calculations performed in this work, thus providing a complete picture of the methods implemented and our reasoning for using them.