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For any ground state, the gradient of each vibrational mode is equal to zero, the second derivatives are positive, indicating that all of the vibrational modes are real. In the case of transition states, one of the frequencies is imaginary, which defines the reaction coordinate. In practical terms, the second derivative from the energy gradients variation is calculated numerically from small displacements of each atom. The Hessian matrix is determined from the forces, its eigenvalues are the vibrational frequencies, and the eigenvectors are the vibrational normal modes of the system. 2.2.4 Boundary Conditions and Modelling Defects in Solids

The approximations and methods highlighted in the previous sections provide the tools to optimise and analyse the structures of interest. Practically, these methods are size limited to systems of no more than a 1000 – 2000 atoms. While improvements in computational resources have drastically improved this picture, 10 years ago it would be unthinkable to tackle systems of more than a few hundred atoms. This situation, while sufficient for molecular systems in the gas phase, fall far short of the 1023 atoms that would be appropriate in the macroscopic context.

There are a number of approaches that are used to treat large scale solid systems, the principle technique involves taking advantage of the periodic nature of crystalline solids. This has the advantage of allowing a small region made up of one

or more unit cells to be extended, then via Bloch theorem, to calculate the electronic structure of the infinite solid.[196]

2.2.4.1 Supercells for Defect Calculations

One of the main challenges when calculating defects within the confines of periodic models is that of an artificially high defect concentration. Taking advantage of periodic boundary conditions, the primitive cell can be repeated to generate large supercells that are then themselves repeated, allowing defects to be added to larger cells, leading to a marked decrease in defect concentration. If a defect was introduced to the primitive cell, which in the 4H-SiC case contains 8 atoms, the application of the periodic boundary conditions would map that defect into every neighbouring cell in three dimensions giving an unrealistically huge defect concentration (one in every eight atoms). By generating supercells, the defect concentration is reduced which in the 3x3x3 example shown schematically in Figure 2.1 would relate to a 96-atom cell for 4H-SiC giving a vastly reduced defect concentration (although still very high). The supercell size is selected to allow the defect relaxation to be described within the simulation cell ensuring that the final structure is not constrained.

A further level of complexity is introduced when charged defects are to be considered. These arise due to the long-range nature of the Coulomb potential, where an unphysical interaction between the defect and its periodic image can be introduced. The principle quantity of interest in defect calculations is the defect formation energy (Ef), as this can be used to infer defect concentration assuming equilibrium has been reached. Ef is the energy required to create an isolated defect in charge state q:

𝐸_J(𝑞) = 𝐸₉ (𝑞) + 𝑛₈𝜇₈− 𝐸_Š¦e•(𝑞) + 𝑞 𝜇_I+ 𝐸_“ + 𝐸_W|VV, (2.31)

where ED is the total energy of the supercell containing the defect, the sum is the energy of the removed or added species s, ns the number of species s, 𝜇8 is the chemical potential of species s, 𝜇_I is the electron chemical potential (Fermi Level), 𝐸_“ is the potential alignment, and Ecorr is a correction term.[96], [102]

There are two major errors that result from the use of periodic boundary conditions, and these must be corrected if accurate defect energies are to be obtained. These errors are image-charge interactions and differences in the reference electrostatic potential. The correction of these problems has led to numerous correction schemes, and vigorous debate as to the best method for a given system type.

The simplest correction scheme of this type is a Madelung type correction proposed by Leslie and Gillian, which is derived from the screened Coulomb interaction between point charges.[197] This idea was extended by Makov and Payne (MP), and until recently was the most common method for correcting the image charge interaction.[198] The correction (Δ𝐸¨}) takes the simple Madelung term and adds an additional term to account for the multipole expansion

Δ𝐸¨} =𝑞 F_𝛼 ¨ 2𝜖𝐿 + 2𝜋𝑞𝑄6 3𝜖𝐿† , (2.32) where 𝛼_¨is the Madelung constant of the super lattice, 𝜖 is the dielectric of the material, Qs is the second radial moment of the charge density, and L is cell length. The MP scheme has been shown to perform well for molecular systems, but the MP scheme has a tendency to overestimate the magnitude of the correction (to the first order approximation). This overestimation comes about from the fact that Qs does not take into account the screening response of the dielectric.

The solution to this overestimation was proposed by Lany and Zunger (Δ𝐸_„«), which has largely become the standard for the correction of image charge interactions.[199] To the first order it is given as

Δ𝐸„« = 1 + 𝐶6¬ 1 − 𝜖h: 𝑞F_𝛼

¨

2𝜖𝐿 , (2.33) where Csh is the shape factor and depends only upon the shape of the supercell, and the dielectric of the material. An alternative approach was proposed by Freysoldt, Neuugebauer, and van de Walle, consideration of this scheme is beyond the scope of this work as the LZ method has been successfully used for the SiC system and allows for comparison with the existing literature.

To correct for the shift in electrostatic potentials due to the divergence of the Madelung sum in charged cells, for a given supercell calculation, the average value of the electrostatic potential is set to zero. This leads to the introduction of an arbitrary constant that is not constant between charge states. This is corrected for by inspection, either via comparison of ions in the defect and bulk region to give an estimation of the shift. Alternatively, the potential shift can be gauged by inspection of the core-levels of ions far from the defect for the neutral and charged systems. It is

important to note that these methods should yield the same value for the potential alignment.