Impact of Charge Transitions at Atomic Defect Sites on Electronic Device Performance

2.2 Defect calculations

2.2.1 Supercell approach

The supercell approach is an established method to calculate the properties of defects in solids [118, 138]. In general, a supercell is constructed by replicating the unit cell of a material system along the three directions of the Cartesian coordinates. Subsequently, the defect of interest is introduced in the host material to create the defect structure. The resulting system is then placed under periodic boundary conditions, defining the new defective unit cell. This allows for the treatment with mathematical methods that require translational periodicity and the usage of highly efficient computer codes, which where designed for periodic solids [117]. The total size of the supercell is limited by the computational power available, as with increasing numbers of atoms, more and more interactions have to be considered. Other established methods for defect calculations are the finite-cluster method [139] and the Green’s function embedding technique [140].

Geometry optimizations were performed in this thesis using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm [141, 142, 143, 144]. Before the defect is introduced into the supercell, first the cell parameters have to be relaxed to find the supercells minimum energy configuration within the framework of the employed computational setup (see Chapter 3.1 for more details). This is necessary to release any internal stress that could impact the structural relaxations related to the defect. The defect system is then constructed by creating an entity of the defect type of interest in the optimized supercell. Subsequently, the geometry of the system is optimized again in different charge states by changing the atomic coordinates until an energy minimum for the defect structure is found. Typically, this leads to considerable structural relaxations of the atoms near the defect site. Defect properties can then be derived from the ground state electronic density of the defective supercell as calculated with density functional theory (DFT). Atomic point defects break the symmetry of the material structure and can introduce localized electronic states in the fundamental electronic band gap of an insulator or a semiconductor. The formation energy of the defect can be calculated in different charge states according to Eq. (2.13) and subsequently the CTL obtained from Eq. (2.12). Several issues have to be considered when calculating defect properties in the supercell approach.

Within this method, rather than modeling single defects, a periodic array of defects is obtained. Thus, due to the limited number of atoms in the supercell, the investigated systems exhibit artificially high defect concentrations, leading to spurious interactions between the defect and its periodic images. These interactions can include magnetic, elastic, electrostatic and other quantum-mechanical effects such as overlaps of electronic wave functions [117], which would vanish for an infinitely large supercell. It is therefore beneficial to increase the supercell size with respect to the number of atoms as much as computationally feasible. Defect-induced electronic levels are broadened due to the finite size of the supercell, resulting in defect bands with a typical width of around 0.1 eV [138]. As a consequence, one electron levels have to be either obtained by averaging over the whole Brillouin zone of the supercell, or by calculating the state only at the $\mathrm{\Gamma }$ point. The latter method is typically sufficiently accurate and computationally efficient for large supercells [145] and was thus used for all defect calculations in this thesis. To reliably evaluate the formation energy of the defect, the supercell has to be large enough to capture all structural relaxations related to the introduced defect in different charge states. Additionally, for charged supercells, electrostatic corrections have to be applied after the first-principle calculations (a posterioi) to remove spurious interactions, as will be described in the following section.

Image charge correction

A detailed review of finite-size supercell corrections schemes for charged defect calculations can be found in [146]. The goal of these corrections schemes is to account for and remove spurious electrostatic interactions, which are unavoidable in DFT calculations of charged periodic supercells. Thereby, the electrostatic interactions of a localized charge with its periodic images and with the neutralizing background charge are removed from the formation energy of a defect by adding the correction term $E^{\mathrm{corr}}$ in Eq. (2.11). The background charge is introduced in periodic DFT calculations to ensure overall charge neutrality and consequently to prevent the divergence of the electrostatic energy due to interactions with the infinite periodic images.

In this thesis, the Freysoldt, Neugebauer, and Van de Walle (FNV) correction scheme [147] is employed for all charged defect calculations. The spurious interactions are evaluated by comparing the electrostatic effects of a defect in a periodic supercell with those of an isolated model charge in a continuous medium. Additionally, the electrostatic potentials of the reference and defect system are aligned far away from the defect to ensure comparability. The corrections to the formation energy were calculated using the software package sxdefectalign [148], which takes inputs directly from first-principle calculations. Recently, corrections schemes for optical transitions at defect site were proposed to treat the difference between static and purely electronic screening screening [149, 150]. Since the compatibility of these formalisms with the lineshape function and non-radiative charge transitions in the context of a 1D CCD is still under discussion [151, 152], such corrections where not employed for calculations of relaxation energies in this thesis.