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

Chapter E Computational setup

In this appendix, details about the computational setup of all DFT calculations in this thesis, including the employed software packages, basis sets, pseudopotentials and energy cutoffs are presented. The references to the respective publications are given in the linked chapters. All DFT calculations were performed with the Gaussian Plane Wave (GPW) method [187] as implemented in the CP2K code [186]. The MD calculations were performed using the LAMMPS code [192] in conjunction with the QUIP package.

Computational setup of Section 4.1

All DFT calculations were carried out in a single structure of a-SiO2 containing 216 atoms with three-dimensional periodic boundary conditions applied to the system. The Goedecker-Teter-Hutter (GTH) pseudopotentials [185] were used in combination with a double-ζ Gaussian basis set [277] and an energy cutoff of 800 Ry for the plane wave expansion of the electron density was employed. To accurately calculate the electronic structure and to minimize the errors of electronic state calculations in the bandgap of the defect system, the non-local hybrid functional PBE0_TC_LRC [180] was used for all calculations. Additionally, the auxiliary basis set pFIT3 was used to mitigate computational costs of calculating the Hartree-Fock exchange [189]. The total energy of the system was converged self-consistently up to 2.7×10−6 eV. Energy barriers were calculated with the climbing image nudged elastic band method (CI-NEB) [136] employing the PBE functional and using 7 intermediate images each.

Computational setup of Section 4.2

Defect calculations were carried out in a monolayer WSe2 supercell containing 432 atoms with 40 Å of vacuum perpendicular to the monolayer to minimize spurious interactions with periodic images. The PBE0_TC_LRC hybrid functional [180] was used with the default mixing parameter of 0.25 to accurately describe the localization of charge and the electronic interactions, including exchange and correlation effects. The electronic wave functions were described with double-zeta valence-polarized Goedecker–Teter–Hutter basis sets and auxiliary basis sets of type cFIT for calculations of the Hartree-Fock exchange. Cell parameters were relaxed with the hybrid functional to reduce the internal stress to <0.01 GPa. The defect systems were self-consistently relaxed down to a residual maximum force of 20 meV/Å for each atom with a convergence criteria for the total energy of 13.6 µeV. Finite size corrections of the total energy compensating for electrostatic potential offsets and spurious interactions due to periodic boundary conditions in charged supercells were carried out using the CoFFEE code [240], which implements the FNV-charge correction scheme [148] specifically for 2D systems.

Computational setup of Section 5.1

The MLIP used for the MD simulations was specifically trained to create amorphous structures following a melt-and-quench procedure [194] as also discussed in Section 3.2.2. To account for the structural variety of amorphous systems, the MLIP training set consisted of energies, forces and stress tensors calculated with DFT or more than 1600 different Si3N4 structures. The structures of the training data set included amorphous systems with different mass densities, dimers and different crystalline phases of Si3N4. For the MD calculations, a timestep of 0.5 fs for the Verlet integration (see also Section 3.2) was used and a Langevin thermostat was employed to control the temperature of the system as discussed in Section 3.2.2. To accurately describe the electronic wave function, a double-zeta Gaussian basis set was used in conjunction with the non-local hybrid functional PBE0_TC_LRC [180] and GTH pseudopotentials were employed for the treatment of the core electrons. Furthermore, the auxiliary basis set pFIT3, which was successfully employed in recent studies to investigate intrinsic charge trapping sites [278], was used to reduce the computational costs of calculating the Hartree–Fock exchange [189]. The total energy of geometry optimizations and single point calculations was converged self-consistently down to 2.7 µeV.

Computational setup of Section 5.2

A double-ζ Gaussian basis set was used for the DFT calculations in combination with the GTH pseudopotentials and an energy cutoff of 800 Ry for the plane wave expansion of the electron density. The auxiliary basis set pFIT3 is employed to mitigate computational costs of calculating the Hartree–Fock exchange. The cell parameters of each amorphous structure were initially relaxed to reduce the internal stress down to 0.01 GPa. For the geometry optimizations of the structures in different charge states, the non-local hybrid functional PBE0_TC_LRC [180] is employed to accurately describe the electronic wavefunction and the localization of charge. The importance of using the hybrid functional PBE0_TC_LRC over PBE-only for studying the localization of charge is discussed in detail in Appendix D. The systems were self-consistently relaxed down to 2.7 µeV with a force convergence criterion of 0.02 eV/Å  for each atom. Bond orders were calculated from the valence electron density from DFT with the DDEC6 method as implemented in [250].

Computational setup of Section 6

The range-separated PBE0_TC_LRC hybrid functional [180] was used to accurately describe the electronic structure, in particular the localization of charge at a defect site. The mixing parameter of this functional, which regulates the portion of the Hartree-Fock exchange, was set to 0.27, resulting in a band gap of 8.9 eV to match the experimental gap of corundum at room temperature (8.8-9.2 eV [279, 97]). A cutoff of 800 Ry was used for the expansion in the plane-wave basis set. To obtain a precise description of the vibrational modes of the defect structures, very tight convergence criteria were applied for the energies and forces of 0.5 µeV and 0.8 meVÅ−1, respectively.