Abstract
Formation entropy of point defects is one of the last crucial elements required to fully describe the temperature dependence of point defect formation. However, while many attempts have been made to compute them for very complicated systems, very few works have been carried out such as to assess the different effects of finite size effects and precision on such quantity. Large discrepancies can be found in the literature for a system as primitive as the silicon vacancy. In this work, we have proposed a systematic study of formation entropy for silicon vacancy in its 3 stable charge states: neutral, +2 and –2 for supercells with size not below 432 atoms. Rationalization of the formation entropy is presented, highlighting importance of finite size error and the difficulty to compute such quantities due to high numerical requirement. It is proposed that the direct calculation of formation entropy of VSi using first principles methods will be plagued by very high computational workload (or large numerical errors) and finite size dependent results.
Highlights
Regarding the temperature dependence of formation free energy, temperature effects have been routinely accounted for in (i) the chemical potentials of elements whose standard thermodynamical state is gaseous such as oxygen [11, 12], (ii) electronic formation entropy in metallic state [13] or (iii) the configurational entropy term which allows one to calculate concentration of point defects from formation free energy
While less common than the calculations of formation enthalpies, the calculations of formation entropy ΔSf of point defects have become more and more popular in recent years and have already been carried out for variety of complex systems such as oxides [14], chalcogenides [15], semiconductors [16, 17], metals [18, 19], employing various techniques ranging from first principles including either Density Functional Perturbation Theory (DFPT) or Frozen Phonon (FP) [16, 17, 20,21,22] to Monte Carlo (MC) [23, 24] or Molecular Dynamics (MD) coupled with empirical potentials [25,26]
Large discrepancies are observed considering either the general form of the vibrational formation entropy or its minimum value at 1000 K being −4 kB (11 kB). Such differences cannot be assigned unambiguously to the many diverse technicalities of the calculation: supercell size, ensemble type considered (NVT versus NPT) for the dynamics, techniques used to compute the phonon spectrum (MD versus FP versus DFPT), restriction applied to phonon modes to be considered, type of Hamiltonian considered, or the local symmetry of the silicon vacancy investigated
Summary
Regarding the temperature dependence of formation free energy, temperature effects have been routinely accounted for in (i) the chemical potentials of elements whose standard thermodynamical state is gaseous such as oxygen [11, 12], (ii) electronic formation entropy in metallic state [13] or (iii) the configurational entropy term which allows one to calculate concentration of point defects from formation free energy. Large discrepancies are observed considering either the general form of the vibrational formation entropy or its minimum (maximum) value at 1000 K being −4 kB (11 kB) Such differences cannot be assigned unambiguously to the many diverse technicalities of the calculation: supercell size (ranging from tenths of thousands to tenths of atoms), ensemble type considered (NVT versus NPT) for the dynamics, techniques used to compute the phonon spectrum (MD versus FP versus DFPT), restriction applied to phonon modes to be considered (local harmonics versus full harmonics), type of Hamiltonian considered (empirical potential versus ab initio), or the local symmetry of the silicon vacancy investigated (unrelaxed Td versus relaxed Td versus C2v versus D2d versus D3d). DFPT and frozen phonon approach (assuming a scaling of O(Ns3c) for DFT) are found to be equivalent in terms of scaling O(Ns4c)
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