Oxygen vacancies in α-(Al x Ga1−x )2O3 alloys: a first-principles study

α-(Al x Ga1−x )2O3 alloys have attracted increasing interest as semiconductors with tunable wide band gaps. We report a systematic analysis of O vacancies in α-Al2O3, α-Ga2O3, and α-(Al x Ga1−x )2O3 alloys using first-principles calculations. The formation energies and electronic levels of the O vacancies are sensitive to not only the nearest-neighbor Al/Ga ratio but also the atomic relaxation around the vacancies. Consequently, the vacancy formation energies vary by up to ∼2 eV, reflecting diverse local atomic environments in the alloys. These results provide insight into further understanding and controlling the properties of α-(Al x Ga1−x )2O3 alloys.

α-(Al x Ga 1−x ) 2 O 3 alloys have attracted increasing interest as semiconductors with tunable wide band gaps. We report a systematic analysis of O vacancies in α-Al 2 O 3 , α-Ga 2 O 3 , and α-(Al x Ga 1−x ) 2 O 3 alloys using first-principles calculations. The formation energies and electronic levels of the O vacancies are sensitive to not only the nearest-neighbor Al/Ga ratio but also the atomic relaxation around the vacancies. Consequently, the vacancy formation energies vary by up to ∼2 eV, reflecting diverse local atomic environments in the alloys. These results provide insight into further understanding and controlling the properties of α-(Al x Ga 1−x ) 2  Supplementary material for this article is available online G allium oxide (Ga 2 O 3 ) has attracted growing attention as a wide-gap semiconductor for power devices and ultraviolet photodetectors. [1][2][3] In addition to β-Ga 2 O 3 with the monoclinic structure, other metastable polymorphs 4) have been the targets of current research. In particular, α-Ga 2 O 3 with the corundum structure is attractive in view of its wider band gap of 5.3 eV 5) than 4.9 eV for β-Ga 2 O 3 . 6) The formation of α-(Al x Ga 1−x ) 2 O 3 alloys between isostructural α-Al 2 O 3 and α-Ga 2 O 3 allows for band gap engineering in a wide range up to 8.6 eV. 7,8) The crystal growth and fundamental properties of α-(Al x Ga 1−x ) 2 O 3 alloys have been studied both experimentally 5,[7][8][9][10][11][12] and theoretically [13][14][15] towards in-depth understanding and full utilization of their functionalities.
The O vacancy is a major native point defect species in most oxides, including Al 2 O 3 and Ga 2 O 3 . [16][17][18][19][20] Previous firstprinciples studies have shown that O vacancies form deep ingap levels in α-Al 2 O 3 , 16) α-Ga 2 O 3 , 19,20) β-Ga 2 O 3 , 17,20) and κ-Ga 2 O 3 . 20) Therefore, they are relevant to the performance of electronic and optoelectronic devices composed of these oxides. In addition, Akiyama  Understanding the environmental dependency of the O vacancies is important to precisely control the alloy properties. In this Letter, we report a systematic investigation into the formation energies and electronic levels of O vacancies in α-(Al x Ga 1−x ) 2 O 3 alloys using first-principles calculations, alongside those in α-Al 2 O 3 and α-Ga 2 O 3 .
The calculations were conducted using the projector augmented-wave method 21) as implemented in the VASP code. [22][23][24] The Perdew-Burke-Ernzerhof semilocal functional tuned for solids (PBEsol) 25) was used in geometry optimization. To correct the valence band maximum (VBM) and the conduction band minimum (CBM), non-self-consistent calculations using the Heyd-Scuseria-Ernzerhof hybrid functional 26,27) were performed on top of the PBEsol results.
The Fock-exchange mixing parameter was set at 0.35 with the screening parameter kept at 0.208 Å −1 to reproduce the band gaps of α-Ga 2 O 3 and α-Al 2 O 3 (supplementary table SI). This approach has been shown to provide band structures and defect formation energies close to those from self-consistent hybrid functional calculations at substantially reduced computational costs, 28,29) allowing for systematic alloy and defect calculations.
To model α-(Al x Ga 1−x ) 2 O 3 alloys and O vacancies, 120atom supercells were constructed by a 2 × 2 × 1 extension of the conventional unit cell of the corundum structure. Special quasi-random structures (SQSs) 30) were generated to describe ideally disordered α-(Al x Ga 1−x ) 2 O 3 alloys with x = 0.25, 0.5, and 0.75 by optimizing correlation functions using the CLUPAN code. 31) Three SQS models were considered for each alloy composition. The geometry optimization including cell parameter relaxation was performed using a 2 × 2 × 2 kpoint mesh and a plane-wave cutoff energy of 520 eV, while a cutoff energy of 400 eV was used in the O vacancy calculations with fixed cell parameters. The point defect calculations and analysis were conducted using the pydefect, 29) vise, 29) and pymatgen 32) codes. Bader charges were evaluated using the code developed by Henkelman et al. 33) More computational details can be found in the supplementary data.
The formation energy of the O vacancy in charge state q is obtained as 34) and E p are the total energies of the supercells with and without an O vacancy in charge state q, respectively. E V q corr [ ] is a cell-size correction term associated with image-charge interactions in charged vacancy supercells (q ≠ 0), for which the extended Freysoldt-Neugebauer-Van de Walle scheme is used to take anisotropic screening into account. 35,36) O m is the O chemical potential at the standard state, which is set at half the total energy of the O 2 molecule here. Static dielectric tensors were obtained on the basis of density functional perturbation theory 37,38) and used for the image-charge corrections of the α-Ga 2 O 3 and α-Al 2 O 3 supercells with charged O vacancies (supplementary table SI). Their composition averages were taken for the corrections of the alloy supercells.
The calculated fundamental properties of α-(Al x Ga 1−x ) 2 O 3 with x = 0, 0.25, 0.5, 0.75, and 1 are shown in Fig. 1. Three SQS alloy models for each composition provide properties similar to each other, particularly for the volume presented in Fig. 1(b). The formation energies of the alloys shown in Fig. 1(a) are obtained with respect to α-Ga 2 O 3 and α-Al 2 O 3 , where α-Ga 2 O 3 is a metastable phase. [39][40][41] The values of the alloy formation energies are close to those in previous theoretical studies. 14,15) Although the α-(Al x Ga 1−x ) 2 O 3 alloys have positive formation energies, configurational entropy tends to stabilize these disordered alloys as the temperature increases. Figure 1 also shows that the volume of α-(Al x Ga 1−x ) 2 O 3 decreases almost linearly with increasing the Al composition ratio [ Fig. 1(b)], while the band gap has a sizable nonlinear composition dependency [ Fig. 1(c)]. The band gap of a pseudo-binary system is often represented as  11) 0.84 and 1.62 eV for the direct and indirect gaps, respectively, 12) and 1.31 and 1.61 eV for the perpendicular and parallel electric field polarization to the c direction in spectroscopic ellipsometry measurements, respectively. 9) Our theoretical bowing parameter is in reasonable agreement with these experimental values and the previously reported theoretical values of 1.37 and 0.93 eV for the direct and indirect gaps, respectively, 13) and 1.6 eV. 14) Figure 2 shows the Bader charges of the O atoms in α-(Al x Ga 1−x ) 2 O 3 . The O site is surrounded by four cation sites in the corundum structures. Overall, the absolute value of the Bader charge tends to increase with increasing the Al ratio at the nearest-neighbor sites, as expected from the electronegativities of Al and Ga. In more detail, however, the Bader charges exhibit distributions of ∼0.1 within the same nearest-neighbor Al/Ga ratio. We found that the arrangement of the AlO 6 and GaO 6 octahedra in the corundum structure affects the Bader charges of the O atoms; smaller and larger absolute values of the Bader charges are obtained when the face-sharing octahedra are both AlO 6 and the mixture of AlO 6 and GaO 6 , respectively, as exemplified in Fig. 2.  Fig. 3(a).  In more detail, the formation energies of the neutral O vacancies are found to exhibit up to ∼1 eV differences within the same nearest-neighbor Al/Ga ratio, as detailed below. The neutral O vacancies have electronic states localized on the vacancy sites (supplementary Fig. S1), which are accompanied by inward relaxation of the nearest-neighbor cations. Therefore, their formation energies are rather sensitive to the nearest-neighbor cation species and the atomic relaxation around the vacancies. On the other hand, a previous firstprinciples and machine learning study of O vacancies in various metal oxides indicates that more electrostatics-related factors are involved in determining the O vacancy formation energies for the +2 charge state. 29) The less sensitive behavior to the nearest-neighbor environments may follow this tendency.  Fig. S2). The vacancy formation energy tends to decrease as the absolute value of the Bader charge decreases. In addition, a clearer correlation is found with the volume change of the nearest-neighbor cation tetrahedron due to the vacancy formation in Fig. 4(b). A larger volume change of the cation tetrahedron results in a lower vacancy formation energy. This tendency is explainable by the aforementioned characteristics of the neutral O vacancies, namely the localized electronic states and the inward relaxation of the nearest-neighbor cations. The volume change of the cation tetrahedron tends to be large as the number of nearest-neighbor Ga atoms increases. However, the data points within the same nearest-neighbor Al/Ga ratio are separated into two or three clusters, with up to ∼1 eV differences in the neutral O vacancy formation energy. As shown in Fig. 2 and discussed above, the Bader charge of the O atom depends on not only the nearest-neighbor Al/Ga ratio but also the configuration of the face-sharing AlO 6 and GaO 6 octahedra around the O site. We found that the volume change of the cation tetrahedron is also related to the configuration of the face-sharing octahedra. When the number of nearest-neighbor Ga atoms is 1 (3), the facesharing is limited to the pair of AlO 6 (GaO 6 ) and the combination of AlO 6 and GaO 6 , thereby forming two clusters. All the three configurations are involved in the case that the number of nearest-neighbor Ga atoms is 2, which results in the separation into three clusters. The O vacancy formation energy thus depends on the details of local atomic environments in α-(Al x Ga 1−x ) 2 O 3 alloys.
In summary, we have investigated the fundamental properties of α-(Al x Ga 1−x ) 2 O 3 alloys and O vacancies therein using first-principles calculations, along with those of α-Al 2 O 3 and α-Ga 2 O 3 . The band gap of α-(Al x Ga 1−x ) 2 O 3 increases with increasing Al content x and shows a large bowing, as previously reported theoretically and experimentally. The O atoms in the alloys have effective charges depending on their local environments, specifically the nearest-neighbor Al/Ga ratio and the configuration of the face-sharing AlO 6 and GaO 6 octahedra. The formation energies and electronic levels of the O vacancies vary accordingly, but they are also sensitive to the atomic relaxation around the vacancies. Diverse local atomic environments in the alloys lead to up to a ∼2 eV variation of the neutral O vacancy formation energies. These results provide a useful guideline for further understanding and controlling the properties of α-(Al x Ga 1−x ) 2 O 3 alloys.