Radiation damage effects on electronic and optical properties of β-Ga₂O₃ from first-principles

β-Ga₂O₃ has attracted considerable interest as a semiconductor material for power electronics such as Schottky barrier diodes (SBDs) and solar-blind photodetectors^1–3 and demonstrates a great promise in aerospace applications.^4,5 One of its main advantages is the large bandgap of 4.9 eV, much larger than typical semiconductor materials such as SiC (3.3 eV) and GaN (3.4 eV).⁶ In the aerospace environments accompanying with high temperature and high pressure, such a wide bandgap can make Ga₂O₃-based power devices work steadily. In addition, the β-Ga₂O₃ semiconductor would be exposed to high-energy particles (electrons, protons, heavy ions, and α-particles) irradiation when working in the aerospace, so its irradiation reliability study is extremely important.⁶ Those high-energy particles have various effects on semiconductor devices. Yang et al. observed a small change in the parameters of SBDs fabricated on HVPE-grown Ga₂O₃ epilayer under gamma-ray irradiation at a dose of 100 kGy.² The irradiation influences of a vertical Schottky rectifier with 1.5 MeV electrons at different fluences were reported by Lee et al.³ The effect of neutron-induced defects on β-Ga₂O₃ was studied by Cojocaru,⁷ who attributed the decrease in electrical conductivity to the effect of Ga vacancies introduced during the annealing process. The performance of semiconductor devices would continue to degrade with increasing absorbed dose, which is called the total ionizing dose (TID) effect. However, most of the current studies focus on evaluating the degradation performance of the power devices and there is a lack of detailed study on the radiation damage effects of β-Ga₂O₃ material properties.

The defect formation energy is closely related to the radiation damage effect of the device. Puzyrev et al.⁸ theoretically calculated the defect formation energy of V_Ga-V_N vacancy in AlGaN and combined with experiments to conclude that proton irradiation would lead to the transformation of V_Ga into a V_Ga-V_N vacancy, causing a negative shift in the threshold voltage and an increase in the noise amplitude. Nowadays, there have been some studies about defect formation energies of Ga₂O_3.^9,10 The type of point defects impacts the physical properties of Ga₂O₃ semiconductors. For example, the Ga vacancy acts as a compensating acceptor, which decreases the electrical conductivity.¹¹ Yet there are some controversies about the effects of O vacancies on Ga₂O_3.^12–15 In Ref. 12, the O vacancy is deemed as a shallow donor and is the main reason for the n-type conductivity of Ga₂O₃. While in Refs. 13–15, the O vacancies are found to act as deep donors and do not contribute to n-type conductivity. The Ga vacancy is considered a deep acceptor due to its transition levels of more than 1.5 eV above valence band maximum (VBM) in Ref. 15. Thus, it is necessary to discuss the point defects of β-Ga₂O₃ in detail. In addition to studying the formation energy of Ga₂O₃ with point defects, we investigate the electronic and optical properties of defective Ga₂O₃, and then the effects of radiation damage on β-Ga₂O₃ can be revealed more clearly. We find that Ga₂O₃ transforms from insulator to metal in the infrared spectral range when Ga atoms are lost, and O vacancies would decrease bandgap and make Ga₂O₃ an indirect bandgap semiconductor.

The physical properties and radiation tolerance of Ga₂O₃ can also be affected by external dopants. Si atoms have been doped to Ga₂O₃ to alter their luminescence behavior.¹⁶ In addition, when Al and In atoms are co-doped into β-Ga₂O₃ nanostructures, the higher the doping concentration, the higher the photocatalytic activity.¹⁷ Ions such as Tb³⁺ and Eu³⁺ doped to Ga₂O₃ to achieve green and red emission.¹⁸ At present, there are few studies on whether defects are more likely to form after adding dopants. Yang et al. used first-principles calculations to reveal that Al-N co-doping has shallower transition levels than N mono-doping and lower defect formation energy.¹⁹ Ma et al. investigated the effect of defect formation energy of Al dopants on β-Ga₂O₃ from first-principles, but the electrostatic corrections are not considered.²⁰ Liu et al. studied the defect formation energy of O vacancies after simultaneous doping Al and In atoms to β-Ga₂O₃, but the term electrostatic corrections is not referred in calculation and the research about Ga vacancies is not involved.²¹ In addition, Tuttle et al.²² proposed another method to calculate the atomic-displacement threshold energies to evaluate whether radiation damage is easily produced.

In this work, we study the defect formation energy, and the physical properties of β-Ga₂O₃ structures with point defects through first-principles calculations. We have obtained the effects of separate Ga, O, and complex Ga-O vacancy defects on electronic and optical properties of β-Ga₂O₃. Moreover, the effects of dopant In atom on the defect formation energy of Ga₂O₃ are investigated, seeking to strengthen its irradiation tolerance.

β-Ga₂O₃ has two unequal positions of Ga atoms: Ga(I) (fourfold) with tetrahedral geometry and Ga(II) (sixfold) with octahedral geometry. Similarly, there are three different sites for O atoms, O(I) and O(II) (threefold) and O(III) (fourfold). Therefore, various vacancy defects would be formed after radiation due to the particularity of Ga₂O₃ atom positions, among which the most typical is point defect. The β-Ga₂O₃ unit cell belongs to the space group C2/m (monoclinic system) with three lattice vectors a = 12.46 Å, b = 3.08 Å, c = 5.87 Å, and the angle β = 103.68°, which is consistent with Ref. 10. This unit cell contains 20 atoms and can be transformed to a smaller primitive cell, which only contains 10 atoms,²³ as displayed in Fig. 1 constructed using VESTA.²⁴ The full Brillouin zone can be calculated correctly through the primitive cell. All first-principles calculations were employed using the projector augmented wave (PAW) method²⁵ within the Vienna ab initio simulation package (VASP).²⁶ The exchange-correlation energy was calculated using the generalized gradient approximation (GGA) functional.²⁷ All calculations set the plane-wave basis cutoff energy to 450 eV. The 2 × 2 × 2 supercell (80 atoms) was generated based on the primitive cell (10 atoms) to study the physical properties of the defective Ga₂O₃ structures. The k-mesh was set to 2 × 2 × 2 for geometry relaxation and 8 × 8 × 8 for electronic band structure and optical calculations, respectively. The structural parameters were optimized using a force criterion of 0.01 eV/Å.

From Fig. 2, it is observed that O vacancies have low formation energies in Ga rich conditions. There is no +1 charge state of O vacancies since it is thermodynamically unstable. The compensation effect between O vacancies and acceptor impurities may be the reason why p-type β-Ga₂O₃ is difficult to prepare. When Al atoms are doped in β-Ga₂O₃, the bandgap is enlarged but the doped In atoms will decrease the bandgap.²¹ That is because the bandgap of Ga₂O₃ is larger than In₂O₃ but smaller than Al₂O_3.³¹ For Al, Ga, and In, the conduction band minimum (CBM) is mainly contributed by the s-orbital of the corresponding cation of the oxide, where the s-orbital energy Al > Ga > In. The +2/0 transition level location calculated through GGA in Fig. 2 implies that O vacancy is a shallow donor and Ga vacancy is a shallow acceptor. However, this is not rigorous because the GGA method underestimates the bandgap, and there are corresponding calculations with HSE06 functional about β-Ga₂O₃ in Ref. 9. The transition levels ɛ (+2/0) of O vacancy are located more than 1 eV under CBM and O vacancies should be deep donors. Silicon and hydrogen may be the cause of the observed n-type conductivity in unintentionally doped β-Ga₂O₃.¹³ Similarly, all types of charge states of Ga vacancies should be deep acceptors with more than 1.5 eV transition levels above CBM.⁹ 

The photocatalytic activity of Ga₂O₃ can be increased by doping In atoms.¹⁷ We constructed the In doping model by replacing a Ga atom with one In atom, denoted as InGa₂O₃. The distance between In atom and other atoms is set as r, and in order to ensure that the results are accurate enough, the effect of r is taken into account in this work. The defect formation energy of InGa₂O₃ is calculated, which includes the electrostatic correction. From Ref. 21, we know that when a Ga(II) atom is substituted by one In atom compared with Ga(I), the structure is more stable. The InGa₂O₃ bulk is optimized based on the intrinsic structure and the obtained volume is increased by 1.37% because the ionic radii In³⁺ > Ga³⁺.

As shown in Fig. 3, the variation trend of defect formation energy of InGa₂O₃ with the Fermi level is almost the same as the undoped. The defect formation energy of O (Ga) vacancy in InGa₂O₃ is slightly larger (smaller) than the undoped, which is consistent with the result (about O vacancy) in Ref. 21. The transition level of the two is almost the same. Compared to β-Ga₂O₃, the Ga vacancy is more easily formed which can be attributed to (1) heavier average atomic mass, (2) weaker interatomic bonding, (3) more complex crystal structure with lower symmetry, and (4) increased anharmonicity.³² On the other hand, since it needs to provide energy for volume expansion, the O vacancy corresponds to higher defect formation energy.²¹ Guo et al.⁴ studied the effect of 100 MeV high-energy protons on β-Ga₂O₃ photodetector and they attributed the transformation of carriers’ transport to the created O vacancies. Yang et al.³³ reported that the low energy 150 keV proton radiation would also introduce oxygen vacancies in Ga₂O₃ film, and the concentration of vacancies increases evidently with the irradiation dosage. When In atoms are doped into Ga₂O₃, the defect formation energy of O vacancies increases, and its resistance to proton irradiation will be enhanced accordingly. In addition, Wang et al. found that the relative content of oxygen vacancy defects in Ga₂O₃ film would significantly decrease with the introduction of pulsed indium after analyzing the x-ray photoelectron spectroscopy (XPS) results.³⁴ The specific data in Figs. 2 and 3 are provided in the supplementary material.⁵² 

Figure 4 presents the calculated electronic band structures of primitive cell using PBE and HSE06 functional, whose bandgaps are 1.8 and 3.8 eV, respectively, and the effective band structure (EBS)^35,36 of bulk Ga₂O₃ supercell. It is observed that the two band structures are the same except for the bandgap. The effective mass of electrons (m_e*) in CBM at Γ is calculated, and it is almost isotropic and slightly depends on the direction. Ten points near Γ in Γ→X and Γ→M directions are taken for fitting to obtain the effective mass. The average of both directions equals 0.281 m_e, which agrees well with experimental measurements (0.28 m_e, Ref. 37). It is relatively small compared to the hole effective masses (m_h*) in VBM, which is because the upper part of valence bands is flatter than bottom conduction bands. The large m_h* means that the introduced holes are easily captured by the local lattice to become small polarons, and thus cannot easily form free holes to participate in conduction, which makes it difficult to obtain p-type Ga₂O₃. To analyze the impact of defects, we calculate the EBS of bulk Ga₂O₃ supercell (80 atoms) whose bandgap is the same as that of primitive cell. The colormap shows the overlap degree of energy bands.

Nowadays, there have been some studies about O vacancies on the physical properties of Ga₂O_3,¹⁴ but the study of Ga vacancies is relatively rare. The Ga vacancies are easily formed in O rich compared to Ga rich conditions and they have been identified by using electron paramagnetic resonance (EPR).³⁸ So it makes sense to investigate the influence of Ga vacancies on Ga₂O₃.

The vacancies studied in this work are all uncharged (q = 0). The EBS of Ga₂O₃ with two (78 atoms) and four (76 atoms) Ga(I) vacancies is shown in Figs. 5(a) and 5(b). Since the calculated bandgap will not affect the analysis about the physical properties of the defective Ga₂O₃, the PBE functional is used in the following calculations. When Ga(II) atoms are lost, the defective structures also will show the metal properties near the Fermi level like that Ga₂O₃ with Ga(I) vacancies. It is observed from Figs. 5(a) and 5(b) that the Ga(I) vacancies cause dramatic changes of energy bands near VBM. The more Ga(I) atoms lost, the greater the overlap of the energy bands near the Fermi level. It is accompanied by an upward shift in valence bands (the position of the Fermi level is assumed to be fixed here). The bandgap of defective Ga₂O₃ with Ga vacancies is negative and exhibits metallic properties. Nowadays, there have been some studies about the physical properties of vacancy-containing metal oxides.^39–41 Nolan and Elliot³⁹ studied the electronic properties of Cu₂O (a semiconductor) with Cu vacancy through DFT and DFT + U calculations, showing that the defective structure would exhibit metal properties near the Fermi level, which is similar to our results about Ga vacancies. They concluded that the origin of p-type conductivity in Cu₂O is due to the formation of a small concentration of Cu vacancies. Scanlon and Watson⁴⁰ reported that Cu vacancy is the dominant intrinsic p-type defect in CuCrO₂. Zeman et al. ⁴¹ demonstrated that V_Ga defects in Ga₂O₃ will induce p-type doping and lead to the high density of states near the Fermi level, which indicated that defective Ga₂O₃ with V_Ga has metallic properties near the Fermi level. The above reports demonstrate the p-type defects caused by metal vacancies and the metallization of semiconductors is common.

Experimentally, we can achieve a metal-insulator phase transition by adjusting the number of defects (mobility edge position) or by adjusting the position of the Fermi surface. Hibel et al. experimentally found the created metal-insulator transition due to Anderson localization in LuRh₄B₄ as the increasing α-particle damage level.⁴³ Therefore, it is also normal for metal vacancy defects in a semiconductor to undergo a metallization transition.

Figure 7 shows the differential charge density map after the formation of Ga and O vacancies. The O atoms around Ga vacancy lose valence electrons, and these bound electrons would accumulate near vacancy position, forming defective levels. Those lost valence electrons of O atoms form partially occupied bands because O atoms do not lose electrons easily due to large electronegativity. These newly formed energy bands are close to VBM, so electrons in VBM would easily transit to them and create holes, forming the p-type defect. The defect type (p or n) can be clearly distinguished by the positive and negative effective mass. As mentioned earlier, holes are more difficult to generate due to the large m_h*. But we also find that the overlap of energy bands near the Fermi level is increased when Ga vacancies are created, which accelerates the interband transition of electrons.

It is well known that the introduction of impurities also produces a certain number of bound electrons, so the effects of impurities and vacancies on semiconductors are similar. Next, we analyze the effect of defects from the viewpoint of doping. For p-type doping, when the impurities are shallow acceptors, the number of impurity levels is usually small and their positions are near the VBM. When the doping concentration is large enough, the Fermi level will enter the valence bands, which can explain the position of the Fermi level in Fig. 5. The valence bands and conduction bands overlap at the Fermi level, thus showing metallicity and producing strong light absorption in infrared bands. This also accounts for although the Ga vacancy is a deep acceptor, it does not affect the formation of p-type defect here. Because electrons forming the defect states are provided by oxygen atoms rather than Ga vacancies as deep acceptors.

The projected electronic band structures of each element of Ga₂O₃ with two Ga(I) vacancies are depicted in Fig. 8. It is clear that these energy bands are formed by the mutual hybridization of orbital electrons, but the contribution of electrons in different orbits is not the same. The electrons in O-2p orbitals contribute much to the formation of valence bands between −5 eV and Fermi level, while the conduction bands (except the bands that cross the Fermi level) are mainly formed by electrons in Ga-4s orbitals. As shown in Fig. 8(a), in the case of only considering Ga atoms, the valence bands in the energy range of −5 to −2 eV and −2 to 0 eV are mainly formed by Ga-4p and Ga-3d orbital electrons, respectively. The VBM of bulk Ga₂O₃ is dominated by Ga-3d and O-2p electrons. The energy bands mainly formed by Ga-3d electrons are relatively flat and have a small bandwidth because these electrons are more localized. The conduction bands near the Fermi level are formed by large O-2p electrons and small amounts of Ga-3d electrons, which is because these bound electrons are mainly from the valence electrons of O atoms. The results in Fig. 8 also explain why the Ga vacancies have a larger effect on the valence bands than conduction bands in Fig. 5. That is because the formation of valence bands is mainly contributed by electrons of the oxygen atoms. Oxygen atoms will lose their partial valence electrons when there are Ga vacancies in crystal, which affects the formation of valence bands.

β-Ga₂O₃ normally exhibits the characteristics of n-type semiconductor, which has been confirmed by experiments in Ref. 47. From Figs. 1(a) and 2, it finds that the O vacancies will be formed relatively easily in Ga rich conditions. In Refs. 14 and 48, the effects of O vacancies on the structural and optical properties of β-Ga₂O₃ have been studied partially, but the analysis of mechanism is not sufficient. In addition, the effect of charged vacancy on β-Ga₂O₃ has been studied in Ref. 49 and we explored the vacancy influence on the thermal conductivity of β-Ga₂O₃ in Ref. 50. From Fig. 10, we find that the presence of neutral O vacancies reduces the degeneracy of the conduction bands, leading to the symmetry destruction of the interaction potential between atoms. Figures 10(a) and 10(b) depict the EBS of defective Ga₂O₃ with two and four O(III) vacancies.

The defective Ga₂O₃ has changed to an indirect bandgap semiconductor and the bandgap is greatly reduced. Compared to energy bands of bulk Ga₂O₃ in Fig. 4, the valence bands of Ga₂O₃ with O vacancies in Fig. 10(a) move down and their overall structures remain almost unchanged except for generating extra valence bands between the Fermi level and previous VBM. However, the conduction bands are significantly different from non-defective. The shape of the extra valence band with a lower energy level is similar to that of free electrons, indicating a small effective mass and leading to higher mobility. In Fig. 10(b), the number of valence bands formed between −1.7 and 0 eV increases with the number of oxygen atoms lost. The bandgap keeps decreasing as the increased number of O(III) vacancies, yet the band structures below −1.7 eV are hardly affected. The physical properties of defective Ga₂O₃ with O(I) or O(II) vacancies are almost identical to those have O(III) vacancies.

Figure 7(b) shows the differential charge density map between bulk and Ga₂O₃ with O(III) vacancies. For the neutral O vacancy of Ga₂O₃, the Ga atoms bonded to this oxygen atom would lose their valence electrons. Figure 11 shows the projected band structures of each element in Ga₂O₃ with two O(III) vacancies. The O-2p orbital electrons contribute most to the valence bands formation between −5 and −1.7 eV. When the influence of oxygen element is not considered, the conduction bands are mainly formed by Ga-4s orbital electrons. In the energy band from −5 to −1.7 eV, the upper and lower parts of the valence bands are formed by Ga-3d and 4p orbital electrons. The bound electrons in defect levels near the Fermi level are from the valence electrons of Ga atoms, so the contribution of Ga element in Fig. 11(a) is obvious. In addition, the O-2p electrons usually contribute much to the formation of valence bands and their effects also cannot be neglected. The electrons of Ga atoms contribute more to the formation of the conduction bands and they are more likely to lose valence electrons, so the structures of the conduction bands will change obviously when there appear O vacancies in the crystal.

Figure 12 shows the DOS, dielectric functions ɛ₂, and average absorption coefficient α of Ga₂O₃ with two O(III) vacancies. In Ref. 14, the TDOS of bulk Ga₂O₃ in the Fermi level is not equal to 0, which is inconsistent with the experimental value. As shown in Fig. 12(a), the DOS in −0.5 eV almost comes from Ga-4s, Ga-4p, and O-2p states. We can know from Fig. 12(b) that there has an apparent peak near 2.5 eV, whose peak value is almost half of the intrinsic absorption. We conclude that the peak around 2.5 eV is caused by the electron transition between the defective levels below the Fermi level and the CBM, showing the O vacancies are n-type defects. In Fig. 12(c), there also exists absorption peak near 2.5 eV, but it is not as pronounced as the peak in Fig. 12(b). Usually, the peak of ɛ₂ is closely related to the DOS of electrons involved in quantum transitions, but the TDOS in Fig. 12(a) is much smaller than that of bulk in Fig. 9(a). On the one hand, the effective mass of the electrons participating in transition is relatively small, which is beneficial to the electrons transport. What is more, the energy level difference between the defective level and CBM is smaller compared to previous VBM. Dong et al.¹⁴ measured the photoluminescence spectra of β-Ga₂O₃ film with O vacancies at room temperature and found that there exist emission peaks near 2.8 eV, which is almost consistent with the calculated peak position in Fig. 12(b). We calculate the transition dipole moment (TDM) from electrons near the VBM to CBM in Fig. 4(a) (bulk) and Fig. 10(a) [Ga₂O₃ with two O(III) vacancies], the results are shown in Fig. 13 and the ratio of the TDM peak is close to that of the dielectric function.

The formation of O vacancies is similar to the addition of acceptor impurities in semiconductors and is called n-type impurities. When the concentration of n-type impurity is large, the Fermi level will be close to CBM. Similar to the analysis about Ga vacancy, the bound electrons are mainly from Ga atoms, not related to deep transition level ɛ (+2/0) of O vacancy.

From Fig. 2, we know that the defect formation energies of neutral Ga and O vacancies are almost equal under O rich conditions, and at this point we consider that the number of both vacancies is also approximately equal. Figure 14 shows the ɛ₂ and energy band structures of defective Ga₂O₃ with two Ga(I) and two O(III) vacancies. The peaks of ɛ₂ near the Fermi level and in 6–12 eV are smaller than that in Fig. 9(c) [only two Ga(I) vacancies], and also less than the value near 2.5 eV in Fig. 12(b) [only two O(III) vacancies]. These are similar to the self-compensation effect about dopants in Ref. 51 and we conclude the similar compensation effects between Ga vacancies and O vacancies will be created in the crystal. We assume that the O vacancies form before Ga vacancies which can analyze the results better. The bound electrons introduced by O(III) vacancies are attracted to Ga(I) vacancies, which increases the electrons concentration of conduction bands near the Fermi level in Fig. 14(b) [compare with the results of Fig. 5(a)], decreasing the effects of p-type defect. Correspondingly, the number of bound electrons around O(III) vacancies is decreased, leading the two filled bands below the Fermi level in Fig. 11 become partially occupied bands. As shown in Fig. 5(a), the introduction of Ga vacancies will cause the Fermi level to shift toward the valence bands, so the number of conduction bands near the Fermi level is increased (part of the valence bands introduced by O vacancies has become the conduction bands). Moreover, the defective levels caused by O(III) vacancies are close to the Fermi level, resulting in the disappearance of the peak around 2.5 eV in Fig. 12(b). We can find from Fig. 14(b) that the defect levels introduced by O vacancies are disappearing. About intrinsic absorption of Ga₂O₃, the peak is decreased and its position is redshifted, which maybe too many point defects destroy the structure of Ga₂O₃. In addition, the physical properties about Ga₂O₃ crystal containing two Ga(I) and four O(III) vacancies are shown in Fig. S1 in the supplementary material,⁵² where the p-type defect totally disappears due to the compensatory effect.

To summarize, we systematically investigate the vacancy-defects induced by radiation damage on electronic and optical properties of β-Ga₂O₃ using first-principles calculations. The defective β-Ga₂O₃ with Ga vacancies exhibits metallic properties, where a strong absorption of light in infrared band occurs. In addition, the O vacancies decrease the bandgap of β-Ga₂O₃ and make it an indirect bandgap semiconductor, leading to a n-type defect. When both Ga and O vacancies exist in Ga₂O₃ crystal, a self-compensation effect emerges, which reduces the influence of the two defects. Compared to the undoped β-Ga₂O₃, the defect formation energy of InGa₂O₃ changes slightly, where O vacancies are more difficult to form while Ga vacancies are easier to create. We can determine the type of defects based on the physical properties and increase the defect formation energy by doping In thus to increase the irradiation resistance ability of Ga₂O₃ devices.

The authors gratefully acknowledge financial support from the National Natural Science Foundation of China (No. 52076123), Shandong University Outstanding Young Scholar (No. 82363194), and Fundamental Research Funds of Shaanxi Key Laboratory of Artificially-Structured Functional Materials and Devices (No. AFMD-KFJJ-22206).

The authors have no conflicts to disclose.

Xiaoning Zhang: Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal). Xi Liang: Conceptualization (equal). Xing Li: Investigation (equal). Yuan Li: Investigation (equal). Jia-Yue Yang: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal). Linhua Liu: Funding acquisition (equal); Project administration (equal); Supervision (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.