Effects of oxygen vacancies on the structural and optical properties of β-Ga2O3

The structural, electronic, and optical properties of β-Ga2O3 with oxygen vacancies are studied by employing first-principles calculations based on density function theory. Based on the defects formation energies, we conclude the oxygen vacancies are most stable in their fully charge states. The electronic structures and optical properties of β-Ga2O3 are calculated by Generalized Gradient Approximation + U formalisms with the Hubbard U parameters set 7.0 eV and 8.5 eV for Ga and O ions, respectively. The calculated bandgap is 4.92 eV, which is consistent with the experimental value. The static real dielectric constants of the defective structures are increased compared with the intrinsic one, which is attributed to the level caused by the Ga-4s states in the bandgap. Extra peaks are introduced in the absorption spectra, which are related to Ga-4s and O-2p states. Experimentally, β-Ga2O3 films are deposited under different O2 volume percentage with ratio-frequency magnetron sputtering method. The measured results indicate that oxygen vacancies can induce extra emission peaks in the photoluminescence spectrum, the location of these peaks are close to the calculated results. Extra O2 can increase the formation energies of oxygen vacancies and thus reduce oxygen vacancies in β-Ga2O3.

of bandgap in semiconductors [22][23][24][25] . As a result, the defects states caused by the defects are not correctly treated. The accurate electronic structures can be described by more elaborate approaches, such as hybrid Hartree-Fock (HF) density functionals, Heyd-Scuseria-Ernzerhof (HSE) and the screened exchange (sX) 14,26 . However, these accurate methods are limited to the computational resources.
In this paper, first-principles based on all-electron DFT is used to study the atomic structures, formation energies, electronic structures and optical properties of the intrinsic β -Ga 2 O 3 with different oxygen vacancies. In order to get a reasonable result, the GGA+ U approach is used, which is computational frugally compared with other hybrid density functionals and also can give an accurate description by controlling the Hubbard U parameter 27,28 .

Results and Discussion
Structural properties. Monoclinic structural β -Ga 2 O 3 with C2/m symmetry can be described with four lattice parameters, i.e, a, b, c and β . Figure 1(a) shows the optimized conventional cell of β -Ga 2 O 3 . The structural parameters of β -Ga 2 O 3 based on our calculation results and other previous theoretical with experimental results are listed in Table 1. The calculation results are in good agreement with other results derived from different functionals, which indicate our optimization method is reasonable.
There are three types of O sites in β -Ga 2 O 3 cell as shown in Fig. 1(a). As a result, three O vacancies exist, which are denoted as V OI , V OII (both are 3-fold) and V OIII (4-fold), respectively. For V OI , there are two 6-fold Ga ions and one 4-fold Ga ion surrounded, while two 4-fold Ga ions and one 6-fold Ga ion are adjacent to V OII . For Ga ions, there are two nonequivalent sites, the 4-fold one and 6-fold one, respectively. The crystalline structures are described in terms of GaO 6 octahedron and GaO 4 tetrahedron chains. Based on the former optimized cell, we calculate the defective structures with the 1 × 2 × 2 supercell shown in Fig. 1(b).
Defects formation energies. The formation energy of an oxygen vacancy D in β -Ga 2 O 3 with charge state q is given by 30 : Where E D Ga O ( , ) t q 2 3 is the total energy of the supercell with an oxygen vacancy D in charge state q, E Ga O ( ) t 2 3 is the total energy of the β -Ga 2 O 3 perfect supercell structure. μ O is the chemical potential of O which we use the potential of O 2 molecule as a reference, n O denotes the numbers of the vacancies in the supercell. ε F is the Fermi level measured from the top of the valence band E VBM and ΔV is the average potential difference between the defective supercell and the perfect one.  From Eq. (1), the formation energy of an oxygen vacancy in charge state q can be determined. In this method, the chemical environment difference between the different finite defective supercells, which leads to the errors in the formation energy, are taken into account. Besides, E VBM varies with the different O vacancies and charge states, which can lead to unreasonable formation energies results. Therefore, it is necessary to align the band structures of the different supercells 31 . E VBM of the defective supercells is defined by 30 : is the total energy of neutral perfect supercell and E perfect q ( , ) t is the total energy of the perfect supercell in charge state q.
The O defects formation energy is associated with oxygen condition in the synthesis process of β -Ga 2 O 3 , and the oxygen chemical potential varies with the different ambient atmospheres. Both μ O and µ Ga 0 are confined by the phase equilibrium condition of β -Ga 2 O 3 . The range of μ O is between the oxygen poor (µ µ = O O 0 , where µ O 0 is the total energies per atom of molecular O 2 ) and oxygen rich ( 2 3 , where µ Ga 0 is the total energy per atom of metallic Ga). The formation energies for different O vacancies with different charge states as a function of the Fermi energy are obtained by the GGA+ U method, and the results are shown in Fig. 2. Both oxygen-rich and oxygen-poor cases are calculated. For each defect, only the charge state in the most energetically favorable at a given Fermi energy is shown. The defects formation energies vary with the Fermi level ε F . When ε F is close to valence band, the stable charge states correspond to + V OI 2 , + V OII 2 and + V OIII 2 , respectively. With the ε F moving up, the neutral defects are dominant. There are no + 1 charge state of oxygen vacancies, which indicates that + 1 charge state is not stable for all three type vacancies. The followed discussion we will focus only on these stable structures, namely, neutral and + 2 charge states.
Thermodynamic transitions between the different charge states of the same defect are denoted by the kinks as shown in Fig. 2, which are derived from the formation energies. Measured from the valence band maximum (VBM), the ε (+ 2/0) transition level of V OI , V OII and V OIII are 3.2 eV, 3.7 eV and 3.9 eV, respectively. The deep transition levels mean all the oxygen vacancies act as deep donors. When Fermi level ε F locates at the mid-gap around, the charged vacancies are more stable than the neutral ones. Under oxygen-poor atmosphere, the negative oxygen vacancies formation energy make the vacancies easy to form. While in the case of oxygen-rich, the positive formation energy of neutral oxygen vacancies is higher than the ones under oxygen-poor atmosphere. As a result, it is hard to generate a high concentration of oxygen vacancies under this atmosphere. These results indicate the oxygen vacancies are hard to be effective n-type donors, and under oxygen-poor atmosphere, the high When an oxygen ion is removed, a defective β -Ga 2 O 3 model with oxygen vacancy is created. For the neutral oxygen vacancies (V OI 0 , V OII 0 and V OIII 0 ), a defective level occupied by two electrons appears, which are shown in Fig. 4. The total energy of the defective β -Ga 2 O 3 is lowered by the attraction of the surrounded Ga ions, and these ions sites distortion leave the defects level set at the middle of the bandgap. When the vacancies carry with two positive charges ( + V OI 2 , + V OII 2 and + V OIII 2 ), the results are very different from the neutral ones. The defects levels are unoccupied, the outward of surrounded Ga ions make the Ga-O bond strengthen, leading to the decrease of the formation energy. As a result, the defective level moves to the conduction band. These results are similar with the band structures of other defective oxides such as ZnO and Al 2 O 3 30,33 . Figure 5 presents the total density of states (TDOS) of β -Ga 2 O 3 with various oxygen vacancies, and detailed partial density of states (PDOS) results induced by the defects are shown inset. Compared with the TDOS of the intrinsic β -Ga 2 O 3 , new peaks arise in the bandgap. For V OI , with the VBM chosen as the reference, the DOS peak of V OI 0 is 2.77 eV away from the VBM. Based on the PDOS of the Ga ions and O ions shown in Fig. 5(a,b), the unpaired hanging bond of Ga ions around the vacancies make the most contributions to these defective peaks. The DOS of defects almost come from Ga-4s and Ga-4p states along with a few O-2p states. In the presence of V OI shown in Fig. 2(a), it is found that Ga1 and Ga3 ions move 0.21 Å toward the vacancy, while Ga2 slightly move about 0.16 Å away from the oxygen vacancy, which is shown in Fig. 6(a). The movement of the Ga ions can be attributed to the ionic size and crystalline structure. When two electrons move away, the defects level caused by the charged oxygen vacancies move toward the conduction band, the interaction caused by the overlap between the defects level and the conduction bottom level make the conduction band bottom shift down, as shown in Fig. 4(d). The attractive interactions between Ga ions and O vacancy disappear. There will be considerate space left in the vacancy site, which will give the adjacent Ga ions more freedom to disperse. For both equivalent tetrahedral structure Ga1 and Ga3 ions, the space makes them move toward to vacancy after the relaxation. While for Ga2 ion, Ga2 ion is not electrostatically attracted by the vacancy anymore, the closer Ga1 and Ga3 ions product   more electrostatically repulsion to Ga2 ion, these effects leave Ga3 ions move away. However, compared with Ga1 and Ga3 ions, Ga2 ion with an octahedral structure has seven bonds connected to neighboring oxygen ions, which make Ga2 ion more geometrically stable.
Similar to V OI , the band structures and DOS of V OII and V OIII shown in Figs 4 and 5, respectively. Extra levels emerge in the band structures, with the VBM chosen as the reference, the DOS peak of V OII 0 and V OIII 0 is 3.10 eV and 3.40 eV away from the VBM, respectively. The structural relaxations of the supercell models with V OII and V OIII are also calculated, and the variation between the defect and surrounding ions after relaxation are listed in Table 2. Detailed atomic structures relaxations shown in Fig. 6(b,c). It is noted that the 4-fold V OIII 0 connected there octahedron and one tetrahedron, which makes the V OIII 0 more geometrically stable. As a result, when V OIII 0 induced, the variation of adjacent Ga ions are less, the three equivalent 6-fold Ga ions move less away from the V OIII 0 , more relaxation is applied to 4-fold Ga1 ion, the result is Ga1 ion is attracted to the vacancy 0.33 Å closer.
Optical properties. As a promising material for optoelectronic devices, a deep understanding of optical properties of β -Ga 2 O 3 is very necessary. To investigate the optical properties of β -Ga 2 O 3 with oxygen vacancies, the complex dielectric function ε ω ε ω ε ω = +i ( ) ( is calculated. The imaginary part of complex dielectric function ε 2 (ω) is calculated by summing the transitions between occupied and unoccupied electronic states, which is relevant to the electronic band structure that can determine other optical properties of the material. The imaginary part is given by the following equation 34 : Where e is the electron charge, m is the mass of free electrons, ω is the frequency of incident photons, M is the dipole matrix, i and j are the initial and final states, respectively. f i is the fermi distribution function for i-th state with wave function vector k. According to the Kramers-Kronig transformation, the real part ε 1 can derived from the imaginary ε 2 , which is given as follows: Where P denotes the principal value of the integral. The other optical properties such as absorption coefficient α(ω ), reflectivity R(ω ), refractive index n(ω ), extinction coefficient k(ω ), and energy-loss L(ω ) can be derived from the dielectric function and defined by 35 :  Figure 7(a,b) display the real and imaginary parts of the dielectric function ε(ω ) of β -Ga 2 O 3 with different oxygen vacancies, respectively. For the pure β -Ga 2 O 3 , from a general view, our calculation dielectric function ε(ω) are consistent with previous studies in tendency. It is noted that the peak for the imaginary part dielectric function at 8.7 eV is much stronger than other peaks, which is related to the interband transition between O 2p and Ga 4 s states. Taking the oxygen vacancies into consideration, new peaks (3.17eV, 3.37eV and 3.69 eV for V OI 0 , V OII 0 and V OIII 0 , respectively) arise in the low energy region, while the effect is barely noticeable in ultraviolet region. From the electronic structures and DOS results, we concluded that these peaks are caused by the transition from Ga-4s states in the defective level induced by oxygen vacancies to the Ga-4s states in the conduction band. The difference locations among these peaks are consistent with the defective levels in the band structures from our previous calculated results. Fig. 1 (b), +denotes Ga ions move close, while −denotes Ga ions move away)  The static dielectric constant ε 1 (0) is given by the low energy limit of ε 1 (∞ ). The calculated ε 1 (0) of pure β -Ga 2 O 3 is 1.36, which is smaller than the experiment results. The underestimation of ε 1 (0) is due to the low number of conduction bands and overlook of phonon contribution. When oxygen vacancies induced in the β -Ga 2 O 3 , all the values of ε 1 (0) increase, the ε 1 (0) for β -Ga 2 O 3 with V OI 0 , V OII 0 and V OIII 0 are 1.52, 1.46 and 1.64, respectively, which are attributed to the extra levels in the bandgap of β -Ga 2 O 3 caused by oxygen vacancies.

Distance in Å (the Ga coordination number around O vacancy which is shown in
The absorption spectra of all the β -Ga 2 O 3 systems are illustrated in Fig. 7(c). The incident radiation has linear polarization along the (100) direction. The intrinsic absorption edge of β -Ga 2 O 3 is consistent with the bandgap (4.9 eV) with the urgently decreases absorption edge, which means β -Ga 2 O 3 is a promising optical material at DUV region. The intrinsic absorption is related with the interband electron transition between O-2p states in the VBM and Ga-4s states in the CBM. Compared with pure β -Ga 2 O 3 , the absorption coefficient of the structures with oxygen vacancies increase in visible and infrared region while decrease in the deep violet region. New absorption peaks appear at 3.80 eV, 3.52 eV and 3.37 eV for V OI 0 , V OII 0 and V OIII 0 , respectively. According our first-principles calculation results, a schematic diagram of the possible absorption processes of oxygen vacancies in β -Ga 2 O 3 is illustrated in Fig. 8. The intrinsic absorption process is the electron transition from O-2p states in the VBM and Ga-4s states in the CBM. The additional absorption parts are deduced from the electron transition between the defective level caused by oxygen vacancies and the valence band.
The reflectivity, and refractive index are depicted in Fig. 7(d,e). For β -Ga 2 O 3 with oxygen vacancies, the reflectivity and refractive index enhance in the low energy region, which is contributed to the defective levels in the bandgap induced by the oxygen vacancies. The location of reflectivity and refractive index peaks are consistent with the dielectric function. The energy loss function describes the energy lost by an electron passing from a homogeneous dielectric material. It has the advantage of covering the complete energy range, including non-scattered and elastically scattered electrons (zero loss), which excite the electrons of atom's outer shell (valence loss) or valence inter-band transitions. In Fig. 7(f), the energy loss peaks caused by oxygen vacancies locates at 3.96eV, 3.45eV and 3.58 eV for V OI 0 , V OII 0 and V OIII 0 , respectively. These peaks are corresponding to where the reflectivity decrease rapidly. XRD and photoluminescence spectra results. To explore the effects of oxygen vacancies on the structural and optical properties of β -Ga 2 O 3 , β -Ga 2 O 3 films were deposited under different O 2 volume percentage. Figure 9 presents the XRD results of the films deposited under 0% and 1% O 2 volume percentage, respectively. Through XRD patterns, both films exhibit β -Ga 2 O 3 structure. The average crystalline sizes in β -Ga 2 O 3 films were estimated from the (400) peak by using the Scherrer's equation, D = kλ /(B × cosθ ), where k is Scherrer's constant with a value of 0.89, λ is the wavelength of the X-ray radiation, B is the FWHM of (400) peak, and θ is the angle of the diffraction peak. The calculated crystalline sizes for the films deposited under 0% and 1% O 2 volume percentage are 20.15 nm and 22.45 nm, respectively. Room-temperature photoluminescence (PL) spectra of the β -Ga 2 O 3 films excited with 325 nm laser are shown in Fig. 10. The fitting curve of the film deposited under 1% O 2 volume percentage is shown in the inset figure. The emission band can be divided into four Gaussian bands centered at about 380 nm, 416 nm, 442 nm and 464 nm, respectively. The emission peak centers at 380 is in the UV region, this peak is concerned with the transition levels between the oxygen vacancy and unintended N impurities introduced in the N 2 annealed 36 . While for the peaks center at 416 nm, 442 nm (both in the violet region) and 464 nm (blue region), these three emission peaks are originated from the electron-hole recombination formed by oxygen vacancies, or to the recombination of Ga-O vacancy pair 37,38 .
From our previous calculated results, extra oxygen gas induced in the procedure of deposition can increase the formation energy of oxygen vacancies. The higher formation energy can reduce the concentration of oxygen vacancies. From the PL results, the emission peaks related to oxygen vacancies of the film deposited under 1% O 2 volume percentage is higher than that deposited under 0%. According to our first-principles calculation results, the peaks center at 416 nm and 442 nm are most related with V OI , and V OII , while the peak at 464 nm is most attributed to V OIII . When irradiated with high energy photon, the electron at defective levels caused by oxygen

Methods
All calculations were based on the density functional theory with Cambridge Serial Total Energy Package (CASTEP) code 39 . The exchange-correlation potential was described with the Perdew-Burke-Ernzerhof (PBE) functional under the Generalized Gradient Approximation (GGA) exchange-correlation functional 40 . The ultrasoft pseudopotential method was used for the interactions between the electrons and ions. The atomic configuration of Ga was [Ar] 3d 10 , and the 3d 10 electrons were considered as valence electrons, while the atomic configuration of O is [He] 2s 2 2p 4 , the 2s 2 and 2p 4 electrons were treated as valence electrons. Before the electronic and optical calculations, structural relaxation was employed. The lattice parameters and internal coordinates were relaxed with Broyden-Fletcher-Goldfarb-Shanno (BFGS) optimization method 41 . The energy tolerance, the tolerance of the force, maximum stress and maximum displacement were 5 × 10 −6 eV/atom, 0.01 eV/Å, 0.02 Gpa and 5 × 10 −4 Å, respectively. The cutoff energy for the plane wave basis set was 450 eV, and a Monkhorst-Pack 2 × 8 × 4 k-points was used for integrations of the Reduced Brillouin zone 42 . For the defective crystal, a 1 × 2 × 2 supercell of β -Ga 2 O 3 based on the optimized cell with 80 atoms was created to act as the computational model, which is presented in Fig.1 (b). The lattice constants of the defective supercell were fixed, only the internal coordinates can be relaxed.
For the calculations of formation energies, electronic structures and optical properties, the GGA+ U method was adopted. After series tests, the U d value for Ga-3d and U p value for O-2p were set at 7.0 eV and 8.5 eV, respectively. Under this correction, the reasonable results can be obtained. All of our calculations are carried out at the theoretical equilibrium lattice constants, which is essential in order to avoid the spurious effects in the process of relaxation.
β -Ga 2 O 3 films were prepared on sapphire substrates (0001) by the radio-frequency magnetron sputtering method with a high-purity Ga 2 O 3 target (99.995% purity, 50.8 mm diameter). The distance between the target and substrate was about 150 mm, and the output of the ratio source was 60 W. To explore the oxygen vacancies on the properties of β -Ga 2 O 3 films, several experiments were conducted by varying the oxygen concentration in the growth chamber. Before the deposition, the chamber was pumped down to 5 × 10 −6 mTorr as the base pressure. The films were deposited under 5 mTorr atmosphere pressure at room temperature. To exclude any interference caused by the variation of film thickness, the thickness of both films were kept between 220 to 250 nm. To improve the crystalline properties, the as-deposited Ga 2 O 3 sample was subsequently annealed at 1000 °C for 60 min. Pure N 2 was chosen as the annealing atmosphere to avoid any effects of caused by extra oxygen. The oxygen volume percentage was varied by changing the gas volume percentage of oxygen to argon gas introduced into the growth chamber. To investigate the crystalline structures of the films, 2θ scan were conducted using an X-ray diffractometer (SHIMADZU XRD-7000), with Cu Kα radiation (λ = 0.154056 nm). The Photoluminescence (PL) spectra were measured under the excitation by a He-Cd UV laser (325 nm and 20 mW).