Electronic structures and optical properties of Si- and Sn-doped β-Ga2O3: A GGA+U study

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Dang Jun-Ning, Zheng Shu-wen, Chen Lang, Zheng Tao. Electronic structures and optical properties of Si- and Sn-doped β-Ga₂O₃: A GGA+U study. Chinese Physics B, 2019, 28(1): 016301 Copy to clipboard

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Electronic structures and optical properties of Si- and Sn-doped β-Ga₂O₃: A GGA+U study

Dang Jun-Ning, Zheng Shu-wen^†, Chen Lang, Zheng Tao

Institute of Opto-electronic Materials and Technology, South China Normal University, Guangzhou 510631, China

† Corresponding author. E-mail: LED@scnu.edu.cn

Project supported by the Science and Technology Program of Guangdong Province, China (Grant No. 2015B010112002) and the Science and Technology Project of Guangzhou City, China (Grant No. 201607010250).

Abstract

The electronic structures and optical properties of β-Ga₂O₃ and Si- and Sn-doped β-Ga₂O₃ are studied using the GGA + U method based on density functional theory. The calculated bandgap and Ga 3d-state peak of β-Ga₂O₃ are in good agreement with experimental results. Si- and Sn-doped β-Ga₂O₃ tend to form under O-poor conditions, and the formation energy of Si-doped β-Ga₂O₃ is larger than that of Sn-doped β-Ga₂O₃ because of the large bond length variation between Ga–O and Si–O. Si- and Sn-doped β-Ga₂O₃ have wider optical gaps than β-Ga₂O₃, due to the Burstein–Moss effect and the bandgap renormalization effect. Si-doped β-Ga₂O₃ shows better electron conductivity and a higher optical absorption edge than Sn-doped β-Ga₂O₃, so Si is more suitable as a dopant of n-type β-Ga₂O₃, which can be applied in deep-UV photoelectric devices.

PACS: 63.20.dk;73.20.At;74.20.Pq;78.20.Ci

Keyword:density functional theory;GGA + U method;Si-doped β-Ga₂O₃;Sn-doped β-Ga₂O₃;electronic structure;optical property

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1. Introduction

β-Ga₂O₃ is an emerging semiconductor material with a wide bandgap of about 4.9 eV. β-Ga₂O₃ exhibits fascinating physical properties and chemical stability, such as good transparency and a critical breakdown field of up to 8 MV/cm, which has made it a focus of attention in recent years.^[1,2] With these advantages, β-Ga₂O₃ is considered to be a promising transparent conducting material operating from the visible to the deep-ultraviolet spectrum for photodetector and other optoelectronic devices.^[3,4]

However, the poor electrical conductivity of β-Ga₂O₃ hinders its application as a transparent conducting oxide (TCO).^[5] It is well known that doping is an effective way to improve the photoelectric properties of materials, including conductivity, mobility and absorptivity. Si is regarded as a suitable dopant for n-type β-Ga₂O₃ because excess free electrons will be generated after doping. Further, the oxides of Si and Ga have a similar melting point, which is beneficial for adjusting the growth temperature and obtaining a high doping concentration. Zhang et al.^[6] obtained a Si-doped β-Ga₂O₃ film using the pulsed laser deposition (PLD) technique, and the highest doping concentration is 9.1 × 10¹⁹ cm⁻³, but the corresponding conductivity is only 2.0 S · cm⁻¹. The low conductivity is due to the amorphous-like structure of the Si-doped β-Ga₂O₃ and blockage of electronic transmission paths. Takakura et al.^[7] deposited Si-doped β-Ga₂O₃ film by radio-frequency (RF) magnetron sputtering, and found that the optical gap of the Si-doped β-Ga₂O₃ is wider than that of β-Ga₂O₃. Recently, a homoepitaxial Si-doped Ga₂O₃ film produced by PLD, with a high carrier concentration of 1.74 × 10²⁰ cm⁻³ and ultrahigh conductivity of 732 S/cm, was reported,^[8] but its conductivity is still far below that of commercial ITO film or Al-doped ZnO film. Sn is another promising dopant as a donor for improving the conductivity of β-Ga₂O₃ because excess free electrons will be generated when Sn⁴⁺ substitutes Ga³⁺. Orita et al.^[1] deposited n-type Ga₂O₃ film by Sn doping and obtained the best conductivity of 8.2 S · cm^−1. Ohira et al.^[5] prepared Sn-doped β-Ga₂O₃ by the float-zone method, found that Sn was dispersed in β-Ga₂O₃ uniformly, and suggested that Sn be substituted in the Ga site in the β-Ga₂O₃ lattice, but they did not discuss the optical properties of Sn-doped β-Ga₂O₃. Baldini et al.^[9] grew Si- and Sn-doped homoepitaxial β-Ga₂O₃ films by metal organic vapor phase epitaxy (MOVPE), and the highest electron concentrations of 8 × 10¹⁹ cm⁻³ for Si-doped β-Ga₂O₃ and 1 × 10¹⁹ cm⁻³ for Sn-doped β-Ga₂O₃ were obtained. The corresponding electron mobilities were 54.37 cm²/V · s and 32 cm²/V · s, but the conductivities and optical properties were not analyzed.

First-principles calculations based on density functional theory (DFT) have been extensively used for scientific research, and some material properties have been successfully parsed, such as electronic structures, optics and magnetism. Varley et al.^[10] found that a stable structure is obtained when Si⁴⁺ replaces Ga³⁺ in the four-fold coordination according to the calculated total energy of Si-doped Ga₂O₃ by using first principles. Siah et al.^[11] reported that Sn⁴⁺ prefers six-fold coordination and tends to substitute the Ga³⁺ octahedral site for Sn-doped β-Ga₂O₃. Zhang et al.^[12] calculated the electronic structures and optical properties of β-Ga₂O₃ and Sn-doped β-Ga₂O₃ by using the generalized gradient approximation (GGA) exchange–correlation function based on DFT, but the calculated bandgap of β-Ga₂O₃ is only 2.315 eV, which is less than half of the experimental value (4.9 eV^[1]). The underestimation of the bandgap is a well-known drawback of the standard DFT calculation due to the lack of precise consideration of the exchange–correlation potential, but it can be corrected by the scissors approximation,^[13] hybrid-functional,^[14] and GGA + U methods.^[15] In order to obtain calculated bandgaps that are close to the experimental data, Li et al.^[14] calculated the electrical and optical properties of β-Ga₂O₃ and Si-doped β-Ga₂O₃ using the screened hybrid functional HSE06, but the calculated result shows that β-Ga₂O₃ is a direct-bandgap material, which is not consistent with the experimental result.^[16] Lately, Dong et al.^[15] calculated the influences of oxygen vacancies on the structural and optical properties of β-Ga₂O₃ by GGA + U approximation with the Hubbard U parameters of 7.0 eV and 8.5 eV for Ga and O ions, respectively. The calculated bandgap of β-Ga₂O₃ is 4.92 eV, which is in agreement with the experimental value, but the calculated peak of the Ga 3d state is located at about −14.8 eV, which deviates from the experimental result of −17.12 eV,^[17] and so the Hubbard U value requires reconsideration. So far, there have been no reports on the electronic structures and optical properties of Si-doped and Sn-doped β-Ga₂O₃ by using GGA/LDA + U methods. There are also very few comparative research reports on the physical properties of Si-doped β-Ga₂O₃ and Sn-doped β-Ga₂O₃. To further understand the differences in the electronic structures and optical properties of Si-doped β-Ga₂O₃ and Sn-doped β-Ga₂O₃, the energy band structures, density of states and optical properties are calculated in this paper using the GGA + U method based on DFT.

2. Model and method

2.1. Structural model

β-Ga₂O₃ has a monoclinic structure and its space group is C2/m,^[18] β-Ga₂O₃ has double chains of edge-sharing GaO₆ octahedra and a single chain of vertex-sharing GaO₄ tetrahedra along the y axis, which is beneficial to the movement of free electrons.^[19] A 1 × 2 × 2 supercell of β-Ga₂O₃ is used as the father model and is shown in Fig. 1(a). The β-Ga₂O₃ supercell is obtained by expanding a unit cell along the y-axis and z-axis directions, so the supercell has 32 Ga atoms and 48 O atoms. It is noted from Fig. 1(b) that the supercell possesses two inequivalent Ga sites: Ga(1) and Ga(2). The Ga(1) atom bonds with the four nearest oxygen atoms to form a tetrahedral coordinated structure, and the Ga(2) atom bonds with the six nearest oxygen atoms to form an octahedral coordinated structure. When a Si or Sn atom substitutes for a Ga atom in the β-Ga₂O₃ supercell, the Si-doped or Sn-doped β-Ga₂O₃ is obtained and marked as β-Si_0.0625Ga_1.9375O₃ or β-Sn_0.0625Ga_1.9375O₃. To ensure the stability of the doped materials, the structural optimization^[30] and total energy calculation of the four doped materials with Si replacing Ga(1) or Ga(2) and Sn replacing Ga(1) or Ga(2) are all carefully considered. The total low-energy results show that β-Si_0.0625Ga_1.9375O₃ with Si replacing Ga(1), and β-Sn_0.0625Ga_1.9375O₃ with Sn replacing Ga(2), are more stable, which are consistent with earlier findings.^[10,11] Therefore, the two structural models of β-Si_0.0625Ga_1.9375O₃ with Si replacing Ga(1), and β-Sn_0.0625Ga_1.9375O₃ with Sn replacing Ga(2), are adopted in this paper.

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	Fig. 1. (a) The 1 × 2 × 2 supercell of β-Ga₂O₃ and (b) the projection along the y-axis direction.

2.2. Computational method

The electronic structures and optical properties of β-Ga₂O₃ and TM-doped β-Ga₂O₃ (TM = Si,Sn) are calculated using the first-principles plane-wave pseudo-potential method based on density functional theory^[20] as implemented in the CASTEP package.^[21] The exchange–correlation potential is employed through the GGA within the Perdew–Burke–Ernzerhof (PBE) function.^[22] The atomic cores are described by the ultra-soft pseudo-potential, and the valence electron configurations are [Ga] 3d¹⁰4s²4p¹, [O] 2s²2p⁴, [Si] 3s²3p² and [Sn] 5s²5p². In order to obtain the accurate bandgap of β-Ga₂O₃, the GGA + U method is adopted and the parameters of U_d,Ga and U_p,O are set to be 14.0 eV and 8.35 eV, respectively. The energy cutoff of the plane-wave basis is set to be 450 eV for the calculation of electronic structures and 600 eV for the calculation of optical properties. A 2 × 4 × 2 mesh of the Monkhorst–Pack k-points^[23] is used for the Brillouin zone integration in reciprocal space.^[24] Before calculation, the atomic coordinates and cell volume of the models need to be optimized, and the energy tolerance, tolerance of the force, maximum stress and maximum displacement are set to be 1 × 10⁻⁵ eV/atom, 0.03 eV/Å, 0.05 GPa, and 0.001 Å, respectively. The structural optimization is completed when the above four parameters meet or exceed the setting precisions.

3. Results and discussion

3.1. Structural properties

The optimized structure parameters of β-Ga₂O₃ and TM-doped β-Ga₂O₃ (TM = Si,Sn) are listed in Table 1. It is clear that the calculated lattice constants of β-Ga₂O₃ are in good agreement with the experimental results^[37] and other calculated data,^[38] which indicates that the calculation method and models are rational. The lattice constants of β-Si_0.0625Ga_1.9375O₃ and β-Sn_0.0625Ga_1.9375O₃ are larger than those of β-Ga₂O₃, which mainly originates from the competitive effects of the Coulomb repulsion and the ionic radius difference between Si and Ga or Sn and Ga. On the one hand, the excess free electrons will be generated and will enhance the Coulomb repulsion when Ga³⁺ is replaced by Si⁴⁺ or Sn⁴⁺, and the strong effect of the Coulomb repulsion will increase the supercell volume. On the other hand, the larger ionic radius will increase the volume of the materials. For β-Si_0.0625Ga_1.9375O₃, the Si ionic radius (0.026 nm^[27]) is much smaller than that of the Ga(1) (0.047 nm^[28]), but the effect of the Coulomb repulsion plays a more important role than the ionic radius difference, so the result of the competition is the supercell volume of β-Si_0.0625Ga_1.9375O₃ being larger than that of β-Ga₂O₃. For β-Sn_0.0625Ga_1.9375O₃, the Sn ionic radius (0.069 nm^[29]) is larger than that of the Ga(2) (0.062 nm^[28]), and therefore the total superposition makes the supercell volume of β-Sn_0.0625Ga_1.9375O₃ larger than that of β-Ga₂O₃.

Table 1.

The structural parameters and formation energies of β-Ga₂O₃ and TM-doped β-Ga₂O₃ (TM = Si,Sn) under the GGA + U method.

3.2. Doping formation energy

To evaluate the structure stability of TM-doped β-Ga₂O₃ (TM = Si,Sn), the doping formation energies of β-Si_0.0625Ga_1.9375O₃ and β-Sn_0.0625Ga_1.9375O₃ are calculated according to the formula^[30]

where E_{TM_0.0625Ga_1.9375O₃} is the total supercell energy of β-Si_0.0625Ga_1.9375O₃ or β-Sn_0.0625Ga_1.9375O₃, E_{β−Ga₂O₃} is the total supercell energy of β-Ga₂O₃, μ_TM represents the chemical potential of the Si or Sn atom, and μ_Ga is the chemical potential of the Ga atom. The chemical potential is connected with the growth atmosphere of the materials, which further affects the doping formation energies of TM-doped β-Ga₂O₃. Therefore, the doping formation energies of β-Si_0.0625Ga_1.9375O₃ and β-Sn_0.0625Ga_1.9375O₃ are calculated under O-rich and O-poor conditions. To determine the values of μ_Ga and μ_TM, the formation enthalpies ΔH_f of β-Ga₂O₃, SnO₂, and SiO₂ are considered assuming their formations from the metallic Ga, Sn, semiconductive Si, and gaseous O₂. The chemical potentials of μ_Ga and μ_TM are calculated using the following relationships:

where μ_GaO and μ_TMO are the energies of β-Ga₂O₃ and TMO₂ in a single molecule, and μ_Ga(bulk), μ_TM(bulk), and μ_O₂ are the chemical potentials of metallic Ga, Sn, semiconductive Si, and gaseous O₂, respectively. By combining Eqs. (2) and (3), the chemical potentials of Ga and TM are

for O-poor conditions, and

for O-rich conditions. The calculated formation enthalpy ΔH_f(Ga₂O₃) for β-Ga₂O₃ is −10.95 eV, which is close to the experimental value of −11.3 eV.^[10] The doping formation energies of β-Si_0.0625Ga_1.9375O₃ or β-Sn_0.0625Ga_1.9375O₃ under O-poor and O-rich conditions are calculated using Eqs. (4) and (5), and the calculated values are listed in Table 1. As can be seen in Table 1, the doping formation energies of β-Si_0.0625Ga_1.9375O₃ or β-Sn_0.0625Ga_1.9375O₃ under O-poor conditions are lower than those under O-rich conditions, which indicates that the TM-doped β-Ga₂O₃ (TM = Si,Sn) tend to form under O-poor conditions. Moreover, the doping formation energy of β-Si_0.0625Ga_1.9375O₃ is larger than that of β-Sn_0.0625Ga_1.9375O₃ whether under O-rich or O-poor conditions, which reveals that the preparation of Si-doped β-Ga₂O₃ requires a higher growth temperature as compared to Sn-doped β-Ga₂O₃. To explain the difference in the doping formation energy between β-Si0_.0625Ga_1.9375O₃ and β-Sn0_.0625Ga_1.9375O₃, the average bond lengths of Ga(1)–O, Si–O, Ga(2)–O, and Sn–O are given in Table 2. It is well known that the bond length variation is related to the deformation and relaxation of the material structure, which is reflected in the formation energy. It is found from Table 2 that the bond length variation between Ga(1)–O and Si–O is significantly larger than that between Ga(2)–O and Sn–O. Larger bond length variation requires more energy, so the doping formation energy of β-Si_0.0625Ga_1.9375O₃ is larger than that of β-Sn_0.0625Ga_1.9375O₃.

Table 2.

The average bond lengths of Ga–O, Si–O and Sn–O for TM-doped β-Ga₂O₃ (TM = Si,Sn).

3.3. Energy band structure and density of states

To illustrate the effect of the Hubbard U on the band structure and density of states, the calculated band structures and densities of states (DOSs) of β-Ga₂O₃ under GGA and GGA + U methods are shown in Fig. 2. The high symmetry points in the Brillouin zone are located at Γ(0,0,0), A(0,0.5,0), H(0,0.5,0.5) and K(0,0,0.5), and the Fermi level is defined as 0 eV. It can be seen in Fig. 2 that β-Ga₂O₃ has an indirect bandgap with the conduction band minimum (CBM) at Γ point and the valence band maximum (VBM) at M point, which is in agreement with the experimental results.^[16] But the direct Γ⁺−Γ⁻ transition is dominant in the optical spectra because the difference between the direct bandgap (Γ⁺–Γ⁻) and the indirect bandgap (Γ⁺–M) is very small (about 0.023 eV), which is proved by the analysis of the dipole transition matrix element and the experimental results of transmission spectra,^[31] so the direct Γ⁺–Γ⁻ transition of β-Ga₂O₃ merits more attention than the indirect Γ⁺–M transition in optics. As can be seen in Fig. 2(a), the bandgap of β-Ga₂O₃ under the GGA approximation is only 1.949 eV, which deviates significantly from the experimental result of 4.9 eV.^[1] Besides, the upper valence bandwidth is only 6.78 eV, which does not agree with the experimental value of about 7.37 eV.^[16] More importantly, the Ga 3d state peak is located at 11.63 eV below the VBM, which is not consistent with the experimental result of −17.12 eV.^[17] In order to correct the above deviation, the GGA + U method is used and the corrected bandgap of β-Ga₂O₃ is about 4.886 eV (see Fig. 2(b)). At the same time, the upper valence bandwidth is 7.29 eV and the Ga 3d state peak is at −17.01 eV, which are in good agreement with the previous experimental results, so the electronic structures calculated by the GGA + U method are reasonable. Moreover, as can be observed from the partial DOSs in Fig. 2(b), the CBM is dominated by the Ga 4s state and the VBM is occupied by the O 2p state, so the bandgap of β-Ga₂O₃ is determined by O 2p and Ga 4s states, which is consistent with the other calculations.^[32]

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	Fig. 2. Band structures and DOSs of β-Ga₂O₃ obtained by (a) GGA and (b) GGA + U methods.

To further investigate the effect of the doping atom on the electronic structure of β-Ga₂O₃, the band structures and DOSs of β-Si_0.0625Ga_1.9375O₃ and β-Sn_0.0625Ga_1.9375O₃ obtained by the GGA + U method are shown in Fig. 3. The bandgaps E_g of β-Si_0.0625Ga_1.9375O₃ and β-Sn_0.0625Ga_1.9375O₃ are 4.821 eV and 4.613 eV, which are smaller than that of β-Ga₂O₃. But the optical gaps (defined as the vertical distance from the Fermi level to the VBM) of β-Si_0.0625Ga_1.9375O₃ and β-Sn_0.0625Ga_1.9375O₃ are 5.853 eV and 5.586 eV, larger than that of β-Ga₂O₃, which is consistent with the experimental result.^[7] The electron transition of absorption spectra is related to the optical gap, and the optical gap is an important physical parameter for a TCO material, so β-Si_0.0625Ga_1.9375O₃ has the largest optical gap and is more suitable for application in photoelectric devices as a deep-ultraviolet TCO than are β-Ga₂O₃ and β-Sn_0.0625Ga_1.9375O₃.

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	Fig. 3. Band structures and DOSs of (a) β-Si_0.0625Ga_1.9375O₃ and (b) β-Sn_0.0625Ga_1.9375O₃.

The total and the partial densities of states of β-Si_0.0625Ga_1.9375O₃ and β-Sn_0.0625Ga_1.9375O₃ are also given in Fig. 3. For β-Si_0.0625Ga_1.9375O₃, the VBM is occupied by the O 2p state, whereas the CBM is occupied by the Si 3s state, so the bandgap of β-Si_0.0625Ga_1.9375O₃ is determined by O 2p and Si 3s states. It can also be seen in Fig. 3(a) that the DOS of β-Si_0.0625Ga_1.9375O₃ moves along the low energy direction after doping and the Fermi level enters into the conduction band, which indicates the n-type semiconductor feature. Therefore, the conductivity of Si-doped β-Ga₂O₃ is significantly improved compared to that of β-Ga₂O₃, which is consistent with the experiment result.^[33] For β-Sn_0.0625Ga_1.9375O₃ in Fig. 4(b), the VBM is occupied by the O 2p state and the CBM is occupied by the Sn 5s state, so the bandgap of β-Sn_0.0625Ga_1.9375O₃ is determined by O 2p and Sn 5s states. Further, the free electrons from Sn 5s state fill the CBM and improve the conductivity of β-Sn_0.0625Ga_1.9375O₃, which is similar to the scenario for β-Si_0.0625Ga_1.9375O₃.

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	Fig. 4. The imaginary part ε₂(ω) of β-Ga₂O₃, β-Si_0.0625Ga_1.9375O₃, and β-Sn_0.0625Ga_1.9375O₃ in the (a) x-axis, (b) y-axis, and (c) z-axis directions.

3.4. Electron effective mass and conductivity

For a doped semiconductor with high doping concentration (i.e., >10¹⁹ cm⁻³), there are primarily two competing mechanisms affecting its optical gap: the Burstein–Moss effect^[34] and the bandgap renormalization effect.^[35] The Burstein–Moss effect causes bandgap widening due to the Fermi-band filling effect, marked as ΔE_BM. The bandgap renormalization effect causes bandgap narrowing due to the many-body bandgap renormalization effect and the semiconductor–metal transition effect, labeled as ΔE_BGN. The ΔE_BM and ΔE_BGN can be obtained by^[36]

where E_go denotes the bandgap of the undoped β-Ga₂O₃,

is the optical gap, and E_g is the bandgap. According to the band structures in Figs. 2–4, E_go is 4.886 eV, and E_g is 4.821 eV for β-Si_0.0625Ga_1.9375O₃ and 4.613 eV for β-Sn_0.0625Ga_1.9375O₃. By Eqs. (6) and (7), the ΔE_BM of β-Si_0.0625Ga_1.9375O₃ and β-Sn_0.0625Ga_1.9375O₃ are 1.032 eV and 0.973 eV, and the ΔE_BGN of β-Si_0.0625Ga_1.9375O₃ and β-Sn_0.0625Ga_1.9375O₃ are 0.065 eV and 0.273 eV, respectively. Further, the electron effective mass

is related to ΔE_BM by the following formula:^[12]

where h is the Planck constant, and n_e is the electron concentration. Using Eq. (8), the electron effective mass of β-Si_0.0625Ga_1.9375O₃ is 0.383m_e (m_e is the electron mass) and 0.402m_e for β-Sn_0.0625Ga_1.9375O₃. The electron conductivity of the n-type material can be expressed as^[37]

where q is the electron charge, μ is the electron mobility and τ is the relaxation time which is related to the ionized concentration (N_i) and thermodynamic temperature. According to Eq. (9), the electron mobility ratio of β-Si_0.0625Ga_1.9375O₃ to β-Sn_0.0625Ga_1.9375O₃ is 1.035, revealing that the electron mobility of β-Si_0.0625Ga_1.9375O₃ is higher than that of β-Sn_0.0625Ga_1.9375O₃, which is consistent with recent experimental results.^[9] Further, the electron conductivity ratio of β-Si_0.0625Ga_1.9375O₃ to β-Sn_0.0625Ga_1.9375O₃ is 1.065 from Eq. (9), so Si-doped β-Ga₂O₃ has an advantage over Sn-doped β-Ga₂O₃ in terms of conductivity.

3.5. Optical properties

Optical properties are important for TCO materials and are expressed by the complex dielectric function ε(ω) = ε₁(ω) + iε₂(ω). The imaginary part ε₂(ω) of the complex dielectric function is determined by the momentum matrix elements between the occupied and unoccupied wave functions. The real part ε₁(ω) can be evaluated from imaginary part ε₂(ω) using the Kramers–Kronig relationship, and the absorption coefficient α(ω) can be calculated according to the complex dielectric function. The above relationships are described by the following formulas:^[38]

Here, ρ_o represents the principal value of the integral, subscripts C and V represent the conduction band and valence band states, respectively, BZ means the first Brillouin zone, k is the reciprocal lattice vector, |M_CV(k)|² denotes the momentum matrix element, c is a constant, ω stands for the angular frequency, and

and

represent the intrinsic energy levels. The imaginary part ε₂(ω) of the complex dielectric function is the basis for analyzing the optical properties of any materials.

The imaginary parts ε₂(ω) of the complex dielectric function of β-Ga₂O₃, β-Si_0.0625Ga_1.9375O₃ and β-Sn_0.0625Ga_1.9375O₃ in the x, y, and z directions are calculated and shown in Fig. 4. For the ε₂(ω) of β-Ga₂O₃, three main peaks at 9.286 eV, 10.587 eV, and 9.506 eV appear in the x, y, and z directions; the corresponding peak intensities are 1.42, 1.316, and 2.134, respectively, which indicates that β-Ga₂O₃ has significant anisotropy. For Si- and Sn-doped β-Ga₂O₃, there are no significant changes in the position of the three main peaks in the x, y, and z directions, but in the low-energy range, two small peaks (about 0.9 eV for β-Sn_0.0625Ga_1.9375O₃ and about 1.05 eV for β-Si_0.0625Ga_1.9375O₃) appear. It is well known that the imaginary part ε₂(ω) is related to the electron transition between the energy bands, so the peak of ε₂(ω) can be explained by the electronic structures. According to the partial DOS in Fig. 3, the main peak of about 10 eV mainly originates from the interband electron transition between the Ga 4s state of the conduction band and the O 2p state of the valence band, but the main peak edges (near to 5 eV) of β-Ga₂O₃, β-Si_0.0625Ga_1.9375O₃ and β-Sn_0.0625Ga_1.9375O₃ are different because their interband electron transitions are different (the electron transitions of Ga 4s and O 2p for β-Ga₂O₃, Si 3s and O 2p for β-Si_0.0625Ga_1.9375O₃, and Sn 5s and O 2p for β-Sn_0.0625Ga_1.9375O₃). The small peak at about 0.9 eV for β-Sn_0.0625Ga_1.9375O₃ is mainly derived from the electron transition between the Sn 5s state and the O 2p state in the conduction band, and the small peak at about 1.05 eV for β-Si_0.0625Ga_1.9375O₃ is mainly derived from the electron transition between the Si 3s state and the O 2p state in the conduction band. The strength difference of the two small peaks is mainly attributed to the different bonding structures: a Si atom bonds with the four closest oxygen atoms but a Sn atom bonds with the six closest oxygen atoms.

The absorption coefficients α(ω) of β-Ga₂O₃, β-Si_0.0625Ga_1.9375O₃ and β-Sn_0.0625Ga_1.9375O₃ in the x, y and z directions are plotted in Fig. 5, and the insets are the partial enlargement. As shown in Fig. 5, the absorption spectra in the x-, y- and z-axis directions are mainly distributed between 5 eV and 17 eV, which are consistent with the imaginary parts ε₂(ω) of the complex dielectric function. The main peaks of the absorption spectra along the x-, y- and z-axes are located at 11.6 eV, 12.04 eV and 10.03 eV, respectively, and the peak intensity gradually increases, which is slightly different from the main peaks of the imaginary parts ε₂(ω) due to the effect of the real part ε₁(ω) of the complex dielectric function. The main peak shapes and intensities are different in the x, y and z directions due to the anisotropy structure, but the peak shapes of β-Ga₂O₃, β-Si_0.0625Ga_1.9375O₃ and β-Sn_0.0625Ga_1.9375O₃ in the same direction are hardly changed except for the edge of the main peak, which means that the main peak is derived from the same electron transition of the Ga 4s and O 2p states and the edge of the peak is related to Si or Sn doping.

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	Fig. 5. The absorption coefficients of β-Ga₂O₃, β-Si_0.0625Ga_1.9375O₃ and β-Sn_0.0625Ga_1.9375O₃ in the (a) x-axis, (b) y-axis and (c) z-axis directions. The inset is a partial enlargement.

To analyze clearly the difference in the peak edge, the spectra between 4.2 eV and 7 eV are also given in the insets of Fig. 5. It can be observed that the optical absorption edges of β-Ga₂O₃ in the x, y, and z directions are 4.753 eV, 5.152 eV and 4.646 eV, respectively, and this explains the cause of the different experimental bandgaps (4.6–5.1 eV) of β-Ga₂O₃ in the literature.^[39,40] Moreover, the average optical absorption edge of β-Ga₂O₃ is 4.85 eV, which is very close to the bandgap (4.886 eV) of β-Ga₂O₃ from the band structure in Fig. 2. For β-Si_0.0625Ga_1.9375O₃, the optical absorption edges in the x, y and z directions are 5.897 eV, 6.017 eV and 5.64 eV, respectively, and the average optical absorption edge is 5.851 eV, close to the optical gap of 5.853 eV in Fig. 3. For β-Sn_0.0625Ga_1.9375O₃, the optical absorption edges in the x, y and z directions are 5.592 eV, 5.813 eV and 5.282 eV, respectively, and the average optical absorption edge is 5.562 eV, which is consistent with the optical gap (5.586 eV) of β-Sn_0.0625Ga_1.9375O₃, so the optical absorption edge can be described as a joint band structure between the Fermi level and the VBM. Further, it is noted that the optical absorption edge in the y direction is the largest among the three directions, and the average optical absorption edge of β-Si_0.0625Ga_1.9375O₃ is the largest among them. Figure 5 shows that two small absorption peaks of about 1.2 eV in the x and z directions occur in β-Sn_0.0625Ga_1.9375O₃, but no obvious absorption peaks are found in β-Ga₂O₃ and β-Si_0.0625Ga_1.9375O₃. Therefore, β-Si_0.0625Ga_1.9375O₃ has better light transmittance in the deep-ultraviolet wavelength range than do β-Ga₂O₃ and β-Sn_0.0625Ga_1.9375O₃.

4. Conclusion

In summary, the energy band structures, densities of states and optical properties of β-Ga₂O₃ and β-TM_0.0625Ga_1.9375O₃ (TM = Si, Sn) are calculated and compared using the GGA + U method based on DFT. The calculated bandgap, Ga 3d-state peak and upper valence bandwidth of β-Ga₂O₃ are close to the experimental results if the Hubbard U parameters are set to 14 eV and 8.35 eV for the Ga d state and the O p state. Si- and Sn-doped β-Ga₂O₃ tend to form under O-poor conditions, and Si-doped β-Ga₂O₃ requires higher growth temperature compared with Sn-doped β-Ga₂O₃ because of the large bond length variation between Ga–O and Si–O, but Si-doped β-Ga₂O₃ has better electron mobility and conductivity than Sn-doped β-Ga₂O₃. Besides, the optical properties of Si- and Sn-doped β-Ga₂O₃ show significant anisotropy, and the average optical absorption edge of Si-doped β-Ga₂O₃ is larger than that of β-Ga₂O₃ or β-Sn_0.0625Ga_1.9375O₃, so Si-doped β-Ga₂O₃ has advantages over Sn-doped β-Ga₂O₃ for applications in deep-ultraviolet photoelectric devices.

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