Role of Native Defects in Fe-Doped β-Ga2O3

Iron impurities are believed to act as deep acceptors that can compensate for the n-type conductivity in as-grown Ga2O3, but several scientific issues, such as the site occupation of the Fe heteroatom and the complexes of Fe-doped β-Ga2O3 with native defects, are still lacking. In this paper, based on first-principle density functional theory calculations with the generalized gradient approximation approach, the controversy regarding the preferential Fe incorporation on the Ga site in the β-Ga2O3 crystal has been addressed, and our result demonstrates that Fe dopant is energetically favored on the octahedrally coordinated Ga site. The structural stabilities are confirmed by the formation energy calculations, the phonon dispersion relationships, and the strain-dependent analyses. The thermodynamic transition level Fe3+/Fe2+ is located at 0.52 eV below the conduction band minimum, which is consistent with Ingebrigtsen’s theoretical conclusion, but slightly smaller than some experimental values between 0.78 eV and 1.2 eV. In order to provide direct guidance for material synthesis and property design in Fe-doped β-Ga2O3, the defect formation energies, charge transitional levels, and optical properties of the defective complexes with different kinds of native defects are investigated. Our results show that VGa and Oi can be easily formed for the Fe-doped β-Ga2O3 crystals under O-rich conditions, where the +3 charge state FeGaGai and −2 charge state FeGaOi are energetically favorable when the Fermi level approaches the valence and conduction band edges, respectively. Optical absorption shows that the complexes of FeGaGai and FeGaVGa can significantly enhance the optical absorption in the visible-infrared region, while the energy-loss function in the β-Ga2O3 material is almost negligible after the extra introduction of various intrinsic defects.


Introduction
Gallium oxides (Ga 2 O 3 ) have received a lot of attention due to their exceptional physical and chemical features with a variety of applications such as solar-blind ultraviolet photodetectors [1,2], high-power transistors [3,4], Schottky diodes [5,6], as well as photocatalysts [7].Due to the inevitable insertion of native defects (such as Ga i [8,9]) and extrinsic impurities (such as Si [10], H [11]) during the growth of materials, perfect Ga 2 O 3 exhibits n-type conductivity, which severely impedes its further applications.Doping engineering, in general, can be a valuable approach to manipulating conductivity, which influences electrical and optical performance by modifying the microscopic crystalline structure [12][13][14][15][16].As a result, studies of acceptors in β-Ga 2 O 3 materials are required.The n-type conductivity in perfect β-Ga 2 O 3 can be compensated by the introduction of deep acceptors, such as Fe dopant.Fe impurity is one of the most attractive dopants because it ever, the correlations among the Fe-doped β-Ga 2 O 3 with native defects, the local crystal structure, and the electronic and optical properties have not been extensively studied.
Herein, we performed density functional theory (DFT) calculations to investigate the defect formation energies, charge transitional levels, electronic structures, and optical properties of Fe-doped β-Ga 2 O 3 , as well as Fe-doped β-Ga 2 O 3 with different kinds of native defects.Our results address the controversies regarding the preferential Fe incorporation on the tetrahedrally or octahedrally coordinated Ga site, as mentioned above.Moreover, since the absence of relevant reports in Fe-doped β-Ga 2 O 3 lattices with native defects from theoretical studies, the defect formation energies, charge transitional levels, and optical properties of the defective complexes of Fe-doped β-Ga 2 O 3 with different kinds of native defects, i.e., oxygen vacancy (V O ), gallium vacancy (V Ga ), oxygen interstitial (O i ), and gallium interstitial (Ga i ), are investigated.Our studies are beneficial for understanding the ground state properties of Fe-doped β-Ga 2 O 3 , as well as for providing theoretical guidance on the design of β-Ga 2 O 3 -based functional materials and the promising applications of β-Ga 2 O 3 for innovative spin-electronic and optoelectronic devices.

Computational Details
To implement the first-principles calculations, we use the Vienna ab initio Simulation Package (VASP) [27,28] based on DFT [29] with projected augmented wave (PAW) potentials.To characterize the exchange-correlation interactions, the generalized gradient approximation (GGA) parameterized by Perdew-Burke-Ernzerhof (PBE) [30] is used.The kinetic energy cutoff for the plane-wave basis set is 450 eV, the energy convergence criterion for the calculations is set to 1 × 10 −5 eV/atom for the interactions between the electrons and ions, and all the atomic positions are fully optimized.When all components of the residual forces are less than 0.01 eV/Å, the relaxation will be terminated.A 4 × 4 × 2 Monkhost-Pack grid is utilized for structural relaxation, whereas a 9 × 9 × 4 Monkhost-Pack grid is used for the calculations of density of states (DOS) and optical properties.The so-called density function perturbation (DFPT) calculated method for phonon calculations is adopted in this work.Usually, phonon dispersion is needed to expand the supercell.However, we do not enlarge the supercell in this work considering the time-consuming nature, which may not influence our conclusions qualitatively.A 2 × 4 × 2 Monkhost-Pack grid and a 1 × 10 −6 eV/atom energy criterion have been used for the calculation of phonon dispersion and mechanical properties.The valence electronic configurations for Ga, O, and Fe are [Ar] 3d 10 4s 2 4p 1 , [He] 2s 2 2p 4 , and [Ar] 3d 7 4s 1 , respectively.
A 1 × 2 × 2 β-Ga 2 O 3 supercell of 32 Ga atoms and 48 O atoms is modeled in this study, with one Fe impurity replacing the Ga atom, corresponding to a doping concentration of 3.125%, as shown in Figure 1a.β-Ga 2 O 3 possesses two inequivalent Ga positions.Fe impurity incorporation on the tetrahedrally and octahedrally coordinated Ga sites is labeled 1 and 2, respectively.Different kinds of native defects in terms of oxygen vacancy, gallium vacancy, oxygen interstitial, and gallium interstitial in the β-Ga 2 O 3 supercell are considered, which are denoted as V O , V Ga , O i , and Ga i , respectively.For the atomic positions of V O and V Ga , we use the results by Dong et al. [31], i.e., the positions of 3 and 4 in Figure 1a, respectively.For the low-energy O i and Ga i doping sites in gallium oxide, we adopt the results given by Zacherle et al. [32], where the two interstitial sites are located at the same position (0.683, 0.500, 0.459) in the supercell before relaxation and labeled as 5 in Figure 1a.For simplicity, Fe impurities replacing tetrahedral and octahedral Ga atoms are named Fe GaI and Fe GaΠ , respectively.Thus, their complexes of Fe GaI with V O , V Ga , O i , and Ga i configurations are named Fe GaI V O , Fe GaI V Ga , Fe GaI O i , and Fe GaI Ga i , respectively, while complexes of Fe GaΠ with V O , V Ga , O i , and Ga i configurations are labeled as Fe GaΠ V O , Fe GaΠ V Ga , Fe GaΠ O i , and Fe GaΠ Ga i , respectively.Besides, we also employ the value of U in accordance with the experimental band gap for the perfect β-Ga 2 O 3 [33], and the U value of 4.09 eV is adopted for the 3d orbital of the Fe dopant as suggested by the literature [34].

Formation Energies, Transitional Levels and Optical Calculations
The formation energy of the defect D in the charge state q is calculated as [36,37] where q , D E and p E denote the total energy of the defect and perfect supercell, respectively.i n represents the number of i atoms added ( 0 n i < ) or removed ( 0 n i > ) from the perfect supercell, and i μ is the corresponding chemical potential.VBM E is energy of the valence band maximum (VBM) for bulk Ga2O3.f E is Fermi level, which is referenced to the VBM in the bulk.corr E is the term that accounts for the finite-size corrections, which is determined by the potential alignment and is given as [36] ) where the potential difference between the charged defect Ga2O3 supercell ( r q , D V ) and perfect Ga2O3 supercell ( r p V ) are calculated from the atomic-sphere-averaged potentials at the atomic sites farther away from the defect employed by the software of VASPKIT Standard Edition 1.3.5 [38].
Note that the chemical potential satisfies the boundary conditions as follows: Chemical potential varies according to different growth conditions.Under O-rich growth condition: Under Ga-rich growth condition:

Formation Energies, Transitional Levels and Optical Calculations
The formation energy of the defect D in the charge state q is calculated as [36,37] where E D,q and E p denote the total energy of the defect and perfect supercell, respectively.n i represents the number of i atoms added (n i < 0) or removed (n i > 0) from the perfect supercell, and µ i is the corresponding chemical potential.E VBM is energy of the valence band maximum (VBM) for bulk Ga 2 O 3 .E f is Fermi level, which is referenced to the VBM in the bulk.E corr is the term that accounts for the finite-size corrections, which is determined by the potential alignment and is given as [36] E corr = q(V r D,q − V r p ) where the potential difference between the charged defect Ga 2 O 3 supercell (V r D,q ) and perfect Ga 2 O 3 supercell (V r p ) are calculated from the atomic-sphere-averaged potentials at the atomic sites farther away from the defect employed by the software of VASPKIT Standard Edition 1.3.5 [38].
Note that the chemical potential satisfies the boundary conditions as follows: Chemical potential varies according to different growth conditions.Under O-rich growth condition: Under Ga-rich growth condition: where, µ The transition energy ε(q 1 /q 2 ) between charge state q 1 and q 2 for defect D doping configuration is calculated as [39] ε(q 1 /q 2 ) = Here, the E represents the formation energy of the defect D in charge state q evaluated at E f = 0.The ε(q 1 /q 2 ) denotes the Fermi-level position where the charge states q 1 and q 2 have equal formation energy.The absorption coefficients in optical properties can be described as [31,40] where ε 1 (ω) and ε 2 (ω) indicate the real and imaginary parts of the dielectric function, respectively.The ε 2 (ω) can be calculated by summing up the transitions between occupied and unoccupied states using the following equation Here, m, e, M, and ω denote the mass of free electrons, the electron charge, the dipole matrix, and the frequency of incident photons, respectively.i, j, f i , and k represent the initial state, the final state, the Fermi distribution function, and the wave function vector, respectively.The ε 2 (ω) is related to the absorption of light and dielectric loss of energy, while ε 1 (ω) is associated with the stored energy.
The energy loss function (ELF) can be described by the following equation [41]

Structural Stability
The calculated lattice parameters of perfect β-Ga 2 O 3 are a = 12.412 Å, b = 3.076 Å, c = 5.872 Å, and the unique angle β = 103.702• , which are in excellent accordance with the theoretically calculated values obtained by PBE [41] and B3PW [42] approaches, as well as with the experimental values [43], as shown in Table 1.The optimized structural parameters for the Fe-doped cases are also summarized in Table 1.The lattice constants of Fe-doped β-Ga 2 O 3 exhibit a slight decrease, which can be ascribed to the comparable ionic radii and local structures between Fe and Ga atoms.The relative difference of the radii between Fe 3+ (Fe 2+ ) and Ga 3+ ions is −1.61% (−11.3%).Fe GaΠ is endowed with smaller lattice parameter variations in terms of all three lattice vectors and the unique angle β compared with Fe GaI , implying that small distortions may be easily formed in the experimental growth.To study the structural stability of Fe-doped β-Ga 2 O 3 supercells, the defect formation energies under different conditions are calculated, as shown in Figure 2.Meanwhile, the transition levels are also employed to assess the ionization energies and the effectiveness of the doped systems.Our calculated value of the band gap for perfect β-Ga 2 O 3 is 2.04 eV, as denoted by the dashed line in Figure 1b, which is consistent with the values obtained by the DFT calculated method but smaller than the experimental values [44].The underestimated band gap for DFT calculation is a common phenomenon; however, it does not affect our conclusions qualitatively [45,46].In addition, Figure 1c illustrates the calculated total density of states (TDOS) and partial density of states (PDOS) for perfect β-Ga 2 O 3 ; the VBM is predominantly composed of O 2p orbital-derived states with minor hybridization with Ga 3d and 4p orbitals, while the CBM is mainly formed by Ga 4s orbitals.
parameter variations in terms of all three lattice vectors and the unique angle β compared with FeGaI, implying that small distortions may be easily formed in the experimental growth.To study the structural stability of Fe-doped β-Ga2O3 supercells, the defect formation energies under different conditions are calculated, as shown in Figure 2.Meanwhile, the transition levels are also employed to assess the ionization energies and the effectiveness of the doped systems.Our calculated value of the band gap for perfect β-Ga2O3 is 2.04 eV, as denoted by the dashed line in Figure 1b, which is consistent with the values obtained by the DFT calculated method but smaller than the experimental values [44].The underestimated band gap for DFT calculation is a common phenomenon; however, it does not affect our conclusions qualitatively [45,46].In addition, Figure 1c illustrates the calculated total density of states (TDOS) and partial density of states (PDOS) for perfect β-Ga2O3; the VBM is predominantly composed of O 2p orbital-derived states with minor hybridization with Ga 3d and 4p orbitals, while the CBM is mainly formed by Ga 4s orbitals.[25] and other experimental results [19,20].The thermodynamic transition level Fe 3+ /Fe 2+ for FeGaП, i.e., ε(0/−), is located at 0.52 eV below CBM, which is comparable to the theoretical value calculated by HSE hybrid functions (0.61 eV or 0.40 eV [25]), but slightly smaller than the reported experimental values (0.78 eV [25], 0.80 eV [26], 0.84 eV [17], 1.2 eV [22]).The transition level Fe 3+ /Fe 2+ for FeGaI measured from Figure 2a indicates the formation energies for Fe-doped β-Ga 2 O 3 under O-rich conditions.Fe GaΠ case has lower formation energy, suggesting the preferential occupation of Fe Ga at the octahedrally coordinated Ga site, which is in agreement with Ingebrigtsen's conclusion by theoretical calculation [25] and other experimental results [19,20].The thermodynamic transition level Fe 3+ /Fe 2+ for Fe GaΠ , i.e., ε(0/−), is located at 0.52 eV below CBM, which is comparable to the theoretical value calculated by HSE hybrid functions (0.61 eV or 0.40 eV [25]), but slightly smaller than the reported experimental values (0.78 eV [25], 0.80 eV [26], 0.84 eV [17], 1.2 eV [22]).The transition level Fe 3+ /Fe 2+ for Fe GaI measured from VBM is 0.24 eV below CBM, thus deep acceptors are expected for both Fe-doped β-Ga 2 O 3 configurations in the n-type β-Ga 2 O 3 conditions, which can compensate for the free electrons caused by native defects or extrinsic impurities.In addition, the tran-sition level ε(+/0) for both Fe GaI and Fe GaΠ is 0.86 eV above the VBM, which demonstrates that both doping cases generate deep donors even in the p-type β-Ga 2 O 3 crystals.For the Ga-rich condition, i.e., the O-poor condition, as shown in Figure 2b, the tendency is the same as for the O-rich atmosphere, with the exception of higher formation energies.This suggests that Fe impurity is more easily substituted for Ga sites under O-rich conditions.
The mechanical characteristics of β-Ga 2 O 3 are evaluated by employing a complete set of elastic constants.There are thirteen independent elastic constants (C 11 , C 22 , C 33 , C 44 , C 55 , C 66 , C 12 , C 13 , C 23 , C 15 , C 25 , C 35 , and C 46 ) in the monoclinic symmetry crystal.The mechanical stability criteria of β-Ga 2 O 3 are described as follows [47]: In this study, the mechanical property calculations are carried out for the perfect and energetically favorable Fe GaΠ doping configuration.The calculated elastic stiffness constants of prefect and Fe-doped β-Ga 2 O 3 are shown in Table 2.For comparisons, available theoretical and experimental results are also listed.The elastic stiffness of prefect and Fedoped β-Ga 2 O 3 meets the mechanical stability criteria presented above, suggesting that prefect and Fe-doped β-Ga 2 O 3 are mechanically stable at ambient conditions.
The formation energies as a function of the biaxial strain with q = 0 under O-rich conditions are shown in Figure 3a to assess the mechanical stability of Fe GaΠ as well.It can be seen that the unstrained Fe GaΠ is endowed with the smallest formation energy of −2.07 eV, indicating the Fe GaΠ structure is in a stable state.Moreover, the formation energies are strongly dependent on the biaxial strain.As the tensile or compressive stress increases, it increases dramatically.When the compressive strain is greater than 2% or the tensile strain is higher than 3%, the defect formation energy is greater than 0, denoting that it may be difficult to materialize in the experiment.The defect formation energy increases more rapidly under compressive strain, which indicates that the defect is more difficult to realize under compressive strain.The formation energies as a function of the biaxial strain with q = 0 under O-rich conditions are shown in Figure 3a to assess the mechanical stability of FeGaП as well.It can be seen that the unstrained FeGaП is endowed with the smallest formation energy of −2.07 eV, indicating the FeGaП structure is in a stable state.Moreover, the formation energies are strongly dependent on the biaxial strain.As the tensile or compressive stress increases, it increases dramatically.When the compressive strain is greater than 2% or the tensile strain is higher than 3%, the defect formation energy is greater than 0, denoting that it may be difficult to materialize in the experiment.The defect formation energy increases more rapidly under compressive strain, which indicates that the defect is more difficult to realize under compressive strain.Phonon analysis has proven to be an effective approach to predicting structural stability [51].The phonon dispersion calculation for FeGaП doping structure is shown in Figure 3b.We observe three small imaginary frequencies, i.e., 0.64, 0.78, and 1.10 cm −1 , locating at the non-gamma point (G).In general, imaginary frequencies at the gamma point can be related to structural instability, whereas the presence of imaginary frequencies at the non-gamma point can be responsible for the finite size of the simulation crystal cell, which can be eliminated by expanding the calculated supercell.Therefore, the Fe-doped Phonon analysis has proven to be an effective approach to predicting structural stability [51].The phonon dispersion calculation for Fe GaΠ doping structure is shown in Figure 3b.We observe three small imaginary frequencies, i.e., 0.64, 0.78, and 1.10 cm −1 , locating at the non-gamma point (G).In general, imaginary frequencies at the gamma point can be related to structural instability, whereas the presence of imaginary frequencies at the non-gamma point can be responsible for the finite size of the simulation crystal cell, which can be eliminated by expanding the calculated supercell.Therefore, the Fe-doped β-Ga 2 O 3 with small imaginary frequencies at the non-gamma point is predicted to be structurally stable, which agrees well with the results as suggested by the low formation energies.
Figure 4a shows the complexes of Fe GaΠ and Fe GaI with native defects under different growth condition limits.For these defective Fe GaΠ complexes under O-rich conditions, positively charged Fe GaΠ Ga i and negatively charged Fe GaΠ O i are energetically favorable when the E f approaches the VBM and CBM, respectively, while positively charged Fe GaΠ Ga i is expected throughout the whole band gap under Ga-rich condition.Moreover, the formation energies of Fe GaΠ Ga i (Ga-rich condition) and Fe GaΠ O i (O-rich condition) are lower compared with those of Fe GaΠ case, suggesting that Fe GaΠ Ga i and Fe GaΠ O i complexes are easily formed under Ga-rich and O-rich conditions during experimental growth, respectively.For the Fe GaΠ V O case, under O-rich conditions, the transition levels ε(+2/+1) and ε(+1/0) are 1.29 and 0.62 eV below CBM, respectively, indicating that the complex acts as a deep donor and cannot contribute to n-type conductivity.Different from the intrinsic Vo defect investigated in literature [10], we observe the +1 charge state rather than +2 and 0 charge states, which may be attributed to the combination of −1 charge state Fe dopant and +2 charge state V O .The similar results are inspected under Ga-rich conditions except for the lower formation energies, suggesting the Fe GaΠ V O complex is easily formed in the O-poor growth atmosphere.For the Fe GaΠ V Ga case, the transition levels ε(0/−2), ε(−2/−3) and ε(−3/−4) are located at 1.34, 0.57, and 2.04 eV below CBM, which demonstrate that the complexes act as deep acceptors with −4 or −3 charge state, respectively.Compared with the Ga-rich condition, the Fe GaΠ V Ga complex possesses lower formation energies in n-type Ga 2 O 3 materials, which demonstrates that the V Ga is more likely produced in Fe-doped Ga 2 O 3 under an O-rich growth atmosphere in experiments.It is in excellent agreement with the reported experimental result by Hany et al. [24].Either under O-rich or Ga-rich conditions, positively charged and negatively charged Fe GaΠ O i is energetically favorable when the Fermi level approaches the VBM and CBM, respectively.Moreover, Fe GaΠ O i complex is more susceptible to being produced under O-rich conditions.For the Fe GaΠ Ga i case, both under O-rich and Ga-rich conditions, positively charged are energetically favorable when the E f is located throughout the whole band gap, which demonstrates that the complex exhibits n-type conductivity.The transition levels ε(+4/+3) and ε(+3/+2) are 1.47 and 3.02 eV above the VBM under O-rich condition and Ga-rich condition, which indicates that the +3 charge state for the Fe GaΠ Ga i complex is expected.In addition, the higher formation energies under the O-rich condition for the Fe GaΠ Ga i illustrate that the complex is more likely to be found under the Ga-rich conditions.It is worth mentioning that the Ga i is the main origin of the native defect to form the n-type conductive β-Ga 2 O 3 crystal, as illustrated in Refs.[8,9], while a low formation energy is gained for the Fe GaΠ Ga i complex.The E f always tends to be positioned at the higher region of the bandgap in β-Ga 2 O 3 and gives rise to the n-type conduction characteristic due to unintentionally introduced native defects during the growth of β-Ga 2 O 3 .Therefore, our calculated results illustrate that the +3 charge state Fe GaΠ Ga i under O-poor condition and −2 charge state Fe GaΠ O i under O-rich condition are easily formed for the growth of β-Ga 2 O 3 crystals.
Figure 4b shows the defect formation energies for the complexes of Fe GaI with native defects.Different from the case of Fe GaΠ complexes, under O-rich conditions, +4 charge state Fe GaI Ga i and −2 charge state Fe GaI O i are dominated when the E f is located near the VBM and CBM, respectively, while −4 charge state Fe GaI Ga i is easily generated throughout the whole band gap under Ga-rich conditions.Thus, different local structures can influence the electron transfer.As shown in Figure 4b, these defective complexes are characterized by similar tendencies with those of Fe GaΠ complexes in exception for different formation energies both under O-rich and Ga-rich conditions.

Optical Property
For wide-band gap semiconductor materials, optical parameters of dielectric function ) (ω ε can be employed to clarify the linear response of the system to electromagnetic radiation, which is crucial to assessing the interactions between photons and electrons.The imaginary part ) ( 2 ω ε of the dielectric constant is related to the absorption of light and the dielectric energy-loss function, while ) ( 1 ω ε is associated with the stored energy.Figure 5 denotes the optical absorption coefficient of perfect, Fe-doped, and various FeGaП/FeGaI complexes in the energy range between 0 and 30 eV. Figure 5b,d exhibits the enlarged plots at the (0-5) eV region.The strong absorption peaks are located at 11.8 and 10.9 eV for perfect β-Ga2O3, as shown in Figure 5a, which originate from the inter-band transitions from O 2p states to Ga 4s states, illustrating that the bulk material is characterized by its deep ultraviolet properties.The calculated data is consistent with Yan and Pan's results [52,53].Compared with perfect β-Ga2O3, the profiles of Fe-doped β-Ga2O3 and various complexes in Figure 5a,c are endowed with similar absorption peaks in the highenergy ultraviolet region, indicating that these dopants can hardly decrease the optical absorption coefficients of β-Ga2O3 in the deep ultraviolet region.The slightly red shift for FeGaПVGa, FeGaIVGa, FeGaПOi, and FeGaIOi can be ascribed to hole doping, while the blue shift for the FeGaПGai, FeGaIGai cases can be associated with the introduction of electrons, which is consistent with our formation energy calculations above.In addition, one can obviously notice that new peaks appear in the low-energy region for these FeGaП/FeGaI complexes, as shown in the amplified plots shown in Figure 5b,d.
The perfect β-Ga2O3 possesses an optical band gap of about 2 eV, which is in good agreement with the value observed from electronic structure calculations in Figure 1b.For the FeGaП and FeGaI cases, the optical absorption spectra remain almost unchanged in the visible region, which can be associated with the deep acceptor doping for the Fe foreigner atom.When introducing extra VO into the β-Ga2O3 crystal, the absorption coefficients become relatively low for both FeGaПVO and FeGaIVO cases in the low-energy region (0-5 eV), whereas a new wide peak for FeGaПVGa configuration is generated, leading to the optical migration from the ultraviolet light region to the visible-infrared region.The new peak for FeGaПVGa configuration originated from the inter-band transitions of O 2p from the VBM to the induced impurity levels.Similarly, a new peak appears at a high energy level of ~1.42 eV and ~0.84 eV for FeGaПGai and FeGaIGai complexes, respectively, which are origi-

Optical Property
For wide-band gap semiconductor materials, optical parameters of dielectric function ε(ω) can be employed to clarify the linear response of the system to electromagnetic radiation, which is crucial to assessing the interactions between photons and electrons.The imaginary part ε 2 (ω) of the dielectric constant is related to the absorption of light and the dielectric energy-loss function, while ε 1 (ω) is associated with the stored energy.Figure 5 denotes the optical absorption coefficient of perfect, Fe-doped, and various Fe GaΠ /Fe GaI complexes in the energy range between 0 and 30 eV. Figure 5b,d exhibits the enlarged plots at the (0-5) eV region.The strong absorption peaks are located at 11.8 and 10.9 eV for perfect β-Ga 2 O 3 , as shown in Figure 5a, which originate from the inter-band transitions from O 2p states to Ga 4s states, illustrating that the bulk material is characterized by its deep ultraviolet properties.The calculated data is consistent with Yan and Pan's results [52,53].Compared with perfect β-Ga 2 O 3 , the profiles of Fe-doped β-Ga 2 O 3 and various complexes in Figure 5a,c are endowed with similar absorption peaks in the high-energy ultraviolet region, indicating that these dopants can hardly decrease the optical absorption coefficients of β-Ga 2 O 3 in the deep ultraviolet region.The slightly red shift for Fe GaΠ V Ga , Fe GaI V Ga , Fe GaΠ O i , and Fe GaI O i can be ascribed to hole doping, while the blue shift for the Fe GaΠ Ga i , Fe GaI Ga i cases can be associated with the introduction of electrons, which is consistent with our formation energy calculations above.In addition, one can obviously notice that new peaks appear in the low-energy region for these Fe GaΠ /Fe GaI complexes, as shown in the amplified plots shown in Figure 5b,d.
The perfect β-Ga 2 O 3 possesses an optical band gap of about 2 eV, which is in good agreement with the value observed from electronic structure calculations in Figure 1b.For the Fe GaΠ and Fe GaI cases, the optical absorption spectra remain almost unchanged in the visible region, which can be associated with the deep acceptor doping for the Fe foreigner atom.When introducing extra V O into the β-Ga 2 O 3 crystal, the absorption coefficients become relatively low for both Fe GaΠ V O and Fe GaI V O cases in the low-energy region (0-5 eV), whereas a new wide peak for Fe GaΠ V Ga configuration is generated, leading to the optical migration from the ultraviolet light region to the visible-infrared region.The new peak for Fe GaΠ V Ga configuration originated from the inter-band transitions of O 2p from the VBM to the induced impurity levels.Similarly, a new peak appears at a high energy level of ~1.42 eV and ~0.84 eV for Fe GaΠ Ga i and Fe GaI Ga i complexes, respectively, which are originated by the transitions from impurity levels to Ga 4s orbitals.These new peaks are expected to benefit the optical transformation from ultraviolet light to the visible-infrared region.In the low-energy region in Figure 5b,d, the Fe GaΠ O i case exhibits the absence of a clear optical absorption peak, while a few small oscillation peaks are present for the Fe GaI O i combination.Therefore, Fe GaΠ Ga i , Fe GaI Ga i , and Fe GaΠ V Ga complexes can significantly enhance the optical absorption in the visible-infrared region.nated by the transitions from impurity levels to Ga 4s orbitals.These new peaks are expected to benefit the optical transformation from ultraviolet light to the visible-infrared region.In the low-energy region in Figure 5b,d, the FeGaПOi case exhibits the absence of a clear optical absorption peak, while a few small oscillation peaks are present for the Fe-GaIOi combination.Therefore, FeGaПGai, FeGaIGai, and FeGaПVGa complexes can significantly enhance the optical absorption in the visible-infrared region.The energy-loss function (ELF) is calculated based on Equation ( 9) from the dynamic dielectric constant at small scattering angles, which can determine the energy loss of free electrons across the material.This ELF function allows a direct comparison between theoretical conclusions and experimental spectroscopy measurements such as EELS [54].
Figure 6 shows the ELF spectra for Fe GaΠ and Fe GaI complexes.The major peak for perfect β-Ga 2 O 3 is located at 16.7 eV.The peak positions remain at the same locations for Fe GaΠ and Fe GaI , while accompanying with higher ELF value.This indicates that the induced Fe dopant tends to increase its energy loss and decreases its emission of peak energy efficiency under the high energy region in the material.The changes in the primary peaks for these Fe GaΠ and Fe GaI complexes are also minor, showing that the energy loss in the β-Ga 2 O 3 material is almost negligible after the extra introduction of various native defects.Additionally, a seemingly little peak develops in the low-energy region for the Fe GaI Ga i case, which may be attributed to the optical absorption peak of ~0.84 eV in Figure 5d.
The energy-loss function (ELF) is calculated based on Equation ( 9) from the dynamic dielectric constant at small scattering angles, which can determine the energy loss of free electrons across the material.This ELF function allows a direct comparison between theoretical conclusions and experimental spectroscopy measurements such as EELS [54]. Figure 6 shows the ELF spectra for FeGaП and FeGaI complexes.The major peak for perfect β-Ga2O3 is located at 16.7 eV.The peak positions remain at the same locations for FeGaП and FeGaI, while accompanying with higher ELF value.This indicates that the induced Fe dopant tends to increase its energy loss and decreases its emission of peak energy efficiency under the high energy region in the material.The changes in the primary peaks for these FeGaП and FeGaI complexes are also minor, showing that the energy loss in the β-Ga2O3 material is almost negligible after the extra introduction of various native defects.Additionally, a seemingly little peak develops in the low-energy region for the FeGaIGai case, which may be attributed to the optical absorption peak of ~0.84 eV in Figure 5d.

GGA + U
We further carried out the GGA + U calculations to gain insight into the influences on the electronic structure of Fe-doped β-Ga2O3, considering the strong electron-electron interactions.GGA + U calculations are usually applied to deal with strong correlations in localized d-or f-electron systems and thus partially solve the band gap underestimation.We employ the values of U in the literature to accord with the experimental band-gap for the perfect β-Ga2O3 [33].Besides, the U value of 4.09 eV has been chosen for the 3d orbital of the Fe dopant based on the literature [34].
The GGA + U calculated spin-up and spin-down band structures of the perfect β-Ga2O3 configuration are shown in Figure 7a,d.The perfect β-Ga2O3 is endowed with a band gap of 4.77 eV accompanied by a direct semiconductor structure, which is in good agreement with the experimental band-gap value of 4.78 eV [55].The much flat valence bands illustrate a large effective mass and low mobility, which prevent the formation of p-type β-Ga2O3.The main orbital compositions of the VBM and CBM are consistent with those from GGA calculations, as mentioned before.Figure 7b,c,e

GGA + U
We further carried out the GGA + U calculations to gain insight into the influences on the electronic structure of Fe-doped β-Ga 2 O 3 , considering the strong electron-electron interactions.GGA + U calculations are usually applied to deal with strong correlations in localized d-or f-electron systems and thus partially solve the band gap underestimation.We employ the values of U in the literature to accord with the experimental band-gap for the perfect β-Ga 2 O 3 [33].Besides, the U value of 4.09 eV has been chosen for the 3d orbital of the Fe dopant based on the literature [34].
The GGA + U calculated spin-up and spin-down band structures of the perfect β-Ga 2 O 3 configuration are shown in Figure 7a,d.The perfect β-Ga 2 O 3 is endowed with a band gap of 4.77 eV accompanied by a direct semiconductor structure, which is in good agreement with the experimental band-gap value of 4.78 eV [55].The much flat valence bands illustrate a large effective mass and low mobility, which prevent the formation of p-type β-Ga 2 O 3 .The main orbital compositions of the VBM and CBM are consistent with those from GGA calculations, as mentioned before.Figure 7b,c,e,f show the spin-up and spin-down band structures of Fe GaΠ and Fe GaI structures, respectively, where the Fe impurity bands of 3d characters are located above the E f near CBM.The Fe GaΠ and Fe GaI exhibit semiconductor ferromagnetic ground states.The magnetic moment mainly originates from the uncompensated spin-down orbitals of Fe impurities, where no extra band has been observed in the spin-up channel.In the spin-down channel, several isolated bands originated from Fe 3d orbitals are located above the E f , which exhibit deep acceptor levels and give rise to a magnetic moment of 5 µ B .The induced deep acceptor levels of Fe 3d orbitals are 0.63 eV lower than the CBM and 3.20 eV higher than the VBM in the Fe GaΠ structure.For the Fe GaI configuration, in the spin-down channel, several isolated bands originated from Fe 3d orbitals are located at 3.31 eV, 3.49 eV, 3.57 eV, 3.59 eV, and 3.70 eV above the E f .The induced deep acceptor levels of Fe 3d orbitals are 1.09 eV lower than the CBM.The larger values below CBM compared with these of GGA calculations can be attributed to the more localized Ga and Fe 3d orbitals under GGA + U calculations, which are reasonable compared with the experimentally measured results, i.e., the EPR results of 0.84 eV observed from Polyakov et al. [26] and 1.2 eV from Bhandari et al. [22].
semiconductor ferromagnetic ground states.The magnetic moment mainly originates from the uncompensated spin-down orbitals of Fe impurities, where no extra band has been observed in the spin-up channel.In the spin-down channel, several isolated bands originated from Fe 3d orbitals are located above the Ef, which exhibit deep acceptor levels and give rise to a magnetic moment of 5 µB.The induced deep acceptor levels of Fe 3d orbitals are 0.63 eV lower than the CBM and 3.20 eV higher than the VBM in the FeGaП structure.For the FeGaI configuration, in the spin-down channel, several isolated bands originated from Fe 3d orbitals are located at 3.31 eV, 3.49 eV, 3.57 eV, 3.59 eV, and 3.70 eV above the Ef.The induced deep acceptor levels of Fe 3d orbitals are 1.09 eV lower than the CBM.The larger values below CBM compared with these of GGA calculations can be attributed to the more localized Ga and Fe 3d orbitals under GGA + U calculations, which are reasonable compared with the experimentally measured results, i.e., the EPR results of 0.84 eV observed from Polyakov et al. [26] and 1.2 eV from Bhandari et al. [22].

Conclusions
Based on first-principle DFT calculations with the GGA approach, we illustrate that Fe dopant is energetically favored for the octahedrally coordinated Ga site in Fe-doped Ga2O3 material.The controversy regarding the preferential Fe incorporation on the Ga site in the β-Ga2O3 crystal has been addressed; our result demonstrates that Fe dopant is energetically favored for the octahedrally coordinated Ga site, while analyses based on phonon dispersion mechanical characteristics, strain-dependent analyses, and formation energies are used to confirm the structural stability.Moreover, Fe impurities are more easily substituted for Ga sites under O-rich conditions.Our calculated results illustrate that the +3 charge state FeGaПGai under O-poor conditions and −2 charge state FeGaПOi under Orich conditions are easily formed for the growth of β-Ga2O3 crystals.The formation energy

Conclusions
Based on first-principle DFT calculations with the GGA approach, we illustrate that Fe dopant is energetically favored for the octahedrally coordinated Ga site in Fe-doped Ga 2 O 3 material.The controversy regarding the preferential Fe incorporation on the Ga site in the β-Ga 2 O 3 crystal has been addressed; our result demonstrates that Fe dopant is energetically favored for the octahedrally coordinated Ga site, while analyses based on phonon dispersion mechanical characteristics, strain-dependent analyses, and formation energies are used to confirm the structural stability.Moreover, Fe impurities are more easily substituted for Ga sites under O-rich conditions.Our calculated results illustrate that the +3 charge state Fe GaΠ Ga i under O-poor conditions and −2 charge state Fe GaΠ O i under O-rich conditions are easily formed for the growth of β-Ga 2 O 3 crystals.The formation energy calculations predict that the V Ga and O i in Fe-doped Ga 2 O 3 are more likely to be formed under an O-rich growth environment.When the E f approaches the valence and conduction band edges, the +3 charge state Fe Ga Ga i and −2 charge state Fe Ga O i are energetically advantageous, respectively.Moreover, V O and Ga i are expected under Garich conditions with the preferred −4 charge sate Fe Ga Ga i complex throughout the whole band gap.Fe GaΠ Ga i , Fe GaI Ga i , and Fe GaΠ V Ga complexes can significantly enhance the optical absorption in the visible-infrared region.The changes in the primary peaks for these Fe GaΠ and Fe GaI complexes are all minor, showing that the energy loss in the β-Ga 2 O 3 material is almost negligible after the introduction of various native defects.The GGA + U calculations show that the induced deep acceptor levels of Fe 3d orbitals are 0.63 eV and 1.09 eV lower than the CBM for Fe GaΠ and Fe GaI configurations, respectively, which are reasonable compared with the experimentally measured results, i.e., the EPR results of 0.84 eV observed from Polyakov et al. [26] and 1.2 eV from Bhandari et al. [22].

Figure 1 .
Figure 1.(a) The calculated complex model of a Fe-doped β-Ga2O3 supercell with native defects obtained from VESTA [35]. 1 and 2 denote the tetrahedrally and octahedrally coordinated Ga sites substituted by an iron atom, respectively.3 and 4 show the vacancy sites for Ga and O, respectively, while 5 represents the interstitial positions for both Ga and O.The a, b, and c axes refer to the crystallographic a, b, and c directions, respectively.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)(b,c) exhibit the band structure and the density of states for perfect β-Ga2O3, respectively.

Figure 1 .
Figure 1.(a) The calculated complex model of a Fe-doped β-Ga 2 O 3 supercell with native defects obtained from VESTA [35]. 1 and 2 denote the tetrahedrally and octahedrally coordinated Ga sites substituted by an iron atom, respectively.3 and 4 show the vacancy sites for Ga and O, respectively, while 5 represents the interstitial positions for both Ga and O.The a, b, and c axes refer to the crystallographic a, b, and c directions, respectively.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)(b,c) exhibit the band structure and the density of states for perfect β-Ga 2 O 3 , respectively.

Figure 2 .
Figure 2. The defect formation energies of Fe-doped β-Ga2O3 under (a) the O-rich and (b) Ga-rich conditions.The dash line represents the calculated band gap of perfect β-Ga2O3.

Figure 2 .
Figure 2. The defect formation energies of Fe-doped β-Ga 2 O 3 under (a) the O-rich and (b) Ga-rich conditions.The dash line represents the calculated band gap of perfect β-Ga 2 O 3 .

Figure 3 .
Figure 3. (a) The formation energies of FeGaП as a function of the biaxial strain with q = 0 under Orich conditions.(b) The phonon dispersion calculations for the FeGaП doping configuration.

Figure 3 .
Figure 3. (a) The formation energies of Fe GaΠ as a function of the biaxial strain with q = 0 under O-rich conditions.(b) The phonon dispersion calculations for the Fe GaΠ doping configuration.

Figure 4 .
Figure 4.The defect formation energies under the O-rich and Ga-rich conditions for the complexes of (a) FeGaП and (b) FeGaI with native defects.The dash line represents the calculated band gap of perfect β-Ga2O3.

Figure 4 .
Figure 4.The defect formation energies under the O-rich and Ga-rich conditions for the complexes of (a) Fe GaΠ and (b) Fe GaI with native defects.The dash line represents the calculated band gap of perfect β-Ga 2 O 3 .

Figure 5 .
Figure 5.Comparison of the optical absorption spectra of perfect, Fe-doped, as well as various FeGaП and FeGaI complexes in energy range from 0-30 eV (a,c).Panels (b,d) show the corresponding amplified spectra at the low-energy region (0-5 eV).

Figure 5 .
Figure 5.Comparison of the optical absorption spectra of perfect, Fe-doped, as well as various Fe GaΠ and Fe GaI complexes in energy range from 0-30 eV (a,c).Panels (b,d) show the corresponding amplified spectra at the low-energy region (0-5 eV).
,f show the spin-up and spin-down band structures of FeGaП and FeGaI structures, respectively, where the Fe impurity bands of 3d characters are located above the Ef near CBM.The FeGaП and FeGaI exhibit

Figure 7 .
Figure 7. GGA + U calculated the band structures of (a) spin-up channel and (d) spin-down channel for perfect β-Ga2O3, (b) spin-up channel and (e) spin-down channel for FeGaП structure, as well as (c) spin-up channel and (f) spin-down channel for perfect FeGaI structure.

Figure 7 .
Figure 7. GGA + U calculated the band structures of (a) spin-up channel and (d) spin-down channel for perfect β-Ga 2 O 3 , (b) spin-up channel and (e) spin-down channel for Fe GaΠ structure, as well as (c) spin-up channel and (f) spin-down channel for perfect Fe GaI structure.
Ga 2 O 3 is the chemical potential of the bulk β-Ga 2 O 3 .The chemical potential of µ Metal Ga and µ Fe are calculated from the energies of the most stable bulk crystal of the Ga and Fe atoms, respectively.µ O represents the chemical potential of O obtained from the energy of O 2 .The chemical potentials of µ O , µ Ga and µ Fe under O-rich condition are −4.92eV, −7.55 eV, −8.24 eV, respectively, while the corresponding values are −8.01 eV, −2.90 eV, −8.24 eV for Ga-rich atmosphere.

Table 1 .
The calculated lattice constants for perfect and Fe-doped β-Ga 2 O 3 .The values in parentheses indicate the changes in lattice parameters compared with these of the perfect β-Ga 2 O 3 .

Table 1 .
The calculated lattice constants for perfect and Fe-doped β-Ga2O3.The values in parentheses indicate the changes in lattice parameters compared with these of the perfect β-Ga2O3.

Table 2 .
Calculated elastic coefficients C ij , bulk modulus B H , Yong modulus E H and shear modulus G H (all in GPa) for perfect and Fe GaΠ structures, as well as the literature values for comparison.The subscript H is responding to Voigt-Reuss-Hill notation.