Role of Native Defects in Fe-Doped β-Ga₂O₃

Abstract

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 V_Ga and O_i can be easily formed for the Fe-doped β-Ga₂O₃ crystals under O-rich conditions, where the +3 charge state Fe_GaGa_i and −2 charge state Fe_GaO_i are energetically favorable when the Fermi level approaches the valence and conduction band edges, respectively. Optical absorption shows that the complexes of Fe_GaGa_i and Fe_GaV_Ga 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.

1. Introduction

Gallium oxides (Ga₂O₃) 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 Ga₂O₃ 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-Ga₂O₃ materials are required. The n-type conductivity in perfect β-Ga₂O₃ can be compensated by the introduction of deep acceptors, such as Fe dopant. Fe impurity is one of the most attractive dopants because it not only exists unintentionally during the synthesis of β-Ga₂O₃ crystals but also possesses the same tri-valence states with comparable ionic radii as host Ga³⁺ [17].

β-Ga₂O₃ exhibits a monoclinic structure with two nonequivalent tetrahedrally and octahedrally coordinated Ga sites. Regarding the replacement of Ga by Fe dopant in the β-Ga₂O₃ system, one general question would be what is the preferential substitution location for Fe, the tetrahedrally or the octahedrally coordinated Ga sites? Previous studies showed that the specified site location and local symmetry may determine the microscopic structure as well as the optical and electronic properties [18]. Zhang et al. reported the preferential occupation of Fe³⁺ ions in the octahedral over the tetrahedral sites in the Fe-doped β-Ga₂O₃ crystal based on electron paramagnetic resonance (EPR) analyses [19]. Trooster et al. reported that Fe-doped Ga_2-xFe_xO₃ powder with high doping concentrations (i.e., x = 0.8, 0.9, 1.08, and 1.15) grown with the flux method of Remeika exhibited ferrimagnetic spin configuration. Based on the Mossbauer measurements of ⁵⁷Fe in different compositions, they concluded that Fe³⁺ ions mainly replaced Ga³⁺ in the octahedral sites, with only a small fraction of Fe ions at the tetrahedral Ga sites [20]. Büscher et al. revealed the occupation of Fe³⁺ in distorted tetrahedral sites in single β-Ga₂O₃ crystals for the first time by employing EPR measurements [21]. Recently, Bhandari et al. showed that the Fe-replaced tetrahedral and octahedral Ga sites were not distinguished since the photon-induced changes at two different Ga sites were the same based on steady-state photo-EPR measurements [22]. This controversy over the specified Fe location in β-Ga₂O₃ will be addressed theoretically as below. In addition, native defects are inevitable in Fe-doping in the β-Ga₂O₃ system. Zhang et al. revealed that, different from the as-grown Fe-doped β-Ga₂O₃ system, the air-annealing treatment can efficiently increase the crystalline quality and reduce oxygen vacancies, along with decreasing the conductivity and halving the spin susceptibility [19]. Zhou et al. found that for the Fe substituting of the Ga site in the β-Ga₂O₃ lattice, the main defects originated from oxygen vacancies at room temperatures, as suggested by the EPR spectra, which led to the high resistivity and the potential application for x-ray detection [23]. Hany et al. also reported the existence of V_O and V_Ga in Fe-doped β-Ga₂O₃ single crystals using optical absorption and temperature-dependent cathodoluminescence (CL) measurements [24].

In terms of the electronic structure variation due to Fe³⁺ substitution, a deep level between (0.78–1.2) eV relative to the conduction band minimum (CBM) has been reported. Ingebrigtsen et al. unfolded two similar deep levels located at 0.78 eV and 0.75 eV, which can be associated with Fe impurities and intrinsic defects, respectively, for both bulk β-Ga₂O₃ crystal growth by hydride vapor phase epitaxy and molecular beam epitaxy methods [25]. Lenyk et al. studied the Fe³⁺/Fe²⁺ level in Fe-doped β-Ga₂O₃ crystals and observed a value of ~0.84 eV below the conduction band using noncontact spectroscopy methods including EPR, infrared absorption, and thermoluminescence [17]. Polyakov et al. ascertained that the Fermi level (E_f) in Fe-doped β-Ga₂O₃ crystals was pinned by the Fe acceptor level near the CBM of ~0.80 eV [26]. Bhandari et al. showed that Fe dopants act as deep acceptors, and the first optically induced change in Fe³⁺ occurred at 1.2 eV in a Fe-doped β-Ga₂O₃ single crystal [22].

To corroborate the experimentally resolved two similar deep levels, Ingebrigtsen et al. performed Heyd-Scuseria-Ernzerhof (HSE) hybrid functional calculations. Their results show that the Fe-replaced Ga site is energetically favored on the octahedral Ga site (Fe_GaII), while Fe substituted for the tetrahedral site (Fe_GaI) exhibits a slightly higher formation energy both under O-rich and Ga-rich conditions compared with the Fe_GaII state. Moreover, the thermodynamic transition level for the Fe³⁺/Fe²⁺ is located at 0.61 eV below the CBM; the level falls to 0.40 eV if assuming the lower screening [25]. We can hardly find more theoretically calculated results for Fe-doped β-Ga₂O₃ to the best of our knowledge. Therefore, it is highly desirable to provide a fundamental understanding of the relationships between the local crystal structure in Fe-doped β-Ga₂O₃ and other electronic properties from different theoretical approaches. Aside from the Fe dopant, intrinsic defects, including vacancies and interstitials, have significant impacts on their physical properties. However, the correlations among the Fe-doped β-Ga₂O₃ 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-Ga₂O₃, as well as Fe-doped β-Ga₂O₃ 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-Ga₂O₃ 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-Ga₂O₃ 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-Ga₂O₃, as well as for providing theoretical guidance on the design of β-Ga₂O₃-based functional materials and the promising applications of β-Ga₂O₃ for innovative spin-electronic and optoelectronic devices.

2. Calculation Methods

2.1. 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] 3d¹⁰4s²4p¹, [He] 2s²2p⁴, and [Ar] 3d⁷4s¹, respectively.

A 1 × 2 × 2 β-Ga₂O₃ 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-Ga₂O₃ 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-Ga₂O₃ 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_GaIV_O, Fe_GaIV_Ga, Fe_GaIO_i, and Fe_GaIGa_i, respectively, while complexes of Fe_GaΠ with V_O, V_Ga, O_i, and Ga_i configurations are labeled as Fe_GaΠV_O, Fe_GaΠV_Ga, Fe_GaΠO_i, and Fe_GaΠGa_i, respectively. Besides, we also employ the value of U in accordance with the experimental band gap for the perfect β-Ga₂O₃ [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].

2.2. Formation Energies, Transitional Levels and Optical Calculations

The formation energy of the defect D in the charge state q is calculated as [36,37]

$${\mathrm{H}}_{\mathrm{D},\mathrm{q}}({\mathrm{E}}_{\mathrm{f}},\mathrm{\mu })=[{\mathrm{E}}_{\mathrm{D},\mathrm{q}}-{\mathrm{E}}_{\mathrm{p}}]+{\displaystyle \sum _{\mathrm{i}}{\mathrm{n}}_{\mathrm{i}}{\mathrm{\mu }}_{\mathrm{i}}}+\mathrm{q}({\mathrm{E}}_{\mathrm{V}\mathrm{B}\mathrm{M}}+{\mathrm{E}}_{\mathrm{f}})+{\mathrm{E}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}}$$

(1)

where ${\mathrm{E}}_{\mathrm{D},\mathrm{q}}$ and ${\mathrm{E}}_{\mathrm{p}}$ denote the total energy of the defect and perfect supercell, respectively. ${\mathrm{n}}_{\mathrm{i}}$ represents the number of i atoms added (${\mathrm{n}}_{\mathrm{i}}<0$) or removed (${\mathrm{n}}_{\mathrm{i}}>0$) from the perfect supercell, and ${\mathrm{\mu }}_{\mathrm{i}}$ is the corresponding chemical potential. ${\mathrm{E}}_{\mathrm{V}\mathrm{B}\mathrm{M}}$ is energy of the valence band maximum (VBM) for bulk Ga₂O₃. ${\mathrm{E}}_{\mathrm{f}}$ is Fermi level, which is referenced to the VBM in the bulk. ${\mathrm{E}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}}$ is the term that accounts for the finite-size corrections, which is determined by the potential alignment and is given as [36]

$${\mathrm{E}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}}=\mathrm{q}({\mathrm{V}}_{\mathrm{D},\mathrm{q}}^{\mathrm{r}}-{\mathrm{V}}_{\mathrm{p}}^{\mathrm{r}})$$

(2)

where the potential difference between the charged defect Ga₂O₃ supercell (${\mathrm{V}}_{\mathrm{D},\mathrm{q}}^{\mathrm{r}}$) and perfect Ga₂O₃ supercell (${\mathrm{V}}_{\mathrm{p}}^{\mathrm{r}}$) 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:

$$2{\mathrm{\mu }}_{\mathrm{G}\mathrm{a}}+3{\mathrm{\mu }}_{\mathrm{O}}={\mathrm{\mu }}_{\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3}, {\mathrm{\mu }}_{\mathrm{G}\mathrm{a}}\le {\mathrm{\mu }}_{\mathrm{G}\mathrm{a}}^{\mathrm{M}\mathrm{e}\mathrm{t}\mathrm{a}\mathrm{l}}, {\mathrm{\mu }}_{\mathrm{O}}\le \frac{1}{2}{\mathrm{\mu }}_{{\mathrm{O}}_2}$$

(3)

Chemical potential varies according to different growth conditions. Under O-rich growth condition:

$${\mathrm{\mu }}_{\mathrm{O}}=\frac{1}{2}{\mathrm{\mu }}_{{\mathrm{O}}_2}, {\mathrm{\mu }}_{\mathrm{Ga}}=\frac{1}{2}({\mathrm{\mu }}_{\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3}-\frac{3}{2}{\mathrm{\mu }}_{{\mathrm{O}}_2})$$

(4)

Under Ga-rich growth condition:

$${\mathrm{\mu }}_{\mathrm{G}\mathrm{a}}={\mathrm{\mu }}_{\mathrm{G}\mathrm{a}}^{\mathrm{M}\mathrm{e}\mathrm{t}\mathrm{a}\mathrm{l}}, {\mathrm{\mu }}_{\mathrm{O}}=\frac{1}{3}({\mathrm{\mu }}_{\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3}-2{\mathrm{\mu }}_{\mathrm{G}\mathrm{a}})$$

(5)

where, ${\mathrm{\mu }}_{\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3}$ is the chemical potential of the bulk β-Ga₂O₃. The chemical potential of ${\mathrm{\mu }}_{\mathrm{G}\mathrm{a}}^{\mathrm{M}\mathrm{e}\mathrm{t}\mathrm{a}\mathrm{l}}$ and ${\mathrm{\mu }}_{\mathrm{Fe}}$ are calculated from the energies of the most stable bulk crystal of the Ga and Fe atoms, respectively. ${\mathrm{\mu }}_{\mathrm{O}}$ represents the chemical potential of O obtained from the energy of O₂. The chemical potentials of ${\mathrm{\mu }}_{\mathrm{O}}$, ${\mathrm{\mu }}_{\mathrm{Ga}}$ and ${\mathrm{\mu }}_{\mathrm{Fe}}$ under O-rich condition are −4.92 eV, −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.

The transition energy $\mathrm{\epsilon }({\mathrm{q}}_1/{\mathrm{q}}_2)$ between charge state q₁ and q₂ for defect D doping configuration is calculated as [39]

$$\mathrm{\epsilon }({\mathrm{q}}_1/{\mathrm{q}}_2)=\frac{{\mathrm{E}}_{\mathrm{D}}^{{\mathrm{q}}_1}{|}_{{\mathrm{E}}_{\mathrm{f}}=0}-{\mathrm{E}}_{\mathrm{D}}^{{\mathrm{q}}_2}{|}_{{\mathrm{E}}_{\mathrm{f}}=0}}{{\mathrm{q}}_2-{\mathrm{q}}_1}$$

(6)

Here, the ${\mathrm{E}}_{\mathrm{D}}^{\mathrm{q}}{|}_{{\mathrm{E}}_{\mathrm{f}}=0}$ represents the formation energy of the defect D in charge state q evaluated at ${\mathrm{E}}_{\mathrm{f}}=0$. The $\mathrm{\epsilon }({\mathrm{q}}_1/{\mathrm{q}}_2)$ denotes the Fermi-level position where the charge states q₁ and q₂ have equal formation energy.

The absorption coefficients in optical properties can be described as [31,40]

$$\mathrm{\alpha }(\mathsf{\omega })=\sqrt{2}\mathsf{\omega }{\left[\sqrt{{\mathrm{\epsilon }}_1^2(\mathsf{\omega })+{\mathrm{\epsilon }}_2^2(\mathsf{\omega })}-{\mathrm{\epsilon }}_1(\mathsf{\omega })\right]}^{ 1/2}$$

(7)

where ${\mathrm{\epsilon }}_1(\mathsf{\omega })$ and ${\mathrm{\epsilon }}_2(\mathsf{\omega })$ indicate the real and imaginary parts of the dielectric function, respectively. The ${\mathrm{\epsilon }}_2(\mathsf{\omega })$ can be calculated by summing up the transitions between occupied and unoccupied states using the following equation

$${\mathrm{\epsilon }}_2(\mathsf{\omega })=\left(\frac{4{\mathrm{\pi }}^2{\mathrm{e}}^2}{\mathrm{m}{\mathsf{\omega }}^2}\right){{\displaystyle \sum _{\mathrm{i},\mathrm{j}}{\displaystyle \int 〈\mathrm{i}\left|\mathrm{M}\left|\mathrm{j}\right.\right.〉}}}^2{\mathrm{f}}_{\mathrm{i}}(1-{\mathrm{f}}_{\mathrm{i}})\times \mathrm{\delta }({\mathrm{E}}_{\mathrm{j}\mathrm{k}}-{\mathrm{E}}_{\mathrm{i}\mathrm{k}}-\mathsf{\omega }){\mathrm{d}}^3\mathrm{k}$$

(8)

Here, $\mathrm{m}$, $\mathrm{e}$, $\mathrm{M}$, and $\mathsf{\omega }$ denote the mass of free electrons, the electron charge, the dipole matrix, and the frequency of incident photons, respectively. $\mathrm{i}$, $\mathrm{j}$, ${\mathrm{f}}_{\mathrm{i}}$, and $\mathrm{k}$ represent the initial state, the final state, the Fermi distribution function, and the wave function vector, respectively. The ${\mathrm{\epsilon }}_2(\mathsf{\omega })$ is related to the absorption of light and dielectric loss of energy, while ${\mathrm{\epsilon }}_1(\mathsf{\omega })$ is associated with the stored energy.

The energy loss function (ELF) can be described by the following equation [41]

$$\mathrm{L}(\mathsf{\omega })=\mathrm{Im}[\frac{-1}{\mathrm{\epsilon }(\mathsf{\omega })}]=\frac{{\mathrm{\epsilon }}_2(\mathsf{\omega })}{{\mathrm{\epsilon }}_1^2(\mathsf{\omega })+{\mathrm{\epsilon }}_2^2(\mathsf{\omega })}$$

(9)

3. Results and Discussions

3.1. Structural Stability

The calculated lattice parameters of perfect β-Ga₂O₃ 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 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.

Lattice Constants	Perfect (This Work)	Perfect (Literature)	FeGaI	FeGaΠ
a (Å)	12.412	12.494 [41]/12.28 [42]/12.214 [43]	12.387 (−0.20%)	12.393 (−0.15%)
b (Å)	3.076	3.096 [41]/3.05 [42]/3.037 [43]	3.076 (0.02%)	3.075 (−0.05%)
c (Å)	5.872	5.898 [41]/5.82 [42]/5.798 [43]	5.888 (0.28%)	5.868 (−0.07%)
β (°)	103.702	103.705 [41]/103.83 [43]	103.835 (0.13%)	103.700 (0%)

. The optimized structural parameters for the Fe-doped cases are also summarized in Table 1. The lattice constants of Fe-doped β-Ga₂O₃ 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 Fe³⁺ (Fe²⁺) and Ga³⁺ 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-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.

Figure 2a indicates the formation energies for Fe-doped β-Ga₂O₃ 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 Fe³⁺/Fe²⁺ 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 Fe³⁺/Fe²⁺ for Fe_GaI measured from VBM is 0.24 eV below CBM, thus deep acceptors are expected for both Fe-doped β-Ga₂O₃ configurations in the n-type β-Ga₂O₃ conditions, which can compensate for the free electrons caused by native defects or extrinsic impurities. In addition, the transition 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-Ga₂O₃ 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-Ga₂O₃ are evaluated by employing a complete set of elastic constants. There are thirteen independent elastic constants (C₁₁, C₂₂, C₃₃, C₄₄, C₅₅, C₆₆, C₁₂, C₁₃, C₂₃, C₁₅, C₂₅, C₃₅, and C₄₆) in the monoclinic symmetry crystal. The mechanical stability criteria of β-Ga₂O₃ are described as follows [47]:

$${\mathrm{C}}_{\mathrm{i}\mathrm{i}}>0, \mathrm{i}=1-6.$$

(10)

$$[{\mathrm{C}}_{11}+{\mathrm{C}}_{22}+{\mathrm{C}}_{33}+2({\mathrm{C}}_{12}+{\mathrm{C}}_{13}+{\mathrm{C}}_{23})]>0$$

(11)

$$({\mathrm{C}}_{33}{\mathrm{C}}_{55}-{\mathrm{C}}_{35}^2)>0$$

(12)

$$({\mathrm{C}}_{44}{\mathrm{C}}_{66}-{\mathrm{C}}_{46}^2)>0$$

(13)

$$({\mathrm{C}}_{22}+{\mathrm{C}}_{33}-2{\mathrm{C}}_{23})>0$$

(14)

$$[{\mathrm{C}}_{22}({\mathrm{C}}_{33}{\mathrm{C}}_{55}-{\mathrm{C}}_{35}^2)+2{\mathrm{C}}_{23}{\mathrm{C}}_{25}{\mathrm{C}}_{35}-{\mathrm{C}}_{23}^2{\mathrm{C}}_{55}-{\mathrm{C}}_{25}^2{\mathrm{C}}_{33}]>0$$

(15)

$$\begin{array}{ll}\mathsf{\Omega }& =2[{\mathrm{C}}_{15}{\mathrm{C}}_{25}({\mathrm{C}}_{33}{\mathrm{C}}_{12}-{\mathrm{C}}_{13}{\mathrm{C}}_{23})+{\mathrm{C}}_{15}{\mathrm{C}}_{35}({\mathrm{C}}_{22}{\mathrm{C}}_{13}-{\mathrm{C}}_{12}{\mathrm{C}}_{23})\\ & +{\mathrm{C}}_{25}{\mathrm{C}}_{35}({\mathrm{C}}_{11}{\mathrm{C}}_{23}-{\mathrm{C}}_{12}{\mathrm{C}}_{13})]-[{\mathrm{C}}_{15}^2({\mathrm{C}}_{22}{\mathrm{C}}_{33}-{\mathrm{C}}_{23}^2)\\ & +{\mathrm{C}}_{25}^2({\mathrm{C}}_{11}{\mathrm{C}}_{33}-{\mathrm{C}}_{13}^2)+{\mathrm{C}}_{35}^2({\mathrm{C}}_{11}{\mathrm{C}}_{22}-{\mathrm{C}}_{12}^2)]+{\mathrm{C}}_{55}\mathrm{g}>0\end{array}$$

(16)

$$\mathrm{g}={\mathrm{C}}_{11}{\mathrm{C}}_{22}{\mathrm{C}}_{33}-{\mathrm{C}}_{11}{\mathrm{C}}_{23}^2-{\mathrm{C}}_{22}{\mathrm{C}}_{13}^2-{\mathrm{C}}_{33}{\mathrm{C}}_{12}^2+2{\mathrm{C}}_{12}{\mathrm{C}}_{13}{\mathrm{C}}_{23}$$

(17)

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-Ga₂O₃ are shown in

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.

		C11	C12	C13	C15	C22	C23	C25	C33
Perfect	This work (PBE)	215	109	119	–15	317	72	14	312
	PBE [48]	208	118	146	0	335	83	0	318
	B3LYP [49]	235	124	138	−13	357	76	7	357
	Exp [50]	238	130	152	−4	359	78	2	346
FeGaΠ		217	107	118	−17	322	72	9	315
		C35	C44	C46	C55	C66	BH	EH	GH
Perfect	This work (PBE)	6	48	14	64	94	159	188	72
	PBE [48]	19	50	9	77	96	171	192	73
	B3LYP [49]	12	55	15	81	101	179	214	82
	Exp [50]	19	49	6	91	107	184	213	82
FeGaΠ		7	51	18	67	90	160	190	73

. For comparisons, available theoretical and experimental results are also listed. The elastic stiffness of prefect and Fe-doped β-Ga₂O₃ meets the mechanical stability criteria presented above, suggesting that prefect and Fe-doped β-Ga₂O₃ 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.

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-Ga₂O₃ 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_GaΠGa_i and negatively charged Fe_GaΠO_i are energetically favorable when the E_f approaches the VBM and CBM, respectively, while positively charged Fe_GaΠGa_i is expected throughout the whole band gap under Ga-rich condition. Moreover, the formation energies of Fe_GaΠGa_i (Ga-rich condition) and Fe_GaΠO_i (O-rich condition) are lower compared with those of Fe_GaΠ case, suggesting that Fe_GaΠGa_i and Fe_GaΠO_i complexes are easily formed under Ga-rich and O-rich conditions during experimental growth, respectively. For the Fe_GaΠV_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_GaΠV_O complex is easily formed in the O-poor growth atmosphere. For the Fe_GaΠV_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_GaΠV_Ga complex possesses lower formation energies in n-type Ga₂O₃ materials, which demonstrates that the V_Ga is more likely produced in Fe-doped Ga₂O₃ 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_GaΠO_i is energetically favorable when the Fermi level approaches the VBM and CBM, respectively. Moreover, Fe_GaΠO_i complex is more susceptible to being produced under O-rich conditions. For the Fe_GaΠGa_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_GaΠGa_i complex is expected. In addition, the higher formation energies under the O-rich condition for the Fe_GaΠGa_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-Ga₂O₃ crystal, as illustrated in Refs. [8,9], while a low formation energy is gained for the Fe_GaΠGa_i complex. The E_f always tends to be positioned at the higher region of the bandgap in β-Ga₂O₃ and gives rise to the n-type conduction characteristic due to unintentionally introduced native defects during the growth of β-Ga₂O₃. Therefore, our calculated results illustrate that the +3 charge state Fe_GaΠGa_i under O-poor condition and −2 charge state Fe_GaΠO_i under O-rich condition are easily formed for the growth of β-Ga₂O₃ 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_GaIGa_i and −2 charge state Fe_GaIO_i are dominated when the E_f is located near the VBM and CBM, respectively, while −4 charge state Fe_GaIGa_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.

3.2. Optical Property

For wide-band gap semiconductor materials, optical parameters of dielectric function $\mathrm{\epsilon }(\mathsf{\omega })$ 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 ${\mathrm{\epsilon }}_2(\mathsf{\omega })$ of the dielectric constant is related to the absorption of light and the dielectric energy-loss function, while ${\mathrm{\epsilon }}_1(\mathsf{\omega })$ 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-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 high-energy 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 Fe_GaΠV_Ga, Fe_GaIV_Ga, Fe_GaΠO_i, and Fe_GaIO_i can be ascribed to hole doping, while the blue shift for the Fe_GaΠGa_i, Fe_GaIGa_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-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 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-Ga₂O₃ crystal, the absorption coefficients become relatively low for both Fe_GaΠV_O and Fe_GaIV_O cases in the low-energy region (0–5 eV), whereas a new wide peak for Fe_GaΠV_Ga configuration is generated, leading to the optical migration from the ultraviolet light region to the visible-infrared region. The new peak for Fe_GaΠV_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_GaΠGa_i and Fe_GaIGa_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_GaΠO_i case exhibits the absence of a clear optical absorption peak, while a few small oscillation peaks are present for the Fe_GaIO_i combination. Therefore, Fe_GaΠGa_i, Fe_GaIGa_i, and Fe_GaΠV_Ga 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-Ga₂O₃ 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-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 Fe_GaIGa_i case, which may be attributed to the optical absorption peak of ~0.84 eV in Figure 5d.

3.3. 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,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].

3.4. 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 Fe_GaΠGa_i under O-poor conditions and −2 charge state Fe_GaΠO_i under O-rich conditions are easily formed for the growth of β-Ga₂O₃ crystals. The formation energy calculations predict that the V_Ga and O_i in Fe-doped Ga₂O₃ 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_GaGa_i and −2 charge state Fe_GaO_i are energetically advantageous, respectively. Moreover, V_O and Ga_i are expected under Ga-rich conditions with the preferred −4 charge sate Fe_GaGa_i complex throughout the whole band gap. Fe_GaΠGa_i, Fe_GaIGa_i, and Fe_GaΠV_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-Ga₂O₃ 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].