Point Defects in Silicon-Doped β-Ga2O3: Hybrid-DFT Calculations

In this work, hybrid density functional theory calculations are used to evaluate the structural and electronic properties and formation energies of Si-doped β-Ga2O3. Overall, eight interstitial (Sii) and two substitutional (SiGa) positions are considered. In general, our results indicate that the formation energy of such systems is significantly influenced by the charge state of the defect. It is confirmed that it is energetically more favorable for the substitution process to proceed under Ga-poor growth conditions than under Ga-rich growth conditions. Furthermore, it is confirmed that the formation of SiGaI with a tetrahedral coordination geometry is more favorable than the formation of SiGaII with an octahedral one. Out of all considered interstitial positions, due to the negative formation energy of the Si +3 charge state at i8 and i9 interstitial positions over the wide range of Fermi energy, this type of defect can be spontaneously stable. Finally, due to a local distortion caused by the presence of the interstitial atom as well as its charge state, these systems obtain a spin-polarized ground state with a noticeable magnetic moment.


INTRODUCTION
In recent years, the monoclinic phase of gallium oxide, known as β-Ga 2 O 3 , has gained significant attention and importance in various fields due to its exceptional properties and versatile applications.With a band gap value typically falling within the range of 4.5−4.9eV, 1−4 the thermally stable β-Ga 2 O 3 phase holds great promise for revolutionizing diverse areas, ranging from electronics to optoelectronics.This phase has found particular significance in the development of cutting-edge devices, including Schottky barrier diodes, 5,6 metal-oxidesemiconductor field-effect transistors (MOSFETs), 7 metal− semiconductor field-effect transistors, 8 solar-blind photodetectors, 9 resistance random access memory devices, 10 gas sensors, 11 and spintronic devices. 12Furthermore, the remarkable properties and potential applications of the monoclinic phase of gallium oxide position it as a game-changing material in the field of power electronics, poised to revolutionize the power electronics industry. 7efects in the crystal structure can significantly affect the performance and reliability of devices based on Ga 2 O 3 . 13In the monoclinic compound with low symmetry, several point defects need to be considered for vacancy and substitution studies, and in addition, interstitial sites, which can be occupied by neutral or charged atoms, must also be considered.According to the literature, some primary defects are considered to be electrically active.For example, gallium vacancies (V Ga ), as well as their complexes with hydrogen, are deep acceptors, but gallium interstitial (with just one of the lowest formation energies the authors have found), Ga i , 14 is a shallow donor.−16 Dopants also play a crucial role in modifying the electrical and optical properties of the material, enabling the creation of specific device functionalities.By carefully selecting and controlling dopants, it is possible to tailor the conductivity type (n-type or p-type) and achieve the desired carrier concentration in gallium oxide.The ability to precisely dope β-Ga 2 O 3 with impurities such as Sn, Si, Ge, and Mg enables the realization of high-quality n-type epitaxial films, offering a wide range of electron densities from 1 × 10 16 to 1 × 10 19 cm −3 .Theoretical calculations indicate that Si, Ge, and Sn serve as the most common shallow donor impurities, with Si predicted to be the shallowest donor among them. 17Recent ab initio study of complexes of substitutional defects shows that doping Ga 2 O 3 with Si could lead to an acceptor with Si O coupled with H Ga or to a donor with Si Ga . 18Despite extensive research on Si doping in β-Ga 2 O 3 and the preference for Si substitution at the Ga I site rather than at the interstitial positions, 15,16 a comprehensive study of the various possible interstitial positions, their formation energy, and the replacement of intrinsic atoms as a result of interstitial doping is still lacking.
One of the reasons for the absence of such studies has been the understanding that in the case of Si interstitials, in a thermodynamically stable condition, only a small concentration of Si i atoms would be present due to the negative formation energy of Si Ga . 18However, the situation could change dramatically when a suitable out-of-equilibrium growth technique, such as ion implantation doping, is used.In this case, due to the ballistic nature of this process, the buildup of a lattice disorder occurs and it is accompanied by the formation of various types of defects in relatively large concentrations.Recent experimental studies of β-Ga 2 O 3 :Si system prepared by ion implantation with Si ions show that Si interstitials occur in large concentration but not as isolated atoms but as complexes of Si i with gallium and/or oxygen vacancies. 19Thus, to better understand the changes in the properties of β-Ga 2 O 3 and gallium oxide-based materials, the most favorable interstitial sites should be identified through further analysis and simulations.
In the study by Blanco et al., 20 a comprehensive theoretical investigation of point defects in β-Ga 2 O 3 was carried out, focusing on the ionic conductivity of the material.The energetics and diffusion properties of both the host lattice and dopant ions in β-Ga 2 O 3 were examined within the framework of the shell model.Overall, eight different optimum configurations were identified for the Ga interstitials.Then, specifically for the Si dopant, the researchers adopted an ionic description considering Si 4+ .Such an approach assumes that the electronic distributions of positively charged ions do not undergo significant changes upon incorporation of ions into the lattice of β-Ga 2 O 3 .This simplification allowed for a more straightforward treatment of the dopant and facilitated the analysis of its behavior within the crystal structure.
In this article, we explore the doping properties of Si in the monoclinic phase of gallium oxide.Specifically, we investigate the formation energies of Si in eight interstitial sites and two distinct substitutional sites (Ga I and Ga II ).Our objective in this work is to analyze the relaxed positions and behavior of the Si dopants, calculate transition energies, and examine their impact on the electronic structure and optical properties of the pristine material.

COMPUTATIONAL METHODS
−24 To accurately capture the structural and electronic properties of β-Ga 2 O 3 , the HSE06 hybrid functional 25 was employed.This functional incorporates a fraction of exact exchange (0.32) 26 and a fixed screening parameter of 0.2 Å −1 .All calculations were spin polarized.The formation energy of an impurity with charge state q can be calculated using 27 Here, E tot (X q ) represents the total energy of the supercell containing the defect in charge state q, and E tot (bulk) denotes the total energy of the defect-free crystal.n i represents the change in the number of atoms due to the formation of a defect: it is negative if the atom of the i-th species is removed and positive in case an atom of the i-th species is added.The term q•(E VBM + E F ) accounts for the energy change upon the removal or addition of electrons during the formation of charged defects, with E F representing the Fermi energy of β-Ga 2 O 3 and E VBM being the valence band maximum (VBM) value.To account for the elimination of false electrostatic interactions between charged defect supercells, the Freysoldt− Neugebauer−Van de Walle correction term, E corr , was included. 28The experimental value of 10 is taken for the dielectric constant. 29The chemical potentials μ i of the elements calculated in this work are −5.69eV for Si (from Si bulk), −3.38 eV for Ga (bulk Ga metal), and −7.187 eV for O (O 2 molecules), enabling the determination of the Ga-rich (O-poor) and Ga-poor (O-rich) limits based on the enthalpy of formation of β-Ga 2 O 3 .
To construct the formation energy curve for a chosen defect, it is necessary to perform calculations according to eq 1 for a set of charge states q (typical values for q are 0, ±1, ±2, and ±3), considering the Fermi energy E F as an independent variable in eq 1.Then, by varying E F , it is possible to identify the charge state q* that has the lowest formation energy E q* f among all of the studied charges at that particular value of E F .By plotting values of E q* f as a function of E F , the formation energy curve for a chosen defect is constructed. 30,31The resulting formation energy curve will be a piecewise linear function, with each segment corresponding to a particular slope associated with the charge that is most stable (favorable) at the given interval of the Fermi energy E F of that segment.
An additional parameter can be extracted from eq 1 and/or from the formation energy curve directly, i.e., a thermodynamic charge transition level (the position of the Fermi energy where q 1 and q 2 charge states have equivalent formation energies), which can be calculated using 32 Here, represents the formation energy of the defect in charge state q considered at E F = 0.

RESULTS AND DISCUSSION
The monoclinic crystal structure of β-Ga 2 O 3 belongs to the C2/m space group and consists of 20 atoms in the conventional unit cell.To determine the optimized lattice constants for the pristine β-Ga 2 O 3 (Figure 1), energy minimization calculations were first performed for different volumes of the 20-atom unit cell.Then, optimization of the lattice constants was carried out under the constant optimized volume condition, yielding values of a = 12.22 Å, b = 3.303 Å, and c = 5.79 Å for pristine β-Ga 2 O 3 .These values, as well as other obtained parameters, are in good agreement with those reported in the literature obtained from ab initio calculations and from experiments (see Table 1).
For the calculation of formation energies of interstitial and substitutional defects, a 160-atoms 1 × 4 × 2 supercell was created that has a size of a = 12.22 Å, b = 12.22 Å, and c = 11.58Å.The choice of such a supercell guaranteed that the average interdefect distance is large enough to ensure that the error in the formation energy is small and, in addition, that this distance is approximately the same in all directions to avoid any unnecessary bias during relaxation. 14,33An energy cutoff of 510 eV was chosen along with a 2 × 8 × 4 Γ-centered Monkhorst−Pack 37 k-point grid for the unit cell.For computations on the supercell, single Γ-point calculations were performed for all interstitial cases, and a 2 × 2 × 2 Γcentered Monkhorst−Pack k-point grid was used for calculations on a few selected cases.The convergence criteria were set so that the forces on each ion were less than 0.03 eV Å −1 and the total energy changes were reduced to less than 1 × 10 −4 eV per atom.
As was mentioned above, the crystal structure of β-Ga 2 O 3 comprises two inequivalent Ga sites (Ga I and Ga II ) and three inequivalent oxygen sites (O I , O II , and O III ). Figure 1 illustrates the coordination environment of these atoms, where Ga I and Ga II are coordinated by four (tetrahedral) and six (octahedral) oxygen atoms, respectively.Note that there are three inequivalent positions for oxygen: O I is coordinated by two Ga II and one Ga I , and O II is coordinated by two Ga I and one Ga II , while O III is coordinated by four atoms.The bond lengths between different atoms are also depicted in Figure 1.
Altogether, there were eight different initial positions, denoted as i 1 , i 2 , i 3 , i 4 , i 5 , i 6 , i 8 , and i 9 , considered for the initial position of interstitial Si atoms (see Supporting Information for a graphical visualization of these eight initial positions used in interstitial calculations).Here, the notation from ref 20 is used to identify the interstitial positions.Note that the initial position i 7 , as well as positions i 10 and i 11 were excluded from consideration as they were shown to be energetically unstable. 20The systems with doped atoms in these eight positions were then relaxed to determine the most stable configuration for interstitial Si atoms in their neutral states in the crystal lattice.Then, using a single Γ-point for a kpoint grid, the formation energies of neutral Si interstitials were computed to identify the most promising cases of interstitial defects for further analysis.Table 2 shows that the formation energies for the Si interstitial at three positions, i 8 , i 9 , and i 5 , are close to each other and have values that are much lower than the values for other cases.These three positions, i 8 , i 9 , and i 5 , were selected for further analysis with more accurate computations (denser k-point grid) and with charged defects.The charge of the supercell was set to take the values 0, +1, +2, +3, and +4 for each defect in these cases.
Let us start the discussion with the formation energy results.As shown in Figure 2, Si substitution at the Ga I site is energetically more favorable than that at the Ga II .For both defects, the charge state +1 is the stable one.The obtained results agree well with the literature data. 16Slight deviations in numerical values are attributed to the different functionals used (HSE06 hybrid functional 25,26 in this work vs PBE0 hybrid functional 38,39 in 16 ) as well as to the different supercells used (160 atoms in this work vs 120 atoms in ref 16).For the considered host matrix, the substitutional Si Ga defects at the Ga I and Ga II sites are energetically more favorable than the interstitial Si i defects over the whole band gap range and for both Ga-rich and Ga-poor conditions.Only in the case of Garich conditions and for Fermi energies very close to the VBM, the formation energy of the interstitial Si at position i 8 is slightly lower than the value for the substitutional Si Ga at the Ga II site.
For Si placed at interstitial positions, our calculations reveal that for all defects studied, the dependence of the formation energy on the Fermi energy is not monotonic, i.e., their charge state changes depending on the current level of the Fermi energy.For example, the charge state +3 is the favorable state for Si placed at the interstitial positions i8 from the VBM to 1.32 eV, and after that the charge state +2 is the favorable state.A similar dependence, with a similar value of the transition level, is observed for Si placed at interstitial positions i 9 .In case Si placed at the interstitial position i 5 , the neutral state could   15.10 a Note that the calculations were performed on a large supercell but with a single Γ-point as k-point grid.also be favorable for the higher E F values.Overall, Si placed at interstitial positions i 8 and i 9 is more energetically favorable than Si placed at interstitial position i 5 over the whole band gap range.Only for Fermi energies very close to the CBM, the formation energy of the interstitial Si at the position i 5 is lower than for the interstitial Si at positions i 8 and i 9 ; hence, the Si i5 case is more favorable.Table 3 indicates the thermodynamic energy transition between different charge states for each considered defect.It is also worth noting that all considered structures correspond to positive charge states in the energy band gap region (and neutral for Si i5 at higher E F values), indicating that these interstitial defects could only act as donors.
To the best of our knowledge, there were only a few theoretical studies on Si interstitial in β-Ga 2 O 3 , 16,18,20 with only one study that looked into different charge states of Si interstitial. 16Qualitatively, the formation energy curves for Si placed at the interstitial positions i 8 and i 9 are similar to the curve for Si interstitial presented in ref 16 (see Figure 1 therein): the formation energy changes from the value of −9 eV at the VBM to the value + 5 eV at the CBM, crossing E F = 0 eV somewhere in the middle of the band gap, and having stable charges +1, +2, and +3.Quantitatively, however, there are changes due to differences in the values of the transition levels +3/+2 and +2/+1, which are attributed to different functionals and supercells used as discussed earlier.
Finally, note also that the formation energy values of neutral defects calculated using a denser k-point grid differ significantly from those calculated using only a single Γ-point.It can also be seen from Table 2 and Figure 2 that even the order of the values is different for Si i5 , Si i8 , and Si i9 .This confirms the fact that DFT calculations should be carefully checked not only with respect to the supercell size but also with respect to the kpoint mesh.One should also keep in mind that, for a chosen Si interstitial case, the relaxation is performed for the neutral state only in this work.The resulting relaxed structures were then used as fixed structures for computations with charged defects.This is a standard approach where it is assumed that the doping procedure or any subsequent treatment steps do not allow for the host matrix to be further relaxed.However, in the future, depending on the conditions, a relaxation procedure may also need to be performed for charged defects.
For all of the studied cases, it was found that the interstitial defects led to some structural reorganization through displacing neighboring O and Ga atoms out of their initial sites.Figure 3 demonstrates the final position of the interstitial atoms and their neighbors for the cases i 5 , i 8 , and i 9 .The case i 5 showed the least change in the initial structure after relaxation.In this case, the Si interstitial did not move from its initial position, and the neighboring atoms of O and Ga were displaced by a maximum of 0.30 Å.In the case i 8 , the Si interstitial moved from its initial position by 0.35 Å; however, Table 3. Transition Level ε(q 1 /q 2 ) for Si-Doped β-Ga 2 O 3 in the Range of E F is Presented in Figure 2    one of the atoms of Ga was displaced by 0.92 Å with the rest of the neighboring atoms of O and Ga by a maximum of 0.35 Å.Finally, in the case i 9 , the Si interstitial was displaced by 0.60 Å from its initial position, which is the largest displacement of Si interstitial among the studied cases.The neighboring atoms of O and Ga were also displaced by a maximum of 0.30 Å, as in the case of i 5 .Some of the coordinates and distances for these cases are given in Table S3 in the Supporting Information.
The calculated density of states (DOS) for the pristine sample, that is, the intrinsic β-Ga 2 O 3 , is shown in Figure 4A, with a band gap of 4.73 eV.The VBM clearly corresponds mainly to the O 2p states.The substitution of Si at the Ga I site (Si GaI ), which is the most favorable of the two substitution cases, causes the material to exhibit metallic behavior, as shown in Figure 4B. Figure 4C,D presents the total DOSs (TDOS) and partial DOSs (PDOS) for Si interstitial at position i 8 with charge states +2 and +3.This choice is based on the fact that case i 8 has the lowest formation energy over almost the entire Fermi energy range from the CBM to the VBM, and charge states +2 and +3 are the most stable (favorable) states near the CBM and the VBM, respectively.The TDOS and PDOS for cases i 9 and i 5 are provided in Supporting Information.Near the VBM, for interstitials at positions i 8 (+3 charge), i 9 (+3 charge), and i 5 (+2 charge), the system remains a semiconductor with E g of 2.076, 1.780, and 1.191 eV, respectively.Near the CBM, for interstitials at positions i 8 (+2 charge), i 9 (+2 charge), and i 5 (neutral), the system is also a semiconductor with E g of 2.24, 2.44, and 1.61 eV, respectively.Finally, it is worth mentioning that the charge states induce magnetic moments of 1.00 μ B for the cases i 8 (+3 charge) and i 9 (+3 charge), 2.00 μ B for the case i 5 (+2 charge) but 0.00 μ B for the case i 8 (+2 charge) and i 9 (+2 charge).The magnetic moment is absent for the case i 5 (neutral).The appearance of magnetic moment can be attributed to the displacement of Ga neighbors, as previously demonstrated in a study by Yang et al., 40 which showed that vacancies of Ga I and Ga II can induce magnetism in the monoclinic phase of Ga 2 O 3 .

CONCLUSIONS
In this work, we utilized a DFT approach with a hybrid functional to study various point defects, such as interstitial and substitutional Si, in the low-symmetric monoclinic phase of Ga 2 O 3 .Our results indicate that in the case of substituting, the Si atom mostly prefers the Ga I site that agrees well with the literature.In the case of eight different interstitial Si point defects studied here, first, the cases i 8 , i 9 , and i 5 were selected based on their lower formation energy of defects at neutral charge computed at Γ-point only.Then, these cases were analyzed at different charges, selecting the most favorable among them.We found that among the studied interstitial defects, the case with Si positioned at i 8 is the most favorable over the entire Fermi level range but could be at different charge states: +3 near the VBM and +2 near the CBM.In addition, our results show that these defects at the charge state induce a noticeable spin polarization, keeping semiconducting properties.

Data Availability Statement
The data that support the findings of this study are publicly available at 10.18150/1MQF0J.
Crystal structure of β-Ga 2 O 3 with marked different initial positions for interstitial atoms; TDOS and PDOS for doped β-Ga 2 O 3 with neutral and charged interstitial defects for different cases; and initial and final afterrelaxation procedure positions of the atoms with the largest displacement for different cases (PDF) ■

Figure 1 .
Figure 1.Unit cell of the monoclinic phase of Ga 2 O 3 .Coordination environment of tetrahedral Ga I and octahedral Ga II positions (purple spheres are Ga atoms; gray spheres are O atoms).

Figure 2 .
Figure 2. Formation energies for Si impurity in β-Ga 2 O 3 plotted against the Fermi energy for (A) Ga-rich and (B) Ga-poor conditions.The boundaries for the nonshaded region correspond to the VBM (E F = 0 eV) and the CBM (E F = E g = 4.73 eV).Note that the calculations were performed on a large supercell with a dense k-point grid.

a
The shaded cells indicate values outside of the band gap range (0 ≤ E F ≤ E g = 4.73 eV).Missing values denote transitions absent within the range of E F presented in Figure2(−1 ≤ E F ≤ 6 eV).

Figure 3 .
Figure 3. Positions of atoms after relaxation for doped β-Ga 2 O 3 with Si interstitial at positions (A) i 5 , (B) i 8 , and (C) i 9 .Note: The red spheres denote Si atoms, gray spheres oxygen (O) atoms, purple spheres gallium (Ga) atoms, and green spheres indicate the Ga atoms shifted most substantially during structural relaxation.

Figure 4 .
Figure 4. TDOS and PDOS for the cases of (A) pristine β-Ga 2 O 3 , (B) substitutional Si at the Ga I site, (C) neutral Si interstitial at the i 8 position, and (D) +3 charge state Si interstitial at the i 8 position (the Fermi energy is set to 0 eV).

Table 1 .
Lattice Constants, Band Gap Energy, and Formation Enthalpy of β-Ga 2 O 3 (ΔE) Resulting from GGA and HSE Calculations from This Work and Compared with Experimental and Theoretical Results Found in the Literature a a note: (a) ref 33 (b) ref 34 (c) ref 35 (d) ref 36 (e) ref 35.

Table 2 .
Formation Energy E f of the Neutral (q = 0) Si Interstitial at Different Locations, Sorted in Ascending Order a