Indium defect complexes in (In x Ga1−x )2O3: a combined experimental and hybrid density functional theory study

Indium defects in small concentration (In x Ga1−x )2O3 were studied using a combination of spectroscopic and magnetic measurements on thin films varying the indium concentration, coupled with hybrid density functional theory simulations using the supercell method. X-ray diffraction spectra along with Tauc plots and density of states plots reveal a decrease (increase) in the electronic band gap (interlayer lattice spacing) due to the inclusion of indium in monoclinic Ga2O3, while room-temperature Hall measurements show an increase in n-type conductivity. Formation energy calculations reveal that the defect complex of substitutional indium at the octahedrally coordinated cation site (InGa) coupled with an indium interstitial (Ini) in the largest Ga–O cavity in the bulk (ia ), where the two impurities are a maximal distance away in the unit cell, results in the lowest formation energy across much of the electronic band gap; near the conduction band edge the single InGa defect becomes the lowest energy defect, though. These calculations help shed light on the impurity band enhanced, n-type conductivity increase due to small concentration indium doping in Ga2O3 as seen in the spectroscopic/magnetic measurements.

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Introduction
Indium-doped gallium oxide (In x Ga 1−x ) 2 O 3 (IGO) is an emergent wide band gap semiconductor with numerous applications in optoelectronic devices.Using an IGO front emitter layer in CdTe solar cells resulted in an increase in efficiency due to the improved conduction band offset through tuning the In:Ga ratio [1].Band gap dependent parameters like the open circuit voltage and fill factor were improved by controlling the indium concentration with optimal device performance occurring near 36% indium content.IGO also performed better than traditional organic-based rectifiers in thin-film transistors (TFTs), resulting in improved output voltages even at lower input voltage [2][3][4][5][6][7].As a bilayer, amorphous IGO (a-IGO) was shown to increase the carrier mobility and current on/off ratio in TFTs [8].a-IGO has also been used in UV photodetector applications to tune the absorption range, which results in an increased responsivity rate, mobility, and threshold voltage [9][10][11].IGO has been shown to function well as a Schottky diode in NiO:IGO heterojunctions, where the inclusion of a small concentration of indium (x ⩽ 0.025, or 5% In) helped to improve the lattice mismatch between NiO and β-Ga 2 O 3 [12].Also, when the traditional polyimide-based alignment layer in liquid crystals was replaced with a thin film of IGO, thermal and optical properties were enhanced [13].Thus, IGO has the potential to improve many next generation optoelectronic technologies.
To advance the development of IGO-based devices, an understanding of the ubiquitous defects present in all devices is required.Experimentally, a variety of spectroscopic methods have been used to study defects in IGO, including x-ray photoemission spectroscopy (XPS), scanning electron microscopy with in-situ cathodoluminescence (SEM + CL), extended x-ray absorption fine structure (EXAFS), and photoluminescence (PL) spectroscopy.XPS and SEM + CL measurements of IGO thin films revealed that O vacancies form easily under indium rich conditions.These defects acted as an emergent deep trap state with energy levels well below the conduction band edge and therefore outside the range to explain n-type conductivity [3].A blue shift was seen in the CL spectra which also confirmed the existence of free carriers due to O vacancies, as well as an ability to tune properties through control over indium concentration [14].Extrinsic hydrogen defects were studied using EXAFS coupled with molecular dynamics simulations, where H acted to reduce the formation of trapped polarons (free charges and their associated lattice distortion) in IGO [15].An increase in the indium concentration as a substitutional defect [16] has also been studied using XPS, showing that multiple phase changes occur as the indium concentration is increased.Experiment, however, appears to be ahead of theory currently, where there is a lack of theoretical understanding of In-based defects in IGO.
To study the stability of defects in IGO using theoretical (first principles) methods, the supercell approach, as implemented within hybrid density functional theory (hDFT) is suitable [17]; this is with the caveat that corrections for the finite supercell size and k-point sampling are taken into account for charged cells [18][19][20][21].Both intrinsic and extrinsic defects in IGO have been studied using the supercell method in systems with a 1:1 Ga:In ratio (x = 0.5 in the stoichiometric formula) [16,17,22].To overcome the electronic band gap issue in earlier studies using local DFT, the inclusion of exact exchange to the local term in screened hybrid functionals helped improve equilibrium lattice properties [23], which resulted in accurate formation energy and transition energy values.With these tools, theory can catch up to experiments and help to explain IGO device behaviors.
We study here the growth and optoelectronic properties of low concentration indium in monoclinic β-Ga 2 O 3 coupled with hDFT calculations of indium-based defects and complexes which mimic the early stages of indium in β-Ga 2 O 3 .At low indium concentration (first stages of doping), indium may substitute at the cation site or move into the interstitials formed by Ga-O cages.As the concentration is increased, there can exist In-based defect complexes where an interstitial and substitutional indium exist within a few unit cells of one another.In this article, we study the effects of In-based defects in low concentration IGO (indium content <5%); this is, a concentration of interest used in pn-junction applications coupled with p-type NiO [12].In those devices, a large n-type conductivity was measured in the IGO materials which is thought to be due, at least in part to defect formation and impurity band conduction.Our hDFT calculations elucidate the electronic and optical properties of low indium concentration IGO to quantify defect stability and show that there are energetically favorable defects which can account for the n-type conductivity seen in experiment.Our paper is outlined as follows: section 2 will detail the theoretical methodology, section 3 will expand upon the results with a discussion of electronic and optical properties of the lowest energy defect configuration, namely the substitutional and interstitial indium defect complex where there is space between the defects, and section 4 will give our concluding remarks with respect to n-type conductivity in low concentration IGO system.

Methods
Pulsed laser deposition was used to grow (In x Ga 1−x ) 2 O 3 thin films on sapphire substrates (α-Al 2 O 3 ) for 2%, 5%, and 10% indium concentration with a base pressure of 1 × 10 −8 Torr, a substrate temperature of 700 • C, and a constant oxygen partial pressure of 1 × 10 −2 Torr.A pulsed KrF excimer laser beam (E = 250 mJ, λ = 248 nm, ν = 10 Hz) impinged onto the sintered (In x Ga 1−x ) 2 O 3 target created a plume of ablated materials which evaporated and expanded towards the substrate.To obtain a uniform thin film the target and substrate were rotated opposite direction of one another.Indium was soldered onto thin film samples as an ohmic contact for electrical resistivity, carrier concentration, and mobility measurements using a van der Pauw (VDP) Hall setup on a Biorad/Nanometrics HL5500 Hall effect system.Structural properties were determined using a Rigaku SmartLab x-ray diffractometer (XRD) while room temperature optical transmittance spectra were measured using a Shimadzu UV-VIS-NIR optical spectrophotometer from which the band gap was determined using Tauc plots.
Our total energy calculations utilized hDFT which implements the screened Heyd, Scuseria, and Ernzerhof hybrid functional revised for solids HSEsol (HSE06 + PBEsol) [23,24].The projector augmented wave method [25] was used as implemented in the VASP [v6.2] [26,27] code with Perdue-Burke-Ernzerhof (PBE) pseudopotentials.The fraction of exact (non-local) exchange was set to 32% to reproduce the correct electronic band gap of β-Ga 2 O 3 in agreement with experiment (Eg = 4.5-4.92eV) [28][29][30][31].The spin-polarized ground state of each 120-atom (1 × 3 × 2), monoclinic supercell was found using a 500 eV plane-wave basis set, a maximum ionic displacement of 0.5 Å, a maximum force between ions of 0.01 eV Å −1 , and a single k-point (k = G) to sample the Brillouin zone; (2 × 2 × 2) k-point meshes were tested and resulted in an energy change of less than 1 meV atom −1 .An anti-ferromagnetic ground state was initialized during relaxation, not taking into account spin-orbit interactions, using [32,33], where µ B is the Bohr magneton; Ga-4s 2 4p 1 3d 10 , O-2s 2 2p 4 , and In-5s 2 5p 1 4d 10 electronic configurations were used as valence.Formation energy calculations were done following the normal formalism [20] with the Kumagai and Oba correction scheme used for finite-size effects as implemented in the Spinney code [18], given by ( Here, E tot [X q ] is the total energy of the supercell containing a defect X in charge state q, E tot [bulk] is the total energy of the defect-free (perfect) supercell, n i is the number of atoms added/removed to form the defect, m i is the corresponding chemical potential related to each n i , E F is the Fermi energy which is varied between the bulk valence band maximum (VBM), E v and conduction band minimum (CBM), ∆V is the core-level shift to align the bulk and defect energy levels, and E corr accounts for the finite supercell size.

Results and discussion
Figure 1 shows the XRD pattern from a 2θ − ω scan comparing thin films of (In 0.1 Ga 0.9 ) 2 O 3 compared with pure β-Ga 2 O 3 at 700 • C; the crystal structure of the IGO structure is shown to be the same as that of monoclinic β-Ga 2 O 3 , with a JCPDS confirmation number of #41-1103 for pure β-Ga 2 O 3 .Measurements on these films reveal an epitaxial characteristic similar to Si doping in β-Ga 2 O 3 [34]; this is opposed to the more polycrystalline structure of thicker films (t > 50 nm) grown at a lower substrate temperature.
Figure 2(a) shows that indium incorporation causes the XRD peaks to shift to a lower angle, similar to what occurs in In x Ga 1−x As relative to GaAs [35]; a lack of broadening with indium inclusion indicates high-quality thin films of (In 0.1 Ga 0.9 )O 3 can be grown with a low incidence of slip dislocations.Film thickness for the rocking curve sample was 62 nm (41.2 nm) for β − Ga 2 O 3 (IGO), with a spectral FWHM of 0.446 ± 0.001 • (0.472 ± 0.002 • ); for hall measurements, samples were ∼400 nm thick.The band gaps and the d-spacings extracted from the experimental measurements are plotted against theoretical values calculated by hDFT, as shown in figure 3. A total of 80 atom supercells were tested with one, two, three, and four neutral In Ga defects, all at octahedral Ga sites; this amounts to 3.125%, 6.25%, 9.375%, and 12.5% indium concentration.Although simulations underestimate the d-spacing and the band gap value by 2%-3%, both show similar trends of a decreasing (increasing) band gap (d-spacing) for increased indium concentration; the discrepancy lies in hDFT being done at T = 0 K (no thermal expansion).This small difference, however, is not large enough to not use hDFT to study defects, as steric effects tend to be minimized as the supercell size is increased.Also, larger simulation supercells were not reoptimized but were created from optimized unit cells elongated in each direction to form the supercell.Thus, there are only two points for dspacing from simulation: one for the 20 atom pure Ga 2 O 3 unit cell and one for a single indium substitution in a 20 atom unit cell, or 12.5% indium content.It is known, however, that in the limit of infinite supercell size, the d-spacing will not change for small concentrations of defects, which is reflected in figure 3   Hall measurements were performed on cleaved VDP structured IGO thin films with indium as an ohmic contact soldered at the four corners of the sample.The room temperature electrical resistivity (ρ), carrier concentration (n), and mobility (µ) measurements done on these films revealed n-type conducting properties with ρ = 5 Wcm, n = 10 18 cm −3 , and µ = 0.34 cm 2 V −1 s −1 respectively.The low mobility, however, may be attributed to the formation of defects in the bulk.As has been reported, the oxygen vacancy alone in β-Ga 2 O 3 [36] and IGO [17] provides deep states within the band gap but is not responsible for n-type conductivity in these materials; the other possibility might be the presence of interstitial indium in the lattice as shown by our hDFT simulations.
Figures 4(a) and (b) show typical n-type conduction as well as impurity band conduction [37].Impurity-band formation is a fundamental effect that occurs when a sufficient defect density (n > 10 18 cm −3 ) is present in a wide band gap material which creates discrete energy states in the band gap.The distance between states decreases at these densities due to a significant overlap of their respective eigenfunctions, which results in charge carrier exchange without involvement of adjacent bands.Figure 4(c) shows the temperature dependent magnetoresistance (MR) measurements which reveals defect induced weak carrier localization and impurity band conduction through the existence of a negative MR signal at low temperature [38][39][40]; similar results were seen in InGaN [41].When the defect density is very high, defect to defect quantummechanical exchange of carriers can occur i.e. tunneling from one trap to another.The effective mass of the carriers within the impurity band is much larger compared to that of the adjacent carrier band [37]; hence, the impurity band mobility is usually very small thereby validating the experimental result.
The optimized unit cell from our previous work was used as a starting point to build the 120 atom (1 × 3 × 2) supercells for our hDFT simulations [42].Varying sized supercells were tested with difficulty in convergence for smaller supercells (n atoms < 120), validating the need for a larger, nearly cubic supercell [19].Thus, the resultant supercell lattice parameters were (a,b,c,β) = (12.29 Å, 9.15 Å, 11.63 Å, 103.78 • ), with an equilibrium unit cell volume of V0 = 211.93Å 3 .Figure 5 shows the two main interstitial regions in β-Ga 2 O 3  where indium can reside (far left structure), where i a (i b ) is the larger (smaller) Ga-O cage; the i a location is found to be the lower energy indium defect location.Also, the octahedrally coordinated Ga (as opposed to the tetrahedral) is the preferred location of the InGa defect.
The standard chemical potentials computed with hDFT are , where E is the ground state energy of the bulk system of n atoms, in the standard crystal structure of each species (given in each bracket).These values were used to compute the growth limited chemical potentials used in equation (1).To calculate the formation energy, the growth limited chemical potentials must be calculated starting with the formation enthalpy.In general, such that each elements chemical potential difference (and thus the growth limited chemical potential) may be isolated and solved for.There exists an upper limit for each chemical potential change such that ∆µ i < 0 or in other words, the growth limited chemical potential must be smaller than the bare chemical potential (µ < µ * ).
One then must define the stoichiometric formula of the system and include variable terms for any defects.For instance, the systems of interest in this paper would have a formula given by In 2x+λ Ga 2−2x O 3 where λ denotes the indium content due to an interstitial, which for a single interstitial in a 120 atom supercell with 24 formula units λ = 1/24.Then, the limiting structures (Ga 2 O 3 and In 2 O 3 ) are used along with the isolated chemical potential differences to define variables which may be varied to determine the growth limited chemical potentials.For IGO, ∆µ Ga = γ∆H f (IGO) and ∆µ O = α∆H f (IGO) where γ and α are the terms which may be varied, given that 2γ ) . Then a is varied from 0 to 1 in both terms to determine which system has a γ that becomes a negative value at the lowest value of α; this occurs in the In i system for α = 0.288, γ = 0.001.Next, to refine these limiting values, one solves for the growth limited chemical potential of Ga, varying α from 0 to 1 again, but now solving for γ in terms of α and using this to find µ Ga = µ * Ga + γ∆H f (IGO) , resulting in α = 0.294, γ = 0.036.Finally, the values for the growth limited chemical potentials in the Ga-poor and Ga-rich conditions may be calculated using these values, where α = 0 for Ga-poor conditions and γ = 0 for Ga-rich conditions.These values are inputs for formation energy calculations using the Spinney code.
The calculated formation energy of each defect as a function of Fermi level is shown in figure 6.We see that the indium defect complex where the In Ga and In i are separated in space In Ga + In i (far) is the lowest energy defect from the VBM energy to 1.0 eV below the CBM energy, where the single In Ga then becomes the most stable defect.Near the VBM all defects can form spontaneously in the +2-charge (doubly ionized) state (except the In Ga which is neutral), while near the CBM only the single In i becomes unstable with a moderate activation energy of 2.0 eV.These trends are the same in both Ga poor and Ga-rich conditions with Ga-poor growth conditions shown to be favorable; the formation energy increases in the rich condition due to the difficulty of introducing indium into a Ga-rich environment.The In i , similar to higher In Ga concentration systems [17] is shown to be the most unstable defect, with a (2+/+) transition near the CBM.However, unlike the higher In Ga concentration systems, in low concentration In Ga systems the +3-charge state is unstable for all defects, with large positive formation energies over 20 eV.These large formation energies for the +3-charge states may be due to the fact the band gap states are all empty and the ionization energy approaches the bulk band gap value.Also, all (+/0) transitions occur at least 0.5 eV above the CBM indicating these states are deeper states in the conduction band.Similar to smaller interstitial dopants like Li, Na, Be or Mg, neutral In i defects have a positive formation energy and thus have an activation energy barrier [43].For the defect complexes, however, the formation energy is favorable for neutral defect formation but the (+/0) transition occurs above the CBM energy.
The isosurface value is set to 10% of the maximum value for both spin (magnetization) density and charge density difference (cdd) plots as shown in figure 7 for the (In i + In Ga (far)) defect complex.The total spin magnetization density in figure 7(a) shows localization of a large hole density (yellow isosurface) surrounding the In i in the defect complex, with smaller density localized on nearby oxygen.The cdd in figure 7(b) shows alternating regions of charge accumulation (yellow isosurface) and depletion (cyan isosurface) surrounding the In i which stabilizes and localizes the charge density around the defect site.In-O bonds on the side of the In i and the In Ga lengthen by 0.05-0.5 Å but decrease by 0.5 Å on the opposite side, shifting the In i out of the center of the 1a vacancy.Also, similar to the single In Ga defect [43], a region of charge depletion surrounds the In i in the defect complex, which is oriented on the side facing the In i .Bader charge analysis reveals that the average Bader charge on each atomic species, taken from the reconstructed all electron density is 2.0, −1.3, and 1.9 for gallium, oxygen, and indium respectively; this can be written as Ga 2+ , O 1.3− , and In 1.9+ .These values are identical to the larger indium concentration IGO system [17], implying the concentration of indium has no effect on how charge moves around the defect; gallium and indium atoms become electron deficient while surrounding oxygen atoms gain charge.
The partial density of states in figure 8 shows that an asymmetric energy state emerges in the bulk band gap due to the doubly ionized In-defect complex (In Ga + In i ) +2 (far).The state is partially occupied with a single electron 0.5 eV below the CBM, while an unoccupied state lies 0.2 eV closer to the CBM.This state may be attributed to the large n-type conductivity seen in experiment as the intermediate state requires much less energy to promote an electron to the conduction band.O-2s and Ga-4s states also contribute to the defect energy level, hybridizing with the In-5s states.The VBM (CBM) remains O-2p (Ga-4s) dominant as in both the higher indium content [17] and the bulk.In-5p and In-5s states form in the valence (conduction) bands below (above) the band edge, while the In-4d electrons reside predominantly deep within the valence bands (E < −14 eV).The ordering of the states in the band gap as shown in figure 8(e) are similar to the In i in higher concentration states [17], along with a (+/0) ionization energy 0.5 eV above the CBM.However, unlike in higher indium concentrations, the In i complex has a negative formation energy for Fermi energies across the full band gap, implying it can form spontaneously under growth conditions.This is in contrast to the higher indium concentration where the formation energy is above 2.5 eV for interstitial formation.Ntype conductivity could not be explained by interstitial formation in high concentration IGO but can in fact be explained in lower indium concentration systems.Increasing the indium content in IGO makes In i formation less favorable due to limiting steric effects from the larger indium atom and explains this difference.
The optical absorption calculated from the frequency dependent dielectric response shown in figure 9 reveals a red shifted absorption onset due to the doubly ionized indium defect complex.The absorption is given in terms of the real and imaginary parts of the dielectric response, such that, where k is the extinction coefficient, w is the frequency, and ε 1 (ε 2 ) is the real (imaginary) part of the complex, frequency .This large dipole moment in the long-axis direction implies strong ionicity of the bonds in this direction, which is also seen in figure 7.
The feature emerging near 3 eV in the absorption is far more pronounced in the low indium concentration system (insert of figure 9(c)), as opposed to the larger indium concentrations [17] where the peak occurs at 3.4 eV.However, as in the larger concentration systems, this peak is due to the transition between occupied O-2p states near the VBM, to the empty cation s-orbital states in the band gap.
Also, as with the larger indium concentration systems, there exist smaller peaks with α (ω) < 1.25 × 10 5 cm −1 in the visible light range.A small peak exists in the [100] direction in the green region of the visible spectrum, and a second in the [001] direction closer to the yellow region.There is also a set of absorption peaks below 1 × 10 5 cm −1 (undertone) in the [101] direction that enhance (increases) the net absorption due to the defect.The absorption coefficient continues to increase with increasing photon energy, with pronounced peaks near the bulk VBM, 8 eV and 12 eV.The upper peaks can be attributed to transitions between Ga 4s and 4p states and O 2p states, as can be seen in the partial density of states plot in 9d.

Conclusions
We have shown the results of hDFT first principles calculations on In-based defects in low indium concentration IGO.Formation energy calculations reveal two low energy defects which can form spontaneously across the electronic band gap energy range: (1) (In Ga + In i ) 2+ (far) [doubly ionized defect complex] from the VBM to 3.8 eV, and (2) In Ga from 3.8 eV to the CBM; all In-based defects can form spontaneously near the VBM while some require an activation energy to form near the CBM energy.Charge density difference and spin density plots reveal localization in the doubly ionized defect complex, surrounding the In i and nearby O. Partial density of states plots for each atomic species show an emergent band gap state due to the doubly ionized defect complex, where an unoccupied state exists within 0.2 eV of the CBM; this state can result in an increase in n-type conductivity due to its proximity to the CBM and the shallow (+/0) transition that occurs 0.5 eV above the CBM.Finally, optical absorption calculations using the frequency dependent dielectric response function reveal a red shift in the optical absorption onset with an emergent peak near 3 eV resulting from transitions between occupied O VBM states and unoccupied cation defect states within the band gap.The increased n-type conductivity in experiment can be explained by emergent unoccupied band gap states near the CBM due to the doubly ionized (In Ga + In i ) 2+ (far) defect complex, where there exists a near band edge transition between the (+/0) states.Also, if one were to wish to mitigate the n-type conductivity, for instance in IGO doped with proposed p-type dopants, Ga-rich growth conditions are favorable along with annealing in an H-rich environments to saturate unoccupied band gap states/dangling bonds.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).
Figure 2(a) also verifies the lattice constant change due to indium incorporation in β-Ga 2 O 3 due to the larger ionic radius of In 3+ (0.80 Å) as compared to Ga 3+ (0.62 Å).
Figure 2(b) shows the band gap lowering of the IGO thin films, using Tauc plots.The band gap of the (In 0.1 Ga 0.9 )O 3 thin films was determined from the Tauc relation [(αhν) 2 − (hν)] as 4.63 eV, as compared to 4.9 eV for pure β-Ga 2 O 3 .
(b) as the change in spacing with increased indium content is ⩽1%.

Figure 3 .
Figure 3.Comparison of experimental and theoretical value of (a) band gap and (b) d-spacing of β − (InxGa 1−x ) 2 O 3 at different indium concentrations (x).

Figure 5 .
Figure 5. Crystal structure of pure Ga 2 O 3 and defect structures near the defect site.The interstitial sites tested are marked in the left-most figure.

Figure 6 .
Figure 6.Formation energy as a function of Fermi level for single and complex indium charge defect states, under both (a) Ga-poor and (b) Ga-rich conditions.The Fermi level is referenced to the VBM and the (2+/+) transition levels are indicated with vertical black lines at the transition energy.

Figure 7 .
Figure 7. Relaxed structure with (a) the total spin (magnetization) density of the (In Ga + In i ) +2 (far) defect complex, and (b) the charge density difference between the (In Ga + In i ) +2 defect complex and the bulk state (ρ cdd = ρ defect − ρ bulk ).The yellow (cyan) isosurface in (a) is the hole (electron) spin density, while the yellow (cyan) isosurface in (b) represent charge accumulation (depletion) regions; the isosurface values are set to 10% of the maximum value, resulting in 0.0006 e Å −3 in (a) and 0.1 e Å −3 in (b).Green, red, and magenta spheres represent Ga, O, and In, respectively.

Figure 8 .
Figure 8. Electronic structure of (In Ga + In i ) +2 (far) defect complex.(a) Bulk β − Ga 2 O 3 and (b) (In Ga + In i ) +2 (far) total density of states, and (c) Ga, O, and (d) In partial density of states in the (In Ga + In i ) +2 (far) defect; horizontal black lines mark the bulk band edges, dashed lines mark the defect Fermi level, and the VBM is set to 0 eV.(e) Shows the quasiparticle eigenstates emerging in the band gap of bulk β − Ga 2 O 3 due to the doubly ionized defect complex.

Figure 9 .
Figure 9. (a) Real and (b) imaginary parts of the energy (frequency) dependent dielectric function, (c) optical absorption and (d) conduction band partial density of states revealing the higher energy conduction band transitions occurring due to the defect.The transparent lines in (a)-(c) are the bulk values while the solid, darker lines represent the defect.The inset in (b) and (c) is a zoomed in view of the visible light region.