Effective Band Structure and Crack Formation Analysis in Pseudomorphic Epitaxial Growth of (InxGa1–x)2O3 Alloys: A First-Principles Study

Ga2O3 is a promising material for power electronic applications. Alloying with In2O3 is used for band gap adjustment and reduction of the lattice mismatch. In this study, we calculate the effective band structure of 160-atom (InxGa1–x)2O3 supercells generated using special quasi-random structures where indium atoms preferentially substitute octahedral gallium sites in β-Ga2O3. We find that the disorder has a minimal effect on the lower conduction bands and does not introduce defect states. Employing the Heyd, Scuseria, and Ernzerhof (HSE06) hybrid functional, we accurately model the band gap, which remains indirect for all considered indium fractions, x, linearly decreasing from 4.8 to 4.24 eV in the range of x ∈ [0, 0.31]. Accordingly, the electron effective mass also decreases slightly and linearly. We determined the critical thickness for epitaxial growth of the alloys over β-Ga2O3 surfaces along the [100], [010], and [001] directions. Our findings offer new insights into site preference, effective band structure, and crack formation within alloys.

The thickness of the slabs is converged with respect to the bulk limit and the convergence of the slab's surface energy (γ) is ensured.This value dictates the surface stability and can be computed using the following equation: Where E slab and E bulk are the total energies of slab and bulk systems respectively, N is the number of unit cells in the supercell, A is the supercell cross-section area, and Γ b is the brittle fracture toughness.In some cases, surface energy does not converge and appears as an oscillation due to differences in computational parameters between bulk and slab, e.g.kpoint sampling.To solve this issue, it is highly recommended to use a linear method, where E bulk is determined through a straightforward linear fit.To avoid this numerical error, we have fitted the slab energy.
Figure S2 shows the linear fit and the surface energy convergence with respect to the unit cell numbers (thickness).

Structure
The transformation matrix M, is used to create 2×2×2 supercells (Figure S1.c) for disordered structures, starting from the primitive unit cell (10 atoms; Figure S1.b).

Effective band structure
Alloys are constructed using the supercell (SC) approach.For this large system, as the number of bands increases, the band structure is folded into a smaller Brillouin zone, becoming quite complex.Compared to the primitive cell (PC), obtaining meaningful information from the electronic band structure can be challenging, even for small supercells.A band unfolding technique is proposed to compute the effective band structure (EBS) of perturbed systems 5,6 enabling a feasible comparison between SC and PC.
As the PC and SC are commensurate, we have that ⃗ where ⃗ K is a wave vector of the SC, ⃗ k i is a wave vector of the PC, and ⃗ G i is a reciprocal lattice vector of the SC.Here N is the ratio of the volumes of the PC and SC Brillouin zones.
Band unfolding is based on the expression of SC eigenvectors as a linear combination of PC eigenvectors.The spectral weight, defined as determines the probability of an SC eigenstate to have the same Bloch character as a PC eigenstate. 7From the knowledge of the spectral weight, the spectral function is obtained, where E − → Km is the energy of the SC eigenvector ψ − → Km .The EBS is given by the spectral function.
In Fig. S5 we validate the method by showing the effective band structure of pristine β-Ga 2 O 3 .The spectral function only gives 0 or 1, as expected from a perfectly conserved Bloch character of the eigenvectors.

Density of states and dielectric function
We computed the density of states and dielectric properties using GGA PBE for a 160-atom supercell.Fig. S5 mainly shows how the band gap decreases as the indium concentration is increased.For all alloys, the peaks around 6 eV are most notably affected by disorder.
The real and imaginary parts of the dielectric function were computed using the independent particle approximation (IPA).Changing the indium fraction x, we observe a redshift of the onset of the imaginary part and an increase of the static dielectric constant (the real part of the dielectric function in the limit of ω → 0 in the inset of Fig. S6.).As the dielectric function in the IPA reflects the joint density of states, the redshift of the onset of absorption corresponds to the decreasing band gap observed in the DOS.The calculations do not seem to suggest that measuring either the absorption spectrum or the static dielectric function may provide a measure of the indium fraction x, since the changes are not monotonic (for example we see a clear change when increasing from x = 0.188 to x = 0.25 but almost no change when increasing from x = 0.25 to x = 0.313).

Elastic constants and stability
The elastic tensor, C ij is determined by applying Hooke's law from the second-order derivative of total energy versus strain for small cell deformations.In practice, as implemented in the VASPKIT code, 8 small strains, ε i were applied to the equilibrium lattice constants.Elastic energy, ∆E, is determined from the total energies of the distorted, E (V, {ε i }), and undistorted, E (V 0 , 0), cells C ij ε i ε j .
V and V 0 are the volumes of the distorted and undistorted cells, respectively.
The thirteen independent elastic constants for the monoclinic structures are computed These structures are mechanically stable, as all elastic stability criteria are met. 11From an experimental point of view, these alloys are known to be stable since have already been synthesized at different concentrations, indicating their stability as powder, films, and ceramics, see Fig. 2 of the main text.
In-plane strain components are computed directly from the mismatch between the lattice constants of the substrate (Ga 2 O 3 ) and films (alloys), and out-of-plane components are computed using a minimization algorithm based on elastic energy.

Figure S2 :
Figure S2: (a,b): A linear model is used to compute the surface energy using linear fitting.Blue circles represent the calculated slab energy (eV) as a function of the unit cell numbers, and the red line shows the fitting line.(c,d): The calculated surface energies (J/m 2 ) as a function of the number of unit cells show a plateau indicating convergence.

Figure S3 :
Figure S3: (a) The 20-atom conventional unit cell of monoclinic β-Ga 2 O 3 .(b) The 10-atom primitive unit cell.(c) The 160-atom 2×2×2 supercell.The inequivalent sites are labelled as Ga 1 and Ga 2 , with Ga atoms in green and O atoms in red spheres.The transformation matrix M1 is used to convert the 10-atom primitive unit cell to the 160-atom supercell.

Figure S4 :
Figure S4: Calculated effective band structure of pristine β-Ga 2 O 3 using a 160-atom supercell, plotted for the N ΓX Brillouin zone path.The valence band maximum (VBM) is set to zero.

Figure S5 :
Figure S5: Calculated density of states for β-(In x Ga 1−x ) 2 O 3 alloys for different indium contents ranging from 0 to 0.31.For clarity, the density of states in the conduction band is scaled by a factor of 3. The zero on the energy axis is placed at the top of the valence band.

Figure S6 :
Figure S6: Calculated and real parts of the dielectric function for β-(In x Ga 1−x ) 2 O 3 alloys, employing the independent particle approximation, within a supercell containing 160 atoms, for various indium contents ranging from 0 to 0.31.
using a 1×2×2 supercell, utilising the SCAN functional.The C ij values are listed for pure β-Ga 2 O 3 and other alloys for [0, 0.31] indium content.For comparison, other experimental and theoretical results are presented.

Figure S7 :
Figure S7: The variation of total energy with respect to different K grid values using the primitive unit cell.

Table S1 :
Calculated elastic constant for β-(In x Ga 1−x ) 2 O 3 alloys for different indium content [0, 0.31].Pure β-Ga 2 O 3 elastic constants are compared with previous experimental and theoretical results.β-Ga 2 O 3 β-(In x Ga 1−x ) 2 O 3We utilized a 0.03 Å separation for the reciprocal k-point grid sampling in our investigation.
12x12x6Total Energy vs K grid