The Possibility of Layered Non-Van Der Waals Boron Group Oxides: A First-Principles Perspective

: Two-dimensional (2D) metal oxides have broad prospective applications in the ﬁelds of catalysis, electronic devices, sensors, and detectors. However, non-van der Waals 2D metal oxides have rarely been studied because they are hard to peel off or synthesize. In this work, taking alumina (Al 2 O 3 ) as a typical representative of 2D boron group oxides, the structural stability and electrical properties of 2D Al 2 O 3 are investigated through ﬁrst-principles calculations. The thinnest Al 2 O 3 structure is a bilayer, and the band gap of Al 2 O 3 is found to decrease with decreasing layer thickness because of the giant surface reconstruction. The band gap of bilayer X 2 O 3 (X = Al, Ga, and In) decreases with increasing atomic radius. Our ﬁndings provide theoretical support for the preparation of non-van der Waals 2D boron group oxide semiconductors.


Introduction
Since the discovery of graphene in 2004 [1], two-dimensional (2D) materials have been receiving increasing attention because of their excellent thermal, mechanical, optical, and electrical properties. Besides graphene, other 2D materials, such as black phosphorus, covalent organic frameworks, hexagonal boron nitride, layered double hydroxides, metals, metal oxides, metal-organic frameworks, transition metal carbides/nitrides (MXenes), and transition metal dichalcogenides [2][3][4], have been discovered and investigated extensively. Because metal oxides are abundant in nature, 2D metal oxides are expected to integrate with other 2D materials for electronic, spintronic, and optoelectronic applications [5][6][7][8][9][10][11]. Although some van der Waals (vdW) layered metal oxides, such as SnO and PbO [12,13], have been predicted, the corresponding experimental realization of vdW layered metal oxides is sparse [14]. Therefore, many attempts have recently been made to prepare non-vdW layered metal oxides, such as WO 3 and MoO 3 [15][16][17][18][19]. Non-vdW metal oxides are difficult to exfoliate into a few layers due to their strong chemical bonds in three dimensions, and the bottom-up synthesis method also poses significant difficulties and challenges [20]. As a result, less has been reported on 2D non-vdW metal oxides, and their corresponding properties have not been fully explored.
Binary non-vdW boron group oxides, such as Al 2 O 3 , Ga 2 O 3 , and In 2 O 3 , are wide-bandgap semiconductors and have been applied in electronic devices and photodetectors [21,22]. Because of its ultra-wide band gap and good thermal conductivity, Al 2 O 3 is often used in ceramic applications [23][24][25]. A bottom-up method to synthesize graphene-like γ-Al 2 O 3 nanosheets has been reported [26]. Using graphene oxide (GO) as template, a homogeneous aluminum sulfate layer was firstly deposited on the GO sheets. Then, the prepared GO-,Al composite sheets were calcined to remove GO and convert basic aluminum sulfate into γ-Al 2 O 3 nanosheets [26]. The γ-Al 2 O 3 nanosheets exhibit faster adsorption kinetics and larger adsorption capacity, which suggests broad applications in catalysis and environmental science. However, the metastable γ-Al 2 O 3 will transform into stable α-Al 2 O 3 under higher temperature. Theoretically, various different monolayer aluminum oxide have been predicted [27][28][29], for example, hexagonal Al 2 O 3 [29] and rectangle AlO 2 [27]. Therefore, exploring the stability of 2D α-Al 2 O 3 is highly essential. Ga 2 O 3 is a transparent semiconductor with wide band gap and high critical electric field, and therefore holds the promising applications in power electronics and solar blind photodetectors [30][31][32][33][34]. In 2 O 3 doped by Sn has been widely used for electronic and optoelectronic devices, such as solar cell [35], optical window [36], film transistor [37], and Schottky contact [38]. Because of its ultra-flat surface, Al 2 O 3 (0001) substrate is considered a promising substrate for 2D materials, such as graphene and transition metal dichalcogenides obtained by chemical vapor deposition [39][40][41][42]. Recently, Ga 2 O 3 is also applied as a substrate to grow transition metal dichalcogenides nanoribbons. Because of their similar structures, 2D Al 2 O 3 , Ga 2 O 3 , and In 2 O 3 are promising when coupled with 2D electronic devices. It has been reported that 2D Fe 2 O 3 was successfully synthesized via the liquid-phase exfoliation of hematite ore in an organic solvent N,N-dimethylformamide (DMF) [43]. Because bulk Fe 2 O 3 has a similar structure to boron group oxides, the synthesis of other non-van der Waals 2D metal oxide materials should thus also be achievable.
In this work, taking 2D Al 2 O 3 as a representative, we design a series of stable 2D boron group oxides using first-principles calculations and investigate their crystal structures, stability, and electronic structures. Two-dimensional Al 2 O 3 is stable down to the bilayer limit and the band gap of few-layer Al 2 O 3 increases with increasing layer thickness. This is the opposite of the band gap thickness trend of vdW layered materials, such as black phosphorus and transition metal dichalcogenides. In addition, we investigate the stability and electronic properties of bilayer Ga 2 O 3 and In 2 O 3 , assuming both bilayers are thermally stable. Our work predicts promising stability for 2D non-vdW boron group oxides and provides theoretical support for the future experimental preparation of 2D boron group oxides.

Methods
All first-principles calculations in this work, such as structural optimization, phonon dispersion, and electronic properties of Al 2 O 3 with different number of layers, were performed using the QUANTUM ESPRESSO package based on density functional theory (DFT) [44,45]. The Perdew-Burke-Ernzerhof (PBE) functional of the generalized gradient approximation (GGA) was adopted for electron exchange and correlation interactions [46]. Ultrasoft pseudopotentials were selected to describe the interaction between electrons and ions. A vacuum spacing of 20 Å was used to minimize the interaction between neighboring slabs. The cut-off energy of 680 eV and the Brillouin zone sampling of a 10 × 10 × 1 Monkhorst-Pack k-point grid [47] were applied for all layered structures. The convergence threshold for self-consistency was set as 10 −11 eV. To ensure the accuracy of the calculation, the structural optimization was continued until the residual forces converged to less than 1.4 × 10 −3 eV/Å and the total energy to less than 1.4 × 10 −4 eV. The phonon dispersions and Raman spectra were calculated using Phonon code within density-functional perturbation theory [48]. We used norm-conserving pseudopotentials (NCPPs) within the local density approximation (LDA) with a plane-wave cutoff energy of 816 eV to describe the interaction between electrons and ions. A 16 × 16 × 1 Monkhorst−Pack k-mesh and a 8 × 8 × 1 Monkhorst−Pack q-mesh were applied for the phonon dispersion calculations. For the Raman spectra calculations, a 10 × 10 × 1 Monkhorst−Pack k-mesh was used. The threshold for self-consistency was set at 1.36 × 10 −13 eV.

Results and Discussions
Pristine bulk Al 2 O 3 is trigonal with the space group R3c (Figure 1a). O 2− ions are arranged in a hexagonal close packing and Al 3+ ions are filled into the octahedral voids. The calculated band gap of bulk Al 2 O 3 is 5.83 eV, which is close to the previously calculated Crystals 2023, 13, 1298 3 of 16 value (6.05 eV) [49]. To model the Al 2 O 3 (0001) surface, we consider the two common surface structures shown in Table 1. The thermodynamic stability of different surface structures can be evaluated based on the surface formation energy, which is defined by the following equation: E f = E surface − E bulk , where E surface is the energy of an optimized surface structure and E bulk is the energy of optimized bulk Al 2 O 3 . The calculated surface formation energies of type I and II surfaces are listed in Table 1. The calculated surface formation energy of the type I surface is about 3.78 eV, which is a quarter of that of the type II surface. Therefore, the type I surface structure is more stable than the type II surface. In the following, all calculations are performed based on the type I surface structure.

Results and Discussions
Pristine bulk Al2O3 is trigonal with the space group R3 _ c ( Figure 1a). O 2− ions are arranged in a hexagonal close packing and Al 3+ ions are filled into the octahedral voids. The calculated band gap of bulk Al2O3 is 5.83 eV, which is close to the previously calculated value (6.05 eV) [49]. To model the Al2O3 (0001) surface, we consider the two common surface structures shown in Table 1. The thermodynamic stability of different surface structures can be evaluated based on the surface formation energy, which is defined by the following equation: Ef = Esurface − Ebulk, where Esurface is the energy of an optimized surface structure and Ebulk is the energy of optimized bulk Al2O3. The calculated surface formation energies of type I and II surfaces are listed in Table 1. The calculated surface formation energy of the type I surface is about 3.78 eV, which is a quarter of that of the type II surface. Therefore, the type I surface structure is more stable than the type II surface. In the following, all calculations are performed based on the type I surface structure.

Before Optimization
After Optimization E f (eV) Type I

Results and Discussions
Pristine bulk Al2O3 is trigonal with the space group R3 _ c ( Figure 1a). O 2− ions are arranged in a hexagonal close packing and Al 3+ ions are filled into the octahedral voids. The calculated band gap of bulk Al2O3 is 5.83 eV, which is close to the previously calculated value (6.05 eV) [49]. To model the Al2O3 (0001) surface, we consider the two common surface structures shown in Table 1. The thermodynamic stability of different surface structures can be evaluated based on the surface formation energy, which is defined by the following equation: Ef = Esurface − Ebulk, where Esurface is the energy of an optimized surface structure and Ebulk is the energy of optimized bulk Al2O3. The calculated surface formation energies of type I and II surfaces are listed in Table 1. The calculated surface formation energy of the type I surface is about 3.78 eV, which is a quarter of that of the type II surface. Therefore, the type I surface structure is more stable than the type II surface. In the following, all calculations are performed based on the type I surface structure.

Results and Discussions
Pristine bulk Al2O3 is trigonal with the space group R3 _ c ( Figure 1a). O 2− ions are arranged in a hexagonal close packing and Al 3+ ions are filled into the octahedral voids. The calculated band gap of bulk Al2O3 is 5.83 eV, which is close to the previously calculated value (6.05 eV) [49]. To model the Al2O3 (0001) surface, we consider the two common surface structures shown in Table 1. The thermodynamic stability of different surface structures can be evaluated based on the surface formation energy, which is defined by the following equation: Ef = Esurface − Ebulk, where Esurface is the energy of an optimized surface structure and Ebulk is the energy of optimized bulk Al2O3. The calculated surface formation energies of type I and II surfaces are listed in Table 1. The calculated surface formation energy of the type I surface is about 3.78 eV, which is a quarter of that of the type II surface. Therefore, the type I surface structure is more stable than the type II surface. In the following, all calculations are performed based on the type I surface structure.

3.78
Type II Pristine bulk Al2O3 is trigonal with the space group R3 _ c ( Figure 1a). O 2− ions are arranged in a hexagonal close packing and Al 3+ ions are filled into the octahedral voids. The calculated band gap of bulk Al2O3 is 5.83 eV, which is close to the previously calculated value (6.05 eV) [49]. To model the Al2O3 (0001) surface, we consider the two common surface structures shown in Table 1. The thermodynamic stability of different surface structures can be evaluated based on the surface formation energy, which is defined by the following equation: Ef = Esurface − Ebulk, where Esurface is the energy of an optimized surface structure and Ebulk is the energy of optimized bulk Al2O3. The calculated surface formation energies of type I and II surfaces are listed in Table 1. The calculated surface formation energy of the type I surface is about 3.78 eV, which is a quarter of that of the type II surface. Therefore, the type I surface structure is more stable than the type II surface. In the following, all calculations are performed based on the type I surface structure.

Before Optimization
After Optimization Ef (eV) Pristine bulk Al2O3 is trigonal with the space group R3 _ c ( Figure 1a). O 2− ions are arranged in a hexagonal close packing and Al 3+ ions are filled into the octahedral voids. The calculated band gap of bulk Al2O3 is 5.83 eV, which is close to the previously calculated value (6.05 eV) [49]. To model the Al2O3 (0001) surface, we consider the two common surface structures shown in Table 1. The thermodynamic stability of different surface structures can be evaluated based on the surface formation energy, which is defined by the following equation: Ef = Esurface − Ebulk, where Esurface is the energy of an optimized surface structure and Ebulk is the energy of optimized bulk Al2O3. The calculated surface formation energies of type I and II surfaces are listed in Table 1. The calculated surface formation energy of the type I surface is about 3.78 eV, which is a quarter of that of the type II surface. Therefore, the type I surface structure is more stable than the type II surface. In the following, all calculations are performed based on the type I surface structure.

Before Optimization After Optimization Ef (eV)
Type Ⅰ 3.78 Type Ⅱ 15.12 15.12 The energetically stable structures of the monolayer, bilayer, trilayer, four-layer, and five-layer Al 2 O 3 are shown in Figure 2a-e. There are five atoms (two Al atoms and three O atoms) in the unit cell of monolayer Al 2 O 3 . By analogy, the bilayer, trilayer, four-layer, and five-layer Al 2 O 3 have 10, 15, 20, and 25 atoms, respectively. During the structural optimization, the Al atoms on the upper and lower surfaces gradually move inward, inducing a flat surface structure. The reconstruction of the surface may further enhance the stability of 2D Al 2 O 3 . Figure 2f shows the in-plane lattice constants of few-layer and bulk Al 2 O 3 . The in-plane lattice constant for monolayer Al 2 O 3 is 5.84 Å, which is 21% larger than that of bulk Al 2 O 3 (4.81 Å). The large change in the lattice constants suggests that the monolayer Al 2 O 3 may be unstable. In addition, the lattices of the bilayer, trilayer, four-layer, and five-layer Al 2 O 3 are similar to that of bulk Al 2 O 3 , indicating that bilayer Al 2 O 3 is the thinnest Al 2 O 3 structure that can be fabricated experimentally.
five-layer Al2O3 are shown in Figure 2a-e. There are five atoms (two Al atoms and three O atoms) in the unit cell of monolayer Al2O3. By analogy, the bilayer, trilayer, four-layer, and five-layer Al2O3 have 10, 15, 20, and 25 atoms, respectively. During the structural optimization, the Al atoms on the upper and lower surfaces gradually move inward, inducing a flat surface structure. The reconstruction of the surface may further enhance the stability of 2D Al2O3. Figure 2f shows the in-plane lattice constants of few-layer and bulk Al2O3. The in-plane lattice constant for monolayer Al2O3 is 5.84 Å, which is 21% larger than that of bulk Al2O3 (4.81 Å). The large change in the lattice constants suggests that the monolayer Al2O3 may be unstable. In addition, the lattices of the bilayer, trilayer, four-layer, and five-layer Al2O3 are similar to that of bulk Al2O3, indicating that bilayer Al2O3 is the thinnest Al2O3 structure that can be fabricated experimentally. The dynamic stability of few-layer Al2O3 can be examined from the phonon dispersions. The absence of negative frequencies in Figure 3a,b indicates that the monolayer and bilayer Al2O3 are dynamically stable and can exist as free-standing 2D structures. Although monolayer Al2O3 is dynamically stable, the giant lattice difference between monolayer and bulk Al2O3 indicates the difficulty of synthesis monolayer Al2O3 by top-down strategy. Considering the small lattice constant change and high dynamic stability of bilayer Al2O3, it can be inferred that the synthesis of bilayer Al2O3 is quite possible by topdown or bottom-up method. Because of the large number of atoms in the cell of trilayer, four-layer, and five-layer Al2O3, the phonon calculation for these structures are extremely heavy. Here, the phonon dispersions of trilayer, four-layer, and five-layer Al2O3 are not present. The in-plane lattice constants of trilayer, four-layer, and five-layer Al2O3 are close to that of bilayer Al2O3, and we claim that the trilayer, four-layer, and five-layer Al2O3 are likely synthesized experimentally.            Figure 1b and shows a direct band gap of 5.83 eV. The values of the band gap for monolayer and bilayer Al 2 O 3 are similar (~3.5 eV), and those for the trilayer, four-layer, and five-layer Al 2 O 3 are also similar (~4.7 eV). It is well known that the band gap is always underestimated and lattice constants are overestimated when using GGA calculation due to the lack of Fock exchange. The hybrid functional will improve band gap and lattice constants because of including the exact exchange correction, but will induce a rise in computational costs [50,51]. In order to obtain the more reliable band gaps, we used the HSE06 hybrid functionals to calculate the band structures of monolayer and bilayer Al 2 O 3 , as shown in Figure 5. The band gaps are 5.08 and 4.95 eV for monolayer and bilayer Al 2 O 3 , respectively. The band gap calculated through using HSE06 hybrid functional is about 1.5 eV larger than that through GGA. However, the band gap of bilayer Al 2 O 3 is also slightly smaller than that of monolayer Al 2 O 3 through using HSE06 hybrid functional, which is consistent with the results through GGA calculation. In addition, the band gap of monolayer Al 2 O 3 is direct, and that of bilayer Al 2 O 3 is indirect when using HSE06 functional, which is also consistent with the result obtained via GGA calculation. Therefore, the band gap calculated using the HSE06 hybrid functional is 1.5 eV larger than that through GGA, but the band gap nature and band gap variation on layer thickness are independent on the functional used. The band gaps of few-layer Al 2 O 3 are wider than 3.0 eV, corresponding to the ultraviolet spectral range. Therefore, the few-layer Al 2 O 3 is suitable for ultraviolet optoelectronic applications. The quantum confinement effect for most semiconductors means that the band gap increases as the thickness decreases. However, for Al 2 O 3 , the band gap narrows as the thickness decreases, indicating that the band gap variation with thickness cannot be explained by the quantum confinement effect. From the partial density of the states of on-surface and interior Al atoms for bilayer Al 2 O 3 (Figure 6), we can find that band edge states, which determine the band gap, mainly originate from on-surface Al atoms. Due to the surface states, the band gap of few-layer Al 2 O 3 is smaller than that of bulk Al 2 O 3 (Figure 4f). Because the surface reconstruction of few-layer Al 2 O 3 varies with thickness ( Figure 2a-e), the band gap changes with thickness accordingly. Therefore, we attribute the band gap variation with the thickness to the different giant surface reconstruction of the few-layer Al 2 O 3 . also similar (~4.7 eV). It is well known that the band gap is always underestimated and lattice constants are overestimated when using GGA calculation due to the lack of Fock exchange. The hybrid functional will improve band gap and lattice constants because of including the exact exchange correction, but will induce a rise in computational costs [50,51]. In order to obtain the more reliable band gaps, we used the HSE06 hybrid functionals to calculate the band structures of monolayer and bilayer Al2O3, as shown in Figure  5. The band gaps are 5.08 and 4.95 eV for monolayer and bilayer Al2O3, respectively. The band gap calculated through using HSE06 hybrid functional is about 1.5 eV larger than that through GGA. However, the band gap of bilayer Al2O3 is also slightly smaller than that of monolayer Al2O3 through using HSE06 hybrid functional, which is consistent with the results through GGA calculation. In addition, the band gap of monolayer Al2O3 is direct, and that of bilayer Al2O3 is indirect when using HSE06 functional, which is also consistent with the result obtained via GGA calculation. Therefore, the band gap calculated using the HSE06 hybrid functional is 1.5 eV larger than that through GGA, but the band gap nature and band gap variation on layer thickness are independent on the functional used. The band gaps of few-layer Al2O3 are wider than 3.0 eV, corresponding to the ultraviolet spectral range. Therefore, the few-layer Al2O3 is suitable for ultraviolet optoelectronic applications. The quantum confinement effect for most semiconductors means that the band gap increases as the thickness decreases. However, for Al2O3, the band gap narrows as the thickness decreases, indicating that the band gap variation with thickness cannot be explained by the quantum confinement effect. From the partial density of the states of on-surface and interior Al atoms for bilayer Al2O3 (Figure 6), we can find that band edge states, which determine the band gap, mainly originate from on-surface Al atoms. Due to the surface states, the band gap of few-layer Al2O3 is smaller than that of bulk Al2O3 (Figure 4f). Because the surface reconstruction of few-layer Al2O3 varies with thickness ( Figure  2a-e), the band gap changes with thickness accordingly. Therefore, we attribute the band gap variation with the thickness to the different giant surface reconstruction of the fewlayer Al2O3.  Raman spectrum is an effective method with which to identify different materials and structures. In order to facilitate future experimental efforts on few-layer Al 2 O 3 , the Raman spectra of monolayer and bilayer Al 2 O 3 were calculated (see Figure 7). Monolayer Al 2 O 3 belongs to the space group P62m with 1 f.u. in the primitive cell (five atoms). The zone-center optical phonons can be classified as the following irreducible representations: Γ opt = E + 3E + A 1 + A 1 + A 2 + A 2 , where E , E , and A 1 are Raman active. For monolayer Al 2 O 3 , the peaks are centered at 387, 754, and 1043 cm −1 , corresponding to the A 1 , E (1) , and E (2) modes, respectively. The corresponding vibration for A 1 , E (1) , and E (2) modes are listed in Table 2. In addition, in Figure 7a, the intensity of A 1 at 387 cm −1 is significantly higher than the other two peaks. Therefore, the A 1 mode can be used as a Raman signature of monolayer Al 2 O 3 . Bilayer Al 2 O 3 belongs to the space group P31m with 2 f.u. (10 atoms) in the primitive cell. Therefore, bilayer Al 2 O 3 has 30 vibrational modes. Among these modes, five E g and three A 1g modes are Raman active. For bilayer Al 2 O 3 , there are five dominant modes at 367, 521, 554, 797, and 881 cm −1 (Figure 7b) corresponding to the A 1g (1) , A 1g (2) , E g (1) , A 1g (3) , and E g (2) modes with apparent intensity, respectively. The corresponding vibrations for A 1g (1) , A 1g (2) , E g (1) , A 1g (3) , and E g (2) modes are shown in Table 3. Furthermore, among these peaks, A 1g (3) at 797 cm −1 has the highest intensity and can be considered as the Raman signature of bilayer Al 2 O 3 . Raman spectrum is an effective method with which to identify different materials and structures. In order to facilitate future experimental efforts on few-layer Al2O3, the Raman spectra of monolayer and bilayer Al2O3 were calculated (see Figure 7). Monolayer Al2O3 belongs to the space group P6 � 2m with 1 f.u. in the primitive cell (five atoms). The zone-center optical phonons can be classified as the following irreducible representations: Γopt = E″ + 3E′ + A″1 + A′1 + A″2 + A′2, where E″, E′, and A′1 are Raman active. For monolayer Al2O3, the peaks are centered at 387, 754, and 1043 cm −1 , corresponding to the A′1, E′ (1) , and E′ (2) modes, respectively. The corresponding vibration for A′1, E′ (1) , and E′ (2) modes are listed in Table 2. In addition, in Figure 7a, the intensity of A′1 at 387 cm −1 is significantly higher than the other two peaks. Therefore, the A′1 mode can be used as a Raman signature of monolayer Al2O3. Bilayer Al2O3 belongs to the space group P3 � 1m with 2 f.u. (10 atoms) in the primitive cell. Therefore, bilayer Al2O3 has 30 vibrational modes. Among these modes, five Eg and three A1g modes are Raman active. For bilayer Al2O3, there are five dominant modes at 367, 521, 554, 797, and 881 cm −1 (Figure 7b) corresponding to the A1g (1) , A1g (2) , Eg (1) , A1g (3) , and Eg (2) modes with apparent intensity, respectively. The corresponding vibrations for A1g (1) , A1g (2) , Eg (1) , A1g (3) , and Eg (2) modes are shown in Table 3. Furthermore, among these peaks, A1g (3) at 797 cm −1 has the highest intensity and can be considered as the Raman signature of bilayer Al2O3.      Figure 7. The calculated Raman spectra of (a) monolayer and (b) bilayer Al2O3.  Figure 7. The calculated Raman spectra of (a) monolayer and (b) bilayer Al2O3.  Figure 7. The calculated Raman spectra of (a) monolayer and (b) bilayer Al2O3.

Mode Frequency (cm −1 ) Side View
A1g (1)  367 Based on the above results, which provide evidence of the stability of bilayer Al 2 O 3 , we can assume that the bilayer Ga 2 O 3 and In 2 O 3 are also stable because Al, Ga, and In are in the same group. The absence of negative frequencies in phonon dispersions (Figure 8a

Mode Frequency (cm −1 ) Side View
A1g (1)  367 A1g (2)  Based on the above results, which provide evidence of the stability of bilayer Al2O3, we can assume that the bilayer Ga2O3 and In2O3 are also stable because Al, Ga, and In are in the same group. The absence of negative frequencies in phonon dispersions (Figure  8a,c) demonstrates the stability of bilayer Ga2O3 and In2O3. Bilayer Ga2O3 and In2O3 are indirect semiconductors (Figure 8b,d) with band gaps of 2.01 and 1.17 eV, respectively, Based on the above results, which provide evidence of the stability of bilayer Al2O3, we can assume that the bilayer Ga2O3 and In2O3 are also stable because Al, Ga, and In are in the same group. The absence of negative frequencies in phonon dispersions (Figure  8a,c) demonstrates the stability of bilayer Ga2O3 and In2O3. Bilayer Ga2O3 and In2O3 are indirect semiconductors (Figure 8b,d)  Based on the above results, which provide evidence of the stability of bilayer Al2O3, we can assume that the bilayer Ga2O3 and In2O3 are also stable because Al, Ga, and In are in the same group. The absence of negative frequencies in phonon dispersions ( Figure  8a,c) demonstrates the stability of bilayer Ga2O3 and In2O3. Bilayer Ga2O3 and In2O3 are indirect semiconductors (Figure 8b,d) with band gaps of 2.01 and 1.17 eV, respectively, indirect semiconductors (Figure 8b,d) with band gaps of 2.01 and 1.17 eV, respectively, which are smaller than that of bilayer Al2O3. The CBM of either bilayer Ga2O3 or In2O3 is located at the Γ point and the VBM at a general point along the K−Γ line. To obtain a variation of band structure on layer thickness, Figures 9 and 10 show the band structures of few-layer Ga2O3 and In2O3, which show a similar band gap variation to that of few-layer Al2O3.     It is also necessary to provide the Raman signature for bilayer Ga2O3 and In2O3. A shown in Figure 11, the Raman spectra of bilayer Ga2O3 and In2O3 were calculated, whic It is also necessary to provide the Raman signature for bilayer Ga 2 O 3 and In 2 O 3 . As shown in Figure 11, the Raman spectra of bilayer Ga 2 O 3 and In 2 O 3 were calculated, which were similar to that of bilayer Al 2 O 3 . Bilayer Ga 2 O 3 and In 2 O 3 belong to the space group P31m with 2 f.u. (10 atoms) in the primitive cell. For bilayer Ga 2 O 3 , the peaks are centered at 196, 320, 410, 664, and 689 cm −1 corresponding to the A 1g (1) , E g (1) , A 1g (2) , E g (2) , and A 1g (3) modes, respectively (Figure 11a). The corresponding vibration for A 1g (1) , E g (1) , A 1g (2) , E g (2) , and A 1g (3) modes are listed in Table 4. The peak centered at 689 cm −1 (A 1g (3) ) with strong intensity should be considered as the Raman signature of bilayer Ga 2 O 3 . As for bilayer In 2 O 3 , Figure 11b shows that there are four major peaks and two minor peaks in the Raman spectra. Six peaks centered at 133, 269, 333, 357, 580, and 607 cm −1 correspond to the A 1g (1) , E g (1) , A 1g (2) , E g (2) , E g (3) , and A 1g (3) modes, respectively. All the Raman active vibrational modes for bilayer In 2 O 3 can be found in Table 5. Similarly, the Raman signature of bilayer In 2 O 3 should be the A 1g (3) peak, which has a pronounced intensity. were similar to that of bilayer Al2O3. Bilayer Ga2O3 and In2O3 belong to the space group P3 � 1m with 2 f.u. (10 atoms) in the primitive cell. For bilayer Ga2O3, the peaks are centered at 196, 320, 410, 664, and 689 cm −1 corresponding to the A1g (1) , Eg (1) , A1g (2) , Eg (2) , and A1g (3) modes, respectively (Figure 11a). The corresponding vibration for A1g (1) , Eg (1) , A1g (2) , Eg (2) , and A1g (3) modes are listed in Table 4. The peak centered at 689 cm −1 (A′1g (3) ) with strong intensity should be considered as the Raman signature of bilayer Ga2O3. As for bilayer In2O3, Figure 11b shows that there are four major peaks and two minor peaks in the Raman spectra. Six peaks centered at 133, 269, 333, 357, 580, and 607 cm −1 correspond to the A1g (1) , Eg (1) , A1g (2) , Eg (2) , Eg (3) , and A1g (3) modes, respectively. All the Raman active vibrational modes for bilayer In2O3 can be found in Table 5. Similarly, the Raman signature of bilayer In2O3 should be the A1g (3) peak, which has a pronounced intensity.

Conclusions
In summary, using first-principles calculations based on DFT, we have investigated the structural stability and electronic properties of 2D boron group oxides. The thinnest stable Al2O3 structure is a bilayer with an indirect band gap of 3.41 eV. Interestingly, we find that the band gap of Al2O3 decreases with decreasing layer thickness, which is opposite to the trend seen in vdW 2D materials, such as MoS2. We attribute this to the giant surface reconstruction of Al2O3. In addition, the phonon dispersion curves and band structures show that both bilayer Ga2O3 and In2O3 are also stable, with band structures similar to that of bilayer Al2O3. From bilayer Al2O3 to bilayer Ga2O3 and In2O3, the band gap decreases with the increasing atomic radius. It is expected to synthesize few layer boron group oxides by top-down strategy, such as liquid exfoliation method. This work provides theoretical support for the discovery of non-vdW 2D boron group oxides. Because of their large band gap, non-vdW 2D boron group oxides hold potential applications in ultraviolet optoelectronics and high-power electronics.

Conclusions
In summary, using first-principles calculations based on DFT, we have investigated the structural stability and electronic properties of 2D boron group oxides. The thinnest stable Al2O3 structure is a bilayer with an indirect band gap of 3.41 eV. Interestingly, we find that the band gap of Al2O3 decreases with decreasing layer thickness, which is opposite to the trend seen in vdW 2D materials, such as MoS2. We attribute this to the giant surface reconstruction of Al2O3. In addition, the phonon dispersion curves and band structures show that both bilayer Ga2O3 and In2O3 are also stable, with band structures similar to that of bilayer Al2O3. From bilayer Al2O3 to bilayer Ga2O3 and In2O3, the band gap decreases with the increasing atomic radius. It is expected to synthesize few layer boron group oxides by top-down strategy, such as liquid exfoliation method. This work provides theoretical support for the discovery of non-vdW 2D boron group oxides. Because of their large band gap, non-vdW 2D boron group oxides hold potential applications in ultraviolet optoelectronics and high-power electronics. Eg (2) 357 Eg (3)  580 A1g (3) 607

Conclusions
In summary, using first-principles calculations based on DFT, we have investigated the structural stability and electronic properties of 2D boron group oxides. The thinnest stable Al2O3 structure is a bilayer with an indirect band gap of 3.41 eV. Interestingly, we find that the band gap of Al2O3 decreases with decreasing layer thickness, which is opposite to the trend seen in vdW 2D materials, such as MoS2. We attribute this to the giant surface reconstruction of Al2O3. In addition, the phonon dispersion curves and band structures show that both bilayer Ga2O3 and In2O3 are also stable, with band structures similar to that of bilayer Al2O3. From bilayer Al2O3 to bilayer Ga2O3 and In2O3, the band gap decreases with the increasing atomic radius. It is expected to synthesize few layer boron group oxides by top-down strategy, such as liquid exfoliation method. This work provides theoretical support for the discovery of non-vdW 2D boron group oxides. Because of their large band gap, non-vdW 2D boron group oxides hold potential applications in ultraviolet optoelectronics and high-power electronics.

Conclusions
In summary, using first-principles calculations based on DFT, we have investigated the structural stability and electronic properties of 2D boron group oxides. The thinnest stable Al 2 O 3 structure is a bilayer with an indirect band gap of 3.41 eV. Interestingly, we find that the band gap of Al 2 O 3 decreases with decreasing layer thickness, which is opposite to the trend seen in vdW 2D materials, such as MoS 2 . We attribute this to the giant surface reconstruction of Al 2 O 3 . In addition, the phonon dispersion curves and band structures show that both bilayer Ga 2 O 3 and In 2 O 3 are also stable, with band structures similar to that of bilayer Al 2 O 3 . From bilayer Al 2 O 3 to bilayer Ga 2 O 3 and In 2 O 3 , the band gap decreases with the increasing atomic radius. It is expected to synthesize few layer boron group oxides by top-down strategy, such as liquid exfoliation method. This work provides theoretical support for the discovery of non-vdW 2D boron group oxides. Because of their large band gap, non-vdW 2D boron group oxides hold potential applications in ultraviolet optoelectronics and high-power electronics.