High Electron Mobility in Si-Doped Two-Dimensional β-Ga2O3 Tuned Using Biaxial Strain

Two-dimensional (2D) semiconductors have attracted much attention regarding their use in flexible electronic and optoelectronic devices, but the inherent poor electron mobility in conventional 2D materials severely restricts their applications. Using first-principles calculations in conjunction with Boltzmann transport theory, we systematically investigated the Si-doped 2D β-Ga2O3 structure mediated by biaxial strain, where the structural stabilities were determined by formation energy, phonon spectrum, and ab initio molecular dynamic simulation. Initially, the band gap values of Si-doped 2D β-Ga2O3 increased slightly, followed by a rapid decrease from 2.46 eV to 1.38 eV accompanied by strain modulations from −8% compressive to +8% tensile, which can be ascribed to the bigger energy elevation of the σ* anti-bonding in the conduction band minimum than that of the π bonding in the valence band maximum. Additionally, band structure calculations resolved a direct-to-indirect transition under the tensile strains. Furthermore, a significantly high electron mobility up to 4911.18 cm2 V−1 s−1 was discovered in Si-doped 2D β-Ga2O3 as the biaxial tensile strain approached 8%, which originated mainly from the decreased quantum confinement effect on the surface. The electrical conductivity was elevated with the increase in tensile strain and the enhancement of temperature from 300 K to 800 K. Our studies demonstrate the tunable electron mobilities and band structures of Si-doped 2D β-Ga2O3 using biaxial strain and shed light on its great potential in nanoscale electronics.


Introduction
Bulk β-gallium oxide (β-Ga 2 O 3 ) possesses an ultrawide band gap (4.8 eV); however, its low mobility significantly limits its applications [1][2][3].Doping engineering is one effective method that is used to tune the conductivity of wide-band gap semiconductors such as β-Ga 2 O 3 [4][5][6][7][8].Among plentiful dopants, the Si dopant is peculiar since it is not only one of the unintentionally introduced impurities during synthesis and device fabrication but also plays a vital role in modifying electron mobility, as reported in three-dimensional (3D) β-Ga 2 O 3 [9].For Si-doped 3D β-Ga 2 O 3 , electron mobilities of 0.1-196 cm 2 V −1 s −1 have been reported.Baldini et al. studied the electron mobility of Si-doped homoepitaxial β-Ga 2 O 3 films ranging from ~50 cm 2 V −1 s −1 for a doping concentration of n = 8 × 10 19 cm −3 to ~130 cm 2 V −1 s −1 for a doping concentration of n = 1 × 10 17 cm −3 [10].Hernandez et al. illustrated that Si-doped homoepitaxial β-Ga 2 O 3 films grown by chemical vapor deposition were characterized by electron mobilities in the range of 0.59-30.59cm 2 V −1 s −1 for Si concentrations ranging from 89.2 to 17.8 nmol/min [11].Recently, Bhattacharyya et al. reported an electron mobility of 196 cm 2 V −1 s −1 (n = 2.3 × 10 16 cm −3 ) measured in a Si-doped β-Ga 2 O 3 film grown on Fe-doped β-Ga 2 O 3 substrates at 810 • C, which is the highest electron mobility value ever reported in β-Ga 2 O 3 epitaxial films, suggesting the great potential of Si dopants in tuning the electron mobility of 3D β-Ga 2 O 3 [12].For these reported heteroepitaxial films, Zhang et al. indicated that the electron mobility was 0.1, 0.1, 0.5, and 5.5 cm 2 V −1 s −1 for 0, 1.1, 4.1, and 10.4% Si-doped β-Ga 2 O 3 thin films on sapphire substrates prepared using pulsed laser deposition (PLD) at 500 • C, respectively [13].Khartsev et al. prepared high-quality Si-doped β-Ga 2 O 3 films on sapphire substrates using PLD and reported an electron mobility of about 2.9 cm 2 V −1 s −1 for n = 2.5 × 10 19 cm −3 [14].Wong et al. reported electron mobilities of 90-100 cm 2 V −1 s −1 at carrier concentrations in the low to mid 10 17 cm −3 range in β-Ga 2 O 3 (010) using Si-ion implantation doping and molecular beam epitaxy [15].Two-dimensional (2D) semiconductors have attracted much attention and are widely used in electronic and optoelectronic devices [16][17][18][19][20][21][22].In the past few years, research interest has been mostly focused on graphene [23], transition metal dichalcogenides (TMDs) [24], and other materials in order to design new nanoelectronic devices, including solar cells and field effect transistors.Although the 2D materials mentioned above usually possess outstanding properties, some of their disadvantages, such as the zero band gap of graphene and the low carrier mobilities of TMDs, make it temporarily difficult to adopt multifunctional applications.Therefore, the discovery and engineering of novel 2D materials with moderate band gaps and high carrier mobility are worthy of research focus.Lately, the fabrication of 2D β-Ga 2 O 3 was reported using mechanical exfoliation, molecular beam epitaxy, and chemical synthesis methods, and its benefits stemmed from its fragile Ga-O bonds along the β-Ga 2 O 3 [100] direction [25,26].Owing to its lower-dimensional nature and its high surface-to-volume ratio, 2D β-Ga 2 O 3 exhibits better physical properties and enormous potential for novel applications.For instance, a monolayer of β-Ga 2 O 3 passivated by H possessed electron mobilities as high as 2685 cm 2 V −1 s −1 and 156 cm 2 V −1 s −1 along the b and c directions, respectively, based on theoretical studies [27].The features of 2D β-Ga 2 O 3 -based solar-blind photodetectors were boosted to nearly three times higher than that of bulk β-Ga 2 O 3 [28][29][30].Therefore, 2D β-Ga 2 O 3 has great potential for innovative nanoscale applications because it combines the benefits of outstanding electron mobility with affordable production techniques.
However, both theoretical and experimental efforts devoted to modifying the carrier mobility in Si-doped 2D β-Ga 2 O 3 materials are lacking.Only limited reports showed the preferred Si atom occupation of the tetrahedral Ga site, which resulted in donor doping behaviors in Si-doped 2D β-Ga 2 O 3 [31].Especially considering the different local crystal structures of 2D and 3D β-Ga 2 O 3 , the results presented regarding 3D β-Ga 2 O 3 are difficult to directly apply to 2D systems, which motivated us to comprehensively analyze the effects of Si dopants on the electronic and transport properties of 2D β-Ga 2 O 3 here.Additionally, previous work showed that the 2D β-Ga 2 O 3 is characterized by low elastic constants; therefore, systematic strain-modulated band structure investigations of 2D β-Ga 2 O 3 shed light on its potential for use in the fabrication of flexible electronic devices [32].Inspired by these concepts, we carried out first-principles calculations using the Perdew-Burke-Ernzerhof (PBE) functional and the Boltzmann transport theory to explore the strainmodulated transport properties of Si-doped 2D β-Ga 2 O 3 .Our work demonstrates that the electron mobilities and band structures of Si-doped 2D β-Ga 2 O 3 are tunable using biaxial strain, highlighting its great potential for use in nanoscale electronics.

Computational Details
Vienna ab initio Simulation Package (VASP) Standard Edition 6.4 was used for firstprinciples calculations [33,34].The generalized gradient approximation (GGA) of the PBE functional was adopted [35,36].In this study, a 3 × 2 × 1 2D β-Ga 2 O 3 supercell including 60 atoms was modeled.The criteria of energy cutoff, energy convergence, and residual forces were 450 eV, 1 × 10 −6 eV/atom, and 0.01 eV/Å, respectively.The k-point grid was set at a resolution of 0.02 × 2π Å −1 .The vacuum thickness was set at 15 Å along the c-axis direction to prevent the spurious interactions between layers.Bulk β-Ga 2 O 3 contains two different Ga coordinations, i.e., octahedral GaI and tetrahedral GaII.Along the β-Ga 2 O 3 [100] direction, bulk β-Ga 2 O 3 can be cleaved into 2D β-Ga 2 O 3 with a low cleavage energy [37].After cleaving from 3D β-Ga 2 O 3 , as shown in Figure 1a, the GaII maintained four coordinations, whereas the GaI changed from six coordinations in 3D β-Ga 2 O 3 to five coordinations in 2D β-Ga 2 O 3 .Thus, one 2D β-Ga 2 O 3 is endowed with two inequivalent Ga sites, namely the square pyramidal GaI and the tetrahedral GaII.The ionic radius of Si 4+ (0.040 nm) dopant is slightly lower than that of host Ga 3+ (0.062 nm), which makes it difficult to be in the interstitial position and easier to substitute Ga atoms.Therefore researchers have focused on its substitution style instead of interstitial doping in 3D and 2D β-Ga 2 O 3 systems theoretically [31,38,39] and experimentally [12] in the literature.In this study, one Si impurity was substituted at the GaI and GaII cation sites labeled as Si GaI and Si GaII , respectively, as illustrated in Figure 1a.Furthermore, in this work, the x and y directions correspond to the crystalline a and b directions, respectively.
direction to prevent the spurious interactions between layers.Bulk β-Ga2O3 contains two different Ga coordinations, i.e., octahedral GaI and tetrahedral GaII.Along the β-Ga2O3 [100] direction, bulk β-Ga2O3 can be cleaved into 2D β-Ga2O3 with a low cleavage energy [37].After cleaving from 3D β-Ga2O3, as shown in Figure 1a, the GaII maintained four coordinations, whereas the GaI changed from six coordinations in 3D β-Ga2O3 to five coordinations in 2D β-Ga2O3.Thus, one 2D β-Ga2O3 is endowed with two inequivalent Ga sites, namely the square pyramidal GaI and the tetrahedral GaII.The ionic radius of Si 4+ (0.040 nm) dopant is slightly lower than that of host Ga 3+ (0.062 nm), which makes it difficult to be in the interstitial position and easier to substitute Ga atoms.Therefore researchers have focused on its substitution style instead of interstitial doping in 3D and 2D β-Ga2O3 systems theoretically [31,38,39] and experimentally [12] in the literature.In this study, one Si impurity was substituted at the GaI and GaII cation sites labeled as SiGaI and SiGaII, respectively, as illustrated in Figure 1a.Furthermore, in this work, the x and y directions correspond to the crystalline a and b directions, respectively.

Formation Energies and Carrier Mobility Calculations
The formation energy f E of Si-doped 2D β-Ga2O3 is defined as [43,44] where defect The dynamical stability was evaluated by phonon dispersion based on density functional perturbation theory (DFPT) within a 6 × 2 × 1 supercell [40].The structural convergence criteria of energy and force were 10 −8 eV and 10 −7 eV/Å, respectively.To gain thermodynamical stability, ab initio molecular dynamic (AIMD) simulation was adopted within a 6 × 2 × 1 supercell.The electron mobility and relaxation time were determined by the deformation potential (DP) theory [41].The Boltzmann theory with the BoltzTraP2 code was utilized to determine the electrical conductivity [42].

Formation Energies and Carrier Mobility Calculations
The formation energy E f of Si-doped 2D β-Ga 2 O 3 is defined as [43,44] where E defect and E Ga 2 O 3 , respectively, represent the energy of the Si-doped and pure 2D β-Ga 2 O 3 configuration.n i demonstrates the quantity of i atom added (n i < 0) or extracted (n i > 0) from the pure β-Ga 2 O 3 system.The chemical potential of µ Si and µ O is gained from the stable bulk Si and O 2 , respectively.The related chemical potential should meet the boundary conditions and fall into the O-rich and Ga-rich cases in terms of the growth conditions, as stated in our previous works [5,43].
Based on the DP theory, the carrier mobility µ is calculated as follows [45]: where k B , h, e, and T are the Boltzmann constant, Planck constant, electric charge, and the temperature (300 K), respectively.The electron mobility µ and relaxation time τ are determined by the deformation potential (DP) theory, where three inconstant parameters are necessary, i.e., the deformation potential constant E 1 , elastic stiffness C 2D , and the electron effective mass m e *.Deformation potential constant E 1 is calculated by E 1 = ∂E edge /∂(∆a/a 0 ), which denotes the shift of the band edge accompanied by uniaxial strain.Here, E edge illustrates the shifted energy of the conduction band minimum (CBM), ∆a/a 0 is the variation in the lattice constant along a certain strained direction, and elastic stiffness C 2D is calculated by Here, E is the total energy of the strained optimized supercell, and S 0 is the surface area of the strained structure.The electron effective mass m e * is calculated based on the equation as m * e = ℏ 2 /(∂ 2 E/∂k 2 ) by fitting the band curves close to the CBM, and k is the electron wave vector.The CBM and valence band maximum (VBM) values are obtained from the band structures of Si-doped 2D β-Ga 2 O 3 .The relaxation time is estimated by τ = µm * e /e.

Structural Stability
Pure 2D β-Ga 2 O 3 possesses the lattice constants of a = 2.97 Å, b = 5.74 Å, which are in accordance with those of previous works [46,47].The calculated lattice constants for Si GaI and Si GaII structures are a = 2.98 Å, b = 5.77 Å and a = 2.99 Å, b = 5.76 Å, respectively.Only a slight structural difference was observed due to the small alteration in local structures and the similar ionic radii between the host Ga and the Si dopant.The formation energies for Si GaI and Si GaII structures are −5.41 and −6.27 eV, respectively, under O-rich conditions, which demonstrates that the Si dopant preferentially incorporates on the tetrahedrally coordinated GaII site.This is consistent with the result of Ref. [31].Moreover, Si GaI and Si GaII structures possess the formation energies of −0.98 and −1.84 eV under a Ga-rich environment, respectively, illustrating that the foreign Si atom also prefers to occupy the GaII site herein.The higher formation energies also suggest the difficulty of introducing Si dopant in a Ga-rich atmosphere.In terms of the lower formation energy of the Si GaII configuration compared with that of Si GaI , we merely studied the Si GaII case under O-rich conditions afterwards.The phonon dispersion spectrum and AIMD simulation of the Si GaII structure are employed in Figure S1a,b (supporting materials), indicating the dynamical stability and thermodynamical stability of the Si GaII system, respectively.

Strain-Engineered Band Structures
Figure 1b shows the band structure of perfect 2D β-Ga 2 O 3 , where the CBM is situated at the G point, whereas the VBM is positioned between the G and X points, demonstrating an indirect band gap semiconductor character.The calculated band gap of 2D β-Ga 2 O 3 is 2.30 eV, which agrees well with previous works [45,47], but it is lower than the value of 3D β-Ga 2 O 3 [48,49].As shown in Figure 2, after substituting the tetrahedrally coordinated GaII with one Si atom, the band gap was slightly reduced to 2.21 eV.Moreover, the shift of VBM from the non-G point (between the G and X points) to the G point resulted in the direct band gap nature.These observations are consistent with the results in ref. [31].Additionally, the orbital-projected band structures in Figure 2a-c illustrate that the majority of the VBM of Si GaI is composed of O-2p orbitals, whilst the CBM is mainly contributed by Ga-4s and small quantities of O-2s and Si-3s orbitals.
The variations in band edges relative to the vacuum level and the band gaps of Si GaII versus biaxial strain are shown in Figure 3a.The band gaps of Si GaII increased slightly at first followed by a rapid decrease when the strain was applied from −8% compressive to +8% tensile cases.We note that this strain interval is of research focus in 2D β-Ga 2 O 3 systems theoretically, which can also be induced by the lattice mismatch, ideally using a different substrate.The orbital-projected band structures of Si GaII in Figure 2 can be used to explore the evolution behaviors of band gaps caused by biaxial strain.The σ* anti-bonding states that predominated in the CBM of Si GaII are formed by the exceptional Ga-s orbitals, tiny O-s, and Si-s orbitals; the π bonding states near VBM of Si GaII are mainly occupied by O-p y orbitals.In general, lengthening the π bonding and σ* anti-bonding (from compressive to tensile strains) gives rise to the energy increases in CBM and VBM, as shown in Figure 3a [50].The changes in energy levels can be attributed to the introduction of the Si dopant, whose orbital energy levels are hybridized with the host Ga-S orbitals, resulting in band gap changes.Consequently, the bigger the energy elevation of the VBM relative to the CBM, the smaller the band gap.The variations in band edges relative to the vacuum level and the band gaps of SiGaII versus biaxial strain are shown in Figure 3a.The band gaps of SiGaII increased slightly at first followed by a rapid decrease when the strain was applied from −8% compressive to +8% tensile cases.We note that this strain interval is of research focus in 2D β-Ga2O3 systems theoretically, which can also be induced by the lattice mismatch, ideally using a different substrate.The orbital-projected band structures of SiGaII in Figure 2 can be used to explore the evolution behaviors of band gaps caused by biaxial strain.The σ* antibonding states that predominated in the CBM of SiGaII are formed by the exceptional Ga-s orbitals, tiny O-s, and Si-s orbitals; the π bonding states near VBM of SiGaII are mainly occupied by O-py orbitals.In general, lengthening the π bonding and σ* anti-bonding (from compressive to tensile strains) gives rise to the energy increases in CBM and VBM, as shown in Figure 3a [50].The changes in energy levels can be attributed to the introduction of the Si dopant, whose orbital energy levels are hybridized with the host Ga-S orbitals, resulting in band gap changes.Consequently, the bigger the energy elevation of the VBM relative to the CBM, the smaller the band gap.
When applying biaxial tensile strain, the VBM moved from the G point to the non-G point, which resulted in a direct-to-indirect band gap transition.According to our findings, the location of VBM is therefore more susceptible to biaxial tensile strain than to biaxial compressive strain.Moreover, the compressive strain is beneficial for maintaining a direct band gap in the SiGaII structure.When biaxial strain was applied, the band gaps of SiGaII were tunable from 2.46 eV (−8% compressive strain) to 1.38 eV (+8% tensile strain).The energy differences between the direct and indirect band gap with respect to the biaxial tensile strains are shown in Figure 3b.One can notice that the energy difference was increased from 8 to 22 meV as the tensile strains increased.

Strain-Engineered Transport Properties
The fluctuations of electronic structures with different strains were further employed to determine the carrier mobility (μ) and effective masses (m*).The electron effective masses (me*) along the G-X and G-Y directions are labeled mex* and mey*, respectively.The calculated mex* and mey* of perfect 2D β-Ga2O3 are 0.29 m0 and 0.27 m0, respectively, which is in good agreement with the results of 0.31 m0 and 0.29 m0 in Ref. [51], as well as 0.36 m0 and 0.36 m0 (isotropy) in Ref. [52].For unstrained SiGaII, the calculated mex* and mey* are 0.30 m0 and 0.28 m0, respectively, as shown in Figure 4a.The detailed variations in mex* and mey* under biaxial strains are exhibited in Figure 4a.It can be shown that the When applying biaxial tensile strain, the VBM moved from the G point to the non-G point, which resulted in a direct-to-indirect band gap transition.According to our findings, the location of VBM is therefore more susceptible to biaxial tensile strain than to biaxial compressive strain.Moreover, the compressive strain is beneficial for maintaining a direct band gap in the Si GaII structure.When biaxial strain was applied, the band gaps of Si GaII were tunable from 2.46 eV (−8% compressive strain) to 1.38 eV (+8% tensile strain).The energy differences between the direct and indirect band gap with respect to the biaxial tensile strains are shown in Figure 3b.One can notice that the energy difference was increased from 8 to 22 meV as the tensile strains increased.

Strain-Engineered Transport Properties
The fluctuations of electronic structures with different strains were further employed to determine the carrier mobility (µ) and effective masses (m*).The electron effective masses (m e *) along the G-X and G-Y directions are labeled m ex * and m ey *, respectively.The calculated m ex * and m ey * of perfect 2D β-Ga 2 O 3 are 0.29 m 0 and 0.27 m 0 , respectively, which is in good agreement with the results of 0.31 m 0 and 0.29 m 0 in Ref. [51], as well as 0.36 m 0 and 0.36 m 0 (isotropy) in Ref. [52].For unstrained Si GaII , the calculated m ex * and m ey * are 0.30 m 0 and 0.28 m 0 , respectively, as shown in Figure 4a.The detailed variations in m ex * and m ey * under biaxial strains are exhibited in Figure 4a.It can be shown that the m ex * and m ey * significantly decreased from biaxial compressive to tensile strains.The small m e * generally implied a higher electron mobility (µ e ).The m ey * decreased faster than m ex * with the transition of biaxial strains from compressive to tensile.Moreover, the anisotropy of the m e * was also enhanced when applying the biaxial tensile strains.The electron mobility was calculated based on the DP theory, and the changes in electron mobility are mainly dependent on the variations in me*, E1, and C2D. Figure 4b illustrates the μe of 2D SiGaII versus the biaxial strains at 300 K, where the μex increased significantly and the μey decreased slightly first and rose from compressive to tensile strains.Therefore, when the biaxial tensile strain was higher than 4%, it had a higher variation in μex than μey, which can be associated with the smaller E1x of CBM, as depicted in Figure S2b.When the biaxial tensile strain was lower than 4% or under compressive strain, the y direction possessed a higher μe.The highest μex and μey were 4911.183 and Different strains ranging from −4% to 4% were employed to calculate the elastic constants (C 2D ). Figure S2a [55] in 3D β-Ga 2 O 3 , respectively.This demonstrates that 2D Si GaII is softer and more susceptible to deformation than bulk β-Ga 2 O 3 material and may have greater potential for applications in flexible electronics.Figure S2b in the supporting materials denotes the calculated deformation potential E 1 of the 2D Si GaII structure.The E 1x and E 1y of the unstrained 2D Si GaII structure are 4.11 and 3.79 eV, respectively.Accompanying the transition of biaxial stains from compressive to tensile, the E 1x of Si GaI increased initially and subsequently decreased, while the E 1y increased monotonously.Figure S3 (Supporting Materials) shows the calculated details of E 1 .
The electron mobility was calculated based on the DP theory, and the changes in electron mobility are mainly dependent on the variations in m e *, E 1 , and C 2D .Figure 4b illustrates the µ e of 2D Si GaII versus the biaxial strains at 300 K, where the µ ex increased significantly and the µ ey decreased slightly first and rose from compressive to tensile strains.Therefore, when the biaxial tensile strain was higher than 4%, it had a higher variation in µ ex than µ ey , which can be associated with the smaller E 1x of CBM, as depicted in Figure S2b.When the biaxial tensile strain was lower than 4% or under compressive strain, the y direction possessed a higher µ e .The highest µ ex and µ ey were 4911.183 and 3434.44 cm 2 V −1 s −1 as the tensile strain reached 8%, which was far greater than those reported in other doped 2D β-Ga 2 O 3 , i.e., H dopant with, respectively, 2684.93 cm 2 V −1 s −1 and 156.25 cm 2 V −1 s −1 along the x and y directions, F of ~300 cm 2 V −1 s −1 , Cl and H co-dopant of ~100 cm 2 V −1 s −1 [56], as well as the H and F co-dopant in our previous work with 4863.05 cm 2 V −1 s −1 under +6% uniaxial tensile strain along the x direction and 2175.37 cm 2 V −1 s −1 under +4% uniaxial tensile strain along the y direction [43].Nevertheless, the µ e tendency was not saturated, which demonstrated that a higher µ e was expected under a larger biaxial strain above 8%.It is noteworthy that, despite the high compressive strain (up to −8%), the µ e remained greater than 1300 cm 2 V −1 s −1 , which surpassed the µ e in other monolayer materials, such as α-In 2 Se 3 (~1000 cm 2 V −1 s −1 ) [57], MoS 2 (~200 cm 2 V −1 s −1 ) [58], and GaN (~300 cm 2 V −1 s −1 ) [59].Thus, in terms of its exceptional µ e , the 2D Si-doped β-Ga 2 O 3 is highly promising for applications in nanoscale optoelectronic devices.
The spatial charge distributions of the unstrained Si GaII structure, as illustrated by the yellow regions, was employed to explain the changed mechanism of electron mobility, as shown in Figure 4c.The wave-function of the CBM is largely embedded in the surface along the a direction, as marked by the blue square box.This means that the production of high µ e is inhibited by the strong quantum confinement effect on the surface.When the biaxial tensile strains were boosted from +4% to +8%, as depicted in Figures 4d and 4e, respectively, the quantum confinement effect became weaker, giving rise to an enhancement of µ e .
We note that biaxial strains can be commonly induced by the lattice mismatch between the substrate and the film according to the following equation: mismatch = (a film − a substrate )/a substrate .Taking different substrates such as sapphire, SiC, Si(111), GaN, and AlN with the same hexagonal symmetry into account, we can calculate that the ideal strains induced by sapphire, SiC, Si(111), GaN, and AlN along the direction of lattice a are 8.85%, −2.95%, −4.63%, −6.24%, and −3.89%, respectively, and along rhe direction of lattice b, they are 4.84%, −6.52%, −8.13%, −9.65%, and −7.43%, respectively.We further note that for β-Ga 2 O 3 on sapphire substrate, the 30 • domain structure was considered from the crystallographic point of view [60].Our calculated results predict that the highest electron mobility is expected in Si-doped β-Ga 2 O 3 deposited on the sapphire.
Figure S4 (supporting materials) shows the relaxation time τ of the Si GaII structure, which possesses the same tendency as µ e .The electrical conductivity (σ) of the Si GaII structure is acquired base on the τ and BoltzTraP2 code [42,61].Figures 5a and 5b show the σ with respect to n-type carrier concentrations at the temperature of 300 K with varying strains along the x and y directions, respectively.Along the x direction, when the doping concentrations varied from 7 × 10 11 to 6 × 10 12 cm −2 at 300 K, the electron mobilities of unstrained and +4%, +8%, −4%, and −8% strained Si GaII ranged from 5.69 × 10 6 to 4.97 × 10 6 cm 2 V −1 s −1 , 9.76 × 10 6 to 9.29 × 10 6 cm 2 V −1 s −1 , 1.10 × 10 7 to 9.83 × 10 6 cm 2 V −1 s −1 , 4.78 × 10 6 to 4.05 × 10 6 cm 2 V −1 s −1 , and 4.66 × 10 6 to 4.01 × 10 6 cm 2 V −1 s −1 , respectively.Therefore, the σ was increased when applying the tensile strains, and declined when applying the compressive strains, which is in accordance with the changing trends of µ ex in Figure 4b.However, different from the trends of µ ey in Figure 4b, the σ was increased when applying the tensile or compressive strain along the y direction.Moreover, the 2D Si GaII with +4% tensile strain possesses the highest σ along the y direction.Motivated by the outstanding µ and σ obtained in the +8% strained Si GaII , the σ values with respect to the different temperatures were considered as well.The fluctuations in σ at different carrier concentrations for +8% strained Si GaII along the x direction are depicted in Figure 5c, which first boosted quickly and then remained essentially steady from 300 K to 800 K. Figure 5d shows the σ along the y direction, which exhibits the same trends as those in Figure 5c.

Conclusions
Using first-principles methods in conjunction with deformation potential (DP) theory and BoltzTraP2 code, we systematically investigated the Si-doped 2D β-Ga2O3 structure mediated by biaxial strain.The structural stabilities of Si-doped 2D β-Ga2O3 were determined by formation energies, phonon spectrum, as well as AIMD simulations.The band gap values of Si-doped 2D β-Ga2O3 initially increased slightly, followed by a rapid

Figure 1 .
Figure 1.(a) The model of Si-doped 2D β-Ga2O3 structure.The O and Ga atoms are represented by the red and green spheres, respectively.The replaced GaI and GaII doping positions with a Si atom are indicated by the blue and magenta colors, respectively.(b) Band structure of undoped 2D β-Ga2O3 unit cell calculated by PBE functional.In this work, G (0, 0, 0), X (0.5, 0, 0), S (0.5, 0.5, 0), and Y (0, 0.5, 0) are set as the high symmetry points, respectively.

Figure 1 .
Figure 1.(a) The model of Si-doped 2D β-Ga 2 O 3 structure.The O and Ga atoms are represented by the red and green spheres, respectively.The replaced GaI and GaII doping positions with a Si atom are indicated by the blue and magenta colors, respectively.(b) Band structure of undoped 2D β-Ga 2 O 3 unit cell calculated by PBE functional.In this work, G (0, 0, 0), X (0.5, 0, 0), S (0.5, 0.5, 0), and Y (0, 0.5, 0) are set as the high symmetry points, respectively.

Materials 2024 ,
17, x FOR PEER REVIEW 5 of 12Additionally, the orbital-projected band structures in Figure2a-c illustrate that the majority of the VBM of SiGaI is composed of O-2p orbitals, whilst the CBM is mainly contributed by Ga-4s and small quantities of O-2s and Si-3s orbitals.

Figure 2 .
Figure 2. Orbital-projected band structures of (a) Ga, (b) O, and (c) Si orbitals for SiGaII without strain.The strengths of the contributions are illustrated by the colored belt.

Figure 2 . 12 Figure 3 .
Figure 2. Orbital-projected band structures of (a) Ga, (b) O, and (c) Si orbitals for Si GaII without strain.The strengths of the contributions are illustrated by the colored belt.Materials 2024, 17, x FOR PEER REVIEW 6 of 12

Figure 3 .
Figure 3. (a) Evolutions of band edges relative to the vacuum level and band gaps of Si GaII with respect to biaxial strains.(b) The energy differences between the indirect and direct band gap versus biaxial tensile strains in Si GaII structure.

Figure 4 .
Figure 4. (a) Electron effective masses of SiGaII versus biaxial strains.(b) Electron mobility of SiGaII relative to biaxial strains.Partial charge distributions at the CBM for SiGaII in three different strain cases: (c) 0%, (d) +4%, and (e) +8%.The isosurface level was measured at 0.001 e/Å 3 .The blue square box highlights the changes of the wave-function of the CBM along a direction.

Figure 4 .
Figure 4. (a) Electron effective masses of Si GaII versus biaxial strains.(b) Electron mobility of Si GaII relative to biaxial strains.Partial charge distributions at the CBM for Si GaII in three different strain cases: (c) 0%, (d) +4%, and (e) +8%.The isosurface level was measured at 0.001 e/Å 3 .The blue square box highlights the changes of the wave-function of the CBM along a direction.
in the supporting materials shows that the C 2Dx and C 2Dy of Si GaII are 204.57and 181.61N m −1 , respectively, smaller than the theoretical values of 324.18 and 329.08 N m −1 [53], and 333.2 and 330 N m −1 [54], as well as experimental values of 343.8 and 347.4 N m −1

Materials 2024 , 12 Figure 5 .
Figure 5.The σ for n-type SiGaII configuration at 300 K along (a) x direction and (b) y direction relative to carrier concentration.The σ of SiGaII configuration with 8% strain along (c) x direction and (d) y direction with different temperatures.

Figure 5 .
Figure 5.The σ for n-type Si GaII configuration at 300 K along (a) x direction and (b) y direction relative to carrier concentration.The σ of Si GaII configuration with 8% strain along (c) x direction and (d) y direction with different temperatures.