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

Abstract

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.

1. Introduction

Bulk β-gallium oxide (β-Ga₂O₃) 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-Ga₂O₃ [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-Ga₂O₃ [9]. For Si-doped 3D β-Ga₂O₃, electron mobilities of 0.1–196 cm² V⁻¹ s⁻¹ have been reported. Baldini et al. studied the electron mobility of Si-doped homoepitaxial β-Ga₂O₃ films ranging from ~50 cm² V⁻¹ s⁻¹ for a doping concentration of n = 8 × 10¹⁹ cm⁻³ to ~130 cm² V⁻¹ s⁻¹ for a doping concentration of n = 1 × 10¹⁷ cm⁻³ [10]. Hernandez et al. illustrated that Si-doped homoepitaxial β-Ga₂O₃ films grown by chemical vapor deposition were characterized by electron mobilities in the range of 0.59–30.59 cm² 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 cm² V⁻¹ s⁻¹ (n = 2.3 × 10¹⁶ cm⁻³) measured in a Si-doped β-Ga₂O₃ film grown on Fe-doped β-Ga₂O₃ substrates at 810 °C, which is the highest electron mobility value ever reported in β-Ga₂O₃ epitaxial films, suggesting the great potential of Si dopants in tuning the electron mobility of 3D β-Ga₂O₃ [12]. For these reported heteroepitaxial films, Zhang et al. indicated that the electron mobility was 0.1, 0.1, 0.5, and 5.5 cm² V⁻¹ s⁻¹ for 0, 1.1, 4.1, and 10.4% Si-doped β-Ga₂O₃ 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-Ga₂O₃ films on sapphire substrates using PLD and reported an electron mobility of about 2.9 cm² V⁻¹ s⁻¹ for n = 2.5 × 10¹⁹ cm⁻³ [14]. Wong et al. reported electron mobilities of 90–100 cm² V⁻¹ s⁻¹ at carrier concentrations in the low to mid 10¹⁷ cm⁻³ range in β-Ga₂O₃ (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-Ga₂O₃ 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-Ga₂O₃ [100] direction [25,26]. Owing to its lower-dimensional nature and its high surface-to-volume ratio, 2D β-Ga₂O₃ exhibits better physical properties and enormous potential for novel applications. For instance, a monolayer of β-Ga₂O₃ passivated by H possessed electron mobilities as high as 2685 cm² V⁻¹ s⁻¹ and 156 cm² V⁻¹ s⁻¹ along the b and c directions, respectively, based on theoretical studies [27]. The features of 2D β-Ga₂O₃-based solar-blind photodetectors were boosted to nearly three times higher than that of bulk β-Ga₂O₃ [28,29,30]. Therefore, 2D β-Ga₂O₃ 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-Ga₂O₃ 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-Ga₂O₃ [31]. Especially considering the different local crystal structures of 2D and 3D β-Ga₂O₃, the results presented regarding 3D β-Ga₂O₃ 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-Ga₂O₃ here. Additionally, previous work showed that the 2D β-Ga₂O₃ is characterized by low elastic constants; therefore, systematic strain-modulated band structure investigations of 2D β-Ga₂O₃ 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 strain-modulated transport properties of Si-doped 2D β-Ga₂O₃. Our work demonstrates that the electron mobilities and band structures of Si-doped 2D β-Ga₂O₃ are tunable using biaxial strain, highlighting its great potential for use in nanoscale electronics.

2. Calculation Methods

2.1. Computational Details

Vienna ab initio Simulation Package (VASP) Standard Edition 6.4 was used for first-principles 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-Ga₂O₃ 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-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 Si⁴⁺ (0.040 nm) dopant is slightly lower than that of host Ga³⁺ (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 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.

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].

2.2. Formation Energies and Carrier Mobility Calculations

The formation energy ${\mathrm{E}}_{\mathrm{f}}$ of Si-doped 2D β-Ga₂O₃ is defined as [43,44]

$${\mathrm{E}}_{\mathrm{f}}={\mathrm{E}}_{\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}}-{\mathrm{E}}_{\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3}-{\displaystyle \sum _{\mathrm{i}}{\mathrm{n}}_{\mathrm{i}}{\mathsf{\mu }}_{\mathrm{i}}}$$

(1)

where ${\mathrm{E}}_{\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}}$ and ${\mathrm{E}}_{\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3}$, respectively, represent the energy of the Si-doped and pure 2D β-Ga₂O₃ configuration. ${\mathrm{n}}_{\mathrm{i}}$ demonstrates the quantity of i atom added (${\mathrm{n}}_{\mathrm{i}}<0$) or extracted (${\mathrm{n}}_{\mathrm{i}}>0$) from the pure β-Ga₂O₃ system. The chemical potential of ${\mathsf{\mu }}_{\mathrm{Si}}$ and ${\mathsf{\mu }}_{\mathrm{O}}$ is gained from the stable bulk Si and O₂, 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]:

$$\mathsf{\mu }={\displaystyle \frac{2\mathrm{e}{\mathrm{h}}^3{\mathrm{C}}_{2\mathrm{D}}}{{(2\mathsf{\pi })}^3\times 3{\mathrm{k}}_{\mathrm{B}}\mathrm{T}{\left|\mathrm{m}*\right|}^2{\mathrm{E}}_1^2}}$$

(2)

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 E₁, elastic stiffness C_2D, and the electron effective mass m_e*. Deformation potential constant E₁ is calculated by ${\mathrm{E}}_1=\partial {\mathrm{E}}_{\mathrm{e}\mathrm{d}\mathrm{g}\mathrm{e}}/\partial (\Delta \mathrm{a}/{\mathrm{a}}_0)$, which denotes the shift of the band edge accompanied by uniaxial strain. Here, ${\mathrm{E}}_{\mathrm{e}\mathrm{d}\mathrm{g}\mathrm{e}}$ illustrates the shifted energy of the conduction band minimum (CBM), $\Delta \mathrm{a}/{\mathrm{a}}_0$ is the variation in the lattice constant along a certain strained direction, and elastic stiffness C_2D is calculated by ${\mathrm{C}}_{2\mathrm{D}}=[{\partial }^2\mathrm{E}/\partial {(\Delta \mathrm{a}/{\mathrm{a}}_0)}^2]/{\mathrm{S}}_0$. Here, $\mathrm{E}$ is the total energy of the strained optimized supercell, and ${\mathrm{S}}_0$ is the surface area of the strained structure. The electron effective mass m_e* is calculated based on the equation as ${\mathrm{m}}_{\mathrm{e}}^{*}={\hslash }^2/({\partial }^2\mathrm{E}/\partial {\mathrm{k}}^2)$ 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-Ga₂O₃. The relaxation time is estimated by $\mathsf{\tau }=\mathsf{\mu }{\mathrm{m}}_{\mathrm{e}}^{*}/\mathrm{e}$.

3. Results and Discussion

3.1. Structural Stability

Pure 2D β-Ga₂O₃ 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.

3.2. Strain-Engineered Band Structures

Figure 1b shows the band structure of perfect 2D β-Ga₂O₃, 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-Ga₂O₃ is 2.30 eV, which agrees well with previous works [45,47], but it is lower than the value of 3D β-Ga₂O₃ [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-Ga₂O₃ 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.

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.

3.3. 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-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 Si_GaII, the calculated m_ex* and m_ey* are 0.30 m₀ and 0.28 m₀, 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.

Different strains ranging from −4% to 4% were employed to calculate the elastic constants (C_2D). Figure S2a in the supporting materials shows that the C_2Dx and C_2Dy of Si_GaII are 204.57 and 181.61 N 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 [55] in 3D β-Ga₂O₃, respectively. This demonstrates that 2D Si_GaII is softer and more susceptible to deformation than bulk β-Ga₂O₃ material and may have greater potential for applications in flexible electronics. Figure S2b in the supporting materials denotes the calculated deformation potential E₁ 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 E₁.

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*, E₁, 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 cm² V⁻¹ s⁻¹ as the tensile strain reached 8%, which was far greater than those reported in other doped 2D β-Ga₂O₃, i.e., H dopant with, respectively, 2684.93 cm² V⁻¹ s⁻¹ and 156.25 cm² V⁻¹ s⁻¹ along the x and y directions, F of ~300 cm² V⁻¹ s⁻¹, Cl and H co-dopant of ~100 cm² V⁻¹ s⁻¹ [56], as well as the H and F co-dopant in our previous work with 4863.05 cm² V⁻¹ s⁻¹ under +6% uniaxial tensile strain along the x direction and 2175.37 cm² 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 cm² V⁻¹ s⁻¹, which surpassed the μ_e in other monolayer materials, such as α-In₂Se₃ (~1000 cm² V⁻¹ s⁻¹) [57], MoS₂ (~200 cm² V⁻¹ s⁻¹) [58], and GaN (~300 cm² V⁻¹ s⁻¹) [59]. Thus, in terms of its exceptional μ_e, the 2D Si-doped β-Ga₂O₃ 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 Figure 4d and Figure 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-Ga₂O₃ 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-Ga₂O₃ 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]. Figure 5a and Figure 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 × 10¹¹ to 6 × 10¹² cm⁻² at 300 K, the electron mobilities of unstrained and +4%, +8%, −4%, and −8% strained Si_GaII ranged from 5.69 × 10⁶ to 4.97 × 10⁶ cm² V⁻¹ s⁻¹, 9.76 × 10⁶ to 9.29 × 10⁶ cm² V⁻¹ s⁻¹, 1.10 × 10⁷ to 9.83 × 10⁶ cm² V⁻¹ s⁻¹, 4.78 × 10⁶ to 4.05 × 10⁶ cm² V⁻¹ s⁻¹, and 4.66 × 10⁶ to 4.01 × 10⁶ cm² 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.

3.4. 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 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 CBM than that of π bonding in the VBM. Additionally, band structure calculations resolved a direct-to-indirect transition under the tensile strains. The μ_ex increased significantly and the μ_ey decreased slightly first and rose from compressive to tensile strain. When the biaxial tensile strain was higher than 4%, it had a higher variation in μ_ex than in μ_ey, which can be associated with the smaller E_1x of CBM. 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 cm² V⁻¹ s⁻¹ as the biaxial tensile strain approached 8%, which was far greater than those reported in other doped 2D β-Ga₂O₃. The electrical conductivity was elevated with the increase in tensile strain and the enhancement of temperature from 300 K to 800 K. Our work demonstrates that the electron mobilities and band structures of Si-doped 2D β-Ga₂O₃ are tunable by biaxial strain, highlighting its great potential in nanoscale electronics.