Thermodynamics of Ga2O3 Heteroepitaxy and Material Growth Via Metal Organic Chemical Vapor Deposition

Heteroepitaxy of gallium oxide (Ga2O3) is gaining popularity to address the absence of p-type doping, limited thermal conductivity of Ga2O3 epilayers, and toward realizing high-quality p-n heterojunction. During the growth of β-Ga2O3 on 4H-SiC (0001) substrates using metal–organic chemical vapor deposition, we observed formation of incomplete, misoriented particles when the layer was grown at a temperature between 650 °C and 750 °C. We propose a thermodynamic model for Ga2O3 heteroepitaxy on foreign substrates which shows that the energy cost of growing β-Ga2O3 on 4H-SiC is slightly lower as compared to sapphire substrates, suggesting similar high-temperature growth as sapphire, typically in the range of 850 °C–950 °C, that can be used for the growth of β-Ga2O3 on SiC. A two-step modified growth method was developed where the nucleation layer was grown at 750 °C followed by a buffer layer grown at various temperatures from 920 °C to 950 °C. 2θ–ω scan of X-ray diffraction (XRD) and transmission electron microscope images confirm the β-polymorph of Ga2O3 with dominant peaks in the (−201) direction. The buffer layer grown at 950 °C using a “ramp-growth” technique exhibits root-mean-square surface roughness of 3 nm and full width of half maxima of XRD rocking curve as low as 0.79°, comparable to the most mature β-Ga2O3 heteroepitaxy on sapphire, as predicted by the thermodynamic model. Finally, the interface energy of an average Ga2O3 island grown on 4H-SiC is calculated to be 0.2 J/m2 from the cross-section scanning transmission electron microscope image, following the Wulff-Kaishew theorem of the equilibrium island shape.


■ INTRODUCTION
The ever-advancing landscape of high-power commercial applications and the pursuit of a net-zero society have driven significant evolution in semiconductor materials.−3 Ga 2 O 3 offers distinct advantages over established materials like Gallium Nitride (GaN) and Silicon Carbide (SiC) due to its significantly higher critical electric field.As GaN and SiC technologies mature, Ga 2 O 3 stands out as a favored contender to provide more efficient solutions for high-voltage commercial power applications.While five stable Ga 2 O 3 polymorphs are known (α, β, γ, δ, and κ), recent years have witnessed a surge in interest toward the most stable β polymorph due to its unique and potentially game-changing properties. 28β-Ga 2 O 3 is presently the most considered for power electronic applications as it is the most thermodynamically stable phase; however, the metastable α and κ polymorphs should not be ignored; these exist under specific conditions and can offer advantages.For example, α-Ga 2 O 3 being a corundum phase, boasts a hexagonal crystal structure and the widest bandgap (5.2 eV) among all Ga 2 O 3 polymorphs.This exceptional bandgap makes it highly desirable for power devices operating at high voltages and temperatures. 41,42On the other hand, κ-Ga 2 O 3 possesses an orthorhombic crystal structure, can offer piezoelectric properties similar to how a GaN electronic device work, and can be deposited on commercially available (0001)oriented sapphire substrates using MOVPE. 43Nonetheless, metastable α, κ(ε), and γ-Ga 2 O 3 films seek a lower-energy state by transforming into the more thermodynamically favorable β-Ga 2 O 3 structure by rearranging their atoms to minimize the overall energy.More details on this can be found elsewhere. 45,46owever, challenges persist, including the absence of p-type doping and limited thermal conductivity, leading to excessive heating in high-power devices that can degrade performance, reliability, and lifetime. 2,3The integration of Ga 2 O 3 with high thermal conductivity substrates, such as silicon carbide (SiC) and diamond, has gained prominence for advanced electronic and power device applications to tackle these limitations.Two primary methods, epitaxial growth and direct bonding, have emerged as crucial techniques for this integration.Notable achievements include the formation of rectifying p−n junctions through low-temperature direct bonding, as demonstrated by Sittimart et al., 4 and the development of β-Ga 2 O 3 field-effect transistors (FETs) on diamond substrates using Ga 2 O 3 nanomembranes, as illustrated by Noh et al. 5 Moreover, Song et al. reported the creation of Ga 2 O 3 /4H-SiC composite wafers through a fusion bonding method, leading to metaloxide-semiconductor field-effect transistors (MOSFETs) with significantly reduced channel temperatures and an impressive power figure of merit. 6However, challenges exist in direct bonding, as Cheng et al. pointed out the limitation of weak van der Waals bonding between the substrate and the Ga 2 O 3 layer; this hinders the full utilization of the thermal conductivity in diamond and SiC substrates. 7Recent successes in this domain include the high-quality epitaxial growth of Ga 2 O 3 on diamond achieved by Nandi et al. 8 and Karim et al. 9 Besides, Girolami et al. recently demonstrated growth of κ-Ga 2 O 3 on polycrystalline diamond substrates. 44Furthermore, several reports have outlined the successful heteroepitaxy of Ga 2 O 3 on SiC substrates, opening new avenues for enhanced device performance and thermal management.
Recent research, such as Hrubisak et al.'s work on liquid injection metal−organic chemical vapor deposition (MOCVD) growth of monoclinic β-Ga 2 O 3 films on 4H-SiC, 10 Hu et al.'s achievement of step-flow growth of β-Ga 2 O 3 films on off-axis 4H-SiC via LPCVD, 11 and Xia et al.'s extension to hexagonal phase-pure ε-Ga 2 O 3 films on 6H-SiC using MOCVD, 12 has contributed to the understanding of Ga 2 O 3 growth dynamics on high thermal conductivity substrates.However, improvements are still needed to meet commercial standards, exemplified by commercial GaN on Si heteroepitaxy, where epilayer quality routinely exhibits XRD rocking curve fwhm values in the range of 0.1°−0.11°. 13,14−24 Further improvement in layer quality can be achieved by using offcut substrates.A better understanding of growth dynamics is crucial to achieve the highest quality of Ga 2 O 3 on high thermal conductivity substrates, such as diamond and SiC, for commercial applications.
MOCVD growth is controlled by both thermodynamics at equilibrium and growth kinetics at steady-state.While there are established thermodynamic models for Ga 2 O 3 homoepitaxy, the field of Ga 2 O 3 heteroepitaxy is still in its nascent stages, and as of now, there is no well-established thermodynamic model for this process.When growing Ga 2 O 3 on a substrate with a different lattice constant, significant strain arises in the Ga 2 O 3 film.−33 Once the mismatch surpasses a specific threshold, usually around 1−2%, it leads to the formation of 3D islands of Ga 2 O 3 through elastic deformation, easing some of the strain caused by the lattice mismatch and reducing the stored elastic energy.Another option involves the creation of misfit dislocations to accommodate the strain. 25It is essential to note that the formation of these Ga 2 O 3 nucleation sites comes at the expense of increased surface energy and interface energy. 29In a two-component system like this, where the materials are immiscible, multiple energy factors must be taken into account.These factors encompass the energy related to the free surface of the substrate, the energy of the overlying material's surface, the interfacial energy at the Ga 2 O 3 -substrate boundary, and the strain energy.Moreover, during the growth process, the chemical potential of adatoms on the substrate also plays a crucial role.The interaction between the chemical potential of adatoms and the substrate-mediated strain adds further layers of complexity to this intricate process.
Thermodynamic Model.Let us consider the initial nucleation phase of β-Ga 2 O 3 on a foreign substrate, assuming a Volmer−Weber growth mode where the substrate surface energy is lower than the epilayer surface energy. 34This phase begins with the formation of a trapezoidal 3D island with a length denoted as s and a width as t as illustrated in Figure 1.
As the thickness h of the nucleation layer increases, the island sidewall will incline at an angle θ to the substrate surface.Both the substrate and the Ga 2 O 3 island possess different surface energies, represented as Γ s , Γ t , and Γ e , i.e., the substrate surface energy, the surface energy of the top epilayer, and the surface energy of the inclined sidewall, respectively.In terms of surface energy minimization, the likelihood of the crystallographic orientation of Ga 2 O 3 side-wall is (001) B with a surface energy of 2.37 J/m 2 (Γ e ) as compared to the (−201) top surface with a surface energy of 2.67 J/m 2 (Γ t ).This corresponds to a value of 50°for θ, a balance between surface and interface energy.In general, for isotropic elasticity, the energy of an island per unit volume can be written as 27 where Γ is the relative surface energy and can be expressed by with G the shear modulus and v the Poisson's ratio.The in-plane stress σ b can be determined using 26 2 where σ xx and σ yy denote the xx and yy stress components, respectively, E represents the Young's modulus, and m signifies the lattice mismatch.
The formation of a 2D layer or 3D islands relies on the equilibrium among the surface energy of the substrate and the top epilayer, along with the interface energy, and depends on the wetting factor W where In the context of 2D growth, i.e., Stranski-Krastanov growth, W is negative; conversely, positive W values signify a 3D island, or Volmer−Weber growth.However, adhesion fails entirely when the W ≥ 2Γ t .
For a thorough comparison of heteroepitaxial growth of Ga 2 O 3 on different substrates, assessing the interface energy is crucial, although not straightforward.The Wulff-Kaishew (WK) theorem aids in estimating the interface energy, considering the equilibrium island shape, as shown in Figure 1. 39In equilibrium, the positioning of the stable 3D island's shape balances the interface energy such that with h i representing the distance of the Wulff point from the interface, and h t signifies the distance of the island surface from the interface (also known as Wulff interface), as shown in Figure 1.The Wulff point is defined by a point location in the crystal where the distance of the crystal facet from that point is proportional to the surface energy of that facet, as shown in Figure 1.We can now consider a set of boundary conditions in order to evaluate the energy cost of a Ga 2 O 3 island in equilibrium.
Let us discuss first the onset of 2D growth where h i = −h t ; i.e., Γ i − Γ s = −Γ t .In this case, the wetting factor W is zero and the interface energy is now negative since −Γ s < Γ t .Any value of h i closer to the interface results in a positive wetting factor, leading to 3D island growth.The second boundary condition arises when h i = 0; i.e., the Wulff point lies on the interface itself.In this scenario, the interface energy equals to the substrate surface energy.Furthermore, W is equal to 2Γ t when h i = h t .For any values of h i > h t , adhesion fails entirely.
Hence, for the Volmer−Weber growth mode, the inequality Γ s − Γ t < Γ t < Γ t + Γ s holds.Notably, in the Volmer−Weber growth, where the substrate surface energy is less than the epilayer surface energy, a negative interface energy implies a higher base-to-height ratio of the grown island, fostering lateral growth.In practical scenarios, for any 3D island growth, Γ i > 0. Therefore, for Volmer−Weber growth, we can assume: A lower interface energy promotes greater lateral growth compared to vertical growth, enhancing a smoother surface morphology.Conversely, the interface energy closer to the maximum limit, i.e., Γ t + Γ s , mostly results in smaller islands in size with a rougher surface.Applying these boundary conditions to eq 1, we now can calculate the energy cost of a Ga 2 O 3 island of unit volume on a foreign substrate.Considering two boundary conditions: first, h i = 0, i.e., Γ i = Γ s as discussed above, the calculated energy cost (E/V) amounts to 6.26 kJ/cm 3 on sapphire and 6.14 kJ/ cm 3 on 4H-SiC.Thus, maintaining growth equilibrium on sapphire (001) and 4H-SiC (001) requires an interface energy of 1.85 and 1.7 J/m 2 , respectively.Second, when h i = h t , implying Γ i = Γ t + Γ s , the energy cost amounts to 12.1 kJ/cm 3 on sapphire and 11.9 kJ/cm 3 on 4H-SiC, leading to interface energies of 4.52 and 4.37 J/m 2 on sapphire and 4H-SiC substrate, respectively.Table 1 provides a summary of all of the key parameters utilized in these calculations.
These calculations suggest that a higher energy cost results in higher interface energy and thus smaller island sizes.Interestingly, Ga 2 O 3 islands on 4H-SiC require slightly less interface energy than those on sapphire.Extending this analysis to diamond substrates such as (111), (110), and (100), the surface energy increases here from 2.7 and 4 to 6.6 J/m 2 corresponding to the (111), (110), and (100) surfaces, respectively. 30,40Therefore, it is possible that strained Ga 2 O 3 epilayers completely wet the diamond substrate, following Stranski-Krastanov growth, at least initially since the surface energy of diamond is significantly higher than the β-Ga 2 O 3 (−201) surface energy.Further analysis provides energy costs of −973, −99.8, and 303 J/cm 3 on (100), (110), and (111) diamond substrates, respectively.Notably, despite the lowest surface energy of the diamond (111) substrate, the positive energy cost suggests that our assumption may not favor the (001) side wall facet inclined to the (−201) plane, particularly on the diamond (111) surface.
From the thermodynamic analysis conducted thus far, it is evident that the growth of Ga 2 O 3 islands on sapphire and 4H-SiC should follow a similar pattern, characterized by Volmer− Weber growth with very similar energy costs and interface energies.Therefore, it is thermodynamically favorable to employ high-temperature Ga 2 O 3 growth on 4H-SiC, akin to the methods widely utilized on sapphire substrates.However, the growth of Ga 2 O 3 on low-angle diamond substrates exhibits a preference for Stranski-Krastanov growth, necessitating a distinctly different growth approach compared to Ga 2 O 3 growth on sapphire and 4H−Si as reported elsewhere. 8The subsequent section elaborates on the epitaxial growth and characterization of β-Ga 2 O 3 on 4H-SiC, which is in line with the predictions derived from the preceding thermodynamic analysis.
MOCVD Growth of β-Ga 2 O 3 on 4H-SiC.All the growths in this study were conducted on commercially available semiinsulating 4H-SiC substrates (0001) using an Agnitron Agilis 100 metal−organic chemical vapor deposition (MOCVD) system.−12 Nevertheless, our thermodynamic modeling indicated that the energy cost for growing phase-pure β-Ga 2 O 3 on 4H-SiC is quite similar to that on sapphire.Therefore, the likelihood of nucleation is expected to be comparable.Notably, high-quality β-Ga 2 O 3 growth on sapphire is often reported at temperatures exceeding 800 °C. 15,16However, one of the challenges in growing β-Ga 2 O 3 on SiC substrate with a pure oxygen source at high temperature is the formation of an amorphous SiO 2 layer on the growth surface that can potentially result in a poor crystal quality.Si-terminated 4H-SiC/β-Ga 2 O 3 interface is highly sensitive to oxygen due to its lowest migration energy.O-terminated β-Ga 2 O 3 in the (−201) direction and Siterminated 4H-SiC in the (0001) direction offer the lowest relaxation energy and thus the highest stability by forming covalent bonds between Si−O at the interface, depending on how the oxygen atoms migrate on the 4H-SiC surface during the initial growth phase of β-Ga 2 O 3 .Therefore, the Siterminated SiC surface is highly reactive to the oxygen.Therefore, we adopted a two-step modified growth method where the nucleation layer was grown at 750 °C to prevent the surface oxidation of the SiC substrate, followed by approximately 350 nm of buffer layer grown at various temperature from 750 °C to 950 °C.Four samples, namely, S0, S1, S2, and S3, were compared to understand the effect of growth parameters on the layer quality.S1 and S2 are identical except for the bulk layer where the growth temperature was 920 °C and 950 °C, respectively.As shown in Figure 2(a), the sample S0 grown at 750 °C throughout exhibits a surface morphology as expected from the thermodynamic model, closely resembling those reported by others at similar growth temperature.S1 shows a partly coalesced surface with a few misoriented grains or nanocrystallites as shown in Figure 2(b).
Facet edges approximately 60°and 120°in different directions are also visible, which signify different pseudohexagonal  domains.In contrast, S2 forms a fully coalesced surface without any visible nanocrystallites on the surface as can be seen in Figure 2(c), suggesting a high quality of the grown layer.Furthermore, AFM scans shown in Figure 2(b,c) reveal an average root-mean-square (RMS) roughness of 16 nm and 7 nm in S1 and S2, respectively, across a 5 μm × 5 μm area.Rather than employing a conventional method of growing a thin nucleation layer at a constant lower temperature, we introduced a novel approach called the 'ramp-growth' technique in S3.In this technique, the nucleation layer's growth initiates at 780 °C, and the temperature gradually was increased to 950 °C over the course of 3 min, i.e., a ramp rate of 56 °C/min while the growth process is ongoing to mimic the high-temperature β-Ga 2 O 3 growth on sapphire by avoiding SiO 2 formation.Subsequently, the buffer layer is grown at 950 °C, similar to S2.As can be seen in Figure 2(d), the fully coalesced surface of S3 shows a roughness down to 3 nm, lowest among all the samples.It is noteworthy that the temperature ramp rate slightly affects the surface morphology of the following epilayer and is also influenced by various growth conditions.The temperature ramp rate of 56 °C/min was optimized at a rector pressure of 40 Torr, a TEGa flow of 26 μmol/min, and O 2 flow of 800 sccm in terms of full width at half-maximum (fwhm) obtained from the ω-scan along the (−402) plane and the AFM surface morphology.A post growth cooling rate of 70 °C/min was used for all the samples studied here.As outlined by Girolami et al., the post growth cooling rate may influence the formation of macroscopic defects and the overall layer quality; it was not studied systematically for the samples under study. 35,44RESULTS AND DISCUSSION Figure 3 shows the XRD 2θ−ω scans of the grown samples.All the samples display consistent peaks at 18.9°, 38.4°, and 59.2°, which can be attributed to the (−201), (−402), and (−603) crystallographic planes, respectively, of the beta polymorph of Ga 2 O 3 .It is evident that the (−201) plane is the predominant growth direction on the SiC (001) plane, consistent with previously reported heteroepitaxial Ga 2 O 3 on sapphire, 4H-SiC and diamond.Furthermore, the full width at half-maximum (fwhm) obtained from the ω-scan along the (−402) plane, which serves as an indicator of material quality, is presented in Figure 3(b).S1 and S2 exhibit fwhm values of 1.02 degrees and 0.9 degrees, respectively.In contrast, S3 boasts the lowest fwhm at 0.79°, the lowest among previously reported β-Ga 2 O 3 samples on 4H-SiC and placing it on par with some of the highest-quality Ga 2 O 3 samples grown on sapphire.
In order to find the epitaxial relationship of the grown layer with 4H-SiC, we further looked into the selected area electron diffraction (SAED) patterns from β-Ga 2 O 3 -on-SiC.Figure 4(  Stacking faults (SF) form, associated with partial dislocations as visible in Figure 5(b,d).The presence of compressive or tensile strain significantly impacts the dislocations that emerge, leading to stacking faults.Interestingly, regardless of whether strain is compressive or tensile, both systems should behave similarly in energy minimization.This is because strain contributes to the thermodynamic model with a squared term (i.e., σ b 2 in eq 3), rendering the sign of strain (hence compressive or tensile) irrelevant.
As discussed earlier to estimate the energy cost of island growth, it is important to correctly estimate the interface energy.The proposed thermodynamic model gives us a range of interface energy which varies in the range 0 < Γ i < Γ t + Γ s (eq 4).−38 In Figure 6, a STEM cross-section image of the sample S3 suggests an initial growth phase with 3D-like islands, followed by lateral coalescence during continuous layer growth.As depicted in Figure 6, these Ga 2 O 3 islands can be approximately fitted with trapezoidal island shapes, as predicted by the WK theorem in Figure 1.The height-to-base ratio of the 3D nucleation islands varies among different individual islands significantly, ranging from 0.23 to 0.44, with a mean value of 0.32.This ratio represents how tall the island is relative to its base dimension.Additionally, the angles formed between the side walls and the top surface of these islands exhibit some variance due to partial coalescence during continuous growth.Most commonly, these angles fall within the range of 123°− 135°, with a mean value of 128°.Although the coalescence of islands may affect the precise geometry, these results are consistent with our assumption of (001) side walls with an angle of approximately 130°to the top surface as considered in the thermodynamic model.
As the side wall could be either (001)A or (001)B, it remains indeterminate from Figure 6 where K = h/b, h and b are the height and the base of the trapezoid, respectively. 39Taking the average value of K as 0.  island on 4H-SiC is calculated to be 0.2 J/m 2 .Remarkably, this value closely aligns with the lower limit of the interface energy predicted by eq 4, thereby validating our thermodynamic model.

■ CONCLUSION
In summary, the success of Ga 2 O 3 growth on foreign substrates is influenced not only by lattice mismatch but also by the differences in surface energy.A thermodynamic analysis indicates that the growth of Ga 2 O 3 islands on both sapphire and 4H-SiC follows a comparable pattern, characterized by Volmer−Weber growth with nearly identical energy costs and interface energies.Hence, employing high-temperature Ga 2 O 3 growth on 4H-SiC, similar to widely adopted methods on sapphire substrates, is thermodynamically advantageous.This is successfully illustrated here.To prevent SiC substrate oxidation at high temperatures, however, in contrast to growth on sapphire, a two-step modified growth method was employed.It involved a nucleation layer grown at 750 °C, followed by a buffer layer at various temperatures from 750 °C to 950 °C.Varying the buffer layer temperature resulted in partly coalesced surfaces with misoriented grains at 920 °C and fully coalesced surfaces at 950 °C.XRD analysis and HRTEM images confirmed the phase-pure β-polymorph of Ga 2 O 3 .Furthermore, a new 'ramp-growth' technique was introduced, starting the nucleation layer at 780 °C and gradually increasing the temperature to 950 °C within 3 min.This approach significantly improved surface quality, reducing roughness to 3 nm and lowering the fwhm to 0.79°.These results are comparable to most matured β-Ga 2 O 3 heteroepitaxy on sapphire, promising potential for high-voltage device fabrication with improved thermal properties and realizing a highquality p-n heterojunction.

Figure 1 .
Figure 1.Formation of a 3D Ga 2 O 3 island on a foreign substrate according to the Wulff-Kaishew (WK) construction.The blue dash line indicates the Wulff interface, and the red dot indicates the Wulff point.

Figure 2 .
Figure 2. AFM surface morphology of Ga 2 O 3 layers grown under different conditions.(a) Surface morphology of Ga 2 O 3 layer grown at 750 °C throughout; (b) surface morphology of S1, displaying full coalescence and the presence of misoriented grains; (c) S2 presenting a fully coalesced surface devoid of nanoparticles; and (d) S3, exhibiting a smooth surface with RMS roughness of 3 nm, signifying enhanced quality.
Figure 3(b).S1 and S2 exhibit fwhm values of 1.02 degrees and 0.9 degrees, respectively.In contrast, S3 boasts the lowest fwhm at 0.79°, the lowest among previously reported β-Ga 2 O 3 samples on 4H-SiC and placing it on par with some of the highest-quality Ga 2 O 3 samples grown on sapphire.In order to find the epitaxial relationship of the grown layer with 4H-SiC, we further looked into the selected area electron diffraction (SAED) patterns from β-Ga 2 O 3 -on-SiC.Figure4(a) shows the edge-on bright field TEM image of S3.Panels (b), (c), (d), and (e) show the corresponding SAED patterns originating from SiC aligned with [010] zone axis, Ga 2 O 3 aligned with [−1−3−2] zone axis, Ga 2 O 3 aligned with [010] zone axis, and Ga 2 O 3 and SiC together aligned with [−1−3− 2] of Ga 2 O 3 interfacial plane parallel to [010] of SiC, respectively.For the SiC reflections, depicted in Figure 4(b), the reflections should be multiples of (001) along the c-axis in the electron diffraction pattern with (004) dominant, as this is the allowed reflection.Reflection from (001), (002), and (003) appears through double diffraction.In the perpendicular direction, diffractions from (100), (101), (10−2), etc., are allowed and visible in Figure 4(b).Note that the strong reflection in Figure 4(d) along the (−201) row is the SiC (004) reflection; (002) should be weak or effectively absent along with (001) and (003), which need double diffraction.Taking the relative spacing into account, the (40−2) reflection from Ga 2 O 3 and the (004) reflection from SiC, both spots in Figure 4(e), should be adjacent to the (40−2) reflection and will be strongly excited when there is any SiC in the selected area, which is presumably the bright spot due to the (004) reflection in Figure 4(d) as well.The reflection from SiC (002) in Figure 4(e) just below Ga 2 O 3 (20−1) is rather weak as can be seen also in Figure 4(a).Therefore, the relative diffraction of SiC and Ga 2 O 3 establishes an epitaxial relationship of [010]/[−1−3−2] (−201) β-Ga 2 O 3 || [010] (001) 4H-SiC.Figure 5(a) shows an HR-TEM image of the grown Ga 2 O 3on-SiC epilayer of sample S3.The strain relaxation of Ga 2 O 3 nucleation on SiC primarily hinges on minimizing energy, as captured by the thermodynamic model discussed above by

Figure 5 (
Figure 3(b).S1 and S2 exhibit fwhm values of 1.02 degrees and 0.9 degrees, respectively.In contrast, S3 boasts the lowest fwhm at 0.79°, the lowest among previously reported β-Ga 2 O 3 samples on 4H-SiC and placing it on par with some of the highest-quality Ga 2 O 3 samples grown on sapphire.In order to find the epitaxial relationship of the grown layer with 4H-SiC, we further looked into the selected area electron diffraction (SAED) patterns from β-Ga 2 O 3 -on-SiC.Figure4(a) shows the edge-on bright field TEM image of S3.Panels (b), (c), (d), and (e) show the corresponding SAED patterns originating from SiC aligned with [010] zone axis, Ga 2 O 3 aligned with [−1−3−2] zone axis, Ga 2 O 3 aligned with [010] zone axis, and Ga 2 O 3 and SiC together aligned with [−1−3− 2] of Ga 2 O 3 interfacial plane parallel to [010] of SiC, respectively.For the SiC reflections, depicted in Figure 4(b), the reflections should be multiples of (001) along the c-axis in the electron diffraction pattern with (004) dominant, as this is the allowed reflection.Reflection from (001), (002), and (003) appears through double diffraction.In the perpendicular direction, diffractions from (100), (101), (10−2), etc., are allowed and visible in Figure 4(b).Note that the strong reflection in Figure 4(d) along the (−201) row is the SiC (004) reflection; (002) should be weak or effectively absent along with (001) and (003), which need double diffraction.Taking the relative spacing into account, the (40−2) reflection from Ga 2 O 3 and the (004) reflection from SiC, both spots in Figure 4(e), should be adjacent to the (40−2) reflection and will be strongly excited when there is any SiC in the selected area, which is presumably the bright spot due to the (004) reflection in Figure 4(d) as well.The reflection from SiC (002) in Figure 4(e) just below Ga 2 O 3 (20−1) is rather weak as can be seen also in Figure 4(a).Therefore, the relative diffraction of SiC and Ga 2 O 3 establishes an epitaxial relationship of [010]/[−1−3−2] (−201) β-Ga 2 O 3 || [010] (001) 4H-SiC.Figure 5(a) shows an HR-TEM image of the grown Ga 2 O 3on-SiC epilayer of sample S3.The strain relaxation of Ga 2 O 3 nucleation on SiC primarily hinges on minimizing energy, as captured by the thermodynamic model discussed above by which observed (001) facet has A or B orientation; let us assume (001 B) due to its lower surface energy relative to the top (−201) surface.Using simple geometry and WK theorem, one can write

3 Figure 5 .
Figure 5. (a) HRTEM image of Ga 2 O 3 layer in S3, (b) stacking faults in the grown layer, (c) illustrating the Ga 2 O 3 −SiC interface, and (d) an inverse FFT image of the square region in (c) showing edge-type dislocations.

Figure 6 .
Figure 6.STEM cross-section image of the Ga 2 O 3 epilayer on the 4H-SiC substrate (S3).The yellow dashed lines indicate the shapes of the 3D islands during the initial deposition of Ga 2 O 3 .

Table 1 .
Parameters Used to Calculate the Total Energy Cost of a Ga 2 O 3 Island