Growth, catalysis and faceting of $\alpha$-Ga$_2$O$_3$ and $\alpha$-(In$_x$Ga$_{1-x}$)$_2$O$_3$ on $m$-plane $\alpha$-Al$_2$O$_3$ by molecular beam epitaxy

The growth of $\alpha$-Ga$_2$O$_3$ and $\alpha$-(In$_x$Ga$_{1-x}$)$_2$O$_3$ on $m$-plane $\alpha$-Al$_2$O$_3$(10$\bar{1}$0) by molecular beam epitaxy (MBE) and metal-oxide-catalyzed epitaxy (MOCATAXY) is investigated. By systematically exploring the parameter space accessed by MBE and MOCATAXY, phase-pure $\alpha$-Ga$_2$O$_3$(10$\bar{1}$0) and $\alpha$-(In$_x$Ga$_{1-x}$)$_2$O$_3$(10$\bar{1}$0) thin films are realized. The presence of In on the $\alpha$-Ga$_2$O$_3$ growth surface remarkably expands its growth window far into the metal-rich flux regime and to higher growth temperatures. With increasing O-to-Ga flux ratio ($R_{\text{O}}$), In incorporates into $\alpha$-(In$_x$Ga$_{1-x}$)$_2$O$_3$ up to $x \leq 0.08$. Upon a critical thickness, $\beta$-(In$_x$Ga$_{1-x}$)$_2$O$_3$ nucleates and subsequently heteroepitaxially grows on top of $\alpha$-(In$_x$Ga$_{1-x}$)$_2$O$_3$ facets. Metal-rich MOCATAXY growth conditions, where $\alpha$-Ga$_2$O$_3$ would not conventionally stabilize, lead to single-crystalline $\alpha$-Ga$_2$O$_3$ with negligible In incorporation and improved surface morphology. Higher $T_{\text{G}}$ further results in single-crystalline $\alpha$-Ga$_2$O$_3$ with well-defined terraces and step edges at their surfaces. For $R_{\text{O}} \leq 0.53$, In acts as a surfactant on the $\alpha$-Ga$_2$O$_3$ growth surface by favoring step edges, while for $R_{\text{O}} \geq 0.8$, In incorporates and leads to a-plane $\alpha$-(In$_x$Ga$_{1-x}$)$_2$O$_3$ faceting and the subsequent ($\bar{2}$01) $\beta$-(In$_x$Ga$_{1-x}$)$_2$O$_3$ growth on top. Thin film analysis by STEM reveals highly crystalline $\alpha$-Ga$_2$O$_3$ layers and interfaces. We provide a phase diagram to guide the MBE and MOCATAXY growth of single-crystalline $\alpha$-Ga$_2$O$_3$ on $\alpha$-Al$_2$O$_3$(10$\bar{1}$0).


I. INTRODUCTION
The ultra-wide band gap semiconductor gallium oxide (Ga 2 O 3 ) has experienced tremendous interest for high-power electronics, whose development is essential to reduce energy loss in power converters 1 .Monoclinic (β-) Ga 2 O 3 can be easily -type doped by Sn, Si or Ge.The availability of commercial β-Ga 2 O 3 substrates reduces the material costs for device production compared to other wide band gap materials, such as SiC or GaN 2 .
Other polymorphs of Ga 2 O 3 such as the orthorhombic structure (ϵ/κ-Ga 2 O 3 ) or the corundum structure (α-Ga 2 O 3 ), can also be epitaxially grown, with the latter being the polymorph with the widest band gap, of  g ≈ 5.3 eV 3 , and isostructural to α-Al 2 O 3 .This allows band gap engineering of α-(Al  Ga 1−  ) 2 O 3 on α-Al 2 O 3 over the whole composition range of 0 ≤  ≤ 1 3 as the phase-stability of α-(Al  Ga 1−  ) 2 O 3 on α-Al 2 O 3 increases with increasing Al content 4 .The growth of high-quality β-(Al  Ga 1−  ) 2 O 3 thin a) Electronic mail: marwilli@uni-bremen.de;These authors contributed equally to this work.b) Electronic mail: alonsoor@uni-bremen.de;These authors contributed equally to this work.c) Electronic mail: pvogt@uni-bremen.defilms and the fabrication of high-electron-mobility transistors based on β-(Al  Ga 1−  ) 2 O 3 is limited to  ⪅ 0.3 5,6 .Those features of α-Ga 2 O 3 provide an alternative route to develop high Al-mole fraction (Al  Ga 1−  ) 2 O 3 alloys for high power electronics.
The epitaxial growth of α-Ga 2 O 3 and α-(Al  Ga 1−  ) 2 O 3 on α-Al 2 O 3 has been investigated by molecular beam epitaxy (MBE) 3,4,7,8 , chemical vapor deposition (CVD) 9,10 and pulsed laser deposition (PLD) 11,12 .In conventional MBE (hereafter named as 'MBE'), i.e., by using elemental Ga and active O as source materials, the growth of Ga 2 O 3 is limited by the formation and subsequent desorption of its volatile suboxide Ga 2 O and complex 2-step kinetics 13 .This growth kinetics hampers the adsorption-controlled growth of Ga 2 O 3 in the Ga-rich regime where its growth ceases in the excess of Ga adsorbates 14,15 , thus, Ga 2 O 3 is typically grown in the O-rich regime.
These growth features have a detrimental impact on the crystalline and transport properties of Ga 2 O 3 thin films, for example, Ga vacancies ( Ga ) formed in β-Ga 2 O 3 when grown in the O-rich regime may act as compensating acceptors 16 .To suppress the formation of  Ga and thus to potentially improve the electrical properties of α-Ga 2 O 3 , its growth in the Ga-rich regime with improved crystalline quality is desirable 8,17 .
To overcome the detrimental growth kinetics of Ga 2 O 3 occurring during MBE growth, a new MBE variant has been recently developed: metal-oxide-catalyzed epitaxy (MOCATAXY) 18 .This growth method is based on metalexchange catalysis (MEXCAT) 19,20 and the use of the catalysts In, Sn, In 2 O or SnO 21 .MOCATAXY expands the growth window of β-Ga 2 O 3 , β-(Al  Ga 1−  ) 2 O 3 , α-Ga 2 O 3 , ϵ/κ-Ga 2 O 3 and In 2 O 3 deep into the metal-rich regimes and enables higher  G while generally improving the properties, such as crystalline quality and surface roughness, of the thin films 8,18,19,[21][22][23][24] .MOCATAXY, with In as a catalyst, emerges due to the favorable formation of intermediate and higher reaction efficiencies of In 2 O 3 over Ga 2 O 3 and the subsequent thermodynamically driven exchange of In-O bonds by Ga-O bonds 19 , and is mathematically explained for elemental and molecular catalysts in Ref. 21.
Following Ref. 8, in this work we provide a comprehensive study on the kinetic and thermodynamic growth features of α-Ga 2 O 3 and α-(In  Ga 1−  ) 2 O 3 synthesized by MBE and MOCATAXY.The purpose of this work is to systematically investigate the growth parameter space accessible by MBE and MOCATAXY on the kinetic and thermodynamic growth processes that lead to α-Ga 2 O 3 and α-(In  Ga 1−  ) 2 O 3 formation.As a first approach, we study α-Ga 2 O 3 films of  ≈ 50 nm formed on α-Al 2 O 3 (10 10).We find that both the oxygen-to-gallium flux ratio ( O ) and the indium-togallium flux ratio ( In ) determine the phase formation, the cation composition in α-(In,Ga) 2 O 3 and the surface features of the thin films grown.Single-crystalline α-Ga 2 O 3 (10 10) is achieved in metal-rich conditions and the presence of In, with step edges formed at the surface for higher growth temperatures ( G ) of 825°C.We develop a growth and phase diagram for the growth of phase-pure α-Ga 2 O 3 and α-(In  Ga 1−  ) 2 O 3 by MBE and MOCATAXY.

II. EXPERIMENTAL
α-Ga 2 O 3 thin films were grown by MBE and MO-CATAXY in a Riber Compact 12 system, equipped with an Oxford Applied Research HD25rf plasma source.Ga and In metals (6N purity) were supplied from standard effusion cells.The α-Al 2 O 3 (10 10) substrates were backside sputtercoated with a Ti 0.1 W 0.9 alloy of thickness  ≈ 500 nm to enable radiative substrate heating during growth.All substrates were cleaned with deionized water and rinsed with isopropanol (IPA) to remove contamination from the dicing process, an ultrasonic bath in acetone for one minute, followed by an IPA rinse and dried by N 2 .To eliminate residual surface contamination, a 10-minute plasma cleaning at 800°C, with O flux  O = 0.75 standard cubic centimeters per minute (SCCM) and radio-frequency plasma-power  rf = 300 W was carried out in situ.
G was measured by a thermocouple located within the substrate heater and an optical pyrometer operating at a wavelength of 980 nm.Reflection high-energy electron diffraction (RHEED) was used for in situ growth monitoring and a retractable ion gauge located at the growth position to measure the Ga flux ( Ga ) and In flux ( In ) as beam equivalent pressure (BEP) in mbar.The O flux was supplied in SCCM, and active O ( O ) was generated by the radio frequency plasma source.To convert the measured BEP into physical units and to allow the reproduciblity of our results in other MBE systems, we convert  Ga and  In as mbar → nm min −1 → nm −2 s −1 and  O as SCCM → nm min −1 → nm −2 s −1 (at given  rf = 300 W) using the procedure established in Ref. 28.To achieve this, Ga 2 O 3 and In 2 O 3 calibration films were grown with conditions where all supplied cations and anions are incorporated into the respective thin film, i.e., when the sticking coefficients of Ga, In and O are unity.The density of the desired atoms in the unit cell is determined by crystallographic software (here VESTA 29 ).The particle fluxes of  Ga ,  In and  O can then be calculated as with the growth rate (Γ) in nm min −1 , atomic density (  ) of species  = Ga, In, O, and conversion factor  = 1/60 to convert min → sec.The maximum available active  O for Ga → Ga 2 O 3 and In → In 2 O 3 oxidation can then be extracted from the Γ-peak at given  G , i.e., Γ at stoichiometric growth conditions for Ga 2 O 3 and In 2 O 3 28 .A summary of the calculated fluxes, as well as  O and  In , are given in Table I and Table II.
High-resolution x-ray diffraction (HRXRD) and x-ray reflectometry were performed with a Philips X'Pert Pro-MRD using the Cu K α1 radiation to identify film thickness, crystal phase and determine the composition of α-(In  Ga 1− ) 2 O 3 .Surface morphologies were measured and root-mean squared (RMS) roughnesses determined by atomic force microscopy (AFM) in a Bruker Dimension Icon XR scanning probe microscope.An FEI Nova 200 focused ion beam was utilized to prepare selected samples for cross-sectional structural and chemical analysis.Scanning transmission electron microscopy (STEM) in high-angle annular dark-field imaging (HAADF) mode, using a probe-corrected Thermo Fisher Scientific Spectra 300 operating at an acceleration voltage of 300 kV, was employed to measure the atomic structure of the thin films, formed facets and interfaces.Spatially resolved The active  O in In-mediated catalysis is multiplied by 2.8 8,19 .The growth time was adjusted to control the layer thickness.

Growth
energy-dispersive x-ray spectroscopy (EDX) was performed with the Super-X detection system, to measure the Al, Ga and In concentrations.Micro-Raman (μ-Raman) spectroscopy was performed for further phase identification and analysis using a Kimmon HeCd laser with a wavelength of 442 nm and a LabRAM HR Evolution confocal spectrometer.

A. Growth Kinetics by MBE and MOCATAXY
In Fig. 1 During MOCATAXY, the In as a catalyst provides more active O for metal oxidation to form the metal oxides, such as α-Ga 2 O 3 8 .Quantitatively, the available  O for MOCATAXY-grown Ga 2 O 3 can be 2.8 times larger than the  O available for MBE-grown Ga 2 O 3 8,19,21 .We note that the models shown in Fig. 1 use arbitrary kinetic parameters, similar to the models shown in Ref. 8. The model closely follows the experimental Γ values as a function of  O for the MBE and MOCATAXY samples.We find a very good correspondence to the experimental data when assuming full Ga incorporation; 2 atoms nm −2 s −1 for sample C2 by MO-CATAXY, with the higher oxygen flux.It must be noted that the contribution of a 7 met.%incorporation of In in this sample, as further discussed below, is considered in the model.

B. Surface Morphology
The impact of  In and  O on the surface morphology of the same samples as in Fig. 1 (except A12, B12 and D2) is depicted in Fig. (i.e. as a (anti)surfactant) was previously reported during MOCATAXY growth of Ga 2 O 3 or (Al,Ga) 2 O 3 8,18,30 .In acting as a surfactant in the MOCATAXY regime for  O ≤ 0.53 may be attributed to an enhanced adatom mobility when In is supplied, similar to the well-known In-Ga-N system 31 .

C. Crystalline Phase and In Incorporation
In order to investigate the crystalline phase of the samples in Fig. 2, μ-Raman spectroscopy was employed, as shown in Fig. 3(a).Sample A1 exhibits no Raman modes beyond those of the α-Al 2 O 3 substrate 32 (at higher Raman shifts than the displayed range), because no growth has occurred.Samples A2, A3, B1 and C1 all exhibit additional intense and well-defined Raman peaks at 217 cm −1 , 285 cm −1 and 327 cm −1 , which are characteristic modes of the corundum α-Ga 2 O 3 structure 33 .In samples B2, B3, C2 and C3, the peak positions of the corundum Raman modes present much lower relative intensities, higher full widths at half maximum (FWHM, e.g.11 cm −1 for the 217 cm −1 peak in sample B2, compared to 5 cm −1 in lower  O and  In samples) and Raman redshifts, which can be explained by the substantial amount of In in these α-Ga 2 O 3 films, in agreement with In concentrations extracted from HRXRD discussed below.In addition, all of these samples except B2 display intense peaks at Raman shifts of 199 cm −1 -200 cm −1 and 344 cm −1 , that are assigned to the A (3) g and A (5) g modes of monoclinic β-(In)Ga 2 O 3 , respectively 34,35 .This reveals the presence of β-(In  Ga 1−  ) 2 O 3 in the MOCATAXY samples when high  In and  O are provided.It should be noted that trial growths with lower thicknesses result in phase-pure α-(In  Ga 1−  ) 2 O 3 layers which implies that β-(In  Ga 1− ) 2 O 3 heteroepitaxially grows on α-(In  Ga 1−  ) 2 O 3 after a critical thickness.This is investigated further in the following section.For fully relaxed films, Vegard's law for the a-lattice parameter may be used to extract In concentrations from diffraction peak positions.From HRXRD, the average In concentration in the α-(In  Ga 1− ) 2 O 3 samples is estimated to  ≈ 0.07 for sample B2.No strain is considered for this estimation, as a reciprocal space map (RSM), shown in Fig. S3 in the supplementary material, shows the layer to be fully relaxed.An independent concentration determination using STEM-EDX yielded  = 0.07 ± 0.01 (see Fig. S6b), which is in good agreement with this result.Once  In is further increased at constant  O a gradual shift to higher 2θ angles, up to 64.7°in samples B2-B4, is observed (see Fig. S4), i.e. a lower In incorporation is measured, down to  ≈ 0.03, despite a higher  In supplied.We attribute this behavior to the In solubility limit in α-Ga 2 O 3 being reached for our growth conditions, and excess In forming the suboxide In 2 O that may desorb from the growth surface at these metal-rich flux conditions.A recent investigation in ϵ-(In  Ga 1−  ) 2 O 3 found analogous behavior at high  In , with In concentrations approaching  ≈ 0 30 .The maximum In concentration of  ≈ 0.08 in sample B2 agrees with the reported maximum In concentration for mist CVD-grown phase-pure α-(In  Ga 1− ) 2 O 3 36 .In contrast, MOCATAXY-grown samples A2, A3 and D2 (blue spectra in Fig. 3 roughnesses.This effect can be correlated to the thermodynamics of the In incorporation in β-(In  Ga 1−  ) 2 O 3 , reported in Ref. 14. There, it was found that a low metal-to-oxygen flux ratio ( Me ) leads to β-(In  Ga 1−  ) 2 O 3 , while β-Ga 2 O 3 was obtained when  Me = ( In +  Ga )/ O = 2.This is also the case here; sample D2 was grown with  Me = 2.1 and no observable In incorporation while sample B2 (red spectrum in Fig. 3), with  Me = 1.4,contains ∼7% In.
Rocking curves for representative samples can be found in the supplementary material, Fig. S5.The MBE-grown α-Ga 2 O 3 (B1 and C1) and the MOCATAXY-grown α-Ga 2 O 3 (A2) samples exhibit the same FWHM, suggesting MOCATAXY does not provide a measurable improvement in the mosaicity at these growth conditions.In contrast, α-(In  Ga 1− ) 2 O 3 samples [e.g., B2 in Fig. S5(c)] show markedly increased FWHM values, due to varying In concentrations within layers and greater mosaic spreads.
The AFM and HRXRD data suggest  Me ≥ 2 is required to take full advantage of In being a catalyst and surfactant for the growth of α-Ga 2 O 3 .Conditions with  Me < 2 result in the sub-optimal role of In as a surfactant with In incorporation in the layer up to its solubility limit, while too high  Me results in no growth.These  Me -dependent behaviors closely follow those observed for β-Ga 2 O 3 14 .

D. Faceting and Interfaces
To identify the mechanisms leading to β-(In  Ga 1−  ) 2 O 3 heteroepitaxial growth or phase-pure α-Ga 2 O 3 on α-Al 2 O 3 (10-10), asymmetric HRXRD scans and STEM analysis were performed.In Fig. 4  the HAADF image, we could only resolve the atomic distances between ( 201) planes of β-(In  Ga 1− ) 2 O 3 , but not perpendicular distances.We expect that these perpendicular distances should be resolvable in [010] and [132]-type orientations.Therefore, the most likely epitaxial relation between α-(In  Ga 1−  ) 2 O 3 and β-( Hence, at a film thickness  ≈ 50 nm, nucleation of the β-phase only occurs in the MOCATAXY region, when there is a sufficiently large  O and  In that allows for In incorporation.STEM-EDX is shown for sample C3 in the supplementary material, Fig. S6(a).Delayed In incorporation is observed in the film.The maximum concentration of In measured in C3,  ≈ 0.12, occurs after  ≈ 25 nm.Accordingly, this higher In incorporation occurs in β-Ga 2 O 3 .Higher metal concentra-tions of In in β-(In  Ga 1−  ) 2 O 3 were previously theoretically predicted 38 and experimentally achieved by MBE 39 .
Cross-sectional HAADF images and STEM-EDX for sample B2 is shown in the supplementary material, Fig. S6(b) and Fig. S7.(2 11 0) a-plane facets are also identified in this sample, without the growth of β-(In  Ga 1−  ) 2 O 3 , suggesting that the critical α-(In  Ga 1−  ) 2 O 3 thickness before β-Ga 2 O 3 nucleation is about to be reached for these growth conditions.Such critical thickness,  ≈ 50 nm, is double that of e.g.sample C3, which implies that the a-plane facet formation and subsequent β-(In  Ga 1−  ) 2 O 3 growth is delayed when reducing  In to 0. Figure 5(c) shows the cross-sectional HAADF overview and magnified images at the surface of sample A2.Singlecrystalline material is identified in the observed regions over the range of a few microns.Unlike sample C3, no a-plane facets or secondary phase formation are present in A2.Instead, (10 11 ) facets are observed on the surface, reflecting the morphology observed by AFM.The equivalency of Ga 2 O 3 (10 11) and Ga 2 O 3 (10 11 ), due to symmetry, confirms the noted surface facet orientation we previously reported 8 .STEM-EDX cross-sectional maps and line scans for sample A2 are shown in the supplementary material, Fig. S6(c).In accordance with the HRXRD and μ-Raman analysis, In does not substantially incorporate in the film, i.e., below 1 at.%.  be noted that, similar to a recent PLD study on α-Ga 2 O 3 12 , we have detected inhomogeneous β-(In)Ga 2 O 3 formation on thicker (> 100 nm) samples under these growth conditions, which is an open topic beyond the scope of this work.Our results assert the need to tune the growth parameters, aiming for a smooth layer-by-layer growth that prevents the formation of facets where the secondary phase growth takes place.This is partially achieved in the A2 sample at  G = 825°C, as presented in the next section.

F. Temperature-Dependent Growth Series
To investigate the effect of  G on our optimized phase-pure α-Ga 2 O 3 film (sample A2), a  G -series at such optimal  Ga ,  In and  O is performed.Figure 7  As shown in the symmetric HRXRD spectrum of the (30 30) reflex of these  G -dependent samples, in Fig. S8, no additional phases and no α-Ga 2 O 3 peak shifts above the experimental uncertainty were present, which indicates the dominant role of the In-Ga-O kinetics with respect to thermodynamics in the formation of phase-pure α-Ga 2 O 3 .

IV. CONCLUSIONS
The effect of varying the In and O fluxes on the formation of α-Ga Overview of samples studied in this work by MBE ( In = 0) and MOCATAXY ( In > 0), with  In =  In / Ga ,  O =  O / Ga and constant  Ga = 4.0 nm −2 s −1 .Samples A12, D2 and B12 are intermediate samples, and provide additional granularity to the results provided by primary samples where necessary. O = 0.40  O = 0.53  O = 0.80  O = 1.20 (a), Γ as a function of  In at different  O , at  G = 775°C is plotted.See Table II for the growth parameters of the displayed samples.All layers grown in this study have thicknesses  ≈ 50 nm, achieved by adjusting growth time after extracting Γ from calibration growths, and consulting the models shown in Fig. 1.For the MBE-grown samples, A1 ( O = 0.4), B1 ( O = 0.8) and C1 ( O = 1.2), at  In = 0, Γ increases with increasing  O .At the growth conditions of A1, the nucleation and growth of α-Ga 2 O 3 on α-Al 2 O 3 (10 10) is kinetically forbidden, as all active O is consumed to form the volatile suboxide Ga 2 O which subsequently desorbs from the α-Al 2 O 3 (10 10) surface.With increasing  O , the formation of Ga 2 O becomes less favored and α-Ga 2 O 3 growth sets in for sample B1, with Γ further increasing for sample C1.Γ as a function of  O is shown in Fig. 1(b).The MBEgrown samples (blue points represent experimental Γ, blue line represents modeled Γ) were grown under Ga-rich conditions, indicated by the increasing Γ with increasing  O .Γ plateaus in the O-rich regime, illustrated by the model at  O > 1.5, and becomes limited by the supplied  Ga .To expand the growth window of α-Ga 2 O 3 , In is additionally supplied to the Ga-O growth system and MOCATAXY is employed.At the same  O and  G , Γ increases with  In [indicated by the arrow in Fig. 1(b)] until Γ plateaus again, now limited by the supplied  Ga and  In .Our experiment reveals that the available growth window of α-Ga 2 O 3 (10 10)/α-Al 2 O 3 (10 10) is widened with larger  O or  In , see series A ( O = 0.4), B ( O = 0.8) and C ( O = 1.2) in Fig. 1(a).In Fig. 1(b), the light-gray-shaded area and arrow depict the expansion of the accessible growth window for α-Ga 2 O 3 grown by MBE and MOCATAXY.
FIG. 1.(a) Γ in nm min -1 as a function of  In at different  O .Γ measured at  In = 0 corresponds to Ga 2 O 3 growth by MBE and  In > 0 to MOCATAXY.Lines are model calculations 19 and serve as a guide to the eye.(b) MBE and MOCATAXY models for Γ in nm −2 s −1 as a function of  O .Models are shown for samples A1-C1 (MBE) and A2-C2 (MOCATAXY).The light-gray-shaded area depicts the expansion of the accessible growth window for α-Ga 2 O 3 grown by MBE and MOCATAXY.
11 and  O to 0.8.An HRXRD RSM of sample A2 around the (22 40) α-Ga 2 O 3 reflex is shown in Fig. 5(a).The dashed line intersects the fully-relaxed α-Al 2 O 3 and α-Ga 2 O 3 reciprocal lattice points.The results indicate that the 50 nm α-Ga 2 O 3 layer is fully relaxed.Figure 5(b) shows the substrate-film interface in sample A2, with the presence of a misfit dislocation.The film relaxes at the interface to reduce the elastic strain energy between Ga 2 O 3 (10 10) and Al 2 O 3 (10 10) 40 .The magnitude of the dislocation shown in Fig. 5(b) can be represented by the Burgers vector b = 1/6[30 32].

E
. MBE and MOCATAXY Phase Diagram of Ga 2 O 3 To guide the growth of α-Ga 2 O 3 (10 10)/α-Al 2 O 3 (10 10), Fig. 6(a) shows a phase diagram that encompasses the  ≈ 50 nm films studied in this work (Table II).Up to this thickness, only MOCATAXY-grown samples with high  In and high  O exhibit heteroepitaxial β-(In  Ga 1−  ) 2 O 3 growth.At lower  In = 0.05 and  O = 0.80 [see Fig. S2(a)], In also incorporates into the grown layers but no secondary β-phase is observed.In that sample, the solubility limit of In ( ≈ 0.08) in α-Ga 2 O 3 is already reached.For  O ≤ 0.53, independent of  In , phase-pure α-Ga 2 O 3 layers with In incorporation below 1 at.% are obtained.

FIG. 7 .
FIG. 7. (a) Temperature-dependent growth rate for samples grown with parameters of A2, line fit is a model fit and serves as a guide to the eye.(b) 5 μm × 5 μm AFM image of A2 grown at 825°C, inset shows RHEED pattern taken along [11 20] azimuth, and (c) 500 nm × 500 nm AFM image of the same sample, exhibiting well-defined terraces and step edges.
(a) shows a decrease in Γ with increasing  G due to increased Ga 2 O desorption13 .The rocking curve for the sample grown at  G = 825°C is shown in the supplementary material, Fig.S5(e), indicating an improvement in the layer's mosaicity with respect to the equivalent sample at  G = 775°C, Fig.S5(a).
Figure 7(b) and 7(c) show 5 μm × 5 μm and 0.5 μm × 0.5 μm AFM images of the sample grown at 825°C.Its surface exhibits well-defined terraces and step edges, also observable by the modulated streaky RHEED pattern.The step size can vary by a significant amount across the surface, with most falling in the range 3-5 nm, corresponding to approximately 14-23 atomic planes.Again, revealed by AFM and HRXRD, these features are aligned parallel to the [ 12 10] direction [marked in Fig. 7(b) and 7(c)].Although the surface RMS roughness of this sample is higher than the value for sample A2 (same conditions but   = 775°C), certain areas such as the one shown in Fig. 7(c) possess markedly lower RMS values of 1.0 -1.3 nm, pointing towards the possibility of achieving smoother α-Ga 2 O 3 thin films at this  G .The drastic  Gdriven surface morphology change with respect to sample A2 can be attributed to enhanced adatom surface mobility.

TABLE I .
Summary of values for  Ga ,  In ,  O and  G for samples grown by MBE and MOCATAXY.For the MBE and MOCATAXY grown samples, the conversions from mbar