Determination of anisotropic optical properties of MOCVD grown m-plane α-(Al x Ga1−x )2O3 alloys

The anisotropic dielectric functions (DF) of corundum structured m-plane α-(Al x Ga1−x )2O3 thin films (up to x = 0.76) grown on m-plane sapphire substrate by metalorganic CVD have been investigated. IR and visible–UV spectroscopic ellipsometry yields the DFs, while X-ray diffraction revealed the lattice parameters (a, m, c), showing the samples are almost fully relaxed. Analysis of the IR DFs from 250 to 6000 cm−1 by a complex Lorentz oscillator model yields the anisotropic IR active phonons E u and A 2u and the shift towards higher wavenumbers with increasing Al content. Analyzing the UV DFs from 0.5 to 6.6 eV we find the change in the dielectric limits ε ∞ and the shift of the Γ-point transition energies with increasing Al content. This results in anisotropic bowing parameters for α-(Al x Ga1−x )2O3 of b ⊥ = 2.1 eV and b ∣∣ = 1.7 eV.

2) The benefits of the metastable α-phase are the possibility of growth using sapphire (α-Al 2 O 3 ) as an affordable substrate with the same crystal structure, a higher symmetry and a slightly higher bandgap [15][16][17][18][19] compared to β-Ga 2 O 3 . Also α-Ga 2 O 3 offers the possibility of alloying with other group III elements like In or Al. In 2 O 3 , despite having a stable cubic bixbyite phase (Ia3), 20) as well has a metastable corundum crystal structure. 21) This offers the opportunity of bandgap engineering over the wide range from 3.38 (α-In 2 O 3 ) 21) to 9.2 eV (α-Al 2 O 3 ). 22) It therefore covers a larger UV spectral range than the (Al x Ga 1−x )N system does (AlN E g = 5.96 eV 23) ), where already high electron mobility transistors, 24) laser diodes, 25,26) or solar-blind photodetectors 27) have been realized.
Some attempts to alloy the stable β-Ga 2 O 3 phase with Al have been made, [28][29][30][31][32][33] but this remains challenging since α-Al 2 O 3 does not share the same crystal structure and monoclinic Al 2 O 3 (θ-Al 2 O 3 ) remains obscure. 34) In contrast, alloying α-Ga 2 O 3 with Al by CVD, 35) pulsed laser deposition (PLD), 36) and MBE, 37) can provide single-crystal films especially when grown on m-plane sapphire. 38) c-plane Al 2 O 3 as substrate seems to lead to the formation of a few monolayers α-Ga 2 O 3 followed by β-Ga 2 O 3 , due to the large in-plane lattice mismatch between layer and substrate, observed in metalorganic CVD (MOCVD), MBE and PLD, but not in mist CVD or halide vapour phase epitaxy. 38,39) This is also not the case for m-plane Al 2 O 3 substrates where much thicker single phase α-Ga 2 O 3 layers have been observed. 40) Also, first attempts of doping α-Ga 2 O 3 , 41,42) α-Al 2 O 3 , 43) and the α-(Al x Ga 1−x ) 2 O 3 alloy system 44) have been made. Sn doping of α-Ga 2 O 3 on m-plane Al 2 O 3 showed mobilities much higher than films grown on c-plane Al 2 O 3 . 45) Density-function calculations identified possible candidates for donor doping in α-(Al x Ga 1−x ) 2 O 3 which is fundamental for electronic applications. 46,47) So far, some investigations of the optical properties of α-(Al x Ga 1−x ) 2 O 3 have been made by Ito et al. 48) and Jinno et al. 38) employing optical transmission measurements. Dang et al. 35) analyzed a Tauc plot and Uchida et al. 49) Chen et al. 36) and Xia et al. 50) utilized X-ray photoelectron spectroscopy (XPS) to determine bandgap values and/or bowing parameters. But since the corundum crystal structure is anisotropic, a polarization dependent investigation of the material properties is crucial. Studies of the optical properties of α-(Al x Ga 1−x ) 2 O 3 taking into account the anisotropy are rare. Hilfiker et al. investigated the optical absorption onset 34) and dielectric limits ε ∞ 51) by spectroscopic ellipsometry. Additionally Stokey et al. 52) determined IR-active phonon modes and static dielectric constants with the same technique in the IR.
Here, m-plane α-(Al x Ga 1−x ) 2 O 3 thin films up to x = 0.76 grown by MOCVD on m-plane sapphire substrate have been investigated anisotropically. X-ray diffraction (XRD) yields lattice parameters while IR and visible-UV spectroscopic ellipsometry yields the complex dielectric functions (DF) in both spectral ranges. The IR DF is dominated by the IR active phonons. In contrast to the work by Stokey et al. 52) the Al-rich side (with x = 0.76) is also discussed in the IR part of this study. The UV DF yields the dielectric limit ε ∞ and the Γ-point transition energies. They are in turn used to determine the anisotropic bowing parameters. To underline the novelty of this study, we emphasize that a different approach is used to evaluate the DF compared to Hilfiker et al. 34) Consequently, this leads to deviating results in the bowing parameters and the order of Γ-point transitions.

Experimental
α-(Al x Ga 1−x ) 2 O 3 thin films (∼100 nm) were grown by MOCiVD on m-plane sapphire substrates using a MOCVD reactor (Agitron Agilis). The precursors used were trimethylaluminum (TMAl), triethylgallium (TEGa), and pure O 2 , while Ar was the carrier gas. Beforehand, the substrates were cleaned ex situ with solvents and in situ in the reaction chamber by high temperature annealing at 920°C under O 2 atmosphere. Afterwards the epitaxial growth was initiated in a temperature range of 650°C-880°C and a pressure of 20 to 80 Torr. Details on the growth process, as well as comprehensive material characterization e.g. XRD asymmetrical reciprocal space mapping, atomic force microscopy, Raman spectroscopy, XPS, high angle annular dark field scanning transmission electron microscopy, and energy dispersive X-ray spectroscopy, can be found elsewhere. 53) Crystal quality and the lattice parameters of the samples were determined by XRD measurements. The in-plane c and a values and their FWHM were directly quantified by grazing incidence in-plane diffraction (GIID) at the critical angle of incidence of 0.36°(for Cu-Kα radiation the 2θ angles were at 36.1°(x = 0) for the (1120 ) planes, and at 40.15°(x = 0) for the (0006) planes, respectively) (Seifert/ FPM URD6/GIID). The m values were determined by high-resolution XRD (HRXRD) on the (3030 ) planes using Cu-Kα at 64.8°( Values of α-Ga 2 O 3 by Marezio et al. 54) and α-Al 2 O 3 by Leszczynski et al. 55) were used to apply Vegard's law 56) to the measured lattice parameters: Generalized 57) IR spectroscopic ellipsometry was performed using a Fourier-transform ellipsometer (Woollam IR-VASE) in the range of 250-6000 cm −1 with the resolution set to 4 cm −1 . The measurements were carried out at three angles of incidence Φ of 50°, 60°, and 70°. In generalized spectroscopic ellipsometry, three ratios of the complex reflection coefficients are measured, r pp /r ss , r ps /r pp , and r sp /r ss , which then provide the corresponding ellipsometric angles Ψ and Δ, 58) where Ψ is the amplitude ratio between the parallel and the perpendicular polarization orientation of the reflected light from the sample, with respect to the plane of incidence, and Δ is the phase shift between them. Ψ and Δ can be transformed into the complex refractive index ρ: From that, the pseudo DF can be calculated as In an isotropic sample with only one semi-infinite layer the pseudo DF is identical to the actual DF of the material. In any other case, e.g. in case of thin film samples, the pseudo DF is only the DF of the sample and a multi-layer model has to be used and fitted to disentangle the DF of the layer of interest. Here, the model contains two layers, the sapphire substrate, based on measurements of a m-plane sapphire wafer, and the α-(Al x Ga 1−x ) 2 O 3 layer of interest. Since both, α-(Al x Ga 1−x ) 2 O 3 and the underlying m-plane sapphire are anisotropic crystals in corundum structure, each sample must be measured twice, with the c-axis perpendicular and parallel to the plane of incidence. From this we gain the ordinary DF (ε ⊥ ) with the electric field vector E⊥c and the extraordinary DF (ε || ) with E || c.
The model DFs in the region of the IR active phonons contain a dielectric background ε ∞ and a sum of broadened Lorentzian phonon oscillators, with the phonon frequency ω TO , the broadening parameter γ TO and the amplitude S: with 6 IR active phonons in the rhombohedral corundum structure: 59) This anisotropic multi-layer model is used as a starting-point for a point-by-point (pbp) fit, where the model is fitted numerically to the experimental data at every wavenumber until the best match is obtained. This leads to the final DFs. In a last step, these numerical DFs were fitted with their model DFs [Eqs. (4) and (5)] to determine parameters like the phonon wavenumber. Generalized UV spectroscopic ellipsometry was performed with a variable-angle scanning ellipsometer based on a grating monochromator, equipped with an autoretarder, in the range from 0.5 (≈4000 cm −1 ) to 6.6 eV, thus it overlaps with the IR ellipsometry range. Identical to the IR ellipsometry the UV ellipsometry measures three ratios of the complex reflection coefficients, which yield corresponding ellipsometric angles Ψ and Δ. Also in the UV spectral range, measurements were taken twice to determine both, ε ⊥ and ε || . However, in the UV range surface roughness has to be taken into account for the multi-layer model using an effective medium approximated layer with Bruggemanʼs formalism. 60) The underlying m-plane sapphire is implemented with a model DF from Malitson. 61) From an analysis of the Fabry-Pérot oscillations the thickness of the epitaxial layer can be determined (see Table I). The α-(Al x Ga 1−x ) 2 O 3 layer of interest is modeled using an anisotropic general oscillator model containing the Herzinger-Johs parameterized semiconductor oscillator functions (PSEMI) to describe the lineshape of the experimental results 62,63) based on a model for rplane α-Ga 2 O 3 by Kracht et al. 17) Again consistent with the evaluation of the IR ellipsometry, this multi-layer model is also used as starting point for a pbp-fit in the UV yielding the actual DFs. In the region below the first transition energy (0.5 eV to ∼5 eV) the Re of the DF, which corresponds to the square of the refractive index (while the Im which corresponds to the absorption coefficient is still zero) was fitted by a model from Shokhovets et al.: 64) e w which allows the calculation of the dielectric limit ε ∞ : In the region of the absorption onset, the DFs were fitted to an error function-like shaped model DF in the Im and the corresponding Re is based on a Kramers-Kronig transformation, using the earlier mentioned PSEMI functions, to determine the Γ-point transition energies E CV of α-(Al x Ga 1−x ) 2 O 3 . We then use results from Kracht et al. 17) as fixed parameters of the transition energies and analyze the relative shift with Al content. This is discussed in more detail below (Sect. 3.3). At last, an anisotropic bowing model is used to describe the change of the dielectric limit and the transition energy with increasing Al content x:

Results and discussion
3.1. XRD Results of the XRD measurements, in terms of lattice parameters a, m, and c together with the ω-scan FWHM, using an analyzer in front of the detector, of the (303̅ 0), the (112̅ 0), and the (0006) peak are displayed in Table I. There is an excellent agreement of the lattice parameters with previous results on m-plane Ga 2 O 3 . 65) The change in the lattice parameters with increasing Al content is displayed in Fig. 1 added by Vegard's law based on Eq.
(1). m shows the best match with Vegard's law, which is not surprising since the (303̅ 0) reflection was used to determine the Al content, as discussed by Bhuiyan et al. 53

IR-Ellipsometry
For the x = 0.07 α-(Al x Ga 1−x ) 2 O 3 sample the experimental data (Ψ and Δ) are displayed in the region of the IR active phonon modes (250-600 cm −1 ) in comparison to the pbp-fit for the two measurements with the c-axis parallel and perpendicular to the plane of incidence in Fig. 2. The fit matches the data almost perfectly. This is exemplary for all samples.
Based on the pbp-fit, we obtain the IR pbp-DFs as displayed for the x = 0.07 sample in Fig. 3, with ε ⊥ in the bottom and ε || in the top panel. Equations (4) and (5) are used to fit these pbp-DFs and four out of the allowed six IR active phonon modes [Eq. (6)] could be determined for all samples. The missing phonon E u (1) is expected to appear below the experimentally accessible spectral range starting at 250 cm −1 for all samples till x ≈ 0.26. 52) The fact that it is not found for higher Al concentrations either is probably related to its Table I. Al contents (x), lattice parameters (a, m and c) and FWHM by GIID and high resolution XRD (HRXRD), together with thicknesses (d) by UV spectroscopic ellipsometry (UVSE) for all investigated α-(Al x Ga 1−x ) 2 O 3 samples are listed, along with literature data on Ga 2 O 3 powders, 54) c-plane Ga 2 O 3 , 18) m-plane Ga 2 O 3 , 65) and c-plane Al 2 O 3 . 55) Since Marezio et al. 54) Ning et al. 18) and Leszczynski et al. 55) do not provide data on the m value, we assumed fully relaxed samples and estimated m by a 4 3. Those values are printed italic.     Table II. Phonon wavenumbers (ω 0 ) of the IR active optical phonon modes E u (E⊥c) and A 2u (E || c) by IR spectroscopic ellipsometry for all investigated α-(Al x Ga 1−x ) 2 O 3 samples and a m-plane sapphire wafer (x = 1), along with literature data on c-plane Ga 2 O 3 , 16,18) m-plane Ga 2 O 3 , 65,66) and c,a-, and m-plane Al 2 O 3 . 16,67) Estimated, not measured, values in the literature are printed italic.  Unfortunately, no pbp-DFs but only model DFs were shown there, 52) therefore a comparison on this level is not possible. Also, a product ansatz was used instead of the sum ansatz used here [Eqs. (4) and (5)], which, due to too many free parameters, often provides poorer results. Additionally the values of Table 1 and Fig. 3 (shown here) in Stokey et al. 52) do not align, which leads to confusion on the actually values.
Overall, however, their values fit to our results in most cases.
Only the E u (2) and the E u (3) mode show slight deviations.

UV-Ellipsometry
Experimental results (Ψ and Δ) exemplary from the x = 0.26 sample in the UV spectral range are shown in Fig. 5 with the corresponding pbp-fit for two different measurements one with the c-axis parallel and one perpendicular to the plane of incidence. Here as well the pbp-fit matches the data almost perfectly. The so obtained pbp-DFs are displayed in Fig. 6 for both ε ⊥ (solid) and ε || (dashed) with the Re ε 1 on the left and the Im ε 2 on the right axis in the region around the absorption onset. Note that in the UV spectral range we do not provide DFs of α-Ga 2 O 3 (i.e. x = 0) in this work, but rather use established and already published anisotropic DFs from earlier studies by Kracht et al. 17) The difference in line shape for ε || compared to ε ⊥ at the absorption onset is clearly visible together with a strong blue shift of the absorption onsets with increasing Al content.
In the region of photon energies lower than the absorption onset, the Re of the DF was fitted using the model from Shokhovets et al. 64) and therefore an analytical expression of the refractive index is obtained (see supplement, Fig. S1). Using Eq. (8) the dielectric limit ε ∞ can be determined from the fitted values E G , A G , E H , and A H . The trend of the dielectric limit as a function of Al content x is displayed in Fig. 7 (red and green circles) and compared to results of Hilfiker et al. 51) (black and gray triangles) and their corresponding bowing fit (black and gray dashed lines).
Overall, the results of Hilfiker et al. 51) match well with our results here. ε ∞,⊥ is always higher than ε ∞,|| . Only for the x = 0.07 sample slight deviations are obvious but we assign this to imperfect fitting of the Shokhovets model to the pbp-DF due to artifacts in the DF caused by an imperfect pbp fit in this spectral range (∼2 to 4 eV).
In the next step we now analyze the UV DF in the region of the absorption onsets to obtain the Γ-point transition energies E CV whereby E CV,⊥ corresponds to Γ 1−1 + Γ 1−2 , 19) and E CV,|| to Γ 1−3 . 19) First of all we note that Kracht et al. 17) and Hilfiker et al. 19,34) agree on the line shape of the DF. However, their analysis of these DFs yield very different results. Hilfiker et al. 19,34) obtain a band order E CV,⊥ < E CV,|| while Kracht et al. 17) have the opposite result E CV,⊥ > E CV,|| (see Fig. 8, bottom panel). The contradiction originates from different approaches used to model the absorption onset and consequently very different exciton binding energies. In Kracht et al. 17) 38 meV was obtained for excitons related to Fig. 5. Ellipsometric angles Ψ pp (left) and Δ pp (right) of α-(Al x Ga 1−x ) 2 O 3 for x = 0.26 in the visible and UV spectral range with the c-axis perpendicular to the plane of incidence (bottom) and parallel to the plane of incidence (top) for three different angles of incidence 50°, 60°, and 70°in red, blue, and green respectively, together with their corresponding pbp fit in black. both absorption onsets by using Elliot's classical theory, 68) while Hilfiker et al. 19,34) described excitons as anharmonically broadened Lorentz oscillators. This approach has the advantage of producing ε 1 and ε 2 simultaneously but the disadvantage of negative contributions to absorption curves and very anisotropic exciton binding energies of 10 meV (⊥) and 180 meV (||). 19) No matter how the analysis is performed in detail we observe very similar transition energies for α-Ga 2 O 3 and as well for α-Al 2 O 3 . 22) Therefore, we expect the same to hold true for the ternary alloy system. Our DFs for ternary α-(Al x Ga 1−x ) 2 O 3 are very similar in shape and amplitude to the case of α-Ga 2 O 3 . However, it is increasingly difficult with increasing x to perform a clean Elliot fit due to the fact that our spectral range ends at 6.6 eV where the signal is already relatively noisy. Therefore, we use the values from our earlier result (Kracht et al.: 17) E CV,⊥ = 5.62 eV and E CV,|| = 5.58 eV) as fixed parameters and analyze the relative shift on the photon energy axis for increasing x.
To gain the Γ-point transition energies we use a model fit as explained in Sect. 2 (see also supplement Fig. S2). So obtained results (values for x = 1 i.e. Al 2 O 3 , are taken from Harman et al.: 22) E CV,⊥ = 9.25 eV and E CV,|| = 9.2 eV) are shown in Fig. 8 (upper panel). Independent bowing fits for E CV,⊥ and E CV,|| yield b ⊥ = 2.1 eV and b || = 1.7 eV. Hilfiker et al. 34) published b ⊥ = 1.31 eV and b || = 1.63 eV, which differs significantly in the case of the b ⊥ . Also in Fig. 8 we displayed the results of the not-anisotropic study by Bhuiyan et al. 53) using XPS leading to b = 2.16 eV, close to b ⊥ of this work. It is noticeable that all values by Bhuiyan et al. 53) in Fig. 8 are below than those reported in this work, however the determination by XPS leads to the fundamental, in α-Ga 2 O 3 indirect, bandgap not the fundamental Γ-point transition. Since also Al 2 O 3 possibly has an indirect bandgap, 69) based on our results this seems to hold true for the ternary alloy system as well. Thus, unlike Hilfiker et al., 34) we do not expect a change in the direct/indirect nature of the bandgap in α-(Al x Ga 1−x ) 2 O 3 .

Summary
In conclusion, we investigated α-(Al x Ga 1−x ) 2 O 3 thin films grown by MOCVD on m-plane sapphire, with XRD, IR and visible-UV spectroscopic ellipsometry. We find the lattice parameters and the anisotropic complex DF in the IR and UV spectral regions. The IR DF yields the shift of the IR active phonon modes with Al content. A linear shift to higher wavenumbers with increasing x is found, in agreement to literature. 52) An evaluation of the UV DFs using the Shokhovets model 64) yields the dielectric limits ε ∞ . The trend in ε ∞ with Al content fits to the earlier results, 51) following a bowing model. Finally, at the onset of strong absorption we find the Γ-point transition energies. A strong increase, with increasing x, also in accordance with a bowing model, is shown. We determine anisotropic bowing parameters of b ⊥ = 2.1 eV and b || = 1.7 eV.

Supplement
See the supplementary material for a comparison of the UV pbp DFs in the region of photon energies lower than the absorption onset and their corresponding Shokhovets-fit, used to determine the dielectric limit ε ∞ . Also displayed is a comparison of the UV pbp-DFs and the corresponding model DFs used to extract the Γ-point transition energies displayed in Fig. 8.