Optical properties of corundum-structured In₂O₃

Over the last decade, In₂O₃ and other semiconducting oxides such as Ga₂O₃ have been attracting a growing interest as technologically relevant wide gap materials.^1,2 When doped with tin, In₂O₃ is a transparent conductive oxide exhibiting high electron mobilities with applications in transparent contacts for solar cells,³ thin film transistors,^4,5 and light-emitting diodes and lasers.⁶ While the stable bixbyite structure of In₂O₃ has been extensively investigated,^1,7–10 studies of a metastable rhombohedral polymorph α-In₂O₃ are scarce and preliminary.¹¹ 

Bandgap engineering in the α-In₂O₃–α-Ga₂O₃–α-Al₂O₃ system opens the possibility of tuning the bandgap from 3.8 to 8.8 eV.¹² A corundum-structured Ga₂O₃ on the c-plane of a sapphire substrate was obtained by a mist chemical vapor deposition (MCVD).¹³ Later, growth of α-In₂O₃ by MCVD on sapphire substrates with Fe₂O₃ buffer layers was reported,¹⁴ and more recently, α-In₂O₃ directly grown on sapphire substrate by MCVD has been achieved.^15,16 The vibrational and dielectric properties of high-quality single-crystal α-Ga₂O₃ films grown by MCVD were thoroughly investigated,^17–21 providing insight into the band structure in the form of effective masses.²² A similarly good knowledge of the optical properties of the corundum-structured In₂O₃ phase is required to advance toward novel devices based on this III-oxide system.

Whereas the Raman-active phonons of bixbyite In₂O₃ have been extensively investigated and their symmetry assignment has been performed by polarized Raman scattering,⁷ most Raman spectra of α-In₂O₃ reported in the literature have been obtained from nanocrystalline powders with limited crystallinity,^23,24 which precluded the symmetry analysis of the Raman modes. An unpolarized Raman spectrum of an α-In₂O₃ thin film grown by metal organic chemical vapor deposition (MOCVD) showing a residual bixbyite component was reported,¹¹ but the observed modes were assigned in reference to the modes of the rhombohedric BiFeO₃ crystals, which belongs to a different space group (⁠$R3c$⁠) than α-In₂O₃ (⁠$R3¯c$⁠). On the other hand, the very nature of the bandgap of In₂O₃ has been controversial owing to its strong optical absorption in the ultraviolet at $∼3.8$ eV and weaker absorption below 3 eV, which early interpretations ascribed to an indirect gap. More recent studies have attributed the gradual absorption onset to dipole-forbidden or low-dipole-intensity direct transitions in bixbyite In₂O₃ and to low density of states and nonparabolicity of the conduction band in the case of the corundum-structured In₂O₃.^25,26 Some dispersion of the gap values reported in the literature exists owing to the characteristics of the samples and the methods of analysis.^26,27

In this Letter, we present a study of the optical properties of a high-quality, single-phase α-In₂O₃ epilayer by means of Raman scattering and spectroscopic ellipsometry measurements. An epilayer of thickness around 800 nm was grown on a c-plane sapphire substrate by MCVD. Mist particles were formed by a 2.4 MHz ultrasonic transducer and carried by a flow of O₂ gas to the reaction chamber heated at 550 °C, where the (0001) sapphire substrate was placed on a sample holder. The details of the growth are described in Ref. 16. X-ray characterization of the sample revealed a high crystalline quality, showing $2θ/θ$ scans with a distinct (0006) α-In₂O₃ peak and no trace of bixbyite reflections. Hall effect experiments yielded a free electron concentration of $7.6×1018$ cm⁻³ and a mobility of $151 cm2/V s$⁠. All characterization experiments were carried out at room temperature.

Raman scattering measurements were carried out using a Jobin-Yvon T64000 spectrometer equipped with a LN₂-cooled charge coupled detector (CCD). Confocal micro-Raman measurements with a pinhole aperture of 100 μm at the back focal plane were performed to minimize the signal from the sapphire substrate. The Raman spectra were excited with the 532 nm line of a Nd:YAG laser. The light was collected in backscattering configuration through an 80× objective with a numerical aperture of 0.75, and the polarized Raman spectra were recorded both on the c-face and on a lateral fracture face with a spectral bandwidth of $∼2.2$ cm⁻¹. The frequency and symmetry of the α-In₂O₃ modes were calculated using density functional perturbation theory (DFPT) as implemented in the abinit package.²⁸ The calculations were made in the local density approximation (LDA) with Perdew–Wang exchange correlation, a 6 × 6 × 6 k-point grid, and a plane wave basis set with an energy cutoff of 70 Ha. The frequencies of the phonon modes and their symmetry are listed in Table I.

Spectroscopic ellipsometry was performed using two different instruments spanning together a spectral range from 250 cm^–1 (⁠$≈$30 meV) up to 6.5 eV. The spectral resolution of the Fourier-transform infrared ellipsometer was set to 4 cm^–1 while for the visible and ultraviolet (UV) spectral ranges, an instrument equipped with an autoretarder was used. The spectral ranges of both ellipsometers overlap. Experiments were performed at three different angles of incidence $Φ$ of $50°$⁠, $60°$⁠, and $70°$⁠.

Figure 1(a) shows the Raman spectra of the α-In₂O₃ epilayer in different backscattering configurations, where z is along the c axis and $x′$ is taken to be along the $(101¯0)$ direction.¹⁷ All spectra are dominated by the strong $A1g$ mode at 163.1 cm^–1, which mainly involves In atoms motion along the c axis. The intensity ratio of this low-frequency $A1g$ mode to all other modes is much higher than in α-Ga₂O₃,¹⁷ suggesting a significantly higher In–O bond polarizability. The $A1g$ modes exhibit stronger polarizability than the E_g modes, typically more than one order of magnitude higher, resulting in the Raman spectra being dominated by the $A1g$ modes. This is in good agreement with the Raman intensities derived from the calculated polarizabilities, which are shown in Fig. 1(b) for the $z(xx)z¯$ configuration. Although the polarizability of the high-frequency $A1g$ mode, which predominantly involves motion of the O atoms perpendicularly to the c axis, is notably higher than that of the low-frequency $A1g$ mode, its Raman intensity is lower because the Raman scattering cross section is inversely proportional to the mode frequency,²⁹ and the occupation factor is also lower. Furthermore, the low-frequency $A1g$ Raman peak exhibits a very narrow linewidth, which reflects the long lifetime of this mode as a consequence of the lack of anharmonic decay channels into lower energy phonons. In contrast, the linewidth of the high-frequency $A1g$ mode is $∼17$ cm⁻¹ and the spectral power of the mode is distributed over a wider spectral range.

It should be noted that the diagonal component of the polarizability tensor for the E_g (219.2 cm^–1) mode is exceedingly small, which makes this mode barely visible in parallel polarization spectra. In contrast, the off diagonal component is comparable to those of the other modes and, consequently, the E_g (219.2 cm^–1) mode can be distinctly observed in the $x′(y′z)x′¯$ spectrum. The weak signal from the forbidden $A1g$ modes detected in the crossed polarization spectra may arise from leakage from other polarizations in slightly misaligned samples, particularly in the case of measurements on the lateral face, where fine control of the film orientation is less precise.

The parallel polarization spectra display weak broad bands that are especially noticeable in the 150–300 cm^–1 range. These bands are attributed to second-order scattering. Overtone scattering can be estimated by considering the phonon density of states with a doubled frequency axis and weighted by the statistical factor $(1+nBE)2$⁠, where $nBE$ is the Bose–Einstein distribution function.³⁰ Using a smoothed phonon density of states derived from our DFPT calculations, we obtain the phonon overtone distribution (POD) shown as a dotted line in Fig. 1. The calculated profile exhibits a prominent band in the low frequency range that is in reasonable agreement with the background signal of the Raman spectra.

The phonon dispersion and density of states obtained from our DFPT calculations are shown in Fig. 2. As can be seen from the plots of the projected PDOS, phonons below 250 cm^–1 correspond predominantly to In motion, whereas above 250 cm^–1, the vibrations mostly involve O motion, without any significant phonon gap between both kinds of modes. There is no phonon gap between acoustic and optical modes either, as low-frequency optical modes dominated by In motion cross the acoustic phonon branches and form nondispersive bands. This may contribute to enhance phonon–phonon scattering and limit the thermal conductivity.³¹ These nondispersive bands also play a major role in the second order scattering observed in the spectra [see Fig. 1(a)].

Spectroscopic ellipsometry yields the ellipsometric angles $Ψ$ and Δ, corresponding to amplitude ratio and phase difference of the different polarization orientations of the reflected light. The experimental results were analyzed using a multi-layer model shaped to the sample structure to determine finally the complex dielectric function (DF). In this case, the layer model consists of α-Al₂O₃ to describe the substrate and on top, the α-In₂O₃ layer of interest. In the UV range, an additional effective medium layer that accounts for surface roughness using Bruggeman's³² formalism was added. As demonstrated previously for cubic and hexagonal $InN$^33,34 as well as for bixbyite In₂O₃,⁸ this low refractive index layer also accounts for the electron accumulation at the surface of α-In₂O₃.²⁶ From UV-ellipsometry, the thickness of the α-In₂O₃ film and the effective medium layer were determined to be 796 and 9.1 nm, respectively.

In the infrared, on top of a sapphire substrate model,¹⁹ the DF of the α-In₂O₃ layer was numerically fitted point-by-point (pbp) until best agreement with the experimental ellipsometric angles was achieved. This pbp DF was line shape fitted by an analytical model to determine optical parameters such as phonon wavenumbers in the infrared or transition energies in the UV. Due to the c-plane orientation of the sample, only the ordinary DF $εord$ is accessible by spectroscopic ellipsometry, i.e., with the electric field vector perpendicular to the c-axis of the crystal.

The characteristic phonon energies are extracted from the pbp DF by assuming an analytic form of the DF containing the dielectric limit $ε∞$ and the IR allowed E_u phonons of the corundum-structured crystal structure, using Lorentzian broadened phonon oscillators with the phonon frequency ω_TO and the broadening parameter γ_TO. To account for the free-electron contribution, a Drude term is added to the DF:³⁵ 

It should be noted that out of the four infrared-active TO phonons (E_u) expected in the ordinary DF, only three are visible within our spectral range. According to our DFPT calculations, the fourth mode is expected at lower frequency (see Table I) and it should exhibit an extremely low oscillator strength.

Figure 3 depicts the real (blue) and imaginary (red) parts of the pbp DF together with the best fit of Eq. (1). The phonon frequencies of the E_u phonons detected are in good agreement with the theoretical values (see Table I). Furthermore, the observed intensities also agree qualitatively very well with the calculated oscillator strengths of the respective modes. Similar to α-Ga₂O₃ and α-Al₂O₃, the lowest and the highest wavenumber phonon have a very low intensity.^22,44 The $A1u$ phonons are only allowed in the extraordinary DF, and thus, they are not accessible experimentally in our sample. The fitted plasma frequency $ωP=1413$ cm⁻¹ and the plasma broadening $γP=406$ cm⁻¹ are averaged over the high-electron density accumulation surface layer and the lower carrier concentration bulk α-In₂O₃. Therefore, this result is not directly comparable to the Hall-effect experiment in such a case. The dielectric limit is found to be $ε∞=3.9 ± 0.1$⁠.

For the UV spectral range, the agreement of a model DF to ellipsometric angles $Ψ$ and Δ is shown in Fig. 4. The difference between the experimental data and the model in the range from 0.5 eV to ca. 3.5 eV is only of minor importance, since this is below the absorption edge and the optical properties are, therefore, not significantly affected. The imperfect match of the Fabry–Pérot oscillations leads to the layer thickness value becoming less precise. A possible explanation for the difference could be layer thickness inhomogeneities. The resulting real (blue) and imaginary parts (red) of the DF in the UV are shown in Fig. 5 (continuous curves). The used model function consists of a substrate model³⁷ and a modified version of Elliott's model³⁸ for the In₂O₃ layer, with the imaginary part

including two direct allowed interband transitions, very similar to the case of the ordinary DF in α-Ga₂O₃,³⁶ which is also shown in Fig. 5 as a dashed gray curve for comparison. The real part is obtained by Kramers–Kronig transformation. α-In₂O₃ is a uniaxial material, so especially the region around the onset of strong absorption has to be considered carefully.³⁹ Advanced ellipsometry data analysis frequently makes use of an analytical model to account for the extraordinary dielectric function on c-plane samples.^40,41 In the current work, the line shape of the extraordinary dielectric function of α-In₂O₃ was chosen similar to the case of α-Ga₂O₃, but with the absorption onset lowered to the region of the ordinary DF of α-In₂O₃. While the agreement of model and data was improved, the overall result of the ordinary dielectric function only changed marginally. Thus, the same procedure was not necessary for the IR dielectric function. In the case of α-In₂O₃, the Elliott model yields two distinct absorption steps at characteristic energies of 3.38 and 3.86 eV, respectively. A number of theoretical and experimental values for the absorption onset reported in the literature lie in the region around 3–3.3 eV.^25–27,42 Nevertheless, more recent experiments by Nishinaka and Yoshimoto⁴³ found a value of 3.75 eV. Our result of two absorption steps provides a refinement of the current state of knowledge and fits very well to theoretical findings,²⁵ where also two steps can be seen.

In summary, we have presented a comprehensive study of the optical properties of a single-phase α-In₂O₃ thin film grown by the MCVD method. The Raman spectra attest the high quality of the crystal and they do not show any trace of the stable bixbyite phase. All Raman active modes of the rhombohedric phase are observed and their symmetries identified by means of polarized Raman measurements. The infrared-active modes accessible from the spectroscopic ellipsometry measurements are determined from the pbp-fitted complex DF. The frequencies of both the Raman and infrared-active modes are in excellent agreement with DFPT calculations in the LDA approximation. In the UV region, the imaginary part of the DF exhibits two distinct onsets of absorption at 3.38 and 3.86 eV.

See the supplementary material for the ellipsometric angles $Ψ$ and Δ measured in the IR spectral range at three different angles and the corresponding pbp fits. A comparison of pbp fits based on isotropic and anisotropic models is performed. The extraordinary DF used in the anisotropic model employed in the UV spectral range is plotted in the spectral region around the onset of strong interband absorption.

This work has been partially supported by the Generalitat Valenciana under Grant No. Prometeu/2021/066 and was funded in part by the Leibniz Science Campus GraFOx II. The authors would like to thank Mr. H. Yokoo of Kogakuin University for sample preparation.