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Abstract
We investigate broadband dielectric properties of an undoped β-Ga2O3 (100) crystal by polarization-resolved transmission spectroscopy in the visible to terahertz regions. We observe average transmittances higher than 0.80 together with a slight birefringence for wavelengths below 6.0 μm (wavenumbers above 1660 cm−1), polarization-dependent stopbands for 12.5–65.0 μm (800–154 cm−1) indicative of reststrahlen bands, and a substantial birefringence for frequencies of 2.0–0.29 THz (66–9.8 cm−1). We find that the high transparency is a useful property for optical windows, while infrared vibrations responsible for the stopbands are significantly anisotropic and induce the terahertz birefringence.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Crystalline β-Ga2O3 is a semiconductor with an ultrawide bandgap of 4.5–5.0 eV [1–5] and has found promising applications in high-power electronic devices and ultraviolet optoelectronic devices [6–9]. For a wider range of potential applications, a systematic understanding of optical processes in crystalline β-Ga2O3 above, near, and below the bandgap is highly desirable. So far, anisotropic features originating from its monoclinic structure (with a space group of C2/m, lattice constants of a = 1.223 nm, b = 0.304 nm, and c = 0.580 nm, and a monoclinic angle of β = 103.7°) have been examined in ultraviolet interband transitions [2,3,5,10] and infrared (IR) vibrations [10,11], and these features have been modeled into the dielectric function [12]. More recently, terahertz (THz) spectroscopy has been used to study anisotropic dielectric and conductive properties of crystalline β-Ga2O3 with different surface orientations in insulating and intentionally/unintentionally doped conditions [13–17]. However, the intrinsic dielectric properties of crystalline β-Ga2O3 for various photon energies below the bandgap have not yet been comprehensively understood.
In this paper, we report the polarization-resolved transmission spectra of an undoped β-Ga2O3 (100) crystal that cover the visible to THz regions with more than three orders of magnitude different wavenumbers. The crystal proved to have very different dielectric properties on the larger and smaller wavenumber sides of polarization-dependent stopbands, which were observed for wavenumbers of k = 154–800 cm−1 (wavelengths of λ = 65.0–12.5 μm) and suggest the presence of reststrahlen bands. Average transmittances (i.e., median levels of transmittance between local maxima and minima in interference fringes) for wavenumbers of k ≥ 1660 cm−1 (wavelengths of λ ≤ 6.0 μm) were higher than 0.80 together with a slight birefringence of nc – nb ≈ –0.01, where nb and nc are the refractive indices for polarizations parallel to the b axis and c axis, respectively. Average transmittances for k ≤ 50 cm−1 (ω/2π ≤ 1.5 THz) were significantly lower and polarization-dependent, indicating a substantial birefringence of nc – nb ≈ 0.3. We found that the high transparency is a useful property for visible to mid-IR broadband optical windows, while IR vibrations responsible for the stopbands are significantly anisotropic and induce the THz birefringence.
2. Materials and methods
The original single crystal of undoped β-Ga2O3 in our experiment was grown from a Ga2O3 powder (GAO04PB, Kojundo Chemical Laboratory, with a purity of more than 99.99%) by the floating zone method in a dry air flow with atmospheric pressure [18], where the growth rate was set to 5 mm/h and the upper and lower shafts were rotated in opposite directions with a rate of 15 rpm. The crystal was cleaved into several (100)-oriented flakes; a large thin flake (with a thickness of 45.4 μm mentioned later) was used as the sample for optical measurements after mounted on a stainless-steel holder with a 4.5 mm wide circular hole. A top-view photograph of the sample against graph paper is shown in Fig. 1(a), indicating that the sample was highly transparent for visible light. To evaluate the orientation and crystallinity of the sample, we measured the X-ray diffraction (XRD) of another thicker flake by using a diffractometer equipped with multiple scan modes (SmartLab 9 kW, Rigaku). As shown in Fig. 1(b), the XRD pattern for an ordinary ω−2θ scan confirms that the sample was (100)-oriented; this pattern includes no forbidden odd-order peaks [19], suggesting that the sample was an undoped single crystal with a sufficiently low density of defects. In addition, pole-figure XRD measurements allowed us to find the directions of the b axis and c axis [12,13,18,20].
Fig. 1. (a) Photograph (against graph paper) of an undoped β-Ga2O3 (100) crystal mounted on a stainless-steel holder with a 4.5 mm wide circular hole. (b) X-ray diffraction pattern of the crystal for an ordinary ω−2θ scan, indicating its surface orientation.
We obtained both the transmission and reflection spectra of the sample by using a film-thickness measurement instrument based on a fiber-optic spectrometer (F20, Filmetrics) for wavenumbers of k = 12000–26300 cm−1 (wavelengths of λ = 833–380 nm), where the sample thickness was estimated to be 45.40(26) μm from interference fringes observed. We also measured the transmission spectra in a wide range of k = 40–16700 cm−1 (λ = 250–0.60 μm), exploiting a Fourier transform infrared (FT-IR) spectrometer (VERTEX 70 v, Bruker) with suitable combination of two light sources, three beam splitters, two polarizers, and four detectors. For wavenumbers of k = 9.8–66 cm−1 (frequencies of ω/2π = 0.29–2.0 THz), a home-built THz time-domain spectrometer triggered by a femtosecond laser (Mai Tai SP, Spectra-Physics) allowed us to obtain the complex refractive index spectra ñ(ω) = n(ω) + iκ(ω) [21–24], as well as transmission spectra, from the THz waveforms transmitted through the sample in dry air. Two linear polarizations with electric fields E parallel to the b axis and c axis (E||b and E||c, respectively) were chosen in the transmission measurements, where possible contribution from the off-diagonal components of the dielectric tensor [12,13,20,25] was too small to detect. All the experiments were performed at room temperature.
3. Results and discussion
3.1 Transmission spectra in the visible and near-IR regions
First, we demonstrate high transparency of the sample and estimate its refractive index in the visible and near-IR regions. Figure 2 shows the transmission spectra for polarized light in the visible region measured using the fiber-optic spectrometer, together with the reflection spectrum for unpolarized light. The transmittances T take Tb = 0.80–0.81 and Tc = 0.81–0.82 for the E||b and E||c polarizations (purple and orange curves), respectively; the reflectance R remains approximately 0.18 with interference fringes pronounced at k ∼ 15000 cm−1 (black curve). The sum of T and R reaches 0.99–1.00, indicating that the sample had negligible absorption in the visible region as expected from its ultrawide bandgap nature.
Fig. 2. Transmission spectra of the undoped β-Ga2O3 (100) crystal for visible light polarized parallel to the b axis and c axis, together with the reflection spectrum for unpolarized light. The inset is a magnified view of the transmission spectra for wavenumbers of 12000–13500 cm−1.
The transmission spectra for the E||b and E||c polarizations in the range of k ≤ 16700 cm−1 measured using the FT-IR spectrometer are also shown in Fig. 2 by blue and red curves, respectively. These data contain significantly clearer interference fringes than those observed through optical fibers, with average transmittances (i.e., median levels of transmittance between local maxima and minima) similar to the Tb and Tc values described above. A magnified view of the interference fringes for k = 12000–13500 cm−1 is shown in the inset, where the fringe intervals are somewhat smaller for the E||b polarization than for the E||c polarization. This reveals that the refractive index is slightly larger for the E||b polarization, regardless of the fact that the estimated sample thickness had uncertainty. Analyzing the fringe intervals and average transmittances, we find that the refractive indices for the E||b and E||c polarizations are nb = 1.977(11) and nc = 1.970(11), respectively, at k = 13000 cm−1.
Figure 3 shows the transmission spectra for the E||b and E||c polarizations in the near-IR region, where the combination of optical components in the FT-IR spectrometer was switched at k = 5250 cm−1. The average transmittances in interference fringes remain Tb = 0.80–0.81 and Tc = 0.81–0.82. Analyzing the fringe intervals and average transmittances, we find that nb = 1.955(11) and nc = 1.942(11) at k = 8000 cm−1; nb = 1.942(11) and nc = 1.932(11) at k = 4000 cm−1. Thus, the birefringence in the visible and near-IR regions is estimated to be nc – nb ≈ –0.01, for which the uncertainties of nb and nc tend to cancel each other out. This slight birefringence can be attributed to anisotropic electronic states associated with ultraviolet interband transitions [5]. Note that the normal dispersion seen above is significantly weaker than that reported in [26] presumably because our sample had better crystallinity and fewer mid-gap states.
Fig. 3. Transmission spectra of the undoped β-Ga2O3 (100) crystal for near-IR light polarized parallel to the b axis and c axis.
3.2 Transmission spectra in the mid-IR to THz regions
Next, we examine possible lattice contributions to the transmission spectra expected for smaller wavenumbers. Figure 4 shows the transmission spectra for the E||b and E||c polarizations in the mid-IR region. When the wavenumber k is decreased from 2200 cm−1, the transmittance first decreases gradually from ∼0.81 and then drops rapidly to zero, exhibiting a considerable polarization dependence for k ≤ 1430 cm−1. This suggests the existence of anisotropic IR vibrations, as discussed later in more detail. Stopbands are observed for k ≤ 800 cm−1.
Fig. 4. Transmission spectra of the undoped β-Ga2O3 (100) crystal for mid-IR light polarized parallel to the b axis and c axis.
Figure 5 shows the transmission spectra for the E||b and E||c polarizations in the far-IR and THz regions. Polarization-dependent stopbands are distributed down to ∼154 cm−1; the transmittances in the far-IR data become higher on the smaller wavenumber side of the stopbands and overlap consistently with those in the THz data for both polarizations, exhibiting pronounced interference fringes. The average transmittances for k ≤ 50 cm−1 (ω/2π ≤ 1.5 THz) are roughly estimated to be Tb ∼ 0.57 and Tc ∼ 0.49 by calculating the median levels of the leftmost fringe cycle in the THz data. These Tb and Tc values suggest that the refractive indices in the THz region are significantly larger than those in the visible and near-IR regions and are accompanied by a substantial birefringence with the opposite sign. In the next subsection, we determine nb and nc in the THz region from the original measurement data, taking into account the multiple reflections inside the sample relevant to the interference fringes.
Fig. 5. Transmission spectra of the undoped β-Ga2O3 (100) crystal for far-IR light and THz pulses polarized parallel to the b axis and c axis.
3.3 Birefringence in the THz region
Here, we examine the original transmitted and referenced THz waveforms shown in Fig. 6(a), whose power spectra gave the THz transmittance data shown in Fig. 5, for the E||b and E||c polarizations. The THz electric field E||c is transmitted with a larger phase delay and a smaller amplitude than E||b, indicating that nc > nb. The complex refractive index spectra, obtained from the THz waveforms in the same manner as described previously [21,23,27], are shown in Fig. 6(b). Both nb and nc remain nearly constant versus frequency ω/2π, together with extinction coefficients of nearly zero. Although the spectral shapes were somewhat distorted by residual water vapor in the optical paths of THz pulses, we find that nb ≈ 3.3 and nc ≈ 3.6 at ω/2π ∼ 1.0 THz. Note that such a THz time-domain measurement of an undoped β-Ga2O3 (100) crystal has not yet been reported; Fig. 6(b) gives a reasonable high-frequency extension of the anisotropic permittivity data on an insulating β-Ga2O3 (100) crystal measured earlier in the frequency domain up to 1.0 THz [13]. The birefringence of nc – nb ≈ 0.3 observed here is consistent with those obtained by earlier experimental and theoretical studies [13,20,28,29].
Fig. 6. (a) Transmitted THz waveforms and (b) obtained complex refractive index spectra of the undoped β-Ga2O3 (100) crystal for polarizations parallel to the b axis and c axis.
3.4 Overall spectral features
Finally, let us describe and discuss the overall spectral features of the undoped β-Ga2O3 (100) crystal in the visible to THz regions. Figure 7 shows the polarization-resolved transmission spectra that unify the data in Figs. 2–5 with more than three orders of magnitude different wavenumbers. These broadband transmission spectra reveal that the crystal is transparent on both larger and smaller wavenumber sides of the polarization-dependent stopbands distributed in the range of k = 154–800 cm−1 (λ = 65.0–12.5 μm). The nearly constant average transmittances higher than 0.80 for k ≥ 1660 cm−1 (λ ≤ 6.0 μm) will be a useful property for visible to mid-IR optical windows if the crystal is treated with attention to its slight birefringence nc – nb ≈ –0.01 (nb = 1.94–1.98 and nc = 1.93–1.97).
Fig. 7. Polarization-resolved broadband transmission spectra of the undoped β-Ga2O3 (100) crystal, unifying the spectral data in Figs. 2–5 with the horizontal axes on logarithmic scales.
For k ≤ 50 cm−1 (ω/2π ≤ 1.5 THz), the average transmittances are significantly lower and polarization-dependent, i.e., Tb ∼ 0.57 and Tc ∼ 0.49, owing to the refractive indices nb ≈ 3.3 and nc ≈ 3.6. Since the birefringence nc – nb ≈ 0.3 is 30 times larger than that for k ≥ 1660 cm−1 and has the opposite sign, it is naturally associated with the presence of the polarization-dependent stopbands and attributed to underlying IR vibrations. Indeed, several experimental and theoretical studies have shown the existence of anisotropic optical phonon modes in β-Ga2O3 crystals for k = 140–800 cm−1 [10,20,28,29]. Thus, the off-resonant dielectric responses of such phonon modes to THz driving fields enhance refractive indices anisotropically and lead to the substantial birefringence in the THz region, as described semi-quantitatively by static dielectric constants [20,28,30].
It should be noted that no systematic observation of such broadband transmission spectra including characteristic polarization-dependent stopbands has been reported for crystalline β-Ga2O3. The stopbands can be assigned to reststrahlen bands, which have been predicted theoretically in the case of a different crystal orientation [31] and are likely to appear similarly in the present case. We hope that our polarization-resolved broadband spectroscopy will stimulate further theoretical studies on how electronic states and phonon modes contribute quantitatively to the intrinsic dielectric properties of crystalline β-Ga2O3.
4. Summary
We measured the broadband transmission spectra of an undoped β-Ga2O3 (100) crystal for the E||b and E||c polarizations in the visible to terahertz regions. We revealed that the crystal has very different dielectric properties on the larger and smaller wavenumber sides of polarization-dependent stopbands, which were observed for wavenumbers of k = 154–800 cm−1 (wavelengths of λ = 65.0–12.5 μm) and suggest the presence of reststrahlen bands. Average transmittances (i.e., median levels of transmittance between local maxima and minima in interference fringes) for k = 1660–26300 cm−1 (λ = 6.0–0.38 μm) were Tb = 0.80–0.81 and Tc = 0.81–0.82, together with refractive indices of nb = 1.94–1.98 and nc = 1.93–1.97. This high transparency will be a useful property for visible to mid-IR optical windows if the crystal is treated with attention to its slight birefringence (nc – nb ≈ –0.01).
On the other hand, average transmittances for k ≤ 50 cm−1 (ω/2π ≤ 1.5 THz) were significantly lower and polarization-dependent, i.e., Tb ∼ 0.57 and Tc ∼ 0.49, owing to larger refractive indices of nb ≈ 3.3 and nc ≈ 3.6. Intriguingly, this birefringence (nc – nb ≈ 0.3) is 30 times larger than that for k ≥ 1660 cm−1 and has the opposite sign. The broadband spectral data allowed us to provide the physical interpretation that optical phonon modes responsible for the polarization-dependent stopbands enhance refractive indices anisotropically (through their off-resonant responses to THz driving fields) and lead to the substantial birefringence in the THz region.
Funding
Japan Science and Technology Agency (CREST Grant Number JPMJCR2101); Ministry of Education, Culture, Sports, Science and Technology (X-NICS Grant Number JPJ011438).
Acknowledgments
This work was partly performed under the GIMRT Program of the Institute for Materials Research, Tohoku University (Proposal Number 20G0048). We thank Analysis and Instrumentation Center at Nagaoka University of Technology for letting us use the X-ray diffractometer.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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