Static Dielectric Constant of β-Ga₂O₃ Perpendicular to the Principal Planes (100), (010), and (001)

The β-Ga₂O₃ crystals used in the present study were either grown by the Czochralski (Cz) method at the Leibniz-Institut für Kristallzüchtung or by the edge-defined, film-fed growth method (EFG) at Tamura Corp., Japan. To obtain semi-insulating material, the unintentional n-type doping of β-Ga₂O₃ is compensated by magnesium (Cz) or iron (EFG) doping. A detailed description for the EFG²² and the Cz^23–25 growth can be found elsewhere. The Cz crystals were oriented by Laue diffraction and subsequently sawed, cleaved, and polished to obtain samples with the surface orientations (100) and (001), corresponding to the surface normals a^* and c^*, respectively. For the (010) oriented sample (surface normal b) a substrate wafer from an EFG crystal was used. Figure 1a illustrates the axis assignment with respect to the unit cell of β-Ga₂O₃. Figure 1b shows a scheme of the plate capacitor structure used for the AC capacitance measurements. Contacts were deposited by electron beam evaporation of Ti (20 nm) and Au (50 nm) in vacuum at a pressure of few 10⁻⁶ mbar using circular shadow masks aligned congruently on opposite sites. The area A of the contacts and its uncertainty were determined by polygonal area analysis of microphotographs. The thickness d of the samples was measured using a commercial digital dial indicator (Mitutoyo Corp ID-S112SB) with a resolution of 1 μm and an accuracy of 3 μm. The error of d was calculated by the root mean square of the indicator's accuracy and the statistical error from a number of measurements at different spots on the sample. The capacitance was measured using a HP4284A precision LCR meter (Hewlett Packard). We made sure that the loss tangent tan (δ) was smaller than 0.1 to neglect the influence of the conductance on the AC capacitance measurement. The capacitance C was independent from the frequency in the range from 20 kHz to 1 MHz and also independent from the DC bias between −100 V and +100 V. Thus all prerequisites were fulfilled to use the AC capacitance measuring method. For the determination of the relative static dielectric constant ɛ_r, we measured the capacitance at zero bias, to reduce the leakage current at elevated temperatures. A test frequency of 1 MHz was chosen since the LCR meter has the highest accuracy in this range and the loss tangent tan (δ) is smaller with larger frequencies, which mainly plays a role for higher temperatures.

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The temperature dependence of the capacitance was measured between 25 K and 500 K in a Janis CCS-400H/204N closed cycle refrigerator cryostat. The capacitance of the setup (without sample) was determined to be 0.85 pF and was taken into account for the zero correction.

The experimentally determined values for the contact area, the sample thickness and the capacitance at room temperature on the three different planes (100), (010), and (001) are summarized in Table I. The relative static dielectric constant ɛ_r is calculated via the formula for plate capacitors

where ε₀ = 8.854 × 10^{− 12} F/m is the vacuum permittivity.

The resulting ɛ_r perpendicular to the (100), (010), and (001) plane between 25 K and 500 K is plotted in Figure 2. One-sigma error bars are indicated for each measurement by the shaded areas. The relative error in the sample thickness is dominating the calculated error of ɛ_r. A significant difference of ɛ_r up to 25% between the three different orientations is found. However, only a weak temperature dependence is observed for all orientations with ɛ_r increasing by about 0.5 from 25 K to 450 K (Table II). A reliable measurement of the capacitance above 500 K was not possible due to the increased leakage current leading to a loss tangent tan (δ) > 0.1.

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The resulting ɛ_r perpendicular to the (100), (010), and (001) plane at room temperature is summarized and compared to literature data in Table III. For ɛ_r perpendicular to the (100) plane, we confirm the experimental result by Hoeneisen et al.¹⁵ They determined ɛ_r also using AC capacitance measurements, but for crystals grown by the Verneuil technique as well as for flux grown crystals suggesting that extrinsic effects are of minor importance. ɛ_r perpendicular to the (010) and (001) planes were up to now only determined by ellipsometry¹⁸ and density functional theory calculations^19–21. They used a Cartesian system (a, b, c^*) for the ɛ_r tensor, so that ɛ_r perpendicular to (100) measured by us deviates from their value with E ∥ a, which, in our opinion, does not make too much difference and cannot be the reason for the deviations discussed in the following. As the comparison in Table III shows, these reports and our results fairly agree in the magnitude of ɛ_r. However, the order of the ɛ_r values with respect to the crystal axes does not agree. Considering the deviations in the ɛ_r values based on DFT calculations, it does not make sense to compare the absolute values with ours. The ellipsometry results of Schubert et al.¹⁸ disagree with ours within the error bars. They used a generalized Lyddane-Sachs-Teller relation (LST) to determine the relative static dielectric constant in the monoclinic system. The LST is an indirect approach to determine ɛ_r from the phonon modes of an ionic crystal, which may fail in the presence of free charge carriers. In their study β-Ga₂O₃:Sn single crystals with net donor concentrations of (2 – 9) × 10¹⁸ cm⁻³ were used.¹⁸ Since Sn is a shallow donor in β-Ga₂O₃²⁶ nearly full ionization at room temperature can be assumed. Hence, free charge carriers are present in the samples they used, which could explain the discrepancies with our results. There are also reports on polycrystalline and amorphous films, that give an average value over all orientations of 〈ɛ_r〉 = 9.57,¹⁷ 〈ɛ_r〉 = 9.93 ± 0.39, and 〈ɛ_r〉 = 10.2 ± 0.6.¹⁶ The average of the relative static dielectric constant over all orientations is in our case 〈ɛ_r〉 = 11.2 ± 0.2. The larger value can be explained by the fact that amorphous networks are less densely packed than the crystalline phase resulting in a significantly lower relative static dielectric constant for the amorphous phase. This was shown especially for binary and mixed oxides.²⁷

The relative static dielectric constant ɛ_r of β-Ga₂O₃ perpendicular to the planes (100), (010), and (001) has been determined at room temperature to 10.2 ± 0.2, 10.87 ± 0.08, and 12.4 ± 0.4, respectively. ɛ_r increased by about 0.5 with increasing temperature from 25 K to 450 K for all orientations. ɛ_r was directly determined from AC capacitance measurements and the geometry of correspondingly oriented plate capacitor structures. This makes the orientation assignment clearly comprehensible. Since demonstrator devices were realized on β-Ga₂O₃ surfaces with one of the three principal orientations (100), (010), or (001), the electric field direction in the space charge region underneath the Schottky contacts or the gates is essentially perpendicular to one of these planes. Therefore, the values reported here allow an exact device design.

The Authors express their gratitude to A. Kwasniewski, U. Juda and K. Banse for the technical support. The authors also thank M. Albrecht, H. von Wenckstern and L. Vines for the discussion. One of the authors (R. Schewski) acknowledges funding by German Research Foundation (DFG) (grant No. GA 2057/2-1). This work was performed in the framework of GraFOx, a Leibniz-ScienceCampus partially funded by the Leibniz association.

A. Fiedler 0000-0003-3404-0804