Thermal and Mechanical Properties

Thermal and mechanical properties of p-Ga₂0₃ are important during bulk crystal growth (heat transport, growth orientation, stress, cleaving), epitaxial growth of layers (substrate orientation, heat transport, growth kinetic), as well as fabrication (cleaving) and operation (heat dissipation) of devices. Monoclinic system of p-Ga₂0₃ determines the anisotropy of physical properties, which take a form of tensors instead of single values. Thermal properties (such as thermal conductivity and thermal expansion) can be described by a second rank tensor that requires four independent components. Elasticity must be described by a higher rank (4) tensor with considerably more components.

Melting Point and Density

The MP of p-Ga₂0₃ was typically measured with a pyrometer and resulted values span in a wide range, between 1725°C and 1850°C [153, 207, 225, 288, 289]. More accurate measurements of the MP were carried out by Hoshikawa et al. [233] with the use of type В thermocouple in the Bridgman configuration. The complete melting of p-Ga₂0₃ was recorded at 1793°C.

Liquid density of Ga₂0₃ was measured by Dingwell [290] using an Ir double-bob Archimedean method. The experimental liquid density is described by pi = 4.8374(84) - 0.00065(12)(7' - 1950°C) [g/cm³]. At 1800°C, which is around the MP of Ga₂0₃, the liquid density is 4.8736 g/cm³, which is about 82% of the measured [163] and calculated p-Ga₂0₃ solid density of p_s = 5.96 g/cm³.

Thermal Conductivity, Diffusivity, and Specific Heat

Thermal conductivity, diffusivity, and specific heat of bulk p-Ga₂0₃ grown from the melt were measured in a wide range from a low (25 K) to a high temperature of 1473 К by different techniques, such as laser flash method (Galazka et al. [226], Amgalan et al. [291], Villora et al. [292], Mu et al. [266]), 2co and 3co methods (Handwerg et al. [293, 294], Slomski et al. [295]), and time domain thermoreflectance TDTR (Guo et al. [296], Jiang et al. [297]). The measured crystals were either electrically insulating (Mg- and Fe- doping), or semiconducting (undoped and Sn-doped). It was found, however, that the intentional doping (whether toward low (Fe) or high (Sn) electrical conductivity) has a minor impact on the host thermal conductivity [295]. Measured and calculated [298] values of the thermal properties are listed in Table 4.9. Measured thermal conductivity is highly anisotropic with values at RT of 11-13, 13, 21-29, and 15-21 W/mK along the [100], [201], [010], and [001] directions, respectively. For the same orientation order, undoped Czochralski-grown bulk p-Ga₂0₃ single crystals showed at RT the following values: 12.2, 15, 24.6, and 17.6 W/mK, respectively (Fig. 4.38a). Although there are some differences in values gathered by different methods, the anisotropy trend is the same. The thermal conductivity decreases with temperature and at 1200°C approaches the value of about 40% of that at RT. The specific heat capacity has an inverse relation, and its value at RT equals 0.56 J/gK. The thermal diffusivity follows the anisotropy of the thermal conductivity with the values atRT of 3.7,9.6, and 7.1 mm²/s along the [100], [010], and [001] directions, respectively. The anisotropy factor for both thermal diffusivity D[010]/D[001] and thermal conductivity A[010]/A[001], which equals 1.4, was found to be temperature independent below RT [294].

Table 4.9 Orientation-dependent thermal conductivity of bulk 8-Ga₂0₃ crystals at RT

MMg doped (insisting), ^undoped (n_e = 3 x 10¹⁷ cm"³), 6)sn doped (n_e = 3.5 x 10¹⁸ cm'³), MFe doped (semi-insulating)

Low temperature measurements of Czochralski-grown crystals by Handwerg et al. [293] revealed that the thermal conductivity between 150 and 300 Kfollows phonon-phonon Umklapp scattering (°c T~^l), while at lower temperatures, there is an increasing deviation from the phonon-phonon scattering, as shown in Fig. 4.38b. This deviation is assigned to point defect scattering, which might play an important role in thermal conductivity. On the other hand, Jiang etal.

[297] claimed that the thermal conductivity of EFG-grown crystals between 80 and 400 К follows the «^T¹'³ relation.

Figure 4.38 (a) High-temperature-dependent thermal conductivity (by laser

flash method) of undoped, Czochralski-grown bulk p-Ga₂0₃ single crystals along directions perpendicular to the (100), (010), (001), and (201) surfaces, respectively [291]; (b) low-temperature-dependent thermal conductivity of Mg-doped (squares, circles) and undoped (triangles) bulk p-Ga₂0₃ single crystals along the [100] direction. Dashed line: phonon-phonon scattering for T » 9₀, where 9_D is the Debye temperature. Solid line: phonon-phonon- Umklapp scattering, calculated for T< 9₀. Dot-dashed line: additional scattering process using Matthiessens rule for a temperature-independent conductivity. The regions (I), (II), and (III) correspond to high-, intermediate-, and low- temperature regions, respectively. Reprinted from Ref. [293], Copyright 2015, with permission from IOP Publishing.

Thermal conductivity of (3-Ga₂0₃ has four independent components, which are described in the standard orientation (twofold axis parallel [010]) by a tensor (basing on Refs. [294], [295], and [291]):

Thermal Expansion

The thermal expansion of p-Ga₂0₃ is also anisotropic and can be described as a tensor with four independent components. The measurements were performed at low and elevated temperatures (between 5 and 700 K) using bulk crystals and powders. Several independent studies [272, 292, 299, 300] on the thermal expansion along main directions and the angle between a- and c-axes showed, however, discrepancy, as listed in Table 4.10. Lattice expansion measurements were carried out by powder X-ray diffraction [292, 299], high-resolution dilatometry [272], and synchrotron-based high-resolution X-ray diffraction [300]. At low-temperature region, the thermal expansion coefficients (with respect to those at RT) of

1.8 x 10'⁶ K'¹ along the [100], and 4.2 x 10~⁶ K'¹ along the [010] and [001] directions were measured, meaning a substantially homogenous expansion in the (100) plane (Villora et al. [292]). At an elevated temperature range, the following thermal expansion coefficients were measured (with respect to those at RT): 1.54 x 10'⁶ K'¹, 3.37 x 10'⁶ K'¹, and 3.15 x 10'⁶ K'¹ along the [100], [010], and [001] directions, respectively, while the expansion of the angle p between the a- and c-axes is 2.23 x 10'⁵ K'¹ (Orlandi et al. [299]). Detailed measurements of high-quality single crystals by Cheng et al. [300] revealed much smaller values of the thermal expansion coefficient as compared with other studies and also a nonlinear increase in the expansion coefficients with temperature (in a range of 298-1200 K).

Table 4.10 Thermal expansion coefficient of p-Ga₂0₃

Stiffness and Hardness

Elastic stiffness coefficients for Czochralski-grown p-Ga₂0₃ crystals were measured by plate-resonance technique and resonant ultrasonic spectroscopy, as reported by Miller et al. [272]. The tensor of the coefficients c,у around RT is as follows:

Theoretical values of elastic stiffness coefficients reported by Furthmiiller and Bechstedt [161] are quite close to those obtained experimentally by Miller et al. [272].

Basing on the experimental elastic tensor, the bulk modulus B₀, shear modulus G, Young's modulus E, and Poisson's ratio v can be calculated from the following Voigt relations:

The obtained values are as follows: B₀ = 185 GPa, G = 88 GPa, E = 228 GPa, and v= 0.295.

The above-described thermal properties along with the elastic stiffness coefficients enabled to calculate thermal stress distribution in 2-inch-diameter crystals grown by the Czochralski method (by using real thermal field in a growth furnace) at two different orientations along b- and c-axes [272]. The numerical results revealed that the maximum of von Mieses stress generated in a crystal grown parallel to the [010] direction is about 50% lower than that in a crystal grown parallel to the [001] direction.

Another mechanical property of bulk p-Ga₂0₃ crystals includes hardness. Microhardness of p-Ga₂0₃ measured by Guzilowa et al. [301] revealed an average value of 8.91 GPa on an (OOl)-oriented crystal sample. Microhardness and Mohs hardness reported by Mu etal. [266] have values of 8.32, 6.44, and 10.1 GPa, and of 6.39,5.78, and 6.82 for (100), (010), and (OOl)-oriented samples, respectively.

Cleavage Planes

p-Ga₂0₃ suffers from two easy cleavage planes {100} and {001}, which are formed by O(III) and 0(1) atoms, respectively, as shown in Fig. 4.39. The {100} plane is the lowest energy plane and constitutes the easiest cleavage plane [277].

On one hand, the cleavage planes enable easy preparation of crystal samples for fundamental study, as they are neither contaminated with polishing agents nor have a damaged layer. On the other hand, the cleavage planes make the bulk crystal growth and wafer fabrication more difficult. In both the Czochralski and EFG methods, the growth direction is limited to the <010> direction, which is parallel to both cleavage planes. Otherwise, the weight of a growing crystal combined with the thermal stress would easily break the seed along one of the cleavage planes. Slicing bulk crystals into thin wafers perpendicularly to the cleavage planes is quite difficult, especially from large-diameter crystals (>1 inch). Polishing such wafers is a challenging task as well. Finally, cutting such wafers with devices fabricated thereon into individual components would require much technical attention.

Figure 4.39 Unit cell of monoclinic p-Ga₂0₃ with indicated cleavage planes {100} and {001}. Reprinted from Ref. [8], Copyright 2018, with permission from IOP Publishing.

Conclusion

Thermal and mechanical properties (Table 4.11) of solid p-Ga₂0₃, i.e., thermal conductivity, diffusivity, specific heat, thermal expansion, and stiffness, are anisotropic and described by tensors. The thermal conductivity increases by a factor of about 2.5 when changing the direction from the [100] through [201] and [001] to the [010] direction. This shows that from the point of view of power device applications, the wafer orientation becomes important for heat dissipation. A complete set of thermal properties enables to calculate thermal stresses in growing crystals, which is important for large crystal diameters. The presence of two easy cleavage planes makes the crystal growth and wafer fabrication more challenging.

Table 4.11 Thermal and mechanical properties of p-Ga₂0₃

(^Calculated on the basis of experimental data in the corresponding reference.