Lattice thermal conductivity of β-, α- and κ- Ga2O3: a first-principles computational study

The thermal transport properties of Ga2O3 in different phases remain inadequately explored. We employ first-principles calculations and the phonon Boltzmann equation to systematically study the lattice thermal conductivity of β-, α- and κ-Ga2O3. Our results reveal that κ-Ga2O3 exhibits pronounced phonon anharmonicity due to its complex polyhedral configurations and weak bonding, resulting in significantly lower lattice thermal conductivity compared to β- and α-Ga2O3. This work provides critical knowledge of the fundamental phonon thermal transport properties of different-phase Ga2O3, as well as helpful guidance for the thermal design of Ga2O3-based high-power devices.

complex crystal structures of Ga 2 O 3 complicate theoretical simulations, and obtaining high-quality pure Ga 2 O 3 crystals for measurements is not easy.][26][27][28] For β-Ga 2 O 3 , there exists certain disparities between the lattice thermal conductivity obtained using different experimental and computational methods (see Table II for details).Furthermore, theoretical investigation on the thermal transport properties of α-Ga 2 O 3 and κ-Ga 2 O 3 is scarce.Using first-principles calculations and a fully iterative solver for the phonon Boltzmann equation, this study systematically investigates the thermal transport properties of β-, αand κ-Ga 2 O 3 , which possess intricate polyhedral configurations.This work aims to provide insights into the phonon and thermal transport mechanisms of β-, αand κ-Ga 2 O 3 phases, offering guidance for the thermal design and thermal management of highpower Ga 2 O 3 electronic and optoelectronic devices.
The optimum crystal structures of β-, αand κ-Ga 2 O 3 were determined using the Vienna ab initio simulation package based on density functional theory (DFT). 29)In our calculations, we utilized the generalized gradient approximation with the revised Perdew-Burke-Ernzerhof prescription for solid (PBEsol) for the exchange-correlation functional. 30)he electron-ion interaction was accounted for using the projector augmented wave (PAW) method. 31)A plane-wave cutoff energy of 520 eV was applied for all crystal structures, and the ionic iterations converged at an energy criterion of 10 −8 eV.The Hellmann-Feynman forces on each ion remained below 0.001 eV Å -1 during the full lattice relaxation.The Brillouin zones of the unit cells were represented using the Monkhorst-Pack special k-point scheme with grid meshes of 5 × 20 × 10, 10 × 10 × 4, and 10 × 6 × 6 for β-, αand κ-Ga 2 O 3 , respectively.
The optimized structures of β-, αand κ-Ga 2 O 3 are shown in Fig. 1.The calculated lattice parameters for these DFToptimized structures are in good agreement with both previous computational and experimental data (see Table SⅠ in supplementary data).β-Ga 2 O 3 has a monoclinic structure belonging to space group C2/m.In this structure, two nonequivalent Ga ions occupy distorted octahedral and tetrahedral sites, while three non-equivalent O atoms (small balls of various colors) are arranged in a distorted cubic pack with three distinct sites as shown in Fig. 1(a).There are no shared faces between the polyhedral, and the shortest Ga-Ga distance measures 3.04 Å.The Ga-O distances range from 1.842 to 1.873 Å within the tetrahedra, whereas they are longer within the octahedra, spanning from 1.943 to 2.080 Å.I summarizes the Ga-O and O-O bond lengths and average bond lengths within octahedral and tetrahedral environments for β-, αand κ-Ga 2 O 3 structures.We can clearly observe a continuous increase in the average Ga-O and O-O distances of octahedra/tetrahedra amongst different phases of Ga 2 O 3 , progressing from βto αand then to κ-Ga 2 O 3 .This trend suggests that the κ-Ga 2 O 3 phase possesses a more complex and weaker chemical bond environment.Therefore, it is anticipated that κ-Ga 2 O 3 may exhibit the lowest lattice thermal conductivity, for which the complex crystal structure and bonding environment tend to induce significant lattice anharmonicity and thus phonon scattering.
To further investigate the lattice vibrational properties, we calculated the phonon dispersion curves for β-, αand κ-Ga   Figure 3 illustrates the temperature-dependent κ L values computed along three crystallographic axes within our investigation.Given that β-Ga 2 O 3 possesses a monoclinic structure, the [001] direction does not align with the z-axis.Hence, we employed the methodology outlined by Liu et al. 38)  ( ) whereas κ [100] and κ [010] correspond to the components of the thermal conductivity tensor, κ xx and κ yy , respectively.Notably, our calculations clearly reveal anisotropic characteristics in κ L of β-, αand κ-Ga 2 O 3 , as indicated by the varying sizes of the marked areas (Fig. 3).Among them, β-Ga 2 O 3 shows the most pronounced anisotropy, while κ-Ga 2 O 3 shows the weakest anisotropy.As the temperature increases, the κ L values of all three phases decrease, and the anisotropy in lattice thermal conductivity diminishes concurrently.The calculated κ L values and experimental values for β-, αand κ-Ga 2 O 3 at 300 K or room temperature (RT) are detailed in Table II.The calculated κ L values at 300 K for β-Ga 2 O 3 along the [100], [010] and [001] directions are 10.63, 18.93, and 13.9 W m −1 K −1 , respectively, and these results exhibit a close agreement with experimental measurements. 5,39)In contrast, the calculated κ L values for α-Ga 2 O 3 are κ [100] = κ [010] = 8.95 W m −1 K −1 ; and κ [001] = 12.95 W m −1 K −1 at 300 K, these are in reasonable accordance with the findings reported by Yang et al., 28) although the crystallographic direction corresponding to the maximum κ L differs, which may be attributed to a smaller cutoff radius 28) for force constant calculations.In comparison to β-Ga 2 O 3 and α-Ga 2 O 3 , κ-Ga 2 O 3 exhibits nearly isotropic κ L in the temperature range of 200 to 1000 K, with an even lower average κ L of approximately 3.40 W m −1 K −1 .The notable differences in κ L among different Ga 2 O 3 phases are primarily  Table II.Lattice thermal conductivity of β-, αand κ-Ga 2 O 3 at 300 K (or RT), respectively.The unit is Wm −1 K −1 .
To gain further insight into the phonon heat transport processes, we initially obtained the anharmonic scattering rate and group velocity for the β-, αand κ-Ga 2 O 3 phases at 300 K.This allowed us to elucidate the microscopic origins of the differences in κ L at different Ga 2 O 3 phases.Figure 4(a) illustrates the phonon-phonon scattering rate as a function of phonon frequency, highlighting the varying degrees of lattice anharmonicity in β-, αand κ-Ga 2 O 3 .It is particularly worth noting that in the low-frequency phonon range (0-5 THz), κ-Ga 2 O 3 exhibits significantly higher scattering rates compared to β-Ga 2 O 3 and α-Ga 2 O 3 .Interestingly, there are sudden changes in scattering rates near the maximum phonon frequency of the three Ga 2 O 3 phases, which could be attributed to the presence of phonon flat bands leading to a decrease in interactions with optical phonon branches (see Fig. 2).This observation is consistent with the phonon group velocity trends depicted in Fig. 4(b).Furthermore, the magnitude of phonon group velocity is positively correlated with κ L .It is evident that κ-Ga 2 O 3 , in contrast to β-Ga 2 O 3 and α-Ga 2 O 3 , exhibits notably low phonon group velocities overall.The above observations also provide an intuitive explanation for the remarkably low L in κ-Ga 2 O 3 .
Figure 5 presents the cumulative κ L values for β-, αand κ-Ga 2 O 3 as a function of the phonon mean-free path at 300 K.It is evident from Fig. 5 that phonons with different meanfree paths along the [010] and other directions are responsible for the anisotropic thermal conductivities of β-Ga 2 O 3 .Similar results have been verified in bulk WTe 2 . 42)The maximum phonon mean-free path in β-Ga 2 O 3 (∼705.48nm, [010]) exceeds that of α-Ga 2 O 3 (∼335.16nm, [100] and [010]).However, for κ-Ga 2 O 3 , the maximum phonon mean-free path significantly falls to ∼75.64 nm along [100], which is much smaller than that of β-Ga 2 O 3 and α-Ga 2 O 3 .Given the intrinsic low κ L of Ga 2 O 3 , enhancing the thermal transport capabilities for device applications can be achieved by incorporating high thermal conductivity substrates, such as diamond (>2000 W m −1 K −1 ). 43)Additionally, the film thickness of Ga 2 O 3 cannot be ignored because as noticed from Fig. 5, sample dimensions of β-Ga 2 O 3 and α-Ga 2 O 3 below 100 nm will decrease κ L by over 20% due to increased phonon boundary scattering.
In summary, we have conducted a comprehensive investigation into the lattice dynamics and thermal transport properties of β-, αand κ-Ga 2 O 3 using first-principles calculations.Our findings reveal that β-Ga 2 O 3 exhibits strong κ L anisotropy, with κ L values along the [100], [010] and [001] crystallographic directions being 10.63, 18.93, and 13.9 W m −1 K −1 at 300 K, respectively, in good agreement with the measured data. 5,39)Furthermore, αand κ-Ga   011001-4 © 2023 The Author(s).Published on behalf of The Japan Society of Applied Physics by IOP Publishing Ltd phonon boundary scattering, which has practical implications for thin film Ga 2 O 3 devices with further reduced thermal conductivity.

Figure 1 (
b) shows that α-Ga 2 O 3 belongs to the rhombohedral corundum structure R 3 ¯c, with a unit cell containing 20 atoms and all Ga and O atoms being equivalent.The structure is exclusively characterized by distorted octahedra centered on Ga atoms, where faces, edges, and corners are simultaneously shared.The Ga-O distances within the tetrahedra range from 1.931 to 2.081 Å, while the O-O distances span from 2.659 to 3.030 Å.As shown in Fig. 1(c), κ-Ga 2 O 3 belongs to the orthorhombic phase Pna2 1 , where octahedra share edges or vertices with other octahedra or tetrahedra.The primitive cell comprises 40 atoms, including four nonequivalent Ga atoms and six non-equivalent O atoms.The intricate tetrahedra/octahedra formation involves different Ga atom states (represented by different colors).Table 2 O 3 as shown in Fig. 2. It is evident that the cutoff frequency of the acoustic phonons in κ-Ga 2 O 3 only reached ∼2.95 THz, significantly lower than that of β-Ga 2 O 3 (∼4.60THz) and α-Ga 2 O 3 (∼4.74THz).This difference

Fig. 2 . 2 ©
Fig. 2. (a)-(c) Phonon dispersion and (d)-(f) participation rate for β-, αand κ-Ga 2 O 3. Detailed distribution of one longitudinal acoustic (LA) and two transverse acoustic (ZA and TA) branches are represented by blue dash, red and green lines, respectively.The 0.5 value of the participation ratio is illustrated as a vertical dashed line.
2 O 3 exhibit weaker κ L anisotropy.Specifically, α-Ga 2 O 3 displays κ L values of 8.95, 8.95, and 12.95 W m −1 K −1 at 300 K along the [100], [010] and [001] crystallographic directions, respectively.Notably, κ-Ga 2 O 3 among the three phases, exhibits the lowest thermal conductivity, with κ L values along the [100], [010] and [001] crystallographic directions being 4.31, 3.11, and 2.77 W m −1 K −1 at 300 K.This remarkably low thermal conductivity can be attributed to its complex polyhedral configurations and weak bonding environment.Moreover, βand α-Ga 2 O 3 's longer phonon meanfree paths in comparison to κ-Ga 2 O 3 highlight the importance of accounting for the size effect on thermal transport due to

Fig. 5 .
Fig. 5. Accumulated κ L along three crystallographic directions with respect to the phonon mean-free path for β-, αand κ-Ga 2 O 3 at 300 K, respectively.The vertical black dashed, dotted, and solid lines correspond to the maximum phonon mean-free paths for β-, αand κ-Ga 2 O 3 , respectively.
to calculate κ L in the [001] direction, k =

Table I .
Average distances of Ga-O and O-O bonds in β-, αand κ-Ga 2 O 3 , respectively.