Measurements and numerical calculations of thermal conductivity to evaluate the quality of β-gallium oxide thin films grown on sapphire and silicon carbide by molecular beam epitaxy Free

Future applications such as power electronics for AC/DC conversion or faster wireless networks will require devices with superior power density and power switching capabilities. To achieve this, semiconductors such as GaAs, SiC, and GaN are commercially available and are under development to broaden their applications. On the contrary, Ga₂O₃, an ultra-wide bandgap semiconductor, is expected to show superior properties when used for power switching devices, according to the Baliga figure of merit, that measures the power losses in switching devices.¹ In addition, wafers of Ga₂O₃ are available and can be fabricated with high-volume commercial techniques such as the Czocharlaski method or the edge-defined film-fed crystal growth. However, one of the limitations for the use of Ga₂O₃ in high-frequency and high-power switching applications is its lower thermal conductivity. The maximum reported bulk thermal conductivity of Ga₂O₃ at 300 °K is around 26 W/(m·K) in the (010) direction.^2,3 The low thermal conductivity can hinder the full potential of Ga₂O₃-based devices because high temperatures will accelerate the degradation of these devices.^4,5

Recently, many studies focused on the electro-thermal transport in Ga₂O₃ transistors.^6–10 In the case of thin-films transistors, the layers of Ga₂O₃ can be obtained using two prime methods: mechanical exfoliation of the membranes or epitaxial growth on compatible substrates. In the case of exfoliation, devices with membrane thickness in the range of tens to hundreds of nm have been presented.^11–13 Even though many studies used mechanical exfoliation of Ga₂O₃ to fabricate devices, this process cannot be scaled for industrial production because it is possible to control neither the thickness nor the lateral dimension of the membranes. In addition, the membranes and the substrate are coupled by weak van der Waals forces, which results in a low thermal boundary conductance (TBC). In general, the TBC at the membrane interface that has been transferred to a substrate is one order of magnitude lower than the TBC between thin layers grown or deposited using physical/chemical methods on a substrate.^14,15 For the devices with epitaxial grown thin film layers,^16–18 the thickness of layers was around 200–600 nm. The TBC at the interface of the Ga₂O₃ thin-film and substrate will depend on the growth process followed. Most of the previous work focuses on analyzing the electrical characteristic of the devices, but only a handful of studies investigates the TBC at the interfaces of Ga₂O₃ thin-films and the effect of interfaces on the thermal conductivity of thin-films.^14,19

There are multiple methods for the epitaxial growth of Ga₂O₃ thin layers such as low-pressure chemical vapor deposition (LPCVD), metal-organic chemical vapor deposition (MOCVD), metal-organic chemical vapor phase epitaxy (MOVPE), molecular beam epitaxy (MBE), etc.²⁰ The growth of Ga₂O₃ thin-films on high thermal conductivity foreign substrates can provide a pathway for the development of high-power devices. An interesting alternative for heteroepitaxy growth of Ga₂O₃ thin layers is the homoepitaxy growth on a Ga₂O₃-on-SiC composite wafer.²¹ LPCVD and MOCVD require less expensive equipment and can have higher growth rates. The choice of growth method influences both the thermal conductivity of Ga₂O₃ thin layers and the TBC at their interfaces because different growth methods can create different types and concentrations of defects. For example, Song et al.²² showed that there is a trade-off between the quality of Ga₂O₃ grown on c-plane sapphire using MOVPE and the TBC. Samples grown on 6° off cut c-plane sapphire had higher thermal conductivity (10%–30%), but lower TBC than those grown on 0° off cut c-plane sapphire. Sapphire has relatively low thermal conductivity (∼30 W/mK), and epitaxial growth on high conductivity substrates like SiC is highly desired. Even if the quality of Ga₂O₃ on SiC is lower than the other substrates, the thermal performance of this combination can be superior because the thermal conductivity of 4H-SiC is ∼10 times higher than sapphire. MBE can be used for the heteroepitaxy growth of Ga₂O₃ and investigation of the fabrication of thin-films of Ga₂O₃ on SiC.

An important aspect of analyzing Ga₂O₃ grown on foreign substrates is how the polycrystalline nature of the material (grain size) along with the formation of point defects (vacancies) and linear defects (dislocations) will affect its thermal conductivity. The influence of vacancies in the crystal lattice of Ga₂O₃ on its thermal conductivity has been studied by incorporating the defect-induced phonon scattering rate into the solution of the Boltzmann transport equation (BTE)²³ and the molecular dynamic simulations.²⁴ However, neither the influence of linear defects nor the mean grain size of polycrystalline Ga₂O₃ has been considered in the previous studies along with vacancies.

We have published preliminary results on the measurement of cross-plane thermal conductivity of β-Ga₂O₃ and TBC at its interfaces for films grown on c-sapphire and 4H-SiC substrates using MBE.²⁵ In this work, we compared the measured values with theoretical results and used this comparison as a tool to estimate the defect densities (vacancies and dislocations) that might have been created during growth. The measurements were performed using time domain thermoreflectance (TDTR), whereas we used the iterative solution of the Boltzmann transport equation (BTE) to estimate the variation of the thermal conductivity with film thickness and the effect of defects on thermal conductivity. In order to make these estimations, we have used the grain size of Ga₂O₃ measured using atomic force microscopy (AFM) and linear defect density using transmission electron microscopy (TEM), as input to the BTE simulations. The percentage of point defects is one unknown, which is not easy to measure, and we estimated that by comparing experimental and numerical results. In summary, we propose a method to study the effects of different types of defects on the thermal conductivity of epitaxially grown Ga₂O₃, which can be employed to study other materials too. Finally, the diffuse mismatch model (DMM) was used to predict TBC and better explain the experimental results.

Two thin films of undoped Ga₂O₃ were grown on c-sapphire and 4H-SiC by MBE at the U.S. Naval Research Laboratory (NRL). MBE is a physical-vapor epitaxial growth process in high vacuum. For Ga₂O₃ epitaxy, ultra-high pure elemental Ga and reactive oxygen were provided using Ga effusion cell and oxygen plasma, respectively.²⁶ The optimized conditions used to grow Ga₂O₃ on sapphire and SiC, along with further characterization of the films, have been published elsewhere.^25,26 More details about the growth of Ga₂O₃ by MBE can be found in Ref. 27. Also, we used bulk samples of the substrates to measure their thermal conductivities. In both cases (bulk and thin-film samples), we deposited a thin Al transducer (97 and 93 nm for bulk and thin-film samples, respectively) by e-beam evaporation. The TDTR measurements were performed using a Ti:Sapphire laser (wavelength = 800 nm). The frequency modulation of the pump beam can be controlled by an electro-optical modulator and is doubled using a BiBO crystal. On the surface of the sample, the pump radius was ∼9.95 μm, while the probe radius was ∼6.2 μm. Based on the sensitivity analysis, the data obtained at modulation frequencies of 8.8 and 11.6 MHz were used for the samples grown on sapphire and SiC, respectively. Section A of the supplementary material has more details on the TDTR system.

High-resolution transmission electron microscopy (HRTEM) performed on both samples revealed a thickness of 119.4 ± 2.8 and 81.3 ± 1.3 nm for Ga₂O₃ on c-sapphire and 4H-SiC substrates, respectively [see Figs. 1(a) and 1(d)]. In addition, the thickness of the Ga₂O₃ layers was measured by spectroscopic ellipsometry and x-ray reflectometry (XRR). Both HRTEM and x-ray diffraction (XRD) measurements showed β-Ga₂O₃ of orientation (−201) for both samples. Details of the structural characterization can be found in our previous work.²⁶ A bright-field low- and high-resolution phase contrast TEM imaging, selected area electron diffraction (SAED) imaging, and post imaging analysis using fast Fourier transform (FFT) show that Ga₂O₃ thin films on sapphire and 4H-SiC substrate have a crystalline structure [Figs. 1(a)–1(g)]. Defect analysis using images of phase contrast and inverse FFT lattice images revealed that both samples have one-dimensional type of defects (dislocations) and zero-dimensional type of defects. In brief, several TEM images are processed using the FFT to identify the regions of interest, and images of those regions are converted back using the inverse FFT. Then, the linear defects are counted, and this number is divided by the area of the region of interest. More details on the method to estimate defect density can be found in Ref. 28. In this case, the defect density was 2.5 × 10¹² and 1.0 × 10¹² cm⁻² (taken from Ref. 26 as they correspond to the same sample) for the Ga₂O₃/Al₂O₃ and Ga₂O₃/SiC, respectively. More details on the structural characterization of the samples can be found in Sec. B of the supplementary material. The TDTR setup and the data interpretation for this study have been described in Refs. 29 and 30.

We used atomic force microscopy (AFM) to scan the top surface of the samples and analyze the presence of lateral grains (Fig. 2). These measurements revealed that the samples are polycrystalline with a mean lateral grain size of 52 nm. This value was obtained using the intercept technique. In this technique, several random lines are drawn on the micrograph, and the number of grain boundaries intersecting the lines is counted. Then, the average grain size is calculated by dividing the length of the line by the number of grain boundaries. This procedure was repeated for 10 lines to have a representative value for the micrography. It is worth mentioning that Fig. 2 was obtained using the “phase” signal because this signal allows for a better contrast image that reveals the presence of grain boundaries, as opposed to the AFM image presented in Ref. 26, that used the “topographic (height)” signal, which is more suitable to measure the roughness of the sample.

Density functional theory (DFT) calculations in conjunction with Boltzmann transport equations have been used for the estimation of phonon relaxation time and thermal conductivity of Ga₂O₃.³¹ Density functional theory (DFT) calculations were performed using the Vienna ab initio simulation package (VASP) to compute interatomic force constants (IFCs) of Ga₂O₃. A plane wave basis set and the projector augmented-wave (PAW) method were used with the Perdew–Burke–Ernzerhof (PBE) exchange-correlation functional.^32–35 A 500 eV kinetic energy cutoff was used to perform the structure optimization and to calculate the second-order harmonic and third-order anharmonic force constants. The convergence criteria for the energy and force were 10⁻⁹ and −0.001 eV/Å, respectively. The 20-atom β-Ga₂O₃ unit cell was optimized using a 4 × 16 × 8 grid for Brillouin zone sampling. The optimized lattice parameters were a = 12.45 Å, b = 3.08 Å, and c = 5.86 Å, with β = 103.76°, which are in good agreement with recent computational²³ and experimental results.³⁶ The second-order harmonic and third-order anharmonic IFCs were calculated using the finite displacement method with a 1 × 4 × 2 supercell of the optimized 20-atom unit cell.³⁷ The finite displacement distance was 0.01 Å, and a fourth nearest neighbor cutoff was used for computing the third-order IFCs. Using the second- and third-order IFCs, the phonon relaxation times and the anisotropic thermal conductivities of bulk β-Ga₂O₃ were calculated using Fermi's golden rule³⁸ with the iterative solution to the BTE.^39,40 A 5 × 17 × 9 k-space sampling mesh was used for Brillouin zone sampling.

After calculating the IFCs, we solved the linearized form of the BTE using ShengBTE to obtain the thermal conductivity tensor (see Sec. D of the supplementary material for details). The combined effects of sample size,⁴¹ oxygen or gallium vacancies, grain boundaries, and linear defects on the thermal conductivity were considered by adding the phonon boundary scattering rate, the vacancies-induced scattering rate, the grain boundary scattering rate, and the linear defect scattering rate (dislocations), respectively, to the anharmonic phonon scattering rate,⁴² according to the following Matthiessen's rule:

Here, $1/τanh$ is the intrinsic anharmonic phonon scattering rate, $1/τb$ is the phonon boundary scattering rate, $1/τV$ is the phonon scattering rate due to the vacancies, $1/τgb$ is the scattering rate due to grain boundaries, and $1/τld$ is the linear defect scattering rate.

The phonon scattering rate caused by oxygen or gallium vacancies in the crystal can be expressed as^43,44

where $x$ is the density of vacancies, $M$ is the average mass per atom, $MV$ is the mass of the missing atom, $gω$ is the phonon density of states, and $G$ is the number of atoms in the crystal (number of atoms in the unit cell). The grain boundary scattering rate was estimated using the Casimir model:⁴⁵ 

where $p(ω)$ is the specularity parameter, $Davg−1$ is the average grain size of polycrystalline samples, and $vg,a$ is the phonon group velocity along the a direction. In this case, the specularity parameter $p(ω)$ was chosen to be zero, which represents diffusive scattering at grain boundaries. The linear defect scattering rate was estimated using the equation in Ref. 46:

where $a$ is the lattice parameter, $v$ is the phonon velocity, $γ$ is the Grunnessien's constant, $ω$ is the angular frequency, and $Ns$ is the number of linear faults per cm. In contrast to the Debye–Callaway model used by other researchers,¹⁹ using an exact iterative solution of phonon BTE is more accurate and allowed us to obtain the thermal conductivity tensor for layers with different thicknesses and calculate the thermal conductivity along different directions, following the procedure presented in Ref. 47.

For the calculation of the TBC, we used the diffusive mismatch model (DMM) considering not only the acoustic phonon branches but also all branches of the materials because the complex crystalline structure leads to a large number of optical branches and cannot be omitted in the calculations. For example, researchers in Ref. 23 demonstrated that, depending on the orientation of the crystal, optical phonon modes contribute significantly to the thermal conductivity of Ga₂O₃. We implemented the qDMM model, where the integration is performed over the wave vector q.⁴⁸ 

Before determining the thermal conductivity of Ga₂O₃ and TBC at its interfaces with the substrate, we first determined the thermal properties of the sapphire and SiC substrates using two-layered samples. The through-plane and in-plane thermal conductivities of SiC were estimated to be 301.4 ± 36.2 W/(m·K) and 387.3 ± 46.5 W/(m·K), respectively. For sapphire, the thermal conductivity was 27.3 ± 2.0 W/(m·K). These values agreed with the previous studies and were used as constants in the three-layer models. The set of data with the best fitting results was used to estimate the thermal conductivity and the TBC. The samples on c-sapphire (119 nm) had a thermal conductivity of 3.2 ± 0.3 W/(m·K), whereas the thermal conductivity of the sample on 4H-SiC (81 nm) was 3.1 ± 0.5 W/(m·K). The numerically estimated conductivities using BTE for the two thin film samples were 8.9 (119) and 7.9 W/(m·K) (81 nm), which are much higher than the measured values.

The thermal conductivity of crystalline samples of Ga₂O₃ thinner than 120 nm has been hardly reported. Thermal conductivity of thin films of Ga₂O₃ fabricated using PLD has been reported in Ref. 19. The thermal conductivity of the samples fabricated by MBE is slightly higher than those fabricated by PLD with comparable thickness (∼100 nm). It is lower than film grown by MOVPE on sapphire in Ref. 22, but those films were thicker (>164 nm). A comparison of different results is presented in Fig. 3(a). The thermal conductivity of MBE-grown samples could also be compared with single-crystal thin-film samples. For example, the study in Ref. 49 presents the thermal conductivity of monocrystalline Ga₂O₃ thin-films bonded to SiC. H ions were implanted in the Ga₂O₃ crystal before the bonding process, which might have induced strain in the crystal and produced defects. For this reason, the thermal conductivity of a ∼140 nm sample was 2.9 W/(m·K), which is lower than what could be expected for a single crystal sample but comparable with our results. It is likely that the fabrication process and post-fabrication treatment will affect both the thermal conductivity and the TBC.

The phonon dispersion curve for Ga₂O₃ in the (−201) direction was calculated using the second-order harmonic force constants (IFCs) and thermal conductivity from the iterative solution of BTE using second- and third-order IFCs, obtained from the first principles simulations. When comparing the experimental and computational results, we observe that the experimental results in the (−201) direction are ∼2 to 3 times smaller than the results computed from the first principles. This could indicate the presence of unavoidable imperfections during the fabrication of the thin-films of Ga₂O₃. Figure 3(b) shows the variation of the computed thermal conductivity of Ga₂O₃ without imperfections with respect to its thickness for the (−201) direction, where it is evident that the thermal conductivity will reach a plateau, corresponding to the bulk value, similar to the results published in Ref. 19.

To explain the lower thermal conductivity measured from TDTR compared to the numerical predictions, we introduced scattering due to the vacancies of gallium and oxygen, the grain boundaries, and linear defects (stacking faults) while computing conductivities [see Eq. (1)]. The gallium vacancies had a higher impact on the reduction of the thermal conductivity because gallium is heavier than oxygen. Linear defects are first not considered to isolate the effect of vacancies [Fig. 4(a)]. Simulations with 3% oxygen vacancies result in thermal conductivity of 4.2 and 4.0 W/(m·K) for 119 and 81 nm samples, respectively, which is still higher than the measured conductivity and suggests the presence of Ga vacancies in our samples. It was estimated that the 119 nm sample had around 3% of Ga vacancies, whereas the 81 nm sample had around 2.5% of Ga vacancies. Through a combination of high-resolution transmission electron microscopy in combination with DFT predictions⁵⁰ and positron spectroscopy,⁵¹ previous studies determined that the most likely vacancies to occur during fabrication are Ga vacancies, for both bulk and thin films. A high density of linear defects is also probable. For this reason, three levels of linear defects were introduced in the calculations for the 119 nm sample, in addition to Ga vacancies [Fig. 4(b)]. The thermal conductivity for a sample with 1% of Ga vacancies and 10⁶ linear defects/cm was 3 W/(m·K), which is close to the measured value, as opposed to 8.9 W/(m·K) for a perfect crystal [Fig. 4(b)]. Also, the effects of the linear defects are more pronounced when the linear density is at least, 10⁶ defects/cm. The actual linear defects density is 1.5 × 10⁶ defects/cm. This value was obtained by calculating the square root of 2.5 × 10¹² linear defects/cm², which is the linear defect density per unit of area previously reported in Ref. 26.

We measured the TBC at the interface of Ga₂O₃ with the metal transducer and the substrates, SiC and sapphire. Using TDTR, the TBC at the interface of Al/Ga₂O₃ was measured as 125.41 + 15.62/−10.66 MW/(m²·K) for the sample grown on sapphire and 158.18 + 88.37/−41.34 MW/(m²·K) for the sample grown on SiC. The results may indicate that different imperfections that the fabrication method induces in the crystal depend on the substrate itself. The TBC values reported here are in the same order of magnitude of those reported in Ref. 53–55. For the Ga₂O₃/substrate interfaces, a summary of the TBC results is presented in Table I. Details on the methods to calculate the uncertainty can be found in Sec. C of the supplementary material. Due to the low parameter sensitivity, high uncertainty in the estimation of the TBC can be observed, regardless of the calculation method. Our results can be compared with TBC values presented in Ref. 49 for crystalline Ga₂O₃ films bonded to SiC. The bonding of Ga₂O₃ with SiC using surface activation bonding required a coating of the thin layer of Al₂O₃ on Ga₂O₃. For a 30 nm thick Al₂O₃ layer, the reported TBC was ∼70.1 + 68.1/−10.4 MW/(m²·K), whereas for a 10 nm thick Al₂O₃ layer, the reported TBC was ∼100.5 + 93.8/−16.2 MW/(m²·K).

For the theoretical calculations of the TBC at the Ga₂O₃/substrate interfaces following DMM (details in Sec. E of the supplementary material), we used the phonon dispersion curve of Ga₂O₃ obtained from the first principles. For sapphire (orientation 001) and SiC (orientation 001), we used the dispersion curves available in the Materials Project⁵⁸ in the Γ→Ζ and Γ→Α directions, respectively. At 300 K, the calculated TBC using the DMM for the Ga₂O₃/sapphire interface was 294.3 MW/(m²·K), whereas for the Ga₂O₃/SiC interface, it was 357.0 MW/(m²·K). When comparing the experimental and theoretical results, we can see that the experimental mean TBC values are between 2 and 2.5 times smaller than the DMM-based estimations. This difference could be attributed to the defects present at the interface of Ga₂O₃/substrate. One important detail in the estimation of the TBC using DMM to point out is the use of the full dispersion curves for the materials because of the complexity of the crystalline structures and the large number of optical phonons, despite their lower phonon branch velocities. If we had only considered the acoustic phonon branches, the TBC values would have been 51.8 MW/(m²·K) for Ga₂O₃/sapphire and 57.4 MW/(m²·K) for Ga₂O₃/SiC, underestimating the TBC values. Finally, it was suggested that Ga₂O₃ could replace GaN-based devices. Studies in Ref. 59 reported TBC values of 199.8 + 29.3/−30.23 MW/(m²·K) for the GaN/AlN-SiC interface and 224.41 + 22.49/−23.3 MW/(m²·K) for the GaN/SiC interface. Those values are between 30% and 50% higher than the ones we report for Ga₂O₃/SiC, but still lower than the TBC that could be expected for a perfect Ga₂O₃/SiC interface.

We report the thermal conductivity of Ga₂O₃ thin films grown by MBE with thicknesses of 119 and 81 nm, and the TBC at Ga₂O₃/sapphire and Ga₂O₃/SiC interfaces. The measured thermal conductivity is around three times smaller than the numerically calculated conductivity for pristine thin-films of similar thickness with no defects. Calculations of the variation in thermal conductivity with the percentage of vacancies of gallium and oxygen atoms, linear defects in thin-films, and the lateral grain boundaries explain the experimental results. For example, the inclusion of grain boundary scattering, corresponding to grain sizes obtained from AFM, contributed to a reduction of 32% in thermal conductivity compared to a sample with no defects. 1% of Ga vacancies contributed to a further reduction of 28%, and the presence 10⁶ cm⁻¹ linear defects reduced another 30% the thermal conductivity, in the samples grown on sapphire. In all, our results provide reference values of thermal properties of thin-film Ga₂O₃ and its interfaces and suggest the level of defects present in the crystal that could be used to accelerate the design of Ga₂O₃-based electronic devices.

See the supplementary material for a description of the TDTR system, the structural characterization of the samples, details of the uncertainty calculations, the procedure to calculate the theoretical thermal conductivity, and the description of the DMM model for the theoretical calculation of the TBC.

The work at the U.S. Naval Research Laboratory was supported by the Office of Naval Research.

D.V. would like to thank Secretaria Nacional de Ciencia y Tecnologia from Ecuador (SENESCYT) for partially funding his graduate studies.