Investigation of Thermal Properties of β-Ga₂O₃ Nanomembranes on Diamond Heterostructure Using Raman Thermometry

As shown in Fig. 1, the sample preparation started with a (−201) β-Ga₂O₃ (Sn doped with 1 × 10¹⁸ cm⁻¹ donor concentration; Novel Crystal Technology). Firstly, the wafer was cleaved into a small piece, followed by mechanical exfoliation using a well-known Scotch tape method^16,17 (Fig. 1a(i)) In this stage, a typical thickness of β-Ga₂O₃ became several micrometer ranges and such a thin- and freestanding- single crystalline material is called β-Ga₂O₃ NMs.¹⁶ We repeated the exfoliation until the desired thicknesses, namely 4000 nm, 1000 nm, and 100 nm, respectively were achieved. To accurately measure the thickness of β-Ga₂O₃ NMs, a three-dimensional surface profilometer (Profilm3D Filmetric) was used which has a spatial resolution of less than 5 nm (Fig. S1a is available online at stacks.iop.org/JSS/9/055007/mmedia). It should be noted that the surface crystal orientation became (100) after exfoliation because β-Ga₂O₃ has a relatively weaker bonding toward a (100) direction as compared to other directions. When the desired thickness was achieved, free-standing β-Ga₂O₃ NMs were carefully transfer-printed onto a single crystalline diamond substrate (undoped (100) diamond) using an elastomeric stamp (PDMS, Polydimethylsiloxane) as depicted in Fig. 1a(ii)–(v).^25–27 This micro transfer-printing method did not use any adhesive between β-Ga₂O₃ NM and the diamond substrate. Thus, to enable intimate contact between β-Ga₂O₃ NMs and the diamond substrate, it is critical to have an extremely smooth surface roughness. Failure to achieve a smooth diamond surface often results in an incorrect thermal property due to increased phonon scattering at the rough hetero-interface.^28–30 As shown in Fig. S1b in the supporting information, the average root-mean-square surface roughness of both polished diamond substrate and an exfoliated β-Ga₂O₃ NM surface measured using an atomic force microscope (AFM) were found to be below 1 nm, thus it was possible to create a high-quality β-Ga₂O₃ NMs/diamond heterostructure. Also, to further improve the accuracy of the thermal characterization, we focused the Raman laser at the center of β-Ga₂O₃ NMs and tried to avoid any particles which are accidently trapped between β-Ga₂O₃ NMs and diamond. Prior to the transfer-printing step, the diamond substrate was thoroughly cleaned using an ammonium sulphuric acid solution for 30 min at 200 °C and then rinsed in an ammonium hydroxide/hydrogen peroxide solution, followed by a deionized (DI) water rinse to obtain a contaminant-free and native oxide-free surface.³¹ The samples were annealed at 300 °C for 1 min in the rapid thermal annealing (RTA) system to enhance the bonding strength.^32,33 This annealing condition was carefully chosen by considering the thermal expansion coefficient of β-Ga₂O₃ NMs and diamond. Finally, samples were ready to measure thermal conductivity. A micro-Raman spectroscopy (Renishaw Raman spectroscopy) was carried out to inspect the initial quality of β-Ga₂O₃ NM/diamond heterostructures as shown in Fig. 1b. Raman peaks in Fig. 1b that are distributed from 180 cm⁻¹ to 800 cm⁻¹ are responsible for β-Ga₂O₃ NM and a strong peak located at 1332 cm⁻¹ is responsible for single crystal diamond.^17,31 No sign of Raman peak shifting also confirmed that there was no residual strain in β-Ga₂O₃ NM from the integration process.

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The thermal conductivity measurement uses local heating by laser illumination under different power intensities of the green (532 nm) Raman laser (from 0 to 20 mW for a 5× objective lens and from 0 to 10 mW for a 20× objective lens), and corresponding Raman peak shifting under different temperature conditions (from 100 K to 500 K with a 40 K interval). Therefore, it is essential to measure the actual power levels of Raman laser on the sample surface and the heating rate that involves the spot size and corresponding light intensity prior to the actual thermal conductivity measurement. The measurement of the heating rate began with the measurement of the Raman laser spot size and corresponding light intensity. Firstly, laser spot sizes were measured at the edge of a Si wafer under different objective lenses using Raman spectroscopy and an error function,³⁴ yielding laser spot diameters of 1.5 μm, and a 2.5 μm for 20× and 5× objective lenses, respectively. These spot diameters were much smaller than the width of β-Ga₂O₃ NM (∼40 μm); therefore, it was possible to locate the Raman laser beam spot at the center of β-Ga₂O₃ NM during a Raman spectroscopy measurement. To accurately realize the desired power level, a disk-type linear variable filter (Edmond optics) was installed at the bottom of the Raman objective lens to precisely adjust the desired beam power level, which provided a continuous variable transmittance from 0% to 100%. Figure 1c shows a schematic illustration of the Raman thermometry setup used in this experiment. Once actual laser power intensity values were measured, the temperature distribution of β-Ga₂O₃ NM was obtained by the heat diffusion equation in the cylindrical coordinate by Eq. 1³⁵:

$$\begin{eqnarray}&&\displaystyle \frac{1}{r}\displaystyle \frac{d}{dr}\left(r\displaystyle \frac{dT}{dr}\right)-\displaystyle \frac{g}{{K}_{s}t}\left(T-{T}_{a}\right)+\displaystyle \frac{q\prime\prime\prime }{{K}_{s}}=0\end{eqnarray} \tag{ 1 }$$

where r, g, and K_s represent the radius of the beam spot, the interfacial thermal boundary conductance and the thermal conductivity of β-Ga₂O₃ NM. Given bandgap (4.9 eV) of β-Ga₂O₃ NM, a 5% of actual light absorption in β-Ga₂O₃ NM at 532 nm was considered in this calculation from the known absorption coefficient of β-Ga₂O₃.³⁶ We also assume that the heat flux is distributed in a Gaussian profile.^35,37 Thus, the local temperature distribution was calculated from the Gaussian heat distribution using the measured laser spot diameter and Eq. 2³⁵:

$$\begin{eqnarray}&&{T}_{m}=\displaystyle \frac{\displaystyle {\int }_{0}^{\infty }T\left(r\right)\exp \left(-\tfrac{{r}^{2}}{{{r}_{0}}^{2}}\right)rdr}{\displaystyle {\int }_{0}^{\infty }\exp \left(-\tfrac{{r}^{2}}{{{r}_{0}}^{2}}\right)rdr}\end{eqnarray} \tag{ 2 }$$

Assuming a Gaussian profile and a laser spot radius of r₀, this can be used to calculate the volumetric heating power density $q\prime\prime\prime$ as shown in Eq. 3³⁵:

$$\begin{eqnarray}&&q\prime\prime\prime =P\cdot \displaystyle \frac{1}{t}\cdot \displaystyle \frac{1}{\pi {{r}_{0}}^{2}}\exp \left(-\displaystyle \frac{{r}^{2}}{{{r}_{0}}^{2}}\right)\end{eqnarray} \tag{ 3 }$$

Once the laser spot diameters with 5× and 20× objective lenses and the heating rate were determined, the β-Ga₂O₃ NMs/diamond heterostructures with different β-Ga₂O₃ NM thicknesses were placed on the stage in the vacuum chamber of the Raman thermometry system at a base pressure of 1 × 10^–5 mtorr. A series of Raman thermometry was performed under different temperature conditions (from 100 K to 500 K with a 40 K interval). As described in the experimental section, three samples that had different β-Ga₂O₃ NM thicknesses namely, 100 nm, 1000 nm, and 4000 nm (labeled as samples #1, #2, and #3 hereafter) were measured under full laser intensity using 5× and 20× objective lenses at each temperature condition (20 mW for the 5× objective lens and 10 mW for the 20× objective lens respectively). Figures 2a–2c show the Raman spectra from each β-Ga₂O₃ NMs/Diamond heterostructure measured from 100 K (blue) to 500 K (red) with 40 K intervals. After the raw Raman spectra were captured, we traced the shifting of the Raman peak at 199 cm⁻¹, which was caused by a libration of Ga-O bonding as a function of the sample temperature using 5× and 20× objective lenses separately to calculate a degree of Raman shifting as shown in Figs. 2d–2f. It should be noted that the Raman mode at 199 cm⁻¹ was chosen in this experiment because it is one of the most dominant Raman modes in β-Ga₂O₃ NM, however this method also works with other distinctive Raman modes such as A_3g mode at 201 cm⁻¹, A_5g mode at 347 cm⁻¹ and A_6g mode at 416 cm⁻¹.¹⁷ Figures 2d–2f indicate the Raman shift as a function of temperature. The shifting rates of the Raman peak for samples #1 ∼ #3 were −0.7 cm⁻¹/10 K, −0.21 cm⁻¹/10 K, and −0.06 cm⁻¹/10 K with the 5× objective lens and −0.16 cm⁻¹/10 K, −0.12 cm⁻¹/10 K, and −0.03 cm⁻¹/10 K with the 20× objective lens, respectively.

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Then, the Raman shift with respect to the stage temperature with six different laser powers was measured. From a set of Raman spectra measured under different laser powers (from 0 mW to 20 mW for the 5× objective lens and from 0 to 10 mW for the 20× objective lens) and temperature conditions (from 100 K to 500 K) shown in Figs. S2–S4 in supporting information, the rate of the Raman peak shifting was calculated as shown in Fig. 3. Similar to Figs. 2d–2f, Raman shifts measured from samples #1 ∼ #3 with respect to the laser power from 1 mW to 18 mW and from 100 K to 500 K using the 5× and 20× objective lenses were plotted to calculate the slopes. For the Raman spectroscopy with a 5× objective lens, a maximum laser power of 20 mW was used, while a maximum laser power of 10 mW was used with a 20× objective lens due to the different numerical aperture value of the 5× objective lens and the 20× objective lens. The temperature increase in the optically heated β-Ga₂O₃ NM causes the red-shift of the Raman peaks due to Ga–O bond softening.^35,38–40 It should be noted that the use of different objective lenses generates different heating intensity as we discussed in the previous section, thus, the degrees of Raman shifting were also different. The shifting rates of the Raman peak with respect to the laser power for samples #1 ∼ #3 were measured to be −0.05324 cm⁻¹ mW⁻¹, −0.02179 cm⁻¹ mW⁻¹, and −0.01 cm⁻¹ mW⁻¹ with the 5× objective lens and −0.11 cm⁻¹ mW⁻¹, −0.064 cm⁻¹ mW⁻¹, and −0.03 cm⁻¹ mW⁻¹ with the 20× objective lens, respectively.

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After the extraction of two different slope values, namely (1) the Raman peak shifting rate vs. the temperature (denoted as R₁ and R₂ in Fig. 2), and (2) the Raman peak shifting rate vs. the temperature-laser intensity (denoted as R₃ and R₄ in Fig. 3), the calculation for thermal conductivity and interfacial thermal boundary conductance started with Eq. 1, as g and K_s represent the interfacial thermal boundary conductance and the thermal conductivity, respectively, here the thermal conductivity K_s represents the directionally averaged thermal conductivity. For the samples with different β-Ga₂O₃ NM thicknesses, t. In Eq. 4, $q\prime\prime\prime$ denotes a volumetric heating power density in the Gaussian profile.³²

$$\begin{eqnarray}&&q\prime\prime\prime =\displaystyle \frac{{{q}_{0}}^{\prime\prime }}{t}\exp \left(-\displaystyle \frac{{r}^{2}}{{{r}_{0}}^{2}}\right)\end{eqnarray} \tag{ 4 }$$

where q₀‴ is the peak absorbed laser power per unit area at the center of the laser spot. The total absorbed laser power, P, is calculated by Eq. 5³⁵:

$$\begin{eqnarray}&&P=\displaystyle {\int }_{0}^{\infty }{{q}_{0}}^{\prime\prime }\exp \left(-\displaystyle \frac{{r}^{2}}{{{r}_{0}}^{2}}\right)2\pi rdr={{q}_{0}}^{\prime\prime }\pi {{r}_{0}}^{2}\end{eqnarray} \tag{ 5 }$$

and q₀'' can be rewritten as Eq. 6:

$$\begin{eqnarray}&&{{q}_{0}}^{{\prime\prime} }=\displaystyle \frac{P}{\pi {{r}_{0}}^{2}}\end{eqnarray} \tag{ 6 }$$

Then, the temperature increase in the β-Ga₂O₃ NM was calculated using Eq. 2. With the use of $\theta \equiv \left(T-{T}_{a}\right)$ and $z={\left(\tfrac{g}{Ks}t\right)}^{1/2}r,$ Eq. 1 becomes a nonhomogeneous Bessel's equation (Eq. 7)³⁵:

$$\begin{eqnarray}&&\displaystyle \frac{{\partial }^{2}\theta }{\partial {z}^{2}}+\displaystyle \frac{1}{z}\displaystyle \frac{\partial \theta }{\partial z}-\theta =-\displaystyle \frac{{{q}_{0}}^{\prime\prime }}{g}\exp \left(-\displaystyle \frac{{z}^{2}}{{{z}_{0}}^{2}}\right)\end{eqnarray} \tag{ 7 }$$

The solution to Eq. 7 is given as^35,41:

$$\begin{eqnarray}&&\theta \left(z\right)={C}_{1}{I}_{0}\left(z\right)+{C}_{2}{K}_{0}\left(z\right)+{\theta }_{p}\left(z\right)\end{eqnarray} \tag{ 8 }$$

where the two homogeneous solutions I₀(z) and K₀(z) are the zero-order modified Bessel functions of the first and second kinds, respectively. We then define the measured thermal resistance as R_m' = T_m/P. Based on Eqs. 2 and 5, Eq. 9 can be derived^35,41:

$$\begin{eqnarray}&&{R}_{m}=\displaystyle \frac{\displaystyle {\int }_{0}^{\infty }\left(-{I}_{0}\left(z\right)\mathop{\mathrm{lim}}\limits_{z\,\to \,\infty }\tfrac{{\theta }_{p}\left(z\right)}{{I}_{0}\left(z\right)}+{\theta }_{p}\left(z\right)\right)\exp \left(-\tfrac{{r}^{2}}{{{r}_{0}}^{2}}\right)rdr}{\displaystyle {\int }_{0}^{\infty }\exp \left(-\tfrac{{r}^{2}}{{{r}_{0}}^{2}}\right)rdr\displaystyle {\int }_{0}^{\infty }{{q}_{0}}^{\prime\prime }\exp \left(-\tfrac{{r}^{2}}{{{r}_{0}}^{2}}\right)2\pi rdr}\end{eqnarray} \tag{ 9 }$$

Given the input power and laser spot size, compared with the two sets of slopes from Figs. 2 and 3: (1) slopes from Raman shifting with respect to temperature using 5× (R₁) and 20× (R₂) objective lenses and (2) slopes from Raman shifting with respect to light intensity using 5× (R₃) and 20× (R₄) objective lenses. From each sample, we can define R_m' = T_m/P = R₄/R₂ and R_m'' = R₃/R₁, respectively. Based on the ratio of R_m values, we can calculate the ratio of g/K_s, which depends only on the ratio of the two R_m values (R_m'/R_m'').⁴² Once the ratio of g/K_s was calculated, the K_s and g values can be further determined by choosing one set of objective lens' results, which yields the room-temperature g values of 7.58 MW/(m²·K). We repeated the calculation under different temperature conditions and plotted the thermal boundary conductance-temperature relationship from 100 K to 500 K as shown in Fig. 4a. The higher interfacial thermal boundary conductance is attributed to multiple factors.⁴³ It is known that the thermal transport property across the heterostructure, particularly the one formed with a van der Waals interface, is largely affected by the interface quality, namely the lower surface roughness and surface energy because these factors reduce phonon transmission across the heterointerface and, thus lower thermal boundary conductance. Therefore, it is important to achieve intimate contact and maintain sufficiently high surface energy at the heterointerface in order to enhance phonon transmission. In our case, the atomically smooth diamond and β-Ga₂O₃ NM surface helps increase the interfacial thermal boundary conductance and facilitate the heat dissipation of β-Ga₂O₃ NMs through the diamond substrate. The extracted interfacial thermal boundary conductance values suggest that the diamond substrate is highly capable of heat dissipation, which is comparable to epi-structures based on other wide bandgap semiconductors such as GaN-on-SiC (3.3 MW/(m²·K)),⁴⁴ GaN-on-sapphire (12 MW/(m²·K)).⁴⁴ Therefore, the β-Ga₂O₃ NM/diamond heterostructure offers a viable route for dissipating heat from β-Ga₂O₃ NM so as to effectively complement the poor thermal property of β-Ga₂O₃.

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Finally, as shown in Fig. 4b, the thermal conductivity values of samples #1 ∼ #3 with respect to temperature were extracted. The thermal conductivity—temperature relationships of these β-Ga₂O₃ NMs were revealed to be 12 W m⁻¹·K⁻¹, 36.4 W m⁻¹·K⁻¹, and 100.86 W m⁻¹·K⁻¹ at 100 K for 100 nm, 1000 nm, and 4000 nm thick β-Ga₂O₃ NMs, respectively. These were gradually decreased to 3.1 W m⁻¹·K⁻¹, 8.68 W m⁻¹·K⁻¹, and 13.97 W m⁻¹·K⁻¹ at room temperature. This trend can be explained by the phonon–phonon scattering rate is sensitive to temperature because fewer numbers of states are occupied. When thickness of β-Ga₂O₃ NM gets thinner, the additional film boundary forms. In this case, the phonon mean free paths become relatively smaller, thus phonon transport is decided primarily by phonon scattering processes occurring at the boundaries of β-Ga₂O₃ NM, which results in smaller thermal conductivity.

In conclusion, we have successfully measured the thermal conductivity of three types of (−201) β-Ga₂O₃ NMs using Raman thermometry under different temperature conditions from 100 K to 500 K with 40 K intervals. To investigate the thermal conductivity of β-Ga₂O₃ with different thicknesses, three different thicknesses of (100) β-Ga₂O₃ NM—namely, 100 nm, 1000 nm, and 4000 nm—were prepared from a (−201) β-Ga₂O₃ substrate using a mechanical exfoliation method and then transfer-printed onto the single crystalline diamond substrates. The thermal conductivity/temperature relationships of these β-Ga₂O₃ NMs were revealed to be 12 W m⁻¹·K⁻¹, 36.4 W m⁻¹·K⁻¹, and 100.86 W m⁻¹·K⁻¹ at 100 K and gradually decreased to 3.1 W m⁻¹·K⁻¹, 8.68 W m⁻¹·K⁻¹, and 13.97 W m⁻¹·K⁻¹ at room temperature for 100-nm-, 1000-nm-, and 4000-nm-thick β-Ga₂O₃ NMs, respectively. Our results provide benchmark knowledge about the thermal conductivity of β-Ga₂O₃ NMs over a wide temperature range for the design of novel β-Ga₂O₃-based power electronics and optoelectronics.

This work was supported by the National Science Foundation (grant number: ECCS-1809077) and partially by the seed grant by Research and Education in energy, Environment, and Water (RENEW) Institute and Center of Excellence in Materials Informatics (CMI) at the University at Buffalo.