High-temperature annealing of sputtered AlN (Sp-AlN) using a face-to-face configuration is a novel technique that has attracted considerable attention because it can reduce the threading dislocation density of Sp-AlN to 10^{7} cm^{−2}. However, drawbacks such as cracking, residual stress, and wafer curvature remain because of a high annealing temperature of 1700 °C. We previously developed a thermal strain analysis model that uses an elastic multilayer system to describe the elastic behavior of Sp-AlN on sapphire under high-temperature annealing. In this study, we expand this model to consider in-plane anisotropy. By performing thermal strain analysis of the curvature, strain, stress, and strain energy of *c*-plane AlN grown on *c*- and *a*-plane sapphire, our calculation successfully approximates the experimental results, even for an in-plane anisotropic structure. The proposed model is, therefore, useful for quantitative evaluation of the residual strain and can contribute to strain engineering of AlGaN-based deep-ultraviolet light-emitting diodes.

## I. INTRODUCTION

High-temperature annealing of sputtered AlN (Sp-AlN) using a face-to-face configuration (FFA) is a novel technology that has attracted considerable attention because of its ability to reduce the threading dislocation density (TDD) of Sp-AlN. This characteristic aids the production of AlGaN-based deep-ultraviolet light-emitting diodes (DUV LEDs) on sapphire substrates.^{1–5} The driving force for this improved crystallinity is solid-phase epitaxy boosted by an annealing temperature of 1700 °C, which allows columnar AlN grains to coalesce. Thus far, optimization of FFA Sp-AlN fabrication processes has produced TDD values as low as 4.3 × 10^{7} cm^{−2} using a Sp-AlN/sapphire template with a Sp-AlN thickness of 1.2 µm.^{6–8} In addition to conventional *c*-plane sapphire (*c*-Sap) substrates, this method has been successfully applied to heteroepitaxial growth on *a*-plane sapphire (*a*-Sap), SiC, diamond, and nanopatterned sapphire substrates.^{9–13} However, drawbacks such as residual stress, wafer bowing, and film cracking appear due to coefficient of thermal expansion (CTE) mismatches between AlN and the substrate. To suppress cracking, we previously applied strain engineering to Sp-AlN by controlling the sputtering conditions so that the compressive strain from the peening effect compensated for the inherent tensile strain caused by intergrain attraction.^{7} We then succeeded in increasing the crack-free thickness from 160 to 850 nm by reducing the chamber pressure from 0.2 to 0.03 Pa. However, few studies of the elastic mechanics that can describe residual stress and wafer bowing among FFA Sp-AlN templates quantitatively have been conducted thus far. In a previous study,^{14} we calculated thermal strain among AlN/*c*-Sap templates analytically and confirmed that the results agreed with the experimental values for FFA Sp-AlN. Critically, this suggests that fully coalesced Sp-AlN obtained using FFA is subject to negligible strain at 1700 °C due to the formation of an 8/9 arrangement of misfit dislocations at the AlN/sapphire interface.^{15} That is to say, the residual strain is derived mainly from the thermal strain produced during the cooldown process. In this study, we expand the aforementioned model to consider in-plane anisotropy so that structures such as *c*-plane AlN grown on *a*-Sap can be simulated. We expect that the proposed model will be useful for evaluating the residual strain in various epitaxial relationships quantitatively and will be able to contribute to strain engineering of AlGaN-based DUV LEDs.

## II. EXPERIMENTAL

AlN films were grown on a two-inch *a*-Sap substrate with a thickness of 430 µm by sputtering a 99.9% sintered-AlN target with N_{2} gas. The N_{2} gas flow rate was 24 SCCM. Sputtering was performed using a radiofrequency (RF) power of 700 W, a chamber pressure of 0.05 Pa, and a heater temperature of 600 °C. The AlN templates were subsequently stacked in face-to-face geometry and annealed at 1700°C for 3 h under atmospheric pressure in a nitrogen-flow environment. The fabricated AlN templates were characterized using x-ray diffraction (XRD) and optical microscopy. AlN 0002 and 1-102 diffraction rocking curves were measured using a Ge 220 two-bounce hybrid monochromator and a Ge 220 three-bounce analyzer. The *c*- and *a*-axis lattice constants were measured using a two-dimensional solid-state detector. Cracks were observed using an optical microscope. The methods used to fabricate and measure the AlN on the *c*-Sap structure are described in a previous report.^{7}

## III. ANALYSIS

### A. Structural model

The elastic multilayer model that we used to consider in-plane anisotropy was developed by using our previous model and Hsueh’s method.^{14,16} We updated the previous model by considering in-plane anisotropy within the strain–stress relationship, the temperature dependence of the physical constants, and the equilibrium equations. Figure 1 shows a schematic of an analysis model that considers *M* layers.

An arbitrary *m*th layer stores the following physical properties: the thickness *t*^{m}, strain tensor *ϵ*^{m,n}, stress tensor *σ*^{m,n}, stiffness-constant tensor *C*^{m}, and CTE tensor *α*^{m}; here, *n* denotes the type of strain and stress. The height of the *m*th layer is calculated from the bottom using

where *h*^{0} = 0. To associate an orthogonal coordinate system with the axis directions of the AlN/*c*- and *a*-Sap structures, the *x*-, *y*-, and *z*-axes are defined as parallel to [1–100], [11–20], and [0001] of *c*-AlN and to [1–100], [0001], and [11–20] of *a*-Sap, respectively. In this configuration, the shear strains can be ignored (*ϵ*_{yz} = *ϵ*_{zx} = *ϵ*_{xy} = 0). Here, Hooke’s law is given in Voigt notation using

This model considers seven types of strains: residual strain, bending strain, constrained strain, uniform strain, thermal strain, hydrostatic strain induced by oxygen, and as-sputtered AlN strain. These are described as follows:

and

Here, $\kappa =\kappa 1\kappa 20000T$ is the curvature; $\beta =\beta 1\beta 20000T$ is the bending axis at which the bending strain is zero; *ξ*_{i} is the uniform strain, which is independent of the *z* position and is derived from the weighted average of *t*^{m} and *D*^{m,n} given in Eq. (12); *D*^{m,n} is the biaxial modulus matrix; *N* is the maximum value of the number *n*; and *T*_{I} and *T*_{F} are the initial and final annealing temperatures, respectively. Furthermore, $\alpha m=\alpha 1\alpha 2\alpha 3000T$ is the CTE; *a*_{b} and $aON+$ are the *a*-axis lattice constants of bulk AlN and AlN that contains $ON+$, respectively; $O$ is the oxygen concentration; N_{site} is the density of nitrogen sites in AlN; and *a*_{f} is the *a*-axis lattice constant of as-sputtered AlN.

Each type of strain can be described logically as follows. The three strains given in Eqs. (3.5)–(3.7) are categorized as unconstrained strains: the thermal strain is the temperature integral of the CTE, the hydrostatic strain is the expansion of the lattice constant due to the oxygen impurities in AlN,^{17,18} and the strain of the as-sputtered AlN is the sum of the compressive strain caused by the peening effect and the tensile strain due to intergrain attraction.^{19,20} The constrained and uniform strains are determined by balancing the three unconstrained strains in the constrained multilayer structure to match the displacements at the individual layer boundaries. The uniform strain is the average strain of the entire multilayer structure weighted by *t*^{m} and *D*^{m,n}, whereas the constrained strain is the uniform minus all three unconstrained strains. Furthermore, the bending strain emerges upon considering the bending degree of freedom so that the stress and moment caused in the multilayer by the constrained strain are relaxed. Finally, the residual strain is calculated as the sum of the bending and constrained strains. This corresponds to the strain that is measured experimentally via XRD. Positive and negative strains are tensile and compressive, and positive and negative curvatures are concave and convex, respectively. As an exception, the tensile (compressive) strain of the as-sputtered AlN is negative (positive) [Eq. (3.7)] because we define *a*_{f} as the *a*-axis lattice constant of as-sputtered AlN that is already subject to biaxial stress under the substrate constraint. In this definition, the value of *a*_{f} can be measured using XRD and can be compared directly to the experimental results, but the negative sign is required to deal with the as-sputtered AlN strain as the unconstrained strain.

### B. Strain–stress relationship

Given the in-plane stress condition for *n* ≠ 5 (i.e., $\sigma 3m,n=0$ and $\u03f54m,n=\u03f55m,n=\u03f56m,n=0$), the strain–stress relation can be written as follows:

where *D*^{m,n} is the biaxial modulus matrix.

Given the hydrostatic strain condition for *n* = 5 (i.e., $\u03f51m,5=\u03f52m,5=\u03f53m,5$ and $\u03f54m,5=\u03f55m,5=\u03f56m,5=0$), the strain–stress relation can be written as follows:

where *D*^{m,5} is the hydrostatic modulus matrix.

AlN has a hexagonal crystal structure, and sapphire has a rhombohedral crystal structure. Their stiffness-constant tensors are given by

and

The stiffness-constant tensor of *a*-Sap is obtained by rotating that of *c*-Sap around the *x*-axis by $\theta =\pi 2$,

where the rotation matrix ** R** is given by

^{21}

Consequently, the stiffness-constant tensor of *a*-Sap is obtained as follows:

### C. Equilibrium equations

To consider in-plane anisotropy when analyzing a strain distribution, six equilibrium equations must be satisfied. The *x*- and *y*-axis equilibrium equations of the bending force, constrained force, and bending moment are given by

where *i* = 1 and 2 correspond to the *x*- and *y*-axes, respectively. Here, the unknowns to be determined are $\u03f5im,2$, *β*_{i}, and *κ*_{i}. The boundary conditions for these equations, which match the displacements at individual layer interfaces, are given by

Consequently, $\u03f5im,2$ can be derived analytically using Eqs. (3.3), (3.4), (10.2), and (11), where *ξ*_{i} is given by

and

The quantities *β*_{i} and *κ*_{i} can be obtained by numerically solving the following nonlinear equations that result from Eqs. (10.1) and (10.3):

Consequently, $\u03f5im,0$ and $\sigma im,0$ can be obtained using *β*_{i} and *κ*_{i}.

To calculate the crack-formation criteria, we calculate the strain-energy density per unit area, *U* (hereafter referred to as the “strain energy”) using

### D. Strain transition and crack formation during annealing

Difficulties encountered when modeling the strain transition during annealing stem from the limitations of *in situ* strain measurements at high temperatures. As a result, there is insufficient information on this matter. Washiyama *et al.* carefully investigated the annealing-time dependence of TDD reduction in an AlN buffer grown by metal–organic vapor-phase epitaxy (MOVPE) on *c*-Sap. They identified the reduction mechanism to be dislocations climbing through vacancy core diffusion.^{22} Wang *et al.* measured the transition of the residual strain of sputtered AlN on *c*-Sap by varying the annealing time. They showed experimentally that the strain within as-sputtered AlN relaxes fully for processing times longer than 3 h.^{8} Both studies relied on *ex situ* measurements and assumed no change during ramp-up. To simulate the strain transition in our model, we assumed exponential strain relaxation with no relaxation during ramp-up. As functions of the annealing time *τ*, the thermal (*n* = 4) and as-sputtered AlN (*n* = 6) strains were assumed to start relaxing at the end of the ramp-up (*τ* = *τ*_{1}) and to approach zero at the start of cooldown (*τ* = *τ*_{2}),

where *τ*_{r} is the relaxation time obtained from the experimental results.^{8}^{,}Figure 2 shows an example of such a calculation for an AlN/*c*-Sap bilayer structure with sapphire and AlN thicknesses of 430 *μ*m and 300 nm, respectively. The *a*-axis lattice constants for as-sputtered AlN are *a*_{f} = 3.113 and 3.116 Å, the lattice constant for the bulk is 3.1111 Å, and the oxygen concentration [O] is 6 × 10^{20} cm^{−3} (the value measured using secondary-ion mass spectroscopy) or 0 cm^{−3} (in the case where hydrostatic strain has no effect). The relaxation time *τ*_{r} is 40 min, the ramp-up time is 1.5 h, the holding time at 1700°C is 3 h, and the cooldown time is 1.5 h. As shown in Fig. 2(a), the *a*-axis residual strain of the AlN film starts to accumulate in the tensile (positive) direction during ramp-up, relaxes gradually according to Eq. (15) once the temperature reaches 1700°C, and finally turns to the compressive (negative) direction during cooldown. The effect of the starting lattice constants of 3.113 and 3.116 Å on the final strain after the cooldown is subtle because almost all of the as-sputtered AlN strain relaxes by the end of the process. Unlike with thermal strain and as-sputtered AlN strain, relaxation of the hydrostatic strain is ignored provisionally because it is unclear whether there is a smooth connection between relaxation and emergence of hydrostatic strain [note that *ϵ*^{AlN,2} (constrained strain) ≫ *ϵ*^{AlN,5} (hydrostatic strain) must be satisfied for this approximation]. Consequently, the residual strain is almost zero at the beginning of cooldown for [O] = 0 cm^{−3}, and the slight compressive strain at the beginning of cooldown for [O] = 6 × 10^{20} cm^{−3} is attributed to the unrelaxed hydrostatic strain. The hydrostatic strain itself is tensile, but AlN is subject to a relative compressive strain because it is constrained on a substrate. As shown in Fig. 2(b), the strain energy is highest at the end of the ramp-up under tensile stress. This suggests that the mode-I cracking criterion is satisfied at this point. According to Griffith theory,^{23} a crack forms when the strain energy *U* accumulated in the film exceeds the critical value,

where *Γ* and *Z* are the fracture toughness and a dimensionless parameter that depends on the geometry, respectively. The values of *Z* are 3.951 and 1.976 for surface- and channel-cracking conditions, respectively. We used the latter in this study because the surface cracking is too small to detect using an optical microscope. To avoid cracking, we can reduce the strain energy dramatically by decreasing the tensile strain of the as-sputtered AlN because the strain energy is proportional to the square of the strain. This provides a theoretical explanation of the effectiveness of low-pressure sputtering technology with regard to increasing the crack-free thickness.^{7} As shown in Fig. 2(c), the residual strain at *a*_{f} = 3.116 Å is resolved into six components. Because the thickness of sapphire is substantially larger than that of AlN, the uniform strain almost matches the thermal strain of sapphire. Hence, the constrained strain in Eq. (3.3) becomes approximately

where the practical values at *τ* = 1.5 h are *ϵ*^{Sap,4} − *ϵ*^{AlN,4} = $\u222b20\xb0C1700\xb0C[\alpha a\u2212axisSapT\u2212\alpha a\u2212axisAlNT]dT$ = 3.82 × 10^{−3} (effective thermal strain), −*ϵ*^{AlN,5}= −3.77 × 10^{−4} (hydrostatic strain), −*ϵ*^{AlN,6} = 6.11 × 10^{−4} (strain of the as-sputtered AlN), and *ϵ*^{AlN,2} = 4.05 × 10^{−3} (constrained strain). The sign of the bending strain always opposes that of the constrained strain to compensate for the strain energy accumulated throughout the bilayer structure. However, its amplitude of 10^{−6} is so small that the residual strain is almost identical to the constrained strain.

### E. Comparison with other models

To verify our calculations, we compared the approximation model described by Jou *et al.*^{24,25} to our calculations under the condition of *τ*_{r} = 0 min and [O] = 0 cm^{−3} (i.e., the system is ideally strain-free at the beginning of the cooldown). The results are shown in Fig. 3. In the model reported by Jou *et al.*, as long as the substrate is sufficiently thicker than the film, the curvature of an in-plane anisotropic bilayer can be approximated using

Here, Δ*ϵ*_{i} is calculated using Eqs. (1)–(12). Young’s modulus $Eim$ and Poisson’s ratio $\nu ijm$ can be derived from the relationship between the stiffness *C*^{m} and compliance *S*^{m} tensors,

Thus, $Eimand\nu ijm$ for AlN on *c*- and *a*-Sap are given as follows:

For AlN and *c*-Sap,

For *a*-Sap,

The curvature obtained using our model eventually agrees well with that of the approximation for thicknesses <10 μm. This verifies the correctness of our model. However, the two models diverge for thicknesses >10 μm because the film is no longer thin enough compared to the substrate, and the approximation used to derive Eq. (18.1) is no longer applicable. Therefore, our model helps provide accurate calculations even for structures with films thicker than 10 *μ*m.

## IV. RESULTS

We performed XRD azimuth-angle scans to measure the epitaxial relationships within the fabricated AlN/*a*-Sap samples. The results shown in Fig. 4(a) confirm the epitaxial relationships within AlN[1–100]||Sap[1–100] because the AlN{1–102} peaks are aligned with the Sap{3–300} peaks, as shown in Fig. 4(b). The two epitaxial relationships reported in the literature—AlN[1–100]||Sap[1–100] and AlN[11–20]||Sap[1–100]—relate to materials grown via MOVPE, hydride vapor-phase epitaxy, and sputtering.^{26–29} For the case of sputtering, the reported AlN[1–100]||Sap[1–100] epitaxial relationship corresponds to our results.^{9}

In-plane anisotropy emerges because Sap[0001] and [1–100] have different CTEs. As shown in Figs. 4(c) and 4(d), a crack-free surface is achieved for samples with an AlN thickness of 300 nm, but anisotropic cracks are observed to propagate along AlN[1–210] and AlN[2–1–10] at a thickness of 400 nm. The crack-propagation direction is determined by the accumulation of significant strain energy along Sap[0001]. As shown in Fig. 4(e), the AlN 1-102 rocking-curve width increases with the film thickness. This indicates that, under FFA, the grain size expands via solid-phase epitaxy. At a thickness of 300 nm, we obtain AlN 0002 and 1-102 rocking-curve widths of 23 and 214 arcsec, respectively. These compare favorably with those of *c*-Sap at a similar AlN thickness.^{7} The increased 0002 rocking-curve width at 400 nm suggests that high thermal strain energy and the occurrence of cracking degrade crystallinity.

To assess the in-plane anisotropic strain, we performed reciprocal-space mapping of AlN 1-104 and 11-24 reflections with the x-ray incidence directions aligned along Sap[0001] and [1–100], respectively. Figure 5 plots the experimentally obtained strain, stress, strain energy, and wafer curvature against the calculated values using *τ*_{r} = 40 min. The curvature *κ* is derived from the relationship *κ* = Δ*ω*_{0}/Δ*x*_{0}, where Δ*ω*_{0} is the peak-angle shift of the AlN 0002 rocking curve and Δ*x*_{0} is the shift in the rocking curve measurement position. The agreement between the calculated and experimental values indicates the accuracy of our model. The measured curvature along Sap[1–100] is different from the calculation trend at *t*^{AlN} = 400 nm. We attribute this to crack formation and note that no other conspicuous effect is found in the curvature along Sap[0001] or in the strain.

We used our model, which is consistent with the experimental results, to investigate the criteria for crack formation in AlN/*c*-Sap^{7} analytically, as shown in Fig. 6(a). First, the strain energy *U*^{AlN} at the end of the ramp-up is mapped for various AlN thicknesses and as-sputtered AlN *a*-axis lattice constants. Next, the presence or absence of cracks is plotted using crosses and circles, respectively. Finally, the boundary lines that satisfy

are presented as the criteria for crack formation. We find that *U*^{c} = 3.3 J/m^{2} for the critical strain energy at the boundary between the presence or absence of a crack. We take two values from the literature for the fracture toughness of *m*-plane AlN—*Γ*^{m} = 4.81 and 9 J/m^{2}. These produce *U*^{c} = 1.2 and 2.3 J/m^{2}, respectively. Dreyer *et al.*^{30} calculated *Γ*^{m} to be 4.81 J/m^{2} using density functional theory (DFT). This agrees well with the experimental results for full Al composition Al_{x}Ga_{1−x}N coherently grown on GaN. In contrast, Einfeldt *et al.*^{31} grew 36 samples of Al_{x}Ga_{1−x}N/GaN structures with *x* values of ∼0.1 via molecular-beam epitaxy and obtained a fracture toughness of 9 J/m^{2} via curve fitting. However, these values are lower than the value *Γ*^{m} = 13 J/m^{2} calculated from *U*^{c} = 3.3 J/m^{2}.

We subsequently determined the crack-formation criteria for AlN/*a*-Sap based on the AlN/*c*-Sap result. There are four possible cases. The *m*-plane crack can appear at *θ* = 0 or π/3, and the *a*-plane crack can appear at *θ* = π/6 or π/2, where *θ* is the angle measured counterclockwise from the *x*-axis. From Mohr’s circle of strain and stress, the strain energy is given by

The crack-formation criteria are, therefore, given using Eq. (21), which yields

where *Γ*^{a} is the fracture toughness of *a*-plane AlN. Because we observe an *m*-plane crack at *θ* = π/3 experimentally in this work [Fig. 4(d)], the boundary line that satisfies $\Gamma m2Z=U\theta =\pi /3AlN$ is presented in Fig. 6(b). The supplementary material contains qualitative comparisons with other crack-formation criteria at *θ* = 0, π/6, and π/2. Relative to AlN/*c*-Sap, the crack-free thickness is noticeably reduced in AlN/*a*-Sap because of the significant strain energy accumulated along Sap[0001], e.g., the critical thickness is reduced from 616 to 420 nm at *U*^{c} = 3.3 J/m^{2} when *a*_{f} = 3.116 Å. This may limit further crystallinity improvements at larger AlN thicknesses. In this calculation, we use a single value of *a*_{f} provisionally for the directions along Sap[0001] and [1–100], although *a*_{f} may be different between Sap[0001] and [1–100] due to in-plane anisotropy. Experimental investigation of anisotropic strains of as-sputtered AlN/*a*-Sap and their effect on the presence or absence of cracks is reserved for future work.

## V. DISCUSSION

Despite the excellent agreement shown in Fig. 5, the model can still be improved with regard to clarifying ambiguity related to physical constants at high temperatures. Unlike the physical constants of sapphire, which are well studied, those of AlN, such as the temperature dependence of the stiffness constants, have been investigated insufficiently. In addition, the CTE in the high-temperature region is ambiguous because of the difficulty in performing precise measurements and fitting the measured values to a curve. According to Ruffa,^{32} a correction term in the Debye model helps formulate the CTEs of AlN and sapphire accurately at temperatures approaching their melting points. However, we have not yet implemented this approach in our analytical model. Details of the temperature dependence of the CTE and stiffness constant are elaborated in the supplementary material.

The divergence between our model and the values from the literature shown in Fig. 6(a) can be attributed to overestimation of the strain energy at the end of ramp-up. This occurs because no relaxation is considered during ramp-up. Indeed, the degree of relaxation is governed by the activation energy derived from the Arrhenius plot, but this term is not implemented in our model. Future investigations will require *in situ* XRD measurements of Sp-AlN in an annealing environment in order to determine the time dependence of strain relaxation and its relationship with TDD reduction experimentally. The temperature dependence of the fracture toughness is another factor to be considered.^{33,34}

The condition *U*^{AlN} = 0 J/m^{2} shown in Fig. 6(a) is satisfied at *a*_{f} = 3.1004 Å regardless of the AlN thickness. This result is reasonable from Eq. (17) because the resulting compressive strain of the as-sputtered AlN is large enough to cancel out the tensile thermal strain at the end of the ramp-up, leading to elimination of residual strain. Therefore, no cracking occurs at *a*_{f} < 3.1004 Å because the stress never becomes tensile and the mode-I cracking criterion is avoided during ramp-up; this is indeed a crack-free condition for any AlN thickness. However, no such highly compressed film of as-sputtered AlN has been synthesized experimentally yet. Hence, a double sputtering-and-annealing method has been developed recently to achieve a further increase in the crack-free thickness and reduce the TDD.^{6} This method can be used to grow FFA Sp-AlN with a smaller *a*_{f} than that produced on sapphire under the same sputtering conditions. This approach yields an increase in the crack-free thickness via a single sputtering and annealing process. Thus, we can obtain crack-free, multilayered FFA Sp-AlN via multiple cycles of sputtering and annealing, provided that the mode-I cracking criterion is avoided during each cycle.

## VI. CONCLUSION

We performed thermal strain analysis of FFA Sp-AlN grown on *c*- and *a*-Sap by considering in-plane anisotropy when modeling the relevant elastic multilayer system. Experiments were performed to fabricate FFA Sp-AlN on *a*-Sap, and the AlN[1–100]||Sap[1–100] epitaxial relationship was confirmed via XRD. At a crack-free thickness of 300 nm, we obtained rocking-curve widths of 23 and 214 arcsec for AlN 0002 and 1-102, respectively. These compare favorably with the results obtained on *c*-Sap at a similar thickness. The measured strain, stress, strain energy, and curvature of FFA Sp-AlN grown on *c*- and *a*-Sap are in good agreement with the calculated values from the analytical model. Furthermore, we identified a crack-formation criterion based on Griffith theory and obtained a critical strain energy of 3.3 J/m^{2} from the analysis model.

## SUPPLEMENTARY MATERIAL

See the supplementary material for temperature dependence of physical constants and qualitative comparison of AlN/*a*-plane sapphire crack-formation criteria.

## ACKNOWLEDGMENTS

The authors thank Mr. K. Norimatsu for providing sample preparation support. This work was partially supported by MEXT “Program for Building Regional Innovation Ecosystem,” JSPS KAKENHI (Grant Nos. 16H06415, 16H06423, 19K15025, and 19K15045), and JST CREST (Grant No. 16815710).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article and its supplementary material.