Anisotropic Three-Dimensional Thermal Stress Modeling and

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Anisotropic Three-Dimensional Thermal Stress Modeling and Simulation of Homoepitaxial AlN Single Crystal Growth by Physical Vapor Transport Method Qikun Wang, Jiali Huang, Zhihao Wang, Guangdong He, Dan Lei, Jiawei Gong, and Liang Wu Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/acs.cgd.8b00118 • Publication Date (Web): 05 Apr 2018 Downloaded from http://pubs.acs.org on April 9, 2018

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Crystal Growth & Design

Anisotropic Three-Dimensional Thermal Stress Modeling and Simulation of Homoepitaxial AlN Single Crystal Growth by Physical Vapor Transport Method Qikun Wang, Jiali Huang, Zhihao Wang, Guangdong He, Dan Lei, Jiawei Gong, and Liang Wu* State Key Laboratory of Advanced Special Steel & Shanghai Key Laboratory of Advanced Ferrometallurgy & School of Materials Science and Engineering, Shanghai University, Shanghai 200072, China ABSTRACT: We develop a thermal-elastic stress model using the finite element method to predict 3D anisotropic stress in AlN single crystals homoepitaxially grown by the physical vapor transport process; we also perform numerical experiments for a 1-inch AlN crystal surrounded by different cone-tube designs and grown along various orientations. The influences of the cone-tube shape and the growth orientation on the stresses inside the AlN crystal are investigated in detail. The simulation results show that the von Mises stress exceeds 1.11 GPa under all specified growth conditions, while the anisotropy is negligible. The resolved shear stresses are strongly dependent on the thermal gradient inside the growing crystal and the growth orientation. Strong anisotropy of the resolved shear stress is observed upon tilting of the ̅0〉 primary slip system reveals that the cgrowth orientation. The resolved shear stress along {0001}〈112 axis growing crystal is under tensile stress along all three primary slip directions. Nevertheless, an inversion of the resolved shear stress from tensile to compressive along the -a3 slip direction is observed when changing the growth orientation. The total resolved shear stress shows 6-fold symmetry, reflection symmetry and 2-fold symmetry along [001], [10√3] and [100] growth orientations, respectively.

Liang Wu, PhD Professor, School of Materials Science and Engineering Shanghai University Address: No. 99 ShangDa Road, BaoShan District, Shanghai University, Shanghai 200072, China. Phone: 0086-18516773360 E-mail: [email protected]

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Anisotropic Three-Dimensional Thermal Stress Modeling and Simulation of Homoepitaxial AlN Single Crystal Growth by Physical Vapor Transport Method Qikun Wang, Jiali Huang, Zhihao Wang, Guangdong He, Dan Lei, Jiawei Gong, and Liang Wu* State Key Laboratory of Advanced Special Steel & Shanghai Key Laboratory of Advanced Ferrometallurgy & School of Materials Science and Engineering, Shanghai University, Shanghai 200072, China

ABSTRACT: We develop a thermal-elastic stress model using the finite element method to predict 3D anisotropic stress in AlN single crystals homoepitaxially grown by the physical vapor transport process; we also perform numerical experiments for a 1-inch AlN crystal surrounded by different cone-tube designs and grown along various orientations. The influences of the conetube shape and the growth orientation on the stresses inside the AlN crystal are investigated in detail. The simulation results show that the von Mises stress exceeds 1.11 GPa under all specified growth conditions, while the anisotropy is negligible. The resolved shear stresses are strongly dependent on the thermal gradient inside the growing crystal and the growth orientation. Strong anisotropy of the resolved shear stress is observed upon tilting of the growth orientation.


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The resolved shear stress along {0001}〈112̅0〉 primary slip system reveals that the c-axis growing crystal is under tensile stress along all three primary slip directions. Nevertheless, an inversion of the resolved shear stress from tensile to compressive along the -a3 slip direction is observed when changing the growth orientation. The total resolved shear stress shows 6-fold symmetry, reflection symmetry and 2-fold symmetry along [001], [10√3] and [100] growth orientations, respectively.

1. INTRODUCTION Due to its wide bandgap, high breakdown field and high thermal conductivity, AlN is an excellent candidate for high-temperature, high-frequency, high-power electronic and deep-UV optoelectronic devices.1 In the past several decades, tremendous efforts have been devoted to the development of various processes and models for the growth of AlN bulk crystals, such as hydride vapor phase epitaxy (HVPE),2-5 flux/solution growth6-9 and physical vapor transport (PVT) growth.10 Among these, the PVT method has been shown to be a very promising technique for the growth of large high-quality bulk AlN crystals.11-13 Currently, crack-free AlN crystals up to 2-inch in diameter grown by the PVT method are available with very limited volume.14,15 AlN single crystals with perfect structural quality can only be grown by the spontaneous freestanding technique at near-thermal-equilibrium conditions.16 Using wafers cut from these high-quality crystals as seeds, large-size and high-quality AlN crystals can be obtained through the subsequent homoepitaxial growth iteratively. However, due to the non-equilibrium growth conditions, single crystalline AlN boules obtained by the homoepitaxial seeded PVT growth normally have heterogeneously distributed defects such as basal plane dislocations (BPDs),


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thread edge dislocations (TEDs), and low angle grain boundaries (LAGBs). BPDs are generated during crystal growth due to thermal stresses, while TEDs are grown-in defects replicated from the seed. BPDs are deformation-induced dislocations, while the generation and multiplication of BPDs is mainly caused by the resolved shear stress along the primary slip direction during the crystal growth process.17 BPDs easily occur in the growing crystals when the effective stress exceeds the critical resolved shear stress (CRSS). For example, higher BPDs normally can be found for constrained growth near the LAGBs at the AlN boule edges due to additional stresses.18 It was reported that high dislocation densities in the active layers (derived from the AlN substrates) of optoelectronic devices show an adverse effect on their internal quantum efficiency,19 and accordingly strongly affect the performance, reliability and lifetime of the devices grown on AlN substrates. Therefore, the thermal-elastic stresses in growing AlN crystals should be minimized as much as possible to reduce the generation and multiplication of BPDs. Since any extensive experiments in the hostile crystal growth environment are time consuming and extremely expensive, numerical modeling and simulation has become an essential and indispensable tool for the prediction and optimization of the crystal growth processes. In the past decades, extensive modeling activities on the thermal stress and dislocation inside growing crystals by various growth methods have been reported.17,20-22 However, dedicated research on the modeling and simulation of thermal stress and dislocation for the PVT AlN crystal growth process is very limited. To the best of our knowledge, the relevant literature can be quite comprehensively summarized as follows: Wu et al.23,24 developed integrated 2D thermal stress and CRSS models to predict the overall thermal stress level in a growing thin AlN crystal (with an assumption of a uniformly temperature distribution over the crystal thickness); Lee et al.25


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investigated the residual thermal stress distribution in an AlN single crystal (film) grown on a tungsten crucible using a numerical study. Karvani et al.26 presented a thermal-elasticviscoplastic model to calculate the thermal stress and dislocation density with an analytical temperature field history, and found that the mismatch in the thermal expansion coefficient between the AlN crystal and the W crystal-holder is the primary source of stress. In this paper, we develop a thermal-elastic stress module using the finite element method (FEM) to predict 3D anisotropic stress in a large-size AlN crystal homoepitaxially grown by a 3inch PVT growth reactor. Based on the developed thermal-elastic stress model, a series of numerical experiments are performed for a 1-inch AlN crystal grown along [001] (c-axis), [10√3] (off-axis rotated by θ=30∘ towards [112̅0]) and [100] (a-axis) orientation when the growing crystal is surrounded by three different cone-tube designs. The global temperature fields in the entire growth reactor loaded with three different tube-cone designs are primarily obtained by a FEMAG two-dimensional (2D) axisymmetric model in an inverse manner through the global modeling technique.27 The 2D axisymmetric temperature fields in the growing crystal obtained by FEMAG are projected to the 3D Cartesian system for the purpose of final thermal stress calculation. The resolved shear stress inside the crystal is evaluated by projecting the stress tensor along the AlN {0001}〈112̅0〉 primary slip system. The influences of the cone-tube shape as well as the growth orientation on the distribution of the von Mises stress, resolved shear stress and total resolved shear stress inside the AlN crystal are investigated in detail, and conclusions are drawn accordingly based on the results. 2. MODELING OF 3D THERMAL-ELASTIC STRESSES IN ALN SINGLE CRYSTALS


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Assuming small strains, quasi-static conditions and neglecting body forces, the thermal-elastic behavior is governed by the momentum balance equation together with a kinematic equation relating displacement and strain according to: ∇∙𝝈 ̿ = 0,

(1)

𝜺̿ = 𝜺̿(𝑼) = 𝜺̿𝜃 + 𝜺̿𝑒 ,

(2) 1

where 𝛔 ̿ is the stress tensor which must satisfy equilibrium; 𝜺̿(𝑼) = 2 (∇𝑼 + ∇𝑼𝑇 ); 𝑼 is the displacement vector field; 𝛆̿ is the total strain tensor, which can be sub-divided into elastic strain 𝛆̿e and thermal strain 𝛆̿θ , and ̿ ∙ (𝑇 − 𝑇0 ). 𝛆̿θ = 𝜷

(3)

̿ is the thermal expansion tensor for the hexagonal Here, 𝑇0 is the reference temperature and 𝜷 AlN single crystal. There are only two independent components along the lattice a and c directions, which depend on the orientation of the lattice relative to the global reference frame and which are assumed to be known. If the elastic strain in eq 2 is applied to Hooke’s law for a thermo-elastic solid body, we obtain the following equations: ̿ ∶ 𝜺̿𝑒 , 𝝈 ̿=𝑪 or

(4)

̿ ∶ 𝜺̿𝑒 = 𝑪 ̿ ∶ (𝜺̿ − 𝜺̿𝜃 ). 𝝈 ̿=𝑪

(5)

Here, 𝑪̿ is the fourth order stress-strain tensor of AlN crystals. AlN has a hexagonal structure 4 with the space group 𝐶6𝑣 , and there are five independent components of 𝑪̿ . Therefore, the stress-

strain elasticity tensor 𝑪̿ can be written as:


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𝐶11

𝐶12 𝐶11

𝑪̿ =

𝐶13 𝐶13 𝐶33

𝑆𝑦𝑚𝑚.

0 0 0 𝐶44

0 0 0 0 𝐶44

(

0 0 0 . 0 0 𝐶66 )

(6)

Here, 𝐶66 = (𝐶11 − 𝐶12 )/2, 𝐶11 , 𝐶12 , 𝐶13 , 𝐶33 and 𝐶44 are the AlN elasticity parameters. If we tilt the crystallographic c-plane such that the normal vector of the c-plane is tilted toward the −𝑎3 -direction ([112̅0]), then a special symmetry property arises: the half circle planes left and right of the 𝑎3 -axis will be mirror-symmetric in their displacement and stress behavior. Therefore, we fix the [100]-axis of the Cartesian system on the −𝑎3 -axis. For the tilt of the crystal lattice towards [112̅0], the current y-axis [010] is the rotation axis. By standard tensor ′ of the hexagonal single crystal associated with transformation, the elastic constant tensor 𝐶𝑖𝑗𝑘𝑙

the rotated Cartesian system is related to 𝐶𝑖𝑗𝑘𝑙 of the non-rotated Cartesian system as follows: ′ Cijkl = aim ajn ako alp Cmnop

= C11 (ai1 aj1 ak1 al1 + ai2 aj2 ak2 al2 ) + C12 (ai1 aj1 ak2 al2 + ai2 aj2 ak1 al1 ) +C13 {(ai1 aj1 + ai2 aj2 )ak3 al3 + ai3 aj3 (ak1 al1 + ak2 al2 )} +C33 ai3 aj3 ak3 al3 +C44 {(ai1 aj3 + ai3 aj1 )(ak1 al3 + ak3 al1 ) + (ai2 aj3 + ai3 aj2 )(ak2 al3 + ak3 al2 )} +C66 (ai1 aj2 + ai2 aj1 )(ak1 al2 + ak2 al1 ).

(7)

Here, 𝑎𝑖𝑗 are the direction cosines of 𝑋𝑖′ 𝑋𝑗 . For a hexagonal single crystal, the thermal expansion ̿ in the non-rotated Cartesian system can be written with two independent thermal tensor 𝜷 expansion coefficients 𝛽1 and 𝛽3 by substituting with 𝛽11 =𝛽22 =𝛽1 , 𝛽33 =𝛽3 , 𝛽23 =𝛽31 =𝛽12 =0. Using the standard tensor transformation, we obtain the thermal expansion coefficient tensors in the rotated Cartesian system as follows:


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𝛽𝑖𝑗′ = 𝑎𝑖𝑘 𝑎𝑗𝑙 𝛽𝑘𝑙 = 𝛽11 (𝑎𝑖1 𝑎𝑗1 + 𝑎𝑖2 𝑎𝑗2 ) + 𝛽33 𝑎𝑖3 𝑎𝑗3 = 𝛽1 (𝑎𝑖1 𝑎𝑗1 + 𝑎𝑖2 𝑎𝑗2 ) + 𝛽3 𝑎𝑖3 𝑎𝑗3 .

(8)

By substituting eqs 3 and 5 into eq 1, the final equation for displacement is obtained as follows: ̿ ′ ∙ (𝑇 − 𝑇0 ))} = 0 . ̿ ′ ∶ (𝜺̿′ (𝑼) − 𝜷 𝛻 ∙ {𝑪

(9)

̿ ′ and 𝜺̿′ is the stress-strain tensor, the thermal expansion coefficient tensor and the total ̿′, 𝜷 Here, 𝑪 strain in a rotated Cartesian system, respectively. After obtaining the displacement distribution, the stresses can be obtained from eqs 2 and 5. In practice, the von Mises stress has been used to represent the stress status in a growing crystal, and dislocation generation is presumed to occur in the regions where the von Mises stress exceeds the critical resolved shear stress. The von Mises stress is defined as follows: 𝜎𝑣𝑚

(𝜎𝑥𝑥 −𝜎𝑦𝑦 ) =√

2

2

2 +𝜎 2 +𝜎 2 ) +(𝜎𝑦𝑦 −𝜎𝑧𝑧 ) +(𝜎𝑧𝑧 −𝜎𝑥𝑥 )2 +6(𝜎𝑥𝑦 𝑦𝑧 𝑧𝑥

2

.

(10)

Another way to describe the thermal stress in a growing crystal is the resolved shear stress 𝛼 (𝜏𝑟𝑠𝑠 ), which is computed by projecting the stress tensor 𝛔 ̿ in Cartesian system onto the glide

plane with the normal 𝒏𝜶 and into the glide direction 𝒔𝜶 : 𝛼 𝜏𝑟𝑠𝑠 = 𝒔𝜶 ∙ 𝛔 ̿ ∙ 𝒏𝜶 ,

(11)

where 𝒔𝜶 is the unit vector of the 𝛂 slip direction and 𝒏𝜶 is the unit normal vector of the slip plane in the Cartesian system. In the on-axis situation, the three primary slip directions are −𝑎1 , −𝑎2 , and −𝑎3 , which are defined as [2̅110], [12̅10] and [112̅0] in the hexagonal system and [1̅√30], [1̅̅̅̅̅ √30], and [100] in the Cartesian system, as shown in Figure 1. To project the resulting stress tensor onto the slip directions within the tilted slip plane, rotations of the normal


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vector 𝒏𝜶 and the slip direction 𝒔𝜶 are required, and the detailed transformation expressions can be obtained from.28,29 Muller et al.30 stated that dislocation formation starts at approximately 10–20% of the lower yield point of the material. Jordan et al.31,32 postulated that if the resolved shear stress exceeds the critical value, dislocation generation occurs. The dislocation generation rate is proportional to the value of excess stresses from the critical value for all slip systems. A lower yield point is generally adopted in order to define the critical resolved shear stress versus temperature. Nevertheless, the relationship between the critical resolved shear stress and temperature is not available at high temperature, as noted in ref 24. Considering a small temperature difference (for example, below 10 K in this work) in the growing AlN crystal, the critical resolved shear stress normally can be regarded as a constant. To qualitatively analyze the rate of dislocation generation on the base plane, we mainly focus our attention on the basal plane resolved shear stress along the primary slip system, because the resolved shear stress on the pyramidal plane and prismatic plane in [1̅21̅0] direction are much smaller when compared to the stresses along the primary directions. Accordingly, the physical meaning of the excess stress can be expressed by the total resolved shear stress 𝜏𝑡𝑜𝑡 along the {0001}〈112̅0〉 primary slip system, which is obtained as follows: 𝛼 | 𝜏𝑡𝑜𝑡 = ∑3𝛼=1|𝜏𝑟𝑠𝑠 .

(12)

Boundary conditions must be provided to complete the above-described thermal stress model. A rigid constraint is applied for the crystal seed in contact with the tungsten crucible cap, while a free boundary described by 𝛔 ̿ ∙ 𝒏 = 0 is applied on the growth surface and peripheries of the growing AlN single crystal in this work. 3. GEOMETRY DESCRIPTION AND SIMULATION PARAMETERS


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Figure 2(left) shows the schematic diagram of an in-house PVT sublimation reactor for 1-3 inch homoepitaxial AlN single crystal growth. The growth reactor consists of a 3-inch crucible and two resistant heaters, while other features are quite similar to another in-house growth reactor described by us in ref 33. Figure 2(right) illustrates a 3-inch tungsten crucible system composed of a container, cap, seed holder and cone-tube. The cone-tube is employed to calibrate the temperature distribution around the seed and the growing crystal, and consequently to suppress the parasitic nucleation surrounding the seed as well as to enhance the mass transport toward the seed. A reasonable radial temperature gradient in front of the seed is required in order to achieve crystal diameter enlargement.34 The radial temperature gradient has to be high enough to ensure appreciable peripheral growth rates. Nevertheless, a too large radial temperature gradient will generate additional defects like LAGB or basal-plane dislocations (BPDs) due to anisotropic thermal-elastic stresses. Accordingly, three cone-tubes are deliberately designed to calibrate the radial temperature gradient around the seed. A 1-inch diameter AlN single crystal seed is attached to the crucible cap, and the growing crystal is simplified as a solid truncated cone with the expansion angle of 24.5° ,35 and a flat growth surface with diameter of 34.5 mm is assumed in this work. Prior to predicting the thermal stress inside the growing AlN crystals, the entire thermal behavior of the growth system must be modeled and evaluated accurately. The AlN crystal growth process involves many coupled physical/chemical phenomena, together with continually deforming geometries of the growing crystal and the powder source. All such physical/chemical behaviors lead to a set of highly nonlinear, time-dependent partial differential equations governing the growth process that must be solved in order to model the growth. Considering the very slow growth rate, the transient effect is negligible during the PVT AlN growth process. In


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this work, the temperature distribution in the entire PVT growth chamber is primarily obtained using the FEMAG two-dimensional (2D) axisymmetric model in an inverse manner through the global quasi-steady modeling technique. In addition, gas convection plays a minor role in the global heat and mass transfer, and therefore can be neglected, as noted by refs 33 and 36. The equations governing the radiation and conduction are described in ref 27 and will not be addressed here. Since two resistive heaters are employed in our in-house growth reactor, two temperaturecontrolling points are required to close the global heat transfer problem. Therefore, when solving those governing equations, the temperature control points are chosen to be the bottom center of the crucible at 2613.15 K and the top center of the seed at 2373.15 K, respectively. The purpose of choosing the seed center as one control point is to make sure that the AlN deposition temperature for all three designs when loaded with different orientated seeds is exactly the same, which will facilitate subsequent stress analysis and comparison. The 2D axisymmetric temperature fields in the growing crystal obtained by FEMAG are projected to the 3D Cartesian system for the final thermal stress calculation. The material properties adopted in all simulations are described in ref 33. All simulations are performed under the same operating conditions on exactly the same growth geometries except for the cone-tube, which is deliberately designed in three different shapes (see Fig. 2) in order to investigate the influence of these shapes on the thermal stress in the growing crystal. It is worth noting that 1) the emissivity and thermal conductivity of tungsten and AlN powder/crystal are assumed to be constant, since these properties either play a minor role on the global heat transfer, or are not available at high temperature or strongly dependent on their surface conditions. 2) Similar as reported literatures,23,36,37 the bulk AlN crystal is assumed to be opaque. 3) Considering a very


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low oxygen and carbon concentration inside the AlN crystals grown by our in-house growth reactor loaded with a pure tungsten setup, the influence of impurities on the global heat transfer is also neglected. To predict the thermal-elastic stresses in the growing single crystal, the anisotropic linear thermal expansion coefficient is chosen to be 𝛽𝑎 =7.2× 10−6 K-1 for the a direction, and 𝛽𝑐 =6.2× 10−6 K-1 for the c direction at the AlN growth temperature. These data were evaluated by Reeber et al.38 using available experimental data39,40 and a semi-empirical multi-frequency Einstein model up to 3000 K. Five independent components of the stress-strain elasticity tensor 𝐶̿ employed in this work were completely measured by Kazan et al.41 using three different configurations for Brillouin scattering, which are 𝐶11 =394, 𝐶12 =134, 𝐶13 =95, 𝐶33 =402 and 𝐶44 =121 in GPa. 4. SIMULATION RESULTS AND DISCUSSIONS Figure 3 presents the temperature distribution inside the single crystal when loaded with three tube-cone designs. For each design, one can see that the highest temperature inside the crystal is located at the growth surface periphery. Nevertheless, the thermal gradient in the growing crystal for design A is much higher than that of designs B and C. In particular, the highest thermal gradient for design A is located on the top seed crystal surface (contacting with the crucible cap), while much smaller values are observed at the same location for designs B and C. Furthermore, the radial thermal gradients along the crystal growth surface (bottom surface) are quite similar to each other for the three designs, although the temperature along this surface for design A is slightly higher (by 0.5-1 K) than those of designs B and C. Finally, it is worth noting that the temperature gradient along the crystal side surface for design B is the highest among all three designs, which can be easily seen from the inspection of Figure 3(right).


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Based on the above-described obtained three temperature fields inside the growing single crystal, the von Mises stress, resolved shear stress on {0001}〈112̅0〉 basal slip system and the total resolved shear stress along varied growth directions are calculated, respectively. To clearly observe the anisotropy of the stress, three tilt angles from on-axis towards [112̅0] direction in the hexagonal crystal system i.e., θ=0∘ [001], 𝜃=30∘ [10√3], and θ=90∘ [100] are studied, and the differences of the stress distribution in the growing crystal along these orientations when loaded with three tube-cone designs are revealed accordingly. 4.1. Von Mises stress analysis. As described previously, the stress level in a growing crystal can be evaluated by the von Mises stress, and dislocation generation is presumed to occur in the regions where the von Mises stress exceeds the CRSS, and even crystal cracking may even occur when the von Mises stress exceeds a critical value. Therefore, reducing the von Mises stress during crystal growth would be a feasible approach to minimize the dislocation generation and micro-cracking possibilities in the as-grown AlN crystals Figure 4 shows the von Mises stress distribution in the crystal when loaded with three tubecone designs and grown along [001], [10√3] and [100] orientations, respectively. It can be seen that the von Mises stress in the entire crystal for all results exceeds 1.1 GPa, which is very similar to the value of 1 GPa revealed by Raman spectroscopy42 and of the same order of magnitude as the value predicted in refs 25 and 43. For one specific design, the von Mises stress inside the growing crystal at the [100] growth orientation is the highest compared to the other growth orientations, implying that a higher risk of crystal cracking and dislocation generation occurred during the growth of the AlN crystal along the [100] orientation. The highest von Mises stress (1.171 GPa) appears at the peripheral region of the seed crystal for the growth along the [100] direction when loaded with the cone-tube design B. On average, this maximum value is


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20-25 MPa higher than that for the other designs. Furthermore, the profiles of the von Mises stress in the growing crystal along the growth axis for different growth orientations are quite different. For example, the von Mises stress profile at the [100] growth orientation shows an opposite tendency along the growth axis when compared to [001] and [10√3] growth. For the crystal grown along the [001] direction when loaded with the tube-cone design A, the maximum von Mises appears at the peripheral region of the seed crystal, where a very high thermal gradient exists, as illustrated in Figure 3. By contrast, the lowest von Mises stress is observed in the same region for designs B and C, while the maximum stress for designs B and C is located at the center of the growth surface, where a quite high von Mises stress is present for the design A as well. In addition, for the growth along the [001] direction, the von Mises stress distribution shows axisymmetric characteristics due to the trigonal symmetry of the AlN hexagonal lattice along the c-axis. Nevertheless, a weak anisotropic von Mises stress distribution in [100] and [10√3] growth direction is observed. The anisotropy can be seen when the stress is projected onto the XY plane after horizontal cutting at the crystal heights of 5 mm and 10 mm (or crystal growth surface), respectively, as shown in Figure 5. 4.2. Resolved Shear Stress Analysis. It is known that plastic deformation of the AlN single crystal is mainly caused by BPDs, and the driving force for BPD multiplication is not the von Mises stress but rather the resolved shear stress in the primary slip direction, that is the a/3〈112̅0〉{0001} direction. Based on the CRSS model of Jordan et al.,32 the dislocation density is proportional to the difference of the resolved shear stress and the critical resolved shear stress on the slip plane. Therefore, to provide a qualitative distribution of the BPDs in the growing crystal, it is necessary to analyze the resolved shear stress for the above specified designs and growth conditions.


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Obviously, the distribution of the resolved shear stress 𝜏𝑟𝑠𝑠 in a AlN crystal growing along the c-axis is axisymmetric in the cylindrical coordinate system, because the axisymmetric shear stress is the only contributor to the resolved shear stress in the on-axis case. However, the distribution of the predicted resolved shear stress 𝜏𝑟𝑠𝑠 in the Cartesian coordinate system is anisotropic, as can be observed in Figure 6. In Figure 6, the predicted resolved shear stress 𝜏𝑟𝑠𝑠 in the AlN crystal growing along the [001] orientation is illustrated for three cone-tube designs, and three {0001}〈112̅0〉 slip directions (-a1 , -a2 , -a3 ) are considered, and the predicted resolved shear stress 𝜏𝑟𝑠𝑠 is distributed in exactly the similar way along the three {0001}〈112̅0〉 slip directions, and the same conclusion was drawn in refs 28 and 29 in the 2D axisymmetric case. According the results illustrated in Figure 6, the distributions of the resolved shear stress in the crystal growing along the [001] orientation for the three designs are quite different. For example, the maximum stress appears at the side surface of the growing crystal (close to the seed crystal surface) for designs B and C. On the other hand, for design A, the maximum stress occurs at the peripheral region along the seed crystal surface. Meanwhile, the maximum stress for design B is 22.3% and 62.5% higher than those of designs C and A, respectively. Since the stress level is strongly dependent on the temperature gradient in the as-grown crystal, we attribute the stress distribution and stress level differences to a higher temperature gradient along the crystal side surface (close to the seed crystal surface) for design B, which was noted previously and can be seen in Figure 3(right). The results imply that design B would not be a favorable choice for the homoepitaxial AlN growth along the [001] orientation, because a large amount of BPDs could accumulate between the seed and side surface region of the growing crystal for design B. Figure 7 shows the 3D-resolved shear stress distribution in the growing AlN crystal along the three primary slip directions for design C with the [001], [10√3] and [100] growth orientations


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by viewing from +Z direction (from the seed crystal surface). The three primary slip directions are classified into two families, i.e., the first family along the -a1 and -a2 slip directions, and the second family along the -a3 direction. For a specific family, the trend and magnitude of the resolved shear stress at different slip directions are identical. The maximum resolved shear stress for each growth orientation appears at the crystal side surface in contact with the seed. The resolved shear stress is tensile along all three primary slip directions for the [001]-grown crystal, as seen in Figure 7(a), and the maximum stress reaches ±7.12 MPa for both families. However, for the [10√3]-grown crystal, the stress is compressive for the +a1 /+a2 direction and is tensile for the -a1 /-a2 direction. Nevertheless, an opposite result is observed for the ±a3 slip direction. Meanwhile, the maximum stress reaches -9.54 MPa and +14.18 MPa for the -a1 /-a2 and -a3 directions, respectively. Furthermore, an inversion of the shear stress from tensile to compressive is observed when the growing crystal transitions from [001] growth to [100] growth for the second family, and the maximum stress reaches ±5.54 MPa and ±8.64 MPa for the first and second families, respectively. 4.3. Total Resolved Shear Stress Analysis. The CRSS model proposed by Jordan et al.30 has been applied extensively to predict the dislocation distribution in Si, GaAs, InP crystals grown from liquid phases.20–22 However, the prediction of thermal stress-induced dislocations in hexagonal crystals (SiC, AlN) grown from vapor phases is still rare. Meanwhile, the critical resolved shear stress is a strong function of the temperature and is not available at high temperature for the AlN crystal. Therefore, in this work the total resolved shear stress 𝜏𝑡𝑜𝑡 is employed to qualitatively represent the rate of dislocation generation induced by the thermal stress.


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The distribution of the total resolved shear stress 𝜏𝑡𝑜𝑡 in the growing crystal for design C along the [001], [10√3] and [100] growth orientations is illustrated in Figure 8. Examination of the results presented in Figure 8 shows that the largest total resolved shear stress for each growth orientation is found at the crystal side surface in contact with the seed surface along the x-axis. Nevertheless, the largest stress for the [001] growth orientation also appears along the other slip axes due to the 6-fold symmetry. The overall stress level in the growing crystal increases steadily until reaching a maximum value and then decreases when the growth changes from the [001], [10√3] to the [100] orientation. The maximum stress in the crystal grown along [100] orientation is approximately 17.29 MPa, 27% higher than that for the [001] growth orientation. Furthermore, the stress distribution in the growing crystal shows 6-fold symmetry, reflection symmetry and 2fold symmetry for growth at the [001], [10 √3 ] and [100] orientations, respectively. This transition can be clearly observed upon the projection of the stress onto the XY plane after horizontal cuttings at the crystal heights of 0.2 mm and 5 mm (or horizontal middle cut-plane), respectively, as shown in Figure 9. Figure 10 presents the distribution of the total resolved shear stress in the longitudinal cut for design C along the [001], [10√3] and [100] growth orientations. It can be observed that the maximum stress is located at the distance of 0.1-0.3 mm from the peripheral region of the seed crystal surface for each growth direction. In general, this maximum stress value is 1-2 orders of magnitude higher than the stress magnitude at any other location, implying that the dislocation multiplication rate beneath the periphery of seed crystal is much higher than at other locations, resulting in the aggregation of a large number of dislocations at this location. This could be one of the reasons that a higher dislocation density is normally found at the peripheral region beneath the seed crystal, and even a parasitic polycrystalline rim could form around the grown single


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crystal in practical homoepitaxial AlN crystal growth, as observed in ref 34. In addition, a highstress region near the crystal growth surface is observed for the crystal grown at [10 √3 ] orientation, which means that the dislocation would multiply and aggregate rapidly toward the center of the crystal growth surface. Finally, the stress distribution for the [001] and [100] growth orientations are identical, and are symmetrical with respect to the z-axis, even though the stress level at the [100] growth orientation is slightly higher than that of the [001] growth orientation. Therefore, the homoepitaxial growth along the [001] orientation for design C could be a favorable choice to reduce the BPDs and avoid parasitic polycrystalline grain nucleation. 5. CONCLUSIONS AND PROSPECTS A thermal-elastic stress model using the FEM method is developed for the prediction of 3D anisotropic stress inside growing AlN single crystals obtained by the PVT growth process. The developed thermal-elastic stress model is applied to model a 1-inch AlN crystal homoepitaxial grown by a 3-inch in-house PVT sublimation reactor. Three different cone-tubes surrounding the growing AlN crystal are deliberately designed to calibrate the temperature field around the seed and the growing crystal. A series of numerical experiments are systematically conducted to investigate the influences of the cone-tube shape and crystal growth orientation on the distribution and magnitude of the thermal stress in the growing AlN crystal. The simulation results show that the distribution and magnitude of the stresses depend strongly on the thermal gradient inside the growing crystal as well as on the crystal growth orientation, and strongly anisotropy of the resolved shear stress and total resolved shear stress in the growing hexagonal AlN single crystal is observed, while the anisotropy of the von Mises stress is negligible. Under the specified growth conditions, the von Mises stress inside the crystal exceeds 1.11 GPa, which is in accordance with literature reports and available Raman spectroscopy measurements. The


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magnitude of the von Mises stress implies that a high risk of micro-cracking and dislocation generation is present during the growth of AlN single crystals, particularly for the crystals grown along the [100] orientation. The distribution of the resolved shear stress along the {0001}〈112̅0〉 primary slip system reveals that the crystal growing along the c-axis is under tensile stress along all three primary slip directions, and an inversion of the shear stress from tensile to compressive along the -𝑎3 slip direction is observed when the growth transitions from the [001] orientation to the [100] orientation. The total resolved shear stress shows 6-fold symmetry, reflection symmetry and 2-fold symmetry in the growing AlN crystals along the [001], [10√3] and [100] growth orientations, respectively. A too low or too high radial temperature gradient around the AlN seed is not beneficial to achieve diameter enlargement through homoepitaxial growth by the PVT method. Therefore, a reasonable radial temperature gradient around the seed plays a key role to ensure appropriate peripheral growth rate, avoid additional defect generation (such as BPDs and LAGBs due to anisotropic thermal stress) and parasitic grain growth. In this work, we design three cone-tubes to calibrate the radial temperature gradient along the AlN seed crystal, and quantitatively investigate the thermal-elastic stress inside the crystal grown along different orientations by anisotropic 3D thermal stress modeling and simulation for the first time. We conclude that homoepitaxial growth along the [001] orientation for the cone-tube design C could be an optimal choice for reducing the BPDs in the growing crystal and to avoid parasitic polycrystalline grain nucleation around the seed crystal. Our future work will further extend our developed model for dislocation density prediction. Homoepitaxial AlN single crystal growth experiments will also be conducted based on the obtained simulation results and defect characterization of the AlN single crystals will be performed as well to validate our developed model.


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AUTHOR INFORMATION Corresponding Author Liang Wu, PhD Professor, School of Materials Science and Engineering Shanghai University Address: No. 99 ShangDa Road, BaoShan District, Shanghai University, Shanghai 200072, China. Phone: 0086-18516773360 E-mail: [email protected] Funding Sources The authors cordially acknowledge the research fund partly supported by “the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning” project (Grant no. 60D-110-14-201), China. Notes The authors declare no competing financial interest. REFERENCES (1) Ambacher, O. Growth and applications of Group III-nitrides. J. Phys. D: Appl. Phys. 1998, 31, 2653-2710.


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(2) Kovalenkov, O.; Soukhoveev, V.; Ivantsov, V.; Usikov, A.; Dmitriev, V. Thick AlN layers grown by HVPE. J. Cryst. Growth 2005, 281, 87-92. (3) Katagiri, Y.; Kishino, S.; Okuura, K.; Miyake, H.; Hiramatu, K. Low-pressure HVPE growth of crack-free thick AlN on a trench-patterned AlN template. J. Cryst. Growth 2009, 311, 2831-2833. (4) Kumagai, Y.; Tajima, J.; Ishizuki, M.; Nagashima, T.; Murakami, H.; Takada, K.; Koukitu, A. Self-Separation of a Thick AlN Layer from a Sapphire Substrate via Interfacial Voids Formed by the Decomposition of Sapphire. Appl. Phys. Express 2008, 1, 045003. (5) Freitas, J. A. Properties of the state of the art of bulk III-V nitride substrates and homoepitaxial layers. J. Phys. D: Appl. Phys. 2010, 43, 073001. (6) Kamei, K.; Shirai, Y.; Tanaka, T.; Okada, N.; Yauchi, A.; Amano, H. Solution growth of AlN single crystal using Cu solvent under atmospheric pressure nitrogen. Phys. Status Solidi C 2007, 4, 2211-2214. (7) Bockowski, M. Growth and doping of GaN and AlN single crystals under high nitrogen pressure. Cryst. Res. Technol. 2001, 36, 771-787. (8) Kangawa, Y.; Toki, R.; Yayama, T.; Epelbaum, B. M.; Kakimoto, K. Novel solution growth method of bulk AlN using Al and Li3N solid sources. Appl. Phys. Express 2011, 4, 095501. (9) Yano, M.; Okamoto, M.; Yap, Y. K.; Yoshimura, M.; Mori, Y.; Sasaki, T. Growth of nitride crystals, BN, AlN and GaN by using a Na flux. Diamond Relat. Mater. 2000, 9, 512-515.


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(10) Slack, G. A.; McNelly, T. F. Growth of high purity AlN crystals. J. Cryst. Growth 1976, 34, 263-279. (11) Noveski, V.; Schlesser, R.; Raghothamachar, B.; Dudley, M.; Mahajang, S.; Beaudoin, S.; Sitar, Z. Seeded growth of bulk AlN crystals and grain evolution in polycrystalline AlN boules. J. Cryst. Growth 2005, 279, 13-19. (12) Epelbaum, B. M.; Bickermann, M.; Nagata, S.; Heimann, P.; Filip, O.; Winnacker, A. Similarities and differences in sublimation growth of SiC and AlN. J. Cryst. Growth 2007, 305, 317-325. (13) Sumathi, R. R. Native seeding and silicon doping in bulk growth of AlN single crystals by PVT method. Phys. Status Solidi C 2014, 11, 545-548. (14) Chemekova, T. Y.; Avdeev, O. V.; Barash, I. S.; Mokhov, E. N.; Nagalyuk, S. S.; Roenkov, A. D.; Segal, A. S.; Makarov, Y. N.; Ramm, M. G.; Davis, S.; Huminic, G.; Helava, H. Sublimation growth of 2 inch diameter bulk AlN crystals. Phys. Status Solidi 2008, 5, 1612-1614. (15) Schujman, S. B.; Schowalter, L. J.; Bondokov, R. T.; Morgan, K. E.; Liu, W.; Smart, J. A.; Bettles, T. Structural and surface characterization of large diameter, crystalline AlN substrates for device fabrication. J. Cryst. Growth 2008, 310, 887-890. (16) Hartmann, C.; Wollweber, J.; Dittmar, A.; Irmscher, K.; Kwasniewski, A.; Langhans, F.; Neugut, T.; Bickermann, M. Preparation of Bulk AlN Seeds by Spontaneous Nucleation of Freestanding Crystals. Jpn. J. Appl. Phys. 2013, 52, 279-287.


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(17) Gao, B.; Kakimoto, K. Three-Dimensional Modeling of Basal Plane Dislocations in 4HSiC Single Crystals Grown by the Physical Vapor Transport Method. Cryst. Growth Des. 2014, 14,1272-1278. (18) Raghothamachar, B.; Dalmau, R.; Moody, B.; Craft, S.; Schlesser, R.; Xie, J. Q.; Collazo, R; Dudley, M.; Sitar, Z. Low Defect Density Bulk AlN Substrates for High Performance Electronics and Optoelectronics. Mater. Sci. Forum 2012, 717-720, 1287-1290. (19) Taniyasu, Y.; Kasu, M.; Makimoto, T. An aluminium nitride light-emitting diode with a wavelength of 210 nanometres. Nature 2006, 441, 325-328. (20) Tavakoli, M. H.; Renani, E. K.; Honarmandnia, M.; Ezheiyan, M. Computational analysis of heat transfer, thermal stress and dislocation density during resistively Czochralski growth of germanium single crystal. J. Cryst. Growth 2018, 483, 125-133. (21) Nakano, S.; Gao, B.; Kakimoto, K. Numerical analysis of dislocation density and residual stress in a GaN single crystal during the cooling process. J. Cryst. Growth 2017, 468, 839–844. (22) Gao, B.; Nakano, S.; Harada, H.; Miyamuraa, Y.; Kakimoto, K. Three-dimensional analysis of dislocation multiplication during thermal process of grown silicon with different orientations. J. Cryst. Growth 2017, 474, 121-129. (23) Wu, B.; Ma, R. H.; Zhang, H.; Prasad, V. Modeling and simulation of AlN bulk sublimation growth systems. J. Cryst. Growth 2004, 266, 303-312. (24) Wu, B.; Ma, R. H.; Zhang, H.; Dudley, M.; Schlesser, R.; Sitar, Z. Growth kinetics and thermal stress in AlN bulk crystal growth. J. Cryst. Growth 2003, 253, 326-339.


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(25) Lee, R. G.; Idesman, A.; Nyakiti, L.; Chaudhuri, J. Modeling of residual thermal stresses for aluminum nitride crystal growth by sublimation. J. Appl. Phys. 2007, 102, 227-234. (26) Karvani, P.; Maniatty, A. M. Modeling micromechanical response to thermal history in bulk grown aluminum nitride. Phys. Status Solidi 2015, 12, 345-348. (27) Dupret, F.; Nicodème, P.; Ryckmans, Y.; Wouters, P.; Crochet, M. J. Global modelling of heat transfer in crystal growth furnaces. Int. J. Heat Mass Transfer 1990, 33, 1849-1871. (28) Böttcher, K.; Cliffe, K. A. Three-dimensional thermal stresses in on-axis grown SiC crystals. J. Cryst. Growth 2005, 284, 425-433. (29) Böttcher, K.; Cliffe, K. A. Three-dimensional resolved shear stresses in off-axis grown SiC single crystals. J. Cryst. Growth 2007, 303, 310-313. (30) Muller, G.; Rupp, R.; Volkl, J.; Wolf, H.; Blum, W. Deformation behaviour and dislocation formation in undoped and doped (Zn, S)InP crystals. J. Cryst. Growth 1985, 71, 771781. (31) Jordan, A. S.; Caruso, R.; Neida, A. R. V. A thermoelastic analysis of dislocation generation in pulled GaAs crystals. Bell Syst. Tech. J. 1980, 59, 593-637. (32) Jordan, A. S.; Neida, A. R. V.; Caruso, R.; Nielsen, W. A comparative study of thermal stress induced dislocation generation in pulled GaAs, InP, and Si crystals. J. Appl. Phys. 1981, 52, 3331-3336. (33) Wang, Z. H.; Deng, X. L.; Cao, K.; Wang, J.; Wu, L. Hotzone design and optimization for 2-in. AlN PVT growth process through global heat transfer modeling and simulations. J. Cryst. Growth 2017, 474, 76-80.


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(34) Hartmann, C.; Wollweber, J.; Sintonen, S.; Dittmar, A.; Kirste, L.; Kollowa, S.; Irmscher, K.; Bickermann, M. Preparation of deep UV transparent AlN substrates with high structural perfection for optoelectronic devices. CrystEngComm 2016, 18, 3488-3497. (35) Lu, P.; Collazo, R.; Dalmau, R. F.; Durkaya, G.; Dietz, N.; Raghothamachar, B.; Dudley, M.; Sitar, Z. Seeded growth of AlN bulk crystals in m - and c -orientation. J. Cryst. Growth 2009, 312, 58-63. (36) Wu, B.; Zhang, H. Transport phenomena in an aluminum nitride induction heating sublimation growth system. Int. J. Heat Mass Transfer 2004, 47, 2989-3001. (37) Wu, B.; Noveski, V.; Zhang, H.; Schlesser, R.; Mahajan, S.; Beaudoin, S.; Sitar, Z. Design of an RF-Heated Bulk AlN Growth Reactor: Induction Heating and Heat Transfer Modeling. Cryst. Growth Des. 2005, 5, 1491-1498. (38) Reeber, R. R.; Wang, K. Lattice Parameters and Thermal Expansion of Important Semiconductors and Their Substrates. Mater. Res. Soc. Symp. Proc. 2000, 622, T6.35.1-T6.35.6. (39) Yim, W. M.; Paff, R. J. Thermal expansion of AlN, sapphire, and silicon. J. Appl. Phys. 1974, 45, 1456-1457. (40) Slack, G. A.; Bartram, S. F. Thermal expansion of some diamondlike crystals. J. Appl. Phys. 1975, 46, 89-98. (41) Kazan, M.; Moussaed, E.; Nader, R.; Masri, P. Elastic constants of aluminum nitride. Phys. Status Solidi 2007, 4, 204-207.


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(42) Dalmau, R.; Sitar, Z. AlN Bulk Crystal Growth by Physical Vapor Transport. In Springer Handbook of Crystal Growth; Dhanaraj, G., Byrappa, K., Prasad, V., Dudley, M.; Springer: Berlin, Heidelberg, 2010; chap 24, pp 821-843. (43) Guo, J. J.; Reddy, K. M.; Hirata, A.; Fujita, T.; Gazonas, G. A.; McCauley, J. W.; Chen, M. W. Sample size induced brittle-to-ductile transition of single-crystal aluminum nitride. Acta Mater. 2015, 88, 252-259.


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For Table of Contents Use Only

Anisotropic Three-Dimensional Thermal Stress Modeling and Simulation of Homoepitaxial AlN Single Crystal Growth by Physical Vapor Transport Method Qikun Wang, Jiali Huang, Zhihao Wang, Guangdong He, Dan Lei, Jiawei Gong, and Liang Wu*

The distribution of the total resolved shear stress 𝜏𝑡𝑜𝑡 in the growing AlN crystal along the c-axis, off-axis (rotated by θ=30∘ towards [112̅0]) and a-axis growth orientations. The stress in the growing crystal shows 6-fold symmetry, reflection symmetry and 2-fold symmetry along the caxis, off-axis and a-axis growth orientations, respectively.


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Figure 1. Hexagonal unit cell: hexagonal axes (a1, a2, a3, c) and Cartesian axes (x, y, z).

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Figure 2. Schematic configuration of a 3-inch PVT AlN growth reactor (left) with associated crucible system (right): design A (a), design B (b) and design C (c). 79x44mm (300 x 300 DPI)



Figure 3. Temperature distribution inside the AlN single crystal (left): (a) design A, (b) design B, (c) design C, and temperature profiles along crystal seed/growth/side surface and growth axis for three designs (right). 59x22mm (300 x 300 DPI)


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Figure 4. Von Mises stress distribution in AlN growing crystal along [001] (top), [10√3] (middle), and [100] (bottom) growth orientations for three designs. 92x53mm (300 x 300 DPI)



Figure 5. Von Mises stress on horizontal middle cut-plane (left of each specific design) and growth surface (right of each specific design) for three designs at growth orientation [001] (top), [10√3] (middle), and [100] (bottom) growth orientations.

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Figure 6. 3D resolved shear stress along three slip directions (-a1, -a2, -a3) for design A (a), design B (b) and design C (c) along [001] c-axis growth orientation. 50x16mm (300 x 300 DPI)



Figure 7. 3D resolved shear stress in the growing crystal along three primary slip directions for design C with [001] (a), [10√3] (b) and [100] (c) growth orientation.

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Figure 8. Total resolved shear stress for design C at [001] (a), [10√3] (b), and [100] (c) growth orientation.

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Figure 9. Total resolved shear stress on seed crystal surface (top) and horizontal middle cut-plane (bottom) for design C at [001] (a), [10√3] (b), and [100] (c) growth orientations.

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Figure 10. Total resolved shear stress on longitudinal cut-plane for design C at [001] (a), [10√3] (b), and [100] (c) growth orientation.

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Anisotropic Three-Dimensional Thermal Stress Modeling and

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