AlexanderâHaasen Model of Basal Plane Dislocations in Single

Article pubs.acs.org/crystal

Alexander−Haasen Model of Basal Plane Dislocations in SingleCrystal Sapphire B. Gao,* S. Nakano, N. Miyazaki,‡ and K. Kakimoto Research Institute for Applied Mechanics, Kyushu University, Fukuoka, Fukuoka Prefecture 812-8581, Japan

ABSTRACT: The Alexander−Haasen model was originally used to model the multiplication of the mobile dislocations in crystalline silicon. Here, we extend this model for studying multiplication of basal plane dislocations (BPDs) in single-crystal sapphire. By fitting the Alexander−Haasen model to experimental data, we find that the model accurately describes the plastic deformation of sapphire caused by BPDs. The application of the Alexander−Haasen model to sapphire growth made it possible to minimize the dislocation density and residual stress in growing crystals by optimizing the furnace structure and operation conditions. We apply the Alexander−Haasen model to investigate the dynamical deformation of single-crystal sapphire during the cooling process and examine the effect of the cooling rate on the generation of BPDs and residual stress. Finally, we present the BPD distribution and discuss the main factor that influences the generation of BPDs.

1. INTRODUCTION Single-crystal sapphire has recently garnered significant attention as an important substrate material for fabricating gallium nitride light-emitting diodes (LEDs), which promise great energy savings over both incandescent and compact fluorescent lighting technologies. The market for such lighting is currently expanding at double-digit annual growth rates.1 High-quality gallium nitride requires high-quality sapphire substrates. To enhance the quality of crystalline sapphire (Al2O3), the density of dislocations must be controlled and reduced. To reduce the generation rate of dislocations during the growth of crystalline sapphire, a model that relates dynamic dislocation generation to the practical growth conditions is required. However, no such model that can describe the plastic deformation of sapphire currently exists. To address this shortfall, the dislocation density-based Alexander−Haasen (AH) model,2,3 which was originally used for crystalline silicon, is applied to crystalline sapphire. The AH model has been used extensively to understand the plastic deformation of elemental4−6 and III−V compound semiconductors7,8 over a wide range of temperatures and stresses. Recently, the AH model has been extended to IV−IV compounds such as SiC,9−11 and the simulated results agree well with experimental data. Applying the AH model12−14 to the global modeling of crystal growth of elemental, III−V, and IV−IV compound semiconductors has © 2014 American Chemical Society

resulted in an extraordinary optimization and reduction of dislocations in these materials. Thus, it would be very useful to determine whether the AH model can be extended to crystalline sapphire.

2. DETERMINATION OF ALEXANDER−HAASEN PARAMETERS FROM EXPERIMENTAL DATA Experiments on the tensile deformation of sapphire15 have been performed at various temperatures (1200−1700 °C) and strain rates (10−5 to 10−3 s−1). These experiments15 confirmed that plastic slip occurs mainly in the {0001} basal slip plane and ⟨112̅0⟩ slip direction; the other slip planes, such as prismatic and pyramidal, have high activation energy. Therefore, in this study, we model only basal plane dislocations. We fit the AH model to the experimental stress−strain curves.15 Details of the fitting procedure are explained as follows. In this paper, the following notation is used: the slip direction is α; in the slip direction α, the resolved shear stress is (α) τ(α) resolv and the plastic strain is εpl ; the Schmid factor is Φ; the applied stress in the tensile system is τa, and the plastic strain along the tensile direction is εpl. The satisfied relationships are as follows: Received: May 13, 2014 Revised: June 30, 2014 Published: July 1, 2014 4080

dx.doi.org/10.1021/cg500705t | Cryst. Growth Des. 2014, 14, 4080−4086

Crystal Growth & Design

Article

(α) τresolv = Φτa

(1)

εpl(α) = εpl /Φ

(2)

dislocation density N(α) m , the Schmid factor Φ, the applied stress τa, the elastic constant η of the tensile system, and the material constants k0 and m. The stress−strain relationship is very complex and stems from a dynamical system that is affected by its history of plastic deformation. To close the system of equations, the AH model gives the following rate equation for the mobile dislocation density:

Equations 1 and 2 are easily understood by considering energy relationships if the applied stress only causes plastic deformation in the α direction. The strain energies should be the same, that is, 1 (α) (α) 1 1 E = Vτresolv εpl = V Φτaεpl /Φ = Vτaεpl 2 2 2

dNm(α) (α)λ (α) (α) = Kτeff Nm v dt

(3)

⎛ Q ⎞ = Kk 0Nm(α)⟨Φτa − D Nm(α) ⟩λ+ m exp⎜ − ⎟ ⎝ kbT ⎠

The apparent shear strain ε along the tensile direction can be calculated based on the displacement of the cross head of the tensile machine. The elastic strain, εel, along the tensile direction is as follows: εel = ε − εpl (4)

where K and λ are material constants. The model parameters in eqs 10 and 11 are obtained by a fitting process that minimizes the target function f = ∑i|τa,experiment(εi)| − τa,numerical(εi) using the downhill simplex method.16 Five experimental stress−strain curves15 acquired at temperatures ranging from 1200 to 1700 °C were globally fit. The values obtained from the fit for the AH parameters are as follows:

The applied stress τa is related to the elastic strain as follows: τa = ηεel = η(ε − εpl)

(5)

where η is the elastic constant of total tensile system. The differential ratio of the applied stress to the apparent shear strain is as follows:

k 0 = 1.084 × 10−8, K = 1.660 × 10−3, λ = 0.939,

⎛ dεpl(α) ⎞ η(dε − dεpl) dτa ⎟ = = η⎜⎜1 − Φ ⎟ dε dε d ε ⎝ ⎠ ⎛ dεpl /dt ⎞ ⎟ = η⎜⎜1 − Φ dε /dt ⎟⎠ ⎝

m = 3.000, Q = 4.709 eV

(6)

According to the AH model, the plastic strain rate in the α slip direction is expressed as follows:

N(α) m

(7)

(α)

where and v are the mobile dislocation density and the velocity of dislocations in the α slip direction, respectively, and b(α) is the Burgers vector. The velocity of dislocations is given by ⎛ Q ⎞ (α)m v(α) = k 0τeff exp⎜ − ⎟ ⎝ kBT ⎠

k 0 = 8.5 × 10−15, K = 7.0 × 10−5, m = 2.800

(8)

where k0 and m are material constants, Q is the activation enthalpy of dislocations, kB is Boltzmann’s constant, T is the temperature in kelvin, and τ(α) eff is the effective stress for dislocation multiplication in the α slip direction and is given by (α) (α) τeff = ⟨τresolv − D Nm(α) ⟩

λ = 1.1, Q = 3.3 eV

when

T > 1000 °C

λ = 0.6, Q = 2.6 eV

when

T ≤ 1000 °C

and (13)

Comparison of eq 12 and eq 13 shows that almost all the parameter values for sapphire are larger than those for SiC, which means that the BPD in sapphire is more mobile than that in SiC. For Si crystal, the parameters are21

(9)

where D is the hardening factor. Note that ⟨x⟩ = x if x > 0 and ⟨x⟩ = 0 if x ≤ 0. Substituting eqs 1 and 7−9 into eq 6 gives

k 0 = 8.5794 × 10−4 , K = 3.1 × 10−4 , λ = 1.0,

⎡ Nm(α)k 0b(α)⟨Φτa − D Nm(α) ⟩m dτa = η⎢1 − Φ ⎢ dε dε /dt ⎣ ⎛ Q ⎞⎤⎥ exp⎜ − ⎟ ⎝ kBT ⎠⎥⎦

(12)

The parameter k0 characterizes the velocity, and K characterizes the multiplication rate; the parameter λ characterizes the influence of the effective stress on the multiplication rate, and m characterizes the influence of the effective stress on the velocity; Q is the activation enthalpy for dislocation movement. Roughly speaking, the magnitude of k0 is equal to the velocity of dislocation when an effective stress of 1 Pa imposes on one dislocation with zero activation enthalpy, and the magnitude of K is equal to the increase of dislocation length after one dislocation sweeps unit area under an effective stress of 1 Pa. A comparison with Si or SiC in those parameters is useful for engineers using the AH model. For SiC crystal, the above parameters are9

(α)

dεpl(α)/dt = Nm(α)v(α)b(α)

(11)

m = 1.1, Q = 2.2 eV

(14)

It can be seen that the activation enthalpy for Si crystal is minimal compared with SiC and sapphire crystals. Figure 1 shows the comparison between the fit numerical results and the experimental data. The symbols and the smooth curves denote the experimental data and the fit numerical results, respectively. The experimental data are well fit by the stress−strain curves obtained from the AH model. This result indicates that the AH model is valid for single-crystal sapphire.

(10)

Equation 10 shows that the stress−strain curve is determined by the strain rate dε/dt, the temperature T, the mobile 4081



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Table 1. Parameters for the Dislocation Calculations symbol

description

value (units)

b K k0 Q λ m kb C11 C12 C13 C33 C44 β D T

Burgers vector in basal plane multiplication constant material constant activation enthalpy stress exponential factor stress exponential factor Boltzmann constant elastic constant elastic constant elastic constant elastic constant elastic constant thermal expansion coefficient strain hardening factor temperature

4.76 × 10−10 (m) 1.660 × 10−3 1.084 × 10−8 4.709 0.939 3.0 8.615 × 10−5 (eV/K) −2.64 × 107T + 5.10 × 1011(Pa) 4.56 × 106T + 1.61 × 1011 (Pa) −6.22 × 106T + 1.15 × 1011 (Pa) −2.98 × 107T + 5.13 × 1011 (Pa) −1.96 × 107T + 1.57 × 1011 (Pa) 1.8 × 10−5 (K−1) 15.0 K

track dislocations by avoiding ambiguous boundary conditions between them. The bottom of crystal is taken to be free during the stress calculation. The basic configuration of the furnace18 around the crucible is shown in Figure 2. The furnace is used for a simple annealing process for testing the AH model.

Figure 1. Comparison between fit numerical results and experimental data at temperatures ranging from 1200 to 1700 °C for a strain rate of 1.667 × 10−6 s−1. The symbols denote the experimental data, and the smooth curves denote the fit numerical results.

3. APPLICATION OF THE ALEXANDER−HAASEN MODEL TO SAPPHIRE COOLING PROCESS Now, we apply the AH model to track the rate-dependent process of plastic deformation during the cooling stage of the sapphire-growth process. The detailed calculations are quite similar to those for three-dimensional basal plane dislocations (BPD) in SiC.9 An isotropic assumption of materials can be used, and the elastic constant tensor matrix can be determined by Young’s modulus and Poisson’s ratio; however, for incorporation of anisotropic effect, the elastic constant tensor matrix, which is determined by elastic constants17 C11, C12, C13, C33, and C44, can be used. The anisotropic stress calculation has to be solved in a three-dimensional system, and it needs much computational time. As pointed out in the paper,23,24 two simplifications can be done to reduce the total computational time with a small error of less than 6%. The first simplification is that the stresses σθz and σrθ were considerably smaller than all other stress components and can be neglected.20,21 The second simplification is that all of the terms involving trigonometric function of nθ can be neglected after the drop power transformation of trigonometric functions. Therefore, a simplified 2D stress−strain relation including thermal strain can be used to incorporate anisotropic effect.23,24 The thermal expansion coefficient is set to 1.8 × 10−5 K−1 according to ref 22. For convenience, a table containing all parameters for simulation is provided in Table 1. The primary target of this section is to test the AH model. Thus, to avoid complications, only the cooling process is investigated. The implementation and numerical results are given in the following sections. 3.1. Furnace Configuration and Targeted Temperature History. An axisymmetric and almost dislocation-free sapphire single-crystal ingot was first placed in an axisymmetric crucible. The initial BPD density was set to 1 cm−2 everywhere, which is quite low and does not influence the final dislocation density. The crystal axis was aligned in the [0001] direction. The crystal was 90 mm in diameter and 80 mm in height. The diameter of the crucible was 96 mm. A small gap of 3 mm between the crystal and crucible was maintained to accurately

Figure 2. Basic configuration of furnace around the crucible.

To determine how the cooling process influences the generation rate of dislocations, three different cooling processes were designed, as shown in Figure 3. In the high-temperature regime (greater than 1400 °C) of the cooling process, the cooling rates of cases 1, 2, and 3 were different; however, the cooling rates of these cases were the same at low temperatures (less than 1400 °C). The heating process was the same in all three cases. To homogenize the temperature distribution inside the furnace, we required the temperatures at two monitoring points (point A on the top heater and point B on the side heater, as shown in Figure 2) to be the same and to evolve as per the predesigned curves shown in Figure 3. To control the temperatures on the predesigned curves required, a nonlinear optimization algorithm was used to determine the heater power 4082



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density increases at both the bottom and the side of the crystal. Therefore, the cooling process is a significant factor affecting the generation of BPDs during the growth of single-crystal sapphire. Figure 5 shows the comparison of BPD density distributions for different cooling rates (cases 1−3). It is clear that the fastest cooling [Figure 5c] results in the largest BPD density, whereas the slowest cooling [Figure 5a] results in almost no increase in dislocation density relative to that before cooling [Figure 4a]. Because the main difference between cases 1−3 is the cooling rate in the high-temperature region, it is essential to use a slow cooling rate at high temperatures in order to reduce the rate at which dislocations are generated. Figure 4 shows that the dislocation density already becomes high at the bottom of crystal before the cooling process. The possible reason is that the crystal directly touches the pedestal, through which the main outgoing cooling flux passes. Therefore, it is strong cooling flux causing high dislocation density at the bottom of crystal before the cooling process starts. 3.2.2. Dislocation Density Distribution in (0001) Slices. To clearly detect the dislocation density distribution inside a crystal, we made a series of cuts along the (0001) plane for case 3 (Figure 6). All slices show a six-fold symmetry. The maximum

Figure 3. Predesigned temperature history at monitoring points.

of the top and side heaters. A detailed numerical method for this is available in ref 18. 3.2. Numerical Results for Dislocations. 3.2.1. Dislocation Density Distribution before and after Cooling. Gao et al.14 showed that, for crystalline Si, the density of BPDs can dramatically increase during fast cooling; however, for crystalline SiC,9 the density of BPDs can decrease during fast cooling. How does cooling rate affect the generation of BPDs in sapphire? Figure 4 shows the dislocation density distribution before and after cooling for case 3. Before cooling [Figure 4a], that is,

Figure 6. BPD distribution in a series of (0001) slices from the bottom to the top of crystal. Figure 4. Comparison of BPD distribution (a) before cooling and (b) after cooling for case 3.

dislocation densities are not at the ends of the crystal but are near the ends of the crystal. Moreover, they are not at the edge of crystal, but at the half radius. The dislocation density is low in the middle of the crystal. Why do we find this particular distribution of the BPD density?

just after the heating process, the BPD density increases only at the bottom; however, after cooling [Figure 4b], the BPD

Figure 5. Comparison of BPD distribution after cooling process for (a) case 1, (b) case 2, and (c) case 3. 4083



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Figure 7. Dislocation distribution under (a) radial temperature distribution and (b) axial temperature distribution.

Figure 8. Distribution of stress σrz inside planes passing through the axial line under (a) radial temperature distribution and (b) axial temperature distribution.

significant factor in determining the generation of BPDs.9,20 Figure 8 shows the distribution of σrz in a cross-section across the crystal axis. Figure 8a,b shows the radial and axial temperature distributions, respectively. The distribution of σrz is the same for both the temperature contours. Therefore, it is the similarity of the σrz distributions for different temperature contours causing the similarity of the BPD distributions even for different temperature contours. Furthermore, the maximum value of σrz occurs near the ends and not at the edge of the crystal but at the half radius; the value of σrz is small in the middle of the crystal. The correspondence between the BPD and the σrz distributions shows that it is σrz that causes the particular BPD distribution. Therefore, the most effective method to reduce the BPD density is to reduce the stress component of σrz. The possible reason is that for the c axis growth crystal, the multiplication of the BPDs is triggered by the activation of slip directions in the c plane, which is mainly triggered by the stress component σrz. Before finishing this section, two problems need to be illuminated. The first problem is why the stress component σrz becomes the largest near the top and bottom of crystal; the second is why the same distribution of σrz is established for significantly different temperature distributions. These two questions are essentially the same. A logical derivation or explanation can be given as follows: First, for an idealized temperature distribution along the radial direction [eq 13] or along the axial direction [eq 14], the distribution of any stress component should be symmetric between the upper part and lower part of crystal and also between left part and right part of crystal. Figure 8 shows this kind of symmetry. Second, due to

Since the present design of furnace configuration must give a small radial flux due to the air gap between sapphire and crucible, the main cooling flux in the crystal should be axial, and the generation of dislocations must be due to the axial cooling flux. Thus, a primitive assumption is that this distribution might originate from the different cooling fluxes in the radial or axial directions. To verify this assumption, we imposed the following artificial temperature distribution in which the temperature varies either in the radial direction or in the axial direction: T (r , z) = 2250 + 50r

(13)

T (r , z) = 2250 + 1250z 2

(14)

In eqs 13 and 14, r and z are the radial and axial positions, respectively. Because a linear temperature distribution along the axial direction produces negligible thermal stress,19 we impose a nonlinear temperature distribution along the axial direction to generate thermal stress. The dislocation densities for these two temperature distributions [eqs 13 and 14] are shown in Figure 7. Figure 7a,b shows the radial and axial temperature distributions, respectively. The dislocation density distributions are quite similar, even though they are for totally different temperature contours. Therefore, the hypothesis that a particular distribution of dislocation density originates from different cooling fluxes is not valid. To understand this particular distribution of dislocation density and to determine why it remains the same distribution even when the temperature contours change completely, we investigate the thermal stress component, σrz, which is a 4084



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Figure 9. Dislocation density distribution at different cooling times for case 1 [panels a1−d1] and for case 3 [panels a2−d2]. From panels a−d for both cases 1 and 3, the corresponding temperatures at monitored points are 2000, 1950, 1900, and 1850 °C, respectively.

Figure 10. Average residual stress in cooled crystal for (a) case 1, (b) case 2, and (c) case 3.

free-traction boundary conditions, the stress component σrz on all the surfaces should be zero, that is, σrz = 0. Thus, if there exists a stress value of σrz inside the crystal, it must exist near the top and near the side, as shown in Figure 8. The properties of the symmetry and the free-traction boundary conditions determine that the stress component σrz should exhibit an always-similar distribution, whatever ideal radial or axial temperature distribution imposed. 3.2.3. Time Evolution of BPD and Residual Stress during Cooling Process. The main advantage of the Alexander− Haasen model is that it appropriately represents the ratedependent process of plastic deformation in semiconductors or estimates the residual stress in cooled crystals. Figure 9 shows the time evolution of the BPD density in one slice taken from near the top of the crystal for cases 1 and 3. Figure 9a1−d1 is for case 1, and Figure 9a2−d2 is for case 3. During slow cooling [Figure 9a1−d1], the dislocation density in the slice remains almost constant. However, during fast cooling [Figure 9a2− d2], the dislocation density rapidly increases. Therefore, slow cooling helps to reduce the generation rate of dislocations. After cooling, the von Mises residual stress inside the crystal can be obtained. Figure 10 shows the residual stress for three cooling schemes. The residual stress was averaged along the peripheral direction, and thus, the planes in Figure 10 pass through the axial line. The slowest cooling (case 1) corresponds to the lowest residual stress. As the cooling rate increases, the magnitude of residual stress gradually increases. Therefore, slow cooling helps reduce residual stress. Since the stress component σrz dominantly controls the generation of dislocations, the distribution of dislocation density or residual stress should resemble that of σrz. Figure

10b,c shows a relatively large dislocation density at four interior regions, where the value of σrz is high. Therefore, the stress component σrz indeed has a large contribution on the generation of dislocations and residual stress. However, there are also some differences between Figures 10b,c and 8. Figure 10b,c shows that the residual stress is high near the center of four sides; however, the dominant stress component σrz is low there. The differences might originate from the different temperature distribution. The cases in Figure 10b,c have a more realistic temperature distribution, which has the largest temperature gradient on the left and right sides, and the largest axial cooling flux gradient on the top and bottom sides; all of gradients gradually decrease from surface to the interior. However, the case in Figure 8 has only a constant radial temperature gradient (eq 13) or axial cooling flux gradient (eq 14). From the comparison, it seems that it is the largest radial temperature gradient or axial cooling flux gradient dominantly controlling the maximum residual stress inside the crystal. 3.2.4. Fast Cooling or Slow Cooling for the Reduction of Dislocations. In the present furnace configurations, slow cooling is better for the reduction of dislocations; however, in the paper for SiC PVT growth,9 fast cooling is better for the reduction of BPD dislocations. Why is the conclusion totally different for sapphire and SiC crystal? For SiC PVT growth, the SiC crystal directly touches the graphite crucible, which has high thermal conductivity. When finishing the crystal growth process, the crucible temperature is much higher than the crystal temperature causing very concave temperature contours. When the cooling process starts, fast cooling can cause a fast decrease of temperature at the edge of 4085



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the crystal and make the concave temperature-contour flat and thus reduce the thermal stress. That is the main reason fast cooling causes smaller dislocation density in SiC PVT growth. However, in the present paper, for a simple annealing process, the sapphire crystal was put into a quartz crucible, which has a very small thermal conductivity, and also a small air gap between sapphire crystal and quartz crucible was set for the simplification of boundary conditions. The use of quartz crucible and air gap causes a very small radial flux inside the crystal and causes relatively flat or minor convex temperature contours. However, the axial cooling flux is very large due to the cooling effect of the bottom pedestal. Thus, the main cooling flux inside crystal is axial. When the cooling process starts, fast cooling can rapidly increase the axial cooling flux, causing more convex temperature contours, and thus increases the thermal stress. That is the main reason slow cooling causes smaller dislocation density in the present design of furnace configurations. Therefore, whether fast cooling or slow cooling, the most important is to make the temperature contour flat in the hightemperature regime. If the temperature contours are concave at the beginning of the cooling process, such as SiC PVT growth, it is better to increase radial cooling flux or reduce the axial cooling flux for the reduction of concavity; if the temperature contours are convex at the beginning of the cooling process, such as the present sapphire annealing process, it is better to reduce axial cooling flux or increase the radial cooling flux for the reduction of convexity.

REFERENCES

(1) Zhang, N.; Park, H. G.; Derby, J. J. J. Cryst. Growth 2013, 367, 27−34. (2) Haasen, P. Z. Phys. 1962, 167, 461−467. (3) Yonenaga, I.; Sumino, K. Phys. Status Solidi A 1978, 50, 685−693. (4) Sumino, K.; Yonenaga, I. Phys. Status Solidi A 1993, 138, 573− 581. (5) Miyazaki, N.; Uchida, H.; Munakata, T. J. Cryst. Growth 1992, 125, 102−111. (6) Volkl, J.; Müller, G. J. Cryst. Growth 1989, 97, 136−145. (7) George, A.; Rabier, J. Rev. Phys. Appl. 1987, 22, 941−966. (8) Rabier, J.; George, A. Rev. Phys. Appl. 1987, 22, 1327−1351. (9) Gao, B.; Kakimoto, K. Cryst. Growth Des. 2014, 14, 1272−1278. (10) Gao, B.; Kakimoto, K. J. Cryst. Growth 2014, 386, 215−219. (11) Gao, B.; Kakimoto, K. J. Cryst. Growth 2014, 392, 92−97. (12) Miyazaki, N.; Okuyama, S. J. Cryst. Growth 1998, 183, 81−88. (13) Miyazaki, N.; Kumamoto, A.; Harada, C. J. Cryst. Growth 2004, 271, 358−367. (14) Gao, B.; Nakano, S.; Harada, H.; Miyamura, Y.; Kakimoto, K. Cryst. Growth Des. 2013, 13, 2661−2669. (15) Kronberg, M. L. J. Am. Ceram. Soc. 1962, 45, 274−279. (16) Nelder, J. A.; Mead, R. Comput. J. 1965, 7, 308−313. (17) Dobrovinskaya, E. R.; Lytvynov, L. A.; Pishchik, V. Sapphire: Material, Manufacturing, Applications; Science Bussiness Media, Springer: New York, 2009. (18) Gao, B.; Kakimoto, K. J. Cryst. Growth 2014, 396, 7−13. (19) Garandet, J. P. J. Cryst. Growth 1989, 96, 680−684. (20) Bottcher, K.; Cliffe, K. A. J. Cryst. Growth 2007, 303, 310−313. (21) Gao, B.; Nakano, S.; Kakimoto, K. J. Cryst. Growth 2013, 369, 32−37. (22) Lu, C. W.; Chen, J. C. J. Cryst. Growth 2001, 225, 274−281. (23) Gao, B.; Nakano, S.; Harada, H.; Miyamura, Y.; Sekiguchi, T.; Kakimoto, K. Cryst. Growth Des. 2012, 12, 5708−5714. (24) Lambropoulos, J. C. J. Cryst. Growth 1987, 84, 349−358.

4. CONCLUSIONS The basal plane dislocations in single-crystal sapphire can be described by the Alexander−Haasen (AH) model. For the first time, the parameters of AH model for sapphire crystal have been derived from experimental stress−strain curves. The application of the Alexander−Haasen model to sapphire growth made it possible to minimize the dislocation density and residual stress in growing crystals by optimizing the furnace structure and operation conditions. We applied the Alexander−Haasen model to crystals cooling in a small furnace. The results show that the cooling process significantly influences the generation rate of basal plane dislocations during the growth of single-crystal sapphire. Slow cooling rate from high temperature reduces the basal plane dislocation density and the residual stress. Comparison of cooling rate effect on the generation of BPDs in SiC and sapphire crystals indicates that the cooling rate effect is dependent on the furnace configurations. For a situation with high heating flux during growth or heating process, fast cooling might be better for the reduction of dislocations; however, for a situation with slow radial heating flux during growth or heating process, slow cooling might be better for the reduction of dislocations. The general rule for choosing fast or slow cooling rate is whether it can make the temperature contours inside crystal flat in the high-temperature regime.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. ‡ N.M.: Professor Emeritus, Kyoto University and Kyushu University, Japan. 4086


AlexanderâHaasen Model of Basal Plane Dislocations in Single

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