Analysis of Macrostep Formation during Top Seeded Solution Growth

Article pubs.acs.org/crystal

Analysis of Macrostep Formation during Top Seeded Solution Growth of 4H-SiC Kanaparin Ariyawong, Yun Ji Shin, Jean-Marc Dedulle, and Didier Chaussende* Université Grenoble Alpes, CNRS, LMGP, F-38000 Grenoble, France ABSTRACT: We used numerical and analytical modeling to investigate fluid flow behaviors close to the growing 4H-SiC crystal surface in the top seeded solution growth process. First, we calculated the azimuthal and radial components of the fluid flow in front of the rotation disc. Second, we developed an analytical model describing the interaction between the step flow (of the vicinal crystal surface) and fluid flow components, considering the crystallography of 4H-SiC and introducing a phase parameter. The correlation of both models allows us to describe qualitatively the conditions for which macrosteps form and destabilize. This phenomenological description is in good agreement with the corresponding experimental observations that are also presented in this paper.

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INTRODUCTION Silicon carbide (SiC) is a wide bandgap semiconductor that offers a substantial gain in performance compared to silicon, owing to its superior physical and electronic properties. The recent development of large size and high quality bulk crystals, and epilayers, have made SiC an ideal platform for many power electronic applications.1 4H-SiC wafers of 3−6 in. diameter are industrially produced by a high-temperature seeded sublimation growth process, also called PVT (for physical vapor transport). Besides this well-established vapor phase method, the liquid phase route is currently being reconsidered because of its ability to reach high structural quality.2 This is mainly due to the process operation at lower temperature than the PVT method and conditions closer to the thermodynamic equilibrium.3 The top seeded solution growth (TSSG) approach has emerged as the most realistic configuration; substantial progress was recently achieved, allowing the demonstration of large diameter (3 in.) and centimeter scale length.4 However, a key issue in solution growth, which is less problematic in the vapor phase process, is the carbon transport from the source, i.e., the graphite crucible, to the crystal in a complex fluid convection pattern. Its understanding and control are critical to set a stable growth front over a long growth time. These are also important for further upscaling of the process. Numerical simulation is of great support for such a purpose. It allows the fluid flow pattern and the associated solute transport to be described both qualitatively and quantitatively. The carbon solubility in pure silicon is always very low below 2000 °C;5 the solutal contribution to convection can thus be neglected. With such an assumption, the convective flow pattern is governed by “only” four main contributions, namely, buoyancy, forced convection, Marangoni and electromagnetic convections.6 Despite such complexity, the process has been described by numerical modeling, and the growth rate, thought to be too low © XXXX American Chemical Society

to expect bulk crystal growth in the authors’ configuration, has been evaluated to be in good agreement with the experiments.7 As a rule, the carbon flux is proportional to the fluid velocity toward the crystal surface. The growth rate can thus be enhanced by increasing the fluid velocity with different possible strategies. The use of an external magnetic field has been proposed and investigated theoretically.8 However, the control of electromagnetic forces is technically complex and has not been implemented yet for SiC. The accelerated crucible rotation technique (ACRT) is also an interesting method to enhance the solute transport; it gives rise to a better homogeneity.9 It also showed an improvement in the crystal growth rate.10 The use of high speed rotation of the seed during crystal growth is another alternative.11 This increases the carbon concentration gradient in the boundary layer. The interaction between the liquid flow and the growth surface, which is a common feature of all solution growth conditions, gives rise to stronger process stability issue for the latter case. As a rule, the surface roughness of the growth front increases if the fluid flow direction is parallel to the step flow direction and decreases if they are antiparallel.12 Exploiting this effect, Umezaki et al. reported an improvement of the surface roughness by shifting the rotation axis of the crystal with respect to the symmetry axis of the crucible.13 Such a configuration is interesting for demonstration purposes, but is not really adapted for bulk crystal growth. This paper aims to go one step further in the description and understanding of the interaction between the fluid flow pattern and the surface morphology. By combining modeling (both numerical and analytical) and experimental observations, the Received: January 28, 2016 Revised: April 29, 2016

A

DOI: 10.1021/acs.cgd.6b00155 Cryst. Growth Des. XXXX, XXX, XXX−XXX

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reached. The crystal temperature was fixed at 1700 °C. Buoyancy convection, Marangoni convection, and electromagnetic convection were all computed in the silicon melt. Forced convection due to the crystal rotation was varied by adjusting the seed rotation rate from 0 to 200 rpm. To describe qualitatively the interaction between the directions of fluid flow and step flow, we developed a simple analytical model. First we considered the crystallographic orientation of the seed surface. For simplification, we only investigated a vicinal surface, composed of a regular array of straight steps coming from the miscut angle of the seed surface. This defines the step flow direction of the growing surface. Second, concerning the fluid flow, we examined two components of the fluid velocity tangential to the seed surface: the azimuthal and radial components. For a circular seed mounted at the bottom of a graphite rod, both having the same unique symmetry axis, the schematic representation of tangential flows is shown in Figure 2. At rest, we define the

effect of the anisotropy of the SiC crystal structure and the fluid flow is investigated. More precisely, we will show how the miscut angle of the seed crystal breaks the symmetry of the rotation and how it influences macrostep formation on the growth front.

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EXPERIMENTAL DETAILS

The experimental setup was a Czochralski puller with medium frequency induction heating (around 15 kHz). Details about the growth chamber and crucible elements have already been described.6,7 Briefly, pure liquid silicon was used as solvent and placed in a high density graphite crucible, which acted both as a container and as the carbon source. The liquid volume was typically 5 cm in diameter and 2−3 cm in height. We used a TSSG geometry, meaning that a SiC seed of 10−30 mm diameter was mounted on a graphite rod and dipped into the liquid. The crucible is presented in Figure 1. The

Figure 1. Schematic representation of the TSSG crucible. The external graphite crucible (called furnace) is directly heated by induction. The inner crucible (called crucible) acts both as a container for the liquid and as the carbon source. It is heated by both induction and radiation from the furnace. The red parallelepiped corresponds to the liquid region for which the simulations are represented in the paper. temperature at the top of the inner crucible was measured with an optical pyrometer. For this study, the temperature was kept constant, at about 1700 °C. The seed crystals consisted of carbon face, 4H-SiC (0001) wafers, with a 4° off-cut toward the [11−20] direction. The crystal rotation rate was varied between 0 and 200 rpm.

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CALCULATION DETAILS Numerical simulations were performed by the finite element method (FEM) in a two-dimensional axisymmetric geometry. All the details of the input parameters and the computation of coupled phenomena including induction heating, heat transfer, and fluid dynamics have been presented previously.6 The SiC growth rate was assessed by considering the carbon flux normal to the crystal surface. It is computed from the carbon concentration gradient, as follows: Vg =

MSiC ( −D∇C) ·n ρSiC

Figure 2. Schematic drawing of the fluid flow and the step flow on a circular seed crystal. The surface steps move from left to right during the growth as indicated in (a). The black, single line arrows represent the fluid flow directions tangential to the seed surface. Panel a represents the fluid flow with the azimuthal component only (Δvϕ > 0, Δvr = 0), while panel b concerns the radial component only (Δvϕ = 0, Δvr > 0). The azimuthal (Δvϕ) flow corresponds to the rotation component of the flow, related to the azimuth angle ϕ. The radial (Δvr) flow corresponds to the radial component of the flow, related to the radius r taken from the center of the crystal to the edge. When comparing the step flow to the fluid flow, parallel and antiparallel flows are separated by the boundary lines (thick dashed line), and red and blue lines correspond to the parallel and antiparallel flow areas, respectively. Panel c gathers the contribution of both azimuthal and radial relative velocities. In this case, the boundary line shows a deviation of ξ degrees from the reference case.

(1)

where MSiC is the molar mass of SiC, ρSiC is its density, D is the diffusion coefficient of carbon in the solvent, C is the carbon concentration and n is the unit vector normal to the crystal surface. Within the range of the induction frequency used for heating, typically about 15 kHz, the electromagnetic forces are strong and dominate convection movements.14 The forces are maximum near the crucible walls, they create localized, highly turbulent flows. These result in difficulties to obtain computational convergence. To solve this problem, we first refined the discretization of the liquid close to the crucible walls. This improved the accuracy and the stability of computation. Second, we applied a high value of fluid viscosity at the initial stage of computation, then decreased it until the real value was

fluid velocity relative to the seed velocity (Δv), which will be called from now on the “fluid velocity”. Then, two different configurations occur while comparing the step flow and the fluid flow directions. If the fluid flow and step flow have the same direction, we refer to them as parallel. In the opposite situation, we name them antiparallel flows. B

DOI: 10.1021/acs.cgd.6b00155 Cryst. Growth Des. XXXX, XXX, XXX−XXX

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For a further quantitative description of the flow behavior, we define the phase parameter

The fluid velocity is composed of two main contributions, the azimuthal relative velocity (Δvϕ) and the radial relative velocity (Δvr). Figure 2a represents the fluid flow with the azimuthal component only (Δvϕ > 0, Δvr = 0), while Figure 2b concerns the radial component only (Δvϕ = 0, Δvr > 0). Concerning the azimuth angle ϕ, positive and negative fluid velocities correspond to clockwise and counterclockwise directions, respectively. For the radial component r, the positive value refers to a fluid flow oriented from the seed center to the edge, while the negative value stands for the opposite flow direction. The thick dashed line demarcates the boundary between the surface areas having parallel or antiparallel flows. Without radial flow (Δvr = 0), the boundary line is the horizontal line crossing the seed center in Figure 2a. It is perfectly parallel to the step flow direction. Without the azimuthal component (Δvϕ = 0), the boundary line is the vertical line, perpendicular to the step flow direction (Figure 2b). Defining the boundary line of Figure 2a as the reference, the contribution of a radial flow will generate a deviation of ξ degrees of the boundary line with respect to the reference (Figure 2c). This angle of deviation is derived from a simple trigonometry consideration:

Ψ = sin(θ + α + ξ)

(4)

where θ is the angle defined in a unit circle and is used to designate the position on the seed crystal. α is the angle between the step orientation and the crystal off-cut direction. The positive and negative values of α denote that the angle rotates in the clockwise and counterclockwise direction, respectively. The phase parameter varies from −1 to 1; the value becomes zero if the step flow and fluid flow directions are perpendicular. The positive and negative values of the phase parameter refer to the parallel and antiparallel flows, respectively. For the regular train of steps oriented to the [11−20] direction, α becomes zero. The situation becomes more complex regarding the 6-fold symmetry related to the hexagonal system of 4H-SiC (Figure 3a). Seeds with vicinal surfaces are commonly used to ensure a

⎛ Δv ⎞ ξ(r ) = tan−1⎜⎜ r ⎟⎟ ⎝ Δvϕ ⎠ ⎛ v (r , z ) − v (r , z = 0) ⎞ fluid r seed ⎟⎟ = tan−1⎜⎜ r v ( r , z ) − v ( r , z ⎝ ϕ ϕ fluid seed = 0) ⎠

(2)

where r is the distance from the seed center and z is the position in the liquid below the seed−liquid interface. This formula is verified for Δvr = 0 where ξ is 0° (Δvϕ > 0) or 180° (Δvϕ < 0), and for Δvϕ = 0 where ξ is 90° (Δvr > 0) or −90° (Δvr < 0). The positive and negative values of ξ represent the rotation of the boundary line in the clockwise and counterclockwise directions, respectively. The selection of the z position in the liquid (zfluid) when using eq 2 is an important issue. If the ratio between the solute (δD) and momentum (δv) boundary layer thicknesses is much less than unity, the near solid−liquid interface can be considered as a motionless fluid, and the mass transport in this region is only governed by the diffusion process.15 According to such a criterion, we assume that the velocities at the diffusion layer thickness are representative of fluid velocities near the seed. The ratio between solute and momentum layer thicknesses is related to the Schmidt number (Sc) as16 δD ⎛ D ⎞n ≈ ⎜ ⎟ = Sc−n ⎝υ⎠ δv

Figure 3. (a) Representation of the (0001) 4H-SiC plane in the hexagonal system and (b, c) interaction between the fluid flow and step flow on the circular SiC seed crystal with an off-cut of a few degrees toward the [11−20] direction. The arrows show the fluid flow direction. Red and blue lines on the hexagons represent the parallel and antiparallel flow conditions, respectively. The parallel flow condition is fulfilled only on the (01−10) plane in area A. In area B, parallel flow is obtained for both (10−10) and (01−10) planes. Only the (10−10) plane satisfies the parallel flow in area C. Both (10− 10) and (01−10) planes have an antiparallel flow in area D. The flow behaviors in different areas are separated by boundary lines (dashed lines).

(3)

The exponent n is generally in the range 1 ≥ n ≥ 1/3 for fluids having 0 ≤ Sc ≤ ∞. Using silicon as a solvent, we obtain 0.05 ≤ (δD/δv) ≤ 0.37. The assumption of motionless fluid near the seed−liquid interface verifies if the momentum boundary layer is about an order of magnitude thicker (δD/δv) ≤ 0.1 than the solute boundary layer. This is the case for n = 1 but not for n = 1/3. To avoid any ambiguity concerning the assumption of motionless fluid, we assigned a mean value to the fluid velocity within the solute boundary layer, such that

good homoepitaxy via the step flow growth mode. The standard vicinal surface is obtained by the miscut angle of a few degrees (typically 4°) of the (0001) plane toward the [11−20] direction. When the 4H-SiC crystal grows under a low supersaturation, the average growth rate is higher on the {11−2n} plane than on the {1−10n} one.17 This is also the case for solution growth, as the conditions are close to the thermodynamic equilibrium. Moreover, solution grown vicinal surfaces are strongly bunched. Thus, instead of having a regular train of straight steps with unit height, the surface morphology

δD

v(r , z fluid) = 1/δ D ∫ v(r , z) dz . This is applied for both 0 radial and azimuthal velocity components. C

DOI: 10.1021/acs.cgd.6b00155 Cryst. Growth Des. XXXX, XXX, XXX−XXX

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commonly evolves to bunched steps having a zigzag step edge, the latter being composed of (10−1n) and (01−1n) facets. In this work, only the two-dimensional model will be considered. Thus, the bunched step edges (faceted) in the [10−1n] and [01−1n] directions can be projected onto the c-plane (0001). This allows the step orientation to be considered on the c-plane such that (10−10) and (01−10) planes represent (10−1n) and (01−1n) facets, respectively. Crystal surfaces with different fluid flow patterns are shown in Figure 3b,c. In both cases, two boundary lines can be drawn since the two adjacent {1−100} facets are considered. The surface of the growing crystal is thus divided into four different regions, denoted A, B, C, and D. In area A, the parallel flow condition is realized only for the (01−10) facet. The parallel flow for both (10−10) and (01−10) facets is achieved in area B. Only the (10−10) facet fulfills the parallel flow in area C. Finally, only the antiparallel flow is possible in area D. In Figure 3b, the boundary lines for the case that Δvϕ > 0 and Δvr = 0 are used as reference lines. If the fluid is different from a perfect clockwise flow, the deviation of the boundary lines from the reference lines can be calculated in the same way as eq 2. In that case, α is −30° for (01−10) facets and 30° for (10−10) facets. The case for the positive radial fluid flow is described in Figure 3c. The phase parameter diagram for Figure 3b (ξ = 0°) is plotted in Figure 4 where the corresponding areas A−D are

Figure 5. Computed distribution of (a) temperature, (b) fluid velocity, (c) carbon concentration, and (d) carbon supersaturation in liquid. Red arrows in (b) indicate the normalized fluid velocity vectors, and black solid lines represent velocity streamlines. The black line (S = 0) in (d) is drawn to separate the areas of supersaturated and undersaturated solution.

the high Lorentz force density near the crucible wall, is also predicted. The distribution of carbon concentration results directly from the convection pattern, with the highest concentration close to the symmetry axis (Figure 5c). The supersaturation (S), calculated from the difference between the actual carbon concentration and the equilibrium value in liquid silicon is shown in Figure 5d. The carbon dissolves from the graphite crucible in the under-saturated area where S < 0, transports by convection, and crystallizes in the supersaturated area (S > 0), i.e., at the seed surface. The crystal growth rate calculated from eq 1 is plotted as a function of crystal rotation speed, together with some corresponding experimental data (Figure 6). The growth rate can be effectively enhanced by increasing the seed rotation rate, which is in good agreement with the experiments. On the basis of such a good agreement, we can use the calculated values for further discussion about the fluid/crystal interaction. For example, we plotted the fluid velocity along the crystal radius at high rotation speed, i.e., at 100 rpm in the counterclockwise direction. Both radial and azimuthal components relative to the seed are distinguished (Figure 7). These two components of fluid velocity increase along the seed radius. In such a case, the solute boundary layer thickness is almost constant along the seed radius: δD = 0.2 mm. The angle of deviation ξ defined in eq 2 is almost constant as well. At r = 6 mm, the value of ξ is around 28°. The phase parameter for these growth conditions is plotted in Figure 8 for both (10−10) and (01−10) facets,

Figure 4. Phase parameter diagram for the conditions shown in Figure 3b. Areas A−D are separated by boundary lines (dashed lines).

separated by boundary lines. The parallel flow conditions are fulfilled for {1−100} facets when Ψ of both facets are positive (area B). The individual (10−10) or (01−10) facets fulfill the parallel flow condition in areas C and A, respectively. Only the antiparallel flow was seen in area D for both {1−100} facets.

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RESULTS AND DISCUSSION Figure 5 shows the distribution of a set of physical parameters, i.e. temperature, fluid velocity, carbon concentration, and carbon supersaturation, as obtained by numerical simulation. The hot point is located at the bottom corner of the crucible, while the cold point localizes to the seed crystal (Figure 5a). The axial temperature difference is around 4 °C, corresponding to an axial temperature gradient of about 2 °C/cm. This was designed on purpose as a small axial thermal gradient is most adapted to prevent parasitic crystallization within the liquid.18 As already pointed out in previous reports,6 the mixture of different convective flows contributes to a complex flow pattern, which is difficult both to describe and to control (Figure 5b). In front of the seed surface, there is a direct flow of fluid from the bottom of the crucible toward the seed around the symmetry axis. The highly turbulent flow near the crucible wall, caused by the strong electromagnetic convection due to D

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dominant phase parameter. The predominant macrosteps close to the (10−10) plane are observed in area C where the parallel flow condition is only fulfilled for the (10−10) facet. In area D, only the antiparallel flow is predicted and macrosteps are rarely observed. The phase parameter diagram shows good agreement in correlating the flow conditions with the macrostep formations, with some deviation for −85° and 175°. However, if we neglect the radial flow such as in the diagram for the purely azimuthal flow (Figure 4), the behavior of macrostep formation becomes poorly described. For example, for θ = −40°, Figure 4 shows the only antiparallel flow when macrosteps close to the (10−10) plane were experimentally observed. Another example is the case for θ = 140°. Figure 4 predicts only parallel flows for both (10−10) and (01−10) facets, whereas only macrosteps predominantly close to the (01−10) plane were observed. Moreover, deviations for the cases of −85° and 175° become even larger. Thus, the effects of the fluid flow in both components that are tangential to the seed have to be considered since they provide better descriptions of macrostep orientations. As pointed out by Chernov et al.,19 a critical negative shear rate exists for which the stable surface becomes unstable with respect to step bunching, by increasing the shear rate magnitude. Differently formulated: the fluid flow has a strong effect on the growth front stability in the parallel flow condition if the fluid velocity is high enough to overcome the stability limit. If such a limit involves the surface energy, this type of instability should not affect the crystal surface to the same extent on the Si- and C-faces and also depends on the deviation from such a limit. The surface morphology should be more stable at lower rpm and at the area closer to the seed center, as the tangential component of the fluid velocity is smallest close to the center (Figure 7). We can see that the macrostep shape does not exactly follow the equilibrium shape given by the {1−100} facets. The macroscopic distribution of supersaturation near the crystal surface can be computed as for instance in Figure 5 and shows continuous and slight evolution along a crystal radius. At a microscopic level, the existence of surface roughness coming from the presence of microsteps, and to a larger extent of forming macrosteps, should induce local inhomogeneities in the supersaturation distribution.12 Though difficult to evidence, this

Figure 6. Evolution of the growth rate as a function of seed rotation speed. Seed temperature is 1700 °C. The carbon solubility is that published by Durand et al.5

Figure 7. Profile of the calculated relative fluid velocity, plotted along the seed crystal radius for both radial (r) and azimuthal (φ) components. Seed rotation rate was 100 rpm in the counterclockwise direction.

where areas A−D are separated by boundary lines. The surface of the grown crystals, observed by Nomarski differential interference contrast (NDIC) optical microscopy is compared with the diagram. Each image was taken in the area r > 4 mm where its azimuthal location on the crystal surface is defined by the angle θ relative to the [11−20] direction. The formation of macrosteps (defined here as the dark lines formed by the accumulation of several microsteps) is correlated to the phase parameter diagram. The parallel flow for the (01−10) facet (area A) gives rise to macrostep formation predominantly close to the (01−10) plane. The parallel flow criteria in both (01− 10) and (10−10) facets (area B) result in zigzag-shape macrosteps, while the main feature is still governed by the

Figure 8. Comparison between the macrostep formation observed experimentally (using an NDIC microscope) and their behavior predicted from the phase parameters calculated using combined numerical and analytical modeling. The θ angle shown in each NDIC image refers to the position where the image was taken on the crystal relative to the [11−20] direction. E

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(6) Mercier, F.; Dedulle, J. M.; Chaussende, D.; Pons, M. Coupled heat transfer and fluid dynamics modeling of high-temperature SiC solution growth. J. Cryst. Growth 2010, 312, 155−163. (7) Lefebure, J.; Dedulle, J. M.; Ouisse, T.; Chaussende, D. Modeling of the Growth Rate during Top Seeded Solution Growth of SiC Using Pure Silicon as a Solvent. Cryst. Growth Des. 2012, 12, 909−913. (8) (a) Mercier, F.; Nishizawa, S.-i. Numerical Investigation of the Growth Rate Enhancement of SiC Crystal Growth from Silicon Melts. Jpn. J. Appl. Phys. 2011, 50, 035603. (b) Mercier, F.; Nishizawa, S.-i. Comparative numerical study of the effects of rotating and traveling magnetic fields on the carbon transport in the solution growth of SiC crystals. J. Cryst. Growth 2013, 362, 99−102. (9) Scheel, H. J.; Schulz-Dubois, E. O. Flux growth of large crystals by accelerated crucible-rotation technique. J. Cryst. Growth 1971, 8, 304−306. (10) Kusunoki, K.; Kamei, K.; Okada, N.; Yashiro, N.; Yauchi, A.; Ujihara, T.; Nakajima, K. Solution growth of SiC crystal with high growth rate using accelerated crucible rotation technique. Mater. Sci. Forum 2006, 527−529, 119−122. (11) Umezaki, T.; Koike, D.; Horio, A.; Harada, S.; Ujihara, T. Increase in the Growth Rate by Rotating the Seed Crystal at High Speed during the Solution Growth of SiC. Mater. Sci. Forum 2014, 778−780, 63−66. (12) Zhu, C.; Harada, S.; Seki, K.; Zhang, H.; Niinomi, H.; Tagawa, M.; Ujihara, T. Influence of Solution Flow on Step Bunching in Solution Growth of SiC Crystals. Cryst. Growth Des. 2013, 13, 3691− 3696. (13) Umezaki, T.; Koike, D.; Harada, S.; Ujihara, T. Improvement of Surface Morphology by Solution Flow Control in Solution Growth of SiC on Off-Axis Seeds. Mater. Sci. Forum 2015, 821−823, 31−34. (14) Ariyawong, K.; Dedulle, J. M.; Chaussende, D. Electromagnetic Enhancement of Carbon Transport in SiC Solution Growth Process: A Numerical Modeling Approach. Mater. Sci. Forum 2014, 778−780, 71−74. (15) Bredikhin, V. I.; Malshakova, O. A. Step bunching in crystal growth from solutions: Model of nonstationary diffusion layer, numerical simulation. J. Cryst. Growth 2007, 303, 74−79. (16) Hurle, D. T. J. Handbook of Crystal Growth, Part B: Growth Mechanisms and Dynamics; Elsevier: Amsterdam: North Holland, 1994; Vol. 2. (17) Nordell, N.; Karlsson, S.; Konstantinov, A. O. Equilibrium crystal shapes for 6H and 4H SiC grown on non-planar substrates. Mater. Sci. Eng., B 1999, 61−62, 130−134. (18) Mercier, F. Cubic silicon carbide crystal growth from high temperature solution. Ph.D. Thesis, Institut National Polytechnique de Grenoble - INPG, Grenoble, France, 2009. (19) Chernov, A. A.; Coriell, S. R.; Murray, B. T. Morphological stability of a vicinal face induced by step flow. J. Cryst. Growth 1993, 132, 405−413.

should provoke local variations of growth kinetics and could explain the deviation from the equilibrium shape. Moreover, experimental observations reveal that the surface is more stable when a low rotation rate is used, a condition which drastically reduces the formation of macrosteps. In such cases the fluid velocity is low. This would support the existence of a critical point of stability discussed earlier in this paper.

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CONCLUSION We set out to couple some physical parameters of the TSSG process, extracted from numerical simulation to an analytical model, developed to describe the interaction between step propagation at the growing surface and the fluid flow direction. For that, a phase parameter was introduced. The variation of this phase parameter has been revealed as a good criterion to identify the process conditions that are favorable for the occurrence of macrosteps and surface instabilities. Simulation is in good agreement with experimental results and suggests that the optimum crystal rotation rate should be carefully selected for the enhancement of the crystal growth rate to avoid the formation of macrosteps since this increases the surface roughness and results in poor crystal quality. By extension, the model developed can be used as a promising guideline to control and optimize the quality of SiC crystals grown by the TSSG process.

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AUTHOR INFORMATION

Corresponding Author

*LMGP, Grenoble INP-Minatec, 3 parvis Louis Néel, CS 50257, 38016 Grenoble cedex 1, France. E-mail: didier. [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge the Rhône-Alpes region for the financial support through the ARC Energie 2012 Program and International Collaborative Energy Technology R&D Program (No. GER12082). This work was also financially supported by the EU in the framework of the NetFISiC project (Grant No. PITN-GA-2010-264613).

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ABBREVIATIONS NDIC, Nomarski differential interference contrast; TSSG, top seeded solution growth REFERENCES

(1) Eddy, C. R.; Gaskill, D. K. Silicon Carbide as a Platform for Power Electronics. Science 2009, 324, 1398−1400. (2) Yamamoto, Y.; Harada, S.; Seki, K.; Horio, A.; Mitsuhashi, T.; Koike, D.; Tagawa, M.; Ujihara, T. Low-dislocation-density 4H-SiC crystal growth utilizing dislocation conversion during solution method. Appl. Phys. Express 2014, 7, 065501. (3) Hofmann, D. H.; Muller, M. H. Prospects of the use of liquid phase techniques for the growth of bulk silicon carbide crystals. Mater. Sci. Eng., B 1999, 61−62, 29−39. (4) Kusunoki, K.; Okada, N.; Kamei, K.; Moriguchi, K.; Daikoku, H.; Kado, M.; Sakamoto, H.; Bessho, T.; Ujihara, T. Top-seeded solution growth of three-inch-diameter 4H-SiC using convection control technique. J. Cryst. Growth 2014, 395, 68−73. (5) Durand, F.; Duby, J. C. Carbon solubility in solid and liquid silicon - A review with reference to eutectic equilibrium. J. Phase Equilib. 1999, 20, 61−63. F

DOI: 10.1021/acs.cgd.6b00155 Cryst. Growth Des. XXXX, XXX, XXX−XXX