Numerical Modeling of the Cellular Structure Formation Process in SiC Solution Growth for Suppression of Solvent Inclusions

Abstract

For the solution growth of silicon carbide, solvent inclusions are significant technological issues, and methods to suppress the formation of solvent inclusions are investigated in this study. Experimental observations show that solvent inclusions are formed behind the cellular structures. A phase field model is used to reproduce the formation process of cellular structures and solvent inclusions. Simulation results indicate that slight perturbations of the step front can convert into cellular structures in the case of insufficient supply of carbon, and the overdeveloped cellular structures consequently result in solvent inclusions. Accordingly, several schemes can be suggested by the simulation model to suppress the formation of cellular structures by enhancing the carbon supply. By increasing the carbon diffusion coefficient, cellular structures can be suppressed. Moreover, the step height and the solution flow direction also play an important role in suppressing the cellular structures. This study provides a comprehensive understanding of the formation process of cellular structures and solvent inclusions. A growth process with a high diffusion coefficient and opposite solution flow to the step flow direction was proposed to suppress the formation of cellular structures. The proposed numerical model could be applied in other solution crystal growth methods.

Effect of Solution Components on Solvent Inclusion in SiC Solution Growth

“…The degree of supersaturation can be formulated as σ = Δμ/( k B T ) . k B is the Boltzmann constant, and T denotes temperature, 2073 K. A (θ) corresponds to the anisotropy factor, which is also contingent on the step angle: A ( θ ) = ε ( θ ) ε 1 [ sin false( 2 θ false) false( φ y y − φ x x false) + 2 cos false( 2 θ false) φ x y ] − 1 2 ( ε 1 2 + ε false( θ false) ε 2 ) [ 2 sin false( 2 θ false) φ x y − normal∇ 2 φ − cos false( 2 θ false) false( φ y y − φ x x false) ] The parameters ε 1 , ε 2 , ϕ yy , ϕ xx , and ϕ xy are detailed in previous research …”

“…Using the finite difference to solve the equation and expressing it as φ i , j n + 1 − φ i , j n = β ν false( 1 + ξ cos ( γ θ ) false) 2 N ( φ i + 1 , j n + φ i , j + 1 n + φ i , j − 1 n + φ i − 1 , j n − 4 φ i , j n ) + ν β R 2 N π sin ( π φ i , j n ) + ν × σ false( normalΔ x ∼ false) 2 ( 1 + cos false( π φ i , j n false) ) − ν × A ∼ ( θ ) where ν = Δ t ∼/Δ x ∼ 2 is the Courant number, and scriptR = Δ x ∼ /δ ∼ is the ratio of the grid size to the step width. A ∼ (θ) is the dimensionless form of A (θ), which is detailed in a previous paper . To ensure high calculation accuracy and reduce calculation cost, Δ x ∼ is 0.05 when δ∼ is 0.1.…”

“…Furthermore, aside from the Cr concentration, the influence of the Gibbs–Thomson effect is also taken into account. The relationship between the normalized carbon equilibrium concentration C eq and the local radius of the step curvature is detailed in prior research …”

“…9,10 Prior research indicates that the formation of inclusions can be attributed to step morphological instability. 11,12 Several experimental studies suggest that under specific conditions, the stability of macrosteps can be compromised, resulting in areas with a nonplanar interface shape, leading to the development of cellular structures. 13 These cellular structures become reservoirs for a trapped solution, subsequently leading to the formation of dot-like point inclusions.…”

“…A phase field model is established to discuss the factors affecting the formation of cellular structures, including constitutional supersaturation, solution viscosity, and interfacial energy. Compared to the step phase field in the previous model, 12 the additive Cr diffusion field is calculated and it is coupled with the phase field. The interfacial energy ratio is employed in the phase field equation.…”