InP Crystals Grow: A Small Phase-Field Numerical Experiment
Recently, I started building a simulation of indium phosphide, or InP, crystal growth. I wanted to explore a simple question: how does cooling affect the growth of an initial seed, and when might new crystals begin to appear elsewhere in the melt?
I used a phase-field model discretized with a control-volume method. Instead of explicitly moving a sharp solid–liquid boundary, the model tracks a continuous variable called phi. A value of zero represents liquid, while a value of one represents solid. A narrow transition region between them allows the interface to move naturally as the crystal grows.
The computational domain is a two-dimensional axisymmetric section of a cylindrical crucible, 50 mm in diameter and 100 mm tall. A small seed sits at the center of the bottom surface. The bottom is cooler than the top, and the sidewall follows a linear temperature profile. During cooling, the boundary temperatures decrease together while maintaining a vertical gradient of 15 K/cm.
The model currently includes heat conduction, with different thermal properties for the solid and liquid. It does not include melt convection. To check the numerical treatment of heat-driven interface motion, I also used a separate one-dimensional Stefan benchmark. This provides a useful verification of the heat-transfer calculation, although it does not establish that the chosen growth kinetics accurately represent InP.
I then added stochastic nucleation using a Poisson process. The local nucleation rate increases with undercooling, allowing a new solid region to appear in the liquid even when it is disconnected from the original seed. Once established, that region can grow through the same phase-field dynamics.
To track its contribution, I introduced two solid-origin fractions: one associated with the original seed and the other with newly nucleated material. These fractions follow the advancing fronts and retain their contributions when the growing regions meet.
The cooling-rate comparison was particularly interesting. At 8 K/h, the crystal grew from the bottom seed without any additional nucleation events in this realization. After 12 hours, about 61.5% of the material had solidified. At 32 K/h, three nucleation events occurred, and the domain became almost entirely solid after 8.82 hours. Material descended from the new nuclei accounted for approximately 72% of the final solid.
I also ran a controlled comparison to isolate the effect of latent heat. Both cases used a cooling rate of 8 K/h, with nucleation disabled and identical solid and liquid properties. The only physical difference was whether solidification released latent heat.
Including latent heat reduced the final solid fraction from approximately 62.22% to 61.52%. The interface near the axis was about 1.26 mm lower. Overlaying the two interface contours showed that the difference was not simply a uniform delay: the interface also became more curved, with a larger height difference between the center and the wall.
There are important limits to these results. The nucleation parameters are illustrative rather than experimentally calibrated, and one stochastic realization cannot establish a reliable defect probability. In an axisymmetric model, an off-axis nucleus represents a ring rather than an isolated three-dimensional particle. The origin fractions also describe where solid material came from—not crystal orientation, grain-boundary structure, or crystal perfection.
For now, this is a useful numerical experiment for exploring how cooling, heat release, and nucleation interact. The next challenge is to connect it to measurements, especially interface growth rates and nucleation statistics, so that the simulation can move from illustrating possible behavior toward making useful predictions.
