Crystal-melt Interface Research Articles

Numerical Investigation on the Effect of Thermal Gate Moving Rate on Directional Solidification Process

The temperature field distribution during the growth of crystalline silicon by the directional solidification (DS) method is an important factor affecting the growth rate, the shape of the melt-crystal (m-c) interface, and thermal stress. To solve the problem of m-c interface convexity at the early stage of crystal growth caused by supercooling at the bottom center of silicon ingot during DS. In this paper, a two dimensional (2D) global transient numerical model based on a large-size ALD-G7 (G7) crystalline silicon ingot furnace is established and experimentally verified. Based on the model, the influence of different bottom thermal gate moving process curves on the convexity of the m-c interface at the early stage was studied, with emphasis on the changes in temperature field, m-c interface, and thermal stress at the early stage of crystal growth. We have designed three cases, case 1 uses the original moving process curve of bottom thermal gate, case 2 and case 3 adjust the process curve to 0.95 and 0.9 of the original ratio, respectively. The numerical simulation results show that compared with case 1, the average heat dissipation at the bottom of the silicon ingot in case 2 and case 3 is reduced by 2.7 kW·m−2 and 4 kW·m−2, and the maximum thermal stress is also reduced by 3 MPa and 4 MPa, separately. The maximum convexity of the interface is reduced by 25 mm and 20 mm, and the convexity is reduced by 55% and 44% on average, respectively.

Controlling the heat, flow, and oxygen transport by double-partitions during continuous Czochralski (CCz) silicon crystal growth

The effect of the depths of two partitions on the heat, flow, and oxygen transport during the CCz growth of an 8-inch diameter silicon crystal is numerically investigated. A crucible in which two partitions are immersed in the silicon melt is so-called a triple-crucible. The different depths of two partitions change the effective wall surface area, which is the main source of oxygen at the growth interface, melt motion, and heat transfer within the crucible. The diffusion and convection of oxygen from the partitions and crucible wall towards the free surface and the crystal-melt (c-m) interface are also affected. The amount of oxygen that dissolves from the effective wall surface into the melt is proportional to the surface temperature and the surface area. The choice of at least one long partition will prevent the entry of unmelted granular silicon into the melt region under the c-m interface. Comparison of various cases shows that when the partition near the c-m interface is longer (90 mm) and the partition near the crucible sidewall is shorter (40 mm), the power consumption and the content of oxygen along the growth interface are lower.

Numerical simulation approach to investigate the effect of gas tube design on the impurities distribution and thermal properties of multi-crystalline silicon ingot grown by directional solidification process

Multi-crystalline Silicon (mc-Si) grown by directional solidification (DS) system is considered as a crucial photovoltaic material due to its relatively low cost and high yield. The numerical modelling is very useful for understanding the impurities segregation during mc-Si growth process. In the present paper, numerical simulation is carried out to investigate the impact of Argon gas tube design as shower head. In this Argon gas tube modification act on the non-metallic impurities, melt-crystal interface, temperature distribution and stress levels in the silicon ingot during the different stages of solidification process. The proposed design resulted in remarkable reduction in the back transfer of CO from heater component of the DS furnace. The generated von misses stress is also discussed as the function of different stage of solidification. The results concluded that the stress level at the initial stage of solidification is lower but it is large at above 75 % solidification rate. Also, we found that the modified gas tube design gives better temperature distribution with melt-crystal interface (m-c) shape. The modified argon tube design will be helpful for enhancing the quality of silicon ingot by controlling the chemical reaction between silicon melt and impurity transport.

On the calculation of crystal–melt interface free energy and interface stiffness using Ni as a model system

Both interface free energy and interface stiffness are crucial parameters in governing the interface morphologies of crystal–melt coexistence systems from which different methods were reported separately to determine these parameters in turn. However, the correlation among these methods has not been a concern in previous research studies. In this work, we created two Ni models of crystal–melt coexistence systems based on a basic crystal–melt interface kinetic equation. Then, the interface stiffness, interface free energy, and anisotropy parameters were calculated using three methods, which are the capillary fluctuation method, critical nucleus method, and modified critical nucleus method. The results are in good agreement with each other. Meanwhile, the interface free energies calculated in our work agree well with the values from experiments and existing simulations.

Predicting Melt-Crystal Interface Position and Shape during the Manufacturing Process of Single Crystal via Explainable Machine Learning Models

In recent years, employing machine learning models to predict the process parameters during the manufacturing process of single crystals has gained wide attention as it’s reliable and much faster than traditional numerical simulation approaches. However, most machine learning models used in previous studies are black box models, which don’t provide explainable results. In this paper, we present a feasibility study of applying explainable machine learning models to predict steady-state melt-crystal interface position and deflection with the set-point temperature of 5 heaters in a vertical Bridgman furnace. The dataset used to train and evaluate the machine learning models was generated by 2-D numerical simulation. We experimented with linear regression and random forest algorithms, and then used linear regression coefficient and SHAP value to quantify the impact of each input on the output, from which we inferred a heater control strategy that could potentially improve the crystal growth process. Our encouraging results show that explainable machine learning models can be applied to predict crystal growth process parameters in real-time and generate actionable insights to guide crystal manufacturing practice.

Local collective dynamics at equilibrium BCC crystal-melt interfaces.

We present a classical molecular-dynamics study of the collective dynamical properties of the coexisting liquid phase at equilibrium body-centered cubic (BCC) Fe crystal-melt interfaces. For the three interfacial orientations (100), (110), and (111), the collective dynamics are characterized through the calculation of the intermediate scattering functions, dynamical structure factors, and density relaxation times in a sequential local region of interest. An anisotropic speedup of the collective dynamics in all three BCC crystal-melt interfacial orientations is observed. This trend differs significantly from the previously observed slowing down of the local collective dynamics at the liquid-vapor interface [del Rio and González, Acta Mater. 198, 281 (2020)]. Examining the interfacial density relaxation times, we revisit the validity of the recently developed time-dependent Ginzburg-Landau theory for the solidification crystal-melt interface kinetic coefficients, resulting in excellent agreement with both the magnitude and the kinetic anisotropy of the crystal-melt interface kinetic coefficients measured from the non-equilibrium molecular-dynamics simulations.

Interface diagnostics: In-situ determination of crystal-melt interface shape evolutions via probing growth interface electromotive force

Detecting crystal-melt interface shape during growth is critical to understanding boundary phase-change kinetics and manufacturing high-quality bulk crystalline materials. However, even for the widely used Czochralski system, the detection of this blind spot has not yet been realized. Here, we determine the interface shape evolution in situ by analyzing the correlation between the growth interface electromotive force (GEMF) and seed crystal temperature of a growing boule. Due to the direct response of GEMF to interface thermal behavior, our determinations show good agreement with computational simulations and experimental as-grown boules. On this basis, it is possible to trace the real-time kinetics of crystallization and provide quantitative interface feedback, which help reveal boundary heat and mass transfer, optimize growth conditions, and hence improve crystal quality. This GEMF-based feedback mechanism will be applied to other directional solidification methods for preventing interface instability and homogenizing component distribution in oxide, semiconductor, and metal crystalline materials.

Molecular dynamics determination of Two-dimensional nucleation kinetic coefficient for modeling the faceted growth of Si (1 1 1) from an undercooled melt

The discrepancies between kinetic model predictions and experimental observations of two-dimensional (2D) nucleation-mediated growth of silicon limits modeling reliability for existing and new crystal growth processes. Molecular dynamics (MD) simulations were performed to identify the mechanism of evolution of crystallites on a Si (111) facet and semi-quantitatively describe 2D nucleation kinetics using the forced-velocity solidification (FVS) and free-solidification (FS) MD simulations techniques. Both MD models predicted similar nucleation expressions but gave lesser nucleation energy barriers than predicted from Monte Carlo (MC) nucleation model. The estimated nucleation rate from MD was fitted to a polynuclear growth model to estimate a 2D kinetic model and compared to available experimentally reported growth rates. The Si (111) facet velocity model derived from the kinetic coefficient given in this work generally provided more conservative estimates of undercooling and the minimum undercooling that may result in kinetic roughening transition. In addition, the FVS model implemented in this work provided a unique opportunity for qualitatively describing the behavior of a crystal-melt interface and gave a molecular-level perspective on the interface stability criterion for the growth of single-crystal silicon during the horizontal ribbon growth (HRG) process.

A new method for computing the anisotropic free energy of the crystal-melt interface

Based on the classical nucleation theory and the crystal-melt interface kinetic equation, a new method was proposed to compute the anisotropic free energy of the crystal-melt interface. The method stems from the fact that the interface stiffness, instead of the interface free energy, controls the morphologies of nuclei when they are in equilibrium with the surrounding melts. In addition, the interface stiffness has an anisotropy which is an order of magnitude higher than that of the interface free energy, and the anisotropies of these two quantities are related to each other. Mapping out the relation between the curvature radius on the local interface of the nucleus and the equilibrium undercooling temperature during molecular dynamics simulations, we are able to determine the interface stiffness, the interface free energy, and their anisotropy parameters. The new method was used to compute the anisotropy of interface free energy for pure Cu as an example, and the results are in good agreement with the data from experiments and existing simulations.

Ordering hard-sphere particle suspensions by medium crystallization: Effect of size and interaction strength.

While microstructure in soft materials is usually given by the self-assembly of their constituting building blocks, colloidal assembly can also be obtained via templating a morphology in a disordered suspension of particles by solidification of the melt. This order-templating process is applicable to different soft-matter systems with a variety of characteristic length scales, including particle suspensions in water, liquid-crystal materials, and polymer melts. In this work, we numerically investigate the effect of particle size and solvent size in the process of solidification templating by implementing a coarse-grain model of hard-sphere particles at a moving melt-crystal interface. This approach captures the dependence of crystallization templating on speed of crystal growth, showing the existence of a threshold speed for crystallization templating to occur. Results in this work show that the threshold speed changes following a power form as solvent size and particle size change. Furthermore, this work analyzes and reports the effect of particle-crystal interaction strength in combination with size effects. This scaling study from a numerical perspective sets a starting point for the development of hybrid soft materials via structural templating, allowing solidification-driven particle ordering in different systems with a variety of length scales.

Modified FZ technique for oxides crystal growth

The crucibleless AHP (axial heat flux close to the phase interface) method for growing single crystals of oxides is proposed. The phenomena of wetting and meniscus shaping are considered for the case when a melt from the top of the submerged heater to its bottom close to the melt-crystal interface gets mostly due to the presence of a capillary through-hole in the center of its casing made of noble metal. Both analytical and numerical approaches are applied, in order to describe conditions of the experiment preparation before the solidification process.Based on the example of TeO2, steady-state meniscus stability, as well as the melt flow dynamics in process are considered in the paper. The mathematical 2D axisymmetric model taken into account continuity equation, laminar flow (Navier-Stokes), including surface tension effects, the equation for the transport of the fluid interface. The Level Set Method implemented for a two-phase gas-liquid domain allows the smooth variation of density and dynamic viscosity across this interface. The numerical calculations, from one hand, make it possible to estimate the appropriate value of the heater height, as well as relations of the translating rate for top and bottom rods to provide the modes of enlarged and constant diameter during the crystal growth. The last is very important from the point of view of the process control being the additional feedback parameters to be used in control loop of charge melt feeding.

Interface morphology evolution of azobenzene from the unsteady to steady state under directional solidification

In this work, the melt-crystal interface of azobenzene at the unsteady and the steady state was in-situ investigated via directional solidification. The experimental results show that, at the unstable stage, interface morphology transforms from planar to zigzag facets at low growth rates and then to faceted dendrites at high growth rates, and the corresponding apex angles change from multiple values to one unified. At the steady state, the interface morphology of azobenzene can maintain planar or zigzag facets or faceted dendrites, and 115° is reserved in the zigzag interface or 50° in the faceted dendrites. Besides, the disappearance of fast-growing planes and the retaining of slow-growing planes contribute to the variation of apex angles from 115° to 50°. Our in-situ observation promotes further understanding of the solidification process in faceted materials.

Effects of different partition depths on heat and oxygen transport during continuous Czochralski (CCz) silicon crystal growth

A series of quasi-steady numerical simulations of the CCz growth of 8-in.-diameter silicon crystals are performed. A quartz partition is installed inside the melt to prevent the solid silicon that is being continuously added during growth from being transported to the crystal-melt interface without being melted. The effects of different partition depths on the heat, flow, and oxygen distributions are investigated. The simulation results show that the oxygen content increases when the depth of partition is increased from 20 mm to 80 mm. The trend is reversed for higher partition depths. This is due to the variation in the thermal field and flow structure. As the depth of the partition increases, there is a more significant change in the velocity under the free surface of the melt in the region inside the partition than in the outer melt region.

Optimal Magnetic Graphite Heater Design for Impurity Control in Single-Crystal Si Grower Using Crystal Growth Simulation

In this paper, we report a successfully modified single-crystal Si growth furnace for impurity control. Four types of arbitrary magnetic heater (AMGH) systems with 3, 4, 5, and poly parts were designed in a coil shape and analyzed using crystal growth simulation. The concentration of oxygen impurities in single-crystal Si ingots was compared among the designed AMGHs and a normal graphite heater (NGH). The designed AMGHs were confirmed to be able to control turbulence and convection in a molten state, which created a vortex that influenced the oxygen direction near the melt–crystal interface. It was confirmed that replacing NGH with AMGHs resulted in a reduction in the average oxygen concentration at the Si melt–crystal interface by approximately 4.8%.

Atomistic Simulation Study of the FCC and BCC Crystal-Melt Interface Stresses

In this work, molecular dynamics simulations have been used to undertake a computational study of the equilibrium crystal-melt interface stresses in face-centered-cubic (FCC) Ni and body-centered-cubic (BCC) Fe, BCC Nb, and a model BCC soft-sphere elemental system, for three different interface orientations, i.e., (100), (110), and (111). The sign, magnitude, and anisotropy of the excess interface stresses and their relationships with the corresponding interfacial free energies have been examined. The universality of a few trends regarding the interfacial stresses observed in FCC crystal-melt interfaces has been assessed for the BCC crystal-melt interfaces. The role of the interatomic bonding that affects the shape of the interfacial stress profiles, thus modulating the magnitude or sign of the excess interface stress, has been discussed through inspecting a particular type of crystal-melt interface over different materials. Besides, for the first time, we have demonstrated that the Irving-Kirkwood fine-grained algorithm for depicting microscopic pressure components and stresses in the vicinity of the crystal-melt interface is superior to the previously used per-particle virial stress algorithm. The reported data and new knowledgqe could enrich the accumulation for theory breakthroughs in predicting interface stresses and motivate future studies on the interfacial stresses for more types of solid-liquid interfaces.

Crystal–melt coexistence in fcc and bcc metals: a molecular-dynamics study of kinetic coefficients

As a sequel to the previous paper on the calculation of the crystal–melt interface free energy (2021 Materialia 15 100962), here we report the results on the kinetic coefficients using molecular dynamics simulations performed on six fcc metals and four bcc metals with the intention to compare the crystal structural influence. We found that the calculated kinetic coefficients are well described by the model by Broughton, Gilmer and Jackson (1982 Phys. Rev. Lett. 49 1496), and in particular, they exhibit varying degrees of anisotropy. We reveal that the anisotropies are related to the fluctuation of the crystal–melt interfaces, which causes the increase of the actual interface area in melting or solidification. The kinetic coefficients always display asymmetry between the solidification and melting process, and the difference is much more pronounced for the (111) interfaces in fcc metals which have the highest anisotropy. We found that the atomic mechanisms of the kinetic behaviors of these interfaces are closely related to the formation of twin-crystal domains during solidification, which delays the solidification process and consequently causes a decrease in the calculated kinetic coefficients.

Nucleation-Controlled Phase Selection in Rapid Solidification from Undercooled Melt of DyMnO<sub>3</sub>

The equilibrium crystal structure of LnMnO3 (Ln: lanthanide) has been reported to be orthorhombic when La3+ to Dy3+ are used as Ln3+, and hexagonal when Ho3+ to Lu3+ are used. Whereas Kumar et al. reported a two-phase structure of orthorhombic and hexagonal phases is formed in DyMnO3 when it was solidified from the undercooled melt under containerless state. The reason for the formation of the two-phase structure was not thoroughly addressed and discussed. We investigated the formation mechanism for the two-phase structure from the undercooled melt of DyMnO3 in detail. As a result, the surface morphology, microstructure, and crystal structure of the samples, in which the nucleation was forced at a predetermined temperature with a Mo needle, indicated that the hexagonal and orthorhombic phases are dominant at high and low temperatures, respectively. When the sample was quenched from below 1670 K in a water bath, as-solidified sample consisted of h-DyMnO3 and o-DyMnO3. Whereas a single phase of h-DyMnO3 was obtained in the sample quenched from above 1670 K. This phenomenon can be quantified in terms of nucleation-rate determined phase selection. That is, the activation energy for forming a critical nucleus calculated based on the model of the crystal-melt interface proposed by Turnbull and Speapen suggests that the o-DyMnO3 phase can be heterogeneously nucleated on the interface of the initially formed h-DyMnO3 phase. This Paper was Originally Published in Japanese in J. Japan Inst. Met. Mater. 85 (2021) 155–161. The abstract and captions of Figs. 1–10 are modified.

Numerical and experimental investigation of CdZnTe growth by the boron oxide encapsulated vertical Bridgman method

Cadmium zinc telluride crystals of 4.8 cm in diameter grown by the boron oxide encapsulated vertical Bridgman technique exhibit good structural quality with large grains and low dislocation density. However, the ingots contain Te inclusions, which are trapped at the growth interface during the crystallization process. Global modeling of the furnace was applied in order to investigate the temperature gradients and the evolution of the growth interface shape in this system. Transient computations, which include the crucible movement, show that the crystal-melt interface is concave toward the melt in the conical part of the ampoule, then becomes convex during the growth in the cylindrical part of the ampoule. Concave shapes of the interface and very homogeneous temperature distribution at the beginning of the solidification process promote the growth of a polycrystalline material. A novel hypothesis explaining the mechanism of grain formation at the ampoule tip was formulated. Our calculations show that the anomalous Zn segregation is due to the poor solute mixing at the beginning of the solidification process. The modeling of the standard growth process reveals very low vertical temperature gradients in the crystal (2-4 K/cm). Such low temperature gradients are not favorable to eliminate the Te inclusions in the as grown crystal. A numerical model was employed to analyze the temperature field influence on the migration/size reduction of Te inclusions.

Numerical Investigation of Bottom Grille for Improving Large-Size Silicon Quality in Directional Solidification Process

2D global transient numerical model based on a new large-size ALD-G7 (G7) crystalline silicon ingot furnace is established and experimentally verified. A new pyramid-shaped bottom grille structure is designed. The temperature field, flow field, crystal growth rate, crystal-melt (c-m) interface and thermal stress during the crystal growth process for the furnace with and without pyramid grille are compared and studied. Simulation results indicate that the supercooling of silicon ingot center and excessive crystal growth rate are easily caused when original grille is used, which will lead to the over-convexity of c-m interface and higher thermal stress. The addition of pyramid grille reduces crystal growth rate in the early stage. By reducing heat flux at heat exchange block center, the pyramid grille improves the supercooling of the center area and the shape of c-m interface significantly. Furthermore, the thermal stress at the bottom center and bottom corner of the silicon ingot is reduced by 20.22 and 16.14 % in the modified furnace, respectively.

Transient Simulation on the Growth of Mono-like Silicon Ingot in DS Process Using Crucible with Plano-Concave Bottom for PV Applications

Two-dimensional numerical simulation on axisymmetric directional solidification furnace has been carried out to investigate the growth of mono-like silicon ingot. The silicon ingots were grown in conventional crucible and modified crucibles with Plano-Concave bottom. Four different crucibles with the thickness variation of 5 mm, 10 mm, 15 mm and 20 mm at the bottom center of the crucible were used for the simulation. The thermal field of the grown ingots were analysed during the growth process. The effect of the modification at the crucible bottom on the outward heat flux has been investigated and the melt-crystal interface was analysed during the growth process. The distribution of thermal stress and shear stress in the grown ingots was investigated. The results show that the modified crucible with Plano-Concave bottom gives better melt-crystal interface by enhancing the spot cooling and lead to lower shear stress and lower thermal stress in the grown ingot which can be used for photovoltaic applications.