Investigating the Growth of GaSb Single Crystals through Optimized LEC Method Utilizing Finite Element Simulation and Machine Learning Techniques

Abstract

In this study, CGsim and liquid-encapsulated Czochralski (LEC) growth experiments were employed to handle the challenges associated with growing large-sized compound semiconductor single crystals. CGsim, a simulation software integrating the finite element method with machine learning (ML) techniques, was utilized to optimize the heat flux and crystallization front morphology at the solid–liquid interface during GaSb crystal growth. ML validation, performed across various crucible rotation speeds and crystal position (CP) configurations, enabled the optimization of the moving front shape at the solid–liquid interface, reducing the protrusion angle to 0.086°. The crystal quality of GaSb single crystal slices was evaluated through X-ray double crystal rocking curves and optical microscopy. The results indicate that the optimized growth parameters reduced the dislocation density in 6-in. GaSb single crystals from 1039 to 369 cm^–2 and narrowed the X-ray rocking curve full width at half-maximum (fwhm) from 29 to 28.5 arcsec. The steady-state/unsteady-state simulations conducted in CGsim, combined with ML-optimized growth parameters, significantly lowered the likelihood of defect formation, involving dislocation clustering, vacancy defects, twins, undercooling striations, and small-angle grain boundaries.

1. Introduction

According to elemental semiconductors, semiconductors can be divided into five generations: elemental semiconductors, high-mobility semiconductors, wide-bandgap semiconductors, ultranarrow-bandgap semiconductors, and ultrawide-bandgap semiconductors. The first generation of semiconductors comprises elemental materials, namely silicon (Si) and germanium (Ge). Subsequent high mobility compound semiconductors, such as gallium arsenide (GaAs) and indium phosphide (InP), possess high electron mobility and direct bandgap characteristics. Indium antimonide (InSb), gallium antimonide (GaSb), and indium arsenide (InAs) are narrow-bandgap semiconductors, primarily used in applications such as infrared detection, thermal imaging, and high-speed, low-power electronics. This research focuses on the GaSb crystal, a critical semiconductor material that has attracted considerable attention for its application in mid-infrared photonics.

The Wide bandgap comprises III–V gallium nitride (GaN) and IV silicon carbide (SiC) semiconductors, which are further classified into ultranarrow bandgap materials (such as indium antimonide (InSb) and gallium antimonide (GaSb)) and ultrawide bandgap materials (such as gallium oxide (Ga₂O₃), aluminum nitride (AIN), and diamond (C)). , Table suggests that semiconductors represent materials exhibiting notable advancements compared to the preceding three generations, particularly in material characteristics, performance, and application domains. These materials offer new avenues for reinforced adaptability in extreme environments, improved energy efficiency, and the creation of innovative electronic devices.

1. Generation Division and Material Performance Comparison of Semiconductor Materials from the First to Fourth Generation.

category	material	decompression (MPa)	stacking fault energy (10–3 J/m2)	shear modulus (1011 dyn/cm2)	application field
element	Si			9.784	integrated circuits, the microelectronics field, photodetectors, etc.
	Ge			7.47
high mobility	GaAs	0.9	43	7.55	wireless communication, optoelectronic devices, photoelectric storage, high-speed photodetectors, and photomultiplier tubes, etc.
	InP	2.75	17	7.32
wide bandgap	SiC			44.2	intelligent power grid, new energy vehicles, 5G/6G communication, microwave radio frequency, power electronics devices, etc.
	GaN
ultranarrow bandgap	InSb		43		ultrawide bandgap: high-temperature/high-frequency, high-power density, and high-conversion-efficiency power electronics devices, etc. Ultranarrow band: Detectors, lasers, etc.
	GaSb	0.016	53	5.63
	InAs	0.33	30
ultrawide bandgap	Ga2O3
	AIN
	diamond

GaSb single-crystal wafers serve as essential substrates for the growth of antimonide epitaxial films, quantum structures, and the fabrication of semiconductor devices. Since the performance of devices is directly influenced by the crystal quality of GaSb, the growth of GaSb single crystals is crucial for the development of antimonide semiconductor devices. Antimonide semiconductors, particularly for infrared focal plane detector applications, exhibit significant market potential, stimulating a growing demand for GaSb epitaxial substrates. The methods for growing GaSb single crystals consist of the Vertical Bridgman (VB), Vertical Gradient Freeze (VGF), and Liquid Encapsulated Czochralski (LEC) techniques. The primary method of GaSb single crystal growth is the LEC technique. − The dominant products available in the market are 2-in. and 3-in. GaSb single crystal substrates. With large-scale single-crystal wafers, however, advancements in industry technology have shifted toward device fabrication processes. While substrates with diameters ranging from 4 to 6 in. have been progressively developed and commercialized, these larger substrates still encounter numerous challenges regarding crystal quality, reliability, and stability.

Nonetheless, LEC technology necessitates a relatively high temperature gradient for growth, leading to a significant temperature gradient at the solid–liquid interface during crystallization and thus the accumulation of substantial residual stress in the crystals. , Stress-induced lattice distortions, dislocations, and crystalline defects adversely influence the photoelectric properties of the crystals and hinder their application in primary uses. − As the crystal diameter increases, convective fluctuations in the crucible melt intensify. This exacerbates stress and defect problems related to elevated temperature gradients while substantially increasing the challenges associated with crystal growth and quality control.

With the purpose of achieving high-quality GaSb single crystals, a multiphysics simulation model for LEC-GaSb crystal growth was developed in this study by incorporating both steady-state and unsteady-state simulations using CGSim − and integrating machine learning (ML) techniques. , Specifically, the distribution of thermal and flow fields within the LEC furnace was calculated. The impact of various factors (including natural convection in the melt, forced convection due to crystal and crucible rotation, crucible height within the heater, and nitrogen flow in the furnace chamber) on single crystal growth was analyzed. Furthermore, a CGSim machine learning module (MLM) was implemented by neural network algorithms to refine the crystallization front and heat flux predictions at the solid–liquid interface. This allowed for the swift evaluation of experimental, simulation, and ML outcomes, facilitating intelligent control in reverse process design. The findings clarify various physical and chemical phenomena in the crystal growth process, contributing to optimizing growth conditions, minimizing trial-and-error expenses, and enhancing growth efficiency.

2. Experiment

2.1. Experimental Raw Materials and Preparation Methods

This study focuses on the growth of 6-in. GaSb crystals, involving the processes of material loading, melting, GaSb synthesis, and single-crystal growth. The raw materials required are Ga (99.9999%), Sb (99.9999%), KCl (99.999%), and NaCl (99.999%), with a specified Sb/Ga atomic ratio of 1.001:1 and a KCl/NaCl weight ratio of 1.19. The flux was placed in a GaSb single-crystal furnace, which was then evacuated, and the material was synthesized at 1158 K. As illustrated in Figure a, 300 and 350 g of flux were weighed for the upper crucible and the lower crucible, respectively, and placed accordingly. Besides, 2000 g of high-purity gallium and 3495 g of high-purity antimony were measured. In the upper crucible, high-purity antimony was loaded, followed by gallium, tellurium, and finally the flux. The lower crucible contained only the flux. The dopant used was high-purity tellurium (99.9999%), weighing 750 mg.

1.

Single crystal growth process of GaSb, (a) Material loading, (b) GaSb synthesis, (c) Seed crystal inoculation, (d) Shoulder-broadening crystal of GaSb, (e) Crystal unit cell structure of GaSb, (f) Schematic diagram of tellurium (Te)-doped GaSb crystal unit cell.

Initially, the GaSb seed crystal oriented along the (100) 0° plane was cleaned of grease with ethanol. Afterward, the polished surface was sequentially subjected to acid etching, rinsing with deionized water, dehydration, and ethanol drying. The acid etch solution had a volume ratio of acetic acid: nitric acid: hydrofluoric acid at 10:9:1. Additionally, the system was evacuated and filled with nitrogen to a pressure of 0.06 MPa. Subsequently, the power was heated and gradually increased by 5% increments. As the synthesis temperature was approached, slower adjustments were made. The heating process spanned from room temperature to 1123 K.

As illustrated in Figure b, the pressure was maintained at 0.016 MPa to prevent the decomposition and volatilization of antimony (Sb) during the synthesis of gallium antimonide (GaSb). The synthesis was considered complete when the liquid in the upper crucible transitioned from a yellowish-green color to a transparent, uniform white, and the temperature remained constant for 2 h. After this 2-h isothermal stage at 1138 K, the temperature was lowered to 1123 K. Next, the polycrystalline melt was transferred to the lower crucible for slag removal in preparation for subsequent single crystal growth.

Figure c reveals that the crystal rotation was adjusted to 3 rpm in a counterclockwise direction, and the crucible rotation was adjusted to 8 rpm in a clockwise direction. The temperature gradually decreased to 1053 K, and the seed crystal was positioned approximately 1–2 cm above the flux for preheating. The desired temperature was maintained for 15–20 min. Subsequently, the seed crystal was brought into contact with the flux surface and held for 3–5 min before being gently lowered to engage with the GaSb melt. The emergence of a step-like square halo signifies an optimal temperature. This allowed for a 2–3 min interval before the temperature decreased to 1047 K, and the pulling rate was adjusted to 8–10 mm/h.

As depicted in Figure d, the temperature gradually decreased during the shoulder forming process. Upon achieving the specified shoulder diameter, the process transitioned to the body growth phase, involving an increase in temperature from 1047 to 1049 K and an elevation of the pulling speed to 8 mm/h. Constant-diameter growth was sustained after the completion of the shoulder transition.

During the equiaxed grain growth process, temperature adjustments were conducted based on the crystalline state, with decisions to increase or decrease temperature informed by the halo effect. Typically, each temperature adjustment did not exceed 0.5 K, and adjustments were spaced 20 to 30 min apart.

During the doping process, foreign atoms commonly occupied specific lattice sites within the original crystal structure, preserving its overall stability and electrical neutrality. Te and Sb, which are both Group V elements and neighbors in the periodic table, facilitate Te substitution for Sb during gallium antimonide doping, forming an N-type gallium antimonide crystal. Concerning the preparation of N-type GaSb single crystals, Sb-rich GaSb polycrystalline material is typically employed, with Te, Se, and S as N-type dopants. Among these dopants, Te atoms introduce additional electrons into the crystal by replacing Sb atoms, resulting in its altered conductivity type. The GaSb unit cell and Te-doped GaSb crystal are depicted in Figure e,f, respectively.

2.2. Mathematical-Physical Model

2.2.1. Basic Assumptions

This simulation incorporates the following assumptions. (1) The GaSb crystal growth process with the LEC method was assumed to be quasi-steady-state, implying that the steady-state analysis was performed to examine the global distribution of internal thermal and flow fields at a given moment. Meanwhile, the effects of crucible and crystal lifting and lowering rates were neglected in the steady-state analysis. (2) During the analysis of phase change and flow-field phenomena, GaSb solid was modeled as a substance with extremely high viscosity. Table presents the physical parameters of GaSb solid and melt, which are modeled as piecewise functions varying with temperature. In Table , λ denotes thermal conductivity, ρ indicates density, C represents heat capacity, ε refers to surface emissivity, and μ designates dynamic viscosity.

2. Physical Parameters of Various Materials Related to the Simulation.

material	λ/[W/(m·K]	ρ/(kg/m3)	ε	C/[J/(kg)]	μ/(Pa·S)
graphite	107.824–10.4968e–2·T+47.2e–6·T2+63.3048e–10·T3	2230	0.7	720
graphite felt	4.3e–2+3.24e–5·T+904e–8·T2	250	0.7	1047
quartz	2.0	2650	0.85	1232
GaSb(solid)	50.6–0.134·T+1.48e–4·T2–5.79e–8·T3	5614	0.6	266
GaSb(melt)	17.1	6030–0.578·T	0.6	328	2.31 × 10–3
covering agent	0.6–0.7	1.4–1.55	0.6–0.9	1.2	0.2031 exp (20 420/RT)

2.2.2. Model Setting and Simulation

2.2.2.1. Geometric Structural Model Setting

In this paper, a graphite resistance-heated LEC single-crystal growth furnace was simulated. The simulation software used includes the finite element-based CGSim software and the Machine Learning Module (MLM), , which were developed by the STR team. Soft-Impact Ltd. has been properly identified as the developer of the CGSim software. STR-soft GmbH, as the distributor, provided the software license and technical support for this simulation and has been duly acknowledged. Figure S1a depicts the simulated geometric structure for GaSb single-crystal growth through the LEC method. Crucial components comprise a thermal insulation container, quartz crucible, graphite crucible, graphite heater support base, and essential materials for GaSb growth. Furthermore, a two-dimensional axial symmetry model, rather than a three-dimensional one, was employed to achieve axial symmetry in the thermal field distribution within the crucible and ensure the validity of thermal field calculation results.

2.2.2.2. Material Parameter Settings

Table summarizes the physical parameters of the thermal insulation and crucible materials, GaSb melt and crystals, and encapsulants employed in this study. GaSb existed in a solid state within a temperature range of 293 to 985 K and was subjected to a phase transition to a liquid state between 985 and 1273 K. The kinetic viscosity served as an indicator of the state changes before and after the phase transition.

2.2.2.3. Physical Module and Mesh Generation

The fluid heat transfer, laminar flow, and turbulence modules were employed for modeling purposes. The fluid heat transfer module consists of nodes for both solid heat transfer and phase change processes. The laminar flow module was adopted to analyze the flow characteristics of GaSb melt, and the turbulence module was used for the flow of N₂ within the furnace. Concerning the flow characteristics of GaSb melt, refined mesh segmentation was applied to the GaSb melt region and boundary layers between GaSb and other materials. Additionally, a refined mesh segmentation method was selected for the remaining areas. The mesh structure is depicted in Figure S1b.

2.2.2.4. Steady-state and Unsteady-State Simulation

Steady-state simulation of crystal growth involved various physical and chemical processes under constant conditions. The parameters of the crystal growth system, including temperature, pressure, and flow rate, remained consistent during steady-state simulation. Consequently, the growth morphology, temperature distribution, flow field distribution, and other relevant properties of the crystal were simulated under stable conditions. Figure S1c illustrates the two-dimensional thermal field distribution of GaSb to facilitate subsequent machine learning-based calibration of solid–liquid interface heat flux.

The growth process of GaSb crystals under unsteady-state conditions is exhibited in Figure S1d. The dynamic changes in the GaSb crystal during the growth process were simulated by defining time-varying parameters, including crucible rotation speed, seed rotation speed, temperature gradient, heater power, and pulling speed. The ability of unsteady-state simulation to accurately represent the actual crystal growth process is its primary advantage and has laid a reliable foundation for optimizing the growth conditions. Unsteady-state simulation allowed for the observation of various transient phenomena during crystal growth, comprising the effects of temperature fluctuations and flow rate changes on crystal quality.

2.2.2.5. Machine Learning Processes and Methods

With respect to the existing LEC method for GaSb crystal growth, multiple computational models (CPs) were developed in our study by CGSim, as depicted in Figure . These models were primarily designed to validate and calibrate the heating and cooling capabilities of the positive heater, in conjunction with temperature measurements, so as to ensure the models’ precision. Experimental data, including temperature gradients, thermal stresses, defect densities, and dopant concentrations, were collected to verify and calibrate turbulent flow models, specifically 2D FM STR-K-ε and 3D FM LES/RANS. Additionally, a thorough analysis of the metal/crystal (m/c) interface shape and dopant concentration data was conducted across various growth parameters.

2.

CGSim simulation and machine learning methods.

Furthermore, the influences of using 2D versus 3D methods to calculate dislocation density and dopant concentration were compared, and the benefits of employing 3D methods to reduce growth rate and thermal field fluctuations were explored. Simultaneously, sustained undercooling was kept throughout the experiment. Then, interface geometry data for the Machine Learning Module (MLM) were obtained. With this method, the corrected heat flux values were derived, reflecting a functional relationship between growth parameters and thermal field variables. The basic CGsim+MLM was employed to calculate thermal stress and defect concentration in GaSb crystals, demonstrating the practical application of our findings. Ultimately, the Liquid Encapsulated Czochralski (LEC) method was successfully optimized for GaSb crystal growth, yielding large-sized, low-stress, and low-defect-concentration single crystals.

As illustrated in Figure S2, the convection of turbulent melts was primarily influenced by factors such as melt thickness, crucible thickness, growth rate, crystal rotation speed, crucible rotation speed, nitrogen gas flow rate, as well as axial and radial temperature gradients within the crucible. Figure S3 presents the input and output parameter scheme for the artificial neural network designed in this study. The training input parameter set consists of crystal height, growth rate, crucible rotation speed, crystal rotation speed, and vertical and horizontal temperature drops in the melt. The simulation of GaSb crystal growth with the LEC method represents a vital aspect of machine learning applications. The input and output parameters for the neural network were clarified. When the geometric shape of the crystallization front was set for the LEC method GaSb crystal simulation, the crystallization speed |V _crys|n _x | should remain constant across the entire front in the |n _x | direction. This is because the shape of the crystallization front is determined by the radial heat flux distribution, and the complexity of turbulent heat transfer in the melt makes accurate determination of this distribution challenging.

In summary, integrating machine learning with two-dimensional simulations offers several advantages. First, heat flux correction enables the simulation to achieve a more precise representation of the crystallization front in an efficient and time-saving manner. Second, the heat flux correction method facilitates the modeling of effects that were previously unaccounted for in two-dimensional simulations. Third, the integration of machine learning models with heat flux correction expands the scope of simulating crystal growth, particularly when experimental data is limited and three-dimensional simulations are impractical under their time requirements.

3. Results and Discussion

3.1. Common Defects of GaSb

GaSb demonstrates a zinc-blende crystal structure, composed of alternating gallium (Ga) and antimony (Sb) atoms. At ambient temperature, GaSb belongs to the space group F4̅3m, characterized by a lattice parameter of 6.095 Å. A complete unit cell is presented in Figure a. GaSb vacancies primarily comprise gallium vacancies (V _Ga) and antimony vacancies (V _Sb), which can exist in either neutral or charged states, as illustrated in Figure b. Sb vacancies (V _Sb) are more likely to form in Ga-rich environments, whereas Ga vacancies (V _Ga) are more prevalent in Sb-rich environments. Vacancy defects introduce deep energy levels within the bandgap, with V_Ga potentially functioning as acceptors, provoking p-type conductivity. Conversely, V _Sb may act as donors, leading to n-type conductivity. Vacancy defects can adversely affect carrier mobility, enhance scattering, and degrade crystal quality.

3.

Two × 2 × 2 Supercell model of GaSb single crystal, (a) Lattice-intact, (b) Ga vacancy defect.

GaSb exhibits low stacking fault energy (SFE), allowing it to form twin formation during growth, as displayed in Figure a. The occurrence of undercooling striations in GaSb stems from a combination of solid–liquid interface instability and solute/heat transport mismatch. Tight control over factors such as growth rate, temperature gradient, and doping can effectively mitigate these defects and improve crystal quality. An example of undercooling striations in GaSb crystals during growth is illustrated in Figure b. Figure c displays small-angle grain boundaries in GaSb crystals. Figure d depicts a high density of dislocations. The emergence of small-angle grain boundaries in GaSb crystals is intimately linked to dislocation arrangement and stress distribution during growth. Dislocation glide and aggregation, coupled with thermal stress exceeding the material’s yield strength attributed to temperature gradients, bring about the nucleation of various dislocations, including edge and screw types. Among other similar types, parallel edge dislocations gradually organize into dislocation walls via glide or climb, forming small-angle grain boundaries with misorientations of typically less than 10°. −

4.

Common crystal defects in gallium antimonide: (a) GaSb twins, (b) GaSb undercooling striations, (c) GaSb low-angle grain boundary, (d) EPD, etch pit density.

3.2. CGsim Simulation and Machine Learning Optimization

3.2.1. Frontier Optimization of Crystallization

During the initial growth stage of solid–liquid interface evolution, after 1 h, the melt volume remained large, the thermal field was stable, and the axial temperature gradient was significant. At this stage, the interface bulged toward the melt, with a faster growth rate at the crystal center compared to the edges. Consequently, the interface shape appeared convex, exhibiting a W-shape (Figure a). After 4 h of shoulder widening, the growth of the crystal stimulated a decrease in melt volume, gradual changes in the thermal field, and an increase in the temperature gradient. This contributed to enhanced interface convexity and greater deviation in the W-shape, as illustrated in Figure b. During the later growth stages (22.4 and 30 h), the thermal capacity of the residual melt decreased as the melt neared depletion, causing the interface to flatten (Figure c,d).

5.

Simulation of the shape evolution of the solid–liquid interface during GaSb crystal growth, (a) Shoulder growth stage, grown for 1 h, (b) Constant diameter stage, grown for 4 h, (c) Midgrowth stage, grown for 22.4 h, (d) Final growth stage, grown for 30 h.

During the crystal growth process, factors such as temperature fluctuations, mechanical vibrations, and abrupt changes in pulling speed can induce instantaneous deformation of the growth interface. This may lead to the formation of local protrusions or depressions, which in turn result in growth stripes or facet defects. The interface shape is closely related to crystal defects. − In a convex interface, the faster growth at the center expelled impurities to the edges, triggering radial segregation and high-stress areas. This may lead to dislocation loops or twins. In a concave interface, the faster growth at the edges induced impurity enrichment at the center and potentially contributed to the formation of voids or constitutional supercooling zones, stimulating cellular structures. While a flat interface is ideal, it requires strict thermal field control because the sensitivity to slight fluctuations can bring about local instability. Hence, it is crucial to accurately adjust the heat flux at the solid–liquid interface.

The shape of the crystallization front was determined by various methods, such as experimental data with specific structures, three-dimensional large-eddy simulations, direct numerical simulations, and two-dimensional calculations with the STR k-ε turbulence model in the flow module (Figure ). Without the consideration of melt flow, the CGsim thermal flux was corrected by an equivalent thermal conductivity method to obtain the accurate interface shape in CGSim. The primary advancement in calculating the crystallization front shape involves utilizing thermal flux information from the Flow Module to correct the thermal flux in the melt. Specifically, the difference between the thermal fluxes obtained from Basic CGSim and the Flow Module was first calculated to determine the necessary thermal flux correction. The shape of the crystallization front was influenced by melt convection, heat conduction within the crystal and melt, and radial heat flux distribution.

6.

Correction of heat flux at the solid–liquid interface.

Stefan’s problem formulation is

$$Q_{\mathrm{crys}}(R)-Q_{\mathrm{melt}}(R)=V_{\mathrm{crys}}|n_x|{\rho }_{\mathrm{crys}}\mathrm{\Delta }H$$ 1

where Q _crys(R) represents the heat absorbed by the crystalline side; Q _melt(R) indicates the heat absorbed by the molten side; V _crys denotes the crystallization rate;|n _x | embodies the direction along the normal vector x; ρ_crys designates the density of the crystal; ΔH stands for the latent heat of crystallization.

The geometric shape of the crystal growth front is set such that V _crys|n _x | remains constant along the entire front. Complex turbulent heat transfer phenomena hinder the accurate calculation of heat flux distribution within the melt.

$$\mathrm{\Delta }{Q}_{\mathrm{melt}=}{Q}_{\mathrm{melt}}{|}_{\mathrm{exact}}-Q_{\mathrm{melt}}{|}_{\mathrm{Basic\; CGSim}}$$ 2

where ΔQ _melt represents the corrected heat flux value; Q _melt denotes the exact heat released by the melt; Q _melt|basic CGSim indicates the heat value calculated by the basic version of CGSim.

Figure presents accurate data comprising two-dimensional calculations through the STR k-ε turbulence model in the flow module, experimental data, and three-dimensional flow module calculation results. Basic CGSim represents two-dimensional calculations conducted without the consideration of convective melt flow, instead assuming an effective thermal conductivity for the melt. Basic CGSim+MLM reflects two-dimensional calculations following basic CGSim with an MLM trained on the precise data provided. The training of the machine learning model for thermal flux correction in basic CGSim does not necessitate overly complex neural networks. Figure presents the detailed architecture of the Deep Neural Network (DNN) employed in this study. Ideally, input parameters should encompass multiple scenarios under various conditions to derive the thermal flux correction ΔQ _melt for each. In the field of single-crystal growth, the scarcity of extensive experimental data relies on existing cases for machine learning applications.

In this study, the data set for the machine learning model consisted of 20 high-fidelity finite element simulation cases. The data sets were randomly divided into training and independent test sets with an 80/20 split. Specifically, 16 cases were used for model training, and the remaining 4 cases were reserved as an independent test set to evaluate the model’s generalization performance. The resulting performance metrics (e.g., R² score and RMSE) are presented in Table . With regard to the use of a validation set and k-fold cross-validation, no separate validation set was adopted. Instead, a more robust k-fold cross-validation approach (with k = 5) was applied during model training and hyperparameter tuning. Furthermore, the 16-case training set was partitioned into five mutually exclusive subsets. In each iteration, one subset served as the validation set, and the other four were used for training. This process was repeated five times, and the average performance metrics were utilized for model selection. Particularly, k-fold cross-validation allows more effective utilization of limited data and yields more stable and reliable hyperparameter optimization compared to a single fixed validation set, thereby eliminating the need for a dedicated hold-out validation set. The final model performance was impartially assessed by the completely independent test set, which was not involved in any phase of the training process. The parameter sets (parameter pairs) listed in Table were systematically generated through Design of Experiments (DoE) with Latin Hypercube Sampling (LHS). This method guarantees uniform sampling across the defined ranges of process parameters (such as heater power, pulling rate, and rotation speed) with a limited number of cases. Consequently, the parameter space is broadly covered, and diverse process conditions are representatively incorporated into the training data.

3. Parameters Used to Predict the Interface Shape of the Modified Hot Zone in MLM.

sample	crucible rotation speed	CP position
training	3, 5, 7, 9, 11 rpm	15, 100, 150 mm
test	4, 6, 8, 10 rpm	30, 75, 125 mm

Figure S4 illustrates the calculation of new heat flux corrections (ΔQ _melt) for various simulation scenarios and parameter sets with an MLM. Subsequently, the basic CGSim method was employed to address challenges related to global heat transfer. Besides, multiple available cases, which were divided into training and testing data sets, were utilized to train the network effectively and ensure reliability. The iterative training process utilized “learning” cases, while the quality of learning was assessed through “testing” cases that are not part of the training data set. Additionally, the radial distribution of heat flux corrections was employed as a criterion to evaluate the learning quality. Modifications were required in the hot zone design, heating, and cooling structures to optimize the growth process. The shape of the crystallization front was governed by melt convection and heat conduction within both the crystal and the melt. The 2D model in basic CGSim lacked accuracy in predicting melt flow. Then, 3D DNS or LES calculations, or specialized turbulence models, are necessary for accurate prediction of turbulent melt convection. In resource-limited scenarios, two-dimensional calculations with high-order approximations were performed to ensure that they did not introduce significant errors. Furthermore, simulations at various grid resolutions and comparison of the outcomes were conducted to verify the grid independence of the heat flux corrections obtained from Basic CGSim.

The fixed parameters comprised a crystal diameter of 150 mm, an N₂ flow rate of 150 slm, a crystal growth rate of 8 mm/h, and a counterclockwise crystal rotation speed of 10 rpm. The varying parameters were crucible rotation speeds ranging from 3 to 11 rpm and powers ranging from 11.38 to 11.48 kW. The crystal positions were 30, 50, 70, 90, 110, 130, and 150 mm. Additionally, two-dimensional calculations were conducted by the STR k-ε turbulence model on the flow module, yielding accurate data for the LEC furnace, as plotted in Figure . Furthermore, two-dimensional calculations were performed in the basic CGSim, neglecting the convective flow of the melt and assuming an effective thermal conductivity for the melt.

7.

Modeling of crystal growth front with heat flow correction, (a) Heat flux correction, (b) Obtaining results of crystallization front.

Figure illustrates a model for crystal growth fronts incorporating heat flux correction through machine learning techniques. The machine learning parameters encompass 10–20 neurons, two hidden layers, a learning rate ranging from 3 to 4, and approximately 200,000 iterations. The training input parameters primarily consist of crystal height (CP), growth rate, crucible rotation, crystal rotation, and vertical and horizontal temperature gradients in the melt. The iterative machine learning process continues until the numerical residual (“loss”) stabilizes. Acceptable results can be achieved when the numerical residual (“loss”) is 1 or less.

The model was trained with the Mean Squared Error (MSE) as the loss function, defined as

$$\mathrm{M}\mathrm{E}\mathrm{S}=\frac{1}{n}\sum _{i=1}^n{(y_i-{\hat{y}}_i)}^2$$ 3

where n is the number of samples, y_i is the true value, and ${\hat{y}}_i$ is the predicted value from the model. Before training, all input features (including heater power, crucible position, and temperature gradient) and output targets (such as crystal diameter and dislocation density) underwent Min-Max normalization. This preprocessing step linearly transforms the data into the range of [0, 1].

Normalizing both the input features and output targets ensures uniform scaling and promotes faster convergence during training. Since the target values y_i and predictions ${\hat{y}}_i$ are confined to [0, 1], an MSE value below 1 implies that the average prediction error is smaller than one normalized unit, which can be regarded as a baseline level of acceptable performance. Notably, the primary goal of model training remains the pursuit of minimizing the MSE as far as possible, rather than merely meeting the condition MSE < 1.

Afterward, test samples were employed to verify the accuracy of the heat flux correction. In Figure b, blue signifies the neglect of melt convection, red indicates actual experimental data, and green represents CGSim+MLM, showcasing a flattened solid–liquid interface. The close alignment between the red and green lines in Figure (right) demonstrates strong agreement between the two-dimensional CGSim+ML model and the experimental data.

Multiple “actual” crystallization fronts can be experimentally obtained or derived through rigorous calculations, including three-dimensional unsteady-state heat and mass transfer models based on complex turbulence theories and high-order numerical methods, as well as two-dimensional simulations. For each interface, a two-dimensional model must be constructed with its specific parameters (such as crystal height, growth rate, rotation speed, and melt depth), followed by prompt calculations. Convection in gas or melt was neglected in this process, and power correction was utilized to attain the target growth rate. A polynomial-approximated, fixed “actual” interface was adopted. The outcomes of these calculations were termed “heat flux corrections” in the context of the present scenario. In the LEC growth model, the geometric shape of the crystallization front was set to be a constant. Consequently, the radial heat flux distribution dictates the shape of the crystallization front. The complexity of turbulent heat transfer in the melt poses significant challenges in accurately calculating the heat flux distribution. Thus, the heat flux derived from the melt was adjusted by incorporating the data from three-dimensional simulations to enhance the accuracy of the crystallization front shape calculation. Initially, heat flux data were separately acquired by the basic CGSim and flow modules, and then the difference between them was computed to derive a heat flux correction factor. Subsequently, the correction factor was applied within the basic CGSim to achieve a more realistic representation of the interface shape.

The Flow Module significantly enhanced the accuracy of calculating the crystal growth front shape through the correction of the heat flux derived from the melt. By computing the discrepancy between the heat flux values from the basic CGSim and the Flow Module, a heat flux correction factor was derived and incorporated into the basic CGSim to simulate realistic interface shapes. Figure demonstrates a high degree of consistency between the distributions obtained from two-dimensional calculations and machine learning, suggesting the feasibility of machine learning in this context. In the case of a new scenario, power adjustment and interface correction should be enabled in two-dimensional calculations, with convection effects omitted. The convexity and concavity of the solid–liquid interface during crystal growth are defined by the parameter Δ in eq :

$$\mathrm{\Delta }=\frac{Z_c-Z_e}{d}$$ 4

where Z _c and Z _e denote the axial positions of the solid–liquid interface at the center and edge, respectively; d represents the crystal diameter. A positive value of Δ (Δ > 0) indicates a concave interface with respect to the crystal, whereas a negative value (Δ < 0) corresponds to a convex interface.

The following analysis focuses on the interface morphology during the steady-state growth phase because it is most critical for determining the overall crystal quality. Figure a displays a schematic diagram illustrating the method employed to measure the solid–liquid interface shape of the GaSb crystal. Figure b,c,d present morphology images of the crystal tail after solidification, following growth termination at 50% of the typical crystal length. These micrographs enable direct measurement and observation of the curvature (concavity/convexity) of the solid–liquid interface at this position. Figure demonstrates a case of the solid–liquid interface shape during GaSb crystal growth, with convex shapes and convexities of 0.4, 0.2, and 0.086°, respectively. These parameters were adopted as input for machine learning in the basic CGSim+MLM model. Notably, optimizing the interface shape also entails considerations such as heater power and insulation design. This represents an aspect of the current work that will be explored and reported in future studies.

8.

Shape of the solid–liquid interface in GaSb crystals. (a) Measurement method for convexity of solid–liquid interface shape, (b) Initial experimental solid–liquid interface shape, (c) Experimental results after optimization using CGSim simulation, (d) Experimental results after optimization using basic CGSim combined with MLM.

Figure displays the training outcomes utilizing the parameters outlined in Table . Specifically, it considers three crucible positions (CPs) of 15, 100, and 150 mm, and nine crucible rotations spanning 4, 6, 8, and 10 rpm, with all other parameters held constant. Regarding the generation of parameter sets (parameter pairs) presented in Table , the input parameter combinations were systematically generated by the Design of Experiments (DoE) methodology. More specifically, Latin Hypercube Sampling (LHS) was employed. LHS enables uniform sampling across the defined value ranges of process parameters, such as heater power, pulling rate, and rotation speed, with a limited number of cases. This approach ensures comprehensive coverage of the parameter space and representative inclusion of diverse process conditions in the training data.

9.

MLM trained on data from the standard thermal zone, used to predict the interface shape of the modified thermal zone. (a) Machine learning validation of different crucible rotation rates, (b) Machine learning validation of different crystal positions.

In Figure a, solid lines indicate accurate experimental data, and dashed lines represent simulation outcomes derived from basic CGSim+MLM. As observed from Figure a, variations in crucible rotation and the introduction of physical fields obviate the need for extensive simulation calculations. Machine learning facilitates approximate calculations of axial and radial temperature differences with a reduced sample size, enabling the procurement of precise interface shapes for subsequent defect simulation analyses. Figure b demonstrates the prediction of solid–liquid interface shapes with basic CGSim+MLM, an integrated approach that combines CGSim with MLM. This methodology harnesses the accuracy of computer simulations alongside the data-driven predictive capabilities of MLM. The MLM can discern relationships between interface shapes and various conditions by analyzing extensive experimental and simulation data, facilitating more accurate predictions of interface shapes across diverse scenarios.

3.2.2. Crystal Quality Optimization

The dislocation density of GaSb was dynamically assessed by both the basic CGSim model and the enhanced CGSim+MLM model. The parameters used for the calculations are listed in Table , including a fixed crystal diameter of 150 mm, a growth rate of 8 mm/h, a counterclockwise crystal rotation of 10 rpm, and a crucible rotation starting at 4 rpm, with varying control points (CPs) for ML training ranging from 15 to 150 mm. Significant differences in defect density were observed between the head and tail of the crystal when either the basic CGSim (Figure a) or the enhanced CGSim+MLM was used (Figure d). Under Sb-poor conditions, numerous inherent defects were observed, particularly antisite defects (Ga_Sb) and vacancy defects (V _Ga), which presented low defect formation energies and consequently elevated defect densities. After optimization with both the basic CGSim and enhanced CGSim+MLM models, the half-peak widths of the dislocation and rocking curves for the grown crystal are illustrated in Figure b,c,e,f, respectively. The dislocation density decreased from 1039 to 369 cm^–2, − while the half-peak width decreased from 29 to 28.5°.

4. Parameters for Dynamically Calculating Defects Using MLM.

parameter	crystal diameter	growth rate	crystal rotation speed	crucible rotation speed	start crystal position	stop crystal position
value	150 mm	8 mm/h	–10 rpm	4 rpm	15 mm	150 mm

10.

Quality of GaSb crystals. (a) Basic CGSim simulated defect density, (b) Basic CGSim optimized EPD wafer map, (c) Basic CGS optimized crystal rocking curve, (d) Basic CGSim + MLM simulated defect density, (e) Basic CGSim + MLM optimized EPD wafer map, (f) Basic CGSim + MLM optimized crystal rocking curve.

4. Conclusion

In this study, six-inch Te-doped GaSb single crystals were synthesized by the LEC method. The growth mechanism of the material was investigated by integrating CGsim simulation technology with machine learning algorithms. Interface geometry data were utilized to train the machine learning model (MLM), which generated thermal flux corrections subsequently applied as growth and thermal field parameters in further calculations. Additionally, the solid–liquid interface shape, dislocation density, and vacancy defect concentration in gallium antimonide crystals were calculated with CGsim+ML. Machine learning validation across various crucible rotation rates and crystal position (CP) configurations enabled the optimization of the solid–liquid interface shape in GaSb crystals, achieving a meniscus convexity of 0.086 degrees. This facilitated the swift assessment of experimental, simulation, and ML outcomes, as well as intelligent management of process reverse design. This method significantly decreased the likelihood of twins, undercooled grains, small-angle grain boundaries, and high dislocation densities, contributing to the consistent attainment of a target half-width of 28.5 arcseconds for the rocking curve of single crystals.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.5c08508.

• Experimental, device fabrication, characterizations, computational details, the specific neural network model, with a detailed description of its design to explain the variable number of neurons and the rationale for exploring multiple architectural variants, clarifies whether the “accurate data” were obtained from experiments or from fully detailed simulations, comparing the performance of basic CGSim vs CGSim+MLM across multiple growth runs, simulation process for LEC-GaSb single crystal growth, parameters for controlling convection in flow turbulent melt, input and output parameters of an artificial neural network, crystallization front correction using machine learning, exact data for MLM, computation performance comparison, physical output accuracy comparison (PDF)

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J.H. and B.X. contributed equally to this work.

The authors declare no competing financial interest.