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Next Generation Multiphysical Models for Crystal Growth Processes

Periodic Reporting for period 2 - NEMOCRYS (Next Generation Multiphysical Models for Crystal Growth Processes)

Reporting period: 2021-08-01 to 2023-01-31

The NEMOCRYS project addresses the production process of crystalline materials used in microelectronics, telecommunication and many other fields of modern technology. These processes take place in crystal growth furnaces with high requirements for process control and efficiency. We are developing a new generation of models for crystal growth processes to raise the physical understanding and possibilities for optimization to a qualitatively new level. In the first step, dedicated MODEL EXPERIMENTS are set up with the aim to investigate the heat transfer, fluid flows and other physical aspects of crystal growth processes in a dedicated environment that is easier to access for in-situ measurements than most crystal growth furnaces. In these experiments, model materials with low melting points are applied. In the next step, new NUMERICAL MODELS are developed and validated in terms of the underlying physical assumptions and approximations using the experimental results from the first step. Finally, the improved physical understanding from model experiments and the newly validated numerical models are applied to crystal growth processes of technologically relevant materials such as silicon and gallium oxide. This allows us to optimize the energy efficiency and the yield of the growth processes as well as improve the quality of the produced crystalline material and the resulting devices.
The development of a new EXPERIMENTAL PLATFORM for multiphysical model experiments has been realized in three stages: (1) desktop demonstration; (2) re-purposed test furnace; (3) novel and flexible furnace design - the MultiValidator. This stepwise approach reduces the technological risks and allows us to reach the scientific goals faster. Currently, we are actively working in the second stage for the CZOCHRALSKI growth process, which is the most popular technique for crystal growth from melt both in research and industry. The developed demonstration experiment is a low-cost (under 1000€) setup including full automation with a microcontroller/single-board computer and various sensors for thermal and electromagnetic measurements. This setup has been applied to grow crystals of model materials such as tin under ambient air and temperatures up to 350 °C. The impact of various growth conditions has been investigated both in scientific studies and as training for students. The test furnace adds the possibility of vacuum or inert gas atmosphere but also enables more realistic process geometries. Two cases with induction and resistance heating have been implemented and compared. The test furnace is equipped with comprehensive in-situ measurements. These currently include thermocouples, resistance thermometers, pyrometers, heat flux sensors, infrared and optical cameras as well as sensors for the heater current, voltage, and magnetic field. Recently, a laser-based setup for two-dimensional flow measurements in transparent melts and gases as well as an ultrasound-based setup for flow measurements in opaque melts were acquired and are being adapted for use in crystal growth experiments. In this way, we make the furnace "transparent" for observations of macroscopic physical phenomena during the growth process. The FLOATING ZONE growth process is being applied for crucible-free growth of industrial high-purity silicon with inductive heating and of oxides with optical heating. Currently, we are working on a desktop demonstration (first stage) of the inductive process in air and at low temperatures. The literature on similar experiments is very scarce and several challenges regarding the inductor design and material properties such as surface tension need to be addressed.

For the development of new NUMERICAL MODELS, the Finite Element software Elmer has been mainly applied so far, in particular for thermal and electromagnetic phenomena. While Elmer is a generally ready-to-use open source code with wide multiphysical capabilities, the setup of simulations becomes increasingly difficult for complex geometries and large-scale parametric studies. Therefore, a new Python-based interface pyelmer has been developed and published under an open-source license. pyelmer facilitates the automation of the pre-processing and post-processing phases of simulations, hence saving time and reducing errors in model setup and implementation. As an alternative simulation program, FEniCS has been evaluated and tested. FEniCS offers more flexibility for coupling and modification of the components of multiphysical models, and a steady-state thermal model has been successfully implemented. However, a further extension to an unsteady model with variable crystal diameter would require significant programming resources. Therefore, the use of the readily available models in Elmer is currently preferred. The OpenFOAM software is based on the Finite Volume Method and is applied for modeling the melt and gas flows. Furthermore, models for a simplified description of high-frequency induction heating as well as for free surface shapes are being developed for the floating zone process. The selected simulation programs together with tools for pre-processing (e.g. Gmsh for grid generation) and post-processing (e.g. ParaView for visualization) are integrated into a Python-based SOFTWARE PLATFORM for open-source crystal growth simulation - OpenCGS.

One of the main overarching goals of the project is the VALIDATION of numerical models. To that end, we have developed a new methodology consisting of three steps: (1) sensitivity analysis to identify the most relevant model parameters; (2) parameter adjustment using in-situ measurements during a growth process or using dedicated experimental setups; (3) global accuracy estimation by comparing simulations with growth experiments. Recently, we have applied this approach to thermal modeling of the Czochralski process and developed routines of parameter adjustment for convective heat transfer in the gas and in the melt in particular. This allowed us to reach a new level of accuracy for such practically relevant quantities as global power balance and crystal diameter.
The second stage of the experimental platform for multiphysical model experiments has already achieved an unprecedented in-situ insight into the physics of crystal growth processes (on a macroscopic scale). The MultiValidator as the third stage will enable model experiments closer to technologically relevant growth processes and also extend the scope of physics toward fluid flow phenomena and solid stresses. Furthermore, we will address other crystal growth processes such as FLOATING ZONE growth with inductive or optical heating. In the end, we expect to reach a new level of physical understanding of the crystal growth process, which can be generalized to various materials and growth techniques.

The development of numerical models was preceded by a careful analysis of the current state of the art in the relevant literature to obtain an overview of the open questions and unresolved challenges. We first addressed the effects of convective heat transfer which have historically been neglected or simplified without adequate justification in most cases. We are now working on the analysis of heat generation in crystal growth processes to validate models for complex three-dimensional shapes of inductive and resistive heaters. Finally, after analyzing many other open modeling questions, we expect to develop a "recipe book" of high practical relevance and large physical and technological scope leading to a new generation of multiphysical models for crystal growth processes. All these models, along with the validation experiments, will be made available under open source licenses and documented in open access publications. Our goal is to provide an alternative to the commonly-used commercial multi-physics packages (which require significant adaptations for crystal growth simulations) and to the very few specialized commercial tools for crystal growth.
Crystal growth setups for the validation of multiphysical models