全球半导体晶体生长仿真著名商业软件FEMAG--Dynamic Simulation of the Entire Crystal Growth Process.docx

1全球半导体晶体生长仿真著名商业软件 FEMAGDynamic Simulation of the Entire Crystal Growth Process: Multi-Scale Analysis of Melt Flow TransientsV. Regnier, L. Wu, B. Delsaute,F. Bioul, N. Van den Bogaert, F. DupretCESAME, Université catholique de Louvain, E-mail: fd@mema.ucl.ac.beAbstractThis paper investigates the transient melt flow evolution during a complete Czochralski crystal growth process. Two basic time scales are considered. The short scale concerns the basic transients associated with flow oscillations at different process stages. Accurate understanding of the flow mechanisms at this scale is required to develop an average axisymmetric flow model for complete dynamic simulations. The long time scale is associated with the transients caused by the slower system evolution occurring during the complete growth process. In order to focus on the fundamental effects governing the flow, a model problem is considered where the liquid is placed into a possibly rotating container while a disk of smaller diameter rotates on its top surface. Both the container and the disk are isothermal. Several transient effects are investigated including the effect of disk radius increase or decrease, and abrupt changes of disk or container temperature or rotation rate.Introduction: dynamic modeling of crystal growth by means of the FEMAG softwareThere is increasing demand today for robust, reliable and user-friendly software to model bulk growth techniques such as the Czochralski (Cz), Liquid Encapsulated Czochralski (LEC), Floating Zone (FZ) and Vertical Bridgman (VB) processes. The aim is to help predict, design and control the growth processes, and to better understand the factors affecting crystal quality. However, the growth techniques are more and more complex, and optimization can be achieved only by use of suitable numerical modeling that accounts for the severely non-linear physical phenomena involved as well as for the high system thermal inertia. The 2resulting problem is coupled, global, nonlinear and dynamic. On the other hand, accurate prediction of crystal quality requires both appropriate modeling of the governing physics, and highly accurate dynamic numerical methods for computing the evolution of the solid-liquid interface shape and the temperature field gradient in its vicinity.The FEMAG simulation software developed in the CESAME center of the University of Louvain is currently used by major crystal growth companies. The numerical model is both global and dynamic, and takes the effect of melt convection into account. Diffuse surface radiation is considered. Geometrical unknowns are dynamically coupled to the other unknowns, i.e. temperature field, velocity field, electrical potential, etc., leading to a complex non-linear system of equations whose solution is found by use of a decoupled scheme at every time step of the simulation. Whereas in its first generation FEMAG already performed global quasi-steady or time-dependent simulations, applications were restricted to top cone, shouldering and body growth stages.Both laminar and non-laminar flow models were considered, including or not the effect of axisymmetric magnetic fields. The objective of launching the FEMAG-2 software generation has been to provide a fully automatic simulator predicting the entire growth process while handling correctly the switches between the growth stages, together with coupling dynamic calculations with accurate melt flow prediction.A significant difficulty lay in the important evolution of the system geometry during a complete growth process. Indeed, the solidified region is very small during seeding and subsequently becomes larger and larger, while the volume of the molten region decreases continually and can take a complex shape during tail-end stage. The solution adopted combines several approaches based on a representation of the furnace by means of deforming unstructured meshes together with automatic mesh generation. New geometrical methods were designed to allow easy calculation of the different system free surfaces (solidification front, melt/gas interface including crystal/melt and crucible/melt menisci, and crystal/gas surface). These methods allow performing easy time-dependent simulations even for stages of the process where important geometrical changes occur.Another important difficulty to address in FEMAG-2 development was related to the complexity of dynamic melt flow modeling. Several problems must be solved to accurately couple melt flow predictions with crystal growth process simulation. First, in semi-conductor growth, the melt flow is time-dependent, 3D and weakly turbulent, whereas it can exhibit 3D azimuthal and temporal structured 3oscillations. The use of an axisymmetric quasi-steady flow model is devoted to average the effect of these oscillations, and the principal issue is to determine reliable average flow models, with the corresponding boundary conditions, above the steady laminar regime. Seco。