Influence of furnace design on the thermal stress during directional solidification of multicrystalline silicon H.S. Fang a,n , S. Wang a , L. Zhou b , N.G. Zhou b , M.H. Lin b a School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan 430074, PR China b School of Photovoltaic Engineering, Nanchang University, Nanchang 330031, PR China article info Article history: Received 3 January 2012 Received in revised form 20 February 2012 Accepted 21 February 2012 Communicated by P. Rudolph Available online 3 March 2012 Keywords: A1. Defects A1. Stresses A2. Growth from melt B2. Semiconducting silicon B3. Solar cells abstract Directional solidification is one of the most popular techniques for massive production of multicrystalline silicon (mc-Si). Dislocation is one of the major defects that significantly affect the photovoltaic performance. For the analysis and optimization of stress-induced dislocation, a computational tool has been developed to investigate thermal stress distribution during directional solidification process of multicrystalline silicon. Temperature distribution in the furnace, S/L interface shape and melt flow are simulated. Parametric studies are further conducted to evaluate the effect of furnace design on the interface shape and on the maximum von Mises stress in the growing ingot. To consider the effects of the crucible geometry qualitatively, three-dimensional modeling of the thermal stress is performed with or without the constraint of the crucible. The regions of dislocation multiplication are evaluated by comparing von Mises stress to critical resolved shear stress (CRSS). The results imply that the dislocation in the growing ingot can be reduced by optimizing the design of the directional solidification furnace. & 2012 Elsevier B.V. All rights reserved. 1. Introduction World solar photovoltaic (PV) market installations have been dramatically grown in recent years, reaching 18.2 GW in 2010 with a growth of 139% over the previous year [1]. Multicrystalline silicon (mc-Si), as the most widely used material in solar cells, shares 45% of the market due to its well-balanced high conversion efficiency and low production cost. Multicrystalline silicon suffers many defects, such as dislocation, twins, grain boundaries and impurities that significantly limit its photovoltaic efficiency [2]. Reduction of the dislocation density, as a major silicon defect, is necessary to improve mc-Si quality and its performance. It is well known that the generation of dislocation is related to thermal stresses induced by the inhomogeneous temperature distribution. Traditional theories [3,4] studied the relation between dislocation density and thermal stresses, and indicated the regions with an intensive dislocation multiplication by comparing von Mises stress to critical resolved shear stress (CRSS). They also illustrated that for dislocation gen- eration in silicon the stresses along the {1 1 1} /11 ¯ 0S slip system are the most important. Miyazaki and Okuyama [5], Miyazaki et al. [6] and Kakimoto’s group [7] applied Hassen–Alexander–Sumino model to predict dislocation density in crystals, in which creep strain rate is related to dislocation density. Muiz ˇnieks et al. [8] proposed a concept of a metastable state and three levels of stress to explain the deviation of calculated dislocation density from industrial experience. Dislocation occurrence is strongly related also to other crystal defects, for example, grain boundary (GB). Takahashi et al. [9] investigated generation mechanism of disloca- tions during directional solidification of mc-Si using artificially designed seed, and found that dislocations occur at grain bound- aries and propagate as crystal growth proceeds. In the present paper, we develop a tool for the analysis of thermal-induced stresses during directional solidification of mc- Si, and provide a theoretical prediction of dislocation occurrence using the exceeding stress model. Melt flow, temperature dis- tribution in the furnace, and thermal stress distribution in the growing ingot during a steady growth stage are predicted. Para- metric studies are further conducted to evaluate the effects of furnace design on the interface shape and the maximum von Mises stress. To consider the effects of the ingot geometry qualitatively, three-dimensional modeling of thermal stress is performed after the S/L interface shape is determined by the two-dimensional model. The analysis makes a clear vision how furnace design affects thermal stress distribution, and provides a basis for the improvement of growth conditions to reduce dislocation generation during directional solidification process. 2. Problem description and mathematical model A typical directional solidification furnace is shown in Fig. 1. During growth, raw material of silicon is firstly loaded and melted in the crucible when the system temperature is higher than the Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/jcrysgro Journal of Crystal Growth 0022-0248/$ - see front matter & 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2012.02.032 n Corresponding author. Tel.: þ86 27 87542618. E-mail address: hafang@gmail.com (H.S. Fang). Journal of Crystal Growth 346 (2012) 5–11