* Corresponding author. Fax:#1-617-253-8549. E-mail address: rab@mit.edu (R.A. Brown). Journal of Crystal Growth 216 (2000) 192}203 Hybrid "nite-volume/"nite-element simulation of heat transfer and melt turbulence in Czochralski crystal growth of silicon Aleksey Lipchin, Robert A. Brown* Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Received 6 May 1999; accepted 20 March 2000 Communicated by J.J. Derby Abstract Simulation of heat transfer and oxygen incorporation during Czochralski growth of silicon in industrial scale equipment requires coupled simulation of conductive, convective and radial heat transfer throughout the system, with simulation of turbulent convection in the melt. We present a hybrid "nite-volume/"nite-element method for simulation of heat transfer throughout the Czochralski system and of turbulent convection in the melt. The hybrid method is based on "nite-element simulation of the thermal-capillary model of heat transport, and "nite-volume solution of the low- Reynolds turbulence model of Jones}Launder. Prototypical calculations demonstrate the robustness of the numerical methods and the in#uence of crystal and crucible rotation on heat transfer and the turbulent #ow.  2000 Elsevier Science B.V. All rights reserved. Keywords: Czochralski crystal growth; Turbulence modeling 1. Introduction Czochralski (CZ) crystal growth is the major source of silicon wafers for the microelectronics industry. Today boules of silicon are grown that yield 200 mm diameter wafers from 100 kg melt, with mass production of 300 and 400 mm wafers in development [1,2]. These large crystal diameters and large melt volumes make capital equipment and processing exceedingly expensive, and place new emphasis on quantitative understanding of large-scale CZ growth of silicon. This understand- ing is especially critical to the control of micro- defects in silicon wafers, which are critically in- #uenced by the temperature "eld in the crystal [3,4]. Numerical simulation of heat transport, convec- tion in the melt, and dopant transport can play an important role in this understanding. The goal of this paper is to present a novel hybrid "nite-ele- ment/"nite-volume method for simulation of trans- port processes in Czochralski crystal growth. The "nite-element/"nite-volume method merges numerical simulation methods that are proven to be robust for computing separately heat transfer and turbulent convection in CZ crystal growth. The "nite-element solution of the thermal-capillary model for heat transfer in CZ system has its roots in the methods developed by Derby and Brown [4,5] which accounted for conduction-dominated heat 0022-0248/00/$ - see front matter  2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 0 0 ) 0 0 4 2 8 - 0