Czochralski Growth of Crystals: Simple Models for Growth Rate and Interface Shape

Abstract

A simple model for the crystal growth by the Czochralski (CZ) process has been proposed based on semiquantitative arguments. The model provides empirical relationships for the dependence of the pulling rate and the interface shape on the important process variables such as crystal radius, crucible temperature, height of the melt level, and the height of the exposed portion of the crucible wall. The parameters of the model can be evaluated by matching the results obtained from a detailed mathematical model of the CZ process or from extensive experimental data. The model has, therefore, the potential application for determining the best process conditions and for on‐line control and optimization of the crystal puller to grow crystals with constant diameter and nearly planar interface.

Modeling the Effect of Operating Parameters on Oxygen Content in Czochralski Growth of Silicon

“…The value of 0(0)'0 ranges from 0.0571 to 0.1627 for Pr of 0.01 to 0.025 (11). Equation [16] can then be expressed in terms of the operating parameters as To -T~ = 2.37 • 10-~Q~ 2 [17] Equation [15] for the mass transfer coefficient can then be simplified in SI units as kcw = 5.8 • 10 6Q~176 [18] m Thus the key operating parameters affecting the wall dissolution are the wall heat flux and the melt height.…”

“…These thermal models are based on the assumption that conduction is the dominant mechanism for heat transfer in the melt. Based on the calculated results of these detailed models, Srivastava et al (17) proposed simple, semi-empirical equations for the variation of the growth rate and the crucible temperature. The equation for the growth rate has the form…”

M. P. Duduković: A Pioneer in Multiphase Reaction Engineering

“…From the early 1980s to the early 1990s, Mike's work in multiphase catalytic and noncatalytic systems was coupled with the development of reaction engineering methodologies for new environmentally benign processes and for the manufacture of advanced materials, such as semiconductor-grade silicon, − composites, − and iron particles . In the late 1980s, the CREL group pioneered a prototype system for noninvasive flow mapping of gas−liquid bubble columns using Computer Automated Radioactive Particle Tracking (CARPT) with γ-rays. , The CARPT experimental platform was expanded in the mid-1990s to include a Computer Assisted Tomography (CAT) system for measurement of the local voidage variations in bubble columns. , These two experimental innovations provided the basis for new insights into multiphase flows that were applied to various multiphase reactor types over the past decade, such as bubble columns, liquid−solid risers, gas−solid risers, ebullated beds, stirred tanks, and reactors containing structured packing .…”