Computer Simulation and Controlled Growth of Large Diameter Czochralski Silicon Crystals

Abstract

Computer simulation leading to controlled large diameter Czochralski crystal growth is discussed. A simple mathematical model, which describes the different crystal growth phases including neck-in, fast flat top, r011-over to constant diameter bulk growth, and tail-off is presented. This model, in conjunction with a computer-implemented simulator, is used to simulate silicon crystal growth. Good agreement between siniulation results and experimental crystal growth is obtained.Large diameter silicon single crystals free of dislocations are routinely grown from the melt by using the Czochralski (CZ) technique. This paper presents a simple mathematical model which describes the behavior of the crystal radius during all phases of 125 m m CZ silicon crystal growth. This model is the basis for a computer simulation of the crystal growth process. The simulation includes the initial neck-in growth after seeding, which represents a standard procedure for elimination of dislocations; then fast flat growth, also called shoulder growth, roll-over to constant diameter bulk growth, and finally, the tailoff procedure to terminate crystal growth. The simulation results are compared to actual growth of 125 m m diam silicon crystals. This modeling effort is an integral part of a larger program directed toward complete interactive digital computer control of the CZ crystal growth process. Modeling of 125 mm Diameter Silicon Crystal GrowthFluctuations of crystal diameter during growth are caused by variations both in crystal pull rate and thermal conditions. In principle, it should be possible to .obtain insight into a cause and effect relationship of crystal diameter variations during growth through thermal analysis. To the best of our knowledge, such an analysis has not yet been performed. Thermal analyses of the CZ crystal growth process have been made by several investigators. However, the efforts reported are concerned with other aspects of crystal growth, such as temperature distributions in the crystal a n d / o r at the solid-liquid interface (1-7), and crystal pull rate limits (8,9). This paper provides insight into crystal diameter variations during crystal growth, through development of a mathematical model in conjunction with a computer simulation comprising all phases of the growth process. This insight is gained through a comparison between simulated and actual crystal growth data. The simulation i:s based on the model described in the following sections.Mathematical model.--The sequence of large diameter CZ silicon crystal growth is schematically illustrated in Fig. 1. The radius (R) of the growing crystal is a function of the crystal pull rate (V) and of the temperature (T) of the system. T is defined here as the difference between the crucible wall temperature and the melting point (I412~ of silicon. Experimentally, it is found that R is inversely proportional to V and T. The behavior of R prevails during neck-in, constant diameter bulk growth, a n d tailoff. During fast fiat top growth, R is assumed to increase exp...

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

“…This relationship implies that the pulling rate is inversely proportional to the crystal radius (R) and directly proportional to the AT (=To -Tm). This is similar to the experimental results of Kim et al (5). Further, when AT = 0, the melt is maintained at the uniform temperature of its melting point, and, under these conditions, the maximum pulling rate is obtained…”

“…[3a] provides a more suitable model. It may also be noted that model [3a] provides the same dependency on radius as that of Kim et al (5), although the actual model equations are different (compare Eq. [1] and [3a]).…”

“…Hence, there is considerable incentive for developing simple model representation to the system. Kim et al (5) proposed a simple mathematical model that describes the different crystal growth phases, including neck-in, shoulder growth, and constant diameter bulk growth. This model basically correlates the crystal diameter with the pulling rate and the crucible temperature using two parameters.…”

Automation of Crystal Growth from Melt

“…Detailed discussions on these subjects can be found in Refs [17,22,50,51]. This means that it does not have a steady state operating point, mainly resulting from the fact that the whole system configuration changes during the growth-while one has a crucible filled with melt in the beginning, it is nearly empty at the end of the process and the melt mass is transferred into the crystal.…”

Numerical simulations of bulk crystal growth on different scales: silicon and GeSi

“…For both applications there is a continuing demand for cost reduction while keeping or even increasing the quality of the wafers cut from the crystals grown. Starting in the 1980s, numerical methods have been developed in order to understand and improve the industrial growth of silicon single crystals (first computations were performed at IBM laboratories 5, 6 and in Japan 7, and heat transfer including radiation has been considered 8). About the same time the first molecular dynamics (MD) simulations of the melting and solidification processes of silicon were performed 9.…”