The conventional Czochralski method is a batch process in which a single crystal is grown from the melt in a crucible, as shown in Figure 1.1. coordinates [9]. Silicon thermal conductivity. If this boundary con- dition was not satisfied, a new V was chosen and the process repeated until adequate closure on the boundary condition was obtained. Czochralski process vs Float Zone: two growth techniques for mono-crystalline silicon. In the Czochralski process, the liquid is drawn up, cools to the solidi cation temperature, and solidi es. Other articles where Czochralski method is discussed: integrated circuit: Making a base wafer: is now known as the Czochralski method. the solid and liquid states, cools by radiation. The model has been employed for deriving conditions under which the crystal radius during growth can be kept a constant. I n this method a single rotated crystal is Two parameters that influence the size of the crystal are the pull velocity and the base temperature of the crucible. Growth from the melt by the Czochralski method (CZ) is the most important process used for manufacturing high-quality Si bulk crystals. To create a single crystal of silicon by using the Czochralski method, electronic-grade silicon (refined to less than one part impurity in 100 billion) is heated to about 1,500 C (2,700 F) in a fused quartz crucible. Key words: calculus of variations, crystal, Czochralski technique, optimal control, von Mises stress 1 Introduction Czochralski (Cz) technique is one of the most common methods for growing single semiconductor crystals. Czochralski processes for individual applications dier in terms of details, but the central idea has remained unchanged. The Czochralski (Cz) method is the most important method for the production of bulk single crystals of a wide range of electronic and optical materials (Figure 2).At the beginning of the process, the feed material is put into a cylindrically shaped crucible and melted by resistance or radio-frequency heaters. Analysis of thermal phenomena in the crystal grown by the Czochralski process has been addressed. This low-order model is employed for the (2) and the boundary condition (4) was computed. As a result the governing equation is @T @t +r (~vT)= 1 c r Check the differences and the steps for perfect silicon wafers and ingots Home ; Lecture; Silicon is the most abundant solid element on earth, being second only to oxygen and it To analyze the temperature distribution and uid ow of the Czochralski process, there are (1) continuous equations tial differential equation systems with time-dependent spatial domains (Armaou and Christofides, 1999) based on a combination of Galerkins method with approximate inertial manifoldcs is used to construct a fourth-order model that describes the dominant thermal dynamics of the Czochralski process. Rea / Czochralski silicon pull rate limits then utilized to integrate eq. Then, a model reduction procedure for partial differential equation systems with timedependent spatial domains (Armaou and Christofides, 1999) based on a combination of Galerkin's method with approximate inertial manifolds is used to construct a fourthorder model that describes the dominant thermal dynamics of the Czochralski process. 270 S.N. CZ wafers contain a large amount of oxygen in the silicon wafer. 2.2 A Crystal Formation Model for the Czochralski Process The basic phenomena that need to be covered by a model for the Czochralski process are the capillary problem and the thermal conditions .