Identification of Crystal Growth dynamics in Czochralski Systems

Abstract

An experimental method is established to deduce the transfer function relating crystal radius to power changes in a Czochralski crystal growth system, The experimental method employs pseudo-random signal injection into the power controller of a gallium phosphide crystal puller and uses crystal radius measurements made on the cold crystal after growth is complete. This is the first known application of pseudo-rendom signal testing to crystal growth. Novel data processing techniques ere developed to convert the crystal radius measurements to samples at equal time intervals. Fast Fourier transform processing is used to compute frequency responses for the process. An algorithm is developed to fit transfer functions to modulus frequency response data. A theoretical derivation of the transfer function is performed. This is based on the heat balance equation at the solid-liquid interface. Due to lack of data on gallium phosphide, the error bounds of the prediction are wide but it is shown that they may be reduced by using the measured transfer function. The work has application to the design of control systems for crystal growth and to a more fundamental understanding of the crystal growth process. This is the first know description of a theoretical analysis of Czochralski crystal growth dynamics.