Thermal-capillary analysis of Czochralski and liquid encapsulated Czochralski crystal growth: I. Simulation

The bulk single crystab of semiconductors (e.g. Si, GaAs) and oxides which are at present commercially produced have mostly non-uniform properties in the microscale (e.g. doping striations) and in the macroscale (longitudinal and lateral segregation). Such inhomogeneities are deleterious for the performance of the devices produced from these crystals. This book gives a review of the various origins of inhomogeneities occuring during crystal growth from the melt. It is shown that convection is the major source of the non-uniformities in the technically used growth configurations, e.g. Czochralski-, zone- and Bridgman-methods, because the growth rate is controlled by the heat transport. The formalism of hydrodynamics, especially dimensionless numbers, is used for a modeling of melt growth, giving a correlation between the occurrence of inhomogeneities and relevant growth parameters.

Convection and Inhomogeneities in Crystal Growth from the Melt

An integrated process model for the growth of oxide crystals by the Czochralski method

A magnetic field suppresses turbulence and thermal convection in a Czochralski crystal puller The amounts and distributions of dopants and oxygen in the crystal are determined by the motion of the molten silicon during crystal growth This paper presents analytical solutions for the melt motion in a Czochralski puller with a strong, uniform, axial magnetic field The relatively small electrical conductivity of the crystal plays a key role in determining the flow Certain combinations of crystal and crucible rotation rates lead to flow patterns with a large volume of almost stagnant fluid under most of the crystal face The values of these rotation rates depend on the magnetic field strength

Melt motion in a Czochralski crystal puller with an axial magnetic field: isothermal motion

A finite element algorithm is presented for simultaneous calculation of the steady state, axisymmetric flows and the crystal, melt/crystal and melt/ambient interface shapes in the Czochralski technique for crystal growth from the melt. The analysis is based on mixed Lagrangian finite element approximations to the velocity, temperature and pressure fields and isoparametric approximations to the interface shape. Galerkin's method is used to reduce the problem to a non-linear algebraic set, which is solved by Newton's method. Sample solutions are reported for the thermophysical properties appropriate for silicon, a low-Prandtl-number semiconductor, and for GGG, a high–Prandtl–number oxide material. The algorithm is capable of computing solutions for both materials at realistic values of the Grashof number, and the calculations are convergent with mesh refinement. Flow transitions and interface shapes are calculated as a function of increasing flow intensity and compared for the two material systems. The flow pattern near the melt/gas/crystal tri-junction has the asymptotic form predicted by an inertialess analysis assuming the meniscus and solidification interfaces are fixed.

W. Zulehner

Papers

Czochralski growth of silicon