Numerical modelling of instability and supercritical oscillatory states in a Czochralski model system of oxide melts

Abstract

The motivation for this study is the need for accurate numerical models of melt flow instabilities during Czochralski growth of oxides. Such instabilities can lead to undesirable spiralling shapes of the bulk crystals produced by the growing process. The oxide melts are characterized by Prandtl numbers in the range 5<Pr < 20, which makes the oxide melt flow qualitatively different from the intensively studied flows of semiconductors characterized by smaller Prandtl numbers Pr < 0.1. At the same time, these flows can be modelled experimentally by many transparent test fluids (e.g. water, silicon oils, salt melts), which have similar Prandtl numbers, but allow one to avoid the extremely high melting-point temperatures of the oxide materials. Most previous studies of melt instabilities for Prandtl numbers larger than unity suffer from a lack of accuracy that is caused by the use of coarse grids. Recent convergence studies made for a series of simplified problems and for a hydrodynamic model of Czochralski growth showed that for a second order finite volume method reliable stability results can be obtained on grids having at least 100 nodes in the shortest spatial direction. The obvious numerical difficulties call for an extensive benchmark exercise, which is proposed here on the basis of recently published experimental and numerical data, as well as some preliminary results of this study. The calculations presented are performed by two independent numerical approaches, which are based on second-order finite volume and finite element discretizations. We start our comparison from the steady states, whose parametric dependencies sometimes exhibit turning points and multiplicity. We then compare the critical temperature differences corresponding to the onset of instability, and finally compare calculated supercritical oscillatory states and phase plots.

On experimental and numerical prediction of instabilities in Czochralski melt flow configuration

“…This task is commonly recognized, however proposed benchmark tests related to crystal growth, e.g. [3][4][5], did not attract too much attention. The term ''validation'' of numerical codes relates to comparison of computational results with laboratory experiment.…”

Numerical Modelling of the Czochralski Growth of β-Ga2O3

“…In contrast to typical semi-conductor problems, the melt flow is time-dependent but far away from turbulence. Some researchers looked for the onset of the flow instability in oxide melts [9], but the realistic conditions are far beyond the ones considered there. For more realistic Grashof numbers, 3D computations have been performed but only for the melt applying simple boundary conditions at its side, bottom and top [10,11].…”

Measurement of physical properties of DyScO3 melt

“…rotation rate, temperature gradients, pulling rate) sometimes the crystal shows a spiral pattern and sometimes not. Apparently, there occur hydrodynamic instabilities which are responsible for the initiation of a spiral formation process generated by random disturbances [3][4][5]. Sometimes spiral growth requires that the growth process must be stopped, which dramatically reduces the yield.…”

“…The numerical analysis of the onset of 2D-symmetry breaking is not subject of the current work, but has been considered in [4,5]. Numerical models which describe fluid flow and heat transport require material properties (e.g.…”