Calculation method of convective heat mass transfer in growing crystals from the melt

V.P.Ginkin, O.M.Naumenko

State Scientific Center of the Russian Federation

Institute for Physics and Power Engineering named after A.I.Leipunsky

Bondarenko sq., 1, Obninsk, Kaluga Region, 249033, Russia

 

The three-dimensional non-steady equations convective heat-mass transfer in Boussinesq approximation are considered for the description of crystal growth process from a melt Stefan problem in natural variables considering the impurity segregation at the interface is solved. The energy equation is solved in enthalpy variables. For the solution of hydrodynamic equations the finite volumes method with fixed diverse grids, implicit stabilization method, exponential transformation method and the conjugate gradient method with preconditioning by the incomplete factorization method are used. The method realized on a problem of calculation of heat mass transfer process at semiconductor crystal growth from a melt by Bridgeman method. Two models used for the solution of this problem are compared. The first model uses a hypothesis about anomalous increasing of melt viscosity near the front of crystallization. This model was used for an explanation of effects of abnormal impurity distribution in experiments on crystallization of Ge doped by Ga in space This model results in large shift strains near the melt-crystal interface. The second model is based on the cluster approach simulation of transitional region in the melt near the front of crystallization. The clusters are considered as motionless firm fractions of crystallizing material near the interface. The melt flow in transitional region is simulated by porous medium approach The flow resisting force proportional to the porosity coefficient that is equal to the ratio of liquid and solid state fractions is introduced into a motion equation. The value of this force can be estimated for semiconductor melt on agreement of the calculation results of impurity distribution in the growing crystal with using two considered models. The numerical calculations results on a crystallization of gallium antimonide in space by Bridgeman method on two models are represented, and the estimation of resistance force value in the second model from comparing of calculations results on both models is given. So:

The model presented in the paper allows the description of melt crystallization process in view of moving interface, convection heat and mass transfer and impurity segregation.

For the first time the given hydrodynamical model considers the structural pre-crystallization state of the boundary layer in the interface region. The transient region is for the first time described as a two-phase medium which together with the melt contains the three-dimensional solid phase clusters. The specific feature of the model consists in the description of clusterization processes and those of substance crystallization in enthalpy variables, thus giving the possibility to compare the results with the data obtained with other independent approaches, i.e. physical chemistry, molecular dynamics, etc. So there is a potential for further development and improvement of model representations. The highest effect from the use of the given model can be expected when the crystallization process is described in the conditions of decreased gravitation. That Is the case when the boundary segregation phenomena near the crystallization front the most prominently affect the quality of the crystal under growth.