Survey of Monte Carlo simulations of crystal surfaces and crystal growth

During the Summerschool in Erice one of us (P.B.) presented a series of five lectures in which a survey of crystal growth theories and a comparison with experiments was given.In this chapter we only will give an updated survey on computer simulations, which are relevant for crystal growth since surveys on the other subjects are published elsewhere.Thanks to an exchange program between the Soviet Union and the Netherlands Mrs. Dr. T.A. Cherepanova spent a year in Delft, which enabled us to present this joint survey paper.     

About the effective temperature of crystal surfaces

Temperature dependences of self-diffusion coefficients, scattered electron intensities, lattice parameters and atomic bond energies are considered. All curves representing the dependences are “splitted” into two linear ones: for a surface and for a volume of the crystal. This splitting can be eliminated and the lines representing the two linear dependences be matched, if a new variable T_c = nT (where n is a scale factor and T_e is an effective temperature) is introduced. The value of n is shown to vary over not a wide range (from 1.6 to 8) for all the processes considered and in all cases n > 1. A possible physical sense of the effective temperature has been considered.     

Computer simulation of crystal growth

Monte Carlo calculations of crystal growth rates are presented for a number of different situations, including two-dimensional nucleation with mobile adatoms, spiral growth, growth and evaporation of a crystal containing a small columnar hole, growth with impurities, and growth on three crystallographic faces of the face-centered cubic crystal.     

In-situ observation of crystal surfaces by optical microscopy

In this experimental course, attendees will learn how to obtain useful information about growth processes of crystals using ordinary optical microscopes, which are usually available in laboratories. We will demonstrate how thicknesses of crystals can be estimated from interference colors. We will also show in-situ observations of spiral steps and strain distributions by differential interference contrast microscopy and polarizing microscopy, respectively.     

Cellular-automata-based simulation of anisotropic crystal growth

Extending the simulation of anisotropic etching, a cellular-automata-based simulator is applied to anisotropic crystal growth. This simulator takes advantage of the equivalence between dissolution and growth of crystals. Metalorganic vapour-phase epitaxial growth experiments were performed on patterned (1 0 0)-oriented InP substrates with very deep V-shaped grooves with {1 1 1}A sidewalls to determine the relevant growth rates of InGaAs and InP. The capability of the simulation method is demonstrated by quantitative comparison of simulated and experimental results. In addition, the versatility of the model is shown with area-selective growth.     

Multiatomic step formation on GaAs(001) vicinal surfaces during thermal treatment

We observed the surface morphology of vicinal GaAs(001) after thermal treatment in AsH₃/H₂ atmosphere by atomic force microscopy (AFM). Clear multiatomic steps were formed under the high temperature thermal treatment. Next, we investigated the mechanism of step bunching during thermal treatment by two experiments from the view point of Ga atom evaporation. One is the selective thermal treatment using a partially masked GaAs wafer, and the evaporation amount of Ga atoms was estimated by AFM. The other is the investigation of photoluminescence (PL) peak energy shifts for AlGaAs/GaAs single quantum wells with a thermal treatment process at the top of the GaAs quantum well layer, compared to those without thermal treatment. These results indicate that the evaporation hardly occurs during the thermal treatment process. Therefore, step bunching phenomena on GaAs(001) vicinal surfaces during thermal treatment are probably caused by migration of the atoms detached from upside steps and their re-incorporation to downside steps.     

Time-dependent simulation of Czochralski silicon crystal growth

We have developed a detailed mathematical model and numerical simulation tools based on the streamline upwind/Petrov-Galerkin (SUPG) finite element formulation for the Czochralski silicon crystal growth. In this paper we consider the mathematical modeling and numerical simulation of the time-dependent melt flow and temperature field in a rotationally symmetric crystal growth environment. Heat inside the Czochralski furnace is transferred by conduction, convection and radiation, Radiating surfaces are assumed to be opaque, diffuse and gray. Hence the radiative heat exchange can be modeled with a non-local boundary condition on the radiating part of the surface. The position of the crystal-melt interface is solved by the enthalpy method. The melt flow is assumed to be laminar and governed by the cylindrically symmetric and incompressible Navier-Stokes equations coupled with the calculation of temperature.     

Mathematical model and numerical simulation of faceted crystal growth

A new mathematical macroscopic model is proposed to describe the nonstationary process of faceted crystal growth by the methods of directional crystallization with a slow change in external thermal conditions and low pulling rate of a cell through the growth system. The facet-growth rate is determined by the Stefan condition, integral over the face. Two boundary conditions are set for temperature: the continuity condition and the relation between the heat-flux jump and the supercooling at the facet points. The supercooling is determined by solving the entire heat problem. A facet is selected as a planar part of the phase boundary. The kinetic coefficient at the facet may depend on the supercooling. The energy conservation law is valid within the model developed. Examples of calculations of some model problems are presented.     

Structural changes of NaCl and Si single crystal surfaces under ion bombardment

The surface structural changes under ion bombardment of NaCl and Si monocrystalline specimens are followed using electron microscopic technique. Formations and regions with circular symmetry are registered on the surface of NaCl and Si single crystals. Qualitative interpretation of the generation mechanism of the circular formations is made. The ion bombardment is used for the visualization of the surface structure of the specimens by the method of crystal selfdecoration.     

Three-dimensional simulation of floating-zone crystal growth of oxide crystals

The floating-zone (FZ) method is a popular technique for the growth of high-temperature oxide crystals. However, the growth usually requires skillful control of the zone stability, which is strongly coupled with heat flow, interfaces, and the grown crystal morphology. In this report, we present a three-dimensional self-consistent simulation of floating-zone oxide growth in a mirror furnace by taking these factors into account simultaneously. This model is based on an efficient finite volume method with multigrid acceleration and interface tracking. The steady growth of a YAG crystal in a double-ellipsoid mirror furnace is taken as an example to illustrate the intricate coupling of convection, interfaces, meniscus, and the grown crystal shape.     

Dependence of the step velocity on its length on the (010) face of the orthorhombic lysozyme crystal

The length of a two-dimensional critical nucleus has been measured, and the Gibbs-Thomson formula has been experimentally verified on orthorhombic lysozyme crystals. The step velocity was found to be independent of its length. The critical length initiating the step motion can be determined by a low density of kinks on a step, and not the critical-nucleus size. The invalidity of the Gibbs-Thomson formula in this case is discussed.     

Birefringence simulation of annealed ingot of calcium fluoride single crystal

We developed a method for simulating birefringence of an annealed ingot of calcium fluoride single crystal caused by the residual stress after annealing process. The method comprises the heat conduction analysis that provides the temperature distribution during the ingot annealing, the elastic thermal stress analysis using the assumption of the stress-free temperature that provides the residual stress after annealing, and the birefringence analysis of an annealed ingot induced by the residual stress. The finite element method was applied to the heat conduction analysis and the elastic thermal stress analysis. In these analyses, the temperature dependence of material properties and the crystal anisotropy were taken into account. In the birefringence analysis, the photoelastic effect gives the change of refractive indices, from which the optical path difference in the annealed ingot is calculated by the Jones calculus. The relation between the Jones calculus and the approximate method using the stress components averaged along the optical path is discussed theoretically. It is found that the result of the approximate method agrees very well with that of the Jones calculus in birefringence analysis. The distribution pattern of the optical path difference in the annealed ingot obtained from the present birefringence calculation methods agrees reasonably well with that of the experiment. The calculated values also agree reasonably well with those of the experiment, when a stress-free temperature is adequately selected.     

Birefringence simulation of annealed ingot of magnesium fluoride single crystal

We developed a method for simulating birefringence of an annealed ingot of MgF₂ single crystal for lithography optics. This method provides the optical path difference caused by crystal symmetry and residual stress existing in the crystal. The method consists of the heat conduction analysis, the residual stress analysis and the birefringence analysis. Because there exists no experimental data on the inelastic behavior of MgF₂ single crystal, the residual stress was estimated with the elastic thermal stress analysis using the finite element method by assuming a stress-free temperature. In this analysis, the temperature dependence of material properties and crystal anisotropy were taken into account. In the birefringence analysis, the distributions of optical path difference were calculated by an approximate method using the result of the residual stress analysis. This approximate method uses the average stress along the wave normal and is equivalent to the exact method in case of low stress dealt with the present study. In this analysis, it is possible to consider both the intrinsic birefringence and the stress birefringence in any crystal orientation. The distribution of the optical path difference in the annealed ingot obtained from the present calculation agrees qualitatively with that of the experiment. Its calculated value also agrees reasonably well with that of the experiment, when a stress-free temperature is adequately selected.     

Large eddy simulation of Marangoni convection in Czochralski crystal growth

Large eddy simulation model is used to simulate the fluid flow and heat transfer in an industrial Czochralski crystal growth system. The influence of Marangoni convection on the growth process is discussed. The simulation results agree well with experiment, which indicates that large eddy simulation is capable of capturing the temperature fluctuations in the melt. As the Marangoni number increases, the radial velocity along the free surface is strengthened, which makes the flow pattern shift from circumferential to spiral. At the same time, the surface tension reinforces the natural convection and forces the isotherms to curve downwards. It can also be seen from the simulation that a secondary vortex and the Ekman layer are generated. All these physical phenomena induced by Marangoni convection have great impacts on the shape of the growth interface and thus the quality of the crystal. (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical simulation of MHD rotator action on hydrodynamics and heat transfer in single crystal growth processes

The article presents the results of the mathematical and physical simulations of the influence of a rotating magnetic field (RMF) on the hydrodynamics and heat transfer in processes of large semiconductor single crystal growth in ampoules. Different versions of the RMF are considered, in particular, for symmetric and asymmetric positions of a RMF inductor with regard to the melt in the ampoule, for two counter-rotating magnetic fields, for different geometrical ratios in the “RMF inductor - liquid melt” system, and for different electrical conductivities of the hard walls at their contact with the melt. The interconnection between the distribution of the electromagnetic forces in the liquid volume and the formed velocity patterns, temperature distribution and shape of the solidification front is studied. An original method for the definition of the electromagnetic forces, which considers finite dimensions of the RMF inductor and melt, was used to calculate real conditions of the RMF influence on growth processes. The numerical results obtained are compared to the data of model experiments. Their satisfactory agreement permits us to propose this calculation method for the definition of the optimal parameters of a growth process under specific conditions and to select the most rational type of RMF influence.     

Numerical simulation of thermal and mass transport during Czochralski crystal growth of sapphire

The thermal and flow transport in an inductively heated Czochralski crystal growth furnace during a crystal growth process is investigated numerically. The temperature and flow fields inside the furnace, coupled with the heat generation in the iridium crucible induced by the electromagnetic field generated by the RF coil, are computed. The results indicate that for an RF coil fixed in position during the growth process, although the maximum value of the magnetic, temperature and velocity fields decrease, the convexity of the crystal‐melt interface increases for longer crystal growth lengths. The convexity of the crystal‐melt interface and the power consumption can be reduced by adjusting the relative position between the crucible and the induction coil during growth. (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)