Capillary shaping of crystals pulled downward from a melt by the crucibleless AHP method

A model is proposed for capillary shaping of a crystal in the crucibleless variant of the AHP (axial heat flux close to the phase interface) method when the melt in the form of a film flowing over the AHP heater is fed to a meniscus. The meniscus and the film of the melt are described by the same equation with a discontinuous right-hand side. The dependences of the crystal radius and the thickness of the melt film on the parameters of the process are numerically investigated, and the capillary stability of the pulling process is analyzed. It is demonstrated that, in this method, the thickness of the melt layer between the crystal and the heater can be considerably larger than the capillary constant.     

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.     

Thermal simulation of the Czochralski silicon growth process by three different models and comparison with experimental results

Temperatures were measured within an industrial Czochralski silicon puller and compared with simulation results. The temperatures were measured by thermocouples in the crystal along the axis as well as inside the lateral and bottom insulations. The temperature distribution of the furnace was computed using three different software codes. It could be demonstrated that today's simulation methods are capable of handling such a complex heat transfer simulation task as that encountered in the case of Czochralski silicon growth furnaces, with the exception of the melt convection problem, which has not yet been satisfactorily solved.     

Influence of boron concentration on the oxidation-induced stacking fault ring in Czochralski silicon crystals

The influence of the boron doping level in the range of 1 × 10¹⁵-2 × 10¹⁹ cm⁻³ on the position of the oxidation-induced stacking fault ring (R-OSF) in silicon crystals has been investigated by experiments and numerical simulation. For low boron-doped crystals, the position of the R-OSF is described by a critical value C_crit defined by the ratio of the pull rate and the temperature gradient in the crystal at the solid/liquid interface. Boron concentrations higher than 10¹⁷ cm⁻³ shift the position of the R-OSF towards the wafer center without change of growth parameters. The critical value C_crit converts into a function C_crit(C_B, depending linearly on the boron concentration C_B. Crystal-originated particles (COP) and gate oxide integrity (GOI) yield distributions which are consistent with the R-OSF pattern. A low COP density and a high GOI yield are observed outside the ring; a high COP density and a medium GOI yield in the inner region bordered by the ring. It is assumed that boron atoms modify the thermodynamical properties of vacancies and self-interstitials.     

Numerical simulation of heat conduction for the growth of anisotropic layered GaSe crystals

In this report, we present the usage of a second rank cylindrical conductivity tensor which we derived to simulate the crystal growth processes of a layered compound GaSe in a cylindrical enclosure by directional solidification. Use of such a tensor is inevitable in the simulations of the growth of highly anisotropic crystals having layered structure, since the crystallographic orientation of the grown material is not necessarily aligned with the ampoule symmetry. Using the finite difference control volume approach in 3D, we solved transient heat conduction equation for a highly anisotropic solid in a cylindrical enclosure. We obtained sloped thermal fields and isothermal surfaces and the magnitudes of the slopes are strong functions of both azimuthal angle and growth orientation. The results showed that the orientation of the crystallographic axes of GaSe in the enclosure is quite effective in the steady and the transient fields, isotherms, and axial and radial temperature gradient within the material. Increase of Bi number decreases the magnitude of the slope of isothermal surface. Anisotropy of the conductivity seems to be effective in the orientation of the growth direction of the resulting crystal within the cylindrical ampoule. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Growth morphology of perfect and twinned face-centered-cubic crystals by Monte Carlo simulation

The effect of the relative growth rate between {1 1 1} and {1 0 0} faces on the growth morphology of perfect and twinned face-centered-cubic crystals was investigated with a Monte Carlo simulation considering both first-nearest-neighbor (FNN) and second-nearest-neighbor (SNN) interactions. When the bond energy ratio of SNN to FNN interactions is close to zero, the {1 1 1} twin planes make a reentrant edge, which enhances the growth rate on this plane, leading to a tabular growth shape. When the ratio is 0.25, the ridge side face of a tabular shape has the {1 0 0}/{1 0 0}/{1 1 1} structure instead of the {1 1 1}/{1 1 1} reentrant edge. In spite of disappearance of the reentrant edge, the side face has a higher growth rate than the top face because the {1 0 0} face still grows faster than the {1 1 1} face.     

Time-dependent simulation of the growth of large silicon crystals by the Czochralski technique using a turbulent model for melt convection

A numerical model is developed to perform the dynamic and global simulation of Czochralski growth. The effect of melt convection is taken into account by means of an eddy viscosity flow model, which can represent the mixing effect of flow oscillations on the heat transfer. Our method is used to investigate the dynamics of the growth of a 40 cm silicon crystal.     

Transient and quasi‐stationary simulation of heat and mass transfer in Czochralski silicon crystal growth

The formation of grown‐in defects in silicon crystals is controlled by the concentration of intrinsic point defects. Under steady state conditions the type of the prevailing point defect species is linked to the ratio of pull rate and temperature gradient in the crystal at the solidification front. It has been shown that this ratio as well as computed point defect distributions are in good agreement with experimental data. In this paper we compare a coupled transient heat transfer and transient point defect transport model with quasi steady state simulations at various time steps. Both simulations show the same qualitative results, quantitative differences in temperature are less than 1 %. But already for constant pull rates the defect distributions show qualitative differences between transient and quasi steady state simulations. Therefore, for a detailed understanding how defects are related to growth conditions, the thermal history should not be neglected.     

3D simulation of the effects of growth parameters on the growth of sapphire crystals using heat exchanger method

3D simulations using the commercial CFDRC and FIDAP code, which are based on finite element techniques, were performed to investigate the effects of anisotropic conductivity on the convexity of the melt–crystal interface and the hot spots of sapphire crystal in a heat‐exchanger‐method crystal growth system. The convection boundary conditions of both the energy input to the crucible by the radiation as well as convection inside the furnace and the energy output through the heat exchanger are modeled. The cross‐sectional flow pattern and the shape of the melt–crystal interface are confirmed by comparing the 3‐D modeling results with previous 2D simulation results. In the 3D model, the “hot spots” in the corners of the crucible are donut shaped, and the shape changes with the value of the conductivity of anisotropic crystal. The outline of the crystal becomes more convex as the conductivity in the z direction (k_sz) increases. The outline of melt–crystal interface is elliptical when the anisotropic conductivity is moving in the radial direction (k_sx and k_sy). The portion at the outline touching the bottom of the crucible is smaller than the maximum outline of the crystal, meaning that the shape at the “hot spot”, changes with the value of the conductivities of anisotropic crystal. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Influence of internal radiation on the heat transfer during growth of YAG single crystals by the Czochralski method

Heat and mass transfer taking place during growth of Y₃Al₅O₁₂ (YAG) crystals by the Czochralski method, including inner radiation, is analyzed numerically using a Finite Element Method. For inner radiative heat transfer through the crystal the band approximation model and real transmission characteristics, measured from obtained crystals, are used. The results reveal significant differences in temperature and melt flow for YAG crystals doped with different dopands influencing the optical properties of the crystals. When radiative heat transport through the crystal is taken into account the melt‐crystal interface shape is different from that when the radiative transport is not included. Its deflection remains constant over a wide range of crystal rotation rates until it finally rapidly changes in a narrow range of rotation rates. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical investigation of induction heating and heat transfer in a SiC growth system

A global simulation model is applied for a silicon carbide growth system heated by induction coils. A finite‐volume method (FVM) and a global model are applied to solve the equations for electromagnetic field, conductive and radiative heat transfer. The growth rate is predicted by Hertz‐Knudsen equation and one‐dimensional mass transfer equation. Further, simulations for five different coil positions are carried out to investigate the effects of coil position on temperature distribution in the furnace. The numerical results reveal that the variation of temperature in the radial direction along the substrate surface and the temperature difference between the powder and substrate are greatly affected by the coil position. The predicted growth rate along the substrate surface for five coil positions is also studied. Finally, a reasonable range of coil positions maintaining a balance between large‐diameter crystal, high growth rate, temperature limitation of material and lower electrical power consumption is obtained. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Preparation of GaN crystals by heating a Li3N‐added Ga melt in Na vapor and their photoluminescence

GaN crystals were prepared by heating a Ga melt with 1 at% Li₃N against Ga at 750 °C in Na vapor under N₂ pressures of 0.4–1.0 MPa. The GaN crystals grown at 1.0 MPa of N₂ were colorless and transparent prismatic, having a size of approximately 0.7 mm in length. A secondary ion mass spectrometry (SIMS) showed the contaminant of lithium in the obtained crystals. A large broad yellow band emission peak of 2.28 eV was observed at room temperature in the photoluminescence spectrum in addition to the near band emission peak of GaN at 3.39 eV and a small broad satellite emission at 3.24 eV. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Self-selecting vapour growth of bulk crystals – Principles and applicability

A method of self-selecting vapour growth (SSVG) for bulk binary and multernary crystals of semiconducting materials is reviewed comprehensively for the first time. Although it has been developed over three decades, the method is less well known – even though it is physically distinct from the more widely used ‘Piper–Polich’ and ‘Markov–Davydov’ vapour transport bulk growth methods. The means by which growth takes place on a polycrystalline source to form a crystal free from the walls is described. Modelling and empirical observations have been used to establish the characteristics of the almost isothermal temperature fields that drive the transport in SSVG. It is demonstrated that precise control of thermal radiation is a fundamental requirement for tailoring the temperature distribution—a fact that has been used well in the design of horizontal tube furnace growth rigs. Achievements in the growth of useful PbS, PbSe, PbTe, CdTe and ZnTe compound crystals are described. The SSVG method has proved to be particularly well suited to the growth of solid solutions, and the results of growth experiments, and of compositional and structural analysis, are presented for Pb(Se,S), (Pb,Sn)Se, (Pb,Sn)Te, (Pb,Ge)Te, Cd(Te,Se), Cd(Te,S) and (Cd,Zn)Te. The excellent compositional uniformity delivered is attributed to entropy driven mixing in the low thermal gradients present in SSVG.

To date, most SSVG has been done at the <50 g level for research or small scale production use. Prospects for scaling up the growth are considered, there being no barriers identified in principle. However, there is a limitation in that the shape of the grown crystals is not accurately controlled at present. To overcome this, and to offer an alternative method of scaling up, the use of vertical tube systems is explored. A significant additional advantage of the vertical configuration is that it allows for continuous recycling of the source/crystal mass so as to continuously self-refine the increasingly uniform – and crystalline – product. Achievements to date in growing II–VI and IV–VI crystals are described for prototype vertical SSVG systems. Finally, future prospects for the SSVG method in terms of further developments to the method, and the specific materials that will benefit from it are highlighted.     

Rotating magnetic fields as a means to control the hydrodynamics and heat transfer in single crystal growth processes

The paper discusses a possibility to use different types of rotating magnetic fields (RMF) and combinations of these to control the hydrodynamics and heat/mass transfer in the processes of bulk semiconductor single crystal growth. Some factors contributing to the efficiency of RMF and their influence on different technologies are analyzed. Their specific practical application is illustrated by some examples.     

Numerical study to investigate the effect of partition block and ambient air temperature on interfacial heat transfer in liquid bridges of high Prandtl number fluid

Surface heat transfer at the liquid–air interface in liquid bridges of high Prandtl number fluid is known to affect the transitional characteristics appreciably. The heat transfer characteristics under microgravity conditions become much different from those of normal gravity mainly due to the absence of natural convection. The present study deals with numerical computations of flow and heat transfer characteristics in the liquid and surrounding air and also at the liquid–air interface of thermocapillary flow in liquid bridges of high Prandtl number fluid. The governing equations are solved in the coupled domain of the liquid bridge and the surrounding air with the help of available commercial CFD software. The results obtained for a range of Marangoni numbers indicate that by placing a partition block in the air region under normal gravity conditions, the surface heat transfer characteristics of microgravity conditions could be effectively mimicked. The effect of ambient temperature on the surface heat transfer has also been investigated and it has been found that the behavior of heat transfer at the interface changes from heat loss to heat gain when the ambient temperature is increased. Moreover, the presence of partition block under normal gravity suppresses surface heat loss as well as surface heat gain similar to microgravity conditions. Streamlines and temperature contours have been presented for various conditions in order to clarify the underlying physics more meaningfully. The computed profiles for velocity and temperature at the liquid–air interface have been validated against established experimental results.     

Explicit solutions for boundary value problems in diffusion of heat and mass

The Fourier's infinite series solutions of transient diffusion in finite solids, with convection boundary conditions, contain eigenvalues, which are defined by implicit transcendental equations. The eigenvalue equations have to be evaluated numerically, making the use of the Fourier's solution difficult. Using the successive substitution method, explicit expressions were derived for the first eigenvalue λ₁, in plates, cylinders and spheres, as a function of the Biot number. The derived expressions for λ₁(Bi), together with the one-term Fourier series solution, yield simple explicit solutions for temperature or concentration profiles in plates, cylinders and spheres. The explicit solutions are valid for Fo>0.2.     

Control of multi-zone resistive heater in low temperature gradient BGO Czochralski growth with a weighing feedback,based on the global dynamic heat transfer model

The global heat transfer in a crystallization setup has been optimized to develop a strategy of control over a three-zone heater in the BGO Czochralski process, in order to provide invariable thermal conditions near the solid–liquid interface in the stage of a constant-diameter crystal growth. The functional related to the exactness of the heat balance condition at the crystallization front, i.e., the Stefan problem, was chosen as the target function. The optimization yielded unexpected results. The temperature of the lower heater should be lowered, relative to that of the middle heater, with increasing crystal length, whereas the temperature of the upper heater is to be raised. These recommendations were incorporated into a dynamic model of the oxide Czochralski process with a weighing control and into the control loop of the temperature regulators of a crystallization setup. A comparison of results of the time-dependent simulation with the real growth process confirmed that the new control strategy minimizes the deviation of the solid–liquid interface from the prescribed one, significantly decreases variations of interface shape during the process, and enables growth of high-quality crystals.     

Growth of benzophenone single crystals from solution: A novel approach with 100% solute ‐ crystal conversion efficiency

An unidirectional 60mm diameter benzophenone single crystal was successfully grown by utilizing a novel crystal growth method at room temperature. <110> oriented single crystal ingots were grown out of xylene as solvent and by fixing a seed at the bottom of the ampoule. The obtained benzophenone ingots with the sizes of 10mm, 25mm and 60mm diameter evident that ease in increasing the diameter of the ingot. The orientation of the ingot and the crystalline quality were justified by X‐ray studies. TG and DTA evaluated the thermal properties of the grown crystal. The optical transmission study and the powder SHG measurement show the suitability of the ingot for nonlinear optical applications. The achieved solute‐crystal conversion efficiency of hundred percent shows vital advantage of this technique for cost effectiveness. The microbial growth as in the case of amino acid based growth solutions can be more effectively controlled in the present method since the freshly prepared growth solution can be constantly made available to the growing crystal. © 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Numerical simulation of a new SiC growth system by the dual-directional sublimation method

A new SiC growth system using the dual-directional sublimation method was investigated in this study. Induction heating and thermal conditions were computed and analyzed by using a global simulation model, and then the values of growth rate and shear stress in a growing crystal were calculated and compared with those in a conventional system. The results showed that the growth rate of SiC single crystals can be increased by twofold by using the dual-directional sublimation method with little increase in electrical power consumption and that thermal stresses can be reduced due to no constraint of the crucible lid and low temperature gradient in crystals.     

Growth and properties of KTP crystals from a new flux

High-quality KTiOPO₄ (KTP) crystals were grown by a top seeded solution growth (TSSG) method using K₈P₆O₁₉–BaF₂ as a flux. The volatility of different solvents, such as K₈P₆O₁₉ (K₈), K₈–NaF, K₈–KF, and K₈–BaF₂, was measured. These fluoride additives in K₈ fluxes and their compositional effects on the growth of KTP crystals were studied and discussed. The transmissivity and optical homogeneity of KTP crystals were also measured.