mathematical model of heat and mass transfer

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MATHEMATICAL MODEL OF HEAT AND MASS TRANSFER DURING CRYSTAL GROWTH PROCESS INCLUDING CLUSTER MODEL OF A MELT CONSTITUTION a

a

a

Vladimir Ginkin , Olga Naumenko , Michael Zabudko , Andrey a

Kartavykh & Michael Milvidsky a

b

SSC RF Institute for Physics and Power Engineering, Obninsk, Russia

b

Institute of Chemical Problems for Microelectronic, Moscow, Russia Version of record first published: 24 Feb 2007.

To cite this article: Vladimir Ginkin , Olga Naumenko , Michael Zabudko , Andrey Kartavykh & Michael Milvidsky (2005): MATHEMATICAL MODEL OF HEAT AND MASS TRANSFER DURING CRYSTAL GROWTH PROCESS INCLUDING CLUSTER MODEL OF A MELT CONSTITUTION, Numerical Heat Transfer, Part B: Fundamentals: An International Journal of Computation and Methodology, 47:5, 459-472 To link to this article: http://dx.doi.org/10.1080/10407790590919207

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Numerical Heat Transfer, Part B, 47: 459–472, 2005 Copyright # Taylor & Francis Inc. ISSN: 1040-7790 print=1521-0626 online DOI: 10.1080/10407790590919207

MATHEMATICAL MODEL OF HEAT AND MASS TRANSFER DURING CRYSTAL GROWTH PROCESS INCLUDING CLUSTER MODEL OF A MELT CONSTITUTION Vladimir Ginkin, Olga Naumenko, Michael Zabudko, and Andrey Kartavykh SSC RF Institute for Physics and Power Engineering, Obninsk, Russia

Michael Milvidsky Institute of Chemical Problems for Microelectronic, Moscow, Russia This article gives a brief analysis of current concepts about the processes of melt ordering and structural self-organization at the temperature close to melting point, including the interface area, when growing semiconductor single crystals. A mathematical model of convection mass transfer is proposed as an independent tool of exploration. This model includes the equations of hydrodynamics, impurity transfer, and convection flow in the interface in view of the medium cluster structure. For the first time, the melt structural model near the crystallization front considers the availability of cluster formations which cause resistance to the melt flow.

INTRODUCTION The study of substance structural transformations close to the melting temperature (T ) at phase transitions is of the highest priority for understanding the fundamental processes and mechanisms of crystallization. The difficulties in this study are aggravated by the fact that the transient region in the process of ‘‘ground’’ growth has small geometric sizes and is constantly being destroyed from the melt side by the natural and artificially created convection flows. For instance, the typical interface layer thickness is equal to 0.1–0.5 mm when crystals are grown by the Czochralski method. As the fundamental effects of structure ordering in the transient layer are extremely sensitive to any energy influence—thermal, gravitational or mechanic Received 20 July 2004; accepted 25 September 2004. The work has been done with financial support from RFBR, Grant 03-02-16282a. The work has not been published elsewhere and has not been submitted simultaneously for publication elsewhere. A preliminary version of this article was presented at CHT-04: An ICHMT International Symposium on Advances in Computational Heat Transfer, April 2004, G. de Vahl Davis and E. Leonardi (eds.), CD-ROM Proceedings, ISBN 1-5670-174-2, Begell House, New York, 2004. Address correspondence to Vladimir Ginkin, SSC RF Institute for Physics and Power Engineering, Bondarenko sq. 1, Obninsk, 249033, Kaluga region, Russia. E-mail: [email protected]

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(convection, stirring, vibration), electromagnetic, etc.—the state of microgravitation implemented aboard orbital space units (SUs) must be the ideal medium for their manifestation and study. During crystallization in maximum conditions close to zero gravity and under growth conditions close to diffusion, the interface region thickness in semiconductor melt can achieve 3–6 mm [1–3]. The purposeful study of the melt state in space has not been performed yet. However, there are some experimental facts which indirectly confirm the existence of strong structuring effects in the melt in the conditions of orbital flight. For instance, the metal liquid-phase diffusion (D) coefficients determined according to the data of space experiments are known to be always 2–3 times underestimated as compared to those of ‘‘ground’’ experiments. When studying the diffusion processes in capillaries aboard the Space Shuttle and SU Mir, Smith [4, 5] discovered the abnormal temperature dependence D T for all the systems under study, including Sb-In, Sb-Ga, Bi-Sb, etc. (in all, 11 systems, more than 200 experiments). In the ground control tests the usual Arrenius relation D exp(1=T) was observed. Evidently, diffusion in space occurred in the undisturbed ordered melt, whose structure changed significantly under the temperature. The most used method of modeling ‘‘cluster’’ mechanisms of precrystallization state evolution at the microlevel consists of a molecular dynamical approach [6]. Significant methodological progress has been recently achieved in this direction (e.g., [7]). In particular, in [8], when modeling the process of GaP single-crystal growth from the melt, it was shown that the total cluster concentration in the transient layer in A3B5 compounds can achieve ð2 3Þ 1020 cm3 , including the one with the biggest clusters (with the number of associated atoms being equal to 50–75) 1019 cm3 . This estimation agrees well with typical values of dopant concentration in diffusion layers near the front and testifies that the dopant atoms can really serve as cluster formation centers. However, it should be noted that the calculation results obtained with the molecular dynamics method depend strongly on the choice of the specific type of atom interaction potentials in the melt and crystal. The Lennard-Johns and Born-Mayer potentials are usually used a priori. Mathematical expressions for the description of potentials in an explicit form have a number of uncertain empirical parameters, thus resulting in unjustified simplification during crystallization process modeling. In view of that, it is necessary to check thoroughly the consistency and agreement of calculation and experimental data. Macroscopic changes of melt structure—dependent characteristics in the front layer which reflect the dynamics of cluster formation at the microlevel—give the grounds for developing hydrodynamical models that can become an additional and independent tool of exploration. The idea of developing and testing such a model as applied to the process of ground and orbital semiconductor crystallization is partially implemented by the authors in this work. The model development makes it possible to consider the fundamental peculiarities of the processes of crystal growth from the melt, in particular, to calculate correctly the heat and mass transfer both in the region core and in its interface. In simulating the process of crystallization, one has to solve the Stefan problem together with the equations of hydrodynamics and convectional transfer of impurity in the melt. The phase-transition interface position is not known in advance. It varies with space and time and is dependent on the velocity and pattern of temperature variation in the region over which the solution is to be computed. Furthermore, it


CLUSTER HYDRODYNAMIC MODEL OF CRYSTAL GROWTH

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is known that in the process of crystallization the interface is diffuse rather than localized, which means it has a certain relatively small thickness depending on the solidification velocity. Inside this diffuse boundary region the properties of a substance being crystallized are different from both the properties of the solid phase and those of the liquid phase of the substance. And yet the transitional region near the crystallization front must have a pronounced effect on both the convective melt flow and impurity distribution in the crystal being grown. The model proposed in this article is based on the cluster approach of the simulation of transition area 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 the transition area is simulated by a porous medium approach. The flow resisting strength proportional to the porosity coefficient, which is equal to the ratio of solid- and liquid-state fractions, is introduced into the motion equation. In the model, the three-dimensional Stefan problem in natural variables is solved. The energy equation is solved in enthalpy variables. For the solution of hydrodynamic equations, the method of finite volumes and the implicit stabilization method are used.

MATHEMATICAL MODEL OF MELT CRYSTALLIZATION The laminar flow in the melt is described by the Navier-Stokes equations in their Boussinesq approximation form: qh þ urh ¼ rkrT qt

ð1Þ

qu þ ðurÞu ¼ rp þ rmru q0 ½bT ðT T0 Þ þ bC ðC C0 Þg q0 qt

ð2Þ

ru ¼ 0

ð3Þ

qC þ urC ¼ rDrC qt

ð4Þ

q

In the solid sections of the region u ¼ 0, and Eqs. (1)–(4) take the form qh ¼ rkrT qt

ð5Þ

qC ¼ rDrC qt

ð6Þ

q

In Eqs. (1)–(6), q, m, k, and D are density, viscosity, thermal conductivity, and diffusion coefficients, respectively; bT ; bC are thermal expansion and concentration

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expansion coefficients; q0 is the melt density at T ¼ T0 , C ¼ C0 ; and g is the acceleration due to gravity. Equations (1)–(6) are supplemented by initial and boundary conditions. Let us break down the region over which the solution is to be computed into meshes and average the enthalpy values in each of the meshes. Let us express the average mesh temperature T in terms of enthalpy h: 8 ðh h1 Þ > > T þ > > < cps ðhÞ T ¼ T > > h h1 n > > :T þ cpl ðhÞ

for h h1 for h1 < h < h1 þ n

ð7Þ

for h h1 þ n

where cps ðhÞ is the heat capacity of the solid phase of the substance. cpl ðhÞ is the heat capacity of the liquid phase of the substance. h1 is the maximum value of enthalpy for the solid phase of the substance. n is the phase-transition latent heat. T is the crystallization temperature. Meshes with enthalpy values h1 < h < h1 þ n are in a state that is intermediate between solid and liquid. Let us use eL and eS to denote the fraction of liquid and solid phase, respectively, in the intermediate state region h1 < h < h1 þ n:

eL ¼

8 0 > > > :

n

1 eS ¼ 1 eL

for h h1 1

for h1 < h < h1 þ n

ð8Þ

for h h1 þ n

Then the expression (7) for temperature T can be written as T ¼ T þ

h h1 neL cp ðhÞ

If we use XS to denote a parameter value for the solid phase and XL for the liquid phase, the value of X for the intermediate state can be obtained from the formula X ¼ X S eS þ X L eL

ð9Þ

where the parameter X is q; k; cp ; D. Let us assume that the intermediate melt region is filled with clusters, i.e., a population of atoms (molecules) aggregated in solid structures. The size and number of clusters can be arbitrary. The clusters’ heat movement velocity is significantly



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lower than that for single melt atoms because they have much bigger sizes. The results of molecular dynamic calculations confirm this assumption. The multiatom cluster movement velocity can be neglected within the model. Then the melt flow in the medium which is not tightly filled with motionless clusters can be presented as a liquid flow in a porous two-phase medium. The melt flow in meshes where the substance is in an intermediate state will be described using the porous solid approximation with the porosity coefficient K ¼ eS =eL . To be able to take such a description, let us make Eq. (2) and include a resisting force due to conditions of porosity [9]. This force varies as the melt viscosity, flow velocity, the porosity coefficient K, inversely to the square of mesh normal velocity vector and is opposed to the velocity vector u: F ¼ a

nu K S

where n ¼ m=q0 , S is the square of mesh normal velocity vector u, and a is an empiric constant which together with K describes the porous medium nature. Then Eq. (2) will take the form qu 1 nu þ ðurÞu ¼ rp þ nDu ½bT ðT T0 Þ þ bC ðC C0 Þg a K qt q0 S

ð10Þ

When K ¼ 1, Eq. (10) changes into u ¼ 0; and when K ¼ 0, it goes over to Eq. (2). Impurity transfer in meshes where the substance is in an intermediate state will be described by the following equation [10]: eL

qC qeS þ urC ¼ rDrC þ ð1 Kp ÞC qt qt

ð11Þ

where Kp is the impurity segregation coefficient. When eL ¼ 1, Eq. (11) changes over to Eq. (4). When eL ¼ 0, Eq. (11) goes to Eq. (6). The last term in Eq. (11) refers to impurity segregation at the phasetransition interface, whose velocity is assumed to be equal to the speed with which the solid-phase proportion is changing in the intermediate-state region. So the mathematical model of heat and mass transfer for the whole area where the crystallization process takes place includes Eqs. (1), (3), (10), and (11), in which the temperature T and values of q; k; cp ; D; eL , and eS are single-valued and continuous one-variable functions of the enthalpy h. The hallmark of the proposed model is that there is no localized phasetransition interface. This boundary is dithered, forming a region of meshes whose enthalpy values refer to an intermediate state of the substance. Nevertheless, the crystallization front velocity can be defined as the rate at which the proportion of the solid phase of the substance changes. This approach allows us to describe impurity segregation in the process of crystallization together with the impurity diffusion-convection transfer in the melt. As far as the hydrodynamics equations are concerned, the adhesion condition for the flow velocity is laid down at the boundary between the solid phase and the intermediate state of the substance being

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crystallized, and the additional resistance to the flow in the intermediate state region is described in terms of porous solid approximation in which the porosity coefficient is determined by the liquid phase-to-solid phase ratio.


DEFINITION OF AN EMPIRICAL CONSTANT a As mentioned above, the state of microgravitation implemented aboard orbital space units is the ideal medium for study of melt structure in the transition region near the front of crystallization. This is illustrated by the discovery of anomalies in Ga dopant distribution in Ge single crystals grown by the floating zone (FZ) technique aboard five unmanned Photon spacecraft (SC) [1–3]. The authors found an unusual, unrelated to any terrestrial analog, distribution of dopant along the growth single crystals length, which indicates an interdependence between the distribution coefficient and the solute boundary-layer thickness near the melt=solid interface on the dopant content in the melt. These data, along with regular changes of the crystallization front shape detected when the doping level varies, show the dopant concentration influence in the melt on heat and mass transfer processes there. It was suggested in [2, 3] that the above phenomena could be caused by interaction of thermal and solutal surface tension-driven (STD) kinds of convection. A mathematical model was developed to describe this interaction, and numerical investigations were performed [11]. The mechanism of this interaction follows. After complete zone melting, an intensive flow in the form of two symmetrical vortexes with opposite flow velocities is developed in the molten zone under the influence of thermal STD convection. After crystallization begins, the impurity driven out of the crystal into the melt is captured by the vortex near crystallization front and spreads uniformly over half of the melt region. The driven impurity cannot penetrate into the other half of the region because the axial velocity of the melt is equal to zero at the boundary of the vortexes. As a result, a gradient of concentrations initiating solutal STD convection is formed in the central part of the molten zone. This gradient is opposite to the temperature gradient in the region adjacent to the crystallization front and it has the same direction as the temperature gradient in the region adjacent to the melting front. Therefore the developed solute STD convection decreases the vortex near the crystallization front and increases the vortex near the melting front until it finally becomes absolutely prevailing. The presence and the size of the residual vortex near the crystallization front define the radial nonuniformity in the impurity distribution in the grown crystal. Calculations showed that increase of the impurity concentration in the melt led to a decrease in the residual vortex size and, consequently, to a decrease of radial nonuniformity in the dopant distribution in the crystal bulk. This result agrees qualitatively with experimental data on the growth of Ge(Ga) crystals aboard SC Photon. However, the results obtained do not adequately explain the effect of an anomalous distribution of Ga over the length of space-grown crystals observed in the experiment. The reason is that we do not take into account an additional resistance to the melt flow (and with it a radial drift of the impurity) near the crystallization front resulting from an increase in viscosity of the melt close to the melt=solid interface. Under microgravity conditions, this resistance, especially in the case of low impurity



465

concentrations near the melt=solid interface, can be compared with other forces, so it should be taken into account. Therefore, the authors of [11] introduced into the model temperature dependence for viscosity of the Ge melt near melt=solid interface. It is assumed in the numerical model that viscosity increases by a factor of 60 in the extrapolation region (T þ 10 KÞ near the interface. Parametrical calculations were done taking into account the temperature dependence of viscosity. The calculation results were in agreement with the results from [2, 3], and describe adequately the anomalous effect of the dopant distribution observed experimentally. The results obtained prompted us to suggest that inside the solute boundary layer, as the crystallization front is approached, considerable structural transformations take place in the melt, which cause changes of its physical properties and in viscosity first of all. These changes reveal themselves especially clearly in the case of growing crystals under conditions of weak mixing of the melt (and, correspondingly, of a large thickness of solute boundary layer near the crystallization front), which is characteristic for crystallization of low-doped melts in microgravity. An imperfection of the model described in [11] is that it results in large shift strains near the melt=crystal interface because of increasing viscosity in this region. The model supposed in this article is free from this deficiency. The value of empirical constant a in Eq. (10) was estimated for semiconductor melt based on agreement of the calculation results of the concentration distributions in the crystal growth process in microgravity condition obtained by these two methods. This condition results in the value a ¼ 7 1017 .

COMPUTATION ALGORITHM Let us introduce a time grid with variable pitch s and use an implicit discretization scheme. This yields the following equation system:

raðhÞrTðhÞ þ urTðhÞ þ

nDu þ ðurÞu þ

h h ¼ cp s cp s

ð12Þ

~ u u nu 1 ð13Þ þ a K ¼ rp ½bT ðT T0 Þ þ bC ðC C0 Þg þ s s S q0 ru ¼ 0

rDrC þ urC þ

eS ð1 Kp ÞC þ eL C eS ð1 Kp ÞC þ eL C ¼ s s

ð14Þ

ð15Þ

u; eS ; C refer to the values of where aðhÞ ¼ ½kðhÞ ½qðhÞcp ðhÞ and the symbols h; ~ those quantities at the preceding point of time.

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In Cartesian axials the following equation system can be given:


qW j q j qW j qW j q j qW j ex x ey ux þ uy qx qy qx qx qy qy qW j q j qW j ez þ uz j ¼ 1; . . . ; 5 þ cj W j ¼ F j qz qz qz

ð16Þ

where W j is a common substance variable accepting values ux , uy , uz , h, C, and ejx , ejy , ejz , cj , F j are parameters conforming to them. Convective terms can be eliminated from Eq. (16) as follows [12]. Let us require the following equalities: qW q qW 1 q qW ex ¼ kex qx qx qx k qx qx qW q qW 1 q qW ey ¼ xey uy qy qy qy x qy qy qW q qW 1 q qW ez ¼ gez uz qz qz qz g qy qz

ux

(Here and below, index j is omitted for simplicity.) Hence the expressions for k, x, and g can be given as Rx

k¼e

x0

ux =ex dx

Ry

x¼e

y0

uy =ey dy

Rz

g¼e

z0

uz =ez dz

where x0, y0, z0 are arbitrary values x, y, z from the calculation area. After this transformation, the following equation can be given instead of (16):

1 q qW 1 q qW 1 q qW kex xey gez þ cW ¼ F k qx qx x qy qy g qz qz

ð17Þ

Using diverse grids and an integro-interpolation method, we build a difference analog of Eq. (17). After that, using the Patankar method and finite-difference expression for continuity equation (14), we can derive a difference equation for pressure which can be written implicitly as LðpÞ ¼ 0

ð18Þ

The obtained system of the equations is linearized and solved by the stabilization method at each time step s. For this purpose we introduce into the left side of Eq. (17) the time derivative qW =qt, then we introduce an auxiliary time grid and solve the resulting equations on it using the implicit method until stabilization is reached. After that we proceed to the next time step on the main grid.


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The iteration process of stabilization stops when the following condition is met: qh max qt ;

qC < eps qt


where eps is the required accuracy. The linearized system of equations is transformed to a symmetric form and is solved by the conjugate gradient method with preconditioning by the incomplete factorisation method [13].

MODEL TESTING In order to test the computer code, numerical modeling of the real experiment on crystallization of GaSb alloyed with In in space and on the ground by the Bridgeman method published in [14] was performed. The calculation scheme is given in Figure 1. Calculation was carried out in 2-D (X, Y) geometry. Figure 2 shows the calculated and experimental distributions of In concentration along the crystal axis for the conditions of zero gravity. Good agreement was achieved between calculated and experimental functions, thus indicating a really low level of convection or no convection at all during the experiments aboard the space unit. The values of physical parameters used in the calculations are given in Table 1. Figure 3 shows similar curves for the crystal grown on the earth under the same experimental conditions. The agreement between the calculated results and experimental data is rather satisfactory. The authors of [14] also describe radial distributions of impurity in different cross sections of the grown crystal. They are not uniform and testify that there was still a weak convection in spite of a very low level of residual gravitation. The

Figure 1. The calculation scheme.


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Figure 2. Longitudinal In concentration distribution in GaSb for the conditions of earth gravity: 1, measured [14]; 2, calculated using the clusterization parameter ratio a ¼ 0.

impurity radial distributions had different slopes for different cross sections, thus indicating the change in the direction of residual gravitation vector. In order to check this assumption, the calculation was made of GaSb crystallization at g ¼ 104 g0 when the residual gravity vector was assumed to be perpendicular to the growth axis during the first 5 104 s of crystallization, and during the next 5 104 s it had the opposite direction.

Table 1. Thermal parameters of GaSb Melting temperature, K Density of the melt, kg=m3 Density of the solid phase, kg=m3 Thermal conductivity, Wt=m K Thermal capacity, J=kg K Kinematic viscosity, m2=s Diffusion coefficient of InSb in GaSb, m2=s Segregation coefficient 0.15 Temperature expansion coefficient, K1 Concentration expansion coefficient, m3=kg Prandtl number

T qL qS K cp n D j bT bC Pr

985 6,010 5,600 8 175 0:38 106 1:2 D8 0.15 2:0 104 0 0.05



469

Figure 3. Longitudinal In concentration distribution in GaSb for the conditions of zero gravity: 1, measured [14]; 2, calculated using the clusterization parameter ratio a ¼ 7 1017 .

Figure 4 shows the spatial distributions of In concentration in the solid GaSb being crystallized, as well as concentration isolines obtained as a result of calculation according to the proposed model. The data presented show that isolines first curve toward one side, then are rectified, and finally curve toward the other side, thus confirming the model performance under the given assumption of the existence of residual gravitation and precession of its vector during solidification. Figure 5 shows spatial distributions of flow rate vector components for axes X sfand Y at the time moments of 5 104 s and 105 s in order to illustrate the flow nature in the case considered. These data testify that the vortex reverses direction when the direction of the residual gravitation vector changes. The maximum value of the flow rate module in the first case was equal to 0:34 105 m=s, and in the second case was 0:39 105 m=s.

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


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Figure 4. Distribution of In impurity concentration in C 1019 at=cm3 (a) and equal In concentration lines in the longitudinal crystal section (b).

For the first time, the given hydrodynamical model considers the structural precrystallization state of the boundary layer in the interface region. The transient region is described as a two-phase medium which together with the melt contains the solid-phase clusters. The specific feature of the model consists of the description of clusterization processes and those of substance crystallization in enthalpy (thermodynamical) variables, thus giving the possibility to compare results with 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. In the suggested version of the model in semiempiric form, consideration was given only to the extreme case of melt structure ordering, i.e., the formation of solidassociated particles. Nevertheless, the model gives quite an adequate description of crystallization system response to the change of ambient conditions—temperature, crystallization rate, the size and direction of microacceleration vector, etc. The results of test calculations of In-allowed GaSb crystallization show good agreement with the results of both space and ground experiments.



471

Figure 5. Spatial distribution of longitudinal and transverse components of melt flow rate vector and melt flow rate fields; (a) at the moment of 5 104 s; (b) at the moment of 105 s.

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 most prominently affect the quality of the crystal under growth. REFERENCES 1. A. V. Kartavykh, E. S. Kopeliovich, M. G. Milvidskii, and V. V. Rakov, Analysis of Axial Profiles of Dopant Distribution in Single Crystals Ge < Sb > Grown by a Floating Zone Method in Space, J. Crystallogr., vol. 43, no. 6, pp. 1136–1141, 1998 (in Russian). 2. A. V. Kartavykh, E. S. Kopeliovich, M. G. Milvidskii, and V. V. Rakov, Specific Effects of Ge Single Crystals Doping during their Floating Zone Processing aboard the Spacecrafts, Micrograv. Sci. Technol., vol. XII, no. 1, pp. 16–22, 1999. 3. A. V. Kartavykh, E. S. Kopeliovich, M. G. Milvidskii, and V. V. Rakov, Abnormal Distribution Effects of Dopant Distribution in Single Crystals Ge, Grown by a Floating Zone Method in Requirements of Space Flight, J. Crystallogr., vol. 45, no. 1, pp. 167–174, 2000 (in Russian).


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4. R. W. Smith, X. Zhu, M. C. Tunnicliffe, T. J. N. Smith, L. Misener, and J. Adamson, The Influence of g-Jitter on the Measurement of Solute Diffusion in Dilute Liquid Metals and Metalloids in a Low Earth Orbiting Laboratory, Proc. Int. Conf. ‘‘Spacebound-2000,’’ Banff, Canada, 2000, pp. 96–121. 5. R. W. Smith, X. Zhu, M. C. Tunnicliffe, T. J. N. Smith, L. Misener, and J. Adamson, The Measurement of Precise Solute Diffusion Coefficients in Dilute Liquid Metals and Metalloids in a Low Earth Orbiting Laboratory—the Influence of g-Jitter, Proc. 1st Int. Symp. on Microgravity Research & Applications in Physical Sciences & Biotechnology, Sorrento, Italy, September 10–15, 2000, ESA SP-454, vol. II, pp. 887–893. 6. N. P. Lyakishev (ed.), Basic Researches of Physical Chemistry of Metal Melts, Academkniga, Moscow, 2002 (in Russian). 7. J. R. Chelikowsky, J. Derby, V. Godlewsky, M. Jain, and J. Y. Raty, About the Simulations of Liquid Semiconductors Using the Pseudopotential-Density Functional Method, J. Phys.: Condensed Matter., vol. 13, no. 41, pp. R817–R854, 2001. 8. S. V. Kotov, A. R. Lyutikov, Yu. P. Khukhryanskii, I. N. Arsentyer and E. A. Kuznetsova, About the Mechanism of Crystal Growth from a Melt in Requirements of a Weightlessness, Lett. J. Theor. Phys., vol. 28, no.14, pp. 15–18, 2002. 9. V. P. Ginkin, Porous Solid Model to Describe Heat-Mass Transfer near Phase Transition Interface in Crystal Growth from Melt Simulations, Proc. 1st Int. Conf. on Application of Porous Media, Jerba, Tunisia, 2002, p. 155. 10. V. Timchenko, P. Y. P. Chen, G. de Vahl Davis, E. Leonardi, and R. Abbaschian, A Computational Study of Transient Plane Front Solidification of Alloys in a Bridgman Apparatus under Microgravity Conditions, Int. J. Heat Mass Transfer, vol. 43, pp. 963– 980, 2000. 11. V. K. Artemyev, V. I. Folomeev, V. P. Ginkin, A. V. Kartavykh, M. G. Milridskii, and V. V. Rakov, The Mechanism of Marangoni Convection Influence on Dopant Distribution in Ge Space-Grown Single Crystals, J. Cryst. Growth, 2001, pp. 29–37. 12. V. P. Ginkin, Methods of Solution of Convective Heat Mass Transfer at Single Crystal Growth Problem, Proc. 2nd Int. Symp. on Advances in Computational Heat Transfer, Palm Cove, Queensland, Australia, 20–25 May, 2001, pp. 1161–1168. 13. V. P. Ginkin, A. V. Kulik, and O. M. Naumenko, An Efficient Preconditioning Procedure in the Conjugate Gradient Method for 3D HEX-Z Geometry, Proc. 4th Int. Conf. on Supercomputing in Nuclear Applications (SNA 2000), Tokyo, Japan, September 4–7, 2000. 14. T. Duffar, M. D. Serrano, C. D. Moore, J. Camassel, S. Contreras, P. Dusserre, A. Rivoallant, and B. K. Tanner, Bridgmen Solidification of GaSb in Space, J. Cryst. Growth, no. 192, pp. 63–72, 1998.