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  • 1
    Electronic Resource
    Electronic Resource
    Finite-element methods for analysis of the dynamics and control of Czochralski crystal growth (1987)
    Springer
    Journal of scientific computing 2 (1987), S. 297-343 
    Details
    ISSN: 1573-7691
    Keywords: Czochralski crystal growth ; finite element method ; thermal-capillary model ; moving-boundary problem
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract Numerical methods are presented for solution of the complex moving-boundary problem described by a thermal-capillary model for Czochralski crystal growth, which accounts for conduction through melt, crystal, and crucible and radiation between diffuse-gray body surfaces. Transients are included that are caused by energy transport, by changes in the shapes of the melt-crystal, melt-ambient phase boundaries and the moving crystal, and by the batchwise decrease of the melt volume in the crucible. Finite-element discretizations are used to approximate the moving boundaries and the energy equation in each phase. A two-level, implicit integration algorithm is presented for transient calculations. The temperature fields and moving boundaries are advanced in time by a trapezoid rule approximation with modified Newton's iterations to solve algebraic systems for effective ambient temperatures computed with diffuse-gray radiation. The implicit coupling between radiative exchange, interface shapes, and the temperature field is necessary for preserving the second-order accuracy of the integration method and is achieved by successive iterations between the radiation calculation and solution of the thermal capillary model. Analysis of a quasi-steady-state model (QSSM) demonstrates the inherent stability of the CZ process. Including either diffuse-gray radiation among crystal, melt, and crucible or a simple controller for maintaining constant radius can lead to oscillations in the crystal radius. The effects of these oscillations on batchwise crystal growth are addressed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01061294
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    A finite element method for analysis of fluid flow, heat transfer and free interfaces in Czochralski crystal growth (1989)
    Chichester : Wiley-Blackwell
    International Journal for Numerical Methods in Fluids 9 (1989), S. 453-492 
    Details
    ISSN: 0271-2091
    Keywords: Czochralski crystal growth ; Finite element method ; Free boundary problem ; Incompressible fluid flow ; Heat transfer ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: A finite element algorithm is presented for simultaneous calculation of the steady state, axisymmetric flows and the crystal, melt/crystal and melt/ambient interface shapes in the Czochralski technique for crystal growth from the melt. The analysis is based on mixed Lagrangian finite element approximations to the velocity, temperature and pressure fields and isoparametric approximations to the interface shape. Galerkin's method is used to reduce the problem to a non-linear algebraic set, which is solved by Newton's method. Sample solutions are reported for the thermophysical properties appropriate for silicon, a low-Prandtl-number semiconductor, and for GGG, a high-Prandtl-number oxide material. The algorithm is capable of computing solutions for both materials at realistic values of the Grashof number, and the calculations are convergent with mesh refinement. Flow transitions and interface shapes are calculated as a function of increasing flow intensity and compared for the two material systems. The flow pattern near the melt/gas/crystal tri-junction has the asymptotic form predicted by an inertialess analysis assuming the meniscus and solidification interfaces are fixed.
    Additional Material: 27 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/fld.1650090406
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  • 3
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    Thermal-capillary analysis of Czochralski and liquid encapsulated Czochralski crystal growth (1986)
    In:  Other Sources
    Details
    Publication Date: 2011-08-19
    Description: Results are presented from finite element analysis of the Czochralski (CZ) and Liquid Encapsulated Czochralski (LEC) crystal growth processes based on a thermal-capillary model which governs the heat transfer in the system simultaneously with setting the shapes of the melt/solid interface, the melt and encapsulant menisci, and the radius of a steadily growing crystal. Calculations are performed for the small-scale growth of silicon (CZ) and gallium arsenide (LEC). The effects of melt volume and crucible position relative to the heater on the radius of the crystal and the shape of the melt/solid interface are predicted for the CZ system, and the importance of including an accurate representation of the melt meniscus for modeling the process is demonstrated. The additional effect of an encapsulant layer on heat transfer is treated for the LEC method for the cases of totally transparent and opaque encapsulant. The responses of these LEC prototype systems are examined for changes in pull rate and encapsulant volume.
    Keywords: SOLID-STATE PHYSICS
    Type: Journal of Crystal Growth (ISSN 0022-0248); 74; 605-624
    Format: text
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  • 4
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    Thermal-capillary analysis of Czochralski and liquid encapsulated Czochralski crystal growth. II - Processing strategies (1986)
    In:  Other Sources
    Details
    Publication Date: 2011-08-19
    Description: The pseudosteady-state heat transfer model developed in a previous paper is augmented with constraints for constant crystal radius and melt/solid interface deflection. Combinations of growth rate, and crucible and bottom-heater temperatures are tested as processing parameters for satisfying the constrained thermal-capillary problem over a range of melt volumes corresponding to the sequence occuring during the batchwise Czochralski growth of a small-diameter silicon crystal. The applicability of each processing strategy is judged by the range of existence of the solution, in terms of melt volume and the values of the axial and radial temperature gradients in the crystal.
    Keywords: SOLID-STATE PHYSICS
    Type: Journal of Crystal Growth (ISSN 0022-0248); 75; 227-240
    Format: text
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  • 5
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    Thermal-capillary model for Czochralski growth of semiconductor materials (1985)
    In:  CASI
    Details
    Publication Date: 2016-06-07
    Description: The success of efficiently calculating the temperature field, crystal radius, melt mensicus, and melt/solid interface in the Czochralski crystal growth system by full finite-element solution of the government thermal-capillary model is demonstrated. The model predicts realistic response to changes in pull rate, melt volume, and the thermal field. The experimentally observed phenomena of interface flipping, bumping, and the difficulty maintaining steady-state growth as the melt depth decreases are explained by model results. These calculations will form the basis for the first quantitative picture of Cz crystal growth. The accurate depiction of the melt meniscus is important in calculating the crystal radius and solidification interface. The sensitivity of the results to the equilibrium growth angle place doubt on less sophisticated attempts to model the process without inclusion of the meniscus. Quantitative comparison with experiments should be possible once more representation of the radiation and view factors in the thermal system and the crucible are included. Extensions of the model in these directions are underway.
    Keywords: SOLID-STATE PHYSICS
    Type: JPL Proceedings of the Flat-Plate Solar Array Project Workshop; p 195-214
    Format: application/pdf
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  • 6
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    Dynamics of Bridgman-Stockbarger growth of non-dilute binary alloys (1985)
    In:  Other Sources
    Details
    Publication Date: 2019-07-12
    Description: The coupling between heat and mass transfer in the Bridgman-Stockbarger growth of nondilute crystals induces growth rate transients that are longer than those caused by heat transfer alone. The dynamics of the diffusion-controlled growth for thermophysical properties representative of Pb(1-x)Sn(x)Te and Hg(1-x)Cd(x)Te are computed from a one-dimensional solidification model which includes melt-crystal phase behavior. Results show that the growth rate response to a step change in pull rate for both systems differs from the exponential decay observed for dilute alloys and that growth rates cannot be inferred from axial segregation data in nondilute systems due to the dependence of the segregation coefficient on composition.
    Keywords: SOLID-STATE PHYSICS
    Type: Journal of Crystal Growth (ISSN 0022-0248); 71; 587-596
    Format: text
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  • 7
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    Numerical Optimization of the Thermal Field in Bridgman Detached Growth (2009)
    In:  CASI
    Details
    Publication Date: 2019-07-13
    Description: The global modeling of the thermal field in two vertical Bridgman-like crystal growth configurations, has been performed to get optimal thermal conditions for a successful detached growth of Ge and CdTe crystals. These computations are performed using the CrysMAS code and expand upon our previous analysis [1] that propose a new mechanism involving the thermal field and meniscus position to explain stable conditions for dewetted Bridgman growth. The analysis of the vertical Bridgman configuration with two heaters, used by Palosz et al. for the detached growth of Ge, shows, consistent with their results, that the large wetting angle of germanium on boron nitride surfaces was an important factor to promote a successful detached growth. Our computations predict that by initiating growth much higher into the hot zone of the furnace, the thermal conditions will be favorable for continued detachment even for systems that did not exhibit high contact angles. The computations performed for a vertical gradient freeze configuration with three heaters representative of that used for the detached growth of CdTe, show favorable thermal conditions for dewetting during the entirely growth run described. Improved thermal conditions are also predicted for coated silica crucibles when the solid-liquid interface advances higher into the hot zone during the solidification process. The second set of experiments on CdTe growth described elsewhere has shown the reattachment of the crystal to the crucible after few centimeters of dewetted growth. The thermal modeling of this configuration shows a second solidification front appearing at the top of the sample and approaching the middle line across the third heater. In these conditions, the crystal grows detached from the bottom, but will be attached to the crucible in the upper part because of the solidification without gap in this region. The solidification with two interfaces can be avoided when the top of the sample is positioned below the middle position of the third furnace.
    Keywords: Solid-State Physics
    Type: M09-0622 , M09-0446 , 17th American Conference on Crystal Growth and Epitaxy/14th Biennial Workshop on Organometallic Vapor Phase Epitaxy/6th International Workshop on Modeling in Crystal Growth; Aug 09, 2009 - Aug 14, 2009; Lake Geneva, SI; United States
    Format: application/pdf
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