Facetting during directional growth of oxides from the melt: coupling between thermal fields, kinetics and melt/crystal interface shapes

doi:10.1016/S0022-0248(99)00253-5

Journal of Crystal Growth

Volume 205, Issue 3, September 1999, Pages 333-353

https://doi.org/10.1016/S0022-0248(99)00253-5 Get rights and content

Abstract

Implicit in most large-scale numerical analyses of crystal growth from the melt is the assumption that the melt/crystal interface shape and position are determined by transport phenomena. Although reasonable for many materials under a variety of growth conditions, this assumption is incorrect for a number of practical systems under realistic growth conditions. Specifically, the behavior of systems (e.g. certain oxides) which tend to develop facets along the melt/crystal interface is often affected both by transport phenomena and by interfacial attachment kinetics. We present a new modeling approach which accounts for interfacial kinetic effects during melt growth of large single crystals. The isotherm condition, typically employed at the melt/crystal interface, is replaced by an equation accounting for undercooling due to interface kinetics. A finite-element algorithm, designed to accommodate its numerical mesh to the appearance of facetted interfaces, is applied to this problem. Results are presented for the simulated directional growth of oxide slabs. The interplay between evolving thermal fields and anisotropic interface kinetics is investigated. In particular, the evolution of facets and the dependence of their size on growth conditions is explored. Trends reported here are in qualitative agreement with those appearing in the literature. Discrepancies between quantitative predictions of facet sizes using a theory (see Ann. Rev. Mater. Sci. 3 (1973) 397 and references within) and those calculated in this paper can, in a number of cases, be attributed to the simplifications on which this theory is based.

Introduction

It is well recognized that large-scale numerical analysis of transport-limited crystal growth is a useful tool for the understanding and improvement of melt growth techniques (see e.g. Refs. [1], [2], [3] and references within). Such analyses have been successfully used to complement and guide experimental studies of directional crystal growth. In particular, they can provide detail which is very difficult or even impossible to obtain from experiments. This approach has been successful in the investigation of a variety of phenomena occurring during growth, such as melt convection (e.g. Refs. [4], [5]), radiative heat transport (e.g. Refs. [6], [7], [8], [9]), solute segregation (e.g. Refs. [10], [11], [12], [13]), and more. However, facetting of the melt/crystal interface, often observed in experimental investigations of oxides (e.g. Refs. [14], [15], [16]) as well as other materials (e.g. semiconductors [17], [18], [19]), cannot be explained using models based on transport phenomena alone.

Facets appearing along the melt/crystal interface are associated with a local reduction of the interfacial temperature (undercooling) to values below the equilibrium melting point of the material grown. This local undercooling along the facets can theoretically be attributed to anisotropy in interfacial energy² and/or anisotropy in the interfacial attachment kinetics. Experimental studies of directional melt growth of certain oxides [15], [16], [21] provide evidence that the evolution of facets along the melt/crystal interface during the directional growth of these materials is often a kinetic phenomenon. For example, in Ref. [15] the size of facets was shown to vary with crystal growth rate in a manner suggesting that the growth rate is proportional to the interfacial undercooling raised to the second power. This behavior is most probably a direct manifestation of interfacial growth kinetics. An alternative explanation, relating local changes in thermal fields (due to the dependence of latent heat release on growth rate) to variations in facet size, is compatible with interfacial energy driven facetting. However, this explanation is unlikely since detailed calculations [9] have shown latent heat effects to be negligible in small-diameter systems operated under conditions similar to those reported in Ref. [15]. Additional evidence supporting this line of reasoning is given in Section 4.4.

Theoretical studies of the relation between interfacial undercooling and growth velocity for both rough and flat faces are well documented in the literature (see e.g. Ref. [22] and references within). It has been recognized that rough surfaces grow at a rate proportional to the undercooling (i.e. yielding a kinetic coefficient independent of undercooling). Facetted interfaces, on the other hand, often grow at a rate dependent in a nonlinear manner on the local undercooling. Two classical mechanisms in this case are growth by screw dislocations, which is characterized by a dependence of growth rate on undercooling raised to the second power (V_n∝ΔT²),³ and two-dimensional nucleation-limited growth which requires relatively large undercoolings and is associated with an exponential growth rate dependence on undercooling (V_n∝exp(−A/ΔT)).

Studying coupled anisotropic kinetics and transport in a crystal growth system often requires the application of computational methods. There has been extensive work in this area with respect to the growth of small particles from a supersaturated solution/vapor or from a supercooled melt (see e.g. Refs. [23], [24], [25], [26]). Of special relevance to this work is a study by Yokoyama and Kuroda [23] who reproduced morphologies of snow crystals using a model for the anisotropic kinetics occurring during supersaturation-driven vapor growth. These authors used a nontrivial kinetic coefficient in their analysis, taking into account nonlinear kinetics on singular (facetted) surfaces while accounting for linear kinetics on nonfacetted surfaces. It is also worth noting the contribution of Yokoyama and Sekerka [25] who studied the anisotropic evolution of two-dimensional particles growing from their melt. They demonstrated how isotropic surface energy prevents the appearance of sharp corners along the melt/crystal interface.

Although the effect of kinetics on the growth of particles from supercooled melts has been addressed using computational analysis techniques, the application of these methods to the study of such effects in large-scale directional growth systems has been largely overlooked. Elementary geometric ideas relating facetting phenomena to kinetic undercooling, during the directional growth of single crystals from the melt, have been put forward in Refs. [27], [28]. Such ideas were later quantified into simple algebraic relations in Refs. [29], [30], [31]. These relations are, however, limited by the simplified assumptions on which they are based.

In this study we apply the Galerkin finite element method (GFEM) in an investigation of the effects of interface kinetics on the directional melt growth of oxide single crystals. As indicated above, existing computational studies of the growth of oxides via confined and meniscus-defined techniques (e.g. Refs. [7], [8], [9], [32], [33], [34]) put an emphasis on the better understanding of internal radiative heat transfer and melt convection. These analyses, however, include the assumption that the melt/crystal interface is at the material's melting point therefore ignoring facet formation during growth. In general, the existence of facets during growth may significantly modify temperature fields, melt flow, and solute segregation. In particular, the formation of facets in these materials may adversely affect crystalline quality. For example, in the case of garnets, nonequilibrium solute segregation along the facetted interface may lead to (unwanted) localized strained areas within the crystal [35], [36]. Note that the phenomenon of nonequilibrium segregation on facets, often termed “the facet effect”, has also been observed in nonoxide materials (see e.g. Ref. [37] and references within). It is also worth noting that facet formation along the melt/crystal interface during the growth of certain semiconductors is known to lead to unwanted twinning phenomenon (see e.g. Refs. [38], [39], [40], [41]).

This paper is organized as follows. In Section 2 the model is formulated, giving details of the transient analysis, quasi-steady state equations and model of interfacial kinetics. The numerical scheme is briefly presented in the next section (Section 3), followed by Section 4 which includes a summary of results and their discussion. Conclusions are given in Section 5.

Section snippets

Model formulation

We consider an idealized system for the growth of oxide slabs via the vertical gradient freeze (VGF) method. Using this technique, a slab-shaped molybdenum crucible, filled with starting material is placed in a vertical furnace whose temperature varies linearly between time-dependent hot-zone T_ho(t) and cold-zone T_co(t) values; directional solidification is induced by slowly lowering T_ho and T_co with time.

The mathematical representation of the physical domain and coordinate system we use is

Numerical technique

In this study we employ a numerical technique based on a standard Galerkin finite element method (GFEM) together, in the case of the transient calculations, with a backward Euler variable time-stepping technique; these methods are described in previous studies (e.g. Refs. [9], [46] and references within). For the sake of brevity we will briefly mention our approach, emphasizing the differences between our method and those we have previously employed. For a more detailed description of the

Results and discussion

New algorithms of the type developed here should always be tested by observing their ability to reproduce results available from the literature. In our case, we applied our technique to the anisotropic solidification problem presented in Ref. [25] as well as to the transient melting analysis discussed in Ref. [52]. Following the success of these tests (for details see Ref. [47]), the algorithm was applied to a realistic problem in which kinetics-driven facetting is expected.

We have investigated

Conclusions

The appearance of facets along the melt/crystal interface during the directional growth of single crystals from the melt is a scientifically and technologically important phenomenon. Unfortunately, until now this effect has been overlooked by researchers applying large-scale computational analyses to relevant systems. In this manuscript a method for the computational analysis of facetting is proposed, tested, and applied. The vertical gradient freeze growth of oxide slabs is chosen as a model

Acknowledgements

This Research was supported by The Technion V.P.R. Fund–Henri Gutwirth promotion of research fund and by The Israel Science Foundation administered by The Israel Academy of Sciences and Humanities. A.V. acknowledges partial support by the Israel Absorption Ministry. The authors would like to thank A.A. Chernov, J.J. Derby, R. Ghez and V.V. Voronkov for their very useful remarks.

References (60)

J.J. Derby et al.
Comput. Methods. Appl. Mech. Eng.
(1994)
F. Dupret et al.
Int. J. Heat Mass Transfer
(1990)
S. Brandon et al.
J. Crystal Growth
(1992)
Q. Xiao et al.
J. Crystal Growth
(1993)
S. Brandon et al.
J. Crystal Growth
(1996)
P.M. Adornato et al.
J. Crystal Growth
(1987)
D.H. Kim et al.
J. Crystal Growth
(1989)
S. Kuppurao et al.
J. Crystal Growth
(1996)
C.W. Lan et al.
J. Crystal Growth
(1998)
A.G. Petrosyan et al.
J. Crystal Growth
(1976)

X. Xuewu et al.

J. Crystal Growth

(1993)

T. Abe

J. Crystal Growth

(1974)

J.P. Farges et al.

J. Crystal Growth

(1987)

T.P. Chen et al.

J. Crystal Growth

(1990)

S. O'Hara et al.

J. Crystal Growth

(1968)

A.M. Ovrutsky

J. Crystal Growth

(1992)

E. Yokoyama et al.

J. Crystal Growth

(1992)

J.C. Brice

J. Crystal Growth

(1970)

S. Brandon et al.

J. Crystal Growth

(1991)

Q. Xiao et al.

J. Crystal Growth

(1994)

D.T.J. Hurle

J. Crystal Growth

(1995)

J. Amon et al.

J. Crystal Growth

(1998)

H. Chung et al.

J. Crystal Growth

(1998)

W.C. Carter et al.

Acta Metall. Matter

(1995)

S. Kuppurao et al.

J. Crystal Growth

(1995)

K.N. Christodoulou et al.

J. Comp. Phys.

(1992)

E. Bauser et al.

J. Crystal Growth

(1984)

R.F. Sekerka

J. Crystal Growth

(1993)

R.A. Brown

AICHE J.

(1988)

F. Dupret et al.

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¹: Current address: Department of Chemical Engineering, Tianjin University, Tianjin 300072, People's Republic of China.

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Facetting during directional growth of oxides from the melt: coupling between thermal fields, kinetics and melt/crystal interface shapes

Abstract

Introduction

Section snippets

Model formulation

Numerical technique

Results and discussion

Conclusions

Acknowledgements

Comput. Methods. Appl. Mech. Eng.

Int. J. Heat Mass Transfer

J. Crystal Growth

J. Crystal Growth

J. Crystal Growth

J. Crystal Growth

J. Crystal Growth

J. Crystal Growth

J. Crystal Growth

J. Crystal Growth

J. Crystal Growth

J. Crystal Growth

J. Crystal Growth

J. Crystal Growth

J. Crystal Growth

J. Crystal Growth

J. Crystal Growth

J. Crystal Growth

J. Crystal Growth

J. Crystal Growth

J. Crystal Growth

J. Crystal Growth

J. Crystal Growth

Acta Metall. Matter

J. Crystal Growth

J. Comp. Phys.

J. Crystal Growth

J. Crystal Growth

AICHE J.