Equilibrium and growth shapes of crystals: how do they differ and why should we care?

Abstract

Since the death of Prof. Dr. Jan Czochralski nearly 50 years ago, crystals grown by the Czochralski method have increased remarkably in size and perfection, resulting today in the industrial production of silicon crystals about 30 cm in diameter and two meters in length. The Czochralski method is of great technological and economic importance for semiconductors and optical crystals. Over this same time period, there have been equally dramatic improvements in our theoretical understanding of crystal growth morphology. Today we can compute complex crystal growth shapes from robust models that reproduce most of the features and phenomena observed experimentally. We should care about this because it is likely to result in the development of powerful and economical design tools to enable future progress. Crystal growth morphology results from an interplay of crystallographic anisotropy and growth kinetics by means of interfacial processes and long-range transport. The equilibrium shape of a crystal results from minimizing its anisotropic surface free energy under the constraint of constant volume; it is given by the classical Wulff construction but can also be represented by an analytical formula based on the ξ-vector formalism of Hoffman and Cahn. We now have analytic criteria for missing orientations (sharp corners or edges) on the equilibrium shape, both in two (classical) and three (new) dimensions. Crystals that grow under the control of interfacial kinetic processes tend asymptotically toward a "kinetic Wulff shape," the analogue of the Wulff shape, except it is based on the anisotropic interfacial kinetic coefficient. If it were not for long range transport, crystals would presumably nucleate with their equilibrium shape and then evolve toward their "kinetic Wulff shape." Allowing for long range transport leads to morphological instabilities on the scale of the geometric mean of a transport length (typically a diffusivity divided by the growth speed) and a capillary length (of the order of atomic dimensions). Resulting crystal growth shapes can be cellular or dendritic, but can also exhibit corners and facets related to the underlying crystallographic anisotropy. Within the last decade, powerful phase field models, based on a diffuse interface, have been used to treat simultaneously all of the above phenomena. Computed morphologies can exhibit cells, dendrites and facets, and the geometry of isotherms and isoconcentrates can also be determined. Results of such computations are illustrated in both two and three dimensions.

Jan Czochralski, the pioneer of crystal research

“…As one may know, both kinds of processes are based on one-seed and many-seeds nucleation and growth-phase transformation rules [5]. However, they rely on different competition behaviour of the feeding material, coming either from a solution or from some melt [4,5]. e basic difference concerning an (un)even distribution of the feeding material is illustrated schematically in Figs.…”

“…is results ultimately in crystalline polymorphic forms [6]. Which of the present-time developments are, in my personal opinion, the most important ones when seen in the context of Czochralski's well-known [1][2][3][4][5] accomplishments? It seems to me that there are at least two: (i) e solid-liquid complex crystal-interface dynamics, in general characteristic of (non)linear morphological instabilities and primarily resulting from concerted actions of mass-and thermal diffusion fields [6]; (ii) Not only was Jan Czochralski the first to propose a method of pulling metallic single crystals from a melt, and to successfully quantify their growth rates, he passionately devoted his entire productive life to crystal research, material science and chemistry, to name only a few areas of his expertise [1].…”

“…is describes a diffusive mass and/or heat transport accross a crystalliquid (originally: ice-water) interface [8]. Many attempts have been made since the early 1960s to achieve a robust nonlinear extension of the morphological, though originally linear, crystal-involving (in)stability analysis by Mullins and Sekerka (MS) [4,5]. e MS problem was originally treated as being driven either by a concentration gradient or by undercooling [5].…”

“…e MS problem was originally treated as being driven either by a concentration gradient or by undercooling [5]. e result of the instability analysis of the growth process has been a square-root-of-time evolution of the sphere radius, as well as an exponential time decay of its crystal-surface perturbation amplitude [4]. However, none of the attempts were able to reproduce an accepted universal and robust solution to the problem of crystal-phase advancement into its ambient (solution/melt) phase, irrespective of the fact that the phenomenon had always been proposed to be diffusion-controlled [4].…”

“…In just three pages the author reported on his new findings, namely, that a method of pulling metallic single crystals had been discovered in terms of their growth rates (and nucleation-promoting ionic additives), and that it had been applied for testing crystal growth of three metals: Sn, Pb, and Zn. is raised, in fact, great admiration both in Poland and abroad [1,[3][4][5]. As a consequence of this work, he is still recognized as one of the founders of today's crystal-growth technology and research [3], although originally his method was rather elaborated for the three metals mentioned above.…”