(crystallography) The growth of a crystal, which involves diffusion of the molecules of the crystallizing substance to the surface of the crystal, diffusion of these molecules over the crystal surface to special sites on the surface, incorporation of molecules into the surface at these sites, and diffusion of heat away from the surface.
Crystal growth
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The growth of crystals, which can occur by natural or artificial processes. Crystal growth generally comes about by means of the following processes occurring in series: (1) diffusion of the atoms (or molecules, in the case of molecular crystals such as hydrocarbons or polymers) of the crystallizing substance through the surrounding environment (or solution) to the surface of the crystal, (2) diffusion of these atoms over the surface of the crystal to special sites on the surface, (3) incorporation of atoms into the crystal at these sites, and (4) diffusion of the heat of crystallization away from the crystal surface. The rate of crystal growth may be limited by any of these four steps. The initial formation of the centers from which crystal growth proceeds is known as nucleation. See also Crystallization; Nucleation.
During its growth into a fluid phase, a crystal often develops and maintains a definite polyhedral form which may reflect the characteristic symmetry of the microscopic pattern of atomic arrangement in the crystal. The bounding faces of this form are those which are perpendicular to the directions of slowest growth. How this comes about is illustrated in Fig. 1, in which it is seen that the faces b, normal to the faster-growing direction, disappear, and the faces a, normal to the slower-growing directions, become predominant. Growth forms, like that shown, are not necessarily equilibrium forms, but they are likely to be most regular when the departures from equilibrium are not large. See also Crystal structure.
Schematic representation of cross section of crystal at three stages of regular growth.
The atomic binding sites on the surface of a crystal can be of several kinds. Thus an atom must be more weakly bound on a perfectly developed plane of atoms at the crystal surface (site A) than at a ledge formed by an incomplete plane one atom thick (site B). Atom A binds with only three neighboring atoms, whereas atom B binds to five neighbors. (An atom in the middle of the island monolayer has bonds with nine neighbors.) Therefore, the binding of atoms in an island monolayer on the crystal surface will be less per atom than it would be within a completed surface layer.
The potential energy of a crystal is most likely to be minimum in forms containing the fewest possible ledge sites. This means that, in a regime of regular crystal growth, dilute fluid, and moderate departure from equilibrium, the crystal faces of the growth form are likely to be densely packed and atomically smooth. There will be a critical size of monolayer, which will be a decreasing function of supersaturation, such that all monolayers smaller than the critical size tend to shrink out of existence, and those which are larger will grow to a complete layer. The critical monolayers form by a fluctuation process. Kinetic analyses indicate that the probability of critical fluctuations is so small that in finite systems perfect crystals will not grow, except at substantial departures from equilibrium. That, in ordinary experience, finite crystals do grow in a regular regime only at infinitesimal departures from equilibrium is explained by the screw dislocation theory. According to this theory, growth is sustained by indestructible surface ledges which result from the emergence of screw dislocations in the crystal face. See also Crystal defects.
When the departures from equilibrium (supersaturation or undercooling) are sufficiently large, the more regular growth shapes become unstable and cellular (grasslike) or dendritic (treelike) morphologies develop. Essentially, the development of protuberances on an initially regular crystal permits more efficient removal of latent heat or of impurities, but at the cost of higher interfacial area and the associated excess surface energy. When the supersaturation becomes so great that the energy associated with the increase in interfacial area is unimportant, protuberances proliferate and the crystal grows in a multibranched form that is even more complicated; its shape is characterized by fractal geometry. See also Fractals.
The advent of semiconductor-based technology generated a demand for large, high-quality single crystals, not only of semiconductors but also of associated electronic materials. With increasing sophistication of semiconductor devices, an added degree of freedom in materials properties was obtained by varying the composition of major components of the semiconductor crystal over very short distances. Thin, multilayered single-crystal structures, and even structures that vary in composition both normal and lateral to the growth direction, are often required.
Bulk single crystals are usually grown from a liquid phase. The liquid may have approximately the same composition as the solid; it may be a solution consisting primarily of one component of the crystal; or it may be a solution whose solvent constitutes at most a minor fraction of the crystal's composition. The most important bulk crystal growth technique is the crystal-pulling or Czochralski method, in which a rotating seed crystal is dipped into the melt (Fig. 2). Rotation reduces radial temperature gradients, and slow withdrawal of the rotating seed results in growth of a cylinder of single-crystal material. Crystal diameter and length depend upon the details of the temperature and pulling rate, and the dimensions of the melt container. Crystal quality depends very critically upon minimization of temperature gradients that enhance the formation of dislocations. Pulled silicon crystals 6 in. (15 cm) in diameter are important for the semiconductor industry. Ruby, sapphire, and group 13–15 compound semiconductor crystals are among the many crystals that are routinely grown by the Czochralski technique.
Czochralski crystal growth and temperature distribution.
The evolution of methods for the growth of very thin but very high-quality epitaxial layers has resulted largely from the need for such layers of semiconductors and magnetic garnets. The technique most closely related to the methods used for bulk crystal growth is liquid-phase epitaxy. For a typical binary semiconductor, growth is done onto a substrate single-crystal wafer from a solution rich in the component with the lowest partial pressure. For a binary compound, the grown layers may differ only in impurity concentrations to modify their electrical characteristics. More often, multilayered structures with layers differing in major component composition but having the same crystal structure and lattice parameter are required. The simplest example of liquid-phase epitaxy with major composition changes in layers is the growth of layers of aluminum gallium arsenide (AlxGa1−xAs; 1 > x > 0) on a gallium arsenide (GaAs) substrate.
Growth by liquid-phase epitaxy is done in an apparatus in which the substrate wafer is sequentially brought into contact with solutions that are at the desired compositions and may be supersaturated or cooled to achieve growth. For crystalline solid solutions other than AlxGa1−xAs, very precise control over solution compositions is required to achieve a lattice match. Typically, structures grown by liquid-phase epitaxy have four to six layers ranging widely in composition and having thickness from 10−7 to 10−6 m.
The desirability of highly reproducible growth and even thinner epitaxial layers of 13–15 compounds on large wafer areas has led to the development of molecular-beam epitaxy and several forms of chemical-vapor deposition. Molecular-beam epitaxy is an ultrahigh-vacuum technique in which beams of atoms or molecules of the constituent elements of the crystal provided by heated effusion ovens, impinge upon a heated substrate crystal. It has been used for epitaxial layers as thin as 0.5 nanometer. Molecular-beam epitaxy has also been used for group 12–16 compounds and for silicon. See also Artificially layered structures; Molecular beams; Semiconductor; Semiconductor heterostructures; Vapor deposition.
| Geography Dictionary: crystal growth |
The growth and expansion of crystals of salt or ice along cracks and fissures in a rock. This expansion causes pressure and splits up the rock. It is thus a form of weathering.
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It has been suggested that this article or section be merged with Crystallization and Recrystallization_(chemistry). (Discuss) |
The crystalline state of matter is characterized by a distinct structural rigidity and virtual resistance to deformation (i.e. changes of shape and/or volume). Most crystalline solids have high values both of Young's modulus and of the shear modulus of elasticity. This contrasts with most liquids or fluids, which have a low shear modulus, and typically exhibit the capacity for macroscopic viscous flow.
Crystal growth is a major stage of a crystallization process, which typically follows an initial stage of either homogeneous or heterogeneous (surface catalyzed) nucleation. It occurs from the addition of new atoms, ions, or polymer strings into the characteristic arrangement of a crystalline Bravais lattice.
The action of crystal growth yields a crystalline solid whose atoms or molecules are typically close packed, with fixed positions in space relative to each other. This accounts for the object's structural rigidity. In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. A specific symmetry or crystal structure is composed of a Bravais lattice which is typically represented by a single unit cell. The unit cell is periodically repeated in three dimensions on a lattice. The spacing between unit cells in various directions is called its lattice parameters. The symmetry properties of the crystal are embodied in its space group. A crystal's structure and symmetry play a role in determining many of its physical properties, such as cleavage, electronic band structure, and optical properties.
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Introduction
Crystalline solids are typically formed by cooling and solidification from the molten (or liquid) state. According to the Ehrenfest classification of first-order phase transitions, there is a discontinuous change in volume (and thus a discontinuity in the slope or first derivative with respect to temperature, dV/dT) at the melting point. Within this context, the crystal and melt are distinct phases with an interfacial discontinuity having a surface of tension with a positive surface energy. Thus, a metastable parent phase is always stable with respect to the nucleation of small embryos or droplets from a daughter phase, provided it has a positive surface of tension. Such first-order transitions must proceed by the advancement of an interfacial region whose structure and properties vary discontinuously from the parent phase. [1] [2] [3] [4]
The process of nucleation and growth generally occurs in two different stages. In the first nucleation stage, a small nucleus containing the newly forming crystal is created. Nucleation occurs relatively slowly as the initial crystal components must impinge on each other in the correct orientation and placement for them to adhere and form the crystal. After crystal nucleation, the second stage of growth rapidly ensues. Crystal growth spreads outwards from the nucleating site. In this faster process, the elements which form the motif add to the growing crystal in a prearranged system, the crystal lattice, started in crystal nucleation. As first pointed out by Frank, perfect crystals would only grow exceedingly slowly. Real crystals grow comparatively rapidly because they contain dislocations (and other defects), which provide the necessary growth points, thus providing the necessary catalyst for structural transformation and long-range order formation.[5]
Discontinuity
The conditions of a homogeneous environment are often approximated to but rarely ever realized. Crystal growth always involves some form of transport of matter or heat (or both). And homogeneous conditions for the transport process can only exist for spherical, cylindrical, or infinite plane surfaces. A polyhedral crystal cannot grow (remaining polyhedral) with uniform levels of supersaturation (or supercooling) over its faces. In general, the supersaturation is greatest at its corners. This refutes the assumption that the growth rate is a function of orientation and local supersaturation.
Thus, the crystal face must grow as a whole. The growth rate of the entire face is determined by the driving force (level of supersaturation) at the point of emergence of the predominant point of growth (e.g. a dislocation, a foreign particle acting as catalyst, or crystal twin). The defect-free habit face can thus resist a finite level of supersaturation without any growth at all.
Gibbs himself was the first to point out that in the growth of a perfect crystal, the first derivative of the free energy with respect to mass becomes periodically undefinable — at each time that an additional layer on the crystal face is completed. There is discontinuity in the chemical potential at each such point.
In one sense, the crystal can then be in equilibrium with environments having a range of chemical potentials. In another sense, it is not in equilibrium. There are available states of lower free energy. But any free energy barrier must be passed by a fluctuation, or nucleation process, in order to access it. The fundamental thermodynamic effect of a screw dislocation is to eliminate this discontinuity in the chemical potential, by making it impossible to ever complete a single crystal face.
Nucleation
Main article: Nucleation
Nucleation can be either homogeneous, without the influence of foreign particles, or heterogeneous, with the influence of foreign particles. Generally, heterogeneous nucleation takes place more quickly since the foreign particles act as a scaffold for the crystal to grow on, thus eliminating the necessity of creating a new surface and the incipient surface energy requirements.
Heterogeneous nucleation can take place by several methods. Some of the most typical are small inclusions, or cuts, in the container the crystal is being grown on. This includes scratches on the sides and bottom of glassware. A common practice in crystal growing is to add a foreign substance, such as a string or a rock, to the solution, thereby providing a nucleating site for the project and speeding up the time it will take to grow a crystal.
The number of nucleating sites can also be controlled in this manner. If a brand-new piece of glassware or a plastic container is used, crystals may not form because the container surface is too smooth to allow heterogeneous nucleation. On the other hand, a badly scratched container will result in many lines of small crystals. To achieve a moderate number of medium sized crystals, a container which has a few scratches works best. Likewise, adding small previously made crystals, or seed crystals, to a crystal growing project will provide nucleating sites to the solution. The addition of only one seed crystal should result in a larger single crystal.
Some important features during growth are the arrangement, the origin of growth, the interface form (important for the driving force), and the final size. When origin of growth is only in one direction for all the crystals, it can result in the material becoming very anisotropic (different properties in different directions). The interface form determines the additional free energy for each volume of crystal growth.
Lattice arrangement in metals often takes the structure of body centered cubic, face centered cubic, or hexagonal close packed. The final size of the crystal is important for mechanical properties of materials. (For example, in metals it is widely acknowledged that large crystals can stretch further due to the longer deformation path and thus lower internal stresses.).
Mechanisms of growth
The interface between a crystal and its vapor can be molecularly sharp at temperatures well below the melting point. An ideal crystalline surface grows by the spreading of single layers, or equivalently, by the lateral advance of the growth steps bounding the layers. For perceptible growth rates, this mechanism requires a finite driving force (or degree of supercooling) in order to lower the nucleation barrier sufficiently for nucleation to occur by means of thermal fluctuations.[6] In the theory of crystal growth from the melt, Burton and Cabrera have distinguished between two major mechanisms:[7][8]
- Non-uniform lateral growth. The surface advances by the lateral motion of steps which are one interplanar spacing in height (or some integral multiple thereof). An element of surface undergoes no change and does not advance normal to itself except during the passage of a step, and then it advances by the step height. It is useful to consider the step as the transition between two adjacent regions of a surface which are parallel to each other and thus identical in configuration — displaced from each other by an integral number of lattice planes. Note here the distinct possibility of a step in a diffuse surface, even thought the step height would be much smaller than the thickness of the diffuse surface.
- Uniform normal growth. The surface advances normal to itself without the necessity of a stepwise growth mechanism. This mean that in the presence of a sufficient thermodynamic driving force, every element of surface is capable of a continuous change contributing to the advancement of the interface. For a sharp or discontinuous surface, this continuous change may be more or less uniform over large areas each successive new layer. For a more diffuse surface, a continuous growth mechanism may require change over several successive layers simultaneously.
Non-uniform lateral growth is a geometrical motion of steps — as opposed to motion of the entire surface normal to itself. Alternatively, uniform normal growth is based on the time sequence of an element of surface. In this mode, there is no motion or change except when a step passes via a continual change. The prediction of which mechanism will be operative under any set of given conditions is fundamental to the understanding of crystal growth. Two criteria have been used to make this prediction:
- Whether or not the surface is diffuse. A diffuse surface is one in which the change from one phase to another is continuous, occurring over several atomic planes. This is in contrast to a sharp surface for which the major change in property (e.g. density or composition) is discontinuous, and is generally confined to a depth of one interplanar distance.[9][10]
- Whether or not the surface is singular. A singular surface is one in which the surface tension as a function of orientation has a pointed minimum. Growth of singular surfaces is known to requires steps, whereas it is generally held that non-singular surfaces can continuously advance normal to themselves.[11]
Driving force
Consider next the necessary requirements for the appearance of lateral growth. It is evident that the lateral growth mechanism will be found when any area in the surface can reach a metastable equilibrium in the presence of a driving force. It will then tend to remain in such an equilibrium configuration until the passage of a step. Afterward, the configuration will be identical except that each part of the step will have advanced by the step height. If the surface cannot reach equilibrium in the presence of a driving force, then it will continue to advance without waiting for the lateral motion of steps.
Thus, Cahn concluded that the distinguishing feature is the ability of the surface to reach an equilibrium state in the presence of the driving force. He also concluded that for every surface or interface in a crystalline medium, there exists a critical driving force, which, if exceeded, will enable the surface or interface to advance normal to itself, and, if not exceeded, will require the lateral growth mechanism.
Thus, for sufficiently large driving forces, the interface can move uniformly without the benefit of either a heterogeneous nucleation or screw dislocation mechanism. What constitutes a sufficiently large driving force depends upon the diffuseness of the interface, so that for extremely diffuse interfaces, this critical driving force will be so small that any measurable driving force will exceed it. Alternatively, for sharp interfaces, the critical driving force will be very large, and most growth will occur by the lateral step mechanism.
Note that in a typical solidification or crystallization process, the thermodynamic driving force is dictated by the degree of supercooling.
Morphology
It is generally believed that the mechanical and other properties of the crystal are also pertinent the subject matter, and that crystal morphology provides the missing link between growth kinetics and physical properties. The necessary thermodynamic apparatus was provided by Gibbs study of heterogeneous equilibrium. He provided us with the clear definition of surface energy, by which the concept of surface tension is made applicable to solids as well as liquids. He also appreciated that an anisotropic surface free energy implied a non-spherical equilibrium shape, which should be thermodynamically defined as the shape which minimizes the total surface free energy. [12]
It may be instructional to note that whisker growth provides the link between the mechanical phenomenon of high strength in whiskers and the various growth mechanisms which are responsible for their fibrous morphologies. (Prior to the discovery of carbon nanotubes, single-crystal whiskers had the highest tensile strength of any materials known). Some mechanisms produce defect-free whiskers, while others may have single screw dislocations along the main axis of growth — producing high strength whiskers.
The mechanism behind whisker growth is not well understood, but seems to be encouraged by compressive mechanical stresses including mechanically induced stresses, stresses induced by diffusion of different elements, and thermally induced stresses. Metal whiskers differ from metallic dendrites in several respects. Dendrites are fern-shaped like the branches of a tree, and grow across the surface of the metal. In contrast, whiskers are fibrous and project at a right angle to the surface of growth, or substrate.
Diffusion-control
Very commonly when the supersaturation (or degree of supercooling) is high, and sometimes even when it is not high, growth kinetics may be diffusion-controlled. Under such conditions, the polyhedral crystal form will be unstable, it will sprout protrusions at its corners and edges where the degree of supersaturation is at its highest level. The tips of these protrusions will clearly be the points of highest supersaturation. It is generally believed that the protrusion will become longer (and thinner at the tip) until the effect of interfacial free energy in raising the chemical potential slows the tip growth and maintains a constant value for the tip thickness.
In the subsequent tip-thickening process, there should be s corresponding instability of shape. Minor bumps or "bulges" should be exaggerated — and develop into rapidly growing side branches. In such an unstable (or metastable) situation, minor degrees of anisotropy should be sufficient to determine directions of significant branching and growth. The most appealing aspect of this argument, of course, is that it yields the primary morphological features of dendritic growth.
See also
- Cloud condensation nuclei
- Crystal structure
- Crystallization
- Czochralski process
- Dendrite (crystal)
- Fractional crystallization
- Glass transition
- Ice nucleus
- Laser-heated pedestal growth
- Manganese nodule
- Micro-pulling-down
- Monocrystalline whisker
- Physics of glass
- Protocrystalline
- Recrystallization
- Seed crystal
- Single crystal
References
- ^ Atkins, P.W., Physical Chemistry (W.H. Freeman & Co., New York, 1994)
- ^ Hilliard, J.E. and Cahn, J.W., On the Nature of the Interface Between a Solid Metal and Its Melt, Acta Met., Vol. 6, p. 772 (1958)
- ^ Cahn, J.W., Theory of crystal growth and interface motion in crystalline materials, Acta Met, Vol. 8, p. 554 (1960)
- ^ Cahn, J.W., Hillig, W.B. and Sears, G.W., The molecular mechanism of solidification, Acta Met., Vol. 12, p. 1421 (1964)
- ^ Frank, F.C., The influence of dislocations on crystal growth, Disc. Faraday Soc., Vol. 5, p. 48 (1949)
- ^ Volmer, M., Kinetic der Phasenbildung, T. Steinkopf, Dresden (1939)
- ^ Burton, W.K., Cabrera, N., Crystal growth and surface structure I, Discuss. Faraday Soc., Vol. 5, p. 33 (1949)
- ^ Cabrera, N., Burton, W.K., Crystal growth and surface structure II, Discuss. Faraday Soc., Vol. 5, p. 40 (1949)
- ^ Burton, W.K., Cabrera, N., Frank, F., The Growth of Crystals and the Equilibrium Structure of their Surfaces, Phil. Trans. Royal Soc. Lond., Vol A243, p. 299 (1951)
- ^ Jackson, K.A. in Growth and Perfection of Crystals, Ed. Doremus, R.H., Roberts, B.W. and Turnbull, D. (Wiley, New York, 1958)
- ^ Cabrera, N., The structure of crystal surfaces, Discuss. Faraday Soc., Vol. 28, p. 16 (1959)
- ^ Gibbs, J.W., On the Equilibrium of Heterogeneous Substances, Collected Works, (Longmans, Green & Co., New York, 1928)
Further reading
- "Mechanical and Physical Properties of Whiskers", CRC Handbook of Chemistry and Physics, 55th edition.
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Mentioned in
- twinning (crystallography)
- epitaxy (crystallography)
- growth step (crystallography)
- salt crust (hydrology)
- salt weathering (geology)
- Dendrite
- extrusion
- Sphenoidal
- overgrowth
- transformation twin (crystallography)
- adcumulus (petrology)
- Single crystal
- weathering
- intrusion
