Crystal Growth and Nonlinear Optics

CHAPTER 1
INTRODUCTION TO CRYSTAL GROWTH AND NONLINEAR OPTICS
1.1 INTRODUCTION
Crystal growth is regarded as an ancient subject, owing to the fact that the crystallization of salt and sugar were known to the ancient Indian and Chinese civilizations. The subject of crystal growth was treated as part of crystallography and never had an independent identity until the last century. It has a long history of evolution from “a substance potting art” to a science in its own right which has accelerated by the invention of transistor in 1948, and the subsequent need for high purity semiconductor single crystals.

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Crystals are the unacknowledged pillars of modern technology. The fundamentals of crystal growth was entirely bestowed upon the morphological studies of the naturally occurring crystals. Thus began the scientific approach for this subject during the seventeenth century by Kepler, followed by quite a few others like Nicolous Steno, Descartes, Bartholinus, etc. This type of morphological study slowly led to the understanding of the atomistic process of crystal growth. Recent bursting research on nanostructured materials depend on the crystal growth theory and technology. In the early twentieth century, the crystal growth evolved as a separate branch of science and several theories from Kossel, Donnay-Harker, Volmer and Burton, Cabrera and Frank (BCF) were proposed. Although science of crystal growth originated through the explanations of Nicolous Steno in 1669, the actual impetus to this field began after the BCF theory was formulated and also when there was a great demand for crystals during World War II.
Crystal growth plays an important role in material science and engineering. It is an interdisciplinary subject of physics and chemistry. Initially the natural crystals were adored as gems and museum pieces. Later, a transition of crystals has occurred from museum to technology which stimulated crystal grower community to produce large crystals artificially. In the recent scientific era, the utility of crystals has been extended to novel devices such as nonlinear optical and piezoelectric devices.
Atomic arrangement with periodicity in three dimensional pattern at equally repeated distances are called single crystals. The preparation of single crystal is more difficult than polycrystalline material and extra effort is justified because of outstanding properties of single crystals (Laudise 1970). The single crystal growth has prominent role in the present era because of rapid technical and scientific advancement. The application of crystals has unbounded limits because of its special optical and electrical properties over noncrystalline material. This means that the new crystals have to be grown and fabricated in order to assess their device properties. The main parameters which involve in crystal growth are nucleation, growth rate, stability, crystalline defects, compositional inhomogeneity and thermodynamics of the source of liquid.
The evolution in the crystal growth requires not only scientific understanding, but also the driving force of applied technology which so often provides a significant influence in highlighting the lack of scientific knowledge and need for a more refined evolution of science and indeed the development of new concepts. The studies on the growth and physical properties of single crystals of amino acids and their compounds are of great interest because they possess properties such as piezoelectricity, pyroelectricity and possibly ferroelectricity.
In the recent century, the development of science in many areas has been achieved through the growth of single crystals. The single crystals designed for producing second harmonic generation (SHG) received consistent attention for applications in the field of telecommunication, optical information processing, laser remote sensing and colour displays.
1.2 KINETICS OF CRYSTAL GROWTH
Crystals are solid substances in general which may be obtained from solid, liquid or vapour phase. Except for solid phase, all other phases yield crystals with developed faces, which represent the crystal medium interface during the development of a crystal from the growth medium. Subsequently, the crystal faces contain information about the nature of the interfaces as well as about the phenomena taking place at the interface.
In solid phase growth, some grains grow larger at the expense of others and the interface mainly concave with respect to the growing grain and lies in the interior of the bulk mass.
In melt growth, the interface is forced to take the shape of the isotherm inside the crucible containing the melt. However, in both cases, a free development of the faces is rarely encountered. It is also possible to obtain valuable information about the growth processes by using suitable methods. Elementary processes involved in the development of the micromorphology of as grown surfaces of bulk single crystal and epitaxial layers, and of evaporated and etched surfaces under different experimental conditions are essentially similar irrespective of the type of a material.
When a crystal nucleus attains a critical size, then it grows into crystal of macroscopic dimension with well developed faces. Several theories have been proposed to explain the mechanism of crystal growth. They are:

Surface energy theory,
Adsorption layer theory and
Diffusion theory.

The surface energy theory states that the growing crystal assumes a shape, which has a minimum surface energy. According to adsorption layer theory, a molecule arriving at a crystal surface from the bulk of the supersaturated solution or super cooled melt loses a part of its latent heat. All molecules similar to this move along the surface and join together to form a small two dimensional nucleus due to inelastic collision. Bravious proposed that the growth rate of a crystal face depends on reticular densities of a lattice point of that face. The surface energy is the least when the face possesses the greatest reticular density. The attachment energy is due to Vander Waals force in the case of homopolar crystals and it is due to electrostatic forces in the case of ionic crystals. According to the diffusion theory matter is deposited continuously on a crystal phase at the rate proportional to the difference in concentration between the point of deposition and the bulk of the solution.
In diffusion theory, the molecules in contact with the crystal surface are adsorbed quickly. A concentration gradient is thus produced between the bulk of the solution and the growing crystal surface. The mass transfer from the bulk of the solution to the surface involves molecular diffusion. In general, in any crystal growth process, the following steps are involved:
(i) Generation of reactants
(ii) Transport of reactants to the growth surface
(iii) Adsorption at the growth surface
(iv) Nucleation
(v) Growth and
(vi) Removal of unwanted reaction products from the growth surface
1.2.1 Solution, Solubility and Super Solubility
A solution is a homogeneous mixture of a solute in a solvent. Solute is a component, which is present in a smaller quantity. For a given solute, there may be different solvents. The solvent is chosen taking into account of the following factors to grow crystals from solution:
(i) Good solubility for the given solute
(ii) Good temperature coefficient of solute solubility

Less viscosity

(iv) Less volatility
(v) Less corrosion and non toxicity
(vi) Low vapour pressure and
(vii) Cost advantage
Solubility of the material in a solvent decides the amount of the material, which is available for the growth and hence defines the total size limit. Solubility gradient is another important parameter, which dictates the growth procedure. If the solubility gradient is very small, slow evaporation of the solvent is the best option for crystal growth in order to maintain a constant supersaturation in the solution. Growth of crystals from solution is mainly a diffusion-controlled process. The medium must be viscous enough to enable faster transference of the growth units from the bulk solution by diffusion. Hence, a solvent with less viscosity is preferable. Supersaturation is an important parameter for the solution growth process. The crystal grows by the access of the solute in the solution where the degree of supersaturation is maintained. The solubility data at various temperatures are essential to determine the level of supersaturation. Hence, the solubility of the solute in the chosen solvent must be determined before starting the growth process.
The relationship between the equilibrium concentrations as a function of temperature is represented by the solubility diagram in Figure 1.1 which is known as temperature-concentration diagram. Miers carried out extensive research in the relationship between supersaturation and spontaneous crystallization. The lower continuous line is the normal solubility curve for the salt concerned. Temperature and concentration at which spontaneous crystallization occurs are represented by the upper broken curve, generally referred to as the supersolubility curve. The whole concentration-temperature field is separated by the saturated solution line (solubility curve) into two regions, unsaturated and supersaturated solutions. Saturated solutions are those mixtures, which can retain their equilibrium indefinitely in contact with the solid phase with respect to which they are saturated. The solubility of most substances increase with temperature (the temperature coefficient of the solubility is positive) and crystals can be grown only from supersaturated solutions, which contain an excess of the solute above the equilibrium value. The temperature-concentration diagram is divided into three regions, which are termed as region I, II and III respectively.

Figure 1.1 Miers solubility curve
(i)The stable (undersaturated) zone where crystallization is not possible (Region I).
(ii)The region II is a metastable zone, between the solubility and supersolubility curves, where spontaneous crystallization is improbable. However, if a seed crystal is placed in metastable solution, growth would occur on it.
(iii)The region III is an unstable or labile (supersaturation) zone, where spontaneous crystallization is more probable.
If the solution whose concentration and temperature represented by point A in the Figure. 1.1, is cooled without loss of solvent (Line ABC) spontaneous crystallization cannot occur until conditions represented by point C are reached. At this point, crystallization is spontaneous. Further cooling to some point D will produce spurious nucleation. The evaporation of solvent from the solution results in supersaturation. The line AB’C’ represents an operation carried out at constant temperature. Penetration beyond the supersolubility curve into the labile zone rarely happens, as the surface from which evaporation takes place is usually supersaturated to a greater degree than the bulk of solution. Crystals, which appear on this surface eventually fall into the solution and seed in it. In practice, a combination of cooling and evaporation as represented by the line AB”C” is also adopted.
1.2.2 Expression for Supersaturation
In order to grow crystals, the solution must be supersaturated. Supersaturation is the driving force, which governs the rate of crystal growth. The supersaturation of a system may be expressed in number of ways. The basic units of concentration as well as temperature must be specified.
The degree of supersaturation of a solution is defined using the concept of absolute supersaturation
(1.1)
where C is the concentration of the dissolved substance at a given moment and Co is its solubility limit. The degree of supersaturation can also be defined as the relative supersaturation, which is given by
(1.2)
or as the coefficient of supersaturation.
(1.3)
The quantities α, β and Г are interrelated (Khamshii 1969)
1.3 NUCLEATION
In a supersaturated or super cooled system, few atoms or molecules join together and a change in energy takes place during the formation of clusters. The cluster of atoms or molecules is called embryo. An embryo may grow or disintegrate and disappear completely. If the embryo grows to a particular size, critical size known as critical nucleus, then there is a tendency for the nucleus to grow. Thus, nucleation is an important phenomenon in crystal growth and is the precursor of crystal growth and of the overall crystallization process. The formation of stable nucleus occurs only by the addition of a number of molecules (A1) until a critical cluster is formed. In general
A n-1 + A 1 → A n (Critical) (1.4)
Any further addition to the critical nucleus results in nucleation followed by growth. Once these nucleus grow beyond a certain size, they become stable under the average condition of supersaturation of the solution. Further, the creation of a new phase in the homogeneous solution demands for the expenditure of certain quantity of energy. Once embryos achieve this critical size there is a high probability that they will grow, relatively unhindered, to macroscopic size.
1.3.1 Types of Nucleation
Nucleation may occur spontaneously or may be induced artificially. These two cases are frequently referred to as homogeneous and heterogeneous nucleation respectively. The term primary will be reserved for both the cases of nucleation in the systems that do not contain crystalline matter. On the other hand, the nucleus is often generated in the vicinity of crystals presented in the supersaturated system. This phenomenon is referred to as secondary nucleation. Figure 1.2 shows the classification of nucleation.
The spontaneous formation of crystalline nucleus in the interior of the parent phase is called homogeneous nucleation. If the nucleus forms heterogeneously around ions, impurity molecules or on dust particles, on surfaces or at structural irrgularities such as dislocations or other imperfections is called heterogeneous nucleation.

Figure 1.2 Schematic diagram indicating the classification of nucleation
Nucleation can often be induced by external processes like agitation, friction, mechanical shock, electromagnetic fields, extreme pressure, ultraviolet, X-rays, γ– rays, sonic and ultrasonic radiation and so on (Mullin 2001; Laudise 1975; Gilman 1963; Stringfellow 1979; Sangwal 1987; Jancic & Grootscholten 1984).
1.3.2 Energy of formation of a nucleus
Any isolated droplet of a fluid is most stable when its free energy is maximum and thus its area is minimum. The growth of an embryo or a crystal could be considered as an example of this principle. The total energy of the crystal in equilibrium with its surrounding at constant temperature and pressure would be minimum for a given volume. When a volume free energy per unit volume is considered to be constant
Σai σi = minimum (1.5)
whereai is area of ith face and
σi is surface energy per unit area
Thus considering the nucleus to be spherical, the energy of formation of the nucleus is determined.
1.3.3 Energy of Formation of Spherical Nucleus
The formation of a droplet nucleus due to supersaturation of vapour demands the expenditure of a certain quantity of energy in the creation of new phase. Therefore the total free energy change associated with the formation of homogeneous nucleation may be considered as follows. Let ΔG be the overall excess free energy of the embryo between the two phases. Since the volume and surface free energies, the total free energy associated with the process can be written as
ΔG = ΔG S + ΔG V (1.6)
where ΔGS is the surface free energy change and ΔGV is the volume free energy change. For a spherical nucleus of radius r,
ΔG = πr3 ΔG V + 4 πr2γ (1.7)
The first term expresses the formation of the new surface and the second term expresses the difference in the chemical potential between the crystalline phase and the surrounding mother liquid. Where γ is the interfacial tension and ΔGv is the free energy change per unit volume, which is a negative quantity, r the radius of the nucleus. Since the surface free energy increases with r2 and volume free energy decreases with r3, the total net free energy change increases with increase in size and attains a critical value after which it decreases. The size corresponding to the maximum free energy change is called critical nucleus.