The Dutch physicist Johannes D. van der Waals first proposed the force that binds molecular solids. Any two atoms or molecules have a force of attraction (F) that varies according to the inverse seventh power of the distance R between the centres of the atoms or molecules: F = −C/R7, where C is a constant. The force, known as the van der Waals force, declines rapidly with the distance R and is quite weak. If the atoms or molecules have a net charge, there is a strong force whose strength varies according to Coulomb’s law as the inverse second power of the separation distance: F = −C′/R2, where C′ is a constant. This force provides the binding in ionic crystals and some of the binding in metals. Coulomb’s law does not apply to atoms or molecules without a net charge. Molecules with a dipole moment, such as water, have a strong attractive force owing to the interactions between the dipoles. For atoms and molecules with neither net charges nor dipole moments, the van der Waals force provides the crystal binding. The force of gravity also acts between neutral atoms and molecules, but it is far too weak to bind molecules into crystals.
The van der Waals force is caused by quantum fluctuations. Two neighbouring atoms that are each fluctuating can lower their joint energy by correlating their fluctuations. The van der Waals force arises from correlations in their dipole fluctuations. Electrons bound in atomic orbits are in constant motion around the nucleus, and the distribution of charges in the atom changes constantly as the electrons move, owing to quantum fluctuations. One fluctuation might produce a momentary electric dipole moment (i.e., a separation of charges) on an atom if a majority of its electrons are on one side of the nucleus. The dipole moment creates an electric field on a neighbouring atom; this field will induce a dipole moment on the second atom. The two dipoles attract one another via the van der Waals interaction. Since the force depends on the inverse seventh power of the distance, it declines rapidly with increasing distance. Atoms have a typical radius of one to three angstroms. The van der Waals force binds atoms and molecules within a few angstroms of each other; beyond that distance the force is negligible. Although weak, the van der Waals force is always present and is important in cases where the other forces are absent.
Hydrogen is rarely found as a single atom. Instead it forms diatomic molecules (H2), which are gaseous at room temperature. At lower temperatures the hydrogen becomes a liquid and at about 20 K turns into a solid. The molecule retains its identity in the liquid and solid states. The solid exists as a molecular crystal of covalently bound H2 molecules. The molecules attract one another by van der Waals forces, which provide the crystal binding. Helium, the second element in the periodic table, has two electrons, which constitute a filled atomic shell. In its liquid and solid states, the helium atoms are bound together by van der Waals forces. In fact, all the rare gases (helium, neon, argon, krypton, and xenon) are molecular crystals with the binding provided by van der Waals forces.
Many organic molecules form crystals where the molecules are bound by van der Waals forces. In methane (CH4), a central carbon makes a covalent bond with each hydrogen atom, forming a tetrahedron. In crystalline methane the molecules are arranged in the fcc structure. Benzene (C6H6) has the carbon atoms in a hexagonal ring; each carbon has three coplanar σ bonds, as in graphite, where two bonds are with neighbouring carbon atoms and the third bond is with a hydrogen atom. Crystalline benzene has four molecules per unit cell in a complex arrangement. Fullerene and the rare gas atoms are spherical, and the crystalline arrangement corresponds to the closest packing of spheres. Most organic molecules, however, are not spherical and display irregular shapes. For odd-shaped molecules, the van der Waals interaction depends on the rotational orientation of the two molecules. In order to maximize the force, the molecules in the crystal have unusual arrangements, as in the case of benzene.
Hydrogen bonding
Hydrogen bonding is important in a few crystals, notably in ice. With its lone electron, a hydrogen atom usually forms a single covalent bond with an electronegative atom. In the hydrogen bond the atom is ionized to a proton. The proton sits between two anions and joins them. Hydrogen bonding occurs with only the most electronegative ions: nitrogen, oxygen, and fluorine. In water the hydrogen links pairs of oxygen ions. Water is found in many different crystal structures, but they all have the feature that the hydrogen atoms sit between pairs of oxygen. Another hydrogen-bonded solid is hydrogen fluoride (HF), in which the hydrogen atom (proton) links pairs of fluorines.
Crystal growth
The earliest crystal grower was nature. Many excellent crystals of minerals formed in the geologic past are found in mines and caves throughout the world. Most precious and semiprecious stones are well-formed crystals. Early efforts to produce synthetic crystals were concentrated on making gems. Synthetic ruby was grown by the French scientist Marc Antoine Augustin Gaudin in 1873. Since about 1950 scientists have learned to grow in the laboratory crystals of quality equal or superior to those found in nature. New techniques for growth are continually being developed, and crystals with three or more atoms per unit cell are continually being discovered.
Vapour growth
Crystals can be grown from a vapour when the molecules of the gas attach themselves to a surface and move into the crystal arrangement. Several important conditions must be met for this to occur. At constant temperature and equilibrium conditions, the average number of molecules in the gas and solid states is constant; molecules leave the gas and attach to the surface at the same rate that they leave the surface to become gas molecules. For crystals to grow, the gas-solid chemical system must be in a nonequilibrium state such that there are too many gaseous molecules for the conditions of pressure and temperature. This state is called supersaturation. Molecules are more prone to leave the gas than to rejoin it, so they become deposited on the surface of the container. Supersaturation can be induced by maintaining the crystal at a lower temperature than the gas. A critical stage in the growth of a crystal is seeding, in which a small piece of crystal of the proper structure and orientation, called a seed, is introduced into the container. The gas molecules find the seed a more favourable surface than the walls and preferentially deposit there. Once the molecule is on the surface of the seed, it wanders around this surface to find the preferred site for attachment. Growth proceeds one molecule at a time and one layer at a time. The process is slow; it takes days to grow a small crystal. Crystals are grown at temperatures well below the melting point to reduce the density of defects. The advantage of vapour growth is that very pure crystals can be grown by this method, while the disadvantage is that it is slow.
Most clouds in the atmosphere are ice crystals that form by vapour growth from water molecules. Most raindrops are crystals as they begin descending but thaw during their fall to Earth. Seeding for rain—accomplished by dropping silver iodide crystals from airplanes—is known to induce precipitation. In the laboratory, vapour growth is usually accomplished by flowing a supersaturated gas over a seed crystal. Quite often a chemical reaction at the surface is needed to deposit the atoms. Crystals of silicon can be grown by flowing chlorosilane (SiCl4) and hydrogen (H2) over a seed crystal of silicon. Hydrogen acts as the buffer gas by controlling the temperature and rate of flow. The molecules dissociate on the surface in a chemical reaction that forms hydrogen chloride (HCl) molecules. Hydrogen chloride molecules leave the surface, while silicon atoms remain to grow into a crystal. Binary crystals such as gallium arsenide (GaAs) are grown by a similar method. One process employs gallium chloride (GaCl) as the gallium carrier. Arsenic is provided by molecules such as arsenous chloride (AsCl3), arsine (AsH3), or As4 (yellow arsenic). These molecules, with hydrogen as the buffer gas, grow crystals of gallium arsenide while forming gas molecules such as gallium trichloride (GaCl3) and hydrogen chloride. Trimethylgallium, (CH3)3Ga, is another molecule that can be used to deliver gallium to the surface.
Growth from solution
Large single crystals may be grown from solution. In this technique the seed crystal is immersed in a solvent that contains typically about 10–30 percent of the desired solute. The choice of solvent usually depends on the solubility of the solute. The temperature and pH (a measure of acidity or basicity) of the solution must be well controlled. The method is faster than vapour growth, because there is a higher concentration of molecules at the surface in a liquid as compared to a gas, but it is still relatively slow.
Growth from the melt
This method is the most basic. A gas is cooled until it becomes a liquid, which is then cooled further until it becomes a solid. Polycrystalline solids are typically produced by this method unless special techniques are employed. In any case, the temperature must be controlled carefully. Large crystals can be grown rapidly from the liquid elements using a popular method invented in 1918 by the Polish scientist Jan Czochralski and called crystal pulling. One attaches a seed crystal to the bottom of a vertical arm such that the seed is barely in contact with the material at the surface of the melt. A modern Czochralski apparatus is shown in . The arm is raised slowly, and a crystal grows underneath at the interface between the crystal and the melt. Usually the crystal is rotated slowly, so that inhomogeneities in the liquid are not replicated in the crystal. Large-diameter crystals of silicon are grown in this way for use as computer chips. Based on measurements of the weight of the crystal during the pulling process, computer-controlled apparatuses can vary the pulling rate to produce any desired diameter. Crystal pulling is the least expensive way to grow large amounts of pure crystal. A photograph of a single crystal of stainless steel grown by the Czochralski method is shown in . The original seed is on the right tip. Binary crystals can also be pulled; for example, synthetic sapphire crystals can be pulled from molten alumina. Special care is required to grow binary and other multicomponent crystals; the temperature must be precisely controlled because such crystals may be grown only at a single, extremely high temperature. The melt has a tendency to be inhomogeneous, since the two liquids may try to separate by gravity.
The Bridgman method (named after the American scientist Percy Williams Bridgman) is also widely used for growing large single crystals. The molten material is put into a crucible, often of silica, which has a cylindrical shape with a conical lower end. Heaters maintain the molten state. As the crucible is slowly lowered into a cooler region, a crystal starts growing in the conical tip. The crucible is lowered at a rate that matches the growth of the crystal, so that the interface between crystal and melt is always at the same temperature. The rate of moving the crucible depends on the temperature and the material. When done successfully, the entire molten material in the crucible grows into a single large crystal. One disadvantage of the method is that excess impurities are pushed out of the crystal during growth. A layer of impurities grows at the interface between melt and solid as this surface moves up the melt, and the impurities become concentrated in the higher part of the crystal.
Epitaxy is the technique of growing a crystal, layer by layer, on the atomically flat surface of another crystal. In homoepitaxy a crystal is grown on a substrate of the same material. Silicon layers of different impurity content, for example, are grown on silicon substrates in the manufacture of computer chips. Heteroepitaxy, on the other hand, is the growth of one crystal on the substrate of another. Silicon substrates are often used since they are readily available in atomically smooth form. Many different semiconductor crystals can be grown on silicon, such as gallium arsenide, germanium, cadmium telluride (CdTe), and lead telluride (PbTe). Any flat substrate can be used for epitaxy, however, and insulators such as rock salt (NaCl) and magnesium oxide (MgO) are also used.
Molecular-beam epitaxy, commonly abbreviated as MBE, is a form of vapour growth. The field began when the American scientist John Read Arthur reported in 1968 that gallium arsenide could be grown by sending a beam of gallium atoms and arsenic molecules toward the flat surface of a crystal of the molecule. The amount of gas molecules can be controlled to grow just one layer, or just two, or any desired amount. This method is slow, since molecular beams have low densities of atoms. Chemical vapour deposition (CVD) is another form of epitaxy that makes use of the vapour growth technique. Also known as vapour-phase epitaxy (VPE), it is much faster than MBE since the atoms are delivered in a flowing gas rather than in a molecular beam. Synthetic diamonds are grown by CVD. Rapid growth occurs when methane (CH4) is mixed with atomic hydrogen gas, which serves as a catalyst. Methane dissociates on a heated surface of diamond. The carbon remains on the surface, and the hydrogen leaves as a molecule. Growth rates are several micrometres (1 micrometre is equal to 0.00004 inch) per hour. At that rate, a stone 1 centimetre (0.4 inch) thick is grown in 18 weeks. CVD diamonds are of poor quality as gemstones but are important electronic materials. Because hydrogen is found in nature as a molecule rather than as a single atom, making atomic hydrogen gas is the major expense in growing CVD diamonds. Liquid-phase epitaxy (LPE) uses the solution method to grow crystals on a substrate. The substrate is placed in a solution with a saturated concentration of solute. This technique is used to grow many crystals employed in modern electronics and optoelectronic devices, such as gallium arsenide, gallium aluminum arsenide, and gallium phosphide.
An important concern in successful epitaxy is matching lattice distances. If the spacing between atoms in the substrate is close to that of the top crystal, then that crystal will grow well; a small difference in lattice distance can be accommodated as the top crystal grows. When the lattice distances are different, however, the top crystal becomes deformed, since structural defects such as dislocations appear (see ). Although few crystals share the same lattice distance, a number of examples are known. Aluminum arsenide and gallium arsenide have the same crystal structure and the same lattice parameters to within 0.1 percent; they grow excellent crystals on one another. Such materials, known as superlattices, have a repeated structure of n layers of GaAs, m layers of AlAs, n layers of GaAs, m layers of AlAs, and so forth. Superlattices represent artificially created structures that are thermodynamically stable; they have many applications in the modern electronics industry. Another lattice-matched epitaxial system is mercury telluride (HgTe) and cadmium telluride (CdTe). These two semiconductors form a continuous semiconductor alloy CdxHg1 − xTe, where x is any number between 0 and 1. This alloy is used as a detector of infrared radiation and is incorporated in particular in night-vision goggles.
Dendritic growth
At slow rates of crystal growth, the interface between melt and solid remains planar, and growth occurs uniformly across the surface. At faster rates of crystal growth, instabilities are more likely to occur; this leads to dendritic growth. Solidification releases excess energy in the form of heat at the interface between solid and melt. At slow growth rates, the heat leaves the surface by diffusion. Rapid growth creates more heat, which is dissipated by convection (liquid flow) when diffusion is too slow. Convection breaks the planar symmetry so that crystal growth develops along columns, or “fingers,” rather than along planes. Each crystal has certain directions in which growth is fastest, and dendrites grow in these directions. As the columns grow larger, their surfaces become flatter and more unstable. This feather or tree structure is characteristic of dendritic growth. Snowflakes are an example of crystals that result from dendritic growth.
