Crystalline solids

Crystal Structure

In three dimensions there are three lattice constants, a,b, and c, and three angles: α alpha between b and c, β beta between a and c, and γ lambda between lattice constants a and b. There are 14 Bravais lattices, ranging from the lowest symmetry triclinic type in which all three lattice constants and all three angles differ from each other (a ≠b≠c and α ≠ β ≠ γ ) to the highest symmetry cubic casein which all the lattice constants are equal and all the angles are 90 ◦ (a = b = c and α = β = γ = 90 ◦) . there are three Bravais lattices in the cubic system: a primitive or simple cubic (sc) lattice in which the atoms occupy the eight apices of the cubic unit cell, a body centred cubic (bcc) lattice with lattice points occupied at the apices and in the centre of the unit cell; and a face-centred cubic (fcc) Bravais lattice with atoms at the apices and in the centres of the faces.
In two dimensions the most efficient way to pack identical circles or spheres is the equilateral triangle arrangement, corresponding to the hexagonal Bravais lattice. A second hexagonal layer of spheres can be positioned on top of the first to form the most efficient packing of two layers. For efficient packing, the third layer can be placed either above the first layer with an atom at the location indicated by T or in the third possible arrangement with an atom above the positon marked by X . in the first case a hexagonal lattice structure called hexagonal close-packed hcp is generated, and in the second case a fcc lattice results. The former is easy to identify in a unit cell, but the latter is not so easy to visualize while looking at a unit cell since the lcose packed planes are oriented perpendicular to [111] direction.

In three dimensions the unit cell of the fcc structure is the cube which has a side (lattice constant) a and volume a^3.it has six face-centred atoms, each shard by two unit cells, and eight apical atoms, each shared by eight unit cells, corresponding to a total of four for this individual unit cell........

The pysics and chemistry of Nanosolids, Owen F, Poole C, Wiley



Extending the argument to three dimensions to consider the electronic properties of bulk crystalline solids, it is to explain for a perfectly ordered three dimensional crystal - that the periodic repetition of atoms (or molecules) along the one dimensional linear-chain should be replaced by the periodic repetition of a unit cell in all three dimensions. The unit cell contains atoms arranged in the characteristic configuration of the crystal, such that continuous replication of the unit cell throughout all space is sufficient to generate the entire crystal structure. In other words, the crystal has translational symmetry, and the crystal structure may be generated by translations of the unit cell in all three dimensions. Translation symmetry in a periodic structure is a so called discrete symmetry, because only certain translations – those corresponding to integer multiples of the lattice translation vectors derived from the unit cell – lead to symmetry equivalent points. (this may be contrasted with the case of empty space, which displays a continuous translation symmetry because any translation leads to a symmetry-equivalent point). Common unit cells are simple cubic, face centred cubic, body centred cubic, and the diamond structure, which comprises two interlocking face centred cubic lattices. However, in general the lattice spacing may be different along the different principal axes, giving rise to the orthorhombic and tetragonal unit cells, and sides of the unit cell may not necessarily be orthogonal, such as in the hexagonal unit cell.

Generally, symmetries generate conservation laws; this is known as Noether’s theorem. The continuous translation symmetry of empty space generates the law of momentum conservation; the weaker discrete translation symmetry in crystals leads to a weaker quasi-conservation law for quasi-or crystal momentum. An important consequence of discrete translation symmetry for the electronic properties of crystals is Bloch’s theorem.


Periodicity of crystal lattices
Such wavefunctions represent travelling waves passing through the crystal, but with a form modified periodically by the crystal potential due to each atomic site.


Electronic conduction
….now the total number of valence electrons in the crystal is zNu where z is the number of valence electrons per unit cell. This leads to two very different electronic configurations in a solid. If z is even, then one energy band is completely filled, with the next band being completely empty. The highest filled band is the valence band, and the next, empty band, is the conduction band. The electrons in the valence band cannot participate in electrical conduction, because there are no available states for them to move into consistent with the small increase in energy required by motion in response to an externally applied voltage: hence this configuration results in an insulator or , if the band gap is sufficiently small, a semiconductor. Alternatively, if z is odd, then the highest occupied energy band is only half full. In such a material, there are many vacant states immediately adjacent in energy to the highest occupied states, therefore electrical conduction occurs very efficiently and the material is a metal.

………….if an atom is located at a surface then it is clear that the number of nearest neighbour atoms are reduced giving rise to differences in bonding leading to the well known phenomenon of surface tension or surface energy and electronic structure. In a small isolated nanoparticle, a large proportion of the total number of atoms will be present either at or near th e free surface. For instance, in a 5 nm particle approximately 30-50% of the atoms are influenced by the surface, compared with approx a few percent for a 100 nm particle. similar arguments apply to nanocrystalline materials, where a large proportion of atoms will be either at or near grain boundaries. such structural differences in reduced dimensional systems would be expected to lead to very different properties form the bulk.

Kelsall, Ian Hamley, Mark Geoghegan; Nanoscale science and technology, WILEY


The problem of chemistry and biology can be greatly reduced if our ability to see what we are doing, and to do things on an atomic level, is turned into everyday practice of every educational programme.

Nonetheless, at the end of the day the law of business rather than of science and engineering formulates future development of nanotechnology.



Creating lattice matched interfaces

…..Silicon and silicon-dioxide have very different lattice constants – spacing between their atoms. It is therefore difficult to imagine that the interface between them can be electronically perfect. GaAs and AlAs on the other side have almost equal lattice spacing, and two crystals can be perfectly place on top of each other. The formation of superlattices of such layers of semiconductors has, in fact, been one of the bigger achievements of recent semiconductor technology and was made possible by new techniques of crystal growth (molecular beam epitaxy, metal organic chemical vapour deposition, and the like).
Quantum walls, wires, and dots have been the subjectof extremely interesting research and have enriched quantum physics for by the discovery of the Quantum Hall Effect. Use of such layers has also brought significant progress to semiconductor electronics. The concept of modulation doping – selective doping of layers, particularly involving pseudomorphic InGaAs – has led to modulation-doped transistors that hold the current speed records and are used for microwave applications. The removal of the doping to neighboring layers permitted the creation of the highest possible electron mobilities and velocities. The effect of resonant tunnelling has also been shown to lead to ultra fast devices and applications that reach to infrared frequencies, encompassing in this way both optics and electronics applications……silicon dioxide, as an insulator, is superior to all possible III-V compound materials; and its interface with silicon can be made electronically perfect enough, at least when treated with hydrogen or deuterium.
When it comes to optical applications, however, silicon is inefficient because it is an indirect semiconductor and therefore cannot emit light efficiently. Light generation may be possible by using silicon but, this is limited by the laws of physics and materials science. …..III-V compounds and quantum well layers have been successfully used to create efficient light emitting devices including light emitting and semiconductor laser diodes. These are ubiquitous in every household, eg in CD players and in the back lights of cars. New forms of laser diodes, such as the so called vertical cavity surface emitting laser diodes VCSELs, are even suitable to relatively large integration. One can put thousands and even millions of them on a chip. Optical pattern generation has made great advances by use of selective superlattice intermixing (compositionally disordered III-V compounds and superlattices have a different index of refraction) and by other methods. This is an area in great flux and with many possibilities for miniaturization. Layered semiconductors and quantum well structures have also led to new forms of lasers such as the quantum cascade laser.
A number of ideas are in discussion for new forms of ultra small electronic switching and storage devices. Using the simple fact that it takes a finite energy to bring a single electron from one capacitor plate to the other (and using tunnelling for doing so), single electron transistors have been proposed and built. …it also has been shown that memory cells storing only a few electrons do have some very attractive features. eg if many electrons are stored in a larger volume, a single material defect can lead to unwanted discharge of the whole volume. If on the other hand all these electrons are stored in a larger number of quantum dots (each carrying few electrons), a single defect can discharge only a single dot, and the remainder of the stored charge stays intact. Two electrons stored on a square shaped quantum dot have been proposed as a switching element by researchers…….. The electrons start residing in a pair of opposite corners of t he square and are switched to the other opposite corner. This switching can be effected by the electrons residing in a neighbouring rectangular dot. Domino type effects can thus be achieved.

A new field referred to as spintronics is developing around the spin properties of particles. Spin properties have not been explored in conventional electronics and enter only indirectly, through the Pauli principle, into the equations for transistors. Of particular interest in this new area are particle pairs that exhibit quantum entanglement. Consider a pair of particles in a singlet spin-state sent out to detectors or spin analyzers in opposite directions. Such a pair has the following remarkable properties: measurements of the spin on each side separately give random values of the spin. However, the spin of one side is always correlated to the spin on the other side. If one is up, the other is down. If the spin analyzers are rotated relative to each other, then the result for the spin pair correlation shows rotational symmetry. A theorem of Bell proclaims such results incompatible with Einstein’s relativity and suggests the necessity of instantaneous influences at a distance.

The handbook of nanoscience, Taylor&Francis



Lattice defects

Lattice defects in SrTiO3 single crystals were characterized by X-ray topography and transmission electron microscopy. We examined two groups of crystals whose lapped faces were (001) and (011), respectively. After taking X-ray topographs, crystals which included relatively many defects were chosen for detailed investigation by transmission electron microscopy, which gave the following results: (i) some subgrain boundaries observed by X-ray topography were small-angle tilt boundaries; and (ii) many dislocations were found in the region where thick line contrast was observed in X-ray topographs. Most of them had <100> type Burgers vectors.

Conclusion: It is difficult generally to observe lattice defects by TEM when the density of defeas is low. In this study, we overcame this experimental barrier using XRT together with TEM. The observations of lattice defeas in Verneuil-grown SrT103 single crystals under no external force were conduaed and the following results were obtained, (i) The subgrain boundary that was observed in the X-ray topograph was found to be a small-angle tilt boundary, (ii) Many dislocations existed in the region where the thick line contrast was seen in the X-ray image. Most of them had <100> type Burgers vectors.

........We carried out a competitive experiment using another crystal plate crashed at room temperature. The result of this experiment shows that many dislocations were formed in this crashed specimen, in contrast to the original state.

Characterization of lattice defects in strontium titanate single crystals by X-ray topography and transmission electron microscopy; www.oxfordjournals.org





Precipitate:







Precipitation is the formation of a solid in a solution during a chemical reaction. When the reaction occurs, the solid formed is called the precipitate, and the liquid remaining above the solid is called the supernate.

Precipitation is also useful to isolate the products of a reaction during workup. Ideally, the product of the reaction is insoluble in the reaction solvent. Thus, it precipitates as it is formed, preferably forming pure crystals. An example of this would be the syn-thesis of porphyrins in refluxing propionic acid. By cooling the reaction mixture to room temperature, crystals of the porphyrin precipitate, and are collected by filtration.

Wikipedia
Chemical dictionary http://www.ktf-split.hr/glossary/en_o.php?def=digestion




The regular arrangement of atoms in a material is, of course, the distinguishing feature of a crystal as compared with an amorphous substance. Many of the important properties of solids can be explained only if details of the structure of atoms in crystals are taken into account. ....in a stable configuration of atoms the intertomic separation is usally between 0.2 nm and 0.3 nm. The precise arrangement depends largely on the form of the electron clouds which surround the central positive nucleus of every atom. In many cases the crystal structure of a solid changes at certain temperautes, but following structures are stable at room temperture:

Cubic structures, close packed structues, hexagonal close packed structure, face-centred cubic structure, the diamond structure.





"Love is most nearly itself when here and now cease to matter". T.S. Eliot




The role of vacancies in grapheme layers

When the enrgy of the electron beam exceeds the threshold energy for atom displacements, atoms can be removed from their positions by knock-on collisions. The displacement threshold energy of the electrons is approximately 100 keV in graphite and 200 keV in diamond. Due to momentum conservation, the enrgies transferred to the displaced atoms are much lower: 15eV in graphite and 30 eV in diamond. The displaced atoms migrate as interstitials through the lattice; their migration is thermally activated and governed by an Arrhenius law. Monovacancies in a grapheme layer are stable until recombination with interstitials occurs. An atom displacement adjacent to an already existing vacancy creates a divacancy. Such a divacancy in graphene can vanish just via reconstruction of the network. Removing the two encircled atoms and rearranging the bonds reduces the number of hexagons from six to five, whereas the number of pentagons remains unchanged. The same process starting from a flat network of hexagons is quite complicated and includes many rearrangements; generally, the removal of two atoms from a graphene plane reduces the number of hexagons by one. Such rearrangements ar eassumed to occur by the Stone-Wales mechanism and have been the subject of much theoretical work. Non six membered rings result from the removal of atoms and rearrangements and lead to curvature of the layer. Through the stone-wales mechanism, the arrangement of pentagons and heptagons can alternate, this changes the topology and leads to a high flexibility of the layer. When a finite layer is curved, dangling bonds at the edge promote a strong tendency towards closure; the energetic optimum is reached when complete spherical closure is attained. We can conclude that irradiation of graphitic structures in an electron microscope leads to isolated monovacancies that are open sites in a layer and an only close by recombination with an interstitial carbon atom or, for short periods, with foreign atoms; and divacancies or multiple vacancies with an even number, which can lead to structureal rearrangements and generate topological defects such as curvature of the layers.

Journal of electron microscopy 51, 2002, www.oxfordjournals.org







Covalent bonding

In ionic crystals electrons are transferred from one atom eg. Na to another eg Cl in order that both will thereby have closed shells of electrons. Another type of interaction arises if the electron-cloud geometry is distorted sufficiently so that electrons are continually being shared between neighbouring atoms, thereby forming completed shells on all of them. By sharing we mean that he electron spends as much time near one atom as it does near another one and so it is impossible to say that the electron now belongs to a particular atom. This type of sharing is especially important if each atom has a half filled outer shell eg. The carbon atom has four outer electrons in a shell which can accommodate eight. We have already described that in the diamond lattice each carbon atom many be considered to be at the centre of a tetrahedron with its nearest neighbours at the four corners of the tetrahedron. Each of these four outer atoms can be thought of as sharing one of its electrons with the central atom, thereby making up a closed shell of eight electrons. It should be noted that the central atom which as was pointed out is not in any privileged position, but is actually equivalent to the corner atoms, will share each of its four electrons with the four corner atoms, thereby contributing ot their closed shells of eight electrons.
The calculation of covalent bonding energies and forces is very hazardous because it demands a computation of the amount of overlap of the electron charge clouds for neighbouring atoms – it involves what is called an exchange integral – and it is not possible to calculate this with any confidence. The fact that the amount of overlap, and hence the reduction in energy due to the overlap, varies very rapidly with intertomic spacing implies that the binding forces associated with covalent bonding are usually very strong ones. The crystals are usually hard and they have high melting points.

In diamond the covalent bonding is very effective. In other crystals, and especially in compounds neighbouring atoms might not share their electrons equally, ie an electron from atom A might spend more time near atom B, than an electron from B spends near A. if this occurs then atoms A and B will be partially ionized and there will be electrostatic forces between them. The binding will therefore not be completely covalent but it will be partially ionic in character. The mixed binding is very prevalent in compounds which have the zincblende structure we can envisage a whole spectrum of different crystals in which there is a graduation in the type of binding from being completely ionic (electron transfer, but no sharing) to mixed (unequal sharing) and finally to covalent bonds (equal bonds).





The diamond sturcture

the crystal structue of diamond can be derived from the fcc lattice although it is not itself a close packed arrangement. Formally it may be described as being built up from two interprenetrating fcc lattices which are offset with respect to one another along the body-diagonal of the cube by one quarter of its legth. Whilst this definition accurately describes the position of the atoms, it is not very useful when one tries to envisage the positions of the atoms relative to one anothe. A more useful model is to consider each atom to be at the centre of a tetrahedron with its four nearest neighbours at the four corners of that tetrahedron. If the central atom of the tetrahedron is on one fcc sub-lattice then the four corner atoms are all on the other sub-lattice. It should also be noted that since all the atoms are equivalent each of them can be thoguht of as being the centre atom of a tetrahedron, although it would require a rather extended diagram to show this. The tetrahedral arrangement in the diamond crystal in which each atom has four equally spaced nearest neighbours is a consequenc eof th e special type of electron sharing (covalent bonding) which occurs in these structures. Apart from diamond, the other elements which have the same structure are also tetravalent - germanium and silicon. Tin also has a diamond-structure allotrope grey tin which is stable below about -40 C.
If the two interpenetrating lattices are of two different elements the atoms on different sub-lattices are of course no longer equivalent although they will still yield a similar tetrahedral arrangement. This is then called the zincblende ZnS structure and it is typical of the 3-5 semiconducting crystals such as LnSb, GaAs, GaP.



Hydrogen bonding

Hydrogen has only one electron and it can therefore only be covalently bonded to one other atom. If however, this second atom is strongly electron- negative (fluorine and oxygen are the most important examples) then the electron from the hydrogen spends most of its time on the other atom and the hydrogen is left positively charged. It can then be attracted to another electronegative atom on a neighbouring molecule. It is this attraction between molecules or sometimes between parts of a large molecule which is known as the hydrogen bond. The cohesive forces in ice are due to hydrogen bonding.


Rosenberg, The solid state, Clarendon Press, 1975


Anisotropy

The physical properties of single crystals of some substances depend on the crystallographic direction in which measurements are taken. for example, the elastic modulus, the electrical conductivity, and the index of refraction may have different values in the [100] and [111] directions. This directionality of properties is termed anisotropy, and it is associated with the variance of atomic or ionic spacing with crystallographic direction. substances in which measured properties are independent of the direction of measurement are isotropic. The extent adn magnitude of anisotropic effects in crystalline materials are functions of the symmetry of the crystal structure; the degree of anisotropy increases with decreasing structural symmetry - triclinic structures normally are highly anisotropic.

Callister W D, (1991) Materials science and engineering, WILEY