Solids, nanoparticles, and interfaces
5.1 Solid formation
Decreasing the temperature or increasing the pressure on most liquids will eventually lead to solidification. At one atmosphere of pressure, the temperature of this phase transition is called the normal melting point or the normal temperature of fusion, Tf. At a pressure of 1 bar (the standard state pressure) this temperature is called the standard melting point or the standard temperature of fusion. In some cases a gas can be directly condensed into a solid. This is essential to the process of snowflake formation, and is also how solid CO2 (dry ice) forms at standard pressure. The condensation of a gas directly to a solid phase is called deposition. The reverse process – the conversion of a solid to the gas phase – is called sublimation.
Solids contract with decreasing temperature, as described by their expansion coefficient
For a simple cubic solid, the volume expansion coefficient αV is the cube of the linear expansion coefficient αl, some typical values of which are listed in Table 5.1. As can be seen from these values, the linear expansion coefficient varies by over one order of magnitude even for simple cubic solids.
Table 5.1 Selected parameters of elements in the solid state regarding their melting point Tf, mass density ρs and linear expansion coefficient αl. Values taken from the 96th edition of the CRC Handbook of Chemistry and Physics.
| Symbol | Tf / K | ρs(25 °C)/ kg m−3 | αl × 106/K−1 |
|---|---|---|---|
| Bi | 544.56 | 979 | 13.4 |
| Ga | 302.92 | 591 | 18 |
| Ge | 1211.40 | 532.3 | 5.8 |
| Si | 1687.15 | 232.9 | 2.6 |
| Ag | 1234.93 | 1050 | 18.9 |
| Al | 933.47 | 270 | 23.1 |
| Au | 1337.33 | 1930 | 14.2 |
| Fe | 1811.05 | 787 | 11.8 |
| Pb | 600.61 | 1130 | 28.9 |
| Pt | 2041.35 | 2150 | 8.8 |
| V | 2183.15 | 600 | 8.4 |
Contracting volume leads to an increase in density with decreasing temperature. Equivalently, as shown in Fig. 5.1, the molar volume decreases. Solid water exhibits typical behavior in Fig. 5.1; specifically, the density increases linearly for an extended range below Tf. However, all solids asymptotically achieve a maximum constant density as the temperature approaches absolute zero as a consequence of the third law of thermodynamics.
Figure 5.1 The mass density and molar volume of solid water. Values taken from the 96th edition of the CRC Handbook of Chemistry and Physics.
Solids can be held together by a range of chemical interactions. This leads to four distinct classifications of solid crystalline materials, namely ionic, covalent, metallic, and molecular.
Ionic crystals are held together by the electrostatic interactions that result from full charge transfer between atoms to form cations and anions in the crystal. Large electronegativity differences lead to the formation of ionic crystals. Typical examples are alkali and alkaline earth atoms bound with halides or chalcogenides, such as NaCl and MgO. However, II–VI1 (e.g., CdS, ZnSe and CdTe) and III–V (e.g., GaN, GaAs, InP and InSb) compound semiconductors also form crystals with significant ionic character, as do a number of other metal oxides, sulfides and halides. Ionic crystals exhibit ionic conductivity at elevated temperatures.
Covalent crystals are bound by the sharing of electrons in covalent bonds between atoms. The elemental semiconductors Si and Ge, as well as C-based solids, are examples. Metallic crystals are distinguished from covalent crystals by the delocalized nature of their valence electrons. The kinetic energy of the electrons is the primary binding force of the simple monatomic metals of the first two columns. This also contributes to the binding of transition metals. However, a much greater degree of covalency in their bonding translates into much higher cohesive energies for transition metals compared to the simple metals. The delocalized nature of metal bonding also contributes to the high electrical conductivity of metals. Molecular crystals are bound by weaker intermolecular forces such as van der Waals forces and hydrogen bonding. Thus, they often exhibit low cohesive energies and a lack of electrical and ionic conductivity.
5.2 Electronic structure of solids
An extremely important characteristic of a solid is its electrical resistivity, a parameter that gives us insight into the electronic structure of the solid. Electrical resistivity results from the scattering of electrons as they try to move through the electronic states of a solid. The structure of the electronic states is called its band structure. The electrons of an atom are classified as being either valence or core electrons. Core electrons in atoms, molecules and solids are much the same. They are tightly bound and localized near the nucleus and are little influenced by chemical changes. The valence electrons are strongly influenced by chemical changes and some participate directly in bonding. Valence electrons in solids form delocalized electronic states that are characterized by three-dimensional wavefunctions known as Bloch waves. Valence electrons in molecules are localized in molecular orbitals. Some of these may be extended, such as the π-systems in conjugated chains or aromatic rings, while others are highly localized near just one or two atoms. The valence electrons are sometimes localized (as in insulators) but are frequently delocalized (as in semiconductors, superconductors, and metals).
With reference to the band structure, it is instructive to categorize electrons as occupying states either in the valence band or the conduction band. The valence band is bonding, and states near its maximum energy are in some ways analogous to the highest occupied molecular orbital in a molecule. The conduction band is antibonding, and states near its minimum are analogous to the lowest unoccupied molecular orbital.
The free movement of electrons requires both occupied states (states with an electron in them that act as the source of an electron) and unoccupied states or holes (states into which the electron can move). Either electrons or holes can be thought of as transporting – or ‘carrying’ – electrical current; thus, they are called carriers. The requirement for both results from the Pauli exclusion principle and that electrons are Fermions. As no two electrons can have the same quantum numbers, an empty state is required if an electron is going to hop from one site to the next. Recall that two electrons can occupy one orbital, but when they do so they must have opposite spin quantum numbers.
A gross classification of materials can be made based on their resistivity and its relationship to the band structure shown in Fig. 5.2. A metal conducts electrons readily because its valence band and conduction band overlap. Thus, it has a high density of both occupied and unoccupied states available within kBT of one another – that is, within the range of thermal excitations available to the electrons. We learn from quantum mechanics that electrons follow Fermi–Dirac statistics, and it is the high-energy tail of this distribution that is important for electrical conductivity. As the temperature increases, the electrons scatter more frequently and the resistivity of a metal increases with increasing temperature as a result. The electrons in a metal fill all the states up to an energy called the Fermi energy, EF. Electrons at EF are bound in the metal by an amount known as the work function 
. The work function is the difference in energy between the Fermi energy and the vacuum energy Evac,
Figure 5.2 The presence and size of a gap between electronic states at the Fermi energy, EF, determines whether a material is a metal, semiconductor, or insulator. Eg, band gap; 
, work function, which is equal to the difference between EF and the vacuum energy, Evac.
The vacuum energy is the energy of an electron when it is far enough away from the solid that its energy is no longer influenced by it. Measurement of the work function is complicated by the need to remove the electron from the bulk of the material through its surface and into the vacuum. Work functions of real materials vary with the geometry and composition (cleanliness) of their surfaces. Therefore, in order to accurately measure the work function by the photoelectric effect, the surface of the material must be clean and the material must be held in a vacuum chamber so that it remains clean during the measurement.
A semiconductor has a band gap with an energetic width Eg that separates the valence band maximum from the conduction band minimum. This gap is small enough that thermal excitations can lead to a small density of carriers. Thermal excitation of an electron to the conduction band leaves behind a valence band hole, both of which can then move to transport electrical current. Because the density of these carriers is low, the resistivity of a semiconductor is high compared to that of a metal. Increasing the temperature increases the carrier density and reduces the resistivity – the opposite trend compared to a metal. Another way to control the resistivity of a semiconductor is to ‘dope’ electrons or holes into its band structure by introducing impurity atoms. Doping – the controlled introduction of electrically active impurities into a semiconductor – is achieved when an atom with a different valence is added to the semiconductor and electrons or holes are made available near the band edges. For example, B is from the group before Si. An atom of B has one fewer valence electrons than Si, but when added to the Si lattice it acts as a p-type dopant. It only adds three electrons, not four, to the valence band, like the Si atom it replaced. Hence, a hole is added to the valence band. A hole is positive. The B atom added a positive carrier so it is a p-type dopant. A P atom has five valence electrons, and when substituted into the Si lattice it adds one electron too many. The valence band is nominally full, and therefore this electron has to go into the conduction band. The P atom added an electron, a negative carrier; hence P is known as an n-type dopant.
Semiconductors are invariably composed of materials that exhibit a high degree of covalency in their bonding. Si and Ge are the most prototypical examples. GaAs and GaN represent mixtures of covalent and ionic bonding. A number of oxides are semiconductors with band gaps below 4 eV such as cerium oxides (ceria, CeO2 with a fluorite structure but also Ce2O3), In2O3 (bixbyite-type cubic structure and an indirect band gap of just 0.9–1.1 eV), SnO2 (rutile, 3.6 eV direct band gap), and ZnO (wurzite, 3.37 eV direct band gap). The ability of ceria to form two different valence states allows its surface and near-surface region to exist in a range of stoichiometries between CeO2 and Ce2O3, producing a so-called ‘mixed-valence’ oxide. When used as an additive in the support for automotive catalysis, this allows ceria to act as a sponge that absorbs or releases oxygen atoms, based on the reactive conditions. By substituting approximately 10% Sn into indium oxide, the resulting material – indium tin oxide (ITO) – combines two seemingly mutually exclusive properties: a thin film of ITO is both transparent and electrically conductive. This makes ITO very important in display (i.e., touch-screen) and photovoltaic technology.
If the valence band and conduction band are so far apart – that is, if the material has a wide band gap of greater than about 5 eV – then thermal excitations cannot provide carriers for conductivity. The resistance of wide band gap materials is high and they are called insulators. The electrons in insulators are not free to move because there are no easily accessible empty states. Ionic crystals such as alkali halide salts form insulators. Many oxides are insulators with band gaps in excess of 6 eV, such as SiO2 (this is known to form 35 different crystalline polymorphs including silica and quartz, but also glass), Al2O3 (also with several polymorphs including alumina, corundum or α-alumina, and γ-alumina), and MgO (magnesia with the same structure as NaCl). These oxides are of particular interest in chemistry because all of them are commonly used as substrates for supported heterogeneous catalysts. α-alumina doped with metal impurities is responsible for sapphire (Cr-, Fe- or Ti-doped) and ruby (Cr-doped).
Most macroscopic-sized, bulk elements are metals, and most macroscopic bulk compounds are insulators. Semiconductors are relatively rare. The responses of these three classes of materials to changes in the size of the structure are different. When materials enter the nanoscale regime (somewhere below 100 nm, with the precise behavior depending on the material and the property under consideration) their electronic structure becomes size-dependent once the wavefunctions of the bands start to extend over distances that are comparable to the size of the material. Semiconductors exhibit this behavior most prominently and over the largest size range. The reason why this is so noticeable with semiconductors is because of the size dependence of their band gap.
5.2.1 Directed practice
Why is the position of the Fermi energy important for determining the properties of a semiconductor?
5.3 Geometrical structure of solids
Crystalline solids are composed of atoms or groups of atoms arranged in an orderly fashion. The three-dimensional periodicity of the patterns distinguishes one crystal type from another. When noninteracting hard spheres are packed into the lowest possible volume – that is, into the highest possible density – they will assume a close-packed, face-centered cubic structure. However, the presence of directional chemical interactions between atoms leads to the wide variety of crystal structures. The unit cell is a small building block (in most cases the smallest) that can be translated and repeated through space to generate the structure of the entire crystal. Once the shape of the unit cell is specified, along with the arrangement of all the atoms within the unit cell, then one has a complete structural description of the crystal.
The concept of a space lattice was introduced by Auguste Bravais. The space lattice, a regular collection of points in three dimensions, is used to generate all possible arrangements of unit cells in a crystal. The periodic translational symmetry of a space lattice is described in terms of three primitive, noncoplanar basis vectors a1, a2, a3, defined such that any lattice point r(n1,n2, n3) can be generated from any other lattice point, for instance the origin at r(0,0,0), in the space translation operation
where
R is the translation vector and n1, n2, and n3 are integers. A lattice generated by a collection of such translation operations is called the simple Bravais lattice. The parallelepiped defined by the three basis vectors is the unit cell of the Bravais lattice. Figure 5.3 displays the three basis vectors and the parallelepiped they define, as well as defining three unit cell angles α, β, and γ. The unit cell with the smallest volume and lattice points located only at the vertices of the parallelepiped is called the primitive cell. The unit cell lengths of basis vectors in this primitive cell – a, b, and c – are called the lattice parameters.
Figure 5.3 The unit cell of a Bravais lattice and the relations of the parameters that define it: basis vectors a1, a2, a3, unit cell angles α, β and γ, lattice parameters a, b, and c.
The space lattice can be categorized depending on the relative lengths of the lattice parameters, the cell angles, and the number of lattice point in a unit cell. As detailed in Table 5.2 and Fig. 5.4, there are seven lattice systems and 14 Bravais lattices. The 14 Bravais lattices comprise a complete set of all lattices that form in Nature. They are formed by using all possible arrangements of lattice point in each unit cell. Each Bravais lattice is unique. The complete set is generated from the seven simple lattice systems, plus those cases in which a base-centered, a body-centered, or a face-centered lattice can be formed.
Table 5.2 The seven lattice systems and 14 Bravais lattices. The Bravais lattice is sometimes designated by adding a single-letter abbreviation to the lattice system name: P = simple; C = base-centered; I = body-centered; F = face-centered.
| Lattice system | Lattice constants and angles | Bravais lattices |
|---|---|---|
| Cubic |  | simple, body-centered (bcc), face-centered (fcc) |
| Hexagonal |  | simple (hcp) |
| Monoclinic |  | simple, base-centered |
| Orthorhombic | a ≠ b ≠ c | simple, base-centered, body-centered, face-centered |
| Tetragonal |  | simple, body-centered |
| Triclinic |  | simple |
| Trigonal |  | simple |
Figure 5.4 The 14 Bravais lattices. (a) Simple cubic; (b) bcc ; (c) fcc; (d) hcp; (e) simple monoclinic; (f) base-centered monoclinic; (g) orthorhombic P; (h) orthorhombic C; (i) orthorhombic I; (j) orthorhombic F; (k) rhombohedral; (l) tetragonal P; (m) tetragonal I; (n) triclinic.
A simple Bravais lattice contains lattice points only at the vertices of the parallelepiped. Base-centered means that the Bravais lattice contains a point in the center of the top and bottom faces in addition to points at the vertices. A body-centered Bravais lattice has lattice points at the vertices as well as one in the center of the cell. A face-centered Bravais lattice has lattice points in the center of each of the six faces as well as at the eight vertices.
Each Bravais lattice has different symmetry properties. The symmetry properties of the unit cell determine the symmetry properties of the bulk crystal. For example, a cubic crystal expands isotropically (equally along all three basis vectors), whereas this need not be the case for a triclinic crystal in which the basis vectors are, by definition, inequivalent. Each Bravais lattice must have translational symmetry since this is what was used to define it. In addition, it may also exhibit different degrees of rotational, reflection, inversion and improper rotation (rotation followed by inversion) symmetry. These are the same symmetry operation as described in Chapter 26 for molecules. Rotational symmetry means that rotation about one of the crystal axes by 2π/n radians leads to an equivalent lattice. Such a lattice is said to exhibit n-fold rotational symmetry. A onefold axis is trivial, but the only other allowed values of n are 2, 3, 4, and 6. A fivefold rotational axis is forbidden.
Each Bravais lattice has a distinct number of lattice points and therefore atoms (or constituent units) enclosed in it. To determine the number of atoms in a unit cell, you need to count the number of lattice points in a unit cell, as well as the number of atoms at each lattice point. Corner lattice points are shared by eight adjacent cells. Each only contributes one-eighth of a lattice point to the unit cell. Summing over all eight corners, they contribute just one lattice point in total to the unit cell. Mid-point of edge sites contribute one-quarter of a lattice point. Face-centered sites contribute one-half and body-centered sites contribute one lattice point. Real crystals may have either one or a set of atoms associated with each lattice point. Such a set of atoms is called the basis. A basis of a particular crystal is identical in composition, arrangement, and orientation. For example, I2 forms an orthorhombic crystal with two I atoms at each lattice point. The most stable crystal structures of the elements are detailed in Appendix 8.
As detailed in Table 5.3, the most important crystal structures for semiconductors are the diamond, zinc blende and wurzite structures. Zinc blende is also known as sphalerite. These structures are shown in Fig. 5.5. The diamond lattice is composed of two interpenetrating fcc lattices. The vertex atom of one of the lattices is located at the point (0,0,0). The second lattice is displaced such that its vertex lies at (a/4,a/4,a/4), where a is the lattice constant. The primitive basis of two identical atoms located at (0,0,0) and (a/4,a/4,a/4) is associated with each lattice point of the fcc lattice. For C, the diamond allotrope is metastable compared to graphite. However, for Si and Ge, diamond represents the most stable crystal structure. The zinc blende structure is similar to diamond in being a tetrahedrally coordinated phase. However, the two fcc lattices are occupied by two chemically distinct atoms, for instance, Ga and As in GaAs. Similarly, the wurzite structure is composed of two interpenetrating hexagonal lattices with chemically distinct atoms placed on them. Several compounds, including GaP, ZnS and CdSe, can exhibit either zinc blende or wurzite structures.
Figure 5.5 The structures most commonly exhibited by semiconductors (a) diamond (b) zinc blende and (c) wurzite.
The rocksalt (NaCl) structure is common among binary oxides of the general formula MX, such as MgO and NiO. The fluorite structure is common for MX2 oxides such as CeO2. An oxide of particular interest is TiO2, titania. The two most important crystalline forms of titania are rutile and anatase. Rutile has a band gap of ~3.2 eV, which means that it absorbs in the near-ultraviolet by excitation from filled valence band states localized essentially on an O(2p) orbital to the empty conduction band states primarily composed of Ti(3d) states. Such excitations lead to very interesting photocatalytic properties.
5.3.1 Miller indices
A crystal plane can be determined by three integers known as the Miller indices h, k, and l. Take h’, k’ and l’ to be the intercepts (in units of the lattice constant) of a particular crystal plane on the three crystal axes defined by a1, a2 and a3. The three smallest integers h, k and l that satisfy the relation
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