Solids contain defects because they introduce disorder into an otherwise perfect structure and hence increase its entropy. The Gibbs energy, G = H − TS, of a solid with defects
has contributions from the enthalpy and the entropy of the sample. The formation of defects is normally endothermic because the lattice is disrupted so the enthalpy of the solid increases. However, the term −TS becomes more negative as defects are formed because they introduce disorder into the lattice and the entropy rises. Provided T > 0, therefore, the Gibbs energy will have a minimum at a nonzero concentra- tion of defects and their formation will be spontaneous (Fig.
4.52a). Moreover, as the temperature of the solid is raised, the minimum in G shifts to higher defect concentrations (Fig. 4.52b), so solids have a greater number of defects as their melting points are approached.
(a) Intrinsic point defects
KEY POINTS Schottky defects are site vacancies, formed in cation/
anion pairs, and Frenkel defects are displaced, interstitial atoms; the structure of a solid influences the type of defect that occurs, with Fren- kel defects forming in solids with lower coordination numbers and more covalency and Schottky defects forming in more ionic materials.
The solid-state physicists W. Schottky and J. Frenkel iden- tified two specific types of point defect. A Schottky defect
(Fig. 4.53) is a vacancy in an otherwise perfect arrangement of atoms or ions in a structure. That is, it is a point defect in which an atom or ion is missing from its normal site in the structure. The overall stoichiometry of a solid is not affected by the presence of Schottky defects because, to ensure charge balance, the defects occur in pairs in a compound of stoichiometry MX and there are equal numbers of vacancies at cation and anion sites. In solids of different composition, for example MX2, the defects must occur with balanced charges, so two anion vacancies must be created for each cation lost. Schottky defects occur at low concentrations in purely ionic solids, such as NaCl; they occur most com- monly in structures with high coordination numbers, such as close-packed ions and metals, where the enthalpy penalty of reducing the average coordination number of the remain- ing atoms (from 12 to 11, for instance) is relatively low.
A Frenkel defect (Fig. 4.54) is a point defect in which an atom or ion has been displaced onto an interstitial site. For example, in silver chloride, which has the rock-salt struc- ture, a small number of Ag+ ions reside in tetrahedral sites (1), leaving vacancies elsewhere on octahedral sites normally occupied. The stoichiometry of the compound is unchanged when a Frenkel defect forms, and it is possible to have Frenkel defects involving either one (M or X displaced) or both (some M and some X interstitials) of the ion types in a binary compound, MX. Thus the Frenkel defects that occur in, for example, PbF2 involve the displacement of a small number of F− ions from their normal sites in the fluorite structure, on the tetrahedral holes in the close-packed Pb2+
ion array, to sites that correspond to the octahedral holes. A useful generalization is that Frenkel defects are most often encountered in structures such as wurtzite and sphalerite in which coordination numbers are low (6 or less, 4:4 for these two structures) and the more open structure provides sites FIGURE 4.52 (a) The variation of the enthalpy and entropy of a
crystal as the number of defects increases. The resulting Gibbs energy G = H − TS has a minimum at a nonzero concentration, and hence defect formation is spontaneous. (b) As the temperature is increased, the minimum in the Gibbs energy moves to higher defect concentrations, so more defects are present at equilibrium at higher temperatures than at low temperatures.
Defect concentration
Energy
H TS
G = H – TS
Equilibrium
Defect concentration
Energy
H
TS
G
High T Low T (a)
(b)
Na+ Cl–
FIGURE 4.53 A Schottky defect is the absence of ions on normally occupied sites; for charge neutrality there must be equal numbers of cation and anion vacancies in a 1:1 compound.
FIGURE 4.54 A Frenkel defect forms when an ion moves to an interstitial site.
Cl– Ag+
that can accommodate the interstitial atoms. This is not to say that Frenkel defects are exclusive to such structures; as we have seen, the 8:4-coordination fluorite structure can accommodate such interstitials although some local reposi- tioning of adjacent anions is required to allow for the pres- ence of the displaced anion.
Ag+ Cl–
Ag+
1 Interstitial Ag+
The concentration of Schottky defects varies considerably from one type of compound to the next. The concentration of vacancies is very low in the alkali metal halides, being of the order of 106 cm−3 at 130°C, corresponding to about one defect per 1014 formula units. Conversely, some d-metal oxides, sulfides, and hydrides have very high concentrations of vacancies. An extreme example is the high-temperature form of TiO, which has vacancies on both the cation and anion sites at a concentration corresponding to about one defect per seven formula units.
Defects, when present in large numbers, may affect the density of a solid. Significant numbers of Schottky defects, as vacancies, will lead to a decrease in density. For example, TiO, with 14 per cent of both the anion and cation sites vacant, has a measured density of 4.96 g cm−1, much less than that expected for a perfect TiO structure, 5.81 g cm−1. Frenkel defects have little effect on density as they involve displaced atoms or ions, leaving the number of species in the unit cell unchanged.
EXAMPLE 4.19 Predicting defect types
What type of intrinsic defect would you expect to find in (a) MgO and (b) CdTe?
Answer The type of defect that is formed depends on factors such as the coordination numbers and the level of covalency in the bonding, with high coordination numbers and ionic bonding favouring Schottky defects and low coordination numbers and partial covalency in the bonding favouring Frenkel defects. (a) MgO has the rock-salt structure and the ionic bonding in this compound generally favours Schottky defects. (b) CdTe adopts the wurtzite structure with 4:4 coordination, favouring Frenkel defects.
Self-test 4.19 Predict the most likely type of intrinsic defects for (a) HgS and (b) CsF.
Schottky and Frenkel defects are only two of the many possible types of defect. Another type is an atom-inter- change or anti-site defect, which consists of an interchanged pair of atoms. This type of defect is common in metal alloys with exchange of neutral atoms. It is expected to be very unfavourable for binary ionic compounds on account of the introduction of strongly repulsive interactions between neighbouring similarly charged ions. For example, a copper/
gold alloy of exact overall composition CuAu has extensive disorder at high temperatures, with a significant fraction of Cu and Au atoms interchanged (Fig. 4.55). The interchange of similarly charged species on different sites in ternary and compositionally more complex compounds is common;
thus in spinels (Section 24.6c) the partial swapping of the metal ions between tetrahedral and octahedral sites is often observed.
(b) Extrinsic point defects
KEY POINT Extrinsic defects are defects introduced into a solid as a result of doping with an impurity atom.
Extrinsic defects, those resulting from the presence of impu- rities, are inevitable because perfect purity is unattainable in practice in crystals of any significant size. Such behaviour is commonly seen in naturally occurring minerals. The incor- poration of low levels of Cr into the Al2O3 structure pro- duces the gemstone ruby, whereas replacement of some Al by Fe and Ti results in the blue gemstone sapphire (Box 4.6).
The substituting species normally has a similar atomic or ionic radius to the species which it replaces; Cr3+ in ruby has a similar ionic radius to Al3+. Impurities can also be intro- duced intentionally by doping one material with another.
A dopant is a small level, typically 0.1–5 per cent, of an element that replaces another in a structure; an example is the introduction of As into Si to modify the latter’s semi- conducting properties. Synthetic equivalents of ruby and
Au Cu
FIGURE 4.55 Atom exchange can give rise to a point defect as in CuAu.
BOX 4.6 Why do defects and dopants give colour to gemstones?
Defects and dopant ions are responsible for the colours of many gemstones. Whereas aluminium oxide (Al2O3), silica (SiO2), and fluorite (CaF2) in their pure forms are colourless, brightly coloured materials may be produced by substituting in small levels of dopant ions or producing vacant sites that trap electrons. The impurities and defects are often produced in naturally occurring minerals on account of the geological and environmental conditions under which they are formed.
For example, d-metal ions were often present in the solutions from which the gemstones grew and the presence of ionizing radiation from radioactive species in the natural environment generated electrons that became trapped in their structure.
The most common origin of colour in a gemstone is a d-metal ion dopant (see Table B4.1). Thus ruby is Al2O3 containing around 0.2–1 atom per cent Cr3+ ions in place of the Al3+ ions, and its red colour results from the absorption of green light in the visible spectrum as a result of the excitation of Cr3d electrons (Section 20.4). The same ion is responsible for the green of emeralds; the different colour reflects a different local coordination environment of the dopant. The host structure is beryl (beryllium aluminium silicate, Be3Al2(SiO3)6), and the Cr3+
ion is surrounded by six silicate ions, rather than the six O2−
ions in ruby, producing absorption at a different energy. Other d-metal ions are responsible for the colours of other gemstones.
Iron(II) produces the red of garnets and the yellow-green of peridots. Manganese(II) is responsible for the pink colour of some tourmalines.
In ruby and emerald the colour is caused by excitation of electrons on a single dopant d-metal ion, Cr3+. When more than one dopant species, which may be of different type or oxidation state, is present it is possible to transfer an electron between them. One example of this behaviour is sapphire. Sapphire, like ruby, is alumina but in this gemstone some adjacent pairs of Al3+ ions are replaced by Fe2+ and Ti4+ pairs. This material absorbs visible radiation of a wavelength corresponding to yellow light as an electron is transferred from Fe2+ to Ti4+, so
producing a brilliant blue colour (the complementary colour of yellow).
In other gemstones and minerals, colour is a result of doping a host structure with a species that has a different charge from the ion that it replaces or by the presence of a vacancy (Schottky- type defect). In both cases a colour-centre or F-centre (F from the German word farbe for colour) is formed. As the charge at an F-centre is different from that of a normally occupied site in the same structure, it can easily supply an electron to, or receive an electron from, another ion. This electron can then be excited by absorbing visible light, so producing colour. For instance, in purple fluorite, CaF2, an F-centre is formed from a vacancy on a normally occupied F− ion site. This site then traps an electron, generated by exposure of the mineral to ionizing radiation in the natural environment. Excitation of the electron, which acts like a particle in a box, absorbs visible light in the wavelength range 530–600 nm, producing the violet/purple colours of this mineral.
In amethyst, the purple derivative of quartz, SiO2, some Si4+
ions are substituted by Fe3+ ions. This replacement leaves a hole (one missing electron) and excitation of this hole, by ionizing radiation for instance, traps it by forming Fe4+ or O− in the quartz matrix. Further excitation of the electrons in this material now occurs by the absorption of visible light at 540 nm, producing the observed purple colour. If an amethyst crystal is heated to 450°C the hole is freed from its trap. The colour of the crystal reverts to that typical of iron-doped silica and is a characteristic of the yellow semiprecious gemstone citrine. If citrine is irradiated the trapped-hole is regenerated and the original colour restored.
Colour centres can also be produced by nuclear transformations. An example of such a transformation is the β-decay of 14C in diamond. This decay produces a 14N atom, with an additional valence electron, embedded in the diamond structure. The electron energy levels associated with these N atoms allow absorption in the visible region of the spectrum and produce the colouration of blue and yellow diamonds.
TABLE B4.1 Gemstones and the origin of their colours
Mineral or gemstone Colour Parent formula Dopant or defect responsible for the colour
Ruby Red Al2O3 Cr3+ replacing Al3+ in octahedral sites
Emerald Green Be3Al2(SiO3)6 Cr3+ replacing Al3+ in octahedral sites
Tourmaline Green or pink Na3Li3Al6(BO3)3(SiO3)6F4 Cr3+ or Mn2+ replacing Li+ and Al3+ in octahedral sites, respectively
Garnet Red Mg3Al2(SiO4)3 Fe2+ replacing Mg2+ in 8-coordination sites
Peridot Yellow-green Mg2SiO4 Fe2+ replacing Mg2+ in 6-coordination sites
Sapphire Blue Al2O3 Electron transfer between Fe2+ and Ti4+ replacing
Al3+ in adjacent octahedral sites
Diamond Colourless, pale blue
or yellow
C Colour centres from N
Amethyst Purple SiO2 Colour centre based on Fe3+/Fe4+
Fluorite Purple CaF2 Colour centre based on trapped electron
sapphire can also be synthesized easily in the laboratory by doping small levels of Cr, or Fe and Ti, into the Al2O3 struc- ture, replacing aluminium.
When the dopant species is introduced into the host the latter’s structure remains essentially unchanged. If attempts are made to introduce high levels of the dopant species, a new structure often forms or the dopant species is not incor- porated. This behaviour usually limits the level of extrin- sic point defects to low levels. The composition of ruby is typically (Al0.998Cr0.002)2O3, with 0.2 per cent of metal sites as extrinsic Cr3+ dopant ions. Some solids may tolerate much higher levels of defects (Section 4.17a). Dopants often mod- ify the electronic structure of the solid. Thus, when an As atom replaces an Si atom, the additional electron from each As atom can be thermally promoted into the conduction band, improving the overall conductivity of the semicon- ductor. In the more ionic substance ZrO2, the introduction of Ca2+ dopant ions in place of Zr4+ ions is accompanied by the formation of an O2− ion vacancy to maintain charge neutrality (Fig. 4.56). The induced vacancies allow oxide ions to migrate through the structure, increasing the ionic conductivity of the solid.
Another example of an extrinsic point defect is a colour centre, a generic term for defects responsible for modifica- tions to the IR, visible, or UV absorption characteristics of solids that have been irradiated or exposed to chemical treatment. One type of colour centre is produced by heat- ing an alkali metal halide crystal in the vapour of the alkali metal, and gives a material with a colour characteristic of the system: NaCl becomes orange, KCl violet, and KBr blue-green. The process results in the introduction of an alkali metal cation at a normal cation site and the associated electron from the metal atom occupies a halide ion vacancy.
A colour centre consisting of an electron in a halide ion vacancy is called an F-centre (Fig. 4.57). The colour results from the excitation of the electron in the localized environ- ment of its surrounding ions. An alternative method of pro- ducing F-centres involves exposing a material to an X-ray beam that ionizes electrons into anion vacancies. F-centres and extrinsic defects are important in producing colour in gemstones (Box 4.6).
EXAMPLE 4.20 Predicting possible dopant ions What transition metal ions might substitute for Al3+ in beryl, Be3Al2(SiO3)6, forming extrinsic defects?
Answer We need to identify ions of similar charge and size.
Ionic radii are listed in Resource section 1. Triply charged cations with ionic radii similar to Al3+ (r = 53 pm) should prove to be suitable dopant ions. Candidates could be Fe3+ (r = 55 pm), Mn3+ (r = 65 pm), and Cr3+ (r = 62 pm). Indeed when the extrinsic defect is Cr3+ the material is a bright green beryl, the gemstone emerald. For Mn3+ the material is a red or pink beryl and for Fe3+
it is the yellow beryl heliodor.
Self-test 4.20 What elements other than As might be used to form extrinsic defects in silicon?