8 Crystal Defects: Point Defects

Amarjeet Singh

    Learning Outcomes

 

After studying this module, you shall be able to

  • Understand that the crystalline solids are never defect free
  • Classify different types of defects such as point defects and line defects.
  • Classification of point defects
  • Understand the geometric patterns of different defects.

    Introduction:

 

A perfect crystal is an idealization; there is no such thing in nature. Atom arrangements in real materials do not follow perfect crystalline patterns. Nonetheless, most of the materials that are useful in engineering are crystalline to a very good approximation. There is fundamental physical reason for this. The preferred structures of solids at low temperature are those that minimize the energy. The low-energy atomic configurations are almost invariably crystalline since the regular pattern of the crystal lattice repeats whatever local configuration is most favorable for bonding. There is also a fundamental physical reason why the crystal is imperfect. While a perfect crystalline structure may be preferred energetically, at least in the limit of low temperature, atoms are relatively immobile in solids and it is, therefore, difficult to eliminate whatever imperfections are introduced into the crystal during its growth, processing or use.

 

It is useful to classify crystal lattice defects by their dimension. The 0-dimensional defects affect isolated sites in the crystal structure, and are hence called point defects. An example is a solute or impurity atom, which alters the crystal pattern at a single point. The 1-dimensional defects are called dislocations. They are lines along which the crystal pattern is broken. The 2-dimensional defects are surfaces, such as the external surface and the grain boundaries along which distinct crystallites are joined together. The 3-dimensional defects change the crystal pattern over a finite volume. They include precipitates, which are small volumes of different crystal structure, and also include large voids or inclusions of second-phase particles.

 

POINT DEFECTS

 

A point defect disturbs the crystal pattern at an isolated site. It is useful to distinguish intrinsic defects, which can appear in a pure material, from extrinsic defects, which are caused by solute or impurity atoms.

 

Intrinsic defects

 

An intrinsic defect is formed when an atom is missing from a position that must be filled in the crystal, creating a vacancy, or when an atom occupies an interstitial site where no atom would ordinarily appear, causing an interstitial. The two types of intrinsic point defects are shown in Fig. 7.1.

 

Figure 7.1: Illustration of a vacancy and an interstitial in a two-dimensional hexagonal lattice.

 

Because the interstitial sites in most crystalline solids are small (or have an unfavorable bonding configuration, as, for example, in the diamond lattice) interstitials are high-energy defects that are relatively uncommon. Vacancies, on the other hand, are present in a significant concentration in all crystalline materials. Their most pronounced effect is to govern the migration of atoms on the crystal lattice (solid state diffusion). In order for an atom to move easily from one crystal lattice site to another the target site must be vacant. As we shall see, the rate of diffusion on the crystal lattice is largely governed by the concentration of vacancies.

 

Ordered compounds can have more complex intrinsic defects. In most compounds the different species are charged to at least some degree. An intrinsic defect destroys the local charge balance, which must be restored in some way. The compound defects that preserve charge are easiest to visualize in binary ionic solids like NaCl. An isolated vacancy in an ionic solid creates an excess charge. The excess charge can be compensated by a paired vacancy on the sublattice of the other specie; for example, the excess charge associated with a Na vacancy is balanced if there is a Cl vacancy nearby. A neutral defect that involves paired vacancies on the cation and anion sub-lattices is called a Schottky defect. Alternatively, the charge imbalance caused by the vacancy can be corrected by adding an interstitial of the same specie; a Na vacancy is compensated by a Na interstitial. A neutral defect that is made up of a paired vacancy and interstitial is called a Frenkel defect. In compounds whose atoms are less strongly ionized it is energetically possible for species to exchange sites, so that an A-atom appears on the B sub-lattice or vice versa. This type of point defect is called an anti-site defect, and is fairly common in semiconducting compounds such as GaAs.

 

Extrinsic defects

 

The extrinsic point defects are foreign atoms, which are called solutes if they are intentionally added to the material and are called impurities if they are not. The foreign atom may occupy a lattice site, in which case it is called a substitutional solute (or impurity) or it may fill an interstitial site, in which case it is called an interstitial solute. Since the interstitial sites are relatively small, the type of the solute is largely determined by its size. Small atoms, such as hydrogen, carbon and nitrogen are often found in interstitial sites. Larger atoms are usually substitutional.

 

More complex extrinsic defects appear in compounds. If the valence of a substitutional defect in an ionic solid differs from that of the lattice ion then the excess charge is often compensated by a paired vacancy or interstitial. For example, when Mg++ ions are substituted for Na + in NaCl they tend to be paired with vacancies on the Na sublattice to maintain local charge neutrality. In semiconductors substitutional atoms with the wrong valence acts as electron donors or acceptors, as described below.

 

Extrinsic point defects affect almost all engineering properties, but they are particularly important in semiconducting crystals, where extrinsic defects are used to control electrical properties, and in structural metals and alloys, where extrinsic defects are added to increase mechanical strength. While these properties will be discussed later in the course, it is perhaps useful to identify the characteristics of the point defects that affect them.

 

Thermodynamic aspects of point defects

 

Point defects are intentionally added to semiconductors to control the type and concentration of charge carriers. Consider, for example, boron (valence 3) as a substitutional solute in elemental silicon. The saturated covalent bonds in silicon are shown schematically in Fig. 7.2a, and depend on the availability of four valence electrons per silicon atom. Since the bonds are saturated, silicon has very low conductivity in its pure state; pure silicon can only conduct electricity when electrons are excited into high energy electron states. If boron is added, as in Fig. 7.2b, a valence electron is missing from the immediate environment of the boron atom, causing a hole in the bonding pattern. Electrons can then move from bond to bond by exchanging with the hole. The exchange requires some energy to separate the hole from the boron ion core, but this energy is small compared to that required to excite an electron from a Si-Si bond into a high-energy state. The room-temperature conductivity of Si increases significantly when a small amount of B is added. Electron-deficient solutes like boron that cause holes in the configuration of bonding electrons are called acceptors.

 

 

Figure 7.2: (a) Tetrahedral bonding configuration in Si. (b) Bonding around a B solute, showing a hole (□). (c) Bonding around a P solute, showing an electron in a loose orbital.

 

The conductivity also rises when a solute with an excess of electrons is added to a semiconductor with saturated bonds. For example, let phosphorous (valence 5) be added to Si, as in Fig. 7.2c. The 5 valence electrons of P are sufficient to fill the local covalent bonds with one electron left over. This electron can only go into an excited state, and orbits about the P ion core somewhat as shown in the figure. It requires a relatively small energy increment to free this electron from the P core, in which case it can transport current by moving through the lattice. The conductivity of Si rises dramatically if a small amount of P is added. Electron-excess solutes such as P in Si are called donors. Semiconductors whose electrical properties are controlled by electrically active solutes are called extrinsic semiconductors. Almost all of the semiconductors that are used in engineering devices are extrinsic.

 

The simplest of the point defects is a vacancy from which an atom is missing as seen in figure 7.3. All crystalline solids contain vacancies and, in fact, it is not possible to create such a material that is free of these defects. The necessity of the existence of vacancies is explained using principles of thermodynamics; in essence, the presence of vacancies increases the entropy of the crystal. The equilibrium number of vacancies for a given quantity of material depends on and increases with temperature according to

 

 

 

 

In this expression, N is the total number of atomic sites, Ev is the energy required for the formation of a vacancy, T is the absolute temperature in kelvins, and kB is the Boltzmann’s constant. The value of kB is 1.38 x 10-23 J/atom-K. Thus, the number of vacancies increases exponentially with temperature; that is, as T in the above equation 7.1. For most metals, the fraction of vacancies Nv/N just below the melting temperature is on the order of 10-4; that is, one lattice site out of 10,000 will be empty. A self-interstitial is an atom from the crystal that is crowded into an interstitial site, a small void space that under ordinary circumstances is not occupied. This kind of defect is also represented in Figure 7.1. In metals, a self-interstitial introduces relatively large distortions in the surrounding lattice because the atom is substantially larger than the interstitial position in which it is situated. Consequently, the formation of this defect is not highly probable, and it exists in very small concentrations, which are significantly lower than for vacancies.

 

Figure 7.3: Self interstitial and vacancy in the lattice

 

A pure metal consisting of only one type of atom just is highly impractical; impurity or foreign atoms will always be present, and some will exist as crystalline point defects. In fact, even with relatively sophisticated techniques, it is difficult to refine metals to a purity in excess of 99.9999%. At this level, on the order of 1022 to 1023 impurity atoms will be present in one cubic meter of material. Most familiar metals are not highly pure; rather, they are alloys, in which impurity atoms have been added intentionally to impart specific characteristics to the material. Ordinarily, alloying is used in metals to improve mechanical strength and corrosion resistance. For example, sterling silver is a 92.5% silver-7.5 % copper alloy. In normal ambient environments, pure silver is highly corrosion resistant, but also very soft. Alloying with copper significantly enhances the mechanical strength without depreciating the corrosion resistance appreciably. The addition of impurity atoms to a metal will result in the formation of a solid solution and/or a new second phase, depending on the kinds of impurity, their concentrations, and the temperature of the alloy. Several terms relating to impurities and solid solutions deserve mention. With regard to alloys, solute and solvent are terms that are commonly employed. “Solvent” represents the element or compound that is present in the greatest amount; on occasion, solvent atoms are also called host atoms. “Solute” is used to denote an element or compound present in a minor concentration.

 

A solid solution forms when, as the solute atoms are added to the host material, A solid solution forms when, as the solute atoms are added to the host material, the crystal structure is maintained, and no new structures are formed. Perhaps it is useful to draw an analogy with a liquid solution. If two liquids, soluble in each other (such as water and alcohol) are combined, a liquid solution is produced as the molecules intermix, and its composition is homogeneous throughout. A solid solution is also compositionally homogeneous; the impurity atoms are randomly and uniformly dispersed within the solid.

 

Impurity point defects are found in solid solutions, of which there are two types: substitutional and interstitial. For the substitutional type, solute or impurity atoms replace or substitute for the host atoms (Figure 7.2). There are several features of the solute and solvent atoms that determine the degree to which the former dissolves in the latter, as follows:

  1. Atomic size factor. Appreciable quantities of a solute may be accommodated in this type of solid solution only when the difference in atomic radii between the two atom types is less than about 10-12 %. Otherwise the solute atoms will create substantial lattice distortions and a new phase will form.
  2. Crystal structure. For appreciable solid solubility the crystal structures for metals of both atom types must be the same.
  3. Electronegativity. The more electropositive one element and the more electronegative the other, the greater is the likelihood that they will form an intermetallic compound instead of a substitutional solid solution.
  4. Valences. Other factors being equal, a metal will have more of a tendency to dissolve another metal of higher valency than one of a lower valency.

  An example of a substitutional solid solution is found for copper and nickel. These two elements are completely soluble in one another at all proportions. With regard to the aforementioned rules that govern degree of solubility, the atomic radii for copper and nickel are 0.128 and 0.125 nm, respectively, both have the FCC crystal structure, and their electronegativities are 1.9 and 1.8; finally, the most common valences are for copper and for nickel.

 

For interstitial solid solutions, impurity atoms fill the voids or interstices among the host atoms (Figure 7.3). For metallic materials that have relatively high atomic packing factors, these interstitial positions are relatively small. Consequently, the atomic diameter of an interstitial impurity must be substantially smaller than that of the host atoms. Normally, the maximum allowable concentration of interstitial impurity atoms is low (less than 10%). Even very small impurity atoms are ordinarily larger than the interstitial sites, and as a consequence they introduce some lattice strains on the adjacent host atoms. Carbon forms an interstitial solid solution when added to iron; the maximum concentration of carbon is about 2%.The atomic radius of the carbon atom is much less than that for iron: 0.071 nm versus 0.124 nm.

 

Alloys and Solid Solutions

 

The addition of solute atoms almost always increases the mechanical strength of a solid. The phenomenon is called solution hardening. It is due to the fact that the solute atom is always a bit too large or a bit too small to fit perfectly into the crystal lattice site it is supposed to occupy, and distorts the crystal lattice in its attempt to fit as well as possible. As we shall see later, this distortion impedes the motion of the linear defects (dislocations) that are responsible for plastic deformation and, consequently, hardens the crystal. The distortion due to a substitutional solute is relatively small, though the associated hardening may be large enough to be useful in the engineering sense. The distortions due to interstitial atoms such as carbon and nitrogen are normally much greater because of the small size of the interstitial void in which they must fit. The hardening effect of interstitial solutes is large and technologically important; for example, high strength structural steels are alloys of Fe and C.

 

There is a simple crystallographic reason why interstitial solutes such as C are particularly effective in strengthening BCC metals such as Fe. The carbon atoms occupy octahedral interstitial sites in the BCC structure since an atom in an octahedral void in BCC is closer to two of its neighbors that to the other four, it causes an asymmetric distortion of the lattice. As shown in Fig. 7.4, the octahedron is stretched along its short axis, which is z-axis in the case shown in the figure. The asymmetric distortion of the interstitial site increases its interaction with the dislocations that cause plastic deformation and promotes hardening. In FCC alloys the interstitial sites are symmetric, and the lattice distortion is isotropic. Interstitial solutes are effective in hardening FCC alloys, but are less effective than in BCC alloys.

 

Figure 7.4: A local distortion of the BCC lattice by an interstitial atom in a void.

 

Value Addition:

 

Do You Know?

 

When use the term “defects” – it immediately implies that something is missing or imperfect and a cause of worry. Generally the defects are never desired. However in crystalline solid substances, the defects work as boon for transforming material properties. All extrinsic semiconductors (n-type, p-type) are nothing but impure substances and are so useful in electronic applications. A few controlled mechanisms to introduce defects in solids are well known to tailor their properties. In nanomaterials, the presence of defects remarkably changes the properties of the material. Therefore the study of defects is one of the very important topics in solids state physics

  1. Suggested Reading

    For More Details (on this topic and other topics discussed in Text Module) See

  1. Neil W. Ashcroft and N. David Mermin, Solid State Physics, Thomson Brooks/Cole, Eastern Press Bangalore (India) 2005
  2. Charles Kittel, Introduction to Solid State Physics, John Wiley & Sons, Singapore 1999
  3. William D. Callister, Jr, Material Science and Engineering, an Introduction, John Wiley & Sons, New York 2007
  4. Wikipedia

    Glossary:

 

0-dimension: The absence of atom or a group of atom from the lattice site is referred to as 0-dimansionla defect. The dimension of an atom or molecule has been considered negligible and hence the term, 0-diemsnion.

 

Sustitutional impurity: When an atom is in a lattice is displaced by another atom of almost similar size.

 

Interstitial impurity: When foreign atoms of small size is introduced between the interstitial spaces without replacing atoms are interstitial species.