Comprehensive nuclear materials 1 02 fundamental point defect properties in ceramics

Comprehensive nuclear materials 1 02 fundamental point defect properties in ceramics Comprehensive nuclear materials 1 02 fundamental point defect properties in ceramics Comprehensive nuclear materials 1 02 fundamental point defect properties in ceramics Comprehensive nuclear materials 1 02 fundamental point defect properties in ceramics Comprehensive nuclear materials 1 02 fundamental point defect properties in ceramics

1.02 Fundamental Point Defect Properties in Ceramics A Chroneos University of Cambridge, Cambridge, UK M J D Rushton and R W Grimes Imperial College of Science, London, UK ß 2012 Elsevier Ltd All rights reserved 1.02.1 1.02.2 1.02.2.1 1.02.2.2 1.02.2.3 1.02.2.4 1.02.3 1.02.3.1 1.02.3.2 1.02.3.3 1.02.3.4 1.02.3.5 1.02.3.6 1.02.3.7 1.02.4 1.02.4.1 1.02.4.2 1.02.4.3 1.02.4.4 1.02.5 1.02.6 1.02.6.1 1.02.6.2 1.02.7 References Introduction Intrinsic Point Defects in Ionic Materials Point Defects Compared to Defects of Greater Spatial Extent Intrinsic Disorder Reactions Concentration of Intrinsic Defects KroăgerVink Notation Defect Reactions Intrinsic Defect Concentrations Effect of Doping on Defect Concentrations Decrease of Intrinsic Defect Concentration Through Doping Defect Associations Nonstoichiometry Lattice Response to a Defect Defect Cluster Structures Electronic Defects Formation Concentration of Intrinsic Electrons and Holes Band Gaps Excited States The Brouwer Diagram Transport Through Ceramic Materials Diffusion Mechanisms Diffusion Coefficient Summary 1.02.1 Introduction The mechanical and electronic properties of crystalline ceramics are dependent on the point defects that they contain, and as a consequence, it is necessary to understand their structures, energies, and concentration defects and their interactions.1,2 In terms of their crystallography, it is often convenient to characterize ceramic materials by their anion and cation sublattices Such models lead to some obvious expectations It might, for example, be energetically unfavorable for an anion to occupy a site in the cation sublattice and vice versa This is because it would lead to anions having nearest neighbor anions with a substantial electrostatic energy penalty Further, there should exist an equilibrium between the concentration of intrinsic defects (such as lattice vacancies), extrinsic defects (i.e., dopants), and 47 48 48 48 49 50 51 51 52 52 52 53 55 56 57 57 57 58 58 60 61 61 63 63 64 electronic defects in order to maintain charge neutrality.1,2 Such constraints on the types and concentrations of point defects are the focus of this chapter In the first section, we consider the intrinsic point defects in ionic materials This is followed by a discussion of the defect reactions describing the effect of doping, defect cluster formation, and nonstoichiometry Thereafter, we consider the importance of electronic defects and their influence on ceramic properties In the final section, we examine solidstate diffusion in ceramic materials Examples are used throughout to illustrate the extent and range of the point defects and associated processes occurring in ceramics The subsequent chapters (see Chapter 1.03, Radiation-Induced Effects on Microstructure and Chapter 1.06, The Effects of Helium in Irradiated Structural Alloys) will deal with defects 47 48 Fundamental Point Defect Properties in Ceramics of greater spatial extent, such as dislocations and grain boundaries, in greater detail; here, however, we begin by comparing them with point defects 1.02.2 Intrinsic Point Defects in Ionic Materials 1.02.2.1 Point Defects Compared to Defects of Greater Spatial Extent In crystallography, we learn that the atoms and ions of inorganic materials are, with the exception of glasses, arranged in well-defined planes and rows.3 This is, however, an idealized representation In reality, crystals incorporate many types of imperfections or defects These can be categorized into three types depending on their dimensional extent in the crystal: Point defects, which include missing atoms (i.e., vacancies), incorrectly positioned atoms (e.g., interstitials), and chemically inappropriate atoms (dopants) Point defects may exist as single species or as small clusters consisting of a number of species Line defects or dislocations, which extend through the crystal in a line or chain The dislocation line has a central core of atoms, which are located well away from the usual crystallographic sites (in ceramics, this extends, in cylindrical terms, to a nanometer or so) Most dislocations are of edge, screw, or mixed type.4 Planar defects, which extend in two dimensions and are atomic in only one direction Many different types exist, the most common of which is the grain boundary Other common types include stacking faults, inversion domains, and twins.1,2 The defect types described above are the chemical or simple structural models for the extent of defects It is critical to bear in mind that all defect types, in all materials, may exert an influence via an elastic strain field that extends well beyond the chemical extent of the defect (i.e., beyond the atoms replaced or removed) This is because the lattice atoms surrounding the defect have had their bonds disrupted Consequently, these atoms will accommodate the existence of the defect by moving slightly from their perfect lattice positions These movements in the positions of the neighboring atoms are referred to as lattice relaxation As a result of the elastic strain and electrostatic potential (if the defect is not charge-neutral), defects can affect the mechanical properties of the lattice In addition, defects have a chemical effect, changing the oxidation/reduction properties Defects also provide mechanisms that support or impede the movement of ions through the lattice Finally, defects alter the way in which electrons interact with the lattice, as they can alter the potential energy profile of the lattice (whether or not the defect is charged) For example, this may lead to the trapping of electrons Also, because dopant ions will have a different electronic configuration from that of the host atom, defects may donate an electron to a conduction band, resulting in n-type conduction, or a defect may introduce a hole into the electronic structure, resulting in p-type conductivity 1.02.2.2 Intrinsic Disorder Reactions A number of different point defects can form in all ceramics, but their concentration and distributions are interrelated In the event of the production of a vacancy by the displacement of a lattice atom, this released atom can be either contained within the crystal lattice as an interstitial species (forming a Frenkel pair), or it can migrate to the surface to form part of a new crystal layer (resulting in a Schottky reaction) Figure represents a Frenkel pair: both cations and anions can undergo this type of disorder reaction, resulting in cation Frenkel and anion Frenkel pairs, respectively In ceramic materials, both the vacancy and interstitial defects are usually charged, but the overall reaction is charge-neutral The energy necessary for this reaction to proceed is the energy to create one vacancy by removing an ion from the crystal to infinity plus the energy to create one interstitial ion by taking an ion from infinity and placing it into the crystal The implication of removing and taking ions from infinity implies that the two species are infinitely separated in the crystal (unlike the two species shown in Figure 1) As separated species, these are defects at infinite dilution, a Vacancy Interstitial Figure Schematic representation of a Frenkel pair in a binary crystal lattice Fundamental Point Defect Properties in Ceramics well-defined thermodynamic limit As the two defects are charged, they will interact if not infinitely separated, a point we will return to later As with the Frenkel reaction, the Schottky disorder reaction must be charge-neutral Here, only vacancies are created, but in a stoichiometric ratio Thus, for a material of stoichiometry AB, one A vacancy and one B vacancy are created The displaced ions are removed to create a new piece of lattice It is important to realize that we are dealing with an equilibrium process for the whole crystal Thus, as the temperature changes, many thousands of vacancies are created/destroyed and new material containing many thousands of ions is formed Thus, it is not simply that one new molecule is formed, but there is also an increase in the volume of the crystal, which is why the lattice energy is part of the Schottky reaction The energy for the Schottky disorder reaction to proceed in an AB material is the energy to create one A-site vacancy by removing an ion from the crystal to infinity plus the energy to create one B-site vacancy by removing an ion from the crystal to infinity, plus the lattice energy associated with one unit of the AB compound For example in Al2O3, the energy would be that associated with the sum of two Al vacancies plus three O vacancies plus the lattice energy of one Al2O3 formula unit Again, the vacancy species are assumed to be effectively infinitely separated In a crystalline material with more than one type of atom, each species usually occupies its own sublattice If two different species are swapped, this produces an antisite pair (see Figure 2) For example, in an AB compound, one A atom is swapped with the B atom While this would be of high energy for an AO compound, where A and O are of opposite charge (e.g., Mg2ỵ and O2), in an ABO3 material where A and B may have similar or even identical positive charges, antisite energies can be small Antisite pair 49 In general, the energies needed to form each type of disorder, in a given material, are different Therefore, only one type of intrinsic disorder dominates: this is often described as the intrinsic disorder of the material If one intrinsic process is of much lower energy than the others, it will dominate the equilibrium: this is useful when investigating other defect processes, as we will see later In most metals and metal alloys, Schottky disorder dominates because of the closely packed nature of their crystal structure In ceramics, both Schottky and Frenkel disorders are possible; for example, in NaCl and MgO (both having the rock salt structure), Schottky disorder dominates, but in CaF2 and UO2 (both having the fluorite structure), anion Frenkel disorder is predominant, while antisite disorder is observed to dominate in MgAl2O4 spinel In Al2O3, the situation is too close to call and it is not clear whether Frenkel or Schottky disorder dominates.5 1.02.2.3 Concentration of Intrinsic Defects We start with the assumption that, for a given set of ions, their crystal structure represents the most stable arrangement of those ions in space Thus, there is an enthalpy cost to form atomic defects: energy is expended in forming the defects How then defects form? The answer is related to free energy considerations; that is, the increase in the enthalpy of the system can be balanced with a corresponding increase in the entropy and more particularly, the configurational entropy Point defects in a crystal can therefore be described as entropically stabilized, and as such they are equilibrium defects (dislocations and grain boundaries, on the other hand, are not equilibrium defects) If the enthalpy of forming n Schottky pairs in an AB material is n Dh, the vibrational entropy is nT Ds, where T is temperature (in K) so that n Dgf ẳ n Dh ỵ nT Ds, and the change in the entropy associated with this reaction is DSc; the change in the free energy (DG) of the system (if we ignore pressure volume term effects) is DG ¼ nDgf À T DSc If we assume that the entropy is all associated with configuration DSc ¼ klnO Figure Schematic representation of an antisite pair in a binary crystal lattice where k is Boltzmann’s constant and O is the number of distinct ways that n Schottky pairs can be arranged in the crystal If we assume that there are N ‘A’ lattice sites (in defect chemistry terms, a lattice Fundamental Point Defect Properties in Ceramics site means a position in the crystal that an ion will usually occupy in that crystal structure), the number of ways, OA, of arranging n A-site vacancies is N! OA ẳ n!N nị! As we have n B-site vacancies to distribute over N, B-lattice sites, the total number of configurations is the product of OA and OA:   ! N! N! DSC ¼ k ln n!ðN À nÞ! n!ðN À nÞ!   N! ẳ 2k ln n!N nị! where N and n are large, as they are when dealing with crystals, we can invoke Stirling’s approximation, which states that ln(M!) ¼ M ln(M) M Thus, DSC ẳ 2kẵN lnN ị N nịlnN nị nlnnị Therefore, DGẳnDgf 2kT ẵN lnN ịN nịlnN nịnlnnị    ! N N n ỵnln ẳnDgf 2kT N ln N n n To find the equilibrium number of defects, we need to find the minimum of DG with respect to n (see Figure 3) That is     @DG N Àn ¼0¼Dgf À2kT ln @n T ;P n Assuming that the number of defects is small in comparison to the number of available lattice sites, then N À n % N: Energy nΔgf       n Dgf Dh Ds ¼ exp À exp ¼ exp À N 2kT 2k 2kT Usually, we assume that the energy associated with the change in vibrational entropy is negligible so that the concentration of defects (n/N) is dominated by the enthalpy of reaction:   n Dh ẵn ẳ ¼ exp À N 2kT However, this is not always a valid assumption and care must be taken When defect concentrations are measured experimentally, they are presented on an Arrhenius plot of ln(concentration) versus 1/T, which yields straight lines with slopes that are proportional to the disorder enthalpy (see Figure 4) 1.02.2.4 KroăgerVink Notation It is usual for defects in ceramic materials to be described using a short hand notation after Kroăger and Vink.6 In this, the defect is described by its chemical formula Thus, a sodium ion would be described as Na, whatever its position in whatever lattice A vacancy is designated as ‘V.’ The description is made with respect to the position within the lattice that the defect occupies For example, a vacant Mg site is designated by VMg and an Na substituted at an Mg site is designated by NaMg Interstitial ions are represented by ‘i’ so that an interstitial fluorine ion in any lattice would be Fi The charge on an ion is described with respect to the site that the ion occupies Thus, an Na ion (which has formal charge ỵ) sitting on an Mg site in MgO (which expects to be occupied by a 2ỵ ion) has one too few ỵ charges; it has a relative charge of 1À which is designated as a vertical dash, meaning that it is written as Na0Mg An Al3ỵ ion at an Mg site in G Defect concentration −TΔSc Figure Relationship of terms contributing to the defect-free energy In [n] 50 1/T Figure Disorder enthalpy is proportional to the gradient of a ln [n] versus 1/T graph Fundamental Point Defect Properties in Ceramics h MgO has too high a charge Positive excess charge relative to a site is designated with a dot, thusAlMg Similarly, a vacant Mg site in MgO is designated by V00Mg and an interstitial Mg ion in MgO byMg i Finally, a neutral charge is indicated by a cross ‘Â,’ so that an Mg ion at an Mg site in MgO is Mg Mg Ions such as Fe may assume more than one oxidation state Therefore, in MgO, we might find both Fe2ỵ  and Fe3ỵ ions on Mg sites, that is, Fe Mg andFeMg It is also possible to encounter bound defect pairs or clusters These are indicated using brackets and an indication of the overall cluster charge; for example, an Fe3ỵ ion bound to an Naỵ ion, both substituted at magnen o sium sites, would be FeMg : Na0Mg These cases are summarized in Figure Finally, defect concentrations are indicated using square brackets Thus, the concentration of Fe3ỵ ions substituted at magnesium sites in MgO would be FeMg i When we consider the role of hole and electron species, these are represented as h and e0 , respectively 1.02.3 Defect Reactions 1.02.3.1 Intrinsic Defect Concentrations Introducing a doping agent to a crystal lattice can have a significant effect on the defect concentration within the material As such, doping represents a powerful tool in the engineering of the properties of ceramic materials The concept of a solid solution, in which solute atoms are dispersed within a diluent matrix, is used in many branches of materials science In many respects, the doped lattice can be viewed as a solid solution in which the point defects are dissolved in the host • = Positive charge Defect species Element label, V for a vacancy, h for hole or e for electron S Charge ı = Negative charge ´ = Neutral Site Subscript denotes species in the nondefective lattice at which defect currently sits Interstitial defects are represented by letter ‘i’ Examples Vacancy on an oxygen site with an effective 2+ charge •• Vo ıı Oxygen interstitial with an effective 2- charge Oi • AIMg Substitutional defect in which an aluminum atom is situated on a magnesium site and has an effective 1+ charge ı {FeMg:NaMg }´ Neutral defect cluster containing: Fe on Mg site (1+ charge) and Na on Mg site (1- charge) Braces indicate defect association MgMg + Oo VMg + Voă + MgO Figure Overview of KroăgerVink notation 51 Defect equation showing Schottky defect formation in MgO 52 Fundamental Point Defect Properties in Ceramics lattice The critical issue with such a view is in defining the chemical potential of an element This is straightforward for the dopant species but is less clear when the species is a vacancy This is circumvented by defining a virtual chemical potential, which allows us to write equations similar to those that describe chemical reactions Within these defect equations, it is critical that mass, charge, and site ratio are all conserved Using KroăgerVink notation, we can describe the formation of Schottky defects in MgO thus: Null ! V00Mg ỵ V O or MgMg þ OO ! V00Mg þ V O þ MgO Note that in this case, the equation balances in terms of charge, chemistry, and site As this is a reaction, it may be described by a reaction constant ‘K,’ which is related to defect concentrations by h i à KS ¼ V00Mg V O In the case of the pure MgO Schottky reaction, charge neutrality dictates that h i  à V00Mg ¼ V O So, if Dh is the enthalpy of the Schottky reaction, if we use Àour previous definition of concentration, Á n Dh ¼ exp À N 2kT   h i  à Dh V00Mg ¼ V ¼ exp À O 2kT 1.02.3.2 Effect of Doping on Defect Concentrations Similar reactions can be written for extrinsic defects via the solution energy For example, the solution of CoO in MgO, where the Co ion has a charge of 2ỵ and is therefore isovalent to the host lattice ion, CoO ! Co Mg ỵ OO ỵ MgO Ksolution ¼ h i à  Co Mg OO ½CoOŠ   Dhsol % exp À kT where Dhsol is the solution enthalpy As the concentration of CoO in CoO ¼ 1,   h i à Dhsol  % exp À Co O Mg O kT Consider the solution of Al2O3 in MgO In this case, the Al ions have a higher charge and are termed aliovalent These ions must be charge-compensated by other defects, for example, 00 Al2 O3 ! 2AlMg ỵ 3O O ỵ VMg Then, h i2  Ãh i 00 AlMg Ô O VMg ½Al2 O3 Š   Dhsol ¼ exp À kT h i As electronegativity dictates that AlMg ¼ follows that   h i pffiffiffi Dhsol  AlMg ¼ exp À 3kT h V00Mg i , it In general, the law of mass action6 states that for a reaction aA ỵ bB $ cC ỵ dD   ẵC c ẵDd DG ẳ K ẳ exp reaction kT ½AŠa ½B Šb 1.02.3.3 Decrease of Intrinsic Defect Concentration Through Doping While considering the intrinsic defect reaction forh MgO, i à we wrote the defect reaction  K S V00Mg VO This implies that there is equilibrium between these two defect concentrations Assuming that the enthalpy of the Schottky reaction, Dh ¼ 7.7 eV,   h i à 7:7 00  VMg VO ¼ exp À kT Now, consider the effect that solution of 10 ppm Al2O3 has on MgO The solution reaction implies that a concentration of ppm of V00Mgh hasi been introduced into the lattice, that is, V00Mg ¼  10À6 Therefore,     à 7:7 VO ¼  105 exp À kT   à  Thus, at 1000 C, the VO ¼ 6.4  10–26 compared to an oxygen vacancy concentration in pure MgO of 5.66  10–16 The introduction of the extrinsic defects has therefore lowered the oxygen vacancy concentration by 10 orders of magnitude! 1.02.3.4 Defect Associations So far, we have assumed that when we form a set of defects through some interaction, although the defects reside in the same lattice, somehow they not interact with one another to any significant extent They are termed noninteracting This is valid in the dilute limit approximation; however, as defect concentrations increase, defects tend to form Fundamental Point Defect Properties in Ceramics into pairs, triplets, or possibly even larger clusters Take, for example, the solution of Al2O3 into MgO 00 Al2 O3 ! 2AlMg ỵ 3O O ỵ VMg þ 3MgO When the concentration is great enough, n o0 AlMg ỵ V00Mg ! AlMg : V00Mg the scope of this chapter.7 Therefore, to illustrate the types of relationships n that canoxoccur, we use the 00 cluster resulting example of the binary Ti Mg : VMg from TiO2 solution in MgO via 00 TiO2 ! Ti Mg ỵ 2OO þ VMg þ 2MgO and n o 00  00 Ti ỵ V ! Ti : V Mg Mg Mg Mg As seen for solution energies, using the enthalpy associated with this pair cluster formation (the binding energy Dhbind), the reaction can be analyzed using mass action: hn o0 i   AlMg : V00Mg Dh ih i % exp À bind Kbinding pair ¼ h kT V00 Al and the electroneutrality condition h i h i V00Mg ¼ Ti Mg But since, yields Mg Mg   h i2 h i Dhsol AlMg V00Mg ¼ exp À kT we have the relationship hn AlMg   o0 ih i Dhbind ỵ Dhsol  00 : VMg AlMg ẳ exp À kT which describes the solution process n o0  00 Al2 O3 ! AlMg ỵ 3O ỵ Al : V O Mg Mg Further, itn is possible tooform a neutral triplet defect   , which has a binding cluster, AlMg : V== Mg : AlMg bind with respect to isolated defects enthalpy of DE so that hn o i AlMg : V00Mg : AlMg Kbinding triplet ¼ h i2 h i AlMg V00Mg   Dhbind % exp À kT which leads to   hn o i Dhbind ỵ Dhsol AlMg : V00Mg : AlMg ẳ exp À kT We now investigate the relative significance of defect clusters over isolated defects as a function of temperature for a fixed dopant concentration For most systems, there are a great number of possible isolated and cluster defects, and the equilibria between them quickly become very complex Solving such equilibria requires iterative procedures that are beyond 53 Then, using h ih i hn o i 00 00 Ti Ti Mg VMg Kbinding pair ¼ Mg : VMg hn 00 Ti Mg : VMg o i h i2 ¼ Ti Mg Kbinding pair a If the concentration of titanium ions on magnesium sites is x so that Mg(1–2x)TixO is the formula of the material, then, hn o i h i 00  Ti : V ¼ x À Ti b Mg Mg Mg Substituting b into a yields the quadratic equation, h i2 h i  ỵ Ti Kbinding pair Ti Mg Mg À x ¼ Solvingh thisi in the usual manner allows us to deteras a function of Kbinding energy If we mine Ti Mg now assume that Dhbind ¼ eV (a typical binding energy between charged pairs of defects in oxide ceramics), relationships between the concentration of the clusters and the isolated substitutional ions can be determined as a function of either total dopant concentration, x, or temperature These are shown in Figure 6, which assumes a fixed temperature of 1000 K and Figure 7, which assumes a fixed value of x ¼  10À6 and a range of temperature from 500 to 2000 K 1.02.3.5 Nonstoichiometry Although some materials such as MgO maintain the ratio between Mg and O very close to the stoichiometric ratio 1:1, other crystal structures such as FeO can tolerate large nonstoichiometries; Fe1ÀxO with 0.05 x 0.15 The extent of the deviation from stoichiometry depends on how easily the host ions can assume charge states other than those associated with 54 Fundamental Point Defect Properties in Ceramics 1ϫ10-2 Isolated favored -4 Cluster favored 1ϫ10 Concentration 1ϫ10-6 1ϫ10-8 1ϫ10-10 1ϫ10-12 Cluster ЈЈ ´ [{TiMg:VMg }] •• -14 1ϫ10 Isolated 1ϫ10-16 ·· ] [TiMg 1ϫ10-18 1ϫ10-12 1ϫ10-11 1ϫ10-10 1ϫ10-9 1ϫ10-8 1ϫ10-7 1ϫ10-6 1ϫ10-5 1ϫ10-4 X Figure Cluster and isolated defect concentration as a function of x at a temperature of 1000 K 1ϫ10−4 Isolated favored Cluster favored 1ϫ10−5 Concentration 1ϫ10−6 1ϫ10−7 1ϫ10−8 Cluster ·· :V ЈЈ }ϫ] [{TiMg Mg 1ϫ10−9 1ϫ10−10 Isolated [Ti ·· ] Mg 1ϫ10−11 600 800 1000 1200 1400 1600 1800 2000 T (K) Figure Cluster and isolated defect concentration for x ¼  10–6 as a function of temperature the host material, Fe3ỵ in the last case Usually, this is dependent on how easily the cation can be oxidized or reduced Associated with these reactions is the removal or introduction of oxygen from the atmosphere For example, the reduction reaction follows ! O2 gị ỵ V O ỵ 2e where e represents a spare electron, which will reside somewhere in the lattice For example, in CeO2, the electron is localized on a cation site forming a Ce3ỵ O O ion This is usually written as Ce0Ce Similarly, the oxidation reaction is O2 gị ! O O ỵ 2h where h represents a hole, that is, where an electron has been removed because the new oxygen species requires a charge of 2À Thus in CoO, for example,   00 O2 gị ỵ 2Co Co ! OO ỵ 2CoCo ỵ VCo Fundamental Point Defect Properties in Ceramics If the enthalpy for the oxidation reaction is DhOX,   Ã2  00 à   CoCo VCo ÀDhOX ¼ exp kT ðPO2 Þ1=2 ðPO2 Þis the partial pressure of oxygen, that is, the concentration of oxygen in the atmosphere Since electroneutrality gives us   à  à CoCo ¼ V00Co  00 à VCo % ðPO2 Þ1=6 Now, if the majority of cobalt vacancies are associated with a single charge-compensating Co3ỵ species, that is, we have some defect clustering, then the oxidation reaction will be È  É0  00 O2 gị ỵ 2Co Co ! OO þ CoCo þ CoCo : VCo and   ÃhÀ  Á0 i CoCo CoCo : V00Co ðPO2 Þ1=2 ¼ exp   ÀDhOX kT which, given that  à hÀ Á0 i CoCo ¼ CoCo : V00Co hÀ Ái implies that CoCo : V00Co is proportional to ðPO2 Þ1=4 Similar relations can be formulated for even larger clusters The defect concentration can be determined by measuring the self-diffusion coefficient When this is related to the oxygen partial pressure on a lnPO2 ÂÈ Éà versus ln CoCo : V00Co graph, the slope shows how the material behaves For CoO, the experimentally determined slope is 1=4, showing that the cation vacancy is predominantly associated with a single charge-compensating defect.8 manifested as a volume change arising as a result of the way in which the lattice responds, that is, how the lattice ions relax around the defect For example, vacancies in ionic materials usually result in positive defect volumes Consider the example of a vacancy in MgO The nearest neighbor cations are displaced outward, away from the vacant site, causing an increase in volume, whereas the second neighbor anions move inward, albeit to a lesser extent (see Figure 8) What drives these ion relaxations is the change in the Coulombic interactions due to defect formation We say that the oxygen vacancy carries an effective positive charge because an O2À has been removed and thus, an electrostatic attraction between O2À and Mg2ỵ is removed As ionic forces are balanced in a crystal, the outer O2 ions now attract the Mg2ỵ away from the V O defect site In covalent materials, vacant sites result in atomic relaxations that are due to the formation of an incomplete complement of bonds, often termed ‘dangling bonds.’ In this case, the net result can be different from those in ionic solids and by way of an example, in silicon, a vacancy results in a volume decrease On the other hand, an arsenic substitutional atom causes an increase in volume.10 Finally, in a material such as ZrN or TiN, which exhibits both covalent and metallic bonding, the volume of a nitrogen vacancy is practically zero.11 Clearly, the overall response of the lattice can be rather complicated However, the defect volume can be determined fairly easily by applying the relationship   dfV uP ¼ ÀKT V dV T Mg O 1.02.3.6 55 O Lattice Response to a Defect To formulate quantitative or often even qualitative models for defect processes in materials, it is essential that lattice relaxation be effected Without lattice relaxation, the total energies calculated for defect reactions would be so great that we would have to conclude that no point defects would ever form in the material.9 Each defect has an associated defect volume That is, each defect, when introduced into the lattice, causes a distortion in its surroundings, which is Vo·· Mg O Mg O Mg Figure Schematic of the lattice relaxations around an oxygen vacancy in MgO 56 Fundamental Point Defect Properties in Ceramics where uP is the defect volume (in A˚3); KT, the isothermal compressibility (in eV/A˚3); V, the volume of the unit cell (in A˚3); and fV, the Helmholz free energy of formation of the defect (in eV) Finally, defect associations can also (but not necessarily) have a significant effect on defect volumes for a given solution reaction For example, for the Al2O3 solution in MgO if we assume isolated AlMg and V00Mg hashthe least i effect on lattice parameter as a function of AlMg whereas the formation of neutral AlMg : V00Mg  has the greatest effect (in fact ten times the reduction in lattice parameter).12 3rd 1st 2nd M3+ Defect Cluster Structures So far, we have ignored possible geometric preferences between the constituent defects of a defect cluster Of course, for oppositely charged defects, electrostatic considerations would drive the defects to sit as close as possible to one another, which would be described as a nearest neighbor configuration However, as we saw in the previous section, defects can cause considerable lattice strain Consequently, the most stable defect configuration will be dictated by a balance between electrostatic and strain effects To illustrate cluster geometry preference, we will consider simple defect pairs in the fluorite lattice, specifically in cubic ZrO2 These are formed between a trivalent ion, M3ỵ, that has substituted for a tetravalent lattice ion (i.e.,M0Zr ) and its partially chargecompensating oxygen vacancy (i.e., V O ) This doping process produces a technologically important fast ion-conducting system, with oxygen ion transport via oxygen vacancy migration.2,13 The lowest energy solution reaction that gives rise to the constituent isolated defects14 is  M2 O3 ỵ 2Zr Zr ! 2MZr þ VO þ 2ZrO2 with the pair cluster formation following: ẩ ẫ   M0Zr ỵ V O ! MZr ỵ VO Figure shows the options for the pair cluster geometry, in which, if we fix the trivalent substitutional ion at the bottom left-hand corner, the associated oxygen vacancy can occupy the first near neighbor, the second (or next) near neighbor, or the third near neighbor position Defect energy calculations have been used to predict the binding energy of the pair cluster as a function of the ionic radius15 of the trivalent substitutional Figure First, second, and third neighbor oxygen ion sites with respect to a substitutional ion (M3ỵ) 0.8 AI Cr (Mzr: Vo)binding energy (eV) 1.02.3.7 Ga Fe 0.6 Ce 0.4 Yb Y Gd La Sm Sc In 0.2 0.0 -0.2 0.7 0.8 0.9 1.0 1.1 1.2 Cation radius () Figure 10 Binding energies of M3ỵ dopant cations to an oxygen vacancy: ▪ a first configuration;  second configuration, and ▼ third configuration Open symbols represent calculations that required stabilization to retain the desired configuration Reproduced from Zacate, M O.; Minervini, L.; Bradfield, D J.; Grimes, R W.; Sickafus, K E Solid State Ionics 2000, 128, 243 ion.14 These suggest (see Figure 10) that there is a change in preference from the near neighbor configuration to the second neighbor configuration as the ionic radius of the substitutional ion increases The change occurs close to the Sc3ỵ ion Furthermore, the binding energy of the near neighbor cluster falls as a function of radius; conversely, the binding energy of the second neighbor cluster increases Consequently, the change in preference occurs at a minimum in binding energy The third neighbor cluster is largely independent of ionic radius Interestingly, the minimum coincides with a maximum in the ionic conductivity, perhaps because the trapping of the oxygen vacancies as they move through the lattice is at a minimum.14 Fundamental Point Defect Properties in Ceramics The change in preference for the oxygen vacancy to reside in a first or second neighbor site is a consequence of the balance of two factors: first, the Coulombic attraction between the vacancy and the dopant substitutional ion, which always favors the first neighbor site, and is largely independent of ionic radius, and second, the relaxation of the lattice, a crystallographic effect that always favors the second neighbor position This is because, in the second neighbor configuration, the Zr4ỵ ion adjacent to the oxygen vacancy can relax away from the effectively positive vacancy without moving away from the effectively negative substitutional ion Nevertheless, lattice relaxation in the first neighbor configuration contributes an important energy term However, in the first neighbor configuration, the relaxation of oxygen ions is greatly hindered by the presence of larger trivalent cations, while small trivalent ions provide more space for relaxation Thus, the relaxation preference for the second neighbor site increases in magnitude as the ionic radius increases and consequently, the second neighbor configuration becomes more stable compared to the first.14 This example shows that even in a simple system such as a fluorite, which has a simple defect cluster, the factors that are involved in determining the cluster geometry become highly complex Even so, we have so far only considered structural defects Next, we investigate the properties of electronic defects 1.02.4 Electronic Defects 1.02.4.1 Formation Electronic defects are formed when single or small groups of atoms in a crystal have their electronic structure changed (e.g., electrons removed, added, or excited) In particular, they are formed when an electron is excited from its ground state configuration into a higher energy state Most often this involves a valence electron, although electrons from inner orbits can also be excited if sufficient energy is available In either case, the state left by this transition, which is no longer occupied by an electron, is usually termed a hole These defects can be generated thermally, optically, by radiation or through ion beam damage The excited electron component may be localized on a single atomic site and if the electron is transferred to another center, it is represented as a change in the ionization state of the ion or atom to which it is localized This is sometimes described as a small polaron or trapped electron Such electronic defects 57 might migrate through the lattice via an activated hopping process An example of a small polaron electron is a Ce3ỵ ion in CeO2x.16 Alternatively, the excited electron may be delocalized so that it moves freely through the crystal In this case, the electron occupies a conduction band state, which is formed by the superposition of atomic wave functions from many atoms This is the case with most semiconductor materials Similarly, the hole may also be localized to one atomic center and be represented as a change in the ionization state of the ion or atom Holes may also move via an activated hopping process An example is a Co3ỵ ion in Co1ÀxO Similarly, the hole may also be delocalized Intermediate situations may occur with the hole or electron being localized to a small number of atoms or ions (known as a large polaron) or a specific type of hole state associated with a particular chemical bond The relationship between doping and its influence on electronic defects is of great technological importance in the field of semiconducting materials For example, doping silicon with defect concentrations in the order of parts per million is sufficient for most microelectronic applications Incorporation of a phosphorous atom in silicon results in a shallow state below the conduction band that will easily donate an electron to the conduction band The remaining four valence electrons of the phosphorous dopant will form sp3 hybrid bonds with the four neighboring tetrahedral silicon atoms Recently, it has been suggested that the state from which the electron is removed is associated with the dopant species and the four silicon atoms surrounding it; in other words, it is associated with a cluster.17 1.02.4.2 Concentration of Intrinsic Electrons and Holes Under equilibrium conditions, the number of electronic defects of energy E is given by,2,3 n E ị ẳ N E ị F ðE Þ where N(E) is the volume density of electronic levels that have energy E (known as the density of states) and F(E) is the probability that a given level is occupied, called the Fermi–Dirac distribution function N(E) is a function of energy It is the maximum density of electrons of energy E allowed (per unit volume of crystal) by the Pauli exclusion principle For a semiconductor, this has an approximately parabolic behavior close to the band edges (i.e., N ðE Þ % E 1=2 , refer to Figure 11) 58 Fundamental Point Defect Properties in Ceramics F(E) E Nc(E) T=0 Ec Ef T Ev Nv(E) E Ef N(E) Figure 12 Variation of the Fermi probability function with respect to the electron energy Figure 11 Schematic representation of the density of states function N(E) Eg > kT To determine Nc, the effective conduction band density of states, we need to integrate2,3 ð Nc ðE ÞdE   2pmÃe kT 3=2 % 1019 cmÀ3 Nc ¼ h2 mÃe where is the effective mass of an electron in the conduction band Similarly, the effective valence band density of states is given by2,3   2pmÃh kT 3=2 Nv ¼ % 1019 cmÀ3 h2 where mÃh is the effective mass of a hole in the valence band Note that mÃe and mÃh are between two and ten times greater than the mass of a free electron Also, per volume, these densities are approximately four orders of magnitude less than the typical atom density in a solid The Fermi–Dirac distribution function is given by2,3 F E ị ẳ Ec Ef Ec E f ỵ exp EÀE kT At K, this implies that all energy levels are occupied up to Ef, the Fermi energy This is a step function The Fermi–Dirac function with respect to the energy is represented in Figure 12 At the Fermi energy, F(E) is 1=2 Above K, some energy levels above Ef are occupied This implies that some levels below Ef are empty Eg >> kT Eg Ev Ev Metal Eg upto 1.5 eV ˜ Eg > 3.5 eV Intrinsic semiconductor Insulator Figure 13 Characteristic electron energy band levels for a metal, an intrinsic semiconductor and an insulator, where Ec is the bottom of the conduction band, Ev is the top of the valence band, Eg is the band gap, and Ef is the Fermi level 1.02.4.3 Band Gaps Materials can be classified based on the occupancy of the energy bands (Figure 13) In an insulator or a semiconductor, an energy band gap, Eg, is between the filled valence band, Ev, and the unoccupied (at K) conduction band In metals, the conduction band is partially filled (refer to Figure 13) Typical semiconductors have band gaps up to 1.5 eV; when the band gap exceeds 3.5 eV, the material is considered to be an insulator Table reports the band gaps of some important semiconductors (Ge, Si, GaAs, and SiC) and insulators (UO2, MgO, MgAl2O4, and Al2O3) 1.02.4.4 Excited States The definition of an electronic defect is effectively ‘a deviation from the ground state electronic Fundamental Point Defect Properties in Ceramics Table insulators Band gaps of important semiconductors and Material Band gap (eV) Ge Si GaAs SiC UO2 MgO MgAl2O4 Al2O3 0.66 1.11 1.43 2.9 5.2 7.8 7.8 8.8 3s 2p Ground state Excited state Figure 14 The 2p ! 3s excitation of an oxygen ion in MgO Source: Chiang, Y.-M.; Birnie, D.; Kingery, W D Physical Ceramics; Wiley: New York, 1997 Excited state Energy configuration.’ The defects discussed in Section 1.02.4.2 were holes and electrons Here, we consider defects in which the excited species is localized around the atom by which it was excited If an electron is excited into a higher lying orbital, there must be a difference between the angular momentum of the ground state and the excited state to accommodate the angular momentum of the photon that has been absorbed during the excitation process (conservation of angular momentum) For example, if the ground state is a singlet, then the excited state may be a triplet A simple example would be 2p ! 3s excitation of an oxygen ion in MgO (Figure 14) Notice how the energy levels in Figure 14 alter their energies between the ground state and excited states Therefore, in this case, it is not correct to estimate the energy difference between the ground state and excited states based on the knowledge of only the ground state energy configuration If the excitation energy is calculated based on the ground state ion positions, it is known as the Franck– Condon vertical transition When a photon is absorbed, the energy can be equal to this transition However, the electron in the higher orbital will cause the forces between the ions to be altered Consequently, the ions in the lattice will change their positions slightly, that is, relaxation will occur Such relaxation processes are known as nonradiative, that is light is not emitted Notice that the total energy of the system in the excited state decreases However, if the triplet excited state now decays back to the singlet ground state (a process known as luminescence,18 see Figure 15), locally the ions are no longer in their optimum positions for the ground state That is, the relaxed system in the ground state has become higher The difference between the excitation energy 59 Excitation energy Luminescence Ground state Relaxation Figure 15 The process of luminescence O2− e h Mg2+ Figure 16 A schematic representation of an exciton in MgO and the luminescence energy is known as the Stokes shift.18 Figure 16 represents an example of an excited state electron in MgO, known as a self-trapped exciton.19 The model uses the idea that an exciton is composed of a hole species and an excite electron Notice that the excited electron has an orbit that is between the hole and its nearest neighboring cations Thus, the hole is shielded from the cations This means that the cations not relax to the extent 60 Fundamental Point Defect Properties in Ceramics 1.02.5 The Brouwer Diagram Li e h Cl2- Cl- Figure 17 A model for the exciton in alkali halides The exciton is composed of a hole shared between two halide ions (Vk center) and an excited electron (the so-called Vk þ e model) Interestingly, the two halide ions that comprise the Vk center are not displaced equally from their original lattice positions Reproduced from Shluger, A L.; Harker, A H.; Grimes, R W.; Catlow, C R A Phil Trans R Soc Lond A 1992, 341, 221 they would if there was a bare hole (the small relaxations are indicated by the arrows) Experimentally, the excitation energy in MgO is 7.65 eV, and the luminescence is 6.95 eV, which yields a small Stokes shift of only 0.7 eV.20 In comparison, a model for the exciton in alkali halides is shown in Figure 17 In this case, the exciton is composed of a Vk center (a hole shared between two halide ions) and an excited electron (the so-called Vk ỵ e model) However, it is to be noted that the two halide ions that comprise the Vk center are not displaced equally from their original lattice positions In fact, one of the halide ions is essentially still on its lattice site, while the other is almost in an interstitial site As calculations suggest that the hole is about 80% localized on this interstitial halide ion, it is almost an interstitial atom known as an H-center Also, the electron is shifted away from the hole center and is sited almost completely in the empty halide site (called an F-center) As such, the model is almost a Frenkel pair plus an electron localized at a halide vacancy (the so-called F–H pair model) Whichever model is nearest to reality, Vk ỵ e or FH pair, it is clear that there is considerable lattice relaxation This is reflected in the large Stokes shift In LiCl, the optical excitation energy is 8.67 eV and the p-luminescence energy is only 4.18 eV, leading to a Stokes shift of 4.49 eV.21 Thus far, we have considered both structural and electronic defects In addition, we have derived the relationship between oxygen vacancies and the oxygen partial pressure, PO2 , which gives rise to nonstoichiometry It should therefore not come as any surprise that we now consider the equilibrium between isolated structural defects, electronic defects, and PO2 Of course, we have also considered the equilibrium that exists between isolated structural defects and defect clusters, but defect clusters will not be considered in the present context Nevertheless, defect clustering does play an important role in the equilibrium between electronic and structural defects and cannot, in a research context, be ignored In solving defect equilibria in previous sections, we have generally ignored the role that minority defects might have For example, when considering Schottky disorder in MgO, which we know from experiments is the dominant defect formation process, the effect that oxygen interstitials might have was not taken into account.2 This is certainly reasonable within the context of determining the oxygen vacancy concentration of MgO The oxygen vacancy concentration is the important parameter to know when predictions of the oxygen diffusivity in MgO are required However, minority defects may well play an important role in other physical processes For example, the electrical conductivity or resistivity will depend on the hole or electron concentration; these may be minority defects compared to oxygen vacancies, but understanding them is nevertheless crucial Thus, we must be concerned with four different defect processes2 simultaneously: The dominant intrinsic structural disorder process (e.g., Schottky or Frenkel) The intrinsic electronic disorder reaction The REDOX reaction Dopant and impurity effects Again we begin by considering MgO.6 If we ignore impurity effects, the three reactions are2  00  Mg Mg ỵ OO ! VMg ỵ VO ỵ MgO Null ! e0 ỵ h  O O ! O2 ỵ VO ỵ 2e h i KS ẳ V00Mg V O Kele ẳ ẵe0 ẵh 1=2 Kredox ẳ PO2 ẵe0 V O Fundamental Point Defect Properties in Ceramics Conversely at high PO2 , both oxygen vacancies and their charge-compensating electrons must have relatively low concentrations and therefore, the electroneutrality condition becomes dominated by the h i V00Mg and ½h Š defects so that2 h i ẵh ẳ V00Mg Between these two regimes, the Brouwer approximation depends on whether structural or electronic defects dominate In the case of MgO, we know that Schottky disorder dominates over electronic disorder (as it is a good insulator) and therefore, at intermediate values of PO2 , the appropriate electroneutrality condition is   à h 00 i VO ¼ VMg If the electronic disorder was dominant, this last reaction would be replaced by ½e0 Š ¼ ½h Š We are now in a position to be able to construct a Brouwer diagram, which is usually in the form of ln (defect concentration) versus lnPO2 for various defect components at a constant temperature In the case of MgO, as indicated above, the diagram will clearly · = 2[V ЈЈ ] [h] M [eЈ] 1/2 [V··o ] [VMЈЈ ] (VMЈЈ Vo·· )* Ks [V·· o] [VMЈЈ ] 1/2 Ki · 1/2 [VMЈЈ ] [h]µp o [h]· Stoichiometric crystal To make the problem more tractable, we now introduce the Brouwer approximations, which simplify the form of the electroneutrality condition These effectively concern the availability of defects via the partial pressure of oxygen For example, if the PO2 is very low, the REDOX equilibrium will  à reaction and ½ e Š concentrations are require that the V O relatively high so that these are the dominant positive and negative defect concentrations Therefore, for low PO2,2  ẵe0 ẳ V O Neutrality condition [V·o·] = [VM ] 2[V··o ] = [eЈ] log concentration These equations contain six unknown quantities: four are defect concentrations, the other two variables are the PO2 and the temperature, which are experimental variables and are thus given Of course, we must know the enthalpies of the defect reactions Nevertheless, to solve these equations simultaneously, we need a further relationship This is provided by the electroneutrality condition, which, for MgO states that2 h i  V00Mg ỵ ẵe0 ẳ V O ỵ ẵh 61 [eЈ] 1/2 log po2 Figure 18 The Brouwer diagram for MgO Reproduced from Chiang, Y.-M.; Birnie, D.; Kingery, W D Physical Ceramics; Wiley: New York, 1997 have three regimes corresponding to the three Brouwer conditions (refer to Figure 18 and Chiang et al.2) 1.02.6 Transport Through Ceramic Materials 1.02.6.1 Diffusion Mechanisms Diffusion in ceramic materials is a process enabled by defects and controlled by their concentrations Owing to the existence of separate sublattices, cation and anion diffusion is restricted to taking place separately (i.e., without exchange of anions and cations), which is one of the main differences with respect to diffusion in other materials.22 Therefore, mechanistically, diffusion theory is applied in ceramics by considering the anion and cation sublattices separately Interestingly, it has recently been suggested23 that where there is more than one cation sublattice, cations can move on an alternate sublattice through the formation of cation antisite defects Finally, it can be the case that ion transport in one of the sublattices is more pronounced For example, in oxygen fast ion conductors, oxygen self-diffusion is faster than cation diffusion by orders of magnitude.24–26 62 Fundamental Point Defect Properties in Ceramics Transport in crystalline materials requires the motion of atoms away from their equilibrium positions and, therefore, the role of point defects is significant.22 For example, vacancies provide the space into which neighboring atoms in the lattice can jump,27–29 although it is often the interstitial defects that provide the transport mechanism.22 Diffusion mechanisms refer to the way an atom can move from one position in the lattice to another, generally through an activated process that sees the ion move over an energy barrier The beginning and end points to each jump may be symmetrically identical, providing a contiguous pathway through the crystal, but this need not be so In some cases, the contiguous migration pathway may involve a number of nonidentical steps Nevertheless, in most materials, the motion of an atom is restricted to a few paths There are three main mechanisms that are relevant to most ceramic systems: the interstitial, the vacancy, and the interstitialcy mechanism However, for completeness, we will also briefly describe the collective and the interstitial–substitutional exchange mechanisms, which may be encountered in other classes of materials.22 In the interstitial mechanism, atoms at interstitial sites initially migrate by jumping from one interstitial site to a neighboring one (Figure 19) At the completion of a single jump, there is no permanent displacement of the other ions, although, of course, in the process of diffusion, the extent of lattice relaxation is likely to have become greater to facilitate the saddle point configuration In principle, it is a simple mechanism as it does not require the existence of defects other than the interstitial ion, although it is possible that transient defects are produced if the lattice relaxation is great enough in the course of the jump Interstitial diffusion is not common in ceramic materials but does occur if the interstitial species is small In the vacancy mechanism, a host or substitutional impurity atom diffuses by jumping to a neighboring vacancy (Figure 20) Vacancy-mediated diffusion is common in a number of systems (particularly ceramics with higher atomic density where interstitial defect energies are high) For example, the vacancy mechanism is important for the diffusion of substitutional impurities, for self-diffusion and the transport of n-type dopants in germanium,30,31 and for oxygen self-diffusion in a number of hypostoichiometric perovskite and fluorite-related systems.32 In the vacancy mechanism, the interaction, attractive or repulsive, between the species that undergo transport and the vacancy can be very important Of course, the vacancy mechanism requires the presence of lattice vacancies and therefore, their concentration in the lattice will influence the kinetics.23 In the interstitialcy mechanism, an interstitial atom displaces an atom from its normal substitutional site (Figure 21) The displaced atom, in turn, moves Interstitial (i) (ii) Figure 19 The interstitial mechanism of diffusion The red and blue atoms are lattice species Vacancy (i) (ii) (iii) Figure 20 The vacancy mechanism of diffusion The red and blue ions are lattice species Fundamental Point Defect Properties in Ceramics 1.02.6.2 63 Diffusion Coefficient The temperature dependence of the diffusion coefficient has an Arrhenius form:   Ha D ¼ D0 exp À kT (i) (ii) Figure 21 The interstitialcy mechanism of diffusion The red and blue ions are lattice species, the blue ion with the red perimeter is initially an interstitial species but becomes a lattice species to an interstitial site This mechanism is important for the diffusion of dopants such as boron in silicon.33 In hyperstoichiometric oxides, such as La2NiO4ỵd, it was recently predicted that oxygen diffuses predominantly via an interstitialcy mechanism.26 Collective mechanisms involve the simultaneous transport of a number of atoms They can be found in ion-conducting oxide glasses22 and have been predicted during the annealing of radiation damage.34 Finally, in the interstitial–substitutional exchange mechanism, the impurities can occupy both substitutional and interstitial sites.22 One possibility for the interstitial atom is to migrate in the lattice until it encounters a vacant site, which it then occupies to become a substitutional impurity (dissociative mechanism).22 Another possibility for the impurity interstitial atom is to migrate in the lattice until it displaces an atom from its normal crystallographic site, thus forming a substitutional impurity and a host interstitial atom (kick-out mechanism) The interstitial–substitutional mechanism has been encountered in zinc diffusion in silicon and gallium arsenide.22 Naturally, there are potential energy barriers hindering the motion of atoms in the lattice The activation energy associated with the barriers may be overcome by providing thermal energy to the system The jump frequency o of a defect is given by3   DGm o ¼ n exp À kT where DGm is the free energy required to transport the defect from an initial equilibrium position to a saddle point and n is the vibrational frequency In real materials, the atomic transport may be locally affected by interactions with other defects especially if the defect concentration is high.35–37 where Ha is the activation enthalpy of diffusion, and D0 is the diffusion prefactor that contains all entropy terms and is related to the attempt frequency for migration When diffusion involves only an interstitial migrating from one interstitial site to an adjacent interstitial site, the activation enthalpy of diffusion is composed mainly of the migration enthalpy In comparison, for vacancy-mediated diffusion, dopants are trapped in substitutional positions and form a cluster with one or more vacancies In such a situation, diffusion requires the formation of the cluster that assists in diffusion, migration of the cluster, and finally, the dissociation of the cluster It is common for experimental studies referring to vacancy-mediated diffusion to refer to the activation enthalpy of diffusion The activation enthalpy is the sum of the formation enthalpy and the migration enthalpy The formation energy represents the energetic cost to construct a defect in the lattice (which may well require a complete Frenkel or Schottky process to occur) The formation energy of a defect, Ef (defect), is defined by X Ef defectị ẳ E defectị ỵ qme nj mj j where Ef (defect) is the total energy of the supercell containing the defect; q, the charge state of the defect; me, the electron chemical potential with respect to the top of the valence band of the pure material; nj, the number of atoms of type j; and mj, the chemical potential of atoms of type j It should be noted that in this definition, contributions of entropy and phonons have been neglected The migration energy is the energy barrier between an initial state and a final state of the diffusion process For a system with a complex potential energy landscape, there are a number of different paths that need to be considered 1.02.7 Summary Point defects are ubiquitous: as intrinsic species, they are a consequence of equilibrium, but usually they are far more numerous incorporated as extrinsic species formed as a consequence of fabrication conditions Slow kinetics mean that impurities are trapped 64 Fundamental Point Defect Properties in Ceramics in ceramic materials, typically once temperatures drop below 800 K, although this value is quite material-dependent The intentional incorporation of dopants into a crystal lattice can be used to fundamentally alter a whole range of processes: this includes the transport of ions, electrons, and holes As a result, diffusion rates and electrical conductivity can be manipulated to increase or decrease by many orders of magnitude.1,2 Other mechanical or radiation tolerance-related properties can also be changed radically This chapter has provided the framework for understanding the properties of point defects In particular, the understanding of the concentration of equilibrium-intrinsic species, dopant ions and their interdependence, defect association to form clusters and nonstoichiometry In each case, these defects alter the lattice surrounding them, with atoms being shifted from their perfect lattice positions in response to the specific defect type Electronic defects have been described: not only electrons and holes formed by doping, but also states formed by excitation Structural defects and electronic defects are considered together through Brouwer diagrams Finally, we have also considered the transport of ions through the lattice via different processes, all of which require the formation of point defects 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 References 34 10 11 12 13 Kingery, W D.; Bowen, H K.; Uhlmann, D R Introduction to Ceramics; Wiley: New York, 1976 Chiang, Y M.; Birnie, D.; Kingery, W D Physical Ceramics: Principles for Ceramic Science and Engineering; MIT Press: Cambridge, 1997 Kittel, C Introduction to Solid State Physics; Wiley: New York, 1996 Hull, D.; Bacon, D J Introduction to Dislocations, 4th ed.; Butterworth-Heinemann: Oxford, 2001 Harding, J H.; Atkinson, K J W.; Grimes, R W J Am Ceram Soc 2003, 86, 554 Kroăger F A.; Vink, H J In Solid State Physics; Seitz, F., Turnbull, D., Eds.; Academic Press: New York, 1956; Vol 3, p 307 Ball, J A.; Pirzada, M.; Grimes, R W.; Zacate, M O.; Price, D W.; Uberuaga, 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Chemistry and its Applications; Blackie & Son: Glasgow, 1987 Van Gool, W Principles of Defect Chemistry of Crystalline Solids; Academic Press: New York, 1966  ... defects 1.02. 2 Intrinsic Point Defects in Ionic Materials 1.02. 2.1 Point Defects Compared to Defects of Greater Spatial Extent In crystallography, we learn that the atoms and ions of inorganic materials. .. Fundamental Point Defect Properties in Ceramics of greater spatial extent, such as dislocations and grain boundaries, in greater detail; here, however, we begin by comparing them with point defects... or a defect may introduce a hole into the electronic structure, resulting in p-type conductivity 1.02. 2.2 Intrinsic Disorder Reactions A number of different point defects can form in all ceramics,