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 Kroger-Vink notation, we can describe the formation of Schottky defects in MgO thus:

Null! VMMg + VO* or MgMg + °0 ! VMM g + vo* + 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

Al2°3 ! 2AlMg + 300 + VMg

Then,

it

Ahsol

3kT

3 2 exp

In general, the law of mass action6 states that for a reaction aA + bB $ cC + dD

m£-K. — exp(-AG) b exp kT

[vO*]

Ks

In the case of the pure MgO Schottky reaction, charge neutrality dictates that

[vO*]

7.7′

exp — JT

[v°*]

Now, consider the effect that solution of 10 ppm Al2O3 has on MgO. The solution reaction implies that a concentration of 5 ppm of VMg has been intro­duced into the lattice, that is, jvMg] = 5 x 10-6. Therefore,

[vO*] =2 x 105 exp(-77)

Thus, at 1000 °C, the [V°*] = 6.4 x 10-26 compared to an oxygen vacancy concentration in pure MgO of 5.66 x 10-16. The introduction of the extrinsic defects has therefore lowered the oxygen vacancy concentration by 10 orders of magnitude!

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 2+ and is therefore isovalent to the host lattice ion,

Co0 ! CoMg + °o + MgO

exp

Ksolution

where Ahsol is the solution enthalpy. As the concen­tration of CoO in CoO = 1,

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 do 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

O

Consider the solution of Al203 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,

into pairs, triplets, or possibly even larger clusters. Take, for example, the solution of Al2O3 into MgO.

Al2O3 ! 2AlMg + 3O0 + VMg + 3MgO When the concentration is great enough,

AlMg + VMg! {AlMg : VMg}

the scope of this chapter.7 Therefore, to illustrate the types of relationships that can occur, we use the example of the binary {TiJJg: VMg cluster resulting from TiO2 solution in MgO via

TiO2 ! TiMg + 2O0 + VMg + 2MgO

and

Ti** V" Ti** : V" 0

TiMg T VMg! TiMg : V Mg

Then, using

As seen for solution energies, using the enthalpy associated with this pair cluster formation (the bind­ing energy Ahbind), the reaction can be analyzed using mass action:

exp

AlMg

But since,

yields

exp

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,

we have the relationship

Ahbind T Ahsol

kT

{TiMg : VMg}"

which describes the solution process

Al2O3 ! AlMg + 3O0 + {AlMg: VMg}

Substituting p into a yields the quadratic equation,

x = 0

Further, it is possible to form a neutral triplet defect cluster, {AlMg: VjMg: AlMj, which has a binding enthalpy of AEbind with respect to isolated defects so that

Solving this in the usual manner allows us to deter­mine [TiM’J as a function of Kbinding energy If we now assume that Ah>ind = 1 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 ofeither 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 = 1 X 10-6 and a range of temperature from 500 to 2000 K.

|AlMg : VMg : AlMg}

Kbinding triplet

VII

VMg

Al*Mg

Ahbind

exp[ kF

which leads to

П * II * fX] Ahbind T Ahsol

AlMg : VMg : AlMg = exp — —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

Figure 6 Cluster and isolated defect concentration as a function of x at a temperature of 1000 K.