Kröger–Vink notation

Schottky pairs

A Schottky defect is an intrinsic point defect that creates vacancies on both the cation and anion sublattices. This defect occurs in ionic crystals when one (positively charged) cation and one (negatively charged) anion leave the lattice simultaneously, resulting in two vacant lattice sites. Because the mass, site, and charge must remain balanced, these vacancies always occur in stoichiometric ratios. However, due to the loss of ions within the crystal lattice, these Schottky defects tend to lead to a decrease in the density of the material because vacancies have been created.

Frenkel pairs

Similarly to a Schottky defect, a Frenkel defect is an intrinsic point defect that produces a vacancy site on either the cation or anion sublattice along with an interstitial site on that same lattice. In an ionic crystal, this occurs when a cation or anion leaves its site in the sublattice, creating the vacancy, and moves to another location to create an interstitial. These Frenkel defect pairs maintain a balanced mass, site, and charge ratio throughout the relocation of the ions. Since the movement remains within the material's single lattice, the density remains the same.

Notation

The notation follows the scheme:

MC
S

M corresponds to the species. These can be

atoms – e.g., Si, Ni, O, Cl,

vacancies – V or v (since V is also the symbol for vanadium)

interstitials – i

electrons – e

electron holes – h

S indicates the lattice site that the species occupies. For instance, Ni might occupy a Cu site. In this case, M would be replaced by Ni and S would be replaced by Cu. The site may also be a lattice interstice, in this case the symbol "i" is used. A cation site can be represented by the symbols C or M (for metal), and an anion site can be represented by either an A or X.

C corresponds to the electronic charge of the species relative to the site that it occupies. The charge of the species is calculated by the charge on the current site minus the charge on the original site. To continue the previous example, Ni often has the same valency as Cu, so the relative charge is zero. To indicate null charge, × is used. A single • indicates a net single positive charge, while two would represent two net positive charges. Finally, ′ signifies a net single negative charge, so two would indicate a net double negative charge.

Examples

Al×
Al — an aluminum ion sitting on an aluminum lattice site, with neutral charge.

Ni×
Cu — a nickel ion sitting on a copper lattice site, with neutral charge.

v•
Cl — a chlorine vacancy, with single positive charge.

Ca••
i — a calcium interstitial ion, with double positive charge.

Cl′
i — a chlorine anion on an interstitial site, with single negative charge.

O′′
i — an oxygen anion on an interstitial site, with double negative charge.

e′
— an electron. No site is normally specified.

Procedure

When using Kröger–Vink notation for both intrinsic and extrinsic defects, it is imperative to keep all masses, sites, and charges balanced in each reaction. If any piece is unbalanced, the reactants and the products do not equal the same entity and therefore all quantities are not conserved as they should be. The first step in this process is determining the correct type of defect and reaction that comes along with it; Schottky and Frenkel defects begin with a null reactant (∅) and produce either cation and anion vacancies (Schottky) or cation/anion vacancies and interstitials (Frenkel). Otherwise, a compound is broken down into its respective cation and anion parts for the process to begin on each lattice. From here, depending on the required steps for the desired outcome, several possibilities occur. For example, the defect may result in an ion on its own ion site or a vacancy on the cation site. To complete the reactions, the proper number of each ion must be present (mass balance), an equal number of sites must exist (site balance), and the charges of the reactants and products must also be equivalent (charge balance).

Example usage

∅ ⇌ v′′′′
Ti + 2 v••
O Schottky defect formation in TiO₂.

∅ ⇌ v′′
Ba + v′′′′
Ti + 3 v••
O Schottky defect formation in BaTiO₃.

Mg×
Mg + O×
O ⇌ O′′
i + v••
O + Mg×
Mg Frenkel defect formation in MgO.

Mg×
Mg + O×
O ⇌ v′′
Mg + v••
O + Mg×
surface + O×
surface Schottky defect formation in MgO.

Basic types of defect reactions

Assume that the cation C has +1 charge and anion A has −1 charge.

1. Schottky defect – forming a vacancy pair on both anion and cation sites:
2. Schottky defect (charged) – forming an electron–hole pair:
3. Frenkel defect – forming an interstitial and vacancy pair on an anion or cation site:
4. Associates – forming an entropically favored site, usually depending on temperature. For the two equations shown below, the right side is usually at high temperature as this allows for more movement of electrons. The left side is usually at low temperature as the electrons lose their mobility due to loss in kinetic energy.

Oxidation–reduction tree

The following oxidation–reduction tree of an ionic species shows the various ways in which a substance can be broken down. Depending on the cation-to-anion ratio, the species can either be reduced and therefore classified as n-type, or if the converse is true, the ionic species is classified as p-type. Below, the tree is shown for a further explanation of the pathways and results of each breakdown of the substance. (A is a cation and X is an anion.)

Schematic examples

From the chart above, there are total of four possible chemical reactions using Kröger–Vink Notation depending on the intrinsic deficiency of atoms within the material. Assume the chemical composition is AX, with A being the cation and X being the anion. (The following assumes that X is a diatomic gas, namely, oxygen and therefore A has a +2 charge as a cation. Note that this could be used to make an oxygen sensor.)

1. For the reduced n-type, with the cation in excess on the interstitial sites:
2. For the reduced n-type, with the anion in deficiency on the lattice sites:
3. For the oxidized p-type, with the cation in deficiency on the lattice site:
4. For the oxidized p-type, with the anion in excess on the interstitial site:

Relating chemical reactions to the equilibrium constant

Using the law of mass action, defect concentration can be related to its Gibbs free energy of formation, and the energy terms (enthalpy of formation) can be calculated given the defect concentration or vice versa.

Examples

For a Schottky reaction in MgO, the Kröger–Vink defect reaction can be written as follows:

∅ ⇌ v′′
Mg + v••
O (1)

Note that the vacancy on the Mg sublattice site has a −2 effective charge, and the vacancy on the oxygen sublattice site has a +2 effective charge. Using the law of mass action, the reaction equilibrium constant can be written as (square brackets indicating concntration):

k = [v′′
Mg][v••
O] (2)

According to the reaction, the stoichiometric relation is as follows,

[v′′
Mg] = [v••
O] (3)

Also, the equilibrium constant can be related to the Gibbs free energy of formation ΔG_f according to the following relations,

k = exp(−ΔG_f/k_BT), where k_B is the Boltzmann constant (4) ΔG_f = ΔH_f − TΔS_f (5)

Relating equations 2 and 4, we get:

exp(−ΔG_f/k_BT) = [v′′
Mg]²

Using equation 5, the formula can be simplified into the following form where the enthalpy of formation can be directly calculated:

[v′′
Mg] = exp(−ΔH_f/2k_BT + ΔS_f/2k_B) = A exp(−ΔH_f/2k_BT), where A is a constant containing the entropic term.

Therefore, given a temperature and the formation energy of Schottky defect, the intrinsic Schottky defect concentration can be calculated from the above equation.