Kröger–Vink notation

Kröger–Vink notation is a set of conventions that are used to describe electric charge and lattice position for point defect species in crystals. It is primarily used for ionic crystals and is particularly useful for describing various defect reactions. It was proposed by F. A. Kröger and H. J. Vink.[1][2]

Schottky and Frenkel defects

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

Examples

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

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:
    ∅ ⇌ v′
    C     + v•
    A     ⇌ v′
    M     + v•
    X
  2. Schottky defect (charged) – forming an electron–hole pair:
    ∅ ⇌ e′
     + h•
  3. Frenkel defect – forming an interstitial and vacancy pair on an anion or cation site:
    ∅ ⇌ v′
    C     + i•
    C     ⇌ v′
    M     + i•
    M     (cationic Frenkel defect)
    ∅ ⇌ v•
    A     + i′
    A     ⇌ v•
    X     + i′
    X     (anionic Frenkel defect)
  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.
    M×
    M     + e′
     → M′
    M     (metal site reduced)
    B×
    M     → B•
    M     + e′
     (metal site oxidized, where B is an arbitrary cation having one more positive charge than the original site)

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.)

Tree diagram representation of the oxidation and reduction of intrinsic defects in ionic substances.

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:
    A×
    A     + X×
    X     ⇌ A••
    i     +  1⁄2 X2(g) + 2 e′
  2. For the reduced n-type, with the anion in deficiency on the lattice sites:
    A(s) ⇌ A×
    A     + v••
    X     + 2 e′
  3. For the oxidized p-type, with the cation in deficiency on the lattice site:
     1⁄2 X2(g) ⇌ v′′
    A     + X×
    X     + 2 h•
  4. For the oxidized p-type, with the anion in excess on the interstitial site:
    A×
    A     + X×
    X     ⇌ A(s) + X′′
    i     + 2 h•

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 concentration):

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 ΔGf according to the following relations,

k = exp(−ΔGf/kBT), where kB is the Boltzmann constant (4)
ΔGf = ΔHf − TΔSf (5)

Relating equations 2 and 4, we get:

exp(−ΔGf/kBT) = [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(−ΔHf/2kBT + ΔSf/2kB) = A exp(−ΔHf/2kBT), 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.

References

  1. ↑ Kröger, F. A.; Vink, H. J. (1956). Seitz, F.; Turnbull, D., eds. Solid State Physics. 3. pp. 307–435. doi:10.1016/S0081-1947(08)60135-6.
  2. ↑ Carter, C. Barry; Norton, M. Grant (2007). Ceramic Materials: Science and Engineering. New York: Springer. ISBN 0-387-46270-8.
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