Bandwidth expansion

Kröger–Vink notation is set of conventions 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.

Schottky and Frenkel Defects

Schottky Pairs

A Schottky defect is an intrinsic point defect that creates vacancies on both the cation and anion sub-lattices. This defect occurs in ionic crystals when one positively charged cation and one negatively charged anion leave the lattice simultaneously, resulting in vacant lattice sites. Because mass, site, and charge numbers 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 Schottky defect, a Frenkel defect is an intrinsic point defect that produces a vacancy site on either the cation or anion sub-lattice 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 sub-lattice, creating the vacancy site, and moves to another location to create a cation/anion 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

$M_S^C$

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 single positive charge, while two would represent two positive charges. Finally, ′ signifies a single negative charge, so two would indicate a double negative charge.

Examples

${\mathrm{A}\mathrm{l}}_{\mathrm{A}\mathrm{l}}^{\times }$ = an aluminum ion sitting on an aluminum lattice site, with neutral charge.

${\mathrm{N}\mathrm{i}}_{\mathrm{C}\mathrm{u}}^{\times }$ = a nickel ion sitting on a copper lattice site, with neutral charge.

$V_{\mathrm{C}\mathrm{l}}^{\bullet }$ = a chlorine vacancy, with singular positive charge.

${\mathrm{C}\mathrm{a}}_{\mathrm{i}}^{\bullet \bullet }$ = a calcium interstitial ion, with double positive charge.

${\mathrm{C}\mathrm{l}}_{\mathrm{i}}^{\text{'}}$ = a chlorine anion on an interstitial site, with singular negative charge.

${\mathrm{O}}_{\mathrm{i}}^{{\text{'}}^{\prime }}$ = an oxygen anion on an interstitial site, with double negative charge.

${\mathrm{e}}_{}^{\text{'}}$ = an electron. A site isn't normally specified.

Procedure

When using Kroger-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 being conserved as they should. 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 or Ø 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

Ø $\iff {V_{\mathrm{T}\mathrm{i}}}^{⁗}+2V_{\mathrm{O}}^{\bullet \bullet }$ = A Kröger–Vink representation of Schottky defect formation in TiO₂.

Ø $\iff {V_{\mathrm{B}\mathrm{a}}}^{″}+{V_{\mathrm{T}\mathrm{i}}}^{⁗}+3V_{\mathrm{O}}^{\bullet \bullet }$ = A Kröger–Vink representation of Schottky defect formation in BaTiO₃.

${\mathrm{M}\mathrm{g}}_{\mathrm{M}\mathrm{g}}^{\times }$+${\mathrm{O}}_{\mathrm{O}}^{\times }\iff {\mathrm{O}}_i^{{\text{'}}^{\prime }}$+${\mathrm{V}}_{\mathrm{O}}^{\bullet \bullet }$+${\mathrm{M}\mathrm{g}}_{\mathrm{M}\mathrm{g}}^{\times }$ = A Kröger-Vink representation of Frenkel defect formation in MgO.

${\mathrm{M}\mathrm{g}}_{\mathrm{M}\mathrm{g}}^{\times }$+${\mathrm{O}}_{\mathrm{O}}^{\times }\iff {\mathrm{V}}_{Mg}^{{\text{'}}^{\prime }}$+${\mathrm{V}}_{\mathrm{O}}^{\bullet \bullet }$+${\mathrm{M}\mathrm{g}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}}^{\times }$+${\mathrm{O}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}}^{\times }$ = A Kröger-Vink representation of Schottky defect formation in MgO.

Basic Types of Defect Reactions

• Assume that the cation has +1 charge and anion has -1 charge.

1. Schottky Defect – Forming vacancy pair on both anion and cation sites

Ø $\iff V_C^{\text{'}}+V_A^{\bullet }\iff V_M^{\text{'}}+V_X^{\bullet }$

2. Schottky Defect (Charged) – Forming electron-hole pair

Ø $\iff e^{\text{'}}+h^{\bullet }$

3. Frenkel Defect – Forming interstitial and vacancy pair on anion or cation site

Ø $\iff V_C^{\text{'}}+i_C^{\bullet }\iff V_M^{\text{'}}+i_M^{\bullet }$ (Cation Frenkel defect)

Ø $\iff V_A^{\text{'}}+i_A^{\bullet }\iff V_X^{\text{'}}+i_X^{\bullet }$ (Anion Frenkel defect)

4. Associates – Forming entropically favored site, usually depending on temperature. For the two equations shown below, the right side, it is usually at high temperature as they allow for more movement of electrons. On the left side, it is usually at low temperature as the electrons lose its mobility due to loss in kinetic energy.

$M_M^{\times }+e^{\text{'}}=M_M^{\text{'}}$ (Metal Site Reduced)

$B_M^{\bullet }+e^{\text{'}}=B_M^{\times }$ (Metal Site Oxidized, where B is an arbitrary cation having one extra positive charge than the original site.)

Oxidation/Reduction Tree

The following oxidation/reduction tree of an ionic species shows the various ways that 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 = Cation

X = Anion

File:Oxidation-Reduction Tree.jpg

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 Kroger-Vink Notation depending on the intrinsic deficiency of atoms within the material. Assume the chemical composition to be AX with A being the cation and X being the anion. (The following assumed 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 cation in excess on the interstitial site:

$A_A^{\times }+X_X^{\times }\iff A_i^{\bullet \bullet }+\frac{1}{2}{X}_2(g)+2e^{\text{'}}$

2. For the reduced n-type, with anion in deficiency on the lattice site:

$A(s)\iff A_A^{\times }+V_X^{\bullet \bullet }+2e^{\text{'}}$

3. For the oxidized p-type, with cation in deficiency on the lattice site:

$\frac{1}{2}{X}_2(g)\iff V_A^{{\text{'}}^{\prime }}+X_X^{\times }+2h^{\bullet }$

4. For the oxidized p-type, with anion in excess on the interstitial site:

$A_A^{\times }+X_X^{\times }\iff A(s)+X_i^{{\text{'}}^{\prime }}+2h^{\bullet }$

Relating Chemical Reactions to the Equilibrium Constant

Using 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 Schottky Reaction for MgO, the Kroger-Vink defect reaction can be written as follows:

Ø $\iff {V_{\mathrm{M}\mathrm{g}}}^{″}+V_{\mathrm{O}}^{\bullet \bullet }$ (1)

Note that the vacancy on the Mg sub-lattice site has a -2 charge and the vacancy on the oxygen sub-lattice site has a +2 charge, detailed explanation of how charge is calculated is shown above. Using the Law of mass action, the reaction equilibrium constant can be written as,

k = $[{V_{\mathrm{M}\mathrm{g}}}^{″}][V_{\mathrm{O}}^{\bullet \bullet }]$ (2)

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

$[V_{Mg}^{{\text{'}}^{\prime }}]=[V_O^{\bullet \bullet }]$ (3)

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

$k=e^{-\frac{\mathrm{\Delta }{G}_F}{k_{B}T}}$, where $k_B$ is the Boltzmann's Constant. (4)

$\mathrm{\Delta }G=\mathrm{\Delta }H-T\mathrm{\Delta }S$ (5)

Relating equation 2 and 4, it yields

$e^{-\frac{\mathrm{\Delta }{G}_F}{k_{B}T}}=[{V_{\mathrm{M}\mathrm{g}}}^{″}{]}^2$

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

$[{V_{\mathrm{M}\mathrm{g}}}^{″}]=e^{-\frac{\mathrm{\Delta }{H}_F}{2k_{B}T}+\frac{\mathrm{\Delta }S}{2k_B}}=A{e}^{-\frac{\mathrm{\Delta }{H}_F}{2k_{B}T}}$, 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

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