2.1.5 Essentials to Chapter 2.1: Point Defect Equilibrium

In global equilibrium all crystals contain point defects with a concentration c_PD given by an Arrhenius expression of the form:

c_PD	= A · exp –	G_F kT	= A · exp	S_F k	· exp –	H_F kT

A is a constant around (1 ....10), reflecting the geometric possibilities to introduce 1 PD in the crystal (A = 1 for a simple vacancy).

G_F, H_F, S_F are the free energy of formation, enthalpy (or colloquial "energy") of formation, and entropy of formation, respectively, of 1 PD

The entropy of formation reflects the disorder introduced by 1 PD; it is tied to the change in lattice vibrations (circle frequency ω) around a PD and is a measure of the extension of the PD. It must not be confused with the entropy of mixing for many PDs!

S_F	= k ·	∑ i	ln	ω_i ω'_i

Formation enthalpies are roughly around 1 eV for common crystals ("normal" metals"); formation entropies around 1 k.

Small PD clusters (e.g. di-vacancies) are still seen as PDs, their concentration follows from the same considerations as for single PDs to:

c_2V	=	z 2	· exp	S_2V k	·	exp –	H_F(2V) kT

c_2V	=	z 2	· c_1V² · exp	ΔS_2V k	·	exp –	E_2V kT

The constant A for di-vacancies is half the coordination number z (= number of possibilities to arrange the axis of a di-vacancy dumbbell)

The formation enthalpy and entropy of a PD cluster can always be expressed as the sum of these parameters for single PDs minus a binding enthalpy E and a binding entropy ΔS

The term c_1V² or c_1Vⁿ for a cluster of n vacancies makes sure that the concentration of clusters is always far smaller than the concentration of single PDs.

The same relations can be obtained by "making" di-vacancies (or any cluster) by a "chemical" reaction between the PDs and employing the mass action law:

1V + 1V ↔ V_2V + E_2V

(c_1V)² c_2V	=	K(T) =	const · exp –	ΔE kT

There are, however, some pitfalls in using the mass action law; we also loose any information about the factor A

Most important in doing "defect chemistry" with mass action, is a proper definition of the "ingredients" to chemical reaction equations. A vacancy, after all, is not an entity like an atom that can exist on its own. More to that in chapter 2.4.

Note: All of the above is generally valid for all independent PDs: "A" and "B" vacancies, interstitials, antisite defects,. ... .

c_V	= A_V ·	exp	S_F^V k	· exp –	H_F^V kT

c_i	= A_i ·	exp	S_Fⁱ k	· exp –	H_Fⁱ kT

However: If there are additional restraints (like charge neutrality), we may have to consider pairs of (atomic) PDs as one point defect; e.g. Frenkel or Schottky defects.

First principle" calculation show that charge neutrality can only be locally violated on length scales given by the Debye length of the crystal.

Frenkel and Schottky defects are vacancy- interstitial or vacancy^–- vacancy⁺ pairs in ionic crystals.

Frenkel defect: V^– + i⁺
Anti-Frenkel defect: V⁺ + i^-

They are extreme cases of the general "mixed defect case" containing all possible PDs (e.g. V^–, V^–, i⁺, i⁺) while maintaining charge neutrality.

Usually, one finds either Frenkel defects or Schottky defects - if the respective formation enthalpies H_Fre or H_Scho differ by some 0.1 eV, one defect type will dominate.

Schottky defect: V^– + V⁺

It is, however, hard to predict the dominating defect type from "scratch".

Frenkel disorder in: AgCl, AgBr, CaF₂, BaF₂, PbF₂, ZrO₂, UO₂, ...
Schottky disorder in: LiF, LiCl, LiBr, NaCl, KCL, KBr, CsI, MgO, CaO, ...