Heitjans P., Karger J. (Eds.). Diffusion in Condensed Matter: Methods, Materials, Models

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5 Diﬀusion in Oxides 211

(a) (b)

Fig. 5.1. Formation of Schottky defects (a) and Frenkel defects (b) in an oxide AO

(• =cation, =anion).

constants for the three equilibria, depending on temperature

.Inthemass

action laws in (5.1)–(5.3) we have assumed that the site fractions of vacancies

and interstitials (defects), denoted by brackets, are small compared to the site

fractions of regular cations and anions, and that the structure elements in

each sublattice form an ideal mixture.

Incorporation of oxygen from the surrounding atmosphere into the crystal

or removal of oxygen from the crystal results in the formation of a non-

stoichiometric oxide with oxygen excess or oxygen deﬁcit. It can also be

described by a quasi-chemical reaction:

(g) + V

••

 O

+2h

•

]

)

1/2

· [V

••

]

. (5.4)

Oxygen from the gas phase,

(g), occupies an oxygen vacancy, V

••

,and

for charge compensation electron holes, h

•

, are produced (see also Fig. 5.2).

Finally, the electronic equilibrium, resulting in the formation of electrons, e



and electron holes, h

•

, has to be considered:

nil  e



•

=[e



] · [h

•

] . (5.5)

The fractions of all charged structure elements are coupled by the condition

of local electrical neutrality:

2[V



]+2[O



]+[e



]=2[V

••

]+2[A

••

]+[h

•

] . (5.6)

The set of equations is completed by the site balances for each sublattice, e.g.

]+[V



] = 1. Provided the equilibrium constants are known, all defect

fractions and the non-stoichiometry

δ =[V



]+[O



] − [V

••

] − [A

••

] (5.7)

can be calculated as a function of the thermodynamic variables, p, T and

The temperature dependence of the equilibrium constants is given by K =

exp(−∆G

/RT )=exp(∆S

/R) · exp(−∆H

/RT ), where ∆G

, ∆S

and ∆H

are the standard Gibbs energy, entropy and enthalpy of the corresponding quasi-

chemical reaction.

212 Manfred Martin

Fig. 5.2. Incorporation of an oxygen

molecule (=) from the gas phase

into an oxide AO with dominating oxy-

gen vacancies (see (5.4)). For each oxy-

gen atom two electron holes are pro-

duced. For the sake of simplicity they

are localized on cations (•).

5.2.1 Dominating Cation Disorder

Oxygen Activity Dependent Disorder

In oxides with a fcc oxygen sublattice, i.e. cubic close packing of the oxygen

ions, as in most transition metal oxides, the formation of oxygen vacancies

and oxygen interstitials is energetically unfavourable in comparison with the

formation of cation defects. Thus, we may regard the oxygen sublattice as

perfect compared to the cation sublattice. This means that in the exactly

stoichiometric oxide with δ = 0 the Frenkel equilibrium dominates

and (5.2)

and (5.6) simplify to [V



]=[A

••

]=(K

)

1/2

. At ﬁxed temperature, T ,and

pressure, p, the so-called stoichiometric point, δ = 0, corresponds to a well

deﬁned oxygen partial pressure, p

∗

. If we increase the oxygen partial pres-

sure relative to p

∗

, oxygen is incorporated into the crystal. Since oxygen

defects are only minority defects it is more convenient to formulate the incor-

poration of oxygen from the ambient atmosphere in terms of cation defects

which is possible by ‘adding’ (5.4) and (5.1):

(g)  O



+2h

•



] · [h

•

]

)

1/2

. (5.8)

Oxidation of the oxide results in the formation of new lattice sites in both

the oxygen and the cation sublattices, i.e. now the crystal grows, in contrast

to the incorporation of oxygen described in (5.4). While the new oxygen

sublattice sites are occupied by oxygen ions the new cation lattice sites are

empty, i.e. cation vacancies are formed (see Fig. 5.3). They are electrically

compensated by the formation of electron holes.

Thus, for oxygen partial pressures that are suﬃciently large compared to

∗

cation vacancies and electron holes will be the majority defects, which

results in a simpliﬁed condition of electrical neutrality, 2[V



]=[h

•

]. Us-

ing this relation, the p

-dependence of the majority type of defects can be

calculated from (5.8):

Here we have assumed that the fractions of the electronic defects are small com-

pared to the fractions of the Frenkel defects. The opposite case, dominating

electronic disorder, will not be considered.

5 Diﬀusion in Oxides 213

Fig. 5.3. Incorporation of an oxygen

molecule (=) from the gas phase

into an oxide AO with dominating

cation vacancies. For each oxygen atom

a new occupied anion lattice site (),

a new unoccupied cation lattice site ()

and two electron holes (•) are produced.

For the sake of simplicity the electron

holes are localized on cations.



•





1/3

· (p

)

1/6

. (5.9)

On the other hand, cation interstitials, A

••

, and electrons, e



, are only mi-

nority defects, and their fractions decrease with increasing p

(see (5.2) and

(5.5)) according to

••

]=K





−1/3

· (p

)

−1/6



]=K

· (2K

)

−1/3

· (p

)

−1/6

. (5.10)

For oxygen partial pressures that are suﬃciently small compared to p

∗

cation interstitials and electrons will be the majority defects, which results in

a simpliﬁed condition of electrical neutrality [e



]=2[A

••

]. With this relation,

the p

-dependence of the majority type of defects can be calculated from

(5.8) and (5.2) by

••





· K



1/3

· (p

)

−1/6

. (5.11)

In this regime, cation vacancies, V



, and electron holes, h

•

, are the minority

defects and their fractions decrease with decreasing p





· K



1/3

· (p

)

+1/6

•



· K



1/3

· (p

)

+1/6

. (5.12)

Equations (5.9) to (5.12) show that all defect fractions follow a power law

dependence on p

with typical exponents 1/6 and −1/6 for oxygen par-

tial pressures which are suﬃciently larger or smaller than p

∗

.Intheclose

vicinity of p

∗

the fractions of the Frenkel defects do not depend on p



]=[A

••

]=(K

)

1/2

. In contrast, the fractions of the minority defects, e



and h

•

, exhibit p

-dependencies with exponents −1/4 and 1/4, respectively

214 Manfred Martin

Fig. 5.4. Dependence of the defect fractions on the oxygen partial pressure p

(Kr¨oger-Vink-diagram) in an oxide AO with dominating Frenkel disorder at the

stoichiometric point (see (5.9)–(5.12)).

(see (5.5) and (5.8)). In a double-logarithmic plot (Kr¨oger-Vink diagram) we

therefore obtain straight lines (see Fig. 5.4) and three typical regions corre-

sponding to δ<0, δ ≈ 0andδ>0.

Without going into further details we note that other exponents are also

possible, mainly for two reasons:

i) Defects might form associates, e.g. a vacancy might trap an electron hole

resulting in a singly ionised vacancy:



•

 V



. (5.13)

If these vacancies become the majority defects they show a dependence



] ∝ (p

)

1/4

. Similarly, cation interstitials might trap an electron,

••



 A

•

, resulting in a dependence [A

•

] ∝ (p

)

−1/4

ii) Often, there are also oxides of diﬀerent stoichiometry, i.e. A

or A

O. For the oxide of a monovalent metal, e. g. Cu

O, we obtain typ-

ical exponents 1/4 and −1/4, and in the mixed-valent oxide Fe

the

exponents are 2/3 and −2/3.

Doping

Doping an oxide A

1−δ

O with dominant cation disorder with an oxide of a

highervalentmetalB,e.g.B

can be described by

5 Diﬀusion in Oxides 215

→ 2B

•



+3O

(5.14)

where we have assumed that the dopant B occupies regular cation lattice sites.

To conserve the lattice structure of the host oxide AO the incorporation of

two dopant cations, B

•

, and three oxygen ions, O

, demands the formation

of one cation vacancy, V



. Since the dopant is charged it must be considered

in the condition of electrical neutrality in (5.6), and the defect structure of

the doped oxide will be changed:

2[V



]+[e



]=2[A

••

]+[h

•

]+[B

•

] . (5.15)

i) For strong doping, the fraction of cation vacancies is directly given by the

dopant fraction, (see (5.14)), and is independent of the thermodynamic

variables p, T and p

, i.e. the exponent of the p

-dependence is zero.

ii) For small dopant fractions, however, the disorder of the host oxide cannot

be neglected, and the defect structure depends on the thermodynamic

variables p, T , p

and the dopant fraction as well.

iii) Coulombic interaction between the dopant ions, B

•

, and diﬀerently

charged cation vacancies, V



and V



, may lead to the formation of solute-

vacancy pairs:

•





•

, V



]

•

] · [V



]

, (5.16)

•





•

, V





]

•

] · [V



]

. (5.17)

Here we have denoted the solute-vacancy pairs containing singly and dou-

bly ionised vacancies by P

and P

5.2.2 Dominating Oxygen Disorder

In oxides where the oxygen ions do not form a cubic close packing but a

more open substructure, for example, in the perovskites or ﬂuorites, oxygen

defects may be the dominating defects while cation defects are only minority

defects. As before, the defect fractions are determined by (5.1)–(5.7), and

typical exponents are obtained for the diﬀerent disorder types. To increase

the fraction of oxygen vacancies, these oxides are often doped with oxides

of lower valent metals. Then, the condition of electrical neutrality in (5.6)

simpliﬁes to



]=2[V

••

] . (5.18)

Thus, the negative excess charge of the dopant cation, B



, is compensated

by oxygen vacancies, and these oxides are good oxygen ion conductors. Some

well know examples are

216 Manfred Martin

– yttria-stabilized zirconia, (Zr

1−x

2−x /2

(YSZ). Here, doping with

increases the fraction of oxygen vacancies, [Y



]=2[V

••

], and sta-

bilizes the cubic ﬂuorite structure. YSZ is a pure oxygen ion conductor

over many orders of magnitude in p

andisusedassuchinoxygen

sensors and solid oxygen fuel cells (SOFC).

– Sr- and Mg-doped lanthanum gallate, (La

1−x

)(Ga

1−y

3−(x +y )/2

(LSGM), which belongs to the class of perovskites ABO

.InLSGM

oxygen vacancies are produced by co-doping in both cation sublattices,

[Sr



]+[Mg



]=2[V

••

], resulting also in a good oxygen ion conductor.

However, at very low or very high oxygen partial pressures the electronic

disorder can no longer be neglected and the oxide becomes a mixed ionic and

electronic conductor. Examples are

– YSZ and LSGM, both of which become mixed conductors (V

••

and e



)at

low oxygen partial pressures.

– strontium-doped lanthanum-chromate, (La

1−x

)CrO

3−δ

,whichisa

good semiconductor at high oxygen partial pressures and becomes a mixed

conductor at low oxygen partial pressures.

5.3 Self- and Impurity Diﬀusion in Oxides

For the deﬁnition of diﬀusion coeﬃcients and ﬂux equations the reader is

referred to Sects. 1.1 to 1.3 of Chap. 1.

5.3.1 Diﬀusion in Oxides with Dominating Cation Disorder

Since cations are mobile in the cation sublattice by means of cation vacancies

and in the interstitial sublattice as cation interstitials, the cation self-diﬀusion

coeﬃcient, D

, can be written as

= D

· [V

]+D

· [A

] . (5.19)

Here D

and D

are the self-diﬀusion coeﬃcients of cation vacancies and

cation interstitials, and [V

]and[A

] are the corresponding site fractions.

If the self-diﬀusion coeﬃcients of vacancies and interstitials do not depend

on the oxygen partial pressure, the p

-dependence of the cation diﬀusion

coeﬃcient is solely determined by the p

-dependence of the defect fractions

which has been calculated in the previous section. Then, the p

-dependence

of the diﬀusion coeﬃcient can be used to identify the disorder type of the

oxide under investigation. At high p

cation vacancies dominate while at low

cation interstitials are the dominating defects (see Fig. 5.4). We therefore

obtain a minimum of the cation self-diﬀusion coeﬃcient as a function of p

and typical exponents in the p

-dependence left and right of the minimum.

Since D

and D

are usually diﬀerent from each other, the minimum in

the diﬀusion coeﬃcient is shifted relative to the stoichiometric point (δ =0).

5 Diﬀusion in Oxides 217

Cation Self- and Impurity Diﬀusion in Spinels A

3−δ

The best-known oxide where the transition from a vacancy to an intersti-

tial regime was found experimentally is magnetite, Fe

3−δ

[10]. It crystal-

lizes in the spinel structure where the oxygen ions form a cubic close pack-

ing while the cations occupy well deﬁned octahedral and tetrahedral sites.

The observed exponents of the iron diﬀusion coeﬃcients are 2/3 at high

and −2/3 at low p

, as expected for dominating cation vacancies and

cation interstitials, respectively (see Sect. 5.2.1). This typical behaviour of

the cation diﬀusion coeﬃcients remains the same if the spinel consists of

several cations, e.g. (Co,Fe,Mn)

[11, 12]. Another example is manganese-

zinc-ferrite, Mn

1−x

, where part of the Fe-ions in magnetite has

been replaced by Mn- and Zn-ions. Cation tracer diﬀusion coeﬃcients have

been measured with radioactive isotopes in the thin-ﬁlm geometry, using both

the sectioning method (described already in Sect. 1.4.1 in Chap. 1) and the

residual activity method [13]. In the sectioning method a thin layer of the

sample is ground oﬀ and its activity, A

sect

, is counted, while in the residual

activity method the residual activity, A

res

, of the sample after grinding oﬀ

a thin layer is counted. In the ﬁrst case the activity proﬁle is a Gaussian

curve, A

sect

∝ exp(−x

/4D

∗

t), while in the second case an error function is

obtained, A

res

∝ (1 −erf(x/

√

∗

t)). Typical proﬁles from both methods are

shown in Fig. 5.5 for diﬀusion of the radioisotope

Mn in manganese-ferrite.

The diﬀusion coeﬃcients obtained from both methods agree well. Figure 5.6

shows in a double-logarithmic plot results for the tracer diﬀusion coeﬃcients

of Mn, Fe and Zn and the impurity diﬀusion coeﬃcient of Co as a function of

the oxygen partial pressure [13]. All diﬀusion coeﬃcients show a minimum as

a function of p

. The slopes +2/3 and −2/3 at high and low p

indicate that

diﬀusion proceeds via cation vacancies and cation interstitials, respectively.

While all diﬀusion coeﬃcients are nearly the same in the vacancy regime,

the diﬀusion coeﬃcient of zinc is higher in the interstitial regime, resulting in

a minimum of the Zn-diﬀusion coeﬃcient which is shifted to higher oxygen

partial pressures compared to the other cations. A more detailed analysis,

considering that the cation sublattice in the spinel structure consists of two

sublattices with octahedral and tetrahedral sites and that iron, manganese

and cobalt cations exist in two charge states, +2 and +3, can be found in [13].

Cation Self- and Impurity Diﬀusion in Monoxides A

1−δ

In most transition metal monoxides, such as Co

1−δ

O, Ni

1−δ

OorMn

1−δ

the oxide is reduced to the metal before the stoichiometric point, δ =0,is

reached. Thus, only a vacancy regime for cation diﬀusion is observed [14–16].

However, as mentioned before, the typical exponent in the p

-dependence

of the cation diﬀusion coeﬃcient quite often diﬀers from the value 1/6 that is

expected if cation vacancies V



would dominate. Subsequently, two examples

will be discussed, pure cobalt oxide and gallium-doped cobalt oxide.

218 Manfred Martin

(a)

(b)

Fig. 5.5. Tracer sectioning proﬁle (a) and residual activity proﬁle (b) of

Mn in

manganese ferrite [13] and corresponding ﬁts with a Gaussian (a) and an error-

function (b).

5 Diﬀusion in Oxides 219

Fig. 5.6. Tracer diﬀusion coeﬃcients of

Mn,

Fe,

Zn, and

Co in

0.54

0.35

2.11

as a function of the oxygen activity at T = 1200

◦

C (symbols)

and ﬁts (solid lines) [13].

Cobalt Monoxide, Co

1−δ

Cobalt monoxide, Co

1−δ

O, exists only with a cation deﬁcit and is a p-type

semiconductor. As shown by Dieckmann [17], the p

-dependence of three

defect-dependent quantities, the non-stoichiometry, δ, the cation tracer dif-

fusion coeﬃcient, D

∗

, and the electrical conductivity, σ, can be explained by

a defect structure, where diﬀerently charged cation vacancies, V



and V



and electron holes, h

•

, are the dominating defects (see (5.13)). At interme-

diate oxygen partial pressures singly ionised cation vacancies V



dominate,

and all three quantities, δ =[V



] (see (5.7)), σ, which is proportional to

•

], as well as the cation tracer diﬀusion coeﬃcient

, D

∗

= f

· D

· [V



show the same dependence on p

with a typical exponent 1/4. Results for

the tracer diﬀusion coeﬃcient of cobalt, D

∗

, measured with the radiotracer

method [18] and demonstrating this p

-dependence are shown in Fig. 5.7

together with results for impurity diﬀusion of Fe in CoO. The smaller slope

can be explained by impurity-vacancy binding between iron ions and cation

vacancies (see (5.16) and (5.17)). In the exact analysis one must consider that

the charge states of both the iron ions and the cation vacancies change with

decreasing p

[19,20].

Ga-Doped Cobalt Monoxide, (Co

1−x

)

1−δ

In Co

1−δ

OdopedwithGa

, resulting in (Co

1−x

)

1−δ

O, both cation

tracer diﬀusion coeﬃcients, D

∗

and D

∗

, have been measured as a function

is the geometrical correlation factor appearing in tracer diﬀusion (see

Sect. 1.3.1 in Chap. 1).

220 Manfred Martin

Fig. 5.7. Tracer diﬀusion co-

eﬃcients of

Co and

Fe in

1−δ

O as function of the oxy-

gen activity at 1100

◦

C. Dashed

lines are guides to the eyes.

For the diﬀerent experimental

techniques see [18].

of the oxygen partial pressure, p

, and the dopant fraction, x,usingra-

dioisotopes [21]. The results depicted in Fig. 5.8 for a temperature of 1350

◦

show two typical features: (i) The dependence of both diﬀusion coeﬃcients

on the dopant fraction, x, is non-linear. (ii) At high oxygen partial pressures

Co is the faster moving species while at low oxygen partial pressures Ga is

faster than Co. To explain this behaviour we have to consider that the solute,

Ga, and the solvent, Co, move by means of diﬀerent diﬀusion mechanisms.

– In the dilute system, x  1, the Ga ions are distinguishable, as tracer ions

in tracer self-diﬀusion. Therefore, Ga is only mobile via solute-vacancy

pairs, P

and P

(see (5.16) and (5.17)), and the solute diﬀusion coeﬃ-

cient, D

∗

, is proportional to the association degrees p

=[P

]/[Ga] and

=[P

]/[Ga] of solute ions bound in these pairs [22].

∗

= D

Ga,1

· p

+ D

Ga,2

· p

(5.20)

The diﬀusion coeﬃcients per defect, D

Ga,1

and D

Ga,2

, depend only on

temperature and include the mobilities of the defects and all physical

correlation eﬀects arising in solute tracer diﬀusion [22] (see also Sect. 1.9.1

in Chap. 1). The dependence of D

∗

on the defect concentrations, and

hence on x, T and p

is given by the association degrees p

and p

– In contrast, the solvent ion Co is mobile by means of free, i.e. unbound

vacancies, V



and V



, and conceivably by vacancies bound in pairs. Since

Co is the majority component, its tracer diﬀusion coeﬃcient is propor-

tional to the appropriate defect fractions (and not to the association de-

grees, as for the solute):

∗

= D

Co,1

·[V



]+D

Co,2

·[V



]+D

Co,P

·[P

]+D

Co,P

·[P

] . (5.21)

Here we have permitted diﬀerent mobilities for the vacancies V



and V

