handbook of heterogenous kinetics (soustelle/handbook of heterogenous kinetics) || the real solid

Chapter 2

The Real Solid: Structure Elements and Quasi-Chemical Reactions

The concept of structure element is absolutely essential for the study and comprehension of the mechanisms in heterogenous kinetics. As a matter of fact, these elements constitute the intermediate compounds produced and consumed by the steps of the mechanism and they are as important for the reactivity of solids as are the radicals and ions for organic chemistry.

In the majority of crystallized mineral solids, the concept of molecule does not have any physical reality. For example, if we consider zinc oxide, whose formula is written as ZnO, we cannot ascribe a particular oxygen atom to a given zinc atom. Moreover, many solids present defects, compared to a crystal, which we could describe as ideal and made up of zinc ions and oxygen ions regularly arranged in space. For example, the presence of a zinc ion in an abnormal position is described as interstitial because of being placed between the positions of the ions in the ideal crystal. Moreover, many of these compounds present distance to the stoichiometry, that is, do not rigorously obey the chemical formula of the ideal compound. For instance, ZnO can actually present an excess of zinc compared with oxygen (that we call a stoichiometric excess of zinc) and whose rigorous formula should be written as Zn1 + xO; the quantity x can vary under various constraints (oxygen pressure, temperature, etc.).

From all these characteristics, it becomes evident that the description of a solid only by means of its atoms, ions, or molecules cannot give an account of many physical and physicochemical properties. This description must be more precise, and

30 Handbook of Heterogenous Kinetics

we must use the nature of the sites occupied by the atoms to specify their environment. This is called the description of the solid in structure elements.

2.1. Structure elements of a solid

2.1.1. Definition of a structure element

A structure element is an atom, an ion, or a vacancy (empty spaces) on a precise site of the crystal. The concept of structure element associates with one of the chemical species and its environment by the nature of the crystalline site taken into account. A structure element can be:

– an atom of the solid in a normal site (within the meaning of the ideal solid);

– an atom of the solid in an abnormal site (atom in substitution of an atom of different nature, atom in interstitial position, etc.);

– a vacancy (atom missing in the ideal solid);

– an empty interstitial position that we will be able to regard as a vacancy in interstitial position;

– a foreign atom with the solid in a precise site (e.g. chromium in place of zinc in zinc oxide).

The solid can be completely described by the enumeration of all its structure elements assimilating, to be complete, free electrons and electron holes likely to be present, with structure elements.

Structure elements can carry variable charges. The effective charge of a structure element, indicated by qe, is the difference between its real charge, symbolized by qr, and the charge that would have the structure element occupying the same site in the ideal crystal, or normal charge, denoted by qn:

e r n q q q= − [2.1]

An element is known as “not ionized” if its effective charge is null.

The normal structure elements of the solid, which are present in the ideal solid (atom or ion of the solid in a normal site or empty interstitial position), have a null effective charge.

The structure elements, other than the normal ones, of the solid are often called “point defects”.

The Real Solid 31

2.1.2. Binary solids

Most of our study of the real solids concerns the binary solids such as metallic oxides, sulfides, and halides. We will indeed see that the description of more complex solids is very often brought back to the binary compounds (section 2.4).

A solid is known as binary if it comprises ideally at least two structure elements occupied by atoms or ions of different chemical nature. In general, these two chemical elements occupy different sites.

A solid is known as stoichiometric if the ratio of the amounts of the two elements that constitute it remains constant and equal to its value in the ideal solid. In the opposite case, the solid is referred to as non-stoichiometric. Note that the ratios of sites represent the site stoichiometry in an ideal solid as in a real solid. For instance, if we consider a binary solid B2A, the real solid will be stoichiometric if the ratio of the amounts of B and A is 2, but, whether it is stoichiometric or not, in all the cases, the ratio of the B sites to the A sites remains 2 and characterizes the structure of the solid.

2.1.3. Symbolic notation of structure elements

In the same way as in chemistry, the need for a symbolic writing system of the atoms and molecules appeared; it was necessary to have a symbolic system for the representation of structure elements. This system must provide three types of information about an element:

– the chemical element involved;

– the occupied site of the lattice (in reference to the ideal solid);

– the effective charge (or the real charge). The effective charge is chosen rather than the real one, because it is null for all the normal elements.

The International Union of Pure and Applied Chemistry (IUPAC) recommended the use of the notation of Kröger [KRO 73]. Tables 2.1 to 2.5 present the whole of this notation, for the various types of structure elements, applied to the example of alumina (fictitious so that it gathers all the cases).

The symbol ′ set on top right of the element means an effective charge of −1; the symbol ° represents an effective charge of +1; the null effective charge is represented by x; the presence of this sign is not obligatory, and it can be omitted. An index i is assigned to an element in interstitial position.


It will be noted that a vacancy is symbolized by V. Because of possible confusion with the representation of vanadium, which has the same symbol, the IUPAC recommends to modify, in the case of the use of structure elements, the V symbol of vanadium and to replace it by the Va symbol.

Al3+ ion in normal position (qe = 0)…. AlAlx or AlAl

O2− ion in normal position (qe = 0)… … OOx or OO

Empty interstitial position (qe = 0)……………….Vi

Table 2.1. Symbols of the structure elements in normal positions

Free electron (qe = −1)…………e′

Hole of free electrons (qe = 1)..h°

Table 2.2. Symbols of the free charges

Ion Al3+ in interstitial position (qe = 3) … °°°iAl

Ion O2− in interstitial position (qe = −2)….. [ ]AO′′

Al atom in interstitial position (qe = 0)……Ali

Table 2.3. Symbols of the structure elements in interstitial positions

Vacancy of Al3+ ion (qe = −3)………………………….…. AlV′′′

Vacancy of O2− ion (qe = −2) ……………………………… °°OV

Vacancy of Al3+ ion trapping an electron hole (qe = −2).. AlV′′

Vacancy of O2− ion trapping an electron (qe = 1)…………. °OV

Table 2.4. Symbols of the lacunar structure elements

Associations of structure elements are represented by indicating between brackets the associated elements without their effective charge and, after the bracket, the effective charge of the unit. For example, the association of a vacancy of an aluminum ion and a vacancy of an oxygen ion will be noted: Al O(V V ) .′ Note that we would obtain the same entity and thus the same notation of effective charge −1 if these vacancies had, respectively, trapped an electron hole and an electron.

The Real Solid 33

S2−

ion in substitution of an O2− ion (qe = 0)… O

xS or SO

Mg2+

ion in substitution of an Al3+ ion (qe = −1)….. AlMg′

Zr4+

ion in substitution of an Al3+ ion (qe = 1)….... ZrAl

°

Li+ ion in interstitial position (qe = 1)………….….. iLi°

Table 2.5. Symbols of structure elements occupied by foreign atoms

2.1.4. Building unit of a solid

The building unit (or unit of construction) of a solid, a combination of structure elements, such as the addition or the subtraction of such a combination, does not modify the relationship between the numbers of the various sites of the crystal (conservation of the structure). The real solid can be built only by the juxtaposition of variable numbers of such units.

For a compound AB (with an interstitial position per site of A), the building unit can be the sum of suitable structure elements, as shown in the following examples:

A B iA B V+ + ; A B iV V V+ + ; A B iE B V+ +

E being the symbol of a foreign atom other than A and B.

2.1.5. Description and composition of a solid

The introduction of the concept of structure elements makes it possible to regard them as components of a solid solution because in a real solid their proportions are likely to vary continuously (the concentration of a structure element can vary without phase change, which is the characteristic of a component in a solution). We will see (sections 3.2.4 and 3.6) that we are able to apply the thermodynamic concepts of solutions to the solid described in structure elements. For that, we must define variables quantifying the composition of the solid phase of each structure element it contains. Several types of quantities are used for this purpose.

2.1.5.1. Site fraction

This is the amount of structure element of a given type divided by the total amount of sites concerned with this type of structure element (either empty or occupied) for the same volume of the solid.


Take, as an example, the case of barium oxide. As we will see, this oxide presents barium cations in interstitial positions. The fraction of sites for these ions is given by the relationship between the amount (mole numbers) of barium ions in this interstitial position and the total amount of interstitial positions, occupied or not, for the same volume. We will thus represent this fraction of sites by

°°i

°°i

°°i i

BaBa

V Ba

nx

n n=

+ [2.2]

When the structure element taken into account is much diluted, this expression is simplified in

°°i

°°i

i

Ba

BaV

nx

n≅ [2.3]

We will use this quantity for the thermodynamic and kinetic studies using structure elements.

2.1.5.2. Concentration

For the concentration, we will use the same definition as the one given for liquid solutions, that is, the amount of the concerned structure element divided by volume of the phase.

In our example of barium cations in interstitial position in barium oxide, the concentration will be given by

°°iBa

nC

V= [2.4]

2.1.5.3. Atomic fraction

The atomic fraction is the amount of the concerned structure element divided by the total amount of product contained in the same volume. In our example, the atomic fraction in interstitial barium ions will be

°°iBa

Ba O

n

n n+ [2.5]

Ban and On represent, respectively, the total amounts of barium and oxygen atoms.

The Real Solid 35

It is easy to switch from a variable defined earlier to another.

2.2. Structure elements of a stoichiometric binary solid

Consider a binary solid that includes two types of chemical elements that we will indicate by A and B and at least three types of normal sites in the ideal solid: the sites ideally occupied by A elements, the sites ideally occupied by B elements, and the empty interstitial sites. In such normal binary compounds, we can find the following point defects (charged or not charged):

– vacancies of A;

– vacancies of B;

– A atoms in interstitial position;

– B atoms in interstitial position;

– A atoms on B sites;

– B atoms on A sites.

The solid being stoichiometric, the ratio of the number of atoms (B/A) must remain constant. In addition, as the ratio of sites (B sites/A sites) should also remain constant, we must thus have the simultaneous presence of at least two types of defects. This whole of two defects found simultaneously is called a “disorder.” We can see, according to the list of defects described earlier, that theoretically there exist six classes of disorders with two defects. Among these classes, we can distinguish two groups: the symmetrical disorders, which utilize the two sub-lattices of A and B, and the asymmetrical disorders, which utilize only one of the two sub-lattices of A or B. In fact, in practice, only four types of disorders are known. Two are symmetrical: Schottky disorder and antistructure disorder. The other two disorders are asymmetrical: Frenkel disorder and S.A. disorder.

2.2.1. Schottky disorder

The Schottky disorder is the simultaneous presence of vacancies of each chemical species VA and VB , the vacancies being in stoichiometric proportions to keep this relationship constant between the atoms. These vacancies can be formed under the action of temperature, by transfer of atoms of A and B from normal sites in the bulk of the crystal to normal sites located at the surface of the crystal. As we will see in section 2.5.2.2.1, the consequence is the creation of a Schottky disorder, which is accompanied by an increase in sizes of the crystal. The source of this disorder (the place where it can be formed) and its well (the place where it can


disappear) are located on extended defects such as dislocations or the surface of the solid. Of course, the formed vacancies may or may not be ionized, preserving the total electric neutrality. We will quote, as example of solids presenting a Schottky disorder the alkaline halides such as potassium or sodium chloride.

2.2.2. Frenkel disorder

The Frenkel disorder is the simultaneous presence of vacancies and atoms in interstitial positions of the same element, for example, VA and Ai. This is an asymmetrical disorder; one will thus have two possible Frenkel disorders for a binary solid: the disorder on the A atoms and the disorder on the B atoms. As we will see in section 2.5.2.2.2, the well and the source of the Frenkel disorder are purely local; its formation does not require displacement of atoms with long distance. The defects that constitute the disorder can be ionized or not, respecting the electric neutrality. It is for the atom of smaller volume that the Frenkel disorder is most probable because it is easiest to place it in an interstitial position. We will quote, as an example, the Frenkel disorder on silver in silver halides.

2.2.3. Antistructure disorder

The antistructure disorder is the simultaneous presence of two types of exchanged atoms AB and BA. The exchanges are simple because they do not modify stoichiometry (see section 2.5.2.2.3). The well and the source of the disorder are local. We encounter this type of disorder especially if the two chemical species have close properties (comparable volumes, close electronegativities), for example, intermetallic compounds.

2.2.4. S.A. disorder

S.A. disorder is the simultaneous presence of exchanged atoms and vacancies of comparable nature; there will be two possibilities: either VA and AB or VB and BA. The creation of the disorder creates a new building unit and thus increases sizes of the crystal (see section 2.5.2.2.4). S.A. disorder exists primarily for intermetallic compounds such as NiAl.

2.3. Structure elements of a non-stoichiometric binary solid

In non-stoichiometric solids, there exist variations that can be very low (e.g. of the range of about 10−4) compared with the stoichiometry of the ideal solid, but the

The Real Solid 37

relationship between the sites must be preserved constant and equal to the ideal ratio of the stoichiometry. These compounds will be characterized by the presence of free or trapped electrons or free or trapped electron holes, which ensure the total electric neutrality and which confer semiconductor properties on these solids. In fact, it is certain that the concept of stoichiometry is as approximate as the concept of purity of a substance: from what amount of impurity is a substance no longer pure? In the same way, we can raise the question: starting from what difference is a solid no longer stoichiometric? Often this character is not detectable by the chemical methods of analysis of elements. It is appropriate to take into account this non-stoichiometry, not detectable by analysis, only when the modeling of a property of the solid requires it. This will be, for example, the case if the solid has semiconductor properties.

2.3.1. Distance from stoichiometry and structure element

Consider a solid of ideal formula Bm An. The result of an ideal proportioning of the elements B and A shows a difference between this theoretical formula and the real formula: the solid is not stoichiometric. Take the case of an excess of B compared to the theoretical formula. The real formula can be expressed in one of the following forms:

BB Am nδ+ or

B(1 )B Am nδ ′+ or A

B Am n δ− or A(1 )B Am n δ ′−

This distance from stoichiometry can thus be expressed by δB, Bδ ′ , δA, Aδ ′ , and also by the difference between the actual and theoretical values of the ratio of the amount of B divided by the amount of A; thus,

B BB m m m

n n n nδ δ

Δ δ+ ′= − = = [2.6]

We wish to express this distance from stoichiometry in terms of concentrations of point defects. For that, we first express the result of a “chemical analysis” of the elements:

total B

total

[B][A]

mn

δ+= [2.7]

Consider the general case where there exist at the same time in the solid A and B vacancies, A and B interstitial sites, and exchanged atoms of A on B sites and those of B on A sites. (All these species can retain electric charges; they will be gathered


in the various forms independently of the charges.) The conservation of sites between the ideal solid and the real solid gives

B B B

A A A

[B ] [A ] [V ][A ] [B ] [V ]

mn

+ +=

+ + [2.8]

The conservations of elements A and B, respectively, result in

[ ] [ ] [ ] [ ]A B itotalA A A A= + + [2.9]

and

[ ] [ ] [ ] [ ]B A itotalB B B B= + + [2.10]

By combining relations [2.7], [2.8], [2.9], and [2.10], we obtain

[ ] [ ] [ ] [ ]{ } [ ] [ ] [ ] [ ]{ }[ ]

A i B B B i A AB

total

B B A V A A B V

A

n mδ

+ − − − + − −=

then we can write

[ ] [ ] [ ] [ ][ ]

[ ] [ ] [ ] [ ][ ]

A i B B B i A AB

total total

B B A V A A B V

A Am

n n

δΔ+ − − + − −

= = − [2.11]

But if the defects are much diluted, we can write roughly:

[ ][ ]

total

total

A

B

nm

≈ and thus [ ] [ ]total totalA Bn

m=

and expression [2.11] leads to

[ ] [ ] [ ] [ ][ ]

[ ] [ ] [ ] [ ][ ]

A i B B B i A AB

total total

B B A V A A B V

B A

mn n

δΔ⎧ ⎫+ − − + − −⎪ ⎪= = −⎨ ⎬⎪ ⎪⎩ ⎭

[2.12]

Expression [2.12] shows that the distance from stoichiometry can be represented by the difference between the two terms. The first one (first fraction of right-hand side) represents a B excess and the other one (second fraction of right-hand side) an

The Real Solid 39

A excess. If the first term is the largest, we will have an overall excess of B. On the contrary, if the second term is the largest, we will have an overall excess of A.

2.3.2. The approximation of Wagner of the prevalent defect for ionic solids

We assume in what follows that A represents the anion and B the cation.

In ionic compounds, the ions exchanged between the anionic and cationic sites are very improbable; this means [ ]BA 0= and [ ]AB 0= . Within this frame-work, Wagner considered and classified the four limiting cases for which the distance from stoichiometry is due practically only to a single type of defect of atomic origin and that he calls the prevalent defect and which is much diluted compared with the normal elements of the lattice. Such solids are sometimes called “solids of Wagner”. Each case simplifies relation [2.12]. As these are ions, to consider the electric neutrality, one must plan the presence of positive charges (electron holes) or negative charges (electrons) that can be free or trapped on other structure elements. This confers on these solids semiconductor properties with primarily electronic conduction. We now examine the four types of solids of Wagner.

2.3.2.1. Ionic binary solids with interstitial cations B

Ionic binary solids with interstitial cations B are characterized by the following conditions on concentrations: [ ]iB 0≠ and [ ] [ ] [ ]B A iV V A 0;= = = , thus, the simplified expression for the distance from stoichiometry for B becomes, according to equation [2.12],

[ ][ ]

[ ][ ]

i i

Btotal

B B

B Bm mn n

Δ = ≈ [2.13]

The electric neutrality is ensured by free or trapped electrons and conductivity σ is expressed as (indicating the mobility of the electrons by μ)

[ ] eσ μ= ′ [2.14]

Examine as an example barium oxide (BaO). The prevalent defect consists of a barium ion in interstitial position accompanied by two free electrons; then the electric neutrality and the distance from stoichiometry Δ yield

°°ie 2 Ba⎡ ⎤′ = ⎣ ⎦ and

[ ]

°°i

Ba

Ba

BaΔ

⎡ ⎤⎣ ⎦=


These solids are n-type semiconductors and the conductivity, primarily electronic, is expressed by

[ ] °°i e 2 Baσ μ μ ⎡ ⎤= ′ = ⎣ ⎦

Figure 2.1 gives a schematic representation plane of the distribution of the structure elements.

O 2− Ba 2+ O 2−

O2−Ba 2+ Ba 2+

O 2− Ba 2+ O 2−

O 2−Ba 2+ Ba 2+

Ba 2+

e´

e´

Figure 2.1. Schematic representation of barium oxide with interstitial cations and free electrons

2.3.2.2. Ionic binary solids with anion vacancies of A

Ionic binary solids with anion vacancies of A are characterized by the following conditions on concentrations: [ ] 0AV ≠ and [ ] [ ] [ ]B i iV B A 0= = = ; thus, the simplified expression for the distance from stoichiometry for B is, according to equation [2.12],

[ ][ ]

[ ][ ]

A A

Atotal

V V

A A

m mn n

Δ = ≈ [2.15]

The electric neutrality is ensured by free or trapped electrons and conductivity is still represented by relation [2.14].

See as an example cerium oxide (CeO2); the prevalent defect consists of vacancies of oxygen ions, accompanied by electrons trapped on normal cerium(IV) ions, transforming them to cerium(III). Thus, the electric neutrality and the oxygen distance from stoichiometry are given in the following way:

The Real Solid 41

[ ]°°O Ce2 V Ce′⎡ ⎤ =⎣ ⎦ and

[ ]

°°O

O

V

OΔ

⎡ ⎤⎣ ⎦=

These oxides are n-type semiconductors and the conductivity, primarily electronic, is then given as

[ ] °°Ce O Ce 2 Vσ μ μ′ ⎡ ⎤= = ⎣ ⎦

Figure 2.2 gives a schematic representation plane of the distribution of the structure elements in cerium oxide.

Ce3+ Ce 3+ Ce4+

O 2− O2−O2−

Ce+4 Ce +4 Ce+4

O -- + O--+ O2−

O 2− O2− O2−

O2−

Figure 2.2. Schematic representation of cerium oxide with anion vacancies and Ce3+ ions

2.3.2.3. Ionic binary solids with interstitial A anions

Ionic binary solids with interstitial A anions are characterized by the following conditions on concentrations: [ ]iA 0≠ and [ ] [ ] [ ]B A iV V B 0.= = = Thus, the simplified expression for the distance from stoichiometry for B is, according to equation [2.12],

[ ][ ]

[ ][ ]

i i

Atotal

A A

A Am mn n

Δ = − ≈ − [2.16]

The electric neutrality is ensured by free or trapped electron holes and conductivity σ is given as (indicating the mobility of the electron holes by μ)

hσ μ °⎡ ⎤= ⎣ ⎦ [2.17]


Examine as an example uranium dioxide (UO2). The prevalent defect consists of

an oxygen ion in interstitial position, accompanied by two electron holes trapped on uranium(IV) ion transforming it to uranium(VI) ion. Then the electric relation of neutrality and the distance from stoichiometry in oxygen are

[ ]°°U iU = O′′ and

[ ][ ]

i

O

O

OΔ

′′= −

These oxides are p-type semiconductors and the conductivity is then

[ ]i 2 Oσ μ ′′=

Figure 2.3 gives a schematic representation plane of the distribution of the structure elements for uranium dioxide.

Figure 2.3. Schematic representation of uranium oxide with interstitial anions and U6+ ions

2.3.2.4. Ionic binary solids with cation vacancies of B

Ionic binary solids with cation vacancies of B are characterized by the following conditions on concentrations: [ ]BV 0≠ and [ ] [ ] [ ]A i iV B A 0= = = . Then, the simplified expression of the distance from stoichiometry for B is

[ ][ ]

[ ][ ]

B B

Btotal

V V

B Bm mn n

Δ = − ≈ − [2.18]

The electric neutrality is ensured by electron holes, free or trapped, and conductivity σ is still given by equation [2.17].

O2− O2− O2− O2−

U6+

O2− O2− O2−

O2− O2− O2−

U4+ U4+

The Real Solid 43

Examine as example iron oxide (FeO). The prevalent defect consists of iron ion vacancies accompanied by trapped electron holes on normal iron(II) ions transforming them to iron(III). Thus, the electric neutrality and the distance from stoichiometry for iron are

[ ] °Fe Fe2 V Fe′′ ⎡ ⎤= ⎣ ⎦ and

[ ][ ]

Fe

Fe

V

FeΔ

′′= −

These oxides are p-type semiconductors and the conductivity, primarily electronic, is then given as

°Fe Feσ μ ⎡ ⎤= ⎣ ⎦

Figure 2.4 gives a schematic representation plane of distribution of the structure elements for iron oxide.

O 2− Fe2+ O2−

O2−Fe 3+ Fe3+

O 2− O2−

O2−Fe 2+ Fe2+

Figure 2.4. Schematic representation of iron oxide with cation vacancies and Fe3+ ions

Ultimately, in a solid of Wagner, the disorder involves two defects, one of atomic (or ionic) nature and the other of electronic nature (even if it is trapped by an ion). Table 2.6 shows the four cases of Wagner. For each one of them, we specify the sign of the charge carrier and the type of semi-conductivity.

Prevalent atomic defect Sign of charge carrier Type of semiconductor Anion vacancies – N

Interstitial cations – N Cation vacancies + P Interstitial anions + P

Table 2.6. Four cases of Wagner ionic solids


2.3.3. More complex binary compounds

When we cannot anymore be satisfied with the Wagner approximation with only one prevailing defect of atomic nature, we must take into account several defects carrying matter and possibly charges. The description of the solid becomes more complicated. For example, some oxides shift from n-semi-conductivity to p-semi-conductivity with the experimental conditions. Except affirming that they abruptly change their case of Wagner, which does not explain anything, this property is incomprehensible, within the framework of this classification. We will see (section 3.6.2) an explanation taking into account a larger variety of defects in the description of the solid.

For example, the description of tantalum oxide, Ta2O5, requires six types of defects: O O Ta Tae , h ,V ,V ,V ,V .° °′ ′

Applying equation [2.12], the distance from oxygen stoichiometry is given as follows:

[ ][ ]

[ ] [ ][ ]

°O O Ta Ta

O Ta

V V V V5 2 O Ta

Δ⎧ ⎫⎡ ⎤+ ′+⎪ ⎪⎣ ⎦= −⎨ ⎬⎪ ⎪⎩ ⎭

and the electric neutrality has the following form:

[ ] [ ] °Ta Oe V h V°′ ′ ⎡ ⎤ ⎡ ⎤+ = +⎣ ⎦ ⎣ ⎦

2.4. Extension to non-binary compounds

It is easy to imagine that the complexity of description increases with one of the studied solid switching from the binary compounds to ternary, quaternary, etc. ones. Some methods of simplification are used to suitably model the behavior of solids more complex than the binary ones. These methods of “degeneration” make it possible to decrease the number of structure elements taken into account. We will quote two rather general methods.

2.4.1. The pseudo-binary approximation

In a solid, bonds of different nature may exist between atoms, and some stronger of these bonds make it possible to assume that gathered several atoms form only one structure element.

The Real Solid 45

Take the example of a metal carbonate, MCO3. In the ideal crystal, we consider that there are only two types of occupied structure elements: the metal ion in cation position and the complex anion carbonate in anion position. We will thus not distinguish the individual behavior of oxygen or carbon atoms. These compounds are thus regarded as binary ones, from where the name of the pseudo-binary approximation comes. These compounds can have defects, for example, an oxygen ion in the place normally occupied by a carbonate ion (this substitution does not involve any charge deficiency).

2.4.2. Generalization of the approximation of the prevalent defect

As was done by Wagner in the case of the binary solids, we can take into account in a more complex solid only the defects considered to be prevalent and describe the solid using only these defects.

Take the example of the oxy-hydroxides metal such as boehmite (AlOOH). The ideal solid is described using the following structure elements:

AlAl , OHOH , and OO

The real boehmite presents defects related to OH sites, and the disorder due to an oxygen ion in substitution of OH (which we should not confuse with an oxygen ion in oxygen position) is taken as prevalent. It leads to the presence of vacancies of OH because of the electric neutrality (and because this solid is not a semiconductor, it does not have free charges). Thus, electric neutrality is expressed by

[ ]OH OHO V°′ ⎡ ⎤= ⎣ ⎦

and conservation of the sites by the following relation:

[ ] [ ]OH OH OHOH O' V 1°⎢ ⎥+ + =⎣ ⎦

Thus, for the description of the properties of boehmite related to the water evolving, we do not have to be concerned with aluminum and oxygen ions placed in normal sites.


2.5. Quasi-chemical reactions

2.5.1. Definition and characteristics of quasi-chemical reactions

When a solid enters into a reaction, it is admitted that it does it via its structure elements. In the same way, the production or the disappearance of a defect in a solid is the consequence of reactions that necessarily involve these structure elements. All these reactions can be can be represented by a way of writing similar to the one of the traditional chemical reactions, but using the symbolic system of the structure element (Kröger notation) instead of atoms, molecules and ions. We obtain what we call quasi-chemical reactions.

Such reactions must have the following characteristics:

– they should not modify the structure of the solid and must thus preserve the relationship between the numbers of the various sites of the solid; a consequence of this property is that a quasi-chemical reaction can be written by using the building units (we will use this property in section 3.2.4 to define the chemical potential of a structure element);

– they must preserve the electric charges and in particular the effective charges;

– they must preserve the chemical elements.

We will examine, as example, the reaction between oxygen gas and barium oxide in the approximation of Wagner. As we saw before, this oxide is with interstitial cations and includes free electrons and thus a cation excess. This distance in excess from stoichiometry is, at the chemical equilibrium, more or less important according to the fixed oxygen pressure above the solid. That is due to a reaction that we can write with the usual chemical symbolic system in the form:

(1 ) 2Ba O O (1 )BaO2xx x+ + = +

(1 )Ba Ox+ indicates the non-stoichiometric form of barium oxide (x > 0).

Such writing presents several disadvantages:

– it lets us believe that under the action of oxygen, barium oxide automatically becomes stoichiometric, and that there exist two barium oxide forms, one stoichiometric and the other non-stoichiometric;

– it does not highlight which barium ions are directly involved with the reaction since it does not distinguish ions in normal positions from those in interstitial ones;

– it involves stoichiometric numbers (x/2, 1+x) witch are variable with pressure, which is not possible for a stoichiometric number.

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We will thus reject this type of writing systematically, which will still present other disadvantages when we try to apply equilibrium laws as the law of mass action (it is difficult to imagine the role of x in such a possibility). To write the quasi-chemical reaction, let us proceed in the way presented hereafter.

Writing down the ideal stoichiometric oxide in structure elements, we have the following equivalence:

Ba O iBaO Ba O V≡ + +

We proceed in the same way for the non-stoichiometric form:

(1 ) Ba O i iBa O Ba O Ba (1 )V 2 e'x x x x+ ≡ + + + − +

We now substitute these the two oxide forms into the previous equilibrium. After simplification of the elements present in the two members of the equation, we obtain:

1i 2 Ba O i2Ba 2e O Ba O 2V°° ′+ + = + +

This equation clearly shows that the addition of oxygen gas to non-stoichiometric oxide leads to the decrease of the amount of barium ions in interstitial position and free electrons. Note that the x quantity disappears, and at equilibrium, the increase in oxygen pressure decreases the number of interstitial barium ions (displacement of equilibrium toward the right) but the oxide does not become automatically stoichiometric. Note that this writing preserves correctly the ratios of sites, the effective charges, and the elements. We note that the reaction results in the appearance of a new building unit (BaO), that is, by an increase in sizes of the crystal.

The quasi-chemical reactions can intervene either in homogenous phase inside the solid or be heterogenous and proceed at an interface between the solid and another phase. We will examine the various types of quasi-chemical reactions.

2.5.2. Homogenous quasi-chemical reactions in the solid

These reactions proceed inside the solid phase. We will distinguish four categories of homogenous reactions: electronic reactions, reactions of creations of disorders (and their opposites for consumption), ionization reactions, and reactions of additions.


2.5.2.1. Electronic reaction

It is about the recombination between the free electrons and the electron holes, which we formulate as

e h 0°′ + = [2.R.1]

The concerned electrons are those of the band of conduction, the holes are at the top of the valence band in the energy diagram (Figure 2.5), and the Gibbs energy associated with this reaction is represented by the height of the forbidden band (gap).

Figure 2.5. Representation of the electronic reaction on the band diagram

2.5.2.2. Reactions of creations of disorders

Let us consider the four types of disorders listed in section 2.2. We will argue on the following compound with the ideal formula: Bm An.

2.5.2.2.1. Creation of Schottky disorder

If 0 represents the perfect crystal, the reaction of Schottky is written as:

A B0 V Vn m= + [2.R.2]

This reaction creates a new building unit of the crystal and there is an increase in its sizes (conservation of the ratio of sites).

2.5.2.2.2. Creation of Frenkel disorder

The reaction is local. It does not involve m and n and is written simply by exchange. The crystal retains its size and, for example, on the A component, we will have:

A i i AA V A V+ = + [2.R.3]

Conductionband

Valenceband

gap

E

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2.5.2.2.3. Creation of the antistructure disorder

It is still about a simple exchange between two close sites, and thus, this reaction does not involve any variation in sizes of the crystal. It is written, independently of m and n, such as

A B B AA B A B+ = + [2.R.4]

2.5.2.2.4. Creation of S.A. disorder

This creation is not local; it is accompanied by the formation of building units and thus results in an increase in sizes of the crystal. We will write from A, for example:

A B AA A 1 Vnm

⎛ ⎞= + +⎜ ⎟⎝ ⎠

[2.R.5]

One A vacancy comes from the A atom, which switches on a B position. The others appear to preserve the constant ratio of sites in the new building unit.

2.5.2.3. Ionization reaction

Any structure element can a priori be ionized either by trapping an electron (or by releasing an electron hole), for example, for an A vacancy which traps an electron

A AV e V′ ′+ = [2.R.6]

or by trapping an electron hole (or by releasing an electron), for example, for an interstitial atom trapping a hole

i iA h A° °+ = [2.R.7]

Figure 2.6. Representation of the ionization reactions in the band diagram

Conductionband

Valenceband

Donor level

Acceptor level


A defect donor of electron introduces an energy level into the forbidden band close to the band of conduction and ionization is the jump of the electron of the element in this band (Figure 2.6). On the contrary, an acceptor element is in the vicinity of the valence band (Figure 2.6) where it can trap an electron.

2.5.2.4. Reactions of addition

Reactions of addition are the reactions of associations of structure elements. They are obviously very simple to write, for example, the creation of a double vacancy such that

A A A2V (V ,V )= [2.R.8]

Note that ionization can be regarded as a particular addition reaction of a free electron or a hole with a structure element of atomic type.

2.5.3. The interphase reactions

The interphase reactions will create or consume defects in the solid starting from another phase containing at least a common element with the solid.

2.5.3.1. Reactions with a gas

We will write, for example, the various reactions starting from A gas. Of course, each one of these reactions should involve only those structure elements that exist naturally in Bm An.

2.5.3.1.1. Creation of A vacancies

The reaction occurs without consumption or creation of a building unit:

A A gasA V A= + [2.R.9]

2.5.3.1.2. Creation of B vacancies

The gas reaction produces building units and thus modifies the crystal sizes:

gas A BA A Vmn

= + [2.R.10]

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2.5.3.1.3. Creation of atoms of A in interstitial positions

This reaction proceeds without modification of the number of units of construction of the solid:

gas i iA V A+ = [2.R.11]

2.5.3.1.4. Creation of atoms of B in interstitial positions

The reaction that provides gas and the defect induces disappearance of units of construction and thus reduction in the crystal sizes:

A B i i gasA B 1 V B Am m mn n n

⎛ ⎞+ + + = +⎜ ⎟⎝ ⎠

[2.R.12]

2.5.3.1.5. Creation of A atoms in B positions

The gas action produces units of construction:

gas A BA A An mm n m n

= ++ +

[2.R.13]

2.5.3.1.6. Creation of B atoms in A positions

The gas action destroys defects and creates units of construction:

A B A gasA B B Am mn n

+ = + [2.R.14]

2.5.3.2. Reactions with another solid phase

Reactions of the solid with another solid phase are also encountered. We consider an interface between the solid and another solid containing the common element B.

2.5.3.2.1. Creation of atoms of B in interstitial positions

This creation involves B vacancies in the other solid, which we denote as Vm to avoid confusions:

i mB B V= + [2.R.15]


2.5.3.2.2. Creation of B vacancies

By using B vacancy in the other solid, we have:

B m BB +V = B +V [2.R.16]

2.5.3.2.3. Creation of interstitial atoms of A

A B i iA B V A Bmn

+ + = + [2.R.17]

2.5.3.2.4. Creation of A vacancies

B A mB B V Vnm

= + + [2.R.18]

2.5.3.2.5. Creation of B atoms on A sites

A B m1 B B B 1 Vm m mn n n

⎛ ⎞ ⎛ ⎞+ = + + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

[2.R.19]

2.5.4. Reactions of solid destruction

Certain reactions materialize the destruction of a solid phase that is not stable any more for certain amounts of defects. It can lead to another solid phase without defect or containing its own defects. We will give two examples illustrating these types of “decompositions”.

Creation of a solid A Bn m′ ′ with n n′ ≤ and m m′ ≤ . We will indicate by A′ and B′ the A and B sites in the produced solid A Bn m′ ′ :

A A B B A BA ( )V ( )V B A Bn n n m m m n m′ ′′ ′ ′ ′ ′ ′+ − + − + = + [2.R.20]

The produced solid can contain its own defects:

A A A' A'( ' 1)A ( ' 1)V ( ' 1)A Vn n n n− + − + = − + [2.R.21]

All the reactions given earlier were written for non-ionized defects. Of course, similar reactions can be written with ionized defects, if the electric neutrality must

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be considered. For example, reaction [2.R.2] creating the defect of Schottky but in the form of ionized vacancies will be written as

A B0 V V° ′= + [2.R.22]

whereas the reaction of creation of interstitial zinc twice ionized is written taking into account reaction [2.R.12] and the existence of free electrons:

°°10 i Zn O i2 O Zn 2e Zn O 2V′+ + = + +

2.6. Introduction of foreign elements into a solid

2.6.1. Concepts of impurity and doping agent

The foreign atoms, introduced inside the solid lattice (in solution), modify the properties of this solid, from where the practice of the doping results, which consists of adding a foreign element to a solid in a voluntary and controlled way. We can distinguish doping in insertion, in which the foreign element comes to take an interstitial position and gives a solid solution of insertion, from doping in substitution, in which the foreign element replaces a normal element of the lattice and gives a solution of substitution.

A doping agent can be introduced by reaction of the considered solid with another solid, liquid, or gas phase containing the foreign element to be introduced.

Take the example of a BO oxide, which we wish to dope with element A. To reach that point, we should be able to put in contact, at high temperature, the BO solid with an AO oxide; a reaction of interface will allow A to penetrate into the BO lattice and to be homogenized there. (It is possible that this treatment also causes the opposite penetration of B in oxide AO.) Another method, usable if A is volatile, is to put, at high temperature, the BO solid in contact with the vapor of A.

Starting from nitrate dissolved in water, a third method consists of filling the BO pores with the just necessary amount of the nitrate solution. Then, by heating the latter, it is broken up by thermolysis and the A elements penetrate in BO (this technique is known as “dry impregnation”).

Doping is particularly interesting for the ionic compounds if the doping element has a oxidation number different from that of the normal elements, which constitute the solid. We will study such effects on stoichiometric and non-stoichiometric solids.


REMARK.– An impurity in solid solution in the studied phase is also a doping agent whose presence is at the same time involuntary and uncontrolled. Consequently, the study of the effects of the impurities present in the solid phases is identical to that of the doping agents.

2.6.2. The controlled atomic imperfection in stoichiometric solids

To study the effect of a doping agent on a stoichiometric solid, we will argue on an example. Introduce divalent calcium ions, Ca2+, in substitution of the cations in potassium chloride, which is a solid of Schottky. To consider at the same time the electric neutrality and the ratio of number of cationic and anionic sites, there are two possibilities, which are presented in the following text.

The first one is the formation of one vacancy of potassium ion for each calcium ion introduced. This is what is schematized in the left part of Figure 2.7. The quasi-chemical reaction of introduction will thus be written in the following way:

2 K K ClCaCl Ca V 2Cl° ′= + +

K + K+Cl−Cl− Cl−

K+ K+ K+Cl−Cl − Cl−

K + K+ K+Cl−Cl− Cl−

K+ Ca2+ Cl−Cl − Cl−

K+

Ca2+

Cl−

Figure 2.7. Schematic representation of potassium chloride doped with calcium ions

The second possibility is the formation of one chloride ion in interstitial site for each calcium ion introduced. The situation will then be as schematized in the right part of Figure 2.7 and the reaction of introduction will be

2 K Cl iCaCl Ca Cl Cl° ′= + +

In this case, the size of chloride ion is too important to be placed in interstitial position and it is thus the first description that is to be retained. Moreover, it

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considers the Schottky character of potassium chloride. The electric neutrality of the doped solid leads to

[ ]K Cl KCa V V° ° ′⎡ ⎤ ⎡ ⎤+ =⎣ ⎦ ⎣ ⎦

Thus, the concentration of calcium ions will influence the concentrations of the other defects. The addition of calcium thus makes it possible for us to modify the concentration of atomic defect such as potassium vacancies voluntarily. For this reason, this type of doping is named “controlled atomic imperfection”. Moreover, if the concentration of calcium ions is sufficient, the potassium vacancies will be found in an amount practically equal to the calcium addition.

Solid type Z (doping) > Z (cation) Z (doping) < Z (cation)

Schottky BV⎡ ⎤⎣ ⎦ increases [ ]AV decreases BV⎡ ⎤⎣ ⎦ decreases [ ]AV increases

Frenkel iB⎡ ⎤⎣ ⎦ decreases BV⎡ ⎤⎣ ⎦ increases iB⎡ ⎤⎣ ⎦ increases BV⎡ ⎤⎣ ⎦ decreases

Table 2.7. Effects of doping on a stoichiometric solid

Table 2.7 examines the various cases of doping, by cations, of solids of Schottky and Frenkel, defined by an anion A and a cation B. This doping leads to the effects given in Table 2.7 and it depends on valence (Z doping) of the introduced cation and valence (Z cation) of the normal cation.

2.6.3. The controlled electronic imperfection in non-stoichiometric solids

The addition of a foreign element with a valence different from that of the basic components will allow controlling the electronic defect in a non-stoichiometric solid. Take as an example a solid of Wagner with cation vacancies such as the FeO and dope it by lithium, by using the vapor of this metal. We have seen earlier (see section 2.3.2.4) that in FeO, the electric compensation of the anion vacancies was due to electron holes trapped on iron of the lattice. The lithium introduction into the iron vacancies causes a decrease in the number of trivalent ions, that is, the number of electron holes (Figure 2.8). The reaction of introduction is as follows:

Fe Fe Fe FeLi Fe V Li Fe° ′ ′+ + = +


Thus, doping enables us to control the amount of electronic defect (here iron III) from where the name of controlled electronic imperfection comes.

Table 2.8 shows the direction of variation of the concentration of the charge carrier in a semiconductor by substitution doping by cations whose valence (Z doping) is different from that of the basic cations.

Figure 2.8. Schematic representation of iron oxide doped with lithium

Solid type

Z (doping) > Z (cation) Z (doping) < Z (cation)

P Charge carrier concentration

decreases Charge carrier concentration

increases

N Charge carrier concentration

increases Charge carrier concentration

decreases

Table 2.8. Effects of doping on a semiconductor of Wagner

REMARK.– The doping by cations in interstitial positions leads, whatever the charge of the doping agent, to the same result as the one obtained with doping agent in substitution by an element of higher valence. Doping by anions leads to results opposite those caused by doping by cations.

2.6.4. Concept of induced valence

In certain cases, the introduced foreign atom can take several oxidation levels and can thus adapt its charge to those of the normal elements of the lattice. It is the

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phenomenon of induced valence. Thus, for example, if manganese is introduced, starting from a manganese(II) compound, in alumina, under certain conditions of temperature and pressure, we find it in the manganese(III) form, whereas if the same manganese compound is introduced into titanium dioxide, under the same conditions, we find it in the manganese(IV) state. Thus, the valence of the doping agent is induced by the matrix and thus does not have the expected effects on the free electric charges.

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