Defect Chemistry and Electrical Properties of Titanium Dioxide. 1

590

J. Phys. Chem. C 2008, 112, 590-601

Defect Chemistry and Electrical Properties of Titanium Dioxide. 1. Defect Diagrams J. Nowotny,* T. Bak, M. K. Nowotny, and L. R. Sheppard Centre for Materials Research in Energy ConVersion, School of Materials Science and Engineering, The UniVersity of New South Wales, Sydney, NSW 2052, Australia ReceiVed: June 13, 2007; In Final Form: October 2, 2007

The present work reports defect diagrams showing the effect of oxygen activity on the concentration of both electronic and ionic defects in TiO2 and its solid solutions with donor- and acceptor-type ions. These diagrams were determined by using defect equilibrium constants related to defect reactions describing the formation of defects. A good agreement between the derived diagrams and defect-related experimental data, including electrical conductivity and thermoelectric power, was revealed. These diagrams may be used for the selection of the optimal processing conditions, including oxygen activity, temperature, and the concentration of aliovalent ions, in order to obtain TiO2 with controlled semiconducting properties that are desired for specific applications.

1. Introduction The interest in studies of titanium dioxide, TiO2, has been generated due to its wide range of applications, including water splitting,1 water purification,2 self-cleaning coatings of building materials,2 antiseptic coatings of sanitary areas,2 and antipollution coatings of road surfaces.3 Most of these applications are related to the photosensitivity of TiO2 and, especially, its photoreactivity with water.2 Intensive research aims at the modification of the TiO2 properties in order to enhance its performance either as a photoelectrode for photoelectrochemical cells (PECs) or as photocatalysts for water purification.1,4-8 These modifications are mainly based on the changes of its chemical composition, including: (1) incorporation of anions, such as C, F, and Cl, into the oxygen sublattice; (2) incorporation of aliovalent cations of the valency which differs from that of the host lattice cation (Ti4+); (3) formation of composites with other oxides and metals; and (4) changes of microstructure through the application of different processing procedures, including sintering, sol-gel, deposition of thin films, and the variation of grain size. It has been shown that semiconducting properties of TiO2, including electronic structure, charge transport, and photoreactivity, are closely related to its defect disorder.9-13 Therefore, defect chemistry may be used as a framework for the modification of the functional properties that are related to semiconducting properties. Defect disorder of nonstoichiometric oxides, such as TiO2, may be modified within a wide range by the following procedures: (1) changes of intrinsic nonstoichiometry and the related Ti/O ratio by the imposition of controlled oxygen activity and (2) incorporation of aliovalent ions into the TiO2 lattice, leading to the formation of donors or acceptors and resulting in the change of the valency of host lattice ions.6-8,14-16 Effect of Oxygen Nonstoichiometry. Properties of metal oxides are closely related to the content of oxygen in the oxide lattice. This content, and the related defect disorder, may be modified by a change of oxygen activity in the gas phase surrounding the oxide specimen at elevated temperatures. * To whom correspondence should be addressed. E-mail: J.Nowotny@ unsw.edu.au. Phone: +61 2 9385 6459. Fax: +61 2 9385 6467.

Figure 1. The concentrations of point defects and electrons in TiO2 as a function of oxygen partial pressure at 1373 K, according to Kofstad.17

Therefore, defect disorder diagrams for metal oxides, such as TiO2, can be derived in the form of defect concentrations as a function of oxygen activity.9-12,17,18 Effect of Impurities/Dopants. Impurities have a strong influence on properties of oxide semiconductors.17,19 Therefore, well-defined defect disorder diagrams must be derived for the defect-related data obtained for the specimens that are either pure and free of impurities9-12 or where the impurity content is known. Unfortunately, the vast majority of the available experimental defect-related data which have been reported so far was obtained for specimens of unknown content of impurities. This is the reason why comparison of the literature data on defect-related properties, such as electrical properties, is difficult, if possible at all. Consequently, the defect disorder diagrams reported so far17 require a verification for either (i) pure specimens or (ii) the specimens of known concentrations of impurities. There has been an accumulation of data indicating that even small amounts of aliovalent ions result in a substantial change of electrical properties of metal oxides.17-19 So far, however, little is known on the effect of small amounts of impurities,

10.1021/jp074565u CCC: $40.75 © 2008 American Chemical Society Published on Web 12/20/2007

1. Defect Diagrams of TiO2

Figure 2. Arrhenius plot of the concentration of titanium vacancies; line 1, in equilibrium in Kinetic Regime II (with respect to all defects); 2, in operational equilibrium in Kinetic Regime I (with respect to oxygen vacancies and interstitial titanium ions).

added either intentionally (dopants) or unintentionally (impurities), on defect disorder and the related electrical properties. The present study aims at addressing this issue. The effect of impurities, which form extrinsic defects, on properties depends on the concentration of intrinsic defects. The effect of impurities is substantial when their concentration is larger than that of the intrinsic defects. For example, when the nonstoichiometry is very low, such as in is the case for NiO,19,20 even traces of aliovalent ions, present at the level of several parts per million, may lead to substantial changes in properties. Therefore, in most cases, the effect of impurities cannot be ignored unless there is a clear experimental evidence that their presence may be neglected. The effect of impurities on properties of nonstoichiometric oxides, such as electrical properties, is substantial mainly when they are formed of aliovalent ions, the valency of which is different than that of the host lattice ions.17,18 Therefore, defect disorder models may be properly used for the assessment of defect-related properties, such as electrical properties, only if derived for a well-defined concentration of aliovalent ions. TiO2 is not an exception. Defect disorder may be described quantitatively by defect disorder diagrams, plotting the concentration of defects as a function of oxygen activity. An attempt was made by Kofstad17 to derive preliminary diagrams for TiO2 at 1373 and 1773 K. The diagram of Kofstad at 1373 K is shown in Figure 1. As seen, this diagram shows the effect of oxygen partial pressure on the concentrations of the following defects: (1) electrons, (2) oxygen vacancies, (3) trivalent titanium interstitials, and (4) tetravalent titanium interstitials. The diagrams of Kofstad, however, are not consistent with the most recent knowledge of the defect disorder in TiO2. Therefore, these diagrams require verification for the following reasons: (1) The diagrams proposed by Kofstad were derived with the assumption that all defects in TiO2 are positively charged. This assumption, however, is not consistent with the lattice charge neutrality condition, which requires that the solid must be electrically neutral. Consequently, this condition requires that the crystallographic lattice includes both positively and negatively charged defects and their charge must be internally compensated. (2) The effect of oxygen pressure on the concentration of electrons in the diagram of Kofstad is consistent with the data

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Figure 3. Electrical conductivity as a function of time for a TiO2 single crystal at 1323 K, showing two kinetic regimes.12

reported by Blumenthal et al.21 for TiO2 specimens which are not well defined in terms of (i) the impurity content, (ii) achievement of the equilibrium conditions (chemical diffusion data were not reported), and (iii) oxygen pressure, which may differ from oxygen activity. It was recently shown9 that the exponent of p(O2) dependence of electrical conductivity at p(O2) > 1 Pa for high-purity TiO2 is -1/4 rather than -1/6, as reported by Blumenthal. The purpose of the present study is to derive a comprehensive picture of the defect chemistry for TiO2, the relationship between defect disorder and defect-related properties, such as electrical conductivity, and the related charge transport. These diagrams may be used for processing TiO2-based materials with controlled defect disorder. The second part of this study considers the effect of aliovalent ions in the TiO2, added intentionally as dopants or unintentionally as impurities, on the electrical properties.22 The data reported in that work allow prediction of the effect of impurities on the properties of TiO2. 2. Postulation of the Problem TiO2 is a nonstoichiometric compound.17 Its nonstoichiometry, and the related Ti/O ratio, has been frequently described in terms of a formula TiO2-x, where x is the extent of oxygen deficiency. Therefore, its properties may be changed in a wide range of nonstoichiometries. However, a specific nonstoichiometry may be realized by several combinations of defects. Defect disorder of TiO2 may be considered in terms of ionic point defects, including oxygen vacancies, titanium interstitials, and titanium vacancies (for simplicity, the presence of complex defects and defect associates will be ignored). Therefore, an increase of oxygen deficit may be considered in terms of the following three scenarios:9 (1) increase of the concentration of oxygen vacancies, (2) increase of the concentration of titanium interstitials, and (3) decrease of the concentration of titanium vacancies. Knowledge of the effect of oxygen activity, p(O2), on oxygen deficit, determined for example by thermogravimetry, does not allow one to establish a simple relation between x and the concentration of the ionic defects. In analogy, assuming that all defects are fully ionized, an increase of electrical conductivity for n-type TiO2 may also be considered in terms of the above three scenarios. Therefore, the modification of properties of TiO2, through the variation of oxygen content and the related

592 J. Phys. Chem. C, Vol. 112, No. 2, 2008

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TABLE 1: The Outline of the Effective Concentrations of Acceptors, A, Which Were Selected to Derive the Defect Diagrams Shown in Figures 4-18. The Specimens Involving the Effective Concentrations of Acceptors, A, Corresponding to Line 1 in Figure 2, Are Termed as Undoped Specimens; Consequently, the Specimens with A Larger and Lower than that Corresponding to Line 1 Are Termed as Acceptor- and Donor-Doped Specimensa (Donor/Acceptor) A (molar ratio) 1.0 × 10-5 2.3 × 10-4 1.3 × 10-3 1.8 × 10-3 3.2 × 10-3 5.6 × 10-3 9.2 × 10-3 1.0 × 10-2 a

1073 K

1198 K

1223 K

1273 K

1323 K

donor undoped donor donor donor undoped undoped donor acceptor undoped acceptor undoped acceptor acceptor acceptor

• A ) 4[V′′′′ Ti] + [A′] - [D ].

Ti/O ratio, may lead to an imposition of well-defined properties only if its defect disorder is well defined in terms of defect diagrams describing the effect of p(O2) on the concentration of all kinds of defects, including ionic defects and electronic defects. Moreover, these defect diagrams may be used for the assessment of the effect of aliovalent ions, if they are derived at well-defined concentrations of aliovalent ions. So far, such defect disorders are not available. Therefore, there is a need to derive well-defined defect disorder diagrams for TiO2. Availability of such diagrams may allow the assessment of the relationship between oxygen activity during processing and the concentration of both ionic and electronic defects. The diagrams may be used for processing TiO2 with desired properties by the imposition of well-defined defect disorder through the imposition of controlled oxygen content and the concentration of aliovalent ions. Consequently, these diagrams may allow to use TiO2 as a raw material for processing a wide range of TiO2based semiconductors and quasi-metallic conductors. The present work is based on the equilibrium constants for the formation of defects in TiO2, which have been recently reported by the authors for well-defined (high-purity) specimens.23 These equilibrium constants will be used to derive a comprehensive defect disorder diagram for TiO2. 3. Defect Disorder The formation of defects in TiO2 may be represented, using Kro¨ger-Vink notation,18 by the following defect equilibria9

The studies of the equilibration kinetics have shown that the diffusion rate of titanium vacancies is extremely slow.12 Therefore, prolonged oxidation is required for TiO2 to be equilibrated with respect to these defects. It was shown that controlled properties, which are desired for specific applications, may be achieved by an appropriate combination of defects.17 However, the concentrations of defects, including both electronic and ionic defects, must satisfy the full lattice charge neutrality condition, which may be represented in the following form • •••• 2[V••O] + 3[Ti••• i ] + 4[Tii ] + [D ] + p ) n + [A′] + 4[V′′′′ Ti ] (6)

where [D•] and [A′] are the concentrations of singly ionized donor- and acceptor-type foreign ions (as examples), respectively, present as introduced dopants or incidental impurities. The equilibrium constants for the preceding reactions may be expressed according to the following equations, respectively

K1 ) [V••O]n2p(O2)1/2

(7)

3 K2 ) [Ti••• i ]n p(O2)

(8)

4 K3 ) [Ti•••• i ]n p(O2)

(9)

Ki ) np

(10)

4 -1 K4 ) [V′′′′ Ti]p p(O2)

(11)

where square brackets represent molar fractions. Using the combination of eqs 6-11, the concentration of both electronic and ionic defects may be expressed as a function of the p(O2). The equilibrium constants, which have been determined by Kofstad17 and recently by the authors,11,15,23 have been used for derivation of defect disorder diagrams. These equilibrium constants were determined from three defect-related properties (electrical conductivity, thermoelectric power, and thermogravimetry) for a wide range of TiO2 specimens, including undoped TiO2, Nb-doped TiO2, and TiO2 after prolonged oxidation. These equilibrium constants are the key data for the determination of defect disorder diagrams. Knowledge of the equilibrium constants expressed by eqs 7-10 allows one to derive the concentrations of defects as a function of p(O2)

1 OO a V••O + 2e′ + O2 2

(1)

[V••O] ) K1n-2p(O2)-1/2

(12)

2OO + TiTi a Ti••• i + 3e′ + O2

(2)

-3 -1 [Ti••• i ] ) K2n p(O2)

(13)

2OO + TiTi a Ti•••• + 4e′ + O2 i

(3)

-4 -1 [Ti•••• i ] ) K3n p(O2)

(14)

p ) Kin-1

(15)

nil a e′ + h

•

(4)

These reactions correspond to the formation of the point defects, which exhibit relatively high diffusion rates in the TiO2 lattice. Therefore, these defects may reach their equilibrium concentrations very fast. It has been recently shown12 that defect disorder of TiO2 also includes titanium vacancies, which are formed according to the following reaction • O2 a 2OO + V′′′′ Ti + 4h

(5)

The concentrations of all defects, including both electronic and ionic defects, must satisfy the lattice charge neutrality condition, which is represented by eq 6. The condition expressed by eq 6 involves thermodynamically reversible defects, including oxygen vacancies, titanium interstitials, and titanium vacancies, and both electronic defects, of which concentration depends on oxygen activity. It also includes the defects which are thermodynamically irreversible; the latter



TABLE 2: Equilibrium Constants of Defect Reactions for TiO2 (K1, K2, K3, K4, and Ki Are Defined in Text)a equilibrium constant K1 K2 K3 K4 Ki

a

∆H0 [kJ/mol]

∆S0 [J/mol]

493.1

106.5

334.9 879.2 1025.8 354.5 394.5 222.1

49.9 190.8 238.3 -202.1 -378.7 44.6

specimen

methods

undoped TiO 2

electrical conductivity thermoelectric power electrical conductivity thermogravimetry thermogravimetry electrical conductivity electrical conductivity electrical conductivity thermoelectric power thermogravimetry

Nb-doped TiO 2 undoped TiO 2 undoped TiO 2 undoped TiO 2 Nb-doped TiO 2 undoped TiO 2

authors Bak et al., 200623 Sheppard et al., 200615 Kofstad, 196717 Kofstad, 196717 Bak et al., 200623 Sheppard et al., 200615 Bak et al., 200623 The intrinsic electronic equilibrium constant was determined by Bak et al.23 using the thermodynamic data reported by Kofstad,26 Frland,27 Moser et al.,28 Atlas and Schlehman,29 Alcock et al.,30 and Lee et al.31

ln K ) (∆S°/R) - (∆H°/RT).

correspond to both aliovalent foreign ions that are incorporated into the TiO2 lattice. Formally, titanium vacancies are reversible defects; however, they are quenched in the experimental conditions usually applied in the determination of defect-related properties and, consequently, can be treated as irreversible. The total concentration of these irreversible defects, which is termed the effective concentration of acceptors, A, may be expressed as follows • A ) 4[V′′′′ Ti] + [A′] - [D ]

(16)

such as a temperature of 1000-1400 K and an equilibration time of 0.5-1 h, the concentration of titanium vacancies remains constant. It was shown that the time of ∼4000 h is required to bring the titanium vacancies to their equilibrium concentration at 1323 K.12 For high-purity TiO2, the quantity A assumes the simple form

A ) 4[V′′′′ Ti]

(17)

The concentration terms [A′] and [D•] are independent of the experimental conditions, such as temperature and annealing time, if their evaporation may be ignored. The concentration of titanium vacancies is practically independent of p(O2) in the experimental conditions, which have been commonly reported in the literature.9,14,17,24,25 In the typical experimental conditions,

The concentration of titanium vacancies in equilibrium may be determined from the equilibrium constant K4 reported elsewhere.23 It has been shown that titanium vacancies exhibit an extremely low diffusion rate. Therefore, these defects may assume their equilibrium concentration only in the kinetic regime that allows a propagation of these defects after the p(O2) or temperature is changed to a new value.

Figure 4. Concentration of defects as a function of oxygen activity, p(O2), for TiO2 at 1073 K and A ) 1.0 × 10-5 (square brackets denote concentrations).





The next section considers the impact of the gas/solid kinetics for the O2/TiO2 system in terms of the two types of defects, which have the predominant effect on properties; these are oxygen vacancies and titanium vacancies. 4. Kinetic Effects The gas/solid kinetics for the O2/TiO2 system has been considered in terms of the following two kinetic regimes.11,12

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(1) Kinetic Regime I. The equilibration kinetics in this regime is determined by oxygen vacancies, which are the predominant defects in TiO2. Consequently, in the experimental conditions usually applied in the determination of electrical properties, the gas/solid equilibrium should be considered in terms of these defects, while the concentration of titanium vacancies remains




practically constant. The equilibrium which can be achieved in this regime is termed operational equilibrium. (2) Kinetic Regime II. The equilibration kinetics in this regime is determined by titanium vacancies, of which the diffusion rate is extremely slow. Therefore, these defects may assume the equilibrium concentration only after prolonged periods of time.12




The Kinetic Regime I and the Kinetic Regime II are illustrated in Figure 3, showing isothermal changes of the electrical conductivity during equilibration at 1323 K, after oxygen activity has been increased.12 As seen on the left-hand side of the diagram, the system reaches a practically constant value within


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0.5-1 h. This type of kinetics is termed the Kinetic Regime I. Such kinetics has been commonly observed in the studies of electrical properties, carried out in the temperature range 10001700 K. As seen on the right-hand side of this figure, however, the electrical conductivity and thermoelectric power continue to change very slowly when the system is left for a prolonged period of time (4000-6000 h). These changes represent the Kinetic Regime II. Therefore, the difference in the equilibration rate between Kinetic Regimes I and II is approximately 1:6000. The concentration of titanium vacancies in equilibrium (in the Kinetic Regime II) for high-purity TiO2 is shown by line 1 in Figure 2. However, the concentration of titanium vacancies in the operational equilibrium (the Kinetic Regime I) is lower since the equilibrium value is not achieved due to the kinetic reason. The concentration of titanium vacancies in the Kinetic Regime I is shown by line 2 (broken) in Figure 2. Their concentration was determined from the p(O2) values corresponding to the n-p transition at S ) 0 (where S is the thermoelectric power). For high-purity TiO2 ([A′] ) 0, [D•] ) 0), under slightly reducing/oxidizing conditions and ignoring the minority defects (titanium interstitials and electronic defects), the charge neutrality condition may be reduced to the following form

[V••O] ) 2[V′′′′ Ti]

Kinetic Regime II (these data were determined for TiO2 after prolonged oxidation12). Therefore, the concentrations of titanium vacancies represented by line 1 will be considered as representing undoped TiO2 in the thermodynamic equilibrium. The equilibrium values of A corresponding to undoped (pure) TiO2 at 1073, 1198, 1223, 1273, and 1323 K are shown in Table 1. Consequently, the values larger and smaller than that corresponding to line 1 will be considered as related to acceptorand donor-doped TiO2, respectively. The present work will derive the defect diagrams determined for both undoped TiO2 as well as for both donor- and acceptor-doped specimens. The predominant type of defects in n-type TiO2, over a wide range of stoichiometries, are oxygen vacancies (donors), which are compensated by electrons.11 Under extremely reduced conditions, which are difficult to achieve experimentally, the predominant defect may become trivalent titanium interstitials.9 At prolonged oxidizing conditions, the concentration of titanium vacancies assumes their equilibrium concentration values.12,24 The titanium vacancies have been frequently confused with acceptor-type foreign ions present as impurities.25 It becomes clear, however, that the acceptor-type ionic defects, which must be taken into account in the charge neutrality condition, include both titanium vacancies and extrinsic aliovalent ions. These two types of defects have different physical meanings, and therefore, their effect on properties must be distinguished. The titanium vacancies are formed intrinsically and their concentration depends on processing and sample history. On the other hand, the extrinsic defects are introduced either intentionally (dopants) or unintentionally (impurities). Correct interpretation of the defect disorder of TiO2 requires that these two types of defects be distinguished in derivation of defect disorder diagrams. In order to address this issue, defect diagrams should be derived at specific values of the quantity A, involving both types of defects. The concentration of acceptors, A′, and donors, D•, may be assessed experimentally by impurity analysis. Knowledge of the impurity data may then be used for the determination of the concentration of titanium vacancies formed under specific experimental conditions. Alternatively, knowledge of the concentration of titanium vacancies determined by line 1 in Figure 2 may be used for the assessment of the impurity level, if the value of A is known. The latter approach may be applied only for the specimens after prolonged oxidation when the concentration of titanium vacancies is equal to their equilibrium concentration. The aim of the present work is the derivation of defect disorder diagrams, showing isothermal concentrations of mobile defects in TiO2, including oxygen vacancies, titanium interstitials, and electronic defects, as a function of oxygen activity in the range of 1073-1323 K. These diagrams are derived at constant values of the effective concentration of acceptors, A. At each temperature, the following three diagrams will be derived: (1) undoped TiO2 (A ) 4[V′′′′ Ti]), (2) acceptor-doped TiO2 (A > 4[V′′′′ Ti]), and (3) donor-doped TiO2 (A < 4[V′′′′ Ti]).

(18) 5. Procedure

where

Substituting eqs 12-16 into the condition in eq 6, we obtain

[V′′′′ Ti] )

A 4

(19)

In most cases, the defect-related properties of TiO2 have been reported in the operational equilibrium (Kinetic Regime I). On the other hand, line 1 in Figure 2 represents the equilibrium concentration of titanium vacancies, which is established in the

n5 + An4 - Kin3 - 2K1p(O2)-1/2n2 - 3K2p(O2)-1n 4K3p(O2)-1 ) 0 (20) Equation 20 includes equilibrium constants K1, K2, K3, and Ki, which are shown in Table 2.26-31 The solutions of this equation are determined numerically following the gsl-poly-complex





solve procedure of GNU Scientific Library.32 As a solution, we select a positive real valuesusually there is only one such solution. When either more than one solution is available or there is a lack of a solution, this indicates that the given combination of the parameters does not describe the studied physical system. Defect diagrams were determined for several values of the effective concentration of acceptors, A. The range of the values of A were selected in the following manner. (1) The standard values of A at each temperature were selected to be equal to the concentrations of titanium vacancies demarcated by line 1 in Figure 2 (A ) 4[V′′′′ Ti]). In the first approximation, the diagrams related to these values describe undoped, and very pure, TiO2. (2) Two values of A are found in the vicinity of the standard values. The values higher and lower than the standard value of A represent the acceptor- and the donor-doped specimens. The deviation from the standard concentration values (corresponding to line 1 in Figure 2) is determined by the spectrum of impurities (or dopants), their concentrations, and the valency. 6. Defect Diagrams 6.1. Theoretical Defect Disorder Models. The derived defect disorder diagrams of TiO2, showing the isothermal effect of oxygen activity on the concentration of both ionic and electronic defects, are shown in Figures 4-18, including 1073 K (Figures 4-6), 1198 K (Figures 7-9), 1223 K (Figures 10-12), 1273 K (Figures 13-15), and 1323 K (Figures 16-18). The defect diagrams derived in the present work are limited to specific concentrations of the effective concentration of acceptors, A, ranging between 10-2 and 10-5. Table 1 represents the outline of the specific values of temperatures and the effective concentration of acceptors, A. As seen in Figure 2, the effect of A on the semiconducting properties of TiO2 may be considered in terms of an acceptor type or a donor type when its value is


larger or lower than the equilibrium concentration of titanium vacancies shown by line 1. In the latter case, the value of A for pure specimens corresponds to undoped TiO2. As seen in Figures 4-6, the increase of A leads to a shift of the p(O2) corresponding to the n-p transition point toward the lower values by six orders of magnitude. As also seen, the character of the effect of p(O2) on the concentrations of


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electronic charge carriers and oxygen vacancies depends on A, which at the largest value of A (Figure 6) becomes independent of p(O2). As seen in Figure 6, the concentration of electrons in strongly reduced conditions, when p(O2) < 10-5 Pa, is the following function of p(O2)9

n ) (2K1)1/3p(O2)-1/6

(21)

Then, the defect disorder of TiO2 is governed by the following charge neutrality

2[V••O] ) n

(22)

The p(O2) regime in which the defect disorder is described by the relations in eqs 21 and 22 is termed as the strongly reduced regime. At high oxygen activity, when p(O2) > 10-1 Pa, defect disorder is governed by the ionic charge compensation represented by eq 18. By the combination of eqs 7, 10, and 11, the concentration of electrons may be expressed as the following function of p(O2)

n)

( ) Ki4K1 2K4

1/6

p(O2)-1/4

(23)

The p(O2) regime in which the defect disorder is described by the relations in eqs 18 and 23 is termed the reduced regime. As seen in Figure 6, the predominant electronic charge carriers at p(O2) > 10-1 Pa are electron holes. Then, the lattice charge neutrality is still governed by the condition expressed by eq 18; however, TiO2 becomes a p-type semiconductor. Then, the concentration of electron holes is the following function of p(O2)

( )

2K4Ki2 p) K1


1/6

p(O2)1/4

(24)

The p(O2) regime in which the defect disorder is described by the relations in eqs 18 and 24 is termed the oxidized regime. The effect of p(O2) on the concentration of defects at higher temperatures (Figures 7-18) is similar and may be described by the same three p(O2) regimes. As seen, the defect disorder diagrams exhibit the following features. (1) The increase of temperature results in (a) an increase of p(O2) related to the n-p transition point and (b) an increase of p(O2) related to the transition between the strongly reduced regime and the reduced regime. (2) The increase of A results in (a) a decrease of p(O2) related to the n-p transition point and (b) a decrease of p(O2) related to the transition between the strongly reduced regime and the reduced regime. (3) Both temperature and p(O2) have a substantial effect on the concentration of titanium interstitials (both tri- and tetravalent); however, the effect of these defects on properties may be ignored because they are the minority defects. 6.2. Experimental Verification of Defect Diagrams. The most commonly studied defect-related properties of TiO2 are electrical properties, such as electrical conductivity and thermoelectric power. However, since the derived defect disorder diagrams describe the system at elevated temperatures, corresponding to the gas/solid equilibrium, the diagrams may be verified against the experimental data, which were determined in equilibrium.

Figure 19. Thermoelectric power of a high-purity TiO2 single crystal as a function of oxygen activity (1073-1323 K).10

Figures 19 and 20 show the thermoelectric power and electrical conductivity data determined for a high-purity TiO2 single crystal in the range of 1073-1323 K. The most reliable way to verify the defect disorder models against the thermoelectric power (S) data is to compare the critical value of S, which corresponds to the n-p transition point (S ) 0). On the other hand, the p(O2) dependence of the electrical conductivity is the most common way to verify the defect disorder models. In the latter case, the electrical conductivity may be expressed as the following function of p(O2)

σ ) σ0p(O2)(1/mσ

(25)

where σ is electrical conductivity, σ0 is the electrical conductivity in standard conditions (when p(O2) ) 1 Pa), and 1/mσ is the p(O2) exponent determined from the electrical conductivity,



Figure 20. Electrical conductivity of a high-purity TiO2 single crystal as a function of oxygen activity (1073-1323 K).9

Figure 21. The plot of the p(O2), corresponding to the n-p transition point for TiO2 at 1073 K, as a function of the effective concentration of acceptors, A; the thin vertical lines indicate the values of A related to the Kinetic Regime I and the equilibrium; the dashed line indicates the n-p transition point determined experimentally (S ) 0).

which is sensitive to defect disorder. Therefore, the p(O2) exponent in eq 25 may be determined from the slope of the following dependence

1 d log σ ) mσ d log p(O2)

(26)

When the mobility term is independent of p(O2), the exponent 1/mσ may be compared with the slope

d log n 1 ) mn d log p(O2)

(27)

where mn is the parameter that is sensitive to defect disorder. Figures 21-25 show the p(O2) corresponding to the n-p transition as a function of the effective concentration of acceptors, showing the following three values of A. (1) The value corresponding to line 1 in Figure 2. This value corresponds to the gas/solid equilibrium, which in this case may be achieved after prolonged oxidation when titanium vacancies reach their equilibrium concentration. (2) The value corresponding to line 2 in Figure 2. This value corresponds to the gas/solid equilibrium determined by the Kinetic Regime I, in which the concentration of oxygen vacancies corresponds to equilibrium; however, the concentration of titanium vacancies may be somewhat lower



than that in equilibrium due to the kinetic reason. (3) The values of A determined experimentally at the condition S ) 0. Figure 26 represents the result of the verification in terms of the concentration of titanium vacancies, corresponding to the A data determined experimentally at S ) 0 versus 1/T, along line 1 from Figure 2. As seen, there is an excellent agreement between the experiment and the theory in the range of 1198-1323 K. The agreement indicates that all of the experimental data of A correspond to the gas/solid equilibrium. The value of A at 1073 K is lower than the value of A corresponding to line 1 in Figure 2, indicating that at the lowest temperature, the system is not well equilibrated with respect to titanium vacancies. The comparison of the slopes 1/mn in Figures 4-18, which are related to the theoretical model, and the experimental slopes of 1/mσ indicates that the experimental data are consistent with the theoretical models in terms of the slope values and also the p(O2) values related to the observed transitions between the defect disorder regimes, which are governed by specific simplified charge neutrality conditions. The defect diagrams may be used as a guide for the selection of optimal processing conditions for TiO2 with the controlled defect disorder and semiconducting properties. These diagrams



Nowotny et al.

Figure 26. Arrhenius plot of the effective concentration of acceptors, A/4, (the concentration of titanium vacancies) at the n-p transition, where solid line (1) corresponds to equilibrium in the Kinetic Regime II and the open circles were calculated at the n-p transition point determined from the condition S ) 0.

impurities or intentionally added dopants, is reported in the following work.22 (3) The derived defect disorder diagrams may be used for the verification of the experimental data of defect-related properties. Acknowledgment. The present work was supported by the Australian Research Council, Mailmasters Pty Ltd, Brickworks Pty Ltd, Avtronics (Australia) Pty Ltd, and Rio Tinto Ltd. This project was performed as part of the UNSW R&D program on solar-hydrogen. References and Notes Figure 25. The plot of the p(O2), corresponding to the n-p transition point for TiO2 at 1323 K, as a function of the effective concentration of acceptors, A; the thin vertical lines indicate the values of A related to the Kinetic Regime I and the equilibrium; the dashed line indicates the n-p transition point determined experimentally (S ) 0).

may also be used for assessing the concentration of defects in specific TiO2 specimens, if the concentration of aliovalent impurities is known. 7. Conclusions The defect disorder diagrams derived in the present work for TiO2 show the effect of oxygen activity at elevated temperatures (1073-1323 K) on the concentration of point defects in the gas/solid equilibrium. Consequently, the diagrams may be used for the imposition of controlled concentration of defects in TiO2 by its annealing in the gas phase of appropriate oxygen activity. This has important practical consequences. (1) The properties of TiO2, including electronic structure and charge transport, are closely related to defect disorder. Therefore, the diagrams reported in the present work allow the imposition of these properties in a controlled manner. (2) The diagrams in this work were derived for predetermined values of aliovalent ions, which are represented by the effective concentration of acceptors. Therefore, these diagrams, which are in a good agreement with the experimental data, may be used for assessment of the concentration of aliovalent ions. Quantitative analysis of the effect of these ions, present as

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Defect Chemistry and Electrical Properties of Titanium Dioxide. 1

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