Electrical Properties of Niobium-Doped Titanium Dioxide. 1. Defect

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J. Phys. Chem. B 2006, 110, 22447-22454

22447

Electrical Properties of Niobium-Doped Titanium Dioxide. 1. Defect Disorder† L. R. Sheppard, T. Bak, and J. Nowotny* 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 14, 2006; In Final Form: August 28, 2006

The present work reports the electrical conductivity and thermoelectric power for Nb-doped TiO2 at elevated temperatures (1073-1298 K) in the gas phase of controlled oxygen activity, 10-14 Pa < p(O2) < 75 kPa. It is shown that in reduced conditions the Nb-doped TiO2 exhibits metallic-type conductivity. This finding paves the way for the development of high-performance photoelectrodes with substantially reduced internal energy losses during charge transport. The present work also determined the equilibrium constant for the formation of oxygen vacancies and titanium vacancies for Nb-doped TiO2.

1. Introduction Titanium dioxide, TiO2, is a nonstoichiometric compound that exhibits interesting photoelectrochemical properties.1,2 The work of Fujishima and Honda1 showed that TiO2 exhibits outstanding photoelectrochemical properties, which may be utilized for the conversion of solar energy into chemical energy. Specifically, they have demonstrated that TiO2 may be used as a photoelectrode for water splitting. There is a close relationship between defect disorder and electrical properties in nonstoichiometric oxides.3,4 It was shown that TiO2 is an amphoteric semiconductor that exhibits n-type and p-type properties at low and high oxygen activities, respectively.4 Therefore, defect chemistry may be used for the modification of defect-related properties, such as electrical properties and diffusion kinetics. It has been recently shown that photoelectrochemical properties of TiO2 may be tailored by the modification of its defect disorder.2,4 Consequently, defect chemistry may be used as a framework for the imposition of controlled semiconducting and photoelectrochemical properties, which are desired for specific applications. Defect disorder of TiO2 has been considered in terms of oxygen vacancies and titanium interstitials.3 Recent studies have shown that the charge transport and the related properties of TiO2 must also be considered in terms of titanium vacancies.5 It was also shown that, while the equilibrium concentration of these defects is established instantly at the gas/solid interface, their transport from the surface into the bulk is extremely slow.5 Therefore, prolonged times are required for their propagation into the lattice. Recent studies by the authors have shown that the imposition of these defects may lead to p-type conductivity at high oxygen activities.5 The imposition of a well-defined concentration of Ti vacancies in undoped TiO2 requires a prolonged period of time. However, these defects may also be formed through the incorporation of donor-type ions, such as Nb.6 Therefore, the electrical properties of Nb-doped TiO2 are well-defined in terms

of the electrical conductivity component related to Ti vacancies. As a consequence, the electrical conductivity data may be used for the determination of the equilibrium constant of the formation of these defects, as it will be shown below. The basic quantity describing defect equilibria are equilibrium constants. The equilibrium constants for the formation of oxygen vacancies and titanium interstitials in TiO2 were reported by Kofstad,3 and the intrinsic electronic equilibrium constant was reported by Bak et al.7 These data have led to the derivation of the full defect disorder diagram of TiO2.8 This diagram, however, requires a revision for the following reasons: (1) the defect diagram reported by Bak et al.8 is inconsistent with the recently reported data of electrical properties for well-defined (high purity) TiO2, 9 and (2) the recently discovered formation of titanium vacancies in TiO2 5 imposes the need to reassess the defect disorder model by considering the defect disorder for undoped TiO2, including these acceptor-type defects. The purpose of the present work is to use the data of the electrical properties for Nb-doped TiO2 for a reassessment of the defect disorder of TiO2 in terms of defect equilibrium constants, including (1) the equilibrium constant for the formation of oxygen vacancies and (2) the equilibrium constant for the formation of Ti vacancies. The present work will also consider the current state of understanding on defect chemistry for TiO2 and the effect of Nb on the defect disorder. 2. Definitions of Terms 2.1. Defect Equilibria. The defect disorder of undoped TiO2 at elevated temperatures may be considered in terms of oxygen vacancies, titanium interstitials, and titanium vacancies. Their formation may be expressed according to the following equilibria:

1 OO a V••O + 2e′ + O2 2

(1)

2OO + TiTi a Ti••• i + 3e′ + O2

(2)

• O2 a 2OO + V′′′′ Ti + 4h

(3)

nil a e′ + h•

(4)

†

The present work has been performed as part of an ongoing R&D program on solar hydrogen. The program aims at using titanium dioxide as a raw material for processing TiO2-based oxide systems that are required to assemble high-performance photoelectrochemical cells for water splitting. * Corresponding author. Phone: +61 2 9385 6465. Fax: +61 2 9385 6467. E-mail: [email protected].

These equilibria are described by the following equilibrium constants, respectively:

10.1021/jp0637025 CCC: $33.50 © 2006 American Chemical Society Published on Web 10/17/2006

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K1 ) [V••O]n2p(O2)1/2

(5)

3 K2 ) [Ti••• i ]n p(O2)

(6)

4 -1 K3 ) [V′′′′ Ti]p p(O2)

(7)

Ki ) np

(8)

where n and p denote the concentrations of electrons and holes, respectively, and p(O2) is the oxygen activity. The equilibrium constants K1 and K2, estimated from thermogravimetric data, were reported by Kofstad.3 The equilibrium constant Ki was determined from the electrical conductivity data by Bak et al.7 The equilibrium constant K3 has not been reported, so far. The purpose of the present work is to determine the equilibrium constants K1 and K3. Their temperature dependence will then be related to both enthalpy and entropy terms:

ln K )

∆S° ∆H° R RT

(9)

where ∆S° and ∆H° denote the thermodynamic quantities related to entropy and enthalpy, respectively. A consideration of the effect of Nb on the properties of TiO2 will be preceded by a short outline of the effect of p(O2) on defect disorder for undoped TiO2. 2.2. Undoped TiO2. The concentration of electronic charge carriers, and the related electrical properties, may be expressed as a function of p(O2). The p(O2) dependence may be expressed as follows:

∂ log(n,p) 1 ) mn,p ∂ log p(O2)

(10)

where 1/mn,p is the parameter that is sensitive to a specific defect disorder derived in terms of the effect of p(O2) on the concentration of electronic charge carriers. The concentration term, n,p, may be related to defect-related properties, such as electrical conductivity and thermoelectric power. 2.2.1. Strongly Reduced Regime. The predominant defects in this regime are doubly ionized oxygen vacancies. Then, the following simplified charge neutrality applies:

2[V••O] ) n

(11)

Accordingly, the concentration of electrons is the following function of oxygen activity, p(O2):

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

(12)

2.2.2. Reduced Regime. This regime is governed by the following ionic charge compensation:

[V••O] ) 2[V′′′′ Ti]

(13)

Then

n)

( ) Ki4K1 2K3

1/6

p(O2)-1/4

(14)

2.2.3. Oxidized Regime. This regime is governed by the ionic charge compensation expressed by eq 13; however, the pre-

Figure 1. Schematic representation of the defect diagram based on simplified charge neutralities showing the effect of oxygen activity, p(O2), on the concentration of both ionic and electronic defects in undoped TiO2.

dominant electronic defects are electron holes. Their concentration is the following function of p(O2):

p)

( ) 2K3Ki2 K1

1/6

p(O2)1/4

(15)

One should note that eqs 14 and 15 represent the theoretical relations describing the effect of p(O2) on the concentration of individual charge carriers. A schematic representation of the effect of p(O2) on the concentration of both electronic and ionic defects within the above regimes is shown in Figure 1. A similar relationship should be expected for the effect of p(O2) on the electrical conductivity, if the mobility terms are independent of p(O2). It is important to note, however, that the formal fit of, for example, the electrical conductivity data to a certain slope should not be considered as proof of the validity of a specific model. The fit only indicates that the specific model should be taken into account as one of several possible models. Its identification, however, requires confirmation by alternative theoretical and experimental approaches. The relationships between the concentration of defects and p(O2) for undoped TiO2, which are based on simplified charge neutralities, are outlined in Table 1. 2.2.4. Transition Regimes. The recent study of the effect of p(O2) on electrical conductivity, σ, for high-purity TiO2 single crystals have confirmed that the predominant defects are oxygen vacancies, which in the strongly reduced regime are compensated by electrons (1/mσ ) -1/6) and in the reduced regime by Ti vacancies (1/mσ ) -1/4), where 1/mσ is the p(O2) exponent related to electrical conductivity.9 One may expect, however, that the experimentally determined p(O2) exponent, 1/mσ, within these two regimes varies continuously between -1/6 and -1/4,

Properties of Niobium-Doped Titanium Dioxide 1

J. Phys. Chem. B, Vol. 110, No. 45, 2006 22449

TABLE 1: Undoped TiO2sConcentrations of Electronic and Ionic Defects within the Regimes Corresponding to Different Oxygen Activities and Governed by Simplified Charge Neutrality Conditionsa regime

strongly reduced n)

charge neutrality

[V•• O]

(2K1)1/3p(O2)-1/6 Ki/(2K1)1/3 p(O2)1/6 (2K1)4/3K3/Ki4 p(O2)1/3 (K1/4)1/3p(O2)-1/6 K2/2K1 p(O2)-1/2

n p [V′′′′ Ti] [V•• O] [Ti••• i ] a

reduced

2[V•• O]

oxidized [V•• O]

) 2[V′′′′ Ti]

(Ki4K1/2K3)1/6p(O2)-1/4 (2K3Ki2/K1)1/6p(O2)1/4 (K12K3/4Ki4)1/3p(O2)0 (2K12K3/Ki4)1/3p(O2)0 (2K22K3/K1)1/2p(O2)-1/4

) 2[V′′′′ Ti]

(Ki4K1/2K3)1/6p(O2)-1/4 (2K3Ki2/K1)1/6p(O2)1/4 (K12K3/4Ki4)1/3p(O2)0 (2K12K3/Ki4)1/3p(O2)0 (2K22K3/K1)1/2p(O2)-1/4

Equilibrium constants: K1, K2, K3, and Ki are defined by eqs 5, 6, 7, and 8, respectively.

concentration of electrons is identical to that for the strongly reduced regime of undoped TiO2 and is described by eq 12. 2.3.2. Reduced Regime I. Defect disorder in this regime is determined by the mechanism of Nb incorporation into the TiO2 lattice, which can be described by the following reaction:

1 Nb2O5 a 2Nb•Ti + 2e′ + 4OO + O2 2

(19)

This regime is governed by the electronic charge compensation of niobium:

n ) [Nb•Ti] Figure 2. Schematic representation of the log σ vs log p(O2) for TiO2 within the strongly reduced regime, the reduced regime, and the transition regime between the two, showing a transition between -1/6 and -1/4.

so that the apparent p(O2) exponent is 1/mσ ) -1/5, as it is schematically shown in Figure 2. This slope may be confused with the defect disorder that corresponds to tetravalent Ti interstitials, which are formed according to the following equilibrium:

2OO + TiTi a Ti•••• + 4e′ + O2 i

(16)

The charge neutrality expressed by eq 20 indicates that the electrical conductivity in this regime is controlled by the concentration of Nb incorporated into the TiO2 lattice and is essentially independent of the p(O2). 2.3.3. Reduced Regime II. Nb incorporation into the TiO2 lattice in this regime can be described by the following representative reaction:

2Nb2O5 a 4Nb•Ti + V′′′′ Ti + 10OO

n ) (4K4) p(O2)

-1/5

• 4[V′′′′ Ti] ) [NbTi]

n)

(22)

(17)

where K4 is the equilibrium constant of reaction 16. The charge neutrality in this case is

4[Ti•••• i ]

(21)

This regime is governed by the ionic charge compensation of niobium:

Therefore 1/5

(20)

(18)

The defect disorder model involving tetravalent Ti interstitials has been considered to explain the slope 1/mσ ) -1/5.6,10,11 According to Kofstad3, the tetravalent Ti interstitials are the minority defects over a wide range of p(O2) values. The slope 1/mσ ) -1/5 should then be considered in terms of the transition between the strongly reduced and reduced regimes, rather than in terms of the defect disorder that is governed by the charge neutrality expressed by eq 18. 2.3. Nb-Doped TiO2. The effect of donor-type ions incorporated into the TiO2 lattice, such as Nb, on defect disorder and the related electrical properties of TiO2 depends on oxygen activity. Therefore, the defect disorder of Nb-doped TiO2 may be considered within several p(O2) regimes that are governed by simplified charge neutralities. 2.3.1. Strongly Reduced Regime. Defect disorder in this regime is governed by the simplified charge neutrality condition expressed by eq 11. Therefore, the effect of p(O2) on the

The relation between the concentration of electronic charge carriers and oxygen activity is:

n)

(

)

Ki[Nb•Ti] 4K3

1/4

p(O2)-1/4

(23)

where K3 is defined by eq 7. According to eq 23, the concentration of electrons in Nb-doped TiO2 in this regime is a function of both Nb concentration and p(O2). As it can be seen from eqs 19-23, the mechanism of Nb incorporation into the TiO2 lattice depends on p(O2). That is, while Nb is an extrinsic cation dopant, oxygen must be considered as an intrinsic anion dopant. Its content is imposed by the gas phase of appropriate oxygen activity. Consequently, the boundaries between the defect disorder regimes in plots of defect concentration versus p(O2) at specific temperatures are determined by the concentration of niobium.6 2.3.4. Oxidation Regime. Defect disorder in this regime is governed by relationships described by eqs 14 and 15 that are identical to those for undoped TiO2. 2.4. Electrical Conductivity. A common approach to verify the defect disorder is the determination of the electrical conductivity, σ, which, for n-type semiconductors, is the product

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of the concentration of electrons and their mobility:

σ ) enµn

(24)

where e is the elementary charge, n is the concentration of electrons, and µn is their mobility. Assuming that the mobility of electrons is independent of p(O2), the data on the effect of the p(O2) on σ may be used to assess the effect of p(O2) on the concentration of electrons. In this standard case, the following relation applies:

1 ∂ log σ ) mσ ∂ log p(O2)

(25)

where 1/mσ is the p(O2) exponent, which may be used for the verification of specific defect models.3 2.5. Thermoelectric Power. Thermoelectric power, S, can be related to the concentration of electronic charge carriers according to the following expressions for n- and p-type regimes, respectively:12

Sn ) Sp )

(

)

k Nn ln + An e n

(26)

(

(27)

)

k Np ln + Ap e p

where k is the Boltzmann constant, Nn and Np denote the density of states in conduction and the valence bands, respectively, and An and Ap denote the kinetic constants associated with scattering of electrons and holes, respectively. The most common way to verify the defect disorder models of metal oxides is based on the dependence of S on the p(O2):

∂S k 1 ) mS e ∂ log p(O2)

(28)

where mS is a parameter related to the specific defect disorder and the subscript S refers to the case when the parameter is obtained using thermoelectric power data. 3. Experimental Section 3.1. Specimen. Nb-doped TiO2 powder was prepared by precipitation from a solution consisting of (a) high-purity titanium isopropoxide Ti(OC3H5OH) (Sigma-Aldrich) dissolved in ethanol (molar ratio 1:6.4) and (b) high-purity NbCl5 (SigmaAldrich) dissolved in ethanol and HCl (molar ratio 1:300:0.03). The solution was mixed and stirred for 50 h at 323 K. Precipitation was induced by the dropwise addition of (c) ethanol and distilled water (molar ratio 1:5) to the mixture of solutions a and b. The mixture of a, b, and c was stirred for 12 h at room temperature and then heated to 353 K until the viscosity increased to the consistency of honey. The mixture then was dried in air at 383 K for 72 h. The formed xerogel was ground by hand into a fine powder using an agate mortar and pestle. The powder was placed in a platinum boat and calcined at 1173 K in oxygen flowing at 15 mL/min for 2.5 h (heating rate ∼300 K/h). After cooling at 300 K/h, the powder was uniaxially pressed into a bar of dimensions 2 mm × 4 mm × 15 mm, or a disk 4-mm-thick and 20 mm in diameter, at 60 MPa and then isostatically pressed at 400 MPa with a 1 min hold. The bar and the disk were sintered in air at 1773 K for 5 h (heating and cooling rates 300 K/h). The selection of sintering temperature and time was dictated by the preceding determination of the diffusion coefficient of niobium in single-crystal TiO2. These

Figure 3. Scanning electron micrograph of the Nb-doped TiO2 (0.65 atom % Nb) studied in the present work.

data demonstrated that after 5 h at 1773 K niobium propagated a distance of 23 µm. A diamond blade was used for cutting the pellet into an orthorhombic specimen of dimensions 1.74 mm × 3.80 mm × 14.91 mm. The mineralogical homogeneity was determined by X-ray diffraction (XRD; Siemens D5000) using Cu KR radiation over the range 20-80° 2θ with a step size of 0.02°. The XRD pattern shows that the specimen consisted of well-crystallized singlephase rutile. An independent study aimed at the determination of (i) the solubility limit, which was 15 atom % at 1773 K, and (ii) the diffusion rate of Nb in the TiO2 lattice. The results of this study were needed for the selection of (i) the Nb content below the solubility limit and (ii) processing conditions (time and temperature) required for the incorporation of Nb and its homogeneous distribution. Consequently, the niobium dopant level of 0.65 atom % was below the solubility limit in TiO2 under these conditions. A field emission scanning electron micrograph of the sintered specimen is shown in Figure 3. It can be seen that it is essentially fully dense and free of intergranular particulates, with faceted low-energy grain boundaries. 3.2. Equipment. The electrical conductivity and thermoelectric power were determined simultaneously using a hightemperature Seebeck probe.12 The external (current) probes consist of platinum plates attached to both sides of the rectangular-shaped specimen. A spring mechanism, located outside the high-temperature zone, was used to maintain effective galvanic contact between the electrodes and specimen. The voltage electrodes were formed of two platinum wires wrapped around the specimen and welded to the Pt wires connected to a voltmeter. The distance between the voltage electrodes was 12.17 mm. The sample holder was placed in an alumina tube, which was connected to a gas-flow system that imposed gas phases of controlled oxygen activities in the reaction chamber, which was inserted in a resistance furnace. 3.3. Procedure. The required oxygen activity, p(O2), in the reaction chamber was imposed using mixtures of appropriate compositions, flowing at 100 mL/min. The p(O2) at lower levels was imposed using mixtures of hydrogen and water vapor; at the higher levels, it was imposed by argon/oxygen mixtures. The oxygen activities were determined using a zirconia-based electrochemical oxygen probe. Measurements of both electrical conductivity, σ, and thermoelectric power, S, were taken during both oxidation and reduction experiments. Measurements of the electrical conductivity as a function of time, which aimed to monitor the rate of gas/solid equilibration, were done using a dc current of 10 µA applied alternatively in both current flow directions (in order to avoid polarization of

Properties of Niobium-Doped Titanium Dioxide 1

Figure 4. Experimental data monitoring sheet showing the changes in (a) oxygen activity, (b) temperature, and (c) electrical conductivity as a function of time during a reduction experiment for the polycrystalline (PC) specimen of Nb-doped TiO2 (0.65 atom % Nb) at 1223 K.

the specimen). The cycle times were ∼50 ms. The time gap between the cycles was 1 s, while the time gap between subsequent measurements was in the range of 30 s to 15 min. Once equilibrium had been established, the electrical conductivity was determined with a very high level of reliability using 10-15 different applied currents of opposite polarity. Details of the experimental procedures used to determine the electrical conductivity have been reported elsewhere.12 Figure 4 shows a standard sheet for monitoring the equilibration kinetics; the data include the changes in oxygen activity, temperature, and electrical conductivity during a reduction experiment at 1223 K. Figure 4a shows that the imposition of a new gas phase results in a very rapid decrease in the p(O2) to a level of ∼95% in a matter of seconds, followed by adoption of the final equilibrium value within 10 min. Figure 4b shows that the temperature during the experiment remained constant within 0.3 K. Because a change in temperature of 1 K leads to a change in electrical conductivity of ∼1% (at the highest experimental temperature level of 1298 K), then the observed fluctuations in temperature have a negligible effect on the measured electrical conductivity data. Figure 4c shows that constant electrical conductivity was reached within ∼1 h and then remained constant for the following 20 h. The period of time during which the electrical conductivity exhibited dynamic change during oxidation and reduction was ∼20 h and ∼0.5 h at 973 and 1323 K, respectively. These kinetics data were used to determine the chemical diffusion coefficient using a procedure described elsewhere.13,14 4. Results and Discussion 4.1. Effect Of Oxygen Activity. The experimental data of both electrical conductivity and thermoelectric power in the range 1073-1298 K are shown in the upper and the lower part of Figure 5, respectively. The solid line represents the dependence within the experimental data, and the dashed lines is the interpolated dependence. These data exhibit the following behaviors: (i) At high oxygen activities, p(O2) > 10 Pa, the electrical conductivity exhibits the p(O2) exponent 1/mσ ) -1/4. This exponent is consistent with the defect disorder model described by eqs 21-23. (ii) In the moderate values of oxygen activities, 10-7 Pa < p(O2) < 10-5 Pa, the electrical conductivity is independent of

J. Phys. Chem. B, Vol. 110, No. 45, 2006 22451

Figure 5. Electrical conductivity and thermoelectric power for Nbdoped TiO2 (0.65 atom % Nb) vs oxygen activity in the range 10731298 K.

Figure 6. Schematic representation of the effect of oxygen activity on the electrical conductivity of Nb-doped TiO2 showing the boundaries between three conductivity regimes; A and B denote the related oxygen activities.

p(O2). The behavior is well consistent with the defect disorder described by eqs 19 and 20. (iii) At low oxygen activities, p(O2) < 10-8 Pa, the electrical conductivity exhibits the p(O2) exponent 1/mσ ) -1/6. This exponent is consistent with the defect disorder model described by eqs 11-12. The experimental data on the effect of p(O2) on the electrical conductivity, shown in Figure 5, overlap all three of the defect disorder regimes, including (i) the strongly reduced regime, which corresponds to the p(O2) exponent -1/6, (ii) the reduced regime I, in which the electrical conductivity is independent of p(O2), and (iii) the reduced regime II, which corresponds to the p(O2) exponent -1/4. A schematic representation of these three conductivity regimes is given in Figure 6, showing the intersections between the defect regimes, including (i) the intersection between the 1/mσ ) -1/6 and 1/mσ ) 0 slopes (denoted by A) and (ii) the intersection between the 1/mσ ) 0 and 1/mσ ) -1/4 slopes (denoted by B). The intersections between the respective defect regimes are shown in Figure 7. 4.2. Equilibrium Constant for the Formation of Oxygen Vacancies, K1. Assuming that the mobility of electrons is independent of p(O2), the intersections A in Figure 6 (left side) demarcate the p(O2) value at which the concentration of

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Sheppard et al.

Figure 7. Effect of oxygen activity on the electrical conductivity for Nb-doped TiO2, in terms of log σ vs log p(O2), showing the boundary p(O2) between the strongly reduced regime and reduced regime I (A) and between the reduced regime I and reduced regime II (B).

Figure 9. Arrhenius plot of the equilibrium constant K3* determined in the present work for Nb-doped TiO2 along the equilibrium constant K3 determined for undoped TiO2 15 and the activity coefficient for the formation of Ti vacancies in Nb-doped TiO2.

Therefore

K3* )

Figure 8. Arrhenius plot of the equilibrium constant K1 determined in the present work along with those reported by Kofstad3 and Bak et al.15

electrons expressed by eqs 12 and 20 are the same. Therefore, we have

[Nb•Ti] ) (2K1)1/3p(O2)-1/6

(29)

1 K1 ) [Nb•Ti]3p(O2)1/2 2

(30)

Consequently

where K1 is the equilibrium constant of reaction 1. Figure 8 shows the equilibrium constant K1 determined in the present work from the data shown in Figure 5, along with the K1 versus 1/T determined from thermogravimetric data.15 As seen, the absolute values determined in the present work are over 1 order of magnitude larger than those reported by Kofstad.3 4.3. Equilibrium Constant for the Formation of Titanium Vacancies, K3. The intersection B in Figures 6 and 7 demarcates the p(O2) at which the concentration of electrons expressed by eqs 20 and 23 are the same. Accordingly, we obtain the following expression:

( )

[Nb•Ti] ) Ki

[Nb•Ti] 4K3*

1/4

p(O2)-1/4

(31)

Ki4 4[Nb•Ti]3

p(O2)-1

(32)

where K3* is the equilibrium constant of the formation of Ti vacancies for Nb-doped TiO2. Taking into account the relatively high concentration of Ti vacancies in Nb-doped TiO2, the determined equilibrium constant should be considered in terms of activities (K3*) rather than concentrations. This equilibrium constant may be determined from eq 32 using the equilibrium constant Ki determined elsewhere15 and the oxygen activity corresponding to the boundary between reduced regime I and reduced regime II. This boundary p(O2) is marked schematically by B in Figure 6. The related experimental data are shown in Figure 7 (right side). The determined K3* is shown in Figure 9 along the equilibrium constant of the formation of Ti vacancies for undoped TiO2, denoted by K3.15 As seen, these two constants differ by approximately 10 orders of magnitude. The difference between the two may be considered within two models, including (i) the activity model and (ii) the effect of Nb on the intrinsic electronic equilibrium constant, Ki. 4.3.1. The Concentration Versus ActiVities Model. The difference between the two equilibrium constants, K3 and K3*, may be considered in terms of the extent of defect interactions in both cases. The molar concentration of Ti vacancies at 1073 K for undoped TiO2 and Nb-doped TiO2 can be expressed as15 A ) 10-4 and A ) [Nb•Ti]/4 ) 1.63 × 10-3, respectively, where A is the effective concentration of acceptors, defined as • A ) [A′] + 4[V′′′′ Ti] - [D ]

(33)

where [A′] and [D•] denote the concentrations of single-valent acceptor- and donor-type ions, respectively. The extent of interactions between Ti vacancies in Nb-doped TiO2 is expected to be larger than that in undoped TiO2. Consequently, the equilibrium constant of reaction 3 for Nb-doped TiO2 may be considered in terms of activities rather than concentrations. Therefore, assuming that the activity coefficient of electron holes is equal to 1 (their concentration is low), we have

Properties of Niobium-Doped Titanium Dioxide 1

K3* K3

fV )

J. Phys. Chem. B, Vol. 110, No. 45, 2006 22453

(34)

where fV denotes the activity coefficients for Ti vacancies in Nb-doped TiO2, which is also shown in Figure 9. 4.3.2. The Intrinsic Electronic Equilibrium Constant. This model is based on an assumption that the equilibrium constant K3 is the same for both undoped and Nb-doped TiO2; however, the intrinsic electronic equilibrium constant is different for undoped TiO2 and Nb-doped TiO2 because of the effect of Nb on the electronic structure. The intrinsic electronic equilibrium constant for undoped TiO2 is reported elsewhere.15 This equilibrium constant for Nb-doped TiO2 may be determined from the following relation:

Ki* ) (4K3[Nb•Ti]3)1/4p(O2)1/4

(35)

Both equilibrium constants are shown in Figure 10. As seen, the equilibrium constant Ki is 2 orders of magnitude larger than that for undoped TiO2. This model seems to be more feasible because of the expected effect of Nb on the effective band gap required for ionization. The larger value of Ki for Nb-doped TiO2 is consistent with the reduced band gap. 4.4. Mobility of Electrons. The electrical conductivity of Nb-doped TiO2 in reduced regime I is governed by the charge neutrality condition expressed by eq 20. Specifically, it was shown that the concentration of electrons in this regime remains constant and is determined by the content of Nb. Therefore, taking into account the relation expressed by eq 24, the mobility terms is

µn )

σ en

Figure 10. Electronic intrinsic equilibrium constant for undoped TiO2 reported by Bak et al.15 and that for Nb-doped TiO2 determined in the present work.

(36)

Figure 11 shows the mobility of electrons as a function of 1/T in the range 973-1298 K. These plots represent the data determined in the following manner: (i) The data determined during isothermal experiments, shown in Figure 5, in the range 1248-1298 K. The related activation energy is 6.8 kJ/mol. (ii) The data determined during isothermal experiments, shown in Figure 5, in the range 1073-1223 K. The related temperature dependence is -1.8 kJ/mol. These data are consistent with metallic-type conduction. (iii) The data determined during isobaric experiments, at p(O2) ) 10-9 Pa, in the range 973-1223 K. The related temperature dependence is -1.1 kJ/mol. These data confirm the metallictype conduction behavior. As seen in Figure 11, there is a discontinuity of the mobility at 1210-1220 K, indicating that it exhibits a sudden change. This change may be considered in terms of a transition between metallic- and semiconducting-type conduction mechanisms and the related changes in the electronic structure. The mechanism of this transition may be considered in terms of either structural changes leading to Mott-type transition16 or changes in defect disorder leading to Anderson-type16 metal-insulator transition. 5. Conclusions The determined electrical properties for Nb-doped TiO2 at elevated temperatures have been used for the determination of several defect-related quantities, including the following: (1) The equilibrium constant for the formation of oxygen vacancies has been determined. This equilibrium constant is similar to those reported so far.

Figure 11. Arrhenius plots of the mobility of electrons for Nb-doped TiO2 (0.65 atom % Nb).

(2) The equilibrium constant for the formation of titanium vacancies has also been determined. This equilibrium constant differs substantially from that reported for undoped TiO2. The difference is considered in terms of different activities of these defects in undoped and Nb-doped TiO2. (3) The equilibrium constant for intrinsic electronic ionization, which is consistent with the electronic structure of Nb-doped TiO2, has been determined. (4) The mobility of electrons has also been determined. The effect of temperature on this quantity is consistent with metallictype conduction of reduced Nb-doped TiO2. The discovered metallic conduction paves the way for the development of high-performance photoelectrodes with substantially reduced internal energy losses during charge transport. Acknowledgment. The present work was supported by the Australian Research Council, Rio Tinto Ltd., Mailmasters Pty. Ltd., Sialon Pty. Ceramics Ltd, and Avtronics (Australia) Pty. Ltd. The support of the Australian Institute of Nuclear Science and Engineering also is gratefully acknowledged. References and Notes (1) Fujishima, A.; Honda, K. Nature 1972, 238, 37. (2) Nowotny, J.; Sorrell, C. C.; Sheppard, L. R.; Bak, T. Int. J. Hydrogen Energy 2005, 30, 521.

22454 J. Phys. Chem. B, Vol. 110, No. 45, 2006 (3) Kofstad, P. Nonstoichiometry, Diffusion and Electrical ConductiVity of Binary Metal Oxides; Wiley: New York, 1972. (4) Nowotny, J.; Sorrell, C. C.; Bak, T.; Sheppard, L. R. In Materials for Energy ConVersion DeVices; Woodhead: Cambridge, 2005; pp 84116. (5) Nowotny, M. K.; Bak, T.; Nowotny, J.; Sorrell, C. C. Phys. Status Solidi B 2005, 242, R88. (6) Baumard, J.-F.; Tani, E. Phys. Status Solidi 1977, 39, 373. (7) Bak, T.; Nowotny, J.; Rekas, M.; Sorrell, C. C. J. Phys. Chem. Solids 2003, 64, 1043. (8) Bak, T.; Nowotny, J.; Rekas, M.; Sorrell, C. C. J. Phys. Chem. Solids 2003, 64, 1057. (9) Nowotny, M. K.; Bak, T.; Nowotny, J. J. Phys. Chem. B 2006, 110, 16270.

Sheppard et al. (10) Balachndran, U.; Eror, N. G. J. Mater. Sci. 1988, 23, 2676. (11) Lee, D.-K.; Yoo, H.-I. Solid State Ionics 2006, 177, 1. (12) Nowotny, J. In The CRC Handbook of Solid-State Electrochemistry; Gellings, P. J., Bouwmeester, H. J. M., Eds.; CRC Press: Boca Raton, FL, 1997; pp 121-159. (13) Nowotny, J.; Bak, T.; Nowotny, M. K.; Sorrell, C. C. Phys. Status Solidi B 2005, 242, R91. (14) Bak, T.; Nowotny, J.; Sorrell, C. C. J. Phys. Chem. Solids 2004, 65, 1229. (15) Bak, T.; Nowotny, J.; Nowotny, M. K. J. Phys. Chem. B 2006, in press. (16) Cox, P. A. Transition Metal Oxides; Clarendon Press: Oxford, 1992; pp 91-95, 192-200, 225-248