Surface into the Bulk - ACS Publications - American Chemical Society

Article pubs.acs.org/JPCC

Model for Oxygen Interstitial Injection from the Rutile TiO2(110) Surface into the Bulk Kristine M. Pangan-Okimoto, Prashun Gorai, Alice G. Hollister, and Edmund G. Seebauer* Department of Chemical and Biomolecular Engineering, University of Illinois, Urbana, Illinois 61801, United States ABSTRACT: A quantitative microkinetic model is constructed for the injection of oxygen interstitial atoms from the rutile TiO2(110) surface into the bulk. The underlying rate expressions resemble conventional Langmuir kinetics to describe a gas−surface interaction, with numerical parameters determined through a global optimization procedure applied to a set of self-diffusion profiles from isotopic exchange experiments with a labeled gas. Key activation barriers for the interstitial were found to be 2.4 eV for injection, 0.65 eV for hopping diffusion, and 0.2 eV for exchange with the lattice. The standard formation enthalpy for the (−2) ionization state is 3.7 eV, which decreases to 1.1 eV for the experimental n-type material in equilibrium.

■

INTRODUCTION Surfaces offer efficient pathways for the production and destruction1−4 of point defects in the underlying bulk because fewer bonds need to be broken or formed at the twodimensional surface than within the three-dimensional bulk. The bond breakage that attends creation of any surface intrinsically generates the decreased chemical coordination that facilitates reaction with point defects such as interstitial atoms and vacancies. This concept was first demonstrated quantitatively for elemental silicon,5,6 for which the atomically clean (100) surface creates and destroys Si interstitials much faster than comparable bulk processes. More recently, this laboratory has generalized that observation of the creation of oxygen interstitial defects (Oi) by rutile TiO2(110)with such high efficiency that Oi supplants oxygen vacancies as the majority Orelated defect7,8is consistent with quantum-based predictions.9 In fact, the injected Oi seems to largely eliminate O vacancies within the solid. The consequent inclusion of Oi in the analysis of defect disorder in rutile adds a noteworthy new dimension to that literature. Unfortunately, the elementary kinetic expressions for the exchange of point defects with surfaces, including variation with temperature, pressure, and the presence of adsorbates, remain unknown. Yet work in this laboratory for Si5,6 and TiO2,7,8 together with that of Diebold et al.10−13 and Bartelt,14 suggests that the reactions of semiconductor surfaces with defects can exhibit chemistry having richness comparable to corresponding reactions with gases or liquids. Very little attention has been given to developing elementary step rate expressions for this alternative form of surface chemistry. Such attention would offer intrinsic benefits of uncovering a rich domain of surface behavior and instrumental benefits of providing new means for manipulating defect behavior in important applications15,16especially where surface-to-volume ratios are high. © 2015 American Chemical Society

The present work develops such kinetic expressions for Oi reacting with TiO2(110) in the context of a more general microkinetic model for the interaction of Oi with the crystal lattice. Such microkinetic models are typically needed because the concentrations of point defects are difficult to monitor directly either in the bulk or on the surface. In conventional surface chemistry, by contrast, gas or liquid composition is more readily measured (often in real time). In many cases, surface adsorbates can be detected directly with appropriate optical-, electron-, or ion-based techniques. For a point defect, the concentrations, spatial distributions, and movements must be inferred through indirect measurements with techniques such as secondary ion mass spectrometry of a tracer element or isotope, together with a mathematical model that can replicate the panoply of movements and reactions of the defect. Such microkinetic models are well developed for the diffusion of dopants in silicon and employ either Monte Carlo17 or continuum18 approaches. Even for silicon, however, quantitative rate expressions for describing the interaction of defects with the surface are largely lacking, and in metal oxides, microkinetic models are lacking almost entirely. In the present case of Oi in TiO2, the model is expressed as a system of continuum differential equations whose embedded kinetic and thermodynamic parameters are determined from isotopic selfdiffusion experiments together with an iterative global optimization method based on the weighted sum of square errors. Received: March 1, 2015 Revised: April 8, 2015 Published: April 16, 2015 9955

DOI: 10.1021/acs.jpcc.5b02009 J. Phys. Chem. C 2015, 119, 9955−9965

Article

The Journal of Physical Chemistry C

■

CALCULATION METHOD Microkinetic Reaction-Diffusion Model. The primary data originated from a standard gas−solid exchange method, in which rutile TiO2(110) single crystals were annealed (650−800 °C) in isotopically labeled gas (18O2) with PO2 in the range of 5 × 10−6−5 × 10−5 Torr. Such a regime corresponds to O-rich conditions, where PO2 is sufficiently high for oxygen to provide the dominant contribution to the chemical potential of TiO2 formation. The diffused isotopic O concentration profiles were measured ex situ with secondary ion mass spectrometry (SIMS). Procedural details appear in ref 8. Prior work7,8,19,20 has shown that Oi carries the preponderance of the diffusional flux under the O-rich conditions of the experiments. This dominance follows from the fact that Oi is the majority O-related defect in rutile TiO2 under O-rich conditions and that Oi moves more quickly than VO. The shapes of the diffusion profiles are therefore determined by the kinetics of Oi injection, migration, and reaction with the lattice. This kinetic behavior was embodied in continuum equations taking the general form ∂Cj ∂t

=−

∂Jj ∂x

+ Gj

formation of Oi of charge state m, which can be written in Kroger−Vink notation as 1 O2(g) ⇄ Oi m − mh• 2

where h represents a hole in the valence band. We follow the convention of ref 24 to express the total concentration of Oi (CTM) as a function of the standard formation enthalpy (ΔHf) and entropy (ΔSf), the hole concentration p, and the oxygen partial pressure PO2 ⎛ ΔS ⎞ ⎛ ΔHf ⎞ 1/2⎛ p ⎞m T = CST exp⎜ f ⎟ exp⎜ − CM ⎟PO ⎜ ⎟ ⎝ kB ⎠ ⎝ kBT ⎠ 2 ⎝ NV ⎠

∂CM ∂x

(4)

where NV denotes the effective density of states in the valence band. T and kB, respectively, denote temperature and Boltzmann’s constant. In this convention, ΔHf and ΔSf are defined for the Oim in a hypothetical state wherein the Fermi level EF is located at the valence band maximum. Any Fermi level dependence of the interstitial concentration is contained within the term (p/NV)m. We assume that m = −2, in accord with quantum calculations9,25,26 and recent dopant pile-up observations in this laboratory.19,20 The hole concentration p depends mainly upon the concentration of the charged majority defect.27 There is widespread agreement that undoped rutile TiO2 is n-type, but the identity of the dominant defect is still debated for several regimes of temperature and pressure.23,26,28−33 Recent work suggests that under our experimental conditions oxygen interstitials largely quench VO,7,8 leaving Ti interstitials (Tii) as the majority point defect overall. Most literature indicates that the +3 charge state of Tii dominates over the +4 state only at very high temperature or low pressures,29,34 so the present work presupposes the +4 state. The equilibrium reaction for Tii4+ formation is represented by

(1)

where x and t represent the spatial and temporal coordinates, and Cj, Jj, and Gj correspond to the concentration, flux, and net generation rate of species j, respectively. The specific species included both isotopes of oxygen (mass 16 and 18) residing in either mobile interstitial (M) or static substitutional (S) sites. Lattice oxygen was assumed to be immobile (with a hopping diffusivity DS = 0). Indeed, existing literature reports for the activation barrier to Oi site hopping indicate that it is either negligible21 or on the order of 0.7−0.8 eV,8,22 which in either case is lower than the barrier of 1.1−1.5 eV reported for vacancy site hopping.21−23 The flux J for the mobile species obeys Fickian diffusion

J = − DM

(3)

2OO + Ti Ti ⇄ Ti i 4 + + 4e′ + O2

CTi4+i obeys an expression akin to eq 4

(5) 27

⎛ ΔSf,Ti i4+ ⎞ ⎛ ΔHf,Ti i4+ ⎞ −1⎛ n ⎞−4 C Ti i4+ = C TiTi exp⎜ ⎟ exp⎜ − ⎟PO ⎜ ⎟ kBT ⎠ 2 ⎝ NC ⎠ ⎝ kB ⎠ ⎝

(2)

where DM denotes the hopping diffusivity of the mobile species. The present treatment does not consider drift motion due to electric fields in the surface space charge region. Drift leads to pile-up of 18O near the surface.19 However, this pile-up negligibly perturbs the profiles deeper in the bulk and can be considered as a separate phenomenon whose kinetics can be modeled independently and analytically.7,20 The initial conditions for eq 1 presuppose natural abundances for interstitial and lattice concentrations of both isotopes, with C18 = 0.002C16 in both S and M states. The total concentration of lattice atoms, CSTt = CS16t + CS18t = C16 S (t) + 0 0 0 18 22 −3 CS (t) = 6.38 × 10 cm , remains the same throughout the experiment and follows from the density of rutile. The total 18 concentration of interstitial atoms CTM = C16 M + CM is also invariant with time under chemical equilibrium but requires knowledge of the thermodynamics of formation for the interstitials. To determine necessary thermodynamic quantities under the prevailing O-rich conditions, a coupled set of defect equilibrium and charge equilibrium equations must be solved in accordance with an electroneutrality condition. The defect equilibrium expressions begin with the stoichiometric reaction for the

(6)

where NC denotes the effective density of states in the conduction band. The formation entropy and enthalpy for Tii4+ are defined in a hypothetical state with EF at the conduction band minimum. The mass action law for electrons and holes relates p to n according to ⎛ E ⎞ np = NVNC exp⎜ − G ⎟ ⎝ kBT ⎠

(7)

where EG denotes the band gap of rutile. EG, NC, and NV vary significantly with temperature in rutile.33,35 These variations, together with the explicit temperature dependences shown in eqs 6 and 7, contribute to the overall temperature dependence of CTM. Electroneutrality requires that the summation of all electron acceptors and donors as well as the charge carriers, n and p, are equal such that n+ 9956

∑ (mC D

) = p+ ∑ (yC D y+)

m−

(8)


Article

The Journal of Physical Chemistry C Here, C denotes the concentrations of donor defects, Dy+, and acceptor defects, Dm−, and y and m refer to the charge state of each defect. For our case, electroneutrality requires the electrons e′ to have a concentration n nearly equal to the majority defect concentration, CTi4+i , such that n ≈ 4C

Ti i4 +

where kinj is the injection rate constant given by k inj = νinje−Einj/ kBT

with an activation energy Einj and a pre-exponential factor νinj. In line with standard estimates from absolute rate theory, νinj was set equal to 1 × 1013 s−1. By analogy to nondissociative gas adsorption, defect annihilation is assumed to obey a rate expression incorporating the impinging flux and the sticking probability. Neither gas adsorption nor defect annihilation entails the breakage of chemical bonds in the elementary step. Sticking probabilities for nondissociative adsorption typically exhibit little or no dependence upon temperature36 in the absence of a precursor mechanism, and annihilation probabilities probably behave similarly. In a first-order Langmuir model with occupation by only one atom per site, the sticking probability S obeys

(9)

Equations 4, 6, 7, and 9 can be solved simultaneously for the unknowns n, p, CTi4+i , and the desired quantity CTM. In addition, straightforward manipulation of eqs 6 and 7 leads to the relation p ∼ PO21/5. Substitution of this result into eq 4 shows that CTM ∼ PO2(5+2m)/10, which for the (−2) charge state reduces to a pressure dependence of PO21/10. Injection Kinetics. Defect injection represents a boundary condition for 16Oi and 18Oi in eq 1. In line with the principles of microscopic reversibility and detailed balance, and as shown experimentally for the case of silicon,5 the surface provides an efficient pathway for both injection and annihilation. At the conditions of chemical equilibrium that prevailed in these experiments, the total rates of injection and annihilation (encompassing both isotopes) are equal. However, the experiments were designed so that the gaseous isotope changed suddenly at t = 0 from mass 16 to 18 after a suitable chemical equilibration period. Thus, the net injection rate of 18O into the bulk exceeds that for 16O for t > 0, and the net annihilation rate for 16O exceeds that for 18O correspondingly. The surface boundary condition for both 16Oi and 18Oi in eq 1 must therefore each include both injection and annihilation terms. The boundary condition for each isotope j can be written as − DM

∂CM, j ∂x

S = S0(1 − θ )

(10)

where rinj,j and rann,j, respectively, denote the elementary-step injection and annihilation rates. The principle of detailed balance requires that the surface injection sites be identical to the annihilation sites. Under a given set of temperature and pressure conditions at chemical equilibrium, these sites must support a well-defined concentration of injectable oxygen. We will treat injection and annihilation of defects in direct analogy to desorption and adsorption of gaseous species. For example, nondissociative gaseous chemisorption often obeys a Langmuir isotherm at equilibrium, with first-order kinetic expressions governing elementary-step adsorption and desorption. The interaction thermodynamics of bulk defects with surface annihilation/ injection sites is unknown for the present case, as are the elementary-step kinetics for annihilation and injection. Such kinetics can be complicated as in the case of silicon, wherein Si interstitials interact with the surface via a precursor-type mechanism.6 However, as a starting point for rutile TiO2(110), we assume conventional first-order Langmuir thermodynamics and kinetics. Thus, the annihilation/injection sites have an areal concentration nsat and support injectable oxygen with a fractional coverage θ. By analogy with gas desorption, defect injection requires the breakage of chemical bonds so that the rate expression contains a rate constant that incorporates thermal activation. With the assumption of first-order kinetics, the rate expression therefore obeys

rinj = k injnsatθ

(13)

However, the impinging gas flux for adsorption obeys expressions from the kinetic theory of gases and has a weak, non-Arrhenius temperature dependence. By contract, the impinging defect flux from the bulk involves diffusive site-tosite hopping with a mean hop length l.37 The flux therefore has an Arrhenius temperature dependence equaling that for hopping diffusion of the mobile defect. For rutile TiO2(110), the identity of the annihilation/injection sites for Oi is unknown. They could consist of kink sites, or step sites, or any terrace site containing oxygen. For generality, we define a coverage Θsat that represents a fractional coverage of annihilation/injection sites, normalized with respect to the total concentration nsat,max of oxygen-containing sites on the surface (i.e., 5.2 × 1014 cm−2 for rutile TiO2(110)10). That is, for an areal concentration of annihilation/injection sites nsat

= rinj, j − rann, j x=0

(12)

Θsat =

nsat nsat,max

(14)

With this definition, the annihilation rate becomes37 rann =

3DM S0 Θsat(1 − θ )CM, x = 0 l

(15)

Note that ref 37 derives an expression similar to that of eq 15 in the case of a silicon surface but without showing explicitly the quantities θ or Θsat. In the case of Si, the possibility of saturating the annihilation/injection sites was neglectedgiven the lack of any gas-phase participation and the preponderance of net annihilation over net injection in that nonequilibrium ion-implanted system wherein the bulk was supersaturated with interstitials. The isotherm for θ follows directly from eqs 11 and 15. Chemical equilibrium requires the following conditions for oxygen (including both isotopes) nsat

∂C ∂θ = DM M ∂t ∂x

= −rinj + rann = 0 x=0

(16)

These expressions lead to an algebraic expression for θ = θ16 + θ18

(11) 9957


Article

The Journal of Physical Chemistry C Table 1. Initial and Final Parameters for Oia parameter

eq

Ediff,M Do,M Eki (Eko) Aki (Ako) ΔHf ΔSf nsat S0 Einj

25 26 21, 22 21, 22 4 4 16, 17 13 12

a

initial value method fit fit fit fit fit fit fit fit fit

to to to to to to to to to

experimental profiles, modified from DFT21,22 experimental profile; no previous values experimental profile; no previous values experimental profile; no previous values experimental profile, modified from DFT9,25,26 experimental profile; no previous values experimental profile; no previous values experimental profile; no previous values experimental profile; no previous values

initial value

WSSE estimate

0.7 eV 8.0 × 10−2 cm2/s (ΔSdiff,M = 3.93kB) 0.2 eV 6.39 × 10−19 cm3/s 3.5 eV 0.39kB 1.5 × 1014 cm−2 (0.29 ML) 1.0 × 10−5 (*S0Nsat = 1.5 × 109 cm−2) 2.4 eV

0.65 ± 0.01 eV 8.0 × 10(−2.0±0.2) cm2/s (ΔSdiff,M = 3.93kB) 0.2 ± 0.1 eV 7.84 × 10(−19.0±0.2) cm3/s 3.7 ± 0.1 eV 1.54 ± 0.01kB ---, *S0Nsat = 3.0 × 10(9.0±0.1) cm−2 2.4 ± 0.5 eV

S0 and nsat cannot be deconvolved from each other.

θ= =

nsat,max lνinje

3DM S0CM, x = 0 −E inj / kBT

and kki and kko denote rate constants having a thermally activated form akin to eq 12. As there is no chemical distinction between the 16O and 18O isotopes, the forward and reverse rate constants must equal each other, i.e., kki = kko. This laboratory has uncovered evidence that Ti interstitial atoms play a mediating role in catalyzing these exchange reactions of Oi with the lattice.7 However, such effects are embedded within kki and kko and are not considered explicitly here. Diffusion. The hopping diffusivity DM of the mobile intermediate is represented in the conventional way as follows

+ 3DM S0CM, x = 0

1 nsat,max lνinje−Einj/ k BT 3DM S0C M, x = 0

+1

(17)

Despite the conditions of chemical equilibrium, the bulk concentrations of 18Oi and 16Oi evolve temporally after the transition from 16O2 to 18O2 in the gas phase. For example, at t = 0, we assume that the annihilation/injection sites fill instantaneously with 18O so that θ = θ18. This assumption is reasonable, as the characteristic time constants for gas adsorption and desorption based upon published kinetics for TiO2(110)38−43 lie roughly 6 orders of magnitude below the time constants (calculated from Table 1) that characterize exchange of Oi with the surface sites. Thus, elementary-step injection of 16O via eq 10 shuts down for t > 0. The net injected isotopic fluxes F in eq 10 then become F18Oi = −DM

DM = gl 2 Γ

where g represents a geometric factor equal to 6 in three dimensions, and Γ denotes the jump rate. The hop length l is on the order of a lattice spacing (1.78 × 10−8 cm26). The jump rate comprises the attempt frequency ν, a standard entropy of migration ΔSdiff,M, and the activation energy for migration Ediff,M ⎛ ΔSdiff,M ⎞ ⎛ −Ediff,M ⎞ Γ = ν exp⎜ ⎟ ⎟exp⎜ ⎝ kB ⎠ ⎝ kBT ⎠

18 ∂CM

∂x

= k injθnsat

x=0

3DM S0 18 − (1 − θ )ΘsatCM, x=0 l

F16Oi = −DM

∂x

=− x=0

(18)

⎛ −Ediff,M ⎞ DM = Do,M exp⎜ ⎟ ⎝ kBT ⎠

3DM S0 16 (1 − θ )ΘsatCM, x=0 l

where θ is determined from the isotherm of eq 10 with CM = 16 C18 M + CM . Equation 1 for each isotope requires a second boundary condition, which was set to be no-flux deep within the solid i.e., ∂CM/∂x = 0 as x → ∞. Lattice Exchange. The generation terms Gj in eq 1 incorporate lattice exchange reactions for Oi that are assumed to occur via kick-in and kick-out according to

⎛ ΔSdiff,M ⎞ Do,M = 6νl 2 exp⎜ ⎟ ⎝ kB ⎠

(20)

kko

18

The forward kick-in reaction rate rki for Oi therefore obeys 18 rki = k kiCS16CM

(21)

with the reverse kick-out reaction obeying 16 rko = k koCS18CM

(26)

Parameter Estimation via Error Minimization. Activation energies, pre-exponential factors, and other parameters embedded within the system of equations represented by eq 1 were determined via the least-squares technique of weighted sum of square errors (WSSE).44 This common iterative global optimization procedure can be summarized as follows. The system of equations (eq 1) is solved using the FLOOPS simulator,45 with initial estimates of the embedded parameters developed as described further below. The simulations yield computed concentration profiles Csim(x) of 18O for the experimental conditions corresponding to the data set consisting of Nexpt diffusion profiles Cexpt(x). The difference Cexpt(x) − Csim(x) is computed for a set of discrete depths x for each profile, and the WSSE objective function Φ is calculated as

kki

Oi + 16OS ⇄ 18OS + 16Oi

(25)

where

(19)

18

(24)

Combining these two equations yields an expression for DM as the product of a pre-exponential term Do,M and an exponential term

and 16 ∂CM

(23)

(22) 9958


Article

The Journal of Physical Chemistry C Nexpt

Φ=

∑∑ n=1

x

1 [Cexpt(x) − Csim(x)]2 σexpt(x)2

averaged from the literature based upon experimental results24,28,34,48,49 using maximum likelihood estimation.50 The estimates for ΔHf,Tii4+ and ΔSf,Tii4+ in eq 6 were 10.67 ± 0.16 eV and (2.38 ± 0.13) × 10−3 eV/K, respectively. In addition, NC and NV were calculated using estimates of the effective masses of electrons and holes, respectively, from literature estimates.33,51−58 NC and NV contribute nontrivial temperature dependence (with functional form T3/2) to eq 4, with a magnitude near 0.13 eV. The initial values in Table 1 were deemed sufficiently accurate by verifying that the mean interstitial diffusion length λ, net surface flux F, and effective diffusivity Deff described in ref 7 were within two standard deviations of those previously reported.7,8

(27)

In this expression, σexpt represents the standard deviation of the experimental concentration at each point. Φ is then computed for arrays of parameter values within a local neighborhood around each parameter; a decrease in Φ signifies an improvement in the parameter. This procedure is iterated for successively smaller changes in each parameter until Φ no longer varies significantly. For the present data set, there was less than 0.5% change in Φ after two iterations. The WSSE objective function weights each error difference [Cexpt(x) − Csim(x)] by the reciprocal of its standard deviation. Isotopic concentrations originate from ion counts that obey a Poisson distribution.46 For a data point comprising N counts, σexpt = √N. Thus, data points with more noise (typically deeper in the solid for the mass 18 isotope) are normalized and weighted less heavily. Confidence intervals for each parameter are determined as follows. Each computed concentration profile k is linearized with respect to the vector of optimal parameter estimates ϕ*44,47 according to ̃ k(ϕ) ≈ Csim, k(ϕ*) + Yk(ϕ*)(ϕ − ϕ*) Csim,

■

RESULTS Table 1 shows the parameter estimates and confidence intervals resulting from the WSSE method. For no parameter does the method yield a large change from the initial value, suggesting that the initial values based upon manual fitting were already near the optimum. Figure 1 compares representative simulated and experimental concentration profiles at 10−5 Torr. The simulated profiles reproduce the observed profiles fairly accurately.

(28)

where Yk is the Jacobian calculated using central differences Yk =

̃ k ∂Csim, ∂ϕ

ϕ*

(29)

The parameter covariance matrix Vϕ is then computed from the relation Nexpt

Vϕ−1 =

∑ YkTVC,−1kYk k=1

(30)

where VC,k is a profile covariance matrix. Its inverse VC,k−1 consists of only diagonal elements that equal the SIMS error σexpt(x)2. The confidence interval contains two ingredients. The first is the t-statistic value for the desired degree of certainty (100 − κ) that the parameter falls within the confidence interval for degrees of freedom (nd − Nϕ), where nd denotes the total number of SIMS points summed over all experiments and Nϕ the total number of parameters. The t-statistic is shown for κ/2 because the uncertainty incorporates possible values both above and below the average. The boundaries of the confidence interval were set in the present work to a conventional certainty level of 67% (i.e., κ = 33%). Given this confidence interval and the number of degrees of freedom, the t-statistic is calculated and then applied to parameter uncertainty. The second ingredient for the confidence interval is Vϕ. For the j-th parameter, the best value together with its confidence interval is finally ϕj = ϕj* ± t1 − κ /2(nd − Nϕ) Vϕ , jj

Figure 1. Examples of isotopic diffusion profiles computed based on WSSE (lines) compared to corresponding experimental profiles (symbols).

As discussed in more detail below, the magnitudes of the various parameters conspired to produce a condition of θ ≪ 0.1 (eq 17), such that S0 and nsat could not be deconvolved from each other. Therefore, Table 1 reports the value and uncertainty for the combined product S0*nsat. Since the physical interpretation of S0 as an annihilation probability imposes an upper bound of unity, S0*nsat = 3 × 109 cm−2 leads to a lower bound for nsat equal to this same number. Similarly, nsat cannot rise above its geometrically determined maximum of 5.2 × 1014 cm−2 for rutile TiO2(110). This upper bound leads to a lower bound on S0 of 6 × 10−6. The experimental diffusion profiles in this work have exponential shapes, which represent the signature of a mobile intermediate species such as Oi that exchanges with the lattice. The slopes and intercepts of the profiles plotted in semilogarithmic form can be employed59,60 to yield the net surface flux F of the injected marker, its mean diffusion length λ, and the effective diffusivity D eff characterizing the profiles (distinguished from the hopping diffusivity DM of the mobile intermediate). F and λ represent composites of fundamental

(31)

Initial Parameters Values for the WSSE Method. From the system of eqs 1 through 26, nine parameters require initial values to begin the WSSE iterationshown in Table 1. Numbers for most parameters were developed by manual fitting of experimental profiles at 10−5 Torr.8 In a few cases, literature estimates based upon density functional theory (DFT) were available to guide the selection. Necessary values for the enthalpy and entropy of formation for Tii4+ were 9959


Article


Table 2. Activation Energies and Oxygen Partial Pressure Coefficients of Deff and Its Two Components, λ and F, Derived from Figure 2(a)−2(c)a pre-exponential factor @ 10−5 Torr

activation energy (eV)

a

PO2b

parameter

analytical

WSSE

analytical

WSSE

analytical

WSSE

Deff λ F

2.22 ± 0.33 0.28 ± 0.26 1.92 ± 0.27

2.36 ± 0.01 0.23 ± 0.03 2.13 ± 0.07

2.45 × 10−(1.0±1.0) 1.74 × 10(6.0±1.0) 5.37 × 10(22.0±1.0)

1.46 × 10−(2.0±3.2) 1.40 × 10(4.0±1.8) 3.58 × 10(23.0±0.9)

0.29 ± 0.32 none 0.14 ± 0.21

0.10 none 0.10

Analytical results given in refs 7 and 8.

quantities, but F and λ can be determined directly and analytically from each profile. The composite parameter Deff is given by9 Deff =

Fλ [CST t − CS18t ] 0

(32)

0

This analytical approach aggregates the profile data differently from the WSSE method. WSSE considers all the profiles simultaneously in a global way to obtain fundamental quantities such as Einj and Ediff,M, with heavier weighting given to values of C18 S that have smaller standard deviations (and are typically large). Composite parameters such as F and λ can be then calculated by relations such as eq 18 and λ=

DM k kiCS16 t

(33)

0

By contrast, the analytical approach yields F, λ, and Deff on a profile-by-profile basis, with all data points C18 S within the profile given the same weight in a least-squares fit. In turn, the temperature dependencies represented by parameters such as EF and Eλ come from least-squares Arrhenius fits of the profileby-profile numbers for F and λ. Because of these methodological differences in which the raw profile data are aggregated and weighted, the WSSE and analytical results for the temperature and pressure dependencies of F, λ, and Deff are likely to differ. Table 2 compares the results for F, λ, and Deff from the two methods. The confidence intervals for the WSSE-based values employed the intervals of the constituent parameters listed in Table 1 together with a standard propagation-of-uncertainty analysis using eqs 18, 32, and 33. The pre-exponential factors listed in Table 2 correspond to an oxygen pressure of 1 × 10−5 Torr; Table 2 also lists the exponent b for the pressure dependence of the form PO2b. Figure 2 compares the results of the two methods graphically. The values given by the two methods differ for all the parameters listed, although they lie within one another’s confidence intervals. Figure 2 confirms that the agreement between the methods is generally quite good. For the activation energies, WSSE gives much tighter confidence intervals than the analytical method. For the pre-exponential factors, by contrast, the analytical method’s confidence intervals are tighter in two out of three cases. The WSSE method by its nature must assume a particular number for the amount of charge on Oi (leading, for example, to CTM ∼ PO21/10 in eq 4, i.e., b = 1/10), whereas this assumption is not required for the analytical method. To compare WSSE results for different magnitudes of m requires a comparison of the minimized objective functions Φ. We performed this comparison for m = −2 and −1 and found Φ to be lower (but only slightly) for the −2 case. 59,60

Figure 2. Arrhenius plots of (a) mean interstitial diffusion length λ, (b) net injection flux F, and (c) effective diffusivity Deff for analytical7,8 and WSSE approaches to analyzing isotopic diffusion profiles. Analytical results yield discrete data points for each profile (symbols) and a least-squares fit (black line), while the simulated (WSSE) results yield a single blue line. Inset in (b) and (c) shows dependence of Deff and F on PO2.

The results of the WSSE analysis permit evaluation of the concentrations of the major charged species described by eqs 4, 6, 7, and 9 over a range of temperatures and pressures. Figure 3 shows a corresponding diagram as a function of oxygen 9960


Article


Figure 3. Equilibrium defect and carrier concentrations at (a) 750 °C and (b) 10−5 Torr in rutile TiO2 as a function of (a) pressure and (b) temperature, respectively. The right-hand axis shows the corresponding Fermi level (EF).

pressure at 750 °C. The figure confirms that Tii is indeed the majority defect overall under our conditions. To compare the injection barrier with the formation enthalpy of Oi, it is useful to define a formation enthalpy ΔHf* at the value of EF that describes the experimental conditions. Straightforward thermodynamic manipulations show that

ΔHf* = ΔHf + mE F

contrast, Uberuaga and Bai calculated 0.01 eV,21 which seems too low in comparison with corresponding numbers for ZnO, CoO, and UO2.65−68 Yet any of these numbers leads to appreciable room-temperature mobility for Oi. This high mobility is reinforced by the slightly positive ΔSdiff,M near 4kBin accord with simple theories of solid-state site hopping. The migrational work done against the elasticity of the lattice normally decreases with temperature as the lattice expands and can be estimated from the free energy of migration (ΔGdiff,M) by ΔSdiff,M = −d(ΔGdiff,M)/dT.69 The quantity −d(ΔGm)/dT can be crudely approximated as ΔHmelt/Tmelt.70,71 In the present case, this latter quotient is 0.65 eV/2116 K, which yields ΔSdiff,M/kB = 3.6in good agreement with WSSE. The activation energy for exchange of Oi with the lattice is quite small at 0.2 eV and manifests in the mean free path λ. Equation 33 shows that the temperature dependence of λ depends upon the mathematical difference between the activation energy for site-to-site hopping and that for kick-in to the lattice. The corresponding numbers in Table 1 yield a modest activation energy for λ of 0.23 eV, with the activation energy for Ediff,M being more than three times Eki. The value for Eki falls below that of other semiconductor systems. In the case of silicon, for example, the corresponding experimental activation energy for exchange of Sii is 1.02 eV.59 No other experimental reports exist for metal oxides, but a DFT study by Huang et al. gives 0.49 eV for Oi exchanging with the lattice in n-type ZnO.65 The small WSSE magnitude may reflect a distinct mechanism, however. Self-diffusion experiments for Oi in rutile in the presence of surface-adsorbed sulfur suggest that lattice exchange of Oi is catalyzed by Ti interstitials.7 Further evidence for such a mechanism comes from the pre-exponential factor for lattice exchange shown in Table 1. A pseudo-first-order pre-exponential factor can be calculated based on eqs 21 and 33 as the product of the lattice 16 O concentration and the second-order pre-exponential factor shown in Table 1. This product AkiCST lies near 105 s−1, which falls far below the “expected” value near 1013 s−1.72 The low product probably reflects the inadequacy of the rate expression that follows from the noncatalyzed model for lattice exchange (eq 20). Surface Kinetics. On the basis of the parameters in Table 1, Figure 4 shows θ calculated from eq 17 over a broad range of T and PO2. The coverage of injectable oxygen is predicted to move

(34)

Figure 3 indicates that EF = 1.3 eV under our conditions (with EG = 2.6 eV), so eq 34 yields ΔHf* for Oi of 1.1 eV.

■

DISCUSSION Oi Equilibrium. The estimate of ΔHf = 3.7 ± 0.1 eV falls about 25% below the corresponding estimates from quantum calculations, which range from 5 to 6 eV for Oi−2 in a comparable standard state.9,25,26 The experimental number we determine depends upon the value assumed for ΔHf,Ti4+i in eq 6, which was taken from experimental measurements rather than DFT. This contribution accounts for roughly 4.3 eV of the total estimate for ΔHf for Oi. That said, DFT-based formation energies for interstitials in semiconductors often seem to overestimate experimental values by a nontrivial margin, often 20% or more. Similar differences are well-known for Si interstitials, wherein ΔHf = 2.4 eV fell well below the range of 3.3−4.5 eV given in the DFT literature of the time.59 Similar discrepancies plague the formation enthalpy of Tii in rutile, which has been the subject of much study by DFT9,23,26,30,31 and experiment.24,28,34,48,49 Of the DFT reports that calculate the formation energy of Tii4+, three23,26,30 yield numbers ∼1 eV higher than experimental numbers. (One DFT report31 for ΔHf,Ti4+i falls near the experimental number, and another9 falls roughly 1.5 eV lower.) WSSE obtains ΔSf = 1.54 ± 0.01kB. A corresponding literature report for rutile TiO2 does not exist, but this slightly positive value agrees with DFT-predicted formation entropies for similar metal oxides (ZnO, In2O3) between 1 and 2kB.61,62 This magnitude is actually modest, given the large positive entropy of formation often associated with charged defect formation in semiconductors due to local lattice mode softening.63,64 Oi Diffusion and Lattice Exchange. The WSSE-derived activation energy for Oi hopping of 0.65 eV is not far from the value of 0.78 eV calculated by Tsetseris using DFT.22 By 9961


Article


Figure 5. Simulated surface flux F of 18Oi (black solid line) at 10−5 Torr and the component surface reaction rates. F18Oi = −F16Oi = rann,16Oi. For 18O species, injection dominates with rinj (18Oi) exceeding rann (18Oi) by roughly an order of magnitude. The activation energies for the component parts are 2.15 eV for rinj and 2.41 eV for rann. The activation energy for F18Oi given in Table 2 is close to that for rinj.

Figure 4. Coverage θ of filled injection sites as a function of T and PO2. Under experimental conditions used here, θ remains below 0.01, and at 10−5 Torr the effective activation energy describing θ is Eθ = −0.25 eV (inset).

up out of the low-θ regime only at temperatures below roughly 300 °Ceven lower for PO2 appreciably below atmospheric pressure. Coverages above roughly 0.1 would be required to deconvolve the mathematical product of S0 and nsat. Figure 4 represents the “coverage” of injectable surface oxygen in equilibrium with the bulk species in a way that resembles, by analogy, the equilibrium of an adsorbate with a gas species in conventional gas−surface adsorption. Thus, the enthalpy of “adsorption” of the surface species can be computed from eqs 17 and 4, which together lead to Figure 4. For example, the enthalpy of the surface species with respect to bulk Oi equals Einj − Ediff,M − (ΔHf − 2Ep), where Ep describes the temperature dependence of p/NV. Here, the surface species enthalpy is 0.25 eV, which can also be calculated from an Arrhenius plot as shown in the inset of Figure 4. This value is quite small, suggesting that the injectable species resides in a local environment that is not too dissimilar from the bulk interstitial. Unlike the case of gas adsorption, where the enthalpy of adsorption is typically close to the activation energy for desorption, in the present case the enthalpy for “adsorption” of the surface species falls far below the activation energy for injection. We cannot say whether such a difference characterizes injection processes in general or is merely an attribute of the present TiO2(110) surface. Figure 5 shows the elementary-step injection and annihilation rates of 18Oi over the temperature range of interest. The temperature variations of both rinj and rann have effective activation energies falling in the approximate range 2.2−2.4 eV. However, rinj exceeds rann by about an order of magnitude. (This relationship is reversed for the mass 16 isotope.) Thus, the net flux F18 Oi is well approximated by r inj . This approximation also follows from the condition that θ ≪ 0.1 that prevails at the temperatures and pressures of this work. All injection-related terms in eq 18 then cancel out, and the expression for F18Oi thereby simplifies to ⎛ 3D S ⎞ 16 F18Oi = nsat⎜⎜ M 0 ⎟⎟CM, x=0 n l ⎝ sat,max ⎠

Inspection of eqs 18, 32, and 33 shows that λ should remain independent of PO2, while F and Deff should depend upon PO2 through θ, which in turn depends upon CM. In the limit of low θ, the dependence of F and Deff should be the maximum possible: PO21/10. In the limit of high θ, the dependence of F and Deff disappears entirely (i.e., PO20). These predictions accord with those of the analytical approach.7 However, the experimental confirmation of these predictions emerges more naturally from the WSSE approach than the analytical one, as the various parameters in Table 1 conspire to push θ firmly into the low coverage limit and therefore force the exponent b to its asymptotic limit of 1/10. The analytical approach requires profile-by-profile plotting of F and Deff as a function of PO2, which accentuates experimental errors in the determination of such weak pressure dependence. The large error bars on the analytically derived values of b confirm these difficulties. As indicated earlier, the mathematical product of S0 and nsat could not be deconvolved, although lower limits could be established for each constituent parameter. S0 is of special interest, as a value for comparison has already been reported5,37 for Sii at Si(100)-(2 × 1): 7 × 10−4. Such a comparison can be rationalized based on the assumption that the injection sites represent a distinct species of surface sites on TiO2(110)-(2 × 1). Such a distinction is evident from the fact that the coverage of injectable oxygen is small in the present experiments, whereas published gas adsorption data38−40,42,43,73−75 indicate that the coverage of dissociatively adsorbed gaseous oxygen should be near unity. In a simple view of annihilation proposed previously for Sii, interstitial annihilation should be exceptionally facile (with a likelihood of nearly unity) at dangling bond sitesloosely akin to adding a reactive atom to a free radical in the gas phase. As the Si(100)-(2 × 1) reconstruction nominally has no dangling bonds,6 the rather low value of S0 could be interpreted in terms of a small number of low-concentration surface sites supporting dangling bonds: e.g., kinks or steps. An equivalent assumption for TiO2(110) would place the injection site concentration

(35) 9962


Article

The Journal of Physical Chemistry C between roughly 1% and 0.1% of nsat,max. Such an assumption tacitly assumes that under our experimental conditions the surface has sufficient covalent character to make the concept of dangling bonds meaningful and that the surface is in the (1 × 1) reconstruction (which nominally has no dangling bonds76,77). The (2 × 1) surface of TiO2(110) dominates after annealing in ultrahigh vacuum and/or ion bombardment,78,79 but such conditions do not represent the present case. Two models have been proposed for the (2 × 1) reconstruction: one model suggests a Ti location that creates one dangling bond per surface Ti atom, whereas the other model positions the Ti atom such that two dangling bonds per surface Ti atom are formed.78,80 With an injection site concentration between 0.01 and 0.001nsat,max, S0 falls between 6 × 10−4 and 6 × 10−3. This range incorporates the magnitude of S0 observed previously for Si. The value of 2.4 eV determined for Einj is not readily comparable to any literature value, either experimental or computational. Gas−surface adsorption kinetics may provide the next-best point of reference. Literature reports for associative oxygen desorption as O2 range from 1.3 to >2.5 eV,38,39,43,74,75 with an average of 1.7 eV. Einj is considerably lower than the energy barrier for anti-Frenkel pair bulk formation of Oi and VO of ∼7 eV.81 Einj is significantly larger than ΔHf* = 1.1 eV for O-interstitials under the present experimental conditions with the associated Fermi level.

gratefully acknowledges fellowship support from the Dow Chemical Company.

■

(1) Chason, E.; et al. Ion Beams in Silicon Processing and Characterization. J. Appl. Phys. 1997, 81, 6513−6561. (2) Fahey, P. M.; Griffin, P. B.; Plummer, J. D. Point Defects and Dopant Diffusion in Silicon. Rev. Mod. Phys. 1989, 61, 289−384. (3) McCarty, K. F.; Nobel, J. A.; Bartelt, N. C. Vacancies in Solids and the Stability of Surface Morphology. Nature 2001, 412, 622−625. (4) Wachs, I. E.; Weckhuysen, B. M. Structure and Reactivity of Surface Vanadium Oxide Species on Oxide Supports. Appl. Catal. A: Gen. 1997, 157, 67−90. (5) Seebauer, E. G.; Dev, K.; Jung, M. Y. L.; Vaidyanathan, R.; Kwok, C. T. M.; Ager, J. W.; Haller, E. E.; Braatz, R. D. Control of Defect Concentrations within a Semiconductor through Adsorption. Phys. Rev. Lett. 2006, 97, 055503. (6) Zhang, X.; Yu, M.; Kwok, C. T. M.; Vaidyanathan, R.; Braatz, R. D.; Seebauer, E. G., Precursor Mechanism for Interaction of Bulk Interstitial Atoms with Si(100). Phys. Rev. B 2006, 74. (7) Gorai, P.; Hollister, A. G.; Pangan-Okimoto, K.; Seebauer, E. G., Kinetics of Oxygen Interstitial Injection and Lattice Exchange in Rutile TiO2. Appl. Phys. Lett. 2014, 104. (8) Hollister, A. G.; Gorai, P.; Seebauer, E. G. Surface-Based Manipulation of Point Defects in Rutile TiO2. Appl. Phys. Lett. 2013, 102, 231601. (9) Lee, H.-Y.; Clark, S. J.; Robertson, J. Calculation of Point Defects in Rutile TiO2 by the Screened-Exchange Hybrid Functional. Phys. Rev. B 2012, 86, 075209. (10) Diebold, U. The Surface Science of Titanium Dioxide. Surf. Sci. Rep. 2003, 48, 53−229. (11) Li, M.; Hebenstreit, W.; Diebold, U. Oxygen-Induced Restructuring of the Rutile TiO2(110)(1 × 1) Surface. Surf. Sci. 1998, 414, L951−L956. (12) Li, M.; Hebenstreit, W.; Diebold, U. Morphology Change of Oxygen-Restructured TiO2(110) Surfaces by UHV Annealing: Formation of a Low-Temperature (1 × 2) Structure. Phys. Rev. B 2000, 61, 4926−4933. (13) Li, M.; Hebenstreit, W.; Diebold, U.; Tyryshkin, A. M.; Bowman, M. K.; Dunham, G. G.; Henderson, M. A. The Influence of the Bulk Reduction State on the Surface Structure and Morphology of Rutile TiO2(110) Single Crystals. J. Phys. Chem. B 2000, 104, 4944− 4950. (14) McCarty, K. F.; Bartelt, N. C. Role of Bulk Thermal Defects in the Reconstruction Dynamics of the TiO2(110) Surface. Phys. Rev. Lett. 2003, 90, 046104. (15) Batzill, M. Fundamental Aspects of Surface Engineering of Transition Metal Oxide Photocatalysts. Energy Environ. Sci. 2011, 4, 3275−3286. (16) Djurišić, A. B.; Leung, Y. H. Optical Properties of ZnO Nanostructures. Small 2006, 2, 944−961. (17) Pelaz, L.; Marqués, L. A.; Aboy, M. a.; Barbolla, J.; Gilmer, G. H. Atomistic Modeling of Amorphization and Recrystallization in Silicon. Appl. Phys. Lett. 2003, 82, 2038−2040. (18) Jung, M. Y. L.; Gunawan, R.; Braatz, R. D.; Seebauer, E. G. A Simplified Picture for Transient Enhanced Diffusion of Boron in Silicon. J. Electrochem. Soc. 2004, 151, G1−G7. (19) Gorai, P.; Hollister, A. G.; Seebauer, E. G. Electrostatic Drift Effects on near-Surface Defect Distribution in TiO2. Appl. Phys. Lett. 2013, 103, 141601. (20) Gorai, P.; Seebauer, E. G. Kinetic Model for Electric-Field Induced Point Defect Redistribution near Semiconductor Surfaces. Appl. Phys. Lett. 2014, 105, 021604. (21) Uberuaga, B. P.; Bai, X.-M. Defects in Rutile and Anatase Polymorphs of TiO2: Kinetics and Thermodynamics near Grain Boundaries. J. Phys.: Condens. Matter 2011, 23, 435004. (22) Tsetseris, L. Configurations, Electronic Properties, and Diffusion of Carbon and Nitrogen Dopants in Rutile TiO2: A Density Functional Theory Study. Phys. Rev. B 2011, 84, 165201.

■

CONCLUSION The present work represents an initial attempt to construct a quantitative microkinetic model for the injection of point defects from semiconductor surfaces into the bulk. The underlying assumptions are analogous to the use of conventional Langmuir kinetics to describe a gas−surface interactionwith appropriate changes made to account for diffusive hopping of defects in the solid in place of free translation of atoms or molecules in the gas. Even though the model is rather simple, it incorporates a multiplicity of parameters that have been estimated using a global optimization procedure. Given the nature of the available data, this approach permits construction of a more detailed model than a conventional analytical profile-by-profile analysis but also depends more heavily on a priori assumptions about the form of the constituent rate expressions. The low apparent values for the activation energy and pre-exponential factor for lattice exchange of Oi provide an example, wherein likely catalysis by Tii was not taken into account. However, as suitable data sets for other semiconductors and other defect types become available over time, a combination of the two approaches should offer a means to gain important insights into this largely unexplored form of surface chemistry.

■

REFERENCES

AUTHOR INFORMATION

Corresponding Author

*Telephone: (217) 244-9214. Fax: (217) 333-5052. E-mail address: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was partially supported by the National Science Foundation (DMR 10-05720 and DMR 13-06822). P.G. 9963


Article

The Journal of Physical Chemistry C (23) Iddir, H.; Ö ğüt, S.; Zapol, P.; Browning, N. D. Diffusion Mechanisms of Native Point Defects in Rutile TiO2: Ab Initio TotalEnergy Calculations. Phys. Rev. B 2007, 75, 073203. (24) Kofstad, P. Thermogravimetric Studies of the Defect Structure of Rutile (TiO2). J. Phys. Chem. Solids 1962, 23, 1579−1586. (25) Kamisaka, H.; Yamashita, K. Theoretical Study of the Interstitial Oxygen Atom in Anatase and Rutile TiO2: Electron Trapping and Elongation of the R(O−O) Bond. J. Phys. Chem. C 2011, 115, 8265− 8273. (26) Peng, H. First-Principles Study of Native Defects in Rutile TiO2. Phys. Lett. A 2008, 372, 1527−1530. (27) Bak, T.; Nowotny, J.; Nowotny, M. K. Defect Disorder of Titanium Dioxide. J. Phys. Chem. B 2006, 110, 21560−21567. (28) Blumenthal, R. N.; Coburn, J.; Baukus, J.; Hirthe, W. M. Electrical Conductivity of Nonstoichiometric Rutile Single Crystals from 1000° to 1500°C. J. Phys. Chem. Solids 1966, 27, 643−654. (29) Lee, D. K.; Yoo, H. I. Unusual Oxygen Re-Equilibration Kinetics of TiO2−Δ. Solid State Ionics 2006, 177, 1−9. (30) Li, X.; Finnis, M. W.; He, J.; Behera, R. K.; Phillpot, S. R.; Sinnott, S. B.; Dickey, E. C. Energetics of Charged Point Defects in Rutile TiO2 by Density Functional Theory. Acta Mater. 2009, 57, 5882−5891. (31) Mattioli, G.; Alippi, P.; Filippone, F.; Caminiti, R.; Amore Bonapasta, A. Deep Versus Shallow Behavior of Intrinsic Defects in Rutile and Anatase TiO2 Polymorphs. J. Phys. Chem. C 2010, 114, 21694−21704. (32) Nowotny, M. K.; Bak, T.; Nowotny, J. Electrical Properties and Defect Chemistry of TiO2 Single Crystal. I. Electrical Conductivity. J. Phys. Chem. B 2006, 110, 16270−16282. (33) Yahia, J. Dependence of the Electrical Conductivity and Thermoelectric Power of Pure and Aluminum-Doped Rutile on Equilibrium Oxygen Pressure and Temperature. Phys. Rev. 1963, 130, 1711−1719. (34) Blumenthal, R. N.; Baukus, J.; Hirthe, W. M. Studies of the Defect Structure of Nonstoichiometric Rutile, TiO2‑X. J. Electrochem. Soc. 1967, 114, 172−176. (35) Baumard, J. F.; Tani, E. Electrical Conductivity and Charge Compensation in Nb Doped TiO2 Rutile. J. Chem. Phys. 1977, 67, 857−860. (36) Seebauer, E. G.; Kong, A. C. F.; Schmidt, L. D. The Coverage Dependence of the Pre-Exponential Factor for Desorption. Surf. Sci. 1988, 193, 417−436. (37) Kwok, C. T. M.; Dev, K.; Braatz, R. D.; Seebauer, E. G. A Method for Quantifying Annihilation Rates of Bulk Point Defects at Surfaces. J. Appl. Phys. 2005, 98, 013524−7. (38) Chrétien, S.; Metiu, H., O2 Evolution on a Clean Partially Reduced Rutile TiO2(110) Surface and on the Same Surface Precovered with Au1 and Au2: The Importance of Spin Conservation. J. Chem. Phys. 2008, 129. (39) Kimmel, G. A.; Petrik, N. G. Tetraoxygen on Reduced TiO2(110): Oxygen Adsorption and Reactions with Bridging Oxygen Vacancies. Phys. Rev. Lett. 2008, 100, 196102. (40) Rasmussen, M. D.; Molina, L. M.; Hammer, B. Adsorption, Diffusion, and Dissociation of Molecular Oxygen at Defected TiO2(110): A Density Functional Theory Study. J. Chem. Phys. 2004, 120, 988−997. (41) Wang, Y.; Pillay, D.; Hwang, G. S. Dynamics of Oxygen Species on Reduced TiO2(110) Rutile. Phys. Rev. B 2004, 70, 193410. (42) Wendt, S.; et al. Oxygen Vacancies on TiO2(110) and Their Interaction with H2O and O2: A Combined High-Resolution STM and DFT Study. Surf. Sci. 2005, 598, 226−245. (43) Wu, X.; Selloni, A.; Lazzeri, M.; Nayak, S. K. Oxygen Vacancy Mediated Adsorption and Reactions of Molecular Oxygen on the TiO2(110) Surface. Phys. Rev. B 2003, 68, 241402. (44) Rusli, E.; Xue, F.; Drews, T. O.; Vereecken, P. M.; Andricacos, P.; Deligianni, H.; Braatz, R. D.; Alkire, R. C. Effect of Additives on Shape Evolution During Electrodeposition. J. Electrochem. Soc. 2007, 154, D584−D597.

(45) By Mark E. Law of the University of Florida and Al Tasch of the University of Texas at Austin. http://www.flooxs.tec.ufl.edu/ (accessed March 14, 2011). (46) De Souza, R. A.; Zehnpfenning, J.; Martin, M.; Maier, J. Determining Oxygen Isotope Profiles in Oxides with Time-of-Flight SIMS. Solid State Ionics 2005, 176, 1465−1471. (47) Beck, J.; Arnold, K. J. Parameter Estimation in Engineering and Science (Probability & Mathematical Statistics); John Wiley & Sons Inc.: New York, 1977. (48) Akse, J. R.; Whitehurst, H. B. Diffusion of Titanium in Slightly Reduced Rutile. J. Phys. Chem. Solids 1978, 39, 457−465. (49) Marucco, J.-F.; Gautron, J.; Lemasson, P. Thermogravimetric and Electrical Study of Non-Stoichiometric Titanium Dioxide TiO2−X, between 800 and 1100°C. J. Phys. Chem. Solids 1981, 42, 363−367. (50) Jung, M. Y. L.; Gunawan, R.; Braatz, R. D.; Seebauer, E. G. Pair Diffusion and Kick-Out: Contributions to Diffusion of Boron in Silicon. AIChE J. 2004, 50, 3248−3256. (51) Acket, G. A.; Volger, J. On the Electron Mobility and the Donor Centres in Reduced and Lithium-Doped Rutile (TiO2). Physica 1966, 32, 1680−1692. (52) Frederikse, H. Recent Studies on Rutile (TiO2). J. Appl. Phys. 1961, 32, 2211−2215. (53) Itakura, M.; Niizeki, N.; Toyoda, H.; Iwasaki, H. Hall Effect and Thermoelectric Power in Semiconductive TiO2. Jpn. J. Appl. Phys. 1967, 6, 311−317. (54) Kasinski, J. J.; Gomez-Jahn, L. A.; Faran, K. J.; Gracewski, S. M.; Miller, R. J. D. Picosecond Dynamics of Surface Electron Transfer Processes: Surface Restricted Transient Grating Studies of the NTiO2/H2O Interface. J. Chem. Phys. 1989, 90, 1253−1269. (55) Kormann, C.; Bahnemann, D. W.; Hoffmann, M. R. Preparation and Characterization of Quantum-Size Titanium Dioxide. J. Phys. Chem. 1988, 92, 5196−5201. (56) Pascual, J.; Camassel, J.; Mathieu, H. Fine Structure in the Intrinsic Absorption Edge of TiO2. Phys. Rev. B 1978, 18, 5606−5614. (57) Poumellec, B.; Marucco, J. F.; Lagnel, F. Electron Transport in TiO2−X at Intermediate Temperatures 300k < T < 1500k. Phys. Status Solidi (a) 1985, 89, 375−382. (58) Yagi, E.; Hasiguti, R. R.; Aono, M. Electronic Conduction above 4k of Slightly Reduced Oxygen-Deficient Rutile TiO2‑X. Phys. Rev. B 1996, 54, 7945−7956. (59) Vaidyanathan, R.; Jung, M. Y. L.; Seebauer, E. G. Mechanism and Energetics of Self-Interstitial Formation and Diffusion in Silicon. Phys. Rev. B 2007, 75, 195209. (60) Gorai, P.; Hollister, A.; Seebauer, E. G. Measurement of DefectMediated Oxygen Self-Diffusion in Metal Oxides. ECS J. Solid State Sci. Technol. 2012, 1, 4. (61) Agoston, P.; Albe, K. Formation Entropies of Intrinsic Point Defects in Cubic In2O3 from First-Principles Density Functional Theory Calculations. Phys. Chem. Chem. Phys. 2009, 11, 3226−3232. (62) Janotti, A.; Van de Walle, C. G. Native Point Defects in ZnO. Phys. Rev. B 2007, 76, 165202. (63) Dev, K.; Seebauer, E. G. Surface Vacancy Charging on Semiconductors at Nonzero Temperatures. Phys. Rev. B 2003, 67, 035312. (64) Van Vechten, J. A.; Thurmond, C. D. Entropy of Ionization and Temperature Variation of Ionization Levels of Defects in Semiconductors. Phys. Rev. B 1976, 14, 3539−3550. (65) Huang, G.-Y.; Wang, C.-Y.; Wang, J.-T. First-Principles Study of Diffusion of Oxygen Vacancies and Interstitials in ZnO. J. Phys.: Condens. Matter 2009, 21, 195403. (66) Jackson, R. A.; Catlow, C. R. A.; Murray, A. D. Point-Defect Calculations on UO2. J. Chem. Soc., Faraday Trans. 2: Mol. Chem. Phys. 1987, 83, 1171−1176. (67) Roma, G.; Limoge, Y.; Baroni, S. Oxygen Self-Diffusion in αQuartz. Phys. Rev. Lett. 2001, 86, 4564−4567. (68) Tarento, R. J. Oxygen Sublattice Defect in Cobalt Oxide: Formation, Migration, Charge Localization and Thermodynamic Processes. Rev. Phys. Appl. (Paris) 1989, 24, 643−647. 9964


Article

The Journal of Physical Chemistry C (69) Dobson, T. W.; Wager, J. F.; Van Vechten, J. A. Entropy of Migration for Atomic Hopping. Phys. Rev. B 1989, 40, 2962−2967. (70) Shewmon, P. G. Diffusion in Solids, 2nd ed.; The Minerals, Metals, and Materials Society: Warrendale, PA, 1989; p 246. (71) Zener, C. Theory of DO for Atomic Diffusion in Metals. J. Appl. Phys. 1951, 22, 372−375. (72) Wang, Z.; Seebauer, E. G. Estimating Pre-Exponential Factors for Desorption from Semiconductors: Consequences for a Priori Process Modeling. Appl. Surf. Sci. 2001, 181, 111−120. (73) Du, Y.; Dohnalek, Z.; Lyubinetsky, I. Transient Mobility of Oxygen Adatoms Upon O2 Dissociation on Reduced TiO2(110). J. Phys. Chem. C 2008, 112, 2649−2653. (74) Kuznetsov, V. N. Study of Oxygen Adsorption and Reoxidation of Reduced Titanium Dioxide by Thermal Desorption Mass Spectrometry. Kinet. Catal. 2002, 43, 868−873. (75) Onishi, Y. The Catalytic Oxidation of Carbon Monoxide on Titanium Dioxide; Anatase and Rutile. Bull. Chem. Soc. Jpn. 1971, 44, 1460−1462. (76) Diebold, U., Chapter 11 the Structure of TiO2 Surfaces. In The Chemical Physics of Solid Surfaces; Woodruff, D. P., Ed.; Elsevier: New York, 2001; Vol. 9, pp 443−484. (77) Ramamoorthy, M.; Vanderbilt, D.; King-Smith, R. D. FirstPrinciples Calculations of the Energetics of Stoichiometric TiO2 Surfaces. Phys. Rev. B 1994, 49, 16721−16727. (78) Onishi, H.; Iwasawa, Y. Reconstruction of TiO2(110) Surface: STM Study with Atomic-Scale Resolution. Surf. Sci. 1994, 313, L783− L789. (79) Shibata, N.; Goto, A.; Choi, S.-Y.; Mizoguchi, T.; Findlay, S. D.; Yamamoto, T.; Ikuhara, Y. Direct Imaging of Reconstructed Atoms on TiO2 (110) Surfaces. Science 2008, 322, 570−573. (80) Park, K. T.; Pan, M.; Meunier, V.; Plummer, E. W. Reoxidation of TiO2 (110) Via Ti Interstitials and Line Defects. Phys. Rev. B 2007, 75, 245415. (81) He, J.; Behera, R. K.; Finnis, M. W.; Li, X.; Dickey, E. C.; Phillpot, S. R.; Sinnott, S. B. Prediction of High-Temperature Point Defect Formation in TiO 2 from Combined Ab Initio and Thermodynamic Calculations. Acta Mater. 2007, 55, 4325−4337.

9965


Surface into the Bulk - ACS Publications - American Chemical Society

Recommend Documents