Ab initio free energy of vacancy formation and mass-action kinetics in vis-active TiO2

Recent reports have identified bulk defects such as oxygen vacancies as key players in visible-light photoactive TiO2. This would imply greater visible light absorption rates may be possible provided effective defect engineering can be achieved. To further this we have developed methods to simulate vacancy formation in bulk TiO2 using ab initio techniques. Initial results of these methods show an entropic reduction in the free energy of vacancy formation of 2.3 eV over a range of 266 K. The use of this result is illustrated by a ‘toy’ mass-action kinetics model which offers insight into vacancy concentration, rate constants, and enthalpy of reaction. (Some figures in this article are in colour only in the electronic version) The process of defect formation is of considerable interest. Defects play a significant role in crystalline transport, affect mechanical properties through interactions with dislocations and stacking faults, modify surface chemistry, and alter electrical/optical properties. In transition metal oxides, which play an important role in heterogeneous catalysis, photoelectrolysis, and biocompatibility, defects such as bulk and surface oxygen vacancies often dominate the electronic and chemical surface properties [1]. Prototypical among such oxides is TiO2 which has oxygen and titanium vacancies, aliovalent Ti interstitials, antisites, and crystallographic shear planes. The nature and quantity of these defects is a subject of long-standing debate with frequently contradictory experimental investigations [2–6]. As an amphoteric semiconductor, TiO2 has a complex defect phase diagram containing an n–p transition varying with oxygen partial pressure p(O2) and temperature [7]. TiO2 is also widely used in industry. An important application, photocatalysis, occurs when the strong oxidative potential of the positive holes oxidizes organics or even oxygen directly. Because of this property it is added to paints, cements, windows, tiles, and many other materials for its sterilizing, deodorizing, and anti-fouling properties, and is even used for hydrolysis or hydrogen production. One drawback is that UV light is required to activate catalysis due to its wide intrinsic bandgap. Various strategies to improve photo-absorption have been devised to create visible-light activated (vis-active) TiO2 such as anion doping [8] or chromophore attachment [9, 10] for multielectron injection. Recent precursor [11] and N-dopant [12] studies indicate, however, that manipulation of bulk oxygen vacancy concentration may be one of the most effective ways to increase photocatalytic activity in the visible due to an increase in color centers capable of absorbing at these wavelengths. To enable rational defect engineering in TiO2, a detailed understanding of defect behavior under complex conditions (varying oxygen partial pressure and temperature) is necessary. In this work and a companion publication [14] we introduce 0953-8984/08/022202+05$30.00 © 2008 IOP Publishing Ltd Printed in the UK 1 J. Phys.: Condens. Matter 20 (2008) 022202 Fast Track Communication a quantum thermodynamics algorithm to calculate defect parameters and a general statistical mechanics model to gage defect influence on material properties. The quantum methods in particular are difficult to implement because of the large supercells and long-time molecular dynamics runs at various temperatures that are required. However, these are now possible due to the evolution of efficient, linear-scaling electronic structure methods. The overall statistical mechanics of defect behavior can perhaps be best modeled by the grand canonical ensemble [13] which gives for a TiO2 crystal containing primarily vacancy defects, j = jTiO2 + nV F V + n V F V − T sconf. (1) jTiO2 = fTiO2 − μOnO − μTinTi is the grand canonical potential of a perfect TiO2 crystal per total atoms N , fTiO2 is the Helmholtz free energy of the perfect crystal per N , nO (nTi) is the number of oxygen (titanium) atoms per N , nV (n Ti V ) is the number of oxygen (titanium) vacancies per N , and sconf is the configurational entropy per N . This ensemble follows the number concentration of oxygens which dynamically exchange with an oxygen atmosphere. Thus the oxygen chemical potential μO is tuned by its partial pressure. The titanium chemical potential μTi is fixed by thermodynamic stability relations which near stoichiometry may be approximated as μTi + 2μO = μTiO2 but in a more general sense depend on the relative concentration of Ti/O [15]. A key component of this ensemble is the Helmholtz free energy of oxygen vacancy formation FO V which is equivalent to the Gibbs free energy at ambient pressures [16]. This quantity is particularly difficult to determine because of the accurate potentials needed to model vacancies with large electronic restructuring and the difficulty in calculating entropic contributions. Measurements are hampered by the necessity to isolate the effect of removing a single atom. This results in a need for extremely high-quality single crystals and dealing with the effects of material averaging and model limitations when fitting experimental data. In this work we introduce a new approach using powerful ab initio thermodynamic methods that efficiently probe bulk oxygen vacancy free energies of formation. This approach should provide greater confidence than force-field based calculations and a more direct means of ascertaining its magnitude than experimental methods such as macroscopic conductivity measurements. In previous vacancy calculations, the quasi-harmonic approximation to the Helmholtz free energy has frequently been used to calculate FO V [17]. However, this has been shown to be inadequate for much of the temperature span of a solid [18, 19]. Therefore we use nonequilibrium thermodynamic integration [20, 21] (TI), which consists of a molecular dynamics (md) run along a nonphysical path λ between two physical states to find their free energy difference [14, 22]. To apply TI to rutile TiO2 vacancies, one constructs a composite potential energy surface containing interactions from (1) the perfect crystal, (2) a crystal with one vacancy, and (3) an oxygen atom in a chemical reservoir such as an atmosphere at varying temperature and oxygen partial pressure. Thus V (λ) = (1−λ)Vperfect⊕λVvacancy⊕λVOres where λ ∈ [0, 1]. The vacancy is created by removing the nuclear charge and all electronic terms, although a charged vacancy can also be created. Nonequilibrium TI is based on equilibrium TI which performs a series of simulations along λ to find the free energy difference,