Tailoring electronic and optical properties of TiO2: nanostructuring, doping and molecular-oxide interactions
L. Chiodo, J. M. García-Lastra, D. J. Mowbray, A. Iacomino, A. Rubio
aa r X i v : . [ c ond - m a t . m t r l - s c i ] M a r Tailoring electronic and optical properties of TiO : nanostructuring, dopingand molecular-oxide interactions L. Chiodo † , J. M. Garc´ıa-Lastra † , D. J. Mowbray † , A. Iacomino ‡ , and A. Rubio † , §† Nano-Bio Spectroscopy group and ETSF Scientific Development Centre,Dpto. F´ısica de Materiales, Universidad del Pa´ıs Vasco,Centro de F´ısica de Materiales CSIC-UPV/EHU- MPC and DIPC,Av. Tolosa 72, E-20018 San Sebasti´an, Spain ‡ Dipartimento di Fisica “E. Amaldi”, Universit`a degli Studi Roma Tre,Via della Vasca Navale 84, I-00146 Roma, Italy § Friz-Haber-Institut der Max-Planck-Gesellschaft, Berlin, Germany bstract Titanium dioxide is one of the most widely investigated oxides. This is due to its broad range of ap-plications, from catalysis to photocatalysis to photovoltaics. Despite this large interest, many of its bulkproperties have been sparsely investigated using either experimental techniques or ab initio theory. Further,some of TiO ’s most important properties, such as its electronic band gap, the localized character of ex-citons, and the localized nature of states induced by oxygen vacancies, are still under debate. We presenta unified description of the properties of rutile and anatase phases, obtained from ab initio state of the artmethods, ranging from density functional theory (DFT) to many body perturbation theory (MBPT) derivedtechniques. In so doing, we show how advanced computational techniques can be used to quantitativelydescribe the structural, electronic, and optical properties of TiO nanostructures, an area of fundamentalimportance in applied research. Indeed, we address one of the main challenges to TiO -photocatalysis,namely band gap narrowing, by showing how to combine nanostructural changes with doping. With thisaim we compare TiO ’s electronic properties for 0D clusters, 1D nanorods, 2D layers, and 3D bulks usingdifferent approximations within DFT and MBPT calculations. While quantum confinement effects lead to awidening of the energy gap, it has been shown that substitutional doping with boron or nitrogen gives rise to(meta-)stable structures and the introduction of dopant and mid-gap states which effectively reduce the bandgap. Finally, we report how ab initio methods can be applied to understand the important role of TiO aselectron-acceptor in dye-sensitized solar cells. This task is made more difficult by the hybrid organic-oxidestructure of the involved systems. PACS numbers: . INTRODUCTION TiO has been one of the most studied oxides over the past few years. This is due to thebroad range of applications it offers in strategic fields of scientific, technological, environmental,and commercial relevance. In particular, TiO surfaces and nanocrystals provide a rich variety ofsuitably tunable properties from structure to opto-electronics. Special attention has been paid toTiO ’s optical properties. This is because TiO is regarded as one of the best candidate materialsto efficiently produce hydrogen via photocatalysis . TiO nanostructures are also widely used indye-sensitized solar cells, one of the most promising applications in the field of hybrid solar cells .Since the first experimental formation of hydrogen by photocatalysis in the early 1980s, TiO has been the catalyst of choice. Reasons for this include the position of TiO ’s conduction bandabove the energy of hydrogen formation, the relatively long lifetime of excited electrons whichallows them to reach the surface from the bulk, TiO ’s high corrosion resistance compared to othermetal oxides, and its relatively low cost . However, the large optical band gap of bulk TiO ( ≈ based photocatalytic reaction. On the other hand, the difference in energy betweenexcited electrons and holes, i.e. the band gap, must be large enough ( & ε gap ofTiO into the range 1.23 . ε gap . .With this aim, much research has been done on the influence of nanostructure and dopants on TiO photocatalytic activity. For low dimensional nanostructured materials, electronsand holes have to travel shorter distances to reach the surface, allowing for a shorter quasi-particlelifetime. However, due to quantum confinement effects, lower dimensional TiO nanostructurestend to have larger band gaps . On the other hand, although doping may introduce mid-gapstates, recent experimental studies have shown that boron and nitrogen doping of bulk TiO yieldsband gaps smaller than the threshold for water splitting . This suggests that low dimensionalstructures with band gaps larger than about . eV may be a better starting point for doping.Recently, several promising new candidate structures have been proposed . These small ( R . nanotubes, with a hexagonal ABC PtO structure (HexABC), were found to be surpris-ingly stable, even in the boron and nitrogen doped forms. This stability may be attributed to their3tructural similarity to bulk rutile TiO , with the smallest nanotube having the same structure as arutile nanorod.A further difficulty for any photocatalytic system is controlling how electrons and holes travelthrough the system . For this reason, methods for reliably producing both n -type and p -typeTiO semiconducting materials are highly desirable. So far, doped TiO tends to yield only n -type semiconductors. However, it has recently been proposed that p -type TiO semiconductingmaterials may be obtained by nitrogen doping surface sites of low dimensional materials . In thischapter we will discuss in detail the effects of quantum confinement and doping on the opticalproperties of TiO .TiO nanostructures are also one of the main components of hybrid solar cells. In a typicalGr¨atzel cell , TiO nanoparticles with average diameters around 20 nm collect the photoelectrontransferred from a dye molecule adsorbed on the surface . Such processes are favoured by aproper energy level alignment between solid and organic materials, although the dynamic part ofthe process also plays an important role in the charge transfer. Clearly, TiO ’s characteristics oflong quasi-particle lifetimes, high corrosion resistance, and relatively low cost, must be balancedwith control of its energy level alignment with molecular states, and a fast electron injection at theinterface.Despite all the engineering efforts, the main scientific goal remains to optimize the efficiency ofsolar energy conversion into readily available electricity. Different research approaches have beendevoted to benefit from quantum size properties emerging at the nanoscale , find an optimaldonor-acceptor complex , mix nanoparticles and one-dimensional structures, such as nanotubesor nanowires , and control the geometry of the TiO nano-assembly .A clear theoretical understanding of TiO ’s optoelectronic properties is necessary to help un-ravel many fundamental questions concerning the experimental results. In particular, the propertiesof excitons, photo-injected electrons, and surface configuration in TiO nanomaterials may play acritical role in determining their overall behaviour in solar cells. For TiO at both the nanoscaleand macroscale regime, it is necessary to first have a complete picture of the optical properties inorder to clarify the contribution of excitons.Despite the clear importance of its surfaces and nanostructures, investigations of TiO bulk (see1) electronic and optical properties have not provided, so far, a comprehensive description of thematerial. Important characteristics,such as the electronic band gap, are still undetermined. Most ofthe experimental and computational work has been focused on synthesis and analysis of systems4 = 9.50a, b = 3.78a, b = 4.59c = 2.95 R ax R eq R ax = 1.98R eq = 1.93R ax =1.98 R eq =1.95 R ax R eq FIG. 1: Unit cell (left), crystallographic structure (center) and TiO octahedrons (right) of rutile (top) andanatase (bottom). The lattice parameters in ˚A are denoted a , b , and c , while R a x and R e q are the distancesin ˚A between a Ti ion and its nearest and next-nearest neighbour O ions, respectively. In the case ofrutile the interstitial Ti impurity site is shown with a green circle (see left top). with reduced dimensionality.The experimental synthesis and characterization of nanostructured materials is in general acostly and difficult task. However, predictions of a dye or nanostructure’s properties from simula-tions can prove a great boon to experimentalists. Modern large-scale electronic structure calcula-tions have become important tools by providing realistic descriptions and predictions of structureand electronic properties for systems of technological interest.Although it will not be treated in this chapter, it is important to mention the problem of electronlocalization in reduced titania . This will provide a glimpse of the complexity faced, from thetheoretical point of view, when studying transition metal atoms. The localization of d electronsmakes the accurate description of their exchange–correlation interaction a difficult task . Theelectron localization in defective titania has been an open question from both experimental andtheoretical points of view, and caused much controversy during the past few years . Oxygenvacancies are quite common in TiO , and their presence and behaviour can significantly affect5he properties of nanostructures. When an oxygen vacancy is created in TiO , i.e. when TiO is reduced, the two electrons coming from the removed O ion must be redistributed within thestructure. One possibility is that these two extra electrons remain localized onto two Ti ions closeto the O vacancy. In this way a pair of Ti ions become Ti ions. Another option is for the twoextra electrons to delocalize along the whole structure, i.e. they do not localize on any particularTi ion. Finally, an intermediate situation, with one electron localized and the other spread, is alsopossible. Concerning the TiO bulk, conventional density functional theory (DFT) calculationsusing either local density approximations (LDA) or generalized gradient approximations (GGA)for the exchange-correlation (xc)-functionals show a scenario with both electrons delocalized. Onthe other hand, hybrid functionals and Hartree-Fock calculations give rise to a situation with bothelectrons localized. For GGA+U calculations, the results are very sensitive to the value of the Uparameter. For certain values of U both electrons remain localized, while for others there is anintermediate situation .Experimentally, there are electron paramagnetic resonance (EPR) measurements suggestingthat the extra electrons are mainly localized on interstitial Ti ions . These interstitial Ti ionsare impurities placed at the natural interstices of the rutile lattice (see 1) and, similarly to the Ti ions of the pure lattice, they also form TiO octahedrons. Recent STM and PES experiments haveshown that the interstitial Ti ions play a key role in the localization of the electrons when a bridgeoxygen is removed from the TiO (110) surface . These experiments concluded the controversialdiscussion about the localization of electrons in the bridge oxygen defective TiO (110) surface(see Refs. for more details). However, the problem remains unresolved for the bulk case.In summary, in this chapter we first analyze the full ab initio treatment of electronic and opticalproperties in II and III, before applying it to the two most stable bulk phases, rutile and anatasein IV. These are also the phases most easily found when nanostructures are synthesized. We willfocus on their optical properties and excitonic behaviour. We then explore the possibility of tuningthe oxide band gap using quantum confinement effects and dimensionality, by analyzing atomicclusters, nanowires and nanotubes in V. A further component whose effect has to be evaluated isthat of doping, which may further tune the optical behaviour by introducing electronic states inthe gap, as presented in VI.Combining the effects of quantum confinement and doping is hoped toproduce a refined properties control. In VII, we report some details on modeling for dye-sensitizedsolar cells, before providing a summary and our concluding remarks.6 I. GROUND STATE PROPERTIES THROUGH DENSITY FUNCTIONAL THEORY
DFT is a many-body approach, successfully used for many years to calculate the ground-stateelectronic properties of many-electron systems. However, DFT is by definition a ground state the-ory, and is not directly applicable to the study of excited states. To describe these types of physicalphenomena it becomes necessary to include many-body effects not contained in DFT throughGreen’s function theory. The use of many-body perturbation theory , with DFT calculations as azero order approximation, is an approach widely used to obtain quasi-particle excitation energiesand dielectric response in an increasing number of systems, from bulk materials to surfaces andnanostructures. We present a short, general discussion of the theoretical framework, referring thereader to the books and the reviews available in the literature for a complete description (see, forinstance, Refs. and ), before applying these methods to rutile and anatase TiO .As originally introduced by Hohenberg and Kohn (HK) in 1964 , DFT is based on the theoremthat the ground state energy of a system of N interacting electrons in an external potential V ext ( r ) is a unique functional of the ground state electronic density. The Kohn and Sham formulationdemonstrates how the the HK Theorem may be used in practice, by self-consistently solving a setof one-electron equations (KS equations), (cid:20) − ∇ + V KSeff [ ρ ( r )] (cid:21) φ i ( r ) = ε KSi φ i ( r ) , ρ ( r ) = N X i | φ i ( r ) | , (1)where ρ ( r ) is the electronic charge density, φ i ( r ) are the non-interacting KS wavefunctions, and V KSeff [ ρ ( r )] = V ext ( r ) + V H [ ρ ( r )] + V xc [ ρ ( r )] is the effective one-electron potential. Here, V H is the Hartree potential and V xc is the exchange–correlation potential defined in terms of the xc-functional E xc as V xc = δE xc δρ ( r ) , which contains all the many-body effects. V xc is usually calculated ineither LDA or GGA approximations. However, semi-empirical functionals are also available,called hybrids , which somewhat correct the deficiencies of LDA and GGA for describingexchange and correlation. This is accomplished by including an exact exchange contribution.Other than the highest occupied molecular orbital (HOMO), KS DFT electronic levels do notcorrespond to the electronic energies of the many electron system. Indeed, the calculated KS bandgaps of semiconductors and insulators severely underestimate the experimental ones. Experimen-tally, occupied states are accessible by direct photoemission, where an electron is extracted fromthe system ( N − ground state), while unoccupied states are accessible by inverse photoemission,where one electron is added to the system ( N + 1 ground state). For isolated systems with a finite7umber of electrons, the electronic gap may be obtained from the DFT calculated ground-state en-ergies with N + 1 , N , and N − electrons . However, for periodic systems, adding or removingan electron is a non-trivial task. We therefore summarize in the following a rigorous method fordescribing excitations based on the Green’s function approach. This method allows us to properlydescribe the electronic band structure. Further information and details about the Green’s functionapproach may be found elsewhere . III. EXCITED STATES WITHIN MANY BODY PERTURBATION THEORY
When a bare particle, such as an electron or hole, enters an interacting system, it perturbsthe particles in its vicinity. In essence, the particle is “dressed” by a polarization cloud of thesurrounding particles, becoming a so-called quasi-particle. Using this concept, it is possible todescribe the system through a set of quasi-particle equations by introducing a non-local, time-dependent, non-Hermitian operator called the self-energy Σ . This operator takes into account theinteraction of the particle with the system via (cid:20) − ∇ + V ext + V H (cid:21) Ψ i ( r , ω ) + Z Σ( r , r ′ , ω )Ψ i ( r ′ , ω ) d r ′ = E i ( ω )Ψ i ( r ) , (2)where Ψ i ( r ) is the quasi-particle wavefunction.Since the operator Σ is non-Hermitian, the energies E i ( ω ) are in general complex, and theimaginary part of Σ is related to the lifetime of the excited particle . The most often used ap-proximation to calculate the self-energy is the so-called GW method. It may be derived as thefirst-step iterative solution of the Hedin integral equations (see Refs. , , and ), which link theGreen’s function G , the self-energy Σ , the screened Coulomb potential W , the polarization P andthe vertex Γ . In practice 2 is not usually solved directly, since the KS wavefunctions are often verysimilar to the GW ones . For this reason it is often sufficient to calculate the quasi-particle (QP)corrections within first-order perturbation theory ( G W ) . Moreover, the energy dependenceof the self-energy is accounted for by expanding Σ in a Taylor series, so that the QP energies ε G W i are then given by ε G W i = ε KSi + Z i h φ KSi | Σ( ε KSi ) − V KSxc | φ KSi i , (3)where Z i are the quasi-particle renomalization factors described in Refs. and . For a large num-ber of materials, the G W approximation of Σ works quite well at correcting the KS electronicgap from DFT. 8oncerning optical properties, the physical quantity to be determined in order to obtain the opti-cal spectra is the macroscopic dielectric function ǫ M ( ω ). This may be calculated at different levelsof accuracy within a theoretical ab initio approach. A major component in the interpretation ofthe optical measurements of reduced dimensional systems are the local-field effects (LFE). Theseeffects are especially important for inhomogeneous systems. Here, even long wavelength exter-nal perturbations produce microscopic fluctuations of the electric field, which must be taken intoaccount. However, LFE are also important for bulk phases such as anatase and rutile TiO . Theymust be taken into account in the evaluation of optical absorption, and to calculate the screenedinteraction W used in GW . The effect becomes increasingly important when going to lower di-mensional systems.It is well known that for inhomogeneous materials ǫ M ( ω ) is not simply the average of thecorresponding microscopic quantity, but is related to the inverse of the microscopic dielectricmatrix by ǫ M ( ω ) = lim q → ǫ − G = , G ′ = ( q , ω ) . (4)The microscopic dielectric function may be determined within the linear response theory , theindependent-particle picture by the random phase approximation (RPA), and using eigenvaluesand eigenvectors of a one-particle scheme such as DFT or GW . There is also a different formu-lation which includes LFE in the macroscopic dielectric function. This becomes useful when theelectron–hole interaction is included in the polarization function. This formulation allows us toinclude, via the Bethe-Salpeter equation (BSE), excitonic and LFE on the same footing. In sodoing, inverting of the microscopic dielectric matrix is avoided. The complete derivation may befound in Appendix B of Ref. .So far, in RPA, we have treated the quasi-particles as non-interacting. To take into account theelectron–hole interaction, a higher order vertex correction needs to be included in the polarization.In other words, the BSE, which describes the electron–hole pair dynamics, needs to be solved.As explained in Ref. , the BSE may be written as an eigenvalue problem involving the effectivetwo-particle Hamiltonian H ( n ,n ) , ( n ,n ) exc = ( E n − E n ) δ n ,n δ n ,n − i ( f n − f n ) × Z d r d r ′ d r d r ′ φ n ( r ) φ ∗ n ( r ′ ) Ξ( r , r ′ , r , r ′ ) φ ∗ n ( r ) φ n ( r ′ ) . The kernel Ξ contains two contributions: ¯ v , which is the bare Coulomb interaction with the long9ange part corresponding to a vanishing wave vector not included and W , the attractive screenedCoulomb electron–hole interaction. Using this formalism and considering only the resonant partof the excitonic Hamiltonian , the macroscopic dielectric function may be expressed as ǫ M ( ω ) = 1 + lim q → v ( q ) X λ (cid:12)(cid:12)(cid:12)P v,c ; k h v, k − q | e − i qr | c, k i A ( v,c ; k ) λ (cid:12)(cid:12)(cid:12) ( E λ − ω ) . (5)In 5 the dielectric function, differently from the RPA approximation, is given by a mixing ofsingle particle transitions weighted by the excitonic eigenstates A λ . These are obtained by thediagonalization of the excitonic Hamiltonian. Moreover, the excitation energies in the denominatorare changed from ǫ c − ǫ v to E λ . The electronic levels are mixed to produce optical transitions,which are no longer between pairs of independent particles. The excitonic calculation is in general,from the computational point of view, very demanding because the matrix to be diagonalized maybe very large. The relevant parameters which determine its size are the number of k -points in theBrillouin zone, and the number of valence and conduction bands, N v and N c respectively, whichform the basis set of pairs of states.Calculations performed for insulators and semiconductors show that the inclusion of theelectron–hole Coulomb interaction yields a near-quantitative agreement with experiment. Thisis not only true below the electronic gaps, where bound excitons are generally formed, but alsoabove the continuum edge. The same results apply to the titania-based materials investigated here,as shown in the following section. IV. THE BULK PHASES OF TIO : ROLE OF MANY BODY EFFECTS Despite the importance of its surfaces and nanostructures, the most recent measurements ofTiO ’s bulk (see Fig. 1) electronic and optical properties were performed in the 1960s, with a fewexceptions. Here we aim to review the existing results obtained using a variety of experimentaltechniques and ab initio calculations, in order to elucidate the known properties of TiO . Previousdata will be compared with a complete, consistent ab initio description, which includes many bodyeffects when describing electronic and optical properties. Most experimental and theoretical datareported refers to the rutile phase, while anatase in general has been less studied. However, anatasehas received more attention recently, due in part to its greater stability at the nanoscale comparedto rutile. 10s we will see, while a general agreement seems to exist concerning the optical absorptionedge of these materials, values for basic electronic properties such as the band gap still have alarge degree of uncertainty.The electronic properties of valence states of rutile TiO have been investigated experimentallyby angle-resolved photoemission spectroscopy , along the two high symmetry directions ( ∆ and Σ ) in the bulk Brillouin zone. The valence band of TiO consists mainly of O 2 p states partiallyhybridized with Ti 3 d states. The metal 3 d states constitute the conduction band, with a smallamount of mixing with O 2 p states. This photoemission data was compared to calculations per-formed with both pseudopotentials and linear muffin-tin orbital (LMTO) methods, which gave adirect gap of 2 eV in both cases. On the other hand, within the linear combination of atomicorbitals (LCAO) method, a gap of 3 eV was obtained.From the symmetry of the TiO octahedrons (see 1), d states are usually grouped into lowenergy t g and high energy e g sub-bands. It is important to note that, from ultraviolet photoe-mission spectroscopy (UPS) data, it has been deduced that the electronic gap for rutile is at least4 eV. This is the observed binding energy of the first states below the Fermi energy. This is inagreement with previous reported data from electron energy loss spectroscopy , and from otherUPS results. The electronic structure of rutile bulk has also been described using other exper-imental techniques, such as electrical resistivity , electroabsorption , photoconductivity andphotoluminescence , X-ray absorption spectroscopy (XAS) , resonant Raman spectra photoelectrochemical analysis and UPS . All of these experiments have provided many im-portant details of its electronic properties, in particular concerning the hybridization between Ti3 d and O 2 p states. However, the electronic band gap, corresponding to the difference between thevalence band maximum (VBM) and the conduction band minimum (CBM), has not been obtaineddirectly from any experimental data. Although the electronic band gap could be measured usingcombined photoemission and inverse photoemission experiments, such experiments do not appearin the literature.The same discussion is valid for the anatase crystalline phase. Even though there are severalXAS measurements concerning its electronic structure , photoemission data is completelylacking for anatase. In the absence of more recent and refined experimental results for rutile, anddue to the lack of results for anatase, we are left with an estimate of 4 eV for the electronic gap ofrutile TiO . It is this value which we shall use as a reference in the following discussion.We will now review the experimental results for optical properties of TiO . Such measurements11re of great interest for the photocatalytic and photovoltaic applications of this material. From theoptical absorption spectra of both phases , the room temperature optical band gap is found to be3.0 eV for rutile, and 3.2 eV for anatase.The absorption edge has been investigated in detail for rutile by combining absorption, photo-luminescence, and Raman scattering techniques . These techniques provide a value for the edgeof 3.031 eV associated to a 2 p xy exciton, while a lower energy 1 s quadrupolar exciton has beenidentified below 3 eV. The first dipole allowed gap is at 4.2 eV according to the combined resultsof these three techniques. Concerning anatase , the optical spectrum have been recently re-evaluated , confirming the 3.2 eV value for the edge. The fine details of anatase’s spectrum havealso been recently investigated . Data on the Urbach tail has revealed that excitons in anatase areself-trapped in the octahedron of coordination of the titanium atom. This is in contrast to the rutilephase, where excitons are known to be free due to the different packing of rutile’s octahedra. In general, measurements of optical properties can be significantly affected by the presence ofdefects, such as oxygen vacancies, and by phonons. Both defects and phonons will be presentin any experimental sample of the material at finite temperatures. These observations have tobe kept in mind when directly comparing experimental measurements with the theoretical resultspresented in the following. Moreover, there is a general trend in theoretical-computational studiesto compare the theoretical electronic band gaps with the experimental optical gap values derived from the above mentioned experiments.It should be remembered that almost by definition, the optical gap is always smaller than theelectronic band gap. This is because the two types of experiments (photoemission, and opticalabsorption) provide information on two different physical quantities. Reverse photoemission ex-periments involve a change in the total number of electrons in the material ( N → N + 1 ), whileoptical absorption experiments do not ( N → N ∗ ). The latter involves the creation of an electron–hole pair in the material, with the hole stabilizing the excited electron. For this reason, comparisonbetween experimental and theoretical data, and the resulting discussion, must take into accountthe proper quantities.The theoretical investigations presented in the literature of TiO ’s structural, electronic, andoptical properties are at varying levels of theory and thus somewhat inconsistent. A comprehensivedescription of properties of both phases in the same theoretical and computational framework isstill missing. Here we present, in a unified description and by treating with the same method forthe two phases, the electronic and optical properties of the two most stable phases of bulk titania.12he ab intio calculations performed yield results in quite good agreement with the few availablepieces of experimental data.A combination of DFT and many body perturbation theory (MBPT) methods is a reliableand well established toolkit to obtain a complete analysis of electronic and optical properties fora large class of materials and structures. In this DFT + GW + BSE framework, the properties ofthe two bulk phases of TiO may be properly analyzed. Their structural and energetic propertieshave been calculated using DFT, as a well established tool for the description of ground stateproperties. However, the DFT gap is, as expected, significantly smaller than the experimental gap,with the relative positions of the s , p , and d levels also affected by this description. To address this,standard G W calculations may be applied to obtain the quasi-particle corrections to the energylevels, starting from DFT eigenvalues and eigenfunctions. Finally the electron–hole interaction isincluded, to properly describe the optical response of the system.The description of ground state properties performed in the framework of DFT are gen-erally quite good, with the structural description of TiO systems in reasonable agreement withexperiments . The lattice constants are within 2% of experiment, while bulk modulii are within10% of the experimental results , as is often found for DFT. However, DFT incorrectly pre-dicts the anatase phase to be more stable than the rutile one, even for a small energy difference,independently of the xc-functional used .Even if DFT is not an excited state method, the KS wavefunctions are often used to evaluate theband structure along the high symmetry directions ( cf. . The KSelectronic gap, corresponding to the difference between the VBM and the CBM, is 1.93 eV and2.16 eV for rutile (direct gap) and anatase (indirect gap), respectively. These are underestimationsby almost 2 eV of the available experimental data . However, the overall behaviour of the banddispersion of KS levels is reasonable, with valence bands mainly given by O 2 p states, and Ti 3 d states forming the conduction bands.The application of standard GW methods gives gaps of 3.59 and 3.97 eV for rutile andanatase , respectively. The value for rutile is again smaller than the one given by the UPS es-timation , but still close to the experimental value of 4 eV.There exist a number of theoretical works, with calculations performed at different levels ofDFT or including MBPT descriptions, for the electronic gap of rutile and anatase TiO .Therefore in literature it is possible to find for the electronic gap a quite large range of possible13 M X G Z A R Z-6-4-202468 E n e r gy ( e V ) FIG. 2: Electronic band structure of rutile bulk, along the high symmetry directions of the irreducibleBrillouin zone, from GGA calculations ( —— ). and including the G W correction ( ● ). values, attributed to the gap of titanium dioxide, which are often erroneously compared with theexperimental optical gap.The DFT-GGA values calculated are comparable to the ones obtained with a variety of dif-ferent DFT approaches, with different functionals, by using plane waves or localized basis meth-ods, and all-electron or pseudopotential approaches. Only the hybrid PBE0 and B3LYP xc-functionals give larger values.From quasi-particle calculations, the electronic gap of anatase has been estimated to be3.79 eV by G W . However the more refined computational approach, because of its inclusionof a self-consistent evaluation of GW , yields a gap of 3.78 eV for rutile TiO . Moving to optical properties, and by applying the RPA method to both KS and QP energies,we obtain spectra (Fig. 3) that do not in overall behaviour agree with experiment. Differences areclear both in absorption edge determination, and in the overall shape of the spectra. The inclu-sion of quasi-particle corrections at the GW level yields a rigid shift of the absorption spectrum,moving the edge at higher energies, due to the opening of the gap. However, the shape of theabsorption is quite unaffected, because the interaction is still treated using an independent quasi-14 E B E C e ( a r b . un . ) Energy (eV) exp BSE RPA RPA+GW E A FIG. 3: Imaginary part of the dielectric constant for rutile (left), and anatase (right), in-plane polarization,calculated by GGA RPA ( – – – ), using G W on top of GGA ( · · · · · · ), and via the Bethe-Salpeter equation(BSE) ( —— ). The experimental spectrum (— • —) from Refs. and is also shown for comparison. particle approximation. A substantially better agreement may be obtained by solving the BSE,which takes into account both many body interactions and excitonic effects. Indeed, it produces agood description of absorption spectra and excitons, as shown in Fig. 3The optical absorption spectra calculated for the two phases, with polarization along the x -direction of the unit cells, are provided in 3. The spectra given by independent-particle transitionspresent two characteristic features. First, the band edge is underestimated, due to the electronicgap underestimation in DFT. Second, the overall shape of the spectrum is, for both phases, andboth orientations, different from the experiment, in the sense that the oscillator strengths are notcorrect. The inclusion of the quasi-particle description, which should improve the electronic gapdescription, does not improve the overall shape of the spectrum. The absorption edge is, however,15 IG. 4: Spatial distribution (yellow isosurfaces in arbitrary units) of the partial dark, dark, and opticallyactive first three excitons in rutile. The hole position is denoted by the light green dot. shifted at higher energies, even higher than expected from experiments. The description of opticalproperties within the two interacting quasi-particle approximation (by solving the BSE) definitelyimproves the results. The absorption edge is now comparable to the experimental one, with theoptical gap estimated from our calculations in good agreement with the available data. Further, theshape of the spectrum is now well described, with a redistribution of transitions at lower energies.The agreement is generally good for both phases .The nature of the exciton is still under debate in TiO materials . The experimental bindingenergy is of 4 meV and some uncertainty exists for the exact determination of optical edge. More-over, the exciton is localized in one of the two phases, and delocalized in the other one, at leastbased on experimental results. However, an explanation for this behaviour is so far missing. Re-fined measurements give an exciton of 2 p xy character at 3.031 eV, and a 1 s quadrupolar excitonat energy lower than 3 eV. From ab initio calculations with x polarization, two dark (opticallyinactive) or quasi dark excitons are located in rutile at 3.40 and 3.55 eV (denoted by E A , E B in4). At the same time, the optically active exciton, with 4 meV of binding energy, is located at3.59 eV ( E C in 4). The spatial distribution of the first three excitons is plotted in 4. The transitionsare from O 2 p states to Ti 3 d states of the triplet t g , as expected. While the first two opticallyforbidden transitions involve Ti atoms farther away from the excited O atom, the optical activetransition involves states of the nearest neighbour Ti atoms.In this section we have endeavoured to clarify though the application of a consistent description,the properties of the two main crystalline phases of titania. Particular attention has been taken tohow the inclusion of a proper description of exchange and correlation effects can improve thedescription of both electronic and optical properties of TiO .16ow that bulk properties are known, from a theoretical point of view, at the level that thestate-of-the-art ab initio techniques allow us to reach, we can attempt to describe how quantumconfinement induced by reduced dimensionality, and doping both effect the properties of TiO .Our final aim is to demonstrate how we may tune the electronic and optical gap of nanostructuresfor photocatalytic and photovoltaic applications. V. USING NANOSTRUCTURE TO TUNE THE ENERGY GAP
Having described in the previous section the electronic structure of the two bulk TiO phases,we will now turn our attention to the influence of nanostructure on the energy gap. This is anarea which has received considerable attention in recent years , in part due to the inherentlyhigh surface to volume ratio of nanostructures. It is hoped that this will allow materials withshorter quasi-particle lifetimes to function effectively for photocatalytic activities, since excitonsare essentially formed at the material’s surface. However, this advantage is partly countered byquantum confinement effects, which tend to increase the energy gap in nanostructures. Thesecompeting factors make the accurate theoretical determination of energy gaps in nanostructuredmaterials a thing of great interest.However, as shown in the previous section, standard DFT calculations tend to underestimateelectronic band gaps for bulk TiO by approximately 2 eV, due in part to self-interaction errors .These errors arise from an incomplete cancellation of the electron’s Coulomb potential in theexchange-correlation (xc)-functional.This may be partially addressed by the use of hybrid functionals such as B3LYP , whichgenerally seem to improve band gaps for bulk systems . However, such calculations are com-putationally more expensive, due to the added dependence of the xc-functional on the electron’swavefunction. Moreover, B3LYP calculations for TiO clusters largely underestimate the gap rel-ative to the more reliable difference between standard DFT calculated ionization potential I p andelectron affinity E a . For isolated systems such as clusters, the needed energetics of charged speciesare quantitatively described by standard DFT. Also, B3LYP and RPBE calculations provide thesame qualitative description of the trends in the energy gaps for TiO , as seen in 5. Anothermethodology is thus needed to describe the band gaps of periodic systems. G W is probably the most successful and generally applicable method for calculating quasi-particle gaps. For clusters it agrees well with I p − E a , and it has been shows to produce reliable17 xxx xx xxxx xxxxxxxxxxxxxx xxxx xxxx xxxx xx xx xxxx xx xxxx xxxx ( T i O ) C l u s t e r( T i O ) C l u s t e r( T i O ) C l u s t e r( T i O ) C l u s t e r( T i O ) C l u s t e r( T i O ) C l u s t e r( T i O ) C l u s t e r( T i O ) C l u s t e r( , ) N ano r od ( , ) N ano t ube ( , ) N ano t ube H e x AB C La y e r A na t a s e La y e r A na t a s e S u r f a c e A na t a s e B u l k R u t il e B u l k E ne r g y G ap e gap [ e V ] RPBE xxxx I p - E a G W ExpB3LYP
0D 1D 2D 3D
FIG. 5: Energy gap ε gap in eV versus TiO structure for 0D (TiO ) n clusters ( n ≤ ), 1D TiO (2,2)nanorods, (3,3) nanotubes, (4,4) nanotubes, 2D HexABC and anatase layers, and 3D anatase surface, anatasebulk, and rutile bulk phases. DFT calculations using the highest occupied and lowest unoccupied stategaps with the standard GGA RPBE xc-functional ( ) and the hybrid B3LYP xc-functional ( H ), are com-pared with DFT I p − E a calculations ( ● ), G W quasi-particle calculations ( ◮ ), and experimental results( ◆ ) . Schematics of representative structures for each dimensionality are shown above and takenfrom Ref. . results for bulk phases. On the other hand, G W calculations describe an N → N + 1 transitionwhere the number of charges is not conserved, rather than the electron–hole pair induced by pho-toabsorption, which is a neutral process. Indeed, a description in terms of electronic gap cannotbe compared with, or provide direct information on the optical gap, which is the most investigatedquantity, due to its critical importance for photocatalytic processes.5 shows that for both 3D and 2D systems, RPBE gaps underestimate the experimental optical18aps by approximately 1 eV. For 1D and 0D systems, there is a much larger difference of about4 eV and 5 eV respectively, between the RPBE gaps and the I p − E a and G W results. Thisincreasing disparity may be attributed to the greater quantum confinement and charge localizationin the 1D and 0D systems, which yield greater self-interaction effects. The B3LYP gaps also tendto underestimate this effect, simply increasing the RPBE energy gaps for both 0D and 3D systemsby about 1.4 eV.On the other hand, the RPBE gaps reproduce qualitatively the structural dependence of the I p − E a , G W , and experimental results for a given dimensionality, up to a constant shift. This istrue even for 3D bulk systems, where standard DFT does not correctly predict rutile to be the moststable structure .To summarize, quantum confinement effects seem to increase the energy gap significantly forboth 0D and 1D systems, while 2D and 3D systems may be much less affected. This suggests that2D laminar structures are viable candidates for reducing the minimal quasi-particle, while leavingthe band gap nearly unchanged. However, a more accurate description of the photoabsorptionproperties of these novel nanostructures, perhaps using the methodologies recently applied to bulkTiO , still remains to be found. VI. INFLUENCE OF BORON AND NITROGEN DOPING ON TIO ’S ENERGY GAP The doping of TiO nanostructures has received much recent attention as a possible meansfor effectively tuning TiO ’s band gap into the visible range . Recent experiments suggestsubstituting oxygen by boron or nitrogen in the bulk introduces mid-gap states, allowing lowerenergy excitations. However, to model such systems effectively requires large supercells, bothto properly describe the experimental doping fractions of . (2,2) nanorods. The highest occupied state is also shown as isosurfaces of ± e/ ˚A in the side views of the doped structures.As with TiO clusters, the influence of boron dopants on TiO nanorods may be understood interms of boron’s weak electronegativity, especially when compared with the strongly electronega-tive oxygen. Boron prefers to occupy oxygen sites which are 2-fold coordinated to neighbouringtitanium atoms. However, as with the 0D clusters, boron’s relatively electropositive character19 N -1 0 1 2 3 e - e VB [eV]0123 D O S [ e V - / T i O ] Nanorod O FIG. 6: Total density of states (DOS) in eV − per TiO functional unit vs. energy ε in eV for undoped (——),boron doped ( – – – ) and nitrogen doped ( —— ) (2,2) TiO nanorods from standard DFT GGA RPBE xc-functional and (inset) G W quasi-particle calculations. The highest occupied states for boron and nitrogendoped (2,2) TiO nanorods are depicted by isosurfaces of ± e/ ˚A in the structure diagrams to the left. induces significant structural changes in the 1D structures, creating a stronger third bond to aneighbouring three-fold coordinated oxygen via an oxygen dislocation, as shown in 6. This yieldsthree occupied mid-gap states localized on the boron dopant, which overlap both the valence bandO 2 p π and conduction band Ti 3 d xy states, as shown in 6. Boron dopants thus yield donor statesnear the conduction band, which may be photocatalytically active in the visible region. However,the quantum confinement inherent in these 1D structures may stretch these gaps, as found for the G W calculated DOS shown in the inset of 6.On the other hand, nitrogen dopants prefer to occupy oxygen sites which are 3-fold coordinatedto Ti, as was previously found for the rutile TiO surface . This yields one occupied state at thetop of the valence band and one unoccupied mid-gap state in the same spin channel. Both statesare localized on the nitrogen dopant but overlap the valence band O 2 p π states, as shown in 6.Nitrogen dopants thus act as acceptors, providing localized states well above the valence band, asis also found for the G W calculated DOS shown in the inset of 6.Although nitrogen dopants act as acceptors in TiO
1D structures, such large gaps betweenthe valence band and the unoccupied mid-gap states would not yield p -type semiconductors. Thismay be attributed to the substantial quantum confinement in these 1D structures. However, for2D and 3D systems, it is possible to produce both p -type and n -type classical semiconductors, as20iscussed in Ref. . This has recently been shown experimentally in Ref. , where co-doping ofanatase TiO with nitrogen and chromium was found to improve the localization of the acceptorstates, and reduce the effective optical gap. By replacing both Ti and O atoms with dopantsin the same TiO octahedral, it should be possible to “tune” the optical band gap to a much finerdegree.Whether calculated using RPBE, I p − E a or G W , the energy gaps for both boron and nitrogendoped TiO nanostructures are generally narrowed, as shown in 7(a) and (b). However, for nitrogendoped (TiO ) n clusters where nitrogen acts as an acceptor ( n = 5 , , ), the energy gap is actually increased when spin is conserved, compared to the undoped clusters in RPBE. This effect is notproperly described by the N → N + 1 transitions of I p − E a , for which spin is not conserved forthese nitrogen doped clusters. On the other hand, when nitrogen acts as a donor ( n = 7 , ) thesmallest gap between energy levels does conserve spin.The boron doped TiO nanorods and nanotubes have perhaps the most promising energy gapresults of the TiO structures, as seen in 7(a). Boron dopants introduce in the nanorods localizedoccupied states near the conduction band edge in both RPBE ( cf.
6) and G W ( cf. inset of 6)calculations. On the other hand, nitrogen doping of nanorods introduces well defined mid-gapstates, as shown in 6. However, to perform water dissociation, the energy of the excited electronmust be above that for hydrogen evolution, with respect to the vacuum level. This is not the casefor such a mid-gap state. This opens the possibility of a second excitation from the mid-gap stateto the conduction band. However, the cross section for such an excitation may be rather low.For boron doping of 2D and 3D structures, the highest occupied state donates its electron almostentirely to the conduction band, yielding an n -type semiconductor. Thus at very low temperatures,the RPBE band gap is very small. The same is true for n -type nitrogen doped bulk anatase.For these reasons the energy gap between the highest fully occupied state and the conductionband, which may be more relevant for photoabsorption, is also shown. These RPBE gaps are stillgenerally smaller than those for their undoped TiO counterparts, shown in 5.In summary, for both boron and nitrogen doped clusters we find RPBE gaps differ from I p − E a by about 3 eV, while for nitrogen doped anatase the RPBE gap differs from experiment by about0.6 eV. Given the common shift of 1 eV for undoped 2D and 3D structures, this suggests that bothboron and nitrogen doped 2D TiO structures are promising candidates for photocatalysis. Further,the boron and nitrogen doped 1D nanotube results also warrant further experimental investigation.21 x xxxx xxxx xx xxxxxxxx xx xxxx xxxx xxxx xxxxxxxxx RPBE xxxx I p - E a G W xxxx x xxxx xxxx x x xx xxx xxxx xxxx xxxxxxxx xxxxx xxx xxxx ( T i O ) C l u s t e r( T i O ) C l u s t e r( T i O ) C l u s t e r( T i O ) C l u s t e r( T i O ) C l u s t e r( , ) N ano r od ( , ) N ano t ube ( , ) N ano t ube H e x AB C La y e r A na t a s e La y e r A na t a s e S u r f a c e A na t a s e B u l k E ne r g y G ap e gap [ e V ] RPBE xxxx I p - E a G W Exp
0D 1D 2D 3D(a) B-doped(b) N-doped
FIG. 7: Influence of doping on the energy gap ε gap in eV versus TiO structure for 0D (TiO ) n clusters( n ≤ ), 1D TiO (2,2) nanorods, (3,3) nanotubes, (4,4) nanotubes, 2D HexABC and anatase layers, and3D anatase surface and bulk. The energy gaps for (a) boron doped systems from DFT calculations usingthe highest occupied and lowest unoccupied state gaps with the standard GGA RPBE xc-functional ( △ ) arecompared with DFT I p − E a calculations ( N ), and G W quasi-particle calculations ( ◮ ), and (b) nitrogendoped systems from DFT calculations using the highest occupied and lowest unoccupied state gaps with thestandard GGA RPBE xc-functional ( (cid:31) ) are compared with DFT I p − E a calculations ( (cid:4) ), and G W quasi-particle calculations ( ◮ ), and experimental ( ◆ ) results . Small open symbols denote transitionsbetween highest fully occupied states and the conduction band. VII. SOLAR CELLS FROM TIO NANOSTRUCTURES: DYE-SENSITIZED SOLAR CELLS
TiO is so far the most widely used solid material in the development of solar cell devicesbased on hybrid architectures . In these devices, the dye, synthetic or organic, absorbs light,22 .5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 eeee ( a r b . un . ) TD A LD A N 719 gas phase
Energy (eV)
FIG. 8: Absorption spectrum calculated by TD Adiabatic LDA of the Ru-dye N719, whose structure isshown in the inset. and electrons excited by the phonons are injected into the underlying oxide nanostructure. Thehybrid system must therefore satisfy several requirements: (1) a proper absorption range for thedye, (2) a fast charge transfer in the oxide, (3) a slow back-transfer process, and (4) an easycollection and conduction of electrons in the oxide.While absorption properties may be easily tuned at the chemical level by changing or addingfunctional groups, a critical point is to understand, and therefore control, the process at the in-terface. Indeed, simply having a good energy level alignment is not sufficient because the fastelectron injection process is dynamical. Since the experimental characterization of complex sys-tems (networks of nanostructures, with adsorbed dyes, and in solution) is quite complicated, thetheoretical description of such systems can be of fundamental importance in unravelling the pro-cesses governing the behaviour of dye-sensitized solar cells (DSSC).The most popular technique for studying these dynamical processes is time dependent DFT(TDDFT). For a detailed discussion of the success and possible limitations of this method whenapplied to hybrid systems, we refer the reader to Ref. . TDDFT is a generalization of DFT whichallows us to directly describe excited states. For example, it has previously been used successfully23o calculate the optical absorption of large organic molecules (8), such as Ru-dyes, or indolines.This same technique has also been applied recently to hybrid systems used for photovoltaic ap-plications. We want here to highlight that the charge injection transfer has also been modelledfor molecules adsorbed on TiO clusters , giving estimations as fast as 8 fs for the injectiontime . TDDFT is therefore a powerful tool to understand dynamical electronic effects in systemsas large and complex as hybrid organic-oxide solar cells. VIII. CONCLUSION
In this chapter we showed how an oxide of predominant importance in nanotechnological andenvironmental fields, such as photocatalysis and photovoltaics, can be successfully investigatesby state-of-the-art ab initio techniques. The optical and excitonic properties of the bulk phasescan be properly described by MBPT techniques. We also showed how the electronic propertiesof TiO may be “tailored” using nanostructural changes in combination with boron and nitrogendoping. While boron doping tends to produce smaller band gap n -type semiconductors, nitrogendoping produces p -type or n -type semiconductors depending on whether or not nearby oxygenatoms occupy surface sites. This suggests that a p -type TiO semiconductor may be producedusing nitrogen doping in conjunction with surface confinement at the nanoscale. We also showedthat it has been proved how, in the field of photovoltaics, TDDFT is a powerful tool to understandthe mechanism of charge injection at organic-inorganic interfaces. Acknowledgments
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