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Semiconducting transition metal oxides

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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 283203 (18pp)

doi:10.1088/0953-8984/27/28/283203

Topical Review

Semiconducting transition metal oxides Stephan Lany National Renewable Energy Laboratory, Golden, CO 80401, USA E-mail: [email protected] Received 21 November 2014, revised 27 February 2015 Accepted for publication 14 April 2015 Published 30 June 2015 Abstract

Open shell transition metal oxides are usually described as Mott or charge transfer insulators, which are often viewed as being disparate from semiconductors. Based on the premise that the presence of a correlated gap and semiconductivity are not mutually exclusive, this work reviews electronic structure calculations on the binary 3d oxides, so to distill trends and design principles for semiconducting transition metal oxides. This class of materials possesses the potential for discovery, design, and development of novel functional semiconducting compounds, e.g. for energy applications. In order to place the 3d orbitals and the sp bands into an integrated picture, band structure calculations should treat both contributions on the same footing and, at the same time, account fully for electron correlation in the 3d shell. Fundamentally, this is a rather daunting task for electronic structure calculations, but quasiparticle energy calculations in GW approximation offer a viable approach for band structure predictions in these materials. Compared to conventional semiconductors, the inherent multivalent nature of transition metal cations is more likely to cause undesirable localization of electron or hole carriers. Therefore, a quantitative prediction of the carrier self-trapping energy is essential for the assessing the semiconducting properties and to determine whether the transport mechanism is a band-like large-polaron conduction or a small-polaron hopping conduction. An overview is given for the binary 3d oxides on how the hybridization between the 3d crystal field symmetries with the O-p orbitals of the ligands affects the effective masses and the likelihood of electron and hole self-trapping, identifying those situations where small masses and band-like conduction are more likely to be expected. The review concludes with an illustration of the implications of the increased electronic complexity of transition metal cations on the defect physics and doping, using as an example the diversity of possible atomic and magnetic configurations of the O vacancy in TiO2, and the high levels of hole doping in Co2ZnO4 due to a self-doping mechanism that originates from the multivalence of Co. Keywords: transition metal oxides, electronic structure calculation, band structure, polarons, defects (Some figures may appear in colour only in the online journal)

I. Introduction

most importantly light emitting diodes (LED) and laser diodes, whereas non-oxide II–VI semiconductors are found in niche applications, such as infrared detectors and thin-film solar cells. Wide-gap oxides play important roles as secondary device components, such as gate dielectrics, transparent conducting oxides (TCO), and in transparent thin-film transistors (TFT). In conventional semiconductor systems, i.e. the elemental (group 14), III–V, and II–VI systems, the origin of the band

Semiconductivity is arguably the most transformative materials property of the 20th century, as it enabled electronics and optoelectronics. Still, the materials base for semiconductor applications is remarkably small, with Si being by far the most used ‘workhorse’ semiconductor for microelectronics and solar cells. III–V compounds dominate optoelectronic applications, 0953-8984/15/283203+18$33.00

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© 2015 IOP Publishing Ltd  Printed in the UK

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J. Phys.: Condens. Matter 27 (2015) 283203

Figure 1.  Schematic model illustrating how the valence and conduction bands are formed by the hybridization between cation and anion atomic orbitals (s and p), opening a band gap in (a) elemental, (b) III–V, and (c) and II–VI semiconductors. The mixing of additional orbitals (e.g. transition metal d) is illustrated in (d).

gap follows from the octet rule [1], i.e. the electron count of 8 for closed-shell configurations based on the atomic s and p orbitals, as schematically illustrated in figure 1 (note that for simplicity, the energy difference between s and p orbitals is not resolved). As a general trend, the band gaps increase with ionicity due to the increased energy difference between cation and anion orbitals, which also implies that splitting of the dangling bond orbitals increases, moving them closer to the band edges. The presence of additional interactions with occupied or unoccupied orbitals (e.g. d-orbitals of transition metals) gives rise to secondary bonding–antibonding interactions, which, in favorable cases, can remove the dangling bonds from the band gap energy window, as illustrated in figure 1(d). Such situations support defect-tolerance, since lattice defects (e.g. point defects, grain boundaries) are less likely to create deep defect states inside the band gap [2]. The widening of the materials base for semiconductor applications is important in various aspects. For example, it is quite ironic that the lion share of solar cells are manufactured using Si as the active absorber material, given that Si has an indirect band gap and is notoriously defect intolerant, requiring defect densities below 1014–1016 cm−3. It seems there should be a material waiting to be discovered that affords higher efficiencies at lower production cost. Another aspect is the availability of elements. Some technologically important elements, including In and Te, have low annual production rates [3], raising concerns about the scalability of semiconductor technologies based on such elements. From a materials design perspective, the goal is of course the optimization of the material properties (e.g. carrier mobilities, doping, optical properties), and to discover or design new materials for desirable functionalities for which no materials with satisfactory performance are available, (e.g. p-type TCO). However, also the interface with other components is of interest (e.g. bandoffsets, minimization of interface defects), and often there are constraints related to the device fabrication process (e.g. a low thermal budget for deposition so not to damage other components).

Particularly for the large-area applications as inherently needed for solar energy conversion, chemically stable materials are desirable that are amenable to large-scale low-cost production processes. Oxides, many of which occur as rockforming minerals, are a class of compounds with such properties, forming readily and with high chemical stability. As semiconductors, however, only few oxides are used so far, i.e. mostly those of Zn, Ga, In, and Sn, as transparent n-type conducting layers in the aforementioned application in TFT and as TCO [4]. These oxides are difficult, if not impossible, to dope p-type, and other main group oxides like MgO or Al2O3 have very large gaps and are insulators. These considerations point toward transition metal (TM) oxides as a potential materials class of interest for exploration of their semiconducting properties. Current areas of interest for developing novel semiconducting transition metal oxides include ternary oxides [5–11], transition metal alloys [12–14], and the deliberate use of non-equilibrium deposition methods to access compositions [13, 14] or disordered configurations [15] outside the thermodynamically accessible phase space. The aim of the present review is to aid the design and discovery of novel oxides by distilling trends and design principles from available data on the binary 3d oxides. II.  Electronic structure of transition metal oxides II.1.  Mott and charge-transfer insulators versus semiconductivity

While TiO2 and Cu2O as the d0 and d10 end points of the series of 3d transition metal oxides are well known n-type and p-type semiconductors, respectively, the oxides with partially occupied d-shells are generally described as Mott insulators. Perhaps due to the notion that semiconductors are band insulators at low temperature, Mott insulators [16] are often viewed as being disparate from semiconductors, as the latter are associated with band theory and the former being defined by the breakdown thereof. The on-site d–d Coulomb 2

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and exchange interaction U and the charge transfer energy Δ are used to describe the trends of the band gaps [17], i.e. the energy needed to create separated, non-interacting electron– hole pairs. NiO is often used as a prototypical example where this excitation is assumed to correspond either to the reaction 2Ni+IIO−II  →  Ni+IO−II + Ni+IIIO−II (case of Mott insulator) or to Ni+IIO−II  →  Ni+IO−I (case of charge transfer insulator). Of course, either one of these descriptions implies a localization of electrons (Ni+I) and holes (Ni+III or O−I) on an atomic scale, thereby discounting the band picture from the outset. Nevertheless, NiO can exhibit good p-type conduction [18], which makes it attractive as a transparent hole-transport layer, e.g. in organic photovoltaics [19]. On the other hand, a band structure theory that fully accounts for the electron–electron interaction no longer suffers from the shortcomings of the original band theory and is able to describe a band gap irrespective of the electron number. In this case, the s, p, and d orbitals are treated on the same footing and the band gap is eventually determined both by on-site correlations of the d orbitals (including the Coulomb splitting between occupied and unoccupied states and the exchange splitting between the two spin channels), and by the band-energies and band-dispersions arising from the interactions between all orbitals. As with all insulators, Mott and charge transfer insulators have a well-defined valence band maximum (VBM) and conduction band minimum (CBM) with their respective carrier effective masses for band transport. As with all semiconductors, there are additional requirements to be met for useful semiconducting properties, e.g. effective masses, dopability, and a low propensity for defect formation. However, more than conventional semiconductors, TM oxides are prone to localize electron or hole carriers, in which case transport occurs via a small-polaron-hopping instead of a band-conduction mechanism [20]. We will devote closer attention to this important issue below.

The quasi-particle energy calculations presented here were performed using the PAW implementation of the GW approximation [41] (vasp version 5.3), with only slightly modified computational settings compared to [40]. Preceding the GW calculations, and initial density functional theory (DFT) calculation was performed using the generalized gradient approximation (GGA) [42] with a Coulomb parameter [43] U =3 eV for all 3d cations except Cu, for which U =5 eV was used. The GGA + U wavefunctions were kept, and the self-consistent band energies were obtained via 5 consecutive GW iterations. Apart from the inclusion of DFT derived local field effects [44], the response functions were calculated in the random phase approximation. The energy cutoff was 330 eV for the wavefunctions and 200 eV for response functions. A total number of 64  ×  nat was used, where nat is the number of atoms in the unit cell. The k-point density was set to approximately 4000/nat. No attempt was made in the present work to resolve exciton energies and internal d–d transitions; note that the oscillator strength of these excitations is usually small compared to direct-allowed band-to-band transitions. The approach of [40] utilizes an on-site potential Vd for the 3d orbitals to compensate for the tendency of this GW scheme to overestimate the 3d orbital energies compared to the sp bands (Vd = −1.3,−2.8,−4.0, 0,−0.9,−1.2,−0.7,−2.4 eV for the 3d cations Ti, V, Cr, Mn, Fe, Co, Ni, Cu, respectively). The slower convergence behavior of the 3d orbitals compared to the sp orbitals [28] seems to be a main factor for necessitating the use of the empirically derived Vd potential. The present approach was constructed as a compromise providing a feasible route for GW calculations for a wide range of materials within a single scheme. Results for a larger number of materials (few hundred) are included in a database that is open to public access [45]. Table 1 shows the calculated band gaps in comparison to experimental data. (Please see [40] and references therein for the survey of the experimental data. The gap of anatase TiO2 was determined in [46].) In order to characterize the band dispersion and the expected band transport properties, table 1 lists also the ‘density-of-states’ (DOS) effective masses [47] calculated, e.g. for electrons, as

II.2.  Band structure calculations

Band gap and band structure predictions for semiconductors are usually performed by means of quasi-particle energy calculations based on the GW approximation [21]. For main group semiconductors, the reliability of such calculations is now well established [22, 23, 24]. However, the importance of d-orbitals for the band-structure of transition metal compounds poses additional challenges for such calculations in regard of convergence parameters (which are already rather critical for conventional semiconductors), particularly the number of bands included for the dielectric response functions [25, 26, 27] and basis set completeness [28]. Apart from convergence issues, further extensions of the GW approach are active research topics, including vertex corrections in the self-energy [29] and the relevance of ionic contributions to the dielectric screening [30]. Accordingly, band gap predictions for transition metal oxides are currently only emerging [31–38], and ad hoc corrections for d-states are sometimes necessary to obtain reliable band gaps, particularly if the goal is to employ a uniform, computationally efficient scheme for a wide range of different compounds [39, 40].

⎛ ⎞2/3 ⎛ eCBM − en, k ⎞⎟ m e* c0 ⎜ 1 = wk exp⎜ ⎟ , (1) ⎝ ⎠⎟ kBT m0 T ⎜⎝ Ω ⎠ n, k

∑

where m0 is the rest mass of the free electron, c0 = 3.50  ×  10−11 K cm2 is a constant, and Ω is the unit cell volume. The summation is over the conduction band states n and the k-points in the irreducible wedge of the Brillouin zone, and wk are the weights of the respective k-points. Note that since the density-of-states effective mass does not account for band degeneracies, it is larger than the respective band effective masses if the band edge states are degenerate in n or k or both. The numbers given in table  1 were determined using T = 1000 K. Due to the finite k-point density, the discrete summation used here tends to overestimate the masses somewhat, particularly for smaller masses (m */m 0   E rel. In the Cu2O conduction band there are no states with high local DOS, i.e. electrons should exhibit band-like transport as well, as observed experimentally [50].

in [57] found that the tradeoff between Eloc and E rel favors the small polaron state for holes in the rock-salt ground state of MnO, in agreement with the conclusion of experimental characterization [20]. However, these calculations also predicted that the desirable band-like hole conduction would prevail in the hypothetical tetrahedral zinc-blende phase [57]. The favorable hole carrier collection, observed in solar absorbers where the tetrahedral structure was stabilized by alloying of MnO with ZnO [13], corroborates this prediction. • In FeO, the unoccupied minority-spin Fe-d states are lower in energy than in MnO, but still separated from the CBM by a significant energy difference. Hence, electron self-trapping in FeO is unlikely. The valence band top is formed by the Jahn–Teller split Fe-t21− state, which is an open-shell configuration. It is a non-bonding, high-DOS state like the valence band in Cr2O3, suggesting that it could be a heavy mass band conductor. However, its open-shell character might cause instability with respect to the QCS Fe+III configuration, which would imply a self-trapped hole. The latter conclusion is supported by temperature dependent measurements of the electrical properties [104]. • Fe2O3 is a prototypical small polaron conductor for electrons [20, 66, 67], and this feature is also borne out by the calculations of [57], predicting EST = −0.2 eV utilizing the electron state potential and the generalized Koopmans condition discussed above. Here, the Fe-t2− states cause a very steep DOS onset in the conduction band which is reflected in the large electron effective mass (table 1), causing a very small Eloc of less than 0.1 eV, so that even a modest relaxation energy of  −0.3 eV stabilizes the self-trapped electron state. For the hole conduction mechanism, the situation is a bit more complex: The occupied Fe-d +5 states are far too low in energy to support a Fe+IV hole polaron state (figure 2). However, the hole-trapping tendency of O-p states in Fe2O3 is somewhat ambivalent, as they are subject to both an anti-bonding type interaction with the occupied Fe-d+ states and a bonding-type interaction with the unoccupied Fe-d− states. The quantitative analysis of [57] determined that O-I holes are slightly unstable relative to the band like state. • In the rs ground state of CoO, the sharply peaked unoccupied state of the Jahn–Teller split Co-t2 manifold suggests a small Eloc, which likely causes electron-trapping by forming the t23− QCS configuration. However, in the tetrahedrally coordinated zb (or wurtzite) structure, the unoccupied Co-d states lie high enough above the s-like CBM to support band like conductivity with a small effective mass (table 1). Hole trapping in rs-CoO is more difficult to estimate from the density of states (figure 2) alone, since the resulting d +5 t21− high spin state is possibly unstable relative to the d6 low spin state that forms a QCS configuration. In this case, a magnetic energy contribution would enter the self-trapping energy. In their seminal work [20], Bosman and van Daal concluded that CoO is a band conductor for holes. This may also be the

IV.  Defects and doping Defects and impurities are of central importance for semiconductors [105] in their role as shallow dopants that introduce charge carriers [106], as compensating defects [107], which often limit the dopability [108], and as deep centers that act as carrier traps and recombination centers. While, in general terms, these issues are of course of interest also for semiconducting transition metal oxides, there are two aspects that deserve special attention, i.e. the increased complexity of point defects due to the magnetism of the TM cations and the interaction with small polaron states, and the implications of the multivalence of transition metal cations on doping. These two aspects are illustrated in the following by examples in TiO2 and Co2ZnO4, respectively. IV.1.  The complex nature of O vacancies in rutile TiO2

Anion vacancies in compound semiconductors exhibit peculiar large lattice relaxation effects [109], which can cause persistent photoconductivity and metastabilities in the electrical behavior [110, 111]. Before discussing the more complex case 11

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of TiO2, it is helpful to recapitulate the phenomenology of O vacancy (VO) defects in main group oxides, such as ZnO. In transparent conducting oxides, like ZnO, the donor level is often deep inside the band gap [111, 112], but it can come closer to the CBM in some cases, e.g. in In2O3 [113, 114], so that thermal ionization can contribute to the n-type conductivity. The gap state is formed by the symmetric combination of the cation dangling bond orbitals, e.g. an a1 symmetric defect state formed by Zn orbitals [115]. This state can be identified as a metal–metal bond [110], leading to an inward relaxation pattern when it is doubly occupied (charge neutral vacancy) and to an outward relaxation in the fully ionized 2+ charge state. In ZnO, the empty a1 defect state of VO2+ shifts to energies above the CBM, giving rise to a metastable shallow donor (effective-mass like) state, which is caused by the Coulomb potential of the charged vacancy attracting free electrons in the conduction band, and which can be characterized as VO2+ + 2e [111]. In TiO2, the oxygen vacancy defect state is created predominantly by the Ti-d orbitals of the ligands. Therefore, the electronic behavior of the VO defect is strongly influenced by the propensity of Ti to assume a Ti+III oxidation state when the Fermi level lies close to the CBM of TiO2. Similarly as in case of the small polaron state of the free electron discussed above, the description of the VO defect state depends on the ability of the functional to describe the localized and spinpolarized Ti+III state. However, DFT + U and hybrid functional approaches [85–121] draw a fairly heterogeneous picture, where the results vary with details in the approach (particularly, the value of U and hybrid functional parameters), and where a range of different atomic and magnetic configurations can be observed in the calculation. Recent EPR and electron–nuclear double resonance (ENDOR) experiments in rutile TiO2 have described the singly charged [122] and the neutral VO defect [123] as one and two Ti+III states, respectively, next to the defect site, where the two electrons at the neutral vacancy are observed in a triplet (S = 1) state [123, 124, 125]. From point of view of the defect model for O vacancies in main group chalcogenides, this behavior is entirely unexpected: First, in case of the singly charged vacancy, the expected symmetric a11 defect state would have equal amplitude at each of the equivalent nearest neighbors, as observed, e.g. in ZnS [126]. Second, in case of the neutral vacancy, the singlet a12 state is expected to be the ground state, with the triplet a11+t21+ state being much higher in energy (see, e.g. discussion in [111] for ZnO). Several previous DFT + U and hybrid functional calculations described the VO+ defect state as a symmetric state with equal amplitude at two equivalent Ti ligands [90, 118, 120, 121], as seen in figure 7 [90], showing the spin-density isosurface for this defect as obtained by the same Koopmans corrected and empirically band gap corrected functional described above in the context of the polaron states. A computational solution corresponding to the EPR result of a single Ti+III ion can be obtained in hybrid functional calculations when using a different atomic geometry [92], i.e. the structure of the doubly ionized vacancy VO2+ where the distance between

Figure 7.  Isosurface-density plot for the singly charged (a) + (b) and charge neutral (c)–(f) oxygen vacancy in different configurations. Shown is the spin-density, except for the non-spinpolarized case (f), where the electron density of the VO singleparticle defect state is shown instead. In (f), the two spin directions are distinguished by color. The vacancy site is marked by a square.

the two Ti neighbors is about 0.3 Å larger. In this case, the singly charged vacancy can be viewed a complex between VO2+ and a Ti+III electron-polaron [92]. Similar complexes between the O vacancy and small polarons have also been described in [91]. Analogously, the charge neutral O vacancy can assume configurations akin to VO2+ with two bound polarons, whose spin can be either parallel (S = 1) or antiparallel (S = 0) [91, 92]. While the experiments have characterized the two EPR active charge states in great detail, there are still open questions about the ground state configuration. The term ‘ground state’ was used somewhat loosely in [122, 123], and it was pointed out that the experiments could not conclusively determine whether the observed triplet (S = 1) or the singlet (S = 0) state of VO0 would be lower in energy. Another complication comes in due to the existence of energy barriers, which could lead to the observation of metastable states. As predicted by Landau early on [62], the small polaron can localize only after 12

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creation of a lattice distortion. An energy barrier is created because the molecular orbital in which the carrier localizes lies initially resonant inside the band continuum (see figure 4), and a minimum distortion is required to create a corresponding quasi-particle state inside the band gap [62]. For the electron small-polaron in rutile, the existence of such a barrier was verified in the calculations of [92]. In case of carrier trapping at defects, there are now two distinct possibilities [77]: Either, the defect in the initial configuration (before trapping) already creates a localized state inside the gap, in which case the trapping is barrierless, or, it forms a resonant state inside the band continuum, in which case the trapping involves an energy barrier, in complete analogy to the self-trapping process of small polarons. In wide-gap semiconductors, both situations have been identified before: Holes localize at NO defect in ZnO spontaneously, but there is an activated hole capture process with a barrier in the case of the LiZn and MgGa acceptors in ZnO and GaN, respectively [77, 127]. If, in the initial configuration, the defect state lies very deep inside the band continuum, the barrier can be large and sustain persistent photoconductivity, as proposed for VO in ZnO [111]. Previous hybrid functional calculations in rutile TiO2 [118] have found that the position of the a1 defect state of VO varies with the different charge states, lying inside the gap for VO0, but above the CBM for VO2+. Similarly as described before for VO in ZnO [111], this upward shift of the defect state is a result of the increased distance between the cations next to the vacancy. Since, the VO2+ state is the initial state before photo-excitation of electrons in the EPR experiments, and since these experiments are performed at low temperature [122, 123], the possibility is given that a metastable state is observed. Figure 7 shows a range of different possible configurations of the EPR active q  =  1+ and q  =  0 charge states of VO in rutile TiO2, and the respective total energies are given in table  2. For the singly charged vacancy, figure 7(a) shows the ‘classic’ defect configuration, where the electron occupies the a1 defect state formed by symmetric combination of the dangling bond orbitals. However, this state lies higher in energy than the case of a Ti+III electron polaron bound to the vacancy, shown in figure 7(b) (where the polaron is denoted ‘ep’ for distinction from a band-electron). In this configuration, the vacancy has an atomic structure with increased Ti–Ti distance akin to the doubly charged vacancy and, hence, can be described as ‘VO2+ + ep’, [92]. This configuration is observed in the EPR experiment for the singly charged vacancy [122]. In the charge neutral state, where two electrons are associated with the vacancy, the symmetric a12 state (figure 7(c)) is also unstable. The Koopmans corrected DFT functional finds a ‘mixed’ configuration lowest in energy, which can be described as ‘a11 + ep’, i.e. one electron in the a1 defect state and one defect-bound Ti+III polaron (figure 6(d)). The configuration observed in the EPR experiments [123, 125], described as ‘VO2+ + 2ep’ [92], is shown in figure 7(e), and lies just 40 meV higher in energy. However, as discussed above, the electronic structure of VO2+ suggests that there is an energy barrier for trapping of electrons into the a1 defect state. It is, therefore, entirely possible that the configuration observed in

Table 2.  Results of Koopmans-corrected and empirically band gap

corrected DFT calculations in rutile TiO2 [90], i.e. the band gap, the electron self-trapping energy EST, the energies Econfig of the different oxygen vacancy configurations relative to the ground state (gs) for the respective charge state (see figure 7), and the ionization energies of VO with respect to the free Ti+III polaron.

Eg EST (ep) Econfig   VO+ (a11); S = 1/2 S =1/2   VO+ (ep);   VO0 (a12);   VO0 (a11  +  ep); S =1   VO0 (2ep); S =1 S =0   VO0 (2ep); Ei1 (VO0  →  VO+  +  ep) Ei2 (VO+  →  VO2+  +  ep)

units

value

eV eV

3.02 −  0.41

meV meV meV meV meV meV eV eV

66 gs 302 gs 42 41 0.23 0.48

EPR is only a metastable state, created by the particular way the experiments is executed, i.e. by photogeneration of electrons in an oxidized sample where VO2+ is the initial state due to compensation by other impurities. Since the lowest energy ‘mixed’ configuration (figure 7(d)) is also a triplet state, it would be interesting to see experiments in less oxidized samples, where uncompensated neutral VO defects should exist even before illumination. Finally, the singlet state, in which the spins of the two Ti+III polarons are in an anti-parallel alignment (figure 7(f)), is essentially degenerate in energy with the respective triplet state, a finding that is consistent with experiment [123]. Table 2 further lists the first and second ionization energies with respect to the free Ti+III polaron, i.e. the enthalpies of the reactions VO0  →  VO+ + ep and VO+  →  VO2+ + ep, respectively. In view of free carrier activation, it is interesting to note that the site-density for small polarons (e.g. 3   ×   1022 cm−3 Ti sites in TiO2) is much larger than the effective density of states for band-electrons (typically NC = 1018–1019 cm−3). Therefore, for a given ionization energy, dopants release carriers much more readily in polaron conductors than in band conductors. From the values listed in table  2, one can expect that most O vacancies are at least singly ionized at room temperature in TiO2. IV.2.  Self-doping of Co2ZnO4 due to Co multivalence

In traditional semiconductors, impurity substitution with elements of a higher oxidation number than the replaced host atom usually creates an electron donor (n-type dopant), whereas substitution with elements in a lower oxidation state creates an acceptor (p-type dopant). However, the situation becomes more intricate when the dopants are multivalent, in which case they can change their oxidation state depending on the position of the Fermi level. This problem was recently addressed in SnO2 doped with group 15 elements (P, As, Sb, Bi) [128], where it was found that BiSn pins the Fermi level 13

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inside the gap rather than creating free electrons. This pinning occurs at the transition level between a positively charged Bi+Sn donor (Bi+IV oxidation state) and a negatively charged Bi−Sb acceptor (Bi+III state). For Sb dopants, this transition occurs above the CBM, explaining why Sb works well as a donor dopant in SnO2, but not Bi [129]. In transition metal compounds, the multivalent nature of the cations inherently enters the picture for the description, or the prediction, of doping and the electrical properties. As an illuminating example, we are discussing here the example of Co2ZnO4 spinels (also often written as ZnCo2O4), which receive increasing interest as p-type oxide [7] in device applications such as diodes and transistors [130, 131] (where, however, Co2ZnO4 is deposited in an amorphous structure), as hole-transport layers in sensitized solar cells and organic photovoltaics [132], or as anode material for Li-ion batteries [133]. The spinel structure has two octahedral and a tetrahedral cation lattice sites per formula unit. At the ideal stoichiometry, Co occupies the Oh sites and Zn occupies the Td sites, and Co+III assumes a low-spin t6 quasi-closed-shell configuration, leading to a sizable band gap (Eg = 2.29 eV determined by the GW approach of [40], see table 1). Without multivalent elements, an excess of the trivalent cation would normally cause an electron donor in III–II spinels, e.g. GaZn in Ga2ZnO4. In Co2ZnO4, however, Co changes the oxidation state to Co+II on the Td site, so to assume a d5e−2 QCS configuration (see section II), with the net effect that Co-rich stoichiometries stay more or less intrinsic semiconductors. In contrast, an excess of Zn creates ZnCo acceptors on the Oh site and causes p-type doping [10], since Zn remains divalent on the Oh site. The different possible tendencies of the cation to change the oxidation state when changing the lattice site from Oh to Td or vice versa has led to the classification of doping types for A2BO4 spinel oxides [11], illustrated in figure  8. In the ideal (normal) spinel structure, the A cations occupy the Oh sites and the B cations the Td sites. In conventional spinels (type 1), where both cations maintain their oxidation number irrespective of the coordination symmetry, a pair of ATd and BOh antisite defects creates oppositely charged, mutually compensating donor and acceptor defects, but no net doping. In type 2, with Co2ZnO4 being an example, the ATd defect is electrically inactive, but BOh forms an acceptor. Type 3 is analogous to type 2, but for the case of net n-type doping due to an electrically active ATd donor but an inactive BOh defect. In type 4, both cations remain inactive when changing the site, leading to no net doping. A very interesting aspect of the doping type 2 of Co2ZnO4 is that a high hole density can be induced just by swapping the cations between the two sites, but without changing the stoichiometry. This feature enables the control of the electrical properties by manipulating the cation disorder, e.g. by deliberate non-equilibrium growth [15].

Figure 8.  Doping types for III–II spinels A2BO4: In type 1, ATd and

BOh antisites are mutually compensating. In type 2, ATd defects are electrically inactive, and BOh acceptors cause p-type doping. In type 3, BOh defects are electrically inactive, and ATd donors cause n-type doping. In type 4, both ATd and BOh defects are inactive, leading to intrinsic behavior. (Modified after [11]).

can be semiconductors, offering a rich class of materials for exploration, design and discovery of novel functional semiconductors. Compared to conventional semiconducting materials, there are more challenges and obstacles to overcome, but an increasingly predictive theoretical description can guide the design process. In particular, this review discussed the band structure properties with emphasis on the spectral intensities of the density of states of the 3d orbitals within the band continuum. An electronic structure theory that correctly accounts for correlation can seamlessly integrate the occupied and unoccupied d symmetries into the familiar band-structure picture for sp semiconductors. Although still facing significant challenges, GW quasiparticle energy calculations seem to be a promising route for this task. Based on the trends distilled from such calculations, special attention is devoted to ‘quasi-closed-shell’ configurations, where the crystal-field and spin-resolved orbital symmetries are either fully occupied or empty. The band-structure properties exhibit systematic trends due to the hybridization between the 3d and the O-p states, affecting both carrier effective masses and the likelihood of carrier self-trapping, the latter of which causing an usually undesirable small-polaron transport mechanism. A high 3d density of states close to the conduction band or valence band extrema should be avoided, because it tends to cause large effective masses and to support self-trapping. Smaller hole effective masses are often observed when the VBM is formed by the anti-bonding interaction between the O-p states and the occupied d-sublevels for which the interaction is allowed by symmetry, examples being Mn+II (d5) and Cu+I(d10). In cases

V.  Summary and conclusions Open-shell transition metal oxides are traditionally described as Mott or charge-transfer insulators, but they nevertheless 14

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[3] 2014 US Geological Survey, Mineral Commodity Summaries [4] Wager J F, Keszler D A and R.E Presley 2008 Transparent Electronics (New York: Springer) [5] Nagarajan R, Draeseke A, Sleight A and Tate J 2001 p-type conductivity in CuCr1−xMgxO2 films and powders J. Appl. Phys. 89 8022 [6] Scanlon D O and Watson G W 2011 Understanding the p-type defect chemistry of CuCrO2 J. Mater. Chem. 21 3655 [7] Trimarchi G, Peng H, Im J, Freeman A J, Cloet V, Raw A, Poeppelmeier K R, Biswas K, Lany S and Zunger A 2011 Using design principles to systematically plan the synthesis of candidate hole-conducting transparent oxides: Cu3VO4 and Ag3VO4 as a case study Phys. Rev. B 84 165116 [8] Peng H et al 2013 Li-doped Cr2MnO4: a new p-type transparent conducting oxide by computational materials design Adv. Funct. Mater. 23 5267 [9] Dekkers M, Rijnders G and Blank D H A 2007 ZnIr2O4, a p-type transparent oxide semiconductor in the class of spinel zinc-d6-transition metal oxide Appl. Phys. Lett. 90 021903 [10] Perkins J D et al 2011 Inverse design approach to hole doping in ternary oxides: enhancing p-type conductivity in cobalt oxide spinels Phys. Rev. B 84 205207 [11] Paudel T R, Zakutayev A, Lany S, d’Avezac M and Zunger A 2011 Doping rules and doping prototypes in A2BO4 spinel oxides 2011 Adv. Functional Mater. 21 4493 [12] Kanan D K and Carter E A 2012 Band gap engineering of MnO via ZnO alloying: a potential new visible-light photocatalyst J. Phys. Chem. C 116 9876 [13] Peng H, Ndione P F, Ginley D S, Zakutayev A and Lany S 2015 Design of semiconducting tetrahedral Mn1−x–ZnxO alloys and their application to solar water splitting Phys. Rev. X 5 021016 [14] Stevanovic V, Zakutayev A and Lany S 2014 Composition dependence of the band gap and doping in Cu2O-based alloys as predicted by an extension of the dilute-defect model Phys. Rev. Appl. 2 044005 [15] Ndione P F, Shi Y, Stevanovic V, Lany S, Zakutayev A, Parilla P A, Perkins J D, Berry J J, Ginley D S and Toney M F 2014 Control of the electrical properties in spinel oxides by manipulating the cation disorder Adv. Funct. Mater. 24 610 [16] Mott N F 1990 Metal–Insulator Transitions (London: Taylor and Francis) [17] Zaanen J, Sawatzky G A and Allen J W 1985 Band gaps and electronic structure of transition-metal compounds Phys. Rev. Lett. 55 418 [18] Sato H, Minami T, Takata S and Yamada T 1993 Transparent conducting p-type NiO thin films prepared by magnetron sputtering Thin Solid Films 236 27 [19] Steirer K X, Chesin J P, Widjonarko N E, Berry J J, Miedaner A, Ginley D S and Olson D C 2010 Solution deposited NiO thin-films as hole transport layers in organic photovoltaics Org. Electron. 11 1414 [20] Bosman A J and van Daal H J 1970 Small-polaron versus band conduction in some transition-metal oxides Adv. Phys. 19 1 [21] Hedin L 1965 New method for calculating the one-particle Green’s function with application to the electron-gas problem Phys. Rev. 139 A796 [22] van Schilfgaarde M, Kotani T and Faleev S 2006 Quasiparticle self-consistent GW theory Phys. Rev. Lett. 96 226402 [23] Fuchs F, Furthmüller J, Bechstedt F, Shishkin M and Kresse G 2007 Quasiparticle band structure based on a generalized Kohn–Sham scheme Phys. Rev. B 76 115109 [24] Gómez-Abal R, Li X, Scheffler M and Ambrosch-Draxl C 2008 Influence of the core-valence interaction and of the pseudopotential approximation on the electron self-energy in semiconductors Phys. Rev. Lett. 101 106404

where the VBM is formed by non-bonding d-symmetries, e.g. octahedral Cr+III or low-spin Co+III, the hole effective masses are rather large, but the non-bonding character reduces the atomic relaxation effect upon hole-trapping, which is the driving force for small polaron formation. Thus, such oxides can be band-conductors for holes, albeit with rather large effective masses. Small electron masses occur, particularly, when the unoccupied d-orbitals lie high in energy, so that the dispersive s-like conduction band akin to that of conventional semiconductors becomes exposed, e.g. in case of divalent (+II) states of Fe, Mn, Co. As illustrated for oxygen vacancies in TiO2, where the observed and calculated defect configurations strongly deviate from expectations derived from the established picture in main group compounds, the transition metal oxides can exhibit rich new defect physics involving complex magnetic configurations and defect-polaron interactions. As far as doping is concerned, transition metal oxides face the same issues as conventional semiconductors, i.e. the dopant solubility, the ionization energy, and compensation. In addition, the inherent multivalent character of transition metal cations should be considered, since the electrical properties can be strongly affected by changes of the oxidation number, depending on the lineup of the respective transition levels relative to the band edges. The potential challenges for semiconductivity, such as large effective masses, carrier self-trapping, and deep level defects, are maybe more pervasive in transition metal oxides than in conventional semiconductors. However, there is no fundamental difference, as all these effects have been observed also in main group compounds, particularly in widegap oxides and nitrides. Due to the complexity of transition metal oxides, the development of new functional materials will depend more than in the past on design aided by theory, computation, and modeling. Areas for exploration include ternary oxides, alloys, and the deliberate use of non-equilibrium deposition techniques to achieve desirable properties. Acknowledgments This work was supported by the US Department of Energy, Office of Science, Office of Basic Energy Sciences, as part of an Energy Frontier Research Center under contract No. DE-AC36-08GO28308 to NREL. SL thanks collaborators and colleagues for fruitful discussions and interactions on transition metal oxides, including E Arca, D S Ginley, T O Mason, A R Nagaraja, PF Ndione, T R Paudel, H Peng, J D Perkins, N H Perry, K R P Poeppelmeier, V Stevanovic, M F Toney, G Trimarchi, A Zakutayev, A Zunger. References [1] Pauling L 1960 The Nature of the Chemical Bond 3rd edn (Ithaca, NY: Cornell University Press) [2] Zakutayev A, Caskey C M, Fioretti A N, Ginley D S, Vidal J, Stevanovic V, Tea E and Lany S 2014 Defect tolerant semiconductors for solar energy conversion J. Phys. Chem. Lett. 5 1117 15

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[47] Ottaviani G, Reggiani L, Canali C, Nava F and AlberigiQuaranta A 1975 Hole drift velocity in silicon Phys. Rev. B 12 3318 [48] Hirose Y, Yamada N, Nakao S, Hitosugi T, Shimada T and Hasegawa T 2009 Large electron mass anisotropy in a d-electron-based transparent conducting oxide: Nb-doped anatase TiO2 epitaxial films Phys. Rev. B 79 165108 [49] Spear W E and Tannhauser D S 1973 Hole transport in pure NiO crystals Phys. Rev. B 7 831 [50] Hodby J W, Jenkins T E, Schwab C, Tamura H and Trivich D 1976 Cyclotron resonance of electrons and of holes in cuprous oxide, Cu2O J. Phys. C: Solid State Phys. 9 1429 [51] Miyahara S and Teranishi T 1968 Magnetic properties of FeS2 and CoS2 J. Appl. Phys. 39 896 [52] Redman M J and Steward E G 1962 Cobaltous oxide with the zinc blende/wurtzite-type crystal structure Nature 193 867 [53] Risbud A S, Snedeker L P, Elcombe M M, Cheetham A K and Seshadri R 2005 Wurtzite CoO Chem. Mater. 17 834 [54] Schrön A, Rödl C and Bechstedt F 2010 Energetic stability and magnetic properties of MnO in the rocksalt, wurtzite, and zinc-blende structures: influence of exchange and correlation Phys. Rev. B 82 165109 [55] Peng H and Lany S 2013 Polymorphic energy ordering of MgO, ZnO, GaN, and MnO within the random phase approximation Phys. Rev. B 87 174113 [56] Furubayashi Y, Hitosugi T, Yamamoto Y, Inaba K, Kinoda G, Hirose Y, Shimada T and Hasegawa T 2005 A transparent metal: Nb-doped anatase TiO2 Appl. Phys. Lett. 86 252101 [57] Peng H and Lany S 2012 Semiconducting transition-metal oxides based on d5 cations: theory for MnO and Fe2O3 Phys. Rev. B 85 201202(R) [58] Hautier G, Miglio A, Waroquiers D, Rignanese G-M and Gonze X 2014 How does chemistry influence electron effective mass in oxides? A high-throughput computational analysis Chem. Mater. 26 5447 [59] Huy H A, Aradi B, Frauenheim T and Deak P 2011 Calculation of carrier-concentration-dependent effective mass in Nb-doped anatase crystals of TiO2 Phys. Rev. 83 155201 [60] Weckhuysen B M and Schoonheydt R A 1994 Synthesis and chemistry of chromium in CrAPO-5 molecular sieves Zeolites 14 360 [61] Nie X, Wei S H and Zhang S B 2002 First-principles study of transparent p-type conductive SrCu2O2 and related compounds Phys. Rev. B 65 075111 [62] Landau L D 1933 Über die Bewegung der Elektronen im Kristallgitter Phys. Z. Sowjet. 3 644 [63] Fröhlich H 1954 Electrons in lattice fields Adv. Phys. 3 325 [64] Holstein T 1959 Studies of polaron motion: II. The ‘small’ polaron Ann. Phys. (NY) 8 343 [65] Mott N F and Stoneham A M 1977 The lifetime of electrons, holes and excitons before self-trapping J. Phys. C: Solid State Phys. 10 3391 [66] Gleitzer C, Nowotny J and Rekas M 1991 Surface and bulk electrical properties of the hematite phase Fe2O3 Appl. Phys. A 53 310 [67] Rosso K M, Smith D M A and Dupuis M 2003 An ab initio model of electron transport in hematite (α-Fe2O3) basal planes J. Chem. Phys. 118 6455 [68] Kanan D K and Carter E A 2013 Ab initio study of electron and hole transport in pure and doped MnO and MnO:ZnO alloyJ. Mater. Chem. A 1 9246 [69] Schirmer O F 2006 O− bound small polarons in oxide materials J. Phys.: Condens. Matter 18 R667 [70] Lægsgaard J and Stokbro K 2001 Hole trapping at Al impurities in silica: a challenge for density functional theories Phys. Rev. Lett. 86 2834 [71] Nolan M and Watson G W 2006 Hole localization in Al doped silica: a DFT + U description J. Chem. Phys. 125 144701

[25] Shih B C, Xue Y, Zhang P, Cohen M L and Louie S G 2010 Quasiparticle band gap of ZnO: high accuracy from the conventional G0W0 approach Phys. Rev. Lett. 105 146401 [26] Stankovski M, Antonius G, Waroquiers D, Miglio A, Dixit H, Sankaran K, Giantomassi M, Gonze X, Côté M and Rignanese G M 2011 G0W0 band gap of ZnO: effects of plasmon-pole models Phys. Rev. B 84 241201(R) [27] Friedrich C, Müller M C and Blügel S 2011 Band convergence and linearization error correction of all-electron GW calculations: the extreme case of zinc oxide Phys. Rev. B 83 081101(R) [28] Klimes J, Kaltak M and Kresse G 2014 Predictive GW calculations using plane waves and pseudopotentials Phys. Rev. B 90 075125 [29] Grüneis A, Kresse G, Hinuma Y and Oba F 2014 Ionization potentials of solids: The importance of vertex corrections Phys. Rev. Lett. 112 096401 [30] Bechstedt F, Seino K, Hahn P H and Schmidt W G 2005 Quasiparticle bands and optical spectra of highly ionic crystals: AlN and NaCl Phys. Rev. B 72 245114 [31] Faleev S V, van Schilfgaarde M and Kotani T 2004 Allelectron self-consistent GW approximation: application to Si, MnO, and NiO Phys. Rev. Lett. 93 126406 [32] Bruneval F, Vast N, Reining L, Izquierdo M, Sirotti F and Barrett N 2006 Exchange and correlation effects in electronic excitations of Cu2O Phys. Rev. Lett. 97 267601 [33] Rödl C, Fuchs F, J. Furthmüller and Bechstedt F 2009 Quasiparticle band structures of the antiferromagnetic transition-metal oxides MnO, FeO, CoO, and NiO Phys. Rev. B 79 235114 [34] Jiang H, Gomez-Abal R I, Rinke P and Scheffler M 2010 First-principles modeling of localized d states with the GW@LDA + U approach Phys. Rev. B 82 045108 [35] Chiodo L, García-Lastra J M, Iacomino A, Ossicini S, Zhao J, Petek H and Rubio A 2010 Self-energy and excitonic effects in the electronic and optical properties of TiO2 crystalline phases Phys. Rev. B 82 045207 [36] Kang W and Hybertsen M S 2010 Quasiparticle and optical properties of rutile and anatase TiO2 Phys. Rev. B 82 085203 [37] Rödl C and Schleife A 2013 Photoemission spectra and effective masses of n- and p-type oxide semiconductors from first principles: ZnO, CdO, SnO2, MnO, and NiO Phys. Status Solidi A 211 74 [38] Rödl C, Sottile F and Reining L 2015 Quasiparticle excitations in the photoemission spectrum of CuO from first principles: a GW study 2015 Phys. Rev. B 91 045102 [39] Berger R F and Neaton J B 2012 Computational design of low-band-gap double perovskites Phys. Rev. B 86 165211 [40] Lany S 2013 Band-structure calculations for the 3d transition metal oxides in GW Phys. Rev. B 87 085112 [41] Shishkin M and Kresse G 2006 Implementation and performance of the frequency-dependent GW method within the PAW framework Phys. Rev. B 74 035101 [42] Perdew J P, Burke K and Ernzerhof M 1996 Generalized gradient approximation made simple Phys. Rev. Lett. 77 3865 [43] Dudarev S L, Botton G A, Savrasov S Y, Humphreys C J and Sutton A P 1998 Electron-energy-loss spectra and the structural stability of nickel oxide: an LSDA + U study Phys. Rev. B 57 1505 [44] Paier J, Marsman M and Kresse G 2008 Dielectric properties and excitons for extended systems from hybrid functionals Phys. Rev. B 78 121201(R) [45] http://materials.nrel.gov [46] Forro L, Chauvet O, Emin D, Zuppiroli L, Berger H and Levy F 1994 High mobility n-type charge carriers in large single crystals of anatase (TiO2) J. Appl. Phys. 75 633 16

Topical Review

J. Phys.: Condens. Matter 27 (2015) 283203

[94] Yang S, Brant A T, Giles N C and Halliburton L E 2013 Intrinsic small polarons in rutile TiO2 Phys. Rev. B 87 125201 [95] Berger T, Sterrer M, Diwald O, Knözinger E, Panayotov D, Thompson T L and Yates J T 2005 Light-induced charge separation in anatase TiO2 particles J. Phys. Chem. B 109 6061 [96] Nagaraja A R, Perry N H, Mason T O, Tang Y, Grayson M, Paudel T R, Lany S and Zunger A 2012 Band or polaron: the hole conduction mechanism in the p-type spinel Rh2ZnO4 J. Am. Ceram. Soc. 95 269 [97] Varley J B, Janotti A, Franchini C and Van de Walle C G 2012 Role of self-trapping in luminescence and p-type conductivity of wide-band-gap oxides Phys. Rev. B 85 081109(R) [98] Lany S and Zunger A 2009 Accurate prediction of defect properties in density functional supercell calculations Modelling Simul. Mater. Sci. Eng. 17 084002 [99] Lany S, Raebiger H and Zunger A 2008 Magnetic interactions of Cr–Cr and Co–Co impurity pairs in ZnO within a band-gap corrected density-functional approach Phys. Rev. B 77 241201(R) [100] Heifets E N and Shluger A L 1992 The calculation of the self-trapping energy in crystals with mixed valence band J. Phys.: Condens. Matter. 4 8311 [101] Deak P, Aradi B and Frauenheim T 2011 Band lineup and charge carrier separation in mixed rutile-anatase systems J. Phys. Chem. C 115 3443 [102] Scanlon D O et al 2013 Band alignment of rutile and anatase TiO2 Nat. Mater. 12 798 [103] Arca E, Fleischer K and Shvets I V 2011 Magnesium, nitrogen codoped Cr2O3: a p-type transparent conducting oxide Appl. Phys. Lett. 99 111910 [104] Garstein E and Mason T O 1982 Reanalysis of wustite electrical properties Commun. Am. Ceram. Soc. 65 C-24 [105] Pantelides S T 1978 The electronic structure of impurities and other point defects in semiconductors Rev. Mod. Phys. 50 797 [106] Woodyard J R 1944 Nonlinear circuit device utilizing germanium US Patent No 2,530,110 (filed 2 June 1944, awarded 14 November 1950) [107] Mandel G 1964 Self-compensation limited conductivity in binary semiconductors: I. Theory Phys. Rev. 134 A1073 [108] Zhang S B 2002 The microscopic origin of the doping limits in semiconductors and wide-gap materials and recent developments in overcoming these limits: a review J. Phys.: Condens. Matter 14 R881 [109] Garcia A and Northrup J E 1995 Compensation of p-type doping in ZnSe: the role of impurity-native defect complexes Phys. Rev. Lett. 74 1131 [110] Lany S and Zunger A 2004 Metal-dimer atomic reconstruction leading to deep donor states of the anion vacancy in II–VI and chalcopyrite semiconductors Phys. Rev. Lett. 93 156404 [111] Lany S and Zunger A 2005 Anion vacancies as a source of persistent photoconductivity in II–VI and chalcopyrite semiconductors Phys. Rev. B 72 035215 [112] Oba F, Togo A, Tanaka I, Paier J and Kresse G 2008 Defect energetics in ZnO: a hybrid Hartree–Fock density functional study Phys. Rev. B 77 245202 [113] Agoston P, Albe K, Nieminen R M and Puska M J 2009 Intrinsic n-type behavior in transparent conducting oxides: a comparative hybrid-functional study of In2O3, SnO2, and ZnO Phys. Rev. Lett. 103 245501 [114] Lany S and Zunger A 2011 Comment on ‘Intrinsic n-type behavior in transparent conducting oxides: a comparative hybrid-functional study of In2O3, SnO2, and ZnO’ Phys. Rev. Lett. 106 069601

[72] To J, Sokol A A, French S A, Kaltsoyannis N and Catlow C R A 2005 Hole localization in AlO04 defects in silica materials J. Chem. Phys. 122 144704 [73] Becke A D 1993 Density-functional thermochemistry: III. The role of exact exchange J. Chem. Phys. 98 5648 [74] Perdew J P, Ernzerhof M and Burke K 1996 Rationale for mixing exact exchange with density functional approximations J. Chem. Phys. 105 9982 [75] Krukau A V, Vydrov O A, Izmaylov A F and Scuseria G E 2006 Influence of the exchange screening parameter on the performance of screened hybrid functionals J. Chem. Phys. 125 224106 [76] Lany S and Zunger A 2009 Polaronic hole localization and multiple hole binding of acceptors in oxide wide-gap semiconductors Phys. Rev. B 80 085202 [77] Lany S 2011 Predicting polaronic defect states by means of generalized Koopmans density functional calculations Phys. Status Solidi B 248 1052 [78] Nuttall R H D and Weil J A 1981 The magnetic properties of the oxygen-hole aluminum centers in crystalline SiO2: I. AlO04 Can. J. Phys. 59 1696 [79] Pacchioni G 2008 Modeling doped and defective oxides in catalysis with density functional theory methods: room for improvements J. Chem. Phys. 128 182505 [80] Lany S and Zunger A 2010 Generalized Koopmans density functional calculations reveal the deep acceptor state of NO in ZnO Phys. Rev. B 81 205209 [81] Mori-Sánchez P, Cohen A J and Yang W 2008 Localization and delocalization errors in density functional theory and implications for band-gap prediction Phys. Rev. Lett. 100 146401 [82] Perdew J P, Parr R G, Levy M and Balduz J L 1982 Densityfunctional theory for fractional particle number: derivative discontinuities of the energy Phys. Rev. Lett. 49 1691 [83] Perdew J P, Ruzsinszky A, Csonka G I, Vydrov O A, Scuseria G E, Staroverov V N and Tao J 2007 Exchange and correlation in open systems of fluctuating electron number Phys. Rev. A 76 040501(R) [84] Janak J F 1978 Proof that ∂E/∂ni = ei in density functional theory Phys. Rev. B 18 7165 [85] Morgan B J and Watson G W 2009 Polaronic trapping of electrons and holes by native defects in anatase TiO2 Phys. Rev. B 80 233102 [86] Deak P, Aradi B and Frauenheim T 2011 Polaronic effects in TiO2 calculated by the HSE06 hybrid functional: dopant passivation by carrier self-trapping Phys. Rev. B 83 155207 [87] Bogomolov V N and Mirlin D N 1968 Optical absorption by polarons in rutile (TiO2) single crystals Phys. Status Solidi 27 443 [88] Austin I G and Mott N F 2001 Polarons in crystalline and noncrystalline materials Adv. Phys. 50 757 [89] Deskins N A and Dupuis M 2007 Electron transport via polaron hopping in bulk TiO2: a density functional theory characterization Phys. Rev. B 75 195212 [90] Lany S 2010 Electron and hole trapping in rutile versus anatase TiO2 Int. CECAM Workshop ‘Titania for all seasons: Multifunctionality of an undercover semiconductor’ (Universität Bremen, 10–16 September 2010) [91] Deak P, Aradi B and Frauenheim T 2012 Quantitative theory of the oxygen vacancy and carrier self-trapping in bulk TiO2 Phys. Rev. B 86 195206 [92] Janotti A, Franchini C, Varley J B, Kresse G and Van de Walle C G 2013 Dual behavior of excess electrons in rutile TiO2 Phys. Status Solidi RRL 7 199 [93] Setvin M, Franchini C, Hao X, Schmid M, Janotti A, Kaltak M, Van de Walle C G, Kresse G and Diebold U 2014 Direct view at excess electrons in TiO2 rutile and anatase Phys. Rev. Lett. 113 086402 17

Topical Review

J. Phys.: Condens. Matter 27 (2015) 283203

[115] Watkins G D 1991 Electronic Structure and Properties of Semiconductors ed Schröter W (Weinheim: VCH) [116] Finazzi E, Di Valentin C, Pacchioni G and Selloni A 2008 Excess electron states in reduced bulk anatase TiO2: comparison of standard GGA, GGA + U, and hybrid DFT calculations J. Chem. Phys. 129 154113 [117] Di Valentin C, Pacchioni G and Selloni A 2009 Reduced and n-type doped TiO2: nature of Ti3+ species J. Phys. Chem. C 113 20543 [118] Janotti A, Varley J B, Rinke P, Umezawa N, Kresse G and Van de Walle C G 2010 Hybrid functional studies of the oxygen vacancy in TiO2 Phys. Rev. B 81 085212 [119] Stausholm-Møller J, Kristoffersen H H, Hinnemann B, Madsen G K H and Hammer B 2010 DFT + U study of defects in bulk rutile TiO2 J. Chem. Phys. 133 144708 [120] Park S G, Magyari-Köpe B and Nishi Y 2010 Electronic correlation effects in reduced rutile TiO2 within the LDA + U method Phys. Rev. B 82 115109 [121] Lee H Y, Clark S J and Robertson J 2012 Calculation of point defects in rutile TiO2 by the screened-exchange hybrid functional Phys. Rev. B 86 075209 [122] Brant A T, Giles N C, Yang S, Sarker M A R, Watauchi S, Nagao M, Tanaka I, Tryk D A, Manivannan A and Halliburton L E 2013 Ground state of the singly ionized oxygen vacancy in rutile TiO2 J. Appl. Phys. 114 113702 [123] Brant A T et al 2014 Triplet ground state of the neutral oxygen-vacancy donor in rutile TiO2 Phys. Rev. B 89 115206 [124] Yang S, Halliburton L E, Manivannan A, Bunton P H, Baker D B, Klemm M, Horn S and Fujishima A 2009

Photoinduced electron paramagnetic resonance study of electron traps in TiO2 crystals: oxygen vacancies and Ti3+ ions Appl. Phys. Lett. 94 162114 [125] Brandão F D, Pinheiro M V B, Ribeiro G M, MedeirosRibeiro G and Krambrock K 2009 Identification of two light-induced charge states of the oxygen vacancy in single-crystalline rutile TiO2 Phys. Rev. B 80 235204 [126] Schneider J and Räuber A 1967 Electron spin resonance of the F-center in ZnS Solid State Commun. 5 779 [127] Lany S and Zunger A 2010 Dual nature of acceptors in GaN and ZnO: the curious case of the shallow MgGa deep state Appl. Phys. Lett. 96 142114 [128] Peng H, Perkins J D and Lany S 2014 Multivalency of group 15 dopants in SnO2 Chem. Mater. 26 4876 [129] Rastomjee C S, Dale R S, Schaffer R J, Jones F H, Egdell R G, Georgiadis G C, Lee M J, Tate T J and Cao L L 1966 An investigation of doping of SnO2 by ion implantation and application of ion-implanted films as gas sensors Thin Solid Films 279 98 [130]  Klüpfel F J, Schein F L, Lorenz M, Frenzel H, von Wenckstern H and Grundmann M 2013 Comparison of ZnO-Based JFET, MESFET, and MISFET IEEE Trans. Electron Devices 60 1828 [131] Schein F L, Winter M, Böntgen T, von Wenckstern H and Grundmann M 2014 Highly rectifying p-ZnCo2O4/n-ZnO heterojunction diodes Appl. Phys. Lett. 104 022104 [132] Mercado C C, Zakutayer A, Zhu K, Flynn C J, Cahoon J F and Nozile A J 2014 Sensitized zinc–cobalt–oxide spinel p-type photoelectrode J. Phys. Chem. C 118 25340 [133] Ai C, Yin M, Wang C and Sun J 2004 Synthesis and characterization of spinel type ZnCo2O4 as a novel anode material for lithium ion batteries J. Mater. Sci. 39 1077

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