Oxygen-vacancy driven electron localization and itinerancy in rutile-based TiO$_2$

TL;DR

This study combines advanced computational methods to explore how oxygen vacancies influence electron localization and itinerancy in rutile TiO₂, revealing complex correlation effects and phase behavior.

Contribution

It provides a detailed theoretical analysis of oxygen vacancy effects on electronic structure and correlations in TiO₂ using DFT+DMFT, highlighting the role of charge localization and phase stability.

Findings

Isolated oxygen vacancies do not metallize TiO₂ at low temperatures.

In-gap states are identified consistent with experimental data.

Higher vacancy concentrations stabilize a correlated metallic phase.

Abstract

Oxygen-deficient TiO$_2$ in the rutile structure as well as the Ti$_3$O$_5$ Magn{\'e}li phase is investigated within the charge self-consistent combination of density functional theory (DFT) with dynamical mean-field theory (DMFT). It is shown that an isolated oxygen vacancy (V$_{\rm O}$) in titanium dioxide is not sufficient to metallize the system at low temperatures. In a semiconducting phase, an in-gap state is identified at $\varepsilon_{\rm IG}^{\hfill}\sim -0.75\,$eV\, in excellent agreement with experimental data. Band-like impurity levels, resulting from a threefold V$_{\rm O}$-Ti coordination as well as entangled $(t_{2g},e_g)$ states, become localized due to site-dependent electronic correlations. Charge localization and strong orbital polarization occur in the V$_{\rm O}$-near Ti ions, which details can be modified by a variation of the correlated subspace. At higher oxygen vacancy concentration, a correlated metal is stabilized in the Magn{\'e}li phase. A V$_{\rm O}$-defect rutile structure of identical stoichiometry shows key differences in the orbital-resolved character and the spectral properties. Charge disproportionation is vital in the oxygen-deficient compounds, but obvious metal-insulator transitions driven or sustained by charge order are not identified.

Tables

Table 1. Table 1: The titanium t 2 ​ g = ( e g ′ ​ ( 2 ) , e g ′ ​ ( 1 ) , a 1 ​ g ) subscript 𝑡 2 𝑔 superscript subscript 𝑒 𝑔 ′ 2 superscript subscript 𝑒 𝑔 ′ 1 subscript 𝑎 1 𝑔 t_{2g}=(e_{g}^{\prime}(2),e_{g}^{\prime}(1),a_{1g}) and e g = ( e g ​ ( 1 ) , e g ​ ( 2 ) ) subscript 𝑒 𝑔 subscript 𝑒 𝑔 1 subscript 𝑒 𝑔 2 e_{g}=(e_{g}(1),e_{g}(2)) orbitals in TiO 2 with their respective crystal-field level ε CF subscript superscript 𝜀 absent CF \varepsilon^{\hfill}_{\rm CF} (in meV), expressed in terms of cubic harmonics.

orbital	${\epsilon }_{\mathrm{CF}}^{}$	$|z^2⟩$	$|xz⟩$	$|yz⟩$	$|x^2-y^2⟩$	$|xy⟩$
$|e_g^{\prime }\left(2\right)⟩$	2684	0.000	0.000	0.000	1.000	0.000
$|e_g^{\prime }\left(1\right)⟩$	2653	0.000	0.707	0.707	0.000	0.000
$|a_{1g}⟩$	2574	-0.827	0.000	0.000	0.000	-0.562
$|e_g\left(1\right)⟩$	4398	0.000	-0.707	0.707	0.000	0.000
$|e_g\left(2\right)⟩$	4413	0.562	0.000	0.000	0.000	-0.827

Table 2. Table 2: Comparison of the inter-atomic distances (in Å) between Ti ions surroundig a vacancy-designated oxygen site.

	Ti1$-$Ti2	Ti1$-$Ti3	Ti2$-$Ti3
TiO2	3.57	3.57	2.96
TiO1.9	3.53	3.53	2.97

Table 3. Table 3: Effective V O -induced orbitals on the three nearby Ti sites in the rutile structure, with their respective crystal-field level ε CF subscript superscript 𝜀 absent CF \varepsilon^{\hfill}_{\rm CF} (in meV), written in terms of linear combinations of the original ( t 2 ​ g , e g ) subscript 𝑡 2 𝑔 subscript 𝑒 𝑔 (t_{2g},e_{g}) functions.

site	orbital	${\epsilon }_{\mathrm{CF}}^{}$	$|e_g^{\prime }\left(2\right)⟩$	$|e_g^{\prime }\left(1\right)⟩$	$|a_{1g}⟩$	$|e_g\left(1\right)⟩$	$|e_g\left(2\right)⟩$
	$|{\psi }_1⟩$	1102	0.035	-0.999	0.000	0.000	-0.002
Ti1	$|{\psi }_2⟩$	698	0.000	0.000	-0.147	-0.006	0.990
	$|{\psi }_3⟩$	1023	0.000	0.001	0.990	-0.003	0.147
	$|{\phi }_1⟩$	710	-0.032	-0.023	0.837	-0.427	-0.214
Ti2	$|{\phi }_2⟩$	785	0.077	0.073	0.308	0.733	0.473
	$|{\phi }_3⟩$	992	-0.759	-0.642	-0.221	0.096	0.058
	$|{\phi }_1^{\prime }⟩$	709	0.036	-0.027	0.872	0.437	-0.220
Ti3	$|{\phi }_2^{\prime }⟩$	780	0.083	-0.079	-0.488	0.726	-0.470
	$|{\phi }_3^{\prime }⟩$	992	0.758	-0.641	0.018	-0.103	0.063

Table 4. Table 4: Comparison of the 3 ​ d 3 𝑑 3d filling of the different Ti sites in defect-rutile Ti 3 O 5 (see Fig.  8 b for Ti labeling). The DFT+DMFT data is taken at T = 290 𝑇 290 T=290\, K.

	GGA	DFT+DMFT($U=5$eV)
Ti1	0.22	0.20
Ti2	0.26	0.24
Ti3	0.90	0.86
Ti4	1.15	1.23

Full text

I Introduction

From two motivating research directions, the investigation of oxygen-deficient transition-metal oxides has gained enormous renewed interest. First, the emerging field of oxide heterostructures lead to questions concerning the impact of oxygen vacancies on interface properties. Since especially the SrTiO3 band insulator marks an important heterostructure building block, elucidating the role of such vacancies in that compound has recently attracted lots of attention. Luo et al. (2004); Pavlenko et al. (2012); Mitra et al. (2012); Shen et al. (2012); Lechermann et al. (2014, 2016); Altmeyer et al. (2016) Second, on the search of realizing a memristor, Yang et al. (2013) TiO2-δ remains a key material. Kwon et al. (2010); Heckel et al. (2015) Formation and migration of oxygen-vacancy defects are identified to regulate the resistance modulation therein.

Stoichiometric SrTiO3 is characterized as a cubic (perovskite) Ti${}^{4+}(3d^{0})$ compound with crystal-field split $e_{g}$ and $t_{2g}$ states (cf. Fig. 1). The band gap is located between the dominantly O$(2p)$- and the $t_{2g}$ manifold. Due to the strong O$(2p)-e_{g}$ hybridization, the creation of an oxygen vacancy (VO) leads to local Ti${}^{3+}(3d^{1})$ sites and $e_{g}$-dominated in-gap states. Luo et al. (2004) The interplay of Ti4+- and Ti3+-like states gives rise to a competition between electron localization and itinerancy, posing an intriguing many-body problem. Recently, that problem was approached by theory within the combination of density functional theory (DFT) with dynamical mean-field theory (DMFT). Lechermann et al. (2014); Behrmann and Lechermann (2015); Lechermann et al. (2016) Experiments indeed suggest that VOs on the surface of strontium titanate as well as in the interface of LaAlO3/SrTiO3 may be relevant not only for metallicity, but also for emergent magnetic and/or superconducting order.

Oxygen vacancies in TiO2 pose a related intricate problem, yet with a twist. Besides single-defect scenarios, long-range-ordered vacancy structures provided by the TinO2n-1 Magnéli phases Åsbrink and Magnéli (1959); Andersson and Jahnberg (1963); Bursill and Hyde (1972) are an additional point of materials reference. The role of oxygen vacancies in titanium dioxide, with its twofold structural representations of rutile and anatase is a longstanding problem, and has so far been studied in several theoretical works based on conventional DFT, Liborio and Harrison (2008) using hybrid functionals, Janotti et al. (2010); Deák et al. (2012); Janotti et al. (2013); Berger et al. (2015); Vásquez et al. (2016) as well as by treating static electronic correlations within DFT+U. Mattioli et al. (2008, 2010); Morgan and Watson (2010); Lin et al. (2015); Vásquez et al. (2016) Within the latter framework, Mattioli et al. Mattioli et al. (2008) originally showed that isolated VOs in TiO2 can introduce shallow electronic levels only in anatase, while solely deep localized levels are induced in rutile. The more recent oxygen-deficient rutile studies Mattioli et al. (2010); Janotti et al. (2013); Lin et al. (2015) suggest an intricate coexistence of shallow and deep levels.

Electronic structure investigations of various Magnéli phases, which may in fact be derived starting from the rutile structure, Liborio and Harrison (2008) furthermore revealed challenging physics, such as e.g. metal-insulator transitions and charge ordering. Leonov et al. (2006); Padilha et al. (2014); Tanaka et al. (2015) But those assessments are so far limited by the possibilities of DFT(+U) to describe electron correlation.

In this work we want to provide a theoretical account of electron correlations in oxygen-deficient rutile-based TiO2 from a DFT+DMFT perspective. This not only provides a relevant examination of defect-mediated electronic self-energy effects beyond Kohn-Sham exchange-correlation treatments for an oxide compound with high potential for technological applications. It also allows us to compare the characteristics of the induced defect states to e.g. the ones found in the SrTiO3 perovskite. We show that energy-similar in-gap states are emerging upon creation of VOs, but metallicity does only occur above a corresponding concentration threshold.

Rutile is known to be the thermodynamically stable TiO2 phase at all temperatures and pressures, Hanaor and Sorrell (2011) while anatase is metastable but can kinetically be stabilised at low temperatures. We therefore restrict the investigation on the rutile structural case of TiO2 as well as the Ti3O5 Magnéli phase as a higher VO-concentration counterpart.

II Computational Approach

The supercell defect structures of rutile-TiO2-δ as well as the Magnéli Ti3O5 structure are structurally relaxed Heckel et al. (2015) on the DFT level within the generalized-gradient approximation (GGA) using the PBEsol Perdew et al. (2008) functional in the VASP code. Kresse and Hafner (1993, 1994); Kresse and Furthmüller (1996a, b)

Our present charge self-consistent DFT+DMFT framework Savrasov et al. (2001); Pourovskii et al. (2007); Grieger et al. (2012) builds up on the mixed-basis pseudopotential approach Louie et al. (1979); Meyer et al. for the DFT part and the continuous-time quantum-Monte-Carlo method Rubtsov et al. (2005); Werner et al. (2006), as implemented in the TRIQS package, Parcollet et al. (2015); Seth et al. (2016) for the DMFT impurity problem. We utilize the GGA in the PBE Perdew et al. (1996) functional form within the Kohn-Sham cycle.

Locally, threefold effective Ti$(3d)$ functions define the correlated subspace, which as a whole consists of the corresponding sum over the various Ti sites in the defect problem. Projected-local orbitals Amadon et al. (2008); Anisimov et al. (2005); Aichhorn et al. (2009); Haule et al. (2010); Karolak et al. (2011) of $3d$ character provide the effective functions from acting on Kohn-Sham conduction states above the O$(2p)$-dominated band manifold. Note that the resulting effective orbitals are not of exclusive $t_{2g}$- or $e_{g}$ kind, but are defined by the local three-orbital sector lowest in energy, respectively. Each Ti site marks an impurity problem, and the whole number of explicitly treated impurity problems depends on the number of symmetry-inequivalent transition-metal sites in the given supercell. A three-orbital Hubbard Hamiltonian of Slater-Kanamori form, if not otherwise stated parametrized by the Hubbard $U=5\,$eV and the Hund’s exchange $J_{\rm H}=0.7\,$eV, is active on each Ti site. These values for the local Coulomb interactions are in line with previous studies on titanates. Mizokawa and Fujimori (1995); Pavarini et al. (2004); Okamoto et al. (2006); Lechermann and Obermeyer (2015) A double-counting correction of the fully-localized form Anisimov et al. (1993) is utilized in this work. The analytical continuation of the finite-temperature Green’s functions on the Matsubara axis $i\omega$ to real frequencies is performed via the maximum-entropy method.

Note that we study only paramagnetic states without broken spin symmetry, as well as charge-neutral lattice configurations. Yet possible inter-site charge disproportionation is surely allowed.

III Results

III.1 Rutile TiO2

To set the stage for the discussion of oxygen-deficient titanium dioxide, we briefly discuss the electronic structure at stoichiometry. The rutile structure Burdett et al. (1987) (see Fig. 2a) has tetragonal symmetry (space group $P4_{2}/mnm$) with a ratio $c/a=0.64$ and the primitive cell comprises two TiO2 formula units. It consistes of corner- and edge-sharing TiO6 octahedra, such that each oxygen ion is coordinated by three neighboring titanium ions. This is in contrast to common perovskite-based titanates, where the TiO6 octahedra are exclusively corner-sharing and O is twofold Ti-coordinated. Of the three rutile minimal Ti-O bond lengths, the two shorter ones are identical in extent. Nominally, titanium is in the Ti4+ oxidation state with $3d^{0}$ occupation.

The given compound is a band insulator with an experimental (optical) band gap of size $\Delta_{\rm g}\sim 3$ eV. Cronemeyer and Gilleo (1951); Cronemeyer (1952); Pascual et al. (1978) Conventional DFT calculations yield a smaller gap $\Delta_{\rm g}\sim 2$ eV (cf. Fig. 2b). Since TiO2 is a nominal $3d^{0}$ material, the HOMO orbitals are of dominant O$(2p)$ kind and the LUMO orbitals are mainly of Ti$(3d_{t_{2g}})$ character, while dominant Ti$(3d_{e_{g}})$ is located at even higher energies above the band gap. Because of the fact that the $c$-axis and the main TiO6-octahedra axes are locally aligned trigonal, the Ti($t_{2g},e_{g}$) orbitals may be written as linear combinations of cubic harmonics from diagonalization of the orbital-density matrix (see Tab. 1).

Due to the local symmetry, the internal $(t_{2g},e_{g})$ degneracies known from the full octahedral group are lifted, respectively. Comparison of the crystal-field levels $\varepsilon^{\hfill}_{\rm CF}$ marks the $a_{1g}$ level as the lowest one, 110 meV below $e_{g}^{\prime}(2)$. The $t_{2g}$-based states are about $2.2$ eV lower in energy than the $e_{g}$-based ones. Within the effective $t_{2g}$ manifold, which has a bandwdith of about $2.5$ eV, the $e_{g}^{\prime}(2)$ orbital is designated since its lobes point along the in-plane tetragonal axes (see Fig. 2c).

III.2 Oxygen vacancy in rutile TiO2

III.2.1 Structural details and correlated subspaces

As shown in Fig. 3, a supercell five times the size of the primitive cell, i.e. with 10 Ti and 20 O atoms, serves as basis structure for the defect study. A single VO leaves three nearest-neighbor titanium ions behind, here labelled Ti1, Ti2 and Ti3. In the stoichiometric rutile structure, the Ti2-Ti3 distance marks the short side of the given Ti triangle. The nominal VO concentration in this constellation amounts to $c\equiv\delta/2=0.05$, i.e. our modeling describes a TiO1.9 defect case. This represents a large VO concentration, however not unrealistic for the given system. Parker and Siegel (1990) Table 2 shows that the inter-atomic distances are only weakly modified upon structural relaxation, which may be also related to the small supercell size. Still, the obtained pattern describes a shortening of the Ti1-Ti2,3 distances and an elongation of the Ti2-Ti3 bonding, providing a trend to balance the triangle distances trough the VO.

In the following we want to investigate the effect of an VO in the rutile structure in terms of the local-orbital configuration as well as the net electronic structure. We will discuss two different choices for the correlated subspace. First in section III.2.2, that space is formed by all Ti sites in the given structure, which marks the canonical and ground-state-oriented case. Second in section III.2.3, the correlated subspace is restricted to the contribution of the Ti1-3 sites, i.e. it becomes more local. This second choice may be interpreted as treating an excited state of the system, where electrons do not see the explicit Coulomb repulsion on the remaining Ti sites. Since the $3d$ occupation on the original Ti($d^{0}$) sites distant to the VO is expected small, double occupation is there rather rare. If moreover the electrons have gained energy from an excitation process, they can even more easily escape from such double occupations (as well as more efficiently screen the Coulomb penalties). Thus the average effect of local $U$ and $J_{\rm H}$ on the VO-distant Ti sites can then be neglected to a good approximation to obtain a rough picture of the present system on a globally-higher excitation level. This approximation and interpretation is also not in conflict with the definition of a one-particle spectral function. We thus simply term that space ’excited correlated subspace’, and this second choice allows us to shed light on possible changes in the oxidation state of the oxygen vacancy. Still note however that the Coulomb interactions on the Ti sites away from the VOs are crucial to understand the semiconducting character of TiO2-δ, as will be explained in the following section III.2.2.

III.2.2 Canonical correlated subspace formed by all Ti sites

Let us start with the well-defined DFT+DMFT setting of the correlated subspace build by all the Ti sites. On the GGA level, as displayed by plotting the total spectral function $A(\omega)=\sum_{\bf k}A({\bf k},\omega)$ in Fig. 4, the considered system becomes metallic with semimetallic tendencies around the Fermi level $\varepsilon_{\rm F}$ positioned within the $t_{2g}$ manifold. Yet correlations and finite temperature $T$ render the situation more intriguing. At low $T$ the defect structure is semiconducting within DFT+DMFT, with a small charge gap $\Delta\sim 0.06$ eV at $T=145$ K. That gap is filled with rising temperature, marking a bad-metal regime. In addition, there is sizable transfer of spectral weight to a broad in-gap structure centered at $\varepsilon_{\rm IG}^{\hfill}\sim-0.75$ eV. Note that optics measurements detect an absorption peak in reduced semiconducting TiO2 single crystals at 0.75 eV, Cronemeyer and Gilleo (1951); Cronemeyer (1952) commonly used to explain its blue color. Parker and Siegel (1990) Photoemission measurements on the rutile surface report a defect-state peak at $\sim-0.9$ eV, Thomas et al. (2007); Wendt et al. (2008); Yim et al. (2010) while on the anatase surface it is located at an higher energy of $\sim-1.1$ eV Thomas et al. (2007); Rödel et al. (2015) Furthermore, scanning tunneling spectroscopy finds an in-gap state at $\sim-0.7$ eV on the defect-rutile surface Setvin et al. (2014) and x-ray photoelectron spectroscopy on rutile TiO2 nanoparticles reveals a defect state at $\sim-0.8$ eV. Vásquez et al. (2016) It is tempting to relate these experimental findings of deep levels in TiO2-δ to the present satellite peak. On the other hand, $n$-type conductivity with rather high mobilities due to shallow level has been also reported in the literature, Yagi et al. (1996) which might be connected to our small-gap feature.

To gain insight in the nature and characteristics of the Ti-local states near the VO, first Tab. 3 provides the effective orbitals on Ti1-3 written in terms of linear combinations of the original $(t_{2g},e_{g})$ functions from Tab. 1. The $\varphi$ and $\varphi^{\prime}$ orbitals on Ti2,3 behave very similarly, therefore we restrict the discussion to the $\varphi$ branch. It is seen that while the $\psi_{2}$ orbital has strong $e_{g}$ character, the orbitals $\varphi_{1}$, $\varphi_{2}$ have sizable contributions from both original orbital sectors. Thereby $\varphi_{1}$ is $t_{2g}$-dominated and $\varphi_{2}$ is $e_{g}$-dominated. Thus a nearly exclusive $e_{g}$ character of the local defect states, as e.g. given in oxygen-deficient SrTiO3, Luo et al. (2004); Lechermann et al. (2016) does not apply for oxygen vacancies in TiO2. Note furthermore that the $e_{g}^{\prime}(2)$ orbital has almost negligible contribution to the VO-induced physics, as the $e_{g}^{\prime}(2)$-dominated $\varphi_{3}$ orbital remains nearly empty.

Concerning tight-binding parameters, the hopping between both $\varphi_{1}$ on Ti2 and Ti3 is largest with $t_{\varphi_{1}}=-0.2$ eV, while the other hopping amplitudes on the Ti triangle are of absolute value $\leq 0.1$ eV. Non-surprisingly, this marks Ti2,3 as more strongly coupled, thus possibly prone to singlet/triplet formation.

Figure 5 provides the spectra on Ti1-3, which are expected to display (partly) Ti3+ character. Note that although all Ti sites in the supercell contribute to the full correlated subspace, electron occupation on the remaining Ti sites is very small. The filling on the Ti sites farther away from VO is also not significantly raised with temperature. The effective orbitals on Ti2 and Ti3 are nearly equivalent by symmetry and behave here very similarly, so no site differentiation is needed. Of the three correlated orbitals $\{\psi_{m}\}$ on Ti1, only $\psi_{2}$ has sizable filling.

Spurious occupation of $\psi_{1},\psi_{3}$ in GGA is eliminated by correlations. On the contrary, two orbitals, i.e. $\varphi_{1}$ and $\varphi_{2}$ are occupied on Ti2,3. The designation of $\psi_{2}$, $\varphi_{1}$ and $\varphi_{2}$ is already suggested from their favorable crystal-field levels (cf. Tab. 3). In DFT+DMFT the occupations read $(n_{\psi_{2}},\,n_{\varphi_{1}},\,n_{\varphi_{2}})=(0.65,\,0.29,\,0.33)$, thus all three Ti sites have similar occupation, however $n_{\rm Ti1}>n_{\rm Ti2,3}$ holds. Since the site-resolved fillings sum up to $n_{V_{\rm O}}=1.89$ and oxygen is in the O2- oxidation state in the compound, the defect is described as of being close to the nominal neutral vacancy state V${}_{\rm O}^{0}$.

In order to check for the influence of the chosen local Coulomb interactions on the physics, we additionally peformed calculations for $U=3.5$ eV and $U=2.5$ eV, both with $J_{\rm H}=0.5$ eV. The values $U=3.5$ eV, $J_{\rm H}=0.5$ eV were used in a recent DFT+DMFT study of the oxygen-deficient SrTiO3 surface. Lechermann et al. (2016) There are no qualitative differences between the resulting total spectral functions (cf. Fig. 6) for the sets ($U=5$ eV, $J_{\rm H}=0.7$ eV) and ($U=3.5$ eV, $J_{\rm H}=0.5$ eV), hence the detected physics is rather stable within a reasonable range of local Coulomb-interaction parameters. Only at much smaller $U=2.5$ eV the semiconducting gap as well as the deeper in-gap state seem to disappear.

It remains to specify the likely mechanism behind the DFT+DMFT finding. The results suggest that the VO provides electron doping by initially forming rather shallow states below the original $t_{2g}$ mainfold. Because of the higher O-Ti connectivity and the entangled ($e_{g}$,$t_{2g}$) defect signature in rutile, both compared to e.g. surface SrTiO3, Lechermann et al. (2016) the coherency of those states is increased and they seemingly develop band-like character. Sizable Coulomb interactions then lead to a Mott criticality, resulting in band renormalization and formation of Hubbard(-like) bands. The latter give rise to the deep-level in-gap spectra. Importantly, the driving force between the Mott(-like) gap formation is different from a conventional correlated multi-band lattice problem. Since the local states near the VO are connected between different VOs by fragile hopping paths, mainly the local Coulomb interactions on the Ti sites distant from the defects are determinative for driving the doped system again insulating. In other words, the ’interstitial’ Coulomb-repulsive region between the VOs destroys the fragile band-like defect states and localizes the electrons dominantly near the oxygen vacancy.

III.2.3 Excited correlated subspace formed by Ti1, Ti2, Ti3

Letting only the sites Ti1-3 contribute to the correlated subspace serves to goals. First, it gives access to a possibly different VO charging state. Second, it provides a check for our presented mechanism, denoting the Coulomb interactions on Ti away from VO as being mainly to be blamed for the semiconducting character.

In general, the states V${}_{\rm O}^{0}$, V${}_{\rm O}^{+1}$ and V${}_{\rm O}^{+2}$ are discussed for oxygen-deficient TiO2, Yang et al. (2009); Mattioli et al. (2010); Janotti et al. (2013); Lin et al. (2015) which in other works appeared relevant to fit findings of shallow-donor properties and well-localized defect states within a coherent picture. As discussed in section III.2.2, the present DFT+DMFT study already provides means to such a coexistence by revealing small-gap as well as deeper in-gap features. Nonetheless, by performing additional calculations within the local-restriced correlated subspace one may learn further details of the VO energy-level structure. Note that in principle the local Coulomb interactions also change when the correlated subspace is modified, i.e. in the present scenario should be lowered for the smaller subspace. But for simplicity we keep $U=5$ eV and $J_{\rm H}=0.7$ eV on Ti1-3 to reveal the key features of the excited system.

As expected, a metallic solution results from the calculation, with reduced and shifted incoherent weight at higher energy (see Fig. 7a). This is indeed in favor of our ’interstitial Coulomb’ mechanism being relevant for fully localizing electrons near VOs. Interestingly, the Ti1-3 sites now display a rather different local filling in the dominant orbitals, namely $(n_{\psi_{2}},\,n_{\varphi_{1}},\,n_{\varphi_{2}})=(0.24,\,0.19,\,0.43)$, and a total filling of these sites amounting to $n_{V_{\rm O}}=1.56$. This means that in the excited scheme not only the total filling of Ti close to VO is substantially reduced compared to the canonical scheme, but also the local filling symmetry is different. Now the Ti2,3 sites are majorly occupied and the Ti1 site plays the weaker role, also visualized by plotting the local spectral functions in Fig. 7b. The high-energy, incoherent and strongly-localized part of the spectra is now dominantly carried by states on Ti2,3. It appears that in the excited-system calculation, localized charge on Ti1 has been partially transfered to itinerant states.

These findings are indeed reminiscent of the identification of different VO charging states in DFT+U and/or DFT hybrid-functional studies. Mattioli et al. (2010); Janotti et al. (2013); Lin et al. (2015) Also the characterization of two different site-orbital levels has been reported before, yet usually by invoking explicit magnetic ordering. Our two detected charging states with $n^{\rm can}_{V_{\rm O}}=1.89$, reading V${}_{\rm O}^{+0.11}$, and $n^{\rm ex}_{V_{\rm O}}=1.56$, reading V${}_{\rm O}^{+0.44}$, do not very strongly deviate from the neutral-vacancy case. However note that highly-oxidized states like V${}_{\rm O}^{+1}$/V${}_{\rm O}^{+2}$ are usually found in the presence of additional trivalent substitutional impurities on the Ti site, such as Fe3+ and Cr3+. Yang et al. (2009) An established theoretical picture Janotti et al. (2013) describes the vacancy-defect state as a bound object consisting of V${}^{2+}_{\rm O}$ plus two polarons, rendering it charge neutral again. In principle one may try to interprete our results also along such lines, since the present filling scenario with Hubbard-like high-energy spectral parts point to a local $S=1$ spin in the paramagnetic material. On the other hand, as discussed, we do not have to invoke the polaron picture on clear grounds to account for the coexistence of shallow and deep states.

Let us finally note that the overall qualitative picture that we here derived for a single VO in rutile TiO2 is believed to be stable against modified structural relaxations, as e.g. provided by a different(larger) supercell. More dilute VO cases could still lead to a weakening of the shallow levels in favor of the deep level, with an increase of the semiconducting gap. Within the local-level manifold the subtle details of the energy hierachy might be affected by structural issues. But the general notion of the relevant $\psi_{2}$ on Ti1 as well as $\varphi_{1}$ and $\varphi_{2}$ on Ti2,3 appears robust.

III.3 Higher oxygen-vacancy concentration: Ti3O5

III.3.1 Spectral properties and total energies

We now shift attention to the problem of oxygen vacancies at higher concentration. In order to gain insight in the properties of VOs in a designated ordered limit, we study a specific Magnéli compound at Ti3O5 stoichiometry. There is strong interest in the various Magnéli phases at Ti3O5, Åsbrink and Magnéli (1959); Hong and Åsbrink (1982); Onoda (1998); Ohkoshi et al. (2010); Padilha et al. (2014); Tanaka et al. (2015) since e.g. photoreversible phase transitions occur at room temperature. But as we are interested in the main effect of vacancy ordering, we focus on a single allotrop, the so-called $\gamma$ phase. The underlying crystal structure has monoclinic symmetry and can be stabilized at room temperature. Hong and Åsbrink (1982)

Described in simple terms, whereas perovskite SrTiO3 has corner-sharing TiO6 octhedra and rutile TiO2 an elementary alternation of corner- and face-sharing octahedra, the Magnéli phases exhibit more complicated arrangement of those two octahedra-sharing types to accomodate a desired stoichiometry (see Fig. 8a). This may be interpreted as an ordering of vacancies, however importantly, contrary to e.g. our TiO1.9 structure, there is no VO-induced ’destruction’ of local TiO6 octhedra. In terms of a formal oxygen deficiency $\delta$, the given Ti3O5 stoichiometry amounts nonetheless to $\delta=0.33$, i.e. the compound would correspond to TiO1.67 with vacancy concentration $c=0.167$.

In order to compare the electronic characteristics of the optimal-ordered Magnéli structure with VOs in rutile, we in addition performed calculations for defect rutile with stoichiometrty Ti3O5. The corresponding crystal structure (cf. Fig. 8b) was determined by energy minimization of oxygen-vacancy arrangements at the desired composition by making use of the cluster-expansion technique (see Ref. Heckel et al., 2015 for details). In this defect-rutile structure, part of the TiO6 are indeed ’damaged’, resulting in selected fivefold-O coordinated Ti sites,here designated as Ti4. A total-energy comparison on the GGA level, as expected, clearly favors the Magnéli structure. The more clever restoration of ideal 6-fold-oxygen coordination around Ti is appreciated by a substantial $\sim 0.42\,$eV per Ti atom against defect rutile.

Figure 9 documents the differences in the GGA density of states in terms of the partitioning in Ti($t_{2g}$,$e_{g}$)-like contributions. Based on the robust TiO6-octahedral structuring of the Magnéli phase, the electronic structure of $\gamma$-Ti3O5 shows a clear distinction into those $3d$ submanifolds. This is partly also true for defect-rutile Ti3O5, yet the $(t_{2g},e_{g})$ states from the TiO5 polyhedra show there more substantial overlap in energy. In fact, the small $t_{2g}$-$e_{g}$ gap region above 2 eV in $\gamma$-Ti3O5 just becomes filled by such states in the defect-rutile structure.

The total spectral functions obtained within DFT+DMFT are plotted in Fig. 10. Both systems remain metallic with including many-body correlations. A lower Hubbard band located at $\sim-1.1\,$eV and renormalization at low energy are identified for the Magnéli phase. This marks the system as a seemingly ’textbook’ correlated material, with coherent renormalized quasiparticles at low energy and incoherent Hubbard bands at higher energies. Already on the GGA level there is a small spectral dip at the Fermi level, verified also with correlations at lower temperature. Note that the Hubbard band is of course nearly exclusively formed by $t_{2g}$-like states. Satellite structures of $t_{2g}$ kind have recently been detected in a hard x-ray photoelectron spectroscopy study of the structurally different $\beta$-Ti3O5 and $\lambda$-Ti3O5 Magnéli compounds. Kobayashi et al. (2017)

Though the same total filling scenario holds for the defect-rutile phase, the characterization with electronic correlations appears more subtle. First, in view of the semiconducting defect-rutile state discussed in section III.2.2, the metallic response at higher VO concentration is not that surprising. The fragile hopping paths between VO-near electronic states become increasingly robust with growing concentration of vacancies, such that eventually Coulomb repulsion is not anymore capable of establishing a charge-gapped material.

In the low-energy spectrum there is some renormalization due to correlations, but not of significant kind. On a first glance, the spectral function looks furthermore rather monotonic in the occupied part, yet importantly, not necessarily implying that correlations are weak. Because the comparison to the GGA results shows, there is significant spectral-weight transfer to energies far away from the Fermi level, too. However this transfer does not give rise to an obvious lower Hubbard-band peak, but is broadly distributed over a wider energy range (see inset of Fig. 10). This finding is somewhat reminiscent of obvervations made in resonant-photoemission experiments on electron-doped SrTiO3. Ishida et al. (2008) There, broader in-gap weight was assigned to the increased relevance of Ti($3d$)-O($2p$) hybridization. Since the $e_{g}$ character, which is more strongly hybridized with O($2p$) than $t_{2g}$, plays a more prominent role in the defect-rutile case, the present result could possibly point to a similar interpretation. On the other hand, a recent GW+DMFT study suggests the possibility of diminished satellite peaks in some correlated compounds, as well as a reinterpretation of their original character. Boehnke et al. (2016)

The total-energy difference between both Ti3O5 structural types is even slightly increasing with including many-body correlations. At room temperature, the Magnéli phase is favored by $\sim 0.46\,$eV per Ti atom.

III.3.2 Charge disproportionation between Ti sites

So far we did not comment on the local occupations of the respective Ti sites in $\gamma$- and defect-rutile Ti3O5. Since the nominal oxidation state at that stoichiometry amounts to Ti3.33+, charge fluctuations are expected to be more relevant than in many other oxides with nominal integer Ti valence. In fact, charge ordering in connection with a metal-insulator transition is commonly discussed for various Magnéli phases, especially for the Ti4O7 compound. Leonov et al. (2006)

The oxygen deficiency quite naturally introduces symmetry-inequivalent Ti sites with potentially different electron occupation. Figure 8b shows four different Ti sites with especially the Ti4 site surrounded only by five oxygen atoms. The $\gamma$-Ti3O5 structure formally has eight Ti sites different by symmetry, which were also differently treated in our calculations. In the discussion however, to a very good approximation, four Ti classes may be grouped to a single effective class, respectively, since the local-orbital structure only marginally differs. Figure 11 displays the present Magnéli structure in that effective two-Ti-sublattice picturing. An obvious pattern is derived therefrom. The TiO6 octahedra of Ti sites within a given sublattice are corner sharing, whereas the octahedra are edge sharing between both sublattices. Therefore the intra-sublattice Ti-Ti distance is about 0.5 Å  longer than the inter-sublattice pair distance. Both effective sublattices are not of equal-site size, since the Ti2 sublattice (dark-blue octahedra in Fig. 11) covers twice as many Ti sites as the Ti1 sublattice (light-blue octahedra in Fig. 11).

Note that the Ti1 octahedra mark the middle of the stoichiometric undisturbed TiO2 rutile slabs, which consist of three octahedra in the rutile (001) direction and are interrupted by a (121) VO defect plane.

Tables 5 and 5 provide the Ti($3d$) occupations for both considered Ti3O5 structures. Nonsurprisingly in defect rutile, the charge on the Ti4 sites with 5-fold oxygen coordination is largest, marking that site with a Ti2.8+ oxidation state. The nearby Ti3 sites are still close to Ti3+, while Ti1-2 are closer to Ti4+. There are no dramatic differences between the numbers based on GGA and those from DFT+DMFT, only the occupation of Ti4 is still somewhat higher with correlations.

The Magnéli phase $\gamma$-Ti3O5 has a more subtle Ti ordering. The geometrical constraints of keeping the TiO6 octahedra in line with the nominal Ti3.33+ state leads to slightly different charge states on both effective sublattices. The smaller Ti1 sublattice carries already on the GGA level more charge, with a disproportionation $\rho({\rm Ti1})-\rho({\rm Ti2})=0.11$ e*-. Electronic correlations increase this charge splitting only by small amounts, even when invoking a rather large Hubbard $U=8\,$eV. Importantly, the system remains always metallic. Also because a seemingly straightforward Ti3+/Ti4+* splitting, as in principle possible in the Ti4O7 Magnéli system with a nominal Ti3.5+ oxidation state, is not an obvious option. Note however that there is also a metal-insulator transition reported for Magnéli Ti3O5, Tanaka et al. (2015) but from $\gamma$-Ti3O5 to $\delta$-Ti3O5 upon lowering temperature. Thus further structural changes are in addition indispensable to allow for insulating behavior.

III.4 Summary and discussion

We presented a detailed first-principles many-body investigation of the effect of oxygen vacancies in rutile TiO2, both in the lower- and the higher-concentration regime. In the former case, the DFT+DMFT results directly provide the three key ingredients known from experiment for TiO2-δ, namely semiconducting behavior, shallow levels as well as deep levels. Our deep-level (or in-gap) positioning is in excellent agreement with results from various experimental studies. Thereby the many-body perspective provides a different viewpoint on the longstanding discussion of deep vs. shallow levels for oxygen-deficient TiO2. Here, both levels are connected in the many-body sense, similar as a lower Hubbard band and a renormalized quasiparticle state in a conventional moderately correlated metal. Still the correlation-induced gap opening in the band-like initial shallow levels is not of standard Mott type. Key to the electron localization at low VO concentration is the Coulomb-repulsion region between defects, that blocks the fragile hopping paths. Therefore, the resulting charge gap is rather small and does not scale with $U$ in a conventional way.

By invoking an excited correlated subspace we brought the semiconducting solution into the transport regime, therewith showing that indeed the Coulomb interactions on the Ti sites distant from the oxygen vacancies dominantly control the competition between itinerancy and localization. Moreover, the localized states at the VO display a different occupation and energy state upon excitation. This connects our study to other (static-correlation) theory work for TiO2-δ, where e.g. different VO charge states and polaron formation are discussed. Mattioli et al. (2008, 2010); Morgan and Watson (2010); Janotti et al. (2010); Deák et al. (2012); Janotti et al. (2013); Lin et al. (2015); Berger et al. (2015); Vásquez et al. (2016)

In comparison to oxygen-deficient SrTiO3, the defect-rutile problem differs in two crucial point. First, the defect structure with a VO in nearest-neighbor distance to three Ti sites, contrary to two Ti sites in the perovskite structure, fosters a modified oxygen-vacancy impact. The higher connectivity with the defect gives way to the formation of more band-like impurity states on the single-electron level. Second, the higher entanglement between $t_{2g}$ and $e_{g}$ states lifts the electron dichotomy observed on the SrTiO3-δ surface, Lechermann et al. (2016) suppressing both, a decoupled $t_{2g}$ band formation and a rather disconnect straightforward $e_{g}$ deep-level formation. Therefore at small VO concentrations, rutile TiO2 remains insulating (or semiconducting) due to correlation-induced gap opening in ($t_{2g}$, $e_{g}$)-entangled band-like impurity levels. Note in this context however that the very correlation details of bulk SrTiO3-δ still need to be investigated by similar means

In the higher VO-concentration regime we focussed on the Ti3O5 stoichiometry, studying the metallic Magnéli $\gamma$-phase as well as a theory-derived metallic defect-rutile phase. The interplay of Ti coordination and $(t_{2g},e_{g})$ electronic structure properties were discussed and therefrom the differences in total energy and in the spectral features explained. While $\gamma$-Ti3O5 turns out as a ’textbook’-like correlated metal, the defect-rutile version displays spectral-weight transfers to higher energies in a much broader (incoherent) fashion. Charge disproportionation is a natural by-product of the VO ordering in rutile-based TiO2 (which formally includes the Magnéli phases) Liborio and Harrison (2008) and is already captured on the Kohn-Sham level. Electronic correlations beyond the latter provide at least for Ti3O5 only minor changes to the Ti charging states.

Although in direct comparison, defect-rutile Ti3O5 is energetically rather unfavorable compared to the Magnéli phase, its discussion is nonetheless revelant in view of engineering memristive processes and devices. Since there in TiO2, transport of VOs and electrons when starting from the rutile structure is a possible technological aspect. In fact, in realistic close-to-device-like TiO2 materials, the formation of defect-rutile and Magnéli TinO2n-1 phases may happen in parallel within a given materials system. Kwon et al. (2010) Thus understanding the differences between both structural types on a basic level is of crucial importance. In the long run, the present study shall contribute to pave the road for elucidating further engineering options in the interplay between the transport of oxygen vacancies and electrons.

It would also be highly interesting to extent the present investigation to the rutile VO2 system. As the vanadium ion in the $+4$ oxdidation state has $3d^{1}$ occupation, it is known that already the stoichiometric system is prone to strong correlation physics, Zylbersztejn and Mott (1975); Sommers and Doniach (1978); Rice et al. (1994); Biermann et al. (2005); Nájera et al. (2017) possibly giving reason to the hallmark metal-to-insulator transition slightly above room temperature. Morin (1959) Recent doping studies of VO2 via oxygen vacancies display opportunities to tune the competition between the metallic and insulating regime. Jeong et al. (2013); Zhang et al. (2017)

Finally coming back to basic features of VOs in transition-metal oxides, our examination challenges the simplest views on in-gap states, namely the weak-coupling defect-level and the strong-coupling Hubbard-band paradigm. Albeit one encounters features of both original mechanisms in oxygen-deficient TiO2, as in related systems such as SrTiO3-δ or the LaAlO3/SrTiO3 interface, a unique and well-defined picture describing the general nature of defects in transition-metal oxides is still missing. Crystal-field effects, renormalizations, Hubbard-band formation, lifetime effects, $p$-$d$ hybridization, screening, charge transfer, polaron formation, etc. are potentially part of this demanding physics. Thoughts trying to put the problem in an adequate model setting have been put forward from several perspectives, e.g. Ref. Haldane and Anderson, 1976. The future task is to cast those into a sound and solid materials-dependent picture.

Acknowledgements.

We gratefully acknowledge financial upport from the German Science Foundation (DFG) via SFB986 and through FOR1346. Computations were performed at the University of Hamburg and the JURECA Cluster of the Jülich Supercomputing Centre (JSC) under project number hhh08.