DFT+DMFT study of oxygen vacancies in a Mott Insulator

TL;DR

This study uses DFT+DMFT to investigate how oxygen vacancies affect the electronic properties and metal-insulator transition in LaTiO3, revealing the robustness of its Mott insulating state despite vacancy-induced bands.

Contribution

It demonstrates that the Mott insulating state in LaTiO3 remains stable even with oxygen vacancies, using a combined DFT+DMFT approach with a minimal correlated subspace.

Findings

Vacancy-related band forms below Ti-t2g bands

Mott insulator remains robust despite vacancies

Vacancy band stays fully occupied with Coulomb interactions

Abstract

Oxygen vacancies are a common source of excess electrons in complex oxides. In Mott insulators these additional electrons can induce a metal-insulator transition (MIT), fundamentally altering the electronic properties of the system. Here we study the effect of oxygen vacancies in LaTiO3, a prototypical Mott insulator close to the MIT. We show that the introduction of oxygen vacancies creates a vacancy-related band immediately below the partially filled Ti-t2g bands. We study the effect of this additional band on the Mott MIT using a combination of density functional theory and dynamical mean-field theory (DFT+DMFT), employing a minimal correlated subspace consisting of effective Ti-t2g orbitals plus an additional Wannier function centered on the vacancy site. We find that the Mott insulating state in LaTiO3 is robust to the presence of the vacancy band, which remains fully occupied even in the presence of a local Coulomb repulsion.

Full text

Point defects are an unavoidable feature in perovskite oxides at finite temperature. Among them, oxygen vacancies tag (OV) are believed to play a key role in a variety of emergent phenomena, such as superconductivity Cava et al. (1987), the establishment of an interfacial two-dimensional electron gas Kalabukhov et al. (2007), magnetoresistance Kormondy et al. (2018), or blue-light emission at room temperature Hwang (2005); Crespillo et al. (2017). Despite their relevance, a complete picture of the effect of oxygen vacancies in complex functional oxides is still lacking, in part due to the strong coupling of multiple degrees of freedom (structural as well as electronic) in these systems Ganduglia-Pirovano et al. (2007); Eckstein (2007); Backes et al. (2016). While it is well known for some perovskite oxides that OV’s create defect states inside the energy gap, the itinerant or localized nature of these states remains the key open question in this field Janotti et al. (2014); Lin and Demkov (2013); Lechermann et al. (2016).

The Mott metal-to-insulator transition (MIT) is an intriguing phenomenon in complex oxides, where electronic correlation effects play a central role (for a review, see Ref. Imada et al., 1998). Although many aspects of the Mott MIT, for realistic materials, are not fully understood, the use of such materials in novel functional applications within the emergent field of Mottronics Mannhart and Haensch (2012); Inoue and Rozenberg (2008) has immediate relevance. Oxygen vacancies can have a potentially large effect on the MIT, either by changing the stochiometry, disordering the lattice, or introducing chemical strain through lattice expansion Aschauer et al. (2013).

Within a Mott insulator, provided that the vacancy-induced band is partially occupied, one can expect that correlation effects would penalize double occupancy and instead change the valence state of neighboring cations. This could then destabilize the Mott insulating state, similarly to a doped Mott insulator within the Hubbard model, where the doping breaks the commensurability between the number of electrons and the number of sites Amaricci et al. (2017); Werner et al. (2009). Thus, in contrast to an uncorrelated semiconductor, where the amount of carriers is essentially proportional to the amount of defects, in a Mott insulator the defect-induced doping could fundamentally alter the electronic state (by causing a MIT), thereby effectively transforming all valence electrons into carriers. This potential MIT caused by the oxygen deficiency has not been widely explored, due to the complicated interpretation of the experimental resultsLechermann et al. (2016) and also to the lack of a suitable theoretical framework.

In this work, we directly address the latter point by using DFT+DMFT Georges et al. (1996); Anisimov et al. (1997); Held (2007); Rozenberg et al. (1995); Pavarini et al. (2005); Lichtenstein et al. (2001); Pavarini et al. (2004) to investigate the Mott insulating state of the prototypical Mott-insulator LaTiO3 in the presence of oxygen vacancies. LaTiO3 has an orthorhombically-distorted $Pnma$ perovskite structure ($a^{-}b^{+}a^{-}$ distortion in Glazer’s notationGlazer (1972)) and a reported optical gap of $\sim 0.2$ eVOkimoto et al. (1995). The Ti atoms are in a Ti*+3* state, with one electron in the $t_{2g}$ orbitals ($d^{1}$). The relative simplicity of its electronic structure and its proximity to the MIT from the insulating side makes LaTiO3 a perfect model system to study the stability of the Mott insulating state.

To obtain accurate geometries and bandstructures, we perform standard DFT calculations using the projector-augmented wave (PAW) method, as implemented in the “Vienna ab-initio simulation package” (VASP) Kresse and Furthmüller (1996); Kresse and Joubert (1999), version 5.4.1, together with the GGA-PBE exchange-correlation functional Perdew et al. (1996). The valence configurations of the PAW potentials used are La($5s6s5p5d$), Ti($3p4s3d$) and O($2s2p$). The La PAW potential includes the empty $4f$ states, which form a set of narrow bands between the Ti-$t_{2g}$ and the $e_{g}$ bands. We apply $U=6$ eV on the DFT$+U$ level Dudarev et al. (1998) to shift these La-$f$ orbitals up in energy reducing the entanglement with the Ti-$t_{2g}$ bands. To accommodate the $Pnma$ structure of LaTiO3, we use a $20$-atom unit cell, $19$-atom for the OV-defective systems LaTiO2.75 (vacancy concentration of 8.3$\%$). Converged results are obtained by sampling the Brillouin zone with a $11\times 9\times 11$ $\Gamma$-centered k-mesh and using a plane wave energy cutoff of $900$ eV. All structural degrees of freedom are relaxed until all forces are smaller than $10^{-4}$ eV/Å. All calculations are performed without considering spin polarization.

The low-energy correlated subspace for the DMFT calculations is then constructed using a basis of maximally localized Wannier functions (MLWF) Lechermann et al. (2006), employing the Wannier90 code Mostofi et al. (2014). We use the TRIQS/DFTTools package Parcollet et al. (2015); Aichhorn et al. (2016) to perform the paramagnetic DMFT calculations. An effective impurity problem is solved for each inequivalent Ti site plus the vacancy site using the TRIQS/CTHYB solver Seth et al. (2016), while the different impurity problems are coupled through the DMFT self-consistency. The local interaction is modeled using the Hubbard-Kanamori parametrization with spin-flip and pair-hopping terms included, and Held’s formula tag is used to compute the double-counting term. All calculations are performed at room temperature, $\beta=(k_{B}T)^{-1}=40$ eV*-1*, with a Hund’s coupling of $J=0.64$ eV for the Ti sites, and without charge self-consistency. The values for the Hubbard $U$ are varied to analyze the effect on the electronic properties. Real frequency spectral functions, $A(\omega)$, are obtained from the local Green’s functions in imaginary time, $G(\tau)$, using the Maximum Entropy algorithmBryan (1990). The spectral weight around the Fermi energy, $\bar{A}(0)$, is calculated from the impurity Green’s function as $\bar{A}(0)=-\beta/\pi\,G(\beta/2)$.

In the Pnma-distorted perovskite structure, there are two inequivalent sites for the oxygen atoms (Wyckoff positions 4c and 8d), hence two different vacancy sites. We denote as axial/planar vacancy the configurations where the missing oxygen belongs to an O-Ti bond parallel/perpendicular to the long orthorhombic axis (see Figs. 1a and 1b, respectively). The oxygen vacancy lowers the symmetry of the system compared to pure $Pnma$ LaTiO3. In the axial case, the resulting space group (for the 20-atom cell used here) is $Pm$, with two inequivalent Ti sites, one next to the vacancy (its oxygen octahedron is missing one vertex) and one farther away (its oxygen octahedron is still intact). The planar case lowers the symmetry even further ($P1$), rendering all four Ti inequivalent. However, the difference between the two octahedra next to OV (and between the two farther from OV) after the relaxation is very small and thus negligible.

We start by relaxing the structure of pure LaTiO3 using DFT-GGA, for which the calculated geometry shows good agreement with available experiments and previous computer simulations Dymkowski and Ederer (2014); Cwik et al. (2003); Eitel and Greedan (1986). For the cases with OV, the atomic positions are allowed to relax within the cell volume and shape of the pure LaTiO3 case. Test calculations showed that the effects of performing a full relaxation are small (the volume increases by $0.08\%$) and irrelevant for the current discussion.

The calculated DFT bandstructure of pure LaTiO3 shows a band with predominant Ti-$t_{2g}$ character around the Fermi energy, and a Ti-$e_{g}$ band slightly higher in energy, with some level of entanglement between them in the vicinity of the $\Gamma$ point (see, e.g., Ref. Dymkowski and Ederer, 2014; Pavarini et al., 2005). The O-$p$ derived band lies ${\sim}4$ eV below the Fermi energy. Predicting metallic behavior, spin-unpolarized DFT-GGA fails to properly reproduce the correlated nature of the $d$ electrons of LaTiO3.

The bandstructure plot for the axial OV configuration is shown in Fig. 2a. The most prominent difference with respect to the structure of pure LaTiO3 is the presence of a band ${\sim}1.5$ eV below $E_{\rm F}$. A similar feature has been attributed to the OV in other perovskite systems, e.g., in SrTiO3-δ Janotti et al. (2014); Lin and Demkov (2013) or SrVO3-δ Lechermann et al. (2016). In the axial configuration, the vacancy band shows a strong dispersion, crossing the Ti-$t_{2g}$ band around the $R=(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})$ k-point, whereas for the planar configuration (not shown here), the vacancy band remains detached from the Ti-$t_{2g}$ bands across the whole Brillouin zone. We note that the high vacancy concentration used in the present calculations (${\sim}8.3$%) likely produces an overestimation of the OV-band dispersions.

In the following, we investigate how the presence of this OV band affects the Mott-insulating character of LaTiO3(OV). As shown in previous work Pavarini et al. (2005); Dymkowski and Ederer (2014); Imada et al. (1998), a good description of the low-energy physics of an early transition metal oxide like LaTiO3 is obtained by including only the Ti-$t_{2g}$-dominated bands around the Fermi level into the correlated subspace used for the DMFT calculation. In the present case, we extend this correlated subspace to also include the OV band slightly below these $t_{2g}$ bands. Thus, we construct MLWFs for the 12 Ti-$t_{2g}$ bands plus the OV band by defining an appropriate energy window and using initial projectors corresponding to 3 $t_{2g}$ orbitals located at each of the 4 Ti sites within the unit cell and an additional $s$ orbital trial projector centered on the vacancy site. Fig. 2a shows the good agreement between the DFT bands and the bands calculated from the $13$ MLWFs, despite the entanglement with the Ti-$e_{\rm g}$ bands. In Fig. 2b, where the projected density of states of the MLWFs is shown, we can see that the Ti-$t_{2g}$ bands are essentially described by the set of $12$ Ti-centered Wannier functions, with both inequivalent Ti presenting very similar features. We can represent essentially the whole weight of the vacancy Bloch state with a single Wannier function (orange line Fig. 2b). Interesting and crucial to our approach is the fact that in the real-space representation of the corresponding Wannier function (Fig. 2c), the spheroidal charge is centered approximately where the missing oxygen would be in the pure case, with tails on the surrounding atoms. For the rest of this work, we will focus on the case of the axial vacancy since our conclusions regarding the stability of the Mott insulating state are the same for both configurations.

Next, we perform paramagnetic DMFT calculations based on the fixed input electronic structure of the Hamiltonian expressed in the MLWF-basis. As stated earlier, an effective 3-orbital impurity problem is solved for each inequivalent Ti site plus an additional 1-orbital impurity problem for the OV-centered MLWF. Initially, we treat the vacancy as “uncorrelated”, i.e., we set $U($OV$)$ to zero, while keeping $U({\rm Ti})$ non-zero. For simplicity, we use the same value of $U$ on each Ti site. In Fig. 3a and 3b, the calculated spectral weight at the Fermi level and the corresponding orbital occupations are shown as a function of $U({\rm Ti})$ for all inequivalent sites of the system, i.e., two Ti sites and one OV sitetag . Our DMFT calculations for vacancy-free LaTiO3 (solid black line in Fig. 3a) show a clear MIT at $U_{\rm{MIT}}=4.5$ eV, indicated by a sudden drop in the spectral weight to $\bar{A}(0)\approx 0$, and thus a separation between a metallic regime (for $U<U_{\rm MIT}$) and a Mott-insulating regime (for $U>U_{\rm MIT}$). This is consistent with previous DFT+DMFT studies Pavarini et al. (2005); Dymkowski and Ederer (2014), where a value of $U\sim 5.0$ eV, slightly above $U_{\rm{MIT}}$, is often used to give a realistic description of LaTiO3.

The changes in LaTiO3(OV) are subtle with respect to the defect-free material. Importantly, the system still undergoes a MIT with $U_{\rm{MIT}}=4.8$ eV, a slightly higher value (by ${\sim}0.3$ eV) than the stoichiometric case. For all sites (including the vacancy site) the spectral weight $\bar{A}(0)$ drops to zero at essentially the same value of $U_{\rm{MIT}}$. In the metallic state, we find all three Ti-$t_{2g}$ orbitals to be fractionally occupied, while in the insulating state the system exhibits a strong orbital polarization with one $t_{2g}$ orbital nearly completely filled, and the other two nearly empty, again consistent with results for the stoichiometric system Pavarini et al. (2005); Dymkowski and Ederer (2014). However, the orbital polarization in the Mott-insulating state is reduced for the Ti atom next to the vacancy, for which the total charge of one electron is split between two $t_{2g}$ orbitals with occupations $0.8$ and $0.2$ ($U=7.0$ eV). For the whole range of $U$, the vacancy site stays close to doubly occupied, in particular in the insulating regime for $U>U_{\rm MIT}$, with a maximum charge transfer of ${\sim}0.22$ e- to the neighboring Ti (for $U=3.0$ eV). This charge transfer gradually decreases until it nearly vanishes at $U_{\rm MIT}$.

Finally, we perform calculations with a non-zero $U$ on the vacancy site, treating explicitly its correlations, setting $U($OV$)=U($Ti$)$ for simplicity. Given the larger spread shown by the calculated OVWannier function compared to those describing the Ti-$t_{2g}$ states, and the inverse relation between the localization of a Wannier function and its on-site Coulomb energy repulsion, this choice of $U($OV$)$ can be viewed as an upper limit. In Fig. 3c and 3d we see that the changes introduced by the defect are smaller than for the case with $U($OV$)=0$. The critical $U$ value for the MIT is now $U_{\rm{MIT}}=4.7$ eV, closer to the value for the defect-free system. In addition, the amount of charge donated to the neighboring Ti from OV is further reduced, becoming almost negligible even below $U_{\rm{MIT}}$.

These differences are also reflected in the total spectral functions shown in Fig. 4. For a small $U=3$ eV, the main difference between the defect-free case and the two cases with OV is the presence of the additional vacancy band centered around \sim$$-1.5 eV. Approaching $U_{\rm{MIT}}$, a three-peaked structure, the well-known hallmark of a strongly correlated metal, emerges for all three scenarios. Similarly to the case of the correlated metal SrVO3 discussed in Ref. Backes et al., 2016, the lower Hubbard band emerges at approximately the energy where the vacancy band is located, making it experimentally hard to distinguish between the two. The slight increase of $U_{\rm MIT}$ due to the presence of the vacancy is apparent from the $U=4.6$ eV panel where pure LaTiO3 is already gapped, while both LaTiO3(OV) cases still show a prominent quasiparticle peak. The next panel to the right, $U=5.0$ eV, shows that the case of LaTiO3(OV) with $U($OV$)=U({\rm{Ti}})$ (blue data) exhibits a gap only slightly smaller than defect-free LaTiO3, whereas in the case with $U($OV$)=0$ (magenta data) the gap is still minimal. Deeper within the insulating phase, for $U=6.0$ eV, the gap still remains smaller for $U($OV$)=0$, compared to the defect-free material and the case with nonzero $U($OV$)$.

From the localization of the corresponding Wannier functions, we expect that a realistic value for $U$ on the vacancy is somewhere in between the two cases studied here, $0<U($OV$)<U(\rm{Ti})$. We therefore also perform DMFT calculations with $U($Ti$)$ kept constant while $U($OV$)$ is varied. We find that the resulting occupations of all sites are largely independent of $U($OV$)$. Furthermore, the system remains either metallic or insulating, depending on whether $U(\rm{Ti})$ is below or above $U=4.6$ eV. Only for $U($Ti$)=4.6$ eV, where the system is still metallic for both $U($OV$)=0$ and $U($OV$)=U(\rm{Ti})$, see Figs. 3 and 4, does increasing the Hubbard $U$ on OV trigger a transition to the insulating state, albeit for a rather high $U($OV$){\sim}7.0$ eV. Thus, we find that for $U($Ti$)>4.7$ eV the system remains insulating for any $U($OV$)$, while in the narrow range of $4.5$ eV $<U($Ti$)\leq 4.7$ eV the incorporation of oxygen vacancies can potentially destroy the Mott insulating character. However, even in this case, the vacancy band stays essentially fully occupied and the MIT is rather caused by subtle changes in the electronic structure. We emphasize that such a scenario would be completely different from doping of the insulating state by the electrons left behind by the oxygen vacancy, which does not occur for any reasonable value of $U$.

In order to understand why increasing the Coulomb repulsion on the vacancy site does not lead to a depletion of the corresponding states and a charge transfer into the Ti-$t_{2g}$, one has to consider the effect of the double-counting correction in our DFT+DMFT calculations. This correction attempts to subtract the effect of the electron-electron interaction within the correlated subspace that is included both on the DFT and the DMFT level, in order to avoid double counting. In practice, it enters the calculation as a local potential shift that depends linearly on both $U$ and the local occupation, and is applied to all Ti sites as well as to the OV. Due to the higher occupation of the OV compared to the Ti sites, the double-counting shift is larger for the former (for similar $U$ values), i.e., the double-counting term shifts the vacancy states further down in energy, thereby reinforcing the double occupation of these states and preventing any charge transfer into the higher-lying Ti bands.

If the double-counting term for the vacancy site is nullified within the calculation (Fig. 5), the OV site is indeed depleted, reaching the half-filled state for $U($OV$)\gtrsim 3$ eV (even for $U($Ti$)=6.0$ eV, deep inside the insulating regime for the pure system). The system then stays in a metallic regime even for values of $U($Ti$)$ where pure LaTiO3 or LaTiO3*-δ* with full double counting already behaves as a Mott insulator. We note that, while there is uncertainty regarding the most appropriate form of the double-counting correction Karolak et al. (2010), completely neglecting the double counting is clearly unphysical, and is done here only for test purposes. We find that, scaling down the value of the double-counting term applied to OV to ${\sim}75\%$ of its original value still results in a doubly occupied vacancy and a stable Mott-insulating state. We therefore conclude that, in spite of any uncertainties regarding the exact value of the double-counting correction, our DFT+DMFT calculations predict a very weak effect of the vacancy states on the Mott-insulating character of LaTiO3. The charge released by the OV remains mainly on the vacancy sites and does not change the filling of the Ti-$t_{2g}$ bands, thus leaving the Mott insulator essentially unperturbed with respect to the situation in pure LaTiO3.

To conclude, we have performed DFT+DMFT calculations for OV-defective LaTiO3 to investigate the effect of such vacancies on the Mott-insulating state. We find that the presence of the vacancy creates new states at energies slightly below the partially occupied Ti-$t_{2g}$ bands. Provided that the Coulomb repulsion is not extremely close to the critical value for the MIT within the pure compound, these defect states remain doubly-occupied, and therefore will not change the filling of the Ti bands and thus affect the Mott-insulating character of LaTiO3. In spite of the relatively small gap of LaTiO3, reflecting its close vicinity to the MIT, its Mott insulating character is surprisingly robust against the incorporation of oxygen vacancies, that do not dope the system for any value of the interaction strength.

The explicit treatment of the vacancy state developed in this work provides an efficient and physically transparent way to study electronic correlations in defective systems, requiring no prior assumptions about the nature of the defect states. This complements other approaches of defect characterization Lin and Demkov (2013), and will hopefully motivate studies in other Mott materials with different kinds of point defects.

This work was supported by the Swiss National Science Foundation through NCCR-MARVEL. Calculations have been performed on the cluster “Mönch” and “Piz Daint”, both hosted by the Swiss National Supercomputing Centre, and the “Euler” clusters of ETH Zurich.