Influence of impurities on the high temperature conductivity of SrTiO₃

The rich chemistry and wide range of industrial applications of strontium titanate (STO) have inspired decades of research, experimental and theoretical, into its bulk properties. Of particular interest is control over the material's defect chemistry through doping and processing conditions to tune its bulk electrical conductivity. In this vein, an extensive body of experimental high temperature electrical conductivity (HiTEC) data and the associated canonical defect models in barium and strontium titanate were compiled in a recent review.¹ 

Canonical defect models have made great strides in the development of dielectric devices based on perovskite oxides but are often implemented with a limited number of defect reactions. The assumptions of weak complex formation and a single dominant defect make the canonical approach tractable as it becomes unwieldy with many defects and charge states. However, these simplifications can obscure the physical mechanisms responsible for an observed macroscopic response.^2–4 This work aims to describe HiTEC in STO using defect formation energies derived from first principles calculations in a model based on the grand canonical (GC) ensemble. Such models have provided a mechanistic understanding of changes in coloration observed in Fe doped STO,⁵ facilitated extension to multi-defect systems, provided a clear route to address defect complexes,^5,6 and accelerated the development of other electronic grade materials and oxides.^6–10 

Modeling HiTEC in STO presents several challenges, as the material is a mixed ionic/electronic conductor. There are typically a large number of impurities present at or above concentrations of main contributors to HiTEC (electrons, holes, and oxygen vacancies).¹¹ First nearest neighbor (1NN) complexes can trap mobile ionic carriers, reducing their contribution to the bulk conductivity.¹² Electronic charge carriers may localize as small polarons and impact the conductivity.¹³ Furthermore, over the temperature range investigated (⁠$T=800–1000 °C$⁠), STO's bandgap decreases 0.5–0.7 eV from its 0 K value of 3.25 eV.¹⁴ While these are notable issues, of primary concern to this investigation is the impurity content of STO.

Few existing studies on STO have simultaneously quantified the background impurity content and electrical characteristics of samples studied. Chan et al. are one of the few researchers who report the concentrations of impurities present in their polycrystalline STO samples.¹¹ Using Spark Ion Mass Spectroscopy, 18 unintentional impurities were found in concentrations above 10¹⁶ cm^–3 (Table I). Such background impurities are often ignored or lumped into a single effective impurity in canonical defect models to reduce the complexity as it is difficult to separate the effects of individual impurities from experiment. Though previous works showed that the optical response of Fe-doped STO is insensitive to the presence of background impurities during coloration,⁵ bulk electrical conductivity is expected to exhibit a greater sensitivity due to the cumulative influence of all ionized impurities on the position of the equilibrium Fermi level.

In this letter, the effects of background impurities on HiTEC are evaluated using a GC first principles approach in three doping regimes: nominally undoped, donor doped (Nb), and acceptor doped (Al). All results are compared to the experimental measurements of Chan et al. given in Ref. 11 under the assumption that all samples had identical impurity profiles. Qualitative agreement of the model with these measurements improves significantly when 8 of the 18 background impurities detected in their work are included in the nominally undoped case. The influence of background impurities in the Nb and Al doped cases was found to be weaker as both form defects with shallow states, do not form strongly associated 1NN complexes, and were present well in excess of other influential impurities. Additionally, the model captures experimentally observed changes with temperature, chemical environment during annealing, and the primary impurity concentration.

A GC model was constructed using point defect formation energies as determined by density functional theory (DFT) with a screened hybrid exchange correlation functional. In the GC formalism, no a priori assumptions are made about the most prevalent defect in a particular environment. This is determined directly from the defect (or defects) with the lowest formation energy (energies). Formation energies of Si, Fe, Cl, Mg, Ca, Ni, N, Al, and Nb related defects as well as native vacancies, anti-sites, and a small hole polaron on the O-site were simulated as a function of the charge state in the GC formalism (see Table S1 in the supplementary material for a list of all defects). 1NN complexes between M_Ti and v_O (where M = Si, Fe, Mg, Nb, Ni, and Al) were simulated and included in HiTEC simulations. We have simulated 8 of the 18 impurities noted by Chan et al. plus Nb. Simulation of the remaining ten impurities and their possible complexes was intractable due to the computational expense of the large number of possible defects and the use of hybrid exchange correlation functionals. Nevertheless, qualitative agreement is evident when the 8 simulated impurities and their complexes are considered. The GC formalism also allows straightforward connection between the chemical potential of the species and experimentally controlled variables like oxygen partial pressure P(O₂). The temperature dependence of the bandgap was included in the model using an expression derived from T-dependent measurements of STO's optical absorption edge.¹⁴ HiTEC was modeled using the calculated defect and carrier concentrations and carrier mobility data from the literature.

The temperature dependent electron and hole mobilities reported by Moos and Härdtl¹⁵ were used with the oxygen vacancy mobility reported by De Souza et al.¹⁶ Both electronic carrier mobilities were derived from Hall and conductivity measurements taken above $800 °C$ and were recently used in drift diffusion simulations based on canonical defect models.^15,17 The electron mobility was measured in nominally undoped single crystal and La doped ceramic STO and found to be described by the same expression.¹⁵ The hole mobility was measured in Fe doped single crystal STO¹⁸ and corrected by Moos and Härdtl for the nonideal contact geometry used in the original study.¹⁵ An expression for the ionic mobility was determined from a tracer diffusion study on acceptor doped, single crystal STO.¹⁶ Though measured in samples processed in different conditions and with different impurity contents, these values provide a first order approximation to conductivity from our calculated carrier concentrations. As conductivity depends linearly on the mobility, this is not expected to influence predicted qualitative trends.

DFT calculations were performed using projector augmented wave (PAW) pseudopotentials as implemented in the Vienna Ab Initio Simulation Package (VASP) 5.3.3.^19–22 The screened hybrid exchange-correlation functional of Heyd, Scuseria, and Ernzerhof (HSE06)^23,24 with an exact exchange amount of 0.2362 was used, and the planewave basis set was expanded to a cut-off energy of 520 eV. The PAW potentials contained 10, 12, and 6 valence electrons for Sr (4s²4p⁶5s²), Ti (3s²3p⁶4s²3d²), and O (2s²2p⁴), respectively. Collinear spin polarization was accounted for in all calculations. The predicted bulk lattice parameter, and direct (Γ-Γ) and indirect (R-Γ) bandgaps were 3.90 Å, and 3.62 and 3.25 eV, respectively. All are in agreement with measured values.^13,14,25,26 The density of states (DOS) was calculated using a $13×13×13$ k-point mesh and is shown in Fig. S5 (supplementary material). Single defects and defect complexes were simulated in 135 and 180 atom supercells, respectively, using a $2×2×2$ k-point mesh. Atoms within 5 $Å$ of each defect were relaxed until the default convergence criteria were met.

Total energies for the supercells containing a defect $EDqtot$ and pristine bulk $Ebulktot$ for a defect D in charge state q were used to calculate the defect formation energy $EDqf$ as a function of Fermi level μ_e relative to the valence band maximum (E_v) using Eq. (1) as detailed in recent reviews^27,28

A correction for the finite size of the supercell, $ΔV$⁠, was calculated based on the work of Kumagai and Oba²⁹ using a dielectric constant of 300.^30,31 The energy from spurious interactions between a charged defect and its images is calculated by a Madelung sum, treating each defect as a point charge. This energy is subtracted from that of the defective supercell to recover the energy of an isolated defect representative of dilute concentrations. Kumagai and Oba demonstrated that the energy of a charged defect corrected in this way converges more rapidly than the uncorrected energy with the supercell size in a wide range of systems.²⁹ The defect energy is further refined by aligning the image-corrected local potential of the defective supercell with the local potential of the bulk supercell at locations far from the defect and its images. Testing convergence of the electrostatic interaction of periodically repeated defects requires large supercells, which becomes computationally intractable when using hybrid functionals. Therefore, energies of a number of representative charged defects in our system were tested using traditional gradient corrected functionals of the Perdew-Burke-Ernzerhof formalism for unit cells up to 625 atoms. It was found that formation energies determined here were converged within 0.2 eV with respect to the 625-atom cell.

The sum in Eq. (1) represents the number of atoms n_i and chemical potential μ_i of species i added or removed from the bulk. For Sr, Ti, and O, $μi=μio+Δμi$⁠, where $μio$ is the DFT energy per atom of i in its 0 K reference phase. $Δμi$ represents deviations from this state and is bound by dissociation into reference [$Sr(s), Ti(s)$⁠, and $O2(g)$] or competing [$SrO(s)$ and $TiO2(s)$] phases giving the STO stability region in Fig. 1(a). Bulk $Δμi$ were constrained to the dashed line bisecting the STO stability region unless specified otherwise. The value of $ΔμO$ was connected to P(O₂) at a given annealing temperature using data from the Joint Army Navy Air Force thermochemical tables.³² The solid and dotted lines were used to simulate annealing in TiO₂ and SrO rich conditions, respectively. Given a set of bulk chemical potentials, μ_i for each impurity was solved such that the concentration of i matched the values measured by Chan et al. reproduced in Table I.

Free electron and hole concentrations were evaluated by integrating the calculated DOS multiplied by the Fermi-Dirac distribution function over appropriate energy intervals (the conduction band for electrons and valence band for holes). The temperature dependence of the bandgap was modeled using Eq. (2) under the assumption that 2/3 of the reduction was accounted for by a rigid increase in E_v with S = 5.9, $〈ℏω〉$ = 63 meV, where k is the Boltzmann constant.¹⁴ The reduction of the bandgap with an increase in temperature is a result of lattice expansion and electron-phonon interactions. Explicit calculation of this shift is of interest but is computationally intensive.

A self-consistent solution for the condition of charge neutrality was used to identify μ_e and the defect concentrations. The electron, hole, and oxygen vacancy mobilities discussed previously were used together with the defect concentrations to calculate the bulk conductivity as $σtot=e(nμn+pμp+2[vO··]μvO··)$ where all variables have their standard meaning

The effect of successively adding each impurity simulated in order of the decreasing concentration on σ_tot in STO at $800 °C$ is shown in Fig. 1(b). Each profile was generated over P(O₂) = 10^–20–10⁵ atm for comparison with measurements in Ref. 11 at $800 °C$⁠. Concentrations of defects and carriers used in the model are given in Figs. S1–S4 in the supplementary material. Predicted σ-profiles for incorporation of the first three major impurities, Si, Fe, and Cl, fail to match the shape of the measured profile. Si, despite being the dominant impurity, does not alter the predicted conductivity, in agreement with the intuition of Chan et al.¹¹ A plateau appears in reducing conditions on the addition of Fe. The height of the plateau is reduced on the addition of Cl, and the profile is shifted to the left. Mg further reduces the height of the plateau and shifts the profile further left while Ca, like Si, has little to no effect. Agreement improves substantially by adding Ni, N, and Al despite their relatively low measured concentration.

Including all simulated impurities yields a slope close to −1/6 in reducing conditions, which transitions to roughly −1/4 in intermediate conditions, before switching to about +1/4 in oxidizing conditions. Similar changes in the slope are observed in measured σ-profiles with P(O₂).¹¹ The influence of temperature on HiTEC in nominally undoped STO was also investigated. Predicted σ-profiles from $800–1000 °C$ every $50 °C$ are shown in Fig. 1(c). As T increases, σ_min increases and shifts to a higher P(O₂), which agrees well with experiment. The caveats on quantitative agreement discussed previously still apply. Quantitative agreement in terms of the P(O₂) at which the conductivity minimum σ_min occurs is not expected as impurities are missing from the model, and the carrier mobilities used were measured in comparable conditions.

Adding impurities that predominantly form conventional acceptor type defects (Mg, Ni, and Al) shift σ_min to a lower P(O₂). Interestingly, N has a similar influence on σ_min, while Fe behaves more like Cl. The influence of Si and Ca is negligible. Each behavior can be explained from atomistic details of the defect chemistry revealed by virtue of the GC and first principles approach. Both Si and Ca prefer to incorporate as neutral defects on the Ti and Sr sites, respectively, and thus do not significantly influence HiTEC. Cl and N substitute for O but prefer the +1 and −1 charge states, respectively, causing Cl to act as a donor and N as an acceptor. Al, Fe, Mg, Nb, and Ni substitute for Ti. Nb acts as a hydrogenic donor with a preference for +1 across all conditions investigated. The remaining impurities primarily form negatively charged defects over the range of conditions investigated. Al, Mg, and Ni reduce the overall electron concentration as expected of acceptors, but Fe increases the electron concentration based on the predicted σ-profiles in Fig. 1(b).

Although Fe prefers the Ti site, Fe_Ti has a high affinity for native oxygen vacancies and readily forms 1NN complexes. In reducing intermediate oxygen partial pressures, the concentration of Fe_Ti-v_O exceeds or is on par with that of isolated Fe_Ti and thus controls the dopant behavior.⁵ As Fe_Ti -v_O prefers the +1 charge state, its predominance over Fe_Ti in the neutral or −1 charge state causes Fe to behave more like a donor in STO in the conditions studied. Similar behavior is not predicted for Al, Mg, Nb, and Ni as the 1NN complexes are predicted to be well below the concentration of the isolated substitutional defects in the conditions studied.

To simulate annealing in SrO and TiO₂ rich environments, defect concentrations were calculated along different paths in the STO stability diagram shown in Fig. 1(a). The predicted σ-profiles for nominally undoped and Nb doped STO annealed at $1000 °C$ in different chemical environments are shown in Fig. 2(a). Varying the processing environment between the competing phases bounding the STO stability region had a minor effect on the predicted conductivity in the case of nominally undoped STO. Predicted σ-profiles for the Sr-rich and intermediate conditions overlap completely while Ti-rich conditions shift σ_min to a lower P(O₂). Varying the processing environment had a more pronounced effect on the predicted σ-profiles for donor doped STO with 8.4 × 10¹⁸ cm^–3 Nb in addition to the background impurities. Moving from Sr to Ti-rich conditions reduces the predicted conductivity significantly, and a minimum is only discernible in Ti-rich conditions within the P(O₂) range investigated, consistent with experiment.¹¹ The variation observed in nominally undoped STO in Ref. 11 was not as pronounced but did exhibit a greater dependence on Ti content than predicted in Fig. 2(a). Considering the influence Nb has on the observed and predicted conductivity, including the remaining impurities could recover the observed behavior in the nominally undoped case.

The model was also used to investigate the effect of intentionally doping STO with Al, an industrially relevant acceptor. At $1000 °C$⁠, a minor increase in σ_min is predicted as the minimum shifts to a lower P(O₂) on increasing the Al content from 8 (1.4 × 10¹⁷) to 281 (4.7 × 10¹⁸) to 1080 (1.8 × 10¹⁹) ppm (cm^–3). Trends in the simulated σ-profiles for these conditions shown in Fig. 2(b) are in good agreement with measurements from similar conditions.¹¹ The 8 ppm line is equivalent to the nominally undoped case at $1000 °C$ in Fig. 1(c).

Agreement was only achieved by including the temperature dependence of the bandgap with Eq. (2) in our defect equilibrium calculations. Changing the fraction did not significantly influence the shape of the predicted σ-profiles and only shifted them along the P(O₂) axis. Additionally, previous work in Si³³ and barium titanate³ provides support for attributing 2/3 of the reduction to a rigid increase of E_v. Assuming a bandgap fixed at the 0 K value of 3.25 eV yields the σ-profiles in Fig. 2(c) which do not reflect the shape or trends observed with the increasing Al content.¹¹ Future work is needed for a better understanding of how the band edges and defect levels vary with temperature.

Unlike Fe, the dominant, electrically active impurity in the nominally undoped case, Nb and Al form shallow states, interact weakly with native oxygen vacancies, and are present in high concentrations relative to the background impurities simulated. As a result, they screen out the influence of background impurities on the predicted trends shown in Figs. 2(a) and 2(b). However, ignoring the impurities simulated still perturbs the predicted σ-profiles, increasing the conductivity in Nb doped STO in the intermediate and Sr-rich limit and shifting the profiles up and left in the Al doped case. This indicates that the reaction parameters used in canonical defect models may be influenced by background impurities.

Ironically, the work of Chan et al. is often cited to support claims of a solid understanding of the effects of background impurities on the electrical behavior of STO even though this claim is not made in the original work. Using the GC approach, the effects of individual impurities on HiTEC in STO were predicted. The results suggest that HiTEC may not be dominated by a single defect in high concentrations and instead relies on the ensemble of defects formed by incorporating all background impurities. The effects of background impurities can be screened to some degree with high concentrations of shallow dopants that interact weakly with native oxygen vacancies. This work motivates the need for improved quality control in ceramic oxides, if there is interest in studying the properties of individual defects. For ceramic grade materials, our approach offers a method of handling multiple impurities, defect complexes, and different processing conditions which are intractable in the canonical formalism.

See supplementary material for additional information used in this study. This includes the defects and charge states simulated and defect concentration plots for conductivity plots in Figs. 1 and 2.

The authors acknowledge funding provided by AFOSR BRI: FA9550-14-1-0264. E.C. Dickey is acknowledged for her invaluable feedback. P.C. Bowes was supported by the National Defense Science & Engineering Graduate (NDSEG) fellowship program. Computing time was provided by DoD HPMCP.