Defect mechanisms of coloration in Fe-doped SrTiO₃ from first principles

Strontium titanate (STO) has a wide indirect bandgap (3.25 eV), a high dielectric constant, and poor bulk carrier mobilities. It is used industrially to manufacture capacitors and varistors.^1–5 There is a growing research focus on this material for applications in thin-film capacitors, resistive random access memory, gas sensing, and high electron mobility two dimensional electron gas heterostructures.^6–9 It is also commonly used as a growth substrate for superconductors and other oxide thin films.^7,10

Ceramic single-crystal STO, commonly used for dielectric applications, has a significant number of background impurities present at levels of parts per million or higher.¹¹ In an attempt to suppress the behavior of these unintentional impurities, a single dopant element is added at concentrations above the background impurities.¹² Iron is commonly used because it is cheap, readily available, and makes the Fe-doped ceramic STO (Fe:STO) insulating through compensation.

Under thermal and electrical stress, the initial resistance of Fe:STO degrades over time.¹³ Experimental measurements of Fe:STO have found that resistance degradation is accompanied by the formation of a color front that transitions from dark brown at the anode (+ external potential) to transparent at the cathode (– external potential).^7,14,15 Similar absorption peaks to those found in electrocoloration have been observed in Fe:STO annealed at different oxygen pressures. Combined electron paramagnetic resonance (EPR) and absorption measurements provided indirect evidence that neutral iron substituting on the titanium site (⁠$FeTi0$⁠) is likely associated with the brown coloration in oxidized Fe:STO.^16,17 This has led to the prevailing viewpoint that both coloration processes arise from shifts in oxygen vacancy concentration, which changes the ionization of iron substitutionals.¹⁴ 

The defect models typically invoked to understand the physical properties of Fe:STO assume that $FeTi0, FeTi−1, VO2$⁠, n, and p are the only significant species in the system and the interaction between Fe_Ti and V_O can be neglected.^3,12,18–21 These models then adjust the ionization of the Fe dopant until predictions fit a measured physical property. Nevertheless, a number of EPR studies have detected the presence of a first nearest neighbor iron oxygen vacancy complex, which suggests this interaction is non-negligible.^{15,16,22–24} Despite the frequent detection of the complex in experiments, its influence on physical properties remains unclear.

In this article, a combination of experimental and ab initio techniques are used to understand the underlying defect mechanisms important to the coloration of Fe:STO annealed in different processing environments. These calculations explicitly account for the presence of the first nearest neighbor complex identified by EPR measurements. Iron is found to be primarily present in four different forms (complexes and charge states) across accessible $PO2$ ranges, rather than the two forms typically assumed. Furthermore, direct evidence is provided that the $FeTi0$ defect gives rise to two absorption shoulders which cause the brown coloration observed upon oxidation.

The coloration behavior of Fe:STO with 0.01 wt. % Fe from MTI Corp. was investigated experimentally. One set of samples was degraded while other samples were annealed in different partial pressures of oxygen. Absorption spectra taken from different regions of a degraded crystal subjected to a 100 V bias applied across the ∼2.5 mm sample for 19 h at 480 ± 10 K are presented in Fig. 1(a). A separate sample was taken through four anneal(1170 ± 5 K)-quench(300 K) cycles alternating between air and a $PO2=15 Pa$ in a tube furnace, with optical spectra taken at the end of each cycle. Absorption spectra from this annealed sample are presented in Fig. 1(b). The color of Fe:STO was found to switch reversibly when annealed in the different environments from brown (air) to transparent (⁠$PO2=15 Pa$⁠). All spectra show a strong indirect band edge absorption at approximately 380 nm (3.25 eV). The spectra of oxidized Fe:STO (anode and air annealed) show broad absorption shoulders centered at approximately 460 nm (2.70 eV) and 570 nm (2.18 eV). These peaks are consistent with previously reported experimental measurements.^14,25,26

Density functional theory (DFT) calculations with screened hybrid exchange correlation functionals were used to understand the underlying defect mechanisms governing the coloration of Fe:STO. Vacancies, native and impurity interstitials, anti-sites, substitutional impurities, and the first nearest neighbor $FeTi-VO$ complex were simulated. The background impurities simulated were chosen based on available composition data from MTI. All impurities found in the crystals at a level greater than 10¹⁷cm⁻³ were simulated, as were several that appear in concentrations of 10¹⁵–10¹⁶cm⁻³, which includes S, Cl, Si, Al, Na, Mg, Ba, and Ni. Isolated point defects and nearest neighbor complexes were simulated using 3 × 3 × 3 (135 atoms) and 3 × 3 × 4 (180 atoms) repetitions of the primitive cell, respectively, with atoms within 5 Å of the defects free to relax.

All calculations were performed with the HSE06^27,28 exchange correlation functional in VASP 5.3.3 with an exact exchange amount of 0.2362, collinear spin polarization, and a plane wave kinetic energy cutoff of 520 eV.^29–32 Defect calculations and bulk electronic structure calculations used 2×2×2 and 6×6×6 Monkhorst-Pack reciprocal space meshes, respectively. The amount of exact exchange was selected to correct the underestimation of the bandgap common to traditional functionals. With the parameters presented here, a direct bandgap at Γ of 3.62 eV and an indirect bandgap between R and Γ of 3.25 eV was obtained, which are in good agreement with previous experiment and theory.^33–35 In addition to the improvement in the electronic structure, HSE06 also improves structural and thermodynamic predictions. These parameters led to a bulk lattice parameter of 3.901 Å and an enthalpy of formation of −16.53 eV, which are in close agreement with experiment.^36–38 Furthermore, HSE06 also improves the over delocalization of electronic charge that is common for traditional functionals.³⁹ 

The formation energy of a point defect D in charge state q is determined by the following equation:

In this expression, $EDqtot$ and $Ebulktot$ are the total energies of super cells containing defect D in charge state q and the perfect bulk, respectively. In this grand canonical approach, the bulk is assumed to be in thermodynamic equilibrium with respective chemical reservoirs, where μ_i is the chemical potential of species i and n_i is the number of atoms exchanged between the bulk and reservoir. Further, $μi=μio+Δμi$⁠, where $μio$ is the DFT energy per atom of the thermodynamic reference phase of i and Δμ_i represents deviations from this 0 K reference phase. Bounds on Δμ_i can be set to account for stability against precipitation into reference phases or competing phases, and, in the case of impurities, solubility limiting phases.^40–44 Accessible values for Δμ_i are shown graphically for Sr, Ti, and O in Fig. 2(a). Bulk STO is thermodynamically stable in the thin white strip of this triangle. ΔV corrects for the finite size of the cell and was obtained using a method based on the work of Kumagai and Oba with a relative permittivity of 300.^39,45–47 The Fermi level E_f was taken relative to the valence band maximum (VBM) and was used as a free parameter when plotting the formation energies. Results of the defect simulations were imported into a point defects database where post-processing was performed to extract defect properties (e.g., thermodynamic transition levels, optical signatures, etc.), while further analysis was performed by solving relevant charge balance equations as done previously.^48,49 Configurational entropy is accounted for in the defect concentration expression prefactor through the number of identical configurations. Vibrational, electronic, and magnetic entropy are neglected due to their small expected contributions relative to other terms in the formation energy expression and the significant expense required to capture these energies for each charged defect. This is not expected to affect qualitative trends.⁴⁴ 

Formation energies for native vacancies and Fe defects are shown in Fig. 2(b). The formation energies of the Fe defects are plotted with μ_Fe set relative to relevant solubility limiting phases at 0 K. The formation energies in the left pane are for highly reducing conditions while those in the right pane are for highly oxidizing conditions. Our results indicate that native interstitials and anti-sites have higher formation energies than vacancies in all processing regimes and are, therefore, not shown. The most favorable vacancies in reducing and oxidizing conditions are V_O and V_Sr, respectively. V_Ti is more favorable than V_Sr in oxidizing conditions when the Fermi level pins near the conduction band minimum (CBM), but this does not often occur in practice.^3,11 The formation energies and thermodynamic transition levels for vacancies presented here are consistent with those of Janotti et al.³⁴ 

Iron was found to prefer the titanium site, either as an isolated defect or in a first nearest neighbor complex with an oxygen vacancy. The isolated Fe_Ti defect assumes all charge states ranging from +1 to −2 within the bandgap. The predicted Fe_Ti (0|−) midgap state is approximately 1.7 eV above the VBM. In intermediate environments, the formation energy of the $FeTi-VO$ complex is close to or lower than either the Fe_Ti or the V_O. This indicates that it can be present in significant concentrations despite the configurational entropy penalty relative to the isolated defects. This is consistent with previous EPR measurements.^{15,16,22–24} The complex adopts the +4, +2, +1, and 0 charge states as the Fermi level moves from the VBM to the CBM, skipping the +3 charge state. The strong interaction between the constituent defects leads to a large favorable (positive) binding energy across most of the Fermi level, as shown in Fig. 2(c), where the binding energy for Fe_Ti and V_O is given in Eq. (2)^39,44

The absorption and emission energies for each defect in the point defects database were calculated using the Franck-Condon approximation.⁴⁴ For each defect, both defect to CBM as well as VBM to defect transitions were evaluated. The majority of the predicted absorption energies either lie close enough to the band edge to blend into its signal, or are in the infrared, where they would not show up in a UV/vis spectrum. Several defects with absorption peaks in the vicinity of those seen experimentally have high formation energies and were eliminated because of their resultant insignificant concentrations in the material. This down selection left four transitions in the visible spectrum, all of which are related to iron. Two of these four, the $(FeTi-VO)0$ to CBM and $FeTi−1$ to CBM transitions, do not satisfy the spin selection rule, and would thus be expected to be of very low intensity, if present at all. The only transitions remaining that satisfy the spin selection rule were associated with a defect of low formation energy, and were in the vicinity of the experimentally observed peaks were both associated with $FeTi0$⁠. The configuration coordinate diagrams for $FeTi0$ to CBM and VBM to $FeTi0$ are shown in Figs. 2(d) and 2(e), respectively. The $FeTi0$ to CBM transition (Fig. 2(d)) has an energy of 2.72 eV (456 nm), while the VBM to $FeTi0$ transition (Fig. 2(e)) has an energy of 2.17 eV (571 nm). These predicted energies are in good agreement with the experimentally observed absorption shoulders.

The calculated formation energies were used to obtain defect concentrations after an anneal and quench cycle. Defect concentrations are related to the defect formation energies via an Arrhenius expression that depends on the Fermi level and chemical potentials.³⁹ The Fermi level is the chemical potential of the electron in equilibrium and is determined by charge neutrality. Charge neutrality is satisfied when the sum of ionized donors and holes is equal to the sum of ionized acceptors and electrons (i.e., p − n + ΣqD^q= 0). Native chemical potentials are fixed by a path through chemical potential space where the value of Δμ_O is connected to the oxygen partial pressure at the annealing temperature using data from the Joint Army Navy Air Force (JANAF) thermochemical tables.^41–44,50 The values for Δμ_Ti and Δμ_Sr must be within the white strip in Fig. 2(a). Here, a path directly down the middle of this stability region was chosen and is shown in Fig. 2(a). While this is an arbitrary selection, it was found not to influence the major conclusions on coloration and only changes the concentrations of background defects, such as V_Sr and Fe_Sr. This leaves the dopant chemical potential, which can assume values of negative infinity up to the solubility limiting energy. Here, the known dopant concentration was used to back calculate the dopant chemical potential at that concentration. If growth was simulated, care should be taken to ensure that impurity chemical potential limits were not exceeded.^49,51,52 For annealing, it was assumed the temperature would not be high enough to change the dopant content of the crystal. During the anneal, defect concentrations were calculated as a function of chemical potential. At each chemical potential, the Fermi level was determined through charge neutrality. After the anneal and during the quench, the high temperature defect concentration was frozen, and a new Fermi level required to maintain charge neutrality was determined.

For this work, we simulated anneals at 1073 and 1173 K followed by an immediate quench to 300 K with an iron doping level of 0.01 wt. % Fe (2.77 ⋅ 10¹⁹cm⁻³). The predicted defect concentration profiles at annealing and quenched temperatures are shown in Fig. 3. Based on these results, a significant fraction of iron is almost always present as a first nearest neighbor complex.

At annealing temperatures, the results presented in Fig. 3 indicate that four distinct defect chemistry regimes form. In extremely reducing environments (Region A), the beginnings of the classical $VO+2:n$ compensation tail can be seen. In moderately reducing conditions (Region B), electrons and the $(FeTi-VO)+1$ complex compensate each other and are the dominant charged species. In the regime where most processing is done and out to highly oxidizing conditions (Region C), the $FeTi−1$ and $(FeTi-VO)+1$ defects compensate each other as majority defects while low concentrations of $FeTi0$ and $(FeTi-VO)0$ cross each other with changing $PO2$⁠. Finally, in extremely oxidizing conditions (Region D), we see the beginnings of a regime where the $FeTi0$ defect becomes dominant.

As the ensemble is quenched to 300 K, the defect chemistry from the high temperature regime is altered. Electrons and $VO+2$ still compensate each other in extremely reducing conditions (A). As $PO2$ is increased, $(FeTi-VO)0$ becomes the dominant defect (B). In the quenched region B, there is a change in the compensation of minority charged defects from $VO+2:n$ to $(FeTi-VO)+1:FeTi−1$⁠. For quenched regions C and D, there is little change to the dominant defects present as compared to the high temperature compositions.

While the dominant defects do not change character after quenching in the higher $PO2$ regime, there is an abrupt change to the concentration of neutral defects present. Most notably, this occurs in the partial pressure window where samples were annealed experimentally (gray lines in Fig. 3) and the $PO2$ of the abrupt drop changes with annealing temperature. Between the gray lines, there is now a near vertical transition between the $(FeTi-VO)0$ and $FeTi0$⁠. The sharp drop-off in concentration of these neutral defects is a result of a jump in the Fermi level, which is the primary mechanism for the material to maintain charge neutrality when quenched to low temperature. It is worth noting that complexes and isolated defects are not changing concentration, rather the charge on each defect is redistributed during the quench. The calculated Fermi level was found to always reside in the upper half of the bandgap when quenched to room temperature over the entire processing regime when Fe is the dominant defect.

The calculated concentration profiles of the neutral defects are consistent with the experimental coloration measurements and provide insight into the underlying physical mechanism associated with the coloration-transparency transition. The brown coloration is associated with the presence of $FeTi0$⁠, which monotonically increases for a $PO2$ greater than the abrupt coloration transition. This monotonic increase is consistent with the optical absorption measurements of Bieger et al. who annealed Fe:STO at increasing $PO2$ levels and observed an increased absorption, in accordance with Fermi's golden rule.²⁵ Furthermore, the anneal at $PO2=15 Pa$ is below the abrupt transition of $(FeTi-VO)0$ to $FeTi0$⁠. Therefore, there should be no $FeTi0$ present in equilibrium and the sample should be transparent, consistent with the optical trends of Fig. 1(b).

The prediction of the coloration-transparency transition is also consistent with other experimental measurements in the literature.^14,25,26 While this may not seem surprising, it is worth noting that these studies were performed with different iron concentrations and there is no guarantee that either the impurity profiles or the paths followed through phase space during annealing are similar to each other or to this study. It was therefore explored how the position of the abrupt $(FeTi-VO)0$ to $FeTi0$ transition changes as a function of path through phase space, increasing doping levels, and varying concentrations of background impurities. While these factors had slight impacts on the location of the coloration onset, the $PO2$ of the coloration transition remained on the same order of magnitude. Lower concentrations of Fe in STO have a more pronounced influence on the transition. As can be seen in Fig. 3, annealing temperature does have an influence on the position of the abrupt transition when quenched. The only other factor that had a significant effect on the position of the $PO2$ transition was the inclusion of the $(FeTi-VO)$ complex in the charge balance solutions. Omission of this complex led to results inconsistent with experimental observations, with the coloration onset shifting down to an annealing pressure of approximately 10⁻¹⁰Pa. The invariance of this transition to background impurities, path through phase space, and increasing Fe content explains why a consistent coloration onset is observed across multiple samples.

These calculations and experiments have extended existing Fe:STO defect chemistry models to include the first nearest neighbor Fe_Ti -V_O complex. This complex was found to have a large favorable binding energy and its presence was essential in reproducing the experimental $PO2$ coloration transition. The $PO2$ coloration transition was found to be only slightly dependent on the path through phase space and to the concentration of background impurities.

The authors acknowledge financial support from AFOSR BRI Grants Nos. FA9550-14-1-0264, FA9550-14-1-0067, and a DoD NDSEG fellowship. Computer time was provided by the DoD HPCMP. The authors thank Ramòn Collazo and Lew Reynolds for fruitful discussions.