Grain growth in strontium titanate in electric fields: The impact of space-charge on the grain-boundary mobility

This study investigates grain growth in the perovskite oxide strontium titanate in an electric field. The seeded polycrystal technique was chosen as it provides a sensitive and controlled setup to evaluate the impact of different parameters on grain growth due to the well-defined driving force for grain growth. Current blocking electrodes were used to prevent Joule heating. The results show faster grain growth, and thus, higher grain-boundary mobility at the negative electrode. It is argued that the electric field causes point-defect redistribution, resulting in a higher oxygen vacancy concentration at the negative electrode. The local oxygen vacancy concentration is suggested to affect the space-charge potential at the grain boundaries. A thermodynamic treatment of the grain-boundary potential at a grain boundary without field shows that for a high oxygen vacancy concentration less space-charge and less accumulation of cationic defects to the boundary occurs. Therefore, at the negative electrode, a higher oxygen vacancy concentration results in less space-charge and less accumulation of cationic defects. The lower degree of defect accumulation requires less diffusion of segregated defects during grain-boundary migration, so that at the negative electrode faster grain growth is expected, as found in the experiments.     

Grain-Boundary Chemistry of Barium Titanate and Strontium Titanate: I, High-Temperature Equilibrium Space Charge

Direct observations using scanning transmission electron microscopy (STEM) of the grain-boundary chemistry of selectively doped SrTiO₃ and BaTiO₃ show the predominant solute segregation in both systems to be that of acceptors (negative effective charge). Appreciable donor segregation is not observed even at lattice concentrations as high as 10 mol%. Donor and acceptor codoped materials show segregation of the acceptor only. The results are consistent with a grain-boundary space-charge distribution consisting of a positive boundary and negative space charge. All grain boundaries examined also show an excess of Ti relative to the A-site cations, suggesting that the positive boundary charge is at least partially accommodated by an excess of Ti ions. The sign and magnitude of the electrostatic potential appear to be remarkably insensitive to changes in lattice defect structure with solute doping. Grain-boundary chemistry appears dominated by space-charge segregation, in contrast with the predictions of recent atomistic simulations which neglect the space-charge potential.     

Grain-Boundary Characterization in a Nonstoichiometric Fine-Grained Magnesium Aluminate Spinel: Effects of Defect Segregation at the Space-Charge Layers

The grain-boundary chemistry of fine-grained spinel MgO· n Al₂O₃ (mean grain size below micron) has been investigated by STEM microanalysis. We have quantified the concentration of each element across the grain boundaries. Stoichiometry variations are observed from the grain-boundary region to the bulk. The Al/Mg ratio increases from 2.1 in the bulk to 2.35 at the grain-boundary regions. X-ray quantification allows us to reveal and to characterize the space-charge layer in the subgrain boundary. The grain-boundary cores are negatively charged due to     vacancies in excess, and in the subgrain-boundary region, an opposite, positive space-charge layer is obtained. The point defect composition and the characteristic (sign, space-charge potential Φ_∞) of the space-charge layer are discussed.     

Chemical Diffusion in Polycrystalline Al2O3 as Determined from Electrical Conductivity Measurements

By analyzing the changes with time of the electrical conductivity of polycrystalline Al₂O₃ after the O₂ pressure was changed, a defect diffusion coefficient was obtained which was assigned to the Al or O ion, whichever is the faster-diffusing species. Both decreased grain size and MgO addition increase the defect diffusion coefficient. The grain-boundary defect diffusion coefficient for the undoped material was estimated to be: and that for the MgO-doped material was over the range 1100° to 1350°C (δ is the effective thickness of the boundary and s the coefficient of segregation of defects to the boundary region). The mechanism of grain-boundary diffusion is discussed in terms of defect mobility.     

Effect of Electric Field on the Migration of Grain Boundaries in Alumina

The effect of an external electric field on the grain-boundary migration in Al₂O₃ ceramics has been investigated. The boundary migration is dependent on the direction and magnitude of the applied bias, and the observed boundary migration behavior is attributed to the presence of an electrostatic potential that inherently forms at the grain boundaries of Al₂O₃ ceramics. The results give experimental evidence that the boundary phenomena in oxide ceramics are related to the grain-boundary potential.     

Solute Segregation and Grain-Boundary Impedance in High-Purity Stabilized Zirconia

Solute segregation at grain boundaries has been correlated with grain-boundary conductivity in high-purity 15-mol%-CaO-stabilized ZrO₂. STEM measurements of solute coverage show that the segregation of impurity silicon (present at bulk levels <80 ppm) is grain-size dependent. The boundary coverage of silicon can be systematically varied by varying grain size at concentrations low enough that a discrete siliceous film does not form. The cosegregation of calcium and silicon is observed. The grain-boundary solute coverage (T_si+Ca) has been correlated with the specific grain-boundary conductivity (σ^sp_gb) determined using impedance spectroscopy. At monolayer segregation levels, the specific boundary conductivity is less than the bulk conductivity by a factor >10³ at 500°C. At the lowest levels of segregation achieved, <0.1 monolayer, σ^sp_gb remains ∼10² less, and possibly represents an "intrinsic" limiting value for the grain boundary. Comparison with Y₂O₃-doped ZrO₂ suggests similar behavior in this system. The control of grain-boundary segregation through purity, microstructure, and thermal history is discussed from the objective of engineering the grain-boundary impedance of polycrystalline ionic conductors.     

Grain-Boundary Migration in Nonstoichiometric Solid Solutions of Magnesium Aluminate Spinel: II, Effects of Grain-Boundary Nonstoichiometry

The grain-boundary chemistry of magnesium aluminate spinel solid solutions MgO· n Al₂O₃ in which grain growth measurements were reported in part I has been investigated in order to understand the mechanism of grain-boundary migration. It is found that although segregation of impurity Ca and Si is common, much larger deviations in grain-boundary stoichiometry are present. There is an excess of Al and O relative to Mg at grain boundaries in all compositions. Grain-boundary migration appears to be rate-limited by solute drag from intrinsic defects accommodating lattice nonstoichiometry, rather than by extrinsic solutes, consistent with the observed impurity tolerance of grain-boundary mobility. Different rate-limiting defects are proposed for magnesia-rich and alumina-rich spinels.     

Electrical Conductivities of (CeO2)1−x(Y2O3)x Thin Films

Electrical properties of CeO₂ thin films of different Y₂O₃ dopant concentration as prepared earlier were studied using impedance spectroscopy. The ionic conductivities of the films were found to be dominated by grain boundaries of high conductivity as compared with that of the bulk ceramic of the same dopant concentration sintered at 1500°C. The film grain-boundary conductivities were investigated with regard to grain size, grain-boundary impurity segregation, space charge at grain boundaries, and grain-boundary microstructures. Because of the large grain boundary and surface area in thin films, the impurity concentration is insufficient to form a continuous highly resistive Si-rich glassy phase at grain boundaries, such that the resistivity associated with space-charge layers becomes important. The grain-boundary resistance may originate from oxygen-vacancy-trapping near grain boundaries from space-charge layers. High-resolution transmission electron microscopy coupled with a trans-boundary profile of electron energy loss spectroscopy gives strong credence to the space-charged layers. Since the conductivities of the films were observed to be independent of crystallographic texture, the interface misorientation contribution to the grain-boundary resistance is considered to be negligible with respect to those of the impurity layer and space-charge layers.     

Space Charge Segregation at Grain Boundaries in Titanium Dioxide: II, Model Experiments

A quantitative study of space charge solute segregation at grain boundaries in TiO₂ is conducted, using a new STEM method for the measurement of aliovalent solute accumulation. It is shown that the electrostatic potential at grain boundaries can be varied in sign and magnitude with doping, oxygen pressure, and temperature, and that the isoelectric point lies in slightly donor-doped compositions for samples annealed in air. The experimental results closely fit the space charge model in Part I. Space charge solute segregation is found even in defect regimes of high electron concentration. Approximately one in ten grain boundaries are "special" in exhibiting no detectable segregation; in one such instance a twin boundary is identified. Among boundaries with significant amounts of segregation, clear differences in potential also exist. From the potential determined in acceptor- and donor-doped compositions, the Frenkel energy (assumed to be lower than the Schottky energy in TiO₂) can be separated into its individual terms. An average value for the titanium vacancy formation energy of g_vTi = 2.4 eV and an upper limit to the titanium interstitial formation energy of g_Tii = 2.6 eV are obtained.     

Equilibrium Point Defect and Electronic Carrier Distributions near Interfaces in Acceptor-Doped Strontium Titanate

A mathematical model based on Frenkel's theory for charged defect segregation at interfaces is used to calculate the equilibrium grain boundary depletion layer widths and conductivity profiles in acceptor-doped SrTiO₃ ceramics. The calculations examine the effect of oxygen vacancy equilibration during annealing at moderate temperatures (∼1000 K) on the development of interfacial charge that influences grain boundary electrical properties at lower temperatures. Good agreement is demonstrated between the model predictions and experimental results reported in the literature.     

Interfacial Segregation in Perovskites: IV, Internal Boundary Layer Devices

A proposed model for interfacial segregation in perovskites, with induced heterogeneous defect distributions, is extended here to account for the formation of internal boundary layer devices, such as positive temperature coefficient of resistance (PTCR) thermistors and internal boundary layer capacitors (IBLC). Boundary layer effects in doped BaTiO₃ are attributed to factors which contribute to the formation of highly resistive boundary layers by a segregation-induced shift in donor incorporation and/or acceptor segregation, and the inhibiting action of segregated donors on boundary mobility and grain growth. The distribution of space charges, formed by electron transfer from conductive grains to resistive boundary layers, leads to the formation of impedance barriers in the grain-boundary vicinity. Depending on the grain size, and on relative size and spatial distribution of the space charge layer and the resistive layer, a transition from semiconducting properties to insulating properties may take place. This model accounts for the observed PTCR and IBLC phenomena.     

Segregation-controlled densification and grain growth in rare earth-doped Y2O3

Cation doping of Y₂O₃ is an established approach for tailoring densification and grain growth during sintering. However, the segregation of doped cations to the grain boundary and their impact on processing are still not completely understood. Segregation can be driven by electrostatic effects due to charge mismatch with the host lattice or elastic effects induced by ion size mismatch. While segregation is caused by thermodynamics, it impacts diffusion and the kinetics of grain boundaries during densification and microstructure evolution. In this study, we utilize two isovalent dopants (La³⁺ and Gd³⁺), that is we focus on the elastic component of segregation. We investigate the densification as well as the grain growth kinetics of both doped and undoped Y₂O₃ during field-assisted sintering/spark plasma sintering (FAST/SPS). While Gd³⁺ is showing no significant effect on densification, La³⁺ resulted in a strongly reduced sintering activity. Furthermore, the analysis of the grain growth behavior during sintering and on predensified samples revealed a decrease in the grain growth coefficient, with La³⁺ having the strongest impact. The structure and chemistry at the grain boundary were observed by aberration-corrected TEM. While no structural change was caused by doping, the chemical analysis showed a strong segregation of La³⁺ to the grain boundary, which could not be observed for Gd³⁺. The results indicate that segregated La³⁺ causes a drastic decrease in grain boundary migration rates through solute drag as well as much slower sintering kinetics, likely caused by a decrease in the grain boundary self-diffusion due to segregation. This study further underlines the importance of the elastic contribution to cation segregation and establishes a clear relationship to grain growth and sintering kinetics, which are both decreased by segregation.     

Plausible Concepts Necessary and Sufficient for Interpretation of Ceramic Grain-Boundary Phenomena: I, Grain-Boundary Characteristics, Structure, and Electrostatic Potential

Analogies between grain boundaries in metals and nonmetals, direct observations of grain-boundary characteristics, and inferences about the atomic structure of grain boundaries are reviewed. The electrostatic potential and associated space charge are discussed, and new experimental measurements showing the existence and sign of the boundary charge are reported for NaCl and Al₂O₃.     

Energy Variations in Diffusive Cavity Growth

The assignment of boundary values for the chemical potential and the calculation of energy-release rates for the growth of creep cavities along grain boundaries by self-diffusion are discussed. For simplicity, it is assumed that the boundaries are flat and that surface and grain-boundary diffusion are the dominant transport mechanisms. As matter diffuses from the void surface into and along the grain boundary, misfit residual stresses are induced to alleviate the high stress concentration ahead of the cavity apex. As a result, the contribution of strain-energy terms to the chemical potential can be neglected in typical cases. Also, contrary to the Griffith crack-extension model, the energy dissipation incurred by diffusive removal of material from the cavity surface and deposition in the grain boundary is a major term in the energy transfers associated with cavity growth. The primary energy "sink" in diffusive cavity growth is shown to arise from the work done by the grain-boundary normal stress when matter is inserted in the near-tip region by diffusion, not from the loss of strain energy of matter that is removed from the cavity at its tip or from the work of bond separation. Thermodynamic restrictions on the angle formed by the void surfaces at their apex, where they join the grain boundary, are considered. Boundary values for the chemical potential are derived in a manner appropriate for arbitrarily large but elastic distortions of material near the cavity tip and, in contrast to most previous work in the area, the effects of surface tension (i.e. of "surface stress," as distinct from surface energy) are included.     

Ab Initio Calculations of Pristine and Doped Zirconia Σ5 (310)/[001] Tilt Grain Boundaries

The structure of the cubic-ZrO₂ symmetrical tilt Σ5 (310)/[001] grain boundary is examined using density functional theory within the local density and pseudopotential approximations. Several pristine stoichiometric grain-boundary structures are investigated and compared with Z-contrast scanning transmission electron microscopy and electron energy loss spectroscopy results. The lowest-energy grain-boundary structure is found to agree well with the experimental data. When Y³⁺ is substituted for Zr⁴⁺ at various sites in the lowest-energy grain-boundary structure, the calculations indicate that Y³⁺ segregation to the grain boundary is energetically preferred to bulk doping, in agreement with experimental results.     

Low-angle twist grain boundary in SrTiO3 fabricated by spark plasma sintering techniques

A SrTiO₃ bicrystal with a low-angle twist grain boundary was fabricated using the spark plasma sintering (SPS) instrument. The atomic and electronic structure of the grain-boundary core was characterized using scanning transmission electron microscopy techniques. It was determined that the boundary is comprised of 2 types of defects with distinct electronic structures: screw dislocations and dislocations with an [001] edge component. The dislocations with an [001] edge component dissociated into 2 partial dislocations, separated by a stacking fault consisting of 2 Ti–O layers. The screw dislocations are attributed to the twist component of the grain boundary, while dislocations with an [001] edge component are attributed to surface steps on the original (100) SrTiO₃ surfaces prior to diffusion bonding. The observed repeat distances between the dislocations with edge components along the grain-boundary plane are smaller than those discovered during traditional diffusion bonding experiments. The higher planar defect density observed in this study results partly from higher heating rates, lower processing temperatures, and shorter holding times during SPS processing.     

Interfacial Segregation in Perovskites: I, Theory

Based on thermodynamic principles a theory for equilibrium interfacial segregation is proposed for perovskite materials, and this theory is applied to BaTiO₃. An approach developed by Frenkel and refined by Kliewer and Koehler is extended to undoped ternary oxide materials such as BaTiO₃. The approach uses regular solution approximations and considers space charge effects as the major driving force for segregation. The analysis based on this model indicates the presence of a negative space charge potential (−0.1 V at 800°C) at the surface of pure BaTiO₃. The model also predicts cation enrichment at the interface. The thickness of the space charge layer decreases with increasing temperature, and calculated values agree well with experimental results. Since both elastic and electrostatic driving forces are important for dopant/impurity segregation, an approach where the grain boundary is considered to be a two-dimensional phase, in equilibrium with the three-dimensional phase of the grain, proves useful. Solving for the impurity/dopant segregation ratio is case specific and requires knowledge of the charge neutrality conditions as well as the strain energy contribution.     

Grain-Boundary Chemistry of Barium Titanate and Strontium Titanate: II, Origin of Electrical Barriers in Positive-Temperature-Coefficient Thermistors

Scanning transmission electron microscopy (STEM) of positive-temperature-coefficient (PTC) BaTiO₃ thermistors shows that the grain-boundary oxygen content in as-received (oxidatively cooled) materials is slightly enriched compared to quenched samples, and the acceptor-rich space-charge present at high temperatures is retained upon cooling. The defect density of the space charge is approximately equal to the acceptor state density at PTC boundaries determined by electrical measurements. Accordingly, it is proposed that the electrical barrier forms when acceptor defects already segregated in the ionic space charge at high temperature become active interface states when compensating donor defects in the grain-boundary core are oxidized. These acceptor defects appear to be primarily barium vacancies, but need not form upon cooling in the manner proposed by Daniels and Wernicke. Acceptor solutes when present can also contribute to barrier formation through space-charge segregation; the increase in interface state density upon addition of Mn is consistent with the magnitude of the expected segregation.     

Ionic Double Layers at the Surface of Magnesium-Doped Aluminum Oxide: Effect on Segregation Properties

A model involving ionic double layers at the surface has been constructed for magnesia-doped sapphire based on earlier models which were developed for cubic halides. The model takes into account the presence of electrostatic potentials, isolated point defects, defect complexes, and special surface sites which can act as sources and sinks for ions. Equations have been set up for the various defect concentrations, and Poisson's equation has been solved numerically to give depth profiles for defects and corresponding electric fields. The calculations suggest that Mg²⁺ ions can segregate both to the free surface and to the space charge region. The effective (or Langmuir) enthalpy of segregation depends not only on the actual binding energies of the dopant ion, but also on other parameters such as the density of special surface sites. Over the temperature range studied, the variation of the calculated surface magnesium concentration with temperature is found to be approximately Arrhenius in nature, as was observed in segregation experiments.     

Detection of Boron Segregation to Grain Boundaries in Silicon Carbide by Spatially Resolved Electron Energy-Loss Spectroscopy

Boron segregation to grain boundaries in SiC was directly observed for the first time by using spatially resolved electron energy-loss spectroscopy methods. The hot-pressed, fully dense material was doped with 0.3 wt% of boron and was free of other additives, except for 2 wt% of free carbon. The detection of boron was achieved in the difference spectra at all the grain boundaries that were examined. Its interfacial excess was in the range of 15–29 atoms/nm², or approximately one monolayer. Concurrently, silicon depletion occurred at these boundaries, although to a lesser extent (−13.5 atoms/nm² on average), which indicated that boron mainly replaces silicon and bonds with carbon at the grain boundary. These findings validate the dual role of boron at the grain boundary for promoting densification via improved grain-boundary diffusivity while maintaining a covalent grain boundary without an oxide phase.