Cover: Przemysław Spiechowski

Publisher: Karol Zawadzki

Editor: Małgorzata Dąbkowska-Kowalik

Reviewer: dr hab. Maria Połomska, Associate professor; Institute of Molecular Physics, Polish Academy of Sciences Poznań

Ebook: Michał Latusek

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Introduction

1. An Introduction to the Crystal Structures of Oxide Perovskites

1.1. Atomic Displacements

1.2. Octahedral Tilting

1.3. Distortion Modes

References

2. Growth of ABO₃ and BO₂ Crystals

2.1. Theory of Crystal Growth

2.1.1. Equilibrium Conditions

2.2. Phase Diagrams of Model Oxides. Major Challenges in Crystal Growth of Stoichiometric Oxide Crystals

2.2.1. Gibbs Phase Rule

2.2.2. Phase Diagrams

2.2.3. Nonstoichiometric Oxides

2.3. Techniques of Growth of Selected Binary and Ternary Oxide Crystals

2.4. Typical Methods for Evaluation of Quality of Oxide Crystals

2.4.1. Stoichiometric TiO₂

2.4.2. Perovskite Structure

References

3. Defect Chemistry in Binary and Ternary Metal Oxides

3.1. Introduction

3.2. Imperfect Crystals

3.3. Quasi-chemical Defect Reactions

3.4. Types of Point Defects

3.5. Thermodynamic Approach

3.6. Nonstoichiometry

3.7. Aggregations of Point Defects and Extended Defects

3.8. Ideal Point Defect Model

3.8.1. Case of Ni1–yO

3.8.2. Case of BaTiO3–y

3.8.3. Case of Zirconia-based Materials

3.8.4. Defect Structure of YSZ

3.9. Debye-Hückel Defect Model

3.9.1. Co1–yO Case

3.10. Cluster Model

3.10.1. Defect Clustering in the Wustite Phase

3.11. Crystallographic Shearing

3.12. Defect Structure of Titanium Dioxide

3.12.1. Extended Defects in TiO₂

3.13. Hydrogen Defects

Acknowledgements

References

4. Self-Polarization of Ferroelectric Thin Films and Its Influence on the Film Properties

4.1. Introduction

4.2. Free Energy Functional

4.3. Renormalized Free Energy and Properties for One Component Polarization P = PZ

4.4. The Influence of Built-in Electric Field on the Properties of Ferroelectric Thin Films for the Case P = PZ

4.5. Phase Diagrams with Electret State and Properties for the Case of Three Polarization Components

4.6. Comparison with Experiment and Summary of Sec. 5 Results

Conclusion

References

5. Lattice Dynamics of Perovskite Oxides

5.1. Introduction

5.2. Early Attempts to Model Ferroelectric Perovskites

5.3. Polarizability Effects in Perovskites

5.4. The Polarizability Model for Perovskites

5.5. Applications of the Model

5.6. Precursor Effects to the Phase Transitions

Conclusions

References

6. Metrology and Measurement Techniques

6.1. Electrical Properties of Nanoscale Solids

6.1.1. Measurement of Electrical and Magnetic Material Properties

6.1.2. The Piezoelectric Effect

6.2. Design of Measurement Setups

6.2.1. General Considerations

6.2.2. Active Amplifier Circuits

6.2.3. Measurement Methods for Ferroelectric Properties

6.2.4. Laser Interferometers

References

7. Multiferroics

Abstract

7.1. Introduction

7.2. A Brief History of Magnetoelectrics and Multiferroics

7.3. Symmetries and Orderings in Multiferroics

7.4. Theory of Magnetoelectric Coupling in Multiferroics

7.5. Magnetoelectric Multiferroics

Acknowledgements

References

8. Electronic Correlations and Metal-Insulator Transitions

8.1. Introduction: The Meaning of Localization – Delocalization Transitions in Solids with Examples

8.1.2. Crystallization of Liquid ³He as Mott-Hubbard Transition

8.1.2. Localization-Delocalization Transition of Ultracold Atoms

8.1.3. Metal-Insulator Transition in Doped V₂O₃

8.2. Elementary Approach to the Metal-Insulator (Mott-Hubbard) Transitions

8.2.1. Normal metal as a Landau Fermi liquid: basic characteristics

8.2.2. Mott-Wigner Criterion of Localization

8.2.3 Localization on the Lattice: Hubbard Model

8.2.4. Quantitative Discussion of the Metal-Insulator Transition

8.2.5. Quasiparticle Representation of Correlated-Electron System and the Phase Diagram

8.2.6. Mott-Hubbard Localization in Correlated Nanoscopic Systems

8.3. Concluding Remarks

Acknowledgment

References

9. Nature of the Insulator-Metal Transition in Transition Metal Oxides Induced by Chemical Defects – Deviation from Stoichiometry and Electrochemical Alkaline Intercalation

9.1. Insulator-Metal Transition in Transition Metal Oxides Induced by Deviation from Stoichiometry

9.2. Insulator-Metal Transition in Transition Metal Oxides Induced by Electrochemical Alkaline Intercalation/Deintercalation

LixCoO₂

NaxCoO2-y

Acknowledgements

References

10. Structural Transformation of the SrTiO₃ Surface Region due to Electric Fields at Ambient Temperature

10.1. Modifications and Equilibrium of the SrTiO₃ Structure

10.1.1. Compositional Changes: the Quasi-binary System SrO–TiO₂

10.1.2. Stability of SrO(SrTiO₃)n Ruddlesden-Popper Phases

10.1.3. A Look at Symmetry and Property Tensors

10.1.4. Chemical Modifications Beyond the Ternary Composition

10.2. Electric-field Induced Ionic Transport and Anisotropy in SrTiO₃

10.2.1. Electroformation

10.2.2. Redistribution of Ionic Species

10.3. Evidence for Structural Transformations and Related Models

10.3.1. Structural Transitions due to Application of an Electric Field

10.3.2. The Occurence of Ruddlesden-Popper Phases

10.3.3. The Migration-Induced Field-Stabilized Polar (MFP) Phase

10.4. Modification of SrTiO₃ Properties and Related Applications

10.4.1. Oxygen Vacancies and Conductivity

10.4.2. Mechanical Properties

10.4.3. Magnetism

10.4.4. Pyroelectricity

10.4.5. Piezoelectricity

10.4.6. Resistive Switching

10.4.7. All-Solid-State All-in-One Battery

10.4.8. Further Applications

10.5. Conclusion

Acknowledgment

References

11. Chemical Transformation of SrTiO₃ Surface Region Exposed to High Temperature and Other Factors

11.1. Introduction

11.2. Investigated Crystals

11.3. Possible Surface Reconstructions of STO

11.4. Methods of Surface Studies

11.5. Thermal Treatment

11.6. Surface Modification upon Vacuum and Exposure to X-Rays and Electron Beam

11.7. Ion Sputtering

11.8. Deposition of Metals

11.9. Mechanical Stress and Defects

Summary

References

12. Chemical Transformation of TiO₂ Surface Region Exposed to High Temperature and Different Chemical Activity of Oxygen

Acknowledgements

References

13. From Electroformation to Resistive Switching in Single Crystals of Strontium Titanate: How SrTiO₃ Can Be Transformed into a Metallic State Using Electrical Stimuli

13.1. Introduction

13.2. Resistive Switching – General Information

13.3. Some Theoretical Considerations

Methods for the Calculation of the Electronic Structure

Trends Observed from DFT Calculations

13.4. Electroforming of Non-conducting (Insulating) SrTiO₃ Crystals.

Electrical Characterization During Electroformation

Oxygen Evolution During Electroformation

Spatial Inhomogeneity of the Electroformation Process and Filamentary Structures

13.5. Model

References

14. Crystallographic Structure, Electronic Structure, and Chemical Composition on the Nanoscale: Important Role of the SPM, LEED, Photoemission Investigations for the Analysis of the Crystal Geometry, Electronic Structure and Diffusion Phenomena on the Surface of Model Oxides

Acknowledgements

References

15. Investigations of Local Thermal Properties by SThM Microscopy

15.1. Scanning Thermal Microscopy: Principle, Equipment, Operation Modes

15.2. Thermal Imaging: Temperature and Conductivity Contrast Measurements

15.3. Quantitative Thermal Measurements with SThM Equipment – Potentialities and Limitations

15.4. Methods for Improvements of Sensitivity and Stability of Thermal Measurements Using Batch Fabricated Thermal Probes

15.5. Determination of the Thermal Conductivity of Submicron Layers on Thick Substrates

15.6. Application of Qualitative and Quantitative SThM Measurements in Investigation of Perovskites (Thin Films, Bicrystals)

Acknowledgements

References

16. Studying the Local Redox Processes on Transition Metal Oxides Surfaces Using Kelvin Probe Force Microscopy

16.1. Introduction

16.2. Principles of Kelvin Probe Force Microscopy

16.2.1. Basics of KPFM Operation

16.2.2. Forces in Atomic Force Microscopy

16.2.3. Technical Realization of KPFM

16.2.4. Limits of Resolutions in KPFM

16.3. Local Surface Potential of Oxide Metal Surfaces upon Reduction and Oxidation Processes

16.3.1. Titanium Dioxide TiO₂(110) Surface

16.3.2. Strontium Titanate SrTiO₃(100) Surface

16.4. Summary

Acknowledgements

References

Footnotes

Introduction

The oxides, especially the multinary transition metal oxides, are a very important class of inorganic materials being enormously interesting for basic research in solid-state physics, surface physics, defect chemistry, electrochemistry, and catalysis, to mention a few. Moreover, the range of applications for these materials is also very broad. It ranges from nano-electronics as the main component of resistive switching devices in neuronal networks, micro-electro-mechanical systems called MEMS, and ultrasonic transducers with outputs of many kilowatts to catalysis in industrial processes. The versatility of ABO₃ oxides, which are of perovskite structure, where A is alkali or alkaline earth ions, consists mainly in the modification of their physical and chemical properties by changing the valence of transition metal ions B in the center of oxygen octahedron BO₆. Using external stimuli like the voltage, current flow, mechanic stresses, magnetic fields, oxygen activities, annealing, irradiations with photons, bombarding with ions, and doping with isovalent or alliovalent ions can modify the electronic structure and lattice dynamics. Another fascinating property of these multinary oxides is so-called self-doping, in which a change in oxygen stoichiometry can be induced under suitable physical or chemical conditions of the reduction process. This process permits easy incorporation of oxygen vacancies as a result of the effusion of oxygen ions during this process. In this way, about 50 years ago, the superconducting properties were, for the first time, induced in ternary oxides SrTiO₃, which is treated as a model perovskite being in stoichiometric state band an insulator with Eg > 3.2eV. For all this, we decided to write this book.

It would be an illusion to claim that this is a representative book in so quickly developing field of oxides. Often a book of 500-600 pages describes properties of one oxide only. Therefore, the strategy of this book is quite different. The book is aimed at finding a common denominator for properties of multinary, yet very functional, metal oxides. Among others is switching, the main topic of this book, which is generalized as a tailored manipulation on selective materials properties. Our intention is not only to link switching to the influence of different macroscopic stimuli on ferroic order parameters as spontaneous polarization, spontaneous strain or magnetization, but to show that switching originates in the nanoscale and can be treated as a low-size (local) phenomenon. Hence, the book presents surface sensitive methods such as LCAFM, PFM, KPFM by which switching processes at the nanoscale are investigated. Especially the transport phenomena along filaments inside oxides are described in detail being analyzed through resistive switching mechanism from semiconducting to a metallic state. This mechanism is associated with a manipulation based on the redox reaction of the electrical properties of dislocations core in the surface region of two model materials TiO₂ and SrTiO₃. It should be emphasized that most chapters are dealing with the important roles played just by surface layer and surface regions in oxides crystal. An example of that is the complicated transformation of the crystallographic and chemical structure of the surface layer in TiO₂ and SrTiO₃. But the theory of switching from insulating/semiconducting oxides to metallic state is not restricted in this book to the surface or filaments. Presented is also the theory of isolator/metal (IM) transition in the entire volume, as it is described for bulk vanadium and cobalt oxides. Additionally, unconventional behavior of polarization formed at the interface between the surface and volume in ferroelectric crystals, called self-polarization, is presented.

We are aware that the switching effects presented in this book, like complex physical and chemical phenomena in non-homogenous systems, require broad knowledge in the fields of defect chemistry, crystallography, electronic structure, and lattice dynamics. That is why few chapters are sacrificed to these topics as well. There are also chapters explaining principles of measurement methods that are important for experimental studies of switching processes in oxides at the nanoscale. In spite of that, in every single chapter, one can find a short description of other experimental methods or references containing relevant information about them.

We think that this book is a kind of breakthrough in the analysis of the variability of the physical and chemical properties of multinary transition metal oxides. Namely, we state that macroscopic descriptions – which dominate in literature – of switching phenomena, will not be complete if research at the nanoscale is not taken into account. In this sense, the book is in line with the old motto by Goethe, which reads as follows: If the whole is ever to gladden thee, that whole in the smallest thing thou must see (J.W. Goethe, Gedichte. Ausgabe letzter Hand. Gott, Gemüt und Welt, 1827).

1

An Introduction to the Crystal Structures of Oxide Perovskites

Anthony Michael Glazer

Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU & Physics Department, University of Warwick, Coventry CV7 3AL.

Abstract

Perovskites are amongst the most important materials that can show switchable properties. This is because the vast range of structural modifications arise from small changes in the atomic positions.

By far the most important of the ferroelectric oxides come from the family of perovskite materials. Interest in perovskites has been growing over the last 50 or so years at a rapidly expanding rate. For instance, Figure 1.1 shows a plot of the number of papers using the word “perovskite” as a function of years. Back in the 1960s roughly 50–100 papers appeared each year, but the numbers continued to rise. In 1987 there was a jump in the numbers, brought about by the discovery of high-temperature superconductors: although not actually perovskites they were closely related. The rise has since continued near exponentially until 2016 when there was another huge jump to a total in excess of 35,000 publications in one year! This is partly, but not exclusively, due to the recent discovery that certain so-called hybrid perovskites show promise as photovoltaic materials, and this is currently a hot topic internationally. So far, around 500,000 papers on perovskites have been published.

Figure 1.1. Number of publications on perovskites with time (Google Scholar).

This huge number of publications now amounts to approximately 2–3 publications per hour, making it just about impossible for any researcher in this field to maintain total awareness of the latest research.

In this article, I shall confine myself to simple perovskite oxides, as these are by far the most useful practically for their ferroelectric behaviour. But the reader should be aware that there are many materials with related structures, such as tungsten-bronzes, Aurivillius, Ruddlesden-Popper, and Dion-Jacobson phases. An excellent book that describes all the varieties of perovskite-based compounds is that by Mitchell [1]. As I concentrate on the crystallographic aspects here, I shall not discuss the different types of properties of perovskites, such as the multiferroic effects. Furthermore, I refer the reader to the papers by Woodward and colleagues who have done much work on considering cation ordering and the so-called Jahn-teller distortions in perovskites [2]. The latter tend to occur more often in non-oxide perovskites and so are not dealt with here.

So what makes the perovskite compounds so special? To understand this, one needs to consider the crystal structure and how it can be modified to produce different effects. Figure 2 shows what is called the “aristotype” structure (the term due to H.D. Megaw) or “parent” structure. The general formula is ABX₃, where A and B are cations and X is an anion, usually oxygen in most ferroelectric perovskites.

In Figure 1.2, there are two equivalent views of the structure. In Figure 1.2(a) the A cation is placed on the origin of the unit cell, while the B cation is at the unit cell center. Surrounding the B cation are 6 anions forming an octahedral arrangement around the B cation. In (b) the B cation has been placed on the unit cell origin, thus putting the A cation at the unit cell center. We now see that the structure can be described in terms of an infinite framework of corner-linked octahedra with the B cations at the center of each octahedron and the A cation occupying the space between the octahedra. This aristotype structure is cubic in the space group Pm3m.

Such a structure is non-polar and is normally the high-temperature phase of most perovskite materials. On cooling below a phase transition temperature, the structure undergoes certain distortions, sometime creating polar structures that can then exhibit phenomena such as ferroelectricity, pyroelectricity and piezoelectricity. These lower-temperature phases were called by Megaw “hettotypes”.

We now have to consider the types of distortions that are found in perovskites.

Figure 1.2. The structure of the aristotype perovskite: (a) origin on A cation; (b) origin on B cation.

1.1. Atomic Displacements

Suppose the A and B cations are displaced slightly in a certain direction within the unit cell. If they all move in the same direction, then the structure becomes polar, thus allowing for polar properties to be generated. In group theory terms, these displacements can be treated as distortion modes at the center of the Brillouin Zone (see later). The displacements in general are very small, but at the same time cause very large changes in the polar properties. Very often these properties are temperature or pressure-dependent, as the atomic positions respond to the external environment. In some perovskites, such as in CaTiO₃, the displacements are antiparallel, so that no net polarization is obtained, and such compounds are termed antiferroelectrics. The antiparallel displacements result in a doubling or more of the basic aristotype unit cell, which can be seen in the development of superlattice peaks in diffraction patterns. It is interesting to look at the first recognized antiferroelectric perovskite, PbZrO₃[3]. When the unit cell parameters were measured it was originally found that they had the relationship a = b ≠ c with all interaxial angles equal to 90°, and so it was thought to be a tetragonal structure. However, observation using polarized light showed that the (001) section was not optically isotropic. In addition, the diffraction pattern showed the presence of superlattice peaks and so the displacements of the Pb atoms in (001) planes were found to follow the arrangement shown in Figure 1.3.

Figure 1.3. Arrangement of Pb displacements in PbZrO₃[3].

In this diagram, the axes marked ap and bp are referred to as pseudocubic axes as they become the cubic axes in the aristotype phase. In PbZrO₃ they have the same length and appear to be perpendicular to one another. However, X-ray and neutron diffraction showed that the Pb atoms are displaced in the directions of the arrows in the figure and then one can define an orthorhombic unit cell given by the axes ao and bo. In fact, much later it was found that the angle between ap and bp is not quite 90°! This illustrates how dangerous it is to assign a crystal system based on unit cell measurements alone.

Probably the most well-known perovskite structure in which cation displacements have been studied is that of barium titanate BaTiO₃. In fact, barium titanate was the first of the ferroelectric perovskites to be studied. It was first used as a piezoelectric transducer material during the Second World War as part of the British effort in submarine detection, but was later superseded by another important perovskite, PZT (lead zirconate-titanate solid solutions, a material which I have studied for the last 40 or so years!). The crystal structure was first determined by Megaw in 1945 [4] who showed that it had tetragonal symmetry. Since then there have been numerous other studies of the crystal structure [5] but almost all agree with this basic determination. In order to understand its structure, consider Figure 1.4, which has been adapted from information given by Megaw in 1973 [6].

Figure 1.4. Crystal structures of barium titanate: (a) cubic; (b) tetragonal; (c) orthorhombic; (d) rhombohedral [2].

In Figure 1.4(a) is shown the cubic aristotype structure which is found above 135 °C in pure top-seeded crystals, or 120 °C in flux-grown crystals. It is important to be aware that almost all research published on this material before around 1970 was conducted on flux-grown crystals that contained Cl or F ions, and so they were impure. These crystals were grown by the Remeika method [7], and had a slight yellow appearance and formed plates. Later it was possible to grow using top-seeded crystals which grew in blocks and were colorless. A comparison between the two types of crystal, pure and impure. was published using lattice parameter measurements by Clarke [8]. This is shown in Figure 1.5, where it can be seen that in both crystals four different phases are found. Phase I is the cubic aristotype, phase II is tetragonal, phase III orthorhombic and phase IV rhombohedral. Note how the phase transition temperatures are higher in the pure material. The space group changes are

(1.1)

Figure 1.5. Plot of lattice parameters of BaTiO₃ as a function of temperature: (a) pure; (b) grown from KF flux.

Returning to the structure, Figure 1.4 (b) shows schematically the essential features of the tetragonal structure. This crystallizes in space group P 4mm, which is a polar symmetry and is ferroelectric. It is this phase that is used in industrial applications, these days mainly as a dielectric material in electrical capacitors. The arrows in the figure indicate the displacement directions of the ions, with exaggerated magnitudes. In reality, they are very small but sufficient to give rise to the polar properties.

Now, it is not commonly appreciated that in polar structures such as this the choice of origin to describe the structure is entirely arbitrary. We could decide to put the origin on the Ba atom, or on a Ti or on an oxygen atom (or anywhere else for that matter). With each choice of origin, the displacement directions would appear to be different. For example, with the origin chosen on a Ti atom, there would appear to be no Ti displacement!

Megaw proposed that the most sensible choice of origin is at the center of the oxygen octahedron, and this is used in Figure 1.4(b). Using this choice, we see that the Ba, Ti, and O2 atoms are displaced upwards along the direction of polarization whereas O1 is displaced downwards. The largest displacements are for the Ti ions.

Figure 1.6. Tetragonal structure of BaTiO3 showing the effect of different origin choices:
(a) Megaw’s choice with Ti at the corners of the unit cell; (b) Ba undisplaced; (c) Ti undisplaced; (d) Megaw’s choice with Ba at the corners of the unit cell; (e) Ba undisplaced; (f) Ti undisplaced.

Figure 1.6 gives schematic diagrams showing how the atomic displacements appear in the tetragonal structure using different origins (Ti and Ba at the unit cell corners, respectively) and different choices of polar displacements. These are all equivalent descriptions, something that is not often appreciated.

In Figure 1.4(c) the displacements in the orthorhombic phase III are shown. In this case the displacements are directed along [110] directions as opposed to [001] in the tetragonal phase II. Therefore, the polarization vector must change dramatically at the phase transition temperature. The space group Bmm2 is not a subgroup of P4mm, and so the transition has to be of first order, as is observed. Note that Bmm2 is however a subgroup of Pm3m.

In Figure 4(d) the displacements in the rhombohedral phase IV is shown. Here again, at the transition there has to be a discrete change in the polar direction. The space group R3m is not a subgroup of Bmm2, although it is a subgroup of the aristotype space group Pm3m. Therefore, again this must be a first-order transition.

While these structure determinations appear at first sight to be straightforward the observation of x-ray diffuse scattering [9][10] in both BaTiO₃ and in KNbO₃, which have the same set of phase transitions, suggested a difference in the local structure. In general, the intensity of x-ray scattering arises from two main components:

(1.2)

IBragg is the intensity observed at reciprocal lattice nodes in the form of sharp scattering. Determination of crystal structures using the Bragg peaks provides knowledge of the crystal structure averaged over all unit cells in the crystal. Elements of disorder (whether static or dynamic) are simply averaged out. However, IDiffuse represents the extra diffuse scattering that is always seen in any x-ray diffraction pattern. It is generally weaker than the Bragg components and arises from elements of short-range order (static or dynamic) in the positions of the atoms.

Interpretation of the diffuse scattering patterns suggested a short-range order model of the structure (although again it should be emphasized that although the problem can be interpreted in terms of static disorder, the use of the term “fluctuations” by the original workers allowed for the possibility that the disorder was dynamic) which can be summarized as in Figure 1.7. Consider first of all the cubic I phase in Figure 1.7(a). Instead of the Ti atom occupying the center of the unit cell, it is in fact displaced a short distance along the eight <111> directions, so that they average out to being seen as a large central atom. The model invoked ordered chains along each of the <100> directions with one of the eight components in each chain, but with the chains randomly related to one another. The effect of this is to produce planes of diffuse sheets perpendicular to <100>. What this means is that locally the structure is not cubic but actually rhombohedral, since within each short-range ordered region the Ti atoms are directed along individual 3-fold axes: thus at the local scale of a few nanometers, the crystal is actually polar, with the polarization effectively averaged to zero over all unit cells. In the tetragonal phase II the Ti atoms occupy four possible <111> sites so that they average out to give a resultant polarization along the [001] direction. In this case one set of diffuse planes vanishes leaving two sets perpendicular to [100] and [010]. In the orthorhombic phase III, the Ti atoms occupy just two of the <111> sites to give a resultant average polarization along [110] and then only one set of diffuse planes is left. Finally, in the rhombohedral phase IV the diffuse scattering is absent altogether and the structure is fully ordered with the Ti displaced along [111]. This is a very beautiful model which has been studied for many years, but with conflicting views on whether it is actually correct or not. For example in [5] no evidence for the disorder was found. On the other hand, Senn [11] recently decided that the disorder was present. A good paper in which this topic is reviewed is given in [12]. A nice recent paper by Shao and Zuo [13] using convergent beam electron diffraction has confirmed that in reality in the tetragonal phase there are large regions showing a normal tetragonal structure mixed with symmetry-breaking regions. In this picture, the true structure of BaTiO₃ in this phase is actually a mixture of ordered and disordered regions. The important thing to note is that this was the first evidence that in ferroelectric perovskites, attention needs to be paid not only to the average crystal structure but also to what happens at the local scale below the coherence length of the radiation used (e.g. for x-ray, neutron and electron diffraction). Interestingly, the perovskite PbTiO₃ is isostructural with tetragonal BaTiO₃, yet there is no evidence of patterned diffuse scattering, and its high-temperature phase transition to the cubic phase appears to be simply given by an underdamped soft mode.

Figure 1.7. Disordered Ti/Nb atoms: (a) cubic; (b) tetragonal; (c) orthorhombic; (d) rhombohedral. The black dot marks the unit cell center. The arrows mark the direction of spontaneous polarization.

Whereas in the early days, perovskite crystal structures were treated purely in terms of translational periodicity, the inclusion of large regions of disorder is now being found to be of increasing importance. For instance, consider the structures of the important ceramic piezoelectric PbZr1-xTixO₃ (PZT). The phase diagram shows that on the Zr-rich side of the phase diagram the average symmetry of the crystallites is R3m or R3c, depending on composition. For x = 0.48 there is a sudden change to tetragonal P4mm symmetry: the boundary here is known as the Morphotropic Phase Boundary (MPB). It is at the MPB that the piezoelectric activity is highest, and over the last 50 years or so many attempts have been made to understand this.

In recent times, Noheda et al. [14] showed that on the Zr-rich side of the MPB for a narrow composition range the structure is actually monoclinic. However, there appears to be no border to the rhombohedral phases. This problem has recently been solved in reference [15] where it was shown that on the rhombohedral side the symmetry of the unit cells is monoclinic but with the cells arranged in a random orientation in some regions of the crystal with embedded ordered monoclinic regions (Figure 1.8). When averaged over all unit cells, the crystal symmetry appears to be rhombohedral, space group R3m or R3c, depending on composition

Figure 1.8. Schematic diagram showing ordered and random order in PZT.

Taken as a whole the average structure then looks rhombohedral. As the composition changes towards the MPB the ordered monoclinic regions grow at the expense of the disordered regions until they are sufficiently large to be seen as a distinct monoclinic phase. The importance of the monoclinic regions is that the Pb polar displacements lie on a single mirror plane and are thus their directions are free to be changed within the mirror plane by an external field. This so-called polarization rotation model thus allows for an increase in piezoactivity. In a more recent work [16] it has been shown that the piezoactivity is due to a combination of polarization rotation (intrinsic effect) and a domain-dominated (extrinsic) effect. On the Zr-rich side of the MPB both effects coincide, whereas on the Ti-rich side it is mainly extrinsic effects that dominate. This explains the observation that the piezoactivity when plotted against composition changes more slowly on the Zr-rich side but drops off rapidly on the Ti-rich side.

1.2. Octahedral Tilting

The oxygen octahedra are all joined at their vertices to form an infinite framework of corner-linked polyhedra. A common feature of perovskites is that one finds many examples where the octahedra are tilted about different directions. The first analysis of tilted structures was made by Glazer in 1972 [17], see also [18], and further expanded by Woodward [19]. The idea is quite simple. Consider the three pseudocubic axes, ap, bp and cp. Suppose we know rotate an octahedron through a small angle about the cp-axis within an ap-bp layer. Now because it is connected by corner-linking to neighboring octahedra within the ap-bp plane, these octahedra must tilt in the opposite sense. This means that the ap and bp axes become doubled as the repeat between octahedra goes from the first octahedron to the one two places away. Now consider what happens to the layers of octahedra lying above and below the first layer. If the octahedra directly above i.e. along cp rotate in the same sense, we write this tilt system as a⁰a⁰c⁺, the a0 a⁰ terms signifying no tilting about the ap and bp axes. Alternatively the octahedra may alternate their sense of rotation along the cp-axis and this we write as a⁰a⁰c–. Both tilt systems create a tetragonal crystal structure, but a⁰a⁰c+ produces a pseudocubic unit cell of apxapx2cp, while a⁰a⁰c– creates 2apx2apx2cp.

It is then a simple matter to apply this idea about the other two axes to build up two and three-tilt systems. Figure 1.9 shows an example of a three-tilt system labeled a⁺b⁺c⁺, meaning that there are in-phase tilts about each axis of different magnitudes. Interestingly the tilt system a⁺a⁺a⁺ in which the three tilts are of equal magnitude is another, but rare, cubic perovskite structure.

Figure 1.9. An example of a three-tilt system.

Originally, there were suggested to be 23 different tilt systems, but the number was later reduced to 15 by group theory arguments [20]. Table 1.1 shows a complete listing.

Table 1.1. List of tilt systems and symmetries.

Serial No. | Tilt System | Space Group | Sp. Gp. No. | Howard & Stokes
(1) | a⁺b⁺c⁺ | Immm | 71 | √
(2) | a⁺b⁺b⁺ | Immm | 71 |
(3) | a⁺a⁺a⁺ | Im3 | 204 | √
(4) | a⁺b⁺c– | Pmmn | 59 |
(5) | a⁺a⁺c– | P4₂/nmc | 137 | √
(6) | a⁺b⁺b– | Pmmn | 59 |
(7) | a⁺a⁺a– | P4₂/nmc | 137 |
(8) | a⁺b–c– | A2₁/m11 | 11 | √
(9) | a⁺a–c– | A2₁/m11 | 11 |
(10) | a⁺b–b– | Pnma | 62 | √
(11) | a⁺a–a– | Pnma | 62 |
(12) | a–b–c– | F1 |   2 | √
(13) | a–b–b– | I2/a | 15 | √
(14) | a–a–a– | R3c | 167 | √
(15) | a⁰b⁺c⁺ | Immm | 71 |
(16) | a⁰b⁺b⁺ | I4/mmm | 139 | √
(17) | a⁰b⁺c– | Bmmb | 63 | √
(18) | a⁰b⁺b– | Bmmb | 63 |
(19) | a⁰b–c– | F2/m11 | 12 | √
(20) | a⁰b–b– | Imcm | 74 | √
(21) | a⁰a⁰c⁺ | C4/mmb | 127 | √
(22) | a⁰a⁰c– | F4/mmc | 140 | √
(23) | a⁰a⁰a⁰ | Pm3m | 221 | √

There are many examples of perovskites with tilted octahedra, sometimes combined with atomic displacements. Thus, for example, SrTiO₃ has no tilts at room temperature but on lowering to around 105K tilts appear with the tilt system a⁰a⁰c– with space group I4/mcm. In PZT, on the Zr-rich side of the MPB the R3c average structure has the tilt system a–a–a–, the three tilts being equal in magnitude. In the antiferroelectric compound NaNbO₃ there are seven different phases with the room-temperature phase consisting of a mixture of different tilt systems plus antiparallel Na displacements.

1.3. Distortion Modes

The various crystal structures adopted by perovskites are pseudosymmetric, in the sense that they are formed by small distortions of the parent aristotype structure. As we have seen above, these mainly involve atomic displacements and tilted octahedra. These distortions can be thought of as arising from phase transitions from the aristotype structure through condensation of distortion modes. There are two useful software applications that enable one to decide through symmetry arguments which modes are responsible for a particular structure [21, 22].

Generally speaking, ferroelectric structures arise from condensation at the Brillouin zone center while tilting and antiferroelectric structures come from condensation at the zone boundaries. Let me illustrate the concept by some examples. Perhaps the best way to understand how this works is to consider a particular application using ISODISTORT [22].

On opening the initial webpage, we find Get started quickly with a cubic perovskite parent. The default is the cubic perovskite structure, and so we choose this. On the next page, it can be seen that the cations chosen are Sr and Ti and the anion is O. We can change the Sr to Ba if we wish. Actually, for the purposes of a symmetry argument actual atom types are irrelevant, and we can equally well call them something else. Four different methods are available for the next stage. Choose method 2 and select GM, k12,(0,0,0): this refers to a mode whose wave-vector is at the Brillouin zone center. On clicking OK the next page deals with the irreducible representations. The pull-down menu offers a series of different labels for the different representations. Choose the one labeled GM4-. On the next page we find

Pm-3m[0,0,0]GM4- (a,0,0) 99 P4mm, basis={(0,1,0),(0,0,1),(1,0,0)}, origin=(0,0,0), s=1 i=6, ferroelectric, k-active= (0,0,0)

This means that starting with the cubic space group Pm3m the distortion mode labeled GM4- leads to space group P4mm (no 99). The basis has the axes chosen in the order bca with the unit cell origin at (0,0,0). The structure is ferroelectric with a mode at wave-vector k = (0,0,0). If you select View Distortion, a Java applet is opened which draws a sketch of the structure and by using the sliders for the different atoms we see that all atoms can move along [010] under the representation GM4-. Notice that this is consistent with the tetragonal BaTiO₃, except that the unique axis has been unconventionally chosen to be along [010] (this is just an arbitrary choice of axes and does not affect the interpretation). Note also the irreducible representations GM1+ and GM3+ cause the unit cell to distort, corresponding to acoustic modes.

If you repeat this process choosing the R point in the Brillouin zone you obtain under R4+

P1 (a,0,0) 140 I4/mcm, basis={(1,1,0),(-1,1,0),(0,0,2)}, origin=(0,0,0), s=2, i=6, k-active= (1/2,1/2,1/2)

and on viewing the distortion we now see that the distortion mode corresponds to the a⁰a⁰c– tilting system.

Another way of carrying out distortion mode analysis is to input both the aristotype structure and also the hettotype and then find the symmetry modes. While this can be done in ISODISTORT, consider, as an alternative, the use of the AMPLIMODES program in the Bilbao Crystallographic Server [21]. Note that to load the structural information the input file can be in the form of a CIF (Crystallographic Input File), or else entered manually. So, for example, suppose we load the structure file for the BaTiO₃ tetragonal structure as the hettotype. First we have to make sure that the transformation matrix is set up as

Then following the procedure through you finally discover a great deal of information after the symmetry mode analysis (atomic shifts, global shift etc) including the table

Atoms | WP | Modes
O1 | 3c | GM4-(2)
Ti1 | 1b | GM4-(1)
Ba1 | 1a | GM4-(1)

where WP are the Wyckoff positions of each atom type in the aristotype structure. We see that the program has automatically found that the distortion mode is GM4-. Similarly, suppose we enter the following as the hettotype for the structure with tilt system a⁰a⁰c–

and the transformation matrices

Pressing the Show button, we get

Atoms | WP | Modes
O1 | 3c | R5-(1)

Note that the result this time appears to be different from that found using ISODISTORT (above). The reason for this is subtle and is generally unappreciated. When dealing with symmetry modes with k≠0, the actual irreducible representation and its label depend on the choice of origin. In our ISODISTORT example, the origin of the unit cell was on the Ba atom, whereas in our AMPLIMODES example it is on the Ti atom. If the origin in the aristotype section is changed so that the Ba atom is at the origin AMPLIMODES then gives the same result as in our ISODISTORT example. So, anyone publishing irreducible representation symbols for distortion modes should always state the origin choice at the same time. Unfortunately, I often see papers in which this is not done.

The use of distortion mode analysis is a very useful way of exploring new possible structures. A good example is a study of the high-temperature phases of NaNbO₃ [23] where ISODISTORT was used to determine the types of tilt systems that existed. The authors found a complex system in which the magnitudes of the tilts varied along a particular direction so that the repeat distance was increased beyond the usual doubling.

References

[1] R.H. Mitchell, Perovskites: modern and ancient. Almaz Press, 2002.

[2] M.W. Lufaso and P. M. Woodward, “Jahn-Teller distortions, cation ordering and octahedral tilting in perovskites,” Acta Crystallogr. Sect. B Struct. Sci., vol. 60, no. 1, pp. 10–20, 2004.

[3] E. Sawaguchi, H. Maniwa, and S. Hoshino, “Antiferroelectric structure of lead zirconate,” Phys. Rev., vol. 83, p. 1078, 1951.

[4] H.D. Megaw, “Crystal structure of barium titanate,” Nature, p. 484, 1945.

[5] G.H. Kwei, C. Lawson, S.J.L. Billinge, and S.W. Cheong, “Structures of the ferroelectric phases of barium titanate,” J. Phys. Chem., vol. 97, no. 10, pp. 2368–2377, 1993.

[6] H.D. Megaw, Crystal structures: a working approach. 1973.

[7] J.P. Remeika, “A method for growing barium titanate single crystals,” J. Am. Chem. Soc., vol. 76, pp. 940–941, 1954.

[8] R. Clarke, “Phase transition studies of pure and flux-grown barium titanate crystals,” J. Appl. Crystallogr., vol. 9, pp. 335–338, 1976.

[9] M. Lambert and R. Comes, “The chain structure and phase transition of BaTiO3 and KNbO3,” Solid State Commun., vol. 7, no. 2, pp. 305–308, 1969.

[10] R. Comes, M. Lambert, and A. Guinier, “The chain structure of BaTiO3 and KNbO3,” Solid State Commun., vol. 6, no. 10, pp. 715–719, 1968.

[11] M. Senn, “Emergence of long-range order in BaTiO3 from local symmetry-breaking distortions.” [Online]. Available: http://www3.imperial.ac.uk/newsandeventspggrp/imperialcollege/centers/nanoscienceandnanotechnology/eventssummary/event_26-5-2016-16-27-17.

[12] M.B. Smith, K. Page, T. Siegrist, P.L. Redmond, E.C. Walter, R. Seshadri, L.E. Brus, and M. L. Steigerwald, “Crystal Structure and the Paraelectric-to-Ferroelectric Phase Transition of Nanoscale BaTiO 3,” J. Am. Chem. Soc., vol. 130, pp. 6955–6963, 2008.

[13] Y.-T. Shao and J.-M. Zuo, “Nanoscale symmetry fluctuations in ferroelectric barium titanate, BaTiO 3,” Acta Crystallogr. Sect. B Struct. Sci. Cryst. Eng. Mater., vol. 73, no. 4, pp. 708–714, Aug. 2017.

[14] B. Noheda, D.E. Cox, G. Shirane, J.A. Gonzalo, L.E. Cross, and S.-E. Park, “A monoclinic ferroelectric phase in the Pb(Zr1−xTix)O3 solid solution,” Appl. Phys. Lett., vol. 74, no. 14, p. 2059, 1999.

[15] N. Zhang, H. Yokota, A. M. Glazer, Z. Ren, D. A. Keen, D. S. Keeble, P. A. Thomas, and Z.-G. Ye, “The missing boundary in the phase diagram of PbZr1−xTixO3,” Nat. Commun., vol. 5, p. 5231, 2014.

[16] N. Zhang, S. Gorfman, H. Choe, T. Vergentev, V. Dyadkin, H. Yokota, D. Chernyshov, B. Wang, A.M Glazer, W. Ren, Z.-G. Ye, “Probing the intrinsic and extrinsic origins of piezoelectricity in lead zirconate titanate single crystals, ” J. Appl. Cryst, 51, 1396–1403, 2018.

[17] A.M. Glazer, “The classification of tilted octahedra in perovskites,” Acta Crystallographica Section B Structural Crystallography and Crystal Chemistry, vol. 28, no. 11. pp. 3384–3392, 1972.

[18] A.M. Glazer, “A brief history of tilts,” Phase Transitions, vol. 84, no. 5–6, pp. 405–420, 2011.

[19] P.M. Woodward and IUCr, “Octahedral Tilting in Perovskites. I. Geometrical Considerations,” Acta Crystallogr. Sect. B Struct. Sci., vol. 53, no. 1, pp. 32–43, Feb. 1997.

[20] C.J. Howard, H.T. Stokes, and IUCr, “Group-Theoretical Analysis of Octahedral Tilting in Perovskites,” Acta Crystallogr. Sect. B Struct. Sci., vol. 54, no. 6, pp. 782–789, 1998.

[21] D. Orobengoa, C. Capillas, M.I. Aroyo, and J.M. Perez-Mato, “AMPLIMODES: Symmetry-mode analysis on the Bilbao Crystallographic Server,” J. Appl. Crystallogr., vol. 42, no. 5, pp. 820–833, 2009.

[22] H.T. Stokes, B.J. Campbell, and D.M. Hatch, “ISODISTORT.” [Online]. Available: http://stokes.byu.edu/iso/isodistort.php.

[23] M.D. Peel, S.P. Thompson, A. Daoud-Aladine, S.E. Ashbrook, and P. Lightfoot, “New Twists on the Perovskite Theme: Crystal Structures of the Elusive Phases R and S of NaNbO3,” Inorg. Chem., vol. 51, no. 12, pp. 6876–6889, 2012.

2

Growth of ABO₃ and BO₂ Crystals

Sławomir Mielcarek¹, Andrzej Majchrowski²

1 Faculty of Physics, Adam Mickiewicz University, Uniwersytetu Poznańskiego 2, 61-614 Poznan, Poland

² Institute of Applied Physics, Military University of Technology, gen. S. Kaliskiego 2, 00-908 Warsaw, Poland

Abstract

Structural quality as well as stoichiometry of crystalline materials are the main factors influencing their physical properties, and therefore decide on their potential applications. Bulk crystallization of many oxide perovskites suffers from many drawbacks due to properties of melts, existing high-temperature phase transitions, and non-stoichiometry, among others. However, tremendous progress in thin-film and other structure technologies allows, mainly due to vapor techniques, fabrication of high-quality samples for further investigations and applications. Understanding of thermodynamics and kinetics of crystal growth, nucleation processes, and knowledge of proper crystal growth techniques is essential to obtain good quality crystals. The chapter is mainly devoted to crystal growth techniques of chosen oxide crystals. The theory of crystal growth, due to limited space, is described in short. Given references will allow the reader a full view into its complexity and many aspects of phenomena connected with crystallization. In the last part of the chapter methods of analysis of as-grown crystals are described.

2.1. Theory of Crystal Growth

2.1.1. Equilibrium Conditions

Single crystal growth of a given compound can be carried out in a controllable manner only under conditions slightly deviating from the equilibrium. Only in this way such disorders as uncontrolled spontaneous crystallization, dendritic growth, or formation of unwanted phases can be avoided. To achieve this goal one must have precise knowledge on the equilibrium conditions in an investigated crystal growth system.

In a many-particles system, the internal energy U (the so called state function, dependent only on physical parameters of the system and independent on the way in which the system reached the given state) decreases when equilibrium is reached. Other state functions to be named are entropy, enthalpy, and free energy. According to the first law of thermodynamics:

(2.1)

The change in the internal energy of a closed system is equal to the amount of heat Q supplied to the system minus the amount of work W done by the system on its surroundings.

On the other hand, entropy S should increase to reach equilibrium. The entropy is defined by Boltzmann’s formula

(2.2)

where k is the Boltzmann constant, and M is the number of the possible states of the system.

Using U and S as well as temperature T, pressure P, and volume V we can define two state functions useful in describing equilibrium of the many-particle systems. Helmholtz free energy F defined as

(2.3)

while Gibbs free energy G is described as

(2.4)

Here enthalpy H equals to

(2.5)

F goes to the minimum when the many-particle system reaches equilibrium under a constant temperature without changing the volume while G goes to minimum at equilibrium under a constant temperature when pressure does not change. In other words

(2.6)

(2.7)

Any disturbance of the system from the equilibrium diminishes entropy (dS < 0), but increases free energy (dF > 0, dG > 0).

An increment of the Gibbs free energy of the multi-particle system, composed of one kind of particle, by adding of one mol of particles is defined as the chemical potential µ. If the system is composed of different kinds of particles, the chemical potential of a particular kind of i-th component is given as

(2.8)

and the change in free Gibbs energy of a system that is held at constant temperature and pressure is given as

(2.9)

In thermodynamic equilibrium when dG = 0

(2.10)

the above equation allows to formulate conditions of mass equilibrium in a closed system. In the simplest case of one compound existing in two phases the condition of equilibrium of these phases is

(2.11)

and spontaneous mass transport from phase 2 to phase 1 occurs at temperatures or pressures when µ2 > µ₁. It is also possible to change the value of chemical potential, and therefore influence the equilibrium of the system, by adding a component that does not crystallize [1].

Nucleation of the new phase, in our case a cluster of atoms forming solid matter, can be discussed in terms of thermodynamics, when the driving force is Δµ < 0. Taking into account Gibbs free energy of the volume of a growing cluster and its surface, it was shown that for some critical value of radius the summary free energy reaches its maximum. In other words, when the radius of the growing seed crystal is bigger than the critical value, the crystal becomes stable [1, 2]. The value of this critical radius depends not only on thermodynamic parameters of the crystallizing material but also on conditions of crystallization (formation of free crystal, growth on surface of the same kind of crystal – homogeneous growth, or seeding on other surface – heterogeneous growth). The kinetics of seeding (nucleation rate) can be expressed taking into account the frequency and the cross-section of collisions of monomers with growing clusters, and their dissociation rate. The nucleation rate is expressed as the frequency of the appearance of a cluster with dimensions bigger than the critical radius. The further growth on the perfect crystal surface demands high supersaturation that is needed to form two-dimensional seeds forming steps on which further growth is possible. On the other hand, crystal growth under conditions of weak supersaturation is observed – this is growth due to the presence of screw dislocations forming growth pyramids on the surface of the crystal. Based on this assumption such phenomena as kinetics of steps, their instability, formation of bunches of steps, roughening of the surface, normal and lateral crystal growth were broadly discussed in the literature. A detailed description of these theories can be found in [2], [3]

2.2. Phase Diagrams of Model Oxides. Major Challenges in Crystal Growth of Stoichiometric Oxide Crystals

2.2.1. Gibbs Phase Rule

For a given system containing C components the number of the coexisting phases P and the number of degrees of freedom, F, that the system has in equilibrium are connected by the following equation:

(2.12)

In a one-component system it means that at the so called triple point in which three phases exist there is no degree of freedom (F = 0). Eutectic points are also invariant due to the existence of three phases (liquid and two crystalline phases). In the case of two phases in equilibrium, for example solid and liquid, the change of the temperature of the system results in the change of the pressure, therefore only one degree of freedom exists. Gibbs phase rule in its classical form takes pressure and temperature as the independent variables. When constant pressure is kept, the number of independent variables is reduced by one. On the other hand, the Gibbs free energy may depend on other fields such as electric work density of polarized crystal or elastic energy stored in a deformed solid [4]. Crystallization of epitaxial layers of SrTiO₃ on different substrates, due to lattice mismatch, can cause different strain in as-grown thin single crystalline layer. Under relatively small strain and at high T the material remains paraelectric (cubic phase). Under large strain, the material lowers its symmetry and becomes ferroelectric. In the intermediate region coexistence of both phases is observed, despite constant pressure.

2.2.2. Phase Diagrams

Single crystal growth of many compound oxides, due to their properties, imposes using of phase diagrams to define precise conditions of crystallization, especially from melts. In the case of incongruent melting or the presence of high-temperature irreversible/destructive phase transition the crystallization must be carried out below the temperature of such transformation to obtain the desired phase. Also congruently melting compounds with high temperatures of melting are easier to be crystallized when there is a possibility of lowering the temperature of crystallization by the addition of some components. The best solution in all discussed above cases is the so-called self-flux High Temperature Solution Growth (HTSG) in which the solvent used to flux growth is one of the components of a given phase diagram. Such a solvent does not introduce other elements to the crystal growth system and therefore is chosen unless other factors make it impossible. The main obstacle in using self-flux is the formation of other phases before reaching the desirable temperature of crystallization. Sometimes high viscosity of such solvent can also limit its applications. ACerS-NIST Phase Equilibria Diagrams Database gives detailed information on existing compounds and their transformations in a given system of two or more chemical components, and therefore is a very useful tool in carrying out crystal growth experiments [5].

2.2.3. Nonstoichiometric Oxides

An important thermodynamic quantity describing nonstoichiometric oxides is the activity of oxygen in the solid compound. It is the chemical potential of oxygen μO2

(2.13)

Here μ⁰O2 is the chemical potential of oxygen in its standard state and PO2 is the equilibrium oxygen pressure in the atmosphere surrounding the nonstoichiometric oxide. RTlnPO2 is the measure of the relative partial molar free energy of oxygen. These values depend obviously on the temperature but also on the composition in some cases. It was shown [6] that some nonstoichiometric phases consist of a whole series of line phases (Magnéli phases). Point defect structures predominate in nonstoichiometric phases (connected to vacancies and interstitial positions of cations or anions). However, in some compounds, like TiO2–x the excess charge of the electronic species is compensated by extended defects with one- or two-dimensional periodicity [7]. Cation movements into oxygen vacancies positions leads to the formation of planar defects known as shear planes. In rutile-structured TiO2–x it is connected with the change from edge to face sharing in the TiO6 octahedra. These shear planes show a strong tendency to form ordered structures. Compounds of TinO2n–1 formula exist, containing regularly spaced shear planes whose spacing increases with n. For higher values of n (15–18), the interplane separation can reach 200–300 Å [8].

2.3. Techniques of Growth of Selected Binary and Ternary Oxide Crystals

Crystallization of TiO₂, SrTiO₃, BaTiO₃, and PbZrO₃ compounds will be discussed to show different techniques possible to use.

TiO₂ crystallizes in several structures. It melts at 1850 °C. Tetragonal rutile and anatase, alongside orthorhombic brookite are most important phases, all composed of slightly deformed octahedra formed by oxygen atoms placed around six fold coordinated titanium. In this review we will concentrate on the rutile phase due to its applications in resistive switching.

SrTiO₃ melts congruently above 1900 °C and crystallizes in a cubic structure. It does not undergo phase transitions between the temperature of crystallization and RT [5].

BaTiO₃ also melts congruently near 1625 °C and crystallizes in a hexagonal structure. However, it undergoes phase transitions (cubic, tetragonal) between the temperature of crystallization and RT [5].

PbZrO₃ melts incongruently and decomposes around 1600 °C to ZrO₂ crystals and peritectic liquid composed mainly of PbO [5]. Using proper solvents allows crystallization of cubic paraelectric PbZrO₃ phase that transforms to antiferroelectric orthorhombic phase, stable at RT.

Bulk single crystals growth of these compounds may be carried out with the use of numerous techniques.

Verneuil flame method was for the first time used in 1904 for crystallization of rubies. Droplets of congruently melting material, molten in the O₂/H₂ flame, solidify under conditions of relatively high-temperature gradients on the surface of the seed. The growth of first materials, obtained by means of this technique, took place far away from the thermodynamic equilibrium, therefore as-grown single crystals of TiO₂ and SrTiO₃ suffered in the past from nonstoichiometry, high mechanical stress, and high density of dislocations [9, 10, 11]. Improvements of growth conditions, such as optimization of thermal insulation and additional heating zones as well as changes in construction of burners, led to better quality of currently grown SrTiO₃ samples. Commercially available SrTiO₃:Nb (100) single crystals were investigated [12]. Inhomogeneous distribution of Ti³⁺ states was confirmed. However, it was shown that in SrTiO₃ single crystals containing 1.4 at% Nb, used as conductive substrates for epitaxial growth of thin films, Nb distribution is uniform even in nano and atomic scale. Although no Nb segregation on existing dislocations was observed, it was found that dislocations and grain boundaries may trap electrons in SrTiO₃ doped with donors and locally reduce electronic conductivity in this way. Despite the improved quality of available Verneuil single crystals, they contain much more dislocations than single crystals grown by other methods.

Czochralski technique, when crystallization of congruently melting oxide compounds is considered, is in most cases the method of choice. However, such instabilities as foot formation, non-uniform diameter of growing crystals, despite using of sophisticated methods of diameter control, and disconnection of growing crystal from the melt are common difficulties observed during Czochralski crystallization of TiO₂ [13]. Pulling SrTiO₃ single crystals from the melt suffers due to similar instabilities leading to subsequent spiraling. Very low thermal conductivity of strontium titanate (thermal conductivity < 2.2 W/(m ∙ K) at temperatures > 1700 °C and almost total opaqueness to infrared radiation close to the melting point lead to a poor heat transport through the growing crystal. As a result of the thermal and optical properties of the material, the heating power has to be continuously lowered in such an abrupt way that stable Czochralski crystallization is impossible to be carried out. Nevertheless, suitable conditions for the growth of short single crystals (up to 15 mm long) were found for the Czochralski method. However, the structural quality of the obtained crystals was poor [14]. Cold-crucible (skull technique) Czochralski SrTiO₃ single crystals growth also was performed but the crystals obtained had poor quality and high density of etch pits [15].

Edge-defined Film-Fed Growth (EFG) method, due to different mass and heat transport conditions enables the crystallization of better quality single crystals than the Czochralski method. The growth is carried out from oxide melts having stoichiometric composition, like in the case of Czochralski growth. Molten oxide, due to capillary force is distributed on the surface of an iridium or platinum die that allows shaping of growing single crystal. TiO₂ ribbon single crystal, with a (110) surface and 2 mm × 20 mm × 80 mm in dimensions, was successfully grown by EFG method. The ribbon crystals were pulled at about 5 mm/h in an Ar atmosphere. Owing to a uniform temperature distribution and the precise temperature control of the die top the ribbon crystal exhibited no grain boundaries. In the growth of crystals with high melting temperature such as TiO₂, heat transfer is mainly by radiation. The absorption in the growing crystal would have a considerable influence on the heat transfer and the growth stability [16]. It was found that bulk rutile crystals, which were difficult to grow using the Czochralski technique, could be successfully grown using the EFG method. The rutile crystals, just after they start growing, are reduced. In this way, as it was mentioned above, it changes heat transport and causes abrupt lateral growth of the rutile crystals when the melt temperature decreases in Czochralski growth. In the case of EFG growth, the crystal shape is limited by the die edge and the meniscus height on the die top is short. Therefore, it is quite natural that the crystal shape control is much easier using the EFG method compared with the Czochralski method. Additionally, the uniformity of the temperature distribution at the EFG growth interface is the good due to the relatively small radial temperature gradient caused by good thermal conductivity of the die. A uniform rutile boule, 15 mm in diameter and 40 mm in length was grown under modified EFG conditions without crystal rotation [17]. The EFG method was also used to grow cylindrical SrTiO₃ bulk single crystals with diameters of about 15 mm and lengths up to 50 mm from stoichiometric melts. Typical average etch pit density values were in the range of 1.7–2.0 × 105 cm–2. Rocking curve measurements revealed FWHM values between 28′′ and 41′′ [14].

The floating zone (FZ) method allows both crystallization from stoichiometric melts as well as crystallization with the use of traveling solvent. In the case of oxides crystallization infra-red IR heating (IR-FZ) is used. A typical furnace for the IR-FZ method is a single-ellipsoidal-mirror-type image furnace, but two- or even four-ellipsoidal-mirror-type image furnaces are also used, giving more uniform heating. The shape of the molten zone may be influenced using a tilting mirror type image furnace. By properly chosen thermal conditions the convexity of the crystal-side interface may be decreased and stability of the molten zone may be increased. The quality of rutile single crystals grown by means of this technique was improved. A large rutile single crystal of 19 mm diameter was successfully grown under the mirror tilted conditions [18]. FZ method was also used to grow good quality single crystals of SrTiO₃ 15–20 mm in diameter and 40–80 mm in length. Additional crystal heating just below the molten zone by an in-growth annealing furnace was applied in order to lower the temperature gradients and to achieve slower cooling of the grown crystal. The crystal perfection was studied with X-ray topography and double-crystal diffractometry. The most perfect crystals were grown in [001] direction with single grain rocking curve widths of about 30″ and sub-grain misorientations of 1′–3′ over 10 × 10 mm² areas of the boule cross-section [19]. Crack-free and yellowish transparent Ba1– xSrxTiO₃ single crystals 8 mm in diameter and 40 mm in length were grown by FZ method. Substitution of more than 1.2 at% Sr for Ba was effective in preventing the formation of the hexagonal phase. It was found that Ba1–xSrxTiO₃ melts congruently at 1.5 at% Sr [20]. Also, BaTiO₃ single crystals were grown by means of traveling solvent zone technique with the use of rutile surplus as the solvent [21].

Laser Heated Pedestal Growth (LHPG) technique is used to fabricate single crystal fibers of congruently melting materials. The fibers are directly grown from upper parts of green-body feed rods molten by laser radiation. Transparent and colorless SrTiO₃ fibers were grown by means of LHPG technique [22]. Influence of oxygen partial pressure and composition of feeding rods on coloration of LHPG obtained SrTiO₃ single crystal fibers was investigated in [23]. SrTiO₃ single crystals grown in oxygen-rich atmosphere are brown and show high Sr/Ti ratios. Decreasing oxygen partial pressure improves the quality of as-grown crystals. Growth under low oxygen partial pressure gives blueish or greyish crystals. By using a Sr-enriched feed rod in an atmosphere that contains about one vol. % oxygen, colorless STO fibers can be produced. The main reason for the low transmittance in crystals grown in high oxygen partial pressures is probably off-stoichiometry and the formation of scattering centers from clustered vacancies. The formation energy of the metal vacancies is considerably higher in oxygen-poor environments which hinders the formation of the scattering centers. This phenomenon is further prevented by Sr-excess in the feed material that compensates SrO(g) evaporating from the melt.

Single crystal fibers can be also grown by means of the micro-pulling down technique, in which fibers are formed due to crystallization on a seed being in contact with meniscus of molten compound formed below crucible with a small hole in the bottom. Proper conditions and crucible configurations may also lead to crystallization of bulk-shaped samples [24] like in the case of TiO₂ [25].

HTSG technique is the most important method of single crystals growth of complex oxide perovskite compounds, discussed in this chapter. Lowering the temperature of crystallization is realized due to the choice of proper solvent. Historically, the first successful attempt to crystalize BaTiO₃ was made by Remeika [26]. Crystallization was carried out from KF solutions that allows to lower the temperature of BaTiO₃ crystallization down to 1200 °C. However, plate-like BaTiO₃ “butterfly” twin crystals contain 1–2 at % of K and F [27]. The growth of pure single crystals can be carried out when the temperature of crystallization is lowered due to the addition of a component of growing crystal. In the case of BaTiO₃ and SrTiO₃ surplus of TiO₂ while for PbZrO₃ addition of PbO acts as self-flux. Barium titanate should be crystallized below 1432 °C to avoid destructive hexagonal to cubic phase transition. Top Seeded Solution Growth (TSSG) method allows crystallization of bulk BaTiO₃ single crystals, pure and doped with many transition metals and rare-earth ions. Problems encountered in BaTiO₃ TSSG include: high-temperature growth (~ 1400 °C), difficult starting growth temperature determination, strict temperature control requirements, slow pulling, long growth run duration, and the formation of a solution droplet on the bottom of the boule, which causes cracking. Continuous weight determination to control the growth process was developed [28].

In the case of SrTiO₃, the temperature of crystallization should be lowered to avoid partial decomposition of growing crystals resulting in perturbations during crystallization caused by changes in the optical transparency of crystals in the IR region. The common instabilities during SrTiO₃ TSSG are spiraling, foot formation, unstable diameter of growing crystal, and inhomogeneities. In recent years extensive investigations of correlations between oxygen partial pressure, composition, IR-transmittance, and coloration of the crystals were carried out in Leibniz Institute for Crystal Growth in Berlin. To improve the quality of SrTiO₃ single crystals grown at moderate temperatures (< 1535 °C), well below the melting temperature, TSSG method was used. Growth was carried out under low-temperature gradients using TiO₂ as a self-flux. Based on modifications of the growth conditions, virtually mosaicity-free SrTiO₃ single crystals of sizes in the range of 1–2 cm were obtained [29–32].

PbZrO₃ melts incongruently (see Figure 2.3) therefore the use of flux growth is not an option but a necessity. Many flux compositions were investigated, including fluorides and borates, however, self-flux crystallization of PbZrO₃ based on PbO solvent was found to give the best single crystals. The addition of B₂O₃ increases the viscosity of the melt and in this way diminishes the evaporation of PbO during crystallization. The addition of Pb₃O₄ improves the transparency of growing crystals and prevents their coloration. Mainly spontaneous crystallization is used. Both pure and Sn doped PbZrO₃ single crystals of good optical and structural quality were obtained in this way [33, 34, 35]. Spontaneous crystallization of Ti-doped PbZrO₃ (PZT) single crystals gives samples containing a small concentration of Ti despite heavy doping of melt. It was found that TSSG allows to grow PZT crystals from all concentration range, including the morphotropic boundary region. Unfortunately, the quality of as-grown samples is not very good in most cases, and PZT crystals composition is not uniform [36, 37].

Bulk rutile crystals can be also obtained due to chemical transport in closed systems. For example, tellurium tetrachloride was used as a transport agent under conditions of reversible chemical reaction controlled by proper temperature distribution in dissolution and crystallization zones of TiO₂ [38].

Although a detailed discussion of thin crystalline films fabrication is not the subject of this chapter, it should be underlined that sophisticated techniques, allowing controlled crystallization even on the atomic scale, and therefore enabling the formation of complicated multi-component structures having desirable physical properties, are now available. The most useful of these techniques are Molecular Beam Epitaxy (MBE) [39, 40], Metal-Organic Chemical Vapour Deposition (MOCVD) [41, 42], Atomic Layer Deposition (ALD) [43, 44], sputtering [45], sol-gel [46] and Pulsed Laser Deposition [47–49].

2.4. Typical Methods for Evaluation of Quality of Oxide Crystals

The most important methods for the determination of crystal quality include the diffraction methods, microscopic methods, and thermal analysis. Diffraction methods are based on the interference of waves scattered by atoms of a crystal studied and provide full information on the structure of elementary cells. Analysis of elastic wave scattering brings information on the static structure of matter. The diffraction methods employ the fact that for small wavelengths the crystal structure is a three-dimensional diffraction grating and the waves are scattered only in certain defined directions. There are two main methods: X-ray diffraction and neutron diffraction. In the X-ray diffraction of single crystals, on the basis of the spatial distribution of scattered X-rays and their intensities, it is possible to obtain accurate values of elementary cell parameters and its symmetry, using Bragg’s law or Laue conditions [50–52]. According to the Bragg’s law the X-rays reflected from particular crystallographic planes undergo constructive interference if the difference in the optical pathways of the rays reflected from a family of planes is equal to nλ:

(2.14)

where: λ – X-ray wavelength, d – distance between the planes at which the rays are scattered, θ – the angle of rays incidence on a plane. The passage of a parallel beam of X-rays through the crystal results in the appearance of Laue’s diffraction pattern. The symmetry of the diffraction image obtained brings information on the crystallographic system of a given crystal and full analysis of the image is possible making use of the Laue conditions.

In neutron diffraction experiments, the total cross-section for elastic neutron scattering is measured as a function of incident neutrons energy or a beam of neutrons of the same energy is scattered by the sample and the intensity of neutrons scattered is measured as a function of the scattering angle. The maxima appearing in the neutron diffraction pattern correspond to the families of crystallographic planes. The method of neutron diffraction extends the range of X-ray scattering and permits the investigation of structures of compounds containing hydrogen or elements of much or not much different atomic numbers as well as isotopes of the same elements [53]. On the basis of the ideal bulk structure of a crystal it is possible to characterize bulk defects and the surface of a given crystal.

2.4.1. Stoichiometric TiO₂

The most important natural polymorphic varieties of titanium dioxide are the minerals of tetragonal rutile or anatase structures and the mineral of rhombic structure known as brookite. As the most stable is that of rutile structure and the other polymorphic varieties can be transformed to it at 800–900oC, we will focus on this structure. The rutile structure belongs to the P4₂/mnm tetragonal space group, the lattice constants are found to be a = 0.4594 nm, c = 0.2958 nm, and one internal parameter u = 0.305 at 298 K. In this structure the titanium atom is surrounded by a slightly deformed octahedron composed of 6 oxygen atoms with the apical Ti-O bond length (0.198 nm) being slightly longer than the equatorial Ti-O bond length (0.195 nm) [54].

2.4.2. Perovskite Structure

The structure of this numerous family of crystals is represented in nature by the mineral CaTiO₃ known as perovskite and is denoted as ABO₃. We shall concentrate on barium titanate BaTiO₃ (BTO), lead titanate PbTiO₃ (PTO), and strontium titanate SrTiO₃ (STO). The interest in barium titanate started in the 1940s when the anomalous dielectric properties of BTO ceramic were discovered independently by Warnier and Salomon in the United States (1942), Ogawa in Japan (1944), and Wul and Goldman in Russia (1945). The first single crystals were obtained in 1947 in Switzerland [55]. The method for obtaining large, clear single crystals of BTO proposed by Remeika [56] has stimulated a dynamic increase in the studies of this family of crystals. The single crystals of STO were first grown by Merker in 1955 [57] and the single crystals of PTO were first grown in the 1930s by Cole & Espenschied [58].

Each of the above-mentioned three crystals show a structural phase transition from a non-polar cubic phase to a polar tetragonal one. The sequence of structural transitions is most developed in BTO crystal in which with decreasing temperature the cubic phase is followed by tetragonal, orthorhombic, and rhombohedral. Above 120oC, BTO has a cubic symmetry with m3m point group and its lattice constant is 0.3996 nm [59]. Below this temperature, BTO has tetragonal symmetry with 4mm point group and the lattice constants a = 0.3992 nm, and c = 0.40361 nm at 20oC [60]. The next symmetry change takes place at 5oC at which BTO assumes orthorhombic symmetry with mm point group and the lattice constants a = 0.399 nm, b = 0.5669 nm and c = 0.5682 nm at –10oC [61]. Then, at about –90oC the crystal assumes rhombohedral symmetry with 3m point group, the lattice constant a = 0.4001 nm and α = 89o51′ [62]. Later reports [63–64] have specified the temperature of transition from cubic to tetragonal phase as 403 K. The directions of polar axis in particular symmetries are: for tetragonal symmetry – along the fourfold symmetry axis c, for orthorhombic symmetry – one of the 12 equivalent [110] directions of cubic phase, in low temperatures – the polar axis direction coincides with [111] direction of cubic phase, along one of the body diagonals [55], [65].

In PTO crystal the phase transition takes place at 763 K, at which the point group changes from m3m to 4mm. At room temperature, the lattice constants are a = 0.3904 nm and c = 0.4152 nm. It has been reported that this crystal undergoes another phase transition at about 173 K observed under very slow crystal cooling [66–67].

In STO crystal the transition from nonpolar to polar phase takes place at 110 K and is accompanied by the point group change from m3m to 4/mcm. The lattice constant at room temperature is a = 0.3905 nm, while below 110 K a small tetragonal distortion c/a = 1.0005 has been detected but the unit-cell volume remains unchanged through the transition. This transition is not accompanied by any anomalies in the dielectric constant but the lattice-dynamical characteristic of the 110 K transition in STO has been elucidated by inelastic neutron scattering measurements [68–71].

The fundamental methods for verification of the quality of a crystal obtained are thermal analyses. Changes in the physical and chemical properties of a given crystal as a function of temperature should be studied taking into account the thermal history of the crystal, ensuring stable pressure and using properly chosen working gas. Below a short description of the most often methods used is given.

DTA – Differential Thermal Analysis. In this method a direct difference in temperature between a sample studied and a reference sample is measured in the so-called Block-type Measuring System, the temperature sensors are in the samples. The method offers high sensitivity and immediate response to the processes taking place in the sample studied. In another, more universal and more often used solution, the Measuring System with Free-standing Crucibles, the thermocouple junction is in thermal contact with the bottom of the crucible. Its advantages are the lack of possibility of chemical reaction with the sample studied, the possibility of using different sample containers, high sensitivity [72].

DSC – Differential Scanning Calorimetry. In this method a stable measuring system with an exactly defined heat flow pathway is employed. The DSC idea can be realized in two approaches, the heat flux DSC and the power compensation DSC. In more detail, three methods of measurements can be applied: heat-compensating calorimeters in which the measured quantity is the energy required for compensating the heat to be measured, heat-accumulating calorimeters in which the measured quantity is the temperature change in the substance due to the heat to be measured and heat-exchanging calorimeters in which the measured quantity is the heat flow rate between the sample studied and its surroundings caused by the heat to be measured [72].

TGA – Thermogravimetric Analysis is a technique in which the mass of a substance is monitored as a function of temperature. The mass loss is measured by a precision balance [72].

Thermal analysis permits accurate determination of the temperatures of phase transitions in the crystal studied and besides the X-ray analysis, it is applied at the first stage of characterization of the crystal obtained. Because of a large number of methods for obtaining perovskite family crystals the information on the actual method used is often provided in literature reports.

The above methods of structural and thermal analyses are applied also for single crystals of very small size. In the studies of nanocrystalline samples, the questions appear about the phase transition temperature dependence on the crystal size and about the critical size in which it is still possible to observe phase transitions. These questions have been the subjects of particularly intense experimental and theoretical studies for barium titanate crystal which shows ferroelectric properties at room temperature and has a wide range of applications. DSC data revealed changes in phase transition temperatures in this material induced by lattice defects, grain size changes [73], the effect of non-stoichiometry [74], surface charges [75] or aging processes [76]. According to the results of studies on the critical size, they indicate a very wide range of particle diameters in which phase transitions have been observed. Applying additionally X-ray methods, including thermodiffraction and synchrotron X-ray diffraction, and Raman spectroscopy, Smith et al. have shown that in nanocrystals the cubic-to-tetragonal phase transition may occur in a wide temperature range, in contrast to the sharp transition found in the bulk material [64]. Analyzing the ¹³⁷Ba NMR spectra, Sedykh and Michel have evidenced a shift of the phase transition in BaTiO₃ by 3 degrees on decreasing the nanoparticles size from 155 nm to 15 nm [77]. On the basis of the variable-temperature electrostatic force microscopy studies, Spanier et al. have shown that for BaTiO₃ nanowires the particle diameter at which phase transition falls below room temperature is ~3 nm [75]. Studying ultrathin perovskite films of lead titanate, Fong et al. have proved the stability of the ferroelectric phase till the thickness of 3 unit cells, so 1.2 nm [78]. For the bulk materials, preliminary optical characterization is made by optical or polarization microscopes, but the samples smaller than 300 nm are beyond observation. Characterization of smaller particles needs the use of electron microscopy and a transmission electron microscope (TEM) and its modifications [79–81]. The idea of TEM is based on the de Broglie wave conception according to which the length of electron wave λ is related to their energy as:

(2.15)

where: E is the electrons energy in eV, λ is the wavelength in nm. For instance, for electrons of 100 keV a beam of waves of the length slightly below 4 pm is obtained. The development of electron microscopy has been triggered by the studies of Knoll & Rusk [82] on focusing an electron beam. Modern TEM produce very narrow electron beams, typically < 5 nm and at the best < 0.1 nm in diameter. These parameters permit the observation of a very small fragment of the sample. Similarly, as in photography, the problems are the depth of field and the field of view, there is also the problem of electron lenses. One way to improve their performance is to insert very small limiting apertures. These apertures obviously cut down the intensity of the beam, but enhance the depth of field of the specimen and depth of focus of the images [79]. The most serious limitation of TEM is the necessity of investigation of thin specimens, preferably thinner than 100 nm. The appropriate procedures for the preparation of adequately thin samples have been proposed for almost all types of materials. Usually, TEM provides the information on the mean size and shape of the perovskite nanoparticles and their number [64, 73, 83].

TEM and numerous modifications of this technique permit analysis of different signals generated when a high-energy beam of electrons interacts with a thin specimen, i.e. the signals from the beams reflected at different angles from the surface (Auger electrons, backscattered electrons, secondary electrons, characteristic X-ray or visible light) as well as those passing through the sample (elastically scattered electrons, inelastically scattered electrons, bremsstrahlung X-rays). When the sample to be studied is of about 10 nm in thickness, a high-resolution TEM (HRTEM) can be applied. This method offers a highly accurate transformation of the sample structure details from the specimen to the image. This transfer is charged with two main problems, the first is that the optical system is not perfect and the lenses have a finite size (Abbe’s theory), while the second is related to the difficulties in application of the atomistic model for interpretation of the image.

Transformation of the sample features into the best possible image by an electron microscope requires correct use of a number of procedures that could be described in the following way: the optical system transforms a certain point f(x,y) in an extended region in the image g(x,y), preferably in the form of a circular disk, to describe this process the point-spread function h(r) is introduced and the process of transformation is called the convolution of f(r) with h(r). Of course, for two close-lying points in the sample, the image will have a common part obtained as a result of the overlapping of the two circular disks corresponding to these points. In order to get a high resolution, it is necessary to provide high spatial frequencies, which corresponds to very small working distances.

Each point of the image has a characteristic intensity which for two-dimensions can be presented as a sum of two sinus functions and this indicates a possibility of Fourier transform application. This transformation must take into account the contributions due to the apertures, attenuation of the wave, and aberration of the lens through the introduction of appropriate functions so that to obtain a linear relationship between the image and the weak specimen potential. For very thin samples, the electrons passing through the sample undergo a phase shift that permits the approximation of the specimen as a ‘phase object’. On the basis of the model of the weak phase-object approximation, the objective lens transfer function T(u) is derived. If the value of T(u) is negative, the positive phase contrast is obtained and the image of atoms is dark against the light background, while if T(u) is positive, the negative phase contrast is obtained and the image of atoms is bright against a dark background. If T(u) is zero, no details are observed in the image. Literature reports on perovskite crystals present predominantly the images obtained with microscopes working with an electron beam obtained at the voltages 120 kV, 200 kV or even 300 kV [30]. No reliable information on the position of a given atom can be inferred from a single image. It is necessary to analyze the so-called focus series knowing the exit wave function and the aberration of the microscope used. Only then it is possible to determine the correct amplitude and phase to be able to achieve the atoms localization with a precision better than 5 pm [84].

The above-described methods for evaluation of crystal structures, starting from the classical X-ray methods providing information on full symmetry of elementary cell to the advanced HRTEM technique are supplemented with many other subtle techniques providing increasingly accurate data. The interpretation of the data permits concluding with increasing reliability about positions of atoms, their surroundings and function in physicochemical properties of a given material.

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