Switching effects - ebook

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.

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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 . 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 . 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. 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.

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 who showed that it had tetragonal symmetry. Since then there have been numerous other studies of the crystal structure 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 .

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

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 , 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 . 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 directions as opposed to 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 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 direction. In this case one set of diffuse planes vanishes leaving two sets perpendicular to and . In the orthorhombic phase III, the Ti atoms occupy just two of the <111> sites to give a resultant average polarization along 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 . 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 no evidence for the disorder was found. On the other hand, Senn recently decided that the disorder was present. A good paper in which this topic is reviewed is given in . A nice recent paper by Shao and Zuo 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. 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 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 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 , see also , and further expanded by Woodward . 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 . 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 .

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 .

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-3mGM4- (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 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 (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 . 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 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

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

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.

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

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

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.

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

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

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

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.

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