Ferro- and antiferroelectricity: order disorder versus displacive (Structure and Bonding, 124)

124 Structure and Bonding Series Editor: D. M. P. Mingos Editorial Board: P. Day · X. Duan · T. J. Meyer G. Parkin · H...

Author: Naresh S. Dalal | Annette Bussmann-Holder | Robert Blinc

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124 Structure and Bonding Series Editor: D. M. P. Mingos

Editorial Board: P. Day · X. Duan · T. J. Meyer G. Parkin · H. W. Roesky · J.-P. Sauvage

Structure and Bonding Series Editor: D. M. P. Mingos Recently Published and Forthcoming Volumes

Ferro- and Antiferroelectricity Volume Editors: Dalal, N. S., Bussmann-Holder, A. Vol. 124, 2007 Photofunctional Transition Metal Complexes Volume Editor: V. W. W. Yam Vol. 123, 2007 Single-Molecule Magnets and Related Phenomena Volume Editor: Winpenny, R. Vol. 122, 2006 Non-Covalent Multi-Porphyrin Assemblies Synthesis and Properties Volume Editor: Alessio, E. Vol. 121, 2006 Recent Developments in Mercury Sience Volume Editor: Atwood, David A. Vol. 120, 2006 Layered Double Hydroxides Volume Editors: Duan, X., Evans, D. G. Vol. 119, 2005 Semiconductor Nanocrystals and Silicate Nanoparticles Volume Editors: Peng, X., Mingos, D. M. P. Vol. 118, 2005 Magnetic Functions Beyond the Spin-Hamiltonian Volume Editor: Mingos, D. M. P. Vol. 117, 2005 Intermolecular Forces and Clusters II Volume Editor: Wales, D. J. Vol. 116, 2005

Intermolecular Forces and Clusters I Volume Editor: Wales, D. J. Vol. 115, 2005 Superconductivity in Complex Systems Volume Editors: Müller, K. A., Bussmann-Holder, A. Vol. 114, 2005 Principles and Applications of Density Functional Theory in Inorganic Chemistry II Volume Editors: Kaltsoyannis, N., McGrady, J. E. Vol. 113, 2004 Principles and Applications of Density Functional Theory in Inorganic Chemistry I Volume Editors: Kaltsoyannis, N., McGrady, J. E. Vol. 112, 2004 Supramolecular Assembly via Hydrogen Bonds II Volume Editor: Mingos, D. M. P. Vol. 111, 2004 Applications of Evolutionary Computation in Chemistry Volume Editors: Johnston, R. L. Vol. 110, 2004 Fullerene-Based Materials Structures and Properties Volume Editor: Prassides, K. Vol. 109, 2004 Supramolecular Assembly via Hydrogen Bonds I Volume Editor: Mingos, D. M. P. Vol. 108, 2004

Ferro- and Antiferroelectricity Order/Disorder versus Displacive Volume Editors: Naresh S. Dalal · Annette Bussmann-Holder

With contributions by R. Blinc · A. Bussmann-Holder · N. S. Dalal · T. Egami O. Gunaydin-Sen · M. Itoh · R. Kind · D. Merunka · B. Rakvin J. F. Scott · H. Taniguchi

123

The series Structure and Bonding publishes critical reviews on topics of research concerned with chemical structure and bonding. The scope of the series spans the entire Periodic Table. It focuses attention on new and developing areas of modern structural and theoretical chemistry such as nanostructures, molecular electronics, designed molecular solids, surfaces, metal clusters and supramolecular structures. Physical and spectroscopic techniques used to determine, examine and model structures fall within the purview of Structure and Bonding to the extent that the focus is on the scientiﬁc results obtained and not on specialist information concerning the techniques themselves. Issues associated with the development of bonding models and generalizations that illuminate the reactivity pathways and rates of chemical processes are also relevant. As a rule, contributions are specially commissioned. The editors and publishers will, however, always be pleased to receive suggestions and supplementary information. Papers are accepted for Structure and Bonding in English. In references Structure and Bonding is abbeviated Struct Bond and is cited as a journal. Springer WWW home page: springer.com Visit the Struct Bond content at springerlink.com

ISSN 0081-5993 ISBN 978-3-540-49602-1 Springer Berlin Heidelberg New York DOI 10.1007/978-3-540-49604-5

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Volume Editors Naresh S. Dalal

Annette Bussmann-Holder

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Preface

Solid-state systems are frequently classiﬁed according to their physical, structural or chemical properties. Such schemes are extremely helpful since properties related to any such classiﬁcation are typically known and facilitate identifying solids with special material classes. The best-known examples of these schemes are conductivity or resistivity measurements by means of which metals are easily distinguishable from insulators. However, frequently clear-cut decisions between material classes are not possible, since anisotropy, chemical composition, binding forces and local effects wash out distinct properties and lead to competition or coexistence. Such unresolved situations are especially typical for transition metal oxides that exhibit a variety of ground-state properties in a fascinating way. Here chemical substitution, doping, pressure or temperature effects easily inﬂuence the physical properties and may, for instance, induce metal/insulator, antiferromagnet/ferromagnet, insulator/superconductor transitions. This situation is analogous to perovskite ferroelectrics and hydrogen-bonded ferroelectrics, where ferroelectric/antiferroelectric transitions occur with chemical substitutions of one of the constituent sublattices. In addition, glass-like states (dipolar glasses) are observed and relaxor ferroelectricity with a large potential for application frequently occurs. Ferroelectrics and antiferroelectrics have a long history, and their phase transition mechanism has early on been classiﬁed according to experimental observations. In 1960 soft mode dynamics were seen in perovskite oxides by inelastic neutron scattering, following theoretical predictions by Anderson and Cochran in 1959. A long wavelength optic mode frequency slows down with decreasing temperature and freezes out at Tc . The displacement pattern of the ions involved in these dynamics determines the low temperature polar structure. These features have been attributed to displacive dynamics and opposed to order/disorder ones where above Tc a speciﬁc ion many occupy various equivalent lattice sites and hop between those. At Tc the ion orders into either of these sites thus giving rise to the polar state. Both transition mechanisms are based on a double-well potential, however, with the distinction that the barrier height for a displacive transition is much smaller than the one for the order/disorder transition.

X

Preface

In 1979 it had already been suggested by Bruce, Müller and Berlinger that a clear-cut distinction between order/disorder and displacive dynamics is not possible. It has been pointed out that different timescales may be associated with the different dynamics and that speciﬁcally EPR experiments would be an excellent probe to test these ideas. EPR is a local probe and can as such test dynamics associated with cluster formation whereas X-ray and neutron scattering experiments only probe the long wavelength average properties. Even though their subsequent EPR experiments indeed provided evidence for order/disorder dynamics in BaTiO3 and SrTiO3 , these observations induced a controversy since other experiments were in support of the displacive picture. This controversial situation could be resolved theoretically by showing that both dynamics can coexist on different timescales. While the long wavelength properties are dominated by mode softening, the ﬁnite momentum excitations exhibit order/disorder dynamics. Even though this approach has been accepted rapidly for perovskite ferroelectrics, it remained unclear in how far also hydrogen-bonded ferroelectrics needed a novel classiﬁcation scheme. Since ferroelectricity was ﬁrst exclusively discovered in hydrogen-bonded systems, the obvious conclusion was that the hydrogen bonds are actively involved in the phase transition mechanism. The observation that the protons occupy a double-well position above Tc has supported the assumption that the dynamics are order/disorder type, where a spontaneous cooperative freezing of the protons into either of the two sites takes place at Tc . The incompleteness of this approach became clear through high-resolution X-ray diffraction experiments which revealed that Tc correlates with the oxygen–hydrogen–oxygen bond length. This ﬁnding has initiated theoretical work where a new approach was made in order to understand the dynamical details of hydrogen-bonded ferroelectrics. Here aspects that are important in perovskite ferroelectrics, i.e., nonlinear polarizability effects, have been combined with the details of the hydrogen bonds. As a consequence a coupled double-well problem was introduced where displacive features related to the heavy ion sublattice are combined with the order/disorder dynamics of the hydrogen bonds. The predictions of the model were soon tested by ultra-high resolution NMR experiments and a veriﬁcation of the coexistence achieved. In contrast to perovskites, here the combined dynamics obey the same timescales. Since the issue of order/disorder versus (or with) displacive aspects has remained an active ﬁeld of research, most of the chapters presented in this book are devoted to it. In addition, new ﬁelds of applications are reviewed, since material optimization has considerably enlarged this area. A new aspect of ferroelectricity has been discovered recently by the ﬁnding of isotope-induced ferroelectricity in the quantum paraelectric SrTiO3 . Here conclusive ideas about its microscopic origin are still missing and also the experimental situation remains controversial, since the symmetry of the low-temperature phase is unclear. But, there seems to be stringent evidence that polar clusters are

Preface

XI

formed far above Tc . Simultaneously Raman scattering experiments report the observation of a soft mode. Both of these ﬁndings again support the picture of a coexistence of order/disorder and displacive dynamics. The microscopic origin of relaxor ferroelectricity is another open problem in this material class. Even though various phenomenological approaches were able to reproduce some aspects of these interesting materials, microscopically no understanding is achieved. Novel neutron spectroscopic methods are presented in one of the chapters in this book, where hints for the microscopic mechanism can be obtained. It seems clear that far above the smeared-out transition point, ﬂuctuating polar domains are formed that increase in size with decreasing temperature. Even though mostly a polar instability is absent, very high dielectric constants are observed which extend over broad temperature regimes. These speciﬁcs make the material extremely interesting for applications and have considerably enlarged the interest in them. The properties of Slater–Takagi groups and their movements within hydrogen-bonded ferroelectrics are also reviewed in this book. Here, the focus is especially on the regime where a polar glass is realized as has been observed in mixed crystals of RDP and ADP. The competition between ferroelectricity and antiferroelectricity induces this state and leads to frustration at certain mixing levels. Although ferroelectrics and antiferroelectrics have a long history, the present book shows that in spite of considerable experimental and theoretical work the ﬁeld remains full of surprises and new challenges. Tallahasse and Stuttgart, November 2006

Naresh S. Dalal Annette Bussmann-Holder

Contents

Order/Disorder Versus or with Displacive Dynamics in Ferroelectric Systems A. Bussmann-Holder · N. S. Dalal . . . . . . . . . . . . . . . . . . . . . 1 Experimental Evidence for the Coexistence of Order/Disorder and Displacive Behavior of Hydrogen-Bonded Ferroelectrics and Antiferroelectrics N. S. Dalal · O. Gunaydin-Sen · A. Bussmann-Holder . . . . . . . . . . . 23 Order and Disorder in Perovskites and Relaxor Ferroelectrics R. Blinc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

Local Structure and Dynamics of Ferroelectric Solids T. Egami . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

Ferroelectricity of SrTiO3 Induced by Oxygen Isotope Exchange M. Itoh · H. Taniguchi . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

Evidence for Ferroelectric Nucleation Centres in the Pseudo-spin Glass System Rb1-x (ND4 )x D2 PO4 : A 87 Rb NMR Study R. Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Anharmonic and Quantum Effects in KDP-Type Ferroelectrics: Modiﬁed Strong Dipole–Proton Coupling Model D. Merunka · B. Rakvin . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 A Comparison of Magnetic Random Access Memories (MRAMs) and Ferroelectric Random Access Memories (FRAMs) J. F. Scott . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Author Index Volumes 101–124 . . . . . . . . . . . . . . . . . . . . . . 209 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

Contents of Volume 114 Superconductivity in Complex Systems Volume Editors: Karl Alexander Müller, Annette Bussmann-Holder ISBN: 978-3-540-23124-0

Essential Heterogeneities in Hole-Doped Cuprate Superconductors K. A. Müller Two-Component Scenarios for Non-Conventional (Exotic) Superconductors A. Bussmann-Holder · R. Micnas · S. Robaszkiewicz Superconductivity in Alkali-Doped Fullerides O. Gunnarsson · J. E. Han · E. Koch · V. H. Crespi Pairing Mechanisms Viewed from Physics and Chemistry S. Deng · A. Simon · J. Köhler Unconventional Isotope Effects in Cuprate Superconductors H. Keller Neutron Scattering Investigations of Charge Inhomogeneities and the Pseudogap State in High-Temperature Superconductors A. Furrer Nanoscale Properties of Superconducting Cuprates Probed by the Electron Paramagnetic Resonance B. I. Kochelaev · G. B. Teitel’baum Electron-Phonon Coupling in High-T c Superconductors T. Egami Nanoscale Lattice Fluctuations in Cuprates and Manganites A. Bianconi · N. L. Saini Dynamic Inhomogeneity, Pairing and Superconductivity in Cuprates D. Mihailovic · V. V. Kabanov Evidences for Polaron Formation in Cuprates A. Bussmann-Holder · H. Keller · K. A. Müller

Struct Bond (2007) 124: 1–21 DOI 10.1007/430_2006_045 © Springer-Verlag Berlin Heidelberg 2006 Published online: 21 November 2006

Order/Disorder Versus or with Displacive Dynamics in Ferroelectric Systems Annette Bussmann-Holder1 (u) · Naresh Dalal2 1 Max-Planck-Institute

for Solid State Research, Heisenbergstr. 1, 70569 Stuttgart, Germany [email protected]

2 Department

of Chemistry and National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32306, USA

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 2.1 2.2 2.3 2.4 2.5

Theoretical Modelling of the Phase Transition Mechanism of Hydrogen-Bonded Ferroelectrics . . . . . . . . . . . . . General Features of Hydrogen-Bonded Systems . . . . . . . The KDP Family . . . . . . . . . . . . . . . . . . . . . . . . The Proton–Lattice Coupled Model . . . . . . . . . . . . . Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coexistence of Order/Disorder and Displacive Dynamics .

. . . . . .

5 5 6 8 13 15

3

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

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1

Abstract For many years ferroelectrics have been classiﬁed as either being of order/disorder or displacive type and thus are strictly distinguishable by their dynamical behaviour. Even though early experiments using different techniques have come to seemingly contradictory results concerning the phase transition mechanism, this issue has not been addressed in detail. In this contribution we show that the classiﬁcation scheme is mostly invalid in ferroelectrics and that a coexistence of both order/disorder and displacive effects is much more common to these materials than a sharp distinction. The special focus will be on hydrogen-bonded ferroelectrics; however, extensions of these ideas to non-hydrogen-bonded compounds will be presented as well. In addition, experimental methods and results are reviewed and are shown to be excellent tools for testing the dynamics at the phase transition temperature. Keywords Ferroelectrics · Hydrogen-bonded systems · NMR techniques · Phase transition mechanism

1 Introduction More than 80 years ago ferroelectricity was ﬁrst discovered in the rather complex compound Rochelle salt. It was long believed to be a rather exceptional property since no other system with reversible polarization was discovered

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A. Bussmann-Holder · N. Dalal

for another 15 years. Between 1935 and 1938 Busch and Scherrer managed to synthesize a series of ferroelectric compounds based on potassium dihydrogen phosphate KH2 PO4 (KDP), which has a much simpler crystal structure than Rochelle salt. A common link between the KDP family and Rochelle salt was rapidly recognized, since both compounds contain strong hydrogen bonds, which order in the ferroelectric phase. This observation gave rise to the ﬁrst theory of ferroelectricity by Slater [1], who proposed that above the ferroelectric transition temperature Tc the hydrogen atoms sit in a doublewell potential with random occupation of either of the double-well potential sites and order into equivalent sites below Tc . Even though the model was rather simplistic, its basic features are still accepted nowadays, including modiﬁcations of the original model. A severe failure of the model is, however, that it cannot account for the large isotope effect on Tc upon deuteration. This was explained later by Blinc [2] and Blinc et al. [3] using an extended version of the Slater model by postulating that quantum tunnelling within the hydrogen bonds takes place. Since the Slater model of ferroelectricity is based on the existence of hydrogen bonds in a solid, the belief was that the polar instability can only occur in hydrogen-bonded systems. In 1945 it was observed that BaTiO3 has an unusually high dielectric constant [4], which increases substantially with increasing temperature. Ferroelectricity in this material was reported soon afterwards [5]. This ﬁnding clearly showed that hydrogen bonds are not unique to ferroelectrics and that other theories or more general ones have to be considered in order to account for the phenomenon. The simple structure of BaTiO3 , its robust structural stability and its easy fabrication made it one of the best-investigated compounds in solid state physics and chemistry. In addition, the ﬁeld of application of ferroelectrics experienced an enormous expansion since not only single crystals could be used but also their ceramic equivalents. The ﬁnding of ferroelectricity in a perovskite oxide caused an avalanche-like discovery of new perovskites being ferroelectric, and mixed crystal compounds were also observed to exhibit a polar instability [6–8]. Also, this discovery required the introduction of new theoretical concepts, which again were ﬁrst proposed by Slater [9]. He assumed that long range dipolar forces compete with local ones and overcome those to form a new ground state. His major focus was based on the speciﬁc ions that are responsible for the ionic instability. The concept of the “rattling” titanium was introduced, and the rest of the lattice was assumed to be silent. These ideas were abandoned when Cochran [10] and Anderson [11] suggested that lattice dynamics are essential and that all ions are involved in the phase transition mechanism. They introduced the soft mode concept where one speciﬁc mode of the system slows down with decreasing temperature and becomes unstable at Tc . The displacement pattern of this speciﬁc long wave length mode then determines the low temperature polar structure. These ideas have been the origin of discrimination between order/disorder and displacive type phase transitions. Since soft modes have been observed in nearly all perovskite-type

Order/Disorder Versus or with Displacive Dynamics in Ferroelectric Systems

3

ferroelectrics, whereas only marginal evidence for them has been obtained in hydrogen-bonded ones, these material classes were grossly assigned to displacive-type ferroelectrics and order/disorder-type ones, respectively. The modiﬁcations of the Slater model for hydrogen-bonded systems as introduced by Blinc and Blinc et al. [2, 3] have long been believed to describe the essential physics of these compounds. A basic ingredient is that a pseudo-spin representation for the hydrogen atoms is used where tunnelling between the two double-well potential minima takes place. Coupling to the rest of the lattice has been introduced by Kobayashi [12] and Blinc [13] whereby renormalizations of the tunnelling frequency have been introduced, which are essential in describing the isotope effect on Tc . The validity of this pseudo-spin approach has been questioned in numerous works, especially where various experimental data have provided substantial evidence that the precise bond geometry and the interactions with the embedding matrix are of essential importance in understanding the microscopic phase transition mechanism [14–16]. Nearly simultaneously, displacive type systems were also investigated in deeper detail. Even though the soft mode concept has been ﬁrmly established experimentally, microscopic modelling in terms of rigid ion models was unable to reproduce the temperature-dependent characteristics of these compounds in quantitative agreement with experiment. Even though these oxides exhibit a band gap of several electronvolts, crucial interactions between the ionic motions and the valence electrons take place that lead to a redistribution of the electronic charges with decreasing temperature. In particular, it has been shown that the transition metal d-states and the oxygen ion p-states form dynamical covalent bonds with strongly directional character [17]. This anisotropic coupling is a consequence of the very speciﬁc properties of the oxygen ion O2– . This ion is unstable as a free ion in spite of the fact that it exhibits the same closed shell conﬁguration [18, 19] as the F– ion, which is stable as a free ion. In a solid, the consequences of this local instability are that the oxygen ion polarizability is temperature-dependent, volume- and pressuresensitive, and is a function of the surrounding ions, i.e. the Madelung potential [18]. Such a local instability can be partially compensated by delocalizing the outer p-electrons to form covalent bonds with the surrounding electronic states, thus reducing the oxygen ion energy. These observations have been the basis for introducing the phenomenological nonlinear polarizability model [17, 20], where its main ingredients are an anisotropic nonlinear core–shell coupling between the oxygen ion and the transition metal ion. This shell model representation captures the major physics of perovskite ferroelectrics and describes, in quantitative agreement with experimental data, the temperature-dependent properties of these systems [17, 21–25]. Later, ﬁrst principles theories have conﬁrmed the importance of dynamical covalency. The important difference from rigid ion models with anharmonic potentials is the renormalization of the soft mode frequency through electron lattice

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A. Bussmann-Holder · N. Dalal

interaction effects, which gives rise to various temperature regimes of the soft mode with varying critical exponents [20]. The quantum regime with enhanced dimensionality is followed by a dimensionality crossover regime. Only then is the classical regime obtained, which is followed by a saturation regime at high temperatures. The model has long been thought to be applicable to displacive-type phase transitions with soft mode behaviour only, as long as the self-consistent phonon approximation is used. Order/disorder-type systems have been thought to be beyond these ideas and a consequence of local tunnelling effects. This means that a clear distinction between both types of dynamics also arises within the polarizability model. However, the need to improve the microscopic description of hydrogen-bonded ferroelectrics has invoked new ideas, where nonlinear polarizability effects are incorporated into the pseudospin approach in order to account for the large polarizability of the PO4 groups in the KDP family [26]. This was the ﬁrst attempt to combine order/disorder dynamics with displacive ones, and a clear prediction for its coexistence. A veriﬁcation of these predictions has rapidly been achieved experimentally by ultra-high resolution magic angle spinning NMR techniques [27]. It was the ﬁrst observation of the coexistence of order/disorder and displacive dynamics on the same time-scale and has initiated a vast new ﬁeld of research. In perovskite oxides new experimental techniques probing local properties meanwhile revealed that a purely displacive approach is at least incomplete (see [28, 29] and refs therein). Locally the picture of the “rattling” ion seems to be realized, however, being reﬁned as an eight-site hopping model where the central transition metal ion is hopping between eight equivalent sites along the diagonals in the cubic lattice [30]. This observation introduced a controversy between experimentalists since, on one hand, experiments that probe long wave length properties only see the displacive component while, on the other hand, experiments probing local properties only see the order/disorder component. The rather puzzling situation was resolved theoretically by showing that a crossover between displacive and order/disorder dynamics may occur in perovskites and that around the phase transition temperature, coexistence is even possible. However, different time and length scales are obeyed by the two components (see [31] and refs therein). Meanwhile the situation has become much clearer since it has been shown that coexistence is indeed more normal than a clear-cut distinction between the different dynamics. In this chapter we will mainly concentrate on hydrogen-bonded ferroelectrics, which were the ﬁrst class of ferroelectrics where the coexistence of the different dynamics was postulated and observed [26, 27]. In Chap. 2 of this volume (by Dalal et al. [51]) new experimental techniques and their advantages over conventional ones will be described together with their applications to hydrogen-bonded systems. The theoretical concept of bilinear

Order/Disorder Versus or with Displacive Dynamics in Ferroelectric Systems

5

coupling between the hydrogen bonds and the polarizable sublattice will be presented here together with its applications to these systems. The implications of these ideas and their experimental veriﬁcations are summarized at the end of this chapter.

2 Theoretical Modelling of the Phase Transition Mechanism of Hydrogen-Bonded Ferroelectrics 2.1 General Features of Hydrogen-Bonded Systems Hydrogen-bonded systems play an outstandingly important role in biology, physics and chemistry. Understanding the dynamics of protons in O – H – O

Fig. 1 Schematic structures of a DNA, b butynedioic acid and c KDP. Large striped circles represent potassium ions and medium ﬁlled circles oxygen ions. Small ﬁlled circles are protons and checkerboard ﬁlled circles represent phosphorous ions. d Schematic structure of squaric acid

6

A. Bussmann-Holder · N. Dalal

(N – H – N) bonds can be of extreme importance in modelling, e.g. dielectric anomalies, DNA sequences, protein folding. Since proton ordering is common to all these systems, the dynamics of the protons can be studied in rather simple structures with generalizations to more complex inorganic and also organic compounds. Various examples of hydrogen-bonded systems are displayed in Fig. 1. The structures vary from plain water and simple structured squaric acid to extremely complex organic compounds. In all these compounds the exchange of hydrogen by its isotope has decisive consequences on the intrinsic properties of the corresponding system. In ferro- and antiferroelectrics the structural phase transition temperature is nearly doubled upon deuteration. This is rather exceptional since in non-hydrogen-bonded ferroelectrics the phase transition temperature is independent of the ionic masses in the classical limit and only sensitive to isotope exchange in the quantum regime [32]. H vitamin C is known to help stop the growth of cancer cells. D vitamin C, on the other hand, has no inﬂuence on the progression of cancer cells [33]. Also, cells with microtubules that are stabilized by D2 O are unable to progress through their growth cycle and stop at mytosis, in contrast to those stabilized by H2 O where the growth cycle continues [34]. While it is common knowledge that water is an essential ingredient to all higher organisms, heavy water is poison to them. In Si the lifetimes of the H- and D-related vibra√ tional modes are inverted and instead of a 1/ mD decrease for deuterium they show an unusual enhancement of the related frequency [35]. In DNA, the hydroxyl reaction sites are altered when the protons are replaced by their isotope [36]. 2.2 The KDP Family In order to understand these extreme changes in physical and chemical properties of hydrogen-bonded systems, ﬁrst attempts to model their dynamics were related to rather simple structures, as exhibited by the KDP family or squaric acid and its analogues. The isotope effects on their ferro- or antiferroelectric transition temperatures are listed in Table 1 together with the corresponding isotope exponent. As is obvious from the table, Tc is almost doubled upon deuteration. These isotope effects are one of the largest observed in any solid state system. The question arises about isotope effects in non-hydrogen-bonded ferro- and antiferroelectrics. As already mentioned in the Introduction, within a meanﬁeld scheme and in a purely ionic model it was predicted that these systems should not exhibit any isotope effect in the classical limit, which has been veriﬁed experimentally. Correspondingly, there was not much effort to look for these effects here. However, using a nonlinear shell-model representation it was predicted that in the quantum limit an isotope effect should

Order/Disorder Versus or with Displacive Dynamics in Ferroelectric Systems

7

Table 1 Isotope effects and exponent α on the ferroelectric transition temperatures for various hydrogen-bonded systems Isotopes

Tc K

KH2 PO4 KD2 PO4

122 213

RbH2 PO4 RbD2 PO4 KH2 AsO4 KD2 AsO4 NH4 H2 PO4 Nh4 D2 PO4 H2 C4 O4 D2 C4 O4

147 218 97 161 148 212 371 516

α

0.80 0.57 0.73 0.52 0.48

be present [32].The experimental veriﬁcation was consequently achieved by showing that the incipient quantum-paraelectric system SrTiO3 exhibits a real instability upon replacing 16 O by 18 O [37]. Since this effect is present only at very low temperatures the general conclusion concerning hydrogen-bonded systems was that the isotope effect is exclusively caused by the proton dynamics. Assigning pseudospin states to the two equivalent positions that the protons can occupy in their local double-well potential, Blinc and Blinc et al. [2, 3] postulated that above Tc the protons tunnel between these minima, while below Tc they order cooperatively into one of the minima and either become ferro- or antiferroelectric. The model corresponds to an Ising model in a transverse ﬁeld where only the proton dynamics are considered. The isotope effect is here a consequence of large variations in the tunnelling frequency Ω upon deuteration ( ΩH = 144 K, ΩD = 1 K) together with a huge increase in the pseudospin–pseudospin interaction constant J ( JH = 550 K, JD = 850 K). The role played by the heavy ion sublattice has also been addressed by the Blinc et al. [13], where anharmonicity in the ionic motion was incorporated. However, also with these extensions, the isotope effect still remains a consequence of only the proton dynamics. A similar approach has been suggested by Kobayashi [12] who considered spin lattice coupling, again within a rigid ion approach neglecting dynamical covalency and hybridization effects. Analogous to Blinc and coworkers, subtle details of the precise O – H – O bond geometry were neglected and again strong changes in the tunnelling frequency and the pseudospin–pseudospin interaction constant were needed in order to reproduce the experimentally observed isotope effect. First doubts about these approaches came from experimental ﬁndings where a correlation between the O – H – O bond length 2R and the H(D) site separation δ with Tc were reported [14–16]. These observations have been

8

A. Bussmann-Holder · N. Dalal

taken as evidence that the heavy ion sublattice plays a much more crucial role to the phase transition than suggested by the pseudospin approach. The situation has been summarized by Krumhansl as the search for chicken and egg [38]. 2.3 The Proton–Lattice Coupled Model For perovskite oxide ferroelectrics it is known that dynamical covalency and p – d hybridization effects crucially trigger the phase transition [17, 20]. The main ingredient of these models is based on the above-mentioned knowledge that the oxygen ion O2– is unstable in the free state, in contrast to F– which has the same electronic conﬁguration [18, 19]. While the O– state still represents a bound state, the O2– conﬁguration is a resonant state that is stabilized by its surrounding lattice. The consequences of this instability are that small changes in the constituting lattice induce a redistribution of the 2p charge and lead to very large changes of the oxygen ion polarizability as a function of the oxygen ion volume. While in binary oxides these changes are proportional to the volume V(O2– ), this relationship is increased to a V3 dependence in perovskites. In addition, strong temperature and pressure dependencies of the polarizability result [18]. Phenomenologically, these observations have been taken into account within the nonlinear shell model representation where the electronic shell that surrounds the oxygen ion is nonlinearly coupled to its core and exhibits anisotropic properties with respect to its coupling to the A and B neighbours in perovkites ABO3 (A is typically a monovalent or divalent cation whereas B is a transition metal element) [17, 20]. Microscopically this model corresponds to one where lattice-driven hybridization effects between the oxygen ion p-states and the transition metal ion d-states are included, i.e. strong electron–phonon interactions are considered. Such effects have always been thought to be irrelevant to ABO3 since these are insulators with a rather large band gap of several electronvolts. This was also the reason while purely ionic models have mostly been used in modelling these compounds. However, the agreement between experimental data of the typical dynamics of ferroelectrics with theory always remained qualitative and only the application of the nonlinear polarizability model achieved quantitative agreement between theory and experiment. For hydrogen-bonded ferroelectrics, similar effects as those displayed by the nonlinear polarizability model should be relevant as well since the PO4 groups in, for example, KH2 PO4 are highly polarizable and carry the essential dipole moment whereas the contribution to it from the hydrogen bonds is perpendicular to the axis of spontaneous polarization and, correspondingly, negligible. Correspondingly, the model we use for hydrogen-bonded ferro- and antiferoelectrics is a combined pseudospin–nonlinear polarizabilty model, i.e. a coupled double-well potential model [26].

Order/Disorder Versus or with Displacive Dynamics in Ferroelectric Systems

9

Fig. 2 Hydrogen bond geometry (not to scale): + and – are H+ positions δ/2 and – δ/2, respectively. P and S are PO4 2– core and shell; K+ is the potassium ion; dotted lines indicate the contributions to C(δ)

The essentials of this model are that it combines the displacive dynamics of the nonlinear shell model with the order/disorder dynamics of the pseudospin model. In addition, the details of the bond geometry have to be incorporated since Tc correlates with the distance δ between the two equilibrium positions of the protons [14]. Thus the model is based on the interplay between the tunnelling motion of the protons (deuterons), the polarizability changes of the neighbouring PO4 groups and the optical phonon mode of the [K – PO4 ] complexes. In order to deal with an analytically tractable problem, a two-dimensional model of KDP is adopted (Fig. 2). Here a formula unit consists of a K+ -ion, a PO4 – group and one single proton (the situation is analogous to PbHPO4, LDP, which is an experimental model system for the study of hydrogen ordering phase transitions [16]). The PO4 -group is treated as a composite structure that consists of a central core of mass Mp surrounded by its electron shell of mass ms , and neglects the tetrahedral shape of PO4 . The displacement of the electron shell along the z-axis is an additional degree of freedom, which accounts for the nonlinear polarizability of PO4 2– . The hydrogen bond is directed along the line which joins the centres of two neighbouring PO4 groups parallel to the y-axis. The geometry of the H positions in the paraelectric phase is inferred from diffraction experiments. With respect to the centre of the bond, the two equilibrium positions of the protons are (δy /2, δz /2) and (– δy /2, – δz /2). Here, δy = δ cos Ψ and δz = δ sin Ψ , where Ψ is the inclination angle between the y-axis and the proton hopping trajectory. High resolution neutron diffraction experiA in KDP, and Ψ = 7.4◦ , δ = 0.4462 ˚ A ments [16] yield Ψ = 8.8◦ , δ = 0.3647 ˚ in DKDP. The bond geometry is C2h [39, 40]. The Hamiltonian of this model is given by: H = HT + HL + HTL ,

(1)

10

A. Bussmann-Holder · N. Dalal

where HT describes the dynamics of the tunnelling protons, HL those of the host lattice, and HTL refers to the coupling between the protons and the lattice. The tunnelling model of N interacting protons is described by [41]: √ 1 z z (2) HT =– J q S q S – q – ΩSx q = 0 N . 2 q

The equilibrium positions of each proton to the right or to the left of the bond centre are described by the pseudospin value Sz with values + 1, – 1, respectively. In Fourier space J( q) is the proton–proton interaction with wave vector q. The operator Sx accounts for proton tunnelling with tunnel frequency Ω. The host lattice dynamics comprises the displacements along the z-axis of the K+ -ion, of the PO4 cores and of the PO4 shells. Its Hamiltonian is given by: HL = TL + UL , where UL =

2f + g ω2p q + φp+ q φp q + φ q φK q Mp mK K q + 2f g ∗ ∗ + 2f + g u q u q – 2 √ φK q + √ φP q u q mK mP

1 2

(3)

is the potential energy. The kinetic energy is given by: 1 TL = πP+ q πP q + πK+ q πK q + mS u˙ + q u˙ q . 2 q

(4)

(5)

√ √ Here, φP / MP is the displacement of the PO4 core centre of mass, φK / mK + the corresponding one of the K ion and u, u˙ are displacement and velocity of the PO4 shell, respectively. The related masses are MP , mK , mS , and πP , πK are the momenta conjugate to φP , φK . The Hamiltonian HL corresponds to the one of the nonlinear polarizability model [17, 20]. The force constant f couples the PO4 shell displacement to the K+ ion displacement, whereas f is the direct coupling between the PO4 cores and determines the long wavelength acoustic properties. Since ωP ( q) = 4f sin( qa)/MP vanishes in the long wavelength limit, its effect is not considered in the following. The on-site core–shell coupling constant g consists of a harmonic attractive part g2 and a fourth order anharmonic repulsive part g4 , which guarantees the stability of the system. Within the self-consistent phonon approximation g is then given by: φP q φP+ q g4 + u q – √ u q – √ , (6) g = g2 + 3 N MP MP q

Order/Disorder Versus or with Displacive Dynamics in Ferroelectric Systems

11

where the brackets denote the thermal average. The interaction HTL between the heavy ion lattice and the proton subsystem is modelled by two back-to-back Morse potentials between each proton and the shells of the two neighbouring PO4 2– groups [42, 43]. The shell plays alternatively the role of a H+ acceptor or donor. The potential for each proton is given by: 1 + ερ Sz

(7) V= V M |Xρ (r) + V M |Xρ (l) , 2 ρ where ρ = ±1, ε+ = +1 and ε– = –1. The distance between the proton and the centre of the PO4 shell to the right (r) or to the left (l) side of the bond cen 1/2 with εξ = +1, –1 for tre is |X± (ξ)| = (R – R0 ∓ εξ δy /2)2 + [u(ξ) ∓ δz /2]2 ξ = r, l. The Morse potentials are deﬁned as: = D e–2α|X| V M |X| – 2 e–α|X| . (8) Here 2R is the equilibrium distance between the centres of the two PO4 groups; D, αandR0 are potential parameters. For the case of zero displacements in the z-direction, the potential V becomes a double minimum potential [42, 43]. The experimental values are 2R = 2.4946 A, δy = 0.3604 A, δz = 0.0558 A in KDP and 2R = 2.5230 A, δy = 0.4424 A, δz = 0.0575 A in DKDP [14, 15]. Expanding V with respect to the shell displacement and summing over the lattice, the coupling Hamiltonian in the long wave length limit reads: HTL = C δ S+z q u q (9) q

with: δz C δ =– 2

∂V M |X+0 (r)| 1 ∂V M |X–0(r)| + . |X+0 (r)| ∂|X+0 (r)| |X–0 (r)| ∂|X–0 (r)| 1

(10)

0 are taken at u = 0. Although C(δ) Here, the index 0 indicates that X± changes sign under the inversion of δ, physical quantities like Tc depend only on the absolute value of C(δ). These are given in Table 2 for KDP and DKDP. The interaction HTL reﬂects the difference in the electronic structure between the donor and acceptor oxygen ions [38, 39, 44, 45]. Standard Green’s function techniques are used in the following [46] to describe the dynamics of the protons and the ionic displacements. The equations of motions for the retarded Green’s functions A+ ( q); Sz ( q) are obtained from the Hamiltonian Eq. 1 where the operator A denotes φP , φK , u or Sz and Sy , and ω is the external frequency. The PO4 shell displacements are eliminated by the adiabatic approximation. The coupled motion of the K and

12

A. Bussmann-Holder · N. Dalal

Table 2 Absolute values of the coupling C(δ) (units K/˚ A) for potential parameters α (units ˚ A–1 ) and R0 (units ˚ A) Parameters

C|KDP

C|DKDP

r

α = 4.68 R0 = 1.00 α = 4.95 R0 = 1.00 α = 4.95 R0 = 0.95

27 499.0

32 193.9

1.171

29 665.6

35 036.1

1.181

20 915.6

25 209.56

1.201

Values of 2R, δx , δy as given in the text. δ|DKDP /δ|KDP = 1.22, D = 34 120 K, r = C|DKDP /C|KDP

PO4 core masses is transformed to optical displacement coordinates: µ µ ξ q = φP q – φK q , MP mK

(11)

where µ is the reduced heavy ion mass. The tunnelling system is treated within the molecular ﬁeld approximation [3], and in the long wavelength limit the following coupled set of equations is obtained: √ + z 2 c µ g 2f z+ z 2 – S q ; S q ω ω – ω0 ξ q ; S q ω = MP mK g + 2f √ 2 c µ g 2f 2 z+ z x ω –Ω S q ; S q ω = S Ω 1 + – MP mK g + 2f + z × ξ q ; S q ω , (12) where ω20 = 2fg/[µ(2f + g)] is the squared optical mode frequency of the 2 = Ω 2 – Ω Sx [J + C2 /(g + 2f )]. The direct heavy ion sublattice and Ω proton–proton interaction is given by J = J( q = 0) > 0. In the following it is assumed that Ω < T and correspondingly Sx = Ω/4T with T being the tem is modiﬁed by the perature. The renormalized proton tunnelling frequency Ω direct and by the PO4 shell-mediated proton–proton interaction. An instability of the coupled system in the static limit ω = 0 occurs at: C2 Ω Sx µ g 2f 2 2 2 Ω ω0 = – (13) 2 MP mK g + 2f which correspond to a second order phase transition. Since both Sx and g depend on T, the transition temperature Tc is determined by Eq. 13. In the limit C = 0 the proton and PO4 motions are decoupled and two types of in-

Order/Disorder Versus or with Displacive Dynamics in Ferroelectric Systems

13

stabilities result, which are either due to the protons or to the heavy ion sublattice. 2.4 Results For ﬁnite C some analytically obtainable limits exist: For large optical mode frequency, i.e. both g and f are large such that J C2 /(2f + g), the dynamics are determined by the proton tunnelling, and the phase transition is driven by the proton mode condensation. In this case Tc is given by: g 1 2f 2 C2 µ2 Tc = – . (14) J+ 1+ 4 2f + g 2fg MP mK Since it is assumed that g is very large, i.e. a polarizability catastrophe cannot occur, g is effectively temperature-independent. Even though the coupling between protons and the lattice enhances Tc , this is counteracted by the large values of the core–shell couplings and the total effect on Tc is consequently small. At the other limit, when J → 0, the phase transition is driven by the soft optical mode and g becomes small as a consequence of the nonlinear polariz 2 ≈ Ω 2 . The temperature dependence ability. In this case, g < f ; ω20 ≈ g/µ; Ω of the renormalized coupling g is due to the second term on the right hand side of Eq. 6. By eliminating the proton motion from Eq. 12, the mean square displacement of the optical mode becomes: 1 + ξ q ξ q ≈ χ0 T (15) N q

with χ0 being the single particle susceptibility. Since the sum over q suppresses critical behaviour, χ0 can be expressed by a T-independent constant: µ C2 3g4 g(T) = g2 + T+ χ0 (16) µ mK 4f and Eq. 13 reduces to: µ 2 C2 g Tc – =0, mK 4Tc

(17)

where g(Tc ) is deﬁned by the right hand side of Eq. 13 at T = Tc . Deﬁning Cˆ = (µ/mK )(C/2), gˆ4 = 3g4 χ0 /µ, fˆ = (µ/mK )f , Θ = |g2 |/ˆg4 , Eq. 15 can be solved for ﬁnite Tc : 2 2 ˆ ˆ Θ – Cˆ 2 /fˆ + 4Cˆ 2 /ˆg4 Θ – C /f + Tc = . (18) 2

14

A. Bussmann-Holder · N. Dalal

In the absence of coupling between the protons and the heavy ion lattice Tc = Θ, which corresponds to the transition temperature of the nonlinear polarizability model [17, 20]. On the other hand, for ﬁnite coupling, through the dependence of C on the hydrogen bond geometry, it is possible to evaluate an isotope effect on Tc caused by changes in the details of the bond geometry. 1/2 Assuming Cˆ 2 /fˆ < Θ, 4Cˆ 2 /ˆg4 < Θ the right-hand side of Eq. 18 can be expanded in terms of C to obtain Tc = Θ + αCˆ 2 , where α = fˆ – |g2 | / |g2 |fˆ . From the nonlinear polarizability model it can be inferred that fˆ > |g2 | and consequently α > 0. This means that the coupling C leads to an increase in Tc and, since C |DKDP > C |KDP the transition temperature is larger for DKDP than for KDP. The direct proton–proton coupling is negligible for any Tc enhancement effects, while the interaction that is mediated by the coupling between the PO4 shells and the protons is essential. Equation 13 can be solved numerically for Tc as a function of the proton– lattice coupling. The parameters are chosen so as to ﬁt the experimental value of Tc for KDP. For C = 21 732 K/A and g2 , g4 , f close to those used for perovskite oxides, Tc |KDP = 115 K . In Fig. 3 Tc is shown as a function of C with all other parameters ﬁxed. Including the deuteration effects (Table 2), Tc r = C |DKDP /C |KDP ≈ 1.2 . With this estimate Tc |DKDP = 168 K. C itself depends only weakly on J, g2, g4 but a strong dependence on f is observed, which is the coupling between the PO4 shells and the K+ ions. This, on the other hand, should not be dependent on deuteration. The above results also explain the pressure dependence of Tc , which has been observed by neutron diffraction experiments [14–16]. With increasing pressure the proton–proton site separation decreases, which implies a decrease of C and a decrease in Tc in agreement with experiments.

Fig. 3 Variation of Tc with the absolute value of C

Order/Disorder Versus or with Displacive Dynamics in Ferroelectric Systems

15

2.5 Coexistence of Order/Disorder and Displacive Dynamics As has been shown above the model has two extreme limiting results. In the limit of large g, f the dynamics are governed by the proton subsystem whereas in the limit J → 0 the polarizable heavy ion lattice dominates the phase transition mechanism. Both of these limits have further consequences since a phase transition driven by tunnelling protons and cooperative proton ordering is of an order/disorder type. However, if polarizability effects are dominant, soft mode behaviour sets in and the transition is of displacive type. In between, new physics are expected to occur since both displacive and order/disorder dynamics coexist. For perovskites such a coexistence has been observed very early on [31], but was not recognized as such. In hydrogenbonded compounds the situation seemed to be obvious since the pseudospin model clearly predicts that the phase transition is of order/disorder type, associated with random occupation of the two-site distribution of the protons above Tc and the ordering into correlated minima below Tc . Even though early neutron scattering experiments on deuterated KDP provided some evidence that a soft mode is also present in these systems, these experiments were frequently overlooked. In addition, early NMR experiments [47–50] supported the order/disorder mechanism and no search for other dynamics was undertaken. Because of the prediction of coexistence of order/disorder in [26], Dalal et al. [27] reinvestigated the phase transition mechanism with an ultra-high resolution NMR method, which is based on magic angle spinning (MAS) and is described in detail by Dalal et al. in Chap 2 of this volume. The advantages of this method as compared to conventional NMR techniques are that it is: • Sensitive to essentially all atomic displacements • Its dynamical range covers both the slow and the fast motion regime (103 –1011 Hz) • It is easily accessible and simple in probing directly the dynamics and displacements of a given atomic site • The isotropic part of the chemical shift tensor, ρiso , is invariant to any rotational and translational change of a given molecular unit The method has been applied to various hydrogen-bonded ferro- and antiferroelectrics (see Chap 2 in this volume). The ﬁrst results have, however, been obtained for the antiferroelectric compound H2 C4 O4 (squaric acid) and its deuterated analogue D2 C4 O4 . Squaric acid (H2 SQ) has been chosen as a ﬁrst test compound because it has a very simple molecular structure. Planar sheets of the squarate (C4 O4 ) groups are linked to each other in a two-dimensional network through O – H...O bonds (Fig. 1) with weak van der Waals forces [52, 53]. The protons perform an order/disorder motion above the antiferroelectric phase transi-

16

A. Bussmann-Holder · N. Dalal

tion temperature Tc = 373 K [52–55], i.e. the distribution of long O...H and short O – H bonds is random. Below Tc the protons order anitferroelectrically and simultaneously the squarate units distort. Even though H2 SQ has been well investigated theoretically [56] as well as experimentally [52–55], the coexistence of order/disorder and displacive dynamics has not been addressed before. In contrast to previous less-resolved NMR data completely new features were revealed by the high resolution MAS method [27] (see also Chap 2 in this volume): a signiﬁcant step-like (S-shape) increase in the average chemical shift occurs in the vicinity of Tc (Fig. 4). Similar anomalies have subsequently also been observed in various members of the KDP family [57] (see also Chap 2 in this volume). The anomaly is not only related to low temperature peak positions but also present in the high temperature peak position. In the coexistence region, which corresponds to a small regime in the S-shaped anomaly, the average of the four low temperature peaks and the high temperature peak coincide. This is evidence that there is no detectable break in the chemical shift as the crystal changes from the paraelectric to the antiferroelectric phase, as would be expected for a ﬁrst order order/disorder transition. A signature of the ﬁrst order transition still remains as the intensity of the four low temperature peaks suddenly becomes zero before they actually merge on the high temperature peak. Even though the protons order spontaneously at Tc , corresponding to order/disorder dynamics, the data suggest that a coexisting displacive component is present and causes the continuous shift in the NMR peak positions. The data have been analyzed within the described coupled double-well model. The protons are represented by pseudospins whereas the C4 O4 unit displays highly nonlinear polarizability effects, which are taken into account within the framework of the nonlinear polarizability model [17, 20]. As compared to KDP-type ferroelectrics the model is less complex [26, 27] since the

Fig. 4 Average NMR peak position of H2 SQ as a function of temperature. The dashed line corresponds to the calculated shift

Order/Disorder Versus or with Displacive Dynamics in Ferroelectric Systems

17

coupling between the PO4 units and the potassium ions is absent. The frequencies of the coupled lattice pseudospin model are thus slightly modiﬁed as compared to the previous case and are given by [27]: 1 ω2± = ω20 – Cχs + Ω Ω – Jq Sx – C2 Sx /g4 T 2 1 2 2 ± ω0 – Cχs + Ω Ω – Jq Sx – C2 Sx /g4 T 2 1/2 – 4ΩC2 χs Sx /g4 T ,

(19)

where the notations are the same as above. For the uncoupled system (C = 0) two transition temperatures are obtained. One corresponds to the ordering of the protons and is of ﬁrst order and pure order/disorder type, whereas the other is second order and displacive related to the polarizable sublattice (Fig. 5). For small coupling the displacive lattice mode frequency slightly hardens and becomes discontinuous at Tc , while the tunnelling mode already adopts a displacive component, apparent as a rounding in the temperature dependence of this mode at Tc . Also, the transition temperature is raised as compared to the C = 0 case. For strong-enough coupling C both transition temperatures merge together and coincide and a substantial displacive component appears in the tunnel mode frequency. The lattice mode, on the other hand, hardens even more and exhibits a small discontinuity at Tc , evidencing that the transition is ﬁrst order in both subsystems. It is important to note that with increasing coupling C the transition temperature of the proton system increases substantially as compared to the uncoupled or weakly

Fig. 5 Tunnel and lattice mode frequencies as a function of temperature with parameters J/Ω = 18.2, g2 /g4 = –0.8 ˚ A2 and Ω/ωq = 0.46. Solid lines refer to the uncoupled system A) and dash-dotted line (C = 0), dashed linesto the weakly coupled system (C = 1 × 104 K/˚ A) to the strongly coupled system (C = 5.7 × 104 K/˚

18

A. Bussmann-Holder · N. Dalal

Fig. 6 Tc as a function of C. Open circles are data points taken from [54]

coupled case. In the vicinity of Tc the frequencies ω0 , Ω are nearly degenerate, and it is no longer possible to make a clear distinction between the lattice mode and the tunnel mode, i.e. displacive and order/disorder features coexist as observed experimentally. As the proton double-well potential is not rigid above Tc , due to coupling to the soft lattice mode, and becomes temperaturedependent, the isotropic part of the chemical shift tensor can be expressed as a sum of a temperature-independent part σ0 and a temperature-dependent (v) 2 ! (v) 2 ! " quantity σ (T) = σT δQ , with δQ = q /ω coth ω/2kB T . Using the self-consistently obtained solutions of Eq. 19, the calculated chemical shift σiso = σ0 + σ (T) is calculated and compared to the experimental data in Fig. 4. Even though the experimentally observed transition is broader than the calculated one, the agreement between theory and experiment is good. As the discontinuity in the lattice-related mode is small at Tc , where Tc corresponds to ω = 0, the chemical shift does not show a discontinuity at Tc within numerical accuracy. It is important to note here that the S-shape in the σ (T) data is a direct consequence of using the renormalized frequencies as deﬁned in Eq. 19. As above, Tc is determined by the condition ω± = 0. At T = Tc g2 + (v) 2 ! = 0, Tc is approximately given by Tc ∼ g4 δQ = –g2/g4 and Sx = Ω/4Tc . The calculated dependence of Tc on C(δ) is shown in Fig. 6 and compared to experimental data. Importantly, even though C is linear in δ and x, Tc is nonlinearly dependent on C, as is obvious from Eq. 19. In the investigated regime of Tc and C (Fig. 6) a close to linear dependence is observed. This is, however, an exception compared to other H/D mixed systems, where deviations from linear behaviour have been observed [58].

Order/Disorder Versus or with Displacive Dynamics in Ferroelectric Systems

19

An approximate analytical solution (see above) yields that Tc is nearly independent of the tunnel mode frequency. Experimentally, it is found that in H2–x Dx SQ substantial variations in Ω take place with variations in x [54]. This fact does not contradict the above results since the tunnel mode is renormalized through the coupling C as: Ωr = 12 Ω(Ω – Jq Sx – C2 Sx /g4 T). As C increases with H...H, D...D site separation, Ωr decreases in agreement with experiment but still remains unimportant for the absolute value of Tc . Assuming that the proton–proton site separation δ is linear in x in H2–x Dx C4 O4 , the coupling between the protons and the lattice is linear as well. As the soft mode and Tc are substantially renormalized through this coupling, σiso is also renormalized through σ (T).

3 Conclusions Even though hydrogen-bonded ferroelectrics have long been believed to exhibit order/disorder dynamics at the phase transition recently experiments as well as theory have provided substantial evidence that a displacive component coexists with the order/disorder one. In particular, it has been shown that the dynamics of these systems are much more complex than expected within the pseudospin approach as the details of the bond geometry together with the unusual polarizability effects from the heavy ion sublattice have a much more profound inﬂuence on the dynamical behaviour than displayed within the pseudospin model alone. The introduction of coupled double-well potentials gives rise to new dynamics in terms of coexistence of order/disorder and displacive type, but it also accounts for the fact that the spontaneous polarization in these systems is a consequence of the heavy ion sublattice whereas the protons contribute only marginally to it. The modelling does not require introduction of non-physical changes in the tunnel mode frequency and the pseudospin–pseudospin interaction potential in order to obtain the correct isotope effect on Tc , but here it is a self-consistent consequence of the bilinear coupling between protons and the heavy ions. Important and in contrast to perovskite ferroelectrics, is the fact that the coexistence of order/disorder and displacive features happens on the same time scale, which makes them observable within a single experiment. In perovskite oxides the time scales differ substantially and long wavelength testing experiments reveal different dynamics as compared to local tools. Our results for hydrogen-bonded compounds have substantial implications for other and more complex hydrogen-bonded systems. In organic superconductors, for instance, an isotope effect on the superconducting transition temperature is observed as long as ions other than the protons are isotope exchanged. Upon deuteration, the superconducting transition increases slightly in contrast to BCS theory [59, 60]. We attribute this ﬁnding to subtle

20

A. Bussmann-Holder · N. Dalal

changes in the bond geometry in analogy to the above conclusions. Similar effects are thought to happen in more complex organic systems but these are, however, beyond the scheme of the present work.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38.

Slater JC (1941) J Chem Phys 9:16 Blinc R (1960) J Phys Chem Solids 13:204 Blinc R, Svetina S (1966) Phys Rev 147:423–430 Wul B, Goldman IM (1945) C R Acad Sci URSS 46:139 Wul B, Goldman IM (1946) C R Acad Sci URSS 51:21 Matthoas BT (1949) Phys Rev 75:1771 Matthias BT, Remeika JP (1956) Phys Rev 103:262, 76:1886 Shirane G, Hoshino S, Suzuki K (1950) Phys Rev 80:1150 Slater JC (1950) Phys Rev 78:748 Cochran W (1960) Adv Phys 9:387 Anderson PW (1960) In: Skanavi GI (ed) Fizika Dielektrikov. Akad Nauk SSSR, Moscow Kobayashi KK (1968) J Phys Soc Jpn 24:497 Blinc R, Zeks B (1974) Soft modes in ferroelectrics and antiferroelectrics. Elsevier, New York Tibbals JE, Nelmes RJ, McIntyre GJ (1982) J Phys C 15:37 Tun Z, Nelmes RJ, Kuhs WF, Stansﬁeld RDF (1988) J Phys C 21:245 McMahon MI, Nelmes RJ, Kuhs WF, Dorwarth R, Piltz RO, Tun Z (1990) Nature 348:317 Migoni R, Bilz H, Bäuerle D (1976) Phys Rev Lett 37:1155 Tessman GR, Kohn AH, Shockley W (1953) Phys Rev 92:890 Bussmann A, Bilz H, Roenspiess R, Schwarz K (1980) Ferroelectrics 25:343 Bilz H, Benedek G, Bussmann-Holder A (1987) Phys Rev B 35:4840 Bussmann-Holder A, Bilz H, Benedek G (1989) Phys Rev B 39:9214 Perry C, Currat R, Buhay H, Migoni R, Stirling W, Axe J (1989) Phys Rev B 39:8666 Khatib D, Migoni R, Kugel GE, Godefroy L (1991) J Phys Cond Mat 1:9811 Stachiotti M, Migoni R (1990) J Phys Cond Mat 2:4341 Stachiotti M, Migoni R, Höchli U (1991) J Phys Cond Mat 3:3689 Bussmann-Holder A, Michel K (1998) Phys Rev Lett 80:2173 Dalal N, Klymachyov A, Bussmann-Holder A (1998) Phys Rev Lett 81:5924 Müller KA, Luspin Y, Servoin JL, Gervais F (1982) J Phys (Paris) Lett 43:L537 Müller KA, Berlinger W (1986) Phys Rev B 34:6130 Zalar B, Lebar A, Seliger J, Blinc R, Laguta VV, Ith M (2005) Phys Rev B 71:064107 Stachiotti M, Dobry A, Magoni R, Bussmann-Holder A (1993) Phys Rev B 47:2473 Bussmann-Holder A, Büttner H (1990) Phys Rev B 41:9581 Matsumoto T (1997) Radiat Phys Chem 48:547 Alberts B, Bray D, Lewis J, Raff M, Roberts K, Watson JD (1983) Molecular biology of the cell. Garland, New York Budde M et al. (2001) Phys Rev Lett 87:145501 Balasubrami B et al. (1998) Proc Natl Acad Sci USA 95:9738 Itoh M et al. (1999) Phys Rev Lett 82:3540 Krumhansl JA (1990) Nature 348:285

Order/Disorder Versus or with Displacive Dynamics in Ferroelectric Systems 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59.

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Katrusiak A (1995) Phys Rev B 51:589 Katrusiak A (1993) Phys Rev B 48:2992 de Gennes PG (1963) Solid State Comm 1:132 Lawrence MC, Robertson GM (1980) Ferroelectrics 25:343 Robertson GM, Lawrence MC (1981) J Phys C 14:4559 Michel KH (1981) Phys Rev B 24:3998 Lyndon-Bell RM, Michel KH (1994) Rev Mod Phys 66:72 Zubarev DN (1960) Sov Phys USP 3:320 Mehring M, Suwalack D (1979) Phys Rev Lett 42:317 Mehring M, Becker JD (1981) Phys Rev Lett 47:366 Seliger J, Zager V, Blinc R (1984) J Magn Reson 58:359 Fischer G, Petersson J, Michel D (1987) Z Phys B 67:387 Dalal N et al. (2007) Struct Bond (in this volume) Samuelson EJ, Semmingsen D (1975) Solid State Comm 17:217 Semmingsen D, Feder J (1974) Sold State Comm 15:1369 Samara GA, Semmingsen D (1979) J Chem Phys 71:1401 Moritomo Y, Katsufuji T, Tokura Y (1991) J Chem Phys 95:2244 Blinc R, Zeks B, Sampaio JF, Pires A ST, Sa Barreto FC (1979) Phys Rev B 20:1991 Bussmann-Holder A, Dalal N, Fu R, Migoni R (2001) J Phys Cond Mat L231 Moritomo Y, Tokura Y, Nagaosa N, Suzuki T, Kumagai K (1993) Phys Rev Lett 71:2833 Kini AM, Carlson KD, Wang HH, Schlueter JA, Dudek JD, Sirchio SA, Geiser U, Lykke KR, Williams JM (1996) Physica C 264:81 60. Kini AM, Carlson KD, Dudek JD, Hau Wang H, Williams JM (1997) Synth Met 85:1617

Struct Bond (2007) 124: 23–50 DOI 10.1007/430_2006_044 © Springer-Verlag Berlin Heidelberg 2006 Published online: 21 December 2006

Experimental Evidence for the Coexistence of Order/Disorder and Displacive Behavior of Hydrogen-Bonded Ferroelectrics and Antiferroelectrics N. S. Dalal1 (u) · O. Gunaydin-Sen1 · A. Bussmann-Holder2 (u) 1 Department

of Chemistry and Biochemistry and National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32306, USA [email protected] 2 Max-Planck-Institute for Solid State Research, , Heisenbergstr. 1, 70569 Stuttgart, Germany [email protected] 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

2 2.1 2.2 2.3 2.3.1 2.3.2

Experimental Details . Crystal Preparation . . Isotope Labelling . . . Crystal Structures . . Squaric Acid . . . . . KDP Family . . . . . .

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26 26 26 27 27 28

3 3.1

NMR Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magic Angle Spinning and the Use of Single Crystals . . . . . . . . . . . .

29 29

4 4.1

Experimental Results and Discussion . . . . . . . . . . . . . . Basis of Using δISO to Distinguish Between the Order/Disorder and Displacive Mechanisms . . . . . . . . . . . . . . . . . . . . Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . Squaric Acid . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical Modelling of the Isotropic Chemical Shift: Evidence for the Existence of a Displacive Component . . . . . Temperature Dependence of the Average O – H Distance: A Possible Order Parameter . . . . . . . . . . . . . . . . . . . . 17 O NMR of Squaric Acid . . . . . . . . . . . . . . . . . . . . . Conclusions on SQA . . . . . . . . . . . . . . . . . . . . . . . . KDP-Type Ferroelectrics . . . . . . . . . . . . . . . . . . . . . . Comparison of 31 P Static and MAS Spectra of KD2 PO4 and Related Crystals . . . . . . . . . . . . . . . . . . . . . . . . Theoretical Analysis of the NMR Data . . . . . . . . . . . . . . Evidence for the Coexistence of an Order/Disorder and a Displacive Component of the Cationic Sites: 15 N NMR in NH H AsO and NH H PO . . . . . . . . . . . . 4 2 4 4 2 4

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Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.2 4.2.1 4.2.2 4.2.3 4.2.4 4.2.5 4.3 4.3.1 4.3.2 4.3.3

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Abstract This chapter describes the experimental complement of theoretical models of the microscopic mechanism of ferroelectric transitions. We use the hydrogen-bonded compounds as examples, and attempt to show that the new experimental data obtained via recently developed high resolution nuclear magnetic resonance techniques for solids clearly support the hypothesis that the transition mechanism must involve lattice polarizability (i.e. a displacive component), in addition to the order/disorder behaviour of the lattices. Keywords Hydrogen-bonded ferroelectrics · Order/disorder–displacive effects · NMR studies

1 Introduction This chapter addresses two of the most frequently studied families of hydrogen-bonded compounds, one represented by H2 C4 O4 (Squaric acid, henceforth abbreviated as SQA) and the other by KH2 PO4 (KDP). Both of these systems have been considered as prototypes of simple lattices exhibiting the order/disorder phenomenon (for a recent review, see [1]), [2, 3]. SQA consists of a two-dimensional lattice of C4 squares, connected by O – H...O hydrogen bonds [4–7], as shown in Figs. 1 and 2. Its lattice exhibits an antiferroelectric (AFE) transition at TN ∼ 373 K [4–10], while KDP undergoes a ferroelectric (FE) transition at TC ∼ 123 K [2, 3]. In both compounds, the order/disorder motion of the protons in the O – H...O bonds is considered to play an important role in the transition mechanisms, since H → D substitution (i.e. deuteration) causes their TN or TC to increase by about 80% [1–11]. Blinc was the ﬁrst to attribute this large H → D isotope shift to quantum mechanical tunnelling [11], invoking the critical role of the much larger zeropoint energy of the D atom relative to that of the H atom in a harmonic oscillator model. Later, Blinc and coworkers proposed a more reﬁned model, the so-called pseudo-spin model of the order/disorder mechanism, which has been successful in explaining several dynamical aspects [1, 3, 12]. Recently, however, the so-called geometrical models have been advanced, which are able to explain the deuteration effect without explicitly invoking quantum tunnelling [13–15]. While this development has been considered as a signiﬁcant step forward, other questions regarding the microscopic details of the transition mechanism(s) remain unanswered. One such point concerns whether displacive components play any role in the mechanism, in addition to the order/disorder behaviour of the H atoms, and, if so, then are the time scales of the displacive and order/disorder interactions essentially the same or are they quite different. This point was addressed in detail by Bussmann-Holder and Dalal in Chap. 1 of this volume. With the view to providing new insight into this aspect, a decade ago our group initiated studies of these lattices by means of modern high resolution NMR spectroscopy. The

Experimental Evidence for the Coexistence

25

Fig. 1 Two layers of squaric acid crystal at room temperature, one is ﬁlled the other one is clear. Large circles represent oxygen, medium circles carbon and small circles hydrogen atoms. Solid heavy lines indicate covalent bonds while solid thin lines indicate hydrogen bonds. This representation refers to four unit cells [5]

Fig. 2 ab plane projection of the crystal structure of squaric acid in the paraelectric (T > TN ) and antiferroelectric (T < TN ) phases [7]

basic tenet was that one can unambiguously probe this question by measuring the temperature dependence of the isotropic component of the NMR isotropic chemical shift, δISO . This premise is based on the fact that δISO is invariant to rotational and translational changes in the molecule [16–25]; this quantity should thus remain essentially unchanged through the phase transition if the transition is of a purely order/disorder nature. If, on the

26

O. Gunaydin-Sen et al.

other hand, the transition involves electronic structural changes as well, then δISO should exhibit a clear change/anomaly in the vicinity of the phase transition. While other techniques such as X-ray and neutron scattering can yield information on polarization ﬂuctuations (displacive effects) at very high time scales [1], NMR spectroscopy has been found to be uniquely sensitive, because the motional dynamics involved in the phase transitions in hydrogen-bonded solids coincide with that of the NMR techniques, as will be detailed in this chapter. Our goal here is to summarize the experimental NMR data using 13 C and 17 O nuclei for SQA; 31 P for KD2 PO4 (DKDP), RbH2 PO4 (RDP) and NH4 H2 PO4 (ADP); and 15 N in NH4 H2 AsO4 (ADA) and ADP. All these compounds have been found to exhibit a clear anomaly in the measured δISO at their respective phase transitions, which can be attributed to the presence of a displacive component in the phase transition mechanisms. For details the reader is referred to the original papers. The interpretation of the data is closely related to the models discussed in Chap. 1 of this volume.

2 Experimental Details 2.1 Crystal Preparation All the compounds discussed here are commercially available, and are easily crystallized by slow evaporation of saturated aqueous solutions. They all grow as several mm-cube-sized single crystals in tetragonal or prismatic shape. For the KDP family, the longest dimension is the unique direction (c-axis of the tetragonal cell). The SQA crystals are bi-pyramidal, with the square base as the C4 -square plane. Their crystal structures are discussed in the next section. Before going into those details, it seems worthwhile to provide some information on the isotope labelling procedures, which are crucial for carrying out the majority of the high resolution NMR studies in the solids under discussion. 2.2 Isotope Labelling All of the compounds can be deuterated by repeated crystallization from D2 O. For SQA, 17 O labelling can be accomplished by heating the dissolved compound in about 20% 17 O-labelled H2 O, in a closed vessel [26]. Both the labelled water and squaric acid should be refrigerated until used. The 17 O isotope labelling should be conﬁrmed by mass spectrometry or NMR spectroscopy. In general, a 10% label is quite satisfactory for good NMR signal

Experimental Evidence for the Coexistence

27

detection within a few minutes. Independent checks by speciﬁc heat measurements showed that 17 O labelling does not cause any appreciable change in the transition temperatures of either the KDP or the SQA family; any change is well within 1 K of the normal transition temperatures [26]. 15 N labelling is generally obtained by using 15 N-enriched 15 NH NO . 4 3 A suitable amount of 15 NH4 NO3 is added to the nearly saturated aqueous solution of the compound to be labelled, say ADA, before the crystallization is started. The crystals grow with the statistical distribution of the 15 NH4 + ions in the mother liquor. Again, about 10% labelling has been found to be quite satisfactory for convenient detection of 15 N signals in ADA as well as ADP. The labelling should be conﬁrmed; NMR of ADA samples showed that the crystals came out with essentially the percentage of the label in the solution, found to be 9 ± 1% using NMR [27, 28]. Here also independent speciﬁc heat measurements demonstrated that this level of 15 N labelling did not lead to any appreciable change in the transition temperature of either ADA or ADP. The crystal habit also remains unchanged such that axisidentiﬁcation for crystal orientation studies is as easy as for the unlabelled samples. 2.3 Crystal Structures 2.3.1 Squaric Acid Figs. 1 and 2 depict the crystal structures of SQA [4–7]. The structure consists of parallel sheets of C4 squares, the sheets facing such that the directions of the O – H...O bonds criss-cross in each sheet, so as to minimize the repulsive effects and avoid steric interactions. The structure is thus essentially two-dimensional, with the C4 planes held together by weak van der Waals forces. Figure 2 shows the schematic structure in the paraelectric (T > TN ) and antiferroelectric (T < TN ) phases. In the paraelectric phase the time-averaged position of the H atoms lies in the middle of an O – H...O bond, whereas in the antiferroelectric phase, the protons localize close to one or the other O atom. Prior to the recent NMR work [20–25], the largely accepted model of the phase transition was that the phase transition involved only the ordering of the H atoms in the O – H...O bonds, and no changes in the electronic structure of the C4 moieties were considered to take place. The NMR results show that, in addition to the order/disorder motion of the H atoms, the transition also involves a change in the electronic charge distribution and symmetry of the C4 squares.

28

O. Gunaydin-Sen et al.

2.3.2 KDP Family All members of the KDP family are isostructural in their high-temperature paraelectric phase. At room temperature they crystallize in the tetragonal space group, I42d. Figure 3 shows a schematic representation of the crystal structure of the KDP family [1]. The structure is of the AH2 XO4 type, where A is an alkali ion (NH4 , K, Rb or Cs ion), and X = P or As. The A and the XO4 ions alternate along the c-axis at distances c/2. The XO4 anions are held together by O – H...O hydrogen bonds lying essentially in the ab plane. As noted before, KDP, RDP and CsH2 PO4 undergo ferroelectric transitions at 123, 148 and 159 K, respectively. ADA and ADP are antiferroelectric, with transitions at TN ∼ 216 K and ∼ 148 K, respectively, to an orthorhombic P21 21 21 phase [2, 3, 29]. The basic crystal structure consists of PO4 3– (or AsO4 3– ) tetrahedra alternating with the K+ (or NH4 + ) ions along the c-axis. The PO4 3– units are connected by O – H...O hydrogen bonds in the ab plane, forming a threedimensional hydrogen-bonded lattice [2]. In the ferroelectric phases, the H atoms are localized such that the two close protons are both on the top of the oxygen ions of the XO4 units, as depicted in Fig. 4b. In the antiferroelectric

Fig. 3 Schematic representation of the basic structure of KH2 PO4 and RbH2 PO4 , and of NH4 H2 AsO4 and NH4 H2 PO4 [2, 3]. Black circles represent N, K, Rb and the grey circles represent H

Experimental Evidence for the Coexistence

29

Fig. 4 a Antiferroelectric NH4 H2 PO4 and b ferroelectric KH2 PO4 . ab plane projection of the structure of NH4 H2 PO4 (ADP) and KH2 PO4 (KDP), respectively. The arrows point along the dipole moment of the H2 PO4 – unit. The darker circles in the centre are P ions, and the small circles are the protons in the O – H. . .O bonds. The medium-sized grey circles are oxygen atoms [2, 3]

phases of ADA and ADP, the H atoms are localized asymmetrically along one of the O atoms, as shown in Fig. 4a.

3 NMR Methodology 3.1 Magic Angle Spinning and the Use of Single Crystals Magic angle spinning (MAS) has proven to be a highly powerful method for obtaining higher resolution NMR spectra from solids [30]. The technique relies on the fact that most sources of NMR line broadening, such as the magnetic dipolar and chemical shift anisotropy (CSA), are proportional to (3 cos2 θ – 1), where θ is the angle between the direction of the magnetic ﬁeld and the axis of rotation of a powder sample [30]. Thus, the broadening can be minimized by placing a sample in a rotor whose axis of rotation makes an angle of 54◦ 36 , which is the so-called magic angle for which (3 cos2 θ – 1) vanishes. However, for a powder sample, there is an additional source of broadening that depends on the particle size and shape, which is termed the anisotropic bulk magnetic susceptibility (ABMS) broadening [31]. It has been noted by Garroway and coworkers that this broadening could be circumvented by using single crystals in a MAS experiment, although these authors mentioned also that the reduction due to this procedure would be rather small [31]. In 1996 our group found that for materials such as squaric acid, the use of single crystals in MAS measurements led to a dramatic (a factor of at least four) enhancement in spectral resolution [20, 21], which has enabled the detection of effects that were not observable by standard MAS methods (i.e. using powders [27]).

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4 Experimental Results and Discussion 4.1 Basis of Using δISO to Distinguish Between the Order/Disorder and Displacive Mechanisms Before proceeding further, it seems useful to brieﬂy discuss how the temperature dependence of δISO can enable one to distinguish between an order/disorder and a displacive transition. Since δISO is a sensitive function of the total electronic distribution within the whole molecule, any anomalous change in this parameter implies the existence of a parallel distortion in the molecular structure, which means that the displacive component also plays a role in the transition mechanism. The schematic in Fig. 5a [20] shows how the temperature dependence of the isotropic NMR signals for a twosite model are supposed to vary through an order/disorder phase transition. Essentially, there ought to be two peaks in the slow motion regime, when the sites are ordered (below the phase transition). As the temperature is raised toward the transition, the order/disorder dynamics set in and the peaks coalesce at the transition and remain more or less unchanged at higher temperatures. An important point is that the average position of the peaks does not change through the phase transition. Figure 5b shows the case where the transition involves not only the order/disorder motion, but also a displacive component. In this case, the average peak position changes during the phase transition, the exact change depending on the nature of the transition mech-

Fig. 5 Schematic representation of the expected change in δISO at an order/disorder transition (left), and a mixed order/disorder plus displacive transition (right) [20]

Experimental Evidence for the Coexistence

31

anism. In fact, the changes observed in the 13 C spectra for SQA mirror those in Fig. 5b, vide infra. 4.2 Experimental Results 4.2.1 Squaric Acid 4.2.1.1 Resolution Enhancement from MAS Using Single Crystals Clearly the single crystal spectrum exhibits a much higher spectral resolution, a factor of at least four in these spectra [20–24, 26]. Figure 6 shows 13 C CP-MAS spectra of SQA powder (top) and a single crystal (bottom). While only two peaks are observed in the spectrum from the powder, corresponding to the two hydrogen-bonded (C – O – H) and the other two carbonyl (C = O) carbons, the spectrum from the single crystal shows much narrower lines, which can now be assigned to the four different carbons in the C4 skeleton. The assignment of the four peaks to the four respective C atoms in the SQA skeleton is shown at the top of Fig. 7. As discussed in details else-

Fig. 6 Comparison of the 13 C CP-MAS NMR spectra of squaric acid powder (top) and single crystals (bottom). Note that MAS using single crystals leads to a 400% enhancement of the spectral resolution [20]

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Fig. 7 CP-MAS spectra of squaric acid, peak assignment and temperature variation in the close vicinity of its paraelectric–antiferroelectric phase transition [24]. Note the pronounced coexistence of the spectra from the two phases

where [20–24, 26] and shown in Fig. 7, within 2–3 K of the phase transition temperature, the spectra consist of a superposition of features from both phases, attesting to the second-order, but very near ﬁrst-order nature of the phase transition. Figure 8 shows this temperature dependence in detail. Note that as the temperature is raised toward the transition temperature TN , the four lowtemperature peaks ﬁrst move smoothly together, and then coalesce into a single peak, which shows a further small increase at T > TN . Their δISO does not stay constant through the phase transition. An enlarged view of the variation of δISO in the vicinity of TN is shown in Fig. 9. In this plot, the low temperature side corresponds to the average of the data of the four peaks in Fig. 8. Notice that very close to TN , the spectra are a superposition of those from both phases, but around T = 373 K there is a sudden jump in the isotropic value, showing the onset of the near ﬁrstorder phase transition. Here the whole plot is thus S-shaped, the jump in the isotropic value due to the phase transition is indicated as ∆δ in the ﬁgure. The shape is consistent with the theoretical model developed in Chap. 1 in this volume. Some pertinent details are included here for completeness.

Experimental Evidence for the Coexistence

33

Fig. 8 Temperature dependence of the four 13 C NMR peaks of squaric acid [20]. Note how the four peaks coalesce to one above the phase transition, but that the average of the peak positions does not stay constant, as required for a pure order/disorder transition. It increases around the transition temperature, emphasizing an additional displacive component, coexisting with the order/disorder one

Fig. 9 Temperature dependence of δISO for the 13 C NMR peaks in squaric acid in the close vicinity of the phase transition temperature (left panel). Note that the change starts as a smooth curve, followed by a jump caused by the ﬁrst-order character of the phase transition. This is considered evidence for the coexistence of an order/disorder and displacive character in the phase transition mechanism [20]. The right panel gives a comparison between theoretical and experimental data. For details, see text

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4.2.2 Theoretical Modelling of the Isotropic Chemical Shift: Evidence for the Existence of a Displacive Component Even though the theoretical background for the analysis of the above data has already been given in detail in Chap. 1 in this volume, it is summarized here brieﬂy for the convenience of the reader. As detailed before, the pseudo-spin model as developed by Blinc et al. [3] accounts exclusively for a pure order/disorder-driven phase transition since it is based on the assumption that the dynamical behaviour of hydrogen-bonded ferroelectrics is driven by the proton dynamics only. However, various experimental results (for details see references in Chap. 1 in this volume) have provided clear evidence that polarizability effects, i.e. charge transfer, stemming from the heavy ion sublattice play a major role. In addition, it is important to note that the spontaneous polarization arises perpendicular to the hydrogen bridges and is nearly exclusively carried by the heavy ions. In order to account for these effects the pseudo-spin model has been extended and a coupled double-well model has been used where, besides the one stemming from the pseudo-spins, one is also incorporated in the core–shell displacement of the acid units. By using a bilinear coupling between both potentials proportional to the constant C (see Chap. 1 in this volume), displacive dynamics are coupled to the order/disorder dynamics. This approach was the ﬁrst to explain self-consistently the isotope effect on Tc and also the pressure effects on it. Consequently, the experimental data observed for squaric acid have also been analyzed within this model where two critical coupled frequencies ω0 , Ω determine the dynamical behaviour. Here ω0 is mainly dominated by the heavy ion sublattice, whereas Ω is the lattice-modiﬁed frequency of the pseudo-spins. It is important to note that the coupling C between both subsystems inﬂuences the transition temperatures of the individual components by increasing the transition temperature of the proton system substantially as compared to the uncoupled or weakly coupled case. In the vicinity of Tc the frequencies ω0 , Ω are nearly degenerate, and it is no longer possible to make a clear distinction between the lattice mode and the tunnel mode, i.e. displacive and order/disorder features coexist as observed experimentally. As the proton double-well potential is not rigid above Tc , due to coupling to the soft lattice mode, and becomes temperature-dependent, the isotropic part of the chemical shift tensor can be expressed as a sum of a temperature-independent part σ0 and a temperature-dependent quan tity σ (T) = σT (δQ(v) )2 , with (δQ(v) )2 = q [/ωq coth(ωq /2kB T)]. Using the self-consistently obtained solutions as given in Chap. 1 of this volume the calculated chemical shift σiso = σ0 + σ (T) has been calculated and compared to the experimental data (see right-hand panel in Fig. 9). Even though the experimentally observed transition is broader than the calculated one, the

Experimental Evidence for the Coexistence

35

agreement between theory and experiment is good. As the discontinuity in the lattice-related mode is small at Tc , where Tc corresponds to ω = 0, the chemical shift does not show a discontinuity at Tc within numerical accuracy. It is important to note here that the S-shape in the σ (T) data is a direct consequence of using the renormalized frequencies due to the bilinear coupling between both double-well potentials. 4.2.3 Temperature Dependence of the Average O – H Distance: A Possible Order Parameter Measuring the change in the (average) value of the O – H bond distance as a function of temperature, and then examining whether the O – H distance can serve as an order parameter for the phase transition is feasible with the high resolution 13 C data. This was indeed possible due to the fact that there were four values for δISO for 13 C and the four corresponding O – H distances were available from the X-ray and neutron diffraction data. The data of 13 C chemical shift versus temperature were converted to the corresponding O – H distances through a simple nonlinear regression curve ﬁtting procedure as discussed earlier [20]. The results obtained are plotted in Fig. 10 as a function of temperature. By comparison with the plots of the order parameter for

Fig. 10 Temperature dependence of the average O – H bond length as measured via the 13 C isotropic chemical shift for all carbon atoms in squaric acid. The change in the O bond length appears to play the role of an order parameter in the phase transition [20]

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Table 1 Fitted parameters, chemical shifts and proton displacements Fitted parameter

A/C [K]

B/C

Tf – Tp

Ref.

13 C

0.017 0.025 0.018

– 0.46 – 0.375 – 0.42

3 1.4 2.4

17 18 19

chemical shift

Proton displacement

H2 SQ reported in the earlier NMR studies [17–19], it was clear that the plot in Fig. 8 resembles that of an order parameter. The plot of Fig. 8 was therefore ﬁtted to the Landau free energy expansion: F = 1/2Aη2 + 1/4Bη4 + 1/6Cη6 ,

(1)

T0+ ).

The parameters thus obtained are listed in Table 1, towith A = A0 (T – gether with those reported earlier from 13 C measurements [17–19]. The good agreement between the results of the various studies [17–20] conﬁrms the essential correctness of the model. 4.2.4 17 O NMR of Squaric Acid While the 13 C NMR data described above provided strong evidence for the coexistence of an order/disorder and displacive component, it appeared that similar studies using 17 O as a probe might provide a more obvious test, since the H atoms are directly bonded to the O atoms, whereas the C atoms are two bonds further away [26, 28]. It should be noted that 17 O is a quadrupolar (I = 5/2) nucleus where MAS techniques are not effective, i.e. MAS does not average quadrupolar tensor components. It is known, however, that the quadrupole effects can be removed from the central doublet of the 17 O NMR spectrum by using a large Zeeman ﬁeld, since the splitting is a second order perturbation of the Zeeman splitting due to the quadrupolar interaction. The line-broadening caused by the quadrupolar interaction thus varies as the inverse square of the ratio of the quadrupolar coupling to the Zeeman ﬁeld [30]. Therefore 17 O CPMAS studies were carried out using the available highestﬁeld NMR spectrometer, operating at 14.1 Tesla. Again, a further resolution enhancement was obtained by using single crystals. Figure 11 shows typical 17 O NMR spectra from 17 O-labelled samples of SQA powder and a single crystal in the low temperature (T < Tc ) phase. It is seen that both the powder and the crystal show four distinct peaks, marked 1, 2, 3 and 4 in Fig. 11. The remaining peak(s)are the spinning side bands, as veriﬁed by the fact that their relative positions change in proportion to the spinning frequency. The doublet around 250 ppm can be assigned to

Experimental Evidence for the Coexistence

37

Fig. 11 17 O MAS NMR spectra of the central (1/2 →– 1/2) transition of a 17 O-labelled SQA in the form of a powder and b single crystal [26]. Note the narrower peaks in b. The peak numbering is the same as used in Fig. 1. The sharp line at 105 ppm corresponds to a spinning side band

the carbonyl (> C = 17 O) oxygens, while that around 100 ppm can be assigned to the > C – 17 O – H hydroxyl oxygens, in analogy with the 13 C peaks reported earlier [18–24], and theoretical calculations [9]. The factor-of-four sharper peaks observed from the single crystal spinning were again surprising. At ﬁrst, they appeared to be related to the reduction of the anisotropic bulk magnetic susceptibility (ABMS) broadening, as described in general by Van der Hart et al. [31]. However, the powder spectra did not exhibit a signiﬁcant narrowing on dispersing the sample in silica, as would be the case if ABMS were the cause. The exact mechanism by which this narrowing occurs is still unclear and should be a fruitful topic for further studies. Comparison of these spectra with those reported earlier for 13 C [21, 26, 28] clearly conﬁrms the much higher sensitivity of the 17 O peaks to H-dynamics, caused by the direct bonding of the O atom to the H atom. The splitting within a given doublet is about 30 ppm for 17 O, but only about 1 ppm for 13 C. This extra dispersion enabled us to follow the phase transition in a much more precise manner (vide infra) than was possible with 13 C. As reported earlier [26], the 17 O spectra exhibit a strong temperature dependence. Increasing the temperature leads to their merging and ﬁnally to their coalescence. At T > TN , they merge to a narrow doublet with a separation of about 10 ppm. As for to the 13 C data, the 17 O data were analyzed by plotting the average of the four peaks as a function of temperature. Figure 12 shows the corresponding plot. An important observation was that the position of the coalesced peak (doublet) did not coincide with the average of the four low-temperature peaks; this point is discussed in detail in the following section. The doubling was not related to any crystal effects, because measurements with powders exhibited a similar splitting [26]. Disregarding the actual amount of this doublet separation, it is important to note that the presence of this doublet, rather than a singlet, at temperatures

38

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Fig. 12 Temperature dependence of the average 17 O MAS peaks at 14.1 Tesla in the vicinity of the antiferroelectric transition at 372 K. Note that the value above the transition differs considerably from its value below the transition temperature, thus yielding clearer evidence for the role of the displacive component in the transition mechanism [26]

well above TN implies two different electronic environments for the oxygen atoms in the paraelectric phase, i.e. it is direct evidence that the structure of SQA at temperatures up to TN is actually a dynamic average of the low (Ci )symmetry structures, made possible by placing the H atoms in two different O – H...O bonds. It may also be noted that the doublet structure persists to at least 20 K above the TN . Finally, there is the interesting observation in Fig. 11 that over a few degrees around TN , signals from the paraelectric and antiferroelectric phases are simultaneously present. This is reminiscent of what was observed in the earlier 13 C NMR studies [21–24]. The result is consistent with the fact that the SQA phase transition is complex: it is essentially ﬁrst order, but exhibits some features reminiscent of a second-order type phase transition. The peak coalescence discussed above resembles a similar observation of peak coalescence in our earlier 13 C chemical shift data [9]. This can be understood by invoking a time-averaging statistical contribution from a carbonyl and a hydroxyl group with different hydrogen bond environments of the partially disordered SQA structure. This is analogous to the pseudo-rotation mechanism proposed for the proton motion by Semmingsen et al. [5]. Applying this hypothesis to the case of 17 O NMR, the environments of O(2) and O(4) after the proton ﬂip of H2 should correspond closely to those of the ordered O(3) and O(1) sites, respectively. In addition, the initial O(1) and O(3) ions gradually adopt O(4) and O(2) character through such + 90◦ pseudo-rotations. The detailed temperature dependence was fully consistent with the reﬁned site occupancies for the hydrogen atoms [9], obtained via the double-well potential given by Samuelsen et al. [32] as a fractional function of the carbonyl and hydroxyl contributions to the 17 O chemical shifts. It should be noted that the pseudo-rotational model predicts the emergence of

Experimental Evidence for the Coexistence

39

two peaks in the middle zone of the 17 O NMR spectra above TN , supporting the view that the two chains of the compound retain their different character at T > TN [9]. In addition to the double peaks at T > TN , the same theoretical modelling also showed that the average δISO increases steadily as T → TN , and exhibits an anomalous increase of about 13 ppm within 2–3 K of TN [9]. It is thus seen that the high-temperature position of 17 O δISO is different from that expected from the motional averaging of the four low-temperature peaks. This result implies that the chemical structure in the paraelectric phase is not just a timeaverage of the various low symmetry forms, but must include a deﬁnitive change in the molecular geometry. These data again point to a simultaneous effect of an order/disorder and a displacive component in the phase transition mechanism. 4.2.5 Conclusions on SQA Both the 13 C and the 17 O data sets yield at least two major results. First, in cases such as SQA, where a strong ABMS broadening is present in the NMR spectra, MAS using single crystals can afford resolutions several times higher than obtained through the more routine procedure of using powdered samples. In addition, as compared to 13 C, the 17 O nucleus exhibits a higher spectral resolution by a factor of nearly 5. Further, the observation of four clearly resolved 17 O NMR peaks at T < TN provides direct evidence of the presence of two different hydrogen-bonded Ising chains in SQA. The detection of a doublet rather than a singlet at T > TN , together with the fact that the higher-temperature peaks are not located at the algebraic average of the four low-temperature signals, implies that the transition mechanism consists of both an order/disorder and a displacive component. Moreover, since the dynamics are controlled by the correlated jumps of the protons, it can be concluded that the timescale of the order/disorder and the displacive ﬂuctuations is essentially the same. High Resolution NMR thus offers a unique time and length scale window on this phase transition. 4.3 KDP-Type Ferroelectrics Much early pioneering NMR work on the KDP-type materials has been carried out by Blinc and coworkers [1]. In particular, they speciﬁcally searched for a change in the isotropic chemical shift at Tc using 31 P resonance as a probe, as soon as the new generation of high-ﬁeld superconducting magnets became available for high resolution NMR work. Measurements on the SQA family were also performed in parallel by Mehring and coworkers [17, 18]. It turned out that, while these studies led to other important results, unfor-

40

Fig. 13 Temperature dependence of the transition [16]

O. Gunaydin-Sen et al.

31 P

chemical shift on going through the phase

tunately, the resolution of NMR measurements that time was not sufﬁciently high to detect the change in the isotropic part of the chemical shift because the isotropic part was much smaller than the dominant, anisotropic one. MAS techniques were not available then, nor were the high power decoupling techniques of the dipolar broadening due to the protons and other nuclei. Figure 13 shows the 31 P chemical shift data for KD2 PO4 reported by Blinc et al. in 1977 [16]. The reported line widths were about 10–15 ppm, as seen from the error bar in the ﬁgure. The present data using MAS techniques (vide infra) show that the change in the isotropic part is of the order of 1 ppm, beyond the resolution available at that time. The authors noted that the four peaks due to the two possible domains and two sites coalesce to only one above the transition temperature. They observed no change in the average position of the peaks on going through the phase transition, within their accuracy of about 10 ppm. This result was considered direct evidence for an order/disorder driven phase transition, and ruled out the existence of a displacive component. 4.3.1 Comparison of 31 P Static and MAS Spectra of KD2 PO4 and Related Crystals Figure 14 (top) shows a comparison of the static and MAS NMR signals of the 31 P nuclei from single crystals of 90% deuterated KD PO RbH PO at room 2 4 2 4

Experimental Evidence for the Coexistence

41

Fig. 14 Comparison between the static (upper spectra in both ﬁgures) and MAS (lower spectra in both ﬁgures) 31 P NMR spectra for RbH2 PO4 (upper panel) and 80% deuterated KD2 PO4 (lower panel). ∗ indicates MAS spinning sidebands [25]

temperature [25]. For DKDP the upper spectrum is a static one, without any rf decoupling or sample spinning, essentially as reported by Blinc et al. [16]. The line width is about 20 kHz, in agreement with the earlier data [16]. The bottom spectrum corresponds to the spectrum obtained under MAS, with the same crystal, but spinning at 7 kHz. The line width under MAS (and rf decoupling) is only about 70 Hz, thus demonstrating that the spectral resolution obtained under MAS is approximately two orders of magnitude higher. This increased resolution opened the door for measuring the change in 31 P δISO across the phase transition temperatures of KD2 PO4 , and also for RbH2 PO4 and NH4 H2 PO4 (vide infra). Figures 15 and 16 show the temperature dependence of the chemical shift for 80% deuterated KD2 PO4 and RbH2 PO4 . Obviously, δISO varies signiﬁcantly with temperature in the paraelectric phase and shows a clear break at Tc of 202 K for DKDP and 147 K for RbH2 PO4 , respectively. The shift exhibits a distinct discontinuity at Tc while the line width shows an abrupt increase below Tc , in agreement with the (close to) ﬁrst-order nature of the phase transition, and an anticipated pronounced distortion in the PO4 moiety. The cause of the line width increase below Tc has yet to be explained, but is at least partly due to a lack of the increased spinning speed to average out the enhanced chemical shift anisotropy below Tc .

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Fig. 15 Temperature dependence of the 31 P isotropic chemical shift for a 80% deuterated crystal of KD2 PO4 . Notice the anomaly around Tc , pointing to the existence of a displacive component in the transition mechanism [25]

Fig. 16 Temperature dependence of the 31 P isotropic chemical shift (top) and line width (bottom) for RbH2 PO4 (left panel). The right panel shows the temperature-dependent asymmetry of the normalized proton potential (solid lines) in comparison to the normalized data shown in the right panel of Fig. 16 [25]

Similar data have also been observed for the 31 P isotropic chemical shift and the line width for ADP in the authors’ laboratory. We noted that a fully deuterated DKDP crystal, which is known to exhibit no ferroelectric transition, showed an essentially ﬂat chemical shift response over the 200–280 K range [25]. 4.3.2 Theoretical Analysis of the NMR Data The above data have been analyzed theoretically within the same model as described for the chemical shift tensor δISO . In order to understand the chemical shift observed within the KDP family, we concentrate here on the polarizability modiﬁed double-well potential of the protons, which has been modelled by two back-to-back Morse potentials (see Chap. 1 in this volume). The separation between the protons and the centre of the left (l) or right (r) PO4 shell

Experimental Evidence for the Coexistence

43

is given by: |X ± (ξ)| = {(R – R0 ∓ εξ δy /2)2 + (u(ξ) ∓ δz /2)2 }1/2 where εξ = ±1, ξ = r, l and 2R being the equilibrium distance between the PO4 groups. u(ξ) is the shell displacement coordinate along the z-axis. The nonlinear core– shell coupling, as introduced in Chap. 1 in this volume, yields a T-dependent component at the mean-ﬁeld level. The z-component of the shell displacement adds a non-equivalence in the proton two-site distribution modelled here by the two Morse potentials. The temperature-dependent coupling to the PO4 shells is fully equivalent to charge ﬂuctuations, which freeze out at Tc and yield a substantial charge transfer. The temperature dependence of the proton-related double-well potential together with the dynamically induced asymmetry has been calculated self-consistently in order to compare experimental and theoretical data directly. Within this scheme the measured value of δISO has been subtracted by its value at Tc and normalized. The theoretical results, on the other hand, have been analyzed in terms of the asymmetry of the proton potential, which reﬂects the dynamical probability distribution of the protons to stay for a longer period in a preferred well. The temperature-dependent asymmetry ϕ(T) – ϕ(Tc ) is shown in Fig. 16 (right panel) in comparison to experimental data. The good agreement between theory and experiment conﬁrms the role of a pronounced component of displacive character in the phase transition mechanism in the KDP family as well. 4.3.3 Evidence for the Coexistence of an Order/Disorder and a Displacive Component of the Cationic Sites: 15 N NMR in NH H AsO and NH H PO 4 2 4 4 2 4 While the recent high resolution NMR data have clearly established that the PO4 or the AsO4 sites exhibit a coexistent order/disorder and a displacive behavior, there have been no parallel studies for the cationic sites, such as the NH4 + site in the antiferroelectric family of KDP-type compounds: NH4 H2 AsO4 (ADA) and NH4 H2 PO4 (ADP). Our group initiated a study to probe this question. A detailed analysis suggested that the 15 N NMR study of the ADA and ADP lattices, which undergo a transition at 216 and 148 K, might offer the best combination in terms of available MAS techniques at the feasible temperatures, leading to the following results. 4.3.3.1 15 N NMR in NH H AsO 4 2 4 Figure 17 shows a comparison between 15 N signals from CP-MAS spectra of an ADA crystal, without any spinning (Fig. 17a) and with spinning (Fig. 17b) at 5 kHz. The line width decreases from 80 Hz to only about 12 Hz under spinning, again demonstrating the signiﬁcant resolution enhancement provided

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Fig. 17 Comparison of 15 N NMR spectra for a NH4 H2 AsO4 single crystal obtained by (a) static and (b) MAS methods [27]

by the MAS method [30]. The spectrum is a single peak in both the paraelectric (T > TN ) and the antiferroelectric (T < TN ) phase. It is centred around 21.3 ppm, which is typical of 15 N from an NH4 + ion [34]. Figure 18 shows the temperature dependence of the 15 N MAS spectra around the transition temperature TN ∼ 216 K. The evolution of the two phases around the phase transition temperature is shown in Fig. 19. The most signiﬁcant result of the 15 N study is shown in Fig. 20: the temperature dependence of δ 15 iso for N in ADA upon cooling through TN . It is seen that δiso for ADA exhibits an ap-

Fig. 18 Temperature dependence of the 15 N CPMAS spectra of NH4 H2 AsO4 around the antiferroelectric phase transition temperature. The peaks corresponding to the paraelectric and antiferroelectric phases are labelled P and AF, respectively

Experimental Evidence for the Coexistence

45

Fig. 19 Temperature evolution of the paraelectric and antiferroelectric phases around the phase transition temperature of NH4 H2 AsO4 , as obtained from Fig. 18

Fig. 20 Temperature dependence of δiso (left), and expanded view around TN (right) without the jump for ADA around its phase transition temperature at 216 K. Note the large anomaly around TN , indicating the presence of a displacive component in the transition mechanism [27]

proximately linear temperature dependence away from the transition region, both above and below TN , with a step at TN caused by the ﬁrst-order nature of the phase transition, and an almost S-shaped anomaly around the transition temperature. It is this obvious change in the isotropic shift that is an evidence for the displacive behavior, as discussed in the earlier sections on the 13 C studies for squaric acid, and 31 P measurements for the KDP-type crystals. The evidence for the order/disorder component is shown by the spin-lattice relaxation rate (1/T1 ) studies.

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4.3.3.2 Temperature Dependence of Spin-Lattice Relaxation Rate, 1/T 1 The temperature dependence of 1/T1 for 15 N is shown in Fig. 21. The discontinuity in the 1/T1 data near TN ∼ 216 K (highlighted by the arrow) appears at the onset of the antiferroelectric phase transition. Below TN 1/T1 increases abruptly from about 370 ms to 700 ms. We analyzed the temperature dependence of 1/T1 using the semi-classical BPP model for the effect of molecular motion on 1/T1 [32]. In this model, 1/T1 can be related to the values of correlation time, τc , which is the characteristic time between signiﬁcant ﬂuctuations in the local magnetic ﬁeld experienced by a spin due to molecular motions or reorientations of a molecule. As usual, it is assumed that τc follows Arrhenius-like behavior: τc = τ0 exp Ea /RT , (2) where τ0 is the single particle correlation time, R the gas constant, T the absolute temperature and Ea the activation energy (per mol) for the dynamic process. The relationship between the spin relaxation rate and the correlation time is expressed by using the BPP equation for heteronuclear dipole–dipole interaction [32]: τc 1 3τc 6τc =C + + , (3) 2 2 2 T1 1 + (ωH + ωN )2 τc2 1 + ωH – ωN τc 2 1 + ωN τc where τc is the correlation time describing the dynamic process, ωN is the 15 N resonance frequency, ω is the 1 H frequency, and C is the dipole–dipole H

Fig. 21 Temperature dependence of 1/T1 for 15 N in ADA at a resonance frequency of 50.7 MHz (H0 = 11.4 Tesla). Squares are the experimental data points while the curve is the theoretical ﬁt based on Eq. 3. The arrow indicates the discontinuity in the 1/T1 data near TN ∼ 216 K

Experimental Evidence for the Coexistence

47

Table 2 Correlation times for 1 H and 15 N in NH4 H2 AsO4 at 250 K and 105 K Parameter

1 T1

=C

1 T1

=C

τc 1+ωH –ωN τc2

+

3τc 1+ωN 2 τc2

+

6τc 1+(ωH +ωN )2 τc2

τc 1+ωH –ωN τc2

+

3τc 1+ωN 2 τc2

+

6τc 1+(ωH +ωN )2 τc2

interaction constant, which is deﬁned as: 2 3 µ0 γN γH C= 3 20 4π rNH

250 K

105 K

3.0 × 10–11 s

1.6 × 10–8 s

3.4 × 10–11 s

1.5 × 10–8 s

(4)

with γN and γH being the gyromagnetic ratios of 15 N and 1 H, respectively, and rNH is the average N – H bond distance. The calculated value for A taken from C was 7.3 × 1010 1/s2 using the N – H bond length of 1.031 ˚ NH4 H2 PO4 (S. Koval, private communication). Figure 21 shows the experimental and calculated temperature dependencies of Zeeman spin-lattice relaxation rates of nitrogen in ADA. All the experimental data were ﬁtted using Eqs. 1 and 2, as shown by the solid lines in Fig. 21, where Ea and τ0 were treated as variables and C was ﬁxed. The activation energy did not change much from the paraelectric to antiferroelectric phase. The slight increase from 3.4 to 3.7 kcal/mol can be ascribed to the increased barrier for ammonium ion reorientations. In order to examine the mechanism of the spinlattice relaxation rate process, we also determined τc for the motion of NH4 + protons using the 1 H 1/T1 data reported earlier by Grosecu [35], and using the same procedure as for 15 N. A close similarity between the two sets of data is observed in Table 2 implying that the same motion underlies the 1/T1 process for both nuclei, related to the phase transition mechanism. The activation energy (Ea ) values obtained for the H atoms were 3.4 and 3.7 kcal/mol for the paraelectric and antiferroelectric phases, respectively, which are comparable with the corresponding values (3.4 and 3.7 kcal/mol) for 15 N. Note also that τc increases by three orders of magnitude in the antiferroelectric phase, showing that the N – H motion slows down anomalously, likely due to the onset of the antiferroelectric ordering. 4.3.3.3 15 N Results for NH H PO (ADP) 4 2 4 15 N

MAS studies have also been carried out on ADP in order to assess the generality of the conclusions derived from the ADA data. Figure 22 shows the temperature dependence of the 15 N MAS spectra around the transition tem-

48

Fig. 22

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15 N

MAS spectra around TN ∼ 148 K

Fig. 23 Temperature dependence of δISO for ADP around TN ∼ 148 K. Note the anomaly in the isotropic chemical shift around 148 K, emphasizing the role of the displacive component in the phase transition mechanism

perature TN ∼ 148 K, while Fig. 23 depicts the temperature dependence of the δISO . It is seen that δISO for ADP exhibits an approximately linear temperature dependence in all temperature ranges. In contrast to the results for ADA, the ADP peak became undetectable below the phase transition temperature. It is most likely due to a much larger chemical shift anisotropy and the lack of fast magic angle spinning capability below 147 K. Additional measurements with a faster MAS technique, currently under development, should help in understanding this point.

Experimental Evidence for the Coexistence

49

4.3.3.4 Conclusions from the 15 N Measurements on the Role of the NH4 Sites in the Antiferroelectric Transition The high resolution 15 N MAS NMR measurements on ADA clearly show that the isotropic chemical shift shows a large anomaly at the phase transition, pointing to a coexistence of order/disorder and displacive behavior. This change is an indication of the presence of a displacive component in the phase transition stemming from the NH+4 ions. On the other hand, the motional dynamics observed via the T1 measurements indicate the presence of the order/disorder behavior at this site as well. Interestingly, the temperature dependence of T1 shows a sudden change at the phase transition. Correlation times and activation energies for the motion of 1 H and 15 N nuclei are found to be essentially similar, indicating that the same motion underlies the T1 processes for both nuclei. The rotational motion of the NH4 + ion slows down by three orders of magnitude, facilitating the N – H...O bond formation, and the onset of the phase transition. The corresponding results on ADP in the paraelectric phase were similar to those for ADA, allowing the generalization that, like the 31 P sites in KDP and ADP, the cationic sites also undergo a distortion in the molecular structure at the phase transition.

5 Conclusions In this chapter we have shown that the experimental data on the coexistence of the order–disorder and displacive features near the phase transitions of H-bonded ferroelectrics and antiferroelectrics are fully supported by the theoretical models. We have used ultra-high resolution MAS NMR techniques to investigate the dynamics of hydrogen-bonded ferro- and antiferroelectrics in the vicinity of their phase transitions. All sublattice sites and all investigated compounds provide clear evidence that an order/disorder-only model is insufﬁcient to understand the new data. Clearly, a displacive component is present that has been overlooked previously due to the much poorer resolution. Also, we have shown that the use of single crystals improves the data quality considerably. Even though the coexistence of order/disorder and displacive aspects have also been discussed for perovskite ferroelectrics, hydrogen-bonded systems are distinct in the time and length scales of both components. While these are very different in perovskites, they are of nearly the same order of magnitude in hydrogen-bonded systems. Our conclusions are that phase transitions in ferroelectrics cannot be strictly classiﬁed but always represent a mixture of two different aspects of phase transition physics.

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References 1. Blinc R (2002) Ferroelectrics 1966–2001. Ferroelectrics 267:3–22 2. Lines ME, Glass AM (1977) Principles and applications of ferroelectrics and related materials. Clarendon, Oxford, UK 3. Blinc R, Zeks B (1974) Soft modes in ferroelectrics and antiferroelectrics. Elsevier, NY 4. Semmingsen D (1973) Tetrahedron Lett 14:807–808 5. Samuelsen EJ, Semmingsen D (1975) Sol St Commun 17:217–219 6. Semmingsen D, Feder J (1974) Sol St Commun 15:1369–1372 7. Semmingsen D, Tun Z, Nelmes RJ, McMullan RK, Koetzele TF (1995) Zeit Kristall 210:934–940 8. Rovira C, Novoa J, Ballone P (2001) J Chem Phys 115:6406–6417 9. Palomar J, Dalal NS (2002) J Phys Chem B 106:4799–4805 10. Ehrhardt KD, Buchenau U, Samuelsen EJ, Maier HD (1984) Phys Rev B 29:996–1007 11. Blinc R (1960) J Phys Chem Solids 13:204–211 12. Blinc R, Zeks B, Xampiao JF, Pires AST, Barreto FC (1979) Phys Rev B 20:1991–2001 13. Ichikawa M, Motida K, Yamada N (1987) Phys Rev B 36:874–876 14. Nelmes RJ (1988) J Phys C 21:245–258 15. Koval S, Kohanoff J, Migoni RL, Tosatti E (2002) Phys Rev Lett 89:187602 16. Blinc R, Burger M, Rutar V, Seliger J, Zupancic I (1977) Phys Rev Lett 38:92–95 17. Mehring M, Suwaleck D (1979) Phys Rev Lett 42:317–321 18. Mehring M, Becker D (1981) Phys Rev Lett 47:366–370 19. Fischer G, Peterson J, Michel D (1987) Z Phys B 67:387–397 20. Klymachyov N, Dalal NS (1997) Z Phys B 104:651–656 21. Klymachyov AN, Dalal NS (1997) Sol State Nuc Magn Reson 9:85–89 22. Klymachyov AN, Dalal NS (1998) Ferroelectrics 206–207:103–112 23. Dalal N, Klymachyov AN, Bussmann-Holder A (1998) Phys Rev Lett 81:5924–5927 24. Fu R, Klymachyov AN, Bodenhausen GF, Dalal NS (1998) J Phys Chem 102:8732–8735 25. Bussmann-Holder A, Dalal NS, Fu R, Migoni R (2001) J Phys Condens Matter 13:L231L237 26. Dalal NS, Pierce KL, Palomar J, Fu R (2003) J Phys Chem A 107:3471–3475 27. Gunaydin-Sen O, Fu R, Achey R, Dalal NS (2006) Ferroelectrics 337:153–160 28. Dalal NS, Gunaydin-Sen O, Achey R, Pierce KL (2006) Ferroelectrics 337:3–12 29. Fukami T (1990) Phys Stat Sol A121:383–390 30. Fyfe CA (1984) Solid state NMR for chemists. CRC, Boca Raton, FL 31. Van der Hart DL, Earl WL, Garroway AN (1981) J Magn Reson 44:361–401 32. Samuelsen EJ, Buchenau U, Dieter M, Ehrhardt K, Fjaer E, Grimm H (1982) Phys Scripta 25:685–687 33. Bloembergen N, Purcell EM, Pound RV (1947) Phys Rev 73:679–712 34. Andreson-Altmann KL, Grant DM (1993) J Phys Chem 97:11096–11102 35. Grosecu R (1973) Chem Phys Lett 21:80

Struct Bond (2007) 124: 51–67 DOI 10.1007/430_2006_050 © Springer-Verlag Berlin Heidelberg 2007 Published online: 25 January 2007

Order and Disorder in Perovskites and Relaxor Ferroelectrics Robert Blinc J. Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia [email protected] 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Order and Disorder in BaTiO3 . . . . . . . . . . . . . . . . . . . . . . . . .

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Order and Disorder in Relaxor PMN . . . . . . . . . . . . . . . . . . . . .

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1 Introduction Ferroelectrics have been traditionally divided into four different classes: 1. Displacive ferroelectrics where a discrete symmetry group is broken at Tc and the ferroelectric transition can be described as the result of an instability of the anharmonic crystal lattice against soft polar lattice vibration (e.g., BaTiO3 ). 2. Order–disorder-type ferroelectrics where a discrete symmetry group is broken due to the ordering of the ions in a rigid lattice potential (e.g., KH2 PO4 ). 3. Ferroelectric liquid crystals where a continuous symmetry group is broken at Tc and the doubly degenerate relaxational soft mode of the high-temperature phase splits below Tc into an “amplitudon”type soft mode and a symmetry restoring Goldstone (i.e., “phason”) mode [e.g., p-decyloxybenzylidene p -amino-2-methylbutylcinnamate (DOBAMBC)]. 4. Relaxors where there is no macroscopic symmetry breaking and where, in view of “site” and “charge” disorder, there is an extremely broad distribution of correlation times. The longest correlation time diverges at the freezing transition whereas other correlation times are still ﬁnite [e.g., Pb Mg1/3 Nb2/3 O3 ]. Recent work by Dalal and Bussman-Holder has shown that such a description is not appropriate for H-bonded crystals where a displacive component has been found in addition to the order–disorder one. It is the purpose of this

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R. Blinc

chapter to show that the above classiﬁcation is also not appropriate for displacive ferroelectrics where an order–disorder component exists in addition to the displacive one.

2 Order and Disorder in BaTiO3 Whereas the ﬁrst microscopic theory of BaTiO3 [1, 2] was based on order– disorder behavior, later on BaTiO3 was considered as a classical example of displacive soft-mode transitions [3, 4] which can be described by anharmonic lattice dynamics [5] (Fig. 1). BaTiO3 shows three transitions: at around 408 K it undergoes a paraelectric to ferroelectric transition from the cubic Pm3m to the tetragonal P4mm structure; at 278 K it becomes orthorhombic, C2mm; and at 183 K a transition into the rhombohedral low-temperature Rm3 phase occurs. More than 20 years ago, electron paramagnetic resonance (EPR) measurements performed on Mn4+ -, Cr3+ -, and Fe3+ -doped BaTiO3 [6, 7] seriously questioned the pure displacive character of the phase transitions. In the low-temperature rhombohedral phase it was found that Mn4+ —substituting A along the pseudo-cubic 111 equally charged Ti4+ —sits off-center by 0.14 ˚ directions and is cooperatively coupled to the Ti4+ and its dynamics. The disappearance of the Mn4+ EPR spectra at 13 and 19.3 GHz in the hightemperature phases of BaTiO3 let the authors conclude on a reorientational hopping of the Mn4+ between different 111 off-center directions with correlation times of 10–9 to 10–10 s, which is suggestive of an order– disorder scenario of the transitions. Recently, the problem has been studied by quadrupole perturbed 47 Ti and 49 Ti NMR [8, 9]. It was clearly shown that the Ti sits off-center not only in the tetragonal but also in the cubic phase. This off-center scenario conﬁrms theoretical studies [10–13] which showed

Fig. 1 Ti positions in the cubic unit cell in the paraelectric phase according to the displacive (left) and order–disorder (right) scenarios

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a combined displacive and order–disorder character of the transitions in BaTiO3 . Recently, the Girshberg–Yacoby off-center cation model of perovskite ferroelectrics has also been successfully applied to the cubic–tetragonal transition of BaTiO3 [14]. This approach clearly shows the characteristics of a displacive transition, but with a simultaneous ordering of the Ti subsystem as an additional order–disorder feature. The transition temperature Tc of the coupled system into the tetragonally ordered phase is then higher than the instability limit Ts of the uncoupled soft mode. Let us now turn to the scattering and NMR experiments. Inelastic neutron scattering [4] and Raman measurements [15] in BaTiO3 have clearly revealed the softening of a transverse optic phonon near the Brillouin zone center and helped to establish the soft-mode picture of displacive ferroelectric phase transitions. Near the cubic–tetragonal phase transition the soft mode becomes strongly overdamped at small wave vectors. In this temperature region the diffuse X-ray scattering in BaTiO3 is anomalous and has been claimed [16] to show the presence of pretransitional correlated clusters of quasistatic off-center Ti ions, as expected for an order–disorder transition. Later it was suggested that this scattering can also result from the strong anisotropy of the soft phonon dispersion relation. The controversy over whether the “quasistatic” anomalous X-ray diffuse scattering can indeed be explained within the displacive model, or whether a “vibronic” approach [17] predicting off-center potential minima for the Ti ion along the 111 directions is more appropriate, has not been settled until recently [18]. The problem to be solved is whether the potential for the Ti motion in the cubic paraelectric phase exhibits a minimum at the center of the oxygen cage or whether the Ti ion is disordered between several off-center sites. Since the electric ﬁeld gradient (EFG) tensor is zero by symmetry at the central position and nonzero at the off-center sites, the above problem could be deﬁnitely solved by quadrupole perturbed Ti NMR. The 47 Ti (I = 5/2) and 49 Ti (I = 7/2) NMR spectra and relaxation times were measured in a magnetic ﬁeld of 9.2 T corresponding to a Larmor frequency of 21.42 MHz for the two Ti isotopes. The ratio of the quadrupole moments of the two isotopes is 49 Q/47 Q = 0.819 and the difference in their Larmor frequencies in the cubic phase is about 5.7 kHz. As their natural abundances are relatively low (47 Ti: c = 7.3%, 49 Ti: c = 5.5%), between 6000 and 20 000 scans were collected for the spectra in the paraelectric (Pm3m) and the tetragonal ferroelectric (P4mm) phases, respectively. In a recent letter the ﬁrst NMR observation of quadrupole coupling induced 47 Ti and 49 Ti satellites in the cubic phase of an ultrapure BaTiO3 single crystal above the ferroelectric transition was presented [8]. For a nonzero electric ﬁeld gradient tensor V at the Ti sites, we expect for each Ti isotope 2I resolved quadrupole perturbed NMR lines. The frequency

54

shift ∆vM = vM–1↔M – vL of the M – 1 ↔ M transition is given by ∆vM = –vQ (M – 1/2) 3 cos2 θ – 1 + η sin2 θ cos 2φ /2 ,

R. Blinc

(1)

where vQ = 3K/[2I(2I – 1)] denotes the quadrupole frequency. K = e2 qQ/h is the quadrupole coupling constant proportional to the largest eigenvalue Vzz = eq of the EFG tensor, whereas η = (Vxx – Vyy )/Vzz measures the asymmetry. The eigenaxes x, y, and z are assigned according to their respective eigenvalues so that |Vzz | ≥ |Vyy | ≥ |Vxx | and consequently 0 ≤ η ≤ 1. θ and φ are the tilt and the azimuthal angle, respectively, of the direction of the external magnetic ﬁeld in the EFG tensor eigenframe xyz. Instead of sharp ±1/2 ↔ ±3/2, ±3/2 ↔ ±5/2, and ±5/2 ↔ ±7/2 lines for 47 Ti and 49 Ti we observed in BaTiO at T = 420 K, i.e., above T , broad back3 c ground components in addition to the two sharp – 1/2 ↔ 1/2 central lines (Fig. 2). The following facts have to be stressed: 1. The integral intensities of the two central lines and the broad background components are in the ratio 9/26 = 0.35 and 4/17 = 0.24, respectively, for the two Ti isotopes. This agrees with the predicted ratio of the intensities 2 I(I, M) ∝ M|Ix |M – 1 = [I(I + 1) – M(M – 1)] /4 (2) of the central component and the satellites in the 47 Ti and 49 Ti spectra. 2. The width of the background component for 47 Ti is about 2.5 times as large as that for 49 Ti, in agreement with the ratio of their quadrupole frequencies vQ (47 Ti)/vQ (49 Ti) ∼ = 2.5. These two observations show that the broad background is indeed due to unresolved satellites as described by Eq. 1, and that all Ti ions are off-center (Fig. 3) so that there are no undistorted cubic regions. The existence of satel-

Fig. 2 Ti quadrupole perturbed NMR spectrum of BaTiO3 in the cubic PE phase. The sharp 1/2 → –1/2 central lines are sitting on a broad satellite background

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Fig. 3 a Structural transitions in BaTiO3 according to the displacive scenario involving the freezing of the soft TO lattice mode (Cochran 1960) [3]. b Structural transitions in BaTiO3 according to the order–disorder scenario. Different Ti ion colors denote different occupancies (Chaves et al. 1976) [18]

lites demonstrates the presence of a nonzero EFG tensor at the Ti sites and thus the local breaking of the Oh symmetry of the cubic phase by off-center displacements of the Ti ions. The results show that in the paraelectric phase near Tc all Ti ions are disordered between several off-center sites. This agrees with the early view of Slater of a “rattling” Ti ion in BaTiO3 as well as with the original interpretation of the diffuse X-ray scattering [16]. The individual satellite transitions cannot be resolved above Tc due to exchange of Ti ions between off-center sites. Discrimination between tetragonal, orthorhombic, and rhombohedral breaking of the cubic symmetry can be obtained from the angular dependence of the second moments M2 of the satellite background [9]. ˜ 2 (ϕ, ϑ) By introducing the I-independent, dimensionless second moment M via ˜ 2 (ϕ, ϑ) M2 (ϕ, ϑ) = F(I)v2Q M

(3a)

(2I – 1)(2I + 3) 40

(3b)

F(I) =

56

R. Blinc

we ﬁnd for the different possible symmetries of the Ti displacements from the central position the following angular dependencies: 1 ˜ 2C4v (ϕ, ϑ) = 31 + 12 cos 2ϑ + 21 cos 4ϑ + 24 cos 4ϕ sin4 ϑ (4) M 64 1 291 – 54η + 115η2 – 9 + 30η – 7η2 768 × 4 cos 2ϑ + 7 cos 4ϑ + 8 cos 4ϕ sin4 ϑ 1 C2v ˜ 2y 291 + 54η + 115η2 – 9 – 30η – 7η2 (ϕ, ϑ) = M 768 × 4 cos 2ϑ + 7 cos 4ϑ + 8 cos 4ϕ sin4 ϑ 1 C2v ˜ 2z 93 + 22η2 + 9 – 2η2 (ϕ, ϑ) = M 192 × 4 cos 2ϑ + 7 cos 4ϑ + 8 cos 4ϕ sin4 ϑ

C2v ˜ 2x (ϕ, ϑ) = M

(5a)

(5b)

(5c)

1 sin2 ϑ 9 + 7 cos 2ϑ – 2 cos 4ϕ sin2 ϑ . (6) 8 In Fig. 4, we show the experimental angular dependencies of M2 (ϑ) for the spectra depicted in Fig. 2. The angular dependencies of M2 (ϑ) clearly show a tetragonal breaking of the symmetry of the cubic phase. In particular, for 49 Ti we ﬁnd ˜ 2C3v (ϕ, ϑ) = M

BaTiO3 :

vQ = 1.3 kHz ,

T = 450 K > Tc .

(7a)

Below Tc = 415 K the broad components disappear (Fig. 5) and are replaced by well-resolved satellites revealing the existence of three physically nonequivalent 90◦ ferroelectric domains. Instead of two we now have four narrow – 1/2 ↔ 1/2 47,49 Ti lines due to 90◦ domains (Fig. 5). The 49 Ti quadrupole frequency vQ = 3e2 qQ/[h · 2I(2I – 1)] is now vQ (T < Tc ) = 125 kHz .

(7b)

Fig. 4 (Color online) Angular dependencies of the second moment of the 49 Ti spectra ˜ 2C4v , shown in Fig. 2 for cubic BaTiO3 at 450 K. Best ﬁts (solid lines) are obtained with M i.e., with the tetragonal displacements scenario

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Fig. 5 Ti spectrum of BaTiO3 in the ferroelectric P4mm phase at T = 410 K < Tc ≈ 415 K, [001] ||B0 showing the presence of three 90-degree domains

The T2 spin–spin nuclear magnetization relaxation time values demonstrate that the Ti disorder above Tc is dynamic and not static (Fig. 6). In the calculation of the angular dependence of the second moment M2 of the 47 Ti and 49 Ti satellite background in the paraelectric phase of BaTiO3 , we have assumed that the Ti off-center shifts are static. The results are however also valid if we have biased exchange among the off-center sites, and

Fig. 6 (Color online) 49 Ti magnetization decay for a the central transition and b the satellite background in BaTiO3 allowing for a determination of T2 at T = 430 K > Tc . T2s values were obtained by ﬁtting the experimental points to the theoretical model

58

R. Blinc

this exchange is fast on the NMR timescale τNMR . Whereas we expect in the fast motion limit in the unbiased case a single, motionally narrowed line, we obtain in the biased case a frequency distribution which reﬂects the local site-occupation distribution function. In the fast motion limit, the inverse spin–spin relaxation time T2 is proportional to the rigid-lattice second moment: T2–1 ∝ M2 (ϑ, ϕ) . One should also mention that the NMR timescale τNMR is related to the second moment M2 via –1 τNMR (ϑ, ϕ) ∼ = M2 (ϑ, ϕ) . –1 ≤ √3/2v ∼ 1.6 kHz at Speciﬁcally, for 49 Ti NMR in BaTiO3 , we have τNMR Q = –1 ≤ 160 kHz at 410 K < T . Thus, in both the paraelectric 450 K > Tc and τNMR c –1 is several orders of magnitude lower than the and ferroelectric phases, τNMR frequencies of the dielectric relaxation mode (vε ∼ = 102 –103 MHz) associated with the hopping of off-center Ti ions among symmetry-related wells. From the point of view of Ti NMR, the interwell hopping is thus observed in the fast motion limit. Consequently, the existence of a satellite background directly signiﬁes biased 111 hopping among different Ti off-center sites and incomplete restoring of cubic symmetry. In addition to this fast 111 motion, there also seems to exist an ultraslow dynamic exchange of the bias directions which will be discussed below. We have demonstrated that at T > Tc the average effective symmetry of the EFG tensor at the Ti site is tetragonal, as expected if the “static tetragonal distortion” (STD) scenario applies. The individual satellite transitions cannot be resolved from the broad satellite background. This observed smearing could be attributed to dynamic effects, according to the “dynamic tetragonal distortion” (DTD) scenario. Speciﬁcally, there exist three symmetry-related 90◦ tetragonal distortions along a, b, and c, respectively (Fig. 7). In the DTD scenario, unbiased exchange among the three tetragonal states occurs (Fig. 8). Let us call this process “90◦ ﬂipping” and let us characterize the ﬂips, or internanodomain exchange, by the exchange time τ90◦ . We shall stress that this process is to be distinguished from the “111 hopping” among individual 111 Ti off-center sites, which, as the order–disorder model implies, occurs on a much shorter timescale τ111 ∼ = v–1 ε < τ90◦ . The experimental magnetization decay curves are shown in Fig. 7 for both the central transition and the satellite background of BaTiO3 at 450 K and [001] ||B0 . For the central transition a double-exponential decay is found as the central line “sits” on the satellite background. The two decays are characterized by T2c = 18 ms and T2s = 108 µs. Here c stands for “central” and s for “satellite”. When moving off-center and measuring T2 on the satellite background we found a single exponential decay characterized by T2s = 102 µs. Thus T2s < T2c , demonstrating the presence of a slow motion which renders

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Fig. 7 EFG tensors at the Ti site according to the order–disorder scenario showing the presence of nanodomains

Fig. 8 T2 spin–spin relaxation showing the exchange between tetragonally distorted nanodomains

the spectral density J(0) < J(ωL ), J(2ωL ). Since the width of the satellites is homogeneous and not inhomogeneous, this slow motion must also be fast on the NMR timescale. This motion thus results in a homogeneous broadening of the satellites FWHMhom = (πT2s )–1 ∼ = 3 kHz. As the 111 motion is much

60

R. Blinc

faster, the slow motion can be ascribed to 90◦ ﬂipping, that is, due to the slow exchange between the three differently oriented tetragonal nanodomains in the cubic phase of BaTiO3 (Fig. 8). 2 –1 eQ T2 = C0 J(0) + C1 J(ωL ) + C2 J(2ωL ) (– 1/2 ↔ 1/2) : C0 = 0 h Let us now try to relate the above data to the phase transition mechanism in BaTiO3 . According to the hypothesis of Comes et al. [16]—later investigated in detail by Chaves et al. [18]—the Ti ion is at all temperatures above Tc disordered between the eight positions shifted from the center along the 111 body diagonals (Fig. 7). In view of the 180◦ symmetry of the EFG tensor, only four of these EFG tensor orientations are distinguishable by Ti NMR. In the hypothetical cubic phase without tetragonal nanodomains all eight sites are equally populated, n1 = n2 = n3 = n4 = n5 = n6 = n7 = n8 . In the tetragonal phase or in the tetragonal nanodomains of the macroscopically cubic phase we have in a given domain n1 = n2 = n3 = n4 = n5 = n6 = n7 = n8 , so that the Ti ion moves preferentially between the four sites in the x–y plane leading to an effective displacement in the z direction. This is true for one 90◦ domain. All three 90◦ domains together then result in effective Ti displacements along the three cubic unit-cell edges (N = 3) as is indeed observed. In each of these 90◦ domains the Ti EFG tensor is a dynamic average over the four off-center sites located at the body diagonals. In contrast to NMR, extended X-ray absorption ﬁne structure (EXAFS) measurements [19, 20] show the presence of rhombohedral deviations from cubic symmetry. The difference is due to the different averaging times of the two methods (typically 10–6 s for NMR and 10–15 s for EXAFS). EXAFS sees the instantaneous Ti position in contrast to NMR. It should be stressed that both the NMR and the EXAFS results can only be understood if one deals with well-deﬁned off-center sites and not just with a shallow Ti potential. The exchange between the tetragonal nanodomains of the cubic phase (Fig. 9) freezes out below Tc in the tetragonal phase leading to static 90◦ domains. We can conclude that the present data clearly show that the Ti ions are off-center and dynamically disordered in the cubic phase of BaTiO3 . Nevertheless, our experimental data cannot be described by the pure eight-site order–disorder model of the phase transition in its original form [18]. The pure eight-site order–disorder model would, in view of the exchange, give a zero value for the Ti EFG tensor not only in the cubic but also in the tetragonal phase, whereas nonzero values were observed in both of these phases. The predictions of the eight-site Ti off-center model are however compatible with our 47 Ti and 49 Ti quadrupole perturbed NMR data in BaTiO3 if, in addition to the order–disorder mechanism, the displacive component— responsible for the dynamical distortion of the Ti frame in the cubic phase and, on freezing, for the elongation of the unit cell—is also taken into account (Fig. 9). Due to the Ti disorder the cubic phase consists of a large number

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61

Fig. 9 Two–timescale dynamics

of small and randomly oriented “tetragonal” nanoclusters with dynamically elongated unit cells, which transform into macroscopic 90◦ domains on cooling through Tc where the soft mode “freezes” out. The presence of both Ti disorder and the soft mode leads to a special type of phase transition in BaTiO3 with a combined displacive and order–disorder character [14]. It should be noted that the same eight-site combined order–disorder and displacive model also works for the other low-temperature phase transitions of BaTiO3 [21].

3 Order and Disorder in Relaxor PMN Classical relaxors [22, 23] are perovskite solid solutions like PbMg1/3 Nb2/3 O3 (PMN), which exhibit both site and charge disorder resulting in random ﬁelds in addition to random bonds. In contrast to dipolar glasses where the elementary dipole moments exist on the atomic scale, the relaxor state is characterized by the presence of polar clusters of nanometric size. The dynamical properties of relaxor ferroelectrics are determined by the presence of these polar nanoclusters [24]. PMN remains cubic to the lowest temperatures measured. One expects that the “disorder”-type dynamics found in the cubic phase of BaTiO3 , characterized by two timescales, is somehow translated into the

62

R. Blinc

dynamics of relaxors. Instead of relatively long-lived tetragonal nanodomains with rapid biased hopping of Ti ions within a nanodomain, we now expect to ﬁnd polar nanoclusters of different orientations with a lifetime which is long compared to the hopping times of the various constitutive ions between different off-center sites inside the polar nanocluster. The presence of site and charge disorder and the distribution of the dipole moments of the various clusters should result in a merging of the two timescales into a broad correlation time distribution. X-ray and diffuse neutron scattering and diffraction studies of PMN have been interpreted in terms of the spherical layer model of Vakhrushev et al. [25, 26]. The Pb atom is not situated at the (000) position as it should be for an ideal perovskite lattice, but is distributed over a sphere of radius R around this position. Above 293 K there is little difference between a uniform and an asymmetric distribution, so that it is difﬁcult to isolate long-living regions with nonzero average Pb displacement. For T ≤ 224 K, on the other hand, we deﬁnitely have nanoregions with nonzero polarization averaged over the acquisition time and coherence volume (10 nm). The model implies that the crystal consists of N regions with approximately equal volumes where the Pb atoms are displaced, with the directions of the displacements being nearly uniformly distributed in space. Above Tg ≈ 230 K one cannot speak about ferroelectric or compact polar nanoregions. Rather a cluster-inside-cluster-inside cluster ramiﬁed model seems appropriate. More compact clusters appear below the freezing transition Tg . At high temperatures above TB ≈ 617 K PMN behaves like all other simple perovskites. The dynamics of the system is determined by the soft transverse optical (TO) phonon which exhibits a normal dispersion and is underdamped at all wave vectors. Below TB , in addition to the soft mode—which becomes overdamped—a new dielectric dispersion mechanism appears at lower frequencies which can be described by a correlation time distribution function f (τ). As can be seen from Fig. 10, the correlation time distribution function f (τ) for PSN–PZN–PMN (S = scandium, Z = zinc), obtained from broad-band dielectric dispersion data, becomes asymmetric with decreasing temperature and a second maximum at long correlation times appears [27]. Such behavior was also observed for other relaxors, such as PMN, lead lanthanum zirconate titanate (PLZT), and strontium barium niobate (SBN). This second maximum is not seen in dipolar glasses. It seems to be speciﬁc to relaxors and signals the formation of polar nanoclusters. This is supported by the fact that this maximum changes if an external electric ﬁeld E larger than the critical ﬁeld EC is applied along the ferroelectric 111 direction at T < 205 K. The ﬁrst maximum of f (τ) corresponding to shorter correlation times, on the other hand, is similar to the one in the dipolar glasses and is not affected by electric ﬁelds.

Order and Disorder in Perovskites and Relaxor Ferroelectrics

63

Fig. 10 Relaxation time distribution function f (τ) describing the dielectric dispersion in relaxor PMN. The short timescale maximum describes the glassy-type dynamics, whereas the long timescale part refers to the polar cluster dynamics. The same features are obtained in PMN, PLZT, and SBN relaxors

To investigate the nature of the polar nanoclusters and the surrounding glassy matrix on the microscopic level, and to get spatially resolved evidence of the electric ﬁeld response of PMN, we performed ﬁeld cooled (FC) and zero ﬁeld cooled (ZFC) 207 Pb (I = 1/2) NMR experiments [28] at different temperatures and orientations of the crystal in magnetic and electric ﬁelds (Fig. 11a–c). The 207 Pb spectrum at 290 K (Fig. 11a) is isotropic and of a Gaussian line shape. Two-dimensional (2D) separation of interactions experiments show that the spectra are in fact frequency distributions and are composed of a large number of individual 207 Pb lines with different chemical shifts. This is incompatible with the assumption that the Pb ions sit at their high-symmetry cubic sites, as in this case all Pb sites would be equivalent and only a single sharp line is expected. The fact that we see a Gaussian frequency distribution demonstrates that we are dealing with a spherical glass, where all Pb nuclei are displaced but there is no preferential frozen-out orientation or magnitude of displacement. If the Pb ion shifts are only varied in orientation and not in magnitude, a powder-like pattern rather than the observed Gaussian line shapes would be seen. Due to short-range order correlation among the displacements, clusters are formed which ﬂuctuate in time, orientation, and magnitude of the dipole moment. The orientational bias is here spherically symmetric. This agrees with the model proposed by Vakhrushev [25], which states that above the freezing temperature the displacements of the Pb nuclei lie in a spherical shell around the cubic position, as well as with the mesoscopic spherical random bond–random ﬁeld (SRBRF) model [29]. At lower temperatures an anisotropic component appears in the 207 Pb NMR spectrum in addition to the isotropic one (Fig. 11b). Its angular dependence in the external magnetic ﬁeld follows the (3 cos2 ϑ – 1) law. The anistropic component—which is it-

64

R. Blinc

Fig. 11 a Comparison of the observed ﬁeld cooled (FC: red line) and zero ﬁeld cooled (ZFC: black line) 207 Pb NMR line shape in a PMN single crystal and the homogeneous line shape (yellow dots) extracted from T2 measurements, δνhom = πT1 , and the dipolar 2 line width (blue line). b FC 207 Pb NMR spectra of PMN single crystal at 80 K and at θ = 0◦ , 35◦ , and 90◦ showing that the anisotropic component obeys the (ν – ν0 ) ∝ (3 cos2 θ – 1) law, as expected for chemical shift tensor anisotropy. c ZFC and FC 207 Pb NMR spectra at 40 K showing the electric ﬁeld induced enhancement of the intensity of the anisotropic ferroelectric cluster component

self a frequency distribution—corresponds to polar clusters frozen out on the NMR timescale and oriented along the ferroelectric 111 axis. Such a twocomponent line shape is not seen in dipolar glasses. As can be seen from Fig. 11c, the anisotropic frozen polar cluster component increases in intensity if the crystal is cooled at low enough temperatures in an electric ﬁeld larger than the critical ﬁeld and applied along the 111 direction. A transition to the ferroelectric phase is induced for E > EC . The difference between the FC and ZFC Pb NMR spectra is striking and

Order and Disorder in Perovskites and Relaxor Ferroelectrics

65

shows the existence of a two-component behavior—the spherical glassy matrix and the ferroelectric clusters—on a microscopic level. According to the intensities of the NMR lines, about 50% of the Pb nuclei still reside in the spherical glass matrix which does not respond to the electric ﬁeld and 50% in the ferroelectric polar clusters which respond to electric ﬁelds. It should be noted that below Tc = 210 K a sudden increase in the intensity of the anisotropic component is also seen in the ZFC spectra, but the increase in the intensity of the anisotropic component is two times smaller than in the FC spectra at the same temperature. This shows that PMN is an incipient ferroelectric and that in the absence of the electric ﬁeld the concentration of the polar clusters is below the threshold for a percolation-type ferroelectric transition. The present results demonstrate that the basic difference between relaxors and dipolar glasses is their response to applied electric ﬁelds: polar nanoclusters, corresponding to the frozen anisotropic component in the NMR spectra, can be oriented in a strong enough applied electric ﬁeld and a ferroelectric phase can be induced. This is not the case in dipolar glasses, where the response is due to single dipoles which cannot be ordered by applied electric ﬁelds. The isotropic spherical matrix, on the other hand, seems to correspond to dynamic nanoclusters which ﬂuctuate fast on the NMR timescale. However, the anisotropic ferroelectric polar nanoclusters are frozen on the NMR timescale. The temperature dependence of the spin–spin relaxation time T2 of the 207 Pb nuclei conﬁrms the dynamic nature of the nanoclusters (Fig. 12). On cooling down from 400 K, T2 decreases with decreasing temperature, demonstrating the rapid slowing down of the nanocluster dynamics. The decrease in T2 is similar to the decrease in T1 but the values of T2 are three orders of magnitude shorter than the T1 values. This is easy to understand as the ﬂuctuation spectrum is not white, and as T1 measures the spectral density at the 207 Pb Larmor frequency of 79.4 MHz whereas T2 measures the spectral density in the zero frequency limit. A very pronounced BBP T2 minimum appears at 255 K where the inverse average ﬂuctuation time τ –1 reaches the kilohertz region. This closely matches the nominal freezing temperature of PMN, Tf ≈ 250 K, where a maximum of the dielectric susceptibility is found. The same mechanism, i.e., ﬂuctuations in the orientation of the nanocluster polarization, thus determines both the dielectric losses and the T2 data. Two other important points have to be stressed. The ﬁrst is that above Tc = 210 K, T2 does not depend on the frequency offset so that the nanocluster dynamics is similar in different parts of the crystal. Below Tc , on the other hand, T2 depends on the frequency offset and the nanocluster dynamics is different in different parts of the crystal, as expected for a two-component system consisting of ferroelectric regions and a spherical glass-type matrix. The second point is that in the FC T2 data around 140–150 K, a second ﬂat T2

66

R. Blinc

Fig. 12 Temperature dependence of the FC 207 Pb spin–spin relaxation time T2 in PMN. The minimum at 200 K corresponds to the freezing of polar clusters. The shallow minimum at 140 K, which is absent in the ZFC data, is related to the motion of FE domain walls

minimum appears which seems to be due to the motion of the domain walls of the ferroelectric 111 polarized regions. It should be also noted that the dynamic NMR line width ∆vdyn = (πT2 )–1 is always small (Fig. 11a) as compared to the static “glassy” line width induced by the inhomogeneous nature of the spectrum. Therefore the nanocluster dynamics can be locally seen only by T2 measurements, and not by 1D line shape data which reﬂect the static glassy nature of the relaxor state characterized by the Edwards–Anderson order parameter.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Mason WP, Matthias BT (1948) Phys Rev 74:1622 Slater JC (1950) Phys Rev 78:748 Cochran W (1960) Adv Phys 9:387 Harada J, Axe JD, Shirane G (1971) Phys Rev B 4:155 Migoni R, Bäuerle D, Bilz H (1976) Phys Rev Lett 37:1155 Müller KA, Berlinger W (1986) Phys Rev B 34:6130 Müller KA, Berlinger W, Blazey KW, Albers J (1987) Solid State Commun 61:21 Zalar B, Laguta V, Blinc R (2003) Phys Rev Lett 90:037601 Zalar B, Laguta V, Seliger J, Blinc R, Lebar A, Itoh M (2005) Phys Rev B 71:064107 Girshberg Y, Yacoby Y (1999) 11:9807 Girshberg Y (2001) J Phys Condens Matter 13:8817

Order and Disorder in Perovskites and Relaxor Ferroelectrics 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.

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Bussmann-Holder A, Bishop AR (1997) Phys Rev B 56:5297 Zhong W, Vanderbilt D, Rabe KM (1995) Phys Rev B 52:6301 Pirc R, Blinc R (2004) Phys Rev B 70:134107 Fleury PA, Worlock JM (1968) Phys Rev 174:613 Comes R, Lambert M, Guinier A (1968) Solid State Commun 6:715 Bersuker IB (1966) Phys Lett 20:589 Chaves AS, Barreto FCS, Nogueira RA, Zeks B (1976) Phys Rev B 13:207 Ravel B, Stern EA, Vedrinskii R, Kraizman V (1998) Ferroelectrics 407:206 Stern EA (2004) Phys Rev Lett 93:037601 Völkel G (2006) Ampere and AvH workshop, Vilnius, 16–21 September 2006 Smolensky GA (1970) J Phys Soc Jpn (Suppl) 28:26 Cross LE (1987) Ferroelectrics 76:241 Burns G, Dacol FH (1983) Phys Rev B 28:2527 Egami T, Mamontov E, Domowski W, Vakhrushev SB (2003) In: Davies PK, Singh DJ (eds) Fundamental physics of ferroelectrics. American Institute of Physics, CP 677:48 Vakhrushev SB, Okuneva NM (2002) AIP Conf Proc 626:117 Blinc R, Laguta V, Zalar B, Banys J (2006) J Mater Sci 41:27 Blinc R, Laguta V, Zalar B (2003) Phys Rev Lett 91:247601 Pirc R, Blinc R (1999) Phys Rev B 60:13470

Struct Bond (2007) 124: 69–88 DOI 10.1007/430_2006_047 © Springer-Verlag Berlin Heidelberg 2007 Published online: 2 February 2007

Local Structure and Dynamics of Ferroelectric Solids T. Egami Department of Materials Science and Engineering and Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA [email protected] Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

2

Probing the Local Structure and Dynamics by Neutron Scattering . . . . .

71

3

Role of Pb in Ferroelectric Solids . . . . . . . . . . . . . . . . . . . . . . .

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4 4.1 4.2

Local Structure and Dynamics of Relaxor Ferroelectrics . . . . . . . . . . Static Local Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamics of Local Structure . . . . . . . . . . . . . . . . . . . . . . . . . .

81 81 85

5

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract Details of local atomic structure and dynamics play a signiﬁcant role in determining the ferroelectric properties, even though ferroelectricity is a highly collective phenomenon. They are most important in disordered ferroelectrics, the relaxor ferroelectrics in particular. In this chapter we discuss how neutron scattering measurements enable us to see directly the local structure and dynamics and facilitate the understanding of their role through the static and dynamic atomic pair-density function (PDF) analysis. The PDF provides direct information on the local atomic structure, even when the structure is disordered, and complements the conventional crystallographic approaches. Recent advances in the pulsed neutron sources and in spectrometers have made this technique particularly powerful for studies of disordered ferroelectric solids. We focus on the role of Pb2+ ions in mixed ferroelectrics and the atomistic dynamic mechanism of relaxor ferroelectrics. Keywords Local atomic structure · Neutron scattering · Pair-density function · Relaxor ferroelectrics

1 Introduction Ferroelectricity is a highly collective phenomenon involving the long-range internal electric ﬁeld, and occurs only when the size of the system is large enough. Yet local physics does play a major role, as is evidenced by the relevance of the lattice anharmonicity that develops a local double-well potential.

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In addition, many of the ferroelectric solids are mixed ions systems, or alloys, for which local disorder inﬂuences the properties. The effect of disorder is most pronounced in the relaxor ferroelectrics, which show glassy ferroelectric behavior with diffuse phase transition [1]. In this chapter we focus on the effect of local disorder on the ferroelectric solids including the relaxor ferroelectrics. As the means of studying the local structure and dynamics we rely mainly on neutron scattering methods coupled with the real-space pairdensity function (PDF) analysis. For a long time neutron scattering was a special technique practiced by a small group of experts. But the recent development of a number of user facilities have brought this technique much closer to general researchers. It became a particularly powerful tool for the study of oxides, in which the positions of oxygen ions are crucially important in determining their properties. The most popular tool of structural study has been the neutron powder diffraction with the Rietveld reﬁnement. With this technique not only the lattice parameters but the positions of each atom within the unit cell and its thermal factor, or the vibrational amplitude, can be determined with very high accuracy. However, the conventional methods of structural analysis, including the Rietveld reﬁnement, assume perfect periodicity of the lattice structure, while many interesting solids are not perfectly periodic. For instance, a number of ferroelectric solids show strong diffuse scattering, indicating local disorder. Such local disorder will show up in the Rietveld analysis only as an artiﬁcially large thermal factor, and no information on the correlations among the disordered atoms is provided. A better alternative is the method of PDF analysis, which describes the pair-correlation among the atoms. The PDF is obtained as the Fourier transformation of the normalized structure function, S(Q), where Q = 4π sin θ/λ, θ is the diffraction angle, and λ is the wavelength of the probe. The PDF represents the distribution of distances between pairs of atoms in the sample, and has been widely used for the study of liquids and solids [2]. With the advent of synchrotron-based radiation sources it became possible to apply this method on crystals with disorder [3]. In the process of the Fourier transformation to obtain the PDF, not only the Bragg peaks but also the diffuse scattering intensities are all included. For this reason this method is also called the total scattering method. Recently, the PDF method was extended to describe the local dynamics of disordered materials (Dmowski W, Vakhrushev SB, Jeong I-K, Hehlen M, Trouw F, Egami T (2006) Abstracts American conference on neutron scattering, St. Charles, IL, 18–22 June 2006, unpublished). The total PDF is obtained by the powder diffraction method so that S(Q) includes both elastic and inelastic intensities. To determine the dynamics we have to use an inelastic neutron scattering spectrometer and measure the dynamic structure factor, S(Q,ω), over a large Q and ω space, and Fourier-transform along Q to obtain the dynamic PDF (DPDF). While the interpretation of the DPDF is a little

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complex because of the non-linear nature of the structure factor, it gives direct information on local lattice dynamics, which is difﬁcult to obtain by other means. In this chapter we discuss how the static and dynamic PDF measurements facilitated the understanding of the effect of local disorder on the ferroelectric properties of oxides, focusing on the role of the Pb ions and the behavior of relaxor ferroelectrics.

2 Probing the Local Structure and Dynamics by Neutron Scattering Neutrons are an excellent probe of the structure and dynamics of solids, since the de Broglie wavelength and the kinetic energy of thermal neutrons are comparable to the interatomic distances and phonon frequencies of a solid. However, since producing neutrons requires large and expensive facilities, such as a nuclear reactor or a spallation neutron source, initially the use of neutron scattering has been limited to a small number of special experts. But as its power became widely recognized more central user facilities have been constructed, and today the number of neutron facility users is around 5000 in Europe. In the US the number is around 1000, much less than in Europe, because of the delay in building large user facilities after the cancelation of the Advanced Neutron Source project. With the construction of the Spallation Neutron Source (SNS) of Oak Ridge National Laboratory in 2006, however, the US will soon catch up and exceed Europe in neutron scattering capability, and the user base is expected to grow rapidly. Neutron scattering is a particularly powerful technique in the study of ferroelectric materials, since the neutron scattering length of oxygen (0.58 × 10–12 cm) is comparable to those of important cations such as Ti (– 0.34 × 10–12 cm), Nb (0.71 × 10–12 cm), and Pb (0.94 × 10–12 cm) so that oxygen atoms are clearly seen by neutrons. In contrast, the scattering amplitude of X-ray is proportional to the atomic number, so that in X-ray scattering the Pb – Pb correlation overwhelms the O – O correlation by a factor of 105. An additional advantage is the capability of observing the lattice dynamics, as we will discuss later. The measured intensity of scattered neutrons includes components which are irrelevant to the structure determination and have to be removed, such as the scattering from the sample container and the incoherent scattering. In conventional analyses such as the Rietveld analysis these components are removed simply as background, by ﬁtting appropriate smooth functions. When we are concerned about diffuse scattering due to disorder, however, the apparent background can contain an important part of the diffuse scattering. Thus we have to remove the true background not coming from the sample by actually measuring the scattering intensity without the sample, and correct for the absorption by the sample. In order to determine the PDF we also have

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to measure the scattering up to a large momentum transfer as shown below, with better counting statistics than for the regular structural study. Consequently, the measurement of the PDF requires much more neutron beam time than regular structural studies. After subtracting the sample-independent background and correcting for absorption, multiple scattering, and incoherent scattering we obtain the structure function, S(Q), which is normalized to unity at Q → ∞. In terms of the atomic position and Q, S(Q) is given by: 1 bi bj S Q = exp iQrij , (1) 2 N b i,j where N is the total number of atoms in the system, bi is the scattering length of the i-th atom, b is the compositionally averaged scattering length, and rij is the distance between the i-th and j-th atoms. The PDF is obtained by: 1 ρ0 g(r) = ρ0 + 2r 2π

∞

S Q – 1 sin Qr QdQ ,

(2)

0

where ρ0 denotes the number density of atoms, and g(r) is the pairdistribution function: 1 bi bj g(r) = δ r – rij . (3) 2 2 4πNr b i,j Further details are discussed in [2]. Figure 1 shows the PDF of SrTiO3 [4]. The A is due to Ti – O pairs, and is negative because the ﬁrst negative peak at 1.95 ˚ scattering length of Ti is negative. The PDF is given in the absolute units. Thus

Fig. 1 PDF of SrTiO3 determined by pulsed neutron diffraction, compared to the PDF calculated for the model structure. Small oscillations at low r are termination errors [4]

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by integrating over speciﬁc peaks the number of atoms in that shell can be determined. For instance, the ﬁrst peak will give the coordination number, the number of the nearest neighbor atoms. In Eq. 2 the integration should be carried out to inﬁnity, but in practice it has to be terminated below Qmax = 4π/λ. In nuclear reactors, neutrons are continuously produced by nuclear ﬁssion at a steady rate with the kinetic energy of several MV (fast neutrons), and then moderated by a moderator, for instance heavy water, to thermal neutrons. Thermal neutrons with a single wavelength (actually a narrow range of wavelengths) are selected usually by a crystal monochromator. For thermal neutrons of ∼ 25 meV the wavelength A, and Qmax is about 8 ˚ A–1 . This is hardly high enough to comis about 1.5 ˚ plete the integration accurately. Premature termination of the integration in Eq. 2 produces spurious oscillations known as termination errors, and they have been the major source of problems with the PDF method. At the spallation neutron sources, such as the SNS, neutrons are produced by short pulses of proton beam accelerated by a linear accelerator and a synchrotron hitting the target, which is heavy metals such as Hg, Ta, W, or U shaking off (spalling) neutrons. By taking advantage of the pulsed nature of the source, the measurement is usually done by the time-of-ﬂight (TOF) method. The spalled neutrons are also fast neutrons, but in order to keep the pulses short and allow accurate TOF measurement moderation is done only for a short time, leaving a lot of epithermal neutrons with relatively high (eV range) energy. These epithermal neutrons allow S(Q) to be determined up to high values of A–1 or more. Because of the Debye–Waller factor that reﬂects Q, often to 40 ˚ atomic vibration, the oscillations in S(Q) quickly decay at large Q, with the envelope: D Q = exp – Q2 u2 ,

(4)

where u is the amplitude of atomic vibration. Consequently, when Qmax is A–1 termination errors usually become quite small, as shown larger than 40 ˚ in Fig. 1. In X-ray diffraction as well, high values of Qmax can be achieved by using high energy X-rays (> 100 keV) from a synchrotron source. Thus the advent of synchrotron-based radiation sources made the PDF method much more accurate, and applicable even to the structural study of crystals. The Debye–Waller factor, Eq. 4, describes how the uncertainty in real space (u) determines the range of S(Q) in Q space. Now the exact converse happens with respect to the resolution of the measurement in Q space. If the Q resolution of the instrument is ∆Q, the PDF will have an envelope exp(– r2 (∆Q)2 ), and the oscillations in the PDF decay. Therefore in order to determine the PDF up to large distances it is important to use an instrument with high Q resolution. Since the PDF method was initially applied to glasses and liquids in which atomic correlation decays quickly with distance, this point was not

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recognized in the ﬁeld. However, in crystals the PDF oscillates up to large distances. In fact, for the reduced PDF, G(r) = 4πrρ0 [g(r) – 1], oscillation ideally continues up to inﬁnity, but is limited only by the Q resolution. The NPDF spectrometer of the Lujan Center of the Los Alamos National Laboratory, the ﬁrst neutron spectrometer speciﬁcally designed for the high resolution PDF measurement, has the resolution of ∆Q/Q = 1.5 × 10–3 , the highest among the neutron powder diffractometers in the US. Using this instrument the PDF can now be determined up to 300 ˚ A, covering the mesoscopic ranges and connecting the local structure seamlessly to the long-range crystal structure. Figure 2 shows the PDF of LiNiO2 determined with the NPDF [5]. In order to determine the dynamics of atoms we have to carry out an inelastic neutron scattering measurement. With a reactor source this can be done with a triple-axis spectrometer, which has an analyzer crystal. “Tripleaxis” refers to the three axes for the monochromator, sample, and analyzer, all moving independently and controlled by a computer. With a pulsed source we use a mechanical chopper, which is a rotating cylinder with a hole perpendicular to the rotating axis that allows neutrons with a chosen range of velocity to go through. The neutrons scattered by the sample are detected

Fig. 2 PDF of LiNiO2 determined at T = 10 and 585 K with the NPDF of the Lujan Center of the Los Alamos National Laboratory. The x-axis of the PDF at 585 K is scaled by the lattice constant to align the peaks. Interesting temperature dependence reveals the presence of nanodomains [5]

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by a large array of two-dimensional detectors operating in the TOF mode. A modern chopper spectrometer (such as Pharos of the Lujan Center, Los Alamos National Laboratory, USA or MAPS of the ISIS, Rutherford Laboratory, UK) produces a large amount of data approaching GB for each measurement, and it is a non-trivial task to process all these data. After a correction similar to that for the PDF for sample independent scattering, absorption, multiple scattering, and incoherent scattering one obtains a dynamic structure factor: 1 bi bj S Q, ω = exp iQ · ri (0) – rj (t) – iωt dt , (5) 2 N b i,j where ri (t) is the position of the i-th atom at time t. To obtain the DPDF, ρ0 g(r, ω), this is Fourier-transformed by: 1 ρ0 g(r, ω) = ρ0 + 2 2π r

∞

S Q, ω – Sinc Q, ω sin Qr Q dQ ,

(6)

0

where Sinc (Q, ω) is the single atom incoherent dynamic structure factor. Examples of the calculated DPDF are shown in Fig. 3. Figure 3a shows the DPDF for a pair of atoms connected with each other with a harmonic potential. Because of the non-linearity of Eq. 5, higher order phonon harmonics are seen. Figure 3b is the DPDF for a pair of atoms connected with each other

Fig. 3 Dynamic PDF calculated for a diatomic molecule a oscillating harmonically at a frequency ω0 , and b two atoms randomly hopping between two states with different bond lengths. (Dmowski W, Vakhrushev SB, Jeong I-K, Hehlen M, Trouw F, Egami T (2006) Abstracts American conference on neutron scattering, St. Charles, IL, 18–22 June 2006)

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with a double-well potential. Because of the incoherent nature of tunneling or thermally activated hopping over the barrier between the double-wells the pattern is a continuous function of energy. At low energies, however, one can see a crossover from a tri-peak structure to a double-peak structure. This crossover energy is a measure of the tunneling frequency, and gives information on the double-well potential. For the integration in Eq. 6 to be accurate we have to determine the dynamic structure factor up to large values of Q, typically 20 ˚ A–1 . This requires an incident neutron beam energy of at least 200 meV, which is eight times the thermal energy at room temperature. Therefore, it is practically impossible to achieve it with the reactor neutron source, and we have to use the spallation neutron source. Inelastic neutron scattering measurement is slow, and requires a large amount of sample. Using the Pharos spectrometer of the Lujan Center, it takes two days to complete a measurement at one temperature point with 100 g of sample. When the chopper spectrometer of the SNS, ARCS, is in operation the measurement time will be shortened by a factor of 20 or more. Nevertheless, it still is a major undertaking, compared to the elastic PDF measurement.

3 Role of Pb in Ferroelectric Solids Many of the strong ferroelectric oxides include Pb2+ ions. The relaxor ferroelectrics almost always contain Pb [1]. While Pb presents health hazard, it obviously has beneﬁts for its ferroelectric properties. To facilitate the search for ferroelectric materials without Pb, its role in ferroelectricity has to be understood. It is well known that Pb2+ ions have so-called lone-pair 6s-p electrons, which lead to off-centering of Pb2+ in the cage of oxygen neighbors. The critical role of the 6s-p electrons in ferroelectricity was ﬁrst dramatically demonstrated by the ﬁrst-principles calculation by Cohen [6]. They form covalent bonds with oxygen resulting in substantial spatial shift of Pb2+ ions and the electron density. These strong covalent bonds are seen in the structure of PbZrO3 (PZ) [7]. Figure 4 shows the positions of the oxygen neighbors A, around Pb in PbZrO3 [7]. While the average Pb – O distance is about 3.2 ˚ A. These short Pb – O s-p electrons form short Pb – O bonds of about 2.45 ˚ bonds are clearly seen in the PDF of PbZrO3 as a peak at 2.45 ˚ A, as shown in Fig. 5. The PDF was determined using the SEPD of the IPNS, Argonne National Laboratory. While the effective charge of Pb is close to the nominal charge of two, because of the relative displacement against oxygen that A a large local electric dipole moment is formed, amounts to more than 0.5 ˚ contributing to the ferroelectricity. For the effective charge of + 2, the dipolar moment due to Pb off-centering is about 0.2 C m–2 if it is collinear [8], comparable to the total moment of BaTiO3 .

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Fig. 4 Pb and its oxygen neighbors in PbZrO3 [7]

Fig. 5 Pulsed neutron PDF of PbZrO3 [7]

It is easy to determine the environment of Pb in PbZrO3 accurately, since PbZrO3 is a well ordered compound. But many useful ferroelectric oxides, for instance well-known ferroelectric oxides, Pb(Zr,Ti)O3 (PZT), are mixed ion systems in which the crystal sites are occupied by two or more different ions. In such systems it is more difﬁcult to assess the state of Pb off-centering with accuracy using conventional crystallographic analysis, such as the Rietveld reﬁnement. Local Pb displacements are often non-collinear, displaced in different directions from site to site. These local displacements will be observed indirectly only as artiﬁcially large thermal factors, and the average displace-

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ment is much smaller than the real displacements. The PDF analysis provides a much better picture of the real local displacements, since the atomic distances represented by the PDF are real in the sense that no artiﬁcial averaging is involved. The PDF of PZT determined by pulsed neutron diffraction also A, suggesting that the oxygen environment of shows the Pb – O peak at 2.45 ˚ Pb2+ is similar in PZ and PZT. The pulsed neutron PDF of Pb(Mg1/3 Nb2/3 )O3 (PMN), which is a prototypical relaxor ferroelectric, is shown in Fig. 6. Again the peak at 2.45 ˚ A due to short Pb – O distances is clearly seen, even though the average structure, which is the simple cubic perovskite, does not show Pb off-centering. Thus Pb2+ appears to retain a similar oxygen environment in any ferroelectric compound. This is not surprising, since the bonding energy of the Pb offcentering must be quite large. The bonding energy could be estimated in the elastic approximation. The total lattice energy is: E=

K 2 x – gx , 2

(7)

where x is the displacement, K is the elastic modulus, and g represents the electron-lattice coupling. By minimizing Eq. 7 we get: Eb = –

g2 , 2K

xmin =

g , K

(8)

for typical values of xmin = 0.5 ˚ A and K = 5 eV ˚ A–2 , Eb is equal to 0.625 eV, a sizable value much larger than kT. Thus the local Pb off-centering exists at

Fig. 6 Pulsed neutron PDF of PMN at T = 10 K (circles), compared to the PDF expected for the cubic perovskite average structure (solid line) [9]

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79

any temperature, but its direction and the site–site correlation change with temperature and composition. The local direction of the Pb off-centering is best probed by the Pb – (Mg/ A. If Pb is displaced along the [1 0 0] direction, Nb) correlation around 3.5 ˚ four of the Pb – (Mg/Nb) distances become short while four others become long. Thus the Pb – (Mg/Nb) peak in the PDF splits into two subpeaks of nearly equal height. If the displacement is along [1 1 0], two short and two long Pb – (Mg/Nb) distances are produced, leaving four unchanged. So the Pb – (Mg/Nb) peak splits into three. When the displacement is along [1 1 1] one short and one long Pb – (Mg/Nb) distances are seen, leaving six others more or less unchanged. The simulated Pb – (Mg/Nb) PDF peaks are shown in Fig. 7 [10]. In PMN the direction of the local Pb polarization thus determined is along [1 0 0], even though the average polarization in the polar nanoregions (PNRs) is along the [1 1 1]. In Pb(Sc1/2 Ta1/2 )O3 (PST) the direction of the ferroelectric polarization is [1 1 1]. However, the PDF indicates that the local polarization is along [1 0 0], as shown in Fig. 8 [10]. Thus the local polarizations cannot be collinear, and the real structure must be different from the average crystal structure determined by the crystallographic analysis. Indeed the magnitude of the Pb A, whereas the PDF has displacement in the average structure is only 0.06 ˚ A, suggesting the actual local off-centering by 0.5 ˚ A, the a peak again at 2.45 ˚

Fig. 7 Calculated PDF of Pb – B correlation for Pb displacements along [1 0 0], [1 1 0], and [1 1 1] directions [10]. See the peak in the left for discussion in the text

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Fig. 8 PDF of PST compared to the model with local Pb displacement along [1 0 0] [10]

canonical value discussed above. The origin of this non-collinearity of the Pb displacements must be the B-site ions, Sc3+ and Ta5+ . They are large ions, A and Ta5+ 0.86 ˚ A [11], while the with the ionic radius of Sc3+ being 0.745 ˚ A. In PbTiO3 , the Ti4+ ideal radius of B ion in the BO6 octahedron is 0.58 ˚ A [11]) can be displaced along [1 0 0], toward an oxygen in the ion (r = 0.605 ˚ TiO6 octahedral cage. Thus PbTiO3 is polarized in the [1 0 0] direction. But Sc3+ and Ta5+ do not ﬁt in the BO6 octahedral cage, and can be displaced only toward a triangular face of the BO6 octahedral cage, i.e. along the [1 1 1] direction. Then the neighboring Pb ions can be polarized equally along x, y, or z directions, resulting in the average [1 1 1] polarization, even though locally Pb2+ ions are displaced along [1 0 0]. The average [1 1 1] polarization induces rhombohedral piezoelectric distortion along [1 1 1], and this distortion must be keeping the average polarization along one of the [1 1 1] directions, even though the local Pb polarization is along [1 0 0]. The interplay between the local Pb polarization and B-site ions may be described by a simple model. As discussed above the magnitude of the local A, and does not depend on off-centering of Pb2+ ions is always about 0.5 ˚ the environment. Thus it could be described as a pseudo-spin, and justiﬁes the pseudo-spin model with the dipolar–dipolar interaction and the local anisotropy energy: H = Hdip + Haniso Jij Pi · Pj Hdip = – ij

Haniso =

i

m

m Dm O Pi ,

(9)

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81

where Pi is the local dipolar moment of the i-th Pb ion and its oxygen neighbors and Om are the spherical harmonics equivalents of the polarization tensor [8]. For instance: O02 =

1 2 3Pz – P2 . 2

(10)

In this picture, the role of the B-site ions is to induce the coupling between local Pb polarizations, described by the Heisenberg exchange Hamiltonian with J. Haniso describes the local anisotropy energy. If the direction of the local easy axis in the anisotropy energy is random, this leads to the random anisotropy spin-glass state [9, 10]. For instance, in PMN a Pb2+ ion is surrounded by eight B-site ions, one third of which are Mg2+ and two thirds Nb5+ . There is a strong energy difference when the off-centered Pb2+ is pointing toward Mg2+ and toward Nb5+ [12]. This directional dependence creates strong anisotropy up to = 4. Because the lowest energy direction varies from site-to-site, Pb off-centering will not be collinear, and will have a ﬁnite correlation length, giving rise to a glass-like strongly disordered ferroelectric state.

4 Local Structure and Dynamics of Relaxor Ferroelectrics 4.1 Static Local Structure Relaxor ferroelectrics are strongly disordered ferroelectrics with diffuse phase transition [1]. Upon cooling, local dipolar moments freeze into a glassy state below a freezing temperature, Tf , which depends on the frequency of dielectric measurement. Above Tf the system shows high dielectric permittivity, which is only weakly temperature-dependent. Since regular ferroelectrics show high permittivity only in the vicinity of the ferroelectric phase transition temperature, and therefore the permittivity is strongly temperaturedependent, this behavior makes relaxor ferroelectrics very attractive for application. The crystal structure of PMN is the simple cubic perovskite, although sometimes tiny orthorhombic distortion is observed, associated with piezoelectricity. However, the pulsed neutron PDF of PMN determined at 10 K, shown in Fig. 6, is completely different from the PDF expected for the simple A that is perovskite structure. In particular there is a prominent peak at 2.45 ˚ absent in the expected PDF. This peak again is due to Pb off-entering, indicating strong polarization of the PbO12 cluster. The local Pb off-centering does not show up in the crystal structure because the direction of off-centering is not uniform and varies from site-to-site, and averages out to zero off-

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centering. This randomness is the origin of the glassy behavior of the relaxor ferroelectrics. While it is natural to think that this disorder is related to the random mixing of Mg2+ and Nb5+ ions on the same B-site of the perovskite structure, the history of research on PMN went through interesting detours because of the observation of short-range chemical order. A NaCl-type chemical ordering with the coherence length of several nanometers was observed by electron diffraction, and was associated with the Mg/Nb ordering [13]. Since [1 : 1] Mg/Nb ordering is not compatible with composition, it was proposed that the system compositionally phase separates into Mg/Nb [1 : 1] ordered domains and Nb-rich domains. Since these domains are not electrically neutral they cannot grow and remain nanosized, and these nanodomains, polar nanoregions, produce relaxor behavior [14]. However, small-angle X-ray scattering measurements did not detect such compositional segregation [15]. It was also shown that the chemically ordered domains can grow to micron size in a very similar Pb(Mg1/3 Ta2/3 )O3 [16], and the nature of chemical order is to form two NaCl-type sublattices with Ta and Mg2/3 Ta1/3 [17]. Thus, even with the NaCl-type chemical ordering strong chemical disorder remains within the sublattice, and this must be the origin of the glassy behavior.

Fig. 9 Temperature dependence of the local structure of PMN described by the pulsed neutron PDF. The upper inset shows the PDF at 650 K measured (circles) and calculated from the crystal structure data (line). At low temperatures the peak at 2.45 ˚ A shows local A suggest that the direction off-centering of Pb2+ ion, and the split peak at 3.33 and 3.63 ˚ of off-centering is along [1 0 0] [18]

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The temperature dependence of the local structure of PMN up to 1000 K was studied by the pulsed neutron PDF method using the high-resolution neutron powder diffractometer, NPDF of the Lujan Center, Los Alamos National Laboratory. The results are shown in Figs. 9–11 [18]. As we discussed A, allowabove, with this spectrometer the PDF can be determined up to 300 ˚ A, seen in Figs. 6 ing study of the nanoscale structure. The extra peak at 2.45 ˚ and 9, is the signature of local off-centering of Pb2+ ions as discussed above. The Pb – O peak at 2.8 ˚ A is broad even at 650 K, above the so-called Burns temperature, Td , indicating that Pb off-centering exists even above Td , consistent with the argument on the energy of off-centering. The low temperature A is split into two at PDF in Fig. 9 shows that the Pb – (Mg/Nb) peak at 3.5 ˚ A, suggesting that the local displacement of Pb is along [1 0 0], 3.33 and 3.64 ˚ as shown in Fig. 7. But, even though the direction of the local polarization of Pb2+ determined by the Pb – (Mg/Nb) peak is [1 0 0], in the intermediate scale beyond 5 ˚ A, the local polarization is along [1 1 1] with rhombohedral distortion, just as in PST, suggesting that the same mechanism is at work.

Fig. 10 Medium-range PDF of PMN. Temperature dependence (upper panel) is well explained by the transition from rhombohedral phase to the cubic phase (lower panel). The two PDF peaks at 8.5 and 9.09 ˚ A indicate 300 K as a characteristic temperature for crossover [18]

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The PDF is strongly temperature-dependent, much more than expected from the lattice vibration. Modeling of the PDF shows that it is best described as the gradual transition from the rhombohedral phase to a cubic phase, implying that the ferroelectric polarization is not spatially uniform, and the nanoscale rhombohedral phase coexists with unpolarized nanoscale cubic phase. As temperature is increased, the volume fraction of the rhombohedral phase decreases and is replaced by the cubic phase, as shown in Fig. 11(d) [18]. Figure 11(d) shows two transitions, at 300 K and 600 K (∼ Td ). The temperature of freezing transition, Tf , depends on frequency, and at a quasi-static limit it is around 230 K. As the frequency is increased Tf is increased, and it reaches 10 GHz aroundTf∞ = 300 K [19]. One interpretation is that the polar nanoregions (PNRs) are formed at Td , and the inter-PNR correlation develops below 300 K and spreads over a macroscopic length. In order to understand the development above 300 K, however, we need dynamic information.

Fig. 11 PDF of PMN at a 500, b 250, and c 50 K, ﬁt well by the calculated PDF assuming a mixture of the rhombohedral and the cubic phase with varying fraction. d Volume fraction of the rhombohedral phase (fraction of PNR) [18]

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4.2 Dynamics of Local Structure In order to study the local dynamics of PMN, a series of inelastic neutron scattering measurements were performed with the Pharos spectrometer of the Lujan Center. The dynamic structure factor thus determined was Fouriertransformed by Eq. 6 into the DPDF (Dmowski W, Vakhrushev SB, Jeong I-K, Hehlen M, Trouw F, Egami T (2006) Abstracts American conference on neutron scattering, St. Charles, IL, 18–22 June 2006, unpublished). An example of the DPDF is shown in Fig. 12. The energy resolution is better than 4 meV (1 THz). The DPDF data at E = 0 (elastic component) shows the correlation among the averaged atomic positions. The time scale for averaging is determined by the resolution, and in this case it is one picosecond (10–12 s). Anything slower than 1 ps is considered static with this resolution. At T = 450 K the average structure is simple cubic perovskite, and indeed the DPDF peaks A (B – O), 2.8 ˚ A (O – O and Pb – O), 3.5 ˚ A (Pb – B), 4 ˚ A (cubic unit cell), at 2 ˚ where B = Mg/Nb, are consistent with the simple cubic perovskite lattice. But A peak at higher frequencies additional peaks are seen. For instance, the 2.45 ˚ A due to the short Pb – O is seen above 15 meV, and the Pb – B peak at 3.5 ˚ splits into two subpeaks also above 15 meV. These indicate the dynamic Pb off-centering with the characteristic frequencies of 15 meV (4 THz). The DPDF was determined at various temperatures covering Tf (= 230 K) and Td (= 600 K). Figure 13a shows the temperature dependence of the A and 3.3 ˚ A indicating Pb offDPDF at E = 0 (static). The subpeaks at 2.45 ˚

Fig. 12 DPDF of PMN at 450 K. See text for interpretation

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T. Egami

Fig. 13 Temperature dependence of DPDF of PMN, a at E = 0, and b at E = 15 meV

centering are clearly seen only at low temperatures, and disappear around 300 K. This result is consistent with the observations discussed earlier that the temperature-dependent relaxation of the dielectric constant reaches the phonon frequencies around 300 K (= Tf∞ ). It also corresponds well with the results shown in Figs. 10 and 11, which indicate that the collective rhombohedral local distortion indicating local Pb polarization sharply decreases around 300 K. On the other hand, the DPDF at E = 15 meV (Fig. 13b) shows these subpeaks and the split Pb – B peak to persist up to around 600 K (= Td ). Thus the characteristic frequency of Pb off-centering appears to increase from below 1 THz to 4 THz over the temperature range of 300 to 600 K. These results indicate that the relaxor ferroelectricity develops in two stages upon cooling, at Td = 600 K and at Tf∞ = 300 K. As we discussed above, Pb must be locally off-centered even above Td because the energy for Pb off-centering is quite large, but they are uncorrelated in space. It is interesting to note that the energy of the soft phonon branches involving Pb – O displacements is about 5–15 meV when the waterfall behavior is observed [20, 21], a similar energy range to where we see anomalies in Figs. 12 and 13. This coincidence suggests that the resonance between the local dynamic off-centering of Pb ions with these soft phonons produces the dynamic short-range correlation in polarization, resulting in the formation of polar cloud at temperatures below Td . It also must be the origin of the waterfall behavior. Burns temperature marks the start of such dynamic resonance between the local Pb off-centering and the soft phonons. This resonance between the local polar moment and the soft phonons was modeled in recent calculations [22]. In this picture local dipoles, or the breathers (off-centered Pb in the O12 cage), are embedded in highly polarizable media ((Mg1/3 Nb2/3 )O3 matrix). The resonance of the local breathers with the ferroelectric soft phonon mode of the matrix produces a dynamic giant polar state. As the size of the polar cloud grows, the relaxation time of the ﬂuctuating giant dipolar moment increases, leading to spin-glass-like freez-

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ing. This picture well captures the essence of the relaxor phenomena, and explains how the collective interaction of local off-centered Pb ions with the soft phonon modes in the medium results in the radio-frequency relaxation phenomenon. A splits into two subpeaks dyThe Pb – (Mg/Nb) peak of the DPDF at 3.5 ˚ namically below Td , and quasi-statically below Tf∞ . This result is consistent with the local Pb displacement along the [1 0 0] direction. Among the six [1 0 0] type directions the one with most Mg2+ neighbors will be chosen as the local off-centering direction of each Pb2+ ion, so the neighboring Pb ions may choose different off-centering directions. The polar clouds formed below Td but above Tf∞ are purely dynamic, and its polarization also ﬂuctuates with time. Below Tf∞ , however, the local polarization clouds develop sufﬁciently in space so that the quasi-static rhombohedral distortion along [1 1 1] is stabilized. These nanoregions or domains with local rhombohedral distortion must be what are known as polar nanoregions (PNRs). With decreasing temperature the relaxation time of the PNRs becomes slower, and ﬁnally the PNRs freeze into the polar glass state at Tf . The frequency-dependent relaxation phenomena are seen between Tf and Tf∞ . Above Tf∞ the permittivity becomes frequency-independent. Therefore, unlike the conventional picture, the frequency-independent super-paraelectric high permittivity over a wide range of temperature is provided by the PNRs only up to 300 K, and above 300 K PNRs are replaced by the polar clouds with less correlation than the PNRs and no quasi-elastic local distortion.

5 Conclusions Ferroelectric materials exhibit high dielectric permittivity and piezoelectricity, and are very useful functional materials with wide applications. However, the properties of pure ferroelectrics, such as BaTiO3 , are too singular, with high responses seen only in the close vicinity of the Curie temperature. Thus the materials actually used for applications are mixed ion alloys with often complex behavior, such as relaxor ferroelectrics. To understand such complex systems, conventional ideas such as the soft mode are not useful. In relaxor ferroelectrics, such as PMN, the soft phonon modes are strongly damped because of strong randomness, resulting in the disappearance of the mode, called the waterfall behavior. In this chapter we tried to explain these complex behaviors from a local point of view, using the static as well as dynamic PDF approach. The super-paraelectric behavior that makes the relaxor ferroelectric very attractive for application was explained in terms of the local dynamic polarization cloud, which is different from the polar nanoregions. Until recently the local dynamics of the relaxor ferroelectrics were assessed

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only by optical spectroscopy with no spatial resolution. Now, the newly developed pulsed neutron dynamic PDF method provides a much more accurate picture of local lattice dynamics and elucidates the complex behavior of the relaxor ferroelectrics.

References 1. Smolensky GA (1981) Ferroelectrics and related materials. Gordon and Breach, New York 2. Warren BE (1969) X-ray diffraction. Addison-Wesley, Reading, MA, USA 3. Egami T, Billinge SJL (2003) Underneath the Bragg peaks: Structural analysis of complex materials. Pergamon, Oxford 4. Louca D, Egami T (1999) Phys Rev B 59:6193 5. Chung J-H, Proffen Th, Shamoto S-I, Ghorayeb AM, Croguennec L, Tian W, Sales BC, Jin R, Mandrus D, Egami T (2005) Phys Rev B 71:064410 6. Cohen R (1992) Nature 358:136 7. Teslic S, Egami T (1998) Acta Cryst B 54:750 8. Egami T (2001) Proc 12th IEEE Int Symp Appl Ferroelectrics, Hawaii, 30 July–2 Aug 2000, p 273 9. Egami T, Rosenfeld HD, Hu R (1992) Ferroelectrics 136:15 10. Dmowski W, Akbas MK, Davies PK, Egami T (2000) J Phys Chem Solids 61:229 11. Shannon R (1976) Acta Cryst A 32:751 12. Grinberg I, Rappe AM (2004) Phys Rev B 70:220101 13. Husson E, Chubb M, Morell A (1988) Mat Res Bull 23:357 14. Cross LE (1987) Ferroelectrics 76:241 15. Egami T, Teslic S, Davies PK, Chen I-W, Chen H (1998) Ferroelectrics 206–207:231 16. Akbas MA, Davies PK (1997) J Am Ceram Soc 80:2933 17. Dmowski W, Akbas MK, Davies PK, Egami T (2000) J Phys Chem Solids 63:15 18. Jeong I-K, Darling TW, Lee JK, Proffen Th, Heffner RH, Park JS, Hong KS, Dmowski W, Egami T (2005) Phys Rev Lett 94:147602 19. Bovtun V, Kamba S, Pashkin A, Savinov M, Samoukhina P, Petzelt J, Bykov I, Glinchuk M (2004) Ferroelectrics 298:23 20. Gehring PM, Wakimoto S, Ye Z-G, Shirane G (2001) Phys Rev Lett 87:277601 21. Wakimoto S, Stock C, Birgeneau RJ, Ye Z-G, Chen W, Buyers WJL, Gehring PM, Shirane G (2002) Phys Rev B 65:172105 22. Bussmann-Holder A, Bishop AR, Egami T (2005) Europhys Lett 71:249

Struct Bond (2007) 124: 89–118 DOI 10.1007/430_2006_046 © Springer-Verlag Berlin Heidelberg 2007 Published online: 27 January 2007

Ferroelectricity of SrTiO3 Induced by Oxygen Isotope Exchange Mitsuru Itoh (u) · Hiroki Taniguchi Materials and Structures Laboratory, Tokyo Institute of Technology, 4259 Nagatsuta, Midori, 226-8503 Yokohama, Japan [email protected] 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 3.1 3.1.1 3.1.2 3.1.3 3.1.4 3.2 3.2.1 3.2.2 3.2.3 3.3 3.3.1 3.3.2

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dielectric Properties . . . . . . . . . . . . . . . . . . . . . . . . . Composition Dependence of ε versus T under Ambient Pressure ε versus T under Hydrostatic High Pressure . . . . . . . . . . . . Heat Capacity Measurement . . . . . . . . . . . . . . . . . . . . . D–E Loop and Pyroelectricity . . . . . . . . . . . . . . . . . . . . Optical Measurement . . . . . . . . . . . . . . . . . . . . . . . . . Optical Birefringence . . . . . . . . . . . . . . . . . . . . . . . . . Light Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optical Birefringence and Second Harmonic Generation . . . . . Structural Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . Neutron and X-ray Measurements . . . . . . . . . . . . . . . . . . NMR Measurements for Ti and Sr . . . . . . . . . . . . . . . . . .

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Abstract Typical quantum paraelectric SrTiO3 becomes quantum paraelectric by oxygen isotope exchange. Apparently the change in the Curie temperature Tc is explained by the criteria of the theory for quantum ferro/paraelectrics. Tc for an isotope-exchanged sample depends only on the average mass of oxygen. Ferroelectricity for the oxygen isotope-exchanged SrTiO3 results mainly from the condensation of the Slater-type soft mode. However, an order–disorder component exists in addition to the displacive soft mode component. Raman spectra and NMR results suggest that ferroelectric domains do not develop through the crystal-like normal ferroelectrics. The role of the lattice defect must be elucidated and related to the phase transition mechanism in future works. Oxygen isotope exchange is considered the best way to understand the phase transition mechanism of SrTiO3 because this exchange may not accompany the formation of random ﬁelds or defects. Keywords SrTiO3 · Oxygen isotope exchange · Quantum paraelectricity · Quantum ferroelectricity · Soft mode dynamics

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Abbreviations FE Ferroelectric CTO CaTiO3 STO SrTiO3 KTO KTaO3 BTO BaTiO3 STO16 SrTi16 O3 STO18 SrTi18 O3 STO18-100x SrTi(16 O1–x 18 Ox )3 SCT Sr1–x Cax TiO3 SCT(0.007) Sr0.993 Ca0.007 TiO3

1 Introduction The concept of quantum ferroelectricity was ﬁrst proposed by Schneider and coworkers [1, 2] and Opperman and Thomas [3]. Shortly thereafter, quantum paraelectricity was conﬁrmed by researchers in Switzerland [4]. The real part of the dielectric susceptibility of KTO and STO, which are known as incipient ferroelectric compounds, increases when temperature decreases and becomes saturated at low temperature. Both of these materials are known to have ferroelectric soft modes. However, the ferroelectric phase transition is impeded due to the lattice’s zero point vibration. These materials are therefore called quantum paraelectrics, or quantum ferroelectrics if quantum paraelectrics are turned into ferroelectrics by an external ﬁeld or elemental substitution. It is well known that commercially available single crystal contains many defects, which can include a dipolar center in the crystal. These dipolar entities can play a certain role in STO. The polar nanoregion (PNR; originally called the polar microregion) may originate from the coupling of the dipolar entities with the lattice [5–7]. When STO is uniaxially pressed, it turns into ferroelectrics [7]. In 1999, we succeeded in our experiments with oxygen isotope exchange for STO and found that the isotope-exchanged STO (STO18) [8, 9] turned into a ferroelectric. Those studies attracted much attention in the ﬁeld of ferroelectrics and became a milestone of quantum ferroelectrics and quantum paraelectrics. Quantum Paraelectricity According to the concept of the displacive-type ferroelectric phase transition [10], an increase in the dielectric constant corresponds directly to the softening of the IR-active transverse phonon. When the crystal can be regarded as an assembly of the vibrators of normal coordinates, the soft phonon

Ferroelectricity of SrTiO3 Induced by Oxygen Isotope Exchange

frequency is given by: ω2 = ω2b + g 4/ /ωµk nµk + 1/2 nµk = 1/ exp hωµk /kB T – 1 ,

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(1) (2)

where ωb is the bare phonon frequency and is an imaginary number due to the lattice instability. The second term is a contribution of ﬂuctuation and is temperaturedependent. When we represent the ﬂuctuation by one phonon frequency, ω1 = kB T1 /, and use the Lyddane–Sachs–Teller (LST) relation, the dielectric constant is given by: ε = B/ T1 /2 coth T1 /2T – T0 . (3) This equation is known as a Barrett formula. The structural phase transition for a quantum paraelectric was handled by the following model Hamiltonian [2]: H = T + Vharm + Vanharm ,

(4)

with a kinetic energy T: 1 2 T= Pl α , 2M n

l

(5)

α=1

and a harmonic potential energy: A 2 Xlα – C Xlα Xl α – h Xlα . 2 n

Vharm =

l

n

α=1

ll α=1

n

l

(6)

α=1

Here, l denotes the lattice sites, while Plα and Xlα are the α components of the n-dimensional momentum and displacement vectors, respectively. M denotes the mass of particle A, C is the model parameter, and h is a homogeneous external ﬁeld. The bilinear interaction is restricted to the nearest neighbor only. For the anharmonic part of the interaction: n p Bp 2 Vanharm = Xlα , (7) 2pnp–1 l

α=1

the statistical mechanical treatment gives the following results. Parameter S was introduced to describe the various properties effectively by the relation: S = 2CZn – A ,

(8)

where Zn is the number of nearest neighbor sites. S < Sc does not give a ferroelectric phase transition and S = Sc gives Tc = 0 K. When S > Sc , Tc , Ps , and

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ε–1 should follow the relations: 1/2 Tc ∝ S – Sc 1/2 PS ∝ S – Sc –1 ε–1 T = 0 ∝ S – Sc ε–1 S ≈ Sc ∝ T 2 .

(9) (10) (11) (12)

Theory by Kvyatkovskii Recently, perfect softening of the ferroelectric TO mode was conﬁrmed by Takesada et al., so STO18 is considered to be a displacive-type ferroelectric [11]. Kvyakovskii [12] gave an explanation for the phase transition mechanism. Part of the theory is summarized below. The mean atomic mass m = (1 – x)m0 + xml , where m0 and ml are a lighter isotope mass (e.g., m0 = 16) and heavier isotope mass, respectively, ml = m0 + δm, and x is the fraction of the heavier isotope. The Landau expression of the free energy of the displacive FE as a function of temperature is given by using the amplitude of FE mode xf , which is taken as the order parameter: 1 F T, xf = kf T x2f + F ah {xf } , 2

(13)

where kf (T) is a generalized force constant corresponding to the FE soft mode and F ah includes all the anharmonic terms of the expansion. kf (T) is divided into two parts, kh and kah : (14) kf T = kh + kah T, m kah > 0 , where h and ah represent “harmonic” and “anharmonic” subscripts, respectively. kh is independent of temperature and atomic masses, whereas kah depends on both T and m. The anharmonic term kah is separated into two parts: the temperature-independent zero-point motion contribution kzp and the residual terms denoted as ∆kah T : (15) kah T = kzp + ∆kah T ∆kah (0K) = 0 , with kzp > 0 and ∆kah T ≥ 0. The Curie temperature Tc is deﬁned by: kf Tc = 0 or ∆kah Tc = – k0 , (16) where k0 = kh + kzp is the sum of the temperature-independent terms. Equation 16 has a physically reasonable solution only if k0 ≤ 0, i.e., kh ≤ – kzp < 0. The FE instability is given by the inequalities: k0 m0 = kh + kzp m0 > 0 , (17)

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and k0 m0 + δm = kb + kzp m0 + δm < 0 , whence the condition follows: ∂ ln kp k0 δm – > . ∂ ln m m=m0 m0 kzp m=m0

(18)

(19)

Condition Eq. 18 is discussed based on two approaches: the two-sublattice model and the FE soft mode dressed anharmonically, giving the inequality: k0 δm > . (20) 2m0 kp m=m0 Equation 19 can be written in terms of experimental data: ω20 δm > , 2m0 ω2zp

(21)

m=m0

where ω2 = k/m, ω0 = ωf (0 K), and ω2zp = ω2h + ω20 . This condition implies that (i) only the substitution of a heavy isotope for a light one may produce the FE transition, and (ii) this isotope effect can occur only in quantum paraelectrics with complete compensation of the harmonic contribution into ωf by the zero-point motion contribution. In the Slater-type mode, oxygen atoms oscillate with the largest relative amplitude. As a result, we ﬁnd that the criterion of the FE lattice instability in totally oxygen isotope-exchanged STO and KTO is: ω20 1 . < 2 ωzp 16 Using the experimental data from the literature: 1 ω20 for STO 20 ≈ 2 ωzp 16 for KTO 1

(22)

(23)

O

Through the comparison of Eqs. 22 and 23, we ﬁnd that STO can be a ferroelectric by the oxygen isotope exchange. Using the values of constants characterizing the dispersion law of the (f, q) phonon branch, Kvyatkovskii got ωf,m = 448 K and Tc 18 O = 34 K. The effect of oxygen isotope exchange on the FE transition in STO has been also discussed by Bussmann-Holder using the shell model [13]. In this review, we summarize the recent developments in ferroelectricity in oxygen-isotope-exchanged STO.

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2 Experimental Based on the diffusion coefﬁcient data of oxygen, one can calculate the exchange rate of 16 O by 18 O for STO. For example, a 0.3-mm-thick STO plate needs about 700 h at 1273 K for a complete exchange. This was demonstrated by Itoh et al. in 1999 [8]. Due to the structural multidomaining problem below 104.5 K in STO, the STO plate was always cut into a rectangular shape, typically

with the dimensions of 7 mm// [001]c , 2 mm// [110]c and 0.3 mm// 110 c . With this shape, the plate could avoid multidomaining below 104.5 K [14]. For the dielectric measurement, the electrode was formed by the evaporation of about 100 nm of Cu, followed by Au evaporation on it. The temperature dependencies of the dielectric constant and the pyroelectricity were measured using an HP4284A LCR meter and a Keithley 6517 electrometer/high-resistance meter, respectively, in the Quantum Design physical properties measurement system (PPMS). Heat capacity was measured using the Quantum Design PPMS heat capacity module and adiabatic calorimeter of Prof. Oguni’s laboratory at the Tokyo Institute of Technology. Raman spectra were measured using a Jovin Yvon T64000 Raman spectrometer. Optical birefringence was measured by a handmade system based on the Sernant method.

3 Results 3.1 Dielectric Properties 3.1.1 Composition Dependence of ε versus T under Ambient Pressure Figure 1 shows ε (T) and 1/ε (T) plots at 1 bar for both STO18-92 and STO16 over an extended temperature range [15]. For soft-mode ferroelectrics, ε (T) in the paraelectric phase is determined by the T dependence of the soft-mode frequency ωs because the two quantities are connected by an LST relation such that ω2s ε = constant. This relationship is well established for STO. The ferroelectric soft mode in STO, a long-wavelength transverse optic phonon, consists primarily of vibrations of the Ti4+ ions against their surrounding oxygen octahedra. Clearly, replacing 16 O by 18 O should reduce the frequency of this mode. Speciﬁcally, the ratio of the frequencies is related to the ratio of the effective masses (µ) of the Ti – O6 octahedral units by: 1/2 ω18 /ω16 = µ16 /µ18 . (24)

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Fig. 1 The ε (T) responses of STO18 and STO16 at 1 bar over an extended range of T. Also shown is 1/ε (T). The insets show an expanded view of ε (T) at low T and the pressure dependence ε (P) for the two crystals at 293 K [15]

For the Slater mode, it is readily shown that ω18 /ω16 = 0.97, i.e., a 3% decrease in ωs on complete 18 O substitution in STO. Our dielectric data accurately conﬁrm this prediction, where ε is plotted versus pressure at 293 K and the ε curve for STO18 is 6% higher than that of the STO16 curve. Speciﬁcally at 1 bar, ε18 = 348 and ε162 = 328 so that ε18 /ε16 = 1.06, i.e., a 6% enhancement, which, from ε ∝ 1/ωs , implies a 3% decrease in ωs , as calculated. Figure 2 [16] shows the temperature dependence of the dielectric constant ε for STO18-100x and 17 O-containing samples STO18-3/17-58 and STO1828/17-43, where x represents the exchange, ratio of 16 O by 18 O. STO18-3/17-58 represents the sample containing 18 O 3% and 17 O 58%. It is clearly shown that ferroelectricity appears above the critical composition. Figure 3 shows the relationship between average oxygen mass, m0av , and Tc . In this ﬁgure, the data for the 17 O-containing samples are shown by open circles. The Tc s for the 16 O–18 O and 16 O–17 O–18 O systems lie on the same line. Thus all the data were ﬁtted by the single relation: 1/2 Tc = 21.0 m0av – 16.62 .

(25)

Equation 25 ﬁts Eq. 9, being the criteria of quantum ferroelectricity given by Morf et al. [2]. In Fig. 3a, the solid line shows the relationship between x and Tc determined by Bednorz and Müller for the SCT system [17]. In this ﬁgure,

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Fig. 2 Temperature dependence of dielectric constant ε for STO18-100x and containing samples STO18-3/17-58 and STO18-28/17-43 [16]

17 O-

Fig. 3 Variation of Tc with the average mass of oxygen for STO18-100x (closed circles) and 17 O-containing samples (open circles). The broken line shows ﬁtted results. Comparisons with the results of Ca substitution (a) and uniaxial pressure experiment (b) are shown

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97

critical compositions of m0av = 16.62 and x = 0.0018 were adjusted to come to the same point in the transverse axis. It is clear that the two data sets depart just after the critical compositions. Figure 3b shows the relationship between Tc and uniaxial stress for STO determined by Fujii et al. [7]. In contrast to the result in Fig. 3a, both the uniaxial pressure p and average oxygen mass m0av lie on the same line. This may suggest that the ferroelectricity induced by uniaxial pressure and oxygen mass change comes from the same origin, i.e., the structural strain in the ferroelectric phase in oxygen-isotope exchanged samples is nominally identical to that induced by the application of uniaxial stress. The critical exponent γ , which is obtained by ﬁtting ε data on the higher temperature side of Tc in the formula: γ ε–1 = c T – Tc , (26) is shown in Fig. 4a [18]. The γ value increases near xc and decreases with distance from xc . These results suggest that the STO18-100x samples with x > xc are quantum ferroelectrics. Figure 4b shows the relationship between ε–1 (T = 0) and x for the STO18-100x system. This result clearly shows the linear relationship is the same as that in Eq. 11. In addition to STO, we carried out the oxygen isotope exchange for BTO and CTO. In the case of BTO, the exchanged sample showed an increase in Tc of only 0.9 K. In contrast, no change was observed in CTO [8]. We must refer to the sensitiveness of the STO crystal. As mentioned in the previous section, STO crystal is very sensitive to stress. Even though STO

Fig. 4 Comparison of the dependence of the critical exponent γ for ε–1 = c(T – Tc )γ (a) and ε–1 (0 K) on x (b) for the 16 O–18 O system STO18-100x [18]

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Fig. 5 Comparison of the measured dielectric constant using two electrodes of thick ﬁred Au and thin Cu (vaporized) + Au (sputtered) [19]

crystal is cut into the rectangular shape described before, it is easily structurally multidomained below 104.5 K by subtle origins, such as the processes of greasing and electrode formation. This is demonstrated in Fig. 5 [19, 20]. For the STO18-99 sample, we attached two kinds of electrodes, thick Au (0.1 mm) and evaporated Cu + Au (200–300 nm). Figure 5 shows that the thick electrode generates a strain on the STO surface due to the difference in thermal expansion coefﬁcients between the electrode material and STO. The strained sample gives an apparently lower Tc and a subsequently higher ε value. This point is also discussed in Sect. 3.2.1. 3.1.2 ε versus T under Hydrostatic High Pressure Figure 6 shows the inﬂuence of the pressure of ε (T) on both STO18-92(a) and SCT(0.007) [21]. We ﬁrst note the large shift in the transition to lower temperature for STO18. The initial slope is dTc dP ∼ = – 20 K/kbar, a large effect. Second, there is a large decrease in the amplitude of the peak with pressure. At 0.70 kbar, the transition is completely suppressed, and the ε (T) response closely resembles that of STO16 at 1 bar. These pressure effects are characteristic of displacive ferroelectrics in the quantum regime and can be understood in terms of the soft-mode theory. The situation is similar for SCT(0.007), as shown in Fig. 6b. In the case of SCT(0.007), ferroelectricity completely disappears at 0.5 kbar. Figure 7 shows the temperature–pressure phase diagrams for STO18-92 and SCT(0.007) [21]. The data show clear evidence that Tc for STO18 vanishes with an inﬁnite slope, i.e., dTc /dp → – ∞ as Tc → 0 K. This is a requirement of the third law of thermodynamics for both ﬁrst- and second-order phase

Ferroelectricity of SrTiO3 Induced by Oxygen Isotope Exchange

99

Fig. 6 Inﬂuence of pressure on the ε (T) responses of STO18 and SCT(0.007) at 10 kHz [15]

Fig. 7 Temperature–pressure phase diagrams of STO18 and SCT(0.007) [15]

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transitions, and it establishes the nature of the transition. The solid line is the ﬁt of the Tc (p) data to the equation: 1/2 Tc = A 1 – p/pc ,

(27)

with A = 23.9 K and pc = 0.69 kbar. This result is also consistent with Eq. 4. It can be seen that the observed response obeys this expression quite well, again underscoring the normal ferroelectric nature of STO18. A contrasting result is obtained for SCT(0.007), as shown in Fig. 7. The linear relation between Tc and p, that is, the ﬁnite value of dTc /dp at 0 K, is a typical characteristic of the relaxor or glass, such as RADP [22]. One of the consequences of the suppression of the phase transition is the presence of a special critical point, Tc = 0 K. This point, called the quantum displacive limit, is characterized by special critical exponents. Its presence gives rise to classical quantum crossover phenomena. Quantum suppression and the response at and near this limit, Tc = 0 K, have been extensively studied on the basis of lattice dynamic models solved within the framework of both classical and quantum statistical mechanics. Figure 8 is a log–log plot of the ε (T) results for STO18 [15]. The expectation from theory is that in the quantum regime, γ = 2 at 0.7 kbar, after which γ should decrease. The results in Fig. 8 quantitatively show the expected behavior; however, γ is < 2 at 0.70 kbar. Despite the difference in the methods to suppress Tc in STO18, the results in Fig. 4a and Fig. 8 are quite similar. As shown in the results in Fig. 3b, uniaxial pressure also can be a critical parameter S for the evolution of ferroelectricity in STO.

Fig. 8 Log–log plots of (ε – ε∞ ) versus T – Tc for STO18 [15]

Ferroelectricity of SrTiO3 Induced by Oxygen Isotope Exchange

101

In contrast to the results for the systems KTa1–x Nbx O3 and K1–y Nay TaO3 [23], the STO system did not show γ = 2 at the displacive limit by either isotope exchange or pressure. This point should be discussed in the future. 3.1.3 Heat Capacity Measurement Figure 9 shows the heat capacities for STO16 and STO18-95 measured by the relaxation method (M. Itoh, unpublished data). Using both adiabatic and relaxation heat capacity measurements, we tried but failed to detect any thermal anomaly resulting from the phase transition at or around Tc = 25 K. This means that the structural change involved in the ferroelectric transition is inﬁnitesimal at that temperature. This result is consistent with the powder Xray and neutron diffraction structural analyses, in which no clear structural change occurred within the experimental limit. On the other hand, the zone boundary phase transition at 104.5 K for STO16 is found to be enhanced to 108.5 K for STO18. This means that the structural instability of antiferrodistortive distortion increases due to the isotope exchange. This enhancement has been conﬁrmed by both the measurement of heat capacity and the study of optical birefringence.

Fig. 9 Comparison of heat capacities for STO16 and STO18-95. Inset shows enlarged plot of C/T versus T near 100 K

3.1.4 D–E Loop and Pyroelectricity Figure 10 shows the D–E loops for STO18-92 [8]. Figure 11 compares the D–E loops for STO16 and STO18-92 [8]. The D–E loops for STO16 and

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Fig. 10 D–E loops for STO18 measured at 18.1 K (a), 20.2 K (b), 24.2 K (c), 28.6 K (d), 35.0 K (e), and 45.5 K (f). 1 division = 1.04 kV/cm and 0.901 µC/cm2 for E and D, respectively [8]

Fig. 11 Comparison of D–E loops for STO16 (a) and STO18 (b) measured at almost the same conditions. In a, one division = 1.11 kV/cm and 0.849 µC/cm2 for E and D, respectively. In b, one division = 1.04 kV/cm and 0.901 µC/cm2 for E and D, respectively. T = 18.0 K [8]

STO18 reveal enhanced electric polarization in STO18. This is attributed to the evolution of ferroelectricity at 25 K in STO18. Figure 12 shows the temperature variation of the pyroelectricity for STO18-100x [24]. STO crystal

Ferroelectricity of SrTiO3 Induced by Oxygen Isotope Exchange

103

Fig. 12 Pyroelectricity of STO18-100x single crystals. The absolute value of the poling ﬁeld is 24.8 kV/m [24]

itself contains many defects, cation vacancies and oxygen vacancies, and dislocations, and thus it is very sensitive to the external electric ﬁeld. Once the external electric ﬁeld is applied to STO, electric dipoles are introduced around the defects into the crystal and persist to higher temperatures. In the present experiment, the electric ﬁeld was applied just near the temperature, Tc – 2 K < T < Tc + 1 K, during the cooling process. However, large induced polarization persists up to 200 K, as shown in Fig. 12. This result suggests that the induced polarization originating from the defects overlaps onto the contribution from the ferroelectrics. This result suggests that the contribution of the induced polarization must always be taken into account in high-voltage measurements. Finally, the real part of the dielectric constant ε cannot detect the anomaly originating from the defects. However, the imaginary part of the dielectric constant ε is sensitive to such defects. Irrespective of the oxygen isotope exchange, anomalies of ε are observed around 10 and 80 K [24].

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3.2 Optical Measurement 3.2.1 Optical Birefringence Figure 13 shows the optical birefringence measured on the (001)c and (110)c faces [19]. According to the indicatrix deformation approach [25], the optical birefringence, ∆n, for the (110)c face can detect the antiferrodistortive transition, the square of long-range order P2 , and its ﬂuctuation term δP2 in STO by: (28) ∆n = R T + ρ gij P2 + δP2 ,

Fig. 13 ∆n(T) of (110)c (a) and (001)c (b) plates in the range of 4.2 K to 120 K for the specimens with 7 [001] × 2 110 × 0.3 [110] mm3 and 10 [001] × 2 110 × 0.5 [110] mm3 dimensions. The inset in a shows the effects of electrodes attached to the 7 × 0.3 mm3 face of 110 c on ∆n. The inset in b shows the enlarged part near ∆n ≈ 0 [19]

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where gij is the quadratic electrooptic coefﬁcient and R(T) is a regular function of temperature. In the case of the (001)c face: 1 3 (29) ∆n = nx – ny = n0 g11 – g12 Px2 – Py2 . 2 As shown in Fig. 13a, ∆n for the (110)c face is composed of two contributions from the antiferrodistortive phase transition and the ferroelectric transition (see data for 7 × 2 × 0.3 mm3 ). On the other hand, only the ferroelectric transition is seen for the (001)c face. The inequality Px = Py means the breaking of symmetry in the (001)c plane. Therefore, the symmetry of the ferroelectric phase is below orthorhombic. The detail is discussed more in the next section. Another clear result in Fig. 13 is the effect of sample size on ∆n. When we compare two sizes, 7 × 2 × 0.3 mm3 and 10 × 2 × 0.5 mm3 , the following two problems are seen for the latter sample: 1. The depression of ∆n below Tc as shown in Fig. 13a 2. The appearance of a step at Ta and stepping up of ∆n at a higher temperature near Tc , as shown in Fig. 13b Both of these are considered to arise from the different orientation of ferroelastic domains below Ta and local distortion due to multidomaining. The inset in Fig. 13a shows ∆n for the (110)c face measured under three conditions: (i) no electrode, (ii) with sputtered Au, and (iii) with pasted Ag. These results imply that even the sputtered Au on the 7 × 0.3 mm2 side can affect multidomaining below Ta . These results suggest that attention must be paid to ensure the single-domain state below Ta . 3.2.2 Light Scattering It has been widely recognized that the light scattering technique yields essential information on a dynamic mechanism of ferroelectric phase transition because it clearly resolves the dynamics of the ferroelectric soft mode that drives the phase transition. Quantum paraelectricity is caused by the nonfreezing of the soft mode. Therefore, the isotope-exchange effect on the soft mode is the key to elucidating the scenario of isotopically induced ferroelectricity. Figure 14 shows the result of a Brillouin scattering experiment in the vicinity of Tc [11]. Closed circles and open circles below Tc indicate the modes split from the doubly degenerated ferroelectric soft Eu mode. The closed circles above Tc denote the frequency of the doubly degenerated soft Eu mode in the paraelectric phase. The results clearly show a softening of the soft mode toward zero frequency at Tc following the Curie–Weiss law. The soft mode remains underdamped even at Tc . Generally, a soft mode is heavily damped in the vicinity of Tc , e.g., as for PbTiO3 , which are typical displacive-type

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Fig. 14 Temperature dependence of modes observed in STO18. The soft Eu mode (closed circles above Tc ) in the tetragonal D4h phase is divided into two modes presented by open circles (below Tc ) and closed circles (at Tc ). Closed squares and open squares denote the modes split from the doubly degenerated Eg mode. Closed triangles above Tc indicate the Raman-inactive A2u mode observed by local symmetry breakdown [11]

ferroelectrics. The complete softening of the underdamped soft mode observed here indicates an ideal displacive-type ferroelectric phase transition of STO18. The spontaneous polarization of STO18 is induced intrinsically by the softening of the soft polar mode. The two soft modes below Tc are assigned as modes A1 and B2 by the Raman selection rule. The doubly degenerated Eg mode is split into A2 (open squares) and B1 (closed squares) modes with the ferroelectric phase transition. The onset of the splitting is associated with softening of the soft mode. The assignment of all modes is based on the speculation of C2v symmetry in the ferroelectric phase. It should be noted that the Raman-inactive soft mode is observed in the temperature region above Tc . A spectral shape completely different from that of the Lorentz-type peak function indicates the defect-induced Raman scattering (DIRS) in the paraelectric phase of STO18. When centrosymmetry is locally broken in the paraelectric phase, the nominally Raman-inactive soft mode is optically activated locally to induce DIRS in the soft mode. The system of STO18 includes several types of inhomogeneity. Inhomogeneity of the system sometimes has a non-negligible effect on a phasetransition property. Therefore, it is quite important to clarify the role of the inhomogeneity in STO18. The role of inhomogeneity in STO18 was caused by point defects such as oxygen and cation vacancies. The oxygen vacancy breaks local neutrality to induce a local dipole. The high susceptibility of the media enhances the formation of a local non-centrosymmetric area around it.

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The above-mentioned DIRS is generated by this mechanism. The other type of inhomogeneity is the mass distribution of oxygen atoms. In isotopically exchanged SrTiO3 , it seems natural that the oxygen isotope is statistically dispersed in the system and that the local average mass would be unevenly distributed. Even though the ferroelectric phase transition is driven by the soft mode intrinsically, these factors have a non-negligible effect on the phase transition property of STO18. Especially in partially exchanged STO18, the effect of mass distribution appears to be more enhanced. Figure 15 presents the Raman spectra of STO16 and STO18-23 observed in the low-temperature region. The scattering geometry is x(yy)-x in the tetragonal notation. As shown in Fig. 15a [27], broad Raman bands indicated by arrows shift to a low-frequency region with decreasing temperature. These broad components are caused by the DIRS of the Raman-inactive soft Eu mode. The sharp peak observed at 45 cm–1 is a structural soft A1g mode related to the antiferrodistortive phase transition at Ta . In the low-temperature region of STO18-23, the broad component develops dramatically into a sharp

Fig. 15 Raman scattering spectra of a STO16 and b STO18-23 observed in x(yy)-x scattering geometry (tetragonal notation). Arrows in a indicate positions of DIRS signal and its higher harmonic component [27]

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peak as shown in Fig. 15b. Since the isotope exchange rate of STO18-23 is still below the critical rate of 33%, the matrix of STO18-23 remains centrosymmetric until 0 K. The development of the sharp peak indicates the growth of a local symmetry breakdown region (LSBR), because the spectral intensity of the soft Eu mode is proportional to the total volume of LSBR in the radiation area of incident light. The sharpening of the spectra also suggests the enlargement of LSBR as reported in [26]. On the other hand, the spectrum of STO18-23 at 29.5 K exhibits a structure similar to that of STO16 at 29.1 K. It should be noted that the LSBR is deﬁnitely non-centrosymmetric but is not the one expected to be in the ferroelectric phase, since the observed soft mode does not freeze at any ﬁnite temperature. Figure 16 shows Raman spectra observed in STO18-32, whose exchange rate is critically close to, but still less than, the critical one [27]. The spectral intensity of the soft Eu mode, as seen in the ﬁgure, is furthermore enhanced with the increase in the exchange rate in the low-temperature region, indi-

Fig. 16 Raman scattering spectra of STO18-32 observed in x(yy)-x scattering geometry. The inset shows a magniﬁed view of the temperature dependence of the soft Eu mode spectrum in a low-temperature region below 15.5 K. Arrows in the inset indicate peak positions of the soft Eu mode spectra [27]

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cating further development of LSBR. At 1.2 K, the spectral height is about twice that observed in STO18-23. The spectrum at 27.9 K, on the other hand, shows almost the same structure as those of STO16 and STO18-23 in the hightemperature region. In the inset of Fig. 16, the temperature dependence of the soft mode spectrum is presented by an extended view. The peak positions, indicated by arrows, show obviously a hardening behavior of the soft mode in the low-temperature region. Generally, the hardening of the soft mode is observed in the ferroelectric phase. However, the ferroelectric phase is expected not to arise in STO18-32, because the exchange rate is still less than the critical value. Moreover, the observed soft mode does not undergo any freezing until 0 K. In the case of the displacive-type ferroelectric PbTiO3 , whose ferroelectric phase is driven by the ferroelectric soft mode, a strong damping of lattice vibration obscures the perfect softening of the soft mode in the vicinity of Tc . In the case of STO18-32, however, the established sharp peak indicates the underdamped feature of the soft mode, and obviously shows an incomplete softening of the soft mode. The temperature dependencies of the soft Eu mode frequency (closed symbols) and spectral linewidths (open symbols) in STO16, STO18-23, and STO18-32 are presented in Fig. 17a–c [27]. The crosses in Fig. 17a denote the temperature dependence of the soft Eu mode frequency observed by hyperRaman scattering. The results shown in Fig. 17 clearly indicate a decrease in

Fig. 17 Temperature dependencies of the Eu mode observed in STO16 (closed circles), STO18-23 (closed squares), and STO18-32 (closed triangles) observed in the scattering geometry x(yy)-x. Crosses indicate the results for STO16 obtained by the hyper-Raman scattering experiment. The corresponding open symbols denotes the half-width at half maximum of the soft Eu mode spectrum of each specimen [27]

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the frequency of the soft Eu mode due to the isotope exchange. In STO18-23, the minimum frequency is depressed by 3 cm–1 compared to that in STO16, maintaining quantum paraelectric behavior until 0 K. The soft Eu mode in STO18-32 increases its frequency after the minimum value of 7 cm–1 at 8 K without any freezing. The spectral analysis clearly indicates incomplete softening of the soft mode in STO18-32, as pointed out above. The noticeable feature is that the linewidth of the soft Eu mode in STO18-32 undergoes a peak-like anomaly before the frequency minimum at 8 K. This incomplete softening with anomalous spectral broadening implies novel soft mode dynamics at the initial stage of the isotope-induced ferroelectricity, and it would manifest the effect of the mass distribution of oxygen atoms. Figure 18 [27] shows Raman spectra of STO18-23 and STO18-32 observed in the scattering geometry x(yz)-x, which allows us to detect the doubly degenerated Eg mode in D4h tetragonal symmetry. A conspicuous spectral change between STO18-23 and STO18-32 is seen in the spectra in the lowtemperature region. As shown in Fig. 18b, the Eg mode, 15 cm–1 at 17 K, begins shifting to low frequency and increases its spectral intensity as T decreases. Furthermore, a new peak appears at 17 cm–1 , clearly indicating the occurrence of symmetry lowering in the low-temperature region of STO18-32.

Fig. 18 Raman scattering spectra of a STO18-23 and b STO18-32 observed in x(yz)-x scattering geometry in a low-temperature region [27]

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Finally, two established peaks and a small peak between them are observed in the spectrum of STO18-32 at 1.2 K. In the spectra of STO18-23 presented in Fig. 18a, on the other hand, the Eg mode exhibits only a marginal broadening as temperature decreases. Figure 19 shows the temperature dependence of the peaks in STO18-32 observed in the x(yz)-x scattering geometry [28]. Compared to the previous results for the fully exchanged STO18, the present spectral feature is quite similar to that in the ferroelectric phase. This strongly indicates the occurrence of ferroelectric symmetry in the low-temperature region of STO18-32 even though its exchange rate is less than xc . In STO18-32, the occurrence of a ferroelectric region at the ﬁnite temperature is supported by the following results: 1. Hardening of the soft Eu mode in x(yy)-x scattering geometry 2. Shifting of the Eg mode in x(yz)-x scattering geometry 3. Appearance of a new peak in x(yz)-x scattering geometry However, the incomplete softening of the soft mode still raises fundamental questions about the mechanism underlying the occurrence of a ferroelectric region in STO18-32. The most plausible interpretation for this phenomenon is the inhomogeneity of the soft mode dynamics.

Fig. 19 Temperature dependence of the modes observed in x(yz)-x scattering geometry [28]

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In the weakly exchanged SrTiO3 , the soft mode dynamics would become spatially inhomogeneous due to the inhomogeneous mass distribution of oxygen atoms. The softening is considered to be locally enhanced by the reduction of quantum suppression in the region where the local average mass is relatively large. On the other hand, the enhancement is not so large in the other region. In STO18-32, whose exchange rate is critically close to xc , the local average exchange rate xloc would exceed xc . In such a region, the quantum suppression is no longer effective, and so the local ferroelectric region is induced at the ﬁnite temperature. When the radiated area of Raman scattering includes the mass distribution mentioned above, the observed spectra would be composed of a freezing soft mode (in the region of xloc > x) and a non-freezing soft mode (in the region of xloc < x). In the present stage, this mechanism is the most plausible origin of the apparent incomplete (rounded) softening behavior of the soft mode observed in weakly exchanged STO18, as is the case with STO18-32. Indeed, the apparent incomplete softening explains the rounded temperature dependence of the dielectric constant of weakly exchanged STO18. This is why a dielectric constant is related not to individual local dynamics but to spatial summation of the dynamics. The important point here is that ferroelectricity itself is intrinsically induced by a long-range soft phonon, as evidenced by the Brillouin scattering experiment. The long-range phonon dynamics are locally modiﬁed by the mass inhomogeneity in imperfectly exchanged STO18. Embedded local dipolar entities due to, for example, oxygen vacancy, form local symmetry breakdown areas around themselves from far above Tc . 3.2.3 Optical Birefringence and Second Harmonic Generation In the ferroelectric phase of strongly isotope-exchanged STO18-x, though this phase is found to be induced intrinsically by the freezing of the soft Eu mode as reported in the light scattering study, observations of second harmonic generation (SHG) and optical birefringence indicate an underlying structural heterogeneity. Figure 20 shows the angular dependencies of SHG intensity observed at 6 K in various conditions: measured after poling parallel to the [010]t direction (Fig. 20a) and measured at three different positions without poling (Fig. 20b–d) [29]. The SH intensities after poling are well reproduced by the calculated values (solid lines in Fig. 20a) expected from m2m point symmetry. Therefore, the symmetry after poling can be reasonably attributed to the orthorhombic symmetry. In the unpoled state, on the other hand, the SH intensities change drastically, indicating a loss of the high symmetry in the poled state. As shown in Fig. 20b–d, the peaks at Φ = 0 and 180, which are observed in the poled state, are obviously lost. The rigorous tensor analysis assuming eight triclinic polar domains gives a satisfactory ﬁt for the observed

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intensities, as denoted by the solid lines in Fig. 20b–d. This result suggests the existence of a random ﬁeld in strongly exchanged STO18-100x. The temperature dependence of linear birefringence ∆n = nc – na with the application of various electric ﬁelds is presented in Fig. 21a [30]. The insets (Fig. 21b and c) show micrographs observed in an area of 0.6 × 0.5 mm2 . They show that complete compensation of the birefringence is achieved only under oblique orientation of the compensator. This result supports the triclinic symmetry below Tc deduced by the SHG observation. The SH intensity markedly increases by the application of an electric ﬁeld in the low-temperature region below Tc , suggesting the alignment of the fraction of orthorhombic domains on the c plane. Furthermore, the non-saturating intensity even at E = 140 kV/m implies an additional rotation of the polarization. The hightemperature tail above Tc is caused by the tetragonal distortion related to the antiferrodistortive phase transition at Ta . SH images of STO18-95 observed during zero-ﬁeld cooling are shown in Fig. 22 [31]. Macroscopic centrosymmetry of the paraelectric phase does not generate an SH signal. However, an inhomogeneous texture of an SH signal is obviously seen in the temperature region above Tc . This result strongly indicates the development of local symmetry breakdown before the onset of

Fig. 20 Angular dependencies of Siy and Siz at 6 K measured after poling parallel to the [010]t direction (a) and at three different positions without poling (b, c, and d) [29]

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Fig. 21 a Temperature dependence of linear birefringence of STO18 observed within the (001)c /(110)c plane. Curves 1–5 were measured at transverse electric ﬁelds, E//[110]c , with E = 0(1), 35(2), 70(3), 110(4), and 140 kV/m(5). b and c are polarized micrographic images [30]

Fig. 22 SHG images of STO18 observed in the cooling process. Bright areas present SHGactive regions [31]

the ferroelectric phase, and probably is related to the lower symmetry rather than to the orthorhombic one observed in the ferroelectric phase of unpoled STO18-100x.

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3.3 Structural Analysis 3.3.1 Neutron and X-ray Measurements Uesu et al. [29] performed full-neutron Rietveld analysis at 300 K, 50 K, and 1.5 K for STO18-88 powder. At room temperature, they obtained a cubic Pm3m structure, with associated normal thermal parameters. At 1.5 K and 50 K, the patterns could be reﬁned using the tetragonal antiferrodistortive structure (I4/mcm). No anomaly in thermal parameters, which could be an indication of some possible disorder, was evidenced. Powder neutron diffraction data gave no sign of ferroelectric transition below Tc . X-ray diffraction studies of crystals and ceramics for STO18-88 samples also failed to detect any structural change below Tc . In contrast to these results, Noda’s group performed a neutron diffraction experiment for a single crystal of STO18-89. They found clear evidence of a change in Bragg intensity at the phase transition temperature, and suggested an orthorhombic or lower symmetry below Tc . Based on the space group Fm2m, they optimized the structure of the ferroelectric phase. This space group implies the Slater mode condensation at Tc . Recently, Ozaki et al. (unpublished data and [32]) performed white-beam X-ray topographic measurement of spontaneous strains in STO16 and STO18. From the splitting of the distances between the paired black and white stripes, they obtained a spontaneous strain ss = c/a – 1 for STO16 and STO18. When the data for both are compared, we see that ss for STO18 clearly shows a drop at Tc , meaning that the symmetry is lowered to below orthorhombic. At the present time, only the Laue topographic data evidences the structural change at 25 K, i.e., from tetragonal (I4/acd) to a lower symmetry, the distortion is deﬁnitely very small. 3.3.2 NMR Measurements for Ti and Sr Measurements of NMR for 47 Ti, 49 Ti [33], and 87 Sr [34, 35] were carried out for STO16 and STO18-96. Ti and Sr nuclear magnetic resonance spectra provide direct evidence for Ti disorder even in the cubic phase and show that the ferroelectric transition at Tc = 25 K occurs in two steps. Below 70 K, rhombohedral polar clusters are formed in the tetragonal matrix. These clusters subsequently grow in concentration, freeze out, and percolate, leading to an inhomogeneous ferroelectric state below Tc . This shows that the elusive ferroelectric transition in STO18 is indeed connected with local symmetry lowering and implies the existence of an order–disorder component in addition to the displacive soft mode [33–35]. Rhombohedral clusters, Ti disorder, and a two-component state are found in the so-called quantum paraelectric

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state of STO16. The concentration of the rhombohedral cluster is, however, not high enough to allow for percolation. 87 Sr quadruple-perturbed NMR data of STO show that local symmetry breaking also takes place at the Asite of the ABO3 perovskite lattice in the cubic phaseout only at the B-site, as observed before. The onset of the antiferrodistortive transition around Ta is accompanied by a huge splitting of the 87 Sr NMR line due to the formation of macroscopic 90◦ domains and an anomalous shortening of the spin-spin relaxation time T2 of the quadruple satellite transitions that persist in the tetragonal phase down to 30 K. This shows the presence of very low frequency ﬂuctuations of the phase transition order parameter. Rhombohedral polar clusters appear in the tetragonal matrix in both STO16 and STO18 below 70 K and percolate in STO18 below 25 K, leading to an inhomogeneous ferroelectric state.

4 Conclusion The evolution of ferroelectricity in oxygen isotope-exchanged STO was conﬁrmed by electrical, optical, and magnetic resonance methods. Tc for oxygen isotope-exchanged STO was conﬁrmed to follow the criteria of quantum ferroelectricity. The structural change may be very small, so only the optical technique could clearly detect it. Roughly speaking, the Slater-type soft mode condenses at Tc ; however, it is very hard to detect by the conventional structural analysis methods such as neutron and X-ray diffraction techniques. Below Tc , ferroelectric domains do not develop totally over the whole crystal,

Fig. 23 Schematic illustration of the development of LSBR. White background, shaded regions and hatched regions denote the paraelectric regions, non-ferroelectric LSBR and ferroelectric regions, respectively. The area surrounded by solid line in g represents the ferroelectric region with mesoscopic length scale

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possibly due to the random ﬁeld originating from the defects in the crystal. The development of LSBR and ferroelectric regions with a decrease in temperature and an increase in 18 O content is schematically depicted in Fig. 23. The remaining problem to be solved in the future is the symmetry of the ferroelectric phase. At the present time, orthorhombic or triclinic symmetry is proposed for the low temperature phase below Tc [29, 34]. The problem of the structural multidomaining below Ta makes it difﬁcult to reach a deﬁnite conclusion, as shown in Figs. 5 and 13. Measurements for defect-free and stress-free STO samples are indispensable for a deﬁnite conclusion about the symmetry of the ferroelectric phase of STO18. Finally, we can conclude that STO18 may be a typical soft mode ferroelectric. Acknowledgements The authors wish to express their thanks to Dr. Ruiping Wang of AIST for initial experiments on STO18; Profs. T. Yagi, Y. Tsujimi, and M. Takesada of Hokkaido University for the optical scattering experiments; Prof. Y. Noda of Tohoku University for the neutron scattering of single crystals; Prof. Y. Uesu of Waseda University for SHG measurements; Profs. T. Shigenari and K. Abe of the University of ElectroCommunications for Raman scattering measurements; Prof. K. Iio of the Tokyo Institute of Technology for optical birefringence measurements; Prof. W. Kleemann of the University of Duisburg for dielectric measurements; Prof. J. Dec of the University of Silesia for dielectric measurements; Profs. R. Blinc and B. Zalar of J. Stefan Institute for NMR measurements; and Prof. G.A. Samara and Dr. E. Venturini of Sandia National Laboratory for dielectric measurements under high pressure. Part of this work was ﬁnancially supported by a Grant-in-Aid for Scientiﬁc Research from the Ministry of Education, Culture, and Sports, and by the Japan Society for the Promotion of Science.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

Schneider T, Beck H, Stoll E (1976) Phys Rev B 13:1123 Morf R, Schneider T, Stoll E (1977) Phys Rev B 16:462 Oppermann R, Thomas H (1975) Z Phys B 22:387 Hochli UT, Wiebeil HE, Boatner LA (1977) Phys Rev Lett 39:1158 Uwe H, Yamaguchi H, Sakudo T (1989) Ferroelectrics 96:123 Uwe H (1985) Jpn J Appl Phys S24-2:513 Fujii Y, Uwe H, Sakudo T (1987) J Phys Soc Jpn 56:1940 Itoh M, Wang R, Inaguma Y, Yamaguchi T, Shan YJ, Nakamura T (1999) Phys Rev Lett 82:3540 Itoh M, Wang R (2000) Appl Phys Lett 76:221 Chaves AS, Barreto FCS, Ribeiro LAA (1976) Phys Rev Lett 37:618 Takesada M, Itoh M, Yagi T (2006) Phys Rev Lett 96:227602 Kvyatkovskii OE (2001) Solid State Commun 117:455 Bussmann-Holder A, Buttner H, Bishop AR (2000) J Phys Cond Mat 12:L115 Müller KA, Berlinger W, Capizzi M, Granicher H (1970) Solid State Commun 8:549 Venturini EL, Samara GA, Itoh M, Wang R (2004) Phys Rev B 69:184105 Itoh M, Wang R (2003) J Phys Soc Jpn 72:1310 Bednorz JG, Müller KA (1984) Phys Rev Lett 52:2289 Wang R, Itoh M (2001) Phys Rev B 64:174104

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19. Itoh M, Azuma T, Kyômen T, Iio K, Yamanaka K, Wang R (2004) J Phys Soc Jpn 73:1377 20. Azuma T, Iio K, Yamanaka K, Kyomen T, Wang R, Itoh M (2004) Ferroelectrics 304:77 21. Venturini EL, Samara GA, Itoh M, Kleemann W (2003) Fundamental physics of ferroelectrics 2003: (AIP conference proceedings) CP677:1 22. Samara GA (1991) Ferroelectrics 117:347 23. Rytz D (1980) Phys Rev B 22:359 24. Wang R (2000) PhD thesis, Tokyo Institute of Technology 25. Wang R, Itoh M (2001) Ferroelectrics 262:125 26. Yamanaka K, Wang R, Itoh M, Iio K (2001) J Phys Soc Jpn 70:3213 27. Taniguchi H, Yagi T, Takesada M, Itoh M (2005) Phys Rev B 72:064111 28. Taniguchi H (2006) PhD thesis, Hokkaido University 29. Zhang L, Kleemann W, Wang R, Itoh M (2002) Appl Phys Lett 81:3022 30. Dec J, Kleemann W, Boldyreva K, Itoh M (2005) Ferroelectrics 314:7 31. Uesu Y, Nakai R, Kiat JM, Menoret C, Kato N, Itoh M, Kyomen T (2004) J Phys Soc Jpn 73:1139 32. Ozaki T, Kusunose K, Sakaue H, Okamoto H, Kajiwara K, Suzuki Y, Chikaura Y (2005) Nucl Inst Methods Phys Res B 238:255 33. Zalar B, Lebar A, Seliger J, Blinc R, Laguta VV, Itoh M (2005) Phys Rev B 71:064107 34. Blinc R, Zalar B, Laguta VV, Itoh M Phys Rev Lett 94:147601 35. Laguta VV, Blinc R, Itoh M, Seliger J, Zalar B (2005) Phys Rev B 72:214117

Struct Bond (2007) 124: 119–147 DOI 10.1007/430_2006_042 © Springer-Verlag Berlin Heidelberg 2006 Published online: 1 December 2006

Evidence for Ferroelectric Nucleation Centres in the Pseudo-spin Glass System Rb1-x(ND4 )x D2PO4 : A 87Rb NMR Study Raymond Kind Institut für Quantenelektronik, Eidgenössische Technische Hochschule Zürich Hönggerberg, 8093 Zürich, Switzerland [email protected] 1 1.1 1.2 1.3

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Soft Mode Versus Thermally Activated Motion Small Excursion to the Roots . . . . . . . . . . Spin-Lattice Relaxation and Critical Behaviour Comparison with Thermally Activated Motion

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Abstract The transition region between long-range ferroelectric (FE) order and glass order in the phase diagram of the deuteron glass system Rb1–x (ND4 )x D2 PO4 (D-RADP-x) extends from about x = 0.20 to x = 0.32. In this region we have observed phase segregation between the two phase states using 87 Rb NMR. We assume that clusters of FE order nucleate with different local transition temperatures depending on the accidental Rb content at various places in the crystal. The size of the FE clusters must be above a critical value in order to exhibit individual transition temperatures (Tc ), as well as individual soft modes. Soft mode behaviour was unambiguously observed in the temperature dependence of the 87 Rb spin lattice relaxation (T ) well above the average T . We expect a cluster size of the 1 c order of 5 nm, clearly smaller than the optical wavelength, but well above the observation range of the Rb nucleus of about 1.4 nm. From our measurements we have determined the total FE and glass phase volume fractions as a function of temperature. The behaviour of the system is discussed for different crystal compositions in the coexistence region of the two phase states.

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Keywords Order–disorder transitions · Soft mode · Phase segregation · Nucleation · Takagi defects

Abbreviations ADP Ammonium dihydrogen phosphate AFE Anti-ferroelectric D-RADP Fully deuterated RADP EFG Electrical ﬁeld gradient FE Ferroelectric KADP Solid solution of KDP and ADP KDP Potassium dihydrogen phosphate KTN Potassium tantallate-niobate NMR Nuclear magnetic resonance NQR Nuclear quadrupole resonance PE Paraelectric RADP Solid solution of RDP and ADP RDP Rubidium dihydrogen phosphate T0, T4 High energy Takagi defects T1 Mobile Takagi defect with negative charge T2 Slater group T3 Mobile Takagi defect with positive charge

1 Introduction 1.1 General Features Diffuse phase transitions, as for example observed in mixed ferroelectrics, have been studied for the past three decades starting with the discovery of Smolenskii [1]. The most studied systems in this family are the perovskitetype relaxor ferroelectrics, see for example Cross [2, 3] or Kleemann [4]. The crucial question always to be asked in these systems is whether we are dealing with a long range cooperative phenomenon with a single, though smeared, phase transition temperature, or with a whole set of short range cooperative phenomena with a probability distribution of individual local phase transition temperatures. Such Tc distributions were suggested for many systems, e.g. for KTN by Kind and Müller in [5], where the temperature dependence of the dielectric constant was explained in this way. A distribution of local transition temperatures implies the coexistence of the paraelectric (PE) and the ferroelectric (FE) phase states in the transition region. An unambiguous identiﬁcation of phase coexistence can only be obtained from microscopic measurements that are able to detect the spatial correlation of the ordering, e.g. neutron scattering and NMR. Macroscopic methods, such as the

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determination of the spontaneous FE polarization Ps [6, 7], do not allow determination of the volume fractions of the involved phase states because Ps is a weighted average of local saturation polarizations, which can considerably differ from each other due to the varying substitutional disorder. The inﬂuence of defects on the saturation polarization is treated theoretically, e.g. by Levanyuk et al. [8]. 1.2 Static and Pseudo-static Properties Phase coexistence was also reported for the family of mixed hydrogen bonded ferroelectrics and anti-ferroelectrics. The random substitutional disorder between Rb and NH4 in the solid solution Rb1–x (NH4 )x H2 PO4 (RADP) and its deuterated analogue Rb1–x (ND4 )x D2 PO4 (D-RADP) leads to competing ferroelectric (FE) and anti-ferroelectric (AFE) interactions. This results in a phase diagram (Fig. 1) with a common paraelectric (PE) high temperature phase state, but ﬁve distinct low temperature phase states differing in the local conﬁguration and the correlation length of the proton ordering on the hydrogen bonds. The phase diagram is discussed in detail in [9]. In this contribution we focus on the region that is sandwiched between the FE and glass phase states, i.e. the range of x-values 0.20 ≤ x ≤ 0.35. In [10] we have shown that the low temperature glass state (0.35 ≤ x ≤ 0.65) consists predominantly of short range ordered AFE clusters with a mean correlation length of about 1 nm. The fact that no FE clusters were found was explained by the unfavourable ratio of electric surface to volume energy, which makes

Fig. 1 Phase diagram of D-RADP-X as obtained from 87 Rb NMR. The closed circles correspond to the to the mean Tc s. Note that all phase boundaries are diffuse [9]

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small FE clusters unstable. However, for x ≤ 0.30, FE clusters are unambiguously identiﬁed in the 87 Rb NMR spectrum. On lowering the temperature from the PE phase, long range ordered FE islands with individual phase transition temperatures grow out of Rb-rich nucleation centres, as we will demonstrate in this contribution. Furthermore we will show that this growth is limited by the NH4 -richer regions down to x = 0.20, at and below which the FE phase state overrides the whole crystal volume and FE domain walls become visible under the polarizing microscope. 1.3 Dynamic Properties Another interesting feature is the dynamics of the system. In the temperature dependence of 87 Rb spin-lattice relaxation we observe essentially two competing mechanisms: a soft mode component related to the displacive part of the FE phase transition and a thermally activated component that can be assigned to the biased random walk of unpaired Takagi groups T1 (HPO4 ) and T3 (H3 PO4 ) [11, 12]. They can move through the lattice without essential hindering, similar to the ion states HO– and H3 O+ in hexagonal ice. This analogy also brings about the Pauling ice rules, adapted by Slater for the case of KH2 PO4 (KDP) [13]. According to Slater, the hydrogen-bonded network of a KDP-type system in its ground state must be arranged in such a way that only Slater states H2 PO4 (two protons close and two protons far on the four O – H...O bonds linking any considered PO4 with its adjacent PO4 groups) are present in the lattice. At that time Slater had very likely only the two long-range ordered FE lattices with opposite polarity in mind, and maybe also the four different long-range ordered AFE lattices. However, there exist an inﬁnite number of possible Slater lattices that can serve as ground states in the absence of other interactions. The situation in hexagonal ice corresponds to what we call here a “random Slater lattice”. Moving a single proton, or deuteron, respectively, in a perfect Slater lattice to the other side of its bond, creates a Takagi (T1– T3) pair. To do this one has to overcome the ice bias 2Eb , which describes the asymmetry of the double well potential with the hindering potential Ea . For Eb < Ea the potential has two localized states, whereas for Ea < Eb only one minimum is left in the strongly asymmetric potential. Up to now there is no general agreement concerning the value of Eb , as the results depend strongly on the method of determination or estimation. However, we believe that Eb depends strongly on the local situation, i.e. it is probably considerably higher in long-range ordered FE bulk with a strain deformed unit cell than at the border of a FE cluster. In any case, Eb is much larger than the tunnel splitting of the symmetric bond and thus no tunnelling can occur on these bonds, i.e. even phonon-assisted tunnelling is inhibited in a pure Slater lattice. The Takagi (T1–T3) pair described above is not stable and usually decays immediately to the Slater pair it originated from. There is, however,

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a non-vanishing probability that the Takagi pair separates into two unpaired Takagi groups that are stable. The existence of unpaired Takagi groups in a KDP-type lattice was proved by means of 31 P two-dimensional (2D) exchange NMR [11, 12]. Recently the separation of a Takagi pair in the FE bulk of KDP was beautifully demonstrated by means of ab initio electronic structure calculations [14]. Though the authors do not mention this fact explicitly in their contribution, they report the formation of a chain consisting of AFE Slater groups and this chain is just the trace the separating T1 and T3 have left behind. An unpaired Takagi group in an otherwise perfect Slater lattice has three symmetric and one strongly asymmetric hydrogen bond. Moving the proton on one of the symmetric bonds to the opposite side corresponds to moving the T1 or T3 along this bond to the adjacent PO4 group, leaving a Slater group behind. However, moving the proton on the strongly asymmetric bond to the opposite side would create a T0 (PO4 ) or T4 (H4 PO4 ) state, respectively, which is highly improbable. The diffusion of unpaired Takagi groups inverts the ice bias on every hydrogen bond that is passed and thus changes the proton order of the lattice. The number of T1 and T3 is given by the Boltzmann factor resulting from the ice bias 2Eb and the temperature T. The presence of unpaired Takagi groups reopens the possibility for tunnelling because of their symmetric hydrogen bonds. This feature will be discussed in more detail later on.

2 NMR Line Shape Analysis The reason why NMR spectroscopy is so successful in the investigation of the D-RADP-x system is the existence of local and non-local NMR probes in this system. While the covalently bonded deuteron sees only the local polarization, but not the polarization of the neighbouring deuterons, the ionically bonded Rb observes, via the nuclear quadrupole interaction, a range of about ±2 lattice constants [9] and is thus capable of distinguishing between different spatial correlations of deuteron order. The point charges of the neighbouring acid deuterons give here a considerable contribution to the electrical ﬁeld gradient tensor at the site of the Rb nucleus. Thus, changes in the intra-bond positions of these deuterons can be observed [15]. Another interesting nucleus is the 31 P that is located in the centre of the PO4 tetrahedra. Via the chemical shift interaction, 31 P NMR can, for example, distinguish between the six Slater states of its PO4 group and thus observe changes of these conﬁgurations [11, 12]. Furthermore, the characteristic time scale of NMR for time averaging is in the order of milliseconds, and can be extended to the order of 100 s by means of 2D exchange NMR measurements. NMR can thus be considered a quasi-static investigation method.

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2.1 One-Dimensional 87 Rb NMR Spectra The quadrupole perturbed NMR spectrum of 87 Rb (I = 3/2) consists of the central line and two satellite lines. In D-RADP-x the nuclear quadrupole coupling constant decreases linearly with x from 9.6 MHz (x = 0) to 6.6 MHz (x = 0.95) [10], whereas the Larmor frequency amounts to 98.163 MHz in a B0 ﬁeld of 7 T. Due to the substitutional disorder, and the corresponding loss of the translational invariance, the satellite transitions are inhomogeneously broadened that much, that only the central transition can be directly observed. The position of the corresponding resonance line can be calculated using standard NMR-NQR theory [16], where the second order shift is given by: 1 1 2 2 2 2 V + Vyz + Vxy , ∆ν = –2 Vyz + (1) 12νL 4 η ik the EFG tensor elements (in frewhere νL is the Lamor frequency and V quency units) in the coordinate system of the Zeeman interaction. They are obtained by a unitary transformation of the EFG tensor V deﬁned in crystal coordinates: –1 . (2) V = R α, β, γ VR α, β, γ The orientational dependencies of the spectra of the 87 Rb central transition for various ammonium contents x in the different phase states of the DRADP-x system are presented in [10]. They clearly show that NMR can well distinguish between the various phase states. The phase diagram (Fig. 1) was determined using this information. In this contribution, however, we use only one crystal orientation. If the c-axis of the crystal is oriented parallel to the external magnetic ﬁeld B0 , R is the unity operator and we have V = V. The ﬁve irreducible space-averaged EFG tensor elements depend on the average local symmetry at the Rb site. Thus, only Vzz differs from zero in the PE phase, whereas in the FE phase Vη ≡ Vxx – Vyy and Vxy differ from zero, while Vxz and Vyz remain zero [10]. According to Eq. 1 this leads to the observed positive NMR frequency shift at the PE → FE phase transition (Fig. 2). Similarly one observes a negative frequency shift at the PE → AFE phase transition since, in the long range, ordered AFE phase Vxz and Vyz give the dominant contribution. This has been observed in D-RADP-95 which served as reference for the long range ordered AFE phase state. The temperature dependence of the 87 Rb NMR line shape of D-RADP-20, shown in Fig. 2, clearly exhibits the diffuse character of the PE–FE phase transition, since the signals assigned to the two phases are coexisting in a temperature range of about 20 K. That means that we have phase coexistence in this temperature range. In contrast to the case of D-RADP-70 where the signals of the PE and the AFE phase states are well resolved [9], in D-RADP-20 the separation of the signals is difﬁcult because of strong overlapping. There-

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Fig. 2 Temperature dependence of the 87 Rb – 1/2 ↔+ 1/2 NMR spectrum in the region of the PE–FE phase transition for D-RADP-20 (B0 ||c). The positive frequency shift from the pure PE line to the pure FE line amounts to 8 kHz. The change in the line shape clearly indicates the coexistence of both phase states over a wide temperature range [17]

Fig. 3 Temperature dependence of the transverse relaxation time T2 for D-RADP-20 as obtained from a bi-exponential ﬁt to the 87 Rb spin echo decay. The fact that T2 changes discontinuously at the phase transition clearly indicates the ﬁrst order character of the transition

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fore we have measured the spin echo amplitude as a function of the echo delay time t1 as in a normal T2 experiment. However, the t1 dependence of the data in the overlapping region showed two time constants. The results of the corresponding ﬁts with two exponentials are shown in Fig. 3. The discontinuous jump of the transverse relaxation time T2 displays the ﬁrst order character of the phase transition. The fact that the two T2 s differ in the overlapping region by a factor of about three opens the possibility for an accurate line separation by means of a 2D experiment. 2.2 Two-Dimensional T2 Resolved NMR Spectra Accurate measurements of the frequency-resolved transverse spin relaxation (T2 ) of 87 Rb NMR on single crystals of D-RADP-x (x = 0.20, 0.25, 0.30, 0.35) have been performed in a B0 ﬁeld of 7 Tesla as a function of temperature. The probe head was placed in a He gas-ﬂow cryostat with a temperature stability of ±0.1 K. To obtain the spin echo of the 87 Rb – 1/2 ↔+ 1/2 central transition we have used the standard (90x – t1 – 180y – t1 – echo – t2 ) pulse sequence with an appropriate phase-cycling scheme to eliminate quadrature detection errors and unwanted coherences due to pulse imperfections. To avoid sparking in the He gas, the RF-ﬁeld B1 had to be reduced to a level where the π/2-pulse length τ90 equalled 3.5 µs at room temperature. The delay time t1 was incremented in logarithmic steps to have the same weight for each data point in the analysis. To obtain the frequency resolution the accumulated echoes were Fourier transformed in t2 and phase corrected for pure absorption. To visualize the frequency dependence of T2 we have stacked the spectra with t1 as parameter yielding 2D representations with a linear frequency scale and a logarithmic time scale, as shown in Fig. 4 for D-RADP 25 at 65 K. At this temperature the 1D spectra consists of two overlapping inhomogeneously broadened lines corresponding to the FE and glassy ordered phase states. The glass line peaks at the Larmor frequency νL , whereas the FE line has its maximum at (νL + 8 kHz). In the stacked plots the two lines appear as rims decaying with increasing delay time t1 . Very much the same resonance frequencies are observed in pure D-RDP, as well as in D-RADP-20 (Fig. 2), for the narrow PE line above Tc and the broadened FE line below Tc . The broadening of the FE line is due to crystal defects that affect the local spontaneous polarization and lead to a probability distribution of the electric ﬁeld gradient (EFG) tensor elements at the site of the 87 Rb nuclei [18]. Similarly, in the glass phase state, which consists predominantly of AFE clusters of nanometer size, the inhomogeneous broadening is a result of the disorder, i.e. of the broken translational invariance. Figures 5, 6 and 7 show a selection of 2D frequency–time plots for various temperatures in D-RADP-20, D-RADP-25 and D-RADP-30, respectively.

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Fig. 4 Stacked Fourier transforms of the 87 Rb NMR spin echoes in D-RADP-25 versus echo delay time at T = 65 K. In contrast to D-RADP-20 (Fig. 2) there are still two rims present at this low temperature indicating the coexistence of two different phase states. The rim at the Larmor frequency νL originates from Rb spins localized in short range ordered glass clusters, whereas the rim at νL + 8 kHz is produced by spins sitting in FE clusters [17]

For the analysis of the stacked spectra recorded at different temperatures we have used the following function:

– t1 – t 1 glass FE A t1 , ν = A + A (ν) · exp (ν) · exp , (3) glass T2FE T2 where Aglass (ν) and AFE (ν) are the line shape functions corresponding to probability distribution of the transition frequencies in the two phase states glass and T2 and T2FE the corresponding transverse relaxation times. Though the signal-to-noise ratio was excellent, a proper separation of the two exponentials in the overlap region is a very delicate and sometimes even unstable procedure. In a ﬁrst step the four parameters were ﬁtted separately for each frequency in order to examine their frequency dependence for tendencies. glass The resulting T2 (ν) and T2FE (ν) showed strong ﬂuctuations. Since there is no reason for discontinuities in the frequency dependence of the ﬁt parameters we have restricted Aglass (ν) and AFE (ν) to smooth functions. It was found that ﬁxing the distribution function AFE (ν) to a Gaussian centred near + 8 kHz with a standard deviation of 3.5 kHz (see Fig. 8a) led to nearly glass frequency-independent values for the two transverse relaxation times T2 and T2FE (see Fig. 8b). The line shape function Aglass (ν) was obtained by subtracting AFE (ν) from the signal at t1 = 0: A(t1 = 0, ν), which was extrapolated from the measurements as the origin data are corrupted by the dead time of glass the spectrometer. At T = 79 K we have a T2 of the order of 0.20 ms, whereas

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Fig. 5 Stacked Fourier transforms of the 87 Rb NMR spin echoes in D-RADP-20 versus echo delay time for various temperatures. This sequence of 2D plots corresponds to Fig. 2. It is evident that the PE rim decays much faster than the FE rim

T2FE is of the order of 1.2 ms. The fact that the two relaxation times differ from each other proves the existence of regions of the two phase states large enough to have different 87 Rb spin coherences at the same resonance frequencies. The distribution function Aglass (ν) has its peak at νL and shows the typical asymmetric shape found, for example, in D-RADP-0.50 for z||B0 at low temperatures [10]. For a normalized A(t1 = 0, ν) the zeroth moments of Aglass (ν) and AFE (ν) correspond to the glassy and the ferroelectric volume fractions, respectively. Figure 9 shows the temperature dependence of the transverse relaxation times in D-RADP-25 resulting from the ﬁts with Eq. 3. The two T2 s differ by glass a factor of 4 to 6, between 138 K and 60 K. Above 138 K only T2 could be

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Fig. 6 Stacked Fourier transforms of the 87 Rb NMR spin echoes in D-RADP-25 versus echo delay time for various temperatures. In contrast to D-RADP-20 (Fig. 5) the PE rim at νL does not disappear at low temperatures but continuously broadens to become the glass rim

measured. Its value rises slightly with rising temperature and is not affected by the formation or disappearance of the FE clusters. An unexpected result is that both T2 rise on lowering the temperature below 60 K. This is probably due to a decrease in the ﬂuctuation amplitudes of the relevant elements of the 87 Rb EFG tensors. Furthermore, the two relaxation times merge to a single value at 30 K. Hence, a separation of the two overlapping NMR frequency distribution functions is no longer possible below this temperature.

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Fig. 7 Stacked Fourier transforms of the 87 Rb NMR spin echoes in D-RADP-30 versus echo delay time for various temperatures. Note that the relative size of the FE rim is much smaller than for D-RADP-25 (Fig. 6)

The temperature dependence of the FE and glassy volume fractions was determined for the four different compositions mentioned above (Fig. 10). Whereas D-RADP-0.20 exhibits a quasi-continuous sequence of local PE–FE phase transitions with a coexistence range of about 20 K and a pure FE phase state throughout the whole crystal below 135 K [17], in D-RADP-0.25 part of the crystal remains in the PE or glass state, respectively, down to very low temperatures. This is also observed in D-RADP-0.30, while in D-RADP-0.35 (not shown) no FE polarization could be observed at all. For the case of D-RADP-20 we have assumed a Gaussian probability distribution of transition temperatures, and therefore used an error function to ﬁt the transition region. For D-RADP-25 the transition temperatures range from 138 K down to 118 K. In this temperature range, it is not possible to separate

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Fig. 8 Separation of the overlapping NMR spectra belonging to the two different phase states (glass and FE) by ﬁtting Eq. 3 to the 2D time-frequency spectrum of D-RADP-25 at T = 79 K. a Measured spectrum for t1 = 0 (crosses), glass spectrum (open squares) and FE spectrum (closed squares), b corresponding frequency resolved transverse relaxation times

Fig. 9 Transverse relaxations times T2 of the 87 Rb NMR signals of the two coexisting phase states versus temperature in D-RADP-25 resulting from the ﬁt of Eq. 3 to the 2D time frequency spectra. The solid lines are guide lines for the eye

the FE signal from the “glass signal” with sufﬁcient accuracy. At 118 K the FE volume fraction has reached about 25%. We assume that this is the value which can be reached without electric dipole–dipole interaction between the FE clusters. It should be noted that the polarization within the clusters must here be almost saturated to obtain a 87 Rb spin NMR frequency within the Gaussian FE frequency distribution. This follows from symmetry considerations requiring that Vη and Vxy are proportional to the FE order parameter p, i.e. to the FE polarization. According to Eq. 1 the positive NMR frequency shift ∆ν is thus proportional to p2 . On lowering the temperature, the FE volume fraction value increases linearly up to about 45% at 30 K. This increase

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Fig. 10 Volume fractions of the FE phase state for x = 0.20, x = 0.25 and x = 0.30 versus temperature. The solid lines are guide lines for the eye, whereas the dashed line is a ﬁt of an error function to the data

corresponds to a growth of the FE clusters at the expense of the glassy ordered phase. The onset of FE ordering in D-RADP-30 is shifted down to 80 K. Another feature of this composition is that the FE volume fraction grows much less on lowering the temperature than in D-RADP-0.25 and seems to saturate at 30% for temperatures below 40 K. For ammonium concentrations above about 33% no FE domains or clusters are formed from room temperature down to low temperatures in the pseudo-spin glass D-RADP-x.

3 Soft Mode Versus Thermally Activated Motion In order to understand our NMR spin-lattice relaxation experiments we have to revisit some basic features of structural phase transitions. 3.1 Small Excursion to the Roots According to the well-known Landau theory, the eigenvector of the order parameter in any second order solid–solid phase transition transforms according to an irreducible representation of the space group of the parent phase state. Furthermore, the free energy F = U – TS can be expanded around the transition temperature Tc in terms of the scalar order parameter p, which

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can be considered as normalized amplitude of the eigenvector: F = F0 + a T – Tc p2 + bp4 + ...

(4)

For thermal equilibrium we have dF/ dp = 0 and d2 F/ dp2 > 0, and p = 0 for T > Tc . For T < Tc , p = Tc – T a/2b. Since F is an energy and p proportional to a displacement, the second derivative d2 F/ dp2 is proportional to the force constant f of the order parameter eigenvector and hence proportional to ω2 , where ω is the oscillation frequency of p around its equilibrium value, i.e. ω is the eigenfrequency of the order parameter collective coordinates. In the parent phase (T > Tc ), ω2 is proportional to 2a(T – Tc ), i.e. it decreases linearly to zero at Tc . For this property the oscillation is called soft mode, i.e. on lowering the temperature the mode softens above Tc and hardens again below Tc , where ω2 is proportional to 4a(Tc – T). It should be noted that the theory described above is strictly valid only close to Tc for an ideal crystal of inﬁnite size, with translational invariance over the whole volume. Real crystals can only approach this behaviour to a certain extent. Here the crystal quality plays an essential role. Furthermore, the coupling of the order parameter to the macroscopic strain often leads to a positive feedback, which makes the transition discontinuous. In fact, from NMR investigations there is not a single example of a second order phase transition known where the soft mode really has reached zero frequency at Tc . The reason for this might also be technical: It is extremely difﬁcult to achieve a zero temperature gradient throughout the sample, especially close to a phase transition where the transition enthalpy requires a heat ﬂow that can only occur when the temperature gradient is different from zero. The soft mode concept can be extended to all distortive phase transitions (transitions with relatively small atomic displacements), even if they are only close to second order. In the case of a ferro-distortive transition, as for example in BaTiO3 or KDP, the order parameter is proportional to the spontaneous electric polarization Ps . d2 F/ dp2 is not only proportional to ω2 , but also to the dielectric susceptibility. This does not, however, mean that all components of the order parameter eigenvector must contribute to Ps . KDP is special in so far as it is often considered as a pure order–disorder system of hydrogen bonds that can be treated in a rigid lattice model. In these models it is assumed that the protons are localized on either side of symmetric double well potentials. In order to fulﬁl Landau’s symmetry criteria one has to label the potential wells (e.g. with 1 or 2) so that the ferroelectric phases with opposite polarizations are represented by n1 = 1, n2 = 0, and n1 = 0, n2 = 1, respectively, where the ni are the space-averaged probabilities of ﬁnding the proton in the ith well. Note that for a PE–AFE transition another labelling has to be used. Above Tc there exists a dynamic disorder such that the time-averaged probabilities n¯ 1 and n¯ 2 equal 0.50 for all the bonds of the system. In an ergodic system the local time average of n equals

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the space average. This is not the case below Tc where ergodicity is usually broken because of the coexisting domains of the daughter phase. Ergodicity is maintained only within a domain. Therefore, only single domain states of the daughter phase are usually taken into account in the models. The simplest model is a mean ﬁeld approximation based on two-particle interactions. This yields only a harmonic term for U and the free energy density becomes: φ = a n21 + n22 + b n1 n2 – kT n1 ln n1 + n2 ln n2 . (5) With n1 + n2 = 1 and the order parameter deﬁnition η = n1 – n2 we get: 1 1 1 φ η, T = – kT ln 2 + k T – Tc η2 + kTη4 + kTη6 + ... (6) 2 12 30 The tremendous isotope effect of deuteration of the hydrogen bonds on Tc usually serves as proof for the classiﬁcation as order–disorder transition. Nevertheless, the soft mode of the proper symmetry could be observed in most members of the KDP family, which clearly indicates a displacive component that contributes to the free energy density φ, at least with a harmonic term and a bi-linear coupling term in ηp, whereas the contribution to the entropy term is very likely negligible. Such a correction leads, however, just to a renormalization of the transition temperature. A still-open question is whether this is the only inﬂuence of the soft mode. We have a strong suspicion that the soft mode provides the information for the labelling of the potential wells, and that it sort of synchronizes the proton motion with the mode. In the models, the labelling is somehow enforced by nearest neighbour and next-nearest neighbour interactions. These are, however, short range interactions, the inﬂuence of which must somehow propagate through the system, whereas a lattice mode that deserves its name extends at least over several lattice constants. For a pure order–disorder transition it is justiﬁable to use a relaxation-type equation of motion for the order parameter dynamics: ∂φ η, T dη =Γ , (7) – dt ∂η where Γ represents the kinetic coefﬁcient, which is not critical. Expanding the equation of motion with respect to the order parameter ﬂuctuations δη = η – η, yields: δη/τ = Γ

∂2φ δη , ∂η2

(8)

which corresponds to an over-damped mode with zero frequency. The coupling to the soft mode introduces here oscillatory terms that can become important close to Tc . The fact that the amplitude, as well as the coherence length of the order parameter ﬂuctuations, increase on approaching Tc from either side also probably brings higher order coupling terms into

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play. The situation in the KDP family is, however, not that simple. As already mentioned in the Introduction, the ice-bias plays an important role, i.e. the hydrogen bond potentials are asymmetric except for those of the unpaired Takagi groups T1 and T3. The dynamic behaviour of these Takagi groups will be discussed in a later section. 3.2 Spin-Lattice Relaxation and Critical Behaviour According to standard NMR theory, the spin-lattice relaxation is proportional to the spectral density of the relevant spin Hamiltonian ﬂuctuations at the transition frequencies ωi . The spectral density is given by the Fourier transform of the auto-correlation function of the single particle ﬂuctuations. For an exponentially decaying auto-correlation function with auto-correlation time τc , the well-known formula for the spectral density reads as: 2 J τc = Ai i

2τc , 1 + ω2i τc2

(9)

where the Ai are the relevant ﬂuctuations; for quadrupole perturbed Zeeman interaction, usually the ﬂuctuations of the EFG tensor elements. In our case of 87 Rb NMR in D-RADP-x with B0 ||c, Eq. 9 has only two terms 2 2 + V 2 and ω = 2ω , A2 = 1/4 V – V 2 with ω1 = ωL , A21 = Vxz 2 L xx yy + 2Vxy . yz 2 For a ﬁxed value of τc , the frequency dependence of either term is a Lorentzian centred at zero frequency. In the two regimes τc dependence are distinguished: In the fast motion regime ωi τc 1 the spectral density is proportional to τc and does not depend on i, the measuring frequency 2 ω –1 whereas in the slow motion regime ωi τc 1 it is proportional to ωi τc , i.e. the relaxation rate exhibits dispersion. Usually it is assumed that τc is the only temperature-dependent variable in Eq. 9. This might be the case for an order–disorder type rigid lattice model, where the only motion is the intra-bond hopping of the protons, since the hopping distance is assumed to be constant and therefore also A1 and A2 are constant. This holds, however, only for symmetric bonds. Below Tc the hydrogen bonds become asymmetric and the mean square ﬂuctuation amplitudes are reduced by the so-called depopulation factor 1 – η2 and become in this way temperature-dependent also. The temperature dependence of τc in this model is given by Eq. 8, i.e. τc–1 would be zero at Tc , proportional to T – Tc above Tc and proportional to Tc – T below Tc . If, however, the transition is of a pure displacive nature, the ﬂuctuation amplitude of the order parameter is critical and is by no means temperatureindependent. Since the soft mode is an under-damped lattice vibration (at least outside the close vicinity of Tc ), deﬁned by its frequency ωs and damping constant τs , the spectral density is a Lorentzian centred at ωs and the

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relaxation rate (Eq. 9) has to be adapted accordingly: 2 τs τs J ωs = Ai 2 + 2 1 + ωs – ωi τs2 1 + ωs + ωi τs2 i

2τs ≈ A2i . 1 + ω2s τs2 i

R. Kind

(10)

Except for ωi (transition frequencies of the nuclear spin Hamiltonian) all values are temperature-dependent. From the previous subsection the behaviour of ωs is known. From the anomalous contribution to the birefringence which is proportional to δp2 we get the information concerning Ai . If we assume that the damping of the soft mode is non-critical (which is generally accepted), Eq. 10 describes a transition from an under-damped mode to an over-damped one as Tc is approached from either side. In the presence of both order–disorder and displacive, as in the KDP family, the two dynamic concepts have somehow to be merged. It could well be that the damping constant τs becomes somewhat critical too (at least in the over-damped regime of the soft mode), because of the bilinear coupling of η and p. It would, however, lead too far to discuss this here in more detail. The corresponding theory of NMR spin-lattice relaxation for the phase transitions in the KDP family has been worked out by Blinc et al. [19]. Calculation of the spectral density is here based on a collective coordinate representation of the hydrogen bond ﬂuctuations connected with a soft lattice mode. Excellent and comprehensive reviews of the theoretical concepts, as well as of the experimental veriﬁcations can be found in [20, 21]. 3.3 Comparison with Thermally Activated Motion The single particle auto-correlation time τc in Eq. 9 can, of course, exhibit also a non-critical temperature dependence. Consider a set of independent hydrogen bonds with symmetric double well potentials and a barrier Ea between the wells. In this case the motion is thermally activated and τc shows an Arrhenius behaviour: τc = τ∞ exp Ea /kB T . (11) In contrast to critical behaviour, where the NMR relaxation rate shows a maximum (or a corresponding T1 minimum) at Tc , thermally activated slowing down provides a T1 minimum for ωL τc = 1, i.e. at the border between the fast motion and the slow motion regimes. Since according to Eq. 11 ln(τc ) is proportional to T –1 , T1 is usually plotted in logarithmically versus T –1 , as for example shown in Fig. 11a. The slopes above and below the minimum are proportional to the activation energy Ea . In Fig. 11b a typical tempera-

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Fig. 11 Schematic representation of the temperature dependence of the spin-lattice relaxation T1 for thermally activated motion (a) according to Eq. 4 and for correlated motion connected with a soft lattice mode (b)

ture dependence of T1 for a second order phase transition is shown using the same scales. It should be noted that the order parameter ﬂuctuations are always in the fast motion regime, even at Tc . There is not a single NMR experiment known to the author where T1 dispersion was observed close to Tc , i.e. the soft mode never reached the slow motion regime below the Larmor frequency. In a T1 experiment, the magnetization recovery M(t) is measured after a saturation pulse. In our case we have two contributions in Eq. 9 so that M(t) contains two time constants which differ by more than a factor of three from each other so that they can be separated by a corresponding ﬁt procedure. The two time constants τ1 and τ2 correspond to the relaxation rates at ωL and 2ωL .

Fig. 12 87 Rb inverse relaxation rates τ1 and τ2 versus inverse temperature 1/T at νL = 98.163 MHz in D-RADP-15. The solid lines are ﬁts based on a collective coordinate representation of the deuteron intra-bond ﬂuctuations connected with a soft lattice mode

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They can serve therefore as a test for T1 dispersion. In Fig. 12 the relaxation results are shown for D-RADP-15. The solid lines are a ﬁt of the theory [19] to the data. Above Tc the ﬁt is excellent, whereas below Tc it probably suffers from the fact that the phase transition is already diffuse and only nearly of second order. This proves that a soft mode component is needed to explain the data. Furthermore, the fact that the ratio τ1 /τ2 remains unchanged above and below Tc proves that the order parameter ﬂuctuations are in the fast motion regime on both sides of the transition. In contrast to this, the measurements on D-RADP-25 (Fig. 13) clearly show dispersion; below 120 K the relevant ﬂuctuations are in the slow motion regime and exhibit an Arrhenius-type behaviour. For comparison the data for D-RADP-50 are also shown, which show Arrhenius behaviour over the whole measured temperature range. Comparison of the two reveals clearly a critical contribution to the spin-lattice relaxation in D-RADP-25. This critical contribution is, of course, coming from the regions that still undergo PE–FE transitions, whereas the Arrhenius behaviour can be assigned to the coexisting regions with glassy type ordering. The importance of the soft mode in the FE clusters will be discussed in a later section. The value of the activation energy derived from the slopes in Fig. 13 is Ea = 80 meV. However, as we have reported in [12], this is only an apparent value, due to the fact that the biased

Fig. 13 Comparison of the 87 Rb inverse relaxation rates τ1 and τ2 versus 1/T at νL = 98.163 MHz for D-RADP-25 and D-RADP-50. Whereas D-RADP-50 exhibits a pure thermally activated behaviour, in D-RADP-25 a soft mode contribution to the relaxation rate can be observed in the fast motion regime [17]

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random walk of unpaired Takagi groups (T1,T3) is responsible for the dominant contribution to the spin-lattice relaxation, and the apparent value is in fact the sum of the hindering potential for symmetric bonds and the ice bias, i.e. Ea + 2Eb = 80 meV.

4 The Role of Unpaired Takagi Groups For tractability reasons most order–disorder model calculations for the KDP family, including the pseudo-spin model, work with symmetric hydrogen bonds just using an FE or AFE order parameter eigenvector leading to the corresponding long range ordered Slater lattices. This assumption seems to be justiﬁed by the deuteron NMR measurements of Bjorkstam [22], which revealed symmetric potentials for the hydrogen bonds above Tc and asymmetric potentials below Tc . This is in apparent contradiction with the Slater ice rules [13], which predict only asymmetric hydrogen bond potentials. As 1D NMR performs a gliding time average with an integration time of the order of T2 , the ice bias must be reversed stochastically or periodically with a correlation time much shorter than T2 to overcome the contradiction. As known since the early work of Slater [13] and Takagi [23], the corresponding bias ﬂuctuations, which include the exchange between FE and AFE Slater conﬁgurations, can take place at low temperatures exclusively via diffusion of unpaired Takagi groups (HPO4 and H3 PO4 ). It seems that this mechanism is dominant also in the PE high temperature phase. To check the effect of the random walk of an unpaired Takagi group, we have used a rigid lattice model of 6 × 6 × 8 lattice constants in the x, y, z directions containing 1152 PO4 groups. As initial state for all calculations a long range ordered FE Slater lattice was used. In this lattice a Takagi T1–T3 pair was created and separated on an arbitrary position. Then the unpaired Takagi groups could start their random walk through the lattice, while the state of every PO4 group was monitored. Periodic boundary conditions were used so that when a T3 or T1 accidentally left the lattice on one side it was re-entered at the opposite side of the prism. Upon accidental recombination of a pair, a new one was created again at an arbitrary position. After a large number (≈ 107 ) of individual steps a state is reached, which we call a random Slater lattice, where all six Slater conﬁgurations are about equally populated. This state, which corresponds to the situation in hexagonal ice, was used as initial state for any further calculations. The fact that such a state can be reached from an initially long range ordered state by the diffusion of unpaired Takagi groups clearly demonstrates the power of the mechanism. However, the ice-bias is not the only short range interaction in the system, especially when we deal with a random substitution of K or Rb ions by NH4 or ND4 groups, as for example in the solid solution Rb1–x (ND4 )x D2 PO4

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(DRADP-x). Depending on the value of x, either the FE or the AFE Slater conﬁgurations become more probable than the random walk model predicts. To take care of this fact we have introduced a biased random walk of unpaired Takagi groups. The corresponding bias energy corresponds to an average Slater energy 2εav . Starting with a random Slater lattice and depending on this average bias we can reach any ratio from 100% FE to 100% AFE Slater conﬁgurations. For symmetry reasons the two FE conﬁgurations are equally populated, and the four AFE conﬁgurations also appear with equal probabilities so that the macroscopic polarization is always zero and the average tetragonal structure is established. From our NMR T1 measurements we know that the Takagi groups move fast enough down to about 70 K that 1D NMR sees only the time-averaged EFG tensors at the Rb or deuteron sites. Since the Slater energy εs is by itself not symmetry-breaking, but only favouring the FE Slater groups to a certain extent at the expense of the AFE Slater groups, the biased random walk of the unpaired Takagi groups is sufﬁcient to remove the asymmetry caused by the ice-bias completely in the time average. Thus, every T2 visited by a propagating T3 or T1 is changed into a different T2. A given Slater group can be approached via two different bonds by a T3 or a T1, respectively. For the propagation there are also two possibilities and thus there are four possibilities for the resulting T2. Table 1 shows the corresponding transition probabilities. The δs take care of the Boltzmann factor in ﬁrst order, i.e. δ = exp (εs /2kB T) – 1 /6 ≈εs /12kB T. The off-diagonal elements indicate the values for forward propagation, whereas the diagonal stands for the back step probabilities. The zeros indicate transitions that are not possible for geometrical reasons (the ice-bias of all four bonds would have to be inverted), i.e. successive visits of two T3 or two T1 are needed to perform such a transition. It is

Table 1 Transition probabilities for a conﬁguration change among the six Slater groups, induced by a visit of an unpaired Takagi group T1 or T3 → ↑\

AFE⇒

AFE⇐

AFE

AFE⊗

FE⇑

AFE⇒

1 3

0

1 6 1 6 1 3

1 6 1 6

–δ

1 6 1 6 1 6 1 6 1 3

–δ

0

AFE⇐

0

AFE

1 6 1 6 1 6 1 6

AFE⊗ FE⇑ FE⇓

– 2δ

–δ –δ

1 3 1 6 1 6 1 6 1 6

– 2δ

– 2δ

0

–δ

1 3 1 6 1 6

0 –δ –δ

1 6 1 6

–δ

– 2δ

+δ

FE⇓

+δ

1 6 1 6 1 6 1 6

+ 4δ

0

+δ +δ

1 3

+δ +δ +δ +δ

+ 4δ

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thus not possible to invert the polarization of a FE Slater group by the visit of a single T1 or T3. The question arises here concerning how such a system can become critical and undergo a phase transition. One can somehow imagine how the soft mode could synchronize the proton motion on symmetric bonds, but to synchronize a biased random walk is something else. Since we do not know how to solve this problem, we will focus now on the role of the unpaired Takagi groups in the FE ordered phase. In the FE phase state of RDP or KDP the domains consist of either pure FE⇑ or FE⇓ groups. The interesting features are here the domain walls. It is well known that domain walls are found only perpendicular to the a-directions of the facecentered unit cell (Fdd2, Z = 4). They consist entirely of AFE Slater groups and the wall thickness is half a lattice constant. The reason why there are no domain walls perpendicular to the c-axis is that they would have to consist of a pure layer of either T3 or T1 Takagi groups depending on whether a head– head or a tail–tail situation of the polarization vectors is encountered. Also here the thickness of the layer would be half a lattice constant. Such a situation is highly improbable. Where are then the Takagi groups localized, which should be there according to Boltzmann? To answer this question we have built the KDP hydrogen bond network for several lattice constants in all directions by means of an Excel spreadsheet. In the primitive unit cell of the parent phase there are two different PO4 groups. In the lattice this makes up two sets of PO4 groups, each set related by the primitive lattice translations. One set serves as input, i.e. each member of the set can be set to one out of the 16 possible conﬁgurations. The second set serves as output, which is completely deﬁned by the input set. Accordingly, each of its members will take on one out of the 16 conﬁgurations. If the whole input set is set to FE⇑ then the whole output set yields also FE⇑. In this FE ordered lattice one can now introduce a Takagi pair by setting one of the input groups to a T3 and immediately one out of the four neighbouring output groups will change to T1. By a further local manipulation of the proper inputs one can separate the Takagi pair and let the resulting T1 and T3 propagate through the lattice. With every step of propagation T3 climbs by c/4 if the polarization is pointing in the + z-direction, whereas the T1 descends with every step by the same amount. More freedom is in the x- and y-directions; the T3 can for example climb on a helical path or, in the other extreme, move essentially along a [1,1,1]-direction. From Table 1 it follows that the T3 and the T1 must be connected by a string of AFE Slater groups, whereas all other groups remain as FE⇑ in the lattice. This is essentially also observed in the ab initio calculations [14], though it is not absolutely clear why the forced Takagi pair separates immediately and why the T1 and T3 vanish beyond the boundaries of the model. The latter is probably an effect of the periodic boundary conditions. Obviously the string has lower energy than the Takagi pair. On the other hand, every propagation step of the T1 or the T3 has to be paid with the Slater energy εs . Furthermore, the T3 carries

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a positive charge and the T1 a negative one, both feeling the internal electric ﬁeld. It is therefore not clear where the propagation of the unpaired Takagi groups ends in this case. It should be noted that back steps are, of course, possible and even probable but they have to follow exactly the trace laid by the AFE Slater groups. Thus, recombination and annihilation of the Takagi pair can only occur on this line with the result that the original pure FE⇑ state is restored. However, in the presence of a FE⇑–AFE–FE⇓ domain wall, both the T1 and the T3 can penetrate through the wall and recombine on the other side, thus leaving a closed loop of AFE Slater groups behind. It should be noted that the number of changed FE⇑ conﬁgurations amounts exactly to the number of changed FE⇓ conﬁgurations so that the total macroscopic polarization remains unchanged. If two Takagi pairs are created in a FE⇑ lattice it can happen that their AFE traces accidentally cross. At the crossing point we ﬁnd then a FE⇓ conﬁguration, since the corresponding PO4 was visited twice by unpaired Takagi groups originating from different pairs. However, such accidental crossings are not sufﬁcient for the system to become critical. Since we are dealing with a cooperative phenomenon, the actors( here the various T3s and T1s in the system) should also show some cooperation. Instead of accidental crossing, efﬁcient organized crossing is needed. The most efﬁcient multiple crossing involving two Takagi pairs occurs for two corner-sharing narrow helical paths, since they meet at intervals of the c-lattice constant. With this one can create a pile of FE⇓ aligned in c-direction of inﬁnite length. But, for every FE⇓ one has to create six AFE groups (six piles). One can now add more Takagi pairs to this construct to improve this ratio and the efﬁciency. For example, with ﬁve correlated Takagi pairs one can create a narrow helix of FE⇓ (four piles). The AFE shield around (domain walls) consists of 12 piles. It can well be imagined that the length of such piles can be changed by an external electric ﬁeld. Thus we have an efﬁcient mechanism that contributes to the dielectric constant. It would lead too far to discuss more possible examples and strategies here. Our aim was to prove that at least one mechanism exists, on the basis of unpaired Takagi groups alone, that allows the system to become critical, provided that the corresponding motions are correlated to a certain extent.

5 Cluster Size and Inter-cluster Distance The NMR results presented in Sect. 2 allow for D-RADP-x (with x = 0.20, 0.25, 0.30), in fact for no other interpretation than a multitude of local ﬁrst order phase transitions with a probability distribution of transition temperatures. We believe therefore that we deal with a nucleation mechanism. To illustrate this possibility we have to make some assumptions:

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1. Substitution of Rb by ND4 is completely random 2. Nucleation centres are small volume fractions that are free of ND4 3. The density of nucleation centres must be such that the resulting FE islands can reach the proper size Before we proceed with some estimations and calculations we should recall how the ND4 group acts in a lattice. The ND4 forms four N – D...O hydrogen bonds to the four nearest oxygen ions. Two bonds are shorter than the other two with the restriction that the two shorter ones cannot be on the same level, i.e. looking from the c-direction they must be orthogonal to each other. This leaves four possibilities for the orientation of the ND4 group and a corresponding deviation of the N atoms from the fourfold axis. The high temperature symmetry is obtained by a fast thermally activated exchange, which slows down with an activation energy of 160 meV. The transition to the slow motion regime was observed at 160 K by means of NMR T1 measurements on the ammonium deuterons in a B0 ﬁeld of 2.0 T [24]. The system wants to avoid ND4 orientations where the same oxygen atom has a deuteron of the O – D...O bond close to a short N – D...O bond, i.e. such a situation has to be paid with a higher energy. This energy approximately equals the Slater energy εs . This can be concluded from the PE–AFE phase transition temperatures that are similar to the ones of the corresponding PE– FE transitions, as shown in Fig. 1. The restriction comes from the fact that the O – D...O bond network forms loops, each involving six PO4 groups. The ten nearest PO4 groups to a given ND4 form four loops. Lets assume that the ten PO4 groups show all FE⇑ conﬁgurations. Independent of the ND4 orientation we have then an unwanted situation in one of the lower loops. This can be mended only by transforming two adjacent FE⇑ into AFE conﬁgurations. Similarly, in the case of ten FE⇓, the problem occurs in one of the two upper loops, and accordingly two AFE groups have to be created there to reach the state of lowest energy. This feature justiﬁes the second assumption concerning the bulk of the FE nucleation centre. However, at the surface, especially in the corners, ND4 groups can be tolerated to a certain extent without affecting the bulk polarization. So we do not need a complete shielding of the PO4 members of the centre by Rb (e.g. ﬁve PO4 shielded by 14 Rb or 13 PO4 shielded by 34 Rb) and can therefore safely work with sets of complete RbD2 PO4 formula units. We can now start with a guess of the nucleation centre size and take a cube with a = 1.4 nm containing eight units cells with Z = 4, i.e. 32 formula units. The probability W(n, x) of ﬁnding n ND4 groups in the cube is given by: W(n, x) =

N! (1 – x)(N–n) xn n! (N – n)!

with N = 32 and

0≤x≤1.

(12)

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For n = 0 the average distance between neighbouring nucleation centres becomes: –1/3 (nm) . (13) d(n, x) = 0.441 × W(n, x) Equation 13 corresponds to shifting a cubic cage in increments of the shortest Rb–Rb vector through the lattice. Every time the cage contains the proper amount n of ND4 the centre is marked in the lattice. Then one can measure the average distance between neighbouring nucleation centres. The prefactor becomes thus 1.4 × (32)–1/3 = 0.441. The function (Eq. 13) is shown in Fig. 14 with n = 0, 1, 2, 3, 4 as parameter versus the ammonium content x. The request that no ND4 should be within the cube is represented by n = 0. The inter-centre distance increases here so fast with x that the behaviour of the system cannot be explained with ND4 clean cubes. Since the density of nucleation centres should not increase drastically on increasing x, we have either to reduce the clean volume with x, or to allow for some stray ND4 keeping the volume constant. The latter possibility is shown in Fig. 14. The open circles correspond to the values where n is increased by one when x is increased by 0.05, starting with n = 0 for x = 0.15. It should be noted that these points still correspond to Rb-rich areas. This can be seen in Fig. 14 where the minima of the curves indicate the average concentration x = n/32. The spoiling of the nucleation centres has, of course, to be paid with a lower local transition temperature, which is indeed observed in the real system. Going along with this procedure, we would obtain cluster sizes of about 100 formula units for D-RADP-25 and of 80 formula units for D-RADP-30, i.e. of about 2 nm in

Fig. 14 Inter-cluster distances d versus ammonium content x as obtained from Eq. 13. Open circles correspond to the values where n is increased by one when x is increased by 0.05

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diameter. This is certainly at the lower border of the possible cluster sizes, but it would barely fulﬁl the size requirements postulated by means of NMR on the non-local probes. On the other hand, looking at the steps of about 20% in the polarization curves of D-RADP-25 and D-RADP-30 in Fig. 10 and assigning these to the polarization of the bare nucleation centres, our guess about size is adequate since the calculated polarization is 18% and 14%, respectively. Increasing the size of the nucleation centres would make them so rare that the corresponding polarization would quickly drop below 1%. Thus our guess is somewhat more than plausible but still speculation. It all depends on how the polarization spreads after the initial instability. However, this problem is still an open question to be left to further investigations.

6 Conclusions and Perspectives We have shown that the structural phase transitions in D-RADP-x for x = 0.15, 0.20, 0.25 and 0.30 consist of a set of local ﬁrst order transitions with a distribution of transition temperatures. Similar to the case of the PE–AFE transition in D-RADP-78 [9], where a Gaussian probability distribution of Tc could be determined with high precision, we also assume Gaussian distributions on the FE side of the phase diagram, though they are not so easy to determine. Furthermore, we have observed phase segregation between the FE phase state and the glass phase. Our system thus shows all the features of a nucleation process. In the previous section is was demonstrated how the random substitution of Rb by ND4 could provide the necessary nucleation centres, though the details are far from clear. Another important problem we have discussed is the old ice-bias dilemma, which has puzzled scientists since it was brought up by Slater [13]. We could show that the random walk of unpaired Takagi groups inverts all the ice biases and can thus destroy any order completely, though not in a critical way. For the system to become critical we have demonstrated that the random walk must become correlated, at least pair-wise T3–T3 and T1–T1 separately, or in larger groups of the same kind. We believe that this is a general property of the KDP family that is valid also for the pure end members RDP and KDP. This opens new possibilities for the theoretical treatment of the corresponding phase transitions. Another feature that is still puzzling scientists is the role of the soft mode in structural order–disorder transitions. As already mentioned in Sect. 3 the energy U in theoretical treatments is based predominantly on short-range interactions and often the local ﬁelds are replaced by a mean ﬁeld to mimic some long-range properties. In the solid solution D-RADP-x the FE soft mode suffers from the lack of translational invariance, so that only for x < 0.32 sufﬁcient coherence can be achieved to provide a successful contribution to

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the free energy. In this case, the coherence length exceeds the range of the short interactions by so much that we can speak of an intermediate range interaction. We believe that the soft mode couples somehow to the motion of unpaired Takagi groups (which are in the lattice according to Boltzmann) in the sense of collecting them gender-wise and orchestrating their correlated motion. If this does not happen the system has no chance to become ferroelectrically ordered in clusters of intermediate size, as for example in the case of D-RADP-50. The next step then is to form large scale domains with planar domain walls. Here electric inter-cluster interactions and elastic forces probably play a dominant role, though they have to act via the Takagi group motion. In contrast to the pure end members RDP and KDP, the macroscopic strain cannot develop properly in the mixed crystals. In the glass phase regions the system remains pseudo-tetragonal down to very low temperatures, though the local symmetry is “apparently” broken (the system is only pseudo-frozen and restores the symmetry in the long time scale [11, 12]). The strain can only develop when the FE clusters, and with them the coherence length of the soft mode, become wide enough. Then the FE phase state is forced against the resistance of the ND4 through the whole lattice. This takes place in D-RADP20 where we have observed the spread of domain walls under the polarizing microscope. We have demonstrated in this contribution that the structural order– disorder transitions in the KDP family are by far more complex than a pseudo-spin model, for example, could describe. Geometrical constraints and coupling to the soft mode, as well as cluster formation of nanometric size, indicate that here the conventional picture of cooperative phenomena has to be revised. Acknowledgements The author would like to thank his former PhD students Th. Koenig, and N. Korner, who both left science long ago for other tasks, for performing the measurements and some of the model calculations presented here.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Smolenskii GA (1970) J Phys Soc Jpn 28:26 Cross LE (1987) Ferroelectrics 76:241 Cross LE (1994) Ferroelectrics 151:635 Kleemann W, Klössner A (1996) Europhys Lett 35:391 Kind R, Müller KA (1976) Commun Phys 1:223 Schmidt VH (1998) J Korean Phys Soc 32:S803 Pinto J, Schmidt VH (1993) Ferroelectrics 141:207 Levanyuk AP, Minyukov SA, Vallade J (1998) J Korean Phys Soc 32:S62 Korner N, Pfammatter C, Kind R (1993) Phys Rev Lett 70:1283 Korner N, Kind R (1994) Phys Rev B49:5918

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11. Kind R, Cereghetti PM, Jeitziner CA, Zalar B, Dolinsek J, Blinc R (2002) Phys Rev Lett 88:195501-1 12. Kind R (2006) Two-dimensional exchange NMR and relaxation study of the Takagi group dynamics in deuteron glasses. In: Dolinsek J, Vilfan M, Zumer S (eds) Novel NMR and EPR techniques. Lecture notes in Physics, vol 684. Springer, Berlin Heidelberg New York, pp 383–405 13. Slater JC (1941) J Chem Phys 9:16 14. Lasave J, Koval S, Dalal NS, Migoni R (2005) Phys Rev B72:104104-1 15. Kind R, Blinc R, Koren M (1988) Phys Rev B37:4864 16. Abragam A (1961) The principles of magnetic resonance. Clarendon, Oxford 17. Kind R, Korner N, Koenig T, Jeitziner C (1998) J Korean Phys Soc 32:S799 18. Korner N (1993) Dissertation ETH Zürich No. 9952 19. Blinc R, Stepisnik J, Jamsek-Vilfan M, Zumer S (1971) J Chem Phys 54:187 20. Bjorkstam JL (1974) Adv Magn Res 7:1 21. Rigamonti A (1984) Adv Phys 33:155 22. Bjorkstam JL (1967) Phys Rev 153:599 23. Takagi Y (1948) J Phys Soc Jpn 3:273 24. Blinc R, Ailion DC, Günther B, Zumer S (1986) Phys Rev Lett 57:2826

Struct Bond (2007) 124: 149–198 DOI 10.1007/430_2006_048 © Springer-Verlag Berlin Heidelberg 2006 Published online: 22 November 2006

Anharmonic and Quantum Effects in KDP-Type Ferroelectrics: Modified Strong Dipole–Proton Coupling Model Dalibor Merunka · Boris Rakvin (u) Ruder Boˇskovi´c Institute, Bijeniˇcka cesta 54, PO BOX 180, 10002 Zagreb, Croatia [email protected] 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

150

2 2.1 2.2 2.3

Experimental Investigations on KDP-Type Ferroelectrics . Investigations of Structural and Static Dielectric Properties Investigations of Dynamical Properties . . . . . . . . . . . Investigations of Local Properties . . . . . . . . . . . . . .

. . . .

152 152 155 160

3 3.1 3.2

Theoretical Models for KDP-Type Ferroelectrics . . . . . . . . . . . . . . . Brief History of Models and the Proton-Tunneling Model . . . . . . . . . . Development of the Modiﬁed Strong Dipole–Proton Coupling Model and Simulation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163 163 167

4.1 4.2 4.3

Results of the Modiﬁed Strong Dipole–Proton Coupling Model and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . Results for Structural and Static Dielectric Properties . . . . . . Results for Dynamical Properties . . . . . . . . . . . . . . . . . Results for Local Properties . . . . . . . . . . . . . . . . . . . . .

. . . .

174 174 183 190

5

Comments and Concluding Remarks . . . . . . . . . . . . . . . . . . . . .

193

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

196

4

. . . .

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. . . .

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. . . .

. . . .

. . . .

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. . . .

. . . .

. . . .

. . . .

. . . .

Abstract Anharmonic and quantum effects caused by hydrogen bonds appear strongly in ferroelectric crystals of the KH2 PO4 - or KDP-type. Both effects are closely related to the long-standing and still unresolved questions about the ferroelectric phase transition in these crystals: that of the type of phase transition, and that of the origin of strong isotope effects of proton-deuteron replacement. In this chapter, the most relevant experimental studies devoted to these problems are reviewed. The proton-tunneling model, as widely accepted model of the phase transition in these crystals, was questioned in the 1980s. A development of the alternative model, modiﬁed strong dipole–proton coupling (MSDPC) model, is presented. The model assumes that quantum protons in KDP have higher characteristic frequencies than the heavy-ion sublattice represented by the system of three-dimensional dipoles. Therefore, protons being in a ground state adiabatically follow dipole dynamics, as opposed to the proton-tunneling model. Proton tunneling is absent in the model, and ferroelectric ordering and anharmonicities in the dipole system are caused by the proton ground-state energy. Different ground-state energies of protons and deuterons cause isotope effects and make deuterated systems more anharmonic, so that the phase transition in deuterated crystal is predicted as more of an order–disorder type than in non-deuterated crystal. Successes and shortcomings of the model in explaining experimental results are commented upon.

150

D. Merunka · B. Rakvin

Keywords Anharmonic effects · Displacive phase transition · Isotope effects · KDP-type ferroelectrics · Order–disorder phase transition

1 Introduction Apart from being technologically interesting, hydrogen-bonded ferroelectric crystals of KH2 PO4 - or KDP-type are also intriguing from a fundamental point of view [1]. The existence of hydrogen bonds connecting oxygen atoms of neighboring PO4 tetrahedra seems to play the crucial role in the ferroelectric phase transition of these crystals. The anharmonic nature of hydrogen bonds and the quantum nature of protons or deuterons involved in these bonds cause several interesting phenomena and raise some questions related to the phase transition that are still unresolved. The quantum mechanical nature of protons and deuterons is evident from the large effect of isotope exchange of protons with deuterons on several static and dynamic properties of the phase transition. Probably the most interesting isotope effect is the large isotope effect on the critical temperature of the ferroelectric phase transition, Tc . Although ferroelectric crystals of KDP-type have been known for about 70 years, the origin of the large isotope effect on Tc and on other static and dynamic quantities of the crystals is not yet fully understood. In addition, it is not clear whether a phase transition mechanism in these materials is of a displacive or an order–disorder type. The ﬁrst model proposed to explain the isotope effect was the protontunneling model later introduced by Blinc in the early 1960s [2]. According to this model [1–4], the interaction between protons, which reside in a doublewell potential of hydrogen bonds, produces a ferroelectric ordering below Tc . Quantum mechanical tunneling between two wells, which has different frequency for a proton and a deuteron due to the larger mass of the deuteron, was proposed as the cause of isotope effect on Tc in this model. In addition, a soft mode of phase transition, instead of being a relaxational mode caused by stochastic inter-well proton hopping as in an order–disorder type of phase transition, becomes oscillatory mode due to the proton tunneling [1, 4]. In other words, the proton-tunneling model predicts a displacive type of phase transition as concerning the soft-mode dynamics. Partly due to its success in explaining the isotope effects and partly due to its elegance and universality, the proton-tunneling model has been accepted as a valid description of ferroelectricity in KDP-type crystals until now. However, in the 1980s some doubts arose about the validity of this model. Results of several experiments indicated that a model based on order– disorder motion of PO4 groups could be more appropriate to describe the phase transition [3, 4]. Within such a type of a model, the isotope effect on Tc could be explained in terms of a geometric isotope effect. More precisely,

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the hydrogen-bond length, or the distance between oxygen atoms, RO–O , have higher values in deuterated than in non-deuterated crystals. Thus, the interactions between dipoles of PO4 groups are possibly different in deuterated and non-deuterated crystals due to this geometric isotope effect. Moreover, studies of pressure effects on the hydrogen-bonded ferroelectrics revealed that Tc depends on RO–O more strongly than on deuterization of the crystals. In the 1990s, Krumhansl [5] reached two important conclusions from these results: (i) the tunneling between two wells in the hydrogen-bond potential cannot be the origin of isotope effects in KDP, and (ii) protons in hydrogen bonds and the rest of the lattice (heavy-ions sublattice) are not separable subsystems, where only one mechanism, such as tunneling, dominates the transition physics. He also pointed out that these reasons made a situation more demanding for theorists and that a new, more complex theory of ferroelectricity in KDP was required. That the situation might not be too demanding was shown soon after that, when Sugimoto and Ikeda in the early 1990s introduced the strong dipole–proton coupling (SDPC) model, according to which protons could be separated from the rest of the lattice [6]. Also, a general lack of theoretical models based on the order–disorder mechanism of PO4 groups was partly diminished by the introduction of the SDPC model. According to this model, protons and deuterons occupy the ground state of the hydrogen-bond potential, which is in contrast to the proton-tunneling model. Separation of the protons (deuterons) and the heavy-ion sublattice was made by assuming that protons (deuterons) adiabatically follow the dynamics of PO4 dipoles aligned along the ferroelectric c-axis of the tetragonal unit cell. In this model, the phenomenological double-Morse potential was proposed as the potential for the motion of a proton and a deuteron in the hydrogen bond. The calculated difference between ground-state energies of proton and deuteron in this potential was caused partly by their different masses and partly by different hydrogen-bond lengths in deuterated and non-deuterated crystals. Since this difference was suggested to be responsible for the isotope effect on Tc in KDP, both the geometric effect and the effect of different masses were included in this model. By taking into account the components of the PO4 dipole along all three tetragonal directions, the SDPC model has been modiﬁed and improved [7]. One of the purposes of this chapter is to present the main results of this modiﬁed model, the MSDPC model. This chapter starts with a brief summary of the old and novel experimental investigations on KDP-type ferroelectrics that are most relevant for the discussion of the present topic. A brief history of theoretical models for KDP, development of the SDPC and MSDPC models, and applied numerical methods are described in Sect. 3. The most important results of the MSDPC model are presented and discussed in Sect. 4, whereas comments about the model and concluding remarks are given in Sect. 5.

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2 Experimental Investigations on KDP-Type Ferroelectrics 2.1 Investigations of Structural and Static Dielectric Properties KH2 PO4 or KDP, represents a family of isomorphous ferroelectric crystals with general formula AX2 BO4 , where A stands for K or Rb, X stands for H or D, and B stands for P or As [1]. These KDP-type ferroelectrics have a tetragonal structure in the paraelectric or PE phase (T > Tc ), and spontaneous polarization in the ferroelectric or FE phase (T < Tc ) appears along the tetragonal c-axis. Hydrogen bonds, O – H – O, which connect one PO4 (or AsO4 ) tetrahedron with four neighboring tetrahedra in these crystals, lie nearly parallel to the tetragonal a- and b-axes. Special characteristics of hydrogen bonds, i.e., strongly anharmonic potential for the motion of H or D atom (probably of a double-well type) and quantum mechanical character of this motion, play an important, if not crucial, role in a mechanism of the phase transition. Static dielectric measurements [8] show that all crystals in the family exhibit a very large quantum effect of isotope replacement H→D on the critical temperature. This effect can be exempliﬁed by the fact that Tc = 122 K in KDP and Tc = 229 K in KD2 PO4 or DKDP. KDP exhibits a weak ﬁrst-order phase transition, whereas the ﬁrst-order character of phase transition in DKDP is more pronounced. The effect of isotope replacement is also observed for the saturated (near T = 0 K) spontaneous polarization, Ps , which has the value Ps = 5.0 µC cm–2 in KDP and Ps = 6.2 µC cm–2 in DKDP. As can be expected for a ferroelectric phase transition, a decrease in the temperature toward Tc in the PE phase causes a critical increase in longitudinal dielectric constant (along the c-axis) in KDP and DKDP. This increase follows the Curie–Weiss law, εc = C/(T – T1 ), and an isotope effect is observed not only for the Curie– Weiss temperature, T1 ≈ Tc , but also for the Curie constant C (C = 3000 K in KDP and C = 4000 K in DKDP). Isotope effects on the quantities Tc , Ps , and C were successfully explained within the proton-tunneling model as a consequence of different tunneling frequencies of H and D atoms. However, this model can hardly reproduce the Curie–Weiss law for εc . Measurements of transverse dielectric constant (perpendicular to the c-axis) or εa show a strong and abrupt decrease of εa as the temperature drops below Tc in KDP and DKDP. This is expected to be the effect of strong coupling between the longitudinal polarization (or the order parameter variable) and the transverse polarization. This coupling has to be non-linear due to different symmetries of modes that induce longitudinal and transverse polarization, and it can be strong enough to play some role in a whole phase transition mechanism. According to the neutron and X-ray diffraction measurements in KDP and DKDP, spontaneous polarization below Tc is produced by the shifts of the

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heavy ions (K, P, and O) along the c-axis, while the largest shift below Tc is shown by H and D atoms along hydrogen bonds, i.e., perpendicular to the c-axis [1]. This observation is in accordance with the idea that the phase transition in KDP-type ferroelectrics is triggered by the motion of H or D atoms, whose localization in one of two wells in hydrogen-bond potential induces ferroelectric ordering below Tc . A double-well character of hydrogen-bond potential is supported by experimental ﬁndings that thermal distributions of H and D atoms show two peaks in the PE phase, which are separated by δ A for KDP and δ = 0.45 ˚ A for DKDP at along the hydrogen bond (δ = 0.36 ˚ room temperature), and that shifts of both atoms below Tc have the value of half of this distance, δ/2 [9, 10]. The double-peaked character of these distributions in the PE phase can be explained by two opposite interpretations. The ﬁrst, based on order–disorder motion of H and D atoms, assumes that above Tc these atoms randomly occupy one of the two sites along the bond, O – H(D)· · ·O and O· · ·H(D) – O, which causes double-peaked distributions. The other interpretation presupposes that a tunneling motion of H and D atoms in a double-well potential of hydrogen bond causes the double-peaked character of their wave function, which leads to the double-peaked thermal distributions. Since the H and D atoms move along hydrogen bonds, perpendicular to the ferroelectric c-axis, while the displacements of heavy ions induce the spontaneous polarization, it is obvious that the heavy-ion sublattice should also participate in the phase transition mechanism. Anharmonic effects in the heavy-ion sublattice were detected by neutron diffraction measurements of reﬁned thermal parameters for heavy atoms in KDP and DKDP [9]. Although the thermal distribution of displacements along the ferroelectric c-axis of P atom (central atom of PO4 tetrahedron) does not have a double-peaked character in the PE phase, like the distribution of H and D atoms along hydrogen bonds, the linear temperature dependence of the mean-square displacement extrapolates to a signiﬁcantly non-zero intercept at T = 0 K, which is also a strong anharmonic effect. These results indicate moving of the P atom between two sites in the PE phase and thus suggest order–disorder motion of the whole PO4 tetrahedron. Again, the distance between these two sites was found to be larger in DKDP than in KDP (by a factor of about two). Experimental studies [11] of the effects of hydrostatic pressure, p, on static properties of KDP-type ferroelectrics may indicate which mechanism is more important for the phase transition in these crystals – the proton tunneling or the order–disorder motion of PO4 groups. The most important result of these studies is a decrease and ultimate vanishing of Tc with the increase in the pressure from the atmospheric value to the critical value, pc . This result can be easily explained within the proton-tunneling model as a natural consequence of the increase in tunneling frequency due to pressure-induced shortening of the hydrogen bond, followed by a lowering of a barrier in double-well potential.

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However, Ichikawa et al. [12] found a linear relationship between Tc and RO–O , which is common to several KDP-type crystals, irrespective of whether they are deuterated or not. This ﬁnding suggests that the difference in RO–O A for KDP and RO–O = 2.52 ˚ A for between KDP and DKDP (RO–O = 2.50 ˚ DKDP at room temperature [10]) or the geometric isotope effect, rather than the difference in tunneling frequencies, causes the isotope effect on Tc . Later on, the role of each effect was examined more precisely by highpressure neutron diffraction measurements on KDP and DKDP [13, 14]. The linear relationship between Tc and RO–O was conﬁrmed in both KDP and DKDP for Tc > 50 K and the same slope dTc / dRO–O was found in both crystals. However, the line in DKDP was found to be shifted from the line in KDP by about 40 K toward higher values. This result indicates that only part of 40 K (≈ 37%) of the whole isotope effect (TcDKDP – TcKDP = 107 K) is caused by different masses or different tunneling frequencies between H and D, while the other part is due to different hydrogen-bond lengths in KDP and DKDP. On the other hand, the same measurements showed a common linear relationship between Tc and δ for both KDP and DKDP, which was a result that abolished RO–O and established δ as the essential structural determinant of Tc and questioned again the direct inﬂuence of tunneling frequency on Tc . High-pressure experiments are also useful for studying the effects of quantum ﬂuctuations (or zero-point motions) of atoms in the KDP crystals, since these effects become more pronounced when the critical temperature is lowered toward 0 K. The ﬁrst experimentally detected effect of this kind is the deviation from a linear phase boundary in the Tc –p phase diagram of KDP (or the Tc –RO–O phase diagram as RO–O linearly depends on p), which appears at high pressures where Tc < 50 K [11, 14, 15]. This effect is described as a quantum saturation effect, with the estimated value for the saturation temperature Ts = 49 K [15]. A much stronger effect of quantum ﬂuctuations appears at still higher pressures: at pressures higher than but near pc , the critical temperature vanishes because quantum ﬂuctuations suppress the phase transition. This regime, known as the quantum paraelectric regime, has been investigated in both KDP and DKDP by dielectric measurements [16]. It was found that the Einstein temperature, T0 , whose value should be about 2Ts and which corresponds to the frequency of lattice mode that causes quantum mechanical suppression of ferroelectricity, has a magnitude of about 130 K at pc in both KDP and DKDP. Since the proton-tunneling model has a quantum mechanical character due to the directly introduced quantum mechanical tunneling of H and D atoms, this model can explain the quantum saturation effect and the quantum paraelectric regime as a consequence of pressure-increased tunneling splitting. Thus, the tunneling frequency should correspond to the temperatures T0 and 2Ts . However, as was commented in [16], relatively low experimental values of T0 and 2Ts in KDP are too small to correspond to the proton-tunneling frequency, whose change by deuteration is responsible for the isotope effect on Tc . Therefore, this is another ﬁnding suggesting that the

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proton tunneling is not an important effect in the phase transition mechanism of KDP crystals. 2.2 Investigations of Dynamical Properties Generally, a ferroelectric phase transition is accompanied by the condensing of one polar (optically active) mode of the lattice, called a soft mode, at the Brillouin-zone center and at T = Tc . Depending on the degree of anharmonicity of a local potential for the motion of elementary dipoles, the phase transition can be of a displacive type or of an order–disorder type. For the displacive phase transition, the local potential is weakly anharmonic and of single-well type. The soft mode in this case has oscillatory and propagating character, with a frequency that tends to zero at Tc . In the opposite extreme of the order–disorder phase transition, the local potential is of deep double-well type and the soft mode has relaxational and diffusive character. In this case, the soft mode includes thermal hopping of elementary dipoles between two wells and its relaxation time tends to inﬁnity at Tc . However, if the frequency of quantum tunneling between two wells is large enough due to small mass of the elementary dipole, the phase transition may become of the displacive type with oscillatory soft mode, as assumed to be the case in the proton-tunneling model for the KDP-type ferroelectrics. The PE phase of these crystals has a tetragonal structure with D2d pointgroup symmetry, whereas the FE phase has an orthorhombic structure with C2v point-group symmetry. Since both phases have non-centrosymmetric structure, the ferroelectric soft mode can be observed in both Raman and infrared spectra. This mode is polarized along the c-axis; it has B2 symmetry in the PE phase and A1 symmetry in the FE phase. The ﬁrst Raman scattering measurement of the soft mode in KDP was performed by Kaminow and Damen [17]. They described the soft mode, which appears as a broad central peak (CP) in the PE phase spectra, as an overdamped oscillatory mode whose frequency reaches zero value at Tc . However, since the soft mode has the CP shape at all temperatures in the PE phase, it is not clear whether it should be considered as an overdamped oscillatory excitation accompanying the displacive phase transitions or as a relaxational excitation accompanying the order–disorder phase transitions. To clarify this question, Peercy performed high-pressure Raman scattering measurements on KDP [18, 19]. The room temperature soft-mode spectra from Peercy’s measurements show transition from the CP shape to the inelastic-peak shape as pressure increases [18, 19]. This observation indicates that the soft mode has to be considered as an oscillatory mode that becomes underdamped by increase in pressure, supporting thus a displacive type of phase transition and the proton-tunneling model. However, by ﬁtting the experimental soft-mode spectra to the response function of damped harmonic oscillator, the major pressure effect appears as

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a strong decrease in the soft-mode damping, while the soft-mode frequency increases only weakly with the pressure [18, 19]. This is a very surprising result in a view of the proton-tunneling model because the tunneling frequency is expected to be a strong function of hydrogen-bond length. Consequently, the strongest effect of pressure is expected to be an increase in the soft-mode frequency, which is related to the tunneling frequency, rather than a decrease of the soft-mode damping. The soft-mode spectra in the FE phase was investigated within the same study [19] and a well-deﬁned peak (S-peak) was found at 150 cm–1 for T Tc as the lowest frequency peak in the spectra. The frequency of S-mode decreases with the increase in pressure, indicating that the S-mode is the soft mode and that the phase transition is of the displacive type, which is in accordance with the proton-tunneling model. Furthermore, Raman scattering experiments on deuterated crystals showed the disappearance of the S-peak in DKDP [20]. Since this phenomenon can also be explained by the protontunneling model, it is taken as another important piece of evidence for this model. However, the Raman spectra of the longitudinal modes exhibit the S-peak in the FE phase of DKDP at the frequency higher than in KDP and this fact excludes the tunneling as an origin of the S-mode [21]. The results of this study and other detailed studies of Raman scattering spectra [22, 23] correlate well with an alternative interpretation of the character of soft mode in KDP, which was proposed by Tominaga and Urabe [22]. Their interpretation [3, 4, 22, 24] is based on the order–disorder mechanism of PO4 dipoles and according to it, the CP in the PE phase has two parts. A narrower part below 20 cm–1 is the ferroelectric relaxational soft mode (f -mode), caused by polarization ﬂuctuations of distorted PO4 tetrahedra, whereas the broader part ranging over 150 cm–1 , as well as the S-peak in the FE phase, originates from the overdamped librational mode of A1 symmetry. This mode exists in the B2 spectra of the PE phase because the symmetry of the FE phase is locally preserved in the PE phase due to the order–disorder behavior of the distorted PO4 tetrahedra. In this context, we should mention Takagi’s careful analysis and interesting interpretation of the soft-mode spectra in the FE and PE phase of KDP [24]. It is known that a CP appears together with the S-peak in the FE-phase spectra at 30 K below Tc . Takagi observed a critical increase in the intensity of CP as the temperature increased toward Tc , which led him to the conclusion that the real soft mode in the FE phase is not the S-mode, but the mode corresponding to the CP. This ﬁnding contradicts the proton-tunneling model and supports the order–disorder character of the phase transition. Another ﬁnding is that the CP in the PE phase does not consist of two different modes as in the interpretation of Tominaga and Urabe, but of a single mode, oscillatory for temperatures much higher than Tc and relaxational in the vicinity of Tc . According to this ﬁnding, KDP shows mixed behavior with

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crossover from the displacive behavior far from Tc to the order–disorder behavior near Tc . Experimental conﬁrmations of the order–disorder behavior of KDP near Tc were obtained by further studies of high-resolution Raman scattering [25], hyper-Raman scattering [26], and impulsive stimulated Raman scattering [27]. Together with the mentioned Raman scattering studies, several investigations of modes polarized along the c-axis in KDP and DKDP were performed by using infrared techniques [28–31]. Similar to the Raman scattering spectra, a broad CP exists in the infrared reﬂectivity spectra of the PE phase of KDP and disappears as the temperature is decreased to 30–40 K below Tc . By analogy with Kaminow and Damen’s analysis of the Raman scattering spectra [17], Simon, Gervais, and Courtens analyzed these infrared spectra [28] within the terms of the displacive phase transition mechanism, in which the CP was described as an overdamped oscillatory soft mode. Disagreements between the diplacive type of soft-mode response and the experimental infrared spectra at temperatures around Tc in the frequency region below 20 cm–1 were noticed by Bréhat and Wyncke [29]. They explained this disagreement as a result of an additional relaxational mode, which is the real ferroelectric soft mode, in accordance with the order–disorder picture of phase transition proposed by Tominaga and Urabe [22]. The order–disorder mechanism of the phase transition in KDP has been more recently conﬁrmed by more precise infrared reﬂectivity measurements where synchrotron radiation was used [30]. The central result of this study is that the CP consists mainly of the soft relaxational mode, which suggests the order–disorder phase transition. Additionally, and contrary to the previous infrared studies, the S-mode was found in the FE phase as a very weak peak, which persisted in the PE phase as a highly damped mode contributing to the CP. The ﬁnding that the CP consists of two modes provides an even stronger support for the interpretation of the phase transition mechanism given by Tominaga and Urabe [22]. An infrared reﬂectivity study of modes polarized along the c-axis in DKDP [31] has also provided interesting results. First, the CP originating from the soft mode was clearly observed in these spectra. This peak is much narrower than the CP in KDP, which is in accordance with Raman scattering study of partially deuterated KDP crystals [20], and indicates slower relaxational dynamics in DKDP. Impulsive stimulated Raman scattering study [27] conﬁrmed this fact: the relaxation time of soft mode, τ, which shows critical slowing down, i.e., τ = τ0 Tc /(T – Tc ), has a value of τ0 about eight times higher for DKDP than for KDP. Second, in contrast to the Raman scattering spectra of transverse modes [20], but in accordance with the Raman scattering spectra of longitudinal modes [21], the S-peak was found in the infrared spectra of the FE phase of DKDP at a frequency higher than in KDP [31]. Third, while the S-mode in KDP, even if it exists in the PE phase, contributes to the CP in the infrared spectra, the S-peak in DKDP is clearly visible in the infrared spectra of the PE phase, where it is well resolved from the CP [31]. This property of the

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S-mode, which is not expected in the PE phase from symmetry considerations, together with the fact that the CP is much narrower in DKDP than in the KDP, suggests that the phase transition is much more order–disorder-like in DKDP than in KDP. Along with the Raman and infrared studies of modes polarized along the c-axis, a few studies of modes polarized perpendicular to this axis (along a,b-axis) have also been performed [20, 28, 31–33]. These modes are doubly degenerate modes of E symmetry in the PE phase, whereas in the FE phase, they are non-degenerate modes of B1 and B2 symmetry. In both KDP and DKDP, anharmonic effects connected to the phase transition are clearly visible in the spectra of these modes. In addition to the peaks of expected modes, the Raman scattering spectra of KDP in the PE phase show a broad and temperature-independent low-frequency continuum, which ranges from 0 to 100 cm–1 and disappears in the FE phase at 20 K below Tc [20, 32]. Infrared reﬂectivity spectra of KDP also show this low-frequency feature [28, 33], whose origin has not yet been fully clariﬁed, but can be explained in two different ways. According to the ﬁrst explanation, the low-frequency feature is an extra mode, which could be a collective proton mode of E symmetry [33]. The second explanation is that this feature is caused by low-frequency tails of expected lattice modes [28]. Irrespective of which interpretation is correct, it has been shown that the temperature dependence of the low-frequency continuum is responsible for the temperature dependence of the static dielectric constant εa [32]. As the degree of deuterization increases, the continuum in the Raman spectra transforms into the well-deﬁned CP for DKDP [20]. The CP is also found in the infrared reﬂectivity spectra of E symmetry modes in DKDP as an extra mode in the group-theoretical sense [31]. In summary, it can be said that the dynamics of both longitudinal (or c-component) and transverse (or a,b-component) polarization in DKDP is characterized by the preservation of all modes of the FE phase in the PE phase, as well as by the appearance of a clearly resolved extra relaxational mode, when the PE phase is approached from the FE phase. These results and the fact that the relaxational mode of longitudinal polarization dynamics is the soft mode strongly suggest that the phase transition in DKDP has a strong order–disorder character, and that local potential for elementary dipoles has, beside two wells along the c-axis, also the wells along a- and b-axes. Such a pure order–disorder mechanism is not apparent in KDP for two reasons. First, even if the S-mode of longitudinal polarization dynamics exists in the PE phase, it is not clearly resolved from the relaxational mode. Second, it is still not known whether the low-frequency feature of transverse polarization dynamics can be ascribed to the extra relaxational mode. These results suggest that KDP is a less anharmonic crystal than DKDP and that the phase transition in KDP has a mixed displacive and order–disorder character rather than a pure order–disorder character.

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The dynamics of KDP and DKDP lattices were also investigated by inelastic neutron scattering, which is a particularly good probe for studying proton dynamics [34–41]. The ﬁrst reliable assignment of proton vibrational modes in KDP was made by measuring incoherent inelastic neutron scattering over a wide energy range [34]. Two bending modes at 125 and 159 meV and one stretching mode at 161 meV of the proton vibrations were found. Additionally, no drastic change was observed between the PE phase and the FE phase of the part of neutron scattering spectrum that corresponds to the stretching mode. Since the strong differences of hydrogen-bond potential are expected between these phases within the proton-tunneling model, this ﬁnding was recognized as possible evidence against this model. Suspicions about validity of this model became even stronger after a neutron Compton scattering experiment was performed on KDP [35]. The observed proﬁle of the peak corresponding to the Compton scattering from protons at 40 K (T < Tc ) looks the same as that at 300 K (T > Tc ). Generally, the proﬁle of this peak reﬂects the shape of the ground-state wave function of the proton. If the proton-tunneling model were applicable, the proﬁle should drastically change at Tc , because in this case the ground-state wave function of the proton transforms from a single-maximum type at T < Tc to a double-maximum type at T > Tc . In contrast to this experiment, a more extended neutron Compton scattering experiment on KDP showed a considerable change in the momentum distribution of the proton along the hydrogen bond at Tc [36]. From the most likely momentum distribution in the PE phase, the double-peaked groundstate wave function of the proton was obtained, which is in accordance with the proton-tunneling model. However, a very high value of estimated tunneling splitting (∼ 1000 K) suggests insigniﬁcance of the proton tunneling in the phase transition mechanism. By comparing neutron and Raman scattering spectra, two deuteron vibrational modes in DKDP were assigned at 87 and 106 meV [37]. The ratios of frequencies of these deuteron modes and corresponding proton modes, which are close to the ratio for harmonic potential, suggest that these modes are local modes in a weakly anharmonic potential, not connected to the possible tunneling mode. Besides the peaks of the local proton modes typical for hydrogen bond, a sharp peak at 28 meV was observed in KDP [34] and attracted much attention [34, 38, 39]. This peak exists in DKDP at somewhat higher frequency; its intensity decreases in both crystals and its width decreases upon the transition from the FE to the PE phase, without any softening of its frequency [38]. Hence, it is concluded that this mode is connected with the phase transition dynamics, i.e., coupled to the polarization ﬂuctuations. This mode is not the tunneling mode or any local mode of proton or deuteron, but rather some collective optical mode of the lattice that involves substantial proton or deuteron displacement. It has been suggested [38] that this mode corresponds to the mode that has a peak at about 200 cm–1 (25 meV) in Raman scattering and infrared reﬂectivity spectra, and that it is coupled to the soft mode and usually

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described as the external lattice mode of K – PO4 translation [18–21, 26, 28– 31]. In contrast to this suggestion, analysis of the pressure dependence of inelastic incoherent neutron spectra in KDP attributes this mode to the librational mode of the PO4 group about the c-axis [39]. The low-frequency region of proton dynamics in KDP, which corresponds to the frequency region of the CP observed in both Raman and infrared spectra, was also investigated by neutron scattering [40, 41]. There were no inelastic peaks observed in these spectra that could correspond to the tunneling mode, but instead, a strong quasi-elastic component, or a CP, was found to appear in the PE phase and its width to increase from 7 meV at 130 K to 30 meV at room temperature. The origin of this component can be explained within the order–disorder picture as classical thermal over-barrier hopping of the proton between two minima. However, a very fast dynamics of this motion suggests phonon-assisted tunneling motion, which falls between classical hopping and quantum tunneling motion. 2.3 Investigations of Local Properties Generally, investigations of local properties of ferroelectric crystals can provide additional information about the phase transition mechanism. Although some local properties, such as thermal distribution of the motion of a particular atom, could be deduced from the scattering techniques, directspace local techniques, such as nuclear magnetic resonance (NMR), nuclear quadrupole resonance (NQR) and electron paramagnetic resonance (EPR), have some advantages in analysis of strongly anharmonic crystals. One of the advantages is a great sensitivity to atomic displacements, whereas a need for doping of the original crystal is an obvious disadvantage, especially in EPR. ˇ ekˇs [42] reviewed the most relevant NMR and NQR experiments, Blinc and Z while Dalal [43] and Müller [44] reviewed the most relevant EPR experiments, which had all been performed on the KDP-type ferroelectrics and antiferroelectrics before 1985. The most informative studies of local properties of O – H(D) – O hydrogen bond in KDP and DKDP were done with the 17 O NQR spectroscopy in KDP and the deuteron NQR in DKDP, because the electric ﬁeld gradient (EFG) tensor at a nucleus site, which determines the position of NQR lines, is highly sensitive to even small atomic displacements and changes in local symmetry [42]. This can be illustrated by the fact that, when the temperature is decreased below Tc , each NQR line splits due to the appearance of two different domains, i.e., two different conﬁgurations of the hydrogen bond. The main result of these studies is that H and D atoms show clear two-site order– disorder behavior, which follows from the fact that the EFG tensor in the PE phase is temperature-independent and strictly equal to the averaged values of two EFG tensors corresponding to two sets of NQR lines in the FE phase. This

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result suggests that a protonic order–disorder mechanism with possible tunneling contribution may be the main phase transition mechanism in KDP and DKDP. In addition to the NQR measurements of atoms involved in hydrogen bonds, the NQR measurements of central tetrahedral atoms in the KDP-type ferroelectrics also produced interesting results. Since the 31 P nucleus has no quadrupole moment, the 75 As NQR measurements were used for analysis of the KDP-type arsenates [42]. Unlike the 17 O and D EFG tensors, the 75 As EFG tensor in these crystals shows temperature dependence in the PE phase, which appears as an anomalous increase in the 75 As NQR frequency when the temperature decreases to Tc . Since this result contradicts the simple two-state order–disorder mechanism of PO4 (or H2 PO4 ) dipoles, it was interpreted as originating from a protonic order–disorder mechanism with six possible states of four protons around one PO4 tetrahedron [42]. This interpretation corresponds to the Slater model for KDP [45], in which the H2 PO4 group can exist in two polar states and four lateral states with higher energy. According to this interpretation, the EFG tensor is different for lateral and polar states, so that temperature-induced changes in the population between these two states causes a temperature dependence of averaged EFG tensor. More recently, a high-precision NMR technique has been employed to measure 31 P isotropic chemical shift in KDP-type ferroelectrics [46]. The temperature dependence of this quantity shows a S-shape anomaly around Tc , which is another unexpected result if the two-state order–disorder mechanism of PO4 groups holds. More precisely, because of symmetry requirements two polar states of PO4 groups have to have the same values of 31 P isotropic chemical shift; hence, this quantity should show insensitivity on the phase transition if this mechanism holds. In contrast to the above interpretation of the temperature dependence of the 75 As EFG tensor, this experimental ﬁnding was not interpreted in terms of additional states of the H2 PO4 group, but taken as evidence for the existence of the displacive component, apart from the order–disorder one, in the whole phase transition mechanism [46]. This interpretation is based on the theoretical model of Bussmann-Holder and Michel [47] and similar observations on the squaric acid crystal [48]. To apply the EPR technique for studying local properties of KDP-type ferroelectrics, paramagnetic centers have to be artiﬁcially introduced as local probes into these crystals. The most investigated paramagnetic centers are AsO4 4– , SeO4 3– , and CrO4 3– radicals with S = 1/2, which replace the original PO4 3– or AsO4 3– anion and are produced by doping and γ - or X-irradiation of the host crystal [43, 44]. EPR spectra of the AsO4 4– radical in the KDP-ferroelectrics show transformation from the PE phase symmetry to the FE phase symmetry not at Tc , but at the temperature T ∗ well above Tc . The explanation of this ﬁnding is that the probe disturbs the host crystal in the PE phase by forming two lo-

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cally stable ferroelectric conﬁgurations, or polar Slater states. In other words, the AsO4 4– probe in the PE phase shows more order–disorder-like behavior than the original PO4 3– or AsO4 3– groups, so that hopping dynamics of the probe between two states is several orders of magnitude slower (107 –1010 Hz) than the dynamics of the original groups. The hopping rate of the probe increases with temperature in such a way that it is slower than the characteristic EPR frequency below T ∗ and becomes faster than this frequency above T ∗ , and thus the EPR spectrum shows symmetry change at T ∗ instead of Tc . Since it was assumed that the excess electron on the AsO4 4– radical induces locally polar conﬁgurations, the SeO4 3– radical was recognized as a better probe for studying local properties of the KDP-ferroelectrics. However, it was found that this radical also induces polar conﬁgurations, but to a lesser extent. Thus, it can be concluded that the unpaired electron in an antibonding orbital, rather than the excess charge on the radical, induces polar conﬁgurations in the original crystal. In contrast to these two radicals, the CrO4 3– radical induces locally lateral conﬁgurations in KDP-ferroelectrics. In summary, all three probes show some kind of perturbed zone around the paramagnetic defect site with speciﬁc local symmetry, order, and dynamics. The hopping dynamics of the probes causes strong changes of the widths of EPR spectral lines around T ∗ . This effect has been used in a number of studies to estimate the temperature dependence of the inverse hopping rate or the correlation time, τc , for the probe. Several studies have shown that the temperature dependence of τc follows the simple Arrhenius law, τc = τc0 exp(∆E/kB T), where ∆E and τc0 are temperature-independent activation energy and preexponential factor, respectively. The general experimental fact is that ∆E has a higher value, while τc0 has a lower value in the deuterated crystal compared to the non-deuterated crystal. However, the SeO4 3– radical in KDP was found to be one of several systems where the deviation from the Arrhenius law was detected and where this deviation was interpreted in terms of the local pseudo-freeze-out model. In this model [49], the probe is the impurity whose dipole is coupled to the surrounding dipoles stronger than the dipole of original anion, so that locally stable ferroelectric states at a temperature above Tc are induced by the impurity. By lowering the temperature to Tc , the activation energy for jumping of the system between these two states shows a strong and non-linear increase due to the enhanced response of the original crystal to the impurity dipole, which leads to a nonArrhenius behavior of τc . It has been more recently shown that the reported non-Arrhenius behavior of some systems is caused by the omission of two processes in the analysis of EPR spectra. One is the broadening of EPR lines at high temperatures, which depends exponentially on temperature, and the other is inhomogeneous broadening of EPR lines due to the proton spin ﬂip transitions [50–52]. By taking into account these processes, simple Arrhenius behavior of τc was found down to Tc for the SeO4 3– radical in KDP [52]. Since this result contradicts the local pseudo-freeze-out model, a new model is re-

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quired to describe local dynamics of this system. Additionally, such a model should be able to reproduce the isotope effect on parameters ∆E and τc0 . To determine static properties of the SeO3– 4 radical in KDP and DKDP, the temperature dependence of the hyperﬁne interaction between unpaired electron and 77 Se (I = 1/2) nucleus was measured [53]. The hyperﬁne tensor component A|| , where the direction || is along the c-axis, shows an isotope effect, because its value is higher in DKDP than in KDP. Furthermore, its value shows a jump at Tc for DKDP and a considerable temperature dependence in the PE phase of both crystals, approximated by the relation A|| (T) = A|| (0) – B coth(T0 /T), where T0 ≈ 570 K for both crystals. It is interesting to note that A|| , similarly to the 75 As NQR frequency and 31 P isotropic chemical shift, should be constant in the PE phase if the two-state order– disorder mechanism of the corresponding tetrahedron holds. However, while the temperature dependencies of the 75 As NQR frequency and 31 P isotropic chemical shift in the PE phase were explained as originating from a six-state order–disorder mechanism [42] and additional displacive mechanism [46], respectively, here it was assumed that excitation of some extra lattice vibration mode with frequency T0 affects the hyperﬁne tensor components and causes the temperature dependence of A|| .

3 Theoretical Models for KDP-Type Ferroelectrics 3.1 Brief History of Models and the Proton-Tunneling Model Historical development of theoretical models of the ferroelectric phase transition in KDP up to the 1980s has been reviewed by several authors [1, 4, 42, 54]. Historically, the ﬁrst such model was the Slater model, proposed in 1941 and still useful as a basis for more complex theories [1, 4, 42, 45, 54]. The Slater model is based on the assumption that conﬁgurations of protons, where each may occupy one of the two sites along hydrogen bond, determine the dielectric properties of the crystal. The only conﬁgurations of four protons around one PO4 group that are allowed are those that satisfy the ice rule, i.e., H2 PO4 conﬁgurations, where the PO4 group has two close and two far protons. Due to the symmetry of the crystal, two kinds of states for one H2 PO4 group exist: One is a polar state where the two upper or two lower protons with respect to the c-axis are close to the PO4 group, so that the resulting H2 PO4 dipole lies along the +c or –c direction. The other is a lateral state where one upper and one lower proton are close to the PO4 group, so that the resulting H2 PO4 dipole lies in the ab-plane. Since the energy of lateral states is higher by ε than the energy of polar states, the ferroelectric phase transition appears at the critical temperature given by kB Tc = ε/ ln 2. An unusual property of this

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model is that the whole saturated polarization appears immediately at Tc and the model does not allow any proton rearrangements, for which HPO4 and H3 PO4 conﬁgurations are required. To change these properties of the Slater model, all 16 conﬁgurations of protons around PO4 were introduced in the Slater–Takagi model [1, 42, 54, 55]. This model can be easily transformed into the pseudo-Ising form, so that KDP is described as the protonic system with the Hamiltonian: 1 z z Hproton = – Jij Si Sj . (1) 2 i j

Eigenvalues ±1/2 of pseudo-Ising operators Szi correspond to the two equilibrium sites of the i-th proton along the bond, whereas Jij is the interaction constant between neighboring (i and j) protons producing the ferroelectric phase transition. Further improvement of this model was done by Senko, who introduced a long-range dipolar interaction energy [1, 42, 54, 56]. The main limitation of the Slater–Takagi–Senko model for KDP is inefﬁciency in explaining the quantum effects of isotope replacement. The elimination of this limitation of the model was attempted by introduction of the quantum effect of the proton-tunneling motion between two equilibrium sites [1–4, 42]. The tunneling frequency has a value Ωt = ∆/, where ∆ is the tunneling energy splitting between the two lowest ground states of the proton in a double-well potential of the hydrogen bond. Introduction of the proton-tunneling energy into Eq. 1 leads to the Hamiltonian of the proton-tunneling model, which has the following form: 1 z z Hproton = – ∆Sxi – Jij Si Sj . (2) 2 i i j

Within the mean-ﬁeld approximation (MFA), the critical temperature and the saturated polarization are given by the relations: ∆ Jij (3) kB Tc = J0 = –1 2 tanh 2∆/J0 j and Ps = Ps (∆ = 0)

1–

2∆ J0

2 .

(4)

Equations 3 and 4 show that the tunneling motion acts as a disordering mechanism, which reduces the values of Tc and Ps and suppress the phase transition when the tunneling splitting reaches the critical value ∆c = J0 /2. Since the tunneling energy splitting has a larger value for the proton than for the deuteron (∆H > ∆D ), the reduction of Tc and Ps is larger for KDP, i.e., the proton-tunneling model can account for the isotope effect on these quantities. Moreover, the pressure-induced decrease of Tc in KDP and DKDP to the

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value of 0 K at the critical pressure can be easily explained within this model as resulting from the increase in ∆ due to pressure-induced shortening of the hydrogen-bond length and lowering of a tunneling barrier in the hydrogenbond potential. However, there are experimental results that are inconsistent with this model. First, the ﬁnding that Tc in KDP and DKDP does not so strongly depend on the mass of tunneling particle as it does on the geometrical parameters RO–O and δ is difﬁcult to explain within the proton-tunneling model [13, 14]. Second, it is questionable whether the effects of quantum saturation and quantum paraelectricity in KDP and DKDP can be ascribed to quantum tunneling [15, 16]. Finally, in the proton-tunneling model thermal distribution of protons over two levels split by ∆ gives a speciﬁc temperature dependence of dielectric susceptibility in the PE phase, which is in contrast with the experimentally detected Curie–Weiss behavior. Introduction of the tunneling motion changes the dynamical properties of the protonic order–disorder system described by Eq. 1 in such a way that the soft mode becomes a collective proton-tunneling mode with an oscillatory character, as in the displacive phase transition. The oscillatory character of the soft mode of KDP is experimentally supported by ﬁndings that the CP in the soft-mode Raman spectra of the PE phase transforms into an inelastic peak at high pressures and that lowest frequency inelastic S-peak in the softmode Raman spectra of the FE phase exhibits soft-mode behavior [18, 19]. Since disappearance of the S-peak in DKDP [20] can be explained by small tunneling splitting of deuteron, this effect can be taken as further experimental evidence for the proton-tunneling model. Another limitation of the Slater–Takagi–Senko model is the assumption that elementary dipoles are determined only by protonic rearrangements around the PO4 group, while displacements of the heavy K, P, and O ions are not explicitly considered. To avoid this limitation, the coupled tunneling proton-phonon model [1, 3, 4] was introduced. In this model, the heavy-ion sublattice, or simply a lattice, is a harmonic phonon system, which produces electric polarization along the c-axis and is coupled to the tunneling motion of protons. The static properties of this model within the MFA are qualitatively the same as the properties of the pure proton-tunneling model described by Eq. 2. Hence, Eqs. 3 and 4 still hold, but the effective interaction between protons J0 now comprises a phonon-mediated part of interaction in addition to the direct proton–proton part. The dynamics in this model within the random-phase approximation (RPA) is characterized by existence of two branches of coupled lattice-proton modes with frequencies ω– (q) and ω+ (q), where – and + denote modes in which proton and lattice system oscillate in phase and in opposite phase, respectively. The soft mode of the ferroelectric phase transition has the frequency ω– (q = 0), so that the oscillatory character of the soft mode is preserved in the coupled tunneling proton-lattice model. Comparison of experimental and theoretical results became possible by introduction of the coupled tunneling proton-lattice model because two coupled

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modes were observed in Raman scattering and infrared reﬂectivity spectra in the FE phase of KDP [18–21, 26, 28–31]. These modes are the S-mode and the mode at 200 cm–1 , which are assigned to the ω– and ω+ modes of the model. However, several experimental facts about the soft-mode dynamics in KDP and DKDP are inconsistent with this model. First, pressure increase in the PE phase of KDP strongly affects soft-mode damping and weakly affects soft-mode frequency [18, 19], while the opposite is expected from the model. Second, the real soft mode in the FE phase is not oscillatory S-mode, but relaxational mode, which corresponds to the CP [24]. Third, the S-peak exists in the spectra of longitudinal modes for both KDP and DKDP, with a frequency that is higher in DKDP than in KDP [21]. From a theoretical point of view, a few comments about the validity of the MFA (and the RPA) treatment of the coupled tunneling proton-phonon model are worth noting. It has already been mentioned that in such a treatment, the only effect of lattice on the protonic subsystem is additional, phononmediated interaction between protons, so that the basic properties of the proton-tunneling model remain intact. In other words, not only the difference in ∆ between proton and deuteron causes isotope effects in KDP, but the phase transition is governed only by proton dynamics. This means that excitations in the protonic subsystem rather than in the lattice subsystem destroy the ferroelectric state above Tc . Because of that, the soft mode has an oscillatory character with frequency closely related to the tunneling frequency. It seems that such a treatment of the lattice effect is justiﬁed only if the adiabatic approximation for lattice motion is valid. In this approximation, phonon coordinates adapt adiabatically to any change in the proton coordinates; hence, this approximation is suitable in the case where the phonon frequency, ω0 , is much larger than tunneling frequency, i.e., for the limit ω0 Ωt . It can be said that in this approximation the lattice dynamics is “slaved” to the proton dynamics, which triggers the phase transition. However, the limit ω0 Ωt is not conﬁrmed experimentally. On the contrary, a very large value of tunneling splitting (∼ 1000 K), which has been estimated in a more recent neutron Compton scattering experiment in KDP [36], suggests that the opposite limit is valid. As will be shown in the discussion of the SDPC model [6], the assumption that the opposite limit is valid leads to different properties of the coupled lattice-proton system. There is another idea about how to describe the lattice subsystem within the coupled lattice-proton system. It assumes that the lattice responsible for electric polarization is the order–disorder system of PO4 groups rather than the harmonic phonon system. One of the models following this idea is the model of Tokunaga and Matsubara [4, 57]. According to their model, each PO4 group can be in two polar states, with dipoles along + and – c-axis, so that it can be described as an Ising spin. These dipoles are coupled to the tunneling protons and interact among themselves with dipole–dipole interactions, so that the phase transition in this model is driven by all three

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interactions (dipole–dipole, dipole–proton, and proton–proton) and the isotope effect on Tc is caused by isotope change of ∆. It seems that this model can explain the order–disorder-like behavior of P atoms, which was detected by neutron diffraction measurements in KDP and DKDP [9], as well as several results for the soft-mode dynamics that indicate PO4 order–disorder mechanism in these crystals [22–27, 29–31]. In accordance with experimental investigations suggesting that transverse polarization ﬂuctuations substantially contribute to the phase transition mechanism of KDP and DKDP [8, 20, 28, 31–33], Fujii extended the model in such a way that each PO4 group can occupy four lateral states, with dipole along ±a-axis and ±b-axis, beside two polar states [58, 59]. In this model, obviously inspired by the Slater model, direct dipole–dipole and direct proton– proton interactions were assumed to be much weaker than dipole–proton interactions and therefore neglected. The proton-tunneling terms were neglected too, so that the isotope change of the dipole–proton coupling constants between KDP and DKDP due to the geometric isotope effect was proposed to cause the isotope effect on Tc . In this model, excitations in protonic and lattice subsystems, i.e., jumps of the protons between two states and jumps of the dipoles between six states, are equally responsible for destruction of ordered ferroelectric phase at Tc . As a major defect of this model one can note the fact that the values of dipole–proton coupling constants were taken as ﬁtting parameters in the model, without employment of quantum mechanical calculations that could quantitatively explain the difference between KDP and DKDP. Additionally, an explanation of the isotope effect on saturated polarization is lacking in the model. This lack arises from the assumed rigorous order–disorder picture of PO4 dipoles and protons, which is apparently an oversimpliﬁed picture and indicates that introduction of more realistic onsite potentials for protons and dipoles is required. 3.2 Development of the Modified Strong Dipole–Proton Coupling Model and Simulation Methods Requirements for more realistic on-site potentials for the protons and PO4 dipoles within the coupled proton-lattice system of KDP were fulﬁlled by the strong dipole–proton coupling (SDPC) model, introduced by Sugimoto and Ikeda [6, 60–62]. In the SDPC model, similarly to the coupled tunneling proton–phonon model [1, 3, 4], the lattice is a harmonic phonon system. This system consists of N PO4 dipoles, µi (i = 1, ..., N), which are aligned along c-axis and produce electric polarization. The on-site harmonic potential with elastic constant A acts on these dipoles, and they interact among themselves by dipole–dipole interactions. These interactions were approximated by nearest-neighbor interaction of the form Bµi µj /4, where B is the dipole–dipole interaction constant. For the on-site potential of H and D atoms

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within the hydrogen bond, O – H(D) – O, the symmetric double-well potential of double-Morse type was assumed. The coupling between the lattice and H or D atom in the hydrogen bond that links two neighboring PO4 groups (i and j) were introduced by the force, Fij , proportional to the sum of these PO4 dipoles: (5) Fij = K µi + µj , where K is the dipole–proton interaction constant. This force acts along the hydrogen bond, i.e., in the O – O direction, thus producing an asymmetric part – Fij xij in the hydrogen-bond potential, where xij is the position of H or D atom along the bond, relative to its center. In contrast to the theoretical treatments of the coupled tunneling protonphonon model [1, 3, 4], in the SDPC model, two assumptions about the coupled proton-lattice system had been made before further calculations were done. The ﬁrst assumption is adiabatic approximation for motion of the H and D atoms. This approximation can be applied because the relation ω0 ∆ presumably holds, where ∆ and ω0 are characteristic excitation energies of the H and D atoms and of the PO4 dipole, respectively. Therefore, the wave functions of H and D atoms connecting PO4 groups i and j are well described by the adiabatic wave functions, Ψijm (m = 0, 1...). These functions are solutions of the Schrödinger equation for momentary values of µi and µj , and the corresponding energies, Em ij (µi , µj ), contribute to the potential energy of the dipole system. The second assumption is that the relation kB Tc ∆ holds, so that energies of excited states, m = 1, 2..., can be neglected for the temperatures of interest (around Tc ). Consequently, only ground-state energies of the H and D atoms, E0ij (µi , µj ), contribute to the potential energy of the dipole system. The system of N PO4 dipoles connected with 2N hydrogen bonds, which describes KDP and DKDP, was treated classically in the SDPC model. Thus, according to the above discussion, the Hamiltonian of this system has the following form: (6) H = Ekin p1 , ..., pN + Epot µ1 , ..., µN =

N N 2N 2N p2i A 2 B + µi + µi µj + E0ij µi , µj , 2M 2 4 i=1

i=1

ij

ij

where pi is momentum of the PO4 dipole i and M is the effective mass of PO4 dipole. To ﬁnd the values of ground-state energies of H and D atoms in the phenomenological hydrogen-bond potential of double-well type, numerical solving of the Schrödinger equation was performed. It turned out for both atoms that the ground-state wave function has only one peak and that ground-state energy depends non-linearly on Fij given by Eq. 5, according to

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the following relation: E0ij µi , µj = – h2 + I 2 Fij2 + h .

(7)

The ground-state energy terms of this form contribute to the potential energy of the Hamiltonian from Eq. 6, where they act as effective ferroelectric interactions between neighboring PO4 dipoles, thus being responsible for phase A) transition in KDP and DKDP. In the case of KDP (H atom and RO–O = 2.50 ˚ A, calculated values of parameters h and I are hH = 110 meV and I H = 0.22 ˚ while in the case of DKDP (D atom and RO–O = 2.52 ˚ A), they are hD = 58 meV A. and I D = I H = 0.22 ˚ It was assumed that values of other model parameters (A, B, K, and M) are the same for both KDP and DKDP. These values were chosen to reproduce some experimental results for KDP. More precisely, KDP and DKDP systems have the Hamiltonian deﬁned by Eqs. 5–7. In the case of classical PO4 dipoles both systems show a ferroelectric ordered state at T = 0 K where µi = µ holds for each PO4 dipole. It follows from Eq. 6 that the potential energy per dipole in this state is given by: A+B 2 eFE µ = µ – 2 h2 + 4I 2 K 2 µ2 + 2h . (8) 2 This energy has minima at µ = ±µs , where µs is a saturated dipole value, so that the saturated force has a value Fs = 2Kµs . The frequency of smallamplitude ferroelectric mode, where all dipoles vibrate in phase around µs , 1/2 , where e FE (µs ) is the second derivative of is given by ωFE = e FE (µs )/M eFE (µ) with respect to µ at µ = µs . First, from the value of spontaneous polar–30 C m. Second, ization in KDP Ps = 5.0 µC cm–2 , it follows that µH s = 4.8 × 10 –1 ˚ for the saturated force value Fs = 0.6 eV A excitation energies of the H atom and its shift from the center of the bond agree well with the experimental values: bending mode energies of 125 meV and 159 meV, stretching mode energy of 161 meV, and proton shift of 0.18 ˚ A, which were obtained by neutron scattering measurements [9, 10, 34]. Then, from these values of µH s and Fs and by using Eq. 8 and Fs = 2Kµs , the values of parameters A + B and K were obtained. Additionally, by application of the classical Monte Carlo (CMC) simulation for KDP, the value of parameter A was adjusted to reproduce phase transition near Tc = 122 K [60]. Finally, in order to explain the peak at 28 meV in neutron scattering spectra of KDP [34] within the terms of dipole vibrations, the value of the last unknown parameter M was chosen to give the value ωFE = 280 cm–1 (ωFE = 34 meV) for the ferroelectric-mode frequency [61]. In conclusion, according to the above mentioned properties of the SDPC model, high excitation energies of vibrational motion of H and D atoms (> 100 meV), when compared to the excitation energy for the dipolar motion (< 35 meV) and the thermal energy at Tc (< 20 meV), lead to the adiabatic motion of H and D atoms in its ground states, without tunneling or any other vibrational motion. Consequently, all isotope effects originate only from the

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isotope change of the response of the ground-state energy to the force induced by dipole moments of PO4 groups. More precisely, isotope effects are caused by the difference in parameter h between KDP and DKDP, which appears because of both different masses of H and D atoms and different hydrogen-bond lengths in KDP and DKDP. Several isotope effects can be reproduced by the SDPC model. As can be seen from Eq. 8, isotope change hH → hD leads to the increase in saturated H dipole value of DKDP. The calculation [60] gives µD s = 1.2µs , which agrees well with experimentally detected isotope effect on saturated polarization (Ps = 5.0 µC cm–2 in KDP, Ps = 6.2 µC cm–2 in DKDP [8]). Since a higher value of µs implies a higher value of Fs , it seems that the SDPC model can explain a higher value of the shift of D atom in the FE phase of DKDP, compared to the shift of H atom in the FE phase of KDP. Also, Eq. 8 shows that ferroelectric state at T = 0 K has lower energy per dipole in DKDP than in KDP, which indicates that Tc has a higher value in DKDP than in KDP. The CMC simulation conﬁrmed this fact and the value Tc = 310 K was obtained for the critical temperature of the DKDP system [60]. Additionally, isotope change hH → hD induces an increase in the ωFE in DKDP, which can explain the experimental fact that the peak at 28 meV in the neutron scattering spectra of KDP shifts to the higher frequencies in DKDP [61]. Moreover, the SDPC model was applied to explain the following features in the neutron scattering spectra of KDP: distribution of excitation energies of vibrational hydrogen modes [60], existence of the mode at 28 meV [61], and appearance of the CP in the PE phase [62]. In conclusion, it seems that introduction of the SDPC model provided a new and consistent picture of the phase transition in KDP and DKDP, together with a successful explanation of the isotope effects. However, it seems that possibilities of this model to explain some of the experimental results were not fully explored. Also, this model shows some weaknesses. First, it predicts a value of Tc in DKDP (310 K) that is higher than the experimental value (229 K). Second, the calculated rate of the proton transfer in the PE phase is too low to explain the width of the CP in the neutron scattering spectra of KDP [62]. Third, and most important, because the model assumes that dipoles of PO4 groups lie along the c-axis, it cannot to explain static and dynamic properties of the transverse polarization ﬂuctuations, whose importance in the phase transition mechanism has been supported by several experimental results [8, 20, 28, 31–33]. These weaknesses of the SDPC model stimulated Merunka and Rakvin [7, 63] to introduce some modiﬁcations to the model. Inspired by the basic ideas of the Fujii model [58, 59], they decided to represent the PO4 dipole i in this modiﬁed SDPC model, or the MSDPC model, by the vector µi = (µai , µbi , µci ). This vector has additional transverse components along the a- and b-axes, beside the longitudinal component along the c-axis. The corresponding impulse is now pi = (pai , pbi , pci ) and the value of the effective mass of the transverse dipole, M⊥ = Ma = Mb , differs from the value of the effective mass of longi-

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tudinal dipole, M = Mc , so that the kinetic energy of the system of N PO4 dipoles is given by: N pai pb pc + i + i . Ekin p1 , ..., pN = 2M⊥ 2M⊥ 2M 2

2

2

(9)

i=1

In the MSDPC model, the on-site harmonic potential that acts on PO4 dipoles is described by longitudinal and transverse elastic constants, denoted as A and A⊥ , respectively. It is assumed that the most important interaction in the coupled proton-lattice system of KDP is the dipole–proton interaction, so that the direct interaction between PO4 dipoles is neglected in the MSDPC model, as opposed to the SDPC model. However, all assumptions about the excitation energies of PO4 dipoles and of H and D atoms from the SDPC model are supposed to be valid. Moreover, the same form of hydrogen-bond potential from the SDPC model is assumed to be valid, so that the ground-state energies of the H and D atoms, E0ij , have the same dependence on Fij (Eq. 7) and, consequently, these energies have the same parameters h and I. However, since Fij in the MSDPC model depends on all three dipole components of two neighboring PO4 groups, the same is true for E0ij . Therefore, the potential energy in the MSDPC model has the form: N N 2N

A c2 A⊥ a2 b2 µi + µi + µi + E0ij µi , µj . Epot µ1 , ..., µN = 2 2 i=1

i=1

ij

(10) Two different dipole–proton coupling constants occur in the MSDPC model. They are the longitudinal and transverse coupling constants, denoted as K and K⊥ , respectively. As in the SDPC model, the PO4 dipole i induces the force of magnitude Kµci on the four surrounding protons or deuterons when it lies along the c-axis. However, in the MSDPC model this dipole can lie along the a-axis (or the b-axis), in which case it induces the force of magnitude K⊥ µai (or K⊥ µbi ) on four surrounding protons or deuterons. The form of Fij for arbitrary dipole moment vectors of two neighboring PO4 groups is given in [7]. By analogy to the SDPC model, the parameters A, A⊥ , K, K⊥ , M, and M⊥ of the MSDPC model have the same value for KDP and DKDP. For the KDP and DKDP system of classical PO4 dipoles in ferroelectric ordered state at T = 0 K, all PO4 dipoles have components µi = (0, 0, µ). The potential energy per dipole and the ferroelectric-mode frequency in this state are given by the same relations from the SDPC model (Eq. 8 and ωFE = (e FE (µs )/M)1/2 ), where B = 0. To reproduce the same saturated dipole value µs = µH s and the A–1 for KDP, the constants A same saturated force value Fs = 2Kµs = 0.6 eV˚ and K have the same values as do the constants A + B and K in the SDPC model, respectively. Therefore, the isotope effect on µs and Fs has the same

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magnitude as in the SDPC model. Values of two transverse constants A⊥ and K⊥ were adjusted so that experimental values of two quantities in KDP, Tc = 122 K and εa = 59 at T = 140 K, were reproduced in the CMC simulation [7]. In contrast to the SDPC model, the value of M in MSDPC model was not chosen to reproduce the peak at 28 meV from neutron scattering spectra [34], but to reproduce the S-peak from Raman scattering and infrared spectra of KDP [19, 30]. Therefore, the value of M was chosen [63] to obtain –1 ωFE = ωH FE = 150 cm for the ferroelectric-mode frequency. In addition to this mode of dipole vibrations along the c-axis, two doubly degenerate modes of dipole vibrations in transverse directions, denoted as the ω– and ω+ modes, occur in the MSDPC model at the Brillouin-zone center near T = 0 K. In Raman scattering and infrared spectra of modes polarized perpendicular to the c-axis, two peaks should correspond to the ω– and ω+ modes. The value of transverse dipole mass, M⊥ , was adjusted so that the lowest-frequency peak in infrared spectra of KDP [33] corresponds to the lower frequency mode, i.e., ω– = 100 cm–1 [63]. The values of parameters of the MSDPC obtained by the above described procedure are given in [7, 63]. For numerical methods of statistical mechanics to be applied to the MSDPC model, the model system for KDP and DKDP is deﬁned as a supercell of n × n × n unit cells with periodic boundary conditions, where each cell contains 4 PO4 dipoles, so that the whole system contains N = 4n3 PO4 dipoles. The Hamiltonian of the system is H = Ekin + Epot , where Ekin and Epot are given by Eqs. 9 and 10. Dimensionless variables sc (or the order parameter variable) and sa deﬁned as: N 1 c,a sc,a = µi NµH s

(11)

i=1

describe polarization of the system along c- and a-axis, respectively. Spontaneous polarization of the system is proportional to the thermally averaged value |sc | in the thermodynamic limit (N → ∞). For the system of classical PO4 dipoles, the classical Monte Carlo (CMC) method [64] and the classical molecular dynamics simulation (CMDS) method [65] were employed, while for the system of quantum PO4 dipoles, the path-integral quantum Monte Carlo (PI-QMC) method [64] was used. In the classical statistical mechanics, Ekin given by Eq. 9 does not affect the thermal average of some quantity that depends on the dipole variables only, so that only the changes in Epot given by Eq. 10 are considered in the CMC method. To calculate thermal averaged value of some quantity, the Metropolis importance sampling of the space of dipole variables was applied in the CMC method. The susceptibility along c-axis in the PE phase and the susceptibility along the a-axis for the classical system are given by: Nκ 2 s (12) χc,a = T c,a

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in the thermodynamic limit, where κ = 1948 K for KDP and κ = 1925 K for DKDP. In the FE phase, the same relation holds for χc , but with s2c – |sc |2 instead of s2c . In the system of quantum dipoles, dipole and momentum variables have to be replaced by the quantum operators, and quantum statistical mechanics has to be applied. Now, the kinetic energy given by Eq. 9 does affect the thermal average of quantity that depends on dipole variables, due to noncommutivity of dipole and momentum operators. According to the PI-QMC method, a quantum system of N dipoles can be approximated by P coupled classical subsystems of N dipoles, where P is the Trotter number and this approximation becomes exact in the limit P → ∞. Each quantum dipole vector is replaced by a cyclic chain of P classical dipole vectors, or “beads”, i.e., – µi → {µi,1 , ..., µi,P , µi,P+1 = µi,1 }. This classical system of N coupled chains with P “beads” has the Hamiltonian: N P kα α 2 HP = µi,s – µαi,s+1 2 α=a,b,c

(13)

i=1 s=1

P Epot µ1,s , ..., µN,s . + P s=1

Along the i-th chain, the neighboring dipoles (µi,s and µi,s+1 ) are coupled with harmonic springs, where the constants of the spring are ka = kb = M⊥ P(kB T/)2 for the a- and b-directions and kc = MP(kB T/)2 for the cdirection of the classical dipole. The same sampling method from the CMC simulations can be applied to the classical system NP dipoles with the potential energy from Eq. 13 to obtain thermally averaged values of quantities of interest. Thus, thermally averaged value |sc | of the quantum system is given by: P 1 |sc,s | , P→∞ P

|sc | = lim

(14)

s=1

where sc,s is the order parameter variable of the s-th subsystem: N 1 c sc,s = µi,s . NµH s

(15)

i=1

The relation analogous to Eq. 12 for the susceptibility along the c-axis of the quantum system in the PE phase is given by: Nκ χc = lim T P→∞

P s=1

2 sc,s /P

.

(16)

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D. Merunka · B. Rakvin

The CMDS method is based on the Langevin equations of motion of classical PO4 dipoles: ∂Epot d2 µai dµai = – – Γ M + Fia (t) 0 ⊥ dt 2 ∂µai dt ∂Epot d2 µb dµbi + Fib (t) i = 1, ..., N M⊥ 2 i = – – Γ M 0 ⊥ b dt dt ∂µi ∂Epot d2 µci dµci M 2 =– + Fic (t) , – Γ M 0 dt ∂µci dt M⊥

(17)

where the stochastic force and damping terms describe inﬂuence of other degrees of freedom in the crystal on the dipole system. The stochastic force at the temperature T has the property: β (18) Fiα (0)Fj (t) = 2Γ0 Mα kB Tδij δαβ δ(t) α, β = a, b, c and the damping constant Γ0 has the same value in all three directions. The Verlet algorithm was used for numerical solving of these equations in the CMDS method. This method can be applied like the CMC method to calculate static quantities, but also to calculate dynamic quantities, e.g., the correlation function L(0)L(t) of some quantity L and its spectral density: 1 G(L, ω) = 2π

∞ L(0)L(t) eiωt dt

(19)

–∞

4 Results of the Modified Strong Dipole–Proton Coupling Model and Discussion 4.1 Results for Structural and Static Dielectric Properties In order to obtain values of model parameters A⊥ and K⊥ of the MSDPC model, which are common to the KDP and DKDP systems, and to estimate isotope effects when only parameter h changed its value (hH → hD ), the CMC method was used [7]. With the parameter values set to reproduce experimental results Tc ≈ 122 K and εa ≈ 59 at T = 140 K in KDP, temperature dependencies of order parameter, |sc |, longitudinal dielectric constant, εc = 12 + χc , and transverse dielectric constants, εa = 12 + χa were calculated for KDP and DKDP (Fig. 1). The low-temperature value of |sc | in DKDP is about 1.2 times higher than H in KDP (Fig. 1a), which is in accordance with both the relation µD s = 1.2µs and the experimental isotope effect in the saturated spontaneous polariza-

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Struct Bond (2007) 124: 199–207 DOI 10.1007/430_2006_043 © Springer-Verlag Berlin Heidelberg 2006 Published online: 4 November 2006

A Comparison of Magnetic Random Access Memories (MRAMs) and Ferroelectric Random Access Memories (FRAMs) J. F. Scott Centre for Ferroics, Earth Sciences Dept., University of Cambridge, Cambridge CB2 3EQ, UK [email protected] 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Experimental Directions . . . Ferroelectric-Gate FETs . . . . Three-Dimensional Capacitors Electroding . . . . . . . . . . .

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206 206 206

4

Top-Down and Bottom-Up . . . . . . . . . . . . . . . . . . . . . . . . . . .

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract A short review is given of the status of MRAMs and FRAMs, summarizing both industry prototype production and university research. This comparison seems especially timely since this month (July 2006) Freescale have announced the ﬁrst commercial MRAM product (4 Mb), and the race is now fairly even between the Samsung 32 Mb lead zirconate titanate FRAM and the Matsushita 4 Mb strontium bismuth tantalate FRAM.

1 Introduction The nanotechnology report issued in February 2004 by the UK Royal Society makes the general observation that: “Electrical transport properties across interfaces remain poorly understood in terms of science/predictive capability. This affects all nanomaterials”. This observation most keenly summarizes the present state of play for Gbit level random access memories (RAMs), and it is our view that the electrode interface issues may dominate the device physics. Within the nanotech “roadmap”, high-dielectric (“high-K”) materials are strongly emphasized, as are nanotubes and new interconnects. FRAM engineering, particularly for high-density integration, is very well reviewed in [1, 2]. The state of the art is illustrated in Fig. 1 from Samsung.

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Fig. 1 a Micrograph of a cross-sectional view showing a unit-cell structure of the 64 Mbit FRAM, recently developed by Samsung, 2006 (W tungsten, SRO strontium ruthenate). b Schematic circuit diagram of one bit memory of 64 Mb FRAM with 1T-1C cell conﬁguration, consisting of both 1-transistor, one node of which is connected to bit-line (BL) to transfer data by accessing word-line (WL), and 1-capacitor, one of which is connected to plate-line (PL). c Schematic circuit diagram showing 1T-1C memory such as (b), corresponding to the micrograph (a). Samsung Proprietary (used by permission)

This 4 Mb FRAM utilizes lead zirconate titanate (PZT) as the active memory element; a similar prototype has been made by Matsushita/Panasonic using strontium bismuth tantalate (SBT). A 32-Mb SBT memory has more recently been announced by Samsung. Device performances are similar in each case. The PZT device has the advantage that it can be processed at lower temperatures (450 ◦ C, compared with ca. 650 ◦ C for SBT); the SBT devices have the environmental advantage that they are lead-free. In Table 1 we compare performance data for reported MRAM and FRAM prototypes. The small Fujitsu FRAM is not a prototype; it is in large-scale production and found in the memory board of every Sony Playstation 2, as part of the Toshiba memory system. The main advantages of FRAMs over EEPROMs or Flash memory are in the WRITE times (100 ns for FRAM, versus 1 µs for Flash and 10 µs for EEPROM), and energy per 32-bit WRITE (1 nJ for FRAM versus 1 or 2 mJ for EEPROM or Flash). Note that parameters such as READ time or WRITE time for FRAMs are dependent upon actual cell architecture; they are not limited by the intrinsic switching time of the ferroelectric thin ﬁlm, which is typically 220 ps [3].

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Table 1 MRAM/FRAM comparison Company

MRAM NEC/Toshiba IBM Matsushita Sony Cypress Motorola State-of-the-art FRAM Fujitsu (in Toshiba memory for Sony Playstation 2) Samsung Matsushita laboratory ∗

Size

1 Mb 16 Mb 4 Mb 8 kb 256 kb 4 Mb 16 Mb

Design rule (feature size) [microns]

0.18 70 0.09 8F 2

32 kb 32 Mb 4 Mb

Speed (access speed) [ns]

25 100

0.18 (PZT) (SBT)

60 60 0.2∗

Single cell access time

Initially it was thought that MRAMs would offer a great advantage in cell size over FRAMs; however, this prediction was based upon the assumption of raw matrix (cross-tie) arrays. In reality, the problem of cross-talk or halfselect disturb pulses is as acute with MRAMs as with FRAMs, and in each case a space-consuming architecture must be employed with pass-gate transistor isolation of each bit.

2 Experimental Directions 2.1 Ferroelectric-Gate FETs The ideal ferroelectric RAM would not have the ferroelectric sitting aside or atop the transistor in each cell (an “1T-1C” design with one transistor and one capacitor per bit); rather, it would place the ferroelectric ﬁlm into the gate of the FET (termed an FE-FET). In this position, switching its polarization up or down would strongly modulate the source-drain current, and hence the cell would have a non-destructive READ operation that did not require erasing and rewriting the addressed cell. Such an operation is at present required of all commercial FRAMs and can cause problems of fatigue (degradation

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J.F. Scott

in switched charge with repetitive cycling). In many operations a cell would be read many million times but erased and rewritten only a few thousand. Thus, non-destructive READ, merely by monitoring the source-drain current, would extend the lifetime of such a memory by many orders of magnitude. Unfortunately, as pointed out by Ishiwara et al. [4], the depolarization ﬁeld in a ferroelectric gate is inevitably generated when the gate is grounded, and this makes it very difﬁcult to obtain > 10 year data retention in an FEFET. He suggested using a 1T-2C capacitor geometry in which the functions of data retention and READ operation are separated. With this scheme, he and his colleagues achieved an on/off source-drain current ratio of > 1000 for a 150 nm thick SBT ﬁlm in a 5 × 50 micron MOSFET channel, with Pt electrodes. An alternative scheme for FE-FETs is to use a completely different kind of gate ferroelectric from that commonly found in pass-gate 1T-1C arrays, and a different transistor material as well. Although one needs a remanent polarization Pr of order 5–10 µC/cm2 to permit the sense ampliﬁers to discriminate reliably between a stored “1” and a stored “0”, this is much more polarization than is required to charge the channel of a ptype CMOS device, where 0.5 µC/cm2 would be adequate. Hence, largepolarization materials such as PZT, SBT or other perovskite oxides are not required. Considerable progress has been made with this approach by Grekhov’s group in St. Petersburg [5, 6]. Grekhov et al. have achieved a 70% channel modulation and > 2 month retention with his PZT/LSCO/NdGaO3

Fig. 2 All-perovskite ferroelectric FET. S source, G gate, D drain [5]

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203

device (Fig. 2). This all-perovskite FET has considerable advantages over the usual Si-device, and we note that Funakubo et al. are carrying out similar studies in Tokyo (personal communication and [7]). Grekhov’s key point is that the LSCO channel layer must be 5–20 nm thick. Thicker layers produce three-dimensional Stranski–Krastanov island growth, which prevents signiﬁcant channel modulation. Unfortunately Grekhov’s device is too slow for commercial standards. The 2-megohm channel resistance for the “On” state limits the switching speed. This is largely limited by geometry; the very long (320 micron gate length) offset gate structures need to be improved. This is a clear application in future for TFT (thin-ﬁlm transistors) technology. The most recent FRAM development is a large-cell 6T-4C design by Masui et al. [7]; this resembles a static-RAM (SRAM) layout (typically an SRAM has four MOSFET transistors, although sometimes the two depletion-mode MOSFETs are replaced with resistors). Although the area per bit is about ﬁve times that of a more conventional 1T-1C device, this structure offers nondestructive READ operation with < 10 ns access times. It is highly suitable for small arrays and will be used for RF identiﬁcation tags (baggage tags at airports, tags on cattle ears in feedlots, etc.) or smart credit cards. Parts are already being shipped of these 6T-4C devices. Together with small devices such as the Sony Playstation 32 kbit FRAM, FRAM sales worldwide have exceeded 200 million parts, primary NOT of large memory devices; the 4 Mb and 32 Mb parts are not yet being shipped. 2.2 Three-Dimensional Capacitors In any real memory device the capacitors take up most of the chip area; the transistors and resistors are very small. Therefore the FRAM roadmap [8] shown in Table 2 mandates a fully three-dimensional (3D) capacitor structure in the industry by 2008. The state of the art at present is a PZT-lined trench, a Tokyo Institute of Technology–Samsung collaboration that achieves a 6.5 : 1 aspect ratio for the trenches. Ru electrodes are used, prepared from the organic precursor Ru-DER, from Tosoh Corp. A more extreme methodology to achieve this end utilizes ferroelectric nanotubes (Fig. 3) deposited via CSD (chemical solution deposition) into porous Si substrates. With this procedure, Morrison et al. have achieved > 200 : 1 aspect ratios, about 30 times better than the T.I.T.–Samsung team [9, 10]. The ferroelectric nanotubes are typically 100 microns long and have ﬁnegrained (40 nm diameter grains) ceramic walls ca. 50 nm thick. Typically, the coercive ﬁeld for ferroelectric switching is about 60–100 kV/cm. In a 50 nm wall, concentric cylindrical electrodes will produce that at an applied voltage of approximately 0.5 V. Therefore these devices not only meet the standard 5V Si-logic (CMOS) levels, but they are functional at all the lower voltage

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J.F. Scott

Table 2 FRAM roadmap Year of production

2004

2005

2006

2007

FRAM technology node – F (nm) FRAM cell size – area factor a in multiples of F2 FRAM cell size (µm2 ) FRAM cell structure FRAM capacitor structure Ferro capacitor voltage (V) FRAM endurance (read/write cycles) FRAM non-volatile data retention (years)

220 16 0.518 1T1C Stack 1.8 1.00e15 10

180 10 0.324 1T1C Stack 1.5 > 1e16 10

150 10 0.225 1T1C Stack 1.3 > 1e16 10

130 10 0.169 1T1C 3D 1.2 > 1e16 10

Fig. 3 SBT nanotubes [9]

Fig. 4 Electron micrograph of porous Si sacriﬁcial substrates for ferroelectric nanotubes, from KTH (Stockholm); ×9000

levels (3.3, 1.1 and 0.5 V) anticipated in the microelectronics industry for the next decade. It is less likely that MRAMs can function at these lower voltages.

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205

Fig. 5 Electron micrograph of porous Si sacriﬁcial substrates for ferroelectric nanotubes, from the University of Trento; ×9500 The scale bar represents 0.5 µm

At present we use porous silicon substrates from KTH in Stockholm (Fig. 4) and from the University of Trento in Italy (Fig. 5). 2.3 Electroding In any 3D system the key problem is electroding. Morrison et al. use Pdacetate to yield metallic Pd electrodes (Fig. 6) [9, 10]. Probably the ideal combination at present is to use Pd as the outer electrode (which becomes the bottom electrode on the planar FET) and Ru as the inner (top) electrode. The Ru has better wetting characteristics but decomposes for temperatures much above 400 ◦ C; hence it cannot be used as the bottom (outer) electrode for SBT, which requires a much higher temperature anneal. Ru can be used as the top electrode on either PZT or SBT because the device is not subjected to very high temperatures after the top electrode is put down.

Fig. 6 Electrode geometry for concentric cylindrical electrodes on ferroelectric nanotubes. TE top electrode, BE bottom electrode

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J.F. Scott

3 Other Related Issues 3.1 Nanoribbons In addition to 3D nanotubes, the microelectronics industry has some need for nanoribbons. A 1 micron ×2 cm ribbon of PZT from Samsung is illustrated in Fig. 7. This ferroelectric ﬁlm has slightly different electrical characteristics to a similar 100 × 100 micron square of the same material, thickness and processing. Because the Samsung processing involves an etch in hydrogencontaining ambients, the H ions can attack the edges of a nanoribbon and form OH-hydroxides, which are both electrically reorientable and lossy. Hence, nanodevice fabrication must be cognizant of edge effects during processing.

Fig. 7 PZT nanoribbon. a Ribbon capacitor, b cross-sectional view. TE top electrode, FE ferroelectric, BE bottom electrode

3.2 Hafnia Gate Oxides In addition to requiring high dielectric ﬁlms for DRAM capacitors (dynamic random access memories) and for the active memory elements in FRAMs, the microelectronics industry has a stated demand for a replacement for SiO2 gate oxides very soon. The leading candidate is hafnia (HfO2 ), and there are signiﬁcant opportunities for the ferroelectrics community to contribute to the solution of this problem.

4 Top-Down and Bottom-Up Finally, we note that there are two different approaches to an nanodevice science: One can carry out some kind of very small-scale lithography (e-beam direct writing, focussed ion beam FIB, X-ray lithography, etc.), all of which

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207

are termed “top-down”. Or one can utilize a kind of self-assembly (“bottomup”) [11]. Self-assembly works rather well for ferroelectrics and could in principle provide multi-Gbit arrays. However, at present these arrays are not fully registered unless one adds additional processing steps, such as inclusion of micron or submicron polystyrene or Si spheres as spacers.

Note Added in Proof Update on FRAMS: Matushita/Symetrix (MEC) has achieved 25-nm-wide SBT FeRAMs with breakdown of 1.5 MV/cm for the semiconductor 45-nm node. Produced at 25 000 on each 8-inch Si wafer, six million/month are shipped for applications such as Japanese Railroad “smart” fare cards. Ramtron/Fujitsu are at 0.35-µm strap-cell PZT design; MEC has produced 500 million units of thin-ﬁlm integrated ferroelectrics (one billion, including BST-on-GaAs) and is at 0.13-µm stacked four-level-metal design. The Symetrix 5-ns access-time “Trinion” cell is the ﬁrst non-volatile cache memory. Acknowledgements The author would like to acknowledge Prof. Lorenzo Pavesi and Paolo Bettotti at the University of Trento, Italy and Prof. Jann Linnros and Xavier Badel at the Royal institute of Technology (KTH) in Stockholm for providing porous substrates.

References 1. Kim K (2004) High density integration. In: Ishiwara H, Okuyama M, Arimoto Y (eds) Ferroelectric random access memories. Top Appl Phys 93:165–176 2. Arimoto Y, Ishiwara H (2004) MRS Bull 29:823–828 3. Li J, Nagaraj B, Liang H, Cao W, Lee CH, Ramesh R (2004) Appl Phys Lett 84:1174– 1177 4. Yoon SM, Ishiwara H (1998) Abstracts of the international conference on solid state devices and materials (SSDM98), Hiroshima, 7–10 September 1998, pp 222–223 5. Grekhov IV, Delimova L, Liniichuk I, Mashovets D, Veselovsky I (2003) Proceedings of the Japan–Russia conference on ferroelectrics, St. Petersburg, 23 June 2002. Ferroelectrics 286(2003):237–244 6. Grekhov IV, Delimova L, Liniichuk I, Mashovets D, Veselovsky I (2002) Integ Ferroelec 43:101–113 7. Masui S, Yokozeki W, Oura M, Ninomiya T, Mukaida K, Takayama Y, Teramoto T (2003) Proceedings IEEE custom integrated circuits conference, 21–24 September, San Jose, California, pp 403–406 8. International FRAM roadmap http://www.itrs.net 9. Morrison FD, Ramsay L, Scott JF (2003) J Phys Condens Mat 15:L527–530 10. Morrison FD, Luo Y, Szafraniak I, Nagarajan V, Wehrspoon RB, Steinhart M, Wendroff JH, Zakharov ND, Mishina ED, Vorotilov KA, Sigov AS, Nakabayashi Alexe SM, Ramesh R, Scott JF (2003) Rev Adv Mat Sci 4:114–122 11. Dawber M, Szfraniak I, Alexe M, Scott JF (2003) J Phys Condens Mat 15:L667–670

Author Index Volumes 101–124 Author Index Vols. 1–100 see Vol. 100

The volume numbers are printed in italics Alajarin M, see Turner DR (2004) 108: 97–168 Aldinger F, see Seifert HJ (2002) 101: 1–58 Alessio E, see Iengo E (2006) 121: 105–143 Alfredsson M, see Corà F (2004) 113: 171–232 Aliev AE, Harris KDM (2004) Probing Hydrogen Bonding in Solids Using State NMR Spectroscopy 108: 1–54 Alloul H, see Brouet V (2004) 109: 165–199 Amstutz N, see Hauser A (2003) 106: 81–96 Anitha S, Rao KSJ (2003) The Complexity of Aluminium-DNA Interactions: Relevance to Alzheimer’s and Other Neurological Diseases 104: 79–98 Anthon C, Bendix J, Schäffer CE (2004) Elucidation of Ligand-Field Theory. Reformulation and Revival by Density Functional Theory 107: 207–302 Aramburu JA, see Moreno M (2003) 106: 127–152 Arˇcon D, Blinc R (2004) The Jahn-Teller Effect and Fullerene Ferromagnets 109: 231–276 Aromí G, Brechin EK (2006) Synthesis of 3d Metallic Single-Molecule Magnets. 122: 1–67 Atanasov M, Daul CA, Rauzy C (2003) A DFT Based Ligand Field Theory 106: 97–125 Atanasov M, see Reinen D (2004) 107: 159–178 Atwood DA, see Conley B (2003) 104: 181–193 Atwood DA, Hutchison AR, Zhang Y (2003) Compounds Containing Five-Coordinate Group 13 Elements 105: 167–201 Atwood DA, Zaman MK (2006) Mercury Removal from Water 120: 163–182 Autschbach J (2004) The Calculation of NMR Parameters in Transition Metal Complexes 112: 1–48 Baerends EJ, see Rosa A (2004) 112: 49–116 Balch AL (2007) Remarkable Luminescence Behaviors and Structural Variations of TwoCoordinate Gold(I) Complexes. 123 Baranoff E, Barigelletti F, Bonnet S, Collin J-P, Flamigni L, Mobian P, Sauvage J-P (2007) From Photoinduced Charge Separation to Light-driven Molecular Machines. 123 Barbara B, see Curély J (2006) 122: 207–250 Bard AJ, Ding Z, Myung N (2005) Electrochemistry and Electrogenerated Chemiluminescence of Semiconductor Nanocrystals in Solutions and in Films 118: 1–57 Barigelletti F, see Baranoff E (2007) 123 Barriuso MT, see Moreno M (2003) 106: 127–152 Beaulac R, see Nolet MC (2004) 107: 145–158 Bebout DC, Berry SM (2006) Probing Mercury Complex Speciation with Multinuclear NMR 120: 81–105

210

Author Index Volumes 101–124

Bellandi F, see Contreras RR (2003) 106: 71–79 Bendix J, see Anthon C (2004) 107: 207–302 Berend K, van der Voet GB, de Wolff FA (2003) Acute Aluminium Intoxication 104: 1–58 Berry SM, see Bebout DC (2006) 120: 81–105 Bianconi A, Saini NL (2005) Nanoscale Lattice Fluctuations in Cuprates and Manganites 114: 287–330 Blinc R, see Arcˇcon D (2004) 109: 231–276 Blinc R (2007) Order and Disorder in Perovskites and Relaxor Ferroelectrics. 124: 51–67 Boˇca R (2005) Magnetic Parameters and Magnetic Functions in Mononuclear Complexes Beyond the Spin-Hamiltonian Formalism 117: 1–268 Bohrer D, see Schetinger MRC (2003) 104: 99–138 Bonnet S, see Baranoff E (2007) 123 Bouamaied I, Coskun T, Stulz E (2006) Axial Coordination to Metalloporphyrins Leading to Multinuclear Assemblies 121: 1–47 Boulanger AM, see Nolet MC (2004) 107: 145–158 Boulon G (2004) Optical Transitions of Trivalent Neodymium and Chromium Centres in LiNbO3 Crystal Host Material 107: 1–25 Bowlby BE, Di Bartolo B (2003) Spectroscopy of Trivalent Praseodymium in Barium Yttrium Fluoride 106: 193–208 Braga D, Maini L, Polito M, Grepioni F (2004) Hydrogen Bonding Interactions Between Ions: A Powerful Tool in Molecular Crystal Engineering 111: 1–32 Brechin EK, see Aromí G (2006) 122: 1–67 Brouet V, Alloul H, Gàràj S, Forrò L (2004) NMR Studies of Insulating, Metallic, and Superconducting Fullerides: Importance of Correlations and Jahn-Teller Distortions 109: 165– 199 Buddhudu S, see Morita M (2004) 107: 115–144 Budzelaar PHM, Talarico G (2003) Insertion and β-Hydrogen Transfer at Aluminium 105: 141–165 Burrows AD (2004) Crystal Engineering Using Multiple Hydrogen Bonds 108: 55–96 Bussmann-Holder A, Dalal NS (2007) Order/Disorder Versus or with Displacive Dynamics in Ferroelectric Systems. 124: 1–21 Bussmann-Holder A, Keller H, Müller KA (2005) Evidences for Polaron Formation in Cuprates 114: 367–386 Bussmann-Holder A, see Dalal NS (2007) 124: 23–50 Bussmann-Holder A, see Micnas R (2005) 114: 13–69 Canadell E, see Sánchez-Portal D (2004) 113: 103–170 Cancines P, see Contreras RR (2003) 106: 71–79 Caneschi A, see Cornia A (2006) 122: 133–161 Cartwright HM (2004) An Introduction to Evolutionary Computation and Evolutionary Algorithms 110: 1–32 Christie RA, Jordan KD (2005) n-Body Decomposition Approach to the Calculation of Interaction Energies of Water Clusters 116: 27–41 Clérac R, see Coulon C (2006) 122: 163–206 Clot E, Eisenstein O (2004) Agostic Interactions from a Computational Perspective: One Name, Many Interpretations 113: 1–36 Collin J-P, see Baranoff E (2007) 123 Conley B, Atwood DA (2003) Fluoroaluminate Chemistry 104: 181–193 Contakes SM, Le Nguyen YH, Gray HB (2007) Conjugates of Heme-thiolate Enzymes with Photoactive Metal-Diimine Wires. 123

Author Index Volumes 101–124

211

Contreras RR, Suárez T, Reyes M, Bellandi F, Cancines P, Moreno J, Shahgholi M, Di Bilio AJ, Gray HB, Fontal B (2003) Electronic Structures and Reduction Potentials of Cu(II) Complexes of [N,N -Alkyl-bis(ethyl-2-amino-1-cyclopentenecarbothioate)] (Alkyl = Ethyl, Propyl, and Butyl) 106: 71–79 Cooke Andrews J (2006) Mercury Speciation in the Environment Using X-ray Absorption Spectroscopy 120: 1–35 Corà F, Alfredsson M, Mallia G, Middlemiss DS, Mackrodt WC, Dovesi R, Orlando R (2004) The Performance of Hybrid Density Functionals in Solid State Chemistry 113: 171–232 Cornia A, Costantino AF, Zobbi L, Caneschi A, Gatteschi D, Mannini M, Sessoli R (2006) Preparation of Novel Materials Using SMMs. 122: 133–161 Coskun T, see Bouamaied I (2006) 121: 1–47 Costantino AF, see Cornia A (2006) 122: 133–161 Coulon C, Miyasaka H, Clérac R (2006) Single-Chain Magnets: Theoretical Approach and Experimental Systems. 122: 163–206 Crespi VH, see Gunnarson O (2005) 114: 71–101 Curély J, Barbara B (2006) General Theory of Superexchange in Molecules. 122: 207–250 Dalal NS, Gunaydin-Sen O, Bussmann-Holder A (2007) Experimental Evidence for the Coexistence of Order/Disorder and Displacive Behavior of Hydrogen-Bonded Ferroelectrics and Antiferroelectrics. 124: 23–50 Dalal NS, see Bussmann-Holder A (2007) 124: 1–21 Daul CA, see Atanasov M (2003) 106: 97–125 Day P (2003) Whereof Man Cannot Speak: Some Scientiﬁc Vocabulary of Michael Faraday and Klixbüll Jørgensen 106: 7–18 Deeth RJ (2004) Computational Bioinorganic Chemistry 113: 37–69 Delahaye S, see Hauser A (2003) 106: 81–96 Deng S, Simon A, Köhler J (2005) Pairing Mechanisms Viewed from Physics and Chemistry 114: 103–141 Di Bartolo B, see Bowlby BE (2003) 106: 191–208 Di Bilio AJ, see Contreras RR (2003) 106: 71–79 Ding Z, see Bard AJ (2005) 118: 1–57 Dovesi R, see Corà F (2004) 113: 171–232 Duan X, see He J (2005) 119: 89–119 Duan X, see Li F (2005) 119: 193–223 Egami T (2005) Electron-Phonon Coupling in High-Tc Superconductors 114: 267–286 Egami T (2007) Local Structure and Dynamics of Ferroelectric Solids. 124: 69–88 Eisenstein O, see Clot E (2004) 113: 1–36 Ercolani G (2006) Thermodynamics of Metal-Mediated Assemblies of Porphyrins 121: 167– 215 Evans DG, see He J (2005) 119: 89–119 Evans DG, Slade RCT (2005) Structural Aspects of Layered Double Hydroxides 119: 1–87 Ewing GE (2005) H2 O on NaCl: From Single Molecule, to Clusters, to Monolayer, to Thin Film, to Deliquescence 116: 1–25 Flamigni L, Heitz V, Sauvage J-P (2006) Porphyrin Rotaxanes and Catenanes: Copper(I)Templated Synthesis and Photoinduced Processes 121: 217–261 Flamigni L, see Baranoff E (2007) 123 Fontal B, see Contreras RR (2003) 106: 71–79 Forrò L, see Brouet V (2004) 109: 165–199

212

Author Index Volumes 101–124

Fowler PW, see Soncini A (2005) 115: 57–79 Frenking G, see Lein M (2003) 106: 181–191 Frühauf S, see Roewer G (2002) 101: 59–136 Frunzke J, see Lein M (2003) 106: 181–191 Furrer A (2005) Neutron Scattering Investigations of Charge Inhomogeneities and the Pseudogap State in High-Temperature Superconductors 114: 171–204 Gàràj S, see Brouet V (2004) 109: 165–199 Gatteschi D, see Cornia A (2006) 122: 133–161 Gillet VJ (2004) Applications of Evolutionary Computation in Drug Design 110: 133–152 Golden MS, Pichler T, Rudolf P (2004) Charge Transfer and Bonding in Endohedral Fullerenes from High-Energy Spectroscopy 109: 201–229 Gorelesky SI, Lever ABP (2004) 107: 77–114 Grant GJ (2006) Mercury(II) Complexes with Thiacrowns and Related Macrocyclic Ligands 120: 107–141 Grätzel M, see Nazeeruddin MK (2007) 123 Gray HB, see Contreras RR (2003) 106: 71–79 Gray HB, see Contakes SM (2007) 123 Grepioni F, see Braga D (2004) 111: 1–32 Gritsenko O, see Rosa A (2004) 112: 49–116 Güdel HU, see Wenger OS (2003) 106: 59–70 Gunnarsson O, Han JE, Koch E, Crespi VH (2005) Superconductivity in Alkali-Doped Fullerides 114: 71–101 Gunter MJ (2006) Multiporphyrin Arrays Assembled Through Hydrogen Bonding 121: 263– 295 Gunaydin-Sen O, see Dalal NS (2007) 124: 23–50 Gütlich P, van Koningsbruggen PJ, Renz F (2004) Recent Advances in Spin Crossover Research 107: 27–76 Guyot-Sionnest P (2005) Intraband Spectroscopy and Semiconductor Nanocrystals 118: 59–77 Habershon S, see Harris KDM (2004) 110: 55–94 Han JE, see Gunnarson O (2005) 114: 71–101 Hardie MJ (2004) Hydrogen Bonded Network Structures Constructed from Molecular Hosts 111: 139–174 Harris KDM, see Aliev (2004) 108: 1–54 Harris KDM, Johnston RL, Habershon S (2004) Application of Evolutionary Computation in Structure Determination from Diffraction Data 110: 55–94 Hartke B (2004) Application of Evolutionary Algorithms to Global Cluster Geometry Optimization 110: 33–53 Harvey JN (2004) DFT Computation of Relative Spin-State Energetics of Transition Metal Compounds 112: 151–183 Haubner R, Wilhelm M, Weissenbacher R, Lux B (2002) Boron Nitrides – Properties, Synthesis and Applications 102: 1–46 Hauser A, Amstutz N, Delahaye S, Sadki A, Schenker S, Sieber R, Zerara M (2003) Fine Tuning the Electronic Properties of [M(bpy)3 ]2+ Complexes by Chemical Pressure (M = Fe2+ , Ru2+ , Co2+ , bpy = 2,2 -Bipyridine) 106: 81–96 He J, Wei M, Li B, Kang Y, G Evans D, Duan X (2005) Preparation of Layered Double Hydroxides 119: 89–119 Heitz V, see Flamigni L (2006) 121: 217–261

Author Index Volumes 101–124

213

Herrmann M, see Petzow G (2002) 102: 47–166 Herzog U, see Roewer G (2002) 101: 59–136 Hoggard PE (2003) Angular Overlap Model Parameters 106: 37–57 Höpﬂ H (2002) Structure and Bonding in Boron Containing Macrocycles and Cages 103: 1–56 Hubberstey P, Suksangpanya U (2004) Hydrogen-Bonded Supramolecular Chain and Sheet Formation by Coordinated Guranidine Derivatives 111: 33–83 Hupp JT (2006) Rhenium-Linked Multiporphyrin Assemblies: Synthesis and Properties 121: 145–165 Hutchison AR, see Atwood DA (2003) 105: 167–201 Iengo E, Scandola F, Alessio E (2006) Metal-Mediated Multi-Porphyrin Discrete Assemblies and Their Photoinduced Properties 121: 105–143 Itoh M, Taniguchi H (2007) Ferroelectricity of SrTiO3 Induced by Oxygen Isotope Exchange. 124: 89–118 Iwasa Y, see Margadonna S (2004) 109: 127–164 Jansen M, Jäschke B, Jäschke T (2002) Amorphous Multinary Ceramics in the Si-B-N-C System 101: 137–192 Jäschke B, see Jansen M (2002) 101: 137–192 Jäschke T, see Jansen M (2002) 101: 137–192 Jaworska M, Macyk W, Stasicka Z (2003) Structure, Spectroscopy and Photochemistry of the [M(η5 -C5 H5 )(CO)2 ]2 Complexes (M = Fe, Ru) 106: 153–172 Jenneskens LW, see Soncini A (2005) 115: 57–79 Jeziorski B, see Szalewicz K (2005) 116: 43–117 Johnston RL, see Harris KDM (2004) 110: 55–94 Jordan KD, see Christie RA (2005) 116: 27–41 Kabanov VV, see Mihailovic D (2005) 114: 331–365 Kang Y, see He J (2005) 119: 89–119 Keller H (2005) Unconventional Isotope Effects in Cuprate Superconductors 114: 143–169 Keller H, see Bussmann-Holder A (2005) 114: 367–386 Khan AI, see Williams GR (2005) 119: 161–192 Kind R (2007) Evidence for Ferroelectric Nucleation Centres in the Pseudo-spin Glass System Rb1-x (ND4 )x D2 PO4 : A 87 Rb NMR Study. 124: 119–147 Kobuke Y (2006) Porphyrin Supramolecules by Self-Complementary Coordination 121: 49–104 Koch E, see Gunnarson O (2005) 114: 71–101 Kochelaev BI, Teitel’baum GB (2005) Nanoscale Properties of Superconducting Cuprates Probed by the Electron Paramagnetic Resonance 114: 205–266 Köhler J, see Deng (2005) 114: 103–141 van Koningsbruggen, see Gütlich P (2004) 107: 27–76 Kume S, Nishihara H (2007) Metal-based Photoswitches Derived from Photoisomerization. 123 Lein M, Frunzke J, Frenking G (2003) Christian Klixbüll Jørgensen and the Nature of the Chemical Bond in HArF 106: 181–191 Leroux F, see Taviot-Gueho C (2005) 119: 121–159 Lever ABP, Gorelesky SI (2004) Ruthenium Complexes of Non-Innocent Ligands; Aspects of Charge Transfer Spectroscopy 107: 77–114

214

Author Index Volumes 101–124

Li B, see He J (2005) 119: 89–119 Li F, Duan X (2005) Applications of Layered Double Hydroxides 119: 193–223 Liebau F, see Santamaría-Pérez D (2005) 118: 79–135 Linton DJ, Wheatley AEH (2003) The Synthesis and Structural Properties of Aluminium Oxide, Hydroxide and Organooxide Compounds 105: 67–139 Lo KK-W (2007) Luminescent Transition Metal Complexes as Biological Labels and Probes. 123 Lux B, see Haubner R (2002) 102: 1–46 Mackrodt WC, see Corà F (2004) 113: 171–232 Macyk W, see Jaworska M (2003) 106: 153–172 Mahalakshmi L, Stalke D (2002) The R2M+ Group 13 Organometallic Fragment Chelated by P-centered Ligands 103: 85–116 Maini L, see Braga D (2004) 111: 1–32 Mallah T, see Rebilly J-N (2006) 122: 103–131 Mallia G, see Corà F (2004) 113: 171–232 Mannini M, see Cornia A (2006) 122: 133–161 Margadonna S, Iwasa Y, Takenobu T, Prassides K (2004) Structural and Electronic Properties of Selected Fulleride Salts 109: 127–164 Maseras F, see Ujaque G (2004) 112: 117–149 McInnes EJL (2006) Spectroscopy of Single-Molecule Magnets. 122: 69–102 Merunka D, Rakvin B (2007) Anharmonic and Quantum Effects in KDP-Type Ferroelectrics: Modiﬁed Strong Dipole–Proton Coupling Model. 124: 149–198 Micnas R, Robaszkiewicz S, Bussmann-Holder A (2005) Two-Component Scenarios for Non-Conventional (Exotic) Superconstructors 114: 13–69 Middlemiss DS, see Corà F (2004) 113: 171–232 Mihailovic D, Kabanov VV (2005) Dynamic Inhomogeneity, Pairing and Superconductivity in Cuprates 114: 331–365 Millot C (2005) Molecular Dynamics Simulations and Intermolecular Forces 115: 125–148 Miyake T, see Saito (2004) 109: 41–57 Miyasaka H, see Coulon C (2006) 122: 163–206 Mobian P, see Baranoff E (2007) 123 Moreno J, see Contreras RR (2003) 106: 71–79 Moreno M, Aramburu JA, Barriuso MT (2003) Electronic Properties and Bonding in Transition Metal Complexes: Inﬂuence of Pressure 106: 127–152 Morita M, Buddhudu S, Rau D, Murakami S (2004) Photoluminescence and Excitation Energy Transfer of Rare Earth Ions in Nanoporous Xerogel and Sol-Gel SiO2 Glasses 107: 115–143 Morsch VM, see Schetinger MRC (2003) 104: 99–138 Mossin S, Weihe H (2003) Average One-Center Two-Electron Exchange Integrals and Exchange Interactions 106: 173–180 Murakami S, see Morita M (2004) 107: 115–144 Müller E, see Roewer G (2002) 101: 59–136 Müller KA (2005) Essential Heterogeneities in Hole-Doped Cuprate Superconductors 114: 1–11 Müller KA, see Bussmann-Holder A (2005) 114: 367–386 Myung N, see Bard AJ (2005) 118: 1–57 Nazeeruddin MK, Grätzel M (2007) Transition Metal Complexes for Photovoltaic and Light Emitting Applications. 123

Author Index Volumes 101–124

215

Le Nguyen YH, see Contakes SM (2007) 123 Nishibori E, see Takata M (2004) 109: 59–84 Nishihara H, see Kume S (2007) 123 Nolet MC, Beaulac R, Boulanger AM, Reber C (2004) Allowed and Forbidden d-d Bands in Octa-hedral Coordination Compounds: Intensity Borrowing and Interference Dips in Absorption Spectra 107: 145–158 O’Hare D, see Williams GR (2005) 119: 161–192 Ordejón P, see Sánchez-Portal D (2004) 113: 103–170 Orlando R, see Corà F (2004) 113: 171–232 Oshiro S (2003) A New Effect of Aluminium on Iron Metabolism in Mammalian Cells 104: 59–78 Pastor A, see Turner DR (2004) 108: 97–168 Patkowski K, see Szalewicz K (2005) 116: 43–117 Patoˇcka J, see Strunecká A (2003) 104: 139–180 Peng X, Thessing J (2005) Controlled Synthesis of High Quality Semiconductor Nanocrystals 118: 137–177 Petzow G, Hermann M (2002) Silicon Nitride Ceramics 102: 47–166 Pichler T, see Golden MS (2004) 109: 201–229 Polito M, see Braga D (2004) 111: 1–32 Popelier PLA (2005) Quantum Chemical Topology: on Bonds and Potentials 115: 1–56 Power P (2002) Multiple Bonding Between Heavier Group 13 Elements 103: 57–84 Prassides K, see Margadonna S (2004) 109: 127–164 Prato M, see Tagmatarchis N (2004) 109: 1–39 Price LS, see Price SSL (2005) 115: 81–123 Price SSL, Price LS (2005) Modelling Intermolecular Forces for Organic Crystal Structure Prediction 115: 81–123 Rabinovich D (2006) Poly(mercaptoimidazolyl)borate Complexes of Cadmium and Mercury 120: 143–162 Rakvin B, see Merunka D (2007) 124: 149–198 Rao KSJ, see Anitha S (2003) 104: 79–98 Rau D, see Morita M (2004) 107: 115–144 Rauzy C, see Atanasov (2003) 106: 97–125 Reber C, see Nolet MC (2004) 107: 145–158 Rebilly J-N, Mallah T (2006) Synthesis of Single-Molecule Magnets Using Metallocyanates. 122: 103–131 Reinen D, Atanasov M (2004) The Angular Overlap Model and Vibronic Coupling in Treating s-p and d-s Mixing – a DFT Study 107: 159–178 Reisfeld R (2003) Rare Earth Ions: Their Spectroscopy of Cryptates and Related Complexes in Glasses 106: 209–237 Renz F, see Gütlich P (2004) 107: 27–76 Reyes M, see Contreras RR (2003) 106: 71–79 Ricciardi G, see Rosa A (2004) 112: 49–116 Riesen H (2004) Progress in Hole-Burning Spectroscopy of Coordination Compounds 107: 179–205 Robaszkiewicz S, see Micnas R (2005) 114: 13–69 Roewer G, Herzog U, Trommer K, Müller E, Frühauf S (2002) Silicon Carbide – A Survey of Synthetic Approaches, Properties and Applications 101: 59–136

216

Author Index Volumes 101–124

Rosa A, Ricciardi G, Gritsenko O, Baerends EJ (2004) Excitation Energies of Metal Complexes with Time-dependent Density Functional Theory 112: 49–116 Rudolf P, see Golden MS (2004) 109: 201–229 Ruiz E (2004) Theoretical Study of the Exchange Coupling in Large Polynuclear Transition Metal Complexes Using DFT Methods 113: 71–102 Sadki A, see Hauser A (2003) 106: 81–96 Saini NL, see Bianconi A (2005) 114: 287–330 Saito S, Umemoto K, Miyake T (2004) Electronic Structure and Energetics of Fullerites, Fullerides, and Fullerene Polymers 109: 41–57 Sakata M, see Takata M (2004) 109: 59–84 Sánchez-Portal D, Ordejón P, Canadell E (2004) Computing the Properties of Materials from First Principles with SIESTA 113: 103–170 Santamaría-Pérez D, Vegas A, Liebau F (2005) The Zintl–Klemm Concept Applied to Cations in Oxides II. The Structures of Silicates 118: 79–135 Sauvage J-P, see Flamigni L (2006) 121: 217–261 Sauvage J-P, see Baranoff E (2007) 123 Scandola F, see Iengo E (2006) 121: 105–143 Schäffer CE (2003) Axel Christian Klixbüll Jørgensen (1931–2001) 106: 1–5 Schäffer CE, see Anthon C (2004) 107: 207–301 Schenker S, see Hauser A (2003) 106: 81–96 Schetinger MRC, Morsch VM, Bohrer D (2003) Aluminium: Interaction with Nucleotides and Nucleotidases and Analytical Aspects of Determination 104: 99–138 Schmidtke HH (2003) The Variation of Slater-Condon Parameters Fk and Racah Parameters B and C with Chemical Bonding in Transition Group Complexes 106: 19–35 Schubert DM (2003) Borates in Industrial Use 105: 1–40 Schulz S (2002) Synthesis, Structure and Reactivity of Group 13/15 Compounds Containing the Heavier Elements of Group 15, Sb and Bi 103: 117–166 Scott JF (2007) A Comparison of Magnetic Random Access Memories (MRAMs) and Ferroelectric Random Access Memories (FRAMs). 124: 199–207 Seifert HJ, Aldinger F (2002) Phase Equilibria in the Si-B-C-N System 101: 1–58 Sessoli R, see Cornia A (2006) 122: 133–161 Shahgholi M, see Contreras RR (2003) 106: 71–79 Shinohara H, see Takata M (2004) 109: 59–84 Sieber R, see Hauser A (2003) 106: 81–96 Simon A, see Deng (2005) 114: 103–141 Slade RCT, see Evans DG (2005) 119: 1–87 Soncini A, Fowler PW, Jenneskens LW (2005) Angular Momentum and Spectral Decomposition of Ring Currents: Aromaticity and the Annulene Model 115: 57–79 Stalke D, see Mahalakshmi L (2002) 103: 85–116 Stasicka Z, see Jaworska M (2003) 106: 153–172 Steed JW, see Turner DR (2004) 108: 97–168 Strunecká A, Patoˇcka J (2003) Aluminoﬂuoride Complexes in the Etiology of Alzheimer’s Disease 104: 139–180 Stulz E, see Bouamaied I (2006) 121: 1–47 Suárez T, see Contreras RR (2003) 106: 71–79 Suksangpanya U, see Hubberstey (2004) 111: 33–83 Sundqvist B (2004) Polymeric Fullerene Phases Formed Under Pressure 109: 85–126 Szalewicz K, Patkowski K, Jeziorski B (2005) Intermolecular Interactions via Perturbation Theory: From Diatoms to Biomolecules 116: 43–117

Author Index Volumes 101–124

217

Tagmatarchis N, Prato M (2004) Organofullerene Materials 109: 1–39 Takata M, Nishibori E, Sakata M, Shinohara M (2004) Charge Density Level Structures of Endohedral Metallofullerenes by MEM/Rietveld Method 109: 59–84 Takenobu T, see Margadonna S (2004) 109: 127–164 Talarico G, see Budzelaar PHM (2003) 105: 141–165 Taniguchi H, see Itoh M (2007) 124: 89–118 Taviot-Gueho C, Leroux F (2005) In situ Polymerization and Intercalation of Polymers in Layered Double Hydroxides 119: 121–159 Teitel’baum GB, see Kochelaev BI (2005) 114: 205–266 Thessing J, see Peng X (2005) 118: 137–177 Trommer K, see Roewer G (2002) 101: 59–136 Tsuzuki S (2005) Interactions with Aromatic Rings 115: 149–193 Turner DR, Pastor A, Alajarin M, Steed JW (2004) Molecular Containers: Design Approaches and Applications 108: 97–168 Uhl W (2003) Aluminium and Gallium Hydrazides 105: 41–66 Ujaque G, Maseras F (2004) Applications of Hybrid DFT/Molecular Mechanics to Homogeneous Catalysis 112: 117–149 Umemoto K, see Saito S (2004) 109: 41–57 Unger R (2004) The Genetic Algorithm Approach to Protein Structure Prediction 110: 153–175 van der Voet GB, see Berend K (2003) 104: 1–58 Vegas A, see Santamaría-Pérez D (2005) 118: 79–135 Vilar R (2004) Hydrogen-Bonding Templated Assemblies 111: 85–137 Wei M, see He J (2005) 119: 89–119 Weihe H, see Mossin S (2003) 106: 173–180 Weissenbacher R, see Haubner R (2002) 102: 1–46 Wenger OS, Güdel HU (2003) Inﬂuence of Crystal Field Parameters on Near-Infrared to Visible Photon Upconversion in Ti2+ and Ni2+ Doped Halide Lattices 106: 59–70 Wheatley AEH, see Linton DJ (2003) 105: 67–139 Wilhelm M, see Haubner R (2002) 102: 1–46 Williams GR, Khan AI, O’Hare D (2005) Mechanistic and Kinetic Studies of Guest Ion Intercalation into Layered Double Hydroxides Using Time-resolved, In-situ X-ray Powder Diffraction 119: 161–192 de Wolff FA, see Berend K (2003) 104: 1–58 Woodley SM (2004) Prediction of Crystal Structures Using Evolutionary Algorithms and Related Techniques 110: 95–132 Xantheas SS (2005) Interaction Potentials for Water from Accurate Cluster Calculations 116: 119–148 Zaman MK, see Atwood DA (2006) 120: 163–182 Zerara M, see Hauser A (2003) 106: 81–96 Zhang H (2006) Photochemical Redox Reactions of Mercury 120: 37–79 Zhang Y, see Atwood DA (2003) 105: 167–201 Zobbi L, see Cornia A (2006) 122: 133–161

Subject Index

Anharmonic effects 149 Antiferroelectric (AFE) interactions – transition 24 BaTiO3 , order/disorder

121

52

Capacitors, three-dimensional

203

D–E loop 101 D-RADP-X 121 Dielectric properties 94 Dipole–proton coupling model, modiﬁed strong 174 Displacive ferroelectrics 51 Displacive phase transition 149 Dynamical properties 155, 183 Electric ﬁeld gradient 126 Electroding 205 FE-FET 201 Ferroelectric (FE) phase state 120 – transition 24 Ferroelectric liquid crystals 51 Ferroelectric solids, role of Pb 76 Ferroelectrics, hydrogen-bonded 5, 23 –, KDP-type 39, 152, 163 –, order/disorder-type 51 FETs, ferroelectric-gate 201 Field cooled (FC) 63 Hafnia gate oxides 206 Heat capacity 101 Hydrogen-bonded systems

5

Isotope effects 149 Isotropic chemical shift, displacive component 34

KD2 PO4 , MAS spectra 40 KDP family 6, 24, 28, 122 KDP-type ferroelectrics 122, 149, 152 Lead lanthanum zirconate titanate (PLZT) 62 Lead zirconate titanate 200 Light scattering 105 LiNiO2 , PDF 74 Local properties 160, 190 Local structure, dynamics 85 Lyddane–Sachs–Teller (LST) relation 91 Magic angle spinning, single crystals 29 Magnetization recovery 137 Nanoribbon, PZT 206 Nanotubes, ferroelectric 204 Neutron powder diffraction, Rietveld reﬁnement 70 Neutron scattering 69 NH4 H2 PO4 26, 43 NMR, 15 N NMR, NH4 H2 AsO4 26, 43 Nucleation 119 O–H distance, order parameter 35 Optical birefringence 104, 112 Order/disorder, displacive effects 15, 23 –, ferroelectrics 51 –, phase transition 119, 149 Oxygen isotope exchange, SrTiO3 89 Pair-density function 69 Paraelectric (PE) phase state 120 Pauling ice rules 122 PbMg1/3 Nb2/3 O3 61 Pb(Sc1/2 Ta1/2 )O3 (PST) 79 PbZrO3 , PDF 76

220 Pd electrodes 205 Perovskite ferroelectric FET 202 Perovskite-type relaxor ferroelectrics 120 Phase coexistence 121 Phase segregation 119 Polar nanoregions (PNRs) 84 Proton–lattice coupled model 8 Proton-tunneling 163 Pseudo-spin glass system 119 Pseudo-spin model 24 Pyroelectricity 101 PZT nanoribbon 206 Quantum ferro-/paraelectricity, SrTiO3 89 Rattling titanium 2 Rb1–x (ND4 )x D2 PO4 119 RbH2 PO4 26 Relaxor ferroelectrics 69 –, local structure/dynamics 81 Relaxor PMN, order/disorder 61 Relaxors 51 Rochelle salt 1

Subject Index Second harmonic generation 112 Soft mode 119 –, SrTiO3 89 Soft transverse optical (TO) phonon 62 Spallation Neutron Source (SNS) 71 Squaric acid 27, 31, 39 –, 17 O NMR 36 Sr, NMR 115 SrTiO3 89 –, PDF 72 Static dielectric properties 152, 174 Static local structure 81 STO (STO18), isotope-exchanged 90 Strontium barium niobate (SBN) 62 Strontium bismuth tantalate 200 Structural dielectric properties 174 Takagi defects 119, 122 Titanium, NMR 115 –, rattling 2 Zero ﬁeld cooled 63