Correlations between Crystal Structure and Elastic Properties of Piezoelectric Crystals: Tourmalines and Relaxor Ferroelectrics of the
type CaxBa1-xNb2O6 and Ce:CaxBa1-xNb2O6
DISSERTATION
zur Erlangung des akademischen Grades eines “Doktor der Naturwissenschaften” an der Fakultät für Geowissenschaften der Ruhr-Universität Bochum Germany
von Chandra Shekhar Pandey aus Bareilly (India)
Bochum, 2010
1. Gutachter Prof. Dr. Jürgen Schreuer 2. Gutachter Prof. Dr. Ladislav Bohatý
Datum der Vorlage 20.10.2010 Datum der Disputation 17.12.2010
“If we knew what we were doing, it wouldn't be research”
Albert Einstein
This thesis is dedicated to
My Love Arti
&
My sweet ‘n’ cute loving daughter Aditi
Abstract
In this thesis the elastic properties of two distinct groups of piezoelectric crystals are presented: one is natural tourmaline occurring with a vast variability in chemical compositions with a stable phase over a wide temperature range and hence allowing to examine the influence of chemical composition on its physical properties; while the other is synthetically grown relaxor ferroelectric calcium barium niobate (CBN) exhibiting a ferroelectric-paraelectric phase transition and thus invoking the interest to investigate the variation of elastic properties in both phases as well as in the vicinity of the phase transition. To this end the non-destructive innovative method of resonant ultrasound spectroscopy (RUS) was employed. The full sets of elastic, piezoelectric, dielectric constants and coefficients of thermal expansion of five natural single crystal tourmalines of gem quality have been determined between 100 K and 903 K. The main influence of Li and Fe on elastic, piezoelectric and dielectric properties of tourmalines was studied. Some tourmalines showed an unexpected irreversible softening of resonance frequencies during first heating, which is probably due to order/disorder processes on certain cation sites. The anomalous elastic behaviour was observed in Czochralski-grown pure and
cerium doped Ca0.28Ba0.72Nb2O6 (CBN-28) single crystals between room temperature
and 1503 K. Doping of Ce in pure CBN leads to lowering of Curie temperature (Tc) by ~ 70°C, and stiffening of all elastic constants. Below about 900 K all resonances
of the freely vibrating samples show rapid softening when approaching Tc. In the ferroelectric phase ultrasound dissipation effects were too strong to observe enough resonances to refine elastic constants, which are probably related to interactions between sound waves and ferroelectric domain walls. Initiation of Burns temperature was clearly observed in the temperature range of 890-910 K.
Contents
Abbreviations
1 Introduction 1
2 Theoretical background 5 2.1 Elastic properties of materials 5 2.1.1 Generalized Hooke’s law 5 2.1.2 Elastic anisotropy 7 2.1.3 Deviation from Cauchy relations 7 2.1.4 Temperature dependence of the elastic constants 8 2.2 Piezoelectricity 8 2.2.1 Constitutive equations 12 2.2.2 Piezoelectric materials 13 2.3 Ferroelectricity 14 2.4 Relaxor ferroelectrics 17 2.5 Resonant ultrasound spectroscopy 19 2.5.1 The forward problem 22 2.5.2 The inverse problem 24
3 Experimental techniques 25 3.1 Sample preparation 25 3.2 Chemical and structural characterization 26 3.2.1 Electron microprobe analysis 26 3.2.2 X-ray diffraction 26 3.2.3 Density determination 26 x Contents ______
3.3 Thermal analysis 27 3.3.1 Dilatometry 27 3.3.2 Differential scanning calorimetry 27 3.4 Dielectric properties 28 3.4.1 Substitution method 28 3.4.2 Geometric method 29 3.5 Elastic and Piezoelectric constants 30 3.5.1 Resonant ultrasound spectroscopy 30
4 Elastic and piezoelectric properties of tourmalines 33 4.1 Crystal chemistry of tourmalines 33 4.2 Chemical and structural characterization 36 4.3 Dielectric constants 41 4.4 Thermal properties 46 4.4.1 Thermal expansion 46 4.4.2 Specific heat capacity 48 4.5 Elastic and piezoelectric properties 48 4.5.1 Correlation between crystal structure and elastic properties 54 4.5.2 Isotropic elastic moduli and Debye temperature 57 4.5.3 Temperature evolution of elastic and piezoelectric constants 59 4.6 Summary and conclusions 62
5 Anomalous elastic behavior of Ca0.28Ba0.72Nb2O6 single crystal in the paraelectric phase 65 5.1 Thermal expansion 69 5.2 Elastic properties 73 5.2.1 Deviation from Cauchy relations 78 5.2.2 Isotropic elastic properties 80 5.2.3 Temperature evolution of eigen modes 81 5.3 Summary and conclusions 82
6 Elastic properties of Ce:Ca0.28Ba0.72Nb2O6 single crystal in paraelectric phase 83 6.1 Comparison of pure and Ce doped CBN-28 85 Contents xi ______
6.1.1 Thermal expansion 85 6.1.2 Elastic properties 87 6.1.3 Deviation from Cauchy relations 91 6.1.4 Isotropic elastic properties 92 6.1.5 Eigen modes behavior 93 6.2 Hypothesis for Ce position in Ce:CBN 94 6.3 Summary and conclusions 97
7 Summary and outlook 99
Bibliography 103
Abbreviations
ai Crystallographic system α Coefficients of linear thermal expansion B Bulk modulus C Mean elastic stiffness CBN, CBN-28 Calcium Barium Niobate Ce:CBN, Ce:CBN-28 Cerium doped Calcium Barium Niobate cijkl, cij Elastic stiffness constants ei Cartesian reference system
εij Strain tensor
є ij Dielectric constant at constant strain
є ij Dielectric constant at constant stress eijk, eij Piezoelectric stress constants EMPA Electron Micro-Probe Analysis f Frequency G Shear modulus gmn Deviation from Cauchy relations μ Poisson’s ratio Q Quality factor
ρb Buoyancy density
ρg Geometric density RUS Resonant Ultrasound Spectroscopy
σij Stress tensor T Temperature tan δ Dielectric loss
TB Burns temperature xiv Abbreviations ______
Tc Curie temperature
ΘD Debye temperature
Tij Thermoelastic constants
Tm Temperature of maximum dielectric constant
VD Mean wave velocity
VL Longitudinal wave velocity
VS Shear wave velocity Angular frequency Wt % Weight percent XRD X-ray Diffraction
Chapter 1
Introduction
The elastic properties of a solid correspond to its response to an applied stress and are of fundamental importance to science and technology because they are related to the forces acting between the constituent atoms or ions and thus give useful information about the present network of bonding interactions in the material. The evolution of elastic properties with respect to temperature is considered valuable to study phase transitions in a material. Further, the elastic constants are closely connected to thermodynamic properties of the material such as specific heat, thermal expansion, and the Debye temperature. The knowledge of elastic constants is essential for many practical applications related to the mechanical properties of a solid, for example load-deflection, thermoelastic stress, internal strain or residual stress, sound velocities and fracture toughness [1-3]. In piezoelectric crystals elastic waves are additionally accompanied by electric fields giving rise to electromechanical coupling effects which can be used to convert electrical → mechanical energy and vice-versa.
Today, crystal species like LiNbO3, Ca3Ga2Ge4O14 and α-quartz, are well investigated and widely used in piezoelectric devices. However, they suffer from phase transitions or strong ultrasound dissipation effects etc. A promising candidate to overcome these limitations is tourmaline. Tourmaline, a structurally complex ring silicate, has been attractive for researchers due to its promising piezoelectric properties. The use of transparent tourmaline in the studies on polarized light is well known. The application of tourmaline at temperatures up to about 1000 K is not limited by phase transitions, electrical conductivity or strong ultrasound dissipation effects. Tourmaline offers 2 Introduction Chapter 1 ______
certain advantages over most other materials, especially for operation at higher frequencies [4]. For example, (a) the attenuation is extremely low, being about one order of magnitude smaller than in α-quartz [5], (b) for transducer applications, tourmaline can be used up to 1100 MHz [4], while widely used α-quartz only till 800 MHz, and (c) it has high acoustic velocities [4] (relative to those in α-quartz and
LiNbO3) together with a high Debye temperature [6]. Tourmaline can also be used as high-frequency surface-acoustic-wave devices, where the combination of low acoustic loss, high acoustic velocities and moderate piezoelectric effect make it attractive for high frequency work [5]. No success has yet been achieved in growing synthetic tourmaline of more than 1mm in diameter in laboratory [7-8]. Therefore, for any practical application, one has to depend only on natural tourmalines. However, natural tourmalines exist in a broad range of chemical compositions, leading to variations in their physical properties. Thus, it becomes important to study the behavior of the desirable properties of tourmalines with chemical composition. The available literature on elastic constants of tourmalines [9-13] are without any discussion on the chemical composition except the work by [12] and [13], whereas piezo- and dielectric properties are missing in those literatures too. Moreover, a systematic study of elastic, piezoelectric and dielectric properties, their systematic variation with chemical composition and, more importantly, the temperature evolution of these physical properties up to the decomposition temperature have not yet been reported. Therefore, the first aim of this work is to investigate the composition and temperature dependence of the elastic and piezoelectric properties of tourmaline. The other material chosen for this research work was the relaxor ferroelectric material CBN, which unlike tourmalines exhibits a phase transition (ferroelectric ↔ paraelectric) and thus invokes the interest to investigate the variation of elastic properties in both phases as well as in the vicinity of phase transition. Relaxor ferroelectric materials, crystallizing in the partially filled tetragonal tungsten bronze (TTB) structure type, are widely investigated because of their
promising properties. Particularly, the relaxor ferroelectric material Sr1−xBaxNb2O6 (SBN-x) with x ≈ 0.61 (SBN-61) has attracted much interest with regard to basic research on uniaxial relaxor ferroelectrics and different applications [14], because of its outstanding pyroelectric [15], piezoelectric [16], electro-optic [17-18], photorefractive properties [19-22], and surface acoustic wave devices [23]. The Chapter 1 Introduction 3 ______
ferroelectric phase transition temperature of the congruent melting SBN-61 is about 79 °C [10], which implies that SBN crystals may be easily depolarized near room temperature. Therefore, it has become necessary to search new materials possessing
higher Curie temperatures for device applications. CaxBa1−xNb2O6 (CBN-x), which also belongs to the partially filled TTB structure type and which is structurally closely related to SBN but with much higher Curie temperature, ~265 °C, is considered as a potential alternate for SBN. Various authors have focused on different aspects of CBN-28 including studies of dielectric, ferroelectric and optical properties [24-27], ferroelectric domain structure and its evolution using transmission electron microscopy [10], phase diagram analysis [28] and room temperature specific heat together with thermal expansion coefficients in the temperature range 298 K - 573 K [29]. However, the elastic constants of CBN single crystals are unknown till date. An accurate knowledge of elastic constants and their temperature dependences is essential for various possible applications. Hence, the second aim of this work is to gain more insight into the physical properties of CBN-28 focusing on the temperature dependence of the full sets of elastic constants together with thermal expansion in paraelectric phase. Doping in ferroelectric materials is useful to tailor their interesting properties. Much work has been done on SBN-61 doped with rare-earth metals (La, Ce, La+Ce, Nd, Tm, Yb) [30-33] for its excellent holographic [34-39], phase conjugation optics and wave mixing properties [40-41]. It has also been proved that doping as well as the increase in the [Sr]/[Ba] ratio in SBN-61 result in an increase of electro-optical (for Ce:SBN) and the piezoelectric (for La:SBN and Ce:SBN [42]) coefficients. Doping of Ce3+ in SBN is found to increase the photorefractive sensitivity by two orders of magnitude in comparison with pure SBN crystals [43-44]; while, at the same time, it strongly lowers the phase transition temperature inversely proportional to the doping concentration [45-46]. As the Curie temperature of CBN is quite high, it suits better for device applications. However, no literature data is available so far regarding any
physical or chemical properties of Ce doped Ca0.28Ba0.72Nb2O6 (Ce:CBN-28, here after Ce:CBN), not even its doping concentration or crystal structure. Therefore, the third objective of this thesis is to investigate the influence of Ce doping on the elastic properties of pure CBN in paraelectric phase. The contents of this dissertation are divided in seven chapters. Chapter 2 provides the theoretical background focusing on fundamentals of elasticity, 4 Introduction Chapter 1 ______
piezoelectricity, ferroelectricity and relaxor ferroelectrics together with the working principal of the resonant ultrasound spectroscopy (RUS). The RUS method facilitates simultaneous determination of all independent elastic and piezoelectric constants on one sample within the same frequency range. Chapter 3 presents the experimental details regarding sample preparation and various characterization techniques employed in this study. Chapter 4 is concerned with the elastic and piezoelectric properties of tourmalines. Chapter 5 and Chapter 6 are devoted to elastic properties of CBN-28 and cerium doped CBN-28, respectively, in the paraelectric phase. The summary and outlook are presented in Chapter 7.
Chapter 2
Theoretical background
2.1 Elastic properties of materials 2.1.1 Generalized Hooke's Law Robert Hooke in 1676 postulated that the stress is proportional to the strain for small displacements. This scalar stress-strain proportionality relation was extended by Thomas Young to an anisotropic medium [47] according to
σij = cijkl εkl (i ,j ,k, l = 1,2,3) (2.1)
here cijkl are the components of the tensor of elasticity, a fourth rank tensor, σij are the
components of the second rank tensor of mechanical stress (Fig. 2.1 a), and εkl are the components of the second rank strain or deformation tensor (Fig. 2.1 b,c).
(a) (b) (c)
FIG. 2.1: Interpretation of stress and strain tensor components [48]. (a) σij represents the strain caused by a force acting along ei on a face element normal to ej. (b) εii = Δli/l0i describes the longitudinal deformation of a body along the direction ei of the reference system. (c) The components εij ≈ γij / 2 are shear strains. 6 Theoretical background Chapter 2 ______
Introducing Voigt’s notation scheme, i.e. replacing the index pair ij by a single index according to
i for i j ij → 9i j for i j (2.2) the generalized Hook’s law becomes,
σi = cij εj, (2.3) where i, j = 1 to 6.
Using this Voigt notation, the {cij} are usually written in matrix form as:
c11 c12 c13 c14 c15 c16
c21 c22 c23 c24 c25 c26 c c c c c c 31 32 33 34 35 36 c41 c42 c43 c44 c45 c46
c51 c52 c53 c54 c55 c56
c61 c62 c63 c64 c65 c66
Due to the reversibility of the elastic energy, the relation cij = cji holds. Hence, the above written 36 elastic constants are reduced to 21 including 6 diagonal elements and 15 off-diagonal elements in the matrix form. Neumann’s principle further leads to a reduction of the number of independent tensor component depending on the symmetry elements of the crystals point group [49]. For example, the number of independent elastic constants is 21 for samples with triclinic symmetry, but is reduced significantly for materials with higher symmetry. For example: cubic materials have 3, and isotropic materials have only 2 independent elastic constants [50]. All elastic constants can be categorized into four types: longitudinal elastic stiffnesses c11, c22, c33, shear stiffnesses c44, c55, c66, transverse interaction coefficients c12, c13, c23, and coupling coefficients c14, c15, c16, c24, etc. Longitudinal elastic stiffnesses vary in a wide range, by more than three orders of magnitude, and are generally about two times larger than the corresponding values of shear stiffnesses. However, this relation is not followed in all cases [48]. The coupling coefficients are not considered of much importance as they are at least one order of magnitude smaller than others and often vanish due to symmetry constraints. Chapter 2 Theoretical background 7 ______
2.1.2 Elastic anisotropy A material is said to be elastically anisotropic if the velocity of sound in it depends on the direction of propagation. Most of the materials found in nature are elastically anisotropic. For example, even highly symmetric cubic crystals have three
independent elastic constants c11, c12, and c44. In an isotropic medium additionally c44 equals (c11 - c12)/2 and based on this a measure of elastic anisotropy is defined by: 2c 1 44 . (2.4) c11 c12 The elastic anisotropy directly reflects the anisotropy of the crystal’s bonding system. On one hand, strong anisotropy of longitudinal elastic stiffness generally reflects the anisotropic arrangements of strong bond chains, while on the other hand extreme anisotropy of shear resistances is observed in the vicinity of certain phase transitions and is caused by acoustic soft mode behavior. The largest known anisotropy of longitudinal elastic stiffness is observed in graphite [48], which explains well its excellent cleavage properties making it possible to produce mono-layers of carbon, known as graphene.
2.1.3 Deviation from Cauchy relations In an atomistic view, if we neglect temperature effects and consider the inter- atomic forces to be central forces and all lattice particles are located on inversion
centers, then the elastic constants cij obey the Cauchy relations [9, 51]:
c23 = c44; c13 = c55; c12 = c66;
c14 = c56; c25 = c46; c36 = c45. (2.5)
The condition is deviated considerably when the bonding has strong covalent character (e.g. MgO) [1]. The deviations from Cauchy relations are represented by the second rank tensorial invariant:
c c c c c c 23 44 45 63 64 52 g mn c45 c63 c31 c55 c56 c41 , (2.6) c64 c52 c56 c41 c12 c66 8 Theoretical background Chapter 2 ______
which provide information about the predominant type of bonding in the crystal [52]. Experimental data for almost all the measured crystal species show often strong
violation of Cauchy relations. The diagonal components gmm are usually by one order
of magnitude higher than the off-diagonal components gmn with m ≠ n. In general, the
gmn follow two rules: (i) positive deviations from Cauchy relations are observed in crystals with strong ionic bonds, because the transverse interaction coefficients
dominate over the corresponding shear stiffnesses and (ii) small or negative gmm are caused by strong covalent bonds or other bonds with preferential orientation (e.g. hydrogen bonds). These rules allow for a qualitative interpretation of elastic anisotropy with respect to the structural parameters such as nature of bonds. A detailed discussion on the quantitative interpretation is given in literature [48].
2.1.4 Temperature dependence of the elastic constants The evolution of elastic constants with respect to temperature yields valuable information about the stability of the material and thus about possible phase transition. For the interpretation of the temperature dependence of elastic constants often the thermoelastic constants are used as:
1 dcij d logcij Tij . (2.7) cij dT dT
-1 The Tij have the dimension of K .
2.2 Piezoelectricity The phenomenon of piezoelectric effect was discovered more than hundred years ago by the Curie brothers, Pierre and Jaques in certain crystals such as quartz, tourmaline, and Rochelle salt [53]. Lippmann in 1881 was the first who predicted the existence of the converse piezoelectric effect based on thermodynamic principles and was promptly confirmed by the Curies. W. Hankel in 1881 proposed the name ‘piezoelectricity’, where the word ‘piezo-’ was derived from Greek words for ‘press’. Piezoelectricity is an interaction between electrical and mechanical quantities and is one of the basic properties of materials that do not possess a centre of symmetry. The piezoelectric effect is of two types ‘the direct’ and ‘the converse’ piezoelectric effect (Fig 2.2). When a piezoelectric crystal is strained by an applied mechanical stress, an electric polarization is produced within the material, which is Chapter 2 Theoretical background 9 ______proportional to the magnitude and sign of the strain and is known as the direct piezoelectric effect.
FIG. 2.2: Macroscopical illustration of electromechanical coupling in piezoelectric materials.
Closely related to it is the converse effect, whereby a crystal becomes strained when an electric field is applied. A simple atomistic model is shown in Fig 2.3, which explains the generation of an electric charge as the result of an applied force on the material. Before application of some external stress, the dipole moments cancel out. Therefore, an electrically neutral molecule appears. After
FIG. 2.3: Simple molecular model for explaining the piezoelectric effect: a) unperturbed molecule; b) molecule subjected to an external force, and c) polarizing effect on the material surfaces [54]. 10 Theoretical background Chapter 2 ______exerting some pressure on the material, its internal structure will get deformed, which causes the separation of the positive and negative centers of the molecules; consequently little dipoles get generated. Eventually, the facing poles inside the material are mutually cancelled and a distribution of a linked charge appears on the material’s surfaces, and the material is polarized. The electric field generated by this polarization then can be used to transform the mechanical energy used in the material’s deformation into electrical energy.
FIG. 2.4: Relationship between electrical, mechanical and thermal systems, modified after [55].
Piezoelectricity, like elasticity is one among various linear interaction processes possible in a material between any two of the three systems – electrical, mechanical and thermal, as shown Fig. 2.4. Similar diagrams and its extended version can be found in many other text books of crystal physics [49, 56] and journals [57]. In this diagram, the outer quantities are the thermodynamic intensive variables Chapter 2 Theoretical background 11 ______
(generalized forces) and the inner quantities are the extensive variables (generalized displacements) [49, 53, 55-56]. A prerequisite of piezoelectricity is the lack of a center of symmetry [58-59]. Neumann’s law states that the geometrical representation of any physical property contains the symmetry of the point group of the material. As shown in Fig. 2.5, of the 32 crystallographic point groups (or classes), 11 classes are centrosymmetric and 21 classes are noncentrosymmetric. However, one of the 21 classes is not piezoelectric because of the combination of 4- and 3-fold symmetry axis. So, only 20 classes are potentially piezoelectric. In 10 classes out of these 20, polarization can be induced by a mechanical stress, while the remaining 10 classes possess spontaneous polarization, so they are permanently polar and thus can show piezoelectric together with pyroelectric effects. There is a subgroup within these 10 classes that possesses spontaneous and reversible polarization; this subgroup can exhibit all three effects— ferroelectricity, piezoelectricity, and pyroelectricity.
FIG. 2.5: Classification scheme for 32 crystallographic point groups with piezoelectric, pyroelectric and ferroelectric effects [60].
12 Theoretical background Chapter 2 ______
2.2.1 Constitutive equations Piezoelectric materials show an important characteristic of its linear behavior within a certain range. In this section, we have used the notations as per IEEE standard and only the piezoelectric coupling is considered (thermoelectric terms are neglected). The linear theory of piezoelectricity consists of the following equations of motion and charge [61]
2 ji ui Di o fi o 2 , and e (2.8) x j t xi where σij are the components of the stress tensor, o is the mass density, f is the body force per unit mass, u is the displacement vector, D is the electric displacement vector, and e is the body free charge density which is usually zero. Piezoelectric constitutive equations can be derived [61] using mechanical stress σ, strain ε, electric field E, and electric displacement D and can be written as:
E stress-charge form (e-form): ij cijkl kl ekij Ek ,
Di eikl kl є ik Ek . (2.9)
Where cijkl , ekij and єij denote the components of the elastic, piezoelectric and dielectric tensor. The components of the strain tensor ε and of the electric field vector E are related to the displacement u and, respectively, to the electric potential ( ) by:
1 u u i j , E . (2.10) ij i x 2 x j xi i Substituting the values from Equations (2.9) and (2.10), the equation (2.8) can be written for u and as:
2 uk ui cijkl ekij fi 2 and xlj xkj t
uk eikl є ij q . (2.11) xli xij In equation (2.11), we have neglected the superscripts of the material constants and the subscript of the reference mass density. Depending on different independent variables, other forms of constitutive equations can be formulated: Chapter 2 Theoretical background 13 ______
D stress-voltage form (h-form): ij cijkl kl hkij Dk , (2.12)
Ei hikl kl ik Dk ,
E strain-charge form (d-from): ij sijkl kl d kij Ek , (2.13)
Di dikl kl є ik Ek ,
D strain-voltage form (g-form): ij sijkl kl g kij Dk , (2.14)
Ei gikl kl ik Dk .
In all above equations σ is mechanical stress (N/m2), ε is mechanical strain (unit less), E is electrical intensity (V/m), D is electrical displacement (C/m2), є is permittivity (F/m), β is impermeability (m/F), e (C/m2), h (N/C), are the piezoelectric stress constants, d (C/N) and g (m2/C) are the piezoelectric strain constants, c is elastic stiffness (N/m2), s is compliance (m2/N). All the superscripts here represent the constant quatities. Different material constants (elastic, piezoelectric and dielectric) found in all four formulations are interrelated as follows:
E E D D cijmn mnkl cijmn mnkl I є ik β kj = є ik β kj , (2.15)
E dijk = є il g ljk = eilm slmjk , (2.16)
E eijk = є il h ljk = dilmclmjk , (2.17)
D gijk il dljk hilmslmjk , (2.18)
D hijk il eljk gilmclmjk . (2.19) The piezoelectric coupling coefficient serves as a measure of the efficiency of energy conversion and can be calculated as [62] k2 = e2/єε cD = e2 / єσ cE (2.20)
2.2.2 Piezoelectric materials
Rochelle salt, α-quartz, tourmaline, LiNbO3 and Ca3Ga2Ge4O14-type crystals are some examples of piezoelectric crystals [63]. Quartz, because of its converse piezoelectric effect, vibrates when an alternating current is applied to and is the most used piezoelectric crystal. Piezoelectric materials are important because of numerous applications which include signal filters, actuators, transducers, capacitors, and 14 Theoretical background Chapter 2 ______sensors. Latest developments include medical underwater communication, high- displacement piezoelectric actuators, ultrasonic imaging, microelectronics and MEMS applications. More on the important applications of piezoelectric crystals can be found in reference [64]. Quartz is probably the most widely used piezoelectric crystal. A combination of direct and converse piezoelectric phenomena is used in quartz clocks to generate a regularly timed series of electrical pulses. It belongs to crystal class 32. Langasite and some of its isomorphs (langanite and langatate) are emerging piezoelectric crystals which have stronger piezoelectric coupling than quartz and also belong to crystal class 32. Lithium niobate and lithium tantalate have stronger piezoelectric coupling than quartz and for these two crystals the crystal class is 3m. Tourmalines also belong to the same class and will be discussed in detail later in chapter 4.
2.3 Ferroelectricity Ferroelectricity is a property of certain materials possessing spontaneous polarization that can be reversed by the application of an external electric field. It was discovered in 1921 by Valasek in Rochelle salt (NaKC4H4O6·4H2O). Many ferroelectrics are low temperature modifications (ferroelectric phase) of a high temperature and higher symmetry structure (prototype known as paraelectric), which
a) b) c) d)
FIG. 2.6 Unit cells of the four phases of BaTiO3, The dotted lines in (a), (b), and (c) delineate the original cubic cell. Arrows indicate the direction of the spontaneous polarization, Ps, in each phase [65]. has no spontaneous polarization. In general, the spontaneous polarization in a ferroelectric crystal is not uniformly distributed throughout the whole crystal, but rather forms certain regions of a crystal, called domains, with each domain having Chapter 2 Theoretical background 15 ______uniform polarization. The interface between two domains is called the domain wall. Ferroelectric materials may exhibit one or more ferroelectric (polar) phases that show a domain structure in which the individual domain states can be reoriented by an applied field. The origin of the ferroelectric domain can be explained from the energy viewpoint; the poly-domain system may be in the state of minimum energy balanced from both electric and mechanical contributions. The polarization direction of each domain is determined by the crystal symmetry of the ferroelectric crystal (Fig. 2.6). The process of applying an electric field, which exceeds a certain field to orient the domains toward the field direction, is termed as poling.
FIG. 2.7: View of a domain structure, showing antiparallel domains with 180°-walls [26].
First demonstration of the ferroelectric domains came out from a study of the spontaneous birefringence. From a microscopic viewpoint, domains were attributed to the change in the electrostatic forces acting on the crystal’s faces owing to the spontaneous polarization that occurs as the crystal goes through the paraelectric- ferroelectric phase transition. A schematic drawing of the structure of ferroelectric domains is shown in Fig. 2.7. Spontaneous polarization is a characteristic of ferroelectricity and can be observed by measuring the polarization versus the electric field curve (P-E hysteresis curve, Fig. 2.8). When the external field is small, the polarization increases linearly with the field mainly due to field-induced polarization, as the field is not large enough to cause orientation of the domains (portion OA). At fields higher than the external field range, polarization increases nonlinearly with increasing field, as all domains start to orient toward the direction of the external field (portion AB). At high fields, a saturation state is reached (portion BC) when nearly all the domains are aligned toward the direction of the poling field. The linear extrapolation of the segment BC 16 Theoretical background Chapter 2 ______back to polarization axis (at the point E on the vertical axis) represents the value of the spontaneous polarization Ps. Now, if the field is gradually decreased to zero, the polarization will decrease, following the path CBD. OE represents the spontaneous polarization Ps and OD represents the remnant polarization Pr. The value of Pr is smaller than Ps because when the field is reduced to zero, some domains may remain aligned and some may return to their original positions due to domain interactions, thus reducing these domains contribution to the net polarization.
FIG. 2.8 A typical P-E hysteresis loop in ferroelectrics [26].
The remnant polarization Pr can only be removed when the applied field reaches at point F in opposite direction. The field required to bring the polarization to zero is called the coercive field Ec (portion OF on zero polarization axis). Further, increase of the field in the negative direction will cause a complete alignment of the dipoles in this direction and the cycle can be completed by reversing the applied field direction. Thus, the relation between P and E is represented by a hysteresis loop CDFGHC as shown in Fig. 2.8.
For most ferroelectrics there is a temperature, the Curie temperature Tc, above which Ps becomes zero. The non-polar phase above this temperature is the paraelectric phase and this temperature is regarded as the ferroelectric - paraelectric phase transition, and is signaled by a dielectric constant peak [65]. Chapter 2 Theoretical background 17 ______
FIG. 2.9: A cubic unit cell of ABO3 (BaTiO3) perovskite-type [65].
Ferroelectric materials can be divided in two broad categories: first those in which the ferroelectric/paraelectric phase change is associated with an order/disorder transition type for example in potassium dihydrogen phosphate, KH2PO4 (KDP); and second in which the transition is induced by a displacive transition. The case of a displacive ferroelectric transition is more complex and is characteristic for perovskites of the general formula ABO3, for example BaTiO3 (Fig. 2.9). Perovskite is the mineral name of calcium titanate (CaTiO3). Its simplest structure is cubic (space symmetry Pm3m), consisting of corner sharing oxygen octahedra (BO6) arranged in three dimensions with smaller, highly charged cations located in the middle of the octahedra, and lower charged, larger cations in between the octahedra.
2.4 Relaxor Ferroelectrics Relaxors are distinguished from “normal” ferroelectrics by the presence of a broad-diffuse phase transition from the paraelectric to ferroelectric state, a strong frequency dependence of the dielectric constant (i.e. dielectric relaxation) and a weak remnant polarization. The phenomena of diffuse phase transition has come into picture only after Smolenskii’s discovery of ferroelectric phase transition on
Ba(Ti,Sn)3. Relaxor ferroelectrics can be prepared either in polycrystalline form or as single crystals. The peculiar characteristics of relaxor ferroelectrics is shown in Fig. 2.10, using relaxor ferroelectric Pb(Mg1/3,Nb2/3)O3 (PMN) as an example. The characteristic of relaxor ferroelectrics of having giant and temperature-insensitive dielectric constants and large frequency dependence (dielectric relaxation) make them ideal for high permittivity capacitors, electrostrictive actuators and recently E-field induced 18 Theoretical background Chapter 2 ______piezoelectrics for low frequency sonar and high frequency bio-medical transducers [66].
a)
b)
c)
FIG. 2.10: Basic characteristics for relaxor ferroelectric Pb(Mg1/3Nb2/3)O3 (PMN). (a) Dielectric properties, (b) polarization-electric field curve behavior, and (c) X-ray and optical evidence [14].
The major differences between normal and relaxor ferroelectrics are shown in Fig. 2.11. A number of solid solution systems exhibiting diffuse phase transition have been discovered. The widely investigated photorefractive Sr1-xBaxNb6O10 (SBN) and the recently grown CaxBa1-xNb6O10 (CBN), are typical examples of relaxor ferroelectrics and exhibit partially filled tetragonal tungsten bronze (TTB) type structures. Chapter 2 Theoretical background 19 ______
FIG. 2.11: Property difference between relaxor and normal perovskite ferroelectrics [67].
2.5 Resonant Ultrasound Spectroscopy Resonant ultrasound spectroscopy (RUS) allows for the determination of the elastic and piezoelectric constants of solids by measuring the vibrational eigenmodes of samples of well defined shapes, usually a rectangular parallelepiped. A typical schematic diagram is shown in Fig. 2.12.
20 Theoretical background Chapter 2 ______
FIG. 2.12: Schematic diagram of RUS arrangement.
The first successful RUS measurement was originally done by Frasier and LeCraw [68] for a sphere of isotropic material. Later in 1970 Schreiber, Anderson and Soga [1, 69-70] improved this and applied it to small lunar rock samples. Holland [71] applied the Rayleigh-Ritz method to the problem of the vibration of rectangular parallelepipeds and used trigonometric functions as basis functions. Demarest [72] applied this technique to an isotropic specimen with a cube shape. In 1976 Ohno [73] further extended Demarest’s work to a rectangular parallelepiped specimen with orthorhombic symmetry, and was the first to introduce the solution of the inverse problem. Visser et. al. [74] used the Rayleigh-Ritz method using single powers of cartesian coordinates as basis functions (the “xyz” method). Final contributions by Maynard [3], Migliori [2, 25, 75], Visscher et al. extended the use of this method from the geophysics community to the general physics community by introducing the computational methods and gave the term “resonant ultrasound spectroscopy”. This method has certain advantages over other conventional methods for measuring elastic constants. First, many specimen geometries can be used, for example cylinders, spheres, rectangular parallelepipeds, and thin films. Second, measurements can be made on variable sample size (0.1 to >100mm). Third, one can obtain all independent elastic and piezoelectric constants of a single sample in one single run with high internal consistency [28] regardless of the symmetry. Fourth, no bonding between transducers and sample is required; only a negligible contact force is required to hold the sample in place. Therefore, this method is excellently suited for experiments at non-ambient temperatures. Apart from all these advantages there are some major disadvantages also. The foremost is, there is no analytical solution to the “inverse problem”. Chapter 2 Theoretical background 21 ______
(a)
(b) FIG. 2.13: (a) Snapshots of calculated eigenmodes of a rectangular parallelepiped single crystal [3],and (b) examples of the resonance spectra of a Beryl single crystal [76].
The forward problem, of which analytical solutions exist (see sec. 2.5.1) includes the calculation of an object’s resonance spectrum from known parameters (orientation, shape, size, density and elastic, piezoelectric and dielectric constants). The inverse procedure compares the measured frequencies with computed frequencies using a non-linear least-square procedure, which requires an initial set of input parameters. The input parameters for the computation are varied in an iterative process to obtain good agreement between measured and computed frequencies, which requires high computational efforts. However, due to ongoing rapid progress on high speed computers, this is no longer a severe limitation. Fig. 2.13 (a) shows 22 Theoretical background Chapter 2 ______some examples of vibrational eigenmodes of a rectangular parallelepiped of an anisotropic crystalline material. A typical resonance spectrum of a beryl sample, measured by our RUS apparatus, is given in Fig. 2.13 (b).
2.5.1 The forward problem The kinetic energy U and the potential energy P of a system is given by [77]
1 2 2 U = ui , (2.21) 2 i
1 u u and, P = C i k (2.22) ijkl 2 ijkl x j xl where ui (i = 1, 2,3) are the components of the displacement vector u , is the density and is the frequency of a free vibration of the system. In case of piezoelectric crystals, together with potential energy due to strain, we also have to consider a potential energy due to local electric field, hence equation (2.22) can be written as
1 u u 1 u P c i k e k (2.23) ijkl mn mkl i, j,k,l 2 x j xl m,n 2 xm xn k,l,m xm xl where is the electric potential, mn is the dielectric tensor, and emkl is piezoelectric stress tensor. An eigen-mode of the system corresponds to a stationary solution of its Lagrangian L L U P dV . (2.24) V Using Hamilton’s principle and the Rayleigh-Ritz method, the displacement vector and, respectively, the electric potential can be expanded in suitable sets of basis functions as following ui ai and (2.25)
b (2.26) where ai are the expansion coefficients. The choice of ( and = 1, 2,…, T) is arbitrary. For example, Demarest [72] and Ohno [73] used normalised Legendre polynomials, Visser [74] used powers of cartesian coordinates in the form
x y z . (2.27) Chapter 2 Theoretical background 23 ______
Now, by substituting equation (2.25) and (2.26) in equation (2.24) through (2.21) and (2.23), we will get stationary solutions of equation (2.24) by solving the generalized eigen problem G KE 1K T a = ρω2Na (2.28) where G, K, and E depend on the shape, dimension and the electromechanical properties of the vibrating body, and can be given as:
G c dV , (2.29) ii ijij x x V j j
K e dV , (2.30) i iij x x V i j
E dV , and (2.31) ij x x V i j
N ii iidV (2.32) where cijij are the elastic stiffnesses under constant electric field, and ii denotes the Kronecker symbol. To keep the computational effort within reasonable limits, the size of the matrices has to be restricted by the use of a truncated set of appropriate basis
functions. Visscher’s normalized powers of Cartesian coordinates x y z are
a) b)
FIG. 2.14: Approximation of the components of (a) displacement vector, and (b) electrical potential [78]. the most simple choice which provides quick convergence and can be easily applied to various samples with simple shapes. Here , and are positive integers. The 24 Theoretical background Chapter 2 ______truncation condition reads T , which leads to N T 1T 2 T 3 / 6 independent basis function for the approximation of the electrical potential and each component of the displacement vector. Schreuer [78] reduced round off errors by introducing normalized powers of Cartesian coordinates. The larger T, the more accurate the results will be. It is already proven that for a non piezoelectric solid with ~ 30 observed resonances, T = 10 is a reasonable choice. Recently, for piezoelectric solids, Schreuer [78] showed that T = 18 for the displacement vector and T = 14 for the approximation of the electric potential (see Fig. 2.14) are suitable choices if about 140 resonances are used.
2.5.2 The inverse problem Using the method developed by Demarest [72], Ohno [73] and Visser [74] (1991) and explained by Migliori et. al. (1993) [2], one can easily estimate the resonance frequencies of a specimen from its elastic constants, dimensions and density. As no analytical solution is known to solve the inverse problem (determination of elastic properties from measured resonance frequencies), therefore, an iterative non-linear least-squares procedure that matches experimental and calculated resonance frequencies by adjusting values for the independent parameters is required.
n 2 2 2 2 The procedure minimizes the quantity wi i calc i obs for n i1 circular eigen frequencies (i 2 i ) by adjusting the values of the elastic constantscij . i calc is the i-th calculated frequency and i obs is the i-th observed one. The wi are weighting factors calculated by assuming an experimental error of 0.1 kHz for each observed resonance frequency. The convergence of this procedure strongly depends on the initial guess of the values of elastic and piezoelectric constants. The program “rusref v1.9” [79] developed by Schreuer was used for all refinement processes in this research work.
Chapter 3
Experimental techniques
All physical properties of tourmaline and pure and Ce doped CBN-28 reported in this thesis are referred to a Cartesian reference system { ei }. The axes ei are related
* to the corresponding crystallographic axes ai according to e1║X║a1, e2║Y║a 2 , and
* e3║Z║a3. Here, a 2 denotes the second basis vector of the reciprocal crystallographic system.
3.1 Sample preparation All tourmaline samples were large crystals of gem quality. Some of these tourmaline samples were provided by Prof. Dr. Joachim Zang (Brazil). The large Czochralski-grown single crystals of CBN-28 and cerium doped CBN-28 was provided by Prof. Manfred Mühlberg (University of Cologne, Germany). Rectangular parallelepipeds with edges parallel ei and edge length in the range 3.9 − 8 mm were cut from large single crystals using a low-speed diamond saw. The orientation of the samples was controlled by Laue back scattering and Bragg diffraction techniques. Deviations from ideal orientation were less than 0.5°. The faces of the samples were polished flat using 8” diameter extra fine grit diamond disc with mesh size 3000 (from CRYSTALITE) and a flatness of ± 2 µm was achieved. 26 Experimental techniques Chapter 3 ______
3.2 Chemical and structural characterization
3.2.1 Electron microprobe analysis The chemical composition of each sample was determined by Electron microprobe analysis (EMPA) on a CAMECA SX-50 instrument in wavelength- dispersive (WDS) mode. Each sample was a thin section of 30-35µm thickness and carbon coated. Beam conditions for all elements were 15kV, 20 nA and a spot beam diameter of 1µm. The final chemical data is an average of at least 30 analyses taken at different points over each sample. Counting times of peak and background for all elements were 20 seconds. The analytical data were deduced using the PAP model (Pouchou and Pichoir, 1988). The following standards were used: yadeite (Na), K-
glass (K), andradite (Fe, Ca, Si), pyrope (Al, Mg), Cr2O3 (Cr), TiO2 (Ti), V-metal (V),
TiO2 (Ti), Ba-glass (Ba), Nb-metal (Nb), and Ce (Ce). Lithium (Li) was determined by Atomic Absorption Spectroscopy (AAS) using a Spectraa-220 spectrometer unit (Varian). For tourmaline, the structure formula recalculation of the analysed data was done using MINCALC-V5 [80] software on the basis of 31 anions, assuming - stoichiometric amounts of H2O as OH (i.e., OH = 4 - F), B2O3 as B = 3 apfu. All Fe and Mn were assumed to be divalent [81]. The tourmaline compositions have close to 6 apfu Si, so the assumption of OH + F = 4 apfu seems reasonable.
3.2.2 X-ray diffraction Dimensions of the unit cell were derived from X-ray powder patterns collected
on a Philips MPD powder diffractometer with Ni filtered CuKα1 (λ = 1.5405981 Å @ 25 °C) radiation. Pure silicon powder (NIST) was used as an internal standard. Single crystal structure determination of Ce:CBN has been performed on a four- circle kappa diffractometer (Xcalibur from Oxford Diffraction) equipped with a CCD detector and an enhanced X-ray source (graphite monochromatised MoKα radiation).
3.2.3 Density determination Densities of all samples were measured by the buoyancy method on large single crystals in pure water using an Excellence XS balance (from METTLER TOLEDO [82]). Geometric densities of all the samples were calculated from the mass Chapter 3 Experimental techniques 27 ______
M and edge lengths li of the rectangular parallelepipeds according to ρg = M/l1l2l3. Both densities were found in good agreement (within 0.2%). The difference between geometric and buoyancy density serves as a measure for the quality of the sample with respect to geometric errors such as deviations from planarity, parallelism, and orthogonality.
3.3 Thermal analysis
3.3.1 Dilatometry The determination of accurate temperature coefficients of elastic constants requires the correction of changes in sample dimensions and density owing to thermal expansion. Therefore, thermal expansion measurements were performed on the rectangular parallelepipeds using a commercial inductive gauge dilatometer type DIL 402 C from Netzsch [83]. The measurements were performed in two overlapping temperature ranges, 100 K - 673 K in He purge gas atmosphere, and between 296 K - 1523 K in air, using a low-temperature (equipped with liquid nitrogen cooling system, resistance heater, thermocouple type E, sample holder made of fused silica) and a high-temperature furnace (equipped with SiC heating element, thermocouple type S, sample holder made of corundum ceramic), respectively. For low temperature measurements, the setup was calibrated with different standard samples made of fused silica (from Netzsch) in He purge gas environment. For the high temperature measurements, the setup was calibrated in air using corundum ceramic rods (from Netzsch) as standard materials. The expansion of the samples was measured using heating/cooling rates of 1K/min. The flow rate for He purge gas was set to 2 l/h for all low temperature measurements. The thermocouple was tested by measuring the phase transition temperature of selected materials. Each run was repeated at least two times to test the reproducibility of the observed strains. Data from the corresponding low and high-temperature runs were merged for further data analysis.
3.3.2 Differential scanning calorimetry
The heat capacities Cp of all samples were determined quantitatively using differential scanning calorimetry (DSC). DSC is a thermal analysis technique that measures the energy changes that occur when a sample is heated, cooled or held isothermally, together with the temperature at which these changes occur, it also tells 28 Experimental techniques Chapter 3 ______
about how a material’s heat capacity is changed by temperature. This technique is widely used to detect phase transitions. For this research work, the DSC 402F1 Pegasus from Netzsch was employed. Samples with weights ranging from 25 – 45 milligrams were placed in a covered platinum/corundum pan. Data were collected in two overlapping ranges of 150 - 673 K and 296 - 1000 K at a scanning rate of 10 K/min. Dry nitrogen purge gas was used in all measurements. Each set of experimental runs consisted of three steps: (i) A baseline with empty pans placed in the furnace, (ii) following first run but adding a reference (sapphire for low temperature and corundum for high temperature) to the sample pan, and (iii) replace the reference with the sample (CBN/Ce:CBN/tourmaline). The three curves are brought up on the screen, isothermals matched; data subtracted and referenced against the standard. The software package that comes with the DSC 402F1 Pegasus instrument offers all these calculations. The reference disks of sapphire and corundum, which were used for calibration purposes, were supplied by Netzsch. To test the reproducibility of the heat capacity results, each measurement was made two times per sample. The error estimation yield ± 3% or better for Cp.
3.4 Dielectric Properties Highly precise dielectric constants on tourmaline samples at room temperature were obtained using a method described by Andeen et. al. [84], the so called substitution method. Temperature dependence of the dielectric properties was studied using the geometric method at 100 kHz, 400 kHz, and 1 MHz. A combination of both methods is the optimum solution for the determination of reliable temperature coefficients.
3.4.1 Substitution method To study the room temperature dielectic response, the single crystals were cut into plates of thicknesses less than 1mm and diameters of less than 12 mm using a low-speed saw (from Buehler). For each tourmaline two samples parallel and perpendicular to the crystallographic c-axis, have been investigated to determine the independent relative dielectric constants, є 33 and є 11 , respectively. The orientation of the samples was controlled by Laue back scattering and Bragg diffraction techniques. Deviations from ideal orientation were less than 0.5°. The final faces of the plates Chapter 3 Experimental techniques 29 ______
were parallel within ± 1 µm. A schematic diagram of the experimental setup is shown in Fig. 3.1. The capacity was measured by a multi frequency LCR meter (HP 4275A) at 100, 400 and 1000 kHz. The dielectric constants were derived from four
measurements of the capacitance Ci (C1 in air, C2 in air + sample, C3 in liquid only,
and C4 in liquid + sample) between the main electrode and the base electrode without changing the distance between them. The liquid used in this work was xylene. The relative dielectric constant of the plate can finally be calculated using the relation:
' 1 C3 C1 C3 C2 C3 C4 є rel,11 єA (3.1) C1 C2 C3 C4
where єA is the dielectric constant of air (1.00053 at 293 K). The reproducibility has been checked by measuring each sample five times. An error of less than 0.6% was observed.
FIG.3.1: Schematic diagram of the room temperature setup for measuring the dielectric constants [85].
3.4.2 Geometric method The geometric method was used to study the temperature dependence of the dielectric constants. For this, the thin plates used for the substitution method were sputtered with gold (Balzers Union Med 010) on both faces and were placed inside a quartz tube surrounded by a cylindrical heating element. The entire set-up was purged by controlled gaseous nitrogen environment to provide cooling and to protect the electrodes against oxidation at higher temperatures. The contact between the wires and the sample electrodes were made by a gold plate (few micrometer thicknesses) 30 Experimental techniques Chapter 3 ______
and gold wires, respectively, which were mechanically pressed on the surface electrodes. The change of capacity and simultaneously the dielectric loss in terms of tan-δ were collected in two overlapping ranges of 153-300 K and 293-673 K at a scanning rate of 5 K/min using home made low and high temperature setups, respectively. Data from the corresponding low and high-temperature runs were merged using room temperature dielectric constants as fixed points. A schematic diagram of the setup is shown in Fig 3.2.
Transistor Power Supply TNs 70 - 1400 Measuring Chamber Digital Mmultimeter TR 4868
Multifrequency LCR meter, HP 4275A Nitrogen evaporator
Power supply EA 7030-100
Nitrogen Can Computer
FIG. 3.2: Schematic diagram for measuring the temperature dependence of dielectric constants [86].
3.5 Elastic and Piezoelectric measurements 3.5.1 Resonant ultrasound spectroscopy The elastic and piezoelectric properties were studied between room temperature and 1500 °C using the innovative method of resonant ultrasound
E spectroscopy (RUS). The elastic constants cij (in Voigt notation) and the
piezoelectric stress constants eijk were derived from ultrasonic resonance frequencies of free vibrating rectangular parallelepipeds using a high-temperature RUS device built in-house (see Fig 3.3). A schematic sketch of the high temperature RUS setup is shown in Fig 3.4. The RUS device consists of a commercial furnace (type 6.219.1-20 from Netzsch) equipped with two thermocouples of type S, one located close to the sample and the other close to the SiC heating element of the furnace. Temperature control was achieved by a cascading controller (EPC900 from Eurotherm) in Chapter 3 Experimental techniques 31 ______
connection with a 2kVA DC power supply (HP6674A from Agilent). At the position of the sample the temperature was stable within ± 0.1 K and the accuracy was better than ± 10 K at 1503 K. The resonance spectra were collected using a network analyzer (HP 4194A).
Fig. 3.3: Photograph showing high temperature RUS device and sample T003 mounted between the transducers (in inset) inside the furnace of the RUS device.
The electrical signals were amplified by a factor of up to 20 times (in case of a week signal from the sample) using a high speed power amplifier NF 4005 (form NF electronics). The rectangular parallelepipeds were weekly clamped at opposite corners
Temperature controller
Power supply HP 6674A Water cooling PC
Furnace
Sample Ceramic rods Network analyzer Power amplifier HP 4194 A NF 4005 Fig. 3.4: Schematic view of the high temperature RUS setup.
32 Experimental techniques Chapter 3 ______
between two rods made of corundum ceramics which act simultaneously as sample holder and as ultrasonic transducer rods. In order to approach the boundary conditions of a freely vibrating body, the mechanical load on the sample was kept below 0.03 N. At room temperature, at least four resonance spectra of each sample with the sample mounted in different orientations were collected to ensure that all resonances of the sample were excited and could be observed. The elastic and piezoelectric constants at all temperatures were calculated using the program “rusref v1.9” [79], in which the truncation condition T = 20 was used for the approximation of the displacement vector and T = 18 for the approximation of the electric potential. The energy loss (or dissipation) is related to the quality factor Q of the resonance. It can be calculated from the full-width at half-maximum (FWHM) of a resonant frequency f according to Q-1 = FWHM / f .
Chapter 4
Elastic and piezoelectric properties of tourmalines
It is well known that the physical properties systematically vary with chemical composition [87]. Tourmalines are found in nature with a wide range of chemical composition. A complete knowledge of compositional dependence of physical properties together with its temperature evolution is essential for device development and application. A variation in chemical composition may result in some changes in the cation – anion distances, ionic sizes, and the bond angles, which might affect the elastic and piezoelectric properties of tourmalines significantly. Thus, to understand their structural dependence due to various cation substitutions in tourmalines, it will be wise to review the crystal chemistry of tourmalines in some detail. 4.1 Crystal chemistry of tourmalines Tourmaline, belonging to crystal class 3m and space group R3m, is structurally and chemically a complex borosilicate. The chemical formula of tourmaline is best described according to Hawthorne and Henry [88]: [9] [6] [6] [4] [3] X Y3 Z6 ( T6O18) ( BO3)3 V3 W, neglecting trace elements, the composition may vary according to: X = Na+, K+, Ca+, vacancy Y = Li+, Fe2+, Fe3+, Mg2+, Al3+, Mn2+, Cr3+ V3+, (Ti4+) Z = Al3+, Mg2+, Fe2+, Fe3+, V3+, Cr3+ T = Si4+, Al3+, (B3+) B = B3+, (vacancy) V = OH-, O2-[O3], (F) W = OH-, O2-[O(1)], F 34 Elastic and piezoelectric properties of tourmalines Chapter 4 ______
The elements in parentheses are not yet proven to occur at these sites. This characterization of tourmaline’s complex structure and chemistry is a result of long research [89-96], and considerable work is currently in progress. A schematic diagram is shown in Fig. 4.1. The structure of tourmaline mainly consists of two types of octahedral sites: three relatively large Y octahedral sites and six smaller and distorted Z octahedra. The Y and Z octahedra share edges to form brucite-like fragments [95, 98]. Structurally two distinct positions are occupied by OH groups, one at the centre of the hexagonal rings and the other at the corner of the brucite-like fragments. Octahedral Y- and Z- sites are main place for substitution in tourmalines, where Al is replaced by various metal cations (Mg2+, Fe2+, Li+, Fe3+, Al3+, Mn2+, Cr3+, V3+ and Cu2+). These cations are mainly controlled by local valance-sum balance and bond-length requirements in the crystal structure. For example, if the W-site is occupied by O2-, then to balance the valance-sums of cation and anion, a trivalent cation have to occupy the Y-site [95]. When the Y-site is occupied by Fe2+ then Z-site is not allowed to be occupied by Mg2+ [99]. At the same time substitution of transition metals either in major or trace amounts in Y- and Z-site octahedra is responsible for the color (pink, black, brown, colorless, pale brown etc. etc.) and pleochroism in tourmalines [100]. The large highly coordinated X-site (9 or 10 anions) is occupied by Na+ (dominantly together with
FIG. 4.1: (a) Tourmaline crystal structure [97], showing the relative positions of OH, X –site (Na+), Y-site (Li+, Fe2+or Al3+) and Z site (Al3+). (b) (001) projection of the tourmaline structure. Chapter 4 Elastic and piezoelectric properties of tourmalines 35 ______
minor traces of K+), Ca+ and sometime with a vacancy.
The X-site is located on the 3-fold axis between six TO4 tetrahedra, three boron trigonal and of the six membered ring, however the X cation links to three Y octahedral, three boron trigonal, six TO4 tetrahedra, and to the O1 anion. When the O1 site is occupied by OH, then there is a lack of hydrogen bonding with the surrounding oxygens. It’s six-membered tetrahedral (T-site) rings are predominantly occupied by Si, a small substitution of Al3+ cation is also possible [101], while the B3+ substitution
has been found in rare case only. The T-site is coordinated by four anions, O4 to O7, resulting a T-cation centered inside a tetrahedron. The TO4 tetrahedra form a six-
membered T6O18 ring with all the tetrahedra lying in a plane parallel to (001) with the apices of the tetrahedra pointing in the (-c) direction, resulting in a chemical and morphological polarity [95]. This structural polarity results in piezo- and pyro-electric properties in tourmalines. The B-site has a triangular coordination with no known substitution so far. The BO3 groups are parallel to the 3-fold axis with its corner shearing to two Y-octahedra, two Z-octahedra (see Fig. 4.1) and one X-polyhedron.
Table 4.1: Classification for end-members of tourmalines based on X-site occupancy [102].
X Y Z T6O18 (BO3)3 V3 W
X-site vacant tourmalines
Magnesiofoitite □ Mg2Al Al6 (Si6O18) (BO3)3 (OH)3 (OH)
2+ Foitite □ Fe 2Al Al6 (Si6O18) (BO3)3 (OH)3 (OH) Rossmanite □ LIAl2 Al6 (Si6O18) (BO3)3 (OH)3 (OH)
Alkali tourmalines
Dravite Na Mg3 Al6 (Si6O18) (BO3)3 (OH)3 (OH) 2+ Schorl Na Fe 3 Al6 (Si6O18) (BO3)3 (OH)3 (OH) 3+ Chromdravite Na Mg3 Cr 6 (Si6O18) (BO3)3 (OH)3 (OH) 3+ 3+ Povondrite Na Fe 3 Fe 4Mg2 (Si6O18) (BO3)3 (OH)3 O 3+ Buergerite Na Fe 3 Al6 (Si6O18) (BO3)3 O3 F Elbaite Na Li15Al15 Al6 (Si6O18) (BO3)3 (OH)3 (OH) Olenite Na Al3 Al6 (Si6O18) (BO3)3 O2(OH) O
Calcic tourmalines
Uvite Ca Mg3 Al5Mg (Si6O18) (BO3)3 (OH)3 (OH) 2+ Feruvite Ca Fe 3 Al5Mg (Si6O18) (BO3)3 (OH)3 (OH) Liddicoatite Ca Li2Al Al6 (Si6O18) (BO3)3 (OH)3 F
36 Elastic and piezoelectric properties of tourmalines Chapter 4 ______
The most updated classification for end-members of natural tourmaline based on the X-site occupancy is presented in Table 4.1. Natural tourmalines are mainly categorized into two main solid solution series: schorl-dravite and schorl-elbait; the solid solution dravite-elbaite is missing.
4.2 Chemical and structural characterization The Chemical compositions of the studied tourmaline samples with end member classifications are summarized in Table 4.2, and final structure formula are given in Table 4.3. End member classifications were done following the method of [103] and [104]. A closer examination of chemical analysis (Table 4.2) gives the following information: (a) sample T003 is Li rich and Fe free and sample T013 is Fe rich and Li free, (b) samples T007, T005, T004, T016 and T020 are in order of gradually decreasing Li and simultaneously increasing Fe content. Samples T013 and T020 contain a significant amount of Mg, while T003 and T007 possess more Ca than rest of the samples. Thus, as explained above, here the studied tourmalines, possessing a systematic variation in chemical composition, will facilitates us to check the compositional dependence of various physical properties.
100 122 051 80 012
60 220 211 152
Intensity [a. u.] Intensity [a. 40 021 101 603 20 300 110 0 10 20 30 40 50 60 70 80 2 [degree]
FIG. 4.2: Example of powder X-ray pattern of tourmaline T004 with indexing. Chapter 4 Elastic and piezoelectric properties of tourmalines 37 ______
TABLE 4.2: Chemical compositions of the tourmaline samples. The errors are in the order of ± 2 in the last digit for oxide weight percent. For easier interpretation, all samples are arranged in the order of increasing iron content from left to right.
T003 T007 T005 T004 T016 T020 T013 Classifi- Elbaite- Elbaite- Elbaite- Elbaite- Elbaite- Schorl- Schorl- cation→ Schorl Schorl Schorl Schorl Schorl dravite dravite Oxide wt % SiO2 37.22 37.93 37.48 36.90 35.65 36.23 35.21 TiO2 0.03 0.04 0.02 0.08 0.38 0.29 0.37 Al2O3 39.32 38.63 39.68 38.88 36.62 33.42 33.65 V2O3 0.02 0.01 0.02 0.01 0.02 0.01 0.02 Cr2O3 0.02 0.02 0.01 0.01 0.02 0.02 0.02 FeO 0.06 0.30 0.84 2.86 7.35 13.39 13.59 MnO 0.32 0.24 2.81 1.29 0.46 0.08 0.08 MgO 0.01 0.03 0.01 0.20 0.25 1.62 1.80 CaO 2.38 3.36 0.34 0.40 0.10 0.15 0.23 Na2O 1.28 0.92 2.28 2.35 2.65 1.61 1.72 K2O 0.01 0.01 0.02 0.02 0.03 0.04 0.05 F 1.40 1.57 1.33 1.39 1.41 0.45 0.51 Li2O 3.23 2.58 1.87 1.72 1.33 0.37 0 B2O3 11.29 10.85 10.88 10.84 10.60 10.55 10.36 H2O 3.43 3.53 3.49 3.22 2.91 3.10 3.16 Total 100.02 100.02 101.08 100.17 99.78 100.33 100.77 Atoms pfu Na 0.38 0.29 0.71 0.73 0.84 0.51 0.56 Ca 0.39 0.58 0.06 0.07 0.02 0.03 0.04 K tr Tr tr tr 0.01 0.01 0.01 Si 5.81 5.95 5.82 5.92 5.85 5.97 5.91 Al 7.13 7.29 7.48 7.35 7.08 6.49 6.66 Li 2.00 1.66 1.20 1.11 0.88 0.25 0 Fe 0.01 0.04 0.11 0.38 1.01 1.85 1.91 Mn 0.04 0.03 0.38 0.17 0.06 0.01 0.01 Mg tr 0.01 tr 0.05 0.06 0.39 0.45 Ti tr 0.01 tr 0.01 0.05 0.03 0.05 V tr Tr tr tr tr tr tr Cr tr Tr tr tr tr tr tr B 3 3 3 3 3 3 3 F 0.68 0.79 0.67 0.70 0.73 0.25 0.27 OH 3.32 3.21 3.33 3.30 3.27 3.75 3.73
Note: Atomic proportions are computed on the basis of 18 oxygen atoms, boron value was set to 3, final calculation based on 31 anions. tr=trace. 38 Elastic and piezoelectric properties of tourmalines Chapter 4 ______
TABLE. 4.3: Structure formulas of the tourmaline samples (tr = trace, = vacancy).
Samples Optimized formula after EMPA
[9] [6] [6] [4] [3] Ideal X Y3 Z6 T6 O18 ( BO3)3 (V3 W) formula
X Y T003 (Na0.39Ca0.38Ktr0.23-tr) (Li2Al0.94Fe0.01Mn0.04MgtrTitrVtrCrtr) Z T (Al6) (Si5.81Al0.19) O18 (BO3)3 [(OH)3.32F0.68]
X Y T007 (Na0.29Ca0.58Ktr0.13-tr) (Li1.66Al1.24Fe0.04Mn0.03Mg0.01Ti0.01VtrCrtr) Z T (Al6) (Si5.95Al0.05) O18 (BO3)3 [(OH)3.21F0.79]
X Y T005 (Na0.71Ca0.06Ktr0.23-tr) (Li1.20Al1.30Fe0.11Mn0.38MgtrTitrVtrCrtr) Z T (Al6) (Si5.82Al0.18) O18 (BO3)3 [(OH)3.33F0.67]
X Y T004 (Na0.73Ca0.07Ktr0.20-tr) (Li1.11Al1.27Fe0.38Mn0.17Mg0.05Ti0.01VtrCrtr) Z T (Al6) (Si5.92Al0.08) O18 (BO3)3 [(OH)3.30F0.70]
X Y T016 (Na0.84Ca0.02K0.010.13) (Li0.88Al0.93Fe1.01Mn0.06Mg0.06Ti0.05VtrCrtr) Z T (Al6) (Si5.84Al0.16) O18 (BO3)3 [(OH)3.27F0.73]
X Y T020 (Na0.51Ca0.03K0.010.45) (Li0.25Al0.46Fe1.85Mn0.01Mg0.39Ti0.03VtrCrtr) Z T (Al6) (Si5.97Al0.03) O18 (BO3)3 [(OH)3.75F0.25]
X Y T013 (Na0.56Ca0.04K0.010.39) (Al0.57Fe1.91Mn0.01Mg0.45Ti0.05VtrCrtr) Z T (Al6) (Si5.91Al0.09) O18 (BO3)3 [(OH)3.73F0.27]
Further, X-ray powder patterns of all tourmaline samples were obtained at room temperature. Diffraction patterns of all the samples were more or less similar except for small changes in amplitudes of the peaks and minor shifts w.r.t. the position of the peaks. A typical example is shown in Fig. 4.2. The (hkl) values for the peaks of 2θ value (from powder pattern) for schorl, elbaite and dravite have been Chapter 4 Elastic and piezoelectric properties of tourmalines 39 ______
identified from the standard powder diffraction file (PDF 2), released by the international centre for diffraction data (ICDD), USA. The lattice parameters and the volume of the unit cell for all tourmaline samples were determined using a unit-cell program, given by Holland and Redfern [105], and are tabulated in Table 4.4. This table also includes the color of the tourmaline samples, their origin and the densities determined by the buoyancy method.
TABLE 4.4: Lattice parameters of tourmaline samples at room temperature. Uncertainty of the last decimal is in parenthesis. ρb is the buoyancy density and ρx is the density obtained from lattice parameter and chemical composition.
Sample T003 T007 T005 T004 T016 T020 T013 Color Pink Brown- Pink- Green Dark Black Black Transparent grey brown transparent green Locality Unknown Nigeria Nigeria Unknown Brazil Australia Brazil c (Å) 7.102(1) 7.103(4) 7.112(2) 7.113(7) 7.133(9) 7.151(7) 7.150(6) a (Å) 15.834(3) 15.832(2) 15.878(7) 15.874(3) 15.929(9) 15.963(5) 15.964(7) V (Å3) 1542.34(2) 1534.17(1) 1552.79(1) 1553.91(2) 1561.12(3) 1579.80(1) 1580.80(2) c/a 0.4485 0.4487 0.4479 0.4481 0.4479 0.4478 0.4480
ρx(g/cc) 3.004(4) 3.064(3) 3.058(4) 3.070(1) 3.124(3) 3.175(1) 3.195(2)
ρb(g/cc) 3.004(2) 3.056(2) 3.058(2) 3.069(8) 3.114(7) 3.175(3) 3.183(4)
Note: Samples T003, T004, T005 and T007 crystals are opaque along crystallographic c-axis, however perpendicular to c-axis they are transparent.
Epprecht [106] reported a systematic variation of the lattice parameters (a, c) from elbaite to schorl and from schorl to dravite by plotting c/a versus a and c for a number of different tourmaline samples. To identify the location of the studied tourmaline sample (pointed by red diamonds), the lattice parameters a and c versus c/a were plotted together with the data of Epprecht as shown in Fig 4.3. A closer examination of Fig 4.3 and the EMP analysis (Table 4.2) reveals: samples T003, T004, T005, T007 and T016 belong to the elbaite-schorl series, and the two samples T013 and T020 belong to the dravite-schorl series. However, the Li-free samples T013 and T020 are close to pure schorl (> 80 %) with less than 20 % dravite component. Sample T003 is Fe-free, however, the remaining samples T004, T005, T007 and T016 contain up to 45 % schorl. Therefore, the studied samples represent a broad variety of tourmalines which cover a wide range of chemical compositions. 40 Elastic and piezoelectric properties of tourmalines Chapter 4 ______
0.454 Dravite
0.452
0.450
c/a Elbaite T007 T003 T004 0.448 T005 T020 T016 T013
0.446 Schorl 15.80 15.85 15.90 15.95 16.00
a [Å]
0.454 Dravite
0.452
0.450 Elbaite c/a T007 T003 T004 T005 0.448 T020 T016 T013
0.446 Schorl
7.05 7.10 7.15 7.20 7.25 c [Å]
FIG. 4.3: Axial ratio c/a plotted against lattice parameters a (top), and c (bottom). The positions of the studied tourmaline samples are shown with red diamonds, other data are adopted from [106]. Chapter 4 Elastic and piezoelectric properties of tourmalines 41 ______
4.3 Dielectric constants Detailed dielectric properties of tourmalines have not been reported in literature yet. Of interest is, in particular, an investigation of the behavior of dielectric properties with chemical composition, since the natural tourmalines exist in a broad range of chemical compositions. Previous investigations [10, 107-112] on tourmaline report some values of dielectric constants but without any information of the chemical composition of tourmaline samples.
Tourmaline has two independent dielectric constants є11 and є33. The dielectric constants under constant stress were studied on six samples (see Table 4.3, excluding sample T020) at 100 kHz, 400 kHz and 1000 kHz on thin plates of thickness ≈ 0.5 – 0.8 mm and diameter ≈ 5-8 mm. The measured room temperature
dielectric constants, є33 (along [001]) and є11 (perpendicular to [001]), are given in
Table 4.5. The room temperature єij values are almost frequency independent in the range of 100 -1000 kHz. Dielectric losses, as expressed by tan δ, were observed to be less then 0.0004 at room temperature over the entire range of frequencies under investigation. The frequency-independent behavior of dielectric constants and the small value of tan δ at room temperature clearly indicate the high quality of the samples [109]. The accuracy of the measurements at room temperature is ± 0.004 for -4 єij , and ± 10 for tan δ.
TABLE 4.5: Room temperature dielectric constants of tourmalines. Uncertainty of the last decimal is in parenthesis.
Sample↓ T003 T007 T005 T004 T016 T013 є11 7.09(2) 7.12(1) 7.15(3) 7.21(1) 7.57(2) 7.96(2) є33 6.26(3) 6.23(2) 6.17(1) 6.11(4) 6.04(1) 5.99(3)
A strong correlation between є11 and є33 has been observed which can be best
described by an exponential function as shown in Fig. 4.4. Variations of the єij values are likely as tourmaline has a highly variable chemical composition. Further, crystal defects such as chemical inhomogenities, micro cracks and pores may also affect the dielectric properties.
To validate this correlation, the reported literature values [110-111] of є11 and
є33 are also shown in the same figure. It is observed that most of the literature values
of єij are in a good agreement with the proposed єij correlation. The scattering of some 42 Elastic and piezoelectric properties of tourmalines Chapter 4 ______
of the green points from the observed correlation is most likely suffer from larger
experimental errors (either in the measurement method of єij or in the determination of chemical composition, or may be both). Hence, the presented dielectric constants are the most reliable ones. 8.25
Exponential fit 8.00 Data: Data81_11 Model: ExpDec1 Equation: y = A1*exp(-x/t1) + y0 7.75 Weighting:
11 y No weighting e Chi^2/DoF= 0.0013 R^2 = 0.9935 7.50 y0 7.07828 ±0.0319 A1 4.475E36 ±5.189E37
t1 0.07089 ±0.00974 7.25
7.00 5.75 6.00 6.25 6.50 6.75 7.00
e33
FIG. 4.4: Correlation between dielectric constants є11 and є33 of tourmalines. The black data points correspond to the samples used in the present study and green data points are taken from literature [110-114].
It is remarkable that the iron-rich sample T013 shows highest є11 and lowest
є33 values, while the sample T003 (almost iron-free) exhibits the lowest value of є33 and highest value of є11. The єij of all other samples are in between. To understand the
compositional dependence of dielectric constants, the measured є11 and є33 values for all six samples are plotted with respect to the variation in Fe, Li, Na, Ca, Mg and water contents in Fig. 4.5 a and b. As depicted from these figures, the most varying constituents of the measured samples are iron (0.04 – 1.91 pfu) and lithium (0.0 – 2.0 pfu). It seems probable that the significant changes in the dielectric constants of tourmalines are mainly associated with the variation of iron content. It has already been found for other ceramics as well that increasing the iron content indeed results in a lower dielectric constant [113-114]. It is clear from the study that an increase in iron Chapter 4 Elastic and piezoelectric properties of tourmalines 43 ______
content results into decrease in the value of є33, and simultaneously an increase of є11. The variation of Ca2+and/or Na+ does not show any observable effect on the dielectric constants. Increasing Mg2+ content (please refer to Table 4.2) seems to favor a decrease in the dielectric constants; however, more experiments are needed to understand the clear influence of Mg2+ on the dielectric properties of tourmalines.
6.30
T003 6.25 T007 6.20 T005 6.15
33 T004 e 6.10 Ca Fe Na Li OH
6.05 T016
6.00 T013
5.95 0.0 0.5 1.0 1.5 2.0 3 4 5 Elements (apfu)
8.0 T013
7.8
7.6 T016 11 e 7.4
Ca Fe Na Li OH 7.2 T004 T005 T007 T003 7.0
0.0 0.5 1.0 1.5 2.0 3 4 5 Elements (apfu)
FIG. 4.5: The variation of the dielectric constants є33 (up), and є11 (down) of tourmalines with varying chemical composition. 44 Elastic and piezoelectric properties of tourmalines Chapter 4 ______
The presence of water in the samples is consistent with the relatively high dissipation factor of 0.009 at higher temperatures. The exact role of water is not clear as the water concentration is more or less the same for all samples. An unclear behaviour of water was also reported earlier for different materials [115]. Further the impact of other atoms also might be crucial, which could not be derived as no systematic variation was observed for those other than iron and lithium. The dielectric properties of tourmalines were further studied at different temperatures ranging from 153 K to 673 K. The change in dielectric constant with respect to temperature were mostly fitted by 3rd-order polynomials according to
T T 2 2 2 3 3 3 єij = єij 0 + ∂єij/∂T (T-T0) + ∂ єij/∂T (T-T0) + ∂ єij/∂T (T-T0) (4.1)
2 2 3 3 where ∂єij/∂T, ∂ єij/∂T and ∂ єij/∂T are the first, second and third order temperature
derivatives; T0 = 299 K. All temperature derivates of the dielectric constants for all the samples are tabulated in Table 4.6.
TABLE 4.6: Room temperature dielectric constants єij and their temperature coefficients for six samples of tourmaline at 100 kHz, 400 kHz and 1000 kHz. T0 = 299 K. The samples are ordered with increasing iron content from top to bottom.
-4 -1 2 2 -7 -2 3 3 -10 -2 Frequency ∂єij/∂T [10 K ] ∂ єij/∂T [10 K ] ∂ єij/∂T [10 K ] (kHz) → 100 400 1000 100 400 1000 100 400 1000 T003 є11 3.11(2) 3.02(1) 3.07(2) -0.26(3) -1.28(4) -1.84(4) 7.21(3) 6.35(2) 5.25(4) є33 4.61(3) 4.66(3) 4.74(1) NA NA NA NA NA NA T007 є11 3.22(4) 3.16(2) 3.17(4) 0.56(3) 0.39(4) 0.26(3) 7.08(4) 6.15(2) 5.24(1) є33 4.68(1) 4.62(3) 4.58(2) 0.34(2) 0.46(3) 0.38(4) 1.02(3) 0.67(3) 0.98(3) T005 є11 2.92(2) 3.08(4) 3.01(1) 0.82(1) 0.64(2) 0.52(1) 7.67(4) 10.21(1) 3.58(4) є33 4.65(2) 4.59(3) 4.73(1) 0.52(3) 0.25(3) -0.71(2) 1.67(3) 0.69(4) 1.10(4)
T004 є11 2.40(4) 2.50(4) 2.51(3) 0.76(4) 0.51(2) 0.22(1) 10.01(4) 8.16(4) 7.07(4)
є33 4.51(2) 4.32(2) 4.26(3) 0.42(1) 0.26(3) 0.09(4) 6.84(2) 9.21(3) 8.06(2)
T016 є11 2.95(3) 2.88(4) 2.87(4) 1.23(1) 0.84(2) 0.78(5) 14.68(4) 10.08(4) 8.84(2)
є33 4.04(2) 4.16(4) 4.17(3) 0.96(4) 0.80(2) 0.47(1) 6.01(2) 0.98(2) 0.09(1) T013 є11 3.02(1) 2.82(2) 2.81(1) 10.32(2) 6.82(3) 6.75(2) 23.26(1) 10.86(4) 9.45(3) є33 4.28(3) 4.08(4) 4.08(4) 9.44(4) 6.56(2) 4.71(2) 22.44(2) 10.54(2) 7.98(3)
Note: “NA” means not applicable, the digits in parentheses are the uncertainty (standard deviation) in the last digit of the value.
The dielectric constants at constant strain have been calculated according to
є ij є de ikljklij , where ejkl and dikl are the components of the piezoelectric stress and strain tensors, respectively. It was observed that the values of dielectric constants at Chapter 4 Elastic and piezoelectric properties of tourmalines 45 ______
constant strain є ij agree with in experimental error with the dielectric constants at
constant stress є ij .
6.5 100 kHz 400 kHz 6.4 1000 kHz
6.3
6.2
6.1
6.0 Dielectric constant along [001] increasing ironcontent
5.9 100 200 300 400 500 600 700 Temperature (K) (a)
0.006 0.006 100 kHz 100 kHz 400 kHz 400 kHz 0.005 0.005 1000 kHz 1000 kHz
0.004 0.004
0.003 0.003
0.002 0.002 Dileectricalong loss [001]
Dielectric loss along [001] 0.001 0.001
100 200 300 400 500 600 700 100 200 300 400 500 600 700 Temperature Temperature [K]
(b) (c)
FIG 4.6: Variation of dielectric constant є33 as a function of temperature, measured at different frequencies for investigated tourmaline samples with varying iron content; with the corresponding loss factors for iron-free sample in (b) and iron-rich sample in (c), respectively. For easier comparison the scales of the axis in (b) and (c) are chosen identical. 46 Elastic and piezoelectric properties of tourmalines Chapter 4 ______
The temperature dependence of dielectric constants of tourmalines with different iron content is shown in Fig 4.6a, measured for three different frequencies of 100 kHz, 400 kHz, and 1 MHz. It is evident from the figure that iron indeed lowers the dielectric constant. The Li bearing tourmalines always possess a higher dielectric
constant є33 than iron bearing tourmalines in the considered temperature range. Li- bearing and iron-free tourmalines show almost linear dependence of є33 on temperature. However, Li-free and iron-rich one shows a highly non-linear increase of
є33 with temperature. Moreover, the iron-free tourmaline retained its frequency independent behavior of dielectric constants up to the highest measured temperature. As the iron content increases, a decrease in dielectric constants is observed for increasing frequencies. This frequency dependence of dielectric constant for iron-rich samples (where heavier iron substitutes lithium) can be well explained due to the delay of the polarized species which could not follow the alternations of the applied electric fields at higher frequencies [113]. The dielectric losses of the tourmaline samples versus temperature are shown in Fig 4.6 b,c. Here only the two extreme cases with respect to the iron content are shown. In most of the cases, the dielectric losses follow the similar trend as for corresponding dielectric constants with increasing temperature. For iron-free samples, the loss is almost frequency as well as temperature independent, while for iron-rich samples, the loss decreases with increasing frequency.
4.4 Thermal properties 4.4.1 Thermal expansion Tourmaline (point group 3m) has two independent principal thermal
expansion coefficients α11 (= α22) and α33. Thermal expansion measurements, parallel and perpendicular to the crystallographic c axis were performed on all samples (see Table 4.4). Calibration corrected strains are shown in Fig. 4.7, and their thermal expansion coefficients are given in Table 4.7. A nonlinear behavior in the temperature
range 100 - 1523 K has been observed. The observed strains ii can be best approximated by cubic polynomials according to
2 3 /lll 00 iiiiiii 0 ii 0 ii TTTTTT 0 (4.2)
where T0 is the reference temperature and αii, βii, and γii are the corresponding coefficients of linear, quadratic, and cubic order thermal expansion, respectively. The Chapter 4 Elastic and piezoelectric properties of tourmalines 47 ______
li is the edge length at temperature T and l0 is the edge length at the reference temperature.
8 T003 T004 33 6 T007 T016
] T020 -3 4 T013 11
2 Strain [10 Strain
0
-2 0 200 400 600 800 1000 Temperature [K]
FIG. 4.7: Temperature-induced strains 33 and 11 in tourmaline crystals.
It can be seen that the temperature evolution of the strains is higher for Li-rich tourmalines compared to Fe-rich tourmalines in both directions. The temperature dependence of the strain for rest of the tourmalines is in between Li- and Fe-rich tourmalines. TABLE 4.7: Coefficients of thermal expansion at 299 K.
-6 -1 -9 -2 -12 -3 Sample ii αij [10 K ] βij [10 K ] γij [10 K ] Temperature range 100 K to 923 K T003 3.84(2) 5.37(2) -3.25(4) T007 3.94(4) 6.48(3) -3.99(6) T004 11 4.32(2) 3.44(3) -1.00(5) T020 3.15(3) 5.58(1) -2.49(3) T013 3.29(1) 5.43(4) -2.77(1) T003 9.99(4) 7.30(5) -5.36(1) T007 9.05(5) 8.44(3) -5.65(6) T004 33 9.69(0) 6.09(1) -3.51(4) T020 8.31(1) 8.34(5) -5.43(4) T013 8.19(2) 7.11(1) -4.84(2) Temperature range 100 K to 673 K T016 11 3.68(1) 5.20(3) -5.13(1) T016 33 8.81(3) 7.91(0) -8.67(4)
48 Elastic and piezoelectric properties of tourmalines Chapter 4 ______
4.4.2 Specific heat capacity The specific heat measurements were carried out in the temperature range 150 K – 900 K in He purge gas atmosphere. The weights of the investigated samples were in the range of 25 – 35 mg. Temperature evolution of the molar specific heat vs temperature is plotted in Fig. 4.8(a). The behavior of the different tourmalines is quite similar. Li rich tourmalines show a slightly smaller specific heat than Fe rich ones. Comparison of the specific heats of T013 and T003 shows a maximum difference of less than 5 % at high temperatures. The temperature evolution of specific heat follows the Dulong-Petit law at high temperatures. The dehydration temperatures were also determined using DSC, the results are plotted in Fig. 4.8(b). The dehydration temperature varies between 1250 – 1280 K depending upon water and Fe content. Fe-rich tourmalines dehydrate earlier than elbaites. Further, in Fe rich tourmalines a precursor effect appears about 30K before the maximum of the dehydration process. After dehydration all tourmaline crystals are full of visible cracks.
1200
3 1000 T003 T004 T005 2 T007 800 T016 T020
(J/mol K) T013 p
C 600 1 DSC signal[a. u.]
400 Li: 2 apfu 0 Li, Fe: 1, ~1 apfu Fe: ~2 apfu 200 800 1000 1200 1400 100 200 300 400 500 600 700 800 900 b) Temperature [K] a) Temperature [K]
FIG. 4.8: (a) Molar specific heat for selected tourmaline samples and (b) dehydration temperature of all studied samples of tourmalines.
4.5 Elastic and piezoelectric properties For tourmaline there exist six elastic and four piezoelectric symmetry independent tensor coefficients [49] as shown in Fig. 4.9.
Chapter 4 Elastic and piezoelectric properties of tourmalines 49 ______
11 12 13 cccc 14 00 cccc 00 12 11 13 14 0000 15 ee 22 13 13 ccc 33 000 eee 000 2222 15 0 ccc 00 14 14 44 31 eee 3331 000 0000 cc 1444 0000 cc 6614 b) a) where 66 (2/1 ccc 1211 )
FIG. 4.9: Components of (a) Elastic, and (b) piezoelectric matrix for point symmetry group 3m.
Rectangular parallelepiped samples were prepared as described in chapter 3. Final sample dimensions are given in Table 4.7. Conoscopic images taken on a polarizing microscope for three transparent tourmaline samples, shown in Fig. 4.10, indicate the good alignment and high optical quality of the samples.
T003 T007 T004
FIG. 4.10: Conoscopic images of tourmalines (upper row), showing uniaxial view of the oriented tourmaline samples (lower row). For sample dimensions please refer to Table 4.8.
Ultrasonic resonance spectra were collected between room-temperature and 900 K on all samples. At least 65 eigen-frequencies in the range of 200 kHz - 1400 50 Elastic and piezoelectric properties of tourmalines Chapter 4 ______
kHz were extracted from each resonance spectrum and used in non-linear least- squares refinements minimizing the quantity 2 . Sample T005 was excluded from further analysis because the resonance data could not be refined due to crack development. Results are presented in Table 4.8.
TABLE 4.8: Elastic and piezoelectric constants of tourmalines at 297 K. Units of cij is -2 -2 in GPa, and eijk is in 10 C m . As stated in section 4.3 є ij є ij , the dielectric constants used for refinement are the same as given in Table 4.4.
Method Resonant ultrasound spectroscopy
Sample → T003 T007 T004 T016 T020 T013
c11 (GPa) 302.3(2) 302.0(2) 300.7(4) 302.6(1) 306.3(2) 307.4(2)
c33 ” 178.9(3) 179.2(1) 169.3(3) 171.6(1) 161.2(4) 162.5(2)
c44 ” 65.6(1) 65.6(3) 64.0(1) 64.0(2) 63.4(1) 63.6(3) c66 ” 97.3(1) 97.4(1) 96.9(4) 97.4(4) 99.2(3) 99.2(5) c12 ” 107.9(2) 107.2(3) 106.9(1) 107.8(1) 107.9(2) 109.0(2) c13 ” 47.5(4) 46.1(4) 47.0(6) 48.2(5) 51.0(7) 54.5(5) c14 ” -6.0(8) -6.0(9) -8.2(4) -8.1(6) -9.9(7) -9.9(9) C ” 134.8(1) 134.5(2) 132.9(4) 134.0(3) 134.4(1) 135.7(4) g11 ” -18.1 -19.5 -16.9 -15.8 -12.4 -9.1
g33 ” 10.6 9.7 10.0 10.3 8.7 9.8
C´1111(max)/ c´1111(min) 1.694 1.690 1.782 1.769 1.909 1.901
l1 (mm) 5.088 4.481 5.518 6.668 5.441 5.546
l2 (mm) 7.073 5.287 4.956 7.146 5.159 5.239
l3(mm)║c-axis 7.275 5.316 5.310 6.551 3.963 6.264
ρg (g/cc) 3.049(3) 3.056(2) 3.069(8) 3.114(7) 3.175(3) 3.183(4)
ρb (g/cc) 3.049(2) 3.060(1) 3.056(3) 3.101(5) 3.172(1) 3.186(4) -2 -2 e15 (10 .C m ) 23.5(2) 23.7(6) 19.4(3) 22.2(2) 21.9(3) 22.2(3)
e22 ” -4.3(3) -4.3(5) -11.6(4) -3.9(1) -5.2(4) -4.8(3)
e31 ” 21.0(3) 21.0(2) 25.1(2) 15.5(3) 21.7(2) 17.2(9) e33 ” 32.5(3) 29.5(4) 33.9(2) 31.1(2) 33.2(3) 32.8(1) 51 Elastic and piezoelectric properties of tourmalines Chapter 4 ______
TABLE 4.9: Elastic constants of tourmalines together with literature data. Units of cij is in GPa. The results obtained from this work are highlighted in light grey color.
S. No. Reference Origin Density EC → c11 c33 c44 c12 c13 c14 c66 # (gm/cm3) Method ↓ 1. T003 (present work) Unknown 3.049 302.3(2) 178.9(3) 65.6(1) 107.7(2) 47.1(2) -6.0(3) 97.3(1) 2. T007 ” Nigeria 3.060 302.0(2) 179.2(1) 65.6(3) 107.2(3) 46.1(4) -6.0(2) 97.4(1) 3. T004 ” Unknown 3.056 300.7(4) 169.3(3) 64.0(1) 106.9(1) 47.0(2) -8.2(4) 96.9(4) 4. T016 ” Brazil 3.101 RUS 302.6(1) 171.6(1) 64.0(2) 107.8(1) 48.2(1) -8.1(3) 97.4(4) 5. T020 ” Australia 3.175 306.3(2) 161.2(4) 63.4(1) 107.9(2) 51.0(3) -9.9(4) 99.2(3) 6. T013 ” Brazil 3.186 307.4(2) 162.5(2) 63.6(3) 109.0(2) 54.5(1) -10.0(2) 99.2(5) 7. Helme & King [12] Brazil 3.149 Pulse-echo 306.0 174.0 64.6 109.2 53.0 -8.0 98.4 8. Tatli & Özkan [13] Mexico 3.108 305.2 176.4 64.6 108.4 51.0 -6.0 98.4 9. Tatli & Özkan [13] Brazil 3.105 295.8 173.3 63.6 103.2 45.0 -10.0 96.3 10. Tatli & Özkan [13] California 3.046 Pulse-echo 300.0 173.8 65.2 106.2 42.0 -7.0 96.9 11. Tatli & Özkan [13] Unknown 3.051 301.1 174.6 65.4 106.5 43.0 -7.0 97.3 12. Tatli & Özkan [13] Sri-Lanka 3.061 301.6 169.8 65.5 96.0 47.0 -9.0 102.8 13. Bhagwantam [11] Unknown - Wedge 304.0 176.0 65.0 88.0 35.0 -4.0 108.0 14. Bhagwantam [11] Unknown - 263.0 151.0 59.5 61.0 49.0 -9.0 101.0 15. Mason [10] Unknown - Resonance 272.0 165.0 65.0 40.0 35.0 -6.8 116.0 16. Voight [9] Unknown - Static 275.0 163.0 68.0 70.4 9.0 -7.9 102.3
Note:- EC: elastic constants, R: figure of merit of the fully converged refinement in [10-3]. 52 Elastic and piezoelectric properties of tourmalines Chapter 4 ______
The elastic constants of tourmaline have also been reported in literature by five different authors. Elastic constants obtained in this study are tabulated in Table 4.9 and drawn in Fig. 4.11 together with literature data and their measuring techniques. There is considerable disagreement between the literature values. Elastic constants obtained with this study agree well with data from literature [7 – 11] showing only about 2% variation. However, strong disagreement with other literature data [14 – 16] with a difference of more than 15 % is observed. Chemical composition and other relevant information about the samples have also not been reported by previous authors [13-16] (see Table 4.8). This large difference with previous authors [14 – 16] can not be explained on the basis of chemical composition alone. Certainly some other major reasons, crystal imperfections, inclusions or micro cracks, and inaccuracy of the employed technique, for example, may be a reason for that.
300 250
150 ij c 100
c11 50 c33 c44 c12 0 c13 c14
4 3 0 g 1 0 07 1 2 n n #9 10 1 ht n m ------0 0 0 i # # #12 ig tam T003 T T T0 T016 T K zka o n & Ö V Maso a & w i ag me tl h a B Bhagwanta Hel T Reference
FIG. 4.11: Direct comparison of the elastic constants of tourmalines with literature data.
Now looking on the variation in the derived values of elastic constants in Table 4.8 together with the chemical compositions of the corresponding samples in Table 4.8, one can observe that there is a relationship. For example, an increase in Fe concentration (and decrease in Li) increases c11 and decreases c33. For all tourmaline
53 Elastic and piezoelectric properties of tourmalines Chapter 4 ______
309 306 303
11 300 c 297 294 180 175 170 33 c 165 160
66
65 44
c 64
63 104 102 100 66
c 98 96 94 012340369121501234012345 CaO (wt %) FeO (wt %) Li O (wt %) Na O (wt %) 2 2
FIG. 4.12: Compositional dependence of the longitudinal elastic stiffnesses compared with literature values [13]. 54 Elastic and piezoelectric properties of tourmalines Chapter 4 ______
samples c11 > c33 indicates that the atomic bonding within the (001) plane is stronger than along [001] direction. Moreover, an increase in Fe concentration (and decrease in
Li) decreases c44 and increases c66. The correlations can be easily seen in Fig 4.12, where the longitudinal and shear stiffnesses together with the literature data are
plotted vs FeO and Li2O variation.
4.5.1 Correlation between crystal structure and elastic properties A systematic variation in tourmaline’s structure with substitution is already known [116]. There are various cations having different ionic sizes which are located in various sites of the tourmaline structure, therefore a change in the bond lengths and valence sums (strengths) of the bonds between these cations and neighboring anions are also expected, which significantly changes the elastic properties of tourmalines with varying chemical composition. Changes in bond lengths and valence sums are rather small. However, this may already cause a meaningful change in the elastic properties. Lets first begin with T-site tetrahedra, which are mainly occupied by Si, and in
each unit cell of tourmaline, there are six SiO4 tetrahedra forming a closed ring, a main building block of tourmaline. In this tetrahedra, Al → Si substitution is already known [117], in case the Si content is less than 6 apfu. Our EMP analyses indicate a Si content of less than 6 apfu, suggesting a partial substitution of Al in it. Al → Si substitution increases the
radii, and therefore the field strength of all these cations are different. Also, there is a linear correlation between “Na1+ + ” and the
The deviation from Cauchy relations, described by g11 and g33, are given in
Table 4.8 and showing positive values for g33 and negative for g11 (= g22). g33 is the one parallel to [001] and that gives an information about the bonding type in the plane
perpendicular to [001]. A positive value of g33 gives a hint that within the (001) plane non-directional ionic bonding dominates over directional covalent bonding. On the
other hand the negative value of g11 gives a hint of strong covalent interaction in the bond chain parallel to 3-fold axis. The anisotropy of the bonding system does not change significantly with temperature. Fig. 4.13 is showing the anisotropy of the longitudinal elastic stiffnesess at room temperature in (001) and (010) plane for all samples ( Li-rich T003 to Fe-rich T013) to best visualize a systematic variation of the materials anisotropy with chemical composition. In all the samples the behavior is almost isotropic within (001) plane, however there is a pronounced anisotropy perpendicular to (001) plane as 56 Elastic and piezoelectric properties of tourmalines Chapter 4 ______
T003 T007 T004 T016 T020 T013
FIG. 4.13: Anisotropy of longitudinal elastic stiffness on tourmaline samples T003 and T013, showing a small maxima and minima in (001) direction. Chapter 4 Elastic and piezoelectric properties of tourmalines 57 ______
expressed by c´1111(max)/ c´1111(min) ≈ 1.9. This can be attributed to the high field strength of Fe bearing tourmalines which dense the silicate rings lying on the (001) plane. In Fe bearing tourmalines the Fe2+ ions attract more neighboring anions and thus stiffen the structure. However, in Li-rich tourmalines the Li+ ion has only one positive charge and hence less stiffer structure than Fe-ones.
4.5.2 Isotropic elastic moduli and Debye temperature The isotropic elastic constants and the bulk modulus (Table 4.10) were calculated using the average of the Voigt and Reuss approximations [119].
TABLE 4.10: Bulk modulus (K), shear modulus (G), and Poission ratio (μ) calculated from isotropic elastic constants at 299 K together with acoustic wave velocities: VL (longitudinal), VS (shear) and VD (mean) in km/sec. Units of elastic modulies are in GPa. The values of Debye temperature (θD) are in Kelvin. The results obtained from this work (highlighted in light grey color) are also compared with literature data.
Reference Method K G µ VS VL VD θD *T003 119.3(2) 80.0(3) 0.220 5.184 8.776 5.743 790.8(2) *T007 118.5(1) 82.2(2) 0.218 5.184 8.755 5.741 787.3(1) *T004 115.7(4) 80.2(2) 0.218 5.124 8.675 5.676 774.3(1) *T016 RUS 117.3(4) 80.5(2) 0.220 5.096 8.649 5.647 768.3(3) *T020 115.6(1) 79.6(4) 0.221 5.008 8.517 5.551 754.6(2) *T013 118.0(2) 79.4(2) 0.225 4.994 8.540 5.538 751.5(3) [7] 126.0 81.7 0.235 5.1278.694 5.681 765.4 [8] 121.3 79.8 0.229 5.0698.563 5.615 766.1 [9] Pulse- 121.1 81.6 0.224 5.1768.688 5.732 762.1 echo [10] 121.7 81.9 0.225 5.1808.693 5.737 763.0 [11] 122.6 82.6 0.224 5.195 8.720 5.750 764.3
Note: * present work, [7 – 11] from Tatli & Özkan [see Table 4.9]. The digits in parentheses are the uncertainty (standard deviation) in the last digit of the value.
The mean acoustic wave velocity VD can be derived in a good approximation from the isotropic elastic constants [120]. The mean wave velocity is related to the
isotropic longitudinal (c11 ) and shear ( c44 ) stiffness by the following relation: 58 Elastic and piezoelectric properties of tourmalines Chapter 4 ______
31 1 12 c c V , where V 11 & V 44 (4.3) D 33 L S 3 VV LS
ρ is the mass density, VL and VS are the longitudinal and transverse velocities, respectively. Calculated velocities together with bulk modulus, shear modulus and Poisson’s ratio are listed in Table 4.10. The mean wave velocity can now be used to calculate the Debye temperature
D using the following expression [51]:
31 31 6 2 N 6 2 rN A D VD VD (4.4) kb V kb M here, and kb are the Plank’ and Boltzmann constants, respectively, r is the number of atoms in a molecule, M is the molar weight, NA is the Avogadro’s number, ρ the mass density and VD is the average acoustic wave velocity. The Debye temperatures are given in Table 4.10 together with previous literature data. The Debye temperature data for sample T016 is in good agreement with literature data [8]. For the rest of the measured samples it is significantly higher than the literature data. This is a consequence of the higher Li-content of most of the studied tourmalines. Literature values are also found in a very good agreement with the data observed in this study (Fig 4.14).
9 9 V V L 8 L 8
V D 5.5 V D 5.5
V S Velocities [km/s] 5.0 V 5.0 S Velocities [km/s]
4.5 4.5 03691215 0123 FeO [wt%] Li2O [wt%]
FIG. 4.14: Variation of mean wave velocity VD with the concentration of Li and Fe. Red data points in both plots correspond to literature values already referred in Table 4.9.
Chapter 4 Elastic and piezoelectric properties of tourmalines 59 ______
4.5.3 Temperature evolution of elastic and piezoelectric constants An irreversible softening of all resonance frequencies was observed during first heating on almost Fe-free samples (Fig. 4.15). These anomalies are probably related to an increase of configurational entropy caused by order/disorder processes on certain cation sites.
695 heating cooling 690
685
680
675 Frequency [kHz] 670
665 200 300 400 500 600 700 800 900 1000 Temperature [K]
FIG. 4.15: Temperature dependence of selected modes of an elbaite sample (left) and a small irreversible softening of one mode on heating at about 630 K for sample T004.
The well reproducible spectra collected in the second and subsequent heating/cooling cycles were used for the calculation of elastic and piezoelectric constants and of their temperature derivatives. The temperature evolution of the elastic and piezoelectric constants is plotted in Fig. 4.16 and Fig 4.17 respectively. The main elastic constants show an almost linear softening with increasing temperature (except of the coupling coefficients c14).
The gradual decrease of the cij with increasing temperature indicates that tourmaline does not experience any phase transition up to 900 K. Both elastic and piezoelectric constants behave almost linearly in the investigated temperature interval. The temperature evolution of the elastic constants can be described in a sufficient approximation by the thermoelastic constants Tij = dlog cij /dT according to cij(T) = cij
(T0) (1+ Tij (T-T0)). The high-temperature behavior of all Tij is quite normal, i.e. all temperature derivatives are negative (see Table 4.11) and in the order of 10-5 K-1. 60 Elastic and piezoelectric properties of tourmalines Chapter 4 ______
310 180
176
300 172
168 290 [GPa] 164 [GPa] 33 33 11 c c T013 160 280 T020 T016 156 T004 T007 T003 152 270 200 300 400 500 600 700 800 900 1000 200 300 400 500 600 700 800 900 1000 Temperature [K] Temperature [K]
66 100
65 98 64 96 63 94
[GPa] 62 [GPa] 44 66 c 61 c 92
60 90 59 88 200 300 400 500 600 700 800 900 1000 200 300 400 500 600 700 800 900 1000 Temperature [K] Temperature [K] 60
108 56
106 52 [GPa] [GPa] 13
12 104 c c 48
102 44
200 300 400 500 600 700 800 900 1000 200 300 400 500 600 700 800 900 1000 Temperature [K] Temperature [K] -4 120
118
-6 116
114 [GPa] 14 14
c -8 112
Bulk modulus [GPa] 110
-10 108
200 300 400 500 600 700 800 900 1000 200 300 400 500 600 700 800 900 1000 Temperature [K] Temperature [K]
FIG. 4.16: Temperature evolution of the elastic constants of the investigated tourmalines. Chapter 4 Elastic and piezoelectric properties of tourmalines 61 ______
40 35 35
30 30
] 25 ] 2 25 2 20 .C/m -2 20 [C/m 15 ijk [10 e
ijk 10 e15 e 15 e15 -5 e22 -2 e22 -10 e31 -4 e31 -6 -15 e33 -8 e33 -20 -10 200 300 400 500 600 700 800 900 1000 200 300 400 500 600 700 800 900 1000 Temperature [K] T003 Temperature [K] T013
35 35 30 30 25 ] 20 ] 25 2 2
15 [C/m [C/m 20 ijk ijk e e 10 e15 e15 15 -5 e22 e22 -3 e31 e31 -4 -10 e33 e33 -5 200 300 400 500 600 700 800 900 200 300 400 500 600 700 800 900 1000 T016 Temperature [K] T020 Temperature [K]
35 35
30 30
25 25 ] ] 2 2 20 20 [C/m [C/m 15
15 ijk ijk e e e15 e15 e22 -5 e22 -10 e31 e31 e33 e33 -15 -10 -20 200 300 400 500 600 700 800 900 1000 200 300 400 500 600 700 800 900 1000 Temperature [K] Temperature [K] T007 T004
FIG. 4.17: Temperature evolution of the piezoelectric constants of tourmaline samples.
The temperature coefficients of Fe-rich tourmalines are a bit larger than those of the Li-rich ones. Above ~ 910 K all resonances of the freely vibrating samples show more or less rapid irreversible softening. This phenomenon is probably due to the escape of volatiles from fluid inclusions which starts already about 300 K below the dehydration temperature.
62 Elastic and piezoelectric properties of tourmalines Chapter 4 ______
-6 - TABLE 4.11: Thermoelastic constants Tij = d(log cij)/dT at 299 K. Units: Tij (10 K 1). The digits in parentheses are the uncertainty (standard deviation) in the last digit of the value. Fe content increases from T003 to T013 (see Table 4.1).
Tij 11 33 44 66 12 13 14 T003 -9.9(2) -9.9(3) -12.9(1) -12.3(3) -6.4(0) -0.9(6) -57.9(5) T007 -10.2(3) -8.9(2) -13.0(6) -11.7(0) -7.9(3) 0.9(3) -66.6(2) T004 -11.3(1) -11.2(2) -12.7(7) -12.3(8) -8.9(2) -11.7(2) -40.9(4) T016 -9.9(1) -9.9(0) -12.1(5) -11.2(9) -7.2(2) -2.7(4) -48.4(3) T020 -9.8(2) -9.3(1) -9.4(4) -11.1(2) -8.4(2) -7.6(3) -20.7(1) T013 -9.4(3) -7.4(2) -11.4(2) -12.1(1) -4.7(1) 8.7(4) -32.7(1)
Temperature derivatives of all independent piezoelectric stress constants are listed in Table 4.12. A linear behavior of piezoelectric stress constants was observed in the temperature range of 300 – 900 K. Except of T004, the piezoelectric behavior of all the samples are quite similar. The reason for the different evolution of e33 and e31 of T004 is not clear. In contrast to the dielectric and elastic properties, the Fe/Li-content has no influence on the piezoelectric properties within experimental errors.
TABLE 4.12: Temperature derivative for the linear fit of the piezoelectric constants at 299 K. Units are in 10-4 Cm-2 K-1. The digits in parentheses are the uncertainty (standard deviation) in the last digit of the value.
deijk/dT → 113 222 311 333 T003 -16.5(2) -55.6(7) -49.8(2) 18.3(0) T007 6.2(3) 27.2(5) -58.8(2) 14.8(2) T004 4.6(1) -70.8(5) -222.1(6) -110.9(3) T016 -5.4(2) 26.0(2) -87.3(1) -23.5(3) T020 -24.0(1) -59.3(4) 52.3(5) 26.8(1) T013 -53.5(1) -167.3(2) 5.2(1) 1.8(4)
4.6 Summary and conclusions The behaviour of dielectric properties of tourmaline with varying chemical composition was studied for the first time using the substitution method. The єij values at room temperature are frequency independent with a dissipation factor of ≤
0.0004 for all frequencies under investigation. A correlation curve between є11 and є33 Chapter 4 Elastic and piezoelectric properties of tourmalines 63 ______
is derived. Dependence of є11 and є33 on chemical compositions is presented for the first time. Temperature dependence of the elastic and piezoelectric properties was studied in the temperature range of 299 – 900 K. An unexpected irreversible softening of all resonance frequencies was observed during first heating on three of six investigated samples. The anomalies are most probably due to an increase of configurational entropy caused by order/disorder processes on certain cation sites. Temperatures evolution of the elastic properties is quite normal, i.e. all temperature derivatives are negative. The piezoelectric constants behave almost linearly in the investigated temperature interval and the piezoelectric properties are almost independent of the lithium or iron content. As a final note, selection of tourmalines for device application is not affected by the Li or Fe content. However, heating of Li bearing tourmalines in air up to 900 K for about 48 hours is a prerequisite for stable behavior.
Chapter 5
Anomalous elastic behavior of Ca0.28Ba0.72Nb2O6 single crystal in the paraelectric phase
Ca0.28Ba0.72Nb2O6 (hereafter CBN-28), crystallizing in the partially filled tetragonal tungsten bronze (TTB) structure type, belongs to point symmetry 4/mm in its paraelectric phase. The tetragonal tungsten bronze structure consists of a
framework of MO6 octahedra (M is commonly Nb, Ti, W, or Ta) sharing corners in such a way that three types of interstitial site A1, A2 and C result, which can accommodate different metal atoms. The typical tungsten bronze crystal structure is shown in Fig. 5.1, typified by oxygen octahedra linked at the corners in a complex way to yield three types of openings, two of which normally contain the A1 and A2 cations. The B cations, typically niobium Nb4+ or Nb5+ are inside the oxygen octahedra. The ferroelectric tungsten bronze compositions are characterized by the
chemical formulae (Al)2(A2)4(C)4(Bl)2(B2)8O30 or (Al)2(A2)4(Bl)2(B2)8O30, in which cations A1 are in the 12-fold coordinated site, cations A2 are in the 15-fold coordinated site, cations C are in the 9-fold coordinated site and cations B are in two different 6- fold coordinated sites. The compositions are known as “stuffed” bronzes when all the
crystallographic sites are filled, for example in K3Li2Nb5O15. All other tungsten bronzes are called either “filled” or “partially-filled” bronzes in which the 9-fold coordinated site is completely vacant and the 12- and 15-fold coordinated sites are either fully or partially occupied. For example, widely investigated strontium barium 66 Anomalous elastic behavior of Ca0.28Ba0.72Nb2O6 … Chapter 5 ______niobate is “partially-filled” TTB having unit cells containing (Sr,Ba)5Nb10O30 (SBN), where only five of the six available A sites (A1 and A2) are occupied in a way that the 12-fold coordinated A1 site is occupied solely by the smaller Sr2+ (1.12 Å), and Ba2+ predominately occupies the 15-fold coordinated A2 site, since Ba2+ (1.38 Å) is substantially larger than Sr2+ (1.12 Å). The B sites (B1 and B2) are completely filled with Nb5+ cations, however, the C sites are empty. A total of five sites are occupied by five Sr + Ba atoms which are partially occupied, hence there exists a built-in disorder among the A sites. Due to the difference between the ionic radii of Ba2+ and Sr2+, and built-in disorders and uniaxial tunnel structures existing along the c axis, local random fields and polarizations are expected to exist along the c axis.
FIG. 5.1: Projection of the tungsten bronze structure type [121] normal to the 4 fold c axis showing the sites for cation occupancy.
2+ 2+ In CBN-28, the Ca (ri = 1.00 Å) ions take the same position as the Sr (ri = 1.12 Å) ions in SBN because of the similar ionic radius. The A2 sites are solely 2+ occupied by Ba (ri = 1.34 Å) ions. The C sites are empty. The B sites are only occupied by niobium ions. Chapter 5 Anomalous elastic behavior of Ca0.28Ba0.72Nb2O6 … 67 ______
The existence of the solid solution CBN-28 with 0.20 ≤ x ≤ 0.40 was reported by Ismailzade [122] in 1959 for the first time. Large single crystals of CBN-28 were grown successfully by Eßer et. al. [24] using the Czochralski method. Extending the work of Eßer et. al., Burianek et. al. [123] have grown some more CBN single crystals with high optical quality using four different compositions within range 0.22 ≤ xCa ≤ 0.40. One sample of CBN-28 single crystals grown by Burianek et. al. [123] is shown in 5.2a and has been used in this thesis to study their elastic properties.
(a) (b) (c)
FIG. 5.2: Ca0.28Ba0.72Nb2O6 crystals: (a) large size boule of CBN, diameter 15 mm; growth habits typical for tetragonal tungsten bronze crystals: (b) side and (c) top view [125].
Optically, the CBN single crystals are of excellent quality, crack free, transparent and having 24 natural faces. The idealized form of the crystal with natural faces is shown Fig. 5.2 b and c. Two rectangular parallelepipeds (named as CBN-0201 and CBN- 0202) were cut from a raw CBN-28 crystal. One sample is shown in Fig. 5.5 and final sample dimensions are given in Table 5.1. The density was measured by the buoyancy
method in pure water at room temperature. The calculated geometric density ρg =
M/l1l2l3, (M is the sample mass and li are the edge lengths), matches well with the density obtained by the buoyancy method confirming the high quality of the samples. 68 Anomalous elastic behavior of Ca0.28Ba0.72Nb2O6 … Chapter 5 ______
A conoscopic view using a polarizing microscope on one sample is shown in Fig. 5.3 stating the alignment of the sample is of high standard.
FIG. 5.3: Conoscopic image of CBN single crystal under the polarizing microscope indicating the high optical quality of the CBN sample.
TABLE 5.1: Characterisation of CBN-28 samples. Uncertainty of the last decimal is in parenthesis. Lattice parameters from [148]
Sample name → CBN-0201 CBN-0202 Structure type partially filled tetragonal tungsten bronze (TTB) space group P4bm (at room temperature) melting point > 1460 ˚C ferroelectric transition Tc = 264.4 ˚C (4mm ↔ 4/mmm) optical quality Light yellow – transparent inclusion free single crystal a (Å) 12.4491(5) c (Å) 3.95858(7) V (Å) 613.30(4) l1 (mm) 7.493(2) 7.370(1) l2 (mm) 6.407(1) 7.972(1) l3 (mm) ║ c-axis 7.436(1) 7.519(1)
b (gm/cc) 5.302(2)
g (gm/cc) 5.292(5) 5.299(3)
Temperature dependent calorimetric measurements have been performed on DSC 404 Pegasus to see the phase transition temperature of CBN-28. It is clearly seen from Fig. 5.4 (black curve) that the ferroelectric phase transition (4mm ↔ 4/mm)
occurs at 264.4 ˚C (Curie temperature Tc). The transition temperature of the light- yellow and transparent CBN-28 crystals is well reproducible with no significant hysteresis. However, on annealing in He atmosphere at temperatures above about 650
K the crystal color changed to dark blue (see Fig. 5.5 left) and Tc shifted to higher temperatures by about 5.4 K (see Fig 5.4 blue curve).
Chapter 5 Anomalous elastic behavior of Ca0.28Ba0.72Nb2O6 … 69 ______
FIG. 5.4: Shifting of ferroelectric transition temperature of CBN-28 before (black) and after heating (blue) in He purge gas atmosphere.
Both, the shift in Tc and the color change are reversed (see Fig. 5.5 right) after annealing the blue crystal in air at 1000 ˚C for 3 hours. These effects are probably caused by oxygen vacancies leading to a Nb5+ ↔ Nb4+ valence change.
Annealing at 1273 K in air for 3 hrs
Heating above ~ 650 K in He atmosphere
FIG. 5.5: Rectangular parallelepiped of CBN-28 used for RUS measurements. Note the color change.
5.1 Thermal expansion Thermal expansion is a typical anharmonic property of the crystal lattice. It plays for example an important role in the design and fabrication of devices that 70 Anomalous elastic behavior of Ca0.28Ba0.72Nb2O6 … Chapter 5 ______exploit photorefractive or electro-optic properties [125]. Enormous interests have been drawn to thermal expansion studies after the discovery of negative thermal expansion in ZrW2O8 over a wide temperature range [126]. Negative thermal expansion along one crystallographic direction together with a positive coefficient along other crystallographic direction has been reported previously by many authors in many other materials [127-130].
TABLE 5.2: Coefficients of linear thermal expansion ij (slope of tangents on strain −6 −1 curves) and of volume thermal expansion ij 211 33 in 10 K . Uncertainty of the last decimal is in parenthesis.
T [K] 173 473 563 723 843 1003 1323 1503
11 8.0(1) 15.2(1) 13.4(2) 11.5(1) 10.5(1) 8.6(1) 8.0(1) 8.7(1)
33 3.8(0) -14.5(3) 1.4(2) 10.2(1) 11.1(1) 7.0(1) 7.2(0) 8.0(1)
V 19.8 15.9 28.2 33.2 32.1 24.2 23.2 25.4
The ferroelectric and paraelectric phases of CBN-28 are tetragonal and hence have only two independent principal thermal expansion coefficients α11 (= α22) and
α33. Thermal expansion measurements parallel and perpendicular to the tetragonal axis were performed on sample CBN-0201. Calibration corrected strains together with linear thermal expansion coefficients are shown in Fig. 5.6 and linear thermal expansion coefficients at selected temperatures are tabulated in Table 5.2. A highly nonlinear and anisotropic behavior in the temperature range 100 - 1523 K has been observed. The thermal expansion along the crystallographic a (=b)- and c-axes are shown in Fig. 5.6 a and b. The thermal expansion along a-axis shows a ‘normal’ behavior, while that along c-axis shows very ‘unusual’ behavior. The strain along a- axis increases steadily with temperature and changes the slope above about 500 K. Along c-axis, the thermal expansion remains positive up to about 300 K, followed by a negative thermal expansion in the temperature regime of about 300< T< 560 K and positive expansion above 560 K. The unusual thermal expansion in CBN along c-axis corresponds to the relaxor behavior of the ferroelectric phase transition at Tc = 535 K. The negative thermal expansion above room temperature suggests that the structural changes start
Chapter 5 Anomalous elastic behavior of Ca0.28Ba0.72Nb2O6 … 71 ______
~Tm
Tb Tc
a)
Tb
~Tm
Tc
b) FIG. 5.6: Temperature evolution of strain and thermal expansion (a) perpendicular and (b) parallel to the 4 fold- axis of CBN-28. 72 Anomalous elastic behavior of Ca0.28Ba0.72Nb2O6 … Chapter 5 ______already at 300 K and end at about 560 K. These results are very comparable to that observed for SBN crystals by various authors [131-133], who reported that the structural changes corresponding to the onset of ferroelectric phase transition are responsible for the anomalies in thermal expansion. In the case of SBN-39, for example, the onset of the ferroelectric phase transition started at temperatures about 150 K below Curie temperature and ends about 50 K above it. However, CBN-28 shows a narrower temperature regime for the structural changes and instabilities corresponding to its ferroelectric phase transition at 535 K. Above 560 K up to about 900 K the behavior of the thermal expansion can be explained as follows: the main features of relaxor ferroelectrics are connected to their structural (compositional) inhomogeneity or disorder and with the presence of polar nanodomains in a nonpolar matrix. Three peculiar temperatures: Tc, Tm and Tb (explained below) have been revealed after structural investigations and the study of the physical properties of relaxors ferroelectrics [134]. Tm is the temperature at which the measured real part of dielectric constant passes through a maximum, which is above Tc, the Curie temperature. Tb is known as the Burns temperature (normally a few hundred degrees higher then Tm) below which the first nucleation of nanodoamins appear, whose interactions and growth can trigger a transition into a glassy or ordered phase. If the domains are large enough, the sample will undergo a cooperative ferroelectric phase transition at Tc. However, if grown nanodomains are not large enough, they will ultimately exhibit a dynamic slowing down of their fluctuations below Tm, leading to an isotropic relaxor state with random orientation of polar domains [67].The Curie temperature Tc, the temperature for dielectric maxima Tm and the Burns temperature Tb are clearly visible in Fig 5.6 a, b. The estimated region for
Tb for CBN is in the range of 890-910 K, where a deviation from its normal linearity is clearly observed. The formation of polar nano domains below the Burns temperature has been suggested as a common feature of relaxor ferroelectrics [135].
The existence of Tm in the thermal expansion behavior was assumed only after literature considerations that it is generally about 50 °C above the Curie temperature. However, other measurements (for example of dielectric properties) and a detailed investigation are needed to confirm whether this is the case. A completely opposite temperature dependence of strain behavior with respect to the crystallographic axes was reported by Song et. al. [29] on CBN-28 in the Chapter 5 Anomalous elastic behavior of Ca0.28Ba0.72Nb2O6 … 73 ______temperature range 298 - 572 K. Our result on thermal expansion of CBN-28 is the first available in a large temperature range of 100 – 1523 K.
5.2 Elastic properties In the paraelectric phase CBN-28 has tetragonal structure with point symmetry 4/mm, which has six symmetry independent elastic tensor components [49] as shown in Fig. 5.7.
c11 c12 c13 0 0 0 c c c 0 0 0 12 11 13
c13 c13 c33 0 0 0 0 0 0 c44 0 0 0 0 0 0 c 0 44 0 0 0 0 0 c66
where c66 1/ 2(c11 c12 ) FIG. 5.7: Elastic constants matrix for the crystal class 4/mm
The elastic properties of CBN-28 in the paraelectric phase were studied between room temperature and 1503 K using resonant ultrasound spectroscopy (RUS) on samples CBN0201 and CBN0202. The sample was held in isothermal state at each temperature for about 45 min before starting the data collection. The spectra were recorded in intervals of 40 K starting from room temperature up to 1503 K in air. A part of the resonance spectra is shown in Fig 5.8. The ferroelectric phase transition temperature (Tc = 264.6 ˚C) can be seen clearly here. In order to calculate the elastic constants at least 120 eigen-frequencies in the range 200 kHz - 1140 kHz were extracted from each resonance spectrum and used in non-linear least-squares refinements minimizing the quantity 2 . The initial model for the elastic constants in the paraelectric phase of CBN was derived from the elastic constants of SBN [136], taking into account well known rules for the elastic behavior of chemically and structurally related compounds [48, 137]. At least four resonance spectra (heating-cooling-heating-cooling) were collected on each sample at elevated temperatures to check the reproducibility of elastic constants. Fully reproducible resonance spectra were observed. The elastic 74 Anomalous elastic behavior of Ca0.28Ba0.72Nb2O6 … Chapter 5 ______moduli calculated from the observed resonance spectra are shown in Fig 5.9 a, b, c, and individual values at selected temperatures are presented in Table 5.3.
]
K
[
e
r
u
t
a
r
e
p
m
e T
Frequency [kHz]
FIG. 5.8: Part of the resonance spectra of CBN-28.
First, the full sets of elastic constants measured on both investigated samples are in excellent agreement (see Fig 5.9) indicating the high precision of our RUS setup, and high quality of the samples. The elastic constants cij of the individual samples deviate only by 0.4 % from the corresponding averaged values at high temperatures. However, this deviation is slightly larger (0.9 %) near Tc, because of strong ultrasound attenuation. Therefore, in the vicinity of the ferroelectric phase transition and in the ferroelectric phase only 10-12 resonance peaks (see Fig 5.8, the resonance spectra of CBN-28 from room temperature to up to Tc) could be observed. The elastic behavior in the ferroelectric low temperature phase is characterized by strong and highly anisotropic ultrasound attenuation which is probably caused by anelastic interactions between elastic waves and ferroelectric domain walls, i.e. there is an energy loss of ultrasound waves due to domain wall movement forth and back by the elastic waves. Thus, CBN crystals in its raw (as-grown) form are not suitable to measure their elastic properties in their ferroelectric phase. Poling of the CBN-28 single crystal is a potential approach to overcome this problem. Chapter 5 Anomalous elastic behavior of Ca0.28Ba0.72Nb2O6 … 75 ______
240
220
200
[GPa] 180
ij C
160 c11-CBN0202 c11-CBN0201 140 c33-CBN0202 c33-CBN0201 200 400 600 800 1000 1200 1400 1600 Temperature [K] a)
76
74
72 [GPa]
70 ij C 68 c44-CBN0202 c44-CBN0201 66 c66-CBN0202 c66-CBN0201 64 200 400 600 800 1000 1200 1400 1600 b) Temperature [K] 110 c12-CBN0202 c12-CBN0201 100 c13-CBN0202 c13-CBN0201 90
80 [GPa]
ij C 70
60
50 200 400 600 800 1000 1200 1400 1600 c) Temperature [K] FIG. 5.9: Temperature dependence of (a) longitudinal elastic stiffnesses, (b) shear stiffnesses, and (c) transverse interaction coefficients of CBN-28 in the paraelectric phase. The arrows mark the region for the Burn’s temperature. 76 Anomalous elastic behavior of Ca0.28Ba0.72Nb2O6 … Chapter 5 ______
TABLE 5.3: Elastic properties of CBN-28 at selected temperatures. Units: Temperature T in K, elastic constants cij , deviations from Cauchy-relations gij and bulk modulus B in GPa, ratio of maximal and minimal longituinal elastic stiffness −4 −1 cmax cmin is dimensionless, thermoelastic constants Tij = d log cij/dT in 10 K .
T 563 723 843 1003 1243 1503 c11 217.6(2) 221.6(2) 229.1(2) 230.2(1) 223.9(1) 213.8(1) c33 143.5(3) 158.1(2) 170.8(2) 183.3(2) 186.2(2) 181.0(2) c44 69.9(1) 69.8(1) 69.4(1) 68.6(1) 67.2(1) 65.3(1) c66 75.4(2) 76.6(1) 76.8(1) 76.2(1) 73.6(1) 70.6(1) c12 99.9(4) 90.8(2) 85.0(2) 85.5(2) 84.1(2) 81.4(2) c13 60.2(3) 57.0(2) 61.4(2) 67.2(2) 69.1(2) 68.6(2) g11 -9.7 -12.8 -7.9 -1.4 +1.9 +3.3 g33 +24.5 +14.3 +8.2 +9.4 +10.3 +10.8 B 105.3 107.0 112.5 118.1 118.3 115.6 cmax cmin 1.63 1.47 1.37 1.28 1.22 1.21
T11 0.0(2) 2.3(2) 1.85(6) -0.60(1) -1.50(1) -2.08(1)
T33 10(2) 4.5(1) 7.3(2) 2.24(4) -0.51(7) -1.55(9)
T44 -0.3(3) -0.26(1) -0.61(1) -0.77(1) -0.98(1) -1.21(1)
T66 2.1(6) 0.52(1) -0.07(4) -0.91(1) -1.52(1) -1.97(2)
T12 -4.1(5) -7.5(9) -1.3(3) -0.16(7) -0.96(2) -1.51(6)
T13 6.0(9) 1.4(6) 9.7(2) 2.8(1) 0.10(6) -0.51(9)
The temperature evolution of cij in the paraelectric phase (see Fig. 5.10) shows strong and pronounced anomalies up to 900 K. At high temperatures (≈ 900 K - 1503 K) the evolution of the elastic constants of CBN-28 is quite normal, i.e. all temperature derivatives are negative (see Table 5.3). Below about 900 K all resonances of the freely vibrating samples show rapid softening when approaching the
Curie temperature. Except shear stiffness (SS) coefficients (c44 and c66), the longitudinal elastic stiffness (LS) coefficients (c11 and c33) and transverse interaction coefficients (TIC) (c12 and c13) seem to be strongly affected by temperature as approaching the phase transition. Below ~ 900 K c11 shows deviation from linearity and c33 shows a step like drop showing a continuous decrease with lowering temperature and reaching a minimum at Tc. The anomalies in the temperature Chapter 5 Anomalous elastic behavior of Ca0.28Ba0.72Nb2O6 … 77 ______evolution of c11 and c33 below ~ 900 K (in the vicinity of TB) are likely to be attributed to the first nucleation of polar nano regions.
1.25 1.20 1.15
1.10 1.05 (1503K 1.00 ij
c Tc 0.95 c11 (T) / c33
ij 0.90 c c12 0.85 c13 0.80 c44 c66 0.75 200 400 600 800 1000 1200 1400 1600
Temperature [K]
FIG. 5.10: Temperature dependence of the relative elastic constants cij(T)/cij(1503K) of CBN-28 in the paraelectric phase.
The transverse interaction coefficients are strongly affected below 900K. The deviation point is shown by an arrow in Fig. 5.9. One can see clearly this change on a relative scale as shown in Fig. 5.10. The value of cmax cmin (ratio of the max value of longitudinal elastic stiffness to the minimum) gives a clear hint about the elastic anisotropy. The values are given in Table 5.3 state the anomalies increase when approaching towards phase transition. This kind of anomalous behaviour below 900 K is a signature to the initiation of the Burns temperature and is also evident from thermal expansion measurements. Anisotropy of longitudinal elastic stiffness is a useful aid to visualize material’s anisotropy which is provided by the system of principal bond chains described by the principal bond chain (PBC) vectors [48]. Fig. 5.11 shows the anisotropy of longitudinal elastic stiffness of CBN-28 at temperatures 603 K and 1503 K. In the plane perpendicular to the 4-fold axis the behavior is almost isotropic 78 Anomalous elastic behavior of Ca0.28Ba0.72Nb2O6 … Chapter 5 ______at both temperatures. However, approaching from high temperature to Tc the anisotropy increases due to the softening of c33.
603 K 1503 K
FIG. 5.11: Projection of the representation surface of the longitudinal elastic stiffness of CBN-28 along [001] and [010]. The labeling x, y, z of the axes correspond to the axes e1, e2 and e3 of the Cartesian reference system. Units are in GPa.
5.2.1 Deviation from Cauchy relations
An increasing anisotropy upon approaching Tc is also observed for the deviations from Cauchy relations as shown in Fig. 5.12. It can be seen that at high temperatures the anisotropy is quite small and both independent gij values are positive. Up to Burn’s temperature the anisotropy remains quite small, however, below it the Chapter 5 Anomalous elastic behavior of Ca0.28Ba0.72Nb2O6 … 79 ______anisotropy increases significantly. The positive value of g33 near Tc tells that the directional bonding contribution within the (001) plane is reduced and simultaneously non-directional bonding contribution increased. However, the negative value of g11 (= g22) states that parallel to the 4-fold axis the directional bonding contributions are dominant when approaching to Tc.
(a) (b)
30 g33 g11 20
10 GPa
ij 0 g
-10
-20 200 400 600 800 1000 1200 1400 1600 Temperature [K]
FIG. 5.12: The longitudinal effect of the deviation from Cauchy relations at (a) T = 603K and (b) T = 1503 K and (c) temperature dependence of the deviation from Cauchy relations for CBN-28.
It is well known that long range Coulomb forces favor the stability of the ferroelectric state [138]. A closer examination of Fig. 5.12 reveals this phenomenon near Tc where the positive value of g33 indicates the development of strong non- directional bonding (coulomb interaction) within the (001) plane resulting the ferroelectric phase in CBN-28.
80 Anomalous elastic behavior of Ca0.28Ba0.72Nb2O6 … Chapter 5 ______
5.2.2 Isotropic elastic properties Once the single crystal elastic constants of a material are known, these elastic moduli are used to determine the ‘isotropic’ elastic constants using the Voigt and Reuss approximations [119]. Calculated bulk modulus, shear modulus and Poisson’s ratio using Voigt and Reuss approximations are tabulated in Table 5.4.
TABLE 5.4: Calculated isotropic elastic constants of CBN-28 single crystal at 1503K.
Parameter [Units] CBN-28 Bulk modulus [GPa] 115.6(2) Shear modulus [GPa] 66.2(1) Poisson’s ratio 0.26(5)
570 ]
560
550
540 Debye temperature [ Debye 530
200 400 600 800 1000 1200 1400 1600 Temperature[T] FIG. 5.13: Temperature dependence of Debye temperature of CBN-28.
The temperature dependence of the Debye temperature (D ) calculated from the measured isotropic elastic constants (detailed procedure outlined in section 4.5.2 of Chapter 4) is shown in Fig 5.13. The temperature dependent behavior shows a strong depression in ΘD at lower temperatures which indicates the bonding system must get softer in the crystal in that region. The behavior is associated with the phase transition, as the phase transition is associated with some structural instability of the bonding system. There will be a minimum value of ΘD at Tc, which is nothing else than a macroscopic expressions for the mechanical instability of the system.
Chapter 5 Anomalous elastic behavior of Ca0.28Ba0.72Nb2O6 … 81 ______
5.2.3 Temperature evolution of eigen modes The temperature dependence of selected resonance modes of a CBN-28 sample is shown in Fig. 5.14 (a) and (b) together with their inverse quality factors.
1.04 Au-1 Au-2 1.03 Ag B1u 1.02
1.01 T c
1.00 f (T) / f (1503K)
0.99
0.98 200 400 600 800 1000 1200 1400 1600 Temperature [K] a) 15 Au-1 Au-2 Ag B1u 10 ] -4 T c [10 -1 Q 5
0 200 400 600 800 1000 1200 1400 1600 Temperature [K] b)
FIG. 5.14: (a) Temperature evolution of frequency and (b) inverse quality factor of selected free vibrations of a CBN-28 sample (rectangular parallelepiped with dimensions 7.969×7.370 × 7.516 mm3). 82 Anomalous elastic behavior of Ca0.28Ba0.72Nb2O6 … Chapter 5 ______
The observed modes of the CBN samples can roughly be divided in two groups characterized by different behavior around Tc. Modes of the first group display strong softening below ~ 900 K which is accompanied by increasing attenuations (e.g. Ag and B1u modes in Fig 5.14). In the ferroelectric phase the dissipation effects are so strong that these modes are no longer observable. The modes belonging to the second groups (e.g. Au-1 & Au-2 in Fig 5.14) show only moderate softening in the vicinity of
Tc and weak attenuation effects. Most of these modes are still observable in the ferroelectric phase. It is not surprising that the modes of group 2 are dominated by the shear stiffnesses c44, c66 which are nearly unaffected by the mechanical interactions during the phase transitions. Whereas modes which are also effected by c11, c12 or c33, for example, suffer strongly from the phase transition.
5.3 Summary and conclusions Thermal expansion of tetragonal CBN-28 single crystals was studied in the temperature range 100 – 1503 K for the first time. Anisotropic thermal expansion was observed, with the behavior very similar to SBN. Negative thermal expansion along the c-axis was observed for 300 The temperature dependence of the elastic constants cij were measured for the first time in the paraelectric phase. The estimated errors of cij’s are less than 0.7 %. Significant softening of longitudinal stiffnesses and transverse interaction coefficients below about 900 K is a signature of the first nucleation of nanodomains. The onset of softening when approaching the phase transition from high temperatures is frequency dependent, a characteristic for relaxor-ferroelectric materials. Strong anisotropic ultrasound attenuation was observed below Tc which is probably due to elastic waves interaction with mobile domain walls. This makes CBN crystals in its as-grown form unsuitable for elastic measurements in their ferroelectric phase. Therefore, poling of the sample is required. Chapter 6 Elastic properties of Ce:Ca0.28Ba0.72Nb2O6 single crystal in paraelectric phase It is well known that in semiconductors the conductivity can be tailored by doping of suitable impurities. Similarly, doping of partially filled ferroelectric materials can alter their ferroelectric and piezoelectric properties. The ferroelectric transition temperature can be tuned in partially filled TTBs by suitable impurity doping because of its complex structure which allows a wide range of cationic substitutions. The tungsten bronze Ce doped CBN single crystals studied to this end is a good examples of it. Moreover, the doping in relaxor ferroelectrics was reported to result in interestingly improved photorefractive properties. It has already been proved that photorefractive properties of SBN-61 can be improved [46] by cerium doping up to four order of magnitude. Among the photorefractive materials the TTBs are important due to the following main features: firstly, there are no twinning or poling problems because of simple paraelectric-ferroelectric (4/mmm ↔ 4mm) transition, and secondly the availability of five crystallographic sites can enhance the photorefractive properties which can be achieved by varying the site preference for a given dopant. No literature data is available so far regarding any physical or chemical properties of Ce doped Ca0.28Ba0.72Nb2O6 (Ce:CBN-28 here after Ce:CBN), not even its doping concentration or crystal structure. Contributing a little to this end, the effect of cerium doping in the elastic properties of pure CBN-28 in the paraelectric phase 84 Elastic properties of Ce:Ca0.28Ba0.72Nb2O6 … Chapter 6 ______ a) b) c) FIG. 6.1: (a) Cerium doped CBN-28 single crystal with 24 natural facets. Ø 15 mm, showing excellent optical quality, (b) rectangular parallelepiped cut from (a) for RUS measurements and (c) the conoscopic image of (b) showing uniaxial view. together with thermal expansion in the temperature range of 100 – 1323 K were studied for the first time. The Ce-doped CBN crystals are grown by the Czochralski method from a congruent melt, to which 1 wt% of CeO2 was added (the Ce content in the melt was 2.3 gm CeO2 to 230 gm CBN-28). The final Ce:CBN crystal is of high optical quality and of dark red color as shown in Fig. 6.1a. From this Ce:CBN crystal two oriented rectangular parallelepiped samples (one sample is shown in Fig 6.1b) were cut. The final sample names, dimensions and all other relevant parameters are listed in Table 6.1. The conoscopic image of Ce:CBN-01 sample (Fig. 6.1c), showing the uniaxial view, confirms that the sample is well aligned. TABLE 6.1: Crystal data for Ce:CBN single crystals. Uncertainty of the last decimal is in parenthesis. Parent sample name → Ce:CBN Samples Ce:CBN-01 Ce:CBN-02 crystal structure partially filled tetragonal tungsten bronze (TTB) space group P4bm (at room temperature) melting point 1430 ˚C Ferroelectric transition Tc = 197 ˚C (4mm ↔ 4/mmm) optical quality Blood color – transparent inclusion free single crystal a (Å) 12.458(4) c (Å) 3.9514(9) V (Å) 613.32(4) l1 (mm) 4.546(1) 4.410(2) l2 (mm) 4.240(2) 4.421(1) l3 (mm) ║ c-axis 4.312(1) 4.768(1) b (gm/cc) 5.310 (2) g (gm/cc) 5.307(3) 5.302(2) Chapter 6 Elastic properties of Ce:Ca0.28Ba0.72Nb2O6 … 85 ______ The ferroelectric phase transition (4mm ↔ 4/mm) temperature obtained from differential scanning calorimetry was found to be 197 ˚C which is ~ 70 ˚C lower than for pure CBN-28. The melting temperature was observed at 1430 ˚C. In contrast to pure CBN-28, no hint of a color change was observed during annealing of the crystals in He atmosphere up to its melting point. It might be possible that the color change due to annealing was not visible because of its dark red color or it was too small to be detected by either naked eyes or by the microscope. 6.1 Comparison of pure and Ce doped CBN-28 Elastic properties of Ce doped CBN single crystal was studied in the paraelectric phase and the thermal expansion in the temperature range of 100 – 1323 K. In order to compare CBN and Ce:CBN, possessing more or less same type of phase transition, it will be wise to plot the properties with respect to the renormalized temperature scale of T-Tc. In almost all the cases these properties match quite well showing the same behavior for pure and Ce doepd CBN as shown below: 6.1.1 Thermal expansion Calibration corrected strains parallel and perpendicular to crystallographic c- axis together with their linear thermal expansion coefficient are shown in Fig. 6.2a,b and the individual values of the linear thermal expansion of selected temperatures are listed in Table 6.2. Temperature evolution of strains for CBN and Ce:CBN on a renormalized temperature scale is shown in Fig 6.2c. As can be seen from Fig 6.2c, the behavior of the thermal expansion of Ce:CBN is identical to those of pure CBN. TABLE 6.2: Coefficients of linear thermal expansion ij and of volume thermal −6 −1 expansion ij 2 3311 in 10 K . Uncertainty of the last decimal is in parenthesis. T 173 373 483 743 863 1023 1263 1323 11 7.8(1) 16.2(0) 14.9(2) 12.0(4) 11.0(1) 9.2(1) 8.8(1) 9.8(1) 33 1.8(8) -7.4(1) -6.0(1) 10.8(4) 8.8(1) 6.6(1) 6.8(1) 9.4(5) V 17.4 25.0 23.9 34.9 30.8 25.0 24.4 29.2 86 Elastic properties of Ce:Ca0.28Ba0.72Nb2O6 … Chapter 6 ______ 14 Ce:CBN 12 CBN 10 ] 8 -3 6 4 Strain [10 2 0 -2 -4 -600 -400 -200 0 200 400 600 800 1000 1200 T-T [K] c FIG. 6.2: Temperature evolution of strain and thermal thermal expansion coefficients (a) perpendicular and (b) parallel to the 4 fold- axis of Ce:CBN, (c) comparison of temperature dependence of the strains parallel and perpendicular to the crystallographic c axis for CBN (blue color) and Ce:CBN (red color). Chapter 6 Elastic properties of Ce:Ca0.28Ba0.72Nb2O6 … 87 ______ Consequently, the phase transition mechanisms are the same, except of the fact that Ce doping shifts the phase transition temperature by ~ 70 K to lower temperature. There all the interpretation and conclusions drawn in Chapter 5 for the thermal expansion of CBN also applies here to Ce:CBN. The Burns temperature TB for Ce:CBN is equals to 900 ± 10 K. 6.1.2 Elastic properties The elastic properties of Ce:CBN were studied in the paraelectric phase on sample Ce:CBN-01 (Fig. 6.1) employing resonant ultrasound spectroscopy. The spectra were recorded in intervals of 40 K in air. A part of the resonance spectra is shown in Fig 6.3. The ferroelectric phase transition temperature (Tc ~ 197 ˚C) and the paraelectric phases can easily be identified in Fig. 6.6. At least 80 eigen-frequencies in the range 200 kHz - 1400 kHz were extracted from each resonance spectrum and used in non-linear least-squares refinements minimizing the quantity 2 . The initial model of elastic constants at 1323 K was taken directly from pure CBN as reported in chapter 5. FIG. 6.3: A part of the resonance spectra taken on Ce:CBN sample. 88 Elastic properties of Ce:Ca0.28Ba0.72Nb2O6 … Chapter 6 ______ In order to check the reproducibility of the resonance spectrum, two resonance spectra (heating-cooling) were collected at elevated temperatures. Fully reproducible resonance spectra were observed. The elastic constants calculated from the observed resonance spectra are shown in Fig 6.7 a,b,c with pure CBN on T-Tc scale. The individual values of the elastic constants of Ce:CBN together with their temperature derivatives at selected temperatures are given in Table 6.3. Temperature evolution of the elastic constants of Ce:CBN follows CBN as can be seen from Fig 6.4. Therefore, the discussion in Ce:CBN will remain the same like for CBN except that for c66. The reason behind is not clearly understood so far, a focused study is required. TABLE 6.3: Elastic properties of Ce:CBN at selected temperatures. Units: temperature T in K, elastic constants cij and bulk modulus B in GPa, ratio of maximal and minimal longitudinal elastic stiffness max cc min is dimensionless, and −4 −1 thermoelastic constants Tij = d log cij/dT in 10 K . Numbers in parentheses denotes the uncertainty of the last digit according to the standard deviation of the corresponding parameter as obtained by by the averaging procedure. T 483 743 863 1023 1263 1323 c11 219.0(3) 233.3(2) 234.0(1) 233.1(3) 226.6(1) 224.6(1) c33 147.1(5) 173.4(4) 186.4(4) 188.8(2) 187.0(2) 186.9(2) c44 69.0(3) 69.8(1) 69.6(3) 69.0(2) 67.0(1) 66.5(2) c66 62.9(3) 76.2(4) 76.9(4) 76.4(3) 74.3(5) 73.6(2) c12 93.0(4) 80.9(3) 80.0(2) 80.3(2) 77.9(3) 77.4(1) c13 67.2(2) 66.2(1) 68.0(3) 69.2(5) 69.4(2) 69.6(4) B 111.7 114.9 119.5 119.1 118.0 117.8 max cc min 1.64 1.35 1.25 1.22 1.21 1.19 T11 6.4(2) 0.3(6) 0.1(4) -0.6(6) -1.5(6) -1.8(2) T33 16.8(2) 4.3(2) 2.5(3) -0.1(4) -0.1(7) -0.0(9) T44 1.9(9) 0.0(4) -0.4(2) -0.9(8) -1.1(6) -1.1(3) T66 22.6(4) 1.4(3) 0.1(5) -0.8(2) -1.4(4) -2.0(2) T12 -14.0(2) -0.7(3) -0.4(8) -0.3(6) -1.8(4) -0.4(4) T13 -12.2(3) 2.5(0) 1.6(7) 0.3(6) 0.0(2) 0.9(6) A quantitative comparison of elastic parameters is made for pure and Ce doped CBN at 743 K as shown in Table 6.4. Temperature evolution of the mean elastic stiffness ‘C’ is shown in relative scale together with pure CBN in Fig 6.5. Chapter 6 Elastic properties of Ce:Ca0.28Ba0.72Nb2O6 … 89 ______ 240 220 c11 200 c 180 33 160 140 Ce:CBN Longitudinal Elastic Stiffness [GPa] Longitudinal Elastic Stiffness CBN 120 -200 0 200 400 600 800 1000 1200 T-T [K] a) c 80 76 c66 72 68 c44 Shear stiffnesses [GPa] 64 Ce:CBN CBN 60 -200 0 200 400 600 800 1000 1200 T-T [K] b) c 105 100 95 90 c coefficients [GPa] coefficients 85 12 80 75 c13 70 65 60 Ce:CBN CBN Transverse interaction 55 -200 0 200 400 600 800 1000 1200 T-T [K] c) c FIG. 6.4: (a) Temperature dependence of longitudinal elastic stiffnesses, (b) shear stiffnesses, and (c) transverse interaction coefficients of Ce:CBN in the paraelectric phase. 90 Elastic properties of Ce:Ca0.28Ba0.72Nb2O6 … Chapter 6 ______ TABLE 6.4: Elastic properties of pure CBN and Ce:CBN at 743 K. Units: temperature T in K, elastic constants cij, bulk modulus B and mean elastic stiffness in GPa. ND – no difference. Parameters CBN Ce:CBN % change c11 223.3(2) 233.3(1) + 4.5 c33 159.4(3) 173.4(2) + 8.8 c44 69.7(1) 69.8(2) ND c66 76.6(2) 76.2(2) ND c12 89.0(2) 80.9(1) - 9.1 c13 57.1(1) 66.2(4) + 15.9 B 107.6 114.9 + 6.8 C 113.9 118.8 + 4.3 122 120 118 116 114 112 Mean [GPa] elastic stiffness Ce:CBN CBN 110 -200 0 200 400 600 800 1000 1200 T-T [K] c FIG. 6.5: Temperature evolution of the mean elastic stiffness of CBN and Ce:CBN. Anisotropy of longitudinal elastic stiffness is shown in Fig. 6.6 for Ce:CBN at temperatures 543 K and 1323 K together with pure CBN. Chapter 6 Elastic properties of Ce:Ca0.28Ba0.72Nb2O6 … 91 ______ 603K, CBN 1323K, CBN 543K, Ce:CBN 1323K, Ce:CBN FIG. 6.6: Projection of the representation surface of the longitudinal elastic stiffness. 6.1.3 Deviation from Cauchy relations The anisotropy of the bonding type, i.e. the deviation from Cauchy relations, is shown in Fig. 6.7 (a) at 543 K, (b) at 1323 K for Ce:CBN and its comparison with CBN is shown in Fig 6.8. FIG. 6.7: The longitudinal effect of the deviation from Cauchy relations at T = 543K (left) and 1323 K (right). 92 Elastic properties of Ce:Ca0.28Ba0.72Nb2O6 … Chapter 6 ______ 40 30 g33 20 10 [GPa] ij g 0 -10 g11 CBN:Ce CBN -20 200 400 600 800 1000 1200 1400 1600 Temperature [K] FIG. 6.8: A comparison graph of the deviation from Cauchy relations for pure and cerium doped CBN with respect to temperature. 6.1.4 Isotropic elastic properties Temperature evolution of the Young’s modulus E, bulk modulus B, Poisson’s ratio ν, longitudinal and shear wave velocities (calculated from isotropic elastic constants) together with Debye temperature of Ce:CBN is shown in Fig. 6.9 on a recalibrated scale. For making a comparison, the corresponding properties of CBN are also plotted in Fig 6.9. As can be seen from Fig 6.9 the temperature dependent behavior of all above mentioned properties for Ce:CBN matches quite well with pure TABLE 6.5: Calculated isotropic elastic constants, mean sound velocities and Debye temperatures of pure CBN-28 and Ce:CBN single crystal at 1323 K. Parameter [Units] CBN Ce:CBN Bulk modulus [GPa] 117.6(2) 117.8(1) Shear modulus [GPa] 68.1(2) 68.2(1) Poisson’s ratio 0.25(8) 0.25(3) Average velocity [km/sec] 4.54(2) 4.56(1) Debye temp [K] 561.2 ± 0.4 561.4 ± 0.1 Chapter 6 Elastic properties of Ce:Ca0.28Ba0.72Nb2O6 … 93 ______ CBN. Consequently, all explanations as described in Chapter 5 for pure CBN will also apply here. Doping of Ce only shifts the phase transition temperature. A direct comparison of the average wave velocity and Debye temperature for pure and Ce doped CBN-28 can bee seen from Table 6.5. 180 12 176 11 172 (x10)[GPa] 168 K 10 164 0.26 160 Young's modulus [GPa] 0.24 156 Ce:CBN Ce:CBN CBN CBN 0.22 152 -200 0 200 400 600 800 1000 1200 -200 0 200 400 600 800 1000 1200 T-T [K] [K] c T-Tc 6.5 580 Longitudinal 570 6.0 560 550 Shear 3.6 540 Debye temperatureDebye [K] Sound velocity[km/sec] 530 3.4 Ce:CBN Ce:CBN CBN CBN 520 0 200 400 600 800 1000 -200 0 200 400 600 800 1000 1200 T-T [K] T-T [K] c c FIG. 6.9: Temperature evolution of (a) the Young’s modulus, (b) the bulk modulus, and Poisson’s ratio, (c) wave velocities, and (d) Debye temperature for undoped (blue squares) and Cerium doped (red circles) CBN-28. 6.1.5 Eigen modes behavior The temperature evolution of selected eigen modes of Ce:CBN are shown in Fig. 6.10 together with their inverse quality factors. It is clear form Fig 6.10 that the behavior of the modes is continuously distributed between two extremes and can not simply be divided in two groups as in the case of CBN. Below ~ 1000 K modes B2u and Ag-1 show rapid softening (one extreme case), however, modes B1u and B3g show very small softening near the transition temperature (second extreme case), the 94 Elastic properties of Ce:Ca0.28Ba0.72Nb2O6 … Chapter 6 ______rest of the modes Au-1, Au-2 and Ag-2 show intermediate softening between these two extremes. 1.04 Au-1 25 1.02 B2u Tc Ag-1 1.00 20 Ag-2 B1u 0.98 B3g ] 15 0.96 Tc -4 Au-2 Au-1 [10 -1 0.94 B2u Q 10 Ag-1 f (T) / f (1323K) 0.92 Ag-2 5 0.90 B1u B3g 0.88 Au-2 0 200 400 600 800 1000 1200 1400 200 400 600 800 1000 1200 1400 Temperature [K] Temperature [K] FIG. 6.10: Temperature evolution of frequency (a) and inverse quality factor (b) of selected eigen modes of the Ce:CBN sample. The modes Ag-1 and B2u soften significantly when approaching the phase transition and do not show significant increase in the Q-1 value before or after the phase transition. The reason for this is as follows: the shear resistances do not -1 -1 contribute to the Q in CBN; the Q goes up because c33 damped. However, in Ce:CBN modes Ag-1 and B2u show strong softening which is probably due to strong softening of c66. As c66 is not involves in damping therefore these modes are also not damped and hence the B2u mode can still be observed in the low temperature ferroelectric phase. The overall behavior of the resonances is strongly frequency dependent which confirms the relaxor behavior of Ce:CBN. 6.2 Hypothesis for the Ce position in Ce:CBN The position of Ce in doped CBN is a question of high interest. The EMP analysis data for pure and Ce doped CBN-28 is given in Table 6.6. One can clearly see that the Ca and Nb contents are reduced in Ce doped CBN-28. This gives a hint that Ce substitutes both Ca and Nb at the A1 and B sites, respectively. Unfortunately, from EMP analysis it is not possible to distinguish between Ce3+ or Ce4+ and to identify on which sites Ce is located. Chapter 6 Elastic properties of Ce:Ca0.28Ba0.72Nb2O6 … 95 ______ TABLE 6.6: Chemical composition of the pure and Cerium doped CBN-28 after EMPA Ce:CBN CBN Elements Atoms pfu Ce 0.024 - Ca 0.251 0.279 Ba 0.720 0.720 Nb 1.996 2.000 O 6.000 6.000 Further, to identify the exact position of the Ce in Ce:CBN a single crystal structure determination of Ce:CBN has been performed on a four-circle kappa diffractometer (for details see Chapter 3). The crystal structure determination confirms the TTB structure type and CBN structure type. However, the position of the Ce could not reliably be detected. It was impossible to distinguish the atoms with X- rays due to their atomic form factor, i.e. scattering strength of each atoms for X-rays. The scattering strength of the Ce is quite similar to that of Ba and Nb, so there is no contrast in the X-ray scattering strength between these atoms. Also, the amount of Ce is very small. Hence, the exact position of Ce was not clear from the single crystal X- ray structure determination. Therefore, one hypothesis is proposed for the exact position of Ce in Ce:CBN as follows: In well known photorefractive strontium barium niobate the location of rare- earth metal ions in the crystal lattice is not known till date. It was shown that the Ce in strontium barium niobate lattice predominantly occurs in the valence state 3+ [32, 139]. The instrumental neutron activation analysis (INAA) of SBN shows that Ce3+ ions are most likely replace Sr and/or Br at the A site [140]. The uncertainty in the location of the Ce ions is because of the Sr ions are statistically distributed over two structural positions [141]. Ce3+ is stable on A (A1 and A2) sites, however, gets ionized if substituted on the Nb site, as per the reaction [142]: Ce3+ + Nb5+ → Ce4+ + Nb4+ (6.1) The existence of Ce is found in both forms (Ce3+ and Ce4+) by photorefractive studies also [143]. A more than 90 % trivalent charge state for Ce has been found by FIR [144]. In a research on Ce+Cr doped SBN it was shown that Ce3+ ions occupy A sites and Cr3+ ions substitute Nb5+ ions [140], however, the charge compensation mechanism has not been solved yet. 96 Elastic properties of Ce:Ca0.28Ba0.72Nb2O6 … Chapter 6 ______ FIG. 6.11: Crystallographic site preference of a dopant and the range of spectral response in TTBs [145]. Keeping all the above said factors in mind, the following hypothesis can be proposed for the position of Ce dopant inside Ce:CBN lattice. In CBN only five of the six A sites are filled while C sites are empty. Therefore, the partially filled structure of CBN offers plenty of space to accommodate various dopants. It is well known that inside the CBN, the A1 and A2 sites are solely occupied by the Ca2+ and Ba2+ ions, respectively. Our assumption is that inside the Ce:CBN both the Ce3+ and Ce4+ ions are present in roughly 90:10 ratio, respectively. It is more reasonable to assume that 3+ inside Ce:CBN the Ce ions, which have a ionic radii [146] (ri) of 1.02 Å, most 2+ 2+ 5+ probably will replace Ca (ri = 1.00 Å) rather than Ba (ri = 1.34 Å) and Nb (ri = 4+ 4+ 0.69 Å). However, Ce (ri = 0.87 Å) will replace Nb (ri = 0.82 Å). Entries for substitution in the interstitial site C are not possible because each impurity ions is significantly less stable at this site than at the A sites [142]. Similar to Ce doped SBN the charge disorder in Ce:CBN is still a challange. The origin of the dark red color in Ce:CBN can also be explained by the above said hypothesis as follows: An unproved research states that the presence of both Ce3+ and Ce4+ in SBN primarily responsible for the crystal color. Ce4+ is red, whereas Ce3+ are usually white or colorless [147]. Doping with Ce3+ on the 15- or 12- fold coordinated Ba2+ or Sr2+ sites of SBN produces pink-colored crystals with spectral response in the visible region. In addition to that, doping of Ce3+ raised the coupling Chapter 6 Elastic properties of Ce:Ca0.28Ba0.72Nb2O6 … 97 ______coefficient to 45 cm-1 and 20 cm-1 in SBN wafers and crystal cubes, respectively [145]. However, placing the Ce3+ in the 9-fold coordinated site shifts the spectral response from visible to near-IR (0.78 μm) with coupling merely of about 6-7 cm-1 [145]. Placing of Ce3+ in lower coordination sites (6-fold coordinated Nb5+ sites) shifts the spectrum region to longer wavelengths. Although Ce3+ is known to occupy the 6-fold site in perovskites, the placement of Ce3+ in the TTB 6-fold coordinated site requires blocking of the 12-, 15- and 9-fold sites. Litrature reports [20], when Ce3+ occupies the 12-fold coordinated site, the spectral absorption shifts to the visible region and this shifts to the near-IR when Ce3+ is forced into the C site. A relationship between site preference of dopant and corresponding shift in spectral range for TTBs is illustrated in Fig. 6.11. Considering all above said literature, the hypothesis for the color of Ce:CBN is: inside Ce:CBN both Ce3+ and Ce4+ are present. EMP analysis on Ce:CBN shows that ~ 85% Ce is in the form of Ce3+ and ~ 15 % as Ce4+. As Ce3+ has similar ionic radius as of Ca2+ hence will fit to the Ca site. However, Ce4+ will fit to Nb4+ site because of similar ionic radii. The small occupancy of Ce4+ on the B site might result in the dark red color of Ce:CBN. This explanation is in good agreement with the proposed hypothesis for Ce position in Ce:CBN given in section 6.1. However, a detailed investigation is needed to confirm whether this is the case. 6.3 Summary and conclusions All independent elastic constants of Ce doped CBN were obtained in the paraelectric phase (over the temperature range of 470 – 1323 K) for the first time. Cerium acts as a moderate dopant resulting in a significant lowering of the Curie temperature by ~ 70 °C. However, the behavior of the thermal expansion and the elastic properties with respect to temperature is almost similar to that of pure CBN. As the position of Ce dopant was not clear from single crystal structure determination and EMP analysis, one hypothesis was proposed for the cerium position in Ce:CBN which is strengthened by the proposed reason for the red color of Ce:CBN crystal. Chapter 7 Summary and outlook Temperature and compositional dependence of the elastic properties of tourmaline and calcium barium niobate single crystals was studied in detail for the first time. To this end the non-destructive innovative method of resonant ultrasound spectroscopy was employed. Tourmalines The elastic, piezoelectric and dielectric constants of tourmalines were determined successfully in the temperature range 100 – 1000 K. The different chemical compositions of the investigated crystals allow for a systematic study of the main influence of Li and Fe on elastic and piezoelectric properties of tourmalines. The structural formula of all samples has been properly determined after an extensive chemical and X-ray study. Systematic variations in the elastic properties were observed with the influence of the iron and lithium content on the Y-site. Increasing iron content (simultaneously decreasing Li content) leads to an increase in c11, however, decreases c44. Up to 1000 K the temperature evolution of the elastic constants is almost linear without any hint of a phase transition. The dehydration temperature is dependent on chemical composition. Iron rich tourmalines dehydrate earlier than lithium bearing tourmalines. Isotropic elastic properties have also been calculated. A correlation of mean elastic velocities with chemical composition was observed which is in a very agreement with the values reported by other authors. The dielectric constants of all tourmalines show a distinct anisotropy characterized by є11 > є33. The єij values at room temperature are almost frequency 100 Summary and outlook Chapter 7 ______ independent up to 1 MHz. A correlation between dielectric constants and chemical composition has been established for the first time. The tourmalines which belong to the schorl-dravite group show relatively smaller є33 then the tourmalines which belong to the elbaite-schorl group, and hence show relatively large electro-mechanical coupling effects. Moreover, the tourmaline crystals belonging to the elbaite-schorl solid solution series show an irreversible softening at about 600 K in its first heating which is probably related to cation ordering/disordering processes. Annealing up to 900 K is a prerequisite to overcome this problem. Additional experiments are needed in order to understand this phenomenon. In general, tourmalines show only very small elastic and piezoelectric temperature coefficients over an extremely broad temperature range of more than 900 K. Therefore they exhibit excellent high-temperature stability. From this study, tourmalines belonging to the schorl-dravite solid solution series are found to be the best choice for device applications because of their relatively high coupling coefficients. Pure and Ce doped CBN-28 The elastic constants of pure and Ce doped CBN-28 single crystals in the paraelectric phase and temperature range of 470 – 1503 K were determined successfully. In the paraelectric phase high-Q resonances were observed yielding excellent results. Burns temperature is clearly visible in anomalies of thermal expansion and elastic constants which give evidence for the existence of the polar nano-regions in the paraelectric phase of CBN. Doping of Ce in CBN-28 stiffens the material and lowers the Curie temperature. The behavior of cij (also valid for αij) on a reduced temperature scale was observed to be very similar to that of pure CBN in the paraelectric phase. However, the reason for the strange behavior of c66 was not clear. Investigation of some new CBN materials with varying dopant amounts might be helpful to understand this behavior. Below the Curie temperature the interactions between the elastic waves and domain walls are too strong to observe enough resonances to refine elastic constants. This makes CBN crystals in its raw (as-grown) form unsuitable to study their elastic properties in the ferroelectric phase. 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My deepest gratitude goes to him for giving me sufficient freedom in this project. I am greatly indebted to Prof. Dr. Ladislav Bohatý, University of Cologne, for being the co-supervisor of my Ph. D. thesis. I also thank him for providing me dielectric measurement facilities and related fruitful discussions. Many thanks to the DGR&D and the Director SASE for granting me EOL for pursuing my Ph.D work. I owe a special word of thank to Dr. R. N. Sarwade, Mr. A. Ganju, Mr. D. N. Sethi and Mr. D. K. Prashar. I would like to thank Prof. Dr. Manfred Mühlberg, University of Cologne, for providing me CBN-28 and Cerium doped CBN-28 single crystals and related fruitful discussions. Many thank goes to Dr. Sven Jodlauk for his invaluable help in the measurements of dielectric constants. I thank to Prof. Dr. Joachim Zang, Brazil, for providing some tourmaline single crystals of high optical quality. I am thankful to Dr. Heinz-Jürgen Bernhardt for EMPA facility and valuable lectures. It has been a pleasure to discuss with him many aspects of EMPA in a friendly and relaxed atmosphere. I am grateful to Dipl.-Ing. Philipp Nörtershäuser for helping me in the alignment of Ce-CBN samples using the Laue facility in the group of Prof. Dr.-Ing. Gunther Eggeler, Ruhr-Universität, Bochum. 112 Acknowledgements ______ I am thankful to Dr. Olaf Medenbach for his help in making conoscopic images and valuable suggestions. Furthermore, I would like to recognize many valuable contributions from Dr. Heribert Graetsch, Dr. Thomas Reinecke, Dr. Thomas Fockenberg Mr. H. Mamen, Ms Nadine, Mrs Kristin, Mrs Sandra., Mr. Tobias, and Mr. Thomas. My sincere thanks go to Prof. Dr. Herman Gies for his valuable advice in many administrative queries, especially for diffraction lectures in english. I would like to thank Mr. Udo Trombach for solving various minor and major technical problems with his skilful knowledge. Special thanks to my colleagues Thomas Krenzel, Dr. Peter Sondergeld, Felix, Sarah, Kathrin, Pia, Dr. Maren Kahl and Dr. Sara Fanara for distinct helpfulness and for maintaining a wonderful atmosphere. I want to acknowledge to Mrs. Rita Bode, Mrs. Agnes Otto for their kind help on many administrative issues. Special thanks go to Mrs. Birgit Wieser-Brotte for her administrative support. Partial funding from the research department Integrity of Small-Scale Systems / High-Temperature Materials (IS3/HTM) (now Material Research Department) is gratefully acknowledged. On personal level, no words to thank to my respected father Late Sri Narayan Datt Ji Pandey for his affection and placing much importance on my education. I thank to my mother, all my sisters and brothers-in-law for their help and moral support during this research work. I really don’t know how to say thanks to my loving daughter Aditi, who scarifies her childhood in the kindergarten starting at an age of merely nine months because of my research work. Loving Aditi, please forgive me for not giving too much time to you during this research work. Finally, I would like to thank Aditi’s mom, my best friend, my love, my inspirational source, and above all a vary caring & loving wife Dr. Arti Dangwal- Pandey for her unending love, moral support, scientific discussions and encouragements. No words to say a special thanks to Arti especially at times when I had to work over night, weekends and during entire Christmas holidays. She always motivated me to work harder and to do my level best. Arti, I am fortunate to have you. Thank you ! Curriculum Vitae Name: Chandra Shekhar Pandey Date of Birth: December 3, 1976 Place of Birth: Bareilly, India Nationality: Indian Sex, Marital status: Male, Married Email Address [email protected] EDUCATION 1999 – 2001 M. Tech. in Cold Region Science and Engineering, “An Estimation of the Optimum Horizontal Grid Resolution for the Avalanche Prone Areas of the Jammu & Kashmir and Himachal Pradesh” G. B. Pant University of Ag. and Tech., Pantnagar, India, CGPA : 7.7/10, 4th Rank. 1995 - 1997 M. Sc. in Physics Bareilly College (Rohilkhand University) Bareilly, India, Score: 65.5 % 1992 - 1995 B. Sc. in Physics, Mathematics, and Chemistry Bareilly College (Rohilkhand University) Bareilly, India, Score: 72.1 %, Topper in Physics FELLOWSHIPS AND AWARDS 1999 - 2001 Scholarship from the Ministry of Defense, Govt. of India 1996 Gold medal for best student (M. Sc.) 1996 Scholarship for pursuing Master studies in Physics 1995 College award for topper in Physics (B. Sc.) PUBLICATIONS 1. A. Dangwal, C. S. Pandey, G. Müller, S. Karim, T. W. Cornellius, and C. Trautmann, ‘Field emission properties of electrochemically deposited gold nanowires’, Appl. Phys. Lett. 92, 063115 (2008). Manuscript under Preparation: 2. Chandra Shekhar Pandey, Jürgen Schreuer, Manfred Burianek, Manfred Mühlberg. “Anomalous elastic behavior of Ca0.28Ba0.72Nb2O6 single crystal in the paraelectric phase”. 3. Chandra Shekhar Pandey, Juergen Schreuer, Manfred Burianek, Manfred Muehlberg, “Anomalous thermo-elastic properties of Ce:CBN single crystals”. 4. Chandra Shekhar Pandey and Jürgen Schreuer, “Temperature dependence of elastic and dielectric properties of Beryl”. 5. Chandra Shekhar Pandey, Sven Jodlauk and Jürgen Schreuer, “Compositional dependence of the dielectric properties of tourmaline single crystals”. 6. Chandra Shekhar Pandey and Jürgen Schreuer, “Compositional dependence piezoelectric and anomalous elastic properties of tourmalines at non-ambient temperatures”. CONTRIBUTION TO CONFERENCES/ WORKSHOPS 1. Chandra Shekhar Pandey, Jürgen Schreuer, Manfred Burianek, Manfred Mühlberg; The 19th IEEE International Symposium on the Applications of Ferroelectrics (ISAF- ECAPD-2010), August 9-12, Edinburgh, Scotland, UK: ‘Anomalous Elastic Behaviour and Thermal Expansion of Cerium Doped Ca0.28Ba0.72Nb2O6 Single Crystal’, (ORAL). 2. Pandey C. S., Schreuer J.; 2010 IEEE International Frequency Control Symposium (IFCS 2010), June 1-4, Newport Beach, California, USA: ‘Elastic Anomalies in Tourmalines’, (Poster). 3. Chandra Shekhar Pandey, Jürgen Schreuer, Manfred Burianek, Manfred Mühlberg: 26th European Crystallographic Meeting (ECM 26) Aug 29 – Sep 2, 2010, Darmstadt, GERMANY: ‘Elastic anomalies and precursor effects in CBN and Ce:CBN relaxor ferroelectrics’, (ORAL). 4. Chandra Shekhar Pandey, Jürgen Schreuer; 26th European Crystallographic Meeting (ECM 26) August 29 – September 2, 2010, Darmstadt, GERMANY: ‘Elastic anomalies and electromechanical properties of tourmalines’, (Poster). 5. Chandra Shekhar Pandey, Jürgen Schreuer; 88 Jahrestagung der Deutschen Mineralogischen Gesellschaft, (DMG 88) September 19–22, 2010, Münster, GERMANY: ‘Compositional dependence of elastic, piezoelectric and dielectric properties of tourmalines’ (ORAL). 6. Chandra Shekhar Pandey, Jürgen Schreuer, Manfred Burianek and Manfred Mühlberg.; International Workshop on Materials Discovery by Scale-Bridging High- Throughput Experimentation and Modelling, 23–24 November 2010, Bochum, GERMANY: ‘Anomalous Elastic Behaviour and Thermal Expansion of Cerium Doped Ca0.28Ba0.72Nb2O6 Single Crystal’, (Poster). 7. C. S. Pandey, J. Schreuer, M. Burianek and M. Mühlberg. The 18th IEEE International Symposium on the Applications of Ferroelectrics (ISAF 2009), August 23-27, Xian, CHINA: ‘Thermal Expansion and Elastic Properties of CaxBa1-xNb2O6 Single Crystals up to 1500 K’, (ORAL). 8. Chandra Shekhar Pandey, Jürgen Schreuer. 2009 IEEE International Ultrasonics Symposium (IUS 2009), September 19-23, Rome, ITALY: ‘Electromechanical Properties of Tourmaline Single Crystals up to 1050 K’, (Poster). 9. Pandey, C. S., Schreuer, J., Jodlauk, S.; 17 Jahrestagung der Deutschen Gesellschaft für Kristallographie (DGK 2009), March 9–12, Hannover, GERMANY: ‘Temperature Dependence of Electromechanical Properties of Single-Crystal tourmaline’, (ORAL). 10. C. S. Pandey, J. Schreuer, M. Burianek and M. Mühlberg; Materials for Harsh Environments & 7 Materialwissenschaftlicher Tag der Ruhr-Universität Bochum (RD IS³/HTM Workshop), 12-13 November 2009, Bochum, GERMANY: ‘Thermal Expansion and Elastic Properties of CaxBa1-xNb2O6 Single Crystals up to 1500 K’, (Poster). 11. Chandra Shekhar Pandey, Jürgen Schreuer, Manfred Burianek and Manfred Mühlberg; Materials Day, 21 Nov 2010, Bochum, Germany: ‘Anomalous Elastic Behaviour and Thermal Expansion of Cerium Doped Ca0.28Ba0.72Nb2O6 Single Crystal’, (Poster). 12. Chandra Shekhar Pandey, Jürgen Schreuer, Manfred Burianek and Manfred Mühlberg; International Workshop on Materials Discovery by Scale-Bridging High-Throughput Experimentation and Modeling, 22-23 Nov 2010, Bochum, Germany: ‘Anomalous Elastic Behaviour and Thermal Expansion of Cerium Doped Ca0.28Ba0.72Nb2O6 Single Crystal’, (Poster). 13. Dangwal, A., Pandey, C. S., Mueller, G., et al.; 20th International Vacuum Nanoelectronics Conference (IVNC 2007), Chicago, USA: ‘Field Emission properties of gold nanowire cathodes based on polymer ion-track membranes’, (Oral). EXPERIMENTAL SKILLS Characterization techniques: Structural analysis : SEM, EDX, XRD Chemical analysis : EMPA Thermal analysis : DSC, Dilatometer, DTA Stochiometric analysis : FTIR Electrical characterization : FE measurements in UHV, C-V Measurements Microelectronics Fabrication: Thin film Deposition : Ion Beam Sputtering Lithography : Electron Beam and Optical lithography COMPUTER SKILLS LabVIEW 8.5, AutoCAD, Maple-13, MATLAB 2007, FORTRAN, and C++. JOB EXPERIENCE Research Scientist (2010- till present): Material Research Department (MRD), Ruhr- University, Bochum, Germany. Research group of Prof. Dr. Jürgen Schreuer. Research Scientist (2007 – till present): Institute of Geology, Mineralogy and Geophysics, Crystallography, Ruhr-University, Bochum, Germany. Research group of Prof Dr. Jürgen Schreuer Research Scientist (2006 – 2007): Experimental Physik-IV, Research group of Prof. Dr. Dr. h.c. H. Zabel, Ruhr-University, Bochum, Germany. Scientist (2002 – 2006): Research Scientist, Chandigarh, India. Presently on leave (EOL). Lecturer (2001 – 2002): Physics lecturer for master and bachelor students in a degree college affiliated to M.J.P. Rolhilkhand University, Bareilly, India. Lecturer (1997 – 1999): Physics lecturer for master and bachelor students in a degree college affiliated to M.J.P. Rolhilkhand University, Bareilly, India. SPECIAL CONTRIBUTION Developed one cryostat of 18mm diameter (operated at 4K) for low temperature MOKE studies. Please visit: http://www.ep4.ruhr-uni- bochum.de/imperia/md/content/publikationen/jahresberichte/2006/jabi_2006.pdf PROFESSIONAL MEMBERSHIP Institute of electrical and electronics engineers (IEEE) Ultrasonics, Ferroelectrics and Frequency Control (UFFC) EDITOR OF PROFESSIONAL/Non-PROFESSIONAL SOCIETIES Web editor : Ultrasonics (UFFC) Editor (English) : University college magazine LANGUAGE SKILLS English : Fluent German : Moderate Hindi : Mother tongue Declaration I, Chandra Shekhar Pandey, hereby declare that the present work has been done and written independently without any unauthorized help, and that the work was not submitted in this or any other form to any faculty or another university. Bochum, 20.10.2010