CRYSTALLOGRAPHY IN SWITZERLAND 562 CHIMIA 2001,55, No, 6 Chimia 55 (2001) 562-569 © Schweizerische Chemische Gesellschaft ISSN 0009-4293 Exploration of Structure-Property Relationships J(jrgen Schreuer* Abstract: Electrostriction and elasticity provide highly sensitive probes forthe exploration of structure-property relationships. With the aid of modern high-resolution capacitive dilatometers reliable linear and quadratic electrostrictive constants can be determined. The unusual thickness dependence of the converse piezoelectric effect in a-quartz and the electrostrictive behavior of the alkali halides are presented as examples. The close correlations between elastic properties and crystal structures are reflected by several heuristic and empirical rules. Resonant ultrasound spectroscopy (RUS)belongs to the most promising methods for the determination of elastic constants in the acoustic frequency regime. The potential of RUS for the simultaneous investigation of elastic and electrostrictive properties including electromechanical coupling effects is discussed. Keywords: Crystal physics' Elasticity' Electrostriction . Piezoelectricity' Structure-property relationships 1. Introduction was the advent of modern computers that known crystal structure. In the case of eventually made ab initio calculations higher order effects the situation is even Since the establishment of the first lattice feasible. A detailed overview of compu- more dramatic. Among the different rea- theoretical models about 100 years ago, tational methods in crystallography and sons for the large discrepancy between the interpretation of structure-property mineralogy has been recently given by importance and experimental knowledge relationships ranks among the most im- Winkler [1]. of macroscopic physical properties, the portant tasks in solid-state sciences. Besides the fascination for this devel- lack of high quality crystals of sufficient Models possessing high predictive power opment one should bear in mind that the size is certainly the most important. Ad- can help, for instance, to provide struc- interpretation of properties is not exclu- ded hindrances against a broader investi- tural and physical data of solids under sively the ability to predict properties gation of crystal properties are the high extreme conditions which are not acces- from first principles, but rather to reveal technical effort for development and sible by experiment, and to tailor new the relations between chemical composi- maintenance of the mostly self-made ex- materials with special properties. The tion, structure and properties of a broad perimental set-ups, and the high expendi- different modeling approaches either variety of crystalline materials. In this ture of time on preparation and measure- make use of transferable force fields de- context one has to face the following ment of samples. termined by certain adjustable parame- questions: (i) Is the present amount of As second order derivatives of ther- ters or rely on first principles calcula- available experimental data sufficient for modynamical potentials the elastic stiff- tions. While the advantage of empirical the derivation of general relations and nesses are directly connected with quan- models lies in their high computational rules? (ii) Which important properties tities in which the lattice energy per unit efficiency, the often poor accuracy and will not be easily accessible by calcula- volume is involved. The Christoffel de- the limited availability of transferable po- tions in the near future? (iii) Is there a terminants combine the elasticity tensor tentials make the use of quantum me- great difference between a crystal de- with the propagation velocities of sound chanical methods desirable. Although the scribed by computer simulation and a waves and related properties such as De- basic concepts were developed in the real macroscopic individual? These top- bye temperature, thermal conductivity, twenties and thirties of the last century, it ics have been controversially discussed thermal expansion (Griineisen relation), on two workshops [2] organized by the melting temperature (Lindemann formu- Laboratory of Crystallography of the la), infrared resonance frequencies and ETH Zurich. In order to illustrate the mo- dielectric behavior. Therefore a general tivation for our own research in the field qualitative interpretation of elastic prop- 'Correspondence: Dr. J. Schreuer of crystal physics, I would like to address erties - in some favorable cases also Laboratorium fOr Kristallographie these questions brief!y. nearly quantitative - will open access to ETH Zurich Sonneggstr. 5 To date the complete sets of elastic such other properties. With regard to pos- CH-B092 Zurich constants of about 2000 crystal species sible applications, additionally piezo- Tel.: +41 1 6323752 are known. At a first glance this seems to electric properties and electromechanical Fax: +41 1 6321133 E-Mail: [email protected] be a remarkable number. In fact it is less coupling effects are of considerable inter- http://www.kristalf.ethz.ch/LFK than 1% of the crystalline materials with est. Higher order effects provide access CRYSTALLOGRAPHY IN SWITZERLAND 563 CHIMIA 2001,55, NO.6 to the anharmonicity of lattice potentials. Table 1.Typical values for longitudinal deformations.1l of a sample with thickness J == 1mm caused For small and high symmetry structures by thermal expansion, linear and quadratic electrostriction, respectively. calculations from first principles nowa- Property days yield elastic constants which agree Value Force Induced change in length to within a few percent with experimental data [3], However, these calculations are thermal expansion a' II == 50 . 10·8K·1 T == 1K ill == 50 nrn still restricted to the athermal limit. linear electrostriction d'lI) == 2 10 12rnV1 E = 106Vm-1 J1 == 2 nm Moreover, the consideration of tempera- 21 ture effects will remain nearly impossible quadratic electrostriction d'lllI= 5· 10· m2y-2 E = 106Vm-' .jJ = 0.005 nm in the foreseeable future. The answer to the third question is definitely 'Yes'. On the one hand, it is can be neglected in first approximation. the lack of reliable experimental data the intrinsically impossible to obtain a real The interest in quadratic electrostriction interpretation of quadratic electrostric- crystal without defects (the surface is the is further stimulated by their fundamental tion has stagnated in the last decades. Just worst possible defect). On the other hand, relation to some non-linear optical effects recently two research groups, one in the certain electrical and optical properties such as stimulated Brillouin scattering Netherlands [7] and the second in Ger- are known to be strongly influenced by and quadratic electrooptic effects (Kerr many/Switzerland [8], studied the sourc- point defects and dislocations. For many effect). es of several disturbing effects and devel- properties no systematic study of their The mechanical strains Eij induced by oped apparative tools which allow for the dependence on different types of defects an electric field E with components Ek successful investigation of quadratic has been performed at all. Deviations are described phenomenologically by the electrostrictive properties. from the periodic structure are currently Taylor series expansion not tractable by computational methods. 2.1. Experimental Approach: High- Even with a supreme effort only a negli- resolution Capacitive Dilatometer gible, probably non-representative part Among the different approaches to of a real macroscopic crystal, which con- where the coefficients dijb dijkl etc. de- the determination of quadratic electro- sists of about 1023 atoms, can be mod- note the corresponding first, second and strictive constants ([8] and references eled. higher order electrostrictive material therein) the direct strain measurements Therefore, our work focuses on the properties. employing dilatometers based on the fre- improvement and development of dy- The components dijb often referred to quency-modulation technique are the namic methods for the experimental in- as converse piezoelectric constants, form most promising. vestigation of electromechanical proper- a polar third-rank tensor. Therefore, line- The heart of our capacitive dilatome- ties of small samples. Employing these ar electro striction can only occur in crys- ter [8] (Fig. 1) consists of an oscillator techniques we study the elastic and elec- tals with a non-centrosymmetric point with basic frequency in the range be- trostrictive properties of crystals as a symmetry group (PSG), with the excep- tween 60 and 110 MHz. The two elec- function of external forces and of their tion of PSG 43. Linear electrostrictive ef- trodes of the frequency-controlling ca- real structure. fects are in the order of 1O-12mY-l(Table 1) pacitor are supported by transducer rods and can be easily measured employing which are connected with the sample and optical or electromechanical dilatometers with a reference crystal, respectively. 2. Electrostrictive Effects [6]. Due to the important technical appli- Thus, a periodic longitudinal deforma- cations of piezoelectric materials (piezo- tion of one of the crystals leads to a mod- The best criterion to test lattice theo- electric actuators, sensors, oscillators, ulation of the high-frequency signal of retical models is proof of their predictive switches, frequency filters etc.) most of the oscillator. The modulation is elec- power. One of the rare examples where a the recent