# Introduction to measurement methods

Some of the more important methods of characterising piezoelectric and ferroelectric materials are presented in the book described in the previous page. Outlined below are chapter synopses of the methods that are described in much more detail in the book:

Electrical Measurement of Ferroelectric Properties
Ferroelectric materials are defined by the existence of a finite polarisation at zero electric field, the direction of which can be switched by the application of an external field. Measurement of the electrical properties is therefore an important tool in identifying ferroelectricity and in characterising the ferroelectric properties of a material. In discussing the electrostatics of materials it is common to assume that a material becomes polarised only by the application of an electric field and that the polarisation returns to zero on removal of the field. This is not the case for ferroelectrics. This is illustrated by brief review of the electrostatics of a simple capacitor, followed by the application to the measurement of switchable polarisation in a ferroelectric and methods for the measurement thereof.

Piezoelectric Resonance
The phenomenon of resonance is introduced and linked to the evaluation of the piezoelectric matrix by appropriate choice of sample geometry, and data analysis method. The IEEE Piezoelectric standards method and complex coefficient method is explained and compared with a worked example helping the user  understand the stages of measurement and analysis.

For piezoelectric materials, since the material can be excited electrically, applying an AC voltage across the device can induce resonance without the need for external mechanical stimulation. Electrically induced resonance occurs in a piezoelectric material because of the elecromechanical coupling that exists between applied field and induced strain, and that is defined by the complete set of piezoelectric equations [IEEE Standard on Piezoelectricity  and the more recent IEC standard]. This strain oscillates at the same frequency as the imposed field that act to set up sound waves in the material. A resonance is set up in the specimen when the dimensions of the specimen match some integral number of sound wavelengths - depending on the mode of vibration. It is clear then that the electrical impedance of a piezoelectric material measured as a function of frequency exhibits peaks that correspond to electromechanical resonance in the specimen. The form of the response is described by 2 quantities - frequency, and its frequency width at half amplitude, $\delta$, (={f}/Q) with Q called the Quality factor that characterise the distinctive properties of a driven system.

Direct Piezoelectric Measurement - The Berlincourt Method
Although piezoelectric materials are often used as actuators in order to make small precise movements it can be difficult to measure these displacements in an industrial environment. Consequently simpler methods have been sought to measure the piezoelectric activity, such as resonance methods, and measurement of the piezoelectric coefficient d$_{33}$ using the direct method (often called the Berlincourt method). This chapter  examines the advantages and disadvantages of the method in detail and with some experimental validation using typical PZT ceramics examine the validity of using the data from this method to predict the displacement of materials in real conditions. The piezoelectric charge coefficient, d$_{ij}$, is one of the fundamental parameters defining the piezoelectric activity of a material, basically the higher the d$_ij$ the more active the material is.  Consequently, manufacturers, designers, and users want to know the d$_{ij}$ coefficient for the material.

Measurement of the d$_{ij}$ coefficient can be realised in several ways varying in accuracy and simplicity. The most reliable method of determining the d$_{ij}$ coefficient is to electrically excite a resonance in a sample, and from the resonance response - given the dimensions of the sample and the density - a d$_{ij}$ coefficient can be calculated. One problem with this method is that the geometry of the sample must be such that only a pure fundamental resonance mode is produced, and the calculated d$_{ij}$ parameter relates to this resonance mode. This leaves the problem how to determine the d$_{ij}$ parameter for shapes that don't have an ideal resonance geometry, or where the resonance mode is not the mode that will be used. For instance, for thin discs poled in the thickness direction it is easy to excite a resonance in the radial direction, and determine the relevant d$_{ij}$ parameter, but to obtain the d$_{ij}$ coefficient for motion in the thickness direction then longer cylinders are needed.

The d$_{ij}$ coefficient is defined as the charge produced for an applied stress, or the strain for an applied voltage, and these are theoretically equivalent. The latter measurement is more difficult to achieve because of the small strains involved, so measurement techniques have concentrated on the former. In this work, initially, the charge was measured in response to an applied static load, but difficulties with thermal drift led to the measurements being performed quasi statically, at a few hundred hertz.

The quasi static method is straightforward; a small oscillating force is applied to the sample and the charge output is measured and divided by the applied force amplitude. The simplicity of the technique has been its downfall, in that anyone can easily build up their own system, and there are a growing number of commercial systems. There are currently no standards for this measurement method, and consequently each system performs the measurement slightly differently. This means that, although the results from these systems are good for measuring within a batch or batch to batch variability, external comparisons usually produce a large variability.

Pyroelectric Materials
Pyroelectrics form a very broad class of materials.  Any material which has a crystal structure possessing a polar point symmetry - i.e. one which both lacks a centre of symmetry and has a unique axis of symmetry - will possess an intrinsic, or spontaneous, polarisation and show the pyroelectric effect.  The pyroelectric effect is a change in that spontaneous polarisation caused by a change in temperature.  It is manifested as the appearance of free charge at the surfaces of the material, or a flow of current in an external circuit connected to it.  The effect is a simple one, but it has been used in a range of sensing devices, most notably uncooled pyroelectric infra-red (PIR) sensors, and has thus come to be of some engineering and economic significance, enabling a wide range of sensing systems, ranging from burglar alarms through FTIR spectroscopic instruments to thermal imagers.

A wide range of material compositions and types are available for potential exploitation, including single crystals, ceramics, polymers, thin films and liquid crystals.  An important problem for the materials engineer is how to select the most promising material for exploitation in any given device type.  This must be determined by which of the materials properties are relevant to the application being contemplated.  Central to the selection decision will be how to accurately measure these properties. This chapter aims first to describe the main device applications of pyroelectrics and to discuss - on the basis of the physics of how these devices work - the key properties which determine device performance.  It will then discuss the main techniques by which these properties can be measured, including some of the potential pitfalls.  It will finish with a brief review of the properties of some of the materials which have been applied in practical devices and systems.

Interferometry for piezoelectric materials and films
Piezoelectricity is the coupling between the mechanical and electric property of materials, which manifests itself by the generation of electric charge upon a pressure or conversely the produce of strain under an electric field. For the converse effect, the displacement related to this kind of strain is generally very small. Even for the best piezoelectric material, for example PMN-PT and PZN-PT single crystals, the longitudinal piezoelectric coefficient d$_{33}$ is no more than 3000 pC/N. If 1000 volts are applied on a 1 mm PMN-PT, the resulted displacement is only 3 $\mu$m. Other piezoelectric materials, such as PZT,  the most widely used material, have a d$_{33}$ nearly one order of magnitude smaller. Optical interferometry  has long been established as one of the most promising techniques for small displacement measurements, due to its capability of very high resolution and advantages such as no mechanical contact and no need for calibration on the length scale. The development of lasers in the last a few decades has almost eliminated the problems associated with the optical path length coherence and beam density, and many techniques based on laser interferometry have been developed for the characterisation of piezoelectric materials. This chapter will review the applications of laser interferometry to the  characterisation of the piezoelectric bulk and thin film materials and discuss possible problematic issues associated with these techniques.

Temperature dependence of piezoelectrics
Temperature induced changes in the properties of piezoelectric and ferroelectric materials have major implications for the design of devices used in wide variety of technological applications. Examples include valves for fuel injection in diesel engines, pneumatics, and gas control in domestic appliances where the flow rate could drift with temperature and compromise valve performance. Similar considerations apply in layered electronic devices employing piezoelectric materials. There is considerable current interest in multiferroic devices  for spintronics, sensor and memory applications.  Small scale layered devices can provide magnetoelectric functionality through strain coupling between ferroelectric and a magnetostrictive components. Strain coupling will also occur through differential thermal expansion  between the layers, so a knowledge of the thermal behaviour of the components is essential. In many of these applications quite large electrical fields are needed to obtain useful energy density, so the temperature dependency of the high field electromechanical coupling  is important. It is a significant characteristic of this type of device that the strain or position relative to the value at a reference temperature is an important factor. For instance, in a valve, the position of the sealing member relative to the valve seat is the critical parameter in controlling the flow rate. In a multiferroic device, the strain of the piezoelectric material relative to the magnetostrictive  component determines magnetic or electrical state of the device. Thermal expansion may couple mechanically in the same way as the piezoelectric coupling to produce unwanted actuation in response to temperature changes rather than the desired electrical activation. When the piezoelectric material is bonded to a different material to form a unimorph or bimorph actuator,  \index{Actuator} the thermal expansion of the partner materials must also be taken into account.  In general actuator applications, thermal drift can cause unwanted actuation at temperature extremes, or limit the operating temperature range.

Measurement and Modelling of Self-Heating in Piezoelectric Materials and Devices
There are many uses of piezoelectric ceramics where the desire for increased power output means increased drive levels, which subsequently can lead to thermal problems within the device. Applications such as:-

• Ultrasonic Cleaning
• Ultrasonic Welding
• Sonar Transducers
• Diesel Injectors
• Ultrasonic Sewage Treatment

all use piezoelectric materials operated at high drive levels, where thermal loading on the device becomes an issue, and where potentially expensive cooling is needed to maintain device performance. When piezoelectric materials are used as actuators  they make use of the indirect piezoelectric effect, where the application of an electric field gives rise to an internal strain. In this solid-state energy transformation there will always be a balance between electrical energy input and work done by the device. The coupling coefficient, k, is used to describe this efficiency for an ideal case where there are no losses. Here, k is essentially the ratio of the open circuit compliance to the short circuit compliance. For most real piezoelectric materials this conversion process is also associated with losses - both mechanical and dielectric. These losses manifest themselves in the form of heat, causing a temperature rise in the device, which, depending on the thermal boundary conditions can be detrimental to device performance. This self-heating effect is most often encountered in resistive components and is termed 'Joule Heating'. However, it is also seen in non-ideal dielectric materials where the dielectric loss gives rise to internal heat generation. To a first approximation, piezoelectric actuators can be thought of as a non-ideal or lossy dielectric but, because the material is moving, additional mechanical terms are needed to model this behaviour. If the energy loss to the surroundings is greater than the internal power generation, then the sample will eventually reach an equilibrium temperature. If the sample losses are greater than those to the environment, or if the losses increase with increasing temperature, then the sample will heat up until some catastrophic event is reached - such as the soldered connections failing, softening of adhesives, or depolarisation of the material.

Piezoresponse Force Micropscopy - PFM
Scanning probe microscopy (SPM) is a very versatile technique allowing for a large range of sample properties to be measured and manipulated with nanometre spatial resolution. One important SPM mode is piezoresponse force microscopy (PFM). PFM is an invaluable tool for measuring the piezoresponse of functional materials at the nanoscale, allowing for high resolution measurements of the electromechanical coupling of thin films. PFM is based on the standard contact mode AFM setup with the cantilever and tip being electrically conductive, typically either through highly doped Si or metallic coating. The samples measured are piezoelectric and a voltage applied between the tip and a bottom electrode results in sample strains due to the inverse piezoelectric effect. In this chapter we will give a brief overview of SPM, following which PFM will be analysed in some detail.

Indentation Stiffness Analysis of Ferroelectric Thin Films
The physics of size effects in ferroelectric materials influences finite, measurable, changes in the macroscopic functional behaviour of small-scale systems. In these confined geometries the ceramic microstructure, and in particular grain size and domain switching properties, often enhances the importance of extrinsic effects in the ferroelectric response by comparison with the properties of the bulk material. Whilst ferroelectric scaling effects have been discussed for many years, the recent technologically-based drive for sub-micron scale ferroelectric memory applications has resulted in many investigations into thin film systems and 3D micro- and nano-structures. Static scale-size effects are a fundamental, as yet unsolved, problem in nano-ferroelectrics in which destabilization of the spontaneous polarization in ultra-thin films (down to the size of a few unit cells) is predicted theoretically but has not yet been conclusively observed. Whilst this phenomenon occurs on the smallest length scales, size effects relating to the dynamic behaviour of ferroelectrics, such as variation in the Curie temperature, collapse of the dielectric constant and phonon hardening, can be seen at the micron level and are caused by changes in both sample thickness and lateral size. This chapter demonstrates the applicability of a depth sensing indentation technique  for the measurement and interpretation of the elastic properties of ferroelectric thin film material. Ferroelectric thin films are of interest as the active materials in actuators and sensors in MicroElectroMechanical Systems (MEMS). For their applications as actuators it is necessary to know their elastic properties in order to design the devices. For example, the design of many actuators is based on bending cantilevers, such as the active arm of an Atomic Force Microscope (AFM). Because of the lack of availability of data, designers are often relying on the elastic coefficients of bulk materials with the same nominal coefficients. This may lead to large errors not only because of the differences in composition and microstructure, but also because the mechanical and functional behaviour of films can be very different to that of bulk materials.

Historically, the method defined by IEEE  has been used extensively to analyse the resonance spectra of piezoelectric materials. Here, the resonance frequencies from a set of samples of different geometries are measured and along with their dimensions and sample density the piezoelectric coefficients determined. This method is simple to carry out in principle, and the calculations are relatively straightforward (although the wave equations for some of the geometries must be solved empirically through the use of Bessel functions).

It is now relevant to discuss losses prevalent in piezoelectric ceramic compositions since these values are often as important as the functional, dielectric and elastic constants that resonance analysis yields. In reality, a piezoelectric material comprises losses originating from its dielectric response to an electrical field, mechanical response to applied stress or following piezoelectric motion and its piezoelectric (strain) response to an electric field. The impact of these losses on a resonance sweep is a reactive and resistive part to the measured impedance. A material with zero losses would exhibit zero impedance at resonance. The significance of loss results in sample heating or noise production and this is why for many applications an understanding of loss mechanisms and absolute values becomes important. Normally, the mechanical loss at resonance is calculated from the width of the resonant peak and is labelled the mechanical Q or Quality factor. The narrower the resonant peak, the higher its Q. Dielectric losses are normally calculated from the phase angle between observed capacitance and applied field, labelled tan $\delta$. Piezoelectric loss may not normally be calculated from resonance data but may be assessed through strain - electric field response whereby any hysteresis present may be tentatively ascribed to this loss alone - of course, if strain is produced then mechanical loss may also have an additive effect. This issue is contentious and discussed in this chapter.

Dielectric Breakdown in dielectrics and ferroelectric ceramics
Under the application of a large enough electrical field a dielectric material can respond through motion of free charge carriers, injection of mobile charge carriers from electrodes, space charge generation, and dissipation of energy in the material, all of which may contribute to the electrical failure of the material. Various theories have been published going back several decades and include two fundamentally separate materials responses to an electric field; 1. an intrinsic dielectric breakdown phenomena and 2. a defect mediated breakdown phenomena. The former is likely only ascribable to very thin films or some polymeric materials. The latter is the more commonly regarded theory that details the weakest link statistics argument for electrical breakdown in porous materials, for example. The type and source of the free carriers can provide for a profound influence on the nature of the breakdown events and so any discussion or analysis of breakdown in solid materials especially must have an appreciation of the conduction processes occurring in the material. If breakdown of the sample is not determined by some defect or imperfection in the material then one can consider what is the materials' intrinsic electrical breakdown strength. If breakdown is determined by defects or imperfections then this implies a weak link mechanism. In practice, even high density, carefully prepared ceramic alumina exhibits a Weibull weak link statistical response signifying defect induced (pores, voids, inclusions etc) breakdown. In this chapter we introduce some of the concepts that appear to govern dielectric breakdown in dielectric and ferroelectric ceramics and cite various excellent reviews and books which explore these issues. Then, we will present the various standards that already exist and ways in which measurement good practice has been developed in the past few years. The focus is on the measurement of dielectric breakdown of bulk (or thin section bulk) ceramic materials.

Standards for piezoelectric and ferroelectric ceramics
This chapter provides detail of standards-related activities for piezoelectric materials --- mainly bulk ceramic types. The production of standards, particularly international ones, is a long process. The existence or not of a standard in a particular area is not solely because of the need for such a standard, but is due to the concerted effort of individuals. Documents must be kept current, and may be withdrawn or cancelled if there is nobody prepared to maintain it after publication.

There are strong groups for piezoelectric materials standards in Europe (CENELEC) and America (IEEE-UFFC), but some of the most quoted standards have been withdrawn (IEEE 176-1987, IEEE 180-1986 , MIL-STD 1376B (SH)). An updated and regularly reviewed set of standards can be found at http://www.piezoinstitute.com.