Integration of Ferroelectric and Piezoelectric Thin Films

Concepts and Applications for Microsystems

Emmanuel Defaÿ

• Publisher: Wiley-ISTE
• Year: 2013
• Print ISBN: 9781848212398
• eBook ISBN: 9781118616604
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Materials Science
• Index: Technology & Engineering

This book contains four parts. The first one is dedicated to concepts. It starts with the definitions and examples of what is piezo-pyro and ferroelectricity by considering the symmetry of the material. Thereafter, these properties are described within the framework of Thermodynamics. The second part described the way to integrate these materials in Microsystems. The third part is dedicated to characterization: composition, structure and a special focused on electrical behaviors. The last part gives a survey of state of the art applications using integrated piezo or/and ferroelectric films.

Chapter 1

Dielectricity, Piezoelectricity, Pyroelectricity and Ferroelectricity ¹

1.1. Crystal structure

1.1.1. Crystal = lattice + pattern

The notions of piezoelectricity, pyroelectricity and ferroelectricity are closely linked to the crystalline nature of materials. Indeed, the study of the crystal structure of materials enables us to see what arrangements of atoms are susceptible to showing one or other property. We will discuss some core notions of crystallography in order to bring out the main conclusions that can be drawn from this approach [ESN 94].

A crystal is defined as follows: the atoms that make up a crystal form a pattern that periodically repeats itself in the three spatial dimensions. A crystal is an object whose dimensions are large compared to the atoms that constitute it. The periodicity of its structure results in the properties of the crystal being identical, depending on the dimensions and planes given, no matter the initial reference point. This is what we call translational symmetry. This notion is important as it will enable us to implement symmetry operations that will be used to classify the crystals.

As a simple example of a periodicity-based property, let us cite silicon cleavage, which is always split in the same directions, or the hexagonal structure of quartz crystals that is visible to the naked eye.

The notions that constitute the foundations of the study of crystallography are the lattice and pattern of a crystal. The lattice is made up of points called nodes that periodically repeat themselves in space, but the lattice does not contain any atoms. The atoms of the crystal belong to the pattern that is attached to the lattice node.

Figure 1.1 is a diagram representing a crystal with its lattice and its pattern.

Figure 1.1. Lattice + pattern = crystal

To help with comprehension, it is useful to start with a classic example: sodium chloride (NaCl). Crystal packing is made up of a regular succession Na⁺ and Cl⁻ ions in the three spatial dimensions, as represented in Figure 1.2.

Figure 1.2. Crystal packing of NaCl

More specifically, we can say that the NaCl consists of the spatial packing of cubes, such as C (Na⁺₁, Na⁺₂, Na⁺₅, Na⁺₄, Na⁺₁₀, Na⁺₁₁, Na⁺₁₄, Na⁺₁₃). On the other hand, it is not possible from cube (Na⁺₁, Cl⁻₁, Na⁺₃, Cl⁻₂, Cl⁻₅, Na⁺₆, Cl⁻₇, Na⁺₇). Packing can be initiated from several equal points of the lattice. In this case, they are points on the corners of the cube C (Na⁺₁, Na⁺₂, Na⁺₅…) or those that are in the center of the faces of C (Na⁺₃, Na⁺₆, Na⁺₇…).

It is these nodes that constitute the lattice called the translational lattice. A pattern is attached to each node. Here, the latter is made up of a pair of ions Na⁺ and Cl⁻. For example, Na⁺₁−Cl⁻₁ or Na⁺₁−Cl⁻₂ or even Na⁺₁−Cl⁻₅. The structure of NaCl can be described as two interpenetrating face-centered cubic lattices with a gap between them of a distance a/2 in one of the three spatial dimensions, depending on the pattern chosen (x axis if Na⁺₁⁻Cl⁻₁).

This description is not the most compact possible, although it is the simplest. The face-centered cubic lattice constitutes a multiple lattice.

This means that several patterns are contained in one lattice. There are four of them here (each corner node counts for 1/8 and each node on a face counts for 1/2). Nevertheless, it is possible to define, for each packing, an elementary lattice that contains only one pattern.

Figure 1.3. Trigonal elementary lattice of the face-centered cubic packing of NaCl

In the case of NaCl (and face-centered cubic in general), this lattice is illustrated in Figure 1.3.

It is a rhombohedral lattice that stretches out on a large diagonal of face-centered cubes, in which the angles between the directions of the polyhedron are 60° and the lengths of the base vectors of the elementary lattice are equal.

1.1.2. Seven primitive lattices — 14 Bravais lattices

If we now concentrate on the lattice, we can see that there are symmetries from which lattice nodes can be deduced from each other. These are the translational symmetries of the lattice. Going back to the example in Figure 1.3, there are several axes of rotation that leave the lattice unchanged. For example, the “order 3” axis along a large diagonal of cube C allows a 120° rotation without the lattice being modified.

It is the same thing for “order 4” and “order 2” axes for the respective rotations of 90° and 180°. There are planes of symmetry such as (Na⁺₁, Na⁺₅, Na⁺₁₄, Na⁺₁₀). There are also centers of symmetry (here, each lattice point is one). These lattice symmetries must be compatible with spatial periodicity, which limits their number. Therefore, we can show that they are limited to rotations of order 2, 3, 4 and 6 (i.e. angle rotation 2π/n for order n simply expressed n), mirror planes (expressed m) and centers of symmetry (expressed

).

Seven possible lattice types result from this:

– cubic (a=b=c,α=β=γ=90°);

– hexagonal (a=b≠c, α=β=90°, γ=120°);

– tetragonal (a=b≠c,α=β=γ=90°);

– rhombohedral (a=b=c, α=β=γ≠90°);

– orthorhombic (a≠,α≠β=γ=90°);

– monoclinic (a≠b≠c, α=γ=90°≠β);

– triclinic (a≠b≠c, α≠β≠γ,).

Here, a, b, c, α, β and γ are the parameters of each lattice (Latin letters = distance and Greek letters = angle) with α being the angle between axes y and z, β, between x and z and y between x and y.

As we were able to see for NaCl, it is sometimes simpler to define a multiple lattice to describe the structure. Therefore, face-centered cubic lattices are simpler to use than the rhombohedral primitive lattice (see Figure 1.3). This causes us to take into consideration new lattices called Bravais lattices. This results in a total of 14 possible lattices, with or without a multiple lattice. The seven multiple lattices that are added to the primitive lattices above are the centered cubic, face-centered cubics, centered tetragonal, base-centered orthorhombic, centered orthorhombic, face-centered orthorhombic and base-centered monoclinic lattices.

1.1.3. Two-hundred and thirty space groups

Beyond lattice symmetries, it is possible to take microscopic symmetries into consideration if we now consider the pattern attached to the lattice. The final crystal symmetry will therefore be the combination of lattice and pattern symmetries. Crystal symmetry, however, can never exceed that of the lattice, as can be seen in Figure 1.1. The pattern must therefore have a symmetry that is less than or equal to the lattice, otherwise these symmetry elements would be directly in the lattice. We can show that to be compatible with spatial periodicity, the symmetry elements

belonging to the pattern must have a glide: they are helicoid rotations and glide reflections (not presented here). Taking into consideration lattice and pattern symmetries enables us to determine all the mathematically possible combinations for transforming a crystal in itself (group theory). We end up with 230 crystal structure possibilities, which are called “space groups”.

1.1.4. Thirty-two point groups (or crystal classes)

Now, to analyze the macroscopic properties of crystals, the hypothesis of ...

Table of contents

• Cover
• Title Page
• Copyright
• Preface
• General Introduction
• Chapter 1: Dielectricity, Piezoelectricity, Pyroelectricity and Ferroelectricity
• Chapter 2: Thermodynamic Study: a Structural Approach
• Chapter 3: Ferroelectric-paraelectric Phase Transition Thermodynamic Modeling
• Chapter 4: Mechanical Formalism
• Chapter 5: Dielectric Formalism
• Chapter 6: Piezoelectric Formalism
• Chapter 7: Acoustic Formalism
• Chapter 8: Electrostrictive Formalism
• Chapter 9: Electric Characterization
• Chapter 10: Piezoelectric Resonators and Filters
• Chapter 11: High Overtone Bulk Acoustic Resonator (HBAR)
• Chapter 12: Electrostrictive Resonators
• Chapter 13: Thin Film Piezoelectric Transducers
• List of Authors
• Index