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Piezoresponse force microscopy of ferroelectric thin filmsMorelli, Alessio

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Introduction

This chapter presents a concise introduction to the field of ferroelectrics and a summary of the piezoelectric effect.

1.1 History of ferroelectrics

Ferroelectric materials exhibit spontaneous polarization, the direction of which can be switched by applying an external electric field. In fact, ferroelectric materials are a subclass of pyroelectrics and piezoelectrics. Pyroelectricity is known since ancient times (IV century BC) [1] first observed in Tourmaline, and rediscovered much later in the XVIII century being formulated by Brewster in 1824 [2]. Piezoelectricity was demonstrated in 1880 by P. J. and P. Curie [3] in Rochelle salt crystals (sodium potassium tartrate tetrahydrate). The first application was seen in sonar technology during World War I (Langevin, 1917). A few years after the introduction of piezoelectricity, Valasek [4], who studied the dielectric properties in Rochelle salt in analogy with the magnetic properties in ferromagnetics, suggested in 1920 the name of ferroelectrics. During the subsequent decades further theoretical studies were performed and Rochelle salt bimorph pickup head were present in most of the phonographs.

In 1935 another ferroelectric material was considered, the potassium dihydrogen phosphate (KDP), and finally in the 40s the consideration of the

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ferroelectric properties in perovskite oxides, namely barium titanate (BaTiO3), started a new era in ferroelectricity. They established the base for applications on a wider scale (starting with transducers) and better fundamental understanding. Subsequently, in the 50s a couple of perovskite oxide families were discovered, among which the PZT family (PbTi(1-x)ZrxO3) (1952, Takagi and Shirane [5]) was and still is one of the more important as far as applications are concerned. During the next decades the development of new experimental techniques took place, which allowed a better understanding of ferroelectricity and lead to an increasing expansion of the number of applications.

Fig. 1.1: Advantages of FeRAM on DRAM [6].

1.2 Motivation

In the last two decades ferroelectrics became integrated in miniaturized systems and in non-volatile memory devices, pushing research towards size

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effects and nanoscale studies. To date ferroelectric materials are employed in a vast range of applications, going from accelerometers, electro-optical devices, high frequency devices for medical imaging and surface acoustic wave devices, to embedded smart systems and ferroelectric random access memories (FeRAMs). In particular, the recent success of FeRAMs as an alternative to dielectric RAMs (DRAMs) in personal computers (as witnessed by its expansion on this particular market), has directed the research towards this direction. Non-volatility, low energy consumption, and faster writing operation are advantages of FeRAMs versus DRAMs (see Fig. 1.1). However, the storage capacity has still to be improved. At the same time studies are devoted to the superparaelectric limit (“the ultimate volume below which spontaneous polarization ceases”) [7].

At any rate, the lateral size of an individual non-volatile ferroelectric memory cell should be in the mesoscopic range (<100 nm). It is therefore necessary to understand and to control the processes that lead to a deterioration of the prospected performance of ferroelectric structures down to nanoscale. At present however, the understanding of the ferroelectric phenomena at these smaller dimensions is by far from complete and needs further improvement. As is seen in the increasing number of publications in this field during the present decade, nanoscale investigation of ferroelectric properties of thin films from micron to nanoscales has gained high importance. In this regard piezoresponse force microscopy (PFM) is a powerful tool for investigating ferroelectric properties at the nanoscale. PFM is a scanning probe technique based on the converse piezoelectric effect that is present in all ferroelectric materials. This technique allows both the detection and the modification of the ferroelectric state, using the same experimental setup, with a resolution down to 10 nm.

1.3 Ferroelectrics

Any dielectric material develops a polarization vector P by application of an electric field E:

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jiji EP χ= (1.1)

ZKHUH� � LV� the dielectric susceptibility. The magnitude of the polarization vector (P) is defined as the dipole moment per unit volume. The latter arises by spatial separation of the centers of positive and negative charges in the constitutive element of the material (molecule or unit cell).

In the case of crystals the direction of the dipole moment lies along one of the so-called polar axis, which is usually one of the high symmetry directions of the unit cell [8]. According to symmetry, crystals can be grouped in 32 classes. The structure of the materials belonging to 11 of those classes shows a center of symmetry (termed centrosymmetric). Such crystal classes can not develop polar properties. However, as any material, they exhibit the electrostrictive effect, which is the quadratic dependence of strain by the applied electric field. The remaining 21 classes are non-centrosymmetric, and the absence of a center of symmetry makes the presence of polar axis possible. Of those classes, 20 exhibit piezoelectric effect, which is the linear dependence of strain by the applied electric field. From these 20 piezoelectric classes, 10 have a unique polar axis and as a result they can be spontaneously polarized. Since the phenomenon of polarization is temperature dependent, they are called pyroelectrics.

When the polarization of a pyroelectric material can be reversed upon application of an appropriate electric field, the material is called ferroelectric. A characteristic aspect of ferroelectric materials is the occurrence of a ferroelectric hysteresis loop as depicted in Fig. 1.2 [8]. The response of polarization to an external applied field is initially linear, as it is described by the equation for dielectrics (Eq. 1.1). With increasing field, in the domains with polarization direction opposite to the field, polarization starts switching in the direction of the field and as a result the measured charge density increases non-linearly. When all the domains have polarization aligned with the applied field (saturation), the response of the polarization P increases linearly. By decreasing the field strength, P decreases linearly according to the Eq. 1.1, and at zero field it maintains a

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positive value (so called remanent polarization PR). When a negative electric field of sufficient strength is reached (coercive field, EC), the nucleation of reversed polarization domains starts. The polarization P decreases non-linearly until saturation is reached for a negative field. After that, P decreases again linearly and increases linearly for increasing field, being negative at zero field. When a positive coercive field value is reached, a steep increase occurs after which again P increases linearly.

Fig. 1.2: Ferroelectric hysteresis loop schematization [8].

The materials studied in this thesis are PbTiO3 (PTO) and PbTi(1-

x)ZrxO3 (PZT) in the form of thin films. Both are oxides and belong to the perovskite group, whose crystal structure can be represented by the formula ABO3. The representative of that group is CaTiO3, a mineral discovered in 1839 in the Urals by Gustav Rose in 1839 [9] and named perovskite after Count Lev Aleksevich von Perovski, a Russian mineralogist. The crystal structure in the non-polar (or paraelectric) phase is cubic (Fig. 1.3a) with the A cations at the edges, the B cations in the body-center positions, and the

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oxygen anions sitting at the face-centered positions, forming an octahedron. At a certain Curie temperature Tc, the crystal experiences a phase transition changing its structure. During the transition to the polar (or ferroelectric) phase the structure becomes tetragonal (Fig. 1.3b and 1.3c). The unit cell is elongated in one direction (c axis), which is the direction of the polar axis of the crystal. The cell is compressed in the two other directions (a axis). In this configuration the oxygen anions, sitting on the faces parallel to the polar axis, shift above or below the center of the oxygen octahedron. In this way a charge separation is displayed in the unit cell, which results in a dipole moment and as a result to presence of polarization. The fact that the two configurations are in thermodynamic equilibrium (Fig. 1.4) is the reason for the presence of spontaneous polarization, and the application of a suitable electric field switches the unit cells from one configuration to the other and as a result the polarization state. Therefore the crystal in this phase is ferroelectric.

Fig. 1.3: Unit cell of perovskite crystals: paraelectric phase (a), ferroelectric phase with polarization vector pointing upwards (b) and downwards (c)[10].

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Fig. 1.4: Representation of the energy dependence by the displacement of the central cation in the unit cell, characterized by two equilibrium positions.

During the transition from the paraelectric to the ferroelectric phase the polar axis can arise in any of the three principal axis of the cubic configuration. Regions of crystal with polarization oriented in the same direction are called polar domains (or ferroelectric domains). Generally the polar domain configuration in an as-grown film is non-homogeneous. Domains with polarization normal to the surface of a film are named c domains, and a domains is the denomination of domains with polarization lying in the plane of the film. Different polarization domains can arise in an as grown film depending on electrical and mechanical boundary conditions. In the first case the presence of a polarization discontinuity, e.g. the surface of the film, leads to charge formation and generation of depolarizing fields. The direction of the latter is opposite to the polarization direction. If the depolarizing charge is not compensated (e.g. by electrical conduction through the crystal), it leads to the formation of domains with opposite polarization, generating charge of opposite sign where polarization discontinuity takes place and as a result reducing the depolarizing field. This process continues until the equilibrium is reached. Moreover, mechanical stresses arise during the transition from cubic to tetragonal

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phase (e.g. by clamping by the substrate). In this case, the system minimizes the elastic energy in a way that in some regions of the crystal the c axis will develop perpendicular to the stress, while in other regions of the crystal the polarization will be parallel to the stress [11].

Domain walls separate two polar domains. They are named as 180º domain walls in case of domains with opposite polarization direction and 90º domain walls if the domains’ polarization is perpendicular (Fig. 1.5) [8] [11]. The thickness of domain walls has been found to be only a few lattice constants (less than 1nm) [11], which is much lower than the thickness of a domain wall in magnetic materials. This aspect in particular is of interest for memory applications. Such a small domain wall thickness is explained by the fact that the polarization orientation close to a domain wall does not change gradually as in Bloch walls for magnetic materials. In magnetic materials the wall’s thickness results from a competition between the exchange and magnetocrystalline anisotropy energy. The former would lead to domain walls as wide as possible, while the latter has the opposite effect. Since the exchange energy is orders of magnitude higher than the magnetocrystalline anisotropy energy, Bloch walls are hundreds of lattice constants. However, in ferroelectric materials, the domain walls thickness is controlled exclusively by the anisotropy energy. As a result a deviation of the polarization orientation from the polar axis is hardly possible, given the high anisotropy of most ferroelectrics. Thus, polarization in proximity of domain walls gradually decreases its magnitude passing through zero and increasing again with opposite orientation [8] within a domain wall thickness of a few lattice constants.

The domain wall orientations between two allowed polarization directions in a perfect infinite single crystal can be predicted using a criterion established by Fousek and Janovec [12]:

• The crystal distortion arising during the non-polar/polar transition can be predicted from the magnitude and direction of the spontaneous

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polarization, and from piezoelectric and electrostrictive coefficients in the paraelectric phase.

• If two domains with different polarization orientations have a common domain wall. This is possible along the planes matching exactly the lattice distortions. These planes are good candidates for domain walls.

• The domains form in a manner that the normal component of the spontaneous polarization is continuous through the domain walls, so that they remain electrically neutral. This leads to the formation of “head to tail” polarization configuration at the 90º domain walls (Fig. 1.5).

Fig. 1.5: Schematic of domain walls: 90º domain walls lie along the segments AA’ , 180º domain wall along segment BB’ [8].

1.4 Piezoelectric effect

Piezoelectricity is the property of a material to generate an electric potential in response to an external applied mechanical stress. This property, defined direct piezoelectric effect, can be represented by the equation [11]:

ikdirectijki XdP = (1.2)

being Xij the second-rank stress tensor and d piezoelectric modulus or the third rank tensor of piezoelectric coefficients, measured in [C/N]. Inversely,

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the converse piezoelectric effect, first demonstrated theoretically and then verified by the Curie’ s, describes the changes in dimensions of a piezoelectric material in response to an the applied electric field E:

iconverseijkjk Edx = (1.3)

where x is the strain. The coefficients connecting the field and strain in the converse effect are the same as those connecting the stress and the polarization in the direct effect. The proof of this equality is based on thermodynamic reasoning and since kiik xx = , it follows that ijkd is symmetrical in j and k. The converse piezoelectric coefficients are measured in [m/V]. Commonly the coefficient measured in the direction of the applied field is called longitudinal coefficient whereas the one measured perpendicular to the field is the transverse coefficient and the others the shear coefficients. The notation can be simplified, following the matrix notation, so that the indices ij = 11, 22, 33 can be written as i = 1, 2, 3 and the ij = 23 or 32, 13 or 31, 12 or 21 can be written as i = 4, 5, 6 [11]. In this notation the longitudinal coefficient, which is the one of major interest and subject of spread investigation, is referred as d33, according to:

( 1,2,3; 1,..6)conversej ij ix d E i j= = = (1.4)

Fig.1.6: Schematic description of the converse piezoelectric effect. a) Polarization hysteresis loop; b) theoretical butterfly loop of strain vs. field; c) actual butterfly loop [8]. In b) and c), x denotes the uniaxial strain.

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The strain versus applied field behavior appears in the shape of a butterfly loop (Fig. 1.6c) [8]. In an ideal case (Fig. 1.6b), when the polarization changes direction, the sign of the strain changes and when it is stable the strain is linear with the field. The value of the piezoelectric coefficient is obtained by calculating the slope of the linear part. In real crystals the strain versus applied field assumes a characteristic butterfly shape, given by the fact that different polarization orientations are present in the crystal, making the curve smoother in accordance with the change in polarization.

It should be noted that for most ferroelectrics the piezoelectric effect can be considered as the electrostrictive effect biased by spontaneous polarization. Electrostriction is the coupling effect, property of all dielectrics, between strain and the square of electric field. Thus, in contrast with piezoelectric effect, the reversal of electric field does not reverse the direction of the deformation induced by electrostriction. The electrostrictive strain is defined as:

lkijklij EEMx = (1.5)

where Mijkl are the components of the four-rank electrostrictive tensor. In terms of induced polarization the electrostrictive strain is expressed by:

lkijklij PPQx = (1.6)

where Qijkl are the polarization-related electrostrictive coefficients. The latter are related to Mijkl by:

ijklkmijmn QM lnχχ= (1.7)

Finally it is important to point out the relationship between spontaneous polarization and piezoelectric coefficients. This relationship can be derived by a thermodynamic approach [11] leading to:

Smjklmilijk PQd ε2= (1.8)

with the relative permittivity. Moreover, as established by Devonshire [13][14] for the case of BaTiO3, for a single crystal with tetragonal

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symmetry in a monodomain state, the longitudinal coefficient can be related to the spontaneous polarization by adapting Eq. 1.4:

SPQd 3333333 2ε= (1.9)

The latter gives a direct relationship between piezoelectric coefficient and spontaneous polarization.

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References.

[1] E. R. Caley and J. F. C. Richards, Theophrastus on Stones, Graduate School Monographs, Contributions in physical science 1 (1956).

[2] D. Brewster, Observation of pyroelectricity of minerals, Edinburgh, J. Sci. 1, 208 (1824).

[3] P. J. Curie and P. Curie, Development by pressure of polar electricity in hemihedral crystals with inclined faces, Bull. Soc. Min. de France 3, 90 (1880).

[4] J. Valasek, Piezoelectric and allied phenomena in Rochelle salt, Phys. Rev.15, 537 (1920), Phys. Rev.17, 475 (1921)

[5] G. Shirane, A. Takeda, Phase transitions in solid solutions of PbZrO3 and PbTiO3 .1. Small concentrations of PbtiO3,J. Phys. Soc. Jpn. 7, vol. 1, 5 (1952)

[6] J. F. Scott, Ferroelectric memories, Springer (2000) [7] A. Rudiger, T. Schneller, A. Roelofs, S. Tiedke, T. Schmitz, and R.

Waser, Nanosize ferroelectric oxides - tracking down the superparaelectric limit, Appl. Phys. A 80, 1247 (2005)

[8] F. Jona and G. Shirane, Ferroelectric crystals, Clarendon Press (1962) [9] R. V. Gaines, H. C. W. Skinner, E. E. Foord, B. Mason, and A.

Rosenzweig, Dana’s new mineralogy, John Wiley & Sons (1997) [10] K. Momma and F. Izumi, VESTA: a three-dimensional visualization

system for electronic and structural analysis, J. Appl. Crystallogr., 41, 653 (2008)

[11] D. Damjanovic, Ferroelectric, dielectric and piezoelectric properties of ferroelectric thin films and ceramics, Rep. Progr. Phys. 61, vol. 9, 1267 (1998)

[12] J. Fousek and V. Janovec, The orientation of domain walls in twinned ferroelectric crystals, J. Appl. Phys. 40, 135 (1969)

[13] A.F. Devonshire, Theory of barium titanate - part I, Philos. Mag. 40, 1040 (1949)

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[14] A.F. Devonshire, Theory of barium titanate - part II, Philos. Mag. 42, 1065 (1951)