Artificial ferroelectric materials

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Thesis

Reference

Artificial ferroelectric materials

STUCKI, Nicolas

Abstract

Dans ce travail, nous étudions les propriétés de superréseaux de PbTiO₃ et SrTiO₃. Pour des couches épaisses de chacun des constituants, le système se comporte comme prévu par notre modèle électrostatique, en trouvant un équilibre entre les propriétés ferro- et paraélectrique de ces derniers. Cependant, pour des couches très fines (de l'ordre de quelques cellules unités), le système a un comportement différent de celui des matériaux de départ et présente des caractéristiques typiques de ferroélectricité impropre. Grâce à des calculs "ab-initio" effectués par le groupe du Professeur Ghosez à l'Université de Liège, il est démontré que ce comportement est dû à un couplage d'instabilités aux interfaces des deux matériaux. L'état fondamental du système est en effet une combinaison subtile de distorsion ferroélectrique et de rotations des octaèdres d'oxygène de la maille pérovskite. Ce couplage, présent uniquement aux interfaces, guide les propriétés macroscopiques de ces superréseaux. On parle alors d'ingénierie d'interfaces.

STUCKI, Nicolas. Artificial ferroelectric materials. Thèse de doctorat : Univ. Genève, 2008, no. Sc. 3954

URN : urn:nbn:ch:unige-5311

DOI : 10.13097/archive-ouverte/unige:531

Available at:

http://archive-ouverte.unige.ch/unige:531

Disclaimer: layout of this document may differ from the published version.

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Université de Genève Faculté des Sciences

Département de physique Professeur J.-M. Triscone

de la matière condensée

Artificial Ferroelectric Materials

Thèse

présentée à la Faculté des sciences de l’Université de Genève pour obtenir le grade de docteur ès sciences, mention physique

par Nicolas Stucki

de Thalwil (ZH)

Thèse n^◦ 3954

Genève

Atelier de Reproduction de la Section de Physique 2008

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Abbreviations

AFM atomic force microscope/microscopy

STM scanning tunneling microscope/miscroscopy TEM transmission electron microscope/microscopy FWHM full width at half maximum

CE (CV) Capacitance-Electric field (Capacitance-Voltage) PE (PV) Polarization-Electric field (Polarization-Voltage) SE (δz-V) Strain-Electric field (equivalent to δz-Voltage)

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Résumé

Les matériaux ferroélectriques, découverts dans les années 1920, présentent une polarisation électrique spontanée et réversible. Ces matériaux sont également pyroélec- triques et piézoélectriques, indiquant qu’une variation de polarisation est induite par un changement de température et un changement de contrainte mécanique re- spectivement. Les matériaux ferroélectriques sont utilisés depuis longtemps pour ces deux dernières propriétés dans divers senseurs et filtres. Possédant deux états stables de polarisation, pouvant être associés aux deux valeurs d’un bit d’information, ces matériaux ont par ailleurs rapidement été pressentis comme des candidats potentiels pour des mémoires informatiques. Ces matériaux ont bénéficié, ces vingt dernières années, de recherches importantes tant du point de vue théorique qu’expérimental.

En effet, le développement des techniques de croissances consécutif à la découverte de la supraconductivité à haute température en 1986 a permis la fabrication de films d’oxydes de haute qualité. Partant, beaucoup de ferroélectriques à la structure proche de la structure perovskite ont pu être réalisés avec un qualité largement accrue et certains problèmes revisités.

De nos jours, il est possible d’obtenir des couches minces ferroélectriques épitax- iées de très grande qualité cristalline, ainsi que des matériaux artificiels comme des superréseaux, avec des surfaces atomiquement plates et, par conséquent, des interfaces entre matériaux contrôlées à l’échelle atomique. Ces matériaux artificiels permettent l’étude de problèmes fondamentaux, comme des effets de taille finie ou la dynamique de domaines ferroélectriques. En outre, ils peuvent être façonnés dans le but d’obtenir des propriétés bien définie que l’on ne pourrait trouver dans des matériaux purs comme, par exemple, des valeurs spécifiques de polarisation ou de constante diélectrique, nécessaires pour une application particulière.

Dans ce travail, nous présentons une étude sur des superréseaux de PbTiO₃ et SrTiO₃, dans laquelle nous montrons que ce système, simple en apparence, révèle en fait un spectre de propriétés physiques très riche et également inattendu. D’une part, il est démontré que les propriétés électriques de ces matériaux peuvent être contrôlées et modifiées en jouant sur l’épaisseur des couches individuelles de chaque constituant et, au moyen de calculs ab-initiod’autre part, que pour des échantillons comportant des couches très fines de PbTiO₃ et SrTiO₃, des instabilités distinctes

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2 Résumé

présentes dans les matériaux parents peuvent se coupler aux interfaces et donner lieu à un nouveau type de ferroélectricité, appelée ferroélectricité impropre.

Afin de mesurer les différentes propriétés des échantillons, plusieurs techniques de caractérisation ont été utilisées : diffraction par rayons-x, tout d’abord, avec pos- sibilité de mesurer l’évolution des paramètres de réseau en température, permettant de déterminer qualité cristalline, tetragonalité et température de transition des échantillons ; mesures électriques, ensuite, avec l’utilisation d’appareils commer- ciaux permettant de déterminer polarisation et capacitance ; microscopie à sonde locale, finalement, utilisée d’une part pour l’observation des surfaces des échantillons mais également pour la mesure des coefficients piézoélectriques, avec un dispositif comportant un microscope à effet tunnel (STM) dans lequel a été implémenté un circuit Sawyer-Tower, permettant une mesure simultanée de la déformation et de la polarisation de l’échantillon sous application d’un champ électrique externe.

Un modèle électrostatique, basé sur la théorie de Landau et utilisant uniquement les coefficients de Landau pour les matériaux massifs a été développé. Sans aucun paramètre ajustable, ce modèle permet de prédire polarisation, température de transition et tetragonalité des superréseaux en fonction d’un paramètre unique appelé fraction de volume de PbTiO₃ et correspondant au pourcentage de PbTiO₃ présent dans l’échantillon. Ce modèle démontre que la polarisation dans les échantillons est uniforme. Son pouvoir de prédiction est très bon pour des échantillons possédant des couches de PbTiO₃ et SrTiO₃ relativement épaisses. Pour des couches très fines, le modèle ne s’applique plus et les échantillons présentent un caractère typique de ferroélectricité impropre, dans laquelle le paramètre d’ordre principal ne correspond pas à la polarisation mais à une autre distorsion structurale du matériau, dans ce cas précis une rotation des octaèdres d’oxygène, ainsi que l’ont démontré les calculs ab-initio. Ces rotations, présentes uniquement aux interfaces, induisent une polarisation localement, qui s’uniformise dans tout l’échantillon pour diminuer le coût énergétique électrostatique. Ce couplage unique, conduisant à un nouveau type de ferroélectricité, est le résultat principal de cette thèse.

La possibilité du choix de valeurs spécifiques de polarisation ou de constante diélec- trique tout en gardant une structure cristalline parfaite, d’une part, et la modifica- tion d’une propriété macroscopique par un couplage d’instabilités dans les interfaces, d’autre part, font des ces superréseaux des candidats intéressants pour des appli- cations potentielles. Ces matériaux artificiels ouvrent un champ de recherche très large. On peut en effet imaginer créer des matériaux dans lesquels un paramètre d’ordre est utilisé pour en contrôler un autre. Ceci pourrait peut-être conduire à un couplage magnétoélectrique entre un ordre magnétique associé à des rotations des octaèdres d’oxygène (par exemple dans BiFeO₃) contrôlé par un état de polarisation macroscopique.

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Chapter 1 Introduction

Ferroelectric materials, discovered in 1921, are widely used in commercial devices, often for their pyroelectric and piezoelectric properties, for instance in infrared de- tectors or ultrasonic sensors. During the last 20 years, the possibility to use them as memory devices, due to their reversible and stable states of polarization, but also the need for high dielectric constant insulators, motivated a lot of research, both theoretical and experimental, that contributed to a better understanding of their different properties.

Nowadays, growth techniques allow epitaxial ferroelectric thin films under various strains or artificial materials with novel properties to be grown. The effect of strain on perovskite ferroelectric materials has been studied both theoretically and experimentally [1, 2, 3, 4]. In recent impressive examples, ferroelectricity has been induced at room temperature in SrTiO3 [5] and ferroelectric properties of BaTiO3

were stongly modified [3], with a large enhancement of the polarization and of the transition temperature. Another key advantage of these growth techniques is the possibility to create new materials by producing artificial structures based on well defined ultrathin layers of different components with various properties. The properties of the new material can be a simple “mixture” of the properties of the two parent materials, or can show, in certain cases, some completely different and unexpected behaviors. For example, the presence of well-defined interfaces can be the origin of new physics, as demonstrated in Ref. [6], where it was found that the interface between two insulating materials can become superconducting.

In order to highlight the role of strain and interfaces in ferroelectric thin films, we present here two studies on ferroelectric and artificial ferroelectric materials. Firstly, we look at an example of the role of the strain in a simple ferroelectric material, Pb(Zr_0.2Ti_0.8)O₃, before moving to a detailed study on artificial materials made of thin layers of PbTiO3 and SrTiO3. This system, which was originally chosen as a test system for epitaxial ferroelectric oxide superlattice growth, revealed a

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4 Chapter 1. Introduction

completely novel physics that required detailed experimental and theoretical studies.

In this superlattice system, we can tune the different properties of the samples by varying the ratio of the layer ticknesses of the two materials, but we also show that the interfaces between these two insulating ferroelectric and paraelectric oxides are the scene of structural modifications giving rise to a completely new ferroelectric behavior.

The basic concepts that we will need for this work are developed in Chapter 2, where we present the essential background for a good understanding of the further studies. We discuss the ferroelectric perovskite materials, their crystalline structure and basic properties, as well as the Ginzburg-Landau-Devonshire mean-field theory used as a guideline to understand the tuning of the ferroelectric properties in our superlattices. Chapter 3 presents the different materials that we use and the techniques used to grow thin films. We present the different techniques used for structural characterization of our films in Chapter 4, along with advanced characterization of the samples. Chapter 5 is a guide to electrical measurements, with special attention to common artefacts that can give rise to misleading results. Finally, Chapter 6 and 7 are devoted to the main studies on the role of strain on Pb(Zr0.2Ti0.8)O3 and on the properties of PbTiO₃/SrTiO₃ superlattices.

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Chapter 2 Background

Ferroelectric materials possess a switchable spontaneous polarization. They are also pyroelectric and piezoelectric, which means that a variation of temperature and a variation of strain respectively induce a change in the polarization. They can have many different crystalline structures, for example monoclinic for Rochelle salt which is the original compound in which ferroelectricity was discovered in 1921 [7]. The discovery in 1945 [8] of ferroelectricity in BaTiO₃ which has a perovskite structure, marked the origin of more active research and the perovskite family of ferroelectrics remains the most studied at the present time. In this chapter, we will discuss the spe- cific perovskite class of ferroelectric materials and the phenomenological mean-field theory originally developped by Devonshire using the Ginzburg-Landau formalism to describe their macroscopic properties. We will also present a simple electrostatic model that we apply later to study the experimentally measured properties of PbTiO₃/SrTiO₃ superlattices.

2.1 Perovskite ferroelectrics

Perovskite is the name given to a pseudo-cubic structure of the general formula ABO₃. The valence of the A cation ranges from +1 to +3 and the one of the B cation from +3 to +6. Depending on the nature of the elements, this structure can be insulating (paraelectric or ferroelectric, antiferromagnetic, ferromagnetic or with more complex magnetic orders), conducting (metallic with or without anti- or ferromagnetic order) and even superconducting. “Perovskite ferroelectrics” are insulating and have a perovskite cubic structure in the high-symmetry paraelectric phase with space groupP m¯3m. They undergo a structural phase transition, leading to a lowering of the symmetry and to ferroelectricity. Neumann’s principle [9, 10]

tells us that the symmetry of any physical property should be higher or equal to the symmetry of the crystal point group. Therefore, from the 32 crystal point groups,

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6 Chapter 2. Background

only 20¹ allow piezoelectricity, since a necessary condition is the absence of center of symmetry. Pyroelectricity requires a unique polar axis to have a spontaneous polarization in absence of an external field and this requirement reduces the number of possible point groups to 10. Ferroelectric materials are then pyroelectric crystals for which the direction of the spontaneous polarization can be switched by an external field.

Fig. 2.1 shows the typical perovskite structure with the A atoms in the corners and the B cation at the center of six O atoms forming an octahedra.

Figure 2.1: ABO3 perovskite structure with theA atoms (green) in the corners and the oxygen octahedra (red atoms) surrounding the B cation (blue) in the center.

2.1.1 A-site vs B-site driven ferroelectrics

In the materials with the B-site occupied by a Ti atom, that are of particular interest in this work, we can distinguish between different behaviors. The tolerance factor t is defined by t = (RA+RO)/√

2(RB+RO) with RA and RB the ionic radii of the cations and R_O the ionic radius of the anions. For t = 1 we are in the presence of an ideal perovskite structure. However, depending on the ionic radius of the A-site cations, the titanium atom in the B-site can be either too small or too big for its site (t >1ot t <1 respectively). We call the class of materials with t >1B-site driven because the Ti atom has a tendency to move off center. These materials have often a rombohedral structure, like BaTiO3 at low temperature. The materials with t <1, called A-site driven, are often not ferroelectric and find favorable bond lengths by tilting the oxygen octahedra [11], examples include SrTiO₃ or CaTiO₃, which have a tolerance factor very close to but smaller than 1 [12]. However, in the case of PbTiO₃, the A-sites are occupied by Pb²⁺ atoms and the hybridization between the Pb 6s states with the O 2p states leads to a large strain, that stabilizes the tetragonal

1This number is reduced from 21 to 20, since the cubic group 412, although non centrosymmetric, does not allow piezoelectricity due to other symmetry elements.

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2.1. Perovskite ferroelectrics 7

phase, and also leads indirectly to changes in the hybridization between the B-site cation (Ti) and the oxygens, hybridization that is necessary for ferroelectricity [13].

Figure 2.2: Perovskite structure of ferroelectric PbTiO₃with a displacement of theB cation and of the oxygen octahedra with respect to the lead atoms, corresponding to respectively up and down spontaneous polarization.

The symmetry of PbTiO₃ in the ferroelectric phase is then tetragonal and reduces fromP m¯3mtoP4mm. It is also non-centrosymmetric, with an atomic displacement of the B cation relatively to the oxygen octahedra. As can be seen in Fig. 2.2, in the case of PbTiO₃, both the oxygen octahedra and the Ti atoms are displaced in the same direction with respect to the Pb. The displacement of the oxygen octahedra is more important than the one of the Ti atoms. These displacements lead to the creation of a dipolar moment, whose direction is opposite to the one of the displacement.

2.1.2 Spontaneous polarization and domains

Ferroelectric materials have stable states with nonzero polarization in zero applied electric field, that is denoted spontaneous polarization and whose direction can be switched by application of an external field. When the crystal undergoes the ferroelectric phase transition, the direction of this spontaneous polarization may arise in any of the sixth possible directions in the case of a transition from a cubic to a tetragonal structure. Regions of the same polarization direction are called ferroelectric domains and regions between two domains are called domain walls. The preferential direction will depend on the mechanical or electrical boundary conditions. For instance, in the case of c-axis oriented PbTiO₃ films, slightly strained in-plane, the two stable positions of the B cation, shown in Fig. 2.2, correspond to two stable states of polarization. The majority of the domains will be 180^◦ oriented as shown in Fig. 2.3 (A). However, some 90^◦ domains cannot be excluded [15] and ferroelectric materials can present both 180^◦ and 90^◦ domains, which are shown on Fig. 2.3 (B).

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A B

P_s P_s

P_s

Figure 2.3: Domain walls for (A) 180^◦ domains and (B) 90^◦ domains in a tetragonal perovskite ferroelectric. The tetragonal distortion is exagerated. From Ref. [14].

The macroscopic experimental evidence of the switching of the polarization by application of an external field is observed for example in a Polarization-Electric field (PE) loop, shown in Fig. 2.4. Starting at the origin (zero polarization, for instance

P

E +E

-E_c _c

+P_s

-P_r

1

2 3

4

5

6

+P_r

Figure 2.4: PE loop showing the switching of the polarization.

in a polydomain state), we increase the field (1), leading to domain wall motion.

This will increase the size of the domains aligned with the field and new domains can also nucleate and grow. When all the domains are aligned along the field (2), the loop is saturated and the polarization variation corresponds now only to the dielectric charging. When decreasing the field, the domains are still in the same direction and the polarization decreases but remains nonzero for zero applied field (4). This polarization value is called remanent polarization+Pr. Increasing now the field in the opposite direction (negative field), there is suddenly a switching of the

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2.2. Ginzburg-Landau-Devonshire approach 9

polarization direction and the field −E_c at which this occurs is called coercive field (5). Increasing further the field, all the domains are aligned and we observe again the simple dielectric response (6). Upon ramping the field in the positive direction, we observe the same behavior at +E_c and we obtain then the full PE loop. The spontaneous polarization +P_s is usually defined as the extrapolation to zero field of the polarization at high field (charging). P_s is often higher thanP_r in polycrystalline films but can be very close in crystals [16]. The difference can be due for instance to domains switching back after poling or to a non-linear dielectric constant.

2.1.3 Microscopic considerations

The microscopic origin of the structural phase transition has been associated to the softening of a transverse optical phonon whose frequency drops to zero, giving rise to an instability of the structure and its transformation from a cubic centrosymmetric to a non-centrosymmetric structure as disscussed above. The phonon mode whose frequency drops to zero is known as the soft-mode. The driving force for this effect is that the short range forces on the B cation due to the nearest neighbors compete with long-range dipolar electric forces through the structure. In the simplest picture, these two forces contribute to the frequency of the transverse optical phonon and the possible cancellation of the two terms can lead to a structural instability [17, 18]. A full understanding of the driving forces behind soft modes in ferroelectrics requires the nature of the bonding between ions to be taken into account. In particular, in materials like PbTiO₃, the covalent nature of the PbO bonds plays an important role. The complete theory of dynamics of atoms in crystals can be found in [19]

(especially p. 128 and ff. regarding the soft-mode theory), and a complete review of the microscopic theories can be found in [20].

In order to study ferroelectrics from a macroscopic perpsective, we will now discuss the Ginzburg-Landau-Devonshire approach to the problem. More details can be found in [21].

2.2 Ginzburg-Landau-Devonshire approach

The idea is to develop the free energyU in an even polynomial as a function of po- larizationP, assuming that all stresses are zero, and to suppose that this polynomial can describe both the paralectric and the ferroelectric phases,² with the direction of polarization and applied field along one of the crystallographic axes. In the vicinity

2The energy should have the symmetry of the high symmetry phase (paraelectric phase) which is taken as centrosymmetric.

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of the transition (i.e. for temperatures close to the transition temperature):

U = α

2P²+ γ

4P⁴+ δ

6P⁶−EP (2.1)

whereE is the electric field. We can look at the minima of this equation to find the equilibrium configuration:

∂U

∂P = αP +γP³+δP⁵−E = 0,

E = αP +γP³+δP⁵. (2.2)

To describe ferroelectric transitions, we can assume that only the quadratic coefficient α is temperature dependent. Close to the transition temperature T0, it is assumed that α depends linearly on the temperature and we can write for the general case α = β(T −T₀) where β is positive and T close to T₀. The order of the transition will now depend on the sign of γ (first order for γ < 0and second order for γ > 0), since δ is necessarily positive.³ It is convenient to look now separately at the two cases.

2.2.1 First order transition

If γ < 0 in Eq. 2.1, we can define γ⁰ = −γ. The Curie temperature T_C is defined (as can be seen in Fig. 2.5) by the conditionsU = 0 and ∂U/∂P = 0 for P 6= 0. We have:

0 = β

2(T −T₀)P²− γ⁰

4P⁴+ δ

6P⁶, (2.3)

0 = β(T −T₀)P −γ⁰P³+δP⁵. (2.4)

Solving the system for T =T_C and P 6= 0, one obtains:

T_C =T₀+ 3 16

γ⁰²

βδ (2.5)

where we see that T₀ < T_C. At T =T₀, the quadratic term in Eq. 2.3 vanishes and the function has then only two minima corresponding to two stable states withP 6=

0. These minima, describing the equilibrium values ofP at zero field, correspond to a spontaneous polarization P_s. We can get the T dependence of P_s by solving Eq.

2.4, taking the root corresponding to one of the minima whereP 6= 0 and using 2.5:

P_s²(T) = ^γ

0+√

γ⁰²+4δβ(T0−T) 2δ

P_s² = ^3γ_4δ⁰, T =T_C

(2.6)

3Forδ <0, lim

P→∞U =−∞.

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P T > T_C U

T = T

C

T₀< T < T_C

T₀= T < T_C

Figure 2.5: U versus P for a first-order transition with transition temperatureT0and Curie-Weiss temperatureTC.

T_C T

T₀ χ κ

Figure 2.6: κandχ= 1/κas a function of T for a first order transition close toTC.

E P

Figure 2.7: P versus E for a first order transition forT > TC(red curve) and with an hysteresis appearing for T < T0< TC (black curve).

T_C T

P_s

Figure 2.8: Ps as a function ofT for a first order transition close toTC.

P_s is thus zero for T > T_C and jumps to q3γ⁰

4δ when T = T_C. We can notice this typical discontinuity of the order parameter for first order transitions in Fig. 2.8.

We can also calculate the dielectric stiffness κ^X,T = ^∂E_∂P = _χ¹ where X refers to the stress, that is fixed. For the first order transiton, we obtain:

κ^X,T = β(T −T0), T > TC (2.7)

κ^X,T = β(T −T₀)−3γ⁰P_s²+ 5δP_s⁴, T < T_C (2.8) Close to T_C, we can rewrite these equations by substituting 2.5 in 2.7 and 2.8. For the latter, we can make an approximation by doing a Taylor expansion around T_C and we get:

κ^X,T = 3γ⁰²

16δ +β(T −T_C), T →T_C⁺ (2.9)

κ^X,T = 3γ⁰²

4δ + 8β(T −T_C), T →T_C⁻ (2.10)

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which is plotted in Fig. 2.6. Finally, the behavior of P as a function of E for two different temperatures (T > T_C and T < T₀ < T_C) is shown in Fig. 2.7. The black dotted line represents the states with ^∂E_∂P

T < 0 that are not accessible to the system when T < T_C.

2.2.2 Second order transition

When γ > 0 in the polynomial 2.1, the higher order terms can be neglected. We then rewrite ∂U/∂P:

0 =β(T −T₀)P +γP³ (2.11)

Forα >0, this derivative vanishes only for P = 0. For α <0, Eq. 2.11 vanishes for two values ofP 6= 0. The transition occurs therefore whenα= 0, which implies then T =TC =T0. We can calculate the evolution of Ps as a function of temperature by

P U

T > T

C

T = T

C

T < T

C

Figure 2.9: U versus P for a second order transition whereTC=T0.

T_C T

χ

κ

Figure 2.10: κandχ= 1/κas a function of T for a second order transition close toTC.

E P

Figure 2.11: P versus E for a second order transition for T > T_C (red curve) and with an hysteresis appearing for T < T_C (black curve).

T_C T

P_s

Figure 2.12: Ps as a function of T for a second order transition close toT_C.

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solving Eq. 2.11 for P 6= 0:

P_s² = β

γ(T_C −T), P_s →0 (2.12)

where we see that P_s undergoes here a continuous transition and is proportional to (T_C−T)^0.5. Since the polynomial expansion is valid only close to T_C, this equation is then valid only for small values of Ps.

For the dielectric stiffness, we obtain:

κ^X,T = ∂E

∂P =β(T −T_C) + 3γP_s². (2.13)

Using 2.12, we get:

κ^X,T = β(T −TC), T > TC (2.14)

κ^X,T = 2β(T_C−T), T < T_C (2.15)

These functions are plotted in Fig. 2.10, where we see that χ = _κ¹ ∝ _T_−T¹

C obeys then a Curie-Weiss law. In Fig. 2.11, we also note the behavior of P as a function of the applied field E, where the states that are not accessible to the system are represented by the black dotted line in the same way as in the first order case.

2.2.3 Role of strain

In all the previous discussion, we assumed that the stresses were fixed or not present and that the field and the polarization were in the same direction along one crystallographic axis. To go beyond this case, we have to write the energy in a more general form. In this case, the coefficients α, γ and δ will become respectively a_i, a_ij and a_ijk where i, j, k = 1,2,3 correspond to the three crystallographic axes in an orthogonal structure. The temperature dependence will still be assumed to be restricted to the a_i parameter and it is now the sign of a_ij that will determine the order of the transition. We have the correspondance:

α/2→a_i γ/4→a_ij δ/6→a_ijk

(2.16)

Since the high symmetry structure is cubic (paraelectric), we have some conditions on the parameters a_i, a_ij, a_ijk and the free energy can now be written as follows [22, 1, 2]:

U(P_i, e_kl) =U_GLD(P_i) +U_el(e_kl) +U_c(P_i, e_kl) (2.17)

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The first term U_GLD is the energy term taking only into account the main parameter (i.e. polarization P_i) in absence of external applied field (E = 0), U_el is the elastic energy (with parameters e_kl) and U_c the energy term coupling strain and polarization. We can write:

U_GDL(P_i) = a₁(P₁² +P₂²+P₃²) +a₁₁(P₁⁴+P₂⁴+P₃⁴)

+a₁₂(P₁²P₂²+P₁²P₃²+P₂²P₃²) +a₁₁₁(P₁⁶+P₂⁶+P₃⁶) +a112[P₁⁴(P₂²+P₃²) +P₂⁴(P₁²+P₃²) +P₃⁴(P₁² +P₂²)]

+a₁₂₃P₁²P₂²P₃² (2.18)

The second and third terms of Eq. 2.18 are coming from (P₁²+P₂²+P₃²)² separated in “direct terms” (a₁₁) and “cross-terms” (a₁₂). The same development is done for the 6th order terms.

The elastic energy, Uel depends on the linear strain tensor ekl with cij the second order elastic constants (or elastic stiffnesses):

U_el(e_kl) = 1

2c₁₁(e²₁₁+e²₂₂+e²₃₃) +c₁₂(e₁₁e₂₂+e₁₁e₃₃+e₂₂e₃₃) +1

2c₄₄(e²₁₂+e²₁₃+e²₂₃) (2.19) The third term of Eq. 2.17, U_c, corresponds to the energy linked to the coupling between these two parameters (Pi and ekl) withgij the electrostrictive coefficients:

U_c(P_i, e_kl) = −g₁₁(e₁₁P₁²+e₂₂P₂²+e₃₃P₃²)

−g₁₂[e₁₁(P₂²+P₃²) +e₂₂(P₁²+P₃²) +e₃₃(P₁²+P₂²)]

−2g₄₄(e₁₂P₁P₂+e₁₃P₁P₃+e₂₃P₂P₃) (2.20)

Our experiments are restricted to the case of a (001) ferroelectric film grown epi- taxially on a cubic substrate. The polarization will thus be perpendicular to the substrate and the in-plane polarizationsP₁ andP₂ will be zero. We redefineP₃ =P and we can rewrite the free energy expansion as:

U(P, e_kl) = a₁P² +a₁₁P⁴+a₁₁₁P⁶ +1

2c₁₁(e²₁₁+e²₂₂+e²₃₃) +c₁₂(e₁₁e₂₂+e₁₁e₃₃+e₂₂e₃₃) +1

2c₄₄(e²₁₂+e²₁₃+e²₂₃)

−g₁₁e₃₃P²−g₁₂(e₁₁P²+e₂₂P²) (2.21) Additionally, the film is free to relax along the normal direction (3) but constrained in plane (1,2) to the bulk substrate lattice parameter. Assuming that there are no

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dislocations or defects, the in-plane strains (e₁₁ and e₂₂) are controlled by the cubic substrate, so we havee₁₁=e₂₂=u_m withu_mthe misfit strain, and the shear strains (e₁₂, e₁₃ and e₂₃) must be zero. The misfit strain u_m is defined by:

u_m = a_substrate−a_{f ilm}

a_substrate (2.22)

where asubstrate is the the in-plane lattice parameter of the substrate and af ilm, the equivalent cubic cell lattice constant of the free standing film. Taking these conditions into account, the free energy expansion becomes:

U(P, e33) = a1P²+a11P⁴ +a111P⁶ +1

2c₁₁(2u²_m+e²₃₃) +c₁₂(u²_m+ 2u_me₃₃)

−g₁₁e₃₃P²−g₁₂2u_mP² (2.23)

The strain component e₃₃ can be found by the condition of zero out-of-plane stress

∂U

∂e33 = 0:

∂U

∂e₃₃ = c11e33+c122um−g11P² = 0 e₃₃ = g₁₁P²−2c₁₂u_m

c₁₁ (2.24)

Collecting the terms of the same order in P together, we get:

U(P) = a^∗₃P²+a^∗₃₃P⁴+a₁₁₁P⁶+c²₁₁+c₁₁c₁₂−2c²₁₂

c₁₁ u²_m (2.25)

where we see that the effect of the strain is to renormalize the coefficients a1 and a₁₁:

a^∗₃ = a₁+ 2(c₁₂

c₁₁g₁₁−g₁₂)u_m, (2.26)

a^∗₃₃ = a₁₁− g₁₁²

2c₁₁. (2.27)

Pure PbTiO3 with typical Landau parameters is described as first-order. Due to the renormalization of the coefficents with strain, a^{P T O∗}₃₃ is found to be >0 and the transition becomes second order. Then, as explained in Subsect. 2.2.2, the transition will occur when a^{P T O∗}₃ passes through zero (i.e. for T =T_C^{P T O∗}) and we see immediately that one of the effects of the strain will be the renormalization of the transition temperature. Since a^{P T O}₁ =β(T −T_C^{P T O}), we obtain:

0 =a^{P T O∗}₃ =a₁ + 2(c₁₂ c11

g₁₁−g₁₂)u_m =β(T_C^{P T O∗}−T_C^{P T O}) + 2(c₁₂ c11

g₁₁−g₁₂)u_m T_C^{P T O∗} =T_C^{P T O}− 2

β c₁₂

c₁₁g₁₁−g₁₂

u_m (2.28)

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For a negative (compressive) misfit strain, we see that the transition temperature T_C increases if ^c_c¹²

11g₁₁−g₁₂ is positive.

We can also write the tetragonality of the filmc/a. The out-of-plane strain is:

e₃₃ = c_{f ilm}−a_{f ilm}

a_{f ilm} , (2.29)

which can be written as c_{f ilm} = a_{f ilm}(1 +e₃₃). Using 1− u_m = a_{f ilm}/a_substrate and Eq. 2.24, one obtains the tetragonality c_{f ilm}/a_substrate,a_{f ilm} being locked by the substrate:

c_{f ilm}/a_substrate = (1 +e₃₃)(1−u_m)

=

1− 2c₁₂u_m c₁₁

(1−u_m) + g₁₁

c₁₁

(1−u_m)P²

= (c/a)para+const·P² (2.30) where (c/a)_para corresponds to the tetragonality of the paraelectric phase (i.e. for P = 0). One notices here that the measurement of the material tetragonality is telling us something about the polarization.

To get the spontaneous polarization P_s, we can simply calculate the extrema of Eq. 2.25 (cf. Eqs. 2.6 and 2.12):

dU(P) dP

_P6=0

= 0 = 2a^∗₃ + 4a^∗₃₃P²+ 6a₁₁₁P⁴ (2.31)

2.2.4 Piezoelectric properties

We have seen in Eq. 2.24 that the out-of-plane straine₃₃ can be written:

e33 = g₁₁P²−2c₁₂u_m c₁₁ .

Since the total polarization is P = PE +Ps, where PE is the polarization induced by application of an external field E:

P_E =₀χE, (2.32)

we can now define the variation of the strain under application of an external field as,

d₃₃ = de₃₃

dE₃ = ∂e₃₃

∂P

∂E₃ = 2g₁₁P

c₁₁ ₀χ. (2.33)

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The d_ij coefficients are the components of the piezoelectric tensor, d₃₃ being the coefficient relating polarization and variation of the strain in the (3) direction. The direct piezoelectric effect is the appearance of charges on the sample surface (or variation of the polarization) by application of an external stress and is usually expressed in _N^C. The converse piezoelectric effect corresponds to the production of strain by application of an external field and is expressed in ^m_V, which is totaly equivalent to the units of the direct effect. In the ferroelectric samples that are of interest in this work, the spontaneous polarization P_s has two equal and opposite values. Therefore, the term P = P_E +P_s may change sign during polarization switching since P_s is usually very large compared to P_E.

-1 0 1

-1 0 1 A

C D

d33

(a.u.)

voltage

z (a.u.)

voltage

voltage,z (a.u.)

time voltage

z

coercive voltage B

z+z

voltage

f erroelectric f ilm

Figure 2.13: Sketch of the strain response for a sawtooth voltage applied across a ferroelectric film. (A) Sketch of the experimental setup. A voltage V is applied across a ferroelectric film of thickness z, producing a change in thickness δz. (B) Grey, voltage applied across the film and black, variation of the thicknessδz as a function of time. The dotted lines represent the coercive voltage. (C)δz as a function of the applied voltage. This plot is fully equivalent to a plot of strain versus electric field. (D) Derivative ofδz with respect to the applied voltage versus applied voltage.

In Fig. 2.13 (B), we show a schematic view of the variation of thickness δz of a sample of thicknessz induced by a sawtooth voltageV applied across a ferroelectric

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film as sketched in Fig. 2.13 (A). For amplitudes below the coercive voltage, the response δz is proportional to the applied voltage, but its phase depends directly on the direction of the polarization: the material expands or contracts for a voltage applied respectively along or in the opposite direction to the polarization. When the signal reaches the coercive voltage, P_s switches, d₃₃ changes sign and the material, contracting, starts to expand. We see in Fig. 2.13 (B) that in this case the δz response has a frequency that is twice the one of the applied voltage. Since the strain is defined by δz/z and the electric field byV /z, a plot of the strain response δz/z versus applied electric field E is fully equivalent to a plot of δz versus V (see Fig. 2.13 (C)). This plot is also called SE loop or butterfly loop, because of its particular shape. Its derivative, shown in Fig. 2.13 (D), gives the d₃₃ value for any applied voltage.

2.3 Electrostatic model for superlattices

The system we will be looking at is composed of dielectric SrTiO₃ and ferroelectric PbTiO3. The two materials are stacked on top of each other with n_P unit cells of PbTiO₃ andn_S unit cells of SrTiO₃. Superlattices are then created by repeating this bilayer a certain number of times. A simple electrostatic model has been developed to study this system, similar to the one used in [23].

The displacement field D is equal to:

D=₀E+P_E +P_s =₀E(1 +χ) +P_s =₀_rE+P_s (2.34) where _r = 1 +χ. We have also:

∂E

∂P = 1

0χ =κ^X,T (2.35)

The total energy density of a bilayer can be described by:

U_SL(P_P, P_S) =xU_P(P_P) + (1−x)U_S(P_S) +U_elec(P_P, P_S) (2.36) where _P stands for PbTiO₃ and _S for SrTiO₃. The PbTiO₃ volume fraction x is defined by the number of PbTiO3 unit cells over the total number of unit cells in a bilayer, i.e. x = _nⁿ^P

P+nS. U_P(P_P) and U_S(P_S)are the free energies given by Eq. 2.25 for each material at polarization P_P and P_S respectively. The last term of Eq. 2.36 is the electrostatic energy coming from electric fields that would appear within each layer of thickness l_P,S:

U_elec(P_P, P_S) = −l_PE_P ·P_P −l_SE_S·P_S

l_P +l_S (2.37)

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2.3. Electrostatic model for superlattices 19

where P_P,S is the total polarization of each material. Without free charges, the normal component of the displacement field is continuous at the interface between PbTiO3 and SrTiO3, so we have the condition:

D_P =P_P +₀E_P =P_S+₀E_S =D_S. (2.38)

The potential through a bilayer of PbTiO3and SrTiO3 under short-circuit boundary conditions should vanish, so we have:

l_PE_P =−l_SE_S (2.39)

and we can then rewrite:

Uelec(PP, PS) = l_Pl_S

₀(l_P +l_S)²(PS−PP)². (2.40) Considering that the two layers have almost the same c-axis, we have l_P =n_P ·c, l_S =n_S·cand then:

U_elec(P_P, P_S) = n_Pn_Sc²

0c²(nP +nS)²(P_S−P_P)² = x(1−x) 0

(P_S −P_P)². (2.41) This term is always positive and scales as(P_P−P_S)². The prefactor is large compared to the other coefficients in the expansion. This means that, as discussed in references [24,25,26], a difference of polarization in the PbTiO3and SrTiO3layers is very costly in energy. Therefore, the system will try to have a nearly uniform polarizationP [27]

and thus the electrostatic energy term can be neglected. Then replacingP_P andP_S byP, Eq. 2.36 now reads:

U_SL(P) = xU_P(P) + (1−x)U_S(P) (2.42) With this equation, we can now calculate:

∂U_SL(P, x)

∂P P6=0

= 0 = 6xa^{P T O}₁₁₁ P⁴ + 4(xa^{P T O∗}₃₃ + (1−x)a^{ST O∗}₃₃ )P²

+2(xa^{P T O∗}₃ + (1−x)a^{ST O}₁ ) (2.43) which is the analog of Eq. 2.31 for a superlattice and from which we can extract the value of P(x). The misfit strain u^{ST O}_m is zero since SrTiO3 is “clamped” on bulk SrTiO₃. Thus, the quadratic coefficient is a^{ST O}₁ and the fourth order coefficient is the renormalized one, a^{ST O∗}₃₃ .

For the tetragonality, we can rewrite Eq. 2.30 as:

c/a(x) = x(1 +e^{P T O}₃₃ )(1−u^{P T O}_m ) + (1−x)(1 +e^{ST O}₃₃ )(1−u^{ST O}_m ) (2.44)

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and find (c/a)_para(x), by setting P = 0:

c/a_para(x) = 1−xu^{P T O}_m

2c^{P T O}₁₂

c^{P T O}₁₁ (1−u^{P T O}_m ) + 1

. (2.45)

Finally we can write:

c/a(x) = (c/a)_para(x) +P²(cte·x+cte⁰) (2.46) which means that for a given x, c/a= (c/a)_para+cte·P².

Concerning the transition temperature, we know that the transition will occur for T =T^∗ when:

xa^{P T O∗}₃ + (1−x)a^{ST O}₁ = 0.

We have a^{P T O∗}₃ = a^{P T O}₁ +C, with C = 2(c^{P T O}₁₂ g₁₁^{P T O}/c^{P T O}₁₁ g₁₂^{P T O})u^{P T O}_m , and for T = T_C^∗, a^{P T O}₁ =β(T_C^∗ −T_C^{P T O}). We can then write:

0 = x(β(T_C^∗ −T_C^{P T O}) +C) + (1−x)a^{ST O}₁

T_C^∗(x) = T_C^{P T O}− C

β(1 + 1−x

x a^{ST O}₁ ). (2.47)

We see that for x = 1, we find the renormalized temperature of pure strained PbTiO₃, found in Eq. 2.28, which is T_C^{P T O∗} =T_C^{P T O}−C/β.⁴

In order to get d33 as a function of the PTO volume fraction, one has to add a field in the expression of the free energy and minimize it to find P:

∂USL(E, P, x)

∂P P6=0

= 0 = 6xa^{P T O}₁₁₁ P⁵+ 4(xa^{P T O∗}₃₃ + (1−x)a^{ST O∗}₃₃ )P³ +2(xa^{P T O∗}₃ + (1−x)a^{ST O}₁ )P −E. (2.48) Knowing P, it is possible to calculate the strain e^{P T O,ST O}₃₃ in the PbTiO3 and SrTiO₃ layers, using Eq. 2.24, and to get the c-axis of each layer, using c = (1 +e₃₃)cpseudomorphic, as a function of PTO volume fraction and field. d₃₃ is then found by calculating the variation of the c-axis of the bilayerδz, with the application of a voltage through the bilayer.

Using the known coefficients for bulk PbTiO3 and SrTiO3, given in Tab. 3.1 p. 25, one can predictP,c/a,T_C andd₃₃ for finite temperatures as a function of the PTO volume fraction.

4Note that Eq. 2.47 does not apply to x=0. This refers to the fact thatT_C is not defined for SrTiO₃within the theoretical framework outlined here.

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Chapter 3 Constituent materials: basic properties and film growth

The two main studies in this work are based on ferroelectric and paraelectric materials. Firstly, we wanted to study the effect of strain on ferroelectric properties, which motivated the choice of Pb(Zr0.2Ti0.8)O3, a ferroelectric material presenting a rea- sonnably large lattice mismatch with the commonly used SrTiO₃ or niobium doped SrTiO₃ substrates. Secondly, in order to study artificial materials, mixing thin layers of paraelectric and ferroelectric materials with high crystalline quality, we chose two materials presenting a low lattice mismatch with the substrate: PbTiO₃ and SrTiO₃.

= thin film (superlattice) = electrode

Figure 3.1: Combination of layers of ferroelectric and paraelectric materials in a superlattice structure (or simple ferroelectric layer) with a top metallic electrode. The bottom electrode can be either a metallic substrate, or a metallic layer deposited on an insulating substrate.

In order to characterize these samples electrically, we need electrodes. A metallic substrate, such as Nb:SrTiO₃, can be used as a bottom electrode. This role can also be played by a metallic layer deposited on an insulating substrate, such as SrTiO3. The quality of the whole structure depends directly on the crystalline quality of

21

Artificial ferroelectric materials

Thesis

Reference

Artificial ferroelectric materials

Artificial Ferroelectric Materials

Contents

Abbreviations

Résumé

Chapter 1 Introduction

Chapter 2 Background

2.1 Perovskite ferroelectrics

2.1.1 A-site vs B-site driven ferroelectrics

2.1.2 Spontaneous polarization and domains

A B

2.1.3 Microscopic considerations

2.2 Ginzburg-Landau-Devonshire approach

2.2.1 First order transition

2.2.2 Second order transition

2.2.3 Role of strain

2.2.4 Piezoelectric properties

2.3 Electrostatic model for superlattices

Chapter 3

Constituent materials: basic properties and film growth