Combining PbTiO3 and SrTiO3 toward 180° ferroelectric domains

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Thesis

Reference

Combining PbTiO3 and SrTiO3 toward 180° ferroelectric domains

FERNANDEZ, Stéphanie

Abstract

Cette thèse se centre sur l'étude des matériaux ferroélectriques et plus particulièrement sur le PbTiO3, qui se présente sous la structure cristalline pérovskite. Les ferroélectriques sont définis comme des matériaux ayant une polarisation spontanée qui doit pouvoir être renversée par l'application d'un champ électrique. Les ferroélectriques, connus depuis 1920, ont été très étudiés sous forme de céramiques. C'est grâce à l'évolution et au contrôle de méthodes de dépôts permettant la croissance de couches minces de qualité, que de nouvelles propriétés ont été découvertes. Ces avancées ont permis de mettre en évidence l'importance et le rôle des conditions de bords: un mauvais écrantage de la polarisation peut entraîner la formation de domaines de polarisation opposée, appelés domaines à 180°. Ces domaines, pouvant être observés par rayons X et microscopie à force atomique, ont des propriétés étonnantes modifiant la réponse physique du matériau – point central de ce travail.

FERNANDEZ, Stéphanie. Combining PbTiO3 and SrTiO3 toward 180° ferroelectric domains . Thèse de doctorat : Univ. Genève, 2017, no. Sc. 5046

DOI : 10.13097/archive-ouverte/unige:94327 URN : urn:nbn:ch:unige-943275

Available at:

http://archive-ouverte.unige.ch/unige:94327

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

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UNIVERSITÉ DE GENÈVE Department of Quantum Matter Physics

FACULTÉ DES SCIENCES Professeur J.-M. Triscone

Combining PbTiO ₃ and SrTiO ₃ toward 180 ^◦ ferroelectric domains

THÈSE

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

par

Stéphanie FERNÁNDEZ-PEÑA

de Thônex (Suisse)

Thèse n^◦5046

GENÈVE Imprimerie Harder

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To my parents, Consuelo and José-Antonio and

To my love, Raphaël

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Albert Einstein

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Résumé

Cette thèse se centre sur l’étude des matériaux ferroélectriques et plus particulièrement sur le PbTiO3, qui se présente sous la structure cristalline perovskite. Les ferroélectriques sont définis comme des matériaux ayant une polarisation spontanée qui doit pouvoir être renversée par l’application d’un champ électrique. Outre cette propriété, ils sont connus pour avoir une constante diélectrique qui varie notablement en fonction de la tem- pérature, tout particulièrement pour des températures proches de leur transition vers l’état paraélectrique. Les ferroélectriques, connus depuis 1920, ont été très étudiés sous forme de céramiques. C’est grâce à l’évolu- tion et au contrôle de méthodes de dépôts permettant la croissance de couches minces de qualité, que de nouvelles propriétés ont été décou- vertes. Ces avancées ont permis de mettre en évidence l’importance et le rôle des conditions de bords: un mauvais écrantage de la polarisation peut entraîner sa diminution voir même sa disparition, ou encore la formation de domaines de polarisation opposée, appelés domaines à 180^◦. Ces domaines, observés dans des couches minces par rayons X ainsi que par microscopie à force atomique, ont des propriétés étonnantes modifiant la réponse physique du matériau.

Cette thèse s’axe ainsi naturellement autour du comportement de ces domaines en présence d’un champ électrique. Pour ce faire, le PbTiO3 a été étudié sous forme de superréseaux et en solutions solides sous forme de couches minces.

Les superréseaux sont formés de couches alternées de PbTiO3 et de SrTiO3, un diélectrique. Des domaines à 180^◦apparaissent naturellement grâce aux couches diélectriques entourant le PbTiO3. Des mesures synchrotron ont permis de mettre en lumière deux régimes de réponse de ces domaines lors de l’application d’un champ électrique. Lorsque de faibles champs sont appliqués, les domaines ayant la polarisation dans le sens du champ vont croître et les autres décroître, et ce de manière

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réversible. Si toutefois les champs appliqués sont plus forts, un deux- ième régime apparaît impliquant un changement de structure irréversible.

Cette conclusion nous a permis de nous concentrer ensuite sur l’étude du comportement diélectrique des parois de domaines lors de l’application de faibles champs. Ces superréseaux nous ont également permis d’étudier plus en détail les effets des conditions de bord sur l’état de polarisation des couches ferroélectriques. Nous avons ainsi montré qu’en changeant la taille des couches diélectriques entourant le ferroélectrique, nous pouvons influencer son état de polarisation: monodomaines ou polydomaines.

Ces superréseaux nous ont également permis d’étudier l’effet de la ca- pacité négative. Ce phénomène, résultant de la frustration de la couche ferroélectrique, permettrait de concevoir des transistors avec un diélec- trique plus épais permettant de réduire les courants de fuites; large prob- lème résultant de la miniaturisation des transistors. Nous avons montré de façon théorique et expérimentale que les domaines ferroélectriques sont le point clé vers la compréhension de l’effet de capacité négative.

Nous avons ensuite fait croître des solutions solides de Pb_1−xSr_xTiO₃ sous forme de couches minces. Leur caractérisation par rayons X a permis d’étudier l’évolution du paramètre de maille ainsi que de la température critique de transition en fonction de la composition. Ceci permet en effet de choisir sur demande, la constante diélectrique de la couche. De plus, la taille des domaines à 180^◦s’est avérée être telle que leur observation a pu être faite exceptionnellement par rayons X et par microscopie à force atomique. Ainsi nous avons pu comparer les valeurs des tailles des domaines et des périodicités obtenues avec ces deux méthodes de mesure.

De plus, l’effet de ces domaines sur la réponse diélectrique a été inves- tigué pour des couches minces sous forme de capacité. Les domaines, répondant en “vibrant” à une faible excitation électrique, contribuent am- plement à la réponse diélectrique. Son évolution en température a été suivie et un changement du comportement des domaines entre basses et hautes températures a été observé, indiquant potentiellement la présence de polarons. Ces derniers, actifs à hautes températures, seraient respon- sables de fluctuations temporelles du potentiel ressenti par les parois de domaines. Ainsi à basses températures, là où les polarons sont gelés, les parois de domaines suivent un comportement de type Rayleigh, ce qui n’est pas le cas à hautes températures.

Ainsi, qu’il s’agisse de couches minces ou de multicouches, les domaines et leurs parois font de ces matériaux des entités intéressantes non seulement du point de vue de la recherche fondamentale, mais également pour de nombreuses applications futures.

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Chapter 1

Introduction

Technology is nowadays chasing for smaller and smaller functional devices to increase the performance of what became our essential and indispens- able electronic equipment. By constantly reducing the size of certain devices such as transistors which, since their discovery in 1947, have been miniaturized from centimeters down to tenths of nanometers, several lim- itations alter the usual device functioning. On the way to replacing con- ventional gate dielectrics in transistors, a potential avenue is offered by ferroelectrics. Discovered more than ninety years ago, ferroelectric materials, characterized by a spontaneous reversible polarization, were first studied in ceramics and crystals which have been used as transducers, sensors or actuators in industrial and military applications. Following the miniaturization trend, and combined with technological advancements in film growth, a wide field of study around ferroelectric thin films was developed. Entering into thenanoworld in which the size of the films can be obtained with a precision reaching the atomic level, a plethora of amazing new phenomena is being discovered and their interesting functional properties unveiled.

The present study

In this thesis, we are interested in what happens with ferroelectrics when they are in thenanoregime. We study one of the most commonly used and best known ferroelectrics that has a perovskite crystalline structure, namely PbTiO₃. Its intrinsic properties are investigated under different electrical environments and more specifically when they force 180^◦ fer-

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roelectric domains to form. Such domains are at the center of attention throughout this manuscript and studied from many different perspectives.

In the second chapter of this thesis, the concept of ferroelectricity is intro- duced. In chapter 3, the details on sample growth and different characterization techniques used throughout the thesis are discussed. In chapter 4, careful attention is paid to the dielectric measurements. Chapter 5 fo- cuses on PbTiO3/SrTiO3 superlattices. The origin of the180^◦domains in this model system and their evolution under electric field are studied via synchrotron measurements. The emphasis is on the role of the depolarization field and its tuning capabilities naturally introduce chapter 6, which concerns a possible application: the negative capacitance. Another combi- nation of PbTiO3and SrTiO3is a mixed solid solution of Pb1−xSrxTiO3in a thin film form which is fully characterized in the first half of chapter 7 via X-ray diffraction and atomic force microscopy. Specific attention is paid to the presence of domains. In the second half of this chapter, a dielectric study is discussed for Pb1−xSrxTiO3, in which the role of the domain wall response is analyzed at different temperatures.

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Chapter 2

Introduction to Ferroelectricity

Since its discovery in 1920 by Valasek [1, 2], ferroelectricity has attracted lots of attention both from a theoretical and experimental point of view.

A ferroelectric material is defined by its capability to display two or more metastable polarization states at zero bias with the possibility to switch from one to another by the application of an external electric field. Fer- roelectic materials are alsopiezoelectricwhich means that electric charges appear at the surfaces of the material once strain is applied to it, andpy- roelectric as well, meaning that the electrical polarization depends upon temperature. Such fundamental properties are especially appealing from a technological point of view and many routes have been developed towards different applications like ferroelectric random access memories or non-volatile memories among others.

In this chapter, we will focus on ferroelectric oxidesand more specifically on those with the very simple formula of ABO₃ called perovskites.

They appear in nature and the first one discovered was BaTiO₃ (BTO) in 1949. Because of their simple chemical form, perovskites are thoroughly used in applications as well as for theoretical studies. Their relative structural simplicity opened many doors towards the deep understanding of ferroelectricity. From an experimental point of view the ABO3 structure allows a multitude of different engineering structures due to its compati- bilities with commercial substrates. Moreover, the perovskite oxide family is large. Depending on the type of A and B cations, the material can either be conducting or insulating. Those which are ferroelectric, display

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Figure 2.1: Dielectric constant versus temperature along the c and a axis for BaTiO3 single domain crystal showing the different structural phase transitions. Adapted from [3]. There are three ferroelectric states: tetragonal with the polarization oriented along the [001] axis, orthorhombic with the polarization oriented along [011] axis and rhombohedral with the polarization oriented along [111]. Such polar distortions are accom- panied with a small change in size and shape of the unit cell.

common features like being paraelectric in the cubic form and therefore highly symmetric with a lowering of symmetry in the ferroelectric phase.

For instance BTO, the archetype ferroelectric material, undergoes many structural phase transitions going from cubic (and paraelectric) to tetragonal at around 120^◦C, orthorhombic around 0^◦C to finally rhombohedral around -90^◦C while staying ferroelectric below 120^◦C [3] as captured in Fig. 2.1 .

In the perovskite structure, ferroelectricity appears as a distortion of the cubic cell. Indeed, as Goldschmidt described in 1926, the stability of such structures can be understood by the relative size of the cations with respect to the oxygen atoms. A tolerance factortis defined withrXbeing the radius of atom X as follows :

t= rA+rO

√2(rB+rO). (2.1)

Only for values oftclose to one, an ideal cubic perovskite structure can be formed. Whent >1, like it is the case for BaTiO3, the size of the oxygen octahedron is dictated by the size of the atomic bond length between A and O. In order to counterbalance the fact that B is too small, a polar distortion will develop. On the other hand, whent < 1, the cell size will be given by the B to O distance. Hence one way of compensating the frustration

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2.1 Phase transition:

A phenomenological approach of the A cation (of being too small) is to induce tilts and rotations of the octahedra like in SrTiO₃(which is not a ferroelectric).

For the perovskite ferroelectrics, one can visualize the tetragonal non- centrosymmetric states corresponding to the up and down polarization of the ferroelectric due to a displacement of the B cations and the oxygen octahedra with respect to the A cations as shown in Fig. 2.2.

All these microscopic pieces of information about crystallographic sig- natures of perovskite ferroelectrics have been obtained thanks to the combined efforts of theoretical structure modeling and experimental techniques like X-ray diffraction on ceramics or single crystals. One may as well gain a deep understanding on ferroelectricity from a phenomenological and macroscopic point of view by studying the model developed by Devonshire based on the Ginzburg-Landau formalism, as discussed below.

Pb Ti O

Cubic phase Polarization UP Polarization DOWN

Figure 2.2: On the left hand side, a cubic perovskite structure where the Ti atom is sitting right in the middle of the oxygen octahedron. The middle and right structures, correspond to the tetragonal perovskite with up and down polarization states, respectively. The centered cations as well as the oxygen octahedra are displaced with respect to the A atoms.

2.1 Phase transition:

A phenomenological approach

A phase transition is a rather complex phenomenon that induces changes in the physical properties of the matter such as paramagnetic to ferro- magnetic, metal to insulator or structural distortions appearing upon a

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pressure or temperature change. The aim of this section is to shine light on why a material chooses a particular phase at a given pressure or temperature.

Following a thermodynamic approach for a temperature driven transition, which we will now focus on, one can understand a phase transition by studying the evolution of the relative stabilities of the different phases.

For a given set of pressure and temperature the more stable phase is the one that minimizes the Gibbs energy:

G=H−TS= (U+P V)−TS (2.2) where H is the enthalpy, T the temperature, P the pressure, U the internal energy,Sthe entropy and V the volume. At low T, the structure is more compact and often with a lower symmetry and therefore the energy is governed by the enthalpy term. A higher T, the structure is less compact - often in a higher symmetry, the atoms are more weakly linked, and the entropy term dominates. Hence, generally a phase transition occurs when the energy of one phase wins over the energy of the other one.

2.1.1 Landau-Devonshire: a macroscopic theory

In the late thirties [4], Landau captured the non-trivial concepts of phase transitions in a symmetry-based model. The mainansatz in the Landau model is that in the vicinity of the phase transition, the Helmoltz free energy F can be written as a polynomial expansion of an order param- eter. The order parameter is a physical quantity either macroscopic or microscopic that goes from zero to a non-zero value through the transition. From symmetry arguments, the expansion ofF has to be valid for temperatures above and below the transition. The terms that are allowed in the expansion are determined from the phase of the highest symmetry.

The phenomenological approach of the theory comes from the fact that it takes as inputs parameters that are either measured or calculated by first-principle calculations. (Hence, it is important to point out that such macroscopic model can only be as good as the given parameters are.) It is based on the assumption that the fluctuations of the order parameter are small and hence it is a useful model for long range interactions systems such as ferroelectrics. It is in 1949 [3, 5] that A.F. Devonshire was the first in implementing Laudau theory to ferroelectrics. It is important to mention that the Landau-Devonshire model is valid for a bulk system with uniform polarization. In order to introduce domains i.e. spatial varia- tion of the polarization and study the thin films with boundary conditions,

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A phenomenological approach Landau-Ginzburg theory is required. (Laudau theory is valid for displacive materials since by definition of the construction of the Landau theory, the order parameter vanishes uniformly everywhere above the critical transition temperature)

Focusing now on ferroelectric materials with the ABO3chemical structure, it feels natural to use the ferroelectric polarization as the main order parameter. Such structural phase transition comes along with a lattice distortion which should also be taken into account as a secondary order parameter. Nevertheless, for the rest of the discussion we will neglect it for simplicity.

The paraelectric case is centro-symmetric therefore only the even terms are allowed in the expansion. The reason for that, is that the energy should not depend on the orientation of the polarization.

F = α 2P²+γ

4P⁴+δ

6P⁶ (2.3)

To find the polarization state, one should minimizeF with respect to P

∂F

∂P =E (2.4)

giving rise to a simple expression of the electric field as a function of the polarization

E=αP +γP³+δP⁵. (2.5)

In this theory, and close to the phase transition one assumes that αlin- early depends on the temperature, changes sign atT₀ and that the other coefficients are temperature independent;

α=β(T −T0) (2.6)

withβ >0andT0the transition temperature. To ensure the stability of the ferroelectric phase, it is required that eitherγorδare positive. As we will see bellow, the sign of the coefficientγ strongly changes the evolution of the order parameter as a function of the temperature. Ifγis negative, the transition will be called first order (discontinuous) and theδterm should be taken into account and positive. For positiveγ, it is a second order (continuous) phase transition and theδterm can be neglected.

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First order phase transition

The first order phase transition appears whenγ <0in Eq. (2.3), whileδ is positive. The Curie temperatureTC corresponds to the temperature at which the system abruptly develops a polarization and is defined by two conditions: F = 0and∂F/∂P = 0forP 6= 0. Such a discontinuity in the order parameter is the signature of a first order phase transition. We then have the following withE= 0:

F= 0 = β

2(T−T0)P²+γ 4P⁴+δ

6P⁶ (2.7)

and

∂F

∂P = 0 =β(T−T0)P+γP³+δP⁵ (2.8) and by solving we getTC =T0+_16δβ^3γ² implying thatTC> T0.

As sketched in Fig. 2.3, forT > TC, three different wells appear inF with only one being a minimum atP = 0. ForT =TC, the three wells appear as three local minima withF = 0: one state hasP = 0 and the two others haveP 6= 0. ForT₀< T < T_C the state atP = 0is still a local minimum of the free energy. AtT =T₀, only two solutions with non zero polarization are stable and those states correspond to the spontaneous ferroelectric polarization. It is hence possible to look at the temperature dependence of the spontaneous polarizationP_sfrom Eq.2.8.

P_s²(T) = −γ±p

γ²−4δβ(T −T0)

2δ . (2.9)

AtT_C, the three possible polarization states are degenerate. Whether the system will be inP = 0or P 6= 0 state will depend whether T_C is ap- proached from a higher (T_C⁺) or a lower (T_C⁻) temperature. If we are cooling, atT_C the system will stay at zero polarization and upon further cooling, the system will jump to eitherP_s⁺orP_s⁻; this jump being a signature of a first order phase transition.

P_s⁺= r

−3γ

4β ; P_s⁻=− r

−3γ

4β. (2.10)

Upon heating, atT_C, the system will be in either of the two polarizations P_s⁺ orP_s⁻depending on its history.

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A phenomenological approach Dielectric stiffness

Let us now look at the dielectric stiffness defined asκ=_χ¹ =0∂²F

∂P². From Eq. 2.5, we get:

ForT > TC, κ/0=β(T−T0).

ForT < T_C, κ/₀=β(T−T₀) + 3γP_s²+ 5δP_s⁴. (2.11) Second order phase transition

In the case of a second order phase transition sinceγ is positive, higher order terms can be neglected. As a starting point we shall write with E= 0:

∂F

∂P = 0 =β(T−T0)P+γP³. (2.12) Forα >0, only one solution of polarization is found to Eq. 2.12: P = 0.

For α < 0, two solutions of polarization are valid for Eq.2.12. The ferroelectric transition occurscontinuously whenα= 0with a spontaneous polarization appearing atT =T0which by definition is the Curie temperature and therefore for a second order phase transitionT0 =TC. Hence the solution of Eq. 2.12 forP 6= 0givesPs:

P_s²= β

γ(TC−T). (2.13)

At zero field and for T < T_C, the free energy landscape displays two minima withP =±Ps. The energy barrier between the wells determines the minimum electric field required to switch all the dipoles together.

Dielectric stiffness

Let’s now look at the dielectric stiffness:

κ= 1 χ =0

∂²F

∂P² =0

β

γ(TC−T) +03γP_s². (2.14) Hence depending on whether we are above or belowTCand using Eq. 2.13 we get:

ForT > T_C, κ/₀=β(T−T_C).

ForT < TC, κ/0= 2β(TC−T). (2.15)

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T₀<T<T_c F

P T>T_c

T=T_c T₀>T<T_c

T<T_c T=T_c

P F

T>T_c ^P

TC T

First order phase transition

Second order phase transition

TC

κ

T

χ

P

T0 TC T

TC T χ

T0

TC T

χ κ

T0

P

E

P

E

Figure 2.3: The top part of the figure describes the first order phase transition and the bottom is for the second order. For both types the free energy as a function of the polarization is displayed for different temperatures across the phase transition. The dielectric susceptibility is displayed as a function of temperature showing a clear difference between the two types of phase transitions. Finally, the behavior of the polarization across temperatures shows the continuous increase of P for the second order and the sudden jump for the first order atT_C.

The dielectric susceptibility which is defined as χ = 1/κ ∝ _T−T¹

C reflects that a singularity is present atT_C. This behavior is called the Curie- Weiss Law. Note the fact that theαterm is directly linked to the dielectric constant. From Eq. 2.14 we obtain:

ForT > TC, P = 0,→χ= 1 α. ForT < TC, P²= −α

γ →χ=− 1 2α.

(2.16)

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A phenomenological approach

2.1.2 Landau-Devonshire: coupling to strain

The Landau-Devonshire theory is valid for bulk mono-domain (poled) ferroelectrics. Nevertheless the ferroelectric properties are known to strongly depend on strain; therefore we shall add a strain-dependent term to the LD theory in Eq. 2.3. This will be of great interest since later on we will study ferroelectric epitaxial films strained to substrates. The difference in lattice parameter will hence induce a strain state [6]. The polarization is now decomposed into its components with respect to the three crystallographic axis i,j and k. A correspondence will be used between α/2, γ/4 andδ/6withαi,αijandαijk. The allowed terms in the energy expansion should imperatively be permitted by the highest symmetry structure. The free energy can be written as follows:

FLD=FLD(Pi) +Fstrain(Si) +Fstrain−polarization−coupling(Pi, Si)

=α1(P₁²+P₂²+P₃²) +α11(P₁⁴+P₂⁴+P₃⁴) +α111(P₁⁶+P₂⁶+P₃⁶) +α12(P₁²P₂²+P₁²P₃²+P₂²P₃²) +1

2c11(S₁²+S₂²+S₃²) +c12(S1S2+S2S3+S1S3) +1

2c44(S₄²+S₅²+S₆²)

−g11(S1P₁²+S2P₂²+S3P₃²)

−g₁₂[S₁(P₂²+P₃²) +S₂(P₁²+P₃²) +S₃(P₁²+P₂²)]

−2g44(S4P2P3+S5P1P3+S6P1P2)

(2.17) α1,α11andα12are the phenomenological LD coefficients being physically linked to the dielectric stiffness and higher order stiffness coefficient at constant strain. Sn defines the strain tensors with n=1,...,6 defined as:

S= (Sxx, Syy, Szz, Syz, Sxz, Sxy)≡(S1, S2, S3, S4, S5, S6).cnlare the elas- tic stiffnesses andgnl are the electrostrictive tensors, both at constant order parameter (here polarization). (A discussion on the use of Helmholtz or Gibbs energy can be found in Appendix A.)

For the specific case of a tetragonal film grown on a cubic substrate with an out-of-plane polarization, namelyP1 =P2 = 0,P3 =Pz 6= 0, we haveS4=S5=S6= 0due to the absence of any shear [7],S1=S2=Sx

andS₃=S_z. S_x= ^a^sub_a^−a⁰

0 wherea₀is the equivalent cubic in-plane lattice constant of the free standing film.

One can then rewrite Eq. 2.17 in a simpler way:

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FLD =α1P_z²+α11P_z⁴+α111P_z⁶+1

2c11(2S²_x+S²_z) +c12(S_x²+ 2SxSz)

−g11SzP_z²−2g12SxP_z².

(2.18) Let’s consider now the specific case of resulting stresses due to an in- plane clamping (and hence strain) that are such asσ1=σ2andσ3=σ4= σ5= 0. We use the mechanical equation of state: σi=−∂F/∂Si. Since

σz= 0 =−∂F/∂Sz=c11Sz+c122Sx−g11P_z² (2.19) hence,

Sz =g11P_z²−2c12Sx

c₁₁ . (2.20)

Putting this into Eq. 2.18 gives:

FLD =S_x²c²₁₁+c12c11−2c²₁₂

c₁₁ + (α1+2g11c12Sx

c₁₁ −2g12Sx)P_z² + (α11− g₁₁²

2c11

)P_z⁴+α111P_z⁶.

(2.21)

In other words, the strain renormalizes all the Landau coefficients and adds an extra coupling term. We will now call the renormalized termsα^∗. Hence,

α^∗₁=α₁+

2g₁₁c₁₂ c11

−2g₁₂

S_x (2.22)

and

α^∗₁₁=α₁₁− g²₁₁ 2c11

(2.23) FLD=α^∗₁P_z²+α^∗₁₁P_z⁴+α111P_z⁶+S²_xc²₁₁+c₁₂c₁₁−2c²₁₂

c11

. (2.24) A renormalization ofα1straightforwardly implies a renormalization ofTC

sinceα1∼(T−TC)implies thatα₁^∗∼(T−T_C^∗). From these results stems the influence of the strain on the ferroelectric transition temperature but also on the dielectric and piezoelectric properties for instance.

This macroscopic treatment of the strain-dependence in a ferroelectric film clamped on a substrate is a good starting point towards the understanding of the observed physical properties which are usually different

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A phenomenological approach from the ones in the unstrained system. As an example, PbTiO3 as bulk is a ferroelectric for which the transition is first order. Once in thin film form (without considering the strain), theα₁₁term becomes positive indi- cating a second order phase transition, while experimentally the transition appears as first order. Nevertheless once strain is taken into account, the α^∗₁₁is found to be negative, consistent with a first order phase transition.

In other words the first order “character” of the transition for a thin film material is totally due to the clamping to the substrate. (This is shown in the reference [8].)¹

Hysteresis - the ferroelectric signature

When an electric field is applied to a piezoelectric (dielectric) material it gets polarized. ² Within the approximation of smallE, one can express the polarizationP of a ferroelectric as follows

P =P_s+₀χE (2.25)

wherePsis the spontaneous polarization,χthe electric susceptibility and 0the vacuum permittivity.

The definition of ferroelectricity enforces that the polarization should be switchable by the application of an electric field leading to an hysteresis represented in Fig.2.4. Mathematically, such hysteretic behavior is obtained by solving the thermodynamic equationdF/dP =Efor P :

dF_LD

dP |P6=0= 2α^∗₁P_z+ 4α^∗₁₁P_z³+ 6α₁₁₁P_z⁵=E. (2.26) The electric field asymmetrizes the free energy and therefore one state is favored over the other leading to an hysteretical behavior like described in Fig. 2.4. This polarization loop shows that the sample will switch polarization for fields bigger than a threshold value called the coercive fieldEc. If the field increases further, the extra contribution to the polarization is the one due to the dielectric response (dielectric charging). Once the field is reduced to zero, all the dipoles stay aligned and the system retains its polarizationPr. Increasing the field in the opposite direction, the dipoles

1As a curiosity, it is important to notice that here we have stopped the expansion at the fourth order term. This comes again from the first order nature of the transition that we are interested in. Nevertheless, it has been mentioned by Kvasov and Tagantsev [9] that for strained thin films where the sixth order term is needed, higher order of the electromechan- ical coupling (P and strain coupling) are necessary.

2If the material already has a polarization, when a field is applied, the polarization will increase or decrease depending on the direction of the field.

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Figure 2.4: Polarization versus electric field of a ferroelectric material.

Once the applied field is zero, the system is either in a state up or down with a remanent polarization P_r. P_s is an estimate of the spontaneous polarization defined as the extrapolation to zero field of the polarization at higher field. Experimentally, if the voltage cycle is slow enough for the charges to equilibrate, Pr=Ps.

will suddenly switch into the opposite polarization at the coercive value.

This is of course a theoretical view which is not the one happening in real systems due to inhomogeneity of the polarization, defects, domains, etc.

A more realistic scenario for switching is the growth and nucleation of regions with opposite polarizations [10].

2.1.3 Tetragonality and Polarization

In ferrolectric materials the tetragonality along the polarization axis varies with the polarization. Focusing on thin films having a direction of the po- larizationP_z pointing away from the surface, the tetragonality is defined asc/a. Given that the out-of-plane strainS_zcan be expressed as

S_z= cfilm−afilm

a_film (2.27)

and the in-plane misfit strainSxas

S_x= asub−a0

a₀ (2.28)

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2.2 The Modern Theory of Ferroelectricity

we can write: c_film=a_film(S_z+ 1)and using 2.28 and 2.20, one get:

cfilm

a_sub = (1−2Sx)

1−2c12Sx

c₁₁

+g11

c₁₁(1−2Sx)P_z²

= (c/a)_para+cst·P_z²

(2.29)

where the first term is associated with the paraelectric c/avalue corresponding to a zero polarization case. The take home message here is that measuring c gives an indication of the polarization value. As it will be shown later, measuring theclattice constant as a function of temperature is an elegant way of determining the critical temperature of a ferroelectric.

2.2 The Modern Theory of Ferroelectricity

The so-calledModern Theory of Ferroelectricity[11–14] is a model which allows ferroelectricity to be rigorously described. The pioneers of this microscopic theory of polarization are R. Resta, D. Vanderbilt and R.D. King- Smith. They propose an elegant way of definingchangesin the polarization of a periodic solid. This rather challenging problem started by the fact that in a bulk system, when computing the polarization, the obtained value will depend upon the choice of the unit cell and also that non-polar materials would give a non-zero polarization. The trick was to understand that although the polarization could bemulti-valued, changes in polarization are single-valued. Importantly enough, experimentally, the polarization itself is not a measurable quantity and only the difference between two polarization values is.

From a classical point of view, we consider the charges to be localized.

From a quantum mechanical point of view, the electrons are in Bloch- states, namely delocalized states. The beauty of this theory is to keep the classical intuition of localized charges using the Wannier functions. A Wannier function is an integral over the Brillouin zone of the Bloch functions describing the charge density in a solid. As a result, the Wannier functions are localized in space and are a useful basis for electronic calculations. They are often used to visualize the chemical bonds and to describe displacements of charge densities. Therefore, when computing the polarization, one should keep in mind two contributions: the one of the valence electrons, the Wannier one, added to the point-charge ionic one. Within this definition, the ferroelectric spontaneous polarization is given by the difference between the polarization of the polarized system

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and the polarization of the un-polarized state.

In a centro-symmetric solid, when cations are periodically moved within the unit cell, the Wannier centers will also move resulting in a mixed ionic and covalent state. This covalency leads to aneffective charge, Z^∗, larger than the one in the pure ionic picture. This is formally written as follows:

Z_ij^∗ =Ω e

∂Pi

∂dj

(2.30) where Ωis the volume of the unit cell, dis the displacement and ethe electronic charge. This means that when a sub-lattice is displaced byd along theidirection, the polarization will be affected along the same direction but also orthogonally to it. Therefore, a change in polarizationδP results from the displacement of these effective charges: δPi = _Ω^eZ_ij^∗δdj. In other words, the Born effective charge corresponds to the polarization induced by an atomic displacement. The calculated effective charges can be significantly larger than the ionic ones. In PbTiO3, while ionically, lead is +2, the effective charge on the ion is found to be +3.9 [15]³

As a take home message, the combined concepts of the ionic charge and the Wannier electronic centers offer a dynamical way of studying ferroelectricity and the polarization. The dynamical aspect of this model relates to the fact that experimentally one can only measure changes in polarization. More details and rigorous mathematical treatment can be found in the pioneer papers in Refs. [11–14].

Structural instability

Whether the theory of ferroelectricity is derived from a microscopic point or macroscopically the final result is the same. Microscopically, the Cochran approach uses short-range interactions (described by springs in between atoms in a lattice-shell model) and long-range Coulomb (dipole-dipole) interactions to describe the system. This leads to vibrational modes called phonons to which frequenciesω are attributed. In the calculation, when ω² becomes negative, it implies that the mode is unstable. For ferrolec- tricity one of the important modes is the transverse optical phonon mode at the center of the Brillouin zone; theΓ point. It can be shown that the phonon frequency (or actually its squared value ω²) depends upon two terms: short and long range interactions. The way these two contribu-

3An equivalent and useful vision for the Born effective charge is the force induced on an ion by a uniform electric field:Z_ij^∗ =−e^δFⁱ

δEj

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2.2 The Modern Theory of Ferroelectricity

tions are balanced will change the sign ofω²[16].

ω²_{T O}= 1

µ[(Φ−4π Ω

Z^∗2

(_∞+ 2))] (2.31)

where Φ is the effective force between ions, µ the reduced mass of the cation and anion. The first term in Eq. 2.31 is the short range term which is positive and hence stabilizes the crystal structure. The second one is negative (due to the opposite charge of the anion with respect to the cation).

It therefore destabilizes the structure. Hence, when the Coulomb interactions become stronger than the short-range ones,ω becomes imaginary - the definition of an instability. Additionally, calculations show that a reduction of the Born effective charge is observed at the transition from paraelectric to ferroelectric.

Figure 2.5: ω² versus Z_{T i}^∗ reproduced from Ref. [16]. The figure shows the decomposed frequency of the TO phonon mode in the cubic phase into the short-range and long-range contribution as a function of the change of the Born effectective charge for BaTiO3.

It was reported by Cohen and Krakauer [17] that the hybridization of atomic-like states strongly influences the ferroelectric instability in the way that the hybridization reduces the short-range forces. The frequency of the transverse optical mode can be written as:ω²=ω²_SR+ω_DD² , where SRstands for short-range andDDfor dipole-dipole. As shown in Fig. 2.5, in the cubic phase, when the Born effective charge changes, both contributions change. More specifically, when the Born effective charge decreases (which is what happens through the phase transition), theω_DD² will take

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over theSRterm bringing the totalω²below zero. A more detailed discussion can be found in the course manuscript from Philippe Ghosez in Ref. [16].

Outlook

The spectacular advances in materials growth during the last decades allow condensed matter experimentalists to expand their research from ceramics and single crystals towards thin films. Indeed, techniques that will be developed later in this thesis, allow high quality layer by layer growth of such materials as perovskites on a wide range of different substrates. Hence, one can now start thinking of boundary conditions, thickness dependence, strain states and domain walls among many other things leading to new phenomena which were not observed in single crystals or in ceramics. Careful consideration is given in this thesis to domain walls in which properties absent from the domains themselves are often present [18–20]. Therefore, it is first important to focus our understanding on the basics of ferroelectricity in thin films and its size effects.

2.3 Depolarization field as the origin of 180

^◦

ferroelectric domains

2.3.1 The role of the depolarization field

For many years, it has been known that reducing the thickness of ferroelectric materials leads to a degradation of their ferroelectric properties.

This is explained as being due to bound charges at the surfaces of a ferroelectric slab creating an electric field, called thedepolarization field, which points in the opposite direction to the spontaneous polarization and hence destabilizes it. The bound charges can be screened by free charges at the surfaces of the ferroelectric, thereby reducing the depolarization field.

Such free charges can be carriers in the metallic electrodes, ions from the atmosphere or simply mobile charges from within the semiconduct- ing ferroelectric itself. Their screening ability (namely how close they can come to the surfaces modulated by the dielectric constant of the interface layer) is described by an effective screening lengthλ_eff. In a simple model considering the interfaces as parallel plate capacitors, and under short circuit boundary conditions, the depolarization fieldE_d is given by:

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2.3 Depolarization field as the origin of 180^◦ferroelectric domains

E_d∼ −^{P λ}^eff

0t

4, for a film of thicknesstand polarizationP.

Note that throughout this entire thesis, the depolarization field we are referring to, is the field that would be generated by a uniform polarization in a capacitor with a finite screening length, and that serves as the driving force for domain formation; it is not the stray field associated with the actual polydomain ground state.

λeff reflects the screening properties of the electrode, which strongly depend on the quality of the interface between the film and the electrode as well as on interface chemistry [21]. Ideal metallic electrodes would correspond toλeff = 0, resulting in a perfect cancellation of the depolarization field. In reality, however, even for structurally perfect metal-insulator interfaces, the screening charges spread over a finite length λI, and the finite dielectric constant of the interface layer_I contributes to the effective screening length viaλ_eff =λ_I/_I. More details about this model can be found in Refs. [22–25]. Ab initio calculations by Stengel et al. [23]

highlight the importance of interface chemistry in affecting the screening.

For instance, although SrRuO₃ is considered a good metal with a good ionic polarizability, its electronic capabilities have been proved not to be the best, resulting into a so-called dead-layer at the interface. Junquera et al. [26] have shown that BaTiO3 in between SrRuO3 electrodes loses ferroelectricity below 6 unit cells due to the presence of dipoles at the interfaces generating a depolarization field. Such results imply that even though incredible improvements have been made in the growth quality of oxide interfaces, their screening properties are still intrinsically limited.

The calculations show that for screening often the best electrodes to use are single elements such as platinum or gold. Saiet al.[27] have reported an enhancement of the polarization in thin films of PbTiO3 and BaTiO3

with platinum electrodes compared to oxide metallic electrodes. Stengel et al.[21] have theoretically demonstrated the importance of the chemical bonds at the metal-oxide interface. The ferroelectricity is found to be enhanced whenever those bounds are polarizable. Experimentally, in addition to the interface chemistry, strain, defects and/or grain boundaries also contribute to imperfect screening that leads to depolarization fields.

Because of a strong depolarization field, the system might lose its po-

4A potential drop is induced at each interface. Because of short-circuit boundary conditions, this results in a voltage drop across the sample equal to the sum of the voltage drops at each interface (but of opposite sign). This voltage drop depends therefore only on the two interfaces and is independent of the film thickness. It thus results in a stronger electric field for thinner samples (∆V =E·t).

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Figure 2.6: The different responses of a ferroelectric to the presence of a depolarization field. The screening charges from atmospheric adsorbates, metallic electrodes or defects in the film itself can screen the bound charges, reducing the depolarization field. In the absence of screening, the ferroelectric can either lose ferroelectricity (P = 0) or form domains.

Reproduced from Ref. [25].

larization. Nevertheless, even in the absence of good screening, ferroelectrics can find other ways of preserving their polarization as illustrated in Fig. 2.6 [25]. For example, the system could “break” into domains of opposite polarization forming so-called 180^◦ domains known as Kittel or Landau-Lifshitz domains. Such domains were observed in ultathin films of PbTiO₃ using synchrotron X-ray diffraction by S.K. Streiffer et al. in 2002 [28].

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When a sample forms domains, although the polarization within each domain may remain large, its macroscopic properties are significantly al- tered. It is therefore important to understand and control the formation of these domains. For example, the effects of screening on domain formation were studied in samples with different electrodes [28–30], dielectric spacers [31, 32] and ionic adsorbates [33, 34]. Photochemical switching of stripe domains was reported by Takahashi et al. in Ref. [35], whereas Ref. [36] investigated the possibility of polarization switching without domain formation in ultrathin films near the ferroelectric critical temperature TC. More complex PbTiO3-based heterostructures, such as ferroelectric-dielectric (PbTiO3-SrTiO3) superlattices [8, 37–39], have been particularly useful for investigating the response of nanoscale stripe domains to applied fields and their effect on the macroscopic electrical properties [40, 41]. The formation of 180^◦ domains typically leads to the reduction or even suppression of the piezoelectric response and of the macroscopic polarization [29]. In addition, many theoretical studies have addressed the formation - as well as the microscopic and macroscopic properties - of nanoscale stripe domains, revealing complex, inho- mogeneous polarization profiles [42–45] and unusual switching dynam- ics [46–48].

2.3.2 Domain formation - Kittel Law

As mentioned above, ferroelectric thin films can show rather different polarization states depending on their electrical boundary conditions. Addi- tionally, the mechanical boundary conditions are also important. For the case of PbTiO3 on a SrTiO3 substrate, the in-plane lattice parameter of SrTiO3forces the long axis of PbTiO3, and therefore its polarization, to be perpendicular to the surface. The electrical boundary conditions, which lead to depolarization and/or built-in fields, govern whether a sample is monodomain or polydomain, influencing the functional properties of the film. In the case of imperfect screening, the sample might form 180^◦ferroelectric domains with opposite polarization, resulting in an overall surface charge equal to zero and a depolarization field which vanishes (at least away from the interfaces). The domain size (lateral width) is governed by theKittellaw which was originally derived for magnetism [49–51] and then extended to ferroelectricity [52].

The fringing fields, shown in Fig. 2.7, penetrate into the film with an exponential decay over a length scale comparable to the domain width

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Figure 2.7: Schematic of 180^◦ domains and fringing fields present at the interface (in red). Their penetration decays exponentially into the film with a characteristic length comparable to the domain width.

d. The electrostatic energy attributed to these fields is proportional to the domain width as

FP ∼P_s²d. (2.32)

While this would imply that small domains are favored, the smaller the domains, the larger the number of domain walls which are energetically costly. This energy cost is due to the fact that a change in polarization through the domain wall alters the short-range interactions. Such energy is given by :

Fd=σd

t

d (2.33)

withtthe thickness of the film, σ_d the domain wall energy per unit area of the wall and1/dthe domain wall density. ⁵ The total energy is given byF =F_P +F_d. By minimizingF with respect todone gets the famous Kittel law:

d∼√

t. (2.34)

This relation implies that as the ferroelectric film is made thinner, domains get smaller. The coefficient of proportionality betweendand√

tdepends on the material properties⁶. Therefore, when comparing samples of the

5Note thatσ_d∼P³as shown in Ref. [53].

6Note that domains are found to be larger in ferromagnets than in ferroelectrics [54]

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same material with different thicknesses, the electrostatic boundary conditions should be kept the same to avoid large changes in polarization.

Interestingly such a law has been proved to be valid for ferroelectrics with thicknesses spread over 6 orders of magnitude. Additionally, if one ex- tends this to the entire ferroic family, a universal behavior is found for d²/Tversus thickness, whereTis the thickness of the domain wall [55].

Depolarization versus built-in field

The subtleties governing the metal-oxides interfaces are such that they can either destabilize or enhance the polarization of a film [21]. In addition to the depolarization field, a built-in field might also be present. The latter arises again from the interfaces chemistry, and/or simply from the asymmetry of the structure itself. In the end, both of these fields affect the polarization stability and the intrinsic domain structure. To gain intuition about their different effect on the polarization, one should go back to the well-known double well potential characteristic of ferroelectrics. As sketched in Fig. 2.8, the built-in field tends to favor one polarization orientation and shifts the polarization versus field loop to one side, while the depolarization field, which always points opposite to the polarization, decreases it and shrinks the polarization loop reducing the coercive fields.

Figure 2.8: (a) shows the characteristic double well energy versus polarization and (b) represents the polarization versus electric field response of a ferroelectric. The green and blue curves are the ones found in the presence of a built-in field and a depolarization field, respectively. The built-in field tends to favor one polarization orientation and shifts the polarization versus field to one side, while the depolarization field always points opposite to the polarization, decreases it and shrinks the polarization loop, reducing the coercive voltages and the spontaneous polarization.

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Chapter 3

Experimental: Growth and Characterization

Studying the physical properties of materials is something that can either be performed in ceramics, large single crystals or thin films. A tremen- dous amount of research has been done on ceramics and crystals from the early development of techniques allowing for example the stoichiometry or structure of the samples to be determined. However, during the last decades, lots of efforts have led to the development of new growth techniques allowing thin films to be studied. Among the many advantages of thin films let us cite the followings: being able to study the thickness dependence of the material, studying the strain effects imposed by the substrate on their physical properties, engineering heterostructures leading to amazing properties sometimes orthogonal to those of the parent compounds, and many more. In this chapter we present the growth technique that has mostly been used for this work and the different ways to characterize the sample structural and physical properties.

3.1 Growth

Growing thin films can be done by several methods including: molecular beam epitaxy, pulsed laser deposition or sputtering. Each technique has its own advantages and disadvantages and some materials grow better with one method or another. The ferroelectric materials that were grown during this work were deposited usingradio frequency off-axis magnetron sputter-

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Figure 3.1: Picture of the inside of the sputtering chamber used.

ing. The main idea of this method is to start with a stoichiometric target (or slightly off-stoichiometric), bombard it with ions from the plasma and deposit the sputtered species on a substrate. More specifically, because some species such as lead are rather volatile, the highly compressed ce- ramic targets can usually be off-stoichiometric in order to compensate for it; i.e. for PbTiO3, targets with 1.1 Pb concentration are used. The bom- barding is done by positively charged ions; typically argon. In order to achieve the ionization of the gas, an electric field is applied between the target (cathode) and the grounded chamber (anode). Once ionized, the ions will be accelerated towards the target, hit it and transfer some kinetic energy to the surface of the target. Note that the threshold energy that is required to start the sputtering process is material dependent since it will depend on the binding energy of the solid. The surface atoms that get dislodged by the ions transfer their energy to the neighboring atoms leading to collisions and eventually to the ejection of atoms away from the target. Those atoms are thermalized in the background pressurized chamber (∼0.1 Torr) and will eventually condense at the surface of the down facing substrate. The gas pressure in the chamber is optimized in order to avoid ballistic particles/atoms directly hitting either the main chamber, the heater, the substrate or anything else since this would lead to “back- sputtering”. Due to the insulating nature of the materials that we sputter, the presence of a DC field would charge the target and eventually can- cel out the external field (extinguishing the plasma). To eliminate this problem, a high frequency (typically in the MHz regime, RF) AC field is instead applied. The RF field will generate the plasma but at such high

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3.1 Growth

frequencies only the light electrons can respond to it whereas the heavy ions can be considered to be insensitive to it. Due to the geometrical and relative sizes of the cathode with respect to the anode, the field lines will be denser around the target than around the chamber itself. This will lead after a few cycles of excitation to a higher electron concentration near the target leading to a negative bias between the target and the chamber. As a consequence a resulting field called "self-bias" accelerates and attracts the Ar⁺ ions towards the target leading to the desired sputtering effect.

Additionally to argon, some oxygen should be present in the gas in order to fully oxidize the film.

One question that has not been mentioned yet is the geometry and po- sitions of the target with respect of the substrate. Experimental evidence shows that the target facing the substrate leads to resputtering. Therefore off-axis sputtering, namely 90^◦ between the substrate and the target, is the most suited geometry for our purpose leading to smooth and high film quality growth with a rather low deposition rate (lower compared to the on-axis geometry: the direction of the ejected particles is preferentially perpendicular to the target). The substrate, which is facing upside down, is glued with silver paste on a resistive heater. During the growth, the substrate is heated in order to give enough energy to the species that land on the surface to diffuse along the surface and to reorganize themselves into a crystal structure. An important innovation was done with the introduction of a permanent magnet behind the target. Its task, combined with the electric field, is to confine the electrons in a torus-shaped region in front of the target, increasing the ionization rate and therefore the deposition rate. This implementation is called magnetron sputtering.

The growth parameters (partial gas pressures, total gas pressure, temperature and gun power) are the tunable parameters that are optimized.

The typical growth conditions for PbTiO3and SrTiO3were found to be at a pressure of 180 mTorr with an oxygen/argon mixture of 20:29 and a substrate temperature of 540^◦C using a power of 60W. For SrRuO3, the growth conditions are 640^◦C at 100 mTorr of oxygen/argon mixture of 3:60 using a power of 80W. For SrTiO₃and SrRuO₃stoichiometric targets were used, but for PbTiO3, in order to compensate for the lead volatility, a target with 10 % excess Pb was used.

Additionally, rotating shutters are present in front of each target to protect it during the growth of another material. A cooling system is also required to cool down the targets during the growth process, otherwise, their temperature could lead to the melting of the targets and even the magnet sitting behind. A picture of the inside of the deposition cham-

Combining PbTiO3 and SrTiO3 toward 180° ferroelectric domains

Thesis

Reference