Domain formation - Kittel Law

As mentioned above, ferroelectric thin films can show rather different po-larization 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^◦ ferro-electric 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

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]

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

same material with different thicknesses, the electrostatic boundary con-ditions 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 addi-tion to the depolarizaaddi-tion 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 intu-ition 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 ori-entation 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 polar-ization and (b) represents the polarpolar-ization versus electric field response of a ferroelectric. The green and blue curves are the ones found in the pres-ence 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 op-posite to the polarization, decreases it and shrinks the polarization loop, reducing the coercive voltages and the spontaneous polarization.

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 tech-niques 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 lead-ing to amazlead-ing properties sometimes orthogonal to those of the parent compounds, and many more. In this chapter we present the growth tech-nique 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-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 cham-ber (∼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

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 sub-strate 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 introduc-tion 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, tem-perature 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-ber during growth is shown in Fig. 3.1. More details can be found in Ref. [56]. One disadvantage of this approach is the lack ofin-situ monitor-ing (although progress in this direction has been recently realized [57]);

hence many different characterization techniques are usedex-situto fully determine the physical properties of the grown films.