Comparing theoretical and experimental results

Whether the problem is approached with a microscopic theory or a macro-scopic one, the results are qualitatively the same: the PbTiO₃/SrTiO₃

sys-Figure 6.7: Temperature evolution of the tetragonality and the domain satellites for a (5,4)₂₈ PbTiO₃-SrTiO₃superlattice. Each satellite intensity was obtained by integrating the measured intensity of the domain satel-lites and subtracting the minimum integrated intensity in the paraelectric phase. The vertical line marks the intercepts of the linear fits of thec/a.

tem shows a negative capacitance behavior within a certain temperature range. The temperature at which the negative capacitance appears and for which temperature range it holds, depends on the assumptions of the model. For more details on those discrepancies we would refer to the sup-plementary information of Ref. [79]. Let’s point out that it would appear that the Landau-Ginzburg result for the static domain structure bears a closer qualitative resemblance to the experimental data and the atomistic simulation results than does the Kittel model, despite the fact that it is the Kittel model that correctly captures the important contribution of domain-wall motion. However, in the Kittel model, the possibility of domain-domain-wall pinning (by defects or all sorts of pinning centers), which would reduce the domain-wall contribution and could lead to an upturn of f at low temperatures, was not included.

Chapter 7

Pb _x Sr _1−x TiO ₃ thin films

Introduction

With the previous chapter, it is now clear that PTO and STO combined to-gether in superlattices are of great interest. The behavior of the domains under an applied external electric field has revealed that, at least for small electric fields, some domains expand at the expense of others. When ap-plying a small AC voltage, the domain walls are therefore expected to breathe. This breathing is believed to strongly enhance the dielectric re-sponse of the system, since domain wall motion leads to a flow of charges.

In Fig. 7.1 taken from Ref. [40], the experimental dielectric constant of a SL with 180^◦ferroelectric domains is compared to the Landau-Devonshire theoretical value and a clear enhancement with respect to the theory is observed. The discrepancy between the model describing the lattice re-sponse for a monodomain sample, and the data, corresponding to polydo-main samples [40], is attributed to the presence of dopolydo-mains. This is some-how expected since, as it was ssome-hown by Chenet al. in [88], the dielectric response of a polydomain sample is strongly influenced by the change in domain fraction. Similar results in Refs. [89–92] converge toward the idea that domain walls largely contribute to the dielectric response in addition to the usual lattice properties.

Following such reasoning, the idea of this chapter is to investigate a compound which, as we will see, turned out to be a great playground to study domains and domain walls from many perspectives. Wishing to keep the interesting physics from the combination of PbTiO₃ and SrTiO₃, we decided to study them in the form of a mixed solid solution of

Sr-Figure 7.1: Adapted from Ref. [40]. (a) In black the measured dielec-tric constant for a SL of PbTiO3/SrTiO3and in red the Landau-Devonshire theory versus applied bias. (b) Grey curves represent the theoretical di-electric constant at room temperature for different PTO volume fraction x=nPTO/(nPTO+nSTO)and for different fields. The dashed line repre-sents the criticalxcomposition; above it samples are ferroelectric and be-low, paraelectric. Black points are the data and in the ferroelectric region their values are above the theoretical predictions while in the paraelectric region they agree with it. This indicates that domain wall motion can be at the origin of such dielectric enhancement.

substituted PbTiO3or PbxSr1−xTiO3(PST) withxvarying from 0 to 1 in a thin film form.

Compared with other solid solutions containing PbTiO₃, most notably Pb(Zr_xTi_1−x)O₃ (PZT), which are technologically important because of their superior electrical and electromechanical properties, PST has received much less attention due to its simpler phase diagram that lacks the mor-photropic phase boundary of PZT. Bulk PST exhibits a single phase transi-tion from a high temperature cubic (paraelectric) phase to a low-tempera-ture tetragonal (ferroelectric) phase [93–95] with a transition temperalow-tempera-ture that increases linearly with Pb content [96]. On the other hand, in thin films, epitaxial strain can be used to induce new phases [97] as well as to control the ferroelastic domain structure [98].

PST thin films not only allow us to study the instability of the sponta-neous polarization to nanoscale domain formation but offer the possibility of studying their functional properties which are strongly influenced by depolarizing fields that arise from the imperfect screening of the ferroelec-tric polarization.

7.1 Ferroelectric domains in epitaxial thin films of PbxSr1−xTiO3

investigated using XRD and PFM This chapter will be divided into two main parts. The first one aims at describing the structural characterization of the PST films and their do-mains. For the latter, their observation is done by X-ray diffraction and piezoresponse force microscopy. The second part focuses on the dielectric properties of the PST samples.

7.1 Ferroelectric domains in epitaxial thin films of Pb

Sr

TiO

investigated using XRD and PFM

The small domain sizes, which scale according to the Landau-Lifshitz-Kittel law as the square root of the film thickness [20], make local imaging of the domain structure by piezoresponse force microscopy (PFM) challenging for films thinner than∼10 nm [32,99] and thus information on nanoscale domains is usually obtained using X-ray diffraction (XRD). Only within a narrow range of domain sizes can both techniques be applied, as reported here, providing a valuable cross-check between these real and reciprocal space methods [32, 99].

We study compressively strained thin films of PST grown on (001) Nb-doped SrTiO3 (SrTi_1−xNbxO3; Nb:STO) substrates. We focus on a set of PST thin films with different compositions and bottom electrodes allowing the degree of screening of the polarization to be changed. The tetragonal-ity andTCof PST for different compositions are studied using temperature dependent XRD and compared to the predictions of Landau-Devonshire theory, whereas the ferroelectric domain structure is studied using a com-bination of XRD and PFM allowing the domain size to be quantified. The results obtained using these two techniques are compared.