Strain on ferroelectric thin films

In order to carry out a phenomenologically pertinent description of PZT thin films, prior knowledge of its bulk counterpart is mandatory. Bulk PZT temperature–composition phase diagram regained recently interest, requiring us to go briefly over some of the latest developments.

The main modification to the temperature–composition phase diagram established by Jaffe et al. in 1971 [8] on the basis of earlier works by Shirane and Takeda [9] was brought in the vicinity of the morphotropic phase boundary. This line was delimiting the rhombohedral zirconate-rich side of the phase diagram and the tetragonal titanate-rich one and had already been established to lie between PZT 60/40 and PZT 55/45 in 1952 [9]. Instead of a line, a boundary, a morphotropic phase region was established circa 2000. This region is characterized by monoclinic symmetries: Cm [10‐13], Cc [14, 15], and Pm [16, 17], either alone or coexisting with the neighboring rhombohedral and tetragonal symmetries. These monoclinic phases are considered to act as a structural bridge between the rhombohedral and tetragonal sides of the phase diagram, enabling the rotation of the polarization within the monoclinic plane [18‐20]. Noheda and Cox have reviewed recently this topic [21] and a model linking the local structure to the macroscopic symmetries was established [22]. The relation between the morphotropic region of PZT (among others) and piezoelectric properties was also reviewed recently by Bell [23]. The phenomenological implications of these monoclinic phases are that the expansion of the Landau polynomial has to be carried out to the twelfth order of the order parameter [24].

It has been recently reported that the actual symmetry of the zirconate-rich side of the diagram was monoclinic instead of rhombohedral [25]. The low-temperature rhombohedral phase would be Cc and the high-temperature one Cm. The high piezoelectric response at the morphotropic phase boundary was then attributed to an elastic softening at the tetragonal to monoclinic phase transition [26]. These results have been reviewed with previous ones recently [27]. The nature of the monoclinic phase was also questioned through the consideration of the influence of nanodomains. It was showed through diffraction theory that rhombohedral nanodomains could produce reflections similar to the ones generated by a monoclinic symmetry [28, 29], contradicting this approach. The monoclinic symmetry of single domains as narrow as 30 nm in width was, however, found to be indeed monoclinic by convergent beam electron diffraction [30]. More research is still needed to clarify the exact nature of the temperature–composition phase diagram of bulk PZT, especially as it is often used as a model ferroelectric solid solution.

This morphotropic region is of primary interest as the dielectric constant, the piezoelectric and electromechanical coupling coefficients are maximized for these compositions or close to them. Therefore, either depositing thin films of the corresponding compositions or depositing thin films exhibiting the same structure are usual objectives for integrated device with optimized properties. In the bulk, the intrinsic width of this region was calculated to be 0.01 [31] and this width has been reported to be of 0.05 at 20 K [12] using solid-state chemistry. We will see that the composition range over which the monoclinic phases are stable in thin films may be much wider than in the bulk.

Very recently, the critical behavior of Pb(Zr_0.5Ti_0.5)O₃ has been studied by first-principles-based calculations and was found to belong to the 3d-random Ising universality class. This tends to suggest that, despite the much stronger long-range (dipolar) interactions, ferroelectrics adopt the same critical behaviors associated with universality classes as what is typically found in magnets [32]. This is of particular interest in the framework of multiferroics where ferroic orders are coupled, such as in the magnetoelectric effect, and for relaxors.

The end-member antiferroelectric PbZrO₃ was less studied than PbTiO₃. Nevertheless its dielectric [33‐36], piezoelectric [37] and structural [38‐40] properties as well as the ones with <10% of PbTiO₃ [9] were studied as early as the 50s. The existence of a ferroelectric rhombohedral phase on a narrow range of temperature (503–506 K upon heating and 505–500 K upon cooling) between the para- and antiferroelectric phases was confirmed later [41‐43] and a detailed study of the phase transitions by combined X-ray diffraction, dielectric and birefringence measurements determined the displacement of lead atoms (described by Kittel’s theory of antiferroelectricity [44]) as well as the oxygen octahedra rotation [45]. The possibility to induce a ferroelectric phase thanks to (high) electric fields in the nominally antiferroelectric phase of PbZrO₃ was reported and the electric field-temperature phase diagram for PbZrO₃ established by Fesenko et al. [46]. The energetics and structural instabilities of lead zirconate have also been computed recently, underlying the small energy difference between the antiferroelectric and the ferroelectric phases [47] estimated quantitatively in the 50s [48].

If the deposition of a thin film on a substrate gives rise to a 2D stress field, study of 1D and 3D stress fields on PZT solid solution have also been carried out and brought into light stress-accommodating mechanisms that are at play in thin films.

Uniaxial pressure has been applied to zirconate-rich compounds [49, 50] and morphotropic compositions in order to study the reorientation of the domains [51]. Corresponding calculations have been carried out on monodomain lead titanate, phenomenologically [52] as well as by first-principles [53], pointing out the resulting enhancement of the piezoelectric coefficient along the polar axis. These experiments emphasize the fact that the least energetically costly mechanism for accommodating stress is domain wall movement and the consecutive reorganization of domains. If this reorganization is either not energetically favorable (for symmetry reasons for example) or if the stress is applied on a monodomain, it is then the symmetry of the lattice that accommodates the applied stress. A similar effect has been observed in other ferroelectric perovskites (see e.g. [54]), suggesting a common behavior under uniaxial pressure. We shall discuss the question of domain wall movement in thin films in relation to the structural defects in “Critical thickness for dislocations generation” section.

Hydrostatic pressure generates a 3D stress field. From lead zirconate [55‐58] to lead titanate [59‐63] through several intermediate compositions [64‐68], PZT has been studied under pressure. Among the numerous interesting results in this field, a high-pressure form of ferroelectricity has been calculated and observed in PbTiO₃ [59, 60] at room temperature. At low temperature and still for PbTiO₃, a monoclinic phase was calculated [69] and then observed [61], even though this last result and its implication have been questioned [63]. The rotation of the oxygen octahedra as a pressure-accommodating mechanism [70] is a common feature of the observations on lead titanate, as well as for Pb(Zr_1−xTi_x)O₃. We will see that this stress-accommodation has not yet been taken into account in the theoretical descriptions of PZT thin films.

The domain phases in bulk depend upon electrical and elastic boundary conditions. In thin films, these conditions are exacerbated due to the finite thickness and to the clamping onto the substrate.

If one considers an homogeneous polarization within a ferroelectric thin film, then the surfaces charges (σ) accumulate on the interfaces: \(\sigma=-{\bf n}\)·P with n the normal to the interface and P the polarization. These charges create a depolarizing field that counteracts the polarization, possibly suppressing it. In order to prevent the onset of the depolarizing field, the film may self-organize into ferroelectric domains, usually 180° ones, or it may sustain such mono(ferroelectric)domain configuration by screening the surface charges thanks to metallic electrodes, therefore minimizing the depolarizing field.

Ferroelectric domains are created in bulk ferroelectric materials to reduce the electric field. In a similar manner, elastic domains coinciding with ferroelectric ones are created to minimize the elastic energy. The case of a domain wall between adjacent 90° domains is illustrated on Fig. 1. We shall see that these modulated structures are formed as a consequence of phase transition rather than during the film growth.

From an X-ray diffraction point of view, (elastic) domain structures are sometimes not obvious to determine. In the monodomain tetragonal configuration, the lattice parameters are usually straightforward to determine (see Fig. 2) as there is only one out-of-plane and one in-plane lattice parameters. However, to determine precisely the out-of-plane lattice parameter, several (00l) peaks have to be measured, in order to correct for any misalignment. The in-plane lattice parameter is deduced from the out-of-plane one and the measurement of several (h0l) peaks, as in the tetragonal case \(d_{h0l}=1/ \sqrt{h^2/a^2+l^2/c^2}\).

In the polydomain case however, the determination of the lattice parameters is slightly more complicated as the reciprocal lattices of several domain types superimpose. The case of two domains oriented at 90° and having the same lattice parameters is illustrated on Fig. 3a. Note however that this is an approximation of the so-called a/c/a/c domain structure where domains with polarization parallel to the growth direction (c domains) coexist with domains with polarization parallel to the interface with the substrate (a domains). A detailed description of this structure shall be given in “Polydomain films” section.

The two out-of-plane lattice parameters will give rise to two peaks. This is illustrated on Fig. 3b. However, as the unit cell is not cubic, there is a slight misorientation of the two domain types (see Figs. 1 and 3c) that implies to correct the incidence angle (by Δ ω′) in order to measure the maximum intensity from the peak. This is of particular importance if the domain population is of interest. Its determination is indeed based on the ratio of the integrated intensity of the peaks.

In order to determine the in-plane lattice parameters the same technique as for the monodomain case can be employed. However, in the present case reciprocal space maps (RSM) around the (h0l) peaks (see Fig. 3d) are of particular interest. Indeed, in addition to the misorientation due to the tetragonality of the unit cell (Δω′), another difference in the incidence angle (Δω) comes from the superposition of the reciprocal space maps of the two domain types, as illustrated on Fig. 3a. Therefore, RSM are a precious tool to precisely measure the positions of the peaks’ maxima.

Various domain structures have been considered in first-principles calculations and Kornev et al. gave a review of these results recently [72]. This gives us an occasion to remind the reader that here, we consider domains from a structural point of view. Alternating ferroelectric 180° domains, such as the stripes observed in [73], are composed of alternating tetragonal domains and are therefore labeled as monodomain here.

In several PZT polydomain thin films deposited on MgO (i.e., on a cubic substrate with a larger lattice parameter), the volume fraction of the c domains in the film (hereafter α_c) was found to be larger than that of a domains [74, 75]. For the special case of lead titanate deposited on MgO however, the situation is less clear and the observations differ significantly from one author to the other. The α_c’s were calculated in various ways, making the comparison difficult (see [76] for a discussion of this problem). What is more, α_c depends on the deposition conditions, e.g. on the cooling pace [77] or on oxygen partial pressure for pulsed laser deposited films [78]. If an increase of α_c with thickness has been reported [79], most the experiments tend to show that α_c decreases with thickness [76, 80, 81]; On the contrary, its evolution with temperature makes a consensus: the decrease of α_c with temperature has been observed [74, 79, 82, 83] as well as calculated [84, 82].

Roytburd and Alpay, in a series of seminal papers [85‐87], have calculated domain stability maps that indicate the stable phase as a function of the tetragonality of the film and of the relative difference between the substrate and the film’s in-plane lattice constants. These domain stability maps are a generalization of Pompe et al.’s domain pattern maps developed for twin domains [88] in the case of heterophase polydomain structures. The calculations were carried out in the short-circuit case (no depolarizing field) and were mainly considering a film much thicker than the critical thickness for domain formation (see “Critical thickness for dislocations generation” section). These calculations were later applied in the case when this approximation does not hold on the example of Pb(Zr_0.2Ti_0.8)O₃ [89] (see Fig. 4). These calculations consider a lamellar structure in which the period of the domain structure is smaller than the thickness of the film. This way, the domain wall energy is considered negligible compared to the energy within the domains and therein the polarization and strain fields may be regarded as homogeneous. Such a structure is expected in films thicker than 100 nm [90].

Therefore, there are four main points that can be drawn from these calculations are:

The experimental evidence of the pertinence of the domain patterns maps developed by Pompe et al. [88] was brought by Foster et al. [91] with PbTiO₃ thin films deposited on various substrates, and the domain stability map was also experimentally confirmed for tetragonal Pb(Zr_0.2Ti_0.8)O₃ thin films in [92] (where the hypothesis of short-circuit conditions or noninfluence of the depolarizing field was validated) and in [75].

To calculate these phase diagrams in the monodomain case, one has to take into account the elastic energy induced during deposition and the consecutive strain because of the lattice mismatch but also the various coupling between strain and polarization. This implies that symmetries unstable in the bulk temperature–composition phase diagram may be stable in the temperature–misfit strain phase diagram. We will develop this point in the next paragraphs. Another consequence is the possible change of the order of the phase transition, as evidenced in PbTiO₃: the paraelectric to ferroelectric phase transition is of first order in the bulk and second order for thin films [94]. This is the consequence of the positive sign of the renormalized second-order coefficients \(a^*_{ij}\), i.e., the ones multiplying the fourth power of the components of polarization, and not a consequence of a finite size effect [103].

The coordinate system used hereafter is the usual one (see Fig. 5): the x,y and z axes (sometimes labeled with subscripts 1,2 and 3) are linked to the <100> directions of the substrate, with the z (or 3) axis parallel to [001]_substrate. The film is also supposed to be oriented so that [100]_film //[100]_substrate //x (or 1) and [010]_film//[010]_substrate//y (or 2) and, consequently, for the growth direction: [001]_film//[001]_substrate//z (or 3).

A comprehensive review on first-principles modeling of ferroelectric thin films was recently published by Ghosez and Junquera [104] and we refer the reader to it and to the review by Kornev et al. [105] for a complete description of this field of intense research.

Phase field models usually consider both mono- and polydomain phases in their temperature–misfit strain phase diagrams and will therefore be mentioned in “Polydomain films” section.

We shall therefore restrict our consideration to the thermodynamical studies of monodomain ferroelectric thin film, starting by reviewing the main hypothesis that can be made in order to better understand the limitations as well as the possible improvements made.

The monodomain case is the simpler one to address and several misfit strain-misfit strain phase diagrams (at constant temperature) have been calculated by thermodynamic theory for PbTiO₃ [131‐135], Pb(Zr_0.5Ti_0.5)O₃ [134], (Pb_0.35Sr_0.65)TiO₃ [131], BaTiO₃ [132‐134], (Ba_xSr_1−x)TiO₃ (x = 0.5 [136], 0.7 [137]). Experimental reports such as the one about lead titanate thin films deposited on orthorhombic DyScO₃ [138] showed that ac and r phases can be stabilized by anisotropic stress fields. One has to pay attention to the set of parameters used to calculate the phase diagrams as the use of an erroneous set leads to incorrect results (compare, for example, [134] and [132]).

The common conclusion is that anisotropic strain may stabilize phases that are absent from the monodomain isotropic strain case. The phase sequence for the isotropic case corresponds to the diagonal of the misfit strain–misfit strain diagrams and one can check that it remains unchanged, as expected. However, and as pointed out by Zembilgotov et al. [131], when the bulk paraelectric phase is cubic, the symmetry of the film’s paraelectric phase is now centrosymmetric orthorhombic and not centrosymmetric tetragonal as in the case of isotropic strains.

However, experimentalists have to be cautious about the details of the calculations as the modified thermodynamics potential are often based on a sixth-order Landau–Devonshire development [131‐133, 135‐137] that limits both the temperature and the misfit-strain range on which the results are meaningful. See [134] and [139] in the case of BaTiO₃, and [137] for Ba_0.7Sr_0.3TiO₃. In addition, the use of an eight-order polynomial enables to address the question of the stability of the phase in which the three components of the polarization are not nil.

What is more, unless expressly addressed, the in-plane shear strain (u₆ in Voigt’s notation) is taken nil. Taking into account this strain destabilizes the phase where \({\bf P}=(\hbox{P}_{1}=0,\hbox{P}_{2}=0,\hbox{P}_{3}\neq0)\) [134], leading to a shift of the 180° ferroelectric domain walls compared to the hypothetical case where the film would be free from the substrate and therefore free to be strained in the z direction [140]. In addition, the use of an eight-order Landau polynomial leads to the complete destabilization, from a thermodynamic point of view, of the phase in which all components are not nil (P = (P₁ ≠ 0, P₂ ≠ 0, P₃ ≠ 0)).

The application of an external uniaxial mechanical loading on the surface of a thin film was studied in the isotropic case by Pertsev et al. [141]. Thin films grown on orthorhombic substrates and submitted to such external loads show a tendency to stabilize the paraelectric phase and therefore enable a stress-driven phase transition between ferro- and paraelectric states [133]. However, this study was restricted to the case where the two in-plane strains are equal in magnitude but of opposite sign (u_m1 = −u_m2).

The influence of depolarizing fields, the other key player in the determination of the structure of ferroelectric thin films, has been considered recently on 8-nm thin film [135]. Uncompensated charges at the surface/interface of a ferroelectric thin film as well as a gradient of polarization throughout the film generate a depolarizing field. This field can be partially screened by the deposition of electrodes, resulting in a situation, between open-circuit conditions (no charge compensation, maximum depolarizing field) and short-circuit conditions (complete compensation, depolarizing field only generated by the polarization gradient). From a first-principle point of view, the partial screening of the depolarizing field has been modeled through a β coefficient [142]. Under ideal open-circuit conditions, β = 0 the depolarizing field is therefore maximal, whereas for a given β = β_SC the depolarizing field vanishes. Quite naturally, the depolarizing field destabilizes the phase in which \({\bf P}=(\hbox{P}_1=0,\hbox{P}_2=0,\hbox{P}_3\neq0)\) and stabilizes the phases with non-zero in-plane components of the polarization [135]. This first result is similar to what happens under isotropic misfit strain [128]. In addition, depolarizing fields stabilize phases exhibiting only one non-zero in-plane component of the polarization \(({\bf P}=(\hbox{P}_1\neq0,\hbox{P}_2=0,\hbox{P}_3=0)\) or \({\bf P}=-(\hbox{P}_1=0,\hbox{P}_2\neq0,\hbox{P}_3=0))\) in detrimental to the phases with both in- and out-of-plane non-zero components of the polarization, which were stable without any depolarizing field. It is worth noting that a partial screening of the surface charges has a pronounced effect only when the partial compensation of the charges rises above 99% [135]. This seems to indicate than under realistic conditions (i.e., with surface charges partially screened) the real system is closer to the open-circuit conditions than the short-circuit ones.

The consequences on dielectric properties of films [131, 134, 136, 137], their tunability [137] and pyroelectric properties [136, 137] of such strain conditions have been calculated. These comparisons between the phenomenological predictions and the experimental measurements may lead to reasonable agreement as long as sufficient expansion of the Landau polynomial, appropriate electrical boundary conditions, available and reliable set of parameters have been used in the model and, as importantly, accurate measurement of the lattice parameters have been carried out [134].

As for the isotropic strain state, much confusion arises from the use of crystallographic terms to describe phases that are in fact defined by the components of polarization. Table 2 presents the symmetry of the phases one may encounter in misfit strain–misfit strain diagrams.

A phase, where \({\bf P}=(\hbox{P}_1\neq0, \hbox{P}_2=0, \hbox{P}_3=0)\), is not tetragonal as claimed by several authors [133, 136, 137] (not even under an isotropic strain state, as we saw in “Clarification of the notations used in the isotropic biaxial stress field case” section) but orthorhombic. In addition, the phases with two non-zero polarization components are monoclinic, as the polarization is confined within one of the <001>-planes, and not orthorhombic [133, 136, 143]. Finally, the phase with P₁ ≠ 0, P₂ ≠ 0, and P₃ ≠ 0 is triclinic (as pointed out by Zembilgotov et al. [131]) and not rhombohedral [143].

An appropriate description of the crystalline symmetry of the calculated phases would ease comparisons between experimental and theoretical results and therefore be beneficial for the whole community.

In parallel to thermodynamical calculations, phase-field simulations have been carried out very recently in order to calculate temperature–misfit strain phase diagrams under anisotropic biaxial strain fields as well as the misfit strain–misfit strain diagrams at constant temperature. The cases of PbTiO₃ [144, 145] and BaTiO₃ [143] have been considered. In the phase-field simulations, the considered energy is the sum of the bulk free energy, the elastic deformation energy, the domain wall energy, and the electrostatic energy. As for thermodynamic calculations, the order of the expansion of the Landau polynomial has to be carefully checked before comparing either experimental results with theoretical predictions or between the phase diagrams.

Qiu et al. have published recently a thickness versus misfit strain phase diagram for PbTiO₃ thin films from thermodynamical calculations [146]. This study considers the possibility of stabilizing monodomain structures and generating misfit dislocations with realistic electrodes (considered through their screening length in the expression of the depolarizing field). They find that the monoclinic so-called r phase is stable on very narrow ranges of misfit strain and thickness and that the tetragonal c phase or the polydomain a/c/a/c phase are the most probable ones considering typical experimental conditions. Their phase diagrams also exhibit the possibility to stabilize a new phase, called a′c′, in which the polarization components are equal to the ones of the c and a phase, i.e., P^a′c′_[100] = P^a_[100] and P^a′c′_[001] = P^c_[001]. This phase diagram incorporates all the main parameters determining the stable phase. It is however based on the Matthews–Blakeslee criteria which underestimates the critical thickness for dislocation generation (see “Critical thickness for dislocations generation” section) and therefore the absolute value of the mentioned thicknesses should be taken with care.

An exponential profile over the thickness for the stress has been considered in several models [147] and compared to experimental variation with thickness of the lattice parameters for Pb(Zr_0.4Ti_0.6)O₃ [147]. Catalan et al. [148] have used this exponential decay of the strain and X-ray diffraction analysis to calculate the strain profile. They have then introduced this gradient into a phenomenological calculation and calculated the dielectric constant as a function of both temperature and thickness, to finally compare their calculations with their experiments. This shows that strain gradient, through the flexoelectric effect can alter the dielectric properties of thin films even when their thickness is of several hundreds of nanometers [148].

A different behavior was reported recently by Bartasyte et al. on multidomain PbTiO₃ thin films (from 30 to 460 nm thick) on LaAlO₃ [149]. Stress measurements were carried out by X-ray diffraction and Raman spectroscopy, leading to residual stresses between 0.4 and 2 GPa that are in the range of reported values for polycrystalline thin films (see [149] and references therein) and comparable to the ones reported for monodomain thin films [97, 98]. The evolution of the a domain fraction presents a maximum whereas the residual stress decreases linearly. This is contradictory with the temperature–misfit strain phase diagrams proposed in [96]. Indeed, at constant temperature, the domain population is a monotonous function (but not continuous at the transition between the a/c/a/c and a phases according to the domain stability map, see “Domain stability map” section) of the misfit strain. However, considering the room temperature thickness-misfit strain phase diagram developed by Qiu et al. [146] and forgetting about the absolute values of the thickness reported on it, one sees that the domain population evolves in a non linear way in the c/a/c/a phase. From the out-of-plane lattice parameters reported by Bartasyte et al. [149] and considering an elastic deformation of a tetragonal material characterized by a Poisson coefficient of 0.37 [89], one finds that the (thermodynamic) misfit strain goes from approximatively 0 (for the thickest film) to +2% (for the thinnest one). Therefore, a non-linear evolution of the domain population with thickness is not in contradiction with this phase diagram. For clarity sake, let us note that the “misfit strain” evoked in [149] is due to the lattice mismatch between the substrate and the bulk and their “residual stress” is linked to the actual lattice parameter of the film, and therefore to the (thermodynamic) misfit strain. These two are markedly different, as pointed out by Bartasyte et al. as they are of opposite signs.

A surprising result has been reported by Gariglio et al. [101] on monodomain tetragonal Pb(Zr_0.2Ti_0.8)O₃ thin films deposited on (Nb doped) SrTiO₃ substrates. Even though elastic relaxation occurs when thickness is increased from 15 to 123 nm (through the exponential increase of the in-plane lattice parameter and the corresponding decrease of the out-of-plane one) the transition temperature remains constant. This report is in contradiction with other reports by Fong et al. on PbTiO₃ ultrathin films (from 3 to 30 unit cells) deposited on SrTiO₃ [150] or SrRuO₃ [151]. As pointed out by the authors, this seems in contradiction with the temperature–misfit strain phase diagram developed by Pertsev et al. [94]. However, the value of the misfit strain for the thinnest film has been calculated by the authors to be of −23 × 10⁻³ [101], which is not in the range of the misfit strain considered in the phase diagram.

Let us recall that the Landau polynomial used by Pertsev et al. [94] is of sixth order and therefore limits the range of the misfit strains that can be considered. As the misfit strain for the thickest film is still of 9 × 10⁻³ it lies out of the range of misfit reported on Pertsev’s et al. phase diagram [94].

Due to the cooperative nature of ferroelectricity, the question of the minimum thickness in which ferroelectricity can develop is of particular interest. Ferroelectricity has been measured in 5 nm [152], 4 nm [153] and even in a 1.2 nm [150, 154, 155] crystalline films and in a 1 nm polymer [156] film. Taking into account realistic interface between the ferroelectric thin film and the electrodes, i.e., a finite screening length of the electrodes as well as the details of the interface, ab-initio calculations showed that a critical thickness for ferroelectricity still exist [157]. Despite the short circuit conditions imposed on the electrodes, a depolarizing field arises into the film due to the imperfect screening of the surface charges by the electrodes.

Recently, Glinchuk et al. [158] showed, thanks to phenomenological calculations, that the reduction of thickness under nonideal short-circuit conditions leads not to a paraelectric state but to an electret state. This state is characterized by the vanishing of the polarization loop, the variation of the polarization with applied electric field being asymmetrical and shifted by the internal electric field that develops due to the imperfect screening of the surface charges. In the electret state pyroelectric effect remains and the film does not transit to a nonpolar state at higher temperature. A critical thickness under which the film is paraelectric only exists when no internal field exist (under the application of a canceling exterior electric field for example) [158].

Very recently, Sai et al. [159] studied by first-principles ultra-thin improper ferroelectric films, i.e., materials for which the primary order parameter, the one driving the phase transition, is not the polarization. They show for isolated single domain YMnO₃ ultrathin films (down to 2 unit cell thick) that there is no critical thickness under which the paraelectric state is stable, even under open circuit boundary conditions.

With increasing thickness, a transition from a monodomain state (i.e., possibly containing only 180° domains in the tetragonal case) to a polydomain state (with 90° domains in the same symmetry) has been considered theoretically [85‐88, 91]. In a monodomain film, the 180° domains reduce the depolarizing field, forming domain stripes as observed on PbTiO₃ thin films [73]. In a polydomain film, the 90° domain structure is a consequence of the minimization of elastic energies as well as the electrostatic one. The concept of critical thickness for domain formation in ferroelectric thin film was developed by Pompe et al. [88]. In the same article, the thickness dependence of the minimum domain wall thickness was also developed. Their model considered only the competition between elastic energy reduction and increase in energy due to domain wall creation. Electrical conditions, kinetics of domain wall motion, energy linked to defect generation, nucleation of domains were assumed to be negligible. Although these apparent limitations, their calculations of the critical thickness were confirmed on 30 nm PbTiO₃ thin films deposited on DyScO₃ [160] where a critical thickness around 10 nm and a domain structure consisting of alternating wide c domains and narrow a domains (with width down to the minimum value of 6–7 nm) with a period of 27–31 nm. These values are the minimum ones preserving the horizontal coherency of the film. This configuration does not fall into the scope of the theory developed by Roytburd et al. [85‐87] and Pertsev et al. [123] as the domain periodicity has to be smaller than the thickness of the film. Considering Kittel’s law [161], which states that the domain period increases with the square root of the thickness of the film, one sees that Roytburd’s model implies to consider thicker film than the one studied by Vlooswijk et al. [160].

The misorientation of the different variants of the tetragonal domains illustrated in Fig. 1 implies to take a closer look at the situation near the interface with the surface. Instead of parallel domain walls, steps have been observed by transmission electron microscopy [71] in the domain wall profile leading to wedge-shaped a domains. This was later confirmed by calculations [162]. In addition, the domains were observed to start from dislocations standing off the interface [71]. This shape minimizes the strain at the interface due to the clamping on the substrate and enables the misorientation to be compensated. This shape is however a function of the stiffness of the substrate, a higher stiffness leading to a higher interface strain and therefore minimizing the width of the a domain a the interface with substrate [71, 162]. As a consequence, the a domain walls observed on the SrRuO₃ electrode deposited on DyScO₃ were parallel [71]. It has to be noted that domains were intentionally generated in these PbTiO₃ films deposited on SrTiO₃ by modifying the deposition conditions. For tetragonal PbZrTiO₃ films grown under optimal conditions (i.e., as close as possible to the thermodynamic equilibrium), the expected and observed domain structure consists of (180°) c domains, in this thickness range (for PbTiO₃: 100 nm in [98], 110 nm in [71]; for Pb (Zr_0.2Ti_0.8)O₃: 90 nm in [163], 100 nm in [97]).

The difference in thermal expansion coefficients does not give rise to a thermal strain that is sufficient to enable dislocation generation during cooling from the deposition temperature (on the contrary to what was considered [111]). This is confirmed by the thermal evolution of the in-plane lattice parameters of monodomain films that run parallel to the substrate’s one [97, 98, 101, 100]. Additional dislocations would indeed enable the in-plane lattice parameter of the film to relax toward its bulk value. The dislocations are therefore introduced during deposition or at the bulk para-ferroelectric transition.

Frank and van der Merwe [164, 165] introduced 60 years ago the concept of the critical thickness above which misfit dislocation formation was the energetically favored stress-relieving mechanism at the interface of an heteroepitaxial system. More precisely, considering a monolayer, they introduced the notion of critical misfit value above which introducing dislocation was energetically favored. The Matthews-Blakeslee [166] model has been the most used one to calculate this critical thickness. It is based on equilibrium considerations with its equations derived from the search of the total (i.e., elastic strain and dislocation) energy minimum or from equilibrating the forces acting on a dislocation.

From Matthews-Blakeslee’s criteria [166], the linear density of dislocation (ρ) is expected to vary according to:with \(\left|{\bf b}\right|\) the norm of Burgers vector, λ the angle between the Burgers vector, and a line belonging to both the interface and to a plane normal to the dislocation line, \(\epsilon_m=\frac{a_{//}-a_0}{a_0}\) the misfit strain, h the thickness, ν Poisson coefficient. The last term, l(h), is defined as: \(l(h)=\frac{\left|{\bf b}\right|} {8\pi\cos{\lambda}}(1-\nu(\cos{\beta})^2)\ln\left(\frac{\alpha h} {\left|\bf{b}\right|}\right)\) with β the angle between the dislocation line and the Burgers vector, and α a parameter taking into account the nature of the chemical bonding (varying between three for ionic crystals and four for covalent ones). In the case of a monodomain film as well as in the a₁/a₂/a₁/a₂ structure, we would like to repeat that the Burgers vectors lie in the plane of the interface and the domain walls are vertical (see “Limitations” section). Therefore \({\bf b}//[100]\) or [010] \((\left|{\bf b}\right|=\max(a_{\rm s},a_{//}))\), λ = 0, and β = π/2.

Following Speck and Pompe [111], this dislocation density could be approximated by:

However this implies, as pointed out in their paper: l(h)≈ l(h_c), i.e., \(\ln{h}\approx\ln{h_{\rm c}}.\)

This model proved to be adequate to describe the situation in metals. Several models have been developed from Matthews–Blakeslee’s, for metals and semiconductors. Let us cite the one by Fischer et al. [167] who extended Matthews–Blakeslee’s model to take into account the interaction between straight dislocations and the one by Chidambarrao et al. [168] who also extended it by taking into account the effect of Peierls barrier. However, for ferroelectrics, it usually predicts critical thicknesses that are smaller than what has been observed experimentally.

Another model, developed by People and Bean [169] for semiconductors, considered a nucleation barrier that had to be overcome by the misfit strain in order to favor misfit dislocations generation. Venkatesan et al. [71] used this latter model to describe their PbTiO₃ thin films deposited on SrTiO₃. The thickness range covered by their study extends from 22 to 340 nm and a slow growth rate prevented the formation of a domains. High-resolution transmission electron microscopy and X-ray diffraction revealed the absence of dislocations for all thickness up to 340 nm. The critical thickness derived from People and Bean’s model reads [169, 170]:with ε the relative difference between the substrate and bulk lattice parameters. Venkatesan et al. considered the expression of People and Bean’s model reported by Maree et al. [171] for the case plotted by People and Bean in [169] (and corrected in [170]) in which they considered \(\left|{\bf b}\right|\approx2\sqrt{2}a_0\) and considered five <110> atom spacings, hence the slightly different expression reported by Maree et al. and used by Venkatesan et al. Interestingly, following People and Bean’s model, calculations have shown that the use of small seed pads on a substrate enable the critical thickness to tend toward infinity for pads with lateral dimensions below a critical size [172]. Even if the orientation of the Burgers vector and the mean value of the effective interfacial width of isolated dislocations may be different from the one considered, the work by Venkatesan et al. points out meaningfully the discrepancies in critical thickness that can be calculated depending on the model considered.

Their work underlines therefore the need for a model designed specifically for ferroelectrics or, more generally for functional oxides, which is still lacking despite the numerous experimental evidence of the influence of dislocations on the properties of ferroelectric thin films.

Jesser and Fox [173] showed that on substrates not as perfect as silicon ones may be, the propagation of preexisting dislocations (rather than a nucleation barrier) was the limiting parameter. Indeed, this friction stress is much higher in semiconductors than in metals. In addition, the density of dislocations generated on these not-so-perfect substrates is evidently much higher and therefore their interaction may come into play. They then developed [173] a model including frictional stress applicable to both metals and semiconductors.

A comprehensive study of the defect microstructure in tetragonal Pb(Zr_0.2Ti_0.8)O₃ thin films on SrTiO₃ has been reported by Misirlioglu et al. [174]. The investigated thin films were mainly c-oriented with a minority of a₁ and a₂ domains and displayed both threading dislocations as well as misfit dislocations. The density of the former ones is high (over 10¹⁰ cm⁻²) and their origin is attributed, in a Volmer–Weber growth mode, to the coalescence of the islands, forcing the misfit dislocations away from the interface with the substrate [174]. The former explanation of threading dislocations being the continuation of the ones existing in the substrate [166] could only account for a small fraction of them. Misirlioglu et al. pointed out [174] that the domain structure, created to accommodate part of the spontaneous strain associated with the phase transition, nucleates at misfit dislocations. These domains are then pinned because of the microstresses generated by the misfit dislocations and therefore the extrinsic contribution (i.e., domain wall movements) to the piezoelectric coefficients that plays a major role in the bulk is strongly reduced in thin films. This lead the authors to conclude that the electrical and electromechanical properties of polydomain thin films should be inferior to the bulk ones. Thermodynamic calculations further showed that the impact of threading dislocations (with a direction line perpendicular to the interface) was “limited” to the smearing of the transition temperature, and that the strongest impact on the other functional properties was indeed due to the misfit dislocations [174].

This is somehow in contradiction with the measurements of 180° domain wall creeplike velocity in tetragonal PZT films [175, 176], complemented by measurements of domain wall roughness [177]. For 180° domain walls other models describe their motion, such as the dynamic poling model developed by Trolier–McKinstry et al. [178]. However, the clamping of 90° domain walls was shown to be reduced through nanoisland patterning of the film using focused ion beam, resulting in a strong enhancement of the properties [179]. It is worth mentioning that, in addition to dislocations, point defects, such as oxygen vacancies, have a critical influence on the domain wall thickness [180].

The misfit dislocations, through the localized strain field generated around them, induce locally a strong reduction of polarization which is detrimental to the properties of the film (see [174] and references therein for the consequences on polarization loops, dielectric, and piezoelectric properties). This was confirmed recently by the study of polarization and piezoelectric properties on Pb(Zr_0.52Ti_0.48)O₃ thin films [110]. The strong reduction of polarization was also shown recently to reach 48% in a Pb(Zr_0.2Ti_0.8)O₃ film deposited on SrTiO₃ and covered by a SrTiO₃ film [181]. Morelli et al. [182] attributed this effect to the steps observed in the domain walls [71] giving rise to leakage currents that dominate the polarization reversal measured by piezo-force miscroscopy. Interfacial dislocations were also held responsible for the poor piezoresponse observed for Pb(Zr_0.52Ti_0.48)O₃ nanoislands [183]. A comparison between the properties of Pb(Zr_0.2Ti_0.8)O₃ films free from extended defects and films presenting 90° domains have been carried out by Vrejoiu et al. [163]. If the polarization is indeed reduced in the case of defective films, the dielectric constant was found to be higher, indicating that the pinning of the domain walls by misfit dislocations is not as effective as to suppress completely extrinsic contributions to the dielectric properties of ferroelectric thin films as predicted in [174]. A recent review of both the influence of growth conditions and of the microstructure of perovskite films has been published recently by Vrejoiu et al. [184].

Pálová et al. [185] have proposed a model drastically different from the ones mentioned so far. Instead of a strain gradient induced by defects present throughout the whole film, a segregated strain gradient is proposed. This model is shown to reproduce the observed thickness dependence of ferroelectric thin films in a comparable manner as the ones based on inhomogeneous strains. In Pálová et al.’s model, the defects are located at the interface with the substrate, over a given thickness. As a consequence, the internal bias field generated is constant throughout the film (for a given thickness), whereas the bias field created by inhomogeneous strains in the film varies within the film. Hence, an applied electric field should be able to cancel the bias field of the segregated model and the film should exhibit bulk-like properties. This is the benchtop probe proposed by Pálová et al. [185] that remains to be tested to the best of our knowledge.

The misfit strain–misfit strain diagrams enable the experimentalists to compare the structure of the film with the calculations or simulations at room temperature. However, in order to compare the evolution with temperature of the film’s lattice parameters, more temperature-dependent results would be of great interest. In addition, no first-principle based calculations results have been published so far on anisotropic biaxial misfit strain phase diagrams. The influence of different growth directions has indeed already been considered [186], but the strain field applied on the material was taken as isotropic. These calculations, though computer-wise intensive, would provide the community with predictive tools for reduced-size device design.

A usual mechanism of accommodation of strain in the bulk is oxygen octahedra rotation. As pointed out in the introduction, this has been observed and calculated both for low temperature and high pressure. This mechanism has only been taken into account so far in calculations on SrTiO₃ [141, 187], despite its potential importance in all perovskites.

Taking into account the electrical boundary condition in the calculations requires the experimentalists to measure it with great precision.

So far thickness dependent properties relied on series of films with different thickness. The measurement of the actual stress profile within a single film should enable to study the influence of the (top) surface on this profile. As was shown experimentally in PbTiO₃ ultrathin films on SrTiO₃ by Fong et al. [188], the displacement of atoms and the resulting unit cell can evolve in a way much more complex than the simple exponential. The usual tetragonality-polarization direct relation as well as the tetragonality throughout the film was shown by high-resolution transmission electron microscopy to vary through ultrathin (7 nm) tetragonal films [189] in a way not easily compatible with an exponential decay. A better understanding and description of this strain gradient would therefore be of great interest, including to design novel device based on properties such as the flexoelectric effect.