Self-Polarization in PbTiO₃ Crystals Induced by Chemical Inhomogeneity in the Surface Layer

Abstract

We study the so-called self-polarization phenomenon in single, electroded PbTiO₃ crystals. In this case, near the electrodes, surface layers are formed with a chemically modified perovskite structure. This generates a built-in electric polarization, which cannot be switched permanently by an external electric field. While the initial samples, having two such surface layers with opposite directions of built-in polarization, exhibit ordinary symmetric hysteresis loops, the “asymmetric” samples, with one of these surface layers removed, show asymmetric hysteresis loops. To describe our experimental findings, we combine two kinds of models: one is phenomenological, utilizing the above general features; and the other is ab initio, taking into account the actual atomic structure at the bulk ferroelectric–surface layer–electrode interface. Namely, the ab initio calculations show that the electric polarization within the surface layer occurs due to the shifts of the relaxed Ti ions with respect to the oxygen ion octahedra on the PbO-terminated surface. We ascribe the self-polarization effect to the occurrence of the built-in electric field resulting from the formation of Pb-O planes within the surface layer.

1. Introduction

Several attempts have been made to explain some physical effects in ferroelectric (FE) perovskites by assuming that the properties of the surface layer differ from those of the bulk. Namely, that such features as the dielectric permittivity, electrical conductance, optical absorption, and domain wall (DW) properties [1,2,3] would be different. The latter properties of ferroelectrics, which are determined by the interaction between the bulk and the surface layer contributions, have been intensively studied. These are also affected by the extrinsic effects related to the homogeneous or the inhomogeneous chemical composition, the presence of impurities or doping [4,5,6,7], structural defects, the strain–stress effect, as well as the electric polarization variance [8,9,10]. Hence, it is a challenge to determine to what extent manipulation of the physical and chemical features of the surface layer may lead to the tuning of the electrical properties of the surface and bulk. The functional goals to be considered are related to the stability and the switching conditions of the polarization perpendicular to the surface of the FE material. A stable orientation can be obtained in a several nm thin layer, as has been shown by calculations [11,12] and proved by experiments [13,14]. The standard procedure used for the formation of an FE polarization is based on the reorientation of the FE domains, induced by the application of external electric fields that are higher than the coercive field or by a suitable combination of mechanical stress and temperature gradient at the paraelectric (PE)–FE phase transition. In cases where the poling occurs at a temperature higher than the PE–FE phase transition temperature, this may be accompanied by electronic or ionic leakage currents. However, the ionic current may lead to side effects, such as the rearrangement of the structure within the formed material. The conditions for the controlled stabilization and switching of the FE polarization are still being studied and discussed [2,9,15,16,17,18,19].

Most of the models of the self-polarization effect in FE materials have been developed for thin films, whose geometrical configuration is, per se, asymmetrical (i.e., a substrate with deposited thin films [20]), and, therefore, cannot be applied automatically to a single crystal, which represents a self-supported structure with full symmetric boundary conditions. In fact, for a virgin PbTiO₃ crystal, one can observe a perfect hysteresis loop without a so-called imprint [21]. This compound is still the basis for applications where large polarization effects are needed [22,23] and its use ranges from memories to energy materials [24,25]. The goal of our paper is to explain what kind of treatment of a virgin PbTiO₃ single crystal (with perfect FE properties), without electrical poling, allows the creation of a system with a distinguished direction of polarization. We will reach this effect by applying a combination of controlled thermodynamic procedures (here moderate temperature of 600 °C and high activity of oxygen) in order to induce structural and chemical modifications in the surface layer. In this way it is possible to produce a stabilized non-zero net P_S in PbTiO₃ crystals without the preliminary application of an external electric field. Alternatively, this state can be obtained by applying the unbalanced contributions from the chemically modified surface layer.

We suggest that the chemically modified surface layer on PbTiO₃ can be formed by thermal treatment of a single crystal under oxidizing conditions (similar to [2,7,26,27]). Thermal annealing conducted in conditions different from those used for the crystal growth ought to lead to the outward migration of ions. Such chemically inhomogeneous structures may form an electronic or ionic space charge within the surface layer. Consecutively, polarization can be induced within the surface layer. Its orientation is expected to be perpendicular to the surface of the perovskite material. The polarization would be stable since the inhomogeneous chemical effects are frozen yet at a moderate temperature range. Hence, the produced perpendicular alignment of the FE polarization ought to be fixed or resistant to external fields.

The above effects can be described within the so-called self-polarization model. The concept of self-polarization is not new; this scenario had been used about twenty years ago to describe the fact that a significant amount of polarization exists in virgin synthesized bulk material without pre-poling treatment [28]. Self-polarization has been observed in thin ferroelectric films (especially for PbTiO₃ and Pb(Zr_xTi_1−x)O₃ (PZT)) prepared by different deposition methods (see, e.g., [29,30,31,32,33]). A stable self-polarization effect can be obtained in PbTiO₃ in a wide temperature range due to the fact that the FE–PE phase transition occurs at a high temperature of around 490 °C–493 °C [34]. Moreover, the spontaneous polarization, P_S, estimated from the ionic shifts, has a high value 0.73–0.81 C m⁻² [10,34,35]. Comparable magnitudes (P_S = 0.80 C m⁻² [36,37]) can be obtained from standard pyroelectric and P–E hysteresis loop measurements [15]. The FE order in the PbTiO₃ crystals has been observed in the perovskite phase with tetragonal symmetry and unit cell parameters a = 3.900 Å and c = 4.150 Å (tetragonality c/a = 1.064) at room temperature [38].

It has been shown by ab initio calculations, that the spontaneous polarization switching in the thin PbTiO₃ layer is affected by the termination of the crystal structure. This means that both PbO-terminated and TiO₂-terminated surfaces ought to be considered. Moreover, the oxygen vacancy, V_O, defects also affect the electronic structure, the P_S value and its switching characteristics [6,15,39]. One can notice a correlation between the thermally treated PbTiO₃ crystals and the Ruddlesden–Popper (RP) structure that shows a close analogy to the perovskite matrix phase and simultaneously has a stoichiometry shifted towards the AO excess in a layered AO–(ABO₃)_n (n = 1, 2, 3, …, ∞) structure. In thermally treated PbTiO_3−x single crystals, PbO(PbTiO₃)_n phases with n = 1–10 and a lattice parameter similar to PbTiO₃ have been identified [40].

The aim of this work is to confirm whether it is possible to induce self-polarization in the surface of the FE PbTiO₃. For that, we use different experimental techniques, assisted by ab initio and phenomenological theoretical modeling: FE hysteresis loop studies, measurements of the electromotive force’s temperature dependence and thermo-stimulated currents as well as piezoresponse force microscopy (PFM) images. The combination of the above techniques enables us to discern the contributions from the surface layer and from the bulk.

2. Materials and Methods

2.1. Crystal Growth and Characterization

We have used single crystals grown by the flux method (for details, see ref. [41,42,43]). This method was described earlier by Fesenko [44]. The transparent crystals chosen for the experiment were in the form of rectangular plates and exhibited a low electric conductivity of the order of 10⁻¹¹ S cm⁻¹ at room temperature. The ferroelectric domain patterns were observed with the use of a polarizing microscope.

Three series of samples were prepared. The virgin, as-grown crystal samples were prepared for electrical tests. The as-grown and cleaved samples were taken for the atomic force microscopy (AFM) and PFM tests. The third series of the samples were treated at high temperature for several hours (e.g., 10, 12, 15 h) under oxidizing conditions at p(O₂) = 200 Torr. Some samples were tested directly after this treatment, whereas others were cleaved prior to the tests.

The crystal samples, their symmetry, and phase content were first characterized by X-ray diffraction (XRD) tests (see, e.g., [40]). The results for virgin (as-grown) and oxidized crystal were published in an earlier paper [41].

2.2. AFM/PFM Tests

The atomic force microscope JEOL 4210A equipped with Pt/Ir conducting tips (Nanoworld) operating in the contact mode was applied. A DC voltage source and femto-ammeter setup was used to measure the electrical local conduction (LC-AFM). The measurement was conducted for the virgin cleaved crystal and for the crystal oxidized at 650 °C for 12 h. The topography images and the piezoresponse of the crystal surface were recorded at room temperature and ambient air at a pressure lowered to ~5 × 10⁻⁸ bar. The WinSPM program (Scanning Probe Microscopes, JEOL) was applied. The piezoelectric response (PFM) measurement was carried out using the atomic force microscope described above. The PFM signal, obtained in the contact mode, was amplified with two lock-in voltmeters, separately for the in-plane and out-of-plane orientations.

2.3. Electrical Measurements

Rectangular samples with a thickness d ≈ 0.1 mm and painted with Ag paste were prepared. Two different configurations of the samples, i.e., “symmetric” and “asymmetric”, were prepared in order to verify the influence of the surface layer features on the effective macroscopic electric properties of lead titanate crystals.

The “symmetric configuration” corresponds to the virgin sample consisting of the perovskite bulk wrapped up in the surface layer, and formed in the growth process (Figure 1a, see also Figure 2a for a more detailed view). The surface layer was also purposefully modified due to the reaction with ambient air during high temperature treatment (Figure 1b and Figure 2b). The electrodes were placed on the surface layers, and the sequence “electrode–surface layer–bulk–surface layer–electrode” was obtained for both cases (see Figure 1a,b and Figure 2a,b). The “asymmetric” configuration of the sample (Figure 1c and Figure 2c) was obtained after removing the surface layer, produced at high temperatures. This was achieved either by polishing only one side of the crystal sample with diamond paste or by cleaving the sample. After this procedure, the electrodes were deposited. Hence, the sequence “electrode–bulk–surface layer–electrode” could be produced.

The thermo-stimulated currents (TSCs) were recorded at a constant rate of temperature increase dT/dt = 0.1 K s⁻¹ in the vicinity of the FE phase transition, T_C ≈ 490 °C. The TSC from the PbTiO₃ single crystal samples was obtained for: (a) a virgin stoichiometric crystal, (b) a crystal after thermal treatment at oxidizing conditions in the paraelectric phase, and (c) a crystal which has been prepared as in point (b), but with the surface layer removed from one side to obtain an “asymmetric configuration”. Although the samples were not polarized prior to the measurement, in the thermo-stimulated current, contributions from the pyroelectric current should be observed [29,45]. The electric current–electric field (I–E) hysteresis loop [1,46] measurement was performed on the multi-domain crystals both for the “symmetric” as well as the “asymmetric” cases. A triangular voltage sweep at frequency f = 0.01 Hz and strength up to 25 kV cm⁻¹ was applied using a triangular signal generator and a digital oscilloscope for recording. The value of the remnant polarization was estimated from these I–E loops.

The temperature dependence of the electromotive force (EMF) was measured with the use of an electrometer (6430-Keithley). The crystal sample was prepared as follows: the PbTiO₃ sample was oxidized at 650 °C, cleaved to obtain an “asymmetric configuration” (Figure 1e and Figure 2c), and then the electrodes were deposited. In this case, one electrode was deposited on the sample’s surface layer. The other electrode was deposited directly on the bulk after cleaving. The measurement was carried out within the FE phase during heating to a range of ~200–460 °C. The analog electrical data for the hysteresis loop measurements were collected using a system Adwin-Gold (Jäger).

2.4. EDX Investigation

For the compositional analysis of the surface layer of an oxidized PbTiO₃ crystal at the nanoscale, electron dispersion X-ray spectroscopy (EDX-Oxford Instruments ISIS300, SiLi detector with an ultrathin window), combined with an analytical TEM (Philips CM 200, LaB6 cathode, 200 kV), was performed on small particles taken from the surface of a single crystal.

2.5. Calculations

Here we perform two kinds of calculations, ab initio and model. The ab initio calculations were performed using density functional theory (DFT) in the generalized gradient approximation (GGA) proposed by Wu and Cohen [47]. The full-potential linearized augmented plane-wave (LAPW) method [48] was used.

A general theoretical approach to the description of the layered dielectric structures, especially those containing ferroelectric layers, consists in the application of the coupled system of Ginzburg–Landau equations for the ferroelectric layers and Maxwell equations for passive (paraelectric or other non-ferroelectric) layers, augmented by corresponding boundary conditions at the interfaces. All of the above equations contain phenomenological material parameters such as dielectric permittivity, which in such general case is frequency and thickness dependent. The resulting complicated equation of state [49,50,51] contains not only the polarization (both spontaneous in ferroelectric and induced in non-ferroelectric layers) but also its spatial derivatives. This requires more boundary conditions [52,53], severely complicating the situation and, because the analytical solution of the above set of equations in the general case is impossible and the numerical solution (the so-called phase-field method, see ref. [54] and references therein) is computationally extremely intensive, requiring some model considerations. Therefore, in problems of multilayer structures, researchers usually resort to the model calculations, which capture the essential physics (and sometimes even can give a quantitative description of the experiments) surprisingly well [49,50,55,56]. For instance, for sufficiently thin layers, the corresponding boundary conditions can serve as the equation of state (i.e., the equation relating the internal electric field and polarization, [49]) in both blocking and ferroelectric layers. Our subsequent calculations show that such very simple description gives a fairly good coincidence with the experiment.

In our model calculations, we consider a PbTiO₃ layer with built-in polarization (so-called self-polarized layer), which exists at the interface between the FE (also PbTiO₃) layer and the electrode, and that is nonconductive. The latter means that the built-in polarization sustains for a long time. In our model, this polarization cannot be switched by the application of an external electric field. In order to satisfy the continuity of the electric displacement vector, D, and the electric field, E, at the interface, the polarizations of FE and self-polarized layers form “head-to-tail” or “tail-to-head” configurations, as illustrated in the Figure 2d. When the polarization in the FE layer switches its direction by application of a voltage, the self-polarized layers turn to a state with a discontinuity of the polarization, termed a “head-to-head” or “tail-to-tail” structure. The discontinuity of the polarization generates a strong electric field. After removal of external voltage, this field works to switch the FE polarization back to the “up” state to restore the continuity. In Figure 2d, this process is shown to generate the ordinary symmetric hysteresis loop as the built-in polarizations in the upper and lower layers have opposite directions. This is not the case in Figure 2e, where one of the polarization directions in the ferroelectric layer is not compensated by a corresponding built-in polarization. As we shall see below, this generates an asymmetric hysteresis loop.

3. Results

3.1. Piezoresponse in Nanoscale

The multi-domain state of the PbTiO₃ crystals was confirmed with the polarizing microscope test. The rectangular array of the domains indicated the possibility of compensation of the FE domain contribution to the net electric polarization (Figure 3).

The occurrence of the FE domains was tested at the nanoscale by piezoresponse measurements (PFM). Although the polarizing microscope studies show a rectangular network of domains on the nanoscale PFM images (Figure 4), the DWs (in-plane and out-of-plane) are more rounded and not as sharply defined. This effect can be ascribed to the influence of the strong electric field on the DWs, probably resulting from structural defects. Nevertheless, the FE domains remained in the PbTiO₃ crystals, though their shapes were affected.

The search for the origin of self-polarization induced by oxidation of the PbTiO₃ crystal calls for an investigation of the surface’s FE (piezoelectric) properties, where the morphology has been dramatically modified during oxidation. After the recrystalization induced by oxidation, the surface of the crystal is covered with many crystalites (see Figure 5a and ref. [57]). PFM on such a rough surface shows a high lateral inhomogeneity (see Figure 5b,c). One can observe an anticorrelation between areas with high piezoelectric response and regions with good electrical conductivity (Figure 5b–d).

LCAFM measurements performed at 250 °C on an oxidized PbTiO₃ crystal reveal that the conductivity is filamentary and can vary by several orders of magnitude between the filaments and the matrix. The band arrangement observed for the conducting filaments (see Figure 5(d1)) also occurs in the piezoresponse. This form of piezoelectric arrangement may be connected with the creation of PbO(PbTiO₃)_n (Ruddlesden–Popper phases with high n) as the growth of these phases in the upper part of the surface region during oxidation has been proven for different perovskites (BaTiO₃, SrTiO₃, PbTiO₃, KnbO₃) with roentgenographic studies [40].

3.2. Electric Polarization

The thermo-stimulated currents were measured both on the as-grown and the thermally treated samples. The obtained results are shown in Figure 6. In the case of the virgin, as-grown, PbTiO₃, the sample was in the “symmetric configuration”, and it was not polarized by an external field prior to the measurement. Hence, the expected and most probable net “natural” polarization was equal to zero due to its compensation in the FE domains. Therefore, the current peaks in the vicinity of the FE–PE transition (Figure 6a) can be classified as Barkhausen noise.

The current measured in a sample that had been thermally treated within the PE phase (T = 650 °C) in ambient air, is shown in Figure 6b. A flat and narrow anomaly can be discerned, within experimental accuracy, in the vicinity of the FE–PE phase transition. The oxidizing conditions provided effective chemical gradients in the high temperature range, which in turn may have caused the migration of ions from the bulk towards the surface and have resulted in a chemical modification of the surface layer. One can expect that such a deliberately modified layer might affect the properties of the sample. However, the lack of a marked peak in the measured current/temperature dependence would correspond with the compensated net electric polarization. However, an unexpected result occurred when the preliminarily unpolarized crystalline PbTiO₃ sample was prepared in the “asymmetric” configuration. A substantial current peak appears in the vicinity of the FE–PE phase transition temperature, exhibiting a maximum at 496 °C (Figure 6c). The magnitude of the polarization estimated by integration was 0.7 C m⁻². The latter peak might either originate from a defect-generated space charge or from the charge density variation due to FE phase transition. If the effect is related to the deep level traps, introduced by the applied thermal treatment, then it should occur both for the “symmetric” and the “asymmetric” configuration of the samples. As this is not the case, another explanation is proposed. Namely, the mechanical removal of the modified surface layer from only one side of the sample to expose the natural polarization of the bulk to direct contact with the re-deposited electrode. We interpret the occurrence of the peak in Figure 6c as the manifestation of the FE–PE phase transition, because the charge density changes when the spontaneous polarization vanishes. During cooling, an opposite current of similar magnitude appears when the sample goes through the PE–FE phase transition. Hence, the electric polarization of the sample is not compensated macroscopically in the “asymmetric” configuration. This allows us to deduce that the lack of the net polarization compensation is related to the modified surface layer. Moreover, one can consider the observed peak as the pyroelectric current peak [35].

The electric current–electric field (I–E) hysteresis loops recorded for samples of the as-grown crystal and the crystals after oxidation (in the “symmetric” and “asymmetric” configuration) are shown in Figure 7. The measurement of the as-grown sample shows the symmetric I–E hysteresis loop (Figure 7a) under the application of the triangular periodic electric field pulses. The spontaneous polarization estimated from both I–E loop branches, labeled “+” and “−”, are similar in absolute value: P₊ = 0.84 C m⁻² and P₋ = −0.80 C m⁻². This yields values that are comparable to the one reported in the literature i.e., P_S = 0.80 C m⁻² [35,36,37,38]. The coercive field equals E_C = 2.6 kV cm⁻¹. The similarity of P₊ and P₋ is compatible with the spatial features of this sample as the virgin sample consists of the bulk FE perovskite lattice enveloped or wrapped uniformly by a surface layer formed during the growth process. One can notice that the estimated P_S value is comparable to that obtained/calculated from the XRD data. Hence, in this case the surface layer contribution is insignificant.

The sample that had been treated at high temperatures exhibited a diminished and slightly asymmetric I–E hysteresis loop (Figure 7b). The estimated values of polarization were equal to P₊ = 0.003 C m⁻² and P₋ = −0.014 C m⁻². Moreover, a rise in the leakage current was observed when the applied electric field was increased.

The I–E hysteresis loops obtained both for as-grown and oxidized crystals in symmetric configuration (shown in Figure 7a,b) yield similar values of P₊ and P₋, which coincides with the pyroelectric current measurements (compare Figure 7a,b to Figure 6a,b). However, the P_S value, obtained for the as-grown (a) and thermally treated (b) samples, differs by about one order of magnitude. In the case of the as-grown sample the charge bound by the FE bulk was released and measured. In case (b), after thermal treatment of the sample, a decreased value of P_S occurred. We ascribe this effect to the surface layer formed by ionic migration. Such a migration is possible in a high temperature range where the chemical gradients are effective. Therefore, the modified surface layer, which was chemically disturbed and different from the FE bulk, dominated the actual polarization estimated for the multi-component sample. This assumption is supported by results obtained by secondary-ion mass spectrometry (SIMS) results, which are presented in detail below, and by the results obtained in an earlier work [41], where it was shown that the Ruddlesden–Popper phases PbO(PbTiO₃)_n, (n = 1, 2, 3, 4, 5, 6) were formed on the PbTiO₃ crystal surface after similar thermal treatment.

The I–E hysteresis loop for the “asymmetric sample” (c), obtained after the removal of surface layer from one side of the crystal (see Figure 1c and Figure 2c), is shown in Figure 7c. In effect, the asymmetry was induced by depositing one electrode directly onto the PbTiO₃ bulk. The actual values of P₊ and P₋ were estimated as 0.86 C m⁻² and −0.032 C m^−2, respectively. The coercive field values are E_c+ = 3.5 kV cm⁻¹ and E_c− = −2.3 kV cm⁻¹.

Hence, the properties of sample (c) can be described as a superposition of the properties of samples (a) and (b). For the as-grown ABO₃ sample (a), one has a chaotic set of FE domains (Figure 2a). This makes the P_S reorientation possible so that the proper I–E hysteresis loop occurs. In contrast, for sample (b), the P_S can only be fixed partially when the segregation of oxides appears in the surface layer. Moreover, the applied external field was varied up to E_EXT = ±25 kV cm ¹, i.e., it was much higher than the corresponding coercive fields E_C. We have found that the “surface layer” electric field E_SL > E_C develops and establishes within the surface layer as a consequence of the ionic migration forced by the thermal treatment. Hence, E_SL can essentially block the polarization reorientation on one side of the sample. This is manifested by the diminished values of the measured, effective polarization P₋ which is equal to 0.03 C m⁻² (see Figure 7c).

3.3. Electromotive Force $ϵ$ (T)

The measurement of the electromotive force, $ϵ$(T), was conducted both for the oxidized “symmetric” sample (Figure 1b) and the oxidized “asymmetric” sample (Figure 1c). A lack of electromotive force $ϵ$(T) was observed for the “symmetric” sample. In contrast, the “asymmetric” sample produced a pronounced $ϵ$(T) signal dependent on temperature, $dϵ$(T)/dT > 0 (Figure 8). We propose to ascribe a linear $ϵ$(T) = A + BT dependence to the variance in the Pb²⁺/Pb⁰ ratio (see XPS data below) within the surface layer of the PbTiO₃ crystal. We would like to emphasize that the measured signal was not related to the Seebeck effect, as the temperatures of the sample and the electrodes T_el,1 = T_sample = T_el,2 were equal to each other. Therefore, we assume that $ϵ$(T) originates from the chemically inhomogeneous surface layer. The gradient $dϵ$(T)/dL can be estimated. Because the $ϵ$(T) varies from 0.15 V to 0.25 V and the surface layer depth (taken from the SIMS test) is ~20 nm for the Pb-rich layer and/or ~50 nm for the whole Pb–Ti surface layer, we calculated the gradient value to be $dϵ$(T)/dL ≈ 10⁵ V cm⁻¹. The estimated electromotive force gradient was at least one order of magnitude higher than the coercive field E_C~10³ V/cm. Hence, such a strong field, locked within the chemically modified surface layer, could affect the FE features of the PbTiO₃ samples, which were detected by the I–E hysteresis loop and the pyroelectric current measurements.

We obtained similar measurements of the local electromotive force on the nanoscale using AFM potentiometry for the oxidized crystal, which was cleaved in situ (for a schematic of the measurement see Figure 9a). The distribution of the potential in the surface region (Figure 9b) allows, without the information from an SIMS depth profile, calculation of the strength of the local electric field (see Figure 9c). The value is higher than the coercive field for PbTiO₃. Therefore, we can make the assumption that in this region the polarization of PbTiO₃ should be permanently oriented in the direction of the built-in fields.

3.4. Surface Layer, XPS Studies

Before we explain the models that allow a better understanding of the origin of the pinning of the spontaneous polarization in a lead titanate crystal, we analyze the differences in the electronic structure, chemical composition, crystallographic and defect structure between the surface of the crystal as grown (that does not exhibit self-poling effects) and the upper part of these crystals after thermal treatment (where self-poling occurs). A prerequisite is the proof that the used crystals possess excellent quality, primarily because they concern crystals grown by spontaneous crystallization from flux. Although the X-ray diffraction pattern of our crystals shows the typical lattice parameter [40] and the dielectric and Raman measurements a correct sequence of the phase transitions [41,42,58], from the chemical point of view, the composition of the crystal due to the method flux growth shows a surplus of Pb. Most scientific articles dedicated to FE properties or defect chemistry are based on such prepared crystals [44,59,60,61,62,63], although some refinements are available [64]. Using the chemical analysis ICP-OES (for determination of the concentration Ti and Pb) and infrared spectroscopy (for the analysis of the amount of oxygen), it was found that the chemical formula of our crystal can be written as follows: Pb_1.02±0.02Ti_0.96±0.012O_3±0.04. This additional incorporation of a few percent of Pb in the matrix of the crystal can be explained in two ways: either additional PbO will create a stacking fault in the matrix, (PbO)_0.06PbTiO₃ [65,66,67,68], where an equidistant distribution of PbO planes will generate an RP phase [40]; or, if the valence of Pb is reduced to 4+, an anti-structural disorder can be observed in the matrix [41,42]. Due to the similar ionic radii of Ti⁴⁺ and Pb⁴⁺, the lead cations can occupy the position of titanium, and the correct formula, in this case, could be Pb²⁺Ti_0.96Pb⁴⁺_0.04O₃ [41,43]. In fact, the stoichiometry analysis of XPS spectra of in situ cleaved PbTiO₃ crystals shows that the surface layer is enriched by 25% in Pb (compare Figure 10). This discrepancy in fifty/fifty termination between PbO and TiO₂ can only take place if the cleaving plane runs along a stacking fault [67,68].

Our XPS analysis shows the typical shapes of the core lines of PbTiO₃. The O1s line is a singlet, and the shapes of Ti2p and Pb4f are single doublets whose energetical position can be assigned to cations with valence 2+ for Pb and 4+ for Ti (Figure 11; however, no evidence for the presence of Pb⁴⁺ was found. The absence of an occupied state in the valence band (VB) and the correct width of the band gap (BG, here > 3 eV, see lower right panel of Figure 11) allow for the classification of the surface layer as a dielectric. It should be noted that, for the cleaved surface with a conchoidal morphology of the cleavage/fracture, no additional compounds of the O1s line (typical for physical and chemical adsorption) were found, and the absence of the C1s peak indicates that the surface is free from carbon contamination. Such a reference surface was annealed in situ under the UHV conditions and oxidized ex-situ. Using thermogravimetric analysis (TGA), we have determined that the maximal annealing temperature in oxidizing conditions should not be higher than 600–650 °C, otherwise evaporation of PbO from PbTiO₃ would occur.

We have studied the changes induced by thermal treatment of the surface or in the surface region by XPS, EDX/TEM, and AFM. The ex-situ oxidation leads to a dramatic increase in contaminants that can be identified by additional compounds appearing in the O1s and C1s lines. Only the Pb 4f core line, after annealing in an oxygen-rich environment, shows an essential change in its shape (Figure 12). The precise difference analysis, relative to the reference Pb4f line of the cleaved (clean) crystal, shows the existence of two peaks with higher and lower binding energy. A search in the literature [41,42] suggests that additional oxide (PbO) and metallic lead have been created in the surface layer of oxidized crystal.

In fact, after a long time of contact between PbTiO₃ and oxygen at high temperatures, we found (using EDX), in the materials removed from the surface layer/regions via mechanical polishing, that a specific population of micro-crystallites does not possess any Ti ions and that their chemical formula may be typical for PbO or Pb (Figure 13). We have presented a similar conclusion about the surplus of Pb in the upper part of the surface layer by the analysis of the SIMS depth profile [57]. Moreover, we have found an anti-correlation between the geometrical position of the surplus of lead and enrichment with Ti (which can be observed in the deeper region of the surface layer between 20–45 nm). Due to the small non-stoichiometric limit for cations in the matrix of PbTiO₃, these materials cannot maintain the perovskite structure and must recrystallize or remain amorphous. The creation of the crystallites on a cleaved surface with a distinct morphology (which can be observed with AFM [57]) and the change of X-ray diffraction pattern (see our previous operando investigation of the crystallographic structure [40]) does not leave any doubt that the surface region is subject to recrystallization from PbTiO₃ into R–P and TiO phases. These observations about the complicated chemical and crystallographic changes in the surface layer/regions should be considered when analyzing the self-poling’s origin in oxidized PbTiO₃ crystals. Although our data confirm that a chemical transformation occurs in the surface layer, and we have information about the stacking sequence (chemistry) in the upper and deeper layers, we do not know which type of ions (cations/anions) can migrate out-of-plane for those moderate temperatures and generate this solid-state reaction.

The XPS investigation in situ on cleaved crystals showed that at 200 °C segregation of Pb can be observed (Figure 10). Because such migration of Pb in the perfect perovskite matrix is high unlikely for temperatures lower than 500 °C, it is probable that the extremely high density of extended defects in PbTiO₃ crystals [59,67,68], with specific crystallographic and electronic properties, may play an important role in the easy diffusion of Pb cations (see, e.g., ref. [69]).

3.5. DFT Calculations

To quantify the effect of an asymmetric environment on a Ti atom in an octahedral cage of oxygen atoms, we performed DFT calculations using a generalized gradient approximation [47] that can offer a reliable description of PbTiO₃ structural properties. Employing the full-potential LAPW method [48] in the thin-film setup [70], we studied a TiO₂-terminated nine-layer PbTiO₃ film with the optimized lattice parameters of the FE structure but set the polarization to zero. Thus, in each plane, the oxygen and metal atoms are at the same vertical position.

Thereafter, we monitored the total energy of the system as a function of the vertical displacement of the central Ti atom. Due to the strong depolarization field in the structure, the Ti prefers to stay in the plane center (blue curve in Figure 14). To introduce an asymmetry in the system, we then replaced one of the surface TiO₂ layers by PbO (right panel of Figure 14). Allowing the central Ti atom to relax again, we found that it became shifted by 0.03 Å away from the PbO terminated side, resulting in a gain of about 6 meV per in-plane unit cell (red curve in Figure 14). Further relaxation of the whole film resulted in a configuration with a finite, non-switchable dipole moment, corresponding to a polarization of 0.007 C m⁻². This value is compatible with a 0.03 Å shift of the Ti⁴⁺ ions in the TiO₂ layer but is significantly lower than the polarization of bulk PbTiO₃, indicating that, in the actual surface region, more extended defects (e.g., additional TiO₂-rich phases or Pb⁰) must be present.

3.6. Model Calculations

Following the methodology outlined in Section 2.5 we derive here the boundary conditions for the normal component of the vector D on the ferroelectric–self-polarized layer interface. It can be seen from Figure 2d that field and polarization vectors are directed along the z axis, i.e., D = Di_z, E = Ei_z, P = Pi_z. In this case, the boundary conditions for the continuity of the normal component of vector D on the FE–self-polarized layer interface read as follows:

D_b = D_f,

(1)

where D_b and D_f are, respectively, the z-components of electric displacement vectors in the self-polarized (we may call it the blocking layer, hence index b) and FE layers. These are given by the following material equations:

D_b = ε₀ε_bE_b, D_f = ε₀E_f + P,

(2)

where E_b, E_f and P = P_f + P_b are, respectively, the average electric fields in the blocking (self-polarized) and FE layer and the average polarization of the FE layer, consisting of switchable P_f and non-switchable P_b polarizations. Here, ε₀ is the vacuum permittivity in SI units. We note that the seemingly asymmetric nature of the material Equation (2) actually comes from the additional boundary conditions [49,53]. The equation for D_b contains only the blocking layer polarization P_b = χ_bE_b (here χ_b = ε₀ (ε_b − 1) is the dielectric susceptibility of the blocking layer), while that for D_f includes both components, P_b and P_f. This shows that, while the ferroelectric layer dipoles can be spontaneously and coherently oriented (thus forming a spontaneous polarization), those of the non-ferroelectric (blocking in our case) layer cannot. On the other hand, the constant polarization P_b of the blocking layer acts as a form of external electric field, which penetrates into the ferroelectric layer as well. At the same time, it can be shown that the spontaneous polarization does not exist beyond the actual ferroelectric layer.

Equation (2), along with (1), determines the polarization coupling in the blocking and ferroelectric layer, i.e., they describe the mutual influence of the above two layers. To make this more evident, we rewrite expression (1) with respect to (2) explicitly as E_b = (ε₀E_f + P_f + P_b)/(ε₀ε_b). Thus, system (2) should be augmented by the equation for the electric potential across the structure in Figure 2e [51]

dE_b + hE_f = (d + h)E,

(3)

where E is the external applied field, and d and h are the thicknesses of blocking and FE layers, respectively. Equation (3) permits to express E_f as a function of E and E_b

$$E_f=-\frac{d}{h} E_b+\left(1+\frac{d}{b}\right) E.$$

(4)

Combining Equation (2) with respect to (1) and (4), we obtain the set of equations for the description of the hysteresis loop P(E) of the whole structure, having at our disposal the “internal” (i.e., that in the FE layer) P(E_f). These equations read for d/h << 1

$$E=E_f+\frac{d}{h}\frac{P}{{\epsilon }_0 {\epsilon }_b}, P=P_f+P_b.$$

(5)

The “internal loop” P(E_f) can be chosen in the arbitrary form. This form should, however, reflect the property in which an electric field is swept between large positive and negative values. The polarization must approach the spontaneous polarization ±Ps, which is realized when all FE dipoles are completely aligned by the external field. We note here that the value of P_S for our structure would be affected by the non-switchable polarization P_b.

The latter property of the hysteresis loop is obviously present in the Landau expansion aP +bP³ +E_f = 0. However, the problem with this expansion is that, near to the coercive field E_c, the derivative dP/dE goes to infinity, which is unphysical. The reason for this is that, in our experiment, we measure the current loops, i.e., the dependence of current density j = dP/dt on the electric field E. Because we sweep the external electric field linearly with time E = at to measure the hysteresis loop, the derivative dP/dt becomes dP/dt = a dP/dE, i.e., it is proportional to the field derivative dP/dE of polarization. This means that realistic dependence P(E_f) should have finite derivatives dP/dE_f everywhere.

Such a realistic hysteresis loop can be taken as an empirical dependence [71]. We have

$$\begin{array}{c}{P}^{+}\left(E\right)=P_S\tanh \frac{E-E_c}{2\delta }+f\left(E\right),\\ P^{-}\left(E\right)=P_S\tanh \frac{E+E_c}{2\delta }+f\left(E\right), P^{+}\left(E\right)=-P^{-}\left(-E\right)\end{array}$$

(6)

where f(E) is an arbitrary odd function, which is to be chosen from the best fit to experiment. Here δ is the fitting parameter and E_c is the coercive field. The plots of the functions P^±(E) (6) and their derivatives dP^±/dE are reported in Figure 15 for f(E) = 0 (panels (a) and (c)) and for f(E) = aE (panels (b) and (d)). It is seen that hysteresis loops and their derivatives resemble those measured in the experiment. Additionally, the derivatives dP/dE have the same magnitude.

Now we are in a position to calculate the real observable hysteresis loops corresponding to Figure 2d (symmetric situation) and Figure 2e (asymmetric situation). To do so, we observe that spontaneous polarization P_S is affected by the built-in P_b, i.e., we should substitute in Equation (6) P_S → P_S + P_b in P⁺(E) and P_S → P_{S −} P_b in P⁻(E). We note here that, for the symmetric structure depicted in Figure 2d, the term P_b is cancelled as we have P_S + P_b − P_b = P_S for both branches P^±(E) of the hysteretic curve so that it will be symmetric, like that shown in Figure 15. Substitution of the thus-modified loops (6) into Equation (5) generates the desirable “external” loops P(E). These are plotted in Figure 16.

Similar to Figure 15, we plot both P(E) curves and their derivatives dP/dE, proportional to the current density. The case P_b = 0 is reported in panels (a) and (d) of Figure 16. This situation corresponds either to complete compensation (panel (d) of Figure 2) or to the absence of blocking layer. This case, similar to Figure 15, corresponds to the symmetric hysteresis loop and the same amplitudes of the currents in panel (d) of Figure 16. In this case, for E → ±∞, the corresponding asymptotic values of P⁺(E) and P⁻(E) are the same. Panels (b) and (e) of Figure 16 report the situation of the incomplete compensation of P_b, when either P_b in a single layer (panel (e) of Figure 2) is smaller than the saturation (spontaneous) polarization of the FE layer P_S or in the situation in which we do not completely cut one of the blocking layers in our structure. Here we note that, at E → ∞, the branches P⁺(E) and P⁻(E) merge, similarly to the case of P_b = 0 (Figure 15), while at E →−∞ P⁺(E) ≠ P⁻(E), with their difference being related to the presence of P_b. If we change the sign of P_b, our picture will be opposite: at E →−∞ P⁺(E) = P⁻(E), while at E → ∞ we find P⁺(E) ≠ P⁻(E). The current, proportional to derivative dP/dE is now slightly larger for the left-to-right direction of the field sweep, while in the opposite direction it is smaller.

Finally, the panels (c) and (f) of Figure 16 display the experimental situation, related to the panel (e) of Figure 2, i.e., to the fully uncompensated P_b. We choose P_b= 5 to be slightly smaller than P_S = 6. In coincidence with the experimental situation, on panel (f) of Figure 16 we can see a large asymmetry in the current magnitudes in back-and-forth directions of the field sweep. This asymmetry is related to the presence of an uncompensated (i.e., “tail-to-tail” or “head-to-head” oriented) built-in polarization relative to the spontaneous polarization. Panel (c) shows the physics of the process. Namely, for P_b > 0 (as we choose here), it is directed in the positive direction of z axis, being parallel to the external field, E, as shown in Figure 2. Thus, when we sweep the field in Figure 16c from left to right (which corresponds to the positive direction of z axis, as in Figure 15), the oppositely directed polarization in the FE layer adjusts to the field direction so that all polarization in the FE layer has a similar direction to P_b, creating a stable (in this case “head-to-tail”) configuration. This situation is reflected by the red curve in Figure 16c. When we sweep the field back, the built-in polarization P_b stays intact, while the FE polarization tends to align with the (opposite to P_b) field direction, thus creating an energetically unfavorable “tail-to-tail” configuration. This is why, for this sweep direction, the system resists the creation of an energetically unfavorable configuration, i.e., it has a very large coercivity so that the polarization does not change much even for large negative fields. This generates a very weak variation of the polarization in this branch which gives a much smaller (than that of the red curve in the Figure 16c, corresponding to the direct sweep of the external electric field) derivative dP/dE. This, in fact, demonstrates the role of the uncompensated built-in polarization.

4. Discussion

The assumption that the surface layer consists of several components allows for the interpretation of the built-in polarization properties in the PbTiO₃ crystal under investigation. The presence of a built-in polarization is manifested by the asymmetric shape of the I–E hysteresis loop and the pyroelectric current peak.

The standard, as-grown PbTiO₃ samples show a symmetric I–E hysteresis loop, a lack of pyrocurrent, no electromotive force, and a spontaneously formed ferroelectric domain pattern structure.

The suppressed FE properties were detected in the thermally treated “symmetric” samples. In this case the modified surface layer envelopes the crystal bulk. The thicknesses of the surface region with PbO and TiO₂ enrichment can be determined from the SIMS profile analysis shown in our previous paper [57]. Using this value and the knowledge of the macroscopic electromotive force, we can calculate gradients of the electromotive force in the surface layer and thus determine the strength of the local electric field—$dϵ$/dL ≈ 10⁵ V cm⁻¹. Therefore, the E_SL, which is essentially stronger than the FE coercive field E_C, induces the polarization P_SL within the surface PbTiO₃ sublayer. Hence, the P_SL is directed outward and fixed by the surface layer field E_SL. The FE polarization of the bulk is screened by the latter surface layer features, despite the fact that the bulk FE domains preserve their ability to reorient. Moreover, we deduce that the surface layer field E_SL appears in the sample below both electrodes. This means that these effects are responsible for the relatively low value of P_S ≅ 0.05 C m⁻² (see Figure 6b and Figure 7b). As a result of this, the true FE properties of the thermally treated PbTiO₃ samples are hidden below the oxidized surface layer, which has been generated by the thermal treatment. This, in turn, results not only in a change in the distribution of Pb and Ti ions but also in a change in the concentration of Pb²⁺ ions and metallic Pb⁰ in the surface layer (out-of-plane).

The “asymmetric” samples were produced by cleaving. One electrode was then placed on the multi-component surface layer of a sample and the other electrode was deposited directly on the FE bulk (Figure 1c and Figure 2c). For such an “asymmetric” configuration one can obtain the direct access to the bulk as well as to the surface layer. The sample response to the triangular voltage sweep was determined by two contributions: (1) the FE polarization P_FE =P_f (see Section 3.6 devoted to our model calculations) related to the reorientable FE domains in the bulk and (2) the fixed, i.e., non-reorientable P_SL=P_b (see also Section 3.6). When the external field is parallel to the resultant P_FE > 0, then the measured electric current signal is low as there is a negligible change in the density of the bound electric charge. When the external field is anti-parallel to the resultant P_FE > 0, then the measured electric current signal is high due to the substantial change in the above charge density_. Hence, two different values of P_S were actually determined for the “asymmetric” case, i.e., P_S,2 equal to 0.015 C m⁻² and P_S,1 = 0.56 C m⁻². The first value is related to the polarization screened by the field in the surface layer. The second is related to the FE domains in the bulk.

We would like to point out that similar oxidation and reduction procedures were performed for PbZrO₃ crystals, where the Pb and O vacancies amounts had been deliberately varied by thermal treatment [72,73,74]. It has been shown that the FE polarization, determined from the pyroelectric effect and the D–E hysteresis loop, was affected by the subsystem of these defects. However, the temperature range of the FE phase occurrence did not change.

We propose to describe the polarization features of the PbTiO₃ samples with respect to the electromotive force’s temperature dependence $ϵ$(T) = A + BT using the Nernst equation. The Nernst equation is applied to fuel and galvanic or concentration cells, mainly in the case of gaseous species transported in them. However, one can find reports where the migration of metal ions is analyzed [75,76,77]. Hence, we presume that the occurrence of the electromotive force for the PbTiO₃ crystals can be explained by the Nernst equation, as the concentration of Pb ions was different for the near-electrode regions of the “asymmetric” (i.e., electrode|Pb·(PbO·(PbTiO₃)_n)|PbTiO₃|TiO₂|PbTiO₃|electrode) samples. The electromotive force $ϵ$ of a cell can be qualitatively expressed as

$$ϵ={ϵ}_0-\frac{RT}{3F}\mathit{ln}\frac{a_{{\mathrm{Pb}}^0}}{a_{{\mathrm{Pb}}^{2+}{\mathrm{TiO}}_3}}$$

(7)

where R, T, and F denote the gas constant, the absolute temperature, and the Faraday constant, respectively. The $a_{{\mathrm{Pb}}^0}$ and $a_{{\mathrm{Pb}}^{2+}{\mathrm{TiO}}_3}$ denote the Pb⁰ (i.e., in the near-electrode regions of a sample) and PbTiO₃ (sample bulk) activities, respectively. Any quantitative estimation of the electromotive force $ϵ$ would require a separate series of experiments.

The earlier XRD study [41] has shown that the surface layer crystal lattice structure was different from the standard perovskite ABO₃ structure. The oxidized samples consist of a mixture of PbTiO₃ and the PbO·(PbTiO₃)_n phases with n = 1, 2, …, 6, and TiO₂-rich phases [40,57]. Hence it may be concluded that the surface layer transformation, occurring via solid-state phase reaction, results in a successive increase in the number of PbO planes and, at high temperatures, to the reduction of surplus PbO to metallic lead. Such a restructuring of the crystal lattice related to the Pb⁰/Pb²⁺ ratio redistribution can lead to the formation of the concentration cell. One can deduce that the electromotive force $ϵ$~0.3–0.6 V is of electrochemical origin. Because $ϵ$ appears in a thin surface, an essential electric field E_SL~10⁵ V cm⁻¹ can be found. We postulate that E_SL is sufficiently high to induce a fixed (i.e., non-reorientable) electric polarization within the intermediate PbTiO₃ sublayer, which may be identified as lying 20–25 nm below the outer surface of the oxidized sample (see SIMS profile of the oxidized PbTiO₃ crystal [57] and compare Figure 9).

The case of spontaneous non-zero net polarization has been discussed in the literature. We would like to note that V. M. Fridkin reported the effect of “natural” polarization for FE materials, e.g., for ferroelectric SbSI crystals [45]. The features of the internal field distribution, which influence the net polarization, can be determined by two mechanisms. One is related to the participation of the surface levels in screening. The second is related to the charge of the surface levels and causes a curvature of the energy bands [29,45]. This curvature appears at the ends of the crystal and can induce a unipolarity of ferroelectric crystals. The occurrence of the net P_S ≠ 0 is reversible and depends on temperature, i.e., the net P_S disappears after heating the samples above the FE–PE phase transition temperature. The net P_S forms in such a way that it can also be varied by applying an external electric field to the sample. Such FE samples exhibit standard D–E hysteresis loops as well as pyroelectric currents.

However, the case studied in this work is qualitatively different. The samples used in our experiment were prepared in two steps. In the first step, they were thermally treated and, as a result, a chemically modified surface layer was obtained. The fabricated samples were spatially symmetric. In the second step they were adapted by mechanical intervention, i.e., one side of the sample was removed. Consequently, an asymmetric, permanent, and chemically inhomogeneous structure of the lead titanate sample was obtained.

The chemical effects related to the ions’ migration, determined the measured effective polarization of the PbTiO₃ crystals. The microscopic insight into the surface layer composition supports the former XRD study [40] and the XPS spectroscopic measurements. The PbTiO₃ crystal, heated for several hours under oxidizing conditions, showed that PbO-rich Ruddlesden–Popper phases as well as TiO₂-based Magnelli phases are formed in the vicinity of the surface of the samples. In addition, the formation in the surface layer of metallic Pb can be determined using XPS analysis. Hence, the occurrence of the layered structure electrode|surface layer|PbTiO₃-bulk|electrode seems to be justified. These products are consistent with the result of the SIMS test [57] carried out for PbTiO₃, where the termination with the PbO–Pb planes of the crystal can be derived from the XPS analysis. It is noteworthy that our experiment was conducted at a moderate temperature in order to induce ionic migration towards the surface of the crystal as well as to obtain a structurally modified surface layer. We conclude that the strong electric field E_SL, caused by the variation in the Pb⁰/Pb²⁺ metal/ions ratio, induced a strong electric field, which fixed the polarization P_SL within the PbTiO₃ sublayer. Simultaneously, the inner part of the bulk remained unaffected, with the FE order unchanged.

The surface layers, electrodes and interfaces play an important role in the FE properties of the ABO₃ perovskite materials at the micro- and nanoscale [78]. It was found that the FE order stability can persist down to structures with very thin layers (as thin as several nanometers), which corresponds to about 5–10 atomic layers of the perovskite structure [8,10,15,16,75,79]. The structures were influenced not only by the interface appearance but also by the surface chemical composition and the related structural defects such as the oxygen vacancies in it. Note that the influence of the ensemble of defects and impurities on the electronic structure and polarization has been studied experimentally [7,9,26,27,80] and theoretically by ab initio calculations [6,8,81,82,83,84]. The value of the built-in polarization in the surface layer is influenced to a great extent by the type of electrodes, e.g., metallic Pt and oxide-type SrRuO₃. Moreover, the type of termination planes, i.e., PbO and TiO₂, affects the dipole moments within the crystal surface layer. Their interaction with the FE polarization in the bulk determines the stability of the surface layer polarization. It has been shown that the surface layer polarization has a tendency to align inward or outward of the surface plane [10,79,85,86].

From this point of view, the occurrence of polarization in the ABO₃ sublayer, thinner than 10 nm, is fully justified. The presence of such an ordered sublayer has been inferred from the SIMS, XRD, and electrical tests on the thermally treated PbTiO₃ samples. The necessary internal electric field that induces this polarization direction has been determined by the formation of PbO and TiO₂ planes as a result of ion migration which, in turn, has been initiated by thermal treatment. Moreover, the obtained results suggest that the fabricated samples show an arrangement which corresponds to a homo-epitaxial structure, i.e., a PbTiO₃ thin layer covered with PbO planes and interfaced from the PbTiO₃ bulk by TiO₂ planes.

Hence, this contribution to the electric polarization is dominated by the internal electric field. Therefore, we call this phenomenon the “self-polarization” effect. This self-polarization is manifested within the thin sublayer of the perovskite structure. It is induced and sustained by the internal field, originating from the chemically inhomogeneous surface layer, which envelopes or wraps the structurally proper perovskite bulk of the PbTiO₃ crystal. Such a layer appears when the sample is cooled through the PE–FE transition, after high-temperature oxidation. The polarization appears due to the interaction of charged interfaces between the Pb-rich and Ti-rich planes [3,10,29,30,40]. In accordance with DFT ab initio calculations and the asymmetry of the lattice structure, the PbO-terminated surface provides a shift of the relaxed Ti ions with respect to the oxygen octahedra. Hence, this shift corresponds to the occurrence of electric polarization within the surface layer. In the case of the “symmetric” sample configuration, these fields and surface layer polarization compensate each other when the macroscopic, electrical measurements are conducted. Only after removing the surface layer part, are unbalanced contributions from the surface layer on one side and from the unwrapped bulk on the other side obtained. Such an asymmetric structure is manifested by the electromotive force and by the asymmetric I–E hysteresis loop. Hence, we suggest that it is possible to produce a non-zero net polarization in PbTiO₃ crystals without the preliminary application of an external electric field [29,30,32].

5. Conclusions

In summary, using both experimental and theoretical arguments, we have shown that both structural and chemical modification of the PbTiO₃ surface layers can lead to self-polarization. In our studies, we have observed that thermal treatment under oxidizing conditions induces the migration of Pb ions and PbO complexes towards the near-surface regions. Moreover, we have also noticed that new Pb-rich structures, for example, PbO planes and/or Ruddlesden–Popper phases appear to exhibit anti-correlations with the Ti-rich regions within the surface layer. We have also found that the induced gradient in Pb and Ti atomic concentrations in the surface layer leads to a built-in electric field E_SL ≈ 10⁵ V/cm, which is stronger than the coercive field E_C related to the FE bulk of PbTiO₃.

Moreover, we maintain that the “self-polarization” effect appears due to the interaction between the Pb-rich and Ti-rich layers, when the sample is cooled through the PE–FE transition, after oxidation at high temperature.

We postulate that it is feasible to produce a non-zero net polarization in PbTiO₃ crystals without the preliminary application of an external electric field. This is possible due to the unbalanced contribution both from the chemically modified surface layer and the unwrapped FE bulk.