Epitaxial ferroelectric interfacial devices

The current interest in interface phenomena stems from the desire to discover and explore new physical effects emerging at the immediate atomic region forming the boundary between two different materials, in particular, since the change in crystal symmetry allows new physical phenomena that are distinct from those of the respective bulk phases. As a consequence, modifications in the electronic structure, which result from electron transfer or charge confinement effects, can lead to the emergence of novel effects and to new functionalities that are intrinsic to the interface.^1,2 An important class of interfacial systems utilizes a ferroelectric material to modulate the electronic properties of the interfacial system through the ferroelectric polarization. We refer to such device structures as ferroelectric interfacial devices in order to emphasize the fact that such phenomena are characteristic of the interface, with the latter acquiring its own physical, chemical, and electronic intrinsic properties. Two different types of device structures can be distinguished: one based on the ferroelectric field effect approach used to induce large and switchable modulations in the charge carrier density at the interface through charge screening effects,^3–5 and the other based on ferroelectric tunnel junctions, where the changes in the electric potential as a function of the direction of the ferroelectric polarization give rise to large changes in the transport characteristics across the ferroelectric used as a tunnel barrier.^6,7 Important physical requirements for both types of device structures, include sharp interfaces for optimal effects, robust ferroelectric effects at ultrathin thicknesses, and precise control of the channel and ferroelectric layer thickness for ferroelectric field effects and tunneling devices, respectively. Under such conditions, the ferroelectric control of several interfacial physical phenomena have been achieved, including interfacial magnetism, superconductivity, and orbital order in ferroelectric field effect devices, and of the tunneling resistance in various ferroelectric tunnel junction architectures, as illustrated in Fig. 1. The control of those physical phenomena involve a laborious identification of the mechanisms linking the ferroelectric order parameter to that of the interfacing material, often a joint experimental and theoretical endeavor. The progress achieved over the last two decades in our understanding of thin film ferroelectricity^8–14 and of interface phenomena,^1,2,15–28 from both practical and fundamental perspectives, has been tremendous. The aim of this review is to take stock of the recent developments in the field to gauge the challenges for further progress in view of potential technological applications. We pay particular attention to the aspects associated with ultrathin ferroelectricity, since ferroelectric films are one of the fundamental building blocks of interfacial ferroelectric devices, as well as to the physical mechanisms underlying the emergence and control of the interfacial effects between the ferroelectric and the other material of interest.

Ferroelectric materials are characterized by the presence of a spontaneous electric polarization, whose direction can be switched between states that are identical in the crystal structure (enantiomorphous) by means of an applied electric field.^29,30 Ferroelectricity involves space inversion symmetry breaking, and the spontaneous ferroelectric polarization can originate from a number of different microscopic mechanisms, including displacement of ions from high symmetry positions, electronic charge asymmetries (as in lone-pair ferroelectricity), charge ordering and/or disproportionation, and geometrical frustration.^9,31–40 Hence, ferroelectric phenomena tend to be more challenging to investigate than their magnetic counterparts,^35,41 also due to the strong interaction with the electric fields generated by free electric charges and the strong coupling to lattice distortions, including strain, which often make modeling and interpretation of the experimental results difficult. Theoretically also, only recently a consistent theory of the ferroelectric polarization^42–47 (expressed in terms of a Berry phase)⁴⁸ and the development of ab initio methods to treat the electric field termination of ferroelectric slabs^9,31,34 have been made available. From a materials' perspective, ferroelectric systems tend to be multicomponent systems, typically containing three or more elements, with a concomitantly complex defect chemistry that make the preparation of good quality, pure samples difficult, but a task made possible in many instances by the advances in the controlled epitaxial growth of thin ferroelectric films and multilayer heterostructures. The ability to grow high quality thin ferroelectric films is important not only for fundamental studies but also for device applications, where size scaling and a long term retention of the ferroelectric properties at small lateral dimensions and reduced thickness are important requirements, and because the onset of interfacial phenomena between dissimilar materials and the control of those properties by means of external stimuli, such as electric fields, require well defined and abrupt interfaces. In this context, state-of-the-art techniques for ferroelectric thin film growth and optimization of the growth process have resulted in the routine fabrication of epitaxial films of very high structural quality, as we illustrate later for a few prototypical ferroelectric systems (Sec. II B). Also significant has been the development of new experimental techniques dedicated to the study of nanoscale ferroelectricity and interfacial phenomena (Sec. II A); in particular, the development of continuous and operando modes of measurement, where the physical properties of the device are probed while subject to a varying external stimulus, are particularly appealing, since they allow one to follow the changes in physical properties in real time while minimizing the influence of other parameters. The combined progress achieved in these theoretical and experimental fronts has led to a strong increase in interest and in the understanding of physical phenomena in ferroelectrics.

Of almost equally importance as the intrinsic properties of ferroelectrics are the conditions under which the ferroelectric system is physically and electrically connected, particularly for thin films, where such boundary conditions have a tremendous impact on the ferroelectric properties. As illustrated in Fig. 2, both three-terminal ferroelectric field effect devices and two-terminal ferroelectric tunnel junctions require a supporting substrate and electrical connections to the ferroelectric layer and to the interface. Since ferroelectricity is generally linked to local structural distortions in the atomic lattice, the constraints imposed by the substrate where the ferroelectric film is grown strongly impact its ferroelectric properties, including ferroelectric polarization, critical temperature, and piezoelectric response. Electrical boundary conditions are also present in terms of metallic contacts to the ferroelectric film, both to switch the ferroelectric state of the device and to establish the functional interface, for example, with a superconductor in a field effect device or with a ferromagnet in a multiferroic ferroelectric tunnel junction. The metallic electrode acts as a charge reservoir that screens the electric polarization at the dielectric interface. In fact, the dielectric/metallic interface has since long received considerable attention since it constitutes the basis of the capacitor structure through which the properties of dielectrics are often investigated. Surprisingly, a strong dependence of the dielectric properties with the electrode material has been observed, in the extreme case leading to a strong reduction of the dielectric properties of the dielectric at reduced thicknesses, built-in electric fields, and fatigue.^49–57 To a large extent, such phenomena are linked to the physical characteristics of the metal/ferroelectric interface, and the key aspects associated with this interface are discussed in Sec. II C.

The prototypical example of a ferroelectric interfacial device is the ferroelectric field effect transistor, first proposed in the 1950s,^58,59 before the inventions of the integrated circuit and SiO₂ as the transistor gate dielectric. The goal of this invention is to combine the logic function of a CMOS transistor with the switching of the ferroelectric polarization to achieve nonvolatile on/off states. Two challenges need to be overcome in such structures: the control of the interface structure in order to avoid spurious screening of the ferroelectric surface polarization that may arise from impurities or charge traps from bandgap states and ensuring that the (semi)conducting channel can provide the requisite charge screening of the ferroelectric polarization to stabilize ferroelectricity and the orientation of the ferroelectric polarization. Such challenges prevented the immediate success of the ferroelectric field effect transistor (FeFET), but recent efforts have resulted in approaches to using ferroelectrics in fast and energy efficient memory devices.⁶⁰ The FeFET concept remains very appealing and has been adapted to controlling the properties of other material systems susceptible to charge modulation [Fig. 1]. For example, instead of a semiconductor channel, large changes in the channel conductivity can be achieved by controlling the insulator to metal transition of a Mott insulator or the superconducting state of a high-critical temperature superconductor. Here, one controls not simply the conductivity through the charge carrier density but directly the correlated state via its strong dependence on the carrier density. This approach to controlling the correlated state of matter through changes in the charge carrier density using the ferroelectric field effect has proved to be a powerful method to observe new physical interfacial phenomena. That the processes are largely confined to the interface is a result of the relatively large charge density of strongly correlated electron systems and the correspondingly short Thomas-Fermi screening lengths of the order of the unit cell.³ By taking advantage of the very large surface bound charges that are characteristic of ferroelectrics, of up to one electron per unit cell, one can reach modulations in the interfacial carrier density well beyond what is possible with silicon oxide gates.³ The ferroelectric field effect approach has been used to electrostatically control the superconducting state, magnetism, the Mott metal to insulator (MIT) transition, and the orbital state, as we discuss in detail in Sec. III. The challenges for these more exotic field effect devices are increasing the size of the observed effects to above room temperature, as desirable for electronic devices.

While tunneling transport phenomena across dielectrics are now well established, the particular case when the dielectric is also a ferroelectric has been explored only recently, first theoretically,^61–64 and later experimentally,^65–68 to reveal surprisingly large changes in conductivity as a function of the ferroelectric polarization direction, in what is termed tunneling electric resistance (TER) in analogy with the magnetic counterpart, tunneling magnetoresistance (TMR).^69,70 The experimental study of such effects is made possible by the availability of high quality, ultrathin ferroelectric films, as discussed in Sec. II. Several mechanisms are responsible for the TER effect, some related to the bulk of the ferroelectric film, including changes in thickness with the applied electric field, while other contributions are linked to modifications of the electronic potential and changes in bonding at the interface.⁶ We overview in Sec. IV the current status of the field.

An important motivation for the research work in this field is the prospect of applying the physical effects observed at ferroelectric interfaces to functional devices.⁷¹ While such systems offer good scaling properties,⁷² novel functionalities, and robust electric field control over electronic properties, the main challenges to realizing ferroelectric devices are the requirement for sustaining the ferroelectric state at ultrathin thicknesses for ferroelectrics of technological interest (which is presently more a practical rather than a fundamental issue) and of achieving sufficiently large interfacial electronic effects at ambient temperature. Also, rather than competing with binary logic operation, the class of multifunctional devices should aim for more complex operations, including devices with selective or self-enforcing memory suitable for simulating neural networks for artificial intelligence, or to control quantum entanglement for quantum computation.⁷³ For example, while current artificial neural networks are based on classical computing, it can be anticipated that better scaling, speed, and lower power consumption of the millions of interconnected artificial neurons and synapses can be better achieved with memristor devices (systems showing hysteresis in the resistivity response) based on oxide-based material systems.⁷⁴ In this context, resistive switching associated with reversible, redox processes that give rise to a nonvolatile change in resistivity with the applied electric field that develops in many thin films,⁷⁵ including oxides, has been proposed as a potential building block for neural networks. Since they rely on filamentary disruption of the insulating barrier between two conducting electrodes, they can be easily scaled up; however, the control of their onset within a device and the spread in their characteristics remain a challenge. Another device structure capable of the same memristor function is the ferroelectric tunnel junction.^76–78 Compatibility and integration of such types of devices with current CMOS technology would also be desirable.⁷⁹ A brief overview of the proposals being put forward in this area is provided in Sec. V.

Given the size of the field, we restrict ourselves to discussing epitaxial systems, where better control over the interface enables a more targeted control of the electronic effects and minimizes extrinsic factors. In particular, our focus is on oxide materials, where the interfacial properties between ferroelectric oxides and complex oxides have been extensively studied and are better understood. Heterostructures consisting of two-dimensional (2D) materials interfaced with ferroelectrics are not discussed; the physical and chemical properties of 2D materials, such as graphene and the transition metal di- and trichalcogenides, have generated much interest due to the wide range of properties exhibited by these systems, including pseudo-spin in the case of graphene,^80,81 magnetism, ferroelectricity, and optical properties in the chalcogenides.^82–84 An increasing body of work has focused on interfacing such materials with ferroelectrics, which has been the focus of several recent review articles.^85–92 The main approach in such device structures has been to modulate the charge carrier density via electrostatic doping, and a significant effort has been placed at forming a clean interface between these systems and the ferroelectric to achieve the expected response to the ferroelectric charge modulation.^93–95 In these systems, epitaxy does not seem critical to device function, which remains at the level of the 2D system itself.

Our goals in this paper are to provide both a timely appraisal of the current developments on ferroelectric interfacial devices and to stimulate further work by pointing, in our view, to promising approaches, as we discuss succinctly in the conclusions and outlook (Sec. VI). Our overview highlights that the large practical and theoretical efforts in fabricating and understanding thin film ferroelectricity have led to a good overall control of the properties of ferroelectric thin films and to the discovery of new ferroelectric systems with enhanced properties that are imminently suited to device applications; in turn, the growth of high quality ferroelectric/complex oxide interfaces have enabled the discovery and study of new interfacial phenomena that have the potential for new functional devices.

The experimental investigation of ferroelectric phenomena at the nanoscale requires techniques with sufficient sensitivity and spatial resolution to measure the physical properties of systems at reduced volumes. Measurements of the dielectric and ferroelectric response to electric stimuli, such as permittivity, dielectric loss, breakdown voltage, ferroelectric polarization, piezoelectric response, and ferroelectric switching, are often carried out in a parallel plate capacitor geometry with metallic electrodes in direct contact to the ferroelectric system. In this geometry, a time-varying, spatially uniform electric field is applied to the ferroelectric system across the metallic plates and the displacement current is measured as a function of time to yield the dielectric constant for small electric field excitations or the ferroelectric polarization by integration of the displacement current associated with the ferroelectric polarization switching. Commercial ferroelectric testers can be used to obtain polarization vs electric field hysteresis curves that provide not only measurements of the coercive field, saturation, and remanent polarization but also convey information about the ferroelectric switching process, leakage current, built-in electric fields at the contact interfaces, and fatigue (i.e., loss in switchable polarization upon repeated switching) due to defects. Alternatively, the ferroelectric polarization can be obtained through the pyroelectric effect by measuring the charge required to compensate the onset of the ferroelectric polarization when cooling the sample down from the paraelectric state.^96–98 In addition, to probe nanoscale volumes and ferroelectric dynamics, the size of the contacts can be defined to be arbitrary small, for example, down to the nanoscale by means of advanced e-beam lithography techniques and on ultrathin ferroelectric films. However, while conceptually simple, such types of electrical measurements are challenging for a number of reasons: in addition to the small magnitude of the signal, tunneling currents in ultrathin films may overwhelm the ferroelectric displacement currents, and the impact of the metallic electrical contacts needs to be taken into consideration in order not to mask the intrinsic dielectric properties of the ferroelectric system.^99,100 For example, for PZT, it is found that Cu, Au, Ag, and Pd are suitable materials that form Schottky contacts, while Ta and Cr can form Ohmic contacts unsuitable for electric characterization.^101–104 Since the impact of the electric contact/ferroelectric interface is fundamental for ferroelectric field effect devices, we will discuss this aspect in more detail in Sec. II C 2.

A second type of measurement probes local ferroelectric phenomena at the nano- or atomic scale and employs techniques that have high spatial resolution. One such technique is piezoresponse force microscopy (PFM),^105–108 a scanning probe technique that relies on the measurement of the sample piezoelectric response to an alternating electric field applied locally through a conducting tip of an atomic force microscope (in contact mode) and an extended bottom electric contact (typically, a conducting substrate or a conduction layer grown between the substrate and the ferroelectric film). At an elementary level, the amplitude of the response provides a direct measure of the local piezoresponse amplitude, while the phase signal provides direct information about the orientation of the ferroelectric domain (in-phase when the polarization points opposite to the electric field and 180° for the opposite direction). While a powerful technique for obtaining spatially resolved maps of the ferroelectric domain structure and of the local piezoelectric response, instrumental artifacts, such as local charge injection from the metal tip to the ferroelectric film surface, can make interpretation of PFM results difficult.¹⁰⁹ Also, quantitative values for the piezoresponse of thin ferroelectric films can be challenging to extract due to the small vertical displacements induced by the electric field and due to the highly non-homogeneous electric field generated by the sharp metallic tip, and multiple strategies have been developed for overcoming such difficulties.^110–116 With the same proviso, local electric fields can be applied to the sample to directly visualize the ferroelectric switching process.^117,118 Another technique that has been used to visualize ferroelectric domains is photoemission electron microscopy (PEEM), which measures the intensity of photoemitted electrons excited by a light source, such as ultra-violet light, monochromatic electrons, or x-rays.^119–122 Since the photoemitted electron intensity depends strongly on the surface potential, differences in the latter associated with the ferroelectric polarization of the different domains at the sample surface can be measured to provide direct images of the ferroelectric domains.^{121,123–126} For similar reasons, low energy electron diffraction (LEED) can be used also as a tool to derive the surface potential of ferroelectric surfaces, although without the spatial resolution.⁵⁶ 

Techniques based on x-rays, and synchrotron x-ray light in particular, are also powerful probes for ferroelectric properties, providing precise measurements of the local atomic structure, including changes in atomic displacements, periodicity of ferroelectric domains, and ferroelectric switching, as a function of electric field, temperature, and pressure via x-ray diffraction and scattering;^127–129 of the electronic state and band alignment via x-ray photoemission spectroscopy (XPS)^130–134 and of the electronic band structure using x-ray absorption spectroscopy (XAS).^135–141 With the advent of free electron laser sources¹⁴² capable of producing extremely intense, highly transverse coherent, and ultra-short (few fs long) pulsed x-ray beams, it is now possible to investigate ultrafast dynamical processes in ferroelectric systems, including phonon dynamics and ultrafast switching induced by strong THz light beams.¹⁴³ 

Finally, we mention here high resolution scanning transmission electron microscopy (HR-STEM), which is a particularly powerful technique for probing the atomic structure in real space of both ferroelectric films and interfaces and also capable of locally probing some aspects of the electronic structure via electron energy loss spectroscopy (EELS).^28,144 The direction of the ferroelectric polarization can be determined from an accurate measurement of the relative atomic displacement between cation and anion species, such that ferroelectric domains and domain walls can be identified;^145–147 also from the amplitude of the atomic displacement, estimates of the ferroelectric polarization can be obtained from the relation between effective Born charges and ferroelectric polarization.^147–149 In many systems, local electrical contacts to the sample enable one to carry out in situ ferroelectric switching studies,¹⁴⁶ including ferroelectric domain wall dynamics.^{145,150–152} Atomically resolved EELS spectra can provide additional information on the interface atomic structure (including the presence of atomic interdiffusion),^153–155 valence profile across the interface,^{147,156–161} and atomic layer stacking across the interface.¹⁶² 

A challenging aspect in the study of thin film ferroelectricity relates to the preparation of such systems, typically achieved by means of film growth techniques such as molecular beam epitaxy (MBE), pulsed layer deposition (PLD), and rf magnetron sputtering.^163–167 In such deposition techniques, the film grows on a supporting substrate by condensation of the various elements, or molecules, that compose the material. To promote crystalline order, the growth is carried out at elevated temperature, such that atomic and molecular surface diffusion is high but bulk mass mobility is low.¹⁶⁸ Often various stable crystalline phases exist for the same material composition or for a subset of the constituent elements, such that spurious phases may develop during growth which can, in some instances, strongly impact the materials properties or lead to an inconclusive interpretation of the experimental findings. A related aspect is that, in the majority of cases, the window in parameter space for the growth of the material of interest is relatively narrow, making the growth of high quality epitaxial films a demanding task that requires a good control over stoichiometry and oxidation state; in addition, a low density of defects, low surface and interface roughness, and the avoidance of spurious phases should be attained as well.

For ferroelectric interface devices, it is important not only that the ferroelectric interface be sharp but also that the ferroelectric properties are robust, including having a high spontaneous polarization, a high ferroelectric critical temperature, a large breakdown voltage, and that the system can be grown on suitable substrates as high quality thin films. A number of ferroelectric systems fulfill these conditions and have been grown as high quality single crystalline films, a representative subset of which are discussed next and whose bulk properties are listed in Table I.

Among the ferroelectrics, barium titanate (BaTiO₃) has played an important historical role in the study of ferroelectricity, since its crystal structure with a small unit cell permitted a more straightforward understanding of the origin of the spontaneous electric polarization in the solid state as compared to the earlier known ferroelectrics, such as the Rochelle salt with a large unit cell;²⁹ it is, in addition, a technologically important ferroelectric due to its robust ferroelectric and optoelastic properties.^190–193 The high-temperature paraelectric phase of BaTiO₃ (between 1733 and 393 K) crystallizes in the centrosymmetric cubic perovskite structure; at lower temperatures between 393 and 278 K, the active soft-phonon mode associated with the relative displacement of the Ti cation with respect to the oxygen octahedron freezes, leading to the onset of a ferroelectric dipole moment in a tetragonal crystal structure, with the polarization pointing along the c axis. At lower temperatures (between 278 and 183 K), further distortion in the crystal structure to orthorhombic takes place along with a change in the direction of the ferroelectric polarization to the pseudocubic [110]_pc direction; below 183 K, the structure changes further to rhombohedral with the polarization pointing along the pseudocubic [111] direction.^169,194

Although high quality BaTiO₃ single crystals are commercially available, its high cost, its sensitivity to variations in temperature due to the tetrahedral to orthorhombic phase transition at 278 K, and the long switching times and high voltages required to excite bulk crystals has driven the need for the growth of high quality epitaxial films on standard substrate materials for use in device applications. Epitaxial thin films of BaTiO₃ have been grown by MBE, PLD, and sputtering on various substrates, including metal oxides such as SrTiO₃(001),^49,195–214 MgO(001),^{196,215–219} GdScO₃(110),^218,220 DyScO₃(110),²¹⁸ LaAlO₃(100),²¹⁷ and MgAl₂O₄(001)²²¹ as well as on semiconductors, including Si(001)^222–229 and Ge(001).^230–232 In most instances, an intermediate buffer layer is used as an electrical contact or for the release of misfit strain. BaTiO₃ films are typically grown at elevated temperatures, well above the critical temperature, such that they are brought down from the paraelectric state to the ferroelectric state at room temperature upon cooling. The most relevant film orientation for BaTiO₃ in ferroelectric interfacial devices is the [001], i.e., with the polar axis pointing along the out-of-plane direction. It is worth pointing out that for the ferroelectric perovskites, the value of the c/a ratio tends to relate to the amplitude of the electric dipole and, therefore, to the amplitude of the ferroelectric polarization (Sec. II C).^{198,233–240} Hence, a substrate of high interest is SrTiO₃, where a lattice mismatch of –2.18% is expected to lead to a strong compressive strain and to a large c/a ratio.²³⁶ On SrTiO₃(001), BaTiO₃ is found to grow layer by layer in steps of one unit cell (u.c.) up to about 12 u.c., above which the film growth proceeds in a three-dimensional island (Stranski–Krastanov) growth mode;^{195,200,201,205,208,241} at substrate temperatures of the order of 680 °C, typical for oxide film growth, the adatom surface mobility is high, as deduced from the recovery of the RHEED intensity after a break in the deposition process of about 20 s.^195,202,205 However, such adatom surface mobility seems to be detrimental to the growth of smooth and fully strained films: the latter can be achieved at lower deposition temperatures and higher deposition rates, while under opposite conditions early strain relaxation and 3D growth are observed.^204,242,243 For the BaTiO₃/SrTiO₃(001) system, the critical thickness for coherent growth (coherent films have the same in-plane lattice parameter and a symmetry compatible with the substrate) has been found to range from 2 to 10 nm, depending on the growth conditions, above which the in-plane lattice constant starts relaxing toward the bulk value through the onset of misfit dislocations.^{201,205,207,214,243,244} Although Nb-doped conducting SrTiO₃ can be used as a substrate for the BaTiO₃ film growth also to form a bottom contact, an alternative solution consists of using a conducting buffer layer, such as LaNiO₃,²⁰⁶ SrRuO₃,^209,211,245 La_1−xSr_xMnO₃,²¹¹ YBa₂Cu₃O_7−x,^196,246 or Sr₂RuO₄,²¹² in the latter case leading to atomically abrupt interfaces, as shown in Fig. 3(a).

Other oxides used for the epitaxial growth of BaTiO₃ include MgO (rock salt with a lattice constant a = 4.2117 Å)²⁴⁷ and LaAlO₃ (rhombohedral below 800 K with a = 5.365 Å, c = 13.11 Å, and a pseudo-cubic lattice constant $apc=3.79$ Å),²⁴⁸ both of which have a large lattice mismatch to BaTiO₃, respectively, of +5.0% and –5.5%. Likely as a consequence, the BaTiO₃ film quality tends to be poorer in comparison to films grown on SrTiO₃, even when buffer layers are incorporated that act to reduce the lattice misfit.^{196,216,217,219} Metal oxide substrates with smaller lattice misfits include MgAl₂O₄ (cubic spinel with a = 8.0831 Å),²⁴⁸ DyScO₃ (orthorhombic, a = 5.43 Å, b = 5.71 Å, c = 7.89 Å; $apc=3.94$ Å),²⁴⁹ and GdScO₃ (orthorhombic, a = 5.487 Å, b = 5.756 Å, c = 7.925 Å; $apc=3.98$ Å).²⁵⁰ For BaTiO₃ films grown on MgAl₂O₄(001), where the lattice misfit is about 0.79%, it is found that relatively high growth temperatures (900–1050 °C) are required for full epitaxy, but with a defective interface as determined by TEM, possibly as a consequence of surface reconstruction of MgAl₂O₄, surface terraces exposing different surface terminations, and the large thermal misfit.²²¹ GdScO₃(110) and DyScO₃(110) have small lattice misfits to BaTiO₃, respectively, of –1.0% and –1.7% and provide excellent templates for the epitaxial growth of fully strained BaTiO₃(001) films with sharp interfaces, as illustrated by the example shown in Fig. 3(b).^218,220

The growth of complex oxides on semiconducting substrates, particularly Si, has been intensively investigated with the aim of integrating complex oxides in semiconductor platforms to realize multifunctional electronic and electro-optic devices^{167,251–260} and for using high-κ dielectric oxides as alternatives to SiO_x for gate dielectrics in MOSFET transistors.^{170,261–263} The challenge is to achieve epitaxial growth and a sharp interface without the formation of an interfacial oxide amorphous layer that may be formed by exposure of the clean semiconducting surface to the oxygen species required for the metal oxide growth or due to reaction with the oxide layer that could degrade device performance. In the case of Si, sharp and abrupt interfaces, without the formation of an amorphous interfacial silicon oxide layer, were obtained for BaO, SrO, and SrTiO₃.^{252,264–268} In these instances, the process relies on the passivation of the Si surface with a submonolayer-thick alkaline metal layer that protects it from oxidation while enabling the start of the oxide film growth. A recent review of the growth of BaTiO₃ on Si, Ge, and GaAs substrates by MBE has been provided by Mazet et al.²⁵⁴ In most instances reported in the literature, the growth of epitaxial BaTiO₃ on Si(001) proceeds through the use of a mediating SrTiO₃ buffer layer, which has the added benefit of reducing the large misfit between Si (a = 5.431 Å) and BaTiO₃ (of about –3.8%), which otherwise would favor the growth of a-oriented films. It is found that the lattice orientation (a or c) depends strongly on the SrTiO₃ buffer layer thickness, surface quality, and oxygen partial pressure during growth, making the ferroelectric state difficult to control.²⁵⁴ Ge (a = 5.658 Å) has a smaller lattice misfit to BaTiO₃, of about +0.27% at room temperature, on which high quality BaTiO₃ films can be grown directly.²⁶⁵ The in-plane tensile strain leads to a preferred a-orientation; however, this can be circumvented by using buffer layers that induce a compressive in-plane strain, such as Ba_1−xSr_xTiO₃²³⁰ or SrTiO₃.²⁶⁹ One example of the BaTiO₃ film quality that is possible to achieve on Ge(001) is shown in Fig. 4 for a 2.5 u.c. film grown by reactive molecular beam epitaxy by first depositing a 0.5 monolayers of BaO to passivate the Ge surface against oxidation, followed by the deposition of two monolayers each of BaO and TiO₂ to form an amorphous BaTiO₃ layer that is subsequently crystallized upon annealing.²⁷⁰ The TEM image in Fig. 4(a) shows full crystallization of the BaTiO₃ layer in perfect registry to the Ge lattice; the analysis of crystal truncation rod (CTR) x-ray diffraction measurements²⁷¹ together with the results of ab initio calculations could show that, for this thin BaTiO₃ film, rumpling of the BaO interfacial layer occurs in response to Ge dimerization [Figs. 4(b) and 4(c)] and leads to distortions in the BaTiO₃ unit cell away from the tetragonal distortion akin to the orthorhombic phase of BaTiO₃; a 5.5 u.c. BaTiO₃ film grown by additional deposition of three monolyers of BaO and TiO₂ at elevated temperature show the tetragonal structure expected for bulk BaTiO₃ at room temperature, however.²⁷⁰ Another study has demonstrated epitaxial growth of [001]-oriented BaTiO₃ on Ge(001) through the deposition of a 2 nm amorphous BaTiO₃ film that was subsequently crystallized upon annealing and that could be used as a seed for further BaTiO₃ growth.²³² Although GaAs (a = 5.653 25 Å) also has a small lattice misfit to BaTiO₃ of about +0.19%, the lack of structural and chemical stability of the GaAs surface to the high temperatures required for BaTiO₃ growth and interdiffusion precludes direct growth of BaTiO₃ on GaAs, making the use of a buffer layer, such as MgO or SrTiO₃, necessary for high quality single crystalline films.^254,272 In this sense, the problem is reduced to one of growing the buffer oxide layer on GaAs.^254,273

Other important perovskite ferroelectrics are PbTiO₃ and PbTiO₃-based materials. Since the first reports of the crystal structure and ferroelectricity in the 1940s,^274–277 PbTiO₃ has attracted significant interest due to its excellent ferroelectric properties,^60,278,279 including a high Curie temperature and a large ferroelectric polarization. At ambient conditions, PbTiO₃ presents a tetragonal structure with a large $c/a≈1.063$⁠.²⁸⁰ The Ti cations are off-centered from the oxygen octahedra, inducing a ferroelectric polarization in the system. The ferroelectric phase is stable up to 763 K, then undergoes a transition to the paraelectric phase accompanied by a structural transition from tetragonal to cubic symmetry.²⁸¹ The spontaneous polarization of PbTiO₃ crystals is around 75 μC/cm², which is much larger than that of BaTiO₃ (see Table I). In addition to the contribution from hybridization of Ti 3d and O 2p orbitals, the 6s lone electron pair of the Pb²⁺ cations and hybridization between the Pb 6s and O 2p orbitals provide an additional driving force for the structural distortion characteristic of PbTiO₃.^34,282–284 The ferroelectric distortion for PbTiO₃ differs from that of BaTiO₃ in that the oxygen octahedra move in the same direction, but with larger magnitude, than the Ti cation.

Given its outstanding ferroelectric characteristics, the epitaxial growth of PbTiO₃ thin films has been extensively studied for decades. Various deposition methods, such as MBE, PLD, chemical vapor deposition (CVD), and sputtering, have been exploited to achieve high-quality epitaxial thin films.^285–289 The majority of these studies employ [001]-oriented SrTiO₃ substrates. On SrTiO₃(001), PbTiO₃ adatoms exhibit ideal 2D layer-by-layer growth²⁹⁰ or even step-flow growth²⁹¹ to form atomically flat surfaces and interfaces. Coherent growth without undesired a-domain formation has been observed up to a film thickness of 340 nm,²⁹² which is far thicker than the coherent critical thickness expected from the Matthews–Blakeslee model. The minimum thickness for the onset of ferroelectricity is reported to be extremely small (three-unit cells) on SrTiO₃.¹²⁷ These features make PbTiO₃/SrTiO₃(001) an ideal platform for ferroelectric interface devices. The successful synthesis of epitaxial PbTiO₃ films has been also reported on other substrates, such as LaAlO₃,²⁹³ MgO,²⁹⁴ Pt,²⁹⁵ and Si.^286,287

Lead zirconate titanate, PbZr_xTi_1−xO₃ (PZT), is the most widely used ferroelectric and piezoelectric material due to its large ferroelectric spontaneous polarization and piezoelectric coefficient. PZT can be seen as a solid solution of PbTiO₃ and PbZrO₃, an antiferromagnetic rhombohedrally distorted perovskite material. The phase diagram of PZT is characterized by an equilibrium tetragonal phase at a large PbTiO₃ content (⁠$x≲0.5$⁠) and several rhombohedral structures for the PbZrO₃ rich phases (⁠$x≳0.5$⁠).^296,297 At $x≈0.52$⁠, PZT is characterized by the coexistence of tetragonal, rhombohedral, and a more recently identified intermediate monoclinic phase,^298–303 defining the so-called morphotropic phase boundary; such phase coexistence is responsible for the large piezoelectric response of PZT at this composition.^{165,297,304–306} In the context of interfacial ferroelectric devices, the tetragonal phase of PZT is the most interesting on account of the higher Curie temperature and a larger ferroelectric polarization along the tetragonal axis.^{174,235,301,307–310} The epitaxial growth of thin PZT films has been demonstrated on several substrates using various deposition techniques, as discussed in detail in Ref. 165, and we consider here more recent examples reported in the literature. Due to the volatile Pb element, PZT films are grown less often by MBE³¹¹ and instead by rf sputtering or PLD, typically on Nb-doped SrTiO₃ substrates or on undoped SrTiO₃ with a conducting oxide buffer layer, which provide a good template for the growth of tetragonal PZT (x < 0.5).³¹² For x = 0.2, the lattice mismatch to SrTiO₃ is –1.2% and the films can grow coherently up to a thickness of about 15 nm.^313,314 The ferroelectric properties of PZT films have been shown to depend sensitively on the condition of the substrate surface: high quality PbZr_0.2Ti_0.8O₃ films grown by PLD on single terminated, atomically flat SrTiO₃(001) substrates using a 20 nm thick SrRuO₃ buffer layer were found to grow layer-by-layer, as deduced from atomic force microscopy, and possess a high saturation ferroelectric polarization value of 105 μC/cm² (attributed to a slightly higher value of the c/a ratio, 1.06, as compared to the bulk value of 1.051).¹⁸⁶ TEM data for a 90 nm thick film, reproduced in Figs. 5(a)–5(d), show abrupt, defect-free interfaces; in comparison, a PZT film grown on a less perfect SrTiO₃ surface shows the presence of defects, as shown in Fig. 5(e), and is also characterized by a smaller ferroelectric polarization value of 65 μC/cm².¹⁸⁶ PZT films deposited on La_0.8Sr_0.2MnO₃/SrTiO₃(001) exhibit larger ferroelectric polarizations, of up to 85 μC/cm², when the La_0.8Sr_0.2MnO₃ film is grown by MBE on single terminated TiO₂ SrTiO₃ substrates^315,316 as compared to PZT/LSMO films deposited by off-axis rf-sputtering on mixed terminated SrTiO₃ substrates, where the ferroelectric polarization is typically of the order of 45 μC/cm².³¹⁷ 

The PbTiO₃-based relaxor-ferroelectrics are also materials of high interest. Relaxor-PbTiO₃ ferroelectrics, including Pb(Mg_1/3Nb_2/3)O₃-PbTiO₃ (PMN-PT) and Pb(Zn_1/3Nb_2/3)O₃-PbTiO₃ (PZN-PT), have been extensively studied due to their ultra-high piezoelectric response, with piezoelectric coefficient $d33>1500$ pC/N and electromechanical coupling $k33≈90%$⁠,^318,319 which exceed those of high-quality PZT films. Although the origin of the high piezoelectric response remains to be clarified, it makes relaxor-PbTiO₃ ferroelectrics ideal materials for electroacoustic transducers and actuators.³²⁰ While the majority of the reports on relaxor-PbTiO₃ ferroelectrics still focus on single crystals or thick films, thin films of relaxor-PbTiO₃ ferroelectrics have also been extensively studied. Magnetron sputtering and PLD have been used to synthesize epitaxial relaxor-PbTiO₃ thin films on various substrates, such as SrTiO₃,^321,322 LaAlO₃,^293,323,324 MgO,³²⁵ (La,Sr)(Al,Ta)O₃ (LSAT),³²⁶ and Si.^327,328 The ultra-high piezoelectric effect of the relaxor-PbTiO₃ ferroelectric crystal also opens a new route to study strain engineering. Most studies on strain effects are based on the synthesis of films with different thicknesses (with different strain relaxation states) or on films grown on different substrates to induce a misfit strain. These methods, however, have the drawback that spurious effects related to slightly different growth conditions between different samples cannot be excluded. On the other hand, using relaxor-PbTiO₃ ferroelectric crystals as a substrate enables electric control of strain in a single sample, an approach that has been used to study the effect of strain on ferroelectricity^329,330 and ferromagnetism,^331–334 for instance.

BiFeO₃ is a well-known multiferroic material characterized by high ferroelectric and antiferromagnetic (Néel) ordering temperatures of $Tce=1123$ K and $TNm=643$ K, respectively,¹⁶⁹ making it one of the relatively few room-temperature multiferroic compounds. Under ambient conditions, BiFeO₃ has a rhombohedral structure with ferroelectric easy-axis along $⟨111⟩pc$ and a high spontaneous polarization in excess of ∼100 μC/cm².^{176,335–338} Unlike typical displacive ferroelectrics, such as BaTiO₃, where off-centering of the Ti cations within the O cage is at the origin of the ferroelectric polarization, the main contribution to the ferroelectric polarization of BiFeO₃ arises from lone pair Bi 6s orbitals.^339,340 This allows BiFeO₃ to have partially filled d shells on the Fe³⁺ site (and a spin moment) while ferroelectricity arises largely independently from the Bi³⁺ cations. Despite the independent origin of ferroelectric and magnetic order, the presence of Dzyaloshinskii–Moriya interactions results in a sizable magnetoelectric coupling between the two order parameters,^341,342 which has been explored in novel artificial multiferroic heterostrutures for electric field control of magnetism.^343–349 The magnetic structure of BiFeO₃ is locally G-type antiferromagnetic, but the presence of a significant Dzyaloshinskii–Moriya interaction (DMI) associated with the polar displacement^350,351 induces both a spin canting between adjacent spins (resulting in a small local net spin moment) and an incommensurate cycloidal winding of the spins within a cycloidal plane defined by the direction of the ferroelectric direction and the direction of propagation of the cycloid, with a periodicity of about 64 nm,^352–354 leading to an effective zero magnetic moment. For a ferroelectric polarization state pointing along $[111]pc$⁠, the possible propagation directions for the spin cycloid are $[1¯10]pc, [011¯]pc$⁠, and $[101¯]pc$⁠. An order of magnitude smaller second DMI interaction, resulting from the alternate rotation of the FeO₆ octahedra along the $[111]pc$ direction,^351,355 causes the cycloid to be canted about 1^o from the cycloidal plane, inducing a spin density wave.^{354,356–358} Under the effect of epitaxial strain, the spin cycloid can be suppressed or modulated in thin films.^{354,359–363}

The first report of the growth of multiferroic epitaxial BiFeO₃ thin films³⁶⁴ generated a strong interest in this system, well documented in the literature.^365–370 Here, we mention only the case of BiFeO₃ films grown on LaAlO₃ crystals, which, due to the large compressive epitaxial strain (⁠$≈$ 4.5%), have been found to stabilize in a pseudo-tetragonal phase with a very large c/a ratio of 1.25–1.3^239,370,371 characterized by an enhanced spontaneous polarization of over 150 μC/cm².³⁷² The large polarization value of pseudo-tetragonal BiFeO₃ provides great functionality for ferroelectric interface devices.^370,373 In addition, with increasing film thickness, the tetragonal BiFeO₃ phase partially relaxes to a pseudo-rhombohedral phase, resulting in an effective morphotropic phase boundary (MPB) where the two structures coexist (see Fig. 6).^238,374 The MPB exhibits enhanced functionalities, such as enhanced piezoelectric properties and photovoltaic effect.³⁷⁵ The excellent ferroelectric, piezoelectric, and electrooptic^370,376 properties of tetragonal BiFeO₃ make it of high interest as a Pb-free ferroelectric replacement for PZT.

The recent discovery of ferroelectricity in pure and doped HfO₂ ultrathin films^173,377,378 is also of high interest for interfacial ferroelectric devices, given its compatibility with CMOS technology and the robust ferroelectric and dielectric properties at thicknesses in the range from 3 to 30 nm, including high electric coercive fields and high breakdown voltages (see Table I).^379,380 Ferroelectricity in HfO₂ is associated with a metastable polar orthorhombic P_ca2₁ phase that can be stabilized by interfacial strain, growth orientation, small crystallite grain size, doping (such as with Si, Zr, Al, Y, Gd, Sr, and La) or in combination with the application of an external electric field.^{173,378,381–384} It is a displacive ferroelectric where the electric dipole moment arises from a coordinated displacement of four of the eight oxygen ions in the orthorhombic unit cell, defining the c direction of the orthorhombic structure and the direction of the ferroelectric polarization; the ferroelectric phase is chiral (upon a point rotation, the polarization reverses but the structure is not that of the mirror reflected structure), with both chiralities possessing the same energy and reversal of the polarization occurring via the coordinated displacement of the same four oxygen cations along the c direction.^378,381

Most studies on ferroelectric HfO₂ reported in the literature are for polycrystalline films grown by atomic layer deposition (ALD) or by chemical solution deposition, typically on TiN contact layers and requiring an annealing step for film crystallization after metallization,^{173,378,382,383,385,386} where the ferroelectric orthorhombic phase coexists with a non-polar tetragonal phase. However, a growing number of studies have reported the growth of epitaxial ferroelectric HfO₂ films.^387–398 One example is the epitaxial growth of 15–20 nm YO_1.5-substituted HfO₂ films by pulsed laser deposition on [100]-oriented yttrium oxide-stabilized zirconium oxide (YSZ) substrates,³⁸⁷ and on YSZ(110) single crystals using Sn-doped In₂O₃ (ITO) as a bottom electrode,³⁸⁹ where large ferroelectric polarizations and an estimated critical temperature of 723 K were observed. A cross-sectional annular bright field STEM image of a ca. 20 nm (YO_1.5)$0.07$(HfO₂)$0.93$/YSZ(100) is shown in Fig. 7 together with simulations of the patterns expected for the polar and non-polar phases, showing a good correspondence for the polar structure.³⁸⁷ Most recently, first principles calculations and x-ray diffraction measurements of epitaxial single crystals grown in the [111] orientation show stabilization of another predicted ferroelectric ground state Pnm2₁ structure.³⁹⁹ Hence, there is promise that better control over the crystalline growth of ferroelectric HfO₂ thin films could be achieved; the harnessing of its excellent dielectric and ferroelectric properties would be of high interest for the future development of interfacial ferroelectric devices.^{379,400–402}

A related ferroelectric that has been recently discovered is monolayer ZrO₂ grown on silicon.^403,404 One of the stable crystal structures of ZrO₂ is the fluorite structure, in which an individual atomic ZrO₂ plane in the material is polar. In the bulk, this polarization is not switchable, but a combination of first principles calculations and experiment shows that a single ZrO₂ plane grown atomically abruptly on silicon has two stable structures with different polarizations. The ferroelectric structure then consists of a continuous monolayer of ZrO₂ that forms on silicon when grown using molecular beam epitaxy on an atomically clean Si(001) surface. When the polarization is switched by application of an electric field, the atomic configuration switches and shifts the silicon surface potential by 0.6 V.⁴⁰⁴ Capacitance–voltage measurements of the gate stack show a large hysteresis with a direction that follows the ferroelectric switching of the ZrO₂ monolayer. This new class of materials consisting of a single atomic monolayer thick film represents the lower limit of device scaling.

One important aspect in studying thin ferroelectric films concerns the role of the supporting substrate, whose choice has to be considered judiciously. In addition to chemical stability,²⁴⁵ a first criterion is a relatively good lattice match in order to enable epitaxial growth, but equally important is its impact on the ferroelectric properties of the film due to the elastic and electric constraints that it imposes on the ferroelectric system. By acting as mechanical boundary condition, it can impose a strain on the ferroelectric film either due to the lattice and/or the thermal misfit associated with the growth process (respectively, lattice mismatch and difference in thermal expansion between film and substrate). Strain affects strongly the ferroelectric properties, including the critical temperature, piezoelectric constants, and ferroelectric polarization.^12,14,405 For example, the ferroelectric critical temperature and polarization of strained BaTiO₃ thin films can be enhanced by up to 500 K and 250%, respectively,²¹⁸ while the quantum paraelectric SrTiO₃ becomes ferroelectric under tensile or compressive strains of about 1%.^406–408 The substrate also defines one electrical boundary condition, impacting the electrostatic energy of the system. In the case of an insulating substrate, the presence of a surface bound charge layer at the ferroelectric interface would result in a large electrostatic energy, which the system minimizes by breaking into ferroelectric domains, switching to an in-plane ferroelectric state, or in extreme cases, making the paraelectric state energetically more favorable. In the case of a conducting substrate, the free carriers in the latter will screen the electric field associated with the ferroelectric surface polarization and an electric depolarization field across the ferroelectric will be present if the potential difference to the other surface is different from zero (open circuit condition) or not, if there is no potential difference (short-circuit condition). In addition, the presence of charge carriers in the ferroelectric layer induced by either point defects^{177,409–412} or band bending at the interface⁴¹³ can result in inhomogeneous internal electric field profiles across the interface, which can manifest in an imprint field that may favor one orientation of the ferroelectric polarization over the other.⁴¹⁴ At a more subtler level, also the particular atomic termination of the substrate may condition the growth of the ferroelectric film and give rise to preferred orientations of the ferroelectric polarization.^{153,156,160,415,416}

Typically, the growth of single crystalline films requires a single crystal substrate that serves as an atomic template for the oriented crystal growth (epitaxy). In the general case, the equilibrium lattice parameter, crystal symmetry, and chemical bonding of the deposited film material is different from that of the substrate (heteroepitaxy) and the lattice registry imposed by minimization of the bond energy at the interface sets an effective elastic boundary condition in the form of a misfit strain. In particular, the condition for a coherently strained epitaxial film applies insofar as the system can accommodate the associated elastic energy, which increases linearly with film thickness. Above a critical thickness, the build up in elastic energy can be favorably counteracted by mechanisms that act to release strain, including the formation of dislocations,^417–422 surface roughness,^422–425 and strain-induced atomic diffusion at elevated temperatures.^426,427 Hence, epitaxial film growth typically proceeds first through coherent growth, whereby the epitaxial film grows fully strained and in registry with the underlying substrate up to a critical thickness above which strain relaxation processes set in, particularly by the introduction of misfit dislocations. By equating the energy gain through strain release to the strain energy associated with the dislocation core (which involves a disruption of the local crystal structure), expressions for the critical coherence thickness can been deduced to provide estimates of the thickness above which strain starts to be relaxed (which however do not consider kinetic barriers for misfit dislocation formation, such that larger critical coherence thicknesses than those predicted from theory are often observed experimentally).^292,428 For ferroelectrics, epitaxial strain has a strong impact on the ferroelectric properties, given the intimate link between lattice structure and ferroelectricity in many ferroelectric systems.^12,14,405 The effect can be twofold: for coherent strain, uniform changes in the ferroelectric properties may be expected, including changes in the ferroelectric polarization and critical temperature, while in the incoherent thickness regime, the local stresses associated with the cores of the dislocations and the associated strain gradients (such as through the flexoelectric effect)^429–434 can lead to local modulations in the ferroelectric order parameter and to modified ferroelectric properties (Sec. II E).³¹² 

The effect of coherent epitaxial strain on the properties of thin ferroelectric films has been extensively addressed in the literature.^12,236,405 In the case of the perovskite ferroelectrics, the effect of biaxial strain is that of modifying the c/a ratio and rotating the oxygen octahedra.^22,435 Its effect on the ferroelectric polarization can be estimated to linear order in strain in terms of the improper piezoelectric tensor $cαi=∂Pα/∂ϵi$⁠, where ϵ_i are the strain components expressed in the Voigt notation and α stands for the cartesian coordinate. For an in-plane biaxial strain and considering only out of plane polarization, one obtains for the change in polarization²³⁶ 

where $n=−ϵ1/ϵ3$ is the Poisson ratio. The results for a number of systems (BaTiO₃, BiFeO₃, PbTiO₃, and LiNbO₃) are shown in Fig. 8 together with the results of ab initio calculations (symbols).²³⁶ It shows that the strongest variation occurs for BaTiO₃ and PbTiO₃, as a consequence of the large c₃₃ values in combination with relatively low Poisson ratios, while for the other materials, the variation is more modest; one finds also no direct link between crystal structure and polarization susceptibility to in-plane biaxial strain. As expected for the perovskite ferroelectrics, biaxial in-plane compressive strain results in an increase in the c/a ratio and in the ferroelectric polarization, while the opposite is the case for tensile strain. Direct experimental confirmation of enhanced ferroelectric polarization of epitaxially compressively strained films has been reported in the literature, some examples of which are listed in Table II, showing critical ferroelectric temperatures and ferroelectric polarizations well in excess of the bulk value (Table I). For rhombohedral BiFeO₃, the change in ferroelectric polarization has been observed to be largely independent of the average in-plane strain (i.e., below and above the critical thickness for coherent growth),^428,436 in agreement with the theory predictions,^236,437 and a similar behavior was reported for Pb(Zr_0.2Ti_0.8)O₃,⁴³⁸ where the saturation ferroelectric polarization is found to be identical for both a strained 30 nm film (⁠$c/a=1.09$⁠) and a relaxed 100 nm film (⁠$c/a=1.05$⁠), an effect that has been ascribed to a suppressed sensitivity of the A-site cations to the epitaxial strain.⁴³⁸ 

Epitaxial strain can also induce a phase transition to a distinct supertetragonal structure (i.e., with a value of c/a much larger than the bulk value and generally found to exceed 1.2) that may also have a large polarization. Two particular reports are striking, namely, for supertetragonal PbTiO₃ and BiFeO₃, where ferroelectric polarizations of 236 and 130 μC/cm², respectively, have been reported,^240,439 consistent with first principles calculations.^234,440,441

In addition to modifying the ferroelectric polarization, epitaxial strain also affects the ferroelectric critical temperature by hindering the transition to the higher symmetry paraelectric phase, while the order of the phase transition may change from first to second order as a consequence of the reduced electrostrictive coupling in strained ferroelectric films;^442–444 intuitively speaking, epitaxial strain prevents the system to lower the crystal symmetry to cubic across the phase transition, resulting in a continuous change in the order parameter. In addition, epitaxial strain can induce the formation of new crystalline phases that otherwise could not be stabilized in bulk.^{12,218,405,443,445} Phenomenologically, epitaxial strain adds an electrostrictive contribution to the polarization that modifies the onset of ferroelectricity in the film; for a [001]-oriented thin film, the change in critical temperature for an out-of-plane polarized film (tetragonal c phase) is estimated as⁴⁴³ 

while for the onset of in-plane polarization along the $⟨110⟩$ direction (orthorhombic aa phase),

where T₀ is the bulk critical temperature, C is the Curie constant, $ϵ0=8.854187817×10−12$ F/m is the electric permittivity of free space, Q are the electrostrictive coefficients, s are the elastic compliances, and $um=(as−a0)/a0$ is the misfit strain, where a_s is the substrate lattice parameter and a₀ the equilibrium lattice parameter of the free standing film. The above expressions predict that, when $Q12<0$ and $Q11+Q12>0$⁠, as is the case for BaTiO₃ and PbTiO₃, the critical temperature increases (linearly) for both signs of the misfit strain.⁴⁴³ Examples of the effect of epitaxial strain in enhancing the critical temperature are given in Table II for several perovskite ferroelectrics, showing a significant increase in $Tc$ with respect to their bulk values (Table I). The enhanced ferroelectric polarization and critical temperature induced by epitaxial strain are beneficial for ferroelectric interfacial devices, since it is conducive to larger field effects over a larger temperature range (as far as the ferroelectric component is concerned). In BiFeO₃, the presence of both polar displacements and oxygen octahedra rotations yields an anomalous strain dependence of the Curie temperature. It actually decreases with increasing strain, which is captured by effective Hamiltonian calculations.⁴⁴⁶ 

While the role of a uniform strain is largely beneficial and can be employed to tune the characteristics of ferroelectric thin films, strain relaxation in thin films is expected to have a predominantly negative impact on the ferroelectric properties.³¹² For example, calculations based on a thermodynamic analysis predict strong variations in the local ferroelectric polarization near the core of the dislocations that lead to strong depolarization fields that suppress ferroelectricity in a region that can extend over several nanometers from the interface.^447–449 A telling example is the local reduction of the ferroelectric polarization of a PZT film by up to 48% caused by the strain field of a dislocation inside an adjacent SrTiO₃ layer, determined from TEM.⁴⁵⁰ The presence of such a depolarization layer can lead to the formation of ferroelectric dead layers that absorb a large fraction of the applied electric field and that can act as pinning centers for the ferroelectric domain walls.⁴⁵¹ The negative role of misfit dislocations has been identified in several systems, including PZT nanoislands⁴⁵² and thin PZT films.^453–456 

Another mechanism for strain relaxation in thin ferroelectric films is through domain formation, such as domains with either the a-axis or c-axis oriented perpendicular to the surface (a and c domains) in tetragonal ferroelectric systems.^457–459 Also for this case, a critical thickness has been estimated above which it is energetically favorable for the system to develop a mixed domain configuration. A manifestation of this process can be found in thick c-oriented PZT films, where a significant a-domain population is often observed experimentally.^145,456,460 The presence of a-domains is undesirable, as they lower the switchable out-of-plane ferroelectric polarization of the system and contribute to inhomogeneities that are detrimental to device scaling.⁴⁵⁸ 

The substrate also sets an electrical boundary condition for the ferroelectric film with important consequences for the equilibrium ferroelectric configuration of the system. This is due to the role of the electrostatic energy, which, in the absence of free charge, is minimized for zero divergence of the ferroelectric polarization (⁠$∇·P=0$⁠) and no net bound charge at the boundaries (⁠$n·P=0$⁠, where n is the surface unit vector). For thin films, this is achieved for a uniform in-plane configuration of the ferroelectric polarization. However, since the direction of the latter is tied to the lattice, the ferroelectric polarization in epitaxial ferroelectric thin films is not free to orient along the direction that minimizes the electrostatic energy. In fact, for field effect devices, a uniform orientation of the ferroelectric polarization along the direction normal to the film plane is generally the most desirable configuration. The latter can be achieved by constraining the polar axis to be oriented along the out-of-plane direction, as in [001]-oriented tetragonal ferroelectric systems or in rhombohedral [001]-oriented BiFeO₃ ([111]$pc$ of the pseudo-cubic perovskite structure); also, in rhombohedral BiFeO₃ grown on SrTiO₃(001), the polarization makes an angle of about 36^o from the surface, resulting in a large polarization component along the out-of-plane direction.^{355,364,365,461} When the ferroelectric polarization cannot be screened by free charges (such as provided by a conducting substrate and/or polar molecules adsorbed at the free surface),^413,462,463 the electrostatic energy of the system can be minimized by the formation of oppositely poled ferroelectric domains^{127,129,464,465} or by reverting to the paraelectric state at the ultrathin limit.^466–468 

The geometry of ferroelectric interfacial devices is typically that of a capacitor structure, where the ferroelectric layer is sandwiched between two conducting layers, for example, a gate contact and a channel layer in the case of ferroelectric field effect devices. These layers provide not only for device function but also act to screen the ferroelectric polarization at the respective interface and ensure the required stability for the uniformly polarized ferroelectric state. However, at metallic interfaces, screening occurs over a finite length scale of the order of the characteristic Thomas-Fermi screening length, $δTF∼0.339(rs/a0)1/2$ (where $rs=(3/4πn)1/3$ is the mean distance between carriers, n is the carrier density, and $a0=0.529 177$ Å is the Bohr atomic radius).⁴⁶⁹ Over this screening length, the electric field associated with the ferroelectric polarization is non-zero and acts as a dielectric layer of thickness⁴⁶⁵ $d∼2δTF$ that adds to the electrostatic energy and contributes to a series capacitance that absorbs a fraction of the applied electric field. In fact, the depolarizing field associated with the screening length of the metallic contact can be sufficient to destabilize the ferroelectric order of ultrathin films, setting a limit to the minimum thickness required for the onset of ferroelectricity that depends on the electrode material, on the ionic polarizability, and on the ferroelectric band structure.^34,466,467 For example, in PbTiO₃ and BaTiO₃, theoretical ab initio studies have predicted the onset of ferroelectricity at 1 and 6 unit cells, respectively.^34,466,467 Experimentally, ferroelectricity has been confirmed in BaTiO₃, PbTiO₃, Pb(Zr_0.2Ti_0.8)O₃, BiFeO₃, HfO, and KNbO₃ films down to a few unit cells in thickness (Table III).

For device applications, it is important to be able to apply electric fields, for example, to switch the direction of the ferroelectric polarization. Since most oxide ferroelectrics have relatively modest bandgaps (∼3 eV, see Table I) and a significant density of extrinsic charge carriers (originating typically from point defects as a consequence of the complex defect chemistry of 3d metal oxides, where the metal cations can exist in various oxidation states),^{412,476–478} the electrical character of the contact (Ohmic or diode-like) will depend on the difference between metal work function $ϕ$ and the electron affinity of the ferroelectric χ (which defines the height of the Schottky barrier, $Vbi=ϕ−χ$⁠), the charge carrier density in the ferroelectric (which defines the width of the depletion layer) and the presence of interface states (metal-induced gap states or interface defect states), which may pin the position of the Fermi level of the ferroelectric charge carriers.^{171,476–481} The electrostatic physics is illustrated in Fig. 9, showing the band alignment for a perfectly insulating ferroelectric in contact with identical metal electrodes compared to that of a defective ferroelectric layer. The situation is similar to that of a metal/semiconductor contact, where the role of the ferroelectric polarization is that of adding a sheet of bound charge at the interface to the electrostatic configuration.^476,479,482 To account for the presence of a depletion layer in the metallic contact or a ferroelectric dead layer, a thin dielectric layer of thickness δ is added between the ferroelectric and the metal, which is otherwise treated as ideal. To first approximation, the ferroelectric polarization adds a term to the Schottky built-in potential, $Vbi$⁠,^476,479,482

where P is the ferroelectric polarization amplitude, δ is the width of the dielectric layer, $ϵs$ is its static dielectric constant, and the ± sign corresponds to positive or negative directions of the ferroelectric polarization. For $P∼50$μC/cm², $δ∼0.2$ Å, and $ϵs∼2$⁠, the additional contribution to the built-in potential is of the order of 0.5 V, which is considerable. In fact, the experimental determination of the polarization dependence of the Schottky barrier height has been reported for several systems, including BaTiO₃, BiFeO₃, and Pb(Zr_0.2Ti_0.8)O₃, showing indeed that such large variations in the Schottky barrier height are reached, see Table IV. In some instances, the change in the Schottky barrier may be sufficiently high to drive the contact from diode-like to Ohmic, leading to large changes in the conductivity of the system when switching the direction of the ferroelectric polarization.^{104,483–486} A similar interface structure will be present at the other side of the ferroelectric layer, possibly with a different conducting material. Hence, the requirement for being able to apply an electric field of sufficient amplitude for switching the ferroelectric polarization is that both interfaces consist of back-to-back Schottky diode junctions. The presence of charge carriers, either intrinsic or due to extrinsic doping arising from point defects, show that ferroelectrics for the most part are not perfect insulators, but instead exhibit charge transport under an applied electric field. In addition to thermally activated transport of charge carriers across the Schottky barrier, and in common to other insulators, tunneling in ultrathin films, electron hopping, or emission from deep traps are also found to contribute to the conductivity;^411,487 for ferroelectrics, it has been recently discovered that ferroelectric domain walls tend to exhibit significantly different electronic properties as compared to the bulk material,^488,489 including high electric conductivity.^490–495 

Different types of mobile charge carriers may be involved in screening the ferroelectric polarization at the interface.^41,414 For example, a detailed STEM-EELS study by Kim et al.⁴⁹⁶ of the BiFeO₃/La_0.67Sr_0.33MnO₃ interface in a region of the sample containing two ferroelectric domains with opposite polarization shows that the domain with the ferroelectric polarization pointing away from the interface presents an out of plane lattice expansion of ∼5% in the BiFeO₃ interface region, together with a decrease in the Mn valency and a reduction in the O K edge intensity, which is interpreted as due to screening by oxygen vacancies; in contrast, the other polarization direction shows no such effects and screening is inferred to be purely electronic.⁴⁹⁶ 

In addition to screening by mobile charges discussed above, other mechanisms can also contribute to screen the ferroelectric polarization, including ionic screening. For example, Gerra et al.⁵⁰⁰ suggested in their theoretical work that additional ionic compensation to the depolarization field can occur at a ferroelectric/electrode interface. By considering a ferroelectric BaTiO₃ layer sandwiched between SrRuO₃ electrodes as a model system, they show that the ferroelectric ionic displacement of the ferroelectric BaTiO₃ layer may penetrate for several atomic layers into the SrRuO₃ electrode. The induced ionic displacement inside the electrode acts to further stabilize ferroelectricity by pushing the center of mass of free electrons toward the interface. Subsequent theoretical and experimental work has shown that ionic compensation is active in various ferroelectric/metal interfaces and is crucial in stabilizing ferroelectricity down to the nanoscale.^{52,467,471,501} Conversely, the ferroelectric induced ionic displacement inside the electrode can result in changes in its electronic properties, leading to an additional mechanism for controlling the correlated state of the conducting channel (see Sec. IV).

Although we have focused on the impact on the ferroelectric properties of materials with decreasing film thickness down to the atomic scale, additional considerations come to the fore when also the lateral dimension of the ferroelectric system is reduced to the nanoscale, in particular for free standing nanoparticle systems, where new domain states such as flux closure states can occur to minimize the electrostatic energy; a critical size may be required to sustain a single domain state and, at smaller dimensions, to sustain a stable ferroelectric state against thermal excitations.^502–505 

In addition to the constraints imposed by elastic and electrostatic constraints, atomic bonding effects arising from electron exchange at the interface with the substrate or top layer can have a strong effect on the overall properties of the system that are important to consider. The study of such phenomena often requires probing the local electronic structure at the interface, and in this context, the recent advances in high resolution scanning TEM have been invaluable.^28,144 One aspect concerns the chemical bonding between the ferroelectric layer and the adjacent conducting channel, which may impose restrictions on the ferroelectric soft-mode across the heterointerface and affect significantly the ferroelectric properties of the system.^52,506 In addition, the asymmetric environment at the interface can induce changes in the electronic state and a concomitant modification of the ferroelectric energy landscape. A striking example of this effect is a preference for a particular orientation of the ferroelectric polarization (imprint effect) that is found to depend on the termination of the interface layer, for example, in BiFeO₃/LSMO,⁴¹⁵ PZT/LSMO,^156,416 and BaTiO₃/LSMO.¹⁶⁰ In extreme cases, the presence os such in-built bias potential can lead to back-switching of ferroelectric domains and loss of information.^146,507 The role of interface termination and of the energy associated with the polar discontinuity^508–511 on the ferroelectric state has been studied for several perovskite interfaces.⁴¹⁵ For example, in ferroelectric BiFeO₃ and metallic La_0.7Sr_0.3MnO₃ heterostructures, the choice of the interface termination sequence, (BiO)⁺/(MnO₂)$−0.7$ or (FeO₂)⁻/(La_0.7Sr_0.3O)$+0.7$⁠, induces a different charge valence mismatch, +0.3 or –0.3, at the interface, which results in the formation of an interface dipole that affects the polarization switching of the BiFeO₃ layer.^153,415 The interface dipole can be formed even when no charge valence mismatch is expected. In the BaO/RuO₂ interface termination at BaTiO₃/SrRuO₃, the difference in ionic radii of the cations induce a pinned interface dipole that hinders ferroelectric switching of the system.^512,513 The undesirable interface dipole effect can be avoided by changing the BaO/RuO₂ termination to TiO₂/SrO or by inserting a few atomic layers of SrTiO₃.^471,514 Such results emphasize the importance of the proper choice of the interface terminating sequence in designing electronic devices based on nanoscale ferroelectricity. Another effect pertains to atomic-scale structural distortions at the interface layers, not only in terms of uniform epitaxial strain, but also associated with modifications in the oxygen octahedral rotations characteristic of some ferroelectric systems, such as BiFeO₃. For example, Kim et al.¹⁵⁴ have reported the onset of a non-polar BiFeO₃ interface layer at the La_0.8Sr_0.2MnO₃ interface of BiFeO₃/La_0.8Sr_0.2MnO₃/SrTiO₃(001) films, whereby a MnO₂ termination of the La_0.8Sr_0.2MnO₃ leads to a suppression of the octahedral rotation angle at the interface extending up to three atomic layers and whose effect propagates over a 3 nm region, while the LaSrO termination leads also to a reduced octahedral rotation compared to the bulk that extends to the whole of the film, as shown in Fig. 10. As a consequence of the presence of the non-polar BiFeO₃ interface layer, the piezoelectric properties of the MnO₂ terminated film are found to be strongly reduced as compared to the other termination.

We focus in this review on metal oxide materials, for which the lattice parameter is largely determined by the oxygen sublattice and the oxygen ionic radius (in the range from 1.35 to 1.42 Å, increasing with increasing coordination number);^515,516 hence, the growth of oxides on oxides is facilitated by both chemical compatibility and the relatively small lattice mismatches. In particular, we have highlighted ferroelectric systems where highly ordered single crystalline films can be obtained on a number of different substrate materials. However, we have also remarked that the oxide growth process is often associated with the creation of defects due to misfit strain relaxation processes (including the onset of dislocations, film roughening, and domain formation), local chemical reactivity at the interface, or associated with the oxygen defect chemistry in oxides, which is largely a direct consequence of the multiple valence states of the 3d transitions metals.^476–478 By locally altering the strain field, by disturbing the local electronic structure, or by adding free carriers to the system, the presence of defects can affect strongly (mostly negatively) the properties of thin ferrolectric films, including by depressing the amplitude of the local ferroelectric polarization, increasing the coercive field, lowering the breakdown voltage and inducing current leakage, charge traps, and fatigue.^{8,451,517,518} Since it may not be possible to fully eliminate the presence of defects, it is important to understand their role on the properties of ferroelectric thin films and their impact on interfacial properties of ferroelectric devices, in order to determine which ones are more deleterious for a given application.

The mechanisms that lead to the presence of defects are directly linked to the growth process itself. The growth temperature of most ferroelectric films, in the range from 500 to 700 °C, tends to be well above the bulk Curie temperature and it is generally assumed that film growth starts from the paraelectric cubic phase (although, as shown in Table II, under misfit strain, ferroelectricity may be present, or start to develop already at such temperatures above a certain critical thickness).^{292,313,519–521} At this elevated temperature, the lattice misfit is generally different from that at room temperature,^312,522 and above a critical thickness, elastic energy is relieved by the onset of misfit dislocations, whose nucleation is expected to be facilitated by the available thermal energy.^{417–421,458,522} Strain can also make the surface unstable against deformation, whereby elastic energy is reduced by surface roughening at a cost of surface energy.^{424,425,523,524} As the temperature of the film is reduced to ambient temperature after growth, the density of misfit dislocations may change to reduce the thermal misfit (difference in thermal expansion coefficients) for films above the critical thickness for coherent growth and, as the thermal energy for the movement of misfit dislocations becomes insufficient to reach equilibrium, a residual strain may be present in the system at ambient temperature. Finally, the transition to the ferroelectric state can result in the formation of ferroelectric domains, for example, a-domains in c-oriented tetragonal ferroelectric films, as another channel for misfit strain relaxation.^{451,457–459,525,526}

Because of the relatively high Peierls energy in the perovskites (for SrTiO₃ it is estimated to be of the order or 0.5 eV for $⟨110⟩$ {110} edge dislocations),⁵²⁷ gliding of dislocations is increasingly suppressed as the temperature is reduced to room temperature, such that the dislocation network observed at low temperature can be assumed to have formed during film growth.^451,522 A typical consequence of this kinetic barrier is the observation of strain opposite to that expected from the lattice mismatch at room temperature due to thermal misfit between film and substrate.^205,528 The particular misfit dislocation pattern in thin films can be quite complex. For the cubic perovskites, screw and edge dislocations along $⟨110⟩$ within {110} glide planes, with a Burgers vector $(a/2)⟨110⟩$⁠, are energetically more favorable, but dislocations along other directions may also occur at elevated temperatures;^529–531 in thin epitaxial films, only dislocations with Burgers vector parallel to the interface act to effectively reduce strain, although experimentally both edge and screw dislocations are observed.^205,312,454 For example, in thick, [001]-oriented BaTiO₃ films, one observes not only a criss-cross network of misfit dislocations, but also perpendicular screw dislocations and oblique screw dislocations in the {110} glide plane;^205,532 these are thought to originate at point defects at the surface that then glide toward the interface, where they dissociate to form interfacial dislocation lines along $⟨100⟩$ to relieve epitaxial strain, with the perpendicular component eventually propagating to the surface and eliminated there.^205,533 Strategies for minimizing the formation of misfit dislocations include increased film deposition rates,^243,534 annealing the film after growth,⁵³⁵ and employing well conditioned single terminated substrates,³¹² for example. Given the high internal stresses associated with the dislocation core and the direct coupling between internal stress and the ferroelectric order parameter, misfit dislocations result in strong local depolarization fields that modify the amplitude of the local polarization; in thin films with a high density of misfit dislocations, such depolarization fields may result in suppressed ferroelectric polarization (ferroelectric dead layers) and in pinning centers for domain wall motion;^{145,312,447–449,451,453–455} they can also act as preferential sites for atomic diffusion processes, such as oxygen transport (the impact of point defects is discussed below).^536–538 This illustrates the point that different types of defects can interact in complex ways; for instance, strain fields arising from structural defects can lead to modifications to the local composition^145,450,539 and promote atomic diffusion,⁴²⁶ while misfit dislocations can drive interdiffusion at the interface⁴²⁷ or lead to surface roughening and provide preferential nucleation sites for film growth.⁵⁴⁰ Strain gradients, originating from the stress fields issuing from dislocations or from compositional variations with thickness generated during film growth, can also result in depressed properties due to coupling with the ferroelectric polarization via the flexoelectric effect^429–432 that acts as an additional local internal electric field and that provides another mechanism for the presence of imprint effects (seen as an horizontal shift in the P-E hysteresis loop) in defective films; however, such effect has been employed recently to bias the direction of the ferroelectric polarization and therefore to control the polarization state by suitable control of the growth process and film thickness.^{433,434,541,542}

The formation of structural polydomains at the transition from the paramagnetic to ferroelectric phase as the film is cooled down to room temperature from the temperature of deposition is another mechanism for reducing misfit strain energy.^{451,457–459,525,526,543} Although the process here is driven by elastic energy considerations,⁵⁴⁴ the link between structure and direction of ferroelectric polarization implies that the system will break into a ferroelectric multidomain state as well, for example, a and c domains in the tetragonal ferroelectric perovskites, or a more complex domain structure for the rhombohedral perovskites, such as BiFeO₃, where four variant domains can be present in thin films.^365,461,545 For the tetragonal perovskites, the presence of a domains leads to a reduction of the switchable ferroelectric polarization,⁴⁶⁰ since the electric fields required to fully switch an a domain to a c domain, either by coherent rotation or by domain wall displacement (typically strongly pinned at defects such as dislocations),^145,152,451 are much higher than those required to switch the polarization along the c direction. For example, in c-oriented PZT films, it is observed that pinning of a domains is associated with pairs of misfit dislocations with Burgers vectors [100] and [001], as illustrated in the TEM results shown in Figs. 11(a)–11(e). Model calculations shown in Fig. 11(f) of the in-plane strain field associated with the dislocation pair shown in Fig. 11(e) show the presence of a large region where the a domain is favored (red color), indicating that pinning of a domains results primarily from the strong strain field produced by the dislocation pair.¹⁴⁵ 

One growth strategy for minimizing the nucleation of a polydomain state has been to employ stepped substrate surfaces obtained from cutting the wafer off the main index plane by up to a few degrees (miscut or vicinal surfaces). The role of the stepped surface is that of allowing an asymmetrical strain relaxation along the direction of the steps,^546,547 setting a preferential crystallographic direction for strain relaxation and for the growth of systems with lower symmetry than that of the substrate. Such an approach has been used to reduce the number of domain variants of BiFeO₃ films grown on SrTiO₃(001) from four to two, resulting in an improvement in the ferroelectric switching characteristics of the BiFeO₃ film,^548,549 or to tune the crystal structure of Pb(Mg_1/3Nb_2/3)O₃-PbTiO₃ thin films.^550,551 The steps tend also to act as nucleation sites for film growth and can lead to a step flow growth mode and to atomically smooth surfaces;⁵⁵² however, the regions where the steps meet are sites susceptible to the formation of lattice defects, such as strain induced by residual lattice matching between the film and substrate,⁵⁵³ stacking faults,⁵⁵¹ and vertical lattice mismatch as a result of step bunching;⁵⁵⁴ additionally, the atomic steps interact with dislocations and modify their configuration in the film.⁵⁵⁵ 

Defect chemistry also plays an important role at the growth temperature. Intrinsic defects, such as vacant sites (vacancies), atoms occupying interstitial sites (interstitials), Frenkel defects (atoms that move to occupy an interstitial site leaving behind a vacancy), and Schottky defects (paired anion and cation vacancies) are thermodynamically favored and will be always present in the crystal, since they reduce the Gibbs free energy of the system by increasing the entropy.⁵⁵⁶ In the oxide perovskites, double ionized oxygen vacancies are one of the most important type of point defects;⁵⁵⁷ their equilibrium density at a given temperature T and oxygen partial pressure $p[O2]$ can be obtained from the chemical reaction,^{409,557–560}

in the Kröger–Vink notation,⁵⁶¹ where $OOx$ is the lattice oxygen, $VO••$ is the oxygen vacancy, and $e′$ is the electron, describing the release (capture) of a pair of electrons to the conduction band of SrTiO₃ for each oxygen vacancy created (annihilated). The corresponding reaction equilibrium constant K₁ is given by

where $[OOx]≈N0$ is approximately constant and close to the number density of oxygen sites (N₀), $p[O2]$ the oxygen partial pressure, H_a and $ΔS$ are the enthalpy (activation energy) and entropy of reaction, respectively, and $kB$ is the Boltzmann constant. For SrTiO₃, $Ha=5.69$ eV/vacancy,⁵⁵⁸ for BaTiO₃, $Ha=5.96$ eV/vacancy.⁵⁶² One consequence of the presence of oxygen vacancies is that they contribute with charge carriers to the electrical conductivity, $σi=μinie$ (where μ_i is the charge mobility, e the elementary electric charge, and n_i the charge carrier density, where i stands for electrons or hole carriers), with the expression above predicting a variation of $[e′]=2[VO..]∝p[O2]−1/6$⁠. Such variation is in fact observed, for example for SrTiO₃ and BaTiO₃, but only in the regime of high temperatures and strongly reducing conditions.^{409,557,558,562–564} More significant in determining the equilibrium concentration of oxygen vacancies is the effect of impurities (extrinsic point defects, typically present in ppm concentrations), which originate mostly from the source material and substitute for the intended atoms or occupy interstitial positions in the lattice (the latter being not favorable in closed packed lattices, such as the perovskite). Most impurities tend to be low valence cations, such as Na, Mg, and Al, which act as acceptors when they replace cations with higher valencies (and as donors in the opposite case).^{409,557,565,566} Since they tend to be the dominant factor determining the oxygen vacancy concentration, in the reaction equation (5), $VO••$ is approximately constant and the electron carrier density now varies as $p[O2]−1/4$⁠, which is observed experimentally over wide temperature and oxygen pressure ranges.^{409,557,558,562–564} Under oxidizing conditions, excess oxygen fills in part the impurity related vacancies, leading to the introduction of pairs of hole carriers to the valence band (and p-type conductivity) leading to a dependence in hole carrier density $p∝p[O2]+1/4$⁠.⁵⁶⁷ For example, substitutional impurities such as Na, Mg, and Al act as acceptors and make PZT a p-doped ferroelectric (although the much larger mobility of minority electrons still makes the latter responsible for charge transport).^{177,409–411} In the intermediate regime, compensation of electron and hole charge carriers is reached and charge excitation becomes again intrinsic, i.e., determined by excitations over the bandgap.^{557,568–570} In particular, by deliberately doping the ferroelectric system with donors, the oxygen vacancy density can be reduced or fully compensated, resulting in softer ferroelectric properties.^557,568,571 Systems with volatile elements, such as Pb in PbTiO₃, or Bi in BiFeO₃, may also have a significant density of cation vacancies at the growth temperature,^409,572,573 while in systems where the electrons can be captured by multivalent cations, such as Fe in BiFeO₃, the defect chemistry reaction is significantly different and more complex.^571,574,575 In PZT, oxygen is released as PbO, such that formation of oxygen vacancies is accompanied by the formation of Pb vacancies.^409,576

While oxygen vacancies do not always have a negative impact on the properties of oxide materials (controlling the density of oxygen vacancies may in fact provide another knob to controlling their electronic properties⁵⁷⁷ and they are desirable for oxygen diffusion and ionic conductivity in solid electrolytes, such as in stabilized zirconia, YSZ^578–581) they generally have a deleterious impact on the dielectric properties of ferroelectric oxides.⁴⁰⁹ They can contribute to the electrical conductivity (in SrTiO₃ they form an impurity state in the bandgap, just below the conduction band)^573,582 and modify the oxidation state of neighboring cations. Furthermore, oxygen vacancies are mobile, with an activation energy of the order of 0.6–1 eV for the perovskite titanates,^583–585 and can be displaced under the action of an applied electric field, or migrate at high temperature to the interface to screen the ferroelectric polarization,⁵⁸⁶ where the subsequent pileup at one electrode interface and depletion at the other