Epitaxial ferroelectric interfacial devices Free

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 can contribute to resistance degradation, imprint effects, and fatigue in ferroelectric capacitor structures.^583,587 It has been inferred that the high density of free carriers in thin PZT ferroelectric layers may be a consequence of the formation of oxygen vacancies induced at the temperature of growth in order to screen the surface ferroelectric polarization (self-doping).⁴¹² Interfacial layers, by either acting as a sink, source, or by blocking the flow of oxygen, can affect the equilibrium oxygen vacancy density in the ferroelectric layer and impact its properties;^409,572 in particular, by judiciously choosing the interfacial contact layers, improved retention and switching characteristics can be obtained. For example, the use of oxide electrodes in ferroelectric capacitors has been shown to reduce fatigue and imprint effects as compared with metal electrodes, which act to trap the oxygen vacancies within the ferroelectric.⁴⁰⁹ 

Since an oxygen vacancy corresponds to a missing oxygen anion, for example, in the TiO₆ octahedron of BaTiO₃ or Pb(Zr_0.2Ti_0.8)O₃, oxygen vacancies also give rise to local electric dipoles that point along, against, or perpendicular to the local ferroelectric polarization and which strongly perturb the local electric field, affecting the ferroelectric switching properties (by acting as pinning centers for domain walls, for example) and contributing to imprint effects.^571,575,583 For similar reasons, they also cause local distortions in the atomic lattice, resulting in local strains. The impact of the latter will depend on the particular system: one may expect it to be strong in displacive ferroelectrics, such as BaTiO₃, while in lone-pair ferroelectric rhombohedral BiFeO₃ the effect of oxygen vacancies (and of strain) has been estimated to be minor.⁴³⁷ 

Controlling the correlated state of matter is a fundamental topic in condensed matter physics, both because understanding the underlying mechanisms brings insights into the role of electron correlations to the emergence of novel physical phenomena and because those effects can potentially be useful for device applications. In fact, a major difficulty stems from the challenge of accounting fully for electron–electron many body interactions.^588,589 Here we focus mostly on transition metal oxides, which are a prototypical class of systems exhibiting strong electron–electron correlations; this is manifested by the strong deviations in their behavior from that expected from a near-free electron system. Typical of these systems are complex phase diagrams, with several equilibrium states as a function of a given physical parameter, such as pressure or doping, as illustrated in the diagram shown in Fig. 12 representative of a wide class of 3d transition metal oxides. Starting from the insulating parent oxide, the addition of charge carriers does not initially modify strongly the electronic state, but at a critical carrier density, the carriers become delocalized, leading to a new electronic conducting ground state; with further increase in the charge carrier density, electron–electron correlations lead to the opening of a bandgap in the hybridized oxygen 2p-transition metal 3d band at half filling, resulting in a typical dome shape in the phase diagram.³ In fact, the general physical properties of strongly correlated systems may be described grosso modo in terms of the charge carrier density and electron bandwidth contributions of the Hubbard Hamiltonian.⁵⁹⁰ The electronic bandwidth is determined by the cationic crystal field and orbital overlap, while the charge carrier density can be modified through either chemical, ionic, or electrostatic doping. A classic example of the changes in the electronic properties of complex oxides with doping is that of the doped manganites AMnO₃, where the average Mn valency can be modulated by varying the amount of di- or trivalent cations occupying the A sites of the perovskite structure via chemical substitution.^591–595 One drawback of this approach to modulating the charge carrier density is that it is irreversible and involves local changes in the crystal structure induced by the different atomic size of the substituting element.⁵⁹⁶ Another approach to modulating the charge carrier is through ionic doping,^597–599 based on ionic mass transport into or out of the material, for example oxygen or lithium, both of which have relatively high mobility in oxide materials^{559,560,600,601} and which can be driven by electric fields (Li⁺ or charged oxygen vacancies, $VO2+$⁠). While in general, a change in the oxygen content in an oxide material is undesirable (an exception being in ionic conductors, such as stabilized zirconia, as mentioned earlier), it has been used to advantage for example in memristor devices, where different resistive states can be obtained by drastic modifications in the oxygen atom distribution under electric breakdown conditions,⁶⁰² or to control the interfacial oxidation and the interfacial magnetic anisotropy of Co/GdO_x by an applied electric field.⁵⁹⁸ Here, we focus on electrostatic doping, which relies on charge screening effects to modulate the charge carrier density at the interface between a dielectric and a conducting material, as in the classic field effect transistor. Unlike semiconductors, however, which are characterized by relatively low intrinsic carrier densities (⁠$1010−1018$ cm^--3),¹⁸⁹ strongly correlated materials have very high charge carrier densities at the Fermi level, of the order of 10²¹ cm^--3, requiring therefore very large induced surface charge polarizations at the dielectric interface which are not possible to achieve with conventional SiO_x gates. Instead, large reversible modulation of the carrier density can be achieved in field effect device structures using either high dielectric constant materials or ferroelectrics, whose spontaneous surface bound charge is used to induce large modulations in the charge carrier density at the interface in ferroelectric field effect device structures;^{4,8,10,603,604} an alternative is to employ electrolytes, i.e., ionic conductors, where the electric field is screened at the electrode interfaces by mobile ions by forming a so-called double-layer of electric charge within a nanometer of the interface and no electric field within the electrolyte.^605,606

The original proposal for the ferroelectric field effect transistor^{58,59,607–609} aimed at combining the binary memory function of the ferroelectric polarization direction with the field effect control of the conductivity in a semiconductor channel in order to achieve a nonvolatile transistor switch, i.e., to extend the classical function of a field effect transistor. An extension of this concept consists of replacing the semiconducting channel layer by a strongly correlated material in order to control electrostatically its electronic state.^{4,5,85,610,611} The goal is to expand the functionality of such types of device structures by controlling the magnetic state, orbital or charge order, superconductivity, or the metal to insulator transition. A schematic of the ferroelectric field effect device structure and of the basic approach to controlling the electronic state of the channel layer is shown in Fig. 12: by choosing a chemical doping in the phase diagram of a strongly correlated system near the boundary between two different ground states, the system is driven across the boundary (horizontal arrow in the figure) through electrostatic carrier modulation, induced by charge carrier depletion or accumulation at the ferroelectric interface.^3,317 In these device structures, the role of the ferroelectric layer is to provide the required change in the interfacial carrier density through its surface bound charge as the direction of the ferroelectric polarization is reversed by application of a voltage to the gate contact. The practical requisites have been discussed in Sec. II and include, for optimal effects, a low density of defects; an abrupt interface with a low charge trap density and minimal interdiffusion; large ferroelectric polarizations; and Schottky contacts to enable application of electric fields for ferroelectric switching. Alternatively, high dielectric materials such as SrTiO₃ can be employed as a gate dielectric, which are capable of providing large surface polarizations at relatively modest applied electric fields, although, in contrast to ferroelectrics, the effect will be present only as long as the electric field is applied to the system. Based on such device structures, control of several physical properties of correlated materials has been achieved and will be discussed in Secs. III A–III E. These include control of the superconducting state in the cuprates and of the Mott insulator to metal transition in Mott insulators, and of spin, charge and orbital order in several metal oxide systems.

Superconductivity is one classic example of the effect of electron–electron and electron–phonon interactions inducing surprising physical phenomena in the solid state, such as zero resistivity and the expulsion of magnetic field lines known as the Meissner effect. In fact, while a good understanding of type I superconductors has been achieved through the Bardeen–Cooper–Schrieffer (BCS) model,^612,613 a complete physical understanding of type II, high-$Tc$⁠, or unconventional superconductivity, occurring at temperatures where the BCS theory would preclude superconductivity, has still not been reached yet.^614–616 Given the intimate link between superconductivity and charge carrier density, it is not surprising that control of the superconducting state through electrostatic doping was attempted early, with the first reports dating back to the 1960s demonstrating small modulations in the superconducting critical temperature in metals, of the order of 0.1–1 mK.^617–619 In addition to offering one knob for controlling the physics governing the electron pairing mechanism, and a perspective for a better understanding of superconducting phenomena by avoiding compositional structural modifications that come with chemical doping,^3,620 electrostatic control of superconductivity is also of potential interest for device applications, such as superconducting FETs, where in principle, switching the system between the superconducting and normal states could result in very high on/off resistance ratios.^621–625 From the BCS expression for the critical temperature,^{612,613,626,627}

where $kB$ is the Boltzmann constant, ω and V₀ are the parameters of the BCS effective interaction (with $ℏω/kB$ of the order of the Debye temperature), and $N0=g(EF)/2$ is the (free electron) density of electronic levels for a single spin population in the normal metal, it follows that

predicting a proportional increase (decrease) in the critical temperature with electron accumulation (depletion), consistent with earlier observations.^617–619 

In superconductors, in addition to the screening length over which electric fields can extend into the system, two other important characteristic length scales are the London penetration depth, $λL$⁠, over which magnetic fields and electric currents can be present in the superconductor, and the coherence length ξ, over which changes in the order parameter occur. The response to magnetic fields depends strongly on whether the London penetration depth is smaller or larger than the coherence length, ξ, separating superconductors into type I and II, respectively.⁶²⁷ For electric fields, the relevant comparison is between the Thomas-Fermi screening length and the coherence length perpendicular to the surface, $ξ⊥$⁠,^624,628 with stronger field effects expected when $δTF≥ξ⊥$⁠. Since in metals the electron density is high, the Thomas-Fermi screening length and the change in the carrier density within $ξ⊥$ are small, such that the changes in $Tc$ are also correspondingly small. One way to optimize the field effect is to reduce the film thickness down to a minimum where superconductivity is still observed (of the order of $ξ⊥$⁠) and ideally to a value comparable to the Thomas–Fermi screening length. Using such an approach, a gradual transition from the normal to the superconducting state could be observed in a 10 Å thick bismuth film grown on SrTiO₃(001) upon accumulation of charge carriers through electrostatic doping using the substrate as the gate dielectric.⁶²⁹ In this example it was argued that, in addition to the change in the carrier density, the electronic screening (described by V₀) was modulated by the increase in electron density. Reducing the lateral dimensions of the system can also result in larger and new field effects, as illustrated by recent field effect experiments carried out in Al and Ti planar wire structures fabricated on Si substrates, showing an ambipolar modulation of the critical current that is not expected from a conventional modulation in the electron density.^625,630 The results mentioned thus far are for polycrystalline metallic films of type I superconductors, indicating that field effect control of the superconducting state is robust, even if both the size of the effect and the critical temperatures tend to be small in these systems.

To achieve larger perturbations in the superconducting state using field effects, systems with larger screening lengths and shorter perpendicular coherence lengths are the most suited.^631,632 In this context, the discovery of high temperature superconductivity in the cuprates⁶³³ made much larger field effect modulations of superconductivity possible.⁶³² The cuprates consist of electron- or hole-doped layered perovskites with phase diagrams similar to those schematically shown in Fig. 12, exhibiting an antiferromagnetic phase at low doping that competes with the superconducting phase at higher doping, with a critical temperature that peaks at the so-called optimal doping. Between the antiferromagnetic and superconducting region, a pseudo-gap phase emerges for hole doped systems.^590,634,635 These systems offer several advantages for field effect modulation, including much lower carrier densities as compared to metallic superconductors, a much smaller coherence length along the crystalline c direction (superconductivity being essentially two-dimensional, occurring in the CuO₂ planes of the layered structure) and, importantly, much higher critical temperatures of up to 135 K at ambient pressure.⁶³⁶ In addition, the presence of boundaries in the doping phase diagram separating superconducting from normal and antiferromagnetic phases suggests that direct changes in the ground state of the system may be achieved by crossing one such boundary using electrostatic doping.⁶³² Indeed, much larger modulations in superconductivity induced by field effects were reported for films of cuprate superconductors, first by using a gate dielectric to modulate the charge carrier density,^{620,637–645} and later by using ferroelectric gates,^{507,646–654} electrolytes,^655,656 and also ionic doping.⁶⁵⁷ In Table V, we list several examples of systems where the variation in $Tc$ has been obtained through electrostatic doping, together with a figure of merit β that gives a measure of the relative change in $Tc$ for a relative change in carrier density via Eq. (8).

By using dielectric gates to electrostatically control the charge density, large modulations in the critical temperature and/or critical current of high $Tc$ superconductors have been observed. Although the effect is non-persistent, this approach has provided interesting insights into the physics of high-temperature superconductors. For example, using a 7 μm thick Kapton dielectric gate, Fiory et al.⁶³⁷ studied the variation of the effective mass and hole carrier density of YBa₂Cu₃O₇ films, to show that the hole carrier density is constant in temperature (except in a region near $Tc$⁠) and to obtain an estimate of the mass of the Cooper pair, $m*=(5.5±0.5)me$ at T < 70 K.^663,664 The first result showed that the hole carrier density cannot be estimated from the Hall coefficient directly, which instead exhibits a marked variation with temperature in the cuprates.^{507,651,657,660,665} Further, it suggested that the critical temperature of the surface layer (probed by modulating the excitation current) is the same as that of the bulk of the film (30 nm thick). Larger effects could be reached using high dielectric constant oxides, including SrTiO₃ and Ba_xSr_1−xTiO₃, with changes in $Tc$ of a few K. Since the oxygen stoichiometry determines hole doping and $Tc$ in YBa₂Cu₃O_7−x,⁶⁶⁶ an important aspect is whether mobile oxygen vacancies are at the origin of the electric field modulation of superconductivity, a question that was answered in the negative in the study by Frey et al.⁶⁴² by showing no significant differences in the field effect response between YBa₂Cu₃O_7−x and a well oxidized superconductor Bi₂Sr₂CaCu₂O_8+x, ruling out a major contribution from electric field induced oxygen diffusion.⁶⁴³ Dielectric gates have also been employed to control the onset of superconductivity induced by field effects at the LaAlO₃/SrTiO₃ interface.^667–673 In this system, the carrier density could be modulated to map a large region of the temperature-doping space parameter, which exhibits a dome-like shape superconducting state within the normal state and indicating the presence of a quantum critical point.⁶⁶⁷ Rather than a result of a simple change in the sheet carrier density, the main impact has been ascribed to changes in the electron charge mobility as a result of compression or expansion of the electron gas due to band bending.^674,675 An alternative approach to probing the phase space electrostatically is by ionic doping, such as in Bi₂Sr₂CaCu₂O_8+x,⁶⁵⁷ achieved by growing the film on a lime-glass substrate and by changing the carrier density through the diffusion of Na⁺ cations induced by the applied electric field. Here also, the effect of lattice distortions that arise with chemical doping is avoided.

Larger and nonvolatile modulations in $Tc$ are found in ferroelectric gated cuprate superconductors, of up to 30 K in YBa₂Cu₃O_7−x,⁶⁵³ and where, in some instances, switching between the normal metallic and superconducting state could be achieved.^620,645 One strategy to minimize the film thickness has been to employ a suitable buffer layer, such as PrBa₂Cu₃O₇,^649,654 a semiconductor that acts to decouple the superconductor from the SrTiO₃ substrate and adds a first layer of CuO chains and metallicity that are thought to promote superconductivity in the superconducting layer.^676,677 An example is provided in Fig. 13 for BiFeO₃/3 u.c. YBa₂Cu₃O_7−x, showing that the critical temperature can be shifted by 30 K between the two polarization states of BiFeO₃.⁶⁵³ This example also shows that the change in $Tc$ decreases with increasing film thickness, as expected for a field effect and for the screening length of YBa₂Cu₃O_7−x, of about 1 nm.⁶²⁴ One other interesting prospect for ferroelectric field effect devices ensues from the possibility of locally controlling the ferroelectric domain pattern^678,679 to create nanoscale superconducting devices, for example, to create weak links in device structures.⁶⁶⁰ Examples of other systems where ferroelectric control of the superconducting state has been reported include Nb-doped SrTiO₃, where on/off switching of superconductivity has been demonstrated using PZT as a ferroelectric gate.^661,662

The metal–insulator transition refers to a change in the conducting state of the system, between metallic and insulating regimes, driven by a control parameter, such as temperature, pressure, charge doping, magnetic field, or disorder.^{590,680–685} Several physical mechanisms can be responsible for such a transition, including changes in band overlap (Wilson metal–insulator transition), disorder (Anderson metal–insulator transition), charge instabilities (Peierls metal–insulator transition) and electron correlations (Mott metal–insulator transition).⁶⁸⁶ Although we focus here on electric field control of the correlated state, we discuss briefly for completeness the other metal–insulator transition mechanisms. The physics underlying metal-insulator transitions is complex and touches on several fundamental concepts in solid state physics, including the role of electron correlations, disorder, electronic band structure, and underlies important magnetic and transport phenomena such as superconductivity and colossal magnetoresistence.⁵⁹⁰ 

One point of interest in using the metal–insulator transition for logic switches is that of achieving smaller subthreshold swings, $S=dVg/d(log Id)$ (gate voltage V_g needed to reduce the drain current I_d by one decade), to achieve lower switching energies than is currently possible with MOSFETs, where $S≈(1+Cd/Ci)(kBT/q) ln 10$ is essentially limited by the thermal excitation of the charge carriers to $(kBT/q) ln 10≈60$ mV/dec at room temperature (C_d and C_i are the capacitance of the depletion and oxide layers, respectively, T the temperature and q the elementary charge value).^189,687 Hence, one motivation for Mott FET transistors is that by replacing the physical mechanism driving the charge carrier density limited by Boltzmann statistics to a faster and more abrupt process based on a phase transition, one may achieve smaller subthreshold swings.⁶⁸⁷ In this context, it is worth mentioning a new device concept based on “negative capacitance” that seeks to reduce the $m=1+Cd/Ci$ factor by designing the gate structure such that an effective negative value for C_d is obtained.^688–690 

One mechanism for the onset of the metal–insulator transition is due to band overlap (Wilson metal–insulator transition), such as in divalent metals where metallicity is due to overlap between the first and second zones and where an insulating state is expect to set in when the atomic distance is increased to the point where reduced orbital overlap leads to electron localization and conduction becomes possible only through hopping.^683,691 Transitions of this kind occur in several materials under high pressure, where the reduced lattice parameter (larger orbital overlap) leads to band overlap and to the onset of an insulator to metal transition, such as in iodine,^692–695 selenium,^693,694 and oxygen.^696–699 A more familiar example of (electrostatic) control of band overlap in band insulators is in semiconductors, such as in the MOSFET transistor, where band bending under a gate voltage changes the channel layer from a back-to-back diode junction to a conducting inversion layer, where electron confinement gives rise to a two-dimensional electron gas.^189,700 Another example of control of the conduction state in band insulators is that of the LaAlO₃/SrTiO₃(001) system, where at a critical LaAlO₃ thickness of about 4 u.c. the system transitions from insulator to metallic by the onset of a 2D electron gas state that sets in at the interface⁷⁰¹ (which can be further driven into a superconducting state by gating).^667–673 (Various mechanisms have been advanced for explaining the onset of the 2D electron gas state at the interface, including band bending, polar catastrophe, interfacial strain effects, and oxygen vacancies.)⁷⁰² Field effect devices employing SrTiO₃^{687,703–707} and KTaO₃^708,709 as the channel layer have also been shown to display large changes in resistivity as a function of applied gate voltage, with on/off current ratios of up to 10⁵. In one instance, a transition from metallic to insulator state in n-doped SrTiO₃ with applied gate voltage has been observed, where a CaHfO₃ layer is used as the gate insulator (epitaxial at the interface but amorphous above 4 unit cells thickness).⁷⁰⁶ This effect is observed as a change in the temperature variation of the channel conductance, from a resistive to a conductive regime, at a bias voltage of 1.3 V; the conduction mechanism is explained in terms of thermal carrier excitation with an activation energy barrier that is found to collapse (i.e., drop to zero) for bias voltages above 1.3 V, leading to a metallic state (the conductivity in the insulating state being attributed to the presence of in-gap states). Interestingly, from a scaling analysis of the resistivity data, it was concluded that the system behaves as three dimensional.

Another example that could be described as a Wilson insulator to metal transition is found to occur at the PZT/LaNiO₃ interface, where changes in the conductivity of LaNiO₃, a metallic oxide, are found to be much larger than expected based on the charge carrier modulation in the thin LaNiO₃ channel layer (3–4 unit cells).⁷¹⁰ The results of ab initio calculations, shown in Fig. 14, suggest that the PbO interface layer of PZT in contact with the TiO₂ layer of LaNiO₃ is subject to an upward band shift when going from the depletion to the accumulation state that makes it cross the Fermi level, leading to the appearance of a conductive PbO state and to a new conducting channel at the interface.

The presence of disorder, such as atomic disorder, can also lead to the onset of an insulating state (Anderson metal-insulator transition), whereby deviations from the periodic lattice potential result in localization of the electronic states near the Fermi energy and to charge transport through hopping between states separated by varying energy barriers.^680,711 In weakly disordered Anderson insulators, a metallic state can be induced by increasing the charge density such that the Fermi energy rises to above the localized states to reach the metallic states (at the so-called mobility edge)^681,711,712 by electrostatic doping, for example, a process that has been extensively investigated in disordered semiconductors.⁷¹¹ Disorder-induced localization effects can also play an important role in the transport properties of correlated oxides, for example due to the short-range disorder associated with the atomic substitution in mixed valent materials, such as La and Sr site disorder in La_0.5Sr_0.5MnO₃, which has been associated with the onset of a disorder-induced insulating state in thin films at low temperatures.⁷¹³ 

Charge instabilities induced by electron–phonon interactions in low dimensional systems can lead to changes in the atomic and band structure and also result in a transition from a conducting to an insulating state (Peierls metal-insulator transition).^687,714,715 Peierls insulators are typically strongly anisotropic materials characterized by exotic charge transport properties, in that charge can be transported by the charge density wave above an applied voltage threshold.⁷¹⁵ To our knowledge, electrostatic control of the Peierls metal-insulator transition has not been reported in the literature.

In contrast to early expectations of electron band theory, many systems containing an odd number of charge carriers per atom are insulators, such as NiO, a behavior that was explained in terms of the role played by electron correlations (electron–electron Coulomb interaction), which favors electron localization (singly occupied electron states) and a gap opening in an otherwise half-filled electron band, resulting an insulating state (Mott insulators).^590,683 In fact, until recently it was thought that the failure of spin dependent local density functional theory (DFT) to correctly predict the opening of band gaps in many complex materials was associated with the shortcomings in accounting for the electron correlation energy in a mean field approximation; however, recent results indicate that the opening of band gaps in DFT ensues once the early constraints imposed to the simulated system (such as cell size, atomic relaxation, breaks in symmetry) are relaxed, indicating that the exchange functionals used are sufficient to describe the band structure of complex oxides.^716–719 Nevertheless, a common approach to describe the effect of electron correlations in the opening of such a bandgap in systems with half-filled bands has been to use the Hubbard model Hamiltonian, where two competing terms, one favoring electron hopping and another favoring localization, induce a gap opening in the density of states with increasing strength of the intra-site energy U. In this picture, two limit cases are distinguished for metal oxides:^590,720 in one case, typically for light 3d transition metals, the oxygen derived p band lies well below the 3d band and the effect of electron correlations is that of splitting the original 3d band into a fully occupied lower Hubbard band and an empty upper Hubbard band, separated by an energy gap of the order of the intra-site energy U (Mott-Hubbard insulators). A second case occurs in oxides with strong p-d hybridization, such as in the late transition metal oxides, where the wavefunction overlap between cations and the oxygen p-orbitals is strong and U is larger than the energy separation between the d and p bands; the effect of electron correlations in such case is to split the d-band into a filled lower Hubbard band below the (filled) p band and an empty upper Hubbard band separated from the p-derived band by an energy gap Δ ( charge transfer insulators). This division underlines the fact that in Mott-Hubbard insulators charge excitation occurs through direct hopping between cation sites while in the charge transfer insulators, charge excitation occurs primarily through the oxygen anions.⁵⁹⁰ In Table VI, we list the electronic band parameters of several oxide materials following this picture, mostly derived from x-ray spectroscopy measurements, running from the early to the later transition 3d metals, showing a gradual decrease in Δ and an increase in U as one goes down the table (increasing 3d atomic number).

At constant temperature, two approaches can be envisaged for inducing an insulator to metal transition in the Mott insulators, namely, through bandwidth modulation or through band filling.⁵⁹⁰ Bandwidth modulation can be achieved by changing the atomic spacing to control the orbital overlap through mechanical or chemical pressure (by substituting isovalent cations within the compound) and thereby affect the strength of electron correlations; in the doped manganites, it can be controlled by applying a magnetic field, where electron hoping is then modified through the double exchange mechanism,^741,742 a process underlying the so-called “colossal” magnetoresistance (CMR) effect.^743–747 Band filling consists in directly modulating the charge carrier density, for example, through chemical, electrostatic, or ionic doping. In the case of chemical doping, it is found that the insulating state is stable against relatively large doping contents, an effect that is ascribed to carrier localization induced by static random potential and/or electron-lattice interaction;⁵⁹⁰ doping levels of the order of 0.1%–10% are typically required for a transition into a metallic state to occur. This implies that large charge carrier modulations are required to bring a Mott insulator to the metallic state. Doping by chemical substitution has the disadvantage of being irreversible and of introducing local distortions in the atomic lattice that can further perturb the electronic state. An alternative approach is to use field effects to modulate the carrier density, which is simultaneously reversible and does not introduce local distortions to the lattice. In this approach to controlling the Mott metal insulator transition, the channel layer consists of a Mott insulator chosen with a composition on the phase diagram close to the insulator to metal transition. For maximum effect, the channel layer is made as thin as possible, ideally of the order of the Thomas–Fermi screening length of the metallic phase. One important consideration here is that, due to the epitaxial strain or other interface effects, the phase diagram of the Mott insulator may deviate strongly from that of the bulk. One example is La_1−xSr_xMnO₃, where epitaxial strain strongly affects the transport and magnetic properties in ultrathin films.^748,749

One motivation for the electrostatic control of the Mott transition has been as an alternative to CMOS field effect transistors, where the main advantages include high on/off resistance ratios (expected at the metal–insulator transition), good scaling characteristics, and the inherent fast dynamics (since the process is purely electronically driven).^750–753 Prototype Mott transition field effect transistors based on cuprates and using SrTiO₃ as the gate dielectric were reported by Newns et al.,^750,754 where control of the conductivity of the channel layer with gate voltage and a quasi-linear variation of the drain current with the drain–source voltage were demonstrated. Unlike in superconducting FETs, the aim here is to control the metal–insulator transition and the measurements were carried out at room temperature. In another example, electrostatic doping of Sm₂CuO₄ was achieved by using the band bending characteristics when in contact with a Nb-doped SrTiO₃ interface,⁷⁵⁵ an approach that has been extended to other materials systems.⁷⁵⁶ In addition, control of the resistive state of Mott insulators using ionic liquids as the gate dielectric has been reported in the manganites, including control of the electronic state⁷⁴² and of the conduction percolation path.⁷⁵⁷ 

A large number of studies has been carried out on ferroelectric Mott devices based on several channel systems, including cuprates,^758,759 manganites,^{603,760–766} nickelates,^710,767,768 and cobaltates.⁷⁶⁴ Several aspects need consideration. A first aspect is that the resistivity of the doped Mott insulator near the insulator to metallic transition is usually far lower than that of the undoped system. For example, for La_1−xSr_xMnO₃, the 0 K resistivity for x = 0.175, in the metallic regime, is $1.0×10−3$ Ω cm, while for x = 0.15, in the insulating state, it is $2.0×103$ Ω cm, which in turn is orders of magnitude lower than that for x = 0.⁷⁶⁹ As a consequence, screening lengths will be small, requiring small channel layer thicknesses for effective field effect modulation. However, control of the electronic properties of strongly correlated oxides at small thicknesses is challenging, since their properties tend to deviate strongly from the bulk for ultrathin films due to their sensitivity to strain and defects.^749,767 Another aspect relates to changes in other properties accompanying the metal-insulator transition, such as changes in the magnetic and orbital state and phase segregation as a result of competing ground states.^{593,770–772} For example, early ferroelectric field effect transistors employed thick manganite layers, 30–50 nm thick,^603,773 and the large changes in resistivity found, up to 300%, were interpreted within a percolative phase separation picture.⁷⁷³ In another study, on 8.2 nm La_1−xCa_xMnO₃ films using SrTiO₃ as a gate dielectric, an ambipolar effect with the applied electric field was observed,⁷⁷⁴ an effect that was attributed to the development of a pseudogap in the density of states, providing another illustration of the complex behavior of this class of complex oxide materials.

Since large changes in the charge carrier density are required to control Mott insulators, it is advantageous to employ ferroelectric gates with large polarizations. Promising in this regard is BiFeO₃ in its supertetragonal phase, characterized by very large spontaneous ferroelectric polarizations in excess of 100 μC/cm² at room temperature,^239,372,439 and which has been employed for the control of the electric conductivity of CaMnO₃ and Ce-doped CaMnO₃.^765,766 For CaMnO₃, large changes in conductivity are obtained, of 400% at room temperature and a tenfold change at 200 K, although the system remains in the insulating state for the film thicknesses investigated, in the range from 6 to 40 unit cells.⁷⁶⁵ With Ce-doping of 0.04, a metal to insulator transition is observed at around 100 K.⁷⁶⁵ However, ferroelectric field modulation of the resistivity was found to be much less efficient in this system, which was attributed to incomplete ferroelectric polarization reversal at the interface, as determined from local measurements of the ferroelectric polarization with HR-STEM.⁷⁶⁶ 

Control of the conductivity of the rare-earth perovskite nickelates has also generated interest. These materials have electrical properties that depend sensitively on the lattice structure and which are strongly affected by the overlap of Ni 3d and O 2p orbitals, as determined by the Ni–O bond length and Ni-O-Ni bond angle.^775,776 Due to the strong electron-lattice coupling, one route to controlling the electrical properties of the nickelates is by modulating the Ni–O–Ni bond angle using strain⁷⁷⁷ and the surface termination.^778,779 However, such approaches are limited to static control of the Ni–O–Ni bond angle. In this context, a recent report on the ferroelectric PbTiO₃/LaNiO₃ interface suggested a novel way to achieve active control of the electrical properties of the nickelates via interface structure modulation.⁷⁸⁰ In this case, switching the ferroelectric polarization of the PbTiO₃ layer modifies the Ni–O bond length and Ni–O–Ni bond angles at the interfacial LaNiO₃ layer through ferroelectrically induced structural distortions; the latter contribute to the conductivity change of the LaNiO₃ channel by a modification of the mobility, unlike in the canonical field effect.⁷¹⁰ The result shows that the ferroelectric interface structure modulation can provide a novel way to controlling functionality in oxide electronics.

Strategies for optimizing the changes in the conduction state of Mott insulators have been proposed, such as by combining different materials in bilayer stacks in order to better engineer the interface charge modulation. For example, it is observed that changes in the conductivity of Sm_0.5Nd_0.5NiO₃, which exhibits a metal to insulator transition as a thin film grown on LaAlO₃,^767,768 can be enhanced from a 10% to a 300% effect when a thin LSMO layer is added at the substrate interface.⁷⁶⁸ In that study, x-ray spectroscopy measurements supported by ab initio calculations showed that between the interfacial Mn and Ni layers a charge transfer from Mn to Ni of about 0.1 electron per 2D unit cell takes place, which has the effect of making the system more susceptible to the ferroelectric charge modulation.

In the examples discussed above, large changes in the resistivity of the doped Mott insulators are observed, much higher than expected based on the simple modulation of the charge carrier density, a signature of the role of electron correlations; however, the state of the system tends to remain either insulating or conducting, an indication of the difficulty in controlling the Mott insulator to metallic transition. However, for optimized device structures, a genuine control of the conduction state can be achieved. An example is shown in Fig. 15 for PZT/La_0.8Sr_0.2MnO₃, where the La_0.8Sr_0.2MnO₃ film thickness is set at the onset of conductivity (3.8 nm or 10 u.c.) and a change between insulating and conducting states is observed as the system is switched between depletion and accumulation states, respectively.⁷⁶¹ 

Orbital order refers to the cooperative onset of a preferential occupation of different orbital states which otherwise are degenerate in energy.^{683,781–786} When those states carry no orbital moment, such as the e_g states, the degeneracy can be lifted by the Jahn-Teller effect (which posits that orbital degeneracy makes the structure of the molecular complex unstable against distortions),⁷⁸⁷ while in the case of states carrying orbital moment, such as the $t2g$ states, the degeneracy can alternatively be lifted by the spin–orbit coupling.⁷⁸¹ Long-range orbital order arises from a preferential spatial overlap of neighboring orbitals and is dictated by the sign of the exchange integrals, similar to the situation with the spin Hamiltonian, such that an orbital Hamiltonian can be written in terms of orbital pseudo-spins.^781–783 Hence, orbital order corresponds to an additional fundamental degree of freedom in the solid state, with its fundamental collective excitation modes,⁷⁸⁸ and whose control constitutes another knob to controlling the electronic state of the system, including the magnetic state to which it is often intimately linked. One example is the orbital arrangement in LaMnO₃, where the single electron at the e_g level occupies alternately $d3z2−r2$ or $dx2−y2$ orbitals to form a complex ordered orbital structure that is also linked to the antiferromagnetic spin structure.⁷⁸⁸ Also important in the context of the orbital polarization is the role of strain, and of epitaxial strain in particular, since distortions in the crystal structure can profoundly modify the local crystal field and the position of the atomic energy levels, and therefore, the orbital arrangement.

Although orbital order has been identified in many systems, its control remains challenging. A first hurdle relates to probing orbital order, which typically requires techniques that are sensitive to local orbital arrangement, such as resonant x-ray scattering^788,789 or x-ray absorption spectroscopy using the x-ray linear dichroic effect,⁷⁹⁰ although it can also be probed indirectly, such as with polarized visible light.⁷⁹¹ For example, control of the orbital domains in La_1/2Sr_3/2MnO₄ using electric current has been reported, where Joule heating results in melting of charge-orbital ordered domains and the subsequent reduction of the current favoring one particular domain over the other.⁷⁹¹ Electric-field control of the orbital order has been investigated theoretically using first principles calculations in perovskite vanadate⁷⁹² and PbTiO₃/LaTiO₃ superlattices.⁷⁹³ In these instances, the polar distortion (induced in the case of the vanadates by a trilinear coupling with antiferrodistortive motions and/or Jahn–Teller distortions) couples with the non-polar ordered Jahn–Teller distortion modes (leading to orbital order), which in turn couple to the spin configuration via the superexchange interaction.^792,793 In principle, such coupling makes electric field control of orbital and spin ordering in those systems possible.

A related concept, particularly relevant for thin films and interfaces, is that of orbital polarization, which refers to a preferential orbital occupation in the system.²⁷ For example, epitaxial strain in thin films can induce a modification in the energy position of the e_g orbital states in the manganites and a change in the relative orbital occupancy, i.e., in orbital polarization.^{749,794–796} Control of orbital polarization at the interface between ferroelectrics and complex oxides has been reported for LSMO/BiFeO₃,^797,798 LSMO/BaTiO₃,^160,799 and LSMO/PZT.⁸⁰⁰ For the case of LSMO/BiFeO₃,^797,798 a modification of the orbital occupancy of the 3d states at the interface was inferred from x-ray linear dichroic effect measurements due to hybridization between the Mn and Fe orbitals and leading to the presence of a large interfacial magnetic moment in BiFeO₃, exchange-coupled to the La_0.8Sr_0.2MnO₃ magnetization; however, in these studies, the effect of switching the ferroelectric polarization of BiFeO₃ was not addressed, nor the extent of the modified orbital polarization. These latter aspects were studied by Chen et al.⁷⁹⁹ and Preziosi et al.⁸⁰⁰ for La_0.8Sr_0.2MnO₃/BaTiO₃ and La_0.825Sr_0.175MnO₃/PZT, respectively. In the first study, the change in orbital polarization was investigated by first principles calculations, predicting the occurrence of a large change in orbital polarization when going from the accumulation state (which favors occupation of the $dx2−y2$ states) to the depletion state (which favors $d3z2−r2$ states) over a region extending to 3 unit cells from the ferroelectric interface [Fig. 16(a)]. The calculations also predict that the orbital polarization depends only modestly on the ferroelectric polarization amplitude and that the main effect results from a displacement in the atomic position of the Mn cations induced by the ferroelectric polar displacements [Figs. 16(b) and 16(c)], a result that was confirmed experimentally by TEM measurements of the same system [Fig. 16(d)].⁷⁹⁹ Hence, the change in orbital occupation is determined by distortions in the Mn octahedron that, in turn, modulate the energy position of the two e_g bonding states. These results were corroborated by x-ray linear dichroism measurements reported for La_0.825Sr_0.175MnO₃/PZT, where a change in orbital polarization of 1.5% averaged over the whole LSMO film is observed (but expected to be significantly higher in the interface region).⁸⁰⁰ Multiplet calculations suggest as well an inversion on the order of the $dx2−y2$ and $d3z2−r2$ states when switching the ferroelectric polarization (where the latter are favored for the depletion state), pointing also here to a modulation in the Mn octahedron structure (see Fig. 17). Changes in the orbital moment of Mn in La_0.67Sr_0.33MnO₃/BaTiO₃ structures where the direction of the ferroelectric polarization was pre-set by controlling the substrate termination, were also observed and quantified by using XMCD sum rules^801–805 at the Mn L_2,3-edge, providing another demonstration of ferroelectric control of the orbital population.¹⁶⁰ 

In systems with mixed cation valency, a reduction in Coulombic energy can be achieved at half or 1/8 doping through ordered charge localization (charge-ordered state), manifested by an electrically insulating state below a critical temperature. The paradigmatic system where charge ordering was first described is magnetite (Verwey transition),^806–810 while more recently, charge ordering has been reported in another ferrite system, LuFe₂O₄,⁸¹¹ in the doped manganites such as La_1−xCa_xMnO₃,^812–815 PrCaMnO₃,^816–818 and BiSrCaMnO₃,^819–821 and in the cuprates, where charge order is thought to be a ubiquitous phenomenon (here most often seen as ordered charge segregation).^615,822 Probing and establishing charge order in a system is, in general, challenging; early work on the manganites cited above relied on neutron scattering measurements, which are not directly sensitive to charge, but rather to lattice modulations induced by doping,⁸²³ while interpretation of resonant x-ray scattering measurements, which are in principle highly sensitive to charge order, have not been without controversy^824,825 and still largely rely on models to interpret the experimental results.⁸²⁶ For example, a pure ionic picture for charge ordering in the manganites would imply that the x-ray absorption spectrum should consist of a superposition of Mn³⁺ and Mn⁴⁺, while the observed spectrum is found to correspond to that of a mixed valency state;^827–832 instead, x-ray spectroscopy and neutron scattering measurements suggest the presence of two Mn sites with different magnetic moments and a difference in valency (charge disproportionation) of ∼0.2 electrons,^{825,826,828,830,833,834} much less than the expected difference of 1 electron between Mn³⁺ and Mn⁴⁺. A similarly complex picture has emerged for magnetite, where the initial ionic model for charge ordering proposed by Verwey^806–809 was found not to be consistent with the results of detailed x-ray diffraction and x-ray resonant scattering measurements.^835–837 In fact, our understanding of magnetite and of the Verwey transition has improved significantly over the last decade.^{835,836,838–841} The current understanding is that, at the Verwey temperature, a structural transition from cubic to monoclinic occurs, characterized by a complex lattice distortion that leads to an increase by $2×2×2$ of the high temperature cubic unit cell together with a charge localization process whereby the double-exchange active octahedral Fe cations cluster in a superstructure of trimeron complexes with charge and orbital order.^835,836,841 The iron valence state varies by 0.6 electrons from +2.35 to +2.95 in contrast to a mixed 3+ and 2+ valence state. This variation is for octahedral sites belonging to the trimeron complex.⁸³⁵ 

Given the challenge of probing charge order directly, it is perhaps not surprising that only a few studies have attempted to control charge order using external excitations. Examples include the ultrafast excitation of La_1−xCa_xMnO₃ thin films using high intensity 100 fs laser pulses, where the superlattice reflection associated with orbital/charge order was used to follow the temporal evolution of the system after the excitation;⁸⁴² under such strong excitation conditions, melting of the orbital/charge order is found to occur on a timescale faster than 200 fs. Another example is the report of a large birefringence effect that is ascribed to charge order in magnetite, controlled by cooling the system under an applied magnetic field.⁸⁴³ In this study, an intense ultra-fast laser pulse was used to achieve a rapid heating of the system to above the Verwey temperature, with the magnetic field setting a preferential orientation for the monoclinic c axis,⁸⁴⁴ and consequently, of charge order; the fast time response of polarization rotation angle was also used to study the dynamics of charge-order melting, found to be faster than 0.8 ps.⁸⁴³ 

Electric field control of magnetism in the solid state has generated much attention recently,^37,845–855 motivated by the interest in understanding better the physical processes that lead to a coupling between spin and charge degrees of freedom and also by its potential for applications in next generation electronic devices.^{344,856–858} In magnetoelectric and in multiferroic materials with magnetic and ferroelectric order, such coupling is intrinsic to the system and intimately linked to the specific origin of ferroelectric polarization.^{33,36,37,40,845,847,852,859–861} Although exciting materials to study from a basic science level, the potential for applications is severely limited by their scarcity in nature (a consequence of the required time reversal and inversion broken symmetries together with dielectric behavior for ferroelectricity)^{32,33,862,863} and the generally small value of the effective magnetoelectric coupling or critical temperatures, with BiFeO₃ being one outstanding exception.^{365,369,864–866} In order to circumvent such limitations, an alternative route to achieving a strong coupling between magnetic and ferroelectric order parameters consists in designing artificial multiferroic systems obtained by interfacing magnetic with ferroelectric materials.^{846,848,850,854,867–869} To reach large magnetoelectric couplings, both the interface characteristics and the coupling mechanism need to be engineered judiciously. Several interfacial magnetoelectric coupling mechanisms have been devised according to the main interaction process exploited: strain, spin exchange, or charge.^848,850 Strain-mediated coupling relies on the elastic coupling to modify the magnetic state via magnetoelastic and piezoelectric effects,^870–873 while spin exchange-mediated magnetoelectric coupling relies on magnetic exchange interactions, for example, between a ferromagnet and an antiferromagnetic multiferroic via the exchange-bias effect.^{343,345,348,349,848,874–877} More directly related to field effects is the charge-mediated magnetoelectric coupling, whereby the magnetic state of the system is modified by changing the charge carrier density by using charge screening at the ferroelectric boundary to accumulate or deplete charge carriers at the metallic interface. We consider next materials whose charge screening length lie within three distinct regimes: (i) metallic ferromagnets, characterized by very short Thomas-Fermi screening lengths due to the very large electron charge density at the Fermi level; (ii) 3d correlated systems, with screening lengths and electron mean free paths of the order of the unit cell, such as mixed valence 3d transition metal oxides; and (iii) magnetic semiconductors, where charge screening occurs over the characteristic Debye length $LD=ϵsϵ0kBT/(q2n)$⁠,¹⁸⁹ characterized by extended charge screening lengths (⁠$ϵs$ is the static dielectric constant, q the carrier charge, and n the charge carrier density), typically in the nanometer range (ca. 2–20 nm for carrier densities in the range from 10¹⁸ to 10¹⁶ cm^–3 at room temperature).

In metallic ferromagnets, such as the 3d transition metals, the exchange interaction responsible for magnetic order originates from the Pauli exclusion principle and the condition for antisymmetry of the electronic wave function, which directly links the exchange integral term (electron–electron Coulomb energy) to the spin state of the system.^626,878 The Pauli exclusion principle effectively acts to increase electron–electron separation, reducing the Coulomb energy but at a cost of kinetic energy; hence, only in systems where the density of states at the Fermi energy is high it becomes advantageous for a spin-polarized state to emerge, such as in late 3d transition metal elements. The high charge carrier density together with quasi-free electron behavior leads to extremely short Thomas-Fermi screening lengths, of below 1 Å. As a consequence, electrostatic control of magnetism, such as of the exchange interaction or the magnetic moment, which entails a shift in the relative occupancy of the spin up and down subbands,⁸⁷⁹ may be expected to be modest except for the thinnest films. Such modifications in the interfacial magnetic moment have been estimated for bcc Fe(001), Ni(001), and hcp Co(0001) when subjected to a static electric field applied across a vacuum⁸⁸⁰ or dielectric barrier.^881,882 However, modulation of the charge carrier density at the interface can modify strongly the surface magnetic anisotropy via spin–orbit coupling, as a result of a change in the relative electron occupancy of the interfacial orbital states.^{880,882–888} The impact of the surface magnetocrystalline anisotropy can be very strong in 3d transition metal ferromagnetic films and nanoparticles, often dominating the magnetic energy.^889–892 Effectively, the control of the surface magnetic anisotropy through charge screening has been demonstrated recently in a number of systems, including Fe,⁸⁸⁴ FePt,⁸⁹³ Co₄₀Fe₄₀B₂₀,^894–896 Co,^897,898 Ni,⁸⁹⁹ Fe_0.5Pd_0.5,⁹⁰⁰ Fe₈₀Co₂₀,^901–903 and in Fe₈₀Co₂₀/MgO/Fe^904,905 and Fe₄₀Co₄₀B₂₀/MgO/Fe₄₀Co₄₀B₂₀ tunnel junctions.^906,907 A motivation in this context is the electric field control of the relative magnetization alignment of ferromagnetic layers in spin-valve device structures and of the electrical resistivity through the giant (GMR) or tunneling magnetoresistance (TMR) effects for applications in nonvolatile random access memories (MRAM) and logic switches.^344,858,908

Additionally, chemical bonding (or hybridization) at the interface can also play an important role. For instance, density functional theory calculations for Fe/BaTiO₃(001) show a large surface magnetoelectric response whose origin is largely attributed to changes in the chemical bonding at the interface, i.e., to modulations in the Ti 3d, Fe 3d and O 2p orbital overlap upon reversal of the ferroelectric polarization direction due to the displacement of the Ti atoms, which additionally become magnetically polarized.^909–914 Similar effects have been calculated for Co/PZT,⁹¹⁵ and also for oxides such as LSMO/BaTiO₃,^160,799 suggesting that such a process may be quite general at ferroelectric interfaces. In fact, magnetic polarization of the Ti interface layer in contact with ferromagnetic systems has been observed in several systems, including Fe/BaTiO₃(001),⁹¹⁶ LSMO/BaTiO₃,^160,917 and Co/PZT,⁹¹⁸ turning the interface region of the ferroelectric also ferromagnetic, i.e., multiferroic.

In the 3d transition metal oxides, the magnetic exchange interaction between cations is indirect, mediated by the oxygen anions, either through virtual excitations of spin up-down pairs to neighboring cations in the case of superexchange interaction (which tends to favor antiferromagnetic coupling)⁹¹⁹ or through electron hopping in the case of double exchange interaction, which favors ferromagnetic coupling.^{741,920–922} Although indirect, magnetic interactions in many oxides can be strong and result in high magnetic critical temperatures, for example, up to 858 K for the ferrimagnetic spinels NiFe₂O₄ and Fe₃O₄.^840,923,924 In comparison to magnetic metals and magnetic semiconductors, complex oxides tend to exhibit stronger electron–electron correlations and a more diverse electronic behavior as a function of charge carrier density, where magnetism competes or co-exists with other types of order, such as orbital order or superconductivity, leading to complex phase diagrams.^{590,593,784,925–927} Typically, the effect of doping in the 3d metal oxides, for example via chemical substitution, is that of modifying the average valence state of the 3d cations and modifying the charge carrier density. The main approach to electrostatic control of magnetism in such mixed-valence systems is to choose a chemical doping that places the system at a phase boundary between two magnetic ground states of the system and to drive it across the boundary by electrostatic modulation of the charge carrier density. Also here it is important to recall that the phase diagram of thin films can deviate significantly from that of the bulk material, for example, due to epitaxial strain, as illustrated by the changes in the magnetic and conducting state of La_1−xSr_xMnO₃ films near 0.5 doping with strain shown in Fig. 18.^748,749

Among the mixed valence materials, 3d metal oxides crystallizing in the perovskite structure have generated notable interest due to their diverse and important chemical and physical properties.^928–931 Within this class of materials, the manganites have been intensively investigated,^932,933 particularly since the observation of the CMR effect,^743–747 effectively, a magnetic field-driven insulator to metal transition. The phase diagrams of the manganites are characterized by several phase transitions as a function of chemical doping, for example, for La_1−xSr_xMnO₃, from an insulating orbital ordered antiferromagnetic (x = 0), to insulating ferromagnetic (⁠$0<x<0.2$⁠), metallic ferromagnetic (⁠$0.2<x<0.5$⁠), metallic antiferromagnetic (x = 0.5) to antiferromagnetic insulating (x > 0.5).^769,934,935 In many manganite perovskites, the equilibrium phases near the boundaries are quasi-degenerate in energy and coexist,^{593,770–772} suggesting that the equilibrium state of the system can be tilted toward one or the other state with only modest perturbations. The charge carrier density at the boundaries separating such phase transitions is high, typically in the range of 10¹⁹–10²² cm^–3, requiring a correspondingly large charge carrier modulation for controlling the equilibrium state.³ In addition, the system should be as thin as possible for maximal field effect, ideally, of the order of the charge screening length. Experimentally, it is known that the magnetic and electrical properties of thin films of the manganites grown on SrTiO₃ are strongly depressed below a thickness of 1–3 nm,^{862,936–944} while epitaxial strain is also known to strongly affect the magnetic and electrical properties (Fig. 18).^748,749 Hence, the strategy for modulating the magnetic properties of the mixed valence manganites using the ferroelectric field effect has been to setting the thickness of the manganite layer at the transition between insulator to metallic state; for example, for La_1−xSr_xMnO₃/SrTiO₃(001) at x = 0.2 and x = 0.3, this occurs at a film thickness of about 10 unit cells.^760,940,945 Using this approach, demonstration of electric field control of the magnetic properties of the doped manganites in ferroelectric field effect structures could be achieved, including modulations in the critical temperature,^760,761 anisotropic magnetoresistance⁵⁹⁶ and in the magnetic polarization.⁹⁴⁶ 

An important breakthrough was the experimental report of a large modulation of the coercivity, magnetic critical temperature, and magnetic moment of a 4 nm La_0.8Sr_0.2MnO₃/PZT heterostructure as a function of the applied electric field measured by magneto-optic Kerr effect (MOKE) magnetometry, demonstrating the presence of a large magnetoelectric coupling in this artificial multiferroic system that was both hysteretic and nonvolatile.³¹⁷ The results are in qualitative agreement with what is expected from the bulk phase diagram of La_1−xSr_xMnO₃,^769,934,935 including a higher critical temperature and lower magnetic moment with increasing doping. Subsequent x-ray absorption near-edge spectroscopy (XANES) measurements showed that switching the ferroelectric polarization resulted in a shift in the absorption edge of the Mn K-edge by about 0.3 eV, interpreted as a change in the valence state of the interfacial Mn layer at the PZT interface when switching between the accumulation and depletion states.^315,945 Such measurements demonstrated the electronic origin of the magnetoelectric effect in PZT/LSMO and allowed a quantitative comparison of the changes in the magnetic moment expected from a simple modulation of the Mn valence state (of 0.1 $μB$/Mn) and that measured experimentally (0.76 $μB$/Mn), showing that an additional change in the spin configuration at the interface takes place, from a ferromagnetic coupling in the depletion state to an antiferromagnetic coupling in the accumulation state, and attributed to a change in the relative weights of the superexchange and double exchange energy terms as a function of the 3dz² orbital occupancy induced by screening. The presence of such a magnetic reconstruction at the interface is supported by the results of first principles calculations.^947–953 Also significant, the hole accumulation state shows a much higher critical temperature than the depletion state (by about 20 K), such that over a large temperature window on/off switching of magnetism driven by the orientation of the ferroelectric polarization is realized.^954,955 Subsequent studies reported similar modulations of the magnetization in La_1−xSr_xMnO₃-ferroelectric heterostructures at various doping levels and employing different ferroelectric systems, examples of which are summarized in Table VII, showing that the magnetoelectric effect is not very sensitive to the exact amplitude of the ferroelectric polarization and that charge modulations in the range between 25 and 100 μC/cm² at the LSMO interface are sufficient to modify the interfacial magnetic state. However, a number of discrepant results for the magnetic and electronic behavior has been observed, such as the case for BaTiO₃/LSMO/SrTiO₃(001) at x = 0.3 doping, where an opposite change in the saturation moment with the direction of the ferroelectric polarization is observed and no change in the critical temperature,⁹⁵⁶ an effect attributed to a modulation in the relative population of the ferromagnetic metallic and antiferromagnetic insulating phases coexisting in the system. One advantage of La_1−xSr_xMnO₃ at x = 0.3 doping resides in its relatively high bulk Curie temperature, of about 370 K, such that room temperature demonstration of a large change in the magnetic moment would be of high practical interest.