Interfacial strain gradients in oxide epitaxial thin films provide an interesting opportunity to study flexoelectric effects and their potential applications. Oxide epitaxial thin films can exhibit giant and tunable flexoelectric effects, which are six or seven orders of magnitude larger than those in conventional bulk solids. The strain gradient in an oxide epitaxial thin film can generate an electric field above 1 MV m−1 by flexoelectricity, large enough to affect the physical properties of the film. Giant flexoelectric effects on ferroelectric properties are discussed in this overview of recent experimental observations.
Couplings between the electrical and mechanical properties of materials are intriguing physical phenomena and have been exploited in many applications . A simple example of electromechanical coupling is electrostriction. Electrostriction is a quadratic field–strain coupling, and it can induce non-hysteretic electro–strain response. Another example of electromechanical coupling is piezoelectricity in polar systems, in which lattice deformation (or strain) can induce electric fields, and vice versa. Conventional electromechanical coupling, such as the earlier-mentioned electrostriction and piezoelectricity, generally assumes a linear relationship between polarization (or electric field) and homogeneous strain. Particularly, piezoelectricity is limited to 20 crystalline point groups. Even in centrosymmetric systems, an electromechanical coupling, namely flexoelectricity [2–8], can occur in the presence of inhomogeneous strain (i.e. strain gradient). The flexoelectric effect can be understood as the electric field that is generated by a strain gradient. Because strain gradients break inversion symmetry (figure 1), flexoelectricity allows the generation of electric responses from lattice deformations in every dielectric material; it occurs in all 32 crystalline point groups of materials (more than piezoelectricity, which exists in only 20 point groups). The phenomenon is characterized by the tensor relationship 1.1where Pl (El) is the generated polarization (electric field), μijkl is the flexoelectric coefficient (a fourth-rank polar tensor), ε is the dielectric constant, uij is a component of the elastic strain, xk is a position coordinate (related to the direction of the strain gradient), e is the electronic charge, ε0 is the permittivity of free space and a is the lattice parameter.
The strain-gradient-induced electricity in a centrosymmetric crystal was proposed by Kogan in 1964  and experimentally confirmed by Bursian & Zaikovskii in 1968 . This phenomenon was coined ‘flexoelectricity’ by Indenbom et al. in 1981 . On the basis of a simple ionic model, the theory of the flexoelectric effect was proposed by Tagantsev in 1986 [5,6]. Recently, further theoretical studies on flexoelectricity using ab initio calculations have been conducted in some oxides [9,10]. Quite recently, Resta showed theoretically that flexoelectricity may be a purely bulk effect without extrinsic (i.e. surface) contributions . Owing to its intrinsic and universal existence in every dielectric material, flexoelectricity has garnered a wide range of scientific interest and potential applications. Particularly in flexible systems, such as liquid crystals [12,13], low-dimensional crystals (e.g. graphene or carbon nanotubes)  and biological molecular membranes  or hairs, the flexoelectric effect can be quite significant.
2. Flexoelectric effect in solids
In rigid solids, however, there has been little investigation into the flexoelectric effect. One of the major reasons for this is the small magnitude of the flexoelectric effect. The flexoelectric coefficient of ∼e/a is small, and elastic deformation is limited in most bulk solids. Additionally, appropriate control of the strain gradient by applying an external stress to bulk solids is difficult. Ma and Cross investigated flexoelectric effects in ferroelectric perovskite polycrystalline bulk solid [16–20]. They found that strain-gradient-induced dielectric polarization was enhanced for barium strontium titanate. In bulk materials, flexoelectricity has typically been investigated by measuring the induced surface charge in mechanically bent samples (figure 2a,b). In studies using ceramic bulk solids, however, problems can occur due to grain boundaries. In addition to the flexoelectric effect, the grain boundaries can contribute to the measured charge owing to their possibly polar nature or via surface piezoelectricity.
Recently, Zubko et al.  studied the flexoelectric effect in single-crystalline bulk paraelectric strontium titanate using the mechanical bending method (figure 2c). With well-designed experiments, they clearly demonstrated the strain-gradient-induced polarization (i.e. flexoelectric effect) in solid bulk material (figure 2d). However, the strain gradient induced by bending the bulk solid was at most as large as 0.1 m−1. The expected electric field based on the strain-gradient value is just below approximately 1.0 V m−1 (equation (1.1)). Thus, despite recent scientific progress in flexoelectricity, achieving large flexoelectric effects in solids remains a challenge and a bottleneck for continued advancement in this field.
3. Nanoscale strain gradient in epitaxial oxide thin films
(a) Strain engineering
Strain engineering is a conventional method used to study epitaxial thin-film systems and was originally used in semiconductors to enhance device performance. Performance benefits are achieved by modulating strain (or lattice parameters) in the transistor channel, which enhances carrier mobility and thereby conductivity through the channel. Recently, strain engineering has been extended to general materials, including complex oxides, through thin-film epitaxy, resulting in outstanding scientific advances. Using thin-film epitaxy and the misfit strain imposed by an underlying substrate, it is possible to strain thin films by a few per cent—far beyond the point at which bulk materials crack. Such strain is used to enhance the functionalities of thin films and to increase the superconducting, ferromagnetic and ferroelectric transition temperatures .
Recently, thin-film deposition techniques have been advanced, enabling the growth of epitaxial heterostructures with atomically controlled interfaces, such as multi-layers , superlattices [24,25] and ultrathin films [26,27]. Owing to advanced deposition techniques, studies using strain engineering have improved, and it has been shown experimentally and theoretically that such strain can even stabilize systems in novel non-bulk phases [28–32]. Thus, strain and its engineering have provided a viable and active means of exploiting the physical properties of materials.
(b) Strain gradient and its engineering at nanoscale
Strain engineering usually assumes homogeneous strain in films. In reality, however, inhomogeneous strain often occurs in films, where a strain gradient is inevitably present. Catalan et al. [33,34] and Lee et al.  noticed that a large strain gradient could occur in epitaxial thin films. Above a critical thickness of the strained epitaxial thin film, a lattice mismatch between the film and substrate can result in strain relaxation. Strain relaxation usually occurs within a region of tens of nanometres near the film–substrate interface (figure 3a), inducing a large strain gradient. Given that a strain of 1 per cent relaxes within 10 nm, the resultant strain gradient will be 0.01/(10 nm)=106 m−1, which is six or seven orders of magnitude larger than that (approx. 0.1 m−1) induced by mechanical bending of bulk solids. Recently, Lee et al.  measured the strain gradient in oxide epitaxial thin films directly using grazing-incidence in-plane X-ray diffraction and showed that its value was as large as 106 m−1. Also, Chu et al.  and Nagarajan et al.  reported an even larger strain gradient near dislocations; relaxation of strain as high as 5 per cent at dislocations over several nanometres can give rise to strain gradients of 107 m−1. Under the flexoelectric effect, this large strain gradient can also induce an electric field of the order of 1–10 MV m−1, large enough to influence the physical properties of the films.
The strain gradient can be modulated by adjusting the growth conditions of the films. Lee et al.  demonstrated that the strain gradient could be engineered by varying oxygen partial pressure (PO2) conditions (i.e. oxygen content in the films) during film growth; they suggested that crystal volume expansion by oxygen vacancies (VO) could suppress the degree of the strain gradient (figure 3b). According to a general model for the strain profile , independent of the actual relaxation mechanism, the strain u(z) in epitaxial thin films can be expressed as 3.1where u0, u1 and α are constants, t is the thickness of the film and z is the distance from the film surface. This general model equation enables a rough estimation of the strain gradient if the two end values of u are known, i.e. the u values at the interface (z=t) and at the surface (z=0). Usually, the film can be fully strained near the interface; thus, the u value at the interface (z=t) is determined by the misfit strain relative to the substrate (note that, in this case, only the tensile misfit strain (i.e. u≥0) is considered). Near the surface, the misfit strain is usually relaxed, and the u value at the surface (z=0) can converge to zero for sufficiently thick films. That is, the lattice constant a relaxes from asubstrate to abulk (figure 3c). On the other hand, if crystal volume expansion occurs as a result of VO, the u value at the surface (z=0) can converge to a non-zero positive value. That is, a relaxes from asubstrate to an expanded quantity aexpanded (greater than abulk). The amount of VO in the films can be roughly adjusted by varying the PO2 conditions during film growth. Thus, using the simple model equation of strain relaxation and crystal volume expansion, we can explain how the strain gradient can be modulated according to the PO2 conditions. This clearly demonstrates that oxide epitaxial thin films provide an intriguing platform for strain-gradient engineering, with which we can exploit the physical phenomena that occur due to flexoelectricity.
4. Giant flexoelectric effect on ferroelectric properties
(a) Dielectric properties
Catalan et al. [33,34] first emphasized the importance of flexoelectricity in ferroelectric epitaxial thin films. They showed that dielectric properties could be degraded by the flexoelectric effect. To demonstrate this, they used ferroelectric Ba0.5Sr0.5TiO3 epitaxial thin films with different thicknesses and deposited them on a SrRuO3 (bottom electrode)/MgO substrate. They analysed the X-ray diffraction peak broadening and shape (figure 4a). Generally, there are three main contributions to peak broadening: the finite thickness of the sample, the inhomogeneous strain and the instrumental resolution of the diffractometer . With this information, Catalan et al. estimated the degree of inhomogeneous strain and the vertical strain gradient (figure 4b). (Note that this method can be applied using home-made laboratory X-ray diffraction equipment.)
With an estimated profile of the internal strain u(z), the strain-gradient contribution to the dielectric properties can be calculated using an elasto–dielectric free energy expansion, incorporating the flexoelectric contribution below : 4.1Here P is the out-of-plane polarization, sij is the elastic compliance, σ is the in-plane stress, Q13 is the transverse electrostrictive coefficient, E and F are constants related to the energy contributions from polarization and the stress gradient, and γ and η are, respectively, the direct and converse flexoelectric coefficients. The calculation using equation (4.1) showed a large degradation in relative dielectric constants, and the results were similar to those measured experimentally (figure 4c). The degradation of the dielectric properties increased as the film became thinner (i.e. as the strain gradient increased). This study successfully explained the degradation of dielectric properties near the ferroelectric transition temperature by considering the flexoelectric contribution.
(b) Domain configurations
Ferroelectric materials have thermodynamically equivalent spontaneous polarization states, which are differently oriented [1,40]. Those ferroelectric polarizations form small regions called domains in which the local polarization has the same orientation. We refer to domains containing upward polarization as upward domains and those containing downward polarization as downward domains. Ferroelectric domains provide a means of studying intriguing physical phenomena and applications; thus, it is important to fully understand domain-related physics. Domain configurations (i.e. collective patterns of the upward and downward domains) are determined by various contributions from depolarization energy, domain wall energy, (external and/or internal) electric field, elastic energy and so on . Owing to the complex nature of domain configurations, many aspects remain unknown. For example, ferroelectric epitaxial thin films have sometimes shown self-poled domains (with either upward or downward polarization) in their as-grown state . An internal field in the films has been suggested as a possible origin for this self-poling behaviour, but there is little consensus on the source of the internal field.
Lee et al.  explained various domain configurations (e.g. self-poled domains) in as-grown ferroelectric epitaxial thin films with an electric field above 1 MV m−1, generated by flexoelectricity. They found that the flexoelectricity-induced electric field could be large and tunable in ferroelectric HoMnO3 epitaxial thin films. They directly estimated the flexoelectricity-driven electric field using the equation [18,42] 4.2where ∂u/∂z is the strain gradient, and z is the distance from the film surface. (Equation (4.2) is a simplified version of equation (1.1).) By inserting the experimental values of ∂u/∂z into equation (4.2), they estimated Eflex in the films: Eflex=0.7 MV m−1 in the film deposited at PO2=10 mTorr and Eflex=5.0 MV m−1 in the film deposited at PO2=350 mTorr. This means that the Eflex value can be large and can also be tuned according to the PO2 growth conditions, as explained in §3b. The estimated value of Eflex seems small compared with the room-temperature ferroelectric coercive field (e.g. approx. 40 MV m−1 in HoMnO3 films) [43,44]. The coercive field, however, becomes much smaller at temperatures close to the ferroelectric Curie temperature (TC), near which the ferroelectric interaction is weaker. (Note that the temperature dependence of Eflex is expected to be weak when the film and substrate have similar thermal expansion coefficients.) Thus, Eflex can become significant enough to affect the domain configurations near TC.
As shown schematically in figure 5a, Eflex plays an important role in determining the domain configurations at temperatures near TC. For films deposited at low PO2 the strain gradient in the film is low and a mixed polydomain forms in the film. Polydomain formation is typical in ferroelectrics because it can reduce the depolarization energy. For films deposited at high PO2, however, a large strain gradient occurs. For high-PO2 deposition conditions, Eflex can be large enough that a monodomain forms at temperatures near TC. Also, by performing quantitative electrostatic calculations as a function of Eflex and temperature, Lee et al.  showed that, as Eflex increased, the domain configurations changed from polydomain to monodomain near TC. It was also found that a larger Eflex allowed a wider temperature window in which the monodomain formed.
The above expectation is interesting because it suggests that domain configurations can be controlled simply by varying the PO2 conditions during growth. Actually, Lee et al.  observed a large variation in domain configurations in HoMnO3 thin films according to the PO2 conditions. They measured the ferroelectric domains using dark-field transmission electron microscopy. Figure 5d shows the ferroelectric domains in the films deposited at PO2 of 20 and 300 mTorr. The bright and dark regions in figure 5d correspond to upward and downward domains, respectively. From the images and intensity profiles, they obtained the averaged ratios of the upward and downward domain widths, which were wup/wdown=0.8 for PO2=20 mTorr and wup/wdown=3.2 for PO2=300 mTorr. The domain images clearly showed that, as PO2 increased, the domain configuration changed from an up/down mixed pattern to one in which up domains are preferred, illustrating domain control by flexoelectricity, as expected from figure 5a.
Domain control by flexoelectricity, however, needs to be further investigated. The study by Lee et al.  was restricted to films experiencing a tensile-strain gradient. One should verify that the polarization direction of self-poled domains is reversed according to the strain-gradient direction. In other words, ferroelectric epitaxial thin films with compressive-strain gradients should have an opposite self-poled polarization direction compared with that of films with tensile-strain gradients. To confirm this, perovskite ferroelectric materials such as Pb(Zr,Ti)O3 and BiFeO3 would be suitable, because there are many available substrates. The (pseudo)cubic lattice constant of commercial substrates ranges from 3.69 Å (for YAlO3) to 4.02 Å (for NdScO3). On them, we can grow perovskite ferroelectric thin films epitaxially and can freely vary the strain gradient in the films.
(c) Polarization hysteresis loops
Domain control by flexoelectricity becomes more important for ferroelectric materials with high TC owing to the alignment of irreversible defect dipoles (Ddefect). The high-temperature growth used for epitaxial thin films typically results in the formation of Ddefect, which tend to be aligned parallel to the polarization direction because of the ‘site preference of point defects’ [44–47]. Figure 5b shows a schematic diagram that explains how the PO2-modulated flexoelectric effect can affect the alignment of Ddefect in ferroelectric epitaxial thin films. This means that the alignment of Ddefect can also be controlled by varying PO2 during growth. The alignment of Ddefect is generally preserved after cooling to room temperature, and so domains will be pinned according to the high-temperature alignment of Ddefect. Domain switching can be influenced by domain pinning, inducing a modification in the polarization–electric field (P–E) hysteresis loops (figure 5c) [1,44]. Lee et al.  showed that the P–E hysteresis loops of HoMnO3 thin films were very dependent on the PO2 conditions. Figure 5e shows that, as PO2 increases, the P–E curve changes from a double loop for PO2=10 mTorr to an asymmetric double loop for PO2=100 mTorr to a nearly biased single loop for PO2=350 mTorr. This variation in the P–E hysteresis loops is consistent with the prediction described above (figure 5c).
The interlinked effect of flexoelectricity and Ddefect provides a viable unified mechanism to explain all the modified P–E hysteresis loops, including double loops  and imprint , often observed in as-grown ferroelectric epitaxial films. This study provides insight into applications of ferroelectric materials, in which both the domain configuration and the P–E hysteresis loops can be carefully manipulated. Also, because the flexoelectric effect and defects are universally present in all materials, this discovery might provide general guidelines for analysing and tuning the physical properties of materials.
Lastly, it should be noted that these effects should be more important for ferroelectric materials with high TC values, such as HoMnO3 , BiFeO3  and highly strained ferroelectric films . Ddefect forms as a result of the migration of charged point defects, which dominates at high temperatures. Thus, if a material has a ferroelectric phase during high-temperature deposition, Ddefect can form with a specific alignment along the polarization orientation by the site preference of point defects. This Ddefect alignment enables the large flexoelectric effect on ferroelectric domains near TC to be sustained down to room temperature. However, if a material is not ferroelectric (i.e. paraelectric) at high temperatures, Ddefect can only have a random orientation, which cannot induce an observable effect on ferroelectric domain dynamics.
5. Future perspectives
For giant flexoelectric effects on ferroelectric properties, we have discussed three recent examples in ferroelectric epitaxial thin films: degraded dielectric properties [33,34], controllable domain configurations  and greatly modified P–E hysteresis loops. We suggested that oxide ferroelectric epitaxial thin films with high TC represent one of the best model systems to study flexoelectric effects. Using oxide epitaxial thin films, we can realize giant and tunable strain gradients, allowing a new route for materials science: strain-gradient engineering. Also, the flexoelectricity-induced electric field can have a more significant effect on ferroelectric properties at temperatures near TC.
We believe that flexoelectric effects can have a broader impact on diverse physical properties. (i) Even if we focused on the case of epitaxial thin films, a strain gradient at the nanoscale can occur in many cases. For example, in nanostructures such as nanowires and nanorods, strain relaxation is facile, possibly causing a large flexoelectric effect. Sharma and co-workers  have already studied the flexoelectric effect in such nanostructures. In particular, they have predicted that flexoelectricity might be used to enhance pieozoelectric responses above bulk values. (ii) Recently, there have been many studies on the domains and domain walls of BiFeO3 thin films owing to their scientific interest and potential applications [41,51–53]. BiFeO3 thin films with a monoclinic distortion undergo lattice deformation during polarization switching (i.e. ferroelastic switching; figure 6a). Baek et al.  suggested that relaxation of nanoscale-switched domains could occur due to high elastic energy during the ferroelastic switching process. In addition to the simple elastic energy contribution, the flexoelectric effect might provide a more detailed understanding of this nanoscale phenomenon. In the ferroelastic switching process, a large strain gradient can occur in the longitudinal direction (figure 6a). The large strain gradient can generate an electric field by flexoelectricity, and its magnitude might be large enough to induce polarization relaxation to the 180° domain configuration. Similarly, quite recently, Catalan et al.  reported that a very large strain gradient could occur in ferroelastic domains of a PbTiO3 thin film, and the associated flexoelectric effect resulted in a rotation of polarization direction. (iii) Also, we have shown that the magnitude of a flexoelectricity-driven electric field can be more than 1 MV m−1 in epitaxial thin films. This value is comparable to those of an internal field in conventional p–n junctions and/or Schottky diodes. Thus, the general electronic properties of epitaxial thin films can be affected by this flexoelectricity-driven electric field, and we are currently investigating this. (iv) Lastly, the giant strain gradient in epitaxial thin films can achieve a large flexomagnetic effect, describing the net magnetization in the response of the strain gradient (figure 6b) [55,56]. Recent discoveries related to giant strain gradients in epitaxial thin films will enable advanced studies of the flexomagnetic effect.
The authors acknowledge support from the National Research Foundation of Korea and from the Korean Ministry of Education, Science, and Technology through grant no. 2010-0020416.
One contribution of 10 to a Discussion Meeting Issue ‘The new science of oxide interfaces’.
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