Giant linear strain gradient with extremely low elastic energy in a perovskite nanostructure array

Although elastic strains, particularly inhomogeneous strains, are able to tune, enhance or create novel properties of some nanoscale functional materials, potential devices dominated by inhomogeneous strains have not been achieved so far. Here we report a fabrication of inhomogeneous strains with a linear gradient as giant as 106 per metre, featuring an extremely lower elastic energy cost compared with a uniformly strained state. The present strain gradient, resulting from the disclinations in the BiFeO3 nanostructures array grown on LaAlO3 substrates via a high deposition flux, induces a polarization of several microcoulomb per square centimetre. It leads to a large built-in electric field of several megavoltage per metre, and gives rise to a large enhancement of solar absorption. Our results indicate that it is possible to build up large-scale strain-dominated nanostructures with exotic properties, which in turn could be useful in the development of novel devices for electromechanical and photoelectric applications.

W hen an elastic strain is exerted on a nanostructure, some properties of the material are tuned, enhanced or created [1][2][3][4][5][6][7][8][9][10][11][12][13] . For example, straining Si channels by incorporating SiGe sources and drains boosts the performance of silicon transistors without aggressively scaling their dimensions 13,14 . Moreover, inhomogeneous strains are able to break the symmetry of a nanoscale crystal and consequently give rise to some exotic phenomena in the material's properties 15 . It is known that a nonuniform strain is able to continuously tune the bandgaps of a semiconductor, enhance the optoelectronic and energy-harvesting efficiencies for MoS 2 monolayer and ZnO microwires 3,16,17 , and induce novel room temperature metal-insulator transition in VO 2 nanobeams 18 . Particularly, through flexoelectric/piezoelectric couplings 15 , bending strain gradients enable to induce polarization in paraelectric SrTiO 3 cantilever actuators 19,20 . Furthermore, strain gradient may also induce cation non-stoichiometry and Cottrell atmospheres around dislocation cores as reported in perovskite ferroelectric films 21 .
Nevertheless, the above proposals are based on either the theoretical simulations 16,24,25,27 or mechanical bending of nanofibers/cantilevers [17][18][19][20] or limited information near the domain walls of ferroelectrics 30,31 . Although mechanical buckling seems an effective way to pattern stretchable semiconductor and ferroelectric nanoribbons 32,34 , the corresponding attainable strain gradients are relatively lower and failure tends to occur when manipulating brittle ceramics such as ferroelectrics. Two preconditions must be met toward making use of inhomogeneous strains. The first is to find an approach of assembling inhomogeneous strains into a nanostructure; and the second is to make these complex strains precisely measured. To quantitatively measure an inhomogeneous strain is always a great challenge, since peak broadening in diffraction experiments results from not only inhomogeneous strains but also a number of other components such as crystallographic defects, refined grains and similarly orientated domains 35 .
In this work, we report a fabrication of giant linear strain gradients in the lead-free perovskite nanostructures and their real-space measurements at the atomic scale. We choose singlephase room temperature multiferroic BiFeO 3 as a perovskite prototype [36][37][38] . The strain extraction is based on the geometric phase analysis (GPA, refs 39,40) technique. A special array of ao1004 and ao1104 type interfacial dislocations is identified to account for the lattice mismatch of the heteroepitaxial system. Such a relaxation mechanism results in a long-range and giant linear strain gradient across the rhombohedral (R phase) BiFeO 3 nanostructures displaying remarkable disclination characters. We find that the giant strain gradient enables to transfer when the BiFeO 3 /LaAlO 3 films are repeatedly grown, suggesting a controllability of such inhomogeneous strains in abundant lead-free perovskites without the need of taking dimension constrain into account. Our results deviate from the general understanding that giant strain gradients and the resultant flexoelectricity only matter at a nanoscale, and provide an example to quantitatively measure the inhomogeneous strains through directly atomic imaging.

Results
Preparation of LaAlO 3 /BiFeO 3 nanostructures. Self-assembled nanostructures of LaAlO 3 /BiFeO 3 (LAO/BFO) are epitaxially grown on a LAO(001) substrate by pulsed laser deposition with a high deposition flux mode. High frequency 5 Hz is chosen and a fresh surface of the ceramic target (mechanically polished) is used to insure a high growth rate of the first BFO layer. In the further growth of upper LAO and BFO layers, the laser was first focused on the ceramic targets for 30 min pre-sputtering to stabilize the target surfaces. The stabilized targets allow a lower deposition flux (detailed in Supplementary Note 1). The objective of the BFO/LAO/BFO film design here is to study the strain interactions and transfers among the lead-free multilayer structures and corresponding strain induced physical effects.
Observation and strain measurements. High-angle annular dark-field (HAADF) scanning transmission electron microscopic (STEM) imaging is applied to investigate the atomic details. Figure 1 shows an atomic morphology of a LAO/BFO/LAO(001) nanostructure. The high deposition flux in the present study avoids the 2D growth of BFO film and thus self-assembled 3D nanostructures are fabricated (more is shown in Supplementary  Figs 1 and 2). For a comparison, a 2D growth of BFO film on LAO substrate under low deposition flux mode is displayed in Supplementary Fig. 3, where the famous strain-driven tetragonal BFO phase (T phase) is observed and the BFO/LAO interface is free of interfacial dislocations. The intensity of HAADF-STEM image is strongly dependent on the atomic number of corresponding element (proportional to BZ 2 , detailed in Supplementary Note 2) 10,41,42 . The interface between BFO and LAO substrate is clearly seen because of respective atomic number of heavy Bi (83) and lighter La (57) atoms. Interfacial defects are identified at interfaces as labelled with pink and green arrows which indicate different types of misfit dislocations. The dislocation arrays are displayed in a low magnification image shown in Supplementary Fig. 4. To reveal the features of the defects and their effects on the BFO nanostructures, four areas each of which is labelled with a rectangle in Fig. 1a are magnified and seen in Fig. 1b-e, respectively. Compared with the LAO substrate, the lattice of BFO near the left area 1 is parallel to the LAO lattice (Fig. 1b), while the lattice of BFO near the right area 2 obviously rotates in a counterclockwise fashion (Fig. 1c). The rotation angle in 2 is B4°. The interfacial defect is composed of two kinds of interfacial dislocations whose Burgers vectors is a[010] and a[011], respectively, based the Burgers circuits drawn in Fig. 1d,e. Here a indicates the lattice parameter of a cubic perovskite unit-cell. Thus the green and pink arrows in Fig. 1a  . It is proposed that the high deposition flux in the present study triggers the formation of dislocations other than an intergrowth of R and T (tetragonal) phases to relax the mismatch. As is seen in Fig. 1e, the a[001] component of the a[011] dislocation has a giant effect which makes a rotation of the BFO lattice. However, the strain effect of a single dislocation is extremely local, thus the lattice rotation in the BFO (area 2) away from the interface is actually derived from a synergetic effect of the special array of interfacial dislocations.
GPA and directly atomic position mappings are effective and accurate tools which show great potential to extract complex and inhomogeneous strain distributions 8,10,39,40,[43][44][45] . We apply GPA to extract the dislocation configuration and unique strain distributions in the nanostructures since GPA works better for long-range strains 40 . Figure 2a It is worthwhile to mention that the array of these dislocations is in an aperiodic fashion ( Fig. 2 dislocations which contain no out-of-plane component, so their rotation effect is restrained. Nonetheless, at the right side, the three well aligned a[011] dislocations result in giant lattice rotation in the BFO and LAO lattice, which spreads for a long distance across the entire BFO and LAO films. Finally, the right BFO lattice is severely rotated, while the left BFO lattice still holds the same orientation as the LAO substrate. This synergetic effect of the a[001] components resembles the low-angle boundaries (or tilt boundaries) formed by a[001] dislocation array in perovskite bicrystal 47 . In fact, the present situation corresponds to a concept of partial disclination, which is responsible for the giant long-range lattice rotation and behaves different from the well known a[100] dislocation arrays 48 .
As above mentioned, continuous lattice rotation is relevant to a bending deformation, thus the lateral deformation must be accompanied 46 . The e xx strain in the LAO/BFO/LAO lattice show obvious and systematic inhomogeneity, as shown in Fig. 2b. The e xx decreases from the bottom to top, which is consistent with the character of bending deformation of materials 46 . A quantitative line-profile of e xx is shown in Fig. 2d. The strain gradients of e xx can be seen directly as the slopes of strains in the BFO and above LAO lattice. Note the strains in the LAO substrate reference lattice are constant 0 which confirms the validity of strain gradients in above BFO and LAO lattice. The strain gradients of e xx in BFO are estimated by the slopes of the curves, which are well above 10 6 (Fig. 4a). The out-of-plane (e yy ) and shear (e xy ) strain mappings are given in Supplementary Fig. 12. Thus our results suggest a practical approach to integrate such linear strain gradient into perovskite nanoislands without taking into account of the thicknesses of these structures and magnitude of mismatches (as schematically illustrated in Fig. 4d). Our results deviate from the general understanding that giant strain gradients and the resultant flexoelectricity only matter at a nanoscale. The utility of such giant strain gradients can thus be extended as novel gradient functional nanostructures.

Discussions
The formation of linear strain gradient at nanoscale can be understood through an elastic energy consideration, and we find that the elastic energy consumption for producing such a giant strain gradient is unexpectedly low, as shown in Fig. 5a. We consider a BFO nanoisland with dimension l Â t Â h, where l and t are the in-plane dimensions vertical to or along the imaging direction (Fig. 1a), h is the out-of-plane thickness. To compare the elastic energy of a BFO island in a fully 2D strain state with that in a linear strain state, we arbitrarily set l ¼ t ¼ 100 nm in our calculations. The detailed deducing process is shown in Supplementary Note 4 and Supplementary Figs 13  [001] Lattice rotation (°) Strain (%) through the elastic properties of dislocations 49 . Thus the whole energy for producing the observed linear strain gradient is plotted versus h as curve 3 (through 1 þ 2). The elastic energy of the fully strained BFO island displays highly increasing linear distribution with increasing of thickness, which is calculated through the elastic properties of 2D strained films 50 by considering the mismatch (4.5%) between BFO and LAO. It is seen that the elastic energy cost for the linear strain is so negligible that it is even much lower than the dislocation energy, especially when the thickness of the BFO island is small (note misfit dislocations in epitaxial perovskite oxides are commonly seen). Thus the observed giant strain gradients here are elastically feasible. A schematic illustration of the experimentally observed giant strain gradients in LAO/BFO islands is given in Fig. 5b. The present study features general implications for other perovskite oxides. We further formulate the elastic energy consumption distribution versus thickness (h) and the location of neutral plane (y) of the BFO nanostructure under linear strain gradient, as shown in Fig. 5c. By considering the observed strain gradient (B1.24 Â 10 6 m À 1 ) and maximum elastic limit of nanomaterials (B10%, ref. 51), the possible maximum h m is B160 nm (h m ¼ 2 Â 10%/(1.2 Â 10 6 m À 1 )E160 nm, when the neutral plane y is in the middle of h m ). Note that the minimum elastic energy consumption occurs when h ¼ 2y, that is, pure bending (more details can be found in Supplementary Note 4). The elastic energy consumption distribution versus thickness (h) and mismatch of the same size BFO nanostructure under fully 2D strained state is shown in Fig. 5d. By comparing Fig. 5c,d, it is obvious that for large mismatch systems with small thickness, the BFO nanostructure under linear strain gradient exhibits negligible elastic energy consumption compared with the fully 2D strained state. For instance, for an h ¼ 50 nm BFO nanostructure under linear strain gradient, its elastic energy is probably o1 Â 10 À 14 J. However, when the same BFO nanostructure is fully 2D strained on a LAO substrate, its elastic energy is B13 Â 10 À 14 J, which is more than 10 times bigger than the former ones. The elastic energy consumption for linear strain state tends to more negligible when the thickness h decreases.
It is known that the 2D compressive strain strongly determines the domain structures, phase constitutions and piezoelectricity. And many unique properties of BiFeO 3 films grown on substrates with smaller lattice parameters, like LaAlO 3 (refs. 1,2,7,37,52,53), have been found. Except for the well-known strain-driven morphotropic-phase-boundary mechanism 7 , our results reveal a peculiar mismatch relax mechanism in the large mismatch systems via the well aligned alternating array of ao1004 and ao1104 interfacial dislocations. Here the mismatch relax mechanism is dominated by mechanical property (elastic limit) of perovskite materials other than the generally accepted elastic energy considerations of epitaxial films 50,54 . Particularly, this mismatch relax process allows patterning giant linear strain gradient and the resultant flexoelectricity in abundant lead-free perovskite oxides.
The giant linear strain gradient over 10 6 m À 1 (defined as DS) corresponds to a radius of curvature (r) much o1,000 nm, which is probably the biggest gradient attainable in long-range linear strains expect for the local strain gradients observed at ferroelectric domain walls and other interfaces 30,31 . Experimentally, it is known that the measured flexoelectric coefficients ( f ) for perovskite are on the order of 10 À 9 -10 À 8 Cm À 1 (refs 19,20,30). The present giant strain gradient over 10 6 m À 1 may induce several mCcm À 2 flexoelectric polarizations (P f ) as estimated according to P f ¼ f Â DS. Although this value is not so large compared with the spontaneous polarization (Ps) of BFO, an important signification is that the flexoelectric behaviour under the form of giant linear strain gradient is applicable for all dielectric materials when integrated into an electronic devices. In addition, the flexoelectric coupling also leads to a large flexoelectricity-driven electric field (E f ), which can be estimated according to the following formula 29 : where e is the electronic charge (1.602 Â 10 À 19 C), e 0 is the permittivity of free space (8.854 Â 10 À 12 Fm À 1 ), a is the lattice parameter. An E f well exceeding 1MVm À 1 is estimated. This value is comparable to the internal field in the conventional p-n junctions and Schottky diodes 29,55 . Details in terms of how such a large field affects the electronic properties of epitaxial interfaces and its couplings with other order parameters are still open questions. Future studies on BiFeO 3 and other perovskite materials nanostructures as flexible electronics, electromechanical or photoelectric devices could thus be stimulated. It is worthwhile to mention that, although flexoelectricity is generally discussed in terms of dielectric insulators, a very recent study by a three-point bending reveals that the flexoelectric-like coupling is much larger in doped oxide semiconductors than in dielectric insulators 56 . Thus, we propose that, by using doped lead-free perovskite oxides, it is possible to construct nanostructures with giant linear strain gradient where the flexoelectric-like behaviours could be further enhanced for electromechanical applications.
To further verify how the strain gradient affects the macroscopic property of the present perovskite nanostructures, we have The green arrow indicates the thickness termination (B24 nm) of the present BiFeO 3 nanostructure (Fig. 2d). Note that, compared with the fully 2D strained state, the present observed disclination strain states exhibit almost negligible elastic energy consumptions, especially when the thickness of the nanostructure tends to smaller. A schematic illustration of the disclination formed through interfacial dislocation arrays is shown in b. The elastic energy consumption distribution versus thickness (h) and the location of neutral plane (y) of the BiFeO 3 nanostructure under linear strain gradient is shown in c. The elastic energy consumption distribution versus thickness (h) and mismatch of the same size BiFeO 3 nanostructure under fully 2D strained state is shown in d. By comparing c and d, it is obvious that for large mismatch systems with small thickness, the BiFeO 3 nanostructure under linear strain gradient exhibits negligible elastic energy consumption compared with the fully 2D strained state. performed ultraviolet-visible absorption measurements on the multilayered LAO/BFO nanostructures (Fig. 6). For comparison, a uniform BFO/STO(001) film with nearly the same thickness as the BFO layer in the nanostructure, a bare LAO(001) substrate and a bare STO(001) substrate were also comparatively measured. First we can see that the absorption of pure LAO(001) substrate is weak since the electronic structure of Al is insensitive to ultraviolet-visible excitations. In contrast, the pure STO(001) substrate exhibit obvious absorption edge (400-450 nm), which is consistent with previous report 57 . For the uniform BFO/STO(001) film, the absorption spectrum is almost the same as pure BFO crystals 58 , where a B580 nm absorption edge is seen, as indicated by a blue arrow. It is of great interest to find that the absorption spectrum for the strain gradient BFO nanostructure is largely modified compared with the BFO/STO(001) film. There is no obvious absorption edge at 580 nm, and the absorption edge is largely extended towards the infrared direction covering much more visible light spectrum, as marked with the red arrow. This phenomenon suggests that the strain gradient induces a continuous bandgap change in the BFO nanostructures (bandgap gradient), which is responsible for the enhancement of visible light absorption since a constant bandgap only induces a specific absorption edge. We note previous studies indicate that chemical gradient also induces bandgap gradient and enhances solar absorption for TiO 2 photocatalyst 33 . Moreover, theoretical calculations and nanobeam bending experiments further suggest that strain gradient could introduce bandgap gradient in semiconductors 3,16,17 . Thus our results supply a novel strategy to integrate strain gradient in materials which could be used to modify the band structures of materials and enhance the performance of photocatalysts.
In summary, we have artificially produced a giant linear strain gradient in the BiFeO 3 /LaAlO 3 multilayer nanostructures by a controlled pulsed laser deposition via a high deposition flux mode. Aberration-corrected STEM observation shows that the asreceived giant strain gradients are dominated by synergetic interfacial dislocation arrays with ao1004 and ao1104 Burger vectors. The well aligned a[001] Burger vector components severely rotate the BiFeO 3 lattice, result in a long-range giant strain gradient and lead to many exotic properties. The interfacial dislocations herein are very useful modulations other than deleterious ingredients as generally cognized. Our calculations indicate the elastic energy consumption for producing such a giant strain gradient is extremely lower than previously regarded, and our experiments show that the giant strain gradient enables to transfer across a multilayer structure possibly reaching a practical scale. The present results may also stimulate the relevant studies on other epitaxial systems with large lattice mismatch, which are not favoured in the past. Our study provides an opportunity to quantitatively measure the contribution of inhomogeneous strains and assemble them in a nanostructure for the development of novel device concepts, which should involve lead-free electromechanical actuators, high-efficient energy harvesting devices and photocatalysts.

Methods
Material preparation. BiFeO 3 nanostructures were deposited by pulsed laser deposition, using a Lambda Physik LPX 305i KrF (l ¼ 248 nm) excimer laser. A sintered BiFeO 3 ceramic target (3 mol% Bi-enriched) and a stoichiometric LaAlO 3 ceramic target were used. The target-substrate distance was 40 mm. The background pressure was 10 À 5 Pa. Before deposition, all substrates were pre-heated at 750°C for 5 min to clean the substrate surface and then cooled down to the growth temperature (10°C min À 1 ). After deposition, the samples were annealed at their growth temperature in an oxygen pressure of 5 Â 10 4 Pa for 10 min, and then cooled down to room temperature at a cooling rate of B5°C min À 1 . Commercial, one-side polishing LaAlO 3 (001) single-crystal substrates with 10 mm Â 10 mm Â 0.5 mm dimension were used for film deposition.
HAADF-STEM imaging and strain analysis. The samples for the HAADF-STEM experiments were prepared by slicing, gluing, grinding, dimpling and finally ion milling. A Gatan PIPS was used for the final ion milling. Before ion milling, the samples were dimpled down to o20 mm. The final ion milling voltage was o1 kV to reduce ion beam damage. HAADF-STEM images were recorded using aberration-corrected scanning transmission electron microscopes (Titan Cubed 60-300 kV microscope (FEI) fitted with a high-brightness field-emission gun (X-FEG) and double Cs corrector from CEOS, and a monochromator operating at 300 kV). The beam convergence angle is 25 mrad, and thus yields a probe size of o0.10 nm. The diffraction contrast image was recorded using a conventional TEM (Tecnai G2 F30 (FEI) working at 300 kV). Large-scale strain fields were deduced by using custom plugins of GPA under the framework of Gatan DigitalMicrograph software. The visualization of the strains and lattice rotations was carried out using Gatan DigitalMicrograph software.
Lattice rotations resulting from the interfacial dislocation arrays. Lattice rotations (o) are derived from the HAADF-STEM images via GPA, during which LaAlO 3 substrate is chosen as the reference lattice. The observed linear strain gradient resulting from the lattice rotation can be persevered in lead-free perovskite Elastic energy consideration of perovskite under linear strain gradient. The elastic energy consumption for producing the present observed strain gradients is calculated. Two types of energies are involved here. One is the elastic energy consumption; and the other is energies of the interfacial dislocation arrays.
Data availability. The data that support the findings of this study are available from the corresponding author upon request.