Observation of a periodic array of flux-closure quadrants in strained ferroelectric PbTiO3 films

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Science  01 May 2015:
Vol. 348, Issue 6234, pp. 547-551
DOI: 10.1126/science.1259869

Getting closure in ferroelectric films

Ferroelectric materials have a spontaneous electric polarization that can be manipulated for applications. The polarization is usually not uniform throughout the material, and for nanosized ferroelectrics, polarization can be quite complex. Using scanning transmission electron microscopy, Tang et al. found that in thin films of the ferroelectric PbTiO3, the polarization vector rotated in space, forming a closed loop, the so-called flux closure. The flux closure structures formed an array, with the period dependent on the width of the thin film, and caused the buildup of considerable strain within the crystal lattice of the material

Science, this issue p. 547


Nanoscale ferroelectrics are expected to exhibit various exotic domain configurations, such as the full flux-closure pattern that is well known in ferromagnetic materials. Here we observe not only the atomic morphology of the flux-closure quadrant but also a periodic array of flux closures in ferroelectric PbTiO3 films, mediated by tensile strain on a GdScO3 substrate. Using aberration-corrected scanning transmission electron microscopy, we directly visualize an alternating array of clockwise and counterclockwise flux closures, whose periodicity depends on the PbTiO3 film thickness. In the vicinity of the core, the strain is sufficient to rupture the lattice, with strain gradients up to 109 per meter. Engineering strain at the nanoscale may facilitate the development of nanoscale ferroelectric devices.

Atomic-scale information is of critical importance for understanding intrinsic characteristics of advanced functional materials such as ferroelectrics, magnets, superconductors, and catalysts. For example, it is often the small deviations from symmetry in atom positions and the resultant strains that allow ferroelectric oxides, used in computer memory chips, to store charge and information or to resonate with magnets as composite multiferroics. Ferroelectric crystals feature asymmetric or polar structures that are switchable under an external field, holding promise for random access memories, thin-film capacitors, and actuators (1). For integration into silicon chips, practical ferroelectric memories take the form of nanoscale films (1). Nanoscale ferroelectrics have been predicted to undergo unusual phase transitions and exhibit distinctive domain patterns, such as closure quadrants with closed head-tail dipole moments, known as flux closures (27). These flux-closure domains should be switchable and may give rise to an unusually high density of bits (2), and they can undergo vortex-polarization phase transformation (6). These domains are also predicted to be potentially useful as mechanical sensors and transducers (7). Similar domains are well known in ferromagnetic materials (810), and their topological properties and dynamics are under investigation (9, 10). However, in ferroelectric materials, particularly in tetragonal ferroelectrics, the coupling of polarization to spontaneous strain would be so pronounced that formation of a closure quadrant with its resultant severe disclination strains could be impossible (11, 12). Although closure quadrants were reported recently in tetragonal ferroelectric BaTiO3 (1316), PbZr0.42Ti0.58O3 (17), and PZN-12PT (18), they are mostly composed of shape-conserving 90° stripe domains or twins within each quadrant to accommodate the disclination strains. In such cases, the closure quadrants may not always involve continuous dipole rotations, as have been observed directly in half of a closure quadrant in PbZr0.2Ti0.8O3 (19) and BiFeO3 (20) by aberration-correction transmission electron microscopy (TEM) or scanning TEM (STEM). The atomic-scale characterization based on STEM imaging has been validated to be capable of directly displaying ionic displacement maps (1924), whereas approaches based on piezoresponse force microscopy are not able to do so because of lower spatial resolution (11, 1418, 25, 26).

In this study, we have grown PbTiO3/SrTiO3 (PTO/STO) multilayer films on a GdScO3 substrate with a lattice parameter larger than the a value of PTO (27). Using aberration-corrected high-angle annular dark-field (HAADF) Z-contrast STEM imaging, we visualize, at the atomic scale, the existence of periodic twin-free flux-closure quadrants.

PbTiO3 has a tetragonal structure (Fig. 1A). Both the oxygen octahedra and the Ti4+ have displacements from the center of the Pb2+ tetragonal cell that give rise to the spontaneous polarization (Fig. 1, B and C). The shifts of Ti4+ (denoted as δTi in Fig. 1B) can be used to determine the polarizations of PTO unit cells. In HAADF images, the Pb2+ columns appear as the brightest dots because the intensity of atom columns is approximately proportional to Z2, where Z is the atomic number (20, 21, 23, 24). The Ti4+ columns show weaker contrast. The displacement vectors of Ti4+Ti) relative to the center of mass of the four nearest Pb2+ neighbors (Fig. 1C) can be determined by fitting them as two-dimensional (2D) Gaussian peaks, based on the HAADF-STEM images (1924, 27, 28). The δTi vector in each unit cell is opposite to the polarization direction of PTO (Fig. 1 and figs. S1 and S2).

Fig. 1 Long-range periodic disclination pairs in a ferroelectric PbTiO3 layer.

(A) Schematic perspective view of the unit cell of PbTiO3 (yellow, Pb; red, Ti; blue, O). (B) Projection of the unit cell along the [100] direction. (C) HAADF-STEM image of the PbTiO3 crystal along [100]. (D) Low-magnification high-resolution HAADF-STEM image showing the upper PbTiO3 layer in a SrTiO3(10 nm)/PbTiO3(36 nm)/SrTiO3(3 nm)/PbTiO3(28 nm)/GdScO3 multilayer. Boxes labeled with numbers 1 to 4 denote typical areas. (E) GPA analysis of the STEM data reveals the out-of-plane strain εyy. The c domains are associated with an abrupt elongation of the out-of-plane lattice parameter (red areas) and exhibit a sinusoidal array. The domains (in green), with triangular head-to-head configurations having a smaller εyy, are a domains.

A low-magnification high-resolution HAADF-STEM image of a PTO/STO film shows that the contrast in each PTO layer is basically uniform, indicating the homogeneity of chemical compositions (Fig. 1D). Slight variation of the contrast implies the existence of domain walls. The domain structure was characterized by geometric phase analysis (GPA) (21, 27, 29). The out-of-plane strain εyy shown in Fig. 1E displays a clear spatial arrangement of the domain patterns. The domains in the upper PTO layer (in red, c domains, with c axis along the out-of-plane direction) form a 2D periodic sinusoidal array, whereas the domains with triangular configurations give rise to another periodic array (in green, a domains, with c axis along the in-plane direction).

Four typical areas (labeled as 1, 2, 3, and 4 in Fig. 1D) are magnified in Fig. 2A, which shows the atomically resolved HAADF-STEM images corresponding to these four areas. The atomic structures in each image depict the positions of Ti4+ and Pb2+ columns by red and yellow circles, respectively. For example, on the left side of area 2, the δTi is upward, whereas on the right side of the image, the δTi is downward. The yellow arrows denote the reversed δTi directions, which are also the directions of spontaneous polarization (22, 24). The opposite local polarization directions indicate the presence of two domains in a 180° orientation relation; the corresponding domain walls are marked by red dashed lines. At the bottom interface of PTO/STO, in-plane δTi vectors are discernable; thus two 90° domain walls are outlined (blue dashed lines). Such in-plane displacement may play an important role in stabilizing this ferroelectric domain configuration (4, 19). The 90° domain walls for areas 1 and 4 are denoted by green dashed lines. Figure 2B displays a superposition of the –δTi vectors mapping with the atomic images in Fig. 2A. The –δTi vectors are marked with an arrow located at the Ti4+ columns. The combined δTi configurations in areas 1 and 2 form a flux-closure pattern (closure quadrant); the same is true for combined areas 3 and 4. Such a closure configuration is almost the same as that predicted in ferromagnetic materials in (12) and in ferroelectrics in (30). The combination of 1 and 2 forms a counterclockwise flux closure, whereas the combination of 3 and 4 leads to a clockwise flux closure. The same closure patterns can be identified in other areas. Because the order parameter of each flux closure—the toroid moment G, defined by the sum of cross-multiplication of radial vector Ri and local dipole pi—can be uniquely assigned (2), an antiparallel configuration of the G can be determined for a periodic closure pattern. The dynamic interactions of the periodic closures are of technological importance, as explored recently in BaTiO3 platelets, although the closures are still composed of shape-conserving 90° stripe domains within each quadrant to accommodate the disclination strains (16).

Fig. 2 Superposition of HAADF-STEM images and Ti4+ displacement vector maps showing the closure quadrants.

(A) Atomically resolved HAADF-STEM images corresponding to the areas labeled as 1, 2, 3, and 4 in Fig. 1D. The red and yellow circles denote the positions of Ti4+ and Pb2+ columns, respectively. Arrows denote reversed Ti4+ displacement directions. (B) Superposition of reversed δTi vectors with experimental images. The δTi vectors corresponding to PTO unit cells are shown as yellow arrows superimposed on the atomic image of Fig. 2A. The green and blue dashed lines indicate the 90° domain walls in areas 1 and 4, as well as 2 and 3, respectively. The red dashed lines indicate the 180° domain walls.

The long-range strains in these ferroelectric closure quadrants show interesting disclination characteristics. Based on the HAADF-STEM images, the in-plane and out-of-plane strains and lattice rotations [rigid-body rotation of crystal lattice (21, 29)] are determined by GPA (Fig. 3). The c lattice strains (that is, the in-plane strain εxx in the triangle domains in Fig. 3, A to C, and the out-of-plane strain εyy in the sinusoidal array in Fig. 3, D to F) have a strong local maximum around the vertices of the triangle domains, leading to a giant c lattice strain gradient. Away from the vertex, the strain gradient estimated in the triangle domain is ~4 × 106 m–1 (Fig. 3C). The lattice rotations are also spatially inhomogeneous (Fig. 3, G to I); in each single domain, the lattice rotation changes continuously (Fig. 3, H and I). Sudden jumps of lattice rotations (red arrows in Fig. 3, H and I) can be used to locate the 180° domain walls in HAADF-STEM images. 3D illustrations of the c lattice strain mappings (fig. S3) indicate a giant strain gradient that appears like a “volcano” at the vertices of the flux closure.

Fig. 3 Strain mappings of the periodic disclination pairs.

(A to C) In-plane strain (εxx) maps and corresponding line profiles of areas 1 and 2 [labeled in (A)]. (D to F) Out-of-plane strain (εyy) maps and corresponding line profiles of areas 1 and 2 [labeled in (D)]. (G to I) Lattice rotation maps and corresponding line profiles of areas 1 and 2 [labeled in (G)]. Note the very inhomogeneous distribution of the strains and lattice rotations. The continuous, giant c lattice strain gradient estimated in the triangle domain in (C) is ~4 × 106 m–1. Blue arrows in (E), (F), (H), and (I) denote the strain-rotation abrupt change points indicating the 90° domain walls. Red arrows in (H) and (I) denote the rotation maximal points indicating the 180° domain walls.

Strains close to the vertex can be interpreted from the point of view of ferroelectric theory. Strains and strain gradients corresponding to the vertex derived from the HAADF images and calculated using the Landau-Ginsburg-Devonshire (LGD) framework are given in figs. S4 to S10. A strain gradient reaching ~109 m–1 near the vertices can be identified (fig. S5). The analysis of the results suggests that most of the considerable deformation that the material undergoes in the vicinity of the vertices (for instance, the peaks in Fig. 3, B and C) and around domain walls can be attributed to the ferroelectric-electrostrictive contribution, which is not surprising given the large c/a ratio of the PbTiO3. However, the shear strain at the 180° domain wall appears to arise from a flexoelectric contributionEmbedded Imagewhere u5 is the shear strain component, F44 is a component of the flexoelectric tensor, PS is the spontaneous polarization, Rc is the correlation radius (width of a domain wall), and x1 is the coordinate normal to the 180° domain wall. In fact, the flexoelectric coefficient F44 can be directly estimated from the experimental data. The offset, which indicates the relative shift between two neighboring 180° domains, can be expressed asEmbedded Imagewhich is similar to the expression given in (31). Experimentally, we observe offsets as large as ~1.5 to 2 Å (fig. S11); together with the spontaneous polarization of 0.75 C m–2, this gives us an estimate of F44 ≈ 10–10 C–1 m3, which is within an order of magnitude of values previously reported (31, 32).

Moreover, the lattice strain reaches extremely high values left and right of the core of the vertex (Fig. 2). At the core itself, the Pb columns appear to have bifurcated into two partial columns, possibly forming a small sliver of PbO. We expect large disclination stress in this region, the magnitude of which is roughly Eu, where E is Young’s modulus and u is strain. The strain is much larger in the polar direction than in the nonpolar direction, consistent with the anisotropic elastic constants reported in (33). Using an average value of elastic constant c33 ~ 80 GPa in (33, 34) as an estimate of E, with the observed dilatation reaching 25% in the core region, the stresses would appear to reach the order of 20 GPa. The theoretical shear strength σtheor of a material can be estimated fromEmbedded Imagewhere γ is the surface energy and a the lattice parameter. Using γ ~ 1 J m−2 (35) gives σtheor ~ 14 GPa, suggesting that the lattice has ruptured because the disclination stresses exceeded the material’s maximum shear strength. This lattice rupture is a general phenomenon at the vertex in this study (see fig. S12 for other examples).

For the whole multilayer structure, the periodic arrangement of the closure-quadrant arrays is characterized by two fundamental aspects. Structurally, the disclination pairs are arranged in a reversed manner, known as strain-compensating hetero-disclination pairs (36), whereas the clockwise and counterclockwise flux closures are stacked in an alternating fashion. This arrangement helps to relax the long-range strains in the disclinations (36), consequently stabilizing the closure quadrants. Nevertheless, a large strain gradient remains (Fig. 3). The disclination strain at the vertex here is up to 25%, which is more than one order of magnitude larger than that in rhombohedral BiFeO3 ferroelectric, where the maximal disclination strain at the core is 1.5% (26). At the atomic level, our results show convincing evidence of the compatibility of disclinations and closure quadrants in a ferroelectric, even with strong tetragonality.

The periodic array of closure quadrants is occasionally absent in our PTO/STO multilayers. Figure 4A is a low-magnification TEM image of another PTO/STO film displaying an overview of the periodic array of the closure quadrants in both of the 20-nm PTO layers. Figure 4B shows a schematic illustration of the periodic closure quadrants.

Fig. 4 Variation of flux-closure period with PbTiO3 thicknesses.

(A) Cross-sectional TEM image showing an overview of a SrTiO3(10 nm)/PbTiO3(20 nm)/SrTiO3(20 nm)/PbTiO3(20 nm)/GdScO3 multilayer. Two flux-closure arrays align periodically in both PbTiO3 layers. (B) Schematic showing the flux-closure domain configuration based on the experimental data in Figs. 1 to 3. Because not all of the 90° domain walls in Fig. 2A are visible in the strain maps, the 90° domain walls were denoted by green (visible) and blue (invisible) dashed lines, respectively. (C) Linear relationship between the thicknesses (d) and the periods (w) of closure-quadrant arrays in PbTiO3 layers on GdScO3. The linear fit is close to Embedded Image.

By comparing PTO films with various thicknesses, we have derived the stabilization condition for such periodic closure quadrants (27). The thickness of each single PTO layer is the most important factor scaling the periodicity of the closure quadrants in tensile strained PTO/STO multilayers. We find that the thicknesses in which closure quadrants may occur are within the range of 15 to 36 nm. Thinner PTO tends to be fully strained without obvious domain structures, whereas thicker PTO tends to form alternating a/c domains (21) (figs. S13 to S16). In the present PTO/STO multilayer structures, each layer of PTO could be composed of such periodic closure quadrants. For example, in Fig. 4A, each of the two PTO layers is ~20 nm thick, and both layers contain the periodic closure quadrants. This indicates that, by repeatedly introducing STO layers, it is possible to fabricate extremely thick PTO/STO films that are composed of periodic disclination pairs in all PTO sublayers. Meanwhile, we find that the period of the closure-quadrant array w is strongly correlated with the PTO thickness d (Fig. 4C, following a linear relationship). In particular, for each PTO thickness, the ratio of period/thickness is ~1.45; i.e., Embedded Image.

According to Kittel’s law, stripe-domain widths are predominantly determined by the bulk domain energy Ed, the domain-wall energy Ew, and the film thickness d (11, 37). This law was further extended for all ferroics, and a universal square root dependence Embedded Image was deduced (11, 38), where w is the domain period. However, the situation in the present study cannot be understood as simple 90°/180° stripe domains (37, 38) because of the substantial nonuniform disclination strains that result from the configuration of the flux closures. In (39) it was theoretically demonstrated that, for films with a specific thickness range, the energy associated with disclination strains breaks the square root law, and a new linear law Embedded Image (where α and β are experiment-determined specific coefficients) is preferred. Our experimental results support the linear law with α = 0 and Embedded Image.

Our results indicate that giant strain gradients can be preserved within a very thick PTO/STO film by simply increasing the PTO/STO period. This may provide opportunities for exploring various functionalities induced by strain gradients—such as the catalytic efficiency (40) of perovskite oxides—and other material properties that are tunable by elastic strains (41).

The results also extend the potential of employing epitaxial strain for modulating ferroelectric domain patterns. Designs based on controllable ferroelectric closure quadrants could be fabricated for investigating their dynamics and flexoelectric responses and, in turn, may assist future development of nanoscale ferroelectric devices such as high-density memories and high-performance energy-harvesting devices.

Supplementary Materials

Materials and Methods

Supplementary Text

Figs. S1 to S16

Table S1

References (4256)

References and Notes

  1. Materials and methods are available as supplementary materials on Science Online.
  2. Acknowledgments: This work is supported by the National Natural Science Foundation of China (grants 51231007 and 51171190) and the National Basic Research Program of China (grants 2014CB921002 and 2009CB623705). A.N.M. and E.A.E. acknowledge financial support from the National Academy of Science of the Ukraine; A.Y.B. and S.J.P. thank the U.S. Department of Energy, Office of Basic Energy Sciences, Materials Sciences and Engineering Division for financial support. The authors at SYNL are grateful to B. Wu and L. X. Yang of this laboratory for their technical support on the Titan platform of the aberration-corrected scanning transmission electron microscope. Author contributions: Y.L.Z. and X.L.M. conceived the project of interfacial characterization in oxides by using aberration-corrected STEM; Y.L.T. performed the thin-film growth and TEM observations; W.Y.W. provided some of the TEM images used in the supplementary materials; Y.J.W. carried out digital analysis of the STEM data; Y.B.X. and Z.D.Z. contributed to the thin-film growth; A.N.M. and E.A.E. participated in the ferroelectric theoretical analysis based on the LGD framework; and A.Y.B. and S.J.P. contributed to STEM-HAADF imaging data analysis, particularly to the calculation of the strain-gradient components near the vertex core. All authors participated in discussion and interpretation of the data. Correspondence and requests for materials should be addressed to X.L.M.
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