LETTER pubs.acs.org/NanoLett
Unexpected Controllable Pair-Structure in Ferroelectric Nanodomains Yachin Ivry,*,† Daping Chu,‡ James F. Scott,§ Ekhard K. H. Salje,|| and Colm Durkan*,† †
Nanoscience Centre, University of Cambridge, 11 JJ Thomson Avenue, Cambridge, CB3 0FF, U.K. Electrical Engineering Division, University of Cambridge, 9 JJ Thomson Avenue, Cambridge, CB3 0FA, U.K. § Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 0HE, U.K. Department of Earth Sciences, University of Cambridge, Downing Street, Cambridge, CB2 3EQ , U.K.
)
‡
ABSTRACT: The imminent inability of silicon-based memory devices to satisfy Moore’s Law is approaching rapidly. Controllable nanodomains of ferroic systems are anticipated to enable future high-density nonvolatile memory and novel electronic devices. We find via piezoresponse force microscopy (PFM) studies on lead zirconate titanate (PZT) films an unexpected nanostructuring of ferroelectric-ferroelastic domains. These consist of c-nanodomains within a-nanodomains in proximity to a-nanodomains within c-domains. These structures are created and annihilated as pairs, controllably. We treat these as a new kind of vertexantivertex pair and consider them in terms of the Srolovitz-Scott 4-state Potts model, which results in pairwise domain vertex instabilities that resemble the vortexantivortex mechanism in ferromagnetism, as well as dislocation pairs (or disclination pairs) that are well-known in nematic liquid crystals. Finally, we show that these nanopairs can be scaled up to form arrays that are engineered at will, paving the way toward facilitating them to real technologies. KEYWORDS: Ferroelectric, ferroelastic, ferroic, nanodomain, nanotwins, nanopair, nonvolatile memory, vortex, vertex, domain engineering, defect engineering
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he three ferroic systems, ferromagnetism, ferroelectricity, and ferroelasticity, share a common behavior of reversible spontaneous strain: magnetization, electric-polarization, and mechanical-strain, respectively. The inherited behavior of these systems generally stems from common structural properties. Thus, controlling the crystallographic structure of these materials at the nanoscale is essential for a fundamental understanding of these materials and is significant for a wide range of applications, including nonvolatile memory devices.1 Changes in the crystallographic structure are usually accompanied by lattice defects, which by themselves significantly influence the magnetic, mechanical, optical and electrical properties of materials.2 Hence, local control over the crystallographic structure of functional materials is highly desirable.3 It has been shown recently that topological defects such as domain walls in ferroic systems display enhanced conductivity.4 The basic consequence of this is that in order to create a controllable resistance state of a device, one needs to establish repeatable control over the domain walls, and to do so at high-resolution, which is the topic of this article. In addition, the similarities between the different ferroic systems have garnered a great interest in comparing different phenomena that are common to two or more ferroic systems. For instance, it has been known for more than 15 years that in small structures (with dimensions of the order 1100 exchange lengths), the spins of ferromagnetic materials tend to form a vortex structure, giving rise to in-plane field closure.5 Inspired by this, recent studies have demonstrated similar field closure arrangements also with ferroelectric and ferroelastic domains.69 Nevertheless, when making such comparisons, one should bear in mind that the Bloch domain walls of ferromagnetic domains allow r 2011 American Chemical Society
a smoothly continuous field closure that constitutes a true vortex, while the field closure in the two other ferroic systems takes place in integer jumps, which in turn occur at a domain wall. Thus, a more accurate parallelization between the formation and annihilation of field-closure structures in the different ferroic systems should refer to vortex-antivortex in ferromagnetism5 versus vertex-antivertex in the case of ferroelectricity/ferroelasticity,10 while the vertices are the intersections between the neighboring domains that form the field closure. Since the scale at which field-closure can take place is a fundamental restriction that dictates the domain size, in order to control nanodomains one must explore the vertex-antivertex mechanism. Contemporary methods for studying the crystallographic structure can be of a very high resolution, up to subatomic.8,11 Nevertheless, most of them do not allow studying its dynamics. Hence, despite their enormous potential, the study of nanodomains is far from being complete. Biferroic materials, such as lead zirconate titanate (PZT) are ideal candidates for studying nanodomain engineering for the following reasons. First, in such materials ferroelasticity couples the mechanical properties to the local crystallographic structure, which in turn is coupled to the electrical properties via ferroelectricity and therefore can be controlled electrically.6,1215 Second, PZT can already be found in commercial nonvolatile memory devices (e.g., in the Sony PlayStation II). Third, in PZT, ferroelastic domains tend to arrange in parallel stripes (“bundle Received: June 22, 2011 Revised: September 9, 2011 Published: October 03, 2011 4619
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Figure 1. Naturally occurring nanopairs. (a) Topography and (b) E-PFM phase images of an area with ferroelastic (in this case, 90°) bundle domains consisting of periodic stripes, and a 180° ferroelectric domain. The ferroelectric domain boundary is marked with a dashed line. (c) Cross-section of phase (purple curve), topography (solid black curve), and E-PFM amplitude (dashed blue curve, multiplied by 2 to aid visibility) along the solid line indicated in (a). We can see the deformation of the surface by almost 1 nm along the 180° domain wall. (e,g) E-PFM amplitude images at the ferroelectric domain interface revealing two sets of naturally occurring twinned nanopairs, i.e., islands of c-domains in a domain stripes adjacent to islands of a-domains in c-domain stripes. The corresponding topography images are shown in (d) and (f). Scale bars are all 100 nm unless stated otherwise.
domains”).6,7,16 Such an arrangement that occurs spontaneously already upon cooling below the Curie temperature assists the experimental perspective of domain engineering. Furthermore, one can control the width and orientation of such stripes. Similar to Kittel’s scaling law for ferromagnetism,17 ferroelectric and ferroelastic domain width can also be determined by the thickness of the sample.18,19 Likewise, it has been demonstrated recently that the orientation of ferroelastic and
ferroelectric bundle domains can be controlled to some extent, for example, in the case of rhombohedral structure6 as well as in bilayer films.12 These factors imply that individual domains of crystallographic phases may also be switchable, creating a possible route toward nanodomain engineering for real device applications. Coupled with the recent discovery that domain walls have increased conductivity, this paves the way for a new family of electronic devices. However, to date controlling the 4620
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Nano Letters domain structure within a bundle domain has remained a challenge. To study the domain structure within the stripes in PZT films we used the method of enhanced piezoresponse force microscopy (E-PFM).20 This method is an improvement of the conventional PFM method that was developed during the 1990s21,22 and allows nanometre resolution in domain imaging. It should be noted at this point that conventional PFM does not have sufficient spatial resolution to observe the structures reported on in this paper, which would otherwise only be observable (but could not be manipulated) by transmission electron microscopy (TEM). Figure 1 shows enhanced PFM (E-PFM) images of the domain distribution in a 60 nm thick, solgel deposited, predominantly (110) PbZr0.3Ti0.7O3 film, similar to that described elsewhere.20,22 In this type of imaging, the amplitude image shows ferroelastic (non-180°) domains as stripes with alternating contrast, as well as ferroelectric (180°) domain walls, which appear as dark fine lines. On the other hand, opposite contrast in the phase image refers to opposite ferroelectric domains, while ferroelastic domains appear with lower contrast (roughly 90°). Natural Nanopairs. Figure 1a,b are the topography and phase images that were taken from the same large area simultaneously. The topography image contains periodic stripes with alternating height difference of a single unit cell (∼34 Å). That is, this area contains a single ferroelastic bundle of periodic a/c domains. However, the spatial variation in contrast in the phase image implies that the out-of-plane polarization in this area is not constant. Rather, the out-of-plane polarization of the c-domains (every second stripe) on the right-hand side of the image is oriented out of the image plane (“up”), while the polarization of the c-domains on the left-hand side of the image is oriented in to the image plane (“down”). Thus, it is interesting to examine the nanometric domain distribution at the boundary of the two macroscopic ferroelectric domains. As can be seen from the topography image, the film is distorted along the line of the ferroelectric domain (dashed line in Figure 1a). This is further highlighted in the cross-section taken through the simultaneously acquired topography, E-PFM amplitude and E-PFM phase images shown in Figure 1c, where the film has buckled upward by almost 1 nm at the domain boundary. A careful look at the boundary between the two macroscopic ferroelectric domains reveals that the ferroelastic domains in this area are distorted in two ways. First, there is a split in some stripes (e.g., at the lowest point of the circle in Figure 1e). Second, in the stripes that are closest to the ferroelectric domain boundary, the crystallographic structure is not homogeneous. Rather, there are areas of c domains (domains with out-of-plane polarization) within a domain stripes (domains with in-plane polarization), as highlighted in Figure 1dg. That is, these c-domain islands are surrounded by planar defects that separate them from the a-stripe of which they are a part. More interestingly even, each of these islands is paired by a twinned island. That is, each c-domain island that is within an a-domain stripe is accompanied by an a-domain island within the neighboring c-domain stripe. Within these nanopairs, each island has a typical dimension of ∼10 nm, which is similar to the stripe domain width and is at least an order of magnitude smaller than the stripe length. The following calculation explores the origin of such small domains. We know that unlike magnetic domain walls, which can travel at supersonic speed,23 ferroelectric walls cannot exceed the
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Figure 2. Repaired nanopairs. (a,b) Topography and E-PFM amplitude zoom in of the nanopair on the right from Figure 1g. (c,d) Topography and E-PFM amplitude of the same area after applying a positive voltage to the AFM tip to produce a polarization in the same direction as the surrounding c domains, it can be seen that the nanopair has been “repaired”. (e) Schematic of repair process.
speed of sound, which is vs = ∼3km/sec in PZT.24 According to the model of viscous drag caused by pairs of zone-boundary acoustic phonons25 at a frequency of26 f = 100 ( 5 cm1=10e12 Hz, there is a characteristic length given by r = vs/f = ∼5 nm. The physical role of the 100 cm1 acoustic phonon energy is that there will be a domain wall energy, at which the acoustic phonon damping causes the wall to reach a terminal velocity. Repairing Natural Nanopairs. The preferential occurrence of the nanotwins next to a macroscopic ferroelectric domain boundary (out-of-plane polarization reversal) implies that in addition to releasing mechanical strain this effect has also an electric role and hence may respond to an external electric field. In particular, the fact that the domain wall energy of such an island is energetically relatively expensive due to the high surface area to volume ratio suggests that it should be metastable and therefore relatively simple to modify its polarization. This can be done locally using the PFM tip to apply to the islands a dc electric field above the coercive value with a polarity similar to the rest of the stripe. The voltage required for manipulating domains of this scale can be calculated as follows. We know that in PZT, the initial domain wall velocity v, just after a small field is applied is 3 ( 0.1 km/sec at the domain nucleation site.25 If we imagine a nucleating domain spreading outward at ca. 3 km/s, then it will slow down even with no collisions (v is proportional to 1/r, the inverse radius of a roughly spherical nanodomain), but it will slow down more abruptly when it hits a defect. However, even this is subtle, because a domain wall hitting a defect can be compared to waves on water hitting a rock (the wave just goes around it). It is only when a domain wall hits a row of ordered defects that the wave is slowed down. Hence, the nucleation of such domains can be associated with a set of crystallographic defects.27 The most commonly found defects in PZT are oxygen vacancies. Thus, one can roughly estimate the intervacancy distance. Using measured values for PZT films, the average oxygen vacancy concentration within 20 nm of the surface of PZT films is of the order 1021 cm3 and is 5 10e18 cm3 in the interior of a 300 nm film.28 The cube root of that (assuming isotropic structure) is ∼1 nm at the surface and 17 nm in the interior. So within the uncertainty in this estimate, these numbers nicely bracket the measured nanodomain diameters. 4621
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Figure 4. Twinning and vertex evolution in nanopairs. (a) A bundle domain with standard continuous 90° domain periodic stripes. (b) Under the external excitation provided by the electric field from the AFM tip, striped domains are bent so that parts of c domains are pushed into a-domains and vice versa. This gives rise to a set of 3-fold vertices (dots) that are coupled. (c) When the excitation continues, the bent segments are pushed further until they meet the next striped domain. This gives rise to 4-fold vertices that are composed of “packed” 3-fold vertices, following Srolovitz and Scott [ref 10]. (d) When the excitation enforces a certain critical strain value, these forced segments form isolated domain islands. The vertices of these twinned nanopairs are now described by Klein bottles. Figure 3. Reversible switching of nanopairs. E-PFM amplitude images. (a) Nanopairs following manipulation shown in Figure 1, where lefthand side nanopair was unaffected. (b) Same area after applying a positive voltage to the tip while on the c island (indicated by the arrow), as shown in Figure 1 resulted in repair of the nanopair. After scanning the area with a negative voltage, the nanopair has been reinduced, demonstrating reversible switching. Scale bar is 100 nm.
We also know that nucleation is complete at a characteristic time that depends logarithmically on the domain perimeter. The importance of the logarithmic dependence is that it varies slowly with size, and hence can be neglected to lowest order. However, the switching speed varies exponentially with applied field, typically in PZT with an activation field of Ea = 250 kV/cm.29 Given that we require a domain velocity of 3 km/sec as above, the required activation field for the nanopairs is ∼10Ea. Thus, if we assume to first order that the tip and sample form a parallel plate capacitor, the voltage between the tip and the bottom electrode should be ∼10 V, higher than the coercive value (∼3 V). In Figure 2a,b, we take a closer look at the right island of Figure 1dg, while Figure 2c,d shows the same area after being manipulated in the manner described above. That is, scanning the area of Figure 1dg while applying 10 V of the same polarity as of the “broken” c-domain stripe in this area, as illustrated in Figure 2e. The resultant domain distribution (Figure 2c,d) demonstrate that indeed, the islands disappeared and the stripes were “repaired” artificially. Moreover, the twinned island (the c-domain island in the neighboring a-domain stripe) also vanished, although it was not excited directly. This suggests that indeed, the two nanotwin domains are coupled. Repeating the same procedure in the left nanopair gave rise to a similar result (Figure 3a,b), indicating the reproducibility of the process. It should be noted that when the manipulation was done with voltages lower than 10 V, but yet higher than the global coercive value, the domain distribution remained unchanged. Reinducing Nanopairs. The unique minute size of the paired nanodomains suggests that they may be pinned. Similarly to ferromagnetism,30 such pinning can be most reasonably associated
with Barkhausen pulses also in the case of ferroelectricity and ferroelasticity.31 In order to examine whether the nanotwins are accompanied by Barkhausen pulses, we attempted to “switch on” the nanotwins. Such an experiment encompasses also a technological interest, as it will determine whether the switching of these nanodomains is reversible and controllable. The procedure for reinducing them follows the same sequence that was behind the experiment to repair them. That is, an area that contains the original defect was scanned, while now a dc voltage producing a field that imposes polarization opposite to that of the entire stripe was applied. Figure 3c shows the domain distribution in this area after the manipulation was done. Again, not only was the original c-domain island reinduced, but its mirror defect, the twinned a-domain, was also reinduced, even though it was not subjected to the applied electric field directly. Furthermore, the nanopair that is further on the right and that was repaired in the previous manipulation was not affected by the current one. This fact indicates that each set of nanotwinned islands can be controlled locally and individually, without affecting the neighboring nanopair. On the basis of the reinduction of nanopairs, one can deduce that they “leave traces behind”, even when apparently repaired. In fact, their disappearance and reappearance exactly at the same place together with the finding that any pair of islands is coupled strongly support the existence of pinning and Barkhausen pulses. Model: Two-Step Vertex Hopping. The main outcome of the above experiments can be summarized as follows: nanodomains of a-islands within c-domain stripes adjacent to nano c-islands within a-domain stripes can exist in bundle domains. The neighboring islands are coupled in pairs. The nanodomains comprising each pair are coupled and can be switched in a reversible manner, while being addressed individually. Furthermore, the nanotwin switching is accompanied by Barkhausen pulses, and during switching, the domain wall velocity is within the acoustic regime. On the basis of this, we suggest the following mechanism. In the first instance, upon phase transformation, a bundle of 4622
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coherent periodic 90° striped domains is formed18 (Figure 4a). Then, an in-plane stress acts at a certain point within the bundle domain inhomogeneously, for example, due to an anisotropic temperature distribution, or substrate clamping, or due to the stress applied from the neighboring grains. As a result, the stripe that contains the stress point will bend toward the adjacent stripe, while pushing the latter with a force per unit area: F = σ 3 b, where σ is the stress tensor and b is the Burgers vector at the bending point32 (Figure 4b). Beyond a certain critical point that is determined by the Barkhausen effect (Figure 4c), the external excitation ceases bending the domains. Rather, the bent area becomes separated from the originating stripes so that two paired islands are formed (Figure 4d). This means that the nanopair switching can be considered as a process of creation and annihilation of a dislocation, which is a result of a shear-stress, τ, applied on a Frank-Read source.33 Thus, F = τ*bx = 2Gb2, giving rise to a critical minimum value for a switching τ: τ¼
2Gb x
ð1Þ
where x is the distance between the pinning points and G is the shear modulus. Since τ is induced by the external field through the inverse piezoelectric effect, τ = dEG, where d is the corresponding element of the piezoelectric tensor. Thus, the minimum value of the 10 V switching voltage in our sample corresponds to τ ≈ 4 GPa (for PZT d = ∼3 1010 m/V, and G = ∼60 GPa24). Substituting in eq 1 this value for τ as well as b = 3.5 Å for the tetragonal PZT gives rise to x = 10 nm, which is in a perfect agreement with the observations. It should be noted that indeed, such elastic Barkhausen noise (or crackling noise) has been observed recently for highly strained ferroelastic materials,34 while the rearrangement of the domain structures are also typical for strong strain interactions, usually due to substrate clamping.35 The closeness to metastable domain states for breaking stripe domains is demonstrated in many systems with defects or strong order/disorder relaxations. Generating domain patterns by cooling a sample through a ferroelastic phase transition will then usually generate tweed structures with adjacent vortexantivortex pairs.36 In these cases, the tweed or the coarser tartan pattern is transient, however, and will give rise over time to pure stripe domains. The tweed can be stabilized by extrinsic defects or by constraints due to the substrate,37 which may be the reason for the stable structures in the current case. To allow a comparison with a vortexantivortex process, this mechanism should be described also from a topological point of view. First, before the stress is turned on, the stripes are parallel so that there are no vertices (Figure 4a). Then, when the domain begins to bend, a 3-fold vertex is formed (Figure 4b). At the critical point, the vertex becomes 4-fold (Figure 4c). As the excitation continues beyond this stage, the bent areas become separated, forming the paired islands (Figure 4d). This gives rise to a close structure of the paired domains and the stripes that host them. Therefore, the a-domain island is coupled with the broken striped a-domains (with a similar behavior of the c-domains), which is in analogy with a Mobius strip, or more precisely, to a Klein bottlelike structure. Large Scale Engineered Nanopairs. Inspired by these native nanotwins, the next question is whether they can be induced artificially even when they do not appear in the native state. From a technological point of view, it is important to determine whether these defects can be induced artificially at will. If so,
Figure 5. Artificially induced nanopair array. (a) Schematics. When a bundle domain (Figure 3a) is scanned while a voltage is applied and the polarity follows a chessboard pattern scheme with the same periodicity as the a and c stripes, the nanotwins are forced to organize into a 2D array. Thus, the vertices are brought closer into a packed structure. Hence, their proximity may enforce the 4-fold vertex (dots). (b) Amplitude image of the native ferroelastic domain distribution in a 300 300 nm2 area. (c) The topography of the same area. (d) Amplitude E-PFM image shows that nanopairs were successfully induced by the manipulation (some are denoted by arrows). (e) A broader view of the region (E-PFM amplitude superimposed on topography, 440 440 nm2 area) with a lower readout voltage, after being manipulated by the same scheme with a 10 10 chessboard pattern indicating that high-density defect generation is possible.
not only will it mean that the crystallographic structure of a material can be tuned locally, down to areas as small as 10 10 nm2, but it will also advance the potential of applying such patterns in real high-density (>5 Tbit/in2) memory devices. Recent studies have shown that when scanning areas that contain several stripes (>10), while applying a voltage higher than the coercive value, the ferroelastic stripe domains can be “dragged” to support the new ferroelectric domain configuration induced by the tip.8,9,12 Hence, based on the mechanism suggested above, one can use a similar concept to form the twinned nanodomains while bending the stripes beyond the critical point. In the framework of the current study, local stress can be applied at chosen areas when a dc voltage is applied by the tip through the inverse-piezoelectric effect.13 Moreover, the fact that the tip is moving during the excitation results in a local gradient of the electric field that can contribute to the stress also through the inverse-flexoelectric effect.38 It should be noted that dislocation pairs (or disclination pairs) are well-known in nematic liquid crystals,39 and the correspondence with domain vertices has been made before for incommensurate ferroelectrics. However, there is an important difference between incommensurate dielectrics, where these vertices occur at random defect sites (e.g., Na-vacancies40 in Ba2NaNb5O15), and systems such as YMnO3, where they are topologically required (and periodic) even in the absence of defects.41 Using this topological concept, bending the striped domains in an array of stress points gives rise to an array of 3-fold vertices. This means that the 2D four-state Potts model of Srolovitz and Scott can be applied.10 Owing to the proximity of the 3-fold vertices in such an array, they will be driven to form a 4-fold vertex array (Figure 4a). Figure 5b,c shows the native domain distribution in an area with well-behaved ferroelastic stripe domains. This area was divided into a 6 6 square grid, so that the size of each of the squares is 4623
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Nano Letters identical to the width of a pair of a and c stripes. Each square was scanned while applying a voltage larger than the coercive value with an opposite polarity between any two neighboring squares, forming a chessboard pattern. Figure 5d indicates that the manipulation formed an array of nanotwins successfully. One of the reasons that the imaging is enhanced with E-PFM is the relatively high driving voltage. Nevertheless, we found that the artificially induced nanodomains are not as stable as the native nanopairs. Therefore, the artificial domains relax due to the energy they gain from the high readout voltage. Indeed, when the readout voltage is lower, artificially induced nanodomains are stable even over larger areas, as seen in Figure 5e. Comparing the relaxation time constant of the artificially formed nanodomain array and that of a similar array of a larger scale12 indicates that the latter is a much slower process, by at least 1 order of magnitude. This leads to the conclusion that the energy associated with rotating bundles is higher than that of patterned nanodomains. In terms of real device applications, a key point is that some domain walls have increased conductivity. The demonstration here in the level of control over the positioning of these domain walls is a significant first step in generating ultra high density switchable electrical channels for future device applications. Summary. To conclude, we presented a new type of nano ferroelectric-ferroelastic domain in which islands of crystallographic domains occur within larger elastic domains, such as striped domains. We showed that each of such islands is coupled with a mirror island that is in the adjacent elastic domain. We demonstrated that these islands of crystallographic domains can be switched by an external electric field locally, individually, reproducibly, and reversibly. We discussed in detail a mechanism that explains the formation and switching of these domains and we supported the model with further experimental results. In particular, we suggested through both quantitative and topological analyses that they are formed by domain bending that exceeds a critical point. Moreover, the twinned nanoislands are paired so that their switching is accompanied by Barkhausen pulses. Inspired by the natural nanopairs, we showed that these defects can be induced artificially in a large-scale array, so that the local crystallographic structure of the material can be controlled in a systematic manner at the nanometre scale, opening the route for commercial ferroicbased massive-density nonvolatile memory devices. Methods. The 60 nm thick PZT films under examination are similar to those described elsewhere.20,22 The imaging was done with the atomic force microscope (AFM) MFP-3D of Asylum Research, while the imaging conditions followed the E-PFM method. Cantilevers were metal coated from Budget Sensors and a nominal spring constant of 0.2 N/m. The driving ac voltage was mostly 5 V (amplitude) and 0.52 V for the low readout voltage experiments, while the driving frequency was 5760 kHz, which is ∼5% lower than the in-contact resonance frequency of the cantilever and within the linear regime of the E-PFM framework. More detail about E-PFM can be found elsewhere.14,20 The manipulation experiments were done with the help of the IGOR software that is supplied with the Asylum AFM, while usually a dc voltage of 10 V was applied to switch the domains. All images were taken under ambient conditions. The data analysis was done with IGOR, as well as with WSxM.42
’ AUTHOR INFORMATION Corresponding Author
*E-mail: (Y.I.)
[email protected]; (C.D.)
[email protected].
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’ ACKNOWLEDGMENT The authors would like to acknowledge Gil Rosenman for fruitful discussion. Some parts of the work were carried out as part of the Nokia-University of Cambridge collaboration in Nanotechnology. ’ REFERENCES (1) Ramesh, R.; Schlom, D. G. Science 2002, 296 (5575), 1975. (2) (a) Chu, M. W.; Szafraniak, I.; Scholz, R. Nat. Mater. 2004, 3, 87. € ur, U.; € Alivov, Ya. I.; Liu, C. J. Appl. Phys. 2005, 98, 041301. (b) Ozg€ (3) Kainuma, R.; Imano, Y.; Ito, W.; et al. Nature 2006, 439 (7079), 957. (4) Seidel, J.; Martin, L. W.; He, Q.; et al. Nat. Mater. 2009, 8 (3), 229. Wang, J.; Neaton, J. B.; Zheng, H.; et al. Science 2003, 299 (5613), 1719. (5) Cowburn, R. P.; Koltsov, D. K.; Adeyeye, A. O.; et al. Phys. Rev. Lett. 1999, 83 (5), 1042. Matsuda, T.; Harada, K.; Kasai, H.; et al. Science 1996, 271 (5254), 1393. (6) Balke, N.; Choudhury, S.; Jesse, S. Nat. Nanotechnol. 2009, 4, 868–875. (7) Ivry, Y.; Chu, D. P.; Scott, J. F.; et al. Phys. Rev. Lett. 2010, 104, 207602. (8) Jia, C. L.; Urban, K. W.; Alexe, M.; et al. Science 2011, 331 (6023), 1420. (9) Schilling, A.; Byrne, D.; Catalan, G.; et al. Nano Lett. 2009, 9 (9), 3359. (10) Srolovitz, D. J.; Scott, J. F. Phys. Rev. B 1986, 34 (3), 1815. (11) (a) Jia, C. L.; Mi, S. B.; Urban, K.; et al. Nat. Mater. 2008, 7, 57. (b) Nellist, P. D.; Chisholm, M. F.; Dellby, N. Science 2004, 305, 1741. (12) Anbusathaiah, V.; Kan, D.; Kartawidjaja, F. C.; et al. Adv. Mater. 2009, 21 (34), 3497. (13) Chen, L.; Ouyang, J.; Ganpule, C. S.; et al. Appl. Phys. Lett. 2004, 84 (2), 254. (14) Ivry, Y.; Wang, N.; Chu, D. P.; et al. Phys. Rev. B 2010, 81, 174118. (15) Nagarajan, V.; Roytburd, A.; Stanishevsky, A.; et al. Nat. Mater. 2003, 2 (1), 43. (16) (a) Ivry, Y.; Chu, D. P.; Durkan, C Nanotechnology 2009, 21, 065702. (b) McGilly, L. J.; Schilling, A.; Gregg, J. M. Nano Letters 2010, 10 (10), 4200. (17) Kittel, C. Phys. Rev. 1946, 70 (11 & 12), 965. (18) Roytburd, A. L. Phys. Status Solidi A 1976, 37 (1), 329. (19) Schilling, A.; Adams, T. B.; Bowman, R. M.; et al. , Physical Review B (2006) 74 (2). (20) Ivry, Y.; Chu, D.; Durkan, C. Appl. Phys. Lett. 2009, 94, 162903. (21) (a) Alexe, M.; Gruverman, A. Nanoscale Characterisation of Ferrorlectric Materials (Scanning Probe Microscopy Approach); SpringerVerlad: Heidelberg, 2004. (b) Gruverman, A.; Auciello, O.; Tokumoto, H. J. Vac. Sci. Technol. B 1996, 14 (2), 602. (c) G€uthner, P.; Dransfeld, K. Appl. Phys. Lett. 1992, 61 (9), 1137. (22) Durkan, C.; Welland, M. E.; Chu, D. P.; et al. Phys. Rev. B 1999, 60 (23), 16198. (23) Demokritov, S. O.; Kreines, N. M.; Kudinov, V. I. JETP Lett. 1985, 41 (1), 46. (24) Low, T. S.; Guo, W. J. Microelectromech. Syst. 1995, 4 (4), 230. (25) Dawber, M.; Jung, D. J.; Scott, J. F. Appl. Phys. Lett. 2003, 82 (3), 436. (26) Ghosez, P.; Cockayne, E.; Waghmare, U. V.; et al. Phys. Rev. B 1999, 60 (2), 836. (27) Dawber, M.; Scott, J. F. Appl. Phys. Lett. 2000, 76, 3801. (28) Mihara, T.; Watanabe, H.; Yoshimoria, H.; et al. Integr. Ferroelectr. 1994, 1, 269. (29) Scott, J. F.; Kammerdiner, L.; Parris, M.; et al. J. Appl. Phys. 1988, 64 (2), 787. (30) Alessandro, B.; Beatrice, C.; Bertotti, G.; et al. J. Appl. Phys. 1990, 68 (6), 2901. 4624
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