This is the accepted version of the article: Everhardt A.S., Damerio S., Zorn J.A., Zhou S., Domingo N., Catalan G., Salje E.K.H., Chen L.-Q., Noheda B.. Periodicity-Doubling Cascades: Direct Observation in Ferroelastic Materials. Physical Review Letters, (2019). 123. 087603: - . 10.1103/PhysRevLett.123.087603. Available at: https://dx.doi.org/10.1103/PhysRevLett.123.087603 Periodicity doubling cascades: direct observation in ferroelastic materials Arnoud S. Everhardt,1, ∗ Silvia Damerio∗ , 1, † Jacob A. Zorn,2 Silang Zhou,1 Neus Domingo,3 Gustau Catalan,3, 4 Ekhard K. H. Salje,5 Long-Quing Chen,2 and Beatriz Noheda1, 6, ‡ 1 Zernike Institute for Advanced Materials, University of Groningen, The Netherlands 2 Department of Materials Science and Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802, USA 3 Catalan Institute of Nanoscience and Nanotechnology (ICN2), Spain 4 ICREA, 08193 Barcelona, Spain 5 University of Cambridge, UK 6 CogniGron Center, University of Groningen, The Netherlands (Dated: May 6, 2019) Very sensitive responses to external forces are found near phase transitions. However, transition dynamics and pre-equilibrium phenomena are difficult to detect and control. We have observed that the equilibrium domain structure following a phase transition in ferroelectric/ferroelastic BaT iO3 , is attained by halving of the domain periodicity multiple times. The process is reversible, with periodicity doubling as temperature is increased. This observation is reminiscent of the period- doubling cascades generally observed during bifurcation phenomena and, thus, it conforms to the ’spatial chaos’ regime earlier proposed by Jensen and Bak[1] for systems with competing spatial modulations. Keywords: ferroelastic thin films; domain dynamics; domain patterns; periodicity doubling; bifurcation Current interest in adaptable electronics calls for new width (w), is determined by the competition between the paradigms of material systems with multiple metastable elastic energy in the domains and the formation energy states. Functional materials with modulated phases of domain walls and is a function of the thin film thick- bring interesting possibilities in this direction[1]. Ferroic ness (d). For d > w, the relation w = βd1/2 holds, as a materials are good prospective candidates because the particular case of the Kittel’s law [6, 9–11]. In the fer- modulation can be controlled by external magnetic, elec- roelastic case, geometrical effects are also important and tric or stress fields. In magnetic materials, the presence discretization of domain widths, with minimum sizes de- of competing interactions can lead to wealth of modu- termined by the need of lateral lattice coherence at the lated structures, exemplified by the axial next-nearest- domain wall, have also been shown [12, 13]. Recently, an neighbor Ising (ANNNI) model [2]. In ferroelectrics, in- original hydrodynamics-like approach has been put for- teresting modulations in the form of domain patterns in- ward, in which non-equilibrium ferroic domain structures volve not only domains with alternating up and down po- are rationalized with respect to surface folding, wrinkling larization but also vortices [3] and ferroelectric skyrmions and relaxation [14]. [4]. Ferroelectrics are often also ferroelastic [5] and dis- As both misfit strain and domain wall energies change play modulations of the strain. with temperature, the equilibrium patterns are temper- When ferroelastic materials are grown in thin film form ature dependent and the question arises of how does on a suitable crystalline substrate, they can be subjected the system evolves towards (global or local) equilib- to epitaxial strain, which typically relaxes by the forma- rium. While studies of ferroelectric domain formation tion of ferroelastic domains [6, 7]. In the simple case of and switching are common [15–20], less attention has ferroelastic systems with orthogonal lattices grown on a been paid to microscopy studies of temperature-driven cubic substrate, different types of 90◦ domain configura- annihilation of ferroelastic nanodomains [21, 22] and it tions form, either so-called a/c domains (with the long is only recently that experimental developments have al- c-axis alternating in-plane and out-of-plane) or a/b do- lowed the in-situ study of domain dynamics [23–28]. mains (long axis fully in-plane), depending on the sign In the present work, we report the direct observation of the strain imposed on the film by the substrate [8]. of ferroelastic/ferroelectric domain evolution by sequen- In the absence of dislocations or other defects, ferroelas- tial periodicity halving and doubling on BaT iO3 thin tic domains in epitaxial films are expected to alternate films. Moreover, we present the results of phase field periodically [6–8]. simulations [29–32] that support the observations show- The periodicity of this modulation, or the domain ing that domain wall nucleation takes place in the center of existing domains. Period doubling cascades, associ- ated to the proximity of order-to-chaos transitions, have ∗ Equal contribution; Presently at: Materials Science Division, been often observed as frequency doubling sequences in Lawrence Berkeley National Laboratory, Berkeley, CA 94720, the time regime, but they have not yet been reported in USA spatially modulated systems. The results, thus, present † s.damerio@rug.nl strong evidence that ferroelastic systems with competing ‡ b.noheda@rug.nl modulations are at the edge of chaos, as predicted for 2 magnetic materials by Jensen and Bak [1]. The experiments have been performed on BaT iO3 films grown under low epitaxial strain on SrRuO3 - buffered N dScO3 substrates, as described in Ref.[33]. The low-strain condition flattens the energy landscape such that different ferroelectric domain configurations can be accessed within a moderate temperature range [34]. In particular, the films display a paraelectric- ferroelectric phase transition at 130 ◦C and a second transition at about 50 ◦C. The latter takes place in between two quite complex ferroelectric and ferroelas- tic phases (see Supplementary Material-section I for de- tails) but, for the sake of clarity, it can simply be de- scribed as a transition from a high-temperature, pseudo- tetragonal a/c domain structure to a low-temperature, pseudo-orthorhombic a/b domain structure [35]. Unlike in other ferroics, the transformation from a/c to a/b do- mains in these films is slow enough to be followed with piezoresponse force microscopy (PFM) (see Supplemen- tary Material-section II.a for details). Typical domain configurations below and above the transition can be seen in the insets of Fig.1 a and b, at 25 ◦C and 70 ◦C, respectively. Inset in Fig.1 a shows a/b domains with domain walls projections parallel to the [110] direction. Inset in Fig.1 b shows a/c domain walls with projections parallel to the [010] direction. In- vestigation of a large number of samples with thicknesses ranging from 30 nm to 330 nm confirms the robustness of the domain configurations and the expected w = βd1/2 FIG. 1. Domain widths for different film thicknesses, de- scaling law, with β ∼ 10 nm1/2 for both the a/b and termined by Grazing Incidence Diffraction (GID) XRD, off- a/c domains, as shown in Fig.1. Here, the a/c struc- specular XRD around the (204) Bragg peaks or PFM (see ture refers to the domains generated while cooling from ref.[33]). The data are fitted to a square-root law with pa- the paraelectric phase. When the a/c domain configura- rameter β shown in the legend for the a/b (a) and the a/c (b) tion is reached upon heating from the low-temperature domain states. The insets show amplitude LPFM images of a/b configuration, β is reduced to ∼ 7 nm1/2 , reflect- a 80 nm thick BaTiO3 film at 25o C (a) and 70o C (b). The ing the important role of pre-existing domain walls in images are of a 1µmx1µm area with the edges along [100] and the nucleation of new domain walls (see Supplementary [010]. Material-section III for details). In order to investigate the evolution of the domain for- mation, the films were heated to 200 ◦C and measured main configuration has disappeared (see inset of Fig.1a). during cooling. During the cool-down process, the a/c Thus, two sequential periodicity halving events take place structure arises below the para-ferro transition. At a during the transition from the a/c to the a/b domain temperature of 70 ◦C, the first b-domains start appearing configuration on cooling. inside the a/c matrix and, increasing in number as the A similar mechanism, albeit shifted in temperature due temperature lowers. Fig.2 shows images of two stages of to thermal hysteresis, is observed when heating the sam- the domain evolution taken while cooling with 5 ◦C steps. ples through the phase transition, as shown in Fig.3. At 50 ◦C (Fig.2a), a/b and a/c domain walls coexist. The transition from a/b to a/c domains is followed on The a/b domain walls, signalled with dashed lines, reveal heating from 30 ◦C to 70 ◦C at 5 ◦C steps. The initial a domain periodicity of (500 ± 50) nm for this 170 nm a/b domain periodicity at 30 ◦C is w0 = 100 nm for a thick film. The periodicity values are obtained by per- sample with thickness of 90 nm (Fig.3a). The domain forming the Fast Fourier Transform (FFT) of the LPFM walls start to rearrange and at 60 ◦C (Fig.3b), domains images (see Supplementary Material-section II.a for de- with w = 2w0 are visible, as indicated by the green ar- tails). Further cooling down to 30 ◦C (Fig.2b) shows new rows. With increasing temperature up to 65 ◦C (Fig.3c), b-domains appearing halfway in between the existing do- w = 4w0 becomes the most abundant period (blue ar- main walls, halving the domain periodicity to (250 ± 20) rows) (see FFT in Fig.3d). In addition, a domain with a nm. This value is, in turn, approximately double that size of w = 8w0 (violet arrow) can be observed within the found at room temperature, after the sample has equi- area of the image (See Supplementary Material, section librated for several hours and the high-temperature do- III, for the FFT of all three images). So the transition 3 FIG. 2. Amplitude lateral PFM (LPFM) images of a 170 nm thick BaT iO3 film on a N dScO3 substrate. The transition from a/c to a/b domains is followed on cooling and images are shown in the coexistance regime at 50 ◦C (a) and 30 ◦C (b). The dashed lines indicate the a/b domain walls, whose in-plane projections are parallel to the [110] direction; while the solid lines indicate two a/c domain walls, with in-plane projection parallel to the [010] direction. The blue double arrows signal the initial periodicity of w0 ∼ 500 nm. At 30◦C, new a/b domains appear in the center of the existing domains. The green double arrows in (b) indicate the observed a/b domains with w = 0.5w0 ∼ 250nm . FIG. 3. Amplitude of the LPFM signal of a 90 nm thick BaT iO3 film on a N dScO3 substrate measured during heating at 30 ◦C (a), 60 ◦C (b) and 65 ◦C (c). (d) FFT of the images to the a/c phase takes place by sequentially annihilat- in (b) and (c), from which the domain period is determined. ing one every other b-domain and, thus, the apparent The basic domain width w0 = 110 nm is indicated by the domain periodicities follow a Cantor set sequence with white arrows in (a) and (b). Other detected periodicities are w = 2n w0 [36]. Pre-fractal domain patterns following w = 2w0 = 200 nm (green) in (b) and (c), as well as w = Cantor set sequences have been observed in ferroelectrics 4w0 = 400 nm (blue). A domain with size w = 8w0 = 800 under electric field [37]. Subsequent periodicity doubling nm (violet) is also observed in (c). events in these material have also been observed by re- ciprocal space mapping (see Supplementary Materials, section IV, for further details). Our experiments also show the self-repairing of ’wrongly’ nucleated domain walls, indicating that the center of existing domains is the equilibrium position for new domains. Fig.4a shows b-domains right after their formation, displaying unequal widths, probably due to pinning by the local defect structure. However, the elas- tic forces in this situation lead to a movement of the b-domains with respect to each other to reach a position FIG. 4. LPFM amplitude images of a 170 nm thick BaT iO3 equidistant between the neighboring b-domains (Fig.4b). film. (a) shows five different b-domains (darker diagonal It is well known that defects act as preferred nucleation bands) with different separations (a-domain widths). (b) Af- points for new domains [11, 38], as observed in Fig.2a, ter waiting for 1.5 hours, the domain walls re-arrange towards where a defect (two a/c domains merging into one) close achieving a periodic configuration. The dashed lines signal to the center of an a-domain induces the bending of the the position of the b-domains in the ideal periodic case. new wall in Fig.2b. Due to the random nature of these defects, we have occasionally observed deviations from the apparent 2n domain evolution law. high temperatures. Fig.5a shows that the domain struc- Phase-field simulations are performed using the time- ture of the high temperature phase, a/c, agrees with the depedent Ginzburg-Landau equations for BaT iO3 under experimental observations previously captured by Ever- strain (see Supplementary Material-Section II.b for de- hardt et al. [33] with {101} domain walls (thus con- tails) and the results are shown in Fig.5. It is seen that sistent with the [010] projections on the PFM images). the alternating pseudo-a/b phase appears at low temper- These simulations also demonstrate the change in do- atures and an a/c tetragonal structure predominates at main wall orientation from {101} to {110}, also consis- 4 tent with PFM observations, with the ’new’ (orthorhom- and local periodicity disorder [49–51]. Finally, we men- bic) domains forming in the middle of the ’old’ (tetrag- tion that in free crystals periodicity of domain walls, pe- onal) domains. An analysis of the phase-field simulation riod doubling and disorder can be induced by the side- results in a β coefficient of β=7nm1/2 at high temper- ways movement of domains in terms of front propagation ature and β =10nm1/2 at low temperature, supporting [40, 45, 52, 53]. our experimental observations in Fig.1). Recent work [54–56] shows that, during the ferroelectic Investigation of the elastic energies agrees with the phase transition at 120 ◦C, BaT iO3 crystals display tran- experimental observation of stress relaxation being sient intersections between polar ferroelastic/ferroelectric achieved by the formation of new domains at the cen- 90◦ walls and the (001) surface that are electrically ter of old domains. To increase the simulation areas, charged and persist up to temperatures above TC , while we have also performed 2D simulations (see Fig.5b). In the bulk has already transformed into the cubic phase. this case, the orientation of the polarization in the two Interestingly, a sequential period halving evolution in phases does not fully agree with the experiment due to which some domains are missing during the cooling down the simplicity of the model, but successive nucleation of process, most likely due to the local presence of other new domains halfway between existing domain walls is type of defects taking care of the stress relaxation, can clearly observed as the temperature is decreased. In ad- explain the discretization of domain sizes that we have dition, it is shown that the ’new’ domains nucleate near detected in various ferroelastic materials [57–61](see Sup- the film-substrate interface, at the 90◦ (100) domain wall plementary Material- section V). and along the < 110 > directions. In conclusion, we directly observe spatial period dou- bling/halving sequences consistent with those found as Thus, periodicity changes take place by the formation part of the clock-model calculations [62]. This behavior of new stress-relieving elements (singularities) exactly in belongs to a class of scaling phenomena known as period- the center of the old domains, as already predicted by doubling cascades that are mathematically investigated dynamic pattern formation [39, 40]. In the current mea- by means of bifurcation theory and characterize systems surements, a new b-domain (thus, a pair of domain walls) at the ’edge-of-chaos’. Even though a link with spatially conforms such singularity, but other types such as single modulated phases of matter has been made [1], spatial domain walls, dislocations, cracks, etc. are governed by period doubling cascades had not yet been observed ex- similar physics. In the case of ferroelastic domains, in- perimentally. Thus, our observation of domain periodic- creasing the strain (e.g. by decreasing the temperature) ity halving should not be exclusive of the system under would repeat the process between a first-generation sin- investigation, but represents a more general mechanism gularity and a second generation singularity, and so on. of transformation in materials with competing periodic More generally, in addition to boundary conditions and structures. the effect of substrates, effective interactions between do- main walls and evolution of domain wall structures can come from other physical processes. If several order pa- ACKNOWLEDGMENTS rameters are activated to generate a domain wall, the energy landscape is complex with several minima, which The authors are grateful to Janusz Przeslawski, Marty combine to define the domains and the wall structures. Gregg, Jim Scott and Yachin Yvry for useful discussions. Lateral extensions of domain walls would favour the wall- A.S.E., S.D. and B.N. acknowledge financial support wall interactions and such phenomena have been explored from the alumni organization of the University of Gronin- in bi-quadratic or linear quadratic coupling between or- gen, De Aduarderking (Ubbo Emmius Fonds). Parts of der parameters [41–46]. Atomistic approaches are very this research were carried out at the light source Petra limited [47, 48] so far and no firm conclusions can be III at DESY, a member of the Helmholtz Association reached for direct interactions. If no such wall-wall inter- (HGF). G.C. and N.D. acknowledge projects FIS2015- action existed and no external constraints occur, then the 73932-JIN and MAT2016-77100-C2-1-P from the Spanish pattern energy density would depend only on the total MINECO and 2017 -SGR-579 project from the Generali- number of domain walls and their self-energy. tat de Catalunya. All work at ICN2 is also supported by Periodicity can also originate from nucleation. Do- the Severo Ochoa Program (Grant No. SEV-2017-0706). mains nucleate from the surface or an interface and the The work at Penn State is supported by the U.S. Depart- nucleation centres repel each other. 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Three-dimensional (a) and two-dimensional (b) domain structures generated by the phase field method, indicating the misfit strain conditions that led to their formation. In (a), as the temperature decreases from 100◦C to 25◦C, a change in domain wall orientation from {101} to {110} takes place signalling the phase transition. It is also apparent that the new domains form at the half-way point between the original domain walls. In (b), subsequent nucleation of a new domain halfway between existing domain walls (shown by the open vertical arrows) is also shown to take place as the temperature decreases from 80◦C(i) to 60◦C (ii) and to 25◦C (iii). Domain notation: a1 = (P0 , 0, 0), a2 = (0, P0 , 0), c = (0, 0, P0 ), O1+ = (P0 , P0 , 0), O1− = (−P0 , −P0 , 0), O2+ = (P0 , −P0 , 0), O2− = (−P0 , P0 , 0), O4+ = (P0 , 0, −P0 ), O5+ = (0, P0 , P0 ), O5− = (0, −P0 , −P0 ), O6+ = (0, P0 , −P0 ), O6− = (0, −P0 , P0 ) and R1 = (P0 , −P0 , −P0 ).