Nature | Vol 577 | 2 January 2020 | 49findings and other computations given in Methods, demonstrate that
the inverse-transition phenomenon is robust against intrinsic and
extrinsic parameters such as boundary conditions, film thickness, as
well as screening conditions and misfit strain. Naturally, varying the
conditions in the studied systems (Pb(Zr0.4Ti0.6)O 3 and BiFeO 3 ) yields
different transition temperatures, as well as different types of domain
walls (for example, 180° in Pb(Zr0.4Ti0.6)O 3 versus 109° and 71° domain
walls in BiFeO 3 ) and labyrinthine morphologies, with no incidence on
the occurrence of the inverse transition.
Figure 4a provides the evolution with temperature of the free-energy-
like potential associated with the transverse component of dipoles at
the domain walls. Each curve is obtained by averaging over the distri-
butions of 100 labyrinthine realizations. Results are gathered at 10 K,
110 K and 210 K upon heating the labyrinthine states and the transverse
component is taken to be the projection of dipoles onto the domain-
wall normal at each point. At 10 K, the potential features three minima,
the leftmost and rightmost ones being associated with the Néel nature
of the domain walls. The minimum at zero is associated with dipoles–3–2–1^0123–3–2–10123–3–3–2–1 0123–2–10123aqx–3 –2 –1 0123
aqx–3 –2 –1 0123
aqx–3 –2 –1 0123
aqx–3 –2 –1 0123
aqx–3 –2 –1 0 123
aqxaqyaqyaqyaqyaqyaqy–3–2–10123–3–2–10123–3–2–10123abc de f gTTTTinvinv TTTTccStripesLabyrinth Paraelectric100150 200250 300350400 45000.10.20.30.40.50.6T (K)OhvFig. 2 | Inverse transition simulations. a, Temperature dependence of the
orientational order parameter Ohv upon slowly heating the labyrinthine state of
an 80 × 80 × 5 film of Pb(Zr0.4Ti0.6)O 3. b–g, The evolution of the labyrinthine
domain pattern in the middle layer of the film with increasing temperature:
10 K (b), 110 K (c), 185 K (d), 260 K (e), 335 K (f) and 410 K (g). Grey (red) dipoles
are oriented along the [001] ([001]) pseudo-cubic direction. The structure
factor plots obtained by Fourier transformation of the z component of the
corresponding dipolar field are also provided, where aqx and aqy are the x and y
components of the dimensionless wave vector, which take values within the
interval from −π to +π (a is the in-plane lattice constant). The colouring
corresponds to the value of the structure factor, with white (pink) indicating
the lowest (highest) value. The colour scale is the same for all plots.As-grown 773 K 1,023 K1,073 K10 K 200 K 400 K 600 K8 00 K1,000 K1,200 K0360Phase (°)0 Current (pA)a bcTinvTinv30Fig. 3 | Experimental observation and simulations of the inverse transition
in BiFeO 3 films. a, In-plane piezoresponse force microscopy phase images of a
95-nm-thick BiFeO 3 film grown on a (110)-oriented orthorhombic DyScO 3
substrate, for the as-grown sample, and the same sample after annealing at
77 3 K, 1,023 K and 1,073 K. The images are 5 × 5 μm^2. b, In-plane piezoresponse
force microscopy image of a 30-nm-thick BiFeO 3 film grown on SrRuO 3 (10 nm)/
DyScO 3 (110) (top left). Scale bar, 2 μm. Conducting atomic force microscopy
(current mapping) images acquired with Vd.c. = 1.7 V applied on a SrRuO 3 bottom
electrode in periodic stripy areas (bottom left) and defected areas (red dashed
lines) with high-conduction spots at three-fold junctions (top right) and end-
point (bottom right) topological defects. Scale bars, 500 nm. c, Distribution of
the z component of polarization (red to green indicate negative to positive
values) in a middle layer of BiFeO 3 film at different temperatures, as obtained
from 36 × 36 × 10 supercell effective-Hamiltonian-based Monte Carlo
simulations under periodic boundary conditions with ideal short-circuit
screening and isotropic misfit strain of −0.16%. The system was abruptly
quenched from 2,000 K to 10 K and consequently progressively heated up with
40,000 relaxation sweeps at each temperature. In the simulations, we find that
below Tinv, the system exhibits mixed 109° and 71° domain walls, while above
Tinv, only 109° domain walls are observed.