The ultrathin limit of improper ferroelectricity

The secondary nature of polarization in improper ferroelectrics promotes functional properties beyond those of conventional ferroelectrics. In technologically relevant ultrathin films, however, the improper ferroelectric behavior remains largely unexplored. Here, we probe the emergence of the coupled improper polarization and primary distortive order parameter in thin films of hexagonal YMnO3. Combining state-of-the-art in situ characterization techniques separately addressing the improper ferroelectric state and its distortive driving force, we reveal a pronounced thickness dependence of the improper polarization, which we show to originate from the strong modification of the primary order at epitaxial interfaces. Nanoscale confinement effects on the primary order parameter reduce the temperature of the phase transition, which we exploit to visualize its order-disorder character with atomic resolution. Our results advance the understanding of the evolution of improper ferroelectricity within the confinement of ultrathin films, which is essential for their successful implementation in nanoscale applications.


. Strain mapping of a fully relaxed YMnO 3 thin film on YSZ(111). a,
Representative HAADF-STEM image of the film-substrate interface in a 17 unit-cell film obtained as an average of a time series consisting of 15 frames acquired with 1 μs dwell time. Two Burgers circuits, built following the finish-start right-handed convention 10 , are shown by the white arrows. Each Burgers circuit encloses a pair of partial dislocations and displays a global Burgers vector b = 1/3[-1-20] (shown by the yellow arrows), that is, an extra half-plane of atoms is inserted in the YMnO 3 film. b, Corresponding inplane (||[-1-20]) and out-of-plane (||[001]) strain maps (ε yy and ε zz , respectively), obtained by geometric phase analysis 11 , revealing the presence of two pairs of partial dislocations separated by ~5 nm. The color bar gives the strain change in percent. The ε yy strain map shows retention of the YMnO 3 bulk-lattice parameters from the first unit cell mediated by misfit dislocations. The YMnO 3 film clearly shows a smaller in-plane lattice parameter (green color) than YSZ and thus does not adopt the larger lattice parameter from YSZ. In particular, the in-plane lattice parameter measured at the YMnO 3 film is -2.1(±0.6)% compared to YSZ, resulting in a lattice constant of 6.14 Å, in excellent agreement with a fully relaxed YMnO 3 film 12 .
4 Supplementary Fig. 3. ISHG polarization dependence in the ferroelectric phase of YMnO 3 thin films. a, Schematic of the ISHG reflection geometry. The directions of incoming and detected light polarization (E in and E out ) are given by their respective angles with respect to the vertical axis. b-d, Comparison between simulated and experimental SHG polarimetry confirms the polar 6mm symmetry associated to the ferroelectric phase in YMnO 3 , where the amplitude of the SHG light is proportional to P S . The simulations fit to the I ISHG when considering interference with a small background SHG contribution from the surface of the sample. The ISHG polarimetry is measured while: (b) E in and E out are rotated either in parallel (black data points) or orthogonal to each other (red data points), (c) E in is rotated while E out is fixed either at 0° (black data points) or 90° (red data points) and (d) E out is rotated while E in is fixed either at 0° (black data points) or 90° (red data points).

Supplementary Note 1: In-situ optical second harmonic generation (ISHG)
The amplitude of the IISHG signal depends on the incident and detected light polarization, which in the electric-dipole approximation for SHG can be expressed as 1 : (1) The ferroelectric phase of YMnO3 has a 6mm point-group symmetry with the following three independent tensor components of the second-order non-linear susceptibility: where x, y, and z are the Cartesian coordinates of the hexagonal crystal lattice with x || a and z || c.
These tensor components become non-zero with the breaking of inversion symmetry due to a net spontaneous polarization and are thus proportional to the order parameter P s 2 .
To confirm this polar signature of the SHG signal in the YMnO3 thin films below the transition temperature, the light-polarization dependence (i.e. polarimetry) of the SHG signal was compared to the expected 6mm symmetry of the SHG obtained by simulations. The agreement seen between experimental data and simulation in Supplementary Fig. 3b-d was reached when taking the relative tensor components = 1, = 0.16 , = 0.3 for the 6mm symmetry, and a small surface-induced SHG contribution was introduced in the simulation. Because of the high surfaceto-volume ratio that is particular to thin films, SHG due to inversion-symmetry breaking at the surface often visibly interfere with the bulk-like SHG 3 .
Based on the polarimetry in the polar phase of the YMnO3 thin films, we chose a measurement geometry for the temperature-dependent SHG that singles out the dominating -related SHG component while minimizing the observed surface contribution. The samples were therefore oriented such that in tilted incidence (~ 45°), we project vertical light polarization (Ein = 0° in Supplementary Fig. 3) onto the vertical y-axis, and horizontal light polarization (Ein = 90° in Supplementary Fig. 3) onto the tilted (x + z) direction. We chose the polarization configuration Ein = 0° and Eout = 90° for the measurement of TC film . Hence, we could detect the intensity as a function of temperature. This signal was subsequently normalized by the square of the incident light intensity, Iin: to obtain the polarization dependence vs. temperature in the thin films.

Supplementary Note 2: Scanning transmission electron microscopy
Nanobeam electron diffraction patterns were acquired in the STEM mode by using the so-called microprobe mode (with the minicondenser lens excited) enabling a probe semi-convergence angle of 1 mrad. Representative nanobeam diffraction patterns acquired at room temperature, 225 ºC and 250 ºC are shown in Supplementary Fig. 4. At room temperature, the characteristic superlattice reflections arising from the structural trimerization of the ferroelectric YMnO3 lattice are present.
At 225 ºC, the intensity of the superlattice reflections start vanishing, and at 250 ºC only the main reflections due to the basic structure of the non-ferroelectric phase are visible. When cooling down the sample to room temperature the superlattice reflections appear again demonstrating the reversible character of the structural phase transition.

Fitting of the trimerization distortion from STEM images
The fitting of the atomic columns intensities was performed using an asymmetric Gaussian model 4,5 : where I0 is the background, Z is the Gaussian amplitude, (x0 , y0) the coordinates of the center of the peak, xw and yw the Gaussian variances and s the parameter describing the asymmetry of the peak.
The fitting of the trimerization associated with the K3 mode was performed using the model proposed by Holtz et al. 6 : where q is the wave vector of the K3 mode, u the position vector of the Y atom, Q the amplitude of the sinusoidal wave and φ its phase. The primary order parameter of the structural trimerization is hence given by Q = (Q cos φ ,Q sin φ). Q is related to the maximum displacement (1.5Q) between Y atoms and φ is associated to the six possible structural domains generated by the possible permutation of the ↑↑↓ and ↓↓↑ patterns.
The average |<Q>| values given in Fig. 3a and Fig. 4b were obtained by first calculating the average values of the Q vectors and then calculating their moduli. In Fig. 4b the |<Q>| values are shown for each temperature of the heating experiment with error bars given by the standard deviation of |<Q>| at T > TQ (where a homogeneous distribution is expected), thus disentangled from its thickness-dependent distribution at T < TQ, (see Fig. 3a).

Calculation of the ellipticity of STEM images peaks
The expression given in Supplementary Eq. 4 resembles the equation of a two-dimensional elliptical Gaussian, given by: In order to extract quantitative information about the elliptic shape of the peak we need to retrieve the lengths of the ellipse semi-axes (a, b), that are related to the coefficients in Supplementary Eq. 6 by the relations: where θ is the rotation of the ellipse. A convenient way to retrieve the ellipse's parameters that does not require solving second order trigonometric equations (as would be the case using where the parameters of the explicit form can be defined as: It is worth to note that Supplementary Eq. 9 represents the equation of an ellipse if and only if Δ>0 and δ>0, where Δ and δ are defined as: The ellipses semi-axes can be calculated using the formulas 7 : where μ is defined as: The ellipticity (ε) is defined as the ratio between the major and minor semi-axes: 14 with values ε ≥ 1 (where ε = 1 is the degenerate case for the spherical Gaussian).

HAADF-STEM Image simulations
The simulation of HAADF-STEM images was performed using a custom developed multislice frozen-phonon code 8 , setting the beam convergence angle to 18 mrad and the collection angles for the HAADF detector in the range 70-190 mrad. The finite size of the source was set to 0.6 Å. The specimen thickness was set to 12.5 nm.
For the simulation of the low temperature (T<TC) polar phase, we generated the slices starting from the P63cm structure 9 (Supplementary Fig. 6a,c). In this case, the K3 distortion produces the typical trimerization of Y atoms that is visible at all temperatures below TC.
For the phase transformation to the paraelectric (T>TC) phase, we assume an order-disorder model.
In this case, the high-temperature phase corresponds to a continuum of states with different values of φ. In order to simulate this state of the system, we generated 20 different configurations with a random phase within the range [0, 2π) and then averaged the simulated HAADF signals to obtain the continuum of states ( Supplementary Fig. 6b). The superimposition of states with different φ is responsible for the appearance of elongated peaks in the HAADF signal, as described by the increase in the ellipticity for T>TC. The experimental HAADF-STEM signals taken at 448 K and 498 K are shown in Supplementary Fig. 6c,d, superimposed with the simulated images (in the white boxes). The excellent comparison between simulated and experimental images supports an order-disorder mechanism for the phase transformation.