Physical reality of the Preisach model for organic ferroelectrics

The Preisach model has been a cornerstone in the fields of ferromagnetism and ferroelectricity since its inception. It describes a real, non-ideal, ferroic material as the sum of a distribution of ideal ‘hysterons’. However, the physical reality of the model in ferroelectrics has been hard to establish. Here, we experimentally determine the Preisach (hysteron) distribution for two ferroelectric systems and show how its broadening directly relates to the materials’ morphology. We connect the Preisach distribution to measured microscopic switching kinetics that underlay the macroscopic dispersive switching kinetics as commonly observed for practical ferroelectrics. The presented results reveal that the in principle mathematical construct of the Preisach model has a strong physical basis and is a powerful tool to explain polarization switching at all time scales in different types of ferroelectrics. These insights lead to guidelines for further advancement of the ferroelectric materials both for conventional and multi-bit data storage applications.


Supplementary Figures
Supplementary Figure 1   Supplementary Figure 6. Alternative data for fitting to the bivariate Eq. 3b. Fitting is performed using only experimental data for projections of the actual dependence (black circles). An example of BTA.   switching background currents. Two implementation modes (b-d) and (e-g) are possible and would give similar results in terms of switched polarization. (b,e) is the setting pulse sequence, which is used to select a particular part of the Preisach distribution as shown in (c,f). The current density response measured at step 3 is given in (d,g).

Supplementary Tables
Supplementary Table 1. Default parameters as used in the interacting dipole model (see also Supplementary Figure 3).  Figures 6(a) and 7. The top three rows are free parameters; the bottom three rows are given by the experimental conditions. Two slightly different sets of parameters are found for BTA due to slight differences in experimental conditions.

Supplementary Note 1: Details on the Preisach distribution electrostatic model
In the model we reduce the complex hierarchical morphology of BTA molecules to a collection of point dipoles and their interactions. The morphology of the system is shown in Supplementary Figure 3. Each BTA molecule contains three dipoles that tilt out of plane to form a helical structure with the rest of the column, resulting in a macro-dipole. Columns are disordered and contain defects that separate perfectly ordered clusters. For simplicity, we assume that the individual dipole flipping process within a cluster can be ignored and only consider the flipping of clusters as a whole. Furthermore, periodic boundary conditions are used. An overview of typical simulation parameters is given in Supplementary Table 1.
For each dipole i the interactions with neighboring dipoles are calculated up to a certain cut-off range .
Including the applied electric field , the energy of a dipole is then with the dipole moment, the distance vector between dipole i and j, and the permittivity of the material.
On top of the permanent dipoles of the amide groups, we should also consider induced dipoles. Unlike the permanent dipoles, these induced dipoles are not restricted to only two fixed orientations, as they will lie along the direction of the local field ⃗⃗⃗⃗⃗⃗⃗ : with the electronic polarizability.
The Preisach distribution is now simulated by replicating the experiment and sweeping the electric field step by step. At each step in a simulation, all interactions are recalculated, and for each cluster it is determined if it is energetically favorable to flip. At each step, the number of flipped dipoles is registered, which will finally give the integrated PD. This integrated distribution is fitted in the same way as the experimental data. As the model does not comprise kinetic processes and is temperature independent, the simulated fields correspond to intrinsic rather than extrinsic switching and exceed the experimental values by at least an order of magnitude, as predicted by Eq. 2 in the main text with T = 0 K.
The error of the fits to all simulated integrated PDs is shown in Supplementary Figure 4. The fit for the narrow distributions (b and c) is decent with the error within a few percent (e,f). However, discrepancies appear for the broader distribution (a and d), indicating that the simulated distribution deviates from the simple double Gaussian form. This means that either our approximation of the PD with a Gaussian is wrong, or the model cannot completely describe the situation for high intercolumnar interactions. The latter is plausible considering the simplicity of the model. On the other hand, it is also to be expected that in reality the distribution is indeed more complex than just a double Gaussian. Experimentally, a Gaussian is found to best fit the data, but the actual measurement noise would hide deviations of the size found in any of the simulations.

Device fabrication and pre-conditioning
Thin film metal-ferroelectric-metal (MFM) capacitor devices of BTA-C10 were formed by spin-coating (500-1000 rpm) of a 40 mg/ml chloroform solution on a chemically cleaned glass substrate with patterned aluminum (or chromium/gold) bottom electrodes. Before thermal vacuum deposition of the aluminum (or gold) top electrodes, spin-coated films were annealed at 60 °C for 15 min to completely evaporate the solvent. The prepared MFM devices were 0.01-1 mm 2 in area. Typical film thickness was 400-700 nm, as measured by a Bruker Dektak XT profilometer.
In the as-cast organic ferroelectric film molecular columns lie in-plane to the electrode. When molecular dipoles are oriented in this way, no polarization can be measured in the bottom-top electrode geometry. Therefore prior to the electrical measurements, the devices are treated by a field-annealing procedure, when at low viscosity conditions (∼100 °C) with the help of an alternating external field molecular bundles are forced to stand perpendicularly to the electrodes (to some scale, determined by the material's tendency towards disorder). The followed field-cooling freezes the system in this quasi-orderly state. Due to π-stacking of the benzene core and hydrogen bonding of the amide groups, accompanied by alkyl chain freezing, the hexagonal packing remains stable even without external field. This has been previously tested by polarized light optical microscopy (POM) and can be seen from unchanged current transients corresponding to polarization reversal after a long waiting time. Therefore only the polarization switching current, rising from the dipole rotation, reflects in the quasi-statically measured P-E curves (see Electrical characterization), once the device is properly conditioned. P(VDF-TrFE) capacitor devices were prepared from 60 mg/ml cyclohexanone solution, which was stirred overnight and filtered before spin-coating (2000 rpm) on patterned chromium/gold (5/50 nm) electrodes on chemically cleaned glass substrates. Deposited films were annealed at 140 °C for 2 hours to increase crystallinity. The procedure was repeated until desired thickness was obtained (400-700 nm). Top gold electrodes were subsequently deposited by thermal evaporation in vacuum to form cross-bar structure capacitor devices. Ferroelectric devices were further characterized as described below.

Electrical characterization
The input signal waveform is supplied by a Tektronix AFG3000 Arbitrary Function Generator and is amplified by a TREK PZD350A high voltage amplifier. The device response is visualized by a Tektronix TBS1000B Digital Oscilloscope.
The polarization loops are obtained by integration of the switching current transients. We use a quasistatic mode, better known as the Double Wave Method (DWM), where the non-switching current is subtracted from the initial signal to avoid displacement and leakage (if any) inputs in the P-E curves.