Depolarization of multidomain ferroelectric materials

Depolarization in ferroelectric materials has been studied since the 1970s, albeit quasi-statically. The dynamics are described by the empirical Merz law, which gives the polarization switching time as a function of electric field, normalized to the so-called activation field. The Merz law has been used for decades; its origin as domain-wall depinning has recently been corroborated by molecular dynamics simulations. Here we experimentally investigate domain-wall depinning by measuring the dynamics of depolarization. We find that the boundary between thermodynamically stable and depolarizing regimes can be described by a single constant, Pr/ε0εferroEc. Among different multidomain ferroelectric materials the values of coercive field, Ec, dielectric constant, εferro, and remanent polarization, Pr, vary by orders of magnitude; the value for Pr/ε0εferroEc however is comparable, about 15. Using this extracted universal value, we show that the depolarization field is similar to the activation field, which corresponds to the transition from creep to domain-wall flow.


Supplementary Note 1. Introduction
In Supplementary Note 2 we discuss suppression of polarization by a dead interface layer or by a finite screening length in the electrodes. We show that in our experiment using thick ferroelectric films these effects can be disregarded. In Supplementary Note 3 we present how the remanent polarization of a ferroelectric capacitor in series with a linear capacitor can graphically be extracted from the quasi-static hysteresis loop of the ferroelectric-only capacitor.
We have investigated the depolarization dynamics using a linear capacitor in series with a ferroelectric capacitor. We applied a voltage pulse, high enough to fully polarize the ferroelectric capacitor. Then the applied voltage abruptly dropped to 0 V, and we recorded the transient of the electric displacement, D(t). In Supplementary Note 4 we show that the depolarization transients can be calculated with the Kolmogorov-Avrami-Ishibashi (KAI) formalism, adapted to a time-dependent electric field and Merz law. A good agreement is obtained. In Supplementary Note 5 we show that the depolarization transients exhibit a negative differential capacitance (NDC). The value is equal to that of the serial capacitance. In Supplementary Note 6 we determined the temperature dependence of Pr/0ferroEc for P(VDF-TrFE). We show that the value is about 15 and independent of temperature between 213 K and 333 K. The constant value implies that at all temperatures the depolarization field, Edep, is similar to the activation field, Eact. In Supplementary Note 7 we discuss the physics of intrinsic switching and we reproduce the relation between the intrinsic, thermodynamic coercive field and ferroelectric polarization as theoretically derived in the seminal work of Tagantsev et al.
using the Ginzburg-Landau-Devonshire formalism. Subsequently we discuss experimental data on the intrinsic coercive field of BaTiO3 and P(VDF-TrFE).

Supplementary Note 2. Depolarization due to electrode screening and dead layers
In a ferroelectric capacitor the metallic electrodes provide free charges that fully compensate the depolarization field, yielding a zero internal electric field inside the ferroelectric material.
When the depolarization field is not fully compensated, the remanent polarization is suppressed.
Incomplete compensation can in practice be due to the occurrence of a capacitive dead layer between the electrodes and the ferroelectric material 1,2 , or due to the finite screening length in metallic electrodes field 3,4,5 Another example is the ferroelectric field-effect transistor 6 , where the semiconducting layer causes an uncompensated depolarization field that limits data retention.
The presence of a dead layer leads to a large voltage offset along the horizontal (electric field) axis. The result is a deformed hysteresis loop with asymmetric switching characteristics. An analytical bilayer model that describes the mechanism has been reported 7 . In our measurements the hysteresis loops are perfectly symmetrical and, hence, the presence of a dead layer can be disregarded.
Another reason for partial charge compensation is incomplete screening, for which there can be two causes. Firstly, the compensating charges in the electrode form a layer of finite thickness due to Thomas-Fermi screening length. Secondly, the polarization cannot drop abruptly when going from the ferroelectric to the metal, the so-called Kretschmer-Binder effect.
The effect of this depolarization field will become larger as the thickness of the ferroelectric decreases. The depolarization field is especially important in ultrathin films in the order of 10 nm, where it determines the critical thickness and domain structure.
For thick films the depolarization field from incomplete screening can be disregarded. Only when the ferroelectric is a perfect insulator, the incomplete screening leads to a finite depolarization field inside the ferroelectric material. However, due to the large film thickness this internal electric field is much smaller than the coercive field. Secondly, and more importantly, ferroelectric materials are not perfect insulators; tangent delta is finite and not zero.
The uncompensated charges by the ferroelectric-electrode interface are neutralized by charge carriers in the ferroelectric material. Consequently, inside a thick film ferroelectric capacitor under short circuit conditions the internal electric field is zero. This statement is supported by the measured polarization of our samples, which is thickness independent. Furthermore, in ultrathin 15 nm BaTiO3 films sandwiched between SrRuO3 electrodes already 80 % of the remanent polarization is retained 8 . Finally, the remanent polarization of ultra-thin PbTiO3 films saturates above 20 nm 9 . Therefore, in our electrostatic analysis the depolarization field due to incomplete screening in the electrodes can be disregarded.

Supplementary Note 3. Graphical extraction of the suppressed polarization
We connect a linear capacitor, Cser, in series with a ferroelectric capacitor, Cferro. At high bias the displacement is the same as for the ferroelectric-only capacitor, as then nearly all dipoles are aligned along the direction of the external electric field. The hysteresis loops have an identical apparent coercive voltage, independent of the ratio Cferro/Cser, as at zero displacement there are no net free charges in the electrodes and hence the equivalent circuit is that of a ferroelectric-only capacitor. Consequently, the hysteresis loops are tilted, and the remanent polarization is suppressed. The value for the apparent remanent polarization can be graphically determined from the hysteresis D-Eferro loop of the ferroelectric-only capacitor 10 . In a serial circuit the internal electric field in the ferroelectric material, Eferro, is derived in the Methods section of the manuscript as: where Vapp is the applied bias and d is the thickness of the ferroelectric layer. At zero applied bias, the internal electric field is Eferro = -D·(Cferro /Cser)/(0ferro). For a given ratio of Cferro/Cser,

Supplementary Note 4. Calculation of depolarization transients using the generalized KAI model
The switching of ferroelectric polarization is typically described by the model developed by Ishibashi 11 , based on the classical statistical theory of nucleation and unrestricted domain growth, as described by Kolmogorov 12 and Avrami 13 . The change in ferroelectric polarization, P(t), upon applying an electric field, E, is given by the compressed exponential function 26 : where t0 is a characteristic switching time. In the conventional KAI approach, the Avrami index, n, depends on the dimensionality of the domains and takes only integer values: the value is 3 for single crystals and 2 for epitaxial thin films. However for polycrystalline thin films such as P(VDF-TrFE), the Avrami index is typically a non-integer between 1 and 2, depending on the strength of the electric field and the density of nucleation sites 32 . The switching time, t0, follows the empirical Merz law 14 : where E is the applied electric field, Eact is the so-called activation field 15 that is proportional to the domain-wall energy, and t∞ is the switching time at infinite applied electric field. The Merz law is observed in many ferroelectric systems ranging from single crystals 14 , through bulk ceramics 16 , and thin films 17,18 , to organic-ferroelectric composites 19 . We note that Tybel et al. 20 first pointed out that Merz law is a special case of domain-wall motion in generic creep systems, describing propagation of elastic objects driven by an external force in the presence of a pinning potential, such as domains in ferroelectric 20 and magnetic materials 21 and vortices in type-II superconductors 22 . For random-field type disordered ferroelectrics 20 , the domain-wall creep velocity reads: The KAI formalism has been previously adapted for a time-dependent electric field and used to calculate hysteresis loops as a function of ramping frequency 32 . Here we adapt the KAI formalism for a time-dependent depolarization field, Edep(t). After poling the ferroelectric material and removing the applied field, the internal field is derived as: Supplementary Eq. 5 shows that the internal field and the polarization are mutually-dependent.
Therefore both quantities need to be calculated iteratively. Depolarization transients were measured using a voltage pulse, high enough to fully polarize the ferroelectric capacitor. Then the applied voltage abruptly dropped to 0 V, and we recorded the electric displacement as a function of time, D(t). The iterative expression for the polarization reads: where the initial polarization is set equal to the fully reversed polarization, -Psat, as the internal field is opposite to the direction of the polarization. For each iteration the displacement, D(t), is then calculated from: The depolarization transient can then be calculated by iteratively solving Supplementary Eqs.

Supplementary Note 6. Temperature dependence of Pr/0ferroEc for P(VDF-TrFE)
To investigate the temperature dependence of Pr/0ferroEc, we have measured the polarization, dielectric constant and the coercive field of P(VDF-TrFE) ferroelectric-only capacitors at temperatures between 213 K and 333 K.
The real part of the dielectric constant is presented as a function of frequency for various temperatures in Supplementary Figure 4a. The dielectric loss is measured to be 2% or less. The dielectric constant is almost frequency-independent below 10 kHz and slightly decreases at higher frequency. We take for the value of the static dielectric constant, ferro, the dielectric constant at 1 kHz. Supplementary Figure 4a shows that the value of the static dielectric constant increases with temperature from 6 at 213 K to 11 at 333 K.
The temperature dependence of the coercive field, Ec, is presented in Supplementary Figure 4b.
The values of Ec were extracted from quasi-static D-E hysteresis loops, measured in a Sawyer- Pr, which slightly decreases from 8.8 µC/cm 2 at 213 K to 6.6 µC/cm 2 at 333 K. We note that the hysteresis loops indicate that the remanent polarization is almost equal to the saturated polarization; the difference in displacement between high bias and zero bias is dominated by the induced polarization, 0ferroE. Pr, which slightly decreases from 8.8 µC/cm 2 at 213 K to 6.6 µC/cm 2 at 333 K. We note that the hysteresis loops indicate that the remanent polarization is almost equal to the saturated polarization; the difference in displacement between high bias and zero bias is dominated by the induced polarization, 0ferroE.

Supplementary
We This relation is reminiscent of our experimental derived relationship, viz. Pr0ferroEc ≈ 15.
However, as will be explained below, the physical mechanisms are completely different, and cannot a priori be related. Here we have taken the dielectric constant as LD = LD + 1 ≈ LD.
Intrinsic switching is a mean-field process, which dictates collective behavior of the dipoles within the ferroelectric material. This happens when all dipoles are in a homogeneous environment; a necessary condition is that there are no pinning defects, which is practically unrealistic. Nevertheless, at very high electric field, the electrostatic energy dominates, thus influences from defects may become trivial: the pinning force by the defects is overwhelmed by the applied high field; the total electric field that the dipoles feel, either in defect-free regions or in vicinity of pinning defects, are about the same, and the system is effectively a homogeneous system. In that case, even a practical ferroelectric material with a considerable number of pinning defects may favour undergoing intrinsic switching.
Experimentally approaching the intrinsic switching has being a great challenge. To the best of our knowledge, experimental substantiation of intrinsic switching has only been reported in ultrathin films of BaTiO3 and P(VDF-TrFE).
The coercive field was extracted from AFM measurements and reported as a function of layer thickness 28 30 . To unambiguously correlate the depolarization and activation field with the intrinsic coercive field, the comprising physical constants of the same device should be measured, which so far have not been reported. P(VDF-TrFE) is an ideal model system as the dielectric constant hardly depends on microstructure and film thickness, and the complete data set on depolarization-, activation-and intrinsic coercive field is available. Intrinsic switching has been reported in Langmuir-Blodgett films of P(VDF-TrFE) 31 . The ferroelectric films with thickness between 1 nm and 10 nm are thin enough to inhibit nucleation. The intrinsic coercive field is independent of layer thickness and has been presented as a function of temperature. At ambient temperature Eint,c is measured to be about 600 MV/m. Taking  Previously the experimentally extracted activation field of P(VDF-TrFE) as a function of temperature has been reported 32,33 . At ambient temperature a value of about 1000 MV/m was determined. This means that the activation field, depolarization field and intrinsic coercive field are of the same order of magnitude. Within a factor of two we arrive at dep~act~int,c .
We note that this equality can be artificial as the underlying physical mechanisms for intrinsic and extrinsic switching are completely different. Current experiments, in our work as well as in literature, are insufficient to draw a solid conclusion about the relation with the intrinsic coercive field.