Chiral Polarization Textures Induced by the Flexoelectric Effect in Ferroelectric Nanocylinders

Anna N. Morozovska 1*, Riccardo Hertel2†, Salia Cherifi-Hertel2, Victor Yu. Reshetnyak3, Eugene A. Eliseev4, and Dean R. Evans5‡

1 Institute of Physics, National Academy of Sciences of Ukraine,

46, pr. Nauky, 03028 Kyiv, Ukraine

2 Université de Strasbourg, CNRS, Institut de Physique et Chimie des Matériaux de Strasbourg, UMR 7504, 67000 Strasbourg, France

3 Taras Shevchenko National University of Kyiv,

Volodymyrska Street 64, Kyiv, 01601, Ukraine

4 Institute for Problems of Materials Science, National Academy of Sciences of Ukraine,

Krjijanovskogo 3, 03142 Kyiv, Ukraine

5 Air Force Research Laboratory, Materials and Manufacturing Directorate,

Wright-Patterson Air Force Base, Ohio, 45433, USA

Abstract

Polar chiral structures have recently attracted much interest within the scientific community, as they pave the way towards innovative device concepts similar to the developments achieved in nanomagnetism. Despite the growing interest, many fundamental questions related to the mechanisms controlling the appearance and stability of ferroelectric topological structures remain open. In this context, ferroelectric nanoparticles provide a flexible playground for such investigations. Here, we present a theoretical study of ferroelectric polar textures in a cylindrical core-shell nanoparticle. The calculations reveal a chiral polarization structure containing two oppositely oriented diffuse axial domains located near the cylinder ends, separated by a region with a zero-axial polarization. We name this polarization configuration “flexon” to underline the flexoelectric nature of its axial polarization. Analytical calculations and numerical simulation results show that the flexon’s chirality can be switched by reversing the sign of the flexoelectric coefficient. Furthermore, the anisotropy of the flexoelectric coupling is found to critically influence the polarization texture and domain morphology. The flexon rounded shape, combined with its distinct chiral properties and the localization nature near the surface, are reminiscent of Chiral Bobber structures in magnetism. In the azimuthal plane, the flexon displays the polarization state of a vortex with an axially polarized core region, i.e., a meron. The flexoelectric effect, which couples the electric

* Corresponding author 1: [email protected]

† Corresponding author 2: [email protected]

‡ Corresponding author 4: [email protected]

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polarization and elastic strain gradients, plays a determining role in the stabilization of these chiral states. We discuss similarities between this interaction and the recently predicted ferroelectric Dyzaloshinskii-Moriya interaction leading to chiral polarization states.

I. INTRODUCTION

Research on ferroelectric materials has received growing interest over the past years, driven in part by the potential of these material systems for low-power technological applications in a broad spectrum of domains [1, 2], ranging from high-density data storage to optical nano-devices. A central aspect of this field of research is the formation of ferroelectric domain structures [3], and more generally the micro- and nanoscale structure of the polarization field [4]. Traditionally, research on ferroelectrics is centered on the study of bulk materials and thin films [5, 6, 7], but recently ferroelectric nanoparticles have also attracted increasing interest [8, 9, 10, 11, 12, 13, 14]. In ferroelectric thin films and nanoparticles, the polarization structure is strongly affected by electrostatic (depolarizing) fields [15, 16, 17, 18], as well as by strain and strain gradients [19, 20, 21, 22] via the flexoelectric effect [23, 24, 25, 26].

Although the foundations for the theoretical description of ferroelectrics have been established decades ago [27], understanding the complex physical properties of these material systems remains a challenge for fundamental research. Recent progress in this field, achieved to a large extent through advanced imaging techniques [28] and by employing modern numerical simulations [29], includes the discovery of highly complex polarization structures, such as flux closure [5, 30] and bubble domains [31], meandering [32, 33] and/or labyrinthine [11, 34] structures, non-Ising type chiral domain walls [35], polarization vortices in thin layers [36, 37, 38], nanodots [39] or nanopillars [40], or polar skyrmions [41, 42].

While skyrmions and other chiral structures have dominated the past decade of research in magnetism [43], these topological states have received less attention by the ferroelectric community. Only recently a strong interest has emerged in chiral polarization structures, which can be attributed to the observation of skyrmion states in ferroelectrics [41-42]. However, the theoretical understanding of these structures is not as advanced as it is in the case of their magnetic counterparts, and the mechanism that underpins the formation of skyrmions in ferroelectrics is not fully understood. The fundamental interaction stabilizing the magnetic version of these structures in chiral ferromagnets [44, 45] is the Dzyaloshinky-Moriya Interaction (DMI). The DMI favors the formation of helical structures with a well-defined handedness as they occur, e.g., along the radial direction of skyrmions. As scientists working on ferroelectrics hope to replicate the success that chiral structures have witnessed in magnetism, the possibility

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of a “ferroelectric DMI” has recently been discussed [46]. However, Erb and Hlinka [47] showed that only very few exotic ferroelectrics could theoretically sustain an intrinsic DMI-type interaction since it requires particular symmetry properties of the crystal lattice. Here we discuss the flexoelectric coupling as an alternative mechanism that can generate chiral polarization states in ferroelectrics.

The thermodynamic description of the flexoelectric effect is given by the Lifshitz invariant in the free energy expansion [22]. It is known that, in magnetic materials, the occurrence of similar Lifshitz invariants converts directly into an antisymmetric coupling known as the DMI [48, 49], which favors the formation of helicoidal structures with a specific chirality. The existence of a ferroelectric counterpart of the DMI was recently predicted by first-principles simulations [46]. The ferroelectric analogue of the DMI was discussed in the context of Lifshitz invariants by Strukov and Levanyuk [50], and more recently by Erb and Hlinka [47], who argued that a ferroelectric DMI can exist. In addition to the remarkable similarity in the mathematical form of the flexoelectric Lifshitz invariant and DMI, the flexoelectric term appears to have a similar impact as the DMI in terms of the formation of chiral structures.

By means of the finite element modeling (FEM) based on the Landau-Ginzburg-Devonshire (LGD) theory, this paper shows that an anisotropic flexoelectric effect can give rise to a previously unexplored type of polarization state with distinct chiral properties. Remarkably, these homochiral properties are not induced by a DMI term. This finding suggests that the recently discussed DMI in ferroelectrics is not the only possible mechanism for the formation of homochiral polarization states, and that anisotropic flexoelectric effects offer an alternative pathway to stabilize such structures in ferroelectric nanostructures. We discuss common aspects of the DMI and the flexoelectric effect, which are both derived from Lifshitz invariants in the framework of the Landau theory of second-order phase transitions [22].

II. CONSIDERED PROBLEM AND MATERIAL PARAMETERS

Using a LGD phenomenological approach along with electrostatic equations and elasticity theory, we model the polarization, the internal electric field, and the elastic stresses and strains in a core-shell nanoparticle using FEM, where the ferroelectric core is made of BaTiO3 and has a cylindrical shape. The aspect ratio of the nanocylinder radius 𝑅 to its length ℎ is significantly higher than unity. The z-axis is parallel to the cylinder axis (Fig. 1). The shell is an elastically soft paraelectric or high-k semiconductor with a thickness Δ𝑅≪𝑅 and screening length Λ≥ 1 nm. The coverage can be artificial (e.g., a soft organic semiconductor or vacancy-enriched SrTiO3) or natural, and in the latter case it would originate from the polarization

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screening by surrounding media. The core-shell nanoparticle is placed in a very soft elastic medium.

FIGURE 1. A cylindrical ferroelectric nanoparticle (core) of radius 𝑅, covered with an elastically soft semiconducting shell with a thickness Δ𝑅≪𝑅 and screening length Λ of 1 nm, placed in an isotropic elastically soft effective medium. The direction of axial polarization 𝑃3 is shown by the straight orange arrow, and lateral components 𝑃1,2 are shown by the curled red-blue arrow to highlight their vortex-type structure.

The LGD free energy functional G of the nanoparticle

Dynamic control of ferroionic states in ferroelectric nanoparticles☆

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Abstract

The polar states of uniaxial ferroelectric nanoparticles interacting with a surface system of electronic and ionic charges with a broad distribution of mobilities is explored, which corresponds to the experimental case of nanoparticles in solution or ambient conditions. The nonlinear interactions between the ferroelectric dipoles and surface charges with slow relaxation dynamics in an external field lead to the emergence of a broad range of paraelectric-like, antiferroelectric-like ionic, and ferroelectric-like ferroionic states. The crossover between these states can be controlled not only by the static characteristics of the surface charges, but also by their relaxation dynamics in the applied field. Obtained results are not only promising for advanced applications of ferroelectric nanoparticles in nanoelectronics and optoelectronics, they also offer strategies for experimental verification.

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Keywords

Ferroelectric nanoparticles

Dielectric layers

Ions

Surface charge dynamics

Ferroionic states

1. Introduction Dimensionally confined ferroics have emerged as one of the objects of interest for the fundamental research and device applications [1]. In particular, ferroelectric thin films and nanoparticles are a playground for exotic physics driven by the interplay of polarization rotation, depolarization fields, surface charges [2,3], flexoelectricity [4,5], and size effects [6,7]. For electrode surfaces and ferroelectric nanoparticles in a dielectric matrix, extensive theoretical work has been developed [8,9,10]. However, this is not the case for the prototypical case of ferroelectric nanoparticles in an ambient environment or in solutions. Here, despite a general understanding that the surface phenomena at open ferroelectric surfaces are dominated by electrochemical processes and ionic screening [11,12,13], their effects on physical properties and phase transitions in nanoferroics are understood only weakly [14]. For instance, ferroelectric thin films with open surfaces compensated via ionic adsorption [15,16,17], reveal very unusual mixed electrochemical-ferroelectric states [18,19].Previously, a theoretical formalism for the analysis of the ferroelectric behavior in proximity to electrochemically coupled interface was developed by Stephenson and Highland (SH) [20]. Using the Landau-Ginzburg-Devonshire (LGD) and Stephenson-Highland approaches, Morozovska and Kalinin groups derived analytical solutions describing unusual phase states in uniaxial [21,22,23,24] and multiaxial [25] ferroelectric thin films, as well as antiferroelectric thin films with electrochemical polarization switching [26,27]. The analysis [21], [22], [23], [24], [25], [26], [27] leads to the elucidation of ferroionic (FEI) and antiferroionic (AFEI) states, which are the result of a nonlinear electrostatic interaction between the surface charges (ions, vacancies, holes and/or electrons) and ferroelectric dipoles. These states have principal differences from the ferroelectric state of thin films with linear screening, which originate from the strong nonlinear dependence of the screening charge density on the acting electric potential (i.e., electrochemical “overpotential” - the electric potential inside the charged layer that is different from the external voltage due to the self-screening and depolarization field) [21]. This nonlinearity gives rise to additional multi-stable polarization states in ultrathin ferroelectric films. Comparatively, linear screening conditions can lead only to paraelectric (PE) states.The FEI and AFEI states reveal themselves as having specific (e.g., nonmonotonic) temperature dependences of the out-of-plane spontaneous polarization and unusually complex shapes of the polarization hysteresis loops [21], [22], [23], [24], [25], [26], [27]. For instance, there can be four bi-stable states of the spontaneous polarization in an ultra-thin uniaxial ferroelectric film covered with ionic-electronic surface charges, as compared to two possible states in thicker films covered with conducting electrodes [21]. If the polarization hysteresis loop of the thin film covered with ions looks similar to a classical ferroelectric loop, we can classify the state as ferroelectric (FE)-like FEI, and when it looks similar to a double antiferroelectric loop, we can classify the state as AFEI. There are plenty of mixed antiferroionic-ferroionic (AFEI-FEI) states characterized by e.g., pinched and/or constricted asymmetric loops. It should be noted that the appearance of AFEI and FEI states depends not only on the static characteristics surface charges (such as their concentration, formation energies, and the details of surface density of states), but also on their relaxation dynamics in an applied field. However, this dynamic aspect of ionic screening is almost unexplored [21], [22], [23], [24], [25], [26], [27], while ample experimental evidence of its existence is available [28,29,30].Similarly, the influence of the nonlinear ionic-electronic surface screening on the polar properties of ferroic nanoparticles has not been considered, despite that this case seems very interesting for the fundamental research of dimensionally confined objects and promising for various applications of the nanoparticles in optoelectronics [31], ferroelectric random access memories [32], and other information technology components [33].Traditionally, the influence of the surface screening on the polar properties of a single free-standing ferroelectric nanoparticle remains difficult to explore due to the lack of corresponding probing techniques. However, free-standing ferroelectric nanoparticles have become accessible to the experimental investigation with the advent of Piezoresponse Force Microscopy (PFM) [34] and other advanced probing techniques. For example, the existence and reordering behaviors of a surface dipole in a metal halide perovskite system has been observed by utilization of advanced probing techniques [35,36]. The study provides a local, nanoscopic insight for understanding the ferroelectric properties of this class of materials. Nanoparticles of metal halide perovskites have recently come to the forefront of research due to their unique properties for variety of optoelectronic applications [37,38]. A recent study revealed the existence of ferroelectricity in CsPbBr3 nanostructure [39]. It is noted that such ferroelectric phases in the nanostructures can be stabilized and/or controlled by the class of ligand molecules with a wide-range of sizes, shapes, chirality, ionic characteristics, and polarity [40,41]. Each ligand molecule can result in the surface system being identical to a model of nonlinear ionic-electronic surface screening; therefore, metal halide perovskite nanoparticles could be ideal systems to explore the ferroionic properties in the confined nanostructures.At the same time, experimental realizations of ferroelectric nanoparticles placed in highly polarized nonlinear media, such as liquid crystals, are abundant (see e.g., Refs. [42,43,44]), and the sizes of 5 – 50 nm are typical experimental values [45,46,47,48]. There are several studies of ferroelectric nanoparticles fabricated in heptane and oleic acid [49,50,51], producing core-shell nanoparticles, where nonlinear screening can be realized in the ultra-thin shells.This theoretical study aims to fill the gap in knowledge and considers polar states and dielectric properties of uniaxial ferroelectric nanoparticles covered by a layer of mixed ionic-electronic nonlinear surface charge with slow relaxation dynamics in an external field. Using LGD-SH approach, we show that a dynamic transition between the PE-like, AFEI, FEI, and FE-like FEI states takes place in ultra-small ferroelectric nanocubes and nanopillars. The paper is structured as following. Section II contains the formulation of the problem and description of methods. Basic equations and assumptions are introduced in Section III. Results discussion and analysis are presented in Section IV, which consist of three subsections considering the electron-hole and electron-cation surface screening and size effects. A brief summary is given in Section V. Calculations details are listed in Suppl. Mat. [52].

2. Problem description The polarization, internal electric field, and elastic stresses and strains in ferroelectric nanoparticles of different shape, material, size, and structure are modelled using finite element modeling (FEM). In this, we use the constitutive equations for relevant order parameter fields based on the LGD phenomenological approach along with electrostatic equations and elasticity theory. For geometry, we consider a parallelepiped-shape nanoparticle of height

, which consists of a uniaxial ferroelectric core of thickness sandwiched between the dielectric layers of thickness and , as shown in Figs. 1(a)-(b); the dielectric permittivity of the layers is and , respectively. The layers can have the same or different physical nature. Specifically, for PFM geometry, the upper layer can be gaseous, soft matter, or liquid gap between the ferroelectric core and the top (or tip) electrode; while the lower dielectric layer is an ultra-thin “passive” or “dead” layer [53,54,55] with high PE-type dielectric permittivity, which is formed at the interface between the ferroelectric core and conductive substrate. For this case, one can assume that and . The relative background permittivity [54] of a ferroelectric core is

.

📷Download : Download high-res image (1MB) Download : Download full-size image Fig. 1. A parallelepiped-shape nanoparticle of height

, which consists of a uniaxial ferroelectric core of thickness sandwiched between two dielectric layers of thickness and . The direction of the uniaxial polarization is shown by the blue arrow. The side view (a) and the vertical cross-section (b) are shown. The side and top surfaces of the ferroelectric core are covered with a layer of mobile surface charges with density , which is the sum of positive (pink circles) and negative (blue circles) charges, and , respectively. (c) The dependence of the total surface charge (green dashed curve), (blue curve), and (red curves) on the overpotential , which are calculated from Eqs. (2) for 1018 m−2, , , 0.1 eV, and

K. The inset shows the central part of the plot (c).

The z axis is parallel to the polar axis, and the uniaxial polarization

. The aspect ratio of the nanoparticle width to height is varied, such that the width ranges from being twice the height () to significantly greater than the height (). This geometry variation corresponds to the transition from a nanocube to nanopillar shape. The nanoparticle bottom surface, , is in contact with an electron-conducting substrate; and the mismatch strain can exist at the nanoparticle-substrate boundary. The outer surface of the upper dielectric layer,

, is an electrode.

The top and side surfaces of the ferroelectric core are covered with a layer of mobile surface charges (holes and/or electrons, ions, protons, hydroxyl groups, etc.). The surface charge density

depends on the electric potential in a complex nonlinear way, where the total ionic-electronic charge is the sum of positive () and negative () charges, , whose mobilities can vary significantly for different kinds of charges. This mobility difference can lead to the separable dynamics (retardation or outrunning) of and

. One cannot exclude the presence of a surface charge at bottom surface of the ferroelectric core, but its nature and properties can be strongly different from the ionic-electronic charge at the electrically-open top surface.

3. Basic equations and approximations 3.1. Surface charge dynamics and LGD free energy To describe the dynamics of the positive and negative surface charges, we use a linear relaxation model [21,56]:(1)

where the relation between the relaxation times and can be very different due to the dissimilar mobilities of the ionic and electronic charges and . Typically, the relaxation of electrons and/or holes are of the same order, and, at the same time, the relaxation of electrons and holes are much faster than the relaxation of ions. Thus, below we consider that the equality is valid for the case when the surface charges are electrons and holes, while the strong inequalities or

are valid for the case when the surface charges are electrons and cations, or holes and anions, respectively.

The dependence of the equilibrium charge densities,

and , on the potential is known, e.g., as proposed by Stephenson and Highland [13,20]. In this work we will use the following extension of the SH model:(2a)(2b)where is an elementary charge, are the ionization degrees of the surface charges, and are the 2D surface charge concentrations (measured in m−2). Positive parameters and are the free energies of the surface defects formation under normal conditions and zero potential (. Exact values of are poorly known for many important cases, but are regarded as varying over the range (0.03 − 0.3) eV [13,20]. Positive prefactors and

can originate from different mechanisms of the charge formation [57,58].

Note that the equilibrium values of screening charge densities,

and , depend nonlinearly on according to Eq. (2), and this dependence is understood by us as the “nonlinear screening model”. The main reason for this nonlinearity is the activation mechanism of the surface charge appearance. The “linear relaxation model” means the linearity of relaxation equations (1) with respect to and

.

The electroneutrality condition equivalent to the total charge absence,

, should be valid for the pairwise formation of negative and positive surface charges at , and the condition imposes limitations on the charge density parameters in Eqs. (2), namely:(3)Being interested in low frequency dynamics, below we impose condition (3). To fulfil this condition, we can consider the pairwise formation of negative and positive surface charges, when the charges have opposite signs, , and their concentrations are equal . Then, the following relation between the prefactors and formation energies, , should be valid. If the prefactors are equal, , then the formation energies are equal too, . A violation of condition (3) can take place far from equilibrium, e.g., under electrochemical reactions [13,20]. The dependences of , , and on the electric potential

calculated from Eqs. (2) for typical characteristics of the surface charges are shown in Fig. 1(c).

Using condition (3) for small potential values,

, we obtain that(4a)where we introduce the inverse effective screening length:(4b)

The LGD free energy functional G of a uniaxial ferroelectric includes a Landau energy – an expansion on 2-4-6 powers of the polarization component

, ; a polarization gradient energy, ; an electrostatic energy, ; an elastic, electrostriction, and flexoelectric contributions, ; and a surface energy, . It has the form [4]:(5a)(5b)(5c)(5d)(5e)(5f)Here is the ferroelectric core volume. The coefficient linearly depends on temperature T, , where is the inverse Curie-Weiss constant and is the ferroelectric Curie temperature renormalized by electrostriction and surface tension [59]. All other coefficients in Eqs.(5) are regarded as temperature-independent. The coefficient is positive if the ferroelectric material undergoes a second order transition to the paraelectric phase and negative otherwise; and the coefficient . The gradient coefficients tensor are positively defined. In Eq.(5e), is the stress tensor, is the elastic compliances tensor, is the electrostriction tensor, and

is the flexoelectric tensor.

Note that the boundary conditions for the conducting electrodes (either their charges or potential are fixed) determine the type of the free energy, and the sign of the electrostatic energy (Eq.(5d)) depends on these conditions (see e.g., Refs. [60,61]). In the considered case, the potential of the electrodes is fixed, so the electrostatic energy (5d) has a negative sign. However, Marvan and Fousek [61] have shown that the terms

and

in Eq.(5d) partially compensate each other in the absence of an external electric field due to the internal depolarization field pointed against the polarization, such that the resulting electrostatic energy is positive.

3.2. A single-domain state approximation Since the stabilization of a single-domain polarization in ultrathin ferroelectric films takes place due to chemical switching (see e.g. Refs.[11,12,13]), we may assume that the distribution of the out-of-plane component

does not deviate significantly from the value averaged over the nanoparticle volume, namely

.

The ranges of parameters for which the domain formation does not take place can be established by FEM. We performed FEM in COMSOL@MultiPhysics for Sn2P2S6 (SPS) nanoparticles covered with an electronic-ionic charge with surface density given by Eq.(4a). Material parameters of SPS are listed in Table SI in the Supplement. The initial distribution of polarization was chosen to be randomly small.

It appeared that the ultra-small SPS nanocubes and nanopillars (with height between 2 and 20 nm) are either in a paraelectric (PE) or a single-domain ferroelectric (SDFE) state, which is dependent on the temperature and screening length

. The polydomain ferroelectric state (PDFE) can be stable in bigger nanoparticles for , where the critical height corresponds to the point of a certain phase transition, e.g., between the PE and SDFE phases, between the PE and DPFE phases, or between the SDFE and DPFE phases. In the considered case, the phase transition between the PE and ferroelectric phases in the particle core depends not only on its sizes, but also on the screening characteristics, which weakens the depolarization field arising due to the polarization jump on the outer surface of the core. If we consider the phase diagram in the variables particle height and screening length λ, then the dependence of

on λ can be interpreted as an equilibrium curve between different phases at a fixed temperature.

A typical phase diagram of the SPS nanopillars, plotted in the variables particle height

and screening length λ, which contains the regions of PE, SDFE, and PDFE, are shown in Fig. 2(a). A typical polarization distribution inside a 4-nm nanocube is shown in Figs. 2(b)-(d) for several values of the effective screening length . The transition from the PE phase to the PDFE phase and then to the SDFE phase occurs with a decrease in

from 200 pm to 20 pm.

📷Download : Download high-res image (931KB) Download : Download full-size image Fig. 2. (a) Phase diagram of the SPS core of height

sandwiched between two dielectric layers of thickness and . (b)-(d) The equilibrium polarization distribution in the SPS nanoparticle calculated for 4 nm and several values of the effective screening length 100 pm (b), 50 pm (c), and 10 pm (d). The direction of the uniaxial polarization is shown by the blue arrow in (d). Other parameters: nm, 0.4 nm, 1.2 nm, , , +1%, and

K.

The FEM results show that the screening charges at the top and bottom surfaces of the ferroelectric core,

and

, can influence on the phase transitions and polar properties of the nanoparticle core. Note that the ferroelectric polarization cannot rotate in uniaxial ferroelectrics. Hence, the polarization directed along z-axis is parallel to the sidewalls and it does not require any screening at the walls.

In some cases, e.g., for single-domain ellipsoidal particles, it is possible to obtain analytical expressions for the equilibrium phase boundaries [62,63]. However, it turned out to be impossible to obtain the analytical results for the considered cubic nanoparticles. Therefore, the curves of the PE-SDFE, PE-DPFE and SDFE-DPFE phase transitions in Fig. 2(a) are determined numerically. In particular, the dependence of the critical core height

on the screening length

at the PE-SDFE and PE-DPFE phase transitions corresponds to the disappearance of the average polarization in the core with a change in the length λ.

Note that linearizing Eq.(3) for a small electric potential

, we determine that the screening charge density is proportional to the inverse screening length λ according to Eq.(4a), where the inverse effective screening length is given by Eq.(4b). If the strong inequality, , is not satisfied, Eq.(3) cannot be linearized, therefore, Fig. (2a), where λ was used as the X-axis, is rather illustrative. When constructing Figs. (2b)-(2d), linearization with respect to

is not carried out; the values of λ given in the legends are calculated using Eq.(4b). Since for the chosen parameters the polar state in the particle core, shown in Figs. (2b)-(2d), are in good agreement with the phase diagram in Fig. (2a), we can conclude that the diagram correctly reflects the situation in the core-shell nanoparticle.

Below we will use the ranges of nanoparticle sizes and surface charge parameters for which the domain formation does not take place in the core. In a single domain state, the Landau-Khalatnikov equation determining the average polarization

has the form [21]:(6a)Here the coefficients and are renormalized by elastic strains originated from the nanoparticles-substrate lattice mismatch [63,64]. Using the results of Ref.[23] and Appendix A, the overpotential can be determined in a self-consistent manner:(6b) where the effective thickness of dielectric layers is defined as , the reduced core thickness is defined as , and the total surface charge density From Eq.(6b), contains a contribution which is proportional to the surface charge density , a depolarization field contribution which is proportional to and an external potential drop which is proportional to the applied voltage . The surface charge and the depolarization field contributions, and , do not depend on the particle lateral size, , because the polarization cannot rotate in uniaxial ferroelectrics; therefore, the screening charges at the sidewalls cannot influence the polar properties and phase transitions of the nanoparticle core. Below we consider the case of a periodic applied voltage, , where is the amplitude and

is the frequency.

Using Eqs. (1)-(2), the nonlinear dynamics of the positive and negative surface charges obeys relaxation equations:(6c)

(6d)Hence, we obtained a system of four nonlinear coupled equations, Eqs.(4), with three different characteristic time-scales, , , and . These times can be very different from each other for the case of ionic-electronic screening, as well as different from the period of applied voltage in Eq.(6b). Since a polarization relaxation is determined by soft optical phonons, the strong inequality, , is valid far from the Curie temperature; and it makes sense to normalize time in Eqs.(6) to the Khalatnikov time,

. The normalization is used hereinafter.

4. Polarization and susceptibility under the presence of nonlinear surface screening Since the system of the coupled Eqs.(6) does not allow for analytical solutions, we studied the numerical solutions for polarization, screening charges, and dielectric susceptibility voltage dependences in the actual range of the nanoparticle sizes

nm, dielectric layer thicknesses nm, temperatures∼(100 – 350) K, concentrations m−2, formation energies (0.03 − 0.3) eV [18,19,20] of the screening charge, and their relaxation times (in the units of ). The frequency of the applied voltage is very low, , so that its period . As uniaxial ferroelectric we use the SPS core, which parameters are listed in Table SI in Supplement. A tensile mismatch strain

+1% is applied to support the spontaneous polarization in the ferroelectric core. Numerical results were visualized in Mathematica 12.2 [65].

It was found that the behavior of polarization and susceptibility is very sensitive to the concentrations, formation energies, and relaxation times of the screening charges. In general, by increasing the concentrations

one can switch the core state between the PE-like, AFEI, mixed AFEI-FEI, FEI, and FE-like FEI states. The typical features of these transitions are shown schematically in Fig. 3(a)-(b). In addition the dependence of the average polarization on applied voltage, , is symmetric for equal relaxation times [Fig. 3(a)] and becomes strongly asymmetric for

[Fig. 3(b)].

📷Download : Download high-res image (2MB) Download : Download full-size image Fig. 3. The quasi-static voltage dependences of the polarization

(black curves), positive (red curves) and negative (blue curves) surface charges, and , and the screening efficiency (green curves) calculated for different concentrations, formation energies, and relaxation times of the screening charges. The surface charges concentrations increase from 1016 m−2 to 1018 m−2 for (a) and (b), which are calculated for 0.1 eV. The relaxation times for (a), and , for (b). The surface charges formation energies decreases from 0.15 eV to 0.01 eV for part (c). Part (d) corresponds to eV and decreasing from 0.15 eV to 0.01 eV. For parts (c) and (d) 1016 m−2 and . For all plots 4 nm, nm, 0.4 nm, 1.2 nm, , , +1%, and

K.

From Fig. 3(a) the continuous transition between the PE-like, AFEI, AFE-FEI, and FE-like FEI states takes place when the surface charges concentration

increases from 1016 m−2 to 1018 m−2 and . The voltage dependences of the polarization , negative and positive surface charges, and , and their difference - the “screening efficiency”, , are quasilinear and hysteresis-less in the PE-like state, which exists at concentrations m−2. When the concentrations increase up to m−2, an antiferroelectric-type double hysteresis loop appears. The voltage positions of the polarization double loops opening almost coincide with the positions of the “minor” loops openings of (at negative voltages) and (at positive voltages) which are well-separated. The minor loops and open at some critical voltages due to the strongly nonlinear exponential dependence of given by Eqs. (2). The behavior corresponds to the AFEI state induced in the ferroelectric core by the interaction between the ferroelectric dipoles and the surface screening charges at . Further increase of the screening charge concentration above m−2 leads to the appearance of the pinched loops and the overlapping of the and minor loops. The behavior corresponds to the mixed AFEI-FEI state in the ferroelectric core. The FEI state of the core, which is characterized by a single FE-type hysteresis loop , is induced by the screening charges with the concentration above m−2. The loops in the FEI state become seemingly indistinguishable from the FE loops at m−2. However, this is an apparent effect only, because it is the FE-like FEI state supported by the nonlinear dynamics of surface charge, and the state does not exist without the nonlinear screening. The statement is grounded by the complex view of and loops with straight lines, sharp edges, and very sharp maxima at the coercive voltage. These unusual peculiarities only occur at high concentrations

m−2. This suggests that in realistic materials other charge relaxation mechanisms can become active (desorption, charge emission), rendering these states unrealizable.

From Fig. 3(b) the continuous transition between the PE-like and strongly asymmetric FEI states takes place when the surface charges concentration

increases from 1016 m−2 to 1018 m−2 and . The asymmetry originates from the strong retardation of the screening by one type of the carriers with respect to the other, and with respect to the applied voltage (). The retardation of dynamical screening is responsible for both the asymmetry and the significant horizontal shift of the polarization and surface charge loops, as well as for the disappearance of the double AFEI loops. Note that the case leads to the same physical picture as , with the substitution . In particular, left-shifted loops at becomes right-shifted at

, because the dynamic properties of positive and negative screening charges interchange.

Figs. 3(b)-(d) illustrates schematically the dependence of the nanoparticle polar state on the formation energies

of the surface charges. The simultaneous decrease of both formation energies from 0.15eV to 0.01eV leads to the continuous transition from the AFEI to the FEI state in the ferroelectric core [Fig. 3(c)]. The decrease of one of the formation energies, , from 0.15eV to 0.01eV with the other fixed, eV, leads to the mixing of the AFEI and FEI states [Fig. 3(d)]. The hysteresis loops in the mixed AFEI-FEI state are typically strongly asymmetric, shifted horizontally, distorted, and can be significantly pinched. Since the antiferroelectric-like double loops and pinched loops of polarization are often observed in polydomain ferroelectric thin films [14], as well as the fact that they are typically associated with polydomain or vortex-like domain states in ferroelectric nanoparticles [3,10], their appearance in a single-domain ferroelectric core covered with ionic-electronic screening charges seems unusual and requires further studies. In the considered case, the appearance of the antiferroelectric-like loops is caused by the minor loops of the surface charges, and , which are absent at small voltages and open at the critical voltage

depends on the nanoparticles thickness and temperature [see the blue and red loops in Figs. 3(c) and 3(d)]. Note, the convergence of the charge loops to the point of full overlap as we step through the different phases, e.g., from PE-like to FE-like FEI phases [see Figs. 3(a) and 3(c)]. This corroborates the surface charge origin of the FE-like hysteresis loops of polarization in the different phases.

Note that for both cases,

and , where m−2, the polarization, , and the screening efficiency, , follow the same trends, meaning that the total screening charge correlates with the polarization changes in order to partially screen it. At the same time the voltage behavior of and are complementary to each other at , as they can screen the bound charge of the opposite polarity [compare the blue, red, and green curves in Figs. 3(a), (c)-(d)]. Let us underline that the difference

can be measured experimentally as the charge on the one of the electrodes.

To explore the physical origin and quantify the trends shown in Fig. 3, we consider the case

, corresponding to the surface screening by electrons and holes; and the case corresponding to the surface screening by electrons and cations, respectively. Results are presented in two subsections below; figures in these subsections illustrate the typical voltage dependences of the average polarization , negative and positive surface charges, and , and the difference . We also show the derivatives, and , which are directly related with the dielectric susceptibility and “effective” capacitance, respectively. In all figures the frequency of the applied voltage is very low,

; here we only show the stationary loops and omit transient processes. The third subsection explores the role of size effects on the polarization and susceptibility hysteresis loops.

• 4.1. The influence of surface screening electrons and holes on the polar state and dielectric susceptibility

Figs. 4-7 illustrate that the transition between symmetric PE-like, AFEI, FEI, and FE-like states occurs under the increase of the surface charges concentration

from 2⋅1016 m−2 to 1018 m−2. These dependences are calculated for a 4-nm SPS core at room temperature and several values of equal relaxation times , , and corresponding to the surface screening by electrons and holes. The top rows in Figs. 4-7 show voltage dependences of the polarization , positive and negatives surface charges, and , and the difference . The bottom rows show the quasi-static dependences of the dielectric susceptibility and effective capacitance , which can have four or two symmetric maxima indicating on the AFEI or the FEI states, respectively (see Figs. 4-6). The voltage dependences of the susceptibility are much more complex in FE-like FEI states (see Fig. 7). The common peculiarities of the figures are the smearing of the loop details, the significant decrease of the polarization saturation rate as increases, and the strong and nonlinear increase of the coercive field (i.e., loop width) as and

increase. To show the saturated loops whenever it is possible, we use different voltage scales in Figs. 4-7. Also, there are several important differences in Figs. 4-7, which are discussed below.

📷Download : Download high-res image (1MB) Download : Download full-size image Fig. 4. Symmetric paraelectric-like antiferroionic and ferroionic states. (a, b, c) The quasi-static voltage dependences of the polarization

(black curves), positive (red curves) and negative (blue curves) surface charges, and the difference (green curves). (d, e, f) The quasi-static dependences of the dielectric susceptibility (black curves) and effective capacitance (green curves). The dependences are calculated for the surface charge concentrations 1016 m−2 and several values of relaxation times (a, d), (b, e), and (c, f). Other parameters: 4 nm, nm, 0.4 nm, 1.2 nm, , , +1%, and K, , , and

0.1 eV.

In Fig. 4 the magnitude of polarization and the coercive voltage, corresponding to the appearance of very thin AFEI-type loops, do not vary significantly between the plots (a) and (b), regardless of the 10 times increase in

(from 10 to ). The opening of the double loops becomes evident from both the loop shape in Fig. 4(b) and especially from the four well-separated susceptibility maxima in Fig. 4(e), while at 10 times longer relaxation times, , a very slim polarization loop with a small remanent polarization appears [Fig. 4(c)]. The loop has at least a four times higher maximal polarization corresponding to the six times higher voltage amplitude in comparison with the loops shown in Fig. 4(a)-(b). The unsaturated loops, similar to the loop in Fig. 4(c), are typical for ferroelectric relaxors; and a relaxor-like character is seen from the susceptibility plot (f), where the maxima at coercive voltages are very diffuse and shifted towards the central point

.

As one can see from Figs. 5(a)-(c), an increase of the concentrations

by a factor of 2.5 [in comparison with Figs. 4(a)-(c)] leads to an opening of double AFEI-type polarization loops at small relaxation times (), a pinching of single FEI-type polarization loops at longer relaxation times (), and a relatively moderate blowing of the relaxor-like loops at very long relaxation times (). These trends are evident from the comparison of the susceptibility maxima position, shape, and height in Figs. 5(d)-(f) with Figs. 4(d)-(f). Actually, we see four very sharp and well-separated pairwise maxima in Fig. 5(d); these maxima become much wider with a reduced height when is increased by a factor of ten [see Fig. 5(e)]. The voltage separation of the pairwise maxima almost disappears at very long values of

[see Fig. 5(f)]. Note that Fig. 5 has different voltage scales, which are ±0.04 V for the plots (a, d), ±0.1 V for the plots (b, e), and ±0.6 V for the plots (c, f).

📷Download : Download high-res image (1MB) Download : Download full-size image Fig. 5. Symmetric antiferroinic and ferroionic states. (a, b, c) The quasi-static voltage dependences of the polarization

(black curves), positive (red curves) and negative (blue curves) surface charges, and the difference (green curves). (d, e, f) The quasi-static dependences of the dielectric susceptibility (black curves) and effective capacitance (green curves). The dependences are calculated for the surface charge concentrations 1016 m−2 and several values of relaxation times (a, d), (b, e), and (c, f) in the units of

. Other parameters are the same as in Fig. 4.

Note that the doubling of the concentrations

leads to a significant increase in the remanent polarization, its maximal value, and the coercive voltages at [compare Fig. 5(b) with 4(b)]. However, the relaxor-like loops, observed at

, are almost insensitive to the concentration increase [compare Fig. 5(c) with 4(c)]. This can be explained by the very strong retardation of the polarization screening with respect to the polarization changes taking place over very long relaxation times. We believe that the retardation effect leads to the relaxor-like loops shown in Figs. 4(c) and 5(c).

As it can be seen from Figs. 6(a)-(c), a further increase of the concentrations

by a factor of two [in comparison with Figs. 5(a)-(c)] leads to an opening of single FEI-type polarization loops at smaller () and longer () relaxation times, and more pronounced saturated relaxor-like loops at very long relaxation times (). These trends are confirmed by the dielectric susceptibility features, shown in Figs. 6(d)-(f). Opposed to the AFEI state shown in Fig. 5(d), there are only two very sharp and well-separated maxima in Fig. 6(d). The main maxima become much wider and lower with increasing

, while small secondary maxima appear [see Fig. 6(e)]. The separation of the main and secondary maxima disappears for very long relaxation times in Fig. 6(f). Note that different parts in Figs. 6 have ten times different voltage scales, which are ±0.2 V for (a - d) and ±2 V for (c, f).

📷Download : Download high-res image (1MB) Download : Download full-size image Fig. 6. Symmetric ferroionic states. (a, b, c) The quasi-static voltage dependences of the polarization

(black curves), positive (red curves) and negative (blue curves) surface charges, and the difference (green curves). (d, e, f) The quasi-static dependences of the dielectric susceptibility (black curves) and effective capacitance (green curves). The dependences are calculated for the surface charge concentrations 1017 m−2 and several values of relaxation times (a, d), (b, e), and (c, f) in the units of

. Other parameters are the same as in Fig. 4.

As it can be seen from Fig. 7(a)-(b), the increase of the concentrations

by a factor of 10 [in comparison with Fig. 6] at smaller () and longer () relaxation times leads to the appearance of well-saturated FE-like square-shaped polarization loops with a high coercive voltage (∼2 V). At very long relaxation times () the loops are less saturated and more smooth-shaped [see Fig. 7(c)]. The complete saturation of the polarization loops taking place at the voltages 1 V leads to unusual properties of the dielectric susceptibility, shown in Figs. 7(d)-(f). Here, besides two very sharp and well-separated maxima of , we also have wide regions of nearly zero susceptibility, , and very narrow regions of a negative effective capacitance . Both of these peculiarities, and

, can be treated as signatures of the negative capacitance of the core, which can be quasi-steady state since the external field frequency is very low. Note that Fig. 7 has slightly different voltage scales, which are ±3 V for (a - d) and ±4 V for (c, f).

• 4.2. The influence of surface screening by electrons and cations on the polarization state and susceptibility

📷Download : Download high-res image (1MB) Download : Download full-size image Fig. 7. Symmetric ferroelectric-like ferroionic states. (a, b, c) The quasi-static voltage dependences of the polarization

(black curves), positive (red curves) and negative (blue curves) surface charges, and the difference (green curves). (d, e, f) The quasi-static dependences of the dielectric susceptibility (black curves) and effective capacitance (green curves). The dependences are calculated for the surface charge concentrations 1018 m−2 and several values of relaxation times (a, d), (b, e), and (c, f) in the units of

. Other parameters are the same as in Fig. 4.

Figs. 8-11 illustrate that the transition between asymmetric PE-like, AFEI, FEI, and almost symmetric FE-like states occurs with an increase of the surface charges concentrations

from 2⋅1016 m−2 to 1018 m−2 under different relaxation times , corresponding to the surface screening by electrons and cations. These dependences are calculated for a 4-nm SPS core at room temperature, an electron relaxation time , and several values of much higher cation relaxation times , , and . Note that the coercive field (i.e., loop width) increases strongly and nonlinearly with

.

📷Download : Download high-res image (1MB) Download : Download full-size image Fig. 8. Asymmetric paraelectric-like antiferroionic states. (a, b, c) The quasi-static voltage dependences of the polarization

(black curves), positive (red curves) and negative (blue curves) surface charges, and the difference (green curves). (d, e, f) The quasi-static dependences of the dielectric susceptibility (black curves) and effective capacitance (green curves). The dependences are calculated for several values of relaxation times (a, d), (b, e), and (c, f) in the units of . Other parameters: 4 nm, nm, nm, nm, , , +1%, K, , , 0.1 eV, and

1016 m−2.

The top rows show quasi-static voltage dependences of the polarization

, positive and negative surface charges, and , and the difference . The bottom rows show the quasi-static dependences of the dielectric susceptibility and effective capacitance , which can have two or four asymmetric maxima indicating AFEI, AFEI-FEI, or FEI states, respectively (see Figs. 8-10). Furthermore, the concentration of surface charges m−2 gives more typical polarization and susceptibility hysteresis loops [shown in Fig. 10] compared to the other cases. For

m−2, the voltage dependences of the surface charge density and susceptibility are much more complex and unusual, they correspond to the FE-like FEI states shown in Fig. 11.

The common peculiarity of all figures is the smearing of the asymmetric loops with an increasing

. The asymmetry of the left and right parts of the loops are related to the fact that the electron retardation is much smaller than it is for cations. Since the cation density increases and saturates at positive overpotentials [see Fig. 1(c)], most of the loops in Figs. 8-9 are left-shifted and their openings are bigger at positive voltages.

📷Download : Download high-res image (1MB) Download : Download full-size image Fig. 9. Asymmetric antiferroionic states. (a, b, c) The quasi-static voltage dependences of the polarization

(black curves), positive (red curves) and negative (blue curves) surface charges, and the difference (green curves). (d, e, f) The quasi-static dependences of the dielectric susceptibility (black curves) and effective capacitance (green curves). The dependences are calculated for the surface charge concentrations 1016 m−2 and several values of relaxation times (a, d), (b, e), and (c, f) in the units of

. Other parameters are the same as in Fig. 8.

Other comments and analyses made in the previous subsection regarding the symmetric polarization loops and susceptibility maxima shown in Figs. 4-7 remain relevant for their asymmetric and/or shifted voltage dependences shown in Figs. 8-11. Note that the asymmetry of the polarization and susceptibility voltage dependences becomes much weaker with a

increase above 1017 m−2; therefore, Figs. 10 and 11 looks similar to Figs. 6 and 7 except for the horizontal shift of the polarization loops and susceptibility maxima.

📷Download : Download high-res image (1MB) Download : Download full-size image Fig. 10. Asymmetric ferroionic states. (a, b, c) The quasi-static voltage dependences of the polarization

(black curves), positive (red curves) and negative (blue curves) surface charges, and the difference (green curves). (d, e, f) The quasi-static dependences of the dielectric susceptibility (black curves) and effective capacitance (green curves). The dependences are calculated for the surface charge concentrations 1017 m−2 and several values of relaxation times (a, d), (b, e), and (c, f) in the units of

. Other parameters are the same as in Fig. 8.

📷Download : Download high-res image (1MB) Download : Download full-size image Fig. 11. Almost symmetric ferroelectric-like ferroionic states. (a, b, c) The quasi-static voltage dependences of the polarization

(black curves), positive (red curves) and negative (blue curves) surface charges, and the difference (green curves). (d, e, f) The quasi-static dependences of the dielectric susceptibility (black curves) and effective capacitance (green curves). The dependences are calculated for the surface charge concentrations 1018 m−2 and several values of relaxation times (a, d), (b, e), and (c, f) in the units of

. Other parameters are the same as in Fig. 8.

4.3. The role of the size effect on polarization and susceptibility hysteresis loops Figs. 12 and 13 illustrate the influence of the size effect (i.e., core thickness

) on polarization and susceptibility hysteresis loops for the cases and . The top rows show quasi-static voltage dependences of the polarization , positive and negative surface charges, and , and the difference . The overall peculiarity of the figures is the transition from the AFE-type double hysteresis loop to the FE-type single loop with an increase of . However, the quasi-static dependences of the dielectric susceptibility, , and effective capacitance, , still have four symmetric (for ) or four asymmetric (for ) maxima indicating AFEI or mixed AFEI-FEI states, respectively. In particular, Fig. 12 illustrates the size-induced transition from symmetric AFEI to FEI states; and Fig. 13 illustrates the transition from asymmetric AFEI to FEI states with an

increase from 3 nm to 5 nm. One of the most interesting behaviors is the strongly pinched polarization loops and four well-separated maxima of the dielectric susceptibility observed in a 3-nm SPS core, shown in Figs. 11(a) and 11(d), 12(a) and 12(d). The loop pinching in ultra-small ferroelectric nanoparticles, which should be paraelectric under linear screening conditions, is attributed to nonlinear electronic-ionic screening. The effect, in principle, can be an additional contribution for the explanation of the unusual multiple maxima of electric current observed in ferroelectric nanoparticles in a nonpolar fluid suspension, which were attributed to aggregation and disaggregation of multipole particles in an AC field [45].

📷Download : Download high-res image (1MB) Download : Download full-size image Fig. 12. Size-induced transition from symmetric antiferrionic to ferroionic states. (a, b, c) The quasi-static voltage dependences of the polarization

(black curves), positive (red curves) and negative (blue curves) surface charges, and the difference (green curves). (d, e, f) The quasi-static dependences of the dielectric susceptibility (black curves) and effective capacitance (green curves). The dependences are calculated for several thickness of the SPS core nm (a, d), nm (b, e), and nm (c, f). Other parameters: nm, 0.4 nm, 1.2 nm, , , +1%, K, , , 0.1 eV, 1016 m−2 and

.

📷Download : Download high-res image (1MB) Download : Download full-size image Fig. 13. Size-induced transition from asymmetric antiferrionic to ferroionic states. (a, b, c) The quasi-static voltage dependences of the polarization

(black curves), positive (red curves) and negative (blue curves) surface charges, and the difference (green curves). (d, e, f) The quasi-static dependences of the dielectric susceptibility (black curves) and effective capacitance (green curves). The dependences are calculated for several thickness of the SPS core nm (a, d), nm (b, e), and nm (c, f). Relaxation times and

. Other parameters are the same as in Fig. 12.

5. Conclusion The polar states of uniaxial ferroelectric nanoparticles interacting with the surface system of electronic and ionic charges with a broad distribution of mobilities is explored, corresponding to the experimental case of nanoparticles in solution or ambient conditions. Due to the nonlinear electric interaction between the ferroelectric dipoles and surface charges with slow relaxation dynamics in an external field, the transitions between the PE-like, AFEI, and FE-like FEI states emerge.The polarization and susceptibility in these systems is very sensitive to the concentrations, formation energies, and relaxation times of the screening charges. By increasing the surface charge concentrations one can continuously switch the state of the ferroelectric core between PE-like, AFEI, mixed AFEI-FEI, FEI, and FE-like FEI states, for which we establish and analyzed the distinct features of polarization, surface charges, and dynamic dielectric response to a periodic external field. In particular, the voltage dependences of the polarization and negative and positive surface charges are quasilinear and hysteresis-less in the PE-like state, which exists for small concentrations of surface charges. When the concentrations increase, an antiferroelectric-type double hysteresis loop appears. The voltage positions of the double loops opening almost coincide with the opening positions of the positive and negative charges’ “minor” loops, which are well-separated. For the minor loops, the surface charge opens at some critical voltages due to the strongly nonlinear exponential dependence of their density on the electric overpotential. This behavior corresponds to the AFEI state induced in the ferroelectric core by the interaction between ferroelectric dipoles and surface screening charges. Further increase of the screening charge leads to the appearance of the pinched polarization loops. The behavior corresponds to the mixed AFEI-FEI state in the ferroelectric core. The FEI state of the core, which is characterized by a single FE-type hysteresis loop, is induced by screening charges in higher concentrations. The loops in the FEI state become seemingly indistinguishable from the FE loops at high concentrations. However, this is an apparent effect only, because it is the FE-like FEI state supported by the nonlinear dynamics of surface charge, and the state does not exist without the nonlinear screening.The simultaneous decrease of the positive and negative surface charges formation energies leads to the continuous transition from the AFEI to the FEI state in the ferroelectric core. The decrease of one of the formation energies, with the other fixed, leads to the mixing of AFEI and FEI states. The hysteresis loops in the mixed AFEI-FEI state are typically strongly asymmetric, horizontally shifted, distorted, and can be strongly pinched. Since the antiferroelectric-like double loops and pinched loops of polarization are often observed in polydomain ferroelectric thin films, as well as the fact that they are typically related with polydomain or vortex-like domain states in ferroelectric nanoparticles, their appearance in a single-domain ferroelectric core covered with ionic-electronic screening charges seems unusual and requires further studies. In the considered case, the appearance of the antiferroelectric-like loops is caused by the minor loops of the surface charges, which are absent at small voltages and open above the critical voltage.The above-described average polarization and susceptibility dependences on the applied voltage are symmetric for equal relaxation times of the positive and negative surface charges, and becomes strongly asymmetric when these times differ by one or more orders of magnitude. The asymmetry originates from the strong retardation of the screening by one type of charge carrier with respect to the other. The retardation of dynamical screening is responsible for both the asymmetry and the significant horizontal shift of the polarization and surface charge loops, as well as the disappearance of double AFEI loops. Thus, we conclude, that the crossover between different polar states can be controlled not only by the static characteristics of the surface charges, such as concentrations and formation energies, but also by their relaxation dynamics in an applied field.

Authors’ contribution A.N.M. and S.V.K. generated the research idea, analyzed results and wrote the manuscript draft. A.N.M. formulated the problem, jointly with M.Ye. performed analytical calculations and prepared figures. E.A.E. wrote the codes. D.R.E. worked on the results explanation and manuscript improvement. All co-authors discussed the results Eq. 2a,5a,5b,5c,5f,6a,6c,6d.

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements A.N.M. acknowledges EOARD project 9IOE063 and related STCU partner project P751a. E.A.E. acknowledges CNMS2021-B-00843 “Effect of surface ionic screening on polarization reversal scenario in antiferroelectric thin films: analytical theory, machine learning, PFM and cKPFM experiments”. This effort (S.V.K) was supported as part of the center for 3D Ferroelectric Microelectronics (3DFeM), an Energy Frontier Research Center funded by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences under Award Number DE-SC0021118, and performed in part at the Oak Ridge National Laboratory's Center for Nanophase Materials Sciences (CNMS), a U.S. Department of Energy, Office of Science User Facility.

Appendix. Supplementary materials

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