Ferroelectric Domain Reversal: The Role of Domain Wall Conduction

Ferroelectric domain reversal is a vast research area relevant to the fundamental science and applications as well as to macro- and nanoscales. An instantaneous compensation of typically huge depolarizing electric fields is a key element here. On the other hand, the domain wall (DW) conduction is now regarded as a general ferroelectric phenomenon. Being often separated by big surface energy barriers in macroscopic experiments, the DW conduction becomes easily available under the conditions of atomic-force microscopy. We claim that the DW conduction has to be regarded as a crucial and general ingredient of the domain reversal processes. Its importance is in providing an automatic compensation of the depolarizing electric fields. The presence of DW conduction essentially modifies the basics of domain reversal processes, including the relations for domain nucleus energies, the domain grows velocities, and the domain shapes. Concerning atomic-force microscopy applications, domain reversal theories have to include injection models from conductive tip electrodes. Below we provide specific primary results relevant to the basics of DW conduction mediated ferroelectric domain reversal.

1 INTRODUCTION

Domain reversal (engineering) in ferroelectrics is a vast research area that dates back to the 1950s, see, e.g., [1–5] and references therein. An abundance of experimental and theoretical results accumulated during the first several decades is largely related to the macroscopic case when the reversal of the spontaneous polarization $\mathbf{P} = \mathbf{P}(\mathbf{r})$ occurs with the use of macroscopic electrically biased electrodes (the capacitor configuration), see Fig. 1a. Here, the common feature of a real process is that the electric field E, necessary for the reversal, is typically orders of magnitude smaller than the characteristic depolarizing field ${{E}_{{\text{d}}}} = 4\pi {{P}_{s}}{\text{/}}{{\varepsilon }_{\parallel }}$, where ${{P}_{s}} = {\text{|}}\mathbf{P}{\text{|}}$ and ${{\varepsilon }_{\parallel }} = {{\varepsilon }_{{zz}}}$ is the longitudinal dielectric constant. In other words, the real coercive field ${{E}_{c}}$ is much smaller than ${{E}_{{\text{d}}}}$. Here and below, we consider for simplicity uniaxial ferroelectrics where $\mathbf{P} = P(\mathbf{r})\hat {\mathbf{z}}$ with $P(\mathbf{r}) = \pm {{P}_{s}}$ is either parallel or antiparallel to the polar z axis.

Fig. 1.

(Color online) Three basic configurations relevant to ferroelectric domain reversal. (a) Capacitor geometry with applied field ${{E}_{0}} = U{\text{/}}h$; $\hat {\mathbf{z}}$ is the unit vector along the z axis, n the unit surface normal vector, and $\hat {\varepsilon }$ is the dielectric tensor. Counter-domain nuclei of different shapes (needle-like and smooth shaped) can appear at the upper electrode; the domain walls are indicated by violet lines. (b) Side view of AFM domain reversal under a square shaped $U,\tau$ voltage pulse, a is the contact area radius. Initially, domains experience the forward growth and then expand laterally, such that ${{r}_{0}}(U,\tau )$ is the after-pulse domain radius. (c) Top view of the inverted domain after the lateral expansion. Blue color indicates the bound polarization charge $- 2{{P}_{s}}$ to be compensated.

The real polarization reversal process in the capacitor geometry is commonly viewed as nucleation, growth, and confluence of numerous microscopic counter-domains [3, 4]. While compensation of the arising bound polarization charge $\pm 2{{P}_{s}}$ definitely occurs at the electrodes, it is not generally allowed at DWs inside the crystal,¹ see also Fig. 1a. This leads to the generation of depolarizing fields ranging from 0 to ${{E}_{{\text{d}}}}$, i.e., to an apparent inconsistency of the reversal concept. To overcome it, the counter-domains are assumed to be needle-like [3, 4, 6], as illustrated in Fig. 1a. This assumption is satisfactory for an initial stage of the reversal, but it cannot ensure the inequality ${{E}_{{\text{d}}}}{\text{|}}\mathbf{n} \cdot \hat {\mathbf{z}}{\text{|}} < {{E}_{c}}$ in the bulk of the crystal during the whole process. Moreover, there are documented cases where ${{E}_{{\text{d}}}}{\text{|}}\mathbf{n} \cdot \hat {\mathbf{z}}{\text{|}} \gg {{E}_{c}}$ and the concept of polarization reversal not including the charge compensation experiences difficulties, see Section 2 for details.

Advent of atomic force microscopy (AFM) in the beginning of 2000s has revolutionized the area and transferred it to the nanoscale [7, 8]. The AFM methods are relevant both to recording and visualization of domain structures on the nanoscale and also to applications to memory devices and nonlinear optics. Useful examples of application of these methods to domain engineering in different ferroelectric materials can be found in [9–18]. With the radius $a \sim 50$ nm of the tip contact-area² and the applied voltage $U =$ 10–100 V, the field E_z ≈ U/a ~ 1–10 MV/cm strongly exceeds ${{E}_{c}}$ causing rapid forward growth of the inverted domain followed by its lateral expansion. The radius of the inverted area ${{r}_{0}}$, as illustrated in Figs. 1b and 1c, can exceed 1 μm [13, 14]. Such a reversal processes would be impossible without an instantaneous compensation (and over-compensation) of the polarization surface charge density $- 2{{P}_{s}}$ and of the corresponding huge depolarizing field $- 4\pi {{P}_{s}}{\text{/}}{{\varepsilon }_{\parallel }}$ in the non-electroded area ${{r}_{0}} \geqslant r > a$.

Consider now the effect of domain-wall (DW) conduction and its relevance to the domain reversal. This effect (predicted about 50 years ago [19]) incorporates the compensation of the bound polarization charge $\pm 2{{P}_{s}}{\text{|}}\mathbf{n} \cdot \hat {\mathbf{z}}{\text{|}}$ at the generally inclined domain wall separating two 180° domains, see Figs. 1a and b. This compensation can be accomplished by electrons or holes via band bending or/and by localized free charges [20, 21]. The compensating charges, whose concentration can reach ~10²⁰ cm^–3 for $\mathbf{n}\parallel \hat {\mathbf{z}}$, are free to move along the wall causing the DW conduction. Nowadays, it is detected in many materials [21]. Being a general phenomenon, the DW conduction is not always easily accessible because of surface energy barriers causing strong surface voltage drops. The AFM configurations facilitate its accessibility. To describe the domain reversal, it is necessary to combine the theory of two-dimensional DW conduction with the known charge injection models [22].

In this letter, our goals are (i) to show that the idea of the DW conduction compensation can substantially advance the physics and prospects of the domain reversal and (ii) to outline the problems to be solved.

2 PROBLEMS OF COMPENSATION-FREE MODELS

Remarkably, the charge compensation issue is often missed in the modeling of the domain reversal. Here, we elucidate its significance as applied to an important and well investigated ferroelectric, lithium niobate (LN). This highly dielectric material possesses a large spontaneous polarization (${{P}_{s}} \approx 70$ μC/cm²), a high Curie temperature (${{T}_{c}} \approx 1200$°C), two allowed orientations of P (parallel and antiparallel to $\hat {\mathbf{z}}$), and modest (without strong anisotropy) room-temperature dielectric constants, ${{\varepsilon }_{\parallel }} \approx 30$ and ${{\varepsilon }_{ \bot }} \approx 85$. The depolarizing field ${{E}_{{\text{d}}}} = 4\pi {{P}_{s}}{\text{/}}{{\varepsilon }_{\parallel }}$ is here above 2 MV/mm, while the coercive field ${{E}_{c}}$ ranges, depending on the composition, from ≈1 to 10 kV/mm. The recorded domain patterns are robust and reversible. The DW conductivity (both ohmic and diode-like) was reported in [23–25]. It exceeds the bulk conductivity by $\gtrsim 13$ orders of magnitude even for ${\text{|}}\mathbf{n} \cdot \hat {\mathbf{z}}{\text{|}} \approx {{10}^{{ - 2}}}$.

Smoothly shaped (as in Fig. 1a) macroscopic counter-domains relevant to the depolarizing field ${{E}_{{\text{d}}}} \gtrsim {{10}^{2}}{{E}_{c}}$ were recorded on the $+ z$ face using the effect of poling inhibition [26, 27]: the crystal is pre-irradiated with a focused UV beam leading to locally and permanently increased coercive field. The subsequent room-temperature poling with macroscopic electrodes repolarizes the whole crystal except for the pre-irradiated area. The resulting μm sized near-surface domains can be visualized via etching and piezo-AFM. The described observations can hardly be explained without assuming a kind of charge compensation.

Independently, experiments on the AFM domain reversal with application of square-shaped $U,\tau$ voltage pulses to conductive tip were performed with z-cut LN samples [9, 11–15]. A remarkable feature of these experiments is that the inverted domain radius ${{r}_{0}}$ grows with U and τ strongly exceeding the contact area radius $a \lesssim 50$ nm and reaching values above 1 μm, see also Fig. 1c. The depolarizing field ${{E}_{{\text{d}}}}$ exceeds ${{E}_{c}}$ by orders of magnitude within a large non-electroded area $a < r < {{r}_{0}}$, making difficult explanation of the observations without charge-compensation assumptions. Surprisingly, no such assumptions were postulated. To explain the results, it was assumed instead that the field ${{E}_{z}}$ near the crystal surface is not affected by the polarization surface charge density $- 2{{P}_{s}}$, but obeys relations relevant to non-polar dielectrics [28].

3 DOMAIN FORMATION ENERGIES

Expressions for the domain formation energy $\delta \mathcal{E}$ (the energy difference for configurations with and without a domain nucleus) lie in the basis of domain reversal theory [3, 4, 29, 30]. Usually, the domain geometry is characterized by two length parameters. As a function of these parameters, the energy is expected to possess a saddle point $\delta {{\mathcal{E}}_{*}}$; it is the critical activation energy [29, 31]. The values $\delta {{\mathcal{E}}_{*}} \gtrsim 1$ eV make the domain reversal practically forbidden. The main contributions to $\delta \mathcal{E}$ are the surface and electrostatic ones. The surface contribution is given by the integral over the DW surface, $\delta {{\mathcal{E}}_{s}} = \int w{\kern 1pt} dS$, where w is a positive surface density. It is expected to depend on ${\text{|}}\mathbf{n} \cdot \hat {\mathbf{z}}{\text{|}}$ and increase substantially when this factor changes from 0 to 1 [20, 21]; i.e., transition from neutral to strongly charged domain walls occurs. We model it by the relation $w = {{w}_{0}} + {{w}_{1}}{\text{|}}\mathbf{n} \cdot \hat {\mathbf{z}}{\text{|}}$ with ${{w}_{1}}{\text{/}}{{w}_{0}} \gg 1$ leading to

$$\delta {{\mathcal{E}}_{s}} = {{w}_{0}}S + {{w}_{1}}{{S}_{ \bot }},$$

(1)

where S is the domain surface and ${{S}_{ \bot }}$ its cross-section at $z = 0$.

The electrostatic contribution to the domain formation energy $\delta {{\mathcal{E}}_{{{\text{el}}}}}$ needs to be considered in more detail. Using the general relations for thermodynamic potentials of dielectrics [32], assuming that the reversal process runs at a constant applied voltage U, and setting $\mathbf{D} = \hat {\varepsilon }\mathbf{E} + 4\pi \mathbf{P}$ for the induction vector, we have for the differential of the electrostatic energy:

$$d{{\mathcal{E}}_{{{\text{el}}}}} = - (\mathbf{P} \cdot \mathbf{E} + \hat {\varepsilon }{\mathbf{E}^{2}}{\text{/}}8\pi ){\kern 1pt} dV.$$

(2)

The total electrostatic energy is given by the integral over the sample. Spontaneous polarization $\mathbf{P} = P(\mathbf{r})\hat {\mathbf{z}}$ changes sign at the DW, and the electric field is expressed by the electrostatic potential, $\mathbf{E} = - \nabla \varphi$.

Relation (2) is valid for both principal cases with and without DW charge compensation. It is instructive to consider and compare the electrostatic contribution $\delta {{\mathcal{E}}_{{{\text{el}}}}}$ for these cases. Let for definiteness a counter-domain be at the $+ z$ face in the capacitor geometry, as depicted in Fig. 1a. Below we refer the subscripts 0 and 1 to the configurations without and with counter-domain, respectively. Correspondingly, we have ${\mathbf{E}_{0}} = {{E}_{0}}\hat {\mathbf{z}} = U\hat {\mathbf{z}}{\text{/}}h$, where h is the sample thickness, and ${\mathbf{P}_{0}} = - {{P}_{s}}\hat {\mathbf{z}}$. To avoid complications, we consider below mostly the isotropic case ${{\varepsilon }_{ \bot }} = {{\varepsilon }_{\parallel }}$. When necessary, we provide generalizations to ${{\varepsilon }_{ \bot }} \ne {{\varepsilon }_{\parallel }}$. Numerous calculation/technical details can be found in Supplementary materials.

$\bullet$ Let the charge compensation be absent. The potential ${{\varphi }_{1}}$ obeys then the Poisson equation ${{\varepsilon }_{\parallel }}{\boldsymbol{\nabla }^{2}}{{\varphi }_{1}} = - 4\pi {{\nabla }_{z}}P(\mathbf{r})$; its right-hand side is singular at the DW. Using this equation, partial integrations, and the boundary conditions for ${{\varphi }_{{0,1}}}$, we come to

$$\delta {{\mathcal{E}}_{{{\text{e}}l}}} = - 2{{E}_{0}}{{P}_{s}}{{V}_{D}} + \frac{{{{\varepsilon }_{\parallel }}}}{{8\pi }}\int {{({\mathbf{E}_{1}} - {\mathbf{E}_{0}})}^{2}}{\kern 1pt} dV,$$

(3)

where ${{V}_{D}}$ is the domain volume. This corresponds to the general result of [30]. While the first (dipole) contribution is negative, the second one (the depolarization energy) is positive. The only source for the field difference ${\mathbf{E}_{1}} - {\mathbf{E}_{0}}$ is the DW surface charge, this polarization charge can be large. This is why the second contribution is often dominating and the domain formation is unlikely [29]. Remarkably, the first/second contribution in Eq. (3) does not correspond to the first/second term in Eq. (2).

Suppose that the domain is small, such that the influence of the bottom electrode is negligible. The depolarization energy and $\delta {{\mathcal{E}}_{{{\text{e}}l}}}$ can be calculated explicitly for the half-spheroidal domain shape with the symmetry axis along z and semi-axes ${{l}_{ \bot }},{{l}_{z}}$. In this case, the distribution ${\mathbf{E}_{1}}(\mathbf{r}) - {\mathbf{E}_{0}}$ for $z \geqslant 0$ is the same as the field of full spheroid possessing the surface charge $2{{P}_{s}}(\mathbf{n} \cdot \hat {\mathbf{z}})$. We get finally:

$$\delta {{\mathcal{E}}_{{{\text{e}}l}}} = - 2{{E}_{0}}{{P}_{s}}{{V}_{D}} + 8\pi P_{s}^{2}{{f}_{z}}{{V}_{D}}{\text{/}}{{\varepsilon }_{\parallel }},$$

(4)

where ${{V}_{D}} = 2\pi l_{ \bot }^{2}{{l}_{z}}{\text{/}}3$ and ${{f}_{z}} = {{f}_{z}}({{l}_{ \bot }}{\text{/}}{{l}_{z}})$ is the so-called depolarization factor [32]. For ${{l}_{ \bot }}{\text{/}}{{l}_{z}} \ll 1$ and $\gg {\kern 1pt} 1$, we have ${{f}_{z}} \simeq {{({{l}_{ \bot }}{\text{/}}{{l}_{z}})}^{2}}\ln (2{{l}_{z}}{\text{/}}{{l}_{ \bot }})$ and $\simeq {\kern 1pt} 1$, respectively. Generalization to ${{\varepsilon }_{ \bot }} \ne {{\varepsilon }_{\parallel }}$ results in the replacement ${{l}_{ \bot }}{\text{/}}{{l}_{z}} \to {{l}_{ \bot }}{{\sqrt \varepsilon }_{\parallel }}{\text{/}}{{l}_{z}}{{\sqrt \varepsilon }_{ \bot }}$ in the argument of ${{f}_{z}}$. The above expression for $\delta {{\mathcal{E}}_{{{\text{e}}l}}}$ is fully consistent with [29]. For ${{E}_{0}} \ll {{P}_{s}}$, the depolarization energy becomes comparable with the dipole energy value only for ${{l}_{ \bot }} \ll {{l}_{z}}$.

Taking the sum $\delta \mathcal{E} = \delta {{\mathcal{E}}_{s}} + \delta {{\mathcal{E}}_{{{\text{e}}l}}}$ and expressing the surface parameters S and ${{S}_{ \bot }}$ in Eq. (1) by ${{l}_{ \bot }},{{l}_{z}}$, we can investigate the dependence $\delta \mathcal{E}({{l}_{ \bot }},{{l}_{z}})$. Figure 2a exhibits lines $\delta \mathcal{E}({{l}_{ \bot }},{{l}_{z}}) = {\text{const}}$ (in electronvolts) for the LN parameters and representative values ${{E}_{0}} = 4$ kV/mm, ${{w}_{0}} = 3$ and ${{w}_{1}} = 15$ erg/cm². We see a saddle point with $l_{ \bot }^{*} \simeq 1.2$ nm, $l_{z}^{*} \simeq 145$ nm, and $\delta {{\mathcal{E}}_{*}} \simeq 4.8$ eV. The effect of ${{w}_{1}}$ is almost absent here. When changing ${{E}_{0}}$ and ${{w}_{0}}$, the above saddle point parameters scale approximately as ${{w}_{0}}{\text{/}}{{E}_{0}}$, ${{w}_{0}}{\text{/}}E_{0}^{{3/2}}$, and $w_{0}^{3}{\text{/}}E_{0}^{{5/2}}$. To get $\delta {{\mathcal{E}}_{*}} \lesssim 1$ eV, one has to make ${{E}_{0}} \gtrsim 7$ kV/mm leading to overly small values of $l_{ \bot }^{*}$ or to assume that ${{w}_{0}}$ is substantially smaller than 3 erg/cm². The latter is unlikely [6, 11, 31].

Fig. 2.

(Color online) Domain formation energy $\delta \mathcal{E}$ (in electronvolts) versus ${{l}_{ \bot }}$, ${{l}_{z}}$, and ${{E}_{0}}$ for the LN parameters, ${{w}_{0}} = 3$ and ${{w}_{1}} = 15$ erg/cm². (a) Contour lines $\delta \mathcal{E}({{l}_{ \bot }},{{l}_{z}}) = {\text{const}}$ for ${{E}_{0}} = 4$ kV/mm in the absence of charge compensation. At the saddle point $l_{ \bot }^{*} \simeq 1.2$ nm, $l_{z}^{*} \simeq 145$ nm, and $\delta {{\mathcal{E}}_{*}} \simeq 4.8$ eV. (b) Contour lines $\delta \mathcal{E}({{l}_{ \bot }},{{l}_{z}}) = {\text{const}}$ for ${{E}_{0}} = 4$ kV/mm in the presence of DW charge compensation, no saddle points. (c) Lines $\delta \mathcal{E}({{E}_{0}},{{l}_{z}}) = {\text{const}}$ for ${{l}_{ \bot }} = 1$ nm with the DW charge compensation.

$\bullet$ Let the DW conduction charge compensation be present. This means that the metal boundary condition $\varphi ({{{\mathbf{r}}}_{{{\text{DW}}}}}) = U$ has to be applied and ${\mathbf{E}_{1}} = 0$ inside the domain, see also Fig. 1a. Note that it is not necessary for the DW conductivity to be extremely large (metal-like). It is sufficient that the electric equilibrium along the DW occurs fast enough on the scale of the characteristic domain reversal time.

Remarkably, the contribution to $\delta {{\mathcal{E}}_{{{\text{e}}l}}}$ coming from the first term in Eq. (2), namely $\int ({\mathbf{E}_{0}} \cdot {\mathbf{P}_{0}} - {\mathbf{E}_{1}} \cdot {\mathbf{P}_{1}})dV$, turns now exactly to zero, because the integral $\int_0^h {{E}_{{1,z}}}dz \equiv - \int_0^h \partial \varphi {\text{/}}\partial z{\kern 1pt} dz$ is identically $U \equiv {{E}_{0}}h$ for any transverse cross-section point. The physical reason for this vanishing is clear. The polarization charge $2{{P}_{s}}{{S}_{ \bot }}$ stays at a constant potential ${{\varphi }_{{{\text{DW}}}}} = U$, such that the voltage source produces no work. On the other hand, the contribution coming from the second term in Eq. (2) is now negative,

$$\delta {{\mathcal{E}}_{{{\text{e}}l}}} = - \frac{{{{\varepsilon }_{\parallel }}}}{{8\pi }}\int {{({\mathbf{E}_{1}} - {\mathbf{E}_{0}})}^{2}}{\kern 1pt} dV,$$

(5)

and opposite in form to the depolarization energy in Eq. (3), see Supplementary materials for details. The source for the field difference ${\mathbf{E}_{1}} - {\mathbf{E}_{0}}$ are still surface charges at the wall, however, these charges are determined by the metal boundary condition and have no direct relation to the spontaneous polarization. The effect of spontaneous polarization on the domain reversal comes only from the surface contribution $\delta {{\mathcal{E}}_{s}}$. Modification of the electrostatic contribution is thus drastic.

Now we return to the half-spheroidal domain shape with semi-axes ${{l}_{ \bot }},{{l}_{z}}$. We indicate that the distribution of ${\mathbf{E}_{1}}({\mathbf{r}})$ around the domain for $z \geqslant 0$ is the same as the distribution of the field around a full uncharged metal spheroid with semi-axes ${{l}_{ \bot }},{{l}_{z}}$ placed in the field ${\mathbf{E}_{0}} = {{E}_{0}}\hat {\mathbf{z}}$. This external field induces the dipole moment $\mathbf{d} = {{\varepsilon }_{\parallel }}{\mathbf{E}_{0}}{{V}_{D}}{\text{/}}2\pi {{f}_{z}}$ with the same depolarization factor ${{f}_{z}}({{l}_{ \bot }}{\text{/}}{{l}_{z}})$ [32]. Correspondingly, the electrostatic contribution is now

$$\delta {{\mathcal{E}}_{{{\text{e}}l}}} = - {\mathbf{E}_{0}} \cdot \mathbf{d}{\text{/}}2 = - {{\varepsilon }_{\parallel }}E_{0}^{2}{{V}_{D}}{\text{/}}4\pi {{f}_{z}}{\kern 1pt} .$$

(6)

The larger is ${{l}_{z}}{\text{/}}{{l}_{ \bot }}$, the more negative $\delta {{\mathcal{E}}_{{{\text{e}}l}}}$. Account for the anisotropy results again in the replacement ${{l}_{ \bot }}{\text{/}}{{l}_{z}} \to {{l}_{ \bot }}{{\sqrt \varepsilon }_{\parallel }}{\text{/}}{{l}_{z}}{{\sqrt \varepsilon }_{ \bot }}$ in the argument of ${{f}_{z}}$.

Taking the sum $\delta \mathcal{E} = \delta {{\mathcal{E}}_{s}} + \delta {{\mathcal{E}}_{{{\text{e}}l}}}$, one can make sure that the function $\delta \mathcal{E}({{l}_{ \bot }},{{l}_{z}})$ has no saddle points. Instead, it grows with ${{l}_{ \bot }}$ and possesses a maximum in ${{l}_{z}}$. As the negative contribution $\delta {{\mathcal{E}}_{{{\text{e}}l}}}$ is proportional to $E_{0}^{2}$, the fields necessary for the domain reversal are less sensitive to the choice of the surface energy densities ${{w}_{{0,1}}}$ compared to the case of no charge compensation. Also, large values of ${{\varepsilon }_{ \bot }}$ are favorable for the reversal.

Consider the LN case. Figure 2b shows the contour lines $\delta \mathcal{E}({{l}_{ \bot }},{{l}_{z}}) = {\text{const}}$ (in electronvolts) for ${{E}_{0}} =$ 4 kV/mm, ${{w}_{0}} = 3$ and ${{w}_{1}} = 15$ erg/cm². For ${{l}_{ \bot }} \approx 1$ nm, the maximal in ${{l}_{z}}$ values of $\delta \mathcal{E}$ are about 1 eV. Figure 2c shows contour lines $\delta \mathcal{E}({{E}_{0}},{{l}_{z}}) = {\text{const}}$ for ${{l}_{ \bot }} = 1$ nm. An increase in ${{E}_{0}}$ causes a rapid decrease in the values of ${{l}_{z}}$ and $\delta \mathcal{E}({{l}_{z}})$ relevant to the maximum. As the maximum in ${{l}_{z}}$ is overcome, the dependence $\delta \mathcal{E}({{l}_{z}})$ becomes decreasing up to negative values. Clearly, the predictions of Figs. 2b and 2c are beneficial for the domain reversal as compared to those of Fig. 2a.

The above relations can be applied to evaluate $\delta \mathcal{E}$ in the AFM configurations; the requirement ${{l}_{ \bot }} < a$ is not obligating here. On the other hand, the actual applied fields become much larger than the coercive field ${{E}_{c}}$. In accordance with Fig. 2c, this decreases $\delta \mathcal{E}$ and facilitates formation of domains with not very large ratio ${{l}_{z}}{\text{/}}{{l}_{ \bot }}$.

An efficient domain nucleation for the applied field ${{E}_{0}}$ substantially exceeding the coercive field ${{E}_{c}}$ in the AFM configurations is expected to be followed by a fast forward domain growth. Modeling of this growth, which can be influenced by different factors, see the discussion section, is beyond the scope of this paper. However, as soon as the domain has reached the bottom electrode, the effect of DW conduction drops down dramatically, and new challenging problems, relevant to the experimentally investigated and modeled lateral domain expansion, arise.

4 LATERAL DOMAIN GROWTH, CHARGE COMPENSATION

One of the most striking features of the previous modeling of the lateral domain expansion (detected and investigated in AFM configurations under $U,\tau$ pulses) is ignorance of the bound-charge compensation, see Section 2. Such modeling attempts could be dismissed unless an important circumstance: A big amount of experimental data on ${{r}_{0}}(U,\tau )$ can be nicely fitted under two assumptions: (i) ${{E}_{z}} = {{E}_{z}}({{r}_{0}}, + 0) \propto$ $U{\text{/}}{{r}_{0}}$ and (ii) $v \propto \exp ( - {\text{const/}}{{E}_{z}})$. The second assumption represents the empirical Merz law [6] for the lateral expansion velocity $v = d{{r}_{0}}{\text{/}}dt$, while the first one has no visible physical grounds. Together they give the relation $d{{r}_{0}}{\text{/}}d\tau = {{v}_{*}}\exp ( - {{r}_{0}}{{E}_{*}}{\text{/}}U)$ with fit parameters ${{v}_{*}}$ and ${{E}_{*}}$. Figure 3 illustrates the dependence ${{r}_{0}}(U,\tau )$.

Fig. 3.

(Color online) Main results of the fit procedure for the lateral domain growth. The ratio ${{r}_{0}}{\text{/}}a$ versus (a) $U{\text{/}}{{U}_{0}}$ for τ/τ₀ = (1) 10², (2) 10³, and (3) 10⁴ and (b) τ/τ₀ for U/U₀ = (1) 0.2, (2) 0.3, and (3) 0.5. Here, ${{U}_{0}} = a{{E}_{*}}$ and ${{\tau }_{0}} = a{\text{/}}{{v}_{*}}$.

One sees that the U-dependence is almost linear, while the τ-dependence changes from constant to linear on the log-scale within several orders of magnitude. Just these characteristic features were established experimentally [14], and they cannot be set aside.

Thus, we must resolve simultaneously two problems: to provide a charge compensation mechanism and ensure that it gives the necessary field dependence ${{E}_{z}}({{r}_{0}})$. The key question is here: where the compensating charges come from? We suppose that they come from the tip electrode. This is relevant to the application of charge injection models. While charge injection is a common phenomenon in solid state [22], it is known to be sensitive to the actual geometry and material. An essential feature of our charge screening case is its two-dimensional (2D) character. The possibility for the surface charge to migrate is analogous to the assumption of DW conduction. The difference is that instead of two counter-domains we have a single domain with naked surface charge $- 2{{P}_{s}}$, see Fig. 1c. As concerns the Merz law for the lateral expansion, its modeling via domain nucleation at $\mathbf{n} \bot \hat {\mathbf{z}}$ is performed in [30] and widely recognized in [2–4].

A general relation relevant to our 2D charge compensation modeling is as follows: let the total surface charge density on the interface $z = 0$ be $\sigma (r)$, such that ${{E}_{z}} = 2\pi \sigma (r){\text{/}}{{\varepsilon }_{\parallel }}$. Then, using the known relations of electrostatics [33], we express the in-plane electrostatic potential $\varphi (r)$ by $\sigma (r)$:

$$\varphi = \int\limits_0^\infty G {\kern 1pt} (r,r{\kern 1pt} ')\sigma (r{\kern 1pt} '){\kern 1pt} dr{\kern 1pt} '{\kern 1pt} .$$

(7)

Here, the Green function is a dimensionless quantity,

$$G = \frac{{8r{\kern 1pt} '}}{{({{\varepsilon }_{\parallel }} + 1){\kern 1pt} (r + r{\kern 1pt} ')}}K\left( {\frac{{2\sqrt {rr{\kern 1pt} '} }}{{r + r{\kern 1pt} '}}} \right),$$

(8)

that is invariant to rescaling of r and $r{\kern 1pt} '$, and $K(k) = \int_0^{\pi /2} d\alpha {\text{/}}\sqrt {1 - {{k}^{2}}\mathop {\sin }\nolimits^2 \alpha }$ with $0 \leqslant k \leqslant 1$ is the complete elliptic integral of the first kind. The function $K(k)$ has a logarithmic (integrable) singularity at $k = 1$, i.e., at $r = r{\kern 1pt} '$. The radial component of the electric field, which is continuous when crossing the interface, is ${{E}_{r}} = - \partial \varphi {\text{/}}\partial r$. It forces free charges to drift along the interface.

An important application of Eq. (7) is the case of biased metal electrode of radius $a$, such that $\varphi (r) = U$ for $r \leqslant a$ and $\sigma (r) = 0$ for $r > a$. Here we have $\sigma = [({{\varepsilon }_{\parallel }} + 1)U{\text{/}}2{{\pi }^{2}}a]{\kern 1pt} {{(1 - {{r}^{2}}{\text{/}}{{a}^{2}})}^{{ - 1/2}}}$ in accordance with [32, 33]. The density $\sigma (r)$ and the field ${{E}_{z}}(r)$ have thus an integrable square-root singularity for $r \to a$. This generic feature of perfect-metal 2D configurations makes us to be cautious with modeling of the charge screening. In particular, implementation of an extended-electrode model of radius ${{r}_{0}}$ leads to an infinite field ${{E}_{z}}({{r}_{0}})$. On the other hand, the growth of ${{E}_{z}}(r)$ for r approaching ${{r}_{0}}$ from below can be regarded as a positive feature for the domain reversal.

We consider here an obvious possibility for regularization of the singularity which is relevant to diffusion of free charge carriers: as the singularity of $\sigma (r)$ causes infinite gradients of the free-carrier concentration, the diffusion current must be taken into account. This argument is not specific for the domain reversal, it is applicable to the general 2D metal disc case. Remarkably, account for the diffusion leads (for a 2D disk of radius ${{r}_{0}}$) not only to a finite value of $\sigma ({{r}_{0}})$, but (simultaneously) to the dependence $\sigma ({{r}_{0}}) \propto 1{\text{/}}{{r}_{0}}$. To explain it, we indicate first that account for the diffusion means establishment of the Boltzmann distribution in steady state: $\sigma (r) \propto \exp ( - e\varphi {\text{/}}{{k}_{{\text{B}}}}T)$, where e is the elementary charge, T the temperature, and ${{k}_{{\text{B}}}}$ the Boltzmann constant. Accordingly, we represent the surface charge density as $\sigma = (cU{\text{/}}{{r}_{0}})\exp [e(U - \varphi ){\text{/}}{{k}_{{\text{B}}}}T]$, where c is a dimensionless constant to be found. The exponent $\exp (eU{\text{/}}{{k}_{{\text{B}}}}T)$ is introduced for convenience (we expect that $\varphi (r) \simeq U$ for $r \leqslant {{r}_{0}}$ if $eU \gg {{k}_{{\text{B}}}}T$), while separation of the ratio $U{\text{/}}{{r}_{0}}$ is dictated by dimension considerations. Thus, we arrive at the normalized integral equation for the determination of $\hat {\varphi } = \varphi {\text{/}}U$ as a function of $\hat {r} = r{\text{/}}{{r}_{0}}$:

$$\hat {\varphi } = c\int\limits_0^1 G {\kern 1pt} (\hat {r},\hat {r}'){\kern 1pt} \exp [\zeta (1 - \hat {\varphi })]{\kern 1pt} d\hat {r}',$$

(9)

where $\zeta = eU{\text{/}}{{k}_{{\text{B}}}}T \gg 1$. As the potential is not exactly constant now along the electrode, Eq. (9) has to be supplemented by the normalization condition $\hat {\varphi }(0) = 1$. After that solution for $\hat {\varphi }(\hat {r};\zeta )$ becomes unambiguous; it is controlled by the only dimensionless parameter ζ. With $\hat {\varphi }(\hat {r})$ and $c = c(\zeta )$ found, we can calculate $\sigma (r)$. Importantly, it automatically satisfies the condition $\sigma ({{r}_{0}}) \propto 1{\text{/}}{{r}_{0}}$, i.e., leads to ${{E}_{z}}({{r}_{0}}) \propto 1{\text{/}}{{r}_{0}}$.

As the integral in Eq. (9) is singular and strongly nonlinear for $\zeta \gg 1$, its solution is resistant to analytical and numerical tools. However, a simple fit procedure enables us to get a precise approximation. Let us search for the normalized potential in the form:

$$\hat {\varphi } = 1 + \frac{1}{\zeta }{\kern 1pt} \ln \left[ {{{\pi }^{2}}c{\kern 1pt} {{{(1 + \nu - {{{(\hat {r} - \mu )}}^{2}})}}^{{1/2}}}} \right]{\kern 1pt} ,$$

(10)

where μ and ν are small fit parameters ($\mu ,\nu \ll 1$). Using the condition $\hat {\varphi }(0) = 1$, we get c = ${{\pi }^{{ - 2}}}{{(1 + \nu - {{\mu }^{2}})}^{{ - 1/2}}}$. In the limit $\mu ,\nu \to 0$ we return to the known perfect metal solution for φ and σ. Calculating numerically the integral in Eq. (9) and plotting its left- and right-hand sides for different of $\mu$ and $\nu$, one can come to an almost perfect coincidence. This is illustrated by Fig. 4a relevant to $\zeta = 200$ ($U = 5$ V), $\mu = 3.5 \times {{10}^{{ - 4}}}$, and $\nu = 1.7 \times {{10}^{{ - 5}}}$. One sees that the deviations of $\varphi (r)$ from $U$ become significant only near the electrode edge, and $U - \varphi ({{r}_{0}}) \simeq 0.023U \ll U$. Figure 4b illustrates the disappearance of the singularity of ${{E}_{z}}(r)$ at $r = {{r}_{0}}$ for $\zeta = 200$; the deviation between our solution and the primary singular solution becomes notable only for $r{\text{/}}{{r}_{0}} \geqslant 0.99$. In particular, we have $\sigma ({{r}_{0}}) \simeq 3.78{\kern 1pt} U{\text{/}}{{r}_{0}}$ and ${{E}_{z}}({{r}_{0}}) \simeq 23.7U{\text{/}}{{\varepsilon }_{\parallel }}{{r}_{0}}$. Thus, our extended-electrode 2D model provides the necessary ingredients for the description of the lateral domain expansion.

Fig. 4.

(Color online) (a) Dependences $\hat {\varphi }(\hat {r})$ for $\zeta = eU{\text{/}}{{k}_{{\text{B}}}}T$ = 200 ($U \simeq 5$ V) obtained by the fit procedure. The open circles correspond to numerical calculation of the right-hand side of Eq. (9), while the solid red line refers to Eq. (10). (b) Ratio $\pi {{\varepsilon }_{\parallel }}{{E}_{z}}{\text{/}}2U$ versus $\hat {r}$ for $\hat {r} \geqslant 0.98$. The circles and solid red line correspond to Eq. (10) and [32], respectively. For $\hat {r} \to 0$, the ratio is $1$. The fit parameters in (a) and (b) are $\mu = 3.5 \times {{10}^{{ - 4}}}$ and $\nu = 1.7 \times {{10}^{{ - 5}}}$.

5 DISCUSSION

The above considerations do not pretend to be exhaustive. On contrary, they intend to provide some basis and indicate prospects for further studies of the ferroelectric domain reversal incorporating the idea of charge compensation and DW conduction. This idea has firm experimental grounds. Most probably, it is relevant to the most common ferroelectrics, such as LiNbO₃, LiTaO₃, BaTiO₃, and SNB, possessing large values of ${{P}_{s}}$. In our opinion, several AFM related issues are worthy of further explorations:

$\bullet$ An efficient domain nucleation near the tip has to be followed by a fast forward domain growth. It is expected to be controlled by different opposing factors: as the longitudinal domain size ${{l}_{z}}$ grows, see Fig. 1b, the field E acting on the domain apex tends to increase because of proximity of the bottom electrode. This accelerates the grows. On the other hand, increasing ${{l}_{z}}{\text{/}}a$ tends to make the wall almost neutral; this suppresses the DW conduction and can slow down the growth. Thus, depending on U, a, and h, one can expect different scenarios for the forward domain growth.

$\bullet$ The above 2D model of extended electrode has to be regarded as the simplest one for the lateral domain expansion. It is likely that it is insufficient to explain additional experimental data on the anomalous domain reversal, i.e., formation of ring type domains on the upper interface [12, 15]. Further efforts to account not only for 2D, but also for 3D injection from the tip [34] are probably needed.

$\bullet$ It was implied throughout the paper that the DW conduction provides an instantaneous charge compensation during the domain reversal. This assumption can be insufficient in the cases when DWs possess areas with ${\text{|}}\mathbf{n} \cdot \hat {\mathbf{z}}{\text{|}} \ll 1$ (i.e., small conductivity) that serve as bottlenecks for the charge compensation. More complicated dynamic models are anticipated here. Also, we cannot exclude additional charge co-mpensation mechanisms relevant, e.g., to corona discharge, air humidity and electrochemical reactions [8].

$\bullet$ Feedback with experiment would be indispensable. This concerns the data on inverted domains not only for the upper but also for the bottom interface, the data on electric currents through the electrodes, and the data elucidating the notion of the AFM contact radius.

6 CONCLUSIONS

Physical models providing a strong charge compensation during the ferroelectric domain reversal, especially for AFM configurations, are crucial for development of the domain engineering. Domain wall conduction can be regarded as a general ingredient for such a compensation and for the explanation of numerous accumulated experimental data. Particular models are presented to demonstrate the positive impact of this conduction on nucleation and growth of ferroelectric counter-domains. The prospects for further development of DW conduction related models of the domain reversal are outlined.