Nanoscale Investigation of Polycrystalline ...

Chapter 9

Nanoscale Investigation of Polycrystalline Ferroelectric Materials via Piezoresponse Force Microscopy V. V. Shvartsman1, A. L. Kholkin2

9.1

Introduction

Ferroelectrics possess a wide spectrum of functional properties including switchable polarization, piezoelectricity, pyroelectricity, dielectric nonlinearity, and high non-linear optical activity, which make these materials promising for a large number of applications [1]. These include nonvolatile random access memories (FERAM) [2], micro-electromechanical systems (MEMS) [3], infrared detectors, optical modulators and waveguides, and many others [4, 5]. The general trends of miniaturization in modern electronics demand a decrease in the size of the active ferroelectric elements to a submicron scale. This in turn necessitates the development of microscopic techniques allowing for the evaluation of ferroelectric and piezoelectric properties with nanoscale resolution. Several fundamental issues have to be addressed such as the effect of the films thickness and lateral size of the capacitor, or of the single grain on ferroelectric and piezoelectric properties, the relationship between grain/capacitor size and peculiarities of the polarization switching, and mechanisms of degradation effects, such as retention, imprint, and polarization fatigue [2]. To answer these questions both ferroelectric domain structures and their evolution during polarization switching have to be studied at micro- and nanoscales. This can be done using scanning probe microscopy (SPM) techniques, which provide an opportunity for non-destructive visualization of domains in ferroelectric thin films, single crystals and ceramics. SPM has made possible the mapping of the surface potential and charge distribution, evaluation of local 1

Angewandte Physik, University of Duisburg-Essen, Duisburg, Germany [email protected] 2 Dept. of Ceramic and Glass Engineering, CICECO, University of Aveiro, Aveiro, Portugal [email protected]

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V. V. Shvartsman and A. L. Kholkin

electromechanical properties, and measurements of the non-linear dielectric constants. Since the first SPM imaging of the 180° domain walls in Gd2(MoO4)3 [6], a growing number of research papers on nanoscale properties of ferroelectrics studied by SPM have been published (see recent review in [7]). Several novel SPM techniques based on different approaches were adopted or specially developed for these studies [8]. Depending on the type of interaction between the probing tip and the sample – attractive or repulsive – the SPM can operate in non-contact and contact regimes, respectively. In the non-contact regime, the tip is scanned over the surface at a distance of 10-100 nm. The cantilever is mechanically driven to oscillate near its resonance and the feedback loop adjusts the tip-to-sample distance to maintain, for example, the constant amplitude of the oscillation. The tip-sample interaction is dominated by the Van-der-Waals forces and, in the case of polar or charged materials, the electrostatic forces may contribute. In particular, when a small acvoltage is applied to the tip, the electrostatic interaction between the tip and surface charges results in an oscillation of the cantilever. From the amplitude and the phase of this oscillation, the charge density and polarity of the charges may be estimated [9, 10]. This mode of SPM, called electrostatic force microscopy (EFM), may be used for ferroelectric domain imaging by detecting the sign of the surface polarization charges [6, 11, 12, 13, 14, 15, 16]. In another approach, a small dc-bias is applied to the tip mechanically driven at the resonance frequency. The electrostatic force between the tip and the surface results in a change of the cantilever resonant frequency, which is proportional to the force gradient. The frequency shift is collected as the EFM image [17, 18]. In the Kelvin probe force Microscopy (KPFM), a dc-bias and an ac-voltage are applied simultaneously to the tip Vtip=Vdc+Vaccosωt. The capacitive (Maxwell) force acting between the tip and the surface with a potential Vs is

Fcap ( z ) =

1 ∂C (Vtip − Vs ) 2 ∂z 2

(1)

where z is the distance between the tip and the surface and C(z) is the tip-surface capacitance. The first harmonics of this force is 1ω ( z ) = (Vdc − Vs )Vac Fcap

∂C ∂z

(2)

The feedback loop is used to nullify this term by adjusting Vdc=-Vs. Thus mapping of the nullifying potential, Vdc, yields a distribution of the surface potential [19, 20]. This provides important information on the surface electronic properties of ferroelectrics, such as distribution of polarization and screening charges and their evolution during phase transitions [21, 22, 23, 24]. To minimize a possible crosstalk between topography and electrostatic signals, the EFM and KPFM measurements are often done in so-called two-pass technique (Lift Mode) [25]. Each line is scanned twice in this mode. In the first scan the topography of

Nanoscale Investigation of Polycrystalline Ferroelectric Materials

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the surface is determined, and during the second scan the tip is lifted to a certain height above the sample surface. This allows for the reconstruction of the distribution of charge or potential on the surface without topographical contribution. General drawbacks of non-contact methods include low-resolution due to large tip-surface separation, sensitivity to sample surface conditions, and susceptibility to screening effects. Contact modes operate in the repulsive force regime: the tip is in permanent contact with the surface. The feedback loop is adjusted to maintain a constant bending of the cantilever. Ferroelectric domain imaging methods in the contact mode may be divided into static and dynamic ones. Among the static methods are lateral (friction) force microscopy (LFM) and conventional contact atomic force microscopy (AFM). LFM is based on the detection of the torsion deformation of the cantilever due to frictional forces between the tip and the surface. The structural differences between surfaces of oppositely polarized domains modify the surface potential resulting in two different friction coefficients experienced by the tip [26, 27]. Twinning between domains with in-plane and out-of-plane polarization (a- and c-domains, respectively) results in surface corrugations at the 90º domain walls. This allows studying ferroelastic domain patterns in single crystals and epitaxial films by topographic imaging of their surfaces [28, 29, 30]. Contact AFM was also used for the visualization of 180º domains in some single crystals via the detection of static thickness change (shrinkage or expansion), piezoelectrically induced by a dcvoltage applied to the tip during scanning [31, 32]. The dynamic methods include scanning non-linear dielectric microscopy (SNDM), atomic force acoustic microscopy (AFAM), and piezoresponse force microscopy (PFM). In SNDM, the sample is a part of a capacitor in a LC resonator circuit. The voltage applied to the tip is modulated in the microwave frequency range. By detecting the voltage-induced changes in the local capacitance SNDM is able to measure point-to-point variations of the non-linear dielectric response of the sample, which translates the distribution of local ferroelectric polarization [33]. This technique may achieve a sub-nanometre lateral resolution [34]. However, the measured non-linear dielectric response is related to a thin surface layer (>1) is used and high contact forces (10-1000 nN) are applied [57].

9.2.4

Resolution in PFM Experiments

In a typical PFM experiment, the sharp tip plays the role of a movable top electrode. Since usually the thickness of the studied sample is much larger than the tip-sample contact area (5-20 nm), the probing electric field is strongly inhomogeneous and measured PFM response comes from a small volume around the contact point. This provides a high spatial resolution of the PFM method. The natural way to estimate the lateral resolution in the PFM experiment is to measure the width of a domain wall between two antiparallel domains. While the intrinsic width of 180° domain walls in ferroelectrics is expected to be a few unit cells [58, 59, 60], the domain walls measured in a PFM experiment are typically thicker (tens of nm) and, therefore, reflect primarily the spatial resolution of the PFM. Experimentally, the width of the domain wall image, w, is estimated from the profile of the piezoresponse signals across the wall, which is fitted by a suitable function, e.g., by the one used to describe the polarization profile in the mean field theory of ferroelectrics [61].

 ( x − x0 )  PR( x) = PR− tanh    w 

(10)

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Fig. 9.3b illustrates the profile of the PFM signal across the 180° domain wall measured on a [001]-oriented PbTiO3 single crystal. The apparent width of the domain wall obtained from the best fit to Equation 10 is about 60 nm.

Fig. 9.3 a The LPFM image of [001]-cut of a PbTiO3 single crystal. Bright and dark contrast corresponds to domains with the spontaneous polarization oriented left and right in the figure plane, respectively. b The cross-section of the piezoresponse image across the 180° domain wall. The broken line is the best fit to Eq. (10).

The theory of the resolution in PFM was recently considered by Kalinin et al [62]. They have shown that for the system with 180° domain walls, the piezoresponse may be presented as a convolution of a function describing the spatial distribution of material properties and a function related to the probe parameters (it is assumed that piezoelectric and dielectric properties are uniform across the sample thickness). In this case, the contrast formation mechanism may be analyzed using the transfer function theory that allows defining both the resolution and information limit. In the linear transfer function theory, the measured image I(x) (where x is a set of spatial coordinates) is given by the convolution of an ideal image, I0(x-y), with the resolution function, F(y) [62]


∫

I (x) = I 0 (x − y )F (y )dy + N (x)

419

(11)

where N(x) is the noise function. In the PFM experiment, the ideal image is the distribution of piezoelectric and stiffness constants that correlate with the domain structure. The resolution function depends on the tip geometry, lock-in amplifier parameters, and scanning conditions. It may be estimated by analyzing an artificial periodical domain pattern created using a template. The Fourier transform of eq. 11 is

I (q) = I 0 (q) F (q) + N (q )

(12)

where F(q) is called the object transform function. It may be defined from ratio of the intensities of fast Fourier transformation of the experimental images to the ideal images. One of the traditional resolution criteria used in optics is the Rayleigh two-point resolution limit – two Gaussian shaped image features of similar intensity can be resolved, if the intensity at the midpoint between them is less than 81% of the maximum [63]. If the object transfer function has a Gaussian shape, the Rayleigh two-point resolution criterion may be defined as wr=1/qr for which F(qr)=0.58F(0). Kalinin et al. [62] showed that in the PFM experiment, the Rayleigh resolution correlates with the measured width of the domain wall. Moreover, the quantitative determination of material properties from the PFM experiment requires that typical domain size exceed wr. Nevertheless, the features with smaller size may still be resolved by PFM. The minimal feature size detectable against the noise corresponds to the information limit defined from the condition N(q)=F(q). However, the intensity of the PFM signal in that case starts to scale with the feature size and no reliable information about material properties can be obtained. For PFM, the information limit may be considerably smaller than the Rayleigh resolution. The dependencies of the resolution on the tip size, as well as on the sample parameters (thickness, material), were studied experimentally by Jungk et al. [64]. They found that for the metal coated tips, the width of the measured walls scales linearly with the tip radius. For the uncoated Si tip, the broader domain walls were measured. It was explained as an effect of the dielectric SiO2 layer formed on the tip surface. As a result, the probe is electrically separated from the sample surface and the electric field is less localized leading to reduced spatial resolution. No effect of the material parameters (dielectric permittivity, elastic and piezoelectric constant) on the resolution was found. The thinnest domain wall width (measured for the tip with the radius 15 nm) was only 17 nm. Recently, Rodriguez et al. [65] reported that the measured width of the domain wall can be as small as 3 nm when the measurements are done not in the ambient condition (air) but in a liquid environment. They suggested that the mobile ions present in the solution screen the long range electrostatic interactions from the conical parts of a tip, at distances greater than the Debye length enhancing localization of the probing field in the tip-surface junction.

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Due to the strong inhomogeneity of the probing electric field, the signal in the typical PFM experiment is collected mainly from a surface layer, whose thickness is a function of the dielectric permittivity and contact conditions, and is typically unknown. To overcome this limitation, the domain structure may be visualized through the top electrode of a ferroelectric capacitor [66]. In this case, the electric voltage may be applied either via the tip or using an external wire attached to the top electrode. In the latter case, the tip is used only for the detection of the piezoelectric displacement. In this configuration, the probing electric field is uniform and a measured response is generated by the entire sample thickness. This method allows the quantitative study of the dynamics of domain walls and polarization reversal mechanisms in ferroelectric capacitors. The drawbacks of this approach is a substantially smaller lateral resolution and inability to measure lateral (LPFM) signal.

9.3

PFM in Polycrystalline Materials. Effect of Microstructure, Texture, Composition

One of the advantages of PFM is the opportunity to correlate peculiarities of observed domain patterns directly with the microstructure of samples (polycrystalline thin films, ceramics). Large grains (> 1 µm) in conventional bulk ceramics are usually polydomain. The shape of the observed domain pattern depends on the symmetry of the crystalline structure and on the crystallographic orientation of the individual grains. Fig 9.4 shows the PFM images taken on BiFeO3 ceramics [67]. Colours ranging between black and white indicate different directions and orientations of Ps with respect to the normal to the sample. Details of the domain structure are analyzed on two large grains selected in Fig. 9.4 a and b. The patterns differ in their overall contrast and depend strongly on which component (VPFM or LPFM) is measured. While the black/white VPFM contrast at the upper left corner transforms into a nearly unstructured brownish colour in LPFM image, the brown and yellow vertical contrast at the lower right corner is essentially unchanged in its LPFM image. Obviously, two different habit planes are encountered. In the grain at the lower right corner, 6 - 15 µm wide domains have straight boundaries. These are parallel and diagonal (i e. intersecting at angles of 45°) to each other, respectively. The observation of nearly identical vertical and lateral patterns complies with ferroelastic domains (twins), in which both Pz and Px (observed by VPFM and LPFM, respectively) change sign simultaneously from one domain to the other. BiFeO3 has rhombohedral symmetry R3c [68]. In this case, ferroelastic domains are separated either by 109° or 71° domain walls, corresponding to {110}p and {100}p planes (pseudo-cubic unit cell indices). The variation of the VPFM and LPFM contrast indicates that the crystallographic orientation (aab) of this grain (b >> a) is tilted with respect to (001)-plane around the [110] direction (Fig. 9.4c). Peculiarly, however, more irregular stripe patterns on a sub-µm scale are observed within the elastic twin domains. They reflect mere FE twinning, either by 180° or ±(Pz + Px)


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while conserving Py. Being unrestricted by strain compatibility rules, they nevertheless form stripe-like patterns in order to minimize electric stray field energy. The extreme VPFM contrast observed in the upper left corner of panel (a) hints at domains, which are viewed in the (001) habit plane. Indeed, in the rhombohedral phase of perovskite materials, the longitudinal piezoresponse is known to attain extreme values not in the polar, but approximately along the [001]p direction [69]. The virtually vanishing lateral contrast (Fig. 9.4b) indicates that the x-component of the polarization is small, i.e. the polarization vector is parallel to domain walls. The observed domain walls may be either 180° ferroelectric domain walls or 109° ferroelastic ones (Fig. 9.4d). The electrostatically stabilized parallel walls between ±Ps domains are slightly irregular, as expected, in the absence of strict strain compatibility rules. The diagonal wall observed in the left part of the grain corresponds most probably to 71° twins (see Fig. 9.4d) whose in-plane polarization component gives rise to a sizable contrast in the lateral PFM image (Fig. 9.4b).

Fig. 9.4 VPFM a and LPFM b images of BiFeO3 ceramics. Schematic domain configurations in c and d refer to the grains in bottom right and upper left corners, respectively (indicated by rectangles in a and b).

The mechanical boundary condition in polycrystalline samples may result in the existence of domain walls, which are forbidden for free-standing single crystals. For instance, Muñoz-Saldaña et al. [70]. found that in Pb(Zr1-xTix)O3 (PZT) ceramics, another array of domain walls parallel to {210} planes exists

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besides the conventional {110}-oriented 90º and 180º domain walls. The reason for this unconventional domain configuration was explained by the clamping of the crystallites in the polycrystalline material.

Fig. 9.5 The topography a,c and VPFM b,d images taken on PZT 70/30 a,b and 80/20 c,d polycrystalline thin films

Fig. 9.5 shows the PFM images taken on polycrystalline PZT thin films with different Ti content [46]. The domains in PZT 70/30 films are of irregular shape with random orientation of the polarization within the grains, which are either single-domain or are split in two domains. On the contrary, in PZT 20/80, a regular a-c domain structure formed by the 90º domains is observed. A similar regular domain structure was also observed in polycrystalline PbTiO3 thin films [71]. This difference in domain patterns is explained by different unit cell distortions (c/a lattice parameter ratio) of the films. In films with large amount of Ti, the distortion is higher and the mechanical stress that appears upon cooling through the phase transition temperature has to be relieved by the formation of the ferroelastic (90-degree in this case) domains. These domains form regular patterns because of the minimization of elastic energy [72]. On the other hand, in PZT70/30 films having smaller distortion of the unit cell, the domain walls are likely to separate 180° domains. The orientation of these walls is not restricted by strain compatibility rules. Their “random” structure reflects different local electrical conditions and inhomogeneity of defects, rather than stress relief. It is known that with the decrease of the grain size in ceramics, the periodicity of 90º domains changes and finally the grain becomes single domain [73]. In the


423

PZT20/80 thin films, it was found that the relative area occupied by domains with in-plane and out-of-plane polarization depends on grain size. Namely a- to cdomain surface ratio increases and then drops down with grain size [46]. Such a complex behaviour was attributed to the stress relief originating not only from the substrate but also from neighbouring grains. Generally, for non-textured polycrystalline materials, both the piezoresponse contrast and the domain patterns vary among individual grains. Nevertheless, important information on distribution of the local polarization may be obtained from the analysis of piezoresponse histograms, i.e., the number of the pixels on the PFM image corresponding to a given piezoresponse signal [74, 75, 76]. The deconvolution of the piezohistograms in several peaks may provide valuable information on relative population of different domain states. From the peak position, an effective piezoresponse value can be estimated. An important parameter is the half-width of the peak; its broadening may indicate coexistence of various polarization directions, which can be the case for polycrystalline films without texture, or even due to existence of oblique domain walls, which will result in a diffuse piezoresponse contrast.

Fig. 9.6 The VPFM images taken on PZT 54/46 thin films deposited by sputtering stoichiometric a and lead-enriched b targets. c The piezoresponse histograms of stoichiometric (1) and nonstoichiometric (2) PZT 54/64 thin films.

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Fig. 9.13 shows the histograms of PFM images taken on two PZT 54/46 thin films prepared by RF magnetron sputtering from a stoichiometric target and a target containing an excess of lead oxide [77]. For stoichiometric films, the piezoresponse distributions are approximately symmetric relative to domains of both up and down polarity. On the contrary, in non-stoichiometric films a “negative” shoulder exists on the piezoresponse histograms indicating that the regions exhibiting the negative piezosignal (in this case, domains with the polarization oriented towards the bottom electrode) occupy essentially larger area than those of the positive piezoresponse (domains with the polarization oriented towards the free surface). Thus, from analysis of the piezohistograms, it may be concluded that the PZT films obtained from lead-enriched targets have excess of negative polarization, i.e., are selfpolarized. This conclusion agrees well with the macroscopic properties of these films [78]. Self-polarization is often observed in films and is characterized, for instance, by a shift of the polarization hysteresis loops or strong polarization imprint. This phenomenon occurs due to the presence of an internal electric field, which is at least as large as the coercive field at the Curie temperature. This field may have different origins [78, 79, 80, 81]. It was suggested that in the PZT films prepared with excess of lead, the built-in electric field arises due to the negative charges captured by deep traps near the ferroelectric–electrode interfaces [82]. These films have many oxygen vacancies in the perovskite structure, which leads to the n-type conductivity. When the film is cooled after crystallization, the electrons occupy the localized states near the film-electrode interface. The disappearance of selfpolarization after high-temperature treatment (above Tc) [82] and UV-illumination [77] confirms the dominance of such “electrical” mechanism.

9.4

Local Polarization Switching by PFM

One of the major advantages of the PFM method is the opportunity to investigate directly the evolution of domain structures under an external electric or mechanical field. A conductive PFM tip may be used not only for domain visualization but also for a local manipulation with the initial domain structure. In particular, due to a very small tip apex radius, even a moderate dc-voltage applied between the tip and the bottom electrode generates an electric field of several hundred kilovolts per centimetre. Such field is higher that the coercive field of most ferroelectrics and induces local polarization reversal. By applying the positive or negative bias, one can create domains of opposite polarity, which can be hereafter imaged by PFM. Thus, PFM provides both “storage” and “read-out” capabilities. Domain patterns written by PFM may be used for non-volatile ferroelectric random access memory (FERAM) applications [2]. Since the width of 180° domain walls in ferroelectrics is typically very small, the domain recording by PFM potentially allows an extremely high data storage density. Tybell et al. [83] reported 40 nm bit size in epitaxial [001] oriented PbZr0.2Ti0.8O3 thin films. Later, storage density 10 Tbit/inch2 (bit size ~ 8 nm) has been achieved by Cho et al. in LiTaO3 thin films using the scanning


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nonlinear dielectric microscopy [84]. Another interesting application is the fabrication of domain gratings with submicron period by PFM-nanodomain engineering. It may be used in optical nonlinear frequency conversion devices, as an example for backward-propagating quasi-phase matched conversion [85, 86]. The domain patterning by PFM may be applied in ferroelectric lithography, a method that explores the relation between surface chemical reactivity of ferroelectric materials and local polarization direction. It allows the allocation of multiple nanostructures of different materials in pre-defined positions [87, 88]. Applications of ferroelectric domain patterning for data storage, electro-optic devices, and ferroelectric lithography necessitate fundamental studies of the domain switching process, including thermodynamics and kinetics of domain nucleation, growth, and relaxation.

9.4.1

Thermodynamics of PFM Tip-Induced Polarization Reversal

Several approaches have been developed to describe the thermodynamics and kinetics of domain switching in PFM. The switching in the PFM experiment starts from the nucleation of a new domain underneath the tip. The direction of the polarization in this domain coincides with that of the normal component of the applied electric field. The newly-formed domain expands by motion of the domain walls. So far, the electric field is larger than the coercive one, the process of growth of the domain is non-activated, and the size of the domain increases rapidly. At larger distances from the contact point, where the electric field decreases below the coercive one, the movement of domain walls becomes thermally activated and is slowed down. The domain walls continue to move until the inverted domain reaches an equilibrium state. In the first approximation, the electric field of the tip may be considered as the field of a metallic sphere, the radius of which is equal to the tip apex radius R [89, 90, 91]. In the frame of this model, Molotskii obtained the closed form solution for the equilibrium domain shape [91]. The change of the free energy related to the nucleating domain is

∆W = Wd + Ws + Wt

(13)

where Wd is the depolarization energy contribution, Ws is the domain wall surface energy, and Wt describes the electrostatic interaction between the domain and electric field of the tip [91]. Wt term favours the enlargement of the domain, while Wd and Ws contributions hinder domain growth. The shape of the created domain is assumed within the Landauer model [92]: to be a half ellipsoid with the small and large axis rd and ld, respectively (Fig. 9.7). In this geometry

Ws = brd ld

(14)

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Wd =

crd 4 l

(15)

where

b = σ wall π 2 / 2 c=

16π 2 Ps2   2ld ln 3ε a   rd 

εa εc

(16)

   − 1    

(17)

Fig. 9.7 Domain geometry by PFM tip-induced polarization reversal.

Here Ps is the spontaneous polarization, σwall is the domain wall surface energy density, and εc and εa are the values of dielectric permittivity in the directions parallel and perpendicular to the polar axis, respectively. When the domain is formed, the polarization value is changed by 2Ps. Therefore, the energy of the interaction between the domain and the electric field may be presented as ld

r( z)

∫ ∫ P E (r , z)rdr

Wt = −4π dz 0

s

n

(18)

0

where En(r,z) is the normal component of the electric field (parallel to the c- axis), r(z)= rd 1 − z 2 / l 2 By minimization of the free energy, Molotskii found parameters of the equilibrium domain shape as functions of the applied voltage [91]:

for s rd, where s is the distance between the centre of the curvature of the tip and the sample surface. In particular, it was found that the parameter

req3 / 2 leq is an invariant of the equilibrium domain shape. It practically does not depend on PFM experimental parameters and is defined only by the properties of ferroelectrics themselves. This model describes well the experimental results obtained in C(NH2)3Al(SO4)2·6H2O (GASH) [93], triglycine sulfate (NH2CH2OOH)3H2SO4 (TGS) [94] and BaTiO3 [94] single crystals. Interesting results were obtained in LiNbO3 where very long (leq > 200 µm) and relatively thin (req ~ 05-0.8 µm) domains are formed at large applied voltage (Vdc > 3.5 kV) in a good agreement with the aforementioned model [90, 95]. These results are surprising at first sight, since the electric field of the PFM tip rapidly decays from the contact point into the sample and cannot influence directly the elongation of the domain far from the surface. The propagation of the domains in this case is due to decreasing of the depolarization field energy. This process continues until the forces associated with the increase of the domain surface area compensate the driving forces caused by the depolarizing field [95]. The effect of the field created by the tip is indirect. It reveals itself through an increase of the domain radius due to the AFM tip field and a corresponding change of the domain length to satisfy the minimum free energy conditions. Such an effect was called “domain breakdown” since the created domains are similar to the electric breakdown channels [95]. While the model proposed by Molotskii describes well the domains with large size, it is not applicable for description of the polarization switching on length scale comparable to the tip apex radius [96]. In particular, according to the Molotskii model, the field in the vicinity of the tip is infinite and the domain nucleation is induced at arbitrary small bias voltages, which is in contradiction to many experimental observations. At small length scales, the thermodynamics of switching process requires exact electroelastic field structure to be taken into account. The model that uses the rigorously derived electroelastic field was proposed by Kalinin et al. [96]. In particular, it was found that the domain nucleation requires a certain threshold bias 0.1-1 V corresponding to non-zero activation energy for nucleation. Further, this problem was elaborated by Morozovska et al. [97, 98] in the case of semi-infinite materials by taking into account the realistic tip geometry, the effects of surface depolarization energy, the surface screening charges, and the finite Debye screening length of domain nucleation.

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The process of the polarization reversal in thin films was considered by Emelyanov [99] by taking into account tip geometry and interaction of the nucleated domain with the bottom electrode. He defined four stages of the PFMinduced switching. (a) Nucleation: at threshold voltage, Vth, the stable nucleus of hemisphere shape is formed by the polarization reversal in finite volume. (b) Bulk growth: the forward domain growth with a minor lateral expansion l/r >> 1 at Vth