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Thermodynamical approach to nanodomain tailoring in thin ferroelectric-semiconductor films

We propose the thermodynamical theory of nanodomain tailoring with the help of atomic force microscope electric field in thin ferroelectric-semiconductor films. We modified the existing thermodynamical models of domain formation allowing for the Debye screening, recharging of sluggish surface charge layers caused by emission current between the tip apex and sample surface.For the first time we calculated the realistic sizes of nanodomains recorded in BaTiO3, PbZrxTi1-xO3 and LiTaO3 ferroelectric-semiconductor thin films in contrast to the over-estimated ones obtained in the evolved approaches. We have shown that the depolarization field energy of the domain butt, Debye screening effects and field emission at high voltages lead to the essential decrease of the equilibrium domain sizes. For the first time we obtained, that the domain radius does not decrease continuously with applied voltage decrease: the domain appears with non-zero radius at definite critical voltage applied to the tip. This result completely agrees with experimentally observed threshold domain recording in PbZrxTi1-xO3 and LiTaO3 thin films. Such threshold domain formation is similar to the first order phase transition. We hope, that our results will help one to determine the necessary recording conditions and appropriate ferroelectric medium in order to obtain the stable domains with minimum lateral size in a wide range of applied voltages.

So the possibilities of information recording in the ferroelectric media have been open, if only the optimization problem of high-speed writing nanodomains with high density, stability and fully controllable reversibility would be solved. First of all it is necessary to record the stable domain "dots" with minimum width in the appropriate ferroelectric medium. To realize this idea, one has to determine the dependences of domain radius on voltage applied to the AFM tip either empirically or theoretically. To our mind simple modeling seems rather urgent for the correct description of the numerous experimental results, but existing models give incomplete description of the nanodomain tailoring owing to the following reasons.
• Thermodynamical description of the nucleation processes in the perfect ferroelectric-dielectric during polarization switching proposed by Landauer [10] as far back as 1957. At present it should be applied to the domain formation with great care, because in this model the depolarization field is partially screened by the free charges on the metallic electrodes. When modeling the domain formation such upper electrode will completely screen the interior of ferroelectric from the AFM tip electric field, thus no external source would induce the polarization reversal. Only homogeneous external field can be applied to such polar dielectric covered with metallic electrodes (see Fig.1a).
• Theoretical modeling of equilibrium ferroelectric domains recorded by AFM tip proposed by Molotskii [11] considers tip electric field inside the perfect dielectric-ferroelectric with free surface, i.e. without any screening layer or upper electrode, but the semi-ellipsoidal domain depolarization field was calculated in Landauer model as if the perfect external screening expected. As a result significantly over-estimated values of domain radius was obtained at high voltages [3].
• In our recent papers [12], [13] we try to overcome the aforementioned discrepancies, taking into consideration screening layers of immovable surface charges (see Figs 1b,c) and semiconductor properties revealed by the most of ferroelectrics [2], [14], [15]. When the distances R ∆ between the tip apex and sample surface is more than several nm and applied voltage is relatively low, this screening layer maintains its negative charge during the domain formation owing to the traps sluggishness. However, in the most of experiments [3]- [9] nm R 1 < ∆ and thus recharging of this layer is quite possible due to the field emission of trapped carriers caused by the positively charged AFM tip. Therefore recharging of surface traps caused by the emission current should be taken into account at least at high voltages [13]. • Nanodomain tailoring in a thin ferroelectric film with thickness no more than 100-150 nm is much more useful for applications, than domain recording in a relatively thick layer, allowing for possible miniaturization of information elements, which is extremely desirable in order to provide their figure of merits and make them preferable in comparison with existing information optical elements. Taking into account, that observed nanodomains have radiuses ~20-50 nm and cross-talk coupling between them are almost absent during their recording, the perfect possibilities of information recording by nanodomains two-dimensional arrays with extremely high density ~ (4-40)Gbits/cm 2 are opened [5]. As far as we know, theoretical description of nanodomain recording in thin ferroelectric films is in the initial stage: screening effects and depolarization field energy are usually either neglected or overestimated (see e.g. [16]).
In the present research for the first time we consider the influence of all aforementioned effects: depolarization field created by the domain butt, Debye screening effects, field emission at high voltages. We have shown that depolarization field of the domain butt, Debye screening and emission current lead to the essential decrease of the equilibrium domain sizes. As a result we obtained the realistic values of domain radius recorded in Pb(Zr,Ti)O 3 and LiTaO 3 thin films and demonstrate that our modeling could be really useful for the optimization of information recording.

THERMODYNAMICAL DESCRIPTION
Hereinafter we use the model of the rigid ferroelectrics with spontaneous polarization , dielectric permittivity ε , displacement We choose constant spontaneous polarization outside the domain (see Fig.2).
For description of the charged tip electric field we use the spherical model [18], in which the tip is represented by the charged sphere with radius R 0 located at the distance R ∆ from the sample surface. The voltage U is applied between the tip and bottom electrode. The validity of such assumptions for dielectric sample and more sophisticated models are discussed in [18]. Usually  Another important experimental fact should be taken into consideration for correct theoretical description of nanodomains tailoring using AFM tip [6], [16], [22]. Namely, the Durkan et al. [6] reported about the layer of the adsorbed water located below the tip apex. Molotskii [16] assumes that a water meniscus appears between the AFM tip apex and a sample surface due to the air humidity. For instance, scanning tunnelling microscopy measurements on Ti show that within the interval 20%-50% humidity water layer thickness increases linearly from 50 to 100 nm (see [22] and ref. therein). The authors argue that this strong water adsorption is based on an oxide layer on the Ti surface. Thus, it seems quite probable that on the BaTiO 3 , PbTiO 3 or other polar oxide surface thin water layers are quite possible. Moreover, if AFM tip is wettable, its apex with curvature 25-50 nm can be completely covered by water [23]. To our mind, all these effects can be easily explained taking into account that dipolar water molecules are pulled into inhomogeneous electric field [24] and then condensates in the area of tip apex -domain surface (see Fig. 2).
Hereinafter we regard, that the region between the tip apex and domain surface has effective dielectric permittivity e ε .
( ) and thus the screening of electric field is realized by free charges with bulk density and Debye screening radius (see Appendix A). The spatial distribution of the electrostatic potential should be determined from the Maxwell equation In order to apply all the following results to the anisotropic semiconductor one can make the substitution: Here a ε and are anisotropic dielectric permittivity values perpendicular and along the polar axis z. c ε These domains usually grow through the thin film (see e.g. [7] and Fig.2). Our preliminary calculations proved, that equilibrium domain length l nearly always is much greater than film thickness , i.e. they usually grow through the film and rich the bottom electrode. For example domain length l for a thick LiNbO So, nanodomains recorded in thin films have rather cylindrical shape than the semi-ellipsoidal one.
Thus hereinafter we put l .
The electrostatic potential ( ) r ϕ of cylindrical domain grown through the thin film and its thermodynamic potential Φ are derived in Appendix A under the typical recording condition . Our calculations have shown, that interaction energy between the charged tip apex and the domain decreases with film thickness decrease: caused by polarization reversal inside the cylindrical domain grown through the film also decreases with film thickness decreases with film thickness decrease: It should be noted that the similar problem (cylindrical microdomain growing through the thin film) was considered by Molotskii in [16]. However the depolarization field energy dependence on the film thickness and domain radius was chosen as in the case . As it follows from our energy (4) the dependence can be expected in the opposite case (see (A.12)). In the case our calculations give Φ . Since the coefficients of proportionality in (4) and in the expressions (10), (13) used in [16] are same order of magnitude, the depolarization field energy [16] for the films with thickness is strongly overestimated. Moreover for the most of the experiments and approximation like (4) is more useful than limiting expressions The surface energy of domain wall has the form: We regard domain walls as infinitely thin, with homogeneous surface energy density S ψ . This assumption used by many authors [10], [11], [16], [25] was confirmed both by the modern first principle calculations (e.g. [26], [27], [28]) and recent experimental data (e.g. [29], [30]), which revealed that the thickness of the domain walls in perovskite ferroelectrics is about several lattice constants. The value S ψ can be calculated from the first principles [26], [27], [28] or measured experimentally from the rate of domain wall motion, their thickness or curvature (e.g. [31], [32]).
Depending on the sample preparation and experiment conditions the measured energy S ψ differs from its calculated equilibrium value in several to several hundreds of times (e.g. compare extracted from domain walls motion values 9-35 mJ/m 2 for LiTaO 3 [31] with the reconstructed ones from the domain wall curvature 200-400 mJ/m 2 [32]). High applied voltages as well as nonequilibrium carriers (emitted, photo-ionized, UV-or thermo-activated ones) could significantly increase the domain wall velocity [2], whereas the equilibrium conductivity, that determines Debye screening radius , virtually does not effect on the domain walls structure [2]. Domain wall kinetic energy is d R consumed to overcome the wall pinning under the domain growth, thus unavoidable growth defects pin the domain wall and significantly increase domain wall relaxation time. Therefore S ψ values for perfect samples are essentially smaller than for the imperfect ones [33]. Note, that the authors of [28] showed, that the free energy ψ is usually smaller than the domain wall energy, because it includes entropic contribution. Taking into account aforementioned facts, we consider effective surface energy ψ as a fitting parameter varied in the range 5-500 mJ/m Taking into account (1)-(5), free energy of the nanodomain recorded in thin film acquires the form: Similarly to the case of semi-ellipsoidal domains recorded in the thick films [13], it is appeared that free energy (6) has the absolute minimum  − (see e.g. [4], [5], [6]).
It should be noted that the minimal value of domain radius (8) has the same dependence on the film thickness and domain wall energy as the equilibrium lateral size of the periodic stripe domain structure (see [1] and references therein).
In the majority of experiments [3], [4], reversed nanodomains remain unchangeable during many days and weeks. This fact proves their stability and is extremely useful for applications. It has the following explanation within the framework of the proposed model. When applied voltage U is turned off, the external field disappears as proportional to U and the domain depolarization field (see (A.7))) vanishes due to the final recharging of the surface traps with . The remained positive surface energy also determines the domain growth kinetics [5].
Let us discuss the peculiarities of possible inter-influence between domains during their recording. In accordance with conventional rules of binary information recording the distance between the centers of neighbour domain dots must be about 2d (see right inset in the Fig. 2).
Therefore neighbour nanodomains can interact via their depolarization fields . If the time interval t between recording neighbour domains is enough for complete recharging of the surface traps, depolarization field vanishes and domain inter-influence is absent. Thus, during such "equilibrium slow" recording it is possible to obtain independent nanodomains with minimum radius at U (see (8) Such optimization can be realized when nanodomain thermodynamic potential Φ is known (e.g. it can be found from "equilibrium" experimental data). For instance, at chosen recording time t and fixed film thickness h one can calculate the optimal voltage U opt from the conventional relaxation equation [16]. In the next section we'll show, that our thermodynamic potential (6) can be unambiguously reconstructed from experimental dependence d .  3 crystals. The depolarization field energy (4) of the domain butt essentially decreases the domain sizes even at low voltages [12]. We would like to underline, that Debye screening not only decreases depolarization field inside the domain, but also it shields the interior of the sample from the AFM tip electric field. As a result, Debye screening radius decrease leads to the essential decrease of the equilibrium domain sizes (see the lowest curves in the Fig. 4a). At small radius of curvature and distance the critical voltage U are almost independent over value, e.g. corresponds to and Fig. 4a and (7)).

Nanodomain recording in thin Pb(Zr,Ti)O 3 films
Let us apply our theoretical results (6) 8)). It is seen from the figure that the critical voltage of nanodomain appearance is about 5 V, which is in a good agreement with the value reported by the authors [5].
The small free energy, spontaneous polarization and high permittivity fitting values are the same order as the BaTiO 3 parameters (compare with Fig. 4). Such parameters are typical for soften Pb(Zr,Ti)O 3 nano-grained films with chemical composition near the morphotropic boundary.
Really, the authors [6] reported about grain sizes nm 10 ≤ . How could conventional Landauer approach, evolved for the perfect ferroelectric material, be applied to the nano-grained sample? In order to describe this experiment, the author [16] used Landauer free energy (without domain butt depolarization energy) and fitting value "averaged over the grain orientations". To 2 cm / 10 C P S µ ≤ our mind, such solution needs additional basing. In our recent papers [14], [15], we modified Landau-Khalatnikov approach for inhomogeneous ferroelectrics and shown, that polar and dielectric properties of such materials can be described by three coupled equations. Within the framework of our model [14], [15] all the parameters S P , It is clear from the Figs. 5 and 7, that we obtained rather well quantitative agreement between our modeling and domain radiuses recorded in thin Pb(Zr,Ti)O 3 films at different applied voltages.
• First of all we modified the existing models for equilibrium domain formation allowing for Debye screening effects, depolarization energy of the domain butt, bottom electrode influence, recharging of sluggish surface screening layers, field emission between the tip apex and the sample surface originated at high electric fields. We have shown that these effects essentially influence on the equilibrium domain sizes. • For the first time we demonstrated, that the domain radius does not decrease continuously with voltage decrease: the domain appears with non-zero radius at definite critical voltage. Such "threshold" domain formation is similar to the well-known first order phase transition. We hope that our results will help one to determine the necessary recording conditions and appropriate ferroelectric medium in order to obtain the stable domains with minimum lateral size in a wide range of applied voltages. [34] Cross, L.E. Ferroelectric ceramics: Tailoring properties for specific application. In Ferroelectric ceramics. Setter; N.; Ed.; Birkhauser Verlag, Basel, (1993).

APPENDIX A
For the extrinsic semiconductor with donors concentration n d , the free charges bulk density f ρ is determined via electric field potential ϕ(r), as follows: filled by isotropic semiconductor with dielectric permittivity could be found from the boundary problem: The solution of (A.3) can be found with the help of Hankel integral transformation. Finally we obtained that: Hereinafter J 0 is Bessel function of zero order, of depolarization field created by the charges on the surface of sample has to satisfy the following system of equations:  Here the first term is the self-energy of depolarization field; the last one is the energy of interaction between the external field and the polarization of the domain. These two summands of the electrostatic energy can be written as follows:  (A. 18)