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Size effects and depolarization field influence on the phase diagrams of cylindrical ferroelectric nanoparticles

Ferroelectric nanoparticles of different shape and their nanocomposites are actively studied in modern physics. Because of their applications in many fields of nanotechnology, the size effects and the possible disappearance of ferroelectricity at a critical particle volume attract a growing scientific interest. In this paper we study the size effects of the cylindrical nanoparticle phase diagrams allowing for effective surface tension and depolarization field influence. The Euler-Lagrange equations were solved by direct variational method. The approximate analytical expression for the paraelectric-ferroelectric transition temperature dependence on nanoparticle sizes, polarization gradient coefficient, extrapolation length, effective surface tension and electrostriction coefficient was derived. It was shown that the transition temperature could be higher than the one of the bulk material for nanorods and nanowires in contrast to nanodisks, where the decrease takes place. The critical sizes and volume of ferroelectric-paraelectric phase transition are calculated. We proved that among all cylindrical shapes a nanobar reveals the minimal critical volume. We predicted the enhancement of ferroelectric properties in nanorods and nanowires. Obtained results explain the observed ferroelectricity enhancement in nanorods and could be very useful for elaboration of modern nanocomposites with perfect polar properties.


Introduction
Ferroelectric nanoparticles of different shape are actively studied in nano-physics and nanotechnology. Because of miniaturization of devices based on these materials, the study of ferroelectric properties size dependence and the possible disappearance of ferroelectricity at a finite critical volume attract a high scientific interest.
Mishina et al [11] revealed that ferroelectric phase exists in PbZr 0.52 Ti 0.48 O 3 (PZT) nanorods with diameter less than 10-20nm. However, earlier Mishra et al [9] demonstrated that the ceramics prepared from the powders of PbZr 0.52 Ti 0.48 O 3 with size about 100 nm had a pseudo-cubic symmetry, but might exhibit a hump in the temperature variation of dielectric constant. So, it seems that the critical size for the PZT nanorod (if any) is about 10 times smaller than the one for the nanosphere.
Geneste et al [2] studied the size dependence of the ferroelectric properties of BaTiO 3 (BT) nanowires from the first principles. They showed that the ferroelectric distortion along the wire axis disappears below a critical size of about 1.2nm. Note, that BaTiO 3 spherical nanoparticles have the much larger critical size of about several tens nm [10], [12].
Morrison et al [5] demonstrated that ultra-small PbZr 0.52 Ti 0.48 O 3 nanorods and nanotubes (radius 20-30nm, length 50 ) possesses rectangular shape of the piezoelectric hysteresis loop with effective remnant piezoelectric coefficient value compatible with the ones typical for PZT films [13].
This fact unambiguously speaks in favor of spontaneous polarization existence. Also the authors demonstrated that the ferroelectric properties of the free BaTiO m µ 3 nanotubes are perfect. Thus, at the first glance recent experimental results contradict the generally accepted viewpoint that the ferroelectric properties disappear under the system volume decreases below the critical one [14]. Actually the aforementioned facts proved that the shape of nanoparticles (e.g. the spherical or cylindrical one) essentially influences on the minimal sizes necessary for the ferroelectricity conservation [2] possibly owing to the different depolarization field and mechanical boundary conditions [15], [16]. Immediately, one should ask the principal question: What nanoparticle shape posses the minimal critical volume and allows ferroelectricity conservation at higher temperatures?
Could the answer be predicted theoretically?
In theoretical papers [6], [17] the special attention was paid to size effects of nanoparticles, but finite cylinders were not considered. It is well known that depolarization field exists in the majority of confined ferroelectric systems [18] (including the cylindrical nanoparticles) and could cause the aforementioned size-induced ferroelectricity disappearance in insulator single-domain films and ellipsoidal particles [19], [20], [21]. As a result of ferroelectric properties degradation, the phase transition temperature in spherical nanoparticles is significantly lower then the bulk one for most of the cases [6], [22], [21], [8]. Since the depolarization field value depends on the shape of a particle, the enhancement of ferroelectricity can be expected for the ones with smaller depolarization field.
In this paper we study the size effects, surface tension and depolarization field influence on the cylindrical nanoparticles properties. We suppose that a nanoparticle surface is covered with a charged layer consisted of the free carriers adsorbed in the ambient conditions (e.g. air with definite humidity or pores filled with a precursor water solution). For instance a thin water layer condensates on the polar oxide surface in the air with humidity 20-50% [23] The surface charges screen the surrounding medium from the nanoparticle electric field [14], but the depolarization field inside the particle is caused by inhomogeneous polarization distribution. As a matter of fact we calculated the depolarization field inside a cylindrical nanoparticle under the short-circuit conditions proposed by Kretschmer and Binder [19].
For the description of nanodisks, nanorods and nanowires ferroelectric properties we used the Euler-Lagrange equations solved by means of a direct variational method [20]. The approximate analytical expression for paraelectric-ferroelectric transition temperature dependence on the nanoparticle sizes, extrapolation length, effective surface tension coefficient, polarization gradient and electrostriction coupling coefficients etc was derived. We obtained, that the possible reason of the polar properties enhancement in confined ferroelectric nanowires and nanorods is the effective surface pressure coupled with polarization via the electrostriction effect and the decrease of depolarization field value occurred in prolate cylindrical particles.

Free energy with renormalized coefficients
Let us consider ferroelectric cylindrical nanoparticle with radius R, height h and polarization ) , ( z P Z ρ oriented along z -axes. Hereinafter V is the particle volume, the polarization distribution is axisymmetric. The external electric field is Fig. 1).
for the first order phase transitions or β for the second order ones. The coefficient in Eq.(1) should be renormalized by the external stress (see e.g. Ref. 22). In Appendix A we study the influence of the effective surface tension on a cylindrical particle and derived the expression for : Here parameters T , α C T and Q 12 are respectively Curie temperature, inverse Curie constant, and electrostriction coefficient regarded known for the bulk material. The parameter µ is the effective surface tension coefficient between the nanoparticle and interface [24], [10].
Note, that the renormalization of coefficient for a cylindrical nanoparticle differs from the one ( p Q Q R 12 11 2 + + α ) = α obtained for a spherical particle [22] (p is the pressure applied to the particle surface). Both results are clear owing to the fact that stresses p − = σ = 2 1 σ and for a cylinder, whereas for a sphere. Also we do not take into account possible stress relaxation caused by dislocations. This approach used by many authors is valid for the small enough particle sizes [25].
The calculation of the depolarization electric field inside the finite cylindrical nanoparticle appeared to be more complex then for a spherical one. The exact expression for depolarization field inside the cylindrical nanoparticle covered with screening charges is derived in Appendix B (see Eq.(B.6)). Hereinafter we use its Pade approximation: The angular brackets are the spatial averaging on the particle volume V , e.g.
for the prolate cylinder with h R << [18], whereas η for the oblate cylinder with [19]. It should be noted that the π → 4 h R >> depolarization field is absent outside the particles in the framework of our model. Therefore the interaction of such nanoparticles is practically absent due to the screening and they can be considered as the assembly of independent particles.
The surface part of the polarization-dependent free energy S G ∆ is supposed proportional to square of polarization on the particle surface S, namely ∫ λ δ = ∆ S S ds G 2 S P 2 (λ is the extrapolation length [6], [17]). A cylindrical nanoparticle has upper and bottom surfaces 2 h z = , 2 h z − = and a sidewall (see Fig. 1), so its surface energy acquires the form: We introduced longitudinal and lateral extrapolation lengths Let us find the approximate solution of the nonlinear Eq.(5) by using the direct variational method as proposed earlier [20]. Firstly we solved the linearized Eq.(5): Here and are Bessel functions of the zero and first orders respectively. In general case the roots depend over the ratio in accordance with Eq. (7). Under the condition . For the case we used the Bessel functions norm in Eq.(7) [26].
The average polarization induced by the external electric field in paraelectric phase and dielectric permittivity acquire the form: Hereinafter Note, that the expression for paraelectric phase permittivity zz ε obtained from Eq.(9) for the infinite coincides with one derived by Wang et. al [6]. In fact, for the case one can neglect the depolarization field ( 0 → η ) and dependence on z ( ) and the summation on k leads to In order to obtain the solution of Eq.
. The variation amplitudes must be determined by the minimization of the expressions (1)- (4). Integration in these expressions with the aforementioned trial function leads to the following form of the free energy:  Eq. (11) expressed as the algebraic sum of different even powers of polarization components with coefficients dependent on the particles sizes. The exact expressions for the renormalized coefficients in Eq.(11) are rather cumbersome. Note, that hereinafter we suppose that and so 1 , which is typical for the majority of ferroelectric nanoparticles. Really, The average polarization should be calculated as: The coupled equations for the amplitudes should be obtained from the variation .
The spatial distribution of depolarization field and spontaneous polarization for a single- Figs. 2 and 3 respectively.
. Numbers near the curves in part (a) correspond to the values of potential normalized on its maximal value.
It is seen from Fig.2a that the depolarization field potential is zero on the particle surface allowing for the short-circuit conditions provided by screening charges. Closeness of depolarization field potential contours appears near the faces. It is clear that the gradient (i.e. the depolarization field) is maximal near the cylinder faces and minimal in its center 0 (compare Fig.2a with Fig.2 b,c).
One can see from Fig. 2b that along the nanoparticle axis 0 = ρ depolarization field has maximal positive values near the surfaces The depolarization field rapidly decreases inside the particle, reaches zero value at 0 z z ± = , then it becomes negative and rather small at . The depolarization field drastically decreases at 2 / h z ≥ and completely vanishes outside the screening layer that plays a role of an electrode. It is well known that depolarization field decays exponentially in such conductive layers (see e.g. [17], [27]).
The spatial distribution of depolarization field vs. the distance from nanoparticle axis 0 = ρ is depicted in Fig.2c for 2 / h z ± = (faces) and 0 = z (plane of symmetry), the field being positive and negative respectively. It is clear that the field is maximal at ρ and decreases up to zero when reaching the sidewalls R = ρ as it should be expected for the electric field tangential component on conducting surface.
The behavior of for arbitrary x (y=0) and z values can be seen from Fig.3a. The spontaneous polarization is maximal on the particle axis (see values near the contours in Fig.3a).
. Numbers near the contours in part (a) correspond to the values of polarization normalized on its maximal value.
The polarization profile on nanoparticle axis in the cylinder symmetry plane is depicted in Fig. 3c. The profile is qualitatively similar to that of , namely has minimal ) , 0 ( z P Z ) 0 , (ρ Z P value on the surface ρ and reaches the maximum in the central part . However, profile looks like the diffuse maximum contrary to the plateau shown in Fig. 3b. respectively (see Eqs.(6,7)). The same speculations are valid for the explanation of the difference of depolarization field profiles and , since they are proportional to Figs. 2b and 2c).
It is worth to underline, that the profiles and average values of the properties related to spontaneous polarization (e.g. piezoelectric and pyroelectric coefficients) can be calculated with the help of obtained polarization distribution.

Phase diagram
The equation for paraelectric to ferroelectric phase transition temperature T can be obtained from Eq. (9) in the following form: In particular case we derived the interpolation for 1 valid for the nanodisks, nanorods and nanowires: where ( ) The first term in Eq. (14) is the bulk transition temperature, the second term is related to the coupling of surface tension with polarization via electrostriction effect, the third one corresponds to the correlation effects, and the fourth represents the depolarization field contribution. The depolarization field is small enough at (see Eq. (3)). The correlation and depolarization terms contribution can only decrease the transition temperature, whereas the second term In these variables The size effect on the phase diagram for the case when the shape of nanoparticle is fixed ), but its radius r or length l increases are represented in Figs  Radius r It seems natural, that the lateral correlation effect on the size-driven phase transition reveals itself at fixed height, whereas the depolarization field influence on critical sizes and transition temperature is more pronounced at fixed radius (compare curves order for different ratio r l 2 in Figs. 5b and 5a).
Finally, let us answer the important question: "What nanoparticle shape posses the minimal critical volume and allows ferroelectricity conservation at higher temperatures?" For this purpose we have to consider the case when the shape of nanoparticle is fixed ( const r l = 2 ), but its volume increases. The results are represented in Figs. 6. It is follows from the figures that the nanobar ( 1 2 = r l , depolarization factor π ≈ η 2 ) has the smallest critical volume both at and at 0 > . Its ferroelectric-paraelectric transition temperature is high enough up to the values higher than T . Moreover, since 0 12 < µ µ R Q this nanoparticle shape is preferable in comparison with a spherical one, because a sphere has positive parameter and only slightly smaller depolarization factor [21]. The nanorods with the shape 100 ≤ 2r 10 reveal the highest transition temperature and thus the best ferroelectric properties in the region of volumes V at Both for and , the dependence of transition temperature on the nanoparticle volume for different shapes is non-monotonic with respect to the ratio

Discussion
We have studied the dependence of ferroelectric nanorod critical sizes on the nanoparticle shape, polarization gradient coefficient, extrapolation length, effective surface tension and electrostriction coefficient . Analyzing obtained results we could summarize that finite nanorods of radius R~20-200 unit cells posses the best ferroelectric properties when the electrostriction coefficient is negative, namely they exhibit the higher transition temperature and spontaneous polarization in comparison with the bulk sample, whereas a nanobar ( ) reveals the minimal critical volume.
Keeping in mind that the condition 1 << h R leads to the small values of depolarization field ( ) , the aforementioned conservation of ferroelectric properties is possible for a sufficiently long nanorod ( . Note, that the existence of polydomain structure lead to the additional decrease of depolarization field and thus improve the estimation. In the case we obtained, that the increase of 1 max > C CN T T appears at radius when the parameter is negative (see the second term in Eq. (14) and Fig. 5a), i.e. when the effective surface pressure leads to the decrease of the inverse dielectric susceptibility due to the negative electrostriction effect.

Q R
Let us qualitatively compare our results with the ones obtained earlier [1], [2], [11] (see Table   1). Table 1 Experiments [1], [11] and ab initio calculations [2] Parameters calculated in the framework of proposed model  The applicability of our model to the description of phase transition between cubic paraelectric and tetragonal ferroelectric phases in BT and PZT is for certain. However, it can be only the approximation for improper ferroelastic -ferroelectrics like RS. Therefore our consideration can be applied to experiments of Yadlovker and Berger [1] only in the temperature range where RS ferroelectric properties can be described by the phenomenological expansion (1) over polarization powers.
Obtained results explain ferroelectricity enhancement in the Rochelle salt nanorods [1] of radius ~, piezoelectric properties conservation in lead-zirconate-titanate nanorods [11] of radius and are in a good agreement with the first principles calculations in barium titanate nanowires [2] (see Table 1). The predicted effects could be very useful for elaboration of modern nanocomposites with perfect polar properties.
Hereinafter we use Voigt notation σ or matrix notation  [24], [10]. This Lame's problem is discussed in details elsewhere [29]. It is easy to obtain that generalized pressure is directed along the cylinder normal, i.e. .
The conditions of mechanical equilibrium x in the bulk of solid body are the following: for a sphere. Also let us underline that we do not take into account possible stress relaxation caused by dislocations.
This approach used by many authors (see e.g. Refs. 20,15) is valid under the conditions discussed elsewhere [25].

Appendix B
Let us consider the depolarization field distribution for the case of cylindrical particle with arbitrary polarization distribution in the ambient conditions. In the equilibrium the perfect screening can be achieved so that there will be no electric field outside the particle.
The field distribution can be obtained on the basis of the electrostatic Poisson's equation for the electric potential : Here is the given z-component polarization distribution inside the particle, which has the cylindrical symmetry: The boundary conditions on the particle surface has the view: Here R and is the cylinder radius and height respectively, U is the applied voltage. At U the boundary conditions (B.2) corresponds to the short-circuit ones proposed by Kretschmer and Binder [19] for a film.
Note, that (B.9) is exact at and coincides with the one obtained for ferroelectric films [19] at ) . (B.10) The interpolation for the depolarization field that contains the aforementioned particular cases (B.9)-(B.10) acquires the form: