General approach for the description of size effects in ferroelectric nanosystems
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Abstract
We propose general analytical approach for the description of size effect influence on polarization and dielectric susceptibility in ferroelectric nanosystems based on the twoparametric direct variational method and Landau–Ginzburg–Devonshire phenomenology. The essence of the approach is to solve Euler–Largange boundary problem for polarization distribution exactly in paraelectric phase without ferroelectric nonlinearity and then to use the linearized solution for derivation of the approximate analytical expression for spontaneous polarization distribution in ferroelectric phase with the average polarization and characteristic spatial scale as variational parameters. Corresponding polarization distributions calculated within the approach in thin ferroelectric films, nanowires and nanotubes were compared with the available exact solution of Landau–Ginzburg–Devonshire equation or approximate results obtained earlier from the one parametric solution. Perfect agreement between the exact solution and obtained approximate ones is demonstrated. The realization of the proposed scheme of the twoparametric direct variational method seems even simpler than the oneparametric scheme based on the Landau–Ginzburg–Devonshire free energy expansion with renormalized coefficients, while the validity range of twoparametric solution is much wider and the accuracy is higher. So, obtained analytical results have methodological importance for calculation of the phase diagram size effects, polarization distribution, all related polar, dielectric, piezoelectric and pyroelectric properties of singledomain ferroelectric nanoparticles and thin films. The proposed method is applicable to different ferroic nanosystems.
Keywords
Average Polarization Paraelectric Phase Misfit Strain Polarization Distribution Thin Ferroelectric FilmIntroduction
Ferroelectric nanosystems open the way to obtain a variety of new unique electromechanical, electronic and dielectric properties, a lot of which are useful for applications, such as ferroelectric memories, the ability to store and release energy in wellregulated manners, making them very useful for sensors and actuators, compact electronics, pyrosensors and thermal imaging [1, 2, 3].
The substantial progress in synthesis of various ferroelectrics nanosystems, like epitaxial films [4], nanoparticles with controllable sizes [5], arrays of tubes and rods [6, 7, 8, 9], the local characterization of their polar properties [10, 11, 12] and domain structure [13], triggered the renovation of interest to ferroic nanosystems theoretical description. It is worth to note the enormous achievements of both the phenomenological [14] and microscopic [15] theories, their recent advances in different fields like the description of nanorods [16, 17], size effects in thin films [18, 19], ferroelectric nanoparticles [20, 21, 22]; flexoelectric effect influence on the intrinsic properties [23, 24] and response [25, 26, 27] of the nanosystems; the developed analytical model accounting for depolarization field as well as the formation of misfit dislocations [28, 29, 30]. However, despite this progress, the phenomenological theory lacks a general method, suitable for the solution of vast variety of different problems of ferroics description.
The possibility to govern the appearance of phase transitions at any arbitrary temperature has been demonstrated in nanosized materials due to the socalled sizedriven phase transition. Such transitions were observed in many solids, including ferroelectric, ferromagnetic and ferroelastic ones [31]. For instance, it is generally accepted, that the ferroelectric properties disappear when the particle size decreases below the critical one [1, 32, 33]. Actually, it is well known that depolarization electric fields exist in the majority of confined ferroelectric systems [34] and causes the sizeinduced ferroelectricity disappearance in thin films and spherical particles [35, 36].

Firstly the analytical solution of the linearized Euler–Lagrange boundary problem obtained from the LGD free energy functional minimization is derived. This solution corresponds to the polarization distribution in the paraelectric phase of the system, where the nonlinearity can be neglected in the weak external electric field. The average paraelectric susceptibility diverges in the point where the paraelectric phase loses its stability, so corresponding expression for the transition temperature T_{cr} could be found directly from the condition of zero inverse susceptibility.

In order to study the system behavior in ferroelectric phase, the coordinatedependent part of the paraelectric solution is chosen as the trial one with its amplitude as variational parameter. After the integration of LGD free energy functional over the particle volume with the trial function we obtained the renormalized free energy with expansion coefficients depending on temperature T and the particle sizes. The polarization amplitude can be determined from the algebraic equation obtained after the minimization of the renormalized free energy. If the analytical (exact or approximate) integration is possible it leads to the corresponding analytical expressions for renormalized coefficients size dependences.
The main advantage of the oneparametric direct variational method is the principal possibility to obtain analytical results, while the typical disadvantage is lengthy integration of the terms in LGD functional in order to obtain renormalized coefficients.
In the paper we propose general analytical approach for the description of size effect of polarization and dielectric susceptibility in ferroelectric nanosystems based on selfconsistent method of successive approximations. Here the first step is to find the deviation of polarization distribution from its average value. The amplitude and spatial scale of distribution appear to be dependent on average polarization due to the system nonlinearity. Next step is to look for the average value of polarization from the full distribution allowing for deviation in a selfconsistent manner. Mathematically this method is equivalent to the twoparametric direct variational method with the average polarization and the distribution length scale as variational parameters. However the proposed scheme is free from the complex integration of the LGD free energy expansion coefficients, instead we solved the one transcendental equation for average polarization determination. Corresponding polarization distributions calculated within the approach in thin ferroelectric films, nanowires and nanotubes were compared with the available exact solution of LGDequation or approximate results obtained earlier from the oneparametric solution applied to the LGD free energy.
General approach
Strain field in ferroelectric systems
Polarization distribution  Nontrivial strain components  

u _{11}  u _{22}  u _{33}  
Free system  P_{3} = const  \( \frac{{\left( {  c_{12} q_{11} + c_{11} q_{12} } \right)P_{3}^{2} }}{{\left( {c_{11} + 2c_{12} } \right)\left( {c_{11}  c_{12} } \right)}} \)  \( \frac{{\left( {\left( {c_{11} + c_{12} } \right)q_{11}  2c_{12} q_{12} } \right)P_{3}^{2} }}{{\left( {c_{11} + 2c_{12} } \right)\left( {c_{11}  c_{12} } \right)}} \)  
Clamped system  P_{3} = const  0  0  0 
Films with out of plane P  P_{3}(x_{3})  u _{ m}  u _{ m}  \( \frac{{q_{11} }}{{c_{11} }}P_{3}^{2}  \frac{{2c_{12} }}{{c_{11} }}u_{m} \) 
Films with in plane P  P_{3}(x_{1})  \( \frac{{q_{12} }}{{c_{11} }}P_{3}^{2}  \frac{{2c_{12} }}{{c_{11} }}u_{m} \)  u _{ m}  u _{ m} 
Shear strain components are zero in these cases, \( u_{12} = u_{13} = u_{23} = 0. \) It should be noted, that the solutions listed in Table 1 are valid only for the polarization distributions, specified in the second column of the table. In the case of arbitrary distribution of polarization either compatibility or equilibrium conditions could be not satisfied for the elastic fields from Table 1. For instance, in the case of onedimensional distribution of polarization, P_{3}(x_{1}), in the elastically free system components u_{22}, u_{33} from second row should be replaced with their mean values. One of the consequences of such distribution is the stress localization in the vicinity of domain walls (see e.g. papers of Cao and Cross [42] and Zhirnov [43]). It should be noted, that the influence of the misfit dislocation on the misfit strain u_{m} relaxation could be taken into account by the renormalization of u_{m} (see e.g. Speck and Pompe paper [44]).
Free energy expansion coefficients renormalization
α  β  

Free system  \( a_{1}^{u} \)  \( a_{11}^{u}  4\frac{{\left( {q_{11}  q_{12} } \right)^{2} }}{{3\left( {c_{11}  c_{12} } \right)}}  2\frac{{\left( {q_{11} + 2q_{12} } \right)^{2} }}{{3\left( {c_{11} + 2c_{12} } \right)}} \) 
Clamped system  \( a_{1}^{u} \)  \( a_{11}^{u} \) 
Films with out of plane P  \( a_{1}^{u}  2q_{12} u_{m} + q_{11} \frac{{2c_{12} }}{{c_{11} }}u_{m} \)  \( a_{11}^{u}  \frac{{q_{11}^{2} }}{{c_{11} }} \) 
Films with in plane P  \( a_{1}^{u}  \left( {q_{11} + q_{12} } \right)u_{m} + q_{12} \frac{{2c_{12} }}{{c_{11} }}u_{m} \)  \( a_{11}^{u}  \frac{{q_{12}^{2} }}{{c_{11} }} \) 
Polarization distribution in ferroelectric films
The exact solution (4) involves higher transcendental function and is limited to the cases of ferroelectric phase and zero external electric field. The solutions for the first derivatives (susceptibility and pyroelectric coefficient) are also available [47], but they have even more sophisticated structure. These lead us to attempt to find the approximate solution of Eq. 3 in terms of elementary functions, valid in both paraelectric and ferroelectric phase.
The proposed approach to the confined ferroelectric system description is analogous to direct variational method with two variational parameters, namely the average polarization \( \overline{P} \) and the characteristic length scale R_{0}. The dependence of the latter on the average polarization reflects changes of the polarization distribution when approaching the phase transition point, which is the feature, present in the exact solution (4) via parameter m, changing from 0 to 1.
If the polarization is pointed perpendicular to the film surface and the depolarization field is present in the system [49, 50] the proposed method gives essentially the same results as the one parametric variational method did (see Appendix 2), since in this case the characteristic length scale appeared to be practically independent on temperature and film thickness and is determined solely by the depolarization field screening [47].
The advantages of the developed approximate method and its high accuracy encourage one to apply this method for other ferroic system of different geometry where exact solutions are not available.
Polarization distribution in ferroelectric nanowires
In contrast to the thin films on the substrate, the elastic field of the spontaneous strain u_{ij} ~ P _{3} ^{2} inside cylindrical ferroelectric nanoparticles is rather complicated because of the polarization distribution, which leads to the appearance of nonlocal terms, involving the term with polarization mean square value \( P_{3} \overline{{P_{3}^{2} }} \) in the polarization equation of state [51]. The proposed method allows taking into account these terms by involving additional parameter \( \overline{{P_{3}^{2} }} , \) which makes the consideration very cumbersome. At the same time, one could get the quantitatively correct picture of the size effect in ferroelectric nanowires neglecting the distinction of strain field from the one of bulk system (see Table 1). Here we suggest using renormalized expansion coefficients for free system from Table 2 as an approximation for real system.
Susceptibility is renormalized on the value \( \chi_{b} =  {1 \mathord{\left/ {\vphantom {1 {2a_{1} (T)}}} \right. } {2a_{1} (T)}}, \) which is dielectric susceptibility of the bulk material. The drop of polarization and the increase of susceptibility in the vicinity of size—driven phase transition is obvious. Also the maximum of susceptibility near the surface of thick wires could be related to the decrease of polarization in this region. The similar effect was predicted for the ferroelectric films with inplane polarization (i.e. without depolarization field) on the basis of exact solution [47].
It is obvious, that the results for nanorods could be generalized to the case of nanotubes of arbitrary sizes in straightforward way (see Appendix 4). The detailed analysis of the results for ferroelectric nanotubes will be presented elsewhere.
Summary
 I.
To obtain the Euler–Lagrange boundary problem for polarization distribution from the minimization of the LGD free energy functional.
 II.
To linearize the Euler–Lagrange boundary problem near the average value of polarization and to obtain the equation for deviation of polarization from its average value. The solution of this equation could be found by using standard methods and gives the polarization distribution with amplitude and length scale dependent on the average polarization.
 III.
To find the average polarization selfconsistently by the averaging of the Euler–Lagrange equation solution, obtained on the step II and dependent on the average polarization.
Allowing for the step III, the twoparametric scheme of the direct variational method is free from the complex integration in order to obtain the LGD free energy with renormalized expansion coefficients; instead we solved the only one transcendental equation for average polarization determination. Thus, the realization of the twoparametric scheme is simpler than the oneparametric scheme based on the LGD free energy expansion with renormalized coefficients, while we proved that the validity range of twoparametric solution is much wider and the accuracy is higher.
Obtained analytical results have priory methodological importance for calculation of the phase diagram size effects, polarization distribution, polar, dielectric, piezoelectric and pyroelectric properties of singledomain ferroelectric nanoparticles and thin films. The method is also applicable to different ferroic nanosystems.
Footnotes
 1.
The linearized solution for the polarization distribution in paraelectric phase and the averaged polarization was derived earlier in Refs. [32, 35] as \( P_{3} (\rho ) = \frac{{E_{0} }}{\alpha }\,\left( {1  \frac{{J_{0} \left( {{\rho \mathord{\left/ {\vphantom {\rho {R_{c} }}} \right. } {R_{c} }}} \right)}}{{J_{0} \left( {{R \mathord{\left/ {\vphantom {R {R_{c} }}} \right. } {R_{c} }}} \right)  \left( {{\lambda \mathord{\left/ {\vphantom {\lambda {R_{c} }}} \right. } {R_{c} }}} \right)J_{1} \left( {{R \mathord{\left/ {\vphantom {R {R_{c} }}} \right. } {R_{c} }}} \right)}}} \right), \) where \( R_{c}^{{}} = \sqrt {{{  g} \mathord{\left/ {\vphantom {{  g} \alpha }} \right. } \alpha }} . \) However the solution is invalid in ferroelectric phase, since the scale R_{c} is different from R_{0} introduced in Eq. 4.
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