Jeffrey F Webb*
Department of Mechanical, Materials and Manufacturing Engineering, The University of Nottingham Malaysia Campus, Jalan Brog, 43500 Semenyih, Selangor, Malaysia
Received date: January 20, 2015; Accepted date: January 25, 2015; Published date: January 31, 2015
Citation: Webb JF (2015) Nanoscale Ferroelectric Structures: Landau Theory for the Polarization Profile in a Ferroelectric Nano-Box and its Response to an Incident Electromagnetic Field. Global J Technol Optim 6:172. doi: 10.4172/2229-8711.1000172
Copyright: © 2015 Webb JF. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Visit for more related articles at Global Journal of Technology and Optimization
The Landau-Devonshire theory of ferroelectrics to date has concentrated mainly on bulk materials, thin film geometries, spherical geometries or other, essentially one-dimensional situations. By extending such calculations to three dimensions it is possible to consider a ferroelectric in the form of a box in which the influence of all surfaces of the box are included in the theory. This influence is expected to be important as the size of the box decreases and the surface to volume ratio increases. With interest in nanoscale materials burgeoning, it is of interest to study such size effects and how they influence the distribution in a ferroelectric box. This paper will show how Landau-Devonshire theory can be applied to such a box. After showing how spontaneous polarization profiles can be calculated, an outline of how Maxwell's equations together with Landau-Khalatnikov dynamical equations can be brought in to study the interaction of the nano-box with incident electromagnetic waves.
One of the defining characteristics of a ferroelectric is that it has a non-zero spontaneous polarization in the absence of an applied electric field provided that the temperature is lower than a certain critical temperature TC . The other defining feature is that the polarization can be reversed by applying an electric field opposing the direction of the polarization. This allows ferroelectrics to be used in random access memories. Binary information can be stored according to the polarization direction which can be switched between two states by an electric field .
With the increasing miniaturization of devices it becomes important to investigate size-effects in ferroelectrics. Thin film geometries for ferroelectrics, in which one spatial dimension is confined, have been studied fairly extensively [3-8]. In this paper a box geometry in which all three dimensions are confined will be considered. Confinement in the box means that the pairs of oppositely facing surfaces are close enough to the central region of the strip that their presence causes a non-negligible effect on the ferroelectric as a whole, so that the ferroelectrics properties are different from those of a bulk region far from any surfaces. Such effects are expected to be important at the nanoscale, and so here we refer to the boxes as ferroelectric nano-boxes.
The formalism for calculating the spontaneous polarization in the nano-box will be developed using Landau-Devonshire theory. In addition Landau-Khalatnikov equations will be introduced, which will allow the dynamic interaction of the box with incident electromagnetic waves to be studied.
A problem similar to the one of interest here, but applied to spheres and cylinders, has been dealt with by Morison’s et al. , who have given approximate analytical solutions of the equations that define the spontaneous polarization. The work in the present paper, however, is concerned with the formalism for ferroelectric nano-boxes and does not consider approximate analytical solutions. This does not lead to equations which can be solved exactly in analytical terms, but possible numerical schemes for solving the equations are discussed.
The region occupied by the interior of the ferroelectric nano-box is defined, in Cartesian coordinates, by
The sides of the box are given by ; i = 1,2,3.
Other notation that will be useful is: x=(X2, X2, X3), for a position in the ferroelectric crystal; P= (P1,P2,P3), the polarization vector; Pi= (x1, X2, X3), i = 1,2,3; i = 1,2,3.
An important quantity in Landau-Devonshire theory is the Gibbs free energy given as an expansion in powers of the polarization components that are invariant under the symmetry operations of the symmetric (T<TC) phase of the ferroelectric [10,11]. For a cubic ferroelectric crystal (such as lead titanate) the free energy can be written as:
Here x1, X2, and X3 are Cartesian coordinates, the polarization components are Pi, with the αi, βi and γi, and are the terms that form the free energy for a bulk ferroelectric; the δ terms are gradient terms accounting for the change of polarization that occur near surfaces of the ferroelectric ; the final terms involving the Pi account for the depolarization field for the case in which the polarization has components along x1, X2, and X3 [4,5]. The αi, βi and γi coefficients are found from experimental data or in some cases may be calculated from first principles; d is an extrapolation length , which in general can vary over the surfaces of the ferroelectric; ïƒ²0 is the vacuum permittivity and ïƒ²∞ accounts for the contribution of high frequency resonances to the dielectric response . The notation P xi denotes partial derivatives of P with respect to xi , and
which is an averaged polarization, and âŸ¨PâŸ© is defined by
Together with the free energy given in Equation. (1) there is also a surface energy density given by
where S is the entire surface of the ferroelectric box given by
in which the sides of the box at are given by
where σ is a cyclic operator performing the operation From this it follows that
Also, σn denotes σ applied n times, and the definition of the operator is extended to when the xi are arguments of a function h such that
Using \sigma, Equation. (2) can be written in a shortened form that shows the cyclic patterns present in the free energy expression:
Minimizing the Free Energy Function to Obtain the Spontaneous Polarization
In Equation. (2), the functional dependence of the polarization enters explicitly via the first term. For a nano-box the polarization varies in all three space dimensions x1, X2, and X3. So it is useful to consider the free energy for the general case in which the ferroelectric is in the region V bounded by a closed surface S. The free energy in this case is of the form
The spontaneous polarization distribution is that which minimizes the free energy, that is, the problem of minimizing a functional. This will be done using the classical Euler-Lagrange calculus of variations method. Applying this method  results in the following set differential equations for P:
Assuming that (x |s ) d is constant over the surfaces, the boundary conditions are
The spontaneous polarization, which is a function of x1, X2, and X3 is found by solving Equation. (12) subject to the boundary conditions (Equation. (14)), and the solution could be denoted P = Ps. There is no analytical solution and the equations will have to be solved numerically. Such work has yet to be done, but is in progress and will be reported in due course. However, it is likely that a multigrid numerical solution technique will be a good way to solve the equations efficiently.
Here it will be shown how the dynamic response of the ferroelectric nano-box to incident electromagnetic radiation can be calculated. The dynamic response is modeled using Landau-Khalatnikov equations, which are of the form ,
The δ notation on the right hand side of the equation denotes a variational derivative ; the incident electric field is E = (E1, E2, E3); m and γ are mass and damping parameters, respectively. Values of these parameters can be found from experimental data , or in some cases from first-principles calculations .
To find the response of the ferroelectric nano-box to E it is necessary to to solve the Landau-Khalatnikov equation together with the Maxwell wave equation:
When solving these equations the dynamic response is written as the sum of the spontaneous polarization Ps and the changes to it Q(t) caused by the incident field E, that is, P(t) = Ps + Q(t). To make further progress it will first be necessary to calculate Ps and therefore is left for future work to be reported in due course.
The analysis above is suitable for free standing nano-boxes in which there are no appreciable strains other than those directed along the direction of the polarization. In Reference  it is explained that strains in this direction do not change the form of the free energy. Instead they cause a change in the phenomenological values of some of the coefficients without altering the form of the free energy. To account for strains in other directions, an extension of the theory that introduces strain terms into the free energy is required. The starting point for such a development could be the consideration of unequal biaxial misfit strains. These strains could be induced in the nano-box if it were grown on a substrate that gave rise to a lattice mismatch. According to Wang and Zang , for a cubic film on a tetragonal substrate, the mismatch creates two components of misfit strain given by and where b and c are the substrate lattice parameters, and a0 is the cubic cell constant of the film in the free standing paraelectric state. Here the interest is in a nano-box rather than a film, but as a first approximation these strain forms could be assumed, and in the direction normal to the substrate, would give rise to an extra term in the free energy of the form
Renormalization of some the other coefficients in the free energy would also occur—see Wang and Zhang  for further details. But these can be considered as modifying the phenomenological values of the affected constants. So the essential change in form of the free energy is given by the additional term in Equation. (17).
Formally, it has been shown how Landau-Devonshire theory can be used to the calculate the spontaneous polarization in a ferroelectric nano-box. In addition the equations for calculating the dynamic response to an incident electric field have been presented. This opens up a program of future research concerned with solving these equations for a given ferroelectric. The nature of the equations is such that numerical techniques will be required to solve them.
A possible way of incorporating the effects of strain into the theory has also been discussed. The substantial work required to extend this in a more general way appropriate for ferroelectric nano-boxes will also provide an interesting programme of future research activity.