Effect of thickness on the magnitude of spontaneous polarization in thin ferroelectric films

The influence of the thickness of a ferroelectric film and the depolarizing field on its spontaneous polarization and the order parameter has been investigated by means of the Monte-Carlo method. Dependences of the polarization of thethin ferroelectric film on temperature at different values of its thickness and the potential well depth of the Lennard- Jones potential are calculated. The thickness of the “dead” layer is analyzed depending on the temperature and of the potential well depth.


Introduction
The introduction of ferroelectric films in modern microelectronic devices, related with limitations on the reduction in the size of the element base, has revived interest in the question of the possible existence of a critical thickness of the films necessary for the appearance of ferroelectricity [1].
The critical size of the ferroelectric film is usually understood to mean the minimal thickness of the sample, at which its ferroelectric properties disappear, and it becomes a usualdielectric material [2]. To describe the properties of the film vs. its thickness, different models are used, taking into account: а) correlation effects [3,4], b) the presence of a transient dielectric layer ("dead" layer) [5], c) the bulk charge density and the surface charge density near boundaries of ferroelectric [6], d) thermoelastic properties [7,8].
However, these effects cannot be considered isolated, because they all are related each with other.We used the Ising model for computer simulation of ferroelectric systems.This model has a discrete symmetry group. Onsager has been strictly proved that there is a spontaneous polarization in this model of monolayer [9]. However, it has been experimentally established that there are sufficient ferroelectric thin films, in which the spontaneous polarization may be absent [2]. In this paper, the Ising model was modified to take into account thermoelastic properties and a depolarizing field.

Model
To describe properties of the ferroelectric films and tostudy of ordering effects we use a threedimensional lattice model (Fig.1), consisting of N 1 ,N 2 and N 3 nodes along the respective axes of the Cartesian coordinate system.The position of the lattice node is characterized by the set of three numbers 1 2 3 ( , , ) n n n n = r .
In this paper, the interaction energy of dipoles is described by a potential that takes into account orientational interactions of particles (as in the classical Ising model) and the additional termrepresenting the Lennard-Jones potential: ∑ r uu r r uu r r uu r (1) where εis well depth of the Lennard-Jones potential, , i j r r r isthe distance between the dipoles, 0 r is average distance between the dipoles (in the absence of orientational interactions).The second term of equation (1) does not depend on temperature and polarization, in contrast to the first term. The short-range orientational order parameters in the longitudinal and transverse directions µ  and µ ⊥ are determined correspondingly by the following formulas: , , 1, , 1 2 3 1 n n n n n n n  1 2 3  1 2  3  1 2 3  1 2 3 , , , 1, , , , , 1 1 2 3 1 2 n n n n n n n n n n n n n n To simplify the calculations the variable 3 0 r x r = was introduced.The values µ  and µ ⊥ depend on the variable x. As an example, figure 2 shows the dependence of the short-range orientational order parameter µ ⊥ on the variable x at different reduced temperatures.
In contrast to the classical Ising model, the interactions between dipoles in the model considered depend on the distance between them. Therefore, it was necessary to calculate the average value of the distance between dipoles at which the potential energy has a minimum. This problem is reduced to finding the minimum of the function: The differentiation of equation (4) with respect to r leads to the equation: To solve equation (5)  , where m andσ are fitting coefficients.

Results of simulation
Usually, the change of properties of the system under an external action is described as its response to this action.For example, the dielectric susceptibility reflects a change in the polarization upon an electric field. :  Figure 3 shows the dependence of the average distance between the dipoles (a) and the susceptibility of the ferroelectric system (b) on the reduced temperature for different values of the potential well depthin the absence of the depolarizing field. As is seen from Fig.3, if the temperature is increased, the average distance between dipoles near the phase transition point increases abruptly. At high temperatures, the distance 0 r r → .With a decrease in the of the potential well depthof the Lennard-Jones potential, the average distance between the dipoles decreases, that leads to a shift of the phase transition point to higher temperatures.

Influence of depolarizing field
Under the action of internal electric field caused by spontaneous polarization, free particles move to the outer surfaces of the film and create an additional depolarizing field which depends on the value of the long-range orientational order parameter: where 0 E and are constants determined by the number of free carriers in the film, the quantity µ is the long-range order parameter in the ferroelectric film, which value is determined by the following formula:

Conclusion
Therefore, correlation effects related with the ordering in thin ferroelectric films lead to the appearance of a surface charge density, the depolarizing field, and also to the presence of the "dead" layer. It is shown that the depolarizing fielddecreases the polarization of the film and shifts the phase transition point to the region of lower temperatures.In sufficiently thin films, the polarization is not observed even at low temperatures. This result can be explained by the fact that its size is less than doublethickness of the "dead" layer that is considered as the critical size of the film, at which its ferroelectric properties disappear, and the film becomes a usual dielectric material. As shown in figures 5-6, the thickness of the "dead" layer is 2-10 unit cells. The value of the critical thickness C N increases with decrease in the well depth of the potential of Lennard-Jones (Fig.5) and increasing temperature (Fig.6). Experimental techniques allowed to obtainperovskite ferroelectric films with a thickness of 40 Å (ten unit cells) [10]. The authors [11] identified a critical thickness of about three unit cells, below which there was no ordering at low temperatures. In the frame of approach proposed in this paper, such difference in the experimental results can be explained by the difference nthe potential welldepths and the concentrations of free carriers in the materials considered.