12th International Conference on DEVELOPMENT AND APPLICATION SYSTEMS, Suceava, Romania, May 15-17, 2014
Size, shape and temperature effects on Ferro/Antiferro-electric hysteresis loops from Monte Carlo simulations on 2D Ising model Daniel Chiruta1,2,3, Christian Chong1, Pierre-Richard Dahoo4, Yasser Alayli1, Mihai Dimian3, Jorge Linares2* 1
LISV, Université de Versailles Saint Quentin en Yvelines, Vélizy-Villacoublay 78140, France 2 GEMAC, Université de Versailles Saint Quentin en Yvelines, Versailles, 78000, France 3 Stefan cel Mare University of Suceava, Suceava, 720229, Romania 4 Université Versailles St-Quentin; Sorbonne Universités, UPMC Univ. Paris 06; CNRS/INSU, LATMOS-IPSL, Guyancourt, 78280, France *
[email protected] Abstract—A Monte Carlo (MC) simulation of a 2D microscopic Ising model is performed to study the relation between electric and thermal properties in a 2D lattice which exhibits Ferro-Electric (FE) and Anti-Ferro-Electric (AFE) coupling. The main purpose is to determine how the size and the shape of these crystals affect this relation at different temperatures. The different effects are discussed as a function of the relative strength of two parameters the Ferro and Anti-Ferro interaction constants for different applied cycling external electric field. Keywords—Ferroelectric, Hysteresis, Simulations, Phase Transformation.
I.
Monte
Carlo,
INTRODUCTION
In the recent decades an intensive and ever growing research studies have been carried in the field of ferroelectric materials. Their potential applications in electro-optical and ferroelectric memory devices or in waveguides phase conjugated mirrors, microwave filters, piezoelectric sensors and actuators were the main targets for the industrial applications [1-5]. However, although the antiferroelectric to ferroelectric phase transition characteristics represented by a double step Polarisation versus Electric field (P-E) hysteresis loop under an applied field is of great interest for the physicists and chemists community, the phenomenological behavior is still not fully understood. The most interesting class of compounds showing ferroelectricity [6,7] is probably metal oxides, characterized by a chemical composition much simpler than that of the historical Rochelle salt, namely PbZrO3 (PZO), BaTiO3, PbTiO3, LiTaO3 or LiNbO3[8].As ceramics with good thermomechanical properties, they are favorable candidates for use in electronic devices [9]. Among these ferroelectrics compounds PZO which is used
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in ferroelectric memory devices and hydrogen containing compound KH2P4 (KDP) exhibit a particular electric field induced FE to AFE two steps transition with hysteresis. To study ferroelectric domains, different formulations of the system’s Gibbs free energy have been used in the literature and based on various models such as: ab-initio principles [10,11], Ginzburg-Landau equation [12,13], electrostatic potential [14], Potts’s model [15,16], and finally Ising model[17] . Over the years, in order to simulate experimental data, the above mentioned models and the critical behavior of the finite system in the vicinity of a characteristic temperature TC, have been used through different methods that have been successfully applied such as: the mean field approximation [18-20], the transfer matrix method [21] or the Monte Carlo method [17]. Hence, a better correlation between the numerical results given by these models and experimental data is expected to be achieved. Moreover, because it is difficult from an experimental point of view to control simultaneously all the parameters (temperature, electric field, particle size, and those that impact on the dynamics…) which govern the electric field induced phase transition process, significant efforts are deployed on this direction by different research groups. Using MC energy minimization method, Misirlioglu et al. have proposed in 2007 an appropriate Ising model [17] in order to simulate the two steps hysteresis loops. Such phenomena were reproduced from a two level Ising like model with a Hamiltonian including two electric dipole interaction terms: a FE interaction and an AFE interaction and the effect of an external electrical field. A Glauber’s formulation was used for the transition probability of spin flip acceptance. To our knowledge few works have been devoted to the study of the influence of the size, shape and temperature on FE to AFE hysteretic behavior, which may be of interest to the nano-material community involved in downsizing of electronic devices.
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Using the above model with a Metropolis formulation for the transition probability of spin flip acceptance we enlarged the study of Misirlioglu et al [17] to include size and shape effects in the study of the dynamics of the two step hysteresis loops transition. The ferroelectric nature of the material considered arises from the competition between order-disorder due to the structural composition in terms of electric dipoles. The dipole can polarize the region in its vicinity forming micro or nano domains which impacts on the dielectric susceptibility related to the polarization of the material. In order to give an in-depth understanding of the physical phenomena involved regarding such behavior in antiferroelectric crystals and the electrical properties of these crystals in terms of polarization we hereby present results from numerical simulations of such systems The model used for the simulations is presented in the following section. Then numerical results regarding size, shape and temperature, parameters which govern the system’s behavior, are given, the last section being devoted to discussions and conclusions. II.
ISING MODEL AND TRANSITION PROBABILITY
Fig 1 a) A random configuration of the dipoles with antiparallel sites. b) A random configuration where the dipoles are connected by two interactions (JFE >0 and JAFE 0 ⎪⎩e
From this model, Monte Carlo Metropolis algorithm is applied to calculate the dynamic properties of phase transition in PZO systems under an external electric field cycling, taking into account size, shape and temperature effects on the two steps FE to AFE hysteresis loops transition.
(2)
where Ei and Ef are the energy of the initial and final configuration respectively and β is 1/ kBT where kB, is the Boltzmann constant and T the temperature of the system. To perform simulations in the 2D lattice which is shown in Figure 1, electrostatic interactions are set such that the FE interaction (JFE>0) leads to the alignment of two neighboring parallel electric dipoles within the same column and the AFE interaction (JAFE
