Modeling Polarization Switching in Thin Ferroelectric Films
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ling Polarization Switching in Thin Ferroelectric Films O. S. Baruzdinaa, b, *, O. G. Maksimovaa, A. V. Maksimova, and V. I. Egorova aCherepovets
State University, Cherepovets, Vologda oblast, 162600 Russia Voronezh State University, Voronezh, 394018 Russia *e-mail: [email protected]
b
Received April 10, 2020; revised April 29, 2020; accepted May 27, 2020
Abstract—A modified three-dimensional Ising model that considers the contribution from the depolarization field is used to study polarization switching in thin ferroelectric films. The effect the film thickness and the depolarization field have on the strength of the coercive field and the shape of dielectric hysteresis curve is studied. DOI: 10.3103/S1062873820090075
INTRODUCTION Polarization switching perpendicular to the plane of the film is used in a number of thin-film ferroelectric devices. This allows thin films to control polarization at very small voltages [1]. Particular attention is given to films in which the size effect is observed [2]. The main lines of theoretical and experimental studies of the dependence of polarization on external electric field P(E) in ferroelectric thin films focus on studying the breaking of dynamic symmetry [3, 4], changes in the area of the hysteresis loop upon a change in film thickness [5], and the existence of internal displacement fields [6]. Experimental studies show that the phase transition point upon a reduction in film thickness can shift toward both high [7] and low temperatures [8]. The authors of [8] showed there is no spontaneous polarization in sufficiently thin films (i.e., there is a critical thickness value). The use of stationary or free boundary conditions in the Ising model for modeling allows us to confirm the shift of the Curie temperature to high or low temperatures, respectively. However, no critical thickness is detected when modeling the polarization of ferroelectric thin films without allowing for the depolarization field. This problem is solved by including the depolarization field in the model, as it has an appreciable effect only for thin films. Polarization is in this case reduced as the film thins under free boundary conditions, and it falls sharply to zero at a certain critical value [9]. Including the depolarization field lowers the Curie temperature. The aim of this work was a theoretical study of the effect of film thickness on the polarization switching of thin ferroelectric films.
EXPERIMENTAL To study the polarization switching of thin ferroelectric films and describe their properties, we used the three-dimensional Ising model in [10, 11], which includes nodes N1, N2, and N3 along the corresponding axes of the Cartesian coordinate system. It was assumed that N3 unit cells are in the cross section of a film with thickness L. The position of the lattice site is characterized by a set of three numbers: n = (n1, n2, n3 ) (Figs. 1a and 1b). In this work, the potential energy of one dipole at node n is described by the formula [9]
U n = −J
S S m
n m
− pESn − pSn Ed ,
(1)
wher
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