A Two-Dimensional Free Energy Model for Single Crystalline Ferroelectrics
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CC1.7.1
A Two-Dimensional Free Energy Model for Single Crystalline Ferroelectrics Sang-Joo Kim1 , Stefan Seelecke2 ; Brian L. Ball3 , Ralph C. Smith3 and Chang-Hoan Lee4 1 Dept. Mech. & Info. Eng. Univ. of Seoul, Seoul, Korea 2 Dept. Mech. & Aero. Eng. Center for Research in Scienti c Computation, North Carolina State Univ., Raleigh, NC 27695 3 Center for Research in Scienti c Computation, North Carolina State Univ., Raleigh, NC 27695 4 Korea Institute of Science & Technology Information Seoul, Korea ABSTRACT The one-dimensional free energy model for ferroelectric materials developed in [1-3] is generalized to two dimensions. The proposed two-dimensional energy potential consists of four energy wells corresponding to four variants of the material, four saddle points representing the barriers for 900 switching processes, and a local energy maximum across which 1800 -switching processes take place. The free energy potential is combined with the evolution equations based on the theory of thermally activated processes. The prediction of the model is compared with the recent measurements on a BaTiO3 single crystalline ferroelectric in [4]. The responses of the model at various loading frequencies are calculated and the kinetics of 900 and 1800 switching processes are discussed. INTRODUCTION In order to simulate the transient dynamics behavior and the closure of biased minor loops in the hysteresis response of ferroelectric materials, ferroelectric switching has been regarded as a thermally activated process, and a one-dimensional model has been developed based on kinetic switching equations [1-3]. The one-dimensional Helmholtz free energy potential consists of two convex energy wells and a concave energy barrier (spinodal region) between the wells. However, the model features only two energy wells, and as a result it can not model 90 switching processes. This has been improved by Sahota [5], who extended the pure polarization model to include a 1-D strain component and a corresponding 90 -variant, which allows to account for the effect of mechanical stress in one direction. In the present paper, we introduce a 2-D polarization model and discuss the additional complexity introduced by the multi-dimensional energy landscape. An extended set of evolution equations accounting for the increased number of variants is presented, which is subsequently solved numerically for several cases of electric loading. The model is compared to polarization hysteresis loops recently observed by Burcsu et al. [4] for BaTiO3 single crystals, and predictions are made for various loading rates.
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FREE ENERGY FUNCTIONS In the two-dimensional energy model, the Helmholtz free energy function per unit reference volume is given by .P1 ; P2 / D 1 .P1 /
D
2 .P2 / D
8 < : 8 < :
1 .P1 / C
1 2 1 2 1 2
.P1 C PR /2 .PI PR /.P12 =PI .P1 PR /2
1 2 1 2 1 2
.P2 C PR /2 .PI PR /.P22 =PI .P2 PR /2
.P2 / C a P12 P22 ;
(1)
PR /
if if if
P1 PI ; PI P1 PI P1 ;
PI ;
(2)
PR /
if if if
P2 PI ; PI P2 PI P2 ;
PI ;
(3)
where 1 and 2 are the
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