Physics Letters A 373 (2009) 4282–4286 Contents lists available at ScienceDirect Physics Letters A www.elsevier.com/locate/pla Simulation of hysteresis loops for polycrystalline ferroelectrics by an extensive Landau-type model Ying-Long Wang a ,∗ , Xing-Yuan Wang a , Li-Zhi Chu a , Ze-Chao Deng a , Wei-Hua Liang a , Bao-Ting Liu a , Guang-Sheng Fu a , N. Wongdamnern b , T. Sareein b , R. Yimnirun b a College of Physics Science and Technology, Hebei University, Baoding 071002, China b Department of Physics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand a r t i c l e i n f o a b s t r a c t Article history: Received 1 September 2009 Accepted 20 September 2009 Available online 3 October 2009 Communicated by R. Wu PACS: 77.84.-s 77.80.-e 77.80.Dj Keywords: Ferroelectric materials Domain structure Hysteresis Ferroelectricity Landau-type phenomenological theory is extended to the polycrystalline ferroelectricsby defining a volume fraction of polarization to simulate polarization hysteresis loops. Using this simple model, polarization hysteresis loops ofa model Pb(Zr 0.52 Ti 0.48 )O 3 ceramic at different fields or temperatures are simulated,it is found that our results are in very good agreement with the experimentaldata for the saturated as wellas the unsaturated loops. Moreover,this model can accurately predict the coercive fields and remnant polarizations under arbitrary applied fields.  2009 Elsevier B.V. All rights reserved. 1. Introduction There is a significant growing interest in ferroelectric materials characterizing bistable property [1–8] for various potential applica- tions especially for ferroelectric nonvolatile random access memo- ries [1–5]. It is of wide interest and very important to investigate the dynamic hysteresis characteristics of ferroelectrics under vari- ous conditions, such as temperatures and electric fields although the ferroelectric hysteresis loops,including the saturated as well as the unsaturated loops, can be observed experimentally using a standardized ferroelectric tester[9–11]. Very recently, we ob- tained the power-law temperature scaling relations based on the investigation of polarization hysteresis loops (i.e. P –E loops) of a Pb(Zr 0.52 Ti 0.48 )O 3 (PZT) ceramic for various electric fields at a fixed temperature,or temperatures ata fixed field [12]. What is the dynamic mechanism during switching a ferroelectric material? It is believed very difficult to model the hysteresis loop due to its nonlinear and history-dependent electric field effects although a number of models [13–17] have been developed to theoreti- cally simulate the hysteretic loops. The current models use either * Corresponding author. Tel./fax: +86 0312 5977068. E-mail address: hdwangyl@hbu.edu.cn (Y.-L. Wang). oversimplified expressions that are insufficient for the unsaturated loops or numerical algorithms that are too complicated to be used for circuit simulation. Although Landau free energy modelis used to describe many ferroelectric behaviors [18–20], this model cannot perfectly depict the P –E loops of a polycrystalline ferroelectric material[21,22]. In fact, for the homogeneous single-domain model, it is impossi- ble to perfectly depict the domain-switching part in a P –E loop since the system is in a multi-domain state during polarization re- versal [23,24],thus polarization reversallines obtained according to single-domain idea had to be added artificially at the coercive fields to form the close loops [21,25]. For the multi-domain model [19,22,26–35], the previous discussions were mainly focus on some physical properties,e.g.,the phase diagram and the domain wall motion, in a single-crystalline bulk or epitaxial film. Compared with those of single-crystalferroelectrics,the simulation of P –E loops of polycrystalline ferroelectric materials are more difficult since factors (e.g.,grain orientation, grain size, grain boundary), can impact the polarization distributions. In order to solve this problem related to a multi-domain switching process, we propose a model in this Letter, which can accurately simulate the process of polarization reversal, to describe the P –E loops of a polycrystalline system based on the modification of Landau-type phenomenologi- cal theory. 0375-9601/$ – see front matter  2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.09.050