3D and 1D calculation of hysteresis loops and energy products for anisotropic nanocomposite films with perpendicular anisotropy X.H. Yuan a , G.P. Zhao a,n , Ming Yue b , L.N. Ye c , J. Xia a , X.C. Zhang a , J. Chang a a College of Physics and Electronic Engineering & Institute of Physics, Sichuan Normal University, Chengdu 610066, China b College of Materials Science and Engineering, Beijing University of Technology, Beijing 100124, China c School of Economic Information Engineering, Southwestern University of Finance and Economics, Chengdu 610074, China article info Article history: Received 23 January 2013 Received in revised form 8 May 2013 Available online 18 May 2013 Keywords: Hard/soft bilayer Hysteresis loop Micromagnetic calculation Magnetic vortex state abstract In this paper, the magnetic reversal process, hysteresis loops and energy products for exchange-coupled Nd 2 Fe 14 B/α-Fe bilayers are studied systematically by a three-dimensional (3D) model. The 3D calcula- tions are numerically solved using the finite difference method, where the results are carefully compared with those calculated by one-dimensional (1D) model. It is found that the calculated hysteresis loops and energy products based on the two methods are consistent with each other. Both nucleation fields and coercivities decrease monotonically as the soft layer thickness L s increases. In addition, the calculated spatial distributions of magnetization orientations in the thickness direction at various applied fields based on both methods signify a three-step magnetic reversal process, which are nucleation, growth and displacement of the domain wall. The calculated magnetic orientations within the film plane, however, are totally different according to the two methods. The 3D calculation exhibits a process of vortex formation and annihilation. On the other hand, the 1D calculation gives a quasi-coherent one, where magnetization orientation is coherent in the film plane and varies in the thickness direction. This new reversal mechanism displayed in the film plane has a systematic influence on the nucleation fields, coercivity and energy products. & 2013 Elsevier B.V. All rights reserved. 1. Introduction Exchange spring materials, proposed by Kneller and Hawing [1] in 1991, have aroused much interest in the past two decades because of their expected giant energy product. These materials have large coercivity provided by the hard phase and large remanence provided by the soft phase. The hard and soft phases are exchange-coupled in nanoscale so that the composite materi- als exhibit a spring behavior [1–12]. In 1993, Skomski and Coey predicted that the giant energy product can exceed 1 MJ/m 3 [13], which, however, has never been achieved experimentally. For many years, the best result obtained in a laboratory was only around 0.2 MJ/m 3 [14,15], which is much smaller than that of the corresponding single phased hard material. Recently, the NEOMAX Company of Japan achieved an energy product more than 470 kJ/ m 3 for sintered Nd 2 Fe 14 B in the laboratory. Such an outstanding discrepancy between experiment and theory is called energy product paradox in some of the literature. Major progress has been made in the last year, where a large energy product of 486 kJ/m 3 has been achieved by the Hono group [16], which is larger than the best value for the single- phased permanent magnets. Interestingly, this super-energy pro- duct was realized in anisotropic Nd 2 Fe 14 B/FeCo multilayers with perpendicular anisotropy [16], rather than with in-plane aniso- tropy, as predicted by the micromagnetic theory [8,13,17,18,19]. Therefore, present theories regarding hysteresis loops and coer- civity mechanisms of the composite magnets, which are mainly based on a one-dimension (1D) micromagnetic method, need to be re-examined and improved,. The 1D micromagnetic approach was first utilized by Goto et al. [20] in the 1960s to deduce analytically the instability field (nucleation field) and the shape of the magnetization curve under the assumption that the hard layer is perfectly rigid and the soft layer has no anisotropy. Skomski and Coey calculated more accurately the nucleation field and predicted the famous giant energy product based on the anisotropic hard/soft multilayers with in-plane anisotropy [13]. Later, Leineweber and Kronmüller [17], Fullerton et al. [10,11], and Zhao et al. [8,18,21,22] investi- gated the demagnetization process, and hysteresis loops as well as the energy products of hard/soft multilayers, with both in-plane and perpendicular anisotropy. Zhao et al. found that as the soft layer thickness increases, the coercivity mechanism changes from nucleation to pinning [8,21] provided that the soft layer is thick enough. Similar results have been obtained by Asti et al. [23,24], Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/jmmm Journal of Magnetism and Magnetic Materials 0304-8853/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmmm.2013.05.012 n Corresponding author. Tel: +86 15228949580. E-mail addresses: zhaogp@uestc.edu.cn, zapple2004@yahoo.com.(G.P. Zhao) Journal of Magnetism and Magnetic Materials 343 (2013) 245–250