Micromagnetic study of magnetic domain structure and magnetization reversal in amorphous wires with circular anisotropy I. Betancourt a,n , G. Hrkac b , T. Schrefl b,c a Departamento de Materiales Meta ´licos y Cera ´micos, Instituto de Investigaciones en Materiales, Universidad Nacional Auto ´noma de Me´xico, Me´xico D.F. 04510, Mexico b Department of Engineering Materials, University of Sheffield, Mappin St., Sheffield S1 3JD, UK c St. P¨ olten University of Applied Sciences, Austria article info Article history: Received 10 August 2010 Received in revised form 2 November 2010 Available online 10 December 2010 Keywords: Magnetic domains Amorphous wires Micromagnetism Circular anisotropy Magnetization reversal of amorphous wires abstract In this work we present a detailed numerical investigation on the magnetic domain formation and magnetization reversal mechanism in sub-millimeter amorphous wires with negative magnetostriction by means of micromagnetic calculations. The formation of circular magnetic domains surrounding a multidomain axially oriented central nucleus was observed for the micromagnetic model representing the amorphous wire. The magnetization reversal explained by micromagnetic computations for the M–H curve is described in terms of a combined nucleation–propagation  rotational mechanism after the saturated state. Results are interpreted in terms of the effective magnetic anisotropy. & 2010 Elsevier B.V. All rights reserved. 1. Introduction Iron- and cobalt-based amorphous alloys in the form of thin metallic wires with typical diameters between 70 and 150 mm obtained by rapid solidification from the melt by means of the rotating-water-bath-melt-spinning process have been the subject of active research since the beginning of the 1990s due to their peculiar magnetic behavior, including (a) magnetic bistability, i.e. a very well defined rectangular M–H loop afforded by a single Barkhausen jump, and (b) the Mateucci effect, consisting of the generation of a voltage across the ends of a twisted wire when it is placed in an alternating magnetic field [1–4]. Both responses are intimately related with the distinctive, experimentally observed magnetic domain structure that characterizes this kind of materials: a central core with a longitudinal single domain structure surrounded by a sheath of perpendicular magnetization with either radial or circular orientation depending on the wire’s saturation magnetostriction l s value, which contributes to the domain formation by means of the magnetoelastic coupling (l s 40 for radial direction and l s o0 for circular orientation, whereas vanishing l s destroys the single inner domain structure because of the facility to form new domain walls) [1–5]. These magnetic features have afforded the implementation of a variety of technological applications for these amorphous wires as active elements in magnetic sensors and/or transducer devices [1–4]. In this work we present a detailed numerical investigation on the magnetic domain formation and reversal mechanism in amorphous wires with l s o0 by means of the continuum theory of micromagnetism. 2. Numerical method Micromagnetic calculations are based on the dynamic magne- tization description given by the Landau–Lifshitz Gilbert (LLG) equation of motion: @M @t ¼guM  H eff  agu M s MðM  H eff Þ ð1Þ where a is the dimensionless damping constant, g is the electron gyromagnetic ratio and g 0 ¼ g/(1+a 2 ). The effective field H eff comprises the following terms: the exchange field H ex ¼ð2A=M s Þr 2 m (where A is the exchange constant, M s is the saturation magnetization and m is the unit vector along M s ), which keeps neighboring magnetic moments parallel to each other; the anisotropy field H K ¼ ð1=M s Þð@E anis =@mÞ (with E anis the anisotropy energy density), which aligns magnetic moments along specific directions; the magnetostatic field H d ¼ ð1=2Þrf (where f is the scalar potential), which arises from the magnetization distribution itself and the externally applied field H ext [6]. In order to use Eq. (1) for calculating the equilibrium magnetization in a cylindrical model representing an amorphous wire, the model was discretized into tetrahedral finite elements according to the Ritz–Galerkin weak formulation [6,7]. For each node of the finite element mesh, a magnetic moment vector and a magnetic scalar potential is defined. The magnetic scalar potential follows from the magnetostatic boundary value problem. Instead of extending the finite element mesh over a larger region outside the magnet, the boundary Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/jmmm Journal of Magnetism and Magnetic Materials 0304-8853/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2010.11.089 n Corresponding author. Fax: + 52 55 56161371. E-mail address: israelb@correo.unam.mx (I. Betancourt). Journal of Magnetism and Magnetic Materials 323 (2011) 1134–1139