3430 IEEE TRANSACTIONS ON MAGNETICS, VOL. 36, NO. 5, SEPTEMBER 2000 Influence of Particle Size Distribution on Magnetic Properties of Nanocrystalline Soft Magnetic Fe Zr Cu B F. C. S. da Silva, M. Knobel, D. Ugarte, and D. Zanchet Abstract—The effect of particle size distribution on the magnetic properties of nanocrystalline Fe Zr Cu B samples is studied in detail. The analysis applies a realistic particle size distribution, ob- tained from transmission electron microscopy images, to the re- cently developed extended Random Anisotropy Model. The effect of a broad size distribution is found to be more relevant when the tail of the distribution is larger than 20 nm. Index Terms—Nanocrystalline materials, soft magnetic mate- rials, random anisotropy model. I. INTRODUCTION F e-BASED nanocrystalline materials, obtained through devitrification of amorphous precursors, display excellent soft magnetic properties, such as low coercivities and high initial permeabilities, associated with high saturation magne- tization values. Two examples are the Fe Si B Nb Cu alloy, also known as FINEMET [1], and the Fe Zr Cu B alloy [2]. Their outstanding magnetic properties are related to the ultrafine structure of bcc Fe or Fe–Si particles embedded in the residual amorphous matrix. The role played by these magnetic particles was first dis- cussed by Herzer using the Random Anisotropy Model (RAM) [3]. Typically, the correlation length of the exchange interac- tions between Fe atoms inside a Fe–Si crystal is roughly nm, where J/m is the exchange stifness constant, and J/m is the magnetocrys- talline anisotropy constant of Fe. is about 4 times greater than the mean particle size in these systems ( nm). This means that a set of randomly oriented Fe–Si particles, within the exchange volume ( ) will behave as a single particle with a reduced effective anisotropy constant . Here, is the crystalline fraction. This reduction increases the the effective exchange correlation length: , and also the number of particles within the effective exchange volume: . The expressions for , , and can be solved together to give: (1) Manuscript received February 15, 2000; revised May 15, 2000. This work was supported by the Brazilian financial agencies: FAPESP, andCNPq. F. C. S. da Silva and M. Knobel are with the Instituto de Física “Gleb Wataghin,” Universidade Estadual de Campinas (UNICAMP), CP: 6165, 13083-970, Campinas-SP, Brazil (e-mail: fcss@ifi.unicamp.brand). D. Ugarte and D. Zanchet are with the Laboratório Nacional de Luz Sín- crotron (LNLS), CP: 6192, 13.083-970 Campinas-SP, Brazil. Publisher Item Identifier S 0018-9464(00)08866-X. Macroscopic magnetic quantities such as coercive field , and initial permeability are closely related to by [4]: (2) where is the average saturation magnetization of the mate- rial, and and are dimensionless pre-factors close to unity. The dependence of on the sixth power of for confirms that the nanocrystalline structure is responsible for the strong reduction of the macroscopic coercivity observed in nanocrystalline Fe-based samples. Another expression can be deduced for the case where giving a dependence of on . In a recent paper, Herzer further developed an extension of the RAM for the case where a particle size distribu- tion can be considered [5]. Following the procedure discussed above, (1) can be generalized: (3) where the index accounts for each representative particle in the system. The amorphous matrix also plays an important role in nanocrystalline magnetic systems because it mediates the exchange interaction between the grains. In fact, typical effects observed in fine particle systems, like superparamagnetism and blocking, have been reported in nanocrystalline systems heated above the Curie temperature of the amorphous phase ( ) [6]. This two phase character of the nanocrystalline magnetic systems was firstly discussed by Hernando and Kulik [7]. They considered an effective exchange stifness constant which depends on both amorphous (through ) and on crystalline (through ) phases. In this paper we analyze the effect of particle size distribution on the magnetic properties of nanocrystalline Fe Zr Cu B samples. The analysis applies a realistic particle size distribu- tion, obtained from transmission electron microscopy images, to (3). The result is compared with temperature dependent measurements of coercive field, remanence, and saturation magnetization. II. EXPERIMENT Amorphous melt-spun Fe Zr Cu B ribbons were pro- duced by melt-spinning. Nanocrystalline samples were 0018–9464/00$10.00 © 2000 IEEE