d’Albuquerque e Castroet al.Reply:

Abstract

d'Albuquerque e Castro et al. Reply: The first point in the Comment [1] regards the scaling relations we have obtained for the magnetic phase diagram (MPD) of cylindrically shaped clusters. The diagram is based on the relative stability of three internal configurations of the clusters, namely, ferromagnetic in plane, ferromagnetic out of plane, and vortex. It consists of regions in the D H plane (D diameter of the basis, and H height of the cylinders), within which one of the three configurations has the lowest total energy E tot . The latter is given as the sum of the dipolar, exchange, and crystalline anisotropy terms. The boundary between any two regions in the MPD, say, ferromagnetic (F) and vortex (V), is determined by the equationThe main result in our Letter is the finding that the MPD for large clusters and full strength of the exchange coupling J can be obtained from those corresponding to smaller systems and a weaker value of J, with all the other parameters kept fixed, just by scaling the D and H axes. In other words, we have found that the two relations E F tot D; H; xJ E V tot D; H; xJ and E F tot D=x ; H=x ; J E V tot D=x ; H=x ; J hold, with 0:55 (within 1% error) and x < 1. Guslienko and Novosad disagree with the value we have obtained for . They think it should be exactly equal to 0.5, although they do not provide any proof for this claim. Instead, they present arguments which they understand would make their claim plausible. Their arguments go as follows.They keep just the exchange and the dipolar interactions and consider two configurations, namely, ferromagnetic (any of the two) and the vortex one. For the first one, they approximate the dipolar term by that of a continuous distribution of magnetic moments, which they call W m . For the vortex configuration, they consider just the exchange energy (relative to that of the ferromagnetic configuration), which they label W ex . By comparing the expressions in the two equations, Guslienko and Novosad concluded that it would be ''natural to express all dot sizes in dimensionless form in units of L ex 2A=M 2 s

Vortex core size in interacting cylindrical nanodot arrays

“…[10,11,12] Recent studies on such structures have been carried out with the aim of determining the stable magnetized state as a function of the geometry of the particles. [3,4,13,14,15,16] In the case of cylindrical particles, three idealized characteristic configurations have been identified: ferromagnetic with the magnetization parallel to the basis of the cylinder, ferromagnetic with the magnetization parallel to the cylinder axis, and a vortex state in which most of the magnetic moments lie parallel to the basis of the cylinder. The occurrence of each of these configurations depends on geometrical factors, such as the linear dimensions and their aspect ratio τ ≡ H/R, with H the height and R the radius of the cylinder.…”

“…The occurrence of each of these configurations depends on geometrical factors, such as the linear dimensions and their aspect ratio τ ≡ H/R, with H the height and R the radius of the cylinder. [3,4,13,14] Another issue to be considered is the interaction between particles. In this case the interparticle distance, D, is the important parameter.…”

Development of vortex state in circular magnetic nanodots: Theory and experiment

“…This procedure decreases the dipolar field exerted on a particle, and therefore the exchange coupling constant is scaled down to keep the correct balance between magnetostatic and exchange energies, responsible for domain formation and reversal mechanisms. With this procedure 15,18 the magnetic properties of a nanoparticle of dimensions d is equivalent to the one of a smaller particle with dimensions dЈ = dx being x Ͻ 1 and Ϸ 0.55-0.57, if the exchange constant is also scaled as JЈ = xJ. This scaling method, in combination with Monte Carlo simulations, 16,18 provides the correct magnetic state of a single nanoparticle.…”

“…To avoid this problem we use a scaling technique developed by d'Albuquerque e Castro et al 15 in which the number of spins is reduced to a value suitable for numerical calculations. This procedure decreases the dipolar field exerted on a particle, and therefore the exchange coupling constant is scaled down to keep the correct balance between magnetostatic and exchange energies, responsible for domain formation and reversal mechanisms.…”

“…Monte Carlo simulations performed using a scaling technique [15][16][17][18] provide the dependence of several properties within the hysteresis loop on the dot diameter and characterize the core diameter when the vortex is centered in the dot at remanence. To compliment the numerical simulations, we evaluate the core diameter analytically.…”