Journal of Magnetism and Magnetic Materials 203 (1999) 268}270 Monte Carlo simulation of hysteresis loops of single-domain particles with cubic anisotropy and their temperature dependence J. GarcmHa-Otero *, M. Porto , J. Rivas , A. Bunde Facultade de Fn&sicas, Depertmento Fn&sica Aplicada, Universidade de Santiago de Compostela, Campus Sur s/n, E-15706 Santiago de Compostela, Spain Institut fu(r Theoretische Physik III, Justus-Liebig-Universita(t Giessen, Heinrich-Buw-Ring 16, D-35392 Giessen, Germany Abstract By means of Monte Carlo simulation the hysteresis of non-intereacting single-domain magnetic particles presenting cubic crystalline anisotropy are studied. Both signs of the anisotropy constant are considered, and relevant properties, such as remanence and coercivity, are obtained as a function of temperature. 1999 Elsevier Science B.V. All rights reserved. Keywords: Monte Carlo simulation; Hysteresis loop; Anisotropy; Remanence; Coercivity 1. Introduction part of the hysteresis loop has been studied [5] but it is not possible to follow the same process of Stoner and The hysteresis loop of a non-interacting system of Wohlfarth for the whole cycle due to the fact that the randomly aligned single-domain particles presenting magnetization can jump to any of the several local uniaxial anisotropy has been known since the publica- minima [5]. tion of the classical paper by Stoner and Wohlfarth [1]. Recently, Usov and Peschany [6] presented the "rst They obtained the values H "0.479H for the coercive advance for the complete hysteresis loop of cubic aniso- "eld and M"0.5M for the remanence, where M is the tropy particles using a dynamical approach. They de- saturation magnetization and H "sK/M is the anisot- scribed the following upper and lower limits for the ropy "eld. They assumed a coherent magnetization rever- coercitivity: 0.320H (H (0.335H when K'0 and sal model, which is a good approach for a restricted 0.180H (H (0.200H when K(0. Essentially fol- range of particle sizes, such as the micromagnetic model lowing the method of Stoner and Wohlfarth, the problem [2,3] predicts. of the indetermination of the discontinuous jumps was Particles presenting cubic crystalline anisotropy, such solved by a dynamical model of evolution, in which there as iron or nickel, are also very important in the experi- are di!erent probabilities to each adjacent energy min- mental magnetism however, until very recently, their imum. In this work we have chosen a di!erent way. theoretical hysteresis loop was unknown. It has been Using the Monte Carlo (MC) simulation technique (see possible to obtain the value of the remanence analytically e.g. Ref. [7] ), we study in detail the hysteresis loops of [4] (M"0.831M for the case K'0 and M" particles presenting cubic cristalline anisotropy (assum- 0.866 M for the case K(0), and also the reversible ing also homogeneous rotation of magnetization). An extended approach using the micromagnetic model dir- ectly should give valid results for a wider range of par- ticles sizes, but it would result in a huge computational * Corresponding author. Tel.: #34-98156-3100-13011; fax: e!ort. The MC approach has the advantage that exten- #34-981-52-0676. sions to nonzero temperature are straightforward. E-mail address: uscfajgo@cesga.es (J. GarcmHa-Otero) In addition, the method can be used for any kind of 0304-8853/99/$- see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 9 ) 0 0 2 6 8 - 1 J. Garcn&a-Otero et al. / Journal of Magnetism and Magnetic Materials 203 (1999) 268}270 269 distribution of particle orientations, and it allows the independently of the temperature. To perform the com- study of interparticle interactions. plete hysteresis loop a very high "eld is applied initially at very high temperature. Then the system is carefully ther- malized to the desired temperature, and the loop is 2. Model, numerical simulation, and results started by slowly varying the reduced applied "eld in steps of 0.05 (0.02 if h3[!0.5, 0.5] for better accuracy) Usually the cubic anisotropy energy is written as every 2000 MC steps. The process is repeated for a large E "K<(  #  #  )#K<   , number of independent con"gurations to perform an where ( , , ) are the direction cosines of the magnetiz- ensemble average. ation with respect to the crystal directions and K The resulting reduced hysteresis loops for both K  and '0 K and K  are the anisotropy constants, whose values are taken (0 at ¹"0 are shown in Fig. 1. The loop for from the experiments and are usually sensitive functions uniaxial particles is also represented for comparison. For of temperature. Since no interaction between particles is K'0, the obtained reduced remanence M"M/M is considered, each particle is completely de"ned by a set of m"0.831$0.004, which is in perfect agreement with six angles ( the theoretical value [4]. The reduced coercive "eld is , ), ( , ), and ( , ). The "rst four angles de"ning the orientation of the easy axes are kept constant h "0.316$0.002, which is within the lower limit of the during the simulation of a single con"guration. The two range given by Usov [6]. The reduced coercivity is lower remaining angles which de"ne the magnetization direc- for K(0, than in the case with the positive constant, tion will be variable throughout the MC simulation. the reduced remanence on the contrary is slightly Taking into account both anisotropy and the interac- larger. The obtained value m"0.865$0.004 is again in tion with an external "eld in z direction, the total energy good agreement with the exact result [4], and h " (with "1! !  and divided by 2"K 0.183$0.002 is within the limits given by Usov [6]. "< in order to compare the results with the Stoner}Wohlfarth model) The evolution of the reduced hysteresis loops and reads as coercitivity with temperature for K'0 and K(0 are shown in Figs. 2}4 respectively. As in the actual experi- 1 e ments the time of measurement plays an important role. "$2 # ! ! !   More time between changing "eld and measurement K means more time to relax. All the loops shown in the #   (1! ! ) present study are carried out with 2000 MC steps be- K !hcos ,  tween measurements, this sets the blocking temperature where h"H/H and H "2/K/M are as de"ned for ¹ when K'0 around k ¹ /(2K<)"0.02 in reduced uniaxial particles. The plus sign corresponds to the case units. A comparison of simulations done with a di!erent when K is positive, the minus sign to the case when K is negative. Furtheron, we restrict ourselves to the K"0 case only. The MC algorithm is as follows: In every MC step an attempted orientation  of the magnetization is gener- ated. The attempted direction is chosen in a spherical segment around the present orientation , which is used as azimuthal axis, with M3[0, 2 ] and M3[0, ]. Thus the energy di!erence e between the attempted and the current state is calculated. If e)0 the magnetization is changed to . If e'0 the magnetization is changed with probability exp(! e/t) and remains unchanged with probability 1!exp(! e/t). Here, t"k ¹/ (2"K"<) is the reduced temperature. In any case the variable counting the MC steps is increased and the process is continued. Varying the aperture angle , i.e. the maximal jump angle, it is possible to modify the range of acceptance in order to optimize the simulation. Using this kind of local dynamics allows us to detect the con"nement in metastable states responsible for the hys- teresis. Choosing a non-local algorithm and drawing the Fig. 1. Reduced hysteresis loops of non-interacting randomly attempted direction independently to the current one, the aligned single-domain particles. In the "gure the cases for uni- system would always be superparamagnetic since it axial anisotropy (Stoner}Wohlfarth model) and for cubic anisot- would be possible to explore the whole phase space ropy with both signs of the anisotropy constant are shown. 270 J. Garcn&a-Otero et al. / Journal of Magnetism and Magnetic Materials 203 (1999) 268}270 Fig. 2. E!ect of temperature on the hysteresis loops of non- Fig. 4. Temperature dependence of the reduced coercivity for interacting randomly aligned single-domain particles with cubic both signs of the "rst constant of cubic anisotropy. anisotropy, case K'0. At high temperatures the loops become superparamagnetic. Thus, taking this additional dependence into account, obtaining the measured parameters for a given material is straightforward. The simple model studied does not take into account interparticle interactions, but can be of interest in the case of very diluted systems of single-domain particles. The in#uence of dipolar interactions will be our next objective. Acknowledgements J. GarcmHa-Otero wishes to thank the autonomical government of Galicia (Xunta de Galicia). M. Porto acknowledges support from the Deutsche Forschun- gsgemeinschaft. References Fig. 3. E!ect of temperature on the hysteresis loops of non- [1] E.C. Stoner, E.P. Wohlfarth, Philos. Trans. Roy. Soc. A 240 interacting randomly aligned single-domain particles with cubic (1948) 599; reprinted by IEEE Trans. Mag. 27 (4) (1991) anisotropy, case K 3475. (0. At high temperatures the loops become superparamagnetic. [2] H. KronmuKller, in: G.J. Long, F. Grandjean (Eds.), Super- paramagnets, Hard Magnetic Materials, Kluwer Academic number of MC steps between measurements can be car- Publishers, Amsterdam, 1991, pp. 461}498. ried out by rescaling the temperature with the blocking [3] A. Aharoni, Introduction to the Theory of Ferromagnetism, temperature. In whichever case, the functional depend- Clarendon Press, Oxford, 1996. ence of the magnetic parameters will be the same, as will [4] R. Gans, Ann. Phys. 15 (1932) 28. be the shape of the loops. [5] E.W. Lee, J.E.L. Bishop, Proc. Phys. Soc. 89 (1966) 661. [6] N.A. Usov, S.E. Peschany, J. Magn. Magn. Mater. 174 Finally, it should be kept in mind that the parameters (1997) 247. which enter in the de"nition of the reduced magnitudes, [7] K. Binder, D.W. Heermann, Monte Carlo Simulations in e.g., M and especially K, depend strongly on temper- Statistical Physics, Springer Series inSolid State Science, ature, and consequently so does the anisotropy "eld H . Vol. 80, second ed., Springer, Berlin, 1992.