Model of thermally activated magnetization reversal in thin films of amorphous rare-earth – transition-metal alloys A. Lyberatos, J. Earl, and R. W. Chantrell Department of Physics, Keele University, Keele, ST5 5BG, United Kingdom ~Received 31 July 1995! Monte Carlo simulations on a two-dimensional lattice of magnetic dipoles have been performed to investi- gate the magnetic reversal by thermal activation in rare-earth–transition-metal ~RE-TM! alloys. Three mecha- nisms of magnetization reversal were observed: nucleation dominated growth, nucleation followed by the growth of magnetic domains containing no seeds of unreversed magnetization, and nucleation followed by dendritic domain growth by successive branching in the motion of the domain walls. The domain structures are not fractal; however, the fractal dimension of the domain wall was found to be a good measure of the jaggedness of the domain boundary surface during the growth process. The effects of the demagnetizing field on the hysteretic and time-dependent properties of the thin films were studied and some limitations in the application of the Fatuzzo model on magneto-optic media are identified. I. INTRODUCTION The magnetization reversal process in amorphous rare- earth–transition-metal ~RE-TM! alloys 1 is of considerable practical interest in thermomagnetic recording. Thermal ac- tivation is one of the factors that determines the stability of the thermomagnetically grown magnetic domains against collapse or irregularity and is, therefore, of practical rel- evance in defining the signal-to-noise ratio. 2 The magnetization reversal in amorphous magneto-optic media occurs in general by a process of nucleation followed by domain growth. The observation of a finite domain-wall coercivity can be interpreted by postulating the existence of nanoscale structural and magnetic inhomogeneities that act as pinning centers. 3 It is possible to study the origin of the coercivity by computer simulations that employ the dynamic Landau-Lifshitz-Gilbert equation of motion for the magneti- zation in the absence of thermal fluctuations. 3 Such studies have demonstrated that it is important to distinguish the nucleation coercivity H n , which is determined primarily by fluctuations in the perpendicular anisotropy constant K u , from the wall-motion coercivity H w that depends on fluctua- tions in the exchange stiffness constant and the dispersion in easy axes. 3 These findings are consistent with the experimen- tal observation that the nucleation coercivity can be substan- tially larger than the wall-motion coercivity in RE-TM al- loys, resulting in rectangular hysteresis loops. 4 When the external field is close to but less than the nucle- ation coercivity H n , magnetization reversal is still possible by thermal activation over the local energy barriers. Thermo- activated reversal has been observed in a variety of amor- phous RE-TM alloys such as Tb-Fe, 5–10 Tb-Fe-Co, 11–13 Gd-Fe, 14–16 Gd-Tb-Fe, 15,16 and Tb~Co!-based alloys. 16 There is a substantial body of evidence for the presence of a ther- mal magnetic aftereffect: ~1! the observation that the domain-wall motion is not a continuous process but consists of discrete Barkhausen jumps of small sections of the wall. 8 ~2! the slow decay of the magnetization under constant ex- ternal field conditions, 5 and ~3! the exponential dependence of the domain-wall velocity on the external applied field. 6,14,15 The precise form of the time dependence of the magneti- zation M ( t ) in a constant field depends on the relative bal- ance of the nucleation and wall-motion processes during magnetic reversal. A theory by Fatuzzo 17 assumes that the nucleated domains are circular with initial radii r c that grow at a constant velocity v , while the rate of nucleation R re- mains constant. The theory predicts that the shape of the time-dependence curves M ( t ) depends on a single parameter k5v / Rr c . From the shape of the experimental curves M ( t ), approximate values of k can be determined. 16 Fatuzzo’s theory, however, does not account for the presence of a dis- persion in energy barriers that may arise, for example, from local fluctuations of the exchange stiffness constant and the uniaxial anisotropy K u or alternatively from the spatial and temporal variations of the demagnetizing field. A dispersion in energy barriers is necessary to account for the common observation that the nucleated domains are frequently irregu- lar in shape 18,19 and their subsequent growth may be den- dritic, forming a fractal structure. 6,8,10 The stability of the domain shape of cylindrical domains in homogeneous thin films with perpendicular anisotropy but low domain-wall coercivity depends on the wall stiffness, 20 which is determined by the competition between the domain wall and demagnetizing energy. 21 When the demagnetizing energy is relatively large, the stiffness of the domain wall is low and the domain shape is more susceptible to local defor- mation by pinning sites. Dendritic growth occurs when the stiffness of the wall is sufficiently low so that the growth of the reverse magnetization is dominated by the geometry of the pinning sites. 10 The computation 10 or measurement 18 of the fractal dimension of the boundary surface of the mag- netic domains have been suggested as useful methods of characterization of the jaggedness of the domain boundary. The simulation of thermoactivated magnetic reversal un- der constant external field conditions can best be carried out using a Monte Carlo algorithm 22 that simulates accurately the kinetic process as in Ref. 23 Kirby et al. 24 applied the algorithm on thin films for magneto-optic recording and ob- PHYSICAL REVIEW B 1 MARCH 1996-I VOLUME 53, NUMBER 9 53 0163-1829/96/53~9!/5493~12!/$10.00 5493 © 1996 The American Physical Society