PHYSICAL REVIEW B VOLUME 55, NUMBER 18 1 MAY 1997-II Thin ferromagnetic films with competing surfaces: A Monte Carlo study of the classical Heisenberg model Hyunbum Jang and Malcolm J. Grimson Department of Physics, University of Auckland, Auckland, New Zealand Received 3 December 1996 Monte Carlo simulations have been performed for different values of perpendicular anisotropy in a thin ferromagnetic Heisenberg film. In the model competing surface fields with the same magnitude but opposite direction have been used. In the Heisenberg limit, 0, no spontaneous magnetization of the film is observed. Whereas, in the Ising limit, , nonzero magnetization of the film is observed below a critical temperature Tc and a degeneracy in the magnetization profiles exists between states of positive and negative total magne- tization at low temperatures. The results of magnetic relaxation studies indicate that the magnetization decays exponentially with a relaxation time that increases with and decreases with temperature. S0163-1829 97 13217-9 I. INTRODUCTION particular surface depends on magnetization fluctuations in the bulk region and, most especially, the initial spin configu- Phase transitions in thin ferromagnetic films have been ration. The low-temperature magnetization profiles of the investigated experimentally1,2 and theoretically3­9 due to film Mn show a degeneracy between states of negative and their importance for applications in magnetic-recording me- positive total magnetization. This phase transition in the thin dia. The order parameter for the ferromagnetic-paramagnetic ferromagnetic Ising film is only observed for applying sur- phase transition is the spontaneous magnetization vector M, face fields which have the same magnitude but opposite di- rection. which is zero for temperatures above a critical temperature Thus, while Binder and co-workers7­9 have shown the Tc in zero external field.10 However the inclusion of appro- significance of surface effects on the phase behavior of thin priate anisotropies and interactions in the energy can signifi- Ising spin system, it is not clear how general this result is for cantly modify the phase behavior. For a perpendicular an- all ferromagnetic systems, since the Ising model of magne- isotropy with long-range dipole-dipole interactions, Moschel tism uses a highly anisotropic spin-spin interaction. Ising and Usadel3,4 have shown that the direction of M relative to spins do not rotate through all possible orientations, but in- the surface varies with increasing temperature from perpen- stead are restricted to a particular axis, conventionally the z dicular to in-plane in a reorientational transition of a ferro- direction. Below the critical temperature, the spins tend to magnetic Heisenberg ultrathin film. They note that the per- preferably align in the z direction and give rise to a finite pendicular anisotropy favors the spins being directed value of M even in the absence of an external field.11 How- perpendicular to the surface whereas the long-range dipole- ever, in contrast, the classical Heisenberg model of magne- tization, the magnetic spins are very sensitive to the tempera- dipole interactions tend to align the spins in the in-plane ture and only order at zero temperature in the absence of an direction. Moreover they showed that the spin cantings are external field. According to the investigation of Taylor and not only affected by temperature but also by the model an- Gyorffy,12 without any perpendicular anisotropy, there is no isotropy parameters. This indicates that the direction of M in magnetic order at any finite temperature. The ferromagnetic the ferromagnetic Heisenberg thin film is very sensitive to order is destroyed by long-wavelength spin waves.13,14 How- the anisotropy properties. ever different anisotropy constraints acting on spins at the For ferromagnetic Ising thin films, both finite-size and surface and in the bulk can change the behavior of Heisen- surface effects can produce phase transitions as a function of berg spins toward that of Ising-like spins. temperature. Recent simulations7­9 on the thin Ising film This paper investigates the phase behavior and magneti- have shown that phase transitions may occur in the bulk zation profiles of ferromagnetic thin Heisenberg films at dif- region of the film due to the presence of competing surface ferent values of the perpendicular anisotropy and tempera- forces which are external fields acting on the surfaces alone. ture. In the following section the model and simulation In the case of a thin film with two surface fields of the same method are detailed. The equilibrium magnetic phase behav- magnitude but opposite direction and zero bulk field, the ior of the model system is discussed in Sec. III and the tem- surface fields favor a negative magnetization at one surface perature dependence of the magnetic relaxation is investi- and a positive magnetization at the other surface. For suffi- gated in Sec. IV. The paper concludes with a summary of the ciently high temperatures the interface between the regions key findings. of negative and positive magnetization is located in the middle of film. However, Binder and co-workers7­9 found II. THE MODEL that for temperatures below a critical temperature Tc(D) The system under consideration is a three-dimensional which depends on the film thickness D, the interface is ferromagnetic thin film of finite thickness D that is described shifted from the center toward the one of the surfaces. The by the Hamiltonian 0163-1829/97/55 18 /12556 5 /$10.00 55 12 556 © 1997 The American Physical Society 55 THIN FERROMAGNETIC FILMS WITH COMPETING . . . 12 557 H J S z i*Sj Si 2 H1*Si i,j i i surface 1 HD*Si, 1 i surface D where S x y z i (Si ,Si ,Si ) is a unit vector representing the ith spin and the notation i, j means that the sum is restricted to nearest-neighbor pairs of Heisenberg spins, each pair being counted only once. J is a coupling constant characterizing the exchange interaction which has a positive sign for ferro- magnetism and determines the strength of the perpendicu- lar anisotropy which is applied to spins throughout the whole film. As stated by Taylor and Gyorffy,12 in the case of 0, the model is a classical Heisenberg spin system, while for it becomes an Ising model. It should be noted that the quadratic anisotropy term in Eq. 1 forces the spins FIG. 1. Mean magnetization per spin vs time in units of Monte to align along a z direction and minimizes the canting of the Carlo steps per spin for thin ferromagnetic Heisenberg films of size spins for increasing temperature. In this paper the perpen- 16 16 12 for different values of the perpendicular anisotropy in the range 0 0.5 from an initial spin state of Sz 1 for all i at dicular anisotropy was investigated over the range 0 i a temperature T* 1.0. The curves through the points are only 0.5 with larger values of only being used to facilitate guides to the eye. comparison with the Ising model. H1 and HD are the surface fields. 1 We consider a simple cubic lattice of size L L D, in Mz z n 6 units of the lattice spacing, and in the Monte Carlo simula- L2 Si tion apply periodic boundary conditions in the x and y direc- were determined for different values of and temperature tions. Free boundary conditions are applied in the z direction T. The fluctuations in the magnetization were used to calcu- which is of finite thickness D and the system is subject to late the layer susceptibility n which is given by surface fields applied a layer n l and n D of the film z2 z n L2 M n M n 2 /kBT, 7 H1 hz i1 , 2 where kB is Boltzmann's constant. Simulations were per- formed for up to 106 Monte Carlo steps per spin MCS/spin HD hz iD , 3 to ensure equilibration of systems in the Heisenberg limit giving a Hamiltonian ( 0).17 Equilibrium averages were typically taken over 2 105 MCS/spin with initial transients ignored. For systems in the Ising limit ( ), much shorter runs could be per- H J S z i*Sj S formed. i 2 i,j i III. MAGNETIC PHASE BEHAVIOR h Sz z i Si . 4 i surface 1 i surface D The simulations show that the z component of mean mag- netization per spin, M The film thickness D 12 and surface field strength h z , depends on both and tempera- ture T. Figure 1 shows the evolution of M 0.55 were used throughout and the simulations performed z with time from an initially ordered state at a reduced temperature of T* for lattices of size L 16, 32. The Metropolis algorithm15 k was used in the Monte Carlo simulations with trial configu- BT/J 1.0 for different values of from 0 to 0.5. In Fig. 1 MCS/spin is used as a unit of time and an rations generated from Barker-Watts16 spin rotations. The initial spin state Sz 1 was selected. For 0.4 and 0.5, magnitude of the maximum spin rotation was adjusted to i the systems quickly approach equilibrium and equilibrium ensure approximately 50% of trial configurations were re- states of nonzero magnetization of the film persist. The tem- jected in the bulk equilibrium state. For large values of , to perature is well below T ensure a rejection rate of approximately 50%, the Barker- c(D) for the Ising system T* 4.0.8 However for 0, the spins continuously rotate to Watts spin rotation was supplemented by a randomly se- reach equilibrium at zero film magnetization. No spontane- lected spin flip. The z component of the magnetization for ous magnetization is observed even though T T the film c(D) for the Ising system. For 0, i.e., an isotropic spin-spin interac- tion, the model is a classical Heisenberg spin system and the 1 D M z ordered spin states are quickly destroyed at finite tempera- z D Mn , 5 n 1 ture. At intermediate values of , 0 0.4, spontaneous magnetization of the film persists but the magnitude of the and the z component of the magnetization for the nth layer equilibrium magnetization of the film decreases with . Like- of the film wise the time to achieve equilibrium increases with . An- 12 558 HYUNBUM JANG AND MALCOLM J. GRIMSON 55 FIG. 2. Magnetization profiles across the film, Mzn , vs layer FIG. 3. Layer susceptibility n vs layer number n for D 12 at number n for D 12 at a temperature T* 1.0 with surface fields a temperature T* 1.0. For 0,0.3 an initial spin state of Szi H1 /J HD /J 0.55. For 0,0.1,0.3,0.5 an initial spin state 1 for all i was used, while for 0.1,0.2 an initial spin state of of Sz z i 1 for all i was used, while for 0.2,0.4 an initial spin Si 1 for all i was used. Note the change of scale for n with state of Szi 1 for all i was used. 0. The curves drawn are only guides to the eye. other aspect of the data in Fig. 1 for smaller is the pro- The temperature dependence of the magnetization profiles nounced fluctuations in M is shown in Fig. 4 for 0.2. For clarity an initial state z . These arise for Heisenberg spin systems since the probability of spin flips becomes very of Szi 1 for all i is used for temperatures T* small and metastable states occur due to strong magnetiza- 0.8,1.0,1.4, while an initial state of Szi 1 for all i is tion in the x and y directions which averages to zero much used for temperatures T* 0.7,0.9,1.1. At the highest tem- quicker than for Mz .17 perature T* 1.4, we find that the interface is located in the The magnetization profiles across the film, Mz center of the film, between n 6 and n 7, and the mean n , for dif- ferent at a temperature T* 1.0 are shown in Fig. 2. For film magnetization Mz is zero due to the symmetry of clarity the figure shows results for 0, 0.1, 0.3, 0.5 from Mzn about the middle of the film. However, for lower tem- an initial state of Sz z peratures from T* 0.7 to 1.1, the interface is shifted toward i 1 and from an initial state of Si 1 for 0.2 and 0.4. It can be seen that the surface fields locally constrain the spins to align in the negative direction near one surface and positive direction near the other surface. In the bulk, the mean spin orientation of the layers varies smoothly from one surface to the other. For 0, the inter- face between regions of negative and positive magnetization is located in the center of the film. The interface moves from the center toward the surface for 0. The direction of the interface displacement depends on the initial spin configura- tion and a degeneracy exists between states of positive and negative total magnetization. However, for larger 0.4 and 0.5 , the interface disappears and spins are confined to one of the z directions according to their initial states to produce a large value of the film magnetization. The tem- perature is well below Tc(D) for the Ising system ( ). Figure 3 shows the film profiles of susceptibility n at a temperature T* 1.0 for 0,0.3 from an initial configura- tion of Szi 1 for all i and for 0.1,0.2 from an initial configuration of Szi 1 for all i. Here for 0, we find a broad peak centered around the middle of the film. While, for 0 the peaks in the profiles are shifted toward the FIG. 4. Magnetization profiles across the film, Mz vs layer num- surface appropriate to the initial spin configuration. More- n ber n, for D 12 with 0.2 at different temperatures with surface over the peaks in n for each are located in the same layer fields H z 1 /J HD /J 0.55. An initial spin state of Si 1 for as the interfaces in the profiles of Mzn , indicating larger fluc- all i was used for T* 0.8,1.0,1.4, while an initial spin state of tuations of spins in the interface. Szi 1 for all i was used for T* 0.7,0.9,1.1. 55 THIN FERROMAGNETIC FILMS WITH COMPETING . . . 12 559 FIG. 5. Temperature dependence of the reduced energy U* U/UG and the specific heat C U/ T* for 0.2. the surface and a degeneracy exists between two states of the film magnetization. The selected state depends on the initial spin state. The film has a finite value of Mz at these tem- peratures. This behavior can be regarded as a remnant of Ising model behavior seen by Binder and co-workers.7­9 When 0, the model becomes a classical Heisenberg spin system which has no spontaneous magnetization at a nonzero temperature and the interface between negative and positive magnetization is always located in the center of the film. It is of value to consider a critical temperature Tc( ,D) which is equivalent to that of Binder and co-workers,7­9 in the Ising limit . For T Tc the film shows no sponta- FIG. 6. Relaxation of the film magnetization with time for dif- neous magnetization with M ferent values of with zero surface field at a temperature T* z 0, while for T Tc sponta- neous magnetization with M 1.5: a reduced magnetization M z 0 is observed. For z(t)/ M z(0) vs time and b 0.4 we find T ln Mz(t)/Mz(0) vs time. c*( ,D) Tc*( ,D) 4.0, while for 0 we have Tc*(0,D) 0. Figure 5 plots the reduced energy U* U/U surface field. The results of Fig. 6 a also show a faster decay G , where UG is the ground-state energy, and specific heat C U/ T* as a function of temperature for 0.2 of the initial state is observed for smaller but that the time and shows T required to achieve the equilibrium is increased. For c*( 0.2, D 12) 1.3. The critical tempera- ture T 0.8, in the Ising limit , a much shorter time is re- c( ,D) reduces smoothly from the Ising limit value to the Heisenberg value as decreases from 0.4 down to zero. quired to an equilibrium and produce finite value of magne- tization. Figure 6 b shows the magnetic relaxation on a natural logarithm scale, ln Mz(t)/Mz(0) , as function of time IV. DEPENDENCE OF THE MAGNETIC RELAXATION for different . The linear character of the curves for short ON AND ON THE TEMPERATURE times indicates that the initial magnetic relaxation can be The time dependence of the magnetic relaxation is inves- characterized by an exponential decay and the magnetic re- tigated here for different values of and temperature. In laxation can be written as these studies, as elsewhere,18 we focus on the role of and temperature in determining the relative magnetic relaxation TABLE I. Relaxation time for thin Heisenberg film with D behavior of the Heisenberg spin systems and do not attempt 12, temperature T* 1.5 and zero surface field h 0 for perpen- to obtain absolute relaxation times. In Fig. 6 a , for different dicular anisotropy . from 0 to 0.8, the ratio of time-dependent magne- tization to the initial magnetization, M MCS/spin z(t)/ M z(0), is shown as a function of time at temperature T* 1.5 for h 0 from 0 46.0 0.6 an initially ordered state with Szi 1 for all i. Comparison 0.2 53.1 0.6 with the results in Fig. 1 for a similar system with h 0.4 59.2 0.7 0.55 shows that equilibrium was obtained in a shorter time 0.8 75.5 0.6 for h 0. In general the relaxation time increases with the 12 560 HYUNBUM JANG AND MALCOLM J. GRIMSON 55 TABLE II. Relaxation time for thin Heisenberg film with D 12, perpendicular anisotropy 0.2, and zero surface field h 0 at temperature T*. T* MCS/spin 1.0 133.1 1.0 1.2 92.2 1.0 1.4 63.1 1.3 1.6 41.8 0.9 1.8 28.8 0.6 the systems in Fig. 6 with error estimates obtained from ten repetitions with different random number sequences. The temperature dependence of the magnetic relaxation is shown in Fig. 7 a for 0.2. Again the decay of the mag- netization is monitored for h 0 with an initial state of Szi 1 for all i. The rate of decay of the initial state is greater for higher temperatures, but the time to achieve equilibrium also increases with temperature. Once more the initial mag- netic relaxation is governed by an exponential decay as shown in Fig. 7 b . Table II gives the relaxation times for 1.0 T* 1.8 at 0.2 corresponding to the systems in Fig. 7 with error estimates obtained from ten repetitions with dif- ferent random number sequences. V. CONCLUSION We have studied the phase behavior of thin ferromagnetic films with Heisenberg systems and competing surface fields. The perpendicular anisotropy in the Hamiltonian is shown to be an important factor in controlling the phase behavior of the film. For 0, the model is a classical Heisenberg spin system which shows no spontaneous magnetization for T FIG. 7. Relaxation of the film magnetization with time for 0. While for value of 0, the model yields a spontaneous 0.2 with different temperatures and zero surface field: a reduced magnetization of the film at low temperatures. The critical magnetization Mz(t)/Mz(0) vs time and b ln Mz(t)/Mz(0) vs temperature Tc characterizing the phase behavior of the mag- time. netization of the film strongly depends on the magnitude of as does the magnetic relaxation time . These observations Mz t can be expected to be of relevance in studies of the phase M exp t/ ,T ..., 8 behavior and dynamics of thin films of more complex mate- z 0 rials such as ferronematic liquid crystals which also have a where is a relaxation time. Table I gives the relaxation continuous spin system and show spontaneous ordering at times for different values of at T* 1.5 corresponding to low temperatures, but have more complicated Hamiltonians. 1 R. Allenspach and A. Bischof, Phys. Rev. Lett. 69, 3385 10 J. M. Yeomans, Statistical Mechanics of Phase Transitions Clar- 1992 . endon, Oxford, 1992 . 2 D. P. Pappas, K.-P. Ka¨mper, and H. Hopster, Phys. Rev. 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