Abstract
In this work we study the thermodynamic properties of ultrathin ferromagnetic dots using Monte Carlo simulations. We investigate the vortex density as a function of the temperature and the vortex structure in monolayer dots with perpendicular anisotropy and longrange dipole interaction. The interplay between these two terms in the hamiltonian leads to an interesting behavior of the thermodynamic quantities as well as the vortex density.
Monte Carlo; Ultrathin films; Phase transition
Monte Carlo simulations of ultrathin magnetic dots
M. Rapini^{I}; R. A. Dias^{I}; B. V. Costa^{I}; D. P. Landau^{II}
^{I}Laboratório de Simulação  Departamento de Física  ICEX  UFMG 30123970 Belo Horizonte  MG, Brazil
^{II}Center for Simulational Physics, University of Georgia, Athens, Georgia 30602
ABSTRACT
In this work we study the thermodynamic properties of ultrathin ferromagnetic dots using Monte Carlo simulations. We investigate the vortex density as a function of the temperature and the vortex structure in monolayer dots with perpendicular anisotropy and longrange dipole interaction. The interplay between these two terms in the hamiltonian leads to an interesting behavior of the thermodynamic quantities as well as the vortex density.
Keywords: Monte Carlo; Ultrathin films; Phase transition
I. INTRODUCTION
Magnetism at nanoscale, when the size of the structure is comparable to or smaller than both the ferromagnetic (FM) and antiferromagnetic (AF) domain size, offers a great potential for new physics. In the last decade there has been an increasing interest in ultrathin magnetic dots from research groups as well as technological industries. Such an interest is due to numerous unique phenomena related to the lowdimension of these systems.
The modern technology demands techniques capable of producing nanometersized structures over large areas. A good perspective is the use of nanodots of nickel that could store terabyte of data in a computer chip just a few centimeters wide. In particular, ferromagnetic nanodots have been widely studied by use of experimental techniques such as MFM (magnetic force microscopy). In addition, some theoretical models were proposed to explain the physical phenomena observed in the experiments, among them the transition from perpendicular to inplane ordering and the magnetoresistence effect.
Regarding the perpendicular to inplane ordering transition, experiments were done using epitaxial films to investigate its transition temperature and thickness dependence [2] [3]. In addition, many theoretical approaches were developed, for example, treating a twodimensional layer by renormalization group [4]. Some lattice models were proposed to take into account longrange dipolar interactions and surface anisotropy [5].
Based on such models, Monte Carlo simulations have been widely used to study the phase diagram of very thin films [6], the nature of this transition [7] as well as its dependence on the magnetic history of the system [8]. On the other hand, magnetic domains [9] and magnetic structures [10] have also been investigated by using computational methods. A topological excitation, the spin vortex, has been found in experiments and also detected in simulations. Vortex structures are believed to drive a BereziinskiKosterlitzThouless (BKT) phase transition in the two dimensional planarrotator (PR) model [11]. Although vortices are present in thin films with long range interactions, it is not clear if they play any role in the transition.
The model we study is described by the Heisenberg spin hamiltonian with exchange and longrange dipolar interactions as well as singleion anisotropy
where we use classical spins S = 1. Here the first sum is performed over nearest neighbors with exchange coupling strenght, J > 0 , while the second sum runs over all spin pairs in the lattice. The constant of dipole coupling is D, r_{ik} is a vector connecting the i and k sites and A is the singlesite anisotropy constant along the zaxis[5].
The main task in this work is to study the importance of vortices in the physics of the model. Although preliminary, our results indicate an anomalous behavior of the vortex density at the transition temperature for d = ^{D}/_{A} << 1. In the following we present a brief background on the simulation, our results and the conclusions.
Method
The simulations are done in a square lattice of volume L × L with L = 20,40,60 by using the MonteCarlo method with the Metropolis algorithm [12, 13]. Since nanodots are finite per nature we have to use open boundary conditions in our simulations. However, we want to emphasize the long range effects of the dipolar term of the model at the boundary of the structure. For that, we have used periodic boundary conditions in the non dipolar terms while for the dipolar term we have used open conditions.
We have studied the model for three different values of the parameters A and D, d = = 0.1, 1.0 and 9.0 for fixed J = 1. Energy is measured in units of JS^{2} and temperature in units of JS^{2}/k_{B}, where k_{B} is the Boltzman constant. For every temperature the first 10^{5} MC steps per spin were used to lead the system to equilibrium and the next 10^{5} configurations were used to calculate thermal averages of thermodynamical quantities of interest.
II. RESULTS
In the case where d = 0.1, we measured the outofplane (z) and inplane (xy) magnetizations (Shown in Fig. 1).
The system comes from an ordered state at low temperature to a disordered state at high temperature. That behavior indicates an orderdisorder phase transition at T_{c} » 0.55. The inplane magnetization, M_{xy}, grows presenting a maximum close to the orderdisorder critical temperature T_{c}. However, the height of the peak diminishes as L grows, in a clear indicative that it is a finite size artifice.
The magnetic susceptibility is shown in Fig. 2. The position of the maxima give us an estimate for T_{c} ( » 0.55).
We also measured the vortex density in the xy plane as a function of the temperature. Starting from the highest temperature, T = 1.2, the number of vortex decreases and reaches a minimum. Then it starts to increase as the system is cooled down. This behavior is shown in Fig. 3 and the graphics indicate that the ground state of the system has a significant number of vortices and antivortices in the xy plane. Apparently, the minimum of the vortex curve is connected with the transition to inplane magnetization, however, we were not able to establish that connection.
For d = 1.0 the behavior of the inplane and outofplane magnetizations (See Fig. 4), suggest that the ground state is disordered in contrast to earlier works of Santamaria [6] and Vedmedenko [10] that argue that the ground state is for spins ordered in the xy plane. A plot of the susceptibility is shown in Fig. 5 as a function of temperature. Although some authors [6, 10] concluded that this transition is of second order, the curves show well defined maxima that do not seem to indicate any critical behavior.
The vortex density curve in the xy plane is shown in Fig. 6. We see that the number of vortices increases monotonically from zero as a function of temperature. As temperature grows we observed that the spins in the lattice start to disorder, so that pairs vorticesantivortices can unbind inducing a BKT transition. However our results are not refined enough to decide that. In Fig. 7 we show two typical configurations for T = 0.8 and 1.2 where the vortices are indicated by circles and the antivortices by squares.
For systems with larger d, for example, d = 9.0, the spins are preferentially in the xy plane but it does not present any magnetic ordering (See Fig. 8). The vortex density curve is similar to the case where d = 1.0 (See Fig. 9).
III. CONCLUSION
In summary, we investigated the Heisenberg spin model with exchange J and dipolar interactions D and an anisotropic term A for different parameters d = . For small d, (0.1), we observed that the vortex density has a minimum and is nonzero for low temperatures. Apparently, this minimum is connected with the orderdisorder phase transition but this connection has to be studied more carefully. For larger values of d (1.0 and 9.0) the vortex density and the configurations of vortices in the system led us to suspect of a phase transition of the BKT type involving the unbinding of vorticesantivortices pairs. However our results are not refined enough to decide that.
Acknowledgments
BVC would like to thank the Center for Simulational Physics at the University of Georgia for its hospitality. This work was funded by CNPq and FAPEMIG, Brazilian agencies, and CIAMCNPq Process Grant No. 49.0101/038, NASA Grant No. NNC04GB24 and NSF Grants Nos. DMR0341874 and DMR0307082.
Received on 19 September, 2005
 [1] Electronic mail: mrapini@fisica.ufmg.br
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Publication Dates

Publication in this collection
16 Oct 2006 
Date of issue
Sept 2006
History

Received
15 Sept 2005