Phase transition in ultrathin magnetic films with long-range interactions: Monte Carlo simulation of the anisotropic Heisenberg model
PPhase transition in ultrathin magnetic films with long-range interactions:Monte Carlo simulation of the anisotropic Heisenberg model
M. Rapini, * R. A. Dias, † and B. V. Costa ‡ Departamento de Física, Laboratório de Simulação, ICEX, UFMG 30123-970 Belo Horizonte, MG, Brazil (cid:1)
Received 7 April 2006; revised manuscript received 17 October 2006; published 22 January 2007 (cid:2)
Ultrathin magnetic films can be modeled as an anisotropic Heisenberg model with long-range dipolarinteractions. It is believed that the phase diagram presents three phases: An ordered ferromagnetic phase (cid:1) I (cid:2) , aphase characterized by a change from out-of-plane to in-plane in the magnetization (cid:1) II (cid:2) , and a high-temperatureparamagnetic phase (cid:1) III (cid:2) . It is claimed that the border lines from phase I to III and II to III are of second orderand from I to II is first order. In the present work we have performed a very careful Monte Carlo simulation ofthe model. Our results strongly support that the line separating phases II and III is of the BKT type.DOI: 10.1103/PhysRevB.75.014425 PACS number (cid:1) s (cid:2) : 75.10.Hk, 75.30.Kz, 75.40.Mg I. INTRODUCTION
Since the late 1980s there has being an increasing interestin ultrathin magnetic films.
This interest is mainly associ-ated with the development of magnetic-nonmagnetic multi-layers for the purpose of giant magnetoresistenceapplications. In addition, experiments on epitaxial magneticlayers have shown that a huge variety of complex structurescan develop in the system.
Rich magnetic domain struc-tures like stripes, chevrons, labyrinths, and bubbles associ-ated with the competition between dipolar long-range inter-actions and strong anisotropies perpendicular to the plane ofthe film were observed experimentally. A lot of theoreticalwork has been done on the morphology and stability of thesemagnetic structures.
Beside that, it has been observedthe existence of a switching transition from perpendicular toin-plane ordering at low but finite temperature: at lowtemperature the film magnetization is perpendicular to thefilm surface; as temperature rises the magnetization flips toan in-plane configuration. Eventually the out-of-plane andthe in-plane magnetization become zero. The general Hamiltonian describing a prototype for anultrathin magnetic film assumed to lay in the xy plane is H = − J (cid:3) (cid:4) ij (cid:5) S i (cid:1) · S j (cid:1) − A (cid:3) i S iz + D (cid:3) (cid:4) ij (cid:5) (cid:6) S i (cid:1) · S j (cid:1) r ij − 3 (cid:1) S i (cid:1) · r (cid:1) ij (cid:2)(cid:1) S j (cid:1) · r (cid:1) ij (cid:2) r ij (cid:7) . (cid:1) (cid:2) Here J is an exchange interaction, which is assumed to benonzero only for nearest-neighbor interaction, D is the dipo-lar coupling parameter, A is a single-ion anisotropy, and r (cid:1) ij = r (cid:1) j − r (cid:1) i , where r (cid:1) i stands for lattice vectors. The structuresdeveloped in the system depend on the sample geometry andsize. Several situations are discussed in Ref. 14 and citationstherein.Although the structures developed in the system are wellknown, the phase diagram of the model is still being studied.There are several possibilities since we can combine the pa-rameters in many ways. We want to analyze the case J (cid:1) (cid:1) (cid:2) Case D = 0. For D = 0 we recover the two-dimensional (cid:1) (cid:2) anisotropic Heisenberg model. The isotropic case A = 0 is known to present no transition. For A (cid:1) undergoing an order-disorder phase transition whose critical temperature isapproximately T = 2 T ln (cid:1) (cid:2) J / A (cid:2) , (cid:1) (cid:2) where T is the transition temperature of the three-dimensional Heisenberg model T / J (cid:8) A (cid:3)
0, the model is in the xy universality class. In thiscase it is known to have a Berezinskii-Kosterlitz-Thouless (cid:1) BKT (cid:2) phase transition.
This is an unusual magnetic-phase transition characterized by the unbinding of pairs oftopological excitations named vortex-antivortex.
A vor-tex (cid:1) antivortex (cid:2) is a topological excitation in which spins ona closed path around the excitation core precess by2 (cid:2) (cid:1) −2 (cid:2) (cid:2) . Above T BKT the correlation length behaves as (cid:4) (cid:8) exp (cid:1) bt −1/2 (cid:2) , with t (cid:9) (cid:1) T − T BKT (cid:2) / T BKT and (cid:4) → (cid:5) below T BKT . (cid:1) (cid:2) Case D (cid:1)
0. In this case, there is a competition be-tween the dipolar and the anisotropic terms. If D is small FIG. 1. A sketch of the phase diagram for the model (cid:10)
Eq. (cid:1) (cid:2)(cid:11) .Phase I corresponds to an out-of-plane magnetization, phase II hasin-plane magnetization, and phase III is paramagnetic. The borderline between phase I and phase II is believed to be of first order andfrom regions I and II to III to be both of second order.PHYSICAL REVIEW B , 014425 (cid:1) (cid:2) (cid:1) (cid:2) /014425 (cid:1) (cid:2) ©2007 The American Physical Society014425-1 ompared to A we can expect the system to have an Isingbehavior. If D is not too small we can expect a transition ofthe spins from out-of-plane to in-plane configuration. Forlarge enough D out-of-plane configurations become unstablesuch that, the system lowers its energy by turning the spinsinto an in-plane anti-ferromagnetic arrangement. For the pla-nar xy model with pure dipolar interactions, the system or-ders at T c = 1.39± 0.05 (cid:1) Ref. 34 (cid:2) where temperature is in unitsof JS / k B and k B is the Boltzmann constant.Earlier works on this model, which discuss the phase dia-gram, were mostly done using renormalization group ap-proach and numerical Monte Carlo simulation. Theyagree between themselves in the main features. The phasediagram for fixed A and J = 1 is schematically shown in Fig.1 in the space (cid:1) D , T (cid:2) . From Monte Carlo (cid:1) MC (cid:2) results it isfound that there are three regions labeled in Fig. 1 as I, II,and III. Phase I corresponds to an out-of-plane magnetiza-tion, phase II has in-plane magnetization, and phase III is paramagnetic. The border line between phase I to phase II isbelieved to be of first order and from regions I and II to IIIare both second order.Although the different results agree between themselvesabout the character of the different regions, much care has tobe taken because they were obtained by using a cutoff r c inthe dipolar term. The long-range character of the potential islost. As a consequence, it will not be surprising if a differentphase line emerges coming from region II to region III.In this work we use MC simulations to investigate themodel defined by Eq. (cid:1) (cid:2) . We use a cutoff r c in the dipolarinteraction. Our results strongly suggest that the transitionbetween regions II and III is in the BKT universality class,instead of second order, as found in earlier works. II. SIMULATION BACKGROUND
Our simulations are done in a square lattice of volume L (cid:6) L (cid:1) L = 10, 20, 30, 40, 50, 80 (cid:2) with periodic boundary con-ditions. We use the Monte Carlo method with the Metropolisalgorithm. To treat the dipole term we use a cutoff r c = 5 a , where a is the lattice spacing, as suggested in the workof Santamaria and co-workers. We have performed the simulations for temperatures inthe range 0.3 (cid:7) T (cid:7) (cid:8) T = 0.1. When neces-sary this temperature interval is reduced to (cid:8) T = 0.01. Forevery temperature the first 5 (cid:6) MC steps per spin wereused to lead the system to equilibrium. The next 10 configu-rations were used to calculate thermal averages of thermody- FIG. 2. Out-of-plane (cid:1) a (cid:2) and in-plane (cid:1) b (cid:2) magnetization for D =0.1. The ground state is ferromagnetic. There is no in-plane spon-taneous magnetization.FIG. 3. Specific heat as a function of temperature for D =0.1. FIG. 4. Out-of-plane susceptibility as a function of temperaturefor D =0.1.FIG. 5. Binder’s cumulant as a function of temperature for D =0.1.RAPINI, DIAS, AND COSTA PHYSICAL REVIEW B , 014425 (cid:1) (cid:2) amical quantities of interest. We have divided these last 10 configurations in 20 bins from which the error bars are esti-mated from the standard deviation of the averages over thesetwenty runs. The single-site anisotropy constant was fixed as A = 1.0 while the D parameter was set to 0.10, 0.15, and 0.20.In this work the energy is measured in units of JS and tem-perature in units of JS / k B , where k B is the Boltzmann con-stant.To estimate the transition temperatures we use finite-size-scaling (cid:1) FSS (cid:2) analysis to the results of our MC simulations.In the following we summarize the main FSS properties. Ifthe reduced temperature is t = (cid:1) T − T c (cid:2) / T , the singular part ofthe free energy is given by F (cid:1) L , T (cid:2) = L − (cid:1) (cid:9) (cid:2) / (cid:10) F (cid:1) tL (cid:10) (cid:2) (cid:1) (cid:2) for T in the vicinity of the critical temperature and L not toosmall.Appropriate differentiation of F yields the various ther-modynamic properties. For an order disorder transition ex-actly at T c the magnetization M , susceptibility (cid:11) , and specificheat C , behave respectively as M (cid:12) L − (cid:13) / (cid:10) , (cid:11)(cid:12) L − (cid:14) / (cid:10) , C (cid:12) L − (cid:9) / (cid:10) . (cid:1) (cid:2) In addition to these an important quantity is the fourth-orderBinder’s cumulant U = 1 − (cid:4) m (cid:5) (cid:4) m (cid:5) . (cid:1) (cid:2) where m is the magnetization. For large enough L , curves for U (cid:1) T (cid:2) cross the same point U * at T = T c . For a BKT transition the quantities definedabove behave in a different way. There is no spontaneousmagnetization for any finite temperature. The specific heatpresents a peak at a temperature that is slightly higher than T BKT . Beside that, the peak height does not depend on L .Because models presenting a BKT transition have an entirecritical region, the curves for U (cid:1) L (cid:2) just coincide inside thatregion presenting no crosses at all.The vortex density is defined as the number of vorticesper area. In the simulation, we analyze each plaquette of foursites and if the sum of the difference of the angles betweenadjacent spins equals ±2 (cid:2) , we have a vortex (cid:1) antivortex (cid:2) .Below we present MC results for three typical regions.When not indicated the error bars are smaller than the sym-bol sizes. III. SIMULATION RESULTSA. Case D =0.1 For D = 0.1 we measured the dependence of the out-of-plane magnetization M z and the in-plane magnetization M xy as a function of temperature for several values of L (cid:1) see Fig.2 (cid:2) . The figures indicate that in the ground state the system isaligned in the z direction. Approximately at T (cid:8) M z magnetization goes to zero, which gives a rough estimate ofthe critical temperature. The in-plane magnetization has asmall peak close to T (cid:8) L grows, in a clear indicative that it is a finite-size artifice. TABLE I. Critical temperature T cL of the specific heat C , suscep-tibility (cid:11) , and the crosses of the fourth-order Binder’s cumulant U as a function of the lattice size L . Data are for D =0.10. L
10 20 30 40 50 80 C (cid:11) U xy plane for D =0.1. FIG. 7. M z and M xy (cid:1) open and full symbols, respectively (cid:2) for D =0.15. FIG. 8. Specific heat for D =0.15.PHASE TRANSITION IN ULTRATHIN MAGNETIC FILMS … PHYSICAL REVIEW B , 014425 (cid:1) (cid:2) he behavior of the specific heat, susceptibility, and Bind-er’s cumulant are shown in Figs. 3, 4, and 5, respectively.The results indicate an order-disorder phase transition inclear agreement with Refs. 5–17 and 19. The vortex densityin the xy plane (cid:1) Fig. 6 (cid:2) has a very shallow minimum near theestimated critical temperature and is almost independent ofthe lattice size. The growth of the number of vortices whenthe temperature is decreased is related to the disordering inthe plane when the magnetic moments tend to be in the z direction. We have performed a finite-size scaling analysis ofthe data above by plotting the temperature T cL as a function of1 / L for the specific heat, the susceptibility, and the crosses ofthe fourth-order cumulant. The results are shown in Table I.By linear regression we have obtained the critical tempera-ture as T c (cid:5) = 0.682 (cid:1) (cid:2) . An analysis of the maxima of the spe-cific heat C max (cid:1) see Fig. 18 (cid:2) as a function of the lattice sizeshows that it behaves as C max (cid:12) ln L , indicating a second-order phase transition. In the phase diagram we crossed thesecond-order line labeled c . B. Case D =0.15 In this region of the parameters, it was observed a transi-tion from an out-of-plane ordering at low temperatures to anin-plane configuration as described by the magnetization be-havior shown in Fig. 7. We show M z and M xy in the samefigure for comparison. The out-of-plane magnetization goesto zero at T (cid:8) and it is due to the competition betweenthe easy axis anisotropy and the dipolar interaction. The spe-cific heat curve presents two peaks (cid:1) see Fig. 8 (cid:2) . The peak atlow temperature is pronounced and is centered in the tem-perature in which occurs the rapid decrease of the out-of-plane magnetization T (cid:8) T (cid:8) TABLE II. Critical temperature T cL as a function of the linearsize L for the susceptibility (cid:11) xy and D =0.15. T cL L
10 20 30 40 50 80FIG. 9. In-plane (cid:1) a (cid:2) and out-of-plane (cid:1) b (cid:2) susceptibility for D =0.15.FIG. 10. In-plane (cid:1) a (cid:2) and out-of-plane (cid:1) b (cid:2) Binder’s cumulant asa function of temperature for D =0.15. Observe that the in-planecumulant has a minimum at T (cid:8) (cid:1) except the spurious case L =10 (cid:2) . up to T (cid:8) D =0.15.RAPINI, DIAS, AND COSTA PHYSICAL REVIEW B , 014425 (cid:1) (cid:2) n Fig. 9 we show the in-plane and out-of-plane suscepti-bilities. The out-of-plane susceptibility presents a single peakclose to T (cid:8) T (cid:8) T , indicating two phase tran-sitions. The Binder’s cumulant for the in-plane and out-of-plane magnetization are shown in Figs. 10. Except for thecase L = 10 the curves for different values of the lattice sizedo not cross each other indicating a BKT transition at T (cid:8) T . Beside that, the in-plane cumulant has a minimum at T (cid:8) T , which is a characteristic of a first-order phasetransition. The vortex density is shown in Fig. 11. Its behavior issimilar to that one shown in Fig. 6. The maxima of the spe-cific heat are shown in Fig. 18 as a function of L . It is clearthat after a transient behavior it remains constant indicating aBKT transition. A FSS analysis of the susceptibility (cid:1) seeTable II (cid:2) gives the BKT temperature T BKT (cid:5) = 0.613 (cid:1) (cid:2) . In thephase diagram we crossed the first-order line labeled a (cid:1) T (cid:2) and the line labeled b (cid:1) T (cid:2) . C. Case D =0.20 In Fig. 12 we show the in-plane and out-of-plane magne-tization curves for several lattice sizes and D = 0.20. We ob-serve that as the lattice size L goes from L = 10 to L = 80, bothmagnetizations decrease. It can be inferred that as the systemapproaches the thermodynamic limit, the net magnetizationshould be zero. Therefore, the system does not present finitemagnetization for any temperature T (cid:1)
0. The specific heat (cid:1)
Fig. 13 (cid:2) presents a maximum at T (cid:8) L . We observe that the position of themaxima and their heights are not strongly affected by thelattice size, all points falling approximately in the samecurve.In Fig. 14 we show the in-plane and out-of-plane suscep-tibilities, respectively. (cid:11) zz does not present any critical be-havior. (cid:11) xy presents a peak, which increases with L . For theBinder’s cumulant (cid:1) see Fig. 15 (cid:2) there is no unique cross ofthe curves (cid:1) except for the L = 10 curve, which is consideredtoo small to be taken into account (cid:2) . This behavior indicates aBKT transition at T BKT (cid:8) (cid:1) see Table III (cid:2) and the maxima of the specific heat. The spe-cific heat is shown in Figs. 17 and 18. Its behavior indicatesa BKT transition. The analysis of the susceptibility gives T BKT (cid:5) = 0.709 (cid:1) (cid:2) . In the phase diagram we crossed the linelabeled b . In our results we could not detect any other tran- FIG. 12. M xy and M z (cid:1) open and full symbols, respectively (cid:2) for D =0.2.FIG. 13. Specific heat for D =0.2. The line is a guide to theeyes. FIG. 14. In-plane (cid:1) a (cid:2) and out-of-plane (cid:1) b (cid:2) susceptibility for D =0.2. FIG. 15. Fourth-order in-plane cumulant for D =0.2.PHASE TRANSITION IN ULTRATHIN MAGNETIC FILMS … PHYSICAL REVIEW B , 014425 (cid:1) (cid:2) ition for D = 0.20, indicating that the line labeled a endssomewhere in between 0.15 (cid:3) D (cid:3) a occurs at a lower temperature (cid:1) T (cid:3) (cid:2) outside the range ofour simulated data.In a preliminary calculation using a lattice of size L = 40,we have estimated the value of the multicritical point in theintersection of the a , b , and c lines around D = 0.14. Ourestimate agrees with the phase diagram obtained by San-tamaria and co-workers for A = 2.0. Their simulations weredone on a BCC lattice with (cid:1) (cid:2) surfaces while we used asimple cubic lattice. However, the first layer of these twostructures is equivalent. IV. CONCLUSIONS
In earlier studies several authors have claimed that themodel for ultrathin magnetic films defined by Eq. (cid:1) (cid:2) pre-sents three phases. Referring to Fig. 1 it is believed that theline labeled a is of first order. The lines b and c are of secondorder. Those results were obtained by introducing a cutoff inthe long-range interaction of the Hamiltonian. In the presentwork we have used a numerical Monte Carlo approach tostudy the phase diagram of the model for J = A = 1 and D = 0.10, 0.15, and 0.20. In order to compare our results tothose discussed above we have introduced a cutoff in thelong-range dipolar interaction. A finite-size scaling analysisof the magnetization, specific heat, susceptibilities, andBinder’s cumulant clearly indicates that the line labeled a isof first order and the line c is of second order in agreementwith other results. However, the b line is of BKT type. Afteranalyzing the results obtained, some questions come out: (cid:1) (cid:2) Is it possible the existence of a limiting range of in-teraction in the dipolar term beyond which the character ofthe transition changes from BKT to second order? (cid:1) (cid:2) How does the line labeled a end in the phase diagram? (cid:1) (cid:2) What is the character of the intersection point of thethree lines in the phase diagram? As the cutoff r in thedipolar term is increased, the symmetry of the Hamiltonian is not changed. Therefore, we expect that for larger values of r , there would be no qualitative changes in our results ex-cept when the range of the interaction goes to infinity.However, to respond to question (cid:1) (cid:2) it is necessary for amuch more detailed study of the model for several values ofthe cutoff range r c . In a simulation program we have to becareful in taking larger r c values since we have to augmentthe lattice size proportionally to prevent misinterpretations.In a very preliminary calculation, Rapini et al. studiedthe model with true dipolar long-range interactions by usingopen boundary conditions and perfoming the sum without acutoff. Their results led them to suspect a phase transition ofthe BKT type involving the unbinding of vortices-antivortices pairs in the model.In order to estimate the point (cid:1) D , T (cid:2) in the phase diagramwhere the a line ends it would be necessary to study thesystem for T (cid:3) (cid:1) (cid:2) and (cid:1) (cid:2) . TABLE III. Critical temperature T cL as a function of the linearsize L for the susceptibility (cid:11) xy and D =0.20. T cL L
10 20 30 40 50 80 FIG. 17. Specific-heat maxima obtained by using histograms forlattices of sizes L =50, 80, 100, and 120 and systems D =0.20 (cid:1) left (cid:2) and D =0.15 (cid:1) right (cid:2) . Each point is the result of 10 configurations.FIG. 18. Maxima of the specific heat as a function of the latticesize. The diamonds are for D =0.10, the squares for D =0.15, andthe circles for D =0.20. While in the second-order phase transitionthe maxima in the specific heat scale as ln L ; in the BKT phasetransition the finite-size effects are very small.FIG. 16. Vortex density in the xy plane for D =0.2.RAPINI, DIAS, AND COSTA PHYSICAL REVIEW B , 014425 (cid:1) (cid:2) CKNOWLEDGMENTS
Financial support from the Brazilian agencies CNPq,FAPEMIG, and CIAM-0249.0101/03-8 (cid:1)
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