Diagram for vortex formation in quasi-two-dimensional magnetic dots
J. C. S. Rocha, P. Z. Coura, S. A. Leonel, R. A. Dias, B. V. Costa
DDiagram for vortex formation in quasi-two-dimensional magnetic dots
J. C. S. Rocha, (cid:1)
P. Z. Coura, (cid:1)
S. A. Leonel, (cid:1)
R. A. Dias, (cid:1) and B. V. Costa (cid:1) Departamento de Física, Laboratório de Simulação, ICEX, UFMG, 30123-970 Belo Horizonte, MG, Brazil Departamento de Física, ICE, UFJF, 36036-330 Juiz de Fora, MG, Brazil (cid:1)
Received 22 June 2009; accepted 19 January 2010; published online 3 March 2010 (cid:2)
The existence of nonlinear objects of the vortex type in two-dimensional magnetic systems presentsitself as one of the most promising candidates for the construction of nanodevices, useful for storingdata, and for the construction of reading and writing magnetic heads. The vortex appears as theground state of a magnetic nanodisk whose magnetic moments interact via the dipole-dipolepotential (cid:3) D (cid:4)(cid:5) S (cid:1) i · S (cid:1) j − 3 (cid:1) S (cid:1) i · rˆ ij (cid:2) (cid:1) (cid:1) S (cid:1) j · rˆ ij (cid:2)(cid:6) / r ij (cid:7) and the exchange interaction (cid:1) − J (cid:4) S (cid:1) i · S (cid:1) j (cid:2) . In this workit is investigated the conditions for the formation of vortices in nanodisks in triangular, square, andhexagonal lattices as a function of the size of the lattice and of the strength of the dipole interaction D . Our results show that there is a “transition” line separating the vortex state from a capacitorlikestate. This line has a finite size scaling form depending on the size, L , of the system as D c = D + 1 / A (cid:1) BL (cid:2) . This behavior is obeyed by the three types of lattices. Inside the vortex phase it ispossible to identify two types of vortices separated by a constant, D = D c , line: An in-plane and anout-of-plane vortex. We observed that the out-of-plane phase does not appear for the triangularlattice. In a two layer system the extra layer of dipoles works as an effective out-of-plane anisotropyinducing a large S z component at the center of the vortex. Also, we analyzed the mechanism forswitching the out-of-plane vortex component. Contrary to some reported results, we foundevidences that the mechanism is not a creation-annihilation vortex anti-vortex process. © . (cid:5) doi:10.1063/1.3318605 (cid:6) I. INTRODUCTION
The miniaturization of electronic devices has a naturallimit imposed by the thermal fluctuations which determineshow long the magnetization of a ferromagnetic structure sur-vives, or in other words: The long-range ferromagnetic ordervanishes when the energy due to the anisotropy becomescomparable to the thermal fluctuation energy in the system.That is the well known superparamagnetic limit that imposephysical limits in the miniaturization of magnetic devices.Recent developments in nanomagnetic materials has shownthat the development of a vortex in quasi-two-dimensional (cid:1) (cid:2) nanomagnets can help to overcome the superparamag-netic limit. Their expected applications include magnetic ran-dom access memory, high density magnetic recording media,magnetic sensors, and magnetic reading and writingheads. By a vortex we mean a special configuration of magneticmoments similar to the stream lines of a circulating flow in afluid. The magnetic moments precess by (cid:2) (cid:3) on a closedpath around the vortex. In the Fig. 1 we show schematicallythe types of vortices and antivortices that can appear in mag-netic systems. The importance of vortices in magnetic sys-tems is known since the early seventies in connection withthe Berezinskii–Kosterlitz–Thouless (cid:1) BKT (cid:2) phase “transi-tion.” In a seminal work Berezinskii and later Kosterlitz andThouless showed that the easy plane Heisenberg model (cid:1) EPHM (cid:2) in two dimensions undergoes an infinite order phase transition. The EPHM is described by the Hamiltonian H = (cid:4) (cid:4) i , j (cid:5) − JS (cid:1) i · S (cid:1) j + AS (cid:1) iz · S (cid:1) jz , where J is an exchange term and A an easy plane anisotropy. The EPHM has a BKT transition ata temperature T BKT coming from a high-temperature phasewhere the same time space-space correlation function exhib-its an exponential decay to a low-temperature phase with a (cid:2) Electronic addresses: jcsrocha@fisica.ufmg.br and bvc@fisica.ufmg.br. b (cid:2) Electronic addresses: pablo@fisica.ufjf.br, sidiney@fisica.ufjf.br, andradias@fisica.ufjf.br. FIG. 1. (cid:1)
Color online (cid:2)
Show schematically in a square lattice: (cid:1) a (cid:2) type Ivortex, (cid:1) b (cid:2) type I antivortex, (cid:1) c (cid:2) type II vortex, and (cid:1) d (cid:2) type II antivortex. JOURNAL OF APPLIED PHYSICS , 053903 (cid:1) (cid:2) (cid:2) (cid:1) /053903/5/$30.00 © 2010 American Institute of Physics , 053903-1 Downloaded 02 Jun 2010 to 150.164.14.44. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp uasi-long-range order where the correlation function has apower-law decay. This phase transition is believed to bedriven by a vortex-antivortex unbinding mechanism. The en-ergy associated with the pair vortex–antivortex is propor-tional to ln r v − a v , where r v − a v is the distance between thevortex and antivortex centers. The logarithmic behavior ofthe energy prohibits the existence of an odd number of vor-tices or antivortices in the EPHM with free or periodicboundary conditions. That is because an isolated vortex (cid:1) orantivortex (cid:2) has energy ln R , where R is the vortex size. In amagnetic nanodot a magnetic dipole-dipole energy term hasto be considered beside the exchange and the anisotropyterms. The dipole energy competes with the exchange termso that for large enough dipole interaction the continuity ofthe magnetic field in the boundary of the system imposes themagnetic moments to be tangent to the border of thenanodot.
This kind of boundary condition favors the ap-pearing of an isolated vortex at the center of the system (cid:1)
SeeFig. 1 (cid:2) . Although, the vortices and antivortices shown in Fig.1 have the same bulk energy, competition with the borderenergy clearly favors the appearing of the vortex of type I inthe system. Due to the singularity at the vortex core thesystem can lower its energy by developing an out-of-planemagnetization, perpendicular to the plane of the disk (cid:1) the z direction (cid:2) . Experimental observations in circular dots ofPermalloy suggest that states “up” and “down” (cid:1) (cid:2) z (cid:2) are de-generated not depending on the vortex orientation (cid:1) clockwiseor counterclockwise (cid:2) . The existence of this degeneracysuggests that it is possible to store one bit of information inthis degree of freedom.
Experimental results also showthat the effective anisotropy is very small and can beneglected in theoretical modeling. Because the vortex for-mation energy is proportional to ln R the superparamagneticlimit is pushed down opening the possibility of buildingsmaller magnetic devices for storing data than those allowedby the nowadays technology. In this work we report a study of quasi-2D magnetic dotsby using Monte Carlo (cid:1) MC (cid:2) and spin dynamicssimulations. The system is modeled by distributing mag-netic particles over a lattice. The particles interact throughexchange and dipolar potentials. We study the vortex forma-tion in three different types of lattices: hexagonal, square,and triangular. Also, we discuss the stability of the vortex asa function of the strength of the dipole interaction and thesystem size. The paper is organized as follows. In Sec. II wepresent the model we will deal with and some considerationsabout the simulations. In Sec. III we present our results anddiscussions. In Sec. IV our conclusions are presented.
II. MODEL
Theoretically we can write a model Hamiltonian for amagnetic nanodot in a pseudospin language as H = − J (cid:4) (cid:4) i , j (cid:5) S (cid:1) i · S (cid:1) j + D (cid:4) i (cid:3) j (cid:8) S (cid:1) i · S (cid:1) j r ij − 3 (cid:1) S (cid:1) i · r (cid:1) ij (cid:2) (cid:1) (cid:1) S (cid:1) j · r (cid:1) ij (cid:2) r ij (cid:9) . (cid:1) (cid:2) Here J is an exchange coupling constant, S (cid:1) i and S (cid:1) j are spin variables defined on sites i and j , r i , j is the distance betweenspins at i and j , and D is the dipole strength. The sum in thefirst term is over first neighbors and the sum in the secondterm considers a cut-off in the dipolar interaction up to aneighbor r ij (cid:6) r cut . As will be discussed in the following ourresults showed that the cut-off in the dipolar interaction hasto be taken very carefully. We can understand the physicalmodel described by Hamiltonian Eq. (cid:1) (cid:2) as follows: the firstterm (cid:1) the exchange interaction (cid:2) tends to align the spins ofneighboring sites. The second term (cid:1) the dipole interaction (cid:2) isdivided in two parts: the first one tends to align the spinsantiferromagnetically. The second part tends to align thespins along the direction of the unity vector that connects thesites i and j . At the border of the system the magnetic mo-ments are aligned tangent to the border satisfying both, thecondition that minimizes the exchange interaction and thesecond part of the dipole energy term. In a vortex configu-ration the first part of the dipole interaction is only mini-mized for spins at sites in opposing positions in relation tothe center of the vortex.The dipole interaction is in general very difficult to treatin any analytical or computational calculation due to its longrange character. Several works that deal with magneticnanodots use a variation in the Hamiltonian Eq. (cid:1) (cid:2) by con-sidering an anisotropic interaction (cid:4)(cid:1) S (cid:1) i · n (cid:1) i (cid:2) to replace thedipole term. Here, n (cid:1) i represents a unit vector perpendicular tothe surface and to the borderline of the system. This termcontributes positively to the total energy, therefore, it forcesthe spins to be perpendicular to n (cid:1) i , competing with the ex-change term. The effect produced by this anisotropic interac-tion is similar to that one of the dipole term. It favors themagnetic moment into a configuration tangent to the borderof the system and parallel to the disk plane. The energy dueto this term is minimized when the magnetic moments ar-range themselves in a curling vortex structure. The low tem-perature properties of the Hamiltonian with the anisotropicterm is similar to the one obtained by using the long rangedipole interaction. However, the high temperature and thedynamical behaviors are quite different. As we want to ex-plore the model beyond its low temperature properties, wetreat the system by using Eq. (cid:1) (cid:2) . So far, much of the theo-retical and computational work done to understand the be-havior of vortices in magnetic nanosystems consider a 2Dmodel. Although, the results obtained by using this simplifi-cation are in quite good agreement with experimental find-ings, part of the present work is dedicated to discuss theinfluence of an additional layer in the 2D model. The nan-odot is defined as follows. An number of magnetic particlesis distributed over the lattice points of a 2D array. A circle ofsize L centered in a previously chosen cell is drawn over thearray. Here L is measured in units of the lattice parameter a .The sites outside the circle are erased. The sites left insideform the nanodot, that is the object of our interest. If we areinterested in a two layer nanodot the building process issimilar. In this case the nanodot will be a small cylinder ofdiameter L and height a . As a matter of simplification fromnow on distances are measured in units of a , defined as thedistance between first neighbors sites in the lattice.A numerical approach to study this model is always very et al. J. Appl. Phys. , 053903 (cid:2) (cid:1)
Downloaded 02 Jun 2010 to 150.164.14.44. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp ime consuming since we must consider the interactions ofeach spin on a site i with all others in the system. It makesthe computer time prohibitive for large values of L . In orderto reduce the computational time much of the work done sofar considers a cut-off in the dipolar interaction up to aneighbor r cut . However, the introduction of a cut-off createsdistortions in the ground state of the system as will be dis-cussed below. In order to understand the effect of the cut-offwe work the model defined by Hamiltonian Eq. (cid:1) (cid:2) with andwithout a cut-off in the three different lattices: triangular,hexagonal, and square. For each of them we varied the di-pole strength, D / J , in the range (cid:5) (cid:6) , for several sizes, L ,with 10 (cid:4) L (cid:4)
90. Without loss of generality we studied thetriangular and the square lattices using only one layer, (cid:1) z = 1 (cid:2) . Some exploratory results showed that the results for thetwo layer (cid:1) z = 2 (cid:2) system are similar to those of the hexagonallattice.To obtain the magnetic nanodot ground state of themodel we used a numerical Metropolis MC method com-bined with simulated annealing. The simulated annealingapproach is a generalization of the MC method to search forthe ground state of a given system. We start with the systemat a high temperature configuration. Then, a process of cool-ing is done slowly until a very low temperature is reached.As temperature decreases the magnetic moments in the sys-tem organizes themselves in a uniform structure with mini-mal energy. If the system is well behaved enough it is ex-pected that the low temperature configuration approaches theground state as close as we want. In our calculations we takethe initial configuration at random, which corresponds to in-finite temperature. The lower temperature is taken as T = 10 −2 J / k B . From now on we consider J = 1 and the Boltzmanconstant k B = 1 so that the energy and temperature are mea-sured in units of J and J / K B , respectively. As a matter ofcomparison we choose in some cases the initial configurationas a vortex like structure. We found that the results for theground state were quantitatively the same when comparedwith those obtained by using the disordered initial state. Inour plots the error bars are smaller than the symbols whennot shown. The results discussed in this work were obtainedfor the hexagonal lattice when not explicitly written. III. RESULTS AND DISCUSSION
In order to understand the influence of the cut-off and ofthe disk size in the ground state we simulated disks of sev-eral diameters (cid:1) (cid:6) L (cid:6) (cid:2) for different cutoffs. We ob-served that the most stable configuration can be a vortex or acapacitorlike structure (cid:1) see Fig. 2 (cid:2) , depending on the value ofthe dipole interaction, D . Our results are shown in Fig. 3 in aplot of D c as a function of the disk diameter for severalvalues of r cut . Here, D c represents the value of the dipoleinteraction at the crossing over value, when the ground statechanges from the vortex to the capacitor configuration, la-beled as III and I , respectively, in the figure. It is also showna third state, labeled II , where the most stable vortex has anout-of-plane component. As a matter of clarity the regionwhere the out-of-plane vortex appears is shown only for thesimulation with no cut-off (cid:1) r cut = L (cid:2) . As can be seen in the Fig. 3 the choice of the cut-off is important in the determi-nation of the border between the vortex and capacitor re-gions. Because of that, we decided to use no cut-offs in ourcalculations even that implying in a longer CPU time forperforming our calculations.In Fig. 4 we plot the border lines defining the regionswhere the planar vortex, the out-of-plane vortex and capaci-tor configurations are more stable for all three types of lat-tices and r cut = L . The figure can be understood as follows.Based in the model described by the Hamiltonian Eq. (cid:1) (cid:2) it ispossible to find a set of values of D that minimizes the en-ergy of the vortex configuration. To minimize the exchangeinteraction, at the center of the vortex, the magnetic momentstend to align in a direction perpendicular to the plane of thedisk, which in turn maximizes the dipolar interaction. Thevortex configurations of minimum energy can be in-plane orout-of-plane at the center of the vortex. This behavior de-pends on the size of the system L . The value of D for vortexconfigurations with out-of-plane components decreases withincreasing the value of L because the contribution of thedipolar interaction (cid:1) long-range (cid:2) becomes greater than the ex-change interaction.In a two layer system, the additional layer acts as an FIG. 2. (cid:1)
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Schematic view of the vortex and capacitorlike con-figurations are shown in the left and right hand sides, respectively.FIG. 3. (cid:1)
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Diagram for the vortex formation in the case of theone layer hexagonal lattices. It is shown the critical values of the dipoleinteraction strength D c , in units of J , as a function of the lattice size, L , forseveral values of the cut-off in the dipole interaction, r cut . The circles cor-respond to the infinite range interaction. The lines separate two regions withdifferent ground states. Regions I and III have a capacitor and a vortex (cid:1) without out-of-plane component (cid:2) in the ground state, respectively. As r cut increases the lines separating the phases approach the asymptotic line r (cid:7) .The shaded area represents a region (cid:1) II (cid:2) where the most stable configurationhas an out-of-plane component at the center of the vortex. et al. J. Appl. Phys. , 053903 (cid:2) (cid:1)
Downloaded 02 Jun 2010 to 150.164.14.44. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp nisotropy so that the net effect is similar to the introductionof an exchange anisotropy in the system. It is well knownthat in an infinite system described by the anisotropicHeisenberg model in two dimensions, an out-of-plane ex-change anisotropy can cause the effect of lowering the en-ergy necessary to the vortex develop the z component. Ina system with exchange anisotropy A (cid:4) s iz s jz , there is a charac-teristic value of the exchange anisotropy, A c (cid:10) Because of that, we can expect that the out-of-plane phase appears at lower values of L and higher D / J when compared with the one layer case. In Fig. 5 we showthe diagram for a z = 2 system. The results confirm the ex-pected picture. The diagram is similar to the one obtained for z = 1 but with the out-of-plane phase starting at lower valuesof L and higher D / J .We observed that in both cases, z = 1 and 2, the capacitorand the vortex states regions are separated by a transition linethat asymptotically tends to a constant, (cid:1) D (cid:2) . In any case thisline can be very well adjusted by a function D c = D + 1 A (cid:1) BL (cid:2) . (cid:1) (cid:2) The values of the parameters are given in the Table I. Al-though, this is an ad hoc expression, a finite size scalingbehavior of the boundary between the capacitor and vortexphases is clearly indicated. The constant D = D c line separatesthe in-plane and the out-of-plane vortex phases. We can de-scribe this result as follows. Due to the competition betweenthe dipolar interaction responsible for the formation of thevortex and the exchange ferromagnetic interaction, the sys-tem may develop a component in the z direction, perpendicu-lar to the plane of the vortex. The z component is restrict toa region around the center of the vortex. For a small disk ofdiameter L , the influence of the edge is large dominating thebehavior of the system. The energy due to a misalignmentbetween the spins in the edge and in the center of the systemis large enough to compete with the exchange energy. In sucha case the vortex is expected to be planar. However, for evenmoderate disk sizes, the border plays no role. The out-of-plane core extends for only a few lattice constants allowingthe spins to develop a z component in the central region.An important question is the thermodynamic behavior ofthe nanodot. As temperature increases, vortices, and antivor-tices are created in the system. They appear always as pairs.The energy associated with the pair excitation is given ap-proximately by ln r v − a v , where r v − a v is the distance betweenthe vortex and antivortex centers. In Fig. 6 we show two spinconfigurations in a hexagonal lattice of size L = 30 for D / J = 0.1 and T / J = 0.7. Figure 7 shows the vortex density as afunction of temperature. We observe that there is a thresholdbelow which there are no pairs present in the system. FIG. 4. (cid:1)
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Diagram for the vortex formation in the three differ-ent types of the one layer lattices studied. Squares and diamonds correspondto the square and the hexagonal lattices, respectively. The inset shows theresults for the triangular lattice. Region I and III have a capacitor and avortex in the ground state, respectively. The shaded area (cid:1)
Region II (cid:2) repre-sents a region where the most stable configuration has an out-of-plane com-ponent at the center of the vortex. The lines separating the capacitor and thevortex states regions were adjusted using the equation D c = D +1 / A (cid:1) BL (cid:2) with appropriate values of the constants D , A , and B , for each typeof lattice.FIG. 5. (cid:1) Color online (cid:2)
Diagram for the vortex formation in a two layerhexagonal lattice (cid:1) open symbols (cid:2) compared with the one layer hexagonallattice (cid:1) filled symbols (cid:2) with r cut = L . The regions I, II, and III are defined asin Figs. 3 and 4. TABLE I. Parameters used in the Eq. (cid:1) (cid:2) . D A B
Triangular 0.098 3.798 0.002Squared 0.011 4.417 0.002Hexagonal (cid:1) z =1 (cid:2) (cid:1) z =2 (cid:2) (cid:1) Color online (cid:2)
Typical configuration of the vortex-antivortex distri-bution in a hexagonal lattice at T =0.7 for D =0.1 and L =30. Figure 6 (cid:1) a (cid:2) shows the spin configuration and the location of the vortices and antivorti-ces. It is possible to see one unpaired vortex and five pairs of vortex-antivortex (cid:1) plus and minus signs indicate vortex and antivortex, respec-tively (cid:2) . Figure 6 (cid:1) b (cid:2) shows only the vortex and antivortex positions inside thecorresponding skeleton of the hexagonal lattice of Fig. 6 (cid:1) a (cid:2) . et al. J. Appl. Phys. , 053903 (cid:2) (cid:1)
Downloaded 02 Jun 2010 to 150.164.14.44. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp he same behavior is observed for the square and trian-gular lattices. At low temperature only one vortex survives inthe system. By using spin dynamics we have obtained somepreliminary results showing that if the vortex has an out-of-plane component, it can flip at random as temperature in-creases. It happens in a regime far before the first vortex-antivortex pair appears. The vortex density, (cid:8) v − a , is zero upto T / J = 0.45, however, the vortex can flip as early as at T / J = 0.10. This observation is in contrast with the reportedmechanism of creation-annihilation for switching the vortexcore discussed in Refs. 3 and 18. A possible explanation ofthis result is as follows. The measurements performed inRefs. 3 and 18 were taken at very low temperature, in aregime that even the presence of spin wave excitations arenot enough to turn the out-of-plane vortex polarization.When a magnetic pulse is applied, it excites adiabatically thesystem elevating its temperature beyond the pair creationthreshold. At the same time, spin waves are excited. The spinwaves excitations can switch the vortex core. We believe thatthe observed phenomenon can be a fortuitous effect. We alsobelieve that a much more careful simulation has to be donein order to decide about the correct mechanism responsiblefor the switching of the out-of-plane vortex component. Inparticular, a rigorous statistical study is of paramount impor-tance. IV. CONCLUSION
In this work we investigated, via MC simulation, theconditions for vortex formation in quasi-2D magnetic dots.We used a model Hamiltonian with exchange and dipolarinteractions for square, triangular, and hexagonal lattices.Our results showed that a cut-off in the dipolar interactioncan give a good approximation only for large dot size andlarge cut-off radius. Besides that, a finite size scaling, as D c = D + 1 / A (cid:1) BL (cid:2) , is proposed to describe the cross overbetween a capacitor-like state to a vortex state. This behavioris obeyed by the three types of lattices. Inside the vortex phase region it is possible to identify two types of vorticesseparated by a constant D = D c line: An in-plane and an out-of-plane vortex. We observed that the out-of-plane phasedoes not appear for the triangular lattice. In the case of a twolayer system we observed that the extra layer of dipolesworks as an effective out-of-plane anisotropy, inducing alarge z component at the center of the vortex in agreementwith the experimental results reported in Ref. 15. We suggestthat in a real system, where a multilayer dot is considered,the range where the out-of-plane vortex exists can be consid-erably large. Also, we analyzed the mechanism responsiblefor the switching of the out-of-plane vortex component. Incontrast to some reported results, we found that the switch-ing mechanism is different from the creation-annihilationvortex antivortex process. ACKNOWLEDGMENTS
We are grateful to Dr. L. A. S. Mol for very fruitfuldiscussions. This work was partially supported by CNPq andFAPEMIG (cid:1)
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