Micromagnetic studies of vortices leaving and entering square nanoboxesDaniel Dotse and Anthony S. Arrott Citation: Journal of Applied Physics 97, 10E307 (2005); doi: 10.1063/1.1851671 View online: http://dx.doi.org/10.1063/1.1851671 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/97/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Micromagnetic calculation of dynamic susceptibility in ferromagnetic nanorings J. Appl. Phys. 105, 083908 (2009); 10.1063/1.3108537 Three-dimensional micromagnetic finite element simulations including eddy currents J. Appl. Phys. 97, 10E311 (2005); 10.1063/1.1852211 Magnetic normal modes of nanoelements J. Appl. Phys. 97, 10J901 (2005); 10.1063/1.1852191 Micromagnetic simulation studies of ferromagnetic part spheres J. Appl. Phys. 97, 10E305 (2005); 10.1063/1.1850073 Spin-polarized current-driven switching in permalloy nanostructures J. Appl. Phys. 97, 10E302 (2005); 10.1063/1.1847292

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Micromagnetic studies of vortices leaving and entering square nanoboxesDaniel Dotse and Anthony S. Arrotta!

Center for Interactive Micromagnetics, Department of Physics and Chemistry, Virginia State University,Petersburg, Virginia 23806

sPresented on 10 November 2004; published online 6 May 2005d

The passage of the virtual vortex of theC state into a square cross-section nanobox proceeds by thedevelopment of a direction of polarization to break the symmetry. This happens in a dynamictransition for nanoboxes below a certain height, while for higher nanoboxes there exists a stablepolarizedC state over a narrow range of applied fields that increases as the square of the excess ofthe height above the critical height. ©2005 American Institute of Physics.fDOI: 10.1063/1.1851671g

A decade ago Dalhberg and Zhu1 popularized the experi-mental and theoretical synergism between magnetic micros-copy and micromagnetics. Microscopy glimpses the behaviorof patterned materials useful in information technology. Mi-cromagnetics, though simplified by approximations and ne-glect of some details, provides complete descriptions of themagnetic configurations and what might happen in transi-tions between states. Great progress has occurred in bothexperiment and theory in recent years.2 Yet, because this is acomplex field, there continue to be subtleties that have es-caped attention. One of these is the nature of the transitionsbetween two well-known states of a magnetic nanobox ofsquare cross section with the height less than the width; theconfigurations are the simple vortex state and theC state.The subtle effects reported here are more of a computationalcuriosity than of practical importance. Experimentally thesewould be difficult to observe because they involve symmetrybreaking that is sensitive to defects and barriers that aresmall compared to room-temperature thermal activation en-ergies.

The simplest vortex state has the magnetization of thecore of the vortex and of all four vertical edges in the samesense along the shortz axis, about which the magnetizationcirculates. When a field is applied perpendicular to thez axis,e.g., along thex axis, the vortex center moves away from thez axis in they direction. At a critical exit field the vortexbecomes unstable and exits through they face. The exitingvortex leaves behind a virtual vortex state, usually called theC state, with its center outside of the box. There is a range ofheights of the nanobox for which theC state is stable in theexiting field. The field can then be lowered to observe thevirtual vortex reentering the box. Simple micromagnetic cal-culation of the exit field and the field for reentry led to someerratic results for which there is a simple explanation and alesson for practitioners of the art of micromagnetics.

This kind of effect is well established as it has to do withthe relative stability of three states, which could be desig-nated as 1, 0, and −1, or up, down, and in the plane. Newelland Merill have referred to this as a three-tined pitchforkbifurcation.3 Here the problem is that the real vortex is po-

larized, either up or down, but the virtual vortex can be po-larized or unpolarized because it does not have a real core.For the virtual vortex to reenter the box, symmetry must bebroken, for example, by a small applied field along thezaxis. When the vortex leaves, the field is sufficiently highthat the virtual vortex is unpolarized. On lowering thex-axisfield theC state becomes unstable with respect to the vortexreentering the box at a field that depends on the bias appliedalong thez direction. The polarization increases in a dynamicjump as the virtual vortex passes in through they face.Above a critical heightdc, the transition is not directly fromthe unpolarizedC state to the vortex state, but rather througha polarizedC state that is stable even if thez-axis bias fieldis removed. The vortex state, the unpolarizedC state, and thepolarizedC state are compared in Fig. 1.

The energy drop between theC state and the vortex stateat reentry is much larger than the energy barrier for reentry.The latter is so small that it would not be observable at roomtemperature. The smallness of the barriers and the very weaktorques associated with them present difficulties for micro-magnetic calculations, necessitating extrapolation and inter-polation techniques4 to determine the phase boundaries ofthe polarizedC state in a diagram of appliedx-axis field andheightd in the z direction.

We demonstrate this with square boxes of permalloy, Py.The square sides are 54354 nm2, which is large compared

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FIG. 1. The vortex statesleftd, unpolarizedC statescenterd, and the polar-izedC statesrightd, as viewed from the topsupperd and from the sideslowerdfrom which the vortex leaves and enters as theHx field is first increased andthen decreased.

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to the characteristic length of the competition between ex-change and magnetostatic demagnetization energies of Py,for which lex=5 nm. The critical height above which thepolarizedC state is stable isdc=32 nm. Below this the un-polarized virtual vortex enters without first achieving thestable polarized state. The upper limit of consideration isd=38 nm. At this height the vortex state on leaving in thecritical exit field passes through theC state to form a flowerstate from which reentry of the vortex is more complicated.

The vortex exits forHx=485 Oe independent ofd in therange of interest here. The direction of the net magnetizationkMxl of the C state is in the same direction as the field. Thefield for reentry depends strongly ond. At d=36 nm reentryoccurs through the polarizedC state which forms forHx

,360 Oe. The calculated reentry field atHx=222 Oe is sen-sitive to the choice of grid size in the micromagnetic calcu-lations, which were done using Scheinfein’s Landau–Lifshitz–Gilbert sLLGd micromagnetics simulator.5 Theeffect of grid size is shown in Fig. 2, wherekMzl is shown asa function ofHx. If the grid size is too large, the vortex itself,as calculated, is unpolarized, because the most energetic por-tion of the vortex hides at the intersection of four computa-tional cells.

Because we restrict the calculation to cubic grid cells,the choices of grid size are given by 54/n and the heights forcalculation limited to 54m/n, wheren andm are integers. Atd=36 nm one has the range of grid sizes shown in Fig. 2.This calculation is performed by first finding a region ofstability of the polarized grid state in the presence of anHz

=1 Oe bias field. The grid size must be substantially lessthanlex=5 nm to properly assess the range of stability of thepolarizedC state. For a 9-nm grid a vortex is hysteretic on

moving between cells. For the 18-nm grid the calculationgives a vortex with or without polarization.

Without the bias field, the field of reentry of the vortexdepends on waiting for an instability at the level of theround-off error of the computer to grow by over ten orders ofmagnitude. A conventional hysteresis loop calculation wouldfollow the central tine of the pitchfork without falling off theridge of instability. Indeed, it was the erratic behavior of asimple exercise in computing the properties of the box as afunction of the height that led to the discovery of the role ofthe stable polarizedC state.

Once a stable polarizedC state is found, it is used as astarting point for calculations to higher and lowerHx fields tofind the limits of stability. The transition from the polarizedC state to the unpolarizedC state is continuous with thezcomponent of the magnetization going to zero as sqrtsH−Hcd. The vortex enters as a first-order jump.

The range of stability of the polarizedC state is shownin the phase diagram of Fig. 3. Note that it takes a negativefield to drive the vortex in fordz,32.5 nm. For higher boxesa positive field is needed to keep the vortex out. The width ofthe region of stability increases as the square of the increasein height above the critical height that is found by curvefitting to bedc=32.0 nm. Becaused=32 nm was so close tothe critical height for the formation of the polarizedC state,the torques were exceedingly small.

The path method6 is used to find the dependence ofkMzlon Hz at each of several choices ofHx. The basic assumptionof the path method is that the equilibrium configurations arespecified with considerable accuracy by just statingkMzl,almost independent of the fields used to produce the configu-ration. For each configuration all the terms in the energy areevaluated. The sum of the demagnetization energy and theexchange energy is called the internal energy of the configu-ration. sThe anisotropy energy is also included, even thoughit is negligible.d The derivative of the internal energy withrespect to thekMzl V is the field in thez direction, whichwould hold the configuration in equilibriumsstable or notd.Unstable equilibrium solutions are investigated with micro-

FIG. 2. Calculations with different grid sizess18–1 nm cubesd of the netfractional z component of the magnetization for the polarizedC state be-tween the onset of polarization at high fields and the transition at low fieldsto the vortex state with larger polarization. The calculation for the large18-nm grid size produces a vortex that appears to have no polarization eventhough it comes from a polarizedC state.

FIG. 3. Phase diagram for the stability of the polarizedC state, shown as thedashed region between the calculated transition fieldsHx for the unpolarizedto polarizedC state transitionsdiamondsd and the polarizedC state to vortexstate transitionssquaresd for the indicated heightsdz of the square box ofpermalloy with 54-nm sides. The lower curvestrianglesd is the difference intransition fields fit tosdz−dcd2 with dc=32.0 as the best fit.

10E307-2 D. Dotse and A. S. Arrott J. Appl. Phys. 97, 10E307 ~2005!

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magnetics by readjusting the applied field from time to timeto maintain the equilibrium solution for a givenkMzl. Thetorques are so low that the approach to equilibrium is ex-tremely slow. The equilibrium values ofkMzl are found byextrapolation over many time constants. The result is judgedto have converged when the extrapolated values are nolonger changing with calculation time. The magnetizationcomponents and the energy are becoming completely expo-nential in approach to equilibrium after times of the order ofone time constant. All of the short wavelength perturbationsdamp out quickly, while the system as a whole moves toequilibrium at glacial speeds.

The above calculations were carried out for a box ratherthan a circular cylinder to avoid problems with serratededges and their effect on reentry of vortices. But this workwas stimulated by the excellent studies of vortices in cylin-ders by the Regensburg group.7

The Center for Interactive Micromagnetics under the di-rection of Professor Carey E. Stronach is supported by theAFOSR. This work constitutes part of a Masters Thesis byone of the authorssD.D.d in the Department of Chemistryand Physics at Virginia State University.

1E. D. Dahlberg and J. G. Zhu, Phys. Today48s4d, 34 s1995d.2R. Wiesendanger, M. Bode, A. Kubetzka, O. Pietzsch, M. Morgenstern, A.Wachowiak, and J. Wiebe, J. Magn. Magn. Mater.272–276, 2115s2004d.

3A. J. Newell and R. T. Merrill, J. Appl. Phys.84, 4394s1998d; G. Ioossand D. D. Joseph,Elementary Stability and Bifurcation Theory, 2nd ed.sSpringer, New York, 1990d.

4A. S. Arrott, Introduction to Micromagneticsin Ultrathin Magnetic Struc-tures, edited by B. Heinrich and J. A. C. BlandsSpringer, Berlin, 2004d,Vol. 4.

5M. R. Scheinfein and A. S. Arrott, J. Appl. Phys.93, 6802s2003d.6http://llgmicro.home.mindspring.com/7M. Rahm, M. Schneider, J. Biberger, R. Pulwey, J. Zweck, D. Weiss, andV. Umansky, Appl. Phys. Lett.82, 4110s2003d.

10E307-3 D. Dotse and A. S. Arrott J. Appl. Phys. 97, 10E307 ~2005!

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