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Properties of magnetic nanostructures

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Properties of magnetic nanostructures

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Content PROPERTIES OF MAGNETIC NANOSTRUCTURES by Wen Zhang A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PHYSICS) August 2010 Copyright 2010 Wen Zhang Dedication To my family ii Acknowledgments First I would like to express my gratitude to my PhD advisor, Professor Stephan Haas, for his guidance on my research. He is a great mentor who has given me tremendous support and encouragement during my PhD study. It is he who has lead me into the area of nanomagnets and who has brought me a lot of insightful ideas and advice. I would like to thank Professor Aiichiro Nakano for his enlightening classes and many fruitful discussions. I also thank Dr. Noah Bray-Ali, Dr. Rong Yu, Dr. Alioscia Hamma, Letian Ding, Yung-Ching Liang and all my colleagues and friends for many invaluable discussions. I express my love and gratitude to my grandpa, Yuxi Zhang, my parents, Zhao- liang Zhang and Guohong Tang, for their love and support throughout my life. Specially, I express my love and gratitude to my wife, Yaqi Tao, for her continuous encouragement and countless suggestions. I also want to send my special love and gratitude to my grandma, Jinyi Chen, who devoted her last twenty year to my growing. FinallyIwouldliketothankProfessorsAiichiroNakano,GeneBickers,Jianfeng Zhang and Werner D¨ appen for serving on my PhD advisory committee. iii Table of Contents Dedication ii Acknowledgments iii List of Tables vi List of Figures vii Abstract xiii Chapter 1: Introduction 1 Chapter 2: Model and Methods 8 2.1 Model .................................. 8 2.2 Dimensional Analysis and Scaling Approach ............. 14 2.3 Cartesian Coordinates Fast Multipole Method ............ 19 2.3.1 Algorithm ............................ 20 2.3.2 Formalism............................ 23 2.3.3 Performance Optimization ................... 27 2.3.4 Derivation of Eq.2.19...................... 34 2.3.5 Pseudocode for FMM...................... 34 Chapter 3: Phase Diagram of Nanomagnets 37 3.1 Shape, anisotropy, and lattice structure................ 37 3.2 Particles with Core Structure ..................... 41 3.3 Cylindrical Nanorings.......................... 43 3.4 Elliptically Shaped Particles ...................... 46 Chapter 4: Configurational Anisotropy of Square Nanomagnets 51 4.1 Chapter 4 Introduction......................... 51 4.2 Results.................................. 54 4.3 Blocking Temperature ......................... 62 iv Chapter5: MagnetizationReversalProcessesinMagneticNanorings 65 5.1 Chapter 5 Introduction......................... 65 5.2 Magnetic States............................. 69 5.3 Switching Processes........................... 75 5.3.1 One-step or single switching .................. 77 5.3.2 Two-step or double switching ................. 80 5.3.3 Triple switching......................... 82 5.3.4 Vertical O-V-O switching (O v -V-O) ............. 83 5.3.5 Twisted Triple Switching: O-T-V-O ............. 86 5.3.6 Twisted Double Switching (O-T-O) ............. 89 5.3.7 Circular Field .......................... 91 5.4 Stochastic Nature of the Switching Process.............. 92 5.5 Applications and Conclusions ..................... 93 Chapter 6: Conclusions 96 References 99 v List of Tables 2.1 Performance for (p m =3, α m =0.54) ................. 29 2.2 Performance for e=0.05 ........................ 30 vi List of Figures 1.1 (a) SEM images of arrays of nanomagnets with different shapes [21] (b) SEM images of various shapes [103] (c) TEM image for aggre- gated rings. A single ring is shown in the inset for clarity. [102] . . 3 1.2 (color online) (a) Measured anisotropy field of 5nm thick nanomag- nets of various shapes.[21] (b) Hysteresis loops measured as a func- tion of diameter d and thickness t from circular nanomagnets. [17] 4 2.1 Computational complexity for the benchmark system. The black squarerepresentsthebrute-forcecalculationwithcomplexityO(n 2 ), whereas the red dot and green triangles stand for the FMM with two different sets of parameters (α m , p m ). The red line is a least square fit to the data........................... 28 2.2 The optimal number of particles in the smallest cell s o as a function of expansion order p m . The red line is fitted by the data (p m > 3). The number next to each data point is the actual s o . ........ 30 2.3 Optimal computing time versus accuracy. The curve is a guide for the eye. The inset table shows the optimal choice of parameter set(p m , α m ) and the corresponding computing time for each error e.31 2.4 ComparisonofperformanceofCartesiancoordinateFMM(CCFMM) (green solid square: p m =5,α m =0.71; red open square: p m = 3,α m =0.63),sphericalharmonicsFMM(SHFMM)(blackdot: p m = 5,α m =0.71) and FFT method (blue star). The inset is a log scale plot. Using Cartesian coordinates makes the FMM roughly three times faster. With p m =5,α m =0.71, the performance of CCFMM is comparable to the FFT method. When the system is sufficiently large or lower accuracy is satisfactory, CCFMM can be superior to FFT.................................... 32 vii 2.5 the recursive function to create Partners list.............. 35 2.6 the recursive function to create multipole moments for each cell. . . 35 2.7 the recursive function to calculate magnetic field at each point and total energy................................ 36 3.1 (coloronline)(a)Scaledphasediagramsforprismshapednanoparti- cles. The radii R are defined as the distance from the base center to the corner of the polygons. The extracted scaling exponents for the triangle (T), the square (S), the pentagon (P) and the hexagon (H) are 0.556 (T), 0.557 (S), 0.563 (P) and 0.559 (H) respectively. (b) The slope (k) of line separating phase I and II versus the square root of the cross section area of nanodot with unit radius. (c) Phase dia- grams of cylindrical nanoparticles with different anisotropies. Solid squaresrepresentcubicanisotropy(C)ofdifferentmagnitude. Open circles with different colors represent uniaxial anisotropy(U) with Ka 3 /D = 1 showing valid scaling behavior with η =0.56. Solid triangles represent combination of both anisotropies (U+C) with Ka 3 /D = 1. (d) The slope k versus the strength of the uniaxial anisotropy. ............................... 40 3.2 (color online) Scaled phase diagram of a single-domain cylindrical magnetic nanoparticle taking the vortex core into consideration. The black hollow circles represent the phase diagram for cylindrical nanoparticle with core free model taken from Fig. 1. The scaling exponent η=0.5. The inset shows the fitting of the core function to the MC result for the case of J /D = 100. ............. 42 3.3 Phase diagrams of a cylindrical nanoring for two different x. There are two competing ferromagnetic phases at small (R,H) and a vor- tex phase at large (R,H). Because of the finite inner radius R i , the onset of the phase transition line between the two ferromagnetic phases is shifted to finite values of R. Also, the vortex regime is more extended for larger R i . The blue lines in the figure are guides to the eye, indicating that the triple points form approximately a straight line. .............................. 44 viii 3.4 (a) For cylindrical nanorings, the phase transition line H c (R,R i ) separating the two ferromagnetic regimes is not straight, in contrast to the topologically connected objects discussed above. (b) Phase diagram for a cylindrical nanoring. (R i =6.3nm, J/D = 5000). The data “A” represent analytical phase transition lines calculated from (a), which are observed to coincide with the numerical results. (c) Height at the triple point (H t ) versus inner radius (R i ). The best fit for x=0.1 is H t =16.5−5(±0.03)×R i , whereas the best fit for x=0.06 is H t =12.7−4.95(±0.09)×R i . Hence, the two lines are approximately parallel, and can thus be collapsed via scaling with η=0.51. (d) The triple point radius (R t )versusx. ......... 45 3.5 Double vortex configuration for J/D = 10(x=0.002), R a /R b =2 (arrows represent the directions of magnetization). (a) Monte Carlo simulation result, ea 3 /D=21.12 (b) Our parametrization, ea 3 /D = 21.11, F between b and a is 0.990 (c) naive parametrization (two single vortices), ea 3 /D=20.91, F between c and a is 0.974. e is the energy per spin and Fisthefidelitydefinedinthetext. .... 47 3.6 Scaled phase diagram of an elliptically shaped magnetic nanoparti- cle (Ka 3 /D = 1 and Ja 3 /D = 5000) as a function of its semi-major axis(R a )andheight(H)withanaspectratio2. Thefourcompeting phases are (I) out-of-plane ferromagnetism, (II) in-plane ferromag- netism, (III) single vortex state and (IV) double vortex state. The scaling exponent is η=0.55....................... 49 4.1 (color online) (a) The angle between the magnetization and field ω as a function of θ for three different sets of parameter scaled from Ja 3 /D = 20000,l = 368a ∼ 110nm,h=37a ∼ 11nmwithη=0.55. (b) Energy per spin and magnetization as a function of θ (c) The anisotropy field H a calculated from two different quantities proving the validity of the Eq. (d) An conventional way to represent H a in polar plot (data were taken in the range [0, π/2) and plotted thrice to complete the circle because of the symmetry of the lattice and shape.).................................. 55 4.2 (color online) (a) Anisotropy field of 4 typical symmetry types (b) energy landscape correspondingly. The sets of parameters are as follows. F8: l=20, h=3, J=20; D4: l=20, h=2, J=100; E4: l=20, h=15, J=100; G8: l=20, h=10, J=50 . . ............... 57 ix 4.3 (color online) ∆ e versus height with R = 10, J = 10. Insets show the anisotropy field in different cases. ................ 58 4.4 ∆ e versus edge length with h =2, J = 10. Insets show the anisotropy field in different cases.)................... 59 4.5 ∆ e versus exchange constant with R = 10, h = 2. Insets show the anisotropy field in different cases.)................... 60 4.6 ∆ e versusexternal magneticfield withR = 10, h=2 J = 10. Insets show the anisotropy field in different cases.) ............. 61 4.7 Monte Carlo simulation for magnetization versus temperature. x is the scaling factor. The parameter for the three sets of solid dot cor- responds to the three sets in Fig. 4.1. The two sets of open dots are calculated from x=0.02 and x = by scaling T = N C /N AorB . Both of them collapse with x =. T B is determined by the temperature at which M/M s =0.5. .......................... 63 5.1 (color online) Schematic representation of a magnetic vortex state in a disc (a) and a ring (b). The core region where spin points out of plane are exaggerated and represented by the height and green color. R o and R i are outer and inner radius of rings and w is their width. .................................. 67 5.2 Elementary topological defects: (a) edge defect ω=1/2, (b) edge defect ω=-1/2, (c) vortex ω=1, (d) antivortex ω=-1.......... 68 5.3 Schematic geometric magnetic phase diagram for nanorings with R i =6nm. Shaded area is the region where the onion state could be (meta)stable, and thus it is of greatest interest........... 70 5.4 Head-to-head domain wall. (a) onion state with transverse domain wall; (b) onion state with vortex domain wall. The numbers in the figure indicate the type of the topological defect. .......... 72 x 5.5 [69]( ReusedwithpermissionfromM.Laufenberg,AppliedPhysics Letters, 88, 052507 (2006). Copyright 2006, American Institute of Physics.) (color online) “(a) Experimental phase diagram for head- to-head domain walls in NiFe rings at room temperature. Black squares indicate vortex walls and red circles transverse walls. The phase boundaries are shown as solid lines. (b) A comparison of the upper experimental phase boundary (solid line) with results from calculations (dotted line) and micromagnetic simulations (dashed line). Closetothephaseboundaries,bothwalltypescanbeobserved in nominally identical samples due to slight geometrical variations. The thermally activated wall transitions shown were observed for the ring geometry marked with a red cross (W=730 nm, t=7 nm).” 73 5.6 Twisted states: (a) with single 360 o domain wall, (b) with two 360 o domain walls. The numbers in the figure indicate the type of defects. 74 5.7 Switching phase diagrams of magnetic nanorings. As indicated in the inset, h denotes the ring height, R i and R o are the inner and outer ring radii, and w = R o −R i is the width. A magnetic field H is applied in plane horizontally. The magnetic exchange length L ex is defined in the text. Phase diagrams are shown for the cases (a) 0.3L ex <h<L ex , (b)h>L ex ,(c)h< 0.3L ex . The shaded areas in the regime of small w’s represent rings whose remanence states are out-of-plane ferromagnets, and hence not relevant for switching. The color filled areas represent the different switching processes dis- cussed in the text. green: onion - onion (O-O); red: onion - twisted - onion (O-T-O); yellow: vertical onion - vortex - onion (O v -V-O); gray: onion - twisted - vortex - onion (O-T-V-O); cyan: onion - vortex - vortex core - onion (O-V-VC-O); white: onion - vortex - onion (O-V-O). The point ‘A’ indicates the onion - vortex tran- sition for discs. This critical point moves towards larger (w,R o ) values when h decreases. ....................... 77 5.8 Schematic switching process and hysteresis: (a) hysteresis of mag- netic discs, (b) one-step switching of rings, (c) double switching of rings, (d) triple switching of rings. .................. 78 xi 5.9 [118]( Reused with permission from Y. G. Yoo, Applied Physics Letters, 82, 2470 (2003). Copyright 2003, American Institute of Physics.) “Phase diagrams of two-step switching (open circles) and single switching (full circles) as a function the ring geomet- rical parameters. (a) For a ring width of 0.25 mm. (b) For a ring diameter of 2 mm. The solid lines define the boundary between the two different switching regimes.” ................... 79 5.10 Typical hysteresis curve for out-of-plane O v -V-O switching (R o = 70 nm, w = 7 nm, h = 10 nm). The insets are the snapshot during the O v -V transition. The areas with strong out-of-plane compo- nents are highlighted by open red circles. The big blue arrow points to a small step indicating the appearance of out-of-plane domains. . 84 5.11 Single domain to onion state transition with increasing ring diame- ter. Here, the height is fixed at 23 nm, and the width is fixed at 15 nm. The shaded area represents the transition region. ....... 85 5.12 Typical hysteresis curve for quadruple O-T-V-O switching (R o = 150 nm,w = 60 nm, h = 15 nm). The insets (a) and (b) are snap- shots during the transition. (a) is the onion state at remanence, while (b) is a typical twisted state, characterized by a 360 o domain wall. ................................... 86 5.13 Twisted triple switching (O-T-O): (a) Hysteresis curves for ring 1 (R o = 70 nm, w = 18 nm, h = 5 nm, black continuous line) and ring 2(R o = 42 nm, w = 16 nm,h = 14 nm, red dashed line); (b) spin configuration for ring 1 at H = 200 Oe; (c) spin configuration for ring 1 at H = 600 Oe; (d) spin configuration for ring 1 at H = 650 Oe. ................................... 89 5.14 Schematic diagram of twisted state reversal by circular field. .... 91 xii Abstract The study of magnetic nanoparticles has been evolving into a rich and rapidly growing research area during the last two decades, featuring many novel phe- nomena and potential applications. As their characteristic spatial dimensions are sufficiently small, the shape of these nano-particles becomes one of the dominant factors in determining their magnetic properties. A great variety of magnetic con- figurations which do not exist in bulk materials have been observed, and many new phenomena have been discovered recently. In this thesis, we study the static and quasistatic properties of nanomagnets, including their phase diagrams, anisotropy and magnetization reversal processes. A recently developed scaling approach is shown to be effective for nanomagnetic systems, enabling researchers to use small systems to study larger ones. Regarding the numerical method, instead of the widely used Fast Fourier Transform, we have adapted a Cartesian Coordinates based Fast Multipole Method (FMM) for point dipolar magnetic systems and showed its superiority. In this thesis, we first analyze the magnetic phase diagrams of single-domain nanoparticles with various geometric shapes, crystalline anisotropies and lattice structures. Then we mainly focus on two geometries: square and ring shapes. For square nanomagnets, a systematic study of the configurational anisotropy (CA) is performed varying their edge length, height, exchange coupling constant and xiii external magnetic field. The CA is a special type of anisotropy, existing only in nanomagnets, which results from the deviation of the magnetization configura- tion from a uniform distribution. We identified four types of anisotropy symme- try, clarifying some existing confusion in literature. Then, the relation between CA and superparamagnetism is discussed. For ring structures, we have focused on their switching hysteresis curves, namely on their magnetization reversal pro- cesses. We discovered three new types of switching mechanisms and present here complete in-plane uniform field switching phase diagrams. We review all the pos- sible (meta)stable states in ring structures and discuss in detail all six different types of switching process. Finally, their potential data storage applications are discussed. xiv Chapter 1 Introduction Magnetic thin films and nanoparticles have been extensively studied during the past two decades [26, 72, 18], not only because of their great potential for tech- nological applications, but also because of fundamental scientific interest. The continuous downscaling in experimental technology has pushed this research field further and further. On one hand, the resolution for the observation of nanoscale phenomena has been much enhanced, and on the other hand, it is now possible to grow more complicated and miniaturized structures. Many new phenomena come about by imposing geometric restrictions in one [26] or more dimensions [50, 8, 96]. One of the most exciting developments in magnetism is the formation of“zero-dimensional”nanometerscalemagnets. Theseso-callednanomagnetspos- sess magnetic properties very different from their parent bulk materials by virtue of their extremely small size. They have tremendous potential for applications. First, they are promising candidates for high-density data storage [15], as a form of non-volatile magnetic random access memory (MRAM). The ultimate storage density can be as high as 2Tbits/in 2 . This accounted for about 90% of the total nanomagnet market of over US $8 billion in 2007, and the total market revenue is projected to reach $16.4 billion by the end of the year 2012. Second, hybrid integrated magnetic-electronic devices which use the spin of the electrons as much as their charge to perform logic operations and information processing may be seen in the near future [90, 24]. Third, highly sensitive magnetic sensors utilizing super- paramgnetism have been built and are used everywhere, including the automotive 1 industry [18]. Fourth, nanomagnets may ultimately provide a suitable environ- ment for implementing quantum computation[28]. Finally, they have proven to be successful in biomedical applications such as drug delivery, for magnetic resonance imaging and as tumor cell killing agents [87, 89]. On the theory side, micromagnetic theory is well established, but unfortunately only few extremely simple structures, like homogenous spheres or infinite rods, can be evaluated analytically. Therefore, micromagnetic simulation [34] is an attrac- tive and powerful tool to explain experimental results and for device parameter engineering. Therearetwotypesofmodels: oneusescontinuousmediaapproxima- tion, andtheotheroneusesadiscretelatticedescription. Insolvingthecontinuous model, systems are discredited into mesh cells, either in a regular cubic mesh as it is done in the finite-difference method (FDM) or in an irregular mesh as in the finite-element method (FEM). The lattice model, however, assumes that the systems consists of interacting spins on a discrete lattice. A scaling law has been shown to exist for the lattice model. It can be motivated by a dimensional analy- sis [3], which suggests an incomplete similarity in magnetic systems. [25, 31, 109] Withitshelp, onecanusesmallsystemstopredictthepropertiesoflargersystems. The properties of magnetic system can be classified into two categories: (quasi)static and dynamic properties. We will focus on the former, which includes: groundstatespinconfiguration, anisotropyandthemagnetizationreversalprocess. Optimizationmethods(liketheconjugategradientmethod)canbeusedtoidentify the competing (meta)stable states. Monte Carlo simulation is also applied exten- sively for magnetic systems. [26] Another popular approach relies on the numerical integration of the Landau-Lifshitz-Gilbert (LLG) equation [71, 39], where an adi- abatic slow process is assumed. Experimentally, nanomagnetic structures have been produced with a wide range of spatial dimensions, extending from dozens of 2 Figure 1.1: (a) SEM images of arrays of nanomagnets with different shapes [21] (b) SEM images of various shapes [103] (c) TEM image for aggregated rings. A single ring is shown in the inset for clarity. [102] micrometers all the way down to several nanometers [102]. Besides, almost any geometric shapes can be produced, i.e. regular shapes such as disks, polygons, rings and ellipses and exotic shapes (see Fig. 1.1). [18] This freedom sparkles the researchers’imaginationandrevealsalotofnovelphenomena. Ashasbeendemon- strated a long time ago, the shape of the particle becomes one of the dominant factors determining its magnetic properties when the characteristic length scale is sufficiently small. [113] Different magnetic configurations have been observed, including vortex, leaf, flower, twisted, antivortex states etc. [22] In particular, the magneticvortex, alsoknownasanon-localizedsolitonhasrecentlybeenintensively exploredbecauseofitsapplicationpotentialandinterestingdynamics.[14,114,116] Besides creating various spin configurations, the freedom of choosing shapes and 3 materials makes it possible to engineer the anisotropy and hysteresis of these nano- magnets. Anisotropy controls the orientation of spins and suppresses the super- paramagnetismphenomenon. Hysteresisrevealsthemagnetizationreversalprocess under magnetic field and is a crucial property for data storage applications. Fig. 1.2 shows some experimental results of various anisotropy and hysteresis measure- ments. Figure 1.2: (color online) (a) Measured anisotropy field of 5nm thick nanomag- nets of various shapes.[21] (b) Hysteresis loops measured as a function of diameter d and thickness t from circular nanomagnets. [17] The dynamic properties of magnetic system can be described by the LLG equa- tion. There are quite a few open source simulation packages. The most popular one is OOMMF [30]. It has dominated the research field in the past decade, prob- ably due to its relatively easily understood algorithm and friendly user interface. More recent ones includes magpar [98] which allows the description of complex inhomogeneous geometries with adaptive mesh refinement. Even though it is more 4 sophisticated and supposed to be more accurate, it is not as popular as OOMMF. Other packages includes Nmag[32] and ψ-mag [5]. For finite temperatures, the stochastic LLG equation is introduced. Alternatively, temperature is naturally included in MC simulations. Concerning experimental fabrication techniques, interested readers can refer to Martin et al [72] and Kl¨ aui et al [61]. Experimentalists are able to produce two kinds of morphology: polycrystalline and epitaxial. In most cases, patterned nanomagnets are made of polycrystalline permalloy, supermalloy and Co, whose crystallineanisotropyisnegligible. Ontheotherhand,whenonewantstostudythe effects of anisotropy and underlying lattice structure, epitaxial face centered cubic (fcc), face centered tetragonal (fct) and hexagonal close-packed (hcp) structure Co rings can be produced. The imaging techniques can be classified into two groups: (i) intrusive techniques such as magnetic force microscope (MFM): this method affects the magnetic state of the sample because of the external magnetic field of the tip; (ii) non-intrusive techniques such as Photoemission Electron Microscopes (PEEM) and Scanning Electron Microscopy with Polarization Analysis (SEMPA): the magnetic states remain the same after the image is scanned. The spatial resolution nowadays can be as high as 10nm. Atomic resolution is also reported by SP-STM [115, 36]. To measure the hysteresis curve, the most popular techniques used are the magneto-optic Kerr effect (MOKE) and SQUID. Other techniques include measurement of the Hall resistance[105] and of the spin-wave spectrum[38]. In most cases, the hysteresis is measured in arrays of nanorings, typically larger than 10×10. Thus the transition is not sharp, which indicates the transition field distribution among these nanoparticles. A few measurements have been reported for single particles where the transition is sharp, but the size of these particles is fairly big. 5 This dissertation gives a review of current knowledge and our new findings of the (quasi)static properties of nanomagnetic systems. The structure goes as follows. Chapter 2 describes the micromagnetic model and computation methods we use. First, four types of interactions are shown to exist in ferromagnetic sys- tems. The sum of these contributions constitutes the Hamiltonian. The widely used expression in the continuous limit is also shown, and the LLG equation is introduced. Then a scaling law is shown to exist, which allows one to find the properties of a large system by studying its small counterpart. This law can be understood from dimensional analysis. Finally, different computational methods are explained and compared, with emphasis on the fast multipole method (FMM) used extensively in our research. Chapter 3 verifies the validity of the scaling approach and applies it to study the geometric phase diagram of magnetic nanoparticles. The effects of different shapes, crystalline anisotropy and lattice structure on the phase diagrams are analyzed. The influence of the vortex core on the scaling behavior and phase diagram is investigated. Furthermore, the scaling approach is applied to nanorings and elliptically shaped nanoparticles. Chapter 4 studies configurational anisotropy (CA). Anisotropy is one of the most important properties of magnetic systems. It brings about the preference for the magnetization to lie in a particular direction and is ultimately responsible for how a magnetic system behaves, and thus leads to suitable technological applica- tions. In bulk materials, magnetic anisotropy originates from the sample shape, crystalline structure, stress or directed atomic pair ordering. [88] Configurational anisotropy, however, only exist in nanomagnets, resulting from the various devi- ations of the magnetization configuration from the uniform distribution. One of 6 the most attractive features of this anisotropy is that it can be easily engineered, catering to various applications. This chapter discusses the CA in square nano- magnet and finally briefly discusses the blocking temperature which is the critical temperature for the ferromagnetism to superparamagnetism transition. Chapter 5 looks into ferromagnetic nanorings. Nanoring magnets exhibit many novel physical phenomena with promise for potential applications. The chapter focuses on the switching processes which are crucial for data storage applications. Complete switching phase diagrams are presented and the mechanisms behind all six types of magnetization reversal processes are discussed in detail. The chapter is finished by a brief discussion of data storage applications. 7 Chapter 2 Model and Methods In this chapter, a brief overview of the micromagnetic model and various simu- lation method is presented. The discussion starts with the introduction of the microscopic Hamiltonian for magnetic systems. The continuous approximation of the discrete lattice Hamiltonian is shown, and the connection between them is discussed. As a further step, the semiclassical dynamic model for damped gyro- magnetic precession, described by the Landau-Lifshitz-Gilbert equations, is briefly analyzed. Then a scaling law is shown to exist for the discrete lattice model, so that one can extrapolate the properties of a large system by studying its small counterpart. The essence of the scaling approach is discussed in detail. Finally, different computational methods are explained and compared, with emphasis on the fast multipole method (FMM) used extensively in our research. 2.1 Model Fromamicroscopicpointofview,amagneticsystemcanberegardedasacollection ofmagneticpointdipoles, whicharefixedinspaceandfreetorotatetheirdirection. There are generally four types of interactions among these dipoles. 1. Exchange interaction. This interaction needs to be analyzed by quantum theory, but its origin is nothing more than electrostatic interactions between elec- trons. In the 30s, Heisenberg [49] introduced an effective Hamiltonian: 8 H ex = − <i,j> J ij S i · S j (2.1) H ex is the famous Heisenberg model. Strictly speaking, this is a pure quantum model, and S is a spin operator. However, it has been shown that S can be treated classically as a simple vector in good approximation. J ij , named exchange integral, is an overlap of several electron wavefunctions which can be found by first principle calculations like Density Functional Theory (DFT). The exchange integral decays exponentially, so it is a short range interaction. For most materials, J ij is assumed approximately uniform and nonzero only for nearest neighbors, for which J ij = J. WhenJ> 0, thesystemisferromagnetic, whileitisantiferromagneticwhenJ< 0. In this study, I am focusing on the former case. 2. Internal magnetostatic interaction. This interaction represents the way the elementary magnetic moments interact over long distances within the system. For systems modeled by point dipoles, this interaction is solely dipolar. It is a microscopic description of the macroscopic magnetostatic energy in the static electromagnetism theory satisfying Maxwell equations. H dip = D i,j S i · S j −3( S i · ˆ r ij )( S j · ˆ r ij ) r 3 ij (2.2) where D is the dipolar coupling parameter and r ij is the displacement vec- tor between sites i and j. Note that if we let S be a dimensionless unit spin vector, then magnetic moments ( µ = |µ| S). Thus D = µ 2 µ 0 /4π = 8.6 × 10 −54 (µ/µ B ) 2 (Joules · m 3 ). We define the energy unit e.u. = D/a 3 = (µ/µ B ) 2 /(a/a t ) 3 3.1×10 −25 (Joules), where a is the lattice spacing and a t =0.3nm is its typical value. For most materials, the ratio D/Ja 3 falls in the range of 10 −3 9 and 10 −5 . The fact that this interaction is long ranged complicates the calculation. Different methods to tackle this problem is discussed in the end of this section. 3. Magnetocrystalline anisotropy energy. The origin of this term is the spin orbital coupling and the coupling to the anisotropic effective crystal exchange field acting on an atom. It is strongly related to the structure of the lattice. The anisotropy term U k can take various forms[55], among which the most common are uniaxial anisotropy U k = K i sin 2 θ i , where θ i is the angle S i makes with the easy axis, and cubic anisotropy U k = K i [α 2 i β 2 i +β 2 i γ 2 i +α 2 i γ 2 i ], where α i ,β i ,γ i are the direction cosines of S i . Note that K is the single site anisotropy energy (not an energy density). It is highly temperature dependent and typically Ka 3 /D = 1 ∼ 10. 4. External Zeeman energy. Magnetic moments will align with the external magneticfield. ThisphenomenonisdescribedbytheZeemanterm: U z = i S i · H i , where H i = B i ∗|µ| is the adjusted magnetic field. For uniform fields, it is simply U z = H · i S i . Another energy contribution we do not consider here is the elastic potential. It is an interaction between spontaneous magnetization and elastic stresses. [64]. In sum, the Hamiltonian (H) of a magnetic nanoparticle in a magnetic field consists of the above four terms: H = −J <i,j> S i · S j +D i,j S i · S j −3( S i · ˆ r ij )( S j · ˆ r ij ) r 3 ij +U k − i S i · H i (2.3) By simple manipulation, one can easily show that 1 e.u. corresponds to 1 field unit f.u. (µ/µ B )(a/a t ) 3 /0.003(Oe), and to 1 temperature unit t.u. (µ/µ B ) 2 /(a/a t ) 3 0.0225(K). Assume simply µ = µ B and a = a t ,wehave 10 e.u. 3.1×10 −25 Joules f.u. 333.3Oe t.u. 0.0225K (2.4) The competition between the various energy terms in H results in the interest- ing complexity of magnetic nanostructures. Among them, the fist two interactions are the dominant. The exchange interaction tends to align spins in the same direc- tion on a small scale, whereas the dipolar interactions are the source of various intriguingspinconfigurationsduetotheintrinsicanisotropicnature. Therearetwo major effects of dipolar interaction. First, it encourages spins to line up along the boundaries, resulting in the famous shape anisotropy. One well-known example is magnetic bars. The magnetization always align along the elongated direction. For the same reason, spins align in-plane in a flat thin film element. Second, it encour- ages spins to deviate gradually from parallel alignment on a large scale. This effect gives rise to various configurations like flower, leaf, onion, buckle, vortex states etc. Another example is domain walls. In the continuous approximation, the energy of the magnetic system becomes, H = A M 2 s Ω d 3 r|∇ M| 2 + µ 0 2 Ω d 3 r{ M( r)·( H d + H ext )}+U k (2.5) where H d is the demagnetization field (or stray field) and H ext is the external field. H d is calculated from Maxwell equations, satisfying the following equations. 11 ∇· H d = −∇· M ∇× H d =0, n· H d = n· Mat∂Ω n× H d = 0 (2.6) Note that A=2Jz/a connects the microscopic description and the continuous approach, where z is the number of sites in the unit cell. The dynamics of ferromagnetic systems is based on the model proposed by Landau and Lifshitz in 1935 [66], and successively modified by Gilbert in 1955 [40], leading to the famous Landau-Lifshitz-Gilbert (LLG) equation ∂ M ∂t = −γ M × H eff + α M s M × ∂ M ∂t (2.7) = − γ 1+α 2 M × H eff − αγ (1+α 2 )M s M ×( M × H eff ) (2.8) where γ is the absolute value of the gyromagnetic ratio (γ = ge/2m e c=2.21× 10 5 mA −1 s −1 , where g 2 is splitting factor ) and α Gilbert damping constant, depending on the material (typical values are in the range 0.001 to 0.1). H eff is the effective field taking all the interactions into account. When temperature effects are considered, a stochastic random field term H s (t) can be introduced into H eff . [6] It has a Gaussian distribution and is δ correlated in time with mean 0 and variance µ proportional to temperature (µ=2k B Tα/(γM s V cell )). 12 <H s (t) > =0 <H s,i (t)·H s,j (t+t ) > = µδ ij δ(t ) (2.9) Although the micromagnetic theory is well established, only few extremely simple structures can be solved analytically. Thus, the study of nanomagnets is commonly approached by a combination of experiments and numerical simulation. As picosecond and nanometer time and spatial resolution detection techniques are still limited, the microscopic details of switching processes and other dynamic transitions are currently mainly investigated by micromagnetic simulations. The first challenge in the micromagnetic is the evaluation of the dipolar interaction between magnetic moments. Since the interaction is long ranged, the complexity of a brute-force calculation will be O(N 2 ). One popular method to improve this performance is by Fast Fourier Transform (FFT), which reduces the complexity to O(N ∗log(N)). However, FFT suffers from the fact that it requires a regular lattice arrangement of dipoles, and a large number of padding areas have to be added when dealing with exotic geometries and open boundary conditions. Thus, the alternative, fast multipole method (FMM) has attracted increasingly more attention in the past several years [99, 111, 5, 41]. Another challenge lies in the nonlinearity of the LLG equation. Special requirements on the integration time steps need to be satisfied. There are quite a few open source simulation packages: OOMMF [30], magpar [98], Nmag [32] and ψ-mag [5]. Among them, the most popularpackageisOOMMF.Ithasdominatedtheresearchfieldinthepastdecade, probably due to its relatively easily understood algorithm and user interface. 13 2.2 Dimensional Analysis and Scaling Approach In simulating the system with Eq.5.2, the major technical problem is that the number of magnetic moments in systems of physical interest is of the order of 10 9 , which cannot easily be handled, even by high-end supercomputer facilities. To overcome this restriction, a scaling approach was recently proposed and demon- strated for cylinder [25] and cone [31] shaped nanoparticles. They showed that the phase diagram for an artificial small J = xJ (x< 1) could be scaled to the phase diagram for the original J according to L = x η L (η 0.55 and L can be R,H). The phase boundary for small J appears at small sizes which involve less number of spins so that lots of computing time is saved. Specifically, this method involves four steps: (a) pick an exchange coupling constant J = xJ, wherex< 1, and J is the exchange coupling for the real physical system, and compare the internal energy of the competing configurations for certain geometric parameters (R,H)to determine the one with the lowest energy; (b) repeat the above step for all points in the R−H parameter manifold to obtain the phase diagram for the chosen J ; (c) repeat the above two steps for different J (or x) ; (d) find a scaling exponent η to scale all length scales up to a factor (1/x) η , so that all the transition lines in the phase diagram corresponding to different J collapse onto a single line. This proposal is equivalent to dimensional analysis coupled with a statement of incomplete similarity. We seek, for example, to find the height H separating the vortex phase from the ferromagnetic phase(s). In addition to the two governing parameters J,D explicitly appearing in Eq. 5.2, we also have the radius, R of the 14 cylinder and the lattice constant a. Thus we seek a physical law for the critical height of the following form: H = f(J,D,R,a). (2.10) From dimensional analysis, only two of the four governing parameters have independent dimensions. Following convention, we choose the independent param- eters to be J and D, define the exchange length L ex = a Ja 3 /D, and express the scaling law in a dimensionless form: Π=Φ(Π 1 ,Π 2 ), (2.11) where Π = H/L ex ,Π 1 = R/L ex ,Π 2 = a/L ex , and the scaling function Φ does not depend on the governing parameters of independent dimension J,D. Now the typical values a ∼ 0.3nm, L ex ∼ 20nm give Π 2 1. We are tempted to suggest complete similarity with respect to the small, dimensionless governing parameter Π 2 .[3] Hence we consider the limit Π 2 =0: Π = Φ(Π 1 ,0) ≡ Φ 1 (Π 1 ), (2.12) 15 where Φ 1 is independent of J, D and a. By recasting in the original variables, we have that H = L ex Φ 1 (R/L ex ). Notice now the invariance of this relation under the following rescaling of the governing parameters and critical height: J = xJ, D = D R = x 1/2 R a = a H = x 1/2 H, (2.13) where, x is any positive number. One way to see this is to notice that all lengths entering Eq. 2.12 get rescaled by the same amount x 1/2 , thus the dimensionless ratios Π,Π 1 , are invariant. The invariance of Eq. 2.10 under this transformation is a consequence of dimensional analysis combined with complete similarity with respect to the dimensionless governing parameter Π 2 = a/L ex . Interestingly, the numerical calculations under the assumption of a core-free vortex phase, where only the magnetic moment exactly located at the center of the vortex has a component pointing out of the vortex plane, do not obey this scaling.[25, 31] Instead, they exhibit only incomplete similarity with respect to Π 2 = a/L ex .[3] Namely, for small values of Π 2 1, we have Π=Π 1−2η 2 Φ 2 ( Π 1 Π 1−2η 2 ), (2.14) where, the constant η ≈ 0.55 does not follow from dimensional analysis. Here, Φ 2 is independent of J,D, and a. The special case of complete similarity is obtained when η=1/2. This implies that the physical law Eq. 2.10, is not invariant under 16 the transformation in Eq. 2.13. Rather, we find invariance under the modified transformation: J = xJ, D = D R = x η R a = a H = x η H. (2.15) This is precisely the transformation described in Ref. [25], and is a consequence of dimensional analysis combined with incomplete similarity with respect to the small dimensionless parameter Π 2 = a/L ex . This incomplete similarity results from the fact that there is a singularity in the magnetization function. Whenever we are presented with a scaling phenomenon such as Eq. 2.14, we have the opportunity to save considerable computational and experimental effort. The scaling law expresses a physical similarity between systems with different values of the governing parameters, so that we can use one to study the other. In particular, the authors of Ref. [25] suggest that we study small systems with small exchange constant J, which are less computationally intensive to simulate. Then one can use Eq. 2.15 to scale up the results to the large systems with large exchange constant that are of immediate physical and technological interest. The proposal is justified by the incomplete similarity of the physical law Eq. 2.10 with respect to the small, dimensionless parameter Π 2 = a/L ex . Other physical 17 quantities of nanomagnets may satisfy incomplete similarity, including dynamic and thermal properties.[76, 109, 74]. The scaling approach can be understood from another point of view. In some sense, it is similar to the finite difference method when one numerically evaluate Eq.2.5. With discretization, one has E = l 3 { A l 2 <i,j> [sin 2 θ i (φ j −φ i ) 2 +(θ j −θ i ) 2 ]− 1 2 i M i · H d i +K i sin 2 θ i } (2.16) where l is the length of the mesh cell and φ i ,θ i are the spherical polar coordinates of M i . Besides a discretization error for large mesh cells, the results should be invariant for different l’s. From Eq.2.16, one can see the prefactor l 3 is irrelevant if only the ground state is considered and an effective exchange coupling could be defined as J eff = J/l 2 . Meanwhile the system with J eff is equivalent to a shrunk system with smaller mesh cells keeping H d invariant. To better illustrate the above arguments, let’s say we have a system with lattice spacing a, exchange interaction constantJ andtypicalsizeL(Lcouldberadius, heightandetc). Soherel = a.We could solve it by mesh length l = a/ √ x (x< 1) and then J eff = xJ. This system in the meantime is equivalent to a system with mesh length l = a keeping the total number of mesh cells and M s invariant, which is just a system with reduced size L = √ xL and J = xJ. In other words, there is a complete similarity with η=1/2=0.5. However, it is slightly different for lattice model. Since in lattice model, the magnetostaticinteractionisrepresentedbydipolarinteraction, itdeviatesfromthe solution from Maxwell equation a little bit. The demagnetization field won’t keep invariant when the system size is scaled, giving rise to the incomplete similarity. 18 2.3 Cartesian Coordinates Fast Multipole Method The Fast Multipole Method was first introduced by L. Greengard et al [42, 43], and has been used ever since to speed up large scale simulations involving long ranged interactions. It has the charming advantage of O(N) complexity. Further- more, there is no constraint on the distribution of particles and the boundaries of the system. Hence it is extremely useful for magnetic fluid systems and nanomag- nets with exotic geometries. Last but not least, it is a scalable algorithm, i.e. it can be efficiently implemented in parallel [86]. With so many advantages though, the adoption of FMM in magnetic system is not widely applied, probably due to two reasons. One concern is that since there is a huge overhead to achieve O(N) complexity, the FMM becomes faster only for very large N. Actually this is true only when one considers a very high order expansion (say 10). In the context of micromagnetics, we will show that an expansion to the order of no more than 6 will be satisfying, and with some optimization procedure FMM will be superior even in a system of 10 3 dipoles. The other reason is that the standard implementation of the FMM algorithm is based on the well-known spherical harmonics expan- sion [52] of 1/r. For dipolar interaction which decays as 1/r 2 , however, it is not straightforward to apply. Further, the additional complexity of calculating spheri- calharmonicsactsasanotherbarrier. Toovercometheseshortcomings, expansions in Cartesian coordinate were proposed[100, 10], but they were not applied success- fullyuntilrecently[5]. Followingthesestudies, wehaveimplementedandoptimized the performance of the Cartesian coordinate based FMM in dipolar systems. A brief description of the algorithm and supplementary equations are provided first 19 in this section. In particular, we present a simple formula (Eq. 2.19) to calculate the multipole moments for point dipoles. Then the ways to optimize program per- formance in term of speed are shown. As a rule of thumb, the optimal number of dipoles in the smallest cell should be chosen according to Eq. 2.33, and the optimal choices of the expansion order and opening angle under different precision requirements are discussed. (p m =5, α m =0.71) is suggested for nanomagnetic simulation generally. By comparing with spherical harmonics FMM and FFT, we have shown that the Cartesian coordinate based FMM is appropriate for magnetic simulations due to their moderate accuracy requirements. In the appendix, the pseudo code and the derivation of Eq. 2.33 are provided. 2.3.1 Algorithm The crucial ideas of the FMM are: (i) chunk source points together into large cells whose field in remote cells is computed by a multipole expansion of the source points; (ii) use a single Taylor expansion to express the smooth field in a given cell contributed by the multiple expansions of all remote cells. As the details of the algorithm are discussed in previous literature [42, 111], here we will only briefly describe them, stressing those aspects which are special to our implementation. Instead of the regular geometric hierarchical traversals of the system, we follow a simple alternative way[111] by using recursive functions. This method simplifies the program significantly and is especially elegant for system with exotic geome- tries, since one only needs to recursively divide the system into halves. After setting up all the cells, the entire program mainly consists of three recur- sive functions traversing the entire system trice. Pseudocodes are provided in the Appendix. 20 1. Downward Pass: construct “partners” list (similar to the interaction list in the original FMM [42]). For each cell, the partners list contain those cells which are near to its parent but far to itself. Instead of the original geometric rules (near cell pairs are those cells touching each other), near cell pairs refer to those with opening angle α larger than certain value α m (α m < 1). The opening angle is defined as 2r/d (α m ≡ 2r/d), where r is the “radius” of two cells and d is the distance between their centers. Note that “radius” denotes the maximum distance between a corner of one cell and its center. With these definitions, it is easy to construct the partners lists. First, set the partners list of each cell to be empty. Then, for each cell except those cells without children (leaf cells), start from its partners list generated by its parent (for the root cell, this list is empty) and perform the following procedure: (i) add the children of all its near cells it to the partners list of each of its children, and delete these cells from its own partners list; (ii) add one of its children to the other child’s partners list (do it for both children); (iii) call its children to perform these tasks. In this way, we start from the root cell and recursively traverse its children until reaching leaf cells. For each leaf cell, add to another list called “nearPartners” those cells in the partners list which are near to it and delete them from its partners list. Finally put itself in its nearPartners list. 2. Upward Pass: construct the multipole moments for each cell. Start from the leaf cells and explicitly calculate their multipole moments according to Eq. 2.19 below. Then upward traverse all the other cells in binary tree hierarchy whose multipole moments are obtained by shifting the origin of its children’s multipole moments(seeEq. 2.23). Thisprocesscanbeeasilyrealizedbyarecursivefunction. 3. Downward Pass: construct the Taylor expansion of the smooth field in each cell. Start from the root cell and down traverse all the other cells in the binary tree 21 hierarchy. For each cell, inherit the Taylor expansion coefficients of the smooth field from its parents according to Eq. 2.26 and then add to it the contribution from the cells in the partners list according to Eq. 2.27. After these three steps, the only contributions to the magnetostatic field that are not counted come from dipole pairs among the leaf cells which are near to each other or within the same cell. So for each field point in the leaf cell, calculate explicitly Eq. 2.32, whichsum over all thedipolesin those cellsin the nearPartners list except the very dipole located at the field point. In the majority of implementations of the FMM, the potential is expanded in terms of spherical harmonics, which complicates the code and requires long computing time. Previous attempts were made to expand in Cartesian coordinates [100, 10, 111]. Initially, the expansion coefficients were calculated either by hand or in a computationally inefficient way, but they can in fact be obtained very fast nowadays[111] using Eqs. 2.28 and 2.29. The disadvantage of Cartesian expansion is that the number of terms below each order is larger than that with spherical harmonics, but this inefficiency is less than a factor of two up to order 8. We will show later that in most cases in the micromagnetic simulation, an order of no more than 6 is sufficient. 22 2.3.2 Formalism In this section, we provide all the necessary formulae to implement the above algorithm for nanomagnetic simulation, following Ref. [111]. To simplify the formalism, the following shorthand notations are defined: n ≡ (n x ,n y ,n z ) n ≡ n x +n y +n z r n ≡ x nx y ny z nz n! ≡ n x !n y !n z ! (2.17) Define the multipole moments of a given charge distribution q i : Q n = 1 n! i q i r n i . (2.18) The evaluation of Q n is straightforward for monopoles. For dipoles, however, thisisnotthecase. Onewaytoproceedisdividethesystemalwaysintocellswhich contain only one dipole at the center, based on the fact that only Q (1,0,0) , Q (0,1,0) and Q (0,0,1) survive for a single dipole m located at the origin, and they equal m x , m y and m z respectively. Nevertheless, this method is not convenient and even inapplicable for certain systems like ferrofluids. Moreover, having multiple dipoles in each leaf cell has performance advantages. This issue will be discussed in detail in Sec IV. Thus it is important to have an easy formula to calculate multipole moments for dipolar system. Previous studies have shown complicated formulas for spherical harmonics expansion [41]. For the multipole moments in Cartesian coordinate, it becomes much simpler, i.e. 23 Q n = 1 n! i m i ·∇r n i . (2.19) In order to derive this formula, we follow the standard procedure to treat a dipole m( r) as a limit of two point monopoles ±q located at r ± d when d → 0, keeping m=2q d. After Taylor expansion, Eq. 2.19 appears. If 1/| r− r i | is expanded in Taylor series at r i , the potential V( r)= i q i /| r− r i | can be expressed in a compact form in terms of the multipole moments, 1 | r− r i | = n 1 n! D n ( r)r n , (2.20) D n ( r) ≡ ∂ n ∂(−r) n ( 1 | r| ), (2.21) V( r)= n D n ( r)Q n . (2.22) In the upward pass (step 2), one needs to shift the origin of the multipole moments. The moments Q n about the origin O are related to Q n about position c (with respect to O)by Q n = p c p p! Q n−p . (2.23) The Taylor expansion of an arbitrary potential function V( r)is: V( r)= n 1 n! V n r n , (2.24) V n ≡ ∂ n ∂r n V( r)| r=0 . (2.25) 24 In the second downward pass (step 3), one needs to shift the origin of the V n when a child inherits V n from its parent. The child’s V n about position c (with respect to its parent’s origin O) is related to its parent’s V n by V n = p c p p! V n+p (2.26) Besides, one also needs to calculate the Taylor expansion coefficient in a given cell from its partner’s multipole moments. This is achieved by V n =(−1) n p D n+p ( c)Q p , (2.27) where c is the position vector of the center of this cell with respect to the center of its partner whose multipole expansion is Q p . It can be shown [111] that D n ( r) has the form: D n ( r)= 1 r 2n+1 p(p=n) F n (p)r p (2.28) Thus the problem of calculating D n breaks down to creating the table F n (p). Note that by definition of D n , we have: can be calculated by D n . D n+ˆ x ( r)= − ∂ ∂x D n = 1 r 2n+3 p F n (p)(−p x r p−ˆ x+2ˆ y −p x r p−ˆ x+2ˆ z +(2n+1−p x )r p+ˆ x ) (2.29) where ˆ x=(1,0,0). Meanwhile according to Eq. 2.28, 25 D n+ˆ x ( r)= 1 r 2n+3 p (p =n+ˆ x) F n+ˆ x (p )r p (2.30) Similarly one can obtain the such formulas for D n+ˆ y and D n+ˆ z (a permutation through x, y, z in above equations). Therefore each n, F n (p) can clearly be built from F n−ˆ ν (p) recursively starting from F (0,0,0) ((0,0,0)) = 1, where ν is chosen among x, y, z such that n ν > 0. Note that for each n, F n (p) is nonzero only for a few p’s. Thus it is better to store a list of ps with nonzero F n (p)foreach n and a list of F n (p) value accordingly. Once these are set up, D n can be calculated very fast. With Eqs. 2.19, 2.20 and 2.28, we can now discuss the criterion to determine near and far cell pairs which is crucial in the first downward pass (step 1). Suppose we truncate the terms after the p th m order. The error is of the order r pm /(d−r ) pm [111], where r, r are the radius of the source and the field cell and d is the distancebetweentheircenters. Inourimplementationr ≡ r ,thusr pm /(d−r ) pm = [α/(2−α)] pm . Obviously, the series will converge whenα< 1. Thus a maximum α m < 1 is chosen to determine whether two cells are near to each other. In Sec. III, we will discuss how to choose it optimally. Finally the smooth part of magnetostatic field is given by: H s = −∇V = n 1 n! V n ∇r n (2.31) And sum over pairs in the nearPartner list L. H i = H s i + j∈L,j=i m j −3( m i · ˆ r ij )ˆ r ij r 3 ij (2.32) 26 The total magnetostatic energy E = − i m i · H i in unit of µ 0 /4π. 2.3.3 Performance Optimization A benchmark where randomly oriented dipoles are arranged on a simple cubic lattice was adopted to test the performance of our program. All the results below are averaged over 50 random configurations and performed on an dual Intel P4 3.0GHz with 2G RAM. Fig. 2.1 demonstrated that the complexity of our program is O(n), and it begins to outperform the brute-force calculation around N c = 1000. The small deviations from linearity are caused by the change of hierarchy as the size of the systems increases. The two parallel lines with different colors and symbols correspond to two sets of α m and p m achieving the same accuracy. The shift between them indicates that there exists an optimal choice of α m and p m set for a given accuracy. In the following analysis, we choose a 32∗32∗32 cubic system, but the conclusions are independent of system size unless it is too small. Before discussing of the parameter set, there is another important issue which affects the performance significantly and needs to be elucidated first. It is the smallest cell size or the average number of particles (s) in the smallest cell. The value of s determines the number of levels in the binary tree and how many brute- force calculations are required on the finest level. Choosing an inappropriate s can deteriorate the performance by a factor of 10. Thus it is indeed a nontrivial issue. Obviously, the larger s is, the fewer levels and the more brute-force calculations will be required. Because of the fairly large overhead of FMM with respect to the brute-force calculation, it is naturally expected that s = 1 will not be a good solution. And it is also not good to make s too large, for we will lose the power of the FMM then. So there must be an optimal number s o which gives best 27 10 4 10 5 10 6 10 0 10 1 10 2 10 3 τ=0.79*N 1.10 (µs) τ=0.95*N 1.15 (µs) brute-force FMM (p m =2,α m =0.37) FMM (p m =3,α m =0.54) time τ (s) number of particles N τ=620*N 2.01 (µs) Figure 2.1: Computational complexity for the benchmark system. The black square represents the brute-force calculation with complexity O(n 2 ), whereas the reddotandgreentrianglesstandfortheFMMwithtwodifferentsetsofparameters (α m , p m ). The red line is a least square fit to the data. performance for a given parameter set. Meanwhile, since the overhead increases as the expansion order p m increases, the optimal s o is expected to be a function of p m and should increase with p m as well. On the other hand, α m determines the number of cells in the “near” region of each specific cell, but the ratio of the number of cells in the partners list and the number of cells in the nearPartners list should be independent of it. If this ratio remains constant, there is no reason to change s o for fixed p m . Therefore, s o only depends on p m . Though it is not too difficult to come to this conclusion, it is still not clear at all how to choose s o . Here, we offer a solution. Table I shows results for a 32 3 system with p m =3, α m =0.54. In this case, s o = 8. However as for arbitrary system sizes, we cannot divide them freely, the dependence of s o on p m is a discreet function, and different for different system sizes. We overcome this problem by averaging s o over 30 system size from 30 3 to 59 3 . The result is shown in Fig. 2.2, where each number beside the square points is s o for p m from 2 to 12. The line is a linear fit for those points with 28 p m > 3, indicating an approximate power law relationship between s o and p m and the power is approximately 2. As s o for a given system size will never be the value indicated in the graph anyhow, we reach an empirical equation as follows: s o = { 0.74p 2 m p m > 3 8 otherwise (2.33) It is very easy to use this equation to estimate s o . Lastly, for the same s o , one can have different divisions of the system, but it is always better to have the smallest cell as close to a cube as possible. Table 2.1: Performance for (p m =3, α m =0.54) s time 1 34.82 2 14.69 4 8.28 8 6.58 16 9.13 32 13.69 64 20.58 Based on above discussion, we propose a rule of thumb on how to choose the smallest cells: (i) calculate s o according to Eq. 2.33; (ii) the smallest cells do not have to be cubic and should contain as close to s o particles as possible; (iii) under the above two conditions, the smallest cells should be as close to cubic as possible. For example: (i) for 50 3 system with p m = 3, the best smallest cell choice is 1.5625*1.5625*3.125 where each cell on average contains 7.63 particles; (ii) for 40*40*10 with p m = 4, the best choice is 2.5*2.5*2.5 (∼15.6); (iii) for 32 3 with p m = 5, the best choice is 2*2*4(∼16). Now that we know how to divide the system, we can optimize the parameter set (p m , α m ). Use e ≡ [α m /(2 − α m )] pm as the error estimate. Obviously, the 29 2 4 6 8 10 1214 10 100 s o p m τ = 0.74*p m 1.98 7.1 8.6 12.3 17.3 24.6 34.6 44.2 55.2 71.5 89.1 102.8 Figure 2.2: The optimal number of particles in the smallest cell s o as a function of expansion order p m . The red line is fitted by the data (p m > 3). The number next to each data point is the actual s o . Table 2.2: Performance for e=0.05 p m α m time 2 0.36 14.15 3 0.54 6.71 4 0.64 6.80 5 0.71 6.22 6 0.76 7.36 7 0.79 8.14 8 0.81 10.65 error decreases as p m increases and α m decreases. To achieve certain accuracy, one can have various sets of (p m , α m ) combinations. The computational effort, on the other hand, increases as p m increases and α m decreases. This claim is easy to understand for p m , since larger p m results in more terms in the expansion. α m ,on the other hand, determines how many cells there will be in the partners list and nearPartner list. The smaller it is, the more cells there are in the lists, resulting in more computing effort. Therefore the computing effort varies for different sets 30 0.00 0.05 0.10 0.15 0.20 10 20 30 time τ (s) e e p m α m τ 0.2 3 0.74 2.8 0.1 3 0.63 4.2 0.05 5 0.71 6.2 0.01 5 0.57 11.5 0.005 6 0.58 14.6 0.001 6 0.48 21.9 Figure 2.3: Optimal computing time versus accuracy. The curve is a guide for the eye. The inset table shows the optimal choice of parameter set(p m , α m ) and the corresponding computing time for each error e. of (p m , α m ). Table II shows one example for e=0.05. Obviously, the choice of (p m =5, α m =0.71) is optimal in this case. The inset table in Fig. 2.3 gives the optimal choices of (p m , α m ) for different accuracies. The curve shows the optimal computing time as a function of accuracy. The computing time grows exponentially whene< 0.05, so it suggests to choose e=0.05 if the error is acceptable. In fact, the error of magnetostatic energy for e=0.05 is less than 1% except for configurations whose energy are close to zero. This already reaches the requirement of micromagnetics where other uncertainties may make it pointless to go beyond such accuracy. Thus (p m =5, α m =0.71) is suggested, if no other requirements exist. Even if higher accuracies are required, (p m =6, α m =0.48) will definitely be sufficient, since the average percentage error in this case will be 0.02%. Thus, the inefficiency caused by Cartesian coordinates is always less than 2. 31 0.0 5.0x10 5 1.0x10 6 1.5x10 6 2.0x10 6 0 500 1000 1500 2000 2500 10 4 10 5 10 6 10 -1 10 0 10 1 10 2 10 3 CCFMM (p m =5,a m =0.71) CCFMM (p m =3,a m =0.63) SHFMM (p m =5,a m =0.71) FFT Computing time number of particles Figure2.4: ComparisonofperformanceofCartesiancoordinateFMM(CCFMM) (green solid square: p m =5,α m =0.71; red open square: p m =3,α m =0.63), spherical harmonics FMM (SHFMM)(black dot: p m =5,α m =0.71) and FFT method (blue star). The inset is a log scale plot. Using Cartesian coordinates makes the FMM roughly three times faster. With p m =5,α m =0.71, the per- formance of CCFMM is comparable to the FFT method. When the system is sufficiently large or lower accuracy is satisfactory, CCFMM can be superior to FFT. Finally we compare the performance of the Cartesian coordinate FMM (CCFMM) with that of the spherical harmonics FMM (SHFMM) and the FFT method. For SHFMM, we use the same tree hierarchy as our CCFMM so that all the performance difference attributes to the difference of base function in expan- sion: CCFMM uses simple polynomial expansion, while SHFMM uses spherical harmonics, which is much more computationally expensive. Regarding the error boundary, these two methods is very similar as long as we keep the parameter set {s,p m ,α m } the same. They will only differ by a small factor of order 1. As to FFT, the routine given in Numerical Recipes [93] is used. The results are summarized in Fig. 2.4. As expected, the computing time increases linearly for both FMM, and the FFT method gives a typical O(N ∗log(N)) scaling behavior, which can 32 be seen more clearly on the log scale plot. Focusing on the two FMM, CCFMM is clearly superior to SHFMM. This is all due to the expensive calculation of trigono- metric functions of spherical harmonics. Further, cumbersome complex number calculations are avoided by working in Cartesian coordinates. The gain is approx- imately a factor of three given the chosen parameters. This result is comparable to what has been reported in previous literature[5]. However, differences arise because we use a higher expansion order (p m = 5) while earlier authors used (p m = 4). As a rule of thumb: the higher the order, the less favorable the carte- sian expansion. Another factor that results in different speed-ups is the method used to calculate spherical harmonics. In our case, we hard-code the expression of spherical harmonics rather than using recursion relations. Meanwhile we have also minimized the trigonometric functions evaluations. With these optimization of SHFMM though, we found the CCFMM clearly exhibits superior performance at relatively lower expansion (p m < 8). To achieve a decent 1% average error, parameter set {p m =5,α m =0.71} is recommended. In this case, the perfor- mances of FFT and CCFMM are similar. Both the benefit and limitation of FFT is clear. It is an exact method, and it is easy to implement. However, FFT requires a uniform simple cubit grid and a large padding area for exotic structures with open boundary condition. Even though special FFTs can be adapted to nonuni- form systems, they become problem specific and complicated. FMM, on the other hand, is quite generic and natural for nonuniform systems and exotic structures with open boundary condition. What’s more, it can beat FFT when only lower accuracy is needed, as shown by the red open square in Fig.2.4. Further, as the FMM is a parallel scalable algorithm, it can be efficiently implemented in parallel architectures. 33 2.3.4 Derivation of Eq.2.19 Treat a dipole m( r) as a limit of two point monopoles ±q located at r ± d when d → 0, keeping m=2q d. Then the multipole moments for this dipole is Q n = q n! [( r+ d) n −( r− d) n ] = q n! [(x+d x ) nx (y +d y ) ny (z +d z ) nz −(x−d x ) nx (y −d y ) ny (z −d z ) nz ] = lim d→0 qr n n! [(1+ n x d x x )(1+ n y n y y )(z + n z d z z )−(1− n x d x x )(1− n y n y y )(z − n z d z z )] = 2qr n n! [ n x d x x + n y d y y + n z d z z ] = r n n! [ n x m x x + n y m y y + n z m z z ] = m·∇r n! (2.34) Notice that the Q (0,0,0) ≡ 0. 2.3.5 Pseudocode for FMM Here we provide the pseudocode for the three recursive function mentioned in Sec. II. Within the framework of object-oriented programming, all these function are public members of an object called “cell”. 34 Function void cell::createPartners() FOR each cell "A" in the "Partners" list IF the cell A is near to this cell, THEN IF this cell has child, THEN Put the children of A into the ‘‘Partners" list of the children of this cell;\\erase A from the Partners list of this cell; ELSE Put A into the NearPartners list of this cell; Erase A from the Partners list of this cell; END IF END IF END FOR IF this cell has child THEN Put one of its children into the ‘‘Partners" list of the other children (do this for both children); call its children’s CreatePartners(); ELSE Add this cell into the NearPartners list of itself; END IF END FUNCTION Figure 2.5: the recursive function to create Partners list. Function void cell::updateMoment() Clear the multipole moments of this cell; IF this cell is not the smallest cell Then Call its children’s updateMoment(); Sum over its children’s multipole moment with shift of origin; ELSE IF there is dipoles in this smallest cell THEN Ccalculate the multipole moments of this cell directly from the dipole distributions; END IF END FUNCTION Figure 2.6: the recursive function to create multipole moments for each cell. 35 Function double cell::updateField() IF this cell has parent THEN Inherit the Taylor expansion from its parent with shift of origin; END IF FOR each cell in the "Partners" list Add to the Taylor expansion of this cell the field generated by A; END FOR IF this cell has child THEN Call its children’s updateField(); Calculate the total energy of this cell by the sum of the energy of its children; ELSE Calculate smooth local field from the Taylor expansion contributed from all far cells; FOR each cell "A" in the "nearPartners" list Add to the local field from each dipoles in A except when the dipole is located at the field point; END FOR Calculate the energy of this cell by summing over all the dipole in it and divide the energy by 2 to eliminate double inclusion of the interaction energy END IF Return the energy of this cell; END FUNCTION Figure 2.7: the recursive function to calculate magnetic field at each point and total energy. 36 Chapter 3 Phase Diagram of Nanomagnets Nanomegnets are usually generated in arrays, where there are two distinct issues of interest: the spin configuration of the individual nanoelement, and the interactions between them. It has been shown that the interactions can be safely neglected, when the distance between the individual element is larger than twice the charac- teristic size of the individual element. [96, 73] Thus a great deal of attention has been put on the properties of single nanomagnet. In this chapter we will verify the validity of the scaling approach in terms of different shapes, anisotropy and crystalline structure. Incomplete similarity with respect to the lattice constant occurs in magnetic nanoparticles regardless of cross- sectional geometry, crystalline anisotropy, or lattice structure. Also, the effects of these parameters on the phase diagrams are analyzed. The influence of the vortex core on the scaling behavior and phase diagram is investigated. Furthermore, the scaling approach is applied to nanorings and elliptically shaped nanoparticles. The resulting phase diagrams are given, and new and interesting phenomena are discussed. 3.1 Shape, anisotropy, and lattice structure In cylinder and prism shaped nanomagnets, three dominant competing configura- tions have been identified[25]: (I) out-of-plane ferromagnetism with the magneti- zation aligned parallel to the nanodot base; (II) in-plane ferromagnetism with the 37 magnetization perpendicular to the base; (III) a vortex state with the magnetic moments circling in the base plane. Double vortex states in elliptically shaped particles will be discussed in detail later. A typical phase diagram for a cylinder is shown in Fig. 1, which exhibits these three phases as a function of the cylinder radius R and its height H. Note that there can be other metastable configurations, such as the buckle state[79], which are not considered here. To illustrate the use of the scaling procedure, let us first consider the example of a cylindrical nanopar- ticle. By using the 2000-node 15.78 teraflop high-performance supercomputer at the University of Southern California (USC), the energies of the competing phases were evaluated throughout the parameter plane spanned by the cylinder radius R and height H for systems with up to 400,000 sites. The scaling procedure was then used to collapse the resulting phase diagrams with different scaling factors, four of which (x=0.02, 0.04, 0.06, and 0.08) are given in Fig. 1 as examples. Note that there is a triple point (R t ,H t ), which is used to extract the scaling exponent (see the inset of Fig. 1). For the sake of simplicity, a simple cubic underlying lattice structure with cubic crystalline anisotropy and the “core-free” vortex state is adopted. Discussion about other structures and the effect of the core will come later. The scaling exponent η=0.556 is consistent with the previous result [25], suggesting incomplete similarity with respect to the lattice constant in this case. It is observed that the slope of the line separating the two ferromagnetic phases is k=1.811, which is in exact agreement with the analytical solution previously given [2] and argued [45]. Sinceanenormouslywiderangeofmagneticpropertiescanbeobtainedbyusing different geometric shapes[18], it is of great interest to see whether nanoparticles with different cross-sectional geometry exhibit incomplete similarity as well. To answer this question, here we consider prism shaped nanoparticles with triangular, 38 square, pentagonal, and hexagonal cross sections. From the results shown in Fig. 2(a) we find that within an error bar of 2%, these different geometries have the same scaling exponent showing incomplete similarity. In spite of the apparently universal scaling behavior, it is also evident that different geometries do favor different spin configurations. More precisely, the more symmetric the cross section is, the more the vortex phase is favored. Obviously, cylindrical nanodot favors the vortex configuration the most. Another property of interest is the slope k of the line separating the two ferromagnetic (FM) phases. Fig. 2(b) shows this slope as a function of the cross section area. To compare the various polygon shapes, they have been normalized such that the distance from the corner of each polygon to its center is unity. The slope is found to increase with the basal area. This trend is easy to understand, since the two FM configurations are determined by dipolar interactions, i.e. via the demagnetizing field which in turn is related to the surface area. Quantitatively the slope is expected to be approximately proportional to the square root of the area, which is found to be in agreement with the numerical results shown in Fig. 2(b). In the following analysis of the universality of scaling for various crystalline anisotropy and underlying lattice structures, we will focus on cylindrical shapes for the simple reason that these are most commonly found in the existing experi- mental literature. Fig. 2(c) gives the phase diagrams for the different anisotropies. In accordance with intuition, the cubic anisotropy equally favors the two ferro- magnetic phases, i.e. the slope separating these two phases does not depend on Ka 3 /D, and at the same time suppresses vortex formation. Hence, one should consider materials with a small cubic anisotropy if one wishes to stabilize the vor- tex state. Besides the cubic anisotropy, another prevalent type is the uniaxial 39 0 20 40 60 80 100 120 140 x=0.016 x=0.030 x=0.060 cylinder H (nm) (a) T S P H C 1.2 1.4 1.6 1.8 1.0 1.2 1.4 1.6 1.8 Square Root of Area slope, k (b) C T P H S 0 20 40 60 80 100 0 20 40 60 80 100 120 C (K/D=0) C (K/D=1) C (K/D=5) U (x=0.02) U (x=0.04) U (x=0.06) C+U H (nm) R (nm) (c) 012345 0.0 0.5 1.0 1.5 2.0 slope, k Ka 3 /D (d) Figure 3.1: (color online) (a) Scaled phase diagrams for prism shaped nanopar- ticles. The radii R are defined as the distance from the base center to the corner of the polygons. The extracted scaling exponents for the triangle (T), the square (S), the pentagon (P) and the hexagon (H) are 0.556 (T), 0.557 (S), 0.563 (P) and 0.559 (H) respectively. (b) The slope (k) of line separating phase I and II versus the square root of the cross section area of nanodot with unit radius. (c) Phase diagrams of cylindrical nanoparticles with different anisotropies. Solid squares represent cubic anisotropy (C) of different magnitude. Open circles with different colors represent uniaxial anisotropy(U) with Ka 3 /D = 1 showing valid scaling behavior with η=0.56. Solid triangles represent combination of both anisotropies (U+C) with Ka 3 /D = 1. (d) The slope k versus the strength of the uniaxial anisotropy. anisotropy. This anisotropy typically exists in hexagonal close-packed (hcp) lat- tices, but it can also occur in cubic lattices due to coupling to the substrate or other parts of the environment. In our calculation, the easy axis is set to be along the axis of the cylinder. The resulting phase diagram is shown in Fig. 2(c). We observe that uniaxial anisotropy does not affect the scaling behavior and exponent. However, a feature worth mentioning is that uniaxial anisotropy does change the 40 slope of the line separating the two ferromagnetic phases, favoring out-of-plane alignment (phase I). The larger the value ofKa 3 /D is, the smaller the slope is (see Fig. 2(d)). Meanwhile, when both anisotropies are present, the slope is dominated by the uniaxial term. Hence an analysis of this slope can be used to experimentally determine the uniaxial anisotropy, based on the information given in Fig. 2(d). Various lattice structures exists in nature. It is important to know whether the scalingtechniquedependsonlatticestructure. Wecalculatedthephasediagramfor hcp and face centered cubic (fcc) lattices and their variance by rotating the lattice structure in the cylinder. The results remain invariant as long as all parameters (Ja 3 /D, Ka 3 /D and density of spins) are kept the same and x is not too small. The above results indicate that the scaling behavior is robust to details of lattice structure, crystalline anisotropy, and geometric shape. 3.2 Particles with Core Structure Interestingly, magnetic nanoparticles with core structure exhibit complete simi- larity with respect to the lattice constant (See Fig. 3.2). Similar effects were reported by Landeros et. al.[68] To analyze the effect of the core, we choose an ansatz (S z = exp(−2r 2 β 2 )) introduced by Feldtkeller and Thomas[33]. We fit the results of Monte Carlo (MC) simulations with this ansatz and obtain acceptable agreement (see the inset of Fig. 3). From dimensional analysis, the core size 1/β obeys a scaling law of the following form: 1/β = L ex Φ β (R/L ex ,H/L ex ,a/L ex ), (3.1) 41 0 1020 30 4050 60 70 0 20 40 60 80 100 120 123 4 5 0.0 0.2 0.4 0.6 0.8 1.0 w/o core w/ core x=0.01 w/ core x=0.02 w/ core x=0.04 w/ core x=0.08 H (nm) R (nm) MC result fitting result 1/β=1.8 r (nm) <S z > Figure 3.2: (color online) Scaled phase diagram of a single-domain cylindrical magnetic nanoparticle taking the vortex core into consideration. The black hollow circlesrepresentthephasediagramforcylindricalnanoparticlewithcorefreemodel taken from Fig. 1. The scaling exponent η=0.5. The inset shows the fitting of the core function to the MC result for the case of J /D = 100. where L ex is the magnetic exchange length, as before, and Φ β is a scaling function, independent of J,D. Numerically, we find that the scaling function Φ β is approxi- mately independent of all its arguments, roughly giving 1/β ≈ 0.6L ex . We use this as an additional governing parameter in the numerical calculations. In the presence of the core, the critical height now satisfies a physical law of the following form: H = g(J,D,R,a,1/β). (3.2) From dimensional analysis, we find again only two independent governing param- eters, define L ex , and write: H = L ex Φ g (R/L ex ,a/L ex ,1/(βL ex )). (3.3) 42 Numerically, we find that Φ g approaches a constant as its second argument a/L ex becomes small. This is evidenced by the collapse of the phase diagrams in Fig. 3 with η=1/2. The collapse implies invariance under the transformation in Eq. 2.13, andtherebythecompletesimilaritywithrespecttothelatticeconstant. How- ever, this is consistent with the incomplete similarity with respect to an exhibited in the core-free approach, since we have an additional dimensional length 1/β that plays the role of the lattice constant. Similar results are obtained in section 4.3 when we change the topology of the nanoparticle and introduce an inner radius. Astothephasediagramitself, thecoresignificantlystabilizesthevortexconfig- uration, pushing the phase boundary between FM and the vortex phase to smaller values of R and H by about 35%. Similar effects would affect Fig. 2(a)(c) as well. 3.3 Cylindrical Nanorings Next we consider the effects of changes in topology on the phase diagram. More precisely, we investigate the phase diagram of hollow cylinders, i.e. a nanoring structure characterized by an inner radius R i , an outer radius R, and a height H. We find, as in the section 4.2, that the critical height exhibits complete similarity with respect to the lattice constant (η ≈ 1/2). This is a consequence of the additional length R i that plays the role of the lattice constant in regulating the vortex core energy. Fig. 3.3 shows three-dimensional phase diagrams in the (R i ,R,H) parameter manifold of the nanoring topology for two different values of the exchange cou- plings J . Again, one observes two ferromagnetic regimes at small (R,H) values, competing with a vortex phase at larger (R,H). Moreover, one finds that for larger inner radii R i the the vortex phase is more extended. This confirms the idea that 43 0 1 2 0 4 8 12 16 0 4 8 12 16 20 x=0.06 x=0.10 H (nm) R (nm) R i (nm) Figure 3.3: Phase diagrams of a cylindrical nanoring for two different x. There aretwocompetingferromagneticphasesatsmall(R,H)andavortexphaseatlarge (R,H). Because of the finite inner radius R i , the onset of the phase transition line between the two ferromagnetic phases is shifted to finite values of R. Also, the vortex regime is more extended for largerR i . The blue lines in the figure are guides to the eye, indicating that the triple points form approximately a straight line. the ring structure stabilizes the vortex configuration. The reason for this is that the core area, which typically pays a high energy penalty, is deliberately avoided in the ring structure. Another new feature of these phase diagrams is that the line separating the two ferromagnetic phases is not straight anymore. Instead, it now starts at finite R = R i , and its slope smoothly changes to 1.81 as the ratio between R and R i becomes very large. This relationship can be clearly observed in Fig. 3.4(a) which is derived from Eqs(11,13) of Ref.[4]. Here we calculate the relationship between the critical height H c (R,R i ) as a function of (R−R i ), result- ing in the “star” symbols in Fig. 3.4(b) which exactly align with the line of our numerical calculation. Finally, the most surprising feature of the phase diagrams in Fig. 3.3 is that the triple points (R t ,H t ) for different R i approximately form a straight line indicated by the two blue lines. This property is more clearly shown 44 in Fig. 3.4(c), i.e. cylinder height at the triple point (H t ) versus R i . It gives us a critical R ic beyond which no in-plane ferromagnetic phase exists. 10 20 30 40 50 0 20 40 60 0.04 0.06 0.08 0.1 3 4 5 6 0.0 0.2 0.4 0.6 0.8 1.0 1.6 2.0 2.4 0123 4 8 12 16 R(nm) H (nm) x=0.04 x=0.06 x=0.08 x=0.10 A (b) R t (nm) x R t =16.75*x 0.515 (d) H c /(R-R i ) R i /R (a) R i (nm) H t (nm) x=0.06 x=0.10 scaled from x=0.1 with η=0.51 (c) Figure 3.4: (a) For cylindrical nanorings, the phase transition line H c (R,R i ) separating the two ferromagnetic regimes is not straight, in contrast to the topo- logically connected objects discussed above. (b) Phase diagram for a cylindrical nanoring. (R i =6.3nm, J/D = 5000). The data “A” represent analytical phase transitionlinescalculatedfrom (a), whichareobservedtocoincidewith thenumer- ical results. (c) Height at the triple point (H t ) versus inner radius (R i ). The best fit for x=0.1 is H t =16.5 − 5(±0.03) × R i , whereas the best fit for x=0.06 is H t =12.7 − 4.95(±0.09) × R i . Hence, the two lines are approximately parallel, and can thus be collapsed via scaling with η=0.51. (d) The triple point radius (R t ) versus x. Another observation worth mentioning is that the intersect phase diagram in the R i = 0 plane of the cylindrical nanoring does not coincide exactly with the phase diagram of the simply connected cylinder (Fig. 1). It is closer to the case when the core structure is considered (Fig. 3.2). This phenomenon happens for the scaling exponent as well, which will be discussed right below. 45 One last feature to be discussed here is that the line connecting H t (see Fig. 3.3 and Fig. 3.4(c)) is parallel for different values of the exchange coupling J .By comparing the two phase diagrams for different J , we anticipate that a scaling behavior exists here as well, as long as all three coordinates (R i ,R,H) are scaled. However, some difficulties arise since R i should be different for different J s, mean- ing that one would need to know the scaling exponent η in advance. Luckily, we can estimate the value of η from Fig. 3.4(c) as the two straight lines should scale if there is a scaling behavior. Thus we first attempt to scale these two lines and find that they fit best when η 0.51. Then we use this η to scale R i and attempt to see whether the scaling behavior holds. Fig. 3.4(d) shows the result. The scaling exponent is η=0.515 which is within 1% of the estimated value 0.51. It is much closer to 0.5 in the finite core case, implying complete self-similarity, since an additional length R i is added and neglecting the core structure in the vortex state has little effect for ring structure. With these results, we can easily calculate the critical inner radius R ic 11nm for the parameters we choose, above which a flat nanoring is always in the vortex phase. This is quite small compared to typical nanorings experimentally fabricated [12], and suggests that nanorings are generically in the vortex phase, since they are typically flat with height small compared to the width. 3.4 Elliptically Shaped Particles It has recently been observed that there exists a double vortex configuration in elliptically shaped ferromagnetic particles.[108, 107, 53, 110] The full phase dia- gramforthiscaseasafunctionofheight, semi-majoraxis(R a )andsemi-minoraxis (R b ), however, has not yet been calculated. One of the difficulties in determining 46 this phase diagram, using the technique outlined above, lies in finding an adequate parametrization of the double vortex state. The naive approximation of two single vortices is far from satisfying (see Fig. 6(c)). As we will see below, the energy of two single vortices with discontinuous magnetization along the minor axis is significantly higher than that of a true double vortex with continuously varying magnetization (see Fig. 6(a)). Without an accurate parametrization of the double vortex one could only rely on Monte Carlo or micromagnetic simulations which are extremely time consuming, and this would make it impossible to obtain a complete phase diagram. (a) (b) (c) R a R o R b Figure 3.5: Double vortex configuration for J/D = 10(x=0.002), R a /R b =2 (arrows represent the directions of magnetization). (a) Monte Carlo simulation result, ea 3 /D =21.12 (b) Our parametrization, ea 3 /D =21.11, F between b and a is 0.990 (c) naive parametrization (two single vortices), ea 3 /D=20.91, F between c and a is 0.974. e is the energy per spin and F is the fidelity defined in the text. 47 Here we propose a simple function to parametrize the double vortex. In our Monte Carlo simulations, we observe that the shape of the double vortex (Fig. 6(a)) looks much like the equipotential lines of two electric point charges with opposite signs placed at the centers of the vortex cores (Fig. 6(b)). By symmetry these cores should lie on the major axis of the ellipse. Let the distance from the core centers to the center of the ellipse be R o . Then the vector field S( r)isgiven by S(x,y)= −E y ˆ i+E x ˆ j E 2 x +E 2 y (3.4) where, E x = x−R o [(x−R o ) 2 +y 2 ] 3/2 − x+R o [(x+R o ) 2 +y 2 ] 3/2 E y = y [(x−R o ) 2 +y 2 ] 3/2 − y [(x+R o ) 2 +y 2 ] 3/2 Interestingly, the optimal positions of the vortex cores yielding the lowest energy configurations do not coincide with the ellipse foci, but are located at non-trivial positions on the major axis with constant κ = R o /R a . κ almost exclu- sively depends on the aspect ratio (R a /R b ), and only very weakly depends on size and Ja 3 /D. Within the range we examined (H< 40nm,R a < 30nm,J), κ decreases by only 2% as the size is increased. For different aspect ratios we find κ=0.44±0.1. These values coincide with recent experimental results[53, 110, 7]. We choose R a /R b = 2 as an example. In this case, κ=0.44. To quantify the quality of our parametrization of the double vortex, we look at the energy per spin (e) and the fidelity F = N −1 i S i · S i , i.e. defined as the average dot product of spins on each lattice point of two configurations S( r) and S ( r). The energy of our 48 parametrization (Fig. 6(b)) is significantly closer to the energy obtained by Monte Carlo (Fig. 6(a)) and its fidelity is significantly closer to 1 than that of the two single vortex parametrization (Fig. 6(c)). This is important because the energies of the single vortex and the double vortex configurations are very close. If one uses the naive parametrization, the double vortex could never be the ground state. By using the parametrization of the double vortex in Eq.3.4, we now apply the scaling procedure to obtain the phase diagram for elliptically shaped particles (see Fig. 7). Since there is no good description for the core of the double vortex yet, a core-free system is assumed for simplicity. We estimate that the boundary will shift to lower values of R a and H by about 35% when taking the core into consideration. 0 100 200 300 400 500 0 100 200 300 400 500 600 IV III II x=0.008 x=0.010 x=0.020 H (nm) R a (nm) I Figure 3.6: Scaled phase diagram of an elliptically shaped magnetic nanoparticle (Ka 3 /D = 1 and Ja 3 /D = 5000) as a function of its semi-major axis (R a ) and height (H) with an aspect ratio 2. The four competing phases are (I) out-of-plane ferromagnetism, (II) in-plane ferromagnetism, (III) single vortex state and (IV) double vortex state. The scaling exponent is η=0.55. As expected, the double vortex state becomes stable when both the semi-major axis and height of the nanoparticle are increased. In the vicinity of the phase 49 boundary between the single vortex and the double vortex states, the energies for the two configurations are very close, and hence there could be a large metastable region close this phase boundary where both states could exist in nature. This is likely the reason why both these configurations have been observed in experiments onthesameparticle[7]. Regardingthescalingexponent, η=0.55isagainobserved in this core-free consideration, implying incomplete self-similarity. Here we have only focused on the double vortex state. When the system size and the aspect ratio are sufficiently large, it is possible that multivortex states emerge. Besides such complex single domain structures, cross-tie domain walls [65] could exist in these structures as well. It would be highly interesting to know under which condition these configurations could be stabilized. In sum, the magnetic properties of single-domain nanoparticles with differ- ent geometric shapes, crystalline anisotropies and lattice structures have been investigated. A scaling approach was used to obtain phase diagrams of magnetic nanoparticles featuring three competing configurations: in-plane and out-of-plane ferromagnetism and vortex formation. The influence of the vortex core on the scaling behavior and phase diagram was analyzed. Three-dimensional phase dia- grams were obtained for cylindrical nanorings, depending on their height, outer and inner radius. The triple points in these phase diagrams were shown to be in linear relationship with the inner radius of the ring. Elliptically shaped magnetic nanoparticles are also studied. A new parametrization for double vortex configu- rations was proposed, and regions in the phase diagram were identified where the double vortex is a stable ground state. 50 Chapter 4 Configurational Anisotropy of Square Nanomagnets 4.1 Chapter 4 Introduction Oneofthemostimportantpropertiesofamagneticsystemisitsanisotropy,i.e. the presence of direction dependence of some physical property. Magnetic anisotropy brings about the preference for the magnetization to lie in a particular direction in a sample and is ultimately responsible for the behavior of a magnetic system and the suitable technological applications. In bulk material, magnetic anisotropy originated from the sample shape, crystal symmetry, stress or directed atomic pair ordering. [88] In nanomagnets, however, there could be other origins. One of them is the the various deviation of the magnetization configuration from the uniform distribution. The resulting anisotropy is thereby called configurational anisotropy (CA). CA is identified in symmetric prims shaped nanomagnets [19, 21, 23], since no anisotropy would be expected in their bulk counterparts. It depends on the shape and size of the sample, but in a totally different way than their parent material. The artificial superstructures take effects as well.[44] One of the most attractive feature of this anisotropy is that it could be easily engineered catering for various applications.[27, 63] Understanding the influence of anisotropy opens 51 the way to designing new nanostructured magnetic materials where the magnetic properties can be engineered to fits different applications. The objects discussed in this chapter are flat magnetic nanoparticles. Instead of a uniform magnetization, spins in nanomagnets tend to align in complex config- urations. There are certain favorable directions for the magnetization under zero magnetic field. This anisotropy originated from the nonuniform magnetization is called configurational anisotropy. To rule out the effect of crystalline anisotropy, we set K = 0 throughout this chapter. Physically the CA results from the compe- tition between the exchange and dipolar interaction. Nonuniform magnetization sacrifice exchange energy for magnetostatic energy. In order to quantify the CA, experimentally a strong in-plane field H L was applied in direction θ together with a small ac in-plane field ∆ H t perpendicular to H L , and then transverse susceptibility χ t ≡ ∂φ/∂H t is measured by modulated field magneto-optical anisometry (MFMA) [20]. It has been shown that χ −1 t = E (φ) M s +H = H a +H (4.1) where φ is the mean magnetization direction, E (φ) is the second derivative of the energy density with respect to φ, M s is the saturation magnetization, and H a = E (φ)/M s is the so-called anisotropy field. Physically H a is the magnetic field needed to be applied to the direction φ in order to give the same energy profile as E(φ) for small deviations of the magnetization about φ.[18] Notice that it is only valid when the deviation between θ and φ is sufficiently small. To simulate χ t , traditional procedure is to imitate the experimental setup. A strong in-plane field H L was applied in direction θ and let the system to equilibrate. Then a small in-plane field ∆ H t perpendicular to H is applied in two opposite directions 52 and the angle between the magnetization in those two case was given as 2∆ φ. So χ t =∆ φ/∆ H t .[62] This method is easy to understand but not convenient to calculate and wastes a lot of computing time. An easier method is suggested here. Because the two fields (H L ,∆ H t ) exerted are equivalent to a single total field H, all the information will be got if we do an angle sweep of single field H.The most important quantities here is the angle ω between the directions of H and M: ω ≡ φ−θ. The recipe to calculate χ is given by the following equation: ∂φ ∂H t = lim ∆ Ht→0 ∆ φ ∆ H t = lim δ→0 φ(θ+δ)−φ(θ) Hsinδ , = 1 H ∂φ ∂θ = 1 H ( ∂ω ∂θ +1), (4.2) In the lattice model, M s = zv|µ| where v is unitcell volume and z is the number of atom in the unitcell. For simple cubic lattice z = 1 and v = a 3 . Therefore we could write χ −1 t = H L ( ∂ω ∂θ +1) −1 = H L +e (φ(θ)) (4.3) H a = H L ( ∂ω ∂θ +1) −1 −H L = e (φ(θ)) (4.4) wheree(φ(θ))theenergyperspinwhenmagneticfieldisappliedinthedirection of θ. When the anisotropy field shows perfect n-fold symmetry, it has been shown that E=2M s H a V/n 2 and argued that the anisotropy field have the same symme- try as the energy landscape.[18] Notice this conclusion only hold when the energy landscape involves a pure n-fold symmetry. If there is more than two symmetry 53 components, the function E could look significantly different than the function H a . The meaning of the statement will be shown clearer with our results. 4.2 Results In literature, the most widely studied shape of nanomagnets is square, mainly due to the application of such structures to MRAM and spin-valve magnetic field sensors[92]. A systematic study of the effects of four parameters are given here, namely, edge length (l), height (h), exchange coupling constant (J) and external magnetic field (H). First of all, the scaling technique mentioned in the previous chapter is carefully studied. Suppose we have a sample with J = 20000,l = 368a ∼ 110nm,h = 37a ∼ 11nm. Different scaling factor x is applied with various η. It turns out that the scaling hypothesis works perfect when η=0.55. Fig.4.1(a) shows ω versus θ with three sets of parameters which are respectively x 1 =0.005,l 1 =20,h 1 =2, x 2 =0.011,l 2 =30,h 2 = 3 and x 3 =0.018,l 3 =40,h 3 = 4. Amazingly, the three curves perfectly collapse. The surprising thing is η=0.55 =0.5. This conclusion coincides with previous result.[109] From the dimensional analysis language [3, 121], it showesincomplete similarity with respect to lattice constant a. ω is an easy quantity to calculate and understand. The angle with a steepest change in ω is a position of energy extreme. The one with a negative slope is the energy minimum (easy axis). For the sample considered here, the easy axis is along the diagonal of the square. Fig. 4.1(b) shows the energy per spin and magnetization versus angle θ. Expected energy minimum at π/4 is found. For conventional anisotropy, magnetization is small at hard axis and large at easy axis. However, this is not necessarily true in the CA case. As shown in Fig.4.1(b) , it is the very opposite. 54 0 306090 0.000 0.005 0.010 0.015 0.020 0.998 0.999 -0.2 0.0 0.2 -0.2 0.0 0.2 030 60 90 -0.2 -0.1 0.0 0.1 0.2 030 60 90 -2 -1 0 1 2 θ (b) energy per spin M H a (f.u.) (d) H m (1/(dF/dθ+1)-1) d 2 e/dθ 2 H a (f.u.) θ (c) ω=ϕ-θ x 1 =0.005 x 2 =0.011 x 3 =0.018 θ (a) Figure 4.1: (color online) (a) The angle between the magnetization and field ω as a function of θ for three different sets of parameter scaled from Ja 3 /D = 20000,l = 368a ∼ 110nm,h=37a ∼ 11nm with η=0.55. (b) Energy per spin and magnetization as a function of θ (c) The anisotropy field H a calculated from two different quantities proving the validity of the Eq. (d) An conventional way to represent H a in polar plot (data were taken in the range [0, π/2) and plotted thrice to complete the circle because of the symmetry of the lattice and shape.) Fig.4.1(c) is the computed anisotropy field calculated in two different ways. One from e(θ) and the other from ω(θ) according to Eq.4.4. The consistency of the two curve prove that the derivation of previous equations and the approximation made is reasonable. The small discrepancy come from both the calculation error in the second derivative of the energy and the approximation made in the derivation. Fig.4.1(d) showsthe conversionalrepresentation ofH a . It showsa 4-fold symmetry and H a is about ±50Oe. We want to emphasis that the internal energy is a more fundamental quantity, but it is not experimentally measurable. That’s why 55 people resort to the anisotropy field, which is related to the curvature of the energy landscape. However in certain circumstance, H a may not show enough information or even misleading especially when it shows mixed symmetry component. Thanks to the existence of the scaling property, the result for small system is meaningful since we could easily scale them to the revelent realistic big system. We perform a systematic study of square nanomagnets with simple cubic lattice based on this argument. With various parameters, four categories of configura- tional anisotropy are identified, all including a basic 4-fold symmetry. They are illustrated in Fig.4.2 and defined as follows: (i) diagonal 4-fold (D4) symmetry, corresponding to a 4-fold symmetry with easy axis along the diagonals; (ii) edge 4-fold (E4) symmetry, corresponding to a 4-fold symmetry with easy axis along the edges; (iii) genuine 8-fold (G8) symmetry, corresponding to an additional 8-fold symmetry with energy minimum along the easy axis indicated by H a ; (iv) fake 8- fold (F8) symmetry, corresponding to an additional 8-fold symmetry with energy minimum along the secondary “hard” axis indicated by H a . The most interesting feature here is the G8 symmetry. It is surprising that in the 4-fold square shape, there could exist spontaneous 8-fold symmetry. Note that there is two different energy barriers in G8 case: one large and one small (see Fig. 4.2). This is an important fact when the superparamagnetism is considered, since there will be two ”Blocking temperature”, i.e. the critical temperature when the sample loses magnetization. The most tricky feature is the F8 symmetry. As we have discussed earlier, it is the case where H a could not represent the landscape of E and the easy equation connecting them is not valid. This patten is observed previously [19] and argued to have easy axis along the maximum of H a . However, it is somewhat confusing. The usual way to define an easy axis is the direction to which the mag- netization points without any external magnetic field. In this case, this direction 56 is still the diagonals which could also proven by finite temperature MC simulation. The key to understand this phenomenon is noticing that there is two symmetry components: 4-fold and 8-fold. They are essentially two different frequency com- ponent if fourier series expansion is performed to the energy profile. The higher symmetry component would take more weight when you take derivatives of the energy landscape so that even though the 8-fold component is observable in H a ,it is not as important. -0.8 -0.4 0.0 0.4 -0.8 -0.4 0.0 0.4 0 204060 80 100 -0.06 -0.04 -0.02 0.00 0.02 -0.004 -0.002 0.000 0.002 0.004 0.006 Ha (f.u.) F8 E4 G8 D4 (a) e (e.u.) θ D4 E4 G8 (b) F8 Figure 4.2: (color online) (a) Anisotropy field of 4 typical symmetry types (b) energy landscape correspondingly. The sets of parameters are as follows. F8: l=20, h=3, J=20; D4: l=20, h=2, J=100; E4: l=20, h=15, J=100; G8: l=20, h=10, J=50 Now the effect of the four parameters is analyzed one by one. To quantify the anisotropy, we use the energy barrier ∆ e since energy is more fundamental than the artificial anisotropy field. The external magnetic field H will set to be 1 f.u. unless otherwise specified. (a) As the height increases, CA symmetry from D4 or F8 to G8 and then to E4, keeping all other parameter constant. It will be shown in (c) that D4 and F8 are tuned by J and are qualitatively equivalent in term of energy landscape, as discussed above and shown in Fig. 4.2. Within the D4 or F8 range, notice that 57 02468 10 12 14 16 0.000 0.005 0.010 0.015 0.020 0.025 ∆ e (e.u.) h (a) Figure 4.3: (color online) ∆ e versus height with R = 10, J = 10. Insets show the anisotropy field in different cases. ∆ e has a maximum, i.e. the anisotropy strength increase first and then decrease, so that there is a critical height h c1 which has max H a for a certain l. Here h c1 = 3 for l = 10. We tried for different radius, and find that r 1 = h c2 /l 0.15 and this ratio r 1 is independent of J,H L . This tells us that in order to obtain the maximum H a , it is better to have certain thickness. The physics behind it needs more investigation. There is a critical region which separates D4(F8) and E4 symmetry. In this region, G8 is found. It is found that the center of this region is at the ratio r 2 = h c2 /l 0.5 independent of J,H. However the width ∆ h of this region depend on all the other three parameters. Basically, increasing J,H tends to shrink this transition region while increasing l tends to enhance it. All these findingisconsistentwithapreviousanalysisforflatsquarenanomagnets.[22]Ithas been shown that there is two phases in remanent state: leaf and flower. The former 58 corresponds to the D4 (or F8) symmetry and the latter corresponds to the E4 case, while G8 symmetry is related to the so-called pseudo-leaf and pseudo-flower state. -0.4 -0.2 0.0 0.2 -0.4 -0.2 0.0 0.2 10 20 30 40 50 60 70 80 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 l=10 l=20 l=60 l=80 H a (f.u.) ∆ e (e.u.) l (a) Figure 4.4: ∆ eversusedgelengthwithh=2, J = 10. Insetsshowtheanisotropy field in different cases.) (b) As the radius increases, CA symmetry changes from E4 to G8 and then D4 or F8 keeping all other parameter constant. As expected, this is the reverse trend with respect to (a). Generally the anisotropy strength increase as the radius increase. However, it saturates at certain radius. This usually happens when l/h > 25. The total energy barrier ∆ E, however, is increasing when the system becomes larger, though ∆ e keeps constant. We are not sure whether the ratio l/h is a constant or not. It needs more intensive calculation beyond our treatment. We will try to answer this question when we implement new algorithm so that we could calculate much larger system. Even though ∆ E saturates, H a could still change little by little. It results from the fact that the energy landscape becomes shaper and shaper around the square edge. This is important again if 59 superparamagnetism is considered. Larger radius may not be good since it shows a pretty flat energy profile around the easy axis which allows magnetization rotate rather easily in a large angle reducing the net magnetization finally. Notice that e is the energy per spin. Another thing worth mentioning is that there isn’t any non-monotonous behavior as it in the previous case when r = h/l is changed by h.So h and l should still be regarded as independent variables. -0.6 -0.4 -0.2 0.0 0.2 -0.6 -0.4 -0.2 0.0 0.2 10 100 1000 1E-3 0.01 0.1 H a (f.u.) 2 10 20 50 ∆ e (e.u.) J (e.u.) ∆ e=1.3*J -0.92 Figure 4.5: ∆ e versus exchange constant with R = 10, h = 2. Insets show the anisotropy field in different cases.) (c) As the exchange constant J increases, the anisotropy strength decease as expected, since J offer the power to align spin in the same direction lowering the difference in different directions. Large J suppresses anisotropy and support only 4-fold symmetry. Small J enhance anisotropy and support only 8-fold symmetry. However there is two different trends regarding the symmetry. When h/l < 0.45, it changes from D4 to F8. Otherwise it changes from E4 to G8. The most surprising result is that ∆ e decay with J as a power law. Further study is needed. 60 -0.2 0.0 0.2 -0.2 0.0 0.2 0 5 10 15 20 25 30 0.010 0.012 0.014 0.016 0.018 0.020 H a (f.u.) 0.5 2 10 l=10a, h=2a ∆ e (e.u.) H L (f.u.) Figure 4.6: ∆ e versus external magnetic field with R = 10, h=2 J = 10. Insets show the anisotropy field in different cases.) (d) Conventional anisotropy is an intrinsic property of the sample. Configura- tional anisotropy, however, depends on external parameter (H). That why some researchers suggests to call this phenomenon “configurational stability” instead of anisotropy.[46] Like J, large H suppresses anisotropy for a obvious reason. In order to know the zero field anisotropy, one has to extrapolate from smaller and smaller field, but the field could not be too arbitrarily small both experimentally andcomputationally, sinceH L hasbelargeenoughtoovercometheintrinsicenergy barrier so that the magnetization could follow the field. Thus the minimum field to dominate the rotation of magnetization reflect qualitatively the intrinsic energy barrier. Overall speaking, the anisotropy strength decease as H L increases, but up to H L 0.2 ∼ 660Oe, the change is very small and the anisotropy obtained could be treated intrinsic. While H/R is such that there exists 8 fold symmetry, magnetic field can be used to tune the anisotropy strength and symmetry. 61 Ref. [27] showed however a edge 4-fold symmetry when h/l=0.1whichis inconsistent with the conclusion here. They claimed that the χ −1 T curves depend critically on the chosen material parameter J,M s . In our calculation however, D4 could not tuned to E4byJor H L . It seems that micromagnetic calculation still have some ambiguity here. Ref. [19] use finite difference method while [27] use 2D OOMMF solver. We exactly solve the lattice model with a scaling argument. Further studied is necessary to clarify this inconsistency 4.3 Blocking Temperature The anisotropy energy is particularly important because of a phenomenon called “superparamagnetism” [112]. As is known to all, bulk ferromagnetic material becomes paramagnetic after the Curie Temperature (T c ) above which the thermal fluctuation is strong enough to overcome the “sticking” power of the “glue” J.In a very small magnetic system, however, it may become non-ferromagnetic at a so- called Blocking Temperature (T B ) at which it would be ferromagnetic in the bulk. This process is due to the thermal fluctuation overcoming the anisotropy energy barrier so that the total magnet behaves like a giant spin rotating freely in space. Itisveryimportantformagneticmoleculesystem[35]sincetheanisotropyissmall, but is not usually discussed for bulk magnetic systems, since the typical anisotropy energy for bulk is much larger than the energy scale of room temperature. It is also the reason why only ground state information is enough. The probability for spins to overcome energy barrier are usually written as P(t s ) ∼ exp(−t s /τ), where τ is a characteristic time scale. [112] τ = τ 0 exp(∆ E/k B T) (4.5) 62 where τ 0 is a prefactor and ∆ E an energy barrier. Lots of efforts has been put to determine the prefactors. [85, 112, 83, 51, 16] It has been shown that MC is a very effective method studying thermal acti- vated switching effect in magnetic systems.[85] Here wesimulatethe magnetization versus temperature for systems with the same parameters as Fig. 4.1(a). We use single spin flip update so that each spin can only move by small steps and hence, the system are forced to overcome the energy barrier for a complete reversal. By the analysis, when k B T is on the order of ∆ E but much smaller than k B T c , the magnet behaves as a single giant spin rotating freely in space. Though the MC time is not real time, but there is certain relation between them[83]. Fig. 4.7 is our results. Typically each point are averaged over 100 nodes, each performing 1000 MC steps. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0.0 0.2 0.4 0.6 0.8 1.0 M/M s T (10eu/k B ) TB=0.29 x=0.020 T B =0.29 x=0.042 T B =1.00 x=0.071 T B =2.50 scale T Figure 4.7: Monte Carlo simulation for magnetization versus temperature. x is the scaling factor. The parameter for the three sets of solid dot corresponds to the three sets in Fig. 4.1. The two sets of open dots are calculated from x=0.02 and x = by scaling T = N C /N AorB . Both of them collapse with x =. T B is determined by the temperature at which M/M s =0.5. 63 1. As shown in Fig. 4.1(a), H a and ∆ e are exactly the same for those three cases. Then we could easily calculate ∆ E = N∗∆ e. So if we scale the temperature by N/N a,b,c , the result should collapse. It is indeed the case in Fig.4.7. This proves that the scaling argument is not only useful in obtaining ground state information, but they could be applied in thermal problem study as well. 2. To have a better understanding of T B , we define T B to be the temperature at which M/M s =0.5. For simple cubic lattice, T c J. For the x=case∆ E 16, k B T 3. The width of transition ∆ T =1,so∆ T/T B =1/3. It means that the magnet is very easy to overcome the energy barrier in 1000 MC steps when k B T ∼ 1/5∆ E. Though it doesn’t tell us what is the characteristic time, further study could address this problem following the proposal in Ref.[85]. By the scaling argument,k B T B =3 ∗ (J/100) η∗3 with R = 10(J/100) η and H =2(J/100) η .For a sample with J = 20000e.u., R = 184a ∼ 55nm and H =37a ∼ 11nm, T B = 1.88×10 4 ∼ 423K,T c =2×10 4 ∼ 450K. The temperature to begin the coherent rotation is about 282K. Though this sample could show superparamagnetism, it is a good ferromagnet in room temperature. 3. There exist some hysteresis calculations by scaling argument in literature. One needs to be careful choosing the right temperature. If the object, including the downscaled artificial one, show no superparamagnetism, it is fine to scale the temperature with T c . If there is superparamagnetism, however, one needs to be very careful. 64 Chapter 5 Magnetization Reversal Processes in Magnetic Nanorings Ferromagnetic rings exhibit many novel physical phenomena with promise for potential applications. Here, we focus on switching processes which are funda- mental properties of magnetic systems and are especially crucial for data storage applications. First, a brief introduction to the advantages of ring geometries is presented, followed by the review of relevant magnetic states identified up till now. In particular, explanations from a topological point of view are shown to be enlightening. Then complete inplane uniform field switching phase diagrams are given, followed by detailed description of six different types of switching process. A flavor of circular field switching is added afterwards. Finally the data storage applications of nanorings are briefly discussed. 5.1 Chapter 5 Introduction Characterizing the magnetic properties of nanostructures is a challenging task, as their shape significantly influences their physical response. Significant work has been invested into identifying the geometries which offer the simplest, fastest, and most reproducible switching mechanisms. Particular attention has focused on magnetic structures with high-symmetry geometries, such as circular disks and 65 squares, since spin configurations with high symmetry are expected for these ele- ments, which in turn are believed to yield simple and reproducible memory states. In analogy to the traditional approach to encode information in dipolar-like giant spins, quasi-uniform single domain states have been proposed and intensively studied. However, these typically suffer from three fundamental disadvantages: (i) they are sensitive to edge roughness so that the switching field typically has a broad distribution; (ii) because of the long-ranged dipolar interactions between separate elements, high density arrays are hard to achieve; (iii) they cannot be made to be too small due to the superparamagnetism effect. To overcome these problems, use of the magnetic vortex state in disc geometries (see Fig. 5.1(a)) has been suggested, since it is insensitive to edge imperfections and entirely avoids the superparamagnetism effect. Moreover, the zero in-plane stray field opens the possibility of high density storage. Nevertheless, the vortex state is stable only in discs of fairly large sizes (diameter over about 100nm) due to the existence of high energy penalty vortex core regions. What is worse, the switching mechanisms for discs are complex and difficult to control. In seeking a solution, ring geometries (see Fig. 5.1(b)) have been proposed and studied intensively in the last decade. In addition to all the advantages of vortex states in the disc geometry, magnetic rings are completely stray field free and can be stable with diameters as small as 10nm. Taking 10nm spacing into account, ring arrays give an ultimate area storage density of about 0.25Tbits/cm 2 (or 1.6Tbits/in. 2 ), substantially higher than the traditional hard disc area storage density limit. Besides this promising application potential, ring geometries have proven to be a wonderful platform for the investigation of fundamental physical questions concerning domain walls. [56] As many magnetic phenomena involve vortices and domain walls, which are defects from a topological point of view, the topological theory of defects [78, 106, 66 Figure 5.1: (color online) Schematic representation of a magnetic vortex state in a disc (a) and a ring (b). The core region where spin points out of plane are exaggerated and represented by the height and green color. R o and R i are outer and inner radius of rings and w is their width. 119]canindeedhelptounderstandthecomplexcreationandannihilationprocesses during switching. One of the most important principles is the conservation of topological charge. Topological charge is defined as the winding number ω, i.e. a line integral around the defect center: ω = 1 2π ∇θ( r)·d r, (5.1) where θ is the angle between the local magnetic moment and the positive x axis. There are several elementary topological defects in magnetic systems. As is known, a vortex in the bulk has an integer winding number ω = 1 and an antivortex has ω = −1. Furthermore, Tchernyshyov et al [106] identified two edge defects with fractional winding numbers ω = ±1/2. These four types of elementary topological defects are shown in Fig.5.2. All the magnetic intricate 67 textures, including domain walls, are composite objects made of some of the above four elementary defects.[106] These defects have several properties: (i) in analogy to Coulomb interaction, two defects with the same winding number sign repel each other, while they attract each other if they have opposite signs of winding number; (ii) the vortex is repelled by the boundary since it has an image “charge” with the same sign; (iii) edge defects are confined to the edge by an effective confining potential; (iv) direct annihilation of two defects with the same sign is prohibited; (v) edge defects can change sign by introducing a bulk defect; (vi) in sufficiently narrow rings, ω = ±1/2 edge defects are degenerate, i.e. they have the same energy, whereas in thick rings the ω=+1/2 will have substantially higher energy than the ω = −1/2 so that it can decay into a vortex ω = 1 and an edge defect ω = −1/2. (a) ω = 1/2 (b) ω = −1/2 (c) ω = −1 (d) ω = 1 Figure 5.2: Elementary topological defects: (a) edge defect ω=1/2, (b) edge defect ω=-1/2, (c) vortex ω=1, (d) antivortex ω=-1. 68 Experimentally, nanomagnetic rings have been produced with a wide range of spatial dimensions, extending from dozens of micrometers down to nanometer scales[102]. Various states have been observed to be stable or metastable at rema- nence, i.e. in the absence of an applied magnetic field, including the vortex state (V),theonionstate(O),i.e. twoferromagneticdomainsseparatedbydomainwalls atopposingendsoftheringstructure,andthetwisted(orsaddle)state(T)[11]),i.e. a vortex state interrupted by a 360 o domain wall. Several works have been devoted to determining the phase diagram for nanoring structures [4, 67, 122, 69]. Apart from the phase diagram at remanance, magnetization reversal processes driven by external magnetic fields are another important characteristic property of magnetic elements. Several switching processes have already been discovered [120]. Experi- mentally, switching phase diagrams have been constructed for micron-scale rings. [60] However, the properties of nano-scale rings are still under investigation, and a number of new features have recently been discovered.[9, 80, 101] In this chapter, we will discuss switching processes in detail and construct switching phase diagrams for magnetic nanorings. 5.2 Magnetic States In this section, we are reviewing competing stable and metastable spin configu- rations at remanence identified so far. As is discussed in Chapter 3, there are three main competing magnetic states: out-of-plane FM, in-plane FM and vortex. Strictly speaking, the in-plane FM is truly single domain FM only when the ring is super small (R o <L ex ). Otherwise it is in the form of an onion state (see Fig.5.4). This information can be summarized in a geometric phase diagram (see Fig.5.3). 69 0 10 203040 0 10 20 30 40 50 60 h (nm) R o (nm) R i out-of-plane ferromagnetism onion vortex Figure 5.3: Schematic geometric magnetic phase diagram for nanorings with R i =6nm. Shaded area is the region where the onion state could be (meta)stable, and thus it is of greatest interest. Several works have been devoted to determining this phase diagram [4, 67, 122] for ring structures. Fig. 5.3 shows an example with inner radius R i fixed. Spins point out of plane in parallel when the element height h is much larger than width w. The opposite situation is of more interest. When h/w < 2, there are two possible states: (i) the vortex state (V) (or the flux close state), characterized by the circulation of spins; (ii) the onion state (O) (or the quasi uniform state), characterizedbytwohead-to-headdomainwalls. Thevortexstateistheonlysingle domain state in ring structures. Because of the resemblance to strips, magnetic rings hold a lot of multi-domain states, such as onion states and various twisted states discussed in detail below. The onion state has two domains, separated by two head-to-head domain walls. The two domains are sometimes referred to two arms. Since the domain walls in ring geometries are well confined, ring magnets serve as a perfect platform to study the motion of domain walls as well as the interaction between them [61, 56]. 70 All the interesting behavior of magnetic systems stems from the fact that there is a great variety of metastable states. Several states may be stable at remanence depending on the history how remanence is reached. One can see from Fig. 5.3 that the dominant phase for thin films is the vortex state, but the onion state is actually a metastable state in a very large region. If the remanence state is obtained by relaxing from saturation, it is usually an onion state. A remanent state phase diagram is given for Co nanorings [70] where the onion state area is significantly increased. As mentioned above, onion states are characterized by two head-to-head domain walls[29] which have long been studied in magnetic strips. Two kinds of head-to-head domain walls exist: transverse domain walls in thin rings, see Fig. 5.4(a) and vortex walls in wide rings, see Fig. 5.4(b). A geometric phase diagram (see Fig. 5.5) for the two head-to-head domain walls was found by Laufenberg et al [69]. Note that the phase diagram was obtained by relaxing the system from satu- ration, so it does not represent the ground state of the system at zero field and shows no vortex state at all. One can see from the Fig. 5.5(b) that analytical calculations tend to favor vortex walls while simulations tend to favor transverse walls compared with the experimental result. It results from the fact that there is an energy barrier between vortex walls and transverse walls. Analytical calcula- tions give the lower energy state, while real systems can stay in the local minimum with transverse walls. Simulations are performed in zero temperature, so that it is harder to form vortex walls than in the experimental situation, where thermal fluctuation can help the system overcome the barrier. From a topological point of view, the transverse wall is composed of a ω=1/2 defect at the outer edge and a ω = −1/2 defect at the inner edge[106]. The vortex 71 Figure 5.4: Head-to-head domain wall. (a) onion state with transverse domain wall; (b) onion state with vortex domain wall. The numbers in the figure indicate the type of the topological defect. wall is composed of two ω = −1/2 defects respectively at the outer and inner edge together with a ω = +1 vortex defect at the center of the rim.[119] Recently a new category of metastable states has been discovered[9] in rela- tively small rings made of both Co and permalloy with R o ∼ 180 − 520nm,w ∼ 30−200nm,h ∼ 10nm: the twisted states (T) (or saddle state[11]). These states are characterized by 360 o domain walls [82, 81] (see Fig. 5.6), which have previ- ously been reported in narrow thin film strips. [123, 91] They are shown to be stable within a field range of several hundred Os [9]. Furthermore, they have low stray fields and are easy to control by current, which makes them attractive for 72 Figure 5.5: [69]( Reused with permission from M. Laufenberg, Applied Physics Letters, 88, 052507 (2006). Copyright 2006, American Institute of Physics.) (color online) “(a) Experimental phase diagram for head-to-head domain walls in NiFe rings at room temperature. Black squares indicate vortex walls and red circles transverse walls. The phase boundaries are shown as solid lines. (b) A comparison of the upper experimental phase boundary (solid line) with results from calcula- tions(dottedline)andmicromagneticsimulations(dashedline). Closetothephase boundaries, both wall types can be observed in nominally identical samples due to slight geometrical variations. The thermally activated wall transitions shown were observed for the ring geometry marked with a red cross (W=730 nm, t=7 nm).” data storage applications. There can be more than one 360 o domain wall in rings, resulting in a multi-twisted state. The 360 o domain walls can be viewed as two transverse domain walls (Fig. 5.6(b)). The attraction between the two transverse walls occurs because they have opposite senses of rotation. This tendency is bal- anced by the exchange energy in the region between the walls. The existence of this domain wall can also be understood fairly easily by topological arguments, referring to the properties (iv) and (vi) of topological defects in Sec. II. When the width is small, the two transverse domain walls have the same defect on the same side, and they are stable. So these edge defects cannot annihilate, resulting in a 360 o domain wall. When the width is larger, however, one of the ω=+1/2 73 defects becomes unstable and transforms into a ω = −1/2 defect by introducing a vortex into the center of the rim. Then the two transverse domain walls are able to annihilate each other, resulting in a vortex state. Therefore, only a portion of the region of stable transverse wall in the phase diagram Fig. 5.5 can possibly hold twisted states. Twisted states in large rings have almost zero remanence, so magnetization is incapable of distinguishing between vortex and twisted states and other quantities must be used. Toroidal moment[94] and winding number are supplementary order parameter candidates. 1/2 (a) (b) 1/2 -1/2 -1/2 Figure 5.6: Twisted states: (a) with single 360 o domain wall, (b) with two 360 o domain walls. The numbers in the figure indicate the type of defects. In the presence of an external magnetic field, several other states exist: wave statesandshiftedvortexcorestates. Theyusuallyexistinverythickrings,whichis of less interest for practical reasons. Regarding the effect of crystalline anisotropy, it stabilizes quasi uniform states (like onion states) and imposes some additional domain structures. 74 5.3 Switching Processes For data storage applications, understanding switching processes is crucial. The main effort is directed towards identifying simple reproducible switching processes and reliable sensitive detection techniques. Meanwhile, in terms of fundamental physics, the understanding of switching processes is just as important. As picosec- ond and nanometer time and spatial resolution detection techniques are still lim- ited, the microscopic details of switching processes and other dynamic transitions are currently mainly investigated by micromagnetic simulations. One issue which is hard to quantify is the distribution of the switching field. Computationally, the transition in the hysteresis curve is very sharp, while the experimental curves have a broad transition region caused by edge roughness, defects and thermal fluctua- tion. We use the Monte Carlo (MC) combined with scaling technique[76, 75, 77] to study the magnetization reversal processes. Neglecting the crystalline energy, the total energy (E) of a magnetic nanoparticle in a magnetic field consists of three terms: exchange interaction, dipolar interaction, and Zeeman energy, represented respectively by the following expression: E = −J <i,j> S i · S j +D i,j S i · S j −3( S i · ˆ r ij )( S j · ˆ r ij ) r 3 ij − H · i S i , (5.2) InMC,singlespinupdateisadoptedtogivequasitimedependentbehavior.[84] We chooseJa 3 /D = 5000whichisclosetotheparameterofCo. Sincewearetrying to find the generic property of nanomagnets, the absolute value of parameters here is not as important as their relative magnitude and the qualitative behavior. To 75 maketheconclusionmoregeneral,weuseexchangelengthL ex = a Ja 3 /D defined in Chapter 2. This quantity differs with traditional definition by factor of 2 or so, depending on the material. Assume a=0.3nm, L ex ∼ 20nm. Exchange length is very important in determining the property of nanomagnetic systems. It is the measure of the span of magnetic domains. As is shown in in Section 3.3, the diameter of vortex core is 1.3L ex [122]. The temperature for MC simulation is set to be around 50K and at least 10 4 MC steps are performed for each field point. As usual, define R o as outer radius, R i as inner radius, w as width and h as height (see the inset of Fig.5.7(a)). Note that the traditional way of studying switching processes is by applying a uniform magnetic field with changing magnitude. Motivated by the vortex con- figuration, circular fields are also studied in order to switch the two vortex states with opposite circulation. Here we mainly discuss the uniform field switching case and say a few words about the circular field afterwards. We conclude our new findings and previous results in literature in Fig.5.7. It summarizes complete switching phase diagrams for magnetic nanorings. Up till recently, three magnetization reversal processes had been intensively studied and well understood: one-step (or single) switching (O-O, see Fig.5.8(b)); two step or double switching (O-V-O, see Fig. 5.8(c)); triple switching (O-V-VC-O, see Fig.5.8(d)). The above magnetization reversal processes are found in relatively large ring in the micrometer range. It has been shown that in submicro and truly nano range, new phenomena exist and wait to be explored. We have performed simulation for a wide range of parameter and identified three new magnetization reversal processes. Each process is discussed in detail below. 76 w(L ex ) w(L ex ) h/2 R o (L ex ) w(L ex ) 1 2 15 h 1 5 0.3L ex <h<L ex (a) A h R o R i w O v -V-O O-V-O O-V-VC-O O-T-V-O O-T-O as h O-O H R o (L ex ) 12 h 1 5 h<0.3L ex O-(T)- V-O (c) h/2 R o (L ex ) 12 h 1 5 h>L ex O-V-O (b) Figure 5.7: Switching phase diagrams of magnetic nanorings. As indicated in the inset, h denotes the ring height, R i and R o are the inner and outer ring radii, and w = R o −R i is the width. A magnetic field H is applied in plane horizontally. The magnetic exchange length L ex is defined in the text. Phase diagrams are shown for the cases (a) 0.3L ex <h<L ex , (b)h>L ex ,(c)h< 0.3L ex .The shaded areas in the regime of small w’s represent rings whose remanence states are out-of-plane ferromagnets, and hence not relevant for switching. The color filled areas represent the different switching processes discussed in the text. green: onion - onion (O-O); red: onion - twisted - onion (O-T-O); yellow: vertical onion - vortex - onion (O v -V-O); gray: onion - twisted - vortex - onion (O-T-V-O); cyan: onion - vortex - vortex core - onion (O-V-VC-O); white: onion - vortex - onion (O-V-O). The point ‘A’ indicates the onion - vortex transition for discs. This critical point moves towards larger (w,R o ) values when h decreases. 5.3.1 One-step or single switching This is a direct onion-to-onion-state switching (O-O). Fig.5.8(b)) shows a typical hysteresis curve. In this process, an onion state is reversed to the opposite onion 77 -1.0 -0.5 0.0 0.5 1.0 M/M s (a) (b) -2 -1 0 1 2 -1.0 -0.5 0.0 0.5 1.0 M/M s H (10 3 Oe) (c) -2 -1 0 1 2 H (10 3 Oe) (d) Figure 5.8: Schematic switching process and hysteresis: (a) hysteresis of mag- netic discs, (b) one-step switching of rings, (c) double switching of rings, (d) triple switching of rings. state directly. This process is usually only observed experimentally in small rings, especially in the thin film limit (small h). Theoretically speaking, it should happen in much thicker system if the ring is perfectly symmetric, as it is observed in simulations. Forrealsystems, however, therealwaysexistssomesortofasymmetry caused either by defects or by the environment. When asymmetry exists, the system tends to fall into its true ground vortex state, which will result in a double switching process. It is believed that the switching mechanism is simply a coherent rotation, where two domain walls move in the same rotational direction. This can easily be observed computationally if one intentionally introduces some sort of asymme- try into the system. Yoo et al[118] offered experimental phase diagrams for Co rings with constant width and R o (equivalent to D/2 in Fig. 5.9) and varying height 2 ∼ 34nm.They conclude that the main geometric factor is the film thickness. Whenh< 6nm, the 78 Figure 5.9: [118]( Reused with permission from Y. G. Yoo, Applied Physics Letters, 82, 2470 (2003). Copyright 2003, American Institute of Physics.) “Phase diagrams of two-step switching (open circles) and single switching (full circles) as a function the ring geometrical parameters. (a) For a ring width of 0.25 mm. (b) For a ring diameter of 2 mm. The solid lines define the boundary between the two different switching regimes.” above one-step switching happens, while otherwise it is two-step switching. For small radii this is not necessarily true, since they only study R o > 500nm. Rings with large radii resemble straight strips very well. With shrinking radius, however, the story is different. Since magnetic processes including nucleation and annihi- lation depend strongly on the curvature, small rings can behave quite differently from their bigger counterparts. 79 5.3.2 Two-step or double switching The dominant switching process for magnetic rings is O-V-O switching (see Fig. 5.8(c)), stabilized by any spurious spatial asymmetry. Here, two opposite onion states are separated by a vortex state. A large regime in the geometric phase diagram falls into this category (see all the white areas of Fig. ??). In most cases the remanence state starting from positive saturation is an onion state, and both transitions occur at negative field. When the element is very thick, the first transition field is positive so that the remanence state can be a vortex state then.[70, 105] This behavior is found for elements near the boundary between double and triple switching in the switching phase diagrams. The mechanisms of the two steps are discussed separately below. For O-V, two possibilities exist, depending on whether it is initiated by domain wall motion or nucleation. (i) Nucleation and buckling: a strong buckling of the magnetization happens first in one arm of the onion state where the magnetic moments are antiparallel to the external field, followed by a nucleation of a vortex passing through the arm[38,105]. Thisprocessisnottypicalintwosteptransitions. Rather, ithappens in some shifted inner circle rings or relatively wide rings. The transition field can be positive, resulting in a vortex state at remanence. It is similar to the triple switching-process, so it can also be regarded as a transient process. (ii) Nucleation free and domain wall motion: either one wall is pinned stronger than the other or both walls move towards each other due to asymmetry. In either case the two domain walls approach each other with increasing applied magnetic field. When these two domain walls meet, they annihilate each other, provided the ring is sufficiently wide (w> 5L ex ). If not, intermediate metastable twisted states 80 have recently been found to exist for small narrow rings [9]. The process involving twisted state will be discussed in detail below. It is interesting and enlightening to understand this process from the topological point of view. As is shown in Sec.III, each of the transverse domain walls consist of two half-vortices: one ω = −1/2 defect on the outer edge and one ω=+1/2 defect on the inner edge. The vortex state cannot form in arbitrarily thin rings since the half vortices with the same winding number cannot annihilate directly and they cannot move to bulk either. Only when the vortex wall is also an energy minimum, one of the ω=+1/2 edge defects can emit a ω = +1 vortex into the bulk and transform into a ω = −1/2 defect. The emitted vortex will travel to the other side and turn the ω = −1/2 defect into a ω =+1/2 defect.[106, 11] Then the edge defects can annihilate. Overall, the choice of the above processes depends strongly on the ring width, but less so on R o and h (see Fig. 5.9). These two processes both happen for relatively small widths. The V-O transition is sometimes called a first magnetization curve. In the half of the ring with magnetization antiparallel to magnetic field H, a vortex domain wall nucleates at the inner side of the rim and passes through it perpendicular to H. In the meantime two transverse head-to-head domain-wall-like structures appearnexttothevortexandquicklypropagateinoppositedirections, formingthe final onion state. This process is mainly shape dependent and has little to do with anisotropy, because it depends on how easy it is to form a vortex wall on the edge, which can be assisted by the edge imperfection (or roughness). As expected, this process depends also on the local curvature of the ring. Since the local curvature and edge roughness affect small rings more, the distribution of the switching field will be larger in these systems. However the magnitude of transition field H c is insensitive to the radius. It increases with height h[58] since thicker elements favor 81 vortex structures, and decreases with width as the vortex state is more stable in thin rings. The nucleation requires a large twisting of the spins, which is harder to achieve in thin rings. Simulations tend to give higher H c in the absence of defects, but defects are known to reduce H c . This discrepancy is more severe for thin rings. The field distribution of O-V is typically larger than in the V-O case and can mainly be attributed to the variation of intrinsic defects among different rings. As temperature tends to wipe out the effects due to defects, i.e. thermal fluctuation makeiteasiertoovercomethelocalenergy barrierinducedbydefects, thedistribu- tion becomes larger when the temperature goes down. This is a little bit surprising at first glance. However, this conclusion can be used to determine whether defects are responsible for the field distribution. If the distribution depends strongly on temperature, then defects are important. Generally speaking, transitions involving nucleation processes are less prone to defects and thermal fluctuation than pro- cesses involving domain wall or vortex core motion.[57] Since defects always exist, simulations at 0K for perfect rings mimic more, in some sense, the experimental situation at high temperature. 5.3.3 Triple switching Thisprocessinvolvethreeintermediatesteps: O-V-VC-O.[57,105,104]Itissimilar toO-V-Oexceptthatthevortexstatedoesnotdeformintoanonionstateabruptly, but nucleates a vortex core in one arm, and the core then moves slowly to the outer rim. To some extent, it is similar to the switching process of magnetic discs where the core in the vortex state is shifted by a magnetic field and finally exits at the boundary (see Fig. 5.8(a)). To see this, Steiner et al [105] have shown a sequence of hysteresis curves with various inner radii R i from the disc limit (R i = 0). The 82 field distribution of the V-VC transition is affected little by temperature, so it is affected little by defects. On the other hand the VC-O is process affected strongly by temperature, so defects play an important role here as cores can be pinned by defects. The triple switching process only exists for thick and very wide rings. Since these rings lack the advantages of rings mentioned before, this switching process is less interesting and less studied. 5.3.4 Vertical O-V-O switching (O v -V-O) On first sight, this process appears to fall in the same category as O-V-O, since its hysteresis, shown in Fig. 5.10, shares characteristics similar to double switching (Fig.5.8(c)). The switching mechanism behind it, however, is quite different. The onion state here does not consist of two transverse head-to-head domain walls, but rather of two vertical head-to-head domain walls (see the areas highlighted by open red circles in Fig. 5.10), where the spins point out-of-plane rather than in-plane. Hence, we call the state involving vertical head-to-head domain walls a “vertical” onion state (O v ). The big blue arrow in Fig. 5.10 points to a small step in the magnetization curve, indicating the appearance of out-of-plane domains. The two domains may point parallel or antiparallel. However, the parallel configuration is preferred, since the switching of O-V is more controllable and the transition is sharper compared with the antiparallel case. As we can see from Fig. 5.10, when the two domains point in the same direction, theO-V transition is simply achieved by rotating spins out-of-plane in the intermediate region between the two domain walls (highlighted by the red rectangle), which subsequently align with the out- of-plane domain wall spins. In the antiparallel case, the domain wall which moves faster will first rotate to form a normal in-plane head-to-head domain wall. Then 83 the other domain wall rotates to in-plane alignment, such that the two domain walls can easily annihilate. This second process results in a slightly larger and broader transition field, which however can be easily avoided by applying a small out-of-plane magnetic field. -2 -1 0 1 2 -1.0 -0.5 0.0 0.5 1.0 M/Ms H (10 3 Oe) Figure 5.10: Typical hysteresis curve for out-of-plane O v -V-O switching (R o = 70 nm, w = 7 nm, h = 10 nm). The insets are the snapshot during the O v -V transition. The areas with strong out-of-plane components are highlighted by open red circles. The big blue arrow points to a small step indicating the appearance of out-of-plane domains. TheO v -V-O process has not been reported before, as most of the initial atten- tion in the field had been focused on thin film rings (h w). As seen in the phase diagrams of Fig. 5.7, this switching process indeed belongs to rings withw<h (see the yellow regions). Shape anisotropies force the spins in the domain walls to point out-of-plane. This process is crucial when one wants to use vortex states in high-density memory applications. In the thin film case (h<L ex ), vortex states are not easy to form when the ring radius is small (say R o < 100nm). Instead, twisted states will stand in the way, preventing the formation of vortex states, which will be discussed in detail next. The only limit of this O v -V-O process 84 is the single domain limit for the ring. As long as onion states exist, this scheme applies. However, whentheringisextremelysmall, theonionstatewillbereplaced by a single domain state. Roughly speaking, this transition happens at R o ∼ L ex . We use the remanence magnetization as an order parameter to locate this tran- sition, as shown in Fig. 5.11. The onion state is characterized by a relatively small magnetization (approximately 0.68 for the chosen w and h), since most of the spins are aligned along the ring boundary, whereas the single domain state is characterized by a magnetization close to saturation, since in this case nearly all the spins align in the same direction. Obviously, the transition region (shaded area in Fig. 5.11) is R o ≈ 20−40 nm ≈ 1−2L ex . For rings with R o <L ex , the only magnetization reversal process will be one-step switching via coherent single domain rotation. Therefore, the R o of rings has to be at least L ex if the vortex state is desired. 10 20 30 40 50 60 70 80 90 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 M/Ms R o (nm) Single Domain Out-of-plane Onion Figure 5.11: Single domain to onion state transition with increasing ring diam- eter. Here, the height is fixed at 23 nm, and the width is fixed at 15 nm. The shaded area represents the transition region. 85 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 M/Ms H (10 3 Oe) (a) (b) ∆ H 360 o domain wall Onion Twist Vortex Figure 5.12: Typical hysteresis curve for quadruple O-T-V-O switching (R o = 150 nm,w = 60 nm, h = 15 nm). The insets (a) and (b) are snapshots during the transition. (a) is the onion state at remanence, while (b) is a typical twisted state, characterized by a 360 o domain wall. 5.3.5 Twisted Triple Switching: O-T-V-O This switching process consists of four steps. In contrast to regular O-V-O triple switching, an additional step, characterized by the appearance of a twisted state (see Fig. 5.12 inset (b)), is involved. As mentioned above, twisted states are only found in rings with relatively small lateral dimensions [9] (R o < 500nm). In order tounderstandtheconditionsforthetwistedstatetooccur,firstrecalltheprocessof onion-to-vortex (O-V) transition during the regular triple switching: two domain walls move towards each other and annihilate by introducing a vortex core passing through the rim, i.e. entering from the outer edge and exiting through the inner edge. As the diameter of the vortex core is typically equal to L ex , the width of the ring needs to be at least of the order of several exchange lengths in order to accommodate such a process. However, when the ring width is sufficiently small, the annihilation process is hindered. In that case, the two domain walls are stuck 86 together, forming a 360 o domain wall (see Fig. 5.12 inset (b)). The configuration involving this 360 o domain wall is called “twisted state”.[9] It turns out that this state can be stabilized within a fairly large field range ∆ H, e.g. for the specific ring shown in Fig. 5.12 it is stable between 100 Oe and 300 Oe. This result is consistent with previous literature[9, 101]. Moveover, the results in Ref. [9] show precursor behavior of this type of switching, since the vortex state will not give positive magnetization when the magnetic field is negative. From our simulations we find that the field range ∆ H of the twisted state is sensitive to the geometric parameters (R o , w, h) in the following ways. 1. h<L ex : We find that the twisted state is not stable when the ring height is larger than L ex . It has been reported that magnetic domain structures vary significantly along the vertical direction when the height is large.[117] Our simulations show that this is indeed the case whenh>L ex , as the exchange length L ex is a measure of typical domain size. In this case, the vertical stacking of slightly off-set domains eases the introduction of vortices in the annihilationprocess, destabilizingthe360 o domainwall. Furthermore, ∆ H is most sensitive to ring height. The flatter the system, the larger ∆ H, i.e. ∆ H can be doubled with decreased height. On the other hand, vortices cannot be formed if h is too small (< 0.3L ex ), which leads to the last switching process we will discuss in the next subsection. 2. L ex <w< 5L ex : Typically, the field range ∆ H varies with ring width (about 20%), with a peak around w=3L ex . The reason for this is that there is a qualitative difference between thin and wide rings when the 360 o domain wall is destroyed. For rings withw<L ex , the 360 o domain wall is not stable, with the exception of ultra-small rings (R o < 2L ex ). As it is the case for 87 out-of-plane O-V-O switching, the spins in the domain area first point out- of-plane, and then turn in-plane along the field with increasing applied field. Slightly wider rings can support the 360 o domain wall in a small, but finite, field range and then evolve with the same process. For wider rings with w> 5L ex , a vortex can be introduced through the outer edge of the ring and traversethroughtherim,asalreadydescribedfortheO-V-Otripleswitching case. Slightly thinner rings still feature vortices. However, in this case the vortices cannot remain in the area of the ring where they entered when the reversed transverse field is increased. Instead, they move along the rim of the ring and finally exit. Hence, in this case the associated magnetization step in the hysteresis curve has a fairly large slope, indicating that the twisted region is moving with increasing applied field. The competition of these two mechanism gives rise to a peak of the field range where the twisted state is stable, which occurs at w ≈ 3L ex . 3. L ex <R o < 15L ex : Inordertoobservethetwistedstate,theentireringhasto be sufficiently small, but still larger than the single domain limit. The reason why larger rings destabilize the twisted state more easily is that in these structuresvorticescanenterfromtheouterrim. Insmallerrings, vorticesare not as easily accommodated in the domain wall region, because of differences between the arclengths of the outer and inner domain wall boundaries, due to the increased curvature. Another reason is that for large rings, there is a higher chance that defects serve as a catalyst of local vortex formation. ∆ H is almost independent of R o for fixed aspect ratios (κ = R i /R o ). On the other hand, the width of the magnetization step in M(H) where the vortex state is stable decreases quickly when R o is decreased, because the curvature 88 eases the process of introducing a vortex during the V-O process. When κ is large and the ring is ultra-small, the extreme case where no vortex is involved emerges as discussed in detail below. The above conditions are fulfilled in the gray regions of the phase diagrams in Fig. 5.7. -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 M/Ms H (10 3 Oe) (a) (b) (c) (d) 650 Oe 600 Oe (b) (c) (d) Twist 200 Oe Figure 5.13: Twisted triple switching (O-T-O): (a) Hysteresis curves for ring 1 (R o = 70 nm, w = 18 nm, h = 5 nm, black continuous line) and ring 2 (R o =42 nm, w = 16 nm,h = 14 nm, red dashed line); (b) spin configuration for ring 1 at H = 200Oe; (c)spinconfigurationforring1atH = 600Oe; (d)spinconfiguration for ring 1 at H = 650 Oe. 5.3.6 Twisted Double Switching (O-T-O) This process is an extreme limit of the O-T-V-O switching process. Rather than forming a vortex state to lower the energy, the system chooses to move the entire domain wall towards either end of the ring along the magnetic field (see Fig. 89 5.13(c)), when the applied magnetic field is increased. Ultimately, the 360 o domain wall breaks up into two 180 o domain walls, and one of them travels back to the other side (see Fig. 5.13(d)), forming the reversed onion state. For relatively large rings, one can observe a small step in the magnetization curve, indicating the occurrence of the twisted state (see Fig. 5.13(b)), while this step is not notice- able for very small rings. After the twisted state is formed, the magnetization decreases approximately linearly with increasing applied magnetic field, indicat- ing the movement of the domain wall (see Fig. 5.13(c)). Finally, a large jump is observed, indicating the breakup (see Fig. 5.13(d)) of the 360 o domain wall and the formation of the reversed onion state. This process can only be observed under two extreme conditions: 1. Very small narrow rings (typically R o < 3L ex andκ> 0.5), when 0.3L ex < h<L ex (see the small red region in Fig. 5.7(a)). The red dashed line in Fig.5.13(a) is the typical hysteresis in this category. Here it is the curvature of the ring that prohibits the annihilation process. Under such conditions, the domain region typically covers a quarter of the ring. 2. Ultra-thin small narrow rings (typicallyh< 0.3L ex andκ> 0.5). The black solid line in Fig.5.13(a) is the typical hysteresis. Fig.5.13(b)(c)(d) are snapshots of the configuration evolution with increasing applied magnetic field. In this case, it is the strong easy-plane shape anisotropy that hinders the annihilation process. For these ultra-thin systems, this switching process can be observed for rings as large as R o ∼ 100nm (see the fairly large red region in Fig.5.7(c)). Theoretically they may exist on even larger rings. In reality, however, local defects will favor the formation of a local vortices, destroying the 360 o domain wall. 90 Note that the twisted state is stable at remanence once it is formed. Designs utilizing this state for magnetic memory have been proposed[82]. Our results show that such designs can use very small rings, for which there appears to be great potential to build high density storage systems. Regarding the effects of anisotropy, itgenerallysuppressesthefielddistributionandincreasestheswitching field. One curious observation is that ring structures make the hard axis of the cubic anisotropy into the global easy axis[61]. 5.3.7 Circular Field (a) I H (b) (c) (d) H I Figure 5.14: Schematic diagram of twisted state reversal by circular field. Circular fields are interesting because they can be naturally generated by a perpendicular current through the disc center. This design is especially suitable for rings, as there is a hole in the center to deposit a nanodot to conduct current. As a result current-induced switching can be achieved, which is essential for data storage applications. At first, a circular field was proposed to switch the vortex state into opposite circulation[126], but it requires a fairly large current density 91 and involves some fairly complicated transient states. Recently, after the twisted states were discovered, a new scheme was proposed.[81, 82] As is shown in Fig. 5.14, switching is adjusted between two twisted states with the 360 o domain wall located in the left and right rim (or any two positions along a diameter). This switching requires low current and has a very clean transition process which only involves a domain wall movement. In addition, it is very fast. 5.4 Stochastic Nature of the Switching Process Strictly speaking, all the above analysis is valid for ring with sufficiently large asymmetry, asthedynamicsoftheswitchingprocesshassomeinevitablestochastic nature. In 2006, magnetic switching bistability is first reported[124]: the rings with the same size process through different mechanisms described in the previous section. Similar stochastic nature is found in different magnetic systems. [47, 47] The fundamental reason of these phenomenon is defects and thermal fluctuation. For perfect symmetric rings, there always exists switching bistability except those ultrasmall rings (R o <L ex ). Generally, it is single O-O switching versus one of the rest five switching mechanisms. This can be understood when we recall the starting phase of all the switching processes: the two domain walls in the two ends begin to move in the same rotational direction or move towards each other. The former choice results in the O-O single switching, while the latter gives birth to the other five siblings. Even though there is slightly energy difference for these two movement,thermalfluctuationcaneasilydisturbthesystemresultinginalmostthe same chances of the two choices especially for thin rings. As is outlined before[97], any sort of asymmetry will favor the two domain’s moving towards each other, but 92 the effect is different for different asymmetry and different ring structures. The switching bistability is still an open question. 5.5 Applications and Conclusions As is mentioned in the introduction, one of the most important applications of nanomagnets is data storage. Three schemes using nanorings have been proposed[13]: (i) two onion states, similar to the traditional oblong memory ele- ment, except that the switching current is significantly less. However, the stray field is an obstacle for high density storage; (ii) two vortex states with different circulation, studied most intensively but requiring a relatively high current density [126]; (iii) two twisted states, most promising because of the low stray field and low current. [81] For the first choice, Kl¨ aui et al [61] have performed a dynamics switching study. They used magnetic pulses to switch the magnetization. They found that switching is only possible when H ∗∆ t ≈ 5∗10 −12 T ∗s , keepingH> 20mT and ∆ t> 50ps. The switching time is T=0.4ns, so that the possible switching rate is about 1.25GHz (= 1/(2T)). Most attention has been focused on the second choice, for this one utilizes all the benefits of nanorings. Manmade asymmetry offers a way to control the vortex circulation. Several techniques are available: shifted inner circle[125, 95], notches[59], and elongated shapes like ellipses[54, 103]. Some works[125, 48] claim that even when there exists asymmetry, the switching process is still a stochastic process due to the thermal fluctuation. Some rings in an array may follow O- O switching while other rings may follow O-V-O. Even for the same ring the 93 circulation may not be deterministic, so the strength of the asymmetry needed is still an important open question. Recently, increasing attention has been given to the third possibility[13, 81, 82], as it is suggested that the twisted state can be switched very easily by a small circular current and can achieve high speed. Experimental evidence is needed to confirm the idea. Another obstacle to application is that ring arrays always show a wide dis- tribution of switching fields. To make the application realistic, the width of the distribution has to be reduced. The distribution is attributed to both the interior and exterior defects. Though the influence is qualitatively known, quantitative analysis is still missing. Even though one would like to get rid of defects as much as possible in most cases, they can also be used to engineer the switching behavior. [37, 1]. We have reviewed current results in the study of ferromagnetic rings, with emphasis on (meta)stable spin configurations and switching processes. There are six types of magnetization reversal processes in all under uniform field for magnetic nanoringsincluding three novel typesdiscovered recently. Thedouble switching, in particular, enjoys tremendous popularity. By introducing engineered asymmetry, the circulation direction of the vortex state can be easily controlled by a uniform field. The out-of-plane O-V-O switching features a new out-of-plane onion state and easy switching process. Rings in this category haveh>w and R o can be as small as L ex . The quadruple switching (O-T-V-O) and twisted triple switching (O-T-O) feature the twisted state. Rings in these two categories are fairly small (typically with outer radius less than 200nm). Width has to be less than about 4L ex to stabilize the 360 o domain wall. The field range where the twisted state is stable is most sensitive to height. Small height give rise to large field range and 94 there will be no twisted state whenh>L ex . The mechanisms of these switching processeswereexplainedcarefullyindetailandtopologicalargumentwereshownto beenlightening. Ontheotherhand, circularfieldswitchingiseasytogenerate, and is suitable to switch the twisted state. We conclude our findings and results from previous literature into complete switching phase diagrams as shown in Fig.5.7. These will be very helpful in designing magnetic nanorings with specific properties. Finally we introduced an open question regarding stochastic switching bistability, followed by a brief discussion of data storage applications and limits. 95 Chapter 6 Conclusions In this thesis we investigated static and quasi-static properties of magnetic nanos- tructures, i.e. ground states, anisotropy and hysteresis. We gave a brief review of the micromagnetic theory and some popular numerical techniques. In particular, weemphasizedonMonteCarlosimulationsandtheFastMultipolemethod(FMM). WeimplementedanoptimizedCartesiancoordinatebasedFMM(CCFMM),show- ing that it is appropriate for magnetic simulations. Rules of thumb to choose the optimal set of parameters were proposed. By comparing with FFT, we revealed the application scope of the CCFMM. Besides the numerical techniques, we demostrated the existence of a scaling law by dimensional analysis. This scaling lawexpressesaphysicalsimilaritybetweensystemswithdifferentvaluesofthegov- erning parameters. Thus a small system can be studied to predict the properties of its scaled larger counterpart, saving significant computing resource. We investigated the ground state spin configurations in single-domain nanopar- ticles with different geometric shapes, crystalline anisotropies and lattice struc- tures. We summerized our results into phase diagrams of magnetic nanoparti- cles featuring three competing configurations: in-plane and out-of-plane ferromag- netism and vortex formation. The influence of the vortex core on the scaling behavior and phase diagram was analyzed. Three-dimensional phase diagrams are obtained for cylindrical nanorings, depending on their height, outer and inner radius. Elliptically shaped magnetic nanoparticles were also studied. A new 96 parametrization for double vortex configurations was proposed, and regions in the phase diagram were identified where the double vortex is a stable ground state. Then a systematic study of the configurational anisotropy in square nanomag- nets was performed. Edge length, height, exchange coupling constant and external magnetic field were shown to play vital roles in determining the anisotropy. We identified that the anisotropy field can have four types of symmetry and clarified an incorrect recognition of the one we named G8. This anisotropy field has a large 8-fold component and the system align along neither edge or diagonal, but some- where in between. Afterwards, the relationship between anisotropy and blocking temperature is also quantized. Understanding the influence of anisotropy opens the way to designing new nanostructured magnetic materials where the magnetic properties can be engineered to fits different applications. Finally, we reviewed current results in the study of ferromagnetic rings, with emphasis on switching processes. We computed the switching phase diagrams of magnetic nanorings, and identified three new types of magnetization reversal pro- cesses, such that there are now six known types of magnetization reversal processes in all. TheO v -V-O switching features a new out-of-plane onion state and an clean switching process. Rings in this category haveh>w, and R o can be as small as L ex . The quadruple switching (O-T-V-O) and twisted triple switching (O-T-O) feature twisted states. Rings in these two categories are fairly small (typically with outer radius less than 200nm). Their width has to be less than about 4L ex to stabilize the 360 o domain wall. The field range where the twisted state is stable is most sensitive to the ring height. Small heights give rise to large field ranges, and there will be no twisted state whenh>L ex . We have combined our findings with results from previous literature into complete switching phase diagrams and explained all six types in detail with enlightening topological arguments. These 97 results should be helpful in designing magnetic nanorings with specific properties. In the end, three designs for data storage applications were discussed. 98 References [1] Agarwal, N., Smith, D. J., and McCartney, M. R. J. Appl. Phys. 102 (2007), 023911. [2] Aharoni, A. J. Appl. Phys. 68 (1990), 2892. [3] Barenblatt, G. Scaling. Cambridge University Press, 2003, pp. 286–294. [4] Beleggia, M., Lau, J. W., Schofield, M. A., Zhu, Y., Tandon, S., and DeGraef, M. J. Magn. Magn. Mater. 301 (2006), 131. [5] Brown, G., Schulthess, T. 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Abstract (if available)

Abstract The study of magnetic nanoparticles has been evolving into a rich and rapidly growing research area during the last two decades, featuring many novel phenomena and potential applications. As their characteristic spatial dimensions are sufficiently small, the shape of these nanoparticles becomes one of the dominant factors in determining their magnetic properties. A great variety of magnetic configurations which do not exist in bulk materials have been observed, and many new phenomena have been discovered recently.

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Creator Zhang, Wen (author)

Core Title Properties of magnetic nanostructures

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Physics

Degree Conferral Date 2010-08

Publication Date 06/19/2010

Defense Date 01/21/2010

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag anisotropy,fast multipole method,hysteresis,magnetization reversal,nanomagnet,OAI-PMH Harvest,phase diagram,switching

Language English

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Advisor Haas, Stephan W. (committee chair), Bickers, Nelson (committee member), Dappen, Werner (committee member), Nakano, Aiichiro (committee member), Zhang, Jianfeng (committee member)

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