Module 2C: Micromagnetics

Total Page:16

File Type:pdf, Size:1020Kb

Module 2C: Micromagnetics Module 2C: Micromagnetics Debanjan Bhowmik Department of Electrical Engineering Indian Institute of Technology Delhi Abstract In this part of the second module (2C) we will show how the Stoner Wolfarth/ single domain mode of ferromagnetism often fails to match with experimental data. We will introduce domain walls in that context and explain the basic framework of micromagnetics, which can be used to model domain walls. Then we will discuss how an anisotropic exchange interaction, known as Dzyaloshinskii Moriya interaction, can lead to chirality of the domain walls. Then we will introduce another non-uniform magnetic structure known as skyrmion and discuss its stability, based on topological arguments. 1 1 Brown's paradox in ferromagnetic thin films exhibiting perpendicular magnetic anisotropy We study ferromagnetic thin films exhibiting Perpendicular Magnetic Anisotropy (PMA) to demonstrate the failure of the previously discussed Stoner Wolfarth/ single domain model to explain experimentally observed magnetic switching curves. PMA is a heavily sought after property in magnetic materials for memory and logic applications (we will talk about that in details in the next module). This makes our analysis in this section even more relevant. The ferromagnetic layer in the Ta/CoFeB/MgO stack, grown by room temperature sput- tering, exhibits perpendicular magnetic anisotropy with an anisotropy field Hk of around 2kG needed to align the magnetic moment in-plane (Fig. 1a). Thus if the ferromagnetic layer is considered as a giant macro-spin in the Stoner Wolfarth model an energy barrier equivalent to ∼2kG exists between the up (+z) and down state (-z) (Fig. 1b). Yet mea- surement shows that the magnet can be switched by a field, called the coercive field, as small as ∼50 G, which is 2 orders of magnitude smaller than the anisotropy field Hk, as observed in the Vibrating Sample Magnetometry measurement on the stack (Fig. 1a). This significant deviation from the Stoner Wolfarth model is known as Brown's paradox in magnetism literature [1]. Within the Stoner Wolfarth model the magnet needs to cross the Hk 0 in-plane (x-y plane) energy barrier 2 to switch by 180 from up (+z) to down (-z) and as a result a switching field close to Hk will be necessary (Fig. 1b). However if a domain wall is introduced in the system the magnet can switch through domain wall motion at a switching field much smaller than Hk. This is because across the width of the domain wall the magnetic moment changes gradually from up to down. For the wall to move, each moment inside the wall needs to turn only by a small angle, which needs much lower energy than Hk (Fig. 1c). Starting from the magnet saturated in the up (+z) direction such a domain wall can be introduced by applying a magnetic field in the negative direction much smaller in magnitude than Hk. The ferromagnetic layer has several defects where the anisotropy is much lower than rest of the magnet. So reverse domains nucleate at these defects with domain walls surrounding them. Theoretically if the applied magnetic field is infinitesimally small but negative the domain wall can move such that the reverse polarized domains expand and the entire magnet switches from up (+z) to down (-z). However in reality the domain wall gets pinned at defects where the domain wall sits at a local energy minimum and an external field is needed to "depin" the domain wall. This field is called the depinning field. When the externally applied reverse magnetic field exceeds the depinning field in magnitude, the domain wall moves entirely to switch the magnet over (Fig. 1d). Thus under the domain wall depinning based switching mechanism, the depinning field determines the coercivity of the magnet. If the reverse magnetic field is applied at an angle θ with respect to the film normal, the magnet switches when the component of the applied 2 magnetic field along the normal exceeds the depinning field. Thus the coercivity of the 1 1 magnet varies as cos(θ) . Such cos(θ) dependence of coercivity has been observed in anoma- lous Hall effect measurements we performed on Hall bars made from the Ta/CoFeB/MgO stack (Fig. 1e), confirming that the magnetization of ferromagnetic layer in these stacks indeed switch under a magnetic field by nucleation of reverse domain followed by motion of depinned domain walls [2]. 2 Micromagnetics As seen in the previous section, domain wall is a non uniform magnetic structure. Across the thickness of the wall, the magnetization gradually turns from a vertically upward direction to a vertically downward direction. Such non-uniformity can of course not be modeled by single domain model. Hence we need to develop a formalism to model a large number of magnetic moments interacting with each other through different energy terms we have discussed in the previous module- exchange, anisotropy, dipole interaction, etc. This formalism is known as micromagnetics. We first start from the Heisenberg model, we discussed in module 2B: H = −Σi;jJex(S~i:S~j) (1) where i and j correspond to neighboring atoms. Now after normalizing the spin of individual atoms by a saturation magnetization, we have obtained reduced magnetization vectors ~mi and ~mj corresponding to two atoms i and j, separated by a displacement vector ~ri;j. 2 ( ~mi: ~mj) turns out to be cos(φi;j) which can be approximated as 1 − φi;j, given φi;j is 2 very small. Thus, the Hamiltonian and the corresponding energy depends on φi;j. Now, jφi;jj ≈ j ~mi − ~mjj (2) Next, a very important assumption is made which forms the very core of the micromagnetics formalism. Instead of considering the magnetization arising out of individual atoms (in this case atoms i and j) the magnetization can be assumed to be a continuous field. Hence φi;j 3 Figure 1: (a)Vibrating Sample Magnetometry (VSM) measurement on thin films of Si (substrate)/ SiO2 (100 nm)/ Ta (10 nm)/ CoFeB (1 nm)/ MgO (1 nm)/ Ta (2 nm) shows that the stack exhibits perpendicular magnetic anisotropy. As a result, a large field (∼2000 gauss here) is needed to saturate the magnet in the in-plane direction (red plot). However the out of plane hysteresis loop shows that a very small field can switch the magnet in the out of plane direction (black plot). This behavior is observed for any Co/Pt/AlOx stack or Ta/CoFeB/MgO stack that exhibits perpendicular magnetic anisotropy. (b) Energy landscape of a single domain magnet shows that an anisotropy field of ∼2000 gauss is needed to switch the magnet by 1800 in the out of plane direction. (c) The ferromagnetic domain wall moves by a distance δθ through the rotation of each moment in the domain wall by an angle δθ. (d) Switching of a magnet with perpendicular anisotropy from up/ out of the plane (blue color) to down/ into the plane (red color) is shown schematically. Starting from a saturated state (all blue), reversed domain (red circle) nucleates with a small magnetic field into the plane. However, the domain wall is pinned at defects and hence cannot move. When the field is high4 enough to depin the domain wall, it moves and the reverse domain (red circle) expands to switch the magnet to into the plane (red square). (e) Minimum magnetic field needed to switch the magnet is plotted as a function of the angle between the direction of the applied magnetic field and normal to the surface of the film [2]. can be approximated as: jφi;jj ≈ j ~mi − ~mjj = (mi;xx^ + mi;yy^ + mi;zz^) − (mj;xx^ + mj;yy^ + mj;zz^) @m @m @m @m @m @m ≈ ( x ∆x + x ∆y + x ∆z)^x + ( y ∆x + y ∆y + y ∆z)^y @x @y @z @x @y @z @m @m @m + ( z ∆x + z ∆y + z ∆z)^z @x @y @z @ @ @ = (∆x + ∆y + ∆z )(m x^ + m y^ + m z^) @x @y @z x y z @ @ @ = ((∆xx^ + ∆yy^ + ∆zz^):( x^ + y^ + z^))(m x^ + m y^ + m z^) = ( ~r :r~ )~m (3) @x @y @z x y z i;j Energy of the system (exchange energy) can written as: 2 2 Eex = JexS Σi;jj ~ri;j:r~mj = 2 Z JS Z @mx @mx @mx 2 @my @my @my 2 3 ((∆x + ∆y + ∆z ) + (∆x + ∆y + ∆z ) + a V @x @y @z @x @y @z @m @m @m (∆x z + ∆y z + ∆z z )2)dV @x @y @z J S2Z Z @m @m @m @m @m @m @m @m @m = ex (( x + x + x )2 +( y + y + y )2 +( z + z + z )2)dV a V @x @y @z @x @y @z @x @y @z Z 2 2 2 = A ((rmx) + (rmy) + (rmz) )dV (4) V where ∆x = ∆y = ∆z = a: distance between nearest neighboring atoms, Z is the nearest 2 JexS Z number of neighboring atoms for a given atom and A = a is the exchange correlation 3kB Tc constant. From module 2B, Jex is related to Curie Temperature Tc as: Jex = J(J+1)Z . So, J S2Z 3k T S2Z 3k T A = ex = B c = B c (5) a J(J + 1)Z a a −10 Considering Tc = 1000K and a= 2.9 Angstrom,for Fe, we calculate A = 1:38 × 10 J=m. Usually exchange parameter A used in micromagnetic simulations is in the range of 10−11 to 10−10 J/m. The energy in equation (4) is the exchange energy, which is dependent on relative alignment of spins of adjacent atoms. In addition, the anisotropy energy is given by: Z 2 2 2 Eani = (K1mx) + (K2my) + (K3mz)dV (6) V where K1;K2;K3 are the anisotropy constants.
Recommended publications
  • Thermal Fluctuations of Magnetic Nanoparticles: Fifty Years After Brown1)
    THERMAL FLUCTUATIONS OF MAGNETIC NANOPARTICLES: FIFTY YEARS AFTER BROWN1) William T. Coffeya and Yuri P. Kalmykovb a Department of Electronic and Electrical Engineering, Trinity College, Dublin 2, Ireland b Laboratoire de Mathématiques et Physique (LAMPS), Université de Perpignan Via Domitia, 52, Avenue Paul Alduy, F-66860 Perpignan, France The reversal time (superparamagnetic relaxation time) of the magnetization of fine single domain ferromagnetic nanoparticles owing to thermal fluctuations plays a fundamental role in information storage, paleomagnetism, biotechnology, etc. Here a comprehensive tutorial-style review of the achievements of fifty years of development and generalizations of the seminal work of Brown [W.F. Brown, Jr., Phys. Rev., 130, 1677 (1963)] on thermal fluctuations of magnetic nanoparticles is presented. Analytical as well as numerical approaches to the estimation of the damping and temperature dependence of the reversal time based on Brown’s Fokker-Planck equation for the evolution of the magnetic moment orientations on the surface of the unit sphere are critically discussed while the most promising directions for future research are emphasized. I. INTRODUCTION A. THERMAL INSTABILITY OF MAGNETIZATION IN FINE PARTICLES B. KRAMERS ESCAPE RATE THEORY C. SUPERPARAMAGNETIC RELAXATION TIME: BROWN’S APPROACH II. BROWN’S CONTINUOUS DIFFUSION MODEL OF CLASSICAL SPINS A. BASIC EQUATIONS B. EVALUATION OF THE REVERSAL TIME OF THE MAGNETIZATION AND OTHER OBSERVABLES III. REVERSAL TIME IN SUPERPARAMAGNETS WITH AXIALLY-SYMMETRIC MAGNETOCRYSTALLINE ANISOTROPY A. FORMULATION OF THE PROBLEM B. ESTIMATION OF THE REVERSAL TIME VIA KRAMERS’ THEORY C. UNIAXIAL SUPERPARAMAGNET SUBJECTED TO A D.C. BIAS FIELD PARALLEL TO THE EASY AXIS IV. REVERSAL TIME OF THE MAGNETIZATION IN SUPERPARAMAGNETS WITH NONAXIALLY SYMMETRIC ANISOTROPY 1) Published in Applied Physics Reviews Section of the Journal of Applied Physics, 112, 121301 (2012).
    [Show full text]
  • Dynamic Symmetry Loss of High-Frequency Hysteresis Loops in Single-Domain Particles with Uniaxial Anisotropy
    Journal of Magnetism and Magnetic Materials 324 (2012) 466–470 Contents lists available at SciVerse ScienceDirect Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm Dynamic symmetry loss of high-frequency hysteresis loops in single-domain particles with uniaxial anisotropy Gabriel T. Landi Instituto de Fı´sica da Universidade de Sao~ Paulo, 05314-970 Sao~ Paulo, Brazil article info abstract Article history: Understanding how magnetic materials respond to rapidly varying magnetic fields, as in dynamic Received 2 June 2011 hysteresis loops, constitutes a complex and physically interesting problem. But in order to accomplish a Available online 23 August 2011 thorough investigation, one must necessarily consider the effects of thermal fluctuations. Albeit being Keywords: present in all real systems, these are seldom included in numerical studies. The notable exceptions are Single-domain particles the Ising systems, which have been extensively studied in the past, but describe only one of the many Langevin dynamics mechanisms of magnetization reversal known to occur. In this paper we employ the Stochastic Landau– Magnetic hysteresis Lifshitz formalism to study high-frequency hysteresis loops of single-domain particles with uniaxial anisotropy at an arbitrary temperature. We show that in certain conditions the magnetic response may become predominantly out-of-phase and the loops may undergo a dynamic symmetry loss. This is found to be a direct consequence of the competing responses due to the thermal fluctuations and the gyroscopic motion of the magnetization. We have also found the magnetic behavior to be exceedingly sensitive to temperature variations, not only within the superparamagnetic–ferromagnetic transition range usually considered, but specially at even lower temperatures, where the bulk of interesting phenomena is seen to take place.
    [Show full text]
  • Magnetic Switching of a Stoner-Wohlfarth Particle Subjected to a Perpendicular Bias Field
    electronics Article Magnetic Switching of a Stoner-Wohlfarth Particle Subjected to a Perpendicular Bias Field Dong Xue 1,* and Weiguang Ma 2 1 Department of Physics and Astronomy, Texas Tech University, Lubbock, TX 79409-1051, USA 2 Department of Physics, Umeå University, 90187 Umeå, Sweden; [email protected] * Correspondence: [email protected]; Tel.: +1-806-834-4563 Received: 28 February 2019; Accepted: 21 March 2019; Published: 26 March 2019 Abstract: Characterized by uniaxial magnetic anisotropy, the Stoner-Wohlfarth particle experiences a change in magnetization leading to a switch in behavior when tuned by an externally applied field, which relates to the perpendicular bias component (hperp) that remains substantially small in comparison with the constant switching field (h0). The dynamics of the magnetic moment that governs the magnetic switching is studied numerically by solving the Landau-Lifshitz-Gilbert (LLG) equation using the Mathematica code without any physical approximations; the results are compared with the switching time obtained from the analytic method that intricately treats the non-trivial bias field as a perturbation. A good agreement regarding the magnetic switching time (ts) between the numerical calculation and the analytic results is found over a wide initial angle range (0.01 < q0 < 0.3), as h0 and hperp are 1.5 × K and 0.02 × K, where K represents the anisotropy constant. However, the quality of the analytic approximation starts to deteriorate slightly in contrast to the numerical approach when computing ts in terms of the field that satisfies hperp > 0.15 × K and h0 = 1.5 × K. Additionally, existence of a comparably small perpendicular bias field (hperp << h0) causes ts to decrease in a roughly exponential manner when hperp increases.
    [Show full text]
  • Magnetically Multiplexed Heating of Single Domain Nanoparticles
    Magnetically Multiplexed Heating of Single Domain Nanoparticles M. G. Christiansen,1) R. Chen,1) and P. Anikeeva1,a) 1Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139, USA Abstract: Selective hysteretic heating of multiple collocated sets of single domain magnetic nanoparticles (SDMNPs) by alternating magnetic fields (AMFs) may offer a useful tool for biomedical applications. The possibility of “magnetothermal multiplexing” has not yet been realized, in part due to prevalent use of linear response theory to model SDMNP heating in AMFs. Predictive successes of dynamic hysteresis (DH), a more generalized model for heat dissipation by SDMNPs, are observed experimentally with detailed calorimetry measurements performed at varied AMF amplitudes and frequencies. The DH model suggests that specific driving conditions play an underappreciated role in determining optimal material selection strategies for high heat dissipation. Motivated by this observation, magnetothermal multiplexing is theoretically predicted and empirically demonstrated for the first time by selecting SDMNPs with properties that suggest optimal hysteretic heat dissipation at dissimilar AMF driving conditions. This form of multiplexing could effectively create multiple channels for minimally invasive biological signaling applications. Text: Magnetic fields provide a convenient form of noninvasive electronically driven stimulus that can reach deep into the body because of the weak magnetic properties and low
    [Show full text]
  • Ncomms5548.Pdf
    ARTICLE Received 30 Oct 2013 | Accepted 27 Jun 2014 | Published 22 Aug 2014 DOI: 10.1038/ncomms5548 Magnetic force microscopy reveals meta-stable magnetic domain states that prevent reliable absolute palaeointensity experiments Lennart V. de Groot1, Karl Fabian2, Iman A. Bakelaar3 & Mark J. Dekkers1 Obtaining reliable estimates of the absolute palaeointensity of the Earth’s magnetic field is notoriously difficult. The heating of samples in most methods induces magnetic alteration—a process that is still poorly understood, but prevents obtaining correct field values. Here we show induced changes in magnetic domain state directly by imaging the domain configurations of titanomagnetite particles in samples that systematically fail to produce truthful estimates. Magnetic force microscope images were taken before and after a heating step typically used in absolute palaeointensity experiments. For a critical temperature (250 °C), we observe major changes: distinct, blocky domains before heating change into curvier, wavy domains thereafter. These structures appeared unstable over time: after 1-year of storage in a magnetic-field-free environment, the domain states evolved into a viscous remanent magnetization state. Our observations qualitatively explain reported underestimates from otherwise (technically) successful experiments and therefore have major implications for all palaeointensity methods involving heating. 1 Paleomagnetic laboratory Fort Hoofddijk, Department of Earth Sciences, Utrecht University, Budapestlaan 17, 3584 CD Utrecht, The Netherlands. 2 NGU, Geological Survey of Norway, Leiv Eirikssons vei, 7491 Trondheim, Norway. 3 Van’t Hoff Laboratory for Physical and Colloid Chemistry, Department of Chemistry, Utrecht University, Padualaan 8, 3584 CH Utrecht, The Netherlands. Correspondence and requests for materials should be addressed to L.V.dG.
    [Show full text]
  • Magnetic Materials: Hysteresis
    Magnetic Materials: Hysteresis Ferromagnetic and ferrimagnetic materials have non-linear initial magnetisation curves (i.e. the dotted lines in figure 7), as the changing magnetisation with applied field is due to a change in the magnetic domain structure. These materials also show hysteresis and the magnetisation does not return to zero after the application of a magnetic field. Figure 7 shows a typical hysteresis loop; the two loops represent the same data, however, the blue loop is the polarisation (J = µoM = B-µoH) and the red loop is the induction, both plotted against the applied field. Figure 7: A typical hysteresis loop for a ferro- or ferri- magnetic material. Illustrated in the first quadrant of the loop is the initial magnetisation curve (dotted line), which shows the increase in polarisation (and induction) on the application of a field to an unmagnetised sample. In the first quadrant the polarisation and applied field are both positive, i.e. they are in the same direction. The polarisation increases initially by the growth of favourably oriented domains, which will be magnetised in the easy direction of the crystal. When the polarisation can increase no further by the growth of domains, the direction of magnetisation of the domains then rotates away from the easy axis to align with the field. When all of the domains have fully aligned with the applied field saturation is reached and the polarisation can increase no further. If the field is removed the polarisation returns along the solid red line to the y-axis (i.e. H=0), and the domains will return to their easy direction of magnetisation, resulting in a decrease in polarisation.
    [Show full text]
  • Spin Wave Dispersion in a Magnonic Waveguide G
    1 Proposal for a standard micromagnetic problem: Spin wave dispersion in a magnonic waveguide G. Venkat, D. Kumar, M. Franchin, O. Dmytriiev, M. Mruczkiewicz, H. Fangohr, A. Barman, M. Krawczyk, and A. Prabhakar Abstract—We propose a standard micromagnetic problem, of Micromagnetic Framework (OOMMF) [12], LLG [13], Micro- a nanostripe of permalloy. We study the magnetization dynamics magus [14] and Nmag [15]. We rely on the finite difference and describe methods of extracting features from simulations. method (FDM) adopted by OOMMF and the finite element Spin wave dispersion curves, relating frequency and wave vector, are obtained for wave propagation in different directions relative method (FEM) used in Nmag. The latter is more suitable for to the axis of the waveguide and the external applied field. geometries with irregular edges [16]. However, the compu- Simulation results using both finite element (Nmag) and finite tation overhead and management of resources become major difference (OOMMF) methods are compared against analytic issues in FEM simulations. To compare different numerical results, for different ranges of the wave vector. solvers, the Micromagnetic Modeling Activity Group (µMag) Index Terms—Computational micromagnetics, spin wave dis- publishes standard problems for micromagnetism [17]–[19]. persion, exchange dominated spin waves A more recent addition included the effects of spin transfer torque [20]. However, there has thus far been no standard INTRODUCTION problem that includes the calculation of the spin wave dis- There have been steady improvements in computational persion of a magnonic waveguide. We believe that specifying micromagnetics in recent years, both in techniques as well as a standard problem will promote the use of micromagnetic in the use of graphical processing units (GPUs) [1]–[4].
    [Show full text]
  • Advanced Micromagnetics and Atomistic Simulations of Magnets
    Advanced micromagnetics and atomistic simulations of magnets Richard F L Evans ESM 2018 Overview • Micromagnetics • Formulation and approximations • Energetic terms and magnetostatics • Magnetisation dynamics • Atomistic spin models • Foundations and approximations • Monte Carlo methods • Spin Dynamics • Landau-Lifshitz-Bloch micromagnetics (tomorrow) Micromagnetics source: mumax Why do we need magnetic simulations? Demagnetization factors for different shapes N = 0 N = 1/3 N = 1/2 N = 1 Infinite thin film Infinitely long Sphere cylinder Infinitely long cylinder Short cylinder Why do we need magnetic simulations? Jay Shah et al, Nature Communications 9 1173 (2018) Why do we need magnetic simulations? • Most magnetic problems are not solvable analytically • Complex shapes (cube or finite geometric shapes) • Complex structures (polygranular materials, multilayers, devices) • Magnetization dynamics • Thermal effects • Metastable phases (Skyrmions) Analytical micromagnetics • An analytical branch of micromagnetics, treating magnetism on a small (micrometre) length scale • Mathematically messy but elegant • When we talk about micromagnetics, we usually mean numerical micromagnetics Numerical micromagnetics • Treat magnetisation as a continuum approximation <M> • Average over the local atomic moments to give an average moment density (magnetization) that is assumed to be continuous • Then consider a small volume of space (1 nm)3 - (10 nm)3 where the magnetization (and all atomic moments) are assumed to point along the same direction The micromagnetic
    [Show full text]
  • Modeling Shape Effects in Nano Magnetic Materials with Web Based Micromagnetics
    University of New Orleans ScholarWorks@UNO University of New Orleans Theses and Dissertations Dissertations and Theses 5-21-2005 Modeling Shape Effects in Nano Magnetic Materials With Web Based Micromagnetics Zhidong Zhao University of New Orleans Follow this and additional works at: https://scholarworks.uno.edu/td Recommended Citation Zhao, Zhidong, "Modeling Shape Effects in Nano Magnetic Materials With Web Based Micromagnetics" (2005). University of New Orleans Theses and Dissertations. 157. https://scholarworks.uno.edu/td/157 This Dissertation is protected by copyright and/or related rights. It has been brought to you by ScholarWorks@UNO with permission from the rights-holder(s). You are free to use this Dissertation in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/ or on the work itself. This Dissertation has been accepted for inclusion in University of New Orleans Theses and Dissertations by an authorized administrator of ScholarWorks@UNO. For more information, please contact [email protected]. MODELING SHAPE EFFECTS IN NANO MAGNETIC MATERIALS WITH WEB BASED MICROMAGNETICS A Dissertation Submitted to the Graduate Faculty of the University of New Orleans in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Chemistry by Zhidong Zhao B.S., Huazhong University of Science and Technology, 1989 M.S., Beijing Normal University, 1992 May 2004 Acknowledgement I would like to express my sincere thanks to my supervisor, Professor Scott L.
    [Show full text]
  • Stoner Wohlfarth Model for Magneto Anisotropy
    Stoner Wohlfarth Model for Magneto Anisotropy Xiaoshan Xu 2016/10/20 Levels of details for ferromagnets • Atomic level: • Exchange interaction that aligns atomic moments J푖푗푆푖 ⋅ 푆푗 • Micromagnetic level • Smear the individual atoms into continuum, see magnetization as a function of position (domain wall) • Domain level • Domains are separated by walls of zero thickness • Nonlinear level • Average magnetization of the entire magnet Magnetic anisotropy • Magneto-crystalline anisotropy • Microscopic • Single-ion • Symmetry of the atomic local environment • Shape anisotropy • Macrocopic • Shape of the magnet • Depolarization field Magnetic shape anisotropy N S N N S S N Repulsion, S high energy Attraction, low energy Polarize Depolarize each other, each other, Depolarization factor: low energy high energy 1 훼 퐷 = ln 훼 + 훼2 − 1 − 1 푧 훼2−1 훼2−1 퐷푥 + 퐷푦 + 퐷푧 = 1 퐿 훼 > 1 is the aspect ratio 푧 퐿푥 Spheroidal model for anisotropy Mathematically, both magneto-crystalline anisotropy and magnetic shape anisotropy can be described using the anisotropy tensor (symmetric matrix): 퐷푥푥 퐷푥푦 퐷푥푧 푫 = 퐷푥푦 퐷푦푦 퐷푦푧 퐷푥푧 퐷푦푧 퐷푧푧 The combined matrix can be diagonalized and along the principle axis, the tensor looks like 퐷11 0 0 푫 = 0 퐷22 0 . 0 0 퐷33 Geometrically, these matrices can be described using ellipsoids. A simplified case assumes the ellipsoid is spherioid. 퐷11 = 퐷22 Anisotropy energy: 2 2 2 2 퐸퐴 = −푀 ⋅ 푫 ⋅ 푀 = −M sin 휃퐷11 − 푀 cos 휃 퐷33 2 2 2 = −푀 sin 휃 퐷11 − 퐷33 − 푀 퐷22 2 2 퐸퐴 = 퐾 sin 휃 , 퐾 = −푀 퐷11 − 퐷33 Stoner Wohlfarth model: single domain, homogeneous magnetization Anisotropy energy: 푧Ԧ 2 2 푀 퐸퐴 = 퐾 sin 휃 , 퐾 = −푀 퐷11 − 퐷22 휃 퐻 휙 Zeeman energy: 퐸푍 = −퐻푀푐표푠(휙 − 휃) Total energy: 퐸 = 퐾 sin2 휃 + 퐻푀푐표푠(휙 − 휃) • The direction of the magnetization is a result of competition between the anisotropy energy and the Zeeman energy.
    [Show full text]
  • Superparamagnetic Nanoparticle Ensembles
    1 Superparamagnetic nanoparticle ensembles O. Petracic Institute of Experimental Physics/Condensed Matter Physics, Ruhr-University Bochum, 44780 Bochum, Germany Abstract: Magnetic single-domain nanoparticles constitute an important model system in magnetism. In particular ensembles of superparamagnetic nanoparticles can exhibit a rich variety of different behaviors depending on the inter-particle interactions. Starting from isolated single-domain ferro- or ferrimagnetic nanoparticles the magnetization behavior of both non-interacting and interacting particle-ensembles is reviewed. A particular focus is drawn onto the relaxation time of the system. In case of interacting nanoparticles the usual Néel-Brown relaxation law becomes modified. With increasing interactions modified superparamagnetism, spin glass behavior and superferromagnetism are encountered. 1. Introduction Nanomagnetism is a vivid and highly interesting topic of modern solid state magnetism and nanotechnology [1-4]. This is not only due to the ever increasing demand for miniaturization, but also due to novel phenomena and effects which appear only on the nanoscale. That is e.g. superparamagnetism, new types of magnetic domain walls and spin structures, coupling phenomena and interactions between electrical current and magnetism (magneto resistance and current-induced switching) [1-4]. In technology nanomagnetism has become a crucial commercial factor. Modern magnetic data storage builds on principles of nanomagnetism and this tendency will increase in future. Also other areas of nanomagnetism are commercially becoming more and more important, e.g. for sensors [5] or biomedical applications [6]. Many potential future applications are investigated, e.g. magneto-logic devices [7], [8], photonic systems [9, 10] or magnetic refrigeration [11, 12]. 2 In particular magnetic nanoparticles experience a still increasing attention, because they can serve as building blocks for e.g.
    [Show full text]
  • Ki-Suk Lee Class Lab
    Tue Thur 13:00-14:15 (S103) Ch. 7 Micromagnetism, domains and hysteresis Ki-Suk Lee Class Lab. Materials Science and Engineering Nano Materials Engineering Track Goal of chapter 7 The domain structure of ferromagnets and ferrimagnets is a result of minimizing the free energy, which includes a self-energy term due to the dipole field Hd(r). Free energy in micromagnetic theory is expressed in the continuum approximation, where atomic structure is averaged away and M(r) is a smoothly varying function of constant magnitude. Domain formation helps to minimize the energy in most cases. The Stoner–Wohlfarth model is an exactly soluble model for coercivity based on the simplification of coherent reversal in single-domain particles. The concepts of domain-wall pinning and nucleation of reverse domains are central to the explanation of coercivity in real materials. The magnetization processes of a ferromagnet are related to the modification, and eventual elimination of the domain structure with increasing applied magnetic field. A continuum theory for describing the magnetic phenomena Heff ddMMG ()MHM eff dt MS dt dM M E dt H , functional derivative of the energy eff M MHeff EEEEEexch d zeeman ani Magneto- Zeeman Exchange Dipole-dipole cryatalline M coupling interaction Anisotropy The basic premise of micromagnetism is that - A magnet is a mesoscopic continuous medium where atomic-scale structure can be ignored (§2.1): M(r) and Hd (r) are generally nonuniform, but continuously varying functions of r. - M(r) varies in direction only: its magnitude is the spontaneous magnetization Ms . Domains tend to form in the lowest-energy state of all but submicrometre-sized ferrromagetic or ferrimagnetic samples, because the system wants to minimize its total self-energy, which can be written as a volume integral of the energy density Ed , in terms of the demagnetizing field (2.78): Energy minimization is subject to constraints imposed by exchange, anisotropy and magnetostriction.
    [Show full text]