Advertisement

Micromagnetics and spintronics: models and numerical methods

  • Claas AbertEmail author
Open Access
Colloquium
  • 176 Downloads

Abstract

Computational micromagnetics has become an indispensable tool for the theoretical investigation of magnetic structures. Classical micromagnetics has been successfully applied to a wide range of applications including magnetic storage media, magnetic sensors, permanent magnets and more. The recent advent of spintronics devices has led to various extensions to the micromagnetic model in order to account for spin-transport effects. This article aims to give an overview over the analytical micromagnetic model as well as its numerical implementation. The main focus is put on the integration of spin-transport effects with classical micromagnetics.

Graphical abstract

Keywords

Mesoscopic and Nanoscale Systems 

Notes

Acknowledgments

Open access funding provided by University of Vienna.

References

  1. 1.
    T. Schrefl, G. Hrkac, S. Bance, D. Suess, O. Ertl, J. Fidler, Numerical methods in micromagnetics (finite element method), in Handbook of Magnetism and Advanced Magnetic Materials (John Wiley & Sons, NJ, 2007) Google Scholar
  2. 2.
    Y. Huai, Spin-transfer torque MRAM (STT-MRAM): challenges and prospects, AAPPS Bull. 18, 33 (2008) Google Scholar
  3. 3.
    S.S. Parkin, C. Kaiser, A. Panchula, P.M. Rice, B. Hughes, M. Samant, S.-H. Yang, Giant tunnelling magnetoresistance at room temperature with MgO 100 tunnel barriers, Nat. Mater. 3, 862 (2004) CrossRefADSGoogle Scholar
  4. 4.
    W. Granig, C. Kolle, D. Hammerschmidt, B. Schaffer, R. Borgschulze, C. Reidl, J. Zimmer, Integrated gigant magnetic resistance based angle sensor, in Proceedings of the IEEE Sensors (2006), pp. 542–545 Google Scholar
  5. 5.
    W.F. Brown, Jr., Micromagnetics (Interscience Publisher, New York, 1963) Google Scholar
  6. 6.
    W. Döring, Über die trägheit der wände zwischen weißschen bezirken, Z. Naturforsch. A 3, 373 (1948) zbMATHCrossRefADSGoogle Scholar
  7. 7.
    H. Kronmüller, General micromagnetic theory, in Handbook of Magnetism and Advanced Magnetic Materials (John Wiley & Sons, NJ, 2007) Google Scholar
  8. 8.
    J.E. Miltat, M.J. Donahue, Numerical micromagnetics: finite difference methods, in Handbook of Magnetism and Advanced Magnetic Materials (John Wiley & Sons, NJ, 2007) Google Scholar
  9. 9.
    J. Leliaert, M. Dvornik, J. Mulkers, J.D. Clercq, M.V. Milošević, B.V. Waeyenberge, Fast micromagnetic simulations on GPU – recent advances made with mumax3, J. Phys. D 51, 123002 (2018) CrossRefADSGoogle Scholar
  10. 10.
    J.D. Jackson, Classical Electrodynamics (John Wiley & Sons, NJ, 2012) Google Scholar
  11. 11.
    D.J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, New Jersey, 1994) Google Scholar
  12. 12.
    W. Döring, Mikromagnetismus, in Handbuch der Physik, edited by S. Flügge (Springer, Berlin, Heidelberg, 1966), Vol. 18/2, pp. 314–437 Google Scholar
  13. 13.
    A. Hubert, R. Schäfer, Magnetic Domains (Springer, Berlin, 1998) Google Scholar
  14. 14.
    I. Dzyaloshinsky, A thermodynamic theory of weak ferromagnetism of antiferromagnetics, J. Phys. Chem. Solids 4, 241 (1958) CrossRefADSGoogle Scholar
  15. 15.
    T. Moriya, Anisotropic superexchange interaction and weak ferromagnetism, Phys. Rev. 120, 91 (1960) CrossRefADSGoogle Scholar
  16. 16.
    X. Yu, Y. Onose, N. Kanazawa, J. Park, J. Han, Y. Matsui, N. Nagaosa, Y. Tokura, Real-space observation of a two-dimensional skyrmion crystal, Nature 465, 901 (2010) CrossRefADSGoogle Scholar
  17. 17.
    X. Yu, M. Mostovoy, Y. Tokunaga, W. Zhang, K. Kimoto, Y. Matsui, Y. Kaneko, N. Nagaosa, Y. Tokura, Magnetic stripes and skyrmions with helicity reversals, Proc. Natl. Acad. Sci. 109, 8856 (2012) CrossRefADSGoogle Scholar
  18. 18.
    A. Bogdanov, U. Rößler, Chiral symmetry breaking in magnetic thin films and multilayers, Phys. Rev. Lett. 87, 037203 (2001) CrossRefADSGoogle Scholar
  19. 19.
    D. Cortés-Ortuño, P. Landeros, Influence of the Dzyaloshinskii-Moriya interaction on the spin-wave spectra of thin films, J. Phys. Condens. Matter 25, 156001 (2013) CrossRefADSGoogle Scholar
  20. 20.
    M.A. Ruderman, C. Kittel, Indirect exchange coupling of nuclear magnetic moments by conduction electrons, Phys. Rev. 96, 99 (1954) CrossRefADSGoogle Scholar
  21. 21.
    T. Kasuya, A theory of metallic ferro-and antiferromagnetism on Zener’s model, Prog. Theor. Phys. 16, 45 (1956) zbMATHCrossRefADSGoogle Scholar
  22. 22.
    K. Yosida, Magnetic properties of Cu-Mn alloys, Phys. Rev. 106, 893 (1957) CrossRefADSGoogle Scholar
  23. 23.
    K. Fabian, F. Heider, How to include magnetostriction in micromagnetic models of titanomagnetite grains, Geophys. Res. Lett. 23, 2839 (1996) CrossRefADSGoogle Scholar
  24. 24.
    Y. Shu, M. Lin, K. Wu, Micromagnetic modeling of magnetostrictive materials under intrinsic stress, Mech. Mater. 36, 975 (2004) CrossRefGoogle Scholar
  25. 25.
    L. Torres, L. Lopez-Diaz, E. Martinez, O. Alejos, Micromagnetic dynamic computations including eddy currents, IEEE Trans. Magn. 39, 2498 (2003) CrossRefADSGoogle Scholar
  26. 26.
    G. Hrkac, M. Kirschner, F. Dorfbauer, D. Suess, O. Ertl, J. Fidler, T. Schrefl, Three-dimensional micromagnetic finite element simulations including eddy currents, J. Appl. Phys. 97, 10E311 (2005) CrossRefGoogle Scholar
  27. 27.
    R. Hertel, A. Kákay, J. Magn. Magn. Mater. 369, 189 (2014) CrossRefADSGoogle Scholar
  28. 28.
    W. Scholz, J. Fidler, T. Schrefl, D. Suess, H. Forster, V. Tsiantos, et al., Scalable parallel micromagnetic solvers for magnetic nanostructures, Comput. Mater. Sci. 28, 366 (2003) CrossRefGoogle Scholar
  29. 29.
    D.V. Berkov, Fast switching of magnetic nanoparticles: Simulation of thermal noise effects using the Langevin dynamics, IEEE Trans. Magn. 38, 2489 (2002) CrossRefADSGoogle Scholar
  30. 30.
    O. Chubykalo, J. Hannay, M. Wongsam, R. Chantrell, J. Gonzalez, Langevin dynamic simulation of spin waves in a micromagnetic model, Phys. Rev. B 65, 184428 (2002) CrossRefADSGoogle Scholar
  31. 31.
    D.A. Garanin, Fokker-Planck and Landau-Lifshitz-Bloch equations for classical ferromagnets, Phys. Rev. B 55, 3050 (1997) CrossRefADSGoogle Scholar
  32. 32.
    U. Atxitia, O. Chubykalo-Fesenko, N. Kazantseva, D. Hinzke, U. Nowak, R.W. Chantrell, Micromagnetic modeling of laser-induced magnetization dynamics using the Landau-Lifshitz-Bloch equation, Appl. Phys. Lett. 91, 232507 (2007) CrossRefADSGoogle Scholar
  33. 33.
    R.F.L. Evans, D. Hinzke, U. Atxitia, U. Nowak, R.W. Chantrell, O. Chubykalo-Fesenko, Stochastic form of the Landau-Lifshitz-Bloch equation, Phys. Rev. B 85, 014433 (2012) CrossRefADSGoogle Scholar
  34. 34.
    L.D. Landau, E.M. Lifshitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sowjetunion 8, 153 (1935) zbMATHGoogle Scholar
  35. 35.
    T.L. Gilbert, A Lagrangian formulation of the gyromagnetic equation of the magnetic field, Phys. Rev. 100, 1243 (1955) Google Scholar
  36. 36.
    T.L. Gilbert, A phenomenological theory of damping in ferromagnetic materials, IEEE Trans. Magn. 40, 3443 (2004) CrossRefADSGoogle Scholar
  37. 37.
    L.D. Landau, E.M. Lifshitz, Mechanics, in Course of Theoretical Physics (Pergamon Press, Oxford, 1969) Google Scholar
  38. 38.
    J.-E. Wegrowe, M.-C. Ciornei, Magnetization dynamics, gyromagnetic relation, and inertial effects, Am. J. Phys. 80, 607 (2012) CrossRefADSGoogle Scholar
  39. 39.
    N. Bode, L. Arrachea, G.S. Lozano, T.S. Nunner, F. von Oppen, Current-induced switching in transport through anisotropic magnetic molecules, Phys. Rev. B 85, 115440 (2012) CrossRefADSGoogle Scholar
  40. 40.
    M. d’Aquino, C. Serpico, G. Miano, Geometrical integration of Landau-Lifshitz-Gilbert equation based on the mid-point rule, J. Comput. Phys. 209, 730 (2005) MathSciNetzbMATHCrossRefADSGoogle Scholar
  41. 41.
    I. Cimrák, A survey on the numerics and computations for the Landau-Lifshitz equation of micromagnetism, Arch. Comput. Methods Eng. 15, 1 (2007) MathSciNetCrossRefADSGoogle Scholar
  42. 42.
    M.N. Baibich, J.M. Broto, A. Fert, F.N. Van Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friederich, J. Chazelas, Giant magnetoresistance of (001) Fe/(001) Cr magnetic superlattices, Phys. Rev. Lett. 61, 2472 (1988) CrossRefADSGoogle Scholar
  43. 43.
    G. Binasch, P. Grünberg, F. Saurenbach, W. Zinn, Enhanced magnetoresistance in layered magnetic structures with antiferromagnetic interlayer exchange, Phys. Rev. B 39, 4828 1989 CrossRefADSGoogle Scholar
  44. 44.
    J.C. Slonczewski, Current-driven excitation of magnetic multilayers, J. Magn. Magn. Mater. 159, L1 (1996) CrossRefADSGoogle Scholar
  45. 45.
    L. Berger, Emission of spin waves by a magnetic multilayer traversed by a current, Phys. Rev. B 54, 9353 (1996) CrossRefADSGoogle Scholar
  46. 46.
    X. Waintal, E.B. Myers, P.W. Brouwer, D. Ralph, Role of spin-dependent interface scattering in generating current-induced torques in magnetic multilayers, Phys. Rev. B 62, 12317 (2000) CrossRefADSGoogle Scholar
  47. 47.
    D. Worledge, G. Hu, D.W. Abraham, J. Sun, P. Trouilloud, J. Nowak, S. Brown, M. Gaidis, E. O’sullivan, R. Robertazzi, Spin torque switching of perpendicular Ta—CoFeB—MgO-based magnetic tunnel junctions, Appl. Phys. Lett. 98, 022501 (2011) CrossRefADSGoogle Scholar
  48. 48.
    D. Houssameddine, U. Ebels, B. Delaët, B. Rodmacq, I. Firastrau, F. Ponthenier, M. Brunet, C. Thirion, J.-P. Michel, L. Prejbeanu-Buda et al., Spin-torque oscillator using a perpendicular polarizer and a planar free layer, Nat. Mater. 6, 447 (2007) CrossRefADSGoogle Scholar
  49. 49.
    J.-V. Kim, Spin-torque oscillators, Solid State Phys. 63, 217 (2012) CrossRefGoogle Scholar
  50. 50.
    D.C. Ralph, M.D. Stiles, Spin transfer torques, J. Magn. Magn. Mater. 320, 1190 (2008) CrossRefADSGoogle Scholar
  51. 51.
    J. Slonczewski, Currents and torques in metallic magnetic multilayers, J. Magn. Magn. Mater. 247, 324 (2002) CrossRefADSGoogle Scholar
  52. 52.
    J. Xiao, A. Zangwill, M.D. Stiles, Macrospin models of spin transfer dynamics, Phys. Rev. B 72, 014446 (2005) CrossRefADSGoogle Scholar
  53. 53.
    D. Apalkov, M. Pakala, Y. Huai, Micromagnetic simulation of spin transfer torque switching by nanosecond current pulses, J. Appl. Phys. 99, 08B907 (2006) CrossRefGoogle Scholar
  54. 54.
    G.E. Rowlands, I.N. Krivorotov, Magnetization dynamics in a dual free-layer spin-torque nano-oscillator, Phys. Rev. B 86, 094425 (2012) CrossRefADSGoogle Scholar
  55. 55.
    D.V. Berkov, J. Miltat, Spin-torque driven magnetization dynamics: micromagnetic modeling, J. Magn. Magn. Mater. 320, 1238 (2008) CrossRefADSGoogle Scholar
  56. 56.
    S.S. Parkin, M. Hayashi, L. Thomas, Magnetic domain-wall racetrack memory, Science 320, 190 (2008) CrossRefADSGoogle Scholar
  57. 57.
    S. Zhang, Z. Li, Roles of nonequilibrium conduction electrons on the magnetization dynamics of ferromagnets, Phys. Rev. Lett. 93, 127204 (2004) CrossRefADSGoogle Scholar
  58. 58.
    S. Zhang, P. Levy, A. Fert, Mechanisms of spin-polarized current-driven magnetization switching, Phys. Rev. Lett. 88, 236601 (2002) CrossRefADSGoogle Scholar
  59. 59.
    M. Dyakonov, V. Perel, Current-induced spin orientation of electrons in semiconductors, Phys. Lett. A 35, 459 (1971) CrossRefADSGoogle Scholar
  60. 60.
    J. Hirsch, Spin Hall effect, Phys. Rev. Lett. 83, 1834 (1999) CrossRefADSGoogle Scholar
  61. 61.
    S. Murakami, N. Nagaosa, S.-C. Zhang, Dissipationless quantum spin current at room temperature, Science 301, 1348 (2003) CrossRefADSGoogle Scholar
  62. 62.
    J. Sinova, D. Culcer, Q. Niu, N. Sinitsyn, T. Jungwirth, A. MacDonald, Universal intrinsic spin Hall effect, Phys. Rev. Lett. 92, 126603 (2004) CrossRefADSGoogle Scholar
  63. 63.
    M. Dyakonov, Magnetoresistance due to edge spin accumulation, Phys. Rev. Lett. 99, 126601 (2007) CrossRefADSGoogle Scholar
  64. 64.
    C. Petitjean, D. Luc, X. Waintal, Unified drift-diffusion theory for transverse spin currents in spin valves, domain walls, and other textured magnets, Phys. Rev. Lett. 109, 117204 (2012) CrossRefADSGoogle Scholar
  65. 65.
    C.A. Akosa, W.-S. Kim, A. Bisig, M. Kläui, K.-J. Lee, A. Manchon, Role of spin diffusion in current-induced domain wall motion for disordered ferromagnets, Phys. Rev. B 91, 094411 (2015) CrossRefADSGoogle Scholar
  66. 66.
    P.M. Haney, H.-W. Lee, K.-J. Lee, A. Manchon, M.D. Stiles, Current induced torques and interfacial spin-orbit coupling: Semiclassical modeling, Phys. Rev. B 87, 174411 (2013) CrossRefADSGoogle Scholar
  67. 67.
    T. Valet, A. Fert, Theory of the perpendicular magnetoresistance in magnetic multilayers, Phys. Rev. B 48, 7099 1993 CrossRefADSGoogle Scholar
  68. 68.
    Y. Niimi, Y. Kawanishi, D. Wei, C. Deranlot, H. Yang, M. Chshiev, T. Valet, A. Fert, Y. Otani, Giant spin Hall effect induced by skew scattering from bismuth impurities inside thin film CuBi alloys, Phys. Rev. Lett. 109, 156602 (2012) CrossRefADSGoogle Scholar
  69. 69.
    C. Abert, F. Bruckner, C. Vogler, D. Suess, Efficient micromagnetic modelling of spin-transfer torque and spin-orbit torque, AIP Adv. 8, 056008 (2018) CrossRefADSGoogle Scholar
  70. 70.
    Y. Tserkovnyak, A. Brataas, G.E. Bauer, Spin pumping and magnetization dynamics in metallic multilayers, Phys. Rev. B 66, 224403 (2002) CrossRefADSGoogle Scholar
  71. 71.
    J. Mathon, A. Umerski, Theory of tunneling magnetoresistance of an epitaxial Fe/MgO/Fe 001 junction, Phys. Rev. B 63, 220403 (2001) CrossRefADSGoogle Scholar
  72. 72.
    N.M. Caffrey, T. Archer, I. Rungger, S. Sanvito, Prediction of large bias-dependent magnetoresistance in all-oxide magnetic tunnel junctions with a ferroelectric barrier, Phys. Rev. B 83, 125409 (2011) CrossRefADSGoogle Scholar
  73. 73.
    W. Butler, X.-G. Zhang, T. Schulthess, J. MacLaren, Spin-dependent tunneling conductance of Fe–MgO–Fe sandwiches, Phys. Rev. B 63, 054416 (2001) CrossRefADSGoogle Scholar
  74. 74.
    D. Berkov, K. Ramstöck, A. Hubert, Solving micromagnetic problems. towards an optimal numerical method, Phys. Status Solidi a 137, 207 (1993) CrossRefADSGoogle Scholar
  75. 75.
    C. Seberino, H.N. Bertram, Concise, efficient three-dimensional fast multipole method for micromagnetics, IEEE Trans. Magn. 37, 1078 (2001) CrossRefADSGoogle Scholar
  76. 76.
    W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes: The art of Scientific Computing, 3rd edn. (Cambridge University Press, Cambridge, 2007) Google Scholar
  77. 77.
    A.J. Newell, W. Williams, D.J. Dunlop, A generalization of the demagnetizing tensor for nonuniform magnetization, J. Geophys. Res. Solid Earth 98, 9551 (1993) CrossRefGoogle Scholar
  78. 78.
    K.M. Lebecki, M.J. Donahue, M.W. Gutowski, Periodic boundary conditions for demagnetization interactions in micromagnetic simulations, J. Phys. D Appl. Phys. 41, 175005 (2008) CrossRefGoogle Scholar
  79. 79.
    B. Krüger, G. Selke, A. Drews, D. Pfannkuche, Fast and accurate calculation of the demagnetization tensor for systems with periodic boundary conditions, IEEE Trans. Magn. 49, 4749 (2013) CrossRefADSGoogle Scholar
  80. 80.
    Y. Kanai, K. Koyama, M. Ueki, T. Tsukamoto, K. Yoshida, S.J. Greaves, H. Muraoka, Micromagnetic analysis of shielded write heads using symmetric multiprocessing systems, IEEE Trans. Magn. 46, 3337 (2010) CrossRefADSGoogle Scholar
  81. 81.
    C. Abert, G. Selke, B. Krüger, A. Drews, A fast finite-difference method for micromagnetics using the magnetic scalar potential, IEEE Trans. Magn. 48, 1105 (2012) CrossRefADSGoogle Scholar
  82. 82.
    S. Fu, W. Cui, M. Hu, R. Chang, M.J. Donahue, V. Lomakin, Finite-difference micromagnetic solvers with the object-oriented micromagnetic framework on graphics processing units, IEEE Trans. Magn. 52, 1 (2016) Google Scholar
  83. 83.
    C.J. García-Cervera, X.-P. Wang, Spin-polarized currents in ferromagnetic multilayers, J. Comput. Phys. 224, 699 (2007) MathSciNetzbMATHCrossRefADSGoogle Scholar
  84. 84.
    M.J. Donahue, OOMMF user’s guide, version 1.0, Tech. Rep., 1999 Google Scholar
  85. 85.
    D. Cortés-Ortuño, W. Wang, R. Pepper, M.-A. Bisotti, T. Kluyver, M. Vousden, H. Fangohr, Fidimag v2.0. https://github.com/computationalmodelling/fidimag (accessed 2019/02/04)
  86. 86.
    D. Berkov, N. Gorn, MicroMagus–package for micromagneticsimulations (2007). http://www.micromagus.de (accessed 2019/02/04)
  87. 87.
    C. Abert, F. Bruckner, C. Vogler, R. Windl, R. Thanhoffer, D. Suess, A full-fledged micromagnetic code in fewer than 70 lines of NumPy, J. Magn. Magn. Mater. 387, 13 (2015) CrossRefADSGoogle Scholar
  88. 88.
    J. Leliaert, M. Dvornik, J. Mulkers, J. De Clercq, M. Milošević, B. Van Waeyenberge, Fast micromagnetic simulations on GPU–recent advances made with MuMax3, J. Phys. D: Appl. Phys. 51, 123002 (2018) CrossRefADSGoogle Scholar
  89. 89.
    A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, B. Van Waeyenberge, The design and verification of MuMax3, AIP Adv. 4, 107133 (2014) CrossRefADSGoogle Scholar
  90. 90.
    G. Selke, B. Krüger, A. Drews, C. Abert, T. Gerhardt, magnum.fd. (2014). https://github.com/micromagnetics/magnum.fd (accessed 2019/02/04)
  91. 91.
    D. Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics (Cambridge University Press, Cambridge, 2007) Google Scholar
  92. 92.
    Y. Saad, in Iterative Methods for Sparse Linear Systems (SIAM, PA, 2003), Vol. 82 Google Scholar
  93. 93.
    Q. Chen, A. Konrad, A review of finite element open boundary techniques for static and quasi-static electromagnetic field problems, IEEE Trans. Magn. 33, 663 (1997) CrossRefADSGoogle Scholar
  94. 94.
    J. Imhoff, G. Meunier, X. Brunotte, J. Sabonnadiere, An original solution for unbounded electromagnetic 2D- and 3D-problems throughout the finite element method, IEEE Trans. Magn. 26, 1659 (1990) CrossRefADSGoogle Scholar
  95. 95.
    X. Brunotte, G. Meunier, J.-F. Imhoff, Finite element modeling of unbounded problems using transformations: a rigorous, powerful and easy solution, IEEE Trans. Magn. 28, 1663 (1992) CrossRefADSGoogle Scholar
  96. 96.
    F. Henrotte, B. Meys, H. Hedia, P. Dular, W. Legros, Finite element modelling with transformation techniques, IEEE Trans. Magn. 35, 1434 (1999) CrossRefADSGoogle Scholar
  97. 97.
    C. Abert, L. Exl, G. Selke, A. Drews, T. Schrefl, Numerical methods for the stray-field calculation: A comparison of recently developed algorithms, J. Magn. Magn. Mater. 326, 176 (2013) CrossRefADSGoogle Scholar
  98. 98.
    D. Fredkin, T. Koehler, Hybrid method for computing demagnetizing fields, IEEE Trans. Magn. 26, 415 (1990) CrossRefADSGoogle Scholar
  99. 99.
    W. Hackbusch, in Hierarchical Matrices: Algorithms and Analysis (Springer, Berlin, 2015), Vol. 49 Google Scholar
  100. 100.
    N. Popović, D. Praetorius, Applications of H-matrix techniques in micromagnetics, Computing 74, 177 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  101. 101.
    C.J. Garcia-Cervera, A.M. Roma, “Adaptive mesh refinement for micromagnetics simulations, IEEE Trans. Magn. 42, 1648 (2006) CrossRefADSGoogle Scholar
  102. 102.
    C. Geuzaine, J.-F. Remacle, Gmsh: A 3-D finite element mesh generator with built-in pre-and post-processing facilities, Int. J. Numer. Methods Eng. 79, 1309 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  103. 103.
    J. Schöberl, NETGEN an advancing front 2D/3D-mesh generator based on abstract rules, Comput. Vis. Sci. 1, 41 (1997) zbMATHCrossRefGoogle Scholar
  104. 104.
    C. Geuzaine, F. Henrotte, J.-F. Remacle, E. Marchandise, R. Sabariego, Onelab: open numerical engineering laboratory, in 11e Colloque National en Calcul des Structures (2013) Google Scholar
  105. 105.
    J. Schöberl, C++ 11 Implementation of Finite Elements in NGSolve (Institute for Analysis and Scientific Computing, Vienna University of Technology, 2014) Google Scholar
  106. 106.
    L. Gross, P. Cochrane, M. Davies, H. Muhlhaus, J. Smillie, Escript: numerical modelling with python, in Australian Partnership for Advanced Computing (APAC) Conferene, APAC (2005), Vol. 1, p. 31 Google Scholar
  107. 107.
    R. Anderson, A. Barker, J. Bramwell, J. Camier, J. Ceverny, J. Dahm, Y. Dudouit, V. Dobrev, A. Fisher, T. Kolev, D. Medina, M. Stowell, V. Tomov, MFEM: a modular finite element library (2010), DOI:  https://doi.org/10.11578/dc.20171025.1248
  108. 108.
    M.S. Alnæs, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M.E. Rognes, G.N. Wells, The FEniCS project version 1.5, Arch. Numer. Softw. 3, 9 (2015) Google Scholar
  109. 109.
    S. Balay, S. Abhyankar, M. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, V. Eijkhout, W. Gropp, D. Kaushik et al., Petsc users manual revision 3.8, Tech. Rep., Argonne National Lab.(ANL), Argonne, IL, United States, 2017 Google Scholar
  110. 110.
    W. Śmigaj, T. Betcke, S. Arridge, J. Phillips, M. Schweiger, Solving boundary integral problems with bem++, ACM Trans. Math. Softw. (TOMS) 41, 6 (2015) MathSciNetzbMATHCrossRefGoogle Scholar
  111. 111.
    N. Albrecht, C. Börst, D. Boysen, S. Christophersen, S. Börm, H2Lib (2016). http://www.h2lib.org (accessed 2019/02/04)
  112. 112.
    M.-A. Bisotti, M. Beg, W. Wang, M. Albert, D. Chernyshenko, D. Cortés-Ortuño, R.A. Pepper, M. Vousden, R. Carey, H. Fuchs, A. Johansen, G. Balaban, L.B.T. Kluyver, H. Fangohr, FinMag (2018). https://github.com/fangohr/finmag (accessed 2019/02/04)
  113. 113.
    C. Abert, L. Exl, F. Bruckner, A. Drews, D. Suess, magnum.fe: a micromagnetic finite-element simulation code based on FEniCS, J. Magn. Magn. Mater. 345, 29 (2013) CrossRefADSGoogle Scholar
  114. 114.
    C. Abert, M. Ruggeri, F. Bruckner, C. Vogler, G. Hrkac, D. Praetorius, D. Suess, A three-dimensional spin-diffusion model for micromagnetics, Sci. Rep. 5, 14855 (2015) CrossRefADSGoogle Scholar
  115. 115.
    M. Ruggeri, C. Abert, G. Hrkac, D. Suess, D. Praetorius, Coupling of dynamical micromagnetism and a stationary spin drift-diffusion equation: a step towards a fully self-consistent spintronics framework, Phys. B Condens. Matter 486, 88 (2016) CrossRefADSGoogle Scholar
  116. 116.
    C. Abert, M. Ruggeri, F. Bruckner, C. Vogler, A. Manchon, D. Praetorius, D. Suess, A self-consistent spin-diffusion model for micromagnetics, Sci. Rep. 6, 16 (2016) CrossRefADSGoogle Scholar
  117. 117.
    F. Alouges, E. Kritsikis, J.-C. Toussaint, A convergent finite element approximation for Landau-Lifschitz-Gilbert equation, Physica B 407, 1345 (2012) zbMATHCrossRefADSGoogle Scholar
  118. 118.
    M. Sturma, J.-C. Toussaint, D. Gusakova, Geometry effects on magnetization dynamics in circular cross-section wires, J. Appl. Phys. 117, 243901 (2015) CrossRefADSGoogle Scholar
  119. 119.
    W. Scholz, MagPar (2010). http://www.magpar.net/ (accessed 2019/02/04)
  120. 120.
    T. Fischbacher, M. Franchin, G. Bordignon, H. Fangohr, A systematic approach to multiphysics extensions of finite-element-based micromagnetic simulations: Nmag, IEEE Trans. Magn. 43, 2896 (2007) CrossRefADSGoogle Scholar
  121. 121.
    D. Suess, T. Schrefl, FEMME (2018). http://suessco.com/simulations/solutions/femme-software/ (accessed 2019/02/04)
  122. 122.
    A. Kakay, E. Westphal, R. Hertel, Speedup of FEM micromagnetic simulations with graphical processing units, IEEE Trans. Magn. 46, 2303 (2010) CrossRefADSGoogle Scholar
  123. 123.
    R. Chang, S. Li, M. Lubarda, B. Livshitz, V. Lomakin, FastMag: fast micromagnetic simulator for complex magnetic structures, J. Appl. Phys. 109, 07D358 (2011) CrossRefGoogle Scholar
  124. 124.
    E. Kritsikis, A. Vaysset, L. Buda-Prejbeanu, F. Alouges, J.-C. Toussaint, Beyond first-order finite element schemes in micromagnetics, J. Comput. Phys. 256, 357 (2014) MathSciNetzbMATHCrossRefADSGoogle Scholar
  125. 125.
    L. Exl, T. Schrefl, Non-uniform FFT for the finite element computation of the micromagnetic scalar potential, J. Comput. Phys. 270, 490 (2014) MathSciNetzbMATHCrossRefADSGoogle Scholar
  126. 126.
    D. Apalkov, P. Visscher, Fast multipole method for micromagnetic simulation of periodic systems, IEEE Trans. Magn. 39, 3478 (2003) CrossRefADSGoogle Scholar
  127. 127.
    P. Palmesi, L. Exl, F. Bruckner, C. Abert, D. Suess, Highly parallel demagnetization field calculation using the fast multipole method on tetrahedral meshes with continuous sources, J. Magn. Magn. Mater. 442, 409 (2017) CrossRefADSGoogle Scholar
  128. 128.
    L. Exl, W. Auzinger, S. Bance, M. Gusenbauer, F. Reichel, T. Schrefl, Fast stray field computation on tensor grids, J. Comput. Phys. 231, 2840 (2012) MathSciNetzbMATHCrossRefADSGoogle Scholar
  129. 129.
    L. Exl, C. Abert, N.J. Mauser, T. Schrefl, H.P. Stimming, D. Suess, FFT-based Kronecker product approximation to micromagnetic long-range interactions, Math. Models Methods Appl. Sci. 24, 1877 (2014) MathSciNetzbMATHCrossRefGoogle Scholar
  130. 130.
    D. Suess, V. Tsiantos, T. Schrefl, J. Fidler, W. Scholz, H. Forster, R. Dittrich, J. Miles, Time resolved micromagnetics using a preconditioned time integration method, J. Magn. Magn. Mater. 248, 298 (2002) CrossRefADSGoogle Scholar
  131. 131.
    E. Fehlberg, Low-order classical Runge-Kutta formulas with stepsize control and their application to some heat transfer problems, NASA Technical Report, Vol. 315, 1969 Google Scholar
  132. 132.
    J.R. Dormand, P.J. Prince, A family of embedded Runge-Kutta formulae, J. Comput. Appl. Math. 6, 19 (1980) MathSciNetzbMATHCrossRefGoogle Scholar
  133. 133.
    R.L. Burden, J.D. Faires, Numerical Analysis (Cengage Learning, MA, 2010) Google Scholar
  134. 134.
    F. Alouges, A new finite element scheme for landau-lifchitz equations, Discrete Contin. Dyn. Syst. Ser. S 1, 187 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  135. 135.
    P. Goldenits, G. Hrkac, D. Praetorius, D. Suess, An effective integrator for the Landau-Lifshitz-Gilbert equation, in Proceedings of Mathmod 2012 Conference (2012) Google Scholar
  136. 136.
    M. Ruggeri, Coupling and numerical integration of the Landau-Lifshitz-Gilbert equation, Ph.D. thesis, TU Wien, 2016 Google Scholar
  137. 137.
    C. Abert, G. Hrkac, M. Page, D. Praetorius, M. Ruggeri, D. Suess, Spin-polarized transport in ferromagnetic multilayers: An unconditionally convergent FEM integrator, Comput. Math. Appl. 68, 639 (2014) MathSciNetzbMATHCrossRefGoogle Scholar
  138. 138.
    A.C. Hindmarsh, P.N. Brown, K.E. Grant, S.L. Lee, R. Serban, D.E. Shumaker, C.S. Woodward, Sundials: Suite of nonlinear and differential/algebraic equation solvers, ACM Trans. Math. Softw. (TOMS) 31, 363 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  139. 139.
    R. Hertel, Micromagnetic simulations of magnetostatically coupled nickel nanowires, J. Appl. Phys. 90, 5752 (2001) CrossRefADSGoogle Scholar
  140. 140.
    J. Fischbacher, A. Kovacs, H. Oezelt, T. Schrefl, L. Exl, J. Fidler, D. Suess, N. Sakuma, M. Yano, A. Kato et al., Nonlinear conjugate gradient methods in micromagnetics, AIP Adv. 7, 045310 (2017) CrossRefADSGoogle Scholar
  141. 141.
    L. Exl, S. Bance, F. Reichel, T. Schrefl, H. Peter Stimming, N.J. Mauser, LaBonte’s method revisited: An effective steepest descent method for micromagnetic energy minimization, J. Appl. Phys. 115, 17D118 (2014) CrossRefGoogle Scholar
  142. 142.
    E. Weinan, W. Ren, E. Vanden-Eijnden, Simplified and improved string method for computing the minimum energy paths in barrier-crossing events, J. Chem. Phys. 126, 164103 (2007) CrossRefADSGoogle Scholar
  143. 143.
    R. Dittrich, T. Schrefl, D. Suess, W. Scholz, H. Forster, J. Fidler, A path method for finding energy barriers and minimum energy paths in complex micromagnetic systems, J. Magn. Magn. Mater. 250, 12 (2002) CrossRefADSGoogle Scholar
  144. 144.
    μMAG standard problem #4. https://www.ctcms.nist.gov/~rdm/std4/spec4.html (accessed 2019/02/04)
  145. 145.
    μMAG standard problem #5. https://www.ctcms.nist.gov/~rdm/std5/spec5.xhtml (accessed 2019/02/04)
  146. 146.
    M. Najafi, B. Krüger, S. Bohlens, M. Franchin, H. Fangohr, A. Vanhaverbeke, R. Allenspach, M. Bolte, U. Merkt, D. Pfannkuche et al., Proposal for a standard problem for micromagnetic simulations including spin-transfer torque, J. Appl. Phys. 105, 113914 (2009) CrossRefADSGoogle Scholar
  147. 147.
    A. Shpiro, P.M. Levy, S. Zhang, Self-consistent treatment of nonequilibrium spin torques in magnetic multilayers, Phys. Rev. B 67, 104430 (2003) CrossRefADSGoogle Scholar
  148. 148.
    X. Zhu, J.-G. Zhu, Bias-field-free microwave oscillator driven by perpendicularly polarized spin current, IEEE Trans. Magn. 42, 2670 (2006) CrossRefADSGoogle Scholar
  149. 149.
    V. Pribiag, I. Krivorotov, G. Fuchs, P. Braganca, O. Ozatay, J. Sankey, D. Ralph, R. Buhrman, Magnetic vortex oscillator driven by dc spin-polarized current, Nat. Phys. 3, 498 (2007) CrossRefGoogle Scholar
  150. 150.
    I. Firastrau, D. Gusakova, D. Houssameddine, U. Ebels, M.-C. Cyrille, B. Delaet, B. Dieny, O. Redon, J.-C. Toussaint, L. Buda-Prejbeanu, Modeling of the perpendicular polarizer-planar free layer spin torque oscillator: micromagnetic simulations, Phys. Rev. B 78, 024437 (2008) CrossRefADSGoogle Scholar
  151. 151.
    L. Liu, C.-F. Pai, Y. Li, H. Tseng, D. Ralph, R. Buhrman, Spin-torque switching with the giant spin Hall effect of tantalum, Science 336, 555 (2012) CrossRefADSGoogle Scholar
  152. 152.
    M. Cubukcu, O. Boulle, M. Drouard, K. Garello, C. Onur Avci, I. Mihai Miron, J. Langer, B. Ocker, P. Gambardella, G. Gaudin, Spin-orbit torque magnetization switching of a three-terminal perpendicular magnetic tunnel junction, Appl. Phys. Lett. 104, 042406 (2014) CrossRefADSGoogle Scholar

Copyright information

© The Author(s) 2019. This article is published with open access at Springerlink.com 2019

Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Christian Doppler Laboratory for Advanced Magnetic Sensing and Materials, Faculty of Physics, University of ViennaViennaAustria

Personalised recommendations