• Sergej O. DemokritovEmail author
Part of the Springer Series in Solid-State Sciences book series (SSSOL, volume 192)


Spin waves and their quanta magnons are the dynamic eigen-excitations of a magnetic system. They provide the basis for the description of spatial and temporal evolution of the magnetization distribution of a magnetic object. The unique features of spin waves such as the possibility to carry spin information over relatively long distances, the possibility to achieve sub-micrometer wavelength at microwave frequencies, and controllability by electronic signal via magnetic fields make these waves uniquely suited for implementation of novel integrated electronic devices characterized by high speed, low power consumption, and extended functionalities. It is important to notice that contrary to photons and phonons magnons possess an anisotropic dispersion. The energy/frequency of a magnons depends not only on the absolute value of the wave vector of the magnon, but also on its angle relative to the orientation of the static magnetization.


  1. 1.
    F. Bloch, Zur Theorie des Ferromagnetismus. Z. Phys. 61, 206–219 (1930)ADSzbMATHCrossRefGoogle Scholar
  2. 2.
    T. Holstein, H. Primakoff, Field dependence of the intrinsic domain magnetization of a ferromagnet. Phys. Rev. 58, 1098 (1940)ADSzbMATHCrossRefGoogle Scholar
  3. 3.
    F.J. Dyson, General theory of spin-wave interactions. Phys. Rev. 102, 1217 (1956)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    S. Neusser, D. Grundler, Magnonics: spin waves on the nanoscale. Adv. Mater. 21, 2927–2932 (2009)CrossRefGoogle Scholar
  5. 5.
    V.V. Kruglyak, S.O. Demokritov, D. Grundler, Magnonics. J. Phys. D: Appl. Phys. 43, 264001 (2010)ADSCrossRefGoogle Scholar
  6. 6.
    B. Lenk, H. Ulrichs, F. Garbs, M. Mnzenberg, The building blocks of magnonics. Phys. Rep. 507, 107 (2011)ADSCrossRefGoogle Scholar
  7. 7.
    A.V. Chumak, V.I. Vasyuchka, A.A. Serga, B. Hillebrands, Nat. Phys. 11, 453–461 (2015)CrossRefGoogle Scholar
  8. 8.
    J.C. Slonczewski, Current-driven excitation of magnetic multilayers. J. Magn. Magn. Mater. 159, L1–L7 (1996)ADSCrossRefGoogle Scholar
  9. 9.
    L. Berger, Emission of spin waves by a magnetic multilayer traversed by a current. Phys. Rev. B 54, 9353–9358 (1996)ADSCrossRefGoogle Scholar
  10. 10.
    J.C. Slonczewski, Excitation of spin waves by an electric current. J. Magn. Magn. Mater. 195, L261–L268 (1999)ADSCrossRefGoogle Scholar
  11. 11.
    M.I. Dyakonov, V.I. Perel, Possibility of orienting electron spins with current. Sov. Phys. JETP Lett. 13, 467–469 (1971)ADSGoogle Scholar
  12. 12.
    J.E. Hirsch, Spin Hall effect. Phys. Rev. Lett. 83, 1834–1837 (1999)ADSCrossRefGoogle Scholar
  13. 13.
    A. Hoffmann, Spin Hall effects in metals. IEEE Trans. Magn. 49, 5172–5193 (2013)ADSCrossRefGoogle Scholar
  14. 14.
    Y. Kajiwara et al., Transmission of electrical signals by spin-wave interconversion in a magnetic insulator. Nature 464, 262–266 (2010)ADSCrossRefGoogle Scholar
  15. 15.
    Z. Wang, Y. Sun, M. Wu, V. Tiberkevich, A. Slavin, Control of spin waves in a thin film ferromagnetic insulator through interfacial spin scattering. Phys. Rev. Lett. 107, 146602 (2011)ADSCrossRefGoogle Scholar
  16. 16.
    V.E. Demidov, S. Urazhdin, A.B. Rinkevich, G. Reiss, S.O. Demokritov, Spin Hall controlled magnonic microwaveguides. Appl. Phys. Lett. 104, 152402 (2014)ADSCrossRefGoogle Scholar
  17. 17.
    K. An et al., Control of propagating spin waves via spin transfer torque in a metallic bilayer waveguide. Phys. Rev. B 89, 140405(R) (2014)ADSCrossRefGoogle Scholar
  18. 18.
    S. Urazhdin, V.E. Demidov, H. Ulrichs, T. Kendziorczyk, T. Kuhn, J. Leuthold, G. Wilde, S.O. Demokritov, Nanomagnonic devices based on the spin-transfer torque. Nat. Nanotech. 9, 509–513 (2014)ADSCrossRefGoogle Scholar
  19. 19.
    C. Kittel, Excitation of spin waves in a ferromagnet by uniform RF field. Phys. Rev. 110, 1295 (1958)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    M.H. Seavey Jr., E. Tannenwald, Direct observation of spin wave resonance. Phys. Rev. Lett. 1, 168 (1958)ADSCrossRefGoogle Scholar
  21. 21.
    R.W. Damon, J.R. Eshbach, Magnetostatic modes of a ferromagnet slab. J. Phys. Chem. Solids 19, 308–320 (1961)ADSCrossRefGoogle Scholar
  22. 22.
    P. Grunberg, F. Metawe, Light scattering from bulk and surface spin waves in EuO. Phys. Rev. Lett. 39, 1561 (1977)ADSCrossRefGoogle Scholar
  23. 23.
    P. Grunberg, R. Schreiber, Y. Pang, M.B. Brodsky, H. Sowers, Layered magnetic structures: evidence for antiferromagnetic coupling of Fe layers across Cr interlayers. Phys. Rev. Lett. 57, 2442 (1986)ADSCrossRefGoogle Scholar
  24. 24.
    C. Mathieu, J. Jorzick, A. Frank, S.O. Demokritov, A.N. Slavin, B. Hillebrands, B. Bartenlian, C. Chappert, D. Decanini, F. Rousseaux, E. Cambrill, Lateral quantization of spin waves in micron size magnetic wires. Phys. Rev. Lett. 81, 3968 (1998)ADSCrossRefGoogle Scholar
  25. 25.
    J. Jorzick, S.O. Demokritov, B. Hillebrands, D. Berkov, N.L. Gorn, K. Guslienko, A.N. Slavin, Spin wave wells in nonellipsoidal micrometer size magnetic elements. Phys. Rev. Lett. 88, 047204 (2002)ADSCrossRefGoogle Scholar
  26. 26.
    J. Park, D.M. Eames, J. Engebretson, A. Berezovsky, P. Crowell, Phys. Rev. Lett. 89, 277201 (2002)ADSCrossRefGoogle Scholar
  27. 27.
    M. Tsoi et al., Excitation of a magnetic multilayer by an electric current. Phys. Rev. Lett. 80, 4281–4284 (1998)ADSCrossRefGoogle Scholar
  28. 28.
    M. Tsoi, A.G.M. Jansen, J. Bass, W.C. Chiang, V. Tsoi, P. Wyder, Generation and detection of phase-coherent current-driven magnons in magnetic multilayers. Nature 406, 46 (2000)ADSCrossRefGoogle Scholar
  29. 29.
    S.I. Kiselev et al., Microwave oscillations of a nanomagnet driven by a spin-polarized current. Nature 425, 380–383 (2003)ADSCrossRefGoogle Scholar
  30. 30.
    W.H. Rippard, M.R. Pufall, S. Kaka, S.E. Russek, T.J. Silva, Direct-current induced dynamics in Co90Fe10/Ni80Fe20 point contacts. Phys. Rev. Lett. 92, 027201 (2004)ADSCrossRefGoogle Scholar
  31. 31.
    I.N. Krivorotov, N.C. Emley, J.C. Sankey, S.I. Kiselev, D.C. Ralph, R.A. Buhrman, Time-domain measurements of nanomagnet dynamics driven by spin-transfer torques. Science 307, 228 (2005)ADSCrossRefGoogle Scholar
  32. 32.
    V.E. Demidov, S. Urazhdin, S.O. Demokritov, Direct observation and mapping of spin waves emitted by spin-torque nano-oscillators. Nat. Mater. 9, 984–988 (2010)ADSCrossRefGoogle Scholar
  33. 33.
    V.E. Demidov et al., Magnetic nano-oscillator driven by pure spin current. Nat. Mater. 11, 1028–1031 (2012)ADSCrossRefGoogle Scholar
  34. 34.
    S.O. Demokritov, V.E. Demidov, O. Dzyapko, G.A. Melkov, A.A. Serga, B. Hillebrands, A.N. Slavin, Bose-Einstein condensation of quasi-equilibrium magnons at room temperature under pumping. Nature 443, 430–433 (2006)ADSCrossRefGoogle Scholar
  35. 35.
    L.D. Landau, E.M. Lifshitz, Theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys. Z. Sowjet. 8, 153 (1935)zbMATHGoogle Scholar
  36. 36.
    C. Herring, C. Kittel, On the theory of spin waves in ferromagnetic media. Phys. Rev. 81, 869 (1951)ADSzbMATHCrossRefGoogle Scholar
  37. 37.
    B.A. Kalinikos, A.N. Slavin, Theory of dipole-exchange spin wave spectrum for ferromagnetic films with mixed exchange boundary conditions. J. Phys. C 19, 7013 (1986)ADSCrossRefGoogle Scholar
  38. 38.
    B. Heinrich, R. Urban, G. Woltersdorf, J. Appl. Phys. 91, 7523 (2002)ADSCrossRefGoogle Scholar
  39. 39.
    W.K. Hiebert, A. Stankiewicz, M.R. Freeman, Phys. Rev. Lett. 79, 1134 (1997)ADSCrossRefGoogle Scholar
  40. 40.
    A. Barman, V.V. Kruglyak, R.J. Hicken, J.M. Rowe, A. Kundrotaite, J. Scott, M. Rahman, Imaging the dephasing of spin wave modes in a square thin film magnetic element. Phys. Rev. B 69, 174426 (2004)ADSCrossRefGoogle Scholar
  41. 41.
    Y. Acremann, C.H. Back, M. Buess, O. Portmann, A. Vaterlaus, D. Pescia, H. Melchior, Imaging precessional motion of the magnetization vector. Science 290, 492 (2000)ADSCrossRefGoogle Scholar
  42. 42.
    S.O. Demokritov, B. Hillebrands, A.N. Slavin, Brillouin light scattering studies of confined spin waves: linear and nonlinear confinement. Phys. Rep. 348, 441 (2001)ADSCrossRefGoogle Scholar
  43. 43.
    S.O. Demokritov, V.E. Demidov, Micro-Brillouin light scattering spectroscopy of magnetic nanostructures. IEEE Trans. Magn. Adv. Magn. 44, 6 (2008)ADSCrossRefGoogle Scholar
  44. 44.
    V.E. Demidov, S.O. Demokritov, Magnonic waveguides studied by microfocus Brillouin light scattering. IEEE Trans. Magn. 51, 8578 (2015)CrossRefGoogle Scholar
  45. 45.
    B.T.M. Willis, C.J. Carlile, Experimental Neutron Scattering (Oxford University Press, Oxford, 2009)Google Scholar
  46. 46.
    J. Jersch, V.E. Demidov, H. Fuchs, K. Rott, P. Krzysteczko, J. Munchenberger, G. Reiss, S.O. Demokritov, Mapping of localized spin-wave excitations by near-field Brillouin light scattering. Appl. Phys. Lett. 97, 152502 (2010)ADSCrossRefGoogle Scholar
  47. 47.
    A.A. Serga, T. Schneider, B. Hillebrands, S.O. Demokritov, M. Kostylev, Phase-sensitive Brillouin light scattering spectroscopy from spin-wave packets. Appl. Phys. Lett. 89, 063506 (2006)ADSCrossRefGoogle Scholar
  48. 48.
    K. Vogt, H. Schultheiss, S.J. Hermsdoerfer, P. Pirro, A.A. Serga, B. Hillebrands, All-optical detection of phase fronts of propagating spin waves in a Ni81Fe19 microstripe. Appl. Phys. Lett. 95, 182508 (2009)ADSCrossRefGoogle Scholar
  49. 49.
    V.E. Demidov, S. Urazhdin, S.O. Demokritov, Control of spin-wave phase and wavelength by electric current on the microscopic scale; Appl. Phys. Lett. 95, 262509 (2009)Google Scholar
  50. 50.
    J. Jorzick, S.O. Demokritov, C. Mathieu, B. Hillebrands, B. Bartenlian, C. Chappert, F. Rousseaux, A. Slavin, Brillouin light scattering from quantized spin waves in micron-size magnetic wires. Phys. Rev. B 60, 15194 (1999)ADSCrossRefGoogle Scholar
  51. 51.
    G. Gubbiotti, O. Kazakova, G. Carlotti, M. Hanson, P. Vavassori, Spin-wave spectra in nanometric elliptical dots arrays. IEEE Trans. Magn. 39, 2750 (2003)ADSCrossRefGoogle Scholar
  52. 52.
    J. Jorzick, S.O. Demokritov, B. Hillebrands, B. Bartenlian, C. Chappert, D. Decanini, F. Rousseaux, E. Cambril, Spin wave quantization and dynamic coupling in micron-size circular magnetic dots. Appl. Phys. Lett. 75, 3859 (1999)ADSCrossRefGoogle Scholar
  53. 53.
    R.I. Joseph, E. Schlomann, Demagnetizing field in nonellipsoidal bodies. J. Appl. Phys. 36, 1579–1593 (1965)ADSCrossRefGoogle Scholar
  54. 54.
    P. Bryant, H. Suhl, Thin‐film magnetic patterns in an external field. Appl. Phys. Lett. 54, 2224 (1989)ADSCrossRefGoogle Scholar
  55. 55.
    C. Bayer, S.O. Demokritov, B. Hillebrands, A.N. Slavin, Spin wave wells with multiple states created in small magnetic elements. Appl. Phys. Lett. 82, 607 (2003)ADSCrossRefGoogle Scholar
  56. 56.
    V.E. Demidov, S.O. Demokritov, K. Rott, J. Krzysteczko, G. Reiss, Self-focusing of spin waves in Permalloy microstripes. Appl. Phys. Lett. 91, 252504 (2007)ADSCrossRefGoogle Scholar
  57. 57.
    V.E. Demidov, J. Jersch, S.O. Demokritov, K. Rott, J. Krzysteczko, G. Reiss, Transformation of propagating spin-wave modes in microscopic waveguides with variable width. Phys. Rev. B 79, 054417 (2009)ADSCrossRefGoogle Scholar
  58. 58.
    V.E. Demidov, M. Kostylev, K. Rott, J. Mnchenberger, G. Reiss, S.O. Demokritov, Excitation of short-wavelength spin waves in magnonic waveguides. Appl. Phys. Lett. 99, 082507 (2011)ADSCrossRefGoogle Scholar
  59. 59.
    E. Myers, D.C. Ralph, J.A. Katine, R.N. Louie, R.A. Buhrman, Current-induced switching of domains in magnetic multilayer devices. Science 285, 867–870 (1999)CrossRefGoogle Scholar
  60. 60.
    J.A. Katine, F.J. Albert, R.A. Buhrman, E.B. Myers, D.C. Ralph, Current-driven magnetization reversal and spin-wave excitations in Co/Cu/Co pillars. Phys. Rev. Lett. 84, 3149–3152 (2000)ADSCrossRefGoogle Scholar
  61. 61.
    D.C. Ralph, M.D. Stiles, Spin transfer torques. J. Magn. Magn. Mater. 320, 1190–1216 (2008)ADSCrossRefGoogle Scholar
  62. 62.
    T.J. Silva, W.H. Rippard, Developments in nano-oscillators based upon spin-transfer point-contact devices. J. Magn. Magn. Mater. 320, 1260–1271 (2008)ADSCrossRefGoogle Scholar
  63. 63.
    A. Brataas, A.D. Kent, H. Ohno, Current-induced torques in magnetic materials. Nat. Mater. 11, 372 (2012)ADSCrossRefGoogle Scholar
  64. 64.
    N. Locatelli, V. Cros, J. Grollier, Spin-torque building blocks. Nat. Mater. 13, 11 (2014)ADSCrossRefGoogle Scholar
  65. 65.
    T. Chen, R.K. Dumas, A. Eklund, K. Muduli, A. Houshang, A.A. Awad, P. Duerrenfeld, B.G. Malm, A. Rusu, J. Akerman, Spin-torque and spin-Hall nano-oscillators. IEEE Trans. Magn. 99, 1 (2016)Google Scholar
  66. 66.
    M. Madami et al., Direct observation of a propagating spin wave induced by spin-transfer torque. Nat. Nanotech. 6, 635–638 (2011)ADSCrossRefGoogle Scholar
  67. 67.
    A. Houshang, E. Iacocca, P. Duerrenfeld, S.R. Sani, J. Akerman, R.K. Dumas, Spin-wave-beam driven synchronization of nanocontact spin-torque oscillators. Nat. Nanotech. 11, 280–286 (2016)ADSCrossRefGoogle Scholar
  68. 68.
    V.E. Demidov, S. Urazhdin, R. Liu, B. Divinskiy, A. Telegin, S.O. Demokritov, Excitation of coherent propagating spin waves by pure spin currents. Nat. Commun. 7, 10446 (2016)ADSCrossRefGoogle Scholar
  69. 69.
    J. Xiao, G.E.W. Bauer, Spin-wave excitation in magnetic insulators by spin-transfer torque. Phys. Rev. Lett. 108, 217204 (2012)ADSCrossRefGoogle Scholar
  70. 70.
    T. Jungwirth, J. Wunderlich, K. Olejnik, Spin Hall effect devices. Nat. Mater. 11, 382 (2012)ADSCrossRefGoogle Scholar
  71. 71.
    F.J. Jedema, A.T. Filip, B.J. van Wees, Electrical spin injection and accumulation at room temperature in an all-metal mesoscopic spin valve. Nature 410, 345–348 (2001)ADSCrossRefGoogle Scholar
  72. 72.
    Y. Otani, T. Kimura, Manipulation of spin currents in metallic systems. Philos. Trans. R. Soc. Lond. Ser. A 369, 3136–3149 (2011)ADSCrossRefGoogle Scholar
  73. 73.
    K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, E. Saitoh, Observation of the spin Seebeck effect. Nature 455, 778–781 (2008)ADSCrossRefGoogle Scholar
  74. 74.
    K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, G.E.W. Bauer, S. Maekawa, E. Saitoh, Spin Seebeck insulator. Nat. Mater. 9, 894–897 (2010)ADSCrossRefGoogle Scholar
  75. 75.
    V.M. Edelstein, Spin polarization of conduction electrons induced by electric current in two-dimensional asymmetric electron systems. Solid State Commun. 73, 233–235 (1990)ADSCrossRefGoogle Scholar
  76. 76.
    A. Chernyshov, M. Overby, X. Liu, J.K. Furdyna, Y. Lyanda-Geller, L.P. Rokhinson, Evidence for reversible control of magnetization in a ferromagnetic material by means of spin-orbit magnetic field. Nat. Phys. 5, 656–659 (2009)CrossRefGoogle Scholar
  77. 77.
    K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda, S. Maekawa, E. Saitoh, Electric manipulation of spin relaxation using the spin Hall effect. Phys. Rev. Lett. 101, 036601 (2008)ADSCrossRefGoogle Scholar
  78. 78.
    M. Evelt et al., High-efficiency control of spin-wave propagation in ultra-thin Yttrium Iron Garnet by the spin-orbit torque. Appl. Phys. Lett. 108, 172406 (2016)ADSCrossRefGoogle Scholar
  79. 79.
    V.E. Demidov, S. Urazhdin, V. Tiberkevich, A. Slavin, S.O. Demokritov, Control of spin-wave emission from spin-torque nano-oscillators by microwave pumping. Phys. Rev. B 83, 060406 (R) (2011)Google Scholar
  80. 80.
    V.E. Demidov et al., Control of magnetic fluctuations by spin current. Phys. Rev. Lett. 107, 107204 (2011)ADSCrossRefGoogle Scholar
  81. 81.
    A. Slavin, V. Tiberkevich, Spin wave mode excited by spin-polarized current in a magnetic nanocontact is a standing self-localized wave bullet. Phys. Rev. Lett. 95, 237201 (2005)ADSCrossRefGoogle Scholar
  82. 82.
    A. Einstein, Quantentheorie des einatomigen idealen Gases. Sitz. Ber. Preuss. Akad. Wiss. 1, 3–8 (1925)zbMATHGoogle Scholar
  83. 83.
    L.P. Pitaevskii, S. Stringari, Bose-Einstein Condensation. International Series of Monographs on Physics 116 (Clarendon Press, 2003)Google Scholar
  84. 84.
    P. Kapitza, Viscosity of liquid helium below the λ-point. Nature 141, 74 (1938)ADSCrossRefGoogle Scholar
  85. 85.
    J.F. Allen, A.D. Misener, Flow of liquid helium II. Nature 141, 75 (1938)ADSCrossRefGoogle Scholar
  86. 86.
    M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, E.A. Cornell, Observation of Bose-Einstein condensation in a Dilute Atomic Vapor. Science 269, 198–201 (1995)ADSCrossRefGoogle Scholar
  87. 87.
    K.B. Davis et al., Bose-Einstein condensation in a gas of sodium atoms. Phys. Rev. Lett. 75, 3969–3973 (1995)ADSCrossRefGoogle Scholar
  88. 88.
    T. Giamarchi, Ch. Regg, O. Tchernyschyov, Bose-Einstein condensation in magnetic insulators. Nat. Phys. 4, 198–204 (2008)CrossRefGoogle Scholar
  89. 89.
    L.V. Butov, A.L. Ivanov, A. Imamoglu, P.B. Littlewood, A.A. Shashkin, V.T. Dolgopolov, K.L. Campman, A.C. Gossard, Stimulated scattering of indirect excitons in coupled quantum wells: signature of a degenerate Bose-gas of excitons. Phys. Rev. Lett. 86, 5608–5611 (2001)ADSCrossRefGoogle Scholar
  90. 90.
    J. Kasprzak et al., Bose-Einstein condensation of exciton polaritons. Nature 443, 409–414 (2006)ADSCrossRefGoogle Scholar
  91. 91.
    R. Balili, V. Hartwell, D. Snoke, L. Pfeiffer, K. West, Bose-Einstein condensation of microcavity polaritons in a trap. Science 316, 1007–1010 (2007)ADSCrossRefGoogle Scholar
  92. 92.
    A. Amo et al., Collective fluid dynamics of a polariton condensate in a semiconductor microcavity. Nature 457, 291–295 (2009)ADSCrossRefGoogle Scholar
  93. 93.
    YuM Bunkov, G.E. Volovik, Bose-Einstein condensation of magnons in superfluid 3He. J. Low Temp. Phys. 150, 135–144 (2008)ADSCrossRefGoogle Scholar
  94. 94.
    J. Klaers, J. Schmitt, F. Vewinger, M. Weitz, Bose-Einstein condensation of photons in an optical microcavity. Nature 468, 545–548 (2010)ADSCrossRefGoogle Scholar
  95. 95.
    V.E. Demidov, O. Dzyapko, S.O. Demokritov, G.A. Melkov, A.N. Slavin, Thermalization of a parametrically driven magnon gas leading to Bose-Einstein condensation. Phys. Rev. Lett. 99, 037205 (2007)ADSCrossRefGoogle Scholar
  96. 96.
    O. Dzyapko, P. Nowik-Boltyk, B. Koene, V.E. Demidov, J. Jersch, A. Kirilyuk, T. Rasing, S.O. Demokritov, High-resolution magneto-optical Kerr effect spectroscopy of magnon Bose-Einstein condensate. IEEE Magn. Lett. 7, 3501805 (2016)CrossRefGoogle Scholar
  97. 97.
    P. Nowik-Boltyk, O. Dzyapko, V.E. Demidov, N.G. Berloff, S.O. Demokritov, Spatially non-uniform ground state and quantized vortices in a two-component Bose-Einstein condensate of magnons. Sci. Rep. 2, 482 (2012)ADSCrossRefGoogle Scholar
  98. 98.
    V.E. Demidov, O. Dzyapko, M. Buchmeier, T. Stockhoff, G. Schmitz, G.A. Melkov, S.O. Demokritov, Magnon kinetics and Bose-Einstein condensation studied in phase space. Phys. Rev. Lett. 101, 257201 (2008)ADSCrossRefGoogle Scholar
  99. 99.
    M.R. Matthews, B.P. Anderson, P.C. Haljan, D.S. Hall, C.E. Wieman, E.A. Cornell, Vortices in a Bose-Einstein condensate. Phys. Rev. Lett. 83, 2498 (1999)ADSCrossRefGoogle Scholar
  100. 100.
    I.A. Dzyaloshinsky, Thermodynamic theory of ‘weak’ ferromagnetism of antiferromagnetics. J. Phys. Chem. Solids 4, 241–255 (1958)ADSCrossRefGoogle Scholar
  101. 101.
    T. Moriya, New mechanism of anisotropic superexchange interaction. Phys. Rev. Lett. 4, 228–230 (1960)ADSCrossRefGoogle Scholar
  102. 102.
    M. Bode, M. Heide, K. von Bergmann, P. Ferriani, S. Heinze, G. Bihlmayer, A. Kubetzka, O. Pietzsch, S. Blügel, R. Wiesendanger, Chiral magnetic order at surfaces driven by inversion asymmetry. Nature 447, 190–193 (2007)ADSCrossRefGoogle Scholar
  103. 103.
    S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, P. Böni, Skyrmion lattice in a chiral magnet. Science 323, 915–919 (2009)ADSCrossRefGoogle Scholar
  104. 104.
    G. Chen, T. Ma, A.T. N’Diaye, H. Kwon, C. Won, Y. Wu, A.K. Schmid, Tailoring the chirality of magnetic domain walls by interface engineering. Nat. Commun. 4, 2671 (2013)ADSCrossRefGoogle Scholar
  105. 105.
    L. Udvardi, L. Szunyogh, Chiral asymmetry of the spin-wave spectra in ultrathin magnetic films. Phys. Rev. Lett. 102, 207204 (2009)ADSCrossRefGoogle Scholar
  106. 106.
    Kh Zakeri, Y. Zhang, J. Prokop, T.-H. Chuang, N. Sakr, W.X. Tang, J. Kirschner, Asymmetric spin-wave dispersion on Fe(110): direct evidence of the Dzyaloshinskii-Moriya interaction. Phys. Rev. Lett. 104, 137203 (2010)ADSCrossRefGoogle Scholar
  107. 107.
    Y. Onose, T. Ideue, H. Katsura, Y. Shiomi, N. Nagaosa, Y. Tokura, Observation of the magnon Hall effect. Science 329, 297–299 (2010)ADSCrossRefGoogle Scholar
  108. 108.
    R. Matsumoto, S. Murakami, Theoretical prediction of a rotating magnon wave packet in ferromagnets. Phys. Rev. Lett. 106, 197202 (2011)ADSCrossRefGoogle Scholar
  109. 109.
    L. Zhang, J. Ren, J.-S. Wang, B. Li, Topological magnon insulator in insulating ferromagnet. Phys. Rev. B 87, 144101 (2013)ADSCrossRefGoogle Scholar
  110. 110.
    J.D. Joannopoulos, S.G. Johnson, J.N. Winn, R.D. Meade, Photonic Crystals—Molding the Flow of Light (Princeton University Press, Princeton, 2008)zbMATHGoogle Scholar
  111. 111.
    B. Lenk, H. Ulrichs, F. Garbs, M. Münzenberg, The building blocks of magnonics. Phys. Rep. 507, 107–136 (2011)ADSCrossRefGoogle Scholar
  112. 112.
    G. Gubbiotti, S. Tacchi, M. Madami, G. Carlotti, A.O. Adeyeye, M. Kostylev, Brillouin light scattering studies of planar metallic magnonic crystals. J. Phys. D Appl. Phys. 43, 264003 (2010)ADSCrossRefGoogle Scholar
  113. 113.
    S.A. Nikitov, D.V. Kalyabin, I.V. Lisenkov, A. Slavin, YuN Barabanenkov, S.A. Osokin, A.V. Sadovnikov, E.N. Beginin, M.A. Morozova, YuA Filimonov, YuV Khivintsev, S.L. Vysotsky, V.K. Sakharov, E.S. Pavlov, Magnonics: a new research area in spintronics and spin wave electronics. Phys.-Usp. 58, 1002 (2015)ADSCrossRefGoogle Scholar
  114. 114.
    S. Tacchi, G. Gubbiotti, M. Madami, G. Carlotti, Brillouin light scattering studies of 2D magnonic crystals. J. Phys.: Condens. Matter 29, 073001 (2017)ADSGoogle Scholar
  115. 115.
    M. Krawczyk, D. Grundler, Review and prospects of magnonic crystals and devices with reprogrammable band structure. J. Phys.: Condens. Matter 26, 123202 (2014)Google Scholar
  116. 116.
    A.V. Chumak, T. Neumann, A.A. Serga, B. Hillebrands, M.P. Kostylev, Magnonic crystals for data processing. J. Phys. D: Appl. 42, 205005 (2009)ADSCrossRefGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute for Applied Physics and Center for Nanotechnology, University of MünsterMünsterGermany

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