Skip to main content
Log in

Nonreciprocal coherent coupling of nanomagnets by exchange spin waves

  • Research Article
  • Published:
Nano Research Aims and scope Submit manuscript

Abstract

Nanomagnets are widely used to store information in non-volatile spintronic devices. Spin waves can transfer information with low-power consumption as their propagations are independent of charge transport. However, to dynamically couple two distant nanomagnets via spin waves remains a major challenge for magnonics. Here we experimentally demonstrate coherent coupling of two distant Co nanowires by fast propagating spin waves in an yttrium iron garnet thin film with sub-50 nm wavelengths. Magnons in two nanomagnets are unidirectionally phase-locked with phase shifts controlled by magnon spin torque and spin-wave propagation. The coupled system is finally formulated by an analytical theory in terms of an effective non-Hermitian Hamiltonian. Our results are attractive for analog neuromorphic computing that requires unidirectional information transmission.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Wolf, S. A.; Awschalom, D. D.; Buhrman, R. A.; Daughton, J. M.; von Molnár, S.; Roukes, M. L.; Chtchelkanova, A. Y.; Treger, D. M. Spintronics: A spin-based electronics vision for the future. Science 2001, 294, 1488–1495.

    CAS  Google Scholar 

  2. Imre, A.; Csaba, G.; Ji, L.; Orlov, A.; Bernstein, G. H.; Porod, W. Majority logic gate for magnetic quantum-dot cellular automata. Science 2006, 311, 205–208.

    CAS  Google Scholar 

  3. Fert, A. Nobel lecture: Origin, development, and future of spintronics. Rev. Mod. Phys. 2008, 80, 1517–1530.

    CAS  Google Scholar 

  4. Cui, J. Z.; Huang, T. Y.; Luo, Z. C.; Testa, P.; Gu, H. R.; Chen, X. Z.; Nelson, B. J.; Heyderman, L. J. Nanomagnetic encoding of shape-morphing micromachines. Nature 2019, 578, 164–168.

    Google Scholar 

  5. Baibich, M. N.; Broto, J. M.; Fert, A.; Nguyen Van Dau, F.; Petroff, F.; Etienne, P.; Creuzet, G.; Friederich, A.; Chazelas J. Giant magnetoresistance of (001)Fe/(001)Cr magnetic superlattices. Phys. Rev. Lett. 1988, 61, 2472–2475.

    CAS  Google Scholar 

  6. Binasch, G.; Grünberg, P.; Saurenbach, F.; Zinn, W. Enhanced magnetoresistance in layered magnetic structures with antiferromagnetic interlayer exchange. Phys. Rev. B 1989, 39, 4828–4830.

    CAS  Google Scholar 

  7. Parkin, S. S. P.; More, N.; Roche, K. P. Oscillations in exchange coupling and magnetoresistance in metallic superlattice structures: Co/Ru, Co/Cr, and Fe/Cr. Phys. Rev. Lett. 1990, 64, 2304–2307.

    CAS  Google Scholar 

  8. Moodera, J. S.; Kinder, L. R.; Wong, T. M.; Meservey, R. Large magnetoresistance at room temperature in ferromagnetic thin film tunnel junctions. Phys. Rev. Lett. 1995, 74, 3273–3276.

    CAS  Google Scholar 

  9. Wang, W. G.; Li, M. G.; Hageman, S.; Chien, C. L. Electric-field-assisted switching in magnetic tunnel junctions. Nat. Mater. 2012, 11, 64–68.

    CAS  Google Scholar 

  10. Borders, W. A.; Pervaiz, A. Z.; Fukami, S.; Camsari, K. Y.; Ohno, H.; Datta, S. Integer factorization using stochastic magnetic tunnel junctions. Nature 2019, 573, 390–393.

    CAS  Google Scholar 

  11. Luo, Z. C.; Dao, T. P.; Hrabec, A.; Vijayakumar, J.; Kleibert, A.; Baumgartner, M.; Kirk, E.; Cui, J. Z.; Savchenko, T.; Krishnaswamy, G. et al. Chirally coupled nanomagnets. Science 2019, 363, 1435–1439.

    CAS  Google Scholar 

  12. Tserkovnyak, Y.; Brataas, A.; Bauer, G. E. W.; Halperin, B. I. Nonlocal magnetization dynamics in ferromagnetic heterostructures. Rev. Mod. Phys. 2005, 77, 1375–1421.

    CAS  Google Scholar 

  13. Heinrich, B.; Tserkovnyak, Y.; Woltersdorf, G; Brataas A.; Urban, R.; Bauer G. E. W. Dynamic exchange coupling in magnetic bilayers. Phys. Rev. Lett. 2003, 90, 187601.

    Google Scholar 

  14. Pigeau, B.; Hahn, C.; De Loubens, G.; Naletov, V. V.; Klein, O.; Mitsuzuka, K.; Lacour, D.; Hehn, M.; Andrieu, S.; Montaigne, F. Measurement of the dynamical dipolar coupling in a pair of magnetic nanodisks using a ferromagnetic resonance force microscope. Phys. Rev. Lett. 2012, 109, 247602.

    CAS  Google Scholar 

  15. Yu, T.; Bauer, G. E. W. Chiral coupling to magnetodipolar radiation. arXiv: 2001.06821, 2020.

  16. Slonczewski, J. C. Current-driven excitation of magnetic multilayers. J. Magn. Magn. Mater. 1996, 159, L1–L7.

    CAS  Google Scholar 

  17. Berger, L. Emission of spin waves by a magnetic multilayer traversed by a current. Phys. Rev. B 1996, 54, 9353–9358.

    CAS  Google Scholar 

  18. Kaka, S.; Pufall, M. R.; Rippard, W. H.; Silva, T. J.; Russek, S. E.; Katine, J. A. Mutual phase-locking of microwave spin torque nano-oscillators. Nature 2005, 437, 389–392.

    CAS  Google Scholar 

  19. Madami, M.; Bonetti, S.; Consolo, G.; Tacchi, S.; Carlotti, G.; Gubbiotti, G.; Mancoff, F. B.; Yar, M. A.; Akerman, J. Direct observation of a propagating spin wave induced by spin-transfer torque. Nat. Nanotechnol. 2011, 6, 635–638.

    CAS  Google Scholar 

  20. Demidov, V. E.; Urazhdin, S.; Ulrichs, H.; Tiberkevich, V.; Slavin, A.; Baither, D.; Schmitz, G.; Demokritov, S. O. Magnetic nanooscillator driven by pure spin current. Nat. Mater. 2012, 11, 1028–1031.

    CAS  Google Scholar 

  21. Urazhdin, S.; Demidov, V. E.; Ulrichs, H.; Kendziorczyk, T.; Kuhn, T.; Leuthold, J.; Wilde, G.; Demokritov, S. O. Nanomagnonic devices based on the spin-transfer torque. Nat. Nanotechnol. 2014, 9, 509–513.

    CAS  Google Scholar 

  22. Kruglyak, V. V.; Demokritov, S. O.; Grundler, D. Magnonics. J. Phys. D Appl. Phys. 2010, 43, 264001.

    Google Scholar 

  23. Lenk, B.; Ulrichs, H.; Garbs, F.; Münzenberg, M. The building blocks of magnonics. Phys. Rep. 2011, 507, 107–136.

    Google Scholar 

  24. Chumak, A. V.; Vasyuchka, V. I.; Serga, A. A.; Hillebrands, B. Magnon spintronics. Nat. Phys. 2015, 11, 453–461.

    CAS  Google Scholar 

  25. Vlaminck, V.; Bailleul, M. Current-induced spin-wave Doppler shift. Science 2008, 322, 410–413.

    CAS  Google Scholar 

  26. Vogt, K.; Fradin, F. Y.; Pearson, J. E.; Sebastian, T.; Bader, S. D.; Hillebrands, B.; Hoffmann, A.; Schultheiss, H. Realization of a spin-wave multiplexer. Nat. Commun. 2014, 5, 3727.

    CAS  Google Scholar 

  27. Pal, S.; Saha, S.; Kamalakar, M. V.; Barman, A. Field-dependent spin waves in high-aspect-ratio single-crystal ferromagnetic nanowires. Nano Res. 2016, 9, 1426–1433.

    CAS  Google Scholar 

  28. Han, J. H.; Zhang, P. X.; Hou, J. T.; Siddiqui, S. A.; Liu, L. Q. Mutual control of coherent spin waves and magnetic domain walls in a magnonic device. Science 2019, 366, 1121–1125.

    CAS  Google Scholar 

  29. Wang, Y.; Zhu, D. P.; Yang, Y. M.; Lee, K.; Mishra, R.; Go, G.; Oh, S. H.; Kim, D. H.; Cai, K. M.; Liu, E. L. et al. Magnetization switching by magnon-mediated spin torque through an antiferromagnetic insulator. Science 2019, 366, 1121–1125.

    Google Scholar 

  30. Yu, W. C.; Lan, J.; Xiao, J. Magnetic logic gate based on polarized spin waves. Phys. Rev. Appl. 2020, 13, 024055.

    CAS  Google Scholar 

  31. Heinz, B.; Brächer, T.; Schneider, M.; Wang, Q.; Lägel, B.; Friedel, A. M.; Breitbach, D.; Steinert, S.; Meyer, T.; Kewenig, M. et al. Propagation of spin-wave packets in individual nanosized yttrium iron garnet magnonic conduits. Nano Lett. 2020, 20, 4220–4227.

    CAS  Google Scholar 

  32. Demokritov, S. O.; Demidov, V. E.; Dzyapko, O.; Melkov, G. A.; Serga, A. A.; Hillebrands, B.; Slavin, A. N. Bose-Einstein condensation of quasi-equilibrium magnons at room temperature under pumping. Nature 2006, 443, 430–433.

    CAS  Google Scholar 

  33. Schneider, M.; Brächer, T.; Breitbach, D.; Lauer, V.; Pirro, P.; Bozhko, D. A.; Musiienko-Shmarova, H. Y.; Heinz, B.; Wang, Q.; Meyer, T. et al. Bose-Einstein condensation of quasiparticles by rapid cooling. Nat. Nanotechnol. 2020, 15, 457–461.

    CAS  Google Scholar 

  34. Khitun, A.; Bao, M. Q.; Wang, K. L. Magnonic logic circuits. J. Phys. D Appl. Phys. 2010, 43, 264005.

    Google Scholar 

  35. Grundler, D. Spintronics: Nanomagnonics around the corner. Nat. Nanotechnol. 2016, 11, 407–408.

    CAS  Google Scholar 

  36. Wagner, K.; Kákay, A.; Schultheiss, K.; Henschke, A.; Sebastian, T.; Schultheiss, H. Magnetic domain walls as reconfigurable spin-wave nanochannels. Nat. Nanotechnol. 2016, 11, 432–436.

    CAS  Google Scholar 

  37. Haldar, A.; Kumar, D.; Adeyeye, A. O. A reconfigurable waveguide for energy-efficient transmission and local manipulation of information in a nanomagnetic device. Nat. Nanotechnol. 2016, 11, 437–443.

    CAS  Google Scholar 

  38. Csaba, G.; Porod, W. Coupled oscillators for computing: A review and perspective. Appl. Phys. Rev. 2020, 7, 011302.

    CAS  Google Scholar 

  39. Wu, H.; Huang, L.; Fang, C.; Yang, B. S.; Wan, C. H.; Yu, G. Q.; Feng, J. F.; Wei, H. X.; Han, X. F. Magnon valve effect between two magnetic insulators. Phys. Rev. Lett. 2018, 120, 097205.

    CAS  Google Scholar 

  40. Cramer, J.; Fuhrmann, F.; Ritzmann, U.; Gall, V.; Niizeki, T.; Ramos, R.; Qiu, Z. Y.; Hou, D. Z.; Kikkawa, T.; Sinova, J. et al. Magnon detection using a ferroic collinear multilayer spin valve. Nat. Commun. 2018, 9, 1089.

    Google Scholar 

  41. Cornelissen, L. J.; Liu, J.; Van Wees, B. J.; Duine, R. A. Spin-current-controlled modulation of the magnon spin conductance in a three-terminal magnon transistor. Phys. Rev. Lett. 2018, 120, 097702.

    CAS  Google Scholar 

  42. Khalili Amiri, P.; Rejaei, B.; Vroubel, M.; Zhuang, Y. Nonreciprocal spin wave spectroscopy of thin Ni-Fe stripes. Appl. Phys. Lett. 2007, 91, 062502.

    Google Scholar 

  43. Kwon, J. H.; Yoon, J.; Deorani, P.; Lee, J. M.; Sinha, J.; Lee, K. J.; Hayashi, M.; Yang, H. Giant nonreciprocal emission of spin waves in Ta/Py bilayers. Sci. Adv. 2016, 2, e1501892.

    Google Scholar 

  44. Bracher, T.; Boulle, O.; Gaudin, G.; Pirro, P. Creation of unidirectional spin-wave emitters by utilizing interfacial Dzyaloshinskii-Moriya interaction. Phys. Rev. B 2017, 95, 064429.

    Google Scholar 

  45. Henry, Y.; Stoeffle, D.; Kim, J. V.; Bailleul, M. Unidirectional spin-wave channeling along magnetic domain walls of Bloch type. Phys. Rev. B 2019, 100, 024416.

    CAS  Google Scholar 

  46. Sandweg, C. W.; Kajiwara, Y.; Chumak, A. V.; Serga, A. A.; Vasyuchka, V. I.; Jungfleisch, M. B.; Saitoh, E.; Hillebrands, B. Spin pumping by parametrically excited exchange magnons. Phys. Rev. Lett. 2011, 106, 216601.

    CAS  Google Scholar 

  47. Wintz, S.; Tiberkevich, V.; Weigand, M.; Raabe, J.; Lindner, J.; Erbe, A.; Slavin, A.; Fassbender, J. Magnetic vortex cores as tunable spin-wave emitters. Nat. Nanotechnol. 2016, 11, 948–953.

    CAS  Google Scholar 

  48. Hämäläinen, S. J.; Brandl, F.; Franke, K. J. A.; Grundler, D.; van Dijken, S. Tunable short-wavelength spin-wave emission and confinement in anisotropy-modulated multiferroic heterostructures. Phys. Rev. Appl. 2017, 8, 014020.

    Google Scholar 

  49. Brächer, T.; Fabre, M.; Meyer, T.; Fischer, T.; Auffret, S.; Boulle, O.; Ebels, U.; Pirro, P.; Gaudin, G. Detection of short-waved spin waves in individual microscopic spin-wave waveguides using the inverse spin Hall effect. Nano Lett. 2017, 17, 7234–7241.

    Google Scholar 

  50. Liu, C. P.; Chen, J. L.; Liu, T.; Heimbach, F.; Yu, H. M.; Xiao, Y.; Hu, J. F.; Liu, M. C.; Chang, H. C.; Stueckler, T. et al. Long-distance propagation of short-wavelength spin waves. Nat. Commun. 2018, 9, 738.

    Google Scholar 

  51. Dieterle, G.; Förster, J.; Stoll, H.; Semisalova, A. S.; Finizio, S.; Gangwar, A.; Weigand, M.; Noske, M.; Fähnle, M.; Bykova, I. et al. Coherent excitation of heterosymmetric spin waves with ultrashort wavelengths. Phys. Rev. Lett. 2019, 122, 117202.

    CAS  Google Scholar 

  52. Che, P.; Baumgaertl, K.; Kúkol’ová, A.; Dubs, C.; Grundler, D. Efficient wavelength conversion of exchange magnons below 100 nm by magnetic coplanar waveguides. Nat. Commun. 2020, 11, 1445.

    CAS  Google Scholar 

  53. Klingler, S.; Amin, V.; Geprägs, S.; Ganzhorn, K.; Maier-Flaig, H.; Althammer, M.; Huebl, H.; Gross, R.; McMichael, R. D.; Stiles, M. D. et al. Spin-torque excitation of perpendicular standing spin waves in coupled YIG/Co heterostructures. Phys. Rev. Lett. 2018, 120, 127201.

    CAS  Google Scholar 

  54. Chen, J. L.; Liu, C. P.; Liu, T.; Xiao, Y.; Xia, K.; Bauer, G. E. W.; Wu, M. Z.; Yu, H. M. Strong interlayer magnon-magnon coupling in magnetic metal-insulator hybrid nanostructures. Phys. Rev. Lett. 2018, 120, 217202.

    CAS  Google Scholar 

  55. Qin, H. J.; Hämäläinen, S. J.; van Dijken, S. Exchange-torque-induced excitation of perpendicular standing spin waves in nanometer-thick YIG films. Sci. Rep. 2018, 8, 5755.

    Google Scholar 

  56. Li, Y.; Cao, W.; Amin, V. P.; Zhang, Z. Z.; Gibbons, J.; Sklenar, J.; Pearson, J.; Haney, P. M.; Stiles, M. D.; Bailey, W. E. et al. Coherent spin pumping in a strongly coupled magnon-magnon hybrid system. Phys. Rev. Lett. 2020, 124, 117202.

    CAS  Google Scholar 

  57. Chen, J. L.; Yu, T.; Liu, C. P.; Liu, T.; Madami, M.; Shen, K.; Zhang, J. Y.; Tu, S.; Alam, S.; Xia, K. et al. Excitation of unidirectional exchange spin waves by a nanoscale magnetic grating. Phys. Rev. B 2019, 100, 104427.

    CAS  Google Scholar 

  58. Yu, T.; Blanter, Y. M.; Bauer, G. E. W. Chiral pumping of spin waves. Phys. Rev. Lett. 2019, 123, 247202.

    CAS  Google Scholar 

  59. Schneider, T.; Serga, A. A.; Neumann, T.; Hillebrands, B.; Kostylev, M. P. Phase reciprocity of spin-wave excitation by a microstrip antenna. Phys. Rev. B 2008, 77, 214411.

    Google Scholar 

  60. Demidov, V. E.; Kostylev, M. P.; Rott, K.; Krzysteczko, P.; Reiss, G.; Demokritov, S. O. Excitation of microwaveguide modes by a stripe antenna. Appl. Phys. Lett. 2009, 95, 112509.

    Google Scholar 

  61. Sekiguchi, K.; Yamada, K.; Seo, S. M.; Lee, K. J.; Chiba, D.; Kobayashi, K.; Ono, T. Nonreciprocal emission of spin-wave packet in FeNi film. Appl. Phys. Lett. 2010, 97, 022508.

    Google Scholar 

  62. El-Ganainy, R.; Makris, K. G.; Khajavikhan, M.; Musslimani, Z. H.; Rotter, S.; Christodoulides, D. N. Non-Hermitian physics and PT symmetry. Nat. Phys. 2018, 14, 11–19.

    CAS  Google Scholar 

  63. Miri, M. A.; Alù, A. Exceptional points in optics and photonics. Science 2019, 363, eaar7709.

    CAS  Google Scholar 

  64. Carlström, J.; Stålhammar, M.; Budich, J. C.; Bergholtz, E. J. Knotted non-Hermitian metals. Phys. Rev. B 2019, 99, 161115.

    Google Scholar 

  65. Grigoryan, V. L.; Shen, K.; Xia, K. Synchronized spin-photon coupling in a microwave cavity. Phys. Rev. B 2018, 98, 024406.

    CAS  Google Scholar 

  66. Harder, M.; Yang, Y.; Yao, B. M.; Yu, C. H.; Rao, J. W.; Gui, Y. S.; Stamps, R. L.; Hu, C. M. Level attraction due to dissipative magnonphoton coupling. Phys. Rev. Lett. 2018, 121, 137203.

    CAS  Google Scholar 

  67. Yu, W. C.; Wang, J. J.; Yuan, H. Y.; Xiao, J. Prediction of attractive level crossing via a dissipative mode. Phys. Rev. Lett. 2019, 113, 227201.

    Google Scholar 

  68. Chang, H. C.; Li, P.; Zhang, W.; Liu, T.; Hoffmann, A.; Deng, L. J.; Wu, M. Z. Nanometer-thick yttrium iron garnet films with extremely low damping. IEEEMagn. Lett. 2014, 5, 6700104.

    Google Scholar 

  69. Yu, H. M.; d’Allivy Kelly, O.; Cros, V.; Bernard, R.; Bortolotti, P.; Anane, A.; Brandl, F.; Huber, R.; Stasinopoulos, I.; Grundler, D. Magnetic thin-film insulator with ultra-low spin wave damping for coherent nanomagnonics. Sci. Rep. 2014, 4, 6848.

    CAS  Google Scholar 

  70. Ciubotaru, F.; Devolder, T.; Manfrini, M.; Adelmann, C.; Radu, I. P. All electrical propagating spin wave spectroscopy with broadband wavevector capability. Appl. Phys. Lett. 2016, 109, 012403.

    Google Scholar 

  71. Topp, J.; Heitmann, D.; Kostylev, M. P.; Grundler, D. Making a reconfigurable artificial crystal by ordering bistable magnetic nanowires. Phys. Rev. Lett. 2010, 104, 207205.

    Google Scholar 

  72. Ding, J.; Kostylev, M.; Adeyeye, A. O. Magnonic crystal as a medium with tunable disorder on a periodical lattice. Phys. Rev. Lett. 2011, 107, 047205.

    CAS  Google Scholar 

  73. Neusser, S.; Duerr, G.; Bauer, H. G.; Tacchi, S.; Madami, M.; Woltersdorf, G.; Gubbiotti, G.; Back, C. H.; Grundler, D. Anisotropic propagation and damping of spin waves in a nanopatterned antidot lattice. Phys. Rev. Lett. 2010, 105, 067208.

    CAS  Google Scholar 

  74. Sun, Y. Y.; Song, Y. Y.; Chang, H. C.; Kabatek, M.; Jantz, M.; Schneider, W.; Wu, M. Z.; Schultheiss, H.; Hoffmann, A. Growth and ferromagnetic resonance properties of nanometer-thick yttrium iron garnet films. Appl. Phys. Lett. 2012, 101, 152405.

    Google Scholar 

  75. Serga, A. A.; Chumak, A. V.; Hillebrands, B. YIG magnonics. J. Phys. D Appl. Phys. 2010, 43, 264002.

    Google Scholar 

  76. Kalinikos, B. A.; Slavin, A. N. Theory of dipole-exchange spin wave spectrum for ferromagnetic films with mixed exchange boundary conditions. J. Phys. C Solid State Phys. 1986, 19, 7013–7033.

    Google Scholar 

  77. Klingler, S.; Chumak, A. V.; Mewes, T.; Khodadadi, B.; Mewes, C.; Dubs, C.; Surzhenko, O.; Hillebrands, B.; Conca, A. Measurements of the exchange stiffness of YIG films using broadband ferromagnetic resonance techniques. J. Phys. D Appl. Phys. 2015, 48, 015001.

    CAS  Google Scholar 

  78. Gardiner, C. W.; Collett, M. J. Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation. Phys. Rev. A 1985, 31, 3761–3774.

    CAS  Google Scholar 

  79. Clerk, A. A.; Devoret, M. H.; Girvin, S. M.; Marquardt, F.; Schoelkopf, R. J. Introduction to quantum noise, measurement, and amplification. Rev. Mod. Phys. 2010, 82, 1155–1208.

    Google Scholar 

  80. Yu, T.; Bauer, G. E. W. Noncontact spin pumping by microwave evanescent fields. Phys. Rev. Lett. 2020, 124, 236801.

    CAS  Google Scholar 

  81. Wong, K. L.; Bi, L.; Bao, M. Q.; Wen, Q. Y.; Chatelon, J. P.; Lin, Y. T.; Ross, C. A.; Zhang, H. W.; Wang, K. L. Unidirectional propagation of magnetostatic surface spin waves at a magnetic film surface. Appl. Phys. Lett. 2014, 105, 232403.

    Google Scholar 

  82. Yu, T.; Wang, H. C.; Sentef, M. A.; Yu, H. M.; Bauer, G. E. W. Magnon trap by chiral spin pumping. Phys. Rev. B 2020, 102, 054429.

    CAS  Google Scholar 

  83. Rustagi, A.; Bertelli, I.; van der Sar, T.; Upadhyaya, P. Sensing chiral magnetic noise via quantum impurity relaxometry. arXiv: 2009.05060, 2020.

  84. Torrejon, J.; Riou, M.; Araujo, F. A.; Tsunegi, S.; Khalsa, G.; Querlioz, D.; Bortolotti, P.; Cros, V.; Yakushiji, Y.; Fukushima, A. et al. Neuromorphic computing with nanoscale spintronic oscillators. Nature 2017, 547, 428–431.

    CAS  Google Scholar 

  85. Zahedinejad, M.; Awad, A. A.; Muralidhar, S.; Khymyn, R.; Fulara, H.; Mazraati, H.; Dvornik, M.; Akerman, J. Two-dimensional mutually synchronized spin Hall nano-oscillator arrays for neuromorphic computing. Nat. Nanotechnol. 2020, 15, 47–52.

    CAS  Google Scholar 

  86. Kandel, E. R.; Schwartz, J. H.; Jessell, T. M. Principles of Neural Science; 4th ed. McGraw-Hill: New York, 2000.

    Google Scholar 

Download references

Acknowledgements

The authors thank L. Flacke, M. Althammer, M. Weiler, K. Schultheiss, H. Schultheiss, M. Madami, G. Gubbiotti and C. M. Hu for their helpful discussions. Funding: We wish to acknowledge the support by the National Key Research and Development Program of China (Nos. 2016YFA0300802 and 2017YFA0206200), the National Natural Science Foundation of China (NSFC) (Nos. 11674020, 12074026 and U1801661), and the 111 talent program B16001. G. B. was supported by the Netherlands Organization for Scientific Research (NWO) and Japan Society for the Promotion of Science Kakenhi Grants-in-Aid for Scientific Research (No. 19H006450). T. Y. was funded through the Emmy Noether Program of Deutsche Forschungsgemeinschaft (SE 2558/2-1). K. X. thanks the National Key Research and Development Program of China (Nos. 2017YFA0303304 and 2018YFB0407601) and the National Natural Science Foundation of China (Nos. 61774017 and 11734004). K. S. was supported by the Fundamental Research Funds for the Central Universities (No. 2018EYT02). M. Z. W. were supported by the US National Science Foundation (No. EFMA-1641989).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Haiming Yu.

Electronic Supplementary Material

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, H., Chen, J., Yu, T. et al. Nonreciprocal coherent coupling of nanomagnets by exchange spin waves. Nano Res. 14, 2133–2138 (2021). https://doi.org/10.1007/s12274-020-3251-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12274-020-3251-5

Keywords

Navigation