- Physical Review Journals
All JournalsPhysics Magazine

Physical Review B

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Microscopic theory of an atomic spin diode

William J. Huddie and Rembert A. Duine

PDF

Phys. Rev. B 114, 144428 – Published 30 September, 2026

DOI: https://doi.org/10.1103/pbc1-yzvd

Abstract

We present a microscopic theory of an atomic spin diode. Our proposed system consists of two magnetic adatoms deposited on the surface of a two-dimensional electron gas with Rashba spin-orbit coupling. A local s-d type coupling between the local spins and the spins of the electrons induces a nonlocal Ruderman-Kittel-Kasuya-Yosida type interaction and a Dzyaloshinskii-Moriya interaction, in addition to dissipative interactions, between the spins. We derive the effective action for the spins using the Keldysh formalism. From the effective action, we also derive equations of motion for the spins which are shown to be of Landau-Lifshitz-Gilbert type, and give expressions for the effective field and Gilbert damping which appear in this equation. From our microscopic theory, we find that for an in-plane magnetic field perpendicular to the vector connecting the two atoms, the magnitude of the field and the distance between the atoms can always be tuned to engender perfectly diodic coupling. Our findings may pave the way for experimental realization of atomic spin diodes.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (64)

  1. A. Barman, G. Gubbiotti, S. Ladak, A. O. Adeyeye, M. Krawczyk, J. Gräfe, C. Adelmann, S. Cotofana, A. Naeemi, V. I. Vasyuchka, B. Hillebrands, S. A. Nikitov, H. Yu, D. Grundler, A. V. Sadovnikov, A. A. Grachev, S. E. Sheshukova, J.-Y. Duquesne, M. Marangolo, G. Csaba, et al., The 2021 magnonics roadmap, J. Phys.: Condens. Matter 33, 413001 (2021).
  2. B. Flebus, D. Grundler, B. Rana, Y. C. Otani, I. Barsukov, G. Gubbiotti, P. Landeros, Q. Wang, T. van der Sar, et al., The 2024 magnonics roadmap, J. Phys.: Condens. Matter 36, 363501 (2024).
  3. B. Lenk, H. Ulrichs, F. Garbs, and M. Münzenberg, The building blocks of magnonics, Phys. Rep. 507, 107 (2011).
  4. W. Shockley, The theory of p-n junctions in semiconductors and p-n junction transistors, Bell Syst. Tech. J. 28, 435 (1949).
  5. S. M. Sze and K. K. Ng, Physics of Semiconductor Devices, 3rd ed. (Wiley-Interscience, Hoboken, NJ, 2006).
  6. A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Magnon spintronics, Nat. Phys. 11, 453 (2015).
  7. R. W. Damon and J. R. Eshbach, Magnetostatic modes of a ferromagnet slab, J. Phys. Chem. Solids 19, 308 (1961).
  8. A. V. Chumak, A. A. Serga, and B. Hillebrands, Magnon transistor for all-magnon data processing, Nat. Commun. 5, 4700 (2014).
  9. J. Lan, W. Yu, R. Wu, and J. Xiao, Spin-wave diode, Phys. Rev. X 5, 041049 (2015).
  10. H. Y. Yuan, R. Lavrijsen, and R. A. Duine, Unidirectional magnetic coupling induced by chiral interaction and nonlocal damping, Phys. Rev. B 107, 024418 (2023).
  11. J. Zou, S. Bosco, E. Thingstad, J. Klinovaja, and D. Loss, Dissipative spin-wave diode and nonreciprocal magnonic amplifier, Phys. Rev. Lett. 132, 036701 (2024).
  12. A. Metelmann and A. A. Clerk, Nonreciprocal photon transmission and amplification via reservoir engineering, Phys. Rev. X 5, 021025 (2015).
  13. A. A. Khajetoorians, J. Wiebe, B. Chilian, S. Lounis, S. Blügel, and R. Wiesendanger, Atom-by-atom engineering and magnetometry of tailored nanomagnets, Nat. Phys. 8, 497 (2012).
  14. A. A. Khajetoorians, D. Wegner, A. F. Otte, and I. Swart, Creating designer quantum states of matter atom-by-atom, Nat. Rev. Phys. 1, 703 (2019).
  15. D.-J. Choi, N. Lorente, J. Wiebe, K. von Bergmann, A. F. Otte, and A. J. Heinrich, Colloquium: Atomic spin chains on surfaces, Rev. Mod. Phys. 91, 041001 (2019).
  16. A. Spinelli, B. Bryant, F. Delgado, J. Fernández-Rossier, and A. F. Otte, Imaging of spin waves in atomically designed nanomagnets, Nat. Mater. 13, 782 (2014).
  17. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002).
  18. K. Yosida, Theory of Magnetism, Springer Series in Solid-State Sciences Vol. 122 (Springer, Berlin, Heidelberg, 1996).
  19. A. Kamenev, Field Theory of Non-equilibrium Systems (Cambridge University Press, Cambridge, 2011).
  20. L. V. Keldysh, Diagram technique for nonequilibrium processes, Sov. Phys. JETP 20, 1018 (1965).
  21. J. Schwinger, Brownian motion of a quantum oscillator, J. Math. Phys. 2, 407 (1961).
  22. A. Kamenev and A. Levchenko, Keldysh technique and non-linear sigma-model: Basic principles and applications, Adv. Phys. 58, 197 (2009).
  23. J. Rammer and H. Smith, Quantum field-theoretical methods in transport theory of metals, Rev. Mod. Phys. 58, 323 (1986).
  24. L. M. Sieberer, M. Buchhold, and S. Diehl, Keldysh field theory for driven open quantum systems, Rep. Prog. Phys. 79, 096001 (2016).
  25. R. A. Duine, A. S. Núñez, J. Sinova, and A. H. MacDonald, Functional Keldysh theory of spin torques, Phys. Rev. B 75, 214420 (2007).
  26. A. S. Núñez and R. A. Duine, Effective temperature and Gilbert damping of a current-driven localized spin, Phys. Rev. B 77, 054401 (2008).
  27. R. C. Verstraten, T. Ludwig, R. A. Duine, and C. M. Smith, Fractional Landau-Lifshitz-Gilbert equation, Phys. Rev. Res. 5, 033128 (2023).
  28. M. G. Quarenta, M. Tharmalingam, T. Ludwig, H. Y. Yuan, L. Karwacki, R. C. Verstraten, and R. A. Duine, Bath-induced spin inertia, Phys. Rev. Lett. 133, 136701 (2024).
  29. A. Altland and B. Simons, Condensed Matter Field Theory, 2nd ed. (Cambridge University Press, Cambridge, 2010).
  30. J. Kondo, Resistance minimum in dilute magnetic alloys, Prog. Theor. Phys. 32, 37 (1964).
  31. P. W. Anderson, G. Yuval, and D. R. Hamann, Exact results in the Kondo problem. II. Scaling theory, qualitatively correct solution, and some new results on one-dimensional classical statistical models, Phys. Rev. B 1, 4464 (1970).
  32. E. Turco, M. Aapro, S. C. Ganguli, N. Krane, R. Drost, N. Sobrino, A. Bernhardt, M. Juríček, R. Fasel, P. Ruffieux, P. Liljeroth, and D. Jacob, Demonstrating Kondo behavior by temperature-dependent scanning tunneling spectroscopy, Phys. Rev. Res. 6, L022061 (2024).
  33. Y.-h. Zhang, S. Kahle, T. Herden, C. Stroh, M. Mayor, U. Schlickum, M. Ternes, P. Wahl, and K. Kern, Temperature and magnetic field dependence of a Kondo system in the weak coupling regime, Nat. Commun. 4, 2110 (2013).
  34. V. Madhavan, W. Chen, T. Jamneala, M. F. Crommie, and N. S. Wingreen, Local spectroscopy of a Kondo impurity: Co on Au(111), Phys. Rev. B 64, 165412 (2001).
  35. R. Hiraoka, E. Minamitani, R. Arafune, N. Tsukahara, S. Watanabe, M. Kawai, and N. Takagi, Single-molecule quantum dot as a Kondo simulator, Nat. Commun. 8, 16012 (2017).
  36. P. Wahl, P. Simon, L. Diekhöner, V. S. Stepanyuk, P. Bruno, M. A. Schneider, and K. Kern, Exchange interaction between single magnetic adatoms, Phys. Rev. Lett. 98, 056601 (2007).
  37. W. Wan, R. Harsh, A. Meninno, P. Dreher, S. Sajan, H. Guo, I. Errea, F. de Juan, and M. M. Ugeda, Evidence for ground state coherence in a two-dimensional Kondo lattice, Nat. Commun. 14, 7005 (2023).
  38. A. Shnirman, Y. Gefen, A. Saha, I. S. Burmistrov, M. N. Kiselev, and A. Altland, Geometric quantum noise of spin, Phys. Rev. Lett. 114, 176806 (2015).
  39. J. Hubbard, Calculation of partition functions, Phys. Rev. Lett. 3, 77 (1959).
  40. H. T. C. Stoof, K. Gubbels, and D. Dickerscheid, Ultracold Quantum Fields (Springer, Dordrecht, 2009).
  41. S. LaShell, B. A. McDougall, and E. Jensen, Spin splitting of an Au(111) surface state band observed with angle resolved photoelectron spectroscopy, Phys. Rev. Lett. 77, 3419 (1996).
  42. F. Reinert, G. Nicolay, S. Schmidt, D. Ehm, and S. Hüfner, Direct measurements of the l-gap surface states on the (111) face of noble metals by photoelectron spectroscopy, Phys. Rev. B 63, 115415 (2001).
  43. G. Nicolay, F. Reinert, S. Hüfner, and P. Blaha, Spin-orbit splitting of the l-gap surface state on Au(111) and Ag(111), Phys. Rev. B 65, 033407 (2001).
  44. M. Hoesch, M. Muntwiler, V. N. Petrov, M. Hengsberger, L. Patthey, M. Shi, M. Falub, T. Greber, and J. Osterwalder, Spin structure of the Shockley surface state on the (111) surface of gold, Phys. Rev. B 69, 241401(R) (2004).
  45. A. Tamai, W. Meevasana, P. D. C. King, C. W. Nicholson, A. de la Torre, E. Rozbicki, and F. Baumberger, Spin-orbit splitting of the Shockley surface state on cu(111), Phys. Rev. B 87, 075113 (2013).
  46. S. Hüfner, F. Reinert, S. Schmidt, G. Nicolay, and F. Forster, Photoemission investigation of the L−−Gap surface states on clean and rare gas-covered noble metal (111)-surfaces, Z. Phys. Chem. 222, 407 (2008).
  47. C. R. Ast, J. Henk, A. Ernst, L. Moreschini, M. C. Falub, D. Pacilé, P. Bruno, K. Kern, and M. Grioni, Giant spin splitting through surface alloying, Phys. Rev. Lett. 98, 186807 (2007).
  48. I. Gierz, B. Stadtmüller, J. Vuorinen, M. Lindroos, F. Meier, J. H. Dil, K. Kern, and C. R. Ast, The structural influence on the Rashba-type spin-splitting in surface alloys, Phys. Rev. B 81, 245430 (2010).
  49. F. Meier, V. Petrov, S. Guerrero, C. Mudry, L. Patthey, J. Osterwalder, and J. H. Dil, Unconventional Fermi surface spin textures in the BixPb1−x/Ag(111) surface alloy, Phys. Rev. B 79, 241408(R) (2009).
  50. D. Pacilé, C. R. Ast, M. Papagno, C. Dasilva, L. Moreschini, M. Falub, A. P. Seitsonen, and M. Grioni, Electronic structure of an ordered Pb/Ag(111) surface alloy: Theory and experiment, Phys. Rev. B 73, 245429 (2006).
  51. A. Varykhalov, D. Marchenko, M. R. Scholz, E. D. L. Rienks, T. K. Kim, G. Bihlmayer, J. Sánchez-Barriga, and O. Rader, Ir(111) surface state with giant Rashba splitting persists under graphene in air, Phys. Rev. Lett. 108, 066804 (2012).
  52. S. Weinberg, The Quantum Theory of fields. Vol. I: Foundations (Cambridge University Press, Cambridge, 1995).
  53. J. Plemelj, Problems in the Sense of Riemann and Klein, Interscience Tracts in Pure and Applied Mathematics Vol. 16 (Interscience Publishers, John Wiley & Sons, New York, 1964).
  54. F. Reyes-Osorio and B. K. Nikolić, Nonlocal damping of spin waves in a magnetic insulator induced by normal, heavy, or altermagnetic metallic overlayer: A Schwinger-Keldysh field theory approach, Phys. Rev. B 110, 214432 (2024).
  55. M. T. Béal-Monod, Ruderman-Kittel-Kasuya-Yosida indirect interaction in two dimensions, Phys. Rev. B 36, 8835 (1987).
  56. H. Imamura, P. Bruno, and Y. Utsumi, Twisted exchange interaction between localized spins embedded in a one- or two-dimensional electron gas with Rashba spin-orbit coupling, Phys. Rev. B 69, 121303(R) (2004).
  57. T. Moriya, Anisotropic superexchange interaction and weak ferromagnetism, Phys. Rev. 120, 91 (1960).
  58. I. Dzyaloshinsky, A thermodynamic theory of “weak” ferromagnetism of antiferromagnetics, J. Phys. Chem. Solids 4, 241 (1958).
  59. V. Madhavan, W. Chen, T. Jamneala, M. F. Crommie, and N. S. Wingreen, Tunneling into a single magnetic atom: Spectroscopic evidence of the Kondo resonance, Science 280, 567 (1998).
  60. N. Knorr, M. A. Schneider, L. Diekhöner, P. Wahl, and K. Kern, Kondo effect of single Co adatoms on Cu surfaces, Phys. Rev. Lett. 88, 096804 (2002).
  61. T. Jamneala, V. Madhavan, W. Chen, and M. F. Crommie, Scanning tunneling spectroscopy of transition-metal impurities at the surface of gold, Phys. Rev. B 61, 9990 (2000).
  62. L. Limot, E. Pehlke, J. Kröger, and R. Berndt, Surface-state localization at adatoms, Phys. Rev. Lett. 94, 036805 (2005).
  63. https://github.com/toor/asd.
  64. E. H. Hwang and S. D. Sarma, Screening, Kohn anomaly, Friedel oscillation, and RKKY interaction in bilayer graphene, Phys. Rev. Lett. 101, 156802 (2008).

Outline

Information

Phys. Rev. B 114, 144428– Published 30 September, 2026
  • Received 5 March 2026
  • Revised 10 July 2026
  • Accepted 11 September 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation