- 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
  • Letter
  • Open Access

Multivalued dispersion equation for coupling between plasmons and surface optical phonons in a graphene/polar-substrate system

S. M. Kukhtaruk1,* and V. A. Kochelap1,2

  • *Contact author: kukhtaruk@gmail.com
PDF

Phys. Rev. B 110, L241403 – Published 6 December, 2024

DOI: https://doi.org/10.1103/PhysRevB.110.L241403

Abstract

Dispersion equations are a common paradigm of collective excitation physics. However, in some systems, dispersion equations contain multivalued functions and their solutions are ambiguous. As an example, we consider graphene on a polar substrate where Dirac plasmons are coupled with surface optical phonons. The dispersion equation for this system contains square-root singularity. Using the initial value problem resolves this uncertainty and gives a unique solution. Particularly, we found that the lower plasmon-phonon mode, which in terms of dispersion can have a good quality factor, is almost absent in excitation spectra. The physical reason and experimental evidence of the mode collapse are discussed.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (52)

  1. L. D. Landau and E. M. Lifshits, Course of Theoretical Physics: Physical Kinetics (Pergamon, New York, 1981), Vol. 10.
  2. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, New York, 2007).
  3. G. Giuliani and G. Vignale, Quantum Theory of the Electron Liquid (Cambridge University Press, New York, 2008).
  4. V. V. Mitin, V. A. Kochelap, and M. A. Stroscio, Quantum Heterostructures: Microelectronics and Optoelectronics (Cambridge University Press, New York, 1999).
  5. V. V. Mitin, V. A. Kochelap, M. Dutta, and M. A. Stroscio, Introduction to Optical and Optoelectronic Properties of Nanostructures (Cambridge University Press, New York, 2019).
  6. J. M. D. Coey, Magnetism and Magnetic Materials (Cambridge University Press, New York, 2010).
  7. A. Rahimi-Iman, Fundamentals of Polariton Physics (Springer, Cham, 2020), Vol. 229.
  8. L. D. Landau, On the vibrations of the electronic plasma, Akad. Nauk SSSR. Zhurnal Eksper. Teoret. Fiz. 16, 574 (1946) [Acad. Sci. USSR. J. Phys. 10, 25 (1946)].
  9. B. Wunsch, T. Stauber, F. Sols, and F. Guinea, Dynamical polarization of graphene at finite doping, New J. Phys. 8, 318 (2006).
  10. E. H. Hwang and S. Das Sarma, Dielectric function, screening, and plasmons in two-dimensional graphene, Phys. Rev. B 75, 205418 (2007).
  11. A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Rev. Mod. Phys. 81, 109 (2009).
  12. S. M. Kukhtaruk and V. A. Kochelap, Semiclassical analysis of intraband collective excitations in a two-dimensional electron gas with Dirac spectrum, Phys. Rev. B 92, 041409(R) (2015).
  13. É. de Mello Silva, Dynamical class of a two-dimensional plasmonic Dirac system, Phys. Rev. E 92, 042146 (2015).
  14. R. J. Koch, Th. Seyller, and J. A. Schaefer, Strong phonon-plasmon coupled modes in the graphene/silicon carbide heterosystem, Phys. Rev. B 82, 201413(R) (2010).
  15. R. J. Koch, T. Haensel, S. I.-U. Ahmed, Th. Seyller, and J. A. Schaefer, HREELS study of graphene formed on hexagonal silicon carbide, Phys. Status Solidi C 7, 394 (2010).
  16. E. H. Hwang, R. Sensarma, and S. Das Sarma, Plasmon-phonon coupling in graphene, Phys. Rev. B 82, 195406 (2010).
  17. M. Jablan, M. Soljacic, and H. Buljan, Unconventional plasmon-phonon coupling in graphene, Phys. Rev. B 83, 161409(R) (2011).
  18. Z.-Y. Ong and M. V. Fischetti, Theory of interfacial plasmon-phonon scattering in supported graphene, Phys. Rev. B 86, 165422 (2012).
  19. Z.-Y. Ong and M. V. Fischetti, Theory of remote phonon scattering in top-gated single-layer graphene, Phys. Rev. B 88, 045405 (2013).
  20. E. H. Hwang and S. Das Sarma, Surface polar optical phonon interaction induced many-body effects and hot-electron relaxation in graphene, Phys. Rev. B 87, 115432 (2013).
  21. Y. Liu, R. F. Willis, K. V. Emtsev, and Th. Seyller, Plasmon dispersion and damping in electrically isolated two-dimensional charge sheets, Phys. Rev. B 78, 201403(R) (2008).
  22. J. Lu, K. P. Loh, H. Huang, W. Chen, and A. T. S. Wee, Plasmon dispersion on epitaxial graphene studied using high-resolution electron energy-loss spectroscopy, Phys. Rev. B 80, 113410 (2009).
  23. Y. Liu and R. F. Willis, Plasmon-phonon strongly coupled mode in epitaxial graphene, Phys. Rev. B 81, 081406(R) (2010).
  24. T. Langer, J. Baringhaus, H. Pfnür, H. W. Schumacher, and C. Tegenkamp, Plasmon damping below the Landau regime: The role of defects in epitaxial graphene, New J. Phys. 12, 033017 (2010).
  25. C. Tegenkamp, H. Pfnür, T. Langer, J. Baringhaus, and H. W. Schumacher, Plasmon electron-hole resonance in epitaxial graphene, J. Phys.: Condens. Matter 23, 012001 (2011).
  26. M. Freitag, T. Low, W. Zhu, H. Yan, F. Xia, and P. Avouris, Photocurrent in graphene harnessed by tunable intrinsic plasmons, Nat. Commun. 4, 1951 (2013).
  27. F. J. Bezares, A. De Sanctis, J. R. M. Saavedra, A. Woessner, P. Alonso-González, I. Amenabar, J. Chen, T. H. Bointon, S. Dai, M. M. Fogler, D. N. Basov, R. Hillenbrand, M. F. Craciun, F. J. García de Abajo, S. Russo, and F. H. L. Koppens, Intrinsic plasmon–phonon interactions in highly doped graphene: A near-field imaging study, Nano Lett. 17, 5908 (2017).
  28. A. N. Grigorenko, M. Polini, and K. S. Novoselov, Graphene plasmonics, Nat. Photonics 6, 749 (2012).
  29. S. Xiao, X. Zhu, B.-H. Li, and N. A. Mortensen, Graphene-plasmon polaritons: From fundamental properties to potential applications, Front. Phys. 11, 117801 (2016).
  30. S. Huang, C. Song, G. Zhang, and H. Yan, Graphene plasmonics: Physics and potential applications, Nanophotonics 6, 1191 (2017).
  31. S. Ogawa, S. Fukushima, and M. Shimatani, Graphene plasmonics in sensor applications: A review, Sensors 20, 3563 (2020).
  32. D. T. Nurrohman and N.-F. Chiu, A review of graphene-based surface plasmon resonance and surface-enhanced raman scattering biosensors: Current status and future prospects, Nanomaterials 11, 216 (2021).
  33. L. Cui, J. Wang, and M. Sun, Graphene plasmon for optoelectronics, Rev. Phys. 6, 100054 (2021).
  34. Hereafter, we use similar representation for any 3D vector, i.e., V=(V∥,Vz).
  35. M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Clarendon, Oxford, 1954).
  36. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Parts I and II.
  37. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevB.110.L241403 for details of substrate polarization and boundary conditions, surface optical phonons, analytical continuation and Fourier analysis of the signals.
  38. S. Fratini and F. Guinea, Substrate-limited electron dynamics in graphene, Phys. Rev. B 77, 195415 (2008).
  39. N. G. van Kampen, On the theory of stationary waves in plasmas, Physica 21, 949 (1955).
  40. W. Norimatsu, A review on carrier mobilities of epitaxial graphene on silicon carbide, Materials 16, 7668 (2023).
  41. One must consider the square of amplitudes (power) to discuss line shapes. However, in this case, the ω− line will be almost invisible near ω=vFk, therefore we will focus on the amplitudes.
  42. Y. S. Joe, A. M. Satanin, and C. S. Kim, Classical analogy of Fano resonances, Phys. Scr. 74, 259 (2006).
  43. A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, Fano resonances in nanoscale structures, Rev. Mod. Phys. 82, 2257 (2010).
  44. S. M. Kukhtaruk, V. A. Kochelap, V. N. Sokolov, and K. W. Kim, Spatially dispersive dynamical response of hot carriers in doped graphene, Physica E 79, 26 (2016).
  45. Another possible way is to formulate the boundary value problem.
  46. J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, Theory of surface plasmons and surface-plasmon polaritons, Rep. Prog. Phys. 70, 1 (2007).
  47. L. Tsang, Time-harmonic solution of the elastic head wave problem incorporating the influence of Rayleigh poles, J. Acoust Soc. Am. 63, 1302 (1978).
  48. V. Gusev, C. Desmet, W. Lauriks, C. Glorieux, and J. Thoen, Theory of Scholte, leaky Rayleigh, and lateral wave excitation via the laser-induced thermoelastic effect, J. Acoust. Soc. Am. 100, 1514 (1996).
  49. T.-X. Hu, Dong Wu, Z. M. Sheng, and J. Zhang, Exact dispersion relation of the quantum Langmuir wave, Phys. Rev. E 109, 065213 (2024).
  50. L. M. Farrell, C. J. Howls, and D. H. J. O'Dell, Logarithmic catastrophes and Stokes's phenomenon in waves at horizons, J. Phys. A: Math. Theor. 56, 044001 (2023).
  51. C. A. Zen Vasconcellos, P. O. Hess, J. de Freitas Pacheco, D. Hadjimichef, and B. Bodmann, The branch-cut cosmology: evidences and open questions, Astron. Nachr. 344, e220079 (2023).
  52. D. F. G. Fiorillo and G. G. Raffelt, Theory of neutrino fast flavor evolution. Part I. Linear response theory and stability conditions, J. High Energy Phys. 08 (2024) 225 .

Outline

Information

Phys. Rev. B 110, L241403– Published 6 December, 2024
  • Received 24 July 2024
  • Revised 9 October 2024
  • Accepted 19 November 2024
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