- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

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

Fast two-dimensional long-wavelength plasmons in atomically thin MoS2

Chien-Ting Wu1, Chih-En Hsu2, Chih-Ying Huang3,4, Yung-Ning Hsu2, Che-Lun Lee5, Ssu-Yen Huang4,6, Chun-Wei Chen5,6, Hung-Chung Hsueh2,*, and Ming-Wen Chu3,6,†

  • *Contact author: hchsueh@gms.tku.edu.tw
  • †Contact author: chumingwen@ntu.edu.tw
PDF

Phys. Rev. Research 8, 023072 – Published 24 April, 2026

DOI: https://doi.org/10.1103/hr7f-kgdg

Abstract

Plasmons are bosonic wavelike charge oscillations in solids and the plasmon group velocity (vg), the first derivative of momentum (q)-dependent dispersions, depicts the many-body electron motions of critical electronic importance. The profound characteristics of vg in either three-dimensional (3D) or two-dimensional (2D) electron systems are nonetheless short of quantitative scrutiny. Using q-resolved electron energy loss spectroscopy, we tackle the q-dependent vg of π and π + σ plasmons in 3D-bulk and 2D-monolayer (ML) MoS2 semiconductors, with the 2D MoS2 being the model transition-metal dichalcogenide (TMD) for emergent atomically thin electronics. vg along with the underlining electronic parameters of the charge density, effective mass, and screening length in 2D for the plasmon dispersions is quantitatively elaborated. 2D vg is further compared to those evaluated from metallic ML-TMD TaS2 and graphene, and the scaling over the single-particle Fermi velocity (vF) unveils a vg/vF function generally obeyed by the semiconducting and metallic TMD and graphene of distinctly different massive and relativistic characters, respectively. Beyond the pertinent critical q (on the order of 10−2Å−1) where one observes vg=vF, electric-field lines connecting the oscillating 2D charges would turn from vacuum into the atomically thin matters, concomitant with the increasing plasmon damping and reducing vg (vg<vF). Long-wavelength 2D plasmons below the critical q characterize the field lines largely in vacuum and are persistent and fast (vg>vF), with high-speed application merits. Conversely, 3D plasmons in bulk MoS2 never exceed vF.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (48)

  1. D. Pines and D. Bohm, A collective description of electron interactions: II. collective vs individual particle aspects of the interactions, Phys. Rev. 85, 338 (1952).
  2. D. Pines, Electron interaction in solids, Can. J. Phys. 34, 1379 (1956).
  3. P. M. Platzman and P. A. Wolff, Waves and Interactions in Solid State Plasma (Academic Press, New York, 1973).
  4. P. W. Anderson, Plasmons, gauge invariance, and mass, Phys. Rev. 130, 439 (1963).
  5. A. A. Husain, E. W. Huang, M. Mitrano, M. S. Rak, S. I. Rubeck, X. Guo, H. Yang, C. Sow, Y. Maeno, B. Uchoda, T. C. Chiang, P. E. Batson, P. W. Phillips, and P. Abbamonte, Pine’s demon observed as a 3D acoustic plasmon in Sr2RuO4, Nature (London) 621, 66 (2023).
  6. L. Landau, On the vibrations of the electronic plasma, J. Phys. USSR 10, 25 (1946).
  7. W. Y. Liang and S. L. Cundy, Electron energy loss studies of the transition metal dichalcogenides, Philos. Mag. 19, 1031 (1969).
  8. Z. Zeppenfeld, Electron energy losses and optical anisotropy of MoS2 single crystals, Opt. Commun. 1, 377 (1970).
  9. H. Raether, Excitation of Plasmons and Interband Transitions by Electrons (Springer, Berlin, 1980).
  10. I.-T. Wang, T.-L. Chou, C.-E. Hsu, Z. Lei, L.-M. Wang, P.-H. Lin, C.-W. Luo, C.-W. Chen, C.-N. Kuo, C. S. Lue, C. H. Chen, H.-C. Hsueh, and M.-W. Chu, The growing charge-density-wave order in CuTe lightens and speeds up electrons, Nat. Commun. 15, 9345 (2024).
  11. B. Yue, F. Hong, K.-D. Tsuei, N. Hiraoka, Y.-H. Wu, V. M. Silkin, B. Chen, and H.-K. Mao, High-energy electronic excitations in a bulk MoS2 single crystal, Phys. Rev. B 96, 125118 (2017).
  12. F. Stern, Polarizability of a two-dimensional electron gas, Phys. Rev. Lett. 18, 546 (1967).
  13. E. H. Hwang and S. Das Sarma, Dielectric function, screening, and plasmons in two-dimensional graphene, Phys. Rev. B 75, 205418 (2007).
  14. S. Das Sarma and E. H. Hwang, Collective modes of the massless Dirac plasma, Phys. Rev. Lett. 102, 206412 (2009).
  15. F. H. da Jornada, L. Xian, A. Rubio, and S. G. Louie, Universal slow plasmons and giant field enhancement in atomically thin quasi-two-dimensional metals, Nat. Commun. 11, 1013 (2020).
  16. B. Diaconescu, K. Pohl, L. Vattuone, L. Savio, P. Hofmann, V. M. Silkin, J. M. Pitarke, E. V. Chulkov, P. M. Echenique, D. Farías, and M. Rocca, Low-energy acoustic plasmons at metal surfaces, Nature (London) 448, 57 (2007).
  17. 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).
  18. 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).
  19. S. C. Liou, C.-S. Shie, C. H. Chen, R. Breitwieser, W. W. Pai, G. Y. Guo, and M.-W. Chu, π-plasmon dispersion in free-standing graphene by momentum-resolved electron energy-loss spectroscopy, Phys. Rev. B 91, 045418 (2015).
  20. D. C. Elias, R. V. Gorbachev, A. S. Mayorov, S. V. Morozov, A. A. Zhukov, P. Blake, L. A. Ponomarenko, I. V. Grigorieva, K. S. Novoselov, F. Guinea, and A. K. Geim, Dirac cones reshaped by interaction effects in suspended graphene, Nat. Phys. 7, 701 (2011).
  21. T. Tudorovskiy and S. A. Mikhailov, Intervalley plasmons in graphene, Phys. Rev. B 82, 073411 (2010).
  22. H. T. B. Do, M. Zhao, P. Li, Y. W. Soh, J. Rangaraj, B. Liu, T. Jiang, X. Zhang, J. Lu, P. Song, J. Teng, and M. Bosman, Slow and highly confined plasmons observed in atomically thin TaS2, Nat. Commun. 16, 5801 (2025).
  23. S. Ghosh, Y. Zheng, M. Rafiq, H. Ravichandran, Y. Sun, C. Chen, M. Goswami, N. U. Sakib, M. ul Karim Sadaf, A. Pannone, S. Ray, J. M. Redwing, Y. Yang, S. Sahay, and S. Das, A complementary two-dimensional material-based one instruction set computer, Nature (London) 642, 327 (2025).
  24. A. Scholz, T. Stauber, and J. Schliemann, Plasmons and screening in a monolayer MoS2, Phys. Rev. B 88, 035135 (2013).
  25. K. Andersen and K. S. Thygesen, Plasmons in metallic monolayer and bilayer transition metal dichalcogenides, Phys. Rev. B 88, 155128 (2013).
  26. R. E. Groenewald, M. Rösner, G. Schönhoff, S. Haas, and T. O. Wehling, Valley plasmonics in transition metal dichalcogenides, Phys. Rev. B 93, 205145 (2016).
  27. P. Johari and V. B. Shenoy, Tunable dielectric properties of transition metal dichalcogenides, ACS Nano 5, 5903 (2011).
  28. J. N. Coleman, M. Lotya, A. O’Neil, S. D. Bergin, P. J. King, U. Khan, K. Young, A. Gaucher, S. De, R. J. Smith, et al., Two-dimensional nanosheets produced by liquid exfoliation of layered materials, Science 331, 568 (2011).
  29. T. Eberlein, U. Bangert, R. R. Nair, R. Jones, M. Gass, A. L. Bleloch, K. S. Novoselov, A. Geim, and P. R. Briddon, Plasmon spectroscopy of free-standing graphene films, Phys. Rev. B 77, 233406 (2008).
  30. H. C. Nerl, K. T. Winther, F. S. Hage, K. S. Thygesen, L. Houben, C. Backes, J. N. Coleman, Q. M. Ramasse, and B. Nicolosi, Probing the local nature of excitons and plasmons in few-layer MoS2, npj 2D Mater Appl. 1, 2 (2017).
  31. E. Moynihan, S. Rost, E. O’Connell, Q. Ramasse, C. Friedrich, and U. Bangert, Plasmons in MoS2 studied via experimental and theoretical correlation of energy loss spectra, J. Microsc. 279, 256 (2020).
  32. C. Habenicht, M. Knupfer, and B. Büchner, Investigation of the dispersion and the effective masses of excitons in bulk 2H−MoS2 using transmission electron energy-loss spectroscopy, Phys. Rev. B 91, 245203 (2015).
  33. P. A. Midgley, A simple new method to obtain high angular resolution ω−q patterns, Ultramicroscopy 76, 91 (1999).
  34. M.-W. Chu, C. H. Chen, F. J. García de Abajo, J.-P. Deng, and C.-Y. Mou, Surface exciton polaritons in individual Au nanoparticles in the far-ultraviolet spectral regime, Phys. Rev. B 77, 245402 (2008).
  35. See Supplemental Material at http://link.aps.org/supplemental/10.1103/hr7f-kgdg for Table S1 and Figs. S1–S18 in Secs. A–H, which include Supplemental Refs. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23].
  36. P. Giannozzi, O. Baseggio, P. Bonfà, D. Brunato, R. Car, I. Carnimeo, C. Cavazzoni, S. de Gironcoli, P. Delugas, F. F. Ruffino, A. Ferretti, N. Marzari, I. Timrov, A. Urru, and S. Baroni, Quantum EXPRESSO toward the exascale, J. Chem. Phys. 152, 154105 (2020).
  37. J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
  38. V. Petkov, S. J. L. Billinge, P. Larson, S. D. Mahanti, T. Vogt, K. K. Rangan, and M. G. Kanatzidis, Structure of nanocrystalline materials using atomic pair distribution function analysis: Study of LiMoS2, Phys. Rev. B 65, 092105 (2002).
  39. J. Deslippe, G. Samsonidze, D. A. Strubbe, M. Jain, M. L. Cohen, and S. G. Louie, BerkeleyGW: A massively parallel computer package for the calculation of the quasiparticle and optical properties of materials and nanostructures, Comput. Phys. Commun. 183, 1269 (2012).
  40. G. Pizzi, V. Vitale, R. Arita, S. Blügel, F. Freimuth, G. Géranton, M. Gibertini, D. Gresch, C. Johnson, T. Koretsune, et al., Wannier90 as a community code: New features and applications, J. Phys. Condens. Matter 32, 165902 (2020).
  41. D. Y. Qiu, F. H. da Jornada, and S. G. Louie, Screening and many-body effects in two-dimensional crystals: Monolayer MoS2, Phys. Rev. B 93, 235435 (2016).
  42. P. Cudazzo, K. O. Ruotsalainen, C. J. Sahle, A. Al-Zein, H. Berger, E. Navarro-Moratalla, S. Huotari, M. Gatti, and A. Rubio, High-energy collective electronic excitations in layered transition-metal dichalcogenides, Phys. Rev. B 90, 125125 (2014).
  43. P. Cudazzo, I. V. Tokatly, and A. Rubio, Dielectric screening in two-dimensional insulators: Implications for excitonic and impurity states in graphane, Phys. Rev. B 84, 085406 (2011).
  44. M. N. Gjerding, M. Pandey, and K. S. Thygesen, Band structure engineered layered metals for low-loss plasmonics, Nat. Commun. 8, 15133 (2017).
  45. 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).
  46. N. Zibouche, M. Schlipf, and F. Giustino, GW band structure of monolayer MoS2 using the SternheimerGW method and effect of dielectric environment, Phys. Rev. B 103, 125401 (2021).
  47. A. Guandalini, R. Senga, Y.-C. Lin, K. Suenaga, A. Ferretti, D. Varsano, A. Recchia, P. Barone, F. Mauri, T. Pichler, and C. Kramberger, Excitonic effects in energy-loss spectra of freestanding graphene, Nano Lett. 23, 11835 (2023).
  48. C.-T. Wu, C.-E. Hsu, C.-Y. Huang, Y.-N. Hsu, C.-L. Lee, S.-Y. Huang, C.-W. Chen, H.-C. Hsueh, and M.-W. Chu, Fast two-dimensional long-wavelength plasmons in atomically thin MoS2, Phys. Rev. Res. (2026).

Outline

Information

Phys. Rev. Research 8, 023072– Published 24 April, 2026
  • Received 15 August 2025
  • Revised 7 January 2026
  • Accepted 27 March 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