Collective excitations in 2D materials


Research on 2D materials has been one of the fastest-growing fields in condensed matter and materials science research in the past 10 years. The low dimensionality and strong correlations of 2D systems give rise to electronic and structural properties, in the form of collective excitations, that do not have counterparts in ordinary 3D materials used in modern technology. These 2D materials present extraordinary opportunities for new technologies, such as in flexible electronics. In this Review, we focus on plasmons, excitons, phonons and magnons in 2D materials. We discuss the theoretical formalism of these collective excitations and elucidate how they differ from their 3D counterparts.

Key points

  • Reduced screening in 2D materials leads to strong interactions and a breakdown of the single-particle picture.

  • 2D plasmons are soft modes with a tunable dispersion and a high confinement of the electric field.

  • Strong interaction produces various exciton-like quasiparticles in addition to the traditional excitons, such as biexcitons, trions and even quintons.

  • Phonons in 2D possess unique flexural modes not found in 3D materials.

  • Recent experimental advances have demonstrated 2D magnetism and topological excitations.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: Experimental detection of plasmonic modes in graphene.
Fig. 2: Optically excited quasiparticles with the photoluminescence emission energies (ΔX, ΔT, ΔXX, ΔXT, ΔPT, ΔIX) indicated for each quasiparticle.
Fig. 3: Real-space representation of quasiparticle decomposition with the binding energies indicated for each quasiparticle.
Fig. 4: Relative positions of quasiparticle PL emission lines with the binding energies indicated.
Fig. 5: Emergence of new vibrational modes in multi-layer 2D materials.
Fig. 6: Magnon band structure in CrI3, a honeycomb 2D material.
Fig. 7: Chiral magnetic spin textures.


  1. 1.

    Ashcroft, N. W. & Mermin, N. D. Solid State Physics (Cengage Learning, 2011).

  2. 2.

    Nozieres, P. The Theory of Interacting Fermi Systems (Benjamin, 1963).

  3. 3.

    Novoselov, K. S. et al. Electric field effect in atomically thin carbon films. Science 306, 666–669 (2004).

    ADS  Google Scholar 

  4. 4.

    Castro Neto, A. H., Guinea, F., Peres, N. M., Novoselov, K. S. & Geim, A. K. The electronic properties of graphene. Rev. Mod. Phys. 81, 109 (2009).

    ADS  Google Scholar 

  5. 5.

    Anderson, P. More is different. Science 177, 393–396 (1972).

    ADS  Google Scholar 

  6. 6.

    Mahan, G. D. Many-Particle Physics 3rd edn (Springer, 2000).

  7. 7.

    Ando, T., Fowler, A. B. & Stern, F. Electronic properties of two-dimensional systems. Rev. Mod. Phys. 54, 437–672 (1982).

    ADS  Google Scholar 

  8. 8.

    Kotov, V. N., Uchoa, B., Pereira, V. M., Guinea, F. & Castro Neto, A. H. Electron–electron interactions in graphene: current status and perspectives. Rev. Mod. Phys. 84, 1067 (2012).

    ADS  Google Scholar 

  9. 9.

    Mermin, N. D. & Wagner, H. Absence of ferromagnetism or antiferromagnetism in one-or two-dimensional isotropic Heisenberg models. Phys. Rev. Lett. 17, 1133 (1966).

    ADS  Google Scholar 

  10. 10.

    Bruus, H. & Flensberg, K. Many-body Quantum Theory in Condensed Matter Physics: An Introduction 1st edn (Oxford Univ. Press, 2004).

  11. 11.

    Wunsch, B., Stauber, T., Sols, F. & Guinea, F. Dynamical polarization of graphene at finite doping. New J. Phys. 8, 0610630 (2006). Calculation of the dynamic polarization and dielectric functions in graphene, leading to the plasmon dispersion.

    Google Scholar 

  12. 12.

    Hwang, E. H. & Das Sarma, S. Dielectric function, screening, and plasmons in two-dimensional graphene. Phys. Rev. B 75, 1–6 (2007).

    Google Scholar 

  13. 13.

    Low, T. et al. Plasmons and screening in monolayer and multilayer black phosphorus. Phys. Rev. Lett. 113, 5–9 (2014).

    Google Scholar 

  14. 14.

    Rodin, A. S. & Castro Neto, A. H. Collective modes in anisotropic double-layer systems. Phys. Rev. B 91, 075422 (2015).

    ADS  Google Scholar 

  15. 15.

    Marton, L., Simpson, J. A., Fowler, H. A. & Swanson, N. Plural scattering of 20-kev electrons in aluminum. Phys. Rev. 126, 182–192 (1962).

    ADS  Google Scholar 

  16. 16.

    Polini, M. et al. Plasmons and the spectral function of graphene. Phys. Rev. B 77, 3–6 (2008).

    Google Scholar 

  17. 17.

    Bostwick, A. et al. Observation of plasmarons in quasi-freestanding doped graphene. Science 328, 999–1002 (2010).

    ADS  Google Scholar 

  18. 18.

    Hillenbrand, R. & Keilmann, F. Complex optical constants on a subwavelength scale. Phys. Rev. Lett. 85, 3029–3032 (2000).

    ADS  Google Scholar 

  19. 19.

    Fei, Z. et al. Infrared nanoscopy of Dirac plasmons at the graphene–SiO2 interface. Nano Lett. 11, 4701–4705 (2011).

    ADS  Google Scholar 

  20. 20.

    Zhang, L. M. et al. Near-field spectroscopy of silicon dioxide thin films. Phys. Rev. B 85, 0754 (2012).

    Google Scholar 

  21. 21.

    Fei, Z. et al. Gate-tuning of graphene plasmons revealed by infrared nano-imaging. Nature 486, 82–85 (2012). Real-space observation of plasmons in graphene.

    ADS  Google Scholar 

  22. 22.

    Chen, J. et al. Optical nano-imaging of gate-tunable graphene plasmons. Nature 487, 77–81 (2012).

    ADS  Google Scholar 

  23. 23.

    Ni, G. X. et al. Fundamental limits to graphene plasmonics. Nature 557, 530–533 (2018).

    ADS  Google Scholar 

  24. 24.

    Wang, G. et al. Colloquium: Excitons in atomically thin transition metal dichalcogenides. Rev. Mod. Phys. 90, 021001 (2018).

    ADS  MathSciNet  Google Scholar 

  25. 25.

    Mak, K. F. & Shan, J. Photonics and optoelectronics of 2D semiconductor transition metal dichalcogenides. Nat. Photonics 10, 216 (2016).

    ADS  Google Scholar 

  26. 26.

    Sun, D., Lai, J.-W., Ma, J.-C., Wang, Q.-S. & Liu, J. Review of ultrafast spectroscopy studies of valley carrier dynamics in two-dimensional semiconducting transition metal dichalcogenides. Chin. Phys. B 26, 037801 (2017).

    ADS  Google Scholar 

  27. 27.

    Keldysh, L. V. Excitons in semiconductor–dielectric nanostructures. Phys. Status Solidi A 164, 3–12 (1997).

    ADS  Google Scholar 

  28. 28.

    Keldysh, L. V. Coulomb interaction in thin semiconductor and semimetal films. Sov. J. Exp. Theor. Phys. Lett. 29, 658 (1979).

    ADS  Google Scholar 

  29. 29.

    Keldysh, L. V. Coulomb interaction in thin semiconductor and semimetal films. Pis’ma Zh. Eksp. Teor. Fiz. 29, 716–719 (1979). References 28 and 29 are the original and translated versions of a paper by Keldysh, deriving the 2D electron–hole potential usually named after him.

    Google Scholar 

  30. 30.

    Rytova, N. S. The screened potential of a point charge in a thin film. Moscow Univ. Phys. Bull. 22, 18 (1967);

  31. 31.

    Cudazzo, P., Tokatly, I. V. & Rubio, A. Dielectric screening in two-dimensional insulators: implications for excitonic and impurity states in graphane. Phys. Rev. B 84, 085406 (2011).

    ADS  Google Scholar 

  32. 32.

    Berkelbach, T. C., Hybertsen, M. S. & Reichman, D. R. Theory of neutral and charged excitons in monolayer transition metal dichalcogenides. Phys. Rev. B 88, 045318 (2013).

    ADS  Google Scholar 

  33. 33.

    Latini, S., Olsen, T. & Thygesen, K. S. Excitons in van der Waals heterostructures: the important role of dielectric screening. Phys. Rev. B 92, 245123 (2015).

    ADS  Google Scholar 

  34. 34.

    Kylänpää, I. & Komsa, H.-P. Binding energies of exciton complexes in transition metal dichalcogenide monolayers and effect of dielectric environment. Phys. Rev. B 92, 205418 (2015).

    ADS  Google Scholar 

  35. 35.

    Trushin, M. Tightly bound excitons in two-dimensional semiconductors with a flat valence band. Phys. Rev. B 99, 205307 (2019).

    ADS  Google Scholar 

  36. 36.

    Zolyomi, V., Drummond, N. D. & Falko, V. I. Band structure and optical transitions in atomic layers of hexagonal gallium chalcogenides. Phys. Rev. B 87, 195403 (2013).

    ADS  Google Scholar 

  37. 37.

    Rybkovskiy, D. V., Osadchy, A. V. & Obraztsova, E. D. Transition from parabolic to ring-shaped valence band maximum in few-layer GaS, GaSe, and InSe. Phys. Rev. B 90, 235302 (2014).

    ADS  Google Scholar 

  38. 38.

    Zhou, J., Shan, W.-Y., Yao, W. & Xiao, D. Berry phase modification to the energy spectrum of excitons. Phys. Rev. Lett. 115, 166803 (2015).

    ADS  Google Scholar 

  39. 39.

    Srivastava, A. & Imamogglu, A. Signatures of Bloch-band geometry on excitons: nonhydrogenic spectra in transition-metal dichalcogenides. Phys. Rev. Lett. 115, 166802 (2015).

    ADS  Google Scholar 

  40. 40.

    Trushin, M., Goerbig, M. O. & Belzig, W. Model prediction of self-rotating excitons in two-dimensional transition-metal dichalcogenides. Phys. Rev. Lett. 120, 187401 (2018).

    ADS  Google Scholar 

  41. 41.

    Trushin, M., Goerbig, M. O. & Belzig, W. Optical absorption by Dirac excitons in single-layer transition-metal dichalcogenides. Phys. Rev. B 94, 041301 (2016).

    ADS  Google Scholar 

  42. 42.

    Goerbig, M. O., Montambaux, G. & Piéchon, F. Measure of Diracness in two-dimensional semiconductors. EPL 105, 57005 (2014).

    ADS  Google Scholar 

  43. 43.

    Chernikov, A. et al. Exciton binding energy and nonhydrogenic Rydberg series in monolayer WS2. Phys. Rev. Lett. 113, 076802 (2014).

    ADS  Google Scholar 

  44. 44.

    He, K. et al. Tightly bound excitons in monolayer WSe2. Phys. Rev. Lett. 113, 026803 (2014). This and ref.43, both published in summer 2014, report observations of non-hydrogenic exciton spectra in WS2 and WSe2, respectively.

    ADS  Google Scholar 

  45. 45.

    Lin, Y. et al. Dielectric screening of excitons and trions in single-layer MoS2. Nano Lett. 14, 5569–5576 (2014).

    ADS  Google Scholar 

  46. 46.

    Borghardt, S. et al. Engineering of optical and electronic band gaps in transition metal dichalcogenide monolayers through external dielectric screening. Phys. Rev. Mater. 1, 054001 (2017).

    Google Scholar 

  47. 47.

    Gupta, G., Kallatt, S. & Majumdar, K. Direct observation of giant binding energy modulation of exciton complexes in monolayer MoSe2. Phys. Rev. B 96, 081403 (2017).

    ADS  Google Scholar 

  48. 48.

    Rodin, A., Carvalho, A. & Castro Neto, A. H. Excitons in anisotropic two-dimensional semiconducting crystals. Phys. Rev. B 90, 075429 (2014).

    ADS  Google Scholar 

  49. 49.

    Raja, A. et al. Coulomb engineering of the bandgap and excitons in two-dimensional materials. Nat. Commun. 8, 1–7 (2017).

    Google Scholar 

  50. 50.

    Qiu, Z. et al. Giant gate-tunable bandgap renormalization and excitonic effects in a 2D semiconductor. Sci. Adv. 5, eaaw2347 (2019).

    ADS  Google Scholar 

  51. 51.

    Courtade, E. et al. Charged excitons in monolayer WSe2: experiment and theory. Phys. Rev. B 96, 085302 (2017).

    ADS  Google Scholar 

  52. 52.

    Szyniszewski, M., Mostaani, E., Drummond, N. D. & Fal’ko, V. I. Binding energies of trions and biexcitons in two-dimensional semiconductors from diffusion quantum Monte Carlo calculations. Phys. Rev. B 95, 081301(R) (2017).

    ADS  Google Scholar 

  53. 53.

    Ganchev, B., Drummond, N., Aleiner, I. & Fal’ko, V. Three-particle complexes in two-dimensional semiconductors. Phys. Rev. Lett. 114, 107401 (2015).

    ADS  Google Scholar 

  54. 54.

    Amani, M. et al. Near-unity photoluminescence quantum yield in MoS2. Science 350, 1065–1068 (2015).

    ADS  Google Scholar 

  55. 55.

    Lien, D.-H. et al. Electrical suppression of all nonradiative recombination pathways in monolayer semiconductors. Science 364, 468–471 (2019).

    ADS  Google Scholar 

  56. 56.

    Efimkin, D. K. & MacDonald, A. H. Many-body theory of trion absorption features in two-dimensional semiconductors. Phys. Rev. B 95, 035417 (2017).

    ADS  Google Scholar 

  57. 57.

    Sidler, M. et al. Fermi polaron-polaritons in charge-tunable atomically thin semiconductors. Nat. Phys. 13, 255–261 (2017).

    Google Scholar 

  58. 58.

    Mostaani, E. et al. Diffusion quantum Monte Carlo study of excitonic complexes in two-dimensional transition-metal dichalcogenides. Phys. Rev. B 96, 075431 (2017).

    ADS  Google Scholar 

  59. 59.

    Kezerashvili, R. Y. & Tsiklauri, S. M. Trion and biexciton in monolayer transition metal dichalcogenides. Few-Body Syst. 58, 18 (2017).

    ADS  Google Scholar 

  60. 60.

    Barbone, M. et al. Charge-tuneable biexciton complexes in monolayer WSe2. Nat. Commun. 9, 1–6 (2018).

    ADS  Google Scholar 

  61. 61.

    Sun, D. et al. Observation of rapid exciton–exciton annihilation in monolayer molybdenum disulfide. Nano Lett. 14, 5625–5629 (2014).

    ADS  Google Scholar 

  62. 62.

    Yu, Y. et al. Fundamental limits of exciton–exciton annihilation for light emission in transition metal dichalcogenide monolayers. Phys. Rev. B 93, 201111(R) (2016).

    ADS  Google Scholar 

  63. 63.

    Linardy, E. et al. Harnessing exciton–exciton annihilation in two-dimensional semiconductors. Nano Lett. 20, 1647–1653 (2020).

    ADS  Google Scholar 

  64. 64.

    Fröhlich, H. Electrons in lattice fields. Adv. Phys. 3, 325–361 (1954).

    ADS  MATH  Google Scholar 

  65. 65.

    Sohier, T., Calandra, M. & Mauri, F. Two-dimensional Fröhlich interaction in transition-metal dichalcogenide monolayers: theoretical modeling and first-principles calculations. Phys. Rev. B 94, 085415 (2016).

    ADS  Google Scholar 

  66. 66.

    Devreese, J. T. Polarons. Encycl. Appl. Phys. 14, 383–409 (1996).

    Google Scholar 

  67. 67.

    Sarkar, S. et al. Polaronic trions at the MoS2/SrTiO3 interface. Adv. Mater. 31, 1903569 (2019).

    Google Scholar 

  68. 68.

    Kittel, C. Quantum Theory of Solids (Wiley, 1987).

  69. 69.

    Trushin, M. et al. Evidence of rotational Fröhlich coupling in polaronic trions. Phys. Rev. Lett. 125, 086803 (2020).

    ADS  Google Scholar 

  70. 70.

    Rivera, P. et al. Interlayer valley excitons in heterobilayers of transition metal dichalcogenides. Nat. Nanotechnol. 13, 1004–1015 (2018).

    ADS  Google Scholar 

  71. 71.

    Tran, K. et al. Evidence for moiré excitons in van der Waals heterostructures. Nature 567, 71–75 (2019).

    ADS  Google Scholar 

  72. 72.

    Seyler, K. L. et al. Signatures of moiré-trapped valley excitons in MoSe2/WSe2 heterobilayers. Nature 567, 66–70 (2019).

    ADS  Google Scholar 

  73. 73.

    Jin, C. et al. Observation of moiré excitons in WSe2/WS2 heterostructure superlattices. Nature 567, 76–80 (2019).

    ADS  Google Scholar 

  74. 74.

    Alexeev, E. M. et al. Resonantly hybridized excitons in moiré superlattices in van der Waals heterostructures. Nature 567, 81–86 (2019).

    ADS  Google Scholar 

  75. 75.

    Danovich, M. et al. Localized interlayer complexes in heterobilayer transition metal dichalcogenides. Phys. Rev. B 97, 195452 (2018).

    ADS  Google Scholar 

  76. 76.

    Ruiz-Tijerina, D. A. & Fal’ko, V. I. Interlayer hybridization and moiré superlattice minibands for electrons and excitons in heterobilayers of transition-metal dichalcogenides. Phys. Rev. B 99, 125424 (2019).

    ADS  Google Scholar 

  77. 77.

    Rivera, P. et al. Observation of long-lived interlayer excitons in monolayer MoSe2–WSe2 heterostructures. Nat. Commun. 6, 1–6 (2015).

    Google Scholar 

  78. 78.

    Fogler, M., Butov, L. & Novoselov, K. High-temperature superfluidity with indirect excitons in van der Waals heterostructures. Nat. Commun. 5, 1–5 (2014).

    Google Scholar 

  79. 79.

    Berman, O. L. & Kezerashvili, R. Y. Superfluidity of dipolar excitons in a transition metal dichalcogenide double layer. Phys. Rev. B 96, 094502 (2017).

    ADS  Google Scholar 

  80. 80.

    Wang, Z. et al. Evidence of high-temperature exciton condensation in two-dimensional atomic double layers. Nature 574, 76–80 (2019).

    ADS  Google Scholar 

  81. 81.

    Arp, T. B., Pleskot, D., Aji, V. & Gabor, N. M. Electron–hole liquid in a van der Waals heterostructure photocell at room temperature. Nat. Photonics 13, 245–250 (2019).

    ADS  Google Scholar 

  82. 82.

    Tartakovskii, A. Excitons in 2D heterostructures. Nat. Rev. Phys. 2, 8–9 (2020).

    Google Scholar 

  83. 83.

    Shi, H., Pan, H., Zhang, Y.-W. & Yakobson, B. I. Quasiparticle band structures and optical properties of strained monolayer MoS2 and WS2. Phys. Rev. B 87, 155304 (2013).

    ADS  Google Scholar 

  84. 84.

    Laturia, A., Van de Put, M. L. & Vandenberghe, W. G. Dielectric properties of hexagonal boron nitride and transition metal dichalcogenides: from monolayer to bulk. NPJ 2D Mater. Appl. 2, 1–7 (2018).

    Google Scholar 

  85. 85.

    Mak, K. F., He, K., Shan, J. & Heinz, T. F. Control of valley polarization in monolayer MoS2 by optical helicity. Nat. Nanotechnol. 7, 494–498 (2012).

    ADS  Google Scholar 

  86. 86.

    Cao, T. et al. Valley-selective circular dichroism of monolayer molybdenum disulphide. Nat. Commun. 3, 1–5 (2012).

    Google Scholar 

  87. 87.

    Sallen, G. et al. Robust optical emission polarization in MoS2 monolayers through selective valley excitation. Phys. Rev. B 86, 081301 (2012).

    ADS  Google Scholar 

  88. 88.

    Zeng, H., Dai, J., Yao, W., Xiao, D. & Cui, X. Valley polarization in MoS2 monolayers by optical pumping. Nat. Nanotechnol. 7, 490–493 (2012).

    ADS  Google Scholar 

  89. 89.

    Glazov, M. et al. Exciton fine structure and spin decoherence in monolayers of transition metal dichalcogenides. Phys. Rev. B 89, 201302 (2014).

    ADS  Google Scholar 

  90. 90.

    Selig, M. et al. Excitonic linewidth and coherence lifetime in monolayer transition metal dichalcogenides. Nat. Commun. 7, 1–6 (2016).

    ADS  Google Scholar 

  91. 91.

    Plechinger, G. et al. Trion fine structure and coupled spin–valley dynamics in monolayer tungsten disulfide. Nat. Commun. 7, 1–9 (2016).

    Google Scholar 

  92. 92.

    Liu, X. et al. Strong light–matter coupling in two-dimensional atomic crystals. Nat. Photonics 9, 30 (2015).

    ADS  Google Scholar 

  93. 93.

    Flatten, L. C. et al. Room-temperature exciton-polaritons with two-dimensional WS2. Sci. Rep. 6, 33134 (2016).

    ADS  Google Scholar 

  94. 94.

    Liu, X. et al. Control of coherently coupled exciton polaritons in monolayer tungsten disulphide. Phys. Rev. Lett. 119, 027403 (2017).

    ADS  Google Scholar 

  95. 95.

    Cuadra, J. et al. Observation of tunable charged exciton polaritons in hybrid monolayer WS2–plasmonic nanoantenna system. Nano Lett. 18, 1777–1785 (2018).

    ADS  Google Scholar 

  96. 96.

    Hu, F. et al. Imaging exciton-polariton transport in MoSe2 waveguides. Nat. Photonics 11, 356–360 (2017).

    ADS  Google Scholar 

  97. 97.

    Chen, Y.-J., Cain, J. D., Stanev, T. K., Dravid, V. P. & Stern, N. P. Valley-polarized exciton-polaritons in a monolayer semiconductor. Nat. Photonics 11, 431 (2017).

    ADS  Google Scholar 

  98. 98.

    Sun, Z. et al. Optical control of room-temperature valley polaritons. Nat. Photonics 11, 491 (2017).

    Google Scholar 

  99. 99.

    Dufferwiel, S. et al. Valley-addressable polaritons in atomically thin semiconductors. Nat. Photonics 11, 497–501 (2017).

    Google Scholar 

  100. 100.

    Dhara, S. et al. Anomalous dispersion of microcavity trion-polaritons. Nat. Phys. 14, 130–133 (2018).

    Google Scholar 

  101. 101.

    Qiu, D. Y., da Jornada, F. H. & Louie, S. G. Optical spectrum of MoS2: many-body effects and diversity of exciton states. Phys. Rev. Lett. 111, 216805 (2013).

    ADS  Google Scholar 

  102. 102.

    Love, A. E. H. XVI. The small free vibrations and deformation of a thin elastic shell. Phil. Trans. R. Soc. Lond. A 179, 491–546 (1888).

  103. 103.

    Jiang, J.-W., Wang, B.-S., Wang, J.-S. & Park, H. S. A review on the flexural mode of graphene: lattice dynamics, thermal conduction, thermal expansion, elasticity and nanomechanical resonance. J. Phys. Condens. Matter 27, 083001 (2015).

    ADS  Google Scholar 

  104. 104.

    Mariani, E. & Von Oppen, F. Flexural phonons in free-standing graphene. Phys. Rev. Lett. 100, 076801 (2008).

    ADS  Google Scholar 

  105. 105.

    Favron, A. et al. Photooxidation and quantum confinement effects in exfoliated black phosphorus. Nat. Mater. 14, 826–832 (2015).

    ADS  Google Scholar 

  106. 106.

    Song, Q. et al. Physical origin of Davydov splitting and resonant Raman spectroscopy of Davydov components in multilayer MoTe2. Phys. Rev. B 93, 115409 (2016).

    ADS  Google Scholar 

  107. 107.

    Zhao, Y. et al. Interlayer breathing and shear modes in few-trilayer MoS2 and WSe2. Nano Lett. 13, 1007–1015 (2013).

    ADS  Google Scholar 

  108. 108.

    Luo, X. et al. Large frequency change with thickness in interlayer breathing modes — significant interlayer interactions in few layer black phosphorus. Nano Lett. 15, 3931–3938 (2015).

    ADS  Google Scholar 

  109. 109.

    Lui, C. H. & Heinz, T. F. Measurement of layer breathing mode vibrations in few-layer graphene. Phys. Rev. B 87, 121404 (2013).

    ADS  Google Scholar 

  110. 110.

    Shang, J. et al. Observation of low-wavenumber out-of-plane optical phonon in few-layer graphene. J. Raman Spectrosc. 44, 70–74 (2013).

    ADS  Google Scholar 

  111. 111.

    Tan, P. et al. The shear mode of multilayer graphene. Nat. Mater. 11, 294–300 (2012).

    ADS  Google Scholar 

  112. 112.

    Ling, X. et al. Low-frequency interlayer breathing modes in few-layer black phosphorus. Nano Lett. 15, 4080–4088 (2015).

    ADS  Google Scholar 

  113. 113.

    Zhang, X. et al. Raman spectroscopy of shear and layer breathing modes in multilayer MoS2. Phys. Rev. B 87, 115413 (2013).

    ADS  Google Scholar 

  114. 114.

    Zhang, X. et al. Phonon and Raman scattering of two-dimensional transition metal dichalcogenides from monolayer, multilayer to bulk material. Chem. Soc. Rev. 44, 2757–2785 (2015).

    Google Scholar 

  115. 115.

    Liang, L. et al. Low-frequency shear and layer-breathing modes in Raman scattering of two-dimensional materials. ACS Nano 11, 11777–11802 (2017).

    Google Scholar 

  116. 116.

    Bunch, J. S. et al. Impermeable atomic membranes from graphene sheets. Nano Lett. 8, 2458–2462 (2008).

    ADS  Google Scholar 

  117. 117.

    Garcia-Sanchez, D. et al. Imaging mechanical vibrations in suspended graphene sheets. Nano Lett. 8, 1399–1403 (2008).

    ADS  Google Scholar 

  118. 118.

    Song, X., Oksanen, M., Li, J., Hakonen, P. & Sillanpää, M. A. Graphene optomechanics realized at microwave frequencies. Phys. Rev. Lett. 113, 027404 (2014).

    ADS  Google Scholar 

  119. 119.

    De Alba, R. et al. Tunable phonon-cavity coupling in graphene membranes. Nat. Nanotechnol. 11, 741 (2016).

    ADS  Google Scholar 

  120. 120.

    Kaasbjerg, K., Thygesen, K. S. & Jacobsen, K. W. Phonon-limited mobility in n-type single-layer MoS2 from first principles. Phys. Rev. B 85, 115317 (2012).

    ADS  Google Scholar 

  121. 121.

    Sohier, T., Calandra, M. & Mauri, F. Density functional perturbation theory for gated two-dimensional heterostructures: theoretical developments and application to flexural phonons in graphene. Phys. Rev. B 96, 075448 (2017).

    ADS  Google Scholar 

  122. 122.

    Halperin, B. I. On the Hohenberg–Mermin–Wagner theorem and its limitations. J. Stat. Phys. 175, 521–529 (2019).

    ADS  MathSciNet  MATH  Google Scholar 

  123. 123.

    Vaz, C. A. F., Bland, J. A. C. & Lauhoff, G. Magnetism in ultrathin film structures. Rep. Prog. Phys. 71, 056501 (2008).

    ADS  Google Scholar 

  124. 124.

    Skomski, R. & Coey, J. M. D. Permanent Magnetism (CRC, 1999).

  125. 125.

    Cortie, D. L. et al. Two-dimensional magnets: forgotten history and recent progress towards spintronic applications. Adv. Funct. Mater. 1901414 (2019).

  126. 126.

    O’Brien, W. & Tonner, B. Room-temperature magnetic phases of Fe on fcc Co(001) and Ni(001). Phys. Rev. B 52, 15332–15340 (1996).

    ADS  Google Scholar 

  127. 127.

    Gong, C. & Zhang, X. Two-dimensional magnetic crystals and emergent heterostructure devices. Science 363, eaav4450 (2019).

    Google Scholar 

  128. 128.

    Huang, B. et al. Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit. Nature 546, 270 (2017). One of the earliest observations of ferromagnetism in a van der Waals crystal down to the monolayer limit.

    ADS  Google Scholar 

  129. 129.

    Jin, W. et al. Raman fingerprint of two terahertz spin wave branches in a two-dimensional honeycomb Ising ferromagnet. Nat. Commun. 9, 5122 (2018).

    ADS  Google Scholar 

  130. 130.

    Gong, C. et al. Discovery of intrinsic ferromagnetism in two-dimensional van der Waals crystals. Nature 546, 265 (2017).

    ADS  Google Scholar 

  131. 131.

    Lee, J.-U. et al. Ising-type magnetic ordering in atomically thin FePS3. Nano Lett. 16, 7433–7438 (2016).

    ADS  Google Scholar 

  132. 132.

    Lado, J. L. & Fernández-Rossier, J. On the origin of magnetic anisotropy in two dimensional CrI3. 2D Mater. 4, 035002 (2017).

    Google Scholar 

  133. 133.

    Pershoguba, S. S. et al. Dirac magnons in honeycomb ferromagnets. Phys. Rev. X 8, 011010 (2018).

    Google Scholar 

  134. 134.

    Fransson, J., Black-Schaffer, A. M. & Balatsky, A. V. Magnon Dirac materials. Phys. Rev. B 94, 075401 (2016).

    ADS  Google Scholar 

  135. 135.

    Chen, L. et al. Topological spin excitations in honeycomb ferromagnet CrI3. Phys. Rev. X 8, 041028 (2018).

    Google Scholar 

  136. 136.

    Skyrme, T. H. R. in Selected Papers, with Commentary, of Tony Hilton Royle Skyrme, 195–206 (World Scientific, 1994).

  137. 137.

    Brey, L., Fertig, H. A., Côté, R. & MacDonald, A. H. in Strongly Correlated Magnetic and Superconducting Systems (eds Sierra, G. & Martín-Delgado, M. A.) 275–283 (Springer, 1997).

  138. 138.

    Fert, A., Cros, V. & Sampaio, J. Skyrmions on the track. Nat. Nanotechnol. 8, 152–156 (2013).

    ADS  Google Scholar 

  139. 139.

    Kovalev, A. A. & Sandhoefner, S. Skyrmions and antiskyrmions in quasi-two-dimensional magnets. Front. Phys. 6, 98 (2018).

    Google Scholar 

  140. 140.

    Yu, X. et al. Transformation between meron and skyrmion topological spin textures in a chiral magnet. Nature 564, 95 (2018).

    ADS  Google Scholar 

  141. 141.

    Banerjee, S., Rowland, J., Erten, O. & Randeria, M. Enhanced stability of skyrmions in two-dimensional chiral magnets with Rashba spin–orbit coupling. Phys. Rev. X 4, 031045 (2014). Theoretical prediction of the stability region for skyrmions in 2D materials, clarifying its relation with spin–orbit coupling.

    Google Scholar 

  142. 142.

    Sorokin, A. Critical density of topological defects upon a continuous phase transition. Ann. Phys. 411, 167952 (2019).

    MathSciNet  Google Scholar 

  143. 143.

    Seixas, L., Rodin, A., Carvalho, A. & Castro Neto, A. Multiferroic two-dimensional materials. Phys. Rev. Lett. 116, 206803 (2016).

    ADS  Google Scholar 

  144. 144.

    Sharpe, A. L. et al. Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene. Science 365, 605–608 (2019).

    ADS  Google Scholar 

  145. 145.

    Schlottmann, P. Commensurate and incommensurate spin-density waves and the superconductivity dome in heavy electron systems. Phys. Rev. B 92, 045115 (2015).

    ADS  Google Scholar 

  146. 146.

    Tonndorf, P. et al. Photoluminescence emission and Raman response of monolayer MoS2, MoSe2, and WSe2. Opt. Express 21, 4908–4916 (2013).

    ADS  Google Scholar 

  147. 147.

    Lui, C. H., Ye, Z., Keiser, C., Xiao, X. & He, R. Temperature-activated layer-breathing vibrations in few-layer graphene. Nano Lett. 14, 4615–4621 (2014).

    ADS  Google Scholar 

  148. 148.

    Woo, S. Elusive spin textures discovered. Nature 564, 43–44 (2018).

Download references


This work was supported by the National Research Foundation, Prime Minister Office, Singapore, under its Medium Sized Centre Programme and CRP award ‘Novel 2D materials with tailored properties: beyond graphene’ (grant no. R-144-000-295-281). A.R. acknowledges support by Yale-NUS College (through grant no. R-607-265-380-121). M.T. is supported by the Director’s Senior Research Fellowship from the Centre for Advanced 2D Materials.

Author information




The authors contributed equally to all aspects of the article.

Corresponding author

Correspondence to A. H. Castro Neto.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information

Nature Reviews Physics thanks Ursula Wurstbauer and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


Bessel and Struve functions

These are solutions of Bessel’s differential equation naturally occurring in problems with cylindrical symmetry.

Wannier–Mott picture

This is a model that assumes the exciton radius (or, in general, the mean electron–hole distance in exciton-like quasiparticles) to be much larger than the lattice constant, allowing for the effective mass description.

Fröhlich interactions

Interactions that couple electrons with phonons, assuming the continuum approximation and long-range forces.

Moiré potential

A potential resulting from a superposition of two periodic potentials with slightly different lattice constants or misalignment, resulting in a pattern with a much larger periodicity.

Ising model

A statistical model in which spin states at lattice sites are represented by discrete variables (‘up’ or ‘down’).

Heisenberg model

A statistical model in which the atomic spins are represented by vectors (classical Heisenberg model) or by spin or orbital angular momentum operators or their respective matrix representations (quantum Heisenberg model).

Dzyaloshinskii–Moriya interaction

A type of antisymmetric exchange interaction between neighbouring spins.

Domain walls

The boundaries between domains with different orientation of magnetization or other order parameter.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Rodin, A., Trushin, M., Carvalho, A. et al. Collective excitations in 2D materials. Nat Rev Phys 2, 524–537 (2020).

Download citation


Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing