Abstract
We consider a system consists of a doped monolayer phosphorene embedded between two hexagonal boron nitride (hBN) slabs along the heterostructure direction. The wavevector azimuthal angle dependence of the plasmon-polariton and plasmon-phonon-polariton modes of the hybrid system are calculated based on the random-phase approximation at finite temperature. The collective modes illustrate strong anisotropy and strong coupling with phonon modes of the polar media, and furthermore, the Landau damping occurs due to the intraband processes when plasmon enters intraband electron-hole continuum. Our numerical results show that the plasmon mode is highly confined to the surface along the zigzag direction. Owing to the strong electron-phonon interaction, the phonon dispersions in the Reststrahlen bands are also angle-dependent. These results are also in agreement with those of the semiclassical model obtained in our calculations.
Similar content being viewed by others
References
Bridgmann PW (1914) Two new modifications of phosphorus. J Am Chem Soc 36(7):1344–1363
Keyes RW (1953) The electrical properties of black phosphorus. Phys Rev 92:580–584
Asahina H, Morita A (1984) Band structure and optical properties of black phosphorus. J Phys C: Solid State Phys 17(11):1839–1852
Sugai S, Shirotani I (1985) Raman and infrared reflection spectroscopy in black phosphorus. Solid State Commun 53(9):753–755
Liu H, Neal AT, Zhu Z, Luo Z, Xu X, Tománek D, Ye PD (2014) Phosphorene: an unexplored 2D semiconductor with a high hole Mobility. ACS Nano 8(4):4033–4041
Li L, Yu Y, Ye GJ, Ge Q, Ou X, Wu H, Feng D, Chen XH, Zhang Y (2014) Black phosphorus field-effect transistors. Nature Nanotechnol 9:372–377
Tran V, Soklaski R, Liang Y, Yang L (2014) Layer-controlled band gap and anisotropic excitons in few-layer black phosphorus. Phys Rev B 89:235319
Xia F, Wang H, Xiao D, Dubey M, Ramasubramaniam A (2014) Two-dimensional material nanophotonics. Nat Photonics 8:899–907
Qiao J, Kong X, Hu ZX, Yang F, Ji W (2014) High-mobility transport anisotropy and linear dichroism in few-layer black phosphorus. Nature Commun 5:4475
Garcia de Abajo JF (2014) Graphene plasmonics: challenges and opportunities. ACS Photonics 1(3):135–152
Jablan M, Soljačić M, Buljan H (2013) Plasmons in graphene: fundamental properties and potential applications. Proc IEEE 101(7):1689–1704
Gonçalves PAD, Peres NMR (2016) An introduction to graphene plasmonics. World Scientific Publishing, Singapore
Tame MS, McEnery KR, Özdemir SK, Lee J, Maier SA, Kim MS (2013) Quantum plasmonics. Nature Phys 9:329–340
Fei Z, Rodin AS, Andreev GO, Bao W, McLeod AS, Wagner M, Zhang LM, Zhao Z, Thiemens M, Dominguez G, Fogler MM, Castro Neto AH, Lau CN, Keilmann F, Basov DN (2012) Gate-tuning of graphene plasmons revealed by infrared nano-imaging. Nature 487:82–85
Chen J, Badioli M, Alonso-González P, Thongrattanasiri S, Huth F, Osmond J, Spasenović M, Centeno A, Pesquera A, Godignon P, Elorza AZ, Camara N, García de Abajo FJ, Hillenbrand R, Koppens FHL (2012) Optical nano-imaging of gate-tunable graphene plasmons. Nature 487:77–81
Poddubny A, Iorsh I, Belov P, Kivshar Y (2013) Hyperbolic metamaterials. Nat Photonics 7:948–957
Caldwell JD, Kretinin AV, Chen Y, Giannini V, Fogler MM, Francescato Y, Ellis CT, Tischler JG, Woods CR, Giles AJ, Hong M, Watanabe K, Taniguchi T, Maier SA, Novoselov KS (2014) Sub-diffractional volume-confined polaritons in the natural hyperbolic material hexagonal boron nitride. Nature Commun 5:5221
Mahan GD (2000) Many-particle physics, 3rd edn. Springer, New York
Mooradian A, Wright GB (1966) Observation of the interaction of plasmons with longitudinal optical phonons in GaAs. Phys Rev Let 16:999
Liu Y, Willis RF (2010) Plasmon-phonon strongly coupled mode in epitaxial graphene. Phys Rev B 81:081406(R)
Koch RJ, Seyller T, Schaefer JA (2010) Strong phonon-plasmon coupled modes in the graphene/silicon carbide heterosystem. Phys Rev B 82:201413(R)
Luxmoore IJ, Gan CH, Liu PQ, Valmorra F, Li P, Faist J, Nash GR (2014) Strong coupling in the far-infrared between graphene plasmons and the surface optical phonons of silicon dioxide. ACS Photonics 1 (11):1151–1155
Dai S, Ma Q, Liu M, Andersen T, Fei Z, Goldflam MD, Wagner M, Watanabe K, Taniguchi T, Thiemens M, Keilmann F, Janssen GCAM, Zhu S -E, Jarillo-Herrero P, Fogler MM, Basov DN (2015) Graphene on hexagonal boron nitride as a tunable hyperbolic metamaterial. Nature Nanotechnol 10:682–686
Tomadin A, Principi A, Song JCW, Levitov LS, Polini M (2015) Accessing phonon polaritons in hyperbolic crystals by angle-resolved photoemission spectroscopy. Phys Rev Lett 115: 087401
Dai S, Fei Z, Ma Q, Rodin AS, Wagner M, McLeod AS, Liu M, Gannett W, Regan W, Watanabe K, Taniguchi T, Thiemens M, Dominguez G, Castro Neto AH, Zettl A, Keilmann F, Jarillo-Herrero P, Fogler MM, Basov DN (2014) Tunable phonon polaritons in atomically thin Van der Waals crystals of boron nitride. Science 343(6175):1125–1129
Brar VW, Jang MS, Sherrott M, Kim S, Lopez JJ, Kim LB, Choi M, Atwater H (2014) Hybrid surface-phonon-plasmon polariton modes in graphene/monolayer h-BN heterostructures. Nano Lett 14(7):3876–388
Zhou Y, Qi H, Wang Y, Qi DX, Hu Q (2018) Curving h-BN thin films can create extra phonon polariton modes. Opt Lett 43(7):1459–1462
Zhao ZW, Wu HW, Zhou Y (2016) Surface-confined edge phonon polaritons in hexagonal boron nitride thin films and nanoribbons. Opt Exp 24(20):22930–22942
Woessner A, Lundeberg MB, Gao Y, Principi A, Alonso-González P, Carrega M, Watanabe K, Taniguchi T, Vignale G, Polini M, Hone J, Hillenbrand R, Koppens FHL (2015) Highly confined low-loss plasmons in graphene–boron nitride heterostructures. Nat Mater 14:421–425
Xu XG, Jiang JH, Gilburd L, Rensing RG, Burch KS, Zhi C, Bando Y, Golberg D, Walker GC (2014) Mid-infrared polaritonic coupling between boron nitride nanotubes and graphene. ACS Nano 8(11):11305–1
Kumar A, ow T, Fung KH, Avouris P, Fang NX (2015) Tunable light–matter interaction and the role of hyperbolicity in graphene–hBN system. Nano Lett 15:3172
Xu XG, Ghamsari BG, Jiang JH, Gilburd L, Andreev GO, Zhi C, Bando Y, Golberg D, Berini P, Walker GC (2014) One-dimensional surface phonon polaritons in boron nitride nanotubes. Nat Commun 5:478
Dai S, Ma Q, Andersen T, Mcleod AS, Fei Z, Liu MK, Wagner M, Watanabe K, Taniguchi T, Thiemens M, Keilmann F, Jarillo-Herrero P, Fogler MM, Basov DN (2015) Subdiffractional focusing and guiding of polaritonic rays in a natural hyperbolic material. Nature Commun 6:6963
Li P, Lewin M, Kretinin AV, Caldwell JD, Novoselov KS, Taniguchi T, Watanabe K, Gaussmann F, Taubner T (2015) Hyperbolic phonon-polaritons in boron nitride for near-field optical imaging and focusing. Nature Commun 6:7507
Elahi M, Khaliji K, Tabatabaei SM, Pourfath M, Asgari R (2015) Modulation of electronic and mechanical properties of phosphorene through strain. Phys Rev B 91:115412
Zare M, Rameshti BZ, Ghamsari FG, Asgari R (2017) Thermoelectric transport in monolayer phosphorene. Phys Rev B 95:045422
Ezawa M (2014) Topological origin of quasi-flat edge band in phosphorene. New J Phys 115004:16
Rudenko AN, Katsnelson MI (2014) Quasiparticle band structure and tight-binding model for single- and bilayer black phosphorus. Phys Rev B 89:201408(R)
Rodin AS, Carvalho A, Castro Neto AH (2014) Strain-induced gap modification in black phosphorus. Phys Rev Lett 112:176801
Low T, Roldán R, Wang H, Xia F, Avouris P, Moreno LM, Guinea F (2014) Plasmons and screening in monolayer and multilayer black phosphorus. Phys Rev Lett 113:106802
Lundeberg MB, Gao Y, Asgari R, Tan C, Duppen BV, Autore M, Alonso-Gonzalez P, Woessner A, Watanabe K, Taniguchi T, Hillenbrand R, Hone J, Polini M, Koppens FHL (2017) Tuning quantum nonlocal effects in graphene plasmonics. Science 357(6347):187–191
Giuliani GF, Vignale G (2005) Quantum theory of the electron liquid. Cambridge University Press, Cambridge
Ashcroft NW, Mermin ND (1976) Solid state physics. Harcourt College Publishers, Orlando
Torbatian Z, Asgari R (2018) Optical absorption properties of few-layer phosphorene. Phys Rev B 98:205407. Note that in Fig. 4 we use the masses obtained by DFT results, mainly the energy gap is 0.98 eV and the masses are mcx = 0.17, mcy = 1.12, mvx = 0.15, and mvy = 6.35 in units of the electron bare mass
Our numerical results show that the plasmon mode can be fitted with \(\sqrt {2{\pi } n e^{2} q/m_{c x}}\) very well along the armchair and with \(\sqrt {2{\pi } n e^{2} q/m_{c y}}\) quite good along the zigzag direction at the long-wavelength limit
Torbatian Z, Asgari R (2018) Plasmonic physics of 2D crystalline materials. Appl Sci 8(2):238
Badioli M, Woessner A, Tielrooij KJ, Nanot S, Navickaite G, Stauber T, Garcia de Abajo FJ, Koppens FHL (2014) Phonon-mediated mid-infrared photoresponse of graphene. Nano Lett 14(11):6374–6381
Gramotnev DK, Bozhevolnyi SI (2010) Plasmonics beyond the diffraction limit. Nat Photonics 4:83–91
Acknowledgments
We thank M. Polini for very useful discussions. This work is partially supported by Iran Science Elites Federation. R. A acknowledges Scuola Normale Superiore, Pisa for its hospitality during the period in which the final stage of this work is completed.
Funding
This work is partially supported by Iran Science Elites Federation.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Ghohroodi Ghamsari, F., Asgari, R. Plasmon-Phonon-Polaritons in Encapsulated Phosphorene. Plasmonics 15, 1289–1304 (2020). https://doi.org/10.1007/s11468-019-01059-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11468-019-01059-9