Research Paper: Calculation of Surface Plasmons of Two Closely Metallic Nanospheres Using the Hydrodynamic Model

Document Type : Research Paper

Author

Assistant Professor, Department of Physics, Faculty of Science, University of Zabol (UOZ), Zabol, Sistan and Baluchestan, Iran

Abstract

In this study, The surface plasmon excitations in a dimer structure of closely spaced metallic nanospheres with different radii are investigated using the hydrodynamic model. Initially, an expression was derived to calculate this structure's multipolar surface plasmon excitations. Subsequently, the surface plasmon excitations in the dipole approximation were examined.The surface plasmon excitations in a dimer structure of closely spaced metallic nanospheres with different radii are examined using the hydrodynamic model in this study. It has been observed that the energy of in-phase modes is lower than that of out-of-phase modes in longitudinal or transverse excitations. Furthermore, at each separation distance between the nanospheres, the energy difference between in-phase and out-of-phase longitudinal modes is greater than that of transverse modes. The results show that with an increase in the separation distance between the nanospheres, the energies of in-phase modes increase, and the energies of out-of-phase modes decrease. At large separation distances, two plasmon modes are obtained, with the higher energy mode corresponding to the smaller nanosphere and the lower energy mode corresponding to the larger nanosphere. Finally, the local limit results are presented for comparison.

Keywords


[1] Zayats A. V., Smolyaninov I. I, and Maradudin A. A., "Nano-optics of surface plasmon polaritons", Physics Reports, 408, 131-314, 2005. https://doi.org/10.1016/j.physrep.2004.11.001
[2] Maier.S. A., "Plasmonics: Fundamentals and Applications", Springer, New York, 1, 2007. https://doi.org/10.1007/0-387-37825-1  
[3] Moradi A., "Canonical Problems in the Theory of Plasmonics", Springer Cham, Switzerland AG ,1 ,2020. https://doi.org/10.1007/978-3-030-43836-4
[4] Kravets V. G., Kabashin A. V., Barnes W. L., and Grigorenko A. N., "Plasmonic Surface Lattice Resonances: A Review of Properties and Applications", Chem. Rev., 118, 5912–51, 2018. https://doi.org/10.1021/acs.chemrev.8b00243 
[5] Davis T. J., Gómez D. E., and Roberts A., "Plasmonic circuits for manipulating optical information", Nanophotonics, 6, 543–59, 2016. https://doi.org/10.1515/nanoph-2016-0131
[6] Devaraj V, Lee J. M., Kim Y. J., Jeong H., and Oh J. W., "Engineering Efficient Self-Assembled Plasmonic Nanostructures by Configuring Metallic Nanoparticle’s Morphology", Int. J. Mol. Sci., 22, 10595, 2021. https://doi.org/10.3390/ijms221910595 
[7] Crespilho F. N., "Advances in Bioelectrochemistry Volume 5", Springer Cham, Switzerland AG ,1, 2023. https://doi.org/10.1007/978-3-031-10832-7
[8] Lee J.-H., Cho H.-Y., Choi H. K., Lee J.-Y., Choi J.-W., "Application of Gold Nanoparticle to Plasmonic Biosensors", Int. J. Mol. Sci., 19(7), 2021, 2018.  https://doi.org/10.3390/ijms19072021
[9] Brongersma M. L, Hartman J. W., and Atwater H. A., "Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit", Phys. Rev. B, 62, R16356–9, 2000. https://doi.org/10.1103/PhysRevB.62.R16356
[10] Mir M., "Spatial nonlocality effect on the surface plasmon propagation in plasmonic nanospheres waveguide", J. Phys.: Condens. Matter, 35, 205301, 2023. https://doi.org/10.1088/1361-648X/acc15f
[11] Liu H., Ng J., Wang S. B., Hang Z. H., and Zhu S. N., "Strong plasmon coupling between two gold nanospheres on a gold slab", New J. Phys., 13, 073040, 2011. https://doi.org/10.1088/1367-2630/13/7/073040  
[12] Lerch S., and Reinhard B. M., "Effect of interstitial palladium on plasmon-driven charge transfer in nanoparticle dimers", Nat. Commun., 9, 1608-2, 2018. https://doi.org/10.1038/s41467-018-04066-2
[13] Armstrong R. E., Horáček M., and Zijlstra P., "Plasmonic Assemblies for Real-Time Single-Molecule Biosensing", Small, 16, 2003934, 2020. https://doi.org/10.1002/smll.202003934
[14] Liu Y. and Zhang X., "Microfluidics-Based Plasmonic Biosensing System Based on Patterned Plasmonic Nanostructure Arrays", Micromachines, 12, 826, 2021. https://doi.org/10.3390/mi12070826
[15] Tingting Y., Liyong J., and Zexiang S., "Recent progress on photoluminescence from plasmonic nanostructures: Phenomenon, mechanism, and application", Chinese Physics B, 27(9), 097803, 2018. https://doi.org/10.1088/1674-1056/27/9/097803
[16] Jiang N., Zhu T., and Hu Y., "Competitive aptasensor with gold nanoparticle dimers and magnetite nanoparticles for SERS-based determination of thrombin", Microchim Acta, 186, 747 2019. https://doi.org/10.1007/s00604-019-3787-9
[17] Abajo F. J. G., "Nonlocal Effects in the Plasmons of Strongly Interacting Nanoparticles, Dimers, and Waveguides", J. Phys. Chem. C, 112, 17983–17987, 2008. https://doi.org/10.1021/jp807345h
[18] David. C. and Abajo. F. J. G., "Spatial Nonlocality in the Optical Response of Metal Nanoparticles", J. Phys. Chem. C, 115, 19470–19475, 2011. https://doi.org/10.1021/jp204261u
[19] Raza S, Bozhevolnyi S. I., Wubs M., and Mortensen N. A, "Nonlocal optical response in metallic nanostructures", J. Phys.: Condens. Matter, 27, 183204, 2015. https://doi.org/10.1088/0953-8984/27/18/183204
[20] Krall N. A. and Trivelpiece A. w., "Principles of Plasma Physics", IEEE, New York, 1, 1974. https://doi.org/10.1109/TPS.1974.4316834
[21] Wubs M., "Classification of scalar and dyadic nonlocal optical response models", Opt. Express, 23, 31296-31312, 2015. https://doi.org/10.1364/OE.23.031296
[22] Pozhela. J. and Germogenova. O. A., "Plasma and Current Instabilities in Semiconductors", Pergamon Press, Oxford, 1, 1981. https://doi.org/10.1016/C2013-0-05933-6
[23] Lucas A. A. and Ronveaux A., "Van der Waals energy between voids in dielectrics", Phys. Rev. B, 12, 5372, 1975. https://doi.org/10.1103/PhysRevB.12.5372
[24] Ruppin. R., "Surface modes of two spheres", Phys. Rev. B, 26, 3440, 1982. https://doi.org/10.1103/PhysRevB.26.3440
[25] Clippe P., Evrard R., and Lucas A. A., "Aggregation effect on the infrared absorption spectrum of small ionic crystals", Phys. Rev. B, 14, 1715–1721, 1976. https://doi.org/10.1103/PhysRevB.14.1715
[26] Brako R., Sunjic M., and Sips V., "Dispersion interaction between small crystals", Solid State Commun., 19, 161–164, 1975. https://doi.org/10.1016/0038-1098(76)90458-0
[27] Olivares I., Rojas R., and Claro F., "Surface modes of a pair of unequal spheres", Phys. Rev. B, 35, 2453–2455, 1987. https://doi.org/10.1103/PhysRevB.35.2453
[28] Schmeits M. and Dambly L., "Fast-electron scattering by bispherical surface-plasmon modes", Phys. Rev. B, 44, 12706–12712, 1991. https://doi.org/10.1103/PhysRevB.44.12706
[29] Nkoma J. S., "Surface modes of two spheres embedded into a third medium", Surf. Sci., 245, 207–212, 1991. https://doi.org/10.1016/0039-6028(91)90479-C
[30] Nordlander P., Oubre C., Prodan E., Li K., and Stockman M. I., "Plasmon hybridization in nanoparticle dimers", Nano Lett., 4, 899– 903, 2004. https://doi.org/10.1021/nl049681c
[31] Mortensen N. A., Raza S., Wubs M., Søndergaard T., and Bozhevolnyi S. I., "A generalized non-local optical response theory for plasmonic nanostructures", Nat. Commun., 5, 3809, 2014. https://doi.org/10.1038/ncomms4809 
[32] Tserkezis C., Mortensen N. A., and Wubs M., "How nonlocal damping reduces plasmon-enhanced fluorescence in ultranarrow gaps", Phys. Rev. B, 96, 085413, 2017. https://doi.org/10.1103/PhysRevB.96.085413  
[33] Tserkezis C., Maack J., Liu Z., Wubs M., and Mortensen N. A., "Robustness of the far-field response of nonlocal plasmonic ensembles", Sci. Rep., 6, 28441, 2016. https://doi.org/10.1038/srep28441
[34] Tserkezis C., Yan W., Hsieh W., Sun G., Khurgin J. B., Wubs M., and Mortensen N. A., "On the origin of nonlocal damping in plasmonic monomers and dimers", International Journal of Modern Physics B, 31, 24, 2017. https://doi.org/10.1142/S0217979217400057
[35] Fernández-Domínguez A. I., Wiener A., García-Vidal F. J., Maier S. A., and Pendry J. B., "Transformation-Optics Description of Nonlocal Effects in Plasmonic Nanostructures", Phys. Rev. Lett., 108, 106802, 2012. https://doi.org/10.1103/PhysRevLett.108.106802
[36] Huang K.-J., Qin S.-J., Zhang Z.-P., Ding Z., Bai Z.-C., "Nonlocal and Size-Dependent Dielectric Function for Plasmonic Nanoparticles", Appl. Sci., 9, 3083, 2019. https://doi.org/10.3390/app9153083
[37] Gasiorowicz S., "Quantum Physics", John Wiley & Sons, Inc, Hoboken, 2003. https://www.academia.edu/36604374/_Stephen_Gasiorowicz_Quantum_Physics_3rd_Ed 
[38] Priya S. and Dantham V. R, "Effect of Size-Dependent Damping on Plasmon-Hybridized Modes of Asymmetric Nanosphere Dimers: the Role of Nanogap, Size Ratio, Surrounding Medium, and Substrate", Plasmonics, 15, 2033–2042, 2020. https://doi.org/10.1007/s11468-020-01216-5
[39] Zohar N., Chuntonov. L., Haran G., "The simplest plasmonic molecules: Metal nanoparticle dimers and trimers", Journal of Photochemistry and Photobiology C: Photochemistry Reviews, 21, 26–39, 2014. https://doi.org/10.1016/j.jphotochemrev.2014.10.002
[40] Ashcroft N. W. and Mermin N. D., "Solid State Physics, Saunders College", Philadelphia, 1976. https://www.scirp.org/(S(vtj3fa45qm1ean45vvffcz55))/reference/ReferencesPapers.aspx?ReferenceID=1476522
[41] Weber W. H. and Ford G. W., "Propagation of optical excitations by dipolar interactions in metal nanoparticle chains", Phys. Rev. B, 70,125429, 2004. https://doi.org/10.1103/PhysRevB.70.125429
[42] Herrera L. J. M., Arboleda D. M., Schinca D. C., and Scaffardi L. B., "Determination of plasma frequency, damping constant, and size distribution from the complex dielectric function of noble metal nanoparticles", Journal of Applied Physics, 116, 233105, 2014. https://doi.org/10.1063/1.4904349
[43] Deng T. S., Parker J, Yifat Y., Shepherd N, and Scherer N. F., "Dark Plasmon Modes in Symmetric Gold Nanoparticle Dimers Illuminated by Focused Cylindrical Vector Beams", J. Phys. Chem. C, 122, 48, 27662–27672, 2018. https://doi.org/10.1021/acs.jpcc.8b10415