A simple unity theory for three types of polaritons (Phonon polariton, exciton polariton and plasmon polariton) expanded model for surface polaritons

104 JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2017-0037 Mathematical and Physical Sci. 2017, Vol. 62, Iss. 8, pp. 104-110 This paper is available online at A SIMPLE UNITY THEORY FOR THREE TYPES OF POLARITONS (PHONON POLARITON, EXCITON POLARITON AND PLASMON POLARITON) EXPANDED MODEL FOR SURFACE POLARITONS Duong Thi Ha 1 , Dinh Thi Thuy 2 and Nguyen Ai Viet 3 1 Thai Nguyen University of Education 2 Thai Binh University of Medicine 3 Institute of Physics, Vietnam Academy of Science and Technology Abstract. In our previous work which studied the theory of three types of polaritons, we proposed a new simple unity theory for three types of bulk polaritons (phonon polariton, exciton polariton and plasmon polariton). In this work, this theory is expanded for these three types of surface polaritons. We introduce an effective LT-splitting for surface plasmon polariton. Using this assumption, a new simple model for surface plasmon polariton with an effective LT–splitting was constructed. In this way, we obtain a unity theory for these three types of both bulk and surface polaritons: phonon, exciton and plasmon polariton. The effective interaction between photon and plasmon will be discussed. Keywords: surface phonon polariton, surface exciton polariton, surface plasmon polariton. 1. Introduction Polariton can be defined as the strong coupling of a photon with another quasiparticle. Polariton is classified in different types: phonon polariton, exciton polariton and plasmon polariton. Phonon polariton has commanded great attention both experimentally and theoretically due to the unique and well-understood frequency-dependent dielectric function. In ionic crystals, phonon polariton is derived from the interaction between crystal lattice oscillations (horizontal phonon-TO phonons) and electromagnetic waves (photons). Phonon polariton has been studied for the application of infrared spectrum prisms. In addition, the lattice of circular potential leads to the formation of the shortened Brillouin region, thus far the far-infrared Raman laser and Reststrahlen filter (Reststralen filter) can be made from AlAs/GaAs [1]. Another feature of phonon polariton is that the dramatically reduced group velocity can be used in solid-state transmitting devices. Received July 29, 2017. Accepted August 28, 2017. Contact Duong Thi Ha, e-mail: duongha@dhsptn.edu.vn A simple unity theory for three types of polaritons 105 The interaction of photons with exciton was discussed by Hopfield in 1958 who called them exciton polariton. Exciton polariton is bosons that can form condensates, a state of matter of a dilute gas of bosons cooled to temperatures very close to absolute zero, with a large fraction of bosons occuping the lowest quantum state, at which point macroscopic quantum phenomena become apparent. Because of very small effective mass, less than eight times the mass of the atom, the result is exciton polarion capable of condensing Bose-Einstein at high temperatures. For example, Bose-Einstein condensation of exciton polariton can observered at 19 K with exciton polariton in a cavity semiconductor [2]. This leads to the enormous potential of application of exciton polariton in nanotechnology especially in the ultralow threshold polariton lasers. When light interacts with a metal nano-particle, the conduction electrons are shifted by the electric field of incident light and form the collective oscillation of electrons which are called surface plasmon. At the surface plasmon wavelength, the amplitude of the electric field near the nano-particles is greatly enhanced so that the metal nano-particles have large absorption and scattering. This leads to many applications in a wide variety of fields such as SEF [3, 4], SERS [5-7], plasmonic solar cells [8, 9], nano medicine and sensing [10]. In the case of phonon and exciton polariton, both wave and particle theory have built for plasmon polariton, most works have focused on wave theory. Using particle theory, the interaction between photon and phonon or exciton can be described by the LT-splitting. In our previous work [11], we compared the similarities and differences between these three types of polaritons. Based on their similarities, we assume that there is an effective interaction between photon and bulk plasmon. We have built a unity model for bulk polaritons. The meaning of this model is thus far not clear because bulk plasmon polariton is at this time purely academic. In this work, we expand the unity model which was built of bulk polaritons for surface polaritons. We introduce a new model for surface plasmon polartion with effective longitudinal- transverse splitting which describes the interaction of photon and surface plasmon. This is a fundamental step in building a particle theory for surface plasmon polariton which is meaningful because surface plasmon polariton plays an important role and has attracted great attention in polaritonics, a new part of modern nanotechnology that bridges the gap between electronics and photonics. 2. Content 2.1. The electromagnetic theory Polariton features are usually described within classical electrodynamics by looking for solutions of Maxwell equations under appropriate boundary conditions. The full set of Maxwell’s equations in the absence of external sources can be expressed as follows: ⃗ , ⃗ ( ) ( ⃗ ), ⃗ ( ) ( ⃗ ), ⃗ , (1) (2) (3) (4) where ⃗ and ⃗ are the magnetic induction vector and the magnetic field vector, ⃗ is the electric field vector, ⃗ is the electric induction vector. We consider the incident electromagnetic wave which is represented as follows: Duong Thi Ha, Dinh Thi Thuy and Nguyen Ai Viet 106 ⃗ ⃗⃗⃗⃗ [ ( ⃗ )] ⃗ ⃗⃗ ⃗⃗ [ ( ⃗ )]. (5) (6) By replacing ⃗ , ⃗ into equation (2) and (3), we have the dispersion relation of polariton . (7) The dielectric function of the medium can be either positive or negative depend on the frequency of incident light ω. In the range of frequency where the permittivity is positive, the electromagnetic wave can propagate in the medium but in range of the frequency in wich permittivity has a negative value, the electromagnetic wave falls off exponentially and is known as evanescent wave. In this case, it is called surface polariton propagating along the interface between two media. We consider two environments which have the dielectric function and , respectively. Assuming that the electromagnetic wave propagates along the Ox axis, the electric field has two components: parallel to the Ox axis and perpendicular to the interface plane, the magnetic vector perpendicular to the Ox axis and located in the interface. The electric field and magnetic field are expressed as follows: ⃗ ( ) , ⃗ (8) (9) where the index i describes the media, i = 1, 2. Using Maxwell's equations with the continuity of the in-plane electromagnetic field components , we obtain the surface polariton dispersion relation: √ . (10) 2.2. Electromagnetic theory applied to surface phonon polariton Phonons are quantized simple harmonic oscillators. It is possible to consider the interaction of simple harmonic oscillators (SHO) with a radiation field in the form of the plane wave. Using classical electromagnetic theory, the dielectric function of a material can be written as follows: (11) where is the natural vibrational frequency of each SHO or the transversal resonance frequency, and is the longitudinal resonance frequency. We note that the dielectric is negative in the frequency range , so electromagnetic waves can’t propagate in the crystal and there exists an evanescent state in this frequency range. This state is called the surface phonon polariton. The evanescent state lies in the gap between and (LT-splitting), starts at and approaches when . The width of the LT energy gap corresponds to the longitudinal-transverse splitting and presents phonon - photon coupling. We assume that there is an evanescent on the interface between the ionic crystal having dielectric function in the form of expression (12) and vacuum with dielectric constant . The dispersion relation of surface phonon polariton can be obtained by replacing the A simple unity theory for three types of polaritons 107 dielectric function in to equation (11). We have the dispersion relation of surface phonon polariton as follows: { √[ ] }. (12) Figure 1. Dispersion of phonon polariton, black solid lines represent the dispersion relation of phonon polariton in ZnO, the red line represents the dispersion relation of surface phonon polariton on interface between ZnO and vacuum, blue dashed lines indicates the photon line 2.3. Electromagnetic theory applied to surface exciton polariton Exciton polariton is defined as a quasi-particle resulting from strong coupling of photon and exciton. It is an electrically neutral quasi-particle that exists in insulators, semiconductors and in some liquids. The permittivity of media can be expressed: (13) We can see that the permittivity is negative in the range of frequency in which the electromagnetic wave decreases exponentially forming an evanescent wave (the surface exciton polariton wave). Similar to the case of surface phonon polariton, the dispersion curve of the surface exciton polariton also starts at and closes to the line (Figure 2). Figure 2. Dispersion of exciton polariton, the dotted-black line represents the dispersion relation of the exciton polariton in the semiconductor ZnO, the black solid line represents the dispersion relation of the surface exciton polariton on the interface of ZnO, the vacuum lies in LT energy band, the blue dashed line shows the photon line. Duong Thi Ha, Dinh Thi Thuy and Nguyen Ai Viet 108 The width of the LT energy gap is the longitudinal-transverse splitting that represents the intensity of interaction between exciton and photon: . (14) 2.4. Electromagnetic theory applied to surface plasmon polariton Surface plasmon is commonly defined as coherent electron oscillations which form the surface charge density waves at the metal–dielectric interface. Surface plasmon polariton (SPP) is formed by strong coupling of surface plasmon with photons that can propagate along the metal surface until its energy is lost by absorption of metal or radiation into free space. SPP features are usually described within classical electrodynamics, by looking for solutions of Maxwell equations under appropriate boundary conditions. Application of electromagnetic theory to plasmon polariton with the dielectric function of metal is given as follows: , (15) with √ is bulk plasmon frequency wich depends on electron density n, the dielectric constant of the vacuum and the effective mass of electron m. Based on the similarities of three types of polaritons (phonon polariton, exciton polariton and plasmon polariton), we propose a new model for surface plasmon polariton with transverse oscillation frequency of plasmon polariton , longitudinal oscillation frequency of the plasmon polariton and an effective longitudinal-transverse splitting of plasmon polariton as (16) The dielectric function and dispersion relation can be written in the form similar to phonon polariton and exciton polariton: ( ) (17) { √[ ] } (18) The effective LT-splitting of surface plasmon polariton equals bulk plasmon energy at about ten eV. This effective LT - splitting also describes the effective coupling between surface plasmon and photon similar to the case of phonon polariton and exciton polariton. By introducing the effective transverse - longitudinal splitting for surface plasmon polariton, we obtain a unity theory for three types of polaritons. The dispersion of surface plasmon polariton is shown in Figure 4. One can see that the dispersion curve starts at ( ) and approaches to for . It is our assumption that when equation (20) is consistent with the Drude model in which the dispersion relation is written as: A simple unity theory for three types of polaritons 109 √ √ (19) . Figure 3. Dispersion curve of surface plasmon polariton on the interface between metal and vacuum (blue solid line) with , , . 3. Conclusions In this paper, we expanded and proposed a unity theory for surface polaritons: surface phonon polariton, surface exciton polariton and surface plasmon polariton. We construct a new simple model for surface plasmon polariton in which we assume that there exists an effective interaction between phonton and surface plasmon which is described by an effective LT-splitting of surface plasmon polariton . In the case of phonon and exciton, LT-splitting corresponds to the polarization of the media while for surface plasmon polariton, the media have no any LT-splitting, so the origin of the effective LT-splitting is unknown, this being a problem that needs to be studied. Acknowledgment. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 103.01-2015.42. REFERENCES [1] R. Tsu and S. S. Jha, 1972. Phonon and polariton modes in a superlattice. App. Phys. Lett., Vol. 20(1), pp. 16-18. [2] J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J. M. J. Keeling, F. M. Marchetti, M. H. Szyman acuteska, R. Andr, J. L. Staehli, V. Savona, P. B. Littlewood, B. Deveaud and L. S. Dang, 2006. Bose-Einstein condensation of exciton polaritons. Nature, Vol. 443, pp. 409-414. [3] S Khn, U Hkanson, L Rogobete, V Sandoghdar, 2006. Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna. Phys. Rev. Lett., Vol. 97(1), pp. 017402. Duong Thi Ha, Dinh Thi Thuy and Nguyen Ai Viet 110 [4] G. Vecchi, V. Giannini, J. G Rivas, 2009. Shaping the Fluorescent Emission by Lacttice Resonances in Plasmonic Crystals of Nanoantennas. Phys. Rev. Lett., Vol. 102, pp. 146807. [5] S. Zhang, D. A. Genov, Y. Wang, M. Liu, X. Zhang, 2008. Plasmon-induced transparency in metamaterials. Phys. Rev. Lett., Vol. 101, pp. 047401. [6] Y. Fang, H. Wei, F. Hao, P. Nordlander, H. Xu, 2009. Remote-Excitation Surface-enhanced Raman Scattering Using Propagating Ag Nanowire Plasmons. Nano. Lett., Vol. 9(5), pp. 2049-2053. [7] J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S White, 2010. Plasmonics for extreme light concentration and manipulation. Nat. Mater. Vol. 9, pp. 193 - 204. [8] H. A. Atwater, A. Polman, 2010. Plasmonics for improved photovoltaic devices. Nat. Mater., Vol. 9 (3), pp. 205 - 213. [9] P. K. Jain, X. Huang, I. H. El-Sayed, M. A. El-Sayed, 2007. Review of some interesting surface plasmon resonance-enhanced properties of noble metal nanoparticles and their applications to biosystems. Plasmonics 2, Vol. 3, pp. 107 - 108. [10] Y. Chen, T. R. Nielsen, N. Gregersen, P. Lodahl, J. Mrk, 2010. Finite-element modeling of spontaneous emission of a quantum emitter at nanoscale proximity to plasmonic waveguides. Phys. Rev. B , Vol. 81, pp. 125431. [11] D. T. Ha, D. T. Thuy, V. T. Hoa, T. T. T. Van, N. A. Viet, 2017. On the theory of three types of polaritons (phonon, exciton and plasmon polaritons). IOP Conf. Series: Journal of Physics: Conf. Series, Vol. 865, pp. 012007.