An Introduction to Graphene Plasmonics

The following sections are included:

• Dedication
• Foreword
• Preface
• Acknowledgments
• Contents
• List of Figures
• List of Tables

Plasmonics holds many promises [Atwater (2007)], such as plasmonic integrated circuitry [Sorger et al. (2012)], nanophotonics (e.g., plasmonic antennas) [Dionne and Atwater (2012)], all-optical manipulation of small particles [Quidant (2012)], quantum plasmonics [Jacob (2012); Hanson et al. (2015)], spin-polarized plasmonics [Toropov and Shubina (2015)], and plasmonic hot-spots [Weber and Willets (2012)], just to name a few. Whether graphene will be able to fulfil some of these promises remains to be seen, but current prospects are encouraging. This book is about plasmonics in graphene and about the methods to excite and control surface plasmons in this material. Below, we give a brief account of some of the recent developments in the field.

In this chapter, we lay the foundations which will constitute the framework of the problems discussed in the chapters ahead. Here, we shall provide a brief overview on elementary concepts in classical electromagnetism and in graphene physics, including the description of some essential tools to discuss plasmons in the forthcoming chapters.

Before discussing plasmonic-related phenomena in graphene structures, it is instructive to review the classical case of light confinement occurring at dielectric-metal interfaces. In this Chapter, we will apply the formalism developed previously to address the question of how the interface between two media with different optical properties can originate surface modes confined to dimensions smaller than the wavelength. This special solution of Maxwell's equations is known as surface plasmon-polaritons.

In recent years we have been witnessing an impressive outburst in the field of plasmonics, with papers on the subject being published everyday at a great pace. On the other hand, graphene has been one of the most active topics in (but not limited to) the condensed matter physics community since its isolation in 2004, and ten years later our interest in the wonder material still thrives as it continues to present new challenges to us. The prediction of graphene surface plasmon-polaritons (GSPs) and their experimental observation led to the emergence of the field of graphene nanoplasmonics, which combines two of the most vibrant topics in physics today, and promises to deliver a huge impact in future technologies ranging from novel optoelectronic devices [Liang et al. (2014)] to applications in nanomedicine [Zheng et al. (2012)]…

In this chapter we will briefly review several techniques for the excitation of GSPs. Recall from our previous discussion that, in general, it is not possible to couple an incident electromagnetic beam to a SPP in planar, extended graphene, due to a mismatch between the momentum of the impinging radiation and the GSP's momentum—the dispersion curve of the latter always lies to the right of the light line. This limitation is shared amongst graphene and plasmonic metals (and, of course, metals in general). Thus, it should not come as a surprise that the very same mechanisms for exciting SPPs in metals also apply to the case of graphene.

One way of exciting graphene surface plasmon-polaritons (or SPPs in general) is to use a metallic contact placed on top of a graphene sheet. The presence of the contact will scatter an impinging electromagnetic wave and induce plasmons in the nearby graphene. Recently, applying a similar reasoning, Alonso-González et al. have experimentally demonstrated the excitation of GSPs using resonant metal antennas, consisting in micrometersized gold bars covering a graphene layer [Alonso-González et al. (2014)]. When illuminated with resonant light, these behave like an optical dipole which in turn drives the launch of plasmon-polaritons propagating through graphene…

The recent developments in nanofabrication techniques and theoretical/computational modelling have brought plasmonics to the nanoscale, leading to the birth of nanoplasmonics. Recently, the realization of surface plasmon-polaritons in graphene [Ju et al. (2011)] has drawn a lot of attention. Since then, research in graphene plasmonics is intense and likely to remain one of the hottest topics in nanophotonics in the forthcoming years. Particular interest has been given to the excitation of GSPs in periodic graphene-based structures, including corrugated graphene [Bludov et al. (2013); Slipchenko et al. (2013a)], diffraction gratings, graphene plasmonic crystals and metamaterials [Ju et al. (2011); Yan et al. (2013); Nikitin et al. (2012a)], and also conductivity gratings (in which graphene's conductivity is periodically modulated) [Bludov et al. (2012a)]. Among these are included arrays of graphene ribbons arranged periodically [Ju et al. (2011); Yan et al. (2013); Nikitin et al. (2012a)], which will be the main subject throughout this Chapter. This problem has also been analyzed from the point of view of circuit theory [Khavasi and Rejaei (2014); Khavasi (2015)].

The 21st century has undoubtedly brought about the “nano-revolution” that began in the 1990s. It can eventually be argued that this led to an outburst of interest in nanomaterials which may have contributed for the isolation of graphene itself…

The study of gratings and rough surfaces [Maradudin (2007a)] as methods of exciting surface plasmon-polaritons was quite popular in the eighties in the context of noble-metals plasmonics [Farias and Maradudin (1983)]. The interest in this topic has been kept alive due to the problem of light scattering by rough surfaces at the nanoscale [Maradudin (2007b)]. With the birth of graphene plasmonics came a renewed interest in the problem of grids and gratings based on graphene. In Chapter 7 we have discussed the grid problem, which was the first method used to induce plasmons in graphene-based periodic structures. As expected, the excitation of graphene plasmons by gratings soon followed (see Chapter 1). The advantage of gratings over grids is that the quality of the graphene sheet is not affected by the lithographic process, since graphene keeps its integrity. On the other hand, the fabrication of gratings with high dielectric contrast may present some difficulties. This is specially true for square-wave gratings. If the grating has some profile, as the one discussed in this Chapter, a graphene sheet may follow that profile thus making a natural graphene-based grating…

The problem of the emission of an excited dye (or quantum dot) over a graphene sheet has been studied both theoretically [Gómez-Santos and Stauber (2011); Koppens et al. (2011)] and experimentally [Kasry et al. (2012); Gaudreau et al. (2013)]. The case of Förster resonant energy transfer (FRET) between two dyes in the presence of graphene has also been considered [Biehs and Agarwal (2013)]. In this chapter we show that a dye can, in principle, de-excite via the excitation of plasmons in graphene. This possibility has its fingerprint in a resonance in the effective dielectric function of the system, or in the dispersion of the life-time of the excited state of the dye with the distance to the graphene sheet. We first formulate the problem in the electrostatic approximation and use Fermi's golden rule, following [Swathi and Sebastian (2009)]. We show that this approach corresponds to a decaying pathway associated with the induction of particle-hole excitations in graphene. We end the chapter with an exact calculation of the total decaying rate of the dye in the presence of graphene, where the fingerprint of plasmon excitation becomes manifest. We show that the decaying rate of an emitter in the presence of graphene differs substantially from the free space decaying rate, a manifestation of the Purcell effect. This effect has been demonstrated in a variety of physical systems, such as radiating molecules near a metallic-dielectric interface, Rydberg atoms in an optical cavity, and nuclear transitions in small metallic particles, the latter example being the basis of the work of Purcell [Purcell (1995)]. A decaying emitter can couple to surface plasmons when the distance to the metallic interface (which can be doped graphene) is smaller than the emission wavelength (but much larger than typical atomic distances), the reason being that the dipole field contains large in-plane wavevectors, associated with evanescent modes, that do not propagate in the far field but can couple to the SPP's at the surface of the conductor [Toropov and Shubina (2015)]. For distances much smaller than the emission wavelength, the emitter couples to the conductor by inducing either inter-band or intra-band absorption. For neutral graphene only the inter-band transitions pathway exists. This latter case is discussed in the appendix section M.2.

The field of plasmonics is growing, as it can be seen from the data-plot in a recent editorial in Nature Photonics [Editorial (2012)]. In that article it is shown that the number of publications in plasmonics has been growing steadily over the last twenty years. Broadly speaking, we can divide the field of plasmonics according to the nature of the surface-plasmon resonances. Whenever these resonances occur in localized nano-particles we talk about localized surface plasmon-polaritons [Willets and Duyne (2007); Pelton and Bryant (2013); Klimov (2014)]. When they occur at metal/dielectric surfaces one usually denotes them as (propagating) surface plasmon-polaritons [Sarid and Challener (2010)]. In both cases, these plasmonic resonances have wavelengths below the diffraction limit [Armstrong (2012)] and exhibit intense electric fields. For propagating surface plasmon-polaritons we are generally interested in the case of propagation along a thin metal film. Since 2004 the thinnest conductive film available has been a graphene sheet. It is natural to inquire whether graphene can support propagating surface plasmon-polaritons [Low and Avouris (2014)] (which can be excited by different methods; see Chapter 5 and [Bludov et al. (2013); Zhang et al. (2014)]). Indeed, the combination of graphene and plasmonics opened a new avenue to this exciting field. This recent branch of nanophotonics is nowadays known as graphene (nano)plasmonics. For example, graphene plasmons can be used to engineer nanoscale infrared light sources [Manjavacas et al. (2014)], and intrinsic plasmons on graphene have already been used for enhancing photodetection [Kim et al. (2014); Koppens et al. (2014)]…

The following sections are included:

• Appendix A. Derivation of the Susceptibility of Graphene (written with André Chaves)
• Appendix B. Derivation of the Intra- and Inter-band Conductivity of Graphene
• Appendix C. Inhomogeneous Drude Conductivity
• Appendix D. Derivation of the Expression Relating the Longitudinal Conductivity with the Polarizability (written with Bruno Amorim)
• Appendix E. Derivation of the Polarization of Graphene in the Relaxation-Time Approximation
• Appendix F. RPA for Double-Layer Graphene
• Appendix G. Effective Dielectric Constants for Coulomb-Coupled Double-Layer Graphene
• Appendix H. Magneto-Optical Conductivity of Graphene: Boltzmann equation
• Appendix I. Supplementary Material for Chapter 7
• Appendix J. The Method of Toigo
• Appendix K. Boundary Condition in the Dipole Problem (written with Jaime E. Santos)
• Appendix L. Fourier Transform of the Dipole Potential
• Appendix M. Non-Radiative Decaying Rate: Electrostatic Calculation
• Appendix N. Reflection Amplitude of an Electromagnetic Wave due to a Graphene Interface
• Appendix O. Green's Functions of a Rope with Two Different Mass Densities Subject to a Point-Like Excitation (written with Bruno Amorim)
• Appendix P. Derivation of the Transition Rate of a Quantum Emitter Near an Interface (written with Bruno Amorim)

The following sections are included:

• Bibliography
• Index