Article Cite This: ACS Photonics 2017, 4, 2908-2915 pubs.acs.org/journal/apchd5 Strong Plasmon-Phonon Splitting and Hybridization in 2D Materials Revealed through a Self-Energy Approach Mikkel Settnes,†,‡,§ J. R. M. Saavedra,∥ Kristian S. Thygesen,§,⊥ Antti-Pekka Jauho,‡,§ F. Javier García de Abajo,∥,# and N. Asger Mortensen*,§,○,△ †Department of Photonics Engineering, ‡Department of Micro and Nanotechnology, §Center for Nanostructured Graphene, and ⊥Center for Atomic-Scale Materials Design (CAMD), Department of Physics, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark ∥ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain # ICREA-Institució Catalana de Recerca i Estudis Avancats,̧ Passeig Lluıś Companys, 23, 08010 Barcelona, Spain ○Center for Nano Optics and △Danish Institute for Advanced Study, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark *S Supporting Information ABSTRACT: We reveal new aspects of the interaction between plasmons and phonons in 2D materials that go beyond a mere shift and increase in plasmon width due to coupling to either intrinsic vibrational modes of the material or phonons in a supporting substrate. More precisely, we predict strong plasmon splitting due to this coupling, resulting in a characteristic avoided crossing scheme. We base our results on a computationally efficient approach consisting in including many-body interactions through the electron self-energy. We specify this formalism for a description of plasmons based upon a tight-binding electron Hamiltonian combined with the random-phase approximation. This approach is valid provided vertex corrections can be neglected, as is the case in conventional plasmon-supporting metals and Dirac-Fermion systems. We illustrate our method by evaluating plasmonic spectra of doped graphene nanotriangles with varied size, where we predict remarkable peak splittings and other radical modifications in the spectra due to plasmon interactions with intrinsic optical phonons. Our method is equally applicable to other 2D materials and provides a simple approach for investigating coupling of plasmons to phonons, excitons, and other excitations in hybrid thin nanostructures. KEYWORDS: graphene plasmons, quantum plasmonics, nanophotonics, 2D materials, molecular plasmonics, many-body interactions wo-dimensional (2D) materials are receiving an increasing examined in thin polar substrates.26−28 Additionally, carrier T interest in nanophotonics due to their ability to host a mobility depends on the electrostatic environment, and has large variety of polaritons,1,2 such as tunable plasmon- been studied in the context of electrostatic gating of polaritons in graphene3,4 or phonon-polaritons in hexagonal graphene.29 The influence of substrate-hosted phonons is boron nitride (h-BN).5−7 In this context, plasmonics benefits commonly included through the substrate dielectric function,8 from the unique charge-transport properties of graphene,8−11 which qualitatively explains experimental observations rather because the electron mobility in a suspended single-atomic- well.24,25 Unfortunately, only limited progress has been layer can reach 200000 cm2 V−1 s−1.12 Indeed, effects associated reported in the quantum description of plasmon-phonon with phonon scattering in polarizable substrates13,14 have been coupling, essentially restricted to analyses based on tight- found to be one of the factors limiting the carrier mobility in binding (TB) models combined with the random-phase 30−33 graphene.15 Hexagonal boron nitride phonons couple weakly to approximation (RPA). More generally, quantum-plas- 34−36 graphene electrons and this material is therefore suitable for monic phenomena are commonly treated at the single- 37−39 graphene encapsulation compatible with preservation of high particle level using various mean-field models, often mobility.16−18 When the characteristic energy of plasmons without more explicit accounts for many-body interactions. In coincides with that of phonons, we anticipate that plasmon− this manuscript, we develop an RPA-inspired formalism for the phonon hybridization can take place.19,20 Actually, strong quantum plasmonic response in which the interactions are phonon−plasmon coupled modes have been observed in 21,22 23−25 graphene/SiC and graphene/SiO2 interfaces. Likewise, Received: August 17, 2017 coupling between plasmons and surface phonons has been Published: October 4, 2017 © 2017 American Chemical Society 2908 DOI: 10.1021/acsphotonics.7b00928 ACS Photonics 2017, 4, 2908−2915 ACS Photonics Article encoded in the single-electron Green functions through dipolar plasmon resonances.32 In brief, we can calculate the electron−phonon self-energies. As an illustration of our theory, eigenvalue loss-spectrum and corresponding plasmon modes we evaluate the influence of phonons on plasmon resonances in using the eigenvalues and eigenvectors of the dielectric matrix graphene nanotriangles. eq 1. Evidently, this requires an efficient way of calculating Our method is amenable to application in arbitrary graphene χ0(ω). As we now show below, interactions with phonons can geometries and can be directly extended to other plasmon- readily be included in this scheme. supporting 2D materials such as black phosphorus40 and thin It is convenient to express the polarizability in terms of the 41 r noble metals. retarded Green function G0(,rr′ , )and the spectral function A (,rr , ): ■ THEORETICAL FORMALISM 0 ′ * Within the RPA, the dielectric matrix, ϵ(r,r′,ω), can be r ψψnn()rr (′ ) G (,rr , ) expressed in terms of the noninteracting polarizability 0 ′=∑ −n +iη (5) χ0(r,r′,ω) as (see, e.g., ref 42) n ra 0 rr rr rr ϵ′(,rr ,ωχω ) = 1−∑ V ( r − R ) ( Rr , ′ , ) AiGG000(,′ , )= [ (,′− , ) (, ′ , )] R (1) = 2()()(),πψψδrr* ′−n 2 ∑ nn Here, V(r − r′) ∝ e /|r − r′| is the bare Coulomb interaction n (6) and the summation is over the atom positions R. The formal where we also use the advanced Green function, which under on-site divergence (r → r′) is only an apparent issue because of fi the assumption of time-reversal symmetry satisfies the orbitals’ nite extension, which is incorporated through a ar 30,32 self-interaction term of 0.58 atomic units for r = r′. The GG00(,rr′ , )= [ ( r′ , r , )]*. Using eqs 5 and 6 while noninteracting polarizability is (see, e.g., ref 42) exploiting the symmetry A(,rr′ , )=A ( r′ , r , ), we can express eq 2 in a more general form without any reference to 0 χω(,rr′ , )= 2∑ [(ffmn )− ( )] single-particle wave functions, nm 0 1 * * χω(,rr′ , )=d()(,,)fA0 rr′ ψψnn()rr (′ ) ψ mm () rr ψ (′ ) π ∫ × i ra ℏωη+−nm+ (2) ×′[(,,GG00rr+ℏωω )+ (,, r′−ℏ r )] (7) 0 0 0 where f ()is the Fermi−Dirac distribution function and n is We focus on the imaginary part of χ (ω)=χR(ω)+iχI (ω). In 47 the energy associated with the single-electron wave function ψn, the low temperature limit, we thus have while η is an infinitesimal broadening. The leading factor of 2 0 originates in spin degeneracy. χωI (,rr′ , )= In our account for self-sustained charge-density oscillations (plasmons) that obey Poisson’s equation, we here follow the 1 F −′′d(,,)(,,AA00rr rr +ℏω ) ∫ approach of ref 43, that we have previously applied to treat 2π F−ℏω (8) plasmons in monolayer and bilayer transition metal dichalco- 44 32 43 where is the Fermi energy. We note that eq 8 can be genides, graphene, and metallic thin-films and nano- F 45 conveniently written in a convolution form, which allows us to wires. Plasmons can exist where the dielectric matrix has zero ffi determinant and the related potential, ϕ, satisfies43 carry out an e cient numerical implementation. The real part of χ0(ω) can then be calculated using the Kramers−Kronig ff ∑ ϵ(,rR ,ωϕ ) ( R , ω )=0 relations with an 8 eV cuto , as shown in the Supporting R (3) Information (SI). For any system with time-reversal symmetry, the spectral The dielectric matrix may have a finite imaginary part, so eq 3 function corresponds to the imaginary part of the Green cannot be satisfied in general for real ω. Instead, we require that fi function A(,rr′ , )=− 2Im{(,G rr ′ , )}. Consequently, eq the real part vanishes, which de nes the plasmon modes ωn, and the associated potential ϕ via43 8 only requires the Green function of the system. This implies n an interesting possibility to estimate the effect of interactions: the noninteracting spectral functions in eq 8 can be replaced by ∑ ϵ(,rR ,ωϕn )n ( R , ωnn )=iϵ ϕn (, r ωn ) R (4) interacting Green functions with self-energy insertions. Notably, an energy-dependent, but spatially uniform, self- Here, ϵ is the imaginary part of the eigenvalue of the dielectric n energy term does not essentially increase the computational matrix ϵ(ω ). Therefore, the plasmon modes are the n burden. The Green function, including an energy-dependent eigenvectors corresponding to purely imaginary eigenvalues of the dielectric matrix.43 retarded self-energy term Σ()can be symbolically written as Plasmonic spectra can be conveniently analyzed in terms of (we suppress the spatial dependence here) ϵ−1 ω r −1 the electron energy−loss function −Im{ ( )}, which is GH()=+ [iη −−Σ ()] (9) relevant for the probing of plasmons in electron energy-loss 46 ff spectroscopy. Considering the eigenvalues ϵn, we e ectively The term Σ() accounts for the interactions between electrons 48 calculate the plasmon frequency, ωn, as the local maximum of with other degrees of freedom in 2D materials, such as −Im{ϵ−1(ω)}. For a system composed of