Abstract

Recent advances in nanofabrication technology now enable unprecedented control over 2D heterostructures, in which single- or few-atom-thick materials with synergetic optoelectronic properties can be combined to develop next-generation nanophotonic devices. Precise control of light can be achieved at the interface between 2D metal and dielectric layers, where surface plasmon polaritons strongly confine electromagnetic energy. Here we reveal quantum and finite-size effects in hybrid systems consisting of graphene and few-atomic-layer noble metals, based on a quantum description that captures the electronic band structure of these materials. These phenomena are found to play an important role in the metal screening of the plasmonic fields, determining the extent to which they propagate in the graphene layer. In particular, we find that a monoatomic metal layer is capable of pushing graphene plasmons toward the intraband transition region, rendering them acoustic, while the addition of more metal layers only produces minor changes in the dispersion but strongly affects the lifetime. We further find that a quantum approach is required to correctly account for the sizable Landau damping associated with single-particle excitations in the metal. We anticipate that these results will aid in the design of future platforms for extreme light–matter interaction on the nanoscale.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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Corrections

13 May 2019: Typographical corrections were made to paragraph one of page 6 and 10.


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References

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2018 (5)

D. Alcaraz Iranzo, S. Nanot, E. J. C. Dias, I. Epstein, C. Peng, D. K. Efetov, M. B. Lundeberg, R. Parret, J. Osmond, J.-Y. Hong, J. Kong, D. R. Englund, N. M. R. Peres, and F. H. L. Koppens, “Probing the ultimate plasmon confinement limits with a van der Waals heterostructure,” Science 360, 291–295 (2018).
[Crossref]

G. X. Ni, A. S. McLeod, Z. Sun, L. Wang, L. Xiong, K. W. Post, S. S. Sunku, B.-Y. Jiang, J. Hone, C. R. Dean, M. M. Fogler, and D. N. Basov, “Fundamental limits to graphene plasmonics,” Nature 557, 530–533 (2018).
[Crossref]

A. Principi, E. van Loon, M. Polini, and M. I. Katsnelson, “Confining graphene plasmons to the ultimate limit,” Phys. Rev. B 98, 035427 (2018).
[Crossref]

E. J. C. Dias, D. Alcaraz Iranzo, P. A. D. Gonçalves, Y. Hajati, Y. V. Bludov, A.-P. Jauho, N. A. Mortensen, F. H. L. Koppens, and N. M. R. Peres, “Probing nonlocal effects in metals with graphene plasmons,” Phys. Rev. B 97, 245405 (2018).
[Crossref]

D. Shah, A. Catellani, H. Reddy, N. Kinsey, V. Shalaev, A. Boltasseva, and A. Calzolari, “Controlling the plasmonic properties of ultrathin tin films at the atomic level,” ACS Photon. 5, 2816–2824 (2018).
[Crossref]

2017 (6)

I. V. Bondarev and V. M. Shalaev, “Universal features of the optical properties of ultrathin plasmonic films,” Opt. Mater. Express 7, 3731–3740 (2017).
[Crossref]

P. Alonso-González, A. Y. Nikitin, Y. Gao, A. Woessner, M. B. Lundeberg, A. Principi, N. Forcellini, W. Yan, Saül, A. J. Huber, K. Watanabe, T. Taniguchi, F. Casanova, L. E. Hueso, M. Polini, J. Hone, F. H. L. Koppens, and R. Hillenbrand, “Acoustic terahertz graphene plasmons revealed by photocurrent nanoscopy,” Nat. Nanotech. 12, 31–35 (2017).
[Crossref]

I.-T. Lin, C. Fan, and J.-M. Liu, “Propagating and localized graphene surface plasmon polaritons on a grating structure,” IEEE J. Sel. Top. Quantum Electron. 23, 144–147 (2017).
[Crossref]

S. de Vega and F. J. García de Abajo, “Plasmon generation through electron tunneling in graphene,” ACS Photon. 4, 2367–2375 (2017).
[Crossref]

M. B. Lundeberg, Y. Gao, R. Asgari, C. Tan, B. V. Duppen, M. Autore, P. Alonso-González, A. Woessner, K. Watanabe, T. Taniguchi, R. Hillenbrand, J. Hone, M. Polini, and F. H. L. Koppens, “Tuning quantum nonlocal effects in graphene plasmonics,” Science 89, 035004 (2017).
[Crossref]

C. David and J. Christensen, “Extraordinary optical transmission through nonlocal holey metal films,” Appl. Phys. Lett. 110, 261110 (2017).
[Crossref]

2016 (2)

R. Yu, V. Pruneri, and F. J. García de Abajo, “Active modulation of visible light with graphene-loaded ultrathin metal plasmonic antennas,” Sci. Rep. 6, 32144 (2016).
[Crossref]

D. N. Basov, M. M. Fogler, and F. J. García de Abajo, “Polaritons in van der Waals materials,” Science 354, aag1992 (2016).
[Crossref]

2015 (5)

A. Woessner, M. B. Lundeberg, Y. Gao, A. Principi, P. Alonso-González, M. Carrega, K. Watanabe, T. Taniguchi, G. Vignale, M. Polini, J. Hone, R. Hillenbrand, and F. H. Koppens, “Highly confined low-loss plasmons in graphene–boron nitride heterostructures,” Nat. Mater. 14, 421–425 (2015).
[Crossref]

J. B. Khurgin, “How to deal with the loss in plasmonics and metamaterials,” Nat. Nanotech. 10, 2–6 (2015).
[Crossref]

H. Qian, Y. Xiao, D. Lepage, L. Chen, and Z. Liu, “Quantum electrostatic model for optical properties of nanoscale gold films,” Nanophotonics 4, 413–418 (2015).
[Crossref]

F. J. García de Abajo and A. Manjavacas, “Plasmonics in atomically thin materials,” Faraday Discuss. 178, 87–107 (2015).
[Crossref]

I. Silveiro, J. M. Plaza Ortega, and F. J. García de Abajo, “Quantum nonlocal effects in individual and interacting graphene nanoribbons,” Light Sci. Appl. 4, e241 (2015).
[Crossref]

2014 (6)

F. Schiller, Z. M. A. El-Fattah, S. Schirone, J. Lobo-Checa, M. Urdanpilleta, M. Ruiz-Osés, J. Cordón, M. Corso, D. Sánchez-Portal, A. Mugarza, and J. E. Ortega, “Metallic thin films on stepped surfaces: lateral scattering of quantum well states,” New J. Phys. 16, 123025 (2014).
[Crossref]

J. D. Cox and F. J. García de Abajo, “Electrically tunable nonlinear plasmonics in graphene nanoislands,” Nat. Commun. 5, 5725 (2014).
[Crossref]

F. Xia, H. Wang, D. Xiao, M. Dubey, and A. Ramasubramaniam, “Two-dimensional material nanophotonics,” Nat. Photonics 8, 899–907 (2014).
[Crossref]

F. J. García de Abajo, “Graphene plasmonics: challenges and opportunities,” ACS Photon. 1, 135–152 (2014).
[Crossref]

C. David and F. J. García de Abajo, “Surface plasmon dependence on the electron density profile at metal surfaces,” ACS Nano 8, 9558–9566 (2014).
[Crossref]

N. A. Mortensen, S. Raza, M. Wubs, T. Søndergaard, and S. I. Bozhevolnyi, “A generalized non-local optical response theory for plasmonic nanostructures,” Nat. Commun. 5, 3809 (2014).
[Crossref]

2013 (7)

S. Raza, T. Christensen, M. Wubs, S. I. Bozhevolnyi, and N. A. Mortensen, “Nonlocal response in thin-film waveguides: loss versus nonlocality and breaking of complementarity,” Phys. Rev. B 88, 115401 (2013).
[Crossref]

A. Moreau, C. Ciracì, and D. R. Smith, “Impact of nonlocal response on metallodielectric multilayers and optical patch antennas,” Phys. Rev. B 87, 045401 (2013).
[Crossref]

C. David, N. A. Mortensen, and J. Christensen, “Perfect imaging, epsilon-near zero phenomena and waveguiding in the scope of nonlocal effects,” Sci. Rep. 3, 2526 (2013).
[Crossref]

C. Ciraci, J. B. Pendry, and D. R. Smith, “Hydrodynamic model for plasmonics: a macroscopic approach to a microscopic problem,” Chem. Phys. Chem. 14, 1109–1116 (2013).
[Crossref]

A. K. Geim and I. V. Grigorieva, “Van der Waals heterostructures,” Nature 499, 419–425 (2013).
[Crossref]

S. Laref, J. Cao, A. Asaduzzaman, K. Runge, P. Deymier, R. W. Ziolkowski, M. Miyawaki, and K. Muralidharan, “Size-dependent permittivity and intrinsic optical anisotropy of nanometric gold thin films: a density functional theory study,” Opt. Express 21, 11827–11838 (2013).
[Crossref]

D. Golla, K. Chattrakun, K. Watanabe, T. Taniguchi, B. J. LeRoy, and A. Sandhu, “Optical thickness determination of hexagonal boron nitride flakes,” Appl. Phys. Lett. 102, 161906 (2013).
[Crossref]

2012 (5)

R. Esteban, A. G. Borisov, P. Nordlander, and J. Aizpurua, “Bridging quantum and classical plasmonics with a quantum-corrected model,” Nat. Commun. 3, 825 (2012).
[Crossref]

R. E. V. Profumo, R. Asgari, M. Polini, and A. H. MacDonald, “Double-layer graphene and topological insulator thin-film plasmons,” Phys. Rev. B 85, 085443 (2012).
[Crossref]

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Zurutuza Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487, 77–81 (2012).
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Supplementary Material (1)

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Figures (5)

Fig. 1.
Fig. 1. Plasmon interactions between two atomically thin silver films. We consider symmetric structures consisting of two metallic films [see scheme in (a)] formed by either (b)–(d)  N = 1 or (e)–(g)  N = 5 . Ag(111) atomic layers and separated by a dielectric (local permittivity ϵ = 2.1 ) of thickness (b),(e)  d d = 1 nm ; (c),(f)  d d = 2 nm ; or (d),(g)  d d = 4 nm . Quantum-mechanical (QM) results in the ALP model (color density plots) are compared with local dielectric theory for metal films of finite thickness described with the measured silver dielectric function [46] (black dashed curves) and with the analytical expressions described in the main text (blue curves).
Fig. 2.
Fig. 2. Quantum-mechanical models for plasmons in atomically thin gold films and graphene–gold heterostructures. (a) Schematic illustration of several quantum-mechanical models used to describe a thin metallic film: atomic layer potential (ALP), finite barrier model (FBM), and infinite barrier model (IBM). The out-of-plane confinement potential is indicated by colored curves, while the unperturbed electronic charge density ρ 0 is shown by filled gray curves and a wave function of normal energy at the Fermi level by black curves. (b) Plasmon dispersion relation of films comprising N atomic Au(111) layers (0.236 nm thickness per layer). Nearly indistinguishable results are obtained from the ALP, FBM, and IBM models. (c) Full width at half-maximum (FWHM) for the curves in (b). (d) Low-energy acoustic plasmon dispersion relation of monolayer graphene (MG) deposited on the films considered in (b) and (c). The plasmon dispersion of isolated MG (black dashed curve) and the intraband damping region ω v F Q (shaded area) are shown for reference. (e) FWHM of the acoustic plasmons in (d). Curves for different models and gold film thicknesses in (b)–(e) follow the legends of (b). Graphene is modeled in the RPA with Fermi energy E F = 1 eV and an intrinsic lifetime of 500 fs in all cases. The plasmon energies are obtained from the peaks of Im { R } , where R is the reflection coefficient (see Appendix A). For self-standing metal films, we have R = r m .
Fig. 3.
Fig. 3. Thickness and doping dependence of plasmons in graphene–gold films. (a) Schematic illustration of a MG–gold film heterostructure containing N = 2 atomic Au(111) layers. (b) Dispersion relation and (c) FWHM of high-energy (solid curves) and acoustic (dashed curves) plasmons for a graphene Fermi energy E F = 1 eV and different gold film thicknesses [see legend for N in (c)]. (d) Plasmon dispersion and (e) FWHM for N = 2 and various Fermi energies [see legend for E F in (e)]. (f)–(i) Spatial distribution of the in-plane ( E x ) and out-of-plane ( i E z ) components of the plasmon electric field for various metal thicknesses [see legend in (f)] at the wave vector indicated by the black arrows in (b) (i.e., Q = 0.1 nm 1 ). Solid and dashed curves represent the fields for the high-energy and acoustic plasmons, respectively [see (b)]. The vertical scales are in arbitrary units. The graphene layer is approximated as a zero-thickness film (vertical dashed lines, separated d g / 2 from the metal, where d g = 0.33 nm is the nominal graphene thickness) in its interaction with the metal, although the effect of finite out-of-plane extension of its induced charge is incorporated in the evaluation of the near-field by convoluting with the p z orbital charge distribution (see Appendix G); the near-fields of the acoustic plasmons (h) and (i) are mainly concentrated in the graphene–metal region, whereas the higher energy plasmons (f) and (g) exhibit more delocalized field profiles. We describe graphene in the RPA and the metal in the ALP model in all plots.
Fig. 4.
Fig. 4. QM versus classical description of acoustic plasmons in graphene–metal hybrid systems. (a) We consider MG on optically thick gold or silver films. (b) Dispersion relation and (c) associated FWHM for MG–gold with E F = 1 eV doping, as obtained by using different QM and classical (specular-reflection model, SRM) approaches to describe the response of gold [see legend in (c) and details in the main text]. (d),(e) Same as (b) and (c) when gold is replaced by silver. We model graphene in the RPA in all cases. The film thickness is infinite in the classical SRM, whereas we take N = 100 (i.e., 23.6 nm) films in the QM approach. The orange shaded area in (b) denotes the bulk electron–hole pair region in gold (see main text). The lower right insets to (b) and (d) show the dispersion of the high-energy surface plasmon.
Fig. 5.
Fig. 5. Plasmons in metal–graphene–metal structures. We consider graphene directly embedded in silver without dielectric spacers. (a) Illustration of a structure consisting of a finite-thickness silver film [ N = 2 Ag(111) atomic layers in the sketch] deposited on a MG sheet, which in turn rests on an optically thick Ag(111) surface. (b) Plasmon dispersion and (c) FWHM of the structure in (a) for various upper film thicknesses (see color-coded labels N ). The graphene doping level is E F = 1 eV . Three different plasmon branches are observed in each structure: two acoustic ones (lower dashed curves and upper solid curves) and a high-energy plasmon (not shown) that is nearly overlapping the vertical energy axis. (d)–(f) Same as (a)–(c), with the bottom metal surface replaced by a monolayer Ag(111) sheet; here we show the upper energy plasmon (dotted curves), which is no longer separated from the vertical axis. As a reference, we show the plasmon dispersion for isolated MG (black long-dashed curves) and the acoustic plasmon observed in the structures without the upper film (black solid curves). We describe graphene in the RPA and the metal in the ALP model.

Tables (1)

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Table 1. Atomic Layer Parametersa

Equations (23)

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R = r gd + t gd t dg r m 1 r dg r m , T = t gd t m 1 r dg r m ,
r i j = r i + t i 2 r j 1 r i r j ,
t i j = t i t j 1 r i r j .
ϕ ( z ) = ϕ b ext ( z ) + d z v b ( z , z ) ρ ind ( z ) ,
v b ( z , z ) = v b dir ( z , z ) + v b ref ( z , z ) ,
v b dir ( z , z ) = 2 π Q e Q | z z | × { 1 , z , z 0 or z , z > d m 1 / ϵ b , 0 < z , z d m 0 , otherwise
v b ref ( z , z ) = g × { ( 1 ϵ b 2 ) ( e 2 Q d m 1 ) e Q ( z + z ) , d m < z , z 2 [ ( ϵ b + 1 ) e Q ( z z ) + ( ϵ b 1 ) e Q ( z + z ) ] , 0 < z d m < z 4 ϵ b e Q ( z z ) , z 0 and d m < z 2 [ ( ϵ b + 1 ) e Q ( z z ) + ( ϵ b 1 ) e Q ( z + z ) ] , 0 < z d m < z ( 1 / ϵ b ) { ( ϵ b 2 1 ) [ e Q ( z + z ) + e Q ( 2 d m z z ) ] + ( ϵ b 1 ) 2 [ e Q ( 2 d m + z z ) + e Q ( 2 d m z + z ) ] } , 0 < z , z d m 2 [ ( ϵ b + 1 ) e Q ( z z ) + ( ϵ b 1 ) e Q ( 2 d m z z ) ] , z 0 < z d m 4 ϵ b e Q ( z z ) , z 0 and d m < z 2 [ ( ϵ b + 1 ) e Q ( z z ) + ( ϵ b 1 ) e Q ( 2 d m z z ) ] , z 0 < z d m ( 1 ϵ b 2 ) ( 1 e 2 Q d m ) e Q ( z + z ) , z , z 0
g = ( 2 π / Q ) ( ϵ b + 1 ) 2 ( ϵ b 1 ) 2 e 2 Q d m .
ρ ind ( z ) = d z χ ( z , z ) ϕ b ext ( z ) ,
χ = χ 0 · ( 1 v b · χ 0 ) 1 ,
χ 0 ( r , r ) = 2 e 2 i i ( f i f i ) ψ i ( r ) ψ i * ( r ) ψ i * ( r ) ψ i ( r ) ω + i γ ( ε i ε i )
χ 0 ( z , z ) = e 2 2 π 2 j j d 2 k ( f j , | k Q / 2 | f j , | k + Q / 2 | ) ϕ j ( z ) ϕ j * ( z ) ϕ j * ( z ) ϕ j ( z ) ω + i γ [ ε j ε j + ( / m * ) k · Q ]
E F = M [ π d m m * n + j = 1 M ε j ] ,
r m = 1 ϵ s ( Q , ω ) 1 + ϵ s ( Q , ω ) ,
ϵ s ( Q , ω ) = 2 Q π 0 d k Q 2 + k 2 1 ϵ m ( k 2 + Q 2 , ω )
ϵ hydro ( q , ω ) = 1 + ω p 2 2 β 2 q 2 / m e 2 ω ( ω + i γ ) , ϵ Lindhard ( q , ω ) = 1 + 2 m e e 2 k F π 2 q 2 [ 1 + F ( q / k F , ω / E F ) + F ( q / k F , ω / E F ) ] , ϵ Mermin ( q , ω ) = 1 + ( ω + i γ ) [ ϵ Lindhard ( q , ω + i γ ) 1 ] ω + i γ [ ϵ Lindhard ( q , ω + i γ ) 1 ] / [ ϵ Lindhard ( q , 0 ) 1 ] ,
F ( x , y ) = 1 2 x [ 1 ( x 2 + y 2 x ) 2 ] log ( x 2 + 2 x + y x 2 2 x + y ) .
r g 2 D = 1 1 i ω / ( 2 π Q σ ) , t g 2 D = 1 r g 2 D ,
C g = d 3 r ϕ 2 p z 2 ( r ) e Q z .
ϵ i ( ω ) = ϵ i + j = 1 2 s i , j 2 ω i , j 2 ω ( ω + i γ i , j )
r d = ( ϵ 2 1 ) 1 e 2 q d d ( ϵ + 1 ) 2 ( ϵ 1 ) 2 e 2 q d d , t d = 4 ϵ e q d d ( ϵ + 1 ) 2 ( ϵ 1 ) 2 e 2 q d d ,
V surf ( z ) = { 1 4 ( z z i ) [ e λ ( z z i ) 1 ] , z < z i A 3 e α ( z z t ) , z i < z < z t A 20 + A 2 cos ( η z ) , z t < z < 0 A 10 + A 1 cos ( 2 π z / a s ) , z > 0
V ( z ) = { V surf ( z a s / 2 ) , z < d m / 2 V surf ( d m a s / 2 z ) , z > d m / 2