Single-Plasmon Thermo-Optical Switching in Graphene | Nano Letters

Subscriber access provided by La Trobe University Library

Communication

Single-Plasmon Thermo-Optical Switching in Graphene Joel D. Cox, and F. Javier García de Abajo Nano Lett., Just Accepted Manuscript • Publication Date (Web): 22 May 2019 Downloaded from http://pubs.acs.org on May 26, 2019

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

Single-Plasmon Thermo-Optical Switching in Graphene Joel D. Cox∗,†,‡,¶ and F. Javier García de Abajo∗,†,§ †ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain ‡Center for Nano Optics, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark ¶Danish Institute for Advanced Study, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark §ICREA-Institució Catalana de Recerca i Estudis Avançats, Passeig Lluís Companys 23, 08010 Barcelona, Spain E-mail: [email protected]; [email protected]

Abstract While plasmons in noble metal nanostructures enable strong light-matter interactions on commensurate length scales, the overabundance of free electrons in these systems inhibits their tunability by weak external stimuli. Countering this limitation, the linear electronic dispersion in graphene endows the two-dimensional material with both an enhanced sensitivity to doping electron density, enabling active tunability of its highly-confined plasmon resonances, and a very low electronic heat capacity that renders its thermo-optical response extraordinarily large. Here we show that these properties combined enables a substantial optical modulation in graphene nanostructures from the energy associated with just one of their supported plasmons.

1

ACS Paragon Plus Environment

Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

We base our analysis on realistic, complimentary classical and quantum-mechanical simulations, which reveal that the energy of a single plasmon, absorbed in a small, moderately-doped graphene nanoisland, can sufficiently modify its electronic temperature and chemical potential to produce unity-order modulation of the optical response within sub-picosecond timescales, effectively shifting or damping the original plasmon absorption peak and thereby blockading subsequent excitation of a second plasmon. The proposed thermo-optical single-plasmon blockade consists in a viable ultra-low power all-optical switching mechanism for doped graphene nanoislands, while their combination with quantum emitters could yield applications in biological sensing and quantum nano-optics.

Keywords: all-optical switching, graphene plasmonics, optical modulation, thermo-optical response, single-photon devices, quantum optics The ability of plasmons –the collective oscillations of conduction electrons– to intensify the electromagnetic field of light has stimulated fertile research efforts in nonlinear plasmonics to control light by light on the nanoscale. 1 In this context, engineered plasmonic near field enhancement from noble metal nanostructures has been enormously successful in bolstering the weak susceptibilities associated with coherent nonlinear processes while reducing the intensity threshold for optical switching. 2 However, despite the achievement of impressive levels of plasmonic enhancement, the realization of strong nonlinear optical interactions at ultra-low powers, ultimately down to the few- or single-photon regime, remains elusive. 3–5 The enhanced light absorption associated with the resonant excitation of plasmons in noble metals stimulates the generation of energetic (hot) electrons and holes, which thermalize via electron-electron scattering on timescales of 10s of femtoseconds, reaching a local equilibrium state with an elevated electron temperature, followed by cooling at a much slower rate (∼ 1 ps) by coupling to lattice vibrations. 6–11 While rapid thermalization inhibits photochemical 12 and photovoltaic 13 functionalities based on the separation and extraction of energetic electrons and holes, the transient plasmonic response associated with the local

2


Page 2 of 21

Page 3 of 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

thermallized equilibrium electron distribution provides a potentially useful source of ultrafast thermal nonlinearity. 14–16 In this context, electrically doped graphene –the atomically-thin carbon layer– appears as an excellent infrared plasmonic material 17,18,18–24 in which a comparatively small number of electrons rule its optical and thermal properties in the ultrafast regime, thus displaying extraordinary thermo-optical properties that hold strong potential for all-optical modulation. 25–29 Following excitation by an ultrafast optical pump, graphene plasmons decay on typically longer timescales than their noble metal counterparts into energetic electrons and holes, 24,30 which subsequently undergo ultrafast thermalization. 26,27 Owing to its linear electronic dispersion relation, which endows graphene with a low electronic heat capacity, the large amount of optical energy absorbed through resonant excitation of plasmons can dramatically elevate the temperature of conduction electrons in the 2D material by 1000s of degrees. 31,32 The thermalized electronic distribution can be optically probed while electrons cool by gradually transferring their energy to phonons on picosecond timescales (several ps for high-quality graphene), essentially rendering the optical response nonlinear through its dependence on the primary excitation pump intensity. 33–35 In this Letter we propose that the combined large thermo-optical response and very low electronic heat capacity originating from the unique conical dispersion of graphene can enable the modulation of the optical response upon excitation of a single plasmon. In particular, we reveal scenarios in which the energy of a single plasmon quantum (~ωp , where ωp is the plasmon frequency), distributed after it inelastically decays among a finite number of electrons, can modify the local equilibrium temperature and chemical potential of a graphene nanoisland enough to produce substantial changes in its supported plasmon resonances. Remarkably, in small graphene nanoislands doped with only a few electrons, we predict unity-order modulation of the optical response through single-plasmon absorption on ultrafast timescales, extending over picosecond timescales, as determined by the release of electronic energy to lattice vibrations. This phenomenon consists in an incoherent plas-

3


Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

mon blockade, whereby the decay of a single excited plasmon suppresses the probability of exciting additional plasmons. In what follows we outline the theoretical treatments used to describe the thermo-optical response associated with single-plasmon absorption in graphene nanoislands. For structures with dimensions & 10 nm, we adopt a semi-analytical approach rooted in the quasistatic limit of classical electromagnetism, employing a local optical conductivity to describe the graphene response. 36 In smaller graphene structures, we implement an atomistic description of the optical response, accounting for quantum and finite-size effects that become important in this regime. 37 In either case, we estimate the change in optical response associated with single-plasmon absorption by a doped graphene nanostructure using the following procedure: 1. We identify the plasmon resonance frequency ωp from the simulated linear absorption spectum of a graphene nanoisland at zero temperature and doped to a Fermi energy EF . 2. The equilibrium chemical potential µ and electronic temperature T of the nanoisland when it has absorbed the single-plasmon energy ~ωp are computed self-consistently, ensuring conservation of energy and electron population. 3. With the obtained values of µ and T , we re-calculate the graphene nanoisland absorption spectrum and observe the change in optical response relative to zero temperature. This approach estimates the maximum possible change in optical response associated with the absorption of a single plasmon by assuming that it only decays to electron-hole pairs that rapidly thermalize via electron-electron scattering without losing energy. Experimental realization of single-plasmon thermo-optical switching could be achieved by positioning in proximity to the graphene nanoisland a quantum emitter (e.g., a three-level system) radiating at the plasmon resonance of the unperturbed graphene structure (i.e., at zero electron temperature) and excited by an external optical pump at a frequency away from the plasmons of the structure, but tuned to a high-energy transition of the emitter, so 4


Page 4 of 21

Page 5 of 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

that it then quickly decays to an intermediate state with a resonance transition frequency tuned to the plasmon; Purcell enhancement of spontaneous emission will ensure that this process excites the plasmon at a fast rate; then, a weak probe is used to record the absorption spectrum of the structure that has absorbed the energy released by the quantum emitter with a delay time sufficient to thermalize the electrons in the graphene. In this scheme, which is further explored in the Supporting Information (SI), we circumvent the use of intense pump fields (i.e., only a π pump pulse for the emitter is needed), thus averting any undesired heating of the graphene. Alternatively, direct optical excitation of the graphene nanoisland plasmon by a sufficiently-intense ultrashort pulse should lead to thermo-optical modulation that can be recorded by a delayed probe field; however, a weak probe pulse delayed by or of duration more than ∼ 1 ps should experience a weaker effect due to rapid disorder-assisted cooling to the graphene lattice. 35 We also remark that impinging optical pulses of extreme intensity will give rise to additional strong-field non-equilibrium phenomena that demand a theoretical treatment beyond linear response. 38,39 Classical Treatment of Single-Plasmon Absorption in Graphene. Low-energy electrons in graphene follow a conical dispersion relation E(kk ) = ~vF |kk |, where vF ≈ 106 m/s is the Fermi velocity and kk the in-plane electron wave vector. At zero temperature, the optical conductivity at frequency ω is obtained in the local response limit (i.e., for vanishing in-plane optical momentum) of the random-phase approximation (RPA) as 40,41 i ω + iτ −1 − 2EF /~ e2 iEF /(π~) 1 = + Θ(~ω − 2E ) + log F ~ ω + iτ −1 4 4π ω + iτ −1 + 2EF /~ "

σω(1)

!#

,

(1)

where the Fermi energy EF is related to the graphene doping charge-carrier density n ac√ cording to EF = ~vF πn and τ is a phenomenological inelastic scattering rate, for which we choose a conservative value (~τ −1 = 50 meV) throughout this study. The first terms in eq 1 describe the optical response arising from intraband electronic transitions, whereas the remaining two terms correspond to interband transitions.

5


Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 21

We consider a graphene nanostructure occupying the R = (x, y) plane and interacting ˆ +c.c., with linearly-polarized monochromatic light of in-plane external electric field E ext e−iωt x ˆ . If the structure has characteristic size D (e.g., the taken along a symmetry direction x diameter of a disk or the side length of a square) and is doped to a Fermi energy EF , a localized plasmon mode with resonance frequency ωp is well-described when ~ωp . EF and D & 10 nm. In this regime, assuming D to be small compared with the light wavelength, an expansion of the electrostatic potential in orthonormal eigenmodes yields real eigenvalues ηj and their associated plasmon wave functions (PWFs) ρj (R/D) corresponding to modes labeled by j for a specific 2D morphology. 22,36 In terms of PWFs, the linear polarizability can be analytically expressed as 36

αω(1) =

where ξj =

R

X

¯ω ξj2 D3

j

1/ηω − 1/ηj

(1)

,

(2)

d2 R xρj (R/D) quantifies the coupling of mode j with the external field, ηω(1) =

iσω(1) /ω ωD contains the dependence on the local, isotropic optical conductivity σω(1) , and ¯ω is the background permittivity. This expression is also valid for graphene supported on a substrate with ¯ω denoting the average permittivity of the media on either side of the interface. 22 With the tabulated parameters ηj and ξj reported in Ref. 36 for specific 2D geometries and the conductivity entering ηω(1) , the linear polarizability can be computed from eq 2 directly. The situation simplifies further if ωp . EF and we assume zero temperature, so that the effect of interband optical transitions can be reasonably neglected and we retain only the leading Drude-like term in the conductivity of eq 1; identifying plasmons in the poles of eq 2, we obtain the resonance energy associated with the j th eigenmode as ~ωj = q

e EF /(−ηj ¯ω πD). A straightforward extension of the above PWF formalism yields the radiative decay rate Γ of a quantum emitter with transition dipole moment d placed at a

6


Page 7 of 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

point r in the presence of the graphene nanostructure, 



 ¯ [b (r)]2 /D 3  2X ω j Γ = Γ0 + Im  , (1) ~ j 1/ηω − 1/ηj 

(3)

expressed in terms of the free space decay rate Γ0 = 4ω 3 |d|2 /3~c3 and the parameter bj (r) = R

d2 Rρj (R/D)(R − r) · d/|R − r|3 obtained from tabulated PWF data used in Ref. 36 (see

SI for further details). We now consider a scenario in which a graphene layer doped to a Fermi energy EF absorbs some thermal energy Q that elevates the electronic temperature T and conserves the doping charge density 28 gv gs Z ∞ dE E [f (E, µ, T ) + f (−E, µ, T ) − 1] , n= 2π(~vF )2 0

(4)

where gv = gs = 2 are the valley and spin degeneracies and f (E, µ, T ) = [e(E−µ)/kB T + 1]−1 is the Fermi-Dirac distribution for a chemical potential µ. Equation 4 is obtained for electron doping, with the term f in the square brackets accounting for the electron distribution in the partially-filled upper Dirac cone and 1 − f describing the holes in the lower cone. Following a previously established procedure, 28 we can recast eq 4 into an expression that implicitly relates EF , T , and µ as

EF kB T

2

=2

Z 0

∞

1

dx x

ex−µ/kB T + 1

−

1

ex+µ/kB T + 1

h

i

= 2 Li2 (−e−µ/kB T ) − Li2 (−eµ/kB T ) ,

where in the first line we define an integration variable x ≡ E/kB T that is eliminated in the second line by introducing the polylogarithm function Li2 . Using the above result along

7


Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 21

with the graphene heat capacity 28 Q (kB T )3 =β , A (~vF )2 where " # 1 1 2 Z∞ 1 EF 3 2 dx x + β= − π 0 ex+µ/kB T + 1 ex−µ/kB T + 1 3 kB T i o 2n h = −2 Li3 (−e−µ/kB T ) − Li3 (−eµ/kB T ) − EF 3 /3(kB T )3 , π

we obtain values of µ and T conserving the net electronic population specified by the Fermi energy EF when the electrons in a graphene layer with area A absorb an energy Q. For finite electron temperatures we use the local-RPA graphene conductivity 28,42 (

σω(1)

)

Z ∞ e2 i E/|E| D µ − dE f (E, µ, T ) , = 2 −1 2 π~ ω + iτ 1 − 4E / [~2 (ω + iτ −1 )2 ] −∞

(5)

where µD = µ + 2kB T log(1 + e−µ/kB T ) is a Drude-weight energy. This expression reduces to eq 1 in the limit of vanishing temperature (i.e., T = 0, µ = EF ). Quantum-Mechanical Description. We adopt an atomistic tight-binding (TB) description of electron states in the RPA to simulate the optical response of small graphene nanoislands containing a finite number of carbon atoms N . 37 In this so-called TB-RPA approach, the TB single-electron states |ji with energies Ej are expanded in the basis set of carbon 2p orbitals |li located at sites Rl according to |ji =

P

l

ajl |li, with coefficients ajl

weighing the amplitude of orbital |li in state |ji, and used to compute the first-order induced

8


Page 9 of 21

(b)

Thermalization

Cooling

Excitation

High thermo-optical response

Low electronic heat capacity

(c)

Singleplasmon excitation (Q=ℏωp)

(d) µ (eV)

T=0 T(ℏωp) 1.0

EF/eV = 1.0

Relative change (%)

0.8 0.5

0.6 0.4

0.0 0.0

Thermalization to local equilibrium (μ, T)

0.2 0.5

1.0

Cooling to (EF, 0)

1.0

1200

0.5

800

0.0 0

400

T (K)

(a)

σ abs/Area

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

abs (σμ,T -σ0abs)/σ0abs (EF-µ)/EF

-40 -80 0.0

Photon energy (eV)

0.5

1.0

Fermi energy (eV)

Figure 1: Single-plasmon thermo-optical switching. (a) Because of its unique conical dispersion relation, few hot carriers are needed to push a high electron temperature in graphene (low heat electronic capacity), also resulting in a large change of effective Drudeweight (large thermo-optical response). (b) Schematic illustration of an unperturbed, doped graphene nanohexagon at zero temperature (lower image): a single plasmon quantum of energy ~ωp is excited (left image) and absorbed, leading to a local thermal equilibrium state at chemical potential µ and electronic temperature T after a few 10s of fs (right image), slowly returning to the original equilibrium state after transferring the absorbed energy to the carbon lattice (∼ 1 ps). (c) Absorption cross section σ abs predicted in classical electrodynamic simulations (adopting the Drude conductivity with a phenomenological relaxation rate ~τ −1 = 50 meV) for a nanohexagon of diameter D = 10 nm at Fermi energies indicated by the color-coordinates curves; in each case, we plot the response at zero temperature (solid curves), from which we identify the plasmon resonance frequency ωp from maxima in σ abs , along with the response of the thermalized system for an excess energy ~ωp (dashed curves). (d) We present µ and T (upper panel with color-coordinated axes and dashed curve indicating µ = EF ) along with the relative change in absorption and chemical potential (lower panel) upon absorption of the zero-temperature nanohexagon plasmon energy ~ωp as a function of EF .

9


Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 21

charge density

ρind = l

X

(0)

χll0 φl0 ,

(6)

l0

where (0)

χll0 =

ajl a∗j 0 l a∗jl0 aj 0 l0 2e2 X (fj 0 − fj ) ~ jj 0 ω + iτ −1 − (Ej − Ej 0 ) /~

(7)

is the non-interacting RPA susceptibility (the factor of 2 arises from spin degeneracy), involving the state occupation factors fj , and

φl = −Rl · Eext +

X

vll0 ρind l0

l0

is the self-consistent electrostatic potential, with the first and second terms accounting for external and induced contributions, respectively, and vll0 denoting the Coulomb interaction between electrons in 2p carbon orbitals located at Rl and Rl0 . Numerical solution of eq 6 yields the induced charge, from which we characterize the optical response through the induced dipole moment pind =

P

l

ρind l Rl . We remark that the TB-RPA description of the

optical response in small (D . 10 nm) graphene nanoislands is found to be in excellent qualitative agreement with the more rigorous time-dependent density functional theory. 43 Through minor modifications to the above procedure we also compute the Purcell enhancement (i.e., Γ/Γ0 ) experienced by an emitter near a graphene nanoisland, as explained in the SI. The occupation factors fj in eq 7 effectively weigh the single-electron transitions between states |ji and |j 0 i. At zero temperature (fj = fj0 ), states are populated by setting fj0 = 1 for Ej < EF and fj0 = 0 for Ej > EF ; additionally, if the Fermi level is degenerate, fj0 is set to a uniform fraction for states with Ej = EF such that the total number of electrons Ne is preserved. When a heat energy Q is added to the system, we assume thermal equilibrium

10


Page 11 of 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

(fj = f (Ej , µ, T )) and impose conservation of energy, X

fj − fj0 Ej = 0,

(8)

j

and electronic population,

Ne = 2

X

fj = 2

j

X

fj0 .

(9)

j

Upon numerical solution of these eqs 8 and 9, we obtain Q-dependent values for µ and T . In our study of single-plasmon thermo-optical switching we consider small graphene nanoislands with hexagonal geometries that can host exclusively armchair-type edge-terminations, thus avoiding detrimental effects to the strength and tunability of plasmon resonances arising from zero-energy states introduced in the electronic spectrum by zigzag edges. 44,45 We consider in Figure 1 graphene hexagons of diameter D = 10 nm undergoing the process illustrated schematically in Figure 1a,b: a nanohexagon at zero temperature doped to some Fermi energy EF is resonantly excited and a single plasmon quantum with energy ~ωp decays into electron-hole pairs that thermalize to yield a local equilibrium distribution with temperature T and chemical potential µ. We characterize the graphene nanoisland optical response from the absorption cross-section σ abs = (4πω/c)Im{αω(1) }, obtaining the linear polarizability αω(1) analytically from eq 2 by truncating the sum after the dominant j = 1 dipolar plasmon mode and adopting the parameters η1 = −0.0681 and ξ1 = 0.7702 associated with the hexagonal geometry. 36 Throughout this work we assume a highly-conservative phenomenological relaxation rate ~τ −1 = 50 meV for graphene plasmons, unless otherwise specified. In Figure 1c we plot the graphene nanohexagon zero-temperature absorption spectra (solid curves) computed in the classical Drude description (i.e., retaining only the first term in eq 5) at various Fermi energies, yielding the single-plasmon energies ~ωp , along with the corresponding spectra obtained for the thermalized electronic distribution resulting from single-plasmon absorption (dashed curves). Although the energy associated with a single 11


Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

plasmon ~ωp scales precisely with

√

Page 12 of 21

EF in the Drude conductivity considered here, more sig-

nificant spectral shifts arise at lower Fermi energies, when the plasmon energy is distributed among fewer electrons. In Figure 1d we study the chemical potential µ and electronic temperature T (upper panel) along with the relative changes in µ and light absorption at the plasmon energy (lower panel) attained through single-plasmon absorption as functions of the Fermi energy. At very low doping levels, negligible plasmon energies lead to only minor changes in µ and T , while the situation quickly changes as the Fermi energy increases to EF ∼ 0.2 eV, where a maximum electron temperature of ∼ 1300 K is achieved. Significant changes in absorption and chemical potential also take place towards lower Fermi energies but vanish for zero doping, when the plasmon energy also vanishes, and we note that in the Drude conductivity implemented here, the relative change in resonance frequency scales strictly proportionally to the change in chemical potential. We remark that much longer plasmon lifetimes than the value τ ≈ 13 fs considered in Figure 1 have been reported in the literature, leading to narrower spectral features that undergo more dramatic shifts upon single-plasmon absorption, as shown in Figure S1 of the SI, although this advantage of longer-lived plasmons can actually hinder their optical excitation if the bandwidth of the exciting ultrashort pulse is much greater than the plasmon linewidth. The results of Figure 1 indicate that at higher doping levels the benefit of a increased plasmon energy is overcome by the division of this energy among additional electrons. However, as plasmon resonances in doped graphene nanostructures scale according to ∝

q

EF /D,

the energy per doping electron obtained from single-plasmon decay should increase in smaller islands. The electronic structure of graphene islands with dimensions . 10 nm is strongly dependent on quantum finite-size and edge-termination effects, warranting their description in an quantum-mechanical (QM) framework. 37 In Figure 2 we explore the classical-to-quantum transition in single-plasmon thermo-optical switching by comparing nanohexagon absorption spectra (upper-row panels) and radiative decay rate enhancement Γ/Γ0 , calculated at a fixed Fermi energy EF = 1.0 eV, before (solid curves) and after (dashed curves) single-plasmon

12


Page 13 of 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

absorption in the classical (CL, red curves) and QM (blue curves) pictures, where in the former method we adopt the local-RPA graphene conductivity in eqs 1 (zero temperature) and 5 (finite temperature) for improved accuracy. For the largest island under consideration (left panel upper row), with a lateral dimension of ∼ 7.1 nm, we obtain excellent agreement in the plasmon resonance shift and peak absorption change predicted by the CL and QM models, albeit with slight discrepancies in the spectral peak positions that are amplified in the Purcell enhancement by a ω −3 dependence of the decay rate. These discrepancies become much more dramatic in smaller islands (center and right panels, upper row), with suppression of absorption emphasized in the CL model and peak energy shifts in the QM picture. To gain insight for the latter, we analyze in the lower row of Figure 2 the electronic energy-level structure obtained in the QM model upon diagonalization of the tight-binding atomistic Hamiltonian for each nanohexagon and plot the electronic distribution function before and after single-plasmon absorption. Clearly, the re-distribution of electron population among the discrete states manifests in huge shifts and the introduction of new modes in the absorption spectrum upon single-plasmon heating, consistent with spectral changes induced by the addition or removal of individual electrons in small graphene nanoislands. 44,46 The results of Figure 2 portray the onset of a smooth departure between the thermooptical response predicted by CL and QM simulations for graphene islands with ∼ 7 nm lateral size, indicating that the excellent thermal properties endowed by the linear band structure of extended graphene persist in its nanostructured form, while further enhancement of this phenomenon (and disagreement between CL and QM descriptions) with decreasing size is attributed to increasing energy gaps between discrete electronic states. Ultimately, optimal single-plasmon thermo-optical switching is achieved in smaller graphene nanoislands doped with fewer electrons. Indeed, for nanohexagons containing ∼ 1000 or fewer carbon atoms, the Fermi energy does not well describe the initial charging state in the presence of large energy gaps between discrete electronic states (see lower row in Figure 2). Instead, we consider in Figure 3 situations in which an integral number of electrons Ne are added to a

13


Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

graphene nanohexagon, down to doping with only a few additional electrons, and entering the so-called molecular plasmonics regime. 43,46 The absorption spectra shown in the upper row of Figure 3 are dominated by quantum and finite-size effects, including peaks associated with the HOMO-LUMO transition appearing in the spectra even for undoped (Ne = 0) structures. The excitation and decay of a single plasmon quantum at the frequency of this high-energy peak for various charging states results in even higher obtained electronic temperatures (see lower row of Figure 3), which manifest prominently in the shift of the lower-energy plasmonic peak appearing for Ne > 0, which dominates the response when Ne 1. We further show in Figure S2 of the SI that the plasmonic radiative decay rate enhancement in the nanoislands of Figure 3 before and after single-plasmon absorption can vary by several orders of magnitude, suggesting that the initial radiative decay of a proximal quantum emitter can blockade subsequent decays within the time taken for the thermalized electronic distribution in graphene to cool (i.e., within ∼ 1 ps). A grand goal of nonlinear optics is the realization of an all-optical switch to produce light modulations in which the modulated light is phase-locked with the light gate signal. Our proposed switch does not preserve gate-signal phase coherence, but in exchange, it produces order-unity modulations with a single photon in a robust, integrable, solid-state platform: nanostructured graphene. One appealing route towards single-plasmon switching consists in the use of an optically-pumped quantum emitter that rapidly emits via Purcell enhancement into the plasmon mode, whereupon subsequent thermalization of the excited plasmon produced a change in the graphene optical response observed by an auxiliary probe field. The single-plasmon thermo-optical switching mechanism in nanostructured graphene is made possible by the intrinsically-low heat capacity of the 2D material and strong lightmatter coupling of its electrically-tunable plasmon resonances, the latter enabling significant absorption even in very small structures containing only a few doping electrons. These properties are essential to achieve significant modifications in the optical response, which result from the distribution of a finite plasmon energy among a small number of electrons to

14


Page 14 of 21

Page 15 of 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

produce dramatic changes in the electronic temperature and the chemical potential. Here we have considered an optimum scenario in which all of the single plasmon energy is distributed among thermalized electrons; in practice, a portion of this energy could be lost through inelastic scattering or coupling to phonons, thus reducing the magnitude of the effect if the graphene quality is not high. The change in optical response associated with single-plasmon absorption consists in a plasmon-blockade attained without intense external excitation, which in combination with proximal quantum emitters could open new avenues in optical sensing and quantum nano-optics.

Acknowledgments This work has been supported in part by the Spanish MINECO (MAT2017-88492-R and SEV2015- 0522), the European Research Council (Advanced Grant 789104-eNANO), the European Commission (Graphene Flagship 696656), the Catalan CERCA Program, and Fundació Privada Cellex.

References (1) Schuller, J. A.; Barnard, E. S.; Cai, W.; Jun, Y. C.; White, J. S.; Brongersma, M. L. Nat. Mater. 2010, 9, 193–204. (2) Kauranen, M.; Zayats, A. V. Nat. Photon. 2012, 6, 737–748. (3) Boyd, R. W. Nonlinear optics, 3rd ed.; Academic Press: Amsterdam, 2008. (4) Khurgin, J. B.; Sun, G. Opt. Express 2013, 21, 27460–27480. (5) Chang, D. E.; Vuletić, V.; Lukin, M. D. Nat. Photon. 2014, 8, 685–694. (6) Manjavacas, A.; Liu, J. G.; Kulkarni, V.; Nordlander, P. ACS Nano 2014, 8, 7630–7638.

15


Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(7) Brongersma, M. L.; Halas, N. J.; Nordlander, P. Nat. Nanotech. 2015, 10, 25–34. (8) Saavedra, J. R. M.; Asenjo-Garcia, A.; García de Abajo, F. J. ACS Photon. 2016, 3, 1637–1646. (9) Brown, A. M.; Sundararaman, R.; Narang, P.; GoddardIII, W. A.; Atwater, H. A. ACS Nano 2016, 10, 957–966. (10) Hartland, G. V.; Besteiro, L. V.; Johns, P.; Govorov, A. O. ACS Energy Lett. 2017, 2, 1641–1653. (11) Tan, S.; Liu, L.; Dai, Y.; Ren, J.; Zhao, J.; Petek, H. J. Am. Chem. Soc. 2017, 139, 6160–6168 . (12) Baffou, G.; Quidant, R. Chem. Soc. Rev. 2014, 3898–3907. (13) Clavero, C. Nat. Photon. 2014, 8, 95–103. (14) Perner, M.; Bost, P.; Lemmer, U.; von Plessen, G.; Feldmann, J.; Becker, U.; Mennig, M.; Schmitt, M.; Schmidt, H. Phys. Rev. Lett. 1997, 78, 2192–2195. (15) Voisin, C.; Del Fatti, N.; Christofilos, D.; Vallée, F. J. Phys. Chem. B 2001, 105, 2264–2280. (16) Khurgin, J. B.; Sun, G.; Chen, W. T.; Tsai, W.-Y.; Tsai, D. P. Sci. Rep. 2015, 5, 17899. (17) Jablan, M.; Buljan, H.; Soljačić, M. Phys. Rev. B 2009, 80, 245435. (18) Koppens, F. H. L.; Chang, D. E.; García de Abajo, F. J. Nano Lett. 2011, 11, 3370– 3377. (19) Ju, L.; Geng, B.; Horng, J.; Girit, C.; Martin, M.; Hao, Z.; Bechtel, H. A.; Liang, X.; Zettl, A.; Shen, Y. R.; Wang, F. Nat. Nanotech. 2011, 6, 630–634.

16


Page 16 of 21

Page 17 of 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

(20) Fei, Z.; Rodin, A. S.; Andreev, G. O.; Bao, W.; McLeod, A. S.; Wagner, M.; Zhang, L. M.; Zhao, Z.; Thiemens, M.; Dominguez, G.; Fogler, M. M.; Neto, A. H. C.; Lau, C. N.; Keilmann, F.; Basov, D. N. Nature 2012, 487, 82–85. (21) Grigorenko, A. N.; Polini, M.; Novoselov, K. S. Nat. Photon. 2012, 6, 749–758. (22) García de Abajo, F. J. ACS Photon. 2014, 1, 135–152. (23) Gonçalves, P. A. D.; Peres, N. M. R. An Introduction to Graphene Plasmonics; World Scientific: Singapore, 2016. (24) Ni, G. X.; McLeod, A. S.; Sun, Z.; Wang, L.; Xiong, L.; Post, K. W.; Sunku, S. S.; Jiang, B.-Y.; Hone, J.; Dean, C. R.; Fogler, M. M.; Basov, D. N. Nature 2018, 557, 530–533. (25) Vafek, O. Phys. Rev. Lett. 2006, 97, 266406. (26) Ni, G. X.; Wang, L.; Goldflam, M. D.; Wagner, M.; Fei, Z.; McLeod, A. S.; Liu, M. K.; Keilmann, F.; Özyilmaz, B.; Neto, A. H. C.; Hone, J.; Fogler, M. M.; Basov, D. N. Nat. Photon. 2016, 10, 244–248. (27) Tan, S.; Argondizzo, A.; Wang, C.; Cui, X.; Petek, H. Phys. Rev. X 2017, 7, 011004. (28) Yu, R.; Manjavacas, A.; García de Abajo, F. J. Nat. Commun. 2017, 8, 2. (29) Cox, J. D.; García de Abajo, F. J. Optica 2018, 5, 429–433. (30) Woessner, A.; Lundeberg, M. B.; Gao, Y.; Principi, A.; Alonso-González, P.; Carrega, M.; Watanabe, K.; Taniguchi, T.; Vignale, G.; Polini, M.; Hone, J.; Hillenbrand, R.; Koppens, F. H. Nat. Mater. 2015, 14, 421–425. (31) Johannsen, J. C.; Ulstrup, S.; Cilento, F.; Crepaldi, A.; Zacchigna, M.; Cacho, C.; Turcu, I. C. E.; Springate, E.; Fromm, F.; Raidel, C.; Seyller, T.; Parmigiani, F.; Grioni, M.; Hofmann, P. Phys. Rev. Lett. 2013, 111, 027403. 17


Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(32) Gierz, I.; Petersen, J. C.; Mitrano, M.; Cacho, C.; Turcu, I. C. E.; Springate, E.; Stöhr, A.; Köhler, A.; Starke, U.; Cavalleri, A. Nat. Mater 2013, 12, 1119–1124. (33) Shi, S.-F.; Tang, T.-T.; Zeng, B.; Ju, L.; Zhou, Q.; Zettl, A.; Wang, F. Nano Lett. 2014, 14, 1578–1582. (34) Jadidi, M. M.; König-Otto, J. C.; Winnerl, S.; Sushkov, A. B.; Drew, H. D.; Murphy, T. E.; Mittendorff, M. Nano Lett. 2016, 16, 2734–2738. (35) Jadidi, M. M.; Daniels, K. M.; Myers-Ward, R. L.; Gaskill, D. K.; König-Otto, J. C.; Winner, S.; Sushkov, A. B.; Drew, H. D.; Murphy, T. E.; Mittendorff, M. ACS Photon. 2019, 6, 302–307 . (36) Yu, R.; Cox, J. D.; Saavedra, J. R. M.; García de Abajo, F. J. ACS Photon. 2017, 4, 3106–3114. (37) Thongrattanasiri, S.; Manjavacas, A.; García de Abajo, F. J. ACS Nano 2012, 6, 1766–1775. (38) Higuchi, T.; Stockman, M. I.; Hommelhoff, P. Phys. Rev. Lett. 2014, 113, 213901. (39) Higuchi, T.; Heide, C.; Ullmann, K.; Weber, H. B.; Hommelhoff, P. Nature 2017, 550, 224–228 . (40) Wunsch, B.; Stauber, T.; Sols, F.; Guinea, F. New J. Phys. 2006, 8, 318. (41) Hwang, E. H.; Das Sarma, S. Phys. Rev. B 2007, 75, 205418. (42) Gusynin, V. P.; Sharapov, S. G.; Carbotte, J. P. Phys. Rev. Lett. 2006, 96, 256802. (43) Manjavacas, A.; Marchesin, F.; Thongrattanasiri, S.; Koval, P.; Nordlander, P.; Sánchez-Portal, D.; García de Abajo, F. J. ACS Nano 2013, 7, 3635–3643. (44) Manjavacas, A.; Thongrattanasiri, S.; García de Abajo, F. J. Nanophotonics 2013, 2, 139–151. 18


Page 18 of 21

Page 19 of 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

(45) Christensen, T.; Wang, W.; Jauho, A.-P.; Wubs, M.; Mortensen, N. A. Phys. Rev. B 2014, 90, 241414(R). (46) Lauchner, A.; Schlather, A.; Manjavacas, A.; Cui, Y.; McClain, M. J.; Stec, G. J.; García de Abajo, F. J.; Nordlander, P.; Halas, N. J. Nano Lett. 2015, 15, 6208–6214.

19


Nano Letters

CL

σ abs/Area

QM

3 nm

7.1 nm

CL

3 nm

T=0 T(ℏωp)

0.16

CL

QM

Γ/Γ0

QM

104 103 0.50

CL

0.75

1.00

1.25

QM

0.00

106

105

CL

QM

104

104

103

103

102

102 0.75

Photon energy (eV)

1.00

1.25

1.50

0.5

0.5

0.5

1

2

Electronic energy (eV)

3

0.0

-1

0

1

2


QM

1.5

2.0

2.5

Photon energy (eV) 1.0

0

CL

Photon energy (eV) 1.0

-1

QM

101 1.0

1.0

0.0

3 nm

N=114

CL

0.08

0.00

105

CL

105

2.0 nm

N=546

QM

3 nm

0.16

0.12

T=0 T(ℏωp)

106

CL

3 nm

0.24

0.00 107

QM

3 nm

4.5 nm

N=1302

0.32

Occupation factor

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 21

3

0.0

-1

0

1

2

3


Figure 2: Classical and quantum-mechanical pictures of single-plasmon-induced thermo-optical modulation. We plot the absorption cross-section (σ abs , upper-row panels) and the radiative decay rate enhancement (Γ/Γ0 , center-row panels) predicted in classical (CL, red curves) and quantum-mechanical (QM, blue curves) descriptions of graphene nanohexagons illustrated above, with lateral sizes of 7.1 nm (left column), 4.5 nm (center column), and 2.0 nm (right column), before (solid curves) and after (dashed curves) single plasmon absorption. The color-coded illustrations above depict the systems considered in the CL and QM pictures, indicating the impinging electric field polarization used in computing σ abs , the external dipole position and orientation adopted to evaluate Γ/Γ0 , and the number of carbon atoms N in each exclusively armchair-edged island considered in QM simulations. The lower-row panels show the discrete spectrum of electronic states (vertical purple lines) and the electronic occupation factor before (gray filled curve) and after (orange filled curve) single-plasmon absorption in the QM description for the hexagons considered above.

20


N=1302 1.5 N e=

N=546

T=0 K T(ℏωp)

1.5 N e=

24

σ abs/Area

1.0

0.5

0.0

N=114

T=0 K T(ℏωp)

1.5 N e=

12

20

1.0

10

5

1.0

8

4

12

6

3

0.5

8

0.0

0.5

4

0.0 0.5

1.0

1.5

1

0

0.0

1.0

1.5

2.0

5

-0.5

Ne

1

24

0

2

3

Photon energy (eV) 5

0.0

-0.5 16

0

Photon energy (eV)

0.0

8

0

0.0 0.5

Photon energy (eV)

0

2

2

0

T=0 K T(ℏωp)

6

16

4

EF-µ (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

5

0.0

-0.5 0

4

8

12

Ne

0

0

2

4

6

0

Temperature (103 K)

Page 21 of 21

Ne

Figure 3: Thermo-optical switching by molecular plasmons. We simulate the absorption spectra of the graphene nanohexagons considered in Figure 2 doped with integral numbers of electrons Ne (upper row) before (solid curves) and after (dashed curves) the absorption of a single plasmon, with the charging states Ne indicated by the color-coordinated curves in each panel. These results are obtained in the QM description. For each hexagon we study the chemical potential shift EF − µ (blue squares) and electronic temperature obtained by absorbing the energy Q = ~ωp associated with the dominant absorption peak at T = 0 as a function of charging configuration.

21


Single-Plasmon Thermo-Optical Switching in Graphene | Nano Letters

Recommend Documents