Article pubs.acs.org/JPCC
Cite This: J. Phys. Chem. C 2017, 121, 27597−27602
Quantum Plasmon Engineering with Interacting Graphene Nanoflakes David Zs. Manrique,† Jian Wei You,† Hanying Deng,†,‡ Fangwei Ye,‡ and Nicolae C. Panoiu*,† †
Department of Electronic and Electrical Engineering, University College London, Torrington Place, London WC1E 7JE, U.K. School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
‡
S Supporting Information *
ABSTRACT: We investigate the potential merits of using nanometer-sized graphene flakes as building blocks for two-dimensional (2D) quantum metamaterials. The choice of the building blocks of metamaterials is crucial to our ability to design quantum metamaterials with desired properties. In this context, graphene nanostructures are promising candidates to fulfill this role as they can be easily grown either by bottom-up chemical synthesis or topdown electron beam patterning in various shapes, topologies, and sizes, down to the molecular scale. This provides a broad range of parameters to tune the optical properties of graphene-based 2D quantum metamaterials. By using time-dependent density functional theory and quantum chemistry computations, we demonstrate that the graphene-based nanostructures accommodate collective charge oscillations, called quantum plasmons, which are qualitatively different in key aspects from their classical counterparts. In particular, our analysis reveals that the exponents characterizing the power-law scaling of plasmon energy with the size of the graphene flakes are markedly different in the classical and quantum regimes, proving that the quantum plasmons cannot be viewed as a trivial extension of the classical ones to the small-flake limit. In addition, the physical properties of quantum plasmons in graphene nanostructures exhibit significant dependence on their shape and size, and external control can be readily achieved with excess charge. Finally, we find that the energy of the fundamental quantum plasmon mode of triangular nanoflakes is larger than that of hexagonal nanoflakes, whereas in the classical case, the plasmon energy ordering is reversed.
■
INTRODUCTION Over the past few years, graphene has been playing an increasingly important role in quantum plasmonics,1−17 mainly due to its strong interaction with light, its potential for unprecedented optical field confinement, and advancements in nanofabrication techniques that allow one to create graphene structures with nanometer-size features. In particular, graphene plasmons, which are collective charge oscillations, are much more confined and low loss than surface plasmons in conventional plasmonic materials, such as metallic nanoparticles. Inheriting these remarkable properties from the extended graphene sheet, graphene nanoflakes (GNFs), which are nanometer-sized, zero-dimensional (0D) graphene structures, provide an ideal platform for studying the physical properties of localized surface plasmons in ultrasmall plasmonic systems. Similar to extended graphene sheets, these finite-sized graphene structures also exist in a stable form, being either grown by using bottom-up chemical synthesis methods or tailored by top-down electron-beam techniques.18 The recent advancements in growing nanometer-sized GNFs have paved a tantalizing route for systematically engineering GNFs to manipulate and study light−matter interactions on the deep-subwavelength scale. In particular, one expects that quantum effects would become more important by scaling down the size of graphene dots, which among other things would have an increasingly relevant signature on the physical © 2017 American Chemical Society
properties of localized plasmons. This has already been observed in relation to quantum plasmons of nanometersized metallic particles;19,20 however, GNFs provide a much more suitable testing ground for an in-depth investigation of key aspects of the classical-to-quantum transition of localized plasmons. In particular, due to the 2D nature of graphene, a GNF of a certain size contains fewer atoms as compared to a 3D particle of the same size, which significantly reduces the computational time required to simulate such systems. Moreover, physical properties of graphene can be readily tuned via chemical doping, which allows one to easily control the experimental conditions under which plasmons are investigated. In this context, by using time-dependent density functional theory (TDDFT) based simulations,21,22 we demonstrate in this Article that the localized plasmons of molecular-scale GNFs can be accurately engineered and controlled, with their physical properties being qualitatively different from what classical plasmonic physics predicts.4 From a practical point of view, these findings could lead to novel design rules for metamaterials based on molecular GNFs or highly sensitive, ultracompact sensors. Received: September 20, 2017 Revised: November 21, 2017 Published: November 23, 2017 27597
DOI: 10.1021/acs.jpcc.7b09358 J. Phys. Chem. C 2017, 121, 27597−27602
Article
The Journal of Physical Chemistry C If the nanoflake size is larger than the electron mean free path, then the resonances in the optical absorption spectra are the result of classical plasmonic dynamics of confined electrons, which are well described by Maxwell equations. However, on the molecular scale, the electronic motion is restricted by quantum confinement and the spectral features are associated with molecular excitations. The electronic motion on the molecular scale, although extended across the GNF, is not due to metallic conduction but transitions between delocalized quantum states of the molecule. Therefore, the submolecular electronic properties on such a small scale cannot originate from macroscopic material properties, such as electric conductivity and permittivity. Since the underlying governing physics is markedly different in the two physical regimes, namely, of quantum nature for molecular-scale GNFs and classical character described by Maxwell equations in the case of large GNFs, qualitative differences in the optical properties of GNF plasmons are expected. It has been previously shown that both the quantum and classical plasmonic responses of GNFs depend on the size, shape, and chemical and electromagnetic environments.13,23 In particular, triangular GNFs have been explored using tightbinding (TB) approach,15 and in a pioneering study,5 the spectral properties of linear molecules have been experimentally measured in the context of molecular plasmonics. The molecular-sized GNFs have a HOMO−LUMO gap ranging from 0.3 to 3 eV5,6,10,13,14 that partially overlaps with the visible-light spectrum, a property which can be relevant to energy-harvesting applications.24 In the classical regime, on the other hand, the quasi-static plasmon energy scales as ℏω ∝ (κ/ D)1/2, where D is the size of the GNF and κ is a dimensionless shape-dependent coefficient. This classical plasmon energy is typically smaller than 1 eV if D > 50 nm9 and, remarkably, the form factor κ is independent of the size of the graphene dot. By using the full TDDFT approach available in OCTOPUS,21,25 we demonstrate that the size dependence of quantum plasmons in molecular GNFs is markedly different from the classical, quasi-static behavior. In addition, we show that excess charge and intermolecular separation can be used effectively to tune the GNF absorption spectra. Exploring the qualitative differences between the physics on different spatial scales governing the same underlying material structure can lead to novel design rules that must be employed when constructing metamaterials from building blocks that might have both quantum and classical plasmonic features.
Figure 1. Geometrical configuration used in simulations and a few examples GNF structures investigated in this work. Two classes of geometries are studied, namely, hexagonal (H) and triangular (T) structures. The hexagonal and triangular structures are labeled with Hn and Tn, respectively, where n is the number of six-atom hexagons located along one side of the GNF. (a) In all simulations, the planar GNF lies in the x−y plane and the optical excitation is an x-polarized plane wave. (b) Examples of GNF structures. The size of the GNF is measured by the distance D. (c) Dimer structures oriented along the xaxis and the corresponding nomenclature used to identify them, where the prefix di- refers to dimer and the postfix labels A and B refer to variations in the arrangements of GNFs relative to each other.
the TB and Kohn−Sham Hamiltonians, the polarizability is calculated with the approximate noninteracting expression αxx(ω) =
2 ℏ
∑ n→m
ωnm|⟨ϕm|e x̂|ϕn⟩|2 2 ωnm − ω 2 − iω Γ
(1)
where n (m) labels the occupied (unoccupied) states, |ϕn⟩ and En are the nth eigenstate and eigenenergy of the system, respectively, ⟨ϕm|ex̂|ϕn⟩ are the transition dipole moments, ωnm = (Em − En)/ℏ is the excitation energy of the transition, and ℏΓ = 0.1 eV is associated with the lifetime of the excited state, which is chosen to be sufficiently smaller than the energy level spacing of molecular GNFs. In the case of the TDDFT approach, the polarizability is calculated directly from its definition, μx(ω) = αxx(ω)Ex(ω), where μx(ω) and Ex(ω) are the Fourier transformed dipole moment and electric field, respectively. To perform DFT and TDDFT simulations, we used the OCTOPUS code.25 In the DFT case, we used OCTOPUS to compute the transition dipole moments and excitation energies, which were subsequently used to evaluate eq 1. In the TDDFT case, we used OCTOPUS to calculate the molecular charge density evolution, ρ(t), for a period of time T as response to a time-dependent driving electric field. To compute αxx(ω), a Dirac-kicking driving-field was introduced and μx(t) was computed at each time step from the evolved charge density. A more elaborate description of the relevant methods to compute the dynamic polarizability with OCTOPUS can be found in a reference publication describing the code functionality,25 whereas a complete list with the simulation parameters is provided in the Supporting Information. All of our simulations were spin-polarized, and GGA exchange-
■
METHODS In order to characterize the electronic dynamics in molecularscale GNFs, we performed ab initio simulations of various structures, with the relevant ones being schematically shown in Figure 1a−c. The specific configuration used to compute the optical response of GNFs is presented in Figure 1a, where the purple x−z plane illustrates the plane normal onto the GNF that contains the time-periodic driving electric field. Since the molecular sizes are much smaller than the typical optical wavelengths, the spatial variation of the driving field is neglected. In all cases, the GNF structures lie in the x−y plane. We initiated our investigations by computing the dynamical polarizabilities, αxx(ω), based on TB and density functional theory (DFT) Hamiltonians (Kohn−Sham Hamiltonian) and then concluded with the TDDFT approach to account for the many-body excitations and dynamical effects, for which the DFT and TB approaches are known to be deficient.21,22,25 For 27598
DOI: 10.1021/acs.jpcc.7b09358 J. Phys. Chem. C 2017, 121, 27597−27602
Article
The Journal of Physical Chemistry C
Our simulations show that in GNFs the resonances defined by the relation Re(αxx) = 0 are characterized by strong collective charge oscillations. In order to see this, let us consider the spectra in Figure 2a obtained using TDDFT, where we labeled as A and B the two resonances identified to satisfy the condition Re(αxx) = 0. The precise spectral location of the resonance peaks varies by about 1 eV with the computational method used to calculate the spectra; however, qualitatively, all methods show two distinct group of resonances around the peaks labeled by A and B, which we refer to as low- and highenergy resonances, respectively. The low-energy resonance, A, located at ℏω = 3.53 eV, corresponds to the first excitations, which, in the TB approach, is exactly the HOMO−LUMO gap. The higher-energy resonance, B, corresponds to higher-level excitations and has contributions from the transitions from the HOMO state to the LUMO + 1 and from the HOMO − 1 state to the LUMO. In order to explore the nature of charge density oscillations at resonance frequencies, we performed continuous-wave (CW) TDDFT simulations with a driving electric field, Ex(t) = E0cos(ω0t), which models a laser field impinging perpendicularly onto the plane of the GNF, as per Figure 1a. The electric field E0 was slowly ramped up, as shown, for example, in Figure S5 in the Supporting Information, so that the spectrum of the excitation light is narrow enough. The frequency ω0 of the driving electric field was chosen to coincide with the frequency of the specific resonance under investigation. In Figure 2, the two insets A and B show the excess charge with respect to the ground state, Δρ(t) = ρ(t) − ρ(0), superimposed over the underlying H2 geometry. The charge density is calculated at a time at which the dipole moment of the structure reaches its maximum value. Note that the phase of the dipole moment is delayed by π/2 from the phase of the driving electric field, as at resonance the polarizability is purely imaginary. The insets A and B in Figure 2 reveal that the charge densities have different delocalization patterns, as the underlying molecular orbitals participating in the transitions are different. In the case of B, the charge distribution is more extended over the whole surface of the molecule, whereas for A the charge density is more localized at the edges of the molecular structure. In Figure 3, we plot the absorption strength function calculated for triangular and hexagonal GNFs with variable size, whereas the inset plots in this figure show the variation of resonance energy with the number of carbon atoms, NC, in the nanoflake. The size of each bubble is proportional to the amplitude of the resonance. We also schematically depict the three GNFs with the smallest sizes as insets. The curves and the bubbles are color-coded such that the black, red, and green colors correspond to the small, middle, and large sized GNFs, respectively. As illustrated in Figure 3, the energy of the quantum plasmon decreases with the size of the GNF, a result that is valid in a classical context, too. In order to illustrate the qualitative differences between the physical properties of classical and quantum plasmons, we performed optical response simulations based on the Maxwell equations of triangularly and hexagonally shaped GNFs with sizes in the range of 30−300 nm. The simulation details are provided in the Supporting Information. As Figure 3 shows, for a given shape, in the quantum regime there are multiple resonance peaks in the spectra. The number of these resonances increases with the size of GNF, and they start to merge if the size of the GNF is further increased. At a more careful inspection, Figure 3 reveals that in the hexagonal case
correlation potential was used with Perdew−Burke−Ernzerhof (PBE) parametrization.26 In the TB approach, the GNF is described by a single site TB graphene-like Hamiltonian, using nearest-neighbor interaction with a coupling constant of γ = 2.6 eV between neighboring carbon atoms. The hydrogen atoms in this case are neglected. From the nearest-neighbor TB Hamiltonian, the eigenstates and energies are computed and the complex polarizability is calculated using eq 1.
■
RESULTS AND DISCUSSION As illustrated in Figure 1a, the driving electric field is applied in the x-direction; consequently, we computed the polarizability αxx. As elaborated in ref 27, for structures with two equivalent axes αyy is related to αxx. In Figure 2, the results of the
Figure 2. Absorption spectra corresponding to the H2 structure, computed using different quantum mechanical approaches. (a) The imaginary part of the polarizability is plotted as optical absorption 2mω Im[αxx(ω)]. The insets A and B show the strength, S(ω) = πℏe 2 TDDFT computed charge density induced by a CW and correspond to the two marked resonance frequencies. (b) The real part of the 2mω polarizability, normalized as R(ω) = 2 Re[αxx (ω)]. πℏe
calculations for the GNF H2 are shown. The spectra correspond to the same molecule, but they are computed with different approaches as indicated in the legend. In order for our results to be easier to interpret from an experimental point of view, we plot in Figure 2a,b the imaginary and real parts of the polarizability, respectively, multiplied by a scaling factor equal to 2mω2 . Although TDDFT is expected to provide πℏe
the most accurate results out of the three different approaches, qualitatively the spectral landscapes of the three cases are similar. These plots show that the imaginary part reaches a large resonance peak when the real part of the polarizability is zero, which suggests that around the corresponding frequency the spectrum is similar to that of a two-level system of dipole oscillators, that is, similar to that of plasmon excitations. If the identified resonance feature belongs to a strong charge oscillation caused by a transition between largely delocalized states, then the corresponding excitation is referred to as nanoplasmon,28 molecular plasmon,5 or, to emphasize the underlying charge-transfer mechanism, quantum plasmon. We stress that the resonances for which the relation Re(αxx) = 0 holds cannot be due to multilevel electronic transitions because this condition is not satisfied for such transitions (see eq 1). 27599
DOI: 10.1021/acs.jpcc.7b09358 J. Phys. Chem. C 2017, 121, 27597−27602
Article
The Journal of Physical Chemistry C
Figure 4. Comparison of the size-dependence of the classical and the quantum plasmon resonance energies. For the hexagonal triangular shapes, D labels the diameter and the side length, respectively. The lines are linearly fitted to the data points, and the corresponding slopes are listed in Table 1. The dashed lines are the asymptotic extensions of the classically computed size dependence to the quantum regime.
The exponents β computed with different theoretical methods are summarized in Table 1. In the classical regime, Figure 3. Absorption spectra of hexagonal and triangular GNFs with different sizes. (a) Oscillator strength function for three hexagonal GNFs, H2, H3, and H4, whose diameter is given in the inset. The subplot shows the dependence of the resonance peak energies on the number of carbon atoms in a GNF. The size and shade of the bubble are proportional to the amplitude of the resonance peak. The three smallest hexagonal GNFs are color-coded black, red, and green, respectively, with the yellow color belonging to larger GNFs. (b) The same as in (a) but calculated for three triangular GNFs, T5, T6, and T7.
Table 1. Scaling Exponent β for Resonances of Hexagonal and Triangular Shapes Computed with Several Different Methods
there are two identifiable resonance groups, which can be tracked as the size increases; we refer to these resonance groups as the high- and low-energy quantum plasmon resonances. In the triangular case, we selected only the low-energy quantum resonances, which we refer to as the first excitation energy. Correspondingly, in the classical regime that we will discuss later, the low- and high-energy resonances refer to the first- and second-order plasmon resonances, respectively. In Figure 4, the quantum plasmon resonances determined using TDDFT and the classical ones calculated from Maxwell equations are shown as a function of the GNF size. As can be observed in this figure, the size dependence of both the quantum and classical plasmon frequencies can be approximately expressed as ℏω = σDβ, where β is a size-independent exponent and σ is a constant specific to the shape and type of the resonance. While in the classical regime β is independent of the shape of the graphene flake, in agreement with predictions based on calculations performed within the quasi-static approximation, in the quantum regime the value of β can vary with the GNF shape. More importantly, in the quantum regime the low- and high-energy resonances have different exponents, in contrast to the classical first- and second-order plasmons depicted with blue and green markers in Figure 4, respectively. This suggests that at the quantum level there are competing effects that lead to the excitation of plasmons, with the dominant one being dependent on the plasmon energy and molecular structure.
resonance
TB
Kohn−Sham (DFT)
TDDFT
Maxwell eqs
hexagon high energy hexagon low energy triangle low energy
−1.22 −0.67
−1.12 −0.65
−0.19 −1.05 −0.61
−0.50 −0.50 −0.49
β ≈ −1/2, whereas in the quantum regime the values are somewhat dependent on the method used; however, qualitatively they display similar trends, as shown in Figure S12 in the Supporting Information. It is also remarkable that if the low-energy plasmon is defined as the first excitation energy in the quantum regime and the first plasmon energy in the classical regime, then the ordering of the low plasmon resonance energies of the hexagonal and triangular GNFs changes as the flake size crosses from the classical to the quantum regime. Graphene nanoflakes on a nonmetallic substrate can be infused with excess charges.15 Alternatively, a strong donor or acceptor environment can modify the charge state of the GNFs, too. In order to demonstrate the strong effect of excess charge on the properties of molecular-scale GNFs, we computed the absorption spectra of GNF H3 with charges q/|e| = −8, ..., +8 using the DFT method. The results of these calculations are summarized in Figure 5a. The energy shift of the resonances does not vary monotonically with the amount of charge, nor is it symmetric around zero. Furthermore, the low-energy resonance of the neutral GNF splits into multiple resonances if the charge state is changed to slightly negative values and, significantly different, new low-energy resonances emerge. This drastic transformation is attributable to the modified selfconsistent level structure due to the reduced or increased molecular orbital occupations. 27600
DOI: 10.1021/acs.jpcc.7b09358 J. Phys. Chem. C 2017, 121, 27597−27602
Article
The Journal of Physical Chemistry C
qualitative different in the classical and quantum regimes. This scaling behavior implies that the quantum plasmon frequencies cannot be parametrized with macroscopic conductivity or permittivity parameters. Our findings suggest that further exploration is required in order to engineer 2D metamaterials in the cross-scale domain from molecular to macroscopic sized building blocks. From an applications point of view, our study reveals that in the quantum regime the molecular-sized graphene building blocks could be used for 2D metamaterials with novel properties, for ultrasensitive single-molecule detection, and for molecular-scale plasmon-induced electric field enhancement.
■
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.7b09358. Computational details and more detailed results of the simulations (PDF)
■
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
David Zs. Manrique: 0000-0002-6974-9592 Nicolae C. Panoiu: 0000-0001-5666-2116
Figure 5. (a) Absorption dispersion map showing the dependence of resonance locations on the excess charge normalized to the number of carbon atoms in the GNF, NC, calculated for the GNF H3 in vacuum. The red, black, and blue vertical lines correspond to charges of −1, 0, and +1, respectively. (b) Shift of resonances of dimer structures vs the GNF interseparation distance. The values of ℏωp and λp correspond to the resonances of the monomer GNF.
Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS This work was supported by the European Research Council (ERC), Grant Agreement no. ERC-2014-CoG-648328. H.D. was supported by a China Scholarship Council Studentship. F.Y. and H.D. acknowledge financial support from NSFC (No. 61475101). The authors acknowledge the use of the UCL Legion High Performance Computing Facility (Legion@UCL) and associated support services and the ARCHER UK National Supercomputing Service (http://www.archer.ac.uk) in the completion of this work. We thank Martin Weismann for stimulating discussions.
In contrast to electronic structure changes, more subtle variations of the location of resonances are possible via tuning of intermolecular separation. In order to illustrate this idea, we performed simulations on dimer structures with varying intermolecular separation. The conclusions of this analysis are presented in Figure 5b, where we plot the wavelength shift of resonances as a function of intermolecular separation. We found that the intermolecular interactions induce a blue shift of resonances of about 10 nm when the separation distance between two GNFs varies by 1 nm. However, as D and d have commensurate values, the inter-GNF interaction strongly depends on the local electronic structure. This local sensitivity is similar to what is observed in the classical case; however, on the molecular scale, the electromagnetic near-field is determined by the orbital structure of the GNF.
■
REFERENCES
(1) Koppens, F. H. L.; Chang, D. E.; Garcia de Abajo, F. J. Graphene plasmonics: a platform for strong light-matter interactions. Nano Lett. 2011, 11, 3370−3377. (2) Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Zhang, Y.; Dubonos, S. V.; Grigorieva, I. V.; Firsov, A. A. Electric field effect in atomically thin carbon films. Science 2004, 306, 666−669. (3) Wang, T.; Nijhuis, C. A. Molecular electronic plasmonics. Applied Materials Today 2016, 3, 73−86. (4) Thongrattanasiri, S.; Manjavacas, A.; Garcia de Abajo, F. J. Quantum finite-size effects in graphene plasmons. ACS Nano 2012, 6, 1766−1775. (5) Lauchner, A.; Schlather, A. E.; Manjavacas, A.; Cui, Y.; McClain, M. J.; Stec, G. J.; Garcia de Abajo, F. J.; Nordlander, P.; Halas, N. J. Molecular plasmonics. Nano Lett. 2015, 15, 6208−6214. (6) Csaki, A.; Schneider, T.; Wirth, J.; Jahr, N.; Steinbruck, A.; Stranik, O.; Garwe, F.; Muller, R.; Fritzsche, W. Molecular plasmonics: light meets molecules at the nanoscale. Philos. Trans. R. Soc., A 2011, 369, 3483−3496. (7) Thongrattanasiri, S.; Manjavacas, A.; Nordlander, P.; de Abajo, F. J. G. Quantum junction plasmons in graphene dimers. Laser and Photonics Reviews 2013, 7, 297−302.
■
CONCLUSIONS In conclusion, we have demonstrated that although there are no classical plasmons in molecular-sized GNFs, certain quantum transitions between the ground and exited states can be interpreted as quantum plasmons. In these transitions, the collective electron dynamics are controlled by the valence orbital structure, which allows, for example, atomically precise design of the pattern of electronic motion. Moreover, intermolecular coupling between two in-plane nanoflakes can cause a spectral blueshift of quantum plasmon resonances of about 10 nm. Similarly to classical plasmons, the quantum plasmon frequencies in GNFs depend on the size and shape of the GNFs; however, the size dependence of plasmon energy is 27601
DOI: 10.1021/acs.jpcc.7b09358 J. Phys. Chem. C 2017, 121, 27597−27602
Article
The Journal of Physical Chemistry C (8) Manjavacas, A.; Nordlander, P.; Garcia de Abajo, F. J. Plasmon blockade in nanostructured graphene. ACS Nano 2012, 6, 1724−1731. (9) Fang, Z.; Thongrattanasiri, S.; Schlather, A.; Liu, Z.; Ma, L.; Wang, Y.; Ajayan, P. M.; Nordlander, P.; Halas, N. J.; Garcia de Abajo, F. J. Gated tunability and hybridization of localized plasmons in nanostructured graphene. ACS Nano 2013, 7, 2388−2395. (10) Yu, R.; Cox, J. D.; de Abajo, F. J. G. Nonlinear plasmonic sensing with nanographene. Phys. Rev. Lett. 2016, 117, 123904. (11) Manjavacas, A.; Thongrattanasiri, S.; Chang, D. E.; de Abajo, F. J. G. Temporal quantum control with graphene. New J. Phys. 2012, 14, 123020. (12) Rosolen, G.; Maes, B. Asymmetric and connected graphene dimers for a tunable plasmonic response. Phys. Rev. B: Condens. Matter Mater. Phys. 2015, 92, 205405. (13) Mansilla Wettstein, C.; Bonafe, F. P.; Oviedo, M. B.; Sanchez, C. G. Optical properties of graphene nanoflakes: shape matters. J. Chem. Phys. 2016, 144, 224305. (14) Kuc, A.; Heine, T.; Seifert, G. Structural and electronic properties of graphene nanoflakes. Phys. Rev. B: Condens. Matter Mater. Phys. 2010, 81, 085430. (15) Cox, J. D.; de Abajo, F. J. G. Electrically tunable nonlinear plasmonics in graphene nanoislands. Nat. Commun. 2014, 5, 5725. (16) Manjavacas, A.; Thongrattanasiri, S.; de Abajo, F. Plasmons driven by single electrons in graphene nanoislands. Nanophotonics 2013, 2, 139−151. (17) Felbacq, D.; Antezza, M. Quantum metamaterials: a brave new world. SPIE Newsroom 2012, 19, 4296. (18) Shi, Z.; Yang, R.; Zhang, L.; Wang, Y.; Liu, D.; Shi, D.; Wang, E.; Zhang, G. Patterning graphene with zigzag edges by self-aligned anisotropic etching. Adv. Mater. 2011, 23, 3061−3065. (19) Scholl, J. A.; Koh, A. L.; Dionne, J. A. Quantum plasmon resonances of individual metallic nanoparticles. Nature 2012, 483, 421−427. (20) Savage, K. J.; Hawkeye, M. M.; Esteban, R.; Borisov, A. G.; Aizpurua, J.; Baumberg, J. J. Revealing the quantum regime in tunnelling plasmonics. Nature 2012, 491, 574−577. (21) Runge, E.; Gross, E. K. U. Density-functional theory for timedependent systems. Phys. Rev. Lett. 1984, 52, 997−1000. (22) Morton, S. M.; Silverstein, D. W.; Jensen, L. Theoretical studies of plasmonics using electronic structure methods. Chem. Rev. 2011, 111, 3962−3994. (23) Schmidt, F. P.; Ditlbacher, H.; Hofer, F.; Krenn, J. R.; Hohenester, U. Morphing a plasmonic nanodisk into a nanotriangle. Nano Lett. 2014, 14, 4810−4815. (24) Qiao, X.; Li, Q.; Schaugaard, R. N.; Noffke, B. W.; Liu, Y.; Li, D.; Liu, L.; Raghavachari, K.; Li, L. Well-defined nanographeneRhenium complex as an efficient electrocatalyst and photocatalyst for selective CO2 reduction. J. Am. Chem. Soc. 2017, 139, 3934−3937. (25) Castro, A.; Appel, H.; Oliveira, M.; Rozzi, C. A.; Andrade, X.; Lorenzen, F.; Marques, M. A.; Gross, E.; Rubio, A. Octopus: a tool for the application of time-dependent density functional theory. Phys. Status Solidi B 2006, 243, 2465−2488. (26) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 1996, 77, 3865−3868. (27) Oliveira, M. J.; Castro, A.; Marques, M. A.; Rubio, A. On the use of Neumann’s principle for the calculation of the polarizability tensor of nanostructures. J. Nanosci. Nanotechnol. 2008, 8, 3392−3398. (28) Yamamoto, N.; Hu, C.; Hagiwara, S.; Watanabe, K. Nanoplasmon dynamics and field enhancement of graphene flakes by firstprinciples simulations. Appl. Phys. Express 2015, 8, 045103.
27602
DOI: 10.1021/acs.jpcc.7b09358 J. Phys. Chem. C 2017, 121, 27597−27602
