Tunable plasmonic enhancement of light scattering and absorption in graphene-coated subwavelength wires
aa r X i v : . [ phy s i c s . op ti c s ] M a y Tunable plasmonic enhancement of light scatteringand absorption in graphene-coated subwavelengthwires
M´aximo Riso , Mauro Cuevas and Ricardo A. Depine Grupo de Electromagnetismo Aplicado, Departamento de F´ısica, FCEN,Universidad de Buenos Aires and IFIBA, Consejo Nacional de InvestigacionesCient´ıficas y T´ecnicas (CONICET), Ciudad Universitaria, Pabell´on I, C1428EHA,Buenos Aires, Argentina Facultad de Ingenier´ıa y Tecnolog´ıa Inform´atica, Universidad de Belgrano,Villanueva 1324, C1426BMJ, Buenos Aires, Argentina and Consejo Nacional deInvestigaciones Cient´ıficas y T´ecnicas (CONICET)E-mail: [email protected]
Abstract.
The electromagnetic response of subwavelength wires coated with agraphene monolayer illuminated by a linearly polarized plane waves is investigated.The results show that the scattering and extintion cross-sections of the coated wirecan be dramatically enhanced when the incident radiation resonantly excites localizedsurface plasmons. The enhancements occur for p–polarized incident waves and forexcitation frequencies that correspond to complex poles in the coefficients of themultipole expansion for the scattered field. By dynamically tuning the chemicalpotential of graphene, the spectral position of the enhancements can be chosen over awide range.PACS numbers: 78.67.Wj, 73.20.Mf, 78.20.Ci, 81.05.ue, 73.21.-b, 78.68.+m
1. Introduction
Advanced light manipulation relies heavily upon controlling the fundamental processesof light-matter interaction. This paper is concerned with light scattering, a processwhich plays a key role in current light control applications like light trapping [1],Anderson localization [2, 3] or photonic band gaps [4]. Particularly, light scatteringby subwavelength particles has received considerable attention in relation to plasmonic systems, where some exciting experimental and theoretical developments can beproduced when incident radiation is coupled to the collective electron charge oscillationsknown as localized surface plasmons [5,6,7,8,9]. Excitation of localized surface plasmonsby an incident electromagnetic field at the frequency where resonance occurs resultsin an enhancement of the local electromagnetic field as well as in the appearance ofintense absorption bands. Moreover, the efficiency of light scattering, determined by thestrength of the electric field in the material, is also enhanced under resonant excitation lasmonic light scattering in graphene-coated subwavelength wires localized plasmon modes in graphene-coated wires has not beeninvestigated yet. The primary motivation of the work described here is to study theplasmonic response of graphene–coated 2D particles (wires). For metallic or metallic-like substrates, the coating is expected to modify the localized surface plasmons alreadyexisting in the bare particle. On the other hand, for substrates made of nonplasmonicmaterials the coating is expected to introduce localized surface plasmons mechanismswhich were absent in the bare particle. In both cases, the good tunability of thegraphene coating is expected to lead to unprecedented control over the location andmagnitude of the particle plasmonic resonances. In this paper we study the scatteringof electromagnetic waves normally incident on graphene–coated cylinders with circularcross–section, and present results for both dielectric and metallic substrates.The plan of the paper is as follows. In Section 2 we sketch an analytical method ofscattering based on the separation of variables approach and obtain a solution for thescattered fields in the form of an infinite series of cylindrical multipole partial waves.In Section 3 we give examples of scattering and extinction cross–section corresponding lasmonic light scattering in graphene-coated subwavelength wires − iωt )time–dependence is implicit throughout the paper, with ω the angular frequency, t thetime, and i = √−
2. Theory
Figure 1.
Schematic of the scattering problem
We consider a graphene–coated cylinder with circular cross–section (radius R )centered at x =0, y =0 (see Figure 1). The wire substrate may be dielectric or conducting(electric permittivity ε and magnetic permeability µ ) and the coated wire is embeddedin a transparent medium (electric permittivity ε and magnetic permeability µ ). Thegraphene layer is treated as an infinitesimaly thin, local and isotropic two-sided layerwith surface conductivity σ ( ω ) given by the Kubo formula [35, 36]. The wave vector ofthe incident radiation is directed along ˆ x . In this case, the scattering problem can behandled in a scalar way since the solution to any incident polarization can be describedas a linear combination of the solutions obtained in two fundamental scalar problems:electric field parallel to the main section of the cylindrical surface (p polarization or E z =0 modes) and magnetic field parallel to the main section of the cylindrical surface(s polarization or H z =0 modes).To obtain analytical solutions to the scattering problem we closely follow the usualseparation of variables approach [37, 38]. In a first step, the non-zero components ofthe total electromagnetic field along the axis of the cylinder for each polarization case,denoted by F ( ρ, ϕ ), are expanded as series of cylindrical harmonics, one for the internal lasmonic light scattering in graphene-coated subwavelength wires ρ < R , subscript 1) and another one for the external region ( ρ > R , subscript2). When the incident electric field is contained in the x − y plane (p polarization), theexpansions for the total magnetic field along the axis of the cylinder are F ( ρ, ϕ ) = ∞ X n = −∞ c n J n ( k ρ ) exp inϕ , (1) F ( ρ, ϕ ) = ∞ X n = −∞ (cid:2) A i n J n ( k ρ ) + a n H (1) n ( k ρ ) (cid:3) exp inϕ , (2)where a n and c n are unknown complex coefficients, k j = ωc √ ε j µ j ( j = 1 , c is thespeed of light in vacuum, A is the amplitude of the incident magnetic field (parallel toˆ z ) and J n and H (1) n are the n -th Bessel and Hankel functions of the first kind respectively.In the same manner, when the incident magnetic field is contained in the x − y plane(s polarization), the expansions for the total electric field along the axis of the cylinderare F ( ρ, ϕ ) = ∞ X n = −∞ d n J n ( k ρ ) exp inϕ , (3) F ( ρ, ϕ ) = ∞ X n = −∞ (cid:2) B i n J n ( k ρ ) + b n H (1) n ( k ρ ) (cid:3) exp inϕ , (4)where B is the amplitude of the incident electric field (parallel to ˆ z ) and b n and d n unknown complex coefficients.In a second step, the boundary conditions at ρ = R are invoked to obtain theunknown coefficients a n , b n , c n and d n in terms of the incident amplitudes A and B . The boundary conditions for the graphene–coated cylinder are: i) the tangentialcomponent of the total electric field ~E is continuous and ii) the discontinuity of thetangential component of the total magnetic field ~H is proportional in magnitude to thesurface current density (that is, to the graphene surface conductivity). In terms of F ,the boundary conditions at ρ = R can be expressed as1 ε ∂F ∂ρ = 1 ε ∂F ∂ρ and F − F = 4 iπωε σ ∂F ∂ρ , (5)for p polarization, and F = F and 1 µ ∂F ∂ρ − µ ∂F ∂ρ = − iπωc σF , (6)for s polarization. Finally, the amplitudes of the scattered field can be written as a n = − i n h ε k J n ( x ) J ′ n ( x ) − ε k J ′ n ( x ) J n ( x ) + πc σ cω ik k J ′ n ( x ) J ′ n ( x ) i ε k J n ( x ) H ′ (1) n ( x ) − ε k J ′ n ( x ) H (1) n ( x ) + πc σ cω ik k J ′ n ( x ) H ′ (1) n ( x ) A , (7) b n = − i n h ε k J n ( x ) J ′ n ( x ) − ε k J ′ n ( x ) J n ( x ) + πc σ cω ik k J n ( x ) J n ( x ) i ε k J n ( x ) H ′ (1) n ( x ) − ε k J ′ n ( x ) H (1) n ( x ) + πc σ cω ik k J n ( x ) H (1) n ( x ) B , (8) lasmonic light scattering in graphene-coated subwavelength wires c n = ε k i n h J n ( x ) H ′ (1) n ( x ) − J ′ n ( x ) H (1) n ( x ) i ε k J n ( x ) H ′ (1) n ( x ) − ε k J ′ n ( x ) H (1) n ( x ) + πc σ cω ik k J ′ n ( x ) H ′ (1) n ( x ) A , (9) d n = ε k i n h J n ( x ) H ′ (1) n ( x ) − J ′ n ( x ) H (1) n ( x ) i ε k J n ( x ) H ′ (1) n ( x ) − ε k J ′ n ( x ) H (1) n ( x ) + πc σ cω ik k J n ( x ) H (1) n ( x ) B , (10)where x = k R , x = k R .As in the problem of scattering by a wire without a graphene coating, the amplitudesgiven by equations (7)-(10) allow us to obtain the electromagnetic field everywherein space as well as other quantities of interest such as differential, absorption andextinction cross–sections (see [37, 38] for details). The multipole coefficients a n and b n for the scattered field have essentially the same form as those corresponding to abare wire [37, 38], except for additive corrections proportional to σ in numerator anddenominator. Similar additive corrections have been recently reported for graphene-coated spheres [28].
3. Resonant excitation of localized surface plasmons
To simulate the electromagnetic response of graphene-coated cylinders we assume that σ ( ω ), the complex surface conductivity of graphene, is given by the high frequencyexpression derived from Kubo formula (equation (1), Ref. [35]), including interband andintraband transition contributions. Apart from the angular frequency ω , the value of σ depends on the chemical potential µ c (controlled with the help of a gate voltage), theambient temperature T and the carriers scattering rate γ c . For the intraband ( σ intra ) andinterband ( σ inter ) contributions to σ ( ω ) we have used the following expressions [35, 36] σ intra ( ω ) = 2 ie Tπ ~ ( ω + iγ c ) ln [2 cosh( µ c / T )] , (11) σ inter ( ω ) = e ~ (cid:20)
12 + 1 π arctan[( ω − µ c ) / T ] − i π ln ( ω + 2 µ c ) ( ω − µ c ) + (2 T ) (cid:21) , (12)where ~ is the reduced Planck constant and e the elementary charge. In all thecalculations we have used T = 300 ◦ K and γ c = 0 . R = 0 . µ m, made with a nonplasmonic, transparent material( ε = 3 . µ = 1) in a vaccum ( µ = ε = 1). The excitation frequencies are in therange between 5 THz (incident wavelength 60 µm ) and 30 THz (incident wavelength10 µ m), the incident wave is p-polarized and different values of µ c have been considered.In Figure 2a the scattering cross-section spectrum of the bare wire (continuous line)is also given as a reference. The absorption cross-section of the bare wire vanishesidentically and is not shown in Figure 2b. We see that while the scattering cross-sectionspectrum of the bare wire does not show any plasmonic feature in this spectral range, lasmonic light scattering in graphene-coated subwavelength wires
10 20 30 40 50 601E-71E-61E-51E-41E-30.010.1110 Q sc a t ( m) bare wire c = 0.3 eV c = 0.5 eV c = 0.8 eV (a)
10 20 30 40 50 601E-51E-41E-30.010.1110200 Q ab s (b) ( m) Figure 2.
Scattering (a) and absorption (b) cross-section spectra for a graphene-coated cylinder illuminated by a p-polarized plane wave and for different values ofthe chemical potential µ c . The scattering cross-section curve corresponding to theuncoated cylinder is given as a reference. The incident wave is p -polarized, R = 0 . µ m, ε = 3 . µ = µ = ε = 1. lasmonic light scattering in graphene-coated subwavelength wires . µ m ( µ c = 0 . . µ m ( µ c = 0 . . µ m ( µ c = 0 . . µ m, 33 . µ m (for µ c = 0 . . µ m, 25 . µ m, 20 . µ m (for µ c = 0 . . µ m, 20 . µ m, 16 . µ m (for µ c = 0 . a n and c n of the multipole expansions for the electromagnetic field. Thecorrespondence can be clearly seen by noting that the spectral positions of the peaks arein very good agreement with the real part of the complex wavelengths Λ n given in Table1 ( n = 1 , . . . n representing the complex root of the common denominator g n (Λ) in equations (7) and (9), and Λ = 2 πc/ω . For the numerical calculation of Λ n we have used a Newton-Raphson method and the same parameters used for curves inFigure 2. We observe that the scattering cross-sections in Figure 2a display minimaat wavelengths near 36 . µ m ( µ c = 0 . . µ m ( µ c = 0 . . µ m( µ c = 0 . a becomes zero. Taking into account that a minimization of the scatteringcross-section is a necessary condition to obtain invisibility or transparency of an object,and that the scattering cross-section of the bare wire does not exhibit a minimumin this spectral range, the observation of these minima is relevant in the context ofinvisibility cloacks. These results, together with those shown in Figure 2, illustrate agreat advantage of the graphene coating, which provides unprecedented control over thespectral location of the resonances of the wire via changes in the chemical potential µ c ,ultimately controlled through electrostatic gating.The correspondence between plasmonic resonances of the graphene coating andthe enhancement of scattering and absorption in the coated wire is further evidencedwhen the cross-sections of the same systems considered in Figure 2 are calculatedfor s-polarized –instead of p-polarized– illumination. In this case (not shown), noenhancements in the cross-sections of the graphene-coated wire are observed, whichis consistent with the fact that localized surface plasmons are not supported in thispolarization mode, since the electric field in the graphene coating can only induceelectric currents directed along the wire axis, and not along the azimuthal directionˆ ϕ , a necessary condition for the existence of localized surface plasmons in the graphenecircular cylinder.In Figure 3a we plot the spatial distribution of the electric field normalized to theincident amplitude for the wire considered in Figure 2 and µ c = 0 . λ = 28 . µ m, a value equal to the real part of the complex pole Λ (seeTable 1) of the multipole coefficients a and c and for which the strongest maxima inthe scattering and absorption cross-sections for µ c = 0 . lasmonic light scattering in graphene-coated subwavelength wires -1-0.5 0 0.5 1 -1 -0.5 0 0.5 1 y ( µ m ) x( µ m) y ( µ m ) x( µ m) y ( µ m ) x( µ m) Figure 3.
Map of | ~E ( ρ, ϕ ) | for a graphene-coated cylinder illuminated by a p-polarizedplane wave. The incident wavelength is λ = 28 . µ m (top), λ = 20 . µ m (center)and λ = 16 . µ m (bottom). The chemical potential is µ c = 0 . | c n | , n = 0 , , , | a n | ), n = 0 , , , lasmonic light scattering in graphene-coated subwavelength wires n µ c = 0 . µ c = 0 . µ c = 0 . . i . . i . . i . . i . . i . . i . . i . . i . . i . . i . . i . . i . n of g n (Λ) = 0 in , where g n is the common denominator ofthe coefficients a n and c n and Λ = 2 πc/ω . The parameters correspond to those used tocalculate the scattering and absorption cross-sections shown in Figure 2.picture is very similar to that obtained for subwavelength metallic wires: the near fieldsare clearly of an electric dipole nature, with the dipolar moment oriented in the directionof the incident electric field (ˆ y ). However, contrary to the case of subwavelength metallic(impenetrable) wires, where the field cannot penetrate into the interior regions and islimited to a surface layer of approximately one skin depth thick, in the graphene-coateddielectric wire there are regions where the interior field can be greatly enhanced. Takinginto account that the electric field induced in the scatterer is much stronger than thatof incident radiation (see scale in Figure 3a), the far-field intensities are dominated byan electric dipole pattern, as indicated in the inset in Figure 3a, where we have sketchedthe angular distribution of scattered power for this excitation wavelength. When theincident wavelength is λ = 20 . µ m, the scattering cross-section for µ c = 0 . µ c exhibits a strong enhancement (an enhancement factor near three orders ofmagnitude greater than the non-resonant case). At this wavelength, the electric fieldinduced in the scatterer, although not as strong, is again stronger than that of incidentradiation and of a quadrupolar, not dipolar, nature. This is shown in Figure 3b, wherewe plot the map of | ~E ( ρ, ϕ ) | normalized to the incident amplitude and calculated for thesame parameters as in Figure 3a, except the incident wavelength λ = 20 . µ m. Sincethe electric field induced in the scatterer is much stronger than that of incident radiation,the far-field intensities are dominated by an electric quadrupole pattern, as indicatedin the inset in Figure 3a. In Figure 3c we show the map of | ~E ( ρ, ϕ ) | , normalized tothe incident amplitude, for the same parameters as in Figure 3a and 3b, except thatnow the incident wavelength is changed to λ = 16 . µ m. For this value of λ and for µ c = 0 . | a n | and | c n | , n = 0 , , ,
3. Similar results have been obtained for the other valuesof chemical potential considered in Figure 2. lasmonic light scattering in graphene-coated subwavelength wires
10 15 20 25 30 35 400.10.20.30.40.50.60.70.80.91.0 n=4 n=3 n=2 R ( m ) ( m) n=1 log(Q ext /Q ext,min ) Figure 4.
Extinction cross-section in a graphene-coated dielectric wire in vacuum( µ = ε = 1), explored as a function of the wire radius R and the excitation wavelength λ . The dashed lines indicate the resonance position, numerically obtained as the polesof the coefficients a n . The incident wave is p -polarized, µ c = 0 . ε = 3 . µ = 1. To investigate the size dispersion of the localized plasmonic modes of graphene-coated wires, we show in Figure 4 a color map of the extinction cross-section in the R − λ plane. The incident wave is p -polarized, µ c = 0 . ε = 3 . µ = 1. We observe that the enhancements in the extinctioncross-section follow the dashed lines labeled with n . These lines indicate the resonanceposition, numerically obtained as the real part of the pole of the coefficient a n given byequation (7).The examples so far highlight the attractive plasmonic features that a graphenecoating can produce in intrinsically nonplasmonic wires. Next, we consider graphene-coated wires made of metallic or metallic-like substrates, that is, intrinsically plasmonicwires, where the localized surface plasmons of the graphene coating are expected tomodify the localized plasmons already existing in the bare particle. We assume that thedispersive behavior of the interior electric permittivity is described by the Drude model ε ( ω ) = ε ∞ − ω p ω + iγ m ω , (13)with ε ∞ the residual high-frequency response of the material, ω p the metallic plasmafrequency and γ m the optical loss-rate of the Drude material. This model is applied lasmonic light scattering in graphene-coated subwavelength wires ( m) , , bare metal, , graphene-coated metal p =0.7 eV Q S / c ( m -1 ) p =0.5 eV p =0.9 eV Figure 5.
Scattering cross-section spectrum in vacuum for a graphene-coated metallicwire with a radius R = 50 nm illuminated under p polarization. σ ( ω ) is given by theKubo expression with T = 300 ◦ K, γ c = 0 . µ c = 0 . ε ( ω ) is givenby equation (13), with ε ∞ = 1, γ m = 0 .
01 eV and ~ ω p = 0.5eV, 0.7 and 0.9eV. Thescattering cross-sections for the bare metallic cylinders and for an empty graphenecylinder in vacuum (black curve, ε = ε = 1 and µ = µ = 1) are also given as areference. to describe strongly doped semiconductors [39] which allow dynamic manipulation ofcarrier densities and have plasma frequencies in the same range that the realizable Fermienergies of graphene. Here the situation is similar to that reported for graphene-coatedDrude spheres [28] in the sense that the region of induced charges in the graphenecoating is not separated from the region of induced charges in the metallic substrate.For the case considered in this paper, both regions coincide with the cylindrical surfaceof radius R and therefore a single hybridized plasmon is expected, rather than two(bonding and antibonding) hybridized plasmons with different energies, characteristicof systems where induced charges occur in two separated regions. In Figure 5 we plot thescattering cross-section spectrum in vacuum for a graphene-coated metallic wire with aradius R = 50 nm illuminated under p polarization. The interior electric permittivity ε ( ω ) is given by equation (13) with ε ∞ = 1, γ m = 0 .
01 eV and different values of ~ ω p (0.5 eV, 0.7 eV and 0.9 eV), while the values of the complex surface conductivity ofgraphene σ ( ω ) have been obtained from the Kubo expression with T = 300 ◦ K, γ c = 0 . lasmonic light scattering in graphene-coated subwavelength wires µ c = 0 . ε = ε = 1 and µ = µ = 1)are also given as a reference. The excitation frequencies are in the range between 23 . . µm ) and 191 THz (incident wavelength 1 . µ m).The curves in Figure 5 show that the hybridized resonances are always blueshiftedcompared to the resonances of the bare wire. This is consistent with the fact thatthe net effect of the graphene coating is to increase the induced charge density on thecylindrical surface of the metallic wire. The hybridized resonances can be controlledwith the help of a gate voltage, which ultimately determines the value of the chemicalpotential µ c . To explore the tunability of the hybridized resonances, in Figure 6 wegive scattering cross-section spectra for a graphene-coated metallic wire ( ~ ω p = 0.5eV)illuminated under p polarization for different values of the chemical potential µ c (0.3eV,0.5, 0.8 and 1.1 eV). Q S / c ( m -1 ) c = 0.3eV c = 0.5eV c = 0.8eV c = 1.1eV ( m) Figure 6.
Scattering cross-section spectrum in vacuum for a graphene-coated metallicwire ( ~ ω p = 0.7eV) illuminated under p polarization for different values of the chemicalpotential µ c (0.3eV, 0.5, 0.8 and 1.1 eV).
4. Summary
Fueled by recent experiments [29, 30, 31, 32] showing that, thanks to the van der Waalsforce, a graphene sheet can be tightly coated on a fiber surface, in this paper we have lasmonic light scattering in graphene-coated subwavelength wires
Acknowledgments
The authors acknowledge the financial support of Consejo Nacional de InvestigacionesCient´ıficas y T´ecnicas, (CONICET, PIP 1800) and Universidad de Buenos Aires (projectUBA 20020100100327).
References [1] Guo C F, Sun T Y, Cao F, Liu Q and Ren Z F 2014 Metallic nanostructures for light trapping inenergy-harvesting devices
Light: Sci. Appl. e161[2] Lagendijk A, van Tiggelen B and Wiersma D 2009 Fifty years of Anderson localization PhysicsToday (8) 24[3] Hashemi A, Hosseini-Farzad M and Montakhaba A 2010 Emergence of semi-localized Andersonmodes in a disordered photonic crystal as a result of overlap probability Eur. Phys. J. B Phys. Rev.Lett. Light: Sci. Appl. e179[6] Bliokh K Y, Bliokh Y P, Freilikher V, Savelev S and Nori F 2008 Colloquium: Unusual resonators:Plasmonics, metamaterials, and random media Rev. Mod. Phys. Adv. Mater. ACS Nano Nano Lett. Laser & Photonics Reviews Science lasmonic light scattering in graphene-coated subwavelength wires [12] Boltasseva A and Atwater H A 2009 Plasmonics in graphene at infrared frequencies Phys. Rev. B Materials Science and Engineering
R 74
Phys.Rev. B (24) 245435[15] Nikitin A Y, Guinea F, Garc´ıa-Vidal F J and Mart´ın-Moreno L 2011 Edge and waveguide terahertzsurface plasmon modes in graphene microribbons Phys. Rev. B ACSNano (1) 431440[17] Liu P, Zhang X, Ma Z, Cai W, Wang L and Xu J 2013 Surface plasmon modes in graphene wedgeand groove waveguides Opt. Express (26) 3243232440[18] Bludov Y V, Vasilevskiy M I and Peres N M R 2010 Mechanism for graphene-based optoelectronicswitches by tuning surface plasmon-polaritons in monolayer graphene EPL J. Opt. Nano Lett. J. Phys.:Condens. Matter Physica B Optics Communications
Opt. Express Opt. Express Opt. Lett. J. Opt. Phys. Rev. B IEEE Photon. Technol. Lett. (14) 13921394[30] He X, Zhang X, Zhang H and Xu M 2014 Graphene covered on microfiber exhibiting polarizationand polarization-dependent saturable absorption IEEE J. Sel. Top. Quantum Electron. (1)4500107[31] Wu Y, Yao B, Zhang A, Rao Y, Wang Z, Cheng Y, Gong Y, Zhang W, Chen Y and Chiang KS 2014 Graphene-coated microfiber Bragg grating for high-sensitivity gas sensing Opt. Lett. (5) 12351237[32] Li W, Chen B, Meng C, Fang W, Xiao Y, Li X, Hu Z, Xu Y, Tong L, Wang H, Liu W, Bao J andShen Y R 2014 Ultrafast all-optical graphene modulator Nano Lett. (2) 955959[33] Yang H, Hou Z, Zhou N, He B, Cao J and Kuang Y 2014 Graphene-encapsulated SnO2 hollowspheres as high-performance anode materials for lithium ion batteries Ceramics International lasmonic light scattering in graphene-coated subwavelength wires [34] Lee J-S, Kim S-I, Yoon J-C and Jang J-H 2013 Chemical Vapor Deposition of MesoporousGraphene Nanoballs for Supercapacitor ACS Nano Physics Uspekhi Phys. Rev. Lett. Light scattering by small particles (New York: John Wiley & Sons)[38] Bohren C F and Huffman D R 1983
Absorption and scattering of light by small particles (NewYork: John Wiley & Sons)[39] Schimpf A, Thakkar N, Gunthardt C, Masiello D and Gamelin D 2014 Charge-Tunable QuantumPlasmons in Colloidal Semiconductor Nanocrystals
ACS Nano8