Multiple Excitation of Confined Graphene Plasmons by Single Free Electrons
MMultiple Excitation of Confined GraphenePlasmons by Single Free Electrons
F. Javier García de Abajo ∗ , † , ‡ ICFO - Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860Castelldefels (Barcelona), Spain
E-mail: [email protected]
Abstract
We show that free electrons can efficiently excite plasmons in doped graphene withprobabilities of order one per electron. More precisely, we predict multiple excitationsof a single confined plasmon mode in graphene nanostructures. These unprecedentedlylarge electron-plasmon couplings are explained using a simple scaling law and furtherinvestigated through a general quantum description of the electron-plasmon interac-tion. From a fundamental viewpoint, multiple plasmon excitations by a single electronprovides a unique tool for exploring the bosonic quantum nature of these collectivemodes. Our study does not only open a viable path towards multiple excitation of asingle plasmon mode by single electrons, but it also reveals graphene nanostructuresas ideal systems for producing, detecting, and manipulating plasmons using electronprobes.
KEYWORDS: graphene, plasmons, multiple plasmon excitation, electron energy loss,quantum plasmonics, nanophotonics ∗ To whom correspondence should be addressed † ICFO ‡ ICREA - Institució Catalana de Recerca i Estudis Avançats, Barcelona, Spain a r X i v : . [ c ond - m a t . m e s - h a ll ] N ov he existence of surface plasmons was first demonstrated by observing energy losses pro-duced in their interaction with free electrons. Following those pioneering studies, electronbeams have revealed many of the properties of plasmons through energy-loss and cathodo-luminescence spectroscopies, which benefit from the impressive combination of high spatialand spectral resolutions that is currently available in electron microscopes and that allowsmapping plasmon modes in metallic nanoparticles and other nanostructures of practicalinterest.
However, plasmon creation rates are generally low, thus rendering multiple ex-citations of a single plasmon mode by a single electron extremely unlikely.From a fundamental viewpoint, the question arises, what it the maximum excitationprobability of a plasmon by a passing electron? This depends on a number of parameters,such as the interplay between momentum and energy conservation during the exchange withthe electron, the spatial extension of the electromagnetic fields associated with the electronand the plasmon, and the interaction time, which are in turn controlled by the spatial ex-tension of the excitation and the speed of the electron. One expects that highly confinedoptical modes, encompassing a large density of electromagnetic energy, combined with low-energy electrons, which experience long interaction times, provide an optimum answer tothis question. This intuition is corroborated here by examining the interaction between freeelectrons and graphene plasmons. The peculiar electronic structure of this material leadsto the emergence of strongly confined plasmons (size < /
100 of the light wavelength) whenthe carbon layer is doped with charge carriers.
Graphene plasmons have been recentlyobserved and their electrical modulation unambiguously demonstrated through near-fieldspatial mapping and far-field spectroscopy.
These low-energy plasmons, which ap-pear at mid-infrared and lower frequencies, should not be confused with the higher-energy π and σ plasmons that show up in most carbon allotropes, and that have been extensivelystudied through electron energy-loss spectroscopy (EELS) in fullerenes, nanotubes, and graphene. These high-energy plasmons are not electrically tunable. We thus con-centrate on electrically driven low-energy plasmons in graphene. Despite their potential for2uantum optics and light modulation, the small size of the graphene structures relativeto the light wavelength (see below) poses the challenge of controlling their excitation anddetection with suitably fine spatial precision. Using currently available subnanometer-sizedbeam spots, free electrons appear to be a viable solution to create and detect grapheneplasmons with large yield and high spatial resolution. As a first step in this direction,angle-resolved EELS performed with low-energy electrons has been used to map the disper-sion relation of low-energy graphene plasmons, as well as their hybridization with thephonons of a SiC substrate, although this technique has limited spatial resolution.The probabilities of multiple plasmon losses, as observed in EELS and photoemission experiments, are well known to follow Poisson distributions. Previous studies haveconcentrated on plasmon bands, where the electrons simultaneously interact with a largenumber of plasmon modes. We are instead interested in the interaction with a spectrallyisolated single mode.In this article, we show that a single electron can generate graphene plasmons with largeyield of order one. We discuss the excitation of both propagating plasmons in extended car-bon sheets and localized plasmons in nanostructured graphene. The excitation probabilityis shown to reach a maximum value when the interaction time is of the order of the plas-mon period. Our results suggest practical schemes for the excitation of multiple localizedgraphene plasmons using electron beams, thus opening new perspectives for the observationof nonlinear phenomena at the level of a few plasmons excited within a single mode of anindividual graphene structure by a single electron.
RESULTS AND DISCUSSION
Plasmon excitation in extended graphene.
Before discussing confined plasmons, weexplore analytical limits for electrons interacting with a homogeneous graphene layer. Thedispersion relation of free-standing graphene plasmons can be directly obtained from the3 F k F E F E F intraband interband plasmon ω = k || v Parallel wave vector, k || P ho t on ene r g y , ħ ω Im{ r p } ( a ) ( E F / ħ ) × Γ α ( ) − v c ( ) ω E F ( ) Electron energy (eV) Lo ss p r obab ili t y
500 eV electron graphene ( b ) . e V e V - - Figure 1:
Excitation of graphene plasmons by electron beams. (a)
Frequency andparallel-wave-vector dependence of the Fresnel reflection coefficient r p for p polarized light,calculated in the random-phase approximation (RPA) and showing plasmons and inter-band/intraband electron-hole-pair transitions in doped extended graphene. The Fermi en-ergy and intrinsic damping time are E F = 0 . τ = 0 . v couples preferentially to excitations in the k k v ∼ ω region. (b) Electron energy-loss spectra for an electron incident normally to a homogeneous graphenesheet under the same conditions as in (a) for two different electron kinetic energies (brokencurves), compared with the universal analytical expression for the plasmon contribution de-rived in the Drude model (solid curve, ?? ). The inset shows the integral of the latter overthe ¯ hω < E F region as a function of electron energy.4ole of the Fresnel coefficient for p polarization, which in the electrostatic limit reduces to r p = 11 − i ω/ πk k σ . (1)Here, σ ( k k , ω ) is the conductivity, and k k and ω are the light parallel wave-vector and fre-quency. Because of the translational invariance of the carbon layer, the full k k depen-dence of the conductivity σ can be directly incorporated in ?? to account for nonlocaleffects, which include the excitation of electron-hole pairs. The dispersion diagram of 1a,which shows Im { r p } calculated in the random-phase approximation (RPA) under re-alistic doping conditions (Fermi energy E F = 0 . n = ( E F / ¯ hv F ) /π = 1 . × cm − , where v F ≈ m/s is the Fermi velocity), revealsa plasmon band as a sharp feature outside the regions occupied by intra- and interbandelectron-hole-pair transitions. We justify the use of the electrostatic approximation to de-scribe the response of graphene because the plasmon wavelength is much smaller than thelight wavelength ( e.g. , the Fermi wavelength is 2 π/k F = q π/n = 8 . hω = E F = 0 . v has a probability Γ ⊥ ( ω ) = 4 e π ¯ hv Z ∞ k k dk k (cid:16) k k + ω /v (cid:17) Im { r p } (2)of lossing energy ¯ hω (see Methods for more details). Likewise, an electron moving along apath length L parallel to the graphene experiences a loss probability (see Methods)Γ k ( ω ) = 2 e Lπ ¯ hv Z ∞ ωv dk k q k k − ω /v e − k k z Im { r p } , (3)where z is the distance between graphene and the electron. These probabilities are deter-mined by Im { r p } , which is represented in 1a. Clearly, losses around the k k ∼ ω/v region are5avored.It is instructive to evaluate ?? using the Drude model for the graphene conductivity σ ( ω ) = e E F π ¯ h i ω + i τ − , (4)where τ is a phenomenological decay time. The latter determines the plasmon quality factor Q = ωτ ∼ −
60, as measured in recent experiments.
This model works well forphoton energies below the Fermi level (¯ hω < E F ), but neglects interband transitions thattake place at higher energies. In the ωτ (cid:29) { r p } ≈ πk SP k δ ( k k − k SP k ) , where k SP k = (¯ h ω / e E F ) (5)is the plasmon wave vector (broken curve in 1a). Using this approximation in ?? , the plasmoncontribution to the loss probability reduces toΓ ⊥ ( ω ) ≈ hE F s (cid:16) s (cid:17) , (6)Γ k ( ω ) ≈ ¯ hωLvE F θ ( s − √ s − − k SP k z , where s = 12 α vc ¯ hωE F and α ≈ /
137 is the fine-structure constant. For a perpendicular trajectory, ?? has amaximum at s = 1 (1b). Its integral over the ¯ hω < E F region, in which the plasmon iswell defined, shows a maximum probability >
35% for an electron energy ∼
215 eV when wetake E F = 0 . L o ss p r ob a b ilit y ( e V - ) Energy loss (eV) m = m = E F = τ − = D =
100 nm
100 eV100 eV
Figure 2:
Plasmons excitation in a graphene nanodisk.
Electron energy-loss proba-bility for two different electron trajectories exciting plasmon modes of different azimuthalsymmetry m , as shown by labels. Plasmon excitation in graphene nanostructures.
Doped graphene nanoislands cansupport plasmons, as recently observed through optical absorption measurements.
Forsmall islands, the concentration of electromagnetic energy that characterizes these plasmonsis extremely high, therefore producing strong coupling with nearby quantum emitters. Like-wise, the interaction of localized plasmons with a passing electron is expected to be particu-larly intense. This intuition is put to the test in 2, where we consider a 100 eV electron inter-acting with a 100 nm graphene disk doped to a Fermi energy E F = 0 . assuming linear response,and describing the graphene through the Drude conductivity of ?? , which we spread overa thin layer of ∼ . We investigate both a paralleltrajectory, with the electron passing 5 nm away from the carbon sheet, and a perpendiculartrajectory, with the electron crossing the disk center. 2 only shows the lowest-energy modethat is excited for each of these geometries, corresponding to m = 1 and m = 0 azimuthalsymmetries, respectively. An overview of a broader spectral range (see Methods, 7) revealsthat these modes are actually well isolated from higher-order spectral features within eachrespective symmetry. Because we use the Drude model, the width of the plasmon peaks in 2coincides with ¯ hτ − (see below), which is set to 1.6 meV, or equivalently, we consider a mo-bility of 10 ,
000 cm / (V s), which is a moderate value below those measured in suspended and BN-supported high-quality graphene.The area of the plasmon peaks is a τ -independent, dimensionless quantity that cor-responds to the number of plasmons excited per incident electron. For the m = 1 mode(parallel trajectory), we find ∼ . Electrostatic scaling law and maximum plasmon excitation rate.
In electrody-namics, the light wavelength introduces a length scale that renders the solution of specificgeometries size dependent. In contrast, electrostatics admits scale-invariant solutions, whichhave long been recognized to provide convenient mode decompositions, particularly whenstudying electron energy losses.
Modeling graphene as an infinitely thin layer, its elec-8rostatic solutions take a particularly simple form. We provide a comprehensive derivationof the resulting scaling laws in the Methods section, the main results of which are sum-marized next. We focus on a spectrally isolated plasmon of frequency ω p sustained by agraphene nanoisland of characteristic size D ( e.g. , the diameter of a disk) and homogeneousFermi energy E F .Assuming the Drude model for the graphene conductivity ( ?? ), the plasmon frequencyis found to be ω p = γ p e ¯ h s E F D , (7)where γ p is a dimensionless, scale-invariant parameter that only depends on the nanoislandgeometry and plasmon symmetry under consideration (see ?? ). In particular, we have γ p =3 . m = 0 azimuthalsymmetry), which is the lowest-frequency mode excited by an electron moving along thedisk axis. Also, we find γ p = 2 . m = 1 mode, which can be excited inasymmetric configurations. It is important to stress that ?? reveals a linear dependence of ω p on q E F /D .Furthermore, the average number of plasmons excited by the electron ( i.e. , the plasmonyield) reduces to (see Methods, ?? ) P cla p ≈ µ F p (cid:18) µν (cid:19) (8)within this classical theory. Here, we have defined the two dimensionless parameters µ = √ E F De = s E F Dα ¯ hc , (9a) ν = ¯ hve = vαc , (9b)9s well as the dimensionless loss function F p ( x ) = x (cid:12)(cid:12)(cid:12)(cid:12) Z d‘ exp(i γ p x‘ ) f p ( ‘ ) (cid:12)(cid:12)(cid:12)(cid:12) . (10)The integral is over the electron path length ‘ in units of D , whereas f p is the dimensionlessscaled plasmon electric potential defined in ?? , which is calculated once and for all followingthe procedure explained in the Methods section (see ?? and beyond).We show characteristic examples of f p and F p in 3 for electrons crossing the centerof a graphene disk following different oblique trajectories. As f p is proportional to theelectrostatic potential associated with the plasmon, it is a real function of position. Forthe m = 1 mode considered in 3, f p vanishes at the axis of rotational symmetry and takeslarge values near the disk edges, leading to antisymmetric dip-peak patterns (3a). Theresulting loss function F p (3b) exhibits oscillations depending on the relative phase withwhich the potential f p is sampled along the electron trajectory (see ?? ). As anticipatedabove, this depends on the path length traveled by the electron during an optical plasmonperiod ( v/ω p ∼ v/ q E F /D ) relative to the extension of the plasmon ( ∼ D ), the ratio of whichis precisely µ/ν . When exciting symmetric plasmon modes ( e.g. , m = 0), non-oscillating F p profiles are obtained, equally characterized by maxima exceeding 1 at µ/ν ∼ θ ∼ − ◦ ), a maximum plasmon excitation proba-bility P cla p ∼ q α ¯ hc/E F D is reached for an electron velocity v ∼ (1 / √ αcE F D (see 3b). Asan indicative value, for a disk of diameter D = 50 nm doped to a Fermi energy E F = 0 . which can sustain m = 1 plasmons of0 .
21 eV energy, the maximum excitation probability is P cla p ∼ . ∼
50 eV electrons. These magnitudes can be readily computed for other disk size and dopingconditions via the scaling laws for the plasmon frequency ω p ∝ q E F /D ( ?? ), the maximumexcitation probability P cla p ∝ / √ E F D , and the electron energy ∝ E F D . Although arbitrar-10 ( a )( b ) F p ( μ / ν ) P r ob a b ilit y ( e V - ) Energy loss (eV) f p electron path length / D D θ θ=20 º40º60º80º85º μ / ν = ( α cE F D/h ) /v Figure 3:
Scaled potential and loss function. (a)
Scaled plasmon electric potential f p (see Methods, ?? ) sampled by different electron trajectories crossing the center of a graphenedisk with different angles relative to the graphene normal, as shown by labels. (b) Scaledloss function F p for the lowest-oder m = 1 plasmon excited under the trajectories consideredin (a). The inset shows the energy-loss probability for a plasmon width of 1.6 meV.11ly large values of P cla p can be in principle achieved through reducing E F and D (even whilemaintaining their ratio constant, and consequently, also ω p ) the graphene size is limited to D ∼
10 nm, as plasmons in smaller islands are strongly quenched by nonlocal effects. With D = 10 nm and E F = 0 . .
48 eV plasmons that can be excited by 9 eV electronswith P cla p = 1 . Quantum mechanical description.
The above classical formalism follows a long tradi-tion of explaining electron energy-loss spectra within classical theory, under the assumptionthat the total excitation rate is small ( i.e. , P cla p (cid:28) P cla p >
1. Therefore, a quantum treatment of the plasmonsbecomes necessary. We follow a similar approach as in previous studies of multiple plasmonlosses, here adapted to deal with a single plasmon mode. Describing the electron as aclassical external charge density ρ ext ( r , t ) and the plasmon as a bosonic mode, we considerthe Hamiltonian H = ¯ hω p a + a + g ( t ) (cid:16) a + + a (cid:17) , (11)where the operator a ( a + ) annihilates (creates) a plasmon of frequency ω p , and the time-dependent coupling coefficient is defined as g ( t ) = Z d r φ p ( r ) ρ ext ( r , t ) (12)in terms of φ p , the electric potential associated with the plasmon. In the electrostaticapproximation, neglecting the effect of inelastic plasmon decay, it is safe to assume that φ p is real. Notice that g is just the electrostatic energy subtracted or added to the system whenremoving or creating one plasmon ( i.e. , the integral represents the potential energy of theexternal charge in the presence of the potential created by one plasmon). The Hamiltonian12 should be realistic under the condition that both the electron-plasmon interaction timeand the optical cycle are small compared with the plasmon lifetime. In practice, this meansthat the electron behaves a point-like particle, or at least, its wave function is spread over aregion of size (cid:28) vτ . Additionally, we assume the electron kinetic energy to be much largerthan the plasmon energy, so that multiple plasmon excitations do not significantly changethe electron velocity (non-recoil approximation).As the plasmon state evolves under the influence of a linear term in ?? with a classicalcoupling constant g , it should exhibit classical statistics. Indeed, it is easy to verify thatthe plasmon wave function | ψ i = e i χ ( t ) | ξ ( t ) i (13)is a solution of Schrödinger’s equation H | ψ i = i ∂ | ψ i /∂t , where | ξ ( t ) i = exp( −| ξ | / X n ( ξa + ) n n ! e − inω p t | i . (14)is a coherent state with ξ ( t ) = − i¯ h Z t −∞ dt g ( t ) e i ω p t + ξ ( −∞ ) , whereas χ ( t ) = − h Z t −∞ dt g ( t ) Re n ξ ( t ) cos( ω p t ) o + χ ( −∞ )is an overall phase that does not affect the plasmon-number distribution. The averagenumber of plasmons excited at a given time is given by | ξ ( t ) | . We thus conclude that theprobability of exciting n plasmons simutaneoulsy follows a Poissonian distribution P ( n ) p = h ψ | h ( a + ) n a n /n ! i | ψ i = | ξ | n n ! e −| ξ | , (15)which yields a second-order correlation g (2) (0) = 1.13xpressing ξ and g in terms of φ p , noticing that ξ ( −∞ ) = 0 ( i.e. , no plasmons presentbefore interaction with the electron), and using a similar scaling as in the classical theorydiscussed above, we can write the probability of exciting n plasmons as P ( n ) p = (cid:16) P cla p (cid:17) n n ! exp( − P cla p ) , (16)where P cla p = | ξ | = µ − F p ( µ/ν ) is the classical linear probability given by ?? , which coincideswith the average number of excited plasmons per electron, P cla p = P n nP ( n ) p , and can takevalues above 1. µ = E F D / α c D average number of excited plasmons
50 eV 5º P p (1) P p (2) Figure 4:
Double plasmon excitation by a single electron.
The probabilities of excitinga single m = 1 plasmon ( P (1) p , blue curve) and two plasmons simultaneously ( P (2) p , red curve)are compared with the average number of excited plasmons ( P cla p = P n nP ( n ) p , black curve)for a 50 eV electron passing grazingly by the center of a graphene disk. The probability isthe same for straight crossing and specularly reflected trajectories. Inclusion of plasmon losses.
Inelastic losses during the electron-plasmon interactiontime have been so far ignored in the above quantum description. However, for sufficientlyslow electrons or very lossy plasmons, the interaction time can be comparable to the plasmonlifetime τ , so that the above quantum formalism needs to be amended, for example by14ollowing the time evolution of the density matrix ρ , according to its equation of motion dρdt = i¯ h [ ρ, H ] + 12 τ (cid:16) aρ a + − a + a ρ − ρ a + a (cid:17) , (17)where the Hamiltonian H is defined by ?? . The solution to this equation can still be givenin analytical form: ρ = | ξ ih ξ | , (18)where | ξ i is again a coherent state (see ?? ), but we have to redefine ξ ( t ) = − i¯ h Z t −∞ dt g ( t ) e i ω p t − ( t − t ) / τ + ξ ( −∞ ) . (19)It is straightforward to verify that ?? are indeed a solution of ?? .A simultaneous density-matrix description of the electron and plasmon quantum evolu-tions involves a larger configuration space that is beyond the scope of this paper. We canhowever argue that the probabilities P ( n ) p corresponding to the electron lossing energies n ¯ hω p must still follow a Poissonian distribution if we trace out the plasmon mode. At t → ∞ , allplasmons must have decayed, so that we can obtain the average number of plasmon lossesfrom the time integral of the total plasmon decay rate, h a + a i /τ . Again, we find that thisquantity coincides with the classical linear loss probability P cla p , which is now given by P cla p = 1 τ Z dt | ξ ( t ) | = 1¯ h τ Z dω π | ˜ g ( ω ) | ( ω p − ω ) + 1 / τ , where ˜ g ( ω ) is the time-Fourier transform of g ( t ). This expression reduces to P cla p = | ˜ g ( ω p ) / ¯ h | ( i.e. , ?? ) in the limit of high plasmon quality factor Q = ω p τ (cid:29) Multiple plasmon generation by a single electron.
We show in 4 results obtainedby solving ?? under the conditions of the most grazing trajectory from those considered in 3.In particular, the electron energy is 50 eV, which corresponds to ν ≈ .
9. The average numberof plasmons excited by a single electron under these conditions reaches a maximum value15lightly above 1, distributed in a ∼
40% probability of exciting only one plasmon, a ∼ µ ∼ D = 50 nm and E F = 0 .
26 eV), which leads to µ/ν ∼ .
6, slightly to the left ofthe main peak observed within linear theory at µ/ν ∼ n = 0 − µ and ν isshown in 5. For large µ , the results approach the linear regime, only single plasmons areeffectively excited, and the highest probability is peaked around a broad region centered alongthe µ = 2 ν line. At small µ ’s, a more complex behavior is observed. The double-plasmonexcitation probability is above 20% over a broad range of µ ’s, and even the probability ofsimultaneously generating three plasmons takes significant values >
10% up to µ ∼
1. Theprobability is increasingly more confined towards the low µ and ν region when a largernumber of plasmons is considered. Notice however the presence of a dip in that region forsingle-plasmon excitation, which is due to transits towards a larger numbers of plasmonscreated. A similar effect is observed for n = 2 at even lower values of µ and ν . µ = E F D / α c ν = v / α c
0 0.5 1 0 0.2 0.4 0 0.2 0.4 0 0.2 0.4 P (0) P (1) P (2) P (3)
0 10 0 10 0 10 0 10 0 10 p p p p
Figure 5:
Multiple-plasmon excitation by a single electron.
Full µ and ν dependenceof the probabilities P ( n ) p of generating n = 0 − ONCLUSIONS AND OUTLOOK
We predict unprecedentedly high graphene-plasmon excitation rates by relatively low-energyfree electrons. When a plasmon is highly confined down to a small size compared with thelight wavelength, the dipole moment associated with the plasmon is small and this limitsthe strength of its coupling to light. Direct optical excitation becomes inefficient, and onerequires near-field probes to couple to the plasmons.
In fact electrons act as versatilenear-field probes that can be aimed at the desired sample region. Furthermore, electronscarry strongly evanescent electromagnetic fields that couple with high efficiency to confinedoptical modes, thus rendering the observation of multiple plasmon excitation feasible. Ourcalculations show double excitation of a single plasmon mode with efficiencies of up to 20%per incident electron in graphene structures with similar size and doping levels as thoseproduced in recent experiments. Besides its fundamental interest, multiple plasmon excitation is potentially useful toexplore nonlinear optical response at the nanoscale. In particular, quantum nonlinearitiesat the single- or few-plasmon level has been predicted in small graphene islands. As theexcitation of strongly confined plasmons by optical means remains a major challenge, freeelectrons provide the practical means to explore this exotic quantum behavior. In particular,the generation of n plasmons by a single electron produces an energy loss ¯ hnω p , the detectionof which can be used to signal the creation of a plasmon-number state in the graphene island.The electron energies for which plasmon excitation probabilities are high lie in the sub-keV range, which is routinely employed in low-energy EELS studies, as already reportedfor collective excitations in graphene. In practice, one could study electrons that areelastically (and specularly) reflected on a patterned graphene film. Actually, the analysisof energy losses in reflected electrons was already pioneered in the first observation of sur-face plasmons in metals. A similar approach could be followed to reveal multiple grapheneplasmon excitations by individual electrons. Incidentally, low-energy electron diffraction atthe carbon honeycomb lattice could provide additional ways of observing these excitations17hrough elastically diffracted beams, for which plasmons should be the dominant channelof inelastic losses. Inelastic losses in core-level photoelectrons, which have been used tostudy plasmons in semi-infinite and ultrahin metals, as well as (the so-called plas-mon satellites), offer another alternative to resolve multiple plasmon excitations in confinedsystems.In practical experiments, the graphene structures are likely to lie on a substrate, andtheir size and shape must be chosen such that the energies of higher-order plasmons are notmultiples of the targeted plasmon energy. Although for the sake of simplicity the analysiscarried out here is limited to self-standing graphene structures, it can be trivially extendedto carbon islands supported on a substrate of permittivity (cid:15) by simply multiplying boththe graphene conductivity (or equivalently, the Fermi energy in the Drude model) and theexternal electron potential by a factor 2 / (1 + (cid:15) ). This factor incorporates a rigorous correctionto the 1 /r free-space point-charge Coulomb interaction when the charge is instead placedright at the substrate surface. Likewise, the contribution of the image potential leads to atotal external potential at the surface given by 2 / (1 + (cid:15) ) times the bare potential of the movingelectron. The presence of a surface can however attenuate the transmitted electron intensity,so reflection measurements appear as a more suitable configuration. It should be emphasizedthat the energy loss and plasmon excitation probabilities produced by transmitted electronscoincide with those for specularly reflected electrons, and therefore, the present theory isequally applicable to reflection geometries, as indicated in 4. This is due to the smallthickness of the graphene, where induced charges cannot provide information on which sidethe electron is coming from. A detailed analysis consisting in using an external electroncharge ρ ext ( r , t ) for a specularly reflected electron fully confirms this result.High plasmon excitation efficiencies should be also observable in metal nanoparticles. Onecould for example study electrons reflected on a monolayer of nanometer-sized gold colloids,which present a similar degree of mode confinement as graphene. However, plasmons in noblemetals have lower optical quality factors than in graphene, thus compromising the condition18hat vτ be smaller than the electron wave function spread (see above). Additionally, thetrajectories of sub-keV electrons reflected from metal colloids can be dramatically affectedby their stochastic distributions of facets and small degree of surface homogeneity comparedwith graphene, the 2D morphology of which can be tailored with nearly atomic detail. In summary, graphene provides a unique combination of surface quality, tunability, andoptical confinement that makes the detection of multiple plasmon excitations by individualelectrons feasible, thus opening a new avenue to explore fundamental quantum phenonema,nanoscale optical nonlinearities, and efficient mechanisms of plasmon excitation and detec-tion with potential application to opto-electronic nanodevices.
METHODS
In this section, we formulate an electrostatic scaling law and a quantum-mechanical modelthat allow us to describe multiple excitations of graphene plasmons by fast electrons. Themodel agrees with classical theory within first-order perturbation theory, and provides a fast,accurate procedure to compute excitation probabilities for a wide range of electron velocitiesand graphene parameters.
Eigenmode expansion of the classical electrostatic potential near graphene.
The graphene structures under consideration are much smaller than the light wavelengthsassociated with their plasmon frequencies, and therefore, we can describe their response inthe electrostatic limit. The optical electric field can be thus expressed as the gradient of anelectric potential φ in the plane of the graphene. It is convenient to write the self-consistentrelation φ ( r , ω ) = φ ext ( r , ω ) + i ω Z d R | r − R | ∇ R · σ ( R , ω ) ∇ R φ ( R , ω ) , (20)where the integral represents the potential produced by the charge density induced on thegraphene, which in virtue of the continuity equation, is in turn expressed as ( − i /ω ) ∇ R · j ( R , ω ) in terms of the induced current j ( R , ω ) = − σ ( R , ω ) ∇ R φ ( R , ω ). Here, σ is the 2D19raphene conductivity, which we assume to act locally. These expressions involve coordinatevectors R = ( x, y ) in the plane of the graphene, z = 0. Although ?? is valid for any point r = ( R , z ), we take z = 0 to obtain the self-consistent electric potential in the graphenesheet. Incidentally, the abrupt change of σ at the edge of a graphene structure produces adivergent boundary contribution to the integrand of ?? . These types of divergences have beenextensively studied in the context of magneplasmons at the edge of a bounded 2D electrongas, and more recently also in graphene. In practice, we can solve ?? numerically bysmoothing the edge (e.g., though in-plane modal expansions of σ and φ ). This producesconvergent results in agreement with direct solutions of the 3D Poisson equation. However,we only use ?? in this article to derive formal relations and scaling laws involving plasmonmodes, whereas specific numerical computations are performed following a different method,as explained below.Given the lack of absolute length scales in electrostatics, we can recast ?? in scale-invariant form by using the reduced 2D coordinate vectors þ = R /D , where D is a char-acteristic length of the graphene structure ( e.g. , the diameter of a disk). Additionally, weassume that the conductivity can be separated as σ ( R , ω ) = f ( R ) σ ( ω ). For homogeneouslydoped graphene, f ( R ) simply represents a filling factor that takes the value f = 1 in thegraphene and vanishes elsewhere. However, the present formalism can be readily applied tomore realistic inhomogeneous doping profiles by transferring the space-dependence of E F to f . This can be applied to describe inhomogeneously doped graphene, including divergencesin the doping density near the edges. Combining these elements, we obtain φ (þ , ω ) = φ ext (þ , ω ) + η ( ω ) Z d þ | þ − þ | ∇ þ · f (þ ) ∇ þ φ (þ , ω ) , (21)where η ( ω ) = iσ ( ω ) ωD (22)is a dimensionless parameter containing all the physical characteristics of the graphene, such20s the doping level, the temperature dependence, and the rate of inelastic losses, as well asthe dependence on frequency ω . Integrating by parts and taking the in-plane 2D gradienton both sides of ?? , we find the more symmetric expression ~ E (þ , ω ) = ~ E ext (þ , ω ) + η ( ω ) Z d þ M (þ , þ ) · ~ E (þ , ω ) , (23)where ~ E (þ , ω ) = − q f (þ) ∇ þ φ (þ , ω ) and M (þ , þ ) = q f (þ) f (þ ) ∇ þ ⊗ ∇ þ | þ − þ | is a symmetric matrix that is invariant under exchange of its arguments: M (þ , þ ) = M (þ , þ).This implies that M is a real, symmetric operator that admits a complete set of real eigen-values 1 /η j and orthonormalized eigenvectors ~ E j satisfying ~ E j (þ) = η j Z d þ M (þ , þ ) · ~ E j (þ ) , Z d þ ~ E j (þ) · ~ E j (þ) = δ jj , and X j ~ E j (þ) ⊗ ~ E j (þ ) = δ (þ − þ ) I , where I is the 2 × ?? can be expressed in terms of these eigenmodes as ~ E (þ , ω ) = X j c j − η ( ω ) /η j ~ E j (þ) , with expansion coefficients c j = Z d þ ~ E j (þ) · ~ E ext (þ , ω ) . (24)The potential outside the graphene can be constructed from φ (þ , ω ) through the induced21harge ( iσ ( ω ) /ω ) ∇ R · f ( R ) ∇ R φ ( R , ω ) = ( − η/D ) ∇ þ · q f (þ) ~ E (þ , ω ). Following a proceduresimilar to the derivation of ?? , we find φ ind ( u , ω ) = X j c j /η j − /η ( ω ) ϕ j ( u ) , (25)where ϕ j ( u ) = Z d þ | u − θ | ∇ þ · q f (þ ) ~ E j (þ ) (26)and we have defined the reduced 3D coordinate vector u = r /D . Classical screened interaction potential.
The screened interaction W ind ( r , r , ω ) isdefined as the potential produced at r by a unit point charge placed at r and oscillating intime as exp( − i ωt ). We use the formalism introduced in the previous paragraph and considerin ?? the point-charge external potential φ ext (þ , ω ) = (1 /D ) / | þ − u | . Integrating by parts,we find c j = ϕ j ( u ) /D (see ?? ), which upon insertion into ?? yields W ind ( u , u , ω ) = 1 D X j /η j − /η ( ω ) ϕ j ( u ) ϕ j ( u ) . (27)The well-known symmetry W ind ( u , u , ω ) = W ind ( u , u , ω ) is apparent in ?? . For an arbitraryexternal charge distribution ρ ext ( r , t ), which we express in frequency space as ρ ext ( r , ω ) = Z dt e i ωt ρ ext ( r , t ) , (28)the induced potential can be written using the screened interaction as φ ind ( r , ω ) = Z d r W ind ( r , r , ω ) ρ ext ( r , ω ) . (29) Electrostatic scaling law for the plasmon frequency.
The condition η ( ω j ) = η j determines a plasmon frequency of the system ω j for a specific eigenstate j , subject tothe condition η j < D r Figure 6: Electron moving with velocity v and crossing a graphene structure of characteristicsize D .limit, so that all the graphene characteristics, including the size of the structure D , are fullycontained within η ( ω ), and thus, given a certain shape ( e.g. , a disk), the eigenvalues η j canbe calculated once and for all to obtain the plasmon frequency for arbitrary size or doping.A powerful electrostatic law can be formulated by assuming the Drude model for theconductivity of graphene ( ?? ). The plasmon frequency is then given by ω j − i / τ , where ω j = vuut e E F − πη j ¯ h D − τ ≈ γ j e ¯ h s E F D (30)and we have defined the real number (provided η j < γ j = 1 / q − πη j (31)to obtain the rightmost expression in ?? . Classical approach to the electron energy-loss probability.
We consider an elec-tron moving with constant velocity vector v along the straight-line trajectory r = r + v t passing near or through a graphene structure, as shown in 6. The energy transferred fromthe electron to the graphene (∆ E >
0) can be written as ∆ E = Z dω ¯ hω Γ( ω ) , ω ) = eπ ¯ h Z dt Im n e − i ωt φ ind ( r + v t, ω ) o (32)is the loss probability per unit of frequency range, and φ ind is the ω component of thepotential induced by the electron along its path. The external charge density associatedwith the moving electron reduces to ρ ext ( r , t ) = − e δ ( r − r − v t ). Using this expressiontogether with ?? , the loss probability is found to beΓ( ω ) = e π ¯ hω D X j Im ( /η ( ω ) − /η j ) G j ( ζ ) , (33)where ζ = ωD/v , G j ( ζ ) = (cid:12)(cid:12)(cid:12)(cid:12) ζ Z d‘ e i ζ‘ ϕ j ( u + ‘ ˆ v ) (cid:12)(cid:12)(cid:12)(cid:12) , (34)we adopt the notation u = r /D , and the integral is over the path length ‘ (in units of D )along the velocity vector direction.We now concentrate on a specific plasmon resonance j and neglect contributions to theloss probability arising from modes other than this particular one. For simplicity, we workwithin the Drude model and assume a small plasmon width τ − (cid:28) ω j . Using ?? in ?? , andintegrating over ω to cover the plasmon peak area, we find P cla j = Z j dω Γ( ω ) ≈ π γ j e √ E F D G j γ j e √ E F D ¯ hv ! (35)for the probability of exciting plasmon j by the incident electron. Finally, we can recast ?? into the scale-invariant form of ?? , using the definitions of ?? , and further defining thedimensionless scaled plasmon electric potential f j = 1 q πγ j ϕ j . (36)We show in 7 two examples of loss spectra for 100 eV electrons passing near a 100 nm24 =
100 nm L o ss p r ob a b ilit y ( e V - ) Energy loss (eV) m = m = E F = ! ! ! =
100 eV 100 eV
Figure 7: Energy-loss spectra for 100 eV electrons passing near a graphene disk along thetrajectories shown in the inset. The graphene parameters are indicated by labels. Solidcurves are calculated with the boundary-element method using the local-RPA model forthe graphene conductivity. The results obtained from the semi-analytical model of ?? arerepresented by symbols for the lowest-order m = 0 and m = 1 modes, in excellent agreementwith the solid curves (see also 2).graphene disk. The lowest-energy plasmon features for both m = 0 and m = 1 symmetriesare clearly separated from other peaks in their corresponding spectral regions, thus justifyingthe approximation of ?? . We further compare in this figure the full solution of Maxwell’sequations (solid curves) with the result obtained from ?? using a single plasmon term foreach of the lowest-order m = 0 and m = 1 modes, with η j = − .
024 and − . ϕ j calculated from the plasmon potential, which is normalized as explained below. Quantum approach to the screened interaction potential.
The linear screenedinteraction potential can be obtained by solving the density matrix ( ?? ), which yields thesolution ρ = | ξ ih ξ | , with | ξ i given by ?? . Expressing g ( t ) in frequency space ω , we can thenwrite ξ ( t ) = − h Z d r Z dω π φ j ( r ) ρ ext ( r , ω ) e i( ω j − ω ) t ω j − ω − i / τ , where ρ ext ( r , ω ) is defined by ?? . Now, calculating the induced potential from its expectation25alue φ ind ( r , t ) = h ξ | (cid:16) a + + a (cid:17) φ j ( r ) | ξ i , we find φ ind ( r , t ) = Z dω π e − i ωt Z d r W ind ( r , r , ω ) ρ ext ( r , ω )(or equivalently, ?? ), where W ind ( r , r , ω ) = 2 ω j ¯ h φ j ( r ) φ j ( r )( ω + i / τ ) − ω j (37)is the quantum-mechanical counterpart of ?? . Normalization of the plasmon potential.
In the Drude model ( ?? ), assuming adominant plasmon mode contributing to the response with frequency given by ?? , we findthat ?? are identical under the assumption ω j τ (cid:29)
1, provided we take φ j = e E F D ! / f j , (38)where f j is the dimensionless scaled plasmon electric potential defined by ?? , which is inde-pendent of the doping level E F and the size of the structure D . As a self-consistency test, wecalculate the plasmon excitation probability to first-order perturbation from the quantummodel, which yields P j = e ¯ h (cid:12)(cid:12)(cid:12)(cid:12)Z dt e i ω j t φ j ( r + v t ) (cid:12)(cid:12)(cid:12)(cid:12) . (39)Indeed, this equation coincides with the classsical result of ?? , provided ?? is satisfied.In practice we obtain the plasmon potential φ j as follows: first, we calculate the potentialinduced by a dipole placed near the graphene and oscillating at frequency ω j using theboundary-element method (BEM) for fixed values of E F and D ; the resulting potentialmust be equal to φ j times an unknown constant; we deduce this constant by calculating theplasmon excitation probability from ?? and by comparing the result to a well-established26lassical calculation of the loss probability based upon BEM; finally, we use the scalinglaws of ?? to obtain ω j and φ j for any desired values of E F and D , assuming the validityof the Drude model. We have verified that this procedure yields, within the accuracy of theBEM method, the same scaled potentials f j and ϕ j for different initial values of E F and D .Incidentally, once φ j is calculated, ?? provides a fast way of obtaining loss probabilities forarbitrary values of the electron velocity, the size of the structure, and the doping conditions. Analytical expressions for the electron energy-loss probability in homoge-neous graphene.
In the electrostatic limit, the loss probability of electrons moving ei-ther parallel or perpendicularly with respect to an extended sheet of homogeneously dopedgraphene can be expressed in terms of the Fresnel reflection coefficient for p polarized light,as shown in ?? . Indeed, for a parallel trajectory, we can readily use the well-establisheddielectric formalism ( e.g. , Eq. (25) of Ref. in the c → ∞ limit), which directly yields ?? .Likewise, for a perpendicular trajectory, we can write the bare potential of the moving elec-tron in ( k k , ω ) space as φ ext ( k k , z ) = − πev e i ωz/v k k + ω /v . Inserting this expression into ?? and writing the 2D Coulomb interaction as (2 π/k k )e − k k | z | ,we find the induced potential φ ind ( k k , z ) = − r p e − k k | z | φ ind ( k k , , which, together with ?? , allows us to write the loss probability for a perpendicular trajectoryas shown in ?? . It should be noted that the external field produced on the graphene by anelectron that is specularly reflected at the graphene plane is also given by the above expressionfor φ ind ( k k , ?? yields the loss probability for such reflected trajectoryas well. 27 upporting Information We provide further examples of the normalized functions f j and F j , as well as multiple plas-mon excitations for parallel and perpendicular trajectories with respect to a doped graphenedisk. z / D x / D f p f p f p f p D z m = m = m = m = D z D / 20 D x D / 20 D x D / 20 Figure 8: Normalized electrostatic potential f j for the lowest-order m = 0 and m = 1 modesof a homogeneously doped graphene disk and for different sampling trajectories. The m = 0and m = 1 lowest-order modes have energies given by ¯ hω j = γ j e q E F /D with γ j = 3 . γ j = 2 .
0, respectively. The parallel electron trajectory in the right panels crosses the diskaxis at x = 0.Examples of the normalized functions f j and F j for a disk and different electron trajec-tories are offered in Figs. 8 and 9. The highest probability among the trajectories consideredin these figures is F j ≈ .
0, which is obtained in the excitation of the m = 1 mode with µ/ν ≈ . E F = 0 . D = 50 nm, we have µ ≈ .
2, and the excitation28 v F p µ v !" m = m = D D / 20 D D / 20 D / 20 (1) (2) (3) (4) (1) (2) (3) (4) Figure 9: Normalized loss function F j for the lowest-order m = 0 and m = 1 modes of ahomogeneously doped graphene disk and for different sampling trajectories.peak occurs for ν = 2 .
2, or equivalently, for an electron energy ≈
65 eV. The plasmon energy¯ hω j = γ j e q E F /D (see main paper) is 0 .
24 eV. The electron passes at a distance D/
20 = 5 nmfrom the graphene (see Fig. 9, upper inset). The excitation probability P j ≈ (1 /µ ) F j ( µ/ν )is then 73%, which reveals a clear departure from the perturbative regime, and thus thisconfiguration requires a more detailed analysis including multiple-plasmon excitation.Furthermore, we show in Fig. 10 the probability for exciting one-plasmon and two-plasmon states under the same trajectories as considered in Figs. 8 and 9 for an electronenergy of 50 eV (i.e., ν = 1 . Acknowledgement
The author acknowledges helpful and enjoyable discussions with Archie Howie and Darrick E.Chang. He also thanks IQFR-CSIC for providing computers used for numerical simulations.This work has been supported in part by the Spanish MEC (MAT2010-14885) and the EC29 µ m = m = m = m = P r ob a b ilit y P r ob a b ilit y P r ob a b ilit y P r ob a b ilit y ω p ω p ω p ω p ω p ω p ω p ω p Figure 10: Probability of exciting the lowest-order m = 0 and m = 1 modes of a homoge-neously doped graphene disk for the same electron trajectories as considered in Figs. 8 and 9,as a function of µ = (1 /e ) √ E F D . The electron energy is 50 eV. We show the total probability(black curves) and the partial probabilities for exciting one-plasmon ( ω p , blue curves) andtwo-plasmon (2 ω p , red curves) states. 30Graphene Flagship CNECT-ICT-604391). References
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