Probing Plasmons in Graphene by Resonance Energy Transfer
PProbing Plasmons in Graphene by Resonance Energy Transfer
Kirill A. Velizhanin ∗ Center for Nonlinear Studies (CNLS)/T-4, Theoretical Division,Los Alamos National Laboratory, Los Alamos, NM 87545
Anatoly Efimov
Center for Integrated Nanotechnologies, Materials Physics & Applications Division,Los Alamos National Laboratory, Los Alamos, NM 87545 (Dated: December 6, 2018)We theoretically propose an experimental method to probe electronic excitations in graphene –monoatomic layer of carbon – by monitoring the fluorescence quenching of a semiconductor quantumdot (or a dye molecule) due to the resonance energy transfer to the graphene sheet. We show howthe dispersion relation of plasmons in graphene can be accurately extracted by varying the back-gatevoltage and the distance between the quantum dot and graphene.
I. INTRODUCTION
Many appealing properties of graphene – amonoatomic crystalline sheet of carbon – stem from itsunique electronic structure. Specifically, its honeycomblattice combined with the conjugation of π -electronsover the entire sheet results in the electronic spectrumof a zero-gap semiconductor with “ultra-relativistic”electrons and holes. Already this makes grapheneenormously appealing from both basic and applica-tion standpoints. What makes graphene even moreattractive is the possibility to tune these properties ina wide range by patterning, chemical functionalization,doping etc. For example, shifting the Fermi level awayfrom the charge neutrality (Dirac) point by applyingthe back-gate voltage, and, therefore, changing thegraphene’s electrical conductivity, can become useful ingraphene-based electronics.
The improved electrical conductivity of a back-gatedor chemically doped graphene sample leads to qualitativechanges in its optical properties. In particular, collectiveexcitations (plasmons), rather then single-particle exci-tations (electron-hole pairs), are expected to define anelectronic response of graphene to a low-frequency op-tical perturbation. Plasmons in graphene are of greatinterest from the basic perspective, as collective excita-tions in a two-dimensional electron gas with very pecu-liar properties. Besides, the graphene plasmonics holdspromise for, e.g., photonic and optoelectronic devices, aswell as metamaterials.
Plasmons in graphene have beenpredicted and studied, theoretically but experimentalstudies of this phenomenon are still very sparse.
In this paper we propose an experimental techniqueto probe plasmons in graphene. The technique is basedon the F¨orster resonance energy transfer between a fluo-rescent semiconductor quantum dot (or a dye molecule)and a nearby graphene layer. Efficient energy trans-fer between a dye molecule and graphene has been the-oretically predicted by Swathi and Sebastian andlater confirmed experimentally. However, Swathi andSebastian described electronic excitations in graphene on the single-particle level, which is accurate only fora nearly undoped (charge-neutral) graphene. In this pa-per, electronic excitations are treated more accuratelyby adopting the random-phase approximation, which al-lows for recovering the collective electronic behavior. Wedemonstrate that electronic excitations in graphene, bothsingle-particle and collective, can be sensitively probedby studying the fluorescence quenching of the quantumdot due to the energy transfer to graphene. Specifically,we show how the plasmon dispersion can be extractedfrom experiment.The expected advantage of the proposed techniqueover the typically used electron energy loss spectroscopy(EELS) is its intrinsic locality, i.e., plasmons are pro-posed to be probed locally by a semiconductor quantumdot (typically a few nanometers in diameter). In con-trast, EELS averages the plasmonic response over a cer-tain portion of a graphene sample, thus, adding the in-homogeneous broadening due to, e.g., charge puddles, to experimental observables.The paper is organized as follows. The general theoryof quantum dot fluorescence quenching due to the reso-nance energy transfer to graphene is given in Sec. II. Theanalysis of quenching efficiency within the single-particleand random-phase approximation levels is provided inSec. IV and Sec. III, respectively. Sec. V concludes. II. F ¨ORSTER RESONANCE ENERGYTRANSFER
F¨orster resonance energy transfer (FRET) refers tothe transfer of electronic excitation energy betweenchromophores mediated by the nonradiative Coulombcoupling. This process consists of deexcitation of aninitially (optically) excited donor chromophore and thesimultaneous excitation of an acceptor chromophore. Anexample of such a process is the energy transfer between asemiconductor quantum dot (QD) and a non-fluorescentorganic molecule as an acceptor chromophore. As theenergy transfer to the non-fluorescent “dark” molecule(i.e., quencher) competes with the intrinsic fluorescence a r X i v : . [ c ond - m a t . m e s - h a ll ] A ug of the “bright” QD, the FRET rate can be assessedthrough the effective decrease (quenching) of the QD flu-orescence quantum yield, and/or through the shorter ap-parent fluorescence lifetime, measured by time-resolvedfluorescence spectroscopy. Specifically, the apparent flu-orescence lifetime is given by τ = 1 / ( τ − + k q ), where τ is the lifetime of the isolated QD and k q is the FRET(quenching) rate. Accordingly, the decreased fluores-cence quantum yield in the presence of FRET is givenby τ /τ . Therefore, k q can be extracted provided τ and τ are known from experiment.The efficiency of FRET is strongly affected by param-eters of the acceptor’s excitation spectrum. A trivial ex-ample is a vanishing FRET rate in a system where anacceptor chromophore does not have excitations resonantto the lowest excited state of a donor chromophore. Thisallows one to use FRET as a spectroscopic tool to probean electronic structure of a system of interest. An advan-tage of this method is the near-field regime of the donor-acceptor interaction, which allows for probing excitationsforbidden in the far-field (optical) regime due to, e.g., thelarge wavelength mismatch between optical photons andmaterial electronic excitations. Short-wavelength charac-ter of electronic excitations in graphene, especially thatof plasmons, hinders the study of these excitations bystandard far-field optical techniques. We propose to useFRET between QD and a graphene sheet to probe elec-tronic excitations in graphene. The theoretical frame-work for FRET in the QD-graphene complex is developedin the rest of the section. A. Fluorescence quenching efficiency
We define the wavefunctions of the excited and theground states of QD as | e (cid:105) and | g (cid:105) , respectively. The truemany-body ground state and excited states of grapheneare denoted by | (cid:105) and | n (cid:105) , respectively. The quenchingrate, k q , for the energy transfer from QD to graphene isgiven by Fermi’s golden rule as k q = 2 π (cid:126) − (cid:88) n |(cid:104) n |(cid:104) g | ˆ V | e (cid:105)| (cid:105)| δ ( (cid:15) − E n ) , (1)where ˆ V is the operator of Coulomb interaction betweenfluctuating charge densities of QD and grapheneˆ V = (cid:90) d r V ( r ) ( | e (cid:105)(cid:104) g | + | g (cid:105)(cid:104) e | ) ˆ ρ ( r ) . (2)The operator of graphene charge density is given by − e ˆ ρ ( r ) = − e ˆ ϕ † ( r ) ˆ ϕ ( r ), where operators ˆ ϕ † ( r ) and ˆ ϕ ( r )create and destroy an electron at position r within thegraphene sheet, respectively. The absolute value of theelectron charge is denoted by e = | e | . The excitationenergies of QD and graphene are denoted by (cid:15) and E n ,respectively. Vector variables are denoted in bold.For the rest of the paper, we adopt the dipole approx-imation for QD, which results in V ( r ) = − e ( d · r ) /r . The transition dipole of QD is given by d . This approx-imation is accurate if z (cid:29) D , where z is the distancebetween QD and graphene, and D is the QD diameter.The validity of this approximation for the realistic caseof PbSe QD is discussed in Sec. IV.The quenching rate in Eq. (1) can be rewritten throughthe retarded polarization operator of graphene Π r ( q, (cid:15) ) as(a detailed derivation is given in the Appendix) k q = − πe (cid:126) − ( d (cid:107) + 2 d ⊥ ) (cid:90) ∞ qdq Im [Π r ( q, (cid:15) )] e − qz , (3)where d (cid:107) and d ⊥ are the projections of the QD transitiondipole d onto the graphene’s plane and the normal to thisplane, respectively. If we ignore for simplicity that thepopulation of the QD excited state depends on the anglebetween the QD transition dipole and the polarizationof the initial QD-exciting laser pulse, we simply needto average over all the possible orientations of d withrespect to the graphene’s plane which yields (cid:104) d (cid:107) (cid:105) / (cid:104) d ⊥ (cid:105) = d / k q = − πe d (cid:126) (cid:90) ∞ qdq Im [Π r ( q, (cid:15) )] e − qz . (4)We define the quenching efficiency (QE) as ϕ q = k q /k r ,where k r is the fluorescence rate for the isolated QD k r = 4 (cid:15) (cid:126) c d . (5)Substituting Eq. (5) into Eq. (4), one obtains for QE ϕ q = − π (cid:126) c e (cid:15) (cid:90) ∞ qdq Im [Π r ( q, (cid:15) )] e − qz . (6) B. Massless Dirac fermions approximation
Polarization operator Π r ( q, (cid:15) ) for graphene can be ob-tained at various levels of theory. At zeroth order approx-imation with respect to the electron-electron Coulombinteraction within the graphene sheet, it can be evalu-ated adopting the free massless Dirac fermions (MDF)approximation. At this level, the polarization opera-tor, denoted by Π r ( q, (cid:15) ), is a bare polarization bubbledescribing a single non-interacting electron-hole pair ingraphene. Within this approximation, Π r ( q, (cid:15) ) has beenevaluated previously for arbitrary doping level and itsimaginary part is shown in Fig. 1(a). This figure is validfor any level of doping, defined by the chemical poten-tial, µ , measured relative to the Dirac point, since dueto the linear dispersion relation of massless fermions ingraphene, the polarization operator does not change withdoping, if measured in units of µ (cid:126) v f and plotted againstunitless coordinates q/k f and (cid:15)/µ , where v f ≈ k f = µ/ (cid:126) v f . With respect to the wave number, q , the MDF approx-imation is only valid when q (cid:28) a − , where a ≈ (a) (b) FIG. 1: Density plot of (a) Im [Π r ( q, (cid:15) )] and (b)Im [Π rRPA ( q, (cid:15) )] in units of µ (cid:126) v f . Checkerboard pattern em-phasizes regions where the imaginary part of the polarizationoperator vanishes exactly. the lattice constant of graphene. Because of the factor e − qz in the integrand of Eq. (6), this condition is equiv-alent to restricting z (cid:29) a , i.e., to considering the quench-ing efficiency ϕ q only at large QD-graphene distancescompared to the lattice constant of graphene. However,because of the adopted dipole approximation we are al-ready restricted to z (cid:29) D , where the typical QD diame-ter ( D ) is on the order of a few nanometers. Thus, setting z (cid:29) a because of the MDF approximation does not re-strict the range of applicability of Eq. (6) any furthercompared to the one set by the dipole approximation.With respect to the excitation energy, the MDF approx-imation has been shown to be valid for (cid:15) (cid:46) The key features of Im [Π r ( q, (cid:15) )] include (i) the singu-larity along the (cid:15)/µ = q/k f line, which corresponds tosingle-particle excitations with ( q , (cid:15) ) vector lying withinthe surface of the Dirac cone, and (ii) the absence ofsingle-particle excitations, i.e., Im [Π r ( q, (cid:15) )] ≡
0, in re-gions A and B , marked by checkerboard patterns inFig. 1(a). Equation 4 with Π r ( q, (cid:15) ) substituted in is verysimilar to the result of Swathi and Sebastian for the fluorescence quenching of a dye molecule due to single-particle excitations in graphene.The polarization operator within the bare bubble ap-proximation does not include the graphene’s polarizationself-consistently which can become crucial at non-zerodoping levels ( µ > Π rRP A ( q, (cid:15) ) = Π r ( q, (cid:15) )1 − W ( q )Π r ( q, (cid:15) ) , (7)where W ( q ) = 2 πe / ˜ κq is the two-dimensionalFourier transform of the Coulomb potential within thegraphene’s plane. The effective dielectric constant of theenvironment is denoted by ˜ κ . A free-standing graphenesheet in vacuum corresponds to ˜ κ = 1. For graphenelaying on top of a half-space dielectric substrate withdielectric constant κ , the effective constant is given by˜ κ = ( κ + 1) / For example, for a SiO substrate( κ =4) the effective dielectric constant is ˜ κ = 2 .
5. In therest of the paper, except for results shown in Fig. 4, thevacuum conditions (˜ κ = 1) are assumed.The imaginary part of Π rRP A ( q, (cid:15) ) is depicted inFig. 1(b). It is seen that the “single-particle” singularity (cid:15)/µ = q/k f is gone and the new singularity at (cid:15) ∝ q / appears instead. This emergent singularity correspondsto the collective electronic excitation, i.e., plasmon, ingraphene. It is rather pedagogical to see exactly howthis singularity appears from Eq. (7). The naive sub-stitution of Π r ( q, (cid:15) ), shown in Fig. 1, into Eq. (7) re-sults in Im [Π rRP A ( q, (cid:15) )] vanishing exactly in the entireregion A since both Π r ( q, (cid:15) ) and W ( q ) are real in this re-gion. However, the more careful analysis reveals that theimaginary part of Π r ( q, (cid:15) ) is not exactly zero in A , butinfinitesimal instead due to the usual small imaginaryconstant in the denominator of the Lindhard function,which preserves causality. The infinitesimal imaginarypart of the bare polarization operator can be safely ne-glected if Π r ( q, (cid:15) ) is analyzed by itself. However, when1 − W ( q )Re [Π r ( q, (cid:15) )] vanishes in the denominator ofEq. (7), this small imaginary part yields singularity ( δ function) in Im [Π rRP A ( q, (cid:15) )], i.e.,Im [Π rRP A ( q, (cid:15) )] ∝ δ ( q − q p ( (cid:15) )) , (8)where the plasmon dispersion relation is given by q p ( (cid:15) ) ≈ (cid:15) e µ at small (cid:15) , i.e., at (cid:15)/µ (cid:28)
1, and obtained numericallyat higher excitation energies. Therefore, the general pre-scription to assume all the infinitesimal parameters finitetill the very end of calculations proves to be critical inthis case.It is important to note that since there is no single-particle excitations in region A , the plasmon lifetime isinfinite [ δ function in the imaginary part of Π rRP A ( q, (cid:15) )]within this region. Once plasmon “leaves” region A IVIIIIII
FIG. 2: Regimes of asymptotic dependence of quenching ef-ficiency on the QD-graphene distance, z , at z → ∞ (RegimesI, II and III) and at z → ( q/k f (cid:38) . r ( q, (cid:15) ),as well as screened excitations in graphene, described byΠ rRP A ( q, (cid:15) ), respectively. III. QUENCHING BY UNSCREENEDEXCITATIONS
QE due to unscreened single-particle excitations ingraphene can be obtained from Eq. (6) by substitutingΠ r ( q, (cid:15) ) with Π r ( q, (cid:15) ). This approximation is accurate ifeither (i) the effective dielectric constant of a substrateis high, and, therefore, the effective electron-electronCoulomb interaction within graphene is greatly reduced,or (ii) (cid:15) (cid:29) µ , i.e., graphene becomes effectively undoped.In both cases, the screening within the graphene’s layerbecomes ineffective yielding Π rRP A ( q, (cid:15) ) ≈ Π r ( q, (cid:15) ).Four different regimes of the asymptotic dependence ofQE on the QD-graphene distance, z , are shown schemat-ically in Fig. 2. At large distances, exponent e − qz inthe integrand of Eq. (6) decays rapidly with q , whichguarantees that at fixed excitation energy (cid:15) the domi-nant contribution to QE comes from lowest possible q ’swhere the imaginary part of the polarization operator isstill nonzero. The two qualitatively different cases are (i)“gapless”, where the imaginary part of the polarizationoperator is already non-zero even at infinitesimally small q >
0, and (ii) “finite-gap”, where Im [Π r ( q, (cid:15) )] becomesnon-zero only at some finite q , i.e., at q > q ∗ with finite q ∗ >
0. The typical power law dependence of Π r ( q, (cid:15) ) on q at q → r ( q, (cid:15) ) ∝ q α , (9a)and Π r ( q, (cid:15) ) ∝ ( q − q ∗ ) α , (9b)at q → q ∗ + 0 in the finite-gap case, yield ϕ q ∝ z − ( α +2) , (10a)and ϕ q ∝ z − ( α +1) e − q ∗ z , (10b)at z → ∞ , respectively. In what follows, we considerthree different asymptotic regimes of QE at z → ∞ (Regimes I, II and III). The asymptotic behavior of QEat z → Regime I.
At high excitation energies ( (cid:15)/µ >
2) theimaginary part of Π r ( q, (cid:15) ) is seen in Fig. 1(a) to corre-spond to the gapless case. Further, it can be shown thatIm [Π r ( q, (cid:15) )] ∝ q at small q (see Eq. (12) in Ref. 7 forthe long wavelength limit of the polarization operator),which is then combined with Eq. (10a) to give ϕ q ∝ /z . (11)This power-law dependence is shown as Regime I inFig. 2. The asymptotic behavior in Eq. (11) is satis-fied at (cid:15)/µ >
2, and, therefore, also at (cid:15) (cid:29) µ , where µ can be treated as zero, i.e., the graphene sheet be-comes effectively undoped. The asymptotics of 1 /z forthe fluorescence quenching by single-particle excitationsin undoped (and also weakly doped) graphene was firstpredicted by Swathi and Sebastian. Generally, the asymptotic dependence 1 /z is ratherexpected since QE due to the energy transfer from a chro-mophore to a quencher accompanied by the interband ex-citation of single electron-hole pairs in an N -dimensionalquencher typically scales as z N − , where z is the distancebetween the chromophore and the quencher. FRET to0 d quencher (e.g., small organic molecule) correspondsto N = 0, naturally resulting in 1 /z . Nanowire as aquencher ( N = 1) leads to 1 /z asymptotics. Excita-tion of single-electron holes in a dipole-to-surface con-figuration, i.e., when a quencher occupies the half-space( N = 3), yields 1 /z asymptotics. The graphene sheetis a 2 d object, which naturally leads to 1 /z dependenceof QE at large z . Regime II.
At 1 < (cid:15)/µ <
2, the imaginary part ofΠ r ( q, (cid:15) ) vanishes exactly at q < q ∗ , where the thresholdvalue of q ∗ /k f = 2 − (cid:15)/µ marks the onset of interband single-particle excitations. It can be shown that in thisfinite-gap case, Im [Π r ( q, (cid:15) )] is proportional to ( q − q ∗ ) / at q → q ∗ + 0, which yields [by virtue of Eq. (10b)] ϕ q ∝ z − / e − q ∗ z , (12)which is marked as regime II in Fig. 2. Regime III.
At low excitation energies, (cid:15)/µ <
1, thefinite-gap case is again realized with q ∗ /k f = (cid:15)/µ , whichis the onset of intraband single-particle excitations. Inthis regime, Im [Π r ( q, (cid:15) )] ∝ ( q − q ∗ ) − / , resulting in ϕ q ∝ z − / e − q ∗ z , (13)depicted as regime III in Fig. 2. Regime IV.
Finally, at a fixed excitation energy thereare no single-particle excitations with q/k f > (cid:15)/µ , i.e,in region B in Fig. 1. This introduces the natural high- q cutoff for integration in Eq. (6). If, at certain (small) z , exponent e − qz is still ≈ q approaching this cut-off, then QE becomes constant with respect to z . Thisis depicted as regime IV in Fig. 2. However, we expectthis regime to be hardly accessible experimentally. Forexample, at (cid:15) = 0 . µ = 0, the realization of thisregime requires z < Besides, at such distances both dipoleand MDF approximations are likely to break down.
IV. QUENCHING BY SCREENEDEXCITATIONS
To evaluate QE in the case of screened excitations ingraphene, one has to substitute Π r ( q, (cid:15) ) in Eq. (6) withΠ rRP A ( q, (cid:15) ) given by Eq. (7). First, we consider regimeI in Fig. 2. In this regime, the product W ( q )Π r ( q, (cid:15) )becomes proportional to 1 /q × q = q at q →
0, which re-sults in the approximate equality Π rRP A ( q, (cid:15) ) ≈ Π r ( q, (cid:15) )at small q . Thus, the asymptotic behavior of ϕ q ( z ) atlarge z is the same for screened and unscreened excita-tions in regime I; i.e., taking screening into account doesnot lead to qualitative changes in QE. This can be eas-ily understood for (cid:15) (cid:29) µ , where graphene becomes ef-fectively undoped, and, therefore, the small free carrierdensity renders screening within graphene inefficient.The situation is different in regime III ( (cid:15)/µ < A . Equation (8) implies that Im [Π rRP A ( q, (cid:15) )] vanishesexactly at q < q ∗ = q p ( (cid:15) ); i.e., the finite-gap situation isrealized with q ∗ defined by the plasmon dispersion rela-tion. Substituting Eq. (8) into Eq. (6), one obtains ϕ q ( z ) ∝ e − q p ( (cid:15) ) z . (14)The absence of the power-law multiplier in front of theexponent, which was universally present in the finite-gapsituations in the previous section, is related to a delta-functional instead of a power-law singularity of the po-larization operator.The asymptotic behavior, given by Eq. (14), is cor-rect even outside regime III, since the plasmon branch in (a) (b) FIG. 3: (a) Dependence of the quenching efficiency ϕ q onthe QD-graphene distance z . (b) The filled area shows where ϕ q (z) is between 1000 and 1 at a given excitation energy (cid:15)/µ .The red (left) and black (right) circles mark ϕ q ( z )=1000 and ϕ q ( z )=1 contour lines, respectively. Fig. 1(b) remains singular up to (cid:15)/µ ≈ .
3, i.e., wellwithin what used to be regime II in the case of un-screened excitations. For 1 . (cid:46) (cid:15)/µ < rRP A ( q, (cid:15) ) scales as ( q − q ∗ ) / at q → q ∗ + 0with the q gap defined by q ∗ /k f = 2 − (cid:15)/µ , which yields ϕ q ( z ) ∝ z − / e − q ∗ z , i.e., the large- z asymptotics is iden-tical to that of regime II in the case of unscreened exci-tations.To examine how accurately large- z asymptotics for ϕ q ( z ) reproduce exact solutions at finite z , we numeri-cally evaluate the integral in Eq. (6) for the realistic caseof PbSe QD with the excitation energy of (cid:15) = 0 . µ s, suggesting that QE can bedirectly extracted from experiment, since the observablefluorescence quantum yield is given by 1 / (1 + ϕ q ) in thepresence of FRET. Numerically evaluated ϕ q ( z ) for several values of chem-ical potential in the range µ = 0 . − . (cid:15)/µ = 4. The cor-responding ϕ q ( z ), depicted by black circles, is expectedto show 1 /z dependence at large z , and, indeed, demon-strates slowly decaying non-exponential tail.All the other values of chemical potential give (cid:15)/µ < ϕ q ( z ) ∝ e − q p ( (cid:15) ) z and ϕ q ( z ) ∝ z − / e − q ∗ z at large z for (cid:15)/µ (cid:46) . . (cid:46) (cid:15)/µ <
2, respectively, with q ∗ defined by q ∗ /k f =2 − (cid:15)/µ . As expected, all ϕ q ( z )’s, except for the onecorresponding to the lowest value of chemical potential( µ = 0 . FIG. 4: QE decay rate q ∗ plotted vs (cid:15)/µ . Circles (black),squares (red) and triangles (blue) correspond to single-particle approximation, RPA for free-standing graphene (vac-uum) and RPA for graphene on substrate (SiO , κ =4), re-spectively. at large z . The filled area in Fig. 3(b) shows the rangeof QD-graphene distances, where QE is between 1 and1000 – somewhat loosely chosen range where the accurateexperimental measurement of QE is still possible. QE isseen to decay the fastest with z (lowest z at fixed ϕ q )where the q -gap is the largest, i.e., at (cid:15)/µ ≈ .
3, as isseen in Fig. 1(b).The rate of the exponential decay of QE at (cid:15)/µ < rRP A ( q, (cid:15) )]at small q , i.e., the width of the finite q -gap. There-fore, the large- z behavior of QE can be used to extractthe valuable information about electronic excitations ingraphene, e.g., the plasmon dispersion. To illustratehow the plasmon dispersion can be extracted, we plot q ∗ against (cid:15)/µ in Fig. 4, where q ∗ is the decay rate ofQE at large z , obtained by fitting the large- z QE decaywith ϕ q ( z ) ∝ e − q ∗ z . Specifically, q ∗ is plotted for thecase of unscreened excitations (black circles), as well asfor screened excitations in free-standing graphene (redsquares) and graphene laying on top of the SiO sub-strate (blue triangles). Scanning through the range ofvalues of chemical potential using the back-gate, we areable to extract the dispersion relation of singularities ofthe imaginary part of the polarization operator. As isseen, the linear dispersion relation of single-particle ex-citations (black circles at (cid:15)/µ <
1) as well as that of theplasmon in the free-standing graphene (red squares at (cid:15)/µ (cid:46) .
3) in graphene is accurately recovered.As discussed in the beginning of Sec. III, electronicexcitations in graphene are effectively single-particle ei- ther if (i) there is a strong substrate-induced dielectricscreening or (ii) the excitation energy is high, i.e., (cid:15) (cid:29) µ .The first case is illustrated by the plasmon dispersion forgraphene on the SiO substrate (blue triangles), whichis closer to the single-particle excitations (black circles)than that corresponding to the free-standing graphene(red squares). The second case is effectively realized at (cid:15)/µ (cid:38) .
3, where dispersions of electronic excitationswith and without accounting for the in-graphene screen-ing are nearly identical.The proposed method to extract the plasmon dis-persion requires an accurate control over the distancebetween PbSe QD and the graphene sheet. This canbe accomplished by either using the core-shell type-1structures or by growing a dielectric layer on top ofgraphene by atomic layer deposition with the controllablethickness and then depositing QDs on top of that dielec-tric layer. The first approach can be based on PbSe/CdSecore-shell structures, where the large bulk bandgap ofCdSe ( ∼ ∼ This confine-ment guarantees that the shell serves only as an inertspacer between the PbSe core and the graphene layer.Recent advances in core-shell structure fabrication tech-niques allow one to control the thickness of the shell withthe monolayer (subnanometer) precision. Atomic layerdeposition provides an alternative strategy for control-ling the distance between QD and graphene with sub-nanometer resolution. Finally, we wish to discuss the effect of the finite plas-mon propagation length on the applicability of the pro-posed method. So far, we assumed that the plasmon hasinfinite propagation length within region A in Fig. 1(b).However, additional damping channels, not accountedfor in RPA (e.g., defect scattering, electron-phonon cou-pling), can “broaden” the plasmon in region A . It fol-lows from Eq. (6), that if the plasmon width in the q domain, δq , is less then 1 /z , then Im [Π rRP A ( q, (cid:15) )] canstill be treated as δ function, resulting in Eq. (14). Ac-cordingly, deviations from the exponential dependenceare expected at z (cid:38) δq − ∼ l , where l is the plasmonpropagation length.Our numerical tests show that the exponential decayof QE is typically already established at z comparablewith the plasmon wavelength λ . On the other hand, es-timations by various authors suggest that the plasmonpropagation length could be as high as 10-100 λ . Therefore, we expect QE to decay exponentially at z (cid:38) λ ,thus allowing for extraction of plasmon dispersion. Theonset of deviations from this exponential decay at largerQD-graphene distances, z (cid:38) l , naturally provides a goodestimate for the plasmon propagation length. V. CONCLUSION
Based on the detailed analysis of fluorescence quench-ing efficiency in the QD-graphene complex, we have pro-posed a method of probing and studying electronic exci-tations in graphene. The method has been demonstratedto be sufficiently sensitive to allow the extraction of thedispersion relation of plasmon in graphene. We hope thatthis study will stimulate experimental efforts in this di-rection, especially because the proposed method is basedon the QD-graphene complex which can be of interestnot only as a means to probe electronic excitations ingraphene, but also on its own merit as a key componentof hybrid nanostructures with promising properties. Forexample, the ability to excite plasmon locally in grapheneusing a semiconductor quantum dot can become of greatuse in graphene plasmonics. Another recently proposeduse of QD-graphene complexes is in photovoltaics. The RPA-based description of electronic excitationsin graphene, adopted in this paper, does not includesuch many-body effects as exchange and correlation, im-purity and defect scattering, or electron-phonon cou-pling. These effects can affect the plasmonic responseof graphene in two ways. First, there can be deviationsof the actual plasmon dispersion relation from the oneshown in Fig. 1(b). Second, additional channels of plas-mon damping can appear, as discussed in the previoussection. The proposed experimental technique is capableof assessing both the dispersion relation and the damp-ing and, thus, is expected to stimulate the developmentof beyond-RPA theoretical methods by providing a nec-essary experimental validation.
Acknowledgments
This work was performed, in part, at the Center for In-tegrated Nanotechnologies, a U.S. Department of Energy,Office of Basic Energy Sciences user facility. K.A.V. ac-knowledges support by the Center for Nonlinear Studies(CNLS), LANL.
Appendix: Derivation of quenching rate
Evaluating explicitly the QD part of the matrix ele-ment in Eq. (1) one obtains k q = 2 π (cid:126) − (cid:88) n (cid:12)(cid:12)(cid:12)(cid:12) (cid:104) n | (cid:90) g d r V ( r )ˆ ρ ( r ) | (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) δ ( (cid:15) − E n ) . (A.1)This can be rewritten as k q = (cid:126) − (2 π ) (cid:88) n (cid:90) d r d r (cid:48) (cid:90) d q d q (cid:48) V ∗ ( q ) e − i q · r V ( q (cid:48) ) e i q (cid:48) · r (cid:48) × ρ n ( r ) ρ n ( r (cid:48) ) δ ( (cid:15) − E n ) , (A.2) where ρ n ( r ) = (cid:104) n | ˆ ρ ( r ) | (cid:105) and V ( r ) = π ) (cid:82) d q V ( q ) e i q · r . Further, the δ function can be sub-stituted using the identity δ ( (cid:15) − E n ) = (cid:126) − π (cid:82) dte i ( (cid:15) − E n ) t/ (cid:126) yielding k q = i (cid:126) − (2 π ) (cid:90) d r d r (cid:48) (cid:90) d q d q (cid:48) V ∗ ( q ) e − i q · r V ( q (cid:48) ) e i q (cid:48) · r (cid:48) × (cid:90) dt Π > ( r , r (cid:48) ; t ) e i(cid:15)t/ (cid:126) , (A.3)where Π > ( r , r (cid:48) ; t ) = − i (cid:126) − (cid:80) n ρ n ( r ) ρ n ( r (cid:48) ) e − iE n t/ (cid:126) isthe “greater” polarization operator for graphene in theLehmann representation. At sufficiently large distancesbetween the quantum dot and the graphene layer, V ( r )varies smoothly within the graphene’s plane. This makesit possible to average the polarization operator over theunit cell with respect to both r and r (cid:48) . After this av-eraging, the polarization operator becomes insensitiveto variations of electronic density on the scale of thegraphene’s unit cell and, therefore, acquires the isotropyand the continuous translational symmetry instead ofthe discrete one, leading to a possibility to substituteΠ > ( r , r (cid:48) ; t ) → Π > ( | r − r (cid:48) | , t ). Then, integrations over r , r (cid:48) , and t can be interpreted as spatial and time Fouriertransforms, respectively, resulting in k q = i (cid:126) − (2 π ) (cid:90) d q | V ( q ) | Π > ( q, (cid:15) ) , (A.4)whereΠ > ( q, (cid:15) ) = (cid:90) d r (cid:90) dt Π > ( | r | , t ) e − i ( qr − (cid:15)t/ (cid:126) ) . (A.5)Using the relations between real-time correlation and re-sponse functions at equilibrium, which at zero temper-ature yields Π > = 2 i Im[Π r ], we obtain k q = − (cid:126) − (2 π ) (cid:90) d q | V ( q ) | Im [Π r ( q, (cid:15) )] . (A.6)This equation is similar to the well-known result by Metiu[Eq. (2.30) in Ref. 43].Finally, we provide without derivation the two-dimensional Fourier transform (within the graphene’splane) of V ( r ) V ( q ) = 2 πie ( d (cid:107) cos( θ ) + id ⊥ ) e − qz , (A.7)where d (cid:107) is the projection of the QD transition dipoleonto the graphene’s plane, d (cid:107) = | d (cid:107) | being its magnitude.The angle between vectors d (cid:107) and q is denoted by θ . Theprojection of the QD transition dipole onto the normalto the graphene’s plane is denoted by d ⊥ . Substitution of V ( q ) into Eq. (A.6) and the subsequent integration over θ yields Eq. (4). ∗ Electronic address: [email protected] A. K. Geim and K. S. Novoselov, Nature Materials , 183(2007). P. R. Wallace, Phys. Rev. , 622 (1947). C. Berger, Z. Song, T. Li, X. Li, A. Y. Ogbazghi, R. Feng,Z. Dai, A. N. Marchenkov, E. H. Conrad, P. N. First, et al.,J. Phys. Chem. B , 19912 (2004). F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, NaturePhot. , 611 (2010). N. Papasimakis, Z. Luo, Z. X. Shen, F. De Angelis,E. Di Fabrizio, A. E. Nikolaenko, and N. I. Zheludev, Op-tics Express , 8353 (2010). K. W.-K. Shung, Phys. Rev. B , 979 (1986). B. Wunsch, T. Stauber, F. Sols, and F. Guinea, New. J.Phys. , 318 (2006). E. H. Hwang and S. Das Sarma, Phys. Rev. B , 205418(2007). M. Polini, R. Asgari, G. Borghi, Y. Barlas, T. Pereg-Barnea, and A. H. MacDonald, Phys. Rev. B , 081411(2008). A. Hill, S. A. Mikhailov, and K. Ziegler, Eur. Phys. Lett. , 27005 (2009). M. Jablan, H. Buljan, and M. Soljaˇci´c, Phys. Rev. B ,245435 (2009). R. A. Muniz, H. P. Dahal, A. V. Balatsky, and S. Haas,Phys. Rev. B , 081411 (2010). Y. Liu, R. F. Willis, K. V. Emtsev, and T. Seyller, Phys.Rev. B , 201403 (2008). T. Eberlein, U. Bangert, R. R. Nair, R. Jones, M. Gass,A. L. Bleloch, K. S. Novoselov, A. Geim, and P. R. Brid-don, Phys. Rev. B , 233406 (2008). C. Tegenkamp, H. Pfn¨ur, T. Langer, J. Baringhaus, andH. W. Schumacher, J. Phys.: Condens. Matter , 012001(2011). R. S. Swathi and K. L. Sebastian, J. Chem. Phys. ,054703 (2008). R. S. Swathi and K. L. Sebastian, J. Chem. Phys. ,086101 (2009). R. S. Swathi and K. L. Sebastian, J. Chem. Sci. , 777(2009). Z. Chen, S. Berciaud, C. Nuckolls, T. F. Heinz, and L. E.Brus, ACS Nano , 2964 (2010). J. Martin, N. Akerman, G. Ulbricht, T. Lohmann, J. H.Smet, K. von Klitzing, and A. Yacobe, Nature Phys. , 144(2008). Here, the term “chromophore” is used rather loosely refer-ring to an object (e.g., a molecule, semiconductor quantumdot), whose electronic degrees of freedom interact with theexternal electromagnetic field. L. Novotny and B. Hecht,
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