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Sep 24, 2015 - An atomistic description is presented of the excited state dynamics in spherical Au/CdS core/shell nanocrystals up to a diameter of 15 ...
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Exciton Formation and Quenching in an Au/CdS Core/Shell Nano-Structure Dirk Ziemann, and Volkhard May J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.5b01820 • Publication Date (Web): 24 Sep 2015 Downloaded from http://pubs.acs.org on September 27, 2015

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Exciton Formation and Quenching in a Au/CdS Core/Shell Nano-Structure Dirk Ziemannn∗ and Volkhard May† Institute f¨ ur Physik, Humboldt-Universit¨at zu Berlin, Netwonstraße 15, D-12489 Berlin, Germany (Dated: September 24, 2015)

Abstract An atomistic description is presented of the excited state dynamics in spherical Au/CdS core/shell nano-crystals up to a diameter of 15 nm. Au-core excited states are considered in a multipole plasmon scheme while a tight-binding description combined with a configuration interaction approach is used to compute single electron-hole pair excitations in the CdS-shell. The electron-hole pair energy-shift and screening due to an Au-core polarization is found of minor importance. For the studied system the energy transfer coupling can be identified as the essential core-shell interaction. Characterizing the CdS-shell excitons by atomic centered transition charges and the Au-core by its multipole plasmon moments an energy transfer coupling can be introduced which gives a complete microscopic description beyond any dipole-dipole approximation and with values around 10 meV. Together with a considerable plasmon-exciton energy mismatch these coupling values explain the measured 300 ps life-time of shell excitons due to energy transfer to the Au-core.

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TOC Graphic

Key Words core/shell nano-crystal; excitation energy transfer; tight-binding model; Coulomb correlated electron-hole pairs; excitons; plasmons; multipole moments; transfer rates;

A decade ago core-shell systems with a molecular shell surrounding a metal core found some interest as prototypical systems displaying exciton-plasmon hybridization1,2 . Such molecule metal nano-particle (MNP) systems have been also discussed in order to realize a working nano-laser3–6 . To replace a molecular coating of a MNP-core by a semiconductor shell requires more involved preparation techniques. Only recently, these system attracted some interest7–10 . After optical excitation of a metal-core semiconductor-shell system a multitude of processes takes place. Ref.10 reported on the transient spectroscopy of an Au-core CdS-shell system similar to that one shown in Fig. 1. Between both subsystems charge and excitation energy transfer can take place. If the core is excited plasmon states may be formed which immediately decay into hot electrons due to electron-electron scattering (see the recent studies 2

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in11–13 ). Despite that, charge trapping in surface and interfacial states is possible. Direct exciton formation in the semiconductor-shell may also result in charge trapping at the surface and subsequent fast charge transfer into the core. Alternatively, somewhat slower energy transfer is possible14 . It is the specifty of the Au-core CdS-shell system described in Ref.10 that a nonepitaxial association has been achieved at the interface of the two components. The considerable reduction of interfacial defects also diminishes carrier trapping. A rather direct Au-core plasmon CdS-shell exciton coupling can be expected. Moreover, charge transfer seems to be fast but inefficient (spectroscopic signatures as described in ref.14 are absent). Therefore, to start the theoretical analysis of excited state dynamics in such metal-core semiconductorshell systems we focus here on those processes which go along without charge transfer. The specific Au-core CdS-shell system as experimentally studied in Ref.10 will be consider. In particular, it is demonstrated that the restriction to energy transfer processes already allows to explain the observed shell-exciton life-times of about 300 ps. Let us firstly study the CdS-shell in the absence of the Au-core. In order to describe singly excited electron-hole pair states of the shell we follow our earlier work of15,16 based on the methodology of17–22 . These approaches combine a tight-binding description of the individual electron and hole states with a consideration of the Coulomb-attraction in a configuration interaction (CI) scheme (see also Section 1 of the Supporting Information). The semi-empirical tight-binding approach uses a sp3 d5 -basis and parameters according to Ref.23 and the improvement of Ref.24 where spin-orbit coupling has been included. For the inner and outer surface a perfect passivation is assumed which is achieved according to the treatment of25 . Accordingly, surface states are removed due to ligands as well as due to a defectless interface between the Au-core and CdS-shell. Following Ref.26 a dielectric constant of  = 12 was chosen for the CdS-shell. It enables us to match the lowest exciton state of the massive nano-crystal (NC) with different diameters and compositions10,26,27 . To get an impression of the exciton states in the CdS-shell we start with a description of a smaller system as shown in Fig. 1. It more directly displays the interrelation between the actual shape and the energy level position. Let us first consider a massive Cd1631 S1322 NC with a diameter d = 5.2 nm. Shell structures are generated from this NC by removing a core with diameter dcore . We considered different cases: dcore = 1.2 nm (Cd1618 S1293 ), dcore = 1.6 nm (Cd1586 S1238 ), dcore = 2.1 nm (Cd1535 S1156 ) and dcore = 2.6 nm (Cd1453 S1045 ). For all 3

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structures a larger number of Cd-atoms compared to the number of S-atoms is chosen. This is necessary in order to reproduce the data of Ref.10 . Because a uniform and non segmented shell structure as well as less than 3% of impurities could be observed there, a non amorphous pure CdS structure can be taken. Hence, all calculations are based on a wurtzite structure with morphological parameters according to28 . First excited states in all these structures appear as Wannier-Mott like excitons. They are characterized by their energy spectrum Eα and the electron-hole pair amplitude Cα (a¯ a) which describes the correlated motion of the electron with quantum number a and of the hole with quantum number a ¯. Excitation energy transfer coupling to the Au-core is determined by the transition densities related to the various exciton levels ρα (x) (see the Supporting Information). Fig. 2 displays the exciton spectrum of the different types of CdS-shell structures mentioned above (the core is assumed to be empty and characterized by  = 1). Note that the exciton levels have been drawn relative to the lowest energy E0 as ∆Eα = Eα − E 0 . Due to the confinement effect the onset of the exciton spectrum increases with increasing dcore . We find for dcore = 0 the value of E0 = 2.706 eV, for dcore = 1.2 nm E0 = 2.865 eV, for dcore = 1.6 nm E0 = 2.985 eV, for dcore = 2.1 nm E0 = 3.158 eV, and for dcore = 2.6 nm E0 = 3.393 eV. At the same time the number of exciton states increases in the given interval. This behavior can be already understood in describing the electron and the hole as an independent particle moving in an isotropic sphere with constant potential energy. Since the centrifugal barrier ~2 l(l + 1)/2m∗ r2 enters the single-particle Schr¨odinger equation (l denotes the angular moment quantum number and m∗ stands for the effective single particle mass), a l-dependent splitting of the spectrum appears. The change from a massive sphere to a shell with an increasing inner diameter reduces the splitting. As a consequence, the resulting denser single particle spectrum leads to a denser exciton spectrum (see Section 1.1 of the Supporting Information). To achieve an impression of the thermally induced population of the exciton levels the upper panel of Fig. 2 includes the course of the thermal distribution fα at 300 K. Accordingly, only a small region of the exciton spectrum is affected (a state around 80 meV is populated with less than 5% compared to a state at the bottom of the spectrum). Transition charge densities ρα related to some of these exciton levels are displayed in Figs. 3 and 4 (they are represented by isosurfaces deduced from Reρα ). Since they mainly 4

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determine the energy transfer coupling to the plasmon excitations of the Au-core it is more appropriate to discuss the ρα instead of the exciton wave function. The figures suggest a classification of the ρα with respect to their multipole moments. In order to do this we place a fictitious charge e at the shell surface. It’s Coulomb interaction with the charge density ρα is reformulated as the coupling Vαl of the l’th multipole moment of the exciton transition density α (averaged with respect to multipole moments m = −l, ..., l; see Section 1.2 of the Supporting Information). Looking first at the energy scheme related to the massive CdS NC (upper panel of Fig. 2) one notices that it is mainly determined by the lowest exciton levels which split into dark and bright states. These states are almost completely characterized by their transition dipole moments (coupling energy Vαl=1 dominates). Also the dark states gain some oscillator strength due to spin mixing resulting from the spin-orbit interaction. Higher excitons with ∆Eα about 40 meV are characterized by a ρα with a large quadrupole moment. The transition density for the lowest lying bright exciton (exciton labeled by 1 in Fig. 2) and the lowest lying exciton with a non-vanishing quadrupole moment (exciton 2 of Fig. 2) are shown in Fig. 3. For the exciton labeled by 1 the transition density is shaped like an atomic p-orbital and displays directly the behavior of a dipole. Exciton 2 shows a clear quadrupole distribution of the transition density and is shaped like an atomic d-orbital. If the core of the CdS NC is removed the excitons are distinguished by higher transition multipole moments. We take a closer view on the NC with the most narrow shell (case of dcore = 2.6 nm, cf. Fig. 2, lower panel, and Fig. 4). The exciton levels display strong transition quadrupole moments for an energy range with ∆Eα < 10 meV, which is in strong contrast to the massive NC without a core. For larger energies excitons with strong transition dipole moments appear. Around ∆Eα = 23 meV an exciton is formed that is dominated by an octupole moment. Above 50 meV the excitons show also an increasing influence of even higher multipole moments. Related transition densities are shown in Fig. 4. They have multipole moments with l = 2 (exciton 3 in Fig. 2), with l = 1 (exciton 4 in Fig. 2), with l = 3 (exciton 5 in Fig. 2) and with l = 4 (exciton 6 in Fig. 2). After having classified the exciton levels for a CdS-shell structure but without the Aucore we now account for the core-shell coupling. On the one-hand side the CdS-shell may interact with the core via charge exchange processes. Following photo-excitation so-called spill out electrons of the metal core may tunnel into the shell. And, shell electrons and holes 5

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can also penetrate in the core. For the following, however, we will assume that the core and the shell are sufficiently separated from each other and any charge exchange is suppressed. Our considerations will concentrate exclusively on those interactions which allow for energy but not for charge transfer. Once an electron-hole pair has been excited in the shell it may couple to the core via energy transfer and subsequent plasmon formation. This is a coupling where the shell electron-hole pair disappears. It will be discussed below. But there is also a coupling to the Au-core where the shell electron-hole pair persist. It covers an electron and a hole energy shift (self-energy effect) due to the metal core polarization and a core induced screening of the shell electron-hole Coulomb coupling. We assume that both processes can be considered as instantaneous rearrangements of Au-core electrons what allows the description in the framework of image charges (see, for example, Refs.29,30 and also Section 2 and 3 of the Supporting Information). The interaction of the electron with the image charges induced by itself results in an energy level shift (Eq. (S46) of the Supporting Information). The same is valid for the hole. The interaction of the electron with the image charges induced by the hole and the interaction of the hole with the image charges belonging to the electron modifies the electron-hole pair interaction (Eq. (S48) of the Supporting Information). Both couplings alter the exciton states. Treating image charge effect due to the Au-core only via an energy shift of the electron and the hole levels, a reduction of the exciton energy of less than 1 meV results for all discussed structures. Despite the size of the Au-core and the CdS-shell the strength of this effect is also determined by screening in the shell. In the considered system it is rather strong leading, thus, to the small energy shift. But taking the full image charge effect into account an increase of the different exciton energies of more than 5 meV can be observed. For example, a shift from Eα=1 = 3.393 eV to Eα=1 = 3.399 eV is obtained for the structure with dcore = 2.6 nm. The Au-core induced slight weakening of the electron-hole Coulomb coupling explains this increase (cf. Eq. (S48) of the Supporting Information). As in Fig. 2, lower panel, we display in Fig. 5 the exciton spectrum and the multipole separation of the transition charge density for the core-shell structure with dcore = 2.6 nm, but now accounting also for the Au-core induced polarization effects. Qualitatively the spectrum stays the same. But the exciton levels are shifted differently leading to a new order of states. The individual strength of each multipole moment is changed slightly. And, 6

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the number of states is increased in the given energy range. Focusing on the mean multipole moments of the first hundred excitons the mean dipole moment is reduced by 3%, the quadrupole moment by 10% and the octupole moment by 4% in comparison to the system without the Au-core. For the system with dcore = 1.2 nm, however, the reduction is below 1% for all multipole moments. The minor importance of Au-core polarization induced changes of the CdS-shell exciton levels for the considered core-shell systems has to be considered as a particular result of the present studies. Accordingly, the excitation energy transfer coupling dominates the core-shell interaction. Since it is only of intermediate strength (see below) a Golden-rule type rate formula is utilized. The rate for excitation energy transfer from the CdS-shell to the Au-core takes the form kshell→core =

2π X X fα |Jlm,α |2 d(Eα − ~ωl ) . ~ l,m α

(1)

Energy transfer coupling between the core and the shell is described by the matrix elements Jlm,α . These matrix elements stand for the decay of an exciton with quantum number α and the excitation of a plasmon with multipole indices l and m. The Jlm,α are obtained as the coupling between shell exciton transition charges and core plasmon multipole moments (for details see the Supporting Information). Moreover, a thermal distribution fα across the initially excited excitons has been assumed. d(Eα − ~ωl ) accounts for the life-time broadened energy conservation of the transition where ~ωl is the plasmon energy. To match p the experimentally measured dipole plasmon frequency ωd of10 we set ωl = 3l/(2l + 1)ωd instead of directly using the Mie-theory formula (cf., e.g.,31 ). If a Lorentzian type of broadening is assumed the d-function takes the form d(∆E) = 1/π × ~γpl /(∆E 2 + (~γpl )2 ). The resonance condition is considerably weakened due to the plasmon life-time broadening γpl . From the transient absorption spectra of10 a broadening of about γpl = 100 meV can be derived. Because these measurements are performed on an ensemble the slightly different shapes and sizes of the MNP result in different dipole plasmon resonances leading to an inhomogeneous broadening. This was studied for Au MNPs in32 . There, a standard deviation with respect to the plasmon resonance of 20 meV has been found what results in a life-time broadening of around ~γpl = 80 meV. To be comparable with Ref.10 a core-shell system will be considered with an Au-core diameter of dcore = 6.2 nm and a CdS-shell with a thickness of 4.5 nm (about 64.000 atoms 7

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in the shell; cf. Fig. 1). Before discussing the concrete values of the transition rate we take a closer look at the exciton levels of this large CdS-shell. In doing this we note that the image charge effect leads to an increase of the lowest exciton state from E0 = 2.548 eV to E0 = 2.551 eV. This underestimates the experimental value of 2.638 eV by 87 meV10 . We do not relate this discrepancy to the inappropriateness of the tight-binding approach but to the shell geometry, shell lattice structure and Cd to S ratio which is not precisely known from the experiment. Since the measured energy is larger and, thus, somewhat more apart from the dipole plasmon energy we take this experimental value when calculating kshell→core (the replacement avoids an overestimation of the rate). In the energy range up to 100 meV above E0 there are about 1100 exciton levels (cf. also Fig. 6). Below 15 meV the exciton levels are dominated by strong dipole moments, followed successively by levels with strong l = 2-, l = 3-, and l = 4-moments. Above 15 meV the spectrum becomes dense and most of the states are characterized by higher moments. In order to judge the energy transfer coupling a thermal weighted coupling is introduced15 sX fα |Jlm,α |2 . (2) J¯l = α,m

Because of the huge number of excitons which couple to the different plasmon modes, such an uniform measure for the strength of this coupling is useful. In Fig. 7 the quantity J¯l is shown in the presence and in the absence of the image charge effect. The coupling stays below 4 meV and decreases with increasing plasmon multipole mode. A slight reduction of J¯l is observed when the image charge effect is accounted for. This behavior can be related to the reduction of the exciton transition density multipole moments as discussed above for the smaller core-shell systems. Turning to the bare coupling Jlm,α to a particular exciton level values up to 30 meV can be obtained. Nevertheless, the rather small value of the J¯l together with the off-resonant position of the plasmon and exciton levels excludes any distinct effect of exciton-plasmon hybridization. Having discussed the plasmon broadening and the energy transfer coupling we turn to an inspection of the complete rate kshell→core , Eq. (1). A comparison of the dipole plasmon energy ~ωl=1 = 2.156 eV with the lowest exciton level at 2.638 eV points to the off-resonant character of the transition. The energy mismatch attains a value of 482 meV but is somewhat reduced when the coupling to higher multipole plasmons is considered. This indicates the decisive role of the broadening-function d(Eα − ~ωl ) when computing the rate. Respective 8

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results for the l–dependent partial rates kl =

2π X fα |Jlm,α |2 d(Eα − ~ωl ) ~ m,α

(3)

are drawn in Fig. 8. Taking a Lorentzian-type of broadening with γpl = 80 meV a rate kl=1 (from the thermal distributed shell excitons to the dipole plasmon) of about 23 · 109 /s is obtained. The long tail of the Lorentz-curve compensates the huge energetic mismatch. The total rate kshell→core amounts about 80 · 109 /s resulting in a exciton lifetime of 13 ps. This value strongly underestimates the measured value of 300 ps10 . Changing to the inverse transition from an initially excited dipole plasmon to all shell excitons with rate expression kcore→shell =

2π X |Jα,l=1 m |2 d(~ωl=1 − Eα ) , ~ m,α

(4)

a value of 12 · 1012 /s can be deduced what corresponds to a life-time of 86 fs. This value is also not confirmed by the experiments of Ref.10 To overcome this discrepancy we suggest here the use of a Gaussian-broadening for the d-function instead of the Lorentz-type used so far to compute the rate. We take this replacement here simply to have a d-function which becomes smaller in it’s border area. But it is also justified by the fact that in a more involved description where the broadening function enters as a frequency dependent imaginary self-energy the d-function will behave differently from a simple Lorentzian form. Respective results for the partial rates, Eq. (3), and for different γpl varying between 80 and 110 meV are also presented in Fig. 8. The rate of energy transfer into the dipole plasmon is below 106 /s. Larger values for kl are obtained if l increases. Two competing mechanism are obvious. On the one-hand side, the energetic difference between the plasmon and exciton decreases with increasing l. On the other hand side the coupling decreases with increasing l. The largest rate is obtained for l = 3. If l > 7 there are only minor contributions, because of the vanishing coupling. The total rates kshell→core range from 2 · 109 /s for γpl = 80 meV to 17 · 109 /s for γpl = 110 meV. The related life–times vary between 476 and 60 ps. Choosing γpl = 85 meV a life-time of 299 ps is obtained which coincides with the measured value (if image charge effects are ignored all rates are by about 10% larger). For the inverse transfer kcore→shell a rate below 0.2·109 /s is obtained resulting in a life-time of 5 ns. Consequently, a direct transfer from the plasmon to the excitons is very improbable since detected plasmon life-times for Au nano-particles of a similar size are in the range of 9

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4 to 10 fs34 . The resulting hot electrons, however, have a life-time of a few ps35 matching the measured transient absorption spectra of10 . Hence, the transfer from the metal to the semiconductor shell could be attributed to a charge or energy transfer between hot electrons of the metal and the CdS-excitons. Let us summarize our findings on excitation energy transfer in a Au/CdS core/shell nano-structure. We can firstly state that the shell exciton life-time of 300 ps as reported in10 could be explained by exclusively considering an energy transfer coupling. Any charge transfer mechanism seems not necessary for an understanding. To arrive at this conclusion spectra and transition densities for excitons confined in a CdS-shell have been presented. If the inner core is increased while the outer shell radius remains unchanged they display an increase in the density of exciton states close to the band gap as well as an increase in the strength of higher multipole moments. The insertion of a metal core influences the exciton spectrum via image charge effects. The resulting exciton spectrum is somewhat shifted to higher energies. And, the multipole moments as well as the energy transfer coupling to the metal plasmons are reduced. However, the overall influence of the metal core polarization is of minor importance. The dominant core-shell coupling is due to energy transfer. For this process it is essential to consider higher plasmon modes. In contrast, excitation energy transfer from an initially excited dipole plasmon to the shell is very improbable. However, in any case the energy transfer coupling is too small to originate noticeable effects of plasmon-exciton hybridization.

ASSOCIATED CONTENT Supporting Information A description of the theoretical background is available. This information can be found free of charge on the internet at http://pubs.acs.org.

AUTHOR INFORMATION Corresponding Authors Volkhard May E-Mail: [email protected] Notes The authors declare no competing financial interest. 10

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ACKNOWLEDGMENTS Financial support by the Deutsche Forschungsgemeinschaft through Sfb 951 is gratefully acknowledged.



Electronic address: [email protected]



Electronic address: [email protected]

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Korkusinski, M.; Voznyy, O.; Hawrylak, P. Fine Structure and Size Dependence of Exciton and

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Biexciton Optical Spectra in CdSe Nanocrystals. Phys. Rev. B 2010, 82, 245304. 23

D´ıaz, J. G.; Zieli´ nski, M.; Jask´ olski, W.; Bryant, G. W. Tight-Binding Theory of ZnSCdS Nanoscale Heterostructures: Role of Strain and d orbitals. Phys. Rev. B 2006, 74, 205309.

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Sapra, S.; Shanthi, N.; Sarma, D. D. Realistic Tight-Binding Model for The Electronic Structure of II-VI Semiconductors. Phys. Rev. B 2002, 66, 205202.

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Lee, S.; Oyafuso, F.; von Allmen, P.; Klimeck, G. Boundary Conditions for the Electronic Structure of Finite-Extent Embedded Semiconductor Nanostructures. Phys. Rev. B 2004, 69, 045316.

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Tiwari, S.; Tiwari, S. Electrical and Optical Properties of CdS Nanocrystalline Semiconductors. Cryst. Res. Technol. 2006, 41, 78-82.

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Garrett, M. D.; Dukes, A. D., III.; McBride, J. R.; Smith, N. J.; Pennycook, S. J.; Rosenthal, S. J. Band Edge Recombination in CdSe, CdS and CdSx Se1−x Alloy Nanocrystals Observed by Ultrafast Fluorescence Upconversion: The Effect of Surface Trap States. J. Phys. Chem. C 2008, 112, 12736-12746.

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Simmons, B. A.; Li, S.; John, V. T.; McPherson, G. L.; Bose, A.; Zhou, W.; He, J. Morphology of CdS Nanocrystals Synthesized in a Mixed Surfactant System. Nano Lett. 2002, 2, 263-268.

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Sugakov, V. I.; Vertsimakha, G. V. Localized Exciton States with Giant Oscillator Strength in Quantum Well in Vicinity of Metallic Nanoparticle. Phys. Rev. B 2010, 81, 235308.

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Benchmakeh, R.; Gippius, N. A.; Even, J.; Nestoklon, M. O.; Jancu, J.-M.; Ithurria, S.; Dubertret, B.; Efros, A. L.; Voisin, P. Tight-Binding Calculations of Image-Charge Effects in Colloidal Nanoscale Platelets of CdSe. Phys. Rev. B 2014, 89, 035307.

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Zelinskyy, Y.; Zhang, Y.; May, V. Supramolecular Complex Coupled to a Metal Nanoparticle: Computational Studies on the Optical Absorption. J. Phys. Chem. A 2012, 116, 11330-11340.

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Berciaud, S.; Cognet, L.; Tamarat, P.; Lounis, B. Observation of Intrinsic Size Effects in the Optical Response of Individual Gold Nanoparticles. Nano Lett. 2005, 5, 515-518.

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In a more involved description the broadening function may enter as a frequency dependent imaginary self-energy part and the d-function will behave different from a simple Lorentzian form.

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Nilius, N.; Ernst, N.; Freund, H.-J. Tip Influence on Plasmon Excitations in Single Gold Particles in an STM. Phys. Rev. B 2002, 65, 115421.

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Perner, M.; Bost, P.; Lemmer, U.; von Plessen, G.; Feldmann, J.; Becker, U.; Mennig, M,;

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Schmitt, M.; Schmidt, H. Optically Induced Damping of the Surface Plasmon Resonance in Gold Colloids. Phys. Rev. Lett. 1997, 78, 2192-2195.

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FIG. 1: Cross-section of an Au-core CdS-shell structure with total diameter of 15.2 nm. The Au-core diameter dcore amounts 6.2 nm and the CdS shell has a thickness of 4.5 nm (Au: yellow spheres, Cd: grey spheres, S: weak yellow spheres).

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FIG. 2: Strength of the Coulomb coupling between a fictitious charge placed at the CdS-shell surface and the CdS-shell exciton transition charge densities ρα . Drawn are the multipole contributions Vαl to this coupling versus exciton energies ∆Eα = Eα −Eα=0 (see text for details; CdS-shell diameter d = 5.2 nm and empty core of varying radius dcore ). Panel a: dcore = 0, panel b: dcore = 1.2 nm, panel c: dcore = 1.6 nm, panel d: dcore = 2.1 nm, and panel e: dcore = 2.6 nm. Exciton levels are marked by vertical gray lines. Contributions due to the exciton transition density multipole moments are also indicated. Vertical black lines: l = 1-moment, blue circles: l = 2-moment, green rectangles: l = 3-moment, and red triangles; l = 4-moment (for l > 1 the quantity Vαl is enlarged by a factor of two). The dashed red line in the upper panel displays the thermal exciton distribution at 300 K. The different numbers in the upper and lower panel label those exciton states which are represented in Figs. 3 and 4 by their transition densities.

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FIG. 3: Excitonic transition densities ρα for a massive CdS nano-crystal (dcore = 0). Shown are isosurfaces embedded in the nano-crystal lattice structure (tiny grey spheres). The isosurfaces are obtained from Reρα for the exciton levels shown in Fig. 2 and labeled by 1 (upper panel) and 2 (lower panel). Blue/red colored parts indicate positively/negatively charged areas.

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FIG. 4: Isosurfaces related to excitonic transition densities ρα as in Fig. 3 but for the exciton levels labeled by 3 to 6 in Fig. 2 (top down; shell structure with dcore = 2.6 nm).

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FIG. 5: Strength of the Coulomb coupling between a fictitious charge placed at the CdS-shell surface and the CdS-shell exciton transition charge densities ρα like in Fig. 2 (panel e) but including the image charge effect.

FIG. 6: Strength of the Coulomb coupling between a fictitious charge placed at the CdS-shell surface and the CdS-shell exciton transition charge densities ρα like in Fig. 2 but for the larger Au-core CdS-shell system of Fig. 1 (overall nano-crystal diameter d = 15.2 nm and core diameter dcore = 6.2 nm).

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FIG. 7: Mean energy transfer coupling J¯l , Eq. (2), versus plasmon multipole index l for a complex with an overall nano-crystal diameter d = 15.2 nm and core diameter dcore = 6.2 nm (cf. also Figs. 6). Red squares: consideration of image charge contributions, black spheres: neglect of image charge contributions.

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FIG. 8: Energy transfer rate versus plasmon multipole index l for a complex with an overall nanocrystal diameter d = 15.2 nm and core diameter dcore = 6.2 nm. Plasmon induced broadening modeled by Gaussian distributions. Black diamonds: ~γpl = 80 meV, red rectangles: ~γpl = 85 meV, blue circle: ~γpl = 90 meV, green triangle: ~γpl = 100 meV, and brown inverse triangle: ~γpl = 110 meV. Gray pentagons: plasmon induced broadening modeled by a Lorentzian distribution with ~γpl = 80 meV.

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