Supporting information for Metal-semiconductor. nanoparticle hybrids formed by self-organization: a platform to address exciton-plasmon coupling

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1 Supporting information for Metal-semiconductor nanoparticle hybrids formed by self-organization: a platform to address exciton-plasmon coupling Christian Strelow, T. Sverre Theuerholz, Christian Schmidtke, Marten Richter, Jan-Philip Merkl, Hauke Kloust, Ziliang Ye, Horst Weller, Tony F. Heinz, Andreas Knorr, and Holger Lange, Institut für Physikalische Chemie, Universität Hamburg, Germany, Institut für Theoretische Physik, Technische Universität Berlin, Germany, Department of Applied Physics, Stanford University, Stanford, CA, United States, The Hamburg Centre for Ultrafast Imaging, Hamburg, Germany, Department of Chemistry, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia, and SLAC National Accelerator Laboratory, Menlo Park, CA, United States Holger.Lange@chemie.uni-hamburg.de To whom correspondence should be addressed Institut für Physikalische Chemie, Universität Hamburg, Germany Institut für Theoretische Physik, Technische Universität Berlin, Germany Department of Applied Physics, Stanford University, Stanford, CA, United States The Hamburg Centre for Ultrafast Imaging, Hamburg, Germany Department of Chemistry, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia SLAC National Accelerator Laboratory, Menlo Park, CA, United States 1

2 Supplementary information Additional TEM micrographs of hybrid clusters Depending on the QD to AuNP number densities, different structures can be obtained by self-organization. Figure S1) displays an example of 40 nm AuNPs mixed with green QDs in a ratio of 1:2 (AuNP to QD). In this example, the QDs attach to AuNPs on two sites and Figure S1: Green-QD based cluster. TEM micrograph of a hybrid structure of green QDs and 40 nm diameter AuNPs. The scale bar corresponds to 50 nm. promote the formation of a chain. Figure S2 displays 20 nm AuNPs mixed with red QDs (5 nm spacer) in a ratio of 1:5 (AuNP to QD). The excess of QDs with gold affinity promotes the formation of larger clusters. Other spare QDs then saturate the surface of AuNPs. 2

3 Figure S2: Red-QD based cluster. TEM image of a hybrid structure of red QDs and 20 nm diameter AuNPs. The scale bar corresponds to 100 nm. 3

4 Determination of parameters in the theory In the following, we give a detailed overview of the calculation for the parameters of the theory. To determine the parameters used in the signal definition (µ, χ) as well as in the interaction Hamiltonian H ex pl = ga ca v a + h.a. (a ca v is the same as an exciton operator often used in other theories of the Jaynes-Cummings model), the coupling between the constituents, the dipole moments and the damping (Lindblad) parameters have to be determined. The coupling between the exciton and the plasmon (g), occurring in H ex pl, as well as the dipole moment of the plasmons χ can be calculated for the AuNP dipole mode according to 1 by comparing the classical field expression with the quantum mechanical modes. This leads to g = i s α ( d sp + r QD + D Au 2 ) 3 6 ηd 3 Au πε 0 for the coupling strength, with the spacer thickness d sp, the QD radius r QD and the diameter ( of the AuNP D Au. η = d(re[ɛau ) 1 (ω)]) ω=ωsp is the inverse of the gradient of the real part of dω the dielectric function of the metal at the plasma frequency and ε 0 is the permittivity of the surrounding medium. The angle dependent factor is s α = 3cos 2 (θ) 1, where θ is the angle between the axis of the hybrid (the dipole-sphere center line) and the axis of the dipole of the corresponding plasmon mode. Note, that the plasmon mode is three fold degenerate for all three axis, and that we use the value obtained from equation S1 only as a systhematic reference for the coupling strengths distribution of the ensemble in the averaging procedure, so that the angle does not enter the calculations. The resulting coupling strength versus the AuNP diameter is depicted in Figure S6. The damping of the plasmons γ pl (D) in dependence of the AuNP diameter D and their resonance frequency ω pl (D) are calculated by considering continuity relations at the metal dielectric interface for the electromagnetic fields analog to Ref. 2 : The condition to find a nontrivial solution to the continuity relations formulates a dispersion relation, which is solved numerically (S1) 4

5 assuming a Drude-Lorentz-Sommerfeld model for the electric permittivity of gold and a size dependent relaxation rate. The size dependent relaxation rate includes not only damping due to ohmic losses in the metal, but also surface scattering induced by the finite size of the AuNPs. The required optical properties of gold can be found in 3. The pure dephasing γ pure of the 1s-1s exciton transition is taken from 4 and the dipole moment µ for the CdSe QDs from 5. The only parameters that have to be obtained from fits to experimental data are the radiative lifetime γ x of the free (uncoupled) exciton and its resonance frequency ω x. Both are provided by independent experiments. Geometric parameters, the distance between the constituents R and the diameter D of the AuNPs are obtained from TEM images. The excitation pulse is described by a Gaussian pulse ( ) 2 E(t, ω) = [ n exp 1 t toff τ 2 τ iωt + i k (t t 2 off) ], 2 including a linear chirp k to match the experimental pulse in time and frequency domain. The parameters used for the numerics are summarized in Table S1. Table S1: Parameters used in theory including the information how they are determined. Parameter Symbol Value dipole moment QD d 1.84enm 5 radiative dephasing γ x fs 1 (green), fs 1 (red) (Fit) pure dephasing γ pure 0.053fs 1 4 resonance QD ω x 2.37eV (green), 2.08eV(red) (Spectrum) distance QD-AuNP R 12, 5nm up to 36, 5nm depending on spacer (TEM) 1 diameter AuNP d m 12nm, 20nm, 40nm (TEM) resonance AuNP ω pl 2.45eV (12nm), 2.45eV (20nm), 2.42eV (40nm) 2 damping AuNP γ pl 0.093fs 1 (12nm) 0.079fs 1 (20nm), 0.082fs 1 (40nm) 2 dipole moment AuNP χ 8.17enm (12nm), 17.58enm (20nm), 49.72enm (40nm) pulse duration τ 300fs (laser) time offset t off 90ps (IRF) excitation frequency ω 2.37eV (green), 2.08eV (red) n 1 quadratic chirp k fs 1 (spectra) 5

6 Description of the ensemble In the following we will present our approach for modeling a realistic situation of an ensemble of hybrids. In this case, it has to be taken into account that the coupling strength between the constituents depends on the angle between the two dipole moments and on the distance 1. Since the QDs are randomly self assembled around the AuNPs, a range of different couplings will contribute to the signal due to e.g. angle mismatch. We treat this by calculating the time dynamics for different coupled systems with different coupling strengths around a mean value and average over the results. A maximum variation of the center to center distance of ± 2 nm and a variation of the angle depending factor in the coupling strength between 2/3 to 2 were assumed 1. Furthermore we took all three plasmonic modes of the metal sphere into account the exciton can couple to 6. Figure S3 illustrates the numerical method of averaging the hybrid ensemble. The fifty calculated signals are shown as well as the resulting average signal and the average convolved signal. The averaged signal is convolved with the instrumental response function to model the measuring process itself. Variations of the width of the Gaussian function by one order of magnitude do not result in qualitative changes. Calculated dynamics of the resonantly coupled system Please note that large AuNPs are easier to excite due to their large dipole moment. Therefore, the dynamics of system with large AuNPs starts earlier. This shifts the maximum of the excitation in time with increasing AuNP size and also changes the decay trend within the first few hundred femtoseconds. On a nanosecond timescale, this effect vanishes. 6

7 Figure S3: Ensemble averaging. Calculated time dependent far field signals (rainbow colors) for different coupling strength used for the ensemble averaging for the red QDs coupled to AuNPs with a diameter of 40 nm and a spacer of 15 nm. The black dashed curve shows the resulting averaged signal and the purple one the averaged signal convolved with the instrumental response function. The purple one is the one we compare with the experiments. 7

8 Figure S4: Strongly-coupled system. Calculated time-dependent far field signals for the resonant green QD-based systems and plasmon contributions to the far field. The curves were not convolved with the IRF. 8

9 Classical treatment beyond the dipole approximation To be able to compare the dipole approximation used in the full quantum calculations not only the experiment, but also with established classical models beyond the dipole approximation, we have carried out numerical simulations to solve for the electromagnetic fields for the experimental geometries of the nanoparticles. The quantum dot was modeled as a localized current source (diameter = 1 nm), oscillating along the direction from the dot to the sphere, at 524 THz (572 nm), the frequency at which the quantum dot emits. To describe the response of the metal nanoparticle, we used the bulk dielectric function of gold ( i) at the corresponding frequency. The response was treated as local and the fields were determined using the Comsol Multiphysics package. The mesh, especially within the gap, was carefully checked in order to ensure convergence of the calculation. The electromagnetic response of the hybrid system was numerically simulated by calculating the total loss rate for the dipole associated with current flow in the metal nanoparticle. Figure S5 displays an extreme case of the simulation for a AuNP of 50 nm diameter with a spacer of 15 nm thickness. For large AuNPs and short distances, most of the Ohmic loss is near the north pole, close to the QD. In this case, the picture of a dipole in the center of the sphere will be a relatively inaccurate approximation. In the quantum-mechanical model, damping in the AuNP is strong and the rate of energy dissipation scales quadraticaly with the coupling strength g. For comparison, we have we calculated the Ohmic loss for a series of AuNP diameters and spacer thicknesses based on the numerically determined electromagnetic fields (Figure S6). We scale the numerically obtained result to the analytic one in the limit of small nanoparticle diameter and large separation from the dipole. We find that the numerical simulation and the dipole-dipole approximation agree in all general trends, as shown in Figure S6. As expected, we find the largest discrepancies for AuNPs and small separations, although the results differ by less than a factor of two even in this limit. 9

10 Figure S5: Classical calculations. Ohmic loss distribution in a gold sphere of 50 nm diameter, excited at a wavelength of 572 nm by a quantum dot located 15 nm away from the nearest point on the sphere. 10

11 Figure S6: Interaction strength. Distance-dependent exciton-plasmon coupling strength in the dipole-dipole interaction model (solid line) compared to the square root of the Ohmic losses from a numerical calculation of the electromagnetic fields for the corresponding geometry (dots). The results of the numerical calculation were scaled to match the analytical expression for D Au = 1 nm and a spacer of 15 nm. 11

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