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Energy Transfer from Quantum Dots to Graphene and Mos2: the Role Of Energy Transfer from Quantum Dots to Graphene and MoS2: The Role of Absorption and Screening in Two-Dimensional Materials Archana Raja,∗,y Andrés Montoya–Castillo,y,z Johanna Zultak,{,x,z Xiao-Xiao Zhang,{ Ziliang Ye,{ Cyrielle Roquelet,{ Daniel A. Chenet,k Arend M. van der Zande,k,# Pinshane Huang,y,@ Steffen Jockusch,y James Hone,k David R. Reichman,y Louis E. Brus,y and Tony F. Heinz∗,? E-mail: [email protected]; [email protected] ∗To whom correspondence should be addressed yDepartment of Chemistry, Columbia University, New York, NY 10027, USA zThese authors contributed equally to this work {Departments of Physics and Electrical Engineering, Columbia University, New York, NY 10027, USA xPresent Address: Department of Micro- and Nanotechnology, Technical University of Denmark kDepartment of Mechanical Engineering, Columbia University, New York, NY 10027, USA ?Departments of Applied Physics and Photon Science, Stanford University, Stanford, CA 94305, USA and SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA #Present Address: Department of Mechanical Science and Engineering, University of Illinois at Urbana- Champaign, Urbana, IL 61801, USA @Present Address: Department of Materials Science and Engineering, University of Illinois at Urbana- Champaign, Urbana, IL 61801, USA 1 Contents 1 Experiment 3 1.1 Methods . .3 1.2 Photoluminescence lifetime of quantum dots in solution . .4 1.3 TEM characterization of quantum dots . .5 1.4 Raman characterization of MoS2 and graphene . .5 2 Theory 7 2.1 Expressions for the total, non-radiative and radiative decay rates . .7 2.2 Applicability of the theory . 14 2 1 Experiment 1.1 Methods Time-resolved photoluminescence was performed using the frequency-doubled output of a mode-locked Ti:sapphire laser operating at 80 MHz. The sample was excited through a 100x (0.9 NA) air objective and the PL was also collected through the same objective and sent to a fast avalanche photodiode (PicoQuant PDM) after selecting a narrow band of wavelengths using a grating spectrometer (Horiba iHR320, 600 lines/mm). The excitation consisted of a 1.5 µm sized spot of 100 fs pulses at 405 nm. A time-resolved single photon counter (PicoQuant PicoHarp 300) was used to determine the temporal profile of the photolumines- cence (PL) centered at the quantum dot emission. Excitation fluences were sufficiently low (<1 µJcm−2), so as to ensure that the sample absorption was in the linear regime and less than one photon per excitation pulse is incident on the photodiode. The instrument response function (IRF) of the single photon counter was determined using 100 fs pulses, centered at 810 nm from the Ti:sapphire laser. Emission lifetimes were extracted from the time traces after deconvoluting the IRF from the temporal response. For the wide-field imaging of single quantum dots, the sample was excited through a 100x (0.9 NA) air objective using the 514 nm line of a cw Argon ion laser was used as the excitation source. The laser beam was expanded to create a 100 µm large excitation field at low laser intensity (5 Wcm−2). Diffraction limited, single QD PL images were collected using the same objective, filtered through a 586±10 nm band pass filter before collecting the image on an EMCCD detector. The PL intensity was extracted by fitting the QD emission profile to a 2D-Gaussian. Fluorescence lifetimes in toluene solution were measured by time correlated single photon counting on an OB920 spectrometer (Edinburgh Analytical Instruments) in conjunction with a pulsed LED (PicoQuant) as the excitation light source (496 nm). Steady-state luminescence spectra were recorded on a Fluorolog-3 fluorometer (HORIBA Jobin Yvon). 3 1.2 Photoluminescence lifetime of quantum dots in solution To obtain absolute decay rates, it is necessary to obtain a value for the decay rate in vacuum, Γ0. While we do not have direct access to Γ0, we may use the experimentally determined rate of PL decay for the QD in a lossless solvent like toluene, granted that we can obtain the relationship between Γ0 and Γtol. In fact, since toluene is a lossless medium, the same 3 electromagnetic model used in section 2 may be used to obtain the ratio Γ0=Γtol = (n0=ntol) 2 (using Eq. (8.87) in Ref. 1), with the refractive index of toluene taken to be ntol = 1:5. The PL lifetime in toluene is extracted by fitting the decay transient to a single exponential convoluted with the instrument response, as shown in Fig. S1. We obtain a PL lifetime of 15 ns or a decay rate of 0.067 ns−1. Figure S1: Time-resolved photoluminescence of QDs suspended in a solution of toluene. The black line corresponds to 15 ns single exponential fit of the data (blue line). The grey line is the instrument response function. 4 1.3 TEM characterization of quantum dots A monolayer of QDs were imaged through transmission electron microscopy (TEM), as shown in Fig. S2 (a). The diameter of the QDs were determined by analysis of the center-to-center distances, following Akselrod et al.3 A diameter of 10.8 ± 1.0 nm was obtained from a Gaussian fit of the first peak (or nearest neighbor distances) of the distribution, as shown in Fig. S2 (b). This will correspond to a distance of 5.4 nm from the center to the QD to the surface of MoS2 or graphene. Figure S2: (a) Transmission electron micrograph of a sub-monolayer of QDs (b) Histogram of center-to-center inter-QD spacings extracted from (a). The blue line is a Gaussian fit of the first peak of the distribution. 1.4 Raman characterization of MoS2 and graphene Monolayer and few layer graphene and MoS2 were prepared via mechanical exfoliation of bulk crystals on 285 nm SiO2 on Si. MoS2 monolayers were also prepared via chemical vapor deposition on silicon wafers covered by 285 nm of thermal oxide.4 Raman spectroscopy, optical contrast and atomic force microscopy were used to determine the thicknesses of the flakes. The Raman measurements were performed in a Renishaw InVia Raman microscope using 5 a 532 nm laser and a 1800 lines/mm grating. As shown in Fig. S3, the thicknesses of 1 exfoliated mono to few-layer MoS2 were determined using the separation between the E2g 5 and A1g modes; and that of graphene via the ratio between the G and 2D intensities, and the width of the 2D peak.6 Figure S3: Raman spectra of exfoliated bulk and 1-4 layers of (a) MoS2 and (b) graphene 6 2 Theory 2.1 Expressions for the total, non-radiative and radiative decay rates From an electromagnetic perspective, the decay rate of a QD, idealized as a radiating dipole, near a thin film, Γ, normalized by the decay rate for the dipole in vacuum, Γ0, may be written as1 Γ 6π" h i = 1 + 1 Im µ∗ · E (r ) Γ jµj2k3 s 0 0 0 (1) 1 Z 1 x h i = 1 + Re dx e2ik0hx¯ (2x2 − 1)r(p)(x) + r(s)(x) ; 2 0 x¯ where Es(r0) is the electric field scattered from the interfaces of the thin film (and the substrate underneath it) at the position of the radiating dipole, r0, and µ is the transition dipole moment of the QD. The second line, which assumes an isotropic distribution of dipoles near the thin film, reexpresses the modification to the rate coming from the interaction with the lossy thin film as a weighted integral of the generalized Fresnel reflection coefficients for s– and p–polarized light, rp and rs, which are functions of the wavevector associated with excitations in the thin film, k, normalized by the wavevector associated with the excitation energy of the QD, k0 = 2π/λ, such that x = k=k0. In this experiment the wavelength p associated with the QD emission is λ = 590 nm. We also define x¯ ≡ 1 − x2. For the geometry considered in Fig. 3(b) in the main text of the paper, the generalized Fresnel reflection coefficients for s– and p–polarized waves are given by, 2ik2?d r12 + r23e r = 2ik d ; (2) 1 + r12r23e 2? where d is the thickness of the thin film (medium 2 in Fig. 3 of main text), rab is the reflection coefficient between two media, and k2? is the perpendicular component of the excitation wavevector in the thin film. 7 The reflection coefficients between two media for p– and s–polarized light are (p) (p) (p) ka?"bk − kb? "ak rab = (p) (p) ; (3) ka?"bk + kb? "ak (s) (s) (s) ka? − kb? rab = (s) (s) : (4) ka? + kb? The perpendicular component of the wavevectors for p– and s–polarized components of the wavevector for anisotropic media take the following forms,1 (p)2 h "ik 2i 2 ki? (x) = "ik − x k0; (5) "i? (s)2 h 2i 2 ki? (x) = "ik − x k0: (6) As is evident from the equations above, Γ depends non-trivially on properties that are intrinsic and extrinsic to the thin film. In the above expression, the extrinsic properties include the wavevector of the QD excitation energy, k0, the distance between the QD and the thin film, h, and the (real) dielectric constants of the material where the QD resides "1 and of the substrate below the thin film "3. Among the intrinsic properties are the (possibly anisotropic) dielectric function "2;k and "2;? and the thin film thickness t. Because there is no simple way to project out the effect of any of these parameters without making potentially unwarranted approximations, full characterization of the behavior of the rate with respect to these parameters requires solution of Eq.
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