Energy Transfer from Quantum Dots to and

MoS2: The Role of Absorption and Screening in

Two-Dimensional Materials

Archana Raja,∗,† Andrés Montoya–Castillo,†,‡ Johanna Zultak,¶,§,‡ Xiao-Xiao

Zhang,¶ Ziliang Ye,¶ Cyrielle Roquelet,¶ Daniel A. Chenet,k Arend M. van der

Zande,k,# Pinshane Huang,†,@ Steffen Jockusch,† James Hone,k David R.

Reichman,† Louis E. Brus,† and Tony F. Heinz∗,⊥

E-mail: [email protected]; [email protected]

∗To whom correspondence should be addressed †Department of Chemistry, , New York, NY 10027, USA ‡These authors contributed equally to this work ¶Departments of Physics and Electrical Engineering, Columbia University, New York, NY 10027, USA §Present Address: Department of Micro- and , 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 , 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 6πε h i = 1 + 1 Im µ∗ · E (r ) Γ |µ|2k3 s 0 0 0 (1) 1 Z ∞ 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 √ 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⊥εbk − kb⊥ εak rab = (p) (p) , (3) ka⊥εbk + kb⊥ εak (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 εik 2i 2 ki⊥ (x) = εik − x k0, (5) εi⊥ (s)2 h 2i 2 ki⊥ (x) = εik − 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 ε1 and of the substrate below the thin film ε3. Among the intrinsic properties are the (possibly

anisotropic) dielectric function ε2,k and ε2,⊥ 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. (1) for each specific case. For certain cases of

interest (e.g., where ε1 = ε3), however, previous studies have suggested simplified forms for the equations above.7,8 Another important advantage of the electromagnetic approach is that, using the energy

7,9 flux method, one can decompose Eq. (1) into its different (Γ0–normalized) contributions:

the decay rate associated with radiation emitted into the upper half–space Γ↑/Γ0, the radi-

ation emitted into the allowed and forbidden regions below the lossy thin film Γ↓,a/Γ0 and

Γ↓,f /Γ0, respectively (Eqs. (10.48), (10.50), and (10.51) in Ref. 1), and the non-radiative

8 component of the rate, ΓNRET/Γ0. For completeness, we reproduce the expressions for all components of the total rate below. The decay rate of energy going into the upper half–space takes the form,

Γ 1 1 Z 1 xh   i ↑ 2 2 2 2ik0hx¯ = + dx |rp(x)| + |rs(x)| + 2Re (1 + x )rp(x) + rs(x) e . (7) Γ0 2 4 0 x¯

The decay rate of energy going into the half–space below the thin film takes the form,

Γ Γ Γ ↓ = ↓,a + ↓,f Γ0 Γ0 Γ0 q ε (8) r 3 r 1 ε Z ε1 ε h i 00 3 1 2 2 2 2 −2k0hx¯ = dx 1 − x |tp(x)| + (1 + x )|ts(x)| e , 4 ε1 0 ε3

where x¯00 = Im(¯x) and the generalized transmission coefficient takes the form,

ik⊥d t12t23e t = 2ik d , (9) 1 + r12r23e 2⊥ and

(p) r (p) 2ka⊥εbk εak tab = (p) (p) , (10) εbk ka⊥εbk + kb⊥ εak (p) (s) 2ka⊥ tab = (p) (p) . (11) ka⊥ + kb⊥

Noting that ΓNRET = Γ − Γ↑ − Γ↓, Eqs. (1), (7), and (8) provide an expression for

ΓNRET/Γ0. When ε1 = ε3, ΓNRET/Γ0 simplifies significantly, taking the form,

Z ∞ ΓNRET 1 xh 2 (p) (s) i = Re dx (2x − 1)r (x) + r (x) exp[2ik0hx¯]. (12) Γ0 2 1 x¯

Because we account for the SiO2 substrate in our treatment, we maintain all terms in

ΓNRET, but note that the difference between using Eq. (12) and the more general expression for the rate in the presence of the SiO2 substrate is only quantitative (See Fig. S4). Moreover,

9 comparison of the non-radiative component of the rate ΓNRET/Γ0 to the total rate Γ/Γ0 shows that, in the presence of a lossy thin film and in particular for the case of graphene and MoS2, the non-radiative component constitutes > 97% of the total rate (See Fig. S5).

300 (a) 300 (b) 0 Γ

/ 200 200 Anisotropic

NRET ε2 ε2 100 k Γ 100 → ε2 (2ε2 + ε2 )/3 → k ⊥ 0 0 0 5 10 15 20 0 5 10 15 20 80 80 (c) (d) 0 60

Γ 60 / 40 40 NRET

Γ 20 20 0 0 0 5 10 15 20 0 5 10 15 20 Film Thickness [nm] Film Thickness [nm]

Figure S4: Comparison of ΓNRET/Γ0, for graphene (panels (a) and (b)) and MoS2 (panels (c) and (d)) under different conditions and approximations. Panels (a) and (c) correspond to the case considered in the experiment where the QD is in air, ε1 = 1.0, and the the thin film lies on an SiO2 substrate, ε3 = 2.25. Panels (b) and (d) correspond to the cases where the dielectric function of SiO2 is replaced with that of air, ε3 = ε1, allowing for the use of Eq. (12). The three lines in each plot correspond to the fully anisotropic dielectric function for the thin film, or to using an isotropically mapped version of the dielectric function. For the latter case, we considered two physically motivated options: replacement of the dielectric function with the parallel component ε2 → ε2k or by the weighted mean of of the two components, ε2 = (2ε2k + ε2⊥)/3. While the different versions yield quantitatively different answers, we note that all versions consistently reproduce the same qualitative behavior. This lends credence to the use of the air–thin film–air version of ΓNRET/Γ0 given by Eq. (12) for the isotropic phase diagram in Fig. 5 in the main text of the paper, which, in turn, is an acceptable proxy for the anisotropic dielectric constants.

10 1.0 250 (a) Graphene (b) Graphene 0.8 200 Γ↑/Γ0 0

Γ 0.6

/ 150 ↑ x . Γ , a /Γ 0 Γ 100 Full rate 0 4

Non-radiative 0.2 50 ↑ Γ , f /Γ0 0 rate 0.0 0 5 10 15 20 0 5 10 15 20 1.0 80 (c) MoS2 (d)(d) MoS2 0.8 60 0

Γ 0.6 / x 40 .

Γ 0 4 20 0.2 0 0.0 0 5 10 15 20 0 5 10 15 20 Film Thickness [nm] Film Thickness [nm]

Figure S5: Comparison of the Γ0–normalized full and all components of the rate, i.e., ra- diative (into the upper half–space, and into the allowed and forbidden regions of lower half–space). Panels (a) and (b) correspond to graphene, while panels (c) and (d) to MoS2. In panels (a) and (c), the solid cyan line corresponds to the full rate, Γ/Γ0 and the dashed blue line to the non-radiative component of the rate ΓNRET/Γ0. Panels (b) and (d) display the radiative contributions to the rate. The red lines with circles represent the rate of decay going into the upper half–space Γ↑/Γ0, whereas the lines with blue triangles and green dia- monds represent the decay rates for radiation going into the allowed and forbidden regions in the lower half–space, Γ↓,a/Γ0 and Γ↓,f /Γ0, respectively. Comparison of the total rate to the non-radiative component of the rate reveals that the non-radiative component dominates the decay.

11 The simplifications introduced in Eq. (12) and explored in Fig. S4, which included setting

ε3 = ε1 and casting the anisotropic dielectric function to an analogous isotropic version, mainly change the ΓNRET curves quantitatively, but not qualitatively. However, the isotropic version Eq. (12) is still too complex to yield a simple reason for the emergence of the low- thickness peak as a function of the dielectric constant. In the literature,8,9 there exist further simplified versions of the nonradiative rate expression that display a similar transition from graphene- to MoS2-type behavior as a function of the dielectric constant. Particularly amenable to analysis is the expression provided recently by Gordon and Gartstein,8

" Z ∞ −2xδρ # ΓNRET −3 2 −2x 1 − e ≈ [k0h] Im η dx x e 2 −2xδρ , (13) Γ0 0 1 − η e

β−ε1 √ where ρ = d/h, η = , β = ε ε2,⊥, and δ = ε /ε2,⊥; in the isotropic case, β → ε2 and β+ε1 2,k 2,k δ → 1. We also note that for cases where |η| < 1, which corresponds to realizations where Re[β], and Re[δ] > 0, the denominator can be expanded as a geometric series, allowing for an analytical solution to the integral, yielding the following uniformly (and fast) converging infinite sum, " 2 ∞ 2n # ΓNRET η − 1 X η = [k h]−3 Im + η . (14) Γ 0 η (1 + nδρ)3 0 n=1 The above formula is easy to evaluate numerically and clearly shows that the bulk limit of the rate, which corresponds to ρ → ∞, is

bulk ΓNRET Im[η] = 3 , (15) Γ0 [k0h]

which agrees with previous analyses.8,9

Since it is desirable to understand how the emergence of a low-thickness peak in the ΓNRET data arises from the electromagnetic treatment, we restrict our attention to the isotropic

12

η0 = 0.55 η0 = 0.7 η0 = 0.85 1.8

η00 = 0.001i

1.5 η00 = 0.05i

η00 = 0.3i 1.2 bulk NRET Γ

/ 0.9 NRET

Γ 0.6

0.3 (a) (b) (c) 0.0 0 3 6 9 12 0 3 6 9 12 0 3 6 9 12 15 Thickness (nm) Thickness (nm) Thickness (nm)

bulk 0 00 Figure S6: Comparison of ΓNRET/ΓNRET as a function of η = η + iη . The dielectric as 1+η 0 a function of η is given by ε2 = 1−η . Panels (a)-(c) correspond to η = 0.55, 0.7 and 0 0.85, respectively. MoS2-type behavior arises from sufficiently large values of η and gets suppressed, as is most evident in panels (b) and (c), with increasing values of η00. We note |ε|2−1+2iε00 00 00 0 that since η = |ε|2+2ε0+1 , η increases with increasing ε and decreasing ε .

version of Eq. (13) and rewrite the complex part of the integrand as follows,

" # η 1 + |η|2e−2xρ Im = η00 . (16) 1 − η2e−2xρ 1 − 2Re[η2]e−2xρ + |η|4e−4xρ

While arguments based on the derivative with respect to ρ provide a way to deter-

mine whether ΓNRET reaches a maximum before approaching its bulk value, the resulting expressions are quite involved. Instead, we would like to offer a more intuitive approach by inspection of Eq. (16). Clearly, maximizing the 2Re[η2]e−2xρ while minimizing |η|4e−4xρ leads to a small denominator, which helps increase the integrand, leading in turn to a large

ΓNRET/Γ0. One may expect to satisfy the above requirement with values of η such that

2 2 0 00 Re[η ] is large and |η| small, which corresponds to large values of ε2 and small values for ε2.

Re(η2) (|ε| − 1)2 − 4(ε00)2 → . (17) |η|2 (|ε| − 1)2 + 4(ε00)2

Indeed, as Fig. S6 shows, this pattern leads to the emergence of the low-thickness peak

for ΓNRET. This finding is in harmony with our previously used intuition that the larger

screening in MoS2 leads to the maximum in ΓNRET occurring at small thickness.

13 2.2 Applicability of the theory

Figure S7: Comparison of Γ/Γbulk for full QD structure obtained on COMSOL (solid line) and the point dipole approximation (squares on solids line)

Here we discuss the validity of the point dipole approximation. For this approximation to be valid, the wavefunction of the QD excitation must have (at least approximate) spherical symmetry and be well localized.10 For our case, where the QD is spherical and has a core– shell structure, spherical symmetry and localization in the roughly 2 nm core are mostly ensured, lending credence to the application of the electromagnetic approach. Nevertheless, the extended body of the QD can quantitatively modify the distance dependent rate, but the

11 qualitative features remain unchanged. We compare Γ/Γbulk obtained from Eq. (1) to that obtained from the numerical solution of macroscopic Maxwell’s equation for the core–shell structure in COMSOL.12 As is evident from Fig. S7, the extended body of the QD does not alter the observed trends significantly. Finally, we remark that a crucial approximation in the current treatment is the well justified assumption that the dielectric functions of graphene and MoS2 do not change ap- preciably with growing thickness in the energy range of the QD excitation (2.1 eV). In fact,

14 there are several experimental13 and theoretical14,15 works that point to the insensitivity of the dielectric functions of these materials to the growing thickness. However, future applica- tions of the theory in different energy ranges or for different materials requires a model for how dielectric functions evolve from the monolayer to the bulk limit.

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