Supporting Information

Transcription

1 Supporting Information Temperature Dependence of Emission Linewidths from Semiconductor Nanocrystals Reveals Vibronic Contributions to Line Broadening Processes Timothy G. Mack, Lakshay Jethi, Patanjali Kambhampati* Department of Chemistry, McGill University, Montreal, Quebec, H3A 0B8, Canada Corresponding Author: S1

2 S1: Fitted Models The commonly employed approach to fitting the temperature dependence of the linewidth of the emission of direct bandgap semiconductors as a function of temperature is presented below as equation 1. 1 Г T =Г + T+ (1) Where Г 0 is an inhomogeneous broadening term, and γ ac and γ LO are the electron-phonon coupling terms to acoustic and optical phonons respectively. E LO represents the ensemble averaged optical phonon energy. This approach was employed in a recent study for the core state emission data of CdSe. 2 If the second term (the acoustic broadening contribution) is dropped, then we obtain equation 2. 3 Г T =Г + (2) Equation 2 is fitted to data either with all initial parameters unconstrained, or by fixing the value of E LO to be mev, which corresponds to the bulk LO phonon of CdSe. 4-6 S2

3 Figure S1: Panel a) Temperature dependence of the linewidth for the core of R=1.1 nm CdSe NCs, fit to equation (2). Fitting the data to the model developed by Rudin et al, with the acoustic phonon term removed (eq. 1). 1 This was performed in the range of 300K to approximately 40K. The LO phonon for CdSe at 208 cm -1 (25.79 mev) was fixed in one case (blue dashed line) and unfixed in the case of the red solid line. The fixed phonon frequency did not fit the experimental data very well. When the LO phonon parameter was unconstrained, a value of 10±1 mev was S3

4 obtained. Panel b) fits equation (2) to the R=1.1 nm CdSe nanocrystal surface emission linewidth data. In this case, both the constrained and unconstrained fits are satisfactory, with the unconstrained fit yielding a E ph energy of 21±2 mev (183cm -1 ). (Г 0 =328±2 mev) S2 Model Details: Heller s wave-packet formalism to calculate absorbance, emission and resonance Raman excitation profiles has been described elsewhere. 7-8 The involves a convolution with homogeneous inhomogeneous and inhomogeneous broadening terms. 9 = <0 0 > (3) <0 0 >= (4) = ħ (5) Where <0 0(t)> is the dipole-dipole time correlation function, ω is the excitation frequency, Θ is an inhomogeneous broadening parameter, and Λ is the homogeneous linewidth. S is the Huang Rhys parameter. 10 The value of w k is assumed to be the LO phonon mode for bulk CdSe (208 cm -1 (25.79 mev)). is the phonon occupation number. By expanding equation 4 and then Fourier transforming, the lineshape can then be calculated as follows*: =!! 2 +1 (6) This above result can be rewritten more compactly in terms of the following expression S4

5 = (7) Where p(m) are thermal occupancy weighing factors and where are the Franck-Condon factors, given by: i = =!! (8) Where n and m correspond to the vibrational quantum numbers of the ground and excited electronic states, and L(S) is the associated Laguerre polynomial. In the limit that S -> 0, the Franck-Condon probabilties collapse to a single transition between the m=n=0 vibrational states of S 1 - > S 0 transition. In this context, the lineshape is then a single transition probability with an associated lineshape envelope function. The limiting case of the atom can be recovered if the homogeneous broadning term is kbt. The m to n transitions are then thermally weighted by the factors p(m). In this paper, the Franck-Condon factors for all m to n transitions were calculated from m=0 to m=3. Transitions involving higher values of m are neglected as they were shown to be negligible. Each thermally weighted transition is then broadened by an arbitrary lineshape. In this case, a convolution with a Voigt profile was used. The thermally weighted sum of these lineshapes was then computed. The parameter models are listed in tables I and II of the Supporting Information. S5

6 Figure S2: Resulting emission and absorbance Franck-Condon factors, resulting from the plot of equation (7). Here we show the results for m=0, and S=2. The absorbance and emission Franck-Condon factors are mirror images along the m=0 to n=0 transition. S6

7 Figure S3: Top left panel: Plot of weighted Franck-Condon factors for m=0,1 and 2. The sum of the thermally weighed Franck-Condon factors is equal to 1. The other panels show the intensity normalized Franck-Condon factors for m=0, 1 and 2. S7

8 Figure S4: Resulting thermally weighed Franck-Condon factors (300K) (S=24.5) S8

9 Figure S5: Summed and normalized Franck-Condon factors from m=0 to m=3, at 10 and 300K. (S=24.5) S9

10 S3 Semiclassical lineshape model for the surface: Model iii) The third approach uses the solution as implemented by Keil 11 and Huang and Rhys 10. The thermally averaged Franck-Condon factors are mathematically equivalent to model 2, and offers an analytical solution which is convenient to benchmark our code to. The advantage of using the third approach is that it computes the average of all m to n transitions in a single expression. = exp δ + h (9) Here I p is the modified Bessel function with imaginary arguments, and p corresponds to the difference in initial (m) to (n) vibrational states. The lineshape can be weighed by the following semiclassical lineshape: 11. = exp ħ (10) ħ For comparison, the analytical solution combining equations 9 and 10 is also provided: Г T Г 0 coth ħ (11) We overlay the combined calculations using equations 9 and 10, and compare with the expression in equation 11. S10

11 Figure S6: Comparison of Analytical equation and values obtained from code. S11

12 S4 Spectral shift details The Huang-Rhys parameter infuences two observables: the spectral width and the emission red-shift. Both increase as a function of S. The strong electron-phonon coupling is consistent with the large spectral shift of the surface with respect to the core state. 9 The emission spectrum will redshift as S increases, by an amount Sħω. Taking into account an energetic separation between core and surface states, the spectral redshift of the surface with respect to the core is approximately: = ħ + ħ (12) The Huang Rhys parameter thus dictates the amount by which the emission is redshifted. This allows for the measurement of G provided that the value of S is accurately determined from the linewidth and lattice contraction is taken into account. The difficulty in separating the spectral shift due to multiple surface states and the spectral blue-shift stemming from the lattice contraction. S12

13 Figure S7: Schematic illustration of slight spectral shift imparted by addition of two surface states. The shift depends on both the values of G seperating the surface states and their respective values of S. S13

14 S5 Inputs of model parameters in manuscript Table 1: Values associated with model calculations for Fig 3 and 4. Model Huang-Rhys (S) Broadening Parameters State Energies Core State (X C B ) X C B = ϴ = 23 mev Λ=17 mev (5K) - 50 mev (300K) Core and Dark States (X C B + k C D X C D ) X C B = X C D = 0.40 ϴ = 23 mev Λ=17 mev (5K) - 50 mev (300K) X C B - X C D = 7 mev One Surface State (X S B ) X S B = 24 ϴ = 23 mev Λ=17 mev (5K) - 50 mev (300K) Two surface states (X S B + X S2 B + k S D X S D ) X S B = 23 X S2 B = 36 X S D = 26 ϴ = 23 mev Λ=17 mev (5K) - 50 mev (300K) X S2 B - X S B = 20 mev X S B - X S D = 7 mev S14

15 Table 2: Parameters used for models for Fig 5. Sample Huang-Rhys (S) Broadening Parameters State Energies 0.9 nm (TDPA) X S = 28 X SD = nm (TDPA) X S = 28 X SD =26.26 ϴ = 115 mev Λ=17 mev (5K) - 50 mev (300K) ϴ = 150 mev Λ=17 mev (5K) - 50 mev (300K) X S - X SD = 7 mev X S - X SD = 7 mev TDPA X S = 26 X SD =26 ADMT X S = 20 X SD =23 DDT X S = 11 X SD =13.0 ϴ= 120 mev Λ=17 mev (5K) - 50 mev (300K) ϴ= 120 mev Λ=17 mev (5K) - 50 mev (300K) ϴ= 120 mev Λ=17 mev (5K) - 50 mev (300K) X S - X SD = 7 mev X S - X SD = 7 mev X S - X SD = 7 mev S15

16 S6 Non-linear least-squares fitting of model to data in Figures 3 and 4 Table 3: Fitted parameter output and uncertainties associated with data for Fig 3 and 4. Model Huang-Rhys (S) Broadening Parameters State Energies Core and Dark States (X C B + k C D X C D ) X C B = 0.09 ± 0.01 X C D = 0.44 ± 0.03 ϴ = 23 mev Λ=17 mev (5K) - 50 mev (300K) X C B - X C D = 7 mev Two surface states (X S B + X S2 B + k S D X S D ) X S B = 23.7 ± 1.2 X S2 B = 14.4 ± 1.4 X S D = 26.3 ± 0.7 ϴ = 23 mev Λ=17 mev (5K) - 50 mev (300K) X S2 B - X S B = 27 ± 4 mev X S B - X S D = 7 mev The least squares fitting employed the Levenberg-Marquardt algorithm, and held the inhomogeneous and homogeneous broadenings fixed as well as the dark-bright state splitting, while leaving the Huang-Rhys and energy differences unconstrained. S16

17 Figure S8: Fit of the model to the data in Fig 3. S17

18 Figure S9: Fit of the model to the data in Fig 4. S18

19 S7 Characterization Data Characterization raw Data: Typical white light NC (Butylamine capped) (480 nm Band Edge) (Figure 2) = B 1 = τ 1 = B 2 = τ 2 = B 3 = τ 3 = Average weighted lifetime of the core emission <t> = ns S19

20 Raw characterization Data: NN Labs CdSe (480 nm Band Edge) (Figures 3 and 4) Core emission lifetime = B 1 = τ 1 = B 2 = τ 2 = B 3 = τ 3 = Average weighted lifetime of the core emission <t> = 88.5 ns QY = 5.6% The Quantum Yield was determined through the reference dye Rhodamine 6G in Ethanol, assuming a quantum yield of 95% in EtOH. S20

21 Characterization raw Data: Typical white light NC (Phosphonic acid capped (Bare), ligand exchanged to adamantanethiol (ADMT) or dodecanethiol (DDT)) (Figure 6) S21

22 S7 Non-Resonant Excitation Effet on surface FWHM of typical white light QD Figure S10: Illustration of the excitation dependence of the surface FWHM as a function of excitation wavelength. Band edge excitation results in lower inhomogeneous broadening in comparison with higher energy non-resonant excitation. Data is obtained on a typical tetradecylacidphosphonic acid capped CdSe NC. S22

23 References: 1. Rudin, S.; Reinecke, T. L.; Segall, B., Temperature-Dependent Exciton Linewidths in Semiconductors. Phys. Rev. B 1990, 42, Valerini, D.; Cretí, A.; Lomascolo, M.; Manna, L.; Cingolani, R.; Anni, M., Temperature Dependence of the Photoluminescence Properties of Colloidal Cdse/Zns Core/Shell Quantum Dots Embedded in a Polystyrene Matrix. Phys. Rev. B 2005, 71, Diab, H., et al., Narrow Linewidth Excitonic Emission in Organic Inorganic Lead Iodide Perovskite Single Crystals. J. Phys. Chem. Lett. 2016, Mooney, J.; Saari, J. I.; Myers Kelley, A.; Krause, M. M.; Walsh, B. R.; Kambhampati, P., Control of Phonons in Semiconductor Nanocrystals Via Femtosecond Pulse Chirp-Influenced Wavepacket Dynamics and Polarization. J. Phys. Chem. B 2013, 117, Lin, C.; Gong, K.; Kelley, D. F.; Kelley, A. M., Size-Dependent Exciton Phonon Coupling in Cdse Nanocrystals through Resonance Raman Excitation Profile Analysis. J. Phys. Chem. C 2015, 119, Lin, C.; Kelley, D. F.; Rico, M.; Kelley, A. M., The Surface Optical Phonon in Cdse Nanocrystals. ACS Nano 2014, 8, Heller, E. J., The Semiclassical Way to Molecular Spectroscopy. Acc. Chem. Res. 1981, 14, Zink, J. I.; Shin, K.-S. K., Molecular Distortions in Excited Electronic States Determined from Electronic and Resonance Raman Spectroscopy. In Advances in Photochemistry, John Wiley & Sons, Inc.: 2007; pp Mooney, J.; Krause, M. M.; Saari, J. I.; Kambhampati, P., A Microscopic Picture of Surface Charge Trapping in Semiconductor Nanocrystals. J. Chem. Phys 2013, 138, Huang, K.; Rhys, A., Theory of Light Absorption and Non-Radiative Transitions in F-Centres. Proceedings A 1950, 204, Keil, T. H., Shapes of Impurity Absorption Bands in Solids. Physical Review 1965, 140, A601-A de Jong, M.; Seijo, L.; Meijerink, A.; Rabouw, F. T., Resolving the Ambiguity in the Relation between Stokes Shift and Huang-Rhys Parameter. Phys. Chem. Chem. Phys 2015, 17, i * 2009/lecture-notes/MIT5_74s09_lec08.pdf S23