SUPPLEMENTARY INFORMATION

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1 SUPEMENTARY INFORMATION DOI:.38/NNANO.23.9 Bright, long-lived and coherent excitons in carbon nanotube quantum dots Matthias S. Hofmann, Jan T. Glückert, Jonathan Noé, Christian Bourjau, Raphael Dehmel, Alexander Högele June 2, 23 Experimental details. Carbon nanotube samples Carbon nanotubes (CNTs) were synthesized by chemical vapor deposition (CVD) in a standard CVD furnace. The growth was assisted by a bimetallic iron-ruthenium (FeRu) catalyst. The catalyst particles were deposited on carrier substrates from a FeRu suspension either by spin or by drop coating. For synthesis of CNTs the samples were heated to 85 C in an Ar/H 2 (95%/5%) gas mixture and then kept in a constant flow of slm methane and.75 slm hydrogen for minutes before allowing to cool down in a hydrogen and Ar/H 2 gas flow. The growth procedure was optimized to yield a CNT density below µm 2 and narrow-diameter nanotubes with a mean diameter below nm. Diameter distributions were obtained by tapping-mode atomic force microscopy on samples with CNTs on SiO 2. Samples with as-grown CNTs in contact with SiO 2 were used as reference material for density and diameter distributions as well as in cryogenic spectroscopy. Fakultät für Physik and Center for NanoScience (CeNS), Ludwig-Maximilians-Universität München, Geschwister-Scholl-Platz, D-8539 München, Germany; Present address: Department of Physics, University of Cambridge, J.J. Thomson Avenue, Cambridge, CB3 HE, UK; These authors contributed equally to this work. NATURE NANOTECHNOLOGY 23 Macmillan Publishers Limited. All rights reserved.

2 a b 5µm d c 4µm 2nm e 2nm 2nm Figure S: Sample layout with as-grown carbon nanotubes. a and b, Optical microscopy images of a TEM grid used as a carrier substrate. A Si3 N4 membrane with regular hole patterns lies on top of a silicon frame (blue region of the TEM grid). In the central area (yellow region of the TEM grid window) the holes provide for the suspension of nanotubes free of substrate underneath. c, TEM image of a carbon nanotube suspended over a hole. d and e, SEM images of carbon nanotubes suspended over a SiO2 crater (d) and in partial contact with the ground of the crater (e). Suspended CNTs were synthesized on commercial grids typically used in transmission electron microscopy (TEM). The specific grid used in the experiments was perforated with holes of 2 µm diameter and a pitch of 4 µm (Fig. Sa and b) and coated with nm of SiO2 by plasma enhanced CVD. Fig. Sa and Sb show optical micrographs of the sample for two different magnifications. The grids provided for suspension of nanotubes in the region where the hole-perforated Si3 N4 membrane was not supported by the underlying Si frame; this region appears yellowish in Fig. Sa and b. In this region fully suspended nanotubes were identified by TEM imaging (Fig. Sc). In regions where the perforated membrane was in contact with the underlying carrier frame (blue regions in Fig. Sa and Sb) we identified with scanning electron microscopy (SEM) CNTs suspended over the full diameter of the crater (Fig. Sd) as well as CNTs that were suspended only near the walls of the crater (Fig. Se). Commercial CoMoCAT-nanotubes (SouthWest NanoTechnologies) encapsulated in sodium dodecylbenzenesulfonate (SDS) were dispersed out of an aqueous suspension on SiO2 substrates. 23 Macmillan 2 Publishers Limited. All rights reserved.

3 .2 Photoluminescence spectroscopy For spectroscopy we used a home-built confocal microscope with a full-width half-maximum (FWHM) diffraction-limited optical spot size of FWHM λ. The two-lens imaging system was not compensated for chromatic aberrations. The microscope was either operated in a bath cryostat at liquid helium temperature (4.2 K), in a dewar filled with liquid nitrogen (77 K) or at room temperature ( 3 K). A Ti:sapphire operated in continuous wave (cw) mode between 73 nm and 9 nm was used to excite the CNTs. was dispersed in a standard monochromator (5 mm focal length) and detected with a nitrogen cooled silicon-ccd (with 4 µev spectral resolution). Individual craters of the sample were identified by scanning a finite area of a few µm 2 and recording the reflected signal of a laser diode at 95 nm wavelength. Intense was detected at the position of individual craters as shown in Fig. S2. A strong antenna effect in the emission intensity was characteristic for sample regions with individual CNTs (Fig. S3). R µm R µm R µm R µm Figure S2: Confocal reflection and photoluminescence maps. Reflection (R) and corresponding photoluminescence () scans of the sample correlating the spatial positions of hot-spots and craters identified as dark areas in the reflection maps. Dashed lines are guides to the eye. All data were measured at T = 4.2 K. 23 Macmillan 3Publishers Limited. All rights reserved.

4 For time-resolved and pulsed photon correlation in a standard Hanbury-Brown and Twiss setup the Ti:sapphire laser was operated in pulsed mode (3 fs pulse width, 76. MHz repetition rate). was recorded by avalanche photodiodes (APDs) with a temporal resolution of 3 ps. For lifetime measurements of micelle-encapsulated CoMoCAT nanotubes a streak-camera with 7 ps temporal resolution was used. Representative spectra and spectral linewidths of suspended CNTs at temperatures of 4.2 K, 77 K and 3 K are shown in Fig. S4. At 4.2 K the linewidths of all as-grown suspended CNTs was limited by the resolution of our spectrometer to 4 µev. An increase of the full-width at half-maximum (FWHM) linewidth up to several mev was observed at the temperature of 77 K. At room temperature the linewidths are on the order of mev consistent with previous reports for suspended CNTs 2,3. Emission energies of all suspended CNTs studied in this work are summarized in histograms (bin size 6 mev) in Fig. S5. The figure also shows as grey triangles chirality-assigned emission energies at room temperature using values of Weisman and Bachilo 4, that were corrected for temperature-induced energy shifts for 77 K and 4.2 K according to Capaz et al.. Further corrections to the emission energy due to strain or details of the confinement potential were not considered. Note that room temperature values for the emission energies were obtained for surfactantencapsulated nanotubes in aqueous suspension 4. Photoluminescence excitation (E) was studied by tuning the frequency of the Ti:sapphire laser at a constant cw-power. As shown in Fig. S6, several resonances were found in the E spectra in the range of 3 3 mev accessible with the laser for CNTs with in the range a R b P L c ^ P L ll d Figure S3: Antenna effect. a, False-color map of a crater in reflection. b and c, map of the same crater with laser polarization perpendicular (b) and parallel (c) to the nanotube axis (scale bars are µm). d, Polar plot of the intensity as a function of the angle between the laser polarization axis and the the nanotube axis. The red solid line is a fit with a cos 2 -dependence. All data were recorded at T = 4.2 K. 23 Macmillan 4Publishers Limited. All rights reserved.

5 a 3 K b m N o rm a liz e d in te n s ity K K F u ll-w id th h a lf-m a x im u m (e V ) m m µ E n e rg y (e V ) µ T e m p e ra tu re (K ) Figure S4: Temperature dependent photoluminescence. a, Exemplary emission spectra of carbon nanotubes at T = 3 K (upper panel), 77 K (middle panel) and at 4.2 K (lower panel). Note that the spectrum at 3 K was recorded on a different nanotube as the spectra at 77 K and 4.2 K. b, Full-width at half-maximum of emission spectra at 4.2 K, 77 K and 3 K. The linewidths at 4.2 K are at the resolution limit of our spectrometer ( 4 µev) ev. Chromatic aberration of our optical setup makes a quantitative analysis of E intensities difficult. In the presence of chromatic aberration, different focal planes are associated with different laser wavelengths and therefore the excitation intensity varies as a function of the laser wavelength. In consequence, E intensities depend on the specific optical alignment and are not comparable in terms of absolute intensity over the entire laser tuning window. To account a 4.2 K b 7 7 K c (7, ) (8,3 ) (9, ) (6,2 ) (6,4 ) (9, ) (6,2 ) (6,4 ) (9, ) (6,2 ) 3 K C o u n ts 3 2 C o u n ts 3 2 C o u n ts E m is s io n e n e rg y (e V ) E m is s io n e n e rg y (e V ) E m is s io n e n e rg y (e V ) Figure S5: Emission energy histograms. Histograms of emission energies at 4.2 K (a), 77 K (b) and 3 K (c). Grey triangles mark energies estimated for small-diameter nanotubes taking into account the temperature dependent energy shift according to Capaz et al. 23 Macmillan 5Publishers Limited. All rights reserved.

6 ..5. N o rm a liz e d P L in te n s ity E n e rg y (e V ) L a s e r d e tu n in g (m e V ) Figure S6: Photoluminescence excitation spectroscopy. Emission spectra (left panel) and corresponding photoluminescence excitation spectra (right panel) of as-grown suspended carbon nanotubes at T = 4.2 K. Red and orange traces were recorded with optical alignments optimized at wavelengths indicated by the asterisks. The variation in E intensity between the traces is due to chromatic aberration of our optical setup. for this effect, we have performed two sweeps of the laser wavelength (red and orange traces in Fig. S6), each in a setup configuration optimized for the laser wavelength indicated by the asterisks. The data shown for three different CNTs in Fig. S6 demonstrate that E resonances may appear strong or weak depending on the laser wavelength used to define the focal plane. In a given setting, however, both the intensity and the energy positions of individual resonances are reproducible from sweep to sweep. The cryogenic E spectra of our as-grown suspended CNTs are inconsistent with previous reports that assigned absorption resonances below 3 mev to phonon sidebands 5 and require further investigation. 23 Macmillan 6Publishers Limited. All rights reserved.

7 2 Theoretical modelling 2. Exciton lifetimes Following Perebeinos et al. 6,7 we calculate the intrinsic radiative lifetime τ of CNT excitons as the inverse of the radiative decay rate Γ rad as τ = Γ rad = n re 2 E 2 f 2πɛ m h 2 c 3. () Here, ɛ is the dielectric permittivity, m the free electron mass, e the elementary charge, c the speed of light c, and n r = ɛ the refractive index. We used ɛ =.846 for a CNT in vacuum 8. For a CNT with chirality (n, m) and diameter d = (a cc /π) 3(n 2 + m 2 + nm) (a cc =.44 nm is the the C C bond length) 9 the exciton emission energy is approximated 6 by E =.84 evnm/d and the oscillator strength per carbon atom by f =.4 ev E π d σ c /A. Here A = 3 3/4 a 2 cc is the area per carbon atom, and σ c is the coherence length given by the confinement length in units of the exciton Bohr radius 8 σ X. The values for the intrinsic exciton lifetimes obtained from Eq. are in good agreement with ab initio calculations by Spataru et al Saturation of a three-level model system To model the saturation response of a three-level system we denote the crystal ground state by, and the exciton ground and excited states by X and X, respectively. The temporal evolution of the corresponding populations is described by a set of coupled rate equations: d dt ρ = ρ Γ abs + ρ X Γ rad (2) d dt ρ X = ρ Γ abs ρ X γ rel (3) d dt ρ X = ρ X γ rel ρ X Γ rad (4) that we solve for steady-state, dρ i /dt =. In the set of equations ρ i denotes the population of state i with ρ + ρ X + ρ X =, and Γ abs, γ rel, Γ rad are the absorption, relaxation and radiative recombination rates, respectively. For the population of the exciton ground state X we find: ρ X = Γ abs Γ rad + (Γ abs /γ rel )(Γ rad + γ rel ) (5) which reduces in the limit γ rel Γ rad to ρ X = Γ abs Γ rad + Γ abs. (6) 23 Macmillan 7Publishers Limited. All rights reserved.

8 This limit of fast relaxation as compared to the radiative emission is justified by linewidths of E resonances of the order of tens of mev corresponding to sub-picosecond relaxation timescales. The intensity is proportional to the product of the excited state population and the radiative decay rate, I Γ rad ρ X, with the proportionality factor given by the overall detection quantum efficiency. Normalizing the intensity by the constant value in saturation I max detection efficiency and with the Eq. 6 we obtain: eliminates the I I max = Γ abs Γ abs + Γ rad. (7) The saturation is uniquely determined by the ratio of the radiative decay rate to the absorption rate. Finally, by relating the absorption rate Γ abs to the excitation laser intensity I laser = αγ abs through the fitting parameter α we obtain the model response of a three-level system: I = I max I laser I laser + αγ rad, (8) where the radiative recombination rate is experimentally obtained as the inverse lifetime Γ rad = /τ. The resulting fit is shown by the solid line in Fig. 4 of the manuscript. Best fit was obtained with I max = cts (µw s) for α =.25 6 kw s cm Photon bunching and antibunching responses The normalized second-order correlation function g (2) (τ) for the CNT in Fig. 4b of the manuscript under cw excitation was plotted with g (2) (τ) = A e τ τ, (9) where τ is the time delay between two successive photon detection events in the avalanche photon detectors of the Hanbury-Brown and Twiss setup, τ is the exciton lifetime measured with time-resolved photoluminescence (τ = 3.35 ns), and A = g (2) () is a fitting parameter representing the degree of photon antibunching (A =.7 or g (2) () =.3 in Fig. 4b of the manuscript). The photon bunching under pulsed laser excitation the CoMoCAT nanotube in Fig. 4b of the manuscript is a result of intermittence and was modelled based on the work of Santori et al. for blinking quantum dots : g (2) (τ) = + τ off τ on e ( ) τon + τ off τ, () 23 Macmillan 8Publishers Limited. All rights reserved.

9 where τ on and τ off are the time constants for the luminescent and non-luminescent CNT states, respectively. To model the bunching data in Fig. 4b of the manuscript we used best-fit parameters τ on = 34 ns and τ off = 6 ns. 23 Macmillan 9Publishers Limited. All rights reserved.

10 References. Capaz, R. B., Spataru, C. D., Tangney, P., Cohen, M. L. & Louie, S. G. Temperature dependence of the band gap of semiconducting carbon nanotubes. Phys. Rev. Lett. 94, 368 (25). 2. Lefebvre, J., Fraser, J. M., Finnie, P. & Homma, Y. Photoluminescence from an individual single-walled carbon nanotube. Phys. Rev. B 69, 7543 (24). 3. Moritsubo, S. et al. Exciton diffusion in air-suspended single-walled carbon nanotubes. Phys. Rev. Lett. 4, (2). 4. Weisman, R. B. & Bachilo, S. M. Dependence of optical transition energies on structure for single-walled carbon nanotubes in aqueous suspension: An empirical Kataura plot. Nano Lett. 3, (23). 5. Htoon, H., O Connell, M. J., Doorn, S. K. & Klimov, V. I. Single carbon nanotubes probed by photoluminescence excitation spectroscopy: The role of phonon-assisted transitions. Phys. Rev. Lett. 94, 2743 (25). 6. Perebeinos, V., Tersoff, J. & Avouris, P. Scaling of excitons in carbon nanotubes. Phys. Rev. Lett. 92, (24). 7. Perebeinos, V., Tersoff, J. & Avouris, P. Radiative lifetime of excitons in carbon nanotubes. Nano Lett. 5, (25). 8. Capaz, R. B., Spataru, C. D., Ismail-Beigi, S. & Louie, S. G. Diameter and chirality dependence of exciton properties in carbon nanotubes. Phys. Rev. B 74, 24 (26). 9. Saito, R., Dresselhaus, D. & Dresselhaus, M. S. Physical Properties Of Carbon Nanotubes (Imperial College Press, 998).. Spataru, C. D., Ismail-Beigi, S., Capaz, R. B. & Louie, S. G. Theory and ab initio calculation of radiative lifetime of excitons in semiconducting carbon nanotubes. Phys. Rev. Lett. 95, (25).. Santori, C., Pelton, M., Solomon, G., Dale, Y. & Yamamoto, Y. Triggered single photons from a quantum dot. Phys. Rev. Lett. 86, (2). 23 Macmillan Publishers Limited. All rights reserved.