Telegraphic Noise in Transport through Colloidal ... - ACS Publications

Letter pubs.acs.org/NanoLett

Telegraphic Noise in Transport through Colloidal Quantum Dots Dany Lachance-Quirion,*,† Samuel Tremblay,† Sébastien A. Lamarre,† Vincent Méthot,† Daniel Gingras,† Julien Camirand Lemyre,‡ Michel Pioro-Ladrière,‡ and Claudine Nì. Allen*,† †

Centre d’Optique, Photonique et Laser (COPL), Département de Physique, de Génie Physique et d’Optique, Université Laval, Québec, Québec, G1V 0A6, Canada ‡ Département de Physique, Université de Sherbrooke, Sherbrooke, Québec, J1K 2R1, Canada S Supporting Information *

ABSTRACT: We report measurements of electrical transport through single CdSe/CdS core/shell colloidal quantum dots (cQDs) connected to source and drain contacts. We observe telegraphic switching noise showing few plateaus at room temperature. We model and interpret these results as charge trapping of individual trap states, and therefore we resolve individual charge defects in these high-quality low-strain cQDs. The small number of observed defects quantitatively validates the passivation method based on thick CdS shells nearly lattice-matched to CdSe cores first developed to suppress photoluminescence blinking. Finally, we introduce a figure of merit useful to efficiently distinguish telegraphic noise from noise with a Gaussian distribution. KEYWORDS: Colloidal quantum dots, telegraphic noise, electrical transport, charge defects, photoluminescence blinking

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through CdSe/CdS nanorods.15 In this Letter, we study electrical transport through thick-shell CdSe/CdS cQDs, and we report measurements of telegraphic noise with two or more plateaus in two different cQDs. We attribute this random telegraph noise to the charging of a small number of individual trap states and therefore confirm that the thick-shell low-strain passivation of CdSe/CdS cQDs can indeed produce devices almost free of charge defects. Core/shell CdSe/CdS cQDs are synthesized using a successive ion layer adsorption and reaction (SILAR) method.11 First, CdSe cores are synthesized using a method modified from ref 16. The resulting core nanocrystals have a mean radius a of 1.4 ± 0.2 nm, as determined by absorption spectroscopy17,18 (Figure 1b). The shell is subsequently grown according to ref 19 by alternately adding Cd precursor and S precursor to add a monolayer at a time for a total of 15 monolayers of CdS. Figure 1a shows a transmission electron microscope (TEM) image of the synthesized thick-shell cQDs. The inset shows that the mean diameter 2(a + t) of the cQDs is 12.7 ± 0.3 nm, where the thickness t of the 15 monolayer CdS shell is then calculated to be 5.0 ± 0.4 nm. As shown in the Supporting Information, photoluminescence blinking in these cQDs is strongly suppressed. Indeed, more than half of the cQDs are in their bright state more than 90% of the time. These results indicate that the synthesized CdSe/CdS cQDs contain few defects.10,11

olloidal nanostructures have many applications in biology and optoelectronics thanks to their simple and costeffective chemical synthesis. Indeed, colloidal quantum dots (cQDs) have been used as nanometer-sized biological markers1,2 and tunable semiconductors for solar cells.3 However, their high surface-to-volume ratio makes these nanostructures prone to the effects of surface dangling bonds on their optoelectronic properties, which turn out to play an important role in the dynamics of photoluminescence blinking.4−6 Whereas blinking has been recently used to achieve super-resolution imaging,7 many applications require stability of the optical and electronic properties of cQDs.8 Therefore, organic and inorganic passivation methods of surface defects have been successfully developed.8,9 Recently, cadmium selenide cQDs with suppressed photoluminescence blinking have been synthesized by growing around them a relatively thick shell of cadmium sulfide with modest lattice mismatch to the core.10,11 Indeed, this core/shell nanoheterostructure effectively produces a low-strain environment for the charge carriers as in self-assembled quantum dots, where photoluminescence blinking is almost inexistent.12,13 Whereas suppressed photoluminescence blinking suggests only indirectly that these passivated cQDs have few defects, we report here direct and quantitative observations of these trap states. Intermittency in transport through PbSe and CdSe cQDs has been observed, where the probability of the high- and lowcurrent states followed a power law as in photoluminescence blinking.14 This observation implies that several charge defects were involved, as expected for core-only cQDs without passivation. On the other hand, telegraphic noise, characteristic of individual distinct defects, has been observed in transport © XXXX American Chemical Society

Received: November 15, 2013 Revised: January 15, 2014

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dx.doi.org/10.1021/nl404247e | Nano Lett. XXXX, XXX, XXX−XXX

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Information). Transport time traces I(VSD,t) are acquired at different biases by measuring the current N = 100 times with a time interval Δt ∼ 100 ms between each data point. Figure 2a and b shows the time-averaged current I(̃ VSD) = I(VSD) − Iref(VSD) for devices A and B, where I(VSD) and

Figure 1. (a) TEM image of thick-shell CdSe/CdS cQDs. Inset: histogram of the cQD diameters and Gaussian fit, yielding a mean diameter 2(a + t) of (12.7 ± 0.3) nm. (b) Absorption spectra of CdSe cores before (light red) and after (light green) the CdS shell growth. The latter spectrum is offset for clarity, and its low-energy part is magnified in the inset to highlight the first excitonic state. At energies higher than the CdS bulk band gap energy (2.5 eV), the absorption is dominated by the shell material, as it constitutes about 99% of the cQD volume. (c) False-colored SEM image of a device without cQDs. (d) SEM image of device B nanogap filled with a few cQDs (same scale as a).

Figure 2. I−V curves of devices A (blue circles) and B (red squares) indicating transport through a cQD (a) on a linear scale and (b) on a logarithmic scale (device B is offset by 25 pA in (a) for clarity). Gold, green, and purple symbols correspond to one, two, and three current plateaus, respectively, in the transport time traces shown in Figure 3. The estimated values of the first electron and hole energy levels of the cQD addition spectrum, μ+1(0)/e and μ−1(0)/e, respectively, are shown as black vertical dashed lines. Horizontal black arrows point to the transport onsets for device A. Energy diagrams at (c) zero bias VSD = 0 and (d) negative bias VSD < 0 in the CS ≫ CD limit of device A, where hole transport occurs at negative biases.

Electrical transport in nanostructures has proven to be an effective way to probe charge defects.15,20 Transport through single cQDs has been almost exclusively studied using either scanning tunneling microscopes21−24 or conductive atomic force microscopes.14,25 Here, we embedded thick-shell CdSe/ CdS cQDs in three-terminal devices.26−28 The devices used in the experiments are fabricated on p-type Si on which a 100 nm thick SiO2 layer is thermally grown. Microstructures, in which source and drain terminals are still interconnected, are defined in a Cr/Au bilayer of respectively 5 and 35 nm using photolithography and a lift-off process. A focused gallium ion (Ga+) beam (FIB) is then used to reduce the width of the source−drain interconnection and etch a nanogap in a singleline scan.29 Figure 1c shows a tilted scanning electron microscopy (SEM) image of the resulting device before incorporating cQDs. Devices are immersed during ∼24 h in a 5 mM solution of 1,6-hexanedithiol in isopropyl alcohol, then rinsed with isopropyl alcohol, and dried using N2 gas.30 Colloidal quantum dot suspensions in hexanes are purified from excess organic ligands by precipitation and then suspended in toluene. Functionalized devices are immersed for another 24 h in this new cQD suspension and subsequently rinsed using toluene, acetone, and isopropyl alcohol and then they are N2-dried. Figure 1d shows one of the device (device B) with a monolayer of cQDs, including some in the nanogap. Room-temperature two-terminal measurements of the devices are performed in the dark by applying a bias VSD to the source while the current I(VSD) is measured at the drain through a current preamplifier. Initial measurements after FIB milling and without cQDs are performed to characterize current leaks. Bare devices show slightly nonlinear I−V characteristics in the ±3 V bias range, with a typical resistance between 100 GΩ and 200 GΩ at low bias (see the Supporting

Iref(VSD) are respectively the current after and before the incorporation of the cQDs (see the Supporting Information). Also subtracted is a zero-bias current of ∼1 pA probably coming from charges leaking to the terminals from the p-Si substrate. These charges are accumulated in the substrate via photodetection by the Schottky contacts to the p-Si substrate when illuminated by room lights prior to measurements. A zero-current gap is distinguished between approximately ±1 V for device A, while device B has an I−V curve characteristic of Fowler−Nordheim tunneling through the barrier between the metallic reservoirs and the cQD conduction band. This indicates that, for device B, the cQD is coupled directly to the reservoirs.31 The different transport mechanisms between device A and B can be explained by the presence or absence of organic ligands acting as tunnel barriers. Indeed it was found that no transport through cQDs is observed in any device directly after incorporating the cQDs, but only for a few devices after making a SEM observation. The long chains of carbon−carbon bonds of the cQDs’ synthetic ligands are highly insulating.32−34 Imaging the devices with an electron beam at an acceleration voltage of 30 kV breaks down some of the cQD ligands (oleic acid and oleylamine) and linking molecules (1,6-hexanedithiol),35 explaining why transport is observed for devices with cQDs only after SEM observation. In this framework, the B


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Figure 3. Time traces of the current fluctuation δI(t) with one, two, or three current plateaus (gold, green, and purple, respectively) in device A (a and b) and B (c). The histograms of δI extracted from these time traces are presented in d−f. Black arrows indicate amplitudes ± ΔI(VSD)/2 of the current variations as determined by a fit of the corresponding noise power spectral densities to a charge trapping model (Figure 4). In all cases, dashed lines indicate the midrange current Im(VSD). As ΔI depends on the bias, histograms of δI at different biases cannot be combined together.

the thickness of the shell t = 5 nm, yielding a finite value for ε1se even without confinement provided by the core/shell interface for the electron. The energy of the first exciton state calculated with this model, including Coulomb interaction, differs by less than 40 meV from the one observed experimentally in the optical absorption spectrum shown in Figure 1b, indicating the validity of our model within a few tens of meV. The zero-current gap measured in a two-terminal experiment depends strongly on the ratio of the coupling capacitances of the source and the drain.37 In the completely asymmetric (CS ≫ CD or CS ≪ CD) and symmetric (CS = CD) cases, the zerocurrent gap is respectively Eg and 2Eg in the macroscopic limit (see the Supporting Information). Since the observed zerocurrent gap of 2.1 ± 0.1 V in device A is only slightly larger than the CdSe bulk band gap of 1.75 eV, we argue that device A is either in CS ≫ CD or CS ≪ CD limit. Indeed, the cQD can be closer to one of the contacts since the size of the nanogap can be slightly larger than the cQD diameter. Different capacitive couplings could also be caused by the presence of residue of organic material at source or drain contacts. Furthermore, the asymmetry between calculated electron and hole confinement energies (ε1se ≈ 60 meV and ε1sh ≈ 320 meV) enables us to distinguish the two limiting cases. Indeed, for device A, the onset of transport at negative bias is at a larger absolute value than at positive bias (Figure 2b). We therefore argue that CS ≫ CD, as shown in Figure 2d. The biases of the transport onsets are estimated using eq 1 and the bias-dependent electron and holes electrochemical potentials μ±1(VSD) = μ±1(0) − eVSD in the limit of CS ≫ CD.37

tunneling barriers in device A are provided by remaining 1,6hexanedithiols while, for device B, the cQD is in direct contact with the source and drain contacts. The zero-current gap observed in measurements of device A can be understood as a cQD tunnel-coupled to source and drain reservoirs.36 In this framework, the electrochemical potentials at equilibrium (VSD = 0) of the first electron and hole states, respectively μ+1(VSD = 0) and μ−1(VSD = 0), are given by37 ⎛ Eg E ⎞ μ±1(VSD = 0) = ±⎜ + ε1se,h + C ⎟ 2 ⎠ ⎝2

(1)

where Eg is the semiconductor band gap of the cQD core and ε1se,h are respectively the ground state energies of the quantumconfined electron and hole. The total capacitance CΣ between the cQD and the rest of the device determines the charging energy EC = e2/2CΣ and is given by the sum of the capacitances between the cQD and the reservoirs (CS and CD), the gate (Cg), and the p-Si substrate (Cp‑Si).37 Figure 2c shows the energy diagram of the system at equilibrium assuming the cQD Fermi level is at midgap, as expected for intrinsic semiconductors. Using an effective mass model, we numerically calculate ε1se,h assuming a spherical confinement by the CdS shell with a barrier height of 0.75 eV for the holes and unconfined electrons (aligned core and shell conduction bands).38,39 The surrounding environment is modeled as an infinite barrier for both electrons and holes. Unlike previous models,40,41 we consider C


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In this limit, transport onsets are simply given by μ±1(0) and are indicated by vertical dashed lines in Figure 2a and b. The observed values of (−1.20 ± 0.05) V and (+0.90 ± 0.05) V for transport onsets at negative and positive biases respectively are in good agreement with the estimated values of μ−1(0) = −1.20 eV and μ+1(0) = +0.94 eV. Therefore, we argue that current through device A is hole and electron transport at negative and positive biases respectively when |eVSD| ≥ μ±1(0). In both devices, the time-resolved measurements of the current I(VSD,t) show, at some biases VSD, fluctuations characteristic of telegraphic switching noise.42,43 No telegraphic noise is observed in the zero-current gap μ−1 < eVSD < μ+1 of device A as the current is blocked in this bias range. Also, despite the absence of a clear zero-current gap in device B, no telegraphic noise is observed for biases |VSD| lower than ∼1 V. The data of Figure 2 for which telegraphic noise is observed are color-coded according to the number of current plateaus, where green and purple indicate respectively two and three current plateaus. Data in blue and red for device A and B respectively indicate either stable or unstructured time traces, that is, not clearly showing multiple plateaus. For each device, stable time traces within the bias range where telegraphic noise is observed are color-coded in gold rather than blue and will be compared to time traces showing multiple plateaus. To observe telegraphic noise, the energy level of a trap state needs to be in the bias window between the source and drain Fermi levels, at −eVSD and 0, respectively, in order for the trap to be charged and discharged. Also the current from transport through the cQD needs to be nonzero to see the effect of the trapped charges on the transport. Therefore, the energy of trap states that are in the band gap cannot be estimated from a twoterminal experiment. Indeed, the energy levels of the trap states cannot be tuned out of the bias window while a nonzero current is flowing. However, since telegraphic noise is observed, within the resolution of our measurements, at the onset of transport at negative bias for device A, we can argue that the trap states are in the energy gap of the addition spectrum of the system. Furthermore, assuming that the cQD Fermi level is midgap at equilibrium, the energy levels of the trap states are closer to the valence band than the conduction band, and therefore trapped charges are likely to be holes (q = +e). To compare current fluctuations of time traces at different source−drain biases VSD, we define the current variation δI(t) ≡ I(t) − Im relative to the midrange current Im ≡ (Max[I(t)] + Min[I(t)])/2. Some of the time traces presented in Figure 3a− c display telegraphic noise, and the histograms of δI of each of those time traces are shown in Figure 3d−f. The bin size for these histograms is fixed at 100 fA, approximately the measurement setup’s noise floor. The gold, green, and purple time traces and histograms of Figure 3 show respectively one, two, and three current plateaus. In a standard telegraphic noise model,42 a trap state causes two discrete current plateaus I0 = Im ± ΔI/2 and I1 = Im ∓ ΔI/2 depending on the trap being empty or singly charged, where ΔI ≡ |I0 − I1| is the amplitude of the current fluctuations. The sign of the variation of the current from Im depends not only on whether charges from transport and trapped charges have the same sign or not (Coulomb interaction), but also on the details of the transport mechanism. In our measurements, an incomplete knowledge of these parameters makes the choice of sign uncertain. However, a spectral analysis of the telegraphic noise can be made without labeling the different current plateaus.

In a telegraphic noise model, multiple trap states are necessary to explain more than two levels in the transport time traces.43,44 Therefore, green and purple time traces in Figure 3, with their two and three plateaus, are considered to be induced by the charging of one and two trap states, respectively. Indeed, time traces of two trap states causing about the same ΔI would show three distinct current plateaus instead of four. The noise spectral density SI(ω) of a Markovian charge trapping process with charging and discharging time constants τ0 and τ1 is a Lorentzian function of the frequency ω/2π SI (ω) = 4(ΔI )2

τ0τ1

1/T (τ0 + τ1) ω + (1/T )2 2

2

(2)

where 1/T = 1/τ0 + 1/τ1.42 The noise power spectral densities are calculated from the discrete Fourier transform of I(t) and are fitted to eq 2 for ΔI and one of the time constant τ0,1 ≡ τH. As we cannot label the different plateaus according to the trap state occupancy 0 or 1, we simply label them H (high) and L (low) according to |IH| > |IL|. Then, the other time constant τL is related to τH by τL = (pL/pH)τH ≡ (1/η)τH, where pH and pL are the probabilities of occupation of each state and η ≡ τH/τL = pH/pL is their ratio. This probability ratio η is estimated from the current variation histogram as the ratio of all counts for δI < 0 to all those for δI > 0 when Im < 0 and vice versa when Im > 0. The estimation of η from the histogram reduces the number of fitted parameters from three to two, ΔI and τH. Using the amplitude ΔI(VSD) of the current variations extracted from the fit of the noise spectral density of a singletrap time trace, the current variations for empty and charged trap states δI0,1 = δIH,L = ±ΔI/2 are shown in Figure 3d−f as black arrows. The agreement of these amplitudes with roughly the center of the peaks in the histograms suggests that the twoplateau telegraphic noise observed is consistent with a single trap state being charged and discharged according to a Markovian process. We therefore show that the few distinct plateaus observed means that both devices have only a few charge defects. Detailed examples of noise spectral densities are shown in Figure 4, both for devices A and B and for time traces pertaining to one and two trap states. The total rate of charging/discharging, given by 1/T = 1/τH + 1/τL, defines the low-frequency cutoff in the noise spectral density. This rate varies between ∼2 and 8 Hz in device A and between ∼0.5 and 3 Hz in device B. Electrical transport through these charge defects is then undetectable in our measurements as it would lead to a current in the atto-ampere range. Figure 4e and f show noise spectral densities of time traces indicating two trap states in device A. For two independent trap states, the noise spectral density should be the sum of spectral densities of single trap states, given by eq 2. However, since the large number of fitting parameters makes an estimation of these parameters uncertain, only the high-frequency part (ω/2π > 1 Hz) of spectra in Figure 4e and f is fitted to a single trap state spectral density (eq 2) in the high-frequency limit, where SI(ω) ∝ ω−2. The spectrum at VSD = −1.5 V (Figure 4f) shows a deviation from the ω−2 fit at low frequencies, characteristic of the cutoff of a Lorentzian spectrum, while the one at VSD = −2.4 V (Figure 4e) shows no significant deviations. While the absence of a low-frequency cutoff could be explained by characteristic times (τ0,1) much longer than the total measurement time (NΔt), the time trace rather suggests that τ0,1 ≈ NΔt. However, the absence of a low-frequency cutoff can also D


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time constant ratios η and all Gaussian noise relative amplitudes σn/ΔI. Therefore, this figure of merit can be used as a criterion to distinguish time traces δI(t) with telegraphic noise from those with a Gaussian probability distribution of noise. Figure 5 shows θδI calculated from eq 3 with (σn/ΔI = 0.2) and without (σn/ΔI = 0) additional Gaussian noise for different

Figure 5. Figure of merit θδI as a function of the time constant ratio η. Black plain lines and dots respectively show theoretical and simulated figures of merit for time traces of telegraphic noise with (σn/ΔI = 0.2) and without (σn/ΔI = 0) additional Gaussian noise. θδI for time traces of device A (blue circles and triangles) and device B (red squares) exhibiting telegraphic noise with only two plateaus are also included. For these data, the time constant ratio η is redefined such that η ≥ 1 (if η < 1, then η → η−1). Blue triangle data from device A are from time traces not shown in this report. The values of θδI in the limit of noiseless (σn = 0) and symmetric (η = 1) telegraphic noise (θδI = 1) as well as Gaussian noise (θδI = (π/2)1/2) are shown as black horizontal dashed lines.

Figure 4. Noise power spectral density of current time traces I(t) indicating one trap state for device A (a and b) and device B (c and d) and two trap states for device A (e and f). Plain lines are least squares fits of a Markovian process of charge trapping using eq 2 for a−d and a SI(ω) ∝ 1/ω2 fit for ω/2π > 1 Hz for e and f. Dashed lines show the 1/ω2 high-frequency limit of eq 2. Vertical dashed lines in a−d show the cutoff frequency 1/T = 1/τH + 1/τL. The time traces associated with each spectrum are shown as insets.

be explained by the presence of more than one trap state. Indeed, in that case, the cutoff at low frequencies is usually not observed in the spectral density since the superposition of trap states with different characteristic times τ(i) 0,1 and amplitudes ΔI(i) soon leads to a power-law spectrum SI(ω) ∝ ω−α with α ≤ 2, even for a small number of trap states.43,45 Time traces showing multiple current plateaus have been identified by visual inspection. However, an efficient method able to distinguish telegraphic switching noise from other types of noises would be useful for implementing an automatic detection of noisy devices. The bimodal probability distribution of single-trap telegraphic noise (Figure 3d−f) is in stark contrast with the almost perfectly Gaussian distribution of 1/f and thermal and shot noises according to the central limit theorem. We therefore introduce a figure of merit based on this difference in probability distribution. We define this figure of merit as θδI ≡ σδI/|δI(t )|, where σδI and |δI(t )| are respectively the standard deviation and the mean absolute value of the current variation δI(t). This figure of merit gives (π/2)1/2 ≈ 1.25 for Gaussian noise while it is always strictly smaller or equal to one for two-plateau telegraphic noise only. For Gaussian noise of standard deviation σn superposed over twoplateau telegraphic noise of amplitude ΔI, the figure of merit is given by θδI = f (σn /ΔI )

⎛ 2σn ⎞2 4η ⎜ ⎟ + ⎝ ΔI ⎠ (1 + η)2

time constant ratios η, as well as θδI calculated from time traces simulated with the same parameters. The time constant ratios η estimated from the histograms of Figure 3d−f and the figure of merit θδI calculated from the corresponding two-plateau time traces of Figure 3a−c yield the data points in Figure 5, all falling in the range 0 < σn/ΔI < 0.2. This nonzero amplitude Gaussian noise, superimposed to telegraphic noise originating from the cQD trap states, can come from either thermal and shot noises, or also from telegraphic noise averaged out during the measurement time Δt ≈ 100 ms. For more than one trap state, a simple telegraphic noise model can be formulated for independent traps and noiseless plateaus (σn = 0) (see the Supporting Information). In this case, the figure of merit can be higher than (π/2)1/2 for symmetric noise (η = 1), but drops off quickly under the (π/2)1/2 limit for η ≳ 1.8. Thus, in most cases, a figure of merit smaller than the Gaussian limit of (π/ 2)1/2 can be used to identify telegraphic noise with few trap states. In conclusion, we have measured transport through single thick-shell CdSe/CdS cQDs in devices where the source and the drain contacts are separated by a nanogap. Biases of transport onsets are well-described by a cQD tunnel-coupled to source and drain reservoirs. In transport time traces, we resolved telegraphic switching noise with distinct current plateaus, characteristic of very few defects in the devices. Noise spectral densities of two-plateau time traces are welldescribed by a Markovian charge-trapping process. We therefore showed that passivating cQDs with a thick shell nearly lattice-matched to the core in CdSe/CdS cQDs is indeed efficient in reducing strain and the number of trap state defects affecting the cQDs optoelectronic properties. Finally, we

(3)

The function f(σn/ΔI), specified explicitly in the Supporting Information, is close to unity for Gaussian noise of small relative amplitude (σn/ΔI < 0.2). For higher Gaussian noise amplitudes, f(σn/ΔI) decreases, such that θδI ≤ (π/2)1/2 for all E


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introduced a figure of merit able to distinguish telegraphic switching noise from normally distributed noise, which can be used to implement an efficient method to detect telegraphic noise in nanoelectronic devices.

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ASSOCIATED CONTENT

S Supporting Information *

Synthesis and photoluminescence blinking analysis of the thickshell CdSe/CdS cQDs, I−V curves of devices A and B, estimation of the reservoirs’ lever arms, and theoretical description of the figure of merit. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Present Address

D.L.-Q.: Département de Physique, Université de Sherbrooke, Sherbrooke, Québec, J1K 2R1, Canada. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors would like to thank Vincent Michaud-Belleau for help on sample fabrication and Patrick Larochelle for technical support. D.L.-Q., S.T., J.C.L., M.P.-L., and C.Nı ́.A. acknowledge financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC), D.L.-Q. and M.P.-L. from Fonds québécois de la recherche sur la nature et les technologies (FRQ-NT), and M.P.-L. from Canadian Institute for Advanced Research (CIFAR).

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REFERENCES

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