CdS Dot

Jul 21, 2010 - The long biexciton lifetime results into an observed long-living gain having a peak that is red shifted with respect to the lowest ener...
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Suppression of Biexciton Auger Recombination in CdSe/CdS Dot/Rods: Role of the Electronic Structure in the Carrier Dynamics Margherita Zavelani-Rossi,† Maria Grazia Lupo,‡ Francesco Tassone,*,‡ Liberato Manna,§ and Guglielmo Lanzani‡ †

Istituto di Fotonica e Nanotecnologie IFN-CNR, Dipartimento di Fisica, Politecnico di Milano, P.zza Leonardo da Vinci, 32, 20133 Milano, Italy, ‡ Center for Nano-Science and Technology, IIT@POLIMI, via G. Pascoli, 20133 Milano, Italy, and § Istituto Italiano di Tecnologia, via Morego 30, 16163, Genova, Italy ABSTRACT We studied carrier dynamics in semiconductor nanocrystals consisting of a small CdSe dot embedded in an elongated, rod-shaped CdS shell, using the ultrafast pump-probe technique. We found clear evidence of a substantial suppression of the Auger nonradiative recombination in the biexciton regime. Moreover, a simple model of the dynamics in which biexcitons show no Auger recombination, and only holes are localized in the dot, fits well the differential transmission observed at all pump densities. The long biexciton lifetime results into an observed long-living gain having a peak that is red shifted with respect to the lowest energy absorption peak. We argue that the origin of the large relative gain observed at large fillings is related to the peculiar structure of the electronic levels, and in particular, to delocalization of electrons in the rod. KEYWORDS Nanocrystals, core-shell, charge-separation, delocalization, Auger recombination, gain

S

emiconductor nanocrystals (NCs) are promising functional materials for opto-electronic applications. In particular, optical properties of NCs are strongly dependent on their composition, shape, and dimensions, and recent advances in synthesis allow a precise and reproducible control of them.1-3 An interesting class of NCs is characterized by a CdSe spherical core embedded in a CdS shell that can have different size and shape with respect to that of the core. These core/shell NCs show improved photoluminescence quantum yields and can have different optical properties and applications depending on the shape of the shell.4 For example, shells with tetrahedral shape show an emission wavelength that is sensitive to strain and can be used for fabricating sensitive optomechanical devices and for biological force investigations.5 Instead, the “giant dot”, where a thick spherical CdS shell embeds the small CdSe core, shows a strong suppression of the Auger recombination, enabling the observation of multiexciton emission and a reduction of blinking in photoluminescence.6 Moreover, gain in the single-exciton regime could be observed as a consequence to the strong blue shift of the biexciton absorption due to a partial separation of holes and electrons in the core and shell, respectively.7 In elongated NCs in which a CdS rod is grown onto a spherical CdSe dot,8-10 a remarkably long gain lifetime was observed, resulting into

lasing when dot/rods were made to self-assemble to form a microcavity.11 These results are promising for the integration of photonic devices onto flat substrates by simple liquidphase processing of NC solutions. The peculiarities of the CdSe/CdS heterostructures are mainly related to the interface between the two materials, and in particular to the band alignment at these interfaces, which results into a strong confinement of holes and in a weak or no confinement of the electrons inside the CdSe core.7,8,12,13 In this work, we study the ultrafast carrier dynamics of CdSe/CdS dot/rods with comparable length, of about 20 nm, and two different dot diameters, 2.3 nm (QR2.3) and 3.5 nm (QR3.5) respectively, up to 400 ps, at different excitation powers. We study the physical processes underlying the observation of long-living gain reported in a previous work.11 We found that gain starts to appear in the biexciton regime, that is, in the regime where two electron-hole pairs are found in the NC, and that this regime is long-living thanks to an almost complete suppression of the Auger recombination for both dot diameters. Indeed, the time constant for this process is larger or comparable to the photoluminescence decay time, in the nanoseconds range. For larger fillings, higher-order Auger recombination processes start to dominate the dynamics. However, also these processes are largely reduced compared to spherical or simple rod-shaped NCs. We found that for an average electron-hole pair occupation of up to ten, the Auger-recombination time constant is still about 60-80 ps for the larger dot sample and in the 50 ps range for the smaller dot sample. The Auger

* To whom correspondence should be addressed: E-mail: [email protected]. Received for review: 06/1/2010 Published on Web: 07/21/2010 © 2010 American Chemical Society

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nonradiative recombination time constant is then reduced to a few picoseconds only well above these occupation numbers. Comparing the dynamics of the two dot diameter samples we could remark that there is some difference in the optical coupling of the lowest energy hole level to the lower energy electron levels, the latter being presumably delocalized into the CdS shell. In particular, this difference also results into merely attaining transparency at the wavelength of lowest absorption peak for the larger dot diameter sample, whereas some gain is observed for the smaller dot samples at this wavelength for large fillings. Moreover, gain is peaked at longer wavelengths compared to the absorption peak, and gain increases to well above the two-level limit in magnitude at large fillings, giving indications of a contribution from scattering processes in the biexciton regime. We show that a comprehensive explanation of all these results, including the suppression of the Auger recombination for the lowest energy hole level, can be found when we assume that the electron is delocalized over the rod, and the lowest energy hole levels are instead localized in the core. On the basis of this reasonable assumption, we set up a simplified model of the dynamics with which we fit well all the experimental data. The laser system used in this work is based on a Ti: sapphire chirp pulse amplified source, with maximum output of about 800 µJ, 1 kHz repetition rate, and central wavelength of 780 nm. Excitation pulses at 390 nm are obtained by doubling the fundamental wavelength in beta barium borate crystal (BBO) with pulse duration around 150 fs. Pump pulses are focused onto a 130 µm spot. Probing is achieved in the visible region by using white light generated in a thin sapphire plate. Chirp-free transient transmission spectra are collected using a fast optical multichannel analyzer (OMA) with a dechirping algorithm. The measured quantity is the normalized transmission change, ∆T/T, which is proportional to the absorbance change ∆R in the small signal limit, for a diluted sample. The average electron-hole pair generated at different pump intensity was calculated as reported in ref 14. All measurements were performed with rods dispersed in a toluene solution at room temperature. We report in Figure 1a,b the absorption and the differential transmission for the two samples, normalized at the band edge absorption. The pump and probe delay is 5 ps and the excitation pulse gives rise to an average occupation of electron-hole pairs Nave of about 2. The lowest absorption features are related to transitions in the CdSe core, whereas the larger absorption features are related to transitions in the larger volume CdS shells. In particular, we label X0, X1, and so forth the former transitions and Y0 the latter. Given the different size of the core dot and rod diameters for the two samples, we find transitions at lower energy in QR3.5 compared to those in QR2.3. We remark that the spectral features X0, X1, and so forth are also found, unshifted, in the differential transmission spectrum. They are due to bleaching of the optically active transitions from © 2010 American Chemical Society

FIGURE 1. Linear absorption spectrum (red line) and differential transmission spectrum ∆T/T (blue line) for sample QR3.5 (a) and QR2.3 (b) for Nave ∼ 2 and probe delay of 5 ps. Sketches of the energy levels involved in the optical transitions are shown in the insets.

partial filling of electron and hole levels by relaxed injected carriers. We remark that even though only two carriers have been injected in the experiment resulting in Figure 1, not only the lowest energy transition X0 becomes bleached, but also X1, X2, and Y0. We have to conclude that either the relaxation of the carriers to the lowest energy does not take place in all dots, or the Y0 transition too is bleached by the relaxed carriers. In fact, we may write in general

α(λ) )

αS 2N

∑ |〈φi(e)|φj(h)〉|2λijL(λ - λij)(1 - ni(e) - nj(h)) ij

(1)

where λ is the sample absorption, RS is an absorption constant related to the material and structure properties, φ(e) i and φj(h) are the electron and hole envelope functions for 3143

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(h) energy levels i and j, n(e) i and nj are the occupation number of electron and hole states i and j respectively, L is a unit area line width function, and λij is the transition wavelength related to the energy difference between states i and j. The indices i and j run over different energy levels having an allowed optical transition, include all different degeneracy indices, and also run over all the N NCs contained within a (h) 2 macroscopic volume V. We understand that, if |〈φ(e) i |φj 〉| for a given electronic level i is not zero for several hole levels j, then we have optically active transitions at different energies which will be bleached by the filling of the same electron level i. The bleaching of Y0 at low occupations therefore gives a strong indication that some electron levels may be related to both X0 and Y0 transitions, and therefore, as the Y0 transition is related to the rod, that these electron levels are delocalized over the rod. Indeed, detailed ab initio calculations on these type of structure have shown that inclusion of strain effects due to the different lattice constants of CdS and CdSe may result in a strong reduction or even an inversion of the band-offset for electrons at the heterointerface.15 If this is the case, electrons are expected to be delocalized throughout the whole NC. Moreover, as the energy barrier for electrons due to inversion of the bandoffset, if any, will be small, presumably below 80 meV, only few, lowest energy electrons could be pushed out of the CdSe core, whereas other electrons having a somewhat larger energy may enter into the core dot. For a 20 nm long rod, the lowest energy electron levels are expected to be closely spaced just as in a one-dimensional box, and we may roughly estimate an energy sequence of the type ∼4 meV·n2, so that only electrons in the first 3 or 4 levels could be pushed away from the dot. Moreover, given the small electron mass, we may expect this push-out effect to be weaker for a smaller dot. Instead, few hole levels, H0, H1, and so forth, depending on the dot size, are well localized into the core as the confinement potential for holes is larger than 0.5 eV, and the hole mass is much larger than the electron mass. Other hole levels having an energy above the bottom of the CdS band will be delocalized over the rod. These energy levels will also be very closely spaced. As a consequence of this structure of levels and wave functions, we expect the overlap integrals (to which the oscillator (h) 2 strength of the transition is proportional) |〈φ(e) i |φj 〉| to be small for the transition between H0, H1 and the first few electron levels E0, E1, and so forth, and the more so for QR3.5. Moreover, we may also expect that these integrals are larger for H1 compared to H0, as the former wave function is less localized in the dot compared to the latter. For large electron index i, the overlap integrals with H0 and H1 become small, as the electron wave function becomes rapidly oscillating and the integrals will average out. Even when considering a weaker potential barrier or no potential barrier for electrons, the overlap integrals of H0 with the lowest energy electron levels remain weaker compared to the higher levels. In conclusion, we understand that due to © 2010 American Chemical Society

strain effects, only a few electron levels are involved in the X0 and X1 transitions, and the oscillator strength of the H0 to E0, E1 and few other transitions are also weak, while they are larger for higher energy levels. This picture is in agreement with the observed bleaching of Y0 transition at low filling. Indeed, the X0 transition is bleached by filling of H0, and weakly bleached by filling of E0, the Y0 transition is bleached by filling of E0, as this level is delocalized over the rod. A schematic representation of the levels and transitions is given in the insets of Figure 1. We analyzed the time-resolved pump/probe measurements in the two different samples in order to further qualitatively understand the different contributions of electrons and holes to bleaching. In Figure 2a, we report the X0 bleaching kinetics, and in Figure 2b we report the X1 bleaching kinetics up to 400 ps delay, for QR3.5, for different average occupation Nave. At low excitation, when Nave < 1, the X0 and X1 bleachings decay with a time constant of about 2 ns. At higher excitations, with 1 < Nave < 30 the X0 signal saturates in the first few tens of picoseconds, then starts to decay with a time-constant that gradually increases at later times to the nanoseconds time scale found at low density. Instead, the X1 decay is faster in the first 100 ps, and then follows the usual low-excitation decay. The X0 signal shows a similar evolution for all pump excitations, both at short times and at long times. In particular, excluding the first few tens of picoseconds, we could fit all decays with a first time constant of about 100 ps and another constant of 2 ns. This long time constant is also comparable to the decay time constant found in photoluminescence, which is about 3 ns.16 Within the short time span of our measurement, we cannot distinguish between these two values. The bleaching at X0 eventually saturates when increasing the pump excitation. At very high excitations when Nave > 30, the X0 bleaching is slightly reduced due to the onset of strong photoinduced absorption (PA) in this spectral region.17 An interpretation of the dynamics of carriers from these results starts from the consideration that at low excitation and at long times electrons and holes are fully relaxed to H0 and E0 (and eventually E1 is also partially thermally populated). As bleachings at X1, X2, and Y0 are not related to H0 filling, they must be mainly due to filling of the lowest electron states E0, E1. Instead, the X0 bleaching is also due to H0 filling. At larger pumping, also H1 and H2 become gradually filled, contributing to additional bleaching of X1 and X2, and at even larger pumping also delocalized hole levels in the rod start to be filled, contributing to a further increase of the bleaching of Y0. Therefore, we may attribute the faster initial decay of X1 at large occupations to the nonradiative Auger recombination of holes in the H1 levels, while during this time the X0 remains saturated, as the H0 and E0 levels are. At longer times, eventually H1 becomes empty and the low-excitation conditions are recovered. We remark from Figure 2a that even at long times a large ∆T/T is observed for large fillings, remaining sizeably larger 3144

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only the standard recombination observed at low occupations occurs. This result is quite surprising and unexpected, as we do observe Auger recombination for higher energy levels. Of course, not all the holes placed in the dot will eventually populate the H0 level, as some may become trapped in the rod before relaxing into the core. We may therefore be overestimating the number of holes in H0. However, we may exclude such a trapping of carriers to be relevant, that is, only a negligible fraction of the holes is trapped in the rod. In fact, we analyzed normalized differential absorption -∆R/ R0 ) (R0 - R)/R0 at X0, where R0 is the linear absorption at X0 (empty system) and R is the absorption for the excited system, at short times after photoexcitation at X0. We report this quantity in Figure 2c. We first notice that by increasing the pump fluence, at first -∆R/R0 saturates to 1, and for very high fluences, when Nave > 30, it is slightly reduced from saturation as PA sets in. According to our picture of the structure of energy levels and transition, this saturation to 1 only is related to a negligible contribution of electrons to the bleaching of X0, due to a reduced overlap integral of E0 and H0 wave functions, as explained above. Eventually, as higher energy electron levels become filled at large occupations, a larger differential absorption should be observed. However, at very large fillings Auger recombination from higher energy levels sets in, preventing occupation numbers to grow indefinitely. Moreover, PA sets in, effectively reducing differential absorption. We may thus approximate eq 1 considering only H0 filling and obtain

-∆α/α0 ∼

nH0 2

(2)

The number 2 accounts for a degeneracy of 2 for the hole levels. When we consider a Poisson statistics for the occupation probabilities of a dot, and calculate nH0 we find

( (

FIGURE 2. Differential bleaching kinetics ∆T/T at X0 (a) and X1 (b) for different pump fluences and superimposed best fit (solid black line), determined as reported in the text in eqs 4 and 5, for the sample QR3.5. In (a) is also reported (right axes) the corresponding differential absorption value and is evidenced the 1 < Nave < 2 region calculated from the Poisson distribution (see text). (c) Experimental data of normalized transient absorption -∆R/R0 at X0 (black symbol), at short probe delay (2 ps), as a function of the initial average occupation. The red dashed line is the predicted dependence assuming a Poisson distribution of carrier population, as explained in eq 3.

-∆α/α0 ∼ 1 - 1 +

(3)

We report this curve in Figure 2c and remark that it fits well to the experimental data. If a relevant fraction of the carriers were trapped into the rod, they would not contribute to differential absorption, and the abscissa of Figure 2c would have to be scaled. However, in this case we would not obtain a good fit to the experimental data. We also remark that we assumed a simple double degeneracy for the hole level H0. This is only approximate, as a more refined picture would consider the two-particle excitonic picture. Here an almost 5-fold degenerate multiplet is found for the lowest energy transition for small dots of which three are dark.18 Another 3-fold almost degenerate multiplet is split

compared to the Nave ) 1 case. Therefore, we may expect this large differential absorption to be related to an average occupation above unity, and a large fraction of the NCs to be populated by at least two electron-hole pairs. Thus, the nonradiative Auger recombination process originating from two holes in H0 is well suppressed, and to such a level that © 2010 American Chemical Society

) )

Nave -Nave e 2

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to larger energies of above 100 meV in small dots. In conclusion, considering a simple double degeneracy of hole and electron levels would result into 4-fold degenerate lowest energy excitons with two-dark states, we see that our approximation is not rough. Moreover, at this basic level of the theory we additionally need to take into account that presumably the considerable strain at the core/shell interface is not spherical due to the shape of the dot/rods, and this will likely lift some of the approximate 5-fold degeneracy of the lowest energy excitons.18 Even after the first few picoseconds when recombination starts to occur, it was remarked in ref 17 that the population of the NC still follows the Poisson statistics. Then, when -∆R/R0 > 0.7 we have Nave > 1, and we may report for convenience the values of differential transmission corresponding to Nave ) 1 and 2 in Figure 2a. We then remark that we have Nave > 1 at long times for several initial conditions of pumping. When Nave > 1, some dots are occupied with two excitons, and we conclude that nonradiative Auger recombination in this case must be negligible compared to the other recombination processes. Therefore, we may place an estimate of this recombination time above 2 ns. We developed a simplified model of the dynamics to check the compatibility of the strong suppression of the Auger nonradiative recombination in the biexciton regime with the overall dynamics. In particular, we simplified the structure of the quasi-1D rod levels grouping all of them but the lowest two into a single group. The energy separation of the electronic levels was taken as an average energy ∆E ∼ 25 meV, whereas for holes the two lowest states are confined in the dot and have large energy difference with the hole quasi-band of levels. We then considered fast thermalization processes, recombination processes, and Auger nonradiative recombination processes. The dynamics of an ensemble of 1000 dots was then simulated using a Monte Carlo technique, and the average occupations were calculated on the ensemble of NC. From these occupations, differential transmission signals were calculated assuming the following dependences

∆T T

|

X0

∆T T

|

∼ 0.0168 · nH0 + 0.015 · nE1

(4)

∼ 0.004 · nH1 + 0.04 · nE0

(5)

X1

different contributions of electron and hole populations to the saturation of the transitions, as explained before using eq 1. However, given the simplified nature of the model, a quantitative connection to eq 1 and to overlap integrals of the real energy levels is not reasonable. Moreover, we remark that some of the dependence of the X1 differential transmission on E0 population is only effective, as in the experimental data there is some contribution from closeby tails of the X0 transition. Also, the separation of E0, E1, and E2 levels, which was chosen to be 25 meV here, is only an effective average, which is reasonable for a 20 nm rod as explained above considering a one-dimensional confinement potential along the rod length, but certainly it is not quantitatively significant. In conclusion, the calculated occupation of the E0 level in our model is at most qualitative and so is the weight in its contribution to the X1 bleaching. However, we want to remark that the simple kinetic model in which a negligible nonradiative recombination rate for the biexciton is assumed and the interpretation of differential absorption at X0 give a coherent picture of the dynamics in the systems and a reasonable fit of all the data considered here. While the assumption of electron delocalization in the rod allows for a reasonable explanation of the ultrafast differential transmission data, we notice that in a previous work by Sitt et al.12 an opposite conclusion has been proposed, based on the interpretation of experimental results on intensity dependent photoluminescence on similar NCs, having the same rod diameter but different lengths compared to the NCs of this work. In particular, a negative biexciton binding energy for small dots was found, indicating occurrence of charge separation, while for larger core diameters the usual positive binding energy was recovered, again an indication of charge localization. The core diameter at which the transition occurs was tentatively indicated to be 3 nm, a value that is coherent with calculations of the electron wave function carried out within the envelope function approximation and assuming a rather large conduction band offset between the CdSe core and the CdS shell. Moreover, a comparable trend was found in the calculated Coulomb repulsion energies and the measured biexciton binding energy, despite correlation energies (positive binding), were reportedly not included in the calculations. However, we argue that a number of relevant simplifications are present both in our simplistic phenomenological modeling of the kinetics and the structure of electronic levels, both in some assumptions of the calculation in ref 12. For example, assuming a simple, flat confinement potential ignores relevant warping of the conduction band due to interface strain effects, as evidenced in many works on heterogeneous core shell nanocrystals and in particular in ref 15, which addressed a structure that is very similar to our QR3.5. We thus argue that although for large core diameters the electron wave function is certainly localized, the actual core diameter at which this localization occurs

Results of the fit are shown in Figure 2. We remark that a reasonable fit is obtained, given the number of simplifications assumed in the detailed structure of the levels and in the nature of the Auger nonradiative recombination processes. The coefficients given in eqs 4 and 5 reflect the © 2010 American Chemical Society

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very high excitations with Nave > 18, the X0 bleaching is reduced due to the onset of strong photoinduced absorption. In Figure 3b, the normalized differential absorption at X0, at short times after photoexcitation, is reported. We find that it saturates to 1.7 for high pump energies and then is reduced for higher excitations when Nave > 12 due to the onset of PA. As in the QR3.5 case, we tried to fit the differential absorption using the Poisson distribution. In particular, we found that we could not fit the differential absorption using eq 3 as before, as this expression saturates to 1. This indicates some influence of filling of electronic levels. As explained above, also lower energy electrons contribute to the X0 transition for QR2.3. In particular, we tried to fit the differential absorption considering filling of higher electronic levels, E2, E3, and so forth as explained before. Although we expect several electronic levels to participate to the bleaching of the X0 level, we do not have the sensitivity to accurately distinguish between different situations, and we used a simplified model, considering only the E4 level, with a fractional contribution

( (

) )

Nave -Nave e + 2 Nnave N5ave -Nave 0.7 1 - 1 + e + n! 2 · 5! n)1,4

-∆α/α0 ∼ 1 - 1 +

( (

) )

(6)

We remark that the fit is reasonably accurate. Contribution from higher levels is presumably not relevant, as in this case, for larger fillings of the NC, faster Auger processes reduce them to lower levels. Using the assumption that the Poisson statistics is maintained in time as recombination occurs, as we did before, we traced the Nave ) 1 and 2 lines in Figure 3. We remark that also in this case at long times, when decay slows down, Nave > 1 in several cases, when initial population is large enough. As before, we conclude that the Auger nonradiative recombination for biexcitons is well suppressed. Also in this case we carried out a simulation of the dynamics of carriers, using similar Auger rates, and in particular, no Auger nonradiative recombination for biexcitons. We used the following coefficients for the X0 bleaching

FIGURE 3. Differential bleaching kinetics ∆T/T at X0 for different pump fluences and superimposed best fit (solid black line), determined as reported in the text in eq 7, for the sample QR2.3. The corresponding differential absorption value is also reported (right axes) and is evidenced the 1 < Nave < 2 region calculated from the Poisson distribution (see text). (b) Normalized transient absorption -∆R/R0 at X0 (black symbol) at short probe delay (2 ps), as a function of the initial average occupation. The red dashed line is the predicted dependence assuming a Poisson distribution of carrier population as explained in eq 6.

may be somewhat larger than what proposed in ref 15. Indeed, ab initio calculations of the electron wave function in ref 15, which include strain and charge redistribution effect and do not rely on envelope function parametrization, have shown that the lowest electron level is fully delocalized into the rod. We show in Figure 3 the X0 bleaching kinetics for the small QR2.3 at different pump intensities. At low excitation, when Nave < 1, the bleaching decays in the nanoseconds time scale, and at higher excitations, with 1 < Nave < 12, kinetics can be fitted by a first time constant of about 60 ps and a slower one, the same found for low excitations, in the nanoseconds range. At increasing excitation, the relative contribution of the fast component with respect to the slower increases. The bleaching at X0 saturates when Nave ∼ 12. At © 2010 American Chemical Society

∑

∆T T

|

X0

∼ 0.0028 · nH0 + 0.006 · nE1

(7)

and found reasonable fit as shown by the dashed lines in Figure 3. We remark that compared to the QR3.5 case, there is a stronger influence of the electronic levels, as expected from the fit of the differential absorption presented before. Again, the dependence on the higher electronic levels of the bleaching at X0 is embedded in a simple dependence on nE1 3147

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in our model of the dynamics, as for the QR3.5 case. Concerning the dynamics of X1, in this small dot case the X1 transition is spectrally close to Y0, as shown in Figure 1. Therefore, at large occupations, there is a significant spread of the Y0 transitions inside the X1 spectral region, due to an increase of the spectral linewidths of the Y0 transitions, and it was not possible to directly extract reliable data for the differential transmission related to the X1 transitions. The strong suppression of the Auger recombination for multiexcitons results into a relatively long lifetime of these states, to such an extent that we may expect to observe significant long-living gain at the X1, X2, and Y0 transitions. In the following, we consider the spectral features in the differential absorption. In Figure 4a, we report the normalized transient absorption spectra -∆R/R0 at short delay of 3.5 ps for different pump fluences for QR3.5. The data provide clear evidence of strong optical gain compared to absorption in several different wavelength regions related to X0, X1 and Y0. The line -∆R/R0 ) 1 represents gain threshold (transparency). Gain in the 620-630 nm region is clearly related to the X0 transition, although it occurs at much longer wavelengths. In this gain region, R0 is already very small. Therefore, even though large values of the normalized differential gain are observed, absolute gain is not large compared to linear absorption R0(X0). Moreover, even when Nave ) 1, we do observe gain in this spectral region. Clearly, these results cannot be simply explained within a two-level model, or an uncorrelated Fermi theory, which results in eq 1; at large occupations, normalized differential absorptions values of about 3 are observed in the experimental data, and this is more than simple inversion of any transition can explain. We conclude that scattering events (correlation) contribute to the formation of gain in this spectral region. We may intuitively understand appearance of gain below the band-edge as the result of broadening due to scattering of charge carriers. For example, a schematic view of such a process is given in Figure 4c. Here an electron in the lowest energy level E0 is scattered to slightly higher energies, while the other electron radiatively recombines with a hole in H0. The dashed line represents a virtual level with finite lifetime. These scattering events are already relevant at relatively low occupations, such as in the biexciton case, because of the small spacing of electronic levels in this structure. Experimentally these scattering events also result into a broadening of differential absorption at X0 with growing excitation. At larger fillings, optical gain from wavelength regions corresponding to the X1 transitions is observed. We call this gain multiexciton gain to distinguish it from biexciton gain. By further increasing the excitation, all the hole levels in the dot become saturated and evidence of optical gain from CdS develops in the Y0 spectral region. Normalized differential absorption for QR2.3 is reported in Figure 4b. Here again a strongly red-shifted gain peak above X0 develops already at small fillings above 1, and at large fillings gain in the X1 and Y0 regions is also observed. © 2010 American Chemical Society

FIGURE 4. Normalized transient absorption spectra -∆R/R0 for the sample QR3.5 (a) and QR2.3 (b), recorded 3.5 ps after excitation for different pump fluences. Inset: Normalized transient absorption spectra -∆R/R0 for different probe delays at high pump excitation for sample QR3.5 (a) and sample QR2.3 (b). (c) Sketch of stimulated emission due to e-e scattering.

However, differently with the QR3.5 case, another peak in the X0 spectral region is found at large fillings in the normalized differential absorption. In particular, this peak develops at the same fillings where gain in the X1 and Y0 regions is observed, and it is slightly blue shifted with respect 3148

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smaller. Of course, the use of an envelope function at these large wavevectors is rather meaningless and gives only a qualitative understanding of the phenomena, which at these energy scales should be analyzed at the atomic level. Here, we do not have any indication on the abruptness of the core-shell interface potential for holes at this atomic scale. However, given that the Auger recombination in the biexciton case is reduced by several orders of magnitude compared to the bare dot case and that such a large reduction of the overlap of electron and hole wave functions is unexpected, we may argue that smoothing of this potential by an unknown process may indeed occur. Further studies are required to reach an understanding of this point. We remark that even though the Coulomb interaction among charged carriers is at the origin of both Auger nonradiative recombination and carrier scattering, and thus presumably the appearance of gain at long wavelengths as yet said, these two processes have very different dependence on the carrier wave functions. In particular, while large momentum is exchanged in the Auger process as explained above, a small momentum is exchanged in the scattering processes that lead to broadening of the optical transitions. Therefore, it is well possible that carrier scattering remains strong even in conditions where the Auger nonradiative recombination is weak or suppressed. In conclusion, we have shown with a careful estimate of the occupation of the NC that the nonradiative recombination rate for a pair of excitations is negligible compared to the radiative recombination time, both for small and larger cores. Moreover, the nonradiative recombination rate at larger fillings is also suppressed, compared to spherical dots of similar size. We have shown that the dynamics of bleaching at the main transitions may be explained within a simple model in which we assume a relatively dense structure of levels for the electrons, which may be due to their delocalization along the rod. We also argue that the appearance of redshifted gain at relatively low fillings, already in the biexciton regime, is related to such a structure of levels. Indeed, the lifetime of this gain is very long and comparable to the recombination lifetime.

to X0. As this peak saturates to a value of 2, it may be simply related to the filling of the higher lying electronic states that contribute to the X0 transition, as explained before. Interestingly, we do not find such a feature for QR3.5, as it may also be partially hidden by the structure at X1, which is spectrally close. We therefore understand that gain does not simply originate from an unstructured broadening of levels, but it is rather related to some specific underlying dynamical process, such as the electron-electron scattering suggested above. Clearly, there is a need for a comprehensive understanding of the origins of gain in these structures, which is beyond the scope of this work. In the insets of Figure 4a,b, we report the normalized differential absorption spectra at increasing delays for QR3.5 and QR2.3 respectively. Noticeably, at longer wavelengths the gain remains positive for more than 400 ps. Instead, optical gain in the X1 and Y0 regions, resulting from multiexciton populations, remains positive for about 80 ps for X1 and 10 ps for Y0 for QR3.5 and for about 10 ps both for X1 and Y0 for QR2.3. These lifetimes are comparable to the fast kinetics decay times found in differential transmission at X1 and Y0. As explained before, these fast kinetics corresponds to the decay of multiexciton populations. Instead, the long gain lifetime close to the X0 transition is clearly related to the long lifetime of the X0 bleaching in differential transmission, even though we remark that this is not obvious, as it occurs in a different spectral region, substantially red shifted with respect to the absorption peak. Compared to quantum dots,19-21 these experimental data show a strong reduction of the Auger recombination even in the multiexciton regime. The gain lifetime of about 80 ps for multiexciton gain is still relatively long when compared to typical gain lifetimes found in dots, and this reduction is also certainly related to the nearly complete suppression of nonradiative Auger recombination in the biexciton regime which we reported above. The significant reduction or even suppression of the nonradiative Auger recombination, which would otherwise reduce the average occupation in the NCs below unity within a short time, is certainly related to the peculiar structure of these core/shell NCs. In particular, delocalization of the electron wave function along the rod reduces the Coulomb interaction between electrons and holes, contributing to a reduction of the Auger scattering integrals through a reduced overlap of electron and hole wave functions. In ref 22, it has been shown that the magnitude of the Auger nonradiative recombination is much influenced by the extension of the hole wave function in k-space, and in particular to its size at large momenta, corresponding to an energy difference of the order of the band gap (i.e., around 2 eV). It was also shown that the value of the wave function in this largewavevector region is mainly dependent on the abruptness of the confinement potential. For example, for a simple box of size L with infinite walls, the wave functions at large momenta k scale as 1/(kL), whereas for a parabolic potential, they scale as exp(-(kL)2), and they are thus exponentially © 2010 American Chemical Society

Acknowledgment. L. Manna acknowledges financial support from the European Union, through the FP7 program (ERC Starting Grant NANO-ARCH, Contract Number 240111). We acknowledge Luigi Carbone and Sasanka Deka for help with the synthesis of nanocrystals. REFERENCES AND NOTES (1) (2) (3) (4) (5)

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Peng, X.; Manna, L.; W.; Yang, a. J. W.; Scher, E.; Kadavanich, A.; Alivisatos, A. Nature 2000, 404, 59–61. Manna, L.; Scher, E.; Alivisatos, A. P. J. Am. Chem. Soc. 2000, 122, 12700. Manna, L.; Milliron, D. J.; Meisel, A.; Scher, E. C.; Alivisatos, A. P. Nat. Mater. 2003, 2, 382–385. Li, J. J.; Wang, Y. A.; Guo, W.; Keay, J. C.; Mishima, T. D.; Johnson, M. B.; Peng, X. J. J. Am. Chem. Soc. 2003, 125, 12567–12575. Choi, C.; Koski, K.; Sivasankar, S.; Alivisatos, A. Nano Lett. 2009, 9, 3544–3549. DOI: 10.1021/nl101930z | Nano Lett. 2010, 10, 3142-–3150

(6)

(7) (8) (9) (10) (11) (12) (13)

(14) Lupo, M. G.; Della Sala, F.; Carbone, L.; Zavelani-Rossi, M.; Fiore, A.; Luer, L.; Polli, D.; Cingolani, R.; Manna, L.; Lanzani, G. Nano Lett. 2008, 8, 4582–4584. (15) Luo, Y.; Wang, L.-W. ACS Nano 2010, 4, 91–98. (16) Deka, S.; Quarta, A.; Lupo, M. G.; Falqui, A.; Boninelli, S.; Giannini, C.; Morello, G.; Giorgi, M. D.; Lanzani, G.; Spinella, C.; Cingolani, R.; Pellegrino, T.; Manna, L. J. Am. Chem. Soc. 2009, 131, 2948– 2958. (17) Klimov, V. I. Annu. Rev. Phys. Chem. 2007, 58, 635–673. (18) Efros, A.; Rosen, M.; Kuno, M.; Nirmal, M.; Norris, D.; Bawendi, M. Phys. Rev. B 1996, 54, 4843. (19) Klimov, V. I.; Mikhailovsky, A. A.; McBranch, D. W.; Leatherdale, C. A.; Bawendi, M. G. Science 2000, 287, 1011–1013. (20) Htoon, H.; Hollingsworth, J. A.; Dickerson, R.; Klimov, V. I. Phys. Rev. Lett. 2003, 91, 227401-1–227401-4. (21) Htoon, H.; Hollingsworth, J. A.; Dickerson, R.; Klimov, V. I. Appl. Phys. Lett. 2003, 82, 4776–4778. (22) Cragg, G. E.; Efros, A. L. Nano Lett. 2010, 10, 313–317.

Chen, Y.; Vela, J.; Htoon, H.; Casson, J. L.; Werder, D. J.; Bussian, D. A.; Klimov, V. I.; Hollingsworth, J. A. J. Am. Chem. Soc. 2008, 130, 5026–5027. Garcia-Santamaria, F.; Chen, Y.; Vela, J.; Schaller, R. D.; Hollingsworth, J. A.; Klimov, V. I. Nano Lett. 2009, 9, 3482. Talapin, D. V.; Nelson, J. H.; Shevchenko, E. V.; Aloni, S.; Sadtler, B.; Alivisatos, A. P. Nano Lett. 2003, 3, 1677–1681. Carbone, L.; et al. Nano Lett. 2007, 7, 2942–2950. Talapin, D. V.; Nelson, J. H.; Shevchenko, E. V.; Aloni, S.; Sadtler, B.; Alivisatos, A. P. Nano Lett. 2007, 7, 2951–2959. Zavelani-Rossi, M.; Lupo, M. G.; Krahne, R.; Manna, L.; Lanzani, G. Nanoscale 2010, 2, 931-935. Sitt, A.; Della Sala, F.; Menagen, G.; Banin, U. Nano Lett. 2010, 9, 3470–3476. Saba, M.; Minniberger, S.; Quochi, F.; Roither, J.; Marceddu, M.; Gocalinska, A.; Kovalenko, M. V.; Talapin, D. V.; Heiss, W.; Mura, A.; Bongiovanni, G. Adv. Mater. 2009, 21, 4942–4944.

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