1. Introduction

When a semiconductor colloidal quantum dot (QD) [1] is irradiated by photons that their energy is much smaller than the full bandgap energy but larger than its half, one-photon absorption is not possible and the dominant absorption process is the nonresonant two-photon absorption. It is a process of both fundamental and technological importance and, hence, it has previously been investigated in many QDs studies [2–17]. Possible novel applications include ultrafast optical switching for optical communications, data storage and optical limiting applications [2] as well as in-vivo biological imaging [3].

However, these past two-photon studies have been mostly limited to the single-excitonic regime, where no more than one exciton is generated in a single QD. So, one of the unique optical properties of QDs of highly efficient multiple exciton formation (MEF) within a single QD [18] has not been fully addressed in the context of two-photon absorption. In the corresponding MEF process each of the multiple excitons is individually generated via the non-resonant absorption of a pair of photons and, consequently, the whole set of multiple excitons is formed by the absorption of several pairs of photons during the pulse irradiation. For the sake of clear terminology, we emphasize that by MEF we do not refer to what is usually referred to as multiple exciton generation (MEG) [19], that is, the carrier multiplication (CM) [20] process in which the absorption of a single high-energy photon results in creation of several excitons.

Here, we investigate with direct multiexcitonic spectroscopy the corresponding MEF process generated via the nonresonant two-photon absorption. Most importantly, our study extends beyond the MEF regime that is described by Poisson statistics into a regime of non-Poissonian MEF. This is the main innovation of our work.

2. Experiment

The present experiments are conducted at room temperature with CdTe QDs having a mean diameter of 3.9 nm (2.06 eV bandgap energy corresponding to the 1S_e-1S_3/2,h state) and 5% size distribution. CdTe QDs are chosen as the model system for the study due to the applicable band gap for their nonresonant two-photon absorption, excellent stability, good quantum yield, very simple crystal structure, and low degeneracy of the exciton states. The QDs are prepared according to the synthesis procedure given in [21]. The linear optical properties of the QDs are discussed in detail in [22]. After the synthesis, the oleic acid/trioctylphosphine capped CdTe QDs are isolated as a clean powder of high-quality, with nearly spherical shape. Then they are dissolved again in n-hexane and placed into a thin quartz cuvette. Determination of the QDs size and its distribution is performed by the analysis of both the HR-TEM spectrographs and the static one-photon absorption measurements [21, 22].

The CdTe QDs are irradiated with a near-infrared linearly polarized femtosecond pump pulse having a pulse duration of 87 fs (Gaussian with a spectral bandwidth of Δω_FWHM = 19 meV) and a central wavelength of 812 nm (photon energy of 1.53 eV). The pump pulse energy E_pump ranges from 20 μJ to 115 μJ corresponding in our experiment to peak pulse intensity I_0,pump of 108 GW/cm² to 620 GW/cm², respectively. According to its intensity, the pulse generates multiple excitons within each QD via nonresonant two-photon absorption, with 〈N₀〉 as the corresponding average number of excitons per QD in the interaction region. To avoid heating, sample degradation, and multiple pulse effects, the pulse repetition rate is set to a low rate of 10 Hz and the QD sample is stirred during the experiments. The experimental values of 〈N₀〉 are obtained for a set of different pump pulse energies by the analysis of one-photon intraband transient absorption (intraTA) signals measured by a time-delayed weak near-infrared femtosecond probe pulse of 0.5 μJ energy. Except for the energy, the probe pulse is identical to the pump pulse. We have made sure that the probe pulse does not have sufficient intensity to excite the sample via two-photon absorption, so it can only induce intraband one-photon transitions within the conduction or valence bands following the pump pulse excitation. We attribute the present probe pulse absorption mostly to intraband hole excitation from the valence band-edge states, rather than to intraband electron excitation. This is in agreement with previous observations in similar systems [23], and as strongly supported by the consistency of our experimental and theoretical-model results presented below.

3. Experimental and Theoretical-Model Results and Discussion

The intraTA probe signal, Δα(τ), measured at a pump-probe delay time τ is proportional to the average number of excitons per QD, 〈N(τ)〉, at the time of probing. Hence, 〈N(τ)〉 = Δα(τ)/K_{sig_exc}, with K_{sig_exc} being the corresponding proportionality constant. This is true as long as our near-infrared one-photon intraband absorption cross-section per exciton is independent of the number of excitons residing in the QD. Since the one-photon intraband transition induced by the near-infrared probe pulse (photon energy of about 1.53 eV) excites the QD far above the band-edge into a region of very high density of hole states, this situation indeed holds in our case. Figure 1 presents several examples of the experimental results of 〈N(τ)〉 as a function of τ, up to τ =50 ps, for different pump pulse energies, together with the corresponding values obtained for 〈N₀〉. The results clearly show the increase in 〈N₀〉 with the increase in the pulse intensity, and the corresponding evolution from the single-excitonic [〈N₀〉=0.35 in Fig. 1(a)] to the multi-excitonic [〈N₀〉=3.80 in Fig. 1(f)] two-photon absorption regime.

 

Fig. 1 Examples of experimental results for the average number of excitons per QD as a function of the pump-probe delay measured with different pump pulse energies. Each panel indicates the corresponding pump pulse energy and the value obtained for the average number of excitons per QD generated by the femtosecond pump pulse via nonresonant two-photon absorption, 〈N₀〉. Note the different scale of the vertical axis in the different panels.

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The experimental value of K_{sig_exc} is obtained using the saturation behavior of the asymptotic probe signal, Δα_asymp=Δα (τ =45–50ps), that occurs with the increase in pump pulse energy. As explained below, Δα_asymp is measured at pump-probe delays when each excited QD contains a single exciton, which is left after the cascaded decay of the multiple excitons initially generated in the QD. Thus, Δα_asymp is a measure of the total number of excited QDs that increases with the pump-pulse energy, until it reaches saturation when all the QDs in the interaction region are excited. At saturation, Δα_{asymp_sat} corresponds to 〈N〉_asymp=1. Hence, one obtains that K_{sig_exc}=Δα_{asym_sat} and 〈N(τ)〉=Δα (τ)/Δα_{asymp_sat}. Such a saturation behavior of the asymptotic signal is indeed seen in our results of Fig. 1.

The decay behavior of the measured 〈N(τ)〉 traces reflects a non-radiative cascaded mul-tiexcitonic decay of the generated excitons that proceeds as long as there is more than one exciton in the QD. In each step, for m>1, m excitons in the QD decay into m−1 excitons. It is commonly interpreted to result from non-radiative Auger recombination of the charge carriers in the QD [24]. Initially, during approximately the first 2 ps after the excitation, the multiple (hot) excitons generated by the two-photon process undergo a fast intraband relaxation to their lowest multiexcitonic energy level. The end of this relaxation correlates well with the delays at which the measured traces have their highest values. Then, multiexcitonic decay is measured on a timescale of 35–40 ps, after which each excited QD is left with only a single exciton that decays radiatively on nanosecond timescale [22]. Overall, the measured traces of 〈N(τ)〉 exhibit a multi-exponential picosecond decay behavior, where the decay constants of the various components correspond to the various lifetimes of multi-excitons of a different order m. By analyzing the total sum over all the 〈N(τ)〉 data, we have extracted the bi-excitonic (m=2) and tri-excitonic (m=3) lifetimes as τ₂=12.2±2 ps and τ₃=4.5±1.5 ps, respectively. For each trace, the value of 〈N₀〉 and its error are obtained by extrapolating the 〈N(τ)〉 results to time zero. It is done by averaging over the results obtained by numerically extrapolating the measured data up to 10 ps using 2nd-, 3rd-, and 4th-order polynomials.

Based on the presented traces, we conclude that, within the experimental noise and error bars, the transient absorption results are not affected by surface trapping of electrons and/or holes. The signal effect of surface trapping of (cold) carriers residing at the bottom of their band can be ruled out due the time independence of the measured signal at larger delay times (25–50 ps) for low excitation regimes (see Figs. 1(a) and 1(b)). The signal effect of trapping of excited (hot) carriers can be ruled out due to the expected saturation behavior that is observed for the measured asymptotic signal as a function of the pump pulse energy.

Figure 2 presents the full set of the experimental values of 〈N₀〉 as a function of the pump pulse energy E_pump (circles) together with several analysis lines. One is a line of quadratic dependence on the pulse energy and intensity (black line) corresponding to ${〈N_0〉}_{\mathit{quad}\_\mathit{intensity}\_\mathit{dep}}={\tilde{K}}_{TPA}^{CdTe}\times E_{\mathit{pump}}^2=K_{TPA}^{CdTe}\times I_{0,\mathit{pump}}^2$, where I_0,pump is the peak intensity of the pump pulse with energy E_pump. The presented line corresponds to $K_{TPA}^{CdTe}=2.094\times {10}^{-5}$ cm⁴/W² that, based on our experimental parameters, corresponds to the theoretical value of the single-excitonic two-photon absorption cross-section, ${\sigma }_{TPA,\mathit{theo}}^{CdTe,3.9nm}=11500$ GM (Goeppert-Mayer), calculated in [11] for 3.9-nm CdTe QDs sample. This theoretical cross-section value is in excellent agreement with our experimental value of ${\sigma }_{TPA,\mathit{exp}}^{CdTe,3.9nm}=12222\pm 1633$ GM that is obtained from fitting our 〈N₀〉 results at the three lower pulse energies to a quadratic intensity dependence. As seen, the full set of experimental results of 〈N₀〉 significantly deviates from a quadratic dependence on I_0,pump and E_pump; the results fit this dependence only at the region of the lower 〈N₀〉 values. As we explain, the meaning of this observed non-quadratic intensity dependence is the non-Poissonian formation of multiple excitons via nonresonant two-photon absorption.

The instantaneous probability at time t for an intrapulse exciton generation in a given QD via two-photon absorption is p_TPA(t) ∝ s_TPA(t) × [I_pump(t)]², where s_TPA(t) is the instantaneous effective two-photon absorption cross-section and I_pump(t) = g(t) × I_0,pump is the instantaneous pump pulse intensity with g(t) as the pulse shape (here, 87-fs Gaussian). The total exciton generation probability is then obtained from integration over the full pulse as $P_{TPA}=\int p_{TPA}(t)dt\propto \int s_{TPA}(t){\left[g(t)\right]}^2{I}_{0,\mathit{pump}}^{2}dt$. Hence, for a given pulse shape, P_TPA exhibits a quadratic dependence on I_0,pump only if s_TPA(t) is time-independent with a constant value along the entire pulse irradiation. This constant value is by definition equal to the two-photon absorption cross-section for generating the first exciton in the QD, σ_TPA. When an ensemble of QDs is considered, since 〈N₀〉 ∝ P_TPA, the same conclusion also holds for the resulting values of 〈N₀〉. In the single-excitonic regime, s_TPA(t) = σ_TPA always holds since no more than one exciton is generated in a QD, so the corresponding 〈N₀〉 values always exhibit a quadratic intensity dependence [4–15]. However, in the multiexcitonic regime, the occurrence of s_TPA(t) = σ_TPA and quadratic intensity dependence happens only if the generation of a new exciton is unaffected by the presence of other excitons in the QD i.e., it is effectively independent of the number of excitons already existing in the QD. Physically, such independence among different events of exciton generation in the same QD also implies that the multiple excitons generated by a given pulse in a QD’s ensemble follow the Poisson distribution, with $P_{\mathit{Poisson}}(m)=\frac{{〈N_0〉}^m}{m!}\text{exp}(-〈N_0〉)$ being the probability of generating m excitons in a QD when 〈N₀〉 = ∑_mm × P_Poisson(m) is the corresponding mean. Hence, the non-quadratic intensity dependence of 〈N₀〉 measured here reflects multiexcitonic formation by femtosecond nonresonant two-photon absorption with a distribution that significantly deviates from the Poisson distribution. This is the main observation of this research.

 

Fig. 2 Results for the average number of excitons per QD generated by the femtosecond pump pulse, 〈N₀〉, as a function of the pump pulse energy. Shown are: (i) the experimental results (circles), (ii) a line of quadratic intensity dependence corresponding to Poissonian multiexcitonic formation (MEF) (black line), and (iii–iv) the results calculated for the full model of non-Poissonian MEF based on intrapulse electronic state filling (red line) and for this model without the inclusion of multiexcitonic transition shift (blue line). See text for details.

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In order to explain our observation, we present a model of non-Poissonian formation of multiple excitons via two-photon absorption (shown in Fig. 2 by red line). The results of the model are in excellent agreement with the measured values of 〈N₀〉. The main ingredient of the model is the phenomenon of intrapulse state filling of the (few) electronic states corresponding to the excitonic states that are excited by the nonresonant two-photon transitions. The state filling effect, which originates from the Pauli exclusion principle, has previously been observed experimentally for QDs only in one-photon excitation processes (for example, see [18]). It is only the electronic state filling that plays a role here (rather than the filling of the full excitonic states), since it is assumed that a generated hole undergoes here an immediate ultrafast intraband transfer out of its generation state into many neighboring states on a time scale that is much shorter than our pump pulse duration. This assumption originates from the high density of valence-band states existing around the excited states at which the holes are generated by the present two-photon transitions. Differently from the holes, an electron that is excited by the two-photon transitions keeps populating its (isolated) electronic level in the conduction band for the whole pump pulse duration [1]. The state-filling effect acts then as follows. After an exciton is generated in a final possible excitonic state, the electron stays in its conduction-band state, while the hole immediately evacuates its corresponding valence-band state that is refilled with electrons. As a result, due to the electronic state filling, the number of excitonic states available as final states for the next two-photon excitation in the QD is reduced and the corresponding effective cross-section for generating the next exciton gets smaller. For example, for the simple case of two-photon absorption involving a single possible final electronic state of degeneracy DG, the effective cross-section for generating the m^th exciton in the QD is $s_{TPA}^{\left(\mathit{exc}.\#m\right)}=\left[\frac{DG-\left(m-1\right)}{DG}\right]\times {\sigma }_{TPA}$, where σ_TPA is the single-excitonic two-photon absorption cross-section. Hence, for an ensemble of QDs, a given pulse generates a multiexcitonic distribution P(m) that is non-Poissonian [i.e., P(m) ≠ P_Poisson(m)] and its mean, 〈N₀〉 = ∑_mm × P(m), is smaller than the mean of the equivalent Poissonian case, where the entire multiexcitonic formation occurs with a constant cross-section equal to σ_TPA. The corresponding difference increases as the Poissonian value of 〈N₀〉 increases, or, equivalently, as the peak pulse intensity increases.

Another multiexcitonic effect that potentially might also lead to the attenuation of the effective two-photon absorption cross-section is the shift to lower energies that the excitonic transitions experience due to the exciton-exciton interactions in the QD [18, 25, 26]. At each point along the multiexcitonic formation process, such a shift might increase or decrease the (partial) cross-section associated with a given two-photon transition, since it changes the relative position and overlap of the corresponding absorption line and pump pulse spectrum. The corresponding changes in the various partial cross-sections and total cross-section depend, both in sign and magnitude, on the specifics of the QD system and pump pulse. For our case, the results show that the dominant phenomenon for explaining our observation of non-Poissonian multiexcitonic formation is the state-filling effect and not the transition-shifting effect. This is seen in Fig. 2, which presents our model results for the case of zero transition shift (blue line) and for the representative model case with a transition-shift-per-exciton of Δ_{per_exc} = −15 meV [25, 26] (red line). The latter means that with (m − 1) excitons in the QD, i.e., for the generation of the m^th exciton, all the two-photon transition energies are different by an amount of (m − 1)×Δ_{per_exc} from their values with zero excitons in the QD. As seen, the difference between the shifting and no-shifting cases is not significant for the deviation of our measured 〈N₀〉 values (circles) from the values of the Poissonian multiexcitonic formation (black line).

The CdTe information used in our model calculations is taken from the detailed theoretical study of Qu and Ji [11] conducted for the single-excitonic nonresonant two-photon absorption in CdTe QDs. Based on this study, the various single-excitonic quantum states that are resonantly accessed here via two-photon transitions by our pump pulse are 1S_e–2P_3/2,h, 1P_1/2,e– 2S_3/2,h and 1P_3/2,e–2S_3/2,h having electronic degeneracy of DG=2, 2 and 4, respectively. The state-specific information that depends on the QD diameter includes here the transition (excitation) energy and partial single-excitonic two-photon absorption cross-section ( ${\sigma }_{k,TPA}^{CdTe,Dnm}$) corresponding to each of these excitonic states (k). Summing up all the partial cross-sections gives the total single-excitonic two-photon absorption cross-section ${\sigma }_{TPA}^{CdTe,Dnm}={\sum }_k{\sigma }_{k,TPA}^{CdTe,Dnm}$. For 3.9-nm CdTe QD, the transition energies are 2.92, 2.98 and 3.00 eV, respectively. The corresponding total single-excitonic cross-section for the present excitation is ${\sigma }_{TPA}^{CdTe,3.9nm}=11500$ GM (as supported also by our experimental results; see above), with a ratio of 0.07 : 0.07 : 0.86 between the partial cross sections ${\sigma }_{k,TPA}^{CdTe,3.9nm}$ of the various excitonic states, respectively. It is worth mentioning that the cross-sections have been calculated in [11] by considering all the possible two-photon transition pathways to each of the final excitonic states, with their corresponding transition energies, dipole-moment matrix elements and intermediate-state detunings. In addition to all these theoretical characteristics, the homogeneous broadening of our room-temperature experiment is accounted for by applying a homogeneous linewidth of 0.1 eV [21, 22, 27] for each two-photon transition.

Based on the above model and information, the dynamics of the intrapulse multiexcitonic formation in a given CdTe QD of diameter D, which experiences a temporal pump pulse intensity I_pump(t), is numerically calculated over a time grid using the Monte Carlo method by following the time-dependent population of each of the electronic states corresponding to the different excitonic states accessed by the two-photon excitation. The probability p_k,TPA(t) of generating a new exciton at state k at the time interval between t and t+dt is determined by $s_{k,TPA}^{CdTe,Dnm}(t)\hspace{0.17em}\hspace{0.17em}{\left[I_{\mathit{pump}}(t)\right]}^2$, where $s_{k,TPA}^{CdTe,Dnm}(t)$ is the corresponding effective partial two-photon absorption cross-section at time t. Based on our model, it is given by $s_{k,TPA}^{CdTe,Dnm}(t)=\left[\frac{D{G}_k-m_k^{(e)}(t)}{D{G}_k}\right]\times T_k\left(m(t);{\mathrm{\Delta }}_{\mathit{per}\_\mathit{exc}}\right)\times {\sigma }_{k,TPA}^{CdTe,Dnm}$, where DG_k and $m_k^{(e)}(t)$ are, respectively, the degeneracy and instantaneous population of the electronic state corresponding to the excitonic state k. As defined above, ${\sigma }_{k,TPA}^{CdTe,Dnm}$ is the partial single-excitonic two-photon cross-section of state k. The factor T_k originates from the multiexcitonic change in the transition energy of state k that amounts to m(t) × Δ_{per_exc}, where m(t) is the total number of excitons in the QD at time $t\left(m(t)={\sum }_k{m}_k^{(e)}(t)\right)$ ; It is equal to the ratio between the values of the overlap integral of the pump pulse spectrum and absorption line of state k in the shifted-line case and unshifted-line case.

For a given pump pulse energy, such calculations are conducted for an array of CdTe QDs of a given diameter that are uniformly distributed over the radial spatial intensity profile of the pump laser beam, I_pump(t; r) = I_0,pump(r) × g(t) where r is the radial distance from the pump laser beam axis. After the pump pulse ending, the probed 〈N₀〉 value for this array is obtained by averaging over the final number of excitons in the different QDs of the array according to the spatial profile of the probe laser beam, i.e., the contribution from each QD is weighted by the relative intensity of the probe laser beam at the position of the QD. The final 〈N₀〉 value calculated for our experiment with a given pump pulse energy is obtained by applying a proper (weighted) averaging over the results calculated for several representative QD diameters within the size distribution of our sample. The model results in Fig. 2 present such calculated 〈N₀〉 values for different pump pulse energies. Their excellent quantitative agreement with the experimental results strongly supports the identification of the intrapulse electronic state filling as the key element behind the observed non-Poissonian multiexcitonic formation.

4. Conclusion

In conclusion, using direct multiexcitonic spectroscopy, we have experimentally observed for the first time the phenomenon of non-Poissonian formation of multiple excitons by femtosecond nonresonant two-photon absorption process in QDs. The non-Poissonian nature of the process is reflected as a non-quadratic dependence on the exciting pulse intensity of the average number of generated excitons per QD. The corresponding near-infrared pump-probe transient absorption experiments have been conducted with a room-temperature ensemble of 3.9-nm CdTe QDs. The experimental observation is explained by a multiexcitonic formation model that its main ingredient is the phenomenon of intrapulse state filling of the few quantum electronic states accessed by the two-photon transitions. The present study is not limited to a specific system and can be extended to other semiconductor nanocrystals and other excitation wavelengths for further fundamental studies and application development.