In our earlier letter, the phase of the MUPPETS signal was not considered.22 With the addition of the transient-grating data and the full model of the complex 1D signal from section 4.3, an analysis of the complex MUPPETS signal will be carried out in this section. The effects of cross relaxation will also be considered. Two issues are primary and are dealt with in two subsections. In subsection 4.1, the MUPPETS signal, including its time-dependent phase, is shown to be consistent with the 1D data. The fitting will yield the ratio of cross-sections for the exciton and biexciton transitions. In the earlier letter, the difference between the MUPPETS magnitude and the real pump–probe signal was presented as a measure of the biexciton decay.22 In subsection 4.2, the difference of complex 1D and 2D signals is taken to obtain the complex biexciton signal. Both topics require only the τ1 = 0 cut through the MUPPETS data. The full τ1–τ2 dependent data are discussed in section 4.5.
4.4.1 Measuring Exciton and Biexciton Cross-Sections.
A key result from the theoretical analysis of MUPPETS in excitonic systems is that exciton–exciton (Figure 4.2A) and exciton–biexciton (Figure 4.2B) pathways have
opposite signs.66 Because the biexciton decay is faster than the exciton decay, the total signal along τ2 should initially rise as the negative biexciton signal decays.
Experimental MUPPETS data with τ1 = 0 are shown in Figure 4.8. The raw, phase- cycled data are reduced to a complex signal (Figure 4.3). The results for different excitation fluences have been normalized according to eq 105 and are shown in Figure 4.8A–B. There is a strong fluence dependence to the shape of the signals, with the predicted rise of the signal seen only at the lowest fluences. This behavior will be explained in section 5.2 of chapter 5.79 For now, we focus on the low-fluence limit. Because the signal-to-noise ratio deteriorates as this limit is approached, it is particularly important to extrapolate to zero fluence. Results are shown in Figure 4.8C–D (red). The measured decay with the lowest fluence is similar to the extrapolated low-fluence signal
(2) 2 0 ( , 0)
A . This component has a delayed maximum in the magnitude, as expected, but it also has a time-dependent phase that must be explained.
Figure 4.8. MUPPETS τ1 = 0 results. Decomposition of the signal versus pulse energy I (left) into low-fluence and fluence-induced components (right). (A–B) Solid curves: The normalized absorbance A(2)( , 0; )2 I at various pump energies. Dots: Values reconstructed from the reduced results in (C–D). (C–D) The low-fluence A0(2)( , 0)2 (red) and fluence-induced A1(2)( , 0)2 (blue) components of the data in (A–B). Black: Fits to eq 114 and to eq II.168 with δ = 0.
Unlike 1D measurements, the biexciton contribution is intrinsic to MUPPETS, even in the low-fluence limit. Moreover in this limit, the ratio of these two contributions is fixed by the cross-sections of the chromophore. The theoretical expression for the complex, low-fluence MUPPETS absorbance with τ1 = 0 is
12 (2) 1 2 2 1 2 1 2 0 2 2 1 2 2 2 12 ( ,0) ( ) 1 ( ) 1 cos ( ) cos 1 e e i i i e A e C e C e C (114)The results of ref 66 have been adapted to the CdSe system defined in Figure 4.7. The exciton C1′(τ2) and biexciton C2′(τ2) contributions dominate the signal. In the simplest
case of σ01 = σ12 (δ = 1 and Φe = Φ12), the biexciton contribution is exactly one-half the exciton signal. However, the 1D spectroscopy has already indicated that these phases are unequal, so a more detailed approach is needed.
The τ1 = 0 cut of the MUPPETS data is almost entirely determined by quantities measured by 1D experiments. Most of the quantities in eq 114 have already been found in section 4.3: Φe, C1′(τ2), Φ12, and C2′(τ2). The only undetermined quantity is δ. A fit to
eq 114 with δ = 0 is shown in Figure 4.8C–D (red). The cross-relaxation C12( )2 is small, but has been included (see eq 120 below). This value correctly reproduces the size of the peak in the magnitude. It simultaneously reproduces the time dependence of the phase. The phases of the individual transitions are constant; the time dependence of the total signal is due to the changing ratio of the different pathways in Figure 4.2. The consistency of the 1D and MUPPETS results increases our confidence in both the theory and the data. Knowing that δ = 0, i.e., σ′01 = σ′12 (eq 110), the previously known values of Φ12 and Φe allow us to calculate that Φ01 = 59° (see Figure 4.7).
A zero value for δ is predicted by the uncorrelated-electron model, which has been widely used to interpret results in CdSe nanoparticles.43 However, Franceschetti and Zhang have suggested that electron correlation causes strong deviations in the cross- sections that can lead to misinterpretations of fluence-induced data.105 They calculated δ = 1/3 at 300 K. It should be noted that there is a small, but non-negligible, shift between the ground–to–exciton and exciton–to–biexciton transitions. Our pulses have a wide bandwidth (see Figure C1 in the Appendix C), but any failure to cover both transitions equally would introduce a systematic error in our measurement of δ, which is defined by spectrally integrated cross-sections. We have also neglected any simulated emission
from either the exciton or biexciton. Detailed calculations including stimulated emission (unpublished) show no new effects other than to perturb the effective cross-sections and, thus, to alter the measured value of δ. However, stimulated emission is known to be small and red-shifted from the band edge.49, 71 Even recognizing these limitations, our results do not support a strong electron-correlation effect on the cross-sections. Overall, the agreement between 1D measurements and MUPPETS at τ1 = 0 paves the way for analysis of the full MUPPETS data in sec 4.5.