Direct Visualization of Confinement and Many‐Body Correlation Effects in 2D Spectroscopy of Quantum Dots

The size tunable color of colloidal semiconductor quantum dots (QDs) is probably the most elegant illustration of the quantum confinement effect. As explained by the simple “particle‐in‐a‐box” model, the transition energies between the levels increase when the “box” becomes smaller. To investigate quantum confinement effects, typically a well‐defined narrow size distribution of the nanoparticles is needed. In this contribution, how coherent electronic two‐dimensional spectroscopy (2DES) can directly visualize the quantum size effect in a sample with broad size distribution of QDs is demonstrated. The method is based on two features of the 2DES – the ability to resolve inhomogeneous broadening and the capability to reveal correlations between the states. In QD samples, inhomogeneous spectral broadening is mainly caused by the size distribution and leads to elongated diagonal peaks of the spectra. Since the cross peaks correlate the energies of two states, they allow drawing conclusions about the size dependence of the corresponding states. It is also found that the biexciton binding energy changes between 3 and 8 meV with the QD size. Remarkably, the size dependence is non‐monotonic with a clear minimum.


Introduction
Semiconductor nanocrystals with a size, in at least one dimension, smaller than the exciton Bohr radius show quantum confinement-related properties.[3] The best-known quantum confinement effect is the DOI: 10.1002/adom.202302968size tunable color of colloidal QDs.6] The relation between the QD size and spectral features has been thoroughly studied via low-temperature fluorescence excitation spectroscopy. [7]The extracted electronic transition energies are well understood in terms of the effective-mass theory via combined electron and hole quantum states in a sphere, [8][9][10] see Figure 1.While the effective-mass theory well describes the main spectral features, the question about carrier relaxation dynamics in QDs remained.If only one excitation is present in a QD, the electron-hole pair excitation cannot scatter from charge carriers, and the main source for the excited state relaxation is the coupling between the longitudinal optical (LO) phonons and the electron-hole pair.However, the predicted gaps between the electron and hole energy levels are significantly larger than the LO phonons in the material.Since the transitions involving multiple phonons are very unlikely, a so-called phonon bottleneck, a slowdown of hot charge carrier relaxation, was predicted in QDs. [11]The lack of this effect was elegantly explained by the atomistic description of the QD electronic structure which showed that the electronic bands form quasi-continua with numerous states within the relevant spectral region providing ample means for one-phonon Figure 1.Illustration of the confinement effect on the valence and conduction band energies.The lower panel shows the nature of the three lowest exciton transitions for CdSe QDs following the effective-mass model. [42]eps in the relaxation. [12]At the same time, the sum of all possible states appears to follow well the main spectral features of the effective-mass theory.
The large distribution of the QD sizes leads to broad inhomogeneous spectral features with strongly overlapping bands. [13][15][16][17][18][19][20][21][22][23] This provides additional means to disentangle strongly congested spectral regions.In addition, 2DES allows to separate the homogeneous and inhomogeneous broadening. [24]Furthermore, since the method probes the third-order nonlinear response, the spectral features are narrower than what they are in linear spec-troscopy because all line-shapes are narrowed by any additional order.
[27][28][29][30] While initially mainly the fundamental nanoscience aspects were addressed, the emphasis has moved more and more toward finding new optoelectronic applications of these nanomaterials. [31,32]They have been widely studied using steady-state and time-resolved techniques to address and characterize their photophysical properties and map out the dynamics of the processes. [33,34]][37] Since 2DES allows disentangling homogenous and inhomogeneous broadening, the method allows investigating size-dependent optical properties of the QDs over a broad size distribution in a single experiment.
In an intense laser beam, an excited QD can absorb another photon, leading to a double excitation called a biexciton.Excitons are neutral and do not have any direct Coulomb coupling, however, due to the complex many-body (two electrons and holes in a biexciton) correlation effects, multiple excitons in a single QD interact with each other.The interaction can be repulsive or attractive.In the former case, the second excitation would absorb at higher energy than the first, while in the case of attractive coupling, the second excitation has lower energy.The amount of lowering is the biexciton binding energy ΔE. [38]The binding energy has also been obtained as the shift between the exciton and biexciton emission bands. [39]][42][43][44] Also, 2DES studies have been performed in the past years to understand and clarify the biexciton shift in several materials, [44][45][46] including CdSe 1−x S x /ZnS alloyed core/shell QDs. [38]n this article, we revisit the quantum confinement-related size dependence of the electronic transition energies in CdSe QDs by using 2DES at 77 K.The two-dimensional (2D) spectra of sizedistributed QDs show pronounced structure of elongated diagonal and cross-diagonal spectral bands.Those spectra provide an intuitive illustration of the size dependent quantum confinement effect on the electronic structure of QDs.Directly below the diagonal bleach-related band, we observe a pronounced negative ESA band.The energy gap between the bleach and ESA reveals the QD size dependent biexciton binding energy.

Materials and Methods
We use CdSe QDs capped with 3-mercaptopropionic acid with a broad size distribution -the diameter range is from 4 to 8 nm.For the specific synthesis details, please see previous publications. [37,47]The 2DES measurements are performed at 77 K.The CdSe QDs are dispersed in a 1:1 mixture of methanol and ethanol, forming a good glass for optical studies.The energy of each pulse is 1 nJ, a spot diameter of the incident pulses on the sample is 100 μm.The specific setup details have been reported in previous publications. [17,37]

2D Spectra and QD Size-Dependent Ground-State Absorption
Already very early size dependence studies of CdSe QDs revealed a gradual increase in the energy difference between the 1S 3/2 1S e , 2S 3/2 1S e , and 1P 3/2 1P e transitions when the QD size decreases. [7]hese three transitions are covered by the spectrum of our laser pulses and for brevity, in the following, we refer them as X 1 , X 2 , and X 3 , or just the corresponding numbers.We point out that the weak 1S 1/2 1S e transition also slightly contributes to X 3 .In the following section, the discussion is built on absorptive 2D maps ) and a negative peak (N).There are three other negative peaks in the 2D spectrum, which have been discussed in previous studies. [15,37].
recorded at population time T = 270 fs.2D maps at T = 0 fs and early population times have not been considered because of distortion effects given by pulse overlap and non-resonant signals, nor the 2D maps at late population time, where the 2D maps may have changed due to the relaxation processes.The map at 270 fs is a good compromise, but many other 2D maps with the same features at different population time could have been considered.The real part of the total 2D spectrum chosen at the population time T = 270 fs displays several elongated spectral features due to X 1 , X 2, and X 3 transitions, see Figure 2. The size distribution of the QDs leads to the inhomogeneous broadening that elongates the diagonal peak.The cross-peak elongation is the result of the correlation of the size dependencies of the related transitions.The length of the elongation is shortened due to the canceling by the ESA signal at higher energy.Moreover, the cross-peak CP 13 that is expected to appear symmetrical to CP 31 is also canceled by the ESA signal.
We quantify the size-dependent energy gaps between the excited states by the angle between the main diagonal and the respective elongated features.We analyzed the four positive bands highlighted in Figure 2. Fitting of the narrow diagonal peak is straightforward.However, the cross-peaks are less defined compared to the main diagonal feature and assigning the elongation direction is not as simple.Hence, we adopt the following "crosssearch" approach for the analysis.
As displayed in Figure 3, we separately find the maxima for each row (along the excitation axis, red dots) and column (along the detection axis, blue dots) of the data set corresponding to the above 2D spectral features.The points are fitted with a line providing the angle in respect to the main diagonal.The fit results are summarized in Table 1.For more specific details of the fitting, please see the Supporting Information.The fit of the diagonal band is tilted by a few degrees because of the influence of the negative ESA band N. The ESA band also partially overlaps with the CP 21 cross peak, which is therefore slightly less tilted with respect to the main diagonal than the CP 12 peak.If we exclude the points in the proximity of the N feature, the fit becomes very symmetric (see Supporting Information) as expected from theoretical considerations of the QD 2D spectra. [48,49]This confirms that the positive diagonal peaks of the X 1 , X 2, and X 3 transitions fall exactly on the diagonal.The cross-peaks emerge from the common ground state of the three involved transitions, also called state filling in semiconductor literature.
The QD sample under investigation has a wide size distribution.In order to understand how the distribution leads to the observed 2D spectra, we choose two QD sizes and derive the energy levels and the corresponding spectral features, see Figure 4. We first look at a relatively large QD with X 1 , X 2 , and X 3 transition energies of about 1.94, 2.00, and 2.08 eV, respectively.We mark all three diagonal and cross diagonal features with red circles and link them with red dotted lines to form a square.As the second example, we choose a smaller QD so that the X 1 transition energy is 2 eV, thereby coinciding with the X 2 of the first example.For the smaller QD, we also assign all possible diagonal and cross-  1.The red (orange) square represents a larger (smaller) CdSe QD, where each circle (cross) along the main diagonal represents X 1 , X 2 , and X 3 transitions, respectively.The off-diagonal circles (crosses) show the corresponding cross peaks.
diagonal features, which correspond to the three transitions using orange crosses and dotted lines.By interpolating between the corresponding features of the two different size QDs it is easy to understand how the elongated 2D spectral features are formed.Interestingly, the nondiagonal features CP 12 and CP 21 have contributions from both X 1 -X 2 (small QDs) and X 2 -X 3 (large QDs) cross peaks.Our argumentation leads to a perfectly symmetric picture.The slight asymmetry of the experimental spectrum has two origins.First of all, above the diagonal and in the higherenergy region, the negative ESA-related signal cancels out the positive features, some only partially but in some cases entirely.In addition, relaxation from higher to lower energy states leads to additional intensity to the positive features below the diagonal, particularly at longer population times. [15]The details of the transition energies used in Figure 4 are provided in the Supporting Information.

The Biexciton Binding Energy
Below the diagonal of the 2D spectrum at lower energies, there is a negative feature N (see Figure 2.) that is attributable to the biexciton ESA. [33,45,50]This is the lowest biexciton state X 1 X 1 .The negative signal is well explained by the double-sided Feynman diagram in Figure 5.The attractive biexciton binding energy ∆E XX shifts the ESA below the diagonal. [50]The lower left corner of the 2D spectrum is shown in the left panel of Figure 6 as a raster image without interpolation of the points.While the positive ground state bleach (GSB) follows the diagonal, the negative ESA band is not only shifted below but also slightly tilted with respect to the diagonal.The tilt means that ∆E XX depends on the QD size.
In order to quantify the QD size dependence of ∆E XX , we take cuts of the 2D spectrum at different excitation energies, as shown for the population time 270 fs in the left panel of Figure 6.Each slice can be seen as a transient absorption spectrum excited at a certain X 1 transition energy, which can be related to a size of the QDs. [45,51]As spectra, the slices consist of a positive diagonal GSB band and a negative ESA band.Because of a significant overlap of the bands, one should not simply take the positions of the positive and negative signals to determine ∆E XX .Instead, the bands need to be fitted with appropriate line shapes.We fit each slice with two Lorentzian functions.The positive (GSB) is fixed at the diagonal energy and the negative (ESA) is free.In addition, three Gaussian functions were added to represent the positive cross peaks CP 21 , CP 12 , and the negative ESA feature at high detection energies.The CP 31 has not been considered in the fitting procedure since the influence was negligible in determining the line shape of the DP and N features.For the details of fitting, see Supporting Information.
The energy difference between the maxima of the two Lorentzians along the detection axis, E GSB , and E ESA , was determined for each slice and taken as ∆E XX .In this way, the biexciton binding energy ∆E XX as a function of the excitation energy is obtained, see the right panel of Figure 6.At different population times, the fitting results differ and the binding energy for the same X 1 varies by about 2 meV.Reasons for such variation can be attributed to the relaxation processes and vibrational coherences that can influence the shape of the 2D spectra.At the same time, the shape of the size dependence remains largely the same.At all population times the obtained biexciton binding energy ∆E XX has a minimum at X 1 transition energy slightly above 1.98 eV.The increase of the binding energy if moving from 1.98 eV toward higher excitation energies (smaller QDs) is obvious.The excitation energy change of ≈40 meV corresponds to the binding energy change of ≈6 meV independent of the population time.Toward the lower excitation energies from 1.98 eV (large QDs), the increase is not as pronounced but clearly beyond the uncertainty of the analyses.Within an excitation change of ≈20 meV, the binding energy increases by ≈2 meV.
In bulk CdSe, excitons are reported to be weakly attractive with a biexciton binding energy of about 4 meV. [52]Also in CdSe QDs, attractive interaction of the excitons has been reported. [53]Coupling between excitons is a nontrivial 4-body problem depending on the overlaps of the wavefunctions of two electrons and two holes.In case of type-II core-shell structures where the electrons and holes localize to different parts of the QDs, electronelectron and hole-hole Coulomb interaction leads to repulsive exciton-exciton coupling. [54,55]However, in the perfectly symmetric case, the total coupling depends on a delicate interplay between positive and negative interaction terms and the resulting interaction is dominated by the correlation effects. [56]In type-I core-shell-shell CdSe/CdS/ZnS QDs, repulsive exciton-exciton coupling was found, which depended on the inner-shell (CdS) thickness. [39]In our study, we observe attractive biexciton binding with clearly non-monotonic size dependence-the coupling is rising both for smaller and larger QDs having a minimum at the QD diameter of ≈6 nm.Rising of the biexciton binding energy while reducing the CdSe QD size from 6 to 4 nm has been reported before. [53]An increase in any kind of interaction for stronger confinement (smaller size) is naturally expected.Take, for example, the dispersion force which is, perhaps, the most intuitive manybody correlation interaction.It is always attractive and rapidly increases for shorter distances.Interestingly, in reference, [53] the binding energy has a maximum at the QD diameter of ≈3.5 nm and starts to decrease if the QD size is further reduced.Also, in our analyses, at some population times the binding energy turns to lower values for the highest excitation energy (smallest QD size) as if the excitons start repelling each other if confined to too small volumes, see the right panel of Figure 6.However, the size of the QDs in our sample does not reach sufficiently small values to allow us to draw a reliable conclusion here.At the other end of the size distribution (> 6 nm), the increase of the binding energy is very clear in our results.[59][60] When moving toward larger QDs from here, the two excitons will overlap less, approaching the bulk-like conditions.When the overlap of the excitons is decreased, the electron-electron and hole-hole repulsion decreases.Of course, other interactions also change, but the decrease of the repulsion may well be what leads to the observed increase of the total biexciton binding when the QD size becomes larger than the exciton Bohr radius.

Conclusion
We have measured coherent electronic 2D spectra of CdSe QDs with broad size distribution at 77 K. Since 2D spectroscopy can separate inhomogeneous broadening due to the size distribution we can explore the size-dependent electronic transitions in a single experiment avoiding the need to produce multiple samples with varying sizes and conduct separate characterizations for each of them.The results are analyzed in terms of size-dependent confinement effects.We observe elongated 2D spectral features which are related to the QD size-dependent transition energies and correlations between the transitions.Based on the difference between the biexciton and exciton transition energies, we determine the biexciton binding energy and its QD size dependence.We confirm that the binding energy is attractive and that it increases for smaller QDs -an intuitively expected trend for any kind of interaction.For the larger QDs, however, we find that the behavior reverses and there is a slight increase in the binding energy even for the larger QDs that we studied.

Figure 2 .
Figure 2. The real part of the absorptive (or total) 2D spectrum of CdSe QDs (T = 270 fs).The peaks discussed in this work are labelled according to the excitonic transitions with which they are connected: positive diagonal peaks (DP), positive cross peaks (CP 21 , CP 31 , CP 12) and a negative peak (N).There are three other negative peaks in the 2D spectrum, which have been discussed in previous studies.[15,37].

Figure 3 .
Figure 3.The fitting analysis performed on the diagonal and cross peaks of the 2D spectrum at T = 270 fs.The red and blue dots track the maxima along rows and columns, respectively, around each positive peak in the 2D spectrum.For specific details, see the Supporting Information.The black lines are linear fits to the red and blue dots characterizing the elongated features in the 2D spectrum.

Figure 5 .
Figure 5. Left: The double-sided Feynman diagram of the ESA pathway for the N feature in the 2D spectrum.Right: Energy level diagram showing the ground state |0〉 the exciton |X〉, biexciton |XX〉, and two non-interacting excitons |X〉|X〉.

Figure 6 .
Figure 6.Left: lower energy part of the 2D spectrum.The vertical lines show the 1D cuts that are used to determine the biexciton shift.Right: attractive biexciton binding energy as a function of excitation energy (QD size).The error bars correspond to the standard deviation calculated over the binding energies obtained by the analysis at different population times, see Figure S4 (Supporting Information).

Table 1 .
Parameters of the linear fitting curves are shown in Figure3.