# Atomic fluctuations in electronic materials revealed by dephasing

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Edited by Catherine J. Murphy, University of Illinois at Urbana–Champaign, Urbana, IL, and approved April 1, 2020 (received for review September 26, 2019).

## Significance

Understanding how both static inhomogeneities and dynamic fluctuations impact the properties of electronic materials is a necessity in fields ranging from light harvesting to quantum information. We demonstrate the analysis of coherent spectroscopic dynamics is a uniquely powerful tool to study electronic fluctuations by performing multidimensional spectroscopy on the model system of CdSe quantum dot. According to standard continuum models, the energies of the exciton states should be strongly correlated through a common dependence on dot size. This hypothesis is contrary to our observation of fast dephasing between the lowest excited states. Using a model with explicit atoms, we demonstrate dephasing arises from atomic fluctuations intrinsic to the nanocrystal. We expect such fluctuations to be of general importance in nanostructures.

## Abstract

The microscopic origin and timescale of the fluctuations of the energies of electronic states has a significant impact on the properties of interest of electronic materials, with implication in fields ranging from photovoltaic devices to quantum information processing. Spectroscopic investigations of coherent dynamics provide a direct measurement of electronic fluctuations. Modern multidimensional spectroscopy techniques allow the mapping of coherent processes along multiple time or frequency axes and thus allow unprecedented discrimination between different sources of electronic dephasing. Exploiting modern abilities in coherence mapping in both amplitude and phase, we unravel dissipative processes of electronic coherences in the model system of CdSe quantum dots (QDs). The method allows the assignment of the nature of the observed coherence as vibrational or electronic. The expected coherence maps are obtained for the coherent longitudinal optical (LO) phonon, which serves as an internal standard and confirms the sensitivity of the technique. Fast dephasing is observed between the first two exciton states, despite their shared electron state and common environment. This result is contrary to predictions of the standard effective mass model for these materials, in which the exciton levels are strongly correlated through a common size dependence. In contrast, the experiment is in agreement with ab initio molecular dynamics of a single QD. Electronic dephasing in these materials is thus dominated by the realistic electronic structure arising from fluctuations at the atomic level rather than static size distribution. The analysis of electronic dephasing thereby uniquely enables the study of electronic fluctuations in complex materials.

- electronic coherence
- two-dimensional electronic spectroscopy
- quantum coherence
- coherence mapping
- ultrafast spectroscopy

Disorder, including both static inhomogeneities and dynamical fluctuations, impacts significantly the properties of electronic materials. It is an important aspect of material science as it is both a major factor in device fabrication and performance as well as a significant challenge for experimental and theoretical investigations. The reduction of grain heterogeneity and defect density in lead halide perovskite are promising avenues to further increase the performance of these materials in photovoltaic devices (1⇓⇓⇓–5). Local fluctuations of the polarization are also thought to prevent recombination in these materials (6, 7). Similarly, photoluminescence and electrical properties of two-dimensional (2D) transition metal dichalcogenides vary significantly across isolated flakes, depending on lateral position, defects, and local strain (8⇓⇓–11). The disordered nature of organic semiconductors plays a significant role in their conduction properties, with contributions from both static energetic disorder, morphology, and doping (12⇓⇓–15). The ideal behavior predicted by the Schottky–Mott limit was restored by the control of interfacial defects at a metal–semiconductor junction (16). The balance between competing phases in strongly correlated electronic materials, such as high-temperature superconductors and Mott insulators, is highly dependent on the presence and type of electronic disorder (17⇓⇓⇓–21). Inhomogeneities in the energy of electronic states also provide both challenges and opportunities for quantum applications such as quantum information-processing and superfluorescence (22, 23). Therefore, static and dynamic sources of disorder in the electronic properties of materials is a wide area of research with far-reaching implications.

Inhomogeneities in the energy of the electronic states often correlate with structural variations. Electronic materials have hierarchies of structures from micron to nanometer to atomic scale. Structural variations at the atomic scale are successfully probed with the help of electron microscopy and scanning probe techniques, such as scanning tunneling microscopy and atomic force microscopies (24⇓⇓⇓⇓–29). These very powerful techniques directly expose exclusive information about disorder at the atomic and nanoscale. However, most of these techniques reveal structural information and as such usually do not directly reveal inhomogeneities of the electronic properties. Furthermore, scanning probe and electron microscopy techniques are exclusively surface-sensitive and thus cannot probe bulk or dilute samples. The properties of samples buried even a few nanometers below the surface can only be probed using advanced nanotomography variants (30, 31).

Correlation spectroscopies reveal complementary information about the statistics of electronic properties in materials. These techniques measure the spectroscopic response of a material at multiple excitation or emission wavelengths (32⇓⇓–35). The resulting multidimensional signals directly reveal and quantify correlations in the response, which can be due to couplings, relaxation processes, or fluctuations of the energy of the electronic states. Two-dimensional electronic spectroscopy (2DE) is a coherent multidimensional spectroscopy method that measures the correlation between electronic excitations. Fluctuations of the energies of the electronically excited states is revealed by the rate of dephasing of the signals. The fluctuations of the energies of the electronic levels can be characterized by their amplitudes, their dynamics, and their correlations to the fluctuations of other levels. The rate of electronic dephasing is directly related to the amplitude of energy fluctuations: a wider energy distribution corresponds to faster dephasing. The dynamics of the fluctuations can range from femtoseconds (e.g., fast vibrational motion) to the static limit (e.g., nanoparticle size). Furthermore, the rate of dephasing between two states is indicative of the degree of correlation between these states. Specifically, an increased level of correlation between the energetic fluctuation of the states decreases the rate of dephasing (36⇓–38). The study of dephasing between excited states thus provides information on the fluctuations of the energies of the electronic excited states and thus enables the experimental discrimination between different microscopic sources of electronic fluctuations in materials.

Here we apply 2DE spectroscopy to study dissipation in the dynamics of electronic degrees of freedom and thus disentangle the sources of electronic energy fluctuations. We employ CdSe quantum dots (QDs) as a model system. This system is well understood and can be described using a hierarchy of theories, from continuum to atomistic models. We report 2DE spectra with many population times that enable coherence mapping. In coherence mapping, one extracts the 2D amplitude and phase of any coherent signal whether electronic or vibrational in origin. In contrast to 1D experiments, the use of 2D coherence maps in both amplitude and phase uniquely enables the assignment of the microscopic quantum mechanical contributions to the signals (39). The coherence maps confirm coherent optical phonons, which serves as an internal standard and confirms the sensitivity of the technique. The experimental signals show very fast dephasing between the first exciton states. This observation is contrary to the predictions of standard continuum theories, in which the distribution of QD size is the dominant contribution to the distribution of electronic energies. Under this model, the energy of the different exciton states should all be correlated through their dependence on QD size, thus enhancing the duration of an interexcitonic coherence. The observation of fast dephasing is rationalized using atomistic ab initio molecular dynamics where electronic fluctuations arise naturally from atomistic details, even in a single nanostructure. Results on this model system demonstrate the study of dephasing on femtosecond timescale can be used to discriminate between multiple sources of electronic fluctuations in complex electronic materials.

Semiconductor colloidal QDs are an ideal model system to demonstrate these ideas. QDs have a rich electronic structure that has been described at various levels of theory, from continuum to atomistic models (35, 40⇓–42). The prevailing model of QDs is called the multiband _{1}, X_{2}) consist of different hole states (X_{1}: 1S_{3/2}; X_{2}: 2S_{3/2}) but an identical electron state (1S_{e}). The electron state and both hole states are of S-type symmetry. This is illustrated in Fig. 1 *A* and *B*. The multiple exciton states yield a congested absorption spectrum that proves challenging to study. Despite its simplicity, the EMA model has successfully provided the framework for decades of spectroscopic investigations, from fluorescence to multidimensional spectroscopy (35, 43⇓–45). In particular, the size dependence of the exciton energies predicted by the EMA can be used to explain the dominant contribution to 2D lineshapes observed in 2D spectroscopy, a powerful and modern tool (43, 44). More refined theories are available and mature (35, 40⇓–42, 46, 47). Atomistic theories directly represent the atoms of the QD. The electronic properties of these structures are then obtained from computational methods such as density-functional theory (DFT). These calculations usually yield a complicated manifold of discrete states that arise due to thermally accessible fluctuations of the atomic positions (48, 49). Although mature and successful, these theories are more complicated than the EMA, and thus the latter is often preferred.

In the framework provided by the EMA, the largest contribution to the width of the distribution of the electronic energies in QDs arises from size distribution. A reduction of QD size yields an increase of the energy of all excitons with an almost linear relationship between the energy shifts (35). Accordingly, a deviation from the ensemble average of the energy of the X_{1} and X_{2} excitons should be strongly correlated. Due to their shared electron state, only hole dynamics contribute to dephase an

## Results

Coherent dynamics can be studied using 2D spectroscopy, which has been described in detail elsewhere (33, 34, 37, 56). A representative 2D spectrum is shown in Fig. 1*C* and the experimental scheme on Fig. 1*E*. Briefly, this experiment is a three-pulse experiment analogous to a transient absorption (TA) measurement where a pair of phase-stable broadband pulses acts as the pump. The first delay *D*. In the case of CdSe QDs, the shape of the 2D peaks is elongated along the diagonal, an observation that has been successfully explained by a distribution of exciton energies arising from size dispersion, as expected from the EMA (43, 45). Of particular interest, coherent processes are revealed by oscillations of the peak intensities as a function of

In 2DE, coherent oscillations can be reported as a function *A*) or *B*). The latter is not possible in other techniques. Following common procedures, incoherent dynamics are removed by global analysis using a multiexponential decaying model (59). The residuals show a strong contribution from the longitudinal optical (LO) phonon, with a period of ∼160 fs. The phase of the oscillations shifts as a function of both detection and excitation energy, consistent with frequency modulation arising from coupling to the LO phonon. The existence of a substantial phase shift implies that a technique that averages over a large excitation bandwidth will attenuate the relative amplitude of the oscillations (*SI Appendix*, Fig. S3). Pump bandwidth thus further complicates the comparison of TA experiments with varying bandwidths (54).

The residual oscillations can be Fourier-transformed to study directly their spectral components. This operation yields a 3D spectrum *C* shows the amplitude of the oscillations integrated over the region spanning 1.85 to 2.1 eV along both

A more detailed picture can be obtained by taking projections at selected values of *SI Appendix*, section 3).

The model coherence maps were obtained by considering systems simplified with respect to previous work by the Pullerits and coworkers (63, 65) (*SI Appendix*, section 4). Fig. 3 *A* and *B* shows the amplitude and phase of the coherence maps for an electronic coherence. The amplitude (Fig. 3*A*) reveals two symmetric features, with a relative phase shift of π (Fig. 3*B*). This structure arises from the interference of SE (top) and ESA (bottom). Results for the model vibrational coherence are shown in Fig. 3 *C* and *D*. It shows four features on the corners of a square, roughly separated by twice the selected value of

Fig. 3 *E* and *F* shows the experimental coherence maps obtained for

## Discussion

The strongest oscillatory component was assigned to the weak LO phonon, in agreement with the literature (60, 67⇓–69). Most intriguing is that no signatures can be assigned to electronic coherence. A more detailed look at the X_{1}, X_{2} cross-peaks is warranted. Fig. 4*A* shows transients obtained by integrating over square regions 40 meV across around the cross-peaks, which should reveal oscillations due to electronic coherences. The data can be fully accounted for by population decay and the LO phonon (solid lines). Multiple phenomena contribute to the incoherent decay. These phenomena have been discussed in our previous work and are recalled in *SI Appendix*, section 5 (70). Dephasing occurs within 20 fs for the

First, the expected properties of the *A*. Decorrelation of the energy of the states is described using lineshape functions. The lack of spectral dynamics in QDs justifies the use of Bloch dynamics (36⇓–38). In order to reduce the number of parameters, a few experimentally justified assumptions are made. The homogeneous dephasing of the electron and holes are assumed to be uncorrelated and of equal magnitude γ. We emphasize this estimate puts a lower bound on the dephasing time: a first-principles calculation is likely to result in correlation between the energy fluctuations of the two holes as they share a phonon bath (46, 52, 76). Inhomogeneous dephasing in QDs is dominated by size distribution. As a result, inhomogeneous broadening is correlated for the excitonic transitions. The energy of the

Using these assumptions, the lineshape of the two transitions can be modeled using three parameters: γ, r, and *SI Appendix*, section 6), and the results are convolved with the experimental instrument response function (full-width half maximum: 24 fs). The amplitude of the oscillating signal is a substantial part of the total cross-peak amplitude: the oscillating and nonoscillating contributions contain the same transition dipole moments (*SI Appendix*, section 5). The results are shown as a blue curve in Fig. 4*B*. Clearly, taking into account generous estimates for dephasing rates, the cross-peak should be strongly modulated by the interexcitonic coherences for 100 fs. Therefore, the EMA is insufficient to predict the experimentally observed fast dephasing between excited states in CdSe QDs.

The previous analysis ignored the impact of exciton–exciton interactions. Biexcitons in QDs give rise to an ESA feature on the 2D spectra. The red curve on Fig. 4*B* shows the impact of the ESA signal, which also gives rise to coherent signals. In semiconductor QDs, ESA and SE contributions overlap and cancel over most of the cross-peak (77). The net result shown in Fig. 4*B* is a phase shift of the oscillation, dictated by the biexciton binding energy, transition dipole moment and degeneracy, as well as the region of integration.

A higher level of theory including atomic details is required to explain the short interexcitonic dephasing in QDs. Despite its successes, the EMA has been criticized for its lack of microscopic details (78). Beyond the EMA, realistic microscopic models of small QDs are within reach of modern ab initio molecular dynamics (AIMD) (75, 79). Here, a 1.5-ps AIMD trajectory of a Cd_{33}Se_{33} model QD was carried out to evaluate the impact of the realistic electronic structure on dephasing (49, 75, 80, 81). The electronic transition energies were obtained using linear response time-dependent DFT. The linear and third-order optical responses were computed from the time-dependent transition energies using the cumulant expansion (36, 38). The results of the calculations for the cross-peak dynamics are shown as the solid purple curve in Fig. 4*C* (*SI Appendix*, sections 7 and 8). This calculation accounts for the realistic electronic structure as well as the dynamics of the nuclei-induced fluctuations. The AIMD calculations predict fast dephasing of the cross-peak, in agreement with the experimental observations. The AIMD calculations also enable us to evaluate the relative contributions of dynamical effects versus interexcitonic correlation effects. This is an important question, since it yields microscopic insights into the origin of interexcitonic dephasing. To address this point, we perform simplified calculations and take a detailed look at the statistics of the energy fluctuations.

The complexity and dynamical nature of the realistic electronic structure of semiconductor QDs can be witnessed by observing the energy of the exciton levels during the AIMD trajectory, as shown in Fig. 5*A*. The energy regions corresponding to the X_{1} and X_{2} spectroscopic bands are indicated by the shaded areas. The amplitude of the energy fluctuations (standard deviations [SDs] *B* alongside the mean values *C*. The atomistic calculations reveal that multiple states are present in each spectroscopic band. The X_{1} band contains two states, whereas the X_{2} band contains a varying number of states, usually three. The distributions of the energies of the different states shows significant overlap due to their large SDs. According to this model, the dynamics of the cross-peak thus result from the interference of at least six distinct coherences.

The fluctuations of the energies occur on a timescale of hundreds of femtoseconds, much slower than the observed dephasing time. The fluctuations of the transition energies are quantified using the time-dependent frequency–frequency correlation functions (FFCFs) [*D* shows an exemplary FFCF (*SI Appendix*, section 9). This calculation computes the dephasing dynamics in the limit of infinitely slow nuclear motions. This case is referred to as “static limit” in Figs. 4*C* and 5*D*. The resulting coherent dynamics are very similar to the results obtained using the time dependent FFCF, confirming the timescale of dephasing is not dictated by the timescale of the fluctuations but instead by their amplitudes.

Finally, coherences are sensitive to the degree of correlation between the fluctuations of the states. Strongly correlated states yield longer-lived coherences. The normalized correlation matrix [i.e., Pearson coefficient, *E*. The states contained in the X_{1} and X_{2} bands are indicated by horizontal and vertical lines, respectively. The states participating in the *SI Appendix*, section 9]. This case is labeled “correlated states” in Fig. 4*C*. The residual oscillations arise from the interference of the coherences between the states present in each band. In this case, however, the X_{1}, X_{2} cross-peak breaks up into a bundle of distinct subpeaks with long coherence times, contrary to the experiment. This analysis shows that, in contrast to the timescale of the fluctuations, the decorrelation between the fluctuations of the different electronic states thus plays a significant role in the dephasing of the

The previous analysis yields the following picture. The detailed positions of atoms break the symmetry of the system: a realistic QD is not a sphere. This results in a dense manifold of transitions (80). This manifold has an irregular, fluctuating structure and the many states have complicated relationships to one another; the realistic electronic structure is disordered. Each spectroscopic band contains a number of sublevels, each with its unique dynamics and a varying degree of correlation to the other states. The motions of the nuclei give rise to fluctuations of large amplitudes (larger than the energy separations) on timescales of hundreds of femtoseconds to picoseconds. These motions are thermally accessible and thus inherent to the nanocrystal. The rapid dephasing thus occurs even for a single QD in the absence of extrinsic defects such as extra atoms or surface impurities (81). One thus expects dephasing to be similarly fast in a hypothetical single dot experiment, as the nuclear motions randomize the energy levels during the experiment. AIMD studies of multiple chemical composition shows that these intrinsic fluctuations of the electronic energies are a general property of quantum-confined QDs (75, 80). Disorder induced by slow, thermally accessible motions of the atoms, termed atomistic fluctuations, is thus expected to be a dominant source of electronic dephasing in nanostructures.

CdSe QDs have long been a model system in which to study the impact of many-body physics on electron dynamics. Here, we have exploited dephasing to study the fluctuations and inhomogeneities in the electronic states of CdSe QDs. Coherence-mapping experiments with high time resolution and high sensitivity reveal fast dephasing between the excitons. The fast rate of dephasing is contrary to predictions of the standard EMA model, in which size distribution is the dominant source of electronic inhomogeneity. The fast dephasing was successfully modeled using time-domain ab initio calculations. Dephasing was shown to arise from the complicated electronic structure of a realistic QD where each bands consist of multiple substates and where the energy of the states is sensitive to details of the atomic positions. Specifically, dephasing arises from the large amplitudes of the nuclei-induced fluctuations as well as their partial lack of correlation. In semiconductor nanostructures, atomistic details present in single nanostructures are the dominant source of electronic fluctuations, instead of size dispersion. Application of coherence mapping on a model system of semiconductor QD shows that electronic dephasing may be used as a test for models of electronic disorder, whether static or dynamic. The study of electronic dephasing proves to be an even more powerful diagnostic of disorder than the analysis of 2D lineshapes (43, 53). This uniquely powerful technique should be generally applicable to discriminate between different sources of electronic energy fluctuations in complex materials.

## Materials and Methods

CdSe QDs in toluene were purchased from NNLabs. Their band-edge absorption was 640 nm with a Wurtzite lattice and octadecylamine ligands. The experiment was carried out in a 200-µm optical path length glass flow cell (Starna). The samples were constantly flowed during the experiment. Optical density of the samples was 0.3 in the 200-µm cuvette. Further details of the 2D spectrometer, the AIMD calculation, and the spectral modeling are available in *SI Appendix*. The resulting data are available online (58).

### Data Availability Statement.

The experimental data and results of the AIMD calculations are available at https://osf.io/8dpqr (58). The code used to obtain model spectra using the cumulant expansion is available at https://github.com/spalato/Mbo.jl (66).

### Note Added in Proof.

During the review of this manuscript, we learned of a recent review (82) on the original theater for electronic coherence—long-lived coherence in quantum biology. This review broadly discusses some of the same points discussed here: The exquisite delicateness of electronic coherence renders it a sensitive probe of manifold processes of disorder.

## Acknowledgments

This work was supported by the Canada Foundation for Innovation, Natural Sciences and Engineering Research Council of Canada (NSERC), and McGill University. H.S. acknowledges support from the Swiss National Science Foundation. S.P. acknowledges support from Fonds de Recherche du Québec–Nature et Technologies and NSERC. O.P. was supported by Department of Energy Grant DE-SC0014429.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: pat.kambhampati{at}mcgill.ca.

Author contributions: S.P., H.S., O.P., and P.K. designed research; S.P. and H.S. performed research; S.P., H.S., P.N., and O.P. contributed new reagents/analytic tools; S.P. and P.N. analyzed data; and S.P., O.P., and P.K. wrote the paper.

The authors declare no competing interest.

This article is a PNAS Direct Submission.

Data deposition: The data, along with explanatory notes and example scripts, have been made available in Open Science Framework at https://osf.io/8dpqr. The code used to perform model calculation is available in GitHub (https://github.com/spalato/Mbo.jl).

This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1916792117/-/DCSupplemental.

Published under the PNAS license.

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