Abstract
Excitons are realizations of a correlated manyparticle wave function, specifically consisting of electrons and holes in an entangled state. Excitons occur widely in semiconductors and are dominant excitations in semiconducting organic and lowdimensional quantum materials. To efficiently harness the strong optical response and high tuneability of excitons in optoelectronics and in energytransformation processes, access to the full wavefunction of the entangled state is critical, but has so far not been feasible. Here, we show how timeresolved photoemission momentum microscopy can be used to gain access to the entangled wavefunction and to unravel the exciton’s multiorbital electron and hole contributions. For the prototypical organic semiconductor buckminsterfullerene (C_{60}), we exemplify the capabilities of exciton tomography and achieve unprecedented access to key properties of the entangled exciton state including localization, chargetransfer character, and ultrafast exciton formation and relaxation dynamics.
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Introduction
Optical excitations in semiconducting materials deposit energy that can, in the best case, be harnessed in optoelectronic and photovoltaic devices. This potential for energy harvesting holds true over an extremely wide range of semiconducting materials, extending from classical silicon to twodimensional transition metal dichalcogenides, perovskites and organic semiconductors^{1,2,3,4}. Hence, major experimental and theoretical research efforts strive to understand such optical excitations.
On the fundamental level, the primary response to the optical excitation is excitonic: Coulombcorrelated electronhole pairs are created. In the most simple picture, for an organic semiconductor, such an excitation can be understood by the simultaneous creation of an excess electron in the lowest unoccupied molecular orbital (LUMO) and an excess hole in the highest occupied molecular orbital (HOMO). A very fundamental manifestation of the correlated interaction between the electron and hole is the exciton binding energy, which can be observed in optical absorption spectroscopy from the appearance of an absorption feature at one exciton binding energy below the singleparticle band gap^{5}. Hence, the correlation of the manybody wavefunction serves to reduce the required energy to place a single electron into an excited state.
The concept of excitons does not stop at the lowest HOMOLUMO excitations, and it provides a natural description of higher excitations as well. This includes transitions that may be described by an electron in a higher conduction level or a hole in a lower valence level, or the buildup of trions and biexcitons that consist of three or four entangled charged particles^{6,7}. Usually, the exciton wavefunction ψ_{m} is described (in the TammDancoff approximation) by a superposition of multiple electronhole pairs^{8}:
Here ϕ_{v} and χ_{c} are the v^{th} valence and c^{th} conduction states of the groundstate system, respectively. The coefficients \({X}_{vc}^{(m)}\) can create an entangled state where the electron and hole coordinates (r_{e} and r_{h}) cannot be considered independently (Fig. 1). Access to this orbital picture of the excitonic wavefunction is highly valuable^{9}, because imaging of the full entangled state would give direct access to exciton properties such as localization and chargetransfer character. Indeed, this information is particularly critical in the case of organic semiconductors, where it is wellknown that such multiorbital correlated quasiparticles dominate the energy landscape^{10,11}. However, it must be emphasized that conventional optical spectroscopy methods, including absorption and fluorescence spectroscopy, only provide access to the exciton energy Ω_{m}, and do not provide any information about the multiorbital contributions (\({\phi }_{v}^{*}{\chi }_{c}\)) of the exciton (see Fig. 1a, b).
For singleparticle molecular orbitals in organic semiconductors, an imaging of the wavefunction is possible through photoemission orbital tomography^{12,13}. In the last years, this technique was increasingly used to study lightinduced dynamics in organic semiconductors^{14,15} and 2D quantum materials^{16,17,18}, which exemplified the tremendous capabilities of this technique when applied to excitonic states. However, the full potential of photoemission exciton tomography was only recently indicated in a theoretical study by Kern et al., and promises to unravel the entangled singleparticle orbital contributions and realspace properties of the excitons^{19}. In this article, we experimentally introduce photoemission exciton tomography (Fig. 1c) and use it for the first time to characterize the correlated excitonic electronhole state in an organic semiconductor. Specifically, we find that the detected photoelectron kinetic energy provides a sensitive probe of the hole component of the exciton, and furthermore that the photoelectron momentum map probes the spatial properties of the electron component.
Results and discussion
The exciton spectrum of C_{60}
In order to introduce the photoemission exciton tomography approach, we select (C_{60} − I_{h})[5,6]fullerene (C_{60}) as an ideal, widely used^{20,21,22} and prototypical example. In particular, C_{60} shows a series of optical absorption features in multilayer and other aggregated structures^{23}, where different spectroscopy studies have indicated that these optical transitions correspond to the formation of excitons with differing chargetransfer character^{24,25,26,27,28}. Although these indirect results are supported by timedependent density functional theory calculations^{29,30,31}, quantitative access to the multiorbital wavefunction contributions (see Eq. (1) and Fig. 1) has so far not been feasible. Thus, the C_{60} organic semiconductor is an ideal platform to showcase the capabilities of photoemission exciton tomography.
In theory, we obtain the exciton spectrum by employing the manybody framework of GW and BetheSalpeterEquation (GW+BSE) calculations on top of a hybridfunctional density functional theory (DFT) ground state calculation (Fig. 2a, see Methods)^{8,32}. We find that the C_{60} crystal (Fig. 2a) can be accurately described using two symmetryinequivalent C_{60} dimers (see Supplementary Information, Supplementary Figs. S1, S4 and S5 for an experimental determination of the crystal structure and a convergence analysis of the dimer model). The calculated singleparticle energy levels are shown in Fig. 2b, where we group the electron removal and electron addition energies into four bands, denoted according to the parent orbitals of the gasphase C_{60} molecule. Building upon the GW singleparticle energies (Fig. 2b), we solve the BetheSalpeter equation and compute the energies Ω_{m} of all correlated electronhole pairs (excitons), which results in the absorption spectrum shown in the bottom panel of Fig. 2c. Furthermore, we obtain the weights \({X}_{vc}^{(m)}\) on the specific electronhole pairs that buildup the m^{th} exciton state by Eq. (1). This provides a full description of the multiorbital entangled excitons, and thereby contains the full spatial properties of each exciton.
To gain more insight into the character of the excitons ψ_{m}, we qualitatively classify them according to the most dominant orbital contributions that are involved in the transitions. This is visualized in the four subpanels above the absorption spectrum in Fig. 2c. For a given exciton energy Ω_{m}, the black bars in each subpanel show the partial contribution \( {X}_{vc}^{(m)}{ }^{2}\) of characteristic electronhole transitions ϕ_{v}χ_{c} to a given exciton ψ_{m}. Looking at individual subpanels, we see that not only the excitons are commonly composed of multiple characteristic electronhole transitions, but also that single electronhole transitions can belong to different excitons ψ_{m} that have very different exciton energies Ω_{m}. For example, the blue panel in Fig. 2c shows the contributions of HOMO → LUMO (abbreviated H → L) transitions as a function of exciton energy Ω_{m}, and we see that these transitions contribute to excitons that are spread in energy over a scale of more than 1 eV (from Ω_{m} ≈ 1.7 eV–3 eV). This spread of H → L contributions (and also H − n → L+m contributions) is caused by the fact that there are already many orbital energies per dimer (see Fig. 2b) which combine to form excitons with different degrees of localization and delocalization of the electrons and holes on one or more molecules.
We now focus on four exciton bands of the C_{60} film, denoted as S_{1}  S_{4}, which are centered around Ω_{S}_{1}, Ω_{S}_{2}, Ω_{S}_{3} and Ω_{S}_{4} at 1.9, 2.1, 2.8 and 3.6 eV, respectively. Notably, S_{2} and S_{3} were previously found to have chargetransfer character^{24,25}, while S_{4} stands out due to a fundamentally different wavefunction composition. It is important to emphasize that each exciton band S_{1}  S_{4} arises from many individual excitons ψ_{m} with similar exciton energies Ω_{m} within the exciton band. From Fig. 2c, we see that the S_{1} and S_{2} exciton bands are made up of excitons ψ_{m} that are almost exclusively composed of transitions from H → L. On the other hand, S_{3} shows in addition to H → L also weak contributions from H → L+1 transitions (pinkdashed panel). The S_{4} exciton band can be characterized as arising from H → L+1 (pinkdashed panel) and H − 1 → L (orangedashdotted panel) as well as transitions from the HOMO to several higher lying orbitals denoted as H → L+n (yellowdotted panel). Note that the inclusion of electronhole correlations has important consequences on the composition of the exciton wave function ψ_{m}^{33,34}. Specifically, S_{3} is not only composed of H → L transitions but it exhibits also an admixture of H → L+1 transitions despite the calculated ≈1 eV energy separation of quasiparticle LUMO and LUMO+1 levels.
Photoemission signature of multiorbital entangled excitons
In the following, we investigate whether these theoretically predicted multiorbital characteristics of the excitons can also be probed experimentally. Therefore, we take the exciton of Eq. (1) as the initial state and apply the common planewave final state approximation of photoemission orbital tomography^{19}. The photoemission intensity of the exciton ψ_{m} is formulated as
Here A is the vector potential of the incident light field, \({{{{{{{\mathcal{F}}}}}}}}\) the Fourier transform, k the photoelectron momentum, hν the probe photon energy, ε_{v} the v^{th} ionization potential, Ω_{m} the exciton energy, and E_{kin} the energy of the photoemitted electron. Note that ε_{v} directly indicates the finalstate energy of the leftbehind hole. In the context of our present study, delving into Eq. (2) leads to two striking consequences that allow us to disentangle the electron and hole contributions of the exciton, which we discuss in the following.
First, we will discuss the importance of the hole contribution based on the consequences of the multiorbital entangled character for the photoelectron spectrum for the four different exciton bands in C_{60}. In Fig. 2d, we sketch the typical singleparticle energy level diagrams for the HOMO and LUMO states and indicate the contributing orbitals to the twoparticle exciton state by blue holes and red electrons in these states, respectively. For the S_{1} exciton band (left panel), we already found that the main orbital contributions to the band are of H → L character (Fig. 2d, left, and see Fig. 2c, blue panel). To determine the kinetic energy of the photoelectrons originating from the exciton, we have to consider the correlated nature of the electronhole pair. The energy conservation expressed by the delta function in Eq. (2) (see also refs. ^{35,36,37}) requires that the kinetic energy of the photoelectron depends on the ionization energy of the involved HOMO hole state ε_{v} = ε_{H} and the correlated electronhole pair energy Ω ≈ Ω_{S}_{1}. Therefore, we expect to measure a single photoelectron peak, as shown in the lower part of the left panel of Fig. 2d. In the case of the S_{2} exciton the situation is similar, since the main orbital contributions are also of H → L character. However, since the S_{2} exciton band has a higher energy Ω_{S}_{2}, the photoelectron peak is also located at a higher kinetic energy with respect to the S_{1} peak.
In the case of the S_{3} exciton band, we find that in contrast to the S_{1} and S_{2} excitons not only H → L, but also H → L+1 transitions contribute (Fig. 2d, middle panel, and see Fig. 2c, blue and pinkdashed panels, respectively). However, we still expect a single peak in the photoemission, because the same hole states are involved for both transitions (i.e., same ε_{v} = ε_{H} in the sum in Eq. (2)), and all orbital contributions have the same exciton energy ≈Ω_{S}_{3}, even though transitions with electrons in energetically very different singleparticle LUMO and LUMO+1 states contribute. With other words, and somewhat counterintuitively, the singleparticle energies of the electron orbitals (the LUMOs) contributing to the exciton do not enter the energy conservation term in Eq. (2), and thus do not affect the kinetic energy observed in the experiment.
Finally, for the S_{4} exciton band at Ω_{S}_{4} = 3.6 eV, we find three major contributions (Fig. 2d, right panel), where not only the electrons but also the holes are distributed over two energetically different levels, namely the HOMO (see pinkdashed and yellowdotted panels in Fig. 2c, d) and the HOMO1 (see orangedashdotted panels in Fig. 2c, d). Thus, there are two different final states available for the hole, each with a different binding energy. Consequently, the photoemission spectrum of S_{4} is expected to exhibit a doublepeak structure with intensity appearing ≈3.6 eV above the HOMO kinetic energy E_{H}, and ≈3.6 eV above the HOMO1 kinetic energy E_{H−1}, as illustrated in the rightmost panel of Fig. 2d. Relating this specifically to the singleparticle picture of our GW calculations, the two peaks are predicted to have a separation of ε_{H−1} – ε_{H} = (8.1 – 6.7) eV = 1.4 eV. In summary, the photoelectron energy distribution provides access to the multiorbital character of the exciton, because different hole states that contribute to the exciton induce a multipeak structure in the spectrum.
The second consequence of Eq. (2) concerns the electron contribution, which modifies the photoemission momentum distribution. In analogy to conventional photoemission orbital tomography, Eq. (2) provides the theoretical framework for interpreting time and momentumresolved data from excitons. Ground state momentum maps can be easily understood in terms of the Fourier transform \({{{{{{{\mathcal{F}}}}}}}}\) of singleparticle orbitals^{12}. A naive extension to excitons might imply an incoherent, weighted sum of all conduction orbitals χ_{c} contributing to the exciton wavefunction. However, as Eq. (2) shows, such a simple picture proves insufficient. Instead, the momentum pattern of the exciton wavefunction is related to a coherent superposition of the electron orbitals χ_{c} weighted by the electronhole coupling coefficients \({X}_{vc}^{(m)}\). The implications of this finding are sketched in the k_{x}k_{y} plots in Fig. 2d and are most obvious for the S_{3} band. Here, the exciton is composed of transitions with a common hole position, i.e., H → L and H → L+1, leading to a coherent superposition of all 12 electron orbitals from the LUMO and LUMO+1 in the momentum distribution. In summary, multiple hole contributions can be identified in a multipeak structure in the photoemission spectrum, and multiple electron contributions will result in a coherent sum of the electron orbitals that can be identified in the corresponding energymomentum patterns of timeresolved data.
Disentangling multiorbital contributions experimentally
These very strong predictions about multipeaked photoemission spectra due to multiorbital entangled excitons can be directly verified in an experiment on C_{60} by comparing spectra for resonant excitation of either the S_{3} or the S_{4} excitons (see Fig. 2). We employ our recently developed setup for photoelectron momentum microscopy^{38,39} and use ultrashort laser pulses to optically excite the S_{3} and the S_{4} bright excitons in C_{60} thin films that were deposited on Cu(111) (measurement temperature T ≈ 80 K, ppolarized excitation; see Methods). The corresponding timeresolved photoelectron spectra of the electrons that were initially part of the bound electronhole pairs are shown in Fig. 3a and c, respectively. Starting from the excitation of the S_{3} exciton band with hν = 2.9 eV photon energy (which is sufficiently resonant to excite the manifold of exciton states that make up the S_{3} band around Ω_{S}_{3} = 2.8 eV), we can clearly identify the direct excitation (at 0 fs delay) of the exciton S_{3} feature at an energy of E ≈ 2.8 eV above the kinetic energy E_{H} of the HOMO level. Shortly after the excitation, additional photoemission intensity builds up at E − E_{H} ≈ 2.0 eV and ≈ 1.7 eV, which is known to be caused by relaxation to the S_{2} and S_{1} dark exciton states^{25} and is in good agreement with the theoretically predicted energies of E − E_{H} ≈ 2.1 eV and ≈ 1.9 eV (see Fig. 2c, blue panel).
Changing now the pump photon energy to hν = 3.6 eV for direct excitation of the S_{4} exciton band (Fig. 3c), two distinct peaks at ≈3.6 eV above the HOMO and ≈3.6 eV above the HOMO1 are expected from theory. While photoemission intensity at E − E_{H} ≈ 3.6 eV above the HOMO level is readily visible in Fig. 3c, the second feature at 3.6 eV above the HOMO1 is expected at E − E_{H} ≈ 2.2 eV above the HOMO level (corresponding to E − E_{H−1} ≈ 3.6 eV) and thus almost degenerate with the aforementioned S_{2} dark exciton band at about E − E_{H} ≈ 2.0 eV, which appears after the optical excitation due to relaxation processes. We note that a fast relaxation to the band of exciton states between 3.0 and 3.5 eV (see Fig. 2c) is also possible and may contribute to the observed signal. However, these excitons also have contributions from the HOMO and HOMO1 and are thus also predicted to lead to two distinct peaks at somewhat lower ≈3.4 eV above the HOMO and HOMO1, thereby also confirming our expectations from theory. In any case, we therefore need to pinpoint the second H − 1 → L lower energy contribution from either the S_{4} or the exciton band around 3.0–3.5 eV, and we do this by analyzing our data at the earliest time of excitation, i.e. before relaxation to the S_{2} dark exciton band occurs, which is degenerate at this photoelectron energy of ≈2.2 eV. Indeed, we find that a closer look around 0 fs delay shows additional photoemission intensity at this particular energy. Using a difference map (Fig. 3b) and direct comparisons of energydistributioncurves at selected timesteps (Fig. 3d), we clearly find a doublepeak structure corresponding to the energy difference of ≈1.4 eV of the HOMO and HOMO1 levels. Thereby, we have shown that photoelectron spectroscopy, in contrast to other techniques (e.g., absorption spectroscopy), is indeed able to disentangle different orbital contributions of the excitons. In this way, we have validated the theoretically predicted multipeak structure of the multiorbital exciton state that is implied by Eq. (2). We also see that the photoelectron energies in the spectrum turn out to be sensitive probes of the corresponding hole contributions of the correlated exciton states.
We note that the signature of the S_{3} excitons, even if not directly excited with the light pulse in this measurement, is still visible and moreover with significantly higher intensity than the multiorbital signals of the resonantly excited S_{4} exciton band. The explanation for this effect is twofold: first, the timeresolved signature suggests a very fast relaxation of the S_{4} excitons to the S_{3}, with relaxation times well below 50 fs (see Supplementary Fig. S3). Second, a calculation following Eq. (2) predicts an about threefold reduced photoemission matrix element for the S_{4} compared to the S_{3} band, explaining the overall weaker signal.
Timeresolved photoemission exciton tomography
In the full timeresolved photoemission experiment, by following the timeevolution of all electrons that were initially part of bound electronhole pairs, one can observe how the opticallyexcited states relax to energetically lowerlying dark exciton states^{17,18,24,25,35,40,41} (Supplementary Fig. S3) with the expected different localization and chargetransfer character. Importantly, the photoemission momentum microscope collects full time and momentumresolved data (i.e., 4D data set with time, energy and 2D momentum resolution), which we now show is ideal to access the spatial properties of the entangled multiorbital contributions.
We once again excite the S_{3} exciton band in the C_{60} film with hν = 2.9 eV pump energy, and collect the momentum fingerprints of the directly excited S_{3} excitons around 0 fs and the subsequently builtup dark S_{2} and S_{1} excitons that appear in the exciton relaxation cascade in the C_{60} film (see Fig. 4a, where the momentum maps of the lowest energy S_{1} exciton band, the S_{2} and the highest energy S_{3} exciton band are plotted from left to right; see Supplementary Fig. S3 for timeresolved traces of the exciton formation and relaxation dynamics). We note that the collection of the S_{1}, S_{2}, and S_{3} momentum maps already required integration times of up to 70 hours and a summation of the data over all measured timesteps from 200 fs–15 ps (see Methods), so that a measurement of the comparatively lowintensity S_{4} feature when excited with hν = 3.6 eV has not yet proved feasible. For the interpretation of the collected momentum maps from the S_{1}, S_{2}, and S_{3} excitons, we also calculate the expected momentum fingerprints for the wavefunctions obtained from the GW+BSE calculation for both dimers, each rotated to all occurring orientations in the crystal. Finally, for the theoretical momentum maps, we sum up the photoelectron intensities of each electronhole transition in an energy range of 200 meV centered on the exciton band. The results are shown in Fig. 4c below the experimental data for direct comparison.
First, recapitulating from Fig. 2c that the S_{1} and the S_{2} exciton bands are both only comprised of H → L transitions, we expect nearly identical momentum maps (but at different energies). Indeed, the experimental S_{1} and S_{2} momentum maps are largely similar (Fig. 4a), showing six lobes centered at k_{∥} ≈ 1.25 Å^{−1}. These sixlobe features, as well as the energy splitting between S_{1} and S_{2} (see Fig. 2c), are accurately reproduced by the GW+BSE prediction (Fig. 4b). Furthermore, the GW+BSE calculation also shows identical momentum maps for S_{1} and S_{2} confirming that the result of the coherent sum over all H → L transitions (Eq. (2)) is similar, and also indicates that the S_{1} and the S_{2} exhibit a similar spatial structure of the exciton wavefunction. This is in contrast to a naive application of static photoemission orbital tomography to the unoccupied orbitals of the DFT ground state of C_{60}, which does indicate a similar momentum map for the LUMO, but cannot explain a kinetic energy difference in the photoemission signal, nor give any indication of differences in the corresponding exciton wavefunctions. With this agreement between experiment and theory, we now extract the spatial properties of the GW+BSE exciton wavefunctions. To visualize the degree of chargetransfer of these twoparticle exciton wavefunctions ψ_{m}(r_{h}, r_{e}), we integrate the electron probability density over all possible hole positions r_{h}, considering only hole positions at one of the C_{60} molecules in the dimer. This effectively fixes the hole contribution to a particular C_{60} molecule (blue circles in Fig. 4c indicate the boundary of considered hole positions around one molecule, hole distribution not shown), and provides a probability density for the electronic part of the exciton wavefunction in the dimer, which we visualize by a yellow isosurface (see Fig. 4c). For the S_{1} and S_{2}, when the hole position is restricted to one molecule of the dimer, the electronic part of the exciton wavefunction is localized at the same molecule of the dimer. Our calculations thus suggest that the S_{1} and S_{2} excitons are of Frenkellike nature. Their energy difference originates from different excitation symmetries possible for the H → L transition (namely t_{1g}, t_{2g}, and g_{g} for the S_{1} and h_{g} for the S_{2})^{31}.
Compared to the S_{1} and S_{2} exciton bands, the S_{3} band is not only composed of H → L transitions, but also has a minor contribution of H → L+1 transitions (see Fig. 2c). We therefore expect that the S_{3} momentum map cannot be identical to the S_{1} and S_{2} momentum maps, but must show a signature of the H → L+1 contribution in addition to a possibly different coherent sum of all involved H → L transitions. Indeed, a closer look at the experimental data shows a more spokelike structure for S_{3}, which is marked with red arrows for three of the six spokes in the raw data as a guide to the eye (Fig. 4a, top right). An analysis over all different orientations, i.e. effectively symmetrizing the data, makes the spokestructure even better visible (Fig. 4a, bottom right, and Supplementary Fig. S6 for momentum lineouts). Hence, our experimental data clearly confirms the different character of the S_{3} exciton band in comparison to S_{1} and S_{2}. Looking at the theoretical data, we also find differences between the nearly identical S_{1} and S_{2} momentum maps (Fig. 4b) in comparison to the S_{3} momentum map. Once again, the differences are marked with red arrows as a guide to the eye (Fig. 4b, top right and bottom right, respectively; lineout analysis in Fig. S6). However, we also find that the experimentally observed spokelike pattern for S_{3} is different to the S_{3} momentum structure calculated using the dimer model. An indication towards the cause of this discrepancy is found by considering the electronhole separation of the excitons making up the S_{3} band. Here, we find that the positions of the electron and the hole contributions are strongly anticorrelated (Fig. 4c), with the electron confined to the neighboring molecule of the dimer. In fact, the mean electronhole separation is as large as 7.6 Å, which is close to the coretocore distance of the C_{60} molecules. Although these theoretical results confirm the previouslyreported chargetransfer nature of the S_{3} excitons^{24,25}, they also reflect the limitations of the C_{60} dimer approach. Indeed, the dimer represents the minimal model to account for an intermolecular exciton delocalization effect, but it cannot fully account for dispersion effects^{42} (see Supplementary Fig. S1), which are required for a quantitative comparison with experimental data. Besides the discrepancy in the S_{3} momentum map, this could also be an explanation why the S_{2} in the present work is of Frenkellike nature, but could have chargetransfer character according to previous studies^{24,25}. However, future developments will certainly allow scaling up of the cluster size in the calculation and to include periodic boundary conditions, so that exciton wavefunctions with larger electronhole separation can be accurately described. Most importantly, we find that the present dimer GW+BSE calculations are clearly suited to elucidate the multiorbital character of the excitons, which is an indispensable prerequisite for the correct interpretation of time and momentumresolved data of excitons in organic semiconductors.
In conclusion, we introduced photoemission exciton tomography to unravel the multiorbital electron and hole contributions of entangled excitonic states. In a case study on C_{60}, we show how to connect time and angleresolved photoelectron spectroscopy data to the wavefunction of fullyinteracting exciton states. For the hole component of the exciton, the spectral position of the hole is reflected in the photoelectron kinetic energy distribution, leading to the appearance of multiple peaks in the photoelectron spectrum for a multiorbital exciton. At the same time, the momentum fingerprint provides access to the electron states that make up the exciton. Applying this analysis to the observed exciton photoelectron spectrum of C_{60}, we were able to access important key properties including different orbital contributions, the wavefunction localization, and the chargetransfer character. We anticipate that photoemission exciton tomography will contribute to the understanding of exciton dynamics and the harnessing of such particles not only in organic semiconductors, but in general to advanced optoelectronic and photovoltaic devices.
Methods
Femtosecond momentum microscopy of C_{60}/Cu(111)
We apply full multidimensional time and angleresolved photoelectron spectroscopy (trARPES) to a multilayer C_{60} crystal evaporated onto Cu(111), where the film thickness was such that no photoemission signature of the underlying Cu(111) could be observed in our experiment. We verified the sample quality by performing momentum microscopy of the occupied HOMO and HOMO1 states simultaneously to the measurement of the excited states (see Supplementary Fig. S1). Femtosecond exciton dynamics were induced using ≈100 fs, hν = 2.9 eV or ≈100 fs, hν = 3.6 eV laser pulses derived from the frequencydoubled output of a optical parametric amplifier. The exciton dynamics were probed using our custom photoemission momentum microscope with a 500 kHz ultrafast 26.5 eV extreme ultraviolet (EUV) light source^{39} that enables us to map the photoelectron momentum distribution over the full photoemission horizon in a kinetic energy range exceeding 6 eV and an overall time resolution of ≈ 100 fs (the EUV pulse length is about 20 fs). The pump fluence was set to 90(10) μJ/cm^{2} and 20(5) μJ/cm^{2} for the hν = 2.9 eV and the hν = 3.6 eV measurement, respectively. In both cases, we found that ppolarized light excites the material most efficiently, leading to an excitation density on the order of 1 excitation per 1000 molecules. To prevent the free rotation of C_{60} molecules, we cooled the sample down to ≈80 K^{43}. In addition to the resulting longrange periodic ordering of the C_{60} crystal, cooling was also observed to prevent lightinduced polymerization.
Momentum microscopy data preprocessing
In the momentum microscopy experiment, a balance has to be found between sufficiently low pump and probe light intensities to avoid spacecharge effects, but also having sufficient intensity for the optical excitation (pump) and reasonably short integration times (probe). For the present experiment, we estimate that the pump and probe pulses each induce less than 50 photoelectrons over the full (≈200 × 150 μm^{2}) footprint of the beam. A 40 μm diameter spatial selection aperture and a low threshold voltage then eliminate most of the lowenergy photoelectrons and pass less than 1 photoelectron per pulse to the timeofflight detector. For our settings, small spacecharge effects are present in the data, but these do not lead to strong distortions in the bandstructure data and can be easily corrected. Therefore, the first step in the data analysis was to subtract a spacechargeinduced delay and momentumdependent kinetic energy shift which affects the full data set. For this purpose, the central kinetic energy of the HOMO was determined for normalization. To avoid the influence of the C_{60} crystal band structure^{42}, we fitted a twodimensional Lorentzian^{44} and shifted the kinetic energy distribution accordingly, leading to the expected overall flat shape of the molecular orbitals in the ARPES data.
Although we use narrowband multilayer mirrors to select the 11th harmonic at hν = 26.5 eV from our laserbased highharmonic generation spectrum^{39}, we observe subtle replicas of the HOMO, HOMO1, and HOMO2 states in the unoccupied regime of the spectrum that are caused by photoemission from the 13th harmonic at at hν = 31.2 eV. To quantify these replica signals, we fitted the static reference spectrum above E − E_{H} = 1.2 eV (i.e., in the unoccupied regime of the spectrum) with three Gaussianshaped peaks for the HOMO replicas and an exponential function to account for residual photoemission intensity in the unoccupied regime that is caused in this spectral region by the much stronger direct 11^{th} harmonic onephotonphotoemission from the HOMO state. After carrying out this fitting routine, we are able to calculate clean 11^{th} harmonic spectra (static and timeresolved) via subtraction of the fitted 13^{th} harmonic HOMO replicas. Note that we only subtract the replica signals, but not the background signal that is caused by onephotonphotoemission with the 11^{th} harmonic from the HOMO state, because this background is timedependent^{24}, and needs to be explicitly considered in the fitting procedure. The data shown in Fig. 3 of the main text is processed in the way described above.
Fitting procedure for the timeresolved data
From replicafree trPES data for hν = 2.9 eV excitation, we determine the amplitude A_{i}, kinetic energy E_{i}, and bandwidth ΔE_{i} for the i^{th} exciton signature using a global fitting approach. In particular, we apply the model
Here, the last term is needed to account for the abovementioned delaydependent photoemission intensity that is caused by a transient renormalization of the HOMO state, as found in ref. ^{24}.
The fit results of this model applied to the hν = 2.9 eV excitation and momentumintegrated data are shown in Table 1, and Supplementary Fig. S2a for the timeresolved exciton dynamics.
For the measurement with hν = 3.6 eV excitation, we account for the S_{4} exciton band by extending the model in Eq. (3) with a set of Gaussian peaks with identical temporal evolution, given by
which follows the same notation as Eq. (3). Here, we set \({E}_{{S}_{4},{\rm upper}}\) to be close to 3.6 eV, and following the GW+BSE calculation we set \({E}_{{S}_{4},{\rm upper}}{E}_{{S}_{4},{\rm lower}}\) = 1.4 eV. Fitting this model to the momentumintegrated hν = 3.6 eV excitation data, we find \({E}_{{S}_{4},{\rm upper}}\) = 3.59(1) eV, and for the FWHM of the S_{4} we find 0.58(3) eV. The timeresolved amplitudes retrieved using this model are shown in Supplementary Fig. S2b. Furthermore, this analysis was used in Fig. 3 of the main text to subtract the exponential background \({A}_{bg}(t) \exp \left[E/\tau \right]\) related to the transient broadening of the HOMO state.
Fitting procedure for the time and momentumresolved data
In order to analyze the timeresolved data also momentumresolved and thereby retrieve the momentum patterns that are shown in Fig. 4 in the main text, we carry out the fitting routine separately for pixelresolved energydistribution curves in the momentum distribution (1 pixel corresponds to ≈0.02 Å^{2}). Despite our efforts to reduce space charge, a residual noise signal remains near the center of the photoemission horizon, leading to a small region where the low signaltobackground ratio does not allow a reliable fit. We therefore exclude this region in our analysis (gray areas in Fig. 4 in the main text). Also, for the momentumresolved data, the replica HOMO background signals due to the 13^{th} harmonic amounts to 0, 1 or at most 2 counts in the pixelresolved (momentumresolved) energydistribution curves, and can therefore not be fitted and subtracted accurately as described above for the momentumintegrated data. As such, we need to ignore the HOMO replicas from the 13^{th} harmonic in the momentumresolved analysis. We avoid overfitting of the model in Eq. (3) by fixing the energy and bandwidth of the peaks in the fitting routine to the parameters given in Table 1. Thus, the set of free parameters in the momentumresolved fitting procedure is limited to A_{S3}(k), A_{S2}(k), A_{S1}(k) and A_{bg}(k). This approach enables the extraction of reliable momentum distributions also for the partially overlapping energy distributions of the S_{1} and S_{2}. The 1σ errors for the full momentum maps are shown in Supplementary Fig. S3.
We note that the overall photoemission intensity of the S_{4} peak in the hν = 3.6 eV excitation data is comparably low due to the sub50 fs decay to the lowerenergy S_{3} excitons (see Fig. 3 in main text and Supplementary Fig. S2b). Furthermore, with hν = 3.6 eV excitation, twophoton photoemission with 2 \(*\) 3.6 eV = 7.2 eV is sufficient to overcome the work function, so that spacecharge effects could only be avoided by considerably reducing the hν = 3.6 eV pump intensity. Therefore, the signaltonoise ratio in these measurements was not sufficient for a momentumresolved analysis of the S_{4} exciton data.
Calculation of the C_{60} exciton spectrum
The ab initio calculation of the exciton spectrum of the C_{60} film was performed in two steps, using a GW+BSE approach. For the static electronic structure, we perform calculations for two unique C_{60} dimers, which have been extracted from the known structure of the molecular film^{45} (see Fig. 2c, dimers 12 and 1–4 respectively). Starting from KohnSham orbitals and energies of a ground state DFT calculation (6311G*/PBE0+D3)^{46,47,48,49} using ORCA 5.0.1^{50,51}, we employ the Fiesta code^{52} to selfconsistently correct the molecular energy levels by quasiparticle selfenergy calculations with the GW approximation. To account for polarization effects beyond the molecular dimer, we embed the dimer cluster in a discrete polarizable model using the MESCal program^{53,54,55}. We found that mimicking 2 layers of the surrounding C_{60} film in such a way resulted in the convergence of the band gap within 0.1 eV with a removal energy from the highest valence level of 6.65 eV. The close agreement of this quasiparticle energy with the experimentally determined work function of 6.5 eV gives us additional confidence in the choice of our embedding environment. The calculated quasiparticle energy levels are shown in Fig. 2a. Here the finite width of the black bars actually arises from multiple energy levels forming bands on the energy axis. We characterize them according to symmetry^{56} as HOMO1, HOMO, LUMO, and LUMO+1 bands, each consisting of 18, 10, 6 and 6 energy levels per dimer, respectively. Note that the HOMO1 is made up by states from two different irreducible representations of the isolated gas phase C_{60} molecule which are practically forming a single band.
Building upon the GW energies, we compute neutral electronhole excitations by solving the BetheSalpeter equation beyond the TammDancoff approximation (TDA). This yields the excitation energies Ω_{m} and the electronhole coupling coefficients \({X}_{vc}^{(m)}\) for a series of excitons labeled with m. We first analyze the resulting optical absorption spectrum which is shown in Fig. 2c as black solid line. It reveals a prominent absorption band around hν = 3.6 eV that is wellknown from gasphase spectroscopy^{57}. Secondly, the dimer calculation reveals a strong optical absorption at hν = 2.8 eV as well as a weakly dipoleallowed transition at hν = 2 eV. Both of these transitions are known to only appear in aggregated phases of C_{60}^{23}, and cannot be understood by considering only a single C_{60} molecule. We refer to the exciton bands around Ω = 1.9, 2.1, 2.8 and 3.6 eV as S_{1}, S_{2}, S_{3} and S_{4}, respectively.
In line with our classification of the GW energy levels, one can group the composition of the excitons into four categories according to the contributing quasiparticle energy levels. As visualized in Fig. 2c, this shows that S_{1} and S_{2} are almost completely described by HOMO to LUMO transitions. On the other hand, S_{3} is predicted to have a small contribution of HOMO1 character, however this contribution is too small to be reliably measured in our experiment. Finally, for S_{4} we observe a clear and almost equal mixture of HOMO1 and HOMO contributions, which is also confirmed by our measurements. Contributions from lower lying valence bands are negligibly small in the studied energy window.
Calculation of the exciton momentum maps
Based on our KohnSham orbitals and BSE excitation coefficients, we calculate theoretical momentum maps for each exciton according to Eq. (2) following the derivation of Kern et al.^{19}. Note that for better readability, Eqs. (1) and (2) are given within the TDA, however, can in general be extended to include also deexcitation terms^{19}. In the present case, we found the deexcitation contributions to be marginal (below 1%) without affecting the appearance of the momentum maps and our interpretation. The energy conservation term comprises the BSE excitation energies (Ω_{m}), the GW quasiparticle energies for electron removal (i.e. ionization potential ε_{v}), and the probe energy of hν = 26.5 eV in accordance with our experimental setup. Furthermore, we include an inner potential to correct for the photoemission intensity variation of 3D molecules along the moment vector component perpendicular to the surface. Here, we choose a value of 12.5 eV, which has already been shown to match with experimental C_{60} data^{42,58}. (Note that while considering the inner potential of the film is essential to describe the ARPES fingerprint, a variation of the inner potential between 12 and 14 eV indicated no influence on the interpretation of our results).
As we exploited the planewave approximation, the calculated photoemission intensity is modulated by the momentumdependent polarization factor \({\left\vert {A} {k} \right\vert }^{2}\), which we modeled as ppolarized light incoming with 68^{∘} to the surface normal according to experiments. To account for the symmetry of the C_{60} film, the momentum maps were 3fold rotated and mirrored. Finally, application of Eq. (2) provides us with a 3D data set of simulated photoemission intensity as a function of the kinetic energy (E_{kin}) and the momentum components k_{x} and k_{y} for each individual exciton with excitation energy Ω_{m}. Analogous to experiment, we referenced the kinetic energy against the energy of the HOMO. The calculated ionization potential was further used to set the photoemission horizon of the theoretical momentum maps. Next, we estimate the population of the different excitonic states and sum up the corresponding 3D photoelectron intensity distributions. Here, we note that the experimental linewidth of the excitonic features is significantly broadened by various external factors, such as inhomogeneity in the sample and a finite energy resolution of the experiment. Therefore, we assume that the different excitonic states within each band are populated equally. For the S_{1}, this includes all excitons from 1.8–2.0 eV, for S_{2} the 2.0–2.2 eV range, and for S_{3} 2.7–3.0 eV. Finally, to arrive at the theoretical momentum maps shown in Fig. 4, we integrate the total signal in a wide kinetic energy range centered on the respective exciton band.
Data availability
The data sets that support the experimental findings of this study are available on GRO.data with the identifier https://doi.org/10.25625/Q7TCIS^{59}. The python codes to evaluate the theoretical data can be obtained from the authors upon request.
Change history
15 April 2024
A Correction to this paper has been published: https://doi.org/10.1038/s4146702447583z
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Acknowledgements
This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—432680300/SFB 1456, project B01 and 217133147/SFB 1073, projects B07 and B10. G.S.M.J. acknowledges financial support by the Alexander von Humboldt Foundation. A.W., C.S.K., and P.P acknowledge support from the Austrian Science Fund (FWF) project I 4145 and from the European Research Council (ERC) Synergy Grant, Project ID 101071259. The computational results presented were achieved using the Vienna Scientific Cluster (VSC) and the local highperformance resources of the University of Graz. R.H., M.A., and B.S. acknowledge financial support by the DFG  268565370/TRR 173, projects B05 and A02. B.S. acknowledges further support by the Dynamics and Topology Center funded by the State of RhinelandPalatinate. We acknowledge support by the Open Access Publication Funds of the University of Göttingen.
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D.St., M.R., S.S., M.A., B.S., P.P., G.S.M.J. and S.M. conceived the research. W.B., D.Sc. and J.P.B. carried out the timeresolved momentum microscopy experiments. W.B. analyzed the data. W.B. and R.H. prepared the samples. A.W., C.S.K., G.D, X.B. and P.P. performed the calculations and analyzed the theoretical results. All authors discussed the results. G.S.M.J. and S.M. were responsible for the overall project direction and wrote the manuscript with contributions from all coauthors.
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Bennecke, W., Windischbacher, A., Schmitt, D. et al. Disentangling the multiorbital contributions of excitons by photoemission exciton tomography. Nat Commun 15, 1804 (2024). https://doi.org/10.1038/s4146702445973x
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DOI: https://doi.org/10.1038/s4146702445973x
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