Supertransport of excitons in atomically thin organic semiconductors at the 2D quantum limit

Long-range and fast transport of coherent excitons is important for the development of high-speed excitonic circuits and quantum computing applications. However, most of these coherent excitons have only been observed in some low-dimensional semiconductors when coupled with cavities, as there are large inhomogeneous broadening and dephasing effects on the transport of excitons in their native states in materials. Here, by confining coherent excitons at the 2D quantum limit, we first observed molecular aggregation-enabled ‘supertransport’ of excitons in atomically thin two-dimensional (2D) organic semiconductors between coherent states, with a measured high effective exciton diffusion coefficient of ~346.9 cm2/s at room temperature. This value is one to several orders of magnitude higher than the values reported for other organic molecular aggregates and low-dimensional inorganic materials. Without coupling to any optical cavities, the monolayer pentacene sample, a very clean 2D quantum system (~1.2 nm thick) with high crystallinity (J-type aggregation) and minimal interfacial states, showed superradiant emission from Frenkel excitons, which was experimentally confirmed by the temperature-dependent photoluminescence (PL) emission, highly enhanced radiative decay rate, significantly narrowed PL peak width and strongly directional in-plane emission. The coherence in monolayer pentacene samples was observed to be delocalised over ~135 molecules, which is significantly larger than the values (a few molecules) observed for other organic thin films. In addition, the supertransport of excitons in monolayer pentacene samples showed highly anisotropic behaviour. Our results pave the way for the development of future high-speed excitonic circuits, fast OLEDs, and other optoelectronic devices.


Supplementary Information Note 2: Role of molecular packing and temperature on optical properties
The effect of molecular packing on excitonic states was firstly reported by Kasha et al. 2 , who extended the geometry-related FR exciton treatment by Davydov. 3 In our model, we define J to be excitonic coupling between different local FR exciton states located at nearest-neighbour molecules. The oscillator strength of an optical transition is a function of the total TDMs in an aggregate of molecules as shown below theoretically. If the Coulombic repulsion is stronger than the Coulombic attraction, their competition will favour a positive J value (J > 0) and vanishingly small total TDM, which will lead to an optically forbidden lowest excited state (S1) ( Figure S3b) 4 . In contrast, if the Coulombic attraction is stronger than the Coulombic repulsion, their competition will favour a negative J value (J < 0) and large total TDM, which will result in an optically allowed S1 ( Figure S3c). The result of this competition mainly depends on the orientation rather than the distance between molecules. 5 Molecular aggregates are usually considered to be H-type when J > 0 and J-type when J < 0. 5 Generally, side-by-side molecular orientations lead to J > 0, whereas head-to-tail orientations lead to J < 0.
Based on our simulation results, the total TDM and oscillator strength for the lowest excited states (S1) in WL were both zero, which led to non-luminescent S1 states at a temperature of absolute zero. In contrast, the total TDM for S1 in 1L had a large magnitude with the direction along the b axis of its unit cell, and its corresponding oscillator strength for S1 was also nonzero. This resulted in luminescent S1 states at absolute zero. The large magnitude of net TDM corresponds the phase-locking and constructive coupling of TDMs in 1L pentacene cooperatively resulting in sharply enhanced PL emission, which is defined as superradiance. This is supported by our temperature dependent PL measurements shown in Figure 2. At low temperature the emission from 1L is much stronger and sharper as compared to WL, which emission reduces as the temperature decreases to 77 K from room temperature. However, thermal disorder with increasing temperatures will cause fluctuations for both onsite energies and excitonic couplings and accordingly break the symmetry in the aggregation model. This will lead to thermal deactivation and activation for PL emissions in J-and H-type aggregations in 1L and WL pentacene, respectively. This results in decreased emission of PL in 1L at room temperature and an enhanced PL emission from WL.
For a transition from state | ⟩ to another state | ⟩, its transition dipole moment (TDM) is 6 where q(r) is the charge on the particle at an arbitrary position 'r'. TDM is not the oscillator strength (f), but it is the most crucial parameter for determining f 7 .
where, Eb and Ea are the energy levels associated with states 'b' and 'a' respectively.
In short, the magnitude of TDM determines the strength of the optical transition to a large extent. Additionally, its α (= x, y, z or a, b, c) component determines the incident light polarized absorption (or transition) along α direction.

Supplementary Information Note 3: Coherence number determination and effect of vibration and phononic interaction
The high crystallinity obtained in 1L pentacene samples will result in large coherence lengths and high diffusion coefficients but the effect of vibrations or phonon interaction in the lattice cannot be ignored. The phonon or vibrational coupling affects the PL spectra which is characterized by a spectral phononic sideband emission in addition to the sharp 0-0 Frank Condon emission, which is at 680 nm in our case (See Main Text Fig 2). As suggested by Zhao et al. 8 and Tanaka et al. 9 , the coherence number will start to decrease as the vibrations increase and hence resulting in lower diffusion lengths and diffusion coefficients.
To quantify the phonon interaction we have followed the technique reported by Spano et al. 10 to precisely determine the coherence number (Nc) in 1L pentacene at 298 K and 77 K. In effect, this vibration serves as a probe for the coherence length. When the optical transition from the lowest exciton to the vibration less electronic ground state (the "0−0" transition) is symmetryallowed, as in a J-aggregate, the ratio of the 0−0 to 0−1 transition line strengths can be used to determine the coherence number. The coherence number can be very precisely number determined taking into account the spectral line strengths (integrated PL) of the coherent peak and the sideband emissions by the following equation: Here, the I0-0 is referred to the FR emission at 680 nm and I0-1 is the phonon sideband emission caused due to phonon or vibrations in the lattice. Nc is the coherence number and λ 2 is the Huang-Rhys factor in pentacene. At 298 K and 77 K the PL spectra can be fit by two Lorentzian peaks as shown in Figure S6a and S6b respectively.
The Huang-Rhys factor used for our calculation is taken to be 0.574 which has been theoretically determined for pentacene and taken from ref 11 . The extracted values of spectral strengths and 0-0 and 0-1 transitions and subsequent Nc has been tabulated as below. is an order of magnitude higher than previously reported values from thin film organic semiconductors and hence a high value of diffusion coefficient is expected from such a system operating with large coherent regimes to allow for excitons to transport quickly.

Supplementary Information Note 4: Excitonic character in WL and 1L governed by molecular packing
The excitonic nature of organic crystals plays an important role in determining their optoelectronic properties. 4,10,12 In organic semiconductors FR and CT excitons are most commonly observed. 13,14 (See Supplementary Information Note 1). As soon as the energy difference between CT and FR excitons becomes small, both excitons can interact and form new FR-CT mixed states. 14 Intermolecular coupling also influences the complex excitonic character in organic molecules. The mixed states are observed in most bulk and thin film organic semiconductors due to the lack of high crystallinity, scattering and the presence of disorders and interfacial states. [15][16][17] Precise and controlled growth of organic molecular crystals in atomically thin 2D states provides an ideal platform for isolating the smaller and tightly bound FR 18 excitons as they have closer packing of molecules within a unit cell. [18][19][20] In order to reduce the FR-CT mixing and to observe the aggregation-controlled coherence phase locking and subsequent superradiant emission, the crystallinity and molecular aggregation into H-(CT exciton dominated aggregation) and J-type (FR exciton dominated molecular aggregation) has to be controlled, as theoretically predicted by Spano et al. 10 In our case, the highly clean and sharp crystalline system allowed us to observe long-range coherent supertransport from FR excitons which are de-localized over hundreds of pentacene molecules, due to of the minimized defect, disorders and interfacial states in the monolayer regime. WL pentacene samples, which had H-type molecular aggregation, showed in-plane isotropic emissions that were dominated by CT excitons. In contrast, 1L pentacene samples, which had J-type molecular aggregation, showed one sharp and strongly anisotropic PL emission peak (See Supplementary Information Note 2 for theoretical simulation details). The WL region showed much stronger PL emissions than the 1L regions at room temperature ( Figure 1d). Compared with inorganic TMD 2D semiconductors, the WL shows a much broader PL spectra at room temperature, which is due to the strong couplings between FR and CT excitons in organic materials. 18,21 The broad PL spectrum is associated with various band energy levels formed in pentacene due to vibronic coupling between FR and CT states 10 . 1L on the other hand shows a single peak centred around 680 nm which is predominantly FR excitonic emission as explained by our polarization dependent measurements in Figure 2g and also explained below. We grew several samples of pentacene in various growth conditions, giving us different optical properties from 1L depending on growth conditions ( Figure S3). The poorly grown 1L samples also gave us broad high energy CT excitonic peaks in addition to the narrow We also performed PL measurements with an angle-resolved emission polarization ( Figure 2f) to confirm the anisotropic nature of superradiant FR excitonic emissions from 1L. In the experiment, the incident polarization angle was controlled by an angle-variable half-wave plate and was fixed, and the polarization angle of the emission (θ) was determined by using an anglevariable polarizer located in front of the detector. In polarization-dependent PL measurement at 77 K, we found that intensity of PL emission from 1L strongly depended on the emission polarization angle θ and shows a period of 90 degrees. On the contrary the emission from WL was completely anisotropic (Figure 2f). The strongly directional total TDM in 1L pentacene predicted by our simulation ( Figure S3a) was expected to lead to an in-plane anisotropic emission of those FR excitons. The observed strong anisotropy of PL emission from 1L pentacene further confirms the strong optical alignment of molecules in a specific orientation and agrees very well with the prediction from our simulations, which confirms that the superradiant emission from 1L pentacene observed in 1L and 2L pentacene at low temperature is dominated by FR excitons.

Supplementary Information Note 5: Coherent exciton transport and supertransport
The diffusion of FR excitons that are commonly found in organic semiconductors migrate much slower as compared to their Wannier counterparts in inorganic materials. This is because of lower dielectric constant in organic materials and results in lower diffusion lengths in organic materials. 22,23 The energy transfer is dominated by incoherent Förster resonant energy transfer mechanism (FRET) 24 , which involved hopping of charges from one chromophore to the other via dipole moment interactions. This mechanism is based on dipole-dipole electromagnetic interaction and occurs when emission spectrum of donor molecule has significant overlap with absorption of the acceptor molecule. It is a non-radiative transfer and relies on long-range Columbic coupling between the nearby molecules. 23 Due to this incoherent hopping, the diffusion of FR excitons is limited. Incoherent energy transfer is generally determined my intermolecular spacing, defect states, excitonic lifetime and range of columbic coupling also referred to as Förster radius. 22,25 All these factors lead to smaller diffusion of FR excitons inorganic molecules and molecular assemblies.
However, in organic molecules the inter-molecular coupling can become very strong if the crystallinity is strong. 26,27 If the coupling between organic molecules is strong, the FR excitons can be delocalized over multiple molecules forming a delocalized coherent state for the exciton, which eventually leads to superradiant emissions. 28 In this phenomenon interactions between transition dipoles of individual molecules allow coherent delocalization across multiple sites.
The corelated states lead to a macroscopic optical polarization proportional to the number of atoms or molecules comprising the coherent domain. 29,30 This leads to a net enhancement in the optical transitional dipole moment (TDM) value and sharp enhancement of the excitonic radiative decay rate of an ensemble of Nc independent emitters as compared to the radiation decay rate of a single emitter. 31,32 The same principle of coherent delocalized superradiant emission gives rise to an analogous phenomenon called cooperative energy transfer or supertransfer (ST). 33 The resulting enhanced oscillator strength from delocalization over large molecular assemblies can lead to large scale exciton transport. Here, the molecular assemblies

Supplementary Information Note 6: Relationship between crystalline order and diffusion length
As discussed above the charge or energy transfer mechanism in organic molecules requires hopping of excitation from one molecule to other (incoherent) or from one delocalized state to other (coherent). In either case, the distance between interacting molecules or states is critical.
This intermolecular distance is a function of degree of crystallinity in the molecule. 34 Hence, the diffusion of excitons in organic system is a direct function of crystallinity. Lunt et al. 35 and Sim et al. 34 recently experimentally demonstrated that exciton transport diffusion length is a monotonic function of the extent of crystalline order. The reduction in diffusion lengths with decreasing crystallinity is due to increase in non-radiative losses in highly disorders systems, as determined by fluorescence quantum yield measurements as a function of grain size in Perylenetetracarboxylic dianhydride (PTCDA). 36 We observed a similar trend during the growth optimization process of our pentacene samples.
The samples which used h-BN surfaces that were exposed to air for a long time or had uneven morphology (detected optically) gave us low crystalline growth, as characterized by AFM measurements. In contrast the samples grown on homogenous and unexposed h-BN substrates gave a much stronger crystallinity and smooth pentacene growth, confirmed by surface roughness measurements. All the samples were characterized by AFM right after their growth.
During the growth of pentacene samples in our case, we observed that the optical performance was also critically affected by the order of crystallinity in our growth as shown in Figure S5.
The samples that had lower crystallinity showed a PL spectrum similar to earlier reports. 37 The broad PL spectrum obtained even from 1L samples is due to vibronic mixing of CT and FR states (Samples 2-4 in Figure S5). After several rounds of optimization, we could precisely control the growth conditions and orientation of molecules in 1L (Sample 1 in Figure S5). There were no CT peaks reported from highly crystalline samples, demonstrating only a sharp FR excitonic emission at 680 nm. Due to reduced defects, lack of interfacial sites and high crystallinity in our samples we could observe higher diffusion coefficients even for short-lived FR excitonic emissions from 1L pentacene samples.

Supplementary Information Note 7: Dimensionality and oscillator strength
Apart from coherent delocalization and high order crystallinity, the exciton diffusion process in organic molecules is also a function of the oscillator strength of the excitons. 23,26,27,38 The oscillator strength in return is strongly influenced by the effective dimensionality of the solidstate system as theoretically demonstrated by Wu et al. 39 The oscillator strength can be highly enhanced by reducing the effective dimensionality of the system. Thus, it is critical for us to develop a true 2D system that can confine the excitons in 2D quantum limit to further enhance the diffusion of the FR excitons.
The quantum confinement effect plays a key role in determining the spatial diffusion of excitons. 27 For the same reason, several recent attempts to achieve higher diffusion coefficients with molecular aggregates have used quasi-1D structures like tubes 25 or wires 27 to enhance the diffusion using J-type aggregation. In our pentacene WL and 1L system, the excitons were spatially confined in a 2D space, which increases the possibility of directed excitonic energy transfer within the 2D plane with limited out of plane losses and hopping. Further, in 1L as we have established the excitons were found to be anisotropic and aligned along the 'b' axis of the unit cell of pentacene (Figure 2f in main text and associated text). This led to further confinement and transport if excitons are in a quasi-1D system, which eventually led to observation of ultra-high exciton diffusion coefficients in our samples.

Supplementary Information Note 8: Diffusion Length using direct CCD imaging
A robust method of determining exciton diffusion is in steady state which involves modelling of the spread of excitons in a 1-D model. In the limit of low exciton density, the number of excitons n (x, t) as a function of both time and position can be described from the diffusion equation.
where, D is the coefficient of diffusion, is the exciton lifetime measured separately, p is the probability of absorption and I(x) is the spatial profile of excitation. If we assume a Gaussian profile with a standard deviation of  (or full width half maximum-FWHM) , the solution will lead to the following equation: In our measurements in Figure 3 and 4, the profile measured is a steady state profile unlike in measurements in Figure S12 where is a function of time. This steady-state profile is modelled by numerically integrating the above equation over time. This modelled profile is the used to fit the experimental data from Figure 3a, b to extract the diffusion length or s. We have then also fit the excitation laser Gaussian profile using the same fitting function to extract its width.
Both emission and laser profiles were fitted using the 1D Gaussian distribution model and their FWHMs were extracted. The spatial extent of exciton diffusion (diffusion length LD) was extracted by taking the difference of these extracted widths from the fitting. showed 2D hopping transport. The photo-excited charges in the 1L layers had much higher inplane diffusion than those in WL, which led to lower PL efficiencies in 1L pentacene. For the 1L+WL hetero-structure (1L region), the photo-excited charges in WL underneath could be quickly transferred to 1L and diffused laterally in 1L before radiative emission occurred ( Figure   S2b). This led to quenching of the PL intensity. This type of ultrafast interlayer charge transfer phenomena has been observed for both inorganic 2D hetero-structures 40    The emission from bulk pentacene layers is a broad PL spectrum, which changes negligibly with lowering of the temperature. The sharp peak emission at 680 nm is missing from the bulk pentacene. The broad PL spectra can be attributed to the coupling of CT and FR states, arising from the disorders and interfacial states, which has been reported in literature previously as well. 10,12 The lifetime from the bulk pentacene peak at 600 nm emission is 1.241 ns at 298 K, which changes to 1.014 ns at 77 K. The linewidth of the spectra is one order of magnitude higher (>100 nm as compared to 8 nm from 1L pentacene) from bulk pentacene, suggesting a non-coherent emission. The polarization dependent PL measurements from 1L at 77 K. In experiment, the excitation polarization angle θ1 was fixed at 0° and the polarization angle of the emission (θ2) was determined by using an angle-variable polarizer located in front of the detector. We also changed θ1 and repeated the measurements. We observed that PL emission from 1L always shows the maximum PL intensity at θ2 = 0° and the minimum PL intensity at θ2 = 90°, regardless of the excitation polarization angle θ1 used 0° (grey), 45° (magenta) and 90° (blue). regime of PEN samples. The sharp PL emission at 680 nm is visible from 2L but coupled with high-energy CT emissions. The FWHM at 298 K from 2L is ~49 nm and at 77 K is ~30 nm, which is much broader as compared to 1L.