Long lived photogenerated charge carriers in few-layer transition metal dichalcogenides obtained from liquid phase exfoliation

Semiconducting transition metal dichalcogenides are important optoelectronic materials thanks to their intense light–matter interaction and wide selection of fabrication techniques, with potential applications in light harvesting and sensing. Crucially, these applications depend on the lifetimes and recombination dynamics of photogenerated charge carriers, which have primarily been studied in monolayers obtained from labour-intensive mechanical exfoliation or costly chemical vapour deposition. On the other hand, liquid phase exfoliation presents a high throughput and cost-effective method to produce dispersions of mono- and few-layer nanosheets. This approach allows for easy scalability and enables the subsequent processing and formation of macroscopic films directly from the liquid phase. Here, we use transient absorption spectroscopy and spatiotemporally resolved pump–probe microscopy to study the charge carrier dynamics in tiled nanosheet films of MoS2 and WS2 deposited from the liquid phase using an adaptation of the Langmuir–Schaefer technique. We find an efficient photogeneration of charge carriers with lifetimes of several nanoseconds, which we ascribe to stabilisation at nanosheet edges. These findings provide scope for photocatalytic and photodetector applications, where long-lived charge carriers are crucial, and suggest design strategies for photovoltaic devices.


Introduction
Light harvesting processes, such as photovoltaics and photocatalysis, convert the absorbed light energy into other forms, such as electrical or chemical energy.Likewise, light sensing relies on the generation of an electrical signal in response to light.For either of these processes to happen, light needs to be absorbed by one or more active materials, converting the photonic energy into electronic excitations of the absorber via the creation of electron-hole pairs (either as free carriers or bound in the form of excitons).To harness their energy or their encoded information, carriers must remain in an excited state (e.g.electrons in the conduction band and holes in the valence band of a semiconductor) for long enough to catalyse a chemical reaction or to be extracted at the electrodes of a photovoltaic or photodetector device.
2][3][4][5][6] The fabrication methods for TMDs can be classied into two main categories.The rst is bottomup synthesis, [7][8][9][10][11] which oen necessitates expensive equipment and is primarily employed to produce monolayer lms that cover a large area, ranging in the order of hundreds of microns.The second approach is top-down exfoliation, where mono-and few-layer nanosheets, typically with smaller lateral dimensions, are extracted from larger crystals or microcrystallites. 124][15][16][17] Furthermore, LPE offers scalability, making it Nanoscale Advances PAPER suitable for mass production of inks that can be used to print TMD lms onto various substrates.However, compared to monolayers, 18,19 in TMD nanosheets obtained using LPE the non-equilibrium dynamics of photogenerated excitons and charge carriers, which is the foundation of light harvesting and sensing, is much less comprehensively studied and understood.
Here, we study the room temperature dynamics of photogenerated charge carriers in lms of few-layer MoS 2 and WS 2 nanosheets obtained by LPE.To achieve the high lm quality required for optical spectroscopy we adapt the Langmuir-Schaefer deposition technique 20 for fabrication of tiled nanosheet lms on a liquid/liquid interface, which are then transferred onto fused silica substrates.For this purpose, inks comprising nanosheets with varying average sizes and thicknesses have been employed.We investigate the charge dynamics over a temporal window from 1 ns to 100 ns using broad band transient absorption (TA) spectroscopy and the charge diffusion from 300 fs to 13 ns using spatiotemporal pump-probe microscopy.Combining these two techniques, we cover a time scale spanning six orders of magnitudefrom hot carrier diffusion to the recombination of long-lived carriersand combine spectral selectivity with the ability to disentangle the decay and diffusion of excited species.We nd charge carrier lifetimes of several nanoseconds, which we attribute to stabilisation at the nanosheet edges.These lifetimes are orders of magnitudes longer than in TMDs obtained with other fabrication methods, [21][22][23] which is very promising for photocatalysis and photodetector applications and suggest design strategies for photovoltaic devices.

Results and discussion
We prepared dispersions of MoS 2 and WS 2 nanosheets from powders in aqueous solutions of the surfactant sodium cholate using sonication assisted LPE.From the resulting stock dispersion, we isolated three different fractions by liquid cascade centrifugation, 17,24 corresponding to centripetal accelerations of 1-5k g, 5-10k g and 10-30k g, as described in the Methods section.Each of these fractions, named small, medium, and large, contains a different distribution of nanosheet thicknesses (i.e.number of layers, N) and lateral sizes, L that are characterised using statistical atomic force microscopy 20 as depicted in Fig. 1A-F, S1, and S2 † and summarised in Table S1 (see ESI †).For both MoS 2 and WS 2 , the three fractions show different mean <N> and <L> values ranging from 2.2 to 4.4 layers and spanning from 35 to 97 nm, respectively.][28] The absorbance spectra of the three fractions in liquid dispersion, depicted in Fig. 1G and H, show the well-known A and B excitonic resonances. 29,30These excitons are associated with direct transitions between valence band maxima and conduction band minima at the K point, where spin-orbit interaction leads to a splitting in the valence band of a few 100 meV. 31The A-exciton peak energy, which is more precisely determined from the second derivative of the absorbance (Fig. 1G and H, inset), shows the expected blueshi in thinner nanosheets 24,25 (from 1.87 to 1.89 eV for MoS 2 , and from 1.98 to 2.03 eV for WS 2 fractions).While all N and L distributions show only one peak, the medium WS 2 fraction shows a double peak in the absorbance 2nd derivative, indicating two dominant N values.This highlights an important difference in characterising the dispersions with microscopy aer deposition or by their absorbance spectra in solution.The histograms in Fig. 1E, F, S1, and S2 † represent the number of nanosheets that fall into a certain size range.In the absorbance spectrum, the contribution of each nanosheet increases with increasing area (approximately L 2 ) and increasing N.As a consequence, the larger and thicker nanosheets are represented more prominently in the absorbance spectra.Hence, if the absorption peaks are sufficiently narrow, a longer tail of the N distribution towards larger values can yield a second maximum in the absorbance.
We prepared tiled nanosheet lms from the three fractions of MoS 2 and WS 2 , respectively, via a customised Langmuir-Schaefer deposition 20 (aer removing surfactant residues and changing the solvent to isopropyl alcohol (IPA), see Methods section) from a water-hexane interface onto fused silica substrates.In this process, the nanosheets of few-layer thickness obtained from LPE arrange in a mosaic with close lateral packing, but with minimal vertical stacking (see SEM images in Fig. S3 †) and excellent optical quality for time-domain spectroscopic investigation.
We investigated the dynamics of photogenerated charges on time scales up to 100 ns using TA spectroscopy with pump pulses of 3.49 eV (355 nm) and approximately 1 ns duration, and broadband probe pulses covering the spectral range 1.65-2.58eV (480-750 nm).The TA maps for the fractions MoS 2 and WS 2 Large in Fig. 2A and B show the relative differential transmission DT/T as a function of both probe energy and delay time between the pump and probe pulses.They exhibit the typical photobleaching (PB) features of the A-and B-exciton resonances (labelled A PB and B PB), accompanied by a series of photoinduced absorption (PA) peaks (the most dominant of which is labelled low energy PA).3][34][35][36][37] Excited state population also enables or enhances transitions towards higher excited states, seen as PA.Contrary to monolayer TMDs, where the TA signal is dominated by excitons, 18,19 in few-layer TMDs the low energy PA originates from transitions from photogenerated charges towards trions, [32][33][34][35][36][37] i.e. excitons bound to a charge. 38Hence, we associate the temporal evolution of the low energy PA with the dynamics of these photogenerated charges.
The transient response, reported in Fig. 2A and B, shows similar PB and PA features for all investigated fractions of MoS 2 and WS 2 .We nd that the transient spectra, shown in Fig. 2C and D, are blue-shied by a few 10 meV for thinner nanosheets, which is fully consistent with the absorbance spectra discussed previously.The spectral shape does not vary appreciably with pump-probe delay (see Fig. S4 †), indicating that

Paper
Nanoscale Advances photogenerated charge carriers dominate not only the low energy PA feature, but also the other PA and PB features.The dynamics of the low energy PA reported in Fig. 2E and F reveal that the charge populations in both MoS 2 and WS 2 exhibit a signicantly slower decay in the largest nanosheet fraction.The dynamics show a non-exponential decay and they can be accurately described by a model consisting of an instrument-limited rise followed by a second order power law decay (see eqn (1) below).This is typical for processes such as non-geminate recombination 39 and will be discussed in detail below.Notably, the recombination rate tends to be slower for larger akes.
In order to understand the origin of the remarkably slow dynamics, we measure the diffusivity using spatiotemporal pump-probe microscopy 40 on the fractions named WS 2 small and WS 2 large.This allows us to determine if the dynamics are associated with charges or with phonon heat, as the electron system has a much larger diffusivity than phonon heat.The choice of WS 2 is motivated by its larger TA signal, well-separated spectral features, and narrower linewidth compared to MoS 2 .
The fractions at opposite ends of the size selection enable the study of the role of nanosheet size and thickness on the carrier diffusion.We use a pump photon energy of 2.41 eV (515 nm) and a probe energy of 1.99 eV (623 nm), in resonance with the A PB peak and record differential reectivity (DR/R) spatial proles at different time delays, as described in the Methods section and shown in the sketch of the experimental setup depicted in Fig. 3A.The tightly focused pump generates an excited state population with a Gaussian spatial prole of initial s 2 = 0.09 mm 2 (s = 300 nm), whose temporal evolution is shown in Fig. 3B and C. Since the excited region is much larger than the size of the individual nanosheets, hundreds of nanosheets are excited simultaneously and hence the evolution of the spatial prole can contain contributions from heat or charge diffusion both inside each nanosheet and between adjacent nanosheets.
The spreading of the spatial prole of the photoexcited region is shown in Fig. 3D and E as the time-dependence of s 2 of the Gaussian curve.We identify an initial fast spreading during the rst few picoseconds and a gradual crossover to a much Fig. 1 Microscopic and optical characterisation of the three size-selected fractions of MoS 2 and WS 2 obtained by liquid cascade centrifugation (see Methods section), after exfoliation in aqueous solution containing sodium cholate as a stabiliser.Each fraction is defined by two centrifugal acceleration values expressed in "g" units, where "g" is the Earth gravitational acceleration constant.The large fraction is stably dispersed at 1k g and precipitates at 5k g.The analogous boundary values are 5k g and 10k g for the medium, and 10k g and 30k g for the small fractions.AFM images of drop-cast nanosheets: (A) for MoS 2 large, (B) for WS 2 large, (C) for MoS 2 small, and (D) for WS 2 small.The colour encodes the height in nm.Histograms of the nanosheet lateral size distribution for large (dark blue) and small (light blue) fractions of MoS 2 in (E), and for large (dark red) and small (orange) fractions of WS 2 in (F).Absorbance spectra of the three fractions dispersed in aqueous sodium cholate solution normalised at 3.6 eV (large in dark blue, medium in blue, and small in light blue) of MoS 2 in (G), and for WS 2 (large in dark red, medium in red, and small in orange colours) in (H).The insets in (G) and (H) show the second derivative around the A-exciton, computed from smoothed absorbance spectra, for more precise determination of the peak position: 1.86 eV (1.98 eV) for MoS 2 (WS 2 ) large; 1.87 eV (1.99 eV and 2.00 eV) for MoS 2 (WS 2 ) medium; and 1.88 eV (2.03 eV) for MoS 2 (WS 2 ) small.Notice that the second derivative of WS 2 medium fraction shows two peaks, corresponding to well resolved contributions from thicker and thinner nanosheets, respectively.Large, medium, and small fractions are the sediments collected after centrifugation between two boundaries of gravitational acceleration (i.e.1-5k g, 5-10k g, and 10-30k g respectively).Each of those fractions correspond to a distribution of flakes with thicknesses and lateral sizes as reported in Fig. S1 and S2 (see ESI † for further details on the distributions).

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slower spreading, which becomes negligible aer 100 ps.
During the slow spreading, the spatial prole widens only modestly from s 2 = 0.20 mm 2 to s 2 = 0.21 mm 2 (i.e. from s = 450 nm to s = 460 nm) for WS 2 large and from 0.26 mm 2 to 0.29 mm 2 (510 to 540 nm) for WS 2 small.The fact that the slow spreading is limited to a few tens of nanometres, which is the size of the smaller nanosheets within the distribution, suggests that the underlying heat or charge diffusion is limited to the individual nanosheets.The time-dependence of s 2 allows to determine the diffusion coefficients D according to Fick's law.We obtain D = 60 cm 2 s −1 and D = 70 cm 2 s −1 , for WS 2 large and WS 2 small, respectively, for the initial fast diffusion and D = 0.87 cm 2 s −1 and D = 1.7 cm 2 s −1 for the subsequent slower diffusion These are clearly higher than the thermal diffusivity measured in freestanding multilayer TMD sheets (in the range from 0.2 to 0.6 cm 2 s −1 ). 40herefore we conclude that the observed signal is likely not associated with phonon heat, but originates from diffusion of photogenerated charges.On the other hand, the obtained diffusivities are one order of magnitude smaller than in a large WS 2 monolayer sheet (980 cm 2 s −1 for hot carriers, 11 cm 2 s −1 for thermalised carriers), 41 suggesting that the diffusion between nanosheets is negligible or at least signicantly slower than diffusion within a nanosheet.
The time dependence of the diffusion coefficient suggests the following scenario for the photoexcitation dynamics.The pump pulse at 2.41 eV, several 100 meV above the A exciton resonances of MoS 2 and WS 2 , creates a population of hot electron-hole pairs.These have a high mobility, as previously found in WS 2 monolayers. 41Aer several picoseconds, the electrons and holes have thermalised to the conduction band minimum and valence band maximum, respectively, 19 where their diffusion coefficient is greatly reduced.This diffusion proceeds for approximately 100 ps, aer which the carriers are either trapped or cannot spread further due to the highly inefficient internanosheet charge transfer. 42,43rom the results of TA and spatiotemporal microscopy we can develop a more comprehensive scenario of the charge carrier dynamics.Aer thermalisation, the diffusion of carriers is conned to the single nanosheet and impedes spreading of the spatial prole aer about 100 ps.On the other hand, charge recombination continues for tens of nanoseconds.This means that at nanoseconds delays, either the carriers are already paired into slowly recombining momentum indirect excitons, 44 or at least one carrier polarity is still mobile and can recombine with opposite charges, for example in a Langevin-type mechanism if both charges are mobile, 39 or in a Shockley-Read-Hall (SRH) mechanism if one of them is trapped. 457][48] Even in type-II heterostructures, where carrier polarities separate onto different materials, the decay times at room temperature are well below 1 ns. 23(ii) Slow charge recombination, as reported in the literature, is typically associated with stabilisation at defects such as sulphur vacancies, 49,50 heteroatoms, 51,52 molecules adsorbed on the nanosheet surface, 53 and dangling bonds/trapped charges of the substrate. 54The main difference in the nanosheets from LPE is that they may have some residual surfactant to passivate some of these defects, 55 and, being smaller, they have a much larger edge-to-basal plane ratio.Indeed, charge lifetimes of several nanoseconds have previously been observed in monolayers synthesized using a particular laser-assisted synthesis technique that yields a material with strongly red-shied absorption and TA spectra due to internal strain. 56Since our spectra do not show such red-shi, we exclude strain as the main reason for the long charge lifetimes.However, a similarly long charge lifetime has been observed in exfoliated material with a less detailed size selection and consequently larger, thicker nanosheets. 35Combining these factsthe unusually long charge carrier recombination time, the possible stabilisation of charges at defects, the high defect concentration at edges, the particularly high edge-to basal plane ratio of our samples, and the strongly inhibited inter-nanosheet transfer of relaxed charge carrierswe ascribe the observed long charge recombination times to the stabilisation of one charge polarity at the edges, while the other is still mobile.
Since the mobile charges can recombine with any of the trapped charges of opposite polarity, we model the temporal evolution of the carrier population as a non-geminate recombination, whose rate is proportional to both the electron and hole populations, which we assume to be equal, and describe as a single time-dependent charge population C(t): where G(t) describes charge photogeneration by the pump pulse, and g is the recombination coefficient.We add a small population (a few % of the total initial population) that does not decay appreciably within the 100 ns temporal window, as it is consistent with the microsecond carrier lifetimes observed in time-resolved microwave conductivity. 57This simple model was used to t the time traces in Fig. 2E and F where we nd excellent agreement with the experimental data.The values obtained for the recombination coefficient g decrease for increasing nanosheet size (from 2.5 × 10 −11 cm 3 s −1 to 6.6 × 10 −12 cm 3 s −1 for MoS 2 , and from 5.2 × 10 −11 cm 3 s −1 to 5.2 × 10 −12 cm 3 s −1 for WS 2 ).Compared to an exponential population decay, nongeminate charge recombination does not have xed recombination rate k, but rather a rate k(t) = gC(t) that decreases with increasing pump-probe delay.Hence, rather than s = 1/k, as is customary for an exponential decay, we dene as the equivalent carrier lifetime the pump-probe delay at which the population  B) and (C) Temporal evolution of spatiotemporal profiles reported at five selected delays (i.e.−10 ps, 0 ps, 5 ps, 50 ps and 100 ps) for WS 2 large (i.e.flakes distribution of thicknesses up to 12 layers, and lateral sizes up to 225 nm) and WS 2 small (i.e.flakes distribution of thicknesses up to 7 layers, and lateral sizes up to 110 nm), respectively.Since pump and probe pulses have a repetition period of 13 ns, the spatial profile at negative delay is due to excitation by the preceding pump pulse and thus corresponds to a pump-probe delay of 13 ns.(D) and (E) Spatiotemporal profiles expressed as the square width (s 2 ) of a Gaussian fit to the spatial intensity distribution for WS 2 large and WS 2 small, respectively.The error bars represent the standard deviation from the fitting routine.The difference in width at time zero is likely due to the slightly different focussing conditions in the two measurements.Different focuses will have different pump spot sizes and therefore dictate the spatial extent of the initial electronic excitation.

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has decreased to 1/e of its initial value.This value increases for increasing nanosheet size (from 4 ns to 8 ns for MoS 2 , and from 2 ns to 10 ns for WS 2 ).This may seem counterintuitive, since one may expect the effect of charge stabilisation at the edges to scale with the edge-to-basal plane ratio, and hence be more pronounced for smaller nanosheets.However, as long as the edges can stabilize most of the photogenerated carriers of one polarity, any further increase in this ratio will have little impact on the carrier stabilization.On the other hand, in the larger nanosheets, the mobile charges of the other polarity on average spend more time in the basal plane and need to travel over a larger distance before recombining with an edge-stabilised countercharge, resulting in a longer charge carrier lifetime for the larger akes.
A formalism that relates g to the charge diffusion coefficient D is provided by Langevin recombination.However, that model assumes a uniform distribution of both charge polarities over the whole sample, while here one charge polarity is trapped at the nanosheet edges.Hence, on average the mobile charges need to travel much farther before recombining and applying the Langevin formalism to our g values underestimates D by orders of magnitude (see ESI †).Similarly, the SRH mechanism, which explicitly considers traps, but again with a uniform distribution over the sample, does not yield a quantitative result for our situation.In summary, our model is a simplication that reproduces the temporal behaviour of the charge population very well, but the obtained g values are not directly linked to the diffusion constant.

Conclusions and outlook
We have shown long-lived photoexcited charge carriers in mosaic-like thin-lms of semiconducting TMD nanosheets deposited using an adapted Langmuir-Schaefer approach, from inks obtained by sonication-assisted liquid phase exfoliation.Our samples exhibit much longer room-temperature carrier lifetimes than TMD monolayers or even heterobilayers with type-II band alignment, [21][22][23] which are purpose-built for light harvesting applications.We attribute this behaviour to stabilisation of carriers at defect-rich edge regions of the nanosheets, whose lateral sizes range from tens to hundreds of nanometres.Slower recombination is observed for larger nanosheets, which also have a lower diffusion constant D for relaxed carriers.In a broader context, the long-lived carriers are stabilised at nanosheet edges, providing an excellent basis for photocatalytic activity of fabricated Langmuir-type thin-lms. 58The size selection via liquid cascade centrifugation allows to optimise the trade-off between longer carrier lifetimes of thicker nanosheets and the more favourable ratio of edge to basal plane of smaller ones.Trapping at the edges may impede extraction of one charge polarity, which is not an issue for high efficiency photodetectors. 59On the other hand, photovoltaics requires extraction of both charge polarities.We envisage that the recently developed covalent interconnection of nanosheets, 42,43 which enables efficient inter-nanosheet charge transport, will ultimately enable the fabrication of macroscopic photovoltaic cells using tiled lms of 2d semiconductor as the active layer.
While photocurrents have already been demonstrated, 60 photovoltaic performance will require a more in-depth study of the ink processing 61 and charge stabilisation 55,62,63 to nd the optimal balance between long lifetimes and high mobilities of photogenerated carriers.

Methods
Liquid phase exfoliation (LPE) WS 2 and MoS 2 dispersions were prepared by sonicating 1.6 g material powder purchased from Sigma-Aldrich in 80 mL aqueous sodium cholate (SC) solution, (C SC = 2 gL −1 ).The mixture was sonicated by a solid athead tip, model Q500 from Fisher scientic, for 1 h at 60% amplitude with a pulse of 6 s on and 2 s off.Overheating was avoided using a cooling system model F32-HL from Julabo, connected to a jacketed beaker from Ace Glass Incorporated.The temperature of the cooling liquid was kept constant at 5 °C.Dispersions were centrifuged at 3000g (where g = 9.81 ms −2 is the Earth's gravitational acceleration) for 1.5 h in a centrifuge from Thermo scientic.Aer discarding the supernatant, the sediment was collected in 80 mL of fresh solvent and sonicated a second time for 7 h at 60% amplitude with a pulse of 6 s on and 2 s off.The resulting stock dispersion was subsequently fractionated by liquid cascade centrifugation.

Liquid cascade centrifugation
Liquid cascade centrifugation is a multi-step procedure with sequentially increasing centripetal acceleration (2 h per run in a Hettich Mikro220R centrifuge equipped with a xed-angle rotor 1195A, 1.5 mL polypropylene tubes), at constant temperature of 10 °C.Centrifugation conditions are expressed as an average centrifugal eld in units of 10 3 × g (or k g), where g is the gravitational acceleration.Four acceleration values were selected for the cascade: 1k g, 5k g, 10k g, and 30k g.For centrifugation speeds up to 30k g, a Hettich Mikro220R centrifuge equipped with a xed-angle rotor (1195A) was used (1.5 mL polypropylene tubes).First, the dispersion was centrifuged at 1k g.The sediment contained mostly unexfoliated material and was discarded.In the next step, the supernatant was centrifuged at 5k g.The sediment was collected and redispersed in fresh solvent for further analysis.This rst collected sediment is labelled as the fraction "1-5k g", i.e. the fraction of TMD nanosheets that are stably dispersed at 1k g but precipitate at 5k g, to indicate consecutive centrifugation steps.The procedure was reiterated with the supernatant to produce additional fractions "5-10k g" and "10-30k g".A solvent transfer was necessary to remove residues of the surfactant (i.e. sodium cholate).The dispersions were washed out in IPA before lm deposition: the dispersions were centrifugated at 30k g per 2 h, the supernatant (i.e.solvent made of aqueous solution containing sodium cholate) was discarded and the collected sediment was redispersed in fresh IPA solvent to make the deposition step easier, avoiding material aggregates due to surfactant residues.
The three fractions are reported in gures using simple labels referring to the different mean nanosheet lateral size for

Deposition
Formation of tiled lms of single-nanosheet thickness is achieved by a modied Langmuir-Schaefer approach as previously reported. 64A Teon stand (10 cm long) supporting the substrate is placed in a beaker (about 100 mL) of deionised water.About 20 mL of distilled hexane is added onto the surface of the deionised water to create a liquid/liquid interface.Inks of MoS 2 and WS 2 nanosheets in IPA are added (∼140 mL) to the interface.The nanomaterial diffuses along the water/hexane interface and forms a thin-lm as increasing amounts of material are added.This lm is transferred onto a substrate, which is vertically moved through the interface.The lms are le to dry in ambient air for ∼6 h.For analysis of the nanosheets by microscopy, Si/SiO 2 substrates with 300 nm oxide thickness and root-mean-square roughness 0.1 nm were used.For the timeresolved measurements, transparent 1 mm thick fused silica was used.

Atomic force microscopy (AFM)
For AFM measurements, a Dimension ICON3 scanning probe microscope (Bruker AXS S.A.S.) was used in ScanAsyst mode (non-contact) in air under ambient conditions using aluminium coated silicon cantilevers (OLTESPA-R3).The concentrated dispersions were diluted with IPA to optical densities <0.1 at 300 nm.A drop of the dilute dispersions (15 mL) was ashevaporated on pre-heated (175 °C) Si/SiO 2 wafers.Aer deposition, the wafers were rinsed with ∼15 mL of water and ∼15 mL of IPA and dried with compressed nitrogen.Typical image sizes ranged from 12 × 12 mm 2 for larger nanosheets to 3 × 3 mm 2 for small nanosheets at scan rates of 0.5-0.8Hz with 1024 lines per image.Previously published length corrections were used to correct lateral dimensions from cantilever broadening and pixilation effects by calibration using transmission electron microscopy (TEM). 26,65anning electron microscopy (SEM) SEM is performed with a Carl Zeiss Ultra SEM operating at 2-5 kV with a 30 mm aperture.Images are acquired using the secondary electron detector.The sample substrate is a piece of SiO 2 /Si wafer (0.5 × 0.5 cm 2 ) with an oxide layer of 300 nm. 55

Static absorption
The absorption spectra were measured in dispersion using a 10 mm optical length quartz cuvette in a PerkinElmer Lambda 1050 spectrophotometer, equipped with an integrating sphere, with the slit width set to a wavelength resolution of 2 nm, at a step of 1 nm.

Transient absorption
A Pharos femtosecond laser by Light Conversion based on a diode pumped Yb: KGW crystal with fundamental wavelength at 1032 nm, pulse duration of approx.200 fs and repetition rate of 2 kHz was used for white light generation.For this, the laser fundamental was focused into a 2 mm thick sapphire crystal, resulting in a broadband pulse with spectrum ranging between 470 nm and 700 nm, which was used as the probe.The differential transmission signal, dened as DT/T = (T pump on − T pump off )/T pump off was recorded using a fast visible camera (Stresing FLCC3001-FFT) coupled to a Princeton Instruments spectrometer as detector.The pump pulses were provided by a Qswitched Nd: YAG laser (Picolo by InnoLas), emitting pulses with a width of approx. 1 ns at 1064 nm fundamental wavelength.The third harmonic at 3.49 eV (355 nm) of these pulses is sent onto the sample with an 1/e 2 diameter of 400 mm and a uence of 140 mJcm −2 .To synchronise the pump and probe pulses, the trigger of the Pharos laser at 2 kHz was sent to a digital delay generator (model DG 645 from Stanford Research Systems) with less than 25 ps rms jitter, and used to trigger the Picolo to control the relative delay between pump and probe pulses.

Spatiotemporal pump-probe microscopy
We used the set-up as described in ref. 41.A mode-locked laser emits 100 fs pulses centred at 1030 nm at a repetition rate f rep = 78 MHz.Most of the power was used to pump an optical parametric oscillator, which generates signal photons with tuneable wavelengths between 1340 and 2000 nm.The probe beam (l probe = 623 nm) was obtained from the third harmonic of the signal output and scanned over the sample area of interest using a 4f-imaging system equipped with a galvanometric mirror (Optics in Motion OIM 101). 41The pump beam was obtained from the second harmonic of the fundamental laser (l pump = 515 nm), and it was modulated using a mechanical chopper.The pump passed over a mechanical delay line (Newport DL225) and then it combined with the probe beam before the objective lens (NA 0.67) using a dichroic mirror.The re-ected probe beam was measured using a photodetector and the pump-induced reectivity change (DR) was obtained by demodulating at the chopper frequency using a lock in ampli-er.By scanning the position of the probe beam (Dx) with respect to the stationary pump, the spatial prole of the transient reection signal, DR(Dx), was measured.Obtaining these spatial proles at different pump-probe delay times allows for the observation of diffusion.
and innovation program through the ERC project SOPHY under grant agreement no.771528.S. V. acknowledges the support of the Spanish Ministry of Economy through FPI-SO2018.K. S. acknowledges nancial support by the Deutsche For-schungsgemeinscha (DFG) through SY 217/1-1.

Fig. 2
Fig.2TA maps of (A) MoS 2 and (B) WS 2 thin films from the large (i.e.flakes distribution of thicknesses up to 12 layers, and lateral sizes up to 225 nm) fraction up to 100 ns delay.The map shows the differential transmission DT/T as a function of probe energy and relative delay between pump and probe pulses.The pump photon energy was 3.49 eV.Experiments were performed in air at room temperature.The colourmap is used to easily identify an increase of transmission (photobleaching, PB) in red and a decrease of transmission (photoinduced absorption, PA) in blue.The main features observed are the two PB peaks around the A and B exciton resonances, and the PA at low energy, which is a signature of charge carriers.Spectra are cuts of the TA map for MoS 2 in (C) and WS 2 in (D) at a fixed delay of 1 ns.Normalised dynamics are cuts at fixed probe energies; 1.8 eV for MoS 2 in (E) and 1.9 eV for WS 2 in (F).Transparent lines are experimental data, non-transparent lines are fits using eqn (1).

Fig. 3
Fig.3Spatiotemporal pump-probe microscopy.(A) Sketch of the setup used to track charge carrier diffusion.(B) and (C) Temporal evolution of spatiotemporal profiles reported at five selected delays (i.e.−10 ps, 0 ps, 5 ps, 50 ps and 100 ps) for WS 2 large (i.e.flakes distribution of thicknesses up to 12 layers, and lateral sizes up to 225 nm) and WS 2 small (i.e.flakes distribution of thicknesses up to 7 layers, and lateral sizes up to 110 nm), respectively.Since pump and probe pulses have a repetition period of 13 ns, the spatial profile at negative delay is due to excitation by the preceding pump pulse and thus corresponds to a pump-probe delay of 13 ns.(D) and (E) Spatiotemporal profiles expressed as the square width (s 2 ) of a Gaussian fit to the spatial intensity distribution for WS 2 large and WS 2 small, respectively.The error bars represent the standard deviation from the fitting routine.The difference in width at time zero is likely due to the slightly different focussing conditions in the two measurements.Different focuses will have different pump spot sizes and therefore dictate the spatial extent of the initial electronic excitation.