RESEARCH ARTICLE | MARCH 20 2023 A pre-time-zero spatiotemporal microscopy technique for the ultrasensitive determination of the thermal diffusivity of thin films Sebin Varghese ; Jake Dudley Mehew ; Alexander Block ; David Saleta Reig ; Paweł Woźniak ; Roberta Farris ; Zeila Zanolli ; Pablo Ordejón ; Matthieu J. Verstraete ; Niek F. van Hulst ; Klaas-Jan Tielrooij  Rev. Sci. Instrum. 94, 034903 (2023) https://doi.org/10.1063/5.0102855 CrossMark   View Export Online Citation 03 November 2023 15:48:22 Review of ARTICLE scitation.org/journal/rsi Scientific Instruments A pre-time-zero spatiotemporal microscopy technique for the ultrasensitive determination of the thermal diffusivity of thin films Cite as: Rev. Sci. Instrum. 94, 034903 (2023); doi: 10.1063/5.0102855 Submitted: 13 June 2022 • Accepted: 2 March 2023 • Published Online: 20 March 2023 Sebin Varghese,1 Jake Dudley Mehew,1 Alexander Block,1 David Saleta Reig,1 Paweł Woźniak,2 1 3 1 4 Roberta Farris, Zeila Zanolli, Pablo Ordejón, Matthieu J. Verstraete, Niek F. van Hulst2,5 and Klaas-Jan Tielrooij1,a) AFFILIATIONS 1 Catalan Institute of Nanoscience and Nanotechnology (ICN2), BIST & CSIC, Campus UAB, Bellaterra (Barcelona) 08193, Spain 2 ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Barcelona, Spain 3 Chemistry Department and ETSF, Debye Institute for Nanomaterials Science, Utrecht University, Utrecht, The Netherlands 4 Nanomat, Q-Mat, CESAM, and European Theoretical Spectroscopy Facility, Université de Liège, B-4000 Liège, Belgium 5 ICREA-Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain 03 November 2023 15:48:22 a) Author to whom correspondence should be addressed: klaas.tielrooij@icn2.cat ABSTRACT Diffusion is one of the most ubiquitous transport phenomena in nature. Experimentally, it can be tracked by following point spreading in space and time. Here, we introduce a spatiotemporal pump–probe microscopy technique that exploits the residual spatial temperature profile obtained through the transient reflectivity when probe pulses arrive before pump pulses. This corresponds to an effective pump–probe time delay of 13 ns, determined by the repetition rate of our laser system (76 MHz). This pre-time-zero technique enables probing the diffusion of long-lived excitations created by previous pump pulses with nanometer accuracy and is particularly powerful for following in-plane heat diffusion in thin films. The particular advantage of this technique is that it enables quantifying thermal transport without requiring any material input parameters or strong heating. We demonstrate the direct determination of the thermal diffusivities of films with a thickness of around 15 nm, consisting of the layered materials MoSe2 (0.18 cm2 /s), WSe2 (0.20 cm2 /s), MoS2 (0.35 cm2 /s), and WS2 (0.59 cm2 /s). This technique paves the way for observing nanoscale thermal transport phenomena and tracking diffusion of a broad range of species. © 2023 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0102855 INTRODUCTION initially pump-excited spot as a function of pump–probe time delay Δt, where the probe arrives later than the pump. The accuracy Pump–probe measurements, where a probe pulse interro- in resolving spatial broadening is determined by the signal-to- gates a system at a variable time delay after the system has been noise ratio and can therefore be significantly beyond the diffrac- excited by a pump pulse, are powerful and versatile techniques tion limit. Recently, this super-resolution technique has gained to obtain information on dynamical processes with high tempo- increasing attention as an important tool for studying diffusion ral resolution and high sensitivity. In spatiotemporal pump–probe in semiconductors,1,2 organic compounds,4 perovskites,6,7 metals,5 measurements,1–7 pump and probe pulses have a variable relative two-dimensional materials,3,8 and more. However, with current spa- offset in space, in addition to their variable time delay. Typically, tiotemporal pump–probe techniques, it is challenging to resolve one of the beams is scanned over the other, fixed beam, pro- processes with low diffusivities. viding information on spatiotemporal transport phenomena, such One specific slow diffusion process, in comparison with elec- as diffusion. Most commonly, one observes the broadening of an tronic diffusion, is the phonon heat diffusion that governs the Rev. Sci. Instrum. 94, 034903 (2023); doi: 10.1063/5.0102855 94, 034903-1 © Author(s) 2023 Review of ARTICLE scitation.org/journal/rsi Scientific Instruments lattice thermal conductivity.9 Understanding thermal transport is An optical pump pulse excites a small region of the sample. Ini- extremely important, in particular, when materials are used in tech- tially, the spatial distribution of the excited species corresponds to nological applications: avoiding (local) overheating is crucial for the profile of the tightly focused pump beam. As the excited species proper device operation and prolonged device lifetimes in elec- diffuse, the profile broadens spatially. In the case that the diffus- tronic, optical, and optoelectronic devices and systems.10 However, ing excitation lives long enough for a new pump pulse to arrive it is highly challenging for spatiotemporal techniques based on before the system has returned to equilibrium, we can monitor the pump–probe measurements to resolve thermal diffusion because of spatially broadened profile using a probe pulse at a small negative the limited amount of spatial broadening. Silicon, for example, has pump–probe time delay. This delay time thus corresponds to a delay a thermal diffusivity D of around 1 cm2 /s.11 If heat is deposited that is roughly equal to the temporal spacing between subsequent by a pump pulse in a Gaussian spot with a width σ Δt=0 , the two- pump pulses. This enables the study of spatial broadening occur- dimensional spatial broadening as a function of pump–probe delay ring over a longer timescale than the conventional pump–probe 2 time Δt is described by the diffusion equation: σΔt>0 = σΔt=0 2 + 2DΔt. delay time. This means that with an initial spot size of 1 μm, after 100 ps, the Whereas the concept is generally applicable to any long-lived broadening due to thermal diffusion in silicon is just a few nanome- excitations, we now focus on the specific case of quantifying thermal ters. Moreover, most materials have a lower diffusivity than silicon transport in thin films, in order to demonstrate the strengths of pre- and, therefore, show even less spatial broadening. Alternative opti- time-zero spatiotemporal pump–probe microscopy. We exploit the cal techniques to measure thermal transport typically obtain thermal notion that a pump-induced electronic excitation relaxes over time, conductivity rather than diffusivity, require accurate knowledge of typically transferring energy to the lattice subsystem, where phonons several material and interface parameters, and/or need relatively carry the energy as heat. Furthermore, we use the idea that heat strong heating of tens to hundreds of Kelvin. Moreover, the thermal changes the optical properties of a material, which we detect with diffusivities of even the most typical transition metal dichalco- our probe pulses. Since the pump–probe measurements enable the genide (TMD) thin films—MoSe2 , WSe2 , MoS2 , and WS2 —have observation of very small pump-induced changes in transmission or not yet been obtained in a direct experimental way. Finally, there reflection, this technique can detect small amounts of heat, which we is no strong consensus on the values of the thermal conductivity of will quantify below. We fix the position of the pump beam and scan these systems, in particular, toward the monolayer limit, neither in the probe beam in the sample plane, resolving the spatial profile of experiments nor calculations. Novel experimental techniques could the transient reflectivity signal ΔR R (detected using a silicon photodi- 03 November 2023 15:48:22 provide useful insights into solving this debate. ode) as a function of spatial pump–probe offset Δx and/or Δy. Here, Here, we introduce pre-time-zero spatiotemporal microscopy transient reflectivity is ΔR R = RexcR−R , where Rexc and R are the reflec- as a technique that enables the direct determination of the diffusivity tions of the probe with and without the pump pulse, respectively. By of long-lived excitations by examining the spatial profile at a small scanning the position of the probe beam with respect to the pump negative pump–probe delay time, where the probe arrives before the beam with nanometer precision, we acquire accurate spatial tran- pump. In this configuration, the probe is sensitive to the residual sig- sient reflectivity maps ΔR R (Δx, Δy). We will show that the transient nal generated by previous pump pulses from the incident pulse train. reflectivity profile represents the temperature profile ΔT(Δx, Δy), For example, with a repetition rate f rep of 76 MHz, the pre-time- which enables us to extract the thermal diffusivity. zero signal corresponds to an effective time delay of Δt ≈ 13 ns. We In our specific experimental setup (see the supplementary demonstrate the strengths of this technique by quantifying the in- material, Fig. S1), a mode-locked laser ( f rep = 76 MHz) generates plane thermal diffusivity D of free-standing thin films of the TMD pulses centered at 1030 nm. Most of the laser output power is used material family. We use suspended films, such that phonon heat can to pump an optical parametric oscillator (OPO) that has a tun- accumulate due to the absence of a heat-sinking substrate. Further- able signal output between 1320 and 2000 nm. The probe beam more, the temperature sensitivity reaches the sub-Kelvin level, when is either the second or the third harmonic of the signal output, probing with a wavelength close to the exciton resonance, where the depending on the sample under investigation. We direct the probe reflection is particularly sensitive to lattice temperature. We obtain via a scanning mirror system (Optics in Motion OIM101) while thermal diffusivity directly by comparing the experimental spatial the pump beam (515 nm, second harmonic of the fundamental profile to a simple model based on Fourier’s law of heat conduction, laser source) goes onto a variable delay line (Newport DL225) and which does not require any material input parameters. The obtained an optical chopper that regulates the pump modulation frequency experimental results are in agreement with our ab initio simulations f mod . Both beams combine via a dichroic mirror and focus on sub- based on density functional theory (DFT) and with experimental micrometer spot sizes via a microscope objective lens (numerical values in the literature. We will discuss the advantages of this tech- aperture: 0.67). nique and provide an outlook of the important studies this technique Figures 1(b) and 1(c) show exemplary transient reflectivity enables, in terms of (nanoscale) thermal transport, and beyond. maps recorded at two pump–probe time delays Δt. At time-zero (Δt = 0), we observe a narrow profile centered at the pump spot Δx = Δy = 0, which is due to the electronic excitation induced by RESULTS absorption of the pump pulse [see Fig. 1(b)]. The spatial profile looks very different when tuning to a small negative time delay Δt < 0, i.e., Concept and implementation of pre-time-zero spatiotemporal microscopy pre-time-zero, typically a few picoseconds before time-zero. Here, we observe a broader profile with a smaller amplitude [see Fig. 1(c)]. We explain the fundamental idea behind our pre-time-zero We attribute this signal to diffused phonon heat in the sample. The spatiotemporal technique, and the experimental setup; see Fig. 1(a). broader the pre-time-zero profile, the larger we expect the diffusivity Rev. Sci. Instrum. 94, 034903 (2023); doi: 10.1063/5.0102855 94, 034903-2 © Author(s) 2023 Review of ARTICLE scitation.org/journal/rsi Scientific Instruments FIG. 1. Schematic representation of the spatiotemporal pump–probe microscopy setup. (a) An optical pump pulse is focused onto the sample, creating a local excitation. To monitor the diffusion of this excitation, a probe beam scans spatially over the sample around the pump spot. The modulated pump beam passes over a delay line that controls the time delay Δt between the pulses. The probe beam position at the sample plane is controlled by a scanning mirror. Both beams are combined with a dichroic mirror and focused onto the sample through an objective. The reflected probe beam is detected using a silicon photodiode and demodulated at the modulation frequency, typically in the kHz range. A detailed schematic is shown in the supplementary material, Fig. S1. (b) The transient reflectivity profile ΔR R (Δx, Δy) at a time delay Δt = 0 (time-zero) shows a narrow profile that corresponds to the purely electronic response created by photoexcitation from the pump pulse. (c) The profile measured at a small 03 November 2023 15:48:22 negative time delay Δt < 0 (pre-time-zero) shows a broadened response due to the diffusion of excitation created by previous pump pulses. Effectively, this corresponds to a pump–probe delay that is equal to the inverse of the laser repetition rate, typically in the nanosecond regime. As explained in detail in the text, the pre-time-zero signal reflects a temperature-dependent electronic response, which allows us to observe heat spreading. of the phonon heat to be, and this eventually allows us to obtain the the substrate with 5/50 nm of Ti/Au. The measurements are per- diffusivity quantitatively. formed in air (and at room temperature), as going to vacuum led to negligibly small differences (see the supplementary material, Fig. S2). Spatiotemporal pump–probe measurements on TMDs Figures 2(a) and 2(b) show the transient reflectivity signal on suspended and supported regions of the flakes, as a function of We apply our spatiotemporal technique to a prototypical thin- pump–probe delay time. In both regions, the signal peaks at Δt = film system: transition metal dichalcogenide crystals. Like graphite, 0 ps. This is associated with the photoexcitation of charge carri- TMDs are layered crystals with strong in-plane bonds between ers and sub-picosecond exciton formation.3 We first examine the atoms, while adjacent layers are weakly bound by van der Waals substrate-supported region, where the signal decays over picosecond forces. They have attracted significant interest in recent years due to timescales and can be understood as a decrease in exciton popula- their many intriguing optical,12 electrical,13 mechanical,14 and ther- tion.17 The pump–probe signal decays to a value very close to zero mal properties.15 A thorough understanding of their thermal prop- during the time window provided by our delay line (150 ps) and erties is crucial to enable technologies, including (opto)electronic, is zero after ∼13 ns, as observed in the pre-time-zero signal. This thermoelectric, medical, and thermal management applications. means that the system is in equilibrium before the next pump pulse Whereas their thermal conductivities κ have been obtained exper- arrives. In the suspended region, in contrast, there is a clear remnant imentally, their thermal diffusivities D have not been determined signal before time zero [see Figs. 2(a) and 2(b)]. We ascribe this sig- directly, even though the diffusivity arguably describes thermal nal to phonon heat that has accumulated in the sample. This does transport more directly. We study free-standing films of MoSe2 , not occur in the supported regions due to efficient out-of-plane heat WSe2 , MoS2 , and WS2 , of thickness around 15 nm, corresponding sinking to the gold-coated substrate. to ∼20 layers, suspended over gold-coated substrates with circu- In order to confirm that the pre-time-zero signal corresponds lar holes with a diameter of 15 μm (Norcada, NTPR005D-C15). to a phonon-heat induced change in the reflection of our probe, we The sample fabrication, based on polydimethylsiloxane (PDMS)- measure the signal as a function of probe wavelength [see Fig. 2(c)]. assisted dry transfer of mechanically exfoliated TMDs (purchased We observe a clear peak at a wavelength of 635, 675, 770, and from HQ graphene), is described in detail elsewhere.16 To facili- 805 nm that corresponds to the A-exciton resonance of WS2 , MoS2 , tate efficient heat sinking outside the suspended area, we coated WSe2 , and MoSe2 , respectively.18–20 We describe the experimental Rev. Sci. Instrum. 94, 034903 (2023); doi: 10.1063/5.0102855 94, 034903-3 © Author(s) 2023 Review of ARTICLE scitation.org/journal/rsi Scientific Instruments FIG. 2. Pre-time-zero spatiotemporal experiment. Comparison of absolute transient reflectivity ∣ ΔR R ∣ dynamics in suspended (sus) and supported (sup) regions of WS2 and MoS2 (a), WSe2 and MoSe2 (b). Insets show a zoomed-in view of the signal before time zero. (c) Pre-time-zero reflectivity signal −( ΔR R pre ) measured at different probe wavelengths with a constant incident pump fluence of ∼ 0.3 J/m2 . The largest response corresponds to the A exciton resonance of the materials. The solid line is a Lorentzian 03 November 2023 15:48:22 fit to the data. The shaded area represents typical A exciton peaks found in the literature. (d) Simulated reflection spectrum around an exciton resonance at two different temperatures. (e) and (f) Magnitude of the pre-time-zero signal as a function of pump fluence (circles, bottom axis) and temperature rise (solid line, top axis) for WS2 (e) and MoSe2 (f) extracted from the Lorentz oscillator model. For (c), (e), and (f), we average ∼50 data points at different negative time delays from an interval of a few picoseconds. The error bars are smaller than the marker size (68% confidence interval), and the exponent of a power law fit these data is 0.98 (e) and 0.99 (f), indicating that the relationship is very close to linear. data with a Lorentzian function and extract linewidths of 21 nm [see Figs. 2(e) and 2(f)]. We compare this extracted ΔT to the peak (WS2 ), 16 nm (MoS2 ), 44 nm (WSe2 ), and 17 nm (MoSe2 ), sim- temperature increase using the relation ΔT peak = (A ⋅ F)/(h ⋅ Cv ), ilar to the literature reports of exciton linewidths.21 It is known where A is the absorption (see the supplementary material, Fig. S4), h that these linewidths are temperature dependent,22 which provides is the thickness, and Cv is the heat capacity, finding good agreement. the physical connection between the measured optical response and A fluence of 0.4 J/m2 , for example, in WS2 (MoSe2 ) corresponds to phonon heat in the sample: Figure 2(d) shows a representative reflec- a ΔT ∼ 2 K (∼8 K) from the Lorentz model [see Figs. 2(e) and 2(f)] tion profile obtained from the Lorentz oscillator model (see the and a peak temperature rise of ΔT peak ∼ 3 K (∼5 K), using h = 17.2 supplementary material, Note 1) at two different phonon temper- nm (13.8 nm), and Cv from literature (WS2 from Ref. 23 and MoSe2 atures. With an increased lattice temperature, the exciton linewidth from Ref. 24). Importantly, these results show that the technique increases, and this leads to a decrease in reflection, which is what we is sensitive to reflection changes caused by heating by less than a observe. Kelvin. We examine this in more detail by measuring the transient Having established that our probe is highly sensitive to phonon reflectivity signal as a function of pump fluence F [see Figs. 2(e) heat via the exciton linewidth, we now examine the Δt < 0 spatial and 2(f)] in WS2 and MoSe2 . The first thing we notice is the linear profiles at different pump modulation frequencies ( f mod ) between dependence of the transient reflectivity ΔR R on fluence (also see the 4 and 240 kHz, as controlled by an optical chopper or an electro- supplementary material, Fig. S3). Importantly, this justifies the use optic modulator. We position the pump beam in the center of the of the transient reflectivity profile ΔR R (Δx, Δy) as a representative of suspended region of the sample and scan the probe beam with the temperature profile ΔT(Δx, Δy), from which we will extract the respect to the pump spot at Δt = 0 and at a small negative time diffusivity. Second, we extract the temperature increase ΔT from the delay (Δt < 0), with a pump fluence of ∼0.3 J/m2 . Figure 3(a) shows observed transient reflectivity and compare this to the expected ΔT the normalized ΔR R spatial profiles acquired at Δt = 0 and Δt < 0 based on pulse energy and heat capacity. For this, we use the complex for MoSe2 . For the Δt = 0 profile, we subtracted the Δt < 0 back- dielectric function of WS2 and MoSe2 according to a Lorentz oscilla- ground signal (see the supplementary material, Fig. S5), and find tor model of the exciton resonance (see the supplementary material, that it is accurately described using a Gaussian function [Fig. 3(a)], Note 1), and the temperature-dependent exciton linewidth from which corresponds to a convolution of our pump and probe spot Ref. 22, finding values below 10 K up to the largest applied fluence sizes (see the supplementary material, Fig. S6 and Note 2). We Rev. Sci. Instrum. 94, 034903 (2023); doi: 10.1063/5.0102855 94, 034903-4 © Author(s) 2023 Review of ARTICLE scitation.org/journal/rsi Scientific Instruments FIG. 3. Spatial profiles at varying modulation frequencies for MoSe2 . (a) Spatial profiles Δt < 0 for different modulation frequencies. For low modulation frequencies (4–30 kHz), the spatial profiles are similar, indicating temperature saturation due to steady state formation. The solid line for Δt < 0 profile (240 kHz) is the description of the data by the sum of two Gaussians. The narrow profile represents the corresponding Δt = 0 profile (gray data points) and is described by a single Gaussian function. (b) Description of the spatial profile with 240 kHz modulation at Δt = 0 (gray line, single Gaussian) and Δt < 0 (light red line, sum of two Gaussians). The narrow dashed Gaussian (offset vertically) represents the resulting profile due to heat spreading from the previous pump pulse. The broad Gaussian is the resulting profile due to accumu- lative heat spreading from all the previous pump pulses. By analyzing the width of the Δt = 0 profile and the width of the narrow Gaussian, we obtain a D of 0.21 cm2 /s. All profiles are normalized to the peak of the fit. observe that the Δt < 0 spatial profiles broaden with decreasing reaching a maximum at the end of these N pump pulses. The f mod and tend to saturate for the lowest modulation frequencies accumulated heat then decays when the pump pulses are blocked, [see Fig. 3(a)]. This saturation is due to the formation of a steady- and the system reaches thermal equilibrium before the next period state profile and is understood as follows: The rate with which heat with N pump pulses. that is added to the material by absorbed light from the pump The evolution of the spatial reflectivity profiles, which corre- 03 November 2023 15:48:22 pulses is equal to the rate with which heat diffuses to the edge of spond to temperature profiles, is shown in Fig. 4(b). After the first the suspended material and heat-sinks into the gold-coated sub- pump pulse, the initial photoexcitation at pump–probe delay time strate. These spatial profiles can be described by two Gaussians: The Δt = 0 has a narrow Gaussian profile [see profile (i)]. After pump narrow component corresponds to the diffused heat from the pre- excitation and electron–phonon coupling, the heat in the phonon vious pump pulse and is, therefore, associated with a timescale of system starts diffusing. Therefore, at a small negative Δt, just before 1/ f rep = 13 ns. The broader component corresponds to accumulated the next pump pulse arrives, the Gaussian profile broadens [see pro- heat spreading from all the previous pump pulses and is therefore file (ii)]. The second pump pulse then adds a narrow Gaussian profile associated with a timescale of 1/ f mod = 4–250 μs. For lower mod- on top of the broadened Gaussian profile [see profile (iii)]. This pro- ulation frequencies, corresponding to longer excitation with pump file then diffuses, leading to a sum of two broadened Gaussians [see pulses, there is more accumulated heat, making the profile broader. profile (iv)]. This process continues for N pump pulses, leading to We analyze the spatial profile taken at 240 kHz and use its nar- profile (v), just after the last pump pulse. The cumulative photoexci- row component to estimate the thermal diffusivity D of the sample. tation after N pulses finally results in a spatial profile as in (vi). This This is done by comparing the Gaussian width (σ Δt=0 ) of the spa- profile contains a narrow component that corresponds to the dif- tial profile at Δt = 0 and the Gaussian width (σ Δt<0,narrow ) of the fused heat from the previous pump pulse and a broader component narrow part of the Δt < 0 profile. The diffusivity is then given by that corresponds to the accumulated heat from all previous pump D = [σΔt<0,narrow 2 − σΔt=0) 2 ]/2Δt. Using Δt = 13 ns, we obtain a D of pulses. The latter tends to the triangular steady-state solution for 2 0.21 cm /s [see Fig. 3(b)]. We will show below that this is a very continuous heating and cooling through diffusion to a perfect heat reasonable estimate. sink, which is the gold-coated substrate located at 7.5 μm from the center of the suspended region. To understand the shape of the spatial profile quantitatively Simulations of the pre-time-zero spatiotemporal and to obtain an accurate thermal diffusivity, we perform a simple technique simulation based on Fourier’s law of heat conduction, In order to more accurately quantify the diffusivity and under- ∂T ∂2T ∂2T stand the spatial profiles, we schematically show the evolution of = D( 2 + 2 ), the temperature rise in the suspended region of the sample during ∂t ∂x ∂y one pump modulation period [Fig. 4(a)]. When the pump beam is not blocked, a pulse train photoexcites and locally heats the sam- where T is temperature, t is time, x and y are the spatial coordi- ple. Since the system does not return to equilibrium before the nates, and D is the thermal diffusivity of the material. We calculate next pump pulse arrives, heat continuously accumulates in the sys- the two-dimensional diffusion with radial symmetry and obtain the tem, which is excited by N = f rep /2 f mod pump pulses (50% duty temperature increase ΔT(x, y). As the boundary condition, we use cycle modulation). Consequently, the system temperature increases, a perfect heat sink at the edge of the circular sample with radius r0 . Rev. Sci. Instrum. 94, 034903 (2023); doi: 10.1063/5.0102855 94, 034903-5 © Author(s) 2023 Review of ARTICLE scitation.org/journal/rsi Scientific Instruments FIG. 4. Simulations of pre-time-zero pro- files. (a) Heat accumulation in the free- standing region of the sample as a func- tion of time, during one pump modulation period. (b) Spatial heat profiles directly after the first pump pulse (i), after the heat of the first pump pulse, has diffused (ii), directly after the second pump pulse (iii), after the combined heat of the first two pulses has diffused (iv), directly after the Nth pump pulse (v), and after the combined heat of N pulses has diffused (vi). By using a negative pump–probe delay time, the Nth probe pulse senses the effect of the (N − 1)th pump pulse. [(c) and (d)] Simulation results obtained from a straightforward Fourier heat dif- fusion model, as explained in the main text. (c) Temperature increase ΔT as a function of the number of pulses N for different input diffusivities D. (d) Normal- ized average spatial heat profiles after N pulses for a range of D. Broader average profiles correspond to higher diffusivities. √ 03 November 2023 15:48:22 Thus, at position r = x2 + y2 , we have ΔT∣r=r0 = 0 K. In the sim- on modulation frequency. Figures 5(a)–5(d) show the normalized ulation, we let an initial Gaussian temperature increase evolve in Δt = 0 and Δt < 0 spatial profiles for MoSe2 (a), WSe2 (b), MoS2 space and time with a fixed diffusion coefficient. At t = 1/ f rep , we (c), and WS2 (d), measured at a pump fluence of ∼0.3 J/m2 . The add another pump pulse to the remnant heat profile. We repeat this extracted diffusivity is the one that minimizes the squared residuals for all N pulses and obtain the final spatial temperature profile. For between the experimental pre-time-zero profile and the simulated increasing N, we find the occurrence of saturation of the peak tem- profile. The only parameters we need for this are the diameter of the perature at Δx = 0, related to the formation of a steady-state profile, suspended flake (15 μm by design) and the experimental beam width see Fig. 4(c). This is in agreement with our experimental observa- of the pump and the probe (see the supplementary material, Note 2). tion [Fig. 3(a)] that the spatial profile does not change for the lowest In order to match the experimental conditions, we convolute the modulation frequencies (larger number of pulses). The simulations obtained profiles with the probe beam spatial profile. Using this indicate that with a higher diffusivity fewer pulses are needed to procedure, we obtain thermal diffusivities D of 0.18 ± 0.02 cm2 /s reach a steady-state situation, and the resulting peak temperature is for MoSe2 , 0.20 ± 0.03 cm2 /s for WSe2 , 0.35 ± 0.04 cm2 /s for MoS2 , lower. This is because in this case heat diffuses more quickly to the and 0.59 ± 0.07 cm2 /s for WS2 . The uncertainty in D comes from heat sink. two contributions: a statistical error from the least-square fit of the As a final simulation result, we show how the spatial profile data to the model (5%–10%) and a structural error mostly from the changes for different diffusivities. This is important, as it indicates uncertainty of the spot sizes (∼5%) (see the supplementary material, how sensitive our technique is toward obtaining the diffusivity of Figs. S7 and S8). a sample, and how large a range of diffusivities can be obtained. In order to assess the validity of the obtained diffusivities for For this, we performed simulations with different diffusion coeffi- these TMD films, we first compare them to experimental values in cients and plot the resulting average spatial profile after N pulses the literature. Most studies report thermal conductivity (κ), which [see Fig. 4(d)]. Our simulations show that the average spatial profiles relates to diffusivity (D) via the heat capacity (Cv ): κ = D ⋅ Cv . get broader for larger D, verifying our earlier intuitive explana- Therefore, we convert our diffusivity values using the bulk heat tion. Furthermore, we see that this technique is capable of obtaining capacities of MoSe2 from Ref. 24, WSe2 from Ref. 31, MoS2 from diffusivities varying over several orders of magnitude. Ref. 32, and WS2 from Ref. 23. Figure 5(e) shows the compari- son of our experimental values (solid lines) to other experimental reports (symbols). We limit our comparison to literature values Obtaining and evaluating the thermal diffusivities for relatively thick crystals of at least ten nanometers. We find of TMDs good agreement between our results and those from literature, con- We quantify the thermal diffusivity D of four different TMD firming the general trend that the selenides (MoSe2 , WSe2 ) have a samples. We use a modulation frequency of 4.37 kHz, in order to be lower thermal conductivity than the sulfides (MoS2 , WS2 ).15,33 For in the saturation regime, where the spatial profile does not depend bulk MoSe2 and WSe2 , other experimental studies show thermal Rev. Sci. Instrum. 94, 034903 (2023); doi: 10.1063/5.0102855 94, 034903-6 © Author(s) 2023 Review of ARTICLE scitation.org/journal/rsi Scientific Instruments FIG. 5. Obtained thermal diffusivities and benchmarking. [(a)–(d)] Transient reflec- tivity ΔR R profiles for suspended MoSe2 (a), WSe2 (b), MoS2 (c), and WS2 (d), all normalized to the peak of the fit. The narrow profiles represent the time-zero profiles (Δt = 0) and are described by Gaussian functions (solid gray lines) to extract their width. We obtain the thermal diffusivity (0.18 ± 0.02 cm2 /s for MoSe2 , 0.20 ± 0.03 cm2 /s for WSe2 , 0.35 ± 0.04 cm2 /s for MoS2 , and 0.59 ± 0.07 cm2 /s for WS2 ) by describ- ing the pre-time-zero (Δt < 0) profiles with our model based on heat diffusion (solid lines). The insets show the optical microscope image (left side, scale bar 20 μm) of the flakes, and atomic force microscopy profiles, from which we obtain the thickness (right side). (e) Obtained thermal conductivity (κ) from the measured diffusivities compared with bulk experimental literature values: △ (Ref. 15), ○ (Ref. 25), ∗ (Ref. 26), ⋆ (Ref. 27), ◻ (Ref. 28), ⊲ (Ref. 29), and ⊳ (Ref. 30). The thick line shows 03 November 2023 15:48:22 the obtained κ for our samples with a thickness of ∼20 layers, where the gradient limit represents the error bound with 95% confidence interval. The dashed lines represent our calculated κ for bulk. conductivities ranging from 22 to 42 W m−1 K−1 ,15,25,26 which is attribute this to the effect of the thickness, where thinner films in excellent agreement with our obtained κ of 33.5 W m−1 K−1 for typically have smaller thermal conductivities, cf. the recent com- MoSe2 and 39.6 W m−1 K−1 for WSe2 [Fig. 5(e)]. For WS2 , we find bined experimental–theoretical work of Ref. 25. This suggests that a relatively large κ of 113.9 W m−1 K−1 , which is in good agreement 15 nm is a thickness where these layered materials do not yet behave with experimental results.15,30 Our value for MoS2 of 70 W m−1 K−1 completely bulk-like in terms of their thermal transport properties. is in good agreement with those reported in Refs. 15, 28, and 29. Overall, we find good agreement between our measured results and We do find a significantly lower thermal conductivity compared to those obtained through our calculations. the measured value in Ref. 27. This discrepancy is likely the result of the smaller thickness of our sample compared to the one in Ref. 27. Overall, we find good agreement between our experimentally Evaluation of the pre-time-zero spatiotemporal obtained conductivities and the literature values. technique We proceed with comparing our obtained values with cal- culated diffusivities using density functional theory at 300 K We have seen that our pre-time-zero spatiotemporal technique for bulk crystals, by using the SIESTA program and the tem- allows for the accurate determination of the thermal diffusivities of perature dependent effective potential (TDEP) method (see the thin films, and now discuss the main benefits of this technique. First supplementary material, Note 3 and Fig. S9 for details). We find that of all, it offers the advantage of being a non-invasive, contact-free for all four TMDs the experimentally obtained conductivities fol- approach. Another main strength of the technique is that it enables low the same qualitative trend as the calculations with κMoSe2 < κWSe2 the determination of the in-plane thermal diffusivity of a thin film by < κMoS2 < κWS2 . Quantitatively, we find that our ∼15 nm thick films comparing the data to a simple description based on Fourier’s heat give slightly lower values than those calculated for bulk films. We law, where it does not require knowledge of any material parameters, Rev. Sci. Instrum. 94, 034903 (2023); doi: 10.1063/5.0102855 94, 034903-7 © Author(s) 2023 Review of ARTICLE scitation.org/journal/rsi Scientific Instruments such as the temperature coefficient, optical absorption, thickness, contribution that corresponds to electronic diffusion, rather than interfacial thermal conductance, heat capacity, and/or properties thermal diffusion. This, however, means that this technique makes related to transducer layers. The only input parameters we require it possible to observe charge diffusion in semiconducting thin films for the simulations are very well-known experimental conditions: with nanosecond or microsecond carrier lifetimes, which is particu- the diameter of the suspended region of the material, the repetition larly interesting for such materials with low electronic diffusivities. rate of the laser, and the spot sizes of the pump and probe beams. Finally, we point out that our 2D diffusion model assumes uniform Since all these values are known within an error of a few percent or heating along the c-axis (out-of-plane), which is justified given the less, we obtain accurate diffusivities (see the supplementary material, thicknesses of our flakes, which are smaller than the optical pen- Fig. S7). Thus, the proposed method eliminates sources of error etration depth. In the case of thicknesses larger than the optical arising from necessary measurements of intermediate parameters penetration depth, combined with very low out-of-plane thermal that enter the model used for the extraction of unknown thermal conductivities (high thermal anisotropy), the 2D model will have to properties. be modified into a 3D diffusion model. Another important advantage is the sensitivity of the technique. The ability to observe changes in a spatial profile is limited only by the signal-to-noise ratio, which ultimately determines the accu- CONCLUSIONS racy of determining the diffusivity. Being a pump–probe technique, In this work, we have introduced a technique for measuring dif- where the pump-induced changes are detected differentially using fusion processes of long-lived excitations by exploiting the pre-time- the combination of an optical chopper and a synchronized lock-in zero spatial profile. In particular, we have shown that this technique amplifier (Zurich Instruments MFLI), means that we can resolve can be used to obtain the in-plane thermal diffusivity of thin films. very subtle heat-induced changes in permittivity (refractive index) We have determined the thermal diffusivities of suspended MoSe2 , of materials. As a result, it is possible to determine thermal trans- WSe2 , MoS2 , and WS2 thin films. Our technique provides a simple, port properties even if the change in temperature due to heating is sensitive, and accurate way of determining the thermal diffusivity, very small, on the single Kelvin level [see Figs. 2(e) and 2(f)]. This and will enable a deeper understanding of the thermal properties is an important advantage over alternative techniques that require of nanoscale materials. Furthermore, the technique of pre-time-zero stronger heating. spatiotemporal scanning constitutes a powerful tool for studying a Beyond offering these advantages, pre-time-zero spatiotempo- multitude of diffusion processes in a large variety of systems. 03 November 2023 15:48:22 ral pump–probe microscopy will make it possible to explore previ- ously inaccessible physical phenomena and properties. Continuing with thermal transport as the main example, we suggest combining our technique that provides the diffusivity with a complementary SUPPLEMENTARY MATERIAL study that provides the thermal conductivity in order to obtain the See the supplementary material for the detailed schematic of heat capacity in an all-optical fashion. Our technique will also likely pre-time-zero spatiotemporal setup (Fig. S1), comparison between make it possible to access transport regimes where diffusive trans- measurements performed in ambient and vacuum environment port breaks down, and ballistic or hydrodynamic phonon transport (Fig. S2), the linear dependence of pre-time-zero signal with pump occurs. These phenomena have been studied theoretically for two- fluence (Fig. S3), absorption measurements on MoSe2 and WS2 at dimensional materials;34,35 however, there is minimal experimental 515 nm pump wavelength (Fig. S4), background removal to obtain evidence to support these reports.8,36 As our technique is highly sen- time-zero spatial profiles (Fig. S5), knife edge measurements of sitive to diffusive processes that occur over nanometer length scales, pump and probe beam to obtain spot sizes (Fig. S6), simulation to we predict that it will facilitate an advance in the understanding of understand the effect of spot size in quantifying thermal diffusivity non-diffusive transport in low-dimensional materials. Furthermore, (Fig. S7), fitted pre-time-zero spatial profiles at different modulation we point out that our technique enables the study of more com- frequencies (Fig. S8), convergence tests for density functional theory plex structures, such as van der Waals heterostructures, consisting of simulations on bulk MoSe2 (Fig. S9), and the measurements per- combinations of different layered materials, with material specificity formed on a 10 μm suspended MoSe2 sample (Fig. S10). Details of through the material-specific exciton resonances. the Lorentz oscillator model, experimental beam width calculation, Finally, we discuss the limitations of pre-time-zero spatiotem- and density functional theory simulations are described in Note 1, poral microscopy. In our example of thermal diffusion in thin Note 2, and Note 3, respectively. layered films, we benefit from an increased sensitivity due to the temperature-dependent exciton response. Not all materials exhibit exciton resonances and will therefore exhibit the same increased sensitivity. However, the technique will likely work for any dif- ACKNOWLEDGMENTS fusing species that has an optical response associated with it and ICN2 was supported by the Severo Ochoa program from Span- that has a long enough lifetime to be detectable at a negative time ish MINECO Grant No. SEV-2017-0706. S.V. and D.S.R. acknowl- delay. Since temperature-dependent refractive indices are very com- edge the support of the Spanish Ministry of Economy through mon, we expect that the thermal diffusivity of materials without FPI-SO2018 and FPI-SO2019, respectively. K.-J.T. acknowledges an exciton response should also be attainable. Furthermore, we funding from the European Union’s Horizon 2020 Research and note that the thermal diffusivity might not be directly obtainable Innovation program under Grant Agreement No. 804349 (ERC StG in all materials, such as materials with very long-lived photoexcited CUHL), RYC fellowship No. RYC-2017-22330 and IAE Project No. charges, because the pre-time zero signal will contain a significant PID2019-111673GB-I00. ICFO was supported by the Severo Ochoa Rev. Sci. Instrum. 94, 034903 (2023); doi: 10.1063/5.0102855 94, 034903-8 © Author(s) 2023 Review of ARTICLE scitation.org/journal/rsi Scientific Instruments program for Centers of Excellence in R&D (CEX2019-000910-S), (lead); Investigation (lead); Methodology (lead); Project adminis- Fundació Privada Cellex, Fundació Privada Mir-Puig, and the Gen- tration (lead); Supervision (lead); Writing – original draft (lead); eralitat de Catalunya through the CERCA program. P.W. acknowl- Writing – review & editing (lead). edges funding from the European Union’s Horizon 2020 Research and Innovation program under the Marie Skłodowska-Curie Grant DATA AVAILABILITY Agreement No. 754510 (PROBIST). N.F.v.H. acknowledges fund- ing by the European Commission (ERC AdG 670949-LightNet), The data that support the findings of this study are available the Spanish Plan Nacional (PGC2018-096875-BI00), and the Cata- within the article and its supplementary material. Additional data lan AGAUR (2017SGR1369). M.J.V. acknowledges support from related to this paper are available from the corresponding author Fédération Wallonie Bruxelles and ULiège (ARC project DREAMS upon reasonable request. G.A. 21/25-11). Z.Z. acknowledges the research program “Materials for the Quantum Age” (QuMAt) for financial support. This pro- gram (Registration No. 024.005.006) is part of the Gravitation REFERENCES program financed by the Dutch Ministry of Education, Culture 1 B. A. Ruzicka, L. K. 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Instrum. 94, 034903 (2023); doi: 10.1063/5.0102855 94, 034903-10 © Author(s) 2023