www.nature.com/npj2dmaterials ARTICLE OPEN Effect of crystallinity and thickness on thermal transport in layered PtSe2 Alexandros El Sachat 1 ✉, Peng Xiao 1,2, Davide Donadio 3, Frédéric Bonell 4, Marianna Sledzinska 1, Alain Marty 4 , Céline Vergnaud 4, Hervé Boukari4, Matthieu Jamet 4, Guillermo Arregui1, Zekun Chen3, Francesc Alzina 1, Clivia M. Sotomayor Torres 1,5 and Emigdio Chavez-Angel 1 ✉ We present a comparative investigation of the influence of crystallinity and film thickness on the acoustic and thermal properties of layered PtSe2 films of varying thickness (1–40 layers) using frequency-domain thermo-reflectance, low-frequency Raman, and pump-probe coherent phonon spectroscopy. We find ballistic cross-plane heat transport up to ~30 layers PtSe2 and a 35% reduction in the cross-plane thermal conductivity of polycrystalline films with thickness larger than 20 layers compared to the crystalline films of the same thickness. First-principles calculations further reveal a high degree of thermal conductivity anisotropy and a remarkable large contribution of the optical phonons to the thermal conductivity in bulk (~20%) and thin PtSe2 films (~30%). Moreover, we show strong interlayer interactions in PtSe2, short acoustic phonon lifetimes in the range of picoseconds, an out-of- plane elastic constant of 31.8 GPa, and a layer-dependent group velocity ranging from 1340 ms−1 in bilayer to 1873 ms−1 in eight layers of PtSe2. The potential of tuning the lattice thermal conductivity of layered materials with the level of crystallinity and the 1234567890():,; real-time observation of coherent phonon dynamics open a new playground for research in 2D thermoelectric devices and provides guidelines for thermal management in 2D electronics. npj 2D Materials and Applications (2022)6:32 ; https://doi.org/10.1038/s41699-022-00311-x INTRODUCTION potential of PtSe2 in thermoelectric applications, experimental Atomically-thin 2D semiconductors have attracted immense studies of the intrinsic thermal properties17 and phonon dynamics attention in the scientific community due to their exceptional of PtSe218 films are few and still limited. layer-dependent optical, electronic, and thermal properties that In this work, we study the phonon dynamics and thermal open new prospects in the microelectronics industry1. Vertical properties of supported crystalline and polycrystalline PtSe2 thin devices consisting of one-or few-atom thick 2D materials, where films of varying thickness (1 to 40 layers), which were grown by heat transport usually occurs in the vertical direction, have already molecular beam epitaxy (MBE) on zinc oxide (ZnO) substrates. shown excellent performance in diodes2, photodetectors3, tran- First, by using a combination of low-frequency Raman and pump- sistors4,5, and solar cells6. In such devices, the interfacial thermal probe coherent phonon spectroscopies we investigate the layer- properties of atomically-thin layered 2D materials significantly breathing modes (LBM) in PtSe2 thin films and extract an effective vary depending on their thickness, interlayer interactions, and out-of-plane elastic constant, a layer-dependent sound velocity, degree of bonding with the substrate. It is therefore essential to and the acoustic phonon lifetimes. Then, we focus on unraveling understand cross-plane thermal transport and phonon dynamics, the impact of crystallinity and size effects on the cross-plane particularly in noble metal dichalcogenides, like PtSe2, due to their thermal conductivity of supported PtSe2 thin films taking into large potential for integration in future high-performance 2D account the interfacial thermal resistances in our multilayer devices7–10. sample geometry. Finally, we investigate the thermal conductivity PtSe2 exhibits outstanding inherent properties, including high by the first-principles Boltzmann transport equation (BTE) carrier mobility at room temperature, which is eight times larger computing the harmonic and anharmonic force constants by than MoS211, excellent stability to air, and resistance to oxidation, density functional theory (DFT) to reveal the microscopic better than black phosphorous5. Moreover, PtSe2 can be easily mechanism of heat transport in PtSe2 thin films. integrated into practical devices since it can be grown at low temperatures12. Together with its widely tunable bandgap13 and layer-dependent semiconductor-to-semimetal transition beha- RESULTS AND DISCUSSION vior14, it is considered a promising material to be employed in Material growth, structural characterization, and phonon many electronic5, optoelectronic7 and thermoelectric devices15. For dynamics instance, Moon et al.15. have recently shown that band engineering Two PtSe2 wedges were grown under ultrahigh vacuum (base by thickness modulation leads to a 50-fold enhancement of the pressure in the low 10–10 mbar range) in an MBE chamber thermopower in bilayer PtSe2 nanosheets with respect to bulk equipped with a cryo-panel and a reflection high energy electron PtSe2. Moreover, calculated results have shown that in monolayer diffraction (RHEED) setup. For the crystalline PtSe2 wedge, four PtSe2 compressive or tensile strain can induce significantly monolayers of PtSe2 were deposited by co-evaporating Pt and Se enhanced n- or p-type Seebeck coefficients16. Despite the large on the ZnO (0001) substrate kept at 450 °C. The resulting RHEED 1 Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST, Campus UAB, Bellaterra 08193 Barcelona, Spain. 2Departamento de Física, Universidad Autónoma de Barcelona, Bellaterra 08193 Barcelona, Spain. 3Department of Chemistry, University of California, Davis, CA 95616, United States. 4Université Grenoble Alpes, CNRS, CEA, Grenoble INP, IRIG-Spintec, 38054 Grenoble, France. 5ICREA, Passeig Lluis Companys 23, 08010 Barcelona, Spain. ✉email: alexandros.elsachat@icn2.cat; emigdio.chavez@icn2.cat Published in partnership with FCT NOVA with the support of E-MRS A. El Sachat et al. 2 1234567890():,; Fig. 1 Structural characterization and low-frequency Raman spectroscopy of PtSe2. a Side and top views of the PtSe2 crystal structure (left), schematics of the side view of a trilayer PtSe2 crystal structure (middle), and the one-dimensional linear atomic chain model (right). The blue and orange spheres represent Pt and Se atoms, respectively. b HR-STEM images of a 5 ML crystalline PtSe2 film and c the Fast Fourier transform of the respective image, from which we extract the a and b lattice constants, a = b ≈ 0.37 nm. d, e Cross-sectional TEM images of the as-grown 3 nm (5 ML) PtSe2 film, from which we extract the c-lattice constant, c ≈ 0.52 nm. The PtSe2 is layered and each layer is parallel to the underlying ZnO substrate. Thickness-dependent Raman spectra of f crystalline and g polycrystalline PtSe2 films deposited on ZnO. The inset in (f) shows blue and orange spheres that represent Pt and Se atoms, respectively and arrows that point to the direction of the movement of that layer. patterns were anisotropic with a 7° of mosaicity (see Supplemen- co-evaporation at the same temperature. We checked by RHEED tary Fig. 1), demonstrating the good-crystalline character of the that the films retained their respective crystalline and polycrystal- film, and the epitaxy relationship was found to be ZnO (0001) line character at the end of the growth. The final structure is [100]/PtSe2 (111) [100]. Additional characterization of epitaxial shown in Supplementary Fig. 1d. Each PtSe2 monolayer consists of ZnO/PtSe2 is reported in reference 11. For the polycrystalline PtSe2 three atomic sublayers, in which Pt atoms are sandwiched wedge, we first deposited a 2.4-Å-thick Pt film by magnetron between Se atoms (see Fig. 1a). Both samples were annealed at sputtering at room temperature and selenized it in the MBE 750 °C for 10 min under Se flux after the growth in order to chamber by deposition of Se at room temperature and improve their crystalline quality. To avoid the film degradation subsequent annealing at 750 °C under Se flux. The magnetron during air transfer, the films were capped by a ~10-nm-thick sputtering reactor and the MBE chamber are being connected amorphous Se layer deposited at room temperature. under ultrahigh vacuum and PtSe2 films are fully grown in situ. The crystalline, acoustic and morphological characterizations of After selenization, the equivalent PtSe2 thickness was two the samples were studied by Raman spectroscopy and high- monolayers (2 ML). The RHEED patterns were streaky but isotropic, resolution scanning transmission electron microscopy (HR-STEM) showing the in-plane polycrystalline character of the film, with a measurements. The arrangements of Pt and Se atoms in the planar well-defined (0001) surface for all grains but random in-plane HR-STEM image (Fig. 1b) and the symmetric diffraction pattern in crystal orientation. Two more PtSe2 films were finally deposited by the fast Fourier transform (FFT) image (Fig. 1c) further support the co-evaporating Pt and Se at 450 °C to obtain a 4 ML thick good crystal structure nature of the films. In addition, the high- polycrystalline film (see Supplementary Fig. 1c). magnification cross-sectional HR-STEM images (Fig. 1d, e) show the In a second step, we covered the sample with a motorized layered hexagonal honeycomb structure of a five layers PtSe2 film, mechanical mask except for a thin 1.5-mm-large band at the edge where bright Pt atoms are surrounded by six lighter colored Se to deposit 14 ML of PtSe2 by co-evaporating Pt and Se at 450 °C. atoms. The in-plane and out-of-plane lattice parameters were The mask was then retracted at a constant speed and stopped at obtained from the (l 0 0) and (l m 0) planes of the FFT and image 1 mm from the edge to evaporate the 0–22 ML PtSe2 wedge by analysis of cross-plane sections of the film (see Fig. 1e), respectively. npj 2D Materials and Applications (2022) 32 Published in partnership with FCT NOVA with the support of E-MRS A. El Sachat et al. 3 The in-plane lattice constants were found to be a = b ≈ 0.37 nm and per unit area μ = 4.8 × 10−6 kg m−2 connected by a string with an c ≈ 0.52 nm in good agreement with previous studies10,15,19 and in effective interlayer breathing force constant (IBFC) per unit area K⊥ excellent agreement with the ones deduced from our x-ray and separated by a distance d ≈ 0.52 nm given by the interatomic diffraction measurements (see Supplementary Fig. 2). On the other distance between layers (see Fig. 1a). The solution of the linear hand, DFT calculations give a = 0.377 nm and c = 0.486 nm. chain with vanishing stress as a boundary condition on the free Whereas a is in excellent agreement with our measurements, c is surface is given by: significantly underestimated (6.5%). There may be multiple reasons for this discrepancy, among which the difficulty in the DFT sffiffiffiffiffiffiffiffi K? q  calculations to simulate PtSe2 and PtSe compounds for all the N; j f ¼ sin d (1) GGA exchange functionals (e.g., PBEsol, AM05)20 and even van der μπ 2 2 Waals functionals (e.g., vdW-DF-C09 and vdW-DF-CX)21. The layered PtSe2 adopts a T-type hexagonal crystal structure, where f is the vibrational frequency of the mode, qN, j = 2π/λN,j is belonging to the P3m1 space group with a D3d (-m) point group. the acoustic wavevector, λN, j = 2Nd/j is the phonon wavelength, N The primitive PtSe2 cell contains 3 atoms, then its vibrational is the number of layers (N = 2, 3, …) and j is the index of the spectrum includes nine modes: three acoustic (A2u + Eu) and six acoustic mode (j = 1, 2, 3…). The experimental and the fitted optical (A1g + Eg + 2A2u + 2Eu). The optical modes can be thickness-dependence of the first LBM (j = 1) is shown in Fig. 2a. classified as Raman active (Eg, A1g and LO (A2u + Eu)) and infrared The fitted curve was obtained using the IBFC adjustable variable, active (2Eu + 2Au) modes22. Fig. 1f, g show the Raman spectra of which is determined to be K⊥ = 6.0 ± 0.14 × 1019 N m−3. A similar crystalline- and polycrystalline PtSe2 films of different thicknesses, value was also obtained by Chen et al.18 with K⊥ = 6.2 × 1019 N respectively. The high-frequency modes (170–210 cm−1) are m−3 using the same approach. Now, if we multiply the IBFC by the originated from in-plane and out-of-plane vibrations of Se atoms interlayer distance, we can also derive the corresponding C33 corresponding to Eg (~180 cm−1) and A1g (~205 cm−1) modes, component of the elastic constant tensor24,25. respectively. In addition, a weak interlayer longitudinal optical (LO) For our system, we found C33 = 31.8 ± 0.95 GPa, which is in the mode can be also observed at ~235 cm−1. LO is generated by a same range as other 2D materials e.g., 38.7–36.5 GPa for graphite, combination of the in-plane Eu and out-of-plane A2u modes from the vibrations of Pt and Se atoms in the opposite phases. The 54.3 GPa for MoSe226, 24.5 GPa for h-BN27, 52 GPa and 52.1 GPa for Raman peaks located in the low-frequency regions (−30 to MoS2 and WSe2, respectively25. It is interesting to notice that the 30 cm−1) correspond to interlayer vibrations of PtSe2 planes5,18. polycrystalline films also follow the same trend as the crystalline Specifically, the vibrations detected here are related to the out- samples (see Fig. 2a) thus most likely the granular characteristics of-plane displacements of the PtSe2 layers known as layer- of the sample do not affect significantly the interlayers forces. breathing modes (LBM). As shown in Fig. 1f, g, the position of Likewise, it is also possible to obtain the C44 constant using the the LBM shifts to lower frequencies as the number of layers effective interlayer shear force constant (K//). Chen et al.18. increases. The thickness-dependence can be simulated by using a calculated K// = 4.6 × 1019 N m−3 for PtSe2 based on density one-dimensional linear atomic chain model23. This model functional theory simulations from Zhao5, giving C44 = 23 GPa considers each PtSe2 layer as a large atom with an effective mass and vTA = 1553 ms−1. In addition, from Eq. (1), it is also possible to Fig. 2 Phonon dynamics in PtSe2. a Peak position of the first order breathing mode measured in crystalline and polycrystalline PtSe2 films using low-frequency Raman spectroscopy and ASOPS as a function of the number of layers and comparison with other experimental works. b Calculated group velocity as a function of the number of layers. c Representative transmission signal obtained with ASOPS in 5 ML PtSe2 (black open circles) and the best fit (red line). d The extracted phonon decay time (yellow dots) vs the number of layers. The red and black curves are theoretical predictions of the phonon lifetimes based on the model proposed by Ziman30. The region displayed with light blue shows the impact of roughness in the calculated lifetimes. The gray data points show phonon lifetimes measured by Chen et al.18. Published in partnership with FCT NOVA with the support of E-MRS npj 2D Materials and Applications (2022) 32 A. El Sachat et al. 4 extract the group velocity considering that vg = d(2πf)/dq. time-domain data for all measured samples are displayed in the sffiffiffi   Supplementary. K πj Figure 2d shows the measured decay time of LBM as a function vg ¼ d cos (2) μ 2N of the number of layers (yellow dots) and the calculated phonon decay due to boundary scattering contribution based on the In the limit N → ∞ and considering the experimental interlayer model proposed by Ziman30. This model considers the impact of distance d ≈ 0.52 ± 0.01 nm, Eq. (2) gives the bulk limit for the the roughness (η) of the film surface through a phenomenological cross-plane longitudinal velocity vLA = 1873 ± 42 ms−1. A similar parameter p = exp[−16π2η2/λ2], which modifies the effective valueqffiffiffiffiffi can be obtained using an approximation of mean free path Λ = (1+p)/(1-p) Nd/vg of the acoustic waves. The vg ¼ Cρ33 ¼1825 ± 28 ms−1 where ρ = 9540 kg m−3 is the density black and red curves of Fig. 2d were calculated using a thickness- of PtSe2 and C33 = 31.8 GPa. Using a numerical differentiation, it is dependent group velocity (Eq. (2)) and a constant value given by also possible to extract the group velocity as a function of the the bulk limit (1873 m s−1), respectively, with a constant number of layers as shown in Fig. 2b. η = 0.82 nm as measured by AFM (see Supplementary Fig. 3). Furthermore, the phonon dynamics of the crystalline PtSe2 Apart from the good agreement between the theory and the samples were measured by pump-probe coherent phonon experimental data, we observe that the phonon lifetime is not spectroscopy using the asynchronous optical sampling method affected by the group velocity (thickness-dependent or constant). (ASOPS)28,29 (see details in Methods). Figure 2c shows the typical Similar observations of boundary-limited lifetimes have been modulation of the reflectivity (ΔR/R) as a function of the time reported in suspended MoSe2 flakes by Soubelet et al.26. They delay (black open circles) and the corresponding best model fit in measured lifetimes of the order of ~1–41 ps in 2–8 ML limited by a 5 ML PtSe2 crystalline film. The change of reflectivity includes the boundary effects. While in thicker samples (>20 ML), they effects of a fast and slow electronic relaxation process as well as observed longer lifetimes (~0.3–10 ns) limited by phonon- the dynamics of the generated phonons. The time traces are fitted phonon scattering. The large difference in lifetimes measured to a damped harmonic oscillator of the form:18 here (1–4 ps) is mainly attributed to the difference in the phonon X   X   group velocity which is 1.4 to 1.5 times smaller than MoSe2. 2 t 2 t   ΔR=R ¼ Ai exp þ Bj exp sin 2πfj t þ ϕj (3) i¼1 τi j¼1 τj Thermal conductivity and interfacial heat transport where t is the time delay between the pump and the probe lasers, measurements A and B are the amplitudes, τι are the decay times, f the phonon For the thermal measurements, we used a custom-built frequency- frequency, and ϕ is a phase delay. The first term of Eq. (3) domain thermo-reflectance (FDTR) setup, a well-established represents the fast and slow relaxation processes of the excited optical pump-probe technique, capable of measuring heat carriers. The second term describes the damped phonon transport in thin films and across interfaces31–36. A schematic oscillations. The Fast Fourier transforms (FFT) of the representation of the experimental setup is shown in Fig. 3a. Fig. 3 Thermal conductivity and interfacial heat transport measurements. a Schematic illustrations of the FDTR technique and the multilayer system. b Typical FDTR data measured in bulk PtSe2 (red circles), 40 layers (black circles), and monolayer (blue circles) crystalline PtSe2 films and the corresponding best model fits in the whole frequency range. c Cross-plane thermal conductivity kz of crystalline (black spheres) and polycrystalline (green spheres) PtSe2 films versus film thickness t. The uncertainty of the estimated kz was calculated based on error propagation for the input parameters. The blue and red lines in (a) display the bulk cross-plane thermal conductivity values obtained by FDTR and DFT calculations, respectively. The black and green curves are a guide for the eye. d Total thermal resistance, Rtotal = Rint + RPtSe2, of crystalline (black spheres) and polycrystalline (green spheres) PtSe2 thin films, plotted versus film thickness t. The uncertainty region is displayed as a gray rectangle. The total cross-plane thermal resistance measurements of Au/graphene/SiO247 (red diamonds) and Al/MoS2/ SiO241 (purple triangles) interfaces are also reported. npj 2D Materials and Applications (2022) 32 Published in partnership with FCT NOVA with the support of E-MRS A. El Sachat et al. 5 A modulated pump beam with a wavelength of 448 nm is focused agreement with the measurement, while kz is significantly onto the sample, creating a periodic heat flux with a Gaussian overestimated, possibly due to the discrepancy in the equilibrium spatial distribution on the sample surface. A reflected probe beam c lattice parameter mentioned above. Nevertheless, we can still with a wavelength of 532 nm is aligned coaxially with the pump use DFT and BTE to analyse the phonon contributions to cross- beam and focused with the pump spot to monitor the periodic plane transport in PtSe2 bulk and the effect of finite thickness in fluctuations in reflectivity at the sample surface caused by the thin films, at least qualitatively. oscillating sample temperature. All the samples after the MBE To study the impact of crystallinity and film thickness on the growth were coated with a 100 nm thick Au layer, which was cross-plane thermal conductivity of PtSe2 thin films, we performed chosen to maximize the coefficient of thermo-reflectance at the FDTR measurements in large-area crystalline and polycrystalline probe wavelength (∼2.36 × 10−4 K−1). We used a lock-in amplifier PtSe2 films of different thicknesses (1–40 ML) on ZnO substrate. (Zurich Instruments HF2LI) to record the amplitude and phase The wedge sample stacks consist of Au/PtSe2/ZnO (see Fig. 3a and response of the reflected probe beam to the thermal wave, and Supplementary Fig. 1). Typical examples of the recorded phase the phase lag between the pump and probe beam as the signals and the corresponding best model fit for bulk, monolayer, observable quantity. More details on the experimental setup can and 40 ML crystalline PtSe2 are shown in Fig. 3b. Here, the key be found elsewhere35. parameters for the model in our multilayer system are the spot In the present experiments, we obtained frequency-domain sizes of the pump and probe beams, kz, the film thickness t, the C measurements by varying the modulation frequency of the pump of each layer, and the interface thermal conductance (between Au beam over a wide range (20 kHz–40 MHz). The cross-plane thermal and PtSe2, G1, and between PtSe2 and ZnO, G2). The thickness of conductivity (kz) of the PtSe2 thin films was subsequently Au and PtSe2 films were measured by AFM and low-frequency extracted by following a multilayer three-dimensional (3D) heat Raman spectroscopy, respectively (see section above and diffusion model that includes the interface thermal conductance Supplementary). The heat capacities of Au, ZnO, and PtSe2 were between the different layers and anisotropic heat transport31. For taken from the literature10,36,40. Thus, the remaining unknown each experiment, we quantified first the sensitivity of the recorded parameters were the kz value of the PtSe2 films, G1 and G2. phase signal to different parameters: kz, the volumetric heat To extract a unique value of kz from a single measurement we capacity (C), the thermal conductivity anisotropy ratio (kr/kz), and followed a fitting approach similar to that suggested in the the different interface thermal conductances, in a similar manner previous works31,32,35 and supported by our sensitivity analysis to that of Schmidt et al.31. The details about the measurement (see Supplementary Fig. 4). First, we estimate kz by fitting to sensitivity to different combinations of parameters in the model experimental data in a low-frequency range (20 kHz–1 MHz), can be found in Supplementary Fig. 4. In all the experiments we where the measurement sensitivity to the G1, G2, and heat used a high-magnification 50× objective lens that produced a capacity of the films is negligible. Then, we fix kz and fit focused root mean square (rms) spot size (1/e2 radius) of experimental data in a high-frequency range (1–40 MHz) to approximately ~1.5 μm. To reduce the uncertainties associated estimate simultaneously G1 and G2. As an initial guess to with variations in the laser spot size between measurements on determine the value of the free parameters (kz, G1, and G2) we different films, the spot size was measured for each different used previously reported values of similar material systems41–43. experiment using the knife’s edge method, as shown in We also verified that the final fit results are not sensitive to the Supplementary Fig. 6. All the room temperature measurements choice of initial values. The same analysis has been followed to were performed under both ambient and vacuum conditions. Before the thermal measurements on thin PtSe2 films, we extract kz, G1, and G2 values for all the PtSe2 films. In Supplementary Fig. 13 we show all the interface thermal performed FDTR measurements on a bulk PtSe2 crystal coated with 100 nm of Au in order to extract the anisotropy ratio of the resistance measurements, R1 = 1/G1, R2 = 1/G2, and Rint = R1 + R2, thermal conductivities. The thermal conductivity of the deposited extracted by the FDTR experiments. The observed variation of Rint Au film was first measured using electrical conductivity and the might be attributed to differences in the interface characteristics Wiedemann−Franz law, kAu = 195 Wm −1 K−1 (see details in of Au (transducer)/PtSe2 and PtSe2/ZnO (substrate), which could Supplementary Fig. 7). The volumetric specific heat of Au and result in different interfacial energy coupling. The apparent PtSe2 were taken from literature10,36. This leaves us with three increase of R2 with increasing film thickness most likely is due unknowns: the cross-plane and in-plane thermal conductivity of to the reduced mechanical coupling of the films to the underlying the bulk PtSe2 crystal and the Au-PtSe2 interface thermal ZnO resulting from a potential increase in sample stiffness or conductance (GAu-bulk). The in-plane thermal conductivity was changes in the vibrational spectra44. However, we note that most expressed in terms of the anisotropic ratio (α) and the cross-plane of the extracted values are within the experimental uncertainty, thermal conductivity kr = α ·kz. thus it is not clear if there is a systematic increase of R2 with However, from the sensitivity analysis, we found that the thickness. Similarly, a weak thickness-dependent thermal resis- recorded phase signal has low sensitivity to GAu-bulk and only at tance trend has been measured in InSe, MoS2, and SnSe2 thin high frequencies while its sensitivity to the anisotropic heat flow is films35,37,41. high in almost the whole frequency range (see Supplementary Fig. It is interesting to note that, in contrast to the bulk case, in FDTR 5). This allowed us to extract α directly from the model fit of the measurements in thin films, the diameter of the laser spot is experimental data in the low-frequency range (20 kHz–1 MHz) usually large compared to the thermal diffusion length during the using a nonlinear least-squares routine, which requires an initial modulation period of the pump beam. Thus, the heat flow is guess to determine the value of the free parameter (kz) (see expected to be mainly one-dimensional in the cross-plane Supplementary Fig. 11). The inset in Supplementary Fig. 11 shows direction32. This has been confirmed by our sensitivity analysis, the numerical errors of the applied model fits as a function of where we found that the sensitivity of the recorded phase signal different anisotropy ratios. The minimum value of the fitting error, to the in-plane transport is relatively low (see Supplementary Fig. which gives us the best fit of the FDTR data, corresponds to α = 9, 4c). Therefore, since anisotropic differences in thermal conductiv- kz = 4.6 ± 0.7 Wm−1 K−1 and kr ≈ 41.4 Wm−1 K−1. ities as a function of the film thickness cannot be resolved, for the The observed anisotropy ratio between kr and kz in PtSe2 is data analysis we used the anisotropy ratio extracted from the bulk similar to previously reported values of other 2D-layered materials, PtSe2 experiments. The dependence of the thermal anisotropy such as InSe (α ~10)37, SnSe2 (α ~7–8)35,38, and armchair black with the film thickness has been studied recently in similar phosphorous (α ~6–10)39. Our first-principles BTE calculations give material systems, such as supported InSe37 and SnSe235 films, kz = 2.84 W m−1 K−1 and kr = 39.0 W m−1 K−1. kr is in excellent showing a thickness-independent α of 10 and 8.4, respectively. Published in partnership with FCT NOVA with the support of E-MRS npj 2D Materials and Applications (2022) 32 A. El Sachat et al. 6 Figure 3c displays the extracted kz value of all the PtSe2 thin spheres), with values approximately of 28 ± 10 m2 KGW−1, while in films as a function of thickness. In both crystalline (black data polycrystalline samples (see green spheres) Rtot is constant only up points) and polycrystalline films (green data points), we observe a to 12 nm and then starts to increase up to 75 ± 10 m2 KGW−1. The linear increase of kz with increasing the film thickness starting large contribution of the volumetric resistance component, RPtSe2, from 2.4 nm (4 ML) up to 10 and 24 nm, respectively. In to Rtot and the thickness-independent Rtot in crystalline films polycrystalline films with thicknesses t > 12 nm, kz shows a indicate ballistic phonon transport along the c-axis of the films. plateau, and a maximum cross-plane thermal conductivity Similar results has been previously observed in few-layer- reduction of ~35% was observed compared to the crystalline graphene and single-crystal MoS2 flakes that showed a samples. This result highlights the strong impact of crystallinity on thickness-independent Rtot for film thicknesses between kz and suggests that depending on the film crystallinity phonons 0.4–4 nm47 and 20–40 nm41, respectively. with different mean free paths (MFPs) contribute to kz. In The increased values of Rtot in polycrystalline films after 12 nm crystalline PtSe2 films, heat is propagating through coherent indicate a transition from ballistic to diffusive transport regime vibrations (phonon modes) that travel distances at least 24 nm most likely due to phonon grain boundary scattering. For (~40 ML) while in polycrystalline samples, they start to decay comparison, in Fig. 3d we plot our calculated Rtot (black and above 12 nm (~20 ML). This behavior most likely is attributed to green spheres) with previous total cross-plane thermal resistance the enhanced phonon scattering due to the presence of high- measurements on different 2D materials, where similar interfacial density defective grain boundaries randomly oriented in the contributions from the bottom (2D material/substrate) and top polycrystalline film. XRD measurements on polycrystalline films (metal/2D material) interfaces were taken into account41,47. Our provided in Supplementary Fig. 12 confirmed the random values are in line with these estimations and suggest that in orientation of the grains. On the contrary, as the film thickness sufficiently thin films, phonons (especially long-wavelength becomes thinner (t < 12 nm), thermal phonons (especially long- phonons) can directly propagate between the metal and substrate wavelength phonons) are not strongly scattered by the reduced without being strongly scattered by interfaces. number of grain boundaries available within the volume of the To further confirm the robustness of our approach to measuring materials, and both crystalline and polycrystalline films show the intrinsic cross-plane thermal conductivity of thin films, we similar kz values. performed Raman thermometry measurements in a few layers of In Fig. 3c, we also show the kz obtained from first-principles crystalline PtSe2 films (2, 3, and 4 layers of PtSe2, see DFT-BTE calculations, both, for bulk PtSe2 and thickness- Supplementary Fig. 8), where we found kz values in good dependent values as two different sets of data. The first set agreement with the FDTR results (see Supplementary Table 1). accounts solely for the finite thickness of the film and gives an The agreement between Raman and FDTR experiments further overall much larger thermal conductivity than the measured one supports our Rtot calculations and shows that it is possible to at the corresponding thickness (red data points). The over- quantify the interfacial thermal contributions in a multilayer estimation of the calculated thermal conductivity for the whole structure from a single set of FDTR measurements despite the low thickness range can be related to the underestimation of the c- phase sensitivity to G1 and G2. Note that in Raman thermometry axis compared with the experimental data and the assumption of measurements we have only the contribution of R2 to the total a perfect single-crystal structure of the films. In addition, G1 and G2 thermal resistance. were not taken into account in the DFT calculations, which further Next, we estimate the room temperature phonon MFP in the explains the overestimated values in the calculated thickness- cross-plane direction (Λz) using a simplified expression from the dependence of kz. However, we observe that the trend of kinetic theory, kz ∼(1/3)CυsΛz, where υs is the averaged sound thickness-dependent of the thermal conductivity is preserved. velocity of cross-plane acoustic modes. For PtSe2 we used C Finally, correcting the BTE calculation for the observed thickness- ∼1.78 MJ m−3 K−149, while kz q and υs were determined experimen- ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dependent reduction of sound velocity provides a better −1 −1 tally (kz ∼4.6 Wm K , υs ∼ 3 ðv13 þ v23 Þ) through our FDTR and 3 1 agreement between theory and experiments over the whole LA TA range of thickness considered (yellow-black diamonds). ASOPS measurements described above. From this estimation, we The observed linear dependence of kz from four monolayers up found Λz ∼5 nm, which corresponds to a thickness of approxi- to a certain thickness in both sets of samples is consistent with mately ten layers. However, our FDTR experimental data suggest ballistic heat conduction processes, as has been shown in previous that in crystalline PtSe2 films phonons with Λz > 24 nm substan- works, where the thermal conductivity increased linearly with the tially contribute to the thermal conductivity (about 37% of the characteristic length of the system41,45,46. However, in few-layer bulk value, from our BTE-DFT calculations) while in polycrystalline crystalline PtSe2 films (t < 2–3 nm), we observe a deviation from films grain boundary scattering reduces this value to ~12 nm (see the linear thickness-dependence of kz (see Fig. 3c, black spheres). Fig. 3c). This is in close agreement with our DFT calculations that Since interfaces might dominate cross-plane heat transport in very show that phonons with MFP larger than 15 nm contribute to thin films, the apparent deviation might be attributed to variations almost 55% of the total cross-plane bulk thermal conductivity (see in the total thermal resistance per unit area, Rtot, of the film (see Fig. 4a). calculations below), which can be written as the sum of the In particular, Fig. 4a shows the normalized cumulative thermal combined interface thermal resistance, Rint = 1/G1 + 1/G2, and conductivity as a function of frequency (red axis) and phonon volumetric cross-plane thermal resistance, RPtSe2 = t/kz41,47. Note mean free path (black axis) for bulk PtSe2. We observe a quite that this expression is not valid in very thin films and G1 and G2 narrow cross-plane MFP distribution (1–200 nm) but longer than should be treated as one diffusive interface instead of two discrete the rough estimation from the kinetic theory ~5 nm. This ones41,47. Another possible explanation for this deviation is the discrepancy is well known and comes from the fact that each semiconductor-to-semimetal evolution of PtSe2 films after 4 phonon contributes in a different manner to the thermal monolayers (~2 nm)48, which might result in an additional conductivity41,50–52. An averaged value can be misleading and contribution of the electron thermal conductivity to kz and/or underestimate the real contribution of the phonon MFP to the lower contact resistivity between PtSe2 and Au12. total thermal conductivity. It is interesting to note also the rather To quantify the impact of cross-plane ballistic phonon transport large contribution (~30%) of low-frequency phonons (<1 THz) to on the total thermal resistance of multilayer PtSe2 films, we the total thermal conductivity. These phonons modes are sensitive plotted the total thermal resistance, Rtot = RPtSe2 + Rint, versus film to the introduction of additional periodicity and their dispersion thickness (Fig. 3d). We observe that in crystalline PtSe2 Rtot relation can be easily tuned through nanofabrication53,54. This remains almost constant in the entire thickness range (see black opens the possibility of exploring alternative ways of tuning npj 2D Materials and Applications (2022) 32 Published in partnership with FCT NOVA with the support of E-MRS A. El Sachat et al. 7 Fig. 4 First-principles DFT calculations. a Normalized thermal conductivity accumulation (kacc) as a function of the phonon frequency (red curve) and MFP (blue curve) for bulk PtSe2. b Calculated bulk phonon mean free as a function of frequency. c Phonon dispersion relations and d group velocity of bulk PtSe2 along high-symmetry directions in the Brillouin zone. thermal conduction through phonon engineering in two- Last, we showed that acoustic phonons in PtSe2 thin films, which dimensional materials. are the main carriers of heat in semiconductors, have extra- Furthermore, in Fig. 4b we show that the MFPs in the cross- ordinarily short lifetimes in the order of picoseconds. Our results plane direction of the acoustic modes may be as high as 200 nm. provide new insight into the heat transport and phonon dynamics Remarkably, there is a large number of optical modes with MFPs in 2D materials at the nanoscale, with potential implications for between 1 to 10 nm. This leads to a significant contribution of the the future design of 2D-based devices for energy harvesting and optical modes to the thermal conductivity of bulk and PtSe2 films, effective heat dissipation in thermoelectric and optoelectronic as shown in Fig. 4a and Supplementary Fig. 10. Finally, DFT-BTE devices. calculations provide further insight into the heat transport mechanisms in PtSe2. Figure 4c shows the phonon dispersions of bulk PtSe2 along high-symmetry paths in the Brillouin Zone. As METHODS opposed to 2H phases, we observe no gap between the optical Molecular beam epitaxy growth and the acoustic modes. It is worth noting that the high-frequency Pt and Se were respectively evaporated thanks to an e-gun evaporator and optical modes are unusually dispersive in the Γ-A direction if a standard effusion cell. The Pt deposition rate was set to 0.75 Å min−1 and compared to other layered materials, for which optical phonons monitored in real-time by a quartz crystal microbalance. The pressure of Se dispersions are usually flat55. This can be seen in Fig. 4d which at the sample position was measured with a retractable ionization gauge displays the group velocities along the same k-point paths. Figure 4d and set to 1.0 × 10−6 mbar, which corresponds to a Se:Pt flux ratio of about also shows that the speed of sound in the Γ-A direction, averaged 15. The MBE reactor is connected under UHV to a magnetron sputtering over the three acoustic branches, is 1930 m s−1 (vLA = 2120 m s−1). chamber with a Pt target. The Pt deposition rate was set to 4.7 Å min−1 In conclusion, we have studied phonon dynamic properties and thanks to a retractable quartz crystal microbalance, the DC magnetron heat transport in MBE-grown crystalline and polycrystalline PtSe2 power was 1 W, and the argon pressure was 1.2 × 10−2 mbar. The ZnO thin films of varying thickness (1 to 40 ML) using a combination of (0001) substrates (CrysTec GmbH) were first etched for 30 s with an HCl characterization techniques, i.e., FDTR, low-frequency Raman, and 1.8% solution and rinsed with deionized water. This chemical treatment pump-probe coherent phonon spectroscopies, and state-of-the- was followed by annealing for 1 h at 900 °C in an O2 atmosphere, then art DFT calculations. Our work demonstrates the ability to quantify annealing in UHV for 5 min at ~800 °C. the influence of thickness and crystallinity on the cross-plane heat propagation in thin layered PtSe2 films, showing an almost 35% X-ray diffraction measurements reduction in the thermal conductivity of polycrystalline films with Grazing-incidence x-ray diffraction measurements were performed with a thickness larger than 12 nm in comparison with crystalline films of SmartLab Rigaku diffractometer. The source is a rotating anode beam tube the same thickness. Moreover, from the phonon dynamic study in (Cu Kα = 1.54 Å) operating at 45 kV and 200 mA. The diffractometer is crystalline PtSe2, we extract an out-of-plane elastic constant C33 = equipped with a parabolic multilayer mirror and in-plane collimators of 0.5° 31.8 GPa and a layer-dependent group velocity ranging from on both source and detector sides defining the angular resolution. A Kβ 1340 ms−1 in bilayer PtSe2 to 1873 ms−1 in eight layers of PtSe2. filter on the detector side eliminates parasitic radiations. Published in partnership with FCT NOVA with the support of E-MRS npj 2D Materials and Applications (2022) 32 A. El Sachat et al. 8 Density functional theory calculations 8. Liu, Y., Zhang, S., He, J., Wang, Z. M. & Liu, Z. Recent progress in the fabrication, We computed the lattice thermal conductivity of 1 T (octahedral) PtSe2 by properties, and devices of heterostructures based on 2D materials. Nano Micro density functional theory (DFT) and the Boltzmann transport equation Lett. 11, 13 (2019). (BTE). In the DFT calculations, we used a van der Waals density functional 9. Li, D. et al. Recent progress of two-dimensional thermoelectric. Mater. Nano Micro with consistent exchange (vdW-DF-CX) that reliably predicts the structural Lett. 12, 36 (2020). and vibrational properties of several 2D materials, including transition 10. Gong, Y. et al. 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Advanced capabilities for materials modelling with Quantum Open Access This article is licensed under a Creative Commons ESPRESSO. J. Phys. Condens. Matter 29, 465901 (2017). Attribution 4.0 International License, which permits use, sharing, 61. Eriksson, F., Fransson, E. & Erhart, P. The hiphive package for the extraction of high‐ adaptation, distribution and reproduction in any medium or format, as long as you give order force constants by machine learning. Adv. Theory Simul. 2, 1800184 (2019). appropriate credit to the original author(s) and the source, provide a link to the Creative 62. Barbalinardo, G., Chen, Z., Lundgren, N. W. & Donadio, D. Efficient anharmonic Commons license, and indicate if changes were made. The images or other third party lattice dynamics calculations of thermal transport in crystalline and disordered material in this article are included in the article’s Creative Commons license, unless solids. J. Appl. Phys. 128, 135104 (2020). indicated otherwise in a credit line to the material. If material is not included in the 63. Maassen, J. & Lundstrom, M. Steady-state heat transport: ballistic-to-diffusive with article’s Creative Commons license and your intended use is not permitted by statutory Fourier’s law. J. Appl. Phys. 117, 035104 (2015). regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons. org/licenses/by/4.0/. ACKNOWLEDGEMENTS This work has been supported by the Severo Ochoa program, the Spanish Research © The Author(s) 2022 Agency (AEI, grant no. SEV-2017-0706), and the CERCA Program/Generalitat de Catalunya. Published in partnership with FCT NOVA with the support of E-MRS npj 2D Materials and Applications (2022) 32