## Abstract

We present a comparative investigation of the influence of crystallinity and film thickness on the acoustic and thermal properties of layered PtSe_{2} 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 PtSe_{2} 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 PtSe_{2} films (~30%). Moreover, we show strong interlayer interactions in PtSe_{2}, 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 PtSe_{2}. The potential of tuning the lattice thermal conductivity of layered materials with the level of crystallinity and the 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.

## Introduction

Atomically-thin 2D semiconductors have attracted immense attention in the scientific community due to their exceptional layer-dependent optical, electronic, and thermal properties that open new prospects in the microelectronics industry^{1}. Vertical devices consisting of one-or few-atom thick 2D materials, where heat transport usually occurs in the vertical direction, have already shown excellent performance in diodes^{2}, photodetectors^{3}, transistors^{4,5}, and solar cells^{6}. In such devices, the interfacial thermal properties of atomically-thin layered 2D materials significantly vary depending on their thickness, interlayer interactions, and degree of bonding with the substrate. It is therefore essential to understand cross-plane thermal transport and phonon dynamics, particularly in noble metal dichalcogenides, like PtSe_{2}, due to their large potential for integration in future high-performance 2D devices^{7,8,9,10}.

PtSe_{2} exhibits outstanding inherent properties, including high carrier mobility at room temperature, which is eight times larger than MoS_{2}^{11}, excellent stability to air, and resistance to oxidation, better than black phosphorous^{5}. Moreover, PtSe_{2} can be easily integrated into practical devices since it can be grown at low temperatures^{12}. Together with its widely tunable bandgap^{13} and layer-dependent semiconductor-to-semimetal transition behavior^{14}, it is considered a promising material to be employed in many electronic^{5}, optoelectronic^{7} and thermoelectric devices^{15}. For instance, Moon et al.^{15}. have recently shown that band engineering by thickness modulation leads to a 50-fold enhancement of the thermopower in bilayer PtSe_{2} nanosheets with respect to bulk PtSe_{2}. Moreover, calculated results have shown that in monolayer PtSe_{2} compressive or tensile strain can induce significantly enhanced n- or p-type Seebeck coefficients^{16}. Despite the large potential of PtSe_{2} in thermoelectric applications, experimental studies of the intrinsic thermal properties^{17} and phonon dynamics of PtSe_{2}^{18} films are few and still limited.

In this work, we study the phonon dynamics and thermal properties of supported crystalline and polycrystalline PtSe_{2} thin films of varying thickness (1 to 40 layers), which were grown by molecular beam epitaxy (MBE) on zinc oxide (ZnO) substrates. First, by using a combination of low-frequency Raman and pump-probe coherent phonon spectroscopies we investigate the layer-breathing modes (LBM) in PtSe_{2} thin films and extract an effective out-of-plane elastic constant, a layer-dependent sound velocity, and the acoustic phonon lifetimes. Then, we focus on unraveling the impact of crystallinity and size effects on the cross-plane thermal conductivity of supported PtSe_{2} thin films taking into account the interfacial thermal resistances in our multilayer sample geometry. Finally, we investigate the thermal conductivity by the first-principles Boltzmann transport equation (BTE) computing the harmonic and anharmonic force constants by density functional theory (DFT) to reveal the microscopic mechanism of heat transport in PtSe_{2} thin films.

## Results and discussion

### Material growth, structural characterization, and phonon dynamics

Two PtSe_{2} wedges were grown under ultrahigh vacuum (base pressure in the low 10^{–10} mbar range) in an MBE chamber equipped with a cryo-panel and a reflection high energy electron diffraction (RHEED) setup. For the crystalline PtSe_{2} wedge, four monolayers of PtSe_{2} were deposited by co-evaporating Pt and Se on the ZnO (0001) substrate kept at 450 °C. The resulting RHEED patterns were anisotropic with a 7° of mosaicity (see Supplementary Fig. 1), demonstrating the good-crystalline character of the film, and the epitaxy relationship was found to be ZnO (0001) [100]/PtSe_{2} (111) [100]. Additional characterization of epitaxial ZnO/PtSe_{2} is reported in reference ^{11}. For the polycrystalline PtSe_{2} wedge, we first deposited a 2.4-Å-thick Pt film by magnetron sputtering at room temperature and selenized it in the MBE chamber by deposition of Se at room temperature and subsequent annealing at 750 °C under Se flux. The magnetron sputtering reactor and the MBE chamber are being connected under ultrahigh vacuum and PtSe_{2} films are fully grown in situ. After selenization, the equivalent PtSe_{2} thickness was two monolayers (2 ML). The RHEED patterns were streaky but isotropic, showing the in-plane polycrystalline character of the film, with a well-defined (0001) surface for all grains but random in-plane crystal orientation. Two more PtSe_{2} films were finally deposited by co-evaporating Pt and Se at 450 °C to obtain a 4 ML thick polycrystalline film (see Supplementary Fig. 1c).

In a second step, we covered the sample with a motorized mechanical mask except for a thin 1.5-mm-large band at the edge to deposit 14 ML of PtSe_{2} by co-evaporating Pt and Se at 450 °C. The mask was then retracted at a constant speed and stopped at 1 mm from the edge to evaporate the 0–22 ML PtSe_{2} wedge by co-evaporation at the same temperature. We checked by RHEED that the films retained their respective crystalline and polycrystalline character at the end of the growth. The final structure is shown in Supplementary Fig. 1d. Each PtSe_{2} monolayer consists of three atomic sublayers, in which Pt atoms are sandwiched between Se atoms (see Fig. 1a). Both samples were annealed at 750 °C for 10 min under Se flux after the growth in order to improve their crystalline quality. To avoid the film degradation during air transfer, the films were capped by a ~10-nm-thick amorphous Se layer deposited at room temperature.

The crystalline, acoustic and morphological characterizations of the samples were studied by Raman spectroscopy and high-resolution scanning transmission electron microscopy (HR-STEM) measurements. The arrangements of Pt and Se atoms in the planar HR-STEM image (Fig. 1b) and the symmetric diffraction pattern in the fast Fourier transform (FFT) image (Fig. 1c) further support the good crystal structure nature of the films. In addition, the high-magnification cross-sectional HR-STEM images (Fig. 1d, e) show the layered hexagonal honeycomb structure of a five layers PtSe_{2} film, where bright Pt atoms are surrounded by six lighter colored Se atoms. The in-plane and out-of-plane lattice parameters were obtained from the (*l* 0 0) and (*l m* 0) planes of the FFT and image analysis of cross-plane sections of the film (see Fig. 1e), respectively. The in-plane lattice constants were found to be *a* = *b* ≈ 0.37 nm and *c* ≈ 0.52 nm in good agreement with previous studies^{10,15,19} and in excellent agreement with the ones deduced from our x-ray diffraction measurements (see Supplementary Fig. 2). On the other hand, DFT calculations give *a* = 0.377 nm and *c* = 0.486 nm. Whereas *a* is in excellent agreement with our measurements, *c* is significantly underestimated (6.5%). There may be multiple reasons for this discrepancy, among which the difficulty in the DFT calculations to simulate PtSe_{2} and PtSe compounds for all the GGA exchange functionals (e.g., PBEsol, AM05)^{20} and even van der Waals functionals (e.g., vdW-DF-C09 and vdW-DF-CX)^{21}.

The layered PtSe_{2} adopts a T-type hexagonal crystal structure, belonging to the \(P\bar 3m1\) space group with a D_{3d} (-m) point group. The primitive PtSe_{2} cell contains 3 atoms, then its vibrational spectrum includes nine modes: three acoustic (A_{2u} + E_{u}) and six optical (A_{1g} + E_{g} + 2A_{2u} + 2E_{u}). The optical modes can be classified as Raman active (E_{g}, A_{1g} and LO (A_{2u} + E_{u})) and infrared active (2E_{u} + 2A_{u}) modes^{22}. Fig. 1f, g show the Raman spectra of crystalline- and polycrystalline PtSe_{2} films of different thicknesses, respectively. The high-frequency modes (170–210 cm^{−1}) are originated from in-plane and out-of-plane vibrations of Se atoms corresponding to E_{g} (~180 cm^{−1}) and A_{1g} (~205 cm^{−1}) modes, respectively. In addition, a weak interlayer longitudinal optical (LO) mode can be also observed at ~235 cm^{−1}. LO is generated by a combination of the in-plane E_{u} and out-of-plane A_{2u} modes from the vibrations of Pt and Se atoms in the opposite phases. The Raman peaks located in the low-frequency regions (−30 to 30 cm^{−1}) correspond to interlayer vibrations of PtSe_{2} planes^{5,18}.

Specifically, the vibrations detected here are related to the out-of-plane displacements of the PtSe_{2} layers known as layer-breathing modes (LBM). As shown in Fig. 1f, g, the position of the LBM shifts to lower frequencies as the number of layers increases. The thickness-dependence can be simulated by using a one-dimensional linear atomic chain model^{23}. This model considers each PtSe_{2} layer as a large atom with an effective mass per unit area *μ* = 4.8 × 10^{−6} kg m^{−2} connected by a string with an effective interlayer breathing force constant (IBFC) per unit area *K*_{⊥} and separated by a distance *d* ≈ 0.52 nm given by the interatomic distance between layers (see Fig. 1a). The solution of the linear chain with vanishing stress as a boundary condition on the free surface is given by:

where *f* is the vibrational frequency of the mode, *q*_{N, j} = 2*π*/*λ*_{N,j} is the acoustic wavevector, *λ*_{N, j} = 2*Nd*/*j* is the phonon wavelength, *N* is the number of layers (*N* = 2, 3, …) and *j* is the index of the acoustic mode (*j* = 1, 2, 3…). The experimental and the fitted thickness-dependence of the first LBM (*j* = 1) is shown in Fig. 2a. The fitted curve was obtained using the IBFC adjustable variable, which is determined to be *K*_{⊥} = 6.0 ± 0.14 × 10^{19} N m^{−3}. A similar value was also obtained by Chen et al.^{18} with *K*_{⊥} = 6.2 × 10^{19} N m^{−3} using the same approach. Now, if we multiply the IBFC by the interlayer distance, we can also derive the corresponding C_{33} component of the elastic constant tensor^{24,25}.

For our system, we found C_{33} = 31.8 ± 0.95 GPa, which is in the same range as other 2D materials e.g., 38.7–36.5 GPa for graphite, 54.3 GPa for MoSe_{2}^{26}, 24.5 GPa for h-BN^{27}, 52 GPa and 52.1 GPa for MoS_{2} and WSe_{2}, respectively^{25}. It is interesting to notice that the polycrystalline films also follow the same trend as the crystalline samples (see Fig. 2a) thus most likely the granular characteristics of the sample do not affect significantly the interlayers forces. Likewise, it is also possible to obtain the C_{44} constant using the effective interlayer shear force constant (*K*_{//}). Chen et al.^{18}. calculated *K*_{//} = 4.6 × 10^{19} N m^{−3} for PtSe_{2} based on density functional theory simulations from Zhao^{5}, giving C_{44} = 23 GPa and *v*_{TA} = 1553 ms^{−1}. In addition, from Eq. (1), it is also possible to extract the group velocity considering that *v*_{g} = *d*(2π*f*)/*dq*.

In the limit N → ∞ and considering the experimental interlayer distance *d* ≈ 0.52 ± 0.01 nm, Eq. (2) gives the bulk limit for the cross-plane longitudinal velocity *v*_{LA} = 1873 ± 42 ms^{−1}. A similar value can be obtained using an approximation of \(v_g = \sqrt {\frac{{C_{33}}}{\rho }} =\)1825 ± 28 ms^{−1} where *ρ* = 9540 kg m^{−3} is the density of PtSe_{2} and C_{33} = 31.8 GPa. Using a numerical differentiation, it is also possible to extract the group velocity as a function of the number of layers as shown in Fig. 2b.

Furthermore, the phonon dynamics of the crystalline PtSe_{2} samples were measured by pump-probe coherent phonon spectroscopy using the asynchronous optical sampling method (ASOPS)^{28,29} (see details in Methods). Figure 2c shows the typical modulation of the reflectivity (ΔR/R) as a function of the time delay (black open circles) and the corresponding best model fit in a 5 ML PtSe_{2} crystalline film. The change of reflectivity includes the effects of a fast and slow electronic relaxation process as well as the dynamics of the generated phonons. The time traces are fitted to a damped harmonic oscillator of the form:^{18}

where *t* is the time delay between the pump and the probe lasers, A and B are the amplitudes, τ_{ι} are the decay times, *f* the phonon frequency, and *ϕ* is a phase delay. The first term of Eq. (3) represents the fast and slow relaxation processes of the excited carriers. The second term describes the damped phonon oscillations. The Fast Fourier transforms (FFT) of the time-domain data for all measured samples are displayed in the Supplementary.

Figure 2d shows the measured decay time of LBM as a function of the number of layers (yellow dots) and the calculated phonon decay due to boundary scattering contribution based on the model proposed by Ziman^{30}. This model considers the impact of the roughness (*η*) of the film surface through a phenomenological parameter *p* = exp[−16π^{2}*η*^{2}/*λ*^{2}], which modifies the effective mean free path Λ = (1+p)/(1-p) N*d*/*v*_{g} of the acoustic waves. The black and red curves of Fig. 2d were calculated using a thickness-dependent group velocity (Eq. (2)) and a constant value given by the bulk limit (1873 m s^{−1}), respectively, with a constant *η* = 0.82 nm as measured by AFM (see Supplementary Fig. 3).

Apart from the good agreement between the theory and the experimental data, we observe that the phonon lifetime is not affected by the group velocity (thickness-dependent or constant). Similar observations of boundary-limited lifetimes have been reported in suspended MoSe_{2} flakes by Soubelet et al.^{26}. They measured lifetimes of the order of ~1–41 ps in 2–8 ML limited by boundary effects. While in thicker samples (>20 ML), they observed longer lifetimes (~0.3–10 ns) limited by phonon-phonon scattering. The large difference in lifetimes measured here (1–4 ps) is mainly attributed to the difference in the phonon group velocity which is 1.4 to 1.5 times smaller than MoSe_{2.}

### Thermal conductivity and interfacial heat transport measurements

For the thermal measurements, we used a custom-built frequency-domain thermo-reflectance (FDTR) setup, a well-established optical pump-probe technique, capable of measuring heat transport in thin films and across interfaces^{31,32,33,34,35,36}. A schematic representation of the experimental setup is shown in Fig. 3a. A modulated pump beam with a wavelength of 448 nm is focused onto the sample, creating a periodic heat flux with a Gaussian spatial distribution on the sample surface. A reflected probe beam with a wavelength of 532 nm is aligned coaxially with the pump beam and focused with the pump spot to monitor the periodic fluctuations in reflectivity at the sample surface caused by the oscillating sample temperature. All the samples after the MBE growth were coated with a 100 nm thick Au layer, which was chosen to maximize the coefficient of thermo-reflectance at the probe wavelength (∼2.36 × 10^{−4} K^{−1}). We used a lock-in amplifier (Zurich Instruments HF2LI) to record the amplitude and phase response of the reflected probe beam to the thermal wave, and the phase lag between the pump and probe beam as the observable quantity. More details on the experimental setup can be found elsewhere^{35}.

In the present experiments, we obtained frequency-domain measurements by varying the modulation frequency of the pump beam over a wide range (20 kHz–40 MHz). The cross-plane thermal conductivity (*k*_{z}) of the PtSe_{2} thin films was subsequently extracted by following a multilayer three-dimensional (3D) heat diffusion model that includes the interface thermal conductance between the different layers and anisotropic heat transport^{31}. For each experiment, we quantified first the sensitivity of the recorded phase signal to different parameters: *k*_{z}, the volumetric heat capacity (*C*), the thermal conductivity anisotropy ratio (*k*_{r}/*k*_{z}), and the different interface thermal conductances, in a similar manner to that of Schmidt et al.^{31}. The details about the measurement sensitivity to different combinations of parameters in the model can be found in Supplementary Fig. 4. In all the experiments we used a high-magnification 50× objective lens that produced a focused root mean square (rms) spot size (1/e^{2} radius) of approximately ~1.5 μm. To reduce the uncertainties associated with variations in the laser spot size between measurements on different films, the spot size was measured for each different experiment using the knife’s edge method, as shown in Supplementary Fig. 6. All the room temperature measurements were performed under both ambient and vacuum conditions.

Before the thermal measurements on thin PtSe_{2} films, we performed FDTR measurements on a bulk PtSe_{2} crystal coated with 100 nm of Au in order to extract the anisotropy ratio of the thermal conductivities. The thermal conductivity of the deposited Au film was first measured using electrical conductivity and the Wiedemann−Franz law, *k*_{Au} = 195 Wm ^{−1} K^{−1} (see details in Supplementary Fig. 7). The volumetric specific heat of Au and PtSe_{2} were taken from literature^{10,36}. This leaves us with three unknowns: the cross-plane and in-plane thermal conductivity of the bulk PtSe_{2} crystal and the Au-PtSe_{2} interface thermal conductance (*G*_{Au-bulk}). The in-plane thermal conductivity was expressed in terms of the anisotropic ratio (*α*) and the cross-plane thermal conductivity *k*_{r} = *α* ·*k*_{z}.

However, from the sensitivity analysis, we found that the recorded phase signal has low sensitivity to *G*_{Au-bulk} and only at high frequencies while its sensitivity to the anisotropic heat flow is high in almost the whole frequency range (see Supplementary Fig. 5). This allowed us to extract *α* directly from the model fit of the experimental data in the low-frequency range (20 kHz–1 MHz) using a nonlinear least-squares routine, which requires an initial guess to determine the value of the free parameter (*k*_{z}) (see Supplementary Fig. 11). The inset in Supplementary Fig. 11 shows the numerical errors of the applied model fits as a function of different anisotropy ratios. The minimum value of the fitting error, which gives us the best fit of the FDTR data, corresponds to *α* = 9, *k*_{z} = 4.6 ± 0.7 Wm^{−1} K^{−1} and *k*_{r} ≈ 41.4 Wm^{−1} K^{−1}.

The observed anisotropy ratio between *k*_{r} and *k*_{z} in PtSe_{2} is similar to previously reported values of other 2D-layered materials, such as InSe (*α* ~10)^{37}, SnSe_{2} (*α* ~7–8)^{35,38}, and armchair black phosphorous (*α* ~6–10)^{39}. Our first-principles BTE calculations give *k*_{z} = 2.84 W m^{−1} K^{−1} and *k*_{r} = 39.0 W m^{−1} K^{−1}. *k*_{r} is in excellent agreement with the measurement, while *k*_{z} is significantly overestimated, possibly due to the discrepancy in the equilibrium *c* lattice parameter mentioned above. Nevertheless, we can still use DFT and BTE to analyse the phonon contributions to cross-plane transport in PtSe_{2} bulk and the effect of finite thickness in thin films, at least qualitatively.

To study the impact of crystallinity and film thickness on the cross-plane thermal conductivity of PtSe_{2} thin films, we performed FDTR measurements in large-area crystalline and polycrystalline PtSe_{2} films of different thicknesses (1–40 ML) on ZnO substrate. The wedge sample stacks consist of Au/PtSe_{2}/ZnO (see Fig. 3a and Supplementary Fig. 1). Typical examples of the recorded phase signals and the corresponding best model fit for bulk, monolayer, and 40 ML crystalline PtSe_{2} are shown in Fig. 3b. Here, the key parameters for the model in our multilayer system are the spot sizes of the pump and probe beams, *k*_{z}, the film thickness *t*, the *C* of each layer, and the interface thermal conductance (between Au and PtSe_{2}, *G*_{1}, and between PtSe_{2} and ZnO, *G*_{2}). The thickness of Au and PtSe_{2} films were measured by AFM and low-frequency Raman spectroscopy, respectively (see section above and Supplementary). The heat capacities of Au, ZnO, and PtSe_{2} were taken from the literature^{10,36,40}. Thus, the remaining unknown parameters were the *k*_{z} value of the PtSe_{2} films, *G*_{1} and *G*_{2}.

To extract a unique value of *k*_{z} from a single measurement we followed a fitting approach similar to that suggested in the previous works^{31,32,35} and supported by our sensitivity analysis (see Supplementary Fig. 4). First, we estimate *k*_{z} by fitting to experimental data in a low-frequency range (20 kHz–1 MHz), where the measurement sensitivity to the *G*_{1}, *G*_{2}, and heat capacity of the films is negligible. Then, we fix *k*_{z} and fit experimental data in a high-frequency range (1–40 MHz) to estimate simultaneously *G*_{1} and *G*_{2}. As an initial guess to determine the value of the free parameters (*k*_{z}, *G*_{1}, and *G*_{2}) we used previously reported values of similar material systems^{41,42,43}. We also verified that the final fit results are not sensitive to the choice of initial values. The same analysis has been followed to extract *k*_{z}, *G*_{1}, and *G*_{2} values for all the PtSe_{2} films.

In Supplementary Fig. 13 we show all the interface thermal resistance measurements, *R*_{1} = 1/*G*_{1}, *R*_{2} = 1/*G*_{2}, and *R*_{int} = *R*_{1} + *R*_{2}, extracted by the FDTR experiments. The observed variation of *R*_{int} might be attributed to differences in the interface characteristics of Au (transducer)/PtSe_{2} and PtSe_{2}/ZnO (substrate), which could result in different interfacial energy coupling. The apparent increase of *R*_{2} with increasing film thickness most likely is due to the reduced mechanical coupling of the films to the underlying ZnO resulting from a potential increase in sample stiffness or changes in the vibrational spectra^{44}. However, we note that most of the extracted values are within the experimental uncertainty, thus it is not clear if there is a systematic increase of *R*_{2} with thickness. Similarly, a weak thickness-dependent thermal resistance trend has been measured in InSe, MoS_{2}, and SnSe_{2} thin films^{35,37,41}.

It is interesting to note that, in contrast to the bulk case, in FDTR measurements in thin films, the diameter of the laser spot is usually large compared to the thermal diffusion length during the modulation period of the pump beam. Thus, the heat flow is expected to be mainly one-dimensional in the cross-plane direction^{32}. This has been confirmed by our sensitivity analysis, where we found that the sensitivity of the recorded phase signal to the in-plane transport is relatively low (see Supplementary Fig. 4c). Therefore, since anisotropic differences in thermal conductivities as a function of the film thickness cannot be resolved, for the data analysis we used the anisotropy ratio extracted from the bulk PtSe_{2} experiments. The dependence of the thermal anisotropy with the film thickness has been studied recently in similar material systems, such as supported InSe^{37} and SnSe_{2}^{35} films, showing a thickness-independent *α* of 10 and 8.4, respectively.

Figure 3c displays the extracted *k*_{z} value of all the PtSe_{2} thin films as a function of thickness. In both crystalline (black data points) and polycrystalline films (green data points), we observe a linear increase of *k*_{z} with increasing the film thickness starting from 2.4 nm (4 ML) up to 10 and 24 nm, respectively. In polycrystalline films with thicknesses *t* > 12 nm, *k*_{z} shows a plateau, and a maximum cross-plane thermal conductivity reduction of ~35% was observed compared to the crystalline samples. This result highlights the strong impact of crystallinity on *k*_{z} and suggests that depending on the film crystallinity phonons with different mean free paths (MFPs) contribute to *k*_{z}. In crystalline PtSe_{2} films, heat is propagating through coherent vibrations (phonon modes) that travel distances at least 24 nm (~40 ML) while in polycrystalline samples, they start to decay above 12 nm (~20 ML). This behavior most likely is attributed to the enhanced phonon scattering due to the presence of high-density defective grain boundaries randomly oriented in the polycrystalline film. XRD measurements on polycrystalline films provided in Supplementary Fig. 12 confirmed the random orientation of the grains. On the contrary, as the film thickness becomes thinner (*t* < 12 nm), thermal phonons (especially long-wavelength phonons) are not strongly scattered by the reduced number of grain boundaries available within the volume of the materials, and both crystalline and polycrystalline films show similar *k*_{z} values.

In Fig. 3c, we also show the *k*_{z} obtained from first-principles DFT-BTE calculations, both, for bulk PtSe_{2} and thickness-dependent values as two different sets of data. The first set accounts solely for the finite thickness of the film and gives an overall much larger thermal conductivity than the measured one at the corresponding thickness (red data points). The overestimation of the calculated thermal conductivity for the whole thickness range can be related to the underestimation of the *c*-axis compared with the experimental data and the assumption of a perfect single-crystal structure of the films. In addition, *G*_{1} and *G*_{2} were not taken into account in the DFT calculations, which further explains the overestimated values in the calculated thickness-dependence of *k*_{z}. However, we observe that the trend of thickness-dependent of the thermal conductivity is preserved. Finally, correcting the BTE calculation for the observed thickness-dependent reduction of sound velocity provides a better agreement between theory and experiments over the whole range of thickness considered (yellow-black diamonds).

The observed linear dependence of *k*_{z} from four monolayers up to a certain thickness in both sets of samples is consistent with ballistic heat conduction processes, as has been shown in previous works, where the thermal conductivity increased linearly with the characteristic length of the system^{41,45,46}. However, in few-layer crystalline PtSe_{2} films (*t* < 2–3 nm), we observe a deviation from the linear thickness-dependence of *k*_{z} (see Fig. 3c, black spheres). Since interfaces might dominate cross-plane heat transport in very thin films, the apparent deviation might be attributed to variations in the total thermal resistance per unit area, *R*_{tot}, of the film (see calculations below), which can be written as the sum of the combined interface thermal resistance, *R*_{int} = 1/*G*_{1} + 1/*G*_{2}, and volumetric cross-plane thermal resistance, *R*_{PtSe2} = t/*k*_{z}^{41,47}. Note that this expression is not valid in very thin films and *G*_{1} and *G*_{2} should be treated as one diffusive interface instead of two discrete ones^{41,47}. Another possible explanation for this deviation is the semiconductor-to-semimetal evolution of PtSe_{2} films after 4 monolayers (*~*2 nm)^{48}, which might result in an additional contribution of the electron thermal conductivity to *k*_{z} and/or lower contact resistivity between PtSe_{2} and Au^{12}.

To quantify the impact of cross-plane ballistic phonon transport on the total thermal resistance of multilayer PtSe_{2} films, we plotted the total thermal resistance, *R*_{tot} = *R*_{PtSe2} + *R*_{int}, versus film thickness (Fig. 3d). We observe that in crystalline PtSe_{2} *R*_{tot} remains almost constant in the entire thickness range (see black spheres), with values approximately of 28 ± 10 m^{2} KGW^{−1}, while in polycrystalline samples (see green spheres) *R*_{tot} is constant only up to 12 nm and then starts to increase up to 75 ± 10 m^{2} KGW^{−1}. The large contribution of the volumetric resistance component, *R*_{PtSe2}, to *R*_{tot} and the thickness-independent *R*_{tot} in crystalline films indicate ballistic phonon transport along the *c*-axis of the films. Similar results has been previously observed in few-layer-graphene and single-crystal MoS_{2} flakes that showed a thickness-independent *R*_{tot} for film thicknesses between 0.4–4 nm^{47} and 20–40 nm^{41}, respectively.

The increased values of *R*_{tot} in polycrystalline films after 12 nm indicate a transition from ballistic to diffusive transport regime most likely due to phonon grain boundary scattering. For comparison, in Fig. 3d we plot our calculated *R*_{tot} (black and green spheres) with previous total cross-plane thermal resistance measurements on different 2D materials, where similar interfacial contributions from the bottom (2D material/substrate) and top (metal/2D material) interfaces were taken into account^{41,47}. Our values are in line with these estimations and suggest that in sufficiently thin films, phonons (especially long-wavelength phonons) can directly propagate between the metal and substrate without being strongly scattered by interfaces.

To further confirm the robustness of our approach to measuring the intrinsic cross-plane thermal conductivity of thin films, we performed Raman thermometry measurements in a few layers of crystalline PtSe_{2} films (2, 3, and 4 layers of PtSe_{2}, see Supplementary Fig. 8), where we found *k*_{z} values in good agreement with the FDTR results (see Supplementary Table 1). The agreement between Raman and FDTR experiments further supports our *R*_{tot} calculations and shows that it is possible to quantify the interfacial thermal contributions in a multilayer structure from a single set of FDTR measurements despite the low phase sensitivity to *G*_{1} and *G*_{2}. Note that in Raman thermometry measurements we have only the contribution of *R*_{2} to the total thermal resistance.

Next, we estimate the room temperature phonon MFP in the cross-plane direction (*Λ*_{z}) using a simplified expression from the kinetic theory, *k*_{z} ∼(1/3)*Cυ*_{s}*Λ*_{z}, where *υ*_{s} is the averaged sound velocity of cross-plane acoustic modes. For PtSe_{2} we used C ∼1.78 MJ m^{−3} K^{−1}^{49}^{,} while *k*_{z} and *υ*_{s} were determined experimentally (*k*_{z} ∼4.6 Wm^{−1} K^{−1}, *υ*_{s} ∼\(\root {3} \of {{\frac{1}{3}(\frac{1}{{v_{LA}^3}} + \frac{2}{{v_{TA}^3}})}}\)) through our FDTR and ASOPS measurements described above. From this estimation, we found *Λ*_{z} ∼5 nm, which corresponds to a thickness of approximately ten layers. However, our FDTR experimental data suggest that in crystalline PtSe_{2} films phonons with *Λ*_{z} > 24 nm substantially contribute to the thermal conductivity (about 37% of the bulk value, from our BTE-DFT calculations) while in polycrystalline films grain boundary scattering reduces this value to ~12 nm (see Fig. 3c). This is in close agreement with our DFT calculations that show that phonons with MFP larger than 15 nm contribute to almost 55% of the total cross-plane bulk thermal conductivity (see Fig. 4a).

In particular, Fig. 4a shows the normalized cumulative thermal conductivity as a function of frequency (red axis) and phonon mean free path (black axis) for bulk PtSe_{2}. We observe a quite narrow cross-plane MFP distribution (1–200 nm) but longer than the rough estimation from the kinetic theory ~5 nm. This discrepancy is well known and comes from the fact that each phonon contributes in a different manner to the thermal conductivity^{41,50,51,52}. An averaged value can be misleading and underestimate the real contribution of the phonon MFP to the total thermal conductivity. It is interesting to note also the rather large contribution (~30%) of low-frequency phonons (<1 THz) to the total thermal conductivity. These phonons modes are sensitive to the introduction of additional periodicity and their dispersion relation can be easily tuned through nanofabrication^{53,54}. This opens the possibility of exploring alternative ways of tuning thermal conduction through phonon engineering in two-dimensional materials.

Furthermore, in Fig. 4b we show that the MFPs in the cross-plane direction of the acoustic modes may be as high as 200 nm. Remarkably, there is a large number of optical modes with MFPs between 1 to 10 nm. This leads to a significant contribution of the optical modes to the thermal conductivity of bulk and PtSe_{2} films, as shown in Fig. 4a and Supplementary Fig. 10. Finally, DFT-BTE calculations provide further insight into the heat transport mechanisms in PtSe_{2}. Figure 4c shows the phonon dispersions of bulk PtSe_{2} along high-symmetry paths in the Brillouin Zone. As opposed to 2H phases, we observe no gap between the optical and the acoustic modes. It is worth noting that the high-frequency optical modes are unusually dispersive in the Γ-A direction if compared to other layered materials, for which optical phonons dispersions are usually flat^{55}. This can be seen in Fig. 4d which displays the group velocities along the same k-point paths. Figure 4d also shows that the speed of sound in the Γ-A direction, averaged over the three acoustic branches, is 1930 m s^{−1} (*v*_{LA} = 2120 m s^{−1}).

In conclusion, we have studied phonon dynamic properties and heat transport in MBE-grown crystalline and polycrystalline PtSe_{2} thin films of varying thickness (1 to 40 ML) using a combination of characterization techniques, i.e., FDTR, low-frequency Raman, and pump-probe coherent phonon spectroscopies, and state-of-the-art DFT calculations. Our work demonstrates the ability to quantify the influence of thickness and crystallinity on the cross-plane heat propagation in thin layered PtSe_{2} films, showing an almost 35% reduction in the thermal conductivity of polycrystalline films with thickness larger than 12 nm in comparison with crystalline films of the same thickness. Moreover, from the phonon dynamic study in crystalline PtSe_{2}, we extract an out-of-plane elastic constant C_{33} = 31.8 GPa and a layer-dependent group velocity ranging from 1340 ms^{−1} in bilayer PtSe_{2} to 1873 ms^{−1} in eight layers of PtSe_{2}. Last, we showed that acoustic phonons in PtSe_{2} thin films, which are the main carriers of heat in semiconductors, have extraordinarily short lifetimes in the order of picoseconds. Our results provide new insight into the heat transport and phonon dynamics in 2D materials at the nanoscale, with potential implications for the future design of 2D-based devices for energy harvesting and effective heat dissipation in thermoelectric and optoelectronic devices.

## Methods

### Molecular beam epitaxy growth

Pt and Se were respectively evaporated thanks to an e-gun evaporator and a standard effusion cell. The Pt deposition rate was set to 0.75 Å min^{−1} and monitored in real-time by a quartz crystal microbalance. The pressure of Se at the sample position was measured with a retractable ionization gauge and set to 1.0 × 10^{−6} mbar, which corresponds to a Se:Pt flux ratio of about 15. The MBE reactor is connected under UHV to a magnetron sputtering chamber with a Pt target. The Pt deposition rate was set to 4.7 Å min^{−1} thanks to a retractable quartz crystal microbalance, the DC magnetron power was 1 W, and the argon pressure was 1.2 × 10^{−2} mbar. The ZnO(0001) substrates (CrysTec GmbH) were first etched for 30 s with an HCl 1.8% solution and rinsed with deionized water. This chemical treatment was followed by annealing for 1 h at 900 °C in an O_{2} atmosphere, then annealing in UHV for 5 min at ~800 °C.

### X-ray diffraction measurements

Grazing-incidence x-ray diffraction measurements were performed with a SmartLab Rigaku diffractometer. The source is a rotating anode beam tube (Cu Kα = 1.54 Å) operating at 45 kV and 200 mA. The diffractometer is equipped with a parabolic multilayer mirror and in-plane collimators of 0.5° on both source and detector sides defining the angular resolution. A Kβ filter on the detector side eliminates parasitic radiations.

### Density functional theory calculations

We computed the lattice thermal conductivity of 1 T (octahedral) PtSe_{2} by density functional theory (DFT) and the Boltzmann transport equation (BTE). In the DFT calculations, we used a van der Waals density functional with consistent exchange (vdW-DF-CX) that reliably predicts the structural and vibrational properties of several 2D materials, including transition metal dichalcogenides^{56,57}. Valence Kohn-Sham wavefunctions are expanded on plane-waves basis set with a cutoff of 40 Ry, and projector augmented pseudopotentials are used to model screened nuclei^{58}. A uniform 8 × 8 × 6 mesh of k-points was used to integrate the first Brillouin Zone. Phonon dispersion relations were computed by density functional perturbation theory interpolating a uniform 4 × 4 × 3 q-point mesh^{59,60}.

### Boltzmann transport equation

Second-order and third-order force constants for the thermal conductivity calculations were computed by fitting the forces of 80 configurations of the 4 × 4 × 3 PtSe_{2} supercells, in which the atoms are randomly displaced with a standard deviation of 0.01 Å using the hiPhive code^{61}. Second-order and third-order force constants were cut off at 6.9 and 5.13 Å, respectively. Fitting employed recursive feature elimination with a limit of 300 features (out of 961). The fit gives a root mean square error on the forces of 0.0044 eV/A with *R*^{2} = 0.999. The solution of the linearized phonon BTE was computed by directly inverting the scattering tensor on a 21 × 21 × 17 mesh of q-points, as implemented in kALDo^{62}. The effect of the finite thickness of the samples is taken into account by including the boundary conditions in the scattering term as proposed by Maassen and Lundstrom^{63}.

### Asynchronous optical sampling measurements

In this technique, two pulsed lasers (pump and probe beams) are focused on the sample surface. The pump produces a change of reflectivity which is measured by the probe beam as a function of the time delay between the lasers. The absorption of the pump laser causes an increase in the local strain via two separate mechanisms, namely thermal expansion, and the hydrostatic deformation potential. The thermal expansion is a consequence of the anharmonicity of the lattice, whereas the deformation potential is due to the excitation of electrons into binding orbitals. Both mechanisms periodically change the effective volume of the material, which in turn modulates the optical properties of the film that are probed by the second laser. In this experiment, we used 2 fs Ti:sapphire lasers with a repetition rate of ∼1 GHz stabilized via an electronic feedback loop to achieve a small repetition rate difference of 2 kHz. The rate difference produces a time delay between the pump and probe pulses without the need for a mechanical delay stage. The delay between the pump pulses is kept at 1 ns, while the pump and probe pulses coincide every 500 ms. The measurements were done at a fixed central wavelength of λ = 790 nm for the pump (~1 mW) and the probe (~0.15 mW) beams, collinearly focused to a ~1-μm-diameter spot on the sample surface.

## Data availability

All relevant data and codes are available from the corresponding author on request.

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## Acknowledgements

This work has been supported by the Severo Ochoa program, the Spanish Research Agency (AEI, grant no. SEV-2017-0706), and the CERCA Program/Generalitat de Catalunya. The authors acknowledge support from the Spanish MICINN project SIP (PGC2018-101743-B-I00), and the EU project NANOPOLY (GA 289061). The LANEF framework (ANR-10-LABX-51-01) is acknowledged for its support of mutualized infrastructure. PX acknowledges support for the Ph.D. fellowship from the EU Marie Sklodowska-Curie COFUND PREBIST (Grant Agreement 754558). AES acknowledges support by the H2020-MSCA-IF project THERMIC-GA No. 101029727. The authors acknowledge Dr. John Cuffe for his critical comments.

## Author information

### Authors and Affiliations

### Contributions

A.E.S. and E.C.-A., conceived the project. A.E.S. and E.C.-A., built the F.D.T.R., setup and performed the thermal measurements and data analysis. A.E.S. performed the A.F.M. measurements. E.C.-A. and P.X., performed the Raman measurements. E.C.-A. and G.A., performed the A.S.O.P.S., measurements and analysis. F.B., A.M., C.V., H.B. and M.J. fabricated the M.B.E. samples and performed the X.R.D., measurements. P.X., M.S., A.E.S. and E.C.-A., performed the T.E.M., measurements and the structural analysis. D.D. and Z.C., performed the D.F.T., calculations and provided support to the theoretical analysis. All authors reviewed and edited the manuscript and have given approval to the final version of the manuscript. The manuscript was written by A.E.S. and E.C.-A., A.E.S. and P.X., contributed equally.

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### Cite this article

El Sachat, A., Xiao, P., Donadio, D. *et al.* Effect of crystallinity and thickness on thermal transport in layered PtSe_{2}.
*npj 2D Mater Appl* **6**, 32 (2022). https://doi.org/10.1038/s41699-022-00311-x

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DOI: https://doi.org/10.1038/s41699-022-00311-x