It is well known that for two-dimensional (2D) materials, the interlayer interaction is vitally important for the layer-dependent properties, such as band structure, carrier mobility, and thermal conductivity. PtSe2, as one of typical group-10 transition metal dichalcogenides (TMDCs), undergoes a dramatical bandgap shrinking from an indirect bandgap semiconductor for monolayer (∼1.2eV) to a semimetal for bulk [1–6], which is expected to be a promising candidate in homogeneous interconnection electronic circuits [7]. The sharp decline of the bandgap with increasing layer thickness in PtSe2 implies that the interaction between adjacent layers might be very large, which is desirable for future device application like high-frequency [terahertz (∼THz)] micro-mechanical resonators [8]. The interlayer interactions can be reflected via detection of interlayer lattice vibrations. Raman spectroscopy is a general method to observe lattice vibrations [9,10]. Although the low-frequency Raman spectroscopy can determine some relatively high interlayer vibrational modes, the ultralow phonon modes (e.g., 10–23 L PtSe2 in this work) that are vital for studying interlayer coupling will be immersed in the laser line due to the limit of the optical filter in Raman spectroscopy. Here, by utilizing the ultrafast pump–probe technique [11], we observe the real time dynamics of two interlayer coherent acoustic phonon modes coexisting in bilayer and few-layer PtSe2. The THz layer-breathing mode (LBM) is in consistence with our low-frequency Raman spectra, while the standing wave mode (SWM), Raman inactive, is observed directly in group-10 TMDCs. The contribution of the adjacent unit cell to the interlayer interaction can be derived from the LBM, proving that PtSe2 indeed owns strong interlayer van der Waals force, much larger than graphene. Meanwhile, we obtain the out-of-plane sound velocity of PtSe2 experimentally from the SWM, in good agreement with the theoretical calculations [12,13].

Large-scale continuous PtSe2 thin films with different layers were synthesized on sapphire substrate by chemical vapor deposition (CVD) growth as described previously [14], and the size of the samples was approximately 10mm×10mm (Sixcarbon Tech, Shenzhen, China). PtSe2 adopts an octahedral structure (1T phase) in the AA arrangement of stacked layers and belongs to the space group P3¯m1 [15]. The number of layers determined by atomic force microscopy (AFM) is 2, 5, 8, 10, 15, 17, and 23 L, respectively, as shown in Figs. 1(a)–1(c) and Appendix A. The Raman spectra from 10cm−1 to 400cm−1 under 532 nm excitation are shown in Fig. 1(d). All the samples show two clear intralayer peaks, Eg and A1g modes. The Eg mode originates from the in-plane vibration of Se atoms and the A1g mode corresponds to the out-of-plane vibration of Se atoms. In addition, there is a rather weak intralayer longitudinal optical (LO) mode, which is the combination of the in-plane Eu and out-of-plane A2u modes from the vibrations of Pt and Se atoms in opposite phase. The peak positions of the three modes are extracted and plotted in Fig. 1(e), which indicates that the Eg mode has a slight redshift around ∼176.5cm−1 with increasing film thickness and the A1g mode is almost pinned at ∼205.5cm−1, along with the nearly unchanged LO mode. The A1g/Eg intensity ratio in Fig. 1(f) keeps on growing from 2 L to 23 L. The peak positions and the tendency of intralayer Raman shifts are in agreement with previous results [3,16,17]. The absolute linear absorptions from 400 nm to 1100 nm were derived from the formula A=1−R−T in the same way as MoS2 and WS2 [18]. The linear transmission (T) and reflection (R) spectra are shown in Appendix A. We can calculate the linear absorption coefficient with I=I0exp(−αL) [19] and the specific data are listed below. Meanwhile, the layer-dependent Tauc plots derived from the absorption spectra verify that PtSe2 with 2–23 L undergoes a transition from semiconductor to semimetal, as shown in Fig. 1(g).

A setup of degenerate noncollinear ultrafast pump–probe spectroscopy was home-built for studying coherent phonons (CPs). The ultrafast laser produced 380 fs pulses at a photon energy of 1.19 eV (1040 nm) with the repetition rate of 100 kHz. These pulses were split into pump and probe beams with orthogonal polarization in order to eliminate mutual interference. The pump and probe beams are focused on the sample, and the focused spot sizes are 107 μm and 31 μm, respectively. To investigate the layer dependence of CPs in layered PtSe2, we measured the time-resolved differential transmission signals ΔT/T of 2, 5, 8, 10, 15, 17, and 23 L samples, covering the transition of semiconductor to semimetal. PtSe2 with 15 layers (15 L-PtSe2) was taken as an example, as shown in Fig. 2(a). It includes two components: incoherent carrier dynamics and CP dynamics. (1) The increased transmission before the zero delay point is owing to the band filling effect as the bandgap of 15 L-PtSe2 [nearly zero in Fig. 1(g)] is much smaller than the excitation photon energy (1.19 eV). After the zero point, the excited carriers first relax fast to reach a state with ΔT/T=0, and then a subsequent absorption process appears, followed by typical carrier relaxation dynamics in several hundred picoseconds, which has been studied in our previous work [20]. (2) Damped oscillations generated by CPs are superimposed on the relaxation process. From Fig. 2(a), the CP oscillation modulates the transmission initially by approximately 28%, much larger than that of MoS2 (group-6 TMDCs) and graphite [21–24], indicating that the contribution of CPs in PtSe2 is very large and is even comparable to that of excited carriers.

The inset of Fig. 2(a) shows the fast Fourier transform (FFT) of the damped oscillation. It is obvious that there are two vibrational frequencies (0.15 THz and 0.05 THz). It means that the experimental data should be fitted with a two-exponential model including two decaying sinusoidal wave functions [25–27]: ΔTT=∑i=1,2Aiexp(−tτi)+∑j=1,2Bjexp(−tτj)×sin(ωjt+ϕj),(1)in which t is the delay time between pump and probe pulses. The exponential terms marked by footnote i depict the fast and slow relaxation processes of the excited carriers. The two sinusoidal decays describe the damped oscillations, where the amplitude Bj, the decay time τj, the frequency ωj, and the initial phase Φj of two different oscillation modes are marked by footnote j. The pure damped oscillation signal was extracted by subtracting the red curve from the black one, as shown in Fig. 2(b) (green curve). It can be seen that the damped oscillation can be decomposed into two separate modes: the higher-frequency mode 0.15 THz (Mode 1, yellow) and lower-frequency mode 0.05 THz (Mode 2, blue).

The CP oscillations of PtSe2 with other different layers (2, 5, 8, 10, 17, and 23 L) were extracted in the same way, as shown in Fig. 2(c). Figure 2(d) summarizes the layer-dependent FFT results of the oscillations of all samples. Mode 1 varies from 0.78 THz for 2 L to 0.10 THz for 23 L, while Mode 2 softens from 0.24 THz to 0.03 THz. Both of them show a typical redshift with the increment of the number of layers. The detailed values of the frequencies (in the units of THz, cm−1, and picosecond) are listed in Table 1.

For 2D layered materials, lattice vibrations contain high-frequency intralayer vibrations and low-frequency interlayer vibrations. In view of the frequency range in our experiment, it is reasonable to deduce that Mode 1 and Mode 2 should originate from interlayer vibrations, which is in accordance with the following theoretical analysis. Interlayer vibrations can be divided into the out-of-plane LBM, in-plane shear mode, and propagating wave mode [21,28,29]. The LBM is the interlayer motions perpendicular to the PtSe2 plane, corresponding to Bz mode in Raman spectra, and the shear mode is parallel to the plane, corresponding to Sx and Sy modes [28]. It was noted that many interlayer vibrational modes of PtSe2 were Raman active [4,30]. In order to identify the CP oscillation modes, we conducted the low-frequency Raman spectroscopy. Figure 3(a) shows the Raman spectra of 2 L-, 5 L-, and 8 L-PtSe2 from 10cm−1 to 100cm−1 under the excitation of 532 nm wavelength (Renishaw inVia Raman Spectroscope). Its spectral resolution is 1cm−1 and the cutoff band of the high-pass filter is from −10cm−1 to 10cm−1. It can be found in Fig. 3(b) that the Raman peak positions are in good consistency with the frequencies of Mode 1. For an N-layer material, it has N−1 kinds of LBMs and shear modes because each layer vibrates in different amplitudes and opposite directions. As a result, there are fan-like branches of the LBMs and shear modes with the increasing number of layers, as shown in Fig. 3(b). These gray squares indicate the theoretical calculation of the LBMs in PtSe2 on the basis of density functional perturbation theory (DFPT) [4]. We can see that the values and tendency of Mode 1 are very close to the lowest branch of LBMs, which is also Raman active. Thus, in good agreement with both Raman spectra and DFPT calculations, CP oscillation of Mode 1 can be identified as the lowest LBM interlayer vibration in PtSe2, which corresponds to the motion that the top half and the bottom half of the layers vibrate collectively but in the opposite phase [28,29]. The diagram of the vibrations varying with the number of layers is shown in Fig. 4. Other peaks in 5 L and 8 L in Fig. 3(a) were attributed to higher branch of LBMs. The frequencies of thicker samples were cut by the filter and unable to be detected. In addition, we also studied the polarization behavior of the phonon modes, as shown in Fig. 3(c). The amplitude of Mode 1 of 2 L-PtSe2 varies periodically with the polarization of I∝(a×cosθ)2 the incident light. The polarization dependence of intralayer Eg (in-plane) and A1g (out-of-plane) modes is compared in Fig. 3(d). We can find that the polarity of Mode 1 is the same as the intralayer A1g mode. Accordingly, as an out-of-plane mode, it is assigned to have the A1g symmetry [31]. Thus, the amplitude of Mode 1 can also be fitted with I∝(a×cosθ)2, where I is the intensity of the Raman peak, a is a Raman tensor element, and θ is the angle between the polarization of the incident and scattered light [32]. In comparison, the amplitude of the Eg mode in Fig. 3(d) is independent on the polarization of the incident laser and can be fitted by I∝c2, where c is another Raman tensor element [32].

The frequency of Mode 1 of each sample is extracted from Fig. 2(d) and plotted in the unit of cm−1 in Fig. 5(a). For the LBM, in the one-dimensional linear atomic chain model [28,33,34], relative movement between intralayer atoms is neglected. Each layer is moving as a unit and each unit vibrates only in the z direction (perpendicular to the PtSe2 plane). We can imagine interlayer interaction as a spring to connect the adjacent unit. Accordingly, the linear atomic chain model that considers merely the interlayer interaction between nearest neighbors was used to fit the data, as expressed in the formula ωN=K2μπ2c2[1−cos(m−1)πN],(2)where ωN represents the associated vibrational frequency for samples with different number of layers N, the interlayer force constant K between the nearest neighbors is a fitting parameter, c is the speed of light, and the mass per unit area is μ=4.8×10−26kg/Å2. m is just an index (m=2,3,4,…,N) and it is assigned to be 2 here because Mode 1 is the lowest branch as we have just mentioned. The fitting result is shown in Fig. 5(a) and the good agreement suggests that the interlayer interactions are dominated by the interactions between the nearest-neighboring layers. Because each layer vibrates in the z direction that we have just mentioned, the fitted interlayer breathing force constant (IBFC) represents the interlayer interaction in z direction Kz, KzPtSe2=6.2×1019N/m3, while the IBFC of graphene Kzgraphene=11.6×1019N/m3 [31] is larger, which seems that interlayer coupling of graphene is stronger than that of PtSe2. But actually, we have to consider that graphene is a planar structure and each carbon atom plays a role in the interlayer interaction [28]. However, PtSe2 has a trigonal structure (1T phase) and only half of the Se atomic layer contributes to the nearest-neighboring interaction. If we make an approximation that the contribution of each Se atom is equal, the IBFC per effective atom comes out to be Kz/atomPtSe2=7.5N/m, which is 2.5 times larger than that of graphene (3 N/m [28]). It reveals that PtSe2 indeed has stronger interlayer interaction than graphene, which also explains its harder mechanical exfoliation. Furthermore, the good fitting result suggests that the force constant remains almost unchanged within the margin of error as the thickness is increasing, which implies that the softening of the phonons is not due to the change of interlayer coupling, but due to the increased number of layers [35]. The interlayer shear force constant (ISFC), KxPtSe2=4.6×1019N/m3, was calculated based on the data in Ref. [4]. It is also larger than that of WSe2 [29], MoS2 [29,36], graphene [37], and black phosphorus (BP) [28], which can be explained by the existence of Se⋯Se quasi-covalent bonds between adjacent layers [32]. The IBFC and ISFC of PtSe2 are comparable, standing for the nearly isotropic interlayer coupling in PtSe2, different from the anisotropic one in WSe2, MoS2, graphene, and BP. It is noteworthy that the frequency of the LBM shows an obvious dependence on the number of layers. As Raman spectroscopy is unable to determine the ultralow frequency (<10cm−1), the study of interlayer LBMs in CP oscillations widens the way to characterize the number of layers for 2D materials.

The frequencies of Mode 2 are extremely low and mismatch any value in the LBMs and shear modes [4]. Figure 5(b) shows that the periods of Mode 2 vary linearly with the thickness, which is a typical feature of SWM [21]. The phenomenon can be explained by the generation of standing wave, which has been studied in many materials [21,38–40]. The thin film partly absorbs the pump pulse and produces electrons and holes. Electrons will transfer excess energy to the lattice by electron–phonon collisions during the relaxation to the band edge, resulting in an acoustic pulse. Then the acoustic phonons produce a negative elastic stress force [40], leading to a nuclear motion around the new equilibrium position just like a standing wave, which changes the lattice constants of PtSe2 periodically. Figure 4 shows the diagram of the vibrations with the increasing number of layers. The acoustic wave bounces back and forth inside the film, and modulates the bandgap of the film and hence the absorption of the probe beam periodically [38]. We can estimate the penetration depth ξ=α−1 [40], where α is the linear absorption coefficient listed in Table 2. The calculated depth ξ of each film is larger than the actual thickness of the corresponding film. For example, the penetration depth of 15 L-PtSe2 is 22.09 nm, larger than its thickness of 8.98 nm, which means that the stress force can act on the whole film rather than only the external surface. The vibrations in few-layer PtSe2 are relatively strong so that PtSe2 is expected to be applied in a micro-mechanical resonator, surface cleaning, and driving motor, etc. [8,41,42].

According to the standing wave conditions, the laser is incident on the free surface of the sample (the zero-stress boundary condition), and the other surface of the sample is limited by the sapphire substrate (the zero-displacement boundary condition) [21,38–40]. Then, the period T of the oscillation can be defined as T=4d/V,(3)where d is the thickness of the film, and V is the out-of-plane sound velocity. The values of the oscillation period T are listed in Table 1. The sound velocity, 1.72×105cm/s, of PtSe2 can be derived from the slope of Fig. 5(b), in the same order of magnitude as the theoretical calculations in Refs. [12,13]. To the best of our knowledge, this is the first time to determine the sound velocity in layered PtSe2 experimentally. Although PtSe2 undergoes a phase transition from semiconductor to semimetal as the number of layers varies from 2 L to 23 L, the oscillation periods do not show any discontinuity. It is reasonable as the lattice structure of PtSe2 remains almost unchanged in spite of its transition. We also note that no acoustic echo disturbs the CP oscillations. Acoustic echoes are only detected in thick films with several hundred nanometers [21,40,43], while the thickest sample (23 L) here is still less than 15 nm.

In conclusion, the interlayer lattice vibrations of few-layer PtSe2 generated by CPs were directly observed in the time domain. We found the coexistence of the LBM and SWM in such kind of group-10 TMDCs. The vibrational frequencies of the LBM were tightly dependent on the thickness, which can be utilized to characterize the number of layers of 2D materials beyond Raman spectroscopy. The IBFC per effective atom was determined to be 7.5N/m, which was 2.5 times larger than that of graphene. In comparison with the ISFC, PtSe2 was demonstrated to have nearly isotropic interlayer coupling. In addition, the out-of-plane sound velocity was derived to be 1.72×105cm/s from the SWM. The results of interlayer interaction in layered PtSe2 can provide support for its application in nanomechanics like a high-frequency (∼THz) micro-mechanical resonator, surface cleaning, and driving motor.