Thermal expansion coefficient of monolayer MoS 2 determined using temperature-dependent Raman spectroscopy combined with finite element simulations

The thermal expansion coefficient is an important parameter of monolayer MoS 2 that affects the performance of its related optoelectronic devices. To obtain the thermal expansion coefficient of monolayer MoS 2 , suspended and supported MoS 2 are systematically investigated using micro-Raman spectroscopy in a temperature range of 77-557 K. Obvious differences in the temperature-dependent evolution of the Raman peaks between suspended and supported MoS 2 are observed, which result from the thermal expansion coefficient mismatch between MoS 2 and the substrate. With the help of the finite element method, the thermal strain in suspended and supported MoS 2 is calculated and used to deduce the thermal expansion coefficient mismatch-induced Raman shift. By matching the simulation and experimental results, the thermal expansion coefficient of MoS 2 is identified through the numerical inversion calculation. Our results demonstrate that the combination of micro-Raman spectroscopy and finite element simulations is highly effective for identifying the intrinsic thermal expansion coefficient of two-dimensional materials. Raman spectroscopy and numerical simulations. Suspended and supported MoS 2 were systematically investigated using Raman spectroscopy in a temperature range from 77 to 557 K, which exhibited an obvious discrepancy in the evolution of Raman frequency shifts, demonstrating the critical effect due to the TEC mismatch between MoS 2 and the substrate. Finite element simulations were used to calculate the TEC mismatch-induced thermal strain and Raman shift in the suspended and supported MoS 2 . By matching the simulation results to the experimental results, the TEC of MoS 2 was determined through a numerical inversion method. This method proposed in our work is a reasonable and adoptable route for determining the TEC of MoS 2 , which can also be employed to obtain the TEC of 2D materials.


INTRODUCTION
Transition metal dichlorides (TMDs) are a large class of two-dimensional layered materials with a natural band gap structure and have attracted increasing attention in the past decade [1][2][3] . Their intralayer atoms are bound by covalent bonds, while their interlayer atoms are coupled by weak van der Waals force. Each layer of a TMD is three atoms thick, with a triangular or hexagonal transition metal atomic plane sandwiched between two triangular layers of dichloride atoms [4,5] . Monolayer TMDs lack the inversion symmetry of the crystal space group and undergo a transition from an indirect to direct band gap [6,7] . MoS 2 is a typical TMD material with a bulk indirect band gap of 1.29 eV and a direct band gap of 1.9 eV for monolayer MoS 2 . MoS 2 has high carrier mobility and optical absorptivity, which give it promising applications in optoelectronic devices.
As a building block of nanodevices, monolayer MoS 2 is attached to the substrate in practical applications. The differences in the thermal expansion coefficient (TEC) between the MoS 2 layer and the substrate causes thermal strain inside the MoS 2 layer. Because the self-heating effect cannot be avoided during the operation of devices, the thermal strain consequently induced by TEC mismatch should be taken into consideration when studying the electronic and transport properties of devices [8,9] . Thermal strain can affect the performance of devices and even results in cracking failure when it exceeds the bearing limit of MoS 2 [10] . Therefore, determining the TEC of monolayer MoS 2 is important for its future application in devices.
However, direct measurement of the TEC of monolayer MoS 2 is still hindered as it is attached to the substrate. Researchers have performed theoretical calculations to obtain the TEC of MoS 2 [11][12][13][14] . The positive linear TEC of two-dimensional (2D) MoS 2 was estimated using first-principles based on the quasiharmonic approximation [11] . Ding et al. [12] used density functional perturbation theory (DFPT) to calculate the phonon spectra of 2H-MoS 2 structures. Huang et al. [13] proposed a negative-positive crossover in the TEC of monolayer MoS 2 at 20 K, which was attributed to the competition between the modes with negative and positive Grüneisen parameters.
Raman spectroscopy has been demonstrated to be a powerful tool for studying the microstructure and electronic properties of MoS 2 , including the layer number, stress, doping and so on [15][16][17][18] . In recent years, Raman spectroscopy has also been used to study the thermal parameters of MoS 2 [19] . Zhang et al. [20] obtained the TEC of monolayer MoS 2 using a combination of theoretical and experimental methods and by characterizing the unique temperature dependence of the Raman peaks with three different substrates. Recently, the TEC of few-layer MoS 2 was investigated by adopting suspended MoS 2 as a freestanding sample [21] . However, absolute freestanding MoS 2 cannot be achieved practically. Even when MoS 2 is suspended on a hole or a groove, it is very difficult to observe the corresponding free expansion.
To our knowledge, previous studies of TEC have depended either on purely theoretical calculations or a combination of experimental characterization with first-principles modeling. In this work, we propose a new method to determine the TEC of MoS 2 by combining the temperature-dependent Raman spectroscopy with finite element simulations. First, a systematic Raman study is carried out on suspended MoS 2 , compared with substrate-supported MoS 2 , in a temperature range from 77 to 557 K. The finite element method is then used to simulate the thermal strain caused by the TEC mismatch between the substrate and MoS 2 . The corresponding Raman frequency shift obtained by thermal mismatch is also calculated. By ensuring compatibility between the simulation and experimental results, the TEC of monolayer MoS 2 is finally achieved by employing a numerical inversion method. Our work presents a simple method for assessing the TEC of monolayer MoS 2 , which can also be widely applied to study the thermophysical properties of many other 2D materials and thin films.

Sample preparation
Monolayer MoS 2 was obtained using a mechanical exfoliation method that has been demonstrated to be a feasible method for producing 2D materials [22] . This was then transferred onto a 300-nm-thick Si substrate that had been previously cleaned by oxygen plasma. Suspended MoS 2 was achieved by transferring exfoliated MoS 2 on periodic microhole arrays that was prefabricated on a SiO 2 /Si substrate by ultraviolet lithography and reactive ion etching technology. The microholes were 5 µm in diameter and 2 µm in depth. Monolayer MoS 2 was roughly picked out by its color from optical microscopy and further identified using micro-Raman spectroscopy and atomic force microscopy.

Temperature-dependent raman measurements
Raman spectra were collected using a confocal micro-Raman spectrometer (Horiba HR Evolution). A solid-state laser with a wavelength of 532 nm was used as the excitation source. The laser beam was focused using a 50× long working distance objective with a numerical aperture of 0.5. The spot size was ~1.5 µm. To avoid the heating effect, the laser power was set to less than 1 mW. The sample was placed inside a cryostat cell (Linkam, THMS 600) and the temperature was changed from 77 to 557 K with an interval of 20 K.

Finite element simulations
According to the experimental environment of the sample in the low-temperature measurements, the mechanical thermal coupling analysis model was established by the finite element method. Two types of simulation models (supported and suspended MoS 2 ) were built accordingly. As shown in Figure 1A, supported MoS 2 is fully attached to the SiO 2 substrate (SUP-MoS 2 ). In contrast, suspended MoS 2 covers the substrate with a microhole (SUS-MoS 2 ), as shown in Figure 1B. The geometries of the models in the simulation were designed to match the real dimensions of the monolayer MoS 2 and microhole fabricated. The diameter of the microhole was 5 µm and the periodicity was 8 µm. The thickness of monolayer MoS 2 was set as 0.7 nm according to a previous study [23] . For the simplicity of the simulations, the thickness of the SiO 2 layer was set as 100 nm, although the SiO 2 layer in the real sample was 280 nm thick. The physical properties of SiO 2 employed in the simulations were adopted from the built-in parameters in the software and are listed in Supplementary Table 1. In the modeling process, the mesh density was conventional and the maximum and minimum elements were 0.8 and 0.144 µm, respectively. The SiO 2 base mapping and sweep function were obtained and the monolayer MoS 2 was then mapped and swept. A layer of hexahedron meshes was obtained. It was assumed  that the heating effect from the laser can be ignored. The sample was only subjected to the heat transfer response from the ambient temperature in the cryostat. In the process of the analysis and calculation, the solid mechanics and heat transfer module were coupled and transient analysis was adopted. The heat flux boundary condition was used in the solid heat transfer module. This convection heat flux can be described by the convection heat transfer equation: (1) where q 0 represents the heat transfer from the environment to the model, h is the heat transfer coefficient, T ext is the ambient temperature and T is the model temperature. Figure 2A exhibits the optical image of monolayer MoS 2 transferred on a prepatterned SiO 2 /Si substrate with microholes. The typical Raman spectra for suspended and supported monolayer MoS 2 at room temperature (293 K) and 77 K are presented in Figure 2B, in which two Raman modes can be seen clearly. The mode originates from the in-plane vibration of molybdenum and sulfur atoms, while the mode represents the out-of-plane vibration [24] . Regarding the room-temperature spectrum of supported MoS 2 (SUP-293 K), the mode is located at ~385.5 cm -1 , whereas the mode is at ~404.5 cm -1 . The frequency difference between the and modes is smaller than 20 cm -1 , demonstrating that it is monolayer MoS 2 [25] . Moreover, the frequency differences between suspended and supported MoS 2 are also observed in the room-temperature Raman spectra, which could be attributed to the internal strain caused during sample preparation [26] .

RESULTS AND DISCUSSION
In addition to the room-temperature spectra, the Raman spectra of SUP-MoS 2 and SUS-MoS 2 collected at 77 K also exhibit differences. It is well known that the binding force from the substrate interferes with the deformation of the MoS 2 film when the temperature changes because of the difference in the TEC of MoS 2 and the SiO 2 substrate. Although the experimental conditions are the same, supported and suspended MoS 2 are subjected to different strain conditions and the deformation in the microstructure of MoS 2 is different as a result. Consequently, the corresponding atomic lattice vibration is affected by the substrate. Figure 3 presents the Raman spectra of supported and suspended MoS 2 at selected temperatures. With increasing temperature, an obvious redshift and broadening of the Raman peaks are observed for both suspended and supported MoS 2 , which can be attributed to the thermal expansion of the crystal lattice of MoS 2 [27,28] .
For a more detailed analysis of the difference between suspended and supported MoS 2 , the Raman spectra presented in Figure 3 were deconvoluted using the Lorentz/Gaussian mixed function. The peak positions of the and modes are plotted as a function of temperature in Figure 4 for comparison. The temperature- dependent evolutions of the frequency shifts for supported and suspended MoS 2 samples are similar and vary approximately linearly with increasing temperature. It is well known that mechanically exfoliated MoS 2 is transferred and attached on a substrate by weak van der Waals forces. Therefore, the TEC mismatch between MoS 2 and the substrate induces biaxial stress into MoS 2 as the temperature changes and becomes a prominent factor that modulates the frequency shift of the Raman peaks. The Raman frequency evolutions of suspended MoS 2 demonstrate that the substrate effect is exerted on the MoS 2 layer as a whole, although the suspended zone of MoS 2 is not directly in contact with the substrate.
Moreover, as shown in Figure 4, the temperature evolution of the mode is significantly different from that of the mode. For the mode, the temperature-dependent frequency shift for suspended MoS 2 is much closer to that of supported MoS 2 . In sharp contrast, the temperature-dependent evolutions of the mode of suspended MoS 2 are very different from those of supported MoS 2 . The temperature dependences of the and modes are fitted using the Grüneisen model [29,30] :  ω (T) = ω 0 + χ T (2) where ω 0 is the frequency at 0 K and χ is the temperature coefficient. The temperature coefficients of the mode ( ) are -0.0985 and -0.0129 cm -1 /K for suspended and supported MoS 2 , respectively. For the mode, the temperature coefficients ( ) are -0.00952 and -0.00316 cm -1 /K for supported and suspended MoS 2 , respectively. Remarkably, the of suspended MoS 2 is just 34.1% of that of supported MoS 2 . This suggests that the mode is more sensitive to the substrate (or dielectric environment). In addition to the TEC mismatch, charge transfer between the TMD film and the substrate or through interfacial states can also influence the temperature evolution of the peak. Su et al. [31] observed the accelerated redshift of the mode of MoS 2 with increasing temperature, which was attributed to the enhanced charge injection from the substrate into the film and the decomposition of adsorbed contaminants. Through the strong electronphonon interaction, the electron doping effect leads to frequency shifts of the mode [32,33] . In contrast, the electron doping effect can be ignored in the temperature-dependent mode, which could be used to estimate the thermal strain in monolayer MoS 2 .
According to the literature, the temperature-dependent Raman frequency shift of freestanding MoS 2 can be commonly attributed to the thermal expansion of the lattice [Δω E (T)] and the anharmonic effect [Δω A (T)], which changes the phonon self-energy [34] . The intrinsic frequency shift of freestanding MoS 2 [Δω int (T)] can be written as: For the temperature-induced frequency shifts of supported MoS 2 , both common thermal effects and TEC mismatch-induced strains must be taken into consideration. As a result, the frequency shifts of supported MoS 2 can be expressed as [21,34] : The term Δω S (T) represents the TEC mismatch-induced frequency shift, which can be expressed as: (5) where β is the biaxial strain coefficient of Raman modes and and are the temperaturedependent TECs of SiO 2 and MoS 2 , respectively. The value of β can be taken from the literature [35] .
It is known that Δω E (T) and Δω A (T) are the intrinsic frequency shifts of MoS 2 [Δω int (T)], which are independent of the substrate. For the measured temperature-dependent Raman shifts in this work, the discrepancy in the frequency shifts between suspended and supported MoS 2 originates only from the TEC mismatch-induced frequency shift between MoS 2 and the substrate. Therefore, after subtracting Δω S (T), the obtained values for suspended and supported MoS 2 should be the same [Δω int (T)]: Using Equation (6), the value of can be deduced if we can obtain and However, the TEC mismatch-induced frequency shift for suspended MoS 2 is challenging to experimentally determine. Because the MoS 2 flake that surrounds the suspended MoS 2 zone is still attached to the substrate, the TEC mismatch-induced strain can affect the Raman shift of suspended MoS 2 to a certain extent.
Numerical simulations based on finite element theory (FET) provide a route to obtaining the thermal strain in suspended MoS 2 . For the further study of the distribution of the thermal strain field, thermal strains at different temperatures were solved. In order to solve the FET simulation, the TECs for SiO 2 and MoS 2 should be provided as material parameters. The is adopted from the literature, whereas is identified in this work. In the first step, we can use the previously obtained using firstprinciples calculation in the simulations [36] . The simulation results are presented in Supplementary Figure 1. It is assumed that the thermal strain is zero in the MoS 2 samples at room temperature (T 0 = 293 K). In fact, the strain distribution obtained using the FET simulations is composed of two components, namely, the strain due to the thermal expansion of the MoS 2 layer and the strain induced by the TEC mismatch between MoS 2 and the substrate. In order to obtain the strain induced by the TEC mismatch, FET simulations were also performed on a bare MoS 2 flake (see Supplementary Figure 2), which is used to subtract the strain obtained based on the film-substrate model illustrated in Figure 1. The obtained TEC mismatch-induced strains ε (T) for the suspended and supported MoS 2 are listed in Table 1. As demonstrated in the literature, the relationship between the TEC-induced strain and Raman shift can be expressed as [35] : Therefore, the thermal strain-induced Raman shift can be obtained using Equation (7). The calculated TEC mismatch-induced strain and corresponding Raman shifts are listed in Supplementary Table 2. By bringing the and calculated by Equation (7) into Equation (6), it is found that the terms on the left-and right-hand sides cannot be consistent. This implies that the reported in previous work is inconsistent with the experimental results in our work. Therefore, in order to meet the requirements for Equation (6), a numerical inversion method was employed to select an optimum TEC of MoS 2 . A flowchart for the numerical inversion procedure is plotted in Supplementary Figure 3.
With the help of the numerical inversion process, the proper TEC of monolayer MoS 2 that satisfies Equation (6) was obtained. Employing the calculated , the simulation results for 100, 293 and 500 K are plotted in Figure 5 to illustrate the typical evolution of the strain distribution in the suspended and supported MoS 2 . As shown in Figure 5A and B, the strain in both two MoS 2 samples is zero at room temperature. Figure 5C and D show the strain of SUP-MoS 2 and SUS-MoS 2 at 100 K, respectively. It can be seen that the thermal strain in SUP-MoS 2 caused by the thermal stress is uniform, while the deformation on the SUS-MoS 2 mainly occurs on the central circular microhole and gradually decreases in the annular direction with a weak change. From the simulation results, it can be concluded that MoS 2 sustains compressive strain. The deformation of SUS-MoS 2 is larger because of the binding effect without the substrate. Because the TEC of MoS 2 is larger than that of SiO 2 , MoS 2 shrinks faster with decreasing temperature and the deformation of MoS 2 is inhibited by SiO 2 . Therefore, the strain in SUS-MoS 2 is larger than that of SUP-MoS 2 . Figure 5E and F show the strain of SUP-MoS 2 and SUS-MoS 2 at 500 K. From the simulation results, it can be concluded that MoS 2 experiences suppressed tensile strain. Similarly, the SUS-MoS 2 deformation is greater due to the absence of the binding force of the substrate.
Using the same that was adopted in Figure 5, FET simulations were also performed on a bare MoS 2 flake at different temperatures (see Supplementary Figure 4). By employing the simulated thermal strain in bare MoS 2 , the TEC mismatch-induced strains for suspended and supported MoS 2 were calculated. Then, by employing Equation (7), the frequency shifts of the mode for suspended and supported MoS 2 are calculated and listed in Table 1.
The suitable obtained by the combination of the numerical simulation and Raman spectroscopy is plotted in Figure 6. One can see that the sign of is positive in the test temperature range. As shown in Table 2, the value of the obtained in this work is close to the values reported in the literature. Taking the TEC at room temperature for example, the obtained in this work is 7.13 (10 -5 K -1 ), while Ding et al. [12] reported an of just 2.44 (10 -5 K -1 ) obtained by DFPT calculation. In contrast, the obtained in this work is in good agreement with previous results estimated experimentally. Late et al. [29] reported a room-temperature of 8.2 (10 -5 K -1 ) obtained by combining temperature-dependent Raman spectroscopy and first-principles calculations. In addition, it is found that the TEC of MoS2 is much larger than that of SiO2. Therefore, MoS2 suppresses a tensile stress from the substrate, whether the sample was cooled or heated.

CONCLUSIONS
In this work, we proposed a feasible method to assess the TEC of monolayer MoS 2 by combining micro-  Raman spectroscopy and numerical simulations. Suspended and supported MoS 2 were systematically investigated using Raman spectroscopy in a temperature range from 77 to 557 K, which exhibited an obvious discrepancy in the evolution of Raman frequency shifts, demonstrating the critical effect due to the TEC mismatch between MoS 2 and the substrate. Finite element simulations were used to calculate the TEC mismatch-induced thermal strain and Raman shift in the suspended and supported MoS 2 . By matching the simulation results to the experimental results, the TEC of MoS 2 was determined through a numerical inversion method. This method proposed in our work is a reasonable and adoptable route for determining the TEC of MoS 2 , which can also be employed to obtain the TEC of 2D materials.