_{2}determined using temperature-dependent Raman spectroscopy combined with finite element simulations

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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.

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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^{[1-3]}. 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}^{[11-14]}. The positive linear TEC of two-dimensional (2D) MoS_{2} was estimated using first-principles based on the quasiharmonic approximation^{[11]}. Ding ^{[12]} used density functional perturbation theory (DFPT) to calculate the phonon spectra of 2H-MoS_{2} structures. Huang ^{[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^{[15-18]}. In recent years, Raman spectroscopy has also been used to study the thermal parameters of MoS_{2}^{[19]}. Zhang ^{[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.

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.

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.

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 _{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 _{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

Schematic illustration of finite element model for (A) supported and (B) suspended MoS_{2}.

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:

where _{0} represents the heat transfer from the environment to the model, _{ext} is the ambient temperature and

_{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 ^{[24]}. Regarding the room-temperature spectrum of supported MoS_{2} (SUP-293 K), the ^{-1}, whereas the ^{-1}. The frequency difference between the ^{-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]}.

(A) Optical microscopy image of monolayer MoS_{2} transferred on a prepatterned SiO_{2}/Si substrate with 5 μm microholes. (B) Raman spectra of MoS_{2} collected at 77 and 293 K. (Solid lines represent SUP-MoS_{2} and the dotted line represents SUS-MoS_{2}).

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. _{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]}.

Temperature-dependent Raman spectra of (A) supported and (B) suspended MoS_{2}.

For a more detailed analysis of the difference between suspended and supported MoS_{2}, the Raman spectra presented in _{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.

Frequency shifts of (A) _{2} as a function of temperature. The blue and red lines show the fitting results obtained using a linear equation of temperature.

Moreover, as shown in _{2} is much closer to that of supported MoS_{2}. In sharp contrast, the temperature-dependent evolutions of the _{2} are very different from those of supported MoS_{2}. The temperature dependences of the ^{[29,30]}:

where _{0} is the frequency at 0 K and ^{-1}/K for suspended and supported MoS_{2}, respectively. For the ^{-1}/K for supported and suspended MoS_{2}, respectively. Remarkably, the _{2} is just 34.1% of that of supported MoS_{2}. This suggests that the ^{[31]} observed the accelerated redshift of the _{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 electron-phonon interaction, the electron doping effect leads to frequency shifts of the ^{[32,33]}. In contrast, the electron doping effect can be ignored in the temperature-dependent _{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 [Δ^{E}(^{A}(^{[34]}. The intrinsic frequency shift of freestanding MoS_{2} [Δ_{int}(

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]}:

where Δ_{0} = 293 K [_{0})].

The term Δ^{S}(

where _{2} and MoS_{2}, respectively. The value of ^{[35]}.

It is known that Δ^{E}(^{A}(_{2} [Δ_{int}(_{2} originates only from the TEC mismatch-induced frequency shift between MoS_{2} and the substrate. Therefore, after subtracting Δ^{S}(_{2} should be the same [Δ_{int}(

Using _{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 ^{[36]}. The simulation results are presented in _{2} samples at room temperature (_{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 _{2} are listed in ^{[35]}:

Simulated strain and the corresponding Raman shifts of the _{2} at selected temperatures

T (K) | 100 | 200 | 300 | 400 | 500 |
---|---|---|---|---|---|

_{sus} (%) |
0.005434 | 0.004523 | -0.0004271 | -0.007408 | -0.01319 |

_{sup} (%) |
0.006833 | 0.005673 | -0.0005339 | -0.009233 | -0.01640 |

^{-1}) |
-0.7227 | -0.6016 | 0.05680 | 0.9853 | 1.755 |

^{-1}) |
-0.9088 | -0.7545 | 0.07101 | 1.228 | 2.182 |

Therefore, the thermal strain-induced Raman shift can be obtained using _{2}. A flowchart for the numerical inversion procedure is plotted in

With the help of the numerical inversion process, the proper TEC of monolayer MoS_{2} that satisfies _{2}. As shown in _{2} samples is zero at room temperature. _{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}. _{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.

Numerical simulation results of supported and suspended MoS_{2}: (A), (C), (E) corresponding to the strain of SUP-MoS_{2} at 293, 100 and 500 K, respectively; (B), (D), (F) corresponding to the strain of SUS-MoS_{2} at 293, 100 and 500 K, respectively.

Using the same _{2} flake at different temperatures (see _{2}, the TEC mismatch-induced strains for suspended and supported MoS_{2} were calculated. Then, by employing _{2} are calculated and listed in

The suitable ^{-5} K^{-1}), while Ding ^{[12]} reported an ^{-5} K^{-1}) obtained by DFPT calculation. In contrast, the ^{[29]} reported a room-temperature ^{-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.

TEC obtained by combining temperature-dependent Raman spectroscopy and finite element simulations.

Comparison of the TEC of MoS_{2} obtained from this work and those reported in the literature

Ref. | Method | TEC (10^{-5} K^{-1}) |
---|---|---|

Ding ^{[12]} (2015) |
Density functional perturbation theory | 2.44 |

Late ^{[29]} (2014) |
Raman + density functional theory | 8.20 |

Hu ^{[37]} (2018) |
Electron energy-loss spectroscopy | 6.49 |

This work | Raman + finite element method | 7.13 |

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.

Design, writing review and editing: Yang Y, Long LC

Data analysis: Lin ZT, Li RF

Data acquisition: Liu WG, Li YT, Zhu K

Sample fabrication: Tian SB

Not applicable.

This work was supported by the National Key Research and Development Program of China (No. 2018YFB0703500; No.2016YFA0200800), the National Natural Science Foundation of China (No. 11704401; No.12074420; No.11674387; No.61905274), the Key Deployment Project of Centre for Ocean Mega-Research of Science, Chinese academy of science (No. COMS2020J03).

All authors declared that there are no conflicts of interest.

Not applicable.

Not applicable

© The Author(s) 2021.

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