Robust and high-sensitivity thermal probing at the nanoscale based on resonance Raman ratio (R3)

Raman spectroscopy-based temperature sensing usually tracks the change of Raman wavenumber, linewidth and intensity, and has found very broad applications in characterizing the energy and charge transport in nanomaterials over the last decade. The temperature coefficients of these Raman properties are highly material-dependent, and are subjected to local optical scattering influence. As a result, Raman-based temperature sensing usually suffers quite large uncertainties and has low sensitivity. Here, a novel method based on dual resonance Raman phenomenon is developed to precisely measure the absolute temperature rise of nanomaterial (nm WS2 film in this work) from 170 to 470 K. A 532 nm laser (2.33 eV photon energy) is used to conduct the Raman experiment. Its photon energy is very close to the excitonic transition energy of WS2 at temperatures close to room temperature. A parameter, termed resonance Raman ratio (R3) Ω=IA1g/IE2g is introduced to combine the temperature effects on resonance Raman scattering for the A1g and E2g modes. Ω has a change of more than two orders of magnitude from 177 to 477 K, and such change is independent of film thickness and local optical scattering. It is shown that when Ω is varied by 1%, the temperature probing sensitivity is 0.42 K and 1.16 K at low and high temperatures, respectively. Based on Ω, the in-plane thermal conductivity (k) of a ∼25 nm-thick suspended WS2 film is measured using our energy transport state-resolved Raman (ET-Raman). k is found decreasing from 50.0 to 20.0 Wm−1 K−1 when temperature increases from 170 to 470 K. This agrees with previous experimental and theoretical results and the measurement data using our FET-Raman. The R3 technique provides a very robust and high-sensitivity method for temperature probing of nanomaterials and will have broad applications in nanoscale thermal transport characterization, non-destructive evaluation, and manufacturing monitoring.


Introduction
Raman-based methods are among the latest techniques for characterizing the thermal transport properties of twodimensional (2D) materials, such as graphene, transition metal dichalcogenides (TMDs), and nanowires [1][2][3][4][5][6][7][8][9]. These methods are used to determine several thermal transport properties, especially thermal conductivity (k) [10,11]. The steadystate (SS) Raman technique has two main steps to track the temperature response of a 2D material: (a) measurement of laser power (P) dependence of Raman wavenumber (ω) (dω/dP), and (b) measurement of temperature (T) coefficient of ω (dω/dT). Here, the second step works as a calibration tool to find the temperature rise during the laser heating in the first step. The temperature calibration using the measured dω/dT and measurement of the laser power absorption could introduce significant errors [1]. For instance, laser absorption could vary a lot due to the optical properties variation of unknown samples. An energy transport state-resolved Raman (ET-Raman) technique was developed by Wang's group to overcome the drawbacks of SS Raman by generating steady and transient energy transport states and measuring the Raman shift power coefficient (dω/dP) under each of these states [12][13][14][15][16][17]. By normalizing the dω/dP under transient state with its counterpart under SS, a normalized Raman shift power coefficient (Θ) is determined, which is a function of thermal properties of the 2D material. This normalization process eliminates the effects of laser absorption coefficient and Raman shift temperature coefficient measurements. Then, the theoretical Raman intensity weighted average temperature rise under each state is calculated using a three-dimensional (3D) numerical method to determine the theoretical Θ, compare it against the experimental result, and determine a specific property, such as k or hot carrier diffusion coefficient (D) [18,19].
Each of these techniques, i.e. SS or ET-Raman, uses a specific Raman mode of the material to perform the analysis. Although it was shown that depending on the Raman mode selection, the determined thermal property could be different [20]. For instance, it was shown that the resulting interfacial thermal resistance between a MoS 2 thin layer and quartz substrate is smaller when the E 2g Raman mode is used. Also, each of these two main methods needs a considerable amount of time to conduct a large number of Raman measurements to obtain reliable dω/dT and dω/dP. It could be very time consuming to conduct experiments over a wide temperature range to measure the thermal properties at each environmental temperature. Last but not the least, most of the optothermal techniques based on Raman spectroscopy use indirect approaches to find the temperature rise of the sample under laser irradiation, like finding dω/dP instead of absolute temperature rise. This is due to the fact that at µm and nm scales, it is hard to track the absolute temperature rise directly using classical methods with sound accuracy.
In this work, a novel technique based on resonance Raman spectroscopy is developed for high sensitivity temperature measurement. It is shown that WS 2 experiences resonance Raman scattering under a 532 nm laser when the Raman experiment is conducted from high to low temperatures, and its two main Raman modes (E 2g and A 1g ) have different intensity variation behaviors against temperature. The intensity ratio of these two modes is found highly sensitive to temperature, and is independent of sample thickness and optical scattering conditions. This provides a remarkable parameter for unprecedented thermal probing. Using this intensity ratio for thermal sensing, k and α of nm-thick WS 2 are obtained at various temperatures from ∼170 K to ∼470 K without the need of Raman mode selection and laser absorption coefficient. Also, frequency domain ET-Raman (FET-Raman) experiment is conducted for the same sample using both Raman modes and at room temperature (RT) to confirm the measurement results based on Raman mode intensity ratio.

Discovery of universal parameter for high sensitivity thermal probing
Here, the two different samples used in this work are prepared using WS 2 and MoS 2 thin films. The WS 2 film will be used for the later on thermal conductivity measurement, and the MoS 2 film is used as a reference of non-resonance Raman scattering. Both films are prepared using mechanical exfoliation from bulk counterparts [21]. Then, each of them is transferred to a silicon (Si) substrate by a gel film and a 3D micro-stage. Thin films are placed over a 5 µm radius hole to create a suspended region. More details of this process and the hole structure can be found in our previous works [17,22]. Figures S1(a) and (b) (available online at stacks.iop.org/IJEM/4/035201/mmedia) show the atomic force microscopy (AFM) images of these prepared samples. Based on these images, the thickness of WS 2 and MoS 2 films are measured as 25.4 and 61.4 nm, respectively. Also, the root mean square roughness (R q ) of these two samples are measured using these AFM images as 0.96 nm and 1.92 nm, which indicates a uniform contact between the thin film and substrate.
Under laser irradiation with a specific photon energy, the electrons at the ground state of a nm-thick TMD material are excited to a higher energy level (excited state). The energy difference between these two states (∆E or excitonic transition energy) is a function of temperature [23,24], and it could be tuned by varying the environment's temperature. Based on the time-dependent perturbation theory explanation of light's polarizability, resonance Raman happens when ∆E is equal (or fairly close) to the incident photon's energy (hω i ), as shown in figure 1(a) [25][26][27]. According to this theory, the Raman intensity of each mode is related to the incident (E i ) and scattered (E s ) photon energy as [26]: (1) Here, E(T) and Γ(T) are the exciton's temperature dependent transition energies and damping constants, respectively. When E(T) is very close to E i (or E s ), the Raman intensity of the corresponding Raman mode is enhanced significantly. This leads to a significant increase of Raman intensity of the corresponding Raman mode. Figure 1(b) shows this trend of Raman intensity as a function of temperature. When the excitonic transition energy is close to hω i , the Raman intensity reaches maximum. Generally, E decreases with the increased temperature for TMD materials and this is explained by the Varshni empirical relationship under resonance condition (figure 1(c)) [23] as Here E B (0) is the electronic transition energy of B exciton at 0 K. α B and β B are Varshni coefficients and are related to the electron (exciton)-phonon interaction and Debye temperature. In this work, a 532 nm continuous-wave (CW) laser (2.33 eV photon energy) is used to excite the Raman signal, and this energy is close to the excitonic transition energy of B exciton of WS 2 .
The excitonic transition energy of MoS 2 excitons is lower than 2.33 eV over the whole temperature range. Therefore, the MoS 2 sample does not experience any resonance Raman under this laser. In our previous work [28], a multi-temperature Raman experiment has been conducted for several supported WS 2 samples on Si substrate from 77 to 797 K. AFM images and further information about these four samples could be found in that work. Over this temperature range, we could observe the resonance Raman of all WS 2 sample. Figures 1(d) and (e) represent the result of this Raman experiment for both E 2g and A 1g modes, respectively. The thicknesses of first three samples were 13, 38, and 45 nm. Also, the thickness of last WS 2 film was more than 200 nm which was considered as bulk. These two plots clearly indicate the resonance Raman scattering of WS 2 under the 532 nm laser for both Raman modes. For each Raman mode, we can obviously see that the I-T curve is different from sample-to-sample. Here, I is the normalized Raman intensity by the laser power and integration time. This difference is related to multiple factors, such as different interfacial spacing between WS 2 and Si, film thicknesses, variation of local optical interference, and different film profile changes against temperature. These figures also indicate that the Raman intensity of E 2g mode reaches its maximum value at a lower temperature than the A 1g mode. This was attributed to several reasons. First, the acoustic 2LA(M) mode is very close to the E 2g mode, and it is relatively strong at low temperatures. Therefore, it can slightly affect the peak fitting at those temperatures. Second, the effects of thermal expansion on Raman modes are different, and it has been observed that the A 1g mode is more sensitive to this effect [28][29][30]. Another factor could be the variation of optical properties of thin-film due to experimental temperature change. The refractive index of WS 2 film at wavelengths related to each Raman mode is different and could be another factor that contributes to this difference among I-T trends. Another key point is that E 2g and A 1g modes have different excitonic temperatures. Each of these two Raman modes shows different temperature dependent behaviors. Even for a similar peak, the I-T curves vary from sample to sample. Very surprisingly, the ratio of intensity of these two modes, here termed as resonance Raman ratio Ω defined as: Ω = I A 1g /I E 2g , is identical for all four samples regardless of their structural differences. This new parameter reflects the combined effects of temperature on excitonic energies and the resulting Raman resonance extent. Note the bulk material used here has a very smooth surface and this surface is used in testing. The curve shown in figure 1(f) will not be affected by surface roughness since this roughness will have the same effect on E 2g and A 1g peaks, and their ratio Ω = I A 1g /I E 2g completely eliminates this effect. Figure 1(f) indicates this observation for all the four supported WS 2 samples for temperature range of 177-477 K. Our key conclusion is that all the samples follow the same curve (Ω) regardless of the thickness, proving Ω is universal and can be used for thermal probing of samples regardless of their thicknesses, structures, and the varied local optical scattering. It is observed when temperature changes from 177 to 477 K, Ω changes by around two orders of magnitude. This strongly demonstrates the potential of using Ω for very high sensitivity temperature measurement. Based on this analysis, when Ω is varied by 1%, the temperature probing sensitivity is 0.46 K and 1.24 K at 177 K and 477 K, respectively. This strongly confirms the high precision of temperature probing using Ω.

Physical principles
In this section, we perform thermal transport characterization of 2D materials using thermal probing based on Ω. Here, two heating stages are designed to characterize the temperature rise of the nm-thick WS 2 sample based on the resonance Raman. The first one is the calibration stage in which the WS 2 film is supported on a Si substrate (right-hand side of figure 2(a)). The Raman intensity of E 2g (or A 1g ) mode of this sample is collected from low to high temperatures under a low heating power laser. The laser power is minimized to prevent any additional significant temperature rise by laser heating. Here, the two main Raman modes of WS 2 are studied (E 2g and A 1g ), and their Raman intensity is collected during the experiments. Figure 2(b) shows I-T and Ω-T plots schematically. This calibration is intended to build the Ω-T correlation. Then the obtained Ω-T correlation is used as the reference to measure the temperature rise of WS 2 under the other stage: suspended sample. Note in the calibration stage, the sample experiences optical interference within the sample and the sample-substrate spacing while the suspended sample only experiences optical interference within it. However, this will not cause errors in using the Ω-T correlation to measure temperatures of suspended samples. This is because the interference-caused intensity change will be completely eliminated in Ω calculation since it has very close effects on both E 2g and A 1g peak intensities. This point can also be readily proved by the same Ω-T relation shown in figure 1(f) for different-thickness samples that have very different optical interference within them. The suspended stage is shown in figure 2(a) (left hand-side). The WS 2 film is suspended over a hole, and is irradiated by using the same 532 nm laser and same objective lens (i.e. similar local heating area). This sample experiences a higher temperature rise caused by laser heating due to its suspended structure (larger thermal resistance), and its I-T curve reaches the maximum value at a lower temperature compared with the supported film. This is shown with the red curve in figure 2(b). Also, the Ω-T plot for this stage will take larger values compared with the calibration stage ( figure 2(b)). As a result, the absolute temperature rise of the suspended WS 2 film (∆T) is measured directly by comparing these two curves at each similar Ω. Note that the temperature rise of the supported WS 2 due to the local laser heating is minimal. This is because of the very high thermal conductivity of Si substrate.
Next, a frequency domain energy transport state-resolved Raman (FET-Raman) technique is used to characterize the thermal conductivity of the thin film. This technique consists of two heating stages: (a) SS (CW) under the CW laser, and (b) the transient state (F) under the amplitude modulated ( f ) CW laser (figure 2(c)). The multi-temperature resonance Raman experiment is conducted for both stages, and the temperature rise of the thin film (∆T CW and ∆T F ) is measured by comparing it to the calibration stage (i.e. supported sample). For both the CW and F states, the sample is excited and heated with the same laser of 532 nm wavelength. The temperature rise under each state is evaluated using their Ω. Note that the temperature rise under the CW stage is more than the transient one, and their multi-temperature Raman intensity curve will not be identical, as shown in the topright panel of figure 2(c). Based on this, a new experimental parameter (Θ) is obtained as: Θ = ∆T F /∆T CW . By performing this ratio calculation, the effects of laser power absorption and Raman intensity temperature coefficient (∂I/∂T) measurements are eliminated. Also, there is no effect of Raman peak selection on our analysis by obtaining Ω. Based on this analysis, Θ is only a function of k and ρc p . Next, 3D numerical calculation based on the finite volume method is conducted to determine the temperature rise of the suspended thin film and find k at each temperature based on the results of the resonance Raman experiment (see section 5.2 for more information about this calculation). Since the ρc p of the thin film is a function of temperature, the exact ρc p at each specific temperature (from references) is used in our numerical modeling to calculate the theoretical temperature rise. The theoretical Θ (Θ th ) is found using the calculated temperature rise under two heating stages as a function of k, and k at each environmental temperature (T) is determined by interpolating this curve against the measured Θ from resonance Raman experiment (figure 2(d)). The classical FET-Raman measurement (see section 5.3 for more information about this technique) is also conducted at RT for a similar WS 2 sample to compare with resonance Raman experiment results [31,32]. This technique uses the Raman wavenumber shift to probe the temperature rise and measure thermophysical properties.

Multi-temperature resonance Raman experiment
The multi-temperature Raman experiment is conducted for both WS 2 and MoS 2 samples from 170 to 470 K. Over this temperature range, we could observe the resonance Raman of the WS 2 sample for all three heating stages: Supported (or calibration), CW, and transient (F). This is shown in figures 3 and S2. Figures 3(a) and (b) show the 2D Raman intensity contour of suspended WS 2 under CW and F states over this temperature range. The same contour for the calibration stage is presented in figure S2. For all Raman experiments conducted in this work, a 20× objective lens with a radius (r 0 ) of ∼2.02 µm is used. Figure S1(c) shows the laser intensity distribution under this objective, and this figure is analyzed to find r 0 .
Here, the CW laser amplitude is modulated at 300 kHz frequency to construct the F state experiment (more details about this frequency selection in section 5.3). The Raman intensity of both modes is lower under F state than CW for the suspended sample, and this is due to the pulsed heating under F state. Note that the on-time laser power of F state is similar to the CW laser power. I of both modes and three states are shown in figure 3(c). This figure clearly shows the non-monotonic behavior of I versus temperature. Ω under each state is calculated and plotted as figure 4(a) for all three stages. The blue curve corresponds to the calibration with zero local temperature rise, and it is shown that Ω takes larger values for the suspended sample. Here, the Ω is plotted versus T to show this effect more clearly. Next, the Ω of calibration stage is fitted using an allometric function (solid lines in this figure) as: Ω = aT b , where a and b are the fitting parameters corresponding to the supported (calibration) stage. Here, the universal parameters a and b are 5.088 × 10 −11 and 3.860, respectively. This function is used to determine the temperature rise (∆T) of the CW and F states at each T. Figure 4(b) shows the result of this analysis (left axis), and it indicates an increasing trend of ∆T versus T under both stages. Also, the solid lines represent the fitting of each ∆T-T plot. Note this fitting is pure data processing to guide eyes and show the data trend. The right vertical axis of this plot shows the experimental Θ values, and it decreases with the increased temperature. Note that one of the Θ values in this plot is larger than 1 (at ∼190 K), and this is caused by the experimental or fitting uncertainties at very low temperatures. Again, the solid green line shows the Θ calculated from the fitted ∆T values (black and solid red lines). As shown in figure 3(c), the Raman intensity of A 1g mode at low T is fairly small. This could lead to some uncertainties when fitting this peak. These Θ values are used to determine the thermal conductivity of the WS 2 sample.
Over a similar temperature range, the MoS 2 sample does not experience any resonance Raman because its excitonic transition energy (∼1.8 eV) is below the incident photon energy (2.33 eV) [26,33,34]. Figure S3 shows the 2D contours of Raman intensity of both modes of MoS 2 for a supported ( figure S3(a)) and a suspended ( figure S3(b)) region. The extracted Raman intensities of each mode are plotted in  figure S4(c). This plot clearly shows that Ω takes a constant value of ∼2.2 for the supported MoS 2 film. The data corresponding to the suspended region is noisier and it could be related to the Raman peak fitting uncertainties caused by lower Raman intensity of both modes at low temperatures in this region.

Thermal conductivity and diffusivity determination
To determine the thermal properties of WS 2 thin film, a 3D numerical calculation based on the finite volume method is conducted and coupled with the experimental results, which were discussed in previous sections (see section 5.2 for details of this calculation). The theoretical temperature rise of the suspended WS 2 film under both states (∆T CW,th and ∆T F,th ) is calculated using the known ρc p values at each experimental temperature for a wide range of thermal conductivity. Then, the theoretical normalized temperature rise (Θ th ) is found as: Θ th = ∆T F,th /∆T CW,th . Figure S5 shows the c p and ρ that are used to perform this calculation [35]. The red solid line in figure S5 shows the fitted c p values using the Debye model of specific heat. The inset of figure 5 shows the result of this calculation. The y-axis of this 2D contour is the environmental temperature which corresponds to each ρc p value in our numerical simulation. The solid line inside this contour corresponds to Θ th value of 0.81, which is the experimental Θ at 303 K (as shown in figure 4(b)), and two dashed lines represent the uncertainty of this experimental value. Using this contour, the thermal conductivity and its uncertainty are determined at each temperature. The main plane of this figure shows each of these contour lines at three specific temperatures to indicate this process more clearly. The determined k at all temperatures is shown in figure 6(a). Larger fluctuations are observed at lower temperatures. This is due to the fluctuations of Θ exp at this temperature range ( figure 4(b)). However, at temperatures larger than 273 K (shown by a gray rectangle), k takes values with an average of ∼25 Wm −1 K −1 . The solid black line is the k determined using the fitted Θ values from figure 4(b) and shows a clear decreasing trend of k with the increased T. Part (b) of this figure shows the determined thermal diffusivity (α) of this sample as a function of temperature (left vertical axis). Also, ρc p values that are used to find α are plotted in this figure (right vertical axis).

Room temperature thermal conductivity measurement using FET-Raman
In this section, k of WS 2 film is measured at RT using the FET-Raman technique, in order to compare with and assess the measurement accuracy of resonance Raman discussed above. Each of these three curves shows the theoretical calculated Θ at a specific temperature as a function k, and k at each T is found by interpolating these curves with the corresponding experimental Θ from the multi-temperature resonance Raman experiment. The inset of this figure shows the 2D contour of Θ th that is calculated at all related temperatures and k. Under both CW and F states, the Raman shift power coefficient (ψ F and ψ CW ) is measured: ψ = dω/dP. Here, ω and P are Raman wavenumber of E 2g (or A 1g ) mode and laser power, respectively. Obviously, the absolute value of ψ CW is larger than ψ f due to the SS heating nature of CW stage, which shows the higher temperature rise of the suspended film. Based on these two values, the normalized Raman shift power coefficient is determined as: Θ FET-Raman = ψ F /ψ CW . Note that Θ FET-Raman is theoretically equal to the normalized temperature rise that was defined in previous sections (see section 5.3 for more details about FET-Raman technique). Again, Θ FET-Raman is a function of k and ρc p . Figures 7(a) and (b) show the 2D contour of Raman intensity of WS 2, two main peaks as a function of P for CW20 and F20 states, respectively. Note that here the 20× objective lens is used to perform the Raman experiment. These contours indicate the redshift of each mode as laser power takes larger values. Each of these Raman spectra is fitted, and the peak location is obtained to find the experimental values of ψ under both states. Figure 7(c) shows the result of this analysis, and each set of data is fitted using a linear function to find ψ. Based on these values, Θ FET-Raman using E 2g and A 1g modes are obtained as 0.766 ± 0.012 and 0.806 ± 0.017, respectively. Then, using these two values and the result of our numerical simulation (figure 5) at RT, k is found by using each Raman mode. This is shown in figure 7(d). For E 2g and A 1g modes, k is found as 21.8 ± 1.36 Wm −1 K −1 and 26.3 ± 3.26 Wm −1 K −1 , respectively.

Discussion
The measured k at RT using the multi-temperature resonance Raman is 23.2 ± 2.88 Wm −1 K −1 (figure 6(a)), and is very close to the result of the FET-Raman experiment at RT (section 3.4). This clearly indicates the accuracy of the multi-temperature resonance Raman technique. As shown in figure 6(a), k of WS 2 sample decreases with the increased temperature, which indicates that the Umklapp scattering mechanism dominates phonon scattering [36,37]. This is consistent with the theoretical calculation of k of WS 2 conducted by Huang et al [38]. It was shown in their work that k of WS 2 is decreased from 32 Wm −1 K −1 at RT to 23 Wm −1 K −1 at 600 K.
Our novel multi-temperature resonance Raman technique possesses several advantages over classical optothermal techniques. First, the absolute temperature rise of the sample is measured in this method, instead of characterizing the temperature rise indirectly using the Raman shift power coefficients or direct temperature sensing using a thermocouple which introduces significant errors to temperature measurement. The Raman spectra of both E 2g and A 1g peaks are obtained in the same measurement, so their intensity ratio eliminates all the effects induced by optical focusing, laser powers, detection, and optical interference since all these have the same effects on these two peaks. If a material has defects that cause change in its excitonic energy variation with temperature, this can be considered by calibrating Ω change with T. The material used in this work: WS 2 has very weak anisotropy, therefore variation of the incident laser polarization with respect to the material's structure has negligible effect on T measurement. For strongly anisotropic 2D materials, we can either use a non-polarized laser source for Raman excitation or use a fixed angle between the crystographical orientation and the laser polarization to obtain reliable temperature measurement. Second, the measured thermal property, like k or interfacial thermal resistance, is independent of the Raman mode selection. Note that each Raman mode is related to a specific optical phonon branch, and these branches experience different temperature rises under a similar heating condition. This effect is studied in detail in our previous work, and it was shown that interface thermal resistance is different among phonon modes of a supported MoS 2 film [20]. Another advantage of this method is that there is no need for Raman shift temperature coefficient calibration and laser absorption measurement. These two factors are the main sources of errors of the SS Raman technique. Last but not least, the experimental procedure takes a much shorter time compared with previous techniques, and it is possible to run several experiments in a shorter period of time.
This technique could have several other applications besides measuring the thermal conductivity of suspended 2D materials. For instance, WS 2 could be used as a thermal sensor to measure the temperature rise of another 2D material that is not experiencing resonance Raman. Also WS 2 can be deposited on other non-Raman active materials to measure their temperature with high accuracy and spatial resolution [39]. In this work, the suitable temperature range of measurement depends on how the excitonic energy varies with temperature. The sensitivity can be tuned by using a Raman excitation laser

Conclusion
A novel resonance Raman ratio (R3) method was developed for high sensitivity temperature measurement at the nanoscale. For WS 2 studied in this work, the resonance Raman ratio Ω = I A 1g /I E 2g changes by two orders of magnitude from 177 to 477 K, demonstrating a very high sensitivity. For example, under RT experimental conditions, 1% change in Ω will lead to a temperature probing sensitivity of 0.78 K, superior over Raman wavenumber-based temperature probing. Also Ω was found intendent of local optical scattering conditions that could be altered by temperature, local structure profiles, and optical focusing level. Using the R3 technique, the thermal conductivity of a suspended WS 2 sample was measured from 170 K to 470 K by directly evaluating the laser-induced temperature rise under CW and periodically modulated laser heating. The result of this experiment at RT was verified with the results of classical FET-Raman technique that uses the Raman wavenumber shift for thermal probing. Also, it was shown that the ET-Raman based on R3 thermal probing eliminates the errors caused by laser absorption measurement and Raman shift temperature coefficient calibration. These are some of the main sources of error in reported optothermal techniques based on Raman spectroscopy. The R3 technique could be applied to other materials with appropriate laser wavelength selection to measure the temperature rise in a wide temperature range, or to characterize their thermal properties, such as thermal conductivity and interfacial thermal resistance.

Raman spectroscopy measurement
The Raman system used in this work consists of a 532 nm continuous wave (CW) laser (Excelsior-532-150-CDRH, Spectra-Physics), a Raman spectrometer (Voyage, B&W Tek, Inc.), an electro-optics frequency modulator (EOM), a microscope (Olympus BX53), and a multi-temperature environment cell chamber. The laser power is controlled using a motorized neutral density filter which is controlled by LabView software. More information about this Raman system could be found in our previous works [13,[40][41][42].

3D numerical modeling for temperature calculation
The theoretical calculation of temperature rise was conducted using 3D numerical modeling based on the finite volume method. To do so, the actual size of the sample and its thickness, as well as the real laser spot radius (r 0 ) measured from the Raman experiment were used to calculate the temperature rise under each heating state (CW or F). The governing equations are: Here ρc p , k, k ⊥ ,q, I 0 (= 1 mW/πr 2 0 ), and τ L are volumetric heat capacity, in-plane thermal conductivity, out-of-plane thermal conductivity, volumetric Gaussian beam heating, unit laser power per unit area, and laser absorption depth of the thin film. For ease of derivation, here we use the form of exp(−r 2 /r 2 0 ) instead of the traditional exp (−r 2 /(2r 2 0 ) to express our Gaussian beam energy distribution in the radial direction. This will not affect our results as we follow the same form in both modeling and experimental data processing. τ L is calculated as: τ L = λ/4πk L , where λ and k L are laser wavelength (here 532 nm) and extinction coefficient of WS 2 /MoS 2 . For WS 2 and MoS 2 , τ L is 47.1 and 36.5 nm, respectively [43][44][45][46]. Although τ L affects the absolute value of temperature rise under each state, the normalized temperature rise (Θ) is almost independent of the τ L value in our technique. This is well illustrated in our previous work [15]. Note that under CW state, the left-hand side of equation (2) is equal to zero. Using these two equations, the temperature rise (T) at each location (r, z) is obtained. Next, the Raman-intensity weighted average temperature rises (∆T CW and ∆T F ) are determined as: Ie −z/τL dvdt, for CW and F states, respectively. Here, V is the volume of the sample, and the term e −z/τL represents the Raman signal attenuation when the signal leaves the scattering location. Also, I is the laser heating source which was shown byq in equation (3). Finally, the normalized theoretical temperature rise is calculated as: Θ th = ∆T F /∆T CW . In our modeling, the surface reflectivity is not considered. This will not affect our Θ th calculation since it only depends on the ratio of temperature rise under CW and F states.

Physics of FET-Raman
The RT FET-Raman technique was first developed and introduced by Wang's group [31,32,47,48]. In this method, the 2D material sample is irradiated under two modes of SS and frequency-modulated transient state. The sample's temperature rise under the transient state depends on the modulation frequency, and it was shown that the temperature rise ratio (transient state divided by SS) takes a value in the range of ∼0.5-1 [31]. Note that the transient state is created by modulating the same CW laser using a square wave function. Therefore, the heating and cooling (or laser on-time and offtime) periods under transient state are identical. The accuracy of FET-Raman is maximum when the temperature rise ratio is ∼0.75. To find the ratio of local temperature rise, the Raman shift power coefficient (ψ) is collected under both states as: ψ SS = ∂ω/∂P ∝ α(∂ω/∂T)f 1 (k) and ψ Transient = ∂ω/∂P ∝ α(∂ω/∂T)f 2 (k, ρc p ) (Top left of figure 2(c)). Next, the normalized Raman shift power coefficient is found as: Θ = ψ Transient /ψ SS ∝ f 2 (k, ρc p )/f 1 (k). This normalized parameter is equal to the normalized temperature rise of the sample under laser irradiation, i.e. T Transient /T SS , where T SS and T Transient are the local temperature rises of the thin film under CW and amplitude modulated laser, respectively. In this work, a 300 kHz frequency is used to modulate the CW laser, and it was shown by figures 5 and 7 that experimental Θ is around ∼0.75, which guarantees the high accuracy performance of FET-Raman. These expressions are related to a suspended sample structure. If the sample is supported over a substrate, each of them will depend on the interfacial thermal resistance too. Therefore, the FET-Raman is capable of measuring the interfacial thermal resistance, as shown in our previous works. Also, by doing this calculation, the effects of laser absorption measurement are totally eliminated in our analysis, which is one of the main sources of error in optothermal techniques based on Raman spectroscopy.