Strain-Dependent Dielectric and Optical Properties of Monolayer MoS₂ with Phase-Sensitive Surface Plasmon Resonance (SPR) Method

Abstract

Monolayer molybdenum disulfide (MoS₂) holds great promise for strain-tunable optoelectronic devices. The strain-dependent dielectric function is a core parameter to characterize the tunability of optoelectronic properties. However, due to the extremely short light–matter interaction path length for atomically thin materials, measurements are challenging. In this work, we measured the dielectric function of strained monolayer MoS₂ using the surface plasmon resonance (SPR) method with the simulated annealing particle swarm optimization (SAPSO) algorithm. When the applied strain ranged from −0.23% (compressive strain) to +0.20% (tensile strain), the dielectric function at seven characteristic wavelengths around the exciton absorption peaks was extracted. Our results demonstrate that both the real part (ε_2r) and the imaginary part (ε_2i) of the dielectric function evolved almost linearly with the applied strain from −0.23% to +0.20%. Based on these results, we further obtained the strain-induced variations in the refractive index (n) and the extinction coefficient (k). At exciton absorption peak B (600 nm), the strain-induced change rate for n reached a maximum of about −0.0141%⁻¹. At the rising edge of the B exciton absorption (580 nm), the strain-induced change rate for k reached a maximum of about −0.3261%⁻¹. This work presents a quantitative extraction of strain-dependent dielectric function of monolayer MoS₂ over excitonic band-edge wavelengths using phase SPR–SAPSO fitting. The proposed method can be extended to the measurement of other atomically thin materials.

1. Introduction

In recent years, monolayer molybdenum disulfide (MoS₂) has garnered significant attention in the field of strain-tunable optoelectronic devices [1,2,3] due to its direct bandgap (1.8~1.9 eV) [4,5] and excellent mechanical flexibility [6,7]. The optical properties of strained monolayer MoS₂ can be quantitatively characterized by its dielectric function. In prior studies, the strain-dependent dielectric function and optical properties of monolayer MoS₂ were usually studied using first-principles simulations [7,8,9,10,11]. However, the simulation results from different computational methods often exhibit some discrepancies [12,13,14,15,16]. The spectroscopic ellipsometry (SE) method is a common technique for measuring the dielectric function spectra of monolayer MoS₂ [17,18], but it encounters an inherent difficulty when strain is applied. Due to the atomic thinness of the material, the optical response from the substrate (e.g., SiO₂/Si) or metal substrate often overwhelms its intrinsic signal, making it difficult for the SE method to extract the tiny strain-induced changes [19].

To tackle this challenge, we employed the surface plasmon resonance (SPR) method to measure the strain-dependent dielectric function for monolayer MoS₂. The SPR method offers extremely high sensitivity for detecting tiny changes in the dielectric function [20,21,22,23,24] and is expected to overcome the difficulty posed by the weak optical response from atomically thin materials [25]. In this work, we systematically measured the dielectric function of monolayer MoS₂ at seven characteristic wavelengths (580, 600, 617, 633, 645, 654, and 670 nm) when the strain ranged from −0.23% (compressive strain) to +0.20% (tensile strain). Our results show that the real part (ε_2r) of the dielectric function decreased monotonically as the applied strain changed from compressive to tensile while the imaginary part (ε_2i) exhibited opposite trends on the two sides of the exciton absorption edge. Further analysis revealed that the strain response of the refractive index was most pronounced at exciton absorption peak B (600 nm), with a change rate of approximately −0.0141%⁻¹. The strongest strain response of the extinction coefficient was at rising edge D (580 nm), where the change rate reached −0.3261%⁻¹ and the corresponding change rate of the absorption coefficient was about −7.0618 × 10⁴ cm⁻¹%⁻¹. The measurement method proposed in this work can be extended to the characterization of the strain-dependent optical properties of other atomically thin materials.

2. Results and Discussion

SPR technology utilizes p-polarized light to excite the evanescent field and surface plasmon polaritons at the metal–dielectric interface. At the resonance angle, the reflected light intensity drops sharply, accompanied by a phase jump. In contrast, the s-polarized light cannot excite the surface plasmons, and its reflection follows the classical Fresnel equations. By measuring the phase difference between the p- and s-polarized reflected light, the common-mode environmental noise can be effectively suppressed, and the phase response caused solely by changes in interfacial dielectric properties is isolated. Prior studies have demonstrated that the phase SPR detection method offers an ultrahigh sensitivity of 10⁻⁶ refractive index units (RIUs) [25,26,27].

Figure 1a illustrates the basic structure of our SPR sensing unit. It comprises three parts: a prism (BK7), a sensing layer (MoS₂/Au) and a strain-loading holder. Considering that the excellent chemical stability of the Au film and its negligible intrinsic elasto-optic effect [28,29] can effectively eliminate interference from the substrate’s own response during measurements, this experiment utilized the Au film to excite SPR. The Au film was deposited on a BK7 glass substrate by magnetron sputtering. Monolayer MoS₂ flakes were synthesized via chemical vapor deposition (CVD) and transferred onto the gold film surface using a wet transfer method. Detailed preparation and transfer procedures are provided in the Methods section. After transfer, the surface morphology of the MoS₂ was characterized by atomic force microscopy (AFM) (Figure 1b). The root mean square (RMS) of the topographic image was 0.3 nm. The top view of the strain-loading holder is shown in Figure 1c. Two polydimethylsiloxane (PDMS) blocks were placed in the two small grooves; one groove was fixed, while the other could move horizontally within a sliding track. The two grooves were connected by a pair of small rigid springs. The sliding groove displacement distance, Δx, was controlled via a micrometer to control the applied force (F = −kΔx). After bonding the sensing layer together with the strain-loading holder, precise control of the sliding groove, Δx, allowed for the release or compression of the spring, and then F was transmitted to the monolayer MoS₂ through the PDMS. Therefore, the monolayer MoS₂ could be modulated by the tensile and the compressive strain.

Figure 1d illustrates the diagram of the measurement system. It employs a broadband laser (SC-PRO, YSL) with a wavelength modulator as the incident light source. The incident beam path is collimated using a pair of mirrors. The beam is shaped by an aperture and then passed through a polarizer to generate a 45° linearly polarized light. The SPR sensing unit is fixed on a high-precision rotation stage (PRM1Z8, Thorlabs). A mirror mounted on the rotation stage is parallel to the bottom surface of the prism within the SPR sensing unit. It was used to convert the rotational movement of the reflected beam’s central position caused by changes in the incident angle into translational motion, facilitating subsequent perpendicular reception by the polarization detector. The polarization detector (PAX10001R1, Thorlabs) was mounted on a three-axis translation stage and continuously acquired the polarization state information at the center of the reflected beam at 0.05 s intervals. The phase difference, Δϕ = ϕ_p − ϕ_s, represents the phase of the p-component relative to the s-component and could be obtained by measuring the Stokes vector, S (S₀, S₁, S₂, and S₃), of the reflected light:

$$\mathit{S}=\left[\begin{array}{c}{S}_0\\ S_1\\ S_2\\ S_3\end{array}\right]=\left[\begin{array}{c}{\mathit{E}}_{\mathrm{p}}^2+{\mathit{E}}_{\mathrm{s}}^2\\ {\mathit{E}}_{\mathrm{p}}^2-{\mathit{E}}_{\mathrm{s}}^2\\ 2{\mathit{E}}_{\mathrm{p}}{\mathit{E}}_{\mathrm{s}}\mathit{cos}∆\varphi \\ 2{\mathit{E}}_{\mathrm{p}}{\mathit{E}}_{\mathrm{s}}\mathit{sin}∆\varphi \end{array}\right]$$

(1)

where E_p and E_s are the electric field of the p-component and the s-component, respectively. Therefore, the phase difference, Δϕ, can be expressed as

$$\Delta \varphi ={\mathit{tan}}^{-1}\left({\displaystyle \frac{S_3}{S_2}}\right)$$

(2)

The transfer matrix method (TMM) [30,31,32] was used to analyze the recorded phase signal from the MoS₂/Au composite structure. As shown in Figure 2a, the TMM model consists of the prism layer (layer 0), the Au layer (layer 1), the MoS₂ layer (layer 2), and the air layer (layer 3). The critical parameters for each layer include the dielectric function, ε_i (i = 0, 1, 2, 3), and the thickness, d_i (i = 0, 1, 2, 3). We assumed the incident beam was a monochromatic plane wave, and the electric field transmission in each layer is described by its characteristic matrix, M_i (i = 0, 1, 2, 3):

$$M_i=\left[\begin{array}{cc}\mathit{cos}{\delta }_i& -{\displaystyle \frac{i}{Q_i}}\mathit{sin}{\delta }_i\\ {\displaystyle \frac{i}{Q_i}}\mathit{sin}{\delta }_i& \mathit{cos}{\delta }_i\end{array}\right]$$

(3)

The total system transfer matrix, M_t, is the product of the individual layer matrices:

$$M_{\mathrm{t}}=M_0·M_1·M_2·M_3=\left(\begin{array}{cc}{m}_{11}& m_{12}\\ m_{21}& m_{22}\end{array}\right)$$

(4)

The reflection coefficient, r, at the prism–Au film interface was calculated from the elements of the M_t and the boundary conditions:

$$r={\displaystyle \frac{{(m}_{11}+m_{12}{Q}_3)Q_0-{(m}_{21}+m_{22}{Q}_3)}{{(m}_{11}+m_{12}{Q}_3)Q_0+{(m}_{21}+m_{22}{Q}_3)}}$$

(5)

The optical admittance, Q_i (i = 0, 1, 2, 3), for s-polarized light (Q_si) and p-polarized light (Q_pi) could be, respectively, expressed as

$$Q_{\mathrm{s}i}=\sqrt{{\epsilon }_i-{\epsilon }_0{sin}^2{\theta }_0}$$

(6)

$$Q_{\mathrm{p}i}={\displaystyle \frac{{\epsilon }_i}{\sqrt{{\epsilon }_i-{\epsilon }_0{sin}^2{\theta }_0}}}$$

(7)

The phase factor, δ_i, could be expressed using ε_i and d_i:

$${\delta }_i={\displaystyle \frac{2\pi }{\lambda }}·d_i·\sqrt{{\epsilon }_i-{\epsilon }_0{sin}^2{\theta }_0}$$

(8)

where θ₀ and λ are the incident angle in the prism and the incident wavelength. The phase shift was derived from the argument of the r. Therefore, the phase difference, Δϕ, could be extracted by

$$∆\varphi =\mathit{arg}\left(r_{\mathrm{p}}\right)-\mathit{arg}\left(r_{\mathrm{s}}\right)$$

(9)

To ensure that the evanescent field possessed sufficient penetration depth and localized electric field intensity to effectively probe the phase difference from the monolayer MoS₂, the thickness of the Au substrate was selected to be about 45 nm (typically, an Au film thickness of 45~50 nm yields the strongest localized electric field for SPR excitation [33]). In this experiment, the thickness of the Au substrate (45 nm) and monolayer MoS₂ (0.7 nm) were confirmed using a stylus profiler and AFM, as detailed in the Supporting Information, part 1. The exciton A and B absorption spectra of the monolayer MoS₂ over the wavelength range of 560 nm to 680 nm were acquired by the transmission-mode SE method using a differential measurement scheme. By measuring the transmitted light intensity ratio between the samples with and without the MoS₂ layer, the contribution from the Au substrate was effectively subtracted, allowing the intrinsic absorption spectrum of the MoS₂ to be extracted, as shown in Figure 2b. Based on the exciton absorption spectrum, seven characteristic wavelengths—exciton peak A (654 nm), exciton peak B (600 nm), absorption valley C (633 nm), rising edges D and E (580 nm and 645 nm), and falling edges F and G (617 nm and 670 nm)—were selected for subsequent dielectric function measurements. The measurement results of the SPR phase difference as a function of angle at seven characteristic wavelengths are plotted as dots in Figure 2c. In addition, the dielectric function (ε_1 = ε_1r + iε_1i) of the Au substrate was measured by the reflection-mode SE method, as shown in Figure 2d, where the dielectric function spectrum is derived from the polarization state of the light reflected from the gold film surface. With the known dielectric function spectrum and thickness of the Au substrate, as well as the thickness of the monolayer MoS₂, the dielectric function (ε_{2 =} ε_2r + iε_2i) of the MoS₂ was extracted using the simulated annealing particle swarm optimization (SAPSO) algorithm [34,35]. The flowchart of the SAPSO algorithm is provided in the Supporting Information, part 2. Obviously, in Figure 2c, there is excellent agreement between the fit curve and the measurement data; the mean squared error (MSE) of the fit is less than 0.01. Therefore, the dielectric function of the monolayer MoS₂ at seven characteristic wavelengths was extracted, as shown in Figure 2e. The dielectric function extracted by our SPR–SAPSO method is in good agreement with the results reported in the literature, as detailed in the Supporting Information, part 3. Furthermore, the absorption coefficient, α, was derived from the measured dielectric function in Figure 2e. The absorption coefficient spectrum obtained by the SPR method follows the same trend as the absorbance using the transmission-mode SE method, as shown in Figure 2f.

During the monolayer MoS₂ flake transfer process, the lattice mismatch between the monolayer MoS₂ flake and the Au substrate introduced pre-strain. Therefore, Raman spectroscopy was used to quantitatively evaluate this pre-strain. In this Raman-based strain calibration, we implicitly assumed that the applied deformation was purely in-plane. For the monolayer MoS₂, the frequency (ω) of the in-plane vibrational mode ($E_g^1$ mode) was highly sensitive to the in-plane strain while the out-of-plane vibrational mode (A_1g mode) primarily reflected the interlayer coupling information. Consequently, in this experiment, the shift of the $E_g^1$ peak was used to calibrate the value of the strain in the monolayer MoS₂. The $E_g^1$ peak position of the original MoS₂ flake on the SiO₂/Si substrate served as the strain-free reference. According to prior studies [36,37], the strain sensitivity of the $E_g^1$ peak frequency was approximately −4 cm⁻¹%⁻¹. Therefore, the strain (σ) could be calibrated using σ (%) = −Δω/4, where Δω represents the measured peak shift of the $E_g^1$ peak. Obviously, the redshift (negative Δω) of the $E_g^1$ peak corresponded to the tensile strain, while the blueshift (positive Δω) corresponded to the compressive strain. The Raman spectrum of the transferred monolayer MoS₂ sample on the 45 nm-thick Au substrate is shown in Figure 3a. Clearly, the $E_g^1$ peak exhibits a redshift of 4.48 cm⁻¹, which indicates that the pre-strain is a tensile strain with a magnitude of 1.12%.

We used the strain-loading holder to compensate for the transfer-induced pre-strain. Figure 3b shows the Raman spectra of the monolayer MoS₂ acquired during the linear modulation after the pre-strain was eliminated. It is evident that the $E_g^1$ peak of the monolayer MoS₂ transitions from the redshift to the blueshift in response to the applied strain. Based on the shift of the $E_g^1$ peak, we achieved the maximum range of linearly modulated strain of approximately ±0.20% on the monolayer MoS₂ sample. Beyond this range, the strain no longer varied linearly with the sliding distance, Δx, of the groove, as detailed in the Supporting Information, part 4. We also characterized the strain uniformity across the monolayer MoS₂, as detailed in the Supporting Information, part 5. Figure 3c presents the linear fitting between the $E_g^1$ peak shift and the groove displacement distance, Δx. The shift coefficient of the $E_g^1$ peak is about 0.28 cm⁻¹mm⁻¹. The zero point in the figure corresponds to the state where the pre-strain has been eliminated. The positive and negative values of Δx represent the compressive and tensile strain stages, respectively.

Leveraging the steep slope of the phase difference–angle curve at the resonance angle, the dielectric variation could be converted into a significant phase-difference change. In Figure 4a, the resonance angles at seven characteristic wavelengths were determined by measuring the SPR reflectivity versus the incident angle curves of the MoS₂/Au composite structure. When the strain ranged from −0.23% to +0.20%, the relative phase difference variations at the resonance angles for these seven wavelengths were measured (Figure 4b). We clarified the sample-to-sample reproducibility by including data from multiple independent samples, as detailed in the Supporting Information, part 6. Using the SAPSO algorithm, the strain-induced changes in the real part (ε_2r) and imaginary part (ε_2i) of the dielectric function were extracted, as shown in Figure 4c,d, respectively. A clear strain response of the dielectric function was observed at seven characteristic wavelengths. Obviously, ε_2r decreased monotonically as the applied strain changed from compressive to tensile, while ε_2i exhibited opposite trends on the two sides of the absorption edge. The results demonstrate that both ε_2r and ε_2i evolved almost linearly with the applied strain from −0.23% to +0.20%, as shown in Figure 4e,f; Δε_2r and Δε_2i denote the changes in the real and imaginary parts of the dielectric function relative to the strain-free state.

Table 1. Change rate of dielectric function per 1% strain.

λ (nm)	Δε2r/Δσ (%−1)	Δε2i/Δσ (%−1)
580	−0.201	−1.998
600	−0.799	−0.223
617	−1.071	1.512
633	−0.492	−1.075
645	−0.394	−1.453
654	−0.925	−0.168
670	−0.839	1.191

presents the change rate of dielectric function per 1% applied strain (σ), obtained by linear fitting as the slope of the variation with strain. At exciton absorption peaks A (654 nm) and B (600 nm) and at absorption valley C (633 nm), the strain-induced change in the dielectric function is dominated by the strain response of ε_2r, while the response of ε_2i is weak. At rising edges D (580 nm) and E (645 nm) and falling edges F (617 nm) and G (670 nm) of the exciton absorption spectrum, the strain response of ε_2i is significantly enhanced. The strain response of ε_2r is most pronounced at falling edge F, with a change rate of about −1.071%⁻¹. The strain response of ε_2i is most pronounced at rising edge D, with a change rate of about −1.998%⁻¹.

From the extracted dielectric function, the refractive index, n; the extinction coefficient, k; and the absorption coefficient, α, of the monolayer MoS₂ could be obtained:

$$n=\sqrt{{\displaystyle \frac{\sqrt{{\epsilon }_{2\mathrm{r}}^2+{\epsilon }_{2\mathrm{i}}^2}+{\epsilon }_{2\mathrm{r}}}{2}}}$$

(10)

$$k=\sqrt{{\displaystyle \frac{\sqrt{{\epsilon }_{2\mathrm{r}}^2+{\epsilon }_{2i}^2}-{\epsilon }_{2\mathrm{r}}}{2}}}$$

(11)

$$\alpha ={\displaystyle \frac{4\pi }{\lambda }}\cdot k$$

(12)

The change rates of the refractive index (Δn), the extinction coefficient (Δk), and the absorption coefficient (Δα) per 1% applied strain for monolayer MoS₂ are shown in

Table 2. Change rate of optical constants per 1% strain.

λ (nm)	Δn/Δσ (%−1)	Δk/Δσ (%−1)	Δα/Δσ (cm−1%−1)
580	−0.0116	−0.3261	−7.0618 × 104
600	−0.0141	−0.0461	−0.9650 × 104
617	−0.0112	0.3025	6.1579 × 104
633	−0.0093	−0.0232	−0.4603 × 104
645	−0.0109	−0.2417	−4.7071 × 104
654	−0.0139	−0.0498	−0.9564 × 104
670	−0.0121	0.2114	3.9630 × 104

. Obviously, the strain response of n is the most pronounced at exciton absorption peak B (600 nm), reaching a change rate of −0.0141%⁻¹. The strain response of k is the most pronounced at rising edge D (580 nm) of the exciton absorption spectrum, where the change rate reaches −0.3261%⁻¹, and the corresponding change rate for α is about −7.0618 × 10⁴ cm⁻¹%⁻¹.

To understand the mechanism of the variation in the dielectric and optical properties of monolayer MoS₂ under strain, we took a closer look at the strain-induced energy band shift in our prior reported DFT work [11]. At zero strain, monolayer MoS₂ is a direct bandgap semiconductor. Upon applying the compressive or tensile strain, the conduction band minimum decreased while the valence band maximum increased, and then this led to the bandgap narrowing. Consequently, the peaks of ε_2r and ε_2i are red-shifted. The changes for the dielectric function of monolayer MoS₂ can be attributed to the modification of the band structure under these strains, which is closely related to the inter-band optical transition probability and the joint density of states. Specifically, ε_2i directly corresponds to the allowed inter-band optical transition probability in the Brillouin zone. The magnitude of ε_2i is proportional to the joint density of states J_DOS [38]:

$${\epsilon }_{2i}\left(\omega \right)\propto {\displaystyle \frac{1}{{\omega }^2}}·J_{DOS}\left(E\right)$$

(13)

$$J_{DOS}\left(E\right)=\int {\displaystyle \frac{2d^{3}k}{{\left(2\pi \right)}^3}}\delta \left[E_C\left(\mathit{k}\right)-E_V\left(\mathit{k}\right)-E\left(\mathit{k}\right)\right]$$

(14)

where ω, k, δ, E(k), E_C(k) and E_V(k) are the angular frequencies of the incident light, the wave vector, the Dirac delta function, the photon energy, and the energies of the conduction band and the valence band at wave vector k, respectively. Treating strain as a perturbation to the band structure, the strain-induced change in ε_2i at a fixed photon energy can be expressed as

$${\displaystyle \frac{d{\epsilon }_{2i}}{d\sigma }}\propto {\displaystyle \frac{d{J}_{DOS}}{dE}}·{\displaystyle \frac{d\left[E_C\left(\mathit{k}\right)-E_V\left(\mathit{k}\right)\right]}{d\sigma }}$$

(15)

The excitonic effect leads to spin-orbit splitting of the valence band and then induces two absorption peaks at the 600 and 654 nm wavelengths, respectively [39]. Strain would alter the lattice constant of the monolayer MoS₂ and then shift the resonance frequency of exciton. The redshift and blueshift of the resonance frequency would occur under the tensile and compressive strains, respectively. Then, the shift of the resonance frequency of the excitons leads to a linear response of dielectric function at the absorption edges [40]. Strain modulates the bandgap by altering the lattice constant, causing an overall shift in the bands. However, since ∂J_DOS/∂E ≈ 0, the contribution of strain-induced band changes to ε_2i comes only from second-order terms, making ε_2i insensitive to strain. At absorption valley C (633 nm), J_DOS is at a local minimum and varies slowly with energy. Strain-induced band shifts also fail to cause significant changes in ε_2i. At rising and falling edges D (580 nm), E (645 nm), F (617 nm) and G (670 nm), these positions lie near the band edges or thresholds of the band structure, where J_DOS varies rapidly with energy. When strain induces an overall band shift, ε_2i changes significantly. Specifically, at the rising edges, J_DOS decreases with the strain changes from compressive to tensile. Therefore, ε_2i decreases with strain. On the contrary, at the falling edges, J_DOS increases with the strain changes from compressive to tensile. Therefore, ε_2i increases with strain. The sensitivity of ε_2i to strain is proportional to ∂J_DOS/∂E. Therefore, the strain response of ε_2i is most pronounced at the band edges.

The development of ε_2r is related to ε_2i over the entire energy range via the Kramers–Kronig (K–K) relation:

$${\epsilon }_{2r}\left(\omega \right)=1+{\displaystyle \frac{2}{\pi }}\mathcal{P}{\int }_0^{\infty }{\displaystyle \frac{{\omega }_1{\epsilon }_{2i}\left({\omega }_1\right)}{{\omega }_1^2-{\omega }^2}}d{\omega }_1$$

(16)

where $\mathcal{P}$ denotes the Cauchy principal value. At exciton absorption peaks A and B and absorption valley C, ε_2i is insensitive to strain. Here, the strain response of ε_2r mainly originates from the strain-induced overall band shift. This is because the strain changes the lattice structure and thereby induces a shift in the bandgap. As a result, from compressive to tensile strain, ε_2r decreases monotonically. At the rising and falling edges, ε_2i varies rapidly with energy. The strain response of ε_2r is governed by the combined action from the shift in bandgap and the drastic change for ε_2i. At the rising edge, ε_2i decreases with tensile strain, and this contribution tends to further reduce ε_2r. At the falling edge, ε_2i increases with tensile strain, which additionally increases ε_2r. These strain coefficients suggest that even modest mechanical deformations can produce measurable optical responses in practical devices. For example, a typical operational strain of 0.1~0.2% would already yield a finite modulation of the refractive index and extinction coefficient, which translates into detectable changes in transmission, reflection, or phase in integrated MoS₂-based waveguides and cavities. Such a strain-tunable optical response indicates that the proposed platform is promising for compact, flexible strain sensors and mechanically reconfigurable filters or modulators in next-generation photonic circuits.

There are three main potential sources of error in the SPR–SAPSO method. (1) The angular positioning error of the rotation stage is a key factor leading to measurement errors in the resonance angle of the SPR signal. In our experiment, the accuracy of the rotation stage (PRM1Z8, Thorlabs) was approximately 0.005°, which could have led to an uncertainty of about ±0.2 for the ε_2r and ε_2i. (2) The SPR signal is also sensitive to the output wavelength and power of the laser (SC-PRO, YSL). Although a controller was used to maintain the working temperature and driving current, the output of the laser still slightly fluctuated with time and then induced a phase-difference variation of about 0.2°. After real-time processing through a low-pass filter, averaging 100 data sets acquired over a 5 s period could control the phase fluctuation to 0.01°. This residual phase uncertainty would lead to an uncertainty of about ±0.05 for ε_2r and ε_2i. (3) The roughness of the monolayer MoS₂ and Au substrate would induce the errors for thickness measurement. The surface roughness values of the monolayer MoS₂ and Au substrate were approximately 0.3 nm and 0.5 nm, as detailed in the Supporting Information, part 7, which could have induced an uncertainty of about ±0.3 for ε_2r and ε_2i. Based on the above discussion, the total uncertainty for ε_2r and ε_2i is about ±0.55.

3. Conclusions

In this work, the dielectric function of monolayer MoS₂ was systematically measured using the SPR method when the strain ranged from −0.23% to +0.20% at seven characteristic wavelengths (580–670 nm) in the exciton absorption spectrum. Our results show that ε_2r decreased monotonically as the applied strain changed from compressive to tensile, while ε_2i exhibited opposite trends on the two sides of the absorption edge. Moreover, both ε_2r and ε_2i evolved almost linearly with the strain. Based on the measured dielectric function, the strain responses of n, k, and α were further obtained. The strain-induced change rate of n reached a maximum of approximately −0.0141%⁻¹ at exciton absorption peak B (600 nm). The change rate of k was most pronounced at the rising edge of the absorption spectrum (580 nm), reaching −0.3261%⁻¹, with the corresponding change rate of α at about −7.0618 × 10⁴ cm⁻¹%⁻¹. The dielectric measurement method employed in this study is universal and can be readily extended to the measurement of the strain-induced optical properties of other atomically thin materials.

4. Method

Au substrate deposition method: The Au film was deposited by the direct-current magnetron sputtering method. The Au target (99.999%) served as the source material. Prior to deposition, the chamber was evacuated to a base pressure lower than 3 × 10⁻⁴ Pa. High-purity argon gas was then introduced as the working gas, maintaining a process pressure of approximately 2 Pa. A direct current of 20 mA was applied to initiate and sustain the plasma for sputtering.

Growth method for monolayer MoS₂ flake: Molybdenum trioxide (MoO₃, 99.999% purity) was used as the molybdenum source, and solid sulfur (99.999% purity) was used as the sulfur source, with argon serving as the carrier gas. Growth was carried out in a dual-zone tube furnace with an 80 mm tube diameter. The MoO₃ was heated to 650 °C and the sulfur to 180 °C. The growth was conducted at a pressure of 4000 Pa for 10 min on a SiO₂/Si substrate.

Transfer method for monolayer MoS₂ flake: First, a 4% poly(methyl methacrylate) (PMMA) in anisole solution was drop-cast onto the SiO₂/Si substrate with as-grown MoS₂ and cured on a hotplate at 100 °C to form a thick supporting film, which facilitated subsequent handling. Next, the sample was immersed in a 2 mol/L KOH solution for approximately 2 h to safely separate the PMMA film from the substrate. The freed PMMA/MoS₂ film was then lifted with tweezers, rinsed 3–5 times in deionized water to remove residual KOH, and subsequently scooped up from below using the target substrate (the Au film) to bring the MoS₂ into contact. Following transfer, the substrate with the film was placed vertically to air-dry and then annealed on an 80 °C hotplate to improve adhesion. Finally, the PMMA support layer was removed by immersing the sample in acetone. The acetone was replaced 3–5 times, with each soak lasting about 30 min, to ensure complete dissolution of the PMMA. The PMMA support layer was removed repeatedly with acetone to ensure complete dissolution of the PMMA as much as possible.