Measuring the optical permittivity of two-dimensional materials without a priori knowledge of electronic transitions

2.1 Deterministic measurement of the refractive index

We suggest a method to directly measure the refractive index of 2D materials using conventional spectroscopic ellipsometers. These conventional instruments measure the complex ratio of the reflection coefficients for p- and s-polarized light, i.e. ρ=r^(p)/r^(s), within a finite spectral range. Figure 1 shows a schematic of the measurement procedures. Suppose that a thin film of 2D material of thickness d_f and refractive index n_f is prepared on a dielectric substrate of refractive index n_sub. The light beam impinges on the thin film from the upper half-plane with the index n₀. We measure the complex reflection ratio of both the thin 2D film material on the substrate (ρ_f) and the bare substrate alone (ρ_sub). Since the thin 2D film material is atomically thin, we can approximate the complex reflection ratio by

Figure 1:

Principle of the deterministic permittivity measurement.

(A) Schematic of the deterministic permittivity measurement of 2D materials. (B) Complex reflection ratio of the 2D thin film material of thickness d_f on the substrate (ρ_f) and of the bare substrate (ρ_sub). E_f is the reflected field when the incident field E_inc impinges on the sample with the angle of incidence θ₀. Reflectivity can be characterized by the reflection amplitude ratio tan(ψ) and the phase difference Δ.

using the Taylor expansion for the optical path length ϕ=n_fk₀d_fcosθ_f in the 2D thin film material, with the vacuum wavenumber k₀=2π/λ, the wavelength of light λ, the propagating angle of light inside the 2D material θ_F, and the reflection coefficients of the thin 2D film material on the substrate for p-polarized (s-polarized) light rf(p)(rf(s)). The angle θ_F is determined by Snell’s law, n_Fsinθ_F=n₀sinθ₀, with the incidence angle of light θ₀. Note that we used the relation ρsubs=(rf(p)/rf(s))|ϕ=0 to derive Eq. (1). The reflection coefficients rf(p, s) are given by

Then, the terms in the round bracket in Eq. (1) evaluated at ϕ=0 can be written as

where the reflection coefficients at the interface of the two media with the refractive indices n₀ and n are given by

Plugging Eqs. (3–5) into Eq. (1), we can obtain the quadratic equations for the refractive index of the 2D thin film material as

where the function α is given by

The solution of Eq. (6) can be expressed in the form

where δ_ρ=(ρ_f–ρ_sub)/ρ_sub is the fractional change in the complex reflection ratio.

The ambiguity of the sign in Eq. (8) can be resolved by imposing the passivity condition Im(n_f)>0. The expression for the refractive index of 2D materials, Eq. (8), is the key result of this paper. We can directly obtain the refractive index n_f from the measured quantities ρ_f and ρ_sub. It requires neither the fitting procedures for the spectral line shape functions nor a priori knowledge of the electronic transitions of the 2D materials. Using our method, the deterministic solution for the permittivity can be obtained from the given measurement data.

2.2 Refractive indices of monolayer TMDCs

We apply our method to monolayer TMDCs, which are promising 2D materials possessing semiconductor properties, with their direct bandgap in visible wavelengths and their spin-valley properties [5], [18]. Large-area, uniform films of monolayer MoS₂, WS₂, and WSe₂ were purchased from SixCarbon and prepared on sapphire (Al₂O₃) wafers. We measured their complex reflection ratios and the permittivity of the sapphire substrate using a commercial spectroscopic ellipsometer (alpha-SE, JA Woollam), which is used throughout this paper.

Figure 2A–C shows the Raman spectra of MoS₂, WS₂, and WSe₂ monolayers. In Figure 2A, the MoS₂ monolayer presents two sharp peaks at 385.48 (E2g1) and 402.90 cm⁻¹ (A_1g), which correspond to the first-order in-plane and out-of-plane modes at the Brillouin zone center, respectively [19]. The frequency difference between the E2g1 and A_1g modes is Δω=17.42 cm⁻¹, which is indicative of the MoS₂ monolayer [6]. In Figure 2B, the WS₂ monolayer presents a peak at 348.93 cm⁻¹ (A_1g) with a shoulder at 330.32 cm⁻¹ (2LA). The shoulder originates from the second-order longitudinal acoustic (LA) phonon mode, and is indicative of the WS₂ monolayer [20]. In Figure 2C, the WSe₂ monolayer presents a peak emerging at 250.53 cm⁻¹ due to the first-order out-of-plane mode (A_1g). It is known that this strikingly sharp peak disappears when the WSe₂ layer increases in thickness to a few layers more than a bilayer [19] and that the emergence of the sharp A_1g peak is indicative of a WSe₂ monolayer.

Figure 2:

Confirmation of monolayer TMDCs.

Raman spectra of monolayers of (A) MoS₂, (B) WS₂, and (C) WSe₂. The optical contrast of monolayer (D) MoS₂, (E) WS₂, and (F) WSe₂ along the black lines in the optical images in the insets.

We also use the optical contrast of the monolayers on the Si/SiO₂ wafer to confirm that the TMDC monolayers are really monolayers. The optical contrast is defined as (I–I_sub)/I_sub using the image intensities I (I_sub) of the 2D TMDCs (the bare substrate). The interference of light from the SiO₂ spacer layer on the Si substrate allows the number of layers of 2D TMDCs to be confirmed by optical contrast [21]. When the optical contrast of the 2D TMDC flakes lies between 0.25 and 0.4, we can conclude that they are composed of a monolayer [21]. First, the samples of monolayers MoS₂, WS₂, and WSe₂ were transferred to Si/SiO₂ wafers with an oxide thickness of 280 nm, and optical images were then recorded using a charge-coupled device (CCD) camera. Figure 2D–F shows the optical contrast of the monolayers of MoS₂, WS₂, and WSe₂ along with the black lines of the optical images in the insets. The optical contrasts of our samples lie between 0.25 and 0.4, indicating that they are monolayers [21].

Figure 3 shows the permittivity spectra of monolayers of MoS₂, WS₂, and WSe₂ in the visible spectral range, obtained using Eq. (8). For all monolayers TMDCs, the A and B exciton bands can be found using the peaks of the imaginary part of the measured permittivity near the energy of 2.0 eV. At the higher energy band near 3.0 eV, the strong interband transition (peak I) and its fine structure (peak I_f) are observed [8]. For monolayers WS₂ and WSe₂ (Figure 3B,C), their interband transitions and fine structure are clearly resolved, while for monolayer MoS₂ the fine structure is barely visible (Figure 2A). By resolving these peaks, we demonstrate that the permittivity of 2D TMDCs can be obtained without a priori knowledge of the electronic transitions in the visible frequency range. In the next section, we will confirm that our method is not influenced by knowledge of the strong electronic transitions outside the spectral window.

Figure 3:

The permittivity of monolayer TMDCs.

Real (left column) and the imaginary part (right column) of the permittivity of monolayers of (A, B) MoS₂, (C, D) WS₂, and (E, F) WSe₂. Solid and dashed lines, respectively, correspond to the data obtained by our method and by Tauc–Lorentz fitting with a limited number of poles (see Supporting Information for the tabulated data of the permittivity and the refractive index plotted in Figure 3).

2.3 Effect of a priori knowledge of high-energy electronic transitions on permittivity measurements

Conventional spectroscopic ellipsometry methods for semiconductor materials use Tauc–Lorentzian (TL) spectral line shape functions to describe the permittivity of these materials [9], [22]. The imaginary permittivity can be written as the sum of the N TL functions, i.e. Imε(E)=∑k=1NImεk(E) with

with the amplitude A, the broadening C, the photon energy E, the peak transition energy E₀, and the bandgap E_b. The subscript k in Eq. (9) denotes the parameters of the kth pole. Each TL function corresponds to the respective electric transitions of the material. The real part of the permittivity can be obtained by the Kramers–Kronig relation [23]

Equation (9) automatically gives the constraints to the obtained permittivity satisfying the Kramers–Kronig relation, resulting in a causal description of the optical response of the materials. However, this approach requires a priori knowledge of the materials because electronic transitions outside the spectral range of interest can affect the analysis of the permittivity in the finite spectral range. For example, high-energy electronic transitions up to 30 eV should be taken into account for the permittivity of monolayer TMDCs within the narrow visible spectral range of 1.5–3.0 eV [9]. Higher energy electronic transitions are generally very strong, and their off-resonant tails can reach the visible spectral range. If these high-energy electronic transitions are neglected in the analysis of the permittivity, it may lead to multiple results satisfying the same reflectivity spectrum.

In Figure 4, we demonstrate how, when using the N TL spectral functions, high-energy electronic transitions outside the spectral range of interest influence the permittivity analysis, i.e. Eq. (9). We numerically find the TL parameters in Eq. (9) that fit the experimentally measured spectra of the reflection ratio, i.e. ρ=r(p)/r(s)=tan(Ψ)eiΔ, with the reflection amplitude ratio tan(ψ) and the phase difference Δ. The accuracy of the spectral fitting is determined for a given number of TL functions N. In contrast, our deterministic index analysis method, based on Eq. (8), does not depend on N. In Figure 4, we compare the errors made by these two permittivity analysis methods. The error can be evaluated by δψ=ψ_exp–ψ_mod and δΔ=Δ_exp–Δ_mod, where ψ_exp and ψ_mod (Δ_exp and Δ_mod) are the reflection amplitude ratio ψ (the phase difference Δ) of the experimental results and the model results, respectively. Without a priori knowledge of the electronic transitions in the material, the function number N is determined by the number of peaks in the visible frequency range, i.e. N_vis=3 (N_vis=4) for monolayer MoS₂ (WS₂ and WSe₂). As shown in Figure 4, TL function analysis using the function number N=N_vis does not provide a perfect fit for the experimentally measured ψ_exp and Δ_exp. Note that the error made by the TL function analysis is significant even for the A and B exciton bands, which are far from the higher energy bands with interband transitions. This implies that the optical response of monolayer TMDCs across the entire visible frequency range is also influenced by higher energy electronic transitions such as the fine structures of the interband transition, partial plasma resonances, and the full plasma excitation involving all valence electrons [8]. On the other hand, our deterministic analysis method shows reduced error compared to the TL function analysis with the function number N=N_vis. We also note that the errors δψ and δΔ in our method tend to increase in the high-energy bands near 3.0 eV because the thin-film approximation starts to fail, but this does not significantly affect the visible frequency range in which the A and B exciton bands are located. Table 1 summarizes the spectral average of the errors given by 〈δΨ〉={∫ω1ω2δΨ(ω)dω}/(ω2−ω1) and 〈δΔ〉={∫ω1ω2δΔ(ω)dω}/(ω2−ω1) for the two methods. It shows that the spectral averages of the errors of our analysis method are smaller than those of the TL analysis method.

Figure 4:

The errors compared to the conventional method.

(A) Reflection amplitude ratio (ψ) and (B) phase difference Δ. Solid lines: experimental results; white squares: results of our method; black squares: results of TL fitting with N poles. (C) Errors in the amplitude ratio δψ and (D) errors in the phase difference δΔ of monolayers of MoS₂ (left column), WS₂ (middle column), and WSe₂ (right column). Solid lines with white squares show the errors in the results of our method and solid lines with black squares are the errors in the results of the TL fitting method with N poles.

2.4 Effect of anisotropy of the 2D materials

It is also worth noting that the permittivity of the 2D materials can be described in the tensor form

with the in-plane permittivity ε_|| and the out-of-plane permittivity ε_⊥ because of material anisotropy. Using the anisotropic permittivity tensor, we can relate the in-plane and out-of-plane permittivities with the complex reflection ratio. In the same manner as when deriving Eq. (6), the relation is given by

Equation (12) shows that the in-plane permittivity ε_|| and the out-of-plane permittivity ε_⊥ cannot be simultaneously determined by our method. However, we can find that the effect of the out-of-plane permittivity ε_⊥ becomes significant when ε_⊥ is small. The zero out-of-plane permittivity, i.e. ε_⊥=0, falls into the singularity of the term nsub2/ε⊥ in Eq. (12), but only for metallic materials whose permittivity decreases to the point of becoming negative. However, for most 2D materials, the out-of-plane permittivity is unlikely to become metallic [24]. Therefore, we can speculate that the anisotropic effect on permittivity is insignificant for the optical response of 2D materials.