Deterministic reflection contrast ellipsometry for thick multilayer two-dimensional heterostructures

2.1 Summary of our approach

Figure 1 shows the concept of DRCE. DRCE aims to measure the complex refractive index n ̃ 2 D of the topmost 2D layer in the multilayer heterostructure. Analytic determination of the index of the topmost layer on the multilayer is in general not straightforward because of the complicated interference effects within the multilayer. To avoid the complexity, we find that the bottom multilayers can be simplified to the reduced substrate with the effective indices n ̃ eff ( p ) and n ̃ eff ( s ) for the p- and s-polarization states of light, respectively. For example, suppose the topmost 2D layer is sitting on an arbitrary layer (#2, e.g. SiO₂) of the index n ̃ 2 and the thickness d ₂. The superstrate (#1, e.g. air) and the substrate (#3, e.g. Si) have the index n ₁ and n ̃ 3 , respectively. Then, the layer (#2) and the substrate (#3) can be simplified to the reduced substrate of the indices n ̃ eff ( p ) and n ̃ eff ( s ) .

Using the effective substrate reduction, any optically thick multilayer can be replaced with a single substrate. This enables analytic determination of the index of the topmost layer on the multilayer. DRCE measurement combines conventional ellipsometry and RC spectroscopy. As in conventional ellipsometry, DRCE measures the complex reflection ratio ρ ̃ = r ̃ ( p ) / r ̃ ( s ) , where r ̃ ( p ) and r ̃ ( s ) denote the reflection coefficient of the p- and s-polarization states, respectively. Similar to RC spectroscopy, DRCE measures the reflection ratio contrast δ = ( ρ ̃ 2 D − ρ ̃ sub )/ ρ ̃ sub between the topmost 2D layer ( ρ ̃ 2 D ) and the reduced substrate ( ρ ̃ sub ) instead of the reflection contrast. Then, we can analytically determine the complex refractive index n ̃ 2 D of the topmost 2D layer in the multilayer heterostructure. Details on the effective substrate reduction and DRCE are described in the following subsections.

2.2 Effective substrate reduction

The proposed DRCE method is based on effective substrate reduction; light reflection by a single layer on a substrate is equivalent to that of a reduced substrate whose effective refractive index n ̃ eff is different from that of the original layer or substrate (refer to Supplementary Materials for the proof). In other words, the Airy formula for a single layer on a substrate is equivalent to the Fresnel formula for an interface between two semi-infinite media if the effective index is adequately determined.

This equivalence allows the reduction of a N layer system to a (N − 1) layer system if we know all the s-wave ( r ̃ ( s ) ) and p-wave ( r ̃ ( p ) ) reflections at the interface between each layer and the superstrate (e.g., air). Experimentally, this condition for DRCE can be achieved in two ways: (i) a multilayer system with a staircase structure, and each layer has an exposed air/layer interface (Figure 2(a)); many lab-scale 2D heterostructures fall into this category because the sizes of mechanically exfoliated and chemically grown 2D flakes are typically different from each other. (ii) If the staircase multilayer structure cannot be achieved, reflection and layer transfer are alternately performed so that we can determine the reflection at each layer/air interface (Figure 2(b)). 2D heterostructures prepared by the subsequent chemical deposition of layers fall into this category. We note that the theoretical values of the reflection coefficients r ̃ ( s ) and r ̃ ( p ) can be used in DRCE if the permittivity of several ingredient layers is known and their optical characterization is not of interest. For example, for the characterization of a monolayer TMD on a SiO₂/Si substrate, we do not need to measure reflections at the interfaces of air/Si and air/SiO₂/Si substrates because we can calculate the reflection coefficients at such interfaces using the permittivity reported in the literature. We note that two measurement schemes in Figure 2(a) and (b) are applicable to heterostructures with no or weak interlayer coupling because strong interlayer coupling can change the permittivity of each 2D layer.

Figure 2:

Two experimental schemes for deterministic reflection contrast ellipsometry (DRCE): (a) Scheme I: A staircase multilayer structure; reflection ratio ( ρ ̃ ) measurements are performed at the interface of each layer and air. (b) Scheme II: A general multilayer structure; reflection ratio measurements are performed before the next layer preparation. (c) Schematic drawing of the subsequent effective substrate reduction for DRCE. (d) Reflection contrast (RC) spectra δR of monolayer MoS₂ on the SiO₂/Si substrate is plotted as a function of oxide thickness. Exciton energies are depicted in the black dashed line in the magnified inset. Apparent peaks in RC spectra are shifted as oxide thickness varies, resulting in inaccurate measurement of exciton energies using RC spectroscopy. (e) The complex refractive index n ̃ = n + i k of the same monolayer MoS₂ on the SiO₂/Si substrate characterized by DRCE at the angle of incidence of 40°. The reference refractive index is obtained from the literature [20], and it is plotted in the left and right sides of the plots. Exciton energies are not shifted in our method although some errors in their magnitudes depend on the oxide thickness. Figure 1(d) and (e) show theoretical calculations.

Subsequent reduction of the effective substrate yielded a single top layer on the reduced substrate (Figure 2(c)). This suggests that the complicated ellipsometry problem for the N-layer system can always be reduced to a simple RC ellipsometry problem for a single-layer system if we can measure the reflections of the s-wave ( r ̃ ( s ) ) and p-wave ( r ̃ ( p ) ), that is, the complex reflection ratio ρ ̃ = r ̃ ( p ) / r ̃ ( s ) , at the interface between each layer and the superstrate, as shown in Figure 1(c). Note that the measurement of ρ ̃ can be achieved using commercial ellipsometers or a home-made setup, such as a rotating compensator ellipsometer (RCE) [30].

Analytically, the effective refractive indices of the reduced substrate replacing the single layer and its substrate are expressed as follows (see Supplementary Materials for details on the derivation),

for the p- and s-polarizations of light, respectively. Notations in Eqs. (1) and (2) follow Figure 1. Plus-minus sign in Eq. (1) can be determined by continuous and nonvanishing values in n ̃ eff ( p ) because null values in the effective indices result in the failure of DRCE (see Supporting Information for details). The single layer has an index of n ̃ 2 and a thickness of d ₂, while the indices of the semi-infinite superstrate and substrate are n ₁ and n ̃ 3 , respectively. θ ₁ and θ ̃ 2 = s i n − 1 n 1 / n ̃ 2 sin θ 1 denote the angles of incidence and refraction in the layer, respectively. In Eq. (2), function β ̃ is expressed as follows:

where ϕ ̃ 2 = k ̃ 2 d 2 ⁡ cos θ ̃ 2 is the optical path length inside the layer whose thickness is d ₂ and wavenumber is k ̃ 2 , and the Fresnel equation determines r ̃ 23 ( p ) = ( n ̃ 2 ⁡ cos θ ̃ 3 − n ̃ 3 ⁡ cos θ ̃ 2 ) / n ̃ 2 ⁡ cos θ ̃ 3 + n ̃ 3 ⁡ cos θ ̃ 2 and r ̃ 23 ( s ) = ( n ̃ 2 ⁡ cos θ ̃ 2 − n ̃ 3 ⁡ cos θ ̃ 3 ) / ( n ̃ 2 ⁡ cos θ ̃ 2 + n ̃ 3 ⁡ cos θ ̃ 3 ) in Eq. (3). θ ̃ 3 = s i n − 1 n 1 / n ̃ 3 sin θ ̃ 3 is the angle of refraction in the substrate. Note that Eqs. (1) and (2) are reduced to our previous result in limit of the normal angle of incidence [31].

2.3 Deterministic reflection contrast ellipsometry (DRCE)

By the effective substrate reduction, any multilayer system can be reduced to the reduced substrate whose effective indices are given by Eqs. (1) and (2). Let a 2D material layer of index n ̃ 2 D and thickness d _2D is sitting on the reduced substrate. Once we can measure the reflection ratio contrast δ = ( ρ ̃ 2 D − ρ ̃ sub )/ ρ ̃ sub between the top 2D layer ( ρ ̃ 2 D ) and the reduced substrate ( ρ ̃ sub ), the refractive index can be obtained as follows:

where the functions A, B, and α are defined as follows:

where λ is the wavelength of light. Notations in Eqs. (4)–(7) follow Figure 1. Eq. (4) is a key result of this study. Note that n ₁ is the index of the superstrate in Eqs. (4)–(7). All variables in Eqs. (4)–(7), except for the experimentally measured δ, are known for a given sample. It is also noteworthy that Eq. (4) is obtained by the first-order expansion of δ in a function of the optical path length ϕ ̃ 2 D = k ̃ 2 D d 2 D ⁡ cos θ ̃ 2 D , and the second-order expansion can increase the accuracy of DRCE [15].

In DRCE, the angle of incidence should be carefully chosen because Eq. (4) is singular if n ̃ eff ( p ) and n ̃ eff ( s ) become 0 or ±1. We find that the singularity can occur when light interferes destructively. The destructive interference condition is given by 2 m + 1 λ / 4 = n i d i ⁡ cos θ i with the refractive index n _i , the thickness d _i, the integer m, and the angle of propagation in an arbitrary ith layer of the multilayer heterostructure. Therefore, large angle of incidence can push the spectral region of the interference-induced errors to the short wavelengths. Note that one should minimize errors in each measurement step (Figure 2(a)) for the reflection ratio contrast δ because errors can be accumulated, and it results in inaccurate results of the refractive indices.

2.4 Experiment and simulation results

To verify the permittivity characterization using Eq. (4) for a given reflection-ratio contrast δ, we theoretically simulated the DRCE of monolayer MoS₂ on a SiO₂/Si substrate with an oxide thickness of 280 nm. In our previous study, we calculated the reflection ratio contrast δ using the transfer matrix technique [32] and tabulated the permittivity of monolayer MoS₂. We use the monolayer thickness d _2D = 0.65 nm in the following [20]. As illustrated in Figure 3, we compare the permittivity obtained by DRCE at various angles of incidence (color profiles) with the reference permittivity [20] (insets on the left and right sides). First, we identified that the positions of the excitonic peaks remained nearly constant and were the same as the reference permittivity for all angles of incidence. Second, the DRCE results for the permittivity became more accurate over a broad energy range as the angle of incidence increased. To quantify the accuracy of DRCE, we plot its error, ( δ n ̃ = n ̃ ref − n ̃ / n ̃ ref ), in Figure 3(b). The error shifted to a higher-energy region as the angle of incidence increased, as shown in Figure 3(a). The error region follows the destructive interference induced by the oxide layer (black lines in Figure 3(b)).

Figure 3:

Simulation for the DRCE measurement of the refractive index of monolayer MoS_2: (a) Refractive index n ̃ = n + i k of monolayer MoS₂ characterized by DRCE with different angles of incidence. Monolayer MoS₂ is sitting on the SiO₂/Si substrate with oxide thickness of 280 nm. The real (n _ref ) and imaginary parts (k _ref ) of the reference refractive index [20] ( n ̃ ref = n ref + i k ref ) are plotted in the left and right sides, respectively. Exciton energies are displayed by black arrows. (b) Corresponding errors of DRCE for the refractive index ( δ n ref = n ref − n / n ref and δ k ref = k ref − k / k ref ). Black lines in (b) are destructive interference condition of the oxide layer.

In addition, we simulated the DRCE of a 2D heterostructure composed of a monolayer MoS₂/monolayer WSe₂/280 nm-thick SiO₂/Si substrate in Figure 4. The refractive index of each monolayer TMDs can be obtained sequentially by the reflection ratio contrast δ at each layer/air interface. (i) Using the reflection ratio contrast δ between the SiO₂ layer and the substrate, oxide layer and substrate are replaced with the first reduced substrate. (ii) Using the reflection ratio contrast δ between the monolayer WSe₂ and SiO₂/Si substrate (i.e., the first reduced substrate), we can obtain the refractive index of the monolayer WSe₂ using Eq. (4). Moreover, the reflection ratio contrast δ yields the second reduced substrate. (iii) Finally, using the reflection ratio contrast δ between monolayer MoS₂ and the monolayer WSe₂/SiO₂/Si substrate (i.e., the second reduced substrate), we can obtain the refractive index of the monolayer MoS₂ by Eq. (4). This procedure is illustrated in Figure 2(a). Figure 4(a) and (b) show the refractive indices of the two monolayers at an incident angle of 50°. As mentioned in the discussion of Figure 3, the errors are prominent in the short-wavelength region (i.e., the high-energy region). In the long-wavelength region, the refractive index can be accurately measured using the DRCE. We also performed DRCE measurements at higher angles (Figure S2 in the Supplementary Material). Higher angle measurements show that the error region moves to the shorter wavelengths, as also shown in Figure 3(b). We emphasize that multilayered 2D heterostructures with more than two layers can be characterized in the same manner, that is, the subsequent reduction of the reduced substrate and DRCE.

Figure 4:

DRCE characterization of the refractive index of each 2D monolayer TMD in the heterostructure composed of monolayer MoS₂/monolayer WSe₂/280 nm-thick SiO₂/Si substrate. (a) The real (n) and (b) imaginary parts (k) of the refractive index (the orange solid line: DRCE result of monolayer MoS₂, the green solid line: DRCE result of monolayer WSe₂, the red dashed line: reference value for of monolayer MoS₂, and blue dashed line: reference value for of monolayer WSe₂). In Figure 3, the angle of incidence is fixed at 50°. Reference refractive indices of monolayer MoS₂ and WSe₂ are obtained from our previous study [20].

In Figure 5, we performed a DRCE experiment on a chemical vapor deposition (CVD)-grown monolayer MoS₂ on a SiO₂/Si substrate with an oxide thickness of 280 nm (purchased from SixCarbon), which has the same structure as the theoretical simulation of DECR shown in Figure 3. The reflection ratio ρ ̃ in the visible wavelength region was measured using a commercial spectroscopic ellipsometer (J.A. Woollam, V-VASE) at an angle of incidence of 40°. This angle of incidence was chosen because the peaks of the A and B excitons were clearly identified in the theoretical simulation shown in Figure 3. As shown in Figure 5, the refractive indices obtained by DRCE (solid orange line), theoretical simulation (dashed red line), and literature values [20] (dashed blue line) demonstrated good agreement, except in the high-energy error region. Our experiments confirmed the validity of DRCE for 2D heterostructures.

Figure 5:

DRCE experiment for monolayer MoS₂ in the heterostructure composed of CVD-grown monolayer MoS₂/280 nm-thick SiO₂/Si substrate. (a) The real (n) and (b) imaginary parts (k) of refractive index characterized at the incident angle of 40°. (The solid orange line: DRCE experiment results, the dashed red line: DRCE theoretical simulation results, and the dashed blue line: reference refractive index of monolayer MoS₂ obtained from our previous study [20]).