Optical detection of the susceptibility tensor in two-dimensional crystals

Abstract

The out-of-plane optical constants of monolayer two-dimensional materials have proven to be experimentally elusive. Owing to their reduced dimensionality, optical measurements have limited sensitivity to these properties which are hidden by the optical response of the substrate. Therefore, there remains an absence of scientific consensus on how to correctly model these crystals. Here we perform an experiment on the optical response of a single-layer two-dimensional crystal that addresses these problems. We successfully remove the substrate contribution to its optical response by a step deposition of a monolayer crystal inside a thick polydimethylsiloxane prism. This allows for a reliable determination of both the in-plane and the out-of-plane components of its surface susceptibility tensor. Our results prescribe one clear theoretical model for these crystals. This precise characterization of their optical properties will be relevant to future progresses in photonics and optoelectronics with two-dimensional materials.

Introduction

One of the great achievements in materials science is certainly the isolation of individual crystal planes, starting from solids with strong in-plane bonds and weak, van der Waals-like, coupling between layers. In general, when dealing with layered crystals, Maxwell’s equations increase in complexity in order to account for anisotropy. The susceptibility of these materials is no longer described by a constant, but by a tensor and, as regards the optical properties, they are at least uniaxial. Consequently, we expect that the optical spectra of single-layer, two-dimensional (2D) crystals also show out-of-plane anisotropy. In spite of the fact that experimental assessment of the out-of-plane anisotropy for the three-dimensional crystals is manageable, this is not the case for isolated monolayers¹. As a result, fifteen years after the exfoliation of the first atomically thin crystal, the exact description of its optical response remains an active and debated area of research^2,3,4,5,6,7.

There are currently two main models in use for the linear optical description of single-layer crystals, the first is a thin film model that can be either isotropic or anisotropic along the vertical direction^1,5,7,8,9,10. Optical contrast measurements, used to detect single and multiple layers of a 2D crystal, are analyzed by choosing an isotropic thin film^5,7,10. This is justified by the normal incidence configuration: the electric field is parallel to the crystal and the in-plane optical constants dominate the optical response. Surprisingly, the same model applies to spectroscopic ellipsometry, a very sensitive technique that works at any angle of incidence. If the first ellipsometric data for graphene were fitted using a uniaxial thin film model⁸, the subsequent analysis concluded that ellipsometry is only sensitive to the in-plane optical constants¹. This is because the sensitivity to anisotropy is dependent on the path length through the film, which is extremely limited for a monolayer¹. We are aware of only two papers that claim the experimental observations of the out-of-plane optical constants of single-layer transition metal dichalcogenides (TMDC) deposited on some substrate^11,12. These experiments show structures in the out-of-plane spectra that ab-initio calculations exclude¹³, meaning that these measurements face the same experimental limitation that all the other face.^1,8,9,14,15.

In the second model, the monolayer is treated as a 2D surface current without any thickness. In this case, the system is intrinsically anisotropic with both null out-of-plane surface susceptibility (χ_⊥) and conductivity (σ_⊥)^3,6,16,17,18. If on one side the use of two different models is a sign of a physical richness, on the other it poses new conundrums. The two approaches, starting from the measured ellipsometric parameters, provide different in-plane surface susceptibility (χ_∥) and conductivity (σ_∥)¹⁴. For χ_∥, this difference is greater than the experimental error¹⁴. From a theoretical point of view, the choice of setting χ_⊥ = 0 m and σ_⊥ = 0 Ω⁻¹ in the surface current model is arbitrary, yet so is the use of an isotropic thin film model, when we expect the system to be highly anisotropic. Ab-initio many body calculation of the optical spectra of graphite, graphene and bilayer graphene report interesting differences in the simulated out-of-plane properties going from the bulk limit, down to a monolayer, but they exclude a null value¹⁹, this also holds for other 2D crystals like TMDC^13,20.

One main experimental problem in the analysis of the optical response of a 2D crystal is the role of the substrate, which adds a background signal, hiding the small contribution that comes from the out-of-plane optical constants. We measure χ_∥, σ_∥, χ_⊥, and σ_⊥ in a two-step experiment (Fig. 1a, b). First, we extract the ellipsometric data (Ψ_s, Δ_s) from a single-layer 2D crystal deposited on a transparent dielectric substrate, namely polydimethylsiloxane (PDMS). Then, the same crystal is completely immersed in PDMS for a second ellipsometric measurement that provides a new set of data (Ψ_i, Δ_i). By inverting the fundamental equation of ellipsometry $\tan \Psi e^i\Delta = r_p/r_s$, where r_p and r_s are the Fresnel coefficients, it is possible to extract χ_∥, σ_∥, χ_⊥, σ_⊥ from Ψ_s, Δ_s, Ψ_i, Δ_i.

Fig. 1: Experimental set up.

a First step of the experiment: standard spectroscopic ellipsometry on a single-layer 2D crystal deposited on a polydimethylsiloxane (PDMS) substrate. b Second step of the experiment: manual ellipsometric measurement on the same crystal completely immersed in a PDMS prism. c Set up of the manual ellipsometer, at the wavelength of 633 nm.

Results and discussion

Theoretical models

While the reflection coefficients for the anisotropic slab model are reported in ref. ²¹, it is much more difficult to find in the literature a complete and correct generalization of the surface current model to include also the orthogonal polarization. We provide here the essential conceptual steps. The detailed calculations of the Fresnel coefficients are in the Methods section. The parallel $\overrightarrow {P_{||}}$ and perpendicular $\overrightarrow {P_ \bot }$ polarizations to the crystal plane induce two surface currents: $\overrightarrow {J_{||}}$ and $\overrightarrow {J_ \bot }$ respectively. The reflected field is the superposition of the reflected fields from these two currents. We thus solve two set of boundary conditions²², one for $\overrightarrow {J_{||}}$:

$$\begin{array}{l}\hat \kappa \wedge (\overrightarrow {H_2} - \overrightarrow {H_1} ) = \frac{{\partial \overrightarrow {P_{||}} }}{{\partial t}} = \overrightarrow {J_{||}} \\ \hat \kappa \wedge (\overrightarrow {E_2} - \overrightarrow {E_1} ) = 0\end{array}$$

(1)

and one for $\overrightarrow {J_ \bot }$:

$$ \, \hat \kappa \wedge \big( {\overrightarrow {H_2} - \overrightarrow {H_1} } \big) = 0\\ \,\hat \kappa \wedge \big( {\overrightarrow {E_2} - \overrightarrow {E_1} } \big) = - \displaystyle\frac{1}{{\varepsilon _0}}\hat \kappa \wedge grad\big( {P_ \bot } \big) = \overrightarrow {J_ \bot }$$

(2)

These boundary conditions were proposed first for metasurfaces²³ and then for 2D crystals in refs. ^4,24, where a surface current model with non-null χ_⊥, and σ_⊥ was applied to nonlinear graphene optics. Here $\hat \kappa ,$ $\hat i$ are the unitary vectors in the z and x direction (Fig. 1), $\vec H$ is the magnetic field, $\vec E$ the electric field, $\varepsilon _0$ the vacuum permittivity and the subscripts 1 and 2 refer to the media above and below the monolayer. The first set of boundary conditions are discussed in refs. ^3,16,17, for the surface current model with null ${{\vec P}_ \bot }$. The second set has the following simple explanation^22,23,25. In the radiation zone²⁶, the electromagnetic field due to an oscillating electric dipole in the $\hat \kappa ,$ direction is identical to an electromagnetic field due to an oscillating magnetic dipole in the $- \hat i$ direction. This last one would cause a jump in the tangential component of $\vec E$.

Assuming a time dependence $e^{i\omega t}$, (ω is the angular frequency of the light) for a monolayer completely immersed in a dielectric medium of refractive index n we find:

$$r_{ip} = \frac{{(ik\chi_{||} + \sigma_{||}\eta)\cos \theta }}{{\left( {ik\chi_{||} + \sigma _{||}\eta} \right)\cos \theta + 2n}} - \frac{{n\left( {ik\chi_\bot + \sigma _\bot\eta } \right)\tan \theta \sin \theta }}{{2 + n\left( {ik\chi_ \bot + \sigma _ \bot\eta } \right)\tan \theta \sin \theta }}$$

(3)

where the subscript i denotes that the sample is immersed, p is in place of p-polarized light, k is the wave vector of light in vacuum, η is the impedance of vacuum and θ is the angle of incidence. For a monolayer deposited at the interface of vacuum with a dielectric substrate of refractive index n we find:

$$r_{sp} = \, - \frac{{n\,\cos \theta - \cos \theta _t}}{{n\,\cos \theta + \cos \theta _t}} + \frac{{n\,\cos \theta - \cos \theta _t + \left( {ik\chi _{||} + \sigma _{||}\eta } \right)\cos \theta \cos \theta _t}}{{n\,\cos \theta + \cos \theta _t + \left( {ik\chi _{||} + \sigma _{||}\eta } \right)\cos \theta \cos \theta _t}}\\ \, + \frac{{n^2\,\cos \theta - n\cos \theta _t - \left( {i\,k\chi _ \bot + \sigma _ \bot \eta } \right)\sin ^2\theta }}{{n^2\,\cos \theta + n\,\cos \theta _t + \left( {i\,k\chi _ \bot + \sigma _ \bot \eta } \right)\sin ^2\theta }}$$

(4)

where the subscript s denotes the substrate, and θ_t is the propagation angle in the dielectric. For s polarized light the Fresnel coefficients depend only on χ_∥, σ_∥ and are provided in formula (6) of ref. ³.

Experimental procedure

In our experiment, a 7 × 7 mm-size, polycrystalline, single-layer 2D crystal is deposited on a 1 cm thick (to avoid back reflections) PDMS substrate (2 × 2 cm square) by chemical vapor deposition (Fig. 1a). The first ellipsometric measurement (VASE ellipsometer, J. A. Wollam) provides Ψ_s, and Δ_s and the confirmation that we are dealing with a monolayer. We then place our sample in a prism-shaped mold (base 6 × 5 cm, height 3.5 cm), we pour non-polymerized PDMS on it and we wait for complete polymerization. This process successfully produces a 2D crystal immersed in PDMS without any additional interface in between the previous and the newly added material. The prism has two optical quality lateral windows (Fig. 1b). The light reflected by the sample in this second step is 2 orders of magnitude less than in the previous one, because there is no more the substrate contribution. For this reason, we set up a manual ellipsometer at the wavelength of 633 nm (Fig. 1c) to measure Ψ_i and Δ_i.

We studied both monolayer graphene and MoS₂. First, we tested our two steps procedure without deposition of the monolayer to ensure that we do not observe any reflection in between the embedded substrate and the final prism. Ellipsometric parameters Ψ_s and Δ_s are measured at angles of incidence θ_s equal to 65° and 70° (Supplementary Figs. 1 and 2), Ψ_i and Δ_i (Figs. 2a, b and 3 a, b) are taken at angles of incidence θ_i around the pseudo-Brewster angle, where Δ_i varies appreciably and Ψ_i has a minimum. When θ_i becomes too small, Δ_i approaches 180° and noise becomes dominant.

Fig. 2: Experimental results for monolayer graphene immersed in polydimethylsiloxane.

Black dots: experimental data, colored dots: extracted optical constants, lines: theoretical fits for the four theoretical models considered in this paper. a, b Ellipsometric parameters Ψ_i and Δ_i. Neither the iso_slab model nor the curr_0 model fit the data, using as optical constants those extracted from the Ψ_s and Δ_s. c, d Optical constants determined with slab and the curr model. They predict similar χ_∥ and σ_∥, but negative and positive χ_⊥ respectively, and σ_⊥ compatible with 0 Ω⁻¹. The error bars reported are the standard deviation of these values (see Table 1). e Reflectivities for s and p polarized light. Experimental data show a pseudo Brewster angle for R_ip. The iso_slab model fails to predict its position by almost 10°. The curr_0 model does not predict it at all. All models fit R_is reasonably well (for clarity we show only the curr model and the slab model).

Fig. 3: Experimental results for monolayer MoS₂ immersed in polydimethylsiloxane.

Black dots: experimental data, colored dots: extracted optical constants, lines: theoretical fits for the four models considered in this paper. a, b Ellipsometric parameters Ψ_i and Δ_i. Neither the iso_slab model nor the curr_0 model fits the data, using as optical constants those extracted from the Ψ_s and Δ_s. c, d Optical constants determined with slab and the curr model. They predict similar χ_∥ and σ_∥, but negative and positive χ_⊥ respectively, and σ_⊥ compatible with 0 Ω⁻¹. The error bars reported are the standard deviation of these values (see Table 1). e Reflectivities for s and p polarized light. Experimental data show a pseudo Brewster angle for R_ip. The iso_slab model fails to predict its position by almost 7°. The curr_0 model does not predict it at all. All models fit R_is reasonably well (for clarity we show only the curr model and the slab model).

Experimental data and comparison with different theoretical models

Figures 2c, d, and 3c, d report the optical constants χ_∥, σ_∥, χ_⊥, σ_⊥ extracted from the ellipsometric data using the Fresnel coefficients provided by the anisotropic slab model (slab model) and by the surface current model (curr model). With no loss of information, for each θ_i we report the average χ_∥, σ_∥, χ_⊥, σ_⊥ considering the Ψ_s, Δ_s at 65° and 70°. The curr model gives a positive χ_⊥ while the slab model a negative one. Within our present experimental precision, we are not able to discriminate σ_⊥ from zero. Both models provide similar χ_∥ and σ_∥.

It is possible to discriminate between the curr model and the slab model thanks to ab initio calculations that predict a positive χ_⊥¹³. Reference¹⁹ computes only σ_⊥, but Kramers–Kronig relations exclude a negative χ_⊥ in the visible spectrum even in this case. We also performed first-principles calculations explicitly focused on the visible domain (Supplementary Figs. 3 and 4). By considering the available theoretical predictions and the experimental data collected here, the surface current model seems the only one able to fit the data in good agreement with the computed optical constants.

As explained above, ellipsometric experiments are usually performed on samples deposited on a substrate. In this case, being only sensitive to the in-plane optical constants, χ_∥ and σ_∥ are extracted using the isotropic slab model (iso_slab model)^1,8,9,15 or the surface current model with null χ_⊥ and σ_⊥ (curr_0 model)^14,16,17. Starting from the Ψ_s and Δ_s (Supplementary Figs. 1 and 2), here we repeat this procedure to clarify its limits. The iso_slab model provides χ_∥ and σ_∥ that still agree with those obtained in our two-step experiment, while for the curr_0 model only σ_∥ looks acceptable. Table 1 resumes our results. Figures 2a, b, and 3a, b show that the iso_slab model and the curr_0 model are unable to fit the Ψ_i and Δ_i. As expected these models are unable to account for the vertical anisotropy, once this is experimentally accessible.

Table 1 Optical constants for monolayer graphene and MoS₂.

We also measure the reflectivities $R_{is} = \left| {r_{is}^2} \right|$ and $R_{ip} = \big| {r_{ip}^2} \big|$, for the sample immersed in PDMS (Figs. 2e, 3e). The observed values for R_is confirm the reliability of our experimental procedure because, being insensitive to χ_⊥ and σ_⊥, they can already be predicted after the first step of our experiment. This is a strong confirmation that immersion in PDMS does not alter the 2D crystal properties. As expected, for graphene R_is is one order of magnitude smaller than for MoS₂. For p polarization we observe a pseudo-Brewster angle θ_pB. The existence and the value of θ_pB are very important in assessing the vertical anisotropy of single-layer graphene and MoS₂ (Table 1). The iso_slab model predicts a wrong θ_pB, the curr_0 model does not predict it at all.

The study of a 2D crystal deposited on a substrate or immersed in a host material is by far the most common situation in the laboratory. Still, measuring a free-standing monolayer²⁷ will be of great interest. If macroscopically this simply means a 2D crystal immersed in a medium with n = 1 rather than 1.4233, microscopically it addresses the question on how much the host material affects the value of χ_⊥. Our experimental results indicate that the influence of the host material is small. It remains to see if it is experimentally appreciable or not.

Conclusions

We have observed the role of $\vec {P_ \bot }$ in a single-layer 2D crystal, showing that it affects dramatically the position of θ_pB. Our results state that χ_⊥ is a measurable optical constant way different from χ_∥ or zero. These findings will have a deep impact in 2D crystal optics, opening new perspectives for fundamental science and technical developments. Based on present knowledge, we identify a single theoretical description²² for a 2D crystal. Contrarily to what sometimes believed²², the slab and the surface current models are not equivalent even when we take $\vec {P_ \bot }$ into account. Out-of-plane anisotropy in 2D materials should play a role also in nonlinear optics²⁸, surface wave phenomena²⁹ or magneto-optical Kerr effect³⁰ and any other field of 2D crystal optics. In view of the advances in photonics and optoelectronics of 2D semiconductors^31,32, we expect a precise characterization of χ_⊥ and σ_⊥ to become of great importance. A relevant open question is to what extent these quantities can be tuned in hetero-structures for new functionalities³³.

Methods

Fresnel coefficients for the curr model (p-polarization)

Insulator single-layer 2D crystal immersed in a dielectric medium of refractive index n

The electromagnetic fields satisfy: $\frac{\eta }{n}\vec H = \hat s\wedge\vec E$, $\hat s$ being the unit vector along the propagation direction. We compute the reflected field due to $\vec {J_{||}}$ by solving the boundary Eq. (1). In this case, the electric field parallel to the crystal plane is continuous across the crystal itself giving:

$$\vec {P_{||}} = \varepsilon _{0}\chi_{||}\vec {E_{||}} .$$

(5)

Choosing² $\vec H$ along $- \hat i$, Eq. (1) plus (5) run:

$$H_i + H_{r1} - H_{t1} = \, \frac{{ik}}{{\eta\varepsilon _0}}P_{||}\\ H_i - H_{r1} = \, H_{t1}\\ H_{t1} = \, \frac{{nP_{||}}}{{\eta\varepsilon _{0}\chi_{||}\cos \theta }}$$

where the subscripts i, r, t denote the incident, reflected and transmitted fields. We compute the reflected field due to $\vec {J_ \bot }$ by solving the boundary Eq. (2). In this case, the electric field orthogonal to the crystal plane is continuous across the crystal itself giving:

$$\vec {P_ \bot } = \varepsilon _{0 }\chi_{\bot}\vec {E_ \bot } .$$

(6)

Choosing $\vec H$ along $- \hat i$, Eq. (2) plus (6) run:

$$H_i + H_{r2} = \, H_{t2}\\ H_i - H_{r2} - H_{t2} = \,\frac{{ikn^2P_ \bot \tan \theta }}{{\eta\varepsilon_0}}\\ H_{t2} = \, \frac{{nP_ \bot }}{{\eta\varepsilon _{0 }\chi_{\bot}\sin \theta }}$$

In accordance to the superposition principle, the total reflected field is: $H_r = H_{r1} + H_{r2}$, the total transmitted field is: $H_t = H_{t1} + H_{t2} - H_i = H_i - H_{r1} + H_{r2}$. The reflection and the transmission coefficients are respectively: $r_{ip} = \frac{{H_r}}{{H_i}}$; $t_{ip} = \frac{{H_t}}{{H_i}}$. We verify that: $R_{ip} + T_{ip} = 1$, where $T_{ip} = \left| {t_{ip}^2} \right|$.

Insulator single-layer 2D crystal at the vacuum–dielectric medium interface

The electromagnetic fields in vacuum satisfy: $\eta\vec H = \hat s\wedge\vec E$. For $\vec {J_{||}}$ Eq. (1) plus (5) run:

$$H_i + H_{r1} - H_{t1} = \, \frac{{ik}}{{\eta\varepsilon _0}}P_{||}\\ \,(H_i - H_{r1})\cos \theta = \, \frac{{H_{t1}\,\cos \,\theta _t}}{n}\\ \,H_{t1} = \, \frac{{nP_{||}}}{{\eta\varepsilon _{0}\chi_{||}\,\cos \theta _t}}$$

For $\vec {J_ \bot }$ Eq. (2) plus (6) run:

$$H_i + H_{r2} = \, H_{t2}\\ \,\left( {H_i - H_{r2}} \right)\cos \theta - \frac{{H_{t2}\cos \theta _t}}{n} = \,\frac{{ikP_ \bot \sin \theta }}{{\eta \varepsilon _0}}\\ \,H_{t2} = \,\frac{{n^2P_ \bot }}{{\eta\varepsilon _0\chi _ \bot \sin \theta }}$$

In accordance to the superposition principle, the total reflected field is: $H_r = H_{r1} + H_{r2} - H_{rn}$, the total transmitted field is: $H_t = H_{t1} + H_{t2} - H_{tn}$, where $H_{rn}$and $H_{tn}$ are the fields reflected and transmitted without the 2D crystal deposited at the interface. The Fresnel coefficients are defined as above, and from energy flux considerations: $R_{ip} + \frac{{n\cos \theta _t}}{{\cos \theta }}T_{ip} = 1$. Thanks to the application of the superposition principle, our connection of $\vec {P_{||}}$ and $\vec {P_ \bot }$ with the macroscopic field is straightforward. This principle is not usually applied in the literature^4,23,24 forcing to definitions different from (5) and (6). Unfortunately, this last approach does not seem to verify energy flux conservation that we expect valid for insulators.

For conducting 2D crystals we have to modify the definition of $\vec {J_\parallel }$ in (1): $\vec {J_{||}} = \frac{{\partial \vec {P_{||}} }}{{\partial t}} + \vec {J_{\sigma _{||}}}$, where:

$$\vec {J_{\sigma _{||}}} = \sigma _{||}\vec {E_{||}}$$

(7)

and the definition of $\vec {J_ \bot }$ in (2)²⁵: $\vec {J_ \bot } = - \frac{1}{{\varepsilon _0}}\hat \kappa \wedge grad\left( {P_ \bot + \frac{{\vec {J_{\sigma _ \bot }} }}{{i\omega }}} \right)$ where:

$$\vec {J_{\sigma _ \bot }} = \sigma _ \bot \vec {E_ \bot } .$$

(8)

We then compute the reflected and the transmitted fields due to$\vec {J_{||}}$ by solving Eqs. (1), (5), and (7) and those due to $\vec {J_ \bot }$ by solving Eqs. (2), (6), and (8).

Sample preparation

Monolayer graphene growth and transfer

We have prepared one large-area (up to mm), polycrystalline, continuous, single-layer graphene with chemical vapor deposition (CVD). Monolayer graphene grows on commercial Cu foils (50 μm thick, Kunshan luzhifa Electronic Technology Co., Ltd) via a low-pressure CVD system. Cu foil is annealed at 1020 °C under a 500 sccm flow of Ar with a 0.1 sccm flow of O₂ (0.1% of diluted O₂ in Ar) for 30 min. Then, graphene grows under 500 sccm flow of H₂ and a step increased flow rate of CH₄ (0.5 sccm for 20 min, 0.8 sccm for 20 min, and then 1.2 sccm for 40 min)³⁴. For the graphene transfer, poly(methyl-methacrylate) (PMMA) is spin-coated on as-grown graphene on Cu at 2000 rpm and baked at 170 °C for 3 min. The Cu foil is then etched away by NaS₂O₈ (1 mol/L) after the graphene on the other side of Cu (graphene was grown on both sides of Cu foils) is removed by air plasma (10 sccm and power of 100 W). Subsequently, the freestanding PMMA/graphene stack floating on NaS₂O₈ solution is washed with deionized water for four times, and salvaged by the PDMS, forming the PMMA/graphene/PDMS structure. After the as-formed PMMA/graphene/PDMS is dried, the PMMA is removed by immersion in acetone at 80 °C for 10 min and it forms the graphene/PDMS structure.

Monolayer MoS₂ growth and transfer

We have prepared one large-area (up to mm), polycrystalline, continuous, single-layer MoS2 with CVD. Monolayer MoS2 grows on soda-lime glass using a three-zone tube furnace under a low-pressure atmosphere. A piece of Mo foil (Alfa Aesar, 9.95%; 0.025 mm thick) was folded as a “bridge” and placed on top of the glass substrate with a gap of 10 mm. S power (Alfa Aesar, purity 99.5%) was located at upstream of the furnace. Before heating, the system was purged with Ar (80 sccm) for 10 min to get out of the air. Then, Ar (50 sccm) and O₂ (6 sccm) mixed gas flows were introduced into the system to create a stable growth atmosphere. The temperature of the S powder and the glass substrate was set at 100 °C and 720 °C, respectively. The growth time was set at 1–3 min. After growth, the furnace was naturally cooled to room temperature. The as-synthesized MoS₂ monolayer/soda-lime samples were firstly spin-coated with PMMA at 1000 rpm for 1 min, followed by baking at 80 °C for 20 min. The PMMA-supported samples were then inclined into the pure water, and the PMMA/MoS₂ complex was naturally peeled off under the surface tension effect. The PMMA/MoS₂ film was collected by PDMS. Finally, the PMMA film was removed via acetone.

Role of sample doping

It is well known that the transfer process would inevitably induce p-doping in the graphene^35,36. As for MoS₂, n-doping is usually obtained due to S-vacancy. Fortunately, in our experiments, the optical performance agrees well with previous measurements for in-plane optical constants and with ab-initio theoretical predictions. So, we believe that our conclusion is robust, regardless of the doping level or doping type.

PDMS substrate preparation and immersion of the 2D crystal in PDMS

The PDMS (Sylgard 184, by Dow Corning) substrate is prepared by mixing the base elastomer and the curing agent in ratio 10:1, and by 72 h room temperature (RT) polymerization. Then, after the deposition of the monolayer 2D crystal (see above), new pre-polymerized PDMS is poured on the structure, using a 3D printed prism shape structure (ABS filament) as mold. We again wait for RT polymerization. Since the 3D printed structure presents a high roughness that would compromise the optical measurements, glass slides are glued on them. In this way, we obtain a PDMS prism with optical quality lateral windows. To prevent the PDMS bonding to the glass slides during the polymerization process, a fluorinated antiadhesive coating (Trichloro(1H,1H,2H,2H-perfluorooctyl) silane, by Sigma-Aldrich) is applied.

Some strain effects could be there on 2D materials during the PMDS polymerization process. These are difficult to estimate but we find the same in-plane optical constants before and after immersion. For this reason, strain, if present, is not supposed to alter our conclusions. We can also wonder how much the dielectric screening of PDMS can influence the measured optical constants before and after immersion in it (especially for MoS₂). Reference³⁷ shows that these effects are large only when capping with materials having a large dielectric constant (see. Figure 3 of ref. ³⁷). The PDMS dielectric constant is 1.96, and this is the reason why we do not observe a variation of the optical response.

Measurement and determination of the optical constants

Spectroscopic ellipsometry of a 2D crystal deposited on a PDMS substrate

Spectroscopic ellipsometric measurements are performed using a VASE ellipsometer (J. A. Wollam) in ambient conditions at room temperature. We follow the same procedure described in¹⁴. The first step of our analysis is the characterization of the substrate. The ellipsometric Ψ parameter of the substrate alone is perfectly fitted by assuming a refractive index n given by the Sellmeier expression³⁸ minus a constant value of 0.005. At a wavelength of 633 nm we have n = 1.4233. For graphene, the use of this measured value instead of the one reported in the literature³⁸ is responsible for a difference of 0.06 nm, 1·10⁻⁶ Ω⁻¹, 0.06 nm and 9·10⁻⁶ Ω⁻¹ respectively for χ_⊥, σ_⊥, χ_∥, and σ_∥. Only the discrepancy for σ_∥ is bigger than the error in the measurements (Table 1). Although small, this last difference shows the importance of a careful substrate characterization when dealing with monolayers. For MoS₂ these discrepancies are even smaller than for graphene and equal respectively to 0.02 nm, 7·10⁻⁷ Ω⁻¹, 0.03 nm and 9·10⁻⁶ Ω⁻¹.

Supplementary Figures 1 and 2 show the ellipsometric parameter Ψ_s and Δ_s of single-layer graphene and MoS₂ deposited on a PDMS substrate. Starting from these data, ref. ¹⁴ describes how we extract χ_∥ and σ_∥ for the iso_slab model and the curr_0 model.

Manual ellipsometer

The light source is a HeNe laser at a wavelength of 633 nm. A lens (focal length 40 cm) focuses the Gaussian beam to a 1/e² intensity diameter of 120 μm allowing measurements at grazing incidence. We set the first polarizer at an azimuthal angle of 45°. The light intensity after the first polarizer is typically 4 mW. A quarter wave plate (QWP) and a second polarizer (the analyzer) perform ellipsometric measurements. We fully characterize the vibrational ellipse for the electric vector of the light reflected from the sample³⁹. Without the QWP installed, we measure the s and p components of the reflected light. Then we identify the azimuthal angles of the minor and the major axes of the ellipse. Finally, we place the QWP with the fast axis along the minor axis. After this, the azimuthal angle of the analyzer, corresponding to a minimum power on the photodetector, fixes the sense in which the endpoint of the electric vector describes the ellipse (the sign of Δ_i). The power meter used in our measurements has a resolution of 1 nW. Only the 2D crystal introduces a phase difference (Δ_i) in between the s and p components of the incident linearly polarized light. Transmission through the two prism sides only introduces a different power reduction in s and p, important for the correct evaluation of Ψ_i, R_is, and R_ip. The QWP is antireflection coated at 633 nm while the analyzer introduces an identical power reduction in s and p (important for a correct evaluation of R_is and R_ip).

We are unable to estimate σ_⊥ because it is much smaller (Supplementary Fig. 6) than our experimental error. We are limited by the quality of the reflected beam and by some scattering. The poor quality of the reflected beam is a consequence of the macroscopic quality of our 2D crystals. Even if we are dealing with some of the best samples available in the world, any small defect affects the reflection of a highly coherent laser source. A few amounts of unwanted scattering come from the back surface of the prism that is far from perfect because of handling or, for some angles of incidence, from the prism corners.

First-principles calculations

Layer susceptibilities have been evaluated using first-principles many-body perturbation theory calculations of periodic dielectric layers. First, density functional theory (DFT) orbitals were obtained with the so-called PBE exchange and correlation functional⁴⁰ at the theoretical lattice constant using the Quantum-Espresso DFT software package⁴¹. Then, the screened Coulomb interaction W⁴² was found using the G₀W₀ approach (GWL program⁴³). The Brillouin zones were sampled at the sole Γ point albeit adopting supercells comprising 8 atoms for graphene and 12 atoms for MoS₂. An optimal polarizability basis of 800 vectors was applied for developing the polarizability operators. Finally, the in-plane ε_∥ and out-of-plane ε_⊥ components of the complex dielectric tensors relative to the periodic thin dielectric layers were calculated including electron-hole interactions using the Bethe–Salpeter equation scheme^44,45 (simple.x code⁴⁶).

For graphene, we used a 12 × 12 × 12 regular mesh of k-points for sampling the Brillouin zone (relative to the super-cell) which was shifted in order not to contain the Γ point and we included 12 occupied and 48 unoccupied orbitals. A stretching factor of 1.67 was applied to the DFT valence and conduction bands in order to match GW energies in the proximity of the band closure at the K-point. These GW values were obtained extrapolating, to the bulk limit, results for models of 12 and 32 atoms.

For MoS₂, we used a 6 × 6 × 6 regular mesh of k-points for sampling the Brillouin zone (relative to the super-cell) together with a fully relativistic treatment of the spin–orbit coupling. We included 56 occupied and 96 unoccupied orbitals. A scissor factor of 1.30 eV was applied to the DFT band structure in order that the DFT gap matches the bulk limit of the GW one. The latter was evaluated at the scalar relativistic level. Moreover, for MoS₂ we added a constant term of 2.13 to ε_∥ and one of 1.33 to ε_⊥ in order to account for the upper unoccupied orbitals which were not included in the BSE calculation. These factors were chosen in order to match the corresponding components of the dielectric tensor calculated from density functional perturbation theory.

It is worth noting that the calculated BSE susceptibilities exhibit only a mild dependence on the strategy chosen for updating the DFT bands in order to account for GW effects. In particular, the out-of-plane responses are particularly insensitive as it is shown, for MoS₂ in Supplementary Figs. 5 and 6 where we report results obtained using scissor factors of 1.0, 1.3 eV and a scissor factor of 1.0 accompanied by a stretching one of 1.2 for both manifolds as in ref. ⁴⁷.

The in-plane χ_C∥ and out-of-plane χ_C⊥ complex susceptibilities are rigorously defined as the ratio between the induced surface dipole densities with respect to the transmitted electric field. In the case of χ_C∥ the transmitted field is parallel to the 2D layer and is conserved along the simulation cell. It is worth noting⁴³ that in first-principles simulations with periodic boundary conditions (PBC) the directly accessible quantity is the response with respect to the total or internal (electric) field. This is due to incommensurate extension of the field with respect to the length period of the PBC. This makes it possible to obtain χ_C∥ from ε_∥:

$$\chi _{C||} = {{{{{{{\mathrm{L}}}}}}}}(\varepsilon _{||} - 1)$$

(9)

where L is the periodic length of the simulation cell in the direction perpendicular to the 2D layer. This length is big from a microscopic point of view in order to null the interaction between the atomic layers but small from a macroscopic point of view (for instance it is much smaller than the wavelength of the incident light).

For χ_C⊥ the transmitted electric field E_t⊥, which is now perpendicular to the 2D layer, can be found⁴⁸ from ε_⊥:

$$E_{t \bot } = \frac{{E_{cell}}}{{\varepsilon _ \bot }}$$

(10)

where E_cell is the perpendicular long-range electric field accounted for in the calculation of ε_⊥. Hence, the out-of-plane complex susceptibility is given by:

$$\chi _{C \bot } = \frac{{{{{{{{{\mathrm{L}}}}}}}}(\varepsilon _ \bot - 1)}}{{\varepsilon _ \bot }}$$

(11)

Equations 1 and 3 have been reported previously^13,22. Reference¹³ derive them to model a dielectric layer as an infinitesimally thin sheet. Reference²² derives them from the effects of local fields to the dielectric responses.

For evaluating ε_∥ and ε_⊥, we applied first a Lorentzian broadening of 0.1 eV and then a Gaussian one of 0.2 eV¹⁹. The surface susceptibilities measured in this paper are the real part of χ_C∥ and χ_C⊥, the surface conductivities are proportional to the imaginary part of χ_C∥ and χ_C⊥^9,14. Supplementary Figs. 3 and 4 report the computed out-of-plane optical constants in the visible spectrum. The in-plane optical constants are similar to those already reported in refs. ^23,24,42.