Introduction

Two-dimensional (2D) semiconductor materials with electrically tunable optical properties could result in various applications ranging from electro-optical modulators to display screens. Indeed, it was recently established that the optical constants of 2D monolayers, in contrast to the most bulk semiconducting materials such as Si, Ge, GaAs, could vary significantly with electrical field.1,2,3 As a result, 2D atomic semiconductor crystals emerged as a potential alternative to bulk semiconductors for optoelectronic applications compatible with CMOS technology.4,5,6 The most investigated 2D semiconductor—a molybdenum disulphide (MoS2) monolayer—deserves special attention as it hosts a high quality electronic system with a unique band structure and a technologically favourable band gap.7,8,9 A MoS2 monolayer is a direct band semiconductor with Eg ≈ 1.9 eV and is composed of hexagonal planes of S and Mo atoms glued together by ionic-covalent interactions in a trigonal prismatic arrangement.10 A unique feature of the electronic structure of MoS2 monolayers is the significant tunability of their carrier density,11 which is important for optoelectronic applications.12,13 In addition, a MoS2 monolayer is chemically and electrically stable when compared to other 2D materials.

Light absorption in a monolayer MoS2 is produced by dipole transitions between localised d orbitals, which yields an absorption for a single layer MoS2 flake at the level of 2–5% in the visible range. This is already a large value for a single layer of a 2D material since MoS2 thickness is only ~0.65 nm. Another important contribution to MoS2 absorption comes from excitonic excitations which are responsible for an increase in the absorption value by 5–10%.7,10 This excitonic absorption is larger than the broadband absorption of graphene.14 Despite the fact that the optical absorption of MoS2 monolayers in the visible range is relatively large compared to other 2D materials,15 its absolute value is still rather small. Therefore, in order to utilise MoS2 monolayers in optical devices it is essential to create light trapping nanostructures which would significantly enhance light interaction with the 2D semiconductor atomic layer. The light trapping can be realised using different methods. In our previous work,16,17 we suggested and studied hybrid plasmon-waveguide nanostructures which are capable of trapping light and enhancing absorption in 2D layers. It was recently demonstrated that enhancement of absorption from 2D monolayers can be also achieved by coupling to planar nanocavities.18,19 Another strategy to realise resonant light trapping in single crystal 2D monolayers is based on plasmonic active layers20 or Fabry–Perot resonators.21

Here, we study hybrid nanostructures that combine an atomic semiconductor with a plasmonic-metal-dielectric waveguide. We focus on a non-silicon CMOS-compatible insulating layer based on silicon nitride (Si3N4). Silicon nitride is a promising dielectric material because of its low linear loss, CMOS-compatibility, material stability and the ability to engineer favourable optical dispersion.22 We investigate a simple heterostructure that optically couples light to an atomically thin semiconducting monolayer and strongly enhances the absorption of MoS2 making possible an efficient 2D semiconductor light modulator. We demonstrate that the light absorption of a MoS2 monolayer integrated with a Si3N4 waveguide can lead to a significant enhancement of visible-light polarisation-dependent modulation (~10%) in a narrow range of wavelengths near the excitonic bands (Δλ ∼ 150 nm) and at large angles of incidence. The suggested planar metal-dielectric-2D semiconductor nanostructures could provide a simple and cost-effective solution for optoelectronic applications in visible light.

Results

Sample preparation and characterisation

Figure 1 shows the geometry and images of the studied samples. To fabricate hybrid nanostructures covered by a MoS2 monolayer, we have used a dry-transfer technique described in detail in Supplementary Information (SI). First, a 3 nm-thick Cr adhesion layer and a 65 nm Au film were deposited on the top of a 1 mm-thick quartz substrate using electron beam evaporation. A Si3N4 dielectric layer was grown by chemical vapour deposition (CVD) to cover the Au film. Then, a CVD MoS2 monolayer (supplied by 2D Semiconductors) was transferred onto a poly(methyl-methacrylate) membrane (PMMA). The MoS2–PMMA heterostructure was loaded into a micromanipulator and placed face-down onto a chosen Quartz/Au/Si3N4 sandwich after which the PMMA was dissolved producing the final device. The schematic of the device and principle of ellipsometric measurements is shown in Fig. 1a. The optical modes of a similar nanostructure were discussed in detail in our previous work.23

Fig. 1
figure 1

Effect of interference on the electrical modulation of the reflectance in Au/Si3N4/MoS2 hybrid waveguide nanostructure. a Schematic illustration of the device structure and principle of ellipsometric measurements. b The enhancement of the absorption in the MoS2 monolayer due to multiple internal reflections at each interface of the Au/Si3N4/MoS2 hybrid waveguide nanostructure. c Optical microscopy image of typical devices showing the top MoS2 monolayer covering. d Contact-mode AFM of the device confirms monolayer nature of MoS2

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The MoS2 monolayer thickness was first identified by the atomic force microscopy (AFM) measurement with additional verification of thickness by Raman and photoluminescence spectroscopy. AFM analysis is performed in the tapping mode on the sample to examine the surface morphology in a scan area of 17 × 5.0 μm2. AFM image of the nanostructure is shown in Fig. 1d. The average thickness from three scans was estimated as 0.7 nm. AFM microscopy of the device confirms monolayer nature of MoS2. The jump of thickness at edge of MoS2 sheet is due to nanoscroll of several MoS2 monolayers. From atomic force microscopy images the rms value of the surface roughness was found to be less than ∼2.5 nm.

Raman spectra of MoS2 monolayer vs. the gating voltage

Figure 2a and Fig. S1, SI show typical Raman spectra obtained from a CVD MoS2 monolayer measured in our fabricated devices. Raman spectra were recorded with 632.8 nm laser excitation using a Witec confocal spectrometer (100x objective). They exhibit two characteristic peaks: one near 386.7 cm−1 and the other one near 403.4 cm−1. The first one is attributed to an in-plane vibrational mode E12g, and the second one is attributed to an out-of-plane vibrational mode A1g (both modes are connected to the presence of MoS2 monolayer). It is worth noting that the peak position of the E12g mode was blue-shifted from 386.0 cm−1 (observed in mechanically exfoliated MoS224) to 386.7 cm−1, while the peak position of the A1g mode was also blue-shifted from 403.0 cm−1 to 403.4 cm−1 for the studied Au/Si3N4/MoS2 structures. Analogous blue shifts of Raman peaks were observed in a hybrid system comprising MoS2 and gold nanorods.25 The separation Δν between the E12g and A1g modes is ∼17 cm−1, which is close to the previously reported value for CVD-grown and mechanically exfoliated monolayer MoS2.24 The Raman data confirm the high quality of the transferred MoS2 on the Si3N4-based waveguide. Next, we performed complementary studies of MoS2 monolayer Raman signals as a function of the gating voltage, see Fig. 2a. We found that the peak positions of E12g and A1g modes were largely unaffected by the Vg within the experimental error of ±0.2 cm−1 while Raman intensities of both E12g and A1g modes for Au/Si3N4/MoS2 hybrid nanostructure changed significantly with the applied gate voltage. According to previous studies,5 the intensity ratio of the Raman peaks could be used to explore exciton properties (bandgap renormalisation, damping factors, oscillator strengths associated with the excitons) of a MoS2 monolayer. For example, the ratio of intensities of the E12g and A1g peaks (IE2g/IA1g) of MoS2 could be used to identify the number of layers in MoS2 (Fig. 2a and Fig. S1). The relative ratio IE2g/IA1g of the Raman bands of a MoS2 monolayer as a function of Vg is shown in Fig. 2b. We can see that the intensity ratio IE2g/IA1g changes by ~30% with the gate voltage. In contrast, no significant change was observed in the relative ratio IE2g/IA1g at the same excitation conditions as a function of Vg for quartz/Si3N4/MoS2 hybrid nanostructure (not shown here). This suggests that the intensity ratio IE2g/IA1g is changing due to the field-induced charge carrier density Δn in a MoS2 monolayer. Injected carriers Δn mostly affect the optical properties of the MoS2 monolayer due to the Coulomb scattering effect.26,27

Fig. 2
figure 2

Evolution of the vibration and optical spectra of the MoS2 monolayer under gating voltage and changes of Raman spectra of monolayer MoS2 at different gating voltage, Vg. a Gating tunability of vibration modes in monolayer MoS2. b The measured Raman intensity ratio as a function of gating voltage, Vg. Reflection at normal angle of incidence. c Comparison of relative reflection spectra at normal incidence for different Vg. d The reflectivity of Au/Si3N4/MoS2 hybrid waveguide nanostructure at normal incidence

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It is interesting to note that while our measurements show a strong increase of Raman intensities for both the E12g and A1g modes with applied gating voltage they display no dependence of the Raman shifts on Vg for both modes. At the same time, it was demonstrated in ref. 28,29 that a gate-induced electron doping results in the shifts (up to 4 cm−1) of the A1g mode. Such behaviour was attributed to the occupation of the antibonding states in the conducting band of MoS2 from the increased electron concentration.28 The discrepancy between our data and those of refs. 28,29 is a mystery. One possible answer could lie in different excitation wavelengths at which Raman signals were measured. Our Raman spectra were recorded with 632.8 nm laser excitation which is in the resonance with excitation transitions. In contrast Raman data of MoS2 monolayer were measured in ref. 28 with λexc = 514.5 nm and29 with λexc = 488 nm which implies non-resonant Raman response. Note that the Raman spectra of CVD MoS2 are affected by defects (usually an amount of defects in CVD MoS2 monolayer is larger than that in mechanically exfoliated monolayer), and strain existing at the interface between the MoS2 and Si3N4 layers, which depend on the lattice mismatch between above two nanostructures. Main difference between resonant and nonresonant excitation of Raman scattering is next: the resonance condition promotes the system to be excited into a bound exciton electronic states of MoS2 monolayer. We can assume that the gate-dependent Raman intensity, shift and linewidth of MoS2 monolayer are results of the combined effect of charge carrier density in the conducting band and exciton charges in the case of resonant-like excitation. The Raman signals of a MoS2 monolayer (shown in Fig. 2 and Fig. S1), as well as its photoluminescence (Fig. S2), have revealed a strong dependence on an underlying optical structure due to substrate-related optical interference effects.30

Reflection at normal angle of incidence

At normal incidence, an application of Vg affects the reflectivity of the Au/Si3N4/MoS2 hybrid nanostructure at wavelengths associated with the excitonic transitions.7 Electric-field-dependent ratios of the reflection spectra R(Vg)/R(Vg = 0 V) are shown in Fig. 2c. We see that R(Vg)/R(Vg = 0 V) reflection spectra of the structure has a single local maximum at λ ∼ 665 nm for positive values of Vg and a minimum within the same spectral range for negative values of Vg. The amplitude of the R(Vg)/R(Vg = 0 V) features increase as the electrical field rises. At the same time, the change in reflection at Vg = ±100 V is relatively small and can be estimated as ∼2%. The quality of Si3N4 film as insulating layer is quite good as the leakage (gate) current monotonically increases from 0.3 nA to 1 nA at room temperature when the gate voltage reaches 100 V. The reflection spectrum (shown in Fig. 2d) exhibits insertion loses closed to 70%. Spectral ellipsometric and FTIR measurements (Figs. S3 and S4, see SI) also confirm the high quality of CVD Si3N4 insulating-guiding layer.

Ellipsometry measurements

To characterise the field-effect tunability of the refractive index \((\tilde n = n + ik)\) of a MoS2 monolayer, we measured the ellipsometric spectra of the monolayer as a function of gate voltages applied to a Au/Si3N4/MoS2 hybrid nanostructure. The ellipsometric measurements (Fig. S5) were performed in the 240−1700 nm wavelength range with a variable angle spectroscopic ellipsometer (VASE, J.A. Woollam Co., Inc.). Data was collected at angles of incidence from 50° to 75° with a 5° angle interval. To model the measured ellipsometric spectra, we used a Fresnel multilayer model containing a quartz substrate, a Au layer, a Si3N4 dielectric, and a MoS2 monolayer (0.7 nm thickness). The modelling was performed with the help of Woollam WVASE32 software, see SI. In our fitting procedure of ellipsometric parameters Ψ and Δ we used isotropic model and only the in-plane optical constants are derived because the reflectance change stronger for s-component of incident light (in plane of monolayer) and the roughness of substrate (Si3N4 deposited film) exceed the thickness of monolayer about 4 times. The bare quartz/Au/Si3N4 and quartz/Si3N4 structures were also measured and modelled using optical constants for Au and Si3N4 films extracted from separate ellipsometric measurements. The gate induced optical constant dependences for MoS2 monolayer presented in Fig. 3a, b. We observed two prominent peaks in n(λ) and k(λ) spectra in the wavelength range 550–700 nm (Fig. 3a, b), the so-called A and B exciton bands which correspond to excitonic transitions between the valence band split by spin-orbit interaction and the conductance band.7,31 A gate voltage dependent change of the charge carrier density may be mainly responsible for the variation of optical constants in vicinity of exciton bands (600–700 nm). It can be seen that a change of the polarity of the gate voltage leads to change the sign in deviation of optical constants Δn and Δk of MoS2 monolayer respect to the Vg = 0 V. The extracted dielectric function, ε(ω) = (n + ik)2, of a CVD MoS2 monolayer can be modelled by fitting with a multi-Lorentzian function32

$$\varepsilon (\omega ) = 1 + \mathop {\sum}\limits_{k = 1}^N {\frac{{f_k}}{{E_k^2 - E^2 - iE\gamma _k}}} ,$$
(1)

where fk, Ek, and γk are the oscillator strength, the spectral resonance energy Ek, and the spectral width γk of the k-th oscillator, respectively. These values for the model parameters for a MoS2 monolayer were determined by fitting Eq. (1) to the experimental data presented in Fig. 3a, b and S5. A good agreement between the extracted experimental dependences of the complex refractive index \((\tilde n = n + ik)\) and the analytical formula (1) were obtained for the fk, Ek, and γk parameters presented in the (Table 1).

Fig. 3
figure 3

Gate induced changes in the complex refractive index, \((\tilde n = n + ik)\) of MoS2 monolayer extracted from ellipsometric measurements: the real part, n (a) and the imaginary part, k (b). Electrical tuning of the Lorentzian parameters for the complex refractive index of the CVD MoS2 monolayer. c Oscillator strengths, fA,B and (d) Spectral width of A and B excitonic absorption bands, γA,B as a function of density of injected charge carrier density, ne. The solid lines in Fig. 3(c) show the linear fit to the data. Inset: the gate-dependent density of induced charge carriers

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Table 1 Fitting parameters for the complex refractive index of the CVD MoS2 monolayer as a function of gate voltage
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Furthermore, we demonstrate that the extracted real, n, and imaginary parts, k, of the refractive index in the range 600–700 nm decreases with the increasing gate voltage, Vg, which indicates suppression of the contributions from A and B excitonic transitions. With Vg increasing from 0 to 75 V, the absorption intensities (α = 4πk/λ) of both A and B excitons decrease significantly while their spectral positions are slightly red-shifted (Fig. S6). It is important to note that only the fk and γk parameters for exciton A and B change significantly with the applied gate voltage. Figure 3 shows the dependence of optical constants as a function of Vg and results of fitting for oscillator strength fk, and the spectral width γk in the case of negative and positive gating and strong changes of these parameters as a function of electron density ne induced by the gate voltage. From Fig. 3 we conclude that the optical constants of a monolayer MoS2 are gate-tunable in the spectral range from 580 to 750 nm. At the same time, n(λ) and k(λ) are Vg-independent outside the spectral range of MoS2 excitons. Therefore, we are to expect optoelectronic modulation of the reflection from the studied structure in the spectral range of MoS2 excitons.

It is worth noting that Fig. 3a, b reports the first observation (to the best of our knowledge) of an evolution of n(λ) and k(λ) spectra with a gating voltage for a MoS2 monolayer extracted from variable angle of incidence measurements of the ellipsometric parameters Ψ and Δ. When Vg increases (from −100 to +75 V) the A and B exciton intensities become weaker and full widths at half maximum become larger, respectively. Gate-dependent absorption associated with the low-energy A exciton diminishes more rapidly than one of B exciton. When the Vg is varied from −100 to +75 V the exciton energy red-shifts slightly. It is interesting to note that the dependences for the real part of refractive index experience a blue-shifts as Vg increases. These dependences arise from the combined effects of Pauli blocking and many-body interactions.33 The complex refractive index of a MoS2 monolayer as a function of gate voltage in our study is consistent with pioneering experimental results of work,34 in which the complex permittivity of monolayer MoS2 was extracted from differential transmission and reflection spectra. In ref. 33 was shown that the exciton peak A can be split into two resonances in the case of low temperature measurements (10 K), where low-energy feature correspondent to negative trions while the other one associated with neutral A exciton.

Reflection at oblique angles of incidence

More prominent electro-optical modulation of reflection from our structures can be achieved at larger angles of incidence where light can couple to waveguide modes of the device leading to pronounced Fabry-Perot resonances. The reflection measurements of both p-polarised and s-polarised light in the spectral range of 240–1000 nm have been performed with a Woollam ellipsometer that has a focusing spot of about 50 μm. Figure 4a, b show reflectance of the Au/Si3N4/MoS2 hybrid nanostructure for different states of polarised light as a function of angle of incidence (the thickness of Si3N4 layer was 390 nm). We see the excited resonant modes as reflection features for both polarisations. There are narrow interference minima for reflection of s-polarised light and pronounced maxima for reflection of p-polarised light observed in the range of 550–700 nm. For example, at zero gate voltage the reflection spectrum for s-polarised light (Rs) exhibits a pronounce and narrow minimum at λ ∼625 nm for θinc = 80°. Figure 5 shows the polarised reflectance of Au/Si3N4/MoS2 hybrid nanostructure for different angles of incidence at Vg = 0, ±100 V as a function of the photon wavelength. Strong absorption is obtained for narrow spectral range for s-polarisation (Figs. 4 and 5) at larger incident angles (above 70°). The results in Fig. 5 demonstrate that the resonance of Rs(λ) is shifted to longer wavelength as the incidence angle is decreased which leads to reducing of modulation effect. We would like to note that the change of the polarity of the gate voltage lead to change the sign of Rs(λ) at resonance wavelength respect to Vg = 0 V for angles θinc < 75°. When our device illuminated with p-polarised light, the modulated reflectance exhibits a small change (about 2–3%).

Fig. 4
figure 4

Change in polarised reflectance for different gating voltages, Vg. a, b Reflection for s-polarised and p-polarised light in a broad spectral range from Au/Si3N4/MoS2 hybrid waveguide nanostructure as a function of incident angle. c, d Change in s-polarised reflectance in the Au/Si3N4/MoS2 hybrid nanostructure designed for this work in the spectral range of the A and B excitons. e, f Change in p-polarised reflectance in the Au/Si3N4/MoS2 hybrid nanostructure designed for this work in the spectral range of the A and B excitons

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Fig. 5
figure 5

The polarised reflection spectra as a function of incident angle (from 45° to 75°) and polarisation states (p-polarisation and s-polarisation). a-d Change in s-polarised reflectance in the Au/Si3N4/MoS2 hybrid nanostructure in the spectral range of the A and B excitons for 45°, 55°, 65° and 75°. e, f Change in p-polarised reflectance in the Au/Si3N4/MoS2 hybrid nanostructure in the spectral range of the A and B excitons for 45° and 75°

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We found that interaction between light and a MoS2 monolayer in our devices increased for larger angle of incidence resulting in larger electro-optical modulation. Figure 4c, d shows the changes of s-polarised reflectance R(Vg) with gate voltage at the angle of incidence 80°. We can see that in the range of gate voltages between 0 and ±100 V, the absolute value of Rs(Vg) is changing by more than 10%. The relative changes of reflectance (Rs(Vg)/Rs(0)) are much larger and reach 440%. This gives the modulation depth of 6 dB with insertion losses at the level of 15 dB. As was shown experimentally,35,36 it is possible to achieve the modulator depths >15 dB for modulators based on graphene working in the infrared (∼1500 nm). However, it is virtually impossible to extend the working range of graphene modulators to visible light. MoS2 provides a nice alternative to graphene for visible light modulators. Indeed, in this work we have experimentally achieved the modulation depth of ∼6 dB for visible light (red light ∼630 nm) for simple electro-optical modulator based on MoS2 monolayers. This modulation depth observed in our device is among the best for TMDs based modulators.4,5,6,37 The changes observed in Fig. 4c, d are attributed to the Vg induced refractive index changes (Fig. 3a, b) in the MoS2 monolayer combined with the optical resonant features of the device. (Indeed, the modulation at 80° light incidence happens in a narrow resonant band near 625 nm with a half width at half maximum of ∼12 nm.) We also observed field-induced modulation for p-polarised light, albeit the changes in reflection were about 3–4 times smaller than those for s-polarised light, see Fig. 4c–f.

Discussion

Carrier properties (concentration, mobility, etc.) of a MoS2 monolayer are critical parameters that govern the electro-optical properties of 2D materials. Some of these properties can be deduced from Raman spectra of MoS2 flakes.28 Because electron-phonon coupling is stronger for out-of-plane vibration (A1g mode) than for in-plane vibration (E12g mode), the increased electron concentration in a MoS2 monolayer produced by gating voltage causes the reduction and broadening of the A1g mode, see Fig. 2a.

The properties of the excitonic modes A and B which produce absorption peaks and gate-tunability of the refractive index (Fig. 3a, b) could be affected by substrate. Indeed, it is known that negative trions could be formed by neutral excitons after accepting an electron and the A mode can split onto two modes A0 and A−.33,38 Hence, the broadening of the absorption spectra shown in Fig. 3b could be explained by superposition of contributions from neutral excitons and negative trions.38 The dipole vibrations of electron-hole pairs (neutral excitons) and electron-electron-hole clusters (trions) are not necessarily coplanar which would allow light coupling to both p-polarisations and s-polarisations. In-plane dipole moments are prevalent in a MoS2 monolayer due to symmetry, which was confirmed by study of momentum-resolved photoluminescence.39 At the same time, out-of-plane dipole photo-luminescence emission is also present albeit much smaller than that of the in-plane dipole photo-luminescence. We observed narrow resonances for s-polarised reflection produced solely by in-plane dipoles while the response for p-polarised light was relatively weak at large angle of incidence (it contains mostly contributions from out-of-plane oriented dipoles), see Fig. 4a, b.

Next, we focus on the complex refractive index of a MoS2 monolayer. Our measurement shows that the absorption is constant (gate-independent) away from the A and B excitonic bands. The dielectric constant (the refractive index) of the TMDs monolayer can be modelled by an oscillator multi-Lorentzian model which takes into account contributions from excitons.32,40 The change in absorption and electro-optical modulation arise when oscillator constants fk, Ek, and γk (fk, Ek, and γk are the oscillator strength, the resonance energy Ek, and the width γk of the k-the oscillator, respectively) are affected by constant electrical field. The fitting results (Fig. S6) show that only the oscillator strengths and widths for the A and B excitonic transitions exhibit significant variations under external electrical gating (Table 1). The excitonic transition energies E(A) and E(B) do not significantly change with Vg. This is in good agreement with theory41 which predicts a cancellation of screening effects on single-particle band gap renormalisation and exciton binding energy. The observed changes in the refractive index are due to the decreasing of the oscillator strength fA,B and increasing the spectral width γA,B under Vg (Table 1). In the measured gate voltage range 0 < Vg < 75 V, the expected change in carrier density due to the electrical field effect can be estimated as ne = (ε0εstVg)/(de) (where ε0 is the permittivity of free space, εst is the static dielectric constant for Si3N4 film extracted from FTIR measurement (see, SI), d is the thickness of Si3N4 insulating film, and e is the electron charge). Consequently, the Vg induces change in n, causes a broadening of the spectral width of A and B excitonic absorption bands γA,B and decreasing of the oscillator strengths, fA,B, which are presented in Fig. 3c, d. The effective concentration of electron in the MoS2 surface for Vg = 0 was also estimated: n0 = 2(2πm0kT/h2)3/2 = 4.32 × 1012 cm−2 (m0 = 0.5me is the effective electron mass for MoS2 bulk, k is the Boltzmann’s constant, T is the temperature set at 300 K, and h is Planck’s constant).

We suggest a possible mechanism for gate voltage dependent absorption in a MoS2 monolayer in the vicinity of A and B exciton—the Pauli blocking effect.42 In the Pauli blocking effect (phase-space filling) a carrier density increase makes part of the phase space unavailable for exciton formation due to the Pauli exclusion principle.42 This effect can be described with the help of a multiexciton Wannier formalism.43,44 (The Wannier exciton theory should be applicable in our case because the effective Bohr radius of an exciton aB = ħ2εst/(μe2), where μ = 0.195me is the e-h reduced mass in unit of electron mass, and εst = 3.43 is the dielectric constant of the monolayer; this value is around 0.93 nm for MoS2 monolayer which is larger than the in-plane lattice constant43,44). The theory is quite elaborate and suggests that for a MoS2 monolayer the excitons and conduction electrons should occupy different regions of the phase space. This causes a reduction in the exciton oscillator strength with induced electron density which can be roughly described as

$$f_{A,B} = f_{A,B}^0\left( {1 - n/n_{eff}} \right) \approx f_{A,B}^0\left( {1 - n \cdot \pi a_B^2} \right).$$
(2)

To check this dependence, we have linear fitted the extracted oscillator strengths as a function of the induced electron density, see Fig. 3c. Using (2), we can evaluate an effective Bohr radius of an exciton from the linear fit parameters. The linear fit gives neff to be 1.6 × 1013 cm−2 and 2.6 × 1013 cm−2 and aB to be 1.4 nm and 1.1 nm for A and B excitons, respectively. For the realistic case in which the MoS2 monolayer locates between air (with a dielectric permittivity ε(air) = 1) and a dielectric medium Si3N4 (with a dielectric permittivity εst(Si3N4) = 8.5 evaluated from FTIR measurement, see SI, Eq(S1)) the dielectric constant can be calculated in the superpositional approximation: εst=[ε(air) + εst(Si3N4)]/2.45 Using values of electron, me/m0 = 0.35 and hole, mh/m0 = 0.428 effective masses in units of the bare electron masses, m045 and the average of the εst = 4.75, we can estimate the exciton Bohr radius, aB* ≈ 1.3 nm. This theoretical value45 is close to an effective Bohr radius of MoS2 excitons evaluated from ellipsometric measurements.

Another contribution to decreasing of fA,B could come from changes in the exciton orbital wave function due to modification of e-h interaction induced by the presence of other e-h pairs.42,44 This reduces the probability of finding an electron near a hole and also yields a contribution to the oscillator strength fA,B. In addition to the change in fA,B, broadening of the exciton bands (the spectral widths γA,B) could be observed after application of Vg. The broadening has collisional nature being governed by the Coulomb scattering effect, carrier density, and kinetic energy of the bonding e-h pairs and the individual charges.42

The changes in reflectance (R) can be associated with absorbance due to the energy conservation law. Indeed, the frequency-dependent absorption A of an optical structure can be calculated as A = 1 − R–T, where T is the total transmittance. For the studied Au/Si3N4/MoS2 hybrid nanostructure, the Au layer is thick (∼65 nm) so that transmittance of the structure is negligible. Hence, according to the Kirchhoff’s law, the absorbance can be found as A = 1 − R. Therefore, changes in the refractive index of a MoS2 monolayer induced by the electric field effect modify both reflectance of and absorbance of the structure. (It is worth noting that light absorption happens mostly within the top MoS2 monolayer.) The appearance of multiple peaks in light absorbance of the hybrid nanostructure shown in Figs. 4 and 5 is connected with Fabry–Perot resonances produced in the studied waveguide-based heterostructure46 as explained in the modelling section and SI. At large angles of incidence, the reflection coefficients become large which increases the quality and depths of the Fabry–Perot resonances. As a result, photons become trapped for a longer time in the heterostructure and experience more bounces from the MoS2 layer which leads to the enhanced absorption of the structure. The large enhancement of absorption can also be explained in terms of optical interference produced by multi-reflection and multi-refraction at the interfaces. The interference can enhance the local electrical fields around the MoS2 monolayer and hence increase the light modulation produced by changes in the excitonic properties induced by the electric field effect (gating voltage). The concentration of electrical field inside of the Si3N4 layer significantly depends on how close incident angle θ is to θc (where the θc is the angle of TIR at Si3N4/Air interface: θc = arcsin(1/nd) ∼29° at λ ∼625 nm). If the angle of incidence θ inside of Si3N4/MoS2 nanostructure larger than θc, no beam propagated outside of Si3N4. Due to high refraction index of MoS2 monolayer (Fig. 3) the density of trapping light in dielectric cavity becomes higher. Probing light can be confined in the top Si3N4/MoS2 layers by internal reflection at MoS2/Si3N4 and MoS2/Air interfaces. Thus indeed MoS2/Si3N4/Au structure can act like a waveguide and we see multiple dips in the reflectance spectra.

The enhancement of absorbance observed in the measurements can be qualitatively reproduced by transfer matrix calculations and is associated with excitation of Fabry–Perot resonances in the multilayer structure47,48 (see Supplementary Information for details). At first stage the light absorption of MoS2 in the spectral region of excitons was optimised for light incidence with respect to thickness of dielectric Si3N4 layer. Figure S7 shows the calculated s-polarised reflection of MoS2/Si3N4/Au nanostructure as a function of the incident wavelength and the thickness of the dielectric Si3N4 layer, suggesting that the optimised Si3N4 thickness is in the range of 390–415 nm. It is also evident that the linewidth of the resonance changes by varying Si3N4 layer thickness. Our calculation also confirms that for an optimised Au/Si3N4/MoS2 hybrid waveguide nanostructure the absorption enhancement in the spectral range of exciton bands of MoS2 is strong and modulation of reflectance induced by external electrical field produced by gating voltage Vg is large (Fig. 6). Calculated values for p-polarised and s-polarised reflectance corresponding to our experimental conditions are shown in Fig. 6. A good agreement between experiment and calculations is observed for both polarisations. Note that the slight difference between experimental and theoretical dependences exists due to surface roughness and impurities on top of sample after MoS2 transferring (see Inset Fig. 6a). The modelling shows that the multi-pass light interaction introduced by interference produces enhanced excitonic absorption in a 2D single layer. The modelling also confirms that the absorbance and reflectance of our structure can be indeed electrically tuned by 10–15% due to an injection of charge carriers into a MoS2 monolayer resulting in refractive index modulation shown in Fig. 3b.

Fig. 6
figure 6

Numerical simulations of the changes in s-polarised (a) and p-polarised (b) reflectance at gating voltages, Vg using the interference effect and electrical tunability of the refractive index of monolayer MoS2. Inset: experimental and modelled s-polarised reflectance of the Au/Si3N4/MoS2 hybrid waveguide nanostructure at Vg = 0

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In summary, we have demonstrated the optical electric field effect on a MoS2 monolayer combined with a dielectric Si3N4 waveguide layer placed on a gold sublayer. The light modulation due to an applied gate voltage was observed near excitonic bands of a MoS2 monolayer. Using spectroscopic ellipsometry, we have extracted the dependence of the complex refractive index of a MoS2 monolayer as a function of gate voltage (applied electric field). We have shown that the light modulation can be significantly enhanced by utilising Fabry–Perot resonances observed in the demonstrated hybrid nanostructures at large angles of incidence. The enhancement of light modulation is due to interference effects observed in the studied structure. We found that a simple Fresnel theory provides a good fit to the experimental data and allows one to model light modulation in the studied structure with help from the gate-dependent MoS2 refractive index elucidated in our work. In contrast to optoelectronic devices based on other 2D materials (including graphene), the fabricated devices showed excellent durability and survived all the measurements for a period of more than one year (they are still alive). The fabricated low-cost waveguide nanostructure with a MoS2 monolayer showed a large modulation range of ∼6 dB for red light. This approach can find realistic applications in designing optical modulators for the visible region of the spectrum.

Methods

Fabrication of devices

Samples were fabricated using electron beam deposition, chemical vapour deposition, etching procedure and 2D material transfer. First, a 3 nm-thick Cr adhesion layer and 65 nm Au film were deposited on top of a 1 mm-thick quartz substrate using electron beam evaporation. Second, a Si3N4 waveguide layer was grown by chemical vapour deposition (CVD) to cover the Au film. Third, a monolayer of MoS2 (grown by the CVD method on Si/SiO2 substrate and acquired from 2D Semiconductors) was spin coated with a poly(methyl methacrylate) (PMMA) resist. Then, the PMMA/MoS2/SiO2/Si stacks were set onto the scotch type with a window and floated in KOH solution until complete Si/SiO2 etching which happened within a few hours. Next, the PMMA/MoS2 films on the scotch type were rinsed with DI water and fished for transfer procedure. Fourth, the resulting monolayer of MoS2 was transferred on top of quartz/Au/Si3N4 structure.

Measurement and characterisation

Raman and photoluminescence (PL) spectra of the CVD grown MoS2 monolayer was recorded with a Witec confocal Raman spectrometer at the excitation wavelengths of 514 nm and 632.8 nm. Fourier Transform Infra-Red (FTIR) spectroscopy was performed with the help of a Bruker Vertex 80 system and a Hyperion 3000 microscope. A variety of sources and detectors, combined with aluminium coated reflective optics enable this system to be used from visible to mid-IR wavelengths.

We also performed an accurate determination of the complex refractive index and thickness of a MoS2 monolayer using a variable angle Woollam M2000F focused-beam spectroscopic ellipsometer. The spot size on the sample was approximately 30 × 80 μm at ~70–80° angles of incidence. Ellipsometry measures two parameters, Ψ (reflection) and Δ (phase), which are related to the complex amplitude reflectance ratio: \(\tan ({\mathrm{\Psi }}){\mathrm{exp}}\left( {i{\mathrm{\Delta }}} \right) = r_p/r_s\), where rp and rs are the amplitude reflection coefficients for p-polarised and s-polarised light. In addition to ellipsometric parameters Ψ and Δ, the ellipsometer allowed us to separately measure Rp and Rs, the intensity reflections for p-polarised and s-polarised light, respectively, at various angles of incidence with respect to the total light intensity.