AFRL-RX-WP-JA-2018-0101 SURFACE PLASMON ABSORPTION IN MoS 2 AND GRAPHENE-MoS 2 MICRO-GRATINGS AND THE IMPACT OF A LIQUID CRYSTAL SUBSTRATE ( POSTPRINT )

The absorption coefficients of a far-infrared wave are calculated at normal incidence for MoS2 and graphene-MoS2 micro-ribbon gratings placed between a nematic LC and an isotropic dielectric medium. Maxima in the absorption spectra, which are related to the excitation of the surface plasmons in micro-ribbons of these gratings, are observed. The spectral position of absorption maxima depends on the grating spacing, micro-ribbon width, and conductivity of the ribbons. The impact of the 2D electron concentration of the MoS2 ribbons on the plasmon bands is different for a MoS2-grating versus a graphene-MoS2 grating. The influence of the LC orientational state on the absorption spectra of the gratings enables the manipulation of the absorption peak magnitude.


I. INTRODUCTION
A possible way to control terahertz (THz) radiation is based on using micro-ribbon gratings. 1n the mid-infrared and far-infrared (THz) spectral region, electromagnetic radiation cannot couple to plasmon excitations in metallic or semiconductor sheets due to the large wave vector mismatch between the plasmonic and incident electromagnetic waves. 2 However, plasmons can be excited in the gratings comprised of metallic or semiconductor ribbons with dimensions much smaller than the radiation wavelength.In such gratings, plasmons can be controlled by manipulation of the microribbon width or the grating spacing.
][5][6] It allowed developing various devices, in particular tunable absorbers [7][8][9] and polarizers. 10][13][14] The resonant plasmon frequency in grating structures depends on the dielectric properties of the layers neighboring with the grating.6][17] A liquid crystal (LC) can also be used as the dielectric substrate of the grating. 18The LC permittivity is anisotropic and depends on the LC director.It can be changed by applying electric and magnetic fields or other external stimuli. 19n addition to the graphene, a monolayer molybdenum disulphide (MoS 2 ) is also very perspective plasmonic material possessing high field confinement and low losses. 20MoS 2 was recently proposed for exciting surface plasmons in the mid-infrared and far-infrared spectral region. 21Corresponding author e-mail address: v.i.zador@gmail.com2158-3226/2018/8(4)/045024/10 8, 045024-1 © Author(s) 2018 The presence of a bandgap in a MoS 2 monolayer allows the fabrication of transistors that can be turned off and used as switches, logic circuits, and high gain amplifiers. 22Electro-optic modulators, photodetectors, optical filters, and phase shifters can also be developed using a MoS 2 monolayer. 21,23,24Recently, different MoS 2 /graphene hybrid heterostructures have been demonstrated.In particular, the appearance of hexagonal flakes at the chemical vapor deposition of MoS 2 on a graphene surface, 25 formation of the van der Waals heterostructures in a growth process of the MoS 2 on epitaxial graphene, 26 the chemically assembled atomic sized transistors based on singlelayer MoS 2 and graphene heterostructures, 27 parallel stitched MoS 2 -graphene heterostructures as a prototype of metal-semiconductor heterostructures. 28n the present paper, we study the light absorption in MoS 2 and graphene-MoS 2 micro-ribbon gratings.We propose to place a LC slab near the gratings and study the possibility of controlling the absorption of these structures in the THz range by reorienting the LC director with an applied field.
The paper is organized as follows.In Sec.II we introduce a model of the LC-grating-dielectric structure and present equations allowing for the calculation of the absorption coefficient for the system.Results of numerical calculations of the absorption coefficient in MoS 2 and graphene-MoS 2 micro-ribbon gratings and their discussion are presented in Sec.III.In Sec.IV we present some brief conclusions.

II. MODEL AND EQUATIONS FOR THE LIGHT FIELD
Let a micro-grating located in the xy-plane is placed between a nematic LC layer (top substrate) and an isotropic dielectric substrate (Fig. 1).The grating is comprised of single layer ribbons made from two different metals (or semi-metals).The ribbons are directed along the y-axis; d and Λ -d are the widths of the adjacent ribbons, Λ is the grating spacing.A plane electromagnetic wave falls on the grating from the side of the LC and excites the surface plasmon polaritons (SPPs) in the grating ribbons.In order to excite the SPPs the light wave electric vector must have the component directed along the grating perpendicular to its ribbons.We consider a normal incidence of the light wave on the grating.Therefore, we can set the magnetic vector of the incident wave to be directed along the y-axis (TM-wave).To avoid the complicated factors at the calculations we assume: i) the system to be infinite in the y-direction; ii) the substrates are semi-infinite in the z-direction; iii) the LC director is located in the xz-plane.
The SPPs are strongly localized near the ribbons, therefore only the thin layer of the LC adjacent to the ribbons can influence the SPPs.To simplify calculations we set the LC director profile in the whole substrate to be the same as in the layer near the ribbons.It allows considering the LC substrate as a homogeneous anisotropic medium.As the TM-wave in the LC is normally incident on the grating, the electric and magnetic vectors of this wave can be written in the form: Satisfying the Maxwell equations, we obtain that and the wave vector is determined by the dispersion equation with ε 1ij denoting the LC dielectric tensor component.
The system is periodic along the x-axis and for reflected and transmitted waves we can use the Fourier-Floquet expansion.Then, the electric and magnetic vectors of the reflected wave, which satisfy the Maxwell equations in the LC are as follows (see Ref. 18): The corresponding dispersion equation takes the following form: For the transmitted wave in the isotropic dielectric substrate we obtain and the dispersion equation Here a n , b n are the coefficients of the Fourier-Floquet expansions for reflected and transmitted waves, respectively; k n = 2πn/Λ, n is the number of the Fourier-Floquet spatial harmonic, ε 2 is the isotropic substrate dielectric permittivity.The electric and magnetic vectors of the incident, reflected, and transmitted waves satisfy the boundary conditions at the grating (z = 0): where j s = σ(x)E tx (z = 0) e x is a current density in the plane of the grating with σ(x) to be the grating surface conductivity.Using expression for E tx (z = 0) from Eq. ( 6), we get Substitution of Eqs. ( 9) and ( 1)-( 7) for electric and magnetic vectors into the boundary conditions (8) yields the following equations for coefficients a n , b n : Eqs. ( 10) and ( 11) were obtained before in our previous paper 18 in the case of the graphene grating.
Here we use these equations for the case of the more complicated grating comprised of the microribbons made from two different materials.
We represent a function σ(x) in Eq. ( 10) in the form of a step function where σ 1 and σ 2 are the surface conductivities of the adjacent ribbons (see Fig. 1).Using Eq. ( 12) and performing some algebra we arrive at the following equations for coefficients a n , b n : Numerically solving Eqs. ( 13) and ( 14) we can calculate coefficients a n , b n and the electric and magnetic vectors of the reflected and transmitted waves [see Eqs. ( 4) and ( 6)].Here we limit ourselves to calculation of the absorption coefficient A of the system under consideration.For this, we use the relation A = 1 (R + T ) where R and T are the reflection and transmission coefficients, respectively, defined as For numerical calculations we take into account that components of the LC dielectric tensor are connected with the components of the LC director by expression , where n i denotes the components of the director n = (sin ψ, 0, cos ψ).Here ψ is the director angle with the z-axis (see Fig. 1), n o and n e are the refractive indices of the ordinary and extraordinary waves, respectively, 29 i = x, y, z.As an example of the LC we take the nematic LC W1791 with n o ≈ 1.53, n e ≈ 1.94 at λ = 1.064 µm studied in Ref. 30 and used by us in the case of the graphene grating. 18It will be convenient for comparison with the corresponding results obtained for the graphene grating.The refractive indices of W1791, n o and n e , are not known in far-infrared spectral range.For this range we use the extended Cauchy dispersion formulas obtained in Ref. 31 for the LC E7.Such extrapolation into sufficiently remote spectral region can give a substantial error and in this case the obtained numerical values must be considered as an estimation.As an example of the isotropic substrate material, silicon was chosen as it is often used in graphene plasmonics (ε 2 = 11.7). 32

A. MoS 2 micro-ribbon grating
The absorption spectra of the MoS 2 micro-ribbon grating placed between a nematic LC and an isotropic dielectric substrate is studied, as is the influence on these spectra of the LC director orientation.In this case, in Fig. 1 conductivity σ 1 refers to MoS 2 , σ 2 = 0, and d is the MoS 2 ribbon width.
We treat here MoS 2 as a monolayer film like graphene.In the infrared and terahertz spectral regions, the intraband contribution to the MoS 2 conductivity dominates. 24The 2D conductivity of n-doped monolayer MoS 2 can be written in the Drude form as 14,24 where n 2D is the 2D electron concentration of MoS 2 , τ MoS2 is the electron relaxation time, and m * e is the electron effective mass.For monolayer MoS 2 we use the values τ MoS2 = 0.17 ps and m * e = 0.53 m e

24
where m e is the mass of electron.
Figure 2 shows the dependence of absorption spectra of the MoS 2 micro-ribbon grating on the 2D electron concentration in the MoS 2 ribbon.The grating spacing value is fixed and equal to FIG.Absorption spectra of the MoS 2 micro-ribbon grating at different 2D electron concentrations: (1) n 2D = 2.5 × 10 13 cm -2 , (2) 5 × 10 13 cm 2 , (3) 10 14 cm 2 .Grating spacing of Λ = 1 µm, the ribbon aspect ratio d/Λ = 0.5, the director angle ψ = 0 • .Λ = 1 µm, ribbon aspect ratio d/Λ = 0.5, and the director angle ψ = 0 • .Maxima in absorption spectra correspond to the excitation of the plasmon modes in the MoS 2 micro-ribbons.As one can see, the plasmon resonance wavelength shifts with an increase of electron concentration into the short-wavelength side that correlates with a corresponding increase of the plasmon frequency.The magnitude of absorption maxima also increases.
Figure 3 illustrates the change of the absorption spectra of the MoS 2 micro-ribbon grating at the change of the LC director orientation from the homeotropic orientation near the grating (ψ = 0 • ) to the planar one (ψ = 90 • ) for several values of the ribbon width to grating spacing ratio: d/Λ = 0.3, 0.5 and 0.7.The grating spacing value was fixed and equal to Λ = 1 µm (Fig. 3a) and Λ = 0.1 µm (Fig. 3b).
As we can see from Figs. 3a and 3b, a decrease of the grating spacing from Λ = 1 µm to Λ = 0.1 µm shifts the resonance peaks into the short-wavelength side.At a fixed grating spacing Λ, an increase of the ribbon width leads to the resonance peak shift into the long-wavelength side and an increase of the resonance peak magnitude.It should also be noted that the magnitude of each peak at the fixed value of ratio d/Λ is the same for different values of the grating spacing Λ.This behavior of the MoS 2 grating plasmon peaks agrees with the results obtained in papers 18,33 for the graphene grating.In particular, following from the qualitative theory developed in Ref. 33 dependence of the resonance wavelength on the ribbon width, λ r ∼ d, agrees well with data presented in Figs.3a and 3b.
Magnitudes of maxima in Fig. 3 depend on the LC director orientation.The rotation of the LC director by 90 • leads to the change of the maxima magnitude by approximately 9.4-11.9% in Fig. 3a  and 9.9-11% in Fig. 3b.The impact of the director rotation on the absorption peak magnitude is more significant for a greater aspect ratio d/Λ.
Note that our results are obtained assuming that the plasmon bands fall into the transparency region of the LC.If the LC absorption bands are not very wide, one can shift the plasmon bands into the LC transparency region choosing the corresponding value of the MoS 2 grating spacing, the ribbon width, and the isotropic substrate material.

B. Graphene-MoS 2 micro-ribbon grating
In this subsection, the features of the absorption spectra of the grating comprised of alternating ribbons of MoS 2 and graphene placed between a nematic LC and an isotropic dielectric substrate is discussed.In this case, we set conductivities σ 1 and σ 2 (see Fig. 1) relating to graphene and MoS 2 , respectively, with d to be the graphene ribbon width, Λ d to be the MoS 2 ribbon width, and σ 2 defined by Eq. ( 16).
The 2D conductivity of graphene in the infrared and terahertz spectral regions can be written as 2 where E F is the Fermi energy of graphene, τ G is the electron relaxation time, and e is the electron charge.For evaluation of the electron relaxation time we use the formula τ G = µE F /(ev 2 F ), where µ is the carrier mobility, v F = 3 × 10 6 m/s is the Fermi velocity in graphene. 2Setting E F = 0.64 eV and the carrier mobility µ = 0.5 m 2 /(V•s), 2,33 the electron relaxation time τ G = 0.32 ps is obtained.
In Figs.4-6 results of our numerical calculations of absorption spectra of the graphene-MoS 2 micro-grating for various grating parameters are presented.In Figs. 4 and 5 we show the influence of the 2D electron concentration in the MoS 2 ribbons on the absorption spectra at two values of the ribbon width to grating spacing ratio d/Λ = 0.5 (Fig. 4) and d/Λ = 0.7 (Fig. 5) setting the grating spacing to be fixed and equal to Λ = 1 µm.
As seen from Fig. 4a, the wavelength of the main peak (∼37 µm) in the absorption spectra of the graphene-MoS 2 grating at the MoS 2 ribbon electron concentration n 2D = 10 13 cm -2 coincides closely with the wavelength of absorption peak of the graphene grating (see Ref. 18).Referring to the results shown in Fig. 2, an absorption in the MoS 2 micro-ribbon grating can appear only at wavelengths much greater than 37 µm for this electron concentration.The influence of this MoS 2 ribbon electron concentration is therefore small and the main absorption peak in Fig. 4a arises mainly due to the plasmon resonance in the graphene micro-ribbons.At n 2D = 2.5 × 10 13 cm 2 , the absorption spectrum changes significantly (see Fig. 4b) and three absorption peaks of the plasmon modes are observed instead of the main peak at n 2D = 10 13 cm -2 .Further increasing n 2D (see Figs. 4c and 4d), the resonance wavelengths of the plasmon modes shift toward shorter wavelengths, the magnitude of the absorption peaks decreases, and the plasmon mode with the smallest absorption peak disappears.As in the case of the MoS 2 grating, the shorter wavelength shift of the plasmon modes in the graphene-MoS 2 grating correlates with corresponding increase of the plasmon frequency.At the same time, a decrease and disappearance of the peaks take place because the conductivity of the graphene and MoS 2 micro-ribbons equalizes with an increase of the MoS 2 ribbon electron concentration leading to a decrease of the refractive index grating magnitude.At n 2D = 14.4 × 10 13 cm 2 , Im σ 1 = Im σ 2 per calculations.In this case, the refractive index grating as such disappears, plasmons cannot be excited and no plasmon peaks in absorption spectrum of the graphene-MoS 2 structure (Fig. 4e) are observed.The plasmon mode absorption peaks reappear in the shorter wavelength part of absorption spectrum when Im σ 2 > Im σ 1 .Fig. 4f shows the case for n 2D > 14.4 × 10 13 cm 2 .
With an increasing share of graphene on the grating period (d/Λ = 0.7), the position of main resonance peaks in the absorption spectrum shifts towards longer wavelengths (Fig. 5) in comparison with the case d/Λ = 0.5 (Fig. 4).This is similar to the graphene and MoS 2 gratings For this wavelength region the influence of the MoS 2 ribbon electron concentration on the absorption spectrum is more pronounced even at small values of n 2D = 10 13 cm -2 (compare Fig. 4a and Fig. 5, curve 1).With an increase of n 2D , as in the case of d/Λ = 0.5, absorption peaks shift toward shorter wavelengths and their magnitude decreases.The smallest absorption peaks disappear, but it takes place at smaller concentrations n 2D than in the case d/Λ = 0.5 (Fig. 5, curves 2 and 3).
In Fig. 6 we show results of calculations of the graphene-MoS 2 micro-grating absorption spectra for two limiting values of the LC director angle at ψ = 0 • and ψ = 90 • for the graphene ribbon width to grating spacing ratio d/Λ = 0.3, 0.5 and 0.7.The electron concentration in the MoS 2 ribbons is fixed and equal to n 2D = 5 × 10 13 cm -2 , and the grating spacing Λ equals 1 µm in Fig. 6a and  plasmon peaks depend on the grating spacing and the ribbon width.As in the case of the graphene and MoS 2 gratings, values of λ r decrease with a decrease of the grating spacing.However, the character of the λ r dependence on the ribbon width still depends on the MoS 2 ribbon electron concentration n 2D .Figure 6 shows that at n 2D = 5 × 10 13 cm -2 , values of λ r decrease with increasing graphene ribbon width while for smaller MoS 2 ribbon electron concentration (n 2D = 10 13 cm -2 ), this dependence becomes opposite (compare Fig. 4a and Fig. 5, curve 1).To explain this, additional study of gratings with different types of the micro-ribbon materials is needed.
The most intensive plasmon mode absorption peaks are in the spectral range 35-45 µm for Λ = 1 µm (Fig. 6a) and 11.6-13.6µm for Λ = 0.1 µm (Fig. 6b).Reorientation of the LC director from homeotropic orientation (ψ = 0 • ) to planar orientation (ψ = 90 • ) influences only the magnitude of absorption peaks.In particular, the rotation of the LC director by ψ = 90 • to a change of the absorption peak magnitude by approximately 5.7-10.8% in the spectral range 35-45 µm (Fig. 6a) and 8.6-10.7% in the spectral range 11.6-13.6µm (Fig. 6b).Influence of the director rotation on the absorption peak magnitude increases with the aspect ratio the ribbon material, which dominates the conductivity, i.e. the ratio d/Λ in the cases presented in Figs.6a and 6b.This is due to the corresponding increase of the refractive index grating magnitude and agrees with results obtained for graphene 18 and MoS 2 gratings.

IV. CONCLUSIONS
The absorption spectra of MoS 2 and graphene-MoS 2 micro-ribbon gratings placed between a nematic LC and isotropic dielectric media was studied theoretically.In the far-infrared region (6-140 µm) maxima in the absorption spectra related to the excitation of plasmons within the microribbons are observed.For fixed parameters of the micro-ribbon conductivity, the spectral position of absorption maxima depends on the grating spacing and micro-ribbon width.The influence of the 2D electron concentration of the MoS 2 ribbons, n 2D , on the absorption spectra was explored and is shown to be different for the MoS 2 and the graphene-MoS 2 grating cases.For the MoS 2 grating case, a shift of the absorption peaks toward shorter wavelengths and an increase of their magnitude with increasing n 2D was observed.For the graphene-MoS 2 grating case, an increase of n 2D leads first to the appearance of new plasmon modes and a shift of their absorption peaks to shorter wavelengths, and then a gradual decrease of the peak magnitude (to include some mode disappearance).This occurs because the conductivities of MoS 2 and graphene ribbons equalize with increase n 2D resulting in a decrease of the refractive index grating magnitude.When the imaginary parts of the MoS 2 and graphene ribbon conductivity are equal, the refractive index grating disappears, and all plasmon peaks disappear.Upon further increasing of n 2D the absorption peaks appear again in the shorter wavelength part of the spectrum.The spectral shift of absorption maxima dependence on the ribbon width can be on the short-or long-wavelength side depending on n 2D .
We show that the magnitude of absorption peaks in the MoS 2 and the graphene-MoS 2 gratings depends on the LC orientation state.The results indicate manipulation of the absorption maximum value within 10-12% of its magnitude using this change.The influence of the LC orientation state change on the plasmonic resonance increases with an increase of the MoS 2 ribbon width to grating spacing ratio for the MoS 2 grating, and the graphene ribbon width to grating spacing ratio for the graphene-MoS 2 grating.Our results can be used for designing new types of photonic structures coupling the dynamic properties of LC molecules with baseline plasmonic properties of MoS 2 and the graphene-MoS 2 gratings.

FIG. 1 .
FIG. 1. Experimental set-up, showing the TM-wave with the wave vector k i incident from the LC on the grating-dielectric structure.Λ is the grating spacing, d, Λ d and σ 1 , σ 2 are the widths and the conductivity of the ribbons, respectively; ε1 is the LC dielectric tensor, ε 2 is the isotropic substrate permittivity, ψ is the director orientation angle in the xz-plane, e n is a normal to the ribbons plane.