Absorbance and optical responsivity of two-dimensional mechanical resonators at oblique incidence

Physics Letters A 383 (2019) 2755–2760

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Physics Letters A www.elsevier.com/locate/pla

Absorbance and optical responsivity of two-dimensional mechanical resonators at oblique incidence Wenjing Mao a,b , Chen Yang a,b , Heng Lu a,b , Jun Lu a,b , Lin Wan a,b , Fengnan Chen a,b , Ying Yan a,b,∗ , Joel Moser a,b,∗ a

School of Optoelectronic Science and Engineering & Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China b Key Lab of Advanced Optical Manufacturing Technologies of Jiangsu Province & Key Lab of Modern Optical Technologies of Education Ministry of China, Soochow University, Suzhou 215006, China

a r t i c l e

i n f o

Article history: Received 1 April 2019 Received in revised form 28 May 2019 Accepted 28 May 2019 Available online 31 May 2019 Communicated by V.A. Markel Keywords: Nanomechanics Two-dimensional materials Physical optics

a b s t r a c t Optical reflectometry is a well known technique employed to measure the displacement response of mechanical resonators. This technique consists in shining monochromatic light on the resonator at normal incidence and measuring the intensity of reflected light. In the case of resonators fabricated from suspended membranes of two-dimensional materials, fluctuations of distance between the membrane and the underlying substrate are transduced into fluctuations of reflected light intensity. Here we consider an alternate configuration where we shine light at oblique incidence. Within the plane wave model, we calculate the absorbance of the membrane, the reflectance of the device composed of the membrane and the reflective substrate, and the optical responsivity defined as the derivative of reflectance with respect to membrane displacement. We illustrate our calculations with the example of a resonator based on monolayer molybdenum disulfide and find that these quantities can be tuned by changing the angle of incidence, especially in the case of s-polarized light. These results may serve to design experiments where the absorbance and optical responsivity of the resonator are tuned by tilting the device. © 2019 Elsevier B.V. All rights reserved.

1. Introduction By virtue of their low mass, high aspect ratio and high stiffness, mechanical resonators fabricated from atomically-thin membranes of two-dimensional (2-D) materials facilitate the study of a wealth of physical phenomena. These include the interplay of quantum Hall states and flexural vibrations [1,2], strong coupling between vibrational modes [3,4], and unusual energy dissipation enabled by mechanical nonlinearities [5]. A variety of detection schemes, either based on optical or electrical means, have been developed to measure the mechanical response of these resonators. Flexural vibrations of a suspended graphene sheet were first detected optically [6] using a technique employed to measure the vibrations of silicon-based nanomechanical resonators [7,8]. Applied to 2-D resonators, this technique consists in shining monochromatic light on the membrane (the resonator) at normal incidence and measuring the intensity of light reflected off the low-finesse Fabry-Perot

*

Corresponding authors at: School of Optoelectronic Science and Engineering & Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China. E-mail addresses: [email protected] (Y. Yan), [email protected] (J. Moser). https://doi.org/10.1016/j.physleta.2019.05.046 0375-9601/© 2019 Elsevier B.V. All rights reserved.

cavity formed by the membrane and the underlying substrate. In graphene resonators, this optical reflectometry technique made it possible to elucidate the interplay of flexural vibrations and stress distribution [9], revealed the effect of photothermal forces on the dynamics of the resonator [10], and enabled the spatial mapping of vibrational modes [11]. Alternatively, several electrical detection schemes have been developed to measure the flexural vibrations of graphene resonators [12–17]. In their various forms, these schemes rely on the possibility of tuning the density of free charges at the surface of graphene to create a uniform potential difference between graphene and a counter electrode, making it possible to determine the position of graphene through capacitive measurements. Extending such capacitive measurements to mechanical resonators based on semiconducting membranes remains technically challenging but has been recently demonstrated [18]. With the emergence of 2-D semiconductors, optical reflectometry has naturally become a technique of choice to study the mechanical properties of resonators based on atomically-thin membranes of transition metal dichalcogenides (TMDs) [19–24]. Monolayer TMDs constitute a playground for optics, featuring strong absorption in the visible and near-infrared range [25,26] and high intensity photoluminescence [25–27]; hence one may wonder

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whether alternate configurations of the reflectometry technique may contribute to the study of these resonators. One such configuration may be realized by placing the membrane in an optical cavity [28–34] to enhance light-matter interaction, in similar fashion to graphene resonators coupled to microwave cavities [15,16]. Another configuration may consist in shining light with a certain polarization and at oblique incidence on the resonator in order to tune the reflectance R of the device and the absorbance A of the membrane. The latter is the configuration we discuss in this paper. Here, we present calculations of R and A as a function of the angle of incidence of the monochromatic light probe, considering both s and p polarizations of the electric field wave. To illustrate these calculations, we consider a device structured as a monolayer of molybdenum disulfide (MoS2 ) parallel to a perfectly reflective substrate. We show that the optical responsivity of the resonator, defined as the derivative of reflectance with respect to the displacement of the membrane, increases with the angle of incidence for s-polarized waves. These simple results may prove useful to devise alternate experiments about light-matter interaction in 2-D mechanical resonators where R and A are tuned by tilting the device. 2. Method Our calculations are based on the transfer-matrix method [35], which was used to analyze specimen contrast in optical microscopy of few-layer graphene flakes deposited on oxidized silicon substrates [36]. We begin by showing that the two formalisms commonly employed to calculate the reflectance of mechanical resonators based on 2-D materials are equivalent. One of these formalisms (hereafter labelled N) posits that free charges are absent from the surface of the membrane, and that light propagates through the membrane in a medium with complex refractive index [36–38]. The other formalism (hereafter labelled ) assumes that the membrane is a pure surface hosting free charges, and that propagation of light through the membrane is irrelevant as the membrane has zero thickness [10,39]. Both formalisms have been used alternatively, and have been shown to account for the optical responsivity of these mechanical resonators. To convince ourselves that these formalisms are equivalent, we consider the simplest model of an electromagnetic, monochromatic plane wave normally incident on the resonator (see Fig. 1 for incidence angle θ = 0). We assume that the wave originates from the far left side of the membrane. In the absence of free surface charges, the continuity of the tangential components of the electric field and of the magnetic field across the surface of the membrane dictates the relation between the amplitudes of propagating electric field waves on both sides of the membrane (Figs. 1(a), 1(c)). This relation reads



E 1+ E 1−



 = M+ρ · P · M−ρ

E 2 + E 2 −

 (1)

,

where E 1+ (resp. E 1− ) denotes the complex amplitude of forward (resp. backward) propagating electric field wave at the left interface between vacuum and the membrane, and E 2 + and E 2 − similarly denote the amplitudes of propagating electric field waves at the right interface. The matching matrices M±ρ and the propagation matrix P read

M±ρ =

⎡ P=



1 1±ρ



1

±ρ 

exp i 2λπ nd ⎣ 0

±ρ



,

1

⎤ 

0

exp −i 2λπ nd

⎦ ,

(2)

Fig. 1. Plane wave model employed to calculate absorbance, reflectance and optical responsivity of the resonator. In (a) and (c) the membrane is depicted as a red shaded area representing a medium with finite thickness, while in (b) and (d) the membrane is depicted as a vertical line representing a surface with zero thickness. The reflective substrate is depicted as a green shaded area on the right. Wave vectors k, electric fields E and magnetic fields H are assigned subscript + or − depending on whether they are associated with forward (+) or backward (−) propagating electromagnetic waves. The light source is assumed to be on the far left of the membrane. The distance between the membrane and the substrate is L. Light is s-polarized in (a, b) and p-polarized in (c, d). Dotted lines at the right-hand side interface between the membrane and vacuum and at the position of the reflective substrate depict the directions of incident and reflected waves. (For interpretation of the colours in the figures, the reader is referred to the web version of this article.) n where n is the complex refractive index of the membrane, ρ = 11− +n is the Fresnel reflection coefficient at the interface between vacuum and the membrane, d is the thickness of the membrane, and λ is the wavelength of light in vacuum. Equations (1) and (2) form the basis of formalism N. In the presence of surface charges, the tangential components of the magnetic field are not continuous across the surface of the membrane; instead, their difference is equal to the surface current density. This boundary condition for the magnetic field yields a relation between amplitudes of propagating electric field waves on both sides of the membrane that is seemingly different from Eq. (1). Indeed, disregarding the propagation of light through the membrane (d = 0), this relation reads:



E+ E−





=M

 E+  E−



,

M=

1 2



2 + ση

−σ η



ση , 2 − ση

(3)

 are the complex amplitudes of propagating elecwhere E ± and E ± tric field waves on the left and on the right interfaces between the membrane and vacuum (see Figs. 1(b), 1(d) for θ = 0). In Eq. (3), σ is the optical conductivity of the membrane which, as we detail later, arises from the expression for the amplitude of the  + E  ). The wave surface current density J = σ ( E + + E − ) = σ ( E + −

W. Mao et al. / Physics Letters A 383 (2019) 2755–2760

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impedance in vacuum η relates the complex amplitudes of propagating magnetic field waves H ± to E ± as H ± = ± η1 E ± . Given that d ∼ 1 nm and that λ is in the visible range, the arguments of the complex exponentials in P of Eq. (2) are small. Expanding these exponentials to second order in d/λ and introducing the complex dielectric constant of the membrane , we find M+ρ · P · M−ρ = M, with n = ( / 0 )1/2 , 0 the permittivity of vacuum, σ = iωd( − 0 ), and ω the angular frequency of the wave. Hence formalisms N and  are equivalent for zero incidence angle, simply because d  λ. 3. Results and discussion We turn to the reflectance R of the device and to the absorbance A of the membrane at oblique incidence. As an example, we consider a device composed of a monolayer of molybdenum disulfide suspended over a perfectly reflective planar substrate. With reference to Figs. 1(a) and 1(c), the reflectance R is defined as the ratio of the reflected energy density over the incident energy density and reads R = | E 1− |2 /| E 1+ |2 . The absorbance A is defined as the ratio of the absorbed energy density over the incident energy density and reads A = (| E 1+ |2 + | E 2 − |2 − | E 1− |2 −

| E 2 + |2 )/| E 1+ |2 . Because the substrate is assumed to be perfectly

reflective and the membrane is the only dissipative element in the device, conservation of energy yields A = 1 − R. Figs. 1(a)–(b) and 1(c)–(d) depict s-polarized and p-polarized waves, respectively. The membrane lies within the x − y plane. Propagating electric and magnetic fields at both left and right interfaces are represented by vectors whose directions do not presuppose a value for the reflection coefficient E − / E + ; namely, both forward and backward electric fields point in the same direction for θ = 0. We first consider s-polarized waves, for which electric fields are oriented along the y direction in Figs. 1(a) and 1(b). We choose λ = 532 nm, a wavelength commonly used in experiments involving 2-D resonators, and the corresponding dielectric constant measured in [40], namely / 0 = 22.91 − i10.33. Within formalism N, the matching matrices M±ρ feature a modified Fresnel reflection coefficient ρ = (cos θ − n cos θ  )/(cos θ + n cos θ  ), and the arguments of the complex exponentials in the propagation matrix P feature a propagation length d/ cos θ  [35]. The refraction angle θ  is given by Snell’s law. R and A are displayed in Figs. 2(a) and 2(b), left-hand side panels, as a function of L /λ, where L is the distance between the substrate and the membrane. Calculations for θ = 0 (thick blue traces) and for θ = 70◦ (thick red traces) are shown. Within formalism , the boundary condition for the magnetic field reads



−ˆz × H − H  = J yˆ ,  where H = H+ + H− and H  = H+ + H− are the total magnetic fields at the left interface and at the right interface, respectively, and yˆ and zˆ are unit vectors along the y and z directions. Writing this expression in terms of electric fields, H± = η1 kˆ ± × E± and  H± = η1 kˆ ± × E± , with kˆ ± and kˆ ± unit vectors in the propagation directions, yields

1

η







    . E + cos θ − E − cos θ − E + cos θ + E − cos θ = σ E + + E−

Combining the above equation with the boundary condition for the  electric field (E+ + E− ) · yˆ = (E+ + E− ) · yˆ yields a matching matrix M in which σ η in Eq. (3) is replaced by σ η/ cos θ :

M=

1 2



2 + σ η/ cos θ −σ η/ cos θ

σ η/ cos θ 2 − σ η/ cos θ

 .

(4)

Fig. 2. (a) Reflectance R of a device composed of monolayer MoS2 suspended over a perfectly reflective substrate and (b) absorbance A of the monolayer, both as a function of distance L between the monolayer and the substrate in units of wavelength λ, and for s and p-polarized light. We use λ = 532 nm and angle of incidence θ = 0 and 70◦ . Calculations based on formalism N (monolayer treated as a membrane with complex refractive index and zero free surface charge density) are shown as thick traces, while those based on formalism  (monolayer treated as a membrane with zero thickness and nonzero free surface charge density) are shown as thin traces.

R and A = 1 − R are readily calculated within formalism  from R = |(2)/(1)|2 , with

⎡

=M·



exp i 2λπ ⎣

L cos θ

⎤





0

exp −i 2λπ

0

L cos θ

⎦·



1 −1

 ,

(5)

given that the substrate is fully reflective. R and A are shown in Figs. 2(a) and 2(b) as thin traces. Clearly, both N and  formalisms are still equivalent for oblique incidence and s polarization. In addition, R ( L /λ) displays deeper minima at θ = 70◦ than at θ = 0, so A features larger maxima at θ = 70◦ than at θ = 0. This simple observation indicates that adjusting the angle of incidence of s-polarized light tunes the absorbance of the membrane. This may be beneficial for experiments where interaction between light and the membrane needs to be enhanced to facilitate the measurement of physical quantities such as the heat capacity and the thermal conductivity of the membrane [10,30]. We then consider p-polarized waves, as depicted in Figs. 1(c) and 1(d). Within formalism N, the Fresnel reflection coefficient in the matching matrices M±ρ reads ρ = (cos θ  − n cos θ)/(cos θ  + n cos θ) [35]. Within formalism , the boundary condition for the magnetic field reads



−ˆz × H − H  = J xˆ , where xˆ is a unit vector along the x direction. As before, writing this expression in terms of electric field wave amplitudes we obtain

1

η







    = σ E+ E+ − E− − E+ + E− cos θ + E − cos θ .

Combining the above equation with the boundary condition for the  electric field (E+ + E− ) · xˆ = (E+ + E− ) · xˆ yields a matching matrix M in which σ η in Eq. (3) is replaced by σ η cos θ :

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Fig. 3. (a) Responsivity |∂ R /∂ L |θ of a device made of monolayer MoS2 suspended over a perfectly reflective substrate as a function of incidence angle θ and L /λ with λ = 532 nm, for s and p-polarized light. Traces in (b) are extracted from (a) for θ = 0 and 70◦ .

M=

1 2



2 + σ η cos θ −σ η cos θ

σ η cos θ 2 − σ η cos θ

 (6)

.

In turn, R and A calculated within both N and  formalisms are plotted in Figs. 2(a) and 2(b), right-hand side panels, as a function of L /λ. Here as well, both formalisms are equivalent. For oblique incidence, modulations of R and A are weaker for p-polarized waves than for s-polarized waves. This confirms the importance of using s-polarized waves for studying light-matter interaction in these resonators. The optical responsivity |∂ R /∂ L |θ of the device is shown in Fig. 3(a) as a function of L /λ and θ for both s and p polarizations. The responsivity transduces fluctuations of membrane displacement δ z into fluctuations of reflected light intensity δ I = I inc δ z × |∂ R /∂ L |θ , where I inc is the intensity of incident light, which means that a large responsivity facilitates the optical detection of the flexural vibrations of the membrane. Fig. 3(b) displays the responsivity for θ = 0 and θ = 70◦ . While increasing θ has little influence on responsivity maxima for p-polarized waves (right-hand side panel), it significantly increases responsivity maxima for s-polarized waves (left-hand side panel). To elucidate the origin of the strong dependence of the responsivity on θ for s-polarized waves, it is helpful to calculate R = |(2)/(1)|2 in a closed form based on its matricial expression (see e.g. Eq. (5) for s-polarized waves). We find

R s,p (ξ ) = 2 + ηs,p (1 − cos 2ξ )



Fig. 4. (a) Reflectance R at zero incidence angle as a function of distance L between the resonator and a perfectly reflective substrate for λ = 457 nm. The blue trace shows R ( L ) obtained with the optical conductivity σ of monolayer MoS2 . R ( L ) calculated with modified optical conductivities σa = 3Re[σ ] + i Im[σ ] (green trace) and σb = Re[σ ] + i 3Im[σ ] (orange trace) are also shown. (b) R ( L ) for a device based on monolayer MoS2 over a perfectly reflective substrate for θ = 0 (blue trace) and θ = 70◦ , s-polarized wave (red trace). R ( L ) calculated with the modified optical conductivity σc = 3σ and θ = 0 is also shown (grey trace). (c) Calculated responsivities of the MoS2 device with optical conductivity σ are shown for s-polarized waves (left-hand side panel) and for p-polarized waves (right-hand side panel) for θ = 0 (blue traces) and θ = 70◦ (red traces).



ηs,p |σ |2 − 2Re[σ ] − 2ηs,p Im[σ ] sin 2ξ

 , 2 + ηs,p (1 − cos 2ξ ) ηs,p |σ |2 + 2Re[σ ] − 2ηs,p Im[σ ] sin 2ξ (7)

where subscripts s and p denote s and p-polarized waves, respectively, and ηs = η/ cos θ , ηp = η cos θ , and ξ = 2π L /(λ cos θ). Firstly, it is clear from Eq. (7) that the reflectometry technique

is entirely based on the absorbance of the membrane. Indeed, in the absence of energy absorption by the membrane we have Re[σ ] = 0, hence R s,p = 1 and |∂ R s,p /∂ L |θ = 0. Secondly, one can see that devices based on a membrane with large Re[σ ] feature R s,p (ξ ) whose oscillation amplitude is large, while devices based on a membrane with large Im[σ ] present R s,p (ξ ) oscillations that are strongly asymmetric. We illustrate this point in Fig. 4 where reflectance and responsivities as a function of L /λ are shown for λ = 457 nm, at which wavelength the real and imaginary parts of the dielectric constant of monolayer MoS2 , / 0 = 27.68 − i27.86, are larger than at λ = 532 nm used earlier [40]. To visualize the effect of Re[σ ] and Im[σ ] on R s,p , in Fig. 4(a) we plot R s,p ( L /λ) for θ = 0 using σ measured at this wavelength (blue trace), and using modified optical conductivities σa = 3Re[σ ] + i Im[σ ] (green trace) and σb = Re[σ ] + i 3Im[σ ] (orange trace). Increased oscillation amplitude obtained with σa and enhanced asymmetry obtained with σb are apparent in R s,p ( L /λ). For s-polarized waves, σ is effectively amplified by the wave impedance η/ cos θ as θ increases. Indeed, R s,p ( L /λ) obtained with σc = 3σ and θ = 0 (grey trace in Fig. 4(b)) has the same oscillation amplitude and the same asymmetry as the trace obtained with σ and θ = 70◦ for s-polarized waves (red trace in Fig. 4(b)), since 3  1/ cos(70◦ ). In turn, increased oscillation amplitude and enhanced asymmetry both mean that the responsivity |∂ R s /∂ L |θ reaches higher values (Fig. 4(c), left-hand side panel). For the same reason, increasing θ has little effect on the responsivity for ppolarized waves (Fig. 4(c), right-hand side panel). Therefore, shining s-polarized light on the resonator at oblique incidence may enhance the sensitivity of detection setups based on measuring the vibrational amplitude of the membrane, provided of course that only limited technical complexity is introduced into the experimental setup.

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4. Summary and conclusion In summary, we consider a 2-D mechanical resonator exposed to a monochromatic plane wave at oblique incidence. We employ two different formalisms commonly used to interpret optical reflectance measurements of such resonators: one formalism is based on the assumption that the 2-D membrane does not host free surface charges, while the other one centres on the presence of such surface charges. We show that both formalisms are equivalent. We calculate the absorbance A of the membrane and the reflectance R of the device composed of the membrane and the reflective substrate, along with the optical responsivity |∂ R /∂ L |θ as the distance L between the membrane and the substrate is varied. We illustrate our calculations with a membrane of monolayer MoS2 , and show that A and |∂ R /∂ L |θ can be increased for s-polarized waves by increasing the angle of incidence θ . The increase of responsivity is due to the effective amplification of the optical conductivity as θ increases, resulting in R ( L /λ) with increased oscillation amplitude and enhanced asymmetry. Interestingly, a nanomechanical resonator with large Im[σ ] placed in an optical cavity is subjected to a radiation pressure whose dependence on resonator position is also strongly asymmetric, enabling efficient cooling of the resonator [29]. This supports the notion that placing the resonator in an optical cavity and exposing the resonator to monochromatic light at oblique incidence both hold promise for 2-D nanomechanics. We find merit in our simple results as they may help devise experiments where A or |∂ R /∂ L |θ are required to be adjusted in situ, which may be achieved by tilting the device. Funding National Natural Science Foundation of China (61674112, 61505133); International Cooperation and Exchange of the National Natural Science Foundation of China NSFC-STINT (61811530020); Natural Science Foundation of JiangSu Province (BK20150308); Key Projects of Natural Science Research in JiangSu Universities (16KJA140001); JiangSu Province Professorship; Six Talent Peaks Project. References [1] V. Singh, B. Irfan, G. Subramanian, H.S. Solanki, S. Sengupta, S. Dubey, A. Kumar, S. Ramakrishnan, M.M. Deshmukh, Coupling between quantum Hall state and electromechanics in suspended graphene resonator, Appl. Phys. Lett. 100 (2012) 233103. [2] C. Chen, V.V. Deshpande, M. Koshino, S. Lee, A. Gondarenko, A.H. MacDonald, P. Kim, J. Hone, Modulation of mechanical resonance by chemical potential oscillation in graphene, Nat. Phys. 12 (2016) 240–244. [3] R. De Alba, F. Massel, I.R. Storch, T.S. Abhilash, A. Hui, P.L. McEuen, H.G. Craighead, J.M. Parpia, Tunable phonon-cavity coupling in graphene membranes, Nat. Nanotechnol. 11 (2016) 741–746. [4] J.P. Mathew, R.N. Patel, A. Borah, R. Vijay, M.M. Deshmukh, Dynamical strong coupling and parametric amplification of mechanical modes of graphene drums, Nat. Nanotechnol. 11 (2016) 747–751. [5] J. Güttinger, A. Noury, P. Weber, A.M. Eriksson, C. Lagoin, J. Moser, C. Eichler, A. Wallraff, A. Isacsson, A. Bachtold, Energy-dependent path of dissipation in nanomechanical resonators, Nat. Nanotechnol. 12 (2017) 631–636. [6] J.S. Bunch, A.M. van der Zande, S.S. Verbridge, I.W. Frank, D.M. Tanenbaum, J.M. Parpia, H.G. Craighead, P.L. McEuen, Electromechanical resonators from graphene sheets, Science 315 (2007) 490–493. [7] D.W. Carr, H.G. Craighead, Fabrication of nanoelectromechanical systems in single crystal silicon using silicon on insulator substrates and electron beam lithography, J. Vac. Sci. Technol. B 15 (1997) 2760–2763. [8] D. Karabacak, T. Kouh, K.L. Ekinci, Analysis of optical interferometric displacement detection in nanoelectromechanical systems, J. Appl. Phys. 98 (2005) 124309. [9] A. Reserbat-Plantey, L. Marty, O. Arcizet, N. Bendiab, V. Bouchiat, A local optical probe for measuring motion and stress in a nanoelectromechanical system, Nat. Nanotechnol. 7 (2012) 151.

2759

[10] R.A. Barton, I.R. Storch, V.P. Adiga, R. Sakakibara, B.R. Cipriany, B. Ilic, S.-P. Wang, P. Ong, P.L. McEuen, J.M. Parpia, H.G. Craighead, Photothermal selfoscillation and laser cooling of graphene optomechanical systems, Nano Lett. 12 (2012) 4681–4686. [11] D. Davidovikj, J.J. Slim, S.J. Cartamil-Bueno, H.S.J. van der Zant, P.G. Steeneken, W.J. Venstra, Visualizing the motion of graphene nanodrums, Nano Lett. 16 (2016) 2768–2773. [12] C. Chen, S. Rosenblatt, K.I. Bolotin, W. Kalb, P. Kim, I. Kymissis, H.L. Stormer, T.F. Heinz, J. Hone, Performance of monolayer graphene nanomechanical resonators with electrical readout, Nat. Nanotechnol. 4 (2009) 861–867. [13] V. Singh, S. Sengupta, H.S. Solanki, R. Dhall, A. Allain, S. Dhara, P. Pant, M.M. Deshmukh, Probing thermal expansion of graphene and modal dispersion at low-temperature using graphene nanoelectromechanical systems resonators, Nanotechnology 21 (2010) 165204. [14] X. Song, M. Oksanen, M.A. Sillanpää, H.G. Craighead, J.M. Parpia, P.J. Hakonen, Stamp transferred suspended graphene mechanical resonators for radio frequency electrical readout, Nano Lett. 12 (2012) 198–202. [15] V. Singh, S.J. Bosman, B.H. Schneider, Y.M. Blanter, A. Castellanos-Gomez, G.A. Steele, Optomechanical coupling between a multilayer graphene mechanical resonator and a superconducting microwave cavity, Nat. Nanotechnol. 9 (2014) 820–824. [16] P. Weber, J. Güttinger, I. Tsioutsios, D.E. Chang, A. Bachtold, Coupling graphene mechanical resonators to superconducting microwave cavities, Nano Lett. 14 (2014) 2854–2860. [17] G. Luo, Z.-Z. Zhang, G.-W. Deng, H.-O. Li, G. Cao, M. Xiao, G.-C. Guo, L. Tian, G.-P. Guo, Strong indirect coupling between graphene-based mechanical resonators via a phonon cavity, Nat. Commun. 9 (2018) 383. [18] R. Yang, Z. Wang, P.X.-L. Feng, All-electrical readout of atomically-thin MoS2 nanoelectromechanical resonators in the VHF band, in: 2016 IEEE 29th International Conference on Micro Electro Mechanical Systems, MEMS, 2016. [19] A. Castellanos-Gomez, R. van Leeuwen, M. Buscema, H.S.J. van der Zant, G.A. Steele, W.J. Venstra, Single-layer MoS2 mechanical resonators, Adv. Mater. 25 (2013) 6719–6723. [20] J. Lee, Z. Wang, K. He, J. Shan, P.X.-L. Feng, High frequency MoS2 nanomechanical resonators, ACS Nano 7 (2013) 6086–6091. [21] R. van Leeuwen, A. Castellanos-Gomez, G.A. Steele, H.S.J. van der Zant, W.J. Venstra, Time-domain response of atomically thin MoS2 nanomechanical resonators, Appl. Phys. Lett. 105 (2014) 041911. [22] C.-H. Liu, I.S. Kim, L.J. Lauhon, Optical control of mechanical mode-coupling within a MoS2 resonator in the strong-coupling regime, Nano Lett. 16 (2016) 2768–2773. [23] N. Morell, A. Reserbat-Plantey, I. Tsioutsios, K.G. Schädler, F. Dubin, F.H.L. Koppens, A. Bachtold, High quality factor mechanical resonators based on WSe2 monolayers, Nano Lett. 16 (2016) 5102–5108. [24] J. Lee, Z.H. Wang, K.L. He, R. Yang, J. Shan, P.X.-L. Feng, Electrically tunable single- and few-layer MoS2 nanoelectromechanical systems with broad dynamic range, Sci. Adv. 4 (2018) eaao6653. [25] K.F. Mak, C. Lee, J. Hone, J. Shan, T.F. Heinz, Atomically thin MoS2 : a new directgap semiconductor, Phys. Rev. Lett. 105 (2010) 136805. [26] A. Splendiani, L. Sun, Y. Zhang, T. Li, J. Kim, C.-Y. Chim, G. Galli, F. Wang, Emerging photoluminescence in monolayer MoS2 , Nano Lett. 10 (2010) 1271–1275. [27] M. Amani, D.-H. Lien, D. Kiriya, J. Xiao, A. Azcatl, J. Noh, S.R. Madhvapathy, R. Addou, S. KC, M. Dubey, K. Cho, R.M. Wallace, S.-C. Lee, J.-H. He, J.W. Ager III, X. Zhang, E. Yablonovitch, A. Javey, Near-unity photoluminescence quantum yield in MoS2 , Science 350 (2015) 1065–1068. [28] J.D. Thompson, B.M. Zwickl, A.M. Jayich, Florian Marquardt, S.M. Girvin, J.G.E. Harris, Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane, Nature 452 (2008) 72–75. [29] I. Favero, K. Karrai, Cavity cooling of a nanomechanical resonator by light scattering, New J. Phys. 10 (2008) 095006. [30] S. Zaitsev, A.K. Pandey, O. Shtempluck, E. Buks, Forced and self-excited oscillations of an optomechanical cavity, Phys. Rev. E 84 (2011) 046605. [31] S. Stapfner, L. Ost, D. Hunger, J. Reichel, I. Favero, E.M. Weig, Cavity-enhanced optical detection of carbon nanotube Brownian motion, Appl. Phys. Lett. 102 (2013) 151910. [32] T.P. Purdy, R.W. Peterson, C.A. Regal, Observation of radiation pressure shot noise on a macroscopic object, Science 339 (2013) 801–804. [33] C. Schwarz, B. Pigeau, L. Mercier de Lépinay, A.G. Kuhn, D. Kalita, N. Bendiab, L. Marty, V. Bouchiat, O. Arcizet, Deviation from the normal mode expansion in a coupled graphene-nanomechanical system, Phys. Rev. Appl. 6 (2016) 064021. [34] M. Aspelmeyer, T.J. Kippenberg, F. Marquardt, Cavity optomechanics, Rev. Mod. Phys. 86 (2014) 1391. [35] S.J. Orfanidis, Electromagnetic Waves and Antennas, Prentice-Hall, 2006. Also available in online form on Prof. Orfanidis’ webpage www.ece.rutgers.edu/ ~orfanidi/ewa. [36] S. Roddaro, P. Pingue, V. Piazza, V. Pellegrini, F. Beltram, The optical visibility of graphene: interference colors of ultrathin graphite on SiO2 , Nano Lett. 7 (2007) 2707–2710. [37] P. Blake, E.W. Hill, A.H. Castro Neto, K.S. Novoselov, D. Jiang, R. Yang, T.J. Booth, A.K. Geim, Making graphene visible, Appl. Phys. Lett. 91 (2007) 063124.

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W. Mao et al. / Physics Letters A 383 (2019) 2755–2760

[38] Z. Wang, P.X.-L. Feng, Interferometric motion detection in atomic layer 2D nanostructures: visualizing signal transduction efficiency and optimization pathways, Sci. Rep. 6 (2016) 28923. [39] T. Stauber, N.M.R. Peres, A.K. Geim, Optical conductivity of graphene in the visible region of the spectrum, Phys. Rev. B 78 (2008) 085432.

[40] Y. Li, A. Chernikov, X. Zhang, A. Rigosi, H.M. Hill, A.M. van der Zande, D.A. Chenet, E.-M. Shih, J. Hone, T.F. Heinz, Measurement of the optical dielectric function of monolayer transition-metal dichalcogenides: MoS2 , MoSe2 , WS2 , and WSe2 , Phys. Rev. B 90 (2014) 205422.