Negative refraction in molybdenum disulfide

Wenhui Wang; Xudong Cui; Erchan Yang; Quanping Fan; Bin Xiang

doi:10.1364/OE.23.022024

1. Introduction

Negative refraction has attracted considerable interest because of its potential applications in realizing a “superlense” with a subwavelength resolution and observing a reversal of the Doppler shift and Vavilov-Cerenkov radiation [1]. Photonic crystals have been artificially fabricated and employed to meet the requirement of negative permittivity and permeability for the realization of the negative refraction phenomenon. In photonic crystal, the direction of the averaged Poynting vector is generally determined by group velocity while the wave vector direction is determined by Snell's law. Negative group index can be realized through a special dispersion relation of the band-gap edge, leading to a negative refraction in the photonic crystals [2]. However, the complex structural features and complicated nanofabrication processes hinder the photonic crystals in the applications of superlense [3], antenna technique [4], invisible weapons [5] as well as ultra-sensitive detection [6]. Recent research indicates that indefinite anisotropic materials with optical axes tilted to the interface can, for some directions of incidence, negatively refract the incoming wave. Negative refraction and total transmission refraction have been demonstrated at a unique type of interface in uniaxial crystals with twinning structures [7]. P. A. Belov et al. have showed the possibilities of negative refraction without backward waves by employing homogeneous transverse magnetic (TM) plane waves [8]. Liu et al. have calculated the angle size between the incident electromagnetic waves and the normal of the surface to achieve negative refraction in a uniaxial crystal [9]. It has been reported that the maximum incident angle and the bending negative refractive angle are only dependent on an anisotropy parameter [10]. The current uniaxial crystals utilized for negative refraction are generally with a small anisotropy parameter. Therefore, it results in the negative refraction occurring only in a small range of incident angles, limiting its applications.

MoS₂ as the typical representative of two-dimensional layered materials has attracted considerable attentions due to their distinctive electronic [11–13] and optical [14–16] properties. Its monolayer structure consists of a layer Mo atoms sandwiched by two layers of S atoms. The bulk MoS₂ is formed by stacking the sandwiched monolayer structures with weak interlayer van der Waals forces [17]. The strong interaction effect only exists in the basal plane of covalent bonds. As a result, bulk MoS₂ exhibits strong anisotropic behavior. In this paper, we, for the first time, report negative refraction in the bulk MoS₂ in a wide range of incident angles. Using the first-principles, we demonstrate the dielectric functions of bulk MoS₂. Conditions for negative refraction in the interface between isotropic medium and anisotropic crystal have also been obtained. We confirm the negative refraction in bulk MoS₂ by using the Finite-difference time-domain (FDTD) simulations and systematically examine the transmission coefficient of negatively refracted incident beam in the MoS₂ slabs with different thickness.

2. Methods and simulation model

All the first-principle calculations are carried out with projector augmented wave potentials implemented in the Vienna ab initio Simulation Package (VASP) [18,19]. To optimize the structure of the bulk MoS₂ (Fig. 1(a)), a well-converged Monkhorst-Pack k–point set (12 × 12 × 3) is utilized for the calculations. The exchange correlation contributions are treated with Perdew-Burke-Ernzerhof (PBE) function [20]. The plane wave cutoff energy of 500 eV is employed and a conjugate gradient scheme is used to optimize the geometries until the force on every atom is less than 1 meV/Å. However, as a ground-state theory, DFT has certain shortcomings in providing a quantitative description of optical and electronic excitations [21]. In order to obtain more accurate dielectric function and optical spectra in our calculations, we consider the effects of the electron-hole (e-h) interactions by employing a theory of single-particle transition. The electron-electron (e-e) repulsion is included by using the Green’s function (GW) [21] methods while both e-e and e-h interactions are taken into account with the GW-Bethe-Salpeter equation [22] (GW-BSE) formalism. We rely on the GW procedure together with the solution of the Bethe-Salpeter equation in the Tamm-Dancoff approximation. Strong excitonic effects [23,24] in MoS₂ have been taken into account, influencing the optical properties of the MoS₂. In addition, spin-orbit interaction is considered in our calculations to remove the degeneracy at the maximum of the valence band [25,26]. We started to calculate the optical spectra from the Kohn-Sham spinor wave functions, and the energies were calculated with the density-functional theory (DFT) in the general gradient approximation (GGA). The 360-unoccupied bands were used to ensure the accuracy of the dielectric function in the optical calculations. Through the Kohm-Sham wave functions and the quasiparticle energies, the optical-spectra were calculated on the level of the BSE [21]:

Fig. 1 (a) The schematic illustration of the bulk MoS₂ structure. (b) The calculated imaginary parts of MoS₂ permittivity $ϵ_{2} (E)$ along x (black solid), y (red dash) and z (blue solid) axis as a function of incident photon energy. (c) The calculated real parts of MoS₂ permittivity $ϵ_{1} (E)$ along x (black solid), y (red dash) and z (blue solid) axis as a function of incident photon energy. (d) The calculated frequency-dependent refractive indexes n along two principal dielectric axes. Red solid line represents the component parallel to the c axis and black solid line represents the component perpendicular to the c axis.

Download Full Size | PDF

(E_{c k} - E_{v k}) A_{v c k}^{S} + \sum_{k^{'}} {_{v^{'}}}_{c^{'}} 〈 v c k | K_{e h} | k^{'} v^{'} c^{'} 〉 A_{v^{'} c^{'} k^{'}}^{S} = Ω^{S} A_{v c k}^{S},

The electronic excitations can be expressed in a basis of electron-hole pairs at a given K point, from a state in the valence band with quasiparticle energy $E_{vk}$ toconduction-band state with energy $E_{ck}$ . The $A_{vck}^{s}$ is the expansion coefficients of the excitons in the electron-hole basis and the $Ω^{s}$ is the eigenenergy corresponding to the possible excitation energy of the system. The interaction kernel $K_{eh}$ describes the screened Coulomb interaction between electrons and holes, and the exchange interaction from a so-called local field effect. Under an illumination of an external electromagnetic field, the optical responses are generally governed by the Maxwell equations. The dielectric function, $ε (E) = ε_{1} (E) + {i*ε}_{2} (E)$ , bridges the electric displacements with the electric field in the constitutive relations of the Maxwell equations. Where $ε_{1} (E)$ is the real part, $ε_{2} (E)$ is the imaginary part, E is the photon energy. In the framework of BSE method, the imaginary part $ε_{2} (E)$ of the dielectric function can be obtained theoretically from the calculations of momentum matrix elements between the occupied and unoccupied states.

ε_{2} (ℏ ω) \propto \underset{S}{Σ} | \underset{c v k}{Σ} A_{v c k}^{S} \frac{〈 c k | p_{i} | v k 〉}{ξ_{c k} - ξ_{v k}} | δ (Ω^{S} - ℏ ω - Γ),

where $〈 ck|p|vk 〉$ are the dipole matrix elements for electronic transitions from valence-band to conduction-band states. The broadening energy is expressed as $Γ$ and P is the dipole polarization. The real part $ϵ_{1} (ℏ ω)$ can be obtained from the imaginary part $ϵ_{2} (ℏ ω)$ according to the Kramer-Kronig relation [27].

To confirm the negative refraction in bulk MoS₂ crystals, the propagation of the electromagnetic wave was calculated by the FDTD method. The FDTD algorithm, which is direct solution of Maxwell’s time dependent curl equations, was firstly formulated by Yee [28] and further developed by Taflove and others [29,30]. Combing with the frequency-dependent dielectric function calculated by GW-BSE approach, we employed two dimensional (2D) FDTD to simulate the electromagnetic wave propagating from an isotropic medium to anisotropic medium MoS₂. In our FDTD simulations, the boundary conditions were set with perfectly matched layer (PML) [31]. The dielectric function utilized for bulk MoS₂ is based on our calculations. The computation domain size is 9.6 µm *12 µm. The width and the height of the middle layer (size of MoS₂, Calcite and YVO₄ crystals) along x and z axis is 2 and 20 µm, respectively. We set Yee’s cells with a mesh size of dx = dz = 0.0005µm as well as a Gaussian form pulse centered at 1.9 eV with an incidence angle of 45°. To study the efficiency of the negative refracted light in the bulk MoS₂ for the further application in novel optical device, we also calculated the transmittance of negative refraction in the MoS₂ slabs with different thickness.

3. Results and discussions

The optical property of the material is mainly dominated by $ε (E)$ the dielectric function. To investigate the optical properties of the bulk MoS₂, we calculated the dielectric function of the bulk MoS₂ by using the GW-BSE approaches. The components of the imaginary part of dielectric function values $ε_{2} (E)$ along all three directions of x, y and z in bulk MoS₂ crystal are demonstrated in the Fig. 1(b). The real part $ε_{1} (E)$ (Fig. 1(c)) was calculated from the imaginary part of the dielectric function by the Kramers-Kroning equations [27]. There is a huge hexagonal anisotropy existing between x (y) component and z-component for the incident photon energy from 1.3 to 3.8 eV, covering most of visible wavelength region. In brief, we found that in the bulk MoS₂ crystals, $ε_{xx} = ε_{yy} \neq ε_{zz}$ , where $ε_{xx} {,ε}_{yy}$ and $ε_{zz}$ are the diagonal elements of the dielectric matrix $ε_{ij}$ . It indicates that the bulk MoS₂ crystals exhibit a behavior of a uniaxial crystal. Here, we define the different components of the dielectric function perpendicular and parallel to the c axis by $ε^{⊥} (E) = \frac{ε {(E)}_{xx} + ε {(E)}_{yy}}{2}$ and $ε^{∥} = ε {(E)}_{zz}$ , respectively. Then the refraction index can be obtained from the frequency dependent dielectric function $ε (E)$ through the GW-BSE approaches by a function of $n = \sqrt{ε_{1} + {iε}_{2}}$ . Figure 1(d) presents the refractive index components as a function of the incident photon energy in the bulk MoS₂. To demonstrate the anisotropic property in the material, we define an anisotropy parameter as $γ = \frac{n_{o}}{n_{e}}$ , where $n_{o} = \sqrt{ε^{⊥} (E)}$ and $n_{e} = \sqrt{ε^{∥} (E)}$ are ordinary and extraordinary refractive indexes, respectively. In the bulk MoS₂, the calculated value of the γ is larger than 2.5 in the entire range of visible wavelength, much higher than the previous reported value in the traditional uniaxial crystals [10]. The large difference in the refractive indexes between the components perpendicular and parallel to the c axis gives a rise to a pronounced anisotropy. As a result, it could induce a strong modulation in propagation of the electromagnetic wave in the bulk MoS₂.

To further evaluate the anisotropy-induced modulation of the incident electromagnetic wave, we have carried out an analysis of optical response to an external electromagnetic wave in the bulk MoS₂. In our coordinate system, all the wave vectors and the optical axis are in the x-z plane, where the interface is in the x-z plane at z = 0 as shown in Fig. 2. $θ_{i}$ is the incident angle and the angle of the optical axis $φ$ is the angle between the optical axis of anisotropic media and the z direction. In xoz coordinate system, $K_{tX}$ and $K_{tZ}$ are the components of wave vectors of the transmitted light in the direction of x and z axis. XOZ is the coordinate system of optical axis, where $K_{tX}$ and $K_{tZ}$ are the components of wave vectors of the transmitted light in the direction of X and Z axis. The ordinary wave travels in the same propagating direction while the different travelling direction occurs for the extraordinary wave. Therefore we focus on the extraordinary wave effect in the following discussion. The extraordinary wave number vector can be written as:

{\begin{cases} k_{t X} = k_{t z} \cos φ - k_{t x} \sin φ \\ k_{t Z} = - k_{t z} \sin φ - k_{t x} \cos φ \end{cases}

The dispersion relation of the extraordinary wave is:

\frac{{(k_{t y} \cos φ - k_{t x} \sin φ)}^{2}}{n_{o}^{2}} + \frac{{(k_{t y} \sin φ + k_{t z} \cos φ)}^{2}}{n_{e}^{2}} = k_{0}^{2}

where

k_{0}^{2} = \frac{k_{ix}^{2} + k_{iz}^{2}}{n_{i}^{2}}

,

n_{i}^{}

is the refractive index of the isotropic medium on the left-handed in Fig. 2,

k_{ix}^{}

and

k_{iz}^{}

are the components of wave vectors of the incident light in the direction of x and z axis. The incident angle is calculated by the function of

\tan θ_{i} = \frac{k_{ix}}{k_{iz}}

and the refractive angle of the wave vector

θ_{γ}

has a relation of

\tan θ_{γ} = \frac{k_{tx}}{k_{tz}}

. Combining Eq. (3) and Eq. (4), we obtained the relation between incident angle

θ_{i}

and the refractive angle of the extraordinary wave vector

θ_{γ}

. Similarly, the relationship between incident angle

θ_{i}

and the refractive angle of the extraordinary time-averaged Poynting vector

β_{γ}

was also achieved. Due to the anisotropy of uniaxial crystal, Poynting vector

S_{t}

and wave vector

k_{t}

travel in different directions. For a lossless medium, the direction of the time-averaged Poynting vector, defining the direction of the propagation of light, is normal to the k surface at any point. In Fig. 2,

S_{tx} = - E_{tz} \times H_{ty}

and

S_{tz} = E_{tx} \times H_{ty}

are the components of Poynting vectors of the transmitted light in the direction of x and z axis; while

β_{γ}

the refractive angle of Poynting vector expresses as

\tan β_{γ} = \frac{S_{tx}}{S_{tz}}

. Since

\nabla \cdot \overset{⇀}{D} = 0

, we can get the relationship between the components of wave vector and the components of Electric field

E_{tX}

and

E_{tZ}

in the XOZ coordinate system:

n_{o}^{2} k_{t X} E_{t X} + n_{e}^{2} k_{t z} E_{t Z} = 0

Where

E_{tX}

and

E_{tZ}

are the components of Electric field on XOZ coordinate system. The relationships between

E_{tX}

,

E_{tZ}

and

E_{tX}

,

E_{tZ}

can be described as:

{\begin{cases} E_{t X} = - E_{t x} \sin φ - E_{t z} \cos φ \\ E_{t Z} = E_{t x} \cos φ - E_{t z} \sin φ \end{cases}

Combing Eq. (5) and Eq. (6), the relation between incident angle

θ_{i}

and the refractive angle of the extraordinary time-averaged Poynting vector

β_{γ}

can then be obtained. Based on the above analysis, the refractive angle

β_{γ}

for energy flow and transmitted wave angle

θ_{γ}

as a function of incident angle

θ_{i}

were plotted in Fig. 3(a). As demonstrated in Fig. 1(d), the value of

n_{e}

and

n_{o}

are 1.4 and 4.7, respectively at a photon energy of 1.8 eV. As long as the refractive angle

β_{γ}

and transmitted wave angle

θ_{γ}

are both positive or negative, the extraordinary wave exhibits positive refraction as shown in Fig. 3(a); while the refractive angle

β_{γ}

< 0 and transmitted wave vector angle

θ_{γ}

< 0, it leads to negative refraction. We attribute it to the large anisotropy parameterγexisting in the bulk MoS₂ crystal. The negative refraction phenomenon can be very distinct as arranging the optical axial angle and anisotropic parameters appropriately. For eachγvalue and anisotropic refractive index, there is a difference between the corresponding optimal optical axis angle and maximum incident critical angle. The critical incident angle

θ_{ic}

with the given direction of the optical axis can be solved under the condition of

S_{tx}

= 0,10

θ_{i c} = arc \sin (\frac{| n_{o}^{2} - n_{e}^{2} | \sin 2 φ}{2 n_{i} {(n_{o}^{2} \sin^{2} φ + n_{e}^{2} \cos^{2} φ)}^{1 / 2}})

Substituting γ into Eq. (7), the maximum critical incident angle

θ_{ic}

can be obtained as the optical axisφ is chosen by

φ^{o p t} = arc \cos (\sqrt{1 / (γ + 1)})

Therefore, the maximum incident angle

θ_{ic}^{max}

is:

Fig. 2 Schematic illustration for mechanism of negative refraction as an electromagnetic wave travelling from isotropic medium to uniaxial medium. It demonstrates the propagating path of time-averaged Poynting vectors Si, Sr and St together with the corresponding wave vectors Ki, K_γ and K_t.

Download Full Size | PDF

Fig. 3 (a) Refractive angle $β_{γ}$ for energy flow (black line) and transmitted wave vector angle $θ_{γ}$ (red line) as a function of incident angle $θ_{i}$ through a uniaxial crystal of MoS₂. (b) The dependence of the particular angle $θ_{i c}$ as a function of optical axis angle $φ$ in the Calcite (black), YVO₄ (red) and bulk MoS₂ material (blue).

Download Full Size | PDF

θ_{i c}^{\max} = arc \sin (\frac{1}{n_{i}} (n_{o} - n {}_{e}))

The maximum incident angle was calculated to be $θ_{ic}^{max}$ = 90°by the Eq. (9) to yield a negative refraction in the bulk MoS₂ crystal. Figure 3(b) illustrates that the negative refraction occurs in an angle range of 0°to 90°, corresponding to the whole third quadrant of the schematic diagram shown in the Fig. 2. Compared to the bulk MoS₂ ( $n_{o}$ = 4.7, $n_{e}$ = 1.4) with a γ value of 3.35 at a photo energy of 1.8 eV, we calculated the maximum incident angle of the Calcite ( $n_{o}$ = 1.658, $n_{e}$ = 1.486) and YVO₄ ( $n_{o}$ = 1.993, $n_{e}$ = 2.215)10 with γof 1.12 and 0.90, respectively as shown in Fig. 3(b). In contrast with the maximum incident angles of 9.9° in Calcite and 12.9°in YVO₄ crystals, the larger anisotropy behavior in bulk MoS₂ crystal can yield negative refraction at up to a maximum incident angle of 90°. The range of incident angle triggering negative refraction in the bulk MoS₂ is much larger than the previous uniaxial crystals.

To confirm our calculations of the negative refraction in bulk MoS₂ crystal, we employed the 2D FDTD to simulate the electromagnetic wave propagating from an isotropic medium to such an anisotropic medium. The light intensity distribution in Fig. 4 illustrates that the refractive light stays at the same side as the incident light resulting from the negative refraction occurring in bulk MoS₂ crystal; while in Calcite and YVO₄ crystals, the refractive light is at different side of the incident side, indicating a positive refraction. Because most of light intensity is refracted, it is hard to demonstrate an obvious reflection phenomena at the interface. However, a simulation on an electromagnetic wave beam propagating from an isotropic medium to an anisotropic medium of bulk MoS₂ derived as a function of time reveals both the reflection and refraction behaviors at the interface (See Visualization 1). There are zigzag features observed in the light intensity distribution in the middle layers. We attribute it to the standing waves resulted from the reaction between the incidence light and reflected light. For the further application of negative refraction in novel optical device, it is necessary to investigate the utilization rate of the negative refracted light. To study the efficiency of the negative refracted light in bulk MoS₂, we calculated the transmittance of negative refraction in MoS₂ slabs with different thickness as shown in Fig. 5(a). The intensity of the transmitted light was collected at a distance of 3.4 µm along x axis to assure the electromagnetic field totally passing through the bulk MoS₂ crystal. With the increase of slab thickness, light intensity decreases. The incident light of ~59.5% is negatively refracted through a 0.1 µm thick slab as shown in our calculations. Figure 5(b) shows the negative refraction angle β_γ as a function of incident angle θ_i in MoS₂ with the slab thickness of 2 μm. It suggests that bulk MoS₂ materials could provide an efficient way to realize negative refraction and be integrated with future optoelectronic devices. Considering the similarity between optical and acoustic crystal, the anisotropic density of mass and Lam'e coefficients could be utilized to modulate propagation of acoustic Wave [32], possibly finding a more efficient way in sound manipulation.

Fig. 4 Light intensity distribution as an electromagnetic wave propagating from isotropic medium (n = 1) to anisotropic medium of bulk MoS₂, Calcite and YVO₄, respectively.

Download Full Size | PDF

Fig. 5 (a) Transmittance as a function of the slab thickness of bulk MoS₂ collected at a distance of 3.4 $μm$ along the x axis. (b) Negative refraction angle β_γ as a function of incident angle θ_i in MoS₂ with the slab thickness of 2 μm.

Download Full Size | PDF

4. Summary

In summary, the inherent anisotropic optical properties of the bulk MoS₂ were systematically studied by the first-principles, behaving as a uniaxial crystal with a large anisotropy parameter of γ. The region of incident angle inducing negative refraction has been achieved in our calculations, demonstrating the left-handed behavior in MoS₂ crystals with no necessary of a negative permittivity and permeability. The FDTD simulations indicated that the incident light of 59.5% can be negatively refracted in a 0.1µm thick slab. Layered MoS₂ provides an efficient way to realize negative refraction, leading to novel device concepts. It substantially extends the capabilities of MoS₂ photonics in the future.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (21373196, 11434009).

References and links

1. J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]

2. M. Notomi, “Negative refraction in photonic crystals,” Opt. Quantum Electron. 34(1-3), 133–143 (2002). [CrossRef]

3. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. Lett. 91(20), 207401 (2003). [CrossRef] [PubMed]

4. E. R. Brown, C. D. Parker, and E. Yablonovitch, “Radiation properties of a planar antenna on a photonic-crystal substrate,” J. Opt. Soc. Am. B 10(2), 404–407 (1993). [CrossRef]

5. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science 314(5801), 977–980 (2006). [CrossRef] [PubMed]

6. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32(1), 53–55 (2007). [CrossRef] [PubMed]

7. Y. Zhang, B. Fluegel, and A. Mascarenhas, “Total negative refraction in real crystals for ballistic electrons and light,” Phys. Rev. Lett. 91(15), 157404 (2003). [CrossRef] [PubMed]

8. P. A. Belov, “Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis,” Microw. Opt. Technol. Lett. 37(4), 259–263 (2003). [CrossRef]

9. Z. Liu, Z. F. Lin, and S. T. Chui, “Negative refraction and omnidirectional total transmission at a planar interface associated with a uniaxial medium,” Phys. Rev. B 69(11), 115402 (2004). [CrossRef]

10. H. L. Luo, W. Hu, X. N. Yi, H. Y. Liu, and J. Zhu, “Amphoteric refraction at the interface between isotropic and anisotropic media,” Opt. Commun. 254(4–6), 353–360 (2006).

11. H. P. Komsa and A. V. Krasheninnikov, “Effects of confinement and environment on the electronic structure and exciton binding energy of MoS 2 from first principles,” Phys. Rev. B 86(24), 233–243 (2012). [CrossRef]

12. E. S. Kadantsev and P. Hawrylak, “Electronic structure of a single MoS 2 monolayer,” Solid State Commun. 152(10), 909–913 (2012). [CrossRef]

13. K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz, “Atomically thin MoS60: a new direct-gap semiconductor,” Phys. Rev. B 105(13), 474–479 (2010).

14. H. Li, G. Lu, Z. Yin, Q. He, H. Li, Q. Zhang, and H. Zhang, “Optical identification of single- and few-layer MoS₂ sheets,” Small 8(5), 682–686 (2012). [CrossRef] [PubMed]

15. H. L. Shi, H. Pan, Y. W. Zhang, and B. I. Yakobson, “Quasiparticle band structures and optical properties of strained monolayer MoS2 and WS2,” Phys. Rev. B 87(15), 2095–2100 (2012).

16. D. Y. Qiu, F. H. da Jornada, and S. G. Louie, “Optical spectrum of MoS2: many-body effects and diversity of exciton states,” Phys. Rev. Lett. 111(21), 216805 (2013). [CrossRef] [PubMed]

17. S. W. Han, H. Kwon, S. K. Kim, S. Ryu, W. S. Yun, D. H. Kim, J. H. Hwang, J. S. Kang, J. Baik, H. J. Shin, and S. C. Hong, “Band-gap transition induced by interlayer van der Waals interaction in MoS₂,” Phys. Rev. B 84(4), 2593–2598 (2011). [CrossRef]

18. G. Kresse and J. Hafner, “Ab initio molecular dynamics for liquid metals,” Phys. Rev. B Condens. Matter 47(1), 558–561 (1993). [CrossRef] [PubMed]

19. G. Kresse and J. Furthmuller, “Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set,” Comput. Mater. Sci. 6(1), 15–50 (1996). [CrossRef]

20. J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approximation made simple,” Phys. Rev. Lett. 78(7), 1396 (1997). [CrossRef] [PubMed]

21. F. Aryasetiawan and O. Gunnarson, “The GW method,” Rep. Prog. Phys. 61(3), 237–312 (1998). [CrossRef]

22. C. Schwartz, “Solution of a Bethe-Salpeter equation,” Phys. Rev. 137(3B), 717–719 (1965). [CrossRef]

23. A. Ramasubramaniam, “Large excitonic effects in monolayers of molybdenum and tungsten dichalcogenides,” Phys. Rev. B 86(11), 2757–2764 (2012). [CrossRef]

24. S. Sim, J. Park, J. G. Song, C. In, Y. S. Lee, H. Kim, and H. Choi, “Exciton dynamics in atomically thin MoS2: Interexcitonic interaction and broadening kinetics,” Phys. Rev. B 88(7), 4521–4526 (2013). [CrossRef]

25. A. Molina-Sánchez, D. Sangalli, K. Hummer, A. Marini, and L. Wirtz, “Effect of spin-orbit interaction on the optical spectra of single-layer, double-layer, and bulk MoS₂,” Phys. Rev. B 88(4), 2358–2367 (2013). [CrossRef]

26. T. Cheiwchanchamnangij and W. R. Lambrecht, “Quasiparticle band structure calculation of monolayer, bilayer, and bulk MoS2,” Phys. Rev. B 85(20), 1449–1454 (2012). [CrossRef]

27. M. E. Orazem and B. Tribollet, “Electrochemical impedance spectroscopy,” Wiley &Sons. 44(24), 4345–4356 (2008).

28. K. S. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antennas and Propagation 14(3), 302–307 (1966). [CrossRef]

29. G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagnetic Compatibility 23(4), 377–382 (1981). [CrossRef]

30. A. Taflove, “Application of the finite-difference time-domain method to sinusoidal steady-state electromagnetic-penetration problems,” IEEE Trans. Electromagnetic Compatibility 22(3), 191–202 (1980). [CrossRef]

31. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994). [CrossRef]

32. L. Feng, X. P. Liu, M. H. Lu, Y. B. Chen, Y. F. Chen, Y. W. Mao, J. Zi, Y. Y. Zhu, S. N. Zhu, and N. B. Ming, “Acoustic backward-wave negative refractions in the second band of a sonic crystal,” Phys. Rev. Lett. 96(1), 014301 (2006). [CrossRef] [PubMed]

Negative refraction in molybdenum disulfide

Abstract

1. Introduction

2. Methods and simulation model

3. Results and discussions

4. Summary

Acknowledgments

References and links

Supplementary Material (1)

Cited By

Figures (5)

Equations (9)

Optics Express