## Abstract

Both plasmon-phonon-polariton (SPP-PHP) modes and phonon-polariton (PHP) modes supported in graphene-coated hexagon boron nitride (h-BN) single nanowire are presented. The field distributions of the lowest 5 order modes of SPP-PHP modes supported in graphene-coated hexagon boron nitride nanowire pairs (SPP-PHP-GHNP) and the lowest 5 order modes of PHP modes supported in graphene-coated hexagon boron nitride nanowire pairs (GHNP) are also demonstrated and analyzed, respectively. The results of numerical calculation show that SPP-PHP-GHNP mode 0 owns the strongest confinement and lowest loss among the lowest 5 order modes of SPP-PHP-GHNP. Furthermore, the field enhancement of SPP-PHP-GHNP mode 0 can reach over 10^{5} by controlling the geometry parameters of GHNP. Meanwhile, the influence of tuning the Fermi level of graphene on the field enhancement is also presented in the paper. The proposed structure may improve the development of graphene-h-BN-based optoelectronic devices.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In recent years, graphene-based structure has attracted numerous attention for its various applications including absorber [1], plasmonically induced transparency [2], isolator [3], filter [4], demultiplexing [5], switch [6], slow light [7], nanofocusing [8], Fano resonance [9], sensor [10], Kerr nonlinearity [11] and splitter [12]. Besides, many hexagon boron nitride (h-BN)-based devices have also been proposed in different fields such as quantum emitter [13], nanofocusing [14], optical imaging [15], hybrid waveguide [16] and absorption [17] in the past few years.

As we know, surface plasmon polariton (SPP) mode can be supported in graphene and phonon polariton (PHP) mode can be supported in h-BN [18,19]. Recently, researchers have demonstrated that hybrid SPP-PHP mode can be supported by graphene-h-BN system [20,21]. And a lot of graphene-h-BN hybrid structures have been proposed and discussed in different applications including absorption [22] and switch [23] in the past several years. Apart from that, Xu et al. reported an in-plane pressure sensor and a tunneling pressure sensor based on graphene-h-BN heterostructures [24]. Zhu et al. discussed a tapered graphene-h-BN hybrid structure and its application of nanofocusing [25]. Brar et al. experimentally investigated the hybrid surface-phonon-plasmon polariton modes in the graphene-h-BN sheets [26]. Barnard et al. proposed an infrared emitter consisting of graphene encapsulated h-BN [27]. Wu et al. discussed the nonlinear surface-phonon-plasmon-polaritons in the structure of a graphene sandwiched by a nonlinear material and h-BN [28].

In this paper, a graphene-coated h-BN nanowire pairs (GHNP) structure is firstly proposed. Both hybrid SPP-PHP modes and PHP modes can be supported in the proposed GHNP structure. The electric field distribution **|**E**|** and z-component of electric field E_{Z} for the SPP-PHP modes and PHP modes supported in the GHNP structure are illustrated by the numerical calculation. The dependencies of the real part of the effective index and ratio between the real part and imaginary part of the effective index for the SPP-PHP modes and PHP modes supported in the GHNP on the Fermi level of graphene, the distance between the two graphene-coated h-BN nanowire (GHN) waveguides, the radius of substrate and the thickness of h-BN are also presented, respectively. Furthermore, the field enhancement is defined and the relationships between the field enhancement and the four parameters above are also discussed.

## 2. Graphene-coated h-BN nanowire pairs

The scheme of the GHNP structure is shown in Fig. 1(a). The proposed structure could be implemented by the following steps. First, graphene flake could be prepared by chemical vapour deposition (CVD) and exfoliation methods [20]. h-BN film can be obtained by exfoliation techniques [14]. Then, graphene flake and h-BN film could be transferred and wrapped around the silica substrate by micromanipulation [29]. Finally, both graphene and h-BN could be spontaneously wrapped around the silica substrate and the proposed graphene-coated hexagon boron nitride nanowire waveguide could be achieved [29]. The thickness of h-BN is t_{h-BN}. In the simulation, h-BN is modelled as an anisotropic material with permittivities [21]:

*ε*and

_{a}*ε*are the out-of-plane (in radial direction) dielectric permittivity and in-plane (in azimuthal direction and z direction) dielectric permittivity of h-BN, respectively.

_{r}*ε*= 2.95,

_{∞,a}*ε*= 4.87,

_{∞,r}*ω*= 830 cm

_{LO,a}^{−1},

*ω*= 780 cm

_{TO,a}^{−1},

*ω*= 1610 cm

_{LO,r}^{−1},

*ω*= 1370 cm

_{TO,r}^{−1}, Γ

*= 4 cm*

_{a}^{−1}, Γ

*= 5 cm*

_{r}^{−1}. Figure 1(b) shows the dielectric permittivity of h-BN versus wavenumber in the range of 500 cm

^{−1}to 2000 cm

^{−1}. Here, wavenumber is defined as 1/λ, where λ is the wavelength. In this paper, we only discuss the upper Reststrahlen band where the in-plane dielectric permittivity of h-BN is negative and out-of-plane one is positive [21]. The conductivity of the graphene,

*σ*can be expressed as

_{g}*σ*

_{g}= σ_{inter}

*+ σ*

_{intra}, where

*σ*

_{inter}and

*σ*

_{intra}represent the interband part and intraband part, respectively [30]. And they can expressed as [30]:

*e*is the electron charge, T = 300 K is the environment temperature.

*τ*= 0.5 ps is the relaxation time.

*E*is the Fermi level of graphene.

_{f}*ω*and

*k*are the angular frequency and Boltzmann constant, respectively. The permittivity of graphene in radial direction can be expressed as

_{B}*ε*= 2.5. And the permittivities of graphene in azimuthal direction and z direction can be expressed as

_{g_r}*ε*2.5 +

_{g_ar}= ε_{g_z}=*iσ*/

_{g}*ωε*[31], where

_{0}t*ε*is the dielectric constant in vacuum.

_{0}*t*= 0.5 nm is the thickness of graphene. The dielectric permittivity of the silica substrate and air (the surrounding media) are assumed to be

*ε*3.9,

_{si}=*ε*1, respectively [32].

_{air}=## 3. Method and numerical analysis

For a graphene/h-BN film/silica substrate structure, the modal dispersion within the upper Reststrahlen band can be expressed as [21,25]:

*σ*is the conductivity of graphene,

_{g}*ε*and

_{si}*ε*are the dielectric permittivity of silica substrate and air (the surrounding media), respectively.

_{air}*k*

_{0}is the propagation constant in vacuum, Z

_{0}≈377Ω is the free space impedance [25]. t

_{h-BN}is the thickness of h-BN, L = 0, 1, 2, 3…, $\Psi =\sqrt{{\epsilon}_{a}}/\sqrt{{\epsilon}_{r}}/i,$ where

*ε*and

_{a}*ε*are the out-of-plane dielectric permittivity and in-plane dielectric permittivity of h-BN, respectively. And for the graphene-coated h-BN single nanowire structure, the modal dispersion can be approximately written as [33]:where

_{r}*R*

_{0}is the effective radius of the proposed graphene-coated h-BN single nanowire structure, and M = 0, 1, 2, 3…. In our work, the finite-element method (FEM) software COMSOL Multiphysics is used for the numerical calculation of the proposed structures. In the COMSOL FEM model, the mode analysis solver is used to finish the numerical calculation. Mode analysis is a two dimensional eigenvalue solver and it can compute the propagation constants at waveguide cross sections for a given wavenumber in electromagnetics. The size of surrounding media (air) is set as 10000 nm*10000 nm, the mesh size of graphene and h-BN are both extremely fine.

The electric field distribution **|**E**|** and the z-component of electric field E_{Z} for the lowest 3 order modes (M = 0, 1, 2) of the plasmon-phonon-polariton mode (L = 0) supported in the graphene-coated h-BN single nanowire (SPP-PHP-GHN) are shown in Fig. 2(a). Here, the wavenumber is set as 1450 cm^{−1}. Without special explication, the parameters of the GHN are set as follows: the radius of silica substrate R = 90 nm, the Fermi level of graphene E* _{f}* = 0.5 eV, the thickness of h-BN t

_{h-BN}= 10 nm. The real part of the effective index (Re(n

_{eff})) and ratio of real part to imaginary part of the effective index (Re(n

_{eff}) / Im(n

_{eff})) for the lowest 3 order modes of SPP-PHP-GHN (SPP-PHP-GHN mode 0, mode 1 and mode 2) are also presented in Figs. 3(a) and 3(b), respectively. Here, the effective refractive index n

_{eff}is defined as n

_{eff}=

*β*

_{z}/

*k*

_{0}, where

*β*

_{z}is the propagation constant in z direction and

*k*

_{0}is the propagation constant in vacuum. One can see that the both Re(n

_{eff}) and Re(n

_{eff}) / Im(n

_{eff}) for SPP-PHP-GHN mode 0, mode 1 and mode 2 increase as the wavenumber increases from 1450 to 1550 cm

^{−1}. Besides, we can find that Re(n

_{eff}) and Re(n

_{eff}) / Im(n

_{eff}) for SPP-PHP-GHN mode 2 owes the smallest Re(n

_{eff}) and Re(n

_{eff}) / Im(n

_{eff}) among these 3 order modes. Higher order phonon-polariton modes (L = 1, 2, 3…) are not affected by the plasmon-phonon-polariton coupling [21,25]. The electric field distribution

**|**E

**|**and z-component of electric field E

_{Z}for the lowest 3 order modes (M = 0, 1, 2) of the phonon-polariton modes (L = 1) supported in the graphene-coated h-BN nanowire (PHP-GHN) are shown in Fig. 2(b). Here, the wavenumber is set as 1450 cm

^{−1}. Moreover, Re (n

_{eff}) and Re (n

_{eff}) / Im (n

_{eff}) for the lowest 3 order modes of PHP-GHN (PHP-GHN mode 0, mode 1 and mode 2) are described in Figs. 3(c) and 3(d), respectively. Here we can observe that Re(n

_{eff}) for them increase with the wavenumber increasing from 1450 to 1550 cm

^{−1}while Re (n

_{eff}) / Im (n

_{eff}) for them reach a peak first and then drop gradually with rising the wavenumber from 1450 to 1550 cm

^{−1}. Apart from that, larger Re(n

_{eff}) and Re(n

_{eff}) / Im(n

_{eff}) reflects the stronger field confinement and lower propagation loss of the plasmon-phonon-polariton modes and phonon-polariton modes supported in the proposed waveguide, respectively [34,35]. As is shown in Figs. 3(c) and 3(d), Re (n

_{eff}) and Re(n

_{eff}) / Im(n

_{eff}) for PHP-GHN mode 0, mode 1 and mode 2 show a rapid upward trend and a rapid downward trend when the wavenumber increases to approach 1550 cm

^{−1}. This is because the Re (

*ε*) of h-BN shows a nonlinear change in this range of wavenumber. Consequently, stronger field confinement but higher propagation loss are achieved among these 3 order modes as wavenumber increases to approach 1550 cm

_{r}^{−1}. However, as is shown in Figs. 3(a) and 3(b), Re(n

_{eff}) and Re(n

_{eff}) / Im(n

_{eff}) for the lowest 3 order modes of SPP-PHP-GHN display a relatively slow upward trend as wavenumber increases. This is because that phonon-polariton modes couple to the plasmon-polariton modes. And both stronger field confinement and lower loss can be achieved among these 3 orders modes as wavenumber increases. For the plasmon polariton modes supported in graphene-coated silica substrate nanowire structure, the eigen equation can be derived from [36]. Especially, for the lowest order mode of plasmon polariton modes supported in graphene-coated silica substrate nanowire structure, the dispersion equation can be written as [36]:

*ε*

_{1}is the permittivity of silica substrate,

*ε*

_{2}is the permittivity of air (the surrounding media),

*σ*is the conductivity of the graphene. ${\mu}_{1}=\sqrt{{\beta}_{g}^{2}-{\omega}^{2}{\epsilon}_{1}{\mu}_{0}},$${\mu}_{2}=\sqrt{{\beta}_{g}^{2}-{\omega}^{2}{\epsilon}_{2}{\mu}_{0}},$

_{g}*I*(

_{m}*μ*

_{1}

*R*

_{0}) and

*K*(

_{m}*μ*

_{2}

*R*

_{0}) are the modified Bessel functions of the first and second kind (

*m*= 0, 1).

*R*

_{0}is the radius of the silica substrate. The effective refractive index for the lowest order mode of plasmon polariton modes supported in graphene-coated silica substrate nanowire structure can be calculated by

*β*

_{g}/

*k*

_{0}. Here,

*k*

_{0}is the propagation constant in vacuum. Then, the dependences of the effective refractive index for the lowest order mode of plasmon polariton modes supported in graphene-coated silica substrate nanowire structure on the wavenumber can be calculated by solving the Eq. (5). And the influences of plasmon-phonon-polariton coupling on the effective refractive index for the lowest 3 order modes of SPP-PHP-GHN versus wavenumber could be understood.

Furthermore, the numerical calculation results of the GHNP structure shown in Fig. 1(a) are presented. Without special explication, the parameters of the GHNP are set as follows: the radius of silica substrate R = 90 nm, the Fermi level of graphene E* _{f}* = 0.5 eV, the thickness of h-BN t

_{h-BN}= 10 nm, the distance between the two GHN waveguides D = 10 nm. Figures 4(a) and 4(b) illustrate the field distributions of the lowest 5 order modes of the plasmon-phonon-polariton modes supported in the graphene-coated h-BN nanowire pairs (SPP-PHP-GHNP mode 0, mode 1, mode 2, mode 3 and mode 4) and the lowest 5 order modes of the phonon-polariton modes supported in the graphene-coated h-BN nanowire pairs (PHP-GHNP mode 0, mode 1, mode 2, mode 3 and mode 4), respectively. Here, the wavenumber is set as 1450 cm

^{−1}. The orders of these modes discussed above can be explicated and derived from [34]. Besides, Re (n

_{eff}) and Re (n

_{eff}) / Im (n

_{eff}) for the lowest 5 order modes of SPP-PHP-GHNP and the lowest 5 order modes of PHP-GHNP are described in Figs. 5(a)-5(d), respectively. Figures 5(a) and 5(b) show that both Re (n

_{eff}) and Re (n

_{eff}) / Im (n

_{eff}) for the lowest 5 order modes of SPP-PHP-GHNP increase with rising the wavenumber from 1450 to 1550 cm

^{−1}. It is worth noting that SPP-PHP-GHNP mode 0 owns the largest Re (n

_{eff}) and Re (n

_{eff}) / Im (n

_{eff}) among these 5 order modes, which leads to a strongest confinement and lowest loss. And its performance of low loss is better than the results in [33,35]. As is shown in Figs. 5(c) and 5(d), Re (n

_{eff}) for the lowest 5 order modes of PHP-GHNP rise as the wavenumber increases from 1450 to 1550 cm

^{−1}while Re (n

_{eff}) / Im (n

_{eff}) for them firstly show a upward trend and then decline with increasing the wavenumber from 1450 to 1550 cm

^{−1}. Besides, as is shown in Figs. 5(c) and 5(d), Re(n

_{eff}) and Re(n

_{eff}) / Im(n

_{eff}) for lowest 5 order modes of PHP-GHNP display a quick upward trend and a rapid downward trend when the wavenumber increases to approach 1550 cm

^{−1}. Similarly, this is owing to the nonlinear change of Re (

*ε*) of h-BN in this range of wavenumber. And stronger field confinement but higher loss are achieved among these 5 order modes by increasing the wavenumber to approach 1550 cm

_{r}^{−1}. But as is shown in Figs. 5(a) and 5(b), Re(n

_{eff}) and Re(n

_{eff}) / Im(n

_{eff}) for lowest 5 order modes of SPP-PHP-GHNP increase relatively slowly with the increasing of wavenumber. This is also because of the plasmon-phonon-polariton coupling as discussed above. And both stronger field confinement and lower propagation loss can be realized among these 5 order modes with the increasing of wavenumber. Here, one can see that Re(n

_{eff}) and Re(n

_{eff}) / Im(n

_{eff}) for lowest 5 order modes of SPP-PHP-GHNP and PHP-GHNP show the similar trend compared with Re(n

_{eff}) and Re(n

_{eff}) / Im(n

_{eff}) for lowest 5 order modes of SPP-PHP-GHN and PHP-GHN as shown in Figs. 3(a)-3(d) when wavenumber increases from 1450 to 1550 cm

^{−1}.

Next, the dependencies of Re (n_{eff}) and Re (n_{eff}) / Im (n_{eff}) for the lowest 5 order modes of SPP-PHP-GHNP and PHP-GHNP on the Fermi level of graphene E* _{f}* are shown in Figs. 6(a)-6(d), respectively. Here, the wavenumber is set as 1500 cm

^{−1}. Figure 6(a) illustrates that Re (n

_{eff}) for the lowest 5 order modes of SPP-PHP-GHNP drop as E

*increases. However, Re (n*

_{f}_{eff}) / Im (n

_{eff}) for them increase with the rising of E

*as shown in Fig. 6(b). Here, it is obvious that SPP-PHP-GHNP mode 0 displays a larger Re (n*

_{f}_{eff}) and Re (n

_{eff}) / Im (n

_{eff}) than the other 4 order modes, which means that SPP-PHP-GHNP mode 0 remains the strongest confinement and lowest loss with the increasing of E

*. As is presented in Fig. 6(c), Re (n*

_{f}_{eff}) for the lowest 5 order modes of PHP-GHNP show a downward trend when E

*increases from 0.3 to 0.8 eV. Whereas Re (n*

_{f}_{eff}) / Im (n

_{eff}) increase and hit a peak as E

*increases from 0.3 to 0.4 eV and then decrease gradually with the increasing of E*

_{f}*from 0.4 to 0.8 eV as shown in Fig. 6(d). Moreover, as is shown in Figs. 6(a) and 6(b), the influences of E*

_{f}*on the Re(n*

_{f}_{eff}) and Re(n

_{eff}) / Im(n

_{eff}) for the lowest 5 order modes of SPP-PHP-GHNP are obvious owing to the coupling between the plasmon polariton modes and phonon-polariton modes. And we can find that lower propagation loss but weaker field confinement can be realized among these 5 order modes by increasing E

*. Besides, the relationship between E*

_{f}*and the effective index for the lowest order mode of plasmon polariton modes supported in graphene-coated silica substrate nanowire structure can also be derived by solving the Eq. (5). And the influences of plasmon-phonon-polariton coupling on the effective refractive index for the lowest 5 order modes of SPP-PHP-GHNP versus E*

_{f}*could be understood. However, as is illustrated in Figs. 6(c) and 6(d), the influences of E*

_{f}*on the Re(n*

_{f}_{eff}) and Re(n

_{eff}) / Im(n

_{eff}) for lowest 5 order modes of PHP-GHNP are not obvious compared with lowest 5 order modes of SPP-PHP-GHNP. This is because high order phonon-polariton modes are almost not influenced by the plasmon-phonon-polariton coupling.

Figures 7(a) and 7(b) illustrate that both Re (n_{eff}) and Re (n_{eff}) / Im (n_{eff}) for the lowest 5 order modes of SPP-PHP-GHNP display a upward trend when the radius of substrate R increases from 30 to 130 nm. The wavenumber is set as 1500 cm^{−1}. Here, one can see that the SPP-PHP-GHNP mode 0 shows the larger Re (n_{eff}) and Re (n_{eff}) / Im (n_{eff}) than the other 4 order modes, which indicates that the SPP-PHP-GHNP mode 0 has stronger confinement and lower loss than the other 4 order modes. Similarly, as is shown in Figs. 7(c) and 7(d), both Re (n_{eff}) and Re (n_{eff}) / Im (n_{eff}) for the lowest 5 order modes of PHP-GHNP increase with the increasing of R from 30 to 130 nm. Here, the wavenumber is set as 1500 cm^{−1}. One can find that PHP-GHNP mode 0 and mode 1 own the larger Re (n_{eff}) and Re (n_{eff}) / Im (n_{eff}) among these lowest 5 order modes.

The dependencies of Re (n_{eff}) and Re (n_{eff}) / Im (n_{eff}) for the lowest 5 order modes of SPP-PHP-GHNP and PHP-GHNP on the distance between the two GHN waveguides D are shown in Figs. 8(a)-8(d), respectively. Here, the wavenumber is set as 1500 cm^{−1}. In Figs. 8(a) and 8(b), we can see that the influences of D on Re (n_{eff}) and Re (n_{eff}) / Im (n_{eff}) for the SPP-PHP-GHNP mode 1, mode 2 and mode 4 are not obvious compared with SPP-PHP-GHNP mode 0 and mode 3. This is because the fields of SPP-PHP-GHNP mode 1, mode 2 and mode 4 are relatively far from the area between the two GHN waveguides as shown in Fig. 4(a) [34]. Similarly, one can find that the dependencies of Re (n_{eff}) and Re (n_{eff}) / Im (n_{eff}) for the PHP-GHNP mode 3 and mode 4 on D are not obvious compared with PHP-GHNP mode 0, mode 1 and mode 2 as shown in Figs. 8(c) and 8(d). This can also be explained by observing the field distributions of the lowest 5 order modes of PHP-GHNP shown in Fig. 4(b). In Fig. 4(b), we can find that the fields of PHP-GHNP mode 3 and mode 4 are more far from the area between the two GHN waveguides than PHP-GHNP mode 0, mode 1 and mode 2.

Figures 9(a)-9(d) illustrate Re (n_{eff}) and Re (n_{eff}) / Im (n_{eff}) for the lowest 5 order modes of SPP-PHP-GHNP and PHP-GHNP versus the thickness of h-BN t_{h-BN}. Here, the wavenumber is set as 1500 cm^{−1} and R is fixed at 90 nm. As is presented in Figs. 9(a) and 9(b), both Re (n_{eff}) and Re (n_{eff}) / Im (n_{eff}) for the lowest 5 order modes of SPP-PHP-GHNP decline as t_{h-BN} increases from 5 to 15 nm. It is obvious that SPP-PHP-GHNP mode 0 shows the larger Re (n_{eff}) (stronger confinement) and larger Re (n_{eff}) / Im (n_{eff}) (lower loss) among these 5 order modes. Besides, Re (n_{eff}) for the lowest 5 order modes of PHP-GHNP decrease with the rising of t_{h-BN} while Re (n_{eff}) / Im (n_{eff}) for them display a upward trend as t_{h-BN} increases from 5 to 15 nm as shown in Figs. 9(c) and 9(d).

Finally, the field enhancement of the SPP-PHP-GHNP mode 0 is demonstrated because of its strong confinement as discussed above. The field enhancement is defined by the ratio of electric field distribution **|**E**|** at the inner surface of graphene (x = D/2) to that at the outer surface of graphene (x = D/2 + 2R + 2t + 2t_{h-BN}) [37]. The dependencies of field enhancement of the SPP-PHP-GHNP mode 0 on the Fermi level of graphene E* _{f}*, the radius of substrate R, the distance between the two GHN waveguides D, and the thickness of h-BN t

_{h-BN}are shown in Figs. 10(a)-10(d), respectively. Here, the wavenumber is set as 1550 cm

^{−1}. A field enhancement, over 10

^{5}, can be realized by tuning E

*and R. Figure 10(a) shows that the field enhancement declines as E*

_{f}*increases from 0.3 to 0.8 eV. On the contrary, the field enhancement rises as R increases from 30 to 130 nm as shown in Fig. 10(b). Figure 10(c) illustrates that the field enhancement increases and reaches the peak when D increases from 4 to 6 nm and then decreases gradually as D increases from 6 to 14 nm. In Fig. 10(d), we can see that the influence of t*

_{f}_{h-BN}on the field enhancement is not obvious compared with the other three parameters.

## 4. Conclusion

In this paper, the graphene-coated h-BN nanowire pairs structure is proposed and demonstrated. Firstly, the lowest 3 order modes of SPP-PHP modes and PHP modes supported in the graphene-coated hexagon boron nitride single nanowire is discussed. Then, we demonstrated the dependencies of Re (n_{eff}) and Re (n_{eff}) / Im (n_{eff}) for the first 5 order modes of SPP-PHP-GHNP and PHP-GHNP on the Fermi level of graphene E* _{f}*, the distance between the two GHN waveguides D, the radius of substrate R, and the thickness of h-BN t

_{h-BN}. The results indicate that SPP-PHP-GHNP mode 0 remains the strongest confinement and lowest loss among the lowest 5 order modes of SPP-PHP-GHNP with the change of E

*, R, D or t*

_{f}_{h-BN}. Finally, we discussed the dependencies of field enhancement of SPP-PHP-GHNP mode 0 on E

*, D, R and t*

_{f}_{h-BN}. A field enhancement over 10

^{5}can be achieved by controlling E

*or changing R. Moreover, the results show that the field enhancement is less dependent on t*

_{f}_{h-BN}than the other three parameters discussed above. The proposed structure may have a promising application in graphene-h-BN-based optical devices.

## Funding

Fundamental Research Funds for the Central Universities (W17RC00020); China Postdoctoral Science Foundation (2018M631327).

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