1. Introduction

Surface plasmon polaritons (SPPs) are electromagnetic excitations propagating at the interface between a dielectric and a metal, which are formed by coupling of light and electron plasma in the metal. The electromagnetic field of SPPs decays exponentially with the distance away from the surface, it means that the electromagnetic field is confined to the near vicinity of the interface. SPPs can be applied in bio-sensors [1], subwavelength optics, data storage, light generation [2] and nano imaging [3]. Graphene, as a single sheet of graphite, has attracted great attention for its outstanding properties recently. Graphene can surport SPPs in both infrared (IR) and terahertz (THz) frequencies [4,5 ]. Compared with surface plasmons in metal, surface plasmons in graphene have many advantages, such as, the longer propagation length of SPP, the better confinement of the electromagnetic field, and the tunable properties of graphene SPPs by changing the gate voltage [6]. Graphene SPPs can be widely used in optical devices, such as, waveguide [6], terahertz laser [7] and antenna [8], and optical modulator [9]. Furthermore, the optical nonlinearity can be enhanced for the confinement of electromagnetic field near the surface, and more and more researchers have turned their attentions to the nonlinear properties of SPPs [10]. Mihalache et al. showed the dispersion relation at a metal/Kerr-type nonlinear interface with first integral method [11]. Rukhlenko et al. studied the dispersion relation of nonlinear surface plasmon waveguide [12,13 ]. Wurtz et al. found the optial bistability in nonlinear surface plasmon crystal [14]. Wang et al. investigated the influence of nonlinearity of the substrate on the property of the graphene surface plasmons [15]. In [16], TE-polarized surface polaritons along the surface of a nonlinear dielectric medium covered by a graphene layer are studied. Dai et al. studied the optical bistability at terahertz frequencies with graphene surface plasmons based on the nonlinearity of the graphene itself [17].

Surface phonon polaritons (SPhPs) are electromagnetic excitations propagating along the interfaces of polar dielectrics, which originate from the coupling of light and phonon in the polar dielectrics. Compared with SPPs in noble metal, SPhPs work in the IR and THz bands and have less loss, which are good candidates for application in optical nanometer device in the IR and THz bands, such as microscopy [18], thermal emission [19] and data storage [20]. The hexagonal boron nitride (hBN), with natural hyperbolicty, has become a research hotspot recently. Researches showed that SPhPs in hBN had higher quality factor compared with graphene SPPs [21], and it can now be used as substrate with graphene for enhancing mobility [22]. Nevertheless, there are few reports on the nonlinear properties of the SPhPs.

The hybridization of graphene plasmons and the hBN phonons have been studied in [23,24 ]. Yan et al. discussed the phonon-induced transparency in bilayer graphene nanoribbons [25]. Brar et al. [24] investigated the coupling between SPPs in graphene and SPhPs in hBN with patterned graphene on monolayer hBN. Kumar [26] showed the coupling between the graphene SPPs and the SPhPs in two Reststrahlen bands. As far as we now know, the investigation of hybridization of graphene SP³ is limited in the linear area. For the high confinement of the electromagnetic field near the interface, the nonlinearity of the substrate should not been ignored in the procedure. In this paper, we study the hybrid nonlinear SP³ dispersion relations in the structure of graphene-covered interface of nolinear media and hBN, and this will be helpful for the application of graphene-hBN heterostructure in reality.

2. Theory

The system studied in the paper is shown in Fig. 1 , which includes a graphene separating semi-infinite nonlinear material and hBN crystal occupying the space of z>0 and z<0 respectively. The permittivity of the nonlinear material can be written as follows:

 

Fig. 1 The system of a graphene separating semi-infinite nonlienear material and hBN crystal.

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The conductivity of the graphene consists of intraband and interband contributions,

hBN is a kind of anisotropic crystal, the principal dielectric tensor components of hBN can be expressed as

In the structure in Fig. 1, there are only three non-zero fields E_x, E_z and H_y for TM mode having the component $\exp (i\beta x)\cdot \exp (-i\omega t)$. From Maxwell’s equations we get the following equations

Inserting Eq. (6) into Eq. (1), we get

3 Numerical results and discussions

3.1 SP³ in the upper Reststrahlen band of hBN

For simplicity, we take the cladding as air as an example at first. Figures 2(a) and 2(b) show the dispersion relations without and with graphene separately. The parameters of the graphene are chosen as follows: the Fermi energy is 0.4eV, relaxation time is 50fs, and T is 300K. The dispersion relations in Fig. 2(b) can be obtained from Eq. (18). The electromagnetic field should decay with distance z from the surface, so we can obtain the relation ${\beta }^2{\epsilon }_{\perp }/{\epsilon }_{\parallel }-k_0^2{\epsilon }_{\perp }>0$ from Eq. (11) in space z<0, and${\beta }^2-k_0^2>0$ in space z>0. Because of ${\epsilon }_{\perp }<0$ in the upper Reststrahlen band of hBN, the surface phonon only exists in the region between the green and blue dashed line in Fig. 2(a). There is no surface phonon exist in the region above ${\omega }_{LO}$ or below ${\omega }_{TO}$ [29]. But when a graphene placed at the interface, one can see that SPPs exist in the region above ${\omega }_{LO}$ or below ${\omega }_{TO}$ in Fig. 2(b). This is because that the permittivity of grapehne is negative and can support SPPs. Furthermore, we can see that $\mathrm{Re}(\beta )$ is much smaller in the upper Reststrahlen band of hBN than that outside the band in Fig. 2(b), so it can be said that graphene has less influence on the property of surface phonon in the upper Reststrahlen band of hBN.

 

Fig. 2 Dispersion relations in the upper Reststrahlen band of hBN without graphene (a) and with graphene (b). The green line is the light line in air $\beta =\omega /c$, and the blue line is the light line in hBN $\beta ={({\epsilon }_{\parallel })}^{1/2}\omega /c$.

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Further, the cladding air is replaced by the nonlinear material, the parameters are chosen as follows: $\alpha =\pm 6.4\times {10}^{-12}{m}^2{V}^{-2}$, ${\epsilon }_l=2.405$ [30]. We get the dispersion in Fig. 3(a) by using Eqs. (14) and (17) with $\Delta \epsilon =0.5$ separately. We find that the results from two methods are very close, the difference between them is caused by the approximation of Eq. (15), so the methods are validated. Two modes can be found outside the upper Reststrahlen band with s = 1 and s = −1 in Eq. (17), corresponding to high frequency mode and low frequency mode respectively. In the two modes, the high frequency mode is the one corresponding to the higher frequency with the same $\beta$, while the low frequency mode corresponding to the lower frequency. And there is only low frequency mode in the upper Reststrahlen band. The spatial shape of the magnetic fields at frequency 1700$c{m}^{-1}$ perpendicular to the interface of the two media are shown in Figs. 3(b)-3(d). When s = −1, we can get z₀<0 in Eq. (16), the maximum of the magnetic field is at the interface; and when s = 1, the maximun of the field is inside the nonlinear medium with z₀>0. It can also be found that the magnetic fields are discontinuous for the graphene at the interface.

 

Fig. 3 (a) Dispersion with two methods: red line obtained from Eq. (14), blue line obtained from Eq. (17). (b) Normalized magnetic field of low frequency mode. (c) Normalized magnetic field of high frequency mode. (d) is the enlarged view of position D in (c).

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Then the influence of the nonlinear part $\Delta \epsilon$ is investigated. It can been seen from Fig. 4 that there are two modes outside the Reststrahlen band when $\Delta \epsilon$ is not zero. $\mathrm{Re}(\beta )$ and $\mathrm{Im}(\beta )$ increase with $\Delta \epsilon$ for low frequency mode, while they decrease with $\Delta \epsilon$ for high frequency mode. The dispersion relations are also shown in Fig. 5 for clarity, and it is well known that $\mathrm{Im}(\beta )$ means the propagation loss of SP³. Because $\beta$ is much smaller in the upper Reststrahlen band of hBN than that outside the band, we draw the figure in two bands for clarity: wavenumber (frequency) above ${\omega }_{LO}$ and between ${\omega }_{LO}$ and ${\omega }_{TO}$. The dispersion below ${\omega }_{TO}$ is omitted for the similarity with that above ${\omega }_{LO}$. In Figs. 5(a) and 5(b) in the band between ${\omega }_{LO}$ and ${\omega }_{TO}$, $\mathrm{Re}(\beta )$ and $\mathrm{Im}(\beta )$ increase along with the frequency, and the value of $\mathrm{Im}(\beta )$ is very small in general which means that propagation length is very long. In special in the upper Reststrahlen band, $\mathrm{Im}(\beta )$ decreases monotonically with $\Delta \epsilon$ for $\Delta \epsilon >0$ and increases with the absolute value $\Delta \epsilon$ at first for $\Delta \epsilon <0$. The minimal value of $\mathrm{Re}(\beta )$and the maximal value of $\mathrm{Im}(\beta )$ can be obtained near the condition $\Delta \epsilon =-1$. It can be inferred from Figs. 5(a) and 5(b) that the better performance of SP³ can be obtained with higher nonlinear dielectric function ($\Delta \epsilon$) when $\Delta \epsilon >0$. In the band above ${\omega }_{LO}$, as shown in Figs. 5(c) and 5(d), it can be seen that $\mathrm{Re}(\beta )$ and $\mathrm{Im}(\beta )$of low frequency mode increase monotonically along with the frequency and $\Delta \epsilon$when $\Delta \epsilon >0$, while decrease with the absolute value $\Delta \epsilon$ at first, then increase when $\Delta \epsilon <0$. The minimal value can be obtained near the condition $\Delta \epsilon =-1$. However the dependence of $\mathrm{Re}(\beta )$ and $\mathrm{Im}(\beta )$ of high frequency mode on $\Delta \epsilon$ is opposite to those in low frequency mode as shown in Figs. 5(e) and 5(f). And it can also be found that there is no high frequency mode existing when $\Delta \epsilon$ = 0.

 

Fig. 4 The dependence of $\beta$ on wavenumber and $\Delta \epsilon$.

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Fig. 5 The dependence of $\beta$ on $\Delta \epsilon$ and wavenumber. (a), (c) and (e) represent $\mathrm{Re}(\beta )$, and (b), (d) and (f) represent$\mathrm{Im}(\beta )$. (c) and (d) represent low frequency mode, (e) and (f) represent high frequency mode.

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Next, we study the role of the graphene sheets in the structure. The conductivity of graphene can be tuned by means of chemical doping or gate voltage, so it is convenient to change the property of SP³. We will discuss three physical parameters of graphene, Fermi energy, layer number and relaxation time, and study how they affect the property of SP³. In the following discussion, three different optical frequencies: 1500$c{m}^{-1}$, 1200$c{m}^{-1}$ and 1800$c{m}^{-1}$ are chosen presenting the frequency in the upper Reststrahlen band, above ${\omega }_{LO}$, and below ${\omega }_{TO}$ separately.

Firstly, the influence of Fermi energy on SP³ is shown in Fig. 6 . In Figs. 6(a), 6(b), 6(e), and 6(f) in the band above ${\omega }_{LO}$ or below ${\omega }_{TO}$, we can see that the $\mathrm{Re}(\beta )$ and $\mathrm{Im}(\beta )$ of low frequency mode decrease monotonically with the increasing Fermi energy of graphene, but they decrease more and more slowly. In addition, it can be found that the change spans of $\mathrm{Re}(\beta )$ and $\mathrm{Im}(\beta )$ are much larger with lower Fermi energy, or we can say that the structure is more sensitive to $\Delta \epsilon$ with lower Fermi energy. From another point of view, we can find that the positive nonlinear dielectric can enlarge $\mathrm{Re}(\beta )$ and $\mathrm{Im}(\beta )$, which means the enhancing of localization and the decreasing of the propagation length of SP³. The properties of high frequency mode are similar with those of low frequency mode, except that their dependences of $\beta$ on $\Delta \epsilon$ are opposite. And it can be found that there is no high frequency mode existing when $\Delta \epsilon$ = 0. Another interesting trend is that the difference of $\beta$ between two modes become smaller as Fermi energy increases. The influence of the graphene on surface phonon in the band between ${\omega }_{LO}$ and ${\omega }_{TO}$ is not very significant due to the finite graphene surface conductivity in Figs. 6(c) and 6(d). The properties of the surface phonon are mainly controlled by hBN in the band. $\mathrm{Im}(\beta )$is very small in the band, which increases slightly with Fermi energy of graphene, so the propagaton length is very long.

 

Fig. 6 The dependence of $\beta$ on $\Delta \epsilon$ with the different E_F: (a) and (b) are at 1200$c{m}^{-1}$, (c) and (d) are at 1500$c{m}^{-1}$, and (e) and (f) are at 1800$c{m}^{-1}$. Solid lines are low frequency modes, dashed lines are high frequency modes.

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Secondly, the influence of layer number of graphene sheets on SP³ is shown in Fig. 7 . In Figs. 7(a), 7(b), 7(e), and 7(f), at the frequencies above ${\omega }_{LO}$ and below ${\omega }_{TO}$, as the layer number N increases, the difference of $\beta$ between two modes gets smaller, and both $\mathrm{Re}(\beta )$and $\mathrm{Im}(\beta )$ decrease for both two modes. Similarly, it can be found that the change spans of $\mathrm{Re}(\beta )$ and $\mathrm{Im}(\beta )$are much larger with less layer number, in another word, SP³ here is more sensitive to the $\Delta \epsilon$with less layer number. The influence of the graphene on surface phonon in the upper Reststrahlen band between ${\omega }_{LO}$ and ${\omega }_{TO}$ is shown in Figs. 7(c) and 7(d). $\mathrm{Re}(\beta )$keeps nearly invariant in this band, and $\mathrm{Im}(\beta )$ is very small. $\mathrm{Im}(\beta )$ increases with the layer number, which means that more sheets of graphene introduce a little additional loss with SP³.

 

Fig. 7 The dependence of $\beta$ on $\Delta \epsilon$ with the different layer number: (a) and (b) are at 1200$c{m}^{-1}$, (c) and (d) are at 1500$c{m}^{-1}$, (e) and (f) are at 1800$c{m}^{-1}$. Solid lines are low frequency modes, dashed lines are high frequency modes.

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Thirdly, we want to explain the influence of the relaxation time of graphene on SP³. We have shown the dependence of dispersion on $\Delta \epsilon$ with different parameter $\tau$, as shown in Fig. 8 . It is clear that $\mathrm{Im}(\beta )$ decreases for both modes and the difference of $\mathrm{Im}(\beta )$ between two modes gets smaller as relaxation time increases, while $\mathrm{Re}(\beta )$ remains nearly constant at three different frequencies.

 

Fig. 8 The dependence of $\beta$ on $\Delta \epsilon$ with the different relaxation time: (a) and (b) are at 1200$c{m}^{-1}$, (c) and (d) are at 1500$c{m}^{-1}$, (e) and (f) are at 1800$c{m}^{-1}$. Solid lines are low frequency modes, dashed lines are high frequency modes.

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3.2 SP³ in the lower Reststrahlen band of hBN

There is no surface phonon or plasmon in the lower Reststrahlen band of hBN without graphene, unlike those in the upper Reststrahlen band because of the relations ${\epsilon }_{\parallel }<0$ and ${\epsilon }_{\perp }>0$ in the lower band [31]. But when there is a graphene sheet at the interface between hBN and nonlinear medium, the SPPs can exist in the structure. The dispersion relation is shown in Fig. 9 , the parameters of nonlinear medium and graphene are the same as those in Sec. 3.1. Obviously there are two modes outside the lower Reststrahlen band. $\mathrm{Re}(\beta )$ and $\mathrm{Im}(\beta )$ increase with $\Delta \epsilon$ for low frequency mode, while decrease with $\Delta \epsilon$ for high frequency mode. And there is only low frequency mode in the lower Reststrahlen band.

 

Fig. 9 The dependence of $\beta$ on wavenumber and $\Delta \epsilon$.

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The dependence of $\beta$ on $\Delta \epsilon$ and wavenumber are also shown in Fig. 10 , in which the high frequency mode is not shown. It can been seen in Fig. 10(a) that $\mathrm{Re}(\beta )$ increases along with frequency below 780$c{m}^{-1}$, decreases at first and then increases above 780$c{m}^{-1}$. We can also find that $\mathrm{Re}(\beta )$ increases monotonically with $\Delta \epsilon$ when $\Delta \epsilon >0$,while it decreases with the absolute value $\Delta \epsilon$ at first, then increases when $\Delta \epsilon <0$ in the lower Reststrahlen band. The minimal value can be obtained near $\Delta \epsilon =-1$. $\mathrm{Im}(\beta )$ increases along with frequency below 780$c{m}^{-1}$, decreases above 780$c{m}^{-1}$, and it is barely affected by $\Delta \epsilon$ in the lower Reststrahlen band.

 

Fig. 10 The dependence of $\beta$ on $\Delta \epsilon$ and wavenumber for low frequency mode, (a) represents $\mathrm{Re}(\beta )$, and (b) represents $\mathrm{Im}(\beta )$.

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Next, we discuss how the physical parameters of graphene, Fermi energy, layer number and relaxation time, affect the properties of SP³ with the frequency fixed at 700 $c{m}^{-1}$ and 800$c{m}^{-1}$ presenting the frequency below ${\omega }_{TO}$ and in the lower Reststrahlen band $\omega$ separately. SP³ in the frequency above ${\omega }_{LO}=830c{m}^{-1}$ has the similar properties with the frequency below${\omega }_{TO}=1370c{m}^{-1}$, which has been discussed in Sec. 3.1, so it is not mentioned here.

The influence of Fermi energy on nonlinear dispersion is shown in Fig. 11 . We can see that $\mathrm{Re}(\beta )$ and $\mathrm{Im}(\beta )$ of two modes decrease monotonically along with the Fermi energy of graphene, while they decrease more and more slowly, and the difference of $\beta$ between two modes become smaller as Fermi energy increases. The influence of $\Delta \epsilon$ on $\beta$ is opposite for the two modes at frequency blow ${\omega }_{TO}$. The variation range of $\mathrm{Re}(\beta )$ of two modes on nonlinear delectric permittivity in the lower Reststrahlen band is larger than below ${\omega }_{TO}$, in other word $\mathrm{Re}(\beta )$ is more sensitive to $\Delta \epsilon$ in the lower Reststrahlen band than below ${\omega }_{TO}$. The nonlinear delectric permittivity $\Delta \epsilon$ has nearly no effect on $\mathrm{Im}(\beta )$ in the lower Reststrahlen band. The layer number of graphene has the similar effects with Sec. 3.1 on the properties of SP³ for two modes below ${\omega }_{TO}$, as shown in Fig. 12 . $\mathrm{Re}(\beta )$ and $\mathrm{Im}(\beta )$ decrease monotonically along with the layer number of graphene, while $\mathrm{Im}(\beta )$ keeps nearly constant with different $\Delta \epsilon$ in the lower Reststrahlen band, which is different with Sec. 3.1.

 

Fig. 11 The dependence of $\beta$ on $\Delta \epsilon$ with the different E_F: (a) and (b) are at 700$c{m}^{-1}$, and (c) and (d) are at 800$c{m}^{-1}$. Solid lines are low frequency modes, dashed lines are high frequency modes.

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Fig. 12 The dependence of $\beta$ on $\Delta \epsilon$ with the different layer number: (a) and (b) are at 700$c{m}^{-1}$, and (c) and (d) are at 800$c{m}^{-1}$. Solid lines are low frequency modes, dashed lines are high frequency modes.

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The influence of relaxation time of graphene on the properties of SP³ is shown in Fig. 13 . $\mathrm{Im}(\beta )$ decreases monotonically along with the relaxation time of graphene, while $\mathrm{Re}(\beta )$ keeps nearly constant with different relaxation time at 700$c{m}^{-1}$for two mode. And $\mathrm{Re}(\beta )$ decreases with the relaxation time of graphene at 800$c{m}^{-1}$, so it can be seen that better perfermance of SP³ can be obtained with larger relaxation time in the lower Reststrahlen band of hBN.

 

Fig. 13 The dependence of $\beta$ on $\Delta \epsilon$ with the different relaxation time: (a) and (b) are at 700$c{m}^{-1}$, and (c) and (d) are at 800$c{m}^{-1}$. Solid lines are low frequency modes, dashed lines are high frequency modes.

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4 Conclusion

In this work, we have investigated theoretically the properties of hybrid surface-phonon-plasmon-polaritons (SP³) in the structrure of graphene lying between semi-infinite nonlienear material and hBN. At the frequencies outside the upper and lower Reststrahlen band of hBN, it is found that there are two modes when the nonlinear dielectric function is not zero, and the real and imaginary parts of the propagtion constant increase and the nonlinear dielectric permittivity has greater effect on the propagation constant by decreasing the Fermi energy and layer number of graphene, and the imaginary part of the propagtion constant decreases with the relaxation time of graphene while the real part keeps nearly constant with it. At the same time, there is only low frequency mode of SP³ existing in the upper Reststrahlen band of hBN, and the real and imaginary part of the propagation constant of the SP³ are much small. In the lower Reststrahlen band, there is only one mode of SP³, and the real part of the propagation constant increases with positive nonlinear dielectric permittivity while the imaginary part is nearly not affected by the nonlinear dielectric function, however the real part increases with the relaxation time of graphene.