The properties of hybrid nonlinear surface-phonon-plasmon-polaritons (SP3) at the interface of nonlinear medium and graphene-covered hexagonal boron nitride (hBN) are investigated theoretically. It is demonstrated that the hybrid nonlinear SP3 can be tuned by controlling the chemical potential, layer number and relaxation time of graphene. The real and imaginary parts of the propagation constant increase by decreasing the Fermi energy or the layer number of the graphene in the frequency outside of the upper Reststrahlen band of hBN. Moreover, we show that the nonlinear dielectric permittivity has great effect on the propagation constant. The real part of the propagation constant increases with positive nonlinear dielectric permittivity at different frequency for low frequency mode; while the imaginary part of the propagation constant decreases in the upper Reststrahlen band of hBN, keeps nearly constant in the lower band, and increases outside the Reststrahlen band with positive nonlinear dielectric permittivity for low frequency mode.
© 2016 Optical Society of America
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 , subwavelength optics, data storage, light generation  and nano imaging . 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 . Graphene SPPs can be widely used in optical devices, such as, waveguide , terahertz laser  and antenna , and optical modulator . 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 . Mihalache et al. showed the dispersion relation at a metal/Kerr-type nonlinear interface with first integral method . 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 . Wang et al. investigated the influence of nonlinearity of the substrate on the property of the graphene surface plasmons . In , 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 .
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 , thermal emission  and data storage . 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 , and it can now be used as substrate with graphene for enhancing mobility . 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 . Brar et al.  investigated the coupling between SPPs in graphene and SPhPs in hBN with patterned graphene on monolayer hBN. Kumar  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.
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:
The conductivity of the graphene consists of intraband and interband contributions,27]
hBN is a kind of anisotropic crystal, the principal dielectric tensor components of hBN can be expressed as26].
In the structure in Fig. 1, there are only three non-zero fields Ex, Ez and Hy for TM mode having the component . From Maxwell’s equations we get the following equationsEq. (6) Eqs. (7) and (8) 28]Eqs. (8)-(12) 27], Eq. (1) can also be approximately expessed asEq. (12), after some algebra we can get the following dispersion relationEq. (16) should be replaced by sinh, and the dispersion relation can be obtained similarly. If the cladding is linear medium, the dispersion with graphene can be obtained easily as follows 
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 from Eq. (11) in space z<0, and in space z>0. Because of 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 or below . But when a graphene placed at the interface, one can see that SPPs exist in the region above or below in Fig. 2(b). This is because that the permittivity of grapehne is negative and can support SPPs. Furthermore, we can see that 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.
Further, the cladding air is replaced by the nonlinear material, the parameters are chosen as follows: , . We get the dispersion in Fig. 3(a) by using Eqs. (14) and (17) with 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 , 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 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.
Then the influence of the nonlinear part is investigated. It can been seen from Fig. 4 that there are two modes outside the Reststrahlen band when is not zero. and increase with for low frequency mode, while they decrease with for high frequency mode. The dispersion relations are also shown in Fig. 5 for clarity, and it is well known that means the propagation loss of SP3. Because 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 and between and . The dispersion below is omitted for the similarity with that above . In Figs. 5(a) and 5(b) in the band between and , and increase along with the frequency, and the value of is very small in general which means that propagation length is very long. In special in the upper Reststrahlen band, decreases monotonically with for and increases with the absolute value at first for . The minimal value of and the maximal value of can be obtained near the condition . 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 () when . In the band above , as shown in Figs. 5(c) and 5(d), it can be seen that and of low frequency mode increase monotonically along with the frequency and when , while decrease with the absolute value at first, then increase when . The minimal value can be obtained near the condition . However the dependence of and of high frequency mode on 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 = 0.
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, 1200 and 1800 are chosen presenting the frequency in the upper Reststrahlen band, above , and below 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 or below , we can see that the and 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 and are much larger with lower Fermi energy, or we can say that the structure is more sensitive to with lower Fermi energy. From another point of view, we can find that the positive nonlinear dielectric can enlarge and , 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 on are opposite. And it can be found that there is no high frequency mode existing when = 0. Another interesting trend is that the difference of between two modes become smaller as Fermi energy increases. The influence of the graphene on surface phonon in the band between and 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. is very small in the band, which increases slightly with Fermi energy of graphene, so the propagaton length is very long.
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 and below , as the layer number N increases, the difference of between two modes gets smaller, and both and decrease for both two modes. Similarly, it can be found that the change spans of and are much larger with less layer number, in another word, SP3 here is more sensitive to the with less layer number. The influence of the graphene on surface phonon in the upper Reststrahlen band between and is shown in Figs. 7(c) and 7(d). keeps nearly invariant in this band, and is very small. increases with the layer number, which means that more sheets of graphene introduce a little additional loss with SP3.
Thirdly, we want to explain the influence of the relaxation time of graphene on SP3. We have shown the dependence of dispersion on with different parameter , as shown in Fig. 8 . It is clear that decreases for both modes and the difference of between two modes gets smaller as relaxation time increases, while remains nearly constant at three different frequencies.
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 and in the lower band . 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. and increase with for low frequency mode, while decrease with for high frequency mode. And there is only low frequency mode in the lower Reststrahlen band.
The dependence of on 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 increases along with frequency below 780, decreases at first and then increases above 780. We can also find that increases monotonically with when ,while it decreases with the absolute value at first, then increases when in the lower Reststrahlen band. The minimal value can be obtained near . increases along with frequency below 780, decreases above 780, and it is barely affected by in the lower Reststrahlen band.
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 and 800 presenting the frequency below and in the lower Reststrahlen band separately. SP3 in the frequency above has the similar properties with the frequency below, 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 and of two modes decrease monotonically along with the Fermi energy of graphene, while they decrease more and more slowly, and the difference of between two modes become smaller as Fermi energy increases. The influence of on is opposite for the two modes at frequency blow . The variation range of of two modes on nonlinear delectric permittivity in the lower Reststrahlen band is larger than below , in other word is more sensitive to in the lower Reststrahlen band than below . The nonlinear delectric permittivity has nearly no effect on 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 , as shown in Fig. 12 . and decrease monotonically along with the layer number of graphene, while keeps nearly constant with different in the lower Reststrahlen band, which is different with Sec. 3.1.
The influence of relaxation time of graphene on the properties of SP3 is shown in Fig. 13 . decreases monotonically along with the relaxation time of graphene, while keeps nearly constant with different relaxation time at 700for two mode. And decreases with the relaxation time of graphene at 800, so it can be seen that better perfermance of SP3 can be obtained with larger relaxation time in the lower Reststrahlen band of hBN.
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.
This work is partially supported by the National Natural Science Foundation of China (Grant No. 61505111 and 61490713), the Guangdong Natural Science Foundation(Grant No. 2015A030313549), the Science and Technology Innovation Commission of Shenzhen (grant Nos.JCYJ20140828163633996 and JCYJ20150324141711667), and the Natural Science Foundation of SZU (Grant Nos. 827-000051, 827-000052, and 827-000059).
References and links
1. S. A. Maier and H. A. Atwater, “Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98(1), 011101 (2005). [CrossRef]
4. M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infra-red frequencies,” Phys. Rev. B 80(24), 308–310 (2009). [CrossRef]
5. I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Lett. 12(5), 2470–2474 (2012). [CrossRef] [PubMed]
7. V. V. Popov, O. V. Polischuk, A. R. Davoyan, V. Ryzhii, T. Otsuji, and M. S. Shur, “Plasmonic terahertz lasing in an array of graphene nanocavities,” Phys. Rev. B 86(19), 80–82 (2012). [CrossRef]
8. M. Tamagnone, J. Gomez-Diaz, J. R. Mosig, and J. Perruisseau-Carrier, “Reconfigurable terahertz plasmonic antenna concept using a graphene stack,” Appl. Phys. Lett. 101(21), 214102 (2012). [CrossRef]
9. D. R. Andersen, “Graphene-based long-wave infrared TM surface plasmon modulator,” J. Opt. Soc. Am. B 27(4), 818–823 (2010). [CrossRef]
10. M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6(11), 737–748 (2012). [CrossRef]
11. D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, and T. Twardowski, “Exact dispersion relations for transverse magnetic polarized guided waves at a nonlinear interface,” Opt. Lett. 12(3), 187–189 (1987). [CrossRef] [PubMed]
12. I. D. Rukhlenko, A. Pannipitiya, and M. Premaratne, “Dispersion relation for surface plasmon polaritons in metal/nonlinear-dielectric/metal slot waveguides,” Opt. Lett. 36(17), 3374–3376 (2011). [CrossRef] [PubMed]
13. I. D. Rukhlenko, A. Pannipitiya, M. Premaratne, and G. P. Agrawal, “Exact dispersion relation for nonlinear plasmonic waveguides,” Phys. Rev. B 84(11), 113409 (2011). [CrossRef]
16. Y. V. Bludov, D. A. Smirnova, Y. S. Kivshar, N. Peres, and M. I. Vasilevskiy, “Nonlinear TE-polarized surface polaritons on graphene,” Phys. Rev. B 89(3), 035406 (2014). [CrossRef]
19. Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature 444(7120), 740–743 (2006). [CrossRef] [PubMed]
21. S. Dai, Z. Fei, Q. Ma, A. S. Rodin, M. Wagner, A. S. McLeod, M. K. Liu, W. Gannett, W. Regan, K. Watanabe, T. Taniguchi, M. Thiemens, G. Dominguez, A. H. C. Neto, A. Zettl, F. Keilmann, P. Jarillo-Herrero, M. M. Fogler, and D. N. Basov, “Tunable phonon polaritons in atomically thin van der Waals crystals of boron nitride,” Science 343(6175), 1125–1129 (2014). [CrossRef] [PubMed]
22. C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, “Boron nitride substrates for high-quality graphene electronics,” Nat. Nanotechnol. 5(10), 722–726 (2010). [CrossRef] [PubMed]
23. R. Koch, T. Seyller, and J. Schaefer, “Strong phonon-plasmon coupled modes in the graphene/silicon carbide heterosystem,” Phys. Rev. B 82(20), 201413 (2010). [CrossRef]
24. V. W. Brar, M. S. Jang, M. Sherrott, S. Kim, J. J. Lopez, L. B. Kim, M. Choi, and H. Atwater, “Hybrid surface-phonon-plasmon polariton modes in graphene/monolayer h-BN heterostructures,” Nano Lett. 14(7), 3876–3880 (2014). [CrossRef] [PubMed]
26. A. Kumar, T. Low, K.-H. Fung, P. Avouris, and N. X. Fang, “Tunable light-matter interaction and the role of hyperbolicity in graphene-hBN system,” Nano Lett. 15(5), 3172–3180 (2015). [CrossRef] [PubMed]
27. Y. Xiang, X. Dai, J. Guo, S. Wen, and D. Tang, “Tunable optical bistability at the graphene-covered nonlinear interface,” Appl. Phys. Lett. 104(5), 051108 (2014). [CrossRef]
28. Nagaraj and A. A. Krokhin, “Long-range surface plasmons in dielectric-metal-dielectric structure with highly anisotropic substrates,” Phys. Rev. B 81(8), 085426 (2010). [CrossRef]
29. G. Borstel, H. Falge, and A. Otto, “Surface and bulk phonon-polaritons observed by attenuated total reflection,” in Solid-State Physics (Springer, 1974), 107–148.
30. B.-C. Liu, L. Yu, and Z.-X. Lu, “The surface plasmon polariton dispersion relations in a nonlinear-metal-nonlinear dielectric structure of arbitrary nonlinearity,” Chin. Phys. B 20(3), 037302 (2011). [CrossRef]
31. T. Dumelow and D. R. Tilley, “Optical properties of semiconductor superlattices in the far infrared,” J. Opt. Soc. Am. A 10(4), 633–645 (1993). [CrossRef]