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Hybrid nonlinear surface-phonon-plasmon-polaritons at the interface of nolinear medium and graphene-covered hexagonal boron nitride crystal

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Abstract

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

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:

εc=εl+α|E1|2,
whereεland εc are the linear and total dielectric permittivities respectively, αis the Kerr coefficient of the material, and |E1|2=|E1x|2+|E1z|2 is the electric field in the cladding.

 figure: Fig. 1

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,

σtotal(ω)=σintra(ω)+σinter(ω).
the intraband and interband contributions can be separately expressed as follows [27]
σintra(ω)=ie2kBTπ2(ω+iτ1)[EF/kBT+2ln(eEF/kBT+1)],
σinter(ω)=ie24πln(2|EF|(ω+iτ1)2|EF|+(ω+iτ1)),
where EF is the Fermi energy, τis the electron relaxation time, e is the charge of the electron, is the reduced Planck’s constant, ω is the radian frequency, kB is the Boltzmann constant, and T is the temperature in K.

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

εu=ε,u(1+ωLO,u2ωTO,u2ωTO,u2ω2iωΓu),
whereu=,, and represent the transverse and z axis respectively, ωLO and ωTOcorrespond to the LO and TO phonon frequencies respectively, ωLO=1610cm1, ωTO=1370cm1 in the upper Reststrahlen band, and ωLO=830cm1, ωTO=780cm1 in the lower Reststrahlen band, ε corresponds to the high-frequency dielectric permittivity, ε,=4.87, ε,=2.95, Γ is the damping constant, Γ=5cm1, Γ=4cm1 [26].

In the structure in Fig. 1, there are only three non-zero fields Ex, Ez and Hy for TM mode having the component exp(iβx)exp(iωt). From Maxwell’s equations we get the following equations

βHy=ωεε0Ez,dHydz=iωεε0Ex,dExdziβEz=iωμ0Hy,
hereβ is the propagation constant of the TM field propagating along the x derection, ε0 and μ0 are the permittivity and permeability of vacuum respectively, ε is permittivity of the medium, and c=1/(ε0μ0)1/2, c is velocity of light. In the region of z>0, we can get from Eq. (6)
(H1yk02εc)=(β2k02εc1)H1y,
where k0=2π/λ=ω/c.

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

(H1yωε0εc)2=εcεlα(βH1yωε0εc)2.
Then the following relation can be obtained from Eqs. (7) and (8)
H1y2=(ωε0)2εc2β2k02εcεc2εl22α.
hBN occupies the space z<0, the electromagnetic field here is [28]
H2y=H2y(0)eγ2z,
where
γ2=εε||β2k02ε.
The boundary conditions at the interfaces z=0 require that
H2y(0)H1y(0)=σE2x(0),E1x=E2x.
We can obtain the following dispersion relation by using Eqs. (8)-(12)
β2=14(2k02εc+γ22k02εc(εc+εl)(k0ε+iγ2σ(ω)/ε0c)2+β2εc(εc+εl)εc2),
where εc=εc(0). When there is no graphene, we can get
β2=(2εε||εc2(εc+εl)εc2ε||)k024εε||εcεc3εlεc2εε||(εc+εl).
To verify the validity of the method above, the dispersion can be obtained in another way. Base on our previous work [27], Eq. (1) can also be approximately expessed as
εc=εl+α|H1y|2,
where α=α/(εlε02c2), and when α>0 the magnetic field in the Kerr medium can be written as
H1y=2αγ1k0cosh[γ1(zz0)],
Where γ1=β2k02εl, z0 is the center of the cosh function. From the boundary condition in Eq. (12), after some algebra we can get the following dispersion relation
sγ1εc1Δεk022γ12=γ2εσγ2/(iωε0),
where Δε=α|H1(0)|2=εcεl,s=±1 representing different modes seperately. If α<0, the cosh term in Eq. (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 [4]
εcβ2εck02+ε(ε/ε)β2εk02+iσωε0=0.
Furthermore, if the substrate is an isotropic crystal with ε=ε=εs, and there is no graphene, we obtain the famous well-known dispersion relationβ2=k02εsεc/(εs+εc).

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 β2ε/εk02ε>0 from Eq. (11) in space z<0, andβ2k02>0 in space z>0. Because of ε<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 ωLO or below ωTO [29]. But when a graphene placed at the interface, one can see that SPPs exist in the region above ωLO or below ωTO in Fig. 2(b). This is because that the permittivity of grapehne is negative and can support SPPs. Furthermore, we can see that Re(β) 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.

 figure: Fig. 2

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 β=ω/c, and the blue line is the light line in hBN β=(ε)1/2ω/c.

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Further, the cladding air is replaced by the nonlinear material, the parameters are chosen as follows: α=±6.4×1012m2V2, εl=2.405 [30]. We get the dispersion in Fig. 3(a) by using Eqs. (14) and (17) with Δε=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 β, 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 1700cm1 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.

 figure: Fig. 3

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 Δε is investigated. It can been seen from Fig. 4 that there are two modes outside the Reststrahlen band when Δε is not zero. Re(β) and Im(β) increase with Δε for low frequency mode, while they decrease with Δε for high frequency mode. The dispersion relations are also shown in Fig. 5 for clarity, and it is well known that Im(β) means the propagation loss of SP3. Because β 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 ωLO and between ωLO and ωTO. The dispersion below ωTO is omitted for the similarity with that above ωLO. In Figs. 5(a) and 5(b) in the band between ωLO and ωTO, Re(β) and Im(β) increase along with the frequency, and the value of Im(β) is very small in general which means that propagation length is very long. In special in the upper Reststrahlen band, Im(β) decreases monotonically with Δε for Δε>0 and increases with the absolute value Δε at first for Δε<0. The minimal value of Re(β)and the maximal value of Im(β) can be obtained near the condition Δε=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 (Δε) when Δε>0. In the band above ωLO, as shown in Figs. 5(c) and 5(d), it can be seen that Re(β) and Im(β)of low frequency mode increase monotonically along with the frequency and Δεwhen Δε>0, while decrease with the absolute value Δε at first, then increase when Δε<0. The minimal value can be obtained near the condition Δε=1. However the dependence of Re(β) and Im(β) of high frequency mode on Δε 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 Δε = 0.

 figure: Fig. 4

Fig. 4 The dependence of β on wavenumber and Δε.

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 figure: Fig. 5

Fig. 5 The dependence of β on Δε and wavenumber. (a), (c) and (e) represent Re(β), and (b), (d) and (f) representIm(β). (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: 1500cm1, 1200cm1 and 1800cm1 are chosen presenting the frequency in the upper Reststrahlen band, above ωLO, and below ωTO 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 ωLO or below ωTO, we can see that the Re(β) and Im(β) 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 Re(β) and Im(β) are much larger with lower Fermi energy, or we can say that the structure is more sensitive to Δε with lower Fermi energy. From another point of view, we can find that the positive nonlinear dielectric can enlarge Re(β) and Im(β), 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 β on Δε are opposite. And it can be found that there is no high frequency mode existing when Δε = 0. Another interesting trend is that the difference of β between two modes become smaller as Fermi energy increases. The influence of the graphene on surface phonon in the band between ωLO and ωTO 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. Im(β)is very small in the band, which increases slightly with Fermi energy of graphene, so the propagaton length is very long.

 figure: Fig. 6

Fig. 6 The dependence of β on Δε with the different EF: (a) and (b) are at 1200cm1, (c) and (d) are at 1500cm1, and (e) and (f) are at 1800cm1. 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 ωLO and below ωTO, as the layer number N increases, the difference of β between two modes gets smaller, and both Re(β)and Im(β) decrease for both two modes. Similarly, it can be found that the change spans of Re(β) and Im(β)are much larger with less layer number, in another word, SP3 here is more sensitive to the Δεwith less layer number. The influence of the graphene on surface phonon in the upper Reststrahlen band between ωLO and ωTO is shown in Figs. 7(c) and 7(d). Re(β)keeps nearly invariant in this band, and Im(β) is very small. Im(β) increases with the layer number, which means that more sheets of graphene introduce a little additional loss with SP3.

 figure: Fig. 7

Fig. 7 The dependence of β on Δε with the different layer number: (a) and (b) are at 1200cm1, (c) and (d) are at 1500cm1, (e) and (f) are at 1800cm1. 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 Δε with different parameter τ, as shown in Fig. 8 . It is clear that Im(β) decreases for both modes and the difference of Im(β) between two modes gets smaller as relaxation time increases, while Re(β) remains nearly constant at three different frequencies.

 figure: Fig. 8

Fig. 8 The dependence of β on Δε with the different relaxation time: (a) and (b) are at 1200cm1, (c) and (d) are at 1500cm1, (e) and (f) are at 1800cm1. 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 ε<0 and ε>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. Re(β) and Im(β) increase with Δε for low frequency mode, while decrease with Δε for high frequency mode. And there is only low frequency mode in the lower Reststrahlen band.

 figure: Fig. 9

Fig. 9 The dependence of β on wavenumber and Δε.

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The dependence of β on Δε 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 Re(β) increases along with frequency below 780cm1, decreases at first and then increases above 780cm1. We can also find that Re(β) increases monotonically with Δε when Δε>0,while it decreases with the absolute value Δε at first, then increases when Δε<0 in the lower Reststrahlen band. The minimal value can be obtained near Δε=1. Im(β) increases along with frequency below 780cm1, decreases above 780cm1, and it is barely affected by Δε in the lower Reststrahlen band.

 figure: Fig. 10

Fig. 10 The dependence of β on Δε and wavenumber for low frequency mode, (a) represents Re(β), and (b) represents Im(β).

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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 cm1 and 800cm1 presenting the frequency below ωTO and in the lower Reststrahlen band ω separately. SP3 in the frequency above ωLO=830cm1 has the similar properties with the frequency belowωTO=1370cm1, 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 Re(β) and Im(β) of two modes decrease monotonically along with the Fermi energy of graphene, while they decrease more and more slowly, and the difference of β between two modes become smaller as Fermi energy increases. The influence of Δε on β is opposite for the two modes at frequency blow ωTO. The variation range of Re(β) of two modes on nonlinear delectric permittivity in the lower Reststrahlen band is larger than below ωTO, in other word Re(β) is more sensitive to Δε in the lower Reststrahlen band than below ωTO. The nonlinear delectric permittivity Δε has nearly no effect on Im(β) 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 ωTO, as shown in Fig. 12 . Re(β) and Im(β) decrease monotonically along with the layer number of graphene, while Im(β) keeps nearly constant with different Δε in the lower Reststrahlen band, which is different with Sec. 3.1.

 figure: Fig. 11

Fig. 11 The dependence of β on Δε with the different EF: (a) and (b) are at 700cm1, and (c) and (d) are at 800cm1. Solid lines are low frequency modes, dashed lines are high frequency modes.

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 figure: Fig. 12

Fig. 12 The dependence of β on Δε with the different layer number: (a) and (b) are at 700cm1, and (c) and (d) are at 800cm1. 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 . Im(β) decreases monotonically along with the relaxation time of graphene, while Re(β) keeps nearly constant with different relaxation time at 700cm1for two mode. And Re(β) decreases with the relaxation time of graphene at 800cm1, so it can be seen that better perfermance of SP3 can be obtained with larger relaxation time in the lower Reststrahlen band of hBN.

 figure: Fig. 13

Fig. 13 The dependence of β on Δε with the different relaxation time: (a) and (b) are at 700cm1, and (c) and (d) are at 800cm1. 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.

Acknowledgment

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).

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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]  

25. H. Yan, T. Low, F. Guinea, F. Xia, and P. Avouris, “Tunable phonon-induced transparency in bilayer graphene nanoribbons,” Nano Lett. 14(8), 4581–4586 (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]  

References

  • View by:

  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]
  2. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
    [Crossref] [PubMed]
  3. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
    [Crossref] [PubMed]
  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]
  6. A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
    [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]
  14. G. A. Wurtz, R. Pollard, and A. V. Zayats, “Optical bistability in nonlinear surface-plasmon polaritonic crystals,” Phys. Rev. Lett. 97(5), 057402 (2006).
    [Crossref] [PubMed]
  15. L. Wang, W. Cai, X. Zhang, and J. Xu, “Surface plasmons at the interface between graphene and Kerr-type nonlinear media,” Opt. Lett. 37(13), 2730–2732 (2012).
    [Crossref] [PubMed]
  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]
  17. X. Dai, L. Jiang, and Y. Xiang, “Low threshold optical bistability at terahertz frequencies with graphene surface plasmons,” Sci. Rep. 5, 12271 (2015).
    [Crossref] [PubMed]
  18. T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a SiC superlens,” Science 313(5793), 1595 (2006).
    [Crossref] [PubMed]
  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]
  20. N. Ocelic and R. Hillenbrand, “Subwavelength-scale tailoring of surface phonon polaritons by focused ion-beam implantation,” Nat. Mater. 3(9), 606–609 (2004).
    [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]
  25. H. Yan, T. Low, F. Guinea, F. Xia, and P. Avouris, “Tunable phonon-induced transparency in bilayer graphene nanoribbons,” Nano Lett. 14(8), 4581–4586 (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]

2015 (2)

X. Dai, L. Jiang, and Y. Xiang, “Low threshold optical bistability at terahertz frequencies with graphene surface plasmons,” Sci. Rep. 5, 12271 (2015).
[Crossref] [PubMed]

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]

2014 (5)

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]

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]

H. Yan, T. Low, F. Guinea, F. Xia, and P. Avouris, “Tunable phonon-induced transparency in bilayer graphene nanoribbons,” Nano Lett. 14(8), 4581–4586 (2014).
[Crossref] [PubMed]

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]

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]

2012 (5)

L. Wang, W. Cai, X. Zhang, and J. Xu, “Surface plasmons at the interface between graphene and Kerr-type nonlinear media,” Opt. Lett. 37(13), 2730–2732 (2012).
[Crossref] [PubMed]

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6(11), 737–748 (2012).
[Crossref]

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]

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]

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]

2011 (4)

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[Crossref] [PubMed]

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]

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]

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]

2010 (4)

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]

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]

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]

D. R. Andersen, “Graphene-based long-wave infrared TM surface plasmon modulator,” J. Opt. Soc. Am. B 27(4), 818–823 (2010).
[Crossref]

2009 (1)

M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infra-red frequencies,” Phys. Rev. B 80(24), 308–310 (2009).
[Crossref]

2006 (3)

G. A. Wurtz, R. Pollard, and A. V. Zayats, “Optical bistability in nonlinear surface-plasmon polaritonic crystals,” Phys. Rev. Lett. 97(5), 057402 (2006).
[Crossref] [PubMed]

T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a SiC superlens,” Science 313(5793), 1595 (2006).
[Crossref] [PubMed]

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]

2005 (2)

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

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]

2004 (1)

N. Ocelic and R. Hillenbrand, “Subwavelength-scale tailoring of surface phonon polaritons by focused ion-beam implantation,” Nat. Mater. 3(9), 606–609 (2004).
[Crossref] [PubMed]

2003 (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref] [PubMed]

1993 (1)

1987 (1)

Agrawal, G. P.

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]

Andersen, D. R.

Atwater, H.

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]

Atwater, H. A.

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]

Avouris, P.

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]

H. Yan, T. Low, F. Guinea, F. Xia, and P. Avouris, “Tunable phonon-induced transparency in bilayer graphene nanoribbons,” Nano Lett. 14(8), 4581–4586 (2014).
[Crossref] [PubMed]

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref] [PubMed]

Basov, D. N.

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]

Bludov, Y. V.

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]

Boardman, A. D.

Brar, V. W.

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]

Buljan, H.

M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infra-red frequencies,” Phys. Rev. B 80(24), 308–310 (2009).
[Crossref]

Cai, W.

Carminati, R.

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]

Chen, J.

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]

Chen, Y.

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]

Choi, M.

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]

Crassee, I.

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]

Dai, S.

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]

Dai, X.

X. Dai, L. Jiang, and Y. Xiang, “Low threshold optical bistability at terahertz frequencies with graphene surface plasmons,” Sci. Rep. 5, 12271 (2015).
[Crossref] [PubMed]

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]

Davoyan, A. R.

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]

De Wilde, Y.

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]

Dean, C. R.

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]

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref] [PubMed]

Dominguez, G.

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]

Dumelow, T.

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref] [PubMed]

Engheta, N.

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[Crossref] [PubMed]

Fang, N.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

Fang, N. X.

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]

Fei, Z.

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]

Fogler, M. M.

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).
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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).
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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).
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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).
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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).
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H. Yan, T. Low, F. Guinea, F. Xia, and P. Avouris, “Tunable phonon-induced transparency in bilayer graphene nanoribbons,” Nano Lett. 14(8), 4581–4586 (2014).
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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).
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M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infra-red frequencies,” Phys. Rev. B 80(24), 308–310 (2009).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a SiC superlens,” Science 313(5793), 1595 (2006).
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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).
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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).
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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]

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N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

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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).
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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]

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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]

H. Yan, T. Low, F. Guinea, F. Xia, and P. Avouris, “Tunable phonon-induced transparency in bilayer graphene nanoribbons,” Nano Lett. 14(8), 4581–4586 (2014).
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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).
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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).
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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).
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Mosig, J. R.

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).
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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]

Nagaraj,

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]

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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).
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N. Ocelic and R. Hillenbrand, “Subwavelength-scale tailoring of surface phonon polaritons by focused ion-beam implantation,” Nat. Mater. 3(9), 606–609 (2004).
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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]

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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).
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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).
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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).
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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).
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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]

Perruisseau-Carrier, J.

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]

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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).
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[Crossref]

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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).
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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]

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]

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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]

Rodin, A. S.

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]

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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]

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]

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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]

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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]

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Seyller, T.

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]

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]

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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]

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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]

Shur, M. S.

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]

Shvets, G.

T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a SiC superlens,” Science 313(5793), 1595 (2006).
[Crossref] [PubMed]

Smirnova, D. A.

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]

Soljacic, M.

M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infra-red frequencies,” Phys. Rev. B 80(24), 308–310 (2009).
[Crossref]

Sorgenfrei, S.

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]

Stegeman, G. I.

Sun, C.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

Tamagnone, M.

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]

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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]

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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]

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]

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T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a SiC superlens,” Science 313(5793), 1595 (2006).
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Thiemens, M.

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]

Tilley, D. R.

Twardowski, T.

Urzhumov, Y.

T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a SiC superlens,” Science 313(5793), 1595 (2006).
[Crossref] [PubMed]

Vakil, A.

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[Crossref] [PubMed]

Vasilevskiy, M. I.

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]

Wagner, M.

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]

Walter, A. L.

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]

Wang, L.

L. Wang, W. Cai, X. Zhang, and J. Xu, “Surface plasmons at the interface between graphene and Kerr-type nonlinear media,” Opt. Lett. 37(13), 2730–2732 (2012).
[Crossref] [PubMed]

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).
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Watanabe, K.

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).
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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).
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Wen, S.

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]

Wright, E. M.

Wurtz, G. A.

G. A. Wurtz, R. Pollard, and A. V. Zayats, “Optical bistability in nonlinear surface-plasmon polaritonic crystals,” Phys. Rev. Lett. 97(5), 057402 (2006).
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Xia, F.

H. Yan, T. Low, F. Guinea, F. Xia, and P. Avouris, “Tunable phonon-induced transparency in bilayer graphene nanoribbons,” Nano Lett. 14(8), 4581–4586 (2014).
[Crossref] [PubMed]

Xiang, Y.

X. Dai, L. Jiang, and Y. Xiang, “Low threshold optical bistability at terahertz frequencies with graphene surface plasmons,” Sci. Rep. 5, 12271 (2015).
[Crossref] [PubMed]

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]

Xu, J.

Yan, H.

H. Yan, T. Low, F. Guinea, F. Xia, and P. Avouris, “Tunable phonon-induced transparency in bilayer graphene nanoribbons,” Nano Lett. 14(8), 4581–4586 (2014).
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Young, A. F.

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).
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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).
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Zanoni, R.

Zayats, A. V.

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6(11), 737–748 (2012).
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G. A. Wurtz, R. Pollard, and A. V. Zayats, “Optical bistability in nonlinear surface-plasmon polaritonic crystals,” Phys. Rev. Lett. 97(5), 057402 (2006).
[Crossref] [PubMed]

Zettl, A.

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]

Zhang, X.

L. Wang, W. Cai, X. Zhang, and J. Xu, “Surface plasmons at the interface between graphene and Kerr-type nonlinear media,” Opt. Lett. 37(13), 2730–2732 (2012).
[Crossref] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

Appl. Phys. Lett. (2)

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).
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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]

Chin. Phys. B (1)

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]

J. Appl. Phys. (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).
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J. Opt. Soc. Am. A (1)

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Nano Lett. (4)

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]

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]

H. Yan, T. Low, F. Guinea, F. Xia, and P. Avouris, “Tunable phonon-induced transparency in bilayer graphene nanoribbons,” Nano Lett. 14(8), 4581–4586 (2014).
[Crossref] [PubMed]

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).
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Nat. Mater. (1)

N. Ocelic and R. Hillenbrand, “Subwavelength-scale tailoring of surface phonon polaritons by focused ion-beam implantation,” Nat. Mater. 3(9), 606–609 (2004).
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Nat. Nanotechnol. (1)

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]

Nat. Photonics (1)

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6(11), 737–748 (2012).
[Crossref]

Nature (2)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
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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).
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Opt. Lett. (3)

Phys. Rev. B (6)

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).
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M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infra-red frequencies,” Phys. Rev. B 80(24), 308–310 (2009).
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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).
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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).
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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).
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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]

Phys. Rev. Lett. (1)

G. A. Wurtz, R. Pollard, and A. V. Zayats, “Optical bistability in nonlinear surface-plasmon polaritonic crystals,” Phys. Rev. Lett. 97(5), 057402 (2006).
[Crossref] [PubMed]

Sci. Rep. (1)

X. Dai, L. Jiang, and Y. Xiang, “Low threshold optical bistability at terahertz frequencies with graphene surface plasmons,” Sci. Rep. 5, 12271 (2015).
[Crossref] [PubMed]

Science (4)

T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a SiC superlens,” Science 313(5793), 1595 (2006).
[Crossref] [PubMed]

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]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[Crossref] [PubMed]

Other (1)

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.

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Figures (13)

Fig. 1
Fig. 1 The system of a graphene separating semi-infinite nonlienear material and hBN crystal.
Fig. 2
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 β = ω / c , and the blue line is the light line in hBN β = ( ε ) 1 / 2 ω / c .
Fig. 3
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).
Fig. 4
Fig. 4 The dependence of β on wavenumber and Δ ε .
Fig. 5
Fig. 5 The dependence of β on Δ ε and wavenumber. (a), (c) and (e) represent Re ( β ) , and (b), (d) and (f) represent Im ( β ) . (c) and (d) represent low frequency mode, (e) and (f) represent high frequency mode.
Fig. 6
Fig. 6 The dependence of β on Δ ε with the different EF: (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.
Fig. 7
Fig. 7 The dependence of β on Δ ε 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.
Fig. 8
Fig. 8 The dependence of β on Δ ε 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.
Fig. 9
Fig. 9 The dependence of β on wavenumber and Δ ε .
Fig. 10
Fig. 10 The dependence of β on Δ ε and wavenumber for low frequency mode, (a) represents Re ( β ) , and (b) represents Im ( β ) .
Fig. 11
Fig. 11 The dependence of β on Δ ε with the different EF: (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.
Fig. 12
Fig. 12 The dependence of β on Δ ε 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.
Fig. 13
Fig. 13 The dependence of β on Δ ε 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.

Equations (18)

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ε c = ε l + α | E 1 | 2 ,
σ t o t a l ( ω ) = σ i n t r a ( ω ) + σ i n t e r ( ω ) .
σ i n t r a ( ω ) = i e 2 k B T π 2 ( ω + i τ 1 ) [ E F / k B T + 2 ln ( e E F / k B T + 1 ) ] ,
σ i n t e r ( ω ) = i e 2 4 π ln ( 2 | E F | ( ω + i τ 1 ) 2 | E F | + ( ω + i τ 1 ) ) ,
ε u = ε , u ( 1 + ω L O , u 2 ω T O , u 2 ω T O , u 2 ω 2 i ω Γ u ) ,
β H y = ω ε ε 0 E z , d H y d z = i ω ε ε 0 E x , d E x d z i β E z = i ω μ 0 H y ,
( H 1 y k 0 2 ε c ) = ( β 2 k 0 2 ε c 1 ) H 1 y ,
( H 1 y ω ε 0 ε c ) 2 = ε c ε l α ( β H 1 y ω ε 0 ε c ) 2 .
H 1 y 2 = ( ω ε 0 ) 2 ε c 2 β 2 k 0 2 ε c ε c 2 ε l 2 2 α .
H 2 y = H 2 y ( 0 ) e γ 2 z ,
γ 2 = ε ε | | β 2 k 0 2 ε .
H 2 y ( 0 ) H 1 y ( 0 ) = σ E 2 x ( 0 ) , E 1 x = E 2 x .
β 2 = 1 4 ( 2 k 0 2 ε c + γ 2 2 k 0 2 ε c ( ε c + ε l ) ( k 0 ε + i γ 2 σ ( ω ) / ε 0 c ) 2 + β 2 ε c ( ε c + ε l ) ε c 2 ) ,
β 2 = ( 2 ε ε | | ε c 2 ( ε c + ε l ) ε c 2 ε | | ) k 0 2 4 ε ε | | ε c ε c 3 ε l ε c 2 ε ε | | ( ε c + ε l ) .
ε c = ε l + α | H 1 y | 2 ,
H 1 y = 2 α γ 1 k 0 cos h [ γ 1 ( z z 0 ) ] ,
s γ 1 ε c 1 Δ ε k 0 2 2 γ 1 2 = γ 2 ε σ γ 2 / ( i ω ε 0 ) ,
ε c β 2 ε c k 0 2 + ε ( ε / ε ) β 2 ε k 0 2 + i σ ω ε 0 = 0.

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