Effects of dielectric substrate on characteristics of surface plasmon polaritons of a single-layer graphene

The use of a dielectric substrate is unavoidable for feasibility of the design and often enables advanced physical regimes and operating modes. Various electromagnetic phenomena can be efficiently controlled by a proper choice of substrate53,54,55,56,57. Effects exerted by substrate can often be understood through the prism of scaling by assuming that the same characteristics can be obtained at higher or lower frequencies, if the structure is properly modified. The classical scaling rule of resonance frequencies, \(f\propto {\varepsilon }_{d}^{-1/2}\) (ε d is permittivity of the dielectric filling a cavity), is known for lossless cavities58. Prediction of scaling capability of an open resonance structure is a challenging task, since there is no exact boundary of the region occupied by the resonance field. A proportional change of geometrical (and material) parameters enables scaling of an existing transmittive/reflective structure with respect to frequency, provided that dispersion and losses in the used materials are relatively weak59. Besides, partial scaling is possible by varying permittivity of substrate and other dielectric components, while all geometrical sizes are kept fixed57,60. Alongside the cavities and transmittive/reflective configurations, effect of dielectric substrate on propagation of plasmons in silver nanowires shows up as interesting characteristics53. Intuitively, increase of permittivity leads to downscaling (for fixed frequency) or redshift (for fixed geometry). However, the quantifying of these effects is impossible without a detailed numerical study. Therefore, in order to investigate influence of substrate on SPPs of a single-layer graphene, in this section we numerically study the basic effects of variation of permittivity of a lossless, dispersion-free dielectric substrate, ε d , on the SPPs characteristics, i.e., propagation length, localization length, and figure of merit.

The schematic of the studied structure is shown in Fig. 1(d), inset. We consider a sheet of graphene placed at z = 0 on a semi-infinite lossless substrate with permittivity ε s . Taking y component of the magnetic field as

$${H}_{y}(z)=\{\begin{array}{cc}a{e}^{-{q}_{a}(z)} & z > 0\\ s{e}^{{q}_{s}(z)} & z < 0\end{array}\},$$ (1)

and applying the boundary conditions for TM polarization14,61, we arrive at the following well-known dispersion relation of graphene SPPs14,15,16,17,18,19,20,21,22,25:

$${\varepsilon }_{a}/{q}_{a}+{\varepsilon }_{s}/{q}_{s}=\alpha $$ (2)

where \({q}_{i}=\sqrt{{\beta }^{2}-{\varepsilon }_{i}{\beta }_{0}^{2}}\) (i = a, s), β = k x , β 0 = ω/c, and α = σ g /iωε 0 . The optical conductivity of graphene (\({\sigma }_{g}={\sigma }_{g}^{{intra}}+{\sigma }_{g}^{{inter}}\)) can be presented as14

$${\sigma }_{g}^{{intra}}=\frac{{e}^{2}}{4\hslash }\frac{i}{2\pi }\{\frac{16{k}_{B}T}{\hslash {\rm{\Omega }}}\,\mathrm{ln}(2\,{\cosh }(\frac{\mu }{2{k}_{B}T}))\},$$ (3a)

$${\sigma }_{g}^{{inter}}=\frac{{e}^{2}}{4\hslash }\{\frac{1}{2}+\frac{1}{\pi }\,\arctan (\frac{\hslash {\rm{\Omega }}-2\mu }{2{k}_{B}T})-\frac{i}{2\pi }\,\mathrm{ln}\,\frac{{(\hslash {\rm{\Omega }}+2\mu )}^{2}}{{(\hslash {\rm{\Omega }}-2\mu )}^{2}+{(2{k}_{B}T)}^{2}}\},$$ (3b)

with Ω = ω + iτ−1, μ is chemical potential of graphene, e is the electron charge, k B is the Boltzmann constant, ℏ is the Plank constant over 2π, and c is the speed of light in vacuum. Here, we take the electron relaxation time and temperature as τ = 0.2 ps (otherwise stated) and T = 300 K, respectively. Moreover, considering β = β′ + iβ″, we define wavelength, propagation length and localization length of the guided modes as λ sp = 2π/β′, PL = 1/2β″ and LL = 1/2Re(q a + q s ), respectively, and figure of merit as FoM = (PL/λ sp )/LL(μm). For the sake of simplicity, we use the notation λ sp for surface waves of all types studied here, i.e., SPPs, SPhPs and SPPPs.

Figure 1 (a) Dispersion, (b) propagation length, (c) localization length, and (d) figure of merit of the SPPs supported by a single-layer graphene on various dielectric substrates at μ = 0.2 eV; ε s = 2.25 (solid blue line), 11.7 (dotted-dashed red line), 16 (dotted black line), and 35 (dashed pink line). The schematic of the studied structure is depicted in panel (d), inset. Full size image

First, we examine the effect of different, dispersion-free lossless substrates on the characteristics of graphene SPPs for μ = 0.2 eV. The considered substrates include glass (ε s = 2.25), Si (ε s = 11.7), Ge (ε s = 16), and a material with ε s = 3560. It is seen in Fig. 1(a) that by increasing ε s , graphene SPPs are supported for larger values of the wavenumber at fixed frequencies, and, consequently can propagate with smaller λ sp and group velocity, v g . The scaling is applicable to the results in Fig. 1(a), i.e., an \({\varepsilon }_{s}^{-\mathrm{1/2}}\)-like fitting can be introduced similarly to57. Note that such large wavenumbers like those in Fig. 1(a) at ε s = 35 can also be achieved using patterned graphene on conventional substrate12. Figure 1(b,c) show that the increase in ε s leads to decrease in PL and LL. However, as observed in Fig. 1(d), FoM of the plasmonic guided modes becomes considerably larger at f > 5 THz, that originates from decrease in λ sp and LL of the graphene SPPs. Thus, increase of FoM is coherent here with decrease of v g .

It is known that one of the most impactful properties of the SPPs supported by graphene is their tunability by varying μ. In Fig. S1 (see Supplementary Information) by taking ε = 35, we examine the SPP characteristics for different values of μ. We found that the effect of increase of μ is similar to the effect of decrease of ε s for λ sp , but the same cannot be said regarding FoM, for which an optimal value of ε s can exist. Moreover, for a richer insight into the localization of the graphene SPPs, field profiles are also presented in Fig. S2 at four different frequencies.

SPhPs of thin LiF films

According to the results discussed in Sect. 2, the larger value of ε s , the larger values of FoM are obtainable for the graphene SPPs. High-permittivity dispersion-free materials are not accessible for the far-IR. Hence, the choice of polar dielectrics like LiF34,35, NaCl34, and GaAs36,37,38 is natural. These materials are known to show a polaritonic gap (RS band), which appears due to phonon-photon interaction. Desired high permittivity values result from this interaction. Graphene SPPs can be hybridized at the far-IR with phonons in heterostructures composed of graphene and polar materials. From a fabrication point of view, those materials should be used as a buffer layer between a single-layer graphene and a thicker low-loss dielectric substrate. Therefore, a study of the waveguide structures comprising thin films of a polar dielectric on low-loss dielectric substrate is required prior to combining with graphene. In this section, we scrutinize propagation and localization characteristics of SPhPs supported by thin films of LiF on different lossless substrates, within 1 THz to 20 THz. LiF is particularly appropriate due to the wide RS band and a wide range of permittivity variation inside and around this band34,35.

Figure 2(a) schematically shows a LiF film on the top of a substrate with ε s . The considered LiF has thickness t and permittivity given by34

$${\varepsilon }_{LiF}={\varepsilon }_{\infty }(1-\frac{{\omega }_{LO}^{2}-{\omega }_{TO}^{2}}{{\omega }^{2}-{\omega }_{TO}^{2}-i\gamma \omega }),$$ (4)

where ε ∞ = 2.027, ω TO = 2πf TO , ω LO = 2πf LO , f TO = 9.22 THz, f LO = 19.1 THz and γ = 2π × 0.527 THz. Here, f TO , f LO and γ are, respectively, transverse optical frequency, longitudinal optical frequency and damping factor. Real and imaginary parts of ε LiF are shown in Fig. 2(b). It is obvious from Fig. 2(b) that Re(ε LiF ) takes considerably large values at frequencies slightly smaller than f TO . A thin layer of LiF that has thickness t and is bounded by air half-space (ε a = 1) on the one side and by a typical dielectric substrate (ε s ) on the other side can support SPhPs. Taking H y as

$${H}_{y}(z)=\{\begin{array}{cc}a{e}^{-{q}_{a}(z-t/2)} & z > t/2\\ {l}_{1}{e}^{-{q}_{LiF}(z)}+{l}_{2}{e}^{{q}_{LiF}(z)} & -t/2\le z\le t/2\\ s{e}^{{q}_{s}(z+t/2)} & z < -\,t/2\end{array}\},$$ (5)

and then applying the TM-case boundary conditions, we arrive at the following dispersion relation of the SPhPs that are supported by the LiF layer on the dielectric substrate:

$${\tanh }({q}_{LiF}t)=-\,\frac{{{\rm{\Gamma }}}_{a}+{{\rm{\Gamma }}}_{s}}{1+{{\rm{\Gamma }}}_{a}{{\rm{\Gamma }}}_{s}},$$ (6)

where Γ a = q LiF ε a /q a ε LiF , Γ s = q LiF ε s /q s ε LiF , and \({q}_{LiF}=\sqrt{{\beta }^{2}-\varepsilon {\beta }_{0}^{2}}\).

Figure 2 (a) Schematic of the structure comprising a thin film of LiF with thickness t that is centered at z = 0 and placed on a lossless substrate. (b) Real (solid blue line) and imaginary (dashed red line) parts of ε LiF within 1–20 THz; vertical dashed lines correspond to f TO = 9.22 THz and f LO = 19.1 THz of LiF, being the boundaries of the polaritonic gap (RS band). Full size image

First, we investigate characteristics of the SPhPs for different thicknesses of the LiF layer placed on a semi-infinite glass substrate (ε s = 2.25). The obtained results are presented in Fig. 3. It is seen in Fig. 3(a) that SPhPs are supported in the polaritonic gap (RS band) of LiF, in which Re(ε LiF ) < 0. At t = 10 nm, they can propagate with large wavenumbers, Re(β)/β 0 < 640, i.e., with small phonon wavelengths (λ sp ). By increasing t, SPhPs with larger λ sp can be supported, as occurs for t = 150 nm when Re(β)/β 0 < 50. Figure 3(b) illustrates that the larger thickness of the phononic waveguides, the larger PL is achievable for the SPhPs. In particular, the maximum PL is obtained at t = 150 nm, while the minimum value of PL belongs to the case of t = 10 nm. More investigations show that taking t = 200 nm, even larger values of PL can be obtained. In turn, the results presented in Fig. 3(c) indicate that the thinner the phononic waveguide is, the higher confinement of the SPPs can be obtained. In other words, the minimum LL is achieved at t = 10 nm. The FoM is presented in Fig. 3(d). The maximum values of the FoM are obtained at t = 10 nm. By increasing t, one reduces the FoM value. Consequently, as far as supporting the phononic guided modes with lower losses (larger values of PL/λ sp ) and stronger localization (smaller values of LL) within 10–18 THz are concerned, the smaller thickness of the phononic waveguide can be chosen. One more point that should be highlighted here is the support of phononic guided modes with negative slope of dispersion. These modes have been considered here as backward waves, being similar to the ones supported by the left-handed metamaterials62,63, and by metal-insulator-metal64,65 and insulator-metal-insulator66 structures. The total power \(1/2{Re}[\int \,{S}_{x}{d}_{z}]\) of the guided modes with backward propagation is negative (S x is x-component of the Poynting vector, S).

Figure 3 (a) Dispersion, (b) propagation length, (c) localization length, and (d) figure of merit of the SPhPs supported by air/LiF waveguides of thickness t on a glass substrate, respectively; t = 10 nm (solid blue line), t = 50 nm (dashed red line), t = 100 nm (dotted black line), and t = 150 nm (dashed-dotted pink line). Dashed black lines in panel (a) show the edges of the polaritonic gap of LiF. Full size image

Similar to the the previous section, we have also investigated the effect of ε s on the modal characteristics of the LiF waveguide. The results are presented in Fig. S3 (see Supporting Information). To get more insight into the SPhPs characteristics, their FoM for different values of t (t = 50 nm, 100 nm, 150 nm and 200 nm) and ε s are illustrated in Fig. S4. As a complementary discussion, in Fig. S5, we also investigated typical mode profiles of the LiF phononic waveguide for t = 10 nm and t = 100 nm. As comprehensively discussed in the the Supporting Information, we found that the SPhPs of the 10 nm-thick LiF waveguide are as confined as the graphene SPPs in 10–18 THz.

SPPPs of graphene-LiF waveguide on glass substrate

To keep the advantages of tunability of graphene and a wide permittivity range of LiF, we combine them in one hybrid structure so that graphene SPPs could be hybridized with SPhPs. As schematically illustrated in Fig. 4(d), inset, the structure in Fig. 2 is modified now by placing a single layer of graphene on top of the LiF film. Substituting the following equation25,67

$${{\rm{\Gamma }}}_{a}=\frac{{q}_{LiF}}{{\varepsilon }_{LiF}}(\frac{{\varepsilon }_{a}}{{q}_{a}}-\alpha )$$ (7)

into Eq. (6), we obtain the dispersion relation of the coupled plasmonic-phononic modes of this system, which are labeled as surface-phonon-plasmon-polaritons (SPPPs)28. According to the results presented in the first part of the Results and Discussion section, we found that the SPPs of a single layer of graphene possess larger values of FoM for larger values of ε s . Hence, due to the large values of Re(ε LiF ) at the frequencies below but close to f TO , unique modal characteristics are expected to appear. In this section, our consideration is restricted to the case when glass is the substrate of the graphene-LiF heterostructure. As discussed in the previous part, the 10 nm-thick LiF waveguide on glass substrate shows the largest FoM values due to the support of low-loss SPhPs with strong confinement [see Fig. S5(a,b)]. Therefore, in Fig. 4, we present the characteristics of the SPPPs of the waveguide structure at t = 10 nm. In Fig. 4(a), one can see that SPPPs dispersion is noticeably modified as compared with SPhPs dispersion of the 10 nm-thick LiF waveguide in Fig. 3(a) and dispersion of the graphene SPPs. Due to the plasmon-phonon coupling, first, the SPPPs can be supported at considerably smaller wavenumbers and, second, backward SPPPs can propagate inside the waveguide for two frequency ranges: 8.63 THz < f < f TO and 16.83 THz < f < 18.68 THz. These two features cannot be observed in the air/graphene/glass and air/LiF/glass waveguides separately, i.e., without combining them in one structure. Figure 4(b,c) indicate that the combination of highly confined SPhPs of LiF with the localized graphene SPPs leads to a giant increase in the losses of the guided modes, so a considerable decrease/increase in the PL/LL values of the SPPPs occurs as compared to the graphene SPPs. As illustrated in Fig. 4(d), the 10 nm-thick graphene-LiF waveguide supports SPPPs with much smaller FoM values than the graphene SPPs.

Figure 4 (a) Dispersion, (b) propagation length, (c) localization length, and (d) figure of merit of the SPPPs supported by air/graphene/LiF/glass waveguide with t = 10 nm (solid blue lines); the schematic of the studied structure is presented in panel (d), inset. For the sake of comparison, results are also presented for the SPPs of the graphene/glass structure (dashed red lines). Here, μ = 0.2 eV and the dashed black horizontal lines in panel (a) show the edges of the polaritonic gap of LiF. Full size image

From the results in Figs 3 and S4 it was found that by increasing the thickness of the polaritonic core material in the LiF waveguides, FoM of the SPhPs is decreased due to the confinement reduction. Therefore, it is worth to investigate the specifics of coupling between the highly confined graphene SPPs and the lowly confined SPhPs of air/LiF/glass system for two larger values of the thickness of LiF; i.e. t = 100 nm and t = 200 nm. As it was expected from the results shown in Fig. 4(a), the modes with negative slope of dispersion are supported at f > 14.86 THz by the air/graphene/LiF/glass waveguides for both t = 100 nm and t = 200 nm, see Fig. 5(a). Even more interestingly, Fig. 5(a) demonstrates that, in contrast with the case of t = 10 nm, for the larger t, the backward modes that occur for 8.63 THz < f < f TO are turned to be forwardly propagating SPPPs with very small v g and λ sp in the vicinity of 8.63 THz. As is seen from Fig. 5(b,c), despite the SPPPs with f < f TO possess smaller values of PL than those of graphene SPPs, they are more strongly confined; i.e., they also possess smaller values of LL. This means that the hybridization of the highly confined graphene SPPs with the lowly confined SPhPs of air/LiF/glass structure leads to the extremely confined SPPPs at frequencies below f TO , where Re(ε LiF ) is large. This decrease in the localization length of the modes yields a considerable rise in the FoM of the SPPPs at 5.2 THz < f < f TO , compared to graphene SPPs illustrated in Fig. 5(d). Thus, an almost 3.2-fold enhancement of FoM is achieved around 8.5 THz. Therefore, hybridization of graphene SPPs with the lowly confined SPhPs leads to the appearance of SPPPs with larger FoM than a single layer of graphene. It should be noted that since LiF is a lossy material, its presence reduces PL, however decreases LL as well for the cases of t = 100 nm and 200 nm. Similar to first part of the Result and Discussion section, increase of FoM is associated in this regime with the slowing effect of the substrate. In the case we are only interested in supporting SPPPs with larger PL than for graphene SPPs, air/graphene/LiF/glass waveguides with t = 100 nm and t = 200 nm are the appropriate designs for the range of 9.45 THz < f < 13.77 THz. In line with the said above, support of SPPPs with significantly different propagation and localization characteristics within three neighboring frequency bands (i.e., below, inside, and above the polaritonic gap) makes the suggested air/graphene/LiF/glass heterostructure a distinguishable candidate for multifunctional applications including, for instance, waveguiding, sensing, and absorption purposes.

Figure 5 (a) Dispersion, (b) propagation length, (c) localization length, and (d) figure of merit of the SPPPs supported by the air/graphene/LiF/glass waveguide for t = 100 nm (solid blue lines), t = 200 nm (dotted black lines) at μ = 0.2 eV. Results for SPPs of the graphene/glass structure are presented for comparison (dashed red lines). The edges of the polaritonic gap of LiF are shown by dashed black line. Full size image

For a deeper insight into the features of the SPPPs, in Fig. 6 we present field profiles of the SPPP modes for the case of t = 100 nm, at four different frequencies. In agreement with Fig. 5(c), the mode profiles at f = 11 THz [Fig. 6(b)] and f = 18 THz [Fig. 6(c)] are very similar to the ones illustrated in Fig. S5(c,d). This similarity verifies that at these frequencies, which are inside the polaritonic gap of LiF, the SPPPs possess mostly SPhPs characteristics. At the same time, for the frequencies outside the polaritonic gap, i.e., at f = 8 THz [Fig. 6(a)] and f = 18 THz [Fig. 6(d)], the mode profiles resemble those shown in Fig. S2(a,d). In tnis case, SPPPs attain characteristics of graphene SPPs. In order to verify the SPPPs propagation length for the case of t = 100 nm in Fig. 5(b), two-dimensional field distributions of the SPPPs are also illustrated in Fig. 7. As expected, at f > f TO propagation in the waveguide with noticeably small confinement is possible, as shown in Fig. 7(b–d). On the other hand, Fig. 7(a) confirms that the slowly propagating SPPPs at f < f TO are strongly confined inside the waveguide and show high FoM. As mentioned above, tunability of the optical properties via changing μ can be considered as the most advantageous characteristic of graphene over other plasmonic and phononic materials. Therefore, we finally investigate the effect of changes in μ on the characteristics of SPPPs supported by the air/graphene/LiF/glass waveguide with t = 100 nm. The results in Fig. 8(a–e) show that by changing μ, dispersion, propagation and localization characteristics of the SPPPs can be efficiently tuned at frequencies inside and outside of the RS band of LiF. In Fig. 8(d) it is illustrated that the maximal FoM of the forwardly propagating slow SPPPs at f < f TO can be significantly increased by decreasing μ, that is associated with decrease of v g and increase of wavenumber. Moreover, it is possible to tune the frequency of the forward-to-backward (FB) wave transition at f TO < f < f LO . In Fig. 8(a) the transition point is indicated as FB point. Figure 8(e) provides more evidence of the frequency and wavenumber tuning for the FB point. By decreasing μ, it can be shifted toward lower frequencies and larger wavenumbers. From a practical point of view, the presence of substrate may add some additional effects, such as electron impurity and electron losses, on the electronic and finally optical properties of graphene. These effects can be included in the calculations by the phenomenological relaxation time (τ) or the electron mobility as \(e{v}_{F}^{2}\tau /\mu \) where \({v}_{F}=c/\mathrm{300}\). Therefore, in Fig. 8(f) we investigated how different values of τ can affect FoM of the hybrid guided modes supported by the graphene/LiF/glass waveguide for t = 100 nm. As it is observed from Fig. 8(f), a 5-time increase in the relaxation time leads to almost 1.3-time enhancement of the FoM of the modes. More investigations reveal that a reverse trend can be observed by decreasing τ. Moreover, for frequencies larger than the optical phonon frequency (f > 48.4 THz), electron-phonon scattering can considerably affect τ20. The inter-layer coupling is also another mechanism that can considerably amend the relaxation time68,69. One more important point should be highlighted here is that, in general, nonlocal optical conductivity of graphene can be considered in the calculations16,17,18,19,20. As explained in16,17,18,19,20, once the substrate of graphene is low-index and not strongly dispersive, taking the local optical conductivity of graphene in the calculations leads to a precise description of the modal characteristics of the guided waves for low frequencies and small values of the wavenumbers; i.e. for β/β 0 < 500 and f < 20 THz. To the present, most of the studies on the investigation of plasmon-phonon hybridizations in the graphene/polar heterostructures have been done in the local regime28,40,41,42,43,44,45,46,47,48,49,50,51,52. However, in order to gain more practical insight into the hybridization mechanism and the modal characteristics of the guided waves supported by those systems, considering the nonlocal effects in the calculations will be the subject of our future studies.

Figure 6 Profiles of the normalized |E x | (dashed black line) and |H y | (solid blue line) for SPPPs of the air/graphene/LiF/glass waveguide with t = 100 nm. (a) f = 8 THz, (b) f = 11 THz, (c) f = 18 THz, and (d) f = 20 THz. Glass substrate, LiF film, and air regions are highlighted in aqua, light brown and white, respectively. The graphene layer is schematically shown by solid black line at z = t/2. Full size image

Figure 7 Spatial distribution of the normalized |E x | for the SPPPs supported by the air/graphene/LiF/glass waveguide with t = 100 nm; (a) f = 8 THz, (b) f = 11 THz, (c) f = 18 THz, and (d) f = 20 THz. Full size image