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General existence condition for TE graphene plasmons

We focus on the discussion of TE surface plasmons supported by the monolayer graphene. For TE graphene plasmons, their in-plane wavevector q (parallel to the graphene plane) is generally comparable to the wavevector k0 = ω/c of light in free space, where c is the speed of light in free space. As such, the nonlocal response of graphene is negligible, and it is reasonable to use the local Kubo formula to describe the surface conductivity σs of graphene. That is σs = σs,intra + σs,inter, where

$$sigma _{{mathrm{s}},{mathrm{intra}}} = frac{{ie^2k_{mathrm{B}}T}}{{pi hbar ^2left( {omega + frac{i}{tau }} right)}}left( {frac{{mu _{mathrm{c}}}}{{k_{mathrm{B}}T}} + 2ln left( {e^{ – frac{{mu _{mathrm{c}}}}{{k_BT}}} + 1} right)} right)$$

(1)

and (sigma _{{mathrm{s}},{mathrm{inter}}} = frac{{ie^2(omega + frac{i}{tau })}}{{pi hbar ^2}}{int}_0^infty {frac{{f_dleft( { – x} right) – f_d(x)}}{{(omega + frac{i}{tau })^2 – 4(frac{x}{hbar })^2}}} dx). Here σs,intra and σs,inter are the parts of the conductivity related to the intra-band and inter-band transitions, respectively; (f_dleft( x right) = (e^{frac{{x – mu _{mathrm{c}}}}{{k_BT}}} + 1)^{ – 1}) is the Fermi-Dirac distribution function; e is the electron charge, T is the temperature, τ is the relaxation time, and μc is the chemical potential.

Figure 1 shows the general existence condition for TE graphene plasmons. The monolayer graphene is located at the interface between region 1 and region 2 (Fig. 1a). Region 1 with z < 0 (region 2 with z > 0) has the relative permittivity εr1 (εr2) and the relative permeability μr1 (μr2). For conceptual demonstration, we let εr1 = εr2 = εr and μr1 = μr2 = μr. From the classical electromagnetic wave theory (see Supplementary Notes1 and 2), the dispersion for TE graphene plasmons is governed by

$$2k_{mathrm{z}} + omega sigma _{mathrm{s}}mu _0mu _{mathrm{r}} = 0$$

(2)

where (k_{mathrm{z}} = sqrt {frac{{omega ^2}}{{c^2}}varepsilon _{mathrm{r}}mu _{mathrm{r}} – q^2}) is the out-of-plane (perpendicular to the graphene plane) component of wavevector; μ0 is the permeability in free space.

Equation (2) explicitly indicates the existence condition for TE graphene plasmons, as briefly summarized in Fig. 1b, c. To be specific, for the environment with μr > 0 (e.g., positive refractive-index environments), Eq. (2) has solutions only if Im(σs) < 0, where the negative Im(σs) lies in the frequency range of ħω/μc > 1.667 (Fig. 1c). In contrast, for the environment with μr < 0 (e.g., negative refractive-index environments), the existence of TE graphene plasmons becomes to require Im(σs) > 0 from Eq. (2) (Fig. 1b) and lies in the range of ħω/μc < 1.667 at room temperature (Fig. 1c).

On the other hand, Eq. (2) also indicates that the skin depth of TE graphene plasmons is proportional to 1/|Im(σs)| for arbitrary μr, since the skin depth is mathematically defined as δ = 1/Im(kz). In other words, in the frequency range where TE graphene plasmons could exist, a larger value of |Im(σs)| would lead to a smaller skin depth and thus a larger spatial confinement. Note that the value of Im(σs) for the monolayer graphene is dominantly determined by σs,intra especially in the frequency range of ħω/μc 1.667, while it is mainly determined by σs,inter if ħω/μc > 1.667. As a result, the maximum value of |Im(σs)| can reach ~40G0 at ħω/μc < 1.667 due to the contribution of σs,intra (Fig. 1c); in contrast, it is only ~0.3G0 at ħω/μc > 1.667 with the negligible contribution from σs,intra (Fig. 1c). This way, the minimum skin depth of TE graphene plasmons in negative permeability environments would be much smaller than that in positive permeability environments.

Moreover, if ħω/μc 1.667, σs,intra in Eq. (1) is dependent on the temperature T, the relaxation time τ, and the chemical potential μc, besides the angular frequency ω. These parameters (T, τ and μc) provide us extra degrees of freedom to achieve the large value of |Im(σs)|; see for example in Fig. 2. Then in negative permeability environments, these parameters could enable us the capability to flexibly modulate the basic features of TE graphene plasmons (Figs. 2 and 3), including their spatial confinement. Below the influence of these parameters on TE graphene plasmons in negative refractive-index environments is analyzed in detail, where the negative refractive-index environment (μr < 0 and εr < 0) is a typical negative permeability environment (μr < 0). In addition, the loss in the surrounding environment is artificially neglected, since the reasonable amount of loss will not have a drastic influence on the confined TE graphene plasmons.

Fig. 2: TE graphene plasmons in negative refractive-index environments.

For conceptual demonstration, the negative refractive-index environment has the relative permittivity εr = −1 and the relative permeability μr = −1. a Surface conductivity of monolayer graphene for different values of relaxation time τ. b Dispersion of TE graphene plasmons. For each dispersion curve, its peak (highlighted by the solid dot) corresponds to the maximum in-plane wavevector Re(q)peak of TE graphene plasmons at the frequency of ωpeak. c Influence of the relaxation time on Re(q)peak and ωpeak. Here μc = 0.2 eV and T = 300 K. In c the colored dots are extracted from (b); the black solid line (i.e., relation between ωpeak and τ) is the analytical solution of (dleft[ {frac{{{mathrm{Re}}left( q right)}}{{k_0}}} right]{mathrm{/}}domega = 0); the gray solid line (relation between Re(q)peak and τ) is obtained by substituting the analytical solution of ωpeak into Eq. (3).

Fig. 3: Influence of the temperature and the chemical potential on TE graphene plasmons in negative refractive-index systems.
figure3

a In-plane wavevector Re(q) of TE graphene plasmons (normalized by the wavevector of light in free space k0) as a function of the temperature T. b, c Skin depth δand quality factor Re(q)/Im(q) of TE graphene plasmons at different temperatures and chemical potentials (their values are denoted in the figure). λ is the wavelength of light in free space. Here εr = −1, μr = −1, and the electron mobility is μ = 10000 cm2 V−1 s−1. The corresponding relaxation time can be obtained from (tau = mu _{mathrm{c}}mu {mathrm{/}}(ev_F^2)), where vF = 1 × 106 m s−1 is the Fermi velocity.

Influence of the relaxation time on TE graphene plasmons

Figure 2 shows the influence of relaxation time on TE graphene plasmons in negative refractive-index environments, from the perspective of the in-plane wavevector. According to Eq. (2), it is straightforward to derive the expression for the in-plane wavevector q, that is

$$q = sqrt {k_0^2varepsilon _{mathrm{r}}mu _{mathrm{r}} – frac{1}{4}omega ^2mu _0^2mu _r^2sigma _{mathrm{s}}^2}$$

(3)

To facilitate the discussion, the effective refractive index of TE graphene plasmons is denoted as neff,0 = Re(q)/k0 and plotted in Fig. 2; see the information of Im(q)/k0 in Supplementary Fig. 2 in Supplementary Note 3. In addition, since the quality factor Re(q)/Im(q) is oftentimes regarded as a key parameter to characterize the basic feature of surface plasmons, the quality factor of TE graphene plasmons is also briefly discussed in Supplementary Figs. 3 and 4 in Supplementary Note 4. For the monolayer graphene, the maximum positive value of Im(σs) appears in the frequency range of ħω/μc < 0.1 (Fig. 2a). If the relaxation time increases (i.e., the loss in graphene decreases), max(|Im(σs)|) in the interested frequency range increases and can be up to ~100G0 (Fig. 2a). Due to the availability of large positive Im(σs), the maximum neff,0 in negative refractive-index environment is much larger than 1 (e.g., up to neff,0 ≈ 1.4 in Fig. 2b, c; also see Supplementary Note 5), and the maximum value of neff,0 would increase if the relaxation time increases. Such a large value of neff,0 is favored for the practical application of TE graphene plasmons. We emphasize that in positive refractive-index environment, neff,0 is generally very close to 1, such as neff,0 ≈ 1.00006 in Supplementary Fig. 5 in Supplementary Note 6.

Influence of Tand μc on TE graphene plasmons

Figure 3 shows the influence of the temperature T and the chemical potential μc on TE graphene plasmons in negative refractive-index environments. In short, if the temperature or the chemical potential increases, the achievable maximum value of neff,0 for TE graphene plasmons would increase, e.g., up to neff,0 = 1.65 in Fig. 3a. We note that in Fig. 3a, the achievable maximum value of neff,0 at high temperatures is more sensitive to the temperature variation than that at low temperatures. This phenomenon is caused by the fact that in Eq. (1), the temperature-insensitive term of (frac{{ie^2k_{mathrm{B}}T}}{{pi hbar ^2left( {omega + frac{i}{tau }} right)}} cdot frac{{mu _{mathrm{c}}}}{{k_{mathrm{B}}T}} = frac{{ie^2mu _c}}{{pi hbar ^2left( {omega + frac{i}{tau }} right)}}) plays a dominant role at low temperatures for σs,intra, while the temperature-sensitive term of (frac{{ie^2k_{mathrm{B}}T}}{{pi hbar ^2left( {omega + frac{i}{tau }} right)}} cdot 2ln ( {e^{ – frac{{mu _{mathrm{c}}}}{{k_BT}}} + 1} )) becomes important to σs,intra at high temperatures.

Correspondingly, if the temperature or the chemical potential increases, the minimum skin depth of TE graphene plasmons would decrease (Fig. 3b). To be specific, the minimum skin depth of TE graphene plasmons in negative refractive-index environments can readily become subwavelength, such as δ/λ < 1 for the case with μc = 0.2 eV at T = 300 K in Fig. 3b. Furthermore, the skin depth can even be decreased down to the deep-subwavelength scale, such as δ/λ < 0.1 for the case with μc = 0.5 eV at T = 300 K in Fig. 3b. As such, the usage of negative refractive-index environments can largely decrease the minimum skin depth of TE graphene plasmons by at least two orders of magnitude, compared to positive refractive-index environments in which δ/λ > 10 (Supplementary Fig. 1). We emphasize that the enticing subwavelength skin depth of TE graphene plasmons can already be achieved at room temperature, although the temperature’s influence in Fig. 3 is studied in a relatively wide range of temperature and the high temperature such as 3000 K in practical scenarios might lead to the instability of negative refractive-index materials. Moreover, it is worthy to highlight that the phenomenon of the temperature-induced large enhancement of the spatial confinement for TE graphene plasmons is only exists in negative refractive-index environments (Fig. 3) and will not happen for positive refractive-index environments (Supplementary Fig. 5). In addition, TE graphene plasmons in negative refractive-index environments generally have a relatively small quality factor Re(q)/Im(q) (Fig. 3c), due to their high spatial confinement and the large material loss of graphene at the studied frequency range.

Influence of μr and εr on TE graphene plasmons

Figure 4 shows the drastic difference of TE graphene plasmons in positive and negative refractive-index environments from another perspective of view, that is, the influence of |μr| and |εr| on Re(q)/|k|, where (k = k_0sqrt {varepsilon _{mathrm{r}}mu _{mathrm{r}}}) is the wavevector of light in the surrounding environment. Physically, Re(q)/|k| is equivalent to the ratio between the wavelength of light in the surrounding environment λenviron and the wavelength of TE graphene plasmons λplasmon, namely Re(q)/|k| = λenvironplasmon. That is, a large Re(q)/|k| indicates a larger contrast between λenviron and λplasmon. Note that k ≠ k0 where k0 is the wavevector of light in free space, and thus Re(q)/|k| is not the effective refractive index of TE graphene plasmons neff,0 = Re(q)/k0 discussed in Figs. 2 and 3.

Fig. 4: Influence of |μr| and |εr| on TE graphene plasmons in negative refractive-index environments.
figure4

Here the in-plane wavevector Re(q) of TE graphene plasmons is normalized by the wavevector (k = k_0sqrt {varepsilon _rmu _r}) of light in the surrounding environment. The setup of monolayer graphene is the same as that in Fig. 2a with the relaxation time τ = 0.2 ps. For comparison, the influence of |μr| and |εr| on TE graphene plasmons in positive refractive-index environments is also shown. For the negative refractive-index environments, εr < 0 and εr < 0 and T = 300 K; for the positive refractive-index systems, μr > 0, εr > 0 and T = 100 K. The environments with εr near zero or μr near zero can be effectively constructed for example via metamaterials36,41,51, photonic crystals52,53,54,55, and waveguides57,58,59.

From Fig. 4, the variation of both |μr| and |εr| for negative refractive-index environments would have a large impact on Re(q)/|k| than that for positive refractive-index environments. To be specific, in negative refractive-index environments, Re(q)/|k| > 10 is achievable if we increase |μr| and decrease |εr|. As such, a larger contrast between λenviron and λplasmon exists in negative refractive-index environments. In contrast, in positive refractive-index environments, Re(q)/|k| is insensitive to the variation of |μr| and |εr|, and it is always very close to 1. Therefore, there is the negligible contrast between λenviron and λplasmon in positive refractive-index environments. Note that the large value of Re(q)/|k| is favored in practical applications, which can be used, for example, to achieve the extraordinarily large scattering cross section from tiny objects in low-index environments56. More discussion on the influence of |μr| and |εr| can be obtained in Supplementary Figs. 6 and 7 in Supplementary Note 7.

In conclusion, we have theoretically revealed some emerging features of TE graphene plasmons in negative refractive-index environments, including their existence condition of Im(σs) > 0 and their existing frequency range of ħω/μc < 1.667. Importantly, these TE graphene plasmons can become highly confined in the direction perpendicular to the graphene plane. To be specific, their skin depth can decrease down to the deep-subwavelength scale (e.g., δ < λ/10). Then the existence of these highly confined TE graphene plasmons should be robust to various surrounding environments (i.e., the permittivity and/or permeability of the substrate and superstrate can be largely different). Such a feature is drastically different from the weakly confined TE graphene plasmons in the positive refractive-index environment, which exist mainly in the almost symmetric environments (the substrate and superstrate should have the negligible difference in their permittivity or permeability)1,28. Our findings in this work further indicate that the negative refractive-index materials might serve as a versatile platform to enable more practical applications of TE graphene plasmons, such as subwavelength guidance, some exotic scattering phenomena of light, and the exploration of TE plasmons in controlling the free electron radiation (e.g., Cherenkov radiation).



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