Confined transverse-electric graphene plasmons in negative refractive-index systems

Transverse electric graphene plasmons are generally weakly confined in the direction perpendicular to the graphene plane. They are featured by a skin depth δ, namely the penetration depth of their evanescent fields into the surrounding environment, much larger than the wavelength λ in free space (e.g., δ > 10λ). The weak spatial confinement of transverse electric graphene plasmons is now the key drawback that limits their practical applications. Here we report the skin depth of TE graphene plasmons can be largely decreased down to the subwavelength scale (e.g., δ < λ/10) in negative refractive-index environments. The underlying mechanism originates from the different existence conditions for TE graphene plasmons in negative and positive refractive-index environments. To be specific, their existence in negative (positive) refractive-index environments requires Im(σs) > 0 (Im(σs) < 0) and lies in the frequency range of ħω/μc < 1.667 (ħω/μc > 1.667), where σs and μc are the surface conductivity and chemical potential of monolayer graphene, respectively.


INTRODUCTION
In their seminal work in 2007, Mikhailov S. A. and Ziegler K. 1 proposed an exotic electromagnetic mode in the monolayer graphene, namely the transverse electric (TE, or s-polarized) graphene plasmons. The TE graphene plasmons lie in the frequency range of ħω/μ c > 1.667, since their existence requires Im(σ s ) < 0 1 , where σ s and μ c are the surface conductivity and chemical potential of graphene, respectively. Another key feature for TE graphene plasmons is the spatial confinement. Note that highly confined surface plasmons 2,3 , such as the transverse magnetic (TM, or p-polarized) graphene plasmons [4][5][6][7] , can enable the flexible control of light flow in the subwavelength scale and even the extreme nanoscale; as such, they can enable many promising applications, including the on-chip terahertz to X-ray radiation sources 8,9 , miniaturized modulators 10 , subwavelength guidance [11][12][13] , deep-subwavelength imaging [14][15][16][17] , and light energy harvesting and scattering [18][19][20] . The spatial confinement of graphene plasmons in the direction perpendicular to the graphene plane can be quantitatively characterized by the skin depth δ. Here the skin depth 21 is defined as the penetration depth of the evanescent fields carried by graphene plasmons into the surrounding environment. For TE graphene plasmons, their skin depth is inversely proportional to |Im(σ s )| 1 . Due to the small achievable negative-value of Im(σ s ) (i.e., max(|Im(σ s )|)~G 0 (Fig. 1)), TE graphene plasmons are featured by a skin depth at least in the wavelength scale, where G 0 = e 2 /4ħ is the universal optical conductivity. To be specific, we generally have δ > 10λ (Supplementary Fig. 1) for TE plasmons in monolayer graphene, where λ is the wavelength in free space. As severely limited by the weak spatial confinement, only several potential applications of TE graphene plasmons have been reported, such as Brewster effects 22 , polarizers 23 , optical sensors 24 , waveguide phase, and amplitude modulators 25 . On the other hand, rapid progress in nano-photonics has fuelled a quest for highly confined TE graphene plasmons, in addition to the highly confined TM graphene plasmons. This way, highly confined graphene plasmons can be achieved without stringent requirement on the polarization of light and can benefit more practical applications based on TE waves. Such a quest still remains elusive, although many researches of TE polaritons in graphene [26][27][28][29] and other 2D materials [30][31][32][33][34] have been ignited by the pioneering work in 2007.
Here we theoretically reveal a viable way to largely enhance the spatial confinement of TE graphene plasmons by using the environment with the negative permeability or refractive index. As firstly proposed by Veselago in 1968 35 , the negative refractiveindex materials simultaneously have negative permittivity and negative permeability; they have triggered tremendous researches both on fundamental science and practical applications [36][37][38][39][40] , exemplified by the well-known negative refraction 41 , the perfect lens/superlens 37,42 , the inverse Doppler effect [43][44][45][46] and the backward Cherenkov radiation [47][48][49][50] . In principle, the environment with the negative permeability or refractive index can be effectively constructed, for example, by metamaterials 36,41,51 and photonic crystals [52][53][54][55] . We find the existence condition of TE graphene plasmons in negative refractive-index environments is drastically different from that in positive refractive-index environments. In negative refractive-index environments, the existence of TE graphene plasmons become to require Im(σ s ) > 0; as a result, TE graphene plasmons now lie in the frequency range of ħω/μ c < 1.667 at room temperature. Moreover, due to the availability of the large positive value of Im(σ s ) (max(|Im(σ s )|)~10 2 G 0 ), the skin depth of TE graphene plasmons can be largely decreased down to the subwavelength scale (e.g., δ < λ/10). 1

RESULTS AND DISCUSSION
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 inplane wavevector q (parallel to the graphene plane) is generally comparable to the wavevector k 0 = 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 σ s of graphene. That is σ s = σ s,intra + σ s,inter , where and σ s;inter ¼ Here σ s,intra and σ s,inter are the parts of the conductivity related to the intra-band and inter-band transitions, respectively; f d x ð Þ ¼ ðe xÀμ c k B T þ 1Þ À1 is the Fermi-Dirac distribution function; e is the electron charge, T is the temperature, τ is the relaxation time, and μ c 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 ε r1 (ε r2 ) and the relative permeability μ r1 (μ r2 ). For conceptual demonstration, we let ε r1 = ε r2 = ε r and μ r1 = μ r2 = μ r . From the classical electromagnetic wave theory (see Supplementary Notes 1 and 2), the dispersion for TE graphene plasmons is governed by is the out-of-plane (perpendicular to the graphene plane) component of wavevector; μ 0 is the permeability in free space.
On the other hand, Eq. (2) also indicates that the skin depth of TE graphene plasmons is proportional to 1/|Im(σ s )| for arbitrary μ r , since the skin depth is mathematically defined as δ = 1/Im(k z ). In other words, in the frequency range where TE graphene plasmons could exist, a larger value of |Im(σ s )| would lead to a smaller skin depth and thus a larger spatial confinement. Note that the value of Im(σ s ) for the monolayer graphene is dominantly determined by σ s,intra especially in the frequency range of ħω/μ c ≪ 1.667, while it is mainly determined by σ s,inter if ħω/μ c > 1.667. As a result, the maximum value of |Im(σ s )| can reach~40G 0 at ħω/μ c < 1.667 due to the contribution of σ s,intra (Fig. 1c); in contrast, it is only~0.3G 0 at ħω/μ c > 1.667 with the negligible contribution from σ s,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/μ c ≪ 1.667, σ s,intra in Eq. (1) is dependent on the temperature T, the relaxation time τ, and the chemical potential μ c , besides the angular frequency ω. These parameters (T, τ and μ c ) provide us extra degrees of freedom to achieve the large value of |Im(σ s )|; 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 (μ r < 0 and ε r < 0) is a typical negative permeability environment (μ r < 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.
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 To facilitate the discussion, the effective refractive index of TE graphene plasmons is denoted as n eff,0 = Re(q)/k 0 and plotted in Fig. 2; see the information of Im(q)/k 0 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(σ s ) appears in the frequency range of ħω/μ c < 0.1 (Fig. 2a). If the relaxation time increases (i.e., the loss in graphene decreases), max(|Im(σ s )|) in the interested frequency range increases and can be up to~100G 0 (Fig. 2a). Due to the availability of large positive Im(σ s ), the The monolayer graphene is located in a symmetric environment with the relative permittivity ε r1 = ε r2 = ε r and the relative permeability μ r1 = μ r2 = μ r . b Existence condition of TE graphene plasmons in environments with various values of ε r and μ r . c Surface conductivity of monolayer graphene as a function of frequency at 300 K. The emergence of TE graphene plasmons in environments with μ r > 0 requires Im(σ s ) < 0, which exists in the frequency range of ħω/μ c > 1.667. In contrast, their emergence in environments with μ r < 0 requires Im(σ s ) > 0, existing in the range of ħω/μ c < 1.667. Here we set the chemical potential μ c = 0.2 eV and the relaxation time τ = 0.2 ps. G 0 = e 2 /4ħ is the universal optical conductivity.
maximum n eff,0 in negative refractive-index environment is much larger than 1 (e.g., up to n eff,0 ≈ 1.4 in Fig. 2b, c; also see Supplementary Note 5), and the maximum value of n eff,0 would increase if the relaxation time increases. Such a large value of n eff,0 is favored for the practical application of TE graphene plasmons. We emphasize that in positive refractive-index environment, n eff,0 is generally very close to 1, such as n eff,0 ≈ 1.00006 in Supplementary Fig. 5 in Supplementary Note 6.
Influence of Tand μ c on TE graphene plasmons Figure 3 shows the influence of the temperature T and the chemical potential μ c on TE graphene plasmons in negative  refractive-index environments. In short, if the temperature or the chemical potential increases, the achievable maximum value of n eff,0 for TE graphene plasmons would increase, e.g., up to n eff,0 = 1.65 in Fig. 3a. We note that in Fig. 3a, the achievable maximum value of n eff,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 ie 2 kBT 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 μ c = 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 μ c = 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 μ r and ε r 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 |μ r | and |ε r | on Re(q)/|k|, where k ¼ k 0 ffiffiffiffiffiffiffi ffi ε r μ r p 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 λ environ and the wavelength of TE graphene plasmons λ plasmon , namely Re(q)/|k| = λ environ /λ plasmon . That is, a large Re(q)/|k| indicates a larger contrast between λ environ and λ plasmon . Note that k ≠ k 0 where k 0 is the wavevector of light in free space, and thus Re(q)/|k| is not the effective refractive index of TE graphene plasmons n eff,0 = Re(q)/k 0 discussed in Figs. 2 and 3.
From Fig. 4, the variation of both |μ r | and |ε r | 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 |μ r | and decrease |ε r |. As such, a larger contrast between λ environ and λ plasmon exists in negative refractiveindex environments. In contrast, in positive refractive-index environments, Re(q)/|k| is insensitive to the variation of |μ r | and |ε r |, and it is always very close to 1. Therefore, there is the negligible contrast between λ environ and λ plasmon 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 environments 56 . More discussion on the influence of |μ r | and |ε r | 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(σ s ) > 0 and their existing frequency range of ħω/μ c < 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).

Dispersion of TE graphene plasmons
Without loss of generality, the monolayer graphene is located at the interface between region 1 and region 2 (Fig. 1a), where region 1 with z < 0 (region 2 with z > 0) has the relative permittivity ε r1 (ε r2 ) and the relative permeability μ r1 (μ r2 ). For TE graphene plasmons, their electric fields are along the y direction. According to the electromagnetic theory 19 , one can Fig. 4 Influence of |μ r | and |ε r | on TE graphene plasmons in negative refractive-index environments. Here the in-plane wavevector Re(q) of TE graphene plasmons is normalized by the wavevector k ¼ k 0 ffiffiffiffiffiffiffi ffi ε r μ r p 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 |μ r | and |ε r | on TE graphene plasmons in positive refractive-index environments is also shown. For the negative refractive-index environments, ε r < 0 and ε r < 0 and T = 300 K; for the positive refractive-index systems, μ r > 0, ε r > 0 and T = 100 K. The environments with ε r near zero or μ r near zero can be effectively constructed for example via metamaterials 36,41,51 , photonic crystals [52][53][54][55] , and waveguides [57][58][59] .
set the electric fields in each region as E 1 ¼ŷE 1 e iqxÀikz1z (4) E 2 ¼ŷE 2 e iqxþikz2z (5) Accordingly, the relationship between k z1,2 and q is k z1;2 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ω 2 c 2 ε r1;2 μ r1;2 À q 2 r (6) In the above equations, q is the component of the wavevector parallel to the interface, k z1 and k z2 are the component of the wavevector perpendicular to the interface in region 1 and region 2, respectively, and ω is the angular frequency. The magnetic field in each region can be obtained according to ∇ × E = iωμH. By enforcing the boundary conditions: n H 1 À H 2 ð Þ¼σ s Ej boundaryjj (8) we can obtain the dispersion of TE graphene plasmons as where μ 0 is the permeability in free space.

DATA AVAILABILITY
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.