Transverse-electric surface plasmon polaritons in periodically modulated graphene

Transverse-electric (TE) surface plasmon polaritons are unique eigenmodes of a homogeneous graphene layer that are tunable with the chemical potential and temperature. However, as their dispersion curve spectrally lies below the light line, they cannot be resonantly excited by an externally incident wave. Here, we propose a way of exciting the TE modes and tuning their peaks in the transmission by introducing a one-dimensional graphene grating. Using the scattering-matrix formalism, we show that periodic modulation of graphene makes transmission more pronounced, potentially allowing for experimental observation of the TE modes. Furthermore, we propose the use of turbostratic graphene to enhance the role of the surface plasmon polaritons in optical spectra.

Figure 1

Schematic of a graphene grating (a) in a perspective view and (b) in the $xz$-plane side view. An infinitesimally thin graphene layer at $z=0$ is periodically modulated along the $x$ axis but homogeneous along the $y$ axis. Arrows labeled $A_n, B_n, C_n$, and $D_n$ schematically represent incoming or outgoing coefficients of diffracted plane waves with an angle of incidence not necessarily normal.

Figure 2

Electronic dispersion, shown as electronic energy versus two-dimensional electronic momentum, of graphene near the $K$ point with nonzero chemical potential $\mu$ and zero temperature. The darker colored region of surface of cone depicts energies occupied by charge carriers.

Figure 3

(a) Conductivity of graphene at zero temperature ($T=0$), sharing the frequency axis with dispersion in (b). The frequency of the interband dip in imaginary part of conductivity is affected by $\mu$ but not $d$. (b) Real part of the frequency of the TE mode dispersion of the homogeneous graphene structure folded into the first Brillouin zone, shown for $G_1=2$, 2.3, and 2.6 (thin, medium and thick blue lines, respectively). The size of the first Brillouin zone, and consequently the frequency of mode crossing at $Q=0$, is controlled by $\mu d$. The TE mode frequency is close to the light line due to small value of the fine-structure constant. (c) The period of grating, $d$, required for normalized frequency $\mathrm{\Omega }=2$, various chemical potentials $\mu$, and corresponding frequencies $\omega$ (legend) as a function of desired incidence angle $\theta =\arcsin (q/\omega )$ in degrees, using Eq. (21).

Figure 4

Transmission spectra ($1-T_0$) for zeroth diffraction order (close-to-normal incidence) when a TE-polarized light with parallel wave number $Q$ is incident on the grating with $b/d=0.3, \mathrm{\Delta }=0$, for (a) single-layer graphene with $\sigma /2\pi ={\sigma }_{\text{intra}}+{\sigma }_{\text{inter}}$, (b) single-layer graphene without interband conductivity $\sigma /2\pi ={\sigma }_{\text{intra}}$, and (c) turbostratically stacked graphene with $\sigma /2\pi =N({\sigma }_{\text{intra}}+{\sigma }_{\text{inter}})$, for $N=10$. The dashed red line indicates $T_0=0$. Spectra for $Q\ne 0$ are offset by powers of 10 in increasing order of $Q$. Red and black vertical dash-dotted lines indicate the real parts of the bright and dark TE SPP resonance frequencies for $Q=0$, which are solutions of Eqs. (34) and (33), respectively.

Figure 5

Approximation of transmission (red solid line) using Eq. (35) showing consistency with the TE peak for $Q=0$ and $\mathrm{\Delta }=0$ (blue solid line). Red and black dashed lines indicate the real part of the frequencies of the bright and dark TE SPP modes, obtained using Eq. (34) and Eq. (33), respectively. The frequency gap between the dashed lines is approximately $1.47\times {10}^{-4}$.

Figure 6

Transmission spectra ($1-T_0$) for zeroth diffraction order when a TE-polarized light is normally incident ($Q=0$) on grating with $b/d=0.3$, for $\mathrm{\Delta }=0$, shown from zero temperature (red) to finite temperature (green), for (a) single-layer graphene and (b) turbostratically stacked graphene. Red and green dash-dotted lines indicate $T_0=0$ for zero temperature and $T_{\text{el}}=0.3$ K (single) or $T_{\text{el}}=3$ K (turbostratic), respectively.

Figure 7

Transmission spectra for single-layer graphene ($N=1$), for the [(a), (b)] minus-first diffraction order, [(c), (d)] zeroth diffraction order, and [(e), (f)] first diffraction order, for the period $d$ adjusted so that $\mathrm{\Delta }=-1$ ($G_1=1$). Panels (b) and (d) zoom in the spectral features near $\mathrm{\Omega }=G_1$ in (a) and (c), respectively, for $Q=0$. (g) Absorption calculated using Eq. (B1), with waterfall increment of $0.02$. $A(\mathrm{\Omega }<2)=0$ for zero temperature. The gray solid lines correspond to the forbidden frequency range where Eq. (15) is not satisfied, and the curve has the meaning of the near-field amplitude.

Figure 8

As Fig. 7 but for $\mathrm{\Delta }=-2+1.667$ ($G_1=1.667$), illustrating attenuation of near-field and transmission amplitudes.

Figure 9

As Fig. 7 but for $\mathrm{\Delta }=0$ ($G_1=2$), illustrating the amplification of the near-field and transmission amplitudes.

Figure 10

Transmission spectra for turbostratic ($N=10$) graphene, for the [(a), (b)] minus-first diffraction order, [(c), (d)] zeroth diffraction order, and [(e), (f)] first diffraction order for detuning $\mathrm{\Delta }=-1$ ($G_1=1$). Panels (b) and (d) zoom in the spectral features near $\mathrm{\Omega }=G_1$ in (a) and (c), respectively, for $Q=0$. Similarly, panel (f) is a zoom in of panel (e) at $Q$ close to but not zero. Gray regions and curves indicate forbidden frequency regions. (g) Absorption spectrum obtained using Eq. (B1), with waterfall increment of $0.2$.

Figure 11

As Fig. 10 but for $\mathrm{\Delta }=-2+1.667$ ($G_1=1.667$).

Figure 12

As Fig. 10 but for $\mathrm{\Delta }=0$ ($G_1=2$).

Figure 13

As Fig. 10 but for $\mathrm{\Delta }=0.5$ ($G_1=2.5$).