From surface to volume plasmons in hyperbolic metamaterials: General existence conditions for bulk high-$k$ waves in metal-dielectric and graphene-dielectric multilayers

We theoretically investigate general existence conditions for broadband bulk large-wave-vector (high-$k)$ propagating waves (such as volume plasmon polaritons in hyperbolic metamaterials) in subwavelength periodic multilayer structures. Describing the elementary excitation in the unit cell of the structure by a generalized resonance pole of a reflection coefficient and using Bloch's theorem, we derive analytical expressions for the band of large-wave-vector propagating solutions. We apply our formalism to determine the high-$k$ band existence in two important cases: the well-known metal-dielectric and recently introduced graphene-dielectric stacks. We confirm that short-range surface plasmons in thin metal layers can give rise to hyperbolic metamaterial properties and demonstrate that long-range surface plasmons cannot. We also show that graphene-dielectric multilayers tend to support high-$k$ waves and explore the range of parameteres for which this is possible, confirming the prospects of using graphene for materials with hyperbolic dispersion. The approach is applicable to a large variety of structures, such as continuous or structured microwave, terahertz, and optical metamaterials.

Figure 1

Theoretical background on hyperbolic metamaterials (HMMs). (a) Isofrequency surfaces in the dispersion relation [Eq. (1)] for conventional uniaxial medium $\left(0<{\varepsilon }_z<{\varepsilon }_{x,y}\right)$ and HMM $({\varepsilon }_{x,y}<0$ and ${\varepsilon }_z>0)$. (b) An infinite periodic multilayer HMM with geometric notations and wave-vector decomposition used in the paper. (c) The central idea of the paper: the replacement of real metal layers with fictitious layers featuring just a polelike elementary excitation with reflection and transmission coefficients given by Eq. (5). (d) Comparison of the exact multilayer dispersion relation with the one derived from the pole expansion and the effective-medium approximation for the structure with $d_m=2.3$ nm, $d_d=11.4$ nm, and ${\varepsilon }_m=-17.2,{\varepsilon }_d=2.59$ (relevant for lossless Au/${\mathrm{Al}}_2{\mathrm{O}}_3$ structures for $\lambda =715$ nm [1]; see [2] for more detail). The yellow shaded area shows the band of propagating high-$k$ VPPs. (e) Plots of $\mathrm{Re}{k}_B$ and $\mathrm{Im}{k}_B/\mathrm{Re}{k}_B$ in the VPP band for lossy metal $\left({\varepsilon }_m=-17.2+0.8i\right)$.

Figure 2

Behavior of Eq. (9) in different regimes. Light colors (cyan to white to yellow) correspond to the area where $-1<F<1$ and propagating high-$k$ waves exist; dark colors (dark red and blue) indicate regions where $\left|F\right|>1$ and high-$k$ wave propagation is forbidden. (a) Illustration of the limiting behavior of $F(\beta ,x=ln|\xi \left|\right)$ for large $\beta$ and $x$; the dashed lines show the asymptotes given by Eqs. (12) and (13). (b)–(d) An enlarged view of $F(\beta ,x)$ around the asymptote intersection point $(\beta =1,x=\eta {\kappa }_p)$ for $\eta {\kappa }_p$ equal to (b) 4, (c) 1, and (d) 0.1. The insets show the plots in (b) and (d) in the same scale as (a) for comparison.

Figure 3

Formation of the VPP band in metal-dielectric HMMs from SRSPPs and LRSPPs. (a) Dependence of $x$ on the permittivities of metal and dielectric in a thin metal layer $(d_m=5$ nm) for SRSPPs and LRSPPs; the dotted line corresponds to $f=1$. (b) Example dependencies of $F(\beta ,x)$ for SRSPPs and LRSPPs in a structure with $d_m=d_d=5$ nm and material parameters as in Fig. 1. The insets show the enlarged view on the scale of ${\kappa }_p/(\omega /c)$. (c),(d) Comparison between SRSPP and LRSPP cases for (c) the dispersion relation of the VPP band and (d) the dependence of ${\kappa }_p$ and $x$ on $d_m$. The values for $x$ for plots in (b) are taken from the plots in (a) at the corresponding values of parameters (shown as a dot).

Figure 4

Graphene conductivity $\sigma$ in units of the elementary conductivity ${\sigma }_0=e^2/4\hslash =0.061$ mS in logarithmic scale. (a) Real and (b) imaginary part, depending on frequency $\omega$ and electrochemical potential (Fermi level) $E_F$. The dashed line corresponds to the Pauli blocking limit $\hslash \omega =2E_F$. The white region around the dashed line in (b) corresponds to the region of negative $\mathrm{Im}\sigma$. Also shown is the figure of merit (FOM) $\mathrm{Im}\sigma /\mathrm{Re}\sigma$ dependence on frequency and (c) $E_F$ for the damping $\gamma ={10}^{13}{\mathrm{s}}^{-1}$; (d) on the damping $\gamma$ for the fixed Fermi level $E_F=0.2\mathrm{eV}$. We consider the regions with FOM greater than zero suitable for HMMs.

Figure 5

Formation of the VPP band in graphene HMMs. (a) The dependence of $x$ on graphene conductivity and dielectric layer permittivity. (b) Same as Fig. 3 for graphene multilayers with $\sigma =20i{\sigma }_0$ [where ${\sigma }_0=e^2/\left(4\hslash \right)\text{]},d=5$ nm, and $\varepsilon =1.96$. (c),(d) Same as (a),(b) but for TE-polarized graphene plasmons. The values for $x$ for plots in (b),(d) are taken from the plots in (a),(c) at the corresponding values of parameters (shown as a dot).

Figure 6

(a) Propagation-to-attenuation ratio for the high-$k$ propagating waves, defined as $\rho =\mathrm{Re}{k}_B/\mathrm{Im}{k}_B=\mathrm{Re}arccosF/\mathrm{Im}arccosF$ as in Ref. [30], for the same conditions as in Fig. 2 for complex ${\kappa }_p$ with $\mathrm{Im}{\kappa }_p/\mathrm{Re}{\kappa }_p$ equal to 0.005, 0.01, and 0.02. Dashed lines denote the contour of the high-$k$ band in the lossless case. (b),(c) Similarly defined $\rho$ for (b) SRSPP excitations in metal-dielectric multilayer with realistic metal losses and (c) TM plasmon excitations in graphene-dielectric multilayers with $\sigma /{\sigma }_0=0.75+20i$ (FOM of 26.7) and other parameters similar to the insets in Fig. 3 and Fig. 5, respectively, and calculated along the horizontal dashed line in those insets.