Fig. 8
From: Hyperbolic metamaterials: fusing artificial structures to natural 2D materials
b Adapted with permission from ref. [206], copyright 2016, APS. c Adapted with permission from ref. [205], copyright 2014, APS. e, f Adapted with permission from ref. [210], copyright 2014, AAAS. h, i Adapted with permission from ref. [47], copyright 2018, NPG. k, l Adapted with permission from ref. [211], copyright 2021, NPG
Hyperbolic responses in natural materials a The surface conductivity of black phosphorus along two distinct directions. Here, \(\eta _{xx}\) = \(\eta _{yy}\) = 0.01 eV, \(m_{xx}\) = 0.2\(m_0\), \(m_{yy}\) = \(m_0\), \(\omega _{xx}\) = 1 eV, \(\omega _{yy}\) = 0.35 eV, \(s_{xx}\) = 1.7\(\sigma _0\), \(s_{yy}\) = 3.7\(\sigma _{0}\); \(m_0\) is the free electron mass, and \(\sigma _0\) = \(e^2/4\hbar \). The electron density is 10\(^{14}\) \(\text {cm}^{-2}\). b The numerically obtained distribution of HPPs at 0.3 eV (in logscale of \(\left| E \right| \)). c The dispersion along x and y directions, for electron doping of 1013 \(\text {cm}^{-2}\), and q is the momentum along x (right axis) and y (left axis) directions. d Permittivity of hexagonal boron nitride (hBN). e Real-space imaging of HPhPs in hBN at 1.550 \(\text {cm}^{-1}\) for hBN with thickness of 256 nm. The scale bar is 800 nm. f Dispersions of HPhPs in hBN slab with thickness 134 nm on top of the \(\text {SiO}_{{2}}\) substrate. Black dashed curves: theoretical values according to Eq. (9); circles and rectangles: measured data. g Permittivity of alpha-phase molybdenum trioxide (\(\alpha \)-\(\text {MoO}_{{3}}\)). h Real-space imaging of HPhPs in \(\alpha \)-\(\text {MoO}_{{3}}\) nanodisk with thickness of 144 nm. i Experimentally obtained dispersion and analytical dispersions. j The normalized optical conductivity of BP along armchair and zigzag directions. k The loss function of HExPs in monolayer BP. The xx and yy mean the armchair and zigzag directions respectively. l The in-plane dispersions of HExPs in BP at different frequencies.