Tunable hyperbolic dispersion and negative refraction in natural electride materials

Hyperbolic (or indefinite) materials have attracted significant attention due to their unique capabilities for engineering electromagnetic space and controlling light propagation. A current challenge is to find a hyperbolic material with wide working frequency window, low energy loss, and easy controllability. Here, we propose that naturally existing electride materials could serve as high-performance hyperbolic medium. Taking the electride ${\mathrm{Ca}}_2\mathrm{N}$ as a concrete example and using first-principles calculations, we show that the material is hyperbolic over a wide frequency window from short-wavelength infrared to near infrared (from about $3.3\mu \mathrm{m}$ to 880 nm). More importantly, it is almost lossless in the window. We clarify the physical origin of these remarkable properties and show its all-angle negative refraction effect. Moreover, we find that the optical properties can be effectively tuned by strain. With moderate strain, the material can even be switched between elliptic and hyperbolic for a particular frequency. Our result points out a new route toward high-performance natural hyperbolic materials, and it offers realistic materials and novel methods to achieve controllable hyperbolic dispersion with great potential for applications.

Figure 1

(a) Plot of the electron density distribution for states around Fermi energy (within an energy window of 0.1 eV), which shows the anionic electrons distributed between the Ca-N-Ca triple layers. The dashed line marks the primitive unit cell. (b) Electronic band structure of ${\mathrm{Ca}}_2\mathrm{N}$. The band that crosses the Fermi level (with yellow color) is from the anionic electrons mostly distributed in the interlayer regions [as illustrated in (a)]. (c) Brillouin zone with symmetry points labeled.

Figure 2

Real and imaginary parts of the two principal components of permittivity of ${\mathrm{Ca}}_2\mathrm{N}$. The shaded region shows the frequency window, in which the material is hyperbolic.

Figure 3

(a) Plot of equifrequency contours (EFCs) followed from Eq. (1). The blue circle is the EFC of air, and the red hyperbola is the EFC of ${\mathrm{Ca}}_2\mathrm{N}$. Here we take a photon energy of 1.179 eV. (b) Negative refraction occurs when a TM polarized light is incident from air towards ${\mathrm{Ca}}_2\mathrm{N}$. The blue (red) arrow indicates the direction of Poynting vector ${\mathit{S}}_i$ (${\mathit{S}}_r$) of the incident (refracted) light. Note that the crystal $c$ axis is along the $z$ direction, i.e., parallel to the interface. (c) Simulation result explicitly showing the negative refraction effect. The setup follows from (b): air (${\mathrm{Ca}}_2\mathrm{N}$) occupies the region of $x>0$ ($x<0$). The photon energy is taken to be 1.179 eV. The incident angle is ${45}^{\circ }$. The color map shows the electric field strength, and the black arrows indicate the directions of local Poynting vectors.

Figure 4

Strain-stress relation for ${\mathrm{Ca}}_2\mathrm{N}$ with uniaxial strains along the $z$ direction.

Figure 5

Real and imaginary parts of the permittivity of ${\mathrm{Ca}}_2\mathrm{N}$ under (a) $-5$% and (b) $+5$% strains. Note the sizable change in the range of hyperbolic window (the shaded region). (c),(d) Simulation results for the refraction process near the air/${\mathrm{Ca}}_2\mathrm{N}$ interface with ${\mathrm{Ca}}_2\mathrm{N}$ under two different strains: (c) $-5$% and (d) $+5$%. The setup is similar to Fig. 3. The photon energy is taken to be 1.453 eV, and the incident angle is ${45}^{\circ }$. The directions of local Poynting vectors are indicated by black arrows.