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Abstract
The naturally hyperbolic materials that conquer the limitations of artificially structured hyperbolic metamaterials are promising candidates for the emerging devices based on light. However, the variety of natural hyperbolic materials and their hyperbolic frequency regime are presently limited. Here, on the basis of first-principles calculations, we demonstrated a family of natural hyperbolic materials, graphite-like metal diborides, with a broadband hyperbolic region from near-IR (∼2.5µm) to the ultraviolet regime (∼248 nm). The operating hyperbolic window and negative refraction can be effectively modulated by extracting electrons from the materials, offering a promising strategy for regulating the optical properties.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Hyperbolic materials refer to a medium in which the permittivity and permeability tensor elements are of opposite signs [1]. For the electric hyperbolic materials, we only consider the permittivity written as:
Hyperbolic light dispersion has been achieved in artificially engineered metamaterials composed of metallic nanostructures embedded in a dielectric medium [6,7,9,20–22]. However, the performance is limited by the finite size of the metallic components. Compared with the metamaterials, natural hyperbolic materials have obvious advantages [23–28]. They require no complicated nanofabrication processes and avoid internal interfaces associated with imperfections for the electrons to scatter off, which lower the energy loss. Moreover, the hyperbolic frequency window is greatly extended without the constraint of the length scale of metallic component of metamaterials. Recently, hyperbolic dispersion has been predicted in the crystal materials with layered structures, such as MgB2 [23], cuprate and ruthenate [23], graphite [29], the tetradymites Bi2Se3 and Bi2Te3 [30], transition metal dichalcogenides (TMDs) [25]. The strongly anisotropic atomic and electronic structures of these natural hyperbolic materials offer a useful guidance for the subsequent exploration. However, the rarity of the natural hyperbolic materials and the narrow frequency window limit their device applications. Few materials have been reported to have the frequency window ranging from near-infrared to the ultraviolet regime.
In this work, we report on the discovery of a class of natural hyperbolic materials, metal diborides with a broadband of hyperbolic light dispersion from near-infrared to ultraviolet regimes. More interestingly, the hyperbolic frequency window can be effectively tuned by extracting electrons from the materials. The all-angle negative refraction for transverse magnetic (TM) polarized incident waves was also verified in these natural hyperbolic materials. Considering the experimental observation of hyperbolic dispersion in MgB2 with a limited frequency window [31,32], our findings offer not only a number of natural hyperbolic materials superior to MgB2, but also a promising strategy to regulate the hyperbolic properties of the mediums.
2. Method
Our first-principles calculations were performed within the framework of density-functional theory (DFT) as implemented in the Vienna ab initio simulation package (VASP) [33] using the projector augmented-wave method [34]. The energy cut-off for the plane-wave expansion was set to 500 eV. During geometry optimization, the lattice constants and the atomic positions were fully relaxed until the atomic forces on the atoms were less than 0.01 eV/Å and the total energy change was less than 10−5 eV. The van der Waals (vdW) interaction was taken into account by using the DFT-D3 strategy [35]. The Brillouin zone (BZ) integrations were carried out by using Monkhorst-Pack k-point grids [36] of 13×13×10 for geometry optimization, while a denser k-mesh of 38×38×29 was adopted to ensure the convergence of the optical property calculations. The Fermi level regulating was achieved by extracting electrons from the lattice and adding a homogeneous background charge of opposite sign. For the exchange-correlation functional, the generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) [37] was used to relax the structural parameters and calculate the electronic and optical properties, while the hybrid functional (HSE06) [38] was adopted to calculate the electronic properties and permittivities of CaB2 for comparison. The validity of the DFT calculations in reproducing the electronic structures of MgB2 and other related borides has been verified in many pervious literatures [39–41]. Additionally, the hyperbolic frequency regime of MgB2 predicted by our DFT calculations agrees well the experimental data [23]. We also adopted different vdW methods, including vdW-DF [42], optB88-vdW [43], optB86b-vdW [44] and DFT-D2 [45] in our calculations. It was found that different vdW methods have negligible effect on the hyperbolic frequency regime with the average difference less than 0.1 eV. The numerical simulations of negative refraction are confirmed in a 6 μm × 2 μm CaB2 sample by the finite-element method (FEM) implemented in COMSOL MULTIPHYSICS [46,47].
3. Results and discussion
The metal diborides (MB2) considered in this work have layered structures with a point group of p6/mmm, which are similar to that of graphite, as shown in Fig. 1. The atoms in the metal layer form a planar triangular lattice, while the boron layer exhibits a honeycomb configuration. The adjacent layers show an AB stacking pattern with the interlayer distances of about 1.46-2.48 Å. MgB2, a representative of this kind of materials, has been synthesized and extensively investigated, revealing interesting properties such as covalent bonds driven metallic and high-temperature superconductivity [31,32,48–50]. The other fourteen metal diborides considered in this work were constructed by replacing the Mg with group IIA alkaline earth metal atoms or group IIIB-VIIB transition metal atoms. In the following parts, we took CaB2 as an example to discuss the electronic and optical properties of the metal diboride materials.
Fig. 1. The crystal structure of metal diborides (MB2) contains graphite-like layers of B, which are intercalated by hexagonal close-packed layers of metal atoms. Fifteen types of metal atoms as listed in the insets were considered in our calculations.
The equilibrium lattice constants of CaB2 are a = b = 3.193 Å and c = 4.046Å, compared to those of MgB2, a = b = 3.049 Å and c = 3.483 Å. The interlayer distance of CaB2 is longer than that of MgB2 by about 0.28 Å and thus leads to weaker interlayer coupling, due to the different ionic radii, rMg = 0.74 Å and rCa = 1.04 Å. The electronic band structure of CaB2 is plotted in Fig. 2(a). We can see the intrinsic metallic nature of CaB2 with few bands crossing the Fermi level. The orbital-resolved band lines show that the metallic properties of CaB2 are mainly dominated by the σ bands of the px,y orbitals of B and π bands of pz orbitals, analogous to the case of MgB2 [51]. The larger interlayer distance in CaB2 leads to less dispersive pz bands along the z-direction and enhances the anisotropy of the electronic structures. The remarkable anisotropy of electronic band properties of CaB2 will be responsible for the anisotropic dielectric properties as described below.
Fig. 2. (a) Orbital-resolved band structure of CaB2 along the high symmetry in the Brillouin zone. The energy of the Fermi level is set to zero. The contributions of px, py orbitals of B and pz orbitals of B are indicated by red and blue disks, respectively. (b) Real and imaginary part of the permittivities of CaB2 obtained at the PBE level. The blue shaded region shows the hyperbolic frequency window. The triangle symbols indicate the hyperbolic frequency regions obtained from HSE06 functional. Real and imaginary part of the permittivities of (c) SrB2 and (d) CrB2 obtained at the PBE level.
Permittivity which was commonly adopted to represent the optical properties of crystalline solids is composed of a real part and an imaginary part ɛ(ω) = Re ɛ(ω) + iIm ɛ(ω), which is contributed by the intra-band and inter-band transition. The inter-band transition of the imaginary part Im ɛ(ω) can be calculated from the standard DFT calculations, while the real part Re ɛ(ω) can be determined from the Kramers-Konig relation [52]. The contribution of intra-band transition in metallic materials can be approximately described by the Drude model [53],
The two components of the permittivity of CaB2, ${{\varepsilon }_ \bot }({\omega } )$ and ${{\varepsilon }_\parallel }({\omega } )$ along the direction perpendicular and parallel to the anisotropy axis (z-axis) are plotted in Fig. 2(b). We can see two obvious type-II hyperbolic frequency regimes with ${{\varepsilon }_ \bot }({\omega } )$ < 0 and ${{\varepsilon }_\parallel }({\omega } )$ > 0 from 0.80 eV to 1.87 eV and from 2.12 eV to 3.26 eV. The hyperbolic frequency regimes cover the near-infrared, visible and ultraviolet regimes and thus are thus accessible in experiments. In order to verify the reliability of the PBE functional in predicting the hyperbolic frequency regimes, we calculated the permittivities of MgB2 crystal. The type-II hyperbolic frequency regime 2.34-3.54 eV given by our first-principles calculations is very close to the experimental data 2.60-3.72 eV obtained from the spectroscopic ellipsometry measurements at room temperature [23]. Additionally, we found that the anisotropy in plasma frequencies of CaB2, 5.27 eV (in-plane) and 2.78 eV (out-of-plane) is more prominent than those of MgB2, 7.27 eV (in-plane) and 6.85 eV (out-of-plane), which can be attributed to the strongly anisotropic atomic and electronic structures of CaB2. The stronger anisotropy of plasma frequencies of CaB2 leads to a wider hyperbolic frequency regime than MgB2. The permittivities of SrB2 and CrB2 were presented in Figs. 2(c) and 2(d) for comparison. SrB2 has a type-II hyperbolic frequency regime 0.99-2.82 eV. Although SrB2 is isoelectronic to CaB2, the larger ionic radius of Sr leads to more band overlap along the z-direction, which contributes to weaker anisotropy. For CrB2, the type-I hyperbolic frequency regime can be ascribed to the high metallicity along the z-direction. It also has a fairly wide hyperbolic window 2.25-4.70 eV, but suffers from high energy loss.
The hyperbolic frequency regimes of the fifteen metal diborides obtained by our first-principles calculations are plotted in Fig. 3. Hyperbolic frequency windows ranging from the near-infrared regime to the ultraviolet regime can be found in these layered materials. Beside MgB2, CaB2, SrB2, BaB2, VB2 and SrB2 exhibit broadband hyperbolic frequency windows, offering a class of candidates for achieving hyperbolic light dispersion and relevant device applications working in wide-range frequencies.
Fig. 3. The hyperbolic frequency regimes of the 15 metal diborides. The Type I and Type II hyperbolic frequency ranges are shown in green and blue bold lines, respectively.
Another interesting scenario of the metal diborides is the tunability of the electronic properties by regulating the Fermi level of the materials. For example, it has been found that the superconducting properties of MgB2 and other metal diborides can be manipulated by electron or hole doping [51,54–56]. This idea can be naturally extended to the optical properties of the metal diborides. From the band structure of CaB2 in Fig. 2(a), we can see that along the Г-A direction the Fermi level crosses the band only in the region near the Г point, while several bands cross the Fermi level along other directions. As the wave vector along the Г-A direction corresponds to the Bloch wavefunction propagating along the z-direction, the anisotropy of the electronic band structure will be greatly enhanced if the band across with the Fermi level is avoided by lowing the Fermi level. To mimic this effect, we extracted one electron in a unit cell of bulk CaB2, corresponding to a hole doping concentration of 2.86 × 1022 cm-3. The electronic band structure of the hole-doped CaB2 is plotted in Fig. 4(a). The Fermi level is pushed down and has no intersection with the electronic bands along the Г-A direction. The permittivities of the hole-doped CaB2 show a broad hyperbolic frequency regime, 0.5-3.96 eV (0.313 - 2.48 μm) which is greatly extended compared with the undoped CaB2. The energy loss determined by imaginary part of permittivity has been obviously suppressed, as shown in Fig. 4(b).
Fig. 4. (a) The electronic band structure of CaB2 within doping a hole. (b) The permittivities of the hole-doped CaB2.
The hyperbolic frequency regimes of the fifteen metal diborides at the same hole-doping concentration (extracting one electron from a unit cell) are plotted in Fig. 5. These metal diborides exhibit different responses in permittivity to the hole-doping. The hyperbolic frequency windows of MgB2 and CrB2 move to the high energy regime, whereas those of the CaB2, SrB2, ScB2 and YB2 are greatly extended compared with the undoped materials and cover the whole visible light regime. The hyperbolic light dispersion in the visible light regime is highly demanding in light device applications.
Fig. 5. Calculated hyperbolic frequency regimes of 15 metal diborides with doping a hole. The Type I and Type II hyperbolic frequency ranges are shown in green and blue bold lines, respectively.
To reveal the origins of the tunable hyperbolic frequency regimes, we plotted the real part of the permittivity $\textrm{Re}\varepsilon (\omega )$ of pristine and hole-doped CaB2 in Fig. 6(a). In the hyperbolic frequency window, CaB2 has a positive $\textrm{Re}{{\varepsilon }_\parallel }({\omega } )$ and a negative $\textrm{Re}{{\varepsilon }_ \bot }({\omega } )$, and the boundaries of the window are defined by $\textrm{Re}{{\varepsilon }_\parallel }({\omega } )= 0$ and ${\textrm{Re}}{{\varepsilon }_ \bot }({\omega } )= 0$. We can see that the hole-doping push the ${{\varepsilon }_{0\parallel }}({\omega } )$ and ${{\varepsilon }_{0 \bot }}({\omega } )$ to the low and high energy regions, respectively, expanding the hyperbolic frequency regions. In addition, the intra-band and inter-band transitions of electrons contribute differently to the permittivity. The intra-band transition modeled by the free electron theory determines the permittivity at low energy region, resulting in negative values of ${\textrm{Re}}{\varepsilon }({\omega } )$. The inter-band transition, however, dominates the permittivity at high energy region. The decomposed permittivities along the perpendicular and parallel directions of the pristine and hole-doped CaB2 are plotted in Figs. 6(b) and 6(c). We can see that the enlarged hyperbolic frequency window is mainly attributed to the changes of intra-band contributions, i.e., the redshift of $\textrm{Re}{{\varepsilon }_\parallel }^{intra}({\omega } )$ and blueshift of $\textrm{Re}{{\varepsilon }_ \bot }^{intra}({\omega } )$. In the Drude model of intra-band transitions, the plasma frequency ${\omega _{p,\alpha \beta }}$ which characterizes the collective oscillations of the valence electrons plays a decisive role. The magnitude plasma frequency depends on the conducting electron density and the electron velocity $\frac{{\partial {E_{n,k}}}}{{\partial k}}$, according to the formula [48,57]:
Fig. 6. (a) The real parts of the permittivities Re ɛ of the undoped CaB2 (solid lines) and doped CaB2 (dashed lines). (b) The intra-band contributions of Re ɛ. (c) The intra-band contributions of Re ɛ. The red areas (I, II and III) indicate the expanded hyperbolic regions after hole-doping. Black and red lines indicate the permittivities along vertical and parallel directions, respectively.
The inter-band transition dominates the imaginary part of permittivity and thus the energy losses of the materials. The contribution of the inter-band transition to the imaginary part of permittivity can be obtained from the momentum matrix elements between the occupied and unoccupied wavefunctions [58–60]:
Fig. 7. Imaginary parts of permittivities of pristine and hole-doped CaB2 contributed by inter-band transitions along (a) in-plane direction and (b) out-of-plane directions, respectively. Diagrams of inter-band transitions of (c) intrinsic and (d) hole-doped CaB2. The arrows indicate the direction of the transition processes between bands. The yellow shaded area represents the ‘parallel-bands’ effect.
Finally, we demonstrated the all-angle negative refraction in these hyperbolic materials. Considering a transverse magnetic (TM) polarized light incident from air into a type II hyperbolic material. Let the optic axis (z-direction) be parallel to the air/MB2 interface and incident plane be the x-z plane, as shown in Fig. 8(a). We plotted the equifrequency contour (EFC) of the light dispersion,
Fig. 8. (a) The orientation of MB2 and the scheme of negative refraction when a TM polarized light is incident from air to MB2. The direction perpendicular to the layers is taken as the optic axis (z-axis). (b) The equifrequency contour (EFC) of the light dispersion relation. The negative refraction happens at the interface between the air (black circle) and MB2 (red hyperbola). The shaded region describes the incident angles for which negative refraction may occur. The solutions indicated by the dashed arrows are physically incorrect. (c) Simulation results for 756 nm TM light at incident of 30°. The color map shows the distribution of the electric field, and Poynting vectors are marked by black arrows.
In the isotropic medium, e.g. air, the Poynting vector ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over S} _i}$ is in the same direction as the wavevector ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k} _i}$. When the electromagnetic wave enters into a hyperbolic material, however, the Poynting vector of refractive waves ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over S} _r}\; $ and the wavevector ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k} _r}$ are on the opposite side of the interface normal, as marked in Fig. 8(b). This scenario is called negative refraction. From the EFC, we can see that there are two solutions ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k} _r}$ and ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k} _r}^{\prime}$ on both side of ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k} _z}$ satisfying the continuity of the tangential component of ${\vec {\textrm k}}$ but according to the causality principle that the energy must flow away from the interface, i.e., ${S_z} = {\hat{e}_z} \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over S} > 0$ [1,8,61–63], only the ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k} _r}$ is physically correct. The refraction angle can be determined by the Poynting vector as [8,24]
The negative refraction can be mimicked by numerically solving the Maxwell equations with a finite element method (FEM). In the FEM simulation, we simplify the CaB2 crystal as a homogenous slab and put the air/CaB2 interface at x = 0, as shown in Fig. 8(a). The excitation wavelength was set to be 756 nm, corresponding to the chosen photon energy of 1.64 eV, and the incident angle of the transverse Gaussian beam is set to be 30 degrees. The permittivity tensors of CaB2 at this photon energy evaluated from abovementioned strategy are ɛ⊥(ω) = -4.90 + 7.34i and ɛ∥ (ω) = 8.17 + 2.52i, respectively. The electric filed distribution obtained by the FEM simulation is plotted Fig. 8(c). The Poynting vectors signed by arrows confirms the negative refraction effect at the interface.
4. Conclusions
In summary, using first-principle calculations, we demonstrated a class of metal diborides as promising hyperbolic materials with broadband hyperbolic frequency regimes and tunable optical properties. The hyperbolic frequency windows of CaB2, SrB2, VS2 and CrB2 are wider than that of MgB2 achieved in experiments, due to the enhanced structural anisotropy. More interestingly, the hyperbolic frequency window can be tuned by extracting electrons from the materials, achieving a broadband hyperbolic window from near-infrared (∼248 nm) to ultraviolet (∼2.5 µm) regimes in the hold-doped CaB2. The all-angle negative refraction for transverse magnetic polarized incident light was also verified in these natural hyperbolic materials by solving the Maxwell equations with a finite-element method. Our findings offer not only a new class of natural hyperbolic materials with broadband hyperbolic frequency windows but also a promising strategy to regulate the hyperbolic properties of the mediums.
Funding
National Natural Science Foundation of China (11774201, 21433006, 21833004); Basic Research Project of Natural Science Foundation of Shandong Province (ZR2018ZB0751).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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