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Abstract
Hyperbolic isofrequency of materials (referred to as hyperbolic materials) renders an unusual electromagnetic response and has potential applications, such as all-angle negative refraction, sub-diffraction imaging and nano-sensing. Compared with artificially structured hyperbolic metamaterials, natural hyperbolic materials have many obvious advantages. However, present natural hyperbolic materials are facing the limitations of narrow operating frequency intervals and high loss stemming from electron-hole excitations. Using first-principles calculations, we demonstrated that the recently-discovered nodal-line semimetallic yttrium nitride (YN) can be tuned to a type-I natural hyperbolic material with a broad frequency window from near-IR (∼1.4 μm) to the visible regime (∼769 nm) along with ultra-low energy loss, owning to the unique electronic band structure near the Fermi level. The unusual optical properties of YN, such as all-angle negative refraction and anisotropic light propagation were verified. The tunable hyperbolic dispersion can be interpreted in terms of the linear relation between critical frequency and plasma frequency. A branch of plasmon dispersion with strong anisotropy in the low-energy region was also revealed in the electron-doped YN. This work is expected to offer a promising strategy for exploring high-performance hyperbolic materials and regulating plasmon properties.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Recent advances in polaritons-involved light-matter interaction offer new avenues for control of light. Permittivity which determines the interaction of materials with electromagnetic fields is usually expressed as a tensor depending on crystal symmetry. Hyperbolic materials (HM), or indefinite materials, refer to a class of bulk materials whose real parts of permittivity or permeability have diagonal components of different signs [1]. Non-magnetic HMs can be simply classified to type-I ($Re{\varepsilon _ \bot } > 0$, $Re{\varepsilon _{||}} < 0$) and type-II ($Re{\varepsilon _ \bot } < 0$, $Re{\varepsilon _{||}} > 0$) HMs [2–6] where ${\varepsilon _{\textrm{xx}}} = {\varepsilon _{\textrm{yy}}} \equiv {\varepsilon _ \bot }$ and ${\varepsilon _{\textrm{zz}}} \equiv {\varepsilon _{||}}$ (εxx, εyy and εzz are the three diagonal elements of the permittivity tensor). The dispersion relation of hyperbolic materials is set by the equation [7]:
The conventional HMs are nanostructured metamaterials with two components: metallic medium and dielectric medium. Through regulating structural parameters and combining components in different ways, such as multilayers of alternating metal-dielectric layered structures [10,17] and metallic nanowires embedded in a dielectric matrix [9,18], the dielectric properties of hyperbolic metamaterials can be effectively modulated. However, manufacturing high-quality hyperbolic metamaterials remains a challenge, because it requires materials with different crystalline characteristics to grow orderly in the sub-wavelength scale [19]. Moreover, the limited maximum wave vector kmax and relatively high loss caused by electrons scattering off from the metal-dielectric interfaces, lead to short propagation distances and low transmission efficiency [20,21].
Natural materials with intrinsic hyperbolic light dispersion, referred to as natural HMs, have many obvious advantages over artificially structured hyperbolic metamaterials in large kmax and low interface scattering. A number of natural HMs have been recently demonstrated, such as graphite [22], graphite-like materials [23,24], transition metal dichalcogenides [25] and in-plane anisotropic α-MoO3 [26], most of which are type-II HMs [27,28]. The layered components of these materials bring about strongly anisotropic dielectric properties, enabling them potential candidates for natural HMs. However, most of these natural HMs suffer from high energy loss and narrow frequency windows for hyperbolic dispersion [28].
The unique electronic states of topological materials, such as topological insulators [29,30] and topological semimetals [31,32], lead to the nontrivial electron transport properties and fascinating optical response. Here, on the basis of first-principles calculations, we demonstrate that the recently proposed nodal-line semimetallic yttrium nitride (YN) [33] serves as a type-I natural HM with a tunable frequency region under electron doping. The hyperbolic frequency window can be effectively modulated by controlling the electron doping concentration, resulting in a broad frequency window from near-IR (∼1.4 μm) to the visible regime (∼769 nm). Meanwhile, electron doping greatly suppresses the inter-band transition, giving rise to extremely low energy loss in the hyperbolic region. Furthermore, the doping effect also induces a branch of plasmon dispersion in the low energy region with strong anisotropy along the in-plane and out-of-plane directions. Our findings not only expand the family of natural HMs to topological materials, but also offer a promising strategy to explore a high-performance hyperbolic medium and regulate the plasmonic properties.
2. Method and computational details
Our first-principle calculations were performed within the framework of density-functional theory (DFT) as implemented in the Vienna ab simulation package (VASP) [34] and the GPAW code [35], both of which make use of the projector augmented-wave method [36]. The exchange-correlation functional was treated self-consistently with a generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) realization [37]. The cutoff energy for the plane-wave expansion was 500 eV in VASP and GPAW. The Brillouin zones were sampled by Monkhorst-Pack scheme [38]. VASP was used for structure relaxation and electronic properties on the 11×11×11 k-point mesh. 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 electron doping effect was simulated by adding electrons to the materials in a homogeneous background charge of opposite sign. The optical properties were performed using linear response calculations [39] as implemented in the GPAW code. A denser k mesh of 34×34×33 was adopted to converge the optical calculations. The dielectric matrix was calculated using the random phase approximation (RPA) as [39]:
The optical properties of crystals can be described by the permittivity $\varepsilon (\omega )= \textrm{Re}\varepsilon (\omega )+ i\textrm{Im}\varepsilon (\omega )$, where the imaginary part $\textrm{Im}\varepsilon (\omega )$ corresponds to the energy loss due to electron transitions. For metallic materials, both inter-band and intra-band transitions contribute to the permittivity. For the inter-band part, the imaginary term $\textrm{Im}[{\varepsilon (\omega )_{\alpha \beta }^{\textrm{inter}}} ]$ are calculated directly from the inter-band transition, while the real term $\textrm{Re}[{\varepsilon (\omega )_{\alpha \beta }^{\textrm{inter}}} ]$ can be derived from $\textrm{Im}[{\varepsilon (\omega )_{\alpha \beta }^{\textrm{inter}}} ]$ according to the Kramers-Kronig relation [44,45]. The intra-band transitions are usually described using the Drude model [46,47]:
3. Results and discussion
Among the different crystal structures of bulk YN [48], the WC-type hexagonal structure [49] was predicted to have nodal-line semimetal features [33]. The non-centrosymmetric hexagonal YN contains two atoms per unit cell with a space group of $P\bar{6}m2$, as shown in Fig. 1(a). The Wyckoff sites show the positions of atom Y at the 1d Wyckoff site (1/3, 2/3, 1/2) and the atom N at the 1a Wyckoff site (0, 0, 0). The optimized lattice constants given by our DFT calculations are a = b = 3.366 Å and c = 3.062 Å. From the orbital-resolved electronic band structure plotted in Fig. 1(b), one can clearly see that two bands cross along the Г-M and Г-K directions right at the Fermi level, forming a nodal ring with the radius of 0.15 Å-1 around the Г point. The electronic states near the Fermi level are contributed mainly by the ${d_{{z^2}}}$ orbitals of Y and ${p_z}$ orbitals of N. Our results are in good agreement with those reported in a previous literature [33].
Fig. 1. Lattice structure, electronic and optical properties of pristine YN (a) The unit cell of hexagonal YN crystal. Y and N atoms are represented respectively by the large and small balls. (b) The orbital-resolved electronic band structure of YN. The energy at the Fermi level was set to zero. The inset gives the zoom-in band structure of the nodal ring around the Г point. (c) Real and imaginary parts of the permittivity of YN. The shaded region shows the hyperbolic frequency window. The solid lines and dotted lines represent the data obtained by using γ = 0.002 and 0.04 eV, respectively.
The permittivity of pristine YN bulk crystal is plotted in Fig. 1(c), in which ${\varepsilon _ \bot }(\omega )$ and ${\varepsilon _{||}}(\omega )$ represent the permittivities perpendicular and parallel to the anisotropy axis (z-axis), respectively. A narrow hyperbolic frequency region from 0.119 to 0.249 eV appears in the low energy (mid-IR) region. The flat nodal ring leads to a low Fermi velocity and thus small plasma frequencies of ${\omega _{p, \bot }} = \textrm{0}{.34\; {\textrm{eV}}}$ and ${\omega _{p,\textrm{||}}} = \textrm{0}{.81\; {\textrm{eV}}}$. Moreover, the inter-band transitions originated from the nodal ring cause high energy loss reflected by large $\textrm{Im}\varepsilon (\omega )$ in the hyperbolic region, which is disadvantageous for light propagation. Notably, in the conduction band region (0.07 eV-3.0 eV above the nodal ring), the band is almost linearly along the Г-A direction, but quadratically dispersive along the Г-M and K-Г directions. Such strong electronic anisotropy determines the indefinite permittivity over a wide spectrum region. More interestingly, the band is well separated from others in a wide energy range, underlying low energy loss in the corresponding photon energy range. We therefore considered an electron-doped YN in which the Fermi level is pushed upwards to this region. At the electron doping concentration of about 1022 cm-3, the Fermi level is lifted by about 0.9 eV compared with the pristine YN, as shown in Fig. 2(a). The plasma frequencies of the electron-doped YN are greatly increased to ${\omega _{p, \bot }} = \textrm{2}{.42\; {\textrm{eV}}}$ and ${\omega _{p,\textrm{||}}} = \textrm{4}{.75\; {\textrm{eV}}}$. The real and imaginary parts of the permittivity of the electron-doped YN along different directions are shown in Fig. 2(b). Re${\varepsilon _ \bot }(\omega )$ and Re${\varepsilon _{||}}(\omega ) $ cross zero at different frequencies, forming a type-I hyperbolic region of 0.874 - 1.613 eV which is much wider than that of pristine YN. More interestingly, the imaginary parts of the permittivities are almost completely suppressed over a wide frequency region, containing the hyperbolic window. Such a low-loss broadband (from near-IR to visible light) type-I hyperbolic material is quite promising for achieving the relevant optical effects, such as all-angle negative refraction, sub-diffraction imaging and nano-sensing.
Fig. 2. Electronic and optical properties of electron-doped YN (a) The electronic band structure of YN with the Fermi level being modulated by different electron-doping concentrations (ne). The red dotted lines indicate the Fermi levels at ne = 0, 1.0× 1022, 2.0× 1022, and 3.33× 1022 cm-3. (b) Real and imaginary parts of the permittivities of electron-doped YN at ne = 1.0× 1022 cm-3. The solid lines and dotted lines represent the data obtained by using γ = 0.002 and 0.04 eV, respectively. (c) The isofrequency contour for air (blue), doped YN (red) and intrinsic YN (green) in the photon energy of 1.49 eV. (d) Isofrequency curves for real (blue) and imaginary (red) parts of wavevector of the electron-doped YN at ne = 1.0× 1022 cm-3 in the photon energy of 1.49 eV, with ${\varepsilon _ \bot } = 5.\textrm{60} + 0.32i$, ${\varepsilon _{\textrm{||}}} = \textrm{ - }1.67 + 0.39i$. The unit of the wavevector (k) is Å-1.
We take the photon energy of 1.49 eV as an example to demonstrate the unusual optical properties of the electron-doped YN. The permittivities at this photon energy are ε|| = -1.67 + 0.39i and ε⊥ = 5.60 + 0.32i. The isofrequency surfaces of extraordinary waves are plotted in Fig. 2(c), from which we can see a typical type-I hyperboloid with two-fold symmetry [6]. Any real kx corresponds to a solution of kz, indicating that the electromagnetic wave can propagate without any cut-off until reaching the limitation of lattice length scale [27,50]. The ellipsoidal isofrequency surface (Reε||(ω) = 7.62, and Reε⊥(ω) = 8.88) of pristine YN at the same photon energy was also presented for comparison. The transition of YN from an ellipsoidal material to a hyperbolic material by electron doping is quite evident from this figure.
For simplification, we project the incident plane to the x-z plane and set the optic axis (x-direction) parallel to the air/YN interface. The dispersion for incident TM-polarized plane waves can be given by Eq. (1). To show the light propagation properties of this system more clearly, we plotted the k-curves composed of both real (Re(k)) and imaginary (Im(k)) parts of wavevector for TM electromagnetic waves. Re(k) represents how many wavelengths of modes can be supported in various propagation directions, while Im(k) reflects the mode attenuation [51,52]. Figure 2(d) gives the isofrequency curves (ω/c = 1.0 Å-1) of Eq. (1) in hyperbolic dispersion regime of doped YN with εz = -1.67 + 0.39i and εx,y = 5.60 + 0.32i. Due to the presence of loss, the ideal hyperbola is truncated in very large wave-vectors (λ > λc). The material will become a non-ideality hyperbolic medium as λ > λc. Fortunately, the λc of the electron-doped YN is in micrometer order of magnitude which exceeds the sizes of some nanostructures. The supported modes propagate mostly in z-direction, while the propagating direction of the damping modes is along x-direction.
All-angle refraction effect is an interesting scenario of HMs. Usually, type-I HMs have lower reflection and better performance in high-resolution imaging than type-II HMs [6], which means electromagnetic waves can propagate through the HMs with a superior transmission. When a TM light (H is along the y-axis) goes into a uniaxial media with dielectric constant ${\boldsymbol \varepsilon } = {\hat{u}_x}{\varepsilon _ \bot } + {\hat{u}_y}{\varepsilon _ \bot } + {\hat{u}_z}{\varepsilon _{||}}$, the time-averaged Poynting vector S representing the directional energy flux is defined as [8]:
Fig. 3. Hyperbolic dispersion and negative refraction phenomena of electron-doped YN. (a) The equifrequency contour (EFC) projected onto the kx-kz plane. The negative refraction happens at the interface between the air (blue circle) and doped YN (red hyperbola). The refracted wave vectors and Poynting vectors are indicated by the solid blue and the yellow arrows, respectively. The solutions represented by the dashed arrows are physically incorrect. (b) Simulated electric field distribution in the x-z plane for the TM light with λ=832 nm and an incident of 45°. The color map shows the distribution of the electric field, and the Poynting vectors are marked by dark gray arrows.
We further confirmed the negative refraction in the electron-doped YN by numerically solving the Maxwell equations with a FEM. The YN is reduced to a homogeneous slab and only the x-z plane is considered in our simulations. Air and YN are respectively placed in z > 0 and z < 0 region. The excitation wavelength was set to 832 nm, corresponding to the photon energy of 1.49 eV, with the permittivity tensors of ε|| = -1.67 + 0.39i and ε⊥ = 5.60 + 0.32i. The incident angle of the transverse Gaussian beam was set to 45°. The electric field distribution obtained from our simulations are plotted in Fig. 3(b). The negative refraction indicated by the directions of the Poynting vectors is obvious. The refraction angle ${\theta _r}$ obtained from Eq. (6) is 35.5$^\circ $. The waves could propagate with slow attenuation due to the low loss.
The dependence of the optical properties of the electron-doped YN was then investigated. As the electron doping concentration (ne) is increased from 3.33 × 1021 cm-3 to 3.33 × 1022 cm-3, the Fermi level will be pushed upwards by about 0.661 - 1.879 eV relative to the pristine YN, as shown in Fig. 2(a), which is expected to have influence on the electron transitions and thus hyperbolic frequency windows. The variation of the critical frequency ${\omega _c}$ at $\textrm{Re}\varepsilon ({{\omega_c}} )= 0$ as a function of electron doping concentration is plotted in Fig. 4(a). For the longitudinal component of permittivity (${\varepsilon _ \bot }$), the critical frequency increases monotonously with the increase of electron doping concentration, resulting in the blue shift of the hyperbolic frequency window. The critical frequency of the transverse permittivity (ε||) first increases and then decreases with increasing ne. Therefore, the hyperbolic frequency window determined by the frequency interval [ωc⊥, ωc||] attains a maximal value at ne = 1.332 × 1022 cm-3, as shown in Fig. 4(b). This offers a promising strategy for modulating the hyperbolic properties of YN by regulating electron-doping concentration.
Fig. 4. Frequency parameters of electron-doped YN (a) The critical frequency (ωc) where $\textrm{Re}\varepsilon ({{\omega_c}} )= 0$ of YN at different electron-doping concentration (ne). The shadow region indicates the hyperbolic area. (b) The frequency window of hyperbolic dispersion of electron-doped YN at different doping concentration. (c) Variation of plasma frequency (ωp) of electron-doped YN as a functional of electron doping concentration. (d) The relation between critical frequency and plasma frequency. The line indicates the linear fitting expression ωc = 0.3 × ωp + 0.07.
To reveal the origins of the tunable hyperbolic properties of electron-doped YN, we calculated the plasma frequency ${\omega _{p,\alpha \beta }} $ using the formula [56]:
The low-loss feature of the electron-doped YN in the hyperbolic region is preserved in different doping concentrations. We attributed it to the restrained inter-band transition arising from special electronic band structure near the Fermi level. We marked the onset energy of inter-band transitions of YN at the doping concentration of ne = 1.0 ×1022, 2 × 1022 and 3.33 × 1022 cm-3 in Fig. 2(a) as examples. The lowest photon energies for the direct inter-band transition marked by the dashed arrows are respectively 1.844, 2.242, and 1.749 eV at these three electron-doping concentrations. All of them are higher than the corresponding upper limits of the hyperbolic frequency window which are 1.574, 1.617, 1.305 eV. The high onsets of inter-band transitions keep the energy loss away from the operating hyperbolic window. This offers an effective and flexible strategy in controlling the light propagation.
Finally, we investigated the plasmonic properties of the electron-doped YN. Plasmons are self-sustaining collective excitations of electrons accompanied by the presence of charge-density oscillations. The relevant plasmonic properties can be determined by the zeros of the longitudinal dielectric function [57]. To examine the electron-doping effect on plasmonic properties, we calculated the electron energy loss functions L(q,Ω) of pristine and electron-doped YN at ne = 1.0×1022 cm-3 for the momentum (q) along the in-plane and out-of-plane directions in the energy region of 0 - 15 eV, as shown in Fig. 5. Here, q and Ω represent the momentum and energy of plasmon. For the pristine YN, only one set of peaks of L(q, Ω) were found in the high energy region ∼12 eV, as shown in Figs. 5(a) and 5(b). Notably, the high energy plasmon shows small dispersion, long-lived features and weak anisotropy along in-plane and out-of-plane directions. The robust high-energy plasmons might be potential for ultraviolet optical devices [58].
Fig. 5. In-plane and out-of-plane electron energy loss spectra L(q,Ω) of (a),(b) pristine YN and (c),(d) electron-doped YN at ne = 1.0 × 1022 cm-3 as a function of the photon energy Ω and momentum q.
For the electron-doped YN, however, in addition to the robust high energy plasmon excitation, a new set of peaks appear in the low energy region, as shown in Figs. 5(c) and 5(d). The low-energy plasmon modes start from different photon energy for two crystalline directions, which is about 0.88 eV for the in-plane mode and 1.63 eV for the out-of-plane mode, exhibiting obvious anisotropic features. The frequency-momentum dispersion relation Ω(q) of the plasmons space can be visualized by the traces of peaks, as shown in Fig. 6. From this figure, we can see that in contrast to the nearly dispersionless high-energy plasmon excitation, the low-energy plasmon modes exhibit remarkable dispersive features characterized by a typical parabolic dispersion relation [59,60]:
Here, Ω0 corresponds to the plasmon frequency at $q \to 0$ and A is the dispersion coefficient. Notably, the low-energy plasmon excitation has lower intensity and gets more quickly damped. The in-plane and out-of-plane modes fade away at q = 0.444 Å-1 and q = 0.808 Å-1, respectively. The anisotropic low-energy plasmon modes of the nodal-line semimetallic YN induced by electron doping offer a promising platform for the study of nontrivial plasmon behaviors, such as hyperbolic plasmons [61].Fig. 6. The plasmon dispersions of (a) high-energy-excited and (b) low-energy-excited branches extracted from traces of peaks of electron-doped YN at ne = 1.0× 1022 cm-3.
Another interesting issue is the plausibility of such a high electron doping concentration (∼1022 cm-3) in the nodal-line semimetallic YN. Experimentally, electron doping to a metallic material can be achieved by gate bias [62], as demonstrated in many supercapacitor related works [63,64]. The carrier density of the 2D Dirac materials like graphene due to electron doping can be up to 4×1014 cm-2 [62], corresponding to a bulk concentration of ∼1022 cm-3 [65]. Therefore, the electron doping concentration considered in this work is attainable in experiments.
4. Conclusion
In summary, using first-principle calculations, we demonstrated the plausibility of tuning the nodal-line semimetallic YN to a broadband type-I hyperbolic material with ultra-low energy loss by electron doping, which is quite promising for achieving unusual optical effects. The frequency window of hyperbolic dispersion ranging from near-IR (∼1.4 μm) to the visible regime (∼769 nm) can be regulated by controlling the electron doping concentration, owning to the unique electronic band structure near the Fermi level. A linear relation between the plasma frequency and the critical frequency of hyperbolic dispersion was revealed, which is dominated by the electron intra-band transition. Furthermore, electron doping induces a new branch of plasmon dispersion with strong anisotropy in the low energy region, paving a way for the study of nontrivial plasmon behaviors. Our findings not only expand the family of natural hyperbolic materials to topological materials, but also offer a promising strategy to explore a high-performance hyperbolic medium and regulate plasmon properties.
Funding
National Key Research and Development Program of China (2016YFA0301200); National Natural Science Foundation of China (11774201); Taishan Scholar Project of Shandong Province.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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