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Abstract
Hyperbolic materials have wide application prospects, such as all-angle negative refraction, sub-diffraction imaging and nano-sensing, owning to the unusual electromagnetic response characteristics. Compared with artificial hyperbolic metamaterials, natural hyperbolic materials have many advantages. Anisotropic two-dimensional (2D) materials show great potential in the field of optoelectronics due to the intrinsic in-plane anisotropy. Here, the electronic and optical properties of two hyperbolic 2D materials, monolayer CuB6 and CuB3, are investigated using first-principles calculations. They are predicted to have multiple broadband hyperbolic windows with low loss and highly-anisotropic plasmon excitation from infrared to ultraviolet regions. Remarkably, plasmon propagation along the x-direction is almost forbidden in CuB3 monolayer. The hyperbolic windows and plasmonic properties of these 2D copper borides can be effectively regulated by electron (or hole) doping, which offers a promising strategy for tuning the optical properties of the materials.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Hyperbolic materials (HMs) refer to a class of materials whose real parts of permittivity tensors (ɛ) have opposite signs in different principal components [1]. For two-dimensional (2D) materials, the dispersion relation between wave vector k and frequency ω which reflects the interaction of materials with electromagnetic fields is set by the equation:
Most HMs known so far are artificially fabricated metamaterials, composed of metallic media and dielectric media [3,10]. Artificial HMs have wide tunable hyperbolic windows, but they are still facing many challenges. They require complex processes to achieve the finely-controlled growth of different materials on the sub-wavelength scale. The scattering between interfaces results in large losses, and the structural scale further limits the maximum propagation wavelength [11–13]. Compared with artificial HMs, natural hyperbolic materials have obvious advantages. Natural materials with three-dimensional (3D) layered structures, such as graphite [14–16], MgB2 [17,18], transition metal dichalcogenides [19,20], metal-organic frameworks (MOFs) [21], hexagonal boron nitride [3,22], and nodal-line semimetal (e.g. YN) [16] were then perceived to be promising candidates for HMs. They stand out because of their wide and tunable operating windows, and low energy dissipation. Up to now, a number of materials, such as graphite [17], h-BN [22,23], tetradymites [24,25], transition metal chalcogenides [26,27], have been demonstrated to have hyperbolic properties in experiments.
2D materials have atomic layer thickness in one direction and thus high size-confinement effects, which are promising candidates for HMs. Therefore, hyperbolicity naturally exists in the anisotropic 2D materials with high tunability, which is difficult to achieve for artificial structures. Recently, in-plane hyperbolicity in 2D materials attracted tremendous interest, because anisotropic 2D materials can achieve hyperbolic plasmons with large momentum, due to their high electromagnetic confinement and diverging photonic density of states [28]. These interesting properties are promising for applications in enhanced light-matter interaction [29–32] and near-field radiative heat transfer [33]. Despite the abundant structures of 2D materials, only few of them have been predicted to be 2D natural hyperbolic materials, such as MoTe2 [34] and black phosphorus [28,35,36]. Notably, a plasmonic regime with electrically tunable hyperbolic dispersion and bianisotropy has been demonstrated in recent experiments [37]. Therefore, searching for new 2D natural hyperbolic materials with strong anisotropy is timely desirable.
In this contribution, we consider two 2D copper boride structures, CuB6 and CuB3, which were proposed in a recent work [38]. Our calculations demonstrate that both of them are 2D hyperbolic materials with wide hyperbolic windows, covering the spectrum from infrared to ultraviolet regions. For CuB6, the hyperbolic windows are 0.53 - 1.44 eV and 1.66 - 3.50 eV, while in CuB3, the hyperbolic frequency window is 1.58 - 4.27 eV. The plasmon dispersion of both materials extracted from their electron energy loss spectrum (EELS) also show remarkable anisotropy in x- and y-directions. More interestingly, the plasmon propagation along the x-direction is almost forbidden in CuB3. Additionally, the hyperbolic regions and the plasmonic properties of the two 2D materials can be effectively regulated via electron (or hole) doping, offering a promising strategy to tune the hyperbolic properties of natural HMs.
2. Computational and methods
Our density functional theory (DFT) calculations were performed using the Vienna ab simulation package (VASP) [39] and GPAW code [40] based on the projector-augmented wave (PAW) [41] method. The structural optimization and electronic properties calculations were conducted by using the VASP code. The generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) realization [42] was adopted to treat the exchange-correlation functional self-consistently. The cutoff energy for the plane-wave expansion was set to 500 eV for both materials. 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 during the geometry optimization. The Brillouin zones (BZ) were sampled using the Monkhorst-Pack scheme with k-mesh of 34×13×1 for CuB6 and 33×33×1 for CuB3. In order to exclude the interaction between neighboring images, we set a vacuum layer of 20 Å along the z-direction. The electron and hole doping effects were simulated by adding electrons or holes to the materials in homogeneous background charges of opposite signs.
The dielectric function and electron energy loss spectrum (EELS) were computed by using the GPAW code based on the linear response theory [43]. The real-space non-interacting density response function obtained from the Adler-Wiser formula [44,45] is written as
A denser k-mesh scheme is always required to get reliable optical properties of materials in the GPAW calculations. A 113×43×1 k-mesh was therefore adopted for the unit cell of CuB6, while for CuB3, a rectangular supercell was employed to calculate the dielectric properties of CuB3 with a k-mesh of 75×15×1. To account for local field effects, the energy cut-off energy was set to 50 eV in reciprocal space. The broadening parameter η was set to be η = 0.05 eV. A 2D truncated Coulomb kernel was adopted to avoid the interaction between the periodic replicas.
3. Results and discussion
3.1 Crystal and electronic structures
The crystal structures of monolayer CuB6 and CuB3 considered in this work are shown in Fig. 1(a) and 1(c). Both of them have the copper chains and boron ribbons aligned alternatively, leading to high structural anisotropy between the directions along and perpendicular to the chains (taken as x- and y-directions respectively). CuB6 monolayer can be characterized by a rectangular lattice with the space group of D2h, and the optimized lattice constants are a = 2.94 Å and b = 7.71 Å. Along the Cu chains, the nearest Cu-Cu distance is 2.94 Å. The boron ribbons consist of hexagon borons and triangular borons with the B-B bond lengths of 1.59 - 1.72 Å. CuB3 monolayer has a rhomboid primitive cell with the space group of C2h. The optimized lattice constants are a = 4.95 Å and b = 5.36 Å, and the angle between them is 99.3°. Compared with the CuB6 structure, the Cu chains of CuB3 are more compacted with the Cu-Cu distance of 2.42 Å, because the boron ribbons are composed surely of triangular borons. These results are in good agreement with those reported in a previous literature [38]. These two copper borides have not yet been synthesized, but the dynamical stability and synthetic plausibility of CuB6 and CuB3 have been verified on the basis of first-principles calculations [38].
Fig. 1. Atomic and electronic band structures of monolayer copper borides. (a), (b) CuB6 and (c), (d) CuB3. The unit cells and the Brillouin zones are presented as the insets of these figures. The energy at the Fermi level is set to zero.
The electronic band structures of monolayer CuB6 and CuB3 are plotted in Fig. 1(b) and 1(d) respectively. From these figures, one can observe that they are both metallic, with several energy bands crossing the Fermi level. For CuB6 as shown in Fig. 1(b), the bands crossing the Fermi level are highly dispersive along the direction paralleling to the copper chains (Y - S and Γ - X directions). Along the X - S direction, there is a hole pocket with a relatively flat band near the Fermi level. The orbital-resolved band analysis shows that the bands near the Fermi level are dominated mainly by the p orbitals of B atoms, and the ${d_{{x^2} - {y^2}}}$ orbitals of Cu atoms. For CuB3 as shown in Fig. 1(d), the bands across the Fermi level are highly dispersive along the Γ - B, B - D and Γ - Z directions, but a band gap ∼2.5 eV emerges along the D - Z direction. Different from CuB6, the bands near the Fermi level are mainly contributed by the pz orbitals of B atoms. Such anisotropic electronic band structures of CuB6 and CuB3 are expected to contribute to the anisotropic optical properties.
3.2 Optical properties
We then calculated the dielectric function ɛ(ω) of the monolayer copper borides to reveal the optical response. Generally, the dielectric function of materials has a complex form, in which the imaginary part Im[ɛ(ω)] corresponds to the energy loss due to the electron transitions, while the real part Re[ɛ(ω)] could be estimated from imaginary part according to the Kramers-Kronig relation [50]. For a metallic system, the dielectric function consists of two kinds of contributions, inter-band (ɛinter) and the intra-band (ɛintra) transitions. The contribution of intra-band transitions is usually described by the Drude model [51],
The dielectric functions along the directions parallel (x) and perpendicular (y) to the copper chains of CuB6 and CuB3 are plotted in Fig. 2(a) and 2(b), respectively. For CuB6, there are two hyperbolic windows with Re[ɛx] < 0 and Re[ɛy] > 0 as the energy is lower than 5 eV. Re[ɛy] crosses zero from negative values at 0.53 eV and keeps positive in the considered energy range, while Re[ɛx] changes signs at about 1.44 eV, constituting the first hyperbolic window. With the increase of energy, however, Re[ɛx] becomes negative again at 1.66 eV, and keeps negative until 3.50 eV, which leads to the second hyperbolic window. These two frequency windows, 0.53 - 1.44 eV (2340 - 860 nm) and 1.66 - 3.50 eV (775 - 354 nm), cover the frequency from infrared to ultraviolet spectrum regime. For monolayer CuB3, there is a wide hyperbolic frequency window from 1.58 - 4.27 eV (784 - 290 nm) with Re[ɛx] < 0 and Re[ɛy] > 0, also covering the spectrum from visible to ultraviolet regime. Compared with the hyperbolic frequency windows of MgB2 (2.34 - 3.54 eV) [18] and monolayer black phosphorus (1.73 - 1.84 eV) [52], the hyperbolic frequency windows of these two copper borides are extremely wider, making them ideal 2D HMs for the relevant applications. We attribute the wide hyperbolic windows of CuB6 and CuB3 to the different contributions of intra-band transitions along the x- and y- directions. To verify this, we separated the intra-band and inter-band transitions from the dielectric functions, as shown in Fig. 2(c) and 2(d). We can see that the hyperbolic window in the intra-band constituent has a large overlap with that of the full dielectric function. For both materials, Re[ɛx] crosses zero later than Re[ɛy]. This can be ascribed to the larger plasma frequencies of the x-direction (ωp,x) than that of the y-direction (ωp,y) in the Drude model which are ωp,x = 3.13 eV and ωp,y = 0.50 eV for CuB6 and ωp,x = 4.65 eV and ωp,y = 3.34 eV for CuB3. The direction-dependent plasma frequency is given by the expression [18],
Fig. 2. The real part Re[ɛ] and imaginary part Im[ɛ] of the dielectric function of (a) CuB6 and (b) CuB3. The hyperbolic frequency windows are indicated by the yellow shaded regions. The separated contributions of the intra-band and inter-band transitions to the Re[ɛ] of (c) CuB6 and (d) CuB3.
The hyperbolic natures of the copper borides would lead to directional plasmons. We therefore calculated the plasmonic properties of monolayer CuB6 and CuB3. Plasmons are the self-sustaining collective excitations of electrons accompanied by the presence of charge-density oscillations [51,53]. An undamped plasmon should fulfill the conditions that the real part of the dielectric function cross zero (Re[ɛ] = 0) and the imaginary part of dielectric function Im[ɛ] has the same sign as $\omega $ partial of Re[ɛ] (${\mathop{\rm Im}\nolimits} [\varepsilon ]/{\partial _\omega }Re [\varepsilon ] > 0$) at the peak of loss function [15,54]. Single-particle excitations (SPEs) that arise from the intra-band and inter-band transitions dominate the damping processes of plasmons [55]. If Im[ɛ] presents a vanishing small value or a local minimum at the energy of Re[ɛ] = 0, the plasmon lies out of the SPE regions and is identified as an undamped plasmon [54–56].
The propagation behaviors of plasmons along the x- and y-directions in monolayer CuB6 and CuB3 are presented in Fig. 3(a) and 3(b), respectively. For CuB6 in Fig. 3(a), in the energy range of 0 - 2 eV, both directions exist plasmon modes, which start from 0 eV and follow the classical expression of 2D free gas electron $\omega = A\sqrt q $ with $A = \sqrt {\sigma {e^2}/({2{m^\ast }{\varepsilon_0}} )} $ at small momentum transfers. We fit the plasmon dispersion in the small momentum region with $\omega = A\sqrt q $ and obtained Ax = 6.24 and Ay = 2.09. In fact, the anisotropy coefficient Ax/Ay corresponds to the ratio of the square root of the effective mass $\sqrt {m_y^\ast{/}m_x^\ast } $ [57]. Along the x-direction, the plasmon is highly dispersive and the frequency increases rapidly to 1.75 eV at q = 0.19 Å-1 before entering the SPE region. For the y-direction, the plasmon mode is less dispersive compared with that along the x-direction. It can propagate to q = 0.19 Å-1 at the energy of 0.70 eV and then enter the SPE region. For monolayer CuB3 in Fig. 3(b), the well-defined plasmon mode only exist in the y-direction. The plasmon starts from 0 eV and follows a $\omega = A\sqrt q $ behavior at small momentums, with the scaling coefficient Ay = 6.68. Then the plasmon dispersion deviates from the $\sqrt q $ fitting and enters the SPE region at q = 0.24 Å-1 with the highest energy of 1.96 eV. Thus, both CuB6 and CuB3 exhibit strongly anisotropic plasmons along the x- and y-directions. Remarkably, for CuB3, the plasmon propagation is allowed only along the y-direction, whereas the plasmon along the x-direction is almost completely suppressed. The anisotropic plasmon propagation can also be correlated to the orbital-selective interband transition along different directions. From the orbital-resolved band structure of CuB3 (Fig. 1(d)), one can find that in the low energy region, the interband transitions along the Γ - B (x-) direction are mainly dominated by the transitions from Cu ${d_{{x^2} - {y^2}}}$ to B pz orbitals and from B pz to B pz orbitals which are allowed by the symmetry constrains. The plasmons along the x-direction will decay rapidly to single-particle electron-hole pair due to the interband transitions. Along the y-direction, however, the interband transitions are mainly from B py to B pz orbitals, as indicated by the bands along the B - D direction, which are forbidden by the symmetry constrains. Therefore, the plasmon propagation along the y-direction is immune to the damping of inter-band transition. Such anisotropic plasmon would lead to an anomalously large photonic density of states and directional propagating plasmonic rays, which promises intriguing applications for planar photonics [58].
Fig. 3. The EELS of (a) CuB6 and (b) CuB3 along the x- (right panel) and y-direction (left panel). Red circles outline the plasmon modes extracted from the EELS. The green solid lines represent the plasmon dispersion of a classical 2D electron gas given by the expression $A\sqrt {\textrm{q}} $ with Ax = 6.24 and Ay = 2.09 for CuB6 and Ay = 6.68 for CuB3. The blue dashed lines show the SPEs regions. (c–f) Dielectric function and loss function at (c) qx = 0.056 Å-1 and (e) qy = 0.056 Å-1 for CuB6, (d) qx = 0.237 Å-1 and (f) qy = 0.037 Å-1 for CuB3. The blue circles denote the positions of Re(ɛ) = 0.
In order to verify the collective excitation features of the plasmon in these copper borides, we calculated the dielectric function and the corresponding energy loss function at several selected momentums. From Fig. 3(c) and 3(e), we can see that at q = 0.056 Å-1, the Re[ɛx] and the Re[ɛy] of CuB6 cross zero from negative values at about 1.22 eV and 0.46 eV respectively, which correspond to the energies of the EELS peaks. While in the same energy region, Im[ɛx] and Im[ɛy] both present local minima, fulfilling the conditions of an undamped plasmon. For CuB3 at the chosen momentum of q = 0.037 Å-1, all conditions are fulfilled for the plasmon at 1.24 eV along the y-direction as shown in Fig. 3(f). Notably, at q = 0.237 Å-1 along the x-direction, as shown in Fig. 3(d), the definition of a plasmon is unsatisfied, which is consistent with the EELS shown in Fig. 3(b).
3.3 Effects of charge doping
It has been demonstrated the electron (or hole) doping can effectively regulate the hyperbolic features and plasmonic properties [18,21]. We therefore considered the charge-doped monolayer copper borides with additional 0.5 electrons (or holes) per unit cell, corresponding to the doping concentrations of 2.205 ×1014 cm-2 (CuB6) and 1.550 ×1014 cm-2 (CuB3). The carrier density of the 2D Dirac materials like graphene due to electron doping can be up to $4 \times {10^{14}}\textrm{c}{\textrm{m}^{\textrm{ - 2}}}$ [59], the doping concentrations considered in this work are thus attainable in experiments. For CuB6, charge doping alters the electron and hole concentrations and thus modifies the hyperbolic window significantly. Electron doping leads to blueshifts of the first hyperbolic window of CuB6, i.e., the first hyperbolic window moves to 1.21 - 3.46 eV, while the second hyperbolic window disappears. For the hole doping, the first hyperbolic window of CuB6 extends to 0.56 - 1.61 eV, and the second hyperbolic window splits into two smaller hyperbolic windows, which is 1.86 - 2.23 eV and 2.59 - 3.19 eV, as shown in Fig. 4(a) and 4(b). For CuB3, charge doping only affects the positions of the Fermi level and thus has little effect on the hyperbolic region. The hyperbolic window changes to 1.60 - 4.33 eV under electron doping, as shown in Fig. 4(d), while hole doping pushes the hyperbolic window to 1.48 - 4.09 eV, as shown in Fig. 4(e). To demonstrate the switching of the dispersion relation under charge doping, we plotted the iso-frequency curves of CuB6 and CuB3 at the photon energy of 1.74 eV for the doped and undoped cases, with assuming ω/c = 1 Å-1, as shown in Fig. 4(c) and 4(f). For CuB6, the iso-frequency curves get changed from hyperbola to ellipse under hole doping. For CuB3, because the hyperbolic windows show weak doping dependence, the hyperbolic iso-frequency curve is preserved under the considered electron/hole doping. Therefore, charge doping can regulate the hyperbolic windows of monolayer CuB6 effectively, while the hyperbolic properties of CuB3 is robust.
Fig. 4. Real and imaginary parts of the dielectric functions of (a) electron-doped and (b) hole-doped CuB6 monolayer, (d) electron-doped and (e) hole-doped CuB3. Hyperbolic regions are represented by yellow shadows in the figures. (c) and (f) is the iso-frequency contours at 1.74 eV of CuB6 and CuB3 respectively. Here, Re[ɛx] = -0.44 and Re[ɛy] = 1.73 for pristine CuB6; Re[ɛx] = -3.68 and Re[ɛy] = 1.91 for electron-doped CuB6; Re[ɛx] = 1 and Re[ɛy] = 2.23 for hole-doped CuB6. And, Re[ɛx] = -3.30 and Re[ɛy] = 1.18 for pristine CuB3; Re[ɛx] = -3.70 and Re[ɛy] = 0.71 for electron-doped CuB3; Re[ɛx] = -3.34 and Re[ɛy] = 1.48 for hole-doped CuB3.
The EELS of monolayer CuB6 and CuB3 can also be tuned via electron/hole doping, as shown in Fig. 5. It can be seen that the strong anisotropy features of the plasmons in these two materials are preserved upon charge doping. The hole doping to monolayer CuB6 reduces the maximal frequency of the plasmon along the y-direction to 0.59 eV before entering the SPE region at q = 0.11 Å-1, and it improves the maximal frequency along the x-direction to 2.32 eV at q = 0.286 Å-1, as shown in Fig. 5(a). Upon electron doping, however, monolayer CuB6 exhibits acoustic plasmon with lower intensity in the low-energy region and rapidly decays in the SPE region, as shown in Fig. 5(b). Additionally, the maximal energies of the undamped plasmons propagating along the x- and y-directions increase, which are 1.86 eV and 1.53 eV at q = 0.25 Å-1 and q = 0.23 Å-1, respectively. Similarly, for the plasmons along the y-direction of monolayer CuB3, the hole doping reduces the maximal energy to 1.72 eV at q = 0.20 Å-1 before entering the SPE region, as shown in Fig. 5(c), while the electron doping improves the maximal plasmon energy to 2.21 eV at q = 0.28 Å-1, as shown in Fig. 5(d). Considering charge transfer widely exists between 2D materials and substrates, such tunable plasmonic properties offer promising platform for regulating the relevant optical properties.
Fig. 5. The EELS of charged-doped monolayer CuB6 and CuB3 along the x- (right panel) and y-direction (left panel). The EELS of CuB6 with additional (a) 0.5 holes and (b) 0.5 electrons per unit cell. The EELS of CuB3 with additional (c) 0.5 holes and (d) 0.5 electrons per unit cell. Red circles outline the plasmon modes extracted from EELS. The blue dashed lines show the SPE regions.
3. Conclusion
In summary, we demonstrate from the first-principles calculations the tunable hyperbolic and highly-anisotropic plasmons in 2D copper borides (CuB6 and CuB3). Our calculations show that they are both 2D hyperbolic materials with broadband hyperbolic frequency regimes. For CuB6, the hyperbolic frequency windows are 0.53 - 1.44 eV and 1.66 - 3.50 eV, covering the spectrum from infrared to ultraviolet, while CuB3 has the hyperbolic frequency window of 1.58 - 4.27 eV ranging from visible to ultraviolet. The plasmon dispersion of both materials extracted from their electron energy loss spectrum (EELS) also shows remarkable anisotropy along the x- and y-directions. It is worth mentioning that the plasmon propagation of monolayer CuB3 along the x-direction is almost suppressed completely, which is quite promising for directional plasmons. The hyperbolic windows and the plasmonic properties of these 2D copper borides can also be effectively regulated by electron/hole doping. These interesting properties offer a promising platform for the study of unusual optical scenarios of 2D materials, as well as the potentials in optoelectronic devices.
Funding
National Natural Science Foundation of China (No. 12074218); Taishan scholarship of Shandong Province.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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