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Abstract
Recently, semimetallic tungsten ditelluride (WTe2) has been proposed as a natural material that supports hyperbolic plasmonic responses. In this paper, we have theoretically discovered that such hyperbolicity, which is due to strongly anisotropic in-plane interband transition of electrons, exists even in the monolayer and can become elliptic under proper doping. Using density functional theory, the permittivities include both the interband and intraband parts have been calculated, which are then used to derive the in-plane conductivities. Based on two-dimensional conductivity, the dispersion relations of the plasmonic modes in the extended monolayer have been analytically solved. It is surprising that monolayer WTe2 supports both elliptic as well as hyperbolic plasmonic responses in the infrared. Edge-confined modes in the extended monolayer in the elliptic regime and waveguiding modes in nanoribbons in the hyperbolic regime have been numerically investigated. After being doped with electrons, the Fermi level is shifted; it is found that moderate electron doping can change the topology of the plasmonic responses from a hyperbolic to an elliptic one within some frequency range. The effects of band broadening are also discussed and the permittivities are calculated using optimal basis functions to further verify our main conclusions. Then, the states corresponding to large interband transition peaks are marked and the wavefunctions are used to explain the strong in-plane dipole. In the end, the permittivities of bulk WTe2 have been investigated. Our investigations indicate that monolayer WTe2 is a promising platform for plasmonic applications.
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1. Introduction
In recent years, the investigations of surface plasmon polaritons (SPPs) in two-dimensional (2D) materials have attracted much attention [1–3]. The plasmonic responses of 2D materials can be either elliptic or hyperbolic depending on their permittivity/conductive tensors [4–6]. In the elliptic regime, the surface plasmon polaritons can propagate along all directions in the plane with similar properties [5]. The isofrequency contours of the modes in this situation are elliptic, which can be isotropic or anisotropic depending on the material. One typical example of 2D materials with isotropic elliptic plasmonic dispersion relations is graphene, which can be modeled by a scalar conductivity and supports highly confined isotropic SPPs in the infrared [7,8]. Black phosphorus (BP) is an example of 2D materials with anisotropic elliptic plasmonic dispersion relations. After being doped with electrons, the conductivity tensor of BP under principle coordinates has two unequal diagonal terms [9–12]. The SPPs possess different wave vectors depending on the propagation direction. In the elliptic regime, the imaginary parts of the conductivities along two in-plane directions have the same sign, which can be either inductive (both positive) or capacitive (both negative) [5]. Considering the Drude type resonances, the effective permittivities derived from the conductivities have negative real parts, which makes the material similar to metallic thin film; the occurrence of some plasmonic phenomena, such as edge plasmons and plasmonic waveguides, can be easily understood in this manner [13–16]. In the hyperbolic regime, the two diagonal terms of the conductivity tensor under the principle coordinates have opposite signs [5,6]. The isofrequency contour of the SPPs is now a hyperbola. Roughly speaking, the SPPs can only propagate along the direction with negative real permittivity or positive imaginary conductivity, leading to some exotic plasmonic phenomena such as extreme confinement, directional propagation [17–20], etc. Materials with hyperbolic plasmonic responses are rare in nature; usually, researchers design and fabricate hyperbolic materials based on the concept of metamaterials [21]. Artificial sub-wavelength structures of which the effective permittivity is negative along one direction but positive along the perpendicular direction have been proposed, enabling a series of optical applications, including sub-wavelength guiding and imaging, enhancement of spontaneous emission, and so on [22–24]. Despite the fact that hyperbolic materials are promising platforms for planar optics, these metamaterials require complex manufacturing processes and are hardly tunable [25,26]. Unremitting efforts have been made by researchers to overcome these limitations; one way to circumvent these is searching for natural materials that support hyperbolic responses.
Recently, hyperbolic plasmonic signatures have been discovered in bulk transition metal dichalcogenide (TMD) WTe2 [27,28]. Using far infrared absorption spectroscopy, C. Wang et al. have measured the absorption of patterned WTe2 arrays and observed the occurrence of hyperbolic plasmonic responses above certain frequency. Based on the observations on its band structure, bands nesting has been proposed as the mechanism behind the hyperbolicity in the near infrared, which causes strong interband transition of electrons when the polarization of light is along one particular direction but not the perpendicular one [29]. Of course, both intra- and interband transition parts of the permittivity, respectively denoted as ${\varepsilon _{intra}}$ and ${\varepsilon _{inter}}$, should be considered. In the low-frequency end, ${\varepsilon _{intra}}$ dominates, which is always negative and can be seen as the signature of metallic materials. ${\varepsilon _{inter}}$ has resonant peaks corresponding to the “vertical” electron-hole excitation in the material. These resonances can lead to negative permittivity in a narrow frequency window with relatively large imaginary part of the permittivity corresponding to optical losses. Hyperbolic plasmonic responses in monolayer WTe2 can be explained by such strongly anisotropic interband transition. Basically, if strong interband transition of electrons happens when the polarization of light is along one direction, the real part of the permittivity along this direction could become negative, while it is positive along the perpendicular direction, leading to opposite signs of the real parts of the permittivities along two directions, i.e., hyperbolic plasmonic responses.
In this paper, we focus on the semimetallic monolayer WTe2. The permittivities of WTe2 have been calculated using density functional theory (DFT). It is found that WTe2 naturally supports both elliptic and hyperbolic plasmonic responses in the infrared. The 2D conductivity tensor is derived from the permittivities based on a simple model, which is then used to analytically solve the plasmonic modes in extended monolayer. The isofrequency contours of the modes are plotted to show their elliptic or hyperbolic signatures. In the elliptic regime, edge plasmons have been investigated; the electric fields of these modes are confined at the boundary of the monolayer. In the hyperbolic regime, normal edge modes do not exist; instead, we solve the waveguiding plasmonic modes in nanoribbons. Due to the hyperbolicity, standing-wave-like field patterns have been found across the nanoribbons. Then, non-neutral (charged) primitive cell is used to investigate the effects of doping. The Fermi energy is shifted up or down depending on the type of doping, which is a common way to modify the plasmonic responses in 2D materials. It is found that the elliptic as well as the hyperbolic responses are relatively robust against small doping. With slightly large electron concentrations, the first hyperbolic region changes its topology and becomes elliptic. The corresponding isofrequency contours are plotted to show such topological transition behaviour. Then, the effects of band broadening are discussed and the permittivities have been calculated based on optimal basis functions to further verify our results. To explore the origin of the hyperbolicity, the states corresponding to large interband transition peaks are marked out on the band surfaces, and the wavefunctions are used to explain the strong in-plane dipole caused by the interband transition. In the end, the permittivities of bulk WTe2 have been calculated, showing the absence of strong interband transition peaks and higher Drude plasma frequency along the other in-plane direction. Our investigations indicate that monolayer WTe2 has rich plasmonic properties and it is possible to actively control the topology of the plasmonic responses in WTe2 via doping, which might lead to a bunch of exciting applications in plasmonics.
2. Methods
Density functional theory calculations were done using software package QUANTUM ESPRESSO (QE) [30]. Perdew-Burke-Ernzerhof (PBE) functional was used together with optimized norm-conserving pseudopotentials [31], and the van der Waals correction was DFT-D2. The structure was treated as 2D and fully relaxed before any calculations of the bands and permittivities [32]. In the calculations of the bands, the cut-off energies for wavefunction and electron density were respectively set to 120 Ry and 480 Ry, and the Monkhorst-Pack (MP) k-grid was 23×13×1. For the permittivity, a 45×25×1 grid was used together with 100 conduction bands, and the cut-off energies for wavefunction and electron density were respectively increased to 140 Ry and 560 Ry. The smearing parameters for inter- and intraband transitions were both 5 meV, and Fermi-Dirac distribution with 0.0019 Ry broadening was used for electron occupation. The permittivities calculated from the DFT have been volume-averaged, where the thickness of the primitive cell was 30 angstroms. In the discussions of the band broadening, the parameters for both inter- and intraband transitions were sequentially set to 5 meV, 10 meV, 50 meV, and Fermi-Dirac distribution with 0.0038 Ry broadening was used for electron occupation.
The permittivity is a complex number that has real and imaginary parts respectively denoted as ${\mathrm{\varepsilon }_1}$ and ${\mathrm{\varepsilon }_2}$. The 2D conductivity $\sigma = {\sigma _1} + i{\sigma _2}$, where ${\sigma _1}$ and ${\sigma _2}$ are respectively the real and imaginary parts, can be derived from the permittivity as: ${\sigma _2} = ({1 - {\varepsilon_1}} )\omega {\varepsilon _0}t$ and ${\sigma _1} = \omega {\varepsilon _0}t{\varepsilon _2}$, where ω is the angular frequency, ɛ0 is the permittivity of vacuum, and t is the thickness. One benefit of using conductivities instead of permittivities is that the dispersion relations in extended monolayer can be analytically solved. For the modes at the edges and those in nanoribbons, one has to use numerical approaches. In this paper, we used finite element method (COMSOL) [33]. In the frequency domain module, the permittivities were used to set up the refractive indices, and the slab representing the monolayer WTe2 had the same thickness as the cell in DFT calculations, which was 30 angstroms. The plasmonic modes at edges and in nanoribbons were found by solving the complex effective mode index.
To further verify the results, the permittivities were also calculated based on optimal basis (OB) functions [34–36]. The thickness of the cell was expanded to 40 nm. A 46×26×1 grid was used to get the density; then, an uniform coarse grid was used to calculate the states for the generation of the optimal basis functions. The threshold for the construction of the OB functions was 0.005. The commutator of the non-local part of the pseudopotential was also included. The permittivities were calculated on a 46×26×1 grid with proper values of broadening used for Fermi energy, Drude plasma frequency and band transitions. Here, Fermi energy was manually and directly set, other than being determined by self-consistent calculations of a charged (non-neutral) primitive cell, which might give us more accurate results. For the bulk WTe2, the grid was 42×24×12 and the smearing parameters were all set to 25 meV.
3. Results and discussion
Figure 1(a) is the structure of the free-standing monolayer WTe2 [37,38]. Two in-plane sides are denoted as a and b; the axis perpendicular to the monolayer is denoted as c. Orange and blue spheres represent Te and W atoms, respectively. The corresponding electron bands are plotted in Fig. 1(b), where the horizontal dashed line indicates the Fermi energy. Strong anisotropy can already be seen from its band structure. Enlarged plot of the shaded area near the Fermi energy is shown in Fig. 1(c), from which one can expect the occurrence of exotic plasmonic phenomena in the infrared [39–41]. Although DFT has limitations in accurately determining the band structures especially the width of bandgaps in semiconductors, WTe2 here is semimetallic or metallic after doping. The electron-hole interactions are screened by the conduction electrons and bandgap is no longer a problem; thus usual DFT is reasonably accurate and independent particle approximation is enough. Also, the plasmonic responses in metallic materials are usually insensitive to such details. After calculating the permittivity, the conductivity can be easily derived. The conductivities and the permittivities are plotted in Figs. 2(a) and 2(b), respectively.
Fig. 1. (a) The structure of WTe2, where orange and blue spheres are Te and W atoms, respectively. Rectangle denotes the primitive cell, where two in-plane axes are respectively marked as a (or x) and b (or y). (b) Band structures of monolayer WTe2. Dashed line indicates the Fermi energy. The shaded area in (b) is enlarged and plotted in (c).
Fig. 2. (a) Two-dimensional conductivities along the a and b axes. Solid and dashed curves denote ${\sigma _2}$ and ${\sigma _1}$, respectively. (b) Permittivities along the a and b axes. Solid and dashed curves denote ${\varepsilon _1}$ and ${\varepsilon _2}$, respectively. Elliptic and hyperbolic regimes are respectively shaded with red and blue. From low to high frequencies, the four shaded areas are called the first elliptic, the first hyperbolic, the second elliptic and the second hyperbolic regions, respectively. (c) The figure of merit defined as $Im(\sigma )/Re(\sigma )$ along the a and b axes. (d) The figure of merit defined as $Re(\varepsilon )/Im(\varepsilon )$ along the a and b axes.
The 2D conductivities in Fig. 2(a) have been derived from the permittivities using a simple model as described in the Methods section, where solid and dashed curves respectively denote its imaginary (${\sigma _2}$) and the real (${\sigma _1}$) parts. The real part of the conductivity ${\sigma _1}$ corresponds to ohmic losses, which is always positive for passive (no optical gain) materials, while the imaginary part ${\sigma _2}$ can be positive or negative depending on whether this material is metallic or dielectric. Since we are interested in the plasmonic responses, frequency regions with positive ${\sigma _2}$ should be concerned. Figure 2(b) shows the volume-averaged permittivities extracted from the DFT calculations, where solid and dashed curves denote the real and imaginary parts, respectively. From Figs. 2(a) and 2(b), one can find four regions that support SPPs: from low to high frequencies, the first elliptic, the first hyperbolic, the second elliptic, and the second hyperbolic regions. The first elliptic region is roughly below 0.1 eV and is shaded with red. This elliptic region is due to the intraband transition of electrons, where the permittivities and conductivities are clearly Drude-type. The first hyperbolic region, which is roughly near 0.15 eV and shaded with blue. In this hyperbolic region, the permittivity/conductivity along the b axis is negative/positive. Such hyperbolicity is due to strong interband transition of electrons when the polarization of light is along the b axis, which can be known from the peak of ${\varepsilon _2}$ in this region. The second elliptic region is near 0.2 eV, and is again shaded with red; finally, the second hyperbolic region which is around 0.3 eV and is again shaded with blue. The second elliptic and hyperbolic regions depend on how strong the interband transitions are in this material. Since most DFT calculations use independent particle approximation, many effects such as electron-electron and electron-phonon interactions have been neglected [42]; however, within the linear response regime, one can expect the existence of these two regions. The figure of merit of conductivity can be defined as $Im(\sigma )/Re(\sigma )$, and the one for the permittivity is $Re(\varepsilon )/Im(\varepsilon )$; Figs. 2(c) and 2(d) respectively show such quantities for the conductivities and permittivities, giving us clues about the losses of the modes. Also, it is worth mention that bands nesting might not be the reason behind these hyperbolic responses, since the photon energy is well below the vertical electron-hole excitation threshold (about 1 eV) in the nesting area mentioned in Ref. [29]. For completeness, we investigate all four regions in this paper.
The plasmonic dispersion relations in extended monolayer can be analytically solved using the 2D conductivity. Due to the anisotropy, these are hybrid modes consisting of both transverse-electric and transverse-magnetic polarized components [43]. Here, monolayer WTe2 has been treated as a conductive surface with a diagonal conductivity tensor
For simplicity, the x and y axes are set along a and b, respectively. The boundary conditions used are similar to graphene except that now it is anisotropic: (1) the tangential electric fields are continuous across the monolayer; (2) the surface current density induced by the tangential electric fields follows Ohm’s law [44]. After matching the boundary conditions, one can get the following equations describing the dispersion relations of SPPs in extended monolayer WTe2 [6,43]:
The existence of edge-confined plasmonic modes can be regarded as the signature of elliptic plasmonic responses [45–47]. The edge modes in the first elliptic region with the a and b axes in the cross-section are shown in Figs. 4(a) and 4(b), respectively. Black and red solid curves denote the real and imaginary parts of the mode indices, respectively. Mode index here is defined as $\mathrm{\beta }/{k_0}$, which is the wave vector of the plasmonic modes normalized by its corresponding value in vacuum. Insets are the electric field amplitude (denoted as |E|) profiles of the modes indicated by the green arrows on the dispersion curves. The fields are tightly confined at the edges of the monolayer. The effective mode indices are related to the anisotropy and slightly different depending on the propagation direction. The edge modes with the a and b axes in the cross-section corresponding to the second elliptic region are shown in Figs. 4(c) and 4(d), respectively. For all these edge modes, the fields are confined within a extremely small volume with a feature size around 10 nm. Such deep sub-wavelength confinement usually leads to huge optical losses, which explains the relatively large imaginary parts of the indices observed in Fig. 4.
Fig. 3. K-surfaces of the plasmonic modes in extended WTe2 monolayer in (a) the first elliptic, (b) the first hyperbolic, (c) the second elliptic, and (d) the second hyperbolic regions. Colorbar on the right of each figure indicates the figure of merit describing the normalized lifetime of the plasmonic modes. Numbers in the unit of eV are photon energies.
Fig. 4. Edge plasmon polaritons in the elliptic regime. (a) and (b) are respectively the effective mode indices of the edge modes with the propagation directions along the b and the a axes in the first elliptic region. (c) and (d) are respectively the effective mode indices of the edge modes with the propagation directions along the b and the a axes in the second elliptic region. The insets in all figures show the electric field profiles (the amplitude of the electric fields, denoted as |E|) of the modes indicated by the green arrows on the curves.
Edge-confined plasmonic modes like those in Fig. 4 cannot be found in the hyperbolic regime. The reason is that in the hyperbolic regime the monolayer is metallic along one direction but dielectric along the perpendicular one; the modes cannot be totally confined at the edges [48,49]. For this reason, we have investigated the waveguiding modes in nanoribbons. Nanoribbons in this section have their edges cut along the a or b axis, meaning that the permittivity tensor is diagonal. Since the hyperbolic regions are very narrow, dispersion relations are not shown here; instead, we plot the effective mode index as a function of the nanoribbon width. Mode index here is again defined as $\mathrm{\beta }/{k_0}$, which is the wave number of the plasmonic modes normalized by its corresponding value in vacuum. In the hyperbolic regime, plasmonic modes propagate along the direction with negative permittivity. In the first hyperbolic region, the permittivity along the b axis is negative, which indicates that only nanoribbons cut along the b axis (with the a axis in the cross-section) support SPPs. Figure 5(a) shows the indices of the first three modes, respectively denoted as 1st, 2nd, and 3rd, as functions of the nanoribbon width with photon energy E = 0.142 eV. For wide nanoribbons, modes are nearly degenerate; similar real and imaginary parts of the indices have been found for all three modes. As the width decreases, they start to split and the real as well as the imaginary parts increase. The electric field amplitude profiles (denoted as |E|) of the three modes corresponding to a 100 nm-wide nanoribbon (marked by the green dashed rectangle with an arrow in Fig. 5(a)) are plotted in Fig. 5(b). Based on the field distributions, one can see that standing waves form across the nanoribbon, which corresponds the fact that this material is dielectric in the cross-sectional directions. Such field distributions are totally different compared with those modes in the elliptic regime. Figure 5(c) shows the effective mode indices of the first three modes as functions of the nanoribbon width in the second hyperbolic region with photon energy E = 0.325 eV. Similar phenomena have been observed, except that now the nanoribbons are cut along the a axis (with the b axis in the cross-section).
Fig. 5. Plasmon polariton modes in nanoribbons in the hyperbolic regime. (a) The effective indices, both the real (Re) and the imaginary (Im) parts, as functions of the nanoribbon width with photon energy E = 0.142 eV, and the propagation direction is along the b axis. Only the first three modes have been plotted, respectively denoted as 1st, 2nd, and 3rd. (b) Electric field profiles (the amplitude of the electric fields, |E|) of the first three modes in a 100 nm-width nanoribbon, marked by the dashed green rectangle in (a), are plotted, respectively denoted as (I) and (II) and (III). (c) The effective indices, both the real (Re) and the imaginary (Im) parts, as functions of the nanoribbon width with photon energy E = 0.325 eV, and the propagation direction is along the a axis. Only the first three modes have been plotted, respectively denoted as 1st, 2nd, and 3rd. (d) Electric field profiles (the amplitude of the electric fields, |E|) of the first three modes in a 100 nm-width nanoribbon, marked by the dashed green rectangle in (c), are plotted, respectively denoted as (I) and (II) and (III).
One advantage of 2D materials over other traditional plasmonic materials is that their properties can be easily modified via doping, which basically changes the Fermi energy of the material. Now, we investigate the effects of such doping on the plasmonic responses in monolayer WTe2. By setting a charged cell in the DFT calculations, one can derive the properties under electron or hole doping conditions. Figures 6(a) and 6(b) respectively show the permittivities under electron doping with total charges −0.02 and −0.05 per cell; Figs. 6(c) and 6(d) respectively show the permittivities under hole doping with total charges +0.02 and +0.05 per cell. The doping densities corresponding to 0.02 and 0.05 total charges are 4.86 cm−3 and 12.14 cm−3, respectively. Compared with Fig. 2(b), the total permittivities, consisting of both the intraband (denoted as ${\varepsilon _{intra}}$) and interband (denoted as ${\varepsilon _{inter}}$) parts, are clearly modified. Based on our calculations, all elliptic as well as hyperbolic regions are robust against small charges. The first elliptic region is caused by the intraband transition of electrons, and should always exist as long as the material is metallic, which is exactly the case for semimetallic WTe2 [50–55]. As the electron or hole concentration varies, the Fermi energy is shifted up or down; the Drude plasma frequencies as well as ${\varepsilon _{intra}}$ along the a and b axes are significantly modified. The interband transition of electrons depends on the relative position of the Fermi energy in the electron bands; with enough large doping, the interband transition of electrons can be turned on or off, ${\varepsilon _{inter}}$ is then modified. As for WTe2, it seems that most of the changes are from ${\varepsilon _{intra}}$.
Fig. 6. Permittivities along the a and b axes with total charges (a) −0.02 (electron doping, 4.86 cm−3), (b) −0.05 (electron doping, 12.14 cm−3), (c) +0.02 (hole doping, 4.86 cm−3), and (d) +0.05 (electron doping, 12.14 cm−3). Solid and dashed curves denote ${\varepsilon _1}$ and ${\varepsilon _2}$, respectively. Elliptic and hyperbolic regimes are respectively shaded with red and blue.
From Fig. 6(b), one can notice that the first hyperbolic region has vanished and become elliptic, i.e., the topology of plasmonic responses has been changed. This is because the Drude plasma frequency along the a axis increases drastically when doped with electrons. In the low-frequency range, the intraband transition part of the permittivity dominates, which can be written as ${\varepsilon _{intra}} ={-} \omega _D^2/[{\omega ({\omega + i\gamma } )} ]$, where ${\omega _D}$ is the Drude plasma frequency and $\gamma $ is the intraband broadening. Large Drude plasma frequency would push the permittivity downwards, finally changing the hyperbolic region into an elliptic one. The 2D conductivities correspond to Figs. 6(a) and 6(b) are plotted in Fig. 7(a). Solid and dashed curves show the imaginary parts of the conductivities of WTe2 with low and high electron concentrations, respectively. The shaded area denotes the frequency range where the transition of the topology happens. The imaginary parts of the conductivities along the a and b axes originally have opposite signs, but both become positive after being doped with enough electrons. The corresponding isofrequency contours or k-surfaces of the plasmonic modes in extended monolayer are plotted in Fig. 7(b), where blue and red curves respectively denote the contours within the hyperbolic and elliptic regimes. The photon energy E = 0.1511 eV is chosen near the center of this frequency range. Materials with switchable topology of plasmonic responses just like WTe2 are rare in nature, which may lead to interesting applications in plasmonics.
Fig. 7. (a) Conductivities (only the imaginary parts) of the low (solid curve) and high (dashed curve) electron doping conditions; the total charges are −0.02 and −0.05, respectively. With increasing doping level, the imaginary parts of the conductivities become both positive, as shown by the shaded region in (a). (b) The isofrequency contours or k-surfaces at 0.1511 eV in the elliptic (red curve) and the hyperbolic (blue curve) regimes. With increasing doping level, the topology of plasmonic responses changes from hyperbolic to elliptic. For simplicity, ohmic losses have not been considered here.
In this part of the paper, we investigate the effects of band broadening on the plasmonic responses of WTe2. Band broadening here is an empirical parameter, which can not be calculated based on DFT in the single particle picture. Large value of broadening can be induced by, for example, electron-electron scattering, electron-phonon scattering, and even defects. Understanding how such parameters affect the permittivities might be important for real-world applications. Figure 8 shows the permittivities along two directions with different values of band broadening (5 meV, 10 meV, 50 meV). Solid and dashed curves now denote the real parts of permittivities along the a and b axes, respectively. With broadening as large as 50 meV, line shapes caused by strong interband transition of electrons have been partially destroyed; the second elliptic and hyperbolic regions which rely on such strong resonances are no longer obvious in Fig. 8. However, the first elliptic and hyperbolic regions seem to be relatively robust against such broadening. Based on these investigations, reducing the dissipation in monolayer WTe2 is critical to the plasmonic responses. Cryogenic environment is probably necessary if one expects the observations of these plasmons.
Fig. 8. Permittivities (only real parts) along the a and b axes with different values of band broadening: 5 meV, 10 meV and 50 meV. The results are respectively denoted by black, blue and red curves. Large band broadening can destroy the line shapes corresponding to the interband transition of electrons. The first elliptic and hyperbolic regions seem to be relatively robust against such broadening.
To further verify the topological transition behaviour of plasmonic responses in monolayer WTe2, we have calculated the permittivities based on optimal basis functions [34–36]. Different from the calculations of the permittivities above, the Fermi energy here is modified manually and directly, other than being determined by self-consistent calculations of the charged (non-neutral) primitive cell, which we believe might be more accurate. Black solid and dashed curves in Fig. 9 are respectively the permittivities of monolayer WTe2 along the a and b axes, where the broadening used for Drude plasma frequency is 25 meV. Clearly, one can find the four regions (two elliptic as well as two hyperbolic regions) in the figure. With larger broadening, for example 100 meV, the line shapes near the resonances are partially destroyed, as shown by the blue solid and dashed curves in Fig. 9. The first elliptic as well as the first hyperbolic regions seem to be relatively robust. The volume-averaged Drude plasma frequencies along a and b axes in this case are 0.71 eV and 0.54 eV, respectively. Under electron doping, the Fermi energy shifts upwards. Red solid and dashed curves in Fig. 9 respectively denote the permittivities along the a and b axes where the Fermi energy has been increased by 0.6 eV. The transition of the topology of plasmonic responses is quite obvious. Such topological transition can be attributed to rapid increasing of the Drude plasma frequency along one particular direction (the a axis as for WTe2); the Drude plasma frequencies along the a and b axes are now 1.41 eV and 0.58 eV, respectively. The effects of spin-orbit coupling (SOC) have also been considered. The corresponding permittivities are plotted as the inset in Fig. 9; only the cases with large broadening (100 meV) are shown. Red/blue curves denote the permittivities with/without doping. With spin-orbit coupling, these elliptic and hyperbolic regions remain their existences and it is still possible to actively control the topology of the plasmonic responses via doping.
Fig. 9. Permittivities (only real parts) along the a and b axes, respectively plotted using solid and dashed curves. Black and blue curves (solid as well as dashed ones) respectively correspond to the cases with 25 meV and 100 meV broadenings (for Fermi energy). Red curves (solid as well as dashed ones) show the permittivities where the Fermi energy has been shifted upwards by 0.6 eV. Topological transition of the plasmonic responses (from hyperbolic to elliptic) induced by doping is quite obvious. Inset shows the corresponding permittivities with large broadening (100 meV, plotted with blue and red curves) after the consideration of spin-orbit coupling.
The hyperbolic responses in WTe2 is due to strong interband transition of electrons when the polarization of light is along the b axis. The transition intensity is proportional to ${\left|{\left\langle {{\varphi_v}|{\boldsymbol r} |{\varphi_c}} \right\rangle } \right|^2}$, where ${\varphi _c}$ and ${\varphi _v}$ are the wavefunctions of the conduction and valence bands at the same k point; if one of the wavefunctions is asymmetric and the other is symmetric with respect to the b axis, there could be a large transition dipole. Figure 10(a) shows the surfaces of the bands near the Fermi energy on a relatively coarse k-grid on the whole Brillouin zone. Interband transition is “vertical”; thus, we have searched the energies of these states and found two pairs (totally four states marked by green dots) of which the energy differences are between 0.15 eV and 0.2 eV. The corresponding square of the wavefunctions are plotted in Fig. 10(b). Numbers in the parentheses are the coordinates of k in the unit of $2\pi /a$. Roughly speaking, the wavefunctions of the two conduction states in Fig. 10(b) align along the b axis, and the two valence states on the other hand seem to mostly align along the a axis, which may explain the strong dipoles (peaks on the permittivity near 0.2 eV) along the b axis induced by interband transition. We have done the same analysis and found two pairs (totally four states marked by red dots) of which the energy differences are between 0.29 eV and 0.33 eV, as shown by Fig. 10(c). The corresponding square of the wavefunctions are plotted in Fig. 10(d). Roughly speaking, two valence states partially align along the b axis, and two conduction states are distorted and only partially align along the a axis. This may also explain the strong dipoles (peaks on the permittivity near 0.3 eV, not as strong as those near 0.2 eV) along the b axis induced by interband transition.
Fig. 10. (a) Surfaces of the two bands near the Fermi energy. Green dots (as indicated by the two small black arrows) denote the states of which the energy differences are between 0.15 eV and 0.2 eV, corresponding to the peaks of permittivity in this range induced by interband transition. (b) The square of the wavefunctions exported from the data. Numbers in parentheses are the coordinates of k in the unit of $2\pi /a$. (c) Surfaces of the two bands near the Fermi energy. Red dots (as indicated by two small black arrows) denote the states of which the energy differences are between 0.29 eV and 0.33 eV, corresponding to the peaks of permittivity in this range induced by interband transition. (d) The square of the wavefunctions exported from the data. Numbers in parentheses are the coordinates of k in the unit of $2\pi /a$.
For the bulk and multilayer WTe2, our calculations show that strong interband transition peaks like those mentioned above might be suppressed. The total permittivities along two in-plane directions are shown in Fig. 11; both the real (solid curves) and imaginary (dashed curves) parts are plotted. Two insets, the lower one depicts the structure of bulk WTe2, and the upper one is the enlarged plot of the shaded area (gray) covering the region where two solid curves crossing zero. From the structure, two neighboring layers are arranged reversely. Thus, strong peaks corresponding to the dipole moments in the y direction (the direction of the b axis) is canceled; one can notice the absence of strong peaks in Fig. 11. Compared with monolayer, bulk WTe2 only has two frequency ranges supporting in-plane plasmons: one is elliptic and the other is hyperbolic. In the hyperbolic regime, the permittivity along the a axis is negative, and Drude plasma frequency along this direction is also higher indicated by the fact that the curve representing the permittivity along the a axis is below the one representing the permittivity along the b axis. Recent experimental researches support such results [27]. As for multilayer WTe2, since the permittivities are volume-averaged, the peaks would also be weaken.
Fig. 11. Permittivities along the a and b axes in bulk WTe2. Solid and dashed curves represent the real and imaginary parts, respectively. The lower inset depicts the structure of bulk WTe2. The upper inset is the enlarged plot of the shaded area (gray) covering the region where two solid curves crossing zero, where the hyperbolic region is highlighted (sky blue).
4. Conclusion
In this paper, we have theoretically studied the plasmonic responses of monolayer WTe2. Both elliptic as well as hyperbolic regimes have been found in this material; the isofrequency contours of the modes in extended monolayer are analytically derived. Edge-confined modes in the elliptic regime and waveguiding modes in nanoribbons in the hyperbolic regime are calculated using numerical method. The hyperbolicity in the low frequency range is due to strong anisotropic interband transition of electrons other than bands nesting since the photon energy is below the excitation threshold. Further investigations show that moderate electron doping can change the plasmonic responses in specific frequency range from the hyperbolic regime to the elliptic regime, leading to active control over the topology of the plasmonic responses. Then, the effects of band broadening are discussed and the permittivities have been calculated using optimal basis functions to further verify our conclusions. Two strong peaks induced by interband transition are discussed and the related wavefunctions are used to explain the in-plane transition dipole along the b axis. In the end, we point out that due to the arrangement of the layers in the bulk WTe2, strong in-plane dipole along the b axis might be suppressed. The calculated permittivities of the bulk WTe2 show the absence of strong peaks and also the Drude plasma frequency is higher along the other axis. Our investigations indicate that monolayer WTe2 is a promising platform for plasmonic applications in the future.
Funding
National Natural Science Foundation of China (61805062, 12004362, 11804070).
Acknowledgments
M. N. Chen gave us some helpful discussions about this work.
Disclosures
The authors declare no conflicts of interest.
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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