Possible Realization of Hyperbolic Plasmons in Few-Layered Rhenium Disulfide

Abstract

Hyperbolic plasmons are a highly desired property in optoelectronics and biomolecular sensing. The necessary condition to realize hyperbolic plasmons is a significant anisotropy of the principal components of the dielectric function, such that at a certain frequency range, one component is negative and the other is positive, i.e., one component is metallic, and the other one is dielectric. Here, we study the effect of anisotropy in ${\mathrm{ReS}}_2$, and our theory shows that ${\mathrm{ReS}}_2$ can host hyperbolic plasmons in the ultraviolet frequency range. The operating frequency range of the hyperbolic plasmons can be tuned with the number of ${\mathrm{ReS}}_2$ layers. However, we note that the significantly large imaginary part of the macroscopic dielectric response in all layered variants of ReS₂ can result in substantial losses for the hyperbolic plasmons, as in the case with other known hyperbolic materials, with the exception of MoOCl₂. We also note that ReS₂ hosts ultraviolet hyperbolic plasmons while ZrSiSe, WTe₂, and CuS nanocrystals host infrared plasmons, providing a novel platform for optoelectronics in the ultraviolet range.

1. Introduction

The rise of hyperbolic materials [1,2,3,4,5,6,7,8,9,10] in recent years promises important applications in optoelectronics, nanophotonics [11,12,13,14,15], and biomolecular sensing [16]. Light can acquire hyperbolic dispersion while passing through such materials—which occurs in certain frequency ranges—when different principal components of the longitudinal dielectric function (dielectric permittivity) have opposite signs. For an isotropic medium, it behaves as a dielectric, that is, it supports propagating electromagnetic waves, when the sign of the dielectric function is positive. When the dielectric function is negative, the incident light is reflected, with only an exponentially decaying evanescent field penetrating the material, like for metals below the plasma threshold.

Anisotropy in electronic, optical, vibrational, and transport behavior can occur when structural anisotropy is present, and if it is sufficiently strong, the different components of the dielectric permittivity tensor may acquire opposite signs to turn the material hyperbolic. Note that exceptions exist where the optical anisotropy is significant even when the structural anisotropy is low [17]. In two-dimensional crystals, in-plane anisotropy is strong enough, making it hyperbolic; this is a unique situation that can confine short wavelengths (large wave vectors) inside a material, promising smaller sizes for optoelectronic devices. Optical anisotropy in rhenium disulfide (${\mathrm{ReS}}_2$) has been established both for bulk crystals [18,19] and for thin layers [20]. In this work, we predict that ${\mathrm{ReS}}_2$, which appears in a distorted 1T phase, can realize hyperbolic plasmons depending on the number of layers.

It has been suggested that anisotropic 2D materials can be tuned to become hyperbolic via electrostatic tuning, strain, or dimensionality, and can host hyperbolic plasmons [21,22]. Several studies [15,23,24] have investigated hyperbolic plasmons (HPs) and their existence in naturally occurring materials. Previous works [25,26] have studied the band structure and anisotropic optical response of ${\mathrm{ReS}}_2$, but the study of HPs remains largely unexplored. Here, we use the ladder-vertex-corrected and local-field-corrected plasmonic response in ReS₂ within a self-consistent solution of the Bethe–Salpeter equation (BSE), as implemented in Questaal [27].

The rest of this paper is organized as follows. In Section 2.1, we describe the anisotropic atomic structure of ${\mathrm{ReS}}_2$, and in Section 2.2, we briefly describe the primary theoretical methods and provide computational details. In Section 2.3, we provide a detailed overview of the mathematical formulation and its numerical implementation that is involved in calculating the optical properties. In Section 3.1 and Section 3.2, we present our results on the electronic structure and optical properties of ${\mathrm{ReS}}_2$. In Section 4, we briefly summarize our results and conclude the paper.

2. Atomic Structure, Theoretical and Computational Details

2.1. Atomic Structure

${\mathrm{ReS}}_2$ belongs to the family of two-dimensional (2D) layered transition metal dichalcogenides (TMDs) of the form ${\mathrm{MX}}_2$, where M is a transition metal atom (Mo, W, Re, etc.) and X is a chalcogen (S, Se, Te). Unlike other TMDs, which usually have either a 1H or 1T structure in their ground state, ${\mathrm{ReS}}_2$ has neither H nor T characteristics. It crystallizes in a distorted-1T structure with a clustering of Re units forming parallel metal chains along the van der Waals (vdW) plane (see Figure 1).

The compound ${\mathrm{ReS}}_2$ belongs to the triclinic space group P1, resembling a distorted CdCl₂ structure. It comprises three atomic layers, S-Re-S, where covalent bonds join Re and S. The adjacent layers of ${\mathrm{ReS}}_2$ are coupled by weak van der Waals (vdW) forces to form bulk crystals. The unit cell is derived from hexagonal symmetry toward a distorted 1T structure, in which Re atoms group into parallelograms of four Re atoms. The formation of Re chains breaks the hexagonal symmetry and doubles the unit cell size. Hence, the unit cell of the single-layer ${\mathrm{ReS}}_2$ in the distorted-1T phase is composed of four Re and eight S atoms (Figure 1a,b). In pristine ${\mathrm{ReS}}_2$, the valence band maximum is composed of 5d orbitals of Re atoms and 3p orbitals of S atoms, and the conduction band minimum is derived from 5d orbitals of Re atoms. The Brillouin zone of the ${\mathrm{ReS}}_2$ is hexagonal but with unequal sides as a result of the distorted atomic structure. Energy band structures were generated along the symmetry lines shown in Figure 1c. The layered variants of ${\mathrm{ReS}}_2$ have been simulated for these calculations with the parameters shown in

Table 1. Lattice parameters of bulk, monolayer (ML), and bilayer (BL) ${\mathrm{ReS}}_2$.

Structure	a(Å)	b(Å)	c(Å)	$\mathit{\alpha }$(°)	$\mathit{\beta }$(°)	$\mathit{\gamma }$(°)
Bulk	6.41695	6.52047	7.28252	91.8128	103.5630	118.8390
ML	6.41910	6.52306	45	90.7434	95.7909	118.8366
BL	6.41910	6.52306	28.66062	84.0127	89.7412	61.1634

.

2.2. Computational Details

LDA, QSGW, and QSG$\widehat{W}$ Self-Consistency

Single-particle calculations (LDA, and the the quasiparticle self-consistent GW [29] (QSGW) self-energy ${\Sigma }^0\left(k\right)$) were performed on $12\times 12\times 12$ points (Monkhorst pack) for bulk and $12\times 12\times 1$ for ML and BL. An energy cutoff of 400 eV was used and Gaussian smearing was performed with a width of 0.05 eV. Energy and root mean square (RMS) density convergence criteria were set to ${10}^{-5}$ eV and $2\times {10}^{-6}$, respectively. The charge density was made self-consistent for each iteration in the QSGW self-consistency cycle. The QSGW cycle was iterated until the RMS change in ${\Sigma }^0$ reached ${10}^{-5}$ Ry. Thus, the calculation was self-consistent in both ${\Sigma }^0\left(k\right)$ and the density. Numerous checks were made to verify that the self-consistent ${\Sigma }^0\left(k\right)$ (k) was independent of the starting point. For ML-${\mathrm{ReS}}_2$, we performed a rigorous check for vacuum correction to all band gap and dielectric screening by increasing the size from 10 Å to 45 Å. Since we have a vacuum along the z-direction, the dielectric constant, which is the real part of the macroscopic dielectric response at $\omega$ = 0, should be close to unity.

In the present work, the electron-hole two-particle correlations are incorporated within a self-consistent ladder BSE implementation [30,31] with Tamn–Dancoff approximation [32]. Ladder diagrams are included in the polarizability P that makes W, via the solution of a BSE; thus, this form of $GW$ outperforms the RPA in constructing the self-energy $\Sigma =iGW$. The electron-hole attraction from the ladders enhances P, thus reducing W, which in turn reduces the band gap. A static vertex is used to construct P. G and W are calculated self-consistently, in a quasiparticle form [29]; G and W are updated iteratively until all of them converge (QSGW). When ladders are incorporated into W, we denote the process as QSG$\widehat{W}$ to signify that W was computed from the BSE. The macroscopic dielectric function we present here, ${\left[{ϵ}_{\mathbf{G}=0,{\mathbf{G}}^{\prime }=0}^{-1}(q\to 0,\omega )\right]}^{-1}$, is also computed with the BSE.

The tetrahedron method is employed for integration over the Brillouin zone to calculate the optical spectrum. When calculating the dielectric response within BSE, the valence and conduction states that form the two-particle Hamiltonian are increased until the two-particle eigenvalues converge within an accuracy of 10 meV. For ReS₂ the excitons are essentially Wannier–Mott [33] in nature and only the states at the valence-band top and conduction-band bottom contribute to their formation. So, the convergence in the two-particle Hamiltonian size is much faster compared to the cases of CrX₃ [34], where the excitons have Frenkel characteristics and many valence and conduction bands over several electron volts. However, for the present work, our focus is the plasmonic response, and we will note the excitonic binding energies in different layered variants later.

Table 1 contains the lattice parameters for ${\mathrm{ReS}}_2$ used throughout the calculations.

2.3. Theory and Numerical Implementation

The QSGW approach is derived from the many-body perturbative approach proposed and developed by Hedin [35]. The following set of closed coupled equations [35,36,37] are to be solved iteratively in this scheme:

$$\begin{array}{cc}\hfill \Sigma (1,2)& =\mathrm{i}\int \mathrm{d}\left(34\right)G(1,3^{+})W(4,1)\Gamma (3,2,4)\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill G(1,2)& =G_0(1,2)+\int \mathrm{d}\left(34\right)G_0(1,3)\Sigma (3,4)G(4,2)\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill W(1,2)& =v(1,2)+\int \mathrm{d}\left(34\right)v(1,3)P(3,4)W(4,2)\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill P\left(12\right)& =-\mathrm{i}\int \mathrm{d}\left(34\right)G(1,3)G(4,1^{+})\Gamma (3,4,2)\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \Gamma (1,2,3)=& \delta (1,2)\delta (1,3)+\hfill \\ & \int \mathrm{d}\left(4567\right){\displaystyle \frac{\delta \Sigma (1,2)}{\delta G(4,5)}}G(4,6)G(7,5)\Gamma (6,7,3)\hfill \end{array}$$

(5)

where G is the Green’s function, $v(\mathit{r},{\mathit{r}}^{\prime })=1/|\mathit{r}-{\mathit{r}}^{\prime }|$ is the bare Coulomb interaction, W is the screened Coulomb interaction, $\Gamma$ is the irreducible vertex function, P is the irreducible polarizability, and $\Sigma$ is the self-energy operator. In Equations (1) and (4), the indices subsume position and time and the + superscript implies $t^{\prime }=t+\eta$, with $\eta \to 0^{+}$.

The approximation for the irreducible polarization obtained by neglecting the vertex corrections is referred to as the independent particle approximation and the Random-Phase-Approximation (RPA) [38] and underlie the QSGW approach.

The RPA, when used to make W, leads to errors, e.g., a band gap overestimation. To include the vertex corrections in Hedin’s equations, one needs to determine the interaction kernel $\delta \Sigma /\delta G$ in Equation (5). In the $GW$ approximation for $\Sigma$ and assuming that $\delta W/\delta G$ is negligible [38], $\delta \Sigma \left(12\right)/\delta G\left(45\right)=\mathrm{i}W\left(12\right)\delta (1,4)\delta (2,5)$, where W is determined in the $GW$ approximation. This expression is then inserted in Equation (5).

Before the BSE polarization is computed, the two-point polarization needs to be expanded into its four-point counterpart: $P\left(12\right)=P\left(1122\right)=P\left(1324\right)\delta (1,3)\delta (2,4)$. This is a form of polarization that exceeds the RPA using Equations (4) and (5) by adopting the expression for the interaction kernel from above:

$$P\left(12\right)=P^{\mathrm{RPA}}\left(12\right)-\int \mathrm{d}\left(34\right)P^{\mathrm{RPA}}\left(1134\right)W\left(34\right)P\left(3422\right),$$

(6)

where $P^{\mathrm{RPA}}\left(1234\right)=-\mathrm{i}G\left(13\right)G\left(42\right)$. The W that appears in the interaction kernel, $\delta \Sigma /\delta G$, is calculated at the level of the RPA where this is usually assumed to be static, i.e., $\delta \Sigma /\delta G=\mathrm{i}{W}^{\mathrm{RPA}}(\omega =0)$. Using this vertex-corrected polarizability, the macroscopic dielectric response of the system can be computed to provide information about plasmons and excitons. Complete details of the numerical implementation is provided in our two previous work [27,31]. Note that the local field corrections [39] are included in our computed polarizability. The plasmon energies are extracted from zeros of the real part of the macroscopic dielectric response [40,41] which coincide with the peaks of the loss function in the frequency–momentum plane.

3. Results

3.1. Quasiparticle Energies and Band Structure

The nature of the band gap of ${\mathrm{ReS}}_2$ has been widely debated in the literature. In a typical TMD family, the band gap is direct when their thickness is reduced to the monolayer, ensuring that coupling with light is strong. One study from 2014 [42] reported a direct band gap for bulk ReS₂, thus generating considerable interest in the system. However, older studies such as [43,44], and also more recent studies such as [45,46], report that the bulk ${\mathrm{ReS}}_2$ is an indirect-band-gap semiconductor.

The electronic band structures (with spin–orbit coupling included) for bulk, BL, and ML are shown in Figure 2 and the band gaps at different levels of theory are summarized in

Table 2. Bandgap of bulk, BL, and ML variants of ${\mathrm{ReS}}_2$ at different levels of theory (with spin–orbit coupling). The gap increases from LDA to the QSGW level. The effect of screening is only moderately increased when two-particle interactions (via a BSE, W $\to \widehat{W}$) are added, thus only weakly decreasing the QSGW band gap.

Band Gap (eV)
Theory	LDA	QSGW	QSG$\widehat{\mathbf{W}}$
Bulk	1.15	1.75	1.7
ML	1.29	2.75	2.66
BL	1.23	2.35	2.3

. Similar to prior work on monolayers of chromium trihalides [47], we check for convergence and scaling of the band gap and the dielectric constant ${ϵ}_{\infty }$ with vacuum size. We obtain the LDA band gap of ∼1.29 eV, which is significantly lower than the QSGW band gap of ∼2.75 eV. LDA is known to underestimate the band gaps in semiconductors, and the enhancement in the QSGW band gap relative to LDA is standard [29]. QSGW usually overcorrects the gap because W is universally too large within the random phase approximation (RPA), and for the same reason, it underestimates the dielectric constant ${ϵ}_{\infty }$ [30]. Adding ladders largely eliminates both tendencies. In the present case, extending $\mathrm{QS}GW\to \mathrm{QS}G\widehat{W}$ causes only a modest reduction in the gap to ∼2.66 eV (Figure 2a), suggesting insignificant corrections to the self-energy originating from BSE. This result is consistent with previous theoretical work [48] on ML-${\mathrm{ReS}}_2$. The self-energy and the reduced screening increase the band gap and modify the band topology, which is observed in the ML-${\mathrm{ReS}}_2$. The band gap at the level of LDA is direct, but the two-particle interactions lower the valence band maxima (VBM), which were at the $\Gamma$ point by about 150 meV. A similar kind of change in band topology has been observed for ${\mathrm{ReSe}}_2$ in [48] but is absent in other theoretical work [25].

We obtain a similar band topology in BL and bulk variants of ${\mathrm{ReS}}_2$; however, the band gap values are ∼2.3 eV (Figure 2b) and ∼1.7 eV (Figure 2c), respectively. The nature of the band gap in ${\mathrm{ReS}}_2$ is different from more commonly studied semiconducting TMDs (e.g., MoS₂, WS₂, etc.), where the bulk and few-layered variants show that the band gap is indirect while ML is direct. In this work, we observe a direct band gap at the LDA level and an indirect band gap for all the variants at the QSG$\widehat{W}$ level. This nature of the quasiparticle band gap does not conflict with experimental measurements because, unlike our calculated free-standing cases, these measured samples are on substrates and are inevitably doped. The resulting self-energy corrections will be reduced under these conditions, resulting in a slightly indirect band gap for the measured samples. We note that the direct-to-indirect transition may not be sharp because the energy difference between the direct and indirect gaps is small, and external perturbations can affect the conclusion. However, our observations on the indirect nature of the band gap is important for many reasons. In most monolayer TMDs, the gap is direct, making them sufficiently bright; also, the electron-hole radiative lifetimes are extremely small (often in picoseconds). While in systems with indirect band gaps, radiative and non-radiative processes compete since the indirect states have longer lifetimes, making them candidates for optoelectronics and photovoltaics [49].

3.2. Optical Absorption Spectra: Hyperbolic Plasmons

The anisotropic optical absorption has been previously studied using DFT [50]. Experimentally [45], it is demonstrated that the reduced crystal symmetry of ${\mathrm{ReS}}_2$ leads to anisotropic optical properties that persist from the bulk down to the monolayer limit. The absence of excitonic correlations and underestimated band gaps in LDA studies hide several physical consequences in ${\mathrm{ReS}}_2$. Advanced theoretical studies such as [48] tackle anisotropic optical responses at the BSE level for monolayer ${\mathrm{ReS}}_2$, where ladder-vertex-corrected optical properties are computed on top of a single-shot DFT-based G₀W₀ one-particle description.

The large structural anisotropy in 2D materials (for example, a 4:3 anisotropy of the in-plane lattice constants in black phosphorus [21] and solid nitrogen [51,52,53]) makes them perfect candidates for hyperbolic materials and a natural place to look for HP. This offers new possibilities as hyperbolic materials showcase a wide variety of interesting properties, such as modes which transport heat by photon tunneling with a high efficiency close to the theoretical limit [54] and broadband absorption [55]. We first define the condition for hyperbolicity. The hyperbolic region appears when

$${ϵ}_1^{xx}\left(\omega \right)·{ϵ}_1^{yy}\left(\omega \right)<0$$

(7)

where ${ϵ}_1^{xx}\left(\omega \right)$ and ${ϵ}_1^{yy}\left(\omega \right)$ are the real parts of the dielectric response along the x and y directions, respectively. We assume that ${ϵ}_1^{xy}\left(\omega \right)=0$ by symmetry and, thus, x and y are the principal directions of the dielectric permittivity. For different variants of ${\mathrm{ReS}}_2$, the real and imaginary parts of dielectric response are plotted in Figure 3. We observe that there is a significant difference in optical response for incident polarized light along different directions. While ML-${\mathrm{ReS}}_2$ (Figure 3a,d) hosts some strongly bound anisotropic excitons deep inside the one-particle gap, HPs are absent (Figure 4a). For the BL-${\mathrm{ReS}}_2$ (Figure 3b,e), the Re(${ϵ}_{yy}$) becomes negative at $\omega =6.65$ eV (Figure 4b), which results in a hyperbolic region in a 0.43 energy window 6.65–7.08 eV. For bulk-${\mathrm{ReS}}_2$ (Figure 3c,f), this energy window increases to 0.76 eV (Figure 4c). The sign change is key to the appearance of the hyperbolic region and it becomes more apparent in Figure 4 where we plot the product ${ϵ}_1^{xx}\left(\omega \right)\times {ϵ}_1^{yy}\left(\omega \right)$ which becomes negative in the energy window. ${ϵ}_2$ remains large for both bulk and BL in the hyperbolic energy window, suggesting large damping of the HPs. Note that the computed ${ϵ}_2$ only contains electronic scatterings and do not involve any phonon-mediated damping. In the BL, these plasmons are less damped compared to bulk. The HP energy windows and the macroscopic dielectric tensors are summarized in

Table 3. Dielectric constant (real part of the dielectric response at $\omega =0$) calculated using QSG$\widehat{W}$. As the dimensionality of ${\mathrm{ReS}}_2$ is lowered, the frequency range where Re(${ϵ}_{xx}$) · Re(${ϵ}_{yy}$) is less than zero becomes smaller.

Dielectric Constant (eV)
	${\mathbf{ϵ}}_{\infty }^{\mathit{xx}}$	${\mathbf{ϵ}}_{\infty }^{\mathit{yy}}$	${\mathbf{ϵ}}_{\infty }^{\mathit{zz}}$	Plasmonic Frequency Range (eV)	Exciton Binding Energy (eV)
Bulk	9.67	9.37	6.21	6.02–6.78 (0.76)	0.1
BL	6.97	7.09	2.66	6.65–7.08 (0.43)	0.3
ML	2.97	3.19	1.42	-	0.74

. Further, we show the plasmon dispersions along [100] and [010] crystalline directions (see Figure 5). At small momenta (long wavelength limit), the plasmons follow a $\sqrt{q}$ behavior characteristic of a 2D system and deviates at larger momenta.

We note that these HPs in ReS₂ are in the ultraviolet range, in contrast to the infrared HPs in CuS nanocrystals [15] and ZrSiSe [4], and visible-to-infrared HPs in MoOCl₂ [2]. Note that the HPs in MoOCl₂ can propagate for several micrometers, up to about ten cycles at room temperature. This is an exceptional scenario considering that in most other systems, HPs barely survive for one cycle before the plasmons decay. As we explored in our MoOCl₂ work [2], we believe the ultrastability of the plasmons are due to the specific crystal field environment and the possibility to host intra-band transitions in MoOCl₂. However, more work is needed on this front to explore the key parameters behind the stability of HPs in various systems. The only prominent exception is the surface plasmons in Graphene, which are ultrastable. ReS₂ plasmons appear to be similar to ZrSiSe and WTe₂ plasmons, where the large ${ϵ}_2$ appear to be strong enough to damp out the plasmons.

The inherent anisotropy in ReS₂ provides an opportunity to tune its magnitude by applying strain and explore whether this can be used to enhance the hyperbolic windows and increase their stability. For example, we do not expect that realistic strains will be large enough to make the ML hyperbolic. In bulk, we expect the HP window to enhance and become more robust under compressive unidirectional strain along x. We also expect those HPs to have lesser damping compared to the unstrained compound. On the other hand, unidirectional strain along y should reduce the HP window based on our observations. In the BL, on the contrary, we expect the application of strain along x to kill the HP window, while strain along y should enhance the HP window and also host more weakly damped plasmonic modes. However, a rigorous analysis of strain-dependent evolution for HPs in ReS₂ is reserved for future work. We note that recent studies on Dirac and semi-Dirac systems reveal exciting plasmonic properties [56,57,58] which are novel and unconventional. A rigorous analysis of the anisotropic plasmonic properties in ReS₂ is needed to reveal its differences with those observed in Dirac systems.

4. Conclusions

Anisotropy is the key to tuning material properties. Discontinuities at surfaces, residual strains, and metamaterials have been used as platforms for realizing anisotropic optical properties. Naturally occurring structurally anisotropic materials are not necessarily always hyperbolic.

In this work, we show that the structural anisotropy in ${\mathrm{ReS}}_2$, even though much weaker than materials like black phosphorus and solid nitrogen, leads to the theoretical realization of hyperbolic plasmons in a narrow energy window, which appear to be significantly damped. The plasmonic resonances, nevertheless, can be tuned by controlling the number of layers in the far-ultraviolet frequency range. The stability of these theoretically predicted damped hyperbolic modes, with potential applications for ultraviolet optoelectronics, nano-imaging, and biomolecular sensing, remains to be explored through experimentation.