Wandering principal optical axes in van der Waals triclinic materials

Symmetry plays a pivotal role in fundamental laws of nature^{1,2,3,4,5,6,7}, including classical equations of motion, conservation laws, superposition principle, selection rules, and exchange interaction^8,9,10,11. In condensed matter, it governs many of the material’s mechanical, electronic, and optical properties, such as stress tensor, electron mobility, conductivity, refractive index, and allowed nonlinear processes, among others^12,13,14,15. Highly symmetric atomic lattices, such as Al, Ni, and Au, result in isotropy of electronic and optical properties, severely limiting their use^16,17,18. For instance, they lack even-harmonic generation, birefringence, and chirality^4,19,20. On the other hand, reducing the lattice’s symmetry group leads to the emergence of anisotropy – the change of particular property in the observation direction^5,21. The most known consequence is the birefringence phenomenon, wherein a birefringent material doubles an image²². This effect is just one of the numerous implications of anisotropic optical properties, traditionally described via the permittivity tensor²³. It effectively describes the difference in refractive indices along various directions. This anisotropy produces complex isofrequency contours in the reciprocal space²⁴ enabling hyperbolic materials²¹, ghost¹ and shear^10,25 polaritons, negative refraction^26,27, canalization of radiation⁴, and many other intriguing wave phenomena.

Van der Waals (vdW) crystals offer a flexible and highly functional platform with a built-in anisotropy due to their fundamental difference between intralayer covalent and interlayer vdW bonding²⁸. Therefore, such layered materials allow exotic light-matter interactions²⁹, resulting in exciton-³⁰, phonon-³¹, edge-³², and moiré-polaritons³³. In most cases, this anisotropy is purely uniaxial, and the principal optical axes of the permittivity tensor are stationary with wavelength²⁸. Some vdW crystals, however, have biaxial anisotropy because of the in-plane low-symmetry crystal structure^{34,35,36,37,38}. Combined with non-orthogonally polarized in-plane exciton resonances³⁴, rhenium disulfide and rhenium diselenide can enable the wandering (wavelength-dispersive) direction of the principal optical axes of the permittivity tensor. Although the prediction of wavelength-dispersive principal optical axes dates back to 1928³⁹, experimental evidence of the discussed behavior has been elusive in inorganic crystals. We anticipate that more exotic optical responses and applications may be expected in materials with wandering principal optical axes, which can extend the evergrowing phenomena in low-symmetry nanophotonics⁴⁰.

In this work, we experimentally observed the rotation of principal optical axes in triclinic vdW crystals. We explained it via a bi-excitonic model, also recreating the wandering of such principal optical axes with first-principle calculations of the permittivity tensor. Here, only the individual components of the obtained permittivity tensor satisfy the Kramers–Kronig (KK) relations. In contrast, the generalized KK relation for crystallographic axes⁴¹ is not applicable to triclinic rhenium disulfide (and diselenide). Hence, these crystals have extraordinary optical properties that set them apart from vdW and non-vdW crystals. Furthermore, from a practical point of view, our near-field nanoimaging results reveal high wavelength sensitivity of light-matter interaction in triclinic vdW crystals, which can be leveraged for advanced light routing. Thus, triclinic van der Waals rhenium disulfide (and diselenide) offer a platform for anisotropic phenomena and next-generation nanophotonics.

Impact of triclinic crystal structure on optical axes

ReS₂ and ReSe₂ are ideal materials for asymmetry-driven phenomena since they exhibit the lowest symmetry triclinic crystal structure⁴², shown in Fig. 1a–c. Consequently, they received considerable interest in recent works^{34,35,36,37,43,44,45}, which reported a high linear and nonlinear optical anisotropy originating from non-collinear excitons³⁴. In particular, the angle between the polarizations of excitons⁴⁶ is about 70° instead of the expected 90°. It arises from Peierls’ distortion of the 1 T structure (Fig. 1a)³⁶. This feature should, naturally, cause nontrivial optical responses, such as non-orthogonal self-hybridized polaritons⁴⁷. Therefore, a more thorough investigation of the anisotropic dielectric tensor \(\hat{\varepsilon }\) of ReS₂ and ReSe₂ remains a significant challenge both because their dielectric tensors cannot be diagonalized in Cartesian coordinates³⁹ and for their great demand for low-symmetry photonics.

Nevertheless, according to Onsager’s theorem⁴⁸, their dielectric tensors are symmetric (\(\hat{\varepsilon }={\hat{\varepsilon }}^{T}\)). They thus can be divided into Hermitian (Re[\(\hat{\varepsilon }\)]) and skew-Hermitian (Im[\(\hat{\varepsilon }\)]) parts (Fig. 1d–e), primarily responsible for polarization and losses, respectively. It is worth noting that the diagonalization basis for Hermitian and skew-Hermitian tensors can differ and vary with wavelengths, as schematically illustrated in Fig. 1d–e, which can result in wavelength-dispersive principal optical axes. In fact, principal optical axes rotation explains the effects observed in earlier reports^35,38,49 on optical properties of ReS₂ (Supplementary Note 1).

Physical origins of wandering principal optical axes

To visualize this effect, we prepared ReS₂ and ReSe₂ samples (Fig. 2a, b and Supplementary Note 2) and measured polarised transmittance (Fig. 2c) around the exciton resonances. Figure 2c demonstrates how the angle for maximum transmittance shifts for different excitons, showing that the principal optical axes change with exciton resonances. In order to capture its wavelength dependence, we provide polarization spectra in Fig. 2d–e for ReS₂ and in Supplementary Note 2 for ReSe₂. Note that excitonic spectral dips vanish at certain polarizations (Fig. 2d), indicating the orientation of excitons. Of immediate interest are wandering (wavelength-dispersive) principal optical axes, shown in Fig. 2e and Supplementary Note 2. In fact, a recent study⁵⁰ showed that the principal optical axis at 550 and 650 nm tilts by 3° and 2°, respectively, with respect to the b-axis for few-layer ReS₂, which is close to our 7° and 8° observed for bulk ReS₂ (see Fig. 2e). At large wavelengths, the principal optical axes almost coincide with the crystallographic axes (Fig. 2e). However, the principal optical axes vary rapidly at fundamental exciton frequencies and then demonstrate complex behavior for high-energy photons, owing to the material’s rich excitonic structure⁵¹. Still, the crystallographic axes influence the position of the principal optical axes since, at the fundamental exciton resonances, the principal optical axes switch from the crystallographic b-axis to the a-axis (Fig. 2e). At infrared wavelengths, this wandering of principal optical axes reaches 65° whereas, for the whole spectral range, it exceeds 110° change, as seen in Fig. 2e.

Furthermore, this extraordinary optical response influences the Raman spectra (Supplementary Note 3). For instance, polarisation-resolved Raman measurements reveal the change of phonon modes’ preferential direction when the excitation wavelength switches from 532 nm to 633 nm and then to 780 nm (Supplementary Note 3). Although phonon modes’ directions have more complex behavior since they depend not only on the orientation of the principal optical axes but on the phonon modes themself, their dispersion follows a similar pattern to principal optical axes (Supplementary Note 3). This trend is unique to ReS₂ and ReSe₂, as we demonstrate in Supplementary Note 4, exemplifying a highly anisotropic As₂S₃ with static principal optical axes⁵². Indeed, As₂S₃ also has a reduced symmetry, which in principle, may cause a similar effect of wandering principal optical axes. However, unlike ReS₂ and ReSe₂, the crystal structure of As₂S₃ is close to orthorhombic phase with the following crystallographic parameters⁵²: a = 0.42546(4) nm, b = 0.95775(10) nm, c = 1.14148(10) nm, α = 90°, β = 90.442°, and γ = 90°; because the monoclinic angle β differs from 90° by just 0.442(4)°. Moreover, As₂S₃ is transparent in the measured spectral interval (450–1350 nm) implying that its excitons lie below 450 nm, and hence, their effect is negligible⁵². In other words, an illustration of static principal optical axes in As₂S₃ highlights the nontrivial behavior of wandering principal optical axes in ReS₂ and ReSe₂ since the observed effect requires both strongly reduced crystal symmetry and the presence of material’s directional resonances: in our case excitons.

The observed behavior of the principal optical axes of ReS₂ and ReSe₂ can be described with a phenomenological bi-exciton model (Fig. 2f and Supplementary Note 5). According to this model, the permittivity tensor of ReS₂ in the visible range can be expressed as:

where \({\omega }_{{{{{\mathrm{1,2}}}}}}\) is the resonant frequency of the exciton resonance, \({\gamma }_{{{{{\mathrm{1,2}}}}}}\) is its non-radiative decay rate, \({f}_{{{{{\mathrm{1,2}}}}}}\) is the rescaled oscillator strength, and \({{{{{{\boldsymbol{n}}}}}}}_{{{{{{\mathbf{1}}}}}}{{{{,}}}}{{{{{\mathbf{2}}}}}}}={({n}_{x},\, {n}_{y},\, {n}_{z})}^{T}\) is a unit vector describing the in-plane polarization of the corresponding excitonic transition. By varying the parameters of the permittivity model (\({\omega }_{i}\), \({\gamma }_{i}\), \({f}_{i}\)), we managed to find a dielectric tensor (Fig. 2g), which reproduces the wandering effect of ReS₂ principal optical axes (Fig. 2h) within a phenomenological bi-exciton model. The satisfactory agreement between the two-exciton model and the experiment corroborates the leading role of excitons in the observed behavior.

Real-space nanoimaging of wandering principal optical axes

Wandering of ReS₂ and ReSe₂ principal optical axes opens the door to wavelength-switchable optics for efficient light manipulation. As a practical demonstration, we show the effect of wavelength–dispersive principal optical axes on waveguide mode propagation direction using a scattering scanning near-field optical microscopy (s-SNOM) in the transmission scheme, depicted in Fig. 3a. This scheme has no angular rotation (Supplementary Note 6) which makes it advantageous over the reflection scheme. Notably, the principal optical axes vary rapidly at fundamental exciton frequencies (see Fig. 2e and Supplementary Note 2). Therefore, for measurements, we focused on ReSe₂ because it provides a strong variation in the orientation of the principal optical axes within the measured wavelength range of our s-SNOM setup (Methods). To eliminate the edge effect on the near-field image when launching the waveguide modes and launch those modes isotropically, we created a circular hole (the inset in Fig. 3a) inside the ReSe₂ sample. It allows us to visualize the asymmetry of waveguide modes (Fig. 3b–d) caused by material anisotropy only: Fig. 3b–d show elliptical light propagation. As anticipated, these ellipses rotate with wavelength change, as seen from the position of their major axes in Fig. 3e–g (theoretical background of direction change which is provided in Supplementary Notes 7–9). Notably, the observed near-field mode is an interference between the air and waveguide modes. Still, according to our analysis, the air mode’s contribution to the rotation of the mode’s propagation direction is negligible with respect to the wavelength (see Supplementary Note 9). Hence, wandering (wavelength-dispersive) principal optical axes offer a platform to manipulate light without additional structuring and engineering.

Anisotropic dielectric tensors with wandering optical axes

Given the strong wavelength dispersion of the principal optical axes, it is challenging to describe the optical responses of ReS₂ and ReSe₂ correctly. Hence, we fitted the polarized transmittance spectra within the isotropic approximation as the initial step (see Supplementary Note 10). This approach yields a refractive index of about 4 in the infrared range, close to earlier reports^35,38,49, and allows for distinguishing the fundamental excitonic transitions. In the next step, we irradiated samples with unpolarized light to obtain the optical properties averaged over polarization angles. Notably, the resulting optical constants do not follow Kramers–Kronig relations (see Supplementary Note 10) in contrast to other anisotropic vdW materials⁵³. Consequently, ReS₂ and ReSe₂ exhibit anomalous optical responses even for unpolarized light due to the wavelength-dispersive principal optical axes.

To better understand the wandering of principal optical axes, we performed first-principle calculations of monolayer, bilayer, trilayer, and bulk ReS₂ and ReSe₂ anisotropic permittivity tensors, shown in Fig. 4 and Supplementary Note 9. As expected, the off-diagonal tensor elements (Fig. 4a and g) are nonzero, and tensors are not diagonalizable on the orthogonal basis (Supplementary Note 11) because of the crystal’s low symmetry. Nonetheless, we can decouple Hermitian and skew-Hermitian parts of tensors and diagonalize them separately, as shown in Fig. 4 and Supplementary Note 11. The diagonalization process also gives a diagonalization basis, which, in the case of the dielectric tensors, coincides with principal optical axes. Moreover, it allows us to directly observe a dramatic change of principal optical axes orientations from theoretical calculations (Fig. 4c–f and i-l), which agree with the experimental findings in Fig. 2, thereby unambiguously verifying the effect of wandering (wavelength-dispersive) principal optical axes in ReS₂ and ReSe₂. Moreover, the non-straight orientation of the principal optical axes of the permittivity tensor leads to slanted isofrequency surfaces with respect to the global z-axis (Supplementary Figure 21), which may enable interesting transmission phenomena, such as negative refraction and the super-prism effect⁵⁴. Hence, the unique dielectric tensors of ReS₂ and ReSe₂ (Fig. 2 and Supplementary Figure 17) provide great flexibility in optical engineering.

Sample preparation

Bulk ReS₂ and ReSe₂ crystals were purchased from 2D Semiconductors (Scottsdale, USA) and micromechanically cleaved down on top of required substrates (Si/SiO₂ and glass). Those substrates were subsequently decontaminated in acetone, isopropanol alcohol, and deionized water before the cleavage and then subjected to oxygen plasma removing the ambient adsorbates. Following plasma treatment, substrates were subjected to thermal treatment at temperatures of 120 °C and then exposed to scotch-tape from Nitto Denko Corporation (Osaka, Japan) with loaded bulk crystals of ReS₂ and ReSe₂. Eventually, the scotch-tape was removed, completing the cleavage procedure. The thickness of as-papered thin ReS₂ and ReSe₂ crystals was measured by an atomic force microscope (NT-MDT Spectrum Instruments, Ntegra II) in HybriD Mode using HA_NC tips with resonant frequency of 140 kHz and spring constant of 3.5 N/m.

Determination of principal optical axes

We used polarized microtransmittance measurement technique implemented on our Accurion nanofilm_ep4 ellipsometer to determine the principal optical axes. During the measurements, we aligned the polarizer and analyzer of the ellipsometer and fitted the obtained polarized microtransmittance for each wavelength by the expression: \({T}\left(\theta,\, \lambda \right)={a}^{2}{\cos }^{4}\left(\theta -\varphi \right)+{b}^{2}{\sin }^{4}\left(\theta -\varphi \right)+2{ab}{\cos }^{2}\left(\theta -\varphi \right){\sin }^{2}\left(\theta -\varphi \right)\cos \left(\Delta \phi \right)\), where \(T\left(\theta,\, \lambda \right)\) is the polarized microtransmittance, which depends on the polarizer’s/analyzer’s angle \(\theta\), and the incident wavelength \(\lambda\). \({a}^{2}\) and \({b}^{2}\) are the transmittances of beams polarized along in-plane principal optical axes, \(\Delta \phi\) is a phase difference between transmitted rays polarized along principal optical axes, and \(\varphi\) indicates the angular position of the principal optical axis (see blue points in Fig. 2e), whereas another principal optical axis is given by the sum \(\varphi+90^\circ\) (see red points in Fig. 2e).

Scanning near-field optical microscopy

The fabricated hole in ReSe₂ was characterized by the amplitude- and phase-resolved scattering-type scanning near-field optical microscopy (s-SNOM) measurements using the “NeaSNOM” setup (Neaspec GmbH). The s-SNOM works as an atomic force microscope (AFM) in a tapping mode with a Pt-coated silicon tip oscillating at the resonance frequency of Ω ≈ 280 kHz with an amplitude of \(\sim\)50 nm. In the s-SNOM working at transmission configuration, the ReSe₂ hole is illuminated from below by a linearly polarized light at a normal angle to the sample surface focused by a bottom parabolic mirror. As a light source, we used Ti:Sapphire continuous wave tuning laser (TiC, AVESTA Lasers and Optical Systems) with fiber coupling output, working at a wavelength range of λ = 700−1000 nm. While mapping the near-field signal and AFM topography around the hole with a scan area of 10×10 µm², the illumination from the bottom parabolic mirror always remained aligned with the hole due to its synchronization moving with the sample during the scan. A top parabolic mirror collects the tip-scattered near-field signal and directs it into the highly sensitive photodetector. To achieve a clear near-field image, the optical background was suppressed by demodulation of the detected signal at high-order harmonic frequency nΩ (n = 2, 3, 4) and using an interferometric pseudoheterodyne detection scheme with a modulated reference beam via oscillating mirror. In this work, the demodulation signal at the third harmonic (3Ω) was taken, which is enough for background-free near-field detection.

First-principle calculations

Optical constants of the ReS₂ and ReSe₂ crystals were calculated within density functional theory (DFT) and GW approximation, as implemented in VASP package⁶³. First, the atomic positions of both crystals were relaxed until the interatomic forces decreased below 10^-3eV/Å, while their unit cells were fixed. The lattice parameters were \(a=6.378\) Å, \(b=6.417\) Å, \(c=6.461\) Å with \(\alpha=91.62^\circ\), \(\beta=119.07^\circ\), \(\gamma=105.115^\circ\) for ReS₂ and \(a=6.716\) Å, \(b=6.602\) Å, \(c=6.728\) Å with \(\alpha=104.90^\circ\), \(\beta=91.82^\circ\), \(\gamma=118.94^\circ\) for ReSe₂. Next, we obtained ground-state one-electron wavefunctions from DFT and used them to initialize the GW routines. Finally, we calculated the imaginary and real parts of the frequency-dependent dielectric function within GW approximation and derived the refractive indices and extinction coefficients of the material. The cutoff energy for the plane-wave basis was set to 500 eV, while the first Brillouin zone was sampled with a \(\Gamma\)-centred 6\(\times\)6\(\times\)6 grid. The exchange correlation effects were described with a generalized gradient approximation (Perdew-Burke-Ernzerhof functional), and the behavior of wavefunctions in the core region was reconstructed with the projector augmented wave pseudopotentials.

The relevant raw and generated data supporting the key findings of this study are available in the figshare database under accession code https://figshare.com/s/a1edc12b21d3a36315ab (https://doi.org/10.6084/m9.figshare.24967593). All data are available from the corresponding author upon a request.

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Authors and Affiliations

1. Emerging Technologies Research Center, XPANCEO, Dubai Investment Park First, Dubai, United Arab Emirates

   Georgy A. Ermolaev, Dmitriy V. Grudinin, Ilia M. Fradkin, Valentin R. Solovey, Roman V. Kirtaev, Ivan Kruglov, Andrey A. Vyshnevyy, Aleksey V. Arsenin & Valentyn S. Volkov

2. Donostia International Physics Center (DIPC), Donostia/San Sebastián, 20018, Spain

   Kirill V. Voronin

3. Moscow Center for Advanced Studies, Kulakova str. 20, Moscow, 123592, Russia

   Adilet N. Toksumakov, Aleksandr S. Slavich, Mikhail K. Tatmyshevskiy, Dmitry I. Yakubovsky, Sergey M. Novikov, Elena S. Zhukova, Denis G. Baranov & Davit A. Ghazaryan

4. Institute of Materials, École Polytechnique Fédérale de Lausanne, 1015, Lausanne, Switzerland

   Arslan Mazitov

5. Laboratory of Advanced Functional Materials, Yerevan State University, Yerevan, 0025, Armenia

   Davit A. Ghazaryan, Aleksey V. Arsenin & Valentyn S. Volkov

6. Instituto de Nanociencia y Materiales de Aragón (INMA), CSIC-Universidad de Zaragoza, 50009, Zaragoza, Spain

   Luis Martin-Moreno

7. Departamento de Física de la Materia Condensada, Universidad de Zaragoza, 50009, Zaragoza, Spain

   Luis Martin-Moreno

8. National Graphene Institute (NGI), University of Manchester, Manchester, M13 9PL, UK

   Kostya S. Novoselov

9. Department of Materials Science and Engineering, National University of Singapore, Singapore, 03-09, Singapore

   Kostya S. Novoselov

10. Institute for Functional Intelligent Materials, National University of Singapore, 117544, Singapore, Singapore

    Kostya S. Novoselov

Contributions

D.G.B., D.A.G., A.V.A., L. M.-M., K.S.N., and V.S.V. suggested and directed the project. G.A.E., D.V.G., A.S.S., M.K.T., D.I.Y., V.R.S., S.M.N., and E.S.Z. performed the measurements and analyzed the data. A.N.T., R.V.K., and D.A.G. prepared the samples. G.A.E., K.V.V., D.V.G., I.M.F., A.M., I.K., A.A.V., and D.G.B. provided theoretical support. G.A.E. wrote the original manuscript. G.A.E., D.G.B., D.A.G., A.V.A., L. M.-M., K.S.N., and V.S.V. reviewed and edited the paper. All authors contributed to the discussions and commented on the paper.

Corresponding author

Correspondence to Kostya S. Novoselov.

Competing interests

The authors declare no competing interests.

Peer review information

Nature Communications thanks the anonymous reviewers for their contribution to the peer review of this work. A peer review file is available.

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