Raman Tensor Formalism for Optically Anisotropic Crystals

In-Plane Anisotropy in the Layered Topological Insulator Ta2Ni3Te5 Investigated via TEM and Polarized Raman Spectroscopy

Layered Ta2M3Te5 (M = Pd, Ni) has emerged as a platform to study 2D topological insulators, which have exotic properties such as spin-momentum locking and the presence of Dirac fermions for use in conventional and quantum-based electronics. In particular, Ta2Ni3Te5 has been shown to have superconductivity under pressure and is predicted to have second-order topology. Despite being an interesting material with fascinating physics, the detailed crystalline and phononic properties of this material are still unknown. In this study, we use transmission electron microscopy (TEM) and polarized Raman spectroscopy (PRS) to reveal the anisotropic properties of exfoliated few-layer Ta2Ni3Te5. An electron diffraction and TEM study reveals structural anisotropy in the material, with a preferential crystal orientation along the [010] direction. Through Raman spectroscopy, we discovered 15 vibrational modes, 3 of which are ultralow-frequency modes, which show anisotropic response with sample orientation varying with the polarization of the incident beam. Using angle-resolved PRS, we assigned the vibrational symmetries of 11 modes to Ag and two modes to B3g. We also found that linear dichroism plays a role in understanding the Raman signature of this material, which requires the use of complex elements in the Raman tensors. The anisotropy of the Raman scattering also depends on the excitation energies. Our observations reveal the anisotropic nature of Ta2Ni3Te5, establish a quick and nondestructive Raman fingerprint for determining sample orientation, and represent a significant advance in the fundamental understanding of the two-dimensional topological insulator (2DTI) Ta2Ni3Te5 material.

Stacking-Order-Dependent Excitonic Properties Reveal Interlayer Interactions in Bulk ReS2

Rhenium disulfide, a member of the transition metal dichalcogenide family of semiconducting materials, is unique among 2D van der Waals materials due to its anisotropy and, albeit weak, interlayer interactions, confining excitons within single atomic layers and leading to monolayer-like excitonic properties even in bulk crystals. While recent work has established the existence of two stacking modes in bulk, AA and AB, the influence of the different interlayer coupling on the excitonic properties has been poorly explored. Here, we use polarization-dependent optical measurements to elucidate the nature of excitons in AA and AB-stacked rhenium disulfide to obtain insight into the effect of interlayer interactions. We combine polarization-dependent Raman with low-temperature photoluminescence and reflection spectroscopy to show that, while the similar polarization dependence of both stacking orders indicates similar excitonic alignments within the crystal planes, differences in peak width, position, and degree of anisotropy reveal a different degree of interlayer coupling. DFT calculations confirm the very similar band structure of the two stacking orders while revealing a change of the spin-split states at the top of the valence band to possibly underlie their different exciton binding energies. These results suggest that the excitonic properties are largely determined by in-plane interactions, however, strongly modified by the interlayer coupling. These modifications are stronger than those in other 2D semiconductors, making ReS2 an excellent platform for investigating stacking as a tuning parameter for 2D materials. Furthermore, the optical anisotropy makes this material an interesting candidate for polarization-sensitive applications such as photodetectors and polarimetry.

Phonon-Phonon Interactions in the Polarization Dependence of Raman Scattering

We have found that the polarization dependence of Raman scattering in organic crystals at finite temperatures can only be described by a fourth-rank tensor formalism. This generalization of the second-rank Raman tensor stems from the effect of off-diagonal components in the crystal self-energy on the light scattering mechanism. We thus establish a novel manifestation of phonon-phonon interaction in inelastic light scattering, markedly separate from the better-known phonon lifetime.

Anisotropic Phonon Behavior and Phase Transition in Monolayer ReSe2 Discovered by High Pressure Raman Scattering

Re-based transition metal dichalcogenides have attracted extensive attention owing to their anisotropic structure and excellent properties in applications such as optoelectronic devices and electrocatalysis. The present study methodically investigated the evolution of specific Raman phonon mode behaviors and phase transitions in monolayer and bulk ReSe2 under high pressure. Considering the distinctive anisotropic characteristics and the vibration vectors of Re and Se atoms exhibited by monolayer ReSe2, we perform phonon dispersion calculations and propose a methodology utilizing pressure-dependent polarized Raman measurements to explore the precise structural evolution of monolayer ReSe2 under the stress fields. Varied behaviors of the Eg-like and Ag-like modes, along with their specific vector transformations, have been identified in the pressure range 0-14.59 GPa. The present study aims to offer original perspectives on the physical evolution of Re-based transition metal dichalcogenides, elucidating their fundamental anisotropic properties and exploring potential applicability in diverse devices.

Observation of optical anisotropy and a linear dichroism transition in layered silicon phosphide

The investigation of in-plane two-dimensional (2D) anisotropic materials has garnered significant attention due to their exceptional electronic, optical, and mechanical characteristics. The anisotropic optical properties and angle-dependent photodetectors based on 2D anisotropic materials have been extensively studied. However, novel in-plane anisotropic materials still need to be explored to satisfy for distinct environments and devices. Here, we report the remarkable anisotropic behavior of excitons and demonstrate a unique linear-dichroism transition of absorption between ultraviolet and visible light in layered silicon phosphide (SiP) through the analysis of polarization photoluminescence (PL) and absorbance spectra. Its high absorption linear dichroism ratio of 1.16 at 388 nm, 1.15 at 532 nm, and 1.19 at 733 nm is revealed, suggesting the brilliant non-isotropic responses. The robust periodic variation of the A₁ and A₂ Raman modes in 2D SiP materials allows for the determination of their crystal orientation. Furthermore, the presence of indirect excitons with phonon sidebands in the temperature-dependent PL spectra exhibits non-monotonic energy shifts with increasing temperature, which is attributed to an enhanced electron-phonon interaction and thermal expansion. Our findings provide valuable insights into the fundamental physical properties of layered SiP and offer guidelines for designing polarization-sensitive photodetectors and angle-dependent devices based on 2D anisotropic materials.

Optical Effect Modulation in Polarized Raman Spectroscopy of Transparent Layered α-MoO3

Optical anisotropy, which is quantified by birefringence (Δn) and linear dichroism (Δk), can significantly modulate the angle-resolved polarized Raman spectroscopy (ARPRS) response of anisotropic layered materials (ALMs) by external interference. This work studies the separate modulation of birefringence on the ARPRS response and the intrinsic response by selecting transparent birefringent crystal α-MoO₃ as an excellent platform. It is found that there are several anomalous ARPRS responses in α-MoO₃ that cannot be reproduced by the real Raman tensor widely used in non-absorbing materials; however, they can be well explained by considering the birefringence-induced Raman selection rules. Moreover, the systematic thickness-dependent study indicates that birefringence modulates the ARPRS response to render an interference pattern; however, the amplitude of modulation is considerably lower than that by linear dichroism as occurred in black phosphorous. This weak modulation brings convenience to the crystal orientation determination of transparent ALMs. Combining the atomic vibrational pattern and bond polarizability model, the intrinsic ARPRS response of α-MoO₃ is analyzed, giving the physical origins of the Raman anisotropy. This study employs α-MoO₃ as an example, although it is generally applicable to all transparent birefringent ALMs.

Crystal Structure, Raman, FTIR, UV-Vis Absorption, Photoluminescence Spectroscopy, TG-DSC and Dielectric Properties of New Semiorganic Crystals of 2-Methylbenzimidazolium Perchlorate

Single crystals of 2-methylbenzimidazolium perchlorate were prepared for the first time with a slow evaporation method from an aqueous solution of a mixture of 2-methylbenzimidazole (MBI) crystals and perchloric acid HClO₄. The crystal structure was determined by single crystal X-ray diffraction (XRD) and confirmed by XRD of powder. Angle-resolved polarized Raman and Fourier-transform infrared (FTIR) absorption spectra of crystals consist of lines caused by molecular vibrations in MBI molecule and ClO₄^- tetrahedron in the region ν = 200-3500 cm^-1 and lattice vibrations in the region of 0-200 cm^-1. Both XRD and Raman spectroscopy show a protonation of MBI molecule in the crystal. An analysis of ultraviolet-visible (UV-Vis) absorption spectra gives an estimation of an optical gap E_g~3.9 eV in the crystals studied. Photoluminescence spectra of MBI-perchlorate crystals consist of a number of overlapping bands with the main maximum at E_photon ≅ 2.0 eV. Thermogravimetry-differential scanning calorimetry (TG-DSC) revealed the presence of two first-order phase transitions with different temperature hysteresis at temperatures above room temperature. The higher temperature transition corresponds to the melting temperature. Both phase transitions are accompanied by a strong increase in the permittivity and conductivity, especially during melting, which is similar to the effect of an ionic liquid.

High-Field Raman Scattering in an Antiferromagnet Fe3BO6

The detailed Raman scattering investigation of the lattice and spin dynamics of a single crystal of the Fe3BO6 antiferromagnet is reported. Azimuthally resolved polarization measurements provided an unambiguous determination of the symmetry of observed excitations. Low-temperature experiments at T=4.2 K allowed us to reduce anharmonic contribution and deconvolute several overlapping phonon modes. Low-frequency measurements have made it possible to find two excitations at 13.1 and 16.6 cm−1, which were assigned to quasi-ferromagnetic and quasi-antiferromagnetic magnon excitations, respectively. The magnetic field applied along the hard-magnetization axis causes energy shifts of these excitations, but no spin-flip transition was observed up to B=30 T.

Anomalous polarization pattern evolution of Raman modes in few-layer ReS2 by angle-resolved polarized Raman spectroscopy

Low-symmetry of ReS₂ has not only in-plane but also out-of-plane anisotropic light scattering, which is complicated, yet interesting with intrinsic strong electron-phonon coupling. In such a case, the Raman tensor also gets sophisticated with nine non-zero elements, which is layer-dependent for different Raman modes. Herein, we systematically investigated the polarization pattern evolution of both in-plane and out-of-plane Raman modes of few-layer ReS₂ by angle-resolved polarized Raman spectroscopy. We found that in-plane Raman modes with less layer-dependence could be used to determine the crystal orientation (Re-chain direction) due to the weak electron-phonon interaction between layers. However, the out-of-plane and mixed vibration Raman modes demonstrate much evident layer-dependence due to the obvious electron-phonon interaction between layers. As such, the polarization patterns for the out-of-plane vibration Raman modes are distorted with layers in not only petal types but also maximum Raman intensity directions. This distortion is mainly due to the phase difference between Raman elements, which are complex values due to the near bandgap excitation laser. The results reveal that deep insights into anisotropy in low-symmetry two-dimensional materials could afford not only rich physics but also potential polarized optoelectronic devices.

Polarized Raman spectroscopy in low-symmetry 2D materials: angle-resolved experiments and complex number tensor elements

In this perspective review, we discuss the power of polarized Raman spectroscopy to study optically anisotropic 2D materials, belonging to the orthorhombic, monoclinic and triclinic crystal families. We start by showing that the polarization dependence of the peak intensities is described by the Raman tensor that is unique for each phonon mode, and then we discuss how to determine the tensor elements from the angle-resolved polarized measurements by analyzing the intensities in both the parallel- and cross-polarized scattering configurations. We present specific examples of orthorhombic black phosphorus and monoclinic 1T'-MoTe₂, where the Raman tensors have null elements and their principal axes coincide with the crystallographic ones, followed by a discussion on the results for triclinic ReS₂ and ReSe₂, where the axes of the Raman tensor do not coincide with the crystallographic axes and all elements are non-zero. We show that the Raman tensor elements are, in general, given by complex numbers and that phase differences between tensor elements are needed to describe the experimental results. We discuss the dependence of the Raman tensors on the excitation laser energy and thickness of the sample within the framework of the quantum model for the Raman intensities. We show that the wavevector dependence of the electron-phonon interaction is essential for explaining the distinct Raman tensor for each phonon mode. Finally, we close with our concluding remarks and perspectives to be explored using angle-resolved polarized Raman spectroscopy in optically anisotropic 2D materials.

Unambiguous determination of crystal orientation in black phosphorus by angle-resolved polarized Raman spectroscopy

Angle-resolved polarized Raman spectroscopy (ARPRS) is widely used to determine the crystal orientations of anisotropic layered materials (ALMs), which is an essential step to study all of their anisotropic properties. However, the understanding of the ARPRS response of black phosphorous (BP) as a most widely studied ALM is still unsatisfactory. Here, we clarify two key controversies about the physical origin of the intricate ARPRS response and the determination of crystal orientations in BP. Through systematic ARPRS measurements, we show that the degree of anisotropy of the response evolves gradually and periodically with the BP thickness, eventually leading to the intricate response. Meanwhile, we find that using the Raman peak intensity ratio of the two A_g phonon modes, the crystal orientations of BP can be unambiguously distinguished via a concise inequality . Comprehensive analysis and first-principles calculations reveal that the external anisotropic interference effect and the intrinsic electron-phonon coupling are responsible for the observations.

Strongly Anharmonic Octahedral Tilting in Two-Dimensional Hybrid Halide Perovskites

Recent investigations of two-dimensional (2D) hybrid organic-inorganic halide perovskites (HHPs) indicate that their optical and electronic properties are dominated by strong coupling to thermal fluctuations. While the optical properties of 2D-HHPs have been extensively studied, a comprehensive understanding of electron-phonon interactions is limited because little is known about their structural dynamics. This is partially because the unit cells of 2D-HHPs contain many atoms. Therefore, the thermal fluctuations are complex and difficult to elucidate in detail. To overcome this challenge, we use polarization-orientation Raman spectroscopy and ab initio calculations to compare the structural dynamics of the prototypical 2D-HHPs [(BA)2PbI4 and (PhE)2PbI4] to their three-dimensional (3D) counterpart, MAPbI3. Comparison to the simpler, 3D MAPbI3 crystal shows clear similarities with the structural dynamics of (BA)2PbI4 and (PhE)2PbI4 across a wide temperature range. The analogy between the 3D and 2D crystals allows us to isolate the effect of the organic cation on the structural dynamics of the inorganic scaffold of the 2D-HHPs. Furthermore, using this approach, we uncover the mechanism of the order-disorder phase transition of (BA)2PbI4 (274 K) and show that it involves relaxation of octahedral tilting coupled to anharmonic thermal fluctuations. These anharmonic fluctuations are important because they induce charge carrier localization and affect the optoelectronic performance of these materials.

Understanding angle-resolved polarized Raman scattering from black phosphorus at normal and oblique laser incidences

The selection rule for angle-resolved polarized Raman (ARPR) intensity of phonons from standard group-theoretical method in isotropic materials would break down in anisotropic layered materials (ALMs) due to birefringence and linear dichroism effects. The two effects result in depth-dependent polarization and intensity of incident laser and scattered signal inside ALMs and thus make a challenge to predict ARPR intensity at any laser incidence direction. Herein, taking in-plane anisotropic black phosphorus as a prototype, we developed a so-called birefringence-linear-dichroism (BLD) model to quantitatively understand its ARPR intensity at both normal and oblique laser incidences by the same set of real Raman tensors for certain laser excitation. No fitting parameter is needed, once the birefringence and linear dichroism effects are considered with the complex refractive indexes. An approach was proposed to experimentally determine real Raman tensor and complex refractive indexes, respectively, from the relative Raman intensity along its principle axes and incident-angle resolved reflectivity by Fresnel's law. The results suggest that the previously reported ARPR intensity of ultrathin ALM flakes deposited on a multilayered substrate at normal laser incidence can be also understood based on the BLD model by considering the depth-dependent polarization and intensity of incident laser and scattered Raman signal induced by both birefringence and linear dichroism effects within ALM flakes and the interference effects in the multilayered structures, which are dependent on the excitation wavelength, thickness of ALM flakes and dielectric layers of the substrate. This work can be generally applicable to any opaque anisotropic crystals, offering a promising route to predict and manipulate the polarized behaviors of related phonons.

Determination of the Raman polarizability tensor in the optically anisotropic crystal potassium dihydrogen phosphate and its deuterated analog

The Raman tensor of the dominant A₁ modes of the nonlinear optical crystalline material potassium dihydrogen phosphate and its 70% deuterated analog have been ascertained. Challenges in determining the A₁ mode tensor element values based on previous reports have been resolved using a specially designed experimental setup that makes use of spherical crystal samples. This novel experimental design enabled the determination of measurement artifacts, including polarization rotation of the pump and/or scattered light propagating through the sample and the contribution of additional overlapping phonon modes, which have hindered previous efforts. Results confirmed that the polarization tensor is diagonal, and matrix elements were determined with high accuracy.

Polarized Raman spectroscopy for studying two-dimensional materials

Raman spectroscopy has been established as one of the core experimental tools to study two-dimensional materials (2DMs) including graphene, black phosphorus, transitional metal chalcogenides, and other layered materials. If the polarization of the incident photons and the scattered photons are carefully controlled, the selection rules for the Raman scattering from phonon modes allow accurate mode assignments, which is not always possible in Raman scattering measurements using unpolarized light. Furthermore, polarized Raman spectroscopy can be used to determine the crystallographic orientation of isotropic 2DMs with in-plane strain or anisotropic 2DMs. This review explains the basics of polarized Raman spectroscopy, especially in the context of 2DMs research, and survey some of the most important applications of polarized Raman spectroscopy in isotropic and anisotropic 2DMs studies.

Anharmonic Lattice Vibrations in Small-Molecule Organic Semiconductors

The intermolecular lattice vibrations in small-molecule organic semiconductors have a strong impact on their functional properties. Existing models treat the lattice vibrations within the harmonic approximation. In this work, polarization-orientation (PO) Raman measurements are used to monitor the temperature-evolution of the symmetry of lattice vibrations in anthracene and pentacene single crystals. Combined with first-principles calculations, it is shown that at 10 K, the lattice dynamics of the crystals are indeed harmonic. However, as the temperature is increased, specific lattice modes gradually lose their PO dependence and become more liquid-like. This finding is indicative of a dynamic symmetry breaking of the crystal structure and shows clear evidence of the strongly anharmonic nature of these vibrations. Pentacene also shows an apparent phase transition between 80 and 150 K, indicated by a change in the vibrational symmetry of one of the lattice modes. These findings lay the groundwork for accurate predictions of the electronic properties of high-mobility organic semiconductors at room temperature.

Fast and quantitative 2D and 3D orientation mapping using Raman microscopy

Non-destructive orientation mapping is an important characterization tool in materials science and geoscience for understanding and/or improving material properties based on their grain structure. Confocal Raman microscopy is a powerful non-destructive technique for chemical mapping of organic and inorganic materials. Here we demonstrate orientation mapping by means of Polarized Raman Microscopy (PRM). While the concept that PRM is sensitive to orientation changes is known, to our knowledge, an actual quantitative orientation mapping has never been presented before. Using a concept of ambiguity-free orientation determination analysis, we present fast and quantitative single-acquisition Raman-based orientation mapping by simultaneous registration of multiple Raman scattering spectra obtained at different polarizations. We demonstrate applications of this approach for two- and three-dimensional orientation mapping of a multigrain semiconductor, a pharmaceutical tablet formulation and a polycrystalline sapphire sample. This technique can potentially move traditional X-ray and electron diffraction type experiments into conventional optical laboratories.

Although polarized Raman microscopy is sensitive to orientation changes, quantitative information has been missing. Here, the authors use simultaneous registration of multiple Raman scattering spectra obtained at different polarizations and show quantitative orientation mapping

Phase-selective synthesis of 1T' MoS2 monolayers and heterophase bilayers

Two-dimensional (2D) MoS₂, which has great potential for optoelectronic and other applications, is thermodynamically stable and hence easily synthesized in its semiconducting 2H phase. In contrast, growth of its metastable 1T and 1T' phases is hampered by their higher formation energy. Here we use theoretical calculations to design a potassium (K)-assisted chemical vapour deposition method for the phase-selective growth of 1T' MoS₂ monolayers and 1T'/2H heterophase bilayers. This is realized by tuning the concentration of K in the growth products to invert the stability of the 1T' and 2H phases. The synthesis of 1T' MoS₂ monolayers with high phase purity allows us to characterize their intrinsic optical and electrical properties, revealing a characteristic in-plane anisotropy. This phase-controlled bottom-up synthesis offers a simple and efficient way of manipulating the relevant device structures, and provides a general approach for producing other metastable-phase 2D materials with unique properties.

Interference-Enhanced Raman Spectroscopy as a Promising Tool for the Detection of Biomolecules on Raman-Compatible Surfaces

Raman spectroscopy in combination with appropriate sample preparation strategies, for example, enrichment of bacteria on metal surfaces, has been proven to be a promising approach for rapidly diagnosing infectious diseases. Unfortunately, the fabrication of the required chip substrates is usually very challenging due to the lack of feasible instruments that can be used for quality control in the surface modification process. The intrinsically weak Raman signal of the biomolecules, employed for the enrichment of the micro-organisms on the chip surface, does not allow for monitoring of the successful immobilization by means of a Raman spectroscopic approach. Within this contribution, we demonstrate how a simple modification of a plain aluminum surface enables enhancement (or a decrease, if desired) of the Raman signal of molecules deposited on that surface. The manipulation of the Raman signal strength is achieved via exploiting interference effects that occur, if the highly reflective aluminum surface is modified with thin layers of transparent dielectrics like aluminum oxide. The thicknesses of these layers were determined by theoretical considerations and calculations. For the first time, it is shown that the interference effects can be used for the detection of biomolecules as well by investigating the siderophore ferrioxamine B. The observed degree of enhancement was approximately 1 order of magnitude. Moreover, the employed aluminum/aluminum oxide layers have been thoroughly characterized using atomic force and scanning electron microscopy as well as X-ray reflectometry and UV-Vis measurements.

Raman tensor elements of β-Ga2O3

The Raman spectrum and particularly the Raman scattering intensities of monoclinic β-Ga₂O₃ are investigated by experiment and theory. The low symmetry of β-Ga₂O₃ results in a complex dependence of the Raman intensity for the individual phonon modes on the scattering geometry which is additionally affected by birefringence. We measured the Raman spectra in dependence on the polarization direction for backscattering on three crystallographic planes of β-Ga₂O₃ and modelled these dependencies using a modified Raman tensor formalism which takes birefringence into account. The spectral position of all 15 Raman active phonon modes and the Raman tensor elements of 13 modes were determined and are compared to results from ab-initio calculations.

In-Plane Optical Anisotropy of Layered Gallium Telluride

Layered gallium telluride (GaTe) has attracted much attention recently, due to its extremely high photoresponsivity, short response time, and promising thermoelectric performance. Different from most commonly studied two-dimensional (2D) materials, GaTe has in-plane anisotropy and a low symmetry with the C2h(3) space group. Investigating the in-plane optical anisotropy, including the electron-photon and electron-phonon interactions of GaTe is essential in realizing its applications in optoelectronics and thermoelectrics. In this work, the anisotropic light-matter interactions in the low-symmetry material GaTe are studied using anisotropic optical extinction and Raman spectroscopies as probes. Our polarized optical extinction spectroscopy reveals the weak anisotropy in optical extinction spectra for visible light of multilayer GaTe. Polarized Raman spectroscopy proves to be sensitive to the crystalline orientation of GaTe, and shows the intricate dependences of Raman anisotropy on flake thickness, photon and phonon energies. Such intricate dependences can be explained by theoretical analyses employing first-principles calculations and group theory. These studies are a crucial step toward the applications of GaTe especially in optoelectronics and thermoelectrics, and provide a general methodology for the study of the anisotropy of light-matter interactions in 2D layered materials with in-plane anisotropy.

Activation of New Raman Modes by Inversion Symmetry Breaking in Type II Weyl Semimetal Candidate T'-MoTe2

We synthesized distorted octahedral (T') molybdenum ditelluride (MoTe2) and investigated its vibrational properties with Raman spectroscopy, density functional theory, and symmetry analysis. Compared to results from the high-temperature centrosymmetric monoclinic (T'mo) phase, four new Raman bands emerge in the low-temperature orthorhombic (T'or) phase, which was recently predicted to be a type II Weyl semimetal. Crystal-angle-dependent, light-polarization-resolved measurements indicate that all the observed Raman peaks belong to two categories: those vibrating along the zigzag Mo atomic chain (z-modes) and those vibrating in the mirror plane (m-modes) perpendicular to the zigzag chain. Interestingly, the low-energy shear z-mode and shear m-mode, absent from the T'mo spectra, become activated when sample cooling induces a phase transition to the T'or crystal structure. We interpret this observation as a consequence of inversion-symmetry breaking, which is crucial for the existence of Weyl fermions in the layered crystal. Our temperature-dependent Raman measurements further show that both the high-energy m-mode at ∼130 cm(-1) and the low-energy shear m-mode at ∼12 cm(-1) provide useful gauges for monitoring the broken inversion symmetry in the crystal.