Layered materials (LMs), such as graphite, hexagonal boron nitride, and transition-metal dichalcogenides, are at the centre of an ever increasing research effort, due to their scientific and technological relevance. Raman and infrared spectroscopies are accurate, non-destructive, approaches to determine a wide range of properties, including the number of layers and the strength of the interlayer interactions. Here, we present a general approach to predict the complete spectroscopic fan diagrams, i.e., the relations between frequencies and number of layers, $N$, for the optically-active shear and layer-breathing modes of any multilayer comprising $N \geq 2$ identical layers. In order to achieve this, we combine a description of the normal modes in terms of a one-dimensional mechanical model, with symmetry arguments that describe the evolution of the point group as a function of $N$. Group theory is then used to identify which modes are Raman and/or infrared active, and to provide diagrams of the optically-active modes for any stack composed of identical layers. We implement the method and algorithms in an open-source tool directly available on the Materials Cloud portal, to assist any researcher in the prediction and interpretation of such diagrams. Our work will underpin all future efforts on Raman and Infrared characterization of known, and yet not investigated, LMs.
Layered materials have huge potential in various applications due to their extraordinary properties. To determine the interlayer interaction (or equivalently the layer spacing under different perturbations) is of critical importance. In this letter, we focus on one of the most prominent layered materials, graphite, and theoretically quantify the relationship between its interlayer spacing and the vibrational frequencies of its layer breathing and shear modes, which are measures of the interlayer interaction. The method used here to determine the interlayer interaction can be further applied to…
The interlayer breathing and shear modes in few-layer black phosphorus are investigated for their symmetry and lattice dynamical properties. The symmetry groups for the even-layer and odd-layer few-layer black phosphorus are utilized to determine the irreducible representation and the infrared and Raman activity for the interlayer modes. The valence force field model is applied to calculate the eigenvectorw and frequencies for the interlayer breathing and shear modes, which are explained using the atomic chain model. The anisotropic puckered configuration for black phosphorus leads to a highly…
We study by Raman scattering the shear and layer breathing modes in multilayer MoS2. These are identified by polarization measurements and symmetry analysis. Their positions change with the number of layers, with different scaling for odd and even layers. A chain model explains the results, with general applicability to any layered material, and allows one to monitor their thickness.
In this work, we study structural and vibrational properties of multilayer graphene using density-functional theory (DFT) with van der Waals (vdW) functionals. Initially, we analyze how different vdW functionals compare by evaluating the lattice parameters, elastic constants and vibrational frequencies of low energy optical modes of graphite. Our results indicate that the vdW-DF1-optB88 functional has the best overall performance on the description of vibrational properties. Next, we use this functional to study the influence of the vdW interactions on the structural and vibrational properties…
Two-dimensional (2D) layered transition metal dichalcogenides (TMDs) have recently attracted tremendous interest as potential valleytronic and nano-electronic materials, in addition to being well-known as excellent lubricants in the bulk. The interlayer van der Waals (vdW) coupling and low frequency phonon modes, and how they evolve with the number of layers, are important for both the mechanical and electrical properties of 2D TMDs. Here we uncover the ultra-low frequency interlayer breathing and shear modes in few-layer MoS2 and WSe2, prototypical layered TMDs, using both Raman spectroscopy …
Uniaxial and biaxial strain approaches are usually implemented to switch the ferroelastic states, which play a key role in the application of the ferroics and shape memory materials. In this work, by using the first-principles calculations, we found not only uniaxial strain, but also shear strain can induce a novel ferroelastic switching, in which the van der Waals (vdW) layered direction rotates with the ferroelastic transition in layered bulk PdSe2. The shear strain induces ferroelastic switching with three times amplitude smaller than uniaxial strain. The novel three-states ferroelastic swi…
As a new two-dimensional layered material, black phosphorus (BP) is a promising material for nanoelectronics and nano-optoelectronics. We use Raman spectroscopy and first-principles theory to report our findings related to low-frequency (LF) interlayer breathing modes (<100 cm-1) in few-layer BP for the first time. The breathing modes are assigned to Ag symmetry by the laser polarization dependence study and group theory analysis. Compared to the high-frequency (HF) Raman modes, the LF breathing modes are much more sensitive to interlayer coupling and thus their frequencies show much stronger …
Relative out of plane displacements of the constituent layers of two dimensional materials gives rise to unique low frequency breathing modes. By computing the height-height correlation functions in momentum space, we show that, the layer breathing modes (LBMs) can be mapped consistently to vibrations of a simple linear chain model. Our calculated thickness dependence of LBM frequencies for few layer (FL) graphene and molybdenum disulphide (MoS$_{2}$) are in excellent agreement with available experiments. Our results show a redshift of LBM frequency with increase in temperature, which is a dir…
2D layered materials have recently attracted tremendous interest due to their fascinating properties and potential applications. The interlayer interactions are much weaker than the intralayer bonds, allowing the as-synthesized materials to exhibit different stacking sequences (e.g. ABAB, ABCABC), leading to different physical properties. Here, we show that regardless of the space group of the 2D material, the Raman frequencies of the interlayer shear modes observed under the typical configuration blue shift for AB stacked materials, and red shift for ABC stacked materials, as the number of la…
Raman spectroscopy is the prime non-destructive characterization tool for graphene and related layered materials. The shear (C) and layer breathing modes (LBMs) are due to relative motions of the planes, either perpendicular or parallel to their normal. This allows one to directly probe the interlayer interactions in multilayer samples. Graphene and other two-dimensional (2d) crystals can be combined to form various hybrids and heterostructures, creating materials on demand with properties determined by the interlayer interaction. This is the case even for a single material, where multilayer s…
Layered materials such as graphite and transition metal dichalcogenides have extremely anisotropic mechanical properties owing to orders of magnitude difference between in-plane and out-of-plane interatomic interaction strengths. Although effects of mechanical perturbations on either intra- or inter-layer interactions have been extensively investigated, mutual correlations between them have rarely been addressed. Here we show that layered materials have an inevitable coupling between in-plane uniaxial strain and interlayer shear. Because of this, the uniaxial in-plane strain induces an anomalo…
For oblique anti-plane shear waves in periodic layered elastic composites, it is shown that negative energy refraction is accompanied by positive phase-velocity refraction and positive energy refraction is accompanied by negative phase-velocity refraction, and that, this happens over a broad range of frequencies. The composite's unit cell may consist of any number of layers of any variable mass-density and elastic shear modulus (with large discontinuities). Explicit series expressions for displacement, velocity, strain and stress components, and energy-flux fields are given, and group-velocity…
The lattice dynamics of low-frequency rigid-plane modes in metallic (graphene multilayers, GML) and in insulating (hexagonal boron-nitride multilayers, BNML) layered crystals is investigated. The frequencies of shearing and compression (stretching) modes depend on the layer number ${\EuScript N}$ and are presented in the form of fan diagrams. The results for GML and BNML are very similar. In both cases only the interactions (van der Waals and Coulomb) between nearest-neighbor planes are effective, while the interactions between more distant planes are screened. A comparison with recent Raman s…
We investigated interlayer phonon modes of mechanically exfoliated few-layer 2H-SnS2 samples by using room temperature low-frequency micro-Raman spectroscopy. Raman measurements were performed using laser wavelength of 441.6, 514.4, 532 and 632.8 nm with power below 100 uW and inside a vacuum chamber to avoid photo-oxidation. The intralayer Eg and A1g modes are observed at ~206 cm-1 and ~314 cm-1, respectively, but the Eg mode is much weaker for all excitation energies. The A1g mode exhibits strong resonant enhancement for the 532 nm (2.33 eV) laser. In the low-frequency region, interlayer vib…
Cobalt pyrochlore fluoride NaCaCo$_2$F$_7$ is a disordered frustrated magnet composed of Co$^{2+}$ ions with an effective spin-$\frac{1}{2}$ magnetic moment and exhibits spin freezing below $T_f \sim$2.4 K. We perform ultrasound velocity measurements on a single crystal of the cubic NaCaCo$_2$F$_7$. The temperature dependence of the bulk modulus (the breathing elastic mode) exhibits Curie-type softening upon cooling below $\sim$20 K down to $T_f$, which is suppressed by the magnetic field. This Curie-type softening should be a precursor to the enhancement of the strength of exchange disorder v…
The anisotropic two-dimensional (2D) van der Waals (vdW) layered materials, with both scientific interest and potential application, have one more dimension to tune the properties than the isotropic 2D materials. The interlayer vdW coupling determines the properties of 2D multi-layer materials by varying stacking orders. As an important representative anisotropic 2D materials, multilayer rhenium disulfide (ReS2) was expected to be random stacking and lack of interlayer coupling. Here, we demonstrate two stable stacking orders (aa and a-b) of N layer (NL, N>1) ReS2 from ultralow-frequency and h…
The Lennard-Jones potential is widely used to describe the interlayer interactions within layered materials like graphene. However, it is also widely known that this potential strongly underestimates the frictional properties for layered materials. Here we propose to supplement the Lennard-Jones potential by a Gaussian-type potential, which enables more accurate calculations of the frictional properties of two-dimensional layered materials. Furthermore, the Gaussian potential is computationally simple as it introduces only one additional potential parameter that is determined by the interlayer…
Layered inorganic materials, such as the transition metal dichalcogenides (TMDs), have attracted much attention due to their exceptional electronic and optical properties. Reliable synthesis and characterization of these materials must be developed if these properties are to be exploited. Herein, we present low-frequency Raman analysis of MoS2, MoSe2, WSe2 and WS2 grown by chemical vapour deposition (CVD). Raman spectra are acquired over large areas allowing changes in the position and intensity of the shear and layer-breathing modes to be visualized in maps. This allows detailed characterizat…
In liquid-phase exfoliation for the production of 2D nanomaterials fluid forces are used to gently overcome adhesive interlayer forces, leading to single- or few-layer 2D nanomaterials. Predicting accurately the critical fluid shear rate for exfoliation is a crucial challenge. By combining notions of fluid mechanics and fracture mechanics, we analyse a mathematical model of exfoliation, focusing on the $π$-peel regime in which bending forces are much smaller than the applied hydrodynamic forces. We find that in this regime the shear rate is approximately proportional to the adhesion energy, in…
We report a comparative study of rigid layer Raman-active modes in $N$-layer transition metal dichalcogenides. Trigonal prismatic (2Hc, such as MoSe$_2$, MoTe$_2$, WS$_2$, WSe$_2$) and distorted octahedral (1T', such as ReS$_2$ and ReSe$_2$) phases are considered. The Raman-active in-plane interlayer shear modes and out-of-plane interlayer breathing modes appear as well-defined features with wavenumbers in the range 0-40~cm$^{-1}$. These rigid layer modes are well-described by an elementary linear chain model from which the interlayer force constants are readily extracted. Remarkably, these fo…
In the design of materials with low lattice thermal conductivity, compounds with high density, low speed of sound, and complexity at either the atomic, nano- or microstructural level are preferred. The layered compound Mg$_3$Sb$_2$ defies these prevailing paradigms, exhibiting lattice thermal conductivity comparable to PbTe and Bi$_2$Te$_3$, despite its low density and simple structure. The excellent thermoelectric performance ($zT$ $\sim$ 1.5) in $n$-type Mg$_3$Sb$_2$ has thus far been attributed to its multi-valley conduction band, while its anomalous thermal properties have been largely ove…
ReS$_2$ has recently emerged as a new member in the rapidly growing family of two-dimensional materials. Unlike MoS$_2$ or WSe$_2$, the optical and electrical properties of ReS$_2$ are not isotropic due to the reduced symmetry of the crystal. Here, we present layer-dependent Raman measurements of ReS$_2$ samples ranging from monolayers to ten layers in the ultralow frequency regime. We observe layer breathing and shear modes which allow for easy assignment of the number of layers. Polarization-dependent measurements give further insight into the crystal structure and reveal an energetic shift …
Hexagonal layered crystalline materials, such as graphene, boron nitride, tungsten sulfate, and so on, have attracted enormous attentions, due to their unique combination of atomistic structures and superior thermal, mechanical, and physical properties. Making use of mechanical buckling is a promising route to control their structural morphology and thus tune their physical properties, giving rise to many novel applications. In this paper, we employ finite element analysis (FEA), molecular dynamic (MD) simulations and continuum modeling to study the mechanical buckling of a column made of laye…
The electronic band structure of crystals is generally influenced by the periodic arrangement of their constituent atoms. Specifically, the emerging two-dimensional (2D) layered structures have shown different band structures with respect to their stacking configurations. Here, based on first-principles density-functional theory calculations, we demonstrate that the band structure of the recently synthesized 2D Ca$_2$N electride changes little for the stacking sequence as well as the lateral interlayer shift. This intriguing invariance of band structure with respect to geometrical variations c…
We uncover the interlayer shear mode of multi-layer graphene samples, ranging from bilayer-graphene (BLG) to bulk graphite, and show that the corresponding Raman peak measures the interlayer coupling. This peak scales from~43cm-1 in bulk graphite to~31cm-1 in BLG. Its low energy makes it a probe of near-Dirac point quasi-particles, with a Breit-Wigner-Fano lineshape due to resonance with electronic transitions. Similar shear modes are expected in all layered materials, providing a direct probe of interlayer interactions
Few-layer Tellurium, an elementary semiconductor, succeeds most of striking physical properties that black phosphorus (BP) offers and could be feasibly synthesized by simple solution-based methods. It is comprised of non-covalently bound parallel Te chains, among which covalent-like feature appears. This feature is, we believe, another demonstration of the previously found covalent-like quasi-bonding (CLQB) where wavefunction hybridization does occur. The strength of this inter-chain CLQB is comparable with that of intra-chain covalent bonding, leading to closed stability of several Te allotro…
The bulk piezoelectric response, as measured by the piezoelectric modulus tensor (\textbf{d}), is determined by a combination of charge redistribution due to strain and the amount of strain produced by the application of stress (stiffness). Motivated by the notion that less stiff materials could exhibit large piezoelectric responses, herein we investigate the piezoelectric modulus of van der Waals-bonded quasi-2D ionic compounds using first-principles calculations. From a pool of 869 known binary and ternary quasi-2D materials, we have identified 135 non-centrosymmetric crystals of which 48 sy…
Talc is an insulating layered material that is stable at ambient conditions and has high-quality basal cleavage, which is a major advantage for its use in van der Waals heterostructures. Here, we use near-field synchrotron infrared nanospectroscopy, Raman spectroscopy, and first-principles calculations to investigate the structural and vibrational properties of talc crystals, ranging from monolayer to bulk, in the 300-750 cm-1 and <60 cm-1 spectral windows. We observe a symmetry crossover from mono to bilayer talc samples, attributed to the stacking of adjacent layers. The in-plane lattice par…
A numerical study on the elastic response of single- and multi-layer systems formed by alternating pentamode lattices and stiffening plates is presented. Finite element simulations are conducted to analyze the dependence of the effective elastic moduli of such structures upon suitable aspect ratios, which characterize the geometry of the generic pentamode layer at the micro- and macro-scale, and the lamination scheme of the layered structure. The given numerical results highlight that the examined structures exhibit bending-dominated response, and are able to achieve low values of the effectiv…
We investigate the interlayer phonon modes in $N$-layer rhenium diselenide (ReSe$_2$) and rhenium disulfide (ReS$_2$) by means of ultralow-frequency micro-Raman spectroscopy. These transition metal dichalcogenides exhibit a stable distorted octahedral (1T') phase with significant in-plane anisotropy, leading to sizable splitting of the (in-plane) layer shear modes. The fan-diagrams associated with the measured frequencies of the interlayer shear modes and the (out-of-plane) interlayer breathing modes are perfectly described by a finite linear chain model and allow the determination of the inte…
Transition metal dichalcogenides (TMDs) have recently attracted attention due to their interesting electronic and optical properties. Fabrication of these materials in a reliable and facile method is important for future applications, as are methods to characterize material quality. Here we present the chemical vapor deposition of MoSe2 monolayer and few layer crystals. These results show the practicality of using chemical vapor deposition to reliably fabricate these materials. Low frequency Raman spectra and mapping of shear and layer breathing modes of MoSe2 are presented for the first time.…
Wave propagation was investigated in the Bouligand-like structure from within the dactyl club of the Stomatopod, a crustacean that is known to smash their heavily shelled preys with high accelerations. We incorporate the layered nature in a unitary material cell through the propagator matrix formalism while the periodic nature of the material is considered via Bloch boundary conditions as applied in the theory of solid state physics. Our results show that these materials exhibit bandgaps at frequencies related to the stress pulse generated by the impact of the dactyl club to its prey, and ther…
Surfactants are often used to stabilise two-dimensional (2D) materials in environmentally friendly solvents such as water. Aqueous-surfactant solutions prevent agglomeration of nanosheets through steric and electrostatic repulsion, facilitating the production of high concentration nanomaterial dispersions. Turbulent, shear-assisted liquid exfoliation of layered precursor materials produces defect-free nanosheets by promoting mixing and generating sufficiently high shear rates to overcome out-of-plane van der Waals bonds. In the presence of a liquid-gas interface, a consequence of using surfact…
The selective excitation of coherent phonons provides unique capabilities to control fundamental properties of quantum materials on ultrafast time scales. For instance, in the presence of strong electron-phonon coupling, the electronic band structure can become substantially modulated. Recently, it was predicted that by this means even topologically protected states of matter can be manipulated and, ultimately, be destroyed: For the layered transition metal dichalcogenide Td-WTe$_2$, pairs of Weyl points are expected to annihilate as an interlayer shear mode drives the crystalline structure to…
III-VI post-transition metal chalcogenides (InSe and GaSe) are a new class of layered semiconductors, which feature a strong variation of size and type of their band gaps as a function of number of layers (N). Here, we investigate exfoliated layers of InSe and GaSe ranging from bulk crystals down to monolayer, encapsulated in hexagonal boron nitride, using Raman spectroscopy. We present the N-dependence of both intralayer vibrations within each atomic layer, as well as of the interlayer shear and layer breathing modes. A linear chain model can be used to describe the evolution of the peak posi…
Since the stacking order sensitively affects various physical properties of layered materials, accurate determination of the stacking order is important for studying the basic properties of these materials as well as for device applications. Because 2H-molybdenum disulfide (MoS2) is most common in nature, most studies so far have focused on 2H-MoS2. However, we found that the 2H, 3R, and mixed stacking sequences exist in few-layer MoS2 exfoliated from natural molybdenite crystals. The crystal structures are confirmed by HR-TEM measurements. The Raman signatures of different polytypes are inves…
The two-fold valley degeneracy in two-dimensional (2D) semiconducting transition metal dichalcogenides (TMDCs) (Mo,W)(S,Se)2 is suitable for "valleytronics", the storage and manupulation of information utilizing the valley degree of freedom. The conservation of luminescent photon helicity in these 2D crystal monolayers has been widely regarded as a benchmark indicator for charge carrier valley polarization. Here we perform helicity-resolved Raman scattering of the TMDC atomic layers. In drastic contrast to luminescence, the dominant first-order zone-center Raman bands, including the low energy…
Layer-parity-defined surface polarization in Nb$_3$Cl$_8$ for excitonic modulation at van der Waals interfaces
Peking University（北京大学）；Chinese Academy of Sciences（中国科学院）；University of Chinese Academy of Sciences（中国科学院大学）；National Institute for Materials Science（国立材料研究所）；South China Normal University（华南师范大学）；Liaoning Academy of Materials（辽宁省材料研究院）；ShanghaiTech University（上海科技大学）；University of Science and Technology of China（中国科学技术大学）；Songshan Lake Materials Laboratory（松山湖材料实验室）
Any of two or more two-dimensional (2D) materials with similar properties can be alloyed into a new layered material, namely, 2D alloy. Individual monolayer in 2D alloys are kept together by Van der Waals interactions. The property of multilayer alloys is a function of their layer number. Here, we studied the shear (C) and layer-breathing (LB) modes of Mo$_{0.5}$W$_{0.5}$S$_2$ alloy flakes and their link to the layer number of alloy flakes. The study reveals that the disorder effect is absent in the C and LB modes of 2D alloys, and the monatomic chain model can be used to estimate the frequenc…
Atomically thin, two-dimensional (2D) indium selenide (InSe) has attracted considerable attention due to large tunability in the band gap (from 1.4 to 2.6 eV) and high carrier mobility. The intriguingly high dependence of band gap on layer thickness may lead to novel device applications, although its origin remains poorly understood, and generally attributed to quantum confinement effect. In this work, we demonstrate via first-principles calculations that strong interlayer coupling may be mainly responsible for this phenomenon, especially in the fewer-layer region, and it could also be an esse…
