High resolution Brillouin spectroscopy of the surface acoustic waves in Sb₂Te₃ van der Waals single crystals

Abstract

High-resolution Brillouin spectroscopy was employed to investigate the anisotropy in surface wave velocities within a bulk single crystal of Sb₂Te₃, a well-known layered van der Waals material. By leveraging the bulk elastic constants derived from various simulation methods, we were able to theoretically calculate the distribution of surface acoustic phonon velocities on the cleavage plane of the material. Upon analyzing multiple simulation results, it became evident that the most significant discrepancies arose in the calculations of the elastic constant c₃₃, with values ranging from 48 to 98 GPa. Consequently, a direct measurement of the c₃₃ elastic constant for Sb₂Te₃ was attempted. Through our ellipsometry results, we determined both the real and imaginary components of the refractive index, leading to an experimental determination of the c₃₃ elastic constant, which was found to be 47.9 GPa. Additionally the results of the conducted studies enabled the analytical determination of all components of the elastic property tensor of the investigated material.

Introduction

Topological insulators (TIs) manifest unique characteristics, operating as insulators within their bulk while serving as conductors on their surfaces. This dual nature has ignited substantial interest in both experimental and theoretical research¹. TIs, as discussed in², have garnered considerable research enthusiasm due to their extensive applications in quantum spintronic devices³. The distinct properties of topological insulators originate from an energy gap within their bulk, while their surfaces or edges lack such a gap, displaying conductivity with mobile electrons^4,5. These properties are due to the presence of surface electronic states with a Dirac cone energy dispersion located in the bulk band gap. Interactions between valence and conduction states in these materials lead to specific valence band states exchanging positions with states in the conduction band along the material's edges. This phenomenon enables electrons to move freely, imparting conductor-like properties exclusively to the surfaces of such materials.

One of the TIs materials are Bi₂Se₃, Bi₂Te₃, Sb₂Te₃ compounds and their alloys. From the application point of view these materials are important according to the thermoelectric⁶, spintronic and quantum computations devices^7,8.

Antimony telluride (Sb₂Te₃) was one of the first materials recognized as a prominent p- type semiconductor and 3D TI. The electronic structure of Sb₂Te₃ is similar to Bi₂Te₃ and Bi₂Se₃⁹. According to the thermoelectric properties of Sb₂Te₃, antimony telluride is used in power generators, heat pumping, data storage^8,10. Sb₂Te₃ is widely employed in the Phase Change Memory (PCM) technology due to its ability to undergo a phase transition between amorphous and crystalline states¹¹. From a technological standpoint, comprehensive information regarding the elastic properties of antimony telluride is crucial for its practical application in devices like acousto-optic devices. On a fundamental level, precise knowledge of the elastic properties can act as a reference point for validating theoretical models aiming to predict the mechanical characteristics of this group of topological insulators. Additionally, three main issues in the field of phonon engineering are worth highlighting. Firstly, spatial restrictions of phonons in nanostructures impact their dispersion, which in turn modifies acoustic phonon properties like group velocity, polarization, and density of states, altering the interactions between acoustic phonons and other phonons, defects, and electrons. Secondly, phonon thermal conductivity is influenced when the size of the 3D structure is comparable to the phonon mean free path but still much larger than the dominant thermal phonon wavelength, where the phonon dispersion remains bulk-like. In this scenario, the spatial confinement of acoustic phonons allows for a modification of thermal conductivity and electron mobility through phonon spectrum engineering. Lastly, our main interest lies in phonon thermal rectification, a phenomenon predicted for material systems where thermal conduction differs in opposite directions. By controlling phonon transport through structural asymmetry or lattice nonlinearity, new possibilities emerge, such as the practical realization of the thermal diode¹².

That is why the precise determination of the elastic properties of TIs that are directly related to the propagation of phonons is so important. The elastic properties of monocrystalline topological insulators (TIs) are anisotropic, particularly in the case of trigonal symmetry, and are described by six non-zero, independent components of the elastic stiffness tensor c_ij. It is noteworthy that a comprehensive assessment of c_ij for van der Waals (vdW) materials poses a challenge, especially for bulk materials, due to the difficulty in preparing bulk samples with flat surfaces, except for the cleavage plane (vdW). So far, the anisotropic elastic properties of TIs have been partially measured using ultrasonics¹³, lattice dynamics spectroscopy¹⁴, Raman spectroscopy^15,16 and neutron scattering¹⁷.

In this study, we focus on investigating the anisotropic elastic properties of freestanding Sb₂Te₃ using Brillouin light scattering (BLS) technique¹⁸. By employing this non-contact and non-invasive method, we demonstrate that BLS can be a precise measurement technique for assessing the values of elastic tensor (c_ij). The additional motivation was the lack of experimental date covering GHz range of frequency. The elastic properties of Sb₂Te₃ so far have been studied by using stress–strain method¹⁹, DFT method^20,21, molecular dynamics method²². Here, we present experimental investigations of surface acoustic waves in Sb₂Te₃ bulk single crystals. High-resolution Brillouin scattering is employed for the first time to determine the dispersion and anisotropy of surface acoustic waves (SAW)²³. The experimental results are complemented by simulations of surface phonon frequencies and their anisotropy, conducted using finite element method (FEM) and analytical method. This article compares the results of simulations with an experiment involving the investigation of surface phonon anisotropy using the Brillouin scattering method. The simulation methods yield the bulk elastic constants, which in turn allow for the determination of theoretical surface phonon velocities.

Comprehensive information regarding the elastic properties of antimony telluride is crucial for its practical application in devices like acousto-optic devices. On a fundamental level, precise knowledge of the elastic properties can act as a reference point for validating theoretical models aiming to predict the mechanical characteristics of this group of topological insulators.

Materials and methods

The samples

The selection of the flux zone growth technique over conventional chemical vapor transport was based on its capacity to yield precise stoichiometric ratios (2:3) and prevent contamination of Sb₂Te₃ crystals by halides commonly used in chemical vapor transport. Growth was conducted at temperatures below 586 °C under 10^–7 torr pressures, employing slow cooling rates (~ 2 °C/hour) to ensure optimal crystallization. All metal and chalcogen precursors were of 6N purity, in ingot (nugget) form, and sealed in a quartz ampoule. Sb₂Te₃ crystals were sourced from HQ Graphene company.

Antimony telluride, akin to bismuth telluride Bi₂Te₃, comprises hexagonal monoatomic layers, forming five-layer "sandwiches" represented by the scheme: |Te–Sb–Te–Sb–Te||Te–Sb–Te–Sb–Te|²⁴. The weak van der Waals interaction between tellurium layers of adjacent "sandwiches" facilitates easy cleavage perpendicular to the c-axis. The unit cell of Sb₂Te₃ comprises up to 10 atomic layers²⁵, as illustrated in Fig. 1a. The layered structure has 5 atomic layers constructing the unit cell, they are called quintuple layers as can be seen in Fig. 1a. Experimentally determined lattice constants are: a = 4.264 Å, c = 30.458 Å^26,27. Sb₂Te₂ is classified under the R3 ̅m space group²⁸.

Fig. 1

Schematic sketch of the considered system. (a) Scheme of the Sb₂Te₃ crystal lattice, (b) Schematic view of BLS backscattering geometry. Symbols k_i, k_s and q, denote incident light, scattered light and surface acoustic wave vectors, respectively, while $\Theta$ is the incident angle. (c) Schematic illustration of the sample holder for the surface phonon anisotropy investigations.

Experimental setup

For the study of SAWs and to determine elastic tensor we used a six-pass, tandem Brillouin spectrometer (TFP2-HC, JRS) which ensures a contrast of 10¹⁵²⁹. The source of scattering light was a frequency stabilized diode-pumped solid-state (DPSS) laser, which operates at λ₀ = 532 nm (Coherent Verdi V5). The geometry used for those measurements was a 180° backscattering geometry with p–p polarization (incident and scattered light polarization parallel to the plane of incidence) for SAWs (Fig. 1b). The frequency of SAWs is represented by the Brillouin frequency shift of the inelastically scattered laser beam. Changing the angle of light incidence, $\Theta$, concerning the surface of the sample, allows selecting the wave vector $\left( {q = \frac{{4 \cdot \pi \cdot {\text{Sin}} (\Theta )}}{{\lambda_{0} }}} \right)$, common for SAWs and SWs, and determining the dispersion relation^29,30. In our experiments, the wave vector was varied in the range 0.007–0.022 nm⁻¹ with a resolution of about 0.0002 nm⁻¹. The free spectral range (FSR) was 30 GHz with a frequency resolution of about 0.06 GHz. Each spectrum was accumulated for 6000 cycles. A Lorentzian curve was fitted for each peak. A more detailed description of the experimental setup can be found in^31,32,33. A schematic sketch of the sample and scattering geometry is shown in Fig. 1. To obtain high-quality spectra, it was necessary to exfoliate the top layer of the single crystal and identify a flat surface with a minimum size of 100 × 100 μm. The sample was then placed in the center of the sample holder to ensure that the scattered light was collected from the same spot during sample rotation. The laser spot of the incident beam has a diameter of 30 μm. Additionally, our spectrometer is equipped with a camera, allowing us to control the position of the laser spot on the sample.

To determine the optical constants of the bulk Sb₂Te₃, the utilized spectroscopic ellipsometry with a SENTECH GmbH SER800 ellipsometer was used, measuring at an incidence angle of 70°. The spectral range for this analysis was from 300 to 1000 nm, with data collected at 1 nm intervals. The Tauc-Lorentz dispersion function extracted the refractive index and extinction coefficients with one Drude and six Tauc components.

FEM simulations

The dispersion relation for SAWs propagating in the studied sample was simulated using the finite element method (FEM) code of the COMSOL Multiphysics with Acoustic module³⁴. The multilayer structure had been treated as one uniform material with defined elastic constants and mass density. The elastic constants and the mass densities of each material take into account the crystallographic orientation. The idea of calculating the dispersion relation for phononic samples was presented in Ref.^31,35.

Results and discussion

Brillouin spectroscopy allows for the observation of Rayleigh surface acoustic waves (RSAW), pseudo-surface waves (P-SAW), and bulk waves in homogeneous materials. Brillouin spectra measurements were conducted to determine the components of the elastic stiffness tensor for the investigated topological insulator. Figure 2a shows the BLS spectrum recorded in the p–p polarization, which exhibits Rayleigh SAW, a type of wave that propagates in the immediate vicinity of the free surface and contributes predominantly to BLS spectra through the surface ripple mechanism. The full spectrum is presented in the Supplementary Materials (Fig. S1).

Fig. 2

Experimental BLS spectra for RSAW in Sb₂Te₃ obtained for p–p polarization (a). The measured dispersion relation for observed RSAW modes (circles) and their fit (solid lines) for bulk Sb₂Te₃ in two independent directions of propagation (b). Anisotropy of RSAW studied on the (001) plane of the crystal with Brillouin spectra (c). The crystallographic directions are given in dual notation: hexagonal and corresponding cubic.

Figure 2b shows the dispersion relation of RSAWs, with frequency f as a function of q, obtained by varying the angle $\Theta$. The phase velocity in this context was accurately determined for two independent directions of SAW propagation differing by 45° in the sample plane, using a linear fit to the BLS data. Moreover, it is important to consider that for the examined light polarization, it was possible to observe modes characterized by displacement in the sagittal plane. These types of surface waves are characterized by a linear dependence of their frequency on the wave vector for homogeneous materials. These conditions allowed for a clear identification of the observed modes. Significantly different RSAW velocity values were obtained for the examined directions (for the first sample orientation $\upsilon$_RSAW1 = (1537 ± 9 m/s), and for the second, $\upsilon$_RSAW2 = (1390 ± 12 m/s)), highlighting the phenomenon of propagation velocity anisotropy of the mentioned waves. In Brillouin spectroscopy, it is not possible to directly determine the crystallographic direction that is being measured. Initially, we conducted simulations using the Finite Element Method to calculate the velocity of surface acoustic waves (SAW) in the studied samples. This allowed us to identify the crystallographic directions in the materials under investigation.

Considering the symmetry of the studied compound on the (001) surface, a threefold axis of symmetry in relation to the propagation velocity of both surface and bulk waves is to be expected. The results of the investigation into the anisotropy of the Rayleigh surface wave propagation velocity on the examined Sb₂Te₃ plane are shown in Fig. 2c.

The study of Rayleigh wave phase velocity anisotropy on the (001) plane of the investigated material, defined as the ratio of the maximum to the minimum velocity on the examined plane, gives a value of 1.10. This is consistent with the anisotropy coefficient of the atomic arrangement on the (001) plane of the investigated crystal, where the anisotropy coefficient is 1.15. The investigation of the distribution of surface wave propagation velocities on different crystallographic planes is particularly useful for determining the components of the elastic stiffness tensor. In the studied material, according to the symmetry of Sb₂Te₃, there are six independent components of the elastic stiffness tensor:

$$ \begin{array}{*{20}c} {c_{11} } & {c_{12} } & {c_{13} } & {c_{14} } & 0 & 0 \\ {c_{12} } & {c_{11} } & {c_{13} } & { - c_{14} } & 0 & 0 \\ {c_{13} } & {c_{13} } & {c_{33} } & 0 & 0 & 0 \\ {c_{14} } & { - c_{14} } & 0 & {c_{44} } & 0 & 0 \\ 0 & 0 & 0 & 0 & {c_{44} } & {c_{14} } \\ 0 & 0 & 0 & 0 & {c_{14} } & {c_{66} } \\ \end{array} $$

(1)

where ${c}_{66}=\left(\frac{{c}_{11}-{c}_{12}}{2}\right).$ The elastic constants characterizing Sb₂Te₃ have been determined using various methods, mainly analytical techniques or based on DFT by different authors. A comparison of these elastic constants and the corresponding bulk wave velocities can be found in Table 1.

Table 1 Elastic constants characteristic of the Sb₂Te₃ studied by using different methods and the values of the velocity (m/s) of bulk acoustic waves corresponding to the values of elastic constants.

By using Brillouin spectroscopy, we determined elastic tensor coefficient as we presented below. The measurement geometry used allows us to directly calculate the bulk wave velocities in the studied material based on the following relationship:

$${v}_{B}=\frac{f\lambda }{2n}$$

(2)

where f—frequency of bulk modes, λ—wavelength of laser = 532 nm, n—refractive index. It is possible directly for modes propagating in the [001] direction on the examined material. In such a scenario, we deal with a 180° geometry for a scattering angle of zero degrees, enabling the direct determination of the c₃₃ component of the elastic stiffness tensor. To accomplish this, it was necessary to first determine the values of the complex refractive index (n + iκ, where n is the index of refraction and κ is the extinction coefficient) in the studied material. Spectroscopic ellipsometry operating over a wide range of wavelengths was used for this purpose. The obtained values of the total light refractive index are depicted in Fig. 3.

Fig. 3

Complex refractive index for Sb₂Te₃. The arrow indicates the wavelength at which measurements were conducted using Brillouin spectroscopy.

The refractive index value was determined to be n = 1.52 for the Brillouin wavelength used in the experiment, while the extinction coefficient κ was found to be 3.39. Based on these measurements, the propagation velocity of the longitudinal bulk mode traveling in the [001] direction was determined, allowing for the direct calculation of the c₃₃ = ($\rho {{\upsilon_{\text{L}}^{{2}} }}$) component of the elastic stiffness tensor, (where $\rho = 6500\;{\text{kg}}/{\text{m}}^{3}$³⁸, $\upsilon_{{\text{L}}}$—represents the propagation velocity of the longitudinal bulk mode traveling in the [001] direction). To determine the values of the other independent elastic components characterizing the material under study, an approximation method based on solving the equation of motion for acoustic waves on the investigated crystallographic plane was used; in this case, it was the (001) plane of the material under study.

First of all, we calculated the velocities of surface waves in Sb₂Te₃ ab initio, based on the procedure described by Fedorov³⁹. For any elastic mode to exist, Eq. (3) must be fulfilled, with l_i, l_l standing for the direction cosines, δ being the Kronecker delta and α_j meaning the oscillation amplitude in direction j.

$$\left({c}_{ijkl}{l}_{i}{l}_{l}-{\delta }_{ik}\rho {v}^{2}\right){\alpha }_{j}=0$$

(3)

From Eq. (3), we can draw the Christoffel Eq. (4), where ${\Gamma }_{ij}={l}_{i}{l}_{l}{c}_{ijkl}$.

$$\left|{\Gamma }_{jk}-{\delta }_{jk}\rho {v}^{2}\right|=0$$

(4)

In the case of the elastic surface mode, it's assumed that the wave propagates in the x₁x₂ plane, therefore its equation will be a linear combination of terms described by Eq. (5), where α_j is the eigenvector corresponding to the given solution of Eqs. (3) and (4).

$${u}_{j}={\alpha }_{j}\text{exp}\left[ik{l}_{3}{x}_{3}\right]\text{exp}\left[ik\left({l}_{1}{x}_{1}+{l}_{2}{x}_{2}-vt\right)\right]$$

(5)

As the surface elastic mode is expected to decay with the depth in the material, the l₃ must have negative imaginary part. Therefore, for given l₁, l₂, ρ and c_ijkl and for v being the parameter, the Christoffel equation yields at most 3 physically meaningful values of l₃. The solution of the equation of motion will be a sum of three terms, as presented in Eq. (6):

$${u}_{j}=\sum_{n=1}^{3}{C}_{n}{\alpha }_{j}^{\left(n\right)}\text{exp}\left[ik{l}_{3}^{\left(n\right)}{x}_{3}\right]\text{exp}\left[ik\left({l}_{1}{x}_{1}+{l}_{2}{x}_{2}-vt\right)\right].$$

(6)

Such a solution must also fulfil three boundary conditions requiring there is no stress on the surface. With these conditions, we obtain a determinant presented in Eq. (7) (further referred to as the boundary condition determinant, BCD), which must be equal 0 for the Rayleigh surface elastic wave.

$$\left|{d}_{mn}\right|=\left|{c}_{m3kl}{\alpha }_{k}^{\left(n\right)}{l}_{l}^{\left(n\right)}\right|=0$$

(7)

The velocity of the Rayleigh mode (the velocity at which the boundary conditions are satisfied) is typically lower than that of the slowest bulk mode. Brillouin spectroscopy can also detect pseudo-surface elastic waves^40,41. These modes are characterized by their appearance in specific crystallographic directions on the examined surfaces of opaque materials. Additionally, their propagation velocity is higher than that of Rayleigh waves and exceeds the velocity of the slowest bulk transverse wave in the examined crystallographic direction. The direction cosines l_i and l_l of these waves have small imaginary parts, which means that the boundary condition determinant does not fully vanish under the given assumptions. Nevertheless, we can predict the velocity of such a wave, as the boundary condition determinant reaches a local minimum at a velocity close to that of the pseudo-surface mode.

To determine the velocities of the Rayleigh surface wave mode and the pseudo-SAW modes, we developed a simple algorithm using Wolfram Mathematica software. Given a set of elastic constants, a specific direction on the surface, and a wave velocity, the algorithm calculates the value of the boundary condition determinant (BCD). By probing different velocity values, it identifies the local minima of the BCD, which are interpreted as the velocities of the Rayleigh surface mode and pseudo-surface modes. The elastic constants determined from BLS are positive and satisfy the thermodynamic stability criteria.

$$ \begin{aligned} c_{11} & > \left| {c_{12} } \right| , c_{44} > 0, c_{13}^{2} < \frac{1}{2} c_{33} \left( {c_{11} + c_{12} } \right), \\ c_{14}^{2} & < \frac{1}{2} c_{44} \left( {c_{11} - c_{12} } \right) \equiv c_{44} c_{66} . \\ \end{aligned} $$

(8)

As a result of the method proposed above, it was possible to determine the components of the elastic stiffness tensor for the material under study. The obtained values are presented in Table 2.

Table 2 The obtained values of the components of the elastic tensor for antimony telluride.

The presented values of the elastic tensor components were used to calculate the velocities of bulk waves, including quasi-longitudinal (QL) and quasi-transverse (QT) waves, as well as RSAW and PSAW waves propagating on the (001) plane (see Supplementary Materials: Fig. S2, Table S1). The obtained results confirm the three-fold symmetry in the investigated materials and are consistent with experimental findings (see Fig. 4a). This three-fold symmetry of the elastic tensor components is also evident in the 3D imaging of Young's modulus (see Fig. 4b). For materials with trigonal symmetry, Young’s modulus E can be calculated using the following Equation⁴²:

$$ \begin{aligned} E^{ - 1} & = \left[ {1 - \cos^{2} \theta } \right]s_{11} + s_{33} \cos^{4} \theta + \left[ {2s_{12} + s_{66} } \right]\cos^{2} \varphi \sin^{2} \varphi \sin^{4} \theta \\ & \quad + \cos^{2} \theta \left[ {1 - \cos^{2} \theta } \right]\left[ {2s_{13} + s_{44} } \right] \\ & \quad - 2\cos \theta \cos \varphi \sin \theta \left[ {3\sin^{2} \varphi \sin^{2} \theta - \cos^{2} \varphi \sin^{2} \theta } \right]s_{14} \\ \end{aligned} $$

(9)

where ${s}_{11}, {s}_{12},{s}_{13},{s}_{14},{s}_{33},{s}_{44}$ and ${s}_{66}$ are the components of stiffness in the crystal plane of the system and $\varphi$ and $\psi$ are the two arbitrary rotation angles to transform the cubic crystal coordinate system into a lattice plane system. The relations among stiffness components ${s}_{11}, {s}_{12},{s}_{13},{s}_{14},{s}_{33},{s}_{44}$ and ${s}_{66}$ and elastic constants ${c}_{11}, {c}_{12},{c}_{13},{c}_{14},{c}_{33},{c}_{44}$ and ${c}_{66}$ are given as⁴²:

$$ \begin{aligned} & c_{11} + c_{12} = \frac{{s_{33} }}{s}, c_{11} - c_{12} = \frac{{s_{44} }}{{s^{\prime}}}, c_{13} = - \frac{{s_{13} }}{s}, \\ & c_{14} = - \frac{{s_{14} }}{{s^{\prime}}},c_{33} = \frac{{s_{11} + s_{12} }}{s}, c_{44} = \frac{{s_{11} - s_{12} }}{{s^{\prime}}}, \\ \end{aligned} $$

(10)

where

$$ \begin{aligned} s & = s_{33} \left( {s_{11} + s_{12} } \right) - 2s_{13}^{2} \;\;{\text{and}} \\ s^{\prime } & = s_{44} \left( {s_{11} - s_{12} } \right) - s_{14}^{2} \\ \end{aligned} $$

Fig. 4

Anisotropy of the velocities of bulk and surface waves propagating on the (001) plane (a), 3D visualization of the Young's modulus (GPa) for Sb₂Te₃ (b). (exp—experimental points).

Using Eqs. (9)–(10), a 3D plot of Young’s modulus $E$ is shown in Fig. 4b for Sb₂Te₃.

Further, we used ELATE⁴³, which is an open-source online tool for the analysis of elastic tensors, to extract Young’s modulus from the known elastic tensor.

Additionally, it was possible to analytically verify the correctness of determining the components of the elastic stiffness tensor for the studied system based on the Equation⁴⁴.

$${c}_{33}\left({v}_{RSW}^{2}- \frac{{c}_{44}}{\rho }\right){\left({v}_{RSW}^{2}- \frac{{c}_{11}}{\rho }+\frac{{c}_{13}^{2}}{\rho {c}_{33}}\right)}^{2}={c}_{44 }{v}_{RSW}^{2}\left({v}_{RSW}^{2}- \frac{{c}_{11}}{\rho }\right)$$

(11)

The values of the elastic stiffness tensor components (Table 2), obtained through non-invasive high-resolution Brillouin spectroscopy, differ from those previously reported by other authors (Table 1), which were obtained using the DFT method. It is important to note that even the elastic tensor components obtained via DFT can vary significantly between different studies. In contrast, our results were calculated from direct measurements, providing experimental validation and greater reliability in capturing the material's true elastic properties. Nevertheless, it should be noted that the studied material has an exceptionally high extinction coefficient. This coefficient directly defines the depth of material penetration by light waves. Sb₂Te₃ is an opaque material, so one should expect that the penetration depth is extremely small compared to other compounds. We have determined that this depth is approximately 12 nm—see Eq. (12). A schematic representation of Rayleigh SAW propagation is shown in Fig. 5a. From a crystallographic point of view, the unit cell of antimony telluride is characterized by a large lattice constant along the [001] direction—approximately 3 nm. Therefore, Brillouin spectra essentially contain information from an area of about 4 elementary cells. Due to such a small penetration depth of light, it is not possible to directly observe all volume modes in the Brillouin spectrum. Additionally, the intensity of the longitudinal bulk mode propagating in the [001] direction on the examined plane was low compared to the intensity of surface modes.

Fig. 5

A schematic diagram illustrating the displacement planes and the depth of light wave penetration in Sb₂Te₃ (a). The mode profiles show the x, y, and z components of the displacement for the RSAW. The height of the shown unit cells is 150 nm, and the length is 50 nm (b).

As shown in Fig. 5, the RSAW wave profiles for waves with short wavelengths are characterized by a displacement that decreases with the depth of wave penetration, which is defined as:

$$ \delta p = \frac{\lambda }{4\pi n} = 12\; {\text{nm}} $$

(12)

Thus, in reality, the depth of light penetration in the studied materials in relation to the surface wave decay depth is very small. Rayleigh waves are typically assumed to decay within two wavelengths in transparent materials. In our case, based on the simulations shown in Fig. 5b, it is evident that these waves decay to a depth of approximately 150 nm. The visible nodal plane appears at different depths for the u_x and u_y components. Therefore, depending on the direction of wave propagation, anisotropy of the RSAW wave velocity is evident on the examined crystallographic plane. The shallow penetration depth means that the information in the examined materials strictly originates from the near-surface region above the nodal planes. Sharp transitions in the displacement field, especially in u_x and u_y, may indicate areas of high stress concentration or boundary effects in the material. This analysis can be crucial to understanding how waves propagate through a material and how its mechanical properties vary depending on direction.

Given the high extinction coefficient of the studied material, the Rayleigh mode profiles and the used method of investigation, it is challenging to classify the material as to be truly 3D in this context. The values of the elastic stiffness tensor components, particularly the directly measured c₃₃ component, are relatively low compared to theoretical predictions. Such variations in the tensor components are characteristic of materials with reduced dimensionality⁴⁵. In cases where the thickness of the material decreases—transitioning from 3 to 2D structures—a corresponding reduction in the values of the elastic stiffness tensor components is typically observed^45,46,47. It is well-established that compared to 3D materials, 2D materials exhibit extraordinary mechanical properties, such as exceptionally high in-plane stiffness⁴⁸, along with notable bending flexibility⁴⁹. Additionally, their interlayer interactions are governed by van der Waals forces, distinguishing their shear, friction, and fracture behaviors from those of 3D materials.

Anisotropic mechanical properties offer unique applications, including the identification of crystallographic orientations in 2D materials⁵⁰, the fabrication of 1D nanoribbons⁵¹. Therefore, a deep understanding of anisotropic mechanics is fundamental to advancing the functional design and applications of 2D materials in the case of van der Waals materials.

Conclusions

In this study, we employed Brillouin spectroscopy to demonstrate the presence of anisotropy in the propagation velocity of surface acoustic waves (SAWs) in antimony telluride (Sb₂Te₃). Investigations on the (001) surface of this material revealed a significant dependence of wave velocity on the propagation direction, reflecting the differences in the crystal structure along various orientations. Through these studies, we directly determined the c₃₃ component of the elastic stiffness tensor, obtaining a value of 47.9 GPa, which is lower than some theoretical predictions. Additionally, these measurements allowed us to indirectly calculate the remaining components of the elastic stiffness tensor. Accurate determination of the elastic properties of Sb₂Te₃, such as the elastic constants, is crucial for its applications in optoacoustic devices and other technological applications, where understanding phonon propagation and mechanical anisotropy plays a vital role. Given the specific characteristics of the studied material, which is defined by van der Waals bonding and a shallow light penetration depth, it can be inferred that the obtained elastic constants are representative of lower-dimensional materials—in this case, on the order of 12 nm. This observation is particularly significant when considering the potential applications of antimony telluride as a 2D material.