Strain-induced variations in the Raman and infrared spectra of monolayer InSe: A first-principles study

Two-dimensional (2D) materials have opened new possibilities in the field of materials science, and indium selenide (InSe) is one of the most remarkable examples due to its excellent electronic and optical properties.¹ InSe is a III–VI semiconductor with high electron mobility and a tunable direct bandgap, making it suitable for nanoelectronic and optoelectronic applications.² The manipulation of these properties through external factors such as strain is essential for developing novel devices.³ 

Strain engineering in 2D materials effectively alters their electronic band structure and, consequently, their physical properties.^4–6 This method has led to the discovery of new phenomena and the improvement of material functionalities, which are important for integrating 2D materials into electronic and photonic systems.^7–9 Raman spectroscopy, a versatile and non-destructive tool, provides insights into the vibrational modes of materials, which are directly related to their structural and electronic properties.¹⁰ Besides Raman spectroscopy, infrared (IR) spectroscopy serves as a complementary non-destructive technique that provides insights into the vibrational modes associated with the movement of atoms and changes in their dipole moments.¹¹ 

InSe has a hexagonal crystal system, a crystal structure, and bonding characteristics that are important to understand before exploring the effects of strain on its vibrational and electronic properties.^12,13 The monolayer has two indium (In) atoms and two selenium (Se) atoms in a corrugated honeycomb pattern. Each In atom forms covalent bonds with three Se atoms, creating sp²-hybridized orbitals, while each Se atom bonds with one In atom and two lone pairs that are out of plane.¹⁴ Monolayer InSe exhibits an indirect bandgap near the Γ point, which is substantially larger than the bandgap observed in bulk InSe.¹⁵ This difference in bandgap size is attributed to the quantum confinement effect in the monolayer form. The bandgap's magnitude is a pivotal factor in determining the material's suitability for various applications, particularly in optoelectronics.¹⁶ The application of external strain to monolayer InSe can lead to significant changes in its bandgap.^17–19 As mentioned earlier, strain can manipulate the electronic band structure of materials by applying tension or compression force.²⁰ This, in turn, affects the vibrational properties of the material, as evidenced by shifts and intensities in the Raman and IR spectra.²¹ The ability to control the bandgap through strain opens up new possibilities for the use of InSe in strain-tunable devices, such as flexible electronics and sensors.^22,23 Currently, theoretical research on strain-induced Raman and infrared spectra in monolayer InSe has been overlooked. Understanding the fundamental crystal structure and bonding characteristics of monolayer InSe is crucial for appreciating how external strain can influence its electronic and vibrational properties.

This study presents a detailed first-principles analysis of the Raman and IR spectra of monolayer InSe under various strain conditions. The application of tensile and compressive strains can cause shifts in phonon modes and changes in vibration intensity, which can be accurately captured by Raman and infrared spectra. By studying these effects, we aimed to establish a correlation between the strain and the observed Raman and IR features, providing a comprehensive understanding of the strain-induced modifications in monolayer InSe. Based on density functional theory, our computational approach allowed us to predict the phonon dispersion, Raman, and IR intensity profiles for strained monolayer InSe.

In this study, we employed density functional theory (DFT)²⁴ and a suite of computational tools, including Phonopy²⁵ and Phonopy–Spectroscopy,²⁶ to investigate the Raman and infrared spectra of monolayer indium selenide (InSe) under different strains. The combination of these methods provided us with a robust theoretical framework to accurately simulate and predict the vibrational properties of the material.

Initially, we used the Vienna ab initio simulation package to perform DFT calculations for the electronic structure and geometric optimization of monolayer InSe. The DFT software package can deal with complex systems under periodic boundary conditions and accurately calculate force constants and electron density.^27–29 We used the PAW pseudopotentials to treat the electron–core interactions and used the GGA with the PBE functional to describe the exchange–correlation potential. Also, a 20 Å vacuum along the z-axis was used to prevent the interaction between periodic images of slabs. All calculations for planewave cutoff energy were 320 eV, including geometric optimization, force constant, and dielectric tensor. We used the 11 × 11 × 1 Monkhorst–Pack k-mesh for structural relaxation and electronic structure calculations for the unit cell. The convergence for total energy was set as 10⁻⁸ eV, and all the atomic positions and lattice structures were fully relaxed with a force tolerance of 10⁻⁴ eV/Å. At this stage, we mainly focused on the material ground state properties, such as lattice parameters, band structure, and Born charge.

Subsequently, we used Phonopy software to compute and examine the phonon dispersion of monolayer InSe. Phonopy is a post-processing tool that relies on DFT calculations to estimate the vibrational modes and associated thermodynamic properties of materials.²⁵ With Phonopy, we understood the material stability and possible phase transitions. Also, using the finite-difference method, a 4 × 4 × 1 supercell was employed to calculate the harmonic interatomic force constants.

Finally, to simulate the Raman and IR spectra, we employed the Phonopy–Spectroscopy extension.²⁶ This tool calculates the IR intensities and Raman activity tensors from data obtained via the previous processes, providing a powerful method to predict and interpret the spectral features.

By using Phonopy–Spectroscopy, we can determine the Raman and IR activities and intensities. Overall, the integration of these computational methods offered us a comprehensive theoretical framework to study the Raman and IR spectra of monolayer InSe under different strains.

Figure 1(a) illustrates the phonon dispersion and phonon density of states (DOS) for monolayer InSe. The phonon dispersion is calculated along high-symmetry directions, revealing the intricate interplay of acoustic and optical phonons.³⁰ In the primitive unit cell of monolayer InSe, there exist three acoustic phonon branches and nine optical phonon branches. Specifically, the three lowest phonon branches correspond to acoustic modes: the in-plane longitudinal acoustic (LA) and transverse acoustic (TA) branches and the out-of-plane flexural acoustic (ZA) branch. Analyzing the DOS plot, we observe that high-frequency optical phonons (above 100 cm⁻¹) predominantly originate from the lighter Se atoms. In contrast, acoustic and optical phonons below 100 cm⁻¹ are primarily contributed by the heavier In atoms. This nuanced vibrational behavior is essential for understanding the material thermal and electronic properties. The single layer of InSe has a honeycomb structure, consisting of an in-plane and a Se plane arranged alternately. Each In atom is covalently bonded to three Se atoms, forming a flat triangular pyramid configuration.³¹ This structure gives the single layer of InSe a high symmetry, belonging to the D_3h point group of the $P 6 ¯m2$ space group. In Fig. 1(b), the E mode vibrates in-plane, and the A mode vibrates out of the plane. The corresponding irreducible representation of optical phonon modes is $Γ optics=2 E ′ ′+2 A 1 ′+ A 2 ′ ′+ E ′$⁠, where $2 E ′ ′$⁠, $2 A 1 ′$⁠, and $E ′$ modes are Raman-active, and the $A 2 ′ ′$ and $E ′$ phonon modes are IR active.^32,33 $A 2 ′ ′$ has no Raman activity, and it is forbidden or extremely weak by selection rules in such geometry.³⁴ 

Table I lists the theoretical calculations of peak positions, irreducible representations, and activity intensities for the Raman and IR spectra of monolayer InSe at the Γ point. The zero intensities for the A″₂ (192.1 cm⁻¹) Raman mode, the E″ (36.8 and 163.4 cm⁻¹) and A′₁ (103.2 and 222.0 cm⁻¹) IR modes suggest that these modes are not active in the Raman and IR spectra, which is consistent with the selection rules for Raman and IR activity based on changes in the dipole moment. Based on the Raman and IR peaks of monolayer InSe from Table I and Fig. 1(b), different irreducible representations can be observed. In the D_3h point group, the A′₁ vibration modes at 103.2 and 222.0 cm⁻¹ are symmetric because they correspond to the principal axis of symmetry of the molecule; the E′ (167.3 cm⁻¹) and E″ (36.8 and 163.4 cm⁻¹) vibration modes are doubly degenerate [see Fig. 1], and they may contain a combination of symmetric and antisymmetric components. The A″₂ (192.1 cm⁻¹) vibration mode is antisymmetric because they are antisymmetric with respect to the horizontal mirror plane of the molecule. For the IR spectrum, the non-zero intensity for the E′ mode at 167.3 cm⁻¹ with a relatively high intensity of 0.4999 Å⁴ amu⁻¹ indicates a strong IR active mode, likely involving significant in-plane atomic displacements that result in a change in the dipole moment.³⁵ The A″₂ mode at 192.1 cm⁻¹ has a small but non-zero intensity, indicating it is weakly IR active. This could be due to a minor change in the dipole moment during vibration or symmetry-related cancelation effects.³⁶ 

Figure 2 shows the Raman and IR spectra of monolayer InSe during free relaxation. Each spectral peak is broadened with a width of 2 cm⁻¹ through the Gaussian profile. The vertical axis of the two sub-plots has been logarithmized due to the weaker intensity of the E″, E′, and A″₂ modes. The Raman spectrum in Fig. 2(a) exhibits five distinct peaks at 36.8, 103.2, 163.4, 167.3, and 222.0 cm⁻¹, corresponding to the E″, A′₁, E″, E′, and A′₁ modes, respectively. The peak positions are consistent with the previous theoretical calculations and experimental measurements.^32,37 The IR spectrum in Fig. 2(b) displays two peaks at 167.3 and 192.1 cm⁻¹, corresponding to the E′ and A″₂ modes, respectively. The peak positions are slightly different from the previous theoretical predictions,³¹ possibly due to the strain effects or the substrate influence on the monolayer InSe. The Raman and IR spectra of monolayer InSe reflect the vibrational properties and symmetry of the crystal structure. The Raman spectra can be used to identify the number of layers and the stacking order of InSe and probe the strain and doping effects on lattice dynamics.³⁸ The IR spectra can be used to measure the electric dipole moment and polarizability of InSe and investigate the interlayer coupling and dielectric environment of monolayer InSe.³⁹ The comparison of the Raman and IR spectra can also reveal the anisotropy and optical phonon dispersion of InSe, which are related to the electronic and optical properties of the material.³¹ Therefore, Raman and IR spectroscopy are powerful techniques for characterizing the structural and physical properties of monolayer InSe and other layered materials.

In our study, we investigated the effects of biaxial strain on monolayer InSe while preserving its crystal symmetry. This is achieved by modifying the lattice parameters, where δ represents the relative change in lattice constant: δ = (L−L₀)/L₀ × 100%. Here, “L” corresponds to the lattice constant of strained monolayer InSe and “L₀” represents the lattice constant of unstrained monolayer InSe. Hence, negative δ represents compressive strain, while positive δ represents tensile strain. Here, the strain range we studied is −6% to 6%, which is usually achievable.⁴⁰ Biaxial strain has emerged as a powerful parameter for tailoring the properties of two-dimensional materials. By applying external forces, researchers can modify the phonon dispersion relation and lifetimes,⁴¹ which directly impact thermal conductivity and lattice dynamics. These strain-induced lattice distortions also influence the corresponding Raman and IR spectra, leading to shifts in peak positions.⁴² Consequently, Raman and IR spectroscopy are valuable techniques for probing these effects and gaining insights into the material's behavior under strain.

The different softening or hardening rates of these two modes with strain can be attributed to the effectiveness of in-plane strain in stretching or compressing the in-plane bonds compared to the out-of-plane bonds. Consequently, in-plane strain significantly impacts the dynamics of in-plane phonon modes, whereas the out-of-plane phonon modes are less affected. Representative phonon band structures of monolayer InSe at various biaxial tensile and compressive strains are depicted in Fig. 3. Notably, the out-of-plane acoustic mode or the flexural mode, which exhibits quadratic behavior under free strain, tends to become linear with increasing tensile strain. Additionally, the frequency gap between the optical and acoustic branches decreases with both tensile and compressive strain. In addition, the impact of strain on high-frequency optical phonons (above 100 cm⁻¹) is more pronounced. Consequently, the phonon band shifts significantly downward in the low-frequency direction. Conversely, the effect of strain on the acoustic and optical phonons below 100 cm⁻¹ is comparatively weaker. Upon further exploration, this behavior can be attributed to the distinct mass and bonding characteristics of the In and Se atoms. From the phonon DOS in Fig. 1, when strain is applied, the heavier In atoms exhibit weaker vibrational responses, resulting in a less pronounced effect on the phonon band. In contrast, the lighter Se atoms display more significant vibrations under strain, leading to the observed downward shift in the phonon band.

Figure 5 shows the peak position changes of Raman and IR spectra under different biaxial strains. It can be seen that with the increase of strain, all the Raman and IR peaks show a redshift trend; that is, the frequency decreases. This is consistent with the positive values of the Grüneisen parameters in Fig. 4. According to Eq. (1), the magnitude of the Grüneisen parameter reflects the sensitivity of the vibrational mode to the volume change. From Fig. 4(b), it can be seen that the Grüneisen parameter of the A″₂ mode is the largest, which is 1.17, indicating that it is most sensitive to volume change, while the Grüneisen parameter of the E″ mode is the smallest, which is 0.21, suggesting that it is least sensitive to volume change. Therefore, in Fig. 6, we can see that the peak position of the A″₂ mode shifts the most with the increase of strain, while the peak position of the E″ mode shifts the least. Such results also verify our previous assumption that the E″ mode may have split under strain, leading to its increased Raman intensity.

As can be seen from Fig. 5, in the absence of stress, the frequency difference between the E′ and A′₁ modes is 54.731 cm⁻¹, while the frequency difference between the E′ and A″₂ modes is 24.834 cm⁻¹. With the increase of strain, both groups of frequency differences gradually decrease; until at +6% strain, the frequency difference of the E′ and A′₁ modes become 46.590 cm⁻¹, while at +6% strain, the frequencies of the E′ and A″₂ modes will overlap, forming a strong infrared peak. These results indicate that strain can change the coupling degree between different optical modes, affecting their spectral characteristics. The physical mechanism of this coupling may be related to lattice distortion, electronic structure change, and outer electron cloud overlap.

The frequency shifts of the optical modes under different biaxial strains are presented in Fig. 6 based on Fig. 5. Figure 6 shows the frequency shifts between different optical modes with compressive and tensile strains as a function of strain, where the frequency shift with strain (δ) is defined as ω(δ) = ω(δ)−ω(0). The different slopes of the ω(δ) curves reflect the different stiffening or softening behavior of each phonon mode under strain. For the E″ mode with 36.8 cm⁻¹, the slope is the smallest among all the modes, consistent with the smallest Grüneisen parameter γ within all the optical modes. Overall, strain has a more significant impact on the frequency shift of high-frequency vibrational modes [see the DOS in Fig. 1], meaning that the vibrations related to the lighter Se atoms in the outer layers are more pronounced. In contrast, the frequency shift impact on low-frequency vibrational modes is smaller [see the DOS in Fig. 1], indicating that the vibrations related to the heavier In atoms in both inner and outer layers are less affected. The influence of strain on vibrational modes can be understood in terms of the atomic mass and the bond strength within the lattice. High-frequency modes are typically associated with lighter atoms and stronger bonds, making them more sensitive to changes in the lattice parameters due to strain. This sensitivity results in a more noticeable frequency shift for these modes when strain is applied. On the other hand, low-frequency modes are often linked to heavier atoms with weaker bonds. These modes are less sensitive to strain because the heavier atoms require more energy to be displaced, and the weaker bonds are less affected by changes in lattice parameters. As a result, the frequency shifts for low-frequency modes are less pronounced under strain.

Figure 7 depicts the Raman and IR activity intensities of different optical modes under varying biaxial strains. In general, As the strain increases from −6% to 6% in Fig. 10(a), the intensity of the A′₁ modes at 103.2 and 222.0 cm⁻¹ decreases, indicating a negative correlation between strain and Raman activity intensity for these specific modes. The A′₁ mode is typically associated with symmetric vibrations of atoms in the lattice, which may occur more readily in the absence of strain or under minor strains. As the strain increases, the disruption of lattice symmetry may hinder the symmetric vibrations of the A′₁ mode, thereby reducing the change in polarizability and decreasing the Raman spectral intensity. Contrary to the Raman activity of the A′₁ modes, the activity intensities of the Raman E″ mode at 163.4 cm⁻¹ in Fig. 7(a) and the IR E′ mode at 167.3 cm⁻¹ in Fig. 7(c) show a positive correlation with increasing strain, as the intensity increases with the strain from −6% to 6%. The E″ mode is usually associated with asymmetric vibrations of atoms in the lattice, which are more sensitive to lattice asymmetry. With increasing strain, the lattice's asymmetry enhances, which may promote the asymmetric vibrations of the E″ mode, thereby increasing the change in polarizability and increasing the Raman spectral intensity. For the in-plane modes (E″ and E′), the Raman and IR activity intensities increase with increasing strain. This indicates that the in-plane modes become more polarized and more responsive to light as the plane is stretched. One possible reason is that the bond lengths and angles change under strain, leading to an increase in the asymmetry and the dipole moment of the in-plane modes. For the out-of-plane modes (A′₁), the activity intensity of Raman decreases with increasing strain. This indicates that the out-of-plane modes become less polarized and less responsive to Raman scattering as the plane is stretched. One possible reason is that the out-of-plane modes involve a larger displacement of the atoms from their equilibrium positions, resulting in a larger change in the polarization and the dipole moment under strain.

In addition, the Raman intensity for the E′ mode at 167.3 cm⁻¹ in Fig. 7(b) is the lowest at zero strain and increases with both tensile and compressive strain. Raman scattering highly depends on the polarizability of the molecule. Strain can alter the electronic environment of the bonds, affecting their polarizability. Both tensile and compressive strains can change the bond lengths and angles, which in turn can increase the polarizability of tensor components, leading to higher Raman intensity. Strain can break the symmetry of the crystal lattice, leading to the activation of otherwise silent Raman modes or the enhancement of weak modes. This symmetry breaking can increase the number of Raman-active phonon modes and their intensities. The potential energy surface of a molecule is not perfectly harmonic, and anharmonicity can lead to changes in Raman intensity with strain. As the material is strained, the anharmonic terms in the potential energy may become more significant, leading to an increase in the Raman scattering cross section and, thus, higher Raman intensity. For the A″₂ mode at 192.1 cm⁻¹ in Fig. 7(d), the highest infrared intensity is at zero strain and decreases in infrared intensity when tensile and compressive strains are applied. Infrared activity is related to changes in the molecular dipole moment. Without strain, this vibrational mode may cause the greatest change in the dipole moment, resulting in the highest infrared intensity. Strain may alter the molecular geometry, thereby reducing the change in the dipole moment and leading to a decrease in infrared intensity. Strain can change the bond lengths and angles within the molecule, which may affect the infrared activity of the vibrational mode. In some cases, the strain may cause an increase or decrease in bond lengths, thus reducing the infrared absorption strength of the vibrational mode. Similar to the frequency shift in Fig. 6, for the low frequency of the E″ mode at 36.8 cm⁻¹ in Fig. 7(b), its intensity change is also insensitive to the variation of strain. This means that the strain has a very weak effect on the frequency shift and active intensity related to heavier atoms. The insensitivity to strain changes for low-frequency modes typically associated with the motion of heavier atoms can be attributed heavier atoms requiring more energy to be displaced from their equilibrium positions. As a result, the vibrational modes involving these atoms are less affected by external perturbations such as strain. This is because the potential energy wells associated with their vibrational modes are deeper, making the vibrational frequencies less susceptible to changes in atomic spacing caused by strain. Therefore, the intensity of these modes, as observed in infrared or Raman spectroscopy, remains relatively constant despite the application of strain.

This paper studied the structural, vibrational, Raman, and IR spectra properties of monolayer InSe under different biaxial strains by using density functional theory. The absence of imaginary frequencies in the phonon dispersion indicates that the monolayer InSe is stable. According to our calculations, the frequencies and activity intensities of the Raman and IR properties of the monolayer InSe will change with the strains. When the different biaxial strain ranging from −6% to 6% is applied to the monolayer InSe, the Raman and IR spectra of monolayer InSe show the changes in the peak positions and activity intensities of the Raman and IR modes change with strain. The frequency shift curves reflect the different stiffening or softening behaviors of each phonon mode under strain, and the behavior is consistent with the Grüneisen parameter within all the optical modes. In addition, the Raman and IR activity intensities vary with biaxial strain. The A′₁ modes show decreased Raman intensity with increased strain due to hindered symmetric vibrations, while the E″ and E′ modes exhibit increased Raman and IR intensities, suggesting enhanced asymmetric vibrations and polarizability. These observations reflect the sensitivity of in-plane and out-of-plane vibrational modes to strain-induced changes in lattice symmetry and electronic structure. However, the Raman intensity for the E′ mode at 167.3 cm⁻¹ and the IR intensity for the A″₂ mode at 192.1 cm⁻¹ exhibit opposite behaviors with strain: the former increases with both tensile and compressive strain due to enhanced polarizability and symmetry breaking, while the latter decreases, likely due to reduced dipole moment changes. Low-frequency modes, such as E″ at 36.8 cm⁻¹, show strain insensitivity, suggesting that heavier atoms involved in these modes are less affected by strain-induced changes in atomic spacing.

We acknowledge the support of the National Key Research and Development Program of China (No. 2019YFA0307701), the National Natural Science Foundation of China (NNSFC) (Nos. 11674128, 11674124, and 11974138), and the Fundamental Research Project of Chinese State Key Laboratory of Laser Interaction with Matter (No. SKLLIM2204).

The authors have no conflicts to disclose.

Xiangyu Zeng: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Yutong Chen: Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). Yuanfei Jiang: Formal analysis (equal); Investigation (equal); Visualization (equal); Writing – review & editing (equal). Laizhi Sui: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). Anmin Chen: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Resources (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Mingxing Jin: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Resources (equal); Supervision (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.