Influence of Defect Structures on Intervalley Scattering in Two-dimensional WSe₂ Revealed by Double-Resonance Raman Spectroscopy

I.   INTRODUCTION

Two-dimensional (2D) materials such as graphene and transition-metal dichalcogenides (TMDs) have drawn broad interests and stimulated many explorations due to their unique physical properties and promising applications [1−4]. Raman spectroscopy can provide rich information about the chemical structures of 2D materials and have been widely applied in the studies of their fundamental properties [5−8]. Although the first-order Raman effect associated with the phonon scattering at Γ point (with zero momentum q ≈ 0) has been extensively studied to identify the layer numbers, defects, disorders and strains in 2D TMDs [9−11], many spectral features are still found to be related with the excitonic properties and cannot be explained by the first-order Raman scattering [12, 13]. Therefore, the second-order Raman scattering involving the scattering of two phonons (or one defect and one phonon) with non-zero momenta (q ≠ 0) and the resonance with the excitonic transition energy should be considered for the assignment of various Raman features especially for the overtone and combination modes [8, 14].

The double-resonance Raman (DRR) scattering model describes such a second-order Raman process by introducing two distinct electron-phonon scattering pathways to interpret the importance of intravalley or intervalley scattering and phonon momenta [8, 15]. The first process corresponds to the excitation of two phonons with opposite momenta q and −q, which can be observed in both purely and impurely crystalline samples due to the momentum conservation [14, 16]. For example, the combinations of acoustic and optical phonon in the vicinity of K and M points through the intervalley scattering can generate the DRR bands in the Raman spectra of MoS₂ [8]. The second process includes the excitation of a phonon (with momentum q ≠ 0) due to the presence of defects in the sample. The presence of defects can break the translation symmetry in perfect lattice structure of crystal and provide additional momentum due to the defect scattering process [6]. Therefore, phonons away from Γ point will be also involved in the Raman scattering for the defective samples, resulting in the observation of Raman peaks from non-Γ Brillouin zones, which is not allowed in a purely crystalline sample [17, 18]. For example, the well-studied D and D' lines in graphene can be well explained by the DRR model, which is generally regarded as the defect-induced Raman peaks [19, 20]. However, in traditional DRR theory, only the idealized defects without any structural details are considered, and the influences of different defect types on the above DRR process are still illusive [9, 17, 20, 21]. Moreover, the electronic band structures, as well as the phonon dispersion relations, are also unchanged when introducing these idealized defects, which might give not entirely correct physical pictures of the electron-phonon interactions and intervalley scattering processes [22].

In this work, we proposed a theoretical method based on the DRR theory to explain the influence of different types of defects on the second-order Raman spectrum of a tungsten diselenide (WSe₂) monolayer. Four types of defects in the WSe₂ monolayer including a Se vacancy (V_Se), two Se vacancies (V_2Se), a W vacancy (V_W) and an O substituting for a Se atom (O_Se) are considered during the theoretical calculations. The electronic band structures and phonon dispersions of pristine and defective WSe₂ monolayers are obtained from first-principles calculations based on density-functional theory (DFT) [23, 24]. In order to demonstrate the changes in the electronic band structures, especially the bending of valence and conduction bands, the band unfolding method [25] is adopted for all the defective WSe₂ monolayers to show the effective band structure (EBS) within the same Brillouin zones. The phonon dispersion curves of the defective WSe₂ monolayers are also unfolded from the simulated phononic band structures [26, 27], ensuring that the involved phonons are described in the same reciprocal space for different defective monolayers. The second-order Raman spectra are then calculated using the standard DRR theory [20, 28] for the intervalley scattering processes of two phonons based on the above calculated electronic and phononic band structures. The full 2D Brillouin zone of electronic bands and all possible phonons with adequate wave vectors are considered during the electronic summation in Raman tensor calculations. Due to different influences on the electronic and phononic band structures, different phonon contributions following different pathways are found for the WSe₂ monolayers with different defect types, which may provide valuable insights in the mechanisms of high-order Raman scattering for 2D materials.

II.   METHOD

A   Double-resonant Raman theory

Different from the first-order Raman scattering where only the phonons in the vicinity of the Γ point (with zero momentum q ≈ 0) of the Brillouin zone contribute most to the Raman intensities, DRR describes the second-order Raman scattering process that involves the resonant intravalley or intervalley scattering of excited electrons by two phonons or one phonon and one defect with non-zero momenta (q ≠ 0) [5, 14−16, 20, 29]. Although the original DRR theory is developed for explaining the appearance of D, D' and 2D lines in graphene, it can be generalized to other 2D materials and interpret the roles of electron-photon and electron-phonon couplings during the Raman scattering when the excitionic transitions are taken into account [8, 9, 14, 28, 30]. For a representative 2D TMDs of a WSe₂ monolayer shown in FIG. 1(a), different from the first-order Raman scattering where only one electron or hole is scattered by a phonon (FIG. 1(b)), four typical pathways involving two phonons are considering for the DRR scattering: ee process with only conduction electron involved, hh process with only valence hole involved, eh and he processes with both electron and hole involved [20, 28]. As shown in FIG. 1(d), the DRR process begins with an incoming photon creating an electron-hole pair near the K valley. For the ee process, the electron in conduction band is inelastically scattered to the ${\bf{K}}'$ valley (or the Q valley in FIG. 1(d)) with the emission of a phonon with wave vector –q and energy $\hbar$ω_−q. After that, the electron in conduction band of ${\bf{K}}'$ (or Q) valley is inelastically scattered back to the K valley with the emission of another phonon with opposite wave vector q and energy $\hbar$ω_q. Then the electron-hole pair recombines at K valley, emitting a Raman photon with energy E_L−$\hbar$ω_−q−$\hbar$ω_q corresponding to the Stokes scattering process. For the hh process, similar intervalley scattering paths involving two phonons are participated, but for the hole in valence band at K valley instead. For the eh and he processes, both the electron in conduction band and the hole in valence band are inelastically scattered with the emission of two phonons with opposite wave vectors q and –q, respectively. Note that for the K→Q intervalley scattering, the hh, eh and he processes are not allowed because of the forbiddance of the hole scattering through the Q valence band. Taken all these processes into account, the DRR intensity can be expressed in the form as [8, 28]:

Figure  1.  (a) Top view and side view of a pristine WSe₂ monolayer. The cross-sectional side view is schematically plotted with only two rows of W and Se atoms in the dashed box in the top view. (b) Schema of first-order Raman pathways, in which either the electron or the hole is scattered by a phonon. (c) Schemas of four typical DRR pathways including ee, hh, eh and he processes for the intervalley scattering from K to ${\bf{K}}'$. (d) Schema of ee process for the intervalley scattering from K to Q, where all the other processes are not allowed because of the forbiddance of the hole scattering through the Q valence band.

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where ${{M}}_{{ai}}{=}\left.\langle{{\psi}}_{{a}}\left({{\bf{k}}}\right)\right|{{H}}_{{\rm{in}}}\left|{{\psi}}_{{i}}\left({{\bf{k}}}\right)\right.\rangle$ and ${{M}}_{{fc}}{=} \left.\langle{{\psi}}_{{f}}\left({{\bf{k}}}\right)\right|{{H}}_{{\rm{out}}}\left|{{\psi}}_{{c}}\left({{\bf{k}}}\right)\right.\rangle$ are the matrix elements from the electron-photon coupling due to the incoming and outgoing photons, ${{M}}_{{cb}}{=}\left.\langle{{\psi}}_{{c}}\left({\bf k}\right)\right|{{H}}_{{\rm{ep}}}^{{\nu}}\left|{{\psi}}_{{b}}\left({\bf k}{+}{\bf q}\right)\right.\rangle$ and ${{M}}_{{ba}}{=}\left.\langle{{\psi}}_{{b}}\left({\bf k}{+}{\bf q}\right)\right|{{H}}_{{\rm{ep}}}^{{\mu}}\left|{{\psi}}_{{a}}\left({\bf k}\right)\right.\rangle$represent the matrix elements of electron-phonon coupling [8, 28], with μ and ν the phonon branches involved in the intervalley scattering as well as k and q the wave vectors of the electron and the phonon, respectively. $E_{{\bf k}{(}{\bf k+q}{)}}^{{{\rm{C}}({\rm{V}})}}$ corresponds to the energies of the related electronic states in conduction and valence bands, and the damping constant γ is related to the finite lifetime of these states. To calculate the Raman intensities, these matrix elements are held constant [8, 28, 29, 31], and a damping constant γ = 0.02 eV is used [8, 28, 31]. FIG. S1 in Supplementary materials (SM) shows the simulated DRR spectra for the pristine WSe₂ monolayer with the contribution from four pathways, in which the ee and hh processes (aa process) are dominant in total DRR intensity, with the assignment of major combination bands shown by the side of corresponding peaks. The details of phonon combination analysis are thoroughly discussed below.

B   Electronic and phononic band structures

All the first-principles calculations were carried out using Vienna ab initio simulation package (VASP) [32, 33]. For the pristine WSe₂ monolayer, the primitive cell (PC) contains one W atom and two Se atoms, while for the defective WSe₂ monolayers, the 4×4×1 supercells (SCs) are considered. To minimize the van der Waals interactions between adjacent layers, a vacuum of 20 Å was introduced between the layers. The structural optimizations and electronic structure calculations were performed using density functional theory (DFT) with projector augmented wave (PAW) pseudopotentials [34]. The exchange-correlation potential was described by the generalized gradient approximation (GGA) of the Perdew-Burke-Ernzerhof (PBE) type [35]. The cutoff energy for the wave functions was set to 500 eV. The structural relaxation of the systems was carried out until the force on each atom was less than 0.001 eV/Å using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton algorithm [36]. The 9×9×1 Monkhorst-Pack k-point mesh was used in the reciprocal space. To obtain the phonon dispersion relations for the pristine and defective WSe₂ monolayers, density functional perturbation theory (DFPT) [37, 38] calculations were performed using a 3×3×1 k-point mesh, and the force constants were obtained through Fourier transformation of the dynamical matrices by using the Phonopy package [39].

C   Band unfolding method

Since the SCs are used during the above simulations to solve the electronic and phononic band structures due to the lattice symmetry breaking resulted from the introduction of defects, the band dispersion curves are folded into the first Brillouin zone of the supercell (SBZ), resulting in an unrecognizable relationship between the energy E and the wave vector k when comparing with the pristine WSe₂ monolayer. Therefore, a computational method [25−27] is introduced here to map the energy eigenvalues obtained from the calculations for the large SC into an effective band structure (EBS) to recover an approximate E(k) for defects in the same primitive Brillouin zone (PBZ). The main procedure is briefly discussed below.

A wave vector K in the SBZ is firstly unfolded into a set of wave vectors k_i in PBZ if the pairs of k_i and the reciprocal lattice vector G_i meet the following relation:

Then the spectral weight $P_{{\bf K}{m}}\left({{\bf k}}_{{i}}\right)$ for energy $E_{{\bf K}{m}}\left({{\bf k}}_{{i}}\right)$ can be calculated from [25, 40]:

which represents the probability of finding a set of PC states $\left|{{\bf k}}_{{i}}{n}\rangle\right.$ contributing to the SC state $\left|{\bf K}{m}\rangle\right.$, where $\left|{\bf k}{n}\rangle\right.$ and $\left|{\bf K}{m}\rangle\right.$ (n and m stand for the band indices) are the eigenvectors of the electronic system of the PC and the SC, respectively. Based on the linear combination relationship between $\left|{\bf k}{n}\rangle\right.$ and $\left|{\bf K}{m}\rangle\right.$ [40−42], $P_{{\bf K}{m}}\left({{\bf k}}_{{i}}\right)$ can be further expressed as:

where ${\bf g}$, ${\bf{g}}'$ are reciprocal lattice vectors of PC, ${{\bf G}}_{{i}}$ is reciprocal lattice vector of SC, and $C_{{\bf K}{m}}$ is the plane wave expansion coefficient of $\left|{\bf K}{m}\rangle\right.$. To unfold a SC $E\left({\bf K}\right)$ into a PC adapted $E\left({{\bf k}}_{{i}}\right)$, we can calculate the spectral function A(k_i, E) from $P_{{\bf K}{m}}\left({{\bf k}}_{{i}}\right)$ as

Similarly, for phonon dispersion relations, a phonon wave vector ${\bf Q}$ in the SBZ can be unfolded into a set of wave vectors ${{\bf q}}_{{l}}$ in PBZ if the pairs of ${{\bf q}}_{{l}}$ and reciprocal lattice vector ${{\bf G}}_{{l}}$ meet the relation:

Thus the spectral weight $P_{{\bf Q}{m}}\left({{\bf q}}_{{l}}\right)$ of $E_{{\bf Q}{m}}\left({{\bf q}}_{{l}}\right)$ can be obtained by constructing the projection operator ${\hat{P}}_{{l}}$ using a finite number of ${{\bf g}}_{{j}}$ symmetrically located near the Γ point in PBZ as

here

and

${{\phi}}_{{\bf Q}{m}}^{{s}}\left({{\bf r}}_{{i}}\right)$ represents phonon polarizability vector coefficients, ${{\bf r}}_{{i}}$ is the coordinate of the ith atom, and s=1, 2, 3 correspond to x, y, and z directions respectively. N is the total number of atoms in the SC. The spectral function can be obtained by:

III.   RESULTS AND DISCUSSION

A   Unfolded electronic band structures

The atomic structures of four representative defects in the WSe₂ monolayer including an O substituting for a Se atom (O_Se), a Se vacancy (V_Se), two Se vacancies (V_2Se), and a W vacancy (V_W) are shown in FIG. 2(a). In order to demonstrate the influence of different defects, all the electronic band structures of the defective WSe₂ have been unfolded into the same PBZ of pristine WSe₂ monolayer, shown as the effective band structures (EBS) by the blue dots in FIG. 2(b). The amplified EBS near the K valley is shown in FIG. 2(c). Both the intervalley scatterings from K to ${\bf{K}}'$ and from K to Q are allowed if the wavelength of the incident laser is smaller than 561 nm ( > 2.21 eV). For the O_Se substitutional defect, the calculated EBS is almost consistent with the band structure of pristine WSe₂ monolayer in the vicinity of the K valley, when only the valence band is uplifted by 0.10 eV and the band gap decreased to 1.47 eV comparing with pristine WSe₂. The Q conduction band (located in the middle of Γ and K) is also elevated by 0.20 eV, leading to the forbiddance of scatterings from K to Q for the excited electron through the conduction bands under the incident light with the wavelength larger than 618 nm ( < 2.01 eV). By removing one and two Se atoms to create the V_Se and V_2Se defects, noticeable changes in the shapes and energies for the bottom of the conduction band and the top of the valence band are observed, which will significantly affect the intervalley scattering and the DRR spectra. The Q conduction band further moves up to prevent the electron scattering from K to Q through a phonon. The bands related to the defect states are observed in the middle of band gaps (at ~1 eV), which would not be involved in the DRR process since they are far away from the resonance conditions. For the V_W defect by removing a W atom in the WSe₂ monolayer, both the conduction and valence bands are significantly distorted, and the valance band at K valley is lifted up by 0.41 eV, mixing with the flat bands of the defect states and resulting in different intervalley scattering pathways discussed later. The different influences on the electronic band structure from different defects can be understood from the atom-projected EBS with the projected electronic density of states (PDOS) overlapped with the band structures, as shown in FIG. S2 (SM). For the pristine WSe₂ monolayer, both the conduction and valance bands are mainly contributed by the W atoms. Therefore, only substituting or removing Se atoms would not significantly change the primary features of the conduction and valance bands, especially in the vicinity of K valley.

Figure  2.  (a) Top and side views of pristine and defective WSe₂ monolayers, where the WSe₂ monolayer with a Se vacancy on the upper layer is denoted as V_Se, WSe₂ monolayer with an O substituting for a Se on the upper layer is denoted as O_Se, WSe₂ monolayer with two Se vacancies at the same positions on the upper and lower layers is denoted as V_2Se, and WSe₂ monolayer with a W vacancy is denoted as V_W. The defect sites are highlighted with red dotted circles. The side views are all schematically plotted with only two rows of W and Se atoms, as illustrated by the dashed box in the top view of the pristine WSe₂. (b) Calculated unfolded electronic band structures (i.e., EBS) along high-symmetry k points in PBZ for the pristine and defective WSe₂ monolayers, with the Fermi level set as 0 eV. The spectral weights are reflected by the size of the blue dots. The red lines are the calculated electronic band structure from the PC of the pristine WSe₂ monolayer. (c) The conductance and valence bands in EBS near the K and Q valleys, in which the direct bandgap (Δ) is marked.

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B   Double-resonance Raman scattering

FIG. 3(a, b) show the maps of calculated conduction band minimum and valence band maximum in the first Brillouin zone for the pristine WSe₂ monolayer. For the conduction band, the local minima appear at the high symmetric K, ${\bf{K}}'$ and Q points, enabling the intervalley scattering through K→${\bf{K}}'$ and K→Q for electrons. While for the valence band, the local maxima have the K, ${\bf{K}}'$ and Γ points, enabling the intervalley scattering through K→${\bf{K}}'$ and K→Γ for holes. Therefore, for the K→${\bf{K}}'$ intervalley scattering, all four pathways including ee, hh, eh and he processes are allowed. More detailedly, for the ee and eh processes with the laser excitation energies at 594 nm (~2.09 eV), the excited electron with the initial momentum k would form a triangularly distorted closed line surrounding the K point in the conduction band, as the iso-energy contour line shown in FIG. 3(c). Then the electron is scattered by emitting a phonon towards the ${\bf{K}}'$ point in the conduction band, reaching the second triangularly distorted closed line formed around ${\bf{K}}'$. The phonons connecting the pockets around the K and ${\bf{K}}'$ points pledge to have the wave vector q_n. By translating all these nesting vectors q_n to the Γ origin point, it can be found that the phonons that satisfy the K→${\bf{K}}'$ intervalley scattering generally possess the wave vector q_~K, where the symbol “~” means in the vicinity of the given high-symmetric point in PBZ. The electron at the ${\bf{K}}'$ valley in the conduction band can be scattered back to the K valley in the conduction band by emitting a phonon with opposite wave vector −q_~K and then recombined with the hole at K valley in the valence band through the ee process, or directly combined with the phonon-scattered hole at ${\bf{K}}'$ valley in the valence band through the eh process, resulting in the emission of a Stoke Raman photon with energy $E_{{{\rm{L}}}}-{\hbar}{\omega }_{{±q}}^{{\mu}{,}{\nu}}$. For the hh and he processes, the excited hole (instead of electron) is first scattered along the K→${\bf{K}}'$ direction through one phonon, and then scattered back to recombine at the K valley or directly combine with the scattered electron at the ${\bf{K}}'$ valley to generate the Stokes Raman photon. Moreover, based on the collection of all possible phonon wave vectors for the intervalley scattering and the corresponding phonon density of states (PDOS) analyses (FIG. S3 in SM), only those phonons with q ≈ 2k measured from the K point (the outer trigonal ring indicated by the red line in FIG. 3(c)) make the main contributions to the DRR intensities, and the contributions from q ≈ 0 usually vanish due to the quantum interference [8]. The trigonal warping effect presented in FIG. S3 (SM) leads to another significant result: the phonon contributes the most to the K→${\bf{K}}'$ DRR scattering process in the pristine WSe₂ monolayer originating from the KM direction (corresponding to the scattering electrons with initial and final states along the KΓ direction). V_Se and O_Se do not change the direction but increase the degree of trigonal warping, while for V_2Se, the trigonal warping effect nearly disappears, thus the scattering contributions are equal in all directions. V_W induces trigonal warping in the opposite direction, resulting in a greater contribution from the phonons in the KΓ direction, and the initial and final states of the scattering electrons are primarily along the KM direction. In our calculations, all the contributions from possible phonons to the DRR spectra are considered by integrating all possible wave vectors in reciprocal space.

Figure  3.  (a, b) Calculated conduction band (a) and valence band (b) in the PBZ of pristine WSe₂ monolayer. (c, d) The collections of all possible phonon scattering paths contributed in the intervalley scattering K→${\bf{K}}'$ (c) and K→Q (d) by a phonon with the wave vector q for the pristine WSe₂ monolayer. The hexagon is the first Brillouin zone. The triangularly distorted contour around K in red represents the electronic states near K that are excited by the incident laser, while the contour around ${\bf{K}}'$ (c) and M (d) in blue represents the electronic states near ${\bf{K}}'$ and M that are deexcited by the emission of a photon and the generation of two phonons. The green nesting lines represent a set of phonon vectors that satisfy the K→${\bf{K}}'$ (c) and K→Q (d) intervalley scattering processes by translating all these vectors q_n to the Γ point. (e) Calculated unfolded effective conduction (upper panels) and valence (lower panels) bands in the PBZ for the defective WSe₂ monolayers with different defects of V_Se, O_Se, V_2Se, V_W, respectively. The energies in all plots are the related values corresponding to the conduction band minimum (CBM) and the valence band maximum (VBM) at the corresponding K (or ${\bf{K}}'$) points, named as energy above CBM and energy below VBM, respectively.

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On the other hand, for the K→Q intervalley scattering in the pristine WSe₂ monolayer, only the ee process is allowed because a local minimum rather than a local maximum is formed at the Q point in the valence band (FIG. 3(b)), prohibiting those paths involving hole scattering processes between K and Q valleys. As shown in FIG. 3(d), for the ee process, the phonons connecting the pockets around the K and Q points possess the wave vector q_~M. Therefore, different scattering paths and phonon combinations would be involved for the K→Q intervalley scattering. Moreover, for the defective WSe₂ monolayers containing the above mentioned four representative defects, all their Q conduction bands are elevated comparing with pristine WSe₂, as the shallower valleys at Q points shown in the corresponding maps of conduction band minimum in FIG. 3(e). Therefore, for O_Se, V_Se, and V_2Se, which exhibit insignificant reduction in bandgap, the K→Q scattering is always forbidden under incident light with the wavelength larger than 620 nm ( < 2.00 eV). However, when the incident light energy is increased (e.g., 594 nm (2.09 eV) used in the DRR simulation), the K→Q scattering becomes permissible for O_Se and V_Se. Conversely, the introduction of W defects results in a notable reduction in the bandgap, thereby facilitating the K→Q scattering even at lower excitation energies. Especially, for the V_2Se vacancy defect, a new local minimum appears at the Γ point in the conduction band, yielding a new intervalley scattering path through K→Γ for all ee, hh, eh and he processes.

C   Phonon dispersion and DRR spectra for pristine WSe₂ monolayer

In order to analyze the contributions from different phonons in the pristine and the defective WSe₂ monolayers during the DRR process, the corresponding phonon dispersion relations are calculated. FIG. 4(a) shows the phonon dispersion relation of the pristine WSe₂ monolayer, with the irreducible representations given for the Γ-point phonon bands. Three Raman-active phonon modes (${\rm{A}}'_1$, ${\mathrm{E}}'$ and ${\mathrm{E}}''$) at Γ point are shown on the right of FIG. 4(a). The ${\rm{A}}'_1$ mode corresponds to the out-of-plane vibration of two Se atoms, also labelled as the optical branch ZO₂. The ${\mathrm{E}}'$ mode is composed of two in-plane vibrations, which are degenerated at Γ point and split into one longitudinal optical (LO₂) branch and one transverse optical (TO₂) branch, respectively. Similarly, ${\mathrm{E}}''$ corresponds to another in-plane vibrational mode composed of two optical branches (LO₁ and TO₁). Based on above discussions, we will mainly focus on the combination bands of acoustic and optical phonons in the vicinity of K and M points, especially in the region of 300−400 cm⁻¹. It should be noted that although the symmetries and corresponding point groups as well as the related irreducible representations are different at different high-symmetry k points, as labeledin FIG. 4(a), we always use the representations at the Γ point to describe the combination bands for convenience. As shown in FIG. 4(b), for the K→${\bf{K}}'$ intervalley scattering, the main contributions to the DRR spectra come from four phonon bands at K point: TO₂(${\mathrm{E}}'$) + ZA at ~320 cm⁻¹, LO₂(${\mathrm{E}}'$) + TA at ~340 cm⁻¹, ${\rm{A}}''_2$ + ZA at ~360 cm⁻¹ and ZO₂(${\rm{A}}'_1$) + ZA at ~365 cm⁻¹, respectively. While for the K→Q intervalley scattering, as shown in FIG. 4(c), the main contributions to the DRR spectra come from three phonon bands at M point: LO₁(${\mathrm{E}}''$) + LA at ~325 cm⁻¹, LO₂(${\mathrm{E}}'$) + LA at ~355 cm⁻¹ and ZO₂(${\rm{A}}'_1$) + LA at ~380 cm⁻¹. Moreover, based on previous reports, the intervalley scattering at M point would also introduce an overtone mode 2LA(M), which is usually overlapped with the intrinsic ${\rm{A}}'_1$ and ${\mathrm{E}}'$ modes in the first-order Raman spectrum [17, 43]. This overtone mode (e.g. 2LA), together with the contributions from other acoustic combinations (e.g., TA+ZA), also appears in our calculated DRR spectra, as shown in FIG. 4(d, e). It should be noted that each combination mode contributed in the DRR spectra is not only a single peak, but also includes all the contributed phonon bands that satisfy the intervalley scattering conditions in the vicinity of the K or M points.

Figure  4.  (a) The calculated phonon dispersion of the pristine WSe₂ monolayer, with the irreducible representations labelled at high-symmetry points Γ, M and K. The atomic displacements of three Raman-active modes (${\rm{A}}'_1$, ${\mathrm{E}}'$ and ${\mathrm{E}}''$) at Γ point are shown on the right. (b, c) The dispersion curve of combination bands near the K (b) and M (c) points within the range of 300−400 cm⁻¹. Four combination modes with the major contributions to the DRR spectra are highlighted in red, with the assignment of corresponding phonon modes labelled on each band. (d, e) The calculated DRR spectrum corresponding to the K→${\bf{K}}'$ (d) and K→Q (e) intervalley scattering processes within the range of 160−400 cm⁻¹ due to the excitation of an incident laser with the energy of 2.09 eV (594 nm). The representative peaks of combination or overtone modes with relatively high Raman intensities are highlighted by different colors.

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D   Phonon dispersion and DRR spectra for defective WSe₂ monolayers

For the defective WSe₂ monolayers, not only the electronic band structures are influenced (as shown in FIG. 2(b)), but also the phonon dispersion relations are also changed due to the existence of different defects. From the unfolded phonon dispersions of defective WSe₂ shown in FIG. 5(a), the introduction of a single Se vacancy defect has a significant impact on the optical branch at the Γ point, leading to the significant red shifts for ${\mathrm{E}}''$ and ${\mathrm{E}}'$ modes by 10 cm⁻¹. The ${\rm{A}}'_1$ and ${\rm{A}}''_2$ modes are also red-shifted by 4 cm^–1. For the phonon modes at the M and K points, both the acoustic and optical branches would be influenced by the existence of the Se vacancy. For example, the LA mode is red-shifted by 2 cm⁻¹ at the K point, while the LO₂ mode is red-shifted by 8 cm⁻¹ at the M point and 8 cm⁻¹ at the K point. On the contrary, if the Se vacancy is substituted by a single O atom, most of the phonon bands are not shifted. For example, only the ${\rm{A}}''_2$ mode is red-shifted by 4 cm⁻¹ at the Γ and M points. At the K point, both the ${\rm{A}}''_2$ and LO₂ modes are red-shifted by 4 cm⁻¹, while the LA and TA modes are blue-shifted by 2 cm⁻¹. Moreover, a local vibrational mode related with the O atom in the vacancy is observed at ~150 cm⁻¹. By removing two Se atoms to introduce the V_2Se vacancy, the ${\mathrm{E}}'$ and ${\rm{A}}''_2$ modes are red-shifted by 10 cm⁻¹ and 13 cm⁻¹ at the Γ point, while the corresponding LO₂ branch is nearly degenerated with the ${\rm{A}}''_2$ mode at the K point and both are red-shifted by 19 cm⁻¹. The acoustic ZA and TA modes at the K point are red-shifted by 6 cm⁻¹ and 3 cm⁻¹. In above analyses, the major features of the phonon dispersion curves are still preserved, allowing us to identify each phonon branch through the comparisons with the phonon dispersion of the pristine WSe₂ monolayer. However, by removing one W atom to introduce the V_W vacancy, the phonon bands, especially the optical branches, are greatly influenced and mixed with each other due to the defect-induced broken translational symmetry in the PC. Therefore, it would be quite difficult to resolve different phonon band directly from the dispersion relations, and we need to check the vibrational eigenvectors of the phonon bands located near the corresponding phonon modes in the pristine WSe₂ monolayer to make sure they refer to the same vibrations. For example, the ${\rm{A}}''_2$ mode is greatly red-shifted by 20 cm⁻¹ at the Γ point for V_W, thus for the M and K points, those phonon bands with similar red shifts should be checked for identifying the same phonon mode, showing a red-shift of 18 cm⁻¹ and 17 cm⁻¹, respectively. The acoustic branches are also greatly influenced, showing the red-shift of 10 cm⁻¹ for LA and ZA modes at both the M and K points. To further calculate the DRR spectra of these defective structures, frequencies of all the acoustic and optical phonons satisfying the intervalley scattering conditions are obtained from the remapping surfaces of the corresponding branches in the unfolded phonon dispersion relations (FIG. S4 in SM).

Figure  5.  (a) Calculated phonon dispersion relations along high-symmetry k points within the PBZ for four types of defective WSe₂ monolayers by using the band unfolding method. The spectral weights are reflected by the size of the blue dots. The red curves are the calculated phonon dispersion relations for the PC of the pristine WSe₂ monolayer for comparison. (b) The calculated DRR spectra corresponding to the K→K' and K→Q intervalley scattering processes corresponding to different defective WSe₂ monolayers due to the excitation of an incident laser with the energy of 2.09 eV (594 nm). The peaks of combination or overtone modes with the major contributions are highlighted by different colors. Note that for V_2Se, the K→Q intervalley scattering is totally forbidden under the current condition, resulting in the absence of the corresponding DRR spectrum.

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FIG. 5(b) presents the simulated DRR spectra for different types of defects at an incident laser energy of 2.09 eV (594 nm). For the K→${\bf{K}}'$ intervalley scattering, similar to the DRR spectrum of pristine WSe₂ monolayer in FIG. 4(d), the main contributions in the DRR spectra of the defective WSe₂ monolayers in the range of 300–400 cm⁻¹ arise from the four combination modes in the vicinity of the K point: TO₂(${\mathrm{E}}'$) +ZA, LO₂(${\mathrm{E}}'$) +TA, ${\mathrm{A}}''_2$+ZA, and ZO₂(${\rm{A}}'_1$)+ZA. Based on the unfolded phonon dispersion curves in FIG. 5(a), for the Se vacancy (V_Se), the TO₂(K) mode is blue-shifted by 3 cm⁻¹ and the ZA(K) mode is red-shifted by 3 cm⁻¹, thus the combination band TO₂+ZA at the K point is nearly unchanged when comparing with that in the pristine monolayer. While for the LO₂+TA band, it is red-shifted by 8 cm⁻¹ due to the great red shift of the LO₂ mode at the K point. For the O_Se defect, as we have mentioned above, since most of the phonon bands are not shifted, the DRR spectra mainly resemble the spectra of the pristine WSe₂ monolayer. Only slight red-shifts of 4 cm⁻¹ for ${\rm{A}}''_2$+ZA band and 2 cm⁻¹ for LO₂+TA band are observed. For the V_2Se vacancy, the ${\mathrm{A}}''_2$+ZA band and the LO₂+TA band are red-shifted by 30 cm⁻¹ and 20 cm⁻¹, respectively. The overlap of LO₂+TA band and TO₂+ZA band results in the maximum total intensity of the DRR spectra red-shifting to ~320 cm⁻¹, a phenomenon that can also be observed when the W vacancy is introduced. On the contrary, since both the electronic band structure and the phonon dispersion are significantly changed when the W vacancy (V_W) is introduced, the resultant DRR spectra also exhibit distinct features. For example, nearly all the combination bands are red-shifted with broadened linewidths, especially for the LO₂+TA band red-shifted by 28 cm⁻¹ and the ${\rm{A}}''_2$+ZA band red-shifted by 30 cm⁻¹. The broadenings of the peak widths are resulted from the changes of the PDOS distributions for the defective WSe₂ monolayers (FIG. S3 in SM).

For K→Q intervalley scattering, the main contributions to the DRR spectra in the range of 300–400 cm⁻¹ come from three combination bands at the M point: LO₁(${\mathrm{E}}''$) +LA, LO₂(${\mathrm{E}}'$) +LA, and ZO₂(${\rm{A}}'_1$) +LA. For the V_Se defect, since both LO₂ and LA modes are red-shifted at the M point, the LO₂+LA combination band is red-shifted by 10 cm⁻¹ when comparing with that in the pristine monolayer. For the O_Se defect, similar to the K→${\bf{K}}'$ scattering, the DRR spectra mainly resemble the spectra of the pristine WSe₂ monolayer. However, the absolute intensities of the spectra are decreased due to the elevation of the Q conduction band. For the V_2Se defect, the intervalley scattering from K to Q is totally forbidden under 594 nm illumination due to the further uplift of the Q conduction band. For the V_W defect, nearly all the DRR peaks are red-shifted and broadened, the LO₂+LA band and the ZO₂+LA band are red-shifted by 20 cm⁻¹. In addition, for the wavenumber range smaller than 300 cm⁻¹, the overtone mode 2LA(M) and the combination mode TA(M)+LA(M) also contribute to the DRR spectra, which are usually overlapped with the intrinsic peaks in the first-order Raman spectrum [17, 43].

IV.   CONCLUSION

In summary, we have calculated the second-order Raman spectra of pristine and defective WSe₂ monolayers by using the improved double-resonance Raman scattering model. By considering the influences of different types of defects on the electronic band structures and photon dispersion relations based on the DFT calculations and the band unfolding methods, different phonon contributions following different pathways during the intervalley scattering are found for different defective WSe₂ monolayers. Especially, for the K→${\bf{K}}'$ scattering of the WSe₂ monolayer with a W vacancy, both the conduction and valence bands are significantly distorted, resulting in the reverse trigonal warping of the electronic band structures and the changes of phonon scattering directions. As a result, the corresponding DRR spectrum is greatly changed, showing downshifts in vibrational frequencies and asymmetrical broadenings in linewidths for most combination modes. For the K→Q scattering of the WSe₂ monolayer with two Se vacancies, the conduction band at the Q point is greatly elevated due to the introduction of the defect, and the intervalley scattering from K to Q is totally forbidden under current condition. Our work may provide deeper understanding of the electron-phonon couplings and the roles of defect structures during the intervalley scattering, which are beneficial for the studies of the second-order Raman scattering processes in 2D materials.

Supplementary materials: Simulated DRR spectra of pristine WSe₂ monolayer corresponding to K→Q and K→${\bf{K}}'$ intervalley scattering processes with the contributions from the ee, hh, eh and he paths are shown in FIG. S1. The atom-projected EBS of the pristine WSe₂ and the defective WSe₂ monolayers with the defect V_Se, O_Se, V_2Se and V_W are shown in FIG. S2. The collection of all possible phonon wave vectors and the radial distribution of phonon density of states (PDOS) for the K→${\bf{K}}'$ intervalley scattering in the pristine WSe₂ and the defective WSe₂ monolayers with the defect V_Se, O_Se, V_2Se and V_W are shown in FIG. S3. Remapping contours of representative acoustic and optical branches in unfolded phonon dispersions for defective WSe₂ monolayers, as well as those for the pristine WSe₂ monolayer, are shown in FIG. S4.