First‐Principle Study of Bandgap Engineering and Optical Properties of Monolayer WSe2 in Second Near‐Infrared Windows

Fluorescence imaging in the second near‐infrared II (NIR‐II) window is opening up new possibilities in bioimaging due to its low scattering rate within the tissue. The integration of 2D materials with NIR‐II fluorescence will enable the development of multifunctional imaging probes. However, there are very few 2D materials that can fluoresce in the NIR‐II range. Monolayer WSe2 is a potential 2D material, but its photoluminescence (PL) around 790 nm is still far from the NIR‐II range due to its bandgap of 1.54 eV. In this study, it is investigated the electronic structures, dielectric functions, and PL spectra for Te, I, and Cr‐doped monolayer WSe2, as well as W and S vacant monolayer WSe2. Most of the defected monolayer WSe2 remain semiconductors, except for a few configurations exhibiting metallic properties after making vacancies. Among the monolayer WSe2 under investigation, the Cr‐doped WSe2 performs the best, exhibiting a strong PL peak in NIR‐II with a decreased bandgap around 1.0 eV. As increasing Cr concentration, the peak shifts further toward the red end of the spectrum due to an enhancement of p–d transition. The results provide a useful guideline for material synthesis applied in NIR‐II bioimaging and other biophysics.


Introduction
Near infrared-II (NIR-II) imaging at 1100-1700 nm has shown great potential in biomedical imaging due to its low tissue scattering, deep penetration, and high resolution. [1][2] New fluorescent probes that can operate in the NIR-II range are highly demanded to enable real-time imaging of deep tissue. While various fluorescent NIR-II probes have been reported, including carbon nanotubes, [3] semiconductor quantum dots, [4][5] rare earth nanoparticles, [6][7] and small clusters, [8] there is still a shortage of bright probes that can emit photoluminescence (PL) in the long wavelength, especially in the NIR-II region. [9][10][11][12][13] Theoretically, PL stems from the relaxation of the excited electrons, whose wavelength and intensity strongly depend on the energy difference between the initial and final states and their joint density of states. In semiconductors, most excited states are located in the bottom of the conduction band. Therefore, manipulating the bandgap remains a significant obstacle in achieving desired PL wavelengths in NIR-II probes.
In bandgap engineering, 2D transition metal dichalcogenides (2D TMDs) are promising candidates for NIR-II imaging at 1100-1700 nm with a bandgap of less than 1.0 eV. The H form of TMDs is the most stable structure, [14] consisting of a single layer of transition metal atoms sandwiched between two layers of chalcogen atoms with prismatic coordination, as depicted in Figure 1a,b. The adjacent layers are coupled by weak van der Waals (vdW) forces and can be exfoliated into ultrathin layers, which allows the bandgaps to be easily tuned by various methods such as doping, [15][16][17] mechanically straining, [18] or stacking in the form of heterostructures. [19] In contrast to graphene, which is unsuitable for optoelectronic devices due to its zero bandgap, 2D TMDs have been widely studied over the past decade because of their unique electronic characteristics, good optical properties, high stability in solution environments, low toxicity, and ultrathin structure. [20][21][22][23][24] As a result of their physicochemical characteristics, 2D TMDs have been explored in variety of applications, including photonics, [25] electrocatalysis, [26] and energy storage. [17] All TMDs-MX 2 (M = Mo, W; X = S, Se) exhibit similar electronic structures and thus similar macroscopic properties. While MoS 2 is commonly used as a representative of TMDs, some studies indicate that the PL quantum yields of WS 2 and WSe 2 monolayers are higher than those of MoS 2 monolayers at room temperature. [27] Luo et al. explored two approaches to increase the maximum PL quantum yield for WSe 2 from 1% to 65%. The first approach utilizes magnetic brightening to convert dark excitons into bright excitons, while the second approach involves directly reducing nonradiative recombination centers by controlling the point defect density. [28] Additionally, the PL can be improved through quantum confinement when the structure of WSe 2 changes from bulk to monolayer. [29][30] Thus, monolayer WSe 2 could be an ideal candidate for NIR-II imaging.
As the thickness of WSe 2 decreases to a monolayer, it undergoes a transition from indirect bandgap semiconductor to a direct one. However, the direct bandgap of 1.65 eV still falls outside the NIR-II upper limit. [31][32] Hence, reducing the bandgap of WSe 2 becomes the foremost task for its applications in NIR-II technology. There are various methods in manipulating the bandgap, the most widely used of which are alloying, doping, and strain-tuning. In this work, we focus on the introduction of doped atoms and vacancies to adjust the bandgap. The use of dopants in MX 2 can extensively alter their physical properties, such as charge transport, magnetism, and optical absorptions. In 2021, Xu et al. studied the effect of Er doping on WSe 2 using pulsed laser deposition and found that the luminescence of 2D nanosheets can be extended to NIR-II window. [33] However, the resulting bandgap narrowing and optical properties have yet to be fully understood. Therefore, it is necessary to investigate the electronic and optical properties of WSe 2 using first-principle methods.
This study employs first-principle calculations to investigate the electronic structures and optical properties of monolayer WSe 2 with various doped atoms and vacancies are calculated using the first-principle method. Point defects can be classified into intrinsic and extrinsic, where intrinsic defects consist of monovacancies, self-interstitials, and antisites, while extrinsic defects comprise substitutional defects and extrinsic-interstitials (doped other atoms). [34] In this work, the substitution of Se atoms by Te or I, and W atoms by Cr in the 4 × 4 × 1 WSe 2 supercell is explored, with various numbers of doped atoms in different atomic planes of WSe 2 . We have also examined the effects of vacancy defects on the bandgap by adjusting their numbers and positions in a 4 × 4 × 1 WSe 2 supercell. Results show that the bandgaps decrease upon doping with the atoms and vacancy defects under consideration, with the Cr-doped and Se vacancy structures displaying the most promising outcomes for NIR-II applications.
The paper is organized as follows. In Section 2, we provide a review of the basic idea involved in calculating optical properties using first-principal methods. In Section 3, we present and analyze the results of our calculations, including the band structures, formation energies, absorption spectra, transition dipole moment, and PL spectra for different extrinsic-interstitials and vacancies defects. We conclude our findings in Section 4. And finally, in Section 5, we provide technical details of the VASP calculations used in the study.

Theories
To evaluate the thermodynamical stability of the doped materials, the formation energy is calculated according to the equation where E doped and E pure are the total energies of the supercells with and without doped atoms, respectively; n m is the number of m atom added (n m < 0) or removed (n m > 0); μ m is the chemical potential of the bulk element m, which is the energy of per atom obtained from the corresponding bulk material. [35][36][37] The plasma frequency and the dynamic dielectric function can be obtained from Vienna Ab initio Simulation Package (VASP) directly, whose imaginary part yields the absorption spectra. Generally, the absorbance of 2D materials is given by [38] 2D ( ) is the in-plane optical conductivity of 2D materials, which is related to the corresponding 3D ( ) component through where L is the slab thickness in the simulation cell. Based on the Maxwell equations, the optical conductivity of 3D materials can be expressed as 3D where is the photon frequency, 0 is the vacuum permittivity, ( ) is the frequency-dependent complex dielectric function ( ) = 1 ( ) + i 2 ( ), with 1 ( ) and 2 ( ) being the real and imaginary parts, respectively. Finally, we obtained the absorption spectra A( ) of 2D materials by It seems straightforward to get absorption spectra from VASP calculation with Equation (3), but the resulting spectra is valid only for zero temperature. A thermodynamic correction as shown in Equation (4) should be applied to take into account the effect of temperature [39] The occupation probabilities in the following manner Here, f v and f c are the occupation probabilities of the valence band and conduction band, respectively, 0K is the absorption coefficient at zero temperature, ℏ is the photon energy, ∆F is the quasi-Fermi level splitting, k B is the Boltzmann's constant and T is the temperature.
To compare with future experimental results, we convert the absorption spectra into PL spectra at room temperature. For a long time, the van Roosbroeck-Shockley (vR-S) equation was used to express the connection between PL and absorption spectra [40] It is inaccurate to use this relationship to describe the interband emission because this cannot be described by the thermal equilibrium condition. As T → 0 K, the emission rate R(ℏ ) goes to zero, but there must be spontaneous emission even at zero temperature, PL measurements are usually done at low temperature, the equation has been modified by later researchers. According to Einstein's theory of spontaneous and stimulated emission, the actual PL spectrum (ℏ ) is related to the probability of the excited state being occupied and the ground state being unoccupied and (ℏ ) through [39,41] where F c (F v ) is the quasi-Fermi energy for electron (hole) distribution in the conduction (valence) band, c is the speed of light, n is the refractive index of the emitting material, and is the density of states.

Foreign Atom-Doped Structures
Chemical doping has been widely used to customize material properties. Figure 1 illustrates the structures of monolayer WSe 2 with different dopants. The supercell volume and the atomic positions are completely relaxed in the structural optimization process. The optimized lattice constants of pure Wse 2 are found to be a = b = 3.29 Å, which agree fairly well with earlier findings, [42][43] indicating that the calculation method is reasonable and subsequent calculation results are reliable. The stability of the doped structures is investigated by calculating their formation energy ( Figure 1l; and Table S1, Supporting Information). The results show that the formation energy of double-doped WSe 2 is always less than twice of that of the single-doped, indicating that sequential doping is energetically more favorable than dispersive doping in the case of low doping concentration. This phenomenon has also been reported in studying the formation of sulfur vacancies in MoS 2 . [44] For the same number of doping atoms, the Cr-doped WSe 2 has the lowest formation energy, making it the most easily prepared in experiments. Figure 2a displays the band structure and atom-projected density of states (pDOS) of pure monolayer WSe 2 . The results indicate a direct bandgap of about 1.54 eV at K point, which is close to the experimental value of ≈1.65 eV [45][46] and aligns with other theoretical calculations [47][48][49] as shown in Table  S2 (Supporting Information). Nevertheless, a slight deviation can be attributed to the choice of exchange-correlation energy. Compared to the Perdew-Burke-Ernzerhof (PBE) results, the Heyd-Scuseria-Ernzerhof (HSE) calculation significantly overestimates the bandgap of pure WSe 2 by 30% (see Figure S1 and Table S2 for details, Supporting Information). Therefore, HSE06 is not an appropriate functional for our simulation. In addition, extending the primitive cell to a supercell leads to the folding of band structure for pure WSe 2 monolayer, which, however, does not affect the bandgap. Figure 2b,c; and Figure S2 (Supporting Information) demonstrate that Te-doping has a trivial effect on the bandgap of monolayer WSe 2 , with a reduction of less than 5%, which will not discuss in the main text. In contrast, doping I atoms led to a significant reduction in the bandgaps, as shown in Figure 2d-f. Specifically, I dopant brings about the flat impurity bands inside the original band, leading to four of the five 2I-doped WSe 2 monolayer under discussion exhibiting bandgaps in the NIR-II range. Moreover, the spin up and spin down of DOS for 1I-doped WSe 2 monolayer are unsymmetrical, making it a magnetic semiconductor. On the other hand, 2I-doped WSe 2 monolayer behaves as a nonmagnetic semiconductor, as illustrated in Figure S3 (Supporting Information). Figure 3 illustrates the effects of Te-and I-doping on the optical properties of monolayer WSe 2 . Specifically, the imaginary part of the dielectric function, which is highly related to the absorption coefficient of the material, is analyzed. In addition, the PL spectra have been calculated using Equation (7). As a benchmark of our calculations, pure WSe 2 monolayer has been tested and the results are in consistent with previous experiments. [46] Monolayer WSe 2 exhibits a peak at 1.62 eV in the imaginary part of the complex dielectric functions (Figure 3a,b), corresponding to the absorption peak at 750 nm (Figure 3c,d). Figure S4 (Supporting Information) compares the interband dielectric functions and the absorption spectra for pure WSe 2 monolayer obtained from using the primitive cell and the supercell, respectively. The results show that the band folding leads to a slight blueshift of interband transitions at high frequencies but does not affect the main peaks in the absorption spectrum, indicating the reliability of the supercell calculation. However, doping of Te hardly affects the electron transitions, resulting in only a slight redshift in absorption due to the small decrease in the bandgap. Similarly, the PL spectra do not show significant changes with Te doping, and no peaks in the NIR-II region are observed, indicating that Te has little effect on modulation the optical properties of WSe 2 . Conversely, doping I atoms can meet the requirements for NIR-II applications ( Figure 3f). The imaginary part of the dielectric function of 2I-2 exhibits a transition peak at 0.79 eV, explaining the improvement in absorption in the NIR-II window, where 2I-2 shows a peak at 1500 nm and stronger absorption in 1000 nm than pure WSe 2 . The PL spectra of I-doped WSe 2 display considerable redshift, with the emission peak moving toward 1200 nm. Although the formation energies of 2I-2 and 2I-4 structures are slightly lower than those of other 2I doping structures, they are almost the same. As a result, it is difficult to control the experiments to synthesize the desired configuration, which limits their application in NIR-II (Figure 1l).
We have also examined the effect of Cr-doping on WSe 2 , which displays the best NIR-II results among the tested structures in this work. Figure 4 presents the band structures and pDOS, while Table S3 (Supporting Information) provides the structural parameters and bandgap of the optimized structures. The formation energies of substituted configurations of doped WSe 2 with various supercell sizes are converged, as shown in Table S4 (Supporting Information). A 4×4×1 supercell is sufficient for the simulations. The Cr band appears as a localized defect band in the intrinsic bandgap, narrowing it, as shown in the band structure. Even though the calculations are limited to a 4×4×1 supercell, the Cr element concentrations can reach the dilute limit. For www.advancedsciencenews.com www.advmatinterfaces.de various dopant concentrations and configurations, the bandgap can be reduced to 1.28, 1.05, and 1.17 eV, respectively, meeting the requirements of NIR-II application. Wang et al. have also investigated Cr-doped monolayer WSe 2 with density functional theory (DFT) calculations and found a reduced bandgap of 1.29 eV. [37] The disparity in bandgap outcomes is caused by different Cr concentrations. In general, the more Cr is doped, the narrower the bandgap becomes. Figure 5 shows the optical properties of the Cr-doped structures. Despite the low concentration of Cr atom, the absorption in NIR-II region is significantly enhanced. Furthermore, we investigated three configurations of Cr-doped WSe 2 , all of which exhibit PL peaks (968, 1024, and 1141 nm, respectively) that shift to the long-wavelength range (Figure 5c). The vertical dashed line in Figure 5c corresponds to the bandgap in Figure 4. Van der Waals (vdW) interactions are known to play an important role in layered materials. [50][51][52] However, in the case of monolayer WSe 2 , the impact of vdW interactions is limited. We have checked the vdW correction to the Cr-doped configurations, which are the best candidates for the NIR-II applications under investigation. Our   Figures S5 and S6 (Supporting Information), indicate the vdW correction does not affect the bandgap or the absorption spectra. While it induces slight modifications in the PL spectra, it only results in a blue shift of the peaks by ≈20 nm. Importantly, these modifications have no substantial impact on the potential of Cr-doped monolayer WSe 2 for NIR-II applications. In addition, we calculated the sum of the squares of the transition dipole moment (sTDM) at different k points to reveal the transition probabilities between the valence band and the conduction band. As shown in Figure 5d, the 1Cr-doped WSe 2 under consideration displays much stronger transitions between conduction and valence band edges than pure WSe 2 , especially around the K point, where the direct bandgap is located. As a result, it is more likely for Cr-doped WSe 2 to exhibit strong NIR-II emission due to an optical transition between the valence band maximum (VBM) and the conduction band minimum (CBM). Figure 5e compares the sTDM between VBM and CBM of all the direct bandgap configurations discussed so far. The intensity of PL spectra is directly proportional to the sTDM. It demonstrates that Cr-doping not only reduces the bandgap of WSe 2 to NIR-II but also provides the best quantum yield in this region.

findings, as shown in
To understand the microscopic mechanism behind the good performance of Cr-doping, we compared the atom orbital PDOS of WSe 2 before and after doping in Figure 6. The PDOS of pure WSe 2 shows that Se-p and W-d orbitals mainly compose the valence band, while the conduction band is mainly composed of W-d orbitals. The dipole transition occurs only between Se-p and W-d orbitals. In addition, the PDOS shows the electronic hybridization behavior among the atoms. Te-doping shows a slight increase in Te-p orbital in both the valence and conduction bands but does not significantly change the total distribution of p orbital compared to pure WSe 2 . In contrast, Cr-doping significantly increases the d-orbital in the conduction band, leading to enhanced dipole transitions and hybridization of orbitals. The change in PDOS also affects the symmetry of partial charge density, as shown in Figure 7. Cr-doping reduces the symmetry of the electron wave function at the W site, particularly in the CBM, reinforcing the dipole transition and boosting the optical response of WSe 2 in the NIR-II region. Moreover, among all foreign-atom doping configurations, Cr-doping has the lowest formation energy (Figure 1l), suggesting that it is the most practical method to make monolayer WSe 2 useful in NIR-II. To further prove this, Figure S7 (Supporting Information) shows optical properties of WSe 2 at higher Cr atom concentrations, which all exhibit PL peaks (1146, 1257, and 1305 nm, respectively) in the NIR-II region. Therefore, a relatively high concentration of Cr atoms is predicted to improve the emission in the NIR-II window.

Vacancy Structure
Besides of foreign-atom doping, we also investigate the effect of introducing vacancies on the bandgap of WSe 2 . Different configurations of monolayer WSe 2 with varying numbers of vacancies are presented in Figure 8. Judging from the formation energy (Figure 8j), V1Se is the most stable configuration among all the vacancy configurations under studied. The results show that Se vacancy is in general easier to form than W vacancy, and the stability of the system decreases as the number of vacancies increases.
As shown in Figure 9, the band structures and pDOS's are calculated to test the effect of Se-vacancy on the electronic structure of the system. The different Se-vacancy defects induce var-ious energy levels that are contributed by W-d orbital and Sep orbital in the bandgap, which leads to a significant reduction in the bandgap compared to pure WSe 2 monolayer. This finding is consistent with previous studies that have also shown a similar reduction in bandgap due to vacancy defects. For instance, Luo et al. investigated the structural, electronic, and magnetic properties of defected monolayer WSe 2 with vacancies and observed the defected WSe 2 could have energy gap of 1.18 eV. [53]   In the presence of Se-vacancy, the imaginary part of the dielectric function (Figure 10a) exhibits a transition peak at energy lower than 1.5 eV, which is invisible for the pure WSe 2 monolayer. Regarding the application of NIR-II, configurations V2Se-2 and V2Se-4 are the best, which show absorbance peaks (Figure 10b) are at 943 and 1276 nm, respectively, and emission peaks (Figure 10c) at 1200 and 1623 nm, respectively. However, these two configurations are not ideal during synthesis as their formation energies are not less than the others. Nonetheless, there is still a possibility of NIR-II luminescence due to Se vacancies, albeit with lower quantum yield.
The band structures and pDOS's of monolayer WSe 2 with different W-vacancy configurations are given in Figure S8 (Supporting Information). The presence of W-vacancy transforms it into a metal, where the Fermi surface is mainly contributed by the Se-p and W-d orbitals. Our results are in good agreement with the work of Tang et al., who also observed the formation of impurity states in WSe 2 with W-vacancies, some of which cross the Fermi level, thereby causing the system to become metallic. [54] The imaginary part of the dielectric function and the absorption spectra of the different configurations of W-vacancy are also shown in Figure S9 (Supporting Information). As the number of W-vacancies increases, the peak of the imaginary part of the dielectric function at 2.60 eV becomes weaker.

Conclusions
To conclude, we used the first-principle method to investigate the electronic structures and the optical properties, including the photoluminescent spectra of defected WSe 2 . The results indicate that Cr-doping is the most effective method for enhancing the NIR-II response of WSe 2 due to the increased contribution of dorbitals in the conduction band, leading to a significant reduction of the bandgap. Se-vacancy is also a viable option for modifying the NIR-II response, although without foreign atom doping. These results suggest that searching for dopant with more d electrons may be a promising direction for NIR-II modification of MX 2 materials.

Computational Method
The calculations in this study were performed using the VASP code. [55][56] The generalized gradient approximation (GGA) in the scheme of projector augmented wave PBE has been used to approximate the exchange-correlation functional. [57] The projector augmented wave (PAW) [58][59] method was used with the valenceelectron configuration of 4s 2 4p 4 and 5p 6 6s 2 5d 4 for Se and W atoms, respectively. The vdW correction (DFT-D3) was included. The HSE06 [60] hybrid functional is also employed to verify the impact of HF exchange on the bandgap and optical properties. In the HSE calculations, the mixing parameter is set to the standard value of = 0.25 and the HF screening parameter μ is set to 0.2 Å −1 . A cut-off energy of 400 eV was used, and a supercell size of 4 × 4 in the x-y plane was used for the monolayer WSe 2 , with a vacuum layer of 15 Å in the z direction to prevent interlayer interac-tions. A Γ-centered 4× 4 × 1 and a Γ-centered 6×6×1 k-point grids were used for geometry optimization and static electronic structure calculations, respectively. [61] Energy convergence and force convergence threshold of 10 −6 eV and 0.02 eV Å −1 , respectively, were used. The geometry optimization allowed for changes in ionic positions, cell volume, and cell shape. Gaussian smearing with the width of 0.05 was used for the monolayer WSe 2 in the self-consistent calculation. Finally, a small complex shift of 0.1 was used in the Kramers-Kronig transformation when calculating the dielectric function.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.