Issue 
Eur. Phys. J. Appl. Phys.
Volume 94, Number 1, April 2021



Article Number  10301  
Number of page(s)  7  
Section  Thin Films  
DOI  https://doi.org/10.1051/epjap/2021210007  
Published online  15 April 2021 
https://doi.org/10.1051/epjap/2021210007
Regular Article
Electronic and optical properties of Janus monolayers MoXB_{2} (X=S, Se): firstprinciples prediction
^{1}
School of Science, Wuhan University of Technology, Wuhan 430070, P.R. China
^{2}
Research Center of Materials Genome Engineering, Wuhan University of Technology, Wuhan 430070, P.R. China
^{*} email: 16941612@qq.com
Received:
8
January
2021
Received in final form:
8
March
2021
Accepted:
18
March
2021
Published online: 15 April 2021
By means of comprehensive firstprinciples calculations, we studied the geometric structure, the stability and electronic properties of the new twodimensional (2D) Janus MoXB_{2} (X=S, Se) monolayers. Our calculations demonstrated that the predicted Janus MoXB_{2} monolayers are all stable semiconductors with direct band gap. In this paper, we focus on impacts upon the electronic and optical properties of the MoXB_{2} monolayers under the different biaxial strains. With the compressive stress increases, the MoXB_{2} monolayers would become indirect band gap semiconductors, and then behave as semimetal. While under tensile strain, MoXB_{2} still maintain direct band gap. In addition, the optical calculation shows that biaxial strain leads to blue shifts in the optical absorption and reflectivity. The result indicates that MoXB_{2} may be promised nano candidate materials in optoelectronic devices.
Note to the reader: Further to the publication of an erratum, the citation of the article was modified on 19 July 2021.
© EDP Sciences, 2021
1 Introduction
Since graphene was prepared, twodimensional atomic thickness materials have attracted wide attention in nanomaterials due to their excellent physical and chemical properties [1,2]. Graphene has excellent properties, such as high specific surface area, high conductivity, good thermal stability, excellent mechanical strength, adjustable electrical and superior optical and superficial properties. Therefore, 2D monolayer materials were considered significant in the design of new optoelectronic devices [3–12]. Due to graphene is a semimetal, the band gap is zero, the conduction band and the valence band intersect in the Brillouin region, so the band gap is difficult to open, which has led to many barriers in applying to electronic devices and the field of semiconductors. Therefore, the new twodimensional semiconductor materials with band gap are the focus of researchers. Recently, the Janus structures are particular interesting because they possess many new properties that could not be found in those of pristine materials [13–22].
The 2D materials are extremely sensitive to structural perfection and geometric symmetry plays a key role in determining their electronic properties. The breaking of mirror symmetry in Janus structures can result in many new properties [23,24]. In recent years, a variety of Janus monolayers have been predicted by breaking of mirror symmetry, such as MoSSe [25,26] monolayer, which the toplayer S atoms of MoS_{2} are fully replaced by Se atoms. Because the hexagonal boron layers in MoB_{4} monolayer [27] have been proved to be stable, based on density functional theory (DFT) and using a firstprinciples calculation in this framework, we designed the new 2D Janus material, namely the MoXB_{2} (X=S, Se) monolayers, which are direct band gap semiconductors. MoXB_{2} (X=S, Se) monolayers can be viewed as the complete replacement of the boron layer in the bottom layer of MoB_{4} monolayer by X atoms to break the mirror symmetry. Afterwards, we systematically investigated their stability and electronic properties. Interestingly, they have good dynamic and mechanical stabilities and tunable electronic properties as a result of exceptional structural characteristics. Thus, we studied the influence of appropriate biaxial strain on the electronic and optical properties of the MoXB_{2} monolayers. These calculated electronic and optical characteristics indicate that the MoXB_{2} monolayers are suitable for making optoelectronic devices.
2 Computational methods
Firstprinciples calculations were performed using the density functional method (DFT) implemented in the Vienna ab initio simulation package (VASP) [28–30]. For the exchange and correlation functional, we use the generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof (PBE) [31]. The cutoff energy is set to 500 eV. The convergence criteria for energy and force are set to 1 × 10^{−5} eV and 0.03 eV/Å, respectively. The vacuum space along the zdirection is set to 15 Å to minimize artificial interactions between neighboring layers. A MonkhorstPack mesh of 9 × 9 × 2 kpoints was used to sample the Brillouin zone of the unit cell for geometry optimizations and properties calculations [32]. The phonon calculations are carried out using Phonopy program implementing density functional perturbation theory (DFPT) method [33].
3 Results and discussion
3.1 Structures and stability of the MoXB_{2} (X=S, Se) monolayers
The optimized structures of MoXB_{2} (X=S, Se) monolayers can be regarded as a hexagonal boron lattice covered by a triangular X layer and a Mo atom under the center of the B hexagonal. Meanwhile, B atoms are arranged in a very slightly buckled honeycomb lattice. Within each unit cell [marked by the dashed line in Figure 1a], it contains two B atoms and one Mo atoms and one X atom. The MoXB_{2} monolayers can also be viewed as a sandwich that the Mo atom is between the X atom and the B atom. The structural parameters, buckling height and bond length of MoXB_{2} are given in Table 1. Because of the large radius of Se atom, the bond length of MoSe (2.51 Å) is longer than that of MoS (2.40 Å), and the Mo−B bond lengths (2.399 and 2.398 Å, respectively) in MoXB_{2} are similar to the Mo−B bond (2.42 Å) in MoB_{2} monolayer [34]. The bond length of BB are 1.75, 1.77 Å in MoXB_{2}, which are about the same as in the TiB_{2} (1.79 Å) [35]. It indicates that the binding ability of bonds are strong in the MoXB_{2}.
To confirm the structural stability, we calculated the cohesive energy , where E_{Mo}, E_{X}, E_{B} and are the total energies of a single Mo atom, individual X atom, an isolated B atom and one unitcell of the MoXB_{2} monolayers, respectively. The calculated values are 6.62 eV per atom, 6.42 eV per atom, respectively, which are larger than the FeB_{2} monolayer (4.87 eV/atom) [36]. The large cohesive energy suggests that the MoXB_{2} monolayers are strongly bonded network.
The dynamic stability of the MoXB_{2} (X=S, Se) monolayers are analyzed by calculating the phonon dispersion. Their phonon spectrum along the path (see Fig. 2) contains no unstable modes, showing that MoXB_{2} are dynamically stable. The highest frequency of the MoXB_{2} (X=S, Se) monolayers reach up to 824, 777 cm^{−1}, respectively. Even far higher than the highest frequency of 473 cm^{−1} in MoS_{2} [37], indicative of robust MoB, MoX and BB bonds in the MoXB_{2} (X=S, Se) monolayers.
To further determine the mechanical stability of the structures of MoXB_{2} (X=S, Se) monolayers, the investigation of mechanical properties began by examining the elastic constants. C_{11}, C_{22}, C_{12}, and C_{66} are the components of the elastic modulus tensor. We first performed calculations for MoSB_{2}, the results are C_{11} = C_{22} = 251.63 N/m, C_{12} = C_{21} = 38.32 N/m and C_{66} = 106.65 N/m. For MoSeB_{2}, the elastic constants are calculated to be C_{11} = C_{22} = 238.42 N/m, C_{12} = C_{21} = 34.47 N/m and C_{66} =101.97 N/m. Based on these tensor components, the isotropic 2D Young's modulus (inplane stiffness) can be estimated by calculating (C_{11}^{2} − C_{12}^{2})/C_{11}, which gives 245.79 and 233.44 N/m for MoSB_{2} and MoSeB_{2}, respectively. Thus, the inplane stiffness of MoXB_{2} are greatly enhanced compared with that of phosphorene (54 N/m) [38], while smaller than that of graphene (344.2 N/m) and hBN (275.8 N/m) [39]. Additionally, the Poisson's ration υ = C_{12}/C_{11} = 0.13,0.14 for MoSB_{2} and MoSeB_{2}, respectively, which are smaller than that of graphene (0.17) and hBN (0.22). It would only slightly shrink in the y direction when MoXB_{2} was stretched in the X direction. All these demonstrate that the MoXB_{2} monolayers have good mechanical properties and could have potential applications in ultrathin highstrength mechanical materials. Notably, the calculated elastic constants of the MoXB_{2} monolayers satisfy the conditions C_{11}C_{22}–C_{12}^{2} > 0 and C_{66} > 0, in agreement with the Born–Huang mechanical stability criteria.
Fig. 1 The structure of MoXB_{2}. (a) Top and (b) side views of relaxed structure. 
Summary of the structural parameters for the MoXB_{2} (X=S, Se) Monolayers. These symbols a, Δ, d1, d2, d3 are the lattice constants, buckling height of boron, bond length of BB, bond length of MoB, bond length of MoX, respectively.
Fig. 2 Phonon dispersion curves of the (a) MoSB_{2} and (b) MoSeB_{2} monolayers. 
3.2 Electronic properties of the MoXB_{2} (X=S, Se) monolayers
The band structure and projected density of states of MoXB_{2} (X=S, Se) monolayers are shown in Figure 3. It is not difficult to analyze from the band structure diagram that MoXB_{2} are direct band gap semiconductors. The band gaps of MoSB_{2} and MoSeB_{2} are 0.769, 0.441 eV, respectively. In addition, the direct band gap at the middle of the ГK, and the valence band is tangent to the Fermi energy level. One of the main factors that determine the electronic properties is the energy distribution of conduction band electrons and valence band electrons, the result can be expressed as a curve that the relationship between projected density of state (PDOS) and energy [3]. The curve indicates that the conduction band near the Fermi level mainly receive contributions from the d orbitals of Mo, and the valence band near the Fermi level mainly attributes the contributions to the p orbitals of B. Lack of inversion symmetry of MoXB_{2} results in splitting of spin degenerate bands due to spin−orbit coupling (see Fig. 4). When the SOC effect is included, the top of the valence band of MoSB_{2} has a significant decrease, and the details of energy bands near the Fermi level of MoSeB_{2} have little influence.
Fig. 3 Electronic band structure and projected density of states (PDOS) for (a) MoSB_{2} and (b) MoSeB_{2}. 
Fig. 4 (a) MoSB_{2} and (b) MoSeB_{2} band structure with spinorbit coupling (SOC) considered. 
4 Strain effect on electronic and optical properties
Analysis of the strain effect on the electronic characteristics of materials is an effective tool for designing electronic structures using 2D materials [40–42]. We applied the biaxial strain by fixing the lattice constant and input a series of acceptable values, which are smaller or larger than the optimized structures of MoXB_{2}. In our computation, the strain value is referred to the variation of the lattice constants under stress and is defined as ε = (l − l_{0})/l_{0},where l_{0} and l are lattice constants of the unstrained and strained, respectively [43]. The MoXB_{2} are direct band gap semiconductors which valence band maximum (VBM) and conduction band minimum (CBM) are between the highsymmetry point Γ and K. Upon compressive strains from 0% to −10% (see Fig. 5), it is evident that increasing the strain value causes the band gap to decrease under compressive strain conditions, the CBM moves gradually to the highsymmetry point Γ, and the CBM moves down linearly. Moreover, the curvature of the band structure increases under compressive strains. Under the strain of −6%, the CBM locates at the highsymmetry point Γ which indicates the characters of the MoXB_{2} monolayers transform from the direct band gap semiconductor to the indirect band gap semiconductor. Remarkably, semiconductorsemimetal transformation occurs in the MoSB_{2} monolayer upon the strain of −10%, which appears the electronhole compensation similar to MoTeB_{2} monolayer [44]. Meanwhile, under the strain of −8%, the CBM and VBM approach and cross the Fermi surface, which also lead to the characters transformation of MoSeB_{2} from semiconductor to semimetal.
In contrast, in the case of the tensile strain, it indicates from Figure 6 that the band gap of MoXB_{2} increase firstly and then decrease with the increase of tensile strains, while they are still direct band gap semiconductors below the strain of 8%. Meanwhile, under the strain of 12%, MoSeB_{2} remains direct band gap as before. Different from compression strain, in the case of the tensile strain, not only the top of valence band will move to the point of high symmetry, but also the bottom of conduction band will move to the same point of high symmetry with the increase of strain. Furthermore, VBM keeps tangent to the Fermi energy level under the limited strain (see Fig. 7).
We next investigated the optical spectra (the parallel polarization light along the aaxis) of the MoXB_{2} monolayers under the biaxial strain. As is known, the basic optical characteristics of materials are expressed by the dielectric function, which is a complex function and defined by ε(ω) = ε_{1}(ω) + іε_{2}(ω). For analyzing the complex optical characteristics of the considered material, the Kramer–Kronig transforms allows us to determine the real part of the optical dielectric function by the help of its imaginary part that can be obtained by the sum of the filled–unfilled transitions [45]. All the other parameters of the optical spectra can be derived from the dielectric function, such as absorption, reflectivity, refractive index, and energy loss function. The reflectivity coefficient can be defined as follows:where n and k are the complex refractive index, they are derived by
The absorption coefficient α(ω) is obtained in terms of the real and imaginary parts of the complex dielectric function as follows:
Here, the corresponding diagrams for the MoXB_{2} monolayers under the presence of the biaxial strain are illustrated in Figure 8. The optical absorption coefficients of the MoXB_{2} monolayers are activated at the incoming energy of about 0.38, 0.70 eV, respectively, which are close to the value of their direct band gap. From the detailed view of Figure 7, a blue shift can be found in the absorption spectrum with the increase of the biaxial compressive strain, while in the reflectivity spectrum, a blue shift occurs with the raise of the biaxial tensile strain. Moreover, MoXB_{2} monolayers exhibit high absorption and reflectivity in the visible region of the electromagnetic spectrum. These optical behaviors suggest that the MoXB_{2} monolayers have potential applications in optical devices.
Fig. 5 Band structures of (a) MoSB_{2} and (b) MoSeB_{2} with biaxial compressive strain from −10% to −2%. 
Fig. 6 Band structures of all biaxial tensile strain states of (a) MoSB_{2} and (b) MoSeB_{2}. 
Fig. 7 Variation of the strain band gap for (a) MoSB_{2} and (b) MoSeB_{2} monolayers. 
Fig. 8 The optical absorption and reflectivity of the (a) MoSB_{2} and (b) MoSeB_{2} monolayers under the biaxial strain, pink represents visible light region. 
5 Conclusions
In summary, we designed two kinds of Janus monolayers: MoSB_{2} and MoSeB_{2}, and systematically examined their geometric structure, stability, electronic properties, optical properties and the strain effect by using a firstprinciples calculation in the framework of DFT. According to the calculations, the MoXB_{2} monolayers have excellent dynamic and mechanical stabilities that proved by phonon mode analysis and mechanical properties calculations. Investigation illustrated that the electronic and optical properties of the MoXB_{2} monolayers are sensitive to the biaxial strain. And the direct band gap semiconductorindirect band gap semiconductorsemimetal transitions occurs when the biaxial compressive strain is applied. Also, the optical absorption and reflectivity of the MoXB_{2} monolayers exhibit a blue shift under the biaxial compressive strain or the biaxial tensile strain. Therefore, the MoXB_{2} monolayers may be suitable candidate materials for designing nano optoelectronic devices.
Author contribution statement
All authors analyzed and discussed the results, and contributed to the final manuscript. Qingwen Lan carried out the search for structure and property calculation, and completed the preparation of the paper. Changpeng Chen provided research direction and proofread the paper. Tian Qin discussed the method of property calculation.
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Cite this article as: Qingwen Lan, Changpeng Chen, Tian Qin, Electronic and optical properties of Janus monolayers MoXB_{2} (X=S, Se): firstprinciples prediction, Eur. Phys. J. Appl. Phys. 94, 10301 (2021)
All Tables
Summary of the structural parameters for the MoXB_{2} (X=S, Se) Monolayers. These symbols a, Δ, d1, d2, d3 are the lattice constants, buckling height of boron, bond length of BB, bond length of MoB, bond length of MoX, respectively.
All Figures
Fig. 1 The structure of MoXB_{2}. (a) Top and (b) side views of relaxed structure. 

In the text 
Fig. 2 Phonon dispersion curves of the (a) MoSB_{2} and (b) MoSeB_{2} monolayers. 

In the text 
Fig. 3 Electronic band structure and projected density of states (PDOS) for (a) MoSB_{2} and (b) MoSeB_{2}. 

In the text 
Fig. 4 (a) MoSB_{2} and (b) MoSeB_{2} band structure with spinorbit coupling (SOC) considered. 

In the text 
Fig. 5 Band structures of (a) MoSB_{2} and (b) MoSeB_{2} with biaxial compressive strain from −10% to −2%. 

In the text 
Fig. 6 Band structures of all biaxial tensile strain states of (a) MoSB_{2} and (b) MoSeB_{2}. 

In the text 
Fig. 7 Variation of the strain band gap for (a) MoSB_{2} and (b) MoSeB_{2} monolayers. 

In the text 
Fig. 8 The optical absorption and reflectivity of the (a) MoSB_{2} and (b) MoSeB_{2} monolayers under the biaxial strain, pink represents visible light region. 

In the text 
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