Strain-Tuned Structural, Mechanical and Electronic Properties of Two-Dimensional Transition Metal Suldes ZrS2: A First Principles Study1

Two-dimensional semiconductor material zirconium disulfide (ZrS 2 ) monolayer is a new promising material with good prospects for nanoscale applications. Recently, a new zirconium disulfide (ZrS 2 ) monolayer with a space group of 59_ Pmmn has been successfully predicted. Using first-principles calculations, this new monolayer ZrS 2 structure is obtained with stable indirect band gaps of 0.65 eV and 1.46 eV at the DFT-PBE (HSE06) functional levels, respectively. Strain engineering studies on ZrS 2 monolayer show effective band gap modulation. The bandgap shows a linear regularity from narrow to wide under applied stresses (strain ranged from -6% to +8%). Young's modulus of elasticity of ZrS 2 rectangular cells along the tensile directions ( x -axis and y -axis) is 83.63 (N/m) and 63.61 (N/m) with Poisson's ratios of 0.09 and 0.07, respectively. The results of carrier mobility show that the electron mobility along the y axis can reach 1.32×10 3 cm 2 V -1 s -1 . Besides, the order of magnitude of the light absorption coefficient in the ultraviolet spectral region is calculated to reach 2.0×10 5 cm 1 for ZrS 2 monolayers. Moreover, by regulating the bandgap under stress, some bandgaps of the stretched energy band exceed the free energy of 1.23 eV and possess a suitable energy band edge position. The results indicates that the new two-dimensional Pmmn -ZrS 2 monolayer is a potential material for photovoltaic devices and photocatalytic water decomposition. meet the conditions of hydrolytic photocatalysis at pH=0 and pH=7. This study shows that the new two-dimensional Pmmn -ZrS 2 monolayer is a potential material for photovoltaic devices and photocatalytic hydrolysis. a Two-dimensional Auxetic Material with Ultralow Lattice Thermal Conductivity and Ultrahigh Hole Mobility.


INTRODUCTION
Since the preparation of graphene [1] by laboratory exfoliation in 2004, twodimensional materials have become a popular research area in the last decade or so.
Currently, two-dimensional materials have great research potential in the direction of electronic components due to their inherent lightweight (low-dimensional) and highperformance (quantitative) characteristics. Besides, low-dimensional materials such as the hexagonal structure of graphene [2,3] or one-dimensional carbon nanotubes have shown excellent performance in electronics, mechanics, and optics, prompting more indepth research [4]. Of course, the two-dimensional materials themselves have some defects, such as the inherent zero bandgap of graphene, which greatly limit their practical application value. Therefore, the search for new 2D materials toward more directions and the analysis of their electronic, mechanical as well as optical properties create possibilities for improving the performance of 2D materials [5][6][7][8][9].
Research on transition metal disulfides began with bulk materials [10] in the fields of electronics, optics, mechanics, chemistry, and catalysis. In recent years, the direct bandgap of the material's elemental family, unique electronic structure, and high species diversity have led to increasing investment in related research [11][12][13][14][15]. Two-dimensional Transition-Metal Dichalcogenides (TMDCs) have a chemical structure in the form of MX2, forming a monolayer compound with multiple atomic layers, known as a pincer or sandwich structure. Most of these compounds have semiconductor properties. In recent years, typical two-dimensional semiconductor materials based on TMD-based monolayers, such as MoX2 [16,17] and WX2 [18], have shown great potential for applications in optoelectronic devices due to their inherent wide bandgap and excellent electronic properties.
In 2011, an experimental synthesis of a new two dimensional TMD, the ZrS2 monolayer has been reported by Zeng et al. [9] In 2014, Li et al. [19] systematically explored the elastic, electronic and optical properties of a D3d point-group symmetric ZrS2 monolayer using properties of the density functional theory. They showed that 2D ZrS2 is an indirect semiconductor material indicates a band gap of 1.12eV(1.93eV) calculated by DFT-PBE (HSE06) method which can be modified by imposing an external strain. Recently, a new two-dimensional orthorhombic structure of monolayer ZrS2 [20] with a space group of 59_Pmmn was theoretically proposed and researchers calculated this new two-dimensional structure with a tunable indirect band gap, which we will subsequently investigated in more depth.
In this work, we systematically investigated the electronic, mechanical and optical properties of the new 2D orthorhombic structure of ZrS2 monolayer with a space group of 59_Pmmn by first principles calculations, in which the carrier mobility of ZrS2 monolayers is further calculated by strain regulation. An applied strain offers a novel way of modifying the band gaps of ZrS2 monolayer over a wide range. The energy band (PBE and HSE06 calculations) under the stress of the deformation from -6% to +8% were obtained by strain engineering to fit and calculate the monolayer ZrS2 transformation situation, and then the carrier mobility of electrons and holes in different directions (x-axis and y-axis) were further obtained by using the elastic constants. The optical absorption coefficients were proposed, and the energy values at the edges of the energy band after subtracting the vacuum energy level were larger than the water oxidation, which predicts that monolayer ZrS2 has a good potential application for photocatalytic water decomposition.

METHODS
The First-principle calculations are performed using the plane wave technique in the Vienna ab initio simulation Package (VASP) [21]. The ion-electron interactions are described by the projector-augmented-wave (PAW) pseudopotentials [22,23]. The electron exchange-correlation function is treated using the generalized gradient approximation (GGA) in the form proposed by Perdew, Burke, and Ernzehof (PBE) [21,24]. In all calculations, the energy cutoff for the plane wave is set to 550 eV. The convergence threshold is set to 10 -6 eV of energy. When the magnitude of the force acting on each atom in the cell is reduced to less than 0.001 eV⁄Å, we consider that the atoms in the cell have reached a position of complete relaxation. A thicker 20 Å vacuum layer was added along the z-direction to avoid the effect of interactions with neighboring layers. A 17 × 17 × 1 Monkhorst-Pack k-point grid is used for geometric optimization and calculation of electronic properties. Fig.1 The atomic structure of ZrS2, Top (a) and side (b) views of the unstressed optimized cell structure. Blue and yellow balls represents Zr and S atoms, respectively. Table 1 The calculated lattice parameters (lattice constant a, b; bond length for Zr-S; Atomic layer S-S and Zr-S distance; angles of Zr-S-Zr, S-Zr-S).

Geometric construction
∠Zr-S-Zr ∠S-Zr-S The optimization results of the monolayer Pmmn-ZrS2 structure are shown in Fig.   1 and Table 1. The geometric structure of the form top and side views of ZrS2 is presented in Fig.1(a) and (b). As shown in Figure 1, in the ZrS2 monolayer, the zirconium and sulfur atoms are arranged with a pleated monolayer structure each Zr atom forms four Zr-S bonds with the four adjacent S atoms. In the top view, ZrS2 shows a graphite-like layered structure with zirconium and sulfur atoms forming a fastened hexagonal network. While from the side view, the layered structure follows the S-Zr-S sandwich. Zirconium atoms are not aligned neatly but are arranged in a wavy layer on the middle atomic plane.
As shown in Table 1  respectively) [20]. And the valence band maximum (VBM) and conduction band minimum (CBM) located between the high symmetry points Y and Γ.

Energy band and density of states of unstressed ZrS2
As shown in Figure 2, The conduction band minimum (CBM) is mainly from the contribution of Zr atoms, while the valance band maximum (VBM) mainly comes from the joint contribution of Zr atoms and S atoms. It is clear from the partial density of states (PDOS) calculations that the CBM is mainly dominated by the d orbitals of the Zr atom, with contributions from other sub-orbital and various sub-orbital of the S atom.
In contrast, the VBM is mainly dominated by the d orbital of the Zr atom and the p orbital of the S atom after hybridization together, with the other orbitals of the Zr and S atoms occupying relatively small contributions. Table 2 The values of VBM, CBM and bandgap by PBE and HSE06 methods at each strain case.

Tuning electronic properties by strain
Because of the efficiency and convenience of applying stress to design and tune the electronic structure properties of two-dimensional materials [25], we further investigated the effect of strain on the band structure of ZrS2 monolayer. We selected   Table 2 and Fig.3, for uniaxial tension (x or y) and biaxial tension, the band gap of the ZrS2 monolayer decreases or increases when the ZrS2 monolayer is subjected to compressive or tensile strain. The compression or stretching of uniaxial tension (xaxis) has more influence on the bandgap than that of uniaxial tension (y-axis). The effect of biaxial compression or stretching on bandgap is similar to that of x-axis compression or stretching.
For the compression or stretching of uniaxial tension (x-axis), the reduction of the bandgap is accelerated as the compressive stress increases. Compared with other materials, such as silicene [26], the bandgap of ZrS2 monolayer is more sensitive to changes in compressive stress, and the bandgap of ZrS2 monolayer decreases faster when the compressive stress reaches 6%. When the compressive stress reaches 6%, the bandgap of the ZrS2 monolayer decreases by about 0.5 eV. While the tensile stress and its band gap almost have a linear trend, and the bandgap becomes wider and wider with the increase of the tensile stress.

Fig.3
Effect of strain engineering on the bandgap of ZrS2 monolayer. Fig.3(a) shows the bandgaps for uniaxial stretching of the x-axis; Fig.3(b) shows the bandgaps for uniaxial stretching of the y-axis; Fig.3(c) shows the bandgaps for biaxial stretching of the ZrS2 monolayer. Based on the elastic solid theory [27], we then calculate the elastic constants of ZrS2 monolayers. For two-dimensional materials, using the standard Voigt notation [28], the strain energy U per unit area is denoted as [29] (ε) = need to satisfy the Born-Huang criteria (C11C22-C12 2 >0 and C66>0) [30]. All the mechanical parameters listed in Table 3 indicate that the ZrS2 monolayer is mechanically stable. The strain energy of ZrS2 monolayer under uniaxial, shear and biaxial in-plane strain are shown in Fig.S2 in Supporting Information. Also, Young's modulus of ZrS2 is relatively small compared to graphene (342 N/m) [28] and MoS2 (330 N/m) [33], indicating that the ZrS2 monolayer is more stretchable and flexible, which is important for applications in the direction of electronic devices and flexible materials. This means that when the material is subjected to a certain degree of tensile deformation, the resistance to deformation along the axis is much weaker than in other directions, especially in the directions of 45°, 135°, 225° and 315° from the axis, where the resistance to deformation is the strongest. The Poisson's ratio of ZrS2 monolayer is somewhat smaller compared to MoS2 monolayer with ν =0.2 [34], which implies a higher potential for the application of ZrS2 monolayer than MoS2 in strain engineering applications. Graphically, the Poisson's ratio in all directions of the ZrS2 monolayer resembles a four-leaf clover, similar to the butterfly-shaped Poisson's ratio of the Penta-X2C family the previous report [35].

Carrier mobility
Carrier mobility is usually referred to as the overall movement of electrons and holes within a semiconductor and is an important physical measure of the performance of semiconductor devices [36][37][38][39][40]. Based on the deformation potential theory proposed by Bardeen and Shockley[41], the carrier mobility of two-dimensional materials can be calculated according to the following equation: Where C2D is the modulus of elasticity of a uniformly deformed crystal, for ZrS2 monolayer, the calculation results are shown in Fig.S3 in Supporting Information.
Where m * is the effective mass in the direction of transmission, T is the temperature, and kB is the Boltzmann constant.   Although compared with the maximum carrier mobilities of 4×10 5 cm 2 V -1 s -1 for graphene and 200 cm 2 V -1 s -1 for MoS2, the monolayer ZrS2 has lower carrier mobility than graphene but it still has better carrier mobility. What's more important is that monolayer ZrS2 has the same characteristics as monolayer PtS2 and PtSe2. Large differences in the values of electron and hole carrier mobilities in different directions, which will lead to the rapid migration of photogenerated electrons and holes. And the numerical difference between the electron and hole carrier mobilities indicates the anisotropy of hole and electron carriers, which facilitates a more efficient separation of electron-hole pairs of electrons. The difference in mobility between electrons and holes allows the monolayer ZrS2 material can to be used for photocatalytic hydrolysis for a long time, maintaining the photocatalytic activity.

Absorption coefficient
From a practical application point of view, we are very concerned about the optical performance of the ZrS2 monolayer. We have calculated absorption properties based on the dielectric function ε( ) = 1 ( ) + 2 ( ) , where is the frequency. The absorption coefficient α( ) was calculated using: [42] α( ) = √2 (√ 1 2 ( ) + 2 2 ( ) − 1 ( )) 1 2 ⁄ where 1 is the real part of the complex dielectric function, which could be obtained from 2 using the Kramer-Kronig relationship. 2 is defined as: [43] 2 ( ) = 4 2 2 Ω lim where α and refer to the x and y directions, and Ω is the volume of the unit cell. The indices c and v refer to the conduction and valence band states, respectively.
corresponds to an eigenstate with wave vector k. x-axis can reach more than 2.0×10 5 cm -1 , and the light absorption coefficient along the y-axis can also reach more than 1.25×10 5 cm -1 . Compared with the visible region, the ZrS2 monolayer performs well in the UV region, which indicates that ZrS2 monolayer may be a promising material for photovoltaic applications. Fig.6 The position of the band edge of the ZrS2 monolayer influenced by strain stretching.

Band edge positions of ZrS2 monolayer
As we know, the most important condition for a two-dimensional semiconductor material to be used for photocatalytic hydrolysis [44][45][46] is to have a bandgap exceeding a certain forbidden bandwidth, the theoretical value of this bandgap must be greater than the electrolytic voltage of water, 1.23 eV. In practice, the 2D semiconductor should be able to withstand a voltage greater than or equal to 1.6eV to ensure that the photocatalytic reaction can be carried out in the visible light range. To ensure that the reduction and oxidation of water can occur, the conduction band should lie at a lower potential than the reduction potential of water (at 0V NHE pH=0) and the valence band should be higher than the oxidation potential of water.
As shown in Fig.6 The results show that the elastic constants of the ZrS2 monolayer is relatively smaller than those of other transition metal sulfide groups, the smaller Young's modulus and Poisson's ratio indicate that ZrS2 monolayer is a relatively soft two-dimensional material that can be used as an ideal material for sensors. By calculating the carrier mobility of the ZrS2 monolayer, we are pleased to find that the electron and hole mobilities in the x and y axis are numerically different and have more pronounced differences. The value of carrier mobility along the y direction reaches 1320 cm 2 V -1 s -1 . The high carrier mobility indicates that the ZrS2 monolayer is valuable for making photovoltaic materials. By calculating the light absorption coefficients in different directions, the absorption value of the ZrS2 monolayer in the UV region reaches 2.5 × 10 5 . In addition, the strain modulation makes it easier for the ZrS2 monolayer to meet the conditions of hydrolytic photocatalysis at pH=0 and pH=7. This study shows that the new two-dimensional Pmmn-ZrS2 monolayer is a potential material for photovoltaic devices and photocatalytic hydrolysis.

Conflicts of interest
There are no conflicts of interest to declare.

Code Availability
All data, models, and code generated or used during the study appear in the submitted article.

Authors Contribution Statement
Qi Song performed the data computation, analysis and wrote the first draft of manuscript; Xin Liu contributed to the data analysis and Program Testing; Hui Wang contributed to the topic selection, data analysis and manuscript revision; Xiaoting Wang helped to the manuscript preparation; Yuxiang Ni helped perform the Program testing; Hongyan Wang helped perform the constructive discussions.