# Monolayer TiNI with Anisotropic Optical and Mechanical Properties

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}. The calculate result of electrical transport properties also indicates the anisotropic electron transport performance of TiNI monolayer. Moreover, the electron transport intensity along the direction b is about six times the conduction intensity along the direction a. The anisotropic mechanical and optical properties, as well as the tunable band gap and special electron transport characteristics, enable a promising future for monolayer TiNI materials in nano-optoelectronics.

## 1. Introduction

_{4}N, Ni

_{3}N, Ni

_{3}FeN, etc. [29,30,31,32]. The related research of TMNs on the first-principles calculation of ternary transition metal nitrides needs further study.

## 2. Methods

^{−6}eV, correspondingly. Γ-centered Monkhorst-Pack k-sampling grids of 9 × 9 × 1 is adopted in the Brillouin zone. The positions of the atoms and the lattice constants are sufficiently optimized using the algorithm of conjugate gradients. In the other hand, a vacuum region of 20 Å is inserted in the z-axis direction, in order to prevent interlayer interactions along the z-axis direction. The conduction characteristics are calculated using the nonequilibrium Green’s function, using the atomically localized basis functions and implemented in OpenMX code [39,40]. For different kinds of elements, we use the PAO basis functions of Ti7.0-s2p2d1, N5.0-s2p2d1, and I7.0-s2p2d1, separately, and use the completely relativistic conformal pseudopotential and pseudo-atomic orbitals (PAOs) to expand the wave functions [41]. Our theoretical work can be obtained experimentally by epitaxial growth and other methods.

## 3. Results and Discussion

#### 3.1. Energy Band and Structural Stability

_{Ti}, E

_{N}, and E

_{I}are the individual Ti, N, and I atoms’ energies, respectively. In addition, E

_{Ti}

_{2N2I2}represents the overall energy of monolayer TiNI. After calculation, the cohesive energy of two-dimensional TiNI is 3.39 eV/Atom. The calculated results are similar to the published cohesion energies of MnNF and black phosphorus [44,45]. The higher the cohesion energy, the easier it is to synthesize in the experiment.

#### 3.2. Mechanical and Optical Properties

_{11}= 160 N·m

^{−1}, C

_{12}= 38 N·m

^{−1}, C

_{22}= 145 N·m

^{−1}and C

_{44}= 65 N·m

^{−1}respectively. The elastic constants of laminated TiNI satisfy the Born-Huang criterion (C

_{11}C

_{22}− C

_{12}

^{2}> 0, C

_{11}> 0, C

_{22}> 0, C

_{44}> 0) [47,48]. This also means that monolayer TiNI meets the equilibrium condition of mechanics, that is, it has mechanical stability. The elastic constants C

_{11}and C

_{22}of laminated TiNI are significantly greater than phosphonene with C

_{11}= 24 N·m

^{−1}and C

_{22}= 103 N·m

^{−1}[49]. On the other hand, the Young’s modulus Y(θ) and Poisson’s ratio υ(θ) along an arbitrary angle θ in the plane satisfy the following equations [47,50]:

_{11}C

_{22}− C

_{12}

^{2})/C

_{44}− 2C

_{12}] and [C

_{11}+ C

_{22}− (C

_{11}C

_{22}− C

_{12}

^{2})/C

_{44}], respectively. From the Figure 2a,b we can see that the laminated TiNI exhibits various anisotropic Poisson’s ratio υ(θ) and Young’s modulus Y(θ). This indicates that the layered TiNI has a strong mechanical anisotropy. This result is in agreement with the anisotropy of the structure of the material. The higher in-plane Young’s modulus represents the fact that the material possesses many atoms and mutual atomic interactions, which are effective in preventing the curling of the film. After calculation we calculated the Young’s modulus of the monolayer TiNI to be 130~160 N/m

^{2}, which is larger than 62 N/m

^{2}of silylene which has been successfully synthesized.

_{1}(ω) + iε

_{2}(ω). Alternatively, the real part of the dielectric function ɛ

_{1}(ω) can be acquired by converting the Kramer–Kronig equation. The real or imaginary components of the dielectric function can be accessed by the following equations:

_{CV}(k)|

^{2}is the element of the dynamics matrix, where C and V are abbreviations for conduction band and valence band, respectively. E

_{C}(k) and E

_{V}(k) are the energy at the CB and the VB, respectively. The equation for the coefficient of optical absorption is given by:

_{0})/a

_{0}, in which a and a

_{0}denote the lattice constants at the strain and equilibrium states, respectively. A variation of energy relations when imposing uniaxial or biaxial strain is shown in Figure 4. It can be seen that two-dimensional TiNI has the lowest energy in the eigenstate (at 0). In combination with the cohesion energy and phonon spectrum calculation discussed before, it can be seen that the eigenstate of the monolayer TiNI after the geometry optimization is the fundamental phase. From the Figure 1, Figure 2 and Figure 3, we can see that the band gap of the monolayer TiNI turns from direct into indirect when a compressive strain of −4% is applied in the a-direction. A band gap change from direct to indirect can also be achieved when stretching strain of 4% is applied in the b-direction along the 2D TiNI. Direct band gap can be transformed into indirect band gap by applying 8% biaxial tensile strain. The result shows that the energy band gap of single-layer TiNI is adjustable through imposing outer strain. This is attributed to the variation of Ti-I bond and Ti-N bond when the crystal structure is compressed or stretched. In addition, the hybridization of atomic orbitals is affected after strain is applied, resulting in the change of band gap of electronic band structure.

#### 3.3. Electronic Transport Properties

## 4. Conclusions

^{2}. The band gap can be regulated by applying −10~10% strain. At the same time, the current can reach 6 μA when the voltage is 2.0 V. Single-layer TiNI transmits six times more current along the b-direction than along the a-direction. Our theoretical study shows that two-dimensional TiNI has promising applications in anisotropic electronic and optical fields.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) Crystal structure diagram of two−dimensional TiNI. (

**b**) Phonon dispersion curves of two−dimensional TiNI. (

**c**) Electronic band diagram. (

**d**) DOS for two−dimensional TiNI.

**Figure 5.**(

**a**) The energy band of single−layer TiNI calculated by VASP shows on left side and the energy band of single−layer TiNI calculated by OpenMX shows on right side (

**b**) Above and below are schematic diagrams of conveying device in a-direction and b-direction, respectively. (

**c**) Voltammetric curves toward the a-direction and the b-direction, with illustrations of local charge densities near Fermi surfaces.

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**MDPI and ACS Style**

Li, S.-J.; Li, M.; Zhang, C.-G.; Shi, K.-Y.; Wang, P.-J.
Monolayer TiNI with Anisotropic Optical and Mechanical Properties. *Crystals* **2022**, *12*, 1202.
https://doi.org/10.3390/cryst12091202

**AMA Style**

Li S-J, Li M, Zhang C-G, Shi K-Y, Wang P-J.
Monolayer TiNI with Anisotropic Optical and Mechanical Properties. *Crystals*. 2022; 12(9):1202.
https://doi.org/10.3390/cryst12091202

**Chicago/Turabian Style**

Li, Shu-Juan, Min Li, Cheng-Gong Zhang, Kun-Yue Shi, and Pei-Ji Wang.
2022. "Monolayer TiNI with Anisotropic Optical and Mechanical Properties" *Crystals* 12, no. 9: 1202.
https://doi.org/10.3390/cryst12091202