First-Principles Prediction of Structure and Properties of the Cu₂TeO₆ Monolayer

Abstract

In this work, first-principles calculations have been utilized to predict the existence of a new Cu₂TeO₆ monolayer. It is shown that the predicted material is dynamically and thermally stable. The Cu₂TeO₆ monolayer is also found to be a narrow band gap semiconductor with a band gap size of 0.20 eV. Considering the obtained properties of the Cu₂TeO₆ monolayer, it is proposed for applications in various nanodevices in electronics and straintronics.

1. Introduction

Two dimensional (2D) materials are atomically thin sheets that exhibit unique properties with lots of potential for technological applications, and they possess an abundance of applications which remain unexplored by fundamental science. Two dimensional materials also serve as building blocks for a variety of the next generation layered and composite materials [1,2]. Since the isolation of the first 2D material, graphene, from a bulk graphite, thousands of 2D materials have been developed. For example, graphene-like 2D materials such as the boron nitride (h-BN) monolayer [3,4] and transition metal dichalcogenides (TMDs), e.g., MoS₂, WS₂, MoSe₂, and WSe₂ [5,6], MXenes [7], mono-layered silicon carbides (Si_xC_y) [8], and monochalcogenides (GaSe) [9], etc.

The reliable synthesis of 2D materials is an important first step towards characterizing the size-related changes in their properties, which provides ways to integrate them into many applications [10]. Synthesis strategies can be divided into top-down and bottom-up methods. In the bottom-up approach, nano-scale materials are constructed from atomic or molecular precursors which grow into complex structures [11,12]. On the other hand, the top-down approach carves nano-scale structures by the removal of materials from bulk solids using various exfoliation methods [13,14,15]. Two dimensional materials synthesized using the methods described above have many unusual properties. For example, MoS₂ has a tunable bandgap, and it can be used as a hole transport layer in solar cells [16], and mono-layered titanium oxide (TiO₂) can be used to enhance the dielectric properties of polyvinylidene fluoride [17], etc.

However, it is difficult to carry out experimental studies on such thin materials due to experimental constraints, as the experiments are costly and time consuming. This means that theoretical and computational studies can be considered as an integral part of the research on 2D materials. At the moment, there are many theoretical methods for studying 2D materials such as machine learning-based approaches [18], first-principles calculations based on the density function theory [19], the Monte Carlo method [20], and finite element simulations [21], etc. More specifically, first-principles calculations have been found to be a powerful tool for predicting the structures and properties of materials due to their consistency with the experimental results. Using first-principles calculations, the structure and properties of various 2D materials such as the novel Dirac material TiZrB₄, with its negative Poisson ratio [22], V₂O₃, with its quantum anomalous Hall effect [23], and high temperature ferromagnetic Co₂Ti₂Sn₂ [24] and BiXenes [25] have been predicted. It is also notable that many 2D materials, in particular, hematene [26], Fe₂O₃ [27] and FeTiO₃ [28], have been synthesized based on first-principles simulations results.

In this work, a new material, the Cu₂TeO₆ monolayer, which is 2D analog of bulk Cu₂TeO₆ [29], is predicted using first-principles calculations. Its dynamic and thermal stability is studied. The comprehensive analysis on the properties of the predicted Cu₂TeO₆ monolayer is also conducted. Particularly, its electronic and mechanical properties are investigated.

2. Materials and Methods

The unit cell structure of the Cu₂TeO₆ monolayer was designed based on the geometry of primitive unit cell of a bulk Cu₂TeO₆, which was available in the Materials Project database (mp-1188594) [30]. The structural and thermodynamic stabilities of the optimized unit cell were defined based on the phonon dispersion spectra using the Phonopy code [31] and ab initio molecular dynamics (AIMD) calculations [32]. Calculations were performed using the plane-wave method and implemented in the Vienna Ab initio Simulation Package [33]. The calculations were performed using the Perdew−Burke−Ernzerhof (PBE) exchange−correlation functional under the generalized gradient approximation (GGA) [34]. The optimization was stopped once all of the components of all of the atomic forces were smaller than 10⁻⁴ eV/Å and the change in total energy was less than 10⁻⁸ eV. The first Brillouin zone was sampled with a 12 × 12 × 1 k-mesh grid, and the kinetic energy cut-off of 520 eV was selected. The periodic boundary conditions were applied for the two in-plane transverse directions, while a vacuum space of 20 Å was introduced to the direction that was perpendicular to the surface plane. The methodology for the calculation of the mechanical properties can be found in previous works [35,36].

3. Results

The atomic structure of the predicted Cu₂TeO₆ monolayer is shown in Figure 1a. It has a monoclinic unit cell, similarly to its bulk counterpart. Structural parameters of the predicted Cu₂TeO₆ monolayer can be found in

Table 1. Structural parameters of the Cu₂TeO₆ monolayer.

a	5.272 Å
b	5.272 Å
α	100.25
β	100.25
γ	112.43

. Firstly, to show a kinetic stability of the predicted Cu₂TeO₆ monolayer, the phonon dispersion spectra along the high symmetry directions Γ→Y→M→A→Γ→L2→Γ→V2 of the Brillouin zone for the Cu₂TeO₆ monolayer were calculated. According to Figure 1b, the phonon dispersion curves of the Cu₂TeO₆ monolayer are positive in whole of the Brillouin zone, and the transverse, longitudinal, and out-of-plane acoustic modes show a normal linear dispersion around the Γ point, which confirms the kinetic stability of the Cu₂TeO₆ monolayer.

Thermal stability of the Cu₂TeO₆ monolayer is verified though the estimation of its formation energy E_form:

E_form = E_tot − N_Cu·E_Cu − N_Te·E_Te − N_O·E_O

(1)

where E_tot is the total energy of the Cu₂TeO₆ monolayer, E_Cu, E_Te, and E_O are the energies of single Cu, Te, and O atoms, and N_Cu, N_Te, and N_O are the numbers of the Cu, Te, and O atoms. The negative E_form = −1.72 eV/atom that was found suggests the thermodynamic stability of the Cu₂TeO₆ monolayer.

To confirm the thermal stability of the materials, AIMD simulations conducted at a room temperature of 300 K are usually used. Figure 2a shows the total energy fluctuation of the Cu₂TeO₆ monolayer system at 300 K for the time of 4 ps. It can be seen that there were no artificial energy fluctuations. In addition, according to Figure 2b, there were no visible changes in the structure of the Cu₂TeO₆ monolayer happening after 4 ps, which suggests its thermal stability. To clarify the application temperature range of Cu₂TeO₆ monolayer, its stability at higher temperature of 320 K for the time of 4 ps was also evaluated via an AIMD simulation. Figure 2c presents the total energy fluctuation of the Cu₂TeO₆ monolayer system at 320 K for the time of 4 ps. It is seen that there are also no artificial energy fluctuations. In addition, according to Figure 2d, there were no structural changes in the Cu₂TeO₆ monolayer after 4 ps, which suggests its thermal stability at 320 K.

To identify the bonding type in the Cu₂TeO₆ monolayer, the electron localization function (ELF) is simulated and analyzed. It is known that the ELF value, which is in the range from 0 to 1, reflects the degree of charge localization in the real space, where the values close to 0 represent the existence a free electronic state, while the values close to 1 represent a perfect electron localization [37,38,39]. The ELF plot for the Cu₂TeO₆ monolayer with the isosurface value of 0.70 is presented in Figure 3a. According to Figure 3a, the electron localization is high at the regions around the sites of the O anions, while at the Cu and Te cations sites the value of ELF is low. A large difference found in the ELF plot proposes the existence of an ionic bonding in the Cu₂TeO₆ monolayer. This conclusion is in line with the fact that an ionic type of bond is characteristic of the bonded metal and non-metal, the difference in the electronegativity of which is higher than 1.5. Particularly, the electronegativity of O (3.5) and Cu (1.9) is higher 1.5, and the electronegativity of Te (2.1) is lower the electronegativity of O (3.5) by 1.4.

The work function (WF) is a critical parameter to characterize a material [40]. It is defined as follows:

WF = E_vac − E_Fermi

(2)

where Evac is the energy level of a stationary electron in the vacuum, and E_Fermi corresponds to the Fermi level of the system. The calculated WF of the Cu₂TeO₆ monolayer is found to be 7.76 eV, as shown in Figure 3b, which is much higher than it is for the most of the 2D materials such as graphene (4.60 eV) [41] and borophene (5.31 eV) [38] and 2D metal chlorides [35] and bulk metals [41] such as Ni (5.23 eV) and Pt (5.65 eV). Importantly, the materials with such a high WF value can be efficiently integrated, for instance, in photovoltaic devices based on a high-WF material/low-WF material hybrid junctions [42].

The calculated band structure and the local density of states (LDOS) of the Cu₂TeO₆ monolayer are shown in Figure 4. It is predicted that the Cu₂TeO₆ monolayer is a direct band gap semiconductor with the conduction band minimum (CBM) and valence band maximum (VBM) located at the Y and M points, and its band gap size is found to be 0.20 eV (Figure 4a). According to the LDOS plot in Figure 4b, both the CBM and VBM of the Cu₂TeO₆ monolayer are formed of the Cu d states and O p states. Such a narrow band gap monolayer semiconductors can be used in infrared nanodetectors [43].

The mechanical stability of the Cu₂TeO₆ monolayer is also evaluated using the conditions presented by Mazdziarz [44]. As the elastic tensor is calculated based on the periodic boundary conditions, an arbitrary vacuum is presented in the out-of-plane direction when one is dealing with the monolayer. Therefore, the output values of the calculated elastic constants should be correct by eliminating the arbitrariness of the vacuum padding. Particularly, the C_ij components (i, j ≠ 3) should be adjusted by multiplying them by the length of the vacuum padding [45]. For the considered Cu₂TeO₆ monolayer elastic constants matrix is further calculated, the values of the obtained elastic constants C_ij are collected in

Table 2. The calculated elastic constants C_ij for the Cu₂TeO₆ monolayer.

C11	101.18 N/m
C22	73.29 N/m
C12	27.26 N/m
C44	34.65 N/m

. According to [44], the mechanical stability criteria for the oblique Bravais lattice are the following:

$$C_{11}>0$$

$$C_{11}{C}_{22} > C_{12}^2$$

(3)

$$\det \left(C_{ij}\right) > 0$$

Based on the data presented in Table 2, the Cu₂TeO₆ monolayer is found to be mechanically stable as it satisfies the criteria in Equation (3). In addition, the mechanical properties of the Cu₂TeO₆ monolayer such as Young’s modulus, Poisson’s ratio, and the shear modulus are calculated.

Young’s modulus of the monolayer for strains in the x and y directions can be calculated as [46,47]:

$$E_{\left[x\right]} =\frac{C_{11}{C}_{22}-C_{12}^2}{C_{11}}, \mathrm{and} E_{\left[y\right]}=\frac{C_{11}{C}_{22}-C_{12}^2}{C_{22}}$$

(4)

The shear modulus of the monolayer can be calculated as [46,47]:

G = C₆₆

(5)

Poisson’s ratio of the monolayer in the x and y directions can be calculated as [46,47]:

$$v_{\left[x\right]} =\frac{C_{12}}{C_{11}}, \mathrm{and} v_{\left[y\right] } =\frac{C_{12}}{C_{22}}$$

(6)

The spatial dependence of Young’s modulus, the shear modulus, and Poisson’s ratio of the Cu₂TeO₆ monolayer are presented in Figure 5. It is found that these quantities are direction dependent. Young’s moduli of the Cu₂TeO₆ monolayer in the x and y directions are found to be E_x = 65.95 N/m and E_y = 91.04 N/m, respectively. The shear modulus of the Cu₂TeO₆ monolayer is found to be G = 34.65 N/m. Poisson’s ratios of the Cu₂TeO₆ monolayer in the x and y directions are found to be v_x = 0.21 and v_y = 0.37, respectively.

4. Discussion and Conclusions

In this work, a novel material, the Cu₂TeO₆ monolayer, is discovered. Its dynamic and thermal stability is studied and confirmed. Importantly, an application of the Cu₂TeO₆ monolayer is possible at various temperature conditions and geographical locations, from Nordic countries to the tropics, as its stability is confirmed at a room temperature of 300 K and at the elevated temperature of 320 K.

It is found that the Cu₂TeO₆ monolayer exhibits a narrow band gap of 0.2 eV and a high WF value of 7.76 eV. In addition, the Cu₂TeO₆ monolayer has anisotropic mechanical properties with the highest Young’s modulus of 91.04 N/m, shear modulus of 34.65 N/m, and Poisson’s ratio of 0.37.

These properties make the Cu₂TeO₆ monolayer a good candidate for applications in various nanodevices. Particularly, due to its exceptionally high WF value, the Cu₂TeO₆ monolayer is a good candidate material for photovoltaic devices [42], metal–insulator–metal capacitors [48], and high-WF anodes [49], etc. In addition, despite the Young’s modulus of the Cu₂TeO₆ monolayer being ~3 time lower than that of graphene [50], which suggests its lower stiffness compared to graphene, the Cu₂TeO₆ monolayer possesses higher elasticity relative to graphene [51]. This offers the application of the Cu₂TeO₆ monolayer in straintronic nanodevices [52].