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Exploring the Structural Stability, Electronic and Thermal Attributes of synthetic 2D Materials and their Heterostructures

Based on first-principles calculations, we have investigated the structural stability, electronic structures, and thermal properties of the monolayer XSi2N4 (X= Ti, Mo, W) and their lateral (LH) and vertical heterostructures (VH). We find that these heterostructures are energetically and dynamically stable due to high cohesive and binding energies, and no negative frequencies in the phonon spectra. The XSi2N4 (X= Ti, Mo, W) monolayers, the TiSi2N4/MoSi2N4-LH, MoSi2N4/WSi2N4-LH, and MoSi2N4/WSi2N4-VH possess a semiconducting nature with an indirect band gap ranging from 0.30 to 2.60 eV. At room temperature, the Cv values are found to be between 100 and 416 J/K.mol for the monolayers and their heterostructures, suggesting the better ability to retain heat with respect to transition metal dichalcogenides. Our study unveils the excellent attributes of XSi2N4 2D monolayers and their heterostructures, proposing them as potential candidates in nanoelectronics and thermoelectric applications.


Introduction
Due to the monolayer limit, the two-dimensional (2D) materials have distinct physical properties and thus are employable in a wide range of device applications [1][2][3][4][5][6][7][8][9]. Graphene, the first 2D material, has been widely studied since its discovery [3,10]. It was expected to be a suitable layered structure for the new generation of nanoelectronic devices thanks to its high carrier mobility. However, graphene's zero band gap reminds us that it cannot be effective in many device applications [11][12][13]. On the other hand, transition metal dichalcogenides (TMDCs) are explored to exhibit tunable band gaps, but on account of practical applications, their relatively low carrier mobilities cannot be neglected [1,14,15]. For instance, MoS2 possesses carrier mobility of approximately 200 cm 2 V -1 s -1 for holes and 72 cm 2 V -1 s -1 for electrons [14], which are much lower in magnitude than that of graphene and even much smaller than Si (480 cm 2 V -1 s -1 for hole and 1350 cm 2 V -1 s -1 for electron) [14,16]. In recent years, from the fundamental and applicative point of view, we have assisted to tremendous efforts in the investigation of the topological and magnetic properties of the 2D materials and quasi-2D materials [17][18][19][20][21][22][23][24][25].
Recently, the discovery of novel 2D materials family, XA2Z4 (X=Transition metal, A=IV element, Z=V element) [26] has attracted broad consideration due to the outstanding properties they demonstrate. The first member of this assembly is the MoSi2N4 that was successfully produced via chemical vapor deposition (CVD) process with a large area of size 15 mm × 15 mm. This semiconducting material with a septuple-atomic-layer configuration (N-Si-N-Mo-N-Si-N) can be regarded as a MoN2 layer sandwiched in two SiN layers, exhibiting an indirect band gap of almost 1.94 eV. The monolayer of MoSi2N4 depicts an elastic modulus that is roughly four times larger than MoS2, and the carrier mobilities of holes and electrons are also four to six times that of a single layer of MoS2. Moreover, a new family, namely XA2Z4 (X=Transition metal, A=IV element, Z=V element) is anticipated by density functional theory (DFT) calculation. So far, several studies have been carried out describing the exceptional features of these materials [27][28][29][30][31][32][33][34].
Designing vertically stacked or laterally stitched two-dimensional heterostructures between 2D materials is an effective technique to unravel new features and extend their applications to electronics, detectors, electroluminescence, and photovoltaics [35][36][37][38][39][40][41]. Several experimental techniques such as physical vapor transport, CVD, plasma-assisted deposition, thermal decomposition, and vapor phase growth method have been utilized to synthesize the lateral heterostructures (LH) and vertical heterostructures (VH) [41][42][43][44][45][46][47]. Recently, the vertically stacked bilayer of MoSi2N4 have been studied using first-principles calculations, which reports the effect of strain on the electronic band gap, the bilayer MoSi2N4 showed a decrease in the band gap as the vertical strain was increased; and at around 22% strain, the bilayer revealed a semiconducting to metal transition [31]. Also, the band gap decreases as a function of the electric field for both the MoSi2N4, and WSi2N4 bilayers, respectively [28]. Likewise, the vertical heterostructures of MoSi2N4 and MoGe2N4 have been reported, where the electronic properties and ferroelectricity are investigated [29,30]. Nonetheless, the van der Waals heterostructures of this class are not explored thoroughly and demand a detailed study. In addition, the laterally stitched heterostructures of this novel family are not reported so far.
In this work using first-principle calculations, we carry out a detailed study of XSi2N4 (

Methodology
We performed first-principles calculation via the Vienna ab initio simulation package (VASP) [48,49] using density functional theory (DFT). Both the Heyd-Scuseria-Ernzerhof (HSE06) and the generalized gradient approximation with PBE form [27] and [28] are adopted to calculate the electronic properties [50,51]. For relaxation, an energy cutoff of 500 eV and a mesh of 15 × 15 × 1 k-points were chosen. The periodic structures in the perpendicular direction of 2D material and heterostructure were separated with a vacuum layer of approximately 20 Å, which would avoid spurious interactions. Our calculations were converged by setting the convergence criteria to 0.01 eV/Å and energy 10 −7 eV. To study the dynamical stability of XSi2N4 monolayers and that of LHs and VHs, PHONOPY code using a 4×4×1 supercell was employed for calculating the phonon band structures [52]. Also, the thermal properties such as the heat capacity, free energy and entropy were explored using phonon calculations.

Structural stability
The optimized structures for XSi2N4 (X=Ti, Mo, W) monolayers and their heterostructures are visualized in Fig. 1, where the (a), (b) and (c) panels depict the monolayer, lateral heterostructure (LH) and vertical heterostructures (VH), respectively. It can be seen that XSi2N4 (X=Ti, Mo, W) possesses a honeycomb structure consisting of Ti/Mo/W, Si and N atoms. The side view illustrates that this structure is stacked by seven atomic layers of N-Si-N-X-N-Si-N that exhibits a sandwich structure of MoN2 layer by two SiN bilayers. We noticed the optimized lattice constants for XSi2N4 (X=Ti, Mo, W) monolayers to be a = b = 2.932, 2.909 and 2.915 Å, respectively. These lattice constants are in perfect agreement with the calculated lattice parameters in the literature [26,27,53,54]. Here, we consider the lateral AA-stitching for the LHs (Fig. 1b) and vertical AB-stacking for the VHs (Fig. 1c), the heterostructures obtained are TiSi2N4/MoSi2N4-LH, TiSi2N4/MoSi2N4-VH, MoSi2N4/WSi2N4-LH, and MoSi2N4/WSi2N4-VH. In the AA-stacking of VHs, all atoms i.e. the X, Si and N atoms of one monolayer of XSi2N4 are located exactly above the X, Si and N atoms of the other XSi2N4 layer. Whereas, in AB-stacking, all the atoms in the one XSi2N4 layer are staggered with respect to another XSi2N4 monolayer, as depicted in Fig. 1c. Based on total ground state energy calculations, the ground state is AB stacking [26,28,29].
In order to check the stability of XSi2N4 monolayers and that of LHs and VHs, we used the total ground state energies to compute the cohesive energies. In addition, the phonon dispersions are calculated to determine their dynamical stability, which can enable the use of the studied systems in the device application. In the monolayer case of XSi2N4, the cohesive energy is calculated by [55,56]; Here, EX, ESi, EN and E XSi 2 N 4 are total ground state energies of isolated X, Si, N atoms and monolayer XSi2N4, respectively. The cohesive energies for TiSi2N4, MoSi2N4 and that of WSi2N4 are evaluated to be 5.562, 5.589 and 5.653 eV/atom, respectively. By comparing these values with that of graphene (7.46 eV/atom), MoS2 (4.98 eV/atom) and phosphorene (3.30 eV/atom) [57][58][59], we noted that they are smaller than graphene but larger than MoS2, which indicate the better stability. For the heterostructures, the binding energies are estimated to determine the stability, it is given by [60]; To examine the dynamical stability of XSi2N4 (X=Ti, Mo, W) monolayers and that of heterostructures, we carried out the phonon dispersion calculations as shown in Fig. 2 and 3. The finite difference method is employed to calculate the phonon spectra using the Phonopy code [52]   TiSi2N4/MoSi2N4-VH, the hardening of this mode can be seen in Fig. 3a and 3b, respectively. The hardening of ~2 THz acoustic mode occurring at the K-point can be ascribed to the enhanced coupling at this wave-vector [62,63].

Electronic properties
To understand the contribution of different orbitals from various elements that characterize our systems' electronic attributes, the total density of states (TDOS) and partial density of states (PDOS) are computed. Figure 4 illustrates the TDOS and PDOS for the respective XSi2N4 monolayers. For the monolayer TiSi2N4, the valence band maximum (VBM) is mainly dominated by N atoms 'p' orbitals with slight contribution from Ti-d orbitals, while the conduction band minima (CBM) is principally originated by the Ti-d orbital, as illustrated in Fig. 4a. In the case of MoSi2N4, and WSi2N4, however, both the VBMs and CBMs maximally arise due to the Mo-/W-d orbitals, which are located in the middle of each monolayer structure (Fig. 4b, c).
In Fig. 5  and WSi2N4, respectively, as shown in Fig. 6a  (CBM) occurs at the Г-point (Fig. 6d). On the other hand, in the cases of MoSi2N4/WSi2N4-LH and MoSi2N4/WSi2N4-VH, the VBM emerges at Г-point, whereas the CBM occurs at the K-point, as can be seen in Fig. 6e, f, suggesting that the indirect band gap nature still prevails. The values of the indirect band gaps are determined to be ~0.30, 2.34, and 2.15 eV for the TiSi2N4/MoSi2N4-LH, MoSi2N4/WSi2N4-LH, and MoSi2N4/WSi2N4-VH, respectively, which could make them promising for the application in solar cells and optoelectronic devices [67][68][69][70]. The calculated band gap values are tabulated in Table 1, which are in agreement with the available literature [27,31,33]. A slight difference in band gaps may arise due to the difference in lattice parameters.

Thermal properties
Based  investigation, we found that is consistent with the entropy we computed here since the variation of entropy corresponds to the integral of ( / ) over a given temperature range.

Conclusions
The structural stability, electronic properties and thermal characteristics of XSi2N4 (X= Ti, Mo, W) monolayers and their lateral and vertical heterostructures are explored using density functional theory. We noticed that interfacing two different 2D structures either laterally or vertically, significantly tunes the physical properties of the parent 2D monolayers that may have technological impact in many potential applications. The structural stability of these systems was confirmed by the cohesive energies which are based on the total ground state energies and the phonon dispersions revealing the dynamical stability. Apart from their stability, our results illustrated that the lateral and vertical heterostructures present outstanding electronic and thermal features. All the three monolayers i.e. TiSi2N4, MoSi2N4, and WSi2N4 revealed indirect band gaps.
Remarkably, a semiconducting to metallic transition is observed in the VH of TiSi2N4/MoSi2N4,