High intrinsic lattice thermal conductivity in monolayer MoSi$_2$N$_4$

Very recently, a novel two-dimension (2D) MXene, MoSi$_2$N$_4$, was successfully synthesized with excellent ambient stability, high carrier mobility, and moderate band gap (Science 369, 670, 2020). In this work, the intrinsic lattice thermal conductivity of monolayer MoSi$_2$N$_4$ is predicted by solving the phonon Boltzmann transport equation based on the first-principles calculations. Despite the heavy atomic mass of Mo and complex crystal structure, the monolayer MoSi$_2$N$_4$ unexpectedly exhibits a quite high lattice thermal conductivity over a wide temperature range between 300 to 800 K. At 300 K, its in-plane lattice thermal conductivity is 224 Wm$^{-1}$K$^{-1}$. The detailed analysis indicates that the large group velocities and small anharmonicity are the main reasons for its high lattice thermal conductivity. We also calculate the lattice thermal conductivity of monolayer WSi$_2$N$_4$, which is only a little smaller than that of MoSi$_2$N$_4$. Our findings suggest that monolayer MoSi$_2$N$_4$ and WSi$_2$N$_4$ are potential 2D materials for thermal transport in future nano-electronic devices.

Recently, a high-quality 2D MXene, MoSi 2 N 4 , was successfully synthesized with excellent ambient stability, moderate band gap, and considerable carrier mobility [25]. Experimental results [25] demonstrated that MoSi 2 N 4 has a band gap of about 1.94 eV. Furthermore, large hole and electron mobilities of monolayer MoSi 2 N 4 are predicted to be about 1200 and 270 cm 2 V -1 s -1 [25], which are 4 to 6 times higher than those of MoS 2 monolayer [26]. Since the synthesis of MoSi 2 N 4 , intensive research efforts have been devoted to unearthing its novel properties [27][28][29][30][31][32]. First principles calculations revealed that the large thermopower of monolayer MoSi 2 N 4 can be obtained in a range of chemical potential from 0 to 1 eV [27]. Cao et al [28] investigated the electrical contact of monolayer MoSi 2 N 4 and an ultralow Schottky barrier height was observed in MoSi 2 N 4 /NbS 2 3 contact [28], which is beneficial for the nano-electronic device applications. The theoretical calculations demonstrated that the piezoelectricity MoSi 2 N 4 enables actuating new electronic components of nanoscale devices [32].
Thermal transport is an important property for materials in many applications including thermal barrier coating [33], heat management [34], and thermoelectric energy conversion [35]. Especially the phonon transport is an essential part of designing all power-dissipating devices [36,37]. 2D materials are ideal platforms to investigate fundamental carrier transport and provide new directions for thermal management and energy control. Phonon transport phenomena are related to various intriguing applications based on 2D materials [34]. In recent years, the lattice thermal conductivities of 2D materials have attracted considerable interest [38][39][40][41][42][43][44][45][46][47][48][49][50]. For these reasons, the study of the thermal transport property of monolayer MoSi 2 N 4 is urgently called to speed up its application.
In this paper, we have systematically investigated the intrinsic thermal transport properties of monolayer MoSi 2 N 4 by iteratively solving the Boltzmann transport equation. It is found that MoSi 2 N 4 unexpectedly exhibits a quite high lattice thermal conductivity despite its great average atomic mass and complex crystal structure. To further explain the mechanism of the high thermal conductivities, we also discuss the phonon lifetimes, group velocities, and Grüneisen parameters of the monolayer MoSi 2 N 4 .

II. THEORETICAL METHODS
The crystal structure of monolayer MoSi 2 N 4 is fully optimized by the Vienna ab initio simulation package(VASP) [51,52] based on the density functional theory (DFT). The projected augmented wave (PAW) method [53,54] and generalized gradient approximation with the  Burke-Ernzerh of exchange-correlation functional [55] are used. The plane-wave cutoff energy of 520 eV is used with a 12 × 12 × 1 k-mesh. Both the lattice constants and internal atomic positions are allowed to relax until the maximal residual Hellmann-Feynman forces are less than 0.0001 eV/Å.
To avoid interactions with other neighboring layers, a vacuum space of 15 Å is taken.
After optimizing the crystal structure, we further perform the calculations of the second-and third-order interatomic force constants (IFCs) with the finite displacement method. The secondorder IFCs in the harmonic approximation and the phonon dispersions of monolayer MoSi 2 N 4 are calculated by using the PHONOPY code [56]. And the third-order IFCs and their lattice thermal conductivities are obtained based on the PHONO3PY code [57], which solves the phonon Boltzmann transport equation by using the iterative self-consistent algorithm. The lattice thermal conductivity is defined as [57] where N and V are the number of unit cells in the crystal, the volume of a unit cell. , and are the group velocity and lifetime of the phonon mode , respectively. The method has already been widely used in the prediction of thermal conductivities for 2D materials [39,41,42,[49][50][51]. A 4 × 4 × 1 supercell (112 atoms) is used to calculate the second-and third-order IFCS in monolayer MoSi 2 N 4 with a cutoff distance of 5.0 Å. And a q-mesh of 30 × 30 × 1 is taken for the calculation of lattice thermal conductivities.

A. Crystal structure and phonon dispersions
The structure of monolayer MoSi 2 N 4 has seven atoms per unit cell. As shown in Fig. 1, the 5 crystal exhibits a sandwiched structure, where the 2H MoS 2 -type MoN 2 layer is sandwiched between two slightly buckled SiN layers. Monolayer MoSi 2 N 4 holds a mirror paralleling to the horizontal plane, inversion asymmetry, and C 3 rotation symmetry. The lattice constants obtained in our calculations are a=b=2.911 Å, which are a little smaller than those of monolayer MoS 2 (3.16 Å) [9]. The distances between Si and its adjacent N (N1 and N2) are 1.748 and 1.755 Å, respectively.
The distance of Mo-Si is 2.093 Å. The thicknesses (L d ) of the vertical MoSi 2 N 4 plane is 7.00 Å.
These results are in good agreement with the previous reports [25,27]. We also check the electronic properties by calculating its electron band structure, which is given in Fig. S1. It is found that monolayer MoSi 2 N 4 exhibits an indirect band gap of 1.77 eV with the valence band maximum (VBM) and conduction band minimum (CBM) located at Γ and K points, respectively, which is also consistent with the previous studies [25,27,29,30]. The band gap could be improved to be 2.30 eV based on the HSE functional calculations [25].

B. Lattice thermal conductivity
We then calculate the temperature-dependent lattice thermal conductivity of monolayer MoSi 2 N 4 , as depicted in Fig. 3. The intrinsic in-plane lattice thermal conductivity decrease with the increase of temperature, which could be explained by the Umklapp scattering mechanism [60]. At 300 K, the lattice thermal conductivity of MoSi 2 N 4 are 224 Wm -1 K -1 , which is much higher than those of the other well-known 2D semiconductors, such as black phosphorene (30.15 Wm -1 K -1 (zigzag), 13.65 Wm -1 K -1 (armchair)) [47], monolayer 2H-MoTe 2 (42.2 Wm -1 K -1 ) [48], MoS 2 (83 Wm -1 K -1 [49] or 23.2 Wm -1 K -1 [61]) and blue phosphorene (106.6 Wm -1 K -1 ) [50], while much lower than that of monolayer hexagonal BN, BP, BAs [62], C 3 N [63], and graphene [64] with low average atomic mass. It is noted that the thermal conductivity of MoSi 2 N 4 is even much higher than those of widely used electronic materials such as Si (142 Wm -1 K -1 ) [65]. Hence, the satisfactory lattice thermal conductivity of MoSi 2 N 4 could guarantee heat removal in the corresponding nano-electronic devices. To deeply understand the lattice thermal conductivity of monolayer MoSi 2 N 4 , we then further calculate the cumulative lattice thermal conductivity of MoSi 2 N 4 at 300 K, given in Fig.4 (a). The cumulative lattice thermal conductivity first increases with the increase of MFP, and then gradually saturates when the phonon mean free path (MFP) is equal to or larger than 1000 nm, which is much longer than those of black phosphorene (66/83 nm) [47], but shorter than graphene [64]. The phonon MFP in MoSi 2 N 4 is much longer than those in the other 2D materials, leading to a much higher thermal conductivity. The representative MFP (rMFP) of materials is useful for studying the size effects on the diffusive or ballistic phonon transport. The rMFP means the phonons whose MFP is smaller than their rMFP contribute to half of the total lattice thermal conductivity. The rMFP of We also calculated the frequency-dependent lattice thermal conductivity of monolayer MoSi 2 N 4 at 300 K, which is presented in Fig. 4(b). The width of each column in the figure is 2.0 THz. The summation of all columns represents the total thermal conductivity. It is found that the phonon below 15 THz contribute most of (96%) the lattice thermal conductivity in monolayer MoSi 2 N 4 . Furthermore, we also analyzed the contribution of acoustic and optical phonons. It is found that the acoustic phonons contribute about 55% and the optical ones contribute about 45% of the thermal conductivity in both directions. The large contribution of the optical phonons is due to the large number of low-frequency optical modes of the complex structure of monolayer MoSi 2 N 4 .

C. Phonon group velocities, lifetimes, and Grüneisen parameters
To understand the underlying mechanism of the high intrinsic lattice thermal conductivity in MoSi 2 N 4 , we further analyze its phonon group velocities, lifetimes, and Grüneisen parameters. The squares of the phonon group velocities are plotted in Fig. 5 (a). The large values of squares of the group velocities almost lie below about 15 THz which could reach more than 100 km 2 /s 2 which are much larger than those of monolayer MoS 2 [61]. Since the lattice thermal conductivity is proportional to the squares of group velocities, therefore the large group velocities in Fig. 5(a) as well as the large sound velocities in Table I are the important reasons for its high thermal conductivity.
The frequency-dependent phonon lifetimes of MoSi 2 N 4 are calculated by using the PHONO3PY code from the third-order anharmonic IFCs, as displayed in Fig. 5

D. Comparative study with monolayer WSi 2 N 4
Finally, we compare MoSi 2 N 4 with another 2D MXene (WSi 2 N 4 ), which has also been successfully synthesized in the recent experiment [25]. Compared with monolayer MoSi 2 N 4 , WSi 2 N 4 has the same crystal structure and similar band characteristics [25], but a wider band gap [25] and higher atomic density. To understand the difference of phonon transport properties between WSi 2 N 4 and MoSi 2 N 4 , we further calculate the lattice thermal conductivity of monolayer WSi 2 N 4 , as given in Fig. S2. Monolayer WSi 2 N 4 also exhibits high lattice thermal conductivity. At 300 K, its lattice thermal conductivity is 219 Wm -1 K -1 . We notice that although the atomic mass of W is much larger than that of Mo, the lattice thermal conductivity of WSi 2 N 4 is only slightly lower than that of 13 MoSi 2 N 4 . The similar thermal conductivity results from their similar phonon spectra, the square of the group velocities, phonon lifetimes, and Gruneisen parameter as shown in Fig. S3.

IV CONCLUSIONS
We investigate the lattice thermal conductivities of monolayer MoSi 2 N 4 based on firstprinciples calculations and the Boltzmann transport equation. Unexpectedly, we find that its intrinsic lattice thermal conductivity (224 Wm -1 K -1 at 300 K) are much higher than those of the other wellknown semiconductors, such as black phosphorene, blue phosphorene, monolayer 2H-MoTe 2, and