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Thermal Properties of 2D Dirac Materials MN4 (M = Be and Mg): A First-Principles Study

Recently, a novel two-dimensional (2D) Dirac material BeN4 monolayer has been fabricated experimentally through high-pressure synthesis. In this work, we investigate the thermal properties of a new class of 2D materials with a chemical formula of MN4 (M = Be and Mg) using first-principles calculations. First, the cohesive energy and phonon dispersion curve confirm the dynamical stability of BeN4 and MgN4 monolayers. Besides, BeN4 and MgN4 monolayers have the anisotropic lattice thermal conductivities of 842.75 (615.97) W m–1 K–1 and 52.66 (21.76) W m–1 K–1 along the armchair (zigzag) direction, respectively. The main contribution of the lattice thermal conductivities of BeN4 and MgN4 monolayers are from the low frequency phonon branches. Moreover, the average phonon heat capacity, phonon group velocity, and phonon lifetime of BeN4 monolayer are 3.54 × 105 J K–1 m–3, 3.61 km s–1, and 13.64 ps, which are larger than those of MgN4 monolayer (3.42 × 105 J K–1 m–3, 3.27 km s–1, and 1.70 ps), indicating the larger lattice thermal conductivities of BeN4 monolayer. Furthermore, the mode weighted accumulative Grüneisen parameters (MWGPs) of BeN4 and MgN4 monolayers are 2.84 and 5.62, which proves that MgN4 monolayer has stronger phonon scattering. This investigation will enhance an understanding of thermal properties of MN4 monolayers and drive the applications of MN4 monolayers in nanoelectronic devices.


INTRODUCTION
The successful preparation of a one-atom-thick two-dimensional (2D) material graphene in 2004 breaks the conventional view that 2D crystal is regarded as an unstable structure in nature. 1 Meanwhile, graphene as a Dirac material has a feature of massless fermions, resulting in outstanding physical properties, such as ultrahigh carrier mobility, 2 half-integer/ fractional/fractal quantum Hall effects, 3−5 high thermal conductivity, 6−8 large Young's modulus, 9 and other novel properties, 10 which has triggered a great deal of attention from researchers. On the other hand, graphene arouses the explorations of other hundreds of 2D materials, including graphene-like 2D materials (graphitic carbon nitride, hexagonal boron nitride, transition metal dichalcogenides, layered metal oxides, and layered double hydroxides), black phosphorus, silicene, antimonene, transition metal carbides and/or nitride, noble metals, metal−organic frameworks, covalent-organic frameworks, and so on. 11,12 However, in these materials, only silicene, germanene, stanene, tinene, graphynes (α-graphyne, β-graphyne, and 6,6,12-graphyne), rectangular carbon and boron allotropes (S-graphene, D-graphene, E-graphene, and Pmmn boron), and some other systems belong to 2D Dirac materials. 13−15 These 2D Dirac materials generally have spatial inversion symmetry and possess the advantages of quantum Hall effect, large Fermi velocity, and high carrier mobility, 15−17 making them a promising candidate for applications in nanoelectronic devices. Hence, there is the need to search the novel Dirac materials in the family of 2D materials.
Very recently, a novel nitrogen-rich Dirac material, beryllonitrene (BeN 4 ) monolayer has been synthesized successfully by employing high-pressure synthesis followed by decompression to ambient conditions. 18 It has been reported that BeN 4 monolayer belongs to a Dirac semimetal, and the electronic band structure near the Fermi energy exhibits the linear dispersion with two overlapped Dirac points, which is similar to graphene monolayer. However, unlike graphene monolayer, BeN 4 monolayer is an oblique crystal system with a lower symmetry, which leads to the fact that the Dirac points are located at the Σ point along the Γ-A direction in the Brillouin zone. Besides, BeN 4 monolayer has essentially anisotropic massless Dirac fermions and Fermi velocities (8 × 10 5 m s −1 along the Σ-Γ direction and 3 × 10 5 m s −1 along the Σ-A direction). 18 These outstanding electronic properties suggest that BeN 4 monolayer can be widely used in nanoelectronic devices. Apart from BeN 4 monolayer, another nitrogen-rich Dirac material, MgN 4 monolayer, belonging to the same class of MN 4 (M = Be and Mg) monolayers is examined by first-principles calculations. 19 Mortazavi et al. investigated the stability and the electronic nature of MN 4 monolayers and proved that the melting points of two monolayers were over 1000 K. Meanwhile, the corresponding tensile strengths of BeN 4 and MgN 4 monolayers were 100(43) and 74(39) GPa along the armchair (zigzag) direction, respectively. The corresponding elastic moduli were 946(590) and 582(353) GPa, indicating high lattice thermal conductivities of BeN 4 monolayer, which has been proven also in other work. 20 In practice, some materials with high lattice thermal conductivities are useful for enhancing the heat dissipation in miniaturized and high-power nanoelectronic devices, which plays a fundamental role in thermal management. Therefore, it is crucial to investigate the lattice thermal conductivity in BeN 4 and MgN 4 monolayers. However, to our knowledge, the thermal properties of BeN 4 and MgN 4 monolayers are yet to be broadly investigated. Hence, in the present work, we identify the lattice thermal conductivities and phonon behaviors of MN 4 monolayers.
In this work, the thermal properties of BeN 4 and MgN 4 monolayers are investigated systematically by utilizing firstprinciples calculations. First, we calculate the cohesive energies and phonon dispersion curves of BeN 4 and MgN 4 monolayers to demonstrate the dynamical stabilities. Moreover, the lattice thermal conductivities of BeN 4 and MgN 4 monolayers and phonon contributions are calculated. Furthermore, the phonon behaviors including the phonon heat capacity, the phonon group velocity, the phonon lifetime, the phonon mean free path (MFP), and the cumulative lattice thermal conductivity are studied. Especially, the mode weighted accumulative Gruneisen parameter (MWGP) and the phonon scattering phase space (P 3 ) are applied to interpret the strength and number of phonon scattering channels. Our investigation can provide deep insights into thermal properties of MN 4 monolayers.

METHODOLOGY
In this investigation, all calculations are carried out by the Vienna Ab initio Simulation Package (VASP) within the framework of density functional theory (DFT). 21 The electron−ion interaction is described by the projector augmented wave (PAW) method, and the generalized gradient approximation (GGA) of Perdew−Burke−Ernzerhof (PBE) is used to model the exchange-correlation functional. 22 For both monolayers, we set an energy cutoff for the plane waves to 500 eV. During the structure optimization, the Hellman-Feynman force and energy drops are lower than 10 −3 eV Å −1 and 10 −6 eV with a 15 × 13 × 1 Monkhorst−Pack k-point mesh, respectively. Meanwhile, a vacuum thickness of 20 Å along the z direction is applied to avoid interactions between the periodic images.
Thermal properties of BeN 4 and MgN 4 monolayers can be studied by utilizing the ShengBTE package, 23 which is widely used in the investigation of thermal properties of 2D materials (As 2 Te 3 , WS 2 , WSe 2 , and AlP 3 ). 24 (1) where λ means phonon branch, α represents the Cartesian direction (x, y, and z), C λ is the phonon heat capacity, v λα and τ λα are the phonon group velocity and phonon lifetime, respectively. The phonon heat capacity C λ can be given by: 2) where N is the total number of q points in a discretization of Brillouin zone, V means the system volume, ω λ represents the phonon angular frequency, and T is the temperature. The parameters of k B and ℏ are the Boltzmann constant and reduced Planck constant, which are the physical constants. The phonon group velocity v λα can be obtained from the following expression: where q is the wave vector. The phonon lifetime τ λ describing the three-phonon processes can be calculated by using the following formula: where W + and W − are three-phonon scattering rates during the absorption and emission processes of phonons, respectively. 23 The ShengBTE package requires interatomic force constants (IFCs) as input files. The second order and third order interatomic force constants (IFCs) are obtained by using DFT calculations, which are also known as harmonic and anharmonic IFCs. For harmonic IFCs, the dynamical matrix is calculated by using the density functional perturbation theory (DFPT). 28 The phonon harmonic properties can be extracted using Phonopy package with a 3 × 3 × 1 supercell and a 5 × 4 × 1 q mesh. 29 The anharmonic IFCs are obtained by using the thirdorder.py script with the same 3 × 3 × 1 supercell and 5 × 4× 1 q mesh. 23 Herein, we consider the interaction cutoff up to ten-nearest-neighbor atoms. Meanwhile, a long-range correction is introduced by calculating the Born effective charge and dielectric tensor, and the isotope scattering has been considered in the calculation of lattice thermal conductivity. A q point grid of 100 × 100 × 1 is selected to study the lattice thermal conductivity. Both thicknesses of BeN 4 and MgN 4 monolayers are set to 3.06 Å. 18

RESULTS AND DISCUSSION
3.1. Crystal Features and Dynamic Stabilities of MN 4 Monolayers. In this section, we first optimize the MN 4 monolayers and measure the crystal features. As shown in Figure 1, the honeycomb-like monolayers exhibit the space group of P2/m (No. 10) and point group of C2h. In both monolayers, each M atom is connected with four N atoms, and each N atom is connected with one M atom and two N atoms. 3.86 Å, 4.88 Å, and 66.69°). 19 For BeN 4 monolayer, the two types of N−N bond lengths are found to be 1.343 and 1.338 Å, the Be−N bond length is determined to be 1.746 Å. For MgN 4 monolayer, three bond lengths are equal to 1.356 Å, 1.349 Å, and 2.055 Å, respectively. These data are consistent with that from previous work. 19,30 Besides, the electron localization functions (ELF) of two monolayers are calculated to further analyze the bonding nature. The ELF results with an isovalue of 0.7 are shown in Figure 2 and are similar to other DFT results. 19 ELF takes a value from 0 to 1 at each point. MN 4 monolayers consist of armchair-shaped polymeric nitrogen chains (N ∞ ). From Figure 2, we can observe that electrons are strongly localized on N ∞ , and on each N pointing toward the M atoms. In BeN 4 and MgN 4 monolayers, the electron cloud of anionic N ∞ is strongly pulled to the Be and Mg atoms, which leads Be−N and Mg−N bonds to have higher covalent character. As shown in Figure 3, we obtain the electronic band structures of BeN 4 and MgN 4 monolayers by using the PBE functional. The electronic band structures of BeN 4 and MgN 4 monolayers are in good agreement with that from previous work, 19 indicating our DFT calculations are correct.
To evaluate the dynamic stability, we evaluate the cohesive energies and phonon dispersion curves of two monolayers. The cohesive energy can be calculated by the following expression:  39 On the other hand, high temperature will monotonically decline the lattice thermal conductivities of these monolayers. As the temperature  The corresponding reductions are 71.94% and 72.13%, 50.11% and 63.43%, respectively. High temperature will activate high frequency phonons and strengthen the Umklapp phonon scattering, generating the lower lattice thermal conductivity. A similar trend can be observed in other 2D materials, such as transition metal dichalcogenide and triphosphides. 26,40 Additionally, we investigate the phonon contributions to the total lattice thermal conductivities of BeN 4 and MgN 4 monolayers, which is displayed in Figure 6. There are two facts that can be found in Figure 6: (1) For BeN 4 and MgN 4 monolayers, the contribution of low frequency phonons at 0−10 THz to the total lattice thermal conductivity is dominant. These low frequency phonons include the acoustic phonon branches and low frequency optical phonon branches. (2) The high frequency optical phonon branches have a small contribution to the total lattice thermal conductivities of BeN 4 and MgN 4 monolayers. This tendency has been proven in a previous study. 34 3.3. Phonon Behaviors of MN 4 Monolayers. To gain a deeper understanding of thermal transport mechanism of BeN 4 and MgN 4 monolayers, the phonon behaviors are evaluated in this section, which includes the phonon heat capacity, the phonon group velocity, the phonon lifetime, the mode weighted accumulative Gruneisen parameter (MWGP), the phonon scattering phase space (P 3 ), the phonon mean free path (MFP), and the cumulative lattice thermal conductivity. We first calculate phonon heat capacities of BeN 4 and MgN 4 monolayers, which are equal to 3.54 ×     respectively. For a more intuitive comparison, the average phonon group velocities of BeN 4 and MgN 4 monolayers are calculated, which are 3.61 km s −1 and 3.27 km s −1 in the whole frequency region, respectively, which are lower than that of graphene monolayer (7.09 km s −1 ). 39 Particularly, the average phonon group velocities of high frequency phonons of BeN 4 and MgN 4 monolayers are 3.04 km s −1 and 2.89 km s −1 , which are smaller than the values of low frequency phonons (6.05 km s −1 and 4.16 km s −1 ), indicating that the low frequency phonons have a large contribution to lattice thermal conductivity. According to eq 1, in comparison with MgN 4 monolayer, BeN 4 monolayer has a higher lattice thermal conductivity because of the higher phonon group velocity.
Apart from phonon heat capacity and phonon group velocity, phonon lifetime is another important factor that affects lattice thermal conductivity. The phonon lifetimes of BeN 4 and MgN 4 monolayers are plotted in Figure 8. For BeN 4 monolayer, the phonon lifetimes of acoustic phonon branches decline with the increasing phonon frequency, which are larger than those of optical phonon branches. This phenomenon can demonstrate that the high frequency optical phonon branches make a small contribution to lattice thermal conductivity. Besides, the ZA phonon branch has the largest phonon lifetime. However, for MgN 4 monolayer, the phonon lifetimes of acoustic phonon branches and optical phonon branches are in the same magnitude. It is noticed that the phonon lifetime of the TA phonon branch is slightly higher than those of other acoustic phonon branches. We can further observe from Figure  8 that the phonon lifetime of BeN 4 monolayer is much higher than that of MgN 4 monolayer, of which the average values are 13.64 and 1.70 ps, respectively. The larger phonon lifetime can   .62, respectively. The larger MWGP means the stronger anharmonicity of the phonon branch, resulting in stronger phonon scattering and lower phonon lifetime. 41 Meanwhile, P 3 will reflect the number of phonon scattering channels, and the larger P 3 indicates stronger phonon scattering. Figure 9 exhibits the P 3 of BeN 4 and MgN 4 monolayers. It can be seen that in the whole frequency region, the P 3 of BeN 4 and MgN 4 monolayers are comparable with each other. The total P 3 of BeN 4 monolayer is 1.94 × 10 −3 , which is slightly smaller than that of MgN 4 monolayer (2.07 × 10 −3 ). The discrepancy of P 3 between BeN 4 and MgN 4 monolayers is negligible, thus the enhancing phonon scattering of MgN 4 monolayer is only ascribed to the strength of the phonon scattering channel. Hence, compared that of to MgN 4 monolayer, the higher lattice thermal conductivity of BeN 4 monolayer can be attributed to the following aspects: (1) the higher phonon heat capacity, (2) the larger phonon group velocity, (3) the longer phonon lifetime due to the stronger phonon scattering.
In addition, the phonon MFP plays a crucial role in phonon thermal transport. We calculate the phonon MFP (l λ ) based on the phonon group velocity and phonon lifetime, that is, l λ = v λ τ λ . As displayed in Figure 10, for BeN 4 monolayer, the phonon MFP generally drops as the frequency increases. The acoustic phonon MFPs of BeN 4 monolayer are larger than those of optical phonon MFPs. Especially, when the frequency is less than 3 THz, the MFP of ZA phonon branch is much higher than those of TA and LA phonon branches. However, for the MgN 4 monolayer, the acoustic phonon MFPs show little deviation from the optical phonon MFPs. At a frequency below 5 THz, the MFP of the ZA phonon branch is 1 order of magnitude higher than those of the other two phonon branches. The average phonon MFPs of BeN 4 and MgN 4 monolayers are 39.30 and 4.25 nm, which demonstrates that BeN 4 monolayer has the higher lattice thermal conductivity. Typically, when the system length is smaller than l λ , phonon transport belongs to the ballistic regime without scattering. When the system length is larger than l λ , the phonon moves diffusively with collision. Thus, the cumulative lattice thermal conductivity should be analyzed to exhibit the phonon MFP distribution. The lattice thermal conductivity as a function of phonon MFP can be given by: 42 The calculated cumulative lattice thermal conductivities of BeN 4 and MgN 4 monolayers are shown in Figure 11. As the phonon MFP increases, the cumulative lattice thermal conductivities of BeN 4 and MgN 4 monolayers increase until reaching the maximum values, and the corresponding phonon MFP can be defined as the longest phonon MFP (L max ). For BeN 4 monolayer, the phonons with large MFP have a significant contribution to lattice thermal conductivity. Nevertheless, the phonons with the small MFP of MgN 4 monolayer dominate lattice thermal conductivity. The L max of BeN 4 monolayer is longer than 10 μm, and the corresponding value of MgN 4 monolayer is smaller than 1 μm, which reveals that the lattice thermal conductivity of BeN 4 monolayer has stronger size effect.   The cumulative lattice thermal conductivities of two monolayers increase gradually until reaching saturation values with the increasing phonon MFP. The L max of BeN 4 monolayer is much longer than that of MgN 4 monolayer, indicating that the lattice thermal conductivity of BeN 4 monolayer exhibits a stronger size dependence. The study explores the thermal properties of novel 2D Dirac materials MN 4 monolayer, which will contribute to a better understanding of thermal management in 2D MN 4 based nanoelectronic devices.