2D planar pentaMN_{2} (M = Pd, Pt) sheets identified through structure search
Received
31st July 2018
, Accepted 26th November 2018
First published on 27th November 2018
Twodimensional (2D) metal dinitrides have attracted increasing attention because of their diverse geometry configurations and extraordinary properties. Using the particle swarm optimization (PSO)based global structure search method combined with first principles calculations, we have identified 2D metal dinitride MN_{2} (M = Pd, Pt) sheets containing N_{2} dimers, pentaMN_{2}, which are planar and composed entirely of pentagons. These pentagonal sheets are not only dynamically, thermally and mechanically stable, but also energetically more favorable over the experimentally synthesized pyrite MN_{2} monolayers. In particular, pentaPtN_{2} can withstand a temperature as high as 2000 K, showing its potential as a refractory material. In addition, due to its unique atomic configuration, pentaMN_{2} exhibits intriguing electronic properties. PentaPdN_{2} is metallic, while pentaPtN_{2} is semiconducting with a direct band gap of 75 meV and an ultrahigh carrier mobility. This study expands the family of 2D metal nitrides, and provides new insight into finding new materials using a reliable structure search algorithm rather than chemical and physical intuition.
1. Introduction
Since the discovery of graphene,^{1} a lot of twodimensional (2D) materials have been experimentally synthesized and theoretically predicted with many fascinating properties, showing great potential for various applications.^{2–8} As compared to the widely studied 2D metal oxides and carbides, much less effort has been devoted to the nitrogen analogues, metal nitrides (MNs). Among the recently studied 2D MNs, the sheets can be in very distinct geometry configurations.^{9–16} For instance, the Ti_{4}N_{3} sheet has a MXene like geometric structure with multiatomic layers and atomic N,^{9} while the tetraMoN_{2} sheet has a square lattice with a singleatomic thickness and atomic N,^{17} and the OYN_{2} sheet possesses a buckled monolayer where the metal atoms are sandwiched between the top and bottom layers of N_{2} dimers.^{18} Additionally, previous studies reported that the pentaAlN_{2} sheet has a buckled Cairo pentagonal tiling geometry with N_{2} dimers,^{19} whereas the BeN_{2} sheet is in distorted honeycomb lattice containing N_{4} cluster units.^{20} These studies demonstrate that the geometric structures of 2D MNs sensitively depend on the metal species and the atomic ratio of metal/N, exhibiting complexities and diversities.
Moreover, the diverse and flexible bonding features of a metal with N/N_{2} lead to rich physical and chemical properties of MNs. For example, the HMoN_{2} sheet displays a robust ferromagnetism with a Curie temperature over 420 K,^{21} the metallic OYN_{2} sheet shows promising potential as an anode material for potassium (K) ion batteries with high K capacity,^{18} and the pentaAlN_{2} sheet is a halfmetal and presents ferromagnetism as well.^{19} These advances inspire us to carry out the present study focusing on 2D MN_{2} (M = Pd, Pt) systems by using the global structure search method and first principles calculations. Our results show that the most stable monolayers are composed of pure pentagons, thus named pentaMN_{2}. They are totally planar, differing from the other hitherto reported 2D pentagonbased materials which have buckled structures.^{8,20,22–29} We further systematically investigated their physical properties, including melting points, inplane Young's modulus, ideal strength, band structure, charge distribution, and carrier mobility. In particular, pentaPtN_{2} possesses ultrahigh melting point over 2000 K and carrier mobility comparable with that of graphene.
2. Computational methods
We use the Crystal structure AnaLYsis by Particle Swarm Optimization (CALYPSO)^{30} code to search for the ground state structures of 2D PdN_{2} and PtN_{2}, and this method has been successfully employed to predict numerous structures ranging from elemental crystals to binary and ternary compounds.^{31} In the global search, the PSO algorithm is used with thirty generations and population size up to thirty, which means a total of 900 structures are produced including the symmetry P4/mbm, Pmmm and so on. It reaches convergence quickly after 14 generations. The unit cells containing 6 atoms and the buffering thickness is set to 0.15 Å to accommodate the buckling of the structures.
Density functional theory (DFT) based firstprinciples calculations are further performed to study the structural and electronic properties as implemented in the Vienna ab initio Simulation Package (VASP)^{32} with the projector augmented wave potential (PAW)^{33} to treat the ionelectron interactions. The generalized gradient approximation (GGA)^{34} with the Perdew–Burke–Ernzerhof (PBE) functional^{35} is used to describe the exchange and correlation energy. All atoms of system are relaxed with the convergence criteria for the force and total energy set to be 10^{−2} eV Å^{−1} and 10^{−4} eV, respectively. The kinetic energy cutoff is set to be 600 eV for all calculations. The Monkhorst–Pack kmesh^{36} is employed to the sample k point of the first Brillouin zone. The kpoint grids are set as 9 × 9 × 1 and 25 × 25 × 1 for the geometry optimization and electronic structure calculations, respectively. For eliminating the interaction between adjacent cells in the vertical orientation, a vacuum distance of 20 Å is used. The PHONOPY package is adopted to calculate the phonon dispersion of the structures and verify their dynamic stability.^{37}
In 2D materials, the carrier mobility can be calculated by using the deformation potential (DP) theory^{38} and given by the expression:

 (1) 
where
m_{e}* is the carrier effective mass, and
m_{d} is the carrier average effective mass defined by
.
T is the temperature, which is set to be 300 K in our calculations.
E_{l} represents the deformation potential constant defined as
E_{l} = Δ
E/(Δ
l_{x}/
l_{x}). Δ
E is the energy change of the
ith band under proper structure compression or dilatation,
l_{x} is the lattice constant along the deformation direction and Δ
l_{x} is the amount of the deformation.
C_{2D} is the elastic modulus of the longitudinal strain in the propagation directions of the longitudinal acoustic wave given by (
E −
E_{0})/
S_{0} =
C_{2D}δ^{2}/2, where
E is the total energy and
S_{0} is the area of the cell and
δ is the applied uniaxial strain.
3. Results and discussion
3.1 Geometric structures
By performing global structure search, we obtain some lowenergy structures of 2D MN_{2} (M = Pt, Pd). Among them, three energetically most stable isomers are displayed in Fig. 1, where the relative energies of the structures are also listed. The groundstate structures of 2D PdN_{2} and PtN_{2} are shown in the bottom panel of Fig. 1. They both have tetragonal structures and P4/mbm symmetry (space group no. 127). From the top views of the structures, one can see that the MN_{2} monolayers are composed entirely of pentagons, which form a Cairo pentagonal tiling,^{39} thus the monolayers are termed pentaMN_{2}. While the side views show that the pentaMN_{2} monolayers are in ideal planar forms which are rare in previously reported pentagonal structures. The optimized lattice constants of pentaPdN_{2} are a = b = 4.873 Å, while those of pentaPtN_{2} are a = b = 4.826 Å. In pentaMN_{2}, all N atoms are in dimers and form the N–N bonds. The N–N bond lengths in pentaPdN_{2} and pentaPtN_{2} are 1.225 Å and 1.260 Å, respectively, both showing the double bond characters. Specifically in pentaMN_{2} sheets, the planar geometry structures and the bond angles θ_{N–M–N} = 90° and θ_{N–N–M} = 120° indicate the square hybridization configurations of M atoms and the regular triangle hybridization configurations of N atoms.

 Fig. 1 The geometrical structures of lowlying (a) PdN_{2} and (b) PtN_{2} and their relative energy. The total energy of the lowestenergy structure (the bottom panel) is set as 0 eV.  
3.2 Stability
To evaluate the energetic stability, we calculate the cohesive energy of pentaMN_{2} by following the formula: E_{coh} = (2E_{M} + 4E_{N} − E_{MN2})/6 (E_{M}, E_{N}, and E_{MN2} are the total energies of a single M atom, a single N atom, and one unit cell of the pentaMN_{2} monolayer, respectively). The calculated cohesive energies of pentaPdN_{2} and pentaPtN_{2} are 4.37 and 5.01 eV per atom, respectively. Using the same calculation methods, we also calculate the cohesive energies of pyrite PdN_{2} and PtN_{2}, which have the buckled pentagonal structures and have been synthesized experimentally.^{40–42} The calculated results are 4.24 and 4.87 eV per atom for pyrite PdN_{2} and PtN_{2}, from which one finds that pentaMN_{2} are energetically more stable than pyrite MN_{2}. In order to evaluate the dynamic stability, we also calculate the phonon dispersion of pentaMN_{2}, with results shown in Fig. 2(a and b). For both of pentaPdN_{2} and pentaPtN_{2}, there are no imaginary modes in the entire Brillouin zone, demonstrating that the pentaMN_{2} monolayers are dynamically stable.

 Fig. 2 (a and b) Phonon band structures of pentaPdN_{2} and pentaPtN_{2}, respectively. (c) Variation of the total potential energy of pentaPdN_{2} with simulation time during AIMD simulation at 700 K and (d) that of pentaPtN_{2} at 2000 K. The insets are the top and side views of the geometrical structures at the end of simulation.  
We then perform ab initio molecular dynamics (AIMD) simulations to study the thermal stability of pentaMN_{2}. The 3 × 3 × 1 supercells are constructed to reduce the constraint of periodic boundary conditions. For pentaPdN_{2}, the simulations are carried out at a temperature of 700 K, for 6 picoseconds (ps) with a time step of 1 femtosecond (fs). The variations of the total potential energies for the 3 × 3 × 1 pentaPdN_{2} supercells with respect to simulation time are plotted in Fig. 2(c). We can see that the average values of the total potential energy remain nearly constants during the entire simulations, confirming that pentaMN_{2} are thermally stable at 700 K. However, when we increase the temperature to 900 K, the structure is found to be seriously disrupted. It suggests that pentaPdN_{2} has a melting point between 700 and 900 K. Much more interesting is that, using the same method, pentaPtN_{2} can withstand temperatures as high as 2000 K (see Fig. 2(d)), and it has a melting point between 2000 and 2500 K. Like the case of other metal nitrides such as IrN_{2} and OsN_{2},^{43} pentaPtN_{2} exhibits potential as a highly refractory material.
It is necessary to estimate the effect of lattice distortion on structural stability as well. Therefore, we calculate the linear elastic constants to investigate the mechanical stability of pentaMN_{2} under small lattice distortion. The elastic constants are obtained by applying the finite distortion method^{44} and the results are summarized in Table 1. Here we use the standard Voigt notation^{45} and the elastic constants C_{11}, C_{22}, C_{12}, and C_{66} are components of the elastic modulus tensor, corresponding to the second partial derivative of strain energy with respect to strain (all other components are strictly zero for 2D structures). According to Born–Huang criteria,^{46} for a mechanically stable 2D sheet, its elastic constants need to satisfy C_{11}C_{22} – C^{2}_{12} > 0, C_{66} > 0. From the results in Table 1, the elastic constants of pentaPdN_{2} and pentaPtN_{2} both meet the criteria, confirming that the pentaMN_{2} monolayers are mechanically stable.
Table 1 Elastic constants C and inplane Young's modulus E in the unit of N m^{−1} and Poisson's ratio ν of pentaMN_{2}
Structure 
C
_{11}

C
_{22}

C
_{12}

C
_{66}

E

ν

PentaPdN_{2} 
129.2 
129.2 
16.9 
28.9 
127 
0.13 
PentaPtN_{2} 
227.7 
227.7 
40.2 
42.2 
220 
0.18 
3.3 Mechanical properties
With the formula E = (C^{2}_{11} − C^{2}_{12})/C_{11}, we obtain the inplane Young's moduli of pentaMN_{2}, which are shown in Table 1 as well. The Young's modulus of pentaPtN_{2} is 220 N m^{−1}, close to 271 N m^{−1} of the hBN monolayer^{45} and 263.8 N m^{−1} of pentagraphene,^{8} indicating the good inplane stiffness of pentaPtN_{2}. Although the Young's modulus of pentaPdN_{2} (127 N m^{−1}) is smaller than that of pentaPtN_{2}, it is larger than those of the MoS_{2} monolayer (120 N m^{−1}), germanene (42 N m^{−1}) and silicene (61 N m^{−1}).^{47,48} The high inplane stiffness implies the strong bonding in pentaMN_{2}, which is beneficial to the structure stability. In addition, we also calculate the Poisson's ratio of pentaMN_{2} by ν = C_{12}/C_{11}, and present the results in Table 1. Besides, the ideal strength is also a very important mechanical property of a nanomaterial. Thus, we apply various biaxial tensile strains to the monolayers and calculate the corresponding stress to study the ideal strength of pentaMN_{2}. The calculated results are plotted in Fig. 3(a and b), which show that the strain at the maximum stress is 17.5% for pentaPdN_{2} and 15.0% for pentaPtN_{2}. The high inplane Young's modulus and high ideal strength indicate that the pentaMN_{2} monolayers exhibit excellent mechanical properties.

 Fig. 3 Stress–strain relationship of (a) pentaPdN_{2} and (b) pentaPtN_{2} under biaxial tensile strain. The black arrows denote the strain at the maximum stress.  
3.4 Electronic properties
To study the charge distribution of pentaMN_{2}, we calculate the deformation charge density which is defined as the total charge density with the contribution of isolated M and N atoms being subtracted. The slices on the (001) plane are shown in Fig. 4(a and b), where the blue and red regions represent the depletion and accumulation of charge, respectively. It is obvious that the electrons transfer from M to N atoms in pentaMN_{2} since M is less electronegative as compared with N. Furthermore, the Bader charge analysis^{49–51} shows that each Pd (Pt) atom transfers 0.66 (0.76) e to each N_{2} dimer, consistent with the results of deformation charge density.

 Fig. 4 Inplane deformation charge density slices of (a) pentaPdN_{2} and (b) pentaPtN_{2}, where the blue and red regions represent depletion and accumulation of charge, respectively. The units of colour bar both are e Å^{−3}. (c and d) are ELF slices of pentaPdN_{2} and pentaPtN_{2} with slices crossing the structure plane.  
In addition, to get the bonding feature of pentaMN_{2}, we also perform the electron localization function (ELF) analysis, which is widely used to analyze the spatial distribution of electrons. ELF is defined as:

 (2) 

 (3) 

 (4) 
where
Φ_{i} is the Kohn–Sham orbital and
ρ is the local density. The value of ELF is renormalized to be in the range of 0.0 to 1.0, where 0.0 represents a very low charge density, while 0.5 and 1.0 represent delocalized and localized electrons, respectively.
Fig. 4(c and d) show that the electrons are localized on the N–N bonds, suggesting the strong covalent bonding in the N
_{2} dimer. In contrast, the electrons on the N–M bonds are localized around N atoms while the electron density is very low near M atoms, showing a feature of ionic bonding.
^{52} These results coincide with the structural characteristics of pentaMN
_{2}, which are composed of a N
_{2} dimer and a transition metal ionic center.
We then calculate the electronic band structures of pentaMN_{2} to explore the electronic properties. The PBE results for pentaPdN_{2} and pentaPtN_{2} are plotted in Fig. 5 and 6, respectively. It is clear that there is no gap between the valence and conduction bands of pentaPdN_{2}, demonstrating its metallic character. However, the band structures of pentaPtN_{2} show that it is a direct bandgap semiconductor with a small band gap of 75 meV. The valance band maximum (VBM) and conduction band minimum (CBM) are both located at the M point of the first Brillouin zone.

 Fig. 5 Electronic band structures and orbitalweighted electronic band structures of pentaPdN_{2}. The color bar indicates the weight of atomic orbital contribution to the band structures. The inset is the detail of band structures near the Fermi level at M point.  

 Fig. 6 Electronic band structures of and orbitalweighted electronic band structures of pentaPtN_{2}. The color bar indicates the weight of atomic orbital contribution to the band structures. The inset is the detail of band structures near the Fermi level at M point.  
To better understand the electronic properties of pentaMN_{2}, their orbital hybridization forms are studied. The atomic orbital contributions to the band structures of pentaPdN_{2} and pentaPtN_{2} are shown in Fig. 5 and 6. The ground state electronic configuration of an isolated N_{2} dimer is (σ_{1s})^{2}(σ_{1s}*)^{2}(σ_{2s})^{2}(σ_{2s}*)^{2}(σ_{2p})^{2}(π_{2p})^{4}. Our results show that for both of pentaPdN_{2} and pentaPtN_{2}, the occupied electronic bands near the Fermi level are mainly contributed by d orbitals of M atoms, because these orbitals have higher energy and transfer electrons to N–N dimers, and they coordinate with unoccupied orbitals of N–N. Meanwhile, the electronic bands in the energy range of −5 to −3 eV for pentaPdN_{2} and −4 to −2 eV for pentaPtN_{2} are contributed by d orbitals of M atoms with slight contributions from the p orbitals of N atoms, suggesting that d orbitals of M atoms can hybridize with the π_{2p} of N–N dimers. Also, the deeper occupied bands in the energy range from −8 to −6 eV are mostly contributed by N–N dimers because their energies are much lower than that of the Md states.
3.5 Carrier mobility
Besides the mechanical and electronic properties, the carrier mobility is also a significant factor to determine the performance of a semiconductor. By using the acousticphononlimited scattering model, we calculate the carrier mobility of pentaPtN_{2}. Due to the tetragonal symmetry of pentaPtN_{2}, the carrier mobility along x and y directions is isotropic. All the calculated results containing effective mass, deformational potential constants, 2D elastic modulus and carrier mobility are summarized in Table 2. We can see that the effective mass of electron and hole are 0.118 m_{0} and 0.198 m_{0} for pentaPtN_{2}. According to equation (a), the light effective mass and high 2D elastic modulus lead to ultrahigh carrier mobilities (μ_{e} = 219 × 10^{3} cm^{2} V^{−1} s^{−1} and μ_{h} = 132 × 10^{3} cm^{2} V^{−1} s^{−1}) which are higher than that of phosphorene (1–26 × 10^{3} cm^{2} V^{−1} s^{−1}) and even are close to that of graphene (∼200 × 10^{3} cm^{2} V^{−1} s^{−1}).^{53–55} The ultrahigh carrier mobility allows pentaPtN_{2} to be of potential application in electronic devices.
Table 2 Effective mass (m*), deformational potential constant (E_{1}), 2D elastic modulus (C_{2D}) and carrier mobility (μ) for electrons and holes of pentaPtN_{2}
Carrier type 
m*/m_{0} 
E
_{1} (eV) 
C
_{2D} (J m^{−2}) 
μ (10^{3} cm^{2} V^{−1} s^{−1}) 
PtN_{2} 
e 
0.118 
1.401 
227.7 
219 
h 
0.198 
1.391 
227.7 
132 
4. Conclusions
In summary, by using the CALYPSO code, we obtain the 2D groundstate structures of MN_{2} (M = Pd, Pt), which are composed entirely of pentagons. DFT calculations confirm their dynamical, thermal and mechanical stabilities. Moreover, compared with the experimentally synthesized pyrite MN_{2}, the pentaMN_{2} monolayers show several distinct characteristics: (1) pentaMN_{2} monolayers are more stable than the pyrite structures in energy; (2) pentaMN_{2} show high melting point. Particularly, pentaPtN_{2} can withstand the temperature over 2000 K before it breaks down; (3) pentaMN_{2} possess high inplane Young's modulus and high ideal strength; (4) pentaPtN_{2} has ultrahigh carrier mobility, with the values of μ_{e} = 219 × 10^{3} cm^{2} V^{−1} s^{−1}, μ_{h} = 132 × 10^{3} cm^{2} V^{−1} s^{−1}, which are higher than those of phosphorene and comparable with graphene. These excellent properties of pentaMN_{2} imply their potential applications in high refractory materials and nanoscale electronic devices. The present study would deepen our understandings on the diversity of geometries and properties of 2D metal nitrides.
Conflicts of interest
There are no conflicts to declare.
Acknowledgements
This work was supported by grants from the National Key Research and Development Program of China (2016YFE0127300, and 2017YFA0205003), the National Natural Science Foundation of China (NSFC51471004, and NSFC21773004), and supported by the High Performance Computing Platform of Peking University, China.
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