First-Principles Study on III-Nitride Polymorphs: AlN/GaN/InN in the Pmn21 Phase

The structural, mechanical, and electronic properties, as well as stability, elastic anisotropy and effective mass of AlN/GaN/InN in the Pmn21 phase were determined using density functional theory (DFT). The phonon dispersion spectra and elastic constants certify the dynamic and mechanical stability at ambient pressure, and the relative enthalpies were lower than those of most proposed III-nitride polymorphs. The mechanical properties reveal that Pmn21-AlN and Pmn21-GaN possess a high Vickers hardness of 16.3 GPa and 12.8 GPa. Pmn21-AlN, Pmn21-GaN and Pmn21-InN are all direct semiconductor materials within the HSE06 hybrid functional, and their calculated energy band gaps are 5.17 eV, 2.77 eV and 0.47 eV, respectively. The calculated direct energy band gaps and mechanical properties of AlN/GaN/InN in the Pmn21 phase reveal that these three polymorphs may possess great potential for industrial applications in the future.


Introduction
Third-generation semiconductor materials have been of great interest in the past decade because of their importance in scientific research and their industrial applications [1][2][3][4][5][6]. Group III-V compound semiconductors are among the most widely used third-generation semiconductor materials, represented by GaN, AlN, SiC and diamond. These semiconductor materials have some common advantages, such as wide band gap, high electron saturation rate, high radiation resistance, high thermal conductivity, and high electric field [7][8][9][10]. Thus, they have important technological applications in optoelectronic devices, light-emitting diodes (LEDs), high-frequency electronic devices, radiation-resistant electronic devices and high-power electronic devices.
First-principles calculations based on density functional theory (DFT) are among the most reliable and popular microscopic theories in material science. This method has a high ability to predict the material structures and properties [11][12][13][14][15][16][17][18][19][20]. Yang et al. [21] predicted a novel high-pressure superhard BN phase at high pressure through a developed particle swarm optimization (PSO) algorithm. The calculations revealed that its Vickers hardness is 47 GPa, which is characteristic of a superhard material. Liu et al. [22] proposed three new metastable phases (P6 4 22, C222, Pbca, and I43d) for AlAs. The electronic band structure calculation reveals I43d-AlAs is a direct semiconductor material with energy band gap of 1.76 eV, whereas C222and P6 4 22-AlAs are indirect semiconductor materials with band gaps of 0.47 eV and 1.36 eV, respectively. Xu et al. [23] calculated the mechanical and thermodynamic properties of AlN/AlP/AlAs compounds in wurtzite, zinc-blende, and rock-salt structures through first-principles calculations. They found the hardness and Debye temperatures both decrease with the X (X = N, P, As) atomic number. Zhang et al. [24] studied the physical properties of four

Computational Methods
For the modeling of the AlN/GaN/InN in the Pmn2 1 phase, we use density functional theory (DFT)-based methods [31] realized in the plane-wave pseudopotential approach in the Cambridge Sequential Total Energy Package (CASTEP) codes [32]. The generalized gradient approximation (GGA) of the Perdew-Burke-Ernzerhof (PBE) [33] scheme and the local density approximation (LDA) [34,35] were used to optimize the geometry and calculate elastic constants.
The values of the cutoff energies are set as 280/330/350 eV for AlN/GaN/InN in the Pmn2 1 phase, with k-point samplings with 0.025 Å −1 (13 × 3 × 7/11 × 3 × 6/13 × 3 × 7) in the first irreducible Brillouin zone for Pmn2 1 -AlN, Pmn2 1 -GaN and Pmn2 1 -InN. The geometry optimization parameters were determined using the Broyden-Fletcher-Goldfarb-Shenno (BFGS) algorithm [36], with the following convergence tolerance: displacement of atoms during the geometry optimization less than 0.0005 Å, energy change less than 5 × 10 −6 eV per atom, stress less than 0.02 GPa, and residual force below 0.01 eV/Å. The phonon frequencies were calculated using linear response theory [37]. We used the Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional [38] to calculate electronic band structures and partial density of state (PDOS) based on the optimized geometry. Figure 1 shows the crystal structure of the predicted Pmn2 1 -AlN/GaN/InN in different views and forms. This structure belongs to the Pmn2 1 space group of the orthorhombic system. The structure of Pmn2 1 -AlN/GaN/InN consists of sp 3 -bonded rings in three different shapes. Figure 1a,c shows the four-, six-, and eight-membered rings consisting of Al/Ga/In atoms and N atoms along two different views. Figure 1b shows the six-membered rings, which can form a honeycomb-like structure. Figure 1c shows that three four-membered Al/Ga/In-N rings are located by the eight-membered ring, and another four-membered Al/Ga/In-N ring is located by the top of the six-membered ring. There are eight atoms in the conventional cell of   The optimized equilibrium lattice parameters for AlN/GaN/InN in the Pmn2 1 phase at zero pressure are listed in Table 1. The results show that the calculated values are in great agreement with other theoretical results and experimental results in Table 1, which shows that the present optimization and calculation are reliable. The results obtained by PBE functional are closer to experimental values, so the results obtained by PBE functional are used in this paper. The bond angles of Al-N, Ga-N and In-N in the Pmn2 1 phase range from 87.62 degrees to 117.91 degrees. In Pmn2 1 -AlN, Pmn2 1 -GaN and Pmn2 1 -InN, the bond lengths range from 1.872 Å to 1.935 Å, 1.939 Å to 2.009 Å and 2.170 Å to 2.243 Å, respectively. For zb-AlN (zinc-blende AlN), zb-GaN, and zb-InN, the bond lengths are 1.905 Å, 1.975 Å and 2.205 Å, respectively. For wz-AlN (wurtzite AlN), wz-GaN, and wz-InN, the bond lengths range from 1.901 Å to 1.913 Å, 1.973 Å to 1.981 Å and 2.202 Å to 2.211 Å, respectively. The densities of Pmn2 1 -AlN, Pmn2 1 -GaN and Pmn2 1 -InN are 3.150 g/cm 3 , 5.742 g/cm 3 and 6.351 g/cm 3 , respectively, which are close to those of AlN/GaN/InN in the wurtzite phase.

Stability and Mechanical Properties
The phonon spectra of AlN/GaN/InN in the Pmn2 1 phase were calculated under ambient conditions (see Figure 2) in this work. There is no imaginary frequency throughout the Brillouin zone, which means Pmn2 1 -AlN, Pmn2 1 -GaN and Pmn2 1 -InN are all dynamically stable. The relative enthalpies at zero pressure are also calculated using the following expression: where the E total is the total energy of Pmn2 1 -XN (X = Al, Ga, In); the n X is the number of Al/Ga/In atoms in the cell; the n N is the number of N atoms in the cell; E X solid is the energy of one X (X = Al, Ga, In) atom in elemental X (X = aluminium, gallium, indium). E N solid is the energy of one nitrogen atom in elemental nitrogen. For AlN, the enthalpies relative to zb-AlN and wz-AlN are 0.004 eV per formula unit and 0.045 eV per formula unit. For GaN, the enthalpies relative to zb-GaN and wz-GaN are 0.055 eV per formula unit and 0.066 eV per formula unit. Finally, for InN, the enthalpies relative to zb-InN and wz-InN are 0.066 eV per formula unit and 0.054 eV per formula unit. Compared with other reported III-nitride polymorphs, Pmn2 1 -AlN (0.045 eV/f.u.) is more favorable than Pnma-AlN (0.232 eV/f.u.) [42], Cmcm-AlN (0.206 eV/f.u.) and Pbca-AlN (0.075 eV/f.u.) [30]. As seen from the enthalpies relative to wz-GaN, Pmn2 1 -GaN (0.066 eV eV/f.u.) is more favorable than Pnma-GaN (0.264 eV/f.u.) [43]. The results reveal that Pmn2 1 -AlN and Pmn2 1 -GaN are more favorable than most proposed polymorphs of AlN and GaN [30,42,43].

Stability and Mechanical Properties
The phonon spectra of AlN/GaN/InN in the Pmn21 phase were calculated under ambient conditions (see Figure 2) in this work. There is no imaginary frequency throughout the Brillouin zone, which means Pmn21-AlN, Pmn21-GaN and Pmn21-InN are all dynamically stable. The relative enthalpies at zero pressure are also calculated using the following expression: where the total E is the total energy of Pmn21-XN (X = Al, Ga, In); the X n is the number of Al/Ga/In atoms in the cell; the N n is the number of N atoms in the cell; X solid E is the energy of one X (X = Al, Ga, In) atom in elemental X (X= aluminium, gallium, indium). N solid E is the energy of one nitrogen atom in elemental nitrogen. For AlN, the enthalpies relative to zb-AlN and wz-AlN are 0.004 eV per formula unit and 0.045 eV per formula unit. For GaN, the enthalpies relative to zb-GaN and wz-GaN are 0.055 eV per formula unit and 0.066 eV per formula unit. Finally, for InN, the enthalpies relative to zb-InN and wz-InN are 0.066 eV per formula unit and 0.054 eV per formula unit. Compared with other reported III-nitride polymorphs, Pmn21-AlN (0.045 eV/f.u.) is more favorable than Pnma-AlN (0.232 eV/f.u.) [42], Cmcm-AlN (0.206 eV/f.u.) and Pbca-AlN (0.075 eV/f.u.) [30]. As seen from the enthalpies relative to wz-GaN, Pmn21-GaN (0.066 eV eV/f.u.) is more favorable than Pnma-GaN (0.264 eV/f.u.) [43]. The results reveal that Pmn21-AlN and Pmn21-GaN are more favorable than most proposed polymorphs of AlN and GaN [30,42,43].  Table 2 shows the elastic constants and elastic modulus for AlN/GaN/InN in the Pmn21 phase, together with reported calculated and experimental results for comparison [44][45][46]. There are nine independent elastic constants for the orthorhombic phase, namely, C11, C12, C13, C22, C23, C33, C44, C55 and C66. The mechanical stability criteria [47] of the orthorhombic structure are given as follows:  Table 2 shows the elastic constants and elastic modulus for AlN/GaN/InN in the Pmn2 1 phase, together with reported calculated and experimental results for comparison [44][45][46]. There are nine Materials 2020, 13, 3212 5 of 12 independent elastic constants for the orthorhombic phase, namely, C 11 , C 12 , C 13 , C 22 , C 23 , C 33 , C 44 , C 55 and C 66 . The mechanical stability criteria [47] of the orthorhombic structure are given as follows: The calculated elastic constants of Pmn2 1 -AlN, Pmn2 1 -GaN, and Pmn2 1 -InN indicate that these structures are mechanically stable. According to the elastic constants listed in Table 2, the C 11 values of Pmn2 1 -AlN, Pmn2 1 -GaN, and Pmn2 1 -InN are larger than those of wz-AlN/GaN/InN and zb-AlN/GaN/InN, which means that all three proposed polymorphs possess a better compression resistance in the x direction than its corresponding III-nitride in the wurtzite phase and zinc-blende phase. Additionally, the C 33 values of Pmn2 1 -AlN/GaN/InN are smaller than those of wz-AlN/GaN/InN and zb-AlN/GaN/InN, which reveal a better deformability along the z direction.
The bulk modulus (B) and shear modulus (G) were calculated by using the Voigt-Reuss-Hill approximation [48], which are defined as Where B V means the Voigt approximation of bulk modulus B; B R is the Reuss approximation of bulk modulus B; G V means the Voigt approximation of shear modulus G; and G R is the Reuss approximation of shear modulus G.
The Young's modulus E is used to offer a measure of the stiffness of a solid. The larger the value of E is, the stiffer the material is [49]. The Young's modulus E and Poisson's ratio v were determined as follows [50]: The obtained results are listed in Table 2. The calculated bulk modulus B and shear modulus G of Pmn2 1 -AlN/GaN/InN are slightly less than those of wz-AlN/GaN/InN and zb-AlN/GaN/InN. The shear modulus is less than the bulk modulus for Pmn2 1 -AlN/GaN/InN. The values of B/G and Poisson's ratio are associated with brittleness or ductility. If B/G > 1.75 [51], a material is characterized as ductile; otherwise, the material has a brittle character. Poisson's ratio <0.26 indicates brittle compounds [52]. Obviously, Pmn2 1 -AlN and Pmn2 1 -GaN exhibit brittle character, whereas Pmn2 1 -InN behaves in a ductile manner. The Vickers hardness (Hv) was calculated by adopting Chen's formula based on an empirical scheme [53]: Materials 2020, 13, 3212 6 of 12 The calculated and experimental hardness values are presented in Table 2. The calculated hardness reveals that the proposed Pmn2 1 -AlN and Pmn2 1 -GaN possess a high Vickers hardness of 16.3 GPa and 12.8 GPa; however, Pmn2 1 -InN possesses a Vickers hardness of 3.9 GPa. The results show that in the Pmn2 1 , zinc-blende and wurtzite phases, AlN possesses the highest hardness among these three polymorphs, the hardness of GaN is slightly lower than that of AlN, and InN possesses the lowest hardness.

Mechanical Anisotropic Properties
It is well-known that the anisotropy of elasticity is an important implication in engineering science and crystal physics. Figure 3 shows the variation in Young's modulus for Pmn2 1 -AlN/GaN/InN with three-dimensional crystallographic directions. The directional dependence of the Young's modulus E for the orthorhombic crystal is [54]: where S ij refers to the elastic compliance constants and m 1 , m 2 , and m 3 are the direct cosines of the [u v w] direction. Figure 3 reveals that Pmn2 1 -AlN possesses the smallest elastic anisotropy and largest Young's modulus among these three polymorphs.

Mechanical Anisotropic Properties
It is well-known that the anisotropy of elasticity is an important implication in engineering science and crystal physics. Figure 3 where Sij refers to the elastic compliance constants and m1, m2, and m3 are the direct cosines of the [u v w] direction.  To further understand the elastic anisotropy of the Young's modulus along different directions, the dependence of the Young's modulus on orientation was investigated by taking the tensile axis within a specific plane. Let α be the angle between [ Two-dimensional representations of the Young's modulus for Pmn21-AlN/GaN/InN are illustrated in Figure 4. The lines representing Pmn21-AlN, Pmn21-GaN, and Pmn21-InN are shown in blue, red, and green, respectively. From Figures 3 and 4, we find that Pmn21-InN exhibits the smallest elastic anisotropy in the Young's modulus and that Pmn21-AlN exhibits the largest elastic anisotropy. E −1 = S 22 sin 4 γ + S 33 cos 4 γ + 2S 33 sin 2 2γ + S 44 sin 2 2γ /4 (9) Two-dimensional representations of the Young's modulus for Pmn2 1 -AlN/GaN/InN are illustrated in Figure 4. The lines representing Pmn2 1 -AlN, Pmn2 1 -GaN, and Pmn2 1 -InN are shown in blue, red, and green, respectively. From Figures 3 and 4, we find that Pmn2 1 -InN exhibits the smallest elastic anisotropy in the Young's modulus and that Pmn2 1 -AlN exhibits the largest elastic anisotropy. For these three primary planes, the maximum values for Pmn2 1 -AlN, Pmn2 1 -GaN, and Pmn2 1 -InN all occur in the xz plane and xy plane, and the minimum values occur in the yz plane. In addition, the xy plane of Pmn2 1 -AlN, Pmn2 1 -GaN, and Pmn2 1 -InN exhibits the smallest elastic anisotropy in the Young's modulus, and the E max /E min ratios are 1.09, 1.13 and 1.14, respectively. The xz plane exhibits the greatest elastic anisotropy in the Young's modulus for Pmn2 1 -AlN/GaN/InN.  In addition, apart from the surface construction and two-dimensional representation of Young's modulus, the universal anisotropic index A U [55] is also calculated for deeper investigation in this work. A U is defined as A U = 5GV/GR + BV/BR − 6, where BV (BR) and GV (GR) represent the symbols of the bulk modulus and shear modulus at Voigt (Reuss) bounds, respectively. The A U of Pmn21-AlN/GaN/InN are 0.0454, 0.0801 and 0.1006, respectively. The calculated A U is similar to the threedimensional and two-dimensional representation of the Young's modulus, it also shows an increasing tendency with the group III element (Al, Ga, In) atomic number.

Electrical and Thermal Properties
The energy band structure of the material determines a variety of properties, especially its electronic and optical properties. The electronic band structure, together with partial density of state (PDOS) of Pmn21-AlN/GaN/InN are shown in Figure 5. All three proposed compounds are semiconductor materials with direct bandgaps at G points of 5.17 eV (Pmn21-AlN), 2.77 eV (Pmn21-GaN) and 0.47 eV (Pmn21-InN), notably Pmn21-AlN and Pmn21-GaN, which are wide bandgap semiconductors [56]. In a light emitting diode, only the direct transition process can produce light, which is the main transition method for direct semiconductors. The wavelength of light is mostly determined by the energy band gap of the semiconductors [57]. The band gap of Pmn21-GaN is 2.77 eV, which is lower than that of wz-GaN (3.4 eV) [58] and corresponds to the blue light region, making it a potential material for blue LEDs without adulteration. The energy band gaps of Pmn21-AlN and Pmn21-InN correspond to the ultraviolet region and infrared region, respectively. This suggests that Pmn21-AlN and Pmn21-InN have the potential to produce optoelectronic devices. In addition, apart from the surface construction and two-dimensional representation of Young's modulus, the universal anisotropic index A U [55] is also calculated for deeper investigation in this work.
and G V (G R ) represent the symbols of the bulk modulus and shear modulus at Voigt (Reuss) bounds, respectively. The A U of Pmn2 1 -AlN/GaN/InN are 0.0454, 0.0801 and 0.1006, respectively. The calculated A U is similar to the three-dimensional and two-dimensional representation of the Young's modulus, it also shows an increasing tendency with the group III element (Al, Ga, In) atomic number.

Electrical and Thermal Properties
The energy band structure of the material determines a variety of properties, especially its electronic and optical properties. The electronic band structure, together with partial density of state (PDOS) of Pmn2 1 -AlN/GaN/InN are shown in Figure 5. All three proposed compounds are semiconductor materials with direct bandgaps at G points of 5.17 eV (Pmn2 1 -AlN), 2.77 eV (Pmn2 1 -GaN) and 0.47 eV (Pmn2 1 -InN), notably Pmn2 1 -AlN and Pmn2 1 -GaN, which are wide bandgap semiconductors [56]. In a light emitting diode, only the direct transition process can produce light, which is the main transition method for direct semiconductors. The wavelength of light is mostly determined by the energy band gap of the semiconductors [57]. The band gap of Pmn2 1 -GaN is 2.77 eV, which is lower than that of wz-GaN (3.4 eV) [58] and corresponds to the blue light region, making it a potential material for blue LEDs without adulteration. The energy band gaps of Pmn2 1 -AlN and Pmn2 1 -InN correspond to the ultraviolet region and infrared region, respectively. This suggests that Pmn2 1 -AlN and Pmn2 1 -InN have the potential to produce optoelectronic devices.
Materials 2020, 13, x FOR PEER REVIEW 8 of 12 The lines represent the total DOS, N-s, N-p, X-s, and X-p (X = Al, Ga, In) are set to purple, black, red, blue and green, respectively. According to the PDOS diagram of Pmn21-AlN/GaN/InN, the density of states mainly comes from N-p orbitals. Below 0 eV, the total DOS in the valence band originates mainly from N-p orbitals for these three compounds. Above 0 eV, the N-p, X-s and X-p orbitals (X = Al, Ga, In) contribute greatly and overlap with each other. In addition, the N-s orbitals contribute the smallest proportion in the valance band and conduction band. For Pmn21-AlN/GaN/InN, the peaks are all present in the energy region close to 0 eV (−2 to 0 eV), and the DOS is mainly due to the contributions from N-p orbitals; the contribution of other electron orbitals is relatively small. These DOS peaks depend on the N-p/X-p (X = Al, Ga, In) bonding orbital contribution. The results show that covalent N-X (X = Al, Ga, In) interactions exist. The effective mass is also calculated by quadratic polynomial fitting of valence and conduction bands along the x, y, and z directions. The effective mass can be determined as follows: . The calculated hole effective mass and electron effective mass of Pmn21-AlN/GaN/InN, zb-AlN/GaN/InN and the experimental values for comparison are listed in Table 3. The electron effective mass of these three proposed III-nitride polymorphs along the x, y and z directions gradually decrease, whereas the electron effective mass along these three directions are almost the same. For Pmn21-AlN/GaN/InN, the largest hole effective mass occurs along the y direction, and the smallest occurs along the z direction. For Pmn21-AlN, the hole effective mass and the electron effective mass along the x, y and z directions are larger than those of Pmn21-GaN, Pmn21-InN and zb-AlN/GaN/InN. For Pmn21-GaN, the hole effective mass along the y direction is close to that of Pmn21-InN. Finally, for Pmn21-InN, the hole effective mass along the z direction is much smaller than that of zb-InN, and the electron effective mass of Pmn21-InN along all directions is close to that of zb-InN. The lines represent the total DOS, N-s, N-p, X-s, and X-p (X = Al, Ga, In) are set to purple, black, red, blue and green, respectively. According to the PDOS diagram of Pmn2 1 -AlN/GaN/InN, the density of states mainly comes from N-p orbitals. Below 0 eV, the total DOS in the valence band originates mainly from N-p orbitals for these three compounds. Above 0 eV, the N-p, X-s and X-p orbitals (X = Al, Ga, In) contribute greatly and overlap with each other. In addition, the N-s orbitals contribute the smallest proportion in the valance band and conduction band. For Pmn2 1 -AlN/GaN/InN, the peaks are all present in the energy region close to 0 eV (−2 to 0 eV), and the DOS is mainly due to the contributions from N-p orbitals; the contribution of other electron orbitals is relatively small. These DOS peaks depend on the N-p/X-p (X = Al, Ga, In) bonding orbital contribution. The results show that covalent N-X (X = Al, Ga, In) interactions exist.
The effective mass is also calculated by quadratic polynomial fitting of valence and conduction bands along the x, y, and z directions. The effective mass can be determined as follows: (m * ) −1 = (1/tsh 2 )(∂ 2 E/∂k 2 ). The calculated hole effective mass and electron effective mass of  Table 3. The electron effective mass of these three proposed III-nitride polymorphs along the x, y and z directions gradually decrease, whereas the electron effective mass along these three directions are almost the same. For Pmn2 1 -AlN/GaN/InN, the largest hole effective mass occurs along the y direction, and the smallest occurs along the z direction. For Pmn2 1 -AlN, the hole effective mass and the electron effective mass along the x, y and z directions are larger than those of Pmn2 1 -GaN, Pmn2 1 -InN and zb-AlN/GaN/InN. For Pmn2 1 -GaN, the hole effective mass along the y direction is close to that of Pmn2 1 -InN. Finally, for Pmn2 1 -InN, the hole effective mass along the z direction is much smaller than that of zb-InN, and the electron effective mass of Pmn2 1 -InN along all directions is close to that of zb-InN.

Conclusions
In summary, we investigated the structural, stability, mechanical and electronic properties of Pmn2 1 -AlN/GaN/InN. Pmn2 1 -AlN/GaN/InN are mechanically and dynamically stable. The relative enthalpies of Pnma-AlN and Pmn2 1 -GaN are more favorable than those of most predicted III-nitride polymorphs. The elastic constants indicate that Pmn2 1 -AlN/GaN/InN possess better deformation resistance properties in the x direction and better deformability along the z direction than wz-AlN/GaN/InN and zb-AlN/GaN/InN. The calculated H v values of Pmn2 1 -AlN and Pmn2 1 -GaN reveal that Pmn2 1 -AlN and Pmn2 1 -GaN possess a high hardness of 16.3 GPa and 12.8 GPa, respectively. Pmn2 1 -AlN, Pmn2 1 -GaN and Pmn2 1 -InN exhibit similar elastic anisotropies. The electron effective mass of Pmn2 1 -InN is smaller along all three directions than that of zb-InN. In addition, the hole effective mass of Pmn2 1 -GaN and Pmn2 1 -InN along the z direction are much smaller than those of zb-GaN and zb-InN. Pmn2 1 -AlN/GaN/InN are direct semiconductor materials with energy band gaps of 5.17 eV (Pmn2 1 -AlN), 2.77 eV (Pmn2 1 -GaN) and 0.47 eV (Pmn2 1 -InN). Ultimately, Pmn2 1 -AlN, Pmn2 1 -GaN and Pmn2 1 -InN may have great potential industrial applications in the future due to their superior electronic and mechanical properties.
Funding: This research was funded by the National Natural Science Foundation of China, grant number 61974116.

Conflicts of Interest:
The authors declare no conflicts of interest.