Novel III-V Nitride Polymorphs in the P4₂/mnm and Pbca Phases

Abstract

In this work, the elastic anisotropy, mechanical stability, and electronic properties for P4₂/mnm XN (XN = BN, AlN, GaN, and InN) and Pbca XN are researched based on density functional theory. Here, the XN in the P4₂/mnm and Pbca phases have a mechanic stability and dynamic stability. Compared with the Pnma phase and Pm-3n phase, the P4₂/mnm and Pbca phases have greater values of bulk modulus and shear modulus. The ratio of the bulk modulus (B), shear modulus (G), and Poisson’s ratio (v) of XN in the P4₂/mnm and Pbca phases are smaller than those for Pnma XN and Pm-3n XN, and larger than those for c-XN, indicating that Pnma XN and Pm-3n XN are more ductile than P4₂/mnm XN and Pbca XN, and that c-XN is more brittle than P4₂/mnm XN and Pbca XN. In addition, in the Pbca phases, XN can be considered a semiconductor material, while in the P4₂/mnm phase, GaN and InN have direct band-gap, and BN and AlN are indirect wide band gap materials. The novel III-V nitride polymorphs in the P4₂/mnm and Pbca phases may have great potential for application in visible light detectors, ultraviolet detectors, infrared detectors, and light-emitting diodes.

1. Introduction

Materials are the basis of human society and guide its development. The discovery and application of every major new material has brought changes to human life. Semiconductor materials are the key factor affecting the development of the semiconductor industry, and they have a vital strategic position. In the middle of the last century, the invention of single-crystal silicon and semiconductor transistors, and the successful development of their silicon-integrated circuits led to a revolution. Since the early 1970s, the successful development of quartz optical fiber materials and optical fibers—materials such as BN, BAs, and other III-V nitrides—and the invention of semiconductor lasers promoted the rapid development of optical fiber communication technology and gradually formed the high-tech industry.

In recent years, novel semiconductor materials have attracted an increasing number of researchers in materials research, such as carbon allotropes [1,2,3,4,5,6,7,8,9], light element compounds [10,11,12], Group IV semiconductor materials and theirs alloys [13,14,15,16,17,18,19,20], and III-V compounds [21,22,23,24,25,26,27,28,29,30,31,32,33,34,35]; Among them, III-V compounds have drawn much attention [21,22,23,24,25,26,27,28,29,30,31,32,33,34,35]. Dai et al. [21] studied type-II BN materials, which are mechanically and thermally stable at a temperature of 1000 K, and their large pore characteristics make them suitable for hydrogen storage. Miao et al. [22] investigated the structural, mechanical stability, and electronic properties of P6₄22 AlP, GaP, and InP based on density functional theory (DFT) [36,37], as well as the thermal properties and elastic anisotropy. P6₄22-AlP, P6₄22-GaP, and P6₄22-InP are dynamically and thermodynamically stable; P6₄22-AlP and P6₄22-GaP exhibit indirect band gaps, while P6₄22-InP has a direct band gap of 0.42 eV and may be applied in infrared detectors. The thermodynamic and elastic properties of β-GaN semiconductors were studied, in detail, using DFT under an ultrasoft pseudopotential scheme by Fan et al. [25]. β-GaN has mechanical stability under a high pressure and temperature, and its v, G, E, and Zener anisotropy index show a greater anisotropy at a high temperature. Recently, Zhang et al. [26] also proposed III-nitride polymorphs Pm-3n XN (XN = BN, AlN, GaN, and InN) and studied their structure, mechanical properties, and electronic properties. Xiong et al. [27] investigated a sp² + sp³ hybrid metal monoclinic 3D BN structure, and donated M-BN. In terms of energy, amid the metal BN structures predicted thus far, M-BN is the most favorable structure, and it is a potential new one-dimensional material, conductive along the y-direction. Additionally, the Vickers hardness of M-BN is 33.7 GPa, indicating that it has the potential to become a hard material.

The invention of the blue light diode won a Nobel Prize in physics. In order to make a blue light diode, another element needs to be doped in gallium nitride. If another phase GaN or another semiconductor material with a band gap can be achieved matching the blue light photon energy, steps can be saved in doping. In this work, based on DFT, the III-V nitride polymorphs XN (X = B, Al, Ga, and In) in P4₂/mnm and Pbca phases are investigated. P4₂/mnm phase GaN can absorb about 484 nm blue light, so it is a potential binary semiconductor material that can be directly made into a blue light diode. In addition, the stability and physical properties of the P4₂/mnm XN and Pbca XN semiconductor materials are estimated and analyzed in this work.

2. Theoretical Methods

DFT is utilized for the theoretical investigations, and all of the phonon spectra of P4₂/mnm XN and Pbca XN are estimated according to the density functional perturbation theory (DFPT) method [38]. In the physical performance prediction and structural geometry optimization calculations, the Perdew–Burke–Ernzerhof (PBE) functional of the exchange-correlation potential and the generalized gradient approximation (GGA) [39] were used, and the DFT calculations were carried out using the ultrasoft pseudopotentials [40] from the Cambridge Sequential Total Energy Package (CASTEP) code [41]. The valence electron structures of the N, B, Al, Ga, and In atoms were 2s²2p³, 2s²2p¹, 3s²3p¹, 4s²4p¹, and 5s²5p¹, respectively. A high k-point separation, less than or approximately 0.025 Å⁻¹ × 2π, was used in the P4₂/mnm XN and Pbca XN, and 9 × 9 × 16, 8 × 8 × 13, 7 × 7 × 12, and 6 × 6 × 11 Monkhorst–Pack meshes [42] were adopted for the conventional cells of XN (X = B, Al, Ga, and In) in the P4₂/mnm phase, and 8 × 8 × 9, 6 × 7 × 8, 6 × 7 × 7, and 6 × 6 × 7 Monkhorst–Pack meshes were adopted for the conventional cells of XN in the Pbca phase, respectively. The geometric optimizations were performed using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) minimization scheme [43]. Additionally, in order to make the energy of the new prediction structure converge and be in the lowest energy state (with a total energy difference between the two times being less than 0.001 eV), we used 500 eV of plane wave cut-off energy for the structural optimization and performance investigation of XN in the P4₂/mnm and Pbca phases, respectively. The electronic energy band structures were estimated using the Heyd–Scuseria–Ernzerhof (HSE06) hybrid functional [44].

3. Results and Discussion

3.1. Structural Properties

The crystal structures of XN in the P4₂/mnm and Pbca phases are illustrated in Figure 1. Blue spheres indicate nitrogen atoms; violet spheres indicate B, Al, Ga, and In atoms, because the four varieties of atoms are in the same positions; and black regions indicate the conventional cell. Both of the crystal structures of P4₂/mnm XN and Pbca XN are composed of sp³-bonded rings, in which there are four-, six-, and eight-membered X-N rings in the P4₂/mnm phase (Figure 1a,e), and six-membered rings in the Pbca phase (Figure 1b). Figure 1e shows that in the P4₂/mnm phase, the X atoms and nitrogen atoms are alternately connected along the ab plane to form four-membered rings and eight-membered rings. It can be found from Figure 1c, that the structure of the P4₂/mnm XN along the ac plane and the structure of Pbca XN along the ab plane are six-membered rings composed of alternately appearing X atoms and nitrogen atoms.

The calculated lattice parameters of the P4₂/mnm XN, Pbca XN, Pm-3n XN, and Pnma XN are presented in

Table 1. Lattice parameters and volumes of Pbca XN, P4₂/mnm XN, c-XN, Pm-3n XN, wurtzite XN, and Pnma XN.

Materials	Methods	a	b	c	c/a	ρ	V
Pbca BN	GGA	5.110	4.434	4.399		3.308	12.459
Pbca AlN	GGA	6.183	5.428	5.291		3.067	22.192
Pbca GaN	GGA	6.428	5.602	5.518		5.599	24.837
Pbca InN	GGA	7.172	6.262	6.150		6.196	34.525
P42/mnm BN	GGA	4.421		2.548		3.309	12.203
P42/mnm AlN	GGA	5.320		3.116		3.087	22.049
P42/mnm GaN	GGA	5.547		3.221		5.612	24.779
P42/mnm InN	GGA	6.184		3.608		6.202	34.493
c-BN	GGA	3.623					11.884
	LDA	3.576					11.436
	Exp. a	3.620
c-AlN	GGA	4.396				3.206	21.229
c-GaN	GGA	4.557				5.878	23.658
	LDA	4.523					23.132
	Exp. b	4.520					23.332
c-InN	GGA	5.088				6.496	32.932
Pm-3n BN	GGA c	4.438				2.829	14.569
Pm-3n AlN	GGA c	5.366				2.643	25.751
Pm-3n GaN	GGA c	5.584				4.793	29.015
Pm-3n InN	GGA c	6.237				5.291	40.428
Pnma BN	GGA d	4.890	2.589	4.284		3.040	13.557
Pnma AlN	GGA e	5.775	3.198	5.123		2.828	24.068
Pnma GaN	GGA e	6.076	3.302	5.421		5.114	27.190
Pnma InN	GGA e	6.777	3.699	6.051		5.642	37.918
wurtzite BN	GGA	2.539		4.200	1.654
	Exp. f	2.550		4.199	1.647
wurtzite AlN	GGA	3.103		5.010	1.615
	Exp. g	3.110		4.980	1.601
wurtzite GaN	GGA	3.186		5.223	1.639
	Exp. g	3.190		5.200	1.630
wurtzite InN	GGA	3.547		5.726	1.614
	Exp. g	3.540		5.710	1.613

^a Ref. [45], ^b Ref. [46], ^c Ref. [26], ^d Ref. [28], ^e Ref. [29]. ^f Ref. [47], ^g Ref. [48].

, where a, b, and c express the lattice parameters, ρ represents the density, and V represents the conventional volume. The obtained lattice parameters in the P4₂/mnm phase and Pbca phase gradually increased from BN to InN. In the P4₂/mnm and Pbca phase, ρ and V gradually increased from BN to AlN along the III-V nitrides because of the atomic radius and relative atomic mass of the X atom increasing gradually. Accordingly, the volume of the XN f.u. (formular unit) in the Pbca phase was greater than that in P4₂/mnm phase, so the ρ of Pbca XN was slightly smaller than that of the P4₂/mnm XN. What is more, the densities of P4₂/mnm XN and Pbca XN were higher than that of Pm-3n XN and Pnma XN, this is also due to the increasing atomic radius and the relative atomic mass of the X atom. It can be found from Table 1 that the crystal volumes of P4₂/mnm XN, Pbca XN, Pm-3n XN, and Pnma XN from large to small followed the order of Pm-3n XN, Pnma XN, Pbca XN, P4₂/mnm XN, and c-XN.

3.2. Stability

Stability is an important physical property of materials. To study the dynamic stability of P4₂/mnm XN and Pbca XN, the phonon spectra of AlN, GaN, and InN in the P4₂/mnm and Pbca phases are presented in Figure 2a–f, respectively. There were no phonons with imaginary frequencies observed in whole Brillouin zone, which proves that aluminum nitride, gallium nitride, and indium nitride in the P4₂/mnm and Pbca phases are all dynamically stable. The frequencies of all of the optical phonons at the Brillouin zone center (Gamma-point) of the P4₂/mnm phase and Pbca phase are given in Tables S1 and S2 in Supporting Information, respectively. Whether the material can be synthesized was predicted by analyzing the enthalpy. The related enthalpies of XN in the P4₂/mnm, Pbca, Pm-3n, and Pnma phases are presented in Figure 2g.

The relative enthalpies of XN in the P4₂/mnm, Pbca, Pm-3n, and Pnma phases were be obtained using the following: ΔH = H_{novel phase}/m-H_{c-XN phase}/n; here, m and n are the amount of BN in the conventional cell. From Figure 2g, all of the enthalpies of XN in the P4₂/mnm phase and Pbca phase XN were less than those in the Pm-3n and Pnma phases, which indicates that they were more favorable than the Pm-3n XN and Pnma XN theoretically, which could be synthesized in experiments. The two new XN phases proposed in this work compared the relative enthalpy of Pbca XN (XN = BN, AlN, and GaN) with that of P4₂/mnm XN (XN = BN, AlN, GaN), and it could be found that the relative enthalpy of Pbca XN was slightly larger, while the Pbca InN was more favorable than P4₂/mnm InN. The P4₂/mnm phase belongs to the tetragonal system, and the Born’s mechanical stability criterion for the tetragonal system can be given as the following equations [49]:

$$C_{11}>0,C_{33}>0,C_{44}>0,C_{66}>0,$$

(1)

$$(C_{11}-C_{12})>0,(C_{11}+C_{33}-2C_{13})>0,$$

(2)

$$[2(C_{11}+C_{12})+C_{33}+4C_{13}]>0.$$

(3)

The Pbca phase belongs to the orthorhombic system, and the Born’s criterion for the orthorhombic system can be indicated by the following equations [49]:

$$C_{11}>0,C_{11}{C}_{22}>C_{12}^2,$$

(4)

$$C_{11}{C}_{22}{C}_{33}+2C_{12}{C}_{13}{C}_{23}-C_{11}{C}_{23}^2-C_{22}{C}_{13}^2-C_{33}{C}_{12}^2>0,$$

(5)

$$C_{44}>0,C_{55}>0,C_{66}>0.$$

(6)

The calculated elastic constants of XN in the P4₂/mnm, Pbca, Pm-3n, Pnma phases and c-XN are listed in

Table 2. Elastic constants (GPa) and elastic moduli (GPa) of Pbca XN, P4₂/mnm, c-XN, Pm-3n XN and Pnma XN.

	C11	C12	C13	C22	C23	C33	C44	C55	C66	B	G	B/G	E	v
Pbca BN	772	135	139	885	92	716	312	257	357	344	316	1.09	718	0.140
Pbca AlN	317	122	92	369	94	275	109	118	103	174	109	1.60	271	0.241
Pbca GaN	268	117	92	320	91	258	84	88	92	160	89	1.80	225	0.265
Pbca InN	170	95	72	202	75	154	45	50	49	111	48	2.31	126	0.311
P42/mnm BN	747	144	99	747		958	332		244	347	318	1.09	731	0.149
P42/mnm AlN	268	116	114	268		384	99		129	177	102	1.74	257	0.258
P42/mnm GaN	251	90	93	251		334	79		93	153	87	1.76	219	0.261
P42/mnm InN	161	59	80	161		206	42		55	106	48	2.21	125	0.303
c-BN	805	187					468			393	396	0.99	889	0.123
	820a	190					480			400
c-AlN	280	146					184			190	123	1.54	304	0.234
	294b	160					189
c-GaN	247	127					150			167	104	1.61	258	0.242
	263c	145					156			184	106	1.74	267	0.258
	285d	161					149			202e	105e
c-InN	152	95					86			114	55	2.07	142	0.292
Pm-3n BN	700 f	85					209			290	244	1.19	572	0.171
Pm-3n AlN	335 f	59					58			151	83	1.82	210	0.268
Pm-3n GaN	238 f	61					58			120	69	1.74	174	0.259
Pm-3n InN	173 f	55					36			95	44	2.16	114	0.299
Pnma BN	392 g	99	256	770	116	675	299	272	187	298	227	1.31	543	0.196
Pnma AlN	160 h	68	133	332	137	282	92	118	82	149	80	1.86	204	0.272
Pnma GaN	152 h	59	116	278	119	267	68	93	67	133	69	1.93	176	0.279
Pnma InN	109 h	48	81	173	92	165	34	52	38	94	38	2.47	100	0.322

^a Ref. [50]-Experimental, ^b Ref. [51]-Experimental, ^c Ref. [25], ^d Ref. [52], ^e calculated by B= (B_V + B_R)/2, B_V = B_R = (C₁₁ + 2C₁₂)/3, G = (G_V + G_R)/2, G_V = (C₁₁ − C₁₂ + 3C₄₄)/5, and G_R = 5(C₁₁ − C₁₂)C₄₄/[4C₄₄ + 3(C₁₁ − C₁₂)], for Cubic system in Ref. [53]; ^f Ref. [26], ^g Ref. [28], ^h Ref. [29].

. The theoretical data of c-XN calculated in this work are excellent agreement with the experimental results and theoretical results reported before [50,51,52,53]. From Table 2, one can see that the elastic constants of XN in the P4₂/mnm phase and Pbca phase satisfy the above Born criterion, thus proving that P4₂/mnm XN and Pbca XN are both mechanically stable. The synthesis methods of these materials can refer to the calculations from References [54,55,56]. These are the structures grown on sapphire substrates. By selecting suitable substrates and matching lattice parameters, new predicted phases may be expected to be grown by MOVPE or MOCVD (Metal-organic Vapor Phase Epitaxy or Metal-organic Chemical Vapor Deposition).

3.3. Mechanical and Anisotropy Properties

The Young’s modulus (E), bulk modulus (B), shear modulus (G), the ratio of B to G, and Poisson’s ratio (v) of III-V nitrides from boron to indium in the P4₂/mnm, Pbca, Pnma, Pm-3n and cubic phases are listed in Table 2. The vales of B and G are estimated using the Voigt–Reuss–Hill approximations [57,58,59]. For a tetragonal system, the following equations are used [53]:

$$B_v=(1/9)[2(C_{11}+C_{12})+C_{33}+4C_{13}],$$

(7)

$$G_V=(1/30)(M+3C_{11}-3C_{12}+12C_{44}+6C_{66}),$$

(8)

$$B_R=C^2/M,$$

(9)

$$G_R=15{\{(18B_V/C^2)+[6/(C_{11}-C_{12})]+(6/C_{44})+(3/C_{66})\}}^{-1},$$

(10)

$$M=C_{11}+C_{12}+2C_{33}-4C_{13},$$

(11)

$$C^2=(C_{11}+C_{12})C_{33}-2C_{13}^2.$$

(12)

For an orthorhombic system, the following equations are used [53]:

$$B_V=(1/9)[C_{11}+C_{22}+C_{33}+2(C_{12}+C_{13}+C_{23})],$$

(13)

$$G_V=(1/15)[C_{11}+C_{22}+C_{33}+3(C_{44}+C_{55}+C_{66})-(C_{12}+C_{13}+C_{23})],$$

(14)

$$\begin{array}{ll}{B}_R=& \mathsf{\Delta }[C_{11}(C_{22}+C_{33}-2C_{23})+C_{22}(C_{33}-2C_{13})-2C_{33}{C}_{12}\\ & +C_{12}(2C_{23}-C_{12})+C_{13}(2C_{12}-C_{13})+C_{23}(2C_{13}-C_{23}){]}^{-1},\end{array}$$

(15)

$$\begin{array}{ll}{G}_R& =15\{4[C_{11}(C_{22}+C_{33}+C_{23})+C_{22}(C_{33}+C_{13})+C_{33}{C}_{12}\\ & -C_{12}(C_{23}+C_{12})-C_{13}(C_{12}+C_{13})-C_{23}(C_{13}+C_{23})]/\mathsf{\Delta }\\ & +3[(1/C_{44})+(1/C_{55})+(1/C_{66})]{\}}^{-1},\end{array}$$

(16)

$$\mathsf{\Delta }=C_{13}(C_{12}{C}_{23}-C_{13}{C}_{22})+C_{23}(C_{12}{C}_{13}-C_{23}{C}_{11})+C_{33}(C_{11}{C}_{22}-C_{12}^2).$$

(17)

The values of B, G, E, and v of the P4₂/mnm XN and Pbca XN are calculated by the following:

$$B=\frac{1}{2}(B_V+B_R),G=\frac{1}{2}(G_V+G_R),$$

(18)

$$E=9BG/(3B+G),v=(3B-2G)/[2(3B+G)].$$

(19)

From Table 2, all of the elastic moduli of P4₂/mnm XN and Pbca XN were greater than those of Pm-3n XN and Pnma XN, but less than those of c-XN. In contrast, the B/G and v of P4₂/mnm XN and Pbca XN were smaller than those of Pm-3n XN and Pnma XN, but greater than those of c-XN. It is obvious that the elastic constants, C₁₁, C₂₂, C₃₃, C₄₄, C₅₅, and C₆₆, and elastic moduli, B, G, and E, of the III-V compounds reduced from the boron atom to indium atom, while the value of v gradually increased in the Pbca and P4₂/mnm phases in Figure 3. Young’s modulus can be regarded as an index to measure the difficulty of elastic deformation. The larger the Young’s modulus, the less likely it is to deform. In the Pbca phase and P4₂/mnm phase, the B, G, and E of BN were the largest, and those of InN were the smallest, which indicates that BN is more difficult to deform than other III-V nitrides. In addition, in the Pbca phase, the elastic moduli of BN were slightly smaller than those in the P4₂/mnm phase, whereas the B and E of Pbca InN were slightly higher than those of P4₂/mnm InN.

Shear modulus is a mechanical property of materials, which characterizes the resistance to plastic deformation of materials. The bulk modulus of the crystal is the macroscopic property of the material, which reflects the resistance of the material to the uniform compression of the outside under the elastic system. Therefore, in order to investigate the brittleness and ductility of materials, the B/G of Pbca XN and P4₂/mnm XN were analyzed. The values of the ratio of B to G and v of XN in the Pbca phase and P4₂/mnm phase are plotted in Figure 4. Taking 1.75 as the dividing line, the B/G value exceeding 1.75 indicates ductility; otherwise, it indicates brittleness. In addition, by comparing the Poisson’s ratio (v) with other elastic constants, a great deal of information about bond strength characteristics can be obtained [60]. The difference between v and B/G is that 0.26 is used as the dividing line. From Figure 4, GaN and AlN in P4₂/mnm phase and Pbca phase were very close to the boundary of B/G and v. The B/G and v for P4₂/mnm AlN and Pbca AlN were both smaller than the dividing line, so they tended to be brittle, the B/G and v for P4₂/mnm GaN and Pbca GaN were both larger than the dividing line, so they tended to be ductile. In addition, the ratio of the bulk modulus to shear modulus and the v increased from BN to InN in the P4₂/mnm phase and Pbca phase; that is to say, BN and AlN are brittle materials, and GaN and InN are ductile materials. From Table 2, the B/G ratio and v of P4₂/mnm XN and Pbca XN were smaller than those of Pnma XN, but higher than those of c-XN, indicating that P4₂/mnm XN and Pbca XN are more brittle than Pnma XN, but weaker than c-XN.

It is known that elastic anisotropy plays a significant role in researching the physical properties of crystals [31]. The three-dimensional (3D) contour can visually display the mechanical anisotropy, because for anisotropic materials, the values of E will change in distinct directions. If the three-dimensional contour is a regular sphere, then the mechanical properties of the material show isotropy; if it is not a regular sphere, then the material exhibits anisotropy of mechanical properties [61,62,63,64,65], and the lower the similarity to a sphere is, the greater the anisotropy. In order to figure out the elasticity anisotropy of Young’s modulus for XN in the P4₂/mnm and Pbca phases in distinct directions, the variations of E for P4₂/mnm XN and Pbca XN in different directions are illustrated in Figure 5. As shown in Figure 5, the values of E_max and E_min in the P4₂/mnm phase are represented by the curved surface of the yellow and the green line, respectively, and in the Pbca phase they are represented by the curved surface of the green and blue line, respectively. The maximum, minimum, and maximum/minimum ratios of P4₂/mnm XN and Pbca XN for Young’s modulus, which are marked as E_max, E_min, and E_max/E_min, respectively. For XN in the Pbca phase, the maximum/minimum ratios of Young’s modulus were 1.34, 1.29, 1.22, and 1.22, respectively, which were smaller than those in the P4₂/mnm phase (the E_max/E_min of P4₂/mnm XN were 1.49, 1.54, 1.40 and 1.31, respectively.). That is, the elastic anisotropy of E decreased in the Pbca phase, in which the Young’s modulus of Pbca BN had the largest elastic anisotropy and Pbca InN had the smallest elastic anisotropy, and the elastic anisotropy of E for Pbca XN was smaller than that of P4₂/mnm XN. The anisotropy in the Pbca phase was less anisotropic than in the other phases, which can be considered to be from the following aspects. The first factor to be considered is that the constituent elements of each phase are all XN—they all have only X and nitrogen elements. The second factor to be considered is the length of the chemical bond, the bond lengths in different phases are all X-N bond lengths; the bond lengths of all X-N are listed in Cif files in Supporting information. The third aspect to consider is the type of chemical bond, they are all sp³ hybrids; both X and nitrogen atoms are coordinated in the Pbca phase, P4₂/mnm phase, Pm-3n phase, and Pnma phase. It can be concluded that in the P4₂/mnm phase, the anisotropy of AlN was the largest, and that of InN was the smallest. Compared with XN in P4₂/mnm phase, the XN in the Pbca phase had less anisotropy from BN to InN; their 3D contours were obviously closer to a sphere in Figure 5. It is worth noting that in the P4₂/mnm phase and Pbca phase, the three-dimensional contour from BN to AlN, GaN, and finally InN became closer to a sphere. In other words, the elastic anisotropy of the Young’s modulus of the XN P4₂/mnm phase and Pbca phase gradually decreased.

3.4. Electronic Properties

The electronic band structure of materials is often used to describe the prohibited or permitted energy of electrons. An analysis of the band structure can reveal more properties in the field of electronics and optics. The energy band structures of XN in the P4₂/mnm phase and Pbca phase are illustrated in Figure 6. From Figure 6a–f, P4₂/mnm XN and Pbca XN can be used as semiconductor materials. Pbca XN belongs to the direct band gap materials, and the band gaps of P4₂/mnm GaN and P4₂/mnm InN had direct band gap materials of 2.57 eV and 2.02 eV, respectively. The BN and AlN in the P4₂/mnm phase had an indirect and wide band gap of 6.13 eV and 4.76 eV, respectively, but the indirect band gap in P4₂/mnm AlN was very close to that of the direct band gap, so the P4₂/mnm AlN belonged to quasi-direct band gap semiconductor materials. For P4₂/mnm AlN, the conduction band minimum (CBM) was located at Γ (0.0000, 0.0000, 0.0000), while the valence band maximum (VBM) was located at (0.0769, 0.0769, 0.0000) along the M–Γ direction (see Figure 6c); it had a direct band gap of 4.802 eV, it was only 0.041 eV different from the indirect band gap. The band gaps of GaN, AlN, and InN in the P4₂/mnm phase were marginally larger than in the Pbca phase. The same compounds had different band gaps in the different phases. For the III-V nitrides, the band gap range in the P4₂/mnm phase (2.02–6.13 eV) was slightly smaller than those in the Pbca (0.95–6.81 eV), Pm-3n (1.04–5.87 eV) [26], and Pnma (0.66–7.18 eV) phases [28,29]. In the P4₂/mnm, Pbca, Pm-3n, and Pnma phases, Pnma BN had the largest band gap, while Pnma InN had the smallest band gap, indicating that the Pnma phase had the largest band gap range. By doping impurities between X atoms or P atoms, the energy band gap could be adjusted, and ternary, quaternary, or higher compounds could be formed. For example, doping impurity Ga atoms into InN in the P4₂/mnm phase could form a ternary compound with a band gap of 2.02 to 2.57 eV. In general, both P4₂/mnm XN and Pbca XN had wide band gaps in the range of 2.02–6.13 eV and 0.95–6.81 eV, respectively. Theoretically, these compounds may have good application prospects in the production of photoelectric semiconductor devices, for example, as blue light emitting diodes (GaN in P4₂/mnm phase) and orange light emitting diodes (GaN in Pbca phase and InN in P4₂/mnm phase).

4. Conclusions

In this manuscript, the structural, stability, anisotropy, and mechanical properties of XN in the P4₂/mnm and Pbca phases are presented. According to the phonon dispersion branches and elastic constants, P4₂/mnm XN (XN = BN, AlN, GaN, and InN) and Pbca XN (XN = BN, AlN, GaN, and InN) are stable. Among the P4₂/mnm phase and Pbca phase, BN has the strongest brittleness, while InN has the strongest ductility. In addition, the E and B of P4₂/mnm XN and Pbca XN are greater than those of Pm-3n XN and Pnma XN. The elastic anisotropy of E decreases gradually as the group III-V nitrides change from BN to InN in the P4₂/mnm phase and Pbca phase, and it is smaller than that of c-XN. The structures of the electronic band of Pbca XN (XN = BN, AlN, GaN, and InN), P4₂/mnm GaN, and P4₂/mnm InN indicate that they can be candidates for semiconductor materials, and they have direct band gaps within HSE06, while BN and Al in the P4₂/mnm phase are semiconductor materials with an indirect and wide band gap. Moreover, for the XN (XN = BN, AlN, GaN, and InN) compounds, the band gap range of the P4₂/mnm phase (2.02–6.13 eV) is slightly smaller than those of the Pbca (0.95–6.81 eV), Pnma (0.66–7.18 eV), and Pm-3n (1.04–5.87 eV) phases. These compounds may have good application prospects in the electronics manufacturing industry.