First-Principles Study on Structural, Mechanical, Anisotropic, Electronic and Thermal Properties of III-Phosphides: XP (X = Al, Ga, or In) in the P6422 Phase

The structural, mechanical, electronic, and thermal properties, as well as the stability and elastic anisotropy, of XP (X = Al, Ga, or In) in the P6422 phase were studied via density functional theory (DFT) in this work. P6422-XP (X = Al, Ga, or In) are dynamically and thermodynamically stable via phonon spectra and enthalpy. At 0 GPa, P6422-XP (X = Al, Ga, or In) are more rigid than F4¯3m-XP (X = Al, Ga, or In), of which P6422-XP (X = Al or Ga) are brittle and P6422-InP is ductile. In the same plane (except for (001)-plane), P6422-AlP and P6422-InP exhibit the smallest and the largest anisotropy, respectively, and P6422-XP (X = Al, Ga, or In) is isotropic in the (001)-plane. In addition, Al, Ga, In, and P bonds bring different electrical properties: P6422-InP exhibits a direct band gap (0.42 eV) with potential application for an infrared detector, whereas P6422-XP (X = Al or Ga) exhibit indirect band gap (1.55 eV and 0.86 eV). At high temperature (approaching the melting point), the theoretical minimum thermal conductivities of P6422-XP (X = Al, Ga, or In) are AlP (1.338 W∙m−1∙K−1) > GaP (1.058 W∙m−1∙K−1) > InP (0.669 W∙m−1∙K−1), and are larger than those of F4¯3m-XP (X = Al, Ga, or In). Thus, P6422-XP (X = Al, Ga, or In) have high potential application at high temperature.

in the Brillouin zone are completed in turn by keeping the cut-off energy and the k-point constant, respectively. As is shown in Figure 1, the plane-wave cut-off energies were finally chosen to be 320, 400, and 420 eV with ultrasoft pseudopotentials for P6422-AlP, P6422-GaP, and P6422-InP, respectively. The k-points in the first irreducible Brillouin zone were set to (11 × 11 × 5; 11 × 11 × 5; 11 × 11 × 5) [15] by using the Monkhorst-Pack scheme [16] for P6422-AlP, P6422-GaP, and P6422-InP. By using the Broyden-Fletcher-Goldfarb-Shenno (BFGS) algorithm [17], structural parameter optimizations were conducted with the following thresholds for the convergent structures: a maximum stress of less than 0.02 GPa, a maximum residual force of less than 0.01 eV/Å, a maximum energy change of less than 5 × 10 −6 eV per atom, and a maximum displacement of atoms for geometry optimization of less than 5 × 10 −4 Å. The phonon spectra were calculated via linear response theory (density functional perturbation theory (DFPT)) [18]. The accurate electronic band-gap structures of P6422-XP (X = Al, Ga, or In) were obtained via the Heyd-Scuseria-Ernzerhof (HSE06) [19,20] screened-exchange hybrid functional base on the previous geometry optimizations via GGA-PBE. The configurations of the valence electrons are 3s 2 3p 3 for P, 3s 2 3p 1 for Al, 3d 10 4s 2 4p 1 for Ga, and 4d 10 5s 2 5p 1 for In.

Structural Properties
The three-dimensional crystal structure of P6 4 22-XP (X = Al, Ga, or In) is illustrated in Figure 2. The 3D crystal structure of P6 4 22-XP (X = Al, Ga, or In) is composed of an sp 3 -bonded network. To evaluate the performance of the theoretical method that is used in this work, the related physical properties of F43m-XP (X = Al, Ga, or In) are also studied via the same method. The lattice parameters of XP (X = Al, Ga, or In) in the P6 4 22 phase and in the F43m phase are listed in Table 1 via GGA-PBE. The lattice parameters and the crystal density of XP (X = Al, Ga, or In) in the F43m phase (sphalerite phase) are very close to other experimental results, namely, the optimization and calculation method that is utilized in this work can provide theoretical support for the results [21][22][23]. In addition, the lattice structure of P6 4 22-and F43m-XP (X = Al, Ga, or In) are also optimized by using DFT-D2 (Grimme) on the basis of GGA-PBE to verify the effect of dispersion on the properties of the material. The results show that the errors between lattice constants a, b, and c of F43m-XP (X = Al, Ga, or In) and experimental values without (with) considering the dispersion action are 0.78% (0.46%), 0.99% (0.72%), 1.77% (0.26%), respectively, which proves our calculation method can provide theoretical support. For P6 4 22-XP (X = Al, Ga, or In), the lattice constants a, b, and c of P6 4 22-AlP change by~1.53% (2.07% for P6 4 22-GaP, 3% for P6 4 22-InP),~1.53% (2.07% for P6 4 22-GaP, 3% for P6 4 22-InP), and~0.16% (0.2% for P6 4 22-GaP, 1.25% for P6 4 22-InP) with considering the dispersive action, indicating that P6 4 22-XP (X = Al, Ga, or In) are insensitive to the dispersive action. Considering the computational cost and accuracy, we adopt the optimized lattice parameters via GGA-PBE for subsequent studies of physical properties. The investigated P6 4 22-XP (X = Al, Ga, or In) has a hexagonal structure with the following equilibrium lattice parameters; a = b = 3.849 Å and c = 8.683 Å for AlP, a = b = 3.899 Å and c = 8.570 Å for GaP, and a = b = 4.190 Å and c = 9.416 Å for InP. For P6 4 22-XP (X = Al, Ga, or In), the P-Al bond length is 2.408 Å, the P-Ga bond length is 2.419 Å, and the P-In bond length is 2.618 Å. As shown in Table 1, in the same crystal structure, the volume per molecule for P6 4 22-XP (X = Al, Ga, or In) increases due to the long bond length and the large lattice constant. In the P6 4 22 phase, the densities of AIP (ρ = 2.591 g/cm 3 ), GaP (ρ = 4.446 g/cm 3 ) and InP (ρ = 5.073 g/cm 3 ) are larger than the corresponding densities in the F43m phase because the corresponding volume per molecule in the P6 4 22 phase is smaller.  [a] Ref. [23]. [b] Ref. [24]. [c] Ref. [25].
In Table 2, the equilibrium volume V 0 and bulk modulus B 0 of P6 4 22-XP (X = Al, Ga, or In) are calculated via GGA-PBE. The calculated total energy (E) per primitive cell for each compound as a function of different cell volumes (V) over a range of 0.9V 0 -1.1V 0 is fitted by the Murnaghan equation of state [EOS] [21,22].
Where B 0 and B are the bulk modulus and their first pressure derivatives at 0 GPa, V 0 is the unit-cell volume at 0 GPa, and E(V) is the total energy under the different cell volume V. The fitted energy vs. volume (E-V) curves are shown in Figure 3. The equation between pressure and volume (P-V in Figure 3) is obtained through the derivation of E(V).

Stability and Mechanical Properties
Dynamic stability is an important property for verifying the existence of new materials. The dynamic stability of P6 4 22-XP (X = Al, Ga, or In) can be determined by studying the phonon spectra. The phonon spectra of P6 4 22-XP (X = Al, Ga, or In) are shown in Figure 4. By observation, the P6 4 22-XP (X = Al, Ga, or In) are dynamically stable because their phonon spectra have no imaginary frequencies in the Brillouin region. The highest vibrational frequencies of P6 4 22-XP (X = Al, Ga, or In) are 13.596 THz at point G, 10.412 THz at point K and 11.298 THz at point K, respectively. The elastic constants and elastic moduli of P6 4 22-and F43m-XP (X = Al, Ga, or In) are listed from 0 GPa to 35 GPa in Table 2. For XP (X = Al, Ga, or In) in the F43m phase, the calculated elastic constants are in good agreement with the reported experimental results, which proves the correctness of the theoretical calculation method. For a hexagonal system, the necessary and sufficient Born criteria for stability can be expressed as follows [26].
Materials 2020, 13, 686 7 of 17  Pressure is a significative physical parameter that has a momentous impact on the Brillouin zone. Enthalpy is an important state parameter in thermodynamics for characterizing the energy of a material system. The lower its energy of matter or a system, the less likely it is to undergo spontaneous processes; therefore, the more stable it is [33].
The relative formation enthalpy curves relative to F4 3m-XP (X = Al, Ga, or In) as functions of the pressure up to 35 GPa for P6422-XP (X = Al, Ga, or In) are plotted in Figure 5. At ambient pressure, F4 3m-XP (X = Al, Ga, or In) are more favorable than any other P6422-XP. Moreover, at 0 GPa, P6422-AlP, P6422-GaP, and P6422-InP have larger enthalpy than F4 3m-XP (X = Al, Ga, or In) (0.418, 0.436, and 0.345 eV per formula (f.u.), respectively). As the pressure increases, P6422-XP (X = Al, Ga, or In) become increasingly stable, and P6422-AlP, P6422-GaP, and P6422-InP become more stable than F4 3m-AlP, F4 3m-GaP, and F4 3m-InP at the pressures that exceed 11.42, 16.60, and 20.91 GPa, respectively. In addition, P6422-InP is the most stable, followed by P6422-AlP and, finally, P6422-GaP. According to the Table 2, the values of the elastic constant, Young's modulus E (GPa), and Poisson's ratio ʋ increase with the pressure.  In Table 2, all the elastic constants of P6 4 22-XP (X = Al, Ga, or In) at 0 GPa satisfy the above stability criteria, namely, P6 4 22-XP (X = Al, Ga, or In) are mechanically stable. The form ability and stability of the alloy can be characterized by the formation enthalpy and the cohesion energy [27]. To study the thermodynamic stability of P6 4 22-XP (X = Al, Ga, or In), its formation enthalpy (∆H) and cohesive energy (E coh ) are also further investigated, and the corresponding formulas [28,29] are described as follows, where E tot is the total energy of P6 4 22-XP (X = Al, Ga, or In) at the equilibrium lattice constant; E X solid and E P solid are the energies per atom of the pure constituents of X (X = Al, Ga, or In) and P, respectively, in the solid states; E X atom and E P atom are the energies from the free atoms of X (X = Al, Ga, or In) and P, respectively; and N X and N p refer to the numbers of X (X = Al, Ga, or In) and P atoms, respectively, in each conventional cell. The calculated formation enthalpies for P6 4   The elastic moduli can be obtained based on the elastic constant. The bulk moduli B and the shear moduli G can be estimated via the Voigt-Reus-Hill approximation [30]. B V , B R , G V and G R can be expressed via the following equations [31], where the subscripts V and R are the Voight and Reuss schemes: Young's modulus E and Poisson's ratio = Al, Ga, or In) are easier to form, where P6422-AlP > P6422-InP > P6422-GaP according to the stability of alloy formation. The cohesion energy is the energy that is needed for decomposing solid materials into isolated atoms. The smaller the value is, the higher the crystal structure stability. The results of E coh for XP (X = Al, Ga, or In) in the P6422 phase are −9.95, −8.21, and −8.74 eV, respectively. P6422-AlP has the highest thermodynamic stability followed by P6422-InP and, finally, P6422-GaP, in a hightemperature environment. The elastic moduli can be obtained based on the elastic constant. The bulk moduli B and the shear moduli G can be estimated via the Voigt-Reus-Hill approximation [30]. V B , R B , V G and R G can be expressed via the following equations [31], where the subscripts V and R are the Voight and Reuss schemes: Young's modulus E and Poisson's ratio ʋ are calculated from B and G as According to Table 2, the elastic constants C11 (147 GPa, 152 GPa, 108 GPa), C22 = C11 (147 GPa, 152 GPa, 108 GPa), and C33 (174 GPa, 144 GPa, 117 GPa) for P6422-XP (X = Al, Ga, or In) are larger than C11 = C22 = C33 (123 GPa, 134 GPa, 96 GPa) of F4 3m-XP (X = Al, Ga, or In); therefore, P6422-XP (X = Al, Ga, or In) have stronger ability to resist elastic deformation along the X-, Y-, and Z-axes. The bulk moduli B and the shear moduli G of P6422-XP (X = Al or In) are larger than those of F4 3m-XP (X = Al or In); thus, the anti-compression and anti-shearing strain abilities of P6422-XP (X = Al or In) are stronger. Furthermore, the B/G ratios [32] of P6422-and F4 3m-XP (X = Al, Ga, or In) at ambient pressure are also shown in Table 2. In the P6422 phase, XP (X = Al or Ga) are brittle (B/G < 1.75) and InP are ductile (B/G > 1.75), and F4 3m-XP (X = Al, Ga, or In) are all brittle (B/G < 1.75).
The calculated Young's modulus E of XP (X = Al, Ga, or In) in the P6422 phase at 0 GPa are 132, 140 and 94 GPa, respectively, which are larger than those (118, 131, and 88 GPa) in the F4 3m phase. Therefore, the stiffness of P6422-XP (X = Al, Ga, or In) are higher, and they are more difficult to deform, especially GaP. There are no significant changes in the calculated values of Poisson's ratio ʋ of XP (X = Al, Ga, or In) between the P6422 phase and F4 3m phase at 0 GPa. The Poisson's ratios ʋ of P6422-AlP and P6422-InP are 0.25 and 0.27, which are slightly larger than that of GaP (0.21) in the P6422 phase. All Poisson's ratios ʋ of P6422-XP (X = Al, Ga, or In) are less than 1; thus, after the P6422-XP (X = Al, Ga, or In) are subjected to uniform longitudinal stress, the transverse deformations are smaller than the longitudinal deformations before plastic deformation occurs, especially for GaP. are calculated from B and G as According to Table 2, the elastic constants C 11 (147 GPa, 152 GPa, 108 GPa), C 22 = C 11 (147 GPa, 152 GPa, 108 GPa), and C 33 (174 GPa, 144 GPa, 117 GPa) for P6 4  The calculated Young's modulus E of XP (X = Al, Ga, or In) in the P6 4 22 phase at 0 GPa are 132, 140 and 94 GPa, respectively, which are larger than those (118, 131, and 88 GPa) in the F43m phase. Therefore, the stiffness of P6 4 22-XP (X = Al, Ga, or In) are higher, and they are more difficult to deform, especially GaP. There are no significant changes in the calculated values of Poisson's ratio G and e following equations [31], where the subscripts V and R are the Voight ( d Poisson's ratio ʋ are calculated from B and G as 9 /(3 ) G and e following equations [31], where the subscripts V and R are the Voight ( d Poisson's ratio ʋ are calculated from B and G as can be expressed via the following equations [31], where the subscripts V and R are the Voight and Reuss schemes: Young's modulus E and Poisson's ratio ʋ are calculated from B and G as 9 /(3 ) According to Table 2 Pressure is a significative physical parameter that has a momentous impact on the Brillouin zone. Enthalpy is an important state parameter in thermodynamics for characterizing the energy of a material system. The lower its energy of matter or a system, the less likely it is to undergo spontaneous processes; therefore, the more stable it is [33].
Young's modulus E and Poisson's ratio ʋ are calculated from B and G as According to Table 2, the elastic constants C11 (147 GPa, 152 GPa, 108 GPa), 152 GPa, 108 GPa), and C33 (174 GPa, 144 GPa, 117 GPa) for P6422-XP (X = Al, G than C11 = C22 = C33 (123 GPa, 134 GPa, 96 GPa) of F4 3m-XP (X = Al, Ga, or In); ther = Al, Ga, or In) have stronger ability to resist elastic deformation along the X-, Y bulk moduli B and the shear moduli G of P6422-XP (X = Al or In) are larger than th = Al or In); thus, the anti-compression and anti-shearing strain abilities of P6422-X stronger. Furthermore, the B/G ratios [32] of P6422-and F4 3m-XP (X = Al, Ga, pressure are also shown in Table 2 The calculated Young's modulus E of XP (X = Al, Ga, or In) in the P6422 phas 140 and 94 GPa, respectively, which are larger than those (118, 131, and 88 GPa) Therefore, the stiffness of P6422-XP (X = Al, Ga, or In) are higher, and they are more especially GaP. There are no significant changes in the calculated values of Poisso = Al, Ga, or In) between the P6422 phase and F4 3m phase at 0 GPa. The Poisson's ra and P6422-InP are 0.25 and 0.27, which are slightly larger than that of GaP (0.21) All Poisson's ratios ʋ of P6422-XP (X = Al, Ga, or In) are less than 1; thus, after the Ga, or In) are subjected to uniform longitudinal stress, the transverse deformation the longitudinal deformations before plastic deformation occurs, especially for Ga [a] Ref. [34]. [b] Ref. [35]. [c] Ref. [36].
The relative formation enthalpy curves relative to F43m-XP (X = Al, Ga, or In) as functions of the pressure up to 35 GPa for P6 4  The elastic moduli can be obtained based on the elastic constant. shear moduli G can be estimated via the Voigt-Reus-Hill approximat R G can be expressed via the following equations [31], where the subscr and Reuss schemes: Young's modulus E and Poisson's ratio ʋ are calculated from B and 9 /(3 ) According to Table 2  [a] Ref. [34].

Mechanical Anisotropic Properties
The universal anisotropic index A U that present the elastic anisotropy of P6 4 22-XP (X = Al, Ga, or In) also calculated for further investigation in this work. The relevant calculation formulas are given in [37]. In Table 2, the A U of P6 4 22-XP (X = Al, Ga, or In) shows an increasing tendency with increasing atomic order (AI < Ga < In) at ambient pressure. The variation tendencies of A U for XP (X = Al, Ga, or In) in the P6 4 22 phase differ from those of Young's modulus E. For example, P6 4 22-InP has the smallest Young's modulus in the P6 4 22-XP (X = Al, Ga, or In) but has the largest universal anisotropic index A U .
The 3D directional constructions and 2D representations of Young's modulus E in the (001)-plane, (011)-plane, (100)-plane, (110)-plane, (010)-plane, and (111)-plane for P6 4 22-XP (X = Al, Ga, or In) are shown in Figure 6. Through observation, along with XY-, XZ-, and YZ-plane, P6 4 22-XP (X = Al, Ga, or In) exhibit strong anisotropy in various planes excluding XY-plane. Compared with the XY-plane, the three-dimensional surface structure in the XZ-plane deviates further from the shape of the sphere; therefore, the XZ-plane has stronger anisotropy than the XY-plane [38]. For P6 4 22-XP (X = Al, Ga, or In), the maximum and minimum values of Young's modulus E are attained in the XZ-and YZ-planes, whereas only the minimum value is attained in the XY-plane because they are isotropic in the (001)-plane. In Figure 6, as Young's modulus has the same properties in the (100)-, (010)-, and (110)-plane, Figure 6 shows only the two-dimensional curve in the (110)-plane.
XY-plane, the three-dimensional surface structure in the XZ-plane deviates further from the shape of the sphere; therefore, the XZ-plane has stronger anisotropy than the XY-plane [38]. For P6422-XP (X = Al, Ga, or In), the maximum and minimum values of Young's modulus E are attained in the XZand YZ-planes, whereas only the minimum value is attained in the XY-plane because they are isotropic in the (001)-plane. In Figure 6, as Young's modulus has the same properties in the (100)-, (010)-, and (110)-plane, Figure 6 shows only the two-dimensional curve in the (110)-plane. The calculated maximum values Emax, minimum values Emin, and ratios Emax/Emin of Young's modulus E in each plane for P6422-XP (X = Al, Ga, or In) are listed in Table 3. It is found that, in the  The calculated maximum values E max , minimum values E min , and ratios E max /E min of Young's modulus E in each plane for P6 4 22-XP (X = Al, Ga, or In) are listed in Table 3. It is found that, in the (001)-plane, the minimum values of E max /E min for P6 4

Electrical and Thermal Properties
In solid-state physics, the electron band structure describes the energy that electrons are prohibited or allowed to carry, which is caused by quantum dynamic electron wave diffraction in periodic lattices [39]. The general characteristics of electron motion in crystals are qualitatively expounded by energy band theory. The orbital projection electronic band structures for P6 4  0.00, 0.00). The band gap of P6422-InP corresponds to a wavelength of 2958.04 nm, which is in the infrared region. P6422-AlP and P6422-GaP show indirect band gap properties with band gaps of 1.55 and 0.86 eV, respectively. The conduction band minimums and the valence band maximums of P6422-AlP are located at point G (0.00, 0.00, 0.00) and point M (0.00, 0.50, 0.00), respectively, whereas the conduction band minimums and the valence band maximums of P6422-GaP are located at point G (0.00, 0.00, 0.00) and point K (0.33, 0.67, 0.00), respectively. The calculated partial atomic site projected densities of states (PDOS) of P6422-XP (X = Al, Ga, or In), which are used to reflect elastic characteristics and the bonding properties and orbital distribution of electrons, are plotted in Figure 8. The main bonding peaks distribute in the range from −15 to 15 eV. Below 0 eV, the PDOS in the valence band consist of three parts: the first part ranges from −5 to −10 eV, where the -s orbital makes a larger contribution to electrical conductivity, and, in this part, the percentages of the -p orbital change minimally with increasing energy; the second part ranges from −10 to −5 eV, where the main contributions to conduct electricity are from the -p orbital for AlP, whereas the main contributions to conduct electricity are from the -s orbital for GaP and InP; and the last part consists of the -p orbital from −5 to 0 eV. Above 0 eV, the PDOS in the conduction band originate mainly consist of the -p orbital. From AlP to XP (X = Ga or In), due to the increase in the atomic volume, the contributions of the -s orbital increase substantially from the Al atom to the X (X = Ga or In) atoms in the range of −10 to −5 eV, and when the energy exceeds −5 eV, the contributions of the -p orbital increase substantially. In addition, in the vast majority of the energy range, the PDOS originate mainly from the -p orbital, namely, strong hybridization from the -p orbital of the P atom and the -p orbital of the X (X = Al, Ga, or In) atoms occurs. These PDOS peaks depend on the X-p/P-p (X = Al, Ga, or In) bonding orbital contribution. The results demonstrate that covalent bonds X-P (X = Al, Ga, or In) interactions occur.
Finally, we examine the theoretical minimum thermal conductivity under high temperature, representing the heat that is transferred through the phonon transmission in a temperature gradient, which depends not only on material thermal conductivity, but also on the temperature at which the material attains the lowest thermal conductivity, namely, the minimal thermal conductivity of the The calculated partial atomic site projected densities of states (PDOS) of P6 4 22-XP (X = Al, Ga, or In), which are used to reflect elastic characteristics and the bonding properties and orbital distribution of electrons, are plotted in Figure 8. The main bonding peaks distribute in the range from −15 to 15 eV. Below 0 eV, the PDOS in the valence band consist of three parts: the first part ranges from −5 to −10 eV, where the −s orbital makes a larger contribution to electrical conductivity, and, in this part, the percentages of the −p orbital change minimally with increasing energy; the second part ranges from −10 to −5 eV, where the main contributions to conduct electricity are from the −p orbital for AlP, whereas the main contributions to conduct electricity are from the −s orbital for GaP and InP; and the last part consists of the −p orbital from −5 to 0 eV. Above 0 eV, the PDOS in the conduction band originate mainly consist of the −p orbital. From AlP to XP (X = Ga or In), due to the increase in the atomic volume, the contributions of the −s orbital increase substantially from the Al atom to the X (X = Ga or In) atoms in the range of −10 to −5 eV, and when the energy exceeds −5 eV, the contributions of the −p orbital increase substantially. In addition, in the vast majority of the energy range, the PDOS originate mainly from the −p orbital, namely, strong hybridization from the −p orbital of the P atom and the −p orbital of the X (X = Al, Ga, or In) atoms occurs. These PDOS peaks depend on the X-p/P-p (X = Al, Ga, or In) bonding orbital contribution. The results demonstrate that covalent bonds X-P (X = Al, Ga, or In) interactions occur.
Young's modulus, the density, the defects in the crystal, and the porosity. In addition, Cahill posits that the wave velocity of the acoustic wave is also closely related to the thermal conductivity of the material, and as the thermal conductivity decreases with the increase of the temperature under hightemperature conditions, its minimum value is of substantial significance to the application of the material under the high-temperature conditions. The theoretical minimum thermal conductivity is calculated via the Clark [41] model and the Cahill [42] model.
In the Clark model, E and ρ represent the Young's modulus and density of the crystal, respectively; kB represents the Boltzmann constant; and Ma = [M/(n • NA)] represents the average mass of the atoms in the lattice, where M is the molar mass of the molecule, n is the number of atoms in the molecule, and NA represents Avogadro's constant. In the Cahill model, p is the number of atoms per unit volume, and ν l and ν t [43] are the average acoustic longitudinal wave and acoustic shear wave, respectively, which can be calculated via the following formulas. Finally, we examine the theoretical minimum thermal conductivity under high temperature, representing the heat that is transferred through the phonon transmission in a temperature gradient, which depends not only on material thermal conductivity, but also on the temperature at which the material attains the lowest thermal conductivity, namely, the minimal thermal conductivity of the material. According to Clark, the main factors that affect it are the average relative atomic mass, the Young's modulus, the density, the defects in the crystal, and the porosity. In addition, Cahill posits that the wave velocity of the acoustic wave is also closely related to the thermal conductivity of the material, and as the thermal conductivity decreases with the increase of the temperature under high-temperature conditions, its minimum value is of substantial significance to the application of the material under the high-temperature conditions. The theoretical minimum thermal conductivity is calculated via the Clark [41] Cahill model: In the Clark model, E and ρ represent the Young's modulus and density of the crystal, respectively; k B represents the Boltzmann constant; and M a = [M/(n · N A )] represents the average mass of the atoms in the lattice, where M is the molar mass of the molecule, n is the number of atoms in the molecule, and N A represents Avogadro's constant. In the Cahill model, p is the number of atoms per unit volume, and ν l and ν t [43] are the average acoustic longitudinal wave and acoustic shear wave, respectively, which can be calculated via the following formulas. v l = (B + 4G/3)/ρ (20) v t = G/ρ The calculation results are presented in Table 4, in accordance with Formulas (18) and (19), and the theoretical minimum thermal conductivities of P6 4  The theoretical minimum thermal conductivities of F43m-XP (X = Al, Ga, or In) at high temperature are lower than those of P6 4 22-XP (X = Al, Ga, or In); therefore, P6 4 22-XP (X = Al, Ga, or In) have stronger thermal conductivity than F43m-XP (X = Al, Ga, or In) at high temperature. Table 4. Average mass per atom, M a /g; the transverse and longitudinal sound velocities, ν t, ν l /(km·s −1 ); the density of number of atom per volume, p; and the minimum thermal conductivity at high temperature, κ min /(W·m −1 ·K −1 ), of P6 4 22-and F43m-XP (X = Al, Ga, or In) base on calculated (GGA-PBE) Young's modulus E, density of the crystal ρ, bulk moduli B, and shear moduli G.

Conclusions
In this study, the related properties of P6 4 22-XP (X = Al, Ga, or In) are investigated via the density functional method, which include structural, mechanical, anisotropy, electrical, and thermal properties. P6 4 22-XP (X = Al, Ga, or In) are dynamically, mechanically, and thermodynamically stable, where P6 4 22-XP (X = Al or In) show stronger anti-compression and anti-shearing strain abilities than F43m-XP (X = Al or In). In the P6 4 22 phase, XP (X = Al or Ga) are brittle, and InP is ductile. The stiffness of P6 4 22-XP (X = Al, Ga, or In) are higher, and they are more difficult to deform than F43m-XP (X = Al, Ga, or In), especially GaP. As the pressure increases, P6 4 22-XP (X = Al, Ga, or In) become increasingly stable.