#
First-Principles Study on Structural, Mechanical, Anisotropic, Electronic and Thermal Properties of III-Phosphides: XP (X = Al, Ga, or In) in the P6_{4}22 Phase

^{*}

## Abstract

**:**

_{4}22 phase were studied via density functional theory (DFT) in this work. P6

_{4}22-XP (X = Al, Ga, or In) are dynamically and thermodynamically stable via phonon spectra and enthalpy. At 0 GPa, P6

_{4}22-XP (X = Al, Ga, or In) are more rigid than F$\overline{4}3$m-XP (X = Al, Ga, or In), of which P6

_{4}22-XP (X = Al or Ga) are brittle and P6

_{4}22-InP is ductile. In the same plane (except for (001)-plane), P6

_{4}22-AlP and P6

_{4}22-InP exhibit the smallest and the largest anisotropy, respectively, and P6

_{4}22-XP (X = Al, Ga, or In) is isotropic in the (001)-plane. In addition, Al, Ga, In, and P bonds bring different electrical properties: P6

_{4}22-InP exhibits a direct band gap (0.42 eV) with potential application for an infrared detector, whereas P6

_{4}22-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 P6

_{4}22-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 F$\overline{4}3$m-XP (X = Al, Ga, or In). Thus, P6

_{4}22-XP (X = Al, Ga, or In) have high potential application at high temperature.

## 1. Introduction

_{4}22, C222, and I$\overline{4}3$d, respectively. The mechanical and dynamic stabilities of these structures were evaluated by calculating the elastic constant and the phonon spectrum. According to first-principle calculations, the hardness of oC12- and hP6-AlAs are larger than that of cI24-AlAs under the same pressure. Under ambient pressure, oC12-, hP6-AlAs, and cI24-AlAs exhibit semiconductor properties and the first two show direct band gap properties (0.468 eV and 1.356 eV), whereas the last exhibits indirect band gap property (1.761 eV).

_{1}-, Pbam-, Pbca-, and bct-AlP) and studied their structures, elastic constants, thermodynamics, and electrical properties based on first-principles. It was found that these four new phases all have semiconductor properties; Pmn2

_{1}-AlP and Pbam-AlP show direct band gap properties with larger electronic advantages than wz-AlP and zb-AlP at ambient pressure; and Pmn2

_{1}-AlP, Pbam-AlP, Pbca-AlP, and bct-AlP are ductile. Pmn2

_{1}-AlP and Pbam-AlP are direct band gap semiconductors (3.22 eV and 3.27 eV), whereas Pbca-AlP and bct-AlP are indirect band gap semiconductors (3.47 eV and 3.04 eV). Based on density functional theory (DFT), A. Baida et al. [8] studied the structural, optical, and electronic properties of indium phosphide (InP) via the augmented plane wave (FP-LAPW) method. The results demonstrated that the phase transitions from zinc-blende phase to Imm2, NiAs, PbO, and CsCl phases are possible at low pressure.

_{S}Cl, d-β-tin, Imm2, Immm, and NiAs of III-phosphide XP (X = Al, Ga, or In) under high pressure. The calculated physical parameters such as the lattice constants and bulk modulus demonstrated that zb-XP (X = Al, Ga, or In) are more stable than these phases and cmcm-XP (X = Al, Ga, or In) have the highest hardness, respectively. The results on pressure transitions demonstrated that GaP will transform from the zb phase to the NaCl phase at 22.19 GPa and into the Imm2 phase above 33.76 GPa. When the pressure changed, zb-AlP and zb-InP will transform into NaCl-AlP (at 11.78 GPa) and NaCl-InP (at 7.35 GPa), respectively, whereas C

_{S}Cl-AlP and C

_{S}Cl-InP transform into the NaCl-AlP (at 64.89 GPa) and NaCl-InP (at 71.79 GPa), respectively.

_{4}22 phase have not been identified to date. Therefore, in this work, the initial geometries of P6

_{4}22-XP (X = Al, Ga, or In) are constructed by atomic substitution base on the structure of hP6-AlAs [6]. The structural, mechanical, thermal, and electronic properties and the stability of P6

_{4}22-XP (X = Al, Ga, or In) have been systematically studied via density functional theory. The results demonstrate that only P6

_{4}22-InP is a direct band gap semiconductor material with potential application in an infrared detector.

## 2. Calculation Methods

_{4}22-XP (X = Al, Ga, or In) were conducted by utilizing density functional theory (DFT) [10,11], which is one of the most commonly used methods for calculating the properties of condensed matter physics based on the CASTEP code [12]. The generalized gradient approximation (GGA) [13] and the Perdew–Burke–Ernzerhof (PBE) [14] exchange-correlation functional were used for geometry optimization and property prediction of the materials. To improve computational precision, the convergence analysis of cut-off energy and the k-point grid allocation 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 P6

_{4}22-AlP, P6

_{4}22-GaP, and P6

_{4}22-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 P6

_{4}22-AlP, P6

_{4}22-GaP, and P6

_{4}22-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 P6

_{4}22-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.

## 3. Results and Discussion

#### 3.1. Structural Properties

_{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 F$\overline{4}3$m-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 F$\overline{4}3$m 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 F$\overline{4}3$m 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 F$\overline{4}3$m-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 F$\overline{4}3$m-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 F$\overline{4}3$m phase because the corresponding volume per molecule in the P6

_{4}22 phase is smaller.

_{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].

_{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).

_{0}, and this minimum energy (−710.776 eV for AlP, −6698.591 eV for GaP, and −5221.333 eV for InP) is in good agreement with the simulation data in Figure 1 (cut-off energy: 320, 400, and 420 eV, K -Points: 11 × 11 × 5, 11 × 11 × 5, 11 × 11 × 5 for P6

_{4}22-XP (X = Al, Ga, or In), respectively). It shows that P6

_{4}22-GaP are more stable than P6

_{4}22-XP (X = Al or In). Through the fitting P–V curve, InP-P6

_{4}22 has the largest volume compressibility: 38.15% (36.55% for AlP and 35.80% for GaP).

#### 3.2. Stability and Mechanical Properties

_{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 F$\overline{4}3$m-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 F$\overline{4}3$m 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].

_{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 (ΔH) and cohesive energy (E

_{coh}) are also further investigated, and the corresponding formulas [28,29] are described as follows,

_{4}22-XP (X = Al, Ga, or In) at the equilibrium lattice constant; ${E}_{\mathrm{solid}}^{\mathrm{X}}$ and ${E}_{\mathrm{solid}}^{\mathrm{P}}$ are the energies per atom of the pure constituents of X (X = Al, Ga, or In) and P, respectively, in the solid states; ${E}_{\mathrm{atom}}^{\mathrm{X}}$ and ${E}_{\mathrm{atom}}^{\mathrm{P}}$ 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}22-AlP, P6

_{4}22-GaP, and P6

_{4}22-InP are −1.72, −0.82, and −1.17 eV, respectively. All the values of formation enthalpies are negative; therefore, the bond energies of P6

_{4}22-XP (X = Al, Ga, or In) are very large and P6

_{4}22-XP (X = Al, Ga, or In) are easier to form, where P6

_{4}22-AlP > P6

_{4}22-InP > P6

_{4}22-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 P6

_{4}22 phase are −9.95, −8.21, and −8.74 eV, respectively. P6

_{4}22-AlP has the highest thermodynamic stability followed by P6

_{4}22-InP and, finally, P6

_{4}22-GaP, in a high-temperature environment.

_{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}22-XP (X = Al, Ga, or In) are larger than C

_{11}= C

_{22}= C

_{33}(123 GPa, 134 GPa, 96 GPa) of F$\overline{4}3$m-XP (X = Al, Ga, or In); therefore, P6

_{4}22-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 P6

_{4}22-XP (X = Al or In) are larger than those of F$\overline{4}3$m-XP (X = Al or In); thus, the anti-compression and anti-shearing strain abilities of P6

_{4}22-XP (X = Al or In) are stronger. Furthermore, the B/G ratios [32] of P6

_{4}22- and F$\overline{4}3$m-XP (X = Al, Ga, or In) at ambient pressure are also shown in Table 2. In the P6

_{4}22 phase, XP (X = Al or Ga) are brittle (B/G < 1.75) and InP are ductile (B/G > 1.75), and F$\overline{4}3$m-XP (X = Al, Ga, or In) are all brittle (B/G < 1.75).

_{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 F$\overline{4}3$m 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 ʋ of XP (X = Al, Ga, or In) between the P6

_{4}22 phase and F$\overline{4}3$m phase at 0 GPa. The Poisson’s ratios ʋ of P6

_{4}22-AlP and P6

_{4}22-InP are 0.25 and 0.27, which are slightly larger than that of GaP (0.21) in the P6

_{4}22 phase. All Poisson’s ratios ʋ of P6

_{4}22-XP (X = Al, Ga, or In) are less than 1; thus, after the P6

_{4}22-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.

_{4}22-XP (X = Al, Ga, or In) are plotted in Figure 5. At ambient pressure, F$\overline{4}3$m-XP (X = Al, Ga, or In) are more favorable than any other P6

_{4}22-XP. Moreover, at 0 GPa, P6

_{4}22-AlP, P6

_{4}22-GaP, and P6

_{4}22-InP have larger enthalpy than F$\overline{4}3$m-XP (X = Al, Ga, or In) (0.418, 0.436, and 0.345 eV per formula (f.u.), respectively). As the pressure increases, P6

_{4}22-XP (X = Al, Ga, or In) become increasingly stable, and P6

_{4}22-AlP, P6

_{4}22-GaP, and P6

_{4}22-InP become more stable than F$\overline{4}3$m-AlP, F$\overline{4}3$m-GaP, and F$\overline{4}3$m-InP at the pressures that exceed 11.42, 16.60, and 20.91 GPa, respectively. In addition, P6

_{4}22-InP is the most stable, followed by P6

_{4}22-AlP and, finally, P6

_{4}22-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.

#### 3.3. Mechanical Anisotropic Properties

^{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}.

_{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.

_{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}22-XP (X = Al, Ga, or In) are all 1.000; thus, P6

_{4}22-XP (X = Al, Ga, or In) are attained with the isotropy in the (001)-plane. The maximum ratio E

_{max}/E

_{min}of Young’s modulus E is 1.206, with the largest anisotropy occurring in the (100)-, (110)-, and (010)-plane for P6

_{4}22-AlP. For P6

_{4}22-GaP, the maximum value of E

_{max}/E

_{min}is 1.273, which is attained in the (100)-, (110)-, and (010)-plane with larger anisotropy. The ratios E

_{max}/E

_{min}for P6

_{4}22-InP are all 1.251 in the (100)-, (110)-, and (010)-plane, which is larger than in the other planes. Therefore, the (100)-, (110)-, and (010)-plane of P6

_{4}22-InP exhibit higher anisotropy. In the (100)-, (110)-, and (010)-plane for P6

_{4}22-XP (X = Al, Ga, or In), the ratios E

_{max}/E

_{min}of Young’s modulus are 1.206, and 1.251, respectively. In the (100)-, (110)-, and (010)-plane, P6

_{4}22-AlP exhibits the smallest anisotropy and P6

_{4}22-GaP exhibits the largest anisotropy. From the (011)-plane to the (111)-plane, P6

_{4}22-InP exhibits the largest anisotropy with E

_{max}/E

_{min}= 1.237, and P6

_{4}22-GaP exhibits the smallest anisotropy with E

_{max}/E

_{min}= 1.147.

#### 3.4. Electrical and Thermal Properties

_{4}22-XP (X = Al, Ga, or In) are plotted in Figure 7. The coordinates of high-symmetry points in the Brillouin zone for P6

_{4}22-XP (X = Al, Ga, or In) are G (0.00, 0.00, 0.00), A (0.00, 0.00, 0.50), H (−0.33, 0.67, 0.50), K (0.33, 0.67, 0.00), G (−0.50, 0.50, 0.00), M (0.00, 0.50, 0.00), L (0.00, 0.50, 0.50), and H (−0.33, 0.67, 0.50). The band structures of P6

_{4}22-XP (X = Al, Ga, or In) are calculated via the HSE06 hybrid functional [40]. In the P6

_{4}22 phase, only InP is a direct band gap semiconductor, which has a band gap of 0.42 eV and the conduction band minimums and the valence band maximums are both located at point G (0.00, 0.00, 0.00). The band gap of P6

_{4}22-InP corresponds to a wavelength of 2958.04 nm, which is in the infrared region. P6

_{4}22-AlP and P6

_{4}22-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 P6

_{4}22-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 P6

_{4}22-GaP are located at point G (0.00, 0.00, 0.00) and point K (0.33, 0.67, 0.00), respectively.

_{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.

_{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.

_{4}22-AlP, P6

_{4}22-GaP, and P6

_{4}22-InP in the Clark model are 1.222 W∙m

^{−1}∙K

^{−1}, 0.972 W∙m

^{−1}∙K

^{−1}, and 0.610 W∙m

^{−1}∙K

^{−1}, respectively. In the Cahill model, the theoretical minimum thermal conductivities for P6

_{4}22-XP (X = Al, Ga, or In) are 1.338 W∙m

^{−1}∙K

^{−1}, 1.058 W∙m

^{−1}∙K

^{−1}and 0.669 W∙m

^{−1}∙K

^{−1}, respectively. According to the calculated values, the theoretical minimum values of the thermal conductivity that are calculated by the Clark model are slightly less than those by the Cahill model. As the contributions of the atomic number density and the phonon spectrum are considered in the Cahill model, whereas the Clark model does not calculate the contribution of the optical phonons [44], the Clark model underestimates the theoretical minimum thermal conductivity and the Cahill model yields a value that is closer to the actual value. The maximum of the theoretical minimum thermal conductivity of P6

_{4}22-XP (X = Al, Ga, or In) corresponds to P6

_{4}22-AlP and the minimum to P6

_{4}22-InP, namely, according to the capacity of heat dissipation at high temperature (approaching the melting point), P6

_{4}22-AlP > P6

_{4}22-GaP > P6

_{4}22-InP. The theoretical minimum thermal conductivities of F$\overline{4}3$m-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 F$\overline{4}3$m-XP (X = Al, Ga, or In) at high temperature.

## 4. Conclusions

_{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 F$\overline{4}3$m-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 F$\overline{4}3$m-XP (X = Al, Ga, or In), especially GaP. As the pressure increases, P6

_{4}22-XP (X = Al, Ga, or In) become increasingly stable. P6

_{4}22-XP (X = Al, Ga, or In) have the largest anisotropy in the (100)-plane and show isotropy in the (001)-plane. P6

_{4}22-InP is a direct band gap semiconductor, which has a band gap of 0.42 eV and potential application as an infrared detector. P6

_{4}22-XP (X = Al or Ga) exhibit indirect band gap properties with band gaps of 1.55 and 0.86 eV, respectively. At high temperature, P6

_{4}22-XP (X = Al, Ga, or In) have stronger thermal conductivity than F$\overline{4}3$m-XP (X = Al, Ga, or In), where maximum and minimum thermal conductivities correspond P6

_{4}22-AlP and P6

_{4}22-InP, respectively. These properties provide a theoretical basis and new ideas for the application of P6

_{4}22-XP (X = Al, Ga, or In) in optoelectronic devices and thermoelectric materials.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The convergence analysis of cut-off energy and the k-point grid allocation in the Brillouin zone.

**Figure 3.**Computed total energy versus unit-cell volume (

**left column**) and the variation of the volume versus pressure (

**right column**) for P6

_{4}22-XP (X = Al, Ga, or In).

**Figure 5.**The relative formation enthalpies curves (relative to F$\overline{4}3$m-XP) as a function of pressure (0 to 35 GPa) for P6

_{4}22-XP (X = Al, Ga, or In); (

**a**,

**b**) are the zoomed in views of selected areas.

**Figure 6.**The 3D directional constructions and 2D representation of Young’s modulus E in the (001)-, (011)-, (100)-, (110)-, and (111)- plane for P6

_{4}22-AlP (

**a**), P6

_{4}22-GaP (

**b**) and P6

_{4}22-InP (

**c**).

**Figure 7.**The electronic band structure for P6

_{4}22-XP (X = Al, Ga, or In), AlP (

**a**), GaP (

**b**), InP (

**c**).

**Figure 8.**The partial densities of states of P6

_{4}22-XP (X = Al, Ga, or In): AlP (

**a**), GaP (

**b**), and InP (

**c**).

**Table 1.**The calculated (GGA-PBE and DFT-D2) lattice parameters and densities of P6

_{4}22- and F$\overline{4}3$m-XP (X = Al, Ga, or In).

Space Group | Methods | a [Å] | c [Å] | V [Å^{3} molecule^{−1}] | ρ [g cm^{−3}] | |
---|---|---|---|---|---|---|

AlP | P6_{4}22 | PBE | 3.849 | 8.683 | 37.139 | 2.591 |

DFT-D2 | 3.790 | 8.669 | 35.942 | 2.678 | ||

F$\overline{4}3$m | PBE | 5.510 | 41.822 | 2.301 | ||

DFT-D2 | 5.442 | 40.297 | 2.388 | |||

F$\overline{4}3$m^{[a]} | Exp. | 5.467 | 40.773 | 2.360 | ||

GaP | P6_{4}22 | PBE | 3.899 | 8.570 | 37.613 | 4.446 |

DFT-D2 | 3.818 | 8.553 | 35.996 | 4.646 | ||

F$\overline{4}3$m | PBE | 5.505 | 41.717 | 4.009 | ||

DFT-D2 | 5.412 | 39.631 | 4.220 | |||

F$\overline{4}3$m^{[b]} | Exp. | 5.451 | 40.488 | 4.130 | ||

InP | P6_{4}22 | PBE | 4.190 | 9.416 | 47.726 | 5.073 |

DFT-D2 | 4.064 | 9.298 | 44.330 | 5.461 | ||

F$\overline{4}3$m | PBE | 5.973 | 53.263 | 4.545 | ||

DFT-D2 | 5.854 | 51.162 | 4.876 | |||

F$\overline{4}3$m^{[c]} | Exp. | 5.869 | 50.540 | 4.790 |

**Table 2.**The calculated elastic constants (C

_{11}, C

_{12}, C

_{13}, C

_{33}, C

_{44}, C

_{66}), bulk moduli B, shear moduli G, Young’s modulus E (GPa), Poisson’s ratios ʋ and universal anisotropic index A

^{U}for P6

_{4}22-XP (X = Al, Ga, or In) when pressure P (GPa) increases from 0 to 35 GPa via the method of GGA-PBE.

Space Group | Methods | P | C_{11} | C_{12} | C_{13} | C_{33} | C_{44} | C_{66} | B | G | B/G | E | ʋ | A^{U} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

P6_{4}22-AlP | PBE | 0 | 147 | 51 | 58 | 174 | 60 | 48 | 88 | 53 | 1.68 | 132 | 0.25 | 0.064 |

5 | 169 | 67 | 77 | 190 | 66 | 51 | 107 | 56 | 1.91 | 143 | 0.28 | 0.095 | ||

10 | 190 | 83 | 96 | 224 | 64 | 53 | 127 | 58 | 2.19 | 151 | 0.30 | 0.060 | ||

15 | 207 | 98 | 114 | 245 | 59 | 54 | 144 | 56 | 2.57 | 149 | 0.33 | 0.032 | ||

20 | 226 | 113 | 132 | 267 | 57 | 56 | 161 | 57 | 2.82 | 153 | 0.34 | 0.029 | ||

25 | 240 | 128 | 148 | 290 | 55 | 56 | 177 | 56 | 3.16 | 152 | 0.36 | 0.039 | ||

30 | 257 | 143 | 166 | 313 | 50 | 57 | 194 | 55 | 3.53 | 151 | 0.37 | 0.064 | ||

35 | 267 | 161 | 181 | 333 | 38 | 53 | 208 | 48 | 4.33 | 134 | 0.39 | 0.235 | ||

F$\overline{4}3$m-AlP | PBE | 0 | 123 | 58 | 60 | 80 | 47 | 1.70 | 118 | 0.25 | 0.494 | |||

F$\overline{4}3$m-AlP ^{[a]} | Exp. | 0 | 129 | 56 | 52 | |||||||||

P6_{4}22-GaP | PBE | 0 | 152 | 37 | 49 | 144 | 67 | 57 | 80 | 58 | 1.38 | 140 | 0.21 | 0.087 |

5 | 178 | 54 | 56 | 150 | 75 | 62 | 92 | 64 | 1.44 | 156 | 0.22 | 0.117 | ||

10 | 213 | 82 | 116 | 228 | 82 | 65 | 140 | 66 | 2.12 | 171 | 0.30 | 0.234 | ||

15 | 230 | 91 | 115 | 222 | 88 | 69 | 147 | 70 | 2.10 | 181 | 0.29 | 0.217 | ||

20 | 250 | 108 | 111 | 207 | 92 | 71 | 151 | 73 | 2.07 | 189 | 0.29 | 0.211 | ||

25 | 271 | 125 | 143 | 248 | 96 | 73 | 179 | 75 | 2.39 | 197 | 0.32 | 0.245 | ||

30 | 291 | 140 | 167 | 273 | 102 | 76 | 200 | 76 | 2.63 | 202 | 0.33 | 0.319 | ||

35 | 310 | 156 | 181 | 285 | 104 | 77 | 216 | 77 | 2.81 | 206 | 0.34 | 0.332 | ||

F$\overline{4}3$m-GaP | PBE | 0 | 134 | 60 | 70 | 80 | 59 | 1.39 | 131 | 0.21 | 0.500 | |||

F$\overline{4}3$m-GaP ^{[b]} | Exp. | 0 | 141 | 62 | 70 | |||||||||

P6_{4}22-InP | PBE | 0 | 108 | 37 | 49 | 117 | 45 | 36 | 67 | 37 | 1.81 | 94 | 0.27 | 0.124 |

5 | 130 | 53 | 64 | 135 | 48 | 38 | 84 | 40 | 2.10 | 104 | 0.29 | 0.120 | ||

10 | 151 | 76 | 94 | 171 | 48 | 38 | 110 | 40 | 2.75 | 107 | 0.34 | 0.157 | ||

15 | 168 | 93 | 108 | 187 | 55 | 37 | 126 | 42 | 3.00 | 113 | 0.35 | 0.257 | ||

20 | 190 | 112 | 129 | 209 | 59 | 39 | 146 | 43 | 3.40 | 117 | 0.37 | 0.298 | ||

25 | 211 | 127 | 148 | 230 | 53 | 42 | 165 | 44 | 3.75 | 121 | 0.38 | 0.164 | ||

30 | 225 | 148 | 168 | 251 | 48 | 38 | 186 | 41 | 4.54 | 115 | 0.40 | 0.118 | ||

35 | 245 | 161 | 188 | 273 | 56 | 42 | 201 | 44 | 4.57 | 123 | 0.40 | 0.233 | ||

F$\overline{4}3$m-InP | PBE | 0 | 96 | 55 | 49 | 59 | 35 | 1.69 | 88 | 0.25 | 0.924 | |||

F$\overline{4}3$m-InP ^{[c]} | Exp. | 0 | 102 | 56 | 47 |

**Table 3.**The calculated maximum values E

_{max}, minimum values E

_{min}and ratios E

_{max}/E

_{min}of XP (X = Al, Ga, or In) in the P6

_{4}22 phase via the method of GGA-PBE.

Planes | Materials | E_{max} | E_{min} | Ratio | Planes | Materials | E_{max} | E_{min} | Ratio |
---|---|---|---|---|---|---|---|---|---|

(001) | P6_{4}22-AlP | 120.333 | 120.333 | 1.000 | (110) | P6_{4}22-AlP | 145.147 | 120.334 | 1.206 |

P6_{4}22-GaP | 132.093 | 132.093 | 1.000 | P6_{4}22-GaP | 151.508 | 119.008 | 1.273 | ||

P6_{4}22-InP | 84.764 | 84.764 | 1.000 | P6_{4}22-InP | 104.849 | 83.797 | 1.251 | ||

(011) | P6_{4}22-AlP | 142.751 | 120.334 | 1.186 | (111) | P6_{4}22-AlP | 145.081 | 120.334 | 1.205 |

P6_{4}22-GaP | 151.508 | 132.093 | 1.147 | P6_{4}22-GaP | 151.508 | 132.093 | 1.147 | ||

P6_{4}22-InP | 104.849 | 84.764 | 1.237 | P6_{4}22-InP | 104.849 | 84.764 | 1.237 | ||

(100) | P6_{4}22-AlP | 145.147 | 120.334 | 1.206 | (010) | P6_{4}22-AlP | 145.147 | 120.334 | 1.206 |

P6_{4}22-GaP | 151.508 | 119.008 | 1.273 | P6_{4}22-GaP | 151.508 | 119.008 | 1.273 | ||

P6_{4}22-InP | 104.849 | 83.797 | 1.251 | P6_{4}22-InP | 104.849 | 83.797 | 1.251 |

**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 F$\overline{4}3$m-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.

Clark | Cahill | |||||
---|---|---|---|---|---|---|

M_{a} × 10^{−23} | k_{min} | ν_{t} | ν_{l} | p × 10^{28} | k_{min} | |

P6_{4}22-AlP | 4.817 | 1.222 | 4.523 | 7.825 | 5.379 | 1.338 |

F$\overline{4}3$m-AlP | 4.817 | 1.132 | 4.520 | 7.874 | 4.777 | 1.240 |

P6_{4}22-GaP | 8.389 | 0.972 | 3.381 | 5.569 | 6.047 | 1.058 |

F$\overline{4}3$m-GaP | 8.389 | 0.904 | 3.836 | 6.291 | 4.779 | 1.024 |

P6_{4}22-InP | 12.126 | 0.610 | 2.885 | 5.115 | 3.666 | 0.669 |

F$\overline{4}3$m-InP | 12.126 | 0.592 | 2.775 | 4.822 | 3.748 | 0.647 |

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**MDPI and ACS Style**

Miao, J.; Chai, C.; Zhang, W.; Song, Y.; Yang, Y.
First-Principles Study on Structural, Mechanical, Anisotropic, Electronic and Thermal Properties of III-Phosphides: *X*P (*X* = Al, Ga, or In) in the *P*6_{4}22 Phase. *Materials* **2020**, *13*, 686.
https://doi.org/10.3390/ma13030686

**AMA Style**

Miao J, Chai C, Zhang W, Song Y, Yang Y.
First-Principles Study on Structural, Mechanical, Anisotropic, Electronic and Thermal Properties of III-Phosphides: *X*P (*X* = Al, Ga, or In) in the *P*6_{4}22 Phase. *Materials*. 2020; 13(3):686.
https://doi.org/10.3390/ma13030686

**Chicago/Turabian Style**

Miao, Junjie, Changchun Chai, Wei Zhang, Yanxing Song, and Yintang Yang.
2020. "First-Principles Study on Structural, Mechanical, Anisotropic, Electronic and Thermal Properties of III-Phosphides: *X*P (*X* = Al, Ga, or In) in the *P*6_{4}22 Phase" *Materials* 13, no. 3: 686.
https://doi.org/10.3390/ma13030686