Nitrides form a specific subgroup of the III–V materials described by wide energy gaps, high ionicity, strong bonds, low compressibility, and high thermal conductivity, and mechanical stability.^[1] Wide energy gap nitride semiconductors such as BN, AlN, InN, and GaN have received a great deal of research interest and achieved important development due to their device applications in blue and ultraviolet wavelengths. They can be utilized in blue and ultraviolet light emitting diodes, room temperature laser diodes (LDs), solar cells, and the field-effect transistors.^[2–6] Recently, the blue light III–V nitride semiconductor laser was first successfully fabricated by Nekamura.^[7] The pressure dependence and temperature dependence of the electronic, optical, and mechanical properties in semiconductors have become the subjects of many studies.^[8–23] These studies can supply further worthy data about the energy band structure and energetic properties of the nitride group.

In this paper, we carry out calculations to determine the electronic band structures, optical and mechanical properties of BN, AlN, InN compounds using the empirical pseudo-potential method (EPM). The studied properties for each of the considered materials are the , n, , , , , , and , and other important parameters such as , , , , , α, β, ζ, and of the studied binary compounds are also investigated under the action of pressure.

The calculations are carried out using the EPM. The pressure-dependent eigenvalues are found out by solving numerically the secular determinant, 1 where 2 is the pressure-dependent pseudo-potential. and are the symmetric and anti-symmetric pressure-dependent form factors, respectively. The EPM involves the fitting of the atomic form factors to experimental values of energy gaps of studied semiconductors. and are the reciprocal lattice vectors, with and is the position vector of each atom in the unit cell with being the pressure-dependent lattice constant.

The pressure-dependent lattice constant is calculated from the relation^[22] 3 where is the bulk modulus, is its pressure derivative, and are the lattice constants at P = 0 and , respectively. Equation (1) is solved numerically by using our MATLAB language program, which is a matrix based on 65 bulk reciprocal lattice vectors 's. These values are corresponding to , 4, and 11 for zinc-Blende-type structure, which satisfy the condition , which gives the non-zero pseudo-potential.^[20]

The valence and conduction energy band gaps of the studied compounds are obtained by arranging the obtained eigenvalues and setting the top of the valence bands to be zero energy. The calculated energy band gaps used in the fitting procedure and the adjusting form factors of BN, AlN, and InN for various pressures are recorded in Table 1.

Table 1.

Table 1.

Table 1. Energy band gaps for BN, AlN, and InN at various pressures. . BN P/kbar 0 20 40 60 80 100 120 eV 11.479 11.619 11.694 11.704 11.677 11.611 11.51 11.5a eV 10.021 10.678 11.146 11.44 11.616 11.691 11.674 10.0a eV 6.1219 6.1125 6.1242 6.1312 6.1442 6.1662 6.1764 6.1a 6.112b 6.124b 6.136b 6.148b 6.16b 6.172b a/Å 3.6155 3.561 3.5155 3.4766 3.4427 3.4126 3.3856 AlN P/kbar 0 20 40 60 80 100 120 eV 8.9013 9.2581 9.6779 10.02 10.349 10.63 10.894 9.07c eV 6.2748 7.0812 8.0067 8.7825 9.529 10.18 10.791 6.2d eV 5.1788 5.1811 5.2687 5.3319 5.4125 5.4827 5.557 5.1e 5.1826b 5.2626b 5.3398b 5.4142b 5.486b 5.555b a/Å 4.37 4.2604 4.1689 4.0951 4.0335 3.9806 3.9345 InN P/kbar 0 20 40 60 80 100 120 eV 5.8141 5.7812 5.7914 5.8102 5.8298 5.8443 5.8609 5.82f eV 1.90 1.9395 1.975 2.0193 2.0586 2.0989 2.1374 1.94f 1.9399b 1.9797b 2.0193b 2.0587b 2.098b 2.1371b eV 2.5098,2.51f 2.1694 2.0199 1.9264 1.8613 1.8038 1.7602 a/Å 4.96 4.8297 4.7676 4.7266 4.6961 4.6718 4.6516 a Ref. [44], b Ref. [22], c Ref. [45], d Ref. [46], e Ref. [47], f Ref. [48].

Table 1. Energy band gaps for BN, AlN, and InN at various pressures. .

Optical properties are of fantastic significance in the design and evaluation of optoelectronic devices. The refractive index is a completely vital parameter in the semiconductor. It represents an essential physical aspect that describes their optical and electronic properties, knowing n is crucial for devices such as photonic crystals, wave guides, solar cells, and detectors.^[24] Numerous efforts have been made to relate the index of refraction to the energy band gap through simple relations.^[25–30] There are different methods to calculate the refractive index; our calculations are obtained by using the Moss formula.^[25] The dielectric characteristic is a key optical quantity to extract all different optical properties like absorption spectrum, energy loss function, refractive index, and reflectivity. The dielectric function depicts the linear response of a system to electromagnetic radiation.^[31] Knowing the refractive index of the considered materials, we calculate the high frequency and static dielectric constants.^[12]

The influence of strain on the electronic property requires the knowledge of the mechanical properties of materials, and precise elastic constants which characterize the response to an applied macroscopic stress.^[32] The cubic crystal has only three independent elastic constants, specifically , , and .^[33–35] The elastic constants of the materials are calculated by knowing the polarities of the studied materials, which are derived from the adjusted symmetric and anti-symmetric form factors at as shown in Ref. [36], 4

The elastic constants , , and of the considered materials are calculated by following the same procedure used by Bouarissa which turned into primarily based essentially on the work of Baranowski^[37,38] as shown below, 5 6 7 where d is the nearest-neighbor distance. The knowledge of these elastic constants permits us to determine the bulk , shear ), and Youn’s ( ) modulus of each of the considered materials as follows:^[34,35] 8 9 10 The bond stretching , bond-bending force constant, and internal-strain parameter of studied compounds are also determined, which are related to the elastic constants as indicated in Refs. [39]–[41], 11 12 13 where is the pressure-dependent linear compressibility that is given by Ref. [22] 14

The parameter is given by Ref. [41] 15 where is the pressure-dependent static dielectric constant, is the pressure-dependent effective charge which was calculated according to the following relation:^[42] 16 where is the pressure-dependent transverse effective charge that is given from the following relation:^[43] 17 where is the change in valence of the studied material.

Other important constants such as Cauchy and Born ratios could also be determined in terms of the elastic constants as^[22] 18 19 The ductility index ( can be calculated according to the empirical formula of Pugh,^[35] 20

The pressure dependent eigenvalues for the considered compound are found by solving the secular equation (1) at room temperature and different pressures. The experimental value of the energy band gap as a function of pressure which is used in the fitting procedure is given from Ref. [22], 21 where a and b represent the pressure coefficients.

The properties at high-pressure values may be different from those under normal conditions. The influence of pressure on the calculated energy band gap of each of BN, AlN, and InN is listed in Table 1 and displayed in Fig. 1. Our calculated results show that BN and AlN are indirect semiconductors under the normal pressure and stay indirect in the studied region of pressure (0 kbar–120 kbar, Pa). In contrast, InN is a direct semiconductor till 40 kbar and becomes indirect (X) in the other region of the studied pressure (40 kbar–120 kbar). Table 1 also contains the lattice constants of the considered materials at different pressures. Our calculated results are compared with the available published data and show excellent accordance.^[23,44–48]

Fig. 1.

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	Fig. 1. (color online) Variations of energy band gaps of BN, AlN, and InN with pressures.

Figure 1 shows the variations of energy band gaps of BN, AlN and InN with pressure. The behaviors of the , , and with pressure could be described by the following polynomials:

For BN 22 23 24

For AlN, 25 26 27

For InN 28 29 30

The final adjusted pseudo-potential form factors of BN, AlN, and InN at different values of pressure are listed in Table 2. From this table we note that the pseudo-potentials increase with pressure increasing.

Table 2.

Table 2.

Table 2. The adjusted form factors of BN, AlN, and InN at different values of pressure. . BN Form factors Ry/P 0 20 40 60 80 100 120 −0.52873 −0.52856 −0.52843 −0.52823 −0.52813 −0.52805 −0.01700 −0.016 −0.016 −0.016 −0.016 −0.016 −0.01609 0.4679 0.5115 0.5531 0.5930 0.6310 0.6700 0.70 0.120014 0.1200 0.1200 0.1200 0.1200 0.1201 0.1201 0.3200 0.3201 0.3202 0.3205 0.3206 0.3208 0.3209 0.1200 0.1201 0.1203 0.1204 0.1207 0.1209 0.12 AlN −0.50300 −0.502 −0.502 −0.502 −0.502 −0.502 −0.502 0.2309 0.2311 0.2312 0.2314 0.2317 0.2318 0.2319 0.029951 0.0867 0.1315 0.1502 0.1676 0.200013 0.2001 0.2003 0.2006 0.2007 0.2009 0.2011 0.322993 0.3231 0.3232 0.3234 0.3237 0.3238 0.3240 0.090077 0.090377 0.090477 0.090677 0.090777 0.090877 0.090977 InN −0.35889 −0.35879 −0.35869 −0.35849 −0.35839 −0.35819 −0.35799 0.253999 0.2542 0.2543 0.2545 0.2546 0.2549 0.2549 0.207999 0.2165 0.2193 0.2215 0.2235 0.2253 0.139967 0.1400 0.1401 0.1403 0.1406 0.1407 0.1408 0.429999 0.4301 0.4302 0.4305 0.4307 0.4308 0.4309 0.3990 0.3992 0.3993 0.3995 0.3996 0.3998 0.3999

Table 2. The adjusted form factors of BN, AlN, and InN at different values of pressure. .

Figure 2 shows the electronic energy band structures for BN, AlN, and InN for two values of pressure, P = 0 kbar (solid lines) and P = 120 kbar (dash lines). Applied hydrostatic pressure affects the electronic band structures of zinc-blende BN, AlN, and InN without changing the shapes of the bands. Note that all the bands are shifted. Each shift is not constant and depends on the k-point and energy. The figure shows that both conduction and valence energy bands are affected by pressure. It is seen that the first conduction energy band is more affected by pressure than the other bands and exhibits greater enhancement at the -high point of symmetry. Also, it is noticed that the InN is less sensitive to pressure than BN and A1N.

Fig. 2.

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	Fig. 2. (color online) Electronic energy band structures for BN, AlN, and InN for pressures P = 0 kbar (solid lines) and P = 120 kbar (dash lines).

Figure 3 shows the variations of the refractive indices of BN, AlN, and InN with pressure. The refractive indices of BN and AlN are approximately the same and less sensitive to pressure, while InN has a large refractive index that decreases almost linearly with pressure increasing.

Fig. 3.

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	Fig. 3. Refractive indices of BN, AlN, and InN versus pressure.

The variations of high frequency and the static dielectric constants for BN, AlN, and InN with pressure are displayed in Fig. 4. The figure shows that the optical dielectric constants for BN and AlN are less sensitive to pressure than InN, which has high values and decreases linearly with pressure increasing. The variations of the static dielectric constants for the considered materials have the same behavior as the high frequency dielectric constants.

Fig. 4.

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	Fig. 4. (color online) Variations of optical and static dielectric constants of BN, AlN, and InN with pressures.

From the final adjusted symmetric and anti-symmetric form factors at , the polarities of the considered materials at various pressures are calculated and shown in Table 3. The polarities of the considered materials slightly increase with pressure increasing. The calculated values of polarities for the considered materials are in excellent accordance with the published values. Knowing the polarities of BN, AlN, and InN, their elastic constants ( , , and at various pressures can be calculated. The calculated values of elastic constants ( , , and and the linear compressibility of BN, AlN, and InN at various pressures are listed in Table 3 and shown in Fig. 5. The elastic constants increase with pressure rising and the increasing rates of BN and AlN is higher than that of InN which has lower sensitivity of pressure. The linear compressibility decreases with pressure rising. The knowing of the elastic constants may assist us in obtaining information about the stability of the crystal over the whole region of pressure. The requirements for mechanical stability of a cubical crystal are , , ,^[32,33] which are fulfilled in our calculations over the whole region of pressure, reflecting that BN, AlN, and InN compounds are more stable in their zinc-blende structures. Our results for , , of the studied material are in good accordance with the published data.

Table 3.

Table 3.

Table 3. Values of polarity , elastic constants , and the linear compressibility of BN, AlN, and InN at different pressures. . BN P/kbar 0 20 40 60 80 10 12 0.22698, 0.226a 0.2270 0.2271 0.2272 0.2273 0.2274 0.2275 dyn/cm 127.0, 83.7c, 120.44a 130.1 146.1 154.45 162.19 169.46 176.32 dyn/cm 54.707, 50.28a, 42.0d 59.02 62.935 66.531 69.868 72.999 75.954 dyn/cm 51.355, 59.19a, 49.3a 55.403 59.079 62.454 65.586 68.525 71.298 cm /dyn) 0.42299, 0.3846a 0.39208 0.36768 0.34781 0.3312 0.317 0.30467 AlN 0.39764, 0.397a 0.3979 0.3983 0.3991 0.3995 0.3999 0.4005 dyn/cm 41.033, 42.99a 46.571 51.875 56.659 61.088 65.211 69.066 /cm 17.871, 18.71a 20.283 22.595 24.68 26.61 28.407 30.088 dyn/cm 16.518, 16.0d 18.747 20.882 22.807 24.589 26.248 27.799 cm /dyn) 1.3025 1.1476 1.0302 0.94322 0.87483 0.8195 0.77375 InN 0.39, 0.39b 0.3904 0.3907 0.3915 0.3925 0.39 0.3935 dyn/cm 22.022, 21.632b 25.143, 25.775e 26.81 27.962 28.843 29.581 30.207 dyn/cm 9.5846, 9.4b 10.943 11.669 12.171 12.556 12.878 13.151 dyn/cm 8.8676, 8.6d 10.124 10.795 11.259 11.613 11.91 12.162 cm /dyn) 2.4277 2.1263 1.9941 1.9119 1.8534 1.8071 1.7696 a Ref. [49], b Ref. [50], c Ref. [51], d Ref. [52], e Ref. [53].

Table 3. Values of polarity , elastic constants , and the linear compressibility of BN, AlN, and InN at different pressures. .

Fig. 5.

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	Fig. 5. (color online) Variations of elastic constants ( , , and ) and of BN, AlN, and InN with pressures.

The determining of the elastic constants , , may help us to proceed with calculating the bulk ( , shear ( , and Young’s ( ) moduli. Table 4 and figure 6 show the calculated values of bulks, shears, and Young’s modul of BN, AlN, and InN and their functions of pressure, respectively. These moduli have the same behaviors under pressure as elastic constants , , and . The Young modulus is desired to supply information about the measure of the stiffness of the material. The stiffness increases with the increase of Young modulus of the material. The present values of Young moduli decrease from BN to InN which indicates that BN is stiffer than AlN and InN. One can estimate the brittle and ductile behaviors of the studied compounds by calculating the values of ductility index ( of the considered materials, according to the empirical formula of Pugh:^[35] this formula states that the value of that separates the ductile behavior and brittle behavior of material is around 1.75. If the material behaves as a ductile material, otherwise the material behaves as a brittle manner.^[32] Table 4 also contains the calculated values of for the considered materials which are approximately constants over the whole region of pressure and greater than 1.75, indicating that the considered materials are ductile materials.

Fig. 6.

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	Fig. 6. Variations mechanical parameters ( , , and of BN, AlN, and InN with pressures.

Table 4.

Table 4.

Table 4. Values of bulk, shear, Young’s moduli, and the ductility indices of BN, AlN, and InN at various pressures. . BN Elastic parameters/P/kbar 0 20 40 60 80 100 120 dyn/cm 78.805, 73.67a, 55.7b 85.017 90.657 95.837 100.64 105.15 109.41 dyn/cm 94.059, 90.61c 101.47 108.21 114.39 120.12 125.51 130.59 dyn/cm 36.147, 31.4c 38.996 41.583 43.959 46.163 48.232 50.184 2.1801 2.1801 2.1801 2.1801 2.1801 2.1801 2.1802 AlN dyn/cm 25.592, 26.81a 29.046 32.355 35.34 8.103 40.675 43.08 dyn/cm 30.189, 29.231d 34.263 38.165 41.683 44.94 47.972 50.805 dyn/cm 11.581, 11.7d 13.144 14.64 15.99 17.239 18.402 19.489 2.2098 2.2098 2.21 2.2101 2.2103 2.2104 2.2105 InN dyn/cm 13.73, 13.8b 15.677 16.716 17.435 17.985 18.445 18.836 dyn/cm 16.209, 23.10c, 11.6e 18.506 19.732 20.58 21.227 21.769 22.229 dyn/cm 6.2188, 6.51c 7.1 7.5704 7.8953 8.1436 8.3515 8.528 2.2078 2.208 2.2081 2.2083 2.2085 2.2086 2.2087 a Ref [49], b Ref. [52], c Ref. [54], d Ref. [55], e Ref. [56].

Table 4. Values of bulk, shear, Young’s moduli, and the ductility indices of BN, AlN, and InN at various pressures. .

Table 5 lists the values of bond-stretching (α), bond-bending ( force constants, internal strain parameter (ζ), transverse effective charge , Cauchy ratio ( ), anisotropy factor ( ), and Poisson ratio ( ) of BN, AlN, and InN at different pressures. From this table, it can be seen that the bond-stretching (α), bond-bending (β) force constants increase with the increase of pressure. Our results of (α, β), at normal pressure agree well with those obtained experimentally.^{[49,50,54–56]} The Poison ratio is also a very significant property for manufacturing investment; it supplies more information about the distinction of the bonding force than any of the other elastic parameters. For the fulfillment of hardness of present materials, the elastic properties, e.g., anisotropy factor, Poisson’s ratio of these materials are measured. From Table 5, one can say that all calculated results except the bond-stretching (α), bond-bending (β) force constants, are slightly influenced by pressure.

Table 5.

Table 5.

Table 5. Values of bond-stretching α, bond-bending (β) force constants, anisotropy factor ( ), Poisson ratio ( ), internal strain parameter (ζ), Cauchy ratio ( ), and transverse effective charge of BN, AlN, and InN at different pressures. . BN P/kbar 0 20 40 60 80 100 120 α/(N/m) 263.14, 261.54e 279.6 294.34 307.71 320 331.42 342.1 β/(N/m) 65.345, 62.2e 69.433 73.093 76.413 79.463 82.298 84.951 0.70386, 1.473a 0.70386 0.70386 0.70386 0.70386 0.70386 0.70386 0.30107, 0.18b 0.30107 0.30107 0.30107 0.30107 0.30107 0.30108 ζ 0.56555, 0.55e 0.56555 0.56555 0.56555 0.56556 0.56556 0.56556 1.0653 1.0653 1.0653 1.0653 1.0653 1.0653 1.0653 1.8634 1.8637 1.8641 1.8644 1.8647 1.865 1.8652 AlN α/(N/m) 103.4, 121.1e 114.41 124.71 133.81 142.1 149.7 156.72 β/(N/m) 25.305, 24.9e 27.999 30.517 32.740 34.767 36.625 38.339 0.70113 0.70112 0.70111 0.70109 0.70108 0.70107 0.70106 0.30339, 0.346b 0.3034 0.30341 0.30342 0.30343 0.30344 0.30345 ζ 0.56971, 0.663c, 0.57e 0.56972 0.56973 0.56976 0.56977 0.56979 0.56981 1.0819 1.082 1.082 1.0821 1.0822 1.0823 1.0823 2.3734 2.3741 2.3753 2.3772 2.3781 2.3793 2.3807 InN α/(N/m) 62.962, 83.3e 69.999 73.680 76.189 78.086 79.670 81.007 β/(N/m) 15.423, 15.5e 17.145 18.046 18.659 19.122 19.508 19.834 0.7013 0.70129 0.70128 0.70126 0.70124 0.70123 0.70122 0.30324, 0.369b 0.30325 0.30326 0.30327 0.30329 0.3033 0.30331 ζ 0.56944, 0.58e 0.56946 0.56947 0.5695 0.56953 0.56955 0.56956 1.0809 1.0809 1.081 1.0811 1.0812 1.0813 1.0813 2.354, 2.35d 2.3551 2.356 2.358 2.3604 2.3617 2.363 a Ref. [54], b Ref. [56], c Ref. [55], d Ref. [50], e Ref. [49].

Table 5. Values of bond-stretching α, bond-bending (β) force constants, anisotropy factor ( ), Poisson ratio ( ), internal strain parameter (ζ), Cauchy ratio ( ), and transverse effective charge of BN, AlN, and InN at different pressures. .

In this study, we use the EPM for performing our calculations. The values of energy band gap, n, , of the nitride semiconductors, BN, AlN, and InN are calculated at different pressures. The polarities of considered materials are determined from the adjusted form factors at and are used to obtain the elastic constants , , and and their relevant elastic parameters namely, bulk ( ), shear ( ), and Young’s ( ) modulus. Other important parameters such as α, β, ζ, , , , , , and A are also calculated. The variations of the studied parameters with pressure are also studied. Our results at normal pressure are compared with the available published data and show they are in satisfactory accordance. The effects of pressure on most of our results have not yet been measured nor calculated, so, our results can serve as a prediction for future investigations.