English

• 0.   Introduction

  Ⅲ-Ⅴ semiconductors composed of group Ⅲ cations and group Ⅴ anions represent a class of optoelectronic materials^[1-4]. Among them, aluminum-based semiconductors such as AlN, AlP, and AlAs, and indium-based semiconductors like InN, InP, and InAs, have been utilized as typical semiconductors^[5-6]. They are characterized by a wide bandgap, low effective mass of electrons, high thermal conductivity, and strong radiation resistance, and most of them are direct bandgap semiconductors, thereby possessing strong optical absorption capabilities^[7]. These characteristics make aluminum-based and indium-based semiconductors the preferred choices for applications in ultraviolet and infrared light-emitting diodes, infrared detectors, sensors, high-power electronic devices, and radiation-resistant electronic devices^[8-9]. For these reasons, these materials have been deeply studied and continuously reported in the electronics manufacturing industry and related fields^[10-14].

  The practical applications of Ⅲ-Ⅴ semiconductors in various fields have prompted researchers to delve into the deeper theoretical mechanisms behind them. In some studies of Ⅲ-Ⅴ semiconductors, it has been reported that doping with different atoms can alter their electronic structure^[15]. Dong et al.^[16] found in their theoretical simulations that doping Mg and Cd into AlN could introduce more vacancies, resulting in improved p-type semiconductors. Nabi et al.^[17] found, through first-principles calculations, that doping AlSb with Sc, P, and Bi caused the bandgap to narrow, thereby further expanding its potential applications in optoelectronics. Wang et al.^[18] doped Be, Mg, Ca, Sr, and Ba into AlP and calculated the electronic structure properties and ferromagnetism of the doped system, indicating its potential as a promising material for applications in spintronics. There have also been reports in the field of heterojunctions and nanosheets. Li et al.^[19] studied the structure and properties of the ZnSe/AlAs/GaAs multilayer heterojunction and found that the multilayer heterojunction has superior optoelectronic properties compared to the AlAs/GaAs and ZnSe/AlAs bilayer structures. Sarmazdeh et al.^[20] found in their computational study that InN nanosheets can serve as a good candidate material for optical communication applications and optoelectronic devices due to their electronic structure. Ali et al.^[21] investigated the optoelectronic properties of AlAs under varying pressures and analyzed the trends. The research found that with pressure increased, the bandgap gradually decreased, but there was no significant change in optical properties. Jalil et al.^[22] used first-principles calculations to study the electronic properties of InSb nanotubes, exploring their potential applications in information storage. Bafekry et al.^[23] started with the fabrication of AlSb monolayer membranes, and employed density functional theory to investigate the optical properties of double-layer honeycomb structures of AlSb and InSb films. They discovered a distinct absorption peak near 5 eV, indicating their potential as ultraviolet-active materials. These studies are significant for applications in novel electronic and optical systems. Salmi et al.^[24] employed first-principles calculations to deeply explore the phase stability, pressure-induced phase transitions, and electronic properties of AlP, AlAs, and AlSb in different phases, which include the zinc blende, rock salt, wurtzite, NiAs, and CsCl structures. These findings lay a theoretical groundwork for the application of these materials in the realms of electronics and optoelectronics. Vurgaftman et al.^[25] have comprehensively compiled band parameters for Ⅲ-Ⅴ compound semiconductors and their alloys, including bandgaps, spin-orbit splitting, and effective masses, providing critical data for the design of electronic and optoelectronic devices. Additionally, this review discusses the correlation between different band structure models and experimental data, laying a theoretical foundation for understanding and simulating the electronic properties of these materials.

  In summary, aluminum-based and indium-based semiconductors have excellent optoelectronic properties. However, most existing literature focuses on optimizing material performance through external conditions^[26-30], and there is relatively less theoretical research and simulation calculation regarding the intrinsic structural characteristics of the materials themselves. Therefore, this paper is based on first-principles calculations and selects eight semiconductors, AlX and InX (X=N, P, As, Sb), as research objects to explore the relationship between their structures and properties.

  1.   Computational methods and model establishment

  This paper is based on first-principles calculations using the density functional theory, employing the Cambridge sequential total energy package (CASTEP) module^[31] to calculate the structure and the most stable geometric shape within the original crystal cell. For AlX and InX, the k-point grid^[32] was less than 0.25 nm^-1, and the plane wave cutoff energies for AlN and InN were 770 eV, for AlP and InP were 500 eV, and for AlAs, AlSb, InAs, and InSb were 330 eV. In this research, the generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE)^[33], local density approximation (LDA)^[34], and Heyd, Scuseria, and Ernzerhof (HSE06)^[35] functionals were used to describe the exchange-correlation potential. The plane wave basis set describing the valence shell of each atom was selected as: Al-3s²3p¹, In-5s²5p¹, N-2s²2p³, P-3s²3p³, As-4s²4p³, and Sb-5s²5p³. The norm-conserving pseudopotential (NCP)^[36] was used to determine the interaction potential between the ionic core and the valence electrons, and other calculation parameters and convergence criteria all used the default values in the CASTEP module. The tests showed that the calculation results obtained with the above parameters are reasonable.

  Fig. 1 shows the crystal model selected in this paper. The lattice constants and cell volumes of the eight semiconductors AlX and InX before and after structural optimization are listed in Table 1. The crystals of AlX and InX are all cubic, with the space group F43m, possessing high symmetry, and being non-polar, with physical properties that are roughly the same in all directions^[37]. The error in the lattice constants of the eight crystals, both before and after geometric structure optimization, was less than 5%, indicating the appropriateness of the calculation parameters selected in this paper.

  Figure 1

  

  Figure 1.  Crystal structure of AlX and InX

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  Table 1

  

  Table 1.  Information on crystal parameters before and after structural optimization of AlX and InX

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  Semiconductor	Space group	Parameter before optimization			Parameter after optimization
  		a=b=c / nm	V / nm3		a=b=c / nm	V / nm3
  AlN	F43m	0.437	0.083 338 2		0.433	0.081 277 8
  AlP		0.547	0.163 934		0.552	0.168 042
  AlAs		0.568	0.182 844		0.564	0.179 363
  AlSb		0.619	0.236 607		0.616	0.233 973
  InN		0.501	0.125 405		0.496	0.121 803
  InP		0.590	0.205 792		0.589	0.204 336
  InAs		0.611	0.227 777		0.602	0.217 732
  InSb		0.663	0.291 859		0.659	0.286 061

  2.   Results and discussion

  2.1   Stability analysis

  To verify the stability of the lattice after geometric structure optimization, the phonon dispersion spectra of AlX and InX were calculated in this paper. Observations from Fig. 2a-2h indicate that across the entire Brillouin zone, the frequencies of all branch wave vectors were greater than zero, with no imaginary frequencies present in the phonon dispersion spectra. This suggests that the crystals can stably exist without external perturbations, as their energy is a non-negative real number, following the minimum energy principle. It was also noted that the smaller the molecular weight of the crystal, the more the vibrational frequency tended towards the high-frequency region, indicating a larger chemical bond energy in the smaller molecular weight crystals.

  Figure 2

  

  Figure 2.  (a-h) Phonon dispersion spectra and (i) cohesive energy of AlX and InX

  (a) AlN, (b) AlP, (c) AlAs, (d) AlSb, (e) InN, (f) InP, (g) InAs, and (h) InSb.

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  The cohesive energy (E_c) of the eight crystals was also calculated to further assess the stability of the optimized crystals and validate the correctness of the parameters chosen for the calculation of the phonon dispersion spectra. The E_c of AlX and InX was calculated using the following formula^[38]:

  $ E_{\mathrm{C}}=E_{\mathrm{tot}}-E_{\mathrm{iso}, 1}-E_{\mathrm{iso} , 2} $

  (1)

  Where E_tot is the total energy of the compounds, E_{iso, 1} is the single-atom energy of Al or In atoms, and E_{iso, 2} is the single-atom energy of N, P, As, or Sb atoms. The more negative the cohesive energy, the better the stability of the crystal. The calculated cohesive energies were plotted in the form of scatter points and are displayed in Fig. 2i. It can be seen that the cohesive energies of AlN, AlP, AlAs, AlSb, InN, InP, InAs, and InSb were -14.576, -26.384, -10.663, -8.617, -10.745, -25.898, -10.198, and -8.658 eV respectively, all of which were less than 0. Cohesive energy indicates that the crystal can stably exist, confirming the rationality of the parameters selected for the phonon dispersion spectrum calculation.

  2.2   Electronic structure

  Fig. 3a-3h show the band structure diagrams of AlX and InX calculated along high-symmetry points in the first Brillouin zone. The orange line in the figure represents the bottom of the conduction band and the top of the valence band, while the blue line represents other energy levels. Fig. 3a-3d show that AlN, AlP, AlAs, and AlSb are all direct bandgap semiconductors, with bandgaps (E_g) of 3.245, 1.635, 1.605, and 1.651 eV, respectively. Fig. 3e-3h show that InN, InP, and InAs are direct bandgap semiconductors, while InSb is an indirect bandgap semiconductor, with bandgap values of 0.706, 1.430, 0.978, and 1.654 eV, respectively. Comparison with the bandgap values reported in the literature^{[25, 39-42]} reveals that our calculated values were slightly smaller, due to the underestimation of the bandgap values by the GGA-PBE functional in the calculations. This study further employed the HSE06 and LDA functionals for computational validation, with the results depicted in Fig. 3i. It was observable that the LDA functional yields results quite similar to those obtained by the GGA functional. However, it was noteworthy that for AlN and InN, the bandgap calculated using the LDA functional was slightly larger than that obtained with the GGA functional. In contrast, the situation was reversed for AlP, AlAs, AlSb, InP, InAs, and InSb. This discrepancy arises because the GGA functional more accurately describes the delocalization of electrons in these materials when calculating AlP, AlAs, AlSb, InP, InAs, and InSb, leading to a lesser underestimation of the bandgap. Conversely, for AlN and InN, the LDA functional more accurately described the localization of electrons, resulting in an overestimation of the bandgap. The HSE06 functional, by incorporating a portion of the Hartree-Fock exchange energy, can improve the prediction of bandgap by the density functional theory (DFT) method, offsetting the convexity error of the LDA/GGA functionals, and enhancing overall computational accuracy. This adjustment brought the bandgap results for AlX and InX closer to experimental values (Table S1, Supporting information). Fig. 3a and 3e show the band structures of AlN and InN, respectively. Apart from the large difference in bandgap, there was also significant variation in the degree of band splitting at the Γ point, due to AlN′s smaller spin-orbit coupling strength compared to InN. Fig. 3b-3d and 3f-3h show that the band structures of these six crystals are similar, with differences primarily around 7-10 eV at the Γ point, where the degree of band splitting varies, with larger atomic sizes corresponding to more active deep-level electrons, thus resulting in a larger degree of band splitting.

  Figure 3

  

  Figure 3.  (a-h) Band structures and (i) comparison of the results obtained from the different functionals of AlX and InX

  (a) AlN, (b) AlP, (c) AlAs, (d) AlSb, (e) InN, (f) InP, (g) InAs, and (h) InSb.

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  To further understand the contributions of each element in the structures of AlX and InX, a comparative analysis of the density of states of the eight crystals is presented in Fig.S1. The total density of states (TDOS) and the partial density of states (PDOS) of the four aluminum-based semiconductors showed a general consistency in the -5-0 eV range, with a peak around -1.5 eV, then gradually decreasing to 0 eV. However, AlP, AlAs, and AlSb exhibited greater activity of the Al3s orbital near -5 eV. In Fig.S1a, the area near 0 eV is primarily contributed to by the N2p and Al3p, indicating that p-p hybridization occurs between Al and N atoms at this point, with the conduction band minimum primarily contributed to by the Al3p orbital. In the range of 8-10 eV, the s and p orbitals of the Al atom make significant contributions, which also differ from the conduction band of AlP, AlAs, and AlSb. In Fig.S1b-S1d, the vicinity of the Fermi level in AlP, AlAs, and AlSb is mainly contributed by the p orbitals of P, As, Sb, and Al atoms. For the conduction band bottom, AlP is primarily contributed by Al3p, AlAs is contributed by Al3p and As4s with Al and As atoms undergoing s-p hybridization, and all atomic orbitals are involved in AlSb. AlP and AlSb exhibited a small peak near 7.5 eV, with AlP′s peak being contributed by a weak Al3p and P3p, while the peak in AlSb is primarily due to Sb5s.

  The TDOS and PDOS for InX are shown in Fig.S2. In the range of -5-0 eV, the contributions from the atomic orbitals of the four indium-based semiconductors show a consistent trend. The p orbitals of X atoms provide the largest contributions, while the In5p orbitals follow closely. The s orbitals of In and X atoms contribute only minimally to this range. Near 0 eV, it is primarily contributed by the Xp and In5p orbitals, and a p-p hybridization occurs between the two atoms near the Fermi level. In the 0.5-1 eV energy range of Fig.S2a, a weak hybridization occurred between the s orbital of In atoms and the p orbital of N atoms, manifesting as a single energy level in the band structure diagram. In the range of 6-10 eV, compared with the other three indium-based semiconductors, InN had more active electron orbitals, mainly contributed by In5s, In5p, and N2p orbitals. This is mainly because InN has the smallest bandgap, allowing more electrons to participate in electron transitions. From Fig.S2b-S2d, it can be seen that the conduction band part of InP is primarily contributed by In5s, In5p, and P3p orbitals, while for InAs and InSb, contributions to the conduction band bottom come from various atomic orbitals to varying degrees, indicating a higher electron activity. In the case of AlP, AlAs, and AlSb, a weak peak was observed near 8 eV; for InP and InAs, it is primarily contributed to by the In5s, In5p, and P3p orbitals, while for InSb, this peak is mainly due to the In5s orbital. This is attributed to its larger lattice constant, which results in a smaller effective mass of electrons, allowing the electrons in the In5s orbital to be more mobile.

  2.3   Optical properties

  The optical properties of a crystal are associated with the transitions of electrons between energy levels. The intensity and probability of electron transitions can be calculated through the dielectric function. The dielectric function ε(ω)=ε₁(ω)+iε₂(ω) was obtained through calculations, and the imaginary part ε₂(ω) is deduced from the momentum matrix elements that describe the interaction between occupied and unoccupied electronic states^[43-44]. The imaginary part was calculated using the following formula:

  $ \begin{gathered} \varepsilon_2(\omega)=\frac{e^2}{2 \mathtt{π} m^2 \omega^2} \sum\limits_{\mathrm{C}, \mathrm{~V}} \int_{\mathrm{BZ}} \mathrm{d}\left|e \cdot M_{\mathrm{CV}}(k)\right|^2 \delta\left[E_{\mathrm{C}, \mathrm{V}}(k)-\hbar \omega\right] \\ \quad=\frac{e^2}{2 \mathtt{π} m^2 \omega^2} \sum\limits_{\mathrm{C}, \mathrm{V}} \int_{E_{\mathrm{C}, \mathrm{V}}(k)=\hbar \omega} \mathrm{d}\left|e \cdot M_{\mathrm{CV}}(k)\right|^2 \frac{\mathrm{d} S}{\left|\nabla_k-E_{\mathrm{C}, \mathrm{V}}(k)\right|} \end{gathered} $

  (2)

  In this formula, C and V represent the conduction band and valence band, respectively; k and ω represent the crystal momentum vector (not reciprocal lattice vector) and angular frequency, respectively; BZ represents the first Brillouin zone; e·M_CV (k) is the momentum curvature matrix element where e denotes the elementary charge; E_C (k) and E_V (k) are the eigenenergy levels of the conduction and valence bands, respectively. The parameter m stands for the effective mass of electrons, ħω corresponds to photon energy, while the δ function δ[E_{C, V}(k)-ħω] enforces energy conservation during interband transitions. The surface integral over E_{C, V}(k)=ħω with dS indicates integration over isoenergy surfaces in k-space, and the denominator |∇_k-E_{C, V}(k)| relates to the joint density of states weighted by band dispersion.

  The relationship between the real part ε₁(ω) and imaginary part was given by the Kramers-Kronig^[45] relations, enabling the calculation of one from the other:

  $ \varepsilon_1(\omega)=1+\frac{2}{\mathtt{π}} P \int_0^{\infty} \frac{\omega^{\prime} \varepsilon_2\left(\omega^{\prime}\right)}{\omega^{\prime 2}-\omega^2} \mathrm{d} \omega^{\prime} $

  (3)

  where P denotes the principal value of the integral. ω and ω′ are the angular frequencies of the final and initial states, respectively.

  2.3.1   Dielectric function, reflectance spectra, and absorption spectra

  Fig. 4a-4b shows the dielectric function of AlX and InX. As shown in Fig. 4a for the real part of the dielectric function, at an energy of 0, the static dielectric constants of AlN, AlP, AlAs, and AlSb were 4.75, 9.08, 9.82, and 10.30 respectively; those of InN, InP, InAs, and InSb were 6.67, 9.72, 10.80, and 10.60 respectively. The static dielectric constants show an increasing trend, which can be attributed to the gradual increase in the system′s total energy and cell volume. Fig. 4b presents the imaginary part of the dielectric function, which mainly reflects the light absorption characteristics of the crystal. Observing the figure reveals that the peak values of AlP, AlAs, AlSb, InP, InAs, and InSb were all greater than those of AlN and InN; however, as the atomic size of X increased, the peak values decreased gradually. In addition, as the cell volume of AlX and InX increased, the imaginary part of the dielectric function as a whole shifted to the low-energy region, exhibiting a redshift. The peak values of the imaginary part for AlP, AlAs, AlSb, InP, InAs, and InSb were all near 4 eV, consistent with their bandgap values. In material design, researchers can adjust the type of X atoms to more precisely control the dielectric constant, meeting the needs of specific applications. For instance, the lower static dielectric constants of AlN and InN may render them more suitable for applications requiring a low dielectric environment, while the higher static dielectric constants of AlSb and InSb can make them perform well in high-frequency electronic devices.

  Figure 4

  

  Figure 4.  (a) Real and (b) imaginary part of the dielectric function, (c) reflectance spectra, and (d) absorption spectra of AlX and InX

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  Fig. 4c displays the reflectance spectra of AlX and InX, demonstrating high reflectivity in the ultraviolet region. As the atomic size of X increased, the ultraviolet region′s peak value strengthened gradually and exhibited a redshift, which was manifested in the band structure diagram as stronger energy level splitting.

  The absorption spectrum represents the degree of light intensity attenuation as light propagates through a crystal medium per unit distance. Fig. 4d illustrates that the absorption peak values of AlP, AlAs, AlSb, InP, InAs, and InSb are concentrated near 6 eV, slightly lower than the absorption peak values of AlN and InN near 13 eV, but still indicate that they all have good absorption performance. In the visible and infrared regions, InN demonstrated superior performance, whereas AlN′s absorption was significantly lower than that of AlP, AlAs, AlSb, InP, InAs, and InSb with larger atomic size crystals. Above 20 eV, the absorption spectra for AlP, AlAs, AlSb, InP, InAs, and InSb gradually approached zero, while AlN and InN only gradually decreased to 0 around 40 eV, the energy range of their absorption spectra was wider, indicating that they are more suitable as key materials for broadband photodetectors^[46].

  2.3.2   Photoconductivity and energy loss function

  When a crystal is exposed to light, some electrons at low energy levels absorb energy and transition to the conduction band, becoming free electrons, which causes a change in electrical conductivity. Photoconductivity is primarily related to the density and mobility of the charge carriers^[47].

  Fig. 5a and 5b show the photoconductivity spectra of AlX and InX. As shown in Fig. 5a, the real part of the photoconductivity for AlX and InX emerged at energies comparable to their bandgap. The peaks for AlP, AlAs, AlSb, InP, InAs, and InSb were around 4 eV, and they gradually decreased as the atomic size of X increased. The peaks for AlN and InN were around 12 eV, and the peak values were smaller than those for AlP, AlAs, AlSb, InP, InAs, and InSb. Overall, crystals with larger atomic sizes exhibit significantly higher photoconductivity in the near ultraviolet region compared to AlN and InN, making them more suitable as optoelectronic materials. Fig. 5b shows the imaginary part of the photoconductivity for the eight crystals, which showed a consistent trend of initially decreasing, then increasing to a peak value, and finally converging. The photoconductivity of AlN and InN was negative from 0 to 12 eV, indicating that the materials absorb alternating light waves in this range without energy dissipation. However, above 12 eV, the photoconductivity became positive, indicating that the materials experience some energy dissipation during the absorption process, releasing energy in the form of heat or non-radiative forms. The inflection point from positive to negative photoconductivity for AlP, AlAs, AlSb, InP, InAs, and InSb was around 6 eV, which was lower than for AlN and InN, further indicating their suitability as optoelectronic materials.

  Figure 5

  

  Figure 5.  (a) Real part and (b) imaginary part photoconductivityies and (c) energy loss functions of AlX and InX

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  The energy loss function describes how the light intensity decreases as light passes through a medium or interacts with it. Studying the loss of light energy is crucial for improving the efficiency of optical devices. The energy loss function can be derived from the dielectric function.

  The energy loss functions for AlX and InX, depicted in Fig. 5c, show maximum peak values associated with plasma oscillations^[48]. The peak values for AlN, AlP, AlAs, and AlSb were 3.16, 7.16, 6.15, and 6.71, respectively. It can be concluded that since the electron density of AlP, AlAs, and AlSb is greater than that of AlN, their plasma frequencies are higher, and the corresponding peak values are also higher. The peak values for InN, InP, InAs, and InSb were 3.24, 6.97, 6.31, and 6.82, respectively, and the conclusions drawn were consistent with those for AlX. However, AlP, AlAs, AlSb, InP, InAs, and InSb exhibited an abnormal decrease in the peak values for AlAs and InAs, which is attributed to the full and stable 3d orbital electrons of the arsenic atom, but the electrons of the phosphorus and antimony atoms were comparatively more active. AlN and InN exhibited lower peak values, due to their lower electron densities, making them more suitable for high-frequency applications. In contrast, AlP, AlAs, AlSb, InP, InAs, and InSb, with their higher electron densities, had higher peak values, which enhanced their effectiveness in electromagnetic shielding and optoelectronic detection. Particularly, the reduced peak values of AlAs and InAs may impact their performance in optoelectronic applications. In addition, as the atomic size of X in AlX and InX increased, the peak position exhibited a redshift.

  2.4   Mechanical properties

  In this paper, the Voigt-Reuss-Hill (VRH) averaging method was used to calculate the mechanical properties of the eight compounds AlX and InX, including Young′s modulus (E), bulk modulus (K), and shear modulus (G). Table S2 presents the calculated values at 0 GPa, where K_V and G_V are the bulk modulus and shear modulus values obtained using the Voigt method^[49], K_R and G_R are the bulk modulus and shear modulus values obtained using the Reuss method^[50], and K and G are the values obtained according to Hill′s theory, with the formulas as follows^[51]:

  $ K=\frac{1}{2}\left(K_{\mathrm{V}}+K_{\mathrm{R}}\right) $

  (4)

  $ G=\frac{1}{2}\left(G_{\mathrm{V}}+G_{\mathrm{R}}\right) $

  (5)

  Young′s modulus, which describes the ability of a material to resist deformation, is calculated using the Hill bulk modulus and shear modulus^[52]:

  $ E=\frac{9 G K}{3 K+G} $

  (6)

  Table S2 lists the bulk modulus, shear modulus, and Young′s modulus of AlX and InX. The shear modulus measures a material′s resistance to shape change when subjected to shear forces. A higher G value corresponds to less deformation and implies increased hardness. The shear modulus order for these eight materials was AlN > InN > AlP > AlAs > InP > InAs > AlSb > InSb, reflecting an increasing hardness with the anion′s growing volume. Young′s modulus can be used to describe the ability of a material to resist deformation. The greater the Young′s modulus, the higher the hardness of the material. The order of Young′s modulus for the eight compounds was AlN > InN > AlP > AlAs > InP > InAs > AlSb > InSb. This trend aligns with the shear modulus, indicating increased hardness with the anion′s increasing volume when the metal cation is constant. The Young′s modulus values for these compounds decreased in the order of AlN, InN, AlP, AlAs, InP, InAs, AlSb, and InSb, correlating well with the shear modulus trend and indicating that hardness increases with the anion′s size when the metal cation is constant.

  It can also be observed from Table S2 that the bulk modulus and shear modulus of AlN and InN exceeded those of the other six crystal types. Furthermore, the higher bulk and shear moduli are usually related to strong bonding, implying significant overlap of the electron clouds between atoms, forming a strong bond, which aligns more closely with the results from the prior phonon dispersion spectrum calculations. Due to the high bulk modulus and shear modulus of AlN and InN, they possess greater lattice rigidity, resulting in lower photoconductivity compared to AlP, AlAs, AlSb, InP, InAs, and InSb in the low energy region due to their superior mechanical properties. AlN and InN have fewer microstructural defects within their lattices, and the peak values of the loss function are lower. The low defect and high modulus characteristics of AlN and InN position them as promising candidates for quantum optical devices, such as quantum dot lasers and single-photon detectors, which are crucial for advancements in quantum communication and quantum computing.

  To further visualize the characteristics of the elastic deformation for AlX and InX under uniform stress, this paper utilized the anisotropy indices for visualization. The anisotropy indices (A_U, A_K, and A_G) are defined as follows: A_U is the general anisotropy index, A_K is the bulk modulus anisotropy index, and A_G is the shear modulus anisotropy index. They are calculated using the formulas presented below^[53-54]:

  $ A_{\mathrm{U}}=\frac{5 G_{\mathrm{V}}}{G_{\mathrm{R}}}+\frac{K_{\mathrm{V}}}{K_{\mathrm{R}}}-6 $

  (7)

  $ A_K=\frac{K_{\mathrm{V}}-K_{\mathrm{R}}}{K_{\mathrm{V}}+K_{\mathrm{R}}} $

  (8)

  $ A_G=\frac{G_{\mathrm{V}}-G_{\mathrm{R}}}{G_{\mathrm{V}}+G_{\mathrm{R}}} $

  (9)

  Fig. 6a-6h show that AlX and InX exhibited cubic shapes, with higher force along the body diagonal, exhibiting an overall isotropic characteristic; the bulk modulus anisotropy index A_K for all eight crystals was zero. Exceptions for AlN and InN, which have relatively higher general anisotropy index and shear modulus anisotropy index, the values of A_U and A_G for AlP, AlAs, AlSb, InP, InAs, and InSb were all lower. The relatively higher anisotropy of AlN and InN results in changes to their polarizability^[55], which is reflected in the optical absorption spectrum as a broader range of absorbed spectral energies. Nonetheless, the anisotropy indices of the eight crystals remained low, indicating that the properties of these crystals are nearly isotropic in applications.

  Figure 6

  

  Figure 6.  (a-h) Young′s modulus diagrams and (i) anisotropy indices for AlX and InX

  (a) AlN, (b) AlP, (c) AlAs, (d) AlSb, (e) InN, (f) InP, (g) InAs, and (h) InSb; E_X, E_Y, and E_Z represent the Young′s modulus in the X, Y, and Z directions, respectively.

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  3.   Conclusions

  In this study, we systematically investigated the electronic structures and optical properties of aluminum-based semiconductors AlX and InX utilizing first-principles calculations. Except for InSb, which exhibited an indirect bandgap, the remaining seven compounds were found to possess direct bandgaps. Employing the HSE06 functional, we determined that InN had the smallest bandgap at merely 1.680 eV, suggesting strong absorption in the visible light spectrum, whereas AlN exhibited the largest bandgap at 4.392 eV. The static dielectric constant for AlX and InX was observed to increase progressively with the enlargement of the atomic size of X in the dielectric function diagrams. Our analysis of the mechanical properties indicates that AlN and InN have the highest elastic modulus and exhibit good ductility, which significantly minimizes the formation of microcracks and defects under stress, thus maintaining their light loss function values at a lower level. Leveraging these characteristics, AlN and InN hold the potential to unlock new application scenarios across various fields.

  Supporting information is available at http://www.wjhxxb.cn

  
  Acknowledgments: This work was completed with the help of the the Xinjiang Key Laboratory under Grant No.2023D04074, the School-level Scientific Research Project of Yili Normal University (Grant No.22XKZZ21), the Yili Normal University Student Innovation Training Project (Grants No.s202210764014 and s202210764016), the Foundation of Xinjiang Tianshan Talents Plan Uygur Autonomous Region of China (2021—2023). Conflicts of interest: There are no conflicts to declare.
  
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• 

  Figure 1  Crystal structure of AlX and InX

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  Figure 2  (a-h) Phonon dispersion spectra and (i) cohesive energy of AlX and InX

  (a) AlN, (b) AlP, (c) AlAs, (d) AlSb, (e) InN, (f) InP, (g) InAs, and (h) InSb.

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  Figure 3  (a-h) Band structures and (i) comparison of the results obtained from the different functionals of AlX and InX

  (a) AlN, (b) AlP, (c) AlAs, (d) AlSb, (e) InN, (f) InP, (g) InAs, and (h) InSb.

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  Figure 4  (a) Real and (b) imaginary part of the dielectric function, (c) reflectance spectra, and (d) absorption spectra of AlX and InX

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  Figure 5  (a) Real part and (b) imaginary part photoconductivityies and (c) energy loss functions of AlX and InX

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  Figure 6  (a-h) Young′s modulus diagrams and (i) anisotropy indices for AlX and InX

  (a) AlN, (b) AlP, (c) AlAs, (d) AlSb, (e) InN, (f) InP, (g) InAs, and (h) InSb; E_X, E_Y, and E_Z represent the Young′s modulus in the X, Y, and Z directions, respectively.

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  Table 1.  Information on crystal parameters before and after structural optimization of AlX and InX

  Semiconductor	Space group	Parameter before optimization			Parameter after optimization
  		a=b=c / nm	V / nm3		a=b=c / nm	V / nm3
  AlN	F43m	0.437	0.083 338 2		0.433	0.081 277 8
  AlP		0.547	0.163 934		0.552	0.168 042
  AlAs		0.568	0.182 844		0.564	0.179 363
  AlSb		0.619	0.236 607		0.616	0.233 973
  InN		0.501	0.125 405		0.496	0.121 803
  InP		0.590	0.205 792		0.589	0.204 336
  InAs		0.611	0.227 777		0.602	0.217 732
  InSb		0.663	0.291 859		0.659	0.286 061

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•