# The Study on Structural and Photoelectric Properties of Zincblende InGaN via First Principles Calculation

^{1}

^{2}

^{3}

^{4}

^{*}

**—**Nano-Theory)

## Abstract

**:**

_{x}Ga

_{1−x}N alloys are systematically calculated and analyzed based on the density functional theory, including the lattice constant, band structure, distribution of electronic states, dielectric function, and absorption coefficient. The calculation results show that with the increase in x, the lattice constants and the supercell volume increase, whereas the bandgap tends to decrease, and In

_{x}Ga

_{1−x}N alloys are direct band gap semiconductor materials. In addition, the imaginary part of the dielectric function and the absorption coefficient are found to redshift with the increase in indium composition, expanding the absorption range of visible light. By analyzing the lattice constants, polarization characteristics, and photoelectric properties of the In

_{x}Ga

_{1−x}N systems, it is observed that zincblende In

_{x}Ga

_{1−x}N can be used as an alternative material to replace the channel layer of wurtzite In

_{x}Ga

_{1−x}N heterojunction high electron mobility transistor (HEMT) devices to achieve the manufacture of HEMT devices with higher power and higher frequency. In addition, it also provides a theoretical reference for the practical application of In

_{x}Ga

_{1−x}N systems in optoelectronic devices.

## 1. Introduction

_{x}Ga

_{1−x}N alloys because they can be used as a candidate material for optoelectronic devices. This was illustrated by Lu, who prepared an ultraviolet detector that showed a good response in the ultraviolet 360–390 nm region, with a peak response rate of 0.15 A/W [6]. Furthermore, the band gap of In

_{x}Ga

_{1−x}N alloys can continuously change from 0.7 to 3.4 eV with the change of x, which is almost perfectly matched to the solar spectrum. Taking this into account, In

_{x}Ga

_{1−x}N alloys have now begun to attract widespread attention as new type of solar cell material [7,8,9,10,11,12].

_{x}Ga

_{1−x}N, the wurtzite structure is more stable than the zincblende structure. Furthermore, and it is difficult to achieve high-quality growth of zincblende structure. Thus, current research mainly focuses on the wurtzite structure. However, due to the strong polarization effect of the wurtzite structure, it is difficult to use zincblende In

_{x}Ga

_{1−x}N to prepare enhanced mode high electron mobility transistor (HEMT) and high-reliability devices compared with the second-generation semiconductor materials, such as III–V compound semiconductors. In addition, the zincblende In

_{x}Ga

_{1−x}N does not have spontaneous polarization, and its smaller effective mass of electrons at the minimum of the conduction band (CB) is beneficial in enhancing the frequency and power of the device [13,14,15]. Mullhauser used radio frequency plasma-assisted molecular beam epitaxy to grow zincblende In

_{0.4}Ga

_{0.6}N, of which the band gap was 2.46 eV [16]. Goldhahn studied the refractive index and energy gap of In

_{x}Ga

_{1−x}N and suggested that the band gap bowing parameter of In

_{x}Ga

_{1−x}N is different when the x is different [17]. In terms of theoretical research, a growing number of groups have conducted research on the band gap bowing parameter of zincblende In

_{x}Ga

_{1−x}N [18,19,20,21,22,23,24], but few reports have been published that discuss in detail the lattice constant, the change of bandgap, density of state, and optical properties. These physical properties are relevant for In

_{x}Ga

_{1−x}N alloy-based heterojunction electronic devices and optoelectronic devices. Furthermore, zincblende In

_{x}Ga

_{1−x}N without the Stark effect helps to improve the luminescence efficiency of optoelectronic devices. Therefore, it is necessary to explore the zincblende In

_{x}Ga

_{1−x}N.

_{x}Ga

_{1−x}N using the first principles, and analyzed the lattice constant, polarization characteristics, the change of bandgap, and optical characteristics. This could provide a theoretical reference for the experimental research of full-spectrum solar cells and HEMT devices with higher frequency and higher power.

## 2. Method of Calculation

_{x}Ga

_{1−x}N, where x is set as 0, 0.125, 0.25, 0.5, 0.75, and 1. To acquire the lattice mismatch of the close-packed planes of zincblende and wurtzite In

_{x}Ga

_{1−x}N, we also calculate the lattice constants of wurtzite In

_{x}Ga

_{1−x}N when x = 0, 0.125, 0.25, 0.5, 0.75, and 1; the calculated structures are shown in Figure 1b–g.

_{x}Ga

_{1−x}N alloys were optimized. The Perdew–Burke–Ernzerhof (PBE) was chosen as the exchange-correlation function and the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm was adopted. The optimization parameters of the maximum interaction force between atoms, the convergence criterion of the maximum displacement, the maximum stress acting on each atom, and the self-consistent accuracy were set as: 0.01 eV/Å, 5.0 × 10

^{−5}nm, 0.01 Gpa, and 5.0 × 10

^{−6}eV, respectively. Because the calculation structures of zincblende and wurtzite In

_{x}Ga

_{1−x}N are different, the setting of their energy cut-off and k-point density are different. For the zincblende In

_{x}Ga

_{1−x}N, energy cut-off was set as 600 eV and k-point density was 3 × 6 × 6. For wurtzite In

_{x}Ga

_{1−x}N, energy cut-off was set as 600 eV and the k-point densities were set as 4 × 4 × 5 for GaN, In

_{0.125}Ga

_{0.875}N, In

_{0.25}Ga

_{0.75}N and InN, and 8 × 8 × 2 for In

_{0.5}Ga

_{0.5}N and In

_{0.75}Ga

_{0.25}N.

## 3. Results and Discussion

#### 3.1. Lattice Constant

_{x}Ga

_{1−x}N when x is set as 0, 0.125, 0.25, 0.5, 0.75, and 1. The broken line formed by the green triangles in Figure 2 is the supercell volume of In

_{x}Ga

_{1−x}N. It can be observed that the supercell volume of In

_{x}Ga

_{1−x}N increases linearly with the increase in x, which is caused by the indium atomic radius being larger than that of gallium. The broken line formed by the magenta square points shows the lattice constants obtained according to Vegard’s law [26], which can be described via the following equation:

_{x}Ga

_{1−x}N, InN, and GaN, respectively. The broken line formed by purple dots represents the optimized lattice constant of the zincblende In

_{x}Ga

_{1−x}N systems. The optimized lattice constant of GaN is 0.4548 nm, which has an error of less than 1% from the experimental lattice constant value of GaN of 0.4520 nm [25]. For other In

_{x}Ga

_{1−x}N structures, the errors between the calculated lattice constants of In

_{x}Ga

_{1−x}N and Vegard’s law are less than 2%, which is related to the pseudopotential used in this calculation.

_{x}Ga

_{1−x}N. The blue, green, and orange curves are the a, c, and ideal axis c

_{0}(c

_{0}= 1.63a) lattice constants of In

_{x}Ga

_{1−x}N, respectively, and all obviously increase with the increase in indium composition. The calculated lattice constants of wurtzite GaN are a = 0.323 nm, c = 0.525 nm, and c/a = 1.626; the differences between these calculated values and the experimental values are: 1.24%, 1.16%, and 0.062%, respectively [27]. In other indium compositions, the lattice constants that are calculated in this paper are consistent with those of the literature [28,29,30]. As evident from the illustration, the lattice constant c is different from the ideal lattice constant c

_{0}, leading to the spontaneous polarization of the wurtzite In

_{x}Ga

_{1−x}N alloys. Moreover, the spontaneous polarization direction of wurtzite In

_{0.5}Ga

_{0.5}N is opposite to that of wurtzite GaN, In

_{0.25}Ga

_{0.75}N, In

_{0.75}Ga

_{0.25}N, and InN, because when indium composition is 0.5, c is greater than c

_{0}, and c is less than c

_{0}for other indium compositions.

_{x}Ga

_{1−x}N:

_{x}Ga

_{1−x}N and the plane formed by ${\stackrel{\rightharpoonup}{a}}_{1}$ and ${\stackrel{\rightharpoonup}{a}}_{2}$. The detailed derivation of spontaneous polarization in In

_{x}Ga

_{1−x}N is described in the Supporting Information (SI).

_{SP}of wurtzite In

_{x}Ga

_{1−x}N was calculated and the results are shown in Table 1. As shown, the spontaneous polarization intensity of wurtzite GaN is −0.030 C/m

^{2}, which only differs from the value in the literatures by −0.001 C/m

^{2}[32,33]. The performance of wurtzite In

_{x}Ga

_{1−x}N heterojunction HEMT devices is limited by the spontaneous polarization in wurtzite In

_{x}Ga

_{1−x}N. Thus, replacing the wurtzite In

_{x}Ga

_{1−x}N channel layer with nonpolarized zincblende In

_{x}Ga

_{1−x}N is a potential choice. However, the difference in lattice constant between wurtzite and zincblende In

_{x}Ga

_{1−x}N must be considered. Thus, we further calculated the close-packed plane lattice constants of the two structures, as shown in Table 1. The results show a negligible (0.23%) variation in lattice constants under all indium compositions. Hence, theoretically, the wurtzite In

_{x}Ga

_{1−x}N channel layer can be replaced with zincblende In

_{x}Ga

_{1−x}N without spontaneous polarization to further improve the performance of HEMT devices.

#### 3.2. Band Structure

#### 3.2.1. Correction of the Energy Gap

_{x}Ga

_{1−x}N alloys with different indium compositions. The green curve in the figure shows the calculated bandgap; the bandgap of GaN is 1.536 eV, which is consistent with the value calculated by Mathieu Cesar et al. using PBE approximation [34]. Note that our calculated band gap of zincblende GaN is different from the 1.69 eV calculated by Poul Georg Moses et al. using PBE approximation [35], this may be due to the structure calculated by Poul Georg Moses et al. is wurtzite GaN structure, while the structure calculated in this paper is zincblende GaN. It is worth mentioning that the calculated band gap of GaN in this paper is smaller than the experimental value of 3.30 eV [12], this is because of the overestimation of the energy of the gallium d state in the calculation, leading to the enhanced interaction between gallium d and nitrogen p orbitals and resulting in broadening of the valence band (VB) [36]. Although this is a common phenomenon in the selection of GGA-PBE exchange-correlation functional calculations, the accurate calculation of the band gap is not important in trend analysis. To be specific, the main topic of this article concerns the same structural system and only changes the incorporation composition of indium [37,38]. Hence, the calculated series of band gaps are still comparable. To make the calculated band gaps of In

_{x}Ga

_{1−x}N alloys closer to the experimental values, the calculated band gaps were corrected based on the experimental values of zincblende GaN and InN [12] using the correction formula as follows [38]:

_{x}Ga

_{1−x}N, InN, and GaN in this work, respectively; and ${E}_{g,InN}^{exp}$ and ${E}_{g,GaN}^{exp}$ are the experimental band gaps of InN and GaN, respectively. The corrected results are shown by the curve composed of yellow triangles in Figure 4.

_{x}Ga

_{1−x}N systems are plotted in Figure 5. The discussion of the energy band diagram is divided into the following points. (1) Type of band gap: According to Figure 5a, we conclude that GaN is a direct band gap semiconductor material; after the indium atoms are doped into GaN, the type of band gap of In

_{x}Ga

_{1−x}N is still a direct gap, and the minimum of the CB and the maximum of the VB are located at the same Γ point in the Brillouin zone (BZ). (2) Degeneracy: Compared with Figure 5a,f, Figure 5 b–f shows an impurity energy level in both the CB and VB, increasing the degeneracy of In

_{x}Ga

_{1−x}N. This is mainly due to the contribution of SP

^{3}hybridization of gallium s/p and indium s/p orbitals. (3) Band gap: It can be clearly seen from Figure 4 that the band gap of In

_{x}Ga

_{1−x}N decrease with the increase in x, which is caused by gallium s/p and indium s/p orbitals in the CB approaching the energy reference point as the indium compositions increase. In addition, it can be seen from Figure 5 that the band gaps of GaN and InN are 3.30 eV and 0.78 eV, respectively; hence, the band gap of In

_{x}Ga

_{1−x}N alloys can continuously vary from 0.78 to 3.30 eV by adjusting the indium compositions. This is almost perfectly matched to the solar energy spectrum, and means that In

_{x}Ga

_{1−x}N can be used to produce photovoltaic devices such as full-spectrum solar cells, by a combination of In

_{x}Ga

_{1−x}N solar cells with various bandgaps.

#### 3.2.2. Mechanism of Bandgap Reduction and Bandgap Bowing Parameter

_{x}Ga

_{1−x}N systems was calculated, and the results are shown in Figure 6; blue indicates low electron density and red indicates high electron density. It can be seen from Figure 6b that when indium atoms replace the gallium atoms, there is an electron enrichment phenomenon around the indium atoms. This indicates that the indium has a stronger ability to bind electrons, which is due to the electronegativity of indium atoms being greater than that of gallium (according to Pauling’s rule, the electronegativities of gallium and indium are 1.6 and 1.7, respectively). Thus, more covalent bond components of indium–nitrogen and the ionic bond components decrease after indium atoms replace gallium. Furthermore, more covalent bond components of the SP

^{3}hybrid bond of indium–nitrogen results in a smaller bond energy of indium–nitrogen compared with the gallium-nitrogen bond, thus, the band gap decreases with the increase in indium compositions.

_{x}Ga

_{1−x}N systems [35,39], therefore, here we discuss and analyze the value and origin of the bowing parameter. The relationship between the energy gap of the ternary alloy and the doping compositions can be expressed by the semi-empirical formula [40]:

_{x}Ga

_{1−x}N using Equation (3) and obtained b = 2.1 ± 0.14 eV as the average bowing parameter when the indium compositions are 0–1, which is caused by the volume deformation, structural relaxation, and charge exchange after the doping of indium atoms into the GaN system. The band gap bowing parameter obtained in this calculation is slightly different from the result in the literature, which is b = 1.9 ± 0.09 eV [19].

#### 3.3. Density of States

_{x}Ga

_{1–x}N and the partial density of states of indium, gallium, and nitrogen. Combining Figure 7b–d, the total density of states of Figure 7a is divided into three parts for discussion, namely: −10 to 5 eV, −5 to 0 eV, and CB. In the range of −10 to 5 eV, its main contribution comes from nitrogen p orbitals and gallium s orbitals. When the indium compositions in the In

_{x}Ga

_{1−x}N systems increase, the electronic states of the gallium s orbitals decrease, while the electronic state of nitrogen p orbitals is almost unchanged, so the TDOS decreases in this range. The density of states in the energy range of −5 to 0 eV is mainly due to the contribution of the nitrogen p orbital and does not change with the increase in doping x. The TDOS in the CB is mainly affected by the SP

^{3}hybridization of gallium s/p orbitals and indium s/p orbitals. Apparently, with the increase in indium compositions, gallium s/p orbitals and indium s/p orbitals move to lower energy, causing the TDOS of the CB to move to the energy reference point and the In

_{x}Ga

_{1−x}N systems to undergo redshift, which is consistent with the conclusion of the band structure.

#### 3.4. Optical Properties

_{1}and ε

_{2}are real and imaginary parts of the dielectric function, respectively, and n and k are refractive index and extinction coefficient, respectively. In

_{x}Ga

_{1−x}N alloys are direct bandgap semiconductor materials, therefore, the dielectric function and absorption parameter $\alpha \left(\omega \right)$ can be derived using the definition of the direct transition probability and the Kramers-Kronig relationship [42,43,44].

_{2}and λ

_{0}are the dielectric constant and wavelength in vacuum, respectively, footnotes C and V represent the CB and VB, BZ is the Brillouin zone,

**K**is the electronic wave vector, ${E}_{C}\left(\mathit{K}\right)$ and ${E}_{V}\left(\mathit{K}\right)$ are the intrinsic energy level of CB and VB, respectively, $\hslash $ is the Planck constant, $\mathit{a}$ is the unit vector of the vector potential

**A**, and ${\mathit{M}}_{V.C}$ is transition matrix element.

_{x}Ga

_{1−x}N alloys when the indium compositions were changed to 0, 0.125, 0.25, 0.5, 0.75, and 1. The calculated results of the imaginary part ε

_{2}of the dielectric function are shown in Figure 8. It is evident from the figure that the ε

_{2}curve of GaN has three peaks, namely C

_{1}, C

_{2}, and C

_{3}, which are located near 8.4, 11, and 13 eV, respectively. Absorption peaks C

_{2}and C

_{3}are primarily caused by the transition of electrons in gallium s/p states to the unoccupied states. The absorption peak C

_{1}reaches the maximum value, which is caused by the direct transition. It can be observed in the graph that the ε

_{2}of In

_{x}Ga

_{1−x}N alloys shift to low energy with the increase in indium atoms. This indicates that the electrons in the In

_{x}Ga

_{1−x}N can undergo transitions even if they absorb photons with lower energy. Additionally, there is an absorption peak near 4 eV energy, which increases and moves to the lower energy direction with the increase in indium compositions. This may be due to the transition of indium s/p state electrons to the unoccupied state.

_{x}Ga

_{1−x}N alloys. There are four peaks, D

_{1}, D

_{2}, D

_{3,}and D

_{4}, in the absorption spectrum of GaN. Absorption peaks D

_{1}, D

_{2}, and D

_{3}are located near 8, 11, and 13 eV, respectively. This is mainly due to the contribution of gallium s/p orbital electronic states. Therefore, the values of the three absorption peaks all decrease when the indium compositions increase. Another absorption peak D

_{4}is located near 26 eV, whose main contribution comes from the gallium s orbital. However, when an indium atom is added, the influence of indium s/p orbitals is greater than that of the gallium s orbital. Hence, with the increase in indium compositions, the peak value increases. The absorption coefficient curve of In

_{x}Ga

_{1−x}N alloys shifts in the low energy direction as x increases. This indicates that the doping of indium atoms improves the absorption of zincblende GaN for visible light, which is consistent with the result of the imaginary part of the dielectric function.

## 4. Conclusions

_{x}Ga

_{1−x}N (x = 0, 0.125, 0.25, 0.5, 0.75, 1) were calculated based on the density functional theory. The calculated results demonstrate that, after doping indium atoms, the nature of the direct bandgap of zincblende In

_{x}Ga

_{1−x}N does not alter and the indium s/p electronic states are introduced near the energy reference point so that the bandgap decreases with the increase in x. By fitting the corrected bandgap, the average energy gap bowing parameter is obtained as b = 2.10 eV. Furthermore, as the indium compositions increases, the indium s/p and gallium s/p orbitals of the CB move to the energy reference point, resulting in a narrowing of the energy window in which electronic states cannot exist. In addition, the imaginary part of the dielectric function and the absorption coefficient of the In

_{x}Ga

_{1−x}N structures shifts to low energy with the increase in x. This enhances the absorption of visible light and provides a theoretical reference for the application of In

_{x}Ga

_{1−x}N alloys in the field of photovoltaic devices such as solar cells.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Park, S.H.; Chuang, S.L. Comparison of zinc-blende and wurtzite GaN semiconductors with spontaneous polarization and piezoelectric field effects. J. Appl. Phys.
**2000**, 87, 353–364. [Google Scholar] [CrossRef] - Xia, S.; Liu, L.; Diao, Y.; Fend, S. Doping process of p-type GaN nanowires: A first principle study. J. Appl. Phys.
**2017**, 122, 135102. [Google Scholar] [CrossRef] - Pan, F.C.; Yang, B.; Lin, X.L. The Study of Magnetism in Un-doped 3C-GaN: The First-Principles Calculations. J. Supercond. Nov. Magn
**2015**, 28, 1617. [Google Scholar] [CrossRef] - Khan, M.J.I.; Liu, J.; Hussain, S.; Usmani, M.N. First Principle Study of Optical Properties of Cu doped zincblende GaN for Novel Optoelectronic Applications. Optik
**2020**, 208, 164529. [Google Scholar] [CrossRef] - Li, J.; Liu, H.; Wu, L. The optical properties of GaN (001) surface modified by intrinsic defects from density functional theory calculation. Optik
**2018**, 154, 378. [Google Scholar] [CrossRef] - Lu, Y.D.; Wang, L.W.; Zhang, Y. Properties of InGaN P-I-N ultraviolet detector. J. Semicond. Optoelectron.
**2014**, 35, 785. [Google Scholar] - Kimura, R.; Shigemori, A.; Shike, J.; Ishida, K.; Takahashi, K. Improvement of cubic GaN film crystal quality by use of an AlN/GaN ordered alloy on GaAs (100) by plasma assisted molecular beam epitaxy. J. Cryst. Growth
**2003**, 251, 455. [Google Scholar] [CrossRef] - Novikov, S.V.; Stanton, N.M.; Campion, R.P.; Morrist, R.D.; Kent, A.J. Growth and characterization of free-standing zinc-blende (cubic) GaN layers and substrates. Semicond. Sci. Technol.
**2008**, 23, 015018. [Google Scholar] [CrossRef] - Liu, S.M.; Xiao, H.L.; Wang, Q.; Yan, J.D.; Zhan, X.M.; Gong, J.M.; Wang, X.L. In
_{x}Ga_{1−x}N/GaN Multiple Quantum Well Solar Cells with Conversion Efficiency of 3.77%. Chin. Phys. Lett.**2015**, 32, 088401. [Google Scholar] [CrossRef] - Jani, O.; Ferguson, I.; Honsberg, C.; Kurtz, S. Design and characterization of GaN/InGaN solar cells. Appl. Phys. Lett.
**2007**, 91, 132117. [Google Scholar] - Neufeld, C.J.; Toledo, N.G.; Cruz, S.C.; Iza, M. High quantum efficiency InGaN/GaN solar cells with 2.95 eV band gap. Appl. Phys. Lett.
**2008**, 93, 1571. [Google Scholar] [CrossRef] - Vurgaftman, I.; Meyer, J.R. Band parameters for nitrogen-containing semiconductors. J. Appl. Phys.
**2003**, 94, 3675. [Google Scholar] [CrossRef] - Jia, W.L.; Zhou, M.; Wang, X.M.; Li, J.W. First-principles study on the optical properties of Fe-doped GaN. Chin. J. Phys.
**2018**, 67, 169. [Google Scholar] - Lu, W. First-Principle Study of Electronic and Optical Properties of Wurtizite Structure GaN. Master’s Thesis, Xidian University, Xi’an, China, 2009. [Google Scholar]
- Shi, P. A Study on Transport Property of Cubic GaN Material and Its Heterojunction. Master’s Thesis, Xidian University, Xi’an, China, 2014. [Google Scholar]
- Mullhauser, J.R.; Brandt, O.; Trampert, A.; Jenichen, B.; Ploog, K.H. Green photoluminescence from cubic In
_{0.4}Ga_{0.6}N grown by radio frequency plasma-assisted molecular beam epitaxy. Appl. Phys. Lett.**1998**, 73, 1230. [Google Scholar] [CrossRef] - Goldhahn, R.; Scheiner, J.; Shokhovets, S.; Frey, T.; Kohler, U.; As, D.J.; Lischka, K. Refractive index and gap energy of cubic In
_{x}Ga_{1−x}N. Appl. Phys. Lett.**2000**, 76, 291. [Google Scholar] [CrossRef] - Wright, A.F.; Nelson, J.S. Bowing parameters for zinc-blende Al
_{x}Ga_{1−x}N and In_{x}Ga_{1−x}N. Appl. Phys. Lett.**1995**, 66, 3051. [Google Scholar] [CrossRef] - Kou, Y.K.; Liou, B.T.; Yen, S.H.; Chu, H.Y. Vegard’s law deviation in lattice constant and band gap bowing parameter of zincblende In
_{x}Ga_{1−x}N. Opt. Commun.**2004**, 237, 363. [Google Scholar] - Ferhat, M.; Furthmuller, J.; Bechstedt, F. Gap bowing and Stoke shift in In
_{x}Ga_{1−x}N alloys: First-principles studies. Appl. Phys. Lett.**2002**, 80, 1394. [Google Scholar] [CrossRef] - Camacho, D.L.A.; Hopper, R.H.; Lin, G.M.; Myers, B.S.; Chen, A.B. Theory of AlN, GaN, InN and their alloys. J. Cryst. Growth
**1997**, 178, 8. [Google Scholar] - Wu, J.; Walukiewicz, W.; Yu, K.M.; Ager, J.W., III; Haller, E.E.; Lu, H.; Schaff, W.J. Small band gap bowing in In
_{x}Ga_{1−x}N alloys. Appl. Phys. Lett.**2002**, 80, 4741. [Google Scholar] [CrossRef] - Caetano, C.; Teles, L.K.; Marques, L.M.; Ferreira, L.G. Phase stability, chemical bonds, and gap bowing of In
_{x}Ga_{1−x}N alloys: Comparison between cubic and wurtzite structures. Phys. Rev. B**2006**, 74, 5215. [Google Scholar] [CrossRef] [Green Version] - Marques, M.; Teles, L.K.; Scolfaro, L.M.R.; Leite, J.R.; Bechstedt, F. Lattice parameter and energy band gap of cubic Al
_{x}GayIn_{1−x−y}N quaternary alloys. Appl. Phys. Lett.**2003**, 83, 890. [Google Scholar] [CrossRef] - Dridi, Z.; Bouhafs, B.; Roterana, P. First-principles investigation of lattice constants and bowing parameters in wurtzite AlxGa1-xN and In
_{x}Al_{1−x}N alloys. Semicond. Sci. Technol.**2003**, 18, 850. [Google Scholar] [CrossRef] - Lei, T.; Moustakas, T.D.; Graham, R.J.; He, Y.; Berkowitz, S.J. Epitaxial growth and characterization of zinc-blende gallium nitride on (001) silicon. J. Appl. Phys.
**1992**, 71, 4933. [Google Scholar] [CrossRef] - Perlin, P.; Jauberthie-Carillon, C.; Itie, J.P.; San, M.A.; Grzegory, I.; Polian, A. Raman scattering and x-ray-absorption spectroscopy ingallium nitride under high pressure. Phys. Rev. B
**1992**, 45, 83. [Google Scholar] [CrossRef] - Davydov, V.Y.; Klochikhin, A.A.; Emtsev, V.V.; Kurdyukov, D.A.; Ivanov, S.V. Band gap of hexagonal InN and InGaN alloys. Phys. Stat. Sol. (b)
**2002**, 234, 787. [Google Scholar] [CrossRef] [Green Version] - Bungaro, C.; Rapcewicz, K. Ab initio phonon dispersions of wurtzite AlN, GaN, and InN. Phys. Rev. B
**2000**, 61, 6720. [Google Scholar] [CrossRef] [Green Version] - Ruan, X.X.; Zhang, F.C. First-Principles Investigation on Electronic Structure and Optical Properties of Wurtzite In
_{x}Ga_{1−x}N Alloys. Rare Met. Mat. Eng.**2015**, 44, 3027. [Google Scholar] - Huang, K. Solid State Physics; Higher Education Press: Beijing, China, 1988; pp. 49–70. [Google Scholar]
- Bernardini, F.; Fiorentini, V.; Vanderbilt, D. Spontaneous polarization and piezoelectric constants of III-V nitrides. Phys. Rev. B
**1997**, 56, 10024–10027. [Google Scholar] [CrossRef] [Green Version] - Hao, Y.; Zhang, J.F. Nitride Wide Band Gap Semiconductor Materials and Electronic, 3rd ed.; Science Press: Beijing, China, 2013; pp. 66–86. [Google Scholar]
- Cesar, M.; Ke, Y.Q.; Ji, W.; Gou, H.; Mi, Z.T. Band gap of In
_{x}Ga_{1−x}N: A first principles analysis. Appl. Phys. Lett.**2011**, 98, 202107. [Google Scholar] [CrossRef] - Moses, P.G.; Walle, C.G.V. Band bowing and band alignment in InGaN alloys. Appl. Phys. Lett.
**2010**, 96, 3675. [Google Scholar] [CrossRef] - Li, J.B. First Principles Study of Electronic Structure and Optical Properties of GaN Doping. Ph.D. Thesis, Xidian University, Xi’an, China, 2019. [Google Scholar]
- Zhang, Y.; Shao, X.H.; Wang, Z.Q. A first principle study on p-type doped 3C-SiC. Acta Phys. Sin.
**2010**, 59, 5652. [Google Scholar] - Zheng, S.W.; Fan, G.H.; Zhang, T.; Su, C.; Song, J.J.; Ding, B.B. First-principles study on the energy bandgap bowing parameter of wurtzite Be
_{x}Zn_{1−x}O. Acta Phys. Sin.**2013**, 62, 305. [Google Scholar] - Moses, P.G.; Miao, M.; Yan, Q.; Walle, C.G.V. Hybrid functional investigations of band gaps and band alignments for AlN, GaN, InN, and InGaN. J. Chem. Phys.
**2011**, 134, 084703. [Google Scholar] [CrossRef] - Hassan, F.E.H.; Hashemifar, S.J.; Akbarzadeh, H. Density functional study of Zn
_{1−x}Mg_{x}SeyTe_{1-y}quaternary semiconductor alloys. Phys. Rev. B**2006**, 73, 195202. [Google Scholar] [CrossRef] - Zhang, Z.D.; Wang, Y.; Huang, Y.B.; Li, Z.H.; Yang, C. First principle study on the electronic and optical properties of Al
_{x}In_{1−x}As. J. At. Mol. Phys.**2018**, 36, 1057. [Google Scholar] - Liu, X.F.; Luo, Z.J.; Zhou, X.; Wei, J.M.; Wang, Y.; Gou, X.; Lang, Q.Z.; Ding, Z. Calculation of electronic and optical properties of surface In
_{x}Ga_{1−x}P and indium-gradient structure on GaP (001). Comput. Mater. Sci.**2018**, 153, 356. [Google Scholar] [CrossRef] - Liu, X.F.; Ding, Z.; Lou, Z.J.; Zhou, X.; Wei, J.M.; Wang, Y.; Gou, X.; Lang, Q.Z. Theoretical study on the electronic and optical properties of bulk and surface (001) In
_{x}Ga_{1−x}As. Physica B**2018**, 537, 68. [Google Scholar] [CrossRef] - Shen, X.C. Semiconductor Spectrum and Optical Properties, 2rd ed.; Science Press: Beijing, China, 2013; pp. 198–237. [Google Scholar]

**Figure 1.**Supercell structure of In

_{x}Ga

_{1−x}N: (

**a**) zincblende GaN structure of 2 × 1 × 1; (

**b**) GaN structure of 2 × 2 × 1; (

**c**) In

_{0.125}Ga

_{0.875}N structure of 2 × 2 × 1; (

**d**) In

_{0.25}Ga

_{0.75}N structure of 2 × 2 × 1; (

**e**) In

_{0.5}Ga

_{0.5}N structure of 1 × 1 × 2; (

**f**) In

_{0.75}Ga

_{0.25}N structure of 1 × 1 × 2; (

**g**) InN structure of 2 × 2 × 1. (From (

**b**) to (

**g**) is wurtzite In

_{x}Ga

_{1−x}N alloys).

**Figure 5.**Band structure of zincblende In

_{x}Ga

_{1−x}N systems: (

**a**) GaN; (

**b**) In

_{0.125}Ga

_{0.875}N; (

**c**) In

_{0.25}Ga

_{0.75}N; (

**d**) In

_{0.5}Ga

_{0.5}N; (

**e**) In

_{0.75}Ga

_{0.25}N; (

**f**) InN.

**Figure 6.**Distribution of electron density difference on the (111) plane of (

**a**) GaN and (

**b**) In

_{x}Ga

_{1−x}N alloys.

**Figure 7.**Density of states of In

_{x}Ga

_{1−x}N alloys: (

**a**) total density of states of In

_{x}Ga

_{1−x}N alloys; (

**b**) density of states of nitrogen s/p orbitals; (

**c**) density of states of gallium s/p orbitals; (

**d**) density of states of indium s/p orbitals.

**Table 1.**Spontaneous polarization intensity of wurtzite In

_{x}Ga

_{1−x}N (P

_{SP}), close-packed plane lattice constants of wurtzite and zincblende In

_{x}Ga

_{1−x}N, and the difference (△L) in lattice constant of close-packed plane wurtzite and zincblende In

_{x}Ga

_{1−x}N with different indium compositions.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Song, J.; Luo, Z.; Liu, X.; Li, E.; Jiang, C.; Huang, Z.; Li, J.; Guo, X.; Ding, Z.; Wang, J.
The Study on Structural and Photoelectric Properties of Zincblende InGaN via First Principles Calculation. *Crystals* **2020**, *10*, 1159.
https://doi.org/10.3390/cryst10121159

**AMA Style**

Song J, Luo Z, Liu X, Li E, Jiang C, Huang Z, Li J, Guo X, Ding Z, Wang J.
The Study on Structural and Photoelectric Properties of Zincblende InGaN via First Principles Calculation. *Crystals*. 2020; 10(12):1159.
https://doi.org/10.3390/cryst10121159

**Chicago/Turabian Style**

Song, Juan, Zijiang Luo, Xuefei Liu, Ershi Li, Chong Jiang, Zechen Huang, Jiawei Li, Xiang Guo, Zhao Ding, and Jihong Wang.
2020. "The Study on Structural and Photoelectric Properties of Zincblende InGaN via First Principles Calculation" *Crystals* 10, no. 12: 1159.
https://doi.org/10.3390/cryst10121159