Structural and Electronic Properties of Hexagonal and Cubic Phase AlGaInN Alloys Investigated Using First Principles Calculations

Structural and electronic properties of hexagonal (h-) and cubic (c-) phase AlGaInN quaternary alloys are investigated using a unified and accurate local-density approximation-1/2 approach under the density-functional theory framework. Lattice bowing parameters of h- (and c-) phase AlGaN, AlInN, InGaN, and AlGaInN alloys are extracted as 0.006 (−0.007), 0.040 (−0.015), 0.014 (−0.011), and −0.082 (0.184) Å, respectively. Bandgap bowing parameters of h- (and c-) phase AlGaN, AlInN, InGaN, and AlGaInN alloys are extracted as 1.775 (0.391), 3.678 (1.464), 1.348 (1.164), and 1.236 (2.406) eV, respectively. Direct-to-indirect bandgap crossover Al mole fractions for c-phase AlGaN and AlInN alloys are determined to be 0.700 and 0.922, respectively. Under virtual crystal approximation, electron effective masses of h- and c-phase AlGaInN alloys are extracted and those of c-phase alloys are observed to be smaller than those of the h-phase alloys. Overall, c-phase AlGaInN alloys are shown to have fundamental material advantages over the h-phase alloys such as smaller bandgaps and smaller effective masses, which motivate their applications in light emitting- and laser diodes.

structure. Recently, Liu et al. have synthesized c-phase GaN with high crystal-quality and high phase-purity using the U-groove hexagonal-to-cubic phase transition approach. A band-to-band emission in the ultraviolet (UV) from the c-phase GaN has shown a record high internal quantum efficiency of 29%, which is higher than that of bulk h-phase GaN at 12% 9 . Further increase of the radiative efficiency using carrier confinement necessitates the use of quantum well structures. Yet, there are limited computational and experimental studies on the structural and electronic properties of these c-phase III-nitrides; there are no numerical expressions for the bandgap, lattice-constant, and effective mass of c-phase III-nitrides as well. In literature, bandgaps, direct-to-indirect bandgap crossover points, and effective masses of h-and c-phase AlN, GaN, and InN are studied using various simulation approaches such as local-density approximation (LDA) 10,11 , G 0 W 0 approximation 12 , and generalized gradient approximation (GGA) approaches 13 , which led to dissimilar bandgap values. For instance, the calculated bandgap of h-phase InN ranges from 0.69 12 and 1.02 13 to 2.00 11 eV; the broad range of bandgap is also reported in experiments, while the widely accepted value is 0.78 eV 14 . The inconsistent bandgap can virtually affect the accuracy in determining the bandgap of In-rich ternary and quaternary III-nitrides, which alters the bowing parameters and the direct-to-indirect bandgap crossover points 15 . In addition, the evolution of electron effective mass in c-phase ternary and quaternary III-nitrides with respect to Al, Ga, and In mole fractions is still unclear.
In this work, lattice constants and bandgaps of h-and c-phase III-nitrides are investigated under the density-functional theory framework. For the crystal relaxation, LDA is applied to approximate the exchange-correlation energy of the many-electron system due to its reliability and low computational cost of determining the ground-state electronic properties. However, it is well-known that LDA underestimates the bandgap since the exchange-correlation potential is not discontinuous between the conduction and valence band. LDA-1/2 approach, on the other hand, is applied to correct the bandgap by half-ionizing electron to conduction band with the same reliability and computational cost as the LDA method. It can retrieve the accurate bandgap of III-V alloys because the relation between the single-particle energy obtained from Kohn-Sham equation and the electron occupation can be linearly approximated 16 . Figure 1(a) illustrates a unit cell of h-phase binary III-nitrides. The red dash lines highlight the primitive cell that contains 2 group III atoms and 2 N atoms and Illustrations of a unit cell of (a) h-and (b) c-phase binary III-nitrides. Black solid lines outline the contour of unit cells; while, the red dash lines highlight the corresponding primitive cells. The primitive cell of h-phase binary III-nitrides contains 2 group III atoms and 2 N atoms; while, the primitive cell of c-phase binary III-nitrides contains 1 group III atom and 1 N atom. Demonstrations of the electronic structure of (c) h-and (d) c-phase GaN along the high-symmetry lines calculated by LDA-1/2 method. E VBM is the energy of valence band maximum.
www.nature.com/scientificreports www.nature.com/scientificreports/ defines the h-phase crystal structure. Similarly, Fig. 1(b) shows a unit cell of c-phase binary III-nitrides, where the primitive cell contains 1 group III atom and 1 N atom. Figure 1(c,d) demonstrate the electronic structure of h-and c-phase GaN along the high-symmetry lines, where the energy states shift with respect to the valence band maximum (E VBM ). The electronic structure of ternary (AlGaN, AlInN, and InGaN) and quaternary (AlGaInN) alloys with different mole fractions are calculated by substituting the group III atoms in a unit cell, which allows an accurate interpolation of direct-to-indirect bandgap crossover points, an extraction of bowing parameters, and an numerical expression of lattice constants and bandgaps. Transverse and longitudinal electron effective masses are also extracted for h-and c-phase III-nitrides.

Methods
First-principles calculations are carried out under the density-functional theory framework implemented in the Vienna ab initio Simulation Package (VASP) 17 and are carried out on binary, ternary and quaternary III-nitrides. Al 3s 2 3p 1 , Ga 4s 2 4p 1 , In 4d 10 5s 2 5p 1 , and N 2s 2 2p 3 valence electrons are characterized using projector augmented wave pseudopotentials (PAW) 18 ; while the cut-off kinetic energy of 500 eV is secured for the plane-wave expansion. All atoms are fully relaxed so that the interatomic forces and energy difference are smaller than 0.01 eVÅ −1 per ion and 10 −6 eV/atom, respectively. Figure 2(a) illustrates 2 × 2 × 1 primitive cell of h-phase binary III-nitrides used to construct and simulate a 16-atom supercell of h-phase quaternary alloys. Similarly, Fig. 2(b) illustrates 2 × 2 × 2 primitive cell of c-phase binary III-nitrides used to construct and simulate a 16-atom supercell of c-phase quaternary alloys. Each supercell contains 8 group III atoms and 8 N atoms. The mole fractions of Al, Ga, and In can be adjusted by substituting the 8 group III atoms in the supercell, which gives the finest mole fraction tunability of 0.125. For instance, the unit cell of Al 0.375 Ga 0.625 N contains 3 Al atoms, 5 Ga atoms, and 8 N atoms; while, the unit cell of Al 0.5 Ga 0.5 N has 4 Al atoms, 4 Ga atoms, and 8 N atoms. Overall, 9 and 45 cases are sampled individually for each phase of ternary (AlGaN, AlInN, and InGaN) and quaternary (AlGaInN) alloys. Although the initial size of supercells used to build ternary and quaternary alloys is identical, the size of supercells changes after the atoms and the cells are relaxed to its equilibrium positions and volumes. For instance, h-and c-phase AlN have the smallest supercells of 163.36 and 163.75 Å 3 ; while h-and c-phase InN have the largest supercells of 240.58 and 240.93 Å 3 , respectively. The Brillouin zone of the supercells is sampled with an 8 × 8 × 8 gamma-centered Monkhorst-Pack set of k-points. Notably, the bandgap of binary alloys (AlN, GaN, and InN) calculated by the supercell approach is consistent with the bandgap calculated using the primitive cells, which indicates the numerical accuracy of the supercell calculations. LDA is performed to calculate the exchange-correlation energy of many-electron system for structural relaxation. To fix the bandgap underestimation of LDA, LDA-1/2 method is utilized to calculate the electronic structure and bandgap of III-nitrides. LDA-1/2 method inherits from Slater half-occupation scheme 16,19 , which assumes that single-particle Kohn-Sham energy linearly depends on the occupation expressed as: where E(N) and E(N−1) are the total system energies under the occupation number of N and N−1, respectively; ε α is the single-particle Kohn-Sham energy at state α. Using the linear dependence of the single-particle Kohn-Sham energy on the occupation number, the ionization energy is approximated by the single-particle Kohn-Sham energy with a half ionization, as shown in the last term. Starting from this postulation, Ferreira et al. further demonstrated that the single-particle Kohn-Sham energy with a half ionization is equivalent to the single-particle Kohn-Sham energy at ground-state, ε α (N), minus a hole self-energy (S α ) at state α expressed as 20 : S α can be formulated by quantum-mechanical average: Figure 2. Illustrations of (a) 2 × 2 × 1 and (b) 2 × 2 × 2 primitive cells of h-and c-phase binary III-nitrides used to construct and simulate 16-atom supercells of h-and c-phase quaternary alloys, respectively. www.nature.com/scientificreports www.nature.com/scientificreports/ where n α , V s , and Θ are the electron density at state α, the self-energy potential, and the trim function, respectively. The trim function is used to cut the Coulomb tail of V s in an infinite crystal, which has a negligible contribution to the self-energy due to the localization of wavefunction, defined by: where CUT and n are the cutoff radius and cutoff sharpness; while, A is the amplitude of trim function, which can be exploited to adjust the amplitude of self-energy and bandgap linearly and semi-empirically. The bandgap (E g ), defined by the energy difference between the system energy that has one electron excited from the valence to the conduction band, E(N−1, 1), and the ground-state system energy, E(N,0), can be formulated as the bandgap calculated by LDA (E g LDA ) with corrections from electron (S c ) and hole (S v ) self-energies: where ɛ c and ɛ v are single-particle Kohn-Sham energies for electron and hole, respectively. According to the suggestions by Ferreira et al., A = 1 and n = 8 are fixed in the first place 20 . The CUT values of 4.40, 1.53, and 3.0Å are benchmarked for Al, Ga, and N ions to make the bandgap of h-phase AlN (6.2 eV) and h-phase GaN (3.5 eV) experimentally-verified 14 . However, several combinations of CUT and n have been benchmarked for In ion, none of them gives a satisfactory result. The optimal bandgap is 1.24 eV, which is similar to other reports 13,15,21 . But, it significantly deviates from the experiment of 0.78 eV. The bandgap overestimation may originate from the assumption that the ionization energy is equal to the single-particle Kohn-Sham energy with a half ionization. In other words, the single-particle Kohn-Sham energy of In ion may not linearly dependent upon the occupation number. To determine the correct self-energy, the trim function amplitude A is benchmarked semi-empirically to adjust the self-energy amplitude and the bandgap linearly. Finally, A = 2.3 and CUT = 2.926Å are benchmarked for In ion.
To model the electronic properties of multinary alloys, the unit cell of multinary alloys is constructed by the supercell method. However, the corresponding Brillouin zone is folded with the increasing size of the supercell. As the consequence, it is challenging to analyze the electronic properties, such as extracting effective mass and determining indirect bandgap, from the E-k dispersion because the energy states outside of the first Brillouin zone are folded into the first Brillouin zone so-called band folding. fold2Bloch utility 22 is employed to unfold the band structure of supercell back to its primitive basis representation by calculating and filtering the Bloch spectral density, the procedure is known as virtual crystal approximation. After unfolding the electronic structure, the curvatures of the lowest conduction band for electron effective mass calculations are extracted using the finite-difference method and parabolic approximation. The lattice constants, bandgaps, transverse ( ⁎ m t ) and longitudinal ( ⁎ m l ) electron effective masses for h-and c-phase III-nitrides with the corresponding corroboration from experiments are tabulated (   Table 2. Since the defectivity and piezoelectric polarization of III-nitrides strongly depend on the lattice mismatch between two substrates, it is essential to estimate the lattice mismatch between the potential materials of quantum well and GaN. The lattice mismatch can be estimated by  www.nature.com/scientificreports www.nature.com/scientificreports/ can be linearly interpolated due to the linear relationship between lattice constants and Al and In mole fractions.

Results and Discussion
The accuracy of LDA-1/2 method on the bandgap calculations of ternary alloys is demonstrated by comparing the simulation results with the experimental measurements [23][24][25][26][27][28][29][30][31][32] , except for c-phase AlInN and In-rich c-phase InGaN due to the lack of experimental reference data. The nonlinear dependence of bandgap on the Al or In mole fraction is summarized using bandgap Vegard's law tabulated in Table 2. The bandgap bowing parameters of h-phase AlGaN, AlInN, and InGaN are extracted as 1.775, 3.678, and 1.348 eV, respectively, which are larger than their c-phase counterparts of 0.391 (AlGaN), 1.464 (AlInN), and 1.164 eV (InGaN). The simulated bandgaps , P AN , and P BN are the property (P) of ternary and binary nitrides, respectively, and b is the bowing parameter.
Scientific RepoRts | (2019) 9:6583 | https://doi.org/10.1038/s41598-019-43113-w www.nature.com/scientificreports www.nature.com/scientificreports/ agree well with the available experiments. It is worth mentioning that most of the exchange-correlation functionals, including highly-accurate hybrid functionals and GW quasi-particle approximation, fail to recover the bandgap of InN 21 , which also leads to the bandgap overestimation of In-rich h-phase InGaN by more than 20% 15 . The similar issue also occurs in the LDA-1/2 method. To address this issue, the self-energy energy of In ion is linearly and semi-empirically increased by 2.3 times so that the bandgap of h-phase InN matches the experiment of 0.78 eV. The numerical adjustment corrects the underestimation of self-energy and, eventually, improves the bandgap overestimation. Therefore, the correction of self-energy using h-phase InN bandgap fixes the bandgap overestimation of In-rich h-phase InGaN. The calculated bandgap of c-phase InN also matches the experiment, which further indicates the consistency and reliability of the semi-empirical correction.
The   Table 3 Table 3, a similar form of Vegard's law is exploited to express the electron effective mass numerically in the unit of m 0 with a bowing parameter of −0.4684  Table 3. Lattice and bandgap bowing parameters of h-and c-phase AlGaInN and effective mass bowing parameter of h-phase Al x Ga y In 1−x−y N. The generic Vegard's law for quaternary alloys is expressed as:  www.nature.com/scientificreports www.nature.com/scientificreports/ m 0 and an R 2 value of 0.990. The ⁎ m t and ⁎ m l of c-phase AlGaInN, except for the Al-rich quaternary alloys, are identical due to the centrosymmetry of zincblende structure. However, both the ⁎ m t and ⁎ m l lift sharply at the crossover points of the direct-to-indirect bandgap transition because the conduction minimum shifts from the Γ-valley to the X-valley at high Al mole fraction, where X-valley has a smaller curvature. At X-valley, the ⁎ m l is significantly greater than the ⁎ m t . The drastic difference between ⁎ m t and ⁎ m l is ascribed to the anisotropic nature of p-orbitals that dominantly contribute to X-valley, as confirmed by the partial density of states. The electron effective masses of c-and h-phase alloys at Γ-valley are similar. However, with the increasing Ga mole fraction, the electron effective masses of the c-phase alloys become increasingly smaller than those of the h-phase alloys. As a result, although the electron effective mass of c-phase InN is 68% larger than that of h-phase InN, the electron effective mass of c-phase GaN is 9.3% smaller than that of h-phase GaN, which evidences an additional advantage of c-phase GaN in electronic transport.

Conclusion
In conclusion, lattice constants, bandgaps, and electron effective masses of binary, ternary, and quaternary III-nitrides have been investigated using the accurate and unified LDA-1/2 approach. C-phase III-nitride alloys have smaller bandgaps and electron effective masses than h-phase nitrides.

Data Availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.